LMFDB - Embedded newform 900.4.j.b.593.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [900,4,Mod(557,900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(900, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 2, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("900.557");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	900.j (of order $4$, degree $2$, 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:	$53.1017190052$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(i)$
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} - 40x^{10} + 607x^{8} - 3980x^{6} + 10171x^{4} + 2180x^{2} + 7225 $ x^12 - 40x^10 + 607x^8 - 3980x^6 + 10171x^4 + 2180*x^2 + 7225
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{19}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			593.5
Root			$-2.57283 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	900.593
Dual form			900.4.j.b.557.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(16.7931 - 16.7931i) q^{7} +O(q^{10})$	$q+(16.7931 - 16.7931i) q^{7} -49.9885i q^{11} +(39.8693 + 39.8693i) q^{13} +(73.3909 + 73.3909i) q^{17} +135.778i q^{19} +(21.1506 - 21.1506i) q^{23} +90.3423 q^{29} -208.236 q^{31} +(173.571 - 173.571i) q^{37} -229.545i q^{41} +(-31.7780 - 31.7780i) q^{43} +(-198.526 - 198.526i) q^{47} -221.014i q^{49} +(226.976 - 226.976i) q^{53} -163.675 q^{59} +524.846 q^{61} +(-52.0093 + 52.0093i) q^{67} +719.776i q^{71} +(727.125 + 727.125i) q^{73} +(-839.460 - 839.460i) q^{77} +665.900i q^{79} +(527.261 - 527.261i) q^{83} +1501.61 q^{89} +1339.06 q^{91} +(-488.389 + 488.389i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 24 q^{7}+O(q^{10}) $ 12 * q + 24 * q^7	$ 12 q + 24 q^{7} + 84 q^{13} + 144 q^{31} + 564 q^{37} + 960 q^{43} + 1920 q^{61} + 2256 q^{67} + 5076 q^{73} + 4944 q^{91} + 2916 q^{97}+O(q^{100}) $ 12 * q + 24 * q^7 + 84 * q^13 + 144 * q^31 + 564 * q^37 + 960 * q^43 + 1920 * q^61 + 2256 * q^67 + 5076 * q^73 + 4944 * q^91 + 2916 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/900\mathbb{Z}\right)^\times$.

$n$	$101$	$451$	$577$
$\chi(n)$	$-1$	$1$	$e\left(\frac{3}{4}\right)$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	16.7931	−	16.7931i	0.906740	−	0.906740i	−0.0892676	−	0.996008i	$-0.528453\pi$
$7$	16.7931	−	16.7931i	0.906740	−	0.906740i	0.996008	+	0.0892676i	$0.0284526\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	49.9885i		−	1.37019i	−0.728454	−	0.685095i	$-0.759761\pi$
$11$		−	49.9885i		−	1.37019i	0.728454	−	0.685095i	$-0.240239\pi$
$12$		0			0
$13$	39.8693	+	39.8693i	0.850597	+	0.850597i	0.990207	−	0.139609i	$-0.0445847\pi$
$13$	39.8693	+	39.8693i	0.850597	+	0.850597i	−0.139609	+	0.990207i	$0.544585\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	73.3909	+	73.3909i	1.04705	+	1.04705i	0.998837	+	0.0482159i	$0.0153536\pi$
$17$	73.3909	+	73.3909i	1.04705	+	1.04705i	0.0482159	+	0.998837i	$0.484646\pi$
$18$		0			0
$19$			135.778i			1.63945i	0.572755	+	0.819727i	$0.305875\pi$
$19$			135.778i			1.63945i	−0.572755	+	0.819727i	$0.694125\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	21.1506	−	21.1506i	0.191748	−	0.191748i	−0.604703	−	0.796451i	$-0.706708\pi$
$23$	21.1506	−	21.1506i	0.191748	−	0.191748i	0.796451	+	0.604703i	$0.206708\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$	90.3423			0.578488			0.289244	−	0.957255i	$-0.406596\pi$
$29$	90.3423			0.578488			0.289244	+	0.957255i	$0.406596\pi$
$30$		0			0
$31$	−208.236			−1.20646			−0.603230	−	0.797568i	$-0.706120\pi$
$31$	−208.236			−1.20646			−0.603230	+	0.797568i	$0.706120\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	173.571	−	173.571i	0.771214	−	0.771214i	−0.207105	−	0.978319i	$-0.566404\pi$
$37$	173.571	−	173.571i	0.771214	−	0.771214i	0.978319	+	0.207105i	$0.0664041\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	229.545i		−	0.874362i	−0.899374	−	0.437181i	$-0.855977\pi$
$41$		−	229.545i		−	0.874362i	0.899374	−	0.437181i	$-0.144023\pi$
$42$		0			0
$43$	−31.7780	−	31.7780i	−0.112700	−	0.112700i	0.648508	−	0.761208i	$-0.275393\pi$
$43$	−31.7780	−	31.7780i	−0.112700	−	0.112700i	−0.761208	+	0.648508i	$0.775393\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−198.526	−	198.526i	−0.616128	−	0.616128i	0.328408	−	0.944536i	$-0.393488\pi$
$47$	−198.526	−	198.526i	−0.616128	−	0.616128i	−0.944536	+	0.328408i	$0.893488\pi$
$48$		0			0
$49$		−	221.014i		−	0.644355i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	226.976	−	226.976i	0.588256	−	0.588256i	−0.348903	−	0.937159i	$-0.613446\pi$
$53$	226.976	−	226.976i	0.588256	−	0.588256i	0.937159	+	0.348903i	$0.113446\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−163.675			−0.361163			−0.180582	−	0.983560i	$-0.557798\pi$
$59$	−163.675			−0.361163			−0.180582	+	0.983560i	$0.557798\pi$
$60$		0			0
$61$	524.846			1.10163			0.550817	−	0.834626i	$-0.314316\pi$
$61$	524.846			1.10163			0.550817	+	0.834626i	$0.314316\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−52.0093	+	52.0093i	−0.0948351	+	0.0948351i	−0.752933	−	0.658098i	$-0.771361\pi$
$67$	−52.0093	+	52.0093i	−0.0948351	+	0.0948351i	0.658098	+	0.752933i	$0.271361\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			719.776i			1.20312i	0.798827	+	0.601561i	$0.205454\pi$
$71$			719.776i			1.20312i	−0.798827	+	0.601561i	$0.794546\pi$
$72$		0			0
$73$	727.125	+	727.125i	1.16580	+	1.16580i	0.983184	+	0.182618i	$0.0584572\pi$
$73$	727.125	+	727.125i	1.16580	+	1.16580i	0.182618	+	0.983184i	$0.441543\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−839.460	−	839.460i	−1.24241	−	1.24241i
$78$		0			0
$79$			665.900i			0.948350i	0.880431	+	0.474175i	$0.157254\pi$
$79$			665.900i			0.948350i	−0.880431	+	0.474175i	$0.842746\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	527.261	−	527.261i	0.697282	−	0.697282i	−0.266541	−	0.963824i	$-0.585881\pi$
$83$	527.261	−	527.261i	0.697282	−	0.697282i	0.963824	+	0.266541i	$0.0858808\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1501.61			1.78843			0.894216	−	0.447636i	$-0.147734\pi$
$89$	1501.61			1.78843			0.894216	+	0.447636i	$0.147734\pi$
$90$		0			0
$91$	1339.06			1.54254
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−488.389	+	488.389i	−0.511221	+	0.511221i	−0.914900	−	0.403680i	$-0.867731\pi$
$97$	−488.389	+	488.389i	−0.511221	+	0.511221i	0.403680	+	0.914900i	$0.367731\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	1198.84i		−	1.18108i	−0.807007	−	0.590542i	$-0.798914\pi$
$101$		−	1198.84i		−	1.18108i	0.807007	−	0.590542i	$-0.201086\pi$
$102$		0			0
$103$	−555.401	−	555.401i	−0.531313	−	0.531313i	0.389650	−	0.920963i	$-0.372596\pi$
$103$	−555.401	−	555.401i	−0.531313	−	0.531313i	−0.920963	+	0.389650i	$0.872596\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−32.8112	−	32.8112i	−0.0296447	−	0.0296447i	0.692129	−	0.721774i	$-0.256673\pi$
$107$	−32.8112	−	32.8112i	−0.0296447	−	0.0296447i	−0.721774	+	0.692129i	$0.756673\pi$
$108$		0			0
$109$		−	192.416i		−	0.169083i	−0.996420	−	0.0845417i	$-0.973057\pi$
$109$		−	192.416i		−	0.169083i	0.996420	−	0.0845417i	$-0.0269426\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1457.11	−	1457.11i	1.21304	−	1.21304i	0.243022	−	0.970021i	$-0.421861\pi$
$113$	1457.11	−	1457.11i	1.21304	−	1.21304i	0.970021	−	0.243022i	$-0.0781386\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	2464.91			1.89881
$120$		0			0
$121$	−1167.85			−0.877422
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	1176.74	−	1176.74i	0.822197	−	0.822197i	−0.164226	−	0.986423i	$-0.552513\pi$
$127$	1176.74	−	1176.74i	0.822197	−	0.822197i	0.986423	+	0.164226i	$0.0525127\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	1756.47i		−	1.17148i	−0.810501	−	0.585738i	$-0.800805\pi$
$131$		−	1756.47i		−	1.17148i	0.810501	−	0.585738i	$-0.199195\pi$
$132$		0			0
$133$	2280.13	+	2280.13i	1.48656	+	1.48656i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−183.842	−	183.842i	−0.114647	−	0.114647i	0.647456	−	0.762103i	$-0.275833\pi$
$137$	−183.842	−	183.842i	−0.114647	−	0.114647i	−0.762103	+	0.647456i	$0.775833\pi$
$138$		0			0
$139$			3000.62i			1.83100i	0.402319	+	0.915499i	$0.368204\pi$
$139$			3000.62i			1.83100i	−0.402319	+	0.915499i	$0.631796\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1993.01	−	1993.01i	1.16548	−	1.16548i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1116.97			−0.614135			−0.307068	−	0.951688i	$-0.599348\pi$
$149$	−1116.97			−0.614135			−0.307068	+	0.951688i	$0.599348\pi$
$150$		0			0
$151$	942.030			0.507691			0.253846	−	0.967245i	$-0.418305\pi$
$151$	942.030			0.507691			0.253846	+	0.967245i	$0.418305\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	1655.46	−	1655.46i	0.841532	−	0.841532i	−0.147526	−	0.989058i	$-0.547131\pi$
$157$	1655.46	−	1655.46i	0.841532	−	0.841532i	0.989058	+	0.147526i	$0.0471312\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	710.365i		−	0.347731i
$162$		0			0
$163$	1313.08	+	1313.08i	0.630972	+	0.630972i	0.948312	−	0.317340i	$-0.102790\pi$
$163$	1313.08	+	1313.08i	0.630972	+	0.630972i	−0.317340	+	0.948312i	$0.602790\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−793.306	−	793.306i	−0.367592	−	0.367592i	0.499006	−	0.866598i	$-0.333698\pi$
$167$	−793.306	−	793.306i	−0.367592	−	0.367592i	−0.866598	+	0.499006i	$0.833698\pi$
$168$		0			0
$169$			982.129i			0.447032i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−2362.02	+	2362.02i	−1.03804	+	1.03804i	−0.0387954	+	0.999247i	$0.512352\pi$
$173$	−2362.02	+	2362.02i	−1.03804	+	1.03804i	−0.999247	+	0.0387954i	$0.987648\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−3239.67			−1.35276			−0.676381	−	0.736552i	$-0.736453\pi$
$179$	−3239.67			−1.35276			−0.676381	+	0.736552i	$0.736453\pi$
$180$		0			0
$181$	−1984.14			−0.814806			−0.407403	−	0.913249i	$-0.633566\pi$
$181$	−1984.14			−0.814806			−0.407403	+	0.913249i	$0.633566\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	3668.70	−	3668.70i	1.43466	−	1.43466i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	1394.33i		−	0.528219i	−0.964493	−	0.264110i	$-0.914922\pi$
$191$		−	1394.33i		−	0.528219i	0.964493	−	0.264110i	$-0.0850780\pi$
$192$		0			0
$193$	−340.514	−	340.514i	−0.126999	−	0.126999i	0.640750	−	0.767749i	$-0.278623\pi$
$193$	−340.514	−	340.514i	−0.126999	−	0.126999i	−0.767749	+	0.640750i	$0.778623\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	390.115	+	390.115i	0.141089	+	0.141089i	0.774124	−	0.633035i	$-0.218191\pi$
$197$	390.115	+	390.115i	0.141089	+	0.141089i	−0.633035	+	0.774124i	$0.718191\pi$
$198$		0			0
$199$		−	2244.48i		−	0.799531i	−0.916617	−	0.399766i	$-0.869092\pi$
$199$		−	2244.48i		−	0.799531i	0.916617	−	0.399766i	$-0.130908\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1517.12	−	1517.12i	0.524538	−	0.524538i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	6787.34			2.24636
$210$		0			0
$211$	−914.862			−0.298492			−0.149246	−	0.988800i	$-0.547685\pi$
$211$	−914.862			−0.298492			−0.149246	+	0.988800i	$0.547685\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−3496.92	+	3496.92i	−1.09395	+	1.09395i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			5852.09i			1.78124i
$222$		0			0
$223$	−915.835	−	915.835i	−0.275017	−	0.275017i	0.556099	−	0.831116i	$-0.312298\pi$
$223$	−915.835	−	915.835i	−0.275017	−	0.275017i	−0.831116	+	0.556099i	$0.812298\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−3727.80	−	3727.80i	−1.08997	−	1.08997i	−0.995531	−	0.0944359i	$-0.969895\pi$
$227$	−3727.80	−	3727.80i	−1.08997	−	1.08997i	−0.0944359	−	0.995531i	$-0.530105\pi$
$228$		0			0
$229$			1907.49i			0.550439i	0.961381	+	0.275220i	$0.0887506\pi$
$229$			1907.49i			0.550439i	−0.961381	+	0.275220i	$0.911249\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	4496.57	−	4496.57i	1.26429	−	1.26429i	0.315300	−	0.948992i	$-0.397895\pi$
$233$	4496.57	−	4496.57i	1.26429	−	1.26429i	0.948992	−	0.315300i	$-0.102105\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−404.513			−0.109480			−0.0547401	−	0.998501i	$-0.517433\pi$
$239$	−404.513			−0.109480			−0.0547401	+	0.998501i	$0.517433\pi$
$240$		0			0
$241$	4593.29			1.22772			0.613859	−	0.789416i	$-0.289616\pi$
$241$	4593.29			1.22772			0.613859	+	0.789416i	$0.289616\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−5413.38	+	5413.38i	−1.39451	+	1.39451i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	4347.78i		−	1.09334i	−0.837347	−	0.546672i	$-0.815894\pi$
$251$		−	4347.78i		−	1.09334i	0.837347	−	0.546672i	$-0.184106\pi$
$252$		0			0
$253$	−1057.28	−	1057.28i	−0.262731	−	0.262731i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	3226.61	+	3226.61i	0.783154	+	0.783154i	0.980362	−	0.197208i	$-0.0631874\pi$
$257$	3226.61	+	3226.61i	0.783154	+	0.783154i	−0.197208	+	0.980362i	$0.563187\pi$
$258$		0			0
$259$		−	5829.58i		−	1.39858i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−4736.52	+	4736.52i	−1.11052	+	1.11052i	−0.117437	+	0.993080i	$0.537468\pi$
$263$	−4736.52	+	4736.52i	−1.11052	+	1.11052i	−0.993080	+	0.117437i	$0.962532\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−626.998			−0.142114			−0.0710571	−	0.997472i	$-0.522637\pi$
$269$	−626.998			−0.142114			−0.0710571	+	0.997472i	$0.522637\pi$
$270$		0			0
$271$	7238.49			1.62253			0.811267	−	0.584675i	$-0.198778\pi$
$271$	7238.49			1.62253			0.811267	+	0.584675i	$0.198778\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−826.689	+	826.689i	−0.179317	+	0.179317i	−0.791058	−	0.611741i	$-0.790470\pi$
$277$	−826.689	+	826.689i	−0.179317	+	0.179317i	0.611741	+	0.791058i	$0.290470\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	257.935i		−	0.0547584i	−0.999625	−	0.0273792i	$-0.991284\pi$
$281$		−	257.935i		−	0.0547584i	0.999625	−	0.0273792i	$-0.00871616\pi$
$282$		0			0
$283$	1773.67	+	1773.67i	0.372557	+	0.372557i	0.868408	−	0.495851i	$-0.165144\pi$
$283$	1773.67	+	1773.67i	0.372557	+	0.372557i	−0.495851	+	0.868408i	$0.665144\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−3854.76	−	3854.76i	−0.792819	−	0.792819i
$288$		0			0
$289$			5859.44i			1.19264i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	3784.93	−	3784.93i	0.754669	−	0.754669i	−0.220678	−	0.975347i	$-0.570827\pi$
$293$	3784.93	−	3784.93i	0.754669	−	0.754669i	0.975347	+	0.220678i	$0.0708269\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1686.52			0.326200
$300$		0			0
$301$	−1067.30			−0.204379
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	1787.21	−	1787.21i	0.332252	−	0.332252i	−0.521189	−	0.853441i	$-0.674511\pi$
$307$	1787.21	−	1787.21i	0.332252	−	0.332252i	0.853441	+	0.521189i	$0.174511\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			3131.30i			0.570932i	0.958389	+	0.285466i	$0.0921484\pi$
$311$			3131.30i			0.570932i	−0.958389	+	0.285466i	$0.907852\pi$
$312$		0			0
$313$	−4466.43	−	4466.43i	−0.806573	−	0.806573i	0.177540	−	0.984113i	$-0.443186\pi$
$313$	−4466.43	−	4466.43i	−0.806573	−	0.806573i	−0.984113	+	0.177540i	$0.943186\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	5820.24	+	5820.24i	1.03122	+	1.03122i	0.999497	+	0.0317255i	$0.0101002\pi$
$317$	5820.24	+	5820.24i	1.03122	+	1.03122i	0.0317255	+	0.999497i	$0.489900\pi$
$318$		0			0
$319$		−	4516.07i		−	0.792638i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−9964.87	+	9964.87i	−1.71659	+	1.71659i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−6667.72			−1.11734
$330$		0			0
$331$	−11136.1			−1.84923			−0.924616	−	0.380900i	$-0.875614\pi$
$331$	−11136.1			−1.84923			−0.924616	+	0.380900i	$0.875614\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	2097.22	−	2097.22i	0.339000	−	0.339000i	−0.516991	−	0.855991i	$-0.672948\pi$
$337$	2097.22	−	2097.22i	0.339000	−	0.339000i	0.855991	+	0.516991i	$0.172948\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			10409.4i			1.65308i
$342$		0			0
$343$	2048.52	+	2048.52i	0.322478	+	0.322478i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	7126.02	+	7126.02i	1.10243	+	1.10243i	0.994116	+	0.108319i	$0.0345467\pi$
$347$	7126.02	+	7126.02i	1.10243	+	1.10243i	0.108319	+	0.994116i	$0.465453\pi$
$348$		0			0
$349$		−	2291.45i		−	0.351457i	−0.984439	−	0.175728i	$-0.943772\pi$
$349$		−	2291.45i		−	0.351457i	0.984439	−	0.175728i	$-0.0562281\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−5495.85	+	5495.85i	−0.828653	+	0.828653i	−0.987331	−	0.158677i	$-0.949277\pi$
$353$	−5495.85	+	5495.85i	−0.828653	+	0.828653i	0.158677	+	0.987331i	$0.449277\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−6576.59			−0.966850			−0.483425	−	0.875386i	$-0.660607\pi$
$359$	−6576.59			−0.966850			−0.483425	+	0.875386i	$0.660607\pi$
$360$		0			0
$361$	−11576.7			−1.68781
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	1117.11	−	1117.11i	0.158891	−	0.158891i	−0.623184	−	0.782075i	$-0.714161\pi$
$367$	1117.11	−	1117.11i	0.158891	−	0.158891i	0.782075	+	0.623184i	$0.214161\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	7623.25i		−	1.06679i
$372$		0			0
$373$	−5732.32	−	5732.32i	−0.795733	−	0.795733i	0.186686	−	0.982420i	$-0.440225\pi$
$373$	−5732.32	−	5732.32i	−0.795733	−	0.795733i	−0.982420	+	0.186686i	$0.940225\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	3601.89	+	3601.89i	0.492060	+	0.492060i
$378$		0			0
$379$			150.195i			0.0203562i	0.999948	+	0.0101781i	$0.00323985\pi$
$379$			150.195i			0.0203562i	−0.999948	+	0.0101781i	$0.996760\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1758.09	+	1758.09i	−0.234554	+	0.234554i	−0.814590	−	0.580037i	$-0.803038\pi$
$383$	−1758.09	+	1758.09i	−0.234554	+	0.234554i	0.580037	+	0.814590i	$0.303038\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−14433.6			−1.88127			−0.940633	−	0.339426i	$-0.889767\pi$
$389$	−14433.6			−1.88127			−0.940633	+	0.339426i	$0.889767\pi$
$390$		0			0
$391$	3104.52			0.401540
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−4301.89	+	4301.89i	−0.543843	+	0.543843i	−0.924653	−	0.380810i	$-0.875645\pi$
$397$	−4301.89	+	4301.89i	−0.543843	+	0.543843i	0.380810	+	0.924653i	$0.375645\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	457.386i		−	0.0569595i	−0.999594	−	0.0284798i	$-0.990933\pi$
$401$		−	457.386i		−	0.0569595i	0.999594	−	0.0284798i	$-0.00906662\pi$
$402$		0			0
$403$	−8302.22	−	8302.22i	−1.02621	−	1.02621i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−8676.56	−	8676.56i	−1.05671	−	1.05671i
$408$		0			0
$409$		−	7075.25i		−	0.855376i	−0.903926	−	0.427688i	$-0.859328\pi$
$409$		−	7075.25i		−	0.855376i	0.903926	−	0.427688i	$-0.140672\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−2748.60	+	2748.60i	−0.327481	+	0.327481i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−5021.08			−0.585432			−0.292716	−	0.956199i	$-0.594559\pi$
$419$	−5021.08			−0.585432			−0.292716	+	0.956199i	$0.594559\pi$
$420$		0			0
$421$	−1611.80			−0.186590			−0.0932951	−	0.995639i	$-0.529740\pi$
$421$	−1611.80			−0.186590			−0.0932951	+	0.995639i	$0.529740\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	8813.77	−	8813.77i	0.998895	−	0.998895i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	1285.67i		−	0.143686i	−0.997416	−	0.0718429i	$-0.977112\pi$
$431$		−	1285.67i		−	0.143686i	0.997416	−	0.0718429i	$-0.0228880\pi$
$432$		0			0
$433$	4453.41	+	4453.41i	0.494266	+	0.494266i	0.909647	−	0.415381i	$-0.136352\pi$
$433$	4453.41	+	4453.41i	0.494266	+	0.494266i	−0.415381	+	0.909647i	$0.636352\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2871.78	+	2871.78i	0.314361	+	0.314361i
$438$		0			0
$439$		−	5640.74i		−	0.613253i	−0.951830	−	0.306626i	$-0.900800\pi$
$439$		−	5640.74i		−	0.613253i	0.951830	−	0.306626i	$-0.0992002\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−4547.25	+	4547.25i	−0.487690	+	0.487690i	−0.907577	−	0.419887i	$-0.862070\pi$
$443$	−4547.25	+	4547.25i	−0.487690	+	0.487690i	0.419887	+	0.907577i	$0.362070\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	3745.76			0.393704			0.196852	−	0.980433i	$-0.436928\pi$
$449$	3745.76			0.393704			0.196852	+	0.980433i	$0.436928\pi$
$450$		0			0
$451$	−11474.6			−1.19804
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−5946.19	+	5946.19i	−0.608645	+	0.608645i	−0.942592	−	0.333947i	$-0.891619\pi$
$457$	−5946.19	+	5946.19i	−0.608645	+	0.608645i	0.333947	+	0.942592i	$0.391619\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			3627.41i			0.366476i	0.983069	+	0.183238i	$0.0586579\pi$
$461$			3627.41i			0.366476i	−0.983069	+	0.183238i	$0.941342\pi$
$462$		0			0
$463$	−6156.53	−	6156.53i	−0.617966	−	0.617966i	0.327044	−	0.945009i	$-0.393948\pi$
$463$	−6156.53	−	6156.53i	−0.617966	−	0.617966i	−0.945009	+	0.327044i	$0.893948\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	12867.8	+	12867.8i	1.27505	+	1.27505i	0.943403	+	0.331648i	$0.107604\pi$
$467$	12867.8	+	12867.8i	1.27505	+	1.27505i	0.331648	+	0.943403i	$0.392396\pi$
$468$		0			0
$469$			1746.79i			0.171982i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−1588.54	+	1588.54i	−0.154421	+	0.154421i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	7170.94			0.684027			0.342013	−	0.939695i	$-0.388891\pi$
$479$	7170.94			0.684027			0.342013	+	0.939695i	$0.388891\pi$
$480$		0			0
$481$	13840.3			1.31199
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−14390.4	+	14390.4i	−1.33899	+	1.33899i	−0.441955	+	0.897037i	$0.645715\pi$
$487$	−14390.4	+	14390.4i	−1.33899	+	1.33899i	−0.897037	+	0.441955i	$0.854285\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			9178.02i			0.843581i	0.906693	+	0.421790i	$0.138598\pi$
$491$			9178.02i			0.843581i	−0.906693	+	0.421790i	$0.861402\pi$
$492$		0			0
$493$	6630.30	+	6630.30i	0.605707	+	0.605707i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	12087.2	+	12087.2i	1.09092	+	1.09092i
$498$		0			0
$499$		−	316.950i		−	0.0284342i	−0.999899	−	0.0142171i	$-0.995474\pi$
$499$		−	316.950i		−	0.0284342i	0.999899	−	0.0142171i	$-0.00452559\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−4171.48	+	4171.48i	−0.369775	+	0.369775i	−0.867395	−	0.497620i	$-0.834207\pi$
$503$	−4171.48	+	4171.48i	−0.369775	+	0.369775i	0.497620	+	0.867395i	$0.334207\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	11704.4			1.01923			0.509617	−	0.860401i	$-0.329787\pi$
$509$	11704.4			1.01923			0.509617	+	0.860401i	$0.329787\pi$
$510$		0			0
$511$	24421.3			2.11416
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−9924.02	+	9924.02i	−0.844213	+	0.844213i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			8724.98i			0.733682i	0.930284	+	0.366841i	$0.119561\pi$
$521$			8724.98i			0.733682i	−0.930284	+	0.366841i	$0.880439\pi$
$522$		0			0
$523$	−11368.3	−	11368.3i	−0.950482	−	0.950482i	0.0483487	−	0.998831i	$-0.484604\pi$
$523$	−11368.3	−	11368.3i	−0.950482	−	0.950482i	−0.998831	+	0.0483487i	$0.984604\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−15282.6	−	15282.6i	−1.26323	−	1.26323i
$528$		0			0
$529$			11272.3i			0.926466i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	9151.79	−	9151.79i	0.743730	−	0.743730i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−11048.1			−0.882889
$540$		0			0
$541$	−16073.2			−1.27734			−0.638672	−	0.769479i	$-0.720516\pi$
$541$	−16073.2			−1.27734			−0.638672	+	0.769479i	$0.720516\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	6402.64	−	6402.64i	0.500470	−	0.500470i	−0.411114	−	0.911584i	$-0.634860\pi$
$547$	6402.64	−	6402.64i	0.500470	−	0.500470i	0.911584	+	0.411114i	$0.134860\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			12266.5i			0.948404i
$552$		0			0
$553$	11182.5	+	11182.5i	0.859907	+	0.859907i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−12663.5	−	12663.5i	−0.963319	−	0.963319i	0.0360316	−	0.999351i	$-0.488528\pi$
$557$	−12663.5	−	12663.5i	−0.963319	−	0.963319i	−0.999351	+	0.0360316i	$0.988528\pi$
$558$		0			0
$559$		−	2533.94i		−	0.191725i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−16769.5	+	16769.5i	−1.25533	+	1.25533i	−0.302030	+	0.953298i	$0.597664\pi$
$563$	−16769.5	+	16769.5i	−1.25533	+	1.25533i	−0.953298	+	0.302030i	$0.902336\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−877.167			−0.0646269			−0.0323135	−	0.999478i	$-0.510287\pi$
$569$	−877.167			−0.0646269			−0.0323135	+	0.999478i	$0.510287\pi$
$570$		0			0
$571$	11607.7			0.850727			0.425364	−	0.905023i	$-0.360146\pi$
$571$	11607.7			0.850727			0.425364	+	0.905023i	$0.360146\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−12156.5	+	12156.5i	−0.877093	+	0.877093i	−0.993233	−	0.116140i	$-0.962948\pi$
$577$	−12156.5	+	12156.5i	−0.877093	+	0.877093i	0.116140	+	0.993233i	$0.462948\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	17708.7i		−	1.26451i
$582$		0			0
$583$	−11346.2	−	11346.2i	−0.806023	−	0.806023i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−6630.07	−	6630.07i	−0.466188	−	0.466188i	0.434489	−	0.900677i	$-0.356929\pi$
$587$	−6630.07	−	6630.07i	−0.466188	−	0.466188i	−0.900677	+	0.434489i	$0.856929\pi$
$588$		0			0
$589$		−	28273.8i		−	1.97793i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	10507.8	−	10507.8i	0.727665	−	0.727665i	−0.242489	−	0.970154i	$-0.577964\pi$
$593$	10507.8	−	10507.8i	0.727665	−	0.727665i	0.970154	+	0.242489i	$0.0779639\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−9783.11			−0.667324			−0.333662	−	0.942693i	$-0.608284\pi$
$599$	−9783.11			−0.667324			−0.333662	+	0.942693i	$0.608284\pi$
$600$		0			0
$601$	−4476.41			−0.303822			−0.151911	−	0.988394i	$-0.548543\pi$
$601$	−4476.41			−0.303822			−0.151911	+	0.988394i	$0.548543\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	20541.1	−	20541.1i	1.37354	−	1.37354i	0.518402	−	0.855137i	$-0.326527\pi$
$607$	20541.1	−	20541.1i	1.37354	−	1.37354i	0.855137	−	0.518402i	$-0.173473\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	15830.2i		−	1.04815i
$612$		0			0
$613$	−3795.79	−	3795.79i	−0.250099	−	0.250099i	0.570912	−	0.821011i	$-0.306590\pi$
$613$	−3795.79	−	3795.79i	−0.250099	−	0.250099i	−0.821011	+	0.570912i	$0.806590\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	3963.54	+	3963.54i	0.258616	+	0.258616i	0.824491	−	0.565875i	$-0.191462\pi$
$617$	3963.54	+	3963.54i	0.258616	+	0.258616i	−0.565875	+	0.824491i	$0.691462\pi$
$618$		0			0
$619$		−	1006.64i		−	0.0653641i	−0.999466	−	0.0326820i	$-0.989595\pi$
$619$		−	1006.64i		−	0.0653641i	0.999466	−	0.0326820i	$-0.0104049\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	25216.6	−	25216.6i	1.62164	−	1.62164i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	25477.1			1.61500
$630$		0			0
$631$	−7959.59			−0.502165			−0.251082	−	0.967966i	$-0.580787\pi$
$631$	−7959.59			−0.502165			−0.251082	+	0.967966i	$0.580787\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	8811.67	−	8811.67i	0.548087	−	0.548087i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			16600.3i			1.02289i	0.859315	+	0.511446i	$0.170890\pi$
$641$			16600.3i			1.02289i	−0.859315	+	0.511446i	$0.829110\pi$
$642$		0			0
$643$	8074.38	+	8074.38i	0.495214	+	0.495214i	0.909944	−	0.414731i	$-0.136124\pi$
$643$	8074.38	+	8074.38i	0.495214	+	0.495214i	−0.414731	+	0.909944i	$0.636124\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−2638.91	−	2638.91i	−0.160350	−	0.160350i	0.622372	−	0.782722i	$-0.286169\pi$
$647$	−2638.91	−	2638.91i	−0.160350	−	0.160350i	−0.782722	+	0.622372i	$0.786169\pi$
$648$		0			0
$649$			8181.85i			0.494862i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−8789.62	+	8789.62i	−0.526745	+	0.526745i	−0.919600	−	0.392855i	$-0.871487\pi$
$653$	−8789.62	+	8789.62i	−0.526745	+	0.526745i	0.392855	+	0.919600i	$0.371487\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−32064.7			−1.89539			−0.947697	−	0.319173i	$-0.896595\pi$
$659$	−32064.7			−1.89539			−0.947697	+	0.319173i	$0.896595\pi$
$660$		0			0
$661$	−17329.5			−1.01972			−0.509862	−	0.860256i	$-0.670304\pi$
$661$	−17329.5			−1.01972			−0.509862	+	0.860256i	$0.670304\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	1910.79	−	1910.79i	0.110924	−	0.110924i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	26236.3i		−	1.50945i
$672$		0			0
$673$	9956.86	+	9956.86i	0.570295	+	0.570295i	0.932211	−	0.361916i	$-0.117877\pi$
$673$	9956.86	+	9956.86i	0.570295	+	0.570295i	−0.361916	+	0.932211i	$0.617877\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	22964.2	+	22964.2i	1.30367	+	1.30367i	0.925898	+	0.377774i	$0.123310\pi$
$677$	22964.2	+	22964.2i	1.30367	+	1.30367i	0.377774	+	0.925898i	$0.376690\pi$
$678$		0			0
$679$			16403.1i			0.927089i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	192.932	−	192.932i	0.0108087	−	0.0108087i	−0.701682	−	0.712490i	$-0.747567\pi$
$683$	192.932	−	192.932i	0.0108087	−	0.0108087i	0.712490	+	0.701682i	$0.247567\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	18098.8			1.00074
$690$		0			0
$691$	−3993.48			−0.219854			−0.109927	−	0.993940i	$-0.535062\pi$
$691$	−3993.48			−0.219854			−0.109927	+	0.993940i	$0.535062\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	16846.5	−	16846.5i	0.915503	−	0.915503i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			3483.40i			0.187684i	0.995587	+	0.0938418i	$0.0299148\pi$
$701$			3483.40i			0.187684i	−0.995587	+	0.0938418i	$0.970085\pi$
$702$		0			0
$703$	23567.1	+	23567.1i	1.26437	+	1.26437i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−20132.3	−	20132.3i	−1.07094	−	1.07094i
$708$		0			0
$709$			28036.3i			1.48508i	0.669800	+	0.742542i	$0.266380\pi$
$709$			28036.3i			1.48508i	−0.669800	+	0.742542i	$0.733620\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−4404.30	+	4404.30i	−0.231336	+	0.231336i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	14513.8			0.752812			0.376406	−	0.926455i	$-0.377160\pi$
$719$	14513.8			0.752812			0.376406	+	0.926455i	$0.377160\pi$
$720$		0			0
$721$	−18653.8			−0.963526
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	19928.0	−	19928.0i	1.01663	−	1.01663i	0.0167662	−	0.999859i	$-0.494663\pi$
$727$	19928.0	−	19928.0i	1.01663	−	1.01663i	0.999859	−	0.0167662i	$-0.00533711\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	4664.44i		−	0.236006i
$732$		0			0
$733$	5116.67	+	5116.67i	0.257829	+	0.257829i	0.824171	−	0.566342i	$-0.191642\pi$
$733$	5116.67	+	5116.67i	0.257829	+	0.257829i	−0.566342	+	0.824171i	$0.691642\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	2599.87	+	2599.87i	0.129942	+	0.129942i
$738$		0			0
$739$			38538.8i			1.91837i	0.282785	+	0.959183i	$0.408742\pi$
$739$			38538.8i			1.91837i	−0.282785	+	0.959183i	$0.591258\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	7440.61	−	7440.61i	0.367388	−	0.367388i	−0.499136	−	0.866524i	$-0.666349\pi$
$743$	7440.61	−	7440.61i	0.367388	−	0.367388i	0.866524	+	0.499136i	$0.166349\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1102.00			−0.0537600
$750$		0			0
$751$	35044.7			1.70280			0.851398	−	0.524520i	$-0.175755\pi$
$751$	35044.7			1.70280			0.851398	+	0.524520i	$0.175755\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−9431.57	+	9431.57i	−0.452835	+	0.452835i	−0.896294	−	0.443459i	$-0.853751\pi$
$757$	−9431.57	+	9431.57i	−0.452835	+	0.452835i	0.443459	+	0.896294i	$0.353751\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			2764.97i			0.131709i	0.997829	+	0.0658543i	$0.0209773\pi$
$761$			2764.97i			0.131709i	−0.997829	+	0.0658543i	$0.979023\pi$
$762$		0			0
$763$	−3231.25	−	3231.25i	−0.153315	−	0.153315i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−6525.60	−	6525.60i	−0.307205	−	0.307205i
$768$		0			0
$769$			6575.70i			0.308356i	0.988043	+	0.154178i	$0.0492730\pi$
$769$			6575.70i			0.308356i	−0.988043	+	0.154178i	$0.950727\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	12868.1	−	12868.1i	0.598748	−	0.598748i	−0.341232	−	0.939979i	$-0.610844\pi$
$773$	12868.1	−	12868.1i	0.598748	−	0.598748i	0.939979	+	0.341232i	$0.110844\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	31167.1			1.43348
$780$		0			0
$781$	35980.5			1.64851
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−9231.75	+	9231.75i	−0.418140	+	0.418140i	−0.884562	−	0.466422i	$-0.845543\pi$
$787$	−9231.75	+	9231.75i	−0.418140	+	0.418140i	0.466422	+	0.884562i	$0.345543\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	48938.8i		−	2.19983i
$792$		0			0
$793$	20925.3	+	20925.3i	0.937047	+	0.937047i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−457.317	−	457.317i	−0.0203250	−	0.0203250i	0.696871	−	0.717196i	$-0.254575\pi$
$797$	−457.317	−	457.317i	−0.0203250	−	0.0203250i	−0.717196	+	0.696871i	$0.754575\pi$
$798$		0			0
$799$		−	29140.0i		−	1.29024i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	36347.9	−	36347.9i	1.59737	−	1.59737i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	13065.8			0.567823			0.283911	−	0.958851i	$-0.408368\pi$
$809$	13065.8			0.567823			0.283911	+	0.958851i	$0.408368\pi$
$810$		0			0
$811$	5474.83			0.237050			0.118525	−	0.992951i	$-0.462183\pi$
$811$	5474.83			0.237050			0.118525	+	0.992951i	$0.462183\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	4314.76	−	4314.76i	0.184767	−	0.184767i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			752.823i			0.0320021i	0.999872	+	0.0160010i	$0.00509350\pi$
$821$			752.823i			0.0320021i	−0.999872	+	0.0160010i	$0.994906\pi$
$822$		0			0
$823$	−6521.25	−	6521.25i	−0.276205	−	0.276205i	0.555387	−	0.831592i	$-0.312570\pi$
$823$	−6521.25	−	6521.25i	−0.276205	−	0.276205i	−0.831592	+	0.555387i	$0.812570\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1405.13	−	1405.13i	−0.0590826	−	0.0590826i	0.676948	−	0.736031i	$-0.263302\pi$
$827$	−1405.13	−	1405.13i	−0.0590826	−	0.0590826i	−0.736031	+	0.676948i	$0.763302\pi$
$828$		0			0
$829$		−	4291.98i		−	0.179815i	−0.995950	−	0.0899075i	$-0.971343\pi$
$829$		−	4291.98i		−	0.179815i	0.995950	−	0.0899075i	$-0.0286571\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	16220.4	−	16220.4i	0.674674	−	0.674674i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	29598.3			1.21794			0.608968	−	0.793195i	$-0.291584\pi$
$839$	29598.3			1.21794			0.608968	+	0.793195i	$0.291584\pi$
$840$		0			0
$841$	−16227.3			−0.665352
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−19611.7	+	19611.7i	−0.795593	+	0.795593i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	7342.25i		−	0.295757i
$852$		0			0
$853$	−7024.10	−	7024.10i	−0.281947	−	0.281947i	0.551938	−	0.833885i	$-0.313888\pi$
$853$	−7024.10	−	7024.10i	−0.281947	−	0.281947i	−0.833885	+	0.551938i	$0.813888\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−25138.6	−	25138.6i	−1.00201	−	1.00201i	−0.999998	−	0.00200715i	$-0.999361\pi$
$857$	−25138.6	−	25138.6i	−1.00201	−	1.00201i	−0.00200715	−	0.999998i	$-0.500639\pi$
$858$		0			0
$859$		−	3277.00i		−	0.130163i	−0.997880	−	0.0650813i	$-0.979269\pi$
$859$		−	3277.00i		−	0.130163i	0.997880	−	0.0650813i	$-0.0207307\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	4771.77	−	4771.77i	0.188219	−	0.188219i	−0.606707	−	0.794926i	$-0.707510\pi$
$863$	4771.77	−	4771.77i	0.188219	−	0.188219i	0.794926	+	0.606707i	$0.207510\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	33287.3			1.29942
$870$		0			0
$871$	−4147.16			−0.161333
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−17526.4	+	17526.4i	−0.674830	+	0.674830i	−0.958826	−	0.283996i	$-0.908340\pi$
$877$	−17526.4	+	17526.4i	−0.674830	+	0.674830i	0.283996	+	0.958826i	$0.408340\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			30643.9i			1.17187i	0.810358	+	0.585935i	$0.199273\pi$
$881$			30643.9i			1.17187i	−0.810358	+	0.585935i	$0.800727\pi$
$882$		0			0
$883$	14497.7	+	14497.7i	0.552534	+	0.552534i	0.927171	−	0.374637i	$-0.122233\pi$
$883$	14497.7	+	14497.7i	0.552534	+	0.552534i	−0.374637	+	0.927171i	$0.622233\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−12319.8	−	12319.8i	−0.466356	−	0.466356i	0.434376	−	0.900732i	$-0.356969\pi$
$887$	−12319.8	−	12319.8i	−0.466356	−	0.466356i	−0.900732	+	0.434376i	$0.856969\pi$
$888$		0			0
$889$		−	39522.2i		−	1.49104i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	26955.5	−	26955.5i	1.01011	−	1.01011i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−18812.5			−0.697922
$900$		0			0
$901$	33315.9			1.23187
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	17480.5	−	17480.5i	0.639946	−	0.639946i	−0.310596	−	0.950542i	$-0.600529\pi$
$907$	17480.5	−	17480.5i	0.639946	−	0.639946i	0.950542	+	0.310596i	$0.100529\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			34851.6i			1.26749i	0.773542	+	0.633746i	$0.218483\pi$
$911$			34851.6i			1.26749i	−0.773542	+	0.633746i	$0.781517\pi$
$912$		0			0
$913$	−26357.0	−	26357.0i	−0.955410	−	0.955410i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−29496.5	−	29496.5i	−1.06222	−	1.06222i
$918$		0			0
$919$			26995.2i			0.968976i	0.874798	+	0.484488i	$0.160994\pi$
$919$			26995.2i			0.968976i	−0.874798	+	0.484488i	$0.839006\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−28697.0	+	28697.0i	−1.02337	+	1.02337i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−44053.4			−1.55581			−0.777903	−	0.628384i	$-0.783717\pi$
$929$	−44053.4			−1.55581			−0.777903	+	0.628384i	$0.783717\pi$
$930$		0			0
$931$	30008.8			1.05639
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−6875.16	+	6875.16i	−0.239703	+	0.239703i	−0.816727	−	0.577024i	$-0.804214\pi$
$937$	−6875.16	+	6875.16i	−0.239703	+	0.239703i	0.577024	+	0.816727i	$0.304214\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			2705.85i			0.0937386i	0.998901	+	0.0468693i	$0.0149244\pi$
$941$			2705.85i			0.0937386i	−0.998901	+	0.0468693i	$0.985076\pi$
$942$		0			0
$943$	−4855.00	−	4855.00i	−0.167657	−	0.167657i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−14147.5	−	14147.5i	−0.485460	−	0.485460i	0.421410	−	0.906870i	$-0.361535\pi$
$947$	−14147.5	−	14147.5i	−0.485460	−	0.485460i	−0.906870	+	0.421410i	$0.861535\pi$
$948$		0			0
$949$			57980.0i			1.98326i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−17218.2	+	17218.2i	−0.585260	+	0.585260i	−0.936344	−	0.351084i	$-0.885813\pi$
$953$	−17218.2	+	17218.2i	−0.585260	+	0.585260i	0.351084	+	0.936344i	$0.385813\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−6174.55			−0.207911
$960$		0			0
$961$	13571.1			0.455544
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	18524.2	−	18524.2i	0.616026	−	0.616026i	−0.328483	−	0.944510i	$-0.606537\pi$
$967$	18524.2	−	18524.2i	0.616026	−	0.616026i	0.944510	+	0.328483i	$0.106537\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			31016.8i			1.02511i	0.858656	+	0.512553i	$0.171300\pi$
$971$			31016.8i			1.02511i	−0.858656	+	0.512553i	$0.828700\pi$
$972$		0			0
$973$	50389.5	+	50389.5i	1.66024	+	1.66024i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	8092.09	+	8092.09i	0.264984	+	0.264984i	0.827075	−	0.562091i	$-0.190003\pi$
$977$	8092.09	+	8092.09i	0.264984	+	0.264984i	−0.562091	+	0.827075i	$0.690003\pi$
$978$		0			0
$979$		−	75063.2i		−	2.45049i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−32991.2	+	32991.2i	−1.07046	+	1.07046i	−0.0731330	+	0.997322i	$0.523300\pi$
$983$	−32991.2	+	32991.2i	−1.07046	+	1.07046i	−0.997322	+	0.0731330i	$0.976700\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1344.25			−0.0432200
$990$		0			0
$991$	−50820.9			−1.62904			−0.814521	−	0.580135i	$-0.803000\pi$
$991$	−50820.9			−1.62904			−0.814521	+	0.580135i	$0.803000\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	21575.2	−	21575.2i	0.685351	−	0.685351i	−0.275850	−	0.961201i	$-0.588959\pi$
$997$	21575.2	−	21575.2i	0.685351	−	0.685351i	0.961201	+	0.275850i	$0.0889593\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.4.j.b.593.5		12
3.2	odd	2	inner	900.4.j.b.593.6		12
5.2	odd	4	inner	900.4.j.b.557.5		12
5.3	odd	4		180.4.j.a.17.3	✓	12
5.4	even	2		180.4.j.a.53.4	yes	12
15.2	even	4	inner	900.4.j.b.557.6		12
15.8	even	4		180.4.j.a.17.4	yes	12
15.14	odd	2		180.4.j.a.53.3	yes	12
20.3	even	4		720.4.w.e.17.3		12
20.19	odd	2		720.4.w.e.593.4		12
60.23	odd	4		720.4.w.e.17.4		12
60.59	even	2		720.4.w.e.593.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.4.j.a.17.3	✓	12	5.3	odd	4
180.4.j.a.17.4	yes	12	15.8	even	4
180.4.j.a.53.3	yes	12	15.14	odd	2
180.4.j.a.53.4	yes	12	5.4	even	2
720.4.w.e.17.3		12	20.3	even	4
720.4.w.e.17.4		12	60.23	odd	4
720.4.w.e.593.3		12	60.59	even	2
720.4.w.e.593.4		12	20.19	odd	2
900.4.j.b.557.5		12	5.2	odd	4	inner
900.4.j.b.557.6		12	15.2	even	4	inner
900.4.j.b.593.5		12	1.1	even	1	trivial
900.4.j.b.593.6		12	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	900.j (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(16.7931 - 16.7931i) q^{7} +O(q^{10})\)	\(q+(16.7931 - 16.7931i) q^{7} -49.9885i q^{11} +(39.8693 + 39.8693i) q^{13} +(73.3909 + 73.3909i) q^{17} +135.778i q^{19} +(21.1506 - 21.1506i) q^{23} +90.3423 q^{29} -208.236 q^{31} +(173.571 - 173.571i) q^{37} -229.545i q^{41} +(-31.7780 - 31.7780i) q^{43} +(-198.526 - 198.526i) q^{47} -221.014i q^{49} +(226.976 - 226.976i) q^{53} -163.675 q^{59} +524.846 q^{61} +(-52.0093 + 52.0093i) q^{67} +719.776i q^{71} +(727.125 + 727.125i) q^{73} +(-839.460 - 839.460i) q^{77} +665.900i q^{79} +(527.261 - 527.261i) q^{83} +1501.61 q^{89} +1339.06 q^{91} +(-488.389 + 488.389i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 24 q^{7}+O(q^{10}) \) 12 * q + 24 * q^7	\( 12 q + 24 q^{7} + 84 q^{13} + 144 q^{31} + 564 q^{37} + 960 q^{43} + 1920 q^{61} + 2256 q^{67} + 5076 q^{73} + 4944 q^{91} + 2916 q^{97}+O(q^{100}) \) 12 * q + 24 * q^7 + 84 * q^13 + 144 * q^31 + 564 * q^37 + 960 * q^43 + 1920 * q^61 + 2256 * q^67 + 5076 * q^73 + 4944 * q^91 + 2916 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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