LMFDB - Embedded newform 675.2.k.b.424.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [675,2,Mod(199,675)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(675, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 3]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("675.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	675.k (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$5.38990213644$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36 $ x^12 - 16x^8 - 24x^7 + 96x^5 + 304x^4 + 384x^3 + 288x^2 + 144*x + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			424.5
Root			$0.583700 + 2.17840i$ of defining polynomial
Character	$\chi$	$=$	675.424
Dual form			675.2.k.b.199.5

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.80664 + 1.04307i) q^{2} +(1.17597 + 2.03684i) q^{4} +(3.53869 + 2.04307i) q^{7} +0.734191i q^{8} +O(q^{10})$	\(q+(1.80664 + 1.04307i) q^{2} +(1.17597 + 2.03684i) q^{4} +(3.53869 + 2.04307i) q^{7} +0.734191i q^{8} +(-0.675970 + 1.17081i) q^{11} +(0.561237 - 0.324030i) q^{13} +(4.26210 + 7.38217i) q^{14} +(1.58613 - 2.74726i) q^{16} -1.35194i q^{17} -0.648061 q^{19} +(-2.44247 + 1.41016i) q^{22} +(-4.14827 + 2.39500i) q^{23} +1.35194 q^{26} +9.61033i q^{28} +(-1.93807 + 3.35683i) q^{29} +(3.84823 + 6.66533i) q^{31} +(7.00279 - 4.04307i) q^{32} +(1.41016 - 2.44247i) q^{34} -7.52420i q^{37} +(-1.17081 - 0.675970i) q^{38} +(-0.0898394 - 0.155606i) q^{41} +(0.710419 + 0.410161i) q^{43} -3.17968 q^{44} -9.99258 q^{46} +(-9.44526 - 5.45323i) q^{47} +(4.84823 + 8.39738i) q^{49} +(1.32000 + 0.762100i) q^{52} -4.17226i q^{53} +(-1.50000 + 2.59808i) q^{56} +(-7.00279 + 4.04307i) q^{58} +(-2.08613 - 3.61328i) q^{59} +(1.91016 - 3.30850i) q^{61} +16.0558i q^{62} +10.5242 q^{64} +(-7.05113 + 4.07097i) q^{67} +(2.75368 - 1.58984i) q^{68} +6.11644 q^{71} -12.3445i q^{73} +(7.84823 - 13.5935i) q^{74} +(-0.762100 - 1.32000i) q^{76} +(-4.78410 + 2.76210i) q^{77} +(5.17226 - 8.95862i) q^{79} -0.374833i q^{82} +(-10.6161 - 6.12920i) q^{83} +(0.855648 + 1.48203i) q^{86} +(-0.859601 - 0.496291i) q^{88} -3.00000 q^{89} +2.64806 q^{91} +(-9.75648 - 5.63290i) q^{92} +(-11.3761 - 19.7041i) q^{94} +(-11.7606 - 6.79001i) q^{97} +20.2281i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 10 q^{4}+O(q^{10}) $ 12 * q + 10 * q^4	$ 12 q + 10 q^{4} - 4 q^{11} + 18 q^{14} - 10 q^{16} - 16 q^{19} + 8 q^{26} + 14 q^{29} - 16 q^{31} - 8 q^{34} - 26 q^{41} - 88 q^{44} - 12 q^{46} - 4 q^{49} - 18 q^{56} + 4 q^{59} - 2 q^{61} + 60 q^{64} + 40 q^{71} + 32 q^{74} + 24 q^{76} + 4 q^{79} + 56 q^{86} - 36 q^{89} + 40 q^{91} - 62 q^{94}+O(q^{100}) $ 12 * q + 10 * q^4 - 4 * q^11 + 18 * q^14 - 10 * q^16 - 16 * q^19 + 8 * q^26 + 14 * q^29 - 16 * q^31 - 8 * q^34 - 26 * q^41 - 88 * q^44 - 12 * q^46 - 4 * q^49 - 18 * q^56 + 4 * q^59 - 2 * q^61 + 60 * q^64 + 40 * q^71 + 32 * q^74 + 24 * q^76 + 4 * q^79 + 56 * q^86 - 36 * q^89 + 40 * q^91 - 62 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/675\mathbb{Z}\right)^\times$.

$n$	$326$	$352$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	1.80664	+	1.04307i	1.27749	+	0.737558i	0.976386	−	0.216033i	$-0.0693120\pi$
$2$	1.80664	+	1.04307i	1.27749	+	0.737558i	0.301103	+	0.953592i	$0.402645\pi$
$3$		0			0
$3$		0			0
$4$	1.17597	+	2.03684i	0.587985	+	1.01842i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	3.53869	+	2.04307i	1.33750	+	0.772206i	0.986436	−	0.164145i	$-0.0524866\pi$
$7$	3.53869	+	2.04307i	1.33750	+	0.772206i	0.351064	+	0.936351i	$0.385820\pi$
$8$			0.734191i			0.259576i
$9$		0			0
$10$		0			0
$11$	−0.675970	+	1.17081i	−0.203813	+	0.353014i	−0.949754	−	0.312998i	$-0.898667\pi$
$11$	−0.675970	+	1.17081i	−0.203813	+	0.353014i	0.745941	+	0.666012i	$0.232000\pi$
$12$		0			0
$13$	0.561237	−	0.324030i	0.155659	−	0.0898699i	−0.420147	−	0.907456i	$-0.638022\pi$
$13$	0.561237	−	0.324030i	0.155659	−	0.0898699i	0.575806	+	0.817586i	$0.304688\pi$
$14$	4.26210	+	7.38217i	1.13909	+	1.97297i
$15$		0			0
$16$	1.58613	−	2.74726i	0.396533	−	0.686815i
$17$		−	1.35194i		−	0.327893i	−0.986469	−	0.163947i	$-0.947577\pi$
$17$		−	1.35194i		−	0.327893i	0.986469	−	0.163947i	$-0.0524225\pi$
$18$		0			0
$19$	−0.648061			−0.148675			−0.0743377	−	0.997233i	$-0.523684\pi$
$19$	−0.648061			−0.148675			−0.0743377	+	0.997233i	$0.523684\pi$
$20$		0			0
$21$		0			0
$22$	−2.44247	+	1.41016i	−0.520736	+	0.300647i
$23$	−4.14827	+	2.39500i	−0.864974	+	0.499393i	−0.865675	−	0.500607i	$-0.833110\pi$
$23$	−4.14827	+	2.39500i	−0.864974	+	0.499393i	0.000700856		1.00000i	$0.499777\pi$
$24$		0			0
$25$		0			0
$26$	1.35194			0.265137
$27$		0			0
$28$			9.61033i			1.81618i
$29$	−1.93807	+	3.35683i	−0.359890	+	0.623349i	−0.987942	−	0.154823i	$-0.950519\pi$
$29$	−1.93807	+	3.35683i	−0.359890	+	0.623349i	0.628052	+	0.778172i	$0.283853\pi$
$30$		0			0
$31$	3.84823	+	6.66533i	0.691163	+	1.19713i	0.971457	+	0.237215i	$0.0762345\pi$
$31$	3.84823	+	6.66533i	0.691163	+	1.19713i	−0.280295	+	0.959914i	$0.590432\pi$
$32$	7.00279	−	4.04307i	1.23793	−	0.714720i
$33$		0			0
$34$	1.41016	−	2.44247i	0.241841	−	0.418880i
$35$		0			0
$36$		0			0
$37$		−	7.52420i		−	1.23697i	−0.785796	−	0.618485i	$-0.787747\pi$
$37$		−	7.52420i		−	1.23697i	0.785796	−	0.618485i	$-0.212253\pi$
$38$	−1.17081	−	0.675970i	−0.189931	−	0.109657i
$39$		0			0
$40$		0			0
$41$	−0.0898394	−	0.155606i	−0.0140306	−	0.0243016i	0.858925	−	0.512102i	$-0.171133\pi$
$41$	−0.0898394	−	0.155606i	−0.0140306	−	0.0243016i	−0.872955	+	0.487800i	$0.837800\pi$
$42$		0			0
$43$	0.710419	+	0.410161i	0.108338	+	0.0625489i	0.553190	−	0.833055i	$-0.313410\pi$
$43$	0.710419	+	0.410161i	0.108338	+	0.0625489i	−0.444852	+	0.895604i	$0.646744\pi$
$44$	−3.17968			−0.479355
$45$		0			0
$46$	−9.99258			−1.47333
$47$	−9.44526	−	5.45323i	−1.37773	−	0.795435i	−0.385847	−	0.922563i	$-0.626091\pi$
$47$	−9.44526	−	5.45323i	−1.37773	−	0.795435i	−0.991886	+	0.127128i	$0.959424\pi$
$48$		0			0
$49$	4.84823	+	8.39738i	0.692604	+	1.19963i
$50$		0			0
$51$		0			0
$52$	1.32000	+	0.762100i	0.183050	+	0.105684i
$53$		−	4.17226i		−	0.573104i	−0.958065	−	0.286552i	$-0.907491\pi$
$53$		−	4.17226i		−	0.573104i	0.958065	−	0.286552i	$-0.0925091\pi$
$54$		0			0
$55$		0			0
$56$	−1.50000	+	2.59808i	−0.200446	+	0.347183i
$57$		0			0
$58$	−7.00279	+	4.04307i	−0.919512	+	0.530880i
$59$	−2.08613	−	3.61328i	−0.271591	−	0.470409i	0.697678	−	0.716411i	$-0.254216\pi$
$59$	−2.08613	−	3.61328i	−0.271591	−	0.470409i	−0.969269	+	0.246002i	$0.920883\pi$
$60$		0			0
$61$	1.91016	−	3.30850i	0.244571	−	0.423609i	−0.717440	−	0.696620i	$-0.754686\pi$
$61$	1.91016	−	3.30850i	0.244571	−	0.423609i	0.962011	+	0.273011i	$0.0880195\pi$
$62$			16.0558i			2.03909i
$63$		0			0
$64$	10.5242			1.31552
$65$		0			0
$66$		0			0
$67$	−7.05113	+	4.07097i	−0.861433	+	0.497349i	−0.864492	−	0.502647i	$-0.832360\pi$
$67$	−7.05113	+	4.07097i	−0.861433	+	0.497349i	0.00305885	+	0.999995i	$0.499026\pi$
$68$	2.75368	−	1.58984i	0.333933	−	0.192796i
$69$		0			0
$70$		0			0
$71$	6.11644			0.725888			0.362944	−	0.931811i	$-0.381772\pi$
$71$	6.11644			0.725888			0.362944	+	0.931811i	$0.381772\pi$
$72$		0			0
$73$		−	12.3445i		−	1.44482i	−0.691467	−	0.722408i	$-0.743035\pi$
$73$		−	12.3445i		−	1.44482i	0.691467	−	0.722408i	$-0.256965\pi$
$74$	7.84823	−	13.5935i	0.912338	−	1.58022i
$75$		0			0
$76$	−0.762100	−	1.32000i	−0.0874188	−	0.151414i
$77$	−4.78410	+	2.76210i	−0.545198	+	0.314770i
$78$		0			0
$79$	5.17226	−	8.95862i	0.581925	−	1.00792i	−0.413326	−	0.910583i	$-0.635633\pi$
$79$	5.17226	−	8.95862i	0.581925	−	1.00792i	0.995251	−	0.0973403i	$-0.0310335\pi$
$80$		0			0
$81$		0			0
$82$		−	0.374833i		−	0.0413934i
$83$	−10.6161	−	6.12920i	−1.16527	−	0.672767i	−0.212706	−	0.977116i	$-0.568228\pi$
$83$	−10.6161	−	6.12920i	−1.16527	−	0.672767i	−0.952560	+	0.304350i	$0.901561\pi$
$84$		0			0
$85$		0			0
$86$	0.855648	+	1.48203i	0.0922669	+	0.159811i
$87$		0			0
$88$	−0.859601	−	0.496291i	−0.0916338	−	0.0529048i
$89$	−3.00000			−0.317999			−0.159000	−	0.987279i	$-0.550827\pi$
$89$	−3.00000			−0.317999			−0.159000	+	0.987279i	$0.550827\pi$
$90$		0			0
$91$	2.64806			0.277592
$92$	−9.75648	−	5.63290i	−1.01718	−	0.587271i
$93$		0			0
$94$	−11.3761	−	19.7041i	−1.17336	−	2.03232i
$95$		0			0
$96$		0			0
$97$	−11.7606	−	6.79001i	−1.19411	−	0.689421i	−0.234876	−	0.972025i	$-0.575468\pi$
$97$	−11.7606	−	6.79001i	−1.19411	−	0.689421i	−0.959237	+	0.282605i	$0.908802\pi$
$98$			20.2281i			2.04334i
$99$		0			0
$100$		0			0
$101$	−0.734191	+	1.27166i	−0.0730547	+	0.126535i	−0.900239	−	0.435397i	$-0.856608\pi$
$101$	−0.734191	+	1.27166i	−0.0730547	+	0.126535i	0.827184	+	0.561931i	$0.189941\pi$
$102$		0			0
$103$	6.51615	−	3.76210i	0.642055	−	0.370691i	−0.143351	−	0.989672i	$-0.545788\pi$
$103$	6.51615	−	3.76210i	0.642055	−	0.370691i	0.785406	+	0.618981i	$0.212454\pi$
$104$	0.237900	+	0.412055i	0.0233280	+	0.0404053i
$105$		0			0
$106$	4.35194	−	7.53778i	0.422698	−	0.732134i
$107$			1.20999i			0.116974i	0.998288	+	0.0584871i	$0.0186277\pi$
$107$			1.20999i			0.116974i	−0.998288	+	0.0584871i	$0.981372\pi$
$108$		0			0
$109$	−14.1042			−1.35094			−0.675469	−	0.737388i	$-0.736059\pi$
$109$	−14.1042			−1.35094			−0.675469	+	0.737388i	$0.736059\pi$
$110$		0			0
$111$		0			0
$112$	11.2257	−	6.48113i	1.06072	−	0.612410i
$113$	10.3270	−	5.96227i	0.971478	−	0.560883i	0.0717915	−	0.997420i	$-0.477128\pi$
$113$	10.3270	−	5.96227i	0.971478	−	0.560883i	0.899687	+	0.436537i	$0.143795\pi$
$114$		0			0
$115$		0			0
$116$	−9.11644			−0.846440
$117$		0			0
$118$		−	8.70388i		−	0.801257i
$119$	2.76210	−	4.78410i	0.253201	−	0.438557i
$120$		0			0
$121$	4.58613	+	7.94341i	0.416921	+	0.722128i
$122$	6.90195	−	3.98484i	0.624873	−	0.360771i
$123$		0			0
$124$	−9.05080	+	15.6765i	−0.812786	+	1.40779i
$125$		0			0
$126$		0			0
$127$			7.07871i			0.628134i	0.949401	+	0.314067i	$0.101692\pi$
$127$			7.07871i			0.628134i	−0.949401	+	0.314067i	$0.898308\pi$
$128$	5.00787	+	2.89130i	0.442637	+	0.255557i
$129$		0			0
$130$		0			0
$131$	−3.00000	−	5.19615i	−0.262111	−	0.453990i	0.704692	−	0.709514i	$-0.251085\pi$
$131$	−3.00000	−	5.19615i	−0.262111	−	0.453990i	−0.966803	+	0.255524i	$0.917752\pi$
$132$		0			0
$133$	−2.29329	−	1.32403i	−0.198853	−	0.114808i
$134$	−16.9852			−1.46729
$135$		0			0
$136$	0.992582			0.0851132
$137$	6.46781	+	3.73419i	0.552582	+	0.319033i	0.750163	−	0.661253i	$-0.229975\pi$
$137$	6.46781	+	3.73419i	0.552582	+	0.319033i	−0.197581	+	0.980287i	$0.563308\pi$
$138$		0			0
$139$	4.00000	+	6.92820i	0.339276	+	0.587643i	0.984297	−	0.176522i	$-0.0564848\pi$
$139$	4.00000	+	6.92820i	0.339276	+	0.587643i	−0.645021	+	0.764165i	$0.723151\pi$
$140$		0			0
$141$		0			0
$142$	11.0502	+	6.37985i	0.927314	+	0.535385i
$143$			0.876139i			0.0732664i
$144$		0			0
$145$		0			0
$146$	12.8761	−	22.3021i	1.06564	−	1.84574i
$147$		0			0
$148$	15.3256	−	8.84823i	1.25976	−	0.727320i
$149$	5.29241	+	9.16673i	0.433571	+	0.750968i	0.997178	−	0.0750759i	$-0.0239199\pi$
$149$	5.29241	+	9.16673i	0.433571	+	0.750968i	−0.563607	+	0.826043i	$0.690587\pi$
$150$		0			0
$151$	−8.84823	+	15.3256i	−0.720059	+	1.24718i	0.240917	+	0.970546i	$0.422552\pi$
$151$	−8.84823	+	15.3256i	−0.720059	+	1.24718i	−0.960976	+	0.276633i	$0.910781\pi$
$152$		−	0.475800i		−	0.0385925i
$153$		0			0
$154$	−11.5242			−0.928646
$155$		0			0
$156$		0			0
$157$	2.19245	−	1.26581i	0.174976	−	0.101023i	−0.409954	−	0.912106i	$-0.634455\pi$
$157$	2.19245	−	1.26581i	0.174976	−	0.101023i	0.584930	+	0.811084i	$0.301122\pi$
$158$	18.6888	−	10.7900i	1.48680	−	0.858407i
$159$		0			0
$160$		0			0
$161$	−19.5726			−1.54254
$162$		0			0
$163$			8.47580i			0.663876i	0.943301	+	0.331938i	$0.107702\pi$
$163$			8.47580i			0.663876i	−0.943301	+	0.331938i	$0.892298\pi$
$164$	0.211297	−	0.365977i	0.0164995	−	0.0285780i
$165$		0			0
$166$	−12.7863	−	22.1465i	−0.992409	−	1.71890i
$167$	11.0281	−	6.36710i	0.853383	−	0.492701i	−0.00840816	−	0.999965i	$-0.502676\pi$
$167$	11.0281	−	6.36710i	0.853383	−	0.492701i	0.861791	+	0.507264i	$0.169343\pi$
$168$		0			0
$169$	−6.29001	+	10.8946i	−0.483847	+	0.838047i
$170$		0			0
$171$		0			0
$172$			1.92935i			0.147111i
$173$	19.9605	+	11.5242i	1.51757	+	0.876169i	0.999787	+	0.0206561i	$0.00657550\pi$
$173$	19.9605	+	11.5242i	1.51757	+	0.876169i	0.517782	+	0.855513i	$0.326758\pi$
$174$		0			0
$175$		0			0
$176$	2.14435	+	3.71413i	0.161637	+	0.279963i
$177$		0			0
$178$	−5.41993	−	3.12920i	−0.406241	−	0.234543i
$179$	2.22808			0.166534			0.0832672	−	0.996527i	$-0.473465\pi$
$179$	2.22808			0.166534			0.0832672	+	0.996527i	$0.473465\pi$
$180$		0			0
$181$	0.468382			0.0348146			0.0174073	−	0.999848i	$-0.494459\pi$
$181$	0.468382			0.0348146			0.0174073	+	0.999848i	$0.494459\pi$
$182$	4.78410	+	2.76210i	0.354621	+	0.204740i
$183$		0			0
$184$	−1.75839	−	3.04562i	−0.129630	−	0.224526i
$185$		0			0
$186$		0			0
$187$	1.58287	+	0.913870i	0.115751	+	0.0668288i
$188$		−	25.6513i		−	1.87081i
$189$		0			0
$190$		0			0
$191$	−10.1140	+	17.5180i	−0.731826	+	1.26756i	0.224276	+	0.974526i	$0.427998\pi$
$191$	−10.1140	+	17.5180i	−0.731826	+	1.26756i	−0.956102	+	0.293034i	$0.905335\pi$
$192$		0			0
$193$	−17.2593	+	9.96467i	−1.24235	+	0.717273i	−0.969573	−	0.244804i	$-0.921277\pi$
$193$	−17.2593	+	9.96467i	−1.24235	+	0.717273i	−0.272780	+	0.962076i	$0.587943\pi$
$194$	−14.1648	−	24.5342i	−1.01698	−	1.76145i
$195$		0			0
$196$	−11.4027	+	19.7501i	−0.814482	+	1.41072i
$197$		−	15.5800i		−	1.11003i	−0.831840	−	0.555015i	$-0.812712\pi$
$197$		−	15.5800i		−	1.11003i	0.831840	−	0.555015i	$-0.187288\pi$
$198$		0			0
$199$	−3.58482			−0.254121			−0.127061	−	0.991895i	$-0.540554\pi$
$199$	−3.58482			−0.254121			−0.127061	+	0.991895i	$0.540554\pi$
$200$		0			0
$201$		0			0
$202$	−2.65284	+	1.53162i	−0.186653	+	0.107764i
$203$	−13.7165	+	7.91920i	−0.962707	+	0.555819i
$204$		0			0
$205$		0			0
$206$	15.6965			1.09362
$207$		0			0
$208$		−	2.05582i		−	0.142545i
$209$	0.438069	−	0.758758i	0.0303019	−	0.0524844i
$210$		0			0
$211$	−7.49629	−	12.9840i	−0.516066	−	0.893852i	−0.999826	−	0.0186518i	$-0.994063\pi$
$211$	−7.49629	−	12.9840i	−0.516066	−	0.893852i	0.483760	−	0.875201i	$-0.339271\pi$
$212$	8.49822	−	4.90645i	0.583660	−	0.336976i
$213$		0			0
$214$	−1.26210	+	2.18602i	−0.0862754	+	0.149433i
$215$		0			0
$216$		0			0
$217$			31.4487i			2.13488i
$218$	−25.4813	−	14.7116i	−1.72581	−	0.996396i
$219$		0			0
$220$		0			0
$221$	−0.438069	−	0.758758i	−0.0294677	−	0.0510396i
$222$		0			0
$223$	23.2363	+	13.4155i	1.55602	+	0.898368i	0.997631	+	0.0687878i	$0.0219131\pi$
$223$	23.2363	+	13.4155i	1.55602	+	0.898368i	0.558388	+	0.829580i	$0.311420\pi$
$224$	33.0410			2.20764
$225$		0			0
$226$	24.8761			1.65474
$227$	−1.17081	−	0.675970i	−0.0777096	−	0.0448657i	0.460642	−	0.887586i	$-0.347619\pi$
$227$	−1.17081	−	0.675970i	−0.0777096	−	0.0448657i	−0.538351	+	0.842721i	$0.680953\pi$
$228$		0			0
$229$	−4.11775	−	7.13215i	−0.272108	−	0.471306i	0.697293	−	0.716786i	$-0.254388\pi$
$229$	−4.11775	−	7.13215i	−0.272108	−	0.471306i	−0.969402	+	0.245480i	$0.921054\pi$
$230$		0			0
$231$		0			0
$232$	−2.46456	−	1.42291i	−0.161806	−	0.0934188i
$233$		−	8.58744i		−	0.562582i	−0.959623	−	0.281291i	$-0.909237\pi$
$233$		−	8.58744i		−	0.562582i	0.959623	−	0.281291i	$-0.0907626\pi$
$234$		0			0
$235$		0			0
$236$	4.90645	−	8.49822i	0.319383	−	0.553187i
$237$		0			0
$238$	9.98025	−	5.76210i	0.646923	−	0.373501i
$239$	11.9623	+	20.7193i	0.773775	+	1.34022i	0.935480	+	0.353378i	$0.114967\pi$
$239$	11.9623	+	20.7193i	0.773775	+	1.34022i	−0.161706	+	0.986839i	$0.551700\pi$
$240$		0			0
$241$	3.12015	−	5.40426i	0.200987	−	0.348119i	−0.747860	−	0.663857i	$-0.768919\pi$
$241$	3.12015	−	5.40426i	0.200987	−	0.348119i	0.948847	+	0.315737i	$0.102252\pi$
$242$			19.1345i			1.23001i
$243$		0			0
$244$	8.98516			0.575216
$245$		0			0
$246$		0			0
$247$	−0.363716	+	0.209991i	−0.0231427	+	0.0133614i
$248$	−4.89363	+	2.82534i	−0.310746	+	0.179409i
$249$		0			0
$250$		0			0
$251$	28.5726			1.80349			0.901743	−	0.432272i	$-0.142288\pi$
$251$	28.5726			1.80349			0.901743	+	0.432272i	$0.142288\pi$
$252$		0			0
$253$		−	6.47580i		−	0.407130i
$254$	−7.38356	+	12.7887i	−0.463286	+	0.802434i
$255$		0			0
$256$	−4.49258	−	7.78138i	−0.280786	−	0.486336i
$257$	−15.5885	+	9.00000i	−0.972381	+	0.561405i	−0.899961	−	0.435970i	$-0.856405\pi$
$257$	−15.5885	+	9.00000i	−0.972381	+	0.561405i	−0.0724199	+	0.997374i	$0.523072\pi$
$258$		0			0
$259$	15.3724	−	26.6258i	0.955196	−	1.65445i
$260$		0			0
$261$		0			0
$262$		−	12.5168i		−	0.773289i
$263$	27.5991	+	15.9344i	1.70183	+	0.982555i	0.943904	+	0.330220i	$0.107123\pi$
$263$	27.5991	+	15.9344i	1.70183	+	0.982555i	0.757931	+	0.652335i	$0.226211\pi$
$264$		0			0
$265$		0			0
$266$	−2.76210	−	4.78410i	−0.169355	−	0.293332i
$267$		0			0
$268$	−16.5838	−	9.57468i	−1.01302	−	0.584867i
$269$	−31.4971			−1.92041			−0.960207	−	0.279289i	$-0.909901\pi$
$269$	−31.4971			−1.92041			−0.960207	+	0.279289i	$0.909901\pi$
$270$		0			0
$271$	−3.24030			−0.196834			−0.0984172	−	0.995145i	$-0.531378\pi$
$271$	−3.24030			−0.196834			−0.0984172	+	0.995145i	$0.531378\pi$
$272$	−3.71413	−	2.14435i	−0.225202	−	0.130020i
$273$		0			0
$274$	7.79001	+	13.4927i	0.470612	+	0.815123i
$275$		0			0
$276$		0			0
$277$	−4.83660	−	2.79241i	−0.290603	−	0.167780i	0.347611	−	0.937639i	$-0.386993\pi$
$277$	−4.83660	−	2.79241i	−0.290603	−	0.167780i	−0.638214	+	0.769859i	$0.720326\pi$
$278$			16.6890i			1.00094i
$279$		0			0
$280$		0			0
$281$	−12.0521	+	20.8749i	−0.718969	+	1.24529i	0.242440	+	0.970166i	$0.422052\pi$
$281$	−12.0521	+	20.8749i	−0.718969	+	1.24529i	−0.961409	+	0.275124i	$0.911281\pi$
$282$		0			0
$283$	−9.12989	+	5.27114i	−0.542715	+	0.313337i	−0.746179	−	0.665746i	$-0.768114\pi$
$283$	−9.12989	+	5.27114i	−0.542715	+	0.313337i	0.203463	+	0.979083i	$0.434780\pi$
$284$	7.19275	+	12.4582i	0.426811	+	0.739259i
$285$		0			0
$286$	−0.913870	+	1.58287i	−0.0540383	+	0.0935970i
$287$		−	0.734191i		−	0.0433379i
$288$		0			0
$289$	15.1723			0.892486
$290$		0			0
$291$		0			0
$292$	25.1438	−	14.5168i	1.47143	−	0.849530i
$293$	−16.4481	+	9.49629i	−0.960906	+	0.554779i	−0.896452	−	0.443141i	$-0.853864\pi$
$293$	−16.4481	+	9.49629i	−0.960906	+	0.554779i	−0.0644541	+	0.997921i	$0.520531\pi$
$294$		0			0
$295$		0			0
$296$	5.52420			0.321088
$297$		0			0
$298$			22.0813i			1.27914i
$299$	−1.55211	+	2.68833i	−0.0897607	+	0.155470i
$300$		0			0
$301$	1.67597	+	2.90286i	0.0966013	+	0.167318i
$302$	−31.9712	+	18.4586i	−1.83973	+	1.06217i
$303$		0			0
$304$	−1.02791	+	1.78039i	−0.0589546	+	0.102112i
$305$		0			0
$306$		0			0
$307$		−	29.4791i		−	1.68246i	−0.540679	−	0.841229i	$-0.681833\pi$
$307$		−	29.4791i		−	1.68246i	0.540679	−	0.841229i	$-0.318167\pi$
$308$	−11.2519	−	6.49629i	−0.641137	−	0.370161i
$309$		0			0
$310$		0			0
$311$	−4.70628	−	8.15152i	−0.266869	−	0.462230i	0.701183	−	0.712982i	$-0.252656\pi$
$311$	−4.70628	−	8.15152i	−0.266869	−	0.462230i	−0.968052	+	0.250751i	$0.919322\pi$
$312$		0			0
$313$	−10.0641	−	5.81050i	−0.568855	−	0.328429i	0.187837	−	0.982200i	$-0.439852\pi$
$313$	−10.0641	−	5.81050i	−0.568855	−	0.328429i	−0.756692	+	0.653771i	$0.773186\pi$
$314$	5.28128			0.298040
$315$		0			0
$316$	24.3297			1.36865
$317$	7.94984	+	4.58984i	0.446507	+	0.257791i	0.706354	−	0.707859i	$-0.250339\pi$
$317$	7.94984	+	4.58984i	0.446507	+	0.257791i	−0.259847	+	0.965650i	$0.583672\pi$
$318$		0			0
$319$	−2.62015	−	4.53824i	−0.146700	−	0.254092i
$320$		0			0
$321$		0			0
$322$	−35.3607	−	20.4155i	−1.97057	−	1.13771i
$323$			0.876139i			0.0487497i
$324$		0			0
$325$		0			0
$326$	−8.84081	+	15.3127i	−0.489647	+	0.848094i
$327$		0			0
$328$	0.114245	−	0.0659593i	0.00630812	−	0.00364199i
$329$	−22.2826	−	38.5946i	−1.22848	−	2.12779i
$330$		0			0
$331$	3.61033	−	6.25327i	0.198442	−	0.343711i	−0.749582	−	0.661912i	$-0.769745\pi$
$331$	3.61033	−	6.25327i	0.198442	−	0.343711i	0.948023	+	0.318201i	$0.103079\pi$
$332$		−	28.8310i		−	1.58231i
$333$		0			0
$334$	26.5652			1.45358
$335$		0			0
$336$		0			0
$337$	−1.97791	+	1.14195i	−0.107744	+	0.0622059i	−0.552904	−	0.833245i	$-0.686480\pi$
$337$	−1.97791	+	1.14195i	−0.107744	+	0.0622059i	0.445160	+	0.895451i	$0.353147\pi$
$338$	−22.7276	+	13.1218i	−1.23622	+	0.713731i
$339$		0			0
$340$		0			0
$341$	−10.4051			−0.563470
$342$		0			0
$343$			11.0181i			0.594921i
$344$	−0.301136	+	0.521583i	−0.0162362	+	0.0281219i
$345$		0			0
$346$	24.0410	+	41.6402i	1.29245	+	2.23859i
$347$	−0.613740	+	0.354343i	−0.0329473	+	0.0190221i	−0.516383	−	0.856358i	$-0.672722\pi$
$347$	−0.613740	+	0.354343i	−0.0329473	+	0.0190221i	0.483436	+	0.875380i	$0.339389\pi$
$348$		0			0
$349$	−10.6723	+	18.4849i	−0.571273	+	0.989474i	0.425163	+	0.905117i	$0.360217\pi$
$349$	−10.6723	+	18.4849i	−0.571273	+	0.989474i	−0.996436	+	0.0843569i	$0.973116\pi$
$350$		0			0
$351$		0			0
$352$			10.9320i			0.582675i
$353$	−8.74408	−	5.04840i	−0.465401	−	0.268699i	0.248912	−	0.968526i	$-0.419927\pi$
$353$	−8.74408	−	5.04840i	−0.465401	−	0.268699i	−0.714312	+	0.699827i	$0.753260\pi$
$354$		0			0
$355$		0			0
$356$	−3.52791	−	6.11052i	−0.186979	−	0.323857i
$357$		0			0
$358$	4.02534	+	2.32403i	0.212746	+	0.122829i
$359$	30.5578			1.61278			0.806388	−	0.591386i	$-0.201419\pi$
$359$	30.5578			1.61278			0.806388	+	0.591386i	$0.201419\pi$
$360$		0			0
$361$	−18.5800			−0.977896
$362$	0.846198	+	0.488553i	0.0444752	+	0.0256778i
$363$		0			0
$364$	3.11404	+	5.39367i	0.163220	+	0.282705i
$365$		0			0
$366$		0			0
$367$	−6.21778	−	3.58984i	−0.324566	−	0.187388i	0.328860	−	0.944379i	$-0.393336\pi$
$367$	−6.21778	−	3.58984i	−0.324566	−	0.187388i	−0.653426	+	0.756991i	$0.726669\pi$
$368$			15.1952i			0.792102i
$369$		0			0
$370$		0			0
$371$	8.52420	−	14.7643i	0.442554	−	0.766527i
$372$		0			0
$373$	−18.9872	+	10.9623i	−0.983120	+	0.567605i	−0.903211	−	0.429197i	$-0.858796\pi$
$373$	−18.9872	+	10.9623i	−0.983120	+	0.567605i	−0.0799096	+	0.996802i	$0.525463\pi$
$374$	1.90645	+	3.30207i	0.0985803	+	0.170746i
$375$		0			0
$376$	4.00371	−	6.93463i	0.206476	−	0.357626i
$377$			2.51197i			0.129373i
$378$		0			0
$379$	17.3929			0.893414			0.446707	−	0.894680i	$-0.352597\pi$
$379$	17.3929			0.893414			0.446707	+	0.894680i	$0.352597\pi$
$380$		0			0
$381$		0			0
$382$	−36.5449	+	21.0992i	−1.86980	+	1.07953i
$383$	−0.412055	+	0.237900i	−0.0210550	+	0.0121561i	−0.510491	−	0.859883i	$-0.670536\pi$
$383$	−0.412055	+	0.237900i	−0.0210550	+	0.0121561i	0.489436	+	0.872039i	$0.337203\pi$
$384$		0			0
$385$		0			0
$386$	−41.5752			−2.11612
$387$		0			0
$388$		−	31.9394i		−	1.62148i
$389$	2.79372	−	4.83886i	0.141647	−	0.245340i	−0.786470	−	0.617629i	$-0.788094\pi$
$389$	2.79372	−	4.83886i	0.141647	−	0.245340i	0.928117	+	0.372289i	$0.121427\pi$
$390$		0			0
$391$	3.23790	+	5.60821i	0.163748	+	0.283619i
$392$	−6.16528	+	3.55953i	−0.311394	+	0.179783i
$393$		0			0
$394$	16.2510	−	28.1475i	0.818712	−	1.41805i
$395$		0			0
$396$		0			0
$397$		−	3.75228i		−	0.188321i	−0.995557	−	0.0941607i	$-0.969983\pi$
$397$		−	3.75228i		−	0.188321i	0.995557	−	0.0941607i	$-0.0300168\pi$
$398$	−6.47649	−	3.73921i	−0.324637	−	0.187429i
$399$		0			0
$400$		0			0
$401$	11.7826	+	20.4080i	0.588394	+	1.01913i	0.994443	+	0.105278i	$0.0335731\pi$
$401$	11.7826	+	20.4080i	0.588394	+	1.01913i	−0.406048	+	0.913852i	$0.633094\pi$
$402$		0			0
$403$	4.31954	+	2.49389i	0.215172	+	0.124229i
$404$	−3.45355			−0.171820
$405$		0			0
$406$	−33.0410			−1.63980
$407$	8.80944	+	5.08613i	0.436668	+	0.252110i
$408$		0			0
$409$	0.524200	+	0.907940i	0.0259200	+	0.0448948i	0.878694	−	0.477385i	$-0.158415\pi$
$409$	0.524200	+	0.907940i	0.0259200	+	0.0448948i	−0.852774	+	0.522279i	$0.825082\pi$
$410$		0			0
$411$		0			0
$412$	15.3256	+	8.84823i	0.755037	+	0.435921i
$413$		−	17.0484i		−	0.838897i
$414$		0			0
$415$		0			0
$416$	2.62015	−	4.53824i	0.128464	−	0.222505i
$417$		0			0
$418$	1.58287	−	0.913870i	0.0774207	−	0.0446988i
$419$	12.9599	+	22.4471i	0.633131	+	1.09661i	0.986908	+	0.161285i	$0.0515638\pi$
$419$	12.9599	+	22.4471i	0.633131	+	1.09661i	−0.353777	+	0.935330i	$0.615103\pi$
$420$		0			0
$421$	−3.82032	+	6.61699i	−0.186191	+	0.322492i	−0.943977	−	0.330011i	$-0.892948\pi$
$421$	−3.82032	+	6.61699i	−0.186191	+	0.322492i	0.757786	+	0.652503i	$0.226281\pi$
$422$		−	31.2765i		−	1.52252i
$423$		0			0
$424$	3.06324			0.148764
$425$		0			0
$426$		0			0
$427$	13.5189	−	7.80516i	0.654227	−	0.377718i
$428$	−2.46456	+	1.42291i	−0.119129	+	0.0687791i
$429$		0			0
$430$		0			0
$431$	−7.98516			−0.384632			−0.192316	−	0.981333i	$-0.561600\pi$
$431$	−7.98516			−0.384632			−0.192316	+	0.981333i	$0.561600\pi$
$432$		0			0
$433$		−	12.5120i		−	0.601287i	−0.953737	−	0.300644i	$-0.902799\pi$
$433$		−	12.5120i		−	0.601287i	0.953737	−	0.300644i	$-0.0972014\pi$
$434$	−32.8031	+	56.8166i	−1.57460	+	2.72728i
$435$		0			0
$436$	−16.5861	−	28.7280i	−0.794332	−	1.37582i
$437$	2.68833	−	1.55211i	0.128600	−	0.0742474i
$438$		0			0
$439$	−4.38225	+	7.59028i	−0.209153	+	0.362264i	−0.951448	−	0.307809i	$-0.900404\pi$
$439$	−4.38225	+	7.59028i	−0.209153	+	0.362264i	0.742295	+	0.670074i	$0.233737\pi$
$440$		0			0
$441$		0			0
$442$		−	1.82774i		−	0.0869367i
$443$	3.17914	+	1.83548i	0.151046	+	0.0872062i	0.573618	−	0.819123i	$-0.305539\pi$
$443$	3.17914	+	1.83548i	0.151046	+	0.0872062i	−0.422572	+	0.906329i	$0.638873\pi$
$444$		0			0
$445$		0			0
$446$	27.9865	+	48.4740i	1.32520	+	2.29531i
$447$		0			0
$448$	37.2419	+	21.5016i	1.75951	+	1.01586i
$449$	28.1723			1.32953			0.664766	−	0.747052i	$-0.268531\pi$
$449$	28.1723			1.32953			0.664766	+	0.747052i	$0.268531\pi$
$450$		0			0
$451$	0.242915			0.0114384
$452$	24.2884	+	14.0229i	1.14243	+	0.659581i
$453$		0			0
$454$	−1.41016	−	2.44247i	−0.0661821	−	0.114631i
$455$		0			0
$456$		0			0
$457$	30.5375	+	17.6308i	1.42848	+	0.824735i	0.997001	−	0.0773867i	$-0.0246576\pi$
$457$	30.5375	+	17.6308i	1.42848	+	0.824735i	0.431482	+	0.902122i	$0.357991\pi$
$458$		−	17.1803i		−	0.802784i
$459$		0			0
$460$		0			0
$461$	17.3384	−	30.0310i	0.807530	−	1.39868i	−0.107039	−	0.994255i	$-0.534137\pi$
$461$	17.3384	−	30.0310i	0.807530	−	1.39868i	0.914570	−	0.404428i	$-0.132530\pi$
$462$		0			0
$463$	6.45080	−	3.72437i	0.299794	−	0.173086i	−0.342556	−	0.939497i	$-0.611293\pi$
$463$	6.45080	−	3.72437i	0.299794	−	0.173086i	0.642350	+	0.766411i	$0.277959\pi$
$464$	6.14806	+	10.6488i	0.285417	+	0.494356i
$465$		0			0
$466$	8.95725	−	15.5144i	0.414937	−	0.718692i
$467$			29.9655i			1.38664i	0.720630	+	0.693319i	$0.243852\pi$
$467$			29.9655i			1.38664i	−0.720630	+	0.693319i	$0.756148\pi$
$468$		0			0
$469$	−33.2691			−1.53622
$470$		0			0
$471$		0			0
$472$	2.65284	−	1.53162i	0.122107	−	0.0704984i
$473$	−0.960443	+	0.554512i	−0.0441612	+	0.0254965i
$474$		0			0
$475$		0			0
$476$	12.9926			0.595514
$477$		0			0
$478$			49.9097i			2.28282i
$479$	3.99258	−	6.91535i	0.182426	−	0.315971i	−0.760280	−	0.649595i	$-0.774938\pi$
$479$	3.99258	−	6.91535i	0.182426	−	0.315971i	0.942706	+	0.333625i	$0.108272\pi$
$480$		0			0
$481$	−2.43807	−	4.22286i	−0.111166	−	0.192546i
$482$	11.2740	−	6.50904i	0.513516	−	0.296479i
$483$		0			0
$484$	−10.7863	+	18.6824i	−0.490286	+	0.849201i
$485$		0			0
$486$		0			0
$487$			11.9442i			0.541243i	0.962686	+	0.270621i	$0.0872291\pi$
$487$			11.9442i			0.541243i	−0.962686	+	0.270621i	$0.912771\pi$
$488$	2.42907	+	1.40242i	0.109959	+	0.0634847i
$489$		0			0
$490$		0			0
$491$	−4.61033	−	7.98533i	−0.208061	−	0.360373i	0.743042	−	0.669244i	$-0.233382\pi$
$491$	−4.61033	−	7.98533i	−0.208061	−	0.360373i	−0.951104	+	0.308872i	$0.900049\pi$
$492$		0			0
$493$	4.53824	+	2.62015i	0.204392	+	0.118006i
$494$	−0.876139			−0.0394193
$495$		0			0
$496$	24.4152			1.09627
$497$	21.6442	+	12.4963i	0.970876	+	0.560535i
$498$		0			0
$499$	15.0861	+	26.1299i	0.675348	+	1.16974i	0.976367	+	0.216119i	$0.0693399\pi$
$499$	15.0861	+	26.1299i	0.675348	+	1.16974i	−0.301019	+	0.953618i	$0.597327\pi$
$500$		0			0
$501$		0			0
$502$	51.6204	+	29.8031i	2.30393	+	1.33018i
$503$			10.5981i			0.472546i	0.971687	+	0.236273i	$0.0759260\pi$
$503$			10.5981i			0.472546i	−0.971687	+	0.236273i	$0.924074\pi$
$504$		0			0
$505$		0			0
$506$	6.75468	−	11.6995i	0.300282	−	0.520104i
$507$		0			0
$508$	−14.4182	+	8.32435i	−0.639704	+	0.369333i
$509$	−14.3761	−	24.9002i	−0.637211	−	1.10368i	−0.986042	−	0.166496i	$-0.946755\pi$
$509$	−14.3761	−	24.9002i	−0.637211	−	1.10368i	0.348831	−	0.937186i	$-0.386579\pi$
$510$		0			0
$511$	25.2207	−	43.6835i	1.11570	−	1.93244i
$512$		−	30.3094i		−	1.33950i
$513$		0			0
$514$	−37.5503			−1.65627
$515$		0			0
$516$		0			0
$517$	12.7694	−	7.37243i	0.561599	−	0.324239i
$518$	55.5449	−	32.0689i	2.44050	−	1.40903i
$519$		0			0
$520$		0			0
$521$	36.0942			1.58132			0.790658	−	0.612259i	$-0.209739\pi$
$521$	36.0942			1.58132			0.790658	+	0.612259i	$0.209739\pi$
$522$		0			0
$523$			11.1297i			0.486669i	0.969942	+	0.243334i	$0.0782412\pi$
$523$			11.1297i			0.486669i	−0.969942	+	0.243334i	$0.921759\pi$
$524$	7.05582	−	12.2210i	0.308235	−	0.533878i
$525$		0			0
$526$	33.2411	+	57.5754i	1.44938	+	2.51041i
$527$	9.01112	−	5.20257i	0.392531	−	0.226628i
$528$		0			0
$529$	−0.0279088	+	0.0483395i	−0.00121343	+	0.00210172i
$530$		0			0
$531$		0			0
$532$		−	6.22808i		−	0.270021i
$533$	−0.100842	−	0.0582214i	−0.00436797	−	0.00252185i
$534$		0			0
$535$		0			0
$536$	−2.98887	−	5.17688i	−0.129100	−	0.223607i
$537$		0			0
$538$	−56.9040	−	32.8536i	−2.45331	−	1.41642i
$539$	−13.1090			−0.564646
$540$		0			0
$541$	−34.7374			−1.49348			−0.746740	−	0.665116i	$-0.768382\pi$
$541$	−34.7374			−1.49348			−0.746740	+	0.665116i	$0.768382\pi$
$542$	−5.85407	−	3.37985i	−0.251454	−	0.145177i
$543$		0			0
$544$	−5.46598	−	9.46735i	−0.234352	−	0.405909i
$545$		0			0
$546$		0			0
$547$	−2.35087	−	1.35727i	−0.100516	−	0.0580328i	0.448899	−	0.893582i	$-0.351816\pi$
$547$	−2.35087	−	1.35727i	−0.100516	−	0.0580328i	−0.549415	+	0.835549i	$0.685149\pi$
$548$			17.5652i			0.750347i
$549$		0			0
$550$		0			0
$551$	1.25599	−	2.17543i	0.0535068	−	0.0926766i
$552$		0			0
$553$	36.6061	−	21.1345i	1.55665	−	0.898732i
$554$	−5.82534	−	10.0898i	−0.247495	−	0.428674i
$555$		0			0
$556$	−9.40776	+	16.2947i	−0.398978	+	0.691050i
$557$		−	8.93676i		−	0.378663i	−0.981913	−	0.189331i	$-0.939368\pi$
$557$		−	8.93676i		−	0.378663i	0.981913	−	0.189331i	$-0.0606321\pi$
$558$		0			0
$559$	0.531618			0.0224850
$560$		0			0
$561$		0			0
$562$	−43.5477	+	25.1423i	−1.83695	+	1.06056i
$563$	8.10826	−	4.68130i	0.341722	−	0.197293i	−0.319311	−	0.947650i	$-0.603451\pi$
$563$	8.10826	−	4.68130i	0.341722	−	0.197293i	0.661033	+	0.750357i	$0.270118\pi$
$564$		0			0
$565$		0			0
$566$	−21.9926			−0.924417
$567$		0			0
$568$			4.49064i			0.188423i
$569$	−17.9368	+	31.0674i	−0.751948	+	1.30241i	0.194929	+	0.980817i	$0.437552\pi$
$569$	−17.9368	+	31.0674i	−0.751948	+	1.30241i	−0.946877	+	0.321595i	$0.895781\pi$
$570$		0			0
$571$	−10.0000	−	17.3205i	−0.418487	−	0.724841i	0.577301	−	0.816532i	$-0.304106\pi$
$571$	−10.0000	−	17.3205i	−0.418487	−	0.724841i	−0.995788	+	0.0916910i	$0.970773\pi$
$572$	−1.78455	+	1.03031i	−0.0746159	+	0.0430795i
$573$		0			0
$574$	0.765809	−	1.32642i	0.0319643	−	0.0553637i
$575$		0			0
$576$		0			0
$577$		−	1.35675i		−	0.0564821i	−0.999601	−	0.0282411i	$-0.991009\pi$
$577$		−	1.35675i		−	0.0564821i	0.999601	−	0.0282411i	$-0.00899060\pi$
$578$	27.4108	+	15.8257i	1.14014	+	0.658260i
$579$		0			0
$580$		0			0
$581$	−25.0447	−	43.3787i	−1.03903	−	1.79965i
$582$		0			0
$583$	4.88494	+	2.82032i	0.202314	+	0.116806i
$584$	9.06324			0.375039
$585$		0			0
$586$	−39.6210			−1.63673
$587$	−24.9329	−	14.3950i	−1.02909	−	0.594145i	−0.112366	−	0.993667i	$-0.535843\pi$
$587$	−24.9329	−	14.3950i	−1.02909	−	0.594145i	−0.916724	+	0.399521i	$0.869176\pi$
$588$		0			0
$589$	−2.49389	−	4.31954i	−0.102759	−	0.177984i
$590$		0			0
$591$		0			0
$592$	−20.6709	−	11.9344i	−0.849570	−	0.490499i
$593$		−	30.9171i		−	1.26961i	−0.772671	−	0.634807i	$-0.781080\pi$
$593$		−	30.9171i		−	1.26961i	0.772671	−	0.634807i	$-0.218920\pi$
$594$		0			0
$595$		0			0
$596$	−12.4474	+	21.5596i	−0.509867	+	0.883115i
$597$		0			0
$598$	−5.60821	+	3.23790i	−0.229337	+	0.132408i
$599$	−0.696460	−	1.20630i	−0.0284566	−	0.0492882i	0.851446	−	0.524442i	$-0.175726\pi$
$599$	−0.696460	−	1.20630i	−0.0284566	−	0.0492882i	−0.879903	+	0.475153i	$0.842393\pi$
$600$		0			0
$601$	−4.41256	+	7.64279i	−0.179992	+	0.311756i	−0.941878	−	0.335956i	$-0.890941\pi$
$601$	−4.41256	+	7.64279i	−0.179992	+	0.311756i	0.761885	+	0.647712i	$0.224274\pi$
$602$			6.99258i			0.284996i
$603$		0			0
$604$	−41.6210			−1.69353
$605$		0			0
$606$		0			0
$607$	1.86783	−	1.07839i	0.0758129	−	0.0437706i	−0.461614	−	0.887081i	$-0.652730\pi$
$607$	1.86783	−	1.07839i	0.0758129	−	0.0437706i	0.537427	+	0.843310i	$0.319396\pi$
$608$	−4.53824	+	2.62015i	−0.184050	+	0.106261i
$609$		0			0
$610$		0			0
$611$	−7.06804			−0.285942
$612$		0			0
$613$		−	9.57521i		−	0.386739i	−0.981126	−	0.193370i	$-0.938058\pi$
$613$		−	9.57521i		−	0.386739i	0.981126	−	0.193370i	$-0.0619416\pi$
$614$	30.7486	−	53.2581i	1.24091	−	2.14932i
$615$		0			0
$616$	−2.02791	−	3.51244i	−0.0817068	−	0.141520i
$617$	−32.6291	+	18.8384i	−1.31360	+	0.758406i	−0.982690	−	0.185258i	$-0.940688\pi$
$617$	−32.6291	+	18.8384i	−1.31360	+	0.758406i	−0.330907	+	0.943663i	$0.607355\pi$
$618$		0			0
$619$	8.55211	−	14.8127i	0.343738	−	0.595372i	−0.641385	−	0.767219i	$-0.721640\pi$
$619$	8.55211	−	14.8127i	0.343738	−	0.595372i	0.985124	+	0.171847i	$0.0549734\pi$
$620$		0			0
$621$		0			0
$622$		−	19.6358i		−	0.787325i
$623$	−10.6161	−	6.12920i	−0.425324	−	0.245561i
$624$		0			0
$625$		0			0
$626$	−12.1215	−	20.9950i	−0.484471	−	0.839128i
$627$		0			0
$628$	5.15650	+	2.97711i	0.205767	+	0.118799i
$629$	−10.1723			−0.405595
$630$		0			0
$631$	33.1090			1.31805			0.659025	−	0.752121i	$-0.270969\pi$
$631$	33.1090			1.31805			0.659025	+	0.752121i	$0.270969\pi$
$632$	6.57734	+	3.79743i	0.261632	+	0.151054i
$633$		0			0
$634$	9.57500	+	16.5844i	0.380272	+	0.658650i
$635$		0			0
$636$		0			0
$637$	5.44201	+	3.14195i	0.215620	+	0.124489i
$638$		−	10.9320i		−	0.432800i
$639$		0			0
$640$		0			0
$641$	−11.5763	+	20.0508i	−0.457237	+	0.791957i	−0.998814	−	0.0486939i	$-0.984494\pi$
$641$	−11.5763	+	20.0508i	−0.457237	+	0.791957i	0.541577	+	0.840651i	$0.317827\pi$
$642$		0			0
$643$	37.2944	−	21.5319i	1.47075	−	0.849137i	0.471287	−	0.881980i	$-0.343789\pi$
$643$	37.2944	−	21.5319i	1.47075	−	0.849137i	0.999461	+	0.0328430i	$0.0104561\pi$
$644$	−23.0168	−	39.8662i	−0.906988	−	1.57095i
$645$		0			0
$646$	−0.913870	+	1.58287i	−0.0359557	+	0.0622771i
$647$		−	20.6439i		−	0.811595i	−0.913963	−	0.405798i	$-0.866994\pi$
$647$		−	20.6439i		−	0.811595i	0.913963	−	0.405798i	$-0.133006\pi$
$648$		0			0
$649$	5.64064			0.221415
$650$		0			0
$651$		0			0
$652$	−17.2638	+	9.96728i	−0.676104	+	0.390349i
$653$	−5.91942	+	3.41758i	−0.231645	+	0.133740i	−0.611331	−	0.791375i	$-0.709365\pi$
$653$	−5.91942	+	3.41758i	−0.231645	+	0.133740i	0.379686	+	0.925116i	$0.376032\pi$
$654$		0			0
$655$		0			0
$656$	−0.569988			−0.0222543
$657$		0			0
$658$		−	92.9688i		−	3.62430i
$659$	13.4307	−	23.2626i	0.523184	−	0.906181i	−0.476452	−	0.879200i	$-0.658077\pi$
$659$	13.4307	−	23.2626i	0.523184	−	0.906181i	0.999636	−	0.0269806i	$-0.00858924\pi$
$660$		0			0
$661$	1.06063	+	1.83706i	0.0412535	+	0.0714532i	0.885915	−	0.463848i	$-0.153532\pi$
$661$	1.06063	+	1.83706i	0.0412535	+	0.0714532i	−0.844661	+	0.535301i	$0.820198\pi$
$662$	13.0451	−	7.53162i	0.507014	−	0.292725i
$663$		0			0
$664$	4.50000	−	7.79423i	0.174634	−	0.302475i
$665$		0			0
$666$		0			0
$667$		−	18.5667i		−	0.718907i
$668$	25.9375	+	14.9750i	1.00355	+	0.579401i
$669$		0			0
$670$		0			0
$671$	2.58242	+	4.47288i	0.0996933	+	0.172674i
$672$		0			0
$673$	−30.1553	−	17.4102i	−1.16240	−	0.671112i	−0.210523	−	0.977589i	$-0.567517\pi$
$673$	−30.1553	−	17.4102i	−1.16240	−	0.671112i	−0.951878	+	0.306477i	$0.900850\pi$
$674$	−4.76450			−0.183522
$675$		0			0
$676$	−29.5874			−1.13798
$677$	21.3772	+	12.3421i	0.821592	+	0.474346i	0.850965	−	0.525222i	$-0.176018\pi$
$677$	21.3772	+	12.3421i	0.821592	+	0.474346i	−0.0293735	+	0.999569i	$0.509351\pi$
$678$		0			0
$679$	−27.7449	−	48.0555i	−1.06475	−	1.84420i
$680$		0			0
$681$		0			0
$682$	−18.7984	−	10.8532i	−0.719827	−	0.415592i
$683$			38.4610i			1.47167i	0.677162	+	0.735834i	$0.263210\pi$
$683$			38.4610i			1.47167i	−0.677162	+	0.735834i	$0.736790\pi$
$684$		0			0
$685$		0			0
$686$	−11.4926	+	19.9057i	−0.438789	+	0.760005i
$687$		0			0
$688$	2.25363	−	1.30114i	0.0859190	−	0.0496054i
$689$	−1.35194	−	2.34163i	−0.0515048	−	0.0892089i
$690$		0			0
$691$	−0.240304	+	0.416219i	−0.00914159	+	0.0158337i	−0.870560	−	0.492062i	$-0.836243\pi$
$691$	−0.240304	+	0.416219i	−0.00914159	+	0.0158337i	0.861418	+	0.507896i	$0.169577\pi$
$692$			54.2084i			2.06070i
$693$		0			0
$694$	−1.47841			−0.0561197
$695$		0			0
$696$		0			0
$697$	−0.210370	+	0.121457i	−0.00796835	+	0.00460053i
$698$	−38.5619	+	22.2637i	−1.45959	+	0.842694i
$699$		0			0
$700$		0			0
$701$	18.1797			0.686637			0.343318	−	0.939219i	$-0.388449\pi$
$701$	18.1797			0.686637			0.343318	+	0.939219i	$0.388449\pi$
$702$		0			0
$703$			4.87614i			0.183907i
$704$	−7.11404	+	12.3219i	−0.268120	+	0.464398i
$705$		0			0
$706$	−10.5316	−	18.2413i	−0.396363	−	0.686520i
$707$	−5.19615	+	3.00000i	−0.195421	+	0.112827i
$708$		0			0
$709$	3.59355	−	6.22421i	0.134959	−	0.233755i	−0.790623	−	0.612303i	$-0.790243\pi$
$709$	3.59355	−	6.22421i	0.134959	−	0.233755i	0.925582	+	0.378548i	$0.123577\pi$
$710$		0			0
$711$		0			0
$712$		−	2.20257i		−	0.0825449i
$713$	−31.9270	−	18.4331i	−1.19568	−	0.690323i
$714$		0			0
$715$		0			0
$716$	2.62015	+	4.53824i	0.0979197	+	0.169602i
$717$		0			0
$718$	55.2069	+	31.8737i	2.06030	+	1.18952i
$719$	12.5168			0.466797			0.233399	−	0.972381i	$-0.425015\pi$
$719$	12.5168			0.466797			0.233399	+	0.972381i	$0.425015\pi$
$720$		0			0
$721$	30.7449			1.14500
$722$	−33.5674	−	19.3802i	−1.24925	−	0.721255i
$723$		0			0
$724$	0.550803	+	0.954019i	0.0204704	+	0.0354558i
$725$		0			0
$726$		0			0
$727$	7.29699	+	4.21292i	0.270631	+	0.156249i	0.629174	−	0.777264i	$-0.283393\pi$
$727$	7.29699	+	4.21292i	0.270631	+	0.156249i	−0.358544	+	0.933513i	$0.616727\pi$
$728$			1.94418i			0.0720562i
$729$		0			0
$730$		0			0
$731$	0.554512	−	0.960443i	0.0205094	−	0.0355233i
$732$		0			0
$733$	−19.0526	+	11.0000i	−0.703722	+	0.406294i	−0.808732	−	0.588177i	$-0.799846\pi$
$733$	−19.0526	+	11.0000i	−0.703722	+	0.406294i	0.105010	+	0.994471i	$0.466513\pi$
$734$	−7.48887	−	12.9711i	−0.276419	−	0.478772i
$735$		0			0
$736$	−19.3663	+	33.5434i	−0.713852	+	1.23643i
$737$		−	11.0074i		−	0.405463i
$738$		0			0
$739$	1.81290			0.0666887			0.0333444	−	0.999444i	$-0.489384\pi$
$739$	1.81290			0.0666887			0.0333444	+	0.999444i	$0.489384\pi$
$740$		0			0
$741$		0			0
$742$	30.8003	−	17.7826i	1.13072	−	0.652819i
$743$	−17.4393	+	10.0686i	−0.639785	+	0.369380i	−0.784532	−	0.620089i	$-0.787097\pi$
$743$	−17.4393	+	10.0686i	−0.639785	+	0.369380i	0.144747	+	0.989469i	$0.453763\pi$
$744$		0			0
$745$		0			0
$746$	−45.7374			−1.67457
$747$		0			0
$748$			4.29873i			0.157177i
$749$	−2.47209	+	4.28179i	−0.0903282	+	0.156453i
$750$		0			0
$751$	−6.10662	−	10.5770i	−0.222834	−	0.385959i	0.732834	−	0.680408i	$-0.238197\pi$
$751$	−6.10662	−	10.5770i	−0.222834	−	0.385959i	−0.955667	+	0.294449i	$0.904864\pi$
$752$	−29.9628	+	17.2991i	−1.09263	+	0.630832i
$753$		0			0
$754$	−2.62015	+	4.53824i	−0.0954203	+	0.165273i
$755$		0			0
$756$		0			0
$757$		−	52.9533i		−	1.92462i	−0.271955	−	0.962310i	$-0.587670\pi$
$757$		−	52.9533i		−	1.92462i	0.271955	−	0.962310i	$-0.412330\pi$
$758$	31.4228	+	18.1419i	1.14133	+	0.658945i
$759$		0			0
$760$		0			0
$761$	−9.22677	−	15.9812i	−0.334470	−	0.579319i	0.648913	−	0.760863i	$-0.275224\pi$
$761$	−9.22677	−	15.9812i	−0.334470	−	0.579319i	−0.983383	+	0.181543i	$0.941891\pi$
$762$		0			0
$763$	−49.9105	−	28.8158i	−1.80688	−	1.04320i
$764$	−47.5752			−1.72121
$765$		0			0
$766$	−0.992582			−0.0358634
$767$	−2.34163	−	1.35194i	−0.0845513	−	0.0488157i
$768$		0			0
$769$	−2.22677	−	3.85688i	−0.0802995	−	0.139083i	0.823079	−	0.567927i	$-0.192254\pi$
$769$	−2.22677	−	3.85688i	−0.0802995	−	0.139083i	−0.903379	+	0.428844i	$0.858921\pi$
$770$		0			0
$771$		0			0
$772$	−40.5929	−	23.4363i	−1.46097	−	0.843491i
$773$		−	38.9368i		−	1.40046i	−0.713918	−	0.700229i	$-0.753081\pi$
$773$		−	38.9368i		−	1.40046i	0.713918	−	0.700229i	$-0.246919\pi$
$774$		0			0
$775$		0			0
$776$	4.98516	−	8.63456i	0.178957	−	0.309962i
$777$		0			0
$778$	10.0945	−	5.82806i	0.361905	−	0.208946i
$779$	0.0582214	+	0.100842i	0.00208600	+	0.00361305i
$780$		0			0
$781$	−4.13453	+	7.16121i	−0.147945	+	0.256248i
$782$			13.5094i			0.483094i
$783$		0			0
$784$	30.7597			1.09856
$785$		0			0
$786$		0			0
$787$	−29.6424	+	17.1140i	−1.05664	+	0.610050i	−0.924500	−	0.381182i	$-0.875517\pi$
$787$	−29.6424	+	17.1140i	−1.05664	+	0.610050i	−0.132137	+	0.991231i	$0.542184\pi$
$788$	31.7340	−	18.3216i	1.13048	−	0.652681i
$789$		0			0
$790$		0			0
$791$	48.7252			1.73247
$792$		0			0
$793$		−	2.47580i		−	0.0879183i
$794$	3.91387	−	6.77902i	0.138898	−	0.240578i
$795$		0			0
$796$	−4.21564	−	7.30171i	−0.149420	−	0.258802i
$797$	20.7547	−	11.9828i	0.735171	−	0.424451i	−0.0851400	−	0.996369i	$-0.527134\pi$
$797$	20.7547	−	11.9828i	0.735171	−	0.424451i	0.820311	+	0.571918i	$0.193800\pi$
$798$		0			0
$799$	−7.37243	+	12.7694i	−0.260818	+	0.451750i
$800$		0			0
$801$		0			0
$802$			49.1600i			1.73590i
$803$	14.4531	+	8.34452i	0.510040	+	0.294472i
$804$		0			0
$805$		0			0
$806$	5.20257	+	9.01112i	0.183253	+	0.317403i
$807$		0			0
$808$	−0.933638	−	0.539036i	−0.0328453	−	0.0189632i
$809$	−0.283896			−0.00998124			−0.00499062	−	0.999988i	$-0.501589\pi$
$809$	−0.283896			−0.00998124			−0.00499062	+	0.999988i	$0.501589\pi$
$810$		0			0
$811$	32.4413			1.13917			0.569584	−	0.821933i	$-0.307104\pi$
$811$	32.4413			1.13917			0.569584	+	0.821933i	$0.307104\pi$
$812$	−32.2603	−	18.6255i	−1.13211	−	0.653626i
$813$		0			0
$814$	10.6103	+	18.3776i	0.371892	+	0.644136i
$815$		0			0
$816$		0			0
$817$	−0.460395	−	0.265809i	−0.0161072	−	0.00929948i
$818$			2.18710i			0.0764701i
$819$		0			0
$820$		0			0
$821$	20.8347	−	36.0868i	0.727136	−	1.25944i	−0.230953	−	0.972965i	$-0.574184\pi$
$821$	20.8347	−	36.0868i	0.727136	−	1.25944i	0.958089	−	0.286472i	$-0.0924824\pi$
$822$		0			0
$823$	−16.7685	+	9.68130i	−0.584514	+	0.337469i	−0.762925	−	0.646487i	$-0.776237\pi$
$823$	−16.7685	+	9.68130i	−0.584514	+	0.337469i	0.178412	+	0.983956i	$0.442904\pi$
$824$	2.76210	+	4.78410i	0.0962223	+	0.166662i
$825$		0			0
$826$	17.7826	−	30.8003i	0.618735	−	1.07168i
$827$			18.8097i			0.654076i	0.945011	+	0.327038i	$0.106050\pi$
$827$			18.8097i			0.654076i	−0.945011	+	0.327038i	$0.893950\pi$
$828$		0			0
$829$	33.1016			1.14967			0.574833	−	0.818271i	$-0.305067\pi$
$829$	33.1016			1.14967			0.574833	+	0.818271i	$0.305067\pi$
$830$		0			0
$831$		0			0
$832$	5.90657	−	3.41016i	0.204774	−	0.118226i
$833$	11.3527	−	6.55451i	0.393349	−	0.227100i
$834$		0			0
$835$		0			0
$836$	2.06063			0.0712682
$837$		0			0
$838$			54.0719i			1.86788i
$839$	19.8482	−	34.3781i	0.685237	−	1.18687i	−0.288125	−	0.957593i	$-0.593032\pi$
$839$	19.8482	−	34.3781i	0.685237	−	1.18687i	0.973362	−	0.229273i	$-0.0736347\pi$
$840$		0			0
$841$	6.98777	+	12.1032i	0.240958	+	0.417351i
$842$	−13.8039	+	7.96969i	−0.475714	+	0.274654i
$843$		0			0
$844$	17.6308	−	30.5375i	0.606878	−	1.05114i
$845$		0			0
$846$		0			0
$847$			37.4791i			1.28780i
$848$	−11.4623	−	6.61775i	−0.393616	−	0.227254i
$849$		0			0
$850$		0			0
$851$	18.0205	+	31.2124i	0.617734	+	1.06995i
$852$		0			0
$853$	3.56494	+	2.05822i	0.122061	+	0.0704722i	0.559788	−	0.828636i	$-0.310883\pi$
$853$	3.56494	+	2.05822i	0.122061	+	0.0704722i	−0.437726	+	0.899108i	$0.644216\pi$
$854$	32.5652			1.11436
$855$		0			0
$856$	−0.888365			−0.0303637
$857$	−12.7694	−	7.37243i	−0.436195	−	0.251837i	0.265787	−	0.964032i	$-0.414368\pi$
$857$	−12.7694	−	7.37243i	−0.436195	−	0.251837i	−0.701982	+	0.712194i	$0.747701\pi$
$858$		0			0
$859$	−18.9269	−	32.7824i	−0.645779	−	1.11852i	−0.984121	−	0.177499i	$-0.943199\pi$
$859$	−18.9269	−	32.7824i	−0.645779	−	1.11852i	0.338342	−	0.941023i	$-0.390134\pi$
$860$		0			0
$861$		0			0
$862$	−14.4263	−	8.32905i	−0.491363	−	0.283688i
$863$		−	26.7704i		−	0.911274i	−0.890166	−	0.455637i	$-0.849412\pi$
$863$		−	26.7704i		−	0.911274i	0.890166	−	0.455637i	$-0.150588\pi$
$864$		0			0
$865$		0			0
$866$	13.0508	−	22.6047i	0.443484	−	0.768137i
$867$		0			0
$868$	−64.0560	+	36.9828i	−2.17420	+	1.25528i
$869$	6.99258	+	12.1115i	0.237207	+	0.410855i
$870$		0			0
$871$	−2.63824	+	4.56956i	−0.0893933	+	0.154834i
$872$		−	10.3552i		−	0.350671i
$873$		0			0
$874$	6.47580			0.219047
$875$		0			0
$876$		0			0
$877$	40.6756	−	23.4841i	1.37352	−	0.793001i	0.382148	−	0.924101i	$-0.375184\pi$
$877$	40.6756	−	23.4841i	1.37352	−	0.793001i	0.991369	+	0.131101i	$0.0418511\pi$
$878$	−15.8343	+	9.14195i	−0.534382	+	0.308526i
$879$		0			0
$880$		0			0
$881$	19.8055			0.667264			0.333632	−	0.942703i	$-0.391726\pi$
$881$	19.8055			0.667264			0.333632	+	0.942703i	$0.391726\pi$
$882$		0			0
$883$		−	6.20257i		−	0.208733i	−0.994539	−	0.104367i	$-0.966718\pi$
$883$		−	6.20257i		−	0.208733i	0.994539	−	0.104367i	$-0.0332815\pi$
$884$	1.03031	−	1.78455i	0.0346532	−	0.0600210i
$885$		0			0
$886$	3.82905	+	6.63210i	0.128639	+	0.222810i
$887$	11.6285	−	6.71370i	0.390446	−	0.225424i	−0.291907	−	0.956447i	$-0.594290\pi$
$887$	11.6285	−	6.71370i	0.390446	−	0.225424i	0.682353	+	0.731023i	$0.260957\pi$
$888$		0			0
$889$	−14.4623	+	25.0494i	−0.485049	+	0.840129i
$890$		0			0
$891$		0			0
$892$			63.1049i			2.11291i
$893$	6.12111	+	3.53402i	0.204835	+	0.118262i
$894$		0			0
$895$		0			0
$896$	11.8142	+	20.4628i	0.394685	+	0.683614i
$897$		0			0
$898$	50.8972	+	29.3855i	1.69846	+	0.980607i
$899$	−29.8325			−0.994971
$900$		0			0
$901$	−5.64064			−0.187917
$902$	0.438860	+	0.253376i	0.0146124	+	0.00843650i
$903$		0			0
$904$	4.37744	+	7.58196i	0.145592	+	0.252172i
$905$		0			0
$906$		0			0
$907$	0.583325	+	0.336783i	0.0193690	+	0.0111827i	0.509653	−	0.860380i	$-0.329774\pi$
$907$	0.583325	+	0.336783i	0.0193690	+	0.0111827i	−0.490284	+	0.871563i	$0.663107\pi$
$908$		−	3.17968i		−	0.105521i
$909$		0			0
$910$		0			0
$911$	−3.95485	+	6.85000i	−0.131030	+	0.226951i	−0.924074	−	0.382214i	$-0.875162\pi$
$911$	−3.95485	+	6.85000i	−0.131030	+	0.226951i	0.793044	+	0.609165i	$0.208495\pi$
$912$		0			0
$913$	14.3523	−	8.28630i	0.474992	−	0.274236i
$914$	36.7802	+	63.7052i	1.21658	+	2.10718i
$915$		0			0
$916$	9.68469	−	16.7744i	0.319991	−	0.554241i
$917$		−	24.5168i		−	0.809615i
$918$		0			0
$919$	8.58263			0.283115			0.141557	−	0.989930i	$-0.454789\pi$
$919$	8.58263			0.283115			0.141557	+	0.989930i	$0.454789\pi$
$920$		0			0
$921$		0			0
$922$	62.6486	−	36.1702i	2.06322	−	1.19120i
$923$	3.43277	−	1.98191i	0.112991	−	0.0652355i
$924$		0			0
$925$		0			0
$926$	15.5390			0.510644
$927$		0			0
$928$			31.3430i			1.02888i
$929$	−14.8081	+	25.6484i	−0.485838	+	0.841496i	−0.999868	−	0.0162766i	$-0.994819\pi$
$929$	−14.8081	+	25.6484i	−0.485838	+	0.841496i	0.514030	+	0.857772i	$0.328152\pi$
$930$		0			0
$931$	−3.14195	−	5.44201i	−0.102973	−	0.178355i
$932$	17.4912	−	10.0986i	0.572944	−	0.330789i
$933$		0			0
$934$	−31.2560	+	54.1370i	−1.02273	+	1.77142i
$935$		0			0
$936$		0			0
$937$			15.2058i			0.496753i	0.968664	+	0.248376i	$0.0798969\pi$
$937$			15.2058i			0.496753i	−0.968664	+	0.248376i	$0.920103\pi$
$938$	−60.1053	−	34.7018i	−1.96251	−	1.13305i
$939$		0			0
$940$		0			0
$941$	2.82643	+	4.89553i	0.0921391	+	0.159590i	0.908411	−	0.418078i	$-0.137296\pi$
$941$	2.82643	+	4.89553i	0.0921391	+	0.159590i	−0.816272	+	0.577668i	$0.803963\pi$
$942$		0			0
$943$	0.745356	+	0.430332i	0.0242721	+	0.0140135i
$944$	−13.2355			−0.430779
$945$		0			0
$946$	−2.31357			−0.0752206
$947$	34.9841	+	20.1981i	1.13683	+	0.656350i	0.945644	−	0.325204i	$-0.105433\pi$
$947$	34.9841	+	20.1981i	1.13683	+	0.656350i	0.191187	+	0.981554i	$0.438766\pi$
$948$		0			0
$949$	−4.00000	−	6.92820i	−0.129845	−	0.224899i
$950$		0			0
$951$		0			0
$952$	3.51244	+	2.02791i	0.113839	+	0.0657249i
$953$			22.9320i			0.742839i	0.928465	+	0.371419i	$0.121129\pi$
$953$			22.9320i			0.742839i	−0.928465	+	0.371419i	$0.878871\pi$
$954$		0			0
$955$		0			0
$956$	−28.1345	+	48.7304i	−0.909936	+	1.57605i
$957$		0			0
$958$	14.4263	−	8.32905i	0.466094	−	0.269099i
$959$	15.2584	+	26.4283i	0.492719	+	0.853415i
$960$		0			0
$961$	−14.1177	+	24.4527i	−0.455411	+	0.788795i
$962$		−	10.1723i		−	0.327967i
$963$		0			0
$964$	14.6768			0.472708
$965$		0			0
$966$		0			0
$967$	−8.98071	+	5.18501i	−0.288800	+	0.166739i	−0.637401	−	0.770533i	$-0.719990\pi$
$967$	−8.98071	+	5.18501i	−0.288800	+	0.166739i	0.348601	+	0.937271i	$0.386657\pi$
$968$	−5.83198	+	3.36710i	−0.187447	+	0.108223i
$969$		0			0
$970$		0			0
$971$	−48.0410			−1.54171			−0.770854	−	0.637012i	$-0.780170\pi$
$971$	−48.0410			−1.54171			−0.770854	+	0.637012i	$0.780170\pi$
$972$		0			0
$973$			32.6890i			1.04796i
$974$	−12.4586	+	21.5789i	−0.399198	+	0.691431i
$975$		0			0
$976$	−6.05953	−	10.4954i	−0.193961	−	0.335950i
$977$	23.4288	−	13.5266i	0.749553	−	0.432754i	−0.0759796	−	0.997109i	$-0.524208\pi$
$977$	23.4288	−	13.5266i	0.749553	−	0.432754i	0.825532	+	0.564355i	$0.190875\pi$
$978$		0			0
$979$	2.02791	−	3.51244i	0.0648122	−	0.112258i
$980$		0			0
$981$		0			0
$982$		−	19.2355i		−	0.613829i
$983$	−19.4697	−	11.2408i	−0.620987	−	0.358527i	0.156266	−	0.987715i	$-0.450054\pi$
$983$	−19.4697	−	11.2408i	−0.620987	−	0.358527i	−0.777253	+	0.629188i	$0.783388\pi$
$984$		0			0
$985$		0			0
$986$	5.46598	+	9.46735i	0.174072	+	0.301502i
$987$		0			0
$988$	−0.855437	−	0.493887i	−0.0272151	−	0.0157126i
$989$	−3.92935			−0.124946
$990$		0			0
$991$	−26.5316			−0.842805			−0.421402	−	0.906874i	$-0.638462\pi$
$991$	−26.5316			−0.842805			−0.421402	+	0.906874i	$0.638462\pi$
$992$	53.8967	+	31.1173i	1.71122	+	0.987975i
$993$		0			0
$994$	26.0689	+	45.1526i	0.826855	+	1.43215i
$995$		0			0
$996$		0			0
$997$	−24.5656	−	14.1829i	−0.777999	−	0.449178i	0.0577217	−	0.998333i	$-0.481616\pi$
$997$	−24.5656	−	14.1829i	−0.777999	−	0.449178i	−0.835721	+	0.549155i	$0.814950\pi$
$998$			62.9433i			1.99243i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.2.k.b.424.5		12
3.2	odd	2		225.2.k.b.124.2		12
5.2	odd	4		135.2.e.b.46.1		6
5.3	odd	4		675.2.e.b.451.3		6
5.4	even	2	inner	675.2.k.b.424.2		12
9.2	odd	6		2025.2.b.l.649.5		6
9.4	even	3	inner	675.2.k.b.199.2		12
9.5	odd	6		225.2.k.b.49.5		12
9.7	even	3		2025.2.b.m.649.2		6
15.2	even	4		45.2.e.b.16.3	✓	6
15.8	even	4		225.2.e.b.151.1		6
15.14	odd	2		225.2.k.b.124.5		12
20.7	even	4		2160.2.q.k.721.3		6
45.2	even	12		405.2.a.j.1.1		3
45.4	even	6	inner	675.2.k.b.199.5		12
45.7	odd	12		405.2.a.i.1.3		3
45.13	odd	12		675.2.e.b.226.3		6
45.14	odd	6		225.2.k.b.49.2		12
45.22	odd	12		135.2.e.b.91.1		6
45.23	even	12		225.2.e.b.76.1		6
45.29	odd	6		2025.2.b.l.649.2		6
45.32	even	12		45.2.e.b.31.3	yes	6
45.34	even	6		2025.2.b.m.649.5		6
45.38	even	12		2025.2.a.n.1.3		3
45.43	odd	12		2025.2.a.o.1.1		3
60.47	odd	4		720.2.q.i.241.3		6
180.7	even	12		6480.2.a.bs.1.1		3
180.47	odd	12		6480.2.a.bv.1.1		3
180.67	even	12		2160.2.q.k.1441.3		6
180.167	odd	12		720.2.q.i.481.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.2.e.b.16.3	✓	6	15.2	even	4
45.2.e.b.31.3	yes	6	45.32	even	12
135.2.e.b.46.1		6	5.2	odd	4
135.2.e.b.91.1		6	45.22	odd	12
225.2.e.b.76.1		6	45.23	even	12
225.2.e.b.151.1		6	15.8	even	4
225.2.k.b.49.2		12	45.14	odd	6
225.2.k.b.49.5		12	9.5	odd	6
225.2.k.b.124.2		12	3.2	odd	2
225.2.k.b.124.5		12	15.14	odd	2
405.2.a.i.1.3		3	45.7	odd	12
405.2.a.j.1.1		3	45.2	even	12
675.2.e.b.226.3		6	45.13	odd	12
675.2.e.b.451.3		6	5.3	odd	4
675.2.k.b.199.2		12	9.4	even	3	inner
675.2.k.b.199.5		12	45.4	even	6	inner
675.2.k.b.424.2		12	5.4	even	2	inner
675.2.k.b.424.5		12	1.1	even	1	trivial
720.2.q.i.241.3		6	60.47	odd	4
720.2.q.i.481.3		6	180.167	odd	12
2025.2.a.n.1.3		3	45.38	even	12
2025.2.a.o.1.1		3	45.43	odd	12
2025.2.b.l.649.2		6	45.29	odd	6
2025.2.b.l.649.5		6	9.2	odd	6
2025.2.b.m.649.2		6	9.7	even	3
2025.2.b.m.649.5		6	45.34	even	6
2160.2.q.k.721.3		6	20.7	even	4
2160.2.q.k.1441.3		6	180.67	even	12
6480.2.a.bs.1.1		3	180.7	even	12
6480.2.a.bv.1.1		3	180.47	odd	12

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	675.k (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.80664 + 1.04307i) q^{2} +(1.17597 + 2.03684i) q^{4} +(3.53869 + 2.04307i) q^{7} +0.734191i q^{8} +O(q^{10})\)	\(q+(1.80664 + 1.04307i) q^{2} +(1.17597 + 2.03684i) q^{4} +(3.53869 + 2.04307i) q^{7} +0.734191i q^{8} +(-0.675970 + 1.17081i) q^{11} +(0.561237 - 0.324030i) q^{13} +(4.26210 + 7.38217i) q^{14} +(1.58613 - 2.74726i) q^{16} -1.35194i q^{17} -0.648061 q^{19} +(-2.44247 + 1.41016i) q^{22} +(-4.14827 + 2.39500i) q^{23} +1.35194 q^{26} +9.61033i q^{28} +(-1.93807 + 3.35683i) q^{29} +(3.84823 + 6.66533i) q^{31} +(7.00279 - 4.04307i) q^{32} +(1.41016 - 2.44247i) q^{34} -7.52420i q^{37} +(-1.17081 - 0.675970i) q^{38} +(-0.0898394 - 0.155606i) q^{41} +(0.710419 + 0.410161i) q^{43} -3.17968 q^{44} -9.99258 q^{46} +(-9.44526 - 5.45323i) q^{47} +(4.84823 + 8.39738i) q^{49} +(1.32000 + 0.762100i) q^{52} -4.17226i q^{53} +(-1.50000 + 2.59808i) q^{56} +(-7.00279 + 4.04307i) q^{58} +(-2.08613 - 3.61328i) q^{59} +(1.91016 - 3.30850i) q^{61} +16.0558i q^{62} +10.5242 q^{64} +(-7.05113 + 4.07097i) q^{67} +(2.75368 - 1.58984i) q^{68} +6.11644 q^{71} -12.3445i q^{73} +(7.84823 - 13.5935i) q^{74} +(-0.762100 - 1.32000i) q^{76} +(-4.78410 + 2.76210i) q^{77} +(5.17226 - 8.95862i) q^{79} -0.374833i q^{82} +(-10.6161 - 6.12920i) q^{83} +(0.855648 + 1.48203i) q^{86} +(-0.859601 - 0.496291i) q^{88} -3.00000 q^{89} +2.64806 q^{91} +(-9.75648 - 5.63290i) q^{92} +(-11.3761 - 19.7041i) q^{94} +(-11.7606 - 6.79001i) q^{97} +20.2281i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 10 q^{4}+O(q^{10}) \) 12 * q + 10 * q^4	\( 12 q + 10 q^{4} - 4 q^{11} + 18 q^{14} - 10 q^{16} - 16 q^{19} + 8 q^{26} + 14 q^{29} - 16 q^{31} - 8 q^{34} - 26 q^{41} - 88 q^{44} - 12 q^{46} - 4 q^{49} - 18 q^{56} + 4 q^{59} - 2 q^{61} + 60 q^{64} + 40 q^{71} + 32 q^{74} + 24 q^{76} + 4 q^{79} + 56 q^{86} - 36 q^{89} + 40 q^{91} - 62 q^{94}+O(q^{100}) \) 12 * q + 10 * q^4 - 4 * q^11 + 18 * q^14 - 10 * q^16 - 16 * q^19 + 8 * q^26 + 14 * q^29 - 16 * q^31 - 8 * q^34 - 26 * q^41 - 88 * q^44 - 12 * q^46 - 4 * q^49 - 18 * q^56 + 4 * q^59 - 2 * q^61 + 60 * q^64 + 40 * q^71 + 32 * q^74 + 24 * q^76 + 4 * q^79 + 56 * q^86 - 36 * q^89 + 40 * q^91 - 62 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more