LMFDB - Embedded newform 1520.2.d.j.609.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1520,2,Mod(609,1520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1520, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

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

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

chi := DirichletCharacter("1520.609");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1520 = 2^{4} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1520.d (of order $2$, degree $1$, 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:	$12.1372611072$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.5161984.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 4x^{3} + 25x^{2} - 20x + 8 $ x^6 - 4x^3 + 25x^2 - 20*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 190)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			609.2
Root			$1.32001 - 1.32001i$ of defining polynomial
Character	$\chi$	$=$	1520.609
Dual form			1520.2.d.j.609.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.12489i q^{3} +(1.80487 - 1.32001i) q^{5} -4.12489i q^{7} -1.51514 q^{9} +O(q^{10})$	$q-2.12489i q^{3} +(1.80487 - 1.32001i) q^{5} -4.12489i q^{7} -1.51514 q^{9} +2.64002 q^{11} -2.51514i q^{13} +(-2.80487 - 3.83515i) q^{15} -0.515138i q^{17} +1.00000 q^{19} -8.76491 q^{21} +3.09461i q^{23} +(1.51514 - 4.76491i) q^{25} -3.15516i q^{27} +7.79518 q^{29} -3.67030 q^{31} -5.60975i q^{33} +(-5.44490 - 7.44490i) q^{35} +10.2498i q^{37} -5.34438 q^{39} +8.88979 q^{41} +8.64002i q^{43} +(-2.73463 + 2.00000i) q^{45} +4.96972i q^{47} -10.0147 q^{49} -1.09461 q^{51} -5.48486i q^{53} +(4.76491 - 3.48486i) q^{55} -2.12489i q^{57} +3.15516 q^{59} -12.6400 q^{61} +6.24977i q^{63} +(-3.32001 - 4.53951i) q^{65} +7.40493i q^{67} +6.57569 q^{69} -11.1396 q^{71} +2.70436i q^{73} +(-10.1249 - 3.21949i) q^{75} -10.8898i q^{77} +16.7493 q^{79} -11.2498 q^{81} +3.28005i q^{83} +(-0.679988 - 0.929759i) q^{85} -16.5639i q^{87} +7.60975 q^{89} -10.3747 q^{91} +7.79897i q^{93} +(1.80487 - 1.32001i) q^{95} +3.93945i q^{97} -4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{5} - 10 q^{9}+O(q^{10}) $ 6 * q + 2 * q^5 - 10 * q^9	$ 6 q + 2 q^{5} - 10 q^{9} - 8 q^{15} + 6 q^{19} - 20 q^{21} + 10 q^{25} + 16 q^{29} - 8 q^{31} - 8 q^{35} + 20 q^{39} + 4 q^{41} + 18 q^{45} + 6 q^{49} + 12 q^{51} - 4 q^{55} + 4 q^{59} - 60 q^{61} - 12 q^{65} + 44 q^{69} + 16 q^{71} - 44 q^{75} - 34 q^{81} - 12 q^{85} + 28 q^{89} - 12 q^{91} + 2 q^{95} - 24 q^{99}+O(q^{100}) $ 6 * q + 2 * q^5 - 10 * q^9 - 8 * q^15 + 6 * q^19 - 20 * q^21 + 10 * q^25 + 16 * q^29 - 8 * q^31 - 8 * q^35 + 20 * q^39 + 4 * q^41 + 18 * q^45 + 6 * q^49 + 12 * q^51 - 4 * q^55 + 4 * q^59 - 60 * q^61 - 12 * q^65 + 44 * q^69 + 16 * q^71 - 44 * q^75 - 34 * q^81 - 12 * q^85 + 28 * q^89 - 12 * q^91 + 2 * q^95 - 24 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times$.

$n$	$191$	$401$	$1141$	$1217$
$\chi(n)$	$1$	$1$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$		−	2.12489i		−	1.22680i	−0.789771	−	0.613402i	$-0.789801\pi$
$3$		−	2.12489i		−	1.22680i	0.789771	−	0.613402i	$-0.210199\pi$
$4$		0			0
$5$	1.80487	−	1.32001i	0.807164	−	0.590327i
$5$	1.80487	−	1.32001i	0.807164	−	0.590327i
$6$		0			0
$7$		−	4.12489i		−	1.55906i	−0.626365	−	0.779530i	$-0.715458\pi$
$7$		−	4.12489i		−	1.55906i	0.626365	−	0.779530i	$-0.284542\pi$
$8$		0			0
$9$	−1.51514			−0.505046
$10$		0			0
$11$	2.64002			0.795997			0.397999	−	0.917386i	$-0.369705\pi$
$11$	2.64002			0.795997			0.397999	+	0.917386i	$0.369705\pi$
$12$		0			0
$13$		−	2.51514i		−	0.697574i	−0.937202	−	0.348787i	$-0.886594\pi$
$13$		−	2.51514i		−	0.697574i	0.937202	−	0.348787i	$-0.113406\pi$
$14$		0			0
$15$	−2.80487	−	3.83515i	−0.724215	−	0.990231i
$16$		0			0
$17$		−	0.515138i		−	0.124939i	−0.998047	−	0.0624697i	$-0.980102\pi$
$17$		−	0.515138i		−	0.124939i	0.998047	−	0.0624697i	$-0.0198977\pi$
$18$		0			0
$19$	1.00000			0.229416
$19$	1.00000			0.229416
$20$		0			0
$21$	−8.76491			−1.91266
$22$		0			0
$23$			3.09461i			0.645271i	0.946523	+	0.322635i	$0.104569\pi$
$23$			3.09461i			0.645271i	−0.946523	+	0.322635i	$0.895431\pi$
$24$		0			0
$25$	1.51514	−	4.76491i	0.303028	−	0.952982i
$26$		0			0
$27$		−	3.15516i		−	0.607211i
$28$		0			0
$29$	7.79518			1.44753			0.723765	−	0.690047i	$-0.242410\pi$
$29$	7.79518			1.44753			0.723765	+	0.690047i	$0.242410\pi$
$30$		0			0
$31$	−3.67030			−0.659205			−0.329603	−	0.944120i	$-0.606915\pi$
$31$	−3.67030			−0.659205			−0.329603	+	0.944120i	$0.606915\pi$
$32$		0			0
$33$		−	5.60975i		−	0.976532i
$34$		0			0
$35$	−5.44490	−	7.44490i	−0.920356	−	1.25842i
$36$		0			0
$37$			10.2498i			1.68505i	0.538656	+	0.842526i	$0.318932\pi$
$37$			10.2498i			1.68505i	−0.538656	+	0.842526i	$0.681068\pi$
$38$		0			0
$39$	−5.34438			−0.855786
$40$		0			0
$41$	8.88979			1.38835			0.694176	−	0.719805i	$-0.255769\pi$
$41$	8.88979			1.38835			0.694176	+	0.719805i	$0.255769\pi$
$42$		0			0
$43$			8.64002i			1.31759i	0.752322	+	0.658796i	$0.228934\pi$
$43$			8.64002i			1.31759i	−0.752322	+	0.658796i	$0.771066\pi$
$44$		0			0
$45$	−2.73463	+	2.00000i	−0.407655	+	0.298142i
$46$		0			0
$47$			4.96972i			0.724909i	0.932002	+	0.362454i	$0.118061\pi$
$47$			4.96972i			0.724909i	−0.932002	+	0.362454i	$0.881939\pi$
$48$		0			0
$49$	−10.0147			−1.43067
$50$		0			0
$51$	−1.09461			−0.153276
$52$		0			0
$53$		−	5.48486i		−	0.753404i	−0.926335	−	0.376702i	$-0.877058\pi$
$53$		−	5.48486i		−	0.753404i	0.926335	−	0.376702i	$-0.122942\pi$
$54$		0			0
$55$	4.76491	−	3.48486i	0.642500	−	0.469899i
$56$		0			0
$57$		−	2.12489i		−	0.281448i
$58$		0			0
$59$	3.15516			0.410767			0.205384	−	0.978682i	$-0.434156\pi$
$59$	3.15516			0.410767			0.205384	+	0.978682i	$0.434156\pi$
$60$		0			0
$61$	−12.6400			−1.61839			−0.809195	−	0.587541i	$-0.800096\pi$
$61$	−12.6400			−1.61839			−0.809195	+	0.587541i	$0.800096\pi$
$62$		0			0
$63$			6.24977i			0.787397i
$64$		0			0
$65$	−3.32001	−	4.53951i	−0.411797	−	0.563056i
$66$		0			0
$67$			7.40493i			0.904656i	0.891852	+	0.452328i	$0.149406\pi$
$67$			7.40493i			0.904656i	−0.891852	+	0.452328i	$0.850594\pi$
$68$		0			0
$69$	6.57569			0.791620
$70$		0			0
$71$	−11.1396			−1.32202			−0.661012	−	0.750376i	$-0.729873\pi$
$71$	−11.1396			−1.32202			−0.661012	+	0.750376i	$0.729873\pi$
$72$		0			0
$73$			2.70436i			0.316521i	0.987397	+	0.158261i	$0.0505886\pi$
$73$			2.70436i			0.316521i	−0.987397	+	0.158261i	$0.949411\pi$
$74$		0			0
$75$	−10.1249	−	3.21949i	−1.16912	−	0.371755i
$76$		0			0
$77$		−	10.8898i		−	1.24101i
$78$		0			0
$79$	16.7493			1.88444			0.942222	−	0.334988i	$-0.108732\pi$
$79$	16.7493			1.88444			0.942222	+	0.334988i	$0.108732\pi$
$80$		0			0
$81$	−11.2498			−1.24997
$82$		0			0
$83$			3.28005i			0.360032i	0.983664	+	0.180016i	$0.0576149\pi$
$83$			3.28005i			0.360032i	−0.983664	+	0.180016i	$0.942385\pi$
$84$		0			0
$85$	−0.679988	−	0.929759i	−0.0737551	−	0.100847i
$86$		0			0
$87$		−	16.5639i		−	1.77583i
$88$		0			0
$89$	7.60975			0.806632			0.403316	−	0.915061i	$-0.367858\pi$
$89$	7.60975			0.806632			0.403316	+	0.915061i	$0.367858\pi$
$90$		0			0
$91$	−10.3747			−1.08756
$92$		0			0
$93$			7.79897i			0.808715i
$94$		0			0
$95$	1.80487	−	1.32001i	0.185176	−	0.135430i
$96$		0			0
$97$			3.93945i			0.399990i	0.979797	+	0.199995i	$0.0640926\pi$
$97$			3.93945i			0.399990i	−0.979797	+	0.199995i	$0.935907\pi$
$98$		0			0
$99$	−4.00000			−0.402015
$100$		0			0
$101$	−13.7990			−1.37305			−0.686524	−	0.727107i	$-0.740864\pi$
$101$	−13.7990			−1.37305			−0.686524	+	0.727107i	$0.740864\pi$
$102$		0			0
$103$			4.57947i			0.451229i	0.974217	+	0.225614i	$0.0724389\pi$
$103$			4.57947i			0.451229i	−0.974217	+	0.225614i	$0.927561\pi$
$104$		0			0
$105$	−15.8196	+	11.5698i	−1.54383	+	1.12910i
$106$		0			0
$107$			10.3747i			1.00296i	0.865170	+	0.501478i	$0.167210\pi$
$107$			10.3747i			1.00296i	−0.865170	+	0.501478i	$0.832790\pi$
$108$		0			0
$109$	−3.01468			−0.288754			−0.144377	−	0.989523i	$-0.546118\pi$
$109$	−3.01468			−0.288754			−0.144377	+	0.989523i	$0.546118\pi$
$110$		0			0
$111$	21.7796			2.06723
$112$		0			0
$113$			19.2001i			1.80620i	0.429435	+	0.903098i	$0.358713\pi$
$113$			19.2001i			1.80620i	−0.429435	+	0.903098i	$0.641287\pi$
$114$		0			0
$115$	4.08492	+	5.58538i	0.380921	+	0.520839i
$116$		0			0
$117$			3.81078i			0.352307i
$118$		0			0
$119$	−2.12489			−0.194788
$120$		0			0
$121$	−4.03028			−0.366389
$122$		0			0
$123$		−	18.8898i		−	1.70324i
$124$		0			0
$125$	−3.55510	−	10.6001i	−0.317978	−	0.948098i
$126$		0			0
$127$		−	14.3103i		−	1.26984i	−0.772580	−	0.634918i	$-0.781034\pi$
$127$		−	14.3103i		−	1.26984i	0.772580	−	0.634918i	$-0.218966\pi$
$128$		0			0
$129$	18.3591			1.61643
$130$		0			0
$131$	6.56009			0.573158			0.286579	−	0.958057i	$-0.407482\pi$
$131$	6.56009			0.573158			0.286579	+	0.958057i	$0.407482\pi$
$132$		0			0
$133$		−	4.12489i		−	0.357673i
$134$		0			0
$135$	−4.16485	−	5.69467i	−0.358453	−	0.490119i
$136$		0			0
$137$		−	6.45459i		−	0.551452i	−0.961236	−	0.275726i	$-0.911082\pi$
$137$		−	6.45459i		−	0.551452i	0.961236	−	0.275726i	$-0.0889183\pi$
$138$		0			0
$139$	−23.0596			−1.95589			−0.977946	−	0.208856i	$-0.933026\pi$
$139$	−23.0596			−1.95589			−0.977946	+	0.208856i	$0.933026\pi$
$140$		0			0
$141$	10.5601			0.889320
$142$		0			0
$143$		−	6.64002i		−	0.555267i
$144$		0			0
$145$	14.0693	−	10.2897i	1.16839	−	0.854516i
$146$		0			0
$147$			21.2800i			1.75515i
$148$		0			0
$149$	−16.0294			−1.31318			−0.656588	−	0.754249i	$-0.728001\pi$
$149$	−16.0294			−1.31318			−0.656588	+	0.754249i	$0.728001\pi$
$150$		0			0
$151$	14.3103			1.16456			0.582279	−	0.812989i	$-0.302161\pi$
$151$	14.3103			1.16456			0.582279	+	0.812989i	$0.302161\pi$
$152$		0			0
$153$			0.780505i			0.0631001i
$154$		0			0
$155$	−6.62443	+	4.84484i	−0.532087	+	0.389147i
$156$		0			0
$157$		−	18.0294i		−	1.43890i	−0.694544	−	0.719450i	$-0.744394\pi$
$157$		−	18.0294i		−	1.43890i	0.694544	−	0.719450i	$-0.255606\pi$
$158$		0			0
$159$	−11.6547			−0.924278
$160$		0			0
$161$	12.7649			1.00602
$162$		0			0
$163$			2.70058i			0.211525i	0.994391	+	0.105763i	$0.0337284\pi$
$163$			2.70058i			0.211525i	−0.994391	+	0.105763i	$0.966272\pi$
$164$		0			0
$165$	−7.40493	−	10.1249i	−0.576473	−	0.788221i
$166$		0			0
$167$		−	8.95035i		−	0.692599i	−0.938124	−	0.346299i	$-0.887438\pi$
$167$		−	8.95035i		−	0.692599i	0.938124	−	0.346299i	$-0.112562\pi$
$168$		0			0
$169$	6.67408			0.513391
$170$		0			0
$171$	−1.51514			−0.115866
$172$		0			0
$173$		−	0.310323i		−	0.0235934i	−0.999930	−	0.0117967i	$-0.996245\pi$
$173$		−	0.310323i		−	0.0235934i	0.999930	−	0.0117967i	$-0.00375510\pi$
$174$		0			0
$175$	−19.6547	−	6.24977i	−1.48576	−	0.472438i
$176$		0			0
$177$		−	6.70436i		−	0.503930i
$178$		0			0
$179$	−1.52982			−0.114344			−0.0571720	−	0.998364i	$-0.518208\pi$
$179$	−1.52982			−0.114344			−0.0571720	+	0.998364i	$0.518208\pi$
$180$		0			0
$181$	−14.7493			−1.09631			−0.548154	−	0.836377i	$-0.684669\pi$
$181$	−14.7493			−1.09631			−0.548154	+	0.836377i	$0.684669\pi$
$182$		0			0
$183$			26.8586i			1.98544i
$184$		0			0
$185$	13.5298	+	18.4995i	0.994732	+	1.36011i
$186$		0			0
$187$		−	1.35998i		−	0.0994513i
$188$		0			0
$189$	−13.0147			−0.946679
$190$		0			0
$191$	−18.1249			−1.31147			−0.655735	−	0.754991i	$-0.727641\pi$
$191$	−18.1249			−1.31147			−0.655735	+	0.754991i	$0.727641\pi$
$192$		0			0
$193$			4.56009i			0.328243i	0.986440	+	0.164121i	$0.0524789\pi$
$193$			4.56009i			0.328243i	−0.986440	+	0.164121i	$0.947521\pi$
$194$		0			0
$195$	−9.64593	+	7.05464i	−0.690759	+	0.505194i
$196$		0			0
$197$		−	2.14048i		−	0.152503i	−0.997089	−	0.0762515i	$-0.975705\pi$
$197$		−	2.14048i		−	0.152503i	0.997089	−	0.0762515i	$-0.0242952\pi$
$198$		0			0
$199$	16.5639			1.17418			0.587091	−	0.809521i	$-0.300273\pi$
$199$	16.5639			1.17418			0.587091	+	0.809521i	$0.300273\pi$
$200$		0			0
$201$	15.7346			1.10984
$202$		0			0
$203$		−	32.1542i		−	2.25679i
$204$		0			0
$205$	16.0450	−	11.7346i	1.12063	−	0.819582i
$206$		0			0
$207$		−	4.68876i		−	0.325891i
$208$		0			0
$209$	2.64002			0.182614
$210$		0			0
$211$	15.2838			1.05218			0.526091	−	0.850428i	$-0.323657\pi$
$211$	15.2838			1.05218			0.526091	+	0.850428i	$0.323657\pi$
$212$		0			0
$213$			23.6703i			1.62186i
$214$		0			0
$215$	11.4049	+	15.5942i	0.777810	+	1.06351i
$216$		0			0
$217$			15.1396i			1.02774i
$218$		0			0
$219$	5.74645			0.388309
$220$		0			0
$221$	−1.29564			−0.0871544
$222$		0			0
$223$		−	3.75023i		−	0.251134i	−0.992085	−	0.125567i	$-0.959925\pi$
$223$		−	3.75023i		−	0.251134i	0.992085	−	0.125567i	$-0.0400750\pi$
$224$		0			0
$225$	−2.29564	+	7.21949i	−0.153043	+	0.481300i
$226$		0			0
$227$			24.1542i			1.60317i	0.597878	+	0.801587i	$0.296011\pi$
$227$			24.1542i			1.60317i	−0.597878	+	0.801587i	$0.703989\pi$
$228$		0			0
$229$	13.0596			0.863005			0.431503	−	0.902112i	$-0.357984\pi$
$229$	13.0596			0.863005			0.431503	+	0.902112i	$0.357984\pi$
$230$		0			0
$231$	−23.1396			−1.52247
$232$		0			0
$233$			6.43899i			0.421832i	0.977504	+	0.210916i	$0.0676447\pi$
$233$			6.43899i			0.421832i	−0.977504	+	0.210916i	$0.932355\pi$
$234$		0			0
$235$	6.56009	+	8.96972i	0.427933	+	0.585120i
$236$		0			0
$237$		−	35.5904i		−	2.31184i
$238$		0			0
$239$	22.8742			1.47961			0.739804	−	0.672822i	$-0.234918\pi$
$239$	22.8742			1.47961			0.739804	+	0.672822i	$0.234918\pi$
$240$		0			0
$241$	4.96972			0.320128			0.160064	−	0.987107i	$-0.448830\pi$
$241$	4.96972			0.320128			0.160064	+	0.987107i	$0.448830\pi$
$242$		0			0
$243$			14.4390i			0.926262i
$244$		0			0
$245$	−18.0752	+	13.2195i	−1.15478	+	0.844563i
$246$		0			0
$247$		−	2.51514i		−	0.160034i
$248$		0			0
$249$	6.96972			0.441688
$250$		0			0
$251$	−15.9201			−1.00487			−0.502433	−	0.864616i	$-0.667562\pi$
$251$	−15.9201			−1.00487			−0.502433	+	0.864616i	$0.667562\pi$
$252$		0			0
$253$			8.16984i			0.513634i
$254$		0			0
$255$	−1.97563	+	1.44490i	−0.123719	+	0.0904830i
$256$		0			0
$257$		−	1.04965i		−	0.0654756i	−0.999464	−	0.0327378i	$-0.989577\pi$
$257$		−	1.04965i		−	0.0654756i	0.999464	−	0.0327378i	$-0.0104226\pi$
$258$		0			0
$259$	42.2791			2.62710
$260$		0			0
$261$	−11.8108			−0.731069
$262$		0			0
$263$		−	11.9394i		−	0.736218i	−0.929783	−	0.368109i	$-0.880005\pi$
$263$		−	11.9394i		−	0.736218i	0.929783	−	0.368109i	$-0.119995\pi$
$264$		0			0
$265$	−7.24008	−	9.89948i	−0.444755	−	0.608120i
$266$		0			0
$267$		−	16.1698i		−	0.989578i
$268$		0			0
$269$	15.9394			0.971845			0.485923	−	0.874002i	$-0.338484\pi$
$269$	15.9394			0.971845			0.485923	+	0.874002i	$0.338484\pi$
$270$		0			0
$271$	−1.46548			−0.0890218			−0.0445109	−	0.999009i	$-0.514173\pi$
$271$	−1.46548			−0.0890218			−0.0445109	+	0.999009i	$0.514173\pi$
$272$		0			0
$273$			22.0450i			1.33422i
$274$		0			0
$275$	4.00000	−	12.5795i	0.241209	−	0.758571i
$276$		0			0
$277$			32.4995i			1.95271i	0.216177	+	0.976354i	$0.430641\pi$
$277$			32.4995i			1.95271i	−0.216177	+	0.976354i	$0.569359\pi$
$278$		0			0
$279$	5.56101			0.332929
$280$		0			0
$281$	1.04965			0.0626171			0.0313085	−	0.999510i	$-0.490033\pi$
$281$	1.04965			0.0626171			0.0313085	+	0.999510i	$0.490033\pi$
$282$		0			0
$283$		−	9.34060i		−	0.555241i	−0.960691	−	0.277620i	$-0.910454\pi$
$283$		−	9.34060i		−	0.555241i	0.960691	−	0.277620i	$-0.0895458\pi$
$284$		0			0
$285$	−2.80487	−	3.83515i	−0.166146	−	0.227175i
$286$		0			0
$287$		−	36.6694i		−	2.16453i
$288$		0			0
$289$	16.7346			0.984390
$290$		0			0
$291$	8.37088			0.490709
$292$		0			0
$293$		−	25.5748i		−	1.49409i	−0.664771	−	0.747047i	$-0.731471\pi$
$293$		−	25.5748i		−	1.49409i	0.664771	−	0.747047i	$-0.268529\pi$
$294$		0			0
$295$	5.69467	−	4.16485i	0.331556	−	0.242487i
$296$		0			0
$297$		−	8.32970i		−	0.483338i
$298$		0			0
$299$	7.78337			0.450124
$300$		0			0
$301$	35.6391			2.05420
$302$		0			0
$303$			29.3212i			1.68446i
$304$		0			0
$305$	−22.8136	+	16.6850i	−1.30631	+	0.955379i
$306$		0			0
$307$		−	9.43991i		−	0.538764i	−0.963033	−	0.269382i	$-0.913181\pi$
$307$		−	9.43991i		−	0.538764i	0.963033	−	0.269382i	$-0.0868194\pi$
$308$		0			0
$309$	9.73085			0.553569
$310$		0			0
$311$	17.4655			0.990377			0.495188	−	0.868786i	$-0.335099\pi$
$311$	17.4655			0.990377			0.495188	+	0.868786i	$0.335099\pi$
$312$		0			0
$313$			10.4546i			0.590928i	0.955354	+	0.295464i	$0.0954743\pi$
$313$			10.4546i			0.590928i	−0.955354	+	0.295464i	$0.904526\pi$
$314$		0			0
$315$	8.24977	+	11.2800i	0.464822	+	0.635559i
$316$		0			0
$317$			4.33348i			0.243393i	0.992567	+	0.121696i	$0.0388334\pi$
$317$			4.33348i			0.243393i	−0.992567	+	0.121696i	$0.961167\pi$
$318$		0			0
$319$	20.5795			1.15223
$320$		0			0
$321$	22.0450			1.23043
$322$		0			0
$323$		−	0.515138i		−	0.0286630i
$324$		0			0
$325$	−11.9844	−	3.81078i	−0.664775	−	0.211384i
$326$		0			0
$327$			6.40585i			0.354244i
$328$		0			0
$329$	20.4995			1.13018
$330$		0			0
$331$	4.56387			0.250853			0.125427	−	0.992103i	$-0.459970\pi$
$331$	4.56387			0.250853			0.125427	+	0.992103i	$0.459970\pi$
$332$		0			0
$333$		−	15.5298i		−	0.851029i
$334$		0			0
$335$	9.77460	+	13.3650i	0.534043	+	0.730206i
$336$		0			0
$337$			13.0109i			0.708749i	0.935104	+	0.354374i	$0.115306\pi$
$337$			13.0109i			0.708749i	−0.935104	+	0.354374i	$0.884694\pi$
$338$		0			0
$339$	40.7980			2.21585
$340$		0			0
$341$	−9.68968			−0.524725
$342$		0			0
$343$			12.4352i			0.671438i
$344$		0			0
$345$	11.8683	−	8.67999i	0.638967	−	0.467315i
$346$		0			0
$347$		−	13.2195i		−	0.709660i	−0.934931	−	0.354830i	$-0.884539\pi$
$347$		−	13.2195i		−	0.709660i	0.934931	−	0.354830i	$-0.115461\pi$
$348$		0			0
$349$	20.4390			1.09407			0.547037	−	0.837108i	$-0.315756\pi$
$349$	20.4390			1.09407			0.547037	+	0.837108i	$0.315756\pi$
$350$		0			0
$351$	−7.93567			−0.423575
$352$		0			0
$353$		−	10.7044i		−	0.569735i	−0.958567	−	0.284868i	$-0.908050\pi$
$353$		−	10.7044i		−	0.569735i	0.958567	−	0.284868i	$-0.0919497\pi$
$354$		0			0
$355$	−20.1055	+	14.7044i	−1.06709	+	0.780426i
$356$		0			0
$357$			4.51514i			0.238966i
$358$		0			0
$359$	−4.31410			−0.227690			−0.113845	−	0.993499i	$-0.536317\pi$
$359$	−4.31410			−0.227690			−0.113845	+	0.993499i	$0.536317\pi$
$360$		0			0
$361$	1.00000			0.0526316
$362$		0			0
$363$			8.56387i			0.449487i
$364$		0			0
$365$	3.56978	+	4.88102i	0.186851	+	0.255484i
$366$		0			0
$367$			12.8099i			0.668669i	0.942454	+	0.334335i	$0.108512\pi$
$367$			12.8099i			0.668669i	−0.942454	+	0.334335i	$0.891488\pi$
$368$		0			0
$369$	−13.4693			−0.701182
$370$		0			0
$371$	−22.6244			−1.17460
$372$		0			0
$373$		−	0.704357i		−	0.0364702i	−0.999834	−	0.0182351i	$-0.994195\pi$
$373$		−	0.704357i		−	0.0364702i	0.999834	−	0.0182351i	$-0.00580474\pi$
$374$		0			0
$375$	−22.5239	+	7.55419i	−1.16313	+	0.390096i
$376$		0			0
$377$		−	19.6060i		−	1.00976i
$378$		0			0
$379$	6.12489			0.314614			0.157307	−	0.987550i	$-0.449719\pi$
$379$	6.12489			0.314614			0.157307	+	0.987550i	$0.449719\pi$
$380$		0			0
$381$	−30.4078			−1.55784
$382$		0			0
$383$			18.6400i			0.952461i	0.879321	+	0.476230i	$0.157997\pi$
$383$			18.6400i			0.952461i	−0.879321	+	0.476230i	$0.842003\pi$
$384$		0			0
$385$	−14.3747	−	19.6547i	−0.732600	−	1.00170i
$386$		0			0
$387$		−	13.0908i		−	0.665444i
$388$		0			0
$389$	−13.9201			−0.705776			−0.352888	−	0.935666i	$-0.614800\pi$
$389$	−13.9201			−0.705776			−0.352888	+	0.935666i	$0.614800\pi$
$390$		0			0
$391$	1.59415			0.0806197
$392$		0			0
$393$		−	13.9394i		−	0.703152i
$394$		0			0
$395$	30.2304	−	22.1093i	1.52106	−	1.11244i
$396$		0			0
$397$		−	28.3784i		−	1.42427i	−0.702041	−	0.712136i	$-0.747728\pi$
$397$		−	28.3784i		−	1.42427i	0.702041	−	0.712136i	$-0.252272\pi$
$398$		0			0
$399$	−8.76491			−0.438794
$400$		0			0
$401$	5.54920			0.277114			0.138557	−	0.990354i	$-0.455754\pi$
$401$	5.54920			0.277114			0.138557	+	0.990354i	$0.455754\pi$
$402$		0			0
$403$			9.23131i			0.459844i
$404$		0			0
$405$	−20.3044	+	14.8498i	−1.00893	+	0.737894i
$406$		0			0
$407$			27.0596i			1.34130i
$408$		0			0
$409$	5.01090			0.247773			0.123886	−	0.992296i	$-0.460464\pi$
$409$	5.01090			0.247773			0.123886	+	0.992296i	$0.460464\pi$
$410$		0			0
$411$	−13.7153			−0.676524
$412$		0			0
$413$		−	13.0147i		−	0.640411i
$414$		0			0
$415$	4.32970	+	5.92007i	0.212537	+	0.290605i
$416$		0			0
$417$			48.9991i			2.39950i
$418$		0			0
$419$	−15.8889			−0.776222			−0.388111	−	0.921613i	$-0.626872\pi$
$419$	−15.8889			−0.776222			−0.388111	+	0.921613i	$0.626872\pi$
$420$		0			0
$421$	−2.38647			−0.116310			−0.0581548	−	0.998308i	$-0.518522\pi$
$421$	−2.38647			−0.116310			−0.0581548	+	0.998308i	$0.518522\pi$
$422$		0			0
$423$		−	7.52982i		−	0.366112i
$424$		0			0
$425$	−2.45459	−	0.780505i	−0.119065	−	0.0378601i
$426$		0			0
$427$			52.1386i			2.52317i
$428$		0			0
$429$	−14.1093			−0.681203
$430$		0			0
$431$	2.35906			0.113632			0.0568160	−	0.998385i	$-0.481905\pi$
$431$	2.35906			0.113632			0.0568160	+	0.998385i	$0.481905\pi$
$432$		0			0
$433$			0.640023i			0.0307576i	0.999882	+	0.0153788i	$0.00489541\pi$
$433$			0.640023i			0.0307576i	−0.999882	+	0.0153788i	$0.995105\pi$
$434$		0			0
$435$	−21.8645	−	29.8957i	−1.04832	−	1.43339i
$436$		0			0
$437$			3.09461i			0.148035i
$438$		0			0
$439$	−25.4499			−1.21466			−0.607328	−	0.794451i	$-0.707759\pi$
$439$	−25.4499			−1.21466			−0.607328	+	0.794451i	$0.707759\pi$
$440$		0			0
$441$	15.1736			0.722553
$442$		0			0
$443$		−	35.6685i		−	1.69466i	−0.531067	−	0.847330i	$-0.678209\pi$
$443$		−	35.6685i		−	1.69466i	0.531067	−	0.847330i	$-0.321791\pi$
$444$		0			0
$445$	13.7346	−	10.0450i	0.651084	−	0.476177i
$446$		0			0
$447$			34.0606i			1.61101i
$448$		0			0
$449$	11.4399			0.539883			0.269941	−	0.962877i	$-0.412996\pi$
$449$	11.4399			0.539883			0.269941	+	0.962877i	$0.412996\pi$
$450$		0			0
$451$	23.4693			1.10512
$452$		0			0
$453$		−	30.4078i		−	1.42868i
$454$		0			0
$455$	−18.7249	+	13.6947i	−0.877839	+	0.642016i
$456$		0			0
$457$			19.4849i			0.911463i	0.890117	+	0.455732i	$0.150622\pi$
$457$			19.4849i			0.911463i	−0.890117	+	0.455732i	$0.849378\pi$
$458$		0			0
$459$	−1.62534			−0.0758645
$460$		0			0
$461$	−29.3893			−1.36880			−0.684399	−	0.729108i	$-0.739935\pi$
$461$	−29.3893			−1.36880			−0.684399	+	0.729108i	$0.739935\pi$
$462$		0			0
$463$			12.1892i			0.566481i	0.959049	+	0.283241i	$0.0914095\pi$
$463$			12.1892i			0.566481i	−0.959049	+	0.283241i	$0.908591\pi$
$464$		0			0
$465$	10.2947	+	14.0761i	0.477407	+	0.652766i
$466$		0			0
$467$			17.8889i			0.827799i	0.910323	+	0.413899i	$0.135833\pi$
$467$			17.8889i			0.827799i	−0.910323	+	0.413899i	$0.864167\pi$
$468$		0			0
$469$	30.5445			1.41041
$470$		0			0
$471$	−38.3103			−1.76525
$472$		0			0
$473$			22.8099i			1.04880i
$474$		0			0
$475$	1.51514	−	4.76491i	0.0695193	−	0.218629i
$476$		0			0
$477$			8.31032i			0.380504i
$478$		0			0
$479$	−1.15894			−0.0529534			−0.0264767	−	0.999649i	$-0.508429\pi$
$479$	−1.15894			−0.0529534			−0.0264767	+	0.999649i	$0.508429\pi$
$480$		0			0
$481$	25.7796			1.17545
$482$		0			0
$483$		−	27.1240i		−	1.23418i
$484$		0			0
$485$	5.20012	+	7.11021i	0.236125	+	0.322858i
$486$		0			0
$487$		−	30.6888i		−	1.39064i	−0.718700	−	0.695320i	$-0.755263\pi$
$487$		−	30.6888i		−	1.39064i	0.718700	−	0.695320i	$-0.244737\pi$
$488$		0			0
$489$	5.73841			0.259500
$490$		0			0
$491$	−3.67030			−0.165638			−0.0828192	−	0.996565i	$-0.526392\pi$
$491$	−3.67030			−0.165638			−0.0828192	+	0.996565i	$0.526392\pi$
$492$		0			0
$493$		−	4.01560i		−	0.180853i
$494$		0			0
$495$	−7.21949	+	5.28005i	−0.324492	+	0.237320i
$496$		0			0
$497$			45.9494i			2.06111i
$498$		0			0
$499$	31.3893			1.40518			0.702590	−	0.711595i	$-0.252027\pi$
$499$	31.3893			1.40518			0.702590	+	0.711595i	$0.252027\pi$
$500$		0			0
$501$	−19.0185			−0.849682
$502$		0			0
$503$		−	18.1542i		−	0.809458i	−0.914437	−	0.404729i	$-0.867366\pi$
$503$		−	18.1542i		−	0.809458i	0.914437	−	0.404729i	$-0.132634\pi$
$504$		0			0
$505$	−24.9054	+	18.2148i	−1.10828	+	0.810548i
$506$		0			0
$507$		−	14.1817i		−	0.629829i
$508$		0			0
$509$	−11.5298			−0.511050			−0.255525	−	0.966802i	$-0.582248\pi$
$509$	−11.5298			−0.511050			−0.255525	+	0.966802i	$0.582248\pi$
$510$		0			0
$511$	11.1552			0.493475
$512$		0			0
$513$		−	3.15516i		−	0.139304i
$514$		0			0
$515$	6.04496	+	8.26537i	0.266373	+	0.364216i
$516$		0			0
$517$			13.1202i			0.577025i
$518$		0			0
$519$	−0.659401			−0.0289445
$520$		0			0
$521$	−42.5895			−1.86588			−0.932939	−	0.360035i	$-0.882765\pi$
$521$	−42.5895			−1.86588			−0.932939	+	0.360035i	$0.882765\pi$
$522$		0			0
$523$			1.21571i			0.0531594i	0.999647	+	0.0265797i	$0.00846159\pi$
$523$			1.21571i			0.0531594i	−0.999647	+	0.0265797i	$0.991538\pi$
$524$		0			0
$525$	−13.2800	+	41.7640i	−0.579589	+	1.82273i
$526$		0			0
$527$			1.89071i			0.0823607i
$528$		0			0
$529$	13.4234			0.583626
$530$		0			0
$531$	−4.78051			−0.207456
$532$		0			0
$533$		−	22.3591i		−	0.968478i
$534$		0			0
$535$	13.6947	+	18.7249i	0.592072	+	0.809550i
$536$		0			0
$537$			3.25069i			0.140278i
$538$		0			0
$539$	−26.4390			−1.13881
$540$		0			0
$541$	−1.92007			−0.0825503			−0.0412751	−	0.999148i	$-0.513142\pi$
$541$	−1.92007			−0.0825503			−0.0412751	+	0.999148i	$0.513142\pi$
$542$		0			0
$543$			31.3406i			1.34495i
$544$		0			0
$545$	−5.44112	+	3.97941i	−0.233072	+	0.170459i
$546$		0			0
$547$		−	10.9697i		−	0.469032i	−0.972112	−	0.234516i	$-0.924650\pi$
$547$		−	10.9697i		−	0.469032i	0.972112	−	0.234516i	$-0.0753504\pi$
$548$		0			0
$549$	19.1514			0.817361
$550$		0			0
$551$	7.79518			0.332086
$552$		0			0
$553$		−	69.0890i		−	2.93796i
$554$		0			0
$555$	39.3094	−	28.7493i	1.66859	−	1.22034i
$556$		0			0
$557$			23.5005i			0.995746i	0.867250	+	0.497873i	$0.165886\pi$
$557$			23.5005i			0.995746i	−0.867250	+	0.497873i	$0.834114\pi$
$558$		0			0
$559$	21.7309			0.919117
$560$		0			0
$561$	−2.88979			−0.122007
$562$		0			0
$563$		−	7.87890i		−	0.332056i	−0.986121	−	0.166028i	$-0.946906\pi$
$563$		−	7.87890i		−	0.332056i	0.986121	−	0.166028i	$-0.0530942\pi$
$564$		0			0
$565$	25.3444	+	34.6538i	1.06625	+	1.45790i
$566$		0			0
$567$			46.4040i			1.94879i
$568$		0			0
$569$	1.09083			0.0457299			0.0228650	−	0.999739i	$-0.492721\pi$
$569$	1.09083			0.0457299			0.0228650	+	0.999739i	$0.492721\pi$
$570$		0			0
$571$	−21.4886			−0.899272			−0.449636	−	0.893212i	$-0.648446\pi$
$571$	−21.4886			−0.899272			−0.449636	+	0.893212i	$0.648446\pi$
$572$		0			0
$573$			38.5133i			1.60892i
$574$		0			0
$575$	14.7455	+	4.68876i	0.614931	+	0.195535i
$576$		0			0
$577$		−	16.1055i		−	0.670481i	−0.942133	−	0.335241i	$-0.891182\pi$
$577$		−	16.1055i		−	0.670481i	0.942133	−	0.335241i	$-0.108818\pi$
$578$		0			0
$579$	9.68968			0.402689
$580$		0			0
$581$	13.5298			0.561311
$582$		0			0
$583$		−	14.4802i		−	0.599707i
$584$		0			0
$585$	5.03028	+	6.87798i	0.207976	+	0.284369i
$586$		0			0
$587$		−	0.480164i		−	0.0198185i	−0.999951	−	0.00990925i	$-0.996846\pi$
$587$		−	0.480164i		−	0.0198185i	0.999951	−	0.00990925i	$-0.00315426\pi$
$588$		0			0
$589$	−3.67030			−0.151232
$590$		0			0
$591$	−4.54828			−0.187091
$592$		0			0
$593$		−	40.4683i		−	1.66184i	−0.556395	−	0.830918i	$-0.687816\pi$
$593$		−	40.4683i		−	1.66184i	0.556395	−	0.830918i	$-0.312184\pi$
$594$		0			0
$595$	−3.83515	+	2.80487i	−0.157226	+	0.114989i
$596$		0			0
$597$		−	35.1963i		−	1.44049i
$598$		0			0
$599$	−36.1698			−1.47786			−0.738930	−	0.673782i	$-0.764669\pi$
$599$	−36.1698			−1.47786			−0.738930	+	0.673782i	$0.764669\pi$
$600$		0			0
$601$	24.3903			0.994899			0.497450	−	0.867493i	$-0.334270\pi$
$601$	24.3903			0.994899			0.497450	+	0.867493i	$0.334270\pi$
$602$		0			0
$603$		−	11.2195i		−	0.456893i
$604$		0			0
$605$	−7.27414	+	5.32001i	−0.295736	+	0.216289i
$606$		0			0
$607$		−	21.6897i		−	0.880357i	−0.897910	−	0.440178i	$-0.854915\pi$
$607$		−	21.6897i		−	0.880357i	0.897910	−	0.440178i	$-0.145085\pi$
$608$		0			0
$609$	−68.3241			−2.76863
$610$		0			0
$611$	12.4995			0.505677
$612$		0			0
$613$			26.2304i			1.05944i	0.848174	+	0.529718i	$0.177702\pi$
$613$			26.2304i			1.05944i	−0.848174	+	0.529718i	$0.822298\pi$
$614$		0			0
$615$	−24.9348	−	34.0937i	−1.00547	−	1.37479i
$616$		0			0
$617$		−	22.7200i		−	0.914671i	−0.889294	−	0.457335i	$-0.848804\pi$
$617$		−	22.7200i		−	0.914671i	0.889294	−	0.457335i	$-0.151196\pi$
$618$		0			0
$619$	−32.9192			−1.32313			−0.661566	−	0.749887i	$-0.730108\pi$
$619$	−32.9192			−1.32313			−0.661566	+	0.749887i	$0.730108\pi$
$620$		0			0
$621$	9.76399			0.391816
$622$		0			0
$623$		−	31.3893i		−	1.25759i
$624$		0			0
$625$	−20.4087	−	14.4390i	−0.816349	−	0.577560i
$626$		0			0
$627$		−	5.60975i		−	0.224032i
$628$		0			0
$629$	5.28005			0.210529
$630$		0			0
$631$	−17.2876			−0.688209			−0.344104	−	0.938931i	$-0.611817\pi$
$631$	−17.2876			−0.688209			−0.344104	+	0.938931i	$0.611817\pi$
$632$		0			0
$633$		−	32.4764i		−	1.29082i
$634$		0			0
$635$	−18.8898	−	25.8283i	−0.749619	−	1.02497i
$636$		0			0
$637$			25.1883i			0.997997i
$638$		0			0
$639$	16.8780			0.667683
$640$		0			0
$641$	−9.29942			−0.367305			−0.183653	−	0.982991i	$-0.558792\pi$
$641$	−9.29942			−0.367305			−0.183653	+	0.982991i	$0.558792\pi$
$642$		0			0
$643$		−	3.40115i		−	0.134128i	−0.997749	−	0.0670642i	$-0.978637\pi$
$643$		−	3.40115i		−	0.134128i	0.997749	−	0.0670642i	$-0.0213632\pi$
$644$		0			0
$645$	33.1358	−	24.2342i	1.30472	−	0.954220i
$646$		0			0
$647$		−	14.9348i		−	0.587146i	−0.955937	−	0.293573i	$-0.905156\pi$
$647$		−	14.9348i		−	0.587146i	0.955937	−	0.293573i	$-0.0948443\pi$
$648$		0			0
$649$	8.32970			0.326969
$650$		0			0
$651$	32.1698			1.26084
$652$		0			0
$653$		−	21.1202i		−	0.826497i	−0.910618	−	0.413248i	$-0.864394\pi$
$653$		−	21.1202i		−	0.826497i	0.910618	−	0.413248i	$-0.135606\pi$
$654$		0			0
$655$	11.8401	−	8.65940i	0.462633	−	0.338351i
$656$		0			0
$657$		−	4.09747i		−	0.159858i
$658$		0			0
$659$	27.6547			1.07727			0.538637	−	0.842538i	$-0.318939\pi$
$659$	27.6547			1.07727			0.538637	+	0.842538i	$0.318939\pi$
$660$		0			0
$661$	10.2342			0.398063			0.199032	−	0.979993i	$-0.436220\pi$
$661$	10.2342			0.398063			0.199032	+	0.979993i	$0.436220\pi$
$662$		0			0
$663$			2.75309i			0.106921i
$664$		0			0
$665$	−5.44490	−	7.44490i	−0.211144	−	0.288701i
$666$		0			0
$667$			24.1231i			0.934048i
$668$		0			0
$669$	−7.96881			−0.308092
$670$		0			0
$671$	−33.3700			−1.28823
$672$		0			0
$673$			13.4693i			0.519202i	0.965716	+	0.259601i	$0.0835911\pi$
$673$			13.4693i			0.519202i	−0.965716	+	0.259601i	$0.916409\pi$
$674$		0			0
$675$	−15.0341	−	4.78051i	−0.578661	−	0.184002i
$676$		0			0
$677$			15.1433i			0.582006i	0.956722	+	0.291003i	$0.0939890\pi$
$677$			15.1433i			0.582006i	−0.956722	+	0.291003i	$0.906011\pi$
$678$		0			0
$679$	16.2498			0.623609
$680$		0			0
$681$	51.3250			1.96678
$682$		0			0
$683$			15.8789i			0.607589i	0.952738	+	0.303795i	$0.0982536\pi$
$683$			15.8789i			0.607589i	−0.952738	+	0.303795i	$0.901746\pi$
$684$		0			0
$685$	−8.52013	−	11.6497i	−0.325537	−	0.445113i
$686$		0			0
$687$		−	27.7502i		−	1.05874i
$688$		0			0
$689$	−13.7952			−0.525555
$690$		0			0
$691$	−27.3094			−1.03890			−0.519449	−	0.854501i	$-0.673863\pi$
$691$	−27.3094			−1.03890			−0.519449	+	0.854501i	$0.673863\pi$
$692$		0			0
$693$			16.4995i			0.626766i
$694$		0			0
$695$	−41.6197	+	30.4390i	−1.57873	+	1.15462i
$696$		0			0
$697$		−	4.57947i		−	0.173460i
$698$		0			0
$699$	13.6821			0.517505
$700$		0			0
$701$	−20.9697			−0.792016			−0.396008	−	0.918247i	$-0.629605\pi$
$701$	−20.9697			−0.792016			−0.396008	+	0.918247i	$0.629605\pi$
$702$		0			0
$703$			10.2498i			0.386577i
$704$		0			0
$705$	19.0596	−	13.9394i	0.717827	−	0.524990i
$706$		0			0
$707$			56.9192i			2.14067i
$708$		0			0
$709$	−37.5592			−1.41056			−0.705282	−	0.708927i	$-0.749180\pi$
$709$	−37.5592			−1.41056			−0.705282	+	0.708927i	$0.749180\pi$
$710$		0			0
$711$	−25.3775			−0.951731
$712$		0			0
$713$		−	11.3581i		−	0.425366i
$714$		0			0
$715$	−8.76491	−	11.9844i	−0.327789	−	0.448191i
$716$		0			0
$717$		−	48.6050i		−	1.81519i
$718$		0			0
$719$	3.94323			0.147058			0.0735288	−	0.997293i	$-0.476574\pi$
$719$	3.94323			0.147058			0.0735288	+	0.997293i	$0.476574\pi$
$720$		0			0
$721$	18.8898			0.703493
$722$		0			0
$723$		−	10.5601i		−	0.392734i
$724$		0			0
$725$	11.8108	−	37.1433i	0.438641	−	1.37947i
$726$		0			0
$727$		−	33.2139i		−	1.23183i	−0.787811	−	0.615917i	$-0.788786\pi$
$727$		−	33.2139i		−	1.23183i	0.787811	−	0.615917i	$-0.211214\pi$
$728$		0			0
$729$	−3.06811			−0.113634
$730$		0			0
$731$	4.45080			0.164619
$732$		0			0
$733$			8.62065i			0.318411i	0.987245	+	0.159205i	$0.0508932\pi$
$733$			8.62065i			0.318411i	−0.987245	+	0.159205i	$0.949107\pi$
$734$		0			0
$735$	28.0899	+	38.4078i	1.03611	+	1.41669i
$736$		0			0
$737$			19.5492i			0.720104i
$738$		0			0
$739$	−45.1689			−1.66157			−0.830783	−	0.556597i	$-0.812107\pi$
$739$	−45.1689			−1.66157			−0.830783	+	0.556597i	$0.812107\pi$
$740$		0			0
$741$	−5.34438			−0.196331
$742$		0			0
$743$		−	24.7905i		−	0.909475i	−0.890626	−	0.454737i	$-0.849733\pi$
$743$		−	24.7905i		−	0.909475i	0.890626	−	0.454737i	$-0.150267\pi$
$744$		0			0
$745$	−28.9310	+	21.1589i	−1.05995	+	0.775204i
$746$		0			0
$747$		−	4.96972i		−	0.181833i
$748$		0			0
$749$	42.7943			1.56367
$750$		0			0
$751$	6.76869			0.246993			0.123497	−	0.992345i	$-0.460589\pi$
$751$	6.76869			0.246993			0.123497	+	0.992345i	$0.460589\pi$
$752$		0			0
$753$			33.8283i			1.23277i
$754$		0			0
$755$	25.8283	−	18.8898i	0.939989	−	0.687470i
$756$		0			0
$757$			45.2101i			1.64319i	0.570072	+	0.821595i	$0.306915\pi$
$757$			45.2101i			1.64319i	−0.570072	+	0.821595i	$0.693085\pi$
$758$		0			0
$759$	17.3600			0.630127
$760$		0			0
$761$	19.8851			0.720834			0.360417	−	0.932791i	$-0.382634\pi$
$761$	19.8851			0.720834			0.360417	+	0.932791i	$0.382634\pi$
$762$		0			0
$763$			12.4352i			0.450185i
$764$		0			0
$765$	1.03028	+	1.40871i	0.0372497	+	0.0509321i
$766$		0			0
$767$		−	7.93567i		−	0.286540i
$768$		0			0
$769$	−8.07615			−0.291233			−0.145617	−	0.989341i	$-0.546517\pi$
$769$	−8.07615			−0.291233			−0.145617	+	0.989341i	$0.546517\pi$
$770$		0			0
$771$	−2.23039			−0.0803257
$772$		0			0
$773$		−	45.9456i		−	1.65255i	−0.563267	−	0.826275i	$-0.690456\pi$
$773$		−	45.9456i		−	1.65255i	0.563267	−	0.826275i	$-0.309544\pi$
$774$		0			0
$775$	−5.56101	+	17.4886i	−0.199757	+	0.628211i
$776$		0			0
$777$		−	89.8383i		−	3.22293i
$778$		0			0
$779$	8.88979			0.318510
$780$		0			0
$781$	−29.4087			−1.05233
$782$		0			0
$783$		−	24.5951i		−	0.878956i
$784$		0			0
$785$	−23.7990	−	32.5407i	−0.849422	−	1.16143i
$786$		0			0
$787$			17.8439i			0.636067i	0.948079	+	0.318034i	$0.103022\pi$
$787$			17.8439i			0.636067i	−0.948079	+	0.318034i	$0.896978\pi$
$788$		0			0
$789$	−25.3700			−0.903194
$790$		0			0
$791$	79.1983			2.81597
$792$		0			0
$793$			31.7914i			1.12895i
$794$		0			0
$795$	−21.0353	+	15.3843i	−0.746044	+	0.545626i
$796$		0			0
$797$		−	37.3326i		−	1.32239i	−0.750215	−	0.661194i	$-0.770050\pi$
$797$		−	37.3326i		−	1.32239i	0.750215	−	0.661194i	$-0.229950\pi$
$798$		0			0
$799$	2.56009			0.0905696
$800$		0			0
$801$	−11.5298			−0.407386
$802$		0			0
$803$			7.13957i			0.251950i
$804$		0			0
$805$	23.0390	−	16.8498i	0.812020	−	0.593878i
$806$		0			0
$807$		−	33.8695i		−	1.19226i
$808$		0			0
$809$	38.6950			1.36044			0.680221	−	0.733007i	$-0.261884\pi$
$809$	38.6950			1.36044			0.680221	+	0.733007i	$0.261884\pi$
$810$		0			0
$811$	−13.3444			−0.468585			−0.234292	−	0.972166i	$-0.575277\pi$
$811$	−13.3444			−0.468585			−0.234292	+	0.972166i	$0.575277\pi$
$812$		0			0
$813$			3.11399i			0.109212i
$814$		0			0
$815$	3.56479	+	4.87420i	0.124869	+	0.170736i
$816$		0			0
$817$			8.64002i			0.302276i
$818$		0			0
$819$	15.7190			0.549268
$820$		0			0
$821$	−37.3482			−1.30346			−0.651730	−	0.758451i	$-0.725956\pi$
$821$	−37.3482			−1.30346			−0.651730	+	0.758451i	$0.725956\pi$
$822$		0			0
$823$		−	51.5630i		−	1.79737i	−0.438593	−	0.898686i	$-0.644523\pi$
$823$		−	51.5630i		−	1.79737i	0.438593	−	0.898686i	$-0.355477\pi$
$824$		0			0
$825$	−26.7299	−	8.49954i	−0.930617	−	0.295916i
$826$		0			0
$827$			48.5639i			1.68873i	0.535767	+	0.844366i	$0.320022\pi$
$827$			48.5639i			1.68873i	−0.535767	+	0.844366i	$0.679978\pi$
$828$		0			0
$829$	−0.325919			−0.0113196			−0.00565982	−	0.999984i	$-0.501802\pi$
$829$	−0.325919			−0.0113196			−0.00565982	+	0.999984i	$0.501802\pi$
$830$		0			0
$831$	69.0578			2.39559
$832$		0			0
$833$			5.15894i			0.178747i
$834$		0			0
$835$	−11.8146	−	16.1542i	−0.408860	−	0.559041i
$836$		0			0
$837$			11.5804i			0.400277i
$838$		0			0
$839$	−17.1202			−0.591055			−0.295527	−	0.955334i	$-0.595495\pi$
$839$	−17.1202			−0.591055			−0.295527	+	0.955334i	$0.595495\pi$
$840$		0			0
$841$	31.7649			1.09534
$842$		0			0
$843$		−	2.23039i		−	0.0768188i
$844$		0			0
$845$	12.0459	−	8.80986i	0.414391	−	0.303069i
$846$		0			0
$847$			16.6244i			0.571222i
$848$		0			0
$849$	−19.8477			−0.681171
$850$		0			0
$851$	−31.7190			−1.08731
$852$		0			0
$853$			24.9092i			0.852874i	0.904517	+	0.426437i	$0.140231\pi$
$853$			24.9092i			0.852874i	−0.904517	+	0.426437i	$0.859769\pi$
$854$		0			0
$855$	−2.73463	+	2.00000i	−0.0935225	+	0.0683986i
$856$		0			0
$857$		−	11.6509i		−	0.397988i	−0.980001	−	0.198994i	$-0.936233\pi$
$857$		−	11.6509i		−	0.397988i	0.980001	−	0.198994i	$-0.0637674\pi$
$858$		0			0
$859$	−5.35998			−0.182880			−0.0914400	−	0.995811i	$-0.529147\pi$
$859$	−5.35998			−0.182880			−0.0914400	+	0.995811i	$0.529147\pi$
$860$		0			0
$861$	−77.9182			−2.65545
$862$		0			0
$863$		−	41.0790i		−	1.39835i	−0.714953	−	0.699173i	$-0.753552\pi$
$863$		−	41.0790i		−	1.39835i	0.714953	−	0.699173i	$-0.246448\pi$
$864$		0			0
$865$	−0.409630	−	0.560094i	−0.0139278	−	0.0190438i
$866$		0			0
$867$		−	35.5592i		−	1.20765i
$868$		0			0
$869$	44.2186			1.50001
$870$		0			0
$871$	18.6244			0.631065
$872$		0			0
$873$		−	5.96881i		−	0.202014i
$874$		0			0
$875$	−43.7240	+	14.6644i	−1.47814	+	0.495747i
$876$		0			0
$877$			45.9532i			1.55173i	0.630899	+	0.775865i	$0.282686\pi$
$877$			45.9532i			1.55173i	−0.630899	+	0.775865i	$0.717314\pi$
$878$		0			0
$879$	−54.3435			−1.83296
$880$		0			0
$881$	−6.90917			−0.232776			−0.116388	−	0.993204i	$-0.537132\pi$
$881$	−6.90917			−0.232776			−0.116388	+	0.993204i	$0.537132\pi$
$882$		0			0
$883$			48.5213i			1.63287i	0.577435	+	0.816437i	$0.304054\pi$
$883$			48.5213i			1.63287i	−0.577435	+	0.816437i	$0.695946\pi$
$884$		0			0
$885$	−8.84983	−	12.1005i	−0.297484	−	0.406754i
$886$		0			0
$887$		−	23.7990i		−	0.799091i	−0.916713	−	0.399546i	$-0.869168\pi$
$887$		−	23.7990i		−	0.799091i	0.916713	−	0.399546i	$-0.130832\pi$
$888$		0			0
$889$	−59.0284			−1.97975
$890$		0			0
$891$	−29.6997			−0.994976
$892$		0			0
$893$			4.96972i			0.166305i
$894$		0			0
$895$	−2.76113	+	2.01938i	−0.0922943	+	0.0675003i
$896$		0			0
$897$		−	16.5388i		−	0.552213i
$898$		0			0
$899$	−28.6107			−0.954219
$900$		0			0
$901$	−2.82546			−0.0941298
$902$		0			0
$903$		−	75.7290i		−	2.52010i
$904$		0			0
$905$	−26.6206	+	19.4693i	−0.884900	+	0.647180i
$906$		0			0
$907$		−	17.9726i		−	0.596770i	−0.954446	−	0.298385i	$-0.903552\pi$
$907$		−	17.9726i		−	0.596770i	0.954446	−	0.298385i	$-0.0964479\pi$
$908$		0			0
$909$	20.9073			0.693453
$910$		0			0
$911$	−19.5592			−0.648024			−0.324012	−	0.946053i	$-0.605032\pi$
$911$	−19.5592			−0.648024			−0.324012	+	0.946053i	$0.605032\pi$
$912$		0			0
$913$			8.65940i			0.286584i
$914$		0			0
$915$	35.4537	+	48.4764i	1.17206	+	1.60258i
$916$		0			0
$917$		−	27.0596i		−	0.893588i
$918$		0			0
$919$	35.4948			1.17087			0.585433	−	0.810720i	$-0.300924\pi$
$919$	35.4948			1.17087			0.585433	+	0.810720i	$0.300924\pi$
$920$		0			0
$921$	−20.0587			−0.660957
$922$		0			0
$923$			28.0175i			0.922209i
$924$		0			0
$925$	48.8392	+	15.5298i	1.60582	+	0.510617i
$926$		0			0
$927$		−	6.93853i		−	0.227891i
$928$		0			0
$929$	−17.9532			−0.589026			−0.294513	−	0.955648i	$-0.595157\pi$
$929$	−17.9532			−0.589026			−0.294513	+	0.955648i	$0.595157\pi$
$930$		0			0
$931$	−10.0147			−0.328218
$932$		0			0
$933$		−	37.1122i		−	1.21500i
$934$		0			0
$935$	−1.79518	−	2.45459i	−0.0587088	−	0.0802735i
$936$		0			0
$937$		−	5.66652i		−	0.185117i	−0.995707	−	0.0925585i	$-0.970495\pi$
$937$		−	5.66652i		−	0.185117i	0.995707	−	0.0925585i	$-0.0295045\pi$
$938$		0			0
$939$	22.2148			0.724953
$940$		0			0
$941$	−24.7044			−0.805339			−0.402670	−	0.915345i	$-0.631918\pi$
$941$	−24.7044			−0.805339			−0.402670	+	0.915345i	$0.631918\pi$
$942$		0			0
$943$			27.5104i			0.895863i
$944$		0			0
$945$	−23.4899	+	17.1795i	−0.764125	+	0.558850i
$946$		0			0
$947$		−	13.5904i		−	0.441628i	−0.975316	−	0.220814i	$-0.929129\pi$
$947$		−	13.5904i		−	0.441628i	0.975316	−	0.220814i	$-0.0708713\pi$
$948$		0			0
$949$	6.80183			0.220797
$950$		0			0
$951$	9.20815			0.298595
$952$		0			0
$953$		−	24.7375i		−	0.801326i	−0.916225	−	0.400663i	$-0.868780\pi$
$953$		−	24.7375i		−	0.801326i	0.916225	−	0.400663i	$-0.131220\pi$
$954$		0			0
$955$	−32.7131	+	23.9251i	−1.05857	+	0.774197i
$956$		0			0
$957$		−	43.7290i		−	1.41356i
$958$		0			0
$959$	−26.6244			−0.859748
$960$		0			0
$961$	−17.5289			−0.565448
$962$		0			0
$963$		−	15.7190i		−	0.506539i
$964$		0			0
$965$	6.01938	+	8.23039i	0.193771	+	0.264946i
$966$		0			0
$967$			35.6897i			1.14770i	0.818960	+	0.573851i	$0.194551\pi$
$967$			35.6897i			1.14770i	−0.818960	+	0.573851i	$0.805449\pi$
$968$		0			0
$969$	−1.09461			−0.0351639
$970$		0			0
$971$	−16.4995			−0.529495			−0.264748	−	0.964318i	$-0.585289\pi$
$971$	−16.4995			−0.529495			−0.264748	+	0.964318i	$0.585289\pi$
$972$		0			0
$973$			95.1184i			3.04935i
$974$		0			0
$975$	−8.09747	+	25.4655i	−0.259327	+	0.815548i
$976$		0			0
$977$			49.8501i			1.59485i	0.603420	+	0.797423i	$0.293804\pi$
$977$			49.8501i			1.59485i	−0.603420	+	0.797423i	$0.706196\pi$
$978$		0			0
$979$	20.0899			0.642076
$980$		0			0
$981$	4.56766			0.145834
$982$		0			0
$983$		−	5.19014i		−	0.165540i	−0.996569	−	0.0827698i	$-0.973623\pi$
$983$		−	5.19014i		−	0.165540i	0.996569	−	0.0827698i	$-0.0263766\pi$
$984$		0			0
$985$	−2.82546	−	3.86330i	−0.0900267	−	0.123095i
$986$		0			0
$987$		−	43.5592i		−	1.38650i
$988$		0			0
$989$	−26.7375			−0.850203
$990$		0			0
$991$	32.4272			1.03008			0.515042	−	0.857165i	$-0.327776\pi$
$991$	32.4272			1.03008			0.515042	+	0.857165i	$0.327776\pi$
$992$		0			0
$993$		−	9.69771i		−	0.307748i
$994$		0			0
$995$	29.8957	−	21.8645i	0.947757	−	0.693152i
$996$		0			0
$997$			10.1992i			0.323012i	0.986872	+	0.161506i	$0.0516351\pi$
$997$			10.1992i			0.323012i	−0.986872	+	0.161506i	$0.948365\pi$
$998$		0			0
$999$	32.3397			1.02318

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1520.2.d.j.609.2		6
4.3	odd	2		190.2.b.b.39.6	yes	6
5.2	odd	4		7600.2.a.cd.1.1		3
5.3	odd	4		7600.2.a.bi.1.3		3
5.4	even	2	inner	1520.2.d.j.609.5		6
12.11	even	2		1710.2.d.d.1369.1		6
20.3	even	4		950.2.a.n.1.1		3
20.7	even	4		950.2.a.i.1.3		3
20.19	odd	2		190.2.b.b.39.1	✓	6
60.23	odd	4		8550.2.a.ck.1.3		3
60.47	odd	4		8550.2.a.cl.1.1		3
60.59	even	2		1710.2.d.d.1369.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
190.2.b.b.39.1	✓	6	20.19	odd	2
190.2.b.b.39.6	yes	6	4.3	odd	2
950.2.a.i.1.3		3	20.7	even	4
950.2.a.n.1.1		3	20.3	even	4
1520.2.d.j.609.2		6	1.1	even	1	trivial
1520.2.d.j.609.5		6	5.4	even	2	inner
1710.2.d.d.1369.1		6	12.11	even	2
1710.2.d.d.1369.4		6	60.59	even	2
7600.2.a.bi.1.3		3	5.3	odd	4
7600.2.a.cd.1.1		3	5.2	odd	4
8550.2.a.ck.1.3		3	60.23	odd	4
8550.2.a.cl.1.1		3	60.47	odd	4

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:	\( N \)	\(=\)	\( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1520.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.12489i q^{3} +(1.80487 - 1.32001i) q^{5} -4.12489i q^{7} -1.51514 q^{9} +O(q^{10})\)	\(q-2.12489i q^{3} +(1.80487 - 1.32001i) q^{5} -4.12489i q^{7} -1.51514 q^{9} +2.64002 q^{11} -2.51514i q^{13} +(-2.80487 - 3.83515i) q^{15} -0.515138i q^{17} +1.00000 q^{19} -8.76491 q^{21} +3.09461i q^{23} +(1.51514 - 4.76491i) q^{25} -3.15516i q^{27} +7.79518 q^{29} -3.67030 q^{31} -5.60975i q^{33} +(-5.44490 - 7.44490i) q^{35} +10.2498i q^{37} -5.34438 q^{39} +8.88979 q^{41} +8.64002i q^{43} +(-2.73463 + 2.00000i) q^{45} +4.96972i q^{47} -10.0147 q^{49} -1.09461 q^{51} -5.48486i q^{53} +(4.76491 - 3.48486i) q^{55} -2.12489i q^{57} +3.15516 q^{59} -12.6400 q^{61} +6.24977i q^{63} +(-3.32001 - 4.53951i) q^{65} +7.40493i q^{67} +6.57569 q^{69} -11.1396 q^{71} +2.70436i q^{73} +(-10.1249 - 3.21949i) q^{75} -10.8898i q^{77} +16.7493 q^{79} -11.2498 q^{81} +3.28005i q^{83} +(-0.679988 - 0.929759i) q^{85} -16.5639i q^{87} +7.60975 q^{89} -10.3747 q^{91} +7.79897i q^{93} +(1.80487 - 1.32001i) q^{95} +3.93945i q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{5} - 10 q^{9}+O(q^{10}) \) 6 * q + 2 * q^5 - 10 * q^9	\( 6 q + 2 q^{5} - 10 q^{9} - 8 q^{15} + 6 q^{19} - 20 q^{21} + 10 q^{25} + 16 q^{29} - 8 q^{31} - 8 q^{35} + 20 q^{39} + 4 q^{41} + 18 q^{45} + 6 q^{49} + 12 q^{51} - 4 q^{55} + 4 q^{59} - 60 q^{61} - 12 q^{65} + 44 q^{69} + 16 q^{71} - 44 q^{75} - 34 q^{81} - 12 q^{85} + 28 q^{89} - 12 q^{91} + 2 q^{95} - 24 q^{99}+O(q^{100}) \) 6 * q + 2 * q^5 - 10 * q^9 - 8 * q^15 + 6 * q^19 - 20 * q^21 + 10 * q^25 + 16 * q^29 - 8 * q^31 - 8 * q^35 + 20 * q^39 + 4 * q^41 + 18 * q^45 + 6 * q^49 + 12 * q^51 - 4 * q^55 + 4 * q^59 - 60 * q^61 - 12 * q^65 + 44 * q^69 + 16 * q^71 - 44 * q^75 - 34 * q^81 - 12 * q^85 + 28 * q^89 - 12 * q^91 + 2 * q^95 - 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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