LMFDB - Embedded newform 900.6.j.c.593.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [900,6,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, 6, names="a")

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

chi := DirichletCharacter("900.557");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
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:	$144.345437832$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(i)$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			593.10
Character	$\chi$	$=$	900.593
Dual form			900.6.j.c.557.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(102.346 - 102.346i) q^{7} +O(q^{10})$	$q+(102.346 - 102.346i) q^{7} +185.565i q^{11} +(729.569 + 729.569i) q^{13} +(1100.44 + 1100.44i) q^{17} +2038.77i q^{19} +(-1540.78 + 1540.78i) q^{23} -6655.22 q^{29} -1250.59 q^{31} +(-2693.46 + 2693.46i) q^{37} -18563.6i q^{41} +(2448.29 + 2448.29i) q^{43} +(-14042.6 - 14042.6i) q^{47} -4142.61i q^{49} +(-13787.0 + 13787.0i) q^{53} +28911.7 q^{59} -27302.7 q^{61} +(-3315.26 + 3315.26i) q^{67} +57519.2i q^{71} +(11765.5 + 11765.5i) q^{73} +(18991.9 + 18991.9i) q^{77} +6891.24i q^{79} +(41987.7 - 41987.7i) q^{83} -104880. q^{89} +149338. q^{91} +(16778.7 - 16778.7i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q + 13848 q^{31} - 28200 q^{61} + 908328 q^{91}+O(q^{100}) $ 24 * q + 13848 * q^31 - 28200 * q^61 + 908328 * q^91

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
$7$	102.346	−	102.346i	0.789456	−	0.789456i	−0.191949	−	0.981405i	$-0.561481\pi$
$7$	102.346	−	102.346i	0.789456	−	0.789456i	0.981405	+	0.191949i	$0.0614808\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			185.565i			0.462396i	0.972907	+	0.231198i	$0.0742646\pi$
$11$			185.565i			0.462396i	−0.972907	+	0.231198i	$0.925735\pi$
$12$		0			0
$13$	729.569	+	729.569i	1.19731	+	1.19731i	0.974968	+	0.222347i	$0.0713717\pi$
$13$	729.569	+	729.569i	1.19731	+	1.19731i	0.222347	+	0.974968i	$0.428628\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1100.44	+	1100.44i	0.923516	+	0.923516i	0.997276	−	0.0737603i	$-0.0235000\pi$
$17$	1100.44	+	1100.44i	0.923516	+	0.923516i	−0.0737603	+	0.997276i	$0.523500\pi$
$18$		0			0
$19$			2038.77i			1.29564i	0.761794	+	0.647820i	$0.224319\pi$
$19$			2038.77i			1.29564i	−0.761794	+	0.647820i	$0.775681\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1540.78	+	1540.78i	−0.607327	+	0.607327i	−0.942247	−	0.334920i	$-0.891291\pi$
$23$	−1540.78	+	1540.78i	−0.607327	+	0.607327i	0.334920	+	0.942247i	$0.391291\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−6655.22			−1.46949			−0.734746	−	0.678342i	$-0.762699\pi$
$29$	−6655.22			−1.46949			−0.734746	+	0.678342i	$0.762699\pi$
$30$		0			0
$31$	−1250.59			−0.233728			−0.116864	−	0.993148i	$-0.537284\pi$
$31$	−1250.59			−0.233728			−0.116864	+	0.993148i	$0.537284\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−2693.46	+	2693.46i	−0.323450	+	0.323450i	−0.850089	−	0.526639i	$-0.823452\pi$
$37$	−2693.46	+	2693.46i	−0.323450	+	0.323450i	0.526639	+	0.850089i	$0.323452\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	18563.6i		−	1.72465i	−0.506353	−	0.862326i	$-0.669007\pi$
$41$		−	18563.6i		−	1.72465i	0.506353	−	0.862326i	$-0.330993\pi$
$42$		0			0
$43$	2448.29	+	2448.29i	0.201926	+	0.201926i	0.800825	−	0.598899i	$-0.204395\pi$
$43$	2448.29	+	2448.29i	0.201926	+	0.201926i	−0.598899	+	0.800825i	$0.704395\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−14042.6	−	14042.6i	−0.927266	−	0.927266i	0.0702629	−	0.997529i	$-0.477616\pi$
$47$	−14042.6	−	14042.6i	−0.927266	−	0.927266i	−0.997529	+	0.0702629i	$0.977616\pi$
$48$		0			0
$49$		−	4142.61i		−	0.246481i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−13787.0	+	13787.0i	−0.674186	+	0.674186i	−0.958678	−	0.284493i	$-0.908175\pi$
$53$	−13787.0	+	13787.0i	−0.674186	+	0.674186i	0.284493	+	0.958678i	$0.408175\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	28911.7			1.08129			0.540646	−	0.841250i	$-0.318180\pi$
$59$	28911.7			1.08129			0.540646	+	0.841250i	$0.318180\pi$
$60$		0			0
$61$	−27302.7			−0.939466			−0.469733	−	0.882809i	$-0.655650\pi$
$61$	−27302.7			−0.939466			−0.469733	+	0.882809i	$0.655650\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−3315.26	+	3315.26i	−0.0902259	+	0.0902259i	−0.750779	−	0.660553i	$-0.770322\pi$
$67$	−3315.26	+	3315.26i	−0.0902259	+	0.0902259i	0.660553	+	0.750779i	$0.270322\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			57519.2i			1.35415i	0.735914	+	0.677075i	$0.236753\pi$
$71$			57519.2i			1.35415i	−0.735914	+	0.677075i	$0.763247\pi$
$72$		0			0
$73$	11765.5	+	11765.5i	0.258407	+	0.258407i	0.824406	−	0.565999i	$-0.191509\pi$
$73$	11765.5	+	11765.5i	0.258407	+	0.258407i	−0.565999	+	0.824406i	$0.691509\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	18991.9	+	18991.9i	0.365042	+	0.365042i
$78$		0			0
$79$			6891.24i			0.124231i	0.998069	+	0.0621154i	$0.0197847\pi$
$79$			6891.24i			0.124231i	−0.998069	+	0.0621154i	$0.980215\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	41987.7	−	41987.7i	0.669001	−	0.669001i	−0.288484	−	0.957485i	$-0.593151\pi$
$83$	41987.7	−	41987.7i	0.669001	−	0.669001i	0.957485	+	0.288484i	$0.0931511\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−104880.			−1.40351			−0.701755	−	0.712418i	$-0.747600\pi$
$89$	−104880.			−1.40351			−0.701755	+	0.712418i	$0.747600\pi$
$90$		0			0
$91$	149338.			1.89045
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	16778.7	−	16778.7i	0.181063	−	0.181063i	−0.610756	−	0.791819i	$-0.709134\pi$
$97$	16778.7	−	16778.7i	0.181063	−	0.181063i	0.791819	+	0.610756i	$0.209134\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			111047.i			1.08319i	0.840640	+	0.541594i	$0.182179\pi$
$101$			111047.i			1.08319i	−0.840640	+	0.541594i	$0.817821\pi$
$102$		0			0
$103$	−82594.1	−	82594.1i	−0.767107	−	0.767107i	0.210489	−	0.977596i	$-0.432494\pi$
$103$	−82594.1	−	82594.1i	−0.767107	−	0.767107i	−0.977596	+	0.210489i	$0.932494\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−32768.6	−	32768.6i	−0.276694	−	0.276694i	0.555094	−	0.831788i	$-0.312682\pi$
$107$	−32768.6	−	32768.6i	−0.276694	−	0.276694i	−0.831788	+	0.555094i	$0.812682\pi$
$108$		0			0
$109$			35645.8i			0.287370i	0.989623	+	0.143685i	$0.0458953\pi$
$109$			35645.8i			0.287370i	−0.989623	+	0.143685i	$0.954105\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−38460.7	+	38460.7i	−0.283349	+	0.283349i	−0.834443	−	0.551094i	$-0.814211\pi$
$113$	−38460.7	+	38460.7i	−0.283349	+	0.283349i	0.551094	+	0.834443i	$0.314211\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	225252.			1.45815
$120$		0			0
$121$	126617.			0.786190
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−176176.	+	176176.i	−0.969253	+	0.969253i	−0.999541	−	0.0302881i	$-0.990358\pi$
$127$	−176176.	+	176176.i	−0.969253	+	0.969253i	0.0302881	+	0.999541i	$0.490358\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			1051.25i			0.00535217i	0.999996	+	0.00267608i	$0.000851825\pi$
$131$			1051.25i			0.00535217i	−0.999996	+	0.00267608i	$0.999148\pi$
$132$		0			0
$133$	208661.	+	208661.i	1.02285	+	1.02285i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−208521.	−	208521.i	−0.949180	−	0.949180i	0.0495896	−	0.998770i	$-0.484209\pi$
$137$	−208521.	−	208521.i	−0.949180	−	0.949180i	−0.998770	+	0.0495896i	$0.984209\pi$
$138$		0			0
$139$		−	261154.i		−	1.14646i	−0.819393	−	0.573232i	$-0.805690\pi$
$139$		−	261154.i		−	1.14646i	0.819393	−	0.573232i	$-0.194310\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−135383.	+	135383.i	−0.553634	+	0.553634i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−129736.			−0.478735			−0.239368	−	0.970929i	$-0.576940\pi$
$149$	−129736.			−0.478735			−0.239368	+	0.970929i	$0.576940\pi$
$150$		0			0
$151$	−456119.			−1.62793			−0.813964	−	0.580915i	$-0.802695\pi$
$151$	−456119.			−1.62793			−0.813964	+	0.580915i	$0.802695\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−106979.	+	106979.i	−0.346379	+	0.346379i	−0.858759	−	0.512380i	$-0.828764\pi$
$157$	−106979.	+	106979.i	−0.346379	+	0.346379i	0.512380	+	0.858759i	$0.328764\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			315388.i			0.958915i
$162$		0			0
$163$	169537.	+	169537.i	0.499798	+	0.499798i	0.911375	−	0.411577i	$-0.135022\pi$
$163$	169537.	+	169537.i	0.499798	+	0.499798i	−0.411577	+	0.911375i	$0.635022\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−430895.	−	430895.i	−1.19559	−	1.19559i	−0.975475	−	0.220110i	$-0.929358\pi$
$167$	−430895.	−	430895.i	−1.19559	−	1.19559i	−0.220110	−	0.975475i	$-0.570642\pi$
$168$		0			0
$169$			693250.i			1.86712i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−309594.	+	309594.i	−0.786462	+	0.786462i	−0.980912	−	0.194451i	$-0.937708\pi$
$173$	−309594.	+	309594.i	−0.786462	+	0.786462i	0.194451	+	0.980912i	$0.437708\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−689654.			−1.60879			−0.804393	−	0.594097i	$-0.797509\pi$
$179$	−689654.			−1.60879			−0.804393	+	0.594097i	$0.797509\pi$
$180$		0			0
$181$	846782.			1.92121			0.960605	−	0.277916i	$-0.0896438\pi$
$181$	846782.			1.92121			0.960605	+	0.277916i	$0.0896438\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−204203.	+	204203.i	−0.427030	+	0.427030i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			99374.3i			0.197102i	0.995132	+	0.0985510i	$0.0314207\pi$
$191$			99374.3i			0.197102i	−0.995132	+	0.0985510i	$0.968579\pi$
$192$		0			0
$193$	−126273.	−	126273.i	−0.244016	−	0.244016i	0.574493	−	0.818509i	$-0.305199\pi$
$193$	−126273.	−	126273.i	−0.244016	−	0.244016i	−0.818509	+	0.574493i	$0.805199\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	10072.1	+	10072.1i	0.0184908	+	0.0184908i	0.716292	−	0.697801i	$-0.245838\pi$
$197$	10072.1	+	10072.1i	0.0184908	+	0.0184908i	−0.697801	+	0.716292i	$0.745838\pi$
$198$		0			0
$199$			229264.i			0.410396i	0.978720	+	0.205198i	$0.0657838\pi$
$199$			229264.i			0.410396i	−0.978720	+	0.205198i	$0.934216\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−681138.	+	681138.i	−1.16010	+	1.16010i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−378324.			−0.599099
$210$		0			0
$211$	791204.			1.22344			0.611719	−	0.791075i	$-0.290478\pi$
$211$	791204.			1.22344			0.611719	+	0.791075i	$0.290478\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−127994.	+	127994.i	−0.184518	+	0.184518i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			1.60570e6i			2.21148i
$222$		0			0
$223$	948392.	+	948392.i	1.27710	+	1.27710i	0.942282	+	0.334820i	$0.108676\pi$
$223$	948392.	+	948392.i	1.27710	+	1.27710i	0.334820	+	0.942282i	$0.391324\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−806278.	−	806278.i	−1.03853	−	1.03853i	−0.999227	−	0.0393056i	$-0.987485\pi$
$227$	−806278.	−	806278.i	−1.03853	−	1.03853i	−0.0393056	−	0.999227i	$-0.512515\pi$
$228$		0			0
$229$		−	1.06078e6i		−	1.33671i	−0.743842	−	0.668355i	$-0.766999\pi$
$229$		−	1.06078e6i		−	1.33671i	0.743842	−	0.668355i	$-0.233001\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	274390.	−	274390.i	0.331115	−	0.331115i	−0.521895	−	0.853010i	$-0.674775\pi$
$233$	274390.	−	274390.i	0.331115	−	0.331115i	0.853010	+	0.521895i	$0.174775\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1.29872e6			1.47069			0.735345	−	0.677693i	$-0.237020\pi$
$239$	1.29872e6			1.47069			0.735345	+	0.677693i	$0.237020\pi$
$240$		0			0
$241$	398973.			0.442487			0.221244	−	0.975219i	$-0.428988\pi$
$241$	398973.			0.442487			0.221244	+	0.975219i	$0.428988\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−1.48742e6	+	1.48742e6i	−1.55129	+	1.55129i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			1.43701e6i			1.43972i	0.694121	+	0.719858i	$0.255793\pi$
$251$			1.43701e6i			1.43972i	−0.694121	+	0.719858i	$0.744207\pi$
$252$		0			0
$253$	−285916.	−	285916.i	−0.280826	−	0.280826i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1.07873e6	−	1.07873e6i	−1.01877	−	1.01877i	−0.999820	−	0.0189539i	$-0.993966\pi$
$257$	−1.07873e6	−	1.07873e6i	−1.01877	−	1.01877i	−0.0189539	−	0.999820i	$-0.506034\pi$
$258$		0			0
$259$			551333.i			0.510699i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	151679.	−	151679.i	0.135219	−	0.135219i	−0.636258	−	0.771477i	$-0.719518\pi$
$263$	151679.	−	151679.i	0.135219	−	0.135219i	0.771477	+	0.636258i	$0.219518\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	692434.			0.583442			0.291721	−	0.956504i	$-0.405772\pi$
$269$	692434.			0.583442			0.291721	+	0.956504i	$0.405772\pi$
$270$		0			0
$271$	1.39086e6			1.15043			0.575215	−	0.818003i	$-0.304919\pi$
$271$	1.39086e6			1.15043			0.575215	+	0.818003i	$0.304919\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−1.05253e6	+	1.05253e6i	−0.824202	+	0.824202i	−0.986708	−	0.162506i	$-0.948042\pi$
$277$	−1.05253e6	+	1.05253e6i	−0.824202	+	0.824202i	0.162506	+	0.986708i	$0.448042\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			1.31573e6i			0.994037i	0.867740	+	0.497018i	$0.165572\pi$
$281$			1.31573e6i			0.994037i	−0.867740	+	0.497018i	$0.834428\pi$
$282$		0			0
$283$	−249933.	−	249933.i	−0.185506	−	0.185506i	0.608244	−	0.793750i	$-0.291874\pi$
$283$	−249933.	−	249933.i	−0.185506	−	0.185506i	−0.793750	+	0.608244i	$0.791874\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1.89992e6	−	1.89992e6i	−1.36154	−	1.36154i
$288$		0			0
$289$			1.00208e6i			0.705762i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−763241.	+	763241.i	−0.519389	+	0.519389i	−0.917386	−	0.397998i	$-0.869705\pi$
$293$	−763241.	+	763241.i	−0.519389	+	0.519389i	0.397998	+	0.917386i	$0.369705\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2.24822e6			−1.45432
$300$		0			0
$301$	501148.			0.318823
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	1.71035e6	−	1.71035e6i	1.03571	−	1.03571i	0.0363713	−	0.999338i	$-0.488420\pi$
$307$	1.71035e6	−	1.71035e6i	1.03571	−	1.03571i	0.999338	−	0.0363713i	$-0.0115799\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			2.21331e6i			1.29760i	0.760959	+	0.648800i	$0.224729\pi$
$311$			2.21331e6i			1.29760i	−0.760959	+	0.648800i	$0.775271\pi$
$312$		0			0
$313$	1.01618e6	+	1.01618e6i	0.586285	+	0.586285i	0.936623	−	0.350338i	$-0.113933\pi$
$313$	1.01618e6	+	1.01618e6i	0.586285	+	0.586285i	−0.350338	+	0.936623i	$0.613933\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−815478.	−	815478.i	−0.455789	−	0.455789i	0.441481	−	0.897271i	$-0.354453\pi$
$317$	−815478.	−	815478.i	−0.455789	−	0.455789i	−0.897271	+	0.441481i	$0.854453\pi$
$318$		0			0
$319$		−	1.23498e6i		−	0.679488i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−2.24354e6	+	2.24354e6i	−1.19654	+	1.19654i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−2.87443e6			−1.46407
$330$		0			0
$331$	3.31302e6			1.66209			0.831044	−	0.556207i	$-0.187744\pi$
$331$	3.31302e6			1.66209			0.831044	+	0.556207i	$0.187744\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−2.60925e6	+	2.60925e6i	−1.25153	+	1.25153i	−0.296498	+	0.955034i	$0.595819\pi$
$337$	−2.60925e6	+	2.60925e6i	−1.25153	+	1.25153i	−0.955034	+	0.296498i	$0.904181\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	232066.i		−	0.108075i
$342$		0			0
$343$	1.29616e6	+	1.29616e6i	0.594870	+	0.594870i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1.92366e6	+	1.92366e6i	0.857639	+	0.857639i	0.991060	−	0.133421i	$-0.0425961\pi$
$347$	1.92366e6	+	1.92366e6i	0.857639	+	0.857639i	−0.133421	+	0.991060i	$0.542596\pi$
$348$		0			0
$349$			1.70511e6i			0.749355i	0.927155	+	0.374678i	$0.122247\pi$
$349$			1.70511e6i			0.749355i	−0.927155	+	0.374678i	$0.877753\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−2.84606e6	+	2.84606e6i	−1.21565	+	1.21565i	−0.246504	+	0.969142i	$0.579282\pi$
$353$	−2.84606e6	+	2.84606e6i	−1.21565	+	1.21565i	−0.969142	+	0.246504i	$0.920718\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	761041.			0.311654			0.155827	−	0.987784i	$-0.450196\pi$
$359$	761041.			0.311654			0.155827	+	0.987784i	$0.450196\pi$
$360$		0			0
$361$	−1.68048e6			−0.678681
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−577231.	+	577231.i	−0.223710	+	0.223710i	−0.810059	−	0.586349i	$-0.800565\pi$
$367$	−577231.	+	577231.i	−0.223710	+	0.223710i	0.586349	+	0.810059i	$0.300565\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			2.82210e6i			1.06448i
$372$		0			0
$373$	2.51462e6	+	2.51462e6i	0.935836	+	0.935836i	0.998062	−	0.0622263i	$-0.0198200\pi$
$373$	2.51462e6	+	2.51462e6i	0.935836	+	0.935836i	−0.0622263	+	0.998062i	$0.519820\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−4.85544e6	−	4.85544e6i	−1.75944	−	1.75944i
$378$		0			0
$379$			3.61283e6i			1.29196i	0.763353	+	0.645981i	$0.223552\pi$
$379$			3.61283e6i			1.29196i	−0.763353	+	0.645981i	$0.776448\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	719993.	−	719993.i	0.250802	−	0.250802i	−0.570497	−	0.821299i	$-0.693250\pi$
$383$	719993.	−	719993.i	0.250802	−	0.250802i	0.821299	+	0.570497i	$0.193250\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	1.59473e6			0.534336			0.267168	−	0.963650i	$-0.413912\pi$
$389$	1.59473e6			0.534336			0.267168	+	0.963650i	$0.413912\pi$
$390$		0			0
$391$	−3.39108e6			−1.12175
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−1.56837e6	+	1.56837e6i	−0.499428	+	0.499428i	−0.911260	−	0.411832i	$-0.864889\pi$
$397$	−1.56837e6	+	1.56837e6i	−0.499428	+	0.499428i	0.411832	+	0.911260i	$0.364889\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	4.13880e6i		−	1.28533i	−0.766148	−	0.642664i	$-0.777829\pi$
$401$		−	4.13880e6i		−	1.28533i	0.766148	−	0.642664i	$-0.222171\pi$
$402$		0			0
$403$	−912394.	−	912394.i	−0.279846	−	0.279846i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−499813.	−	499813.i	−0.149562	−	0.149562i
$408$		0			0
$409$		−	2.26690e6i		−	0.670075i	−0.942205	−	0.335037i	$-0.891251\pi$
$409$		−	2.26690e6i		−	0.670075i	0.942205	−	0.335037i	$-0.108749\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	2.95901e6	−	2.95901e6i	0.853633	−	0.853633i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−37772.3			−0.0105109			−0.00525544	−	0.999986i	$-0.501673\pi$
$419$	−37772.3			−0.0105109			−0.00525544	+	0.999986i	$0.501673\pi$
$420$		0			0
$421$	5.02315e6			1.38124			0.690622	−	0.723216i	$-0.257337\pi$
$421$	5.02315e6			1.38124			0.690622	+	0.723216i	$0.257337\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−2.79433e6	+	2.79433e6i	−0.741667	+	0.741667i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			947167.i			0.245603i	0.992431	+	0.122801i	$0.0391878\pi$
$431$			947167.i			0.245603i	−0.992431	+	0.122801i	$0.960812\pi$
$432$		0			0
$433$	−1.23165e6	−	1.23165e6i	−0.315696	−	0.315696i	0.531416	−	0.847111i	$-0.321660\pi$
$433$	−1.23165e6	−	1.23165e6i	−0.315696	−	0.315696i	−0.847111	+	0.531416i	$0.821660\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−3.14130e6	−	3.14130e6i	−0.786876	−	0.786876i
$438$		0			0
$439$		−	5.12487e6i		−	1.26918i	−0.772851	−	0.634588i	$-0.781170\pi$
$439$		−	5.12487e6i		−	1.26918i	0.772851	−	0.634588i	$-0.218830\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−606990.	+	606990.i	−0.146951	+	0.146951i	−0.776754	−	0.629804i	$-0.783135\pi$
$443$	−606990.	+	606990.i	−0.146951	+	0.146951i	0.629804	+	0.776754i	$0.283135\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	501014.			0.117283			0.0586414	−	0.998279i	$-0.481323\pi$
$449$	501014.			0.117283			0.0586414	+	0.998279i	$0.481323\pi$
$450$		0			0
$451$	3.44475e6			0.797473
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	4.09178e6	−	4.09178e6i	0.916477	−	0.916477i	−0.0802946	−	0.996771i	$-0.525586\pi$
$457$	4.09178e6	−	4.09178e6i	0.916477	−	0.916477i	0.996771	+	0.0802946i	$0.0255861\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			4.17114e6i			0.914118i	0.889436	+	0.457059i	$0.151097\pi$
$461$			4.17114e6i			0.914118i	−0.889436	+	0.457059i	$0.848903\pi$
$462$		0			0
$463$	−717865.	−	717865.i	−0.155629	−	0.155629i	0.624998	−	0.780627i	$-0.285100\pi$
$463$	−717865.	−	717865.i	−0.155629	−	0.155629i	−0.780627	+	0.624998i	$0.785100\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	5.88634e6	+	5.88634e6i	1.24897	+	1.24897i	0.956174	+	0.292798i	$0.0945863\pi$
$467$	5.88634e6	+	5.88634e6i	1.24897	+	1.24897i	0.292798	+	0.956174i	$0.405414\pi$
$468$		0			0
$469$			678611.i			0.142459i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−454317.	+	454317.i	−0.0933698	+	0.0933698i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	1.24020e6			0.246975			0.123488	−	0.992346i	$-0.460592\pi$
$479$	1.24020e6			0.246975			0.123488	+	0.992346i	$0.460592\pi$
$480$		0			0
$481$	−3.93014e6			−0.774542
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	3.14272e6	−	3.14272e6i	0.600459	−	0.600459i	−0.339975	−	0.940434i	$-0.610419\pi$
$487$	3.14272e6	−	3.14272e6i	0.600459	−	0.600459i	0.940434	+	0.339975i	$0.110419\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			4.31872e6i			0.808446i	0.914661	+	0.404223i	$0.132458\pi$
$491$			4.31872e6i			0.808446i	−0.914661	+	0.404223i	$0.867542\pi$
$492$		0			0
$493$	−7.32367e6	−	7.32367e6i	−1.35710	−	1.35710i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	5.88688e6	+	5.88688e6i	1.06904	+	1.06904i
$498$		0			0
$499$			1.28237e6i			0.230549i	0.993334	+	0.115274i	$0.0367747\pi$
$499$			1.28237e6i			0.230549i	−0.993334	+	0.115274i	$0.963225\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−3.59798e6	+	3.59798e6i	−0.634072	+	0.634072i	−0.949087	−	0.315015i	$-0.897991\pi$
$503$	−3.59798e6	+	3.59798e6i	−0.634072	+	0.634072i	0.315015	+	0.949087i	$0.397991\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−967445.			−0.165513			−0.0827565	−	0.996570i	$-0.526372\pi$
$509$	−967445.			−0.165513			−0.0827565	+	0.996570i	$0.526372\pi$
$510$		0			0
$511$	2.40832e6			0.408002
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	2.60582e6	−	2.60582e6i	0.428764	−	0.428764i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			446507.i			0.0720665i	0.999351	+	0.0360333i	$0.0114722\pi$
$521$			446507.i			0.0720665i	−0.999351	+	0.0360333i	$0.988528\pi$
$522$		0			0
$523$	6.87458e6	+	6.87458e6i	1.09899	+	1.09899i	0.994529	+	0.104456i	$0.0333102\pi$
$523$	6.87458e6	+	6.87458e6i	1.09899	+	1.09899i	0.104456	+	0.994529i	$0.466690\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1.37620e6	−	1.37620e6i	−0.215852	−	0.215852i
$528$		0			0
$529$			1.68831e6i			0.262309i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1.35434e7	−	1.35434e7i	2.06495	−	2.06495i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	768723.			0.113972
$540$		0			0
$541$	−2.07917e6			−0.305419			−0.152709	−	0.988271i	$-0.548800\pi$
$541$	−2.07917e6			−0.305419			−0.152709	+	0.988271i	$0.548800\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	4.54080e6	−	4.54080e6i	0.648880	−	0.648880i	−0.303842	−	0.952722i	$-0.598270\pi$
$547$	4.54080e6	−	4.54080e6i	0.648880	−	0.648880i	0.952722	+	0.303842i	$0.0982697\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	1.35685e7i		−	1.90393i
$552$		0			0
$553$	705294.	+	705294.i	0.0980748	+	0.0980748i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	5.78370e6	+	5.78370e6i	0.789892	+	0.789892i	0.981476	−	0.191585i	$-0.0613626\pi$
$557$	5.78370e6	+	5.78370e6i	0.789892	+	0.789892i	−0.191585	+	0.981476i	$0.561363\pi$
$558$		0			0
$559$			3.57240e6i			0.483537i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	1.16284e6	−	1.16284e6i	0.154613	−	0.154613i	−0.625561	−	0.780175i	$-0.715130\pi$
$563$	1.16284e6	−	1.16284e6i	0.154613	−	0.154613i	0.780175	+	0.625561i	$0.215130\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−1.65453e6			−0.214237			−0.107118	−	0.994246i	$-0.534162\pi$
$569$	−1.65453e6			−0.214237			−0.107118	+	0.994246i	$0.534162\pi$
$570$		0			0
$571$	−1.41367e7			−1.81450			−0.907249	−	0.420593i	$-0.861822\pi$
$571$	−1.41367e7			−1.81450			−0.907249	+	0.420593i	$0.861822\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−7.29389e6	+	7.29389e6i	−0.912052	+	0.912052i	−0.996433	−	0.0843819i	$-0.973108\pi$
$577$	−7.29389e6	+	7.29389e6i	−0.912052	+	0.912052i	0.0843819	+	0.996433i	$0.473108\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	8.59459e6i		−	1.05629i
$582$		0			0
$583$	−2.55838e6	−	2.55838e6i	−0.311741	−	0.311741i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−4.53909e6	−	4.53909e6i	−0.543718	−	0.543718i	0.380899	−	0.924617i	$-0.375615\pi$
$587$	−4.53909e6	−	4.53909e6i	−0.543718	−	0.543718i	−0.924617	+	0.380899i	$0.875615\pi$
$588$		0			0
$589$		−	2.54967e6i		−	0.302828i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−218544.	+	218544.i	−0.0255213	+	0.0255213i	−0.719752	−	0.694231i	$-0.755745\pi$
$593$	−218544.	+	218544.i	−0.0255213	+	0.0255213i	0.694231	+	0.719752i	$0.255745\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	8.29099e6			0.944147			0.472073	−	0.881559i	$-0.343506\pi$
$599$	8.29099e6			0.944147			0.472073	+	0.881559i	$0.343506\pi$
$600$		0			0
$601$	−1.05221e7			−1.18828			−0.594138	−	0.804363i	$-0.702507\pi$
$601$	−1.05221e7			−1.18828			−0.594138	+	0.804363i	$0.702507\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	2.68098e6	−	2.68098e6i	0.295340	−	0.295340i	−0.543846	−	0.839185i	$-0.683032\pi$
$607$	2.68098e6	−	2.68098e6i	0.295340	−	0.295340i	0.839185	+	0.543846i	$0.183032\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	2.04902e7i		−	2.22046i
$612$		0			0
$613$	−9.28465e6	−	9.28465e6i	−0.997963	−	0.997963i	0.00203502	−	0.999998i	$-0.499352\pi$
$613$	−9.28465e6	−	9.28465e6i	−0.997963	−	0.997963i	−0.999998	+	0.00203502i	$0.999352\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	6.08140e6	+	6.08140e6i	0.643118	+	0.643118i	0.951321	−	0.308203i	$-0.0997275\pi$
$617$	6.08140e6	+	6.08140e6i	0.643118	+	0.643118i	−0.308203	+	0.951321i	$0.599728\pi$
$618$		0			0
$619$		−	1.46235e7i		−	1.53399i	−0.641651	−	0.766997i	$-0.721750\pi$
$619$		−	1.46235e7i		−	1.53399i	0.641651	−	0.766997i	$-0.278250\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−1.07341e7	+	1.07341e7i	−1.10801	+	1.10801i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−5.92799e6			−0.597422
$630$		0			0
$631$	−1.74705e6			−0.174676			−0.0873378	−	0.996179i	$-0.527836\pi$
$631$	−1.74705e6			−0.174676			−0.0873378	+	0.996179i	$0.527836\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	3.02232e6	−	3.02232e6i	0.295115	−	0.295115i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	1.62435e7i		−	1.56148i	−0.624857	−	0.780739i	$-0.714843\pi$
$641$		−	1.62435e7i		−	1.56148i	0.624857	−	0.780739i	$-0.285157\pi$
$642$		0			0
$643$	1.13435e6	+	1.13435e6i	0.108198	+	0.108198i	0.759133	−	0.650935i	$-0.225623\pi$
$643$	1.13435e6	+	1.13435e6i	0.108198	+	0.108198i	−0.650935	+	0.759133i	$0.725623\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1.48746e7	+	1.48746e7i	1.39696	+	1.39696i	0.808588	+	0.588376i	$0.200232\pi$
$647$	1.48746e7	+	1.48746e7i	1.39696	+	1.39696i	0.588376	+	0.808588i	$0.299768\pi$
$648$		0			0
$649$			5.36500e6i			0.499986i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	5.81573e6	−	5.81573e6i	0.533730	−	0.533730i	−0.387950	−	0.921680i	$-0.626817\pi$
$653$	5.81573e6	−	5.81573e6i	0.533730	−	0.533730i	0.921680	+	0.387950i	$0.126817\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	1.71899e7			1.54191			0.770957	−	0.636887i	$-0.219778\pi$
$659$	1.71899e7			1.54191			0.770957	+	0.636887i	$0.219778\pi$
$660$		0			0
$661$	6.74176e6			0.600163			0.300082	−	0.953914i	$-0.402986\pi$
$661$	6.74176e6			0.600163			0.300082	+	0.953914i	$0.402986\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	1.02543e7	−	1.02543e7i	0.892462	−	0.892462i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	5.06643e6i		−	0.434406i
$672$		0			0
$673$	7.70835e6	+	7.70835e6i	0.656030	+	0.656030i	0.954438	−	0.298408i	$-0.0964557\pi$
$673$	7.70835e6	+	7.70835e6i	0.656030	+	0.656030i	−0.298408	+	0.954438i	$0.596456\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	4.23937e6	+	4.23937e6i	0.355492	+	0.355492i	0.862148	−	0.506656i	$-0.169119\pi$
$677$	4.23937e6	+	4.23937e6i	0.355492	+	0.355492i	−0.506656	+	0.862148i	$0.669119\pi$
$678$		0			0
$679$		−	3.43449e6i		−	0.285882i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1.17372e7	−	1.17372e7i	0.962748	−	0.962748i	−0.0365828	−	0.999331i	$-0.511647\pi$
$683$	1.17372e7	−	1.17372e7i	0.962748	−	0.962748i	0.999331	+	0.0365828i	$0.0116473\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−2.01171e7			−1.61442
$690$		0			0
$691$	−1.74048e7			−1.38667			−0.693337	−	0.720614i	$-0.743860\pi$
$691$	−1.74048e7			−1.38667			−0.693337	+	0.720614i	$0.743860\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	2.04281e7	−	2.04281e7i	1.59274	−	1.59274i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			1.53152e7i			1.17714i	0.808448	+	0.588568i	$0.200308\pi$
$701$			1.53152e7i			1.17714i	−0.808448	+	0.588568i	$0.799692\pi$
$702$		0			0
$703$	−5.49135e6	−	5.49135e6i	−0.419074	−	0.419074i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1.13653e7	+	1.13653e7i	0.855129	+	0.855129i
$708$		0			0
$709$			2.11374e7i			1.57919i	0.613626	+	0.789597i	$0.289710\pi$
$709$			2.11374e7i			1.57919i	−0.613626	+	0.789597i	$0.710290\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1.92689e6	−	1.92689e6i	0.141949	−	0.141949i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	2.51829e6			0.181670			0.0908350	−	0.995866i	$-0.471046\pi$
$719$	2.51829e6			0.181670			0.0908350	+	0.995866i	$0.471046\pi$
$720$		0			0
$721$	−1.69064e7			−1.21119
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	2.71102e6	−	2.71102e6i	0.190238	−	0.190238i	−0.605561	−	0.795799i	$-0.707051\pi$
$727$	2.71102e6	−	2.71102e6i	0.190238	−	0.190238i	0.795799	+	0.605561i	$0.207051\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			5.38840e6i			0.372963i
$732$		0			0
$733$	−1.80004e7	−	1.80004e7i	−1.23744	−	1.23744i	−0.961045	−	0.276393i	$-0.910861\pi$
$733$	−1.80004e7	−	1.80004e7i	−1.23744	−	1.23744i	−0.276393	−	0.961045i	$-0.589139\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−615197.	−	615197.i	−0.0417201	−	0.0417201i
$738$		0			0
$739$		−	1.73894e7i		−	1.17131i	−0.810559	−	0.585657i	$-0.800837\pi$
$739$		−	1.73894e7i		−	1.17131i	0.810559	−	0.585657i	$-0.199163\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1.83395e7	−	1.83395e7i	1.21875	−	1.21875i	0.250686	−	0.968068i	$-0.419344\pi$
$743$	1.83395e7	−	1.83395e7i	1.21875	−	1.21875i	0.968068	−	0.250686i	$-0.0806563\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−6.70751e6			−0.436875
$750$		0			0
$751$	1.65358e7			1.06986			0.534929	−	0.844897i	$-0.320338\pi$
$751$	1.65358e7			1.06986			0.534929	+	0.844897i	$0.320338\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−6.53883e6	+	6.53883e6i	−0.414725	+	0.414725i	−0.883381	−	0.468656i	$-0.844739\pi$
$757$	−6.53883e6	+	6.53883e6i	−0.414725	+	0.414725i	0.468656	+	0.883381i	$0.344739\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	2.03626e7i		−	1.27459i	−0.770618	−	0.637297i	$-0.780052\pi$
$761$		−	2.03626e7i		−	1.27459i	0.770618	−	0.637297i	$-0.219948\pi$
$762$		0			0
$763$	3.64822e6	+	3.64822e6i	0.226866	+	0.226866i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	2.10931e7	+	2.10931e7i	1.29465	+	1.29465i
$768$		0			0
$769$			1.74233e7i			1.06246i	0.847226	+	0.531232i	$0.178271\pi$
$769$			1.74233e7i			1.06246i	−0.847226	+	0.531232i	$0.821729\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	9.90571e6	−	9.90571e6i	0.596262	−	0.596262i	−0.343054	−	0.939316i	$-0.611461\pi$
$773$	9.90571e6	−	9.90571e6i	0.596262	−	0.596262i	0.939316	+	0.343054i	$0.111461\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	3.78468e7			2.23453
$780$		0			0
$781$	−1.06735e7			−0.626154
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	127508.	−	127508.i	0.00733838	−	0.00733838i	−0.703428	−	0.710766i	$-0.748348\pi$
$787$	127508.	−	127508.i	0.00733838	−	0.00733838i	0.710766	+	0.703428i	$0.248348\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			7.87264e6i			0.447383i
$792$		0			0
$793$	−1.99192e7	−	1.99192e7i	−1.12484	−	1.12484i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1.67329e6	+	1.67329e6i	0.0933092	+	0.0933092i	0.752221	−	0.658911i	$-0.228983\pi$
$797$	1.67329e6	+	1.67329e6i	0.0933092	+	0.0933092i	−0.658911	+	0.752221i	$0.728983\pi$
$798$		0			0
$799$		−	3.09062e7i		−	1.71269i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−2.18327e6	+	2.18327e6i	−0.119486	+	0.119486i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	3.23500e6			0.173781			0.0868906	−	0.996218i	$-0.472307\pi$
$809$	3.23500e6			0.173781			0.0868906	+	0.996218i	$0.472307\pi$
$810$		0			0
$811$	−5.20208e6			−0.277731			−0.138866	−	0.990311i	$-0.544346\pi$
$811$	−5.20208e6			−0.277731			−0.138866	+	0.990311i	$0.544346\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−4.99150e6	+	4.99150e6i	−0.261623	+	0.261623i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	2.41516e7i		−	1.25051i	−0.780420	−	0.625256i	$-0.784994\pi$
$821$		−	2.41516e7i		−	1.25051i	0.780420	−	0.625256i	$-0.215006\pi$
$822$		0			0
$823$	3.58849e6	+	3.58849e6i	0.184677	+	0.184677i	0.793390	−	0.608713i	$-0.208314\pi$
$823$	3.58849e6	+	3.58849e6i	0.184677	+	0.184677i	−0.608713	+	0.793390i	$0.708314\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−9.65182e6	−	9.65182e6i	−0.490733	−	0.490733i	0.417804	−	0.908537i	$-0.362800\pi$
$827$	−9.65182e6	−	9.65182e6i	−0.490733	−	0.490733i	−0.908537	+	0.417804i	$0.862800\pi$
$828$		0			0
$829$		−	3.12345e7i		−	1.57851i	−0.614063	−	0.789257i	$-0.710466\pi$
$829$		−	3.12345e7i		−	1.57851i	0.614063	−	0.789257i	$-0.289534\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	4.55869e6	−	4.55869e6i	0.227629	−	0.227629i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	3.16507e6			0.155231			0.0776155	−	0.996983i	$-0.475269\pi$
$839$	3.16507e6			0.155231			0.0776155	+	0.996983i	$0.475269\pi$
$840$		0			0
$841$	2.37808e7			1.15941
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	1.29588e7	−	1.29588e7i	0.620662	−	0.620662i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	8.30009e6i		−	0.392879i
$852$		0			0
$853$	2.03904e7	+	2.03904e7i	0.959520	+	0.959520i	0.999212	−	0.0396916i	$-0.0126375\pi$
$853$	2.03904e7	+	2.03904e7i	0.959520	+	0.959520i	−0.0396916	+	0.999212i	$0.512638\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1.46770e7	−	1.46770e7i	−0.682632	−	0.682632i	0.277961	−	0.960592i	$-0.410341\pi$
$857$	−1.46770e7	−	1.46770e7i	−0.682632	−	0.682632i	−0.960592	+	0.277961i	$0.910341\pi$
$858$		0			0
$859$		−	3.88161e6i		−	0.179486i	−0.995965	−	0.0897428i	$-0.971396\pi$
$859$		−	3.88161e6i		−	0.179486i	0.995965	−	0.0897428i	$-0.0286045\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−2.89206e7	+	2.89206e7i	−1.32184	+	1.32184i	−0.409558	+	0.912284i	$0.634317\pi$
$863$	−2.89206e7	+	2.89206e7i	−1.32184	+	1.32184i	−0.912284	+	0.409558i	$0.865683\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1.27877e6			−0.0574439
$870$		0			0
$871$	−4.83743e6			−0.216057
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1.02328e7	−	1.02328e7i	0.449259	−	0.449259i	−0.445849	−	0.895108i	$-0.647098\pi$
$877$	1.02328e7	−	1.02328e7i	0.449259	−	0.449259i	0.895108	+	0.445849i	$0.147098\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	3.92745e7i		−	1.70479i	−0.522898	−	0.852396i	$-0.675149\pi$
$881$		−	3.92745e7i		−	1.70479i	0.522898	−	0.852396i	$-0.324851\pi$
$882$		0			0
$883$	−1.09098e7	−	1.09098e7i	−0.470886	−	0.470886i	0.431315	−	0.902201i	$-0.358050\pi$
$883$	−1.09098e7	−	1.09098e7i	−0.470886	−	0.470886i	−0.902201	+	0.431315i	$0.858050\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1.43569e7	+	1.43569e7i	0.612703	+	0.612703i	0.943650	−	0.330946i	$-0.107368\pi$
$887$	1.43569e7	+	1.43569e7i	0.612703	+	0.612703i	−0.330946	+	0.943650i	$0.607368\pi$
$888$		0			0
$889$			3.60620e7i			1.53037i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	2.86297e7	−	2.86297e7i	1.20140	−	1.20140i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	8.32297e6			0.343462
$900$		0			0
$901$	−3.03435e7			−1.24524
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−1.78613e7	+	1.78613e7i	−0.720934	+	0.720934i	−0.968795	−	0.247862i	$-0.920272\pi$
$907$	−1.78613e7	+	1.78613e7i	−0.720934	+	0.720934i	0.247862	+	0.968795i	$0.420272\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			3.72428e7i			1.48678i	0.668859	+	0.743389i	$0.266783\pi$
$911$			3.72428e7i			1.48678i	−0.668859	+	0.743389i	$0.733217\pi$
$912$		0			0
$913$	7.79145e6	+	7.79145e6i	0.309344	+	0.309344i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	107592.	+	107592.i	0.00422530	+	0.00422530i
$918$		0			0
$919$			4.69436e7i			1.83353i	0.399430	+	0.916764i	$0.369208\pi$
$919$			4.69436e7i			1.83353i	−0.399430	+	0.916764i	$0.630792\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−4.19642e7	+	4.19642e7i	−1.62134	+	1.62134i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−4.96687e7			−1.88818			−0.944091	−	0.329685i	$-0.893058\pi$
$929$	−4.96687e7			−1.88818			−0.944091	+	0.329685i	$0.893058\pi$
$930$		0			0
$931$	8.44582e6			0.319350
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−3.19831e7	+	3.19831e7i	−1.19007	+	1.19007i	−0.213017	+	0.977048i	$0.568329\pi$
$937$	−3.19831e7	+	3.19831e7i	−1.19007	+	1.19007i	−0.977048	+	0.213017i	$0.931671\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			3.83446e7i			1.41166i	0.708381	+	0.705831i	$0.249426\pi$
$941$			3.83446e7i			1.41166i	−0.708381	+	0.705831i	$0.750574\pi$
$942$		0			0
$943$	2.86025e7	+	2.86025e7i	1.04743	+	1.04743i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1.12554e7	+	1.12554e7i	0.407836	+	0.407836i	0.880983	−	0.473147i	$-0.156882\pi$
$947$	1.12554e7	+	1.12554e7i	0.407836	+	0.407836i	−0.473147	+	0.880983i	$0.656882\pi$
$948$		0			0
$949$			1.71675e7i			0.618789i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	2.84874e7	−	2.84874e7i	1.01606	−	1.01606i	0.0161930	−	0.999869i	$-0.494845\pi$
$953$	2.84874e7	−	2.84874e7i	1.01606	−	1.01606i	0.999869	−	0.0161930i	$-0.00515461\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−4.26828e7			−1.49867
$960$		0			0
$961$	−2.70652e7			−0.945371
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	3.54405e7	−	3.54405e7i	1.21880	−	1.21880i	0.250753	−	0.968051i	$-0.419322\pi$
$967$	3.54405e7	−	3.54405e7i	1.21880	−	1.21880i	0.968051	−	0.250753i	$-0.0806782\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	1.70005e7i		−	0.578647i	−0.957231	−	0.289323i	$-0.906570\pi$
$971$		−	1.70005e7i		−	0.578647i	0.957231	−	0.289323i	$-0.0934303\pi$
$972$		0			0
$973$	−2.67282e7	−	2.67282e7i	−0.905082	−	0.905082i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1.88321e7	−	1.88321e7i	−0.631193	−	0.631193i	0.317174	−	0.948367i	$-0.397266\pi$
$977$	−1.88321e7	−	1.88321e7i	−0.631193	−	0.631193i	−0.948367	+	0.317174i	$0.897266\pi$
$978$		0			0
$979$		−	1.94620e7i		−	0.648978i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−2.58547e7	+	2.58547e7i	−0.853405	+	0.853405i	−0.990551	−	0.137146i	$-0.956207\pi$
$983$	−2.58547e7	+	2.58547e7i	−0.853405	+	0.853405i	0.137146	+	0.990551i	$0.456207\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−7.54458e6			−0.245270
$990$		0			0
$991$	1.08275e7			0.350223			0.175112	−	0.984549i	$-0.443971\pi$
$991$	1.08275e7			0.350223			0.175112	+	0.984549i	$0.443971\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	1.34443e6	−	1.34443e6i	0.0428352	−	0.0428352i	−0.685365	−	0.728200i	$-0.740357\pi$
$997$	1.34443e6	−	1.34443e6i	0.0428352	−	0.0428352i	0.728200	+	0.685365i	$0.240357\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.6.j.c.593.10	yes	24
3.2	odd	2	inner	900.6.j.c.593.9	yes	24
5.2	odd	4	inner	900.6.j.c.557.9	yes	24
5.3	odd	4	inner	900.6.j.c.557.3	✓	24
5.4	even	2	inner	900.6.j.c.593.4	yes	24
15.2	even	4	inner	900.6.j.c.557.10	yes	24
15.8	even	4	inner	900.6.j.c.557.4	yes	24
15.14	odd	2	inner	900.6.j.c.593.3	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
900.6.j.c.557.3	✓	24	5.3	odd	4	inner
900.6.j.c.557.4	yes	24	15.8	even	4	inner
900.6.j.c.557.9	yes	24	5.2	odd	4	inner
900.6.j.c.557.10	yes	24	15.2	even	4	inner
900.6.j.c.593.3	yes	24	15.14	odd	2	inner
900.6.j.c.593.4	yes	24	5.4	even	2	inner
900.6.j.c.593.9	yes	24	3.2	odd	2	inner
900.6.j.c.593.10	yes	24	1.1	even	1	trivial

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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	900.j (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(102.346 - 102.346i) q^{7} +O(q^{10})\)	\(q+(102.346 - 102.346i) q^{7} +185.565i q^{11} +(729.569 + 729.569i) q^{13} +(1100.44 + 1100.44i) q^{17} +2038.77i q^{19} +(-1540.78 + 1540.78i) q^{23} -6655.22 q^{29} -1250.59 q^{31} +(-2693.46 + 2693.46i) q^{37} -18563.6i q^{41} +(2448.29 + 2448.29i) q^{43} +(-14042.6 - 14042.6i) q^{47} -4142.61i q^{49} +(-13787.0 + 13787.0i) q^{53} +28911.7 q^{59} -27302.7 q^{61} +(-3315.26 + 3315.26i) q^{67} +57519.2i q^{71} +(11765.5 + 11765.5i) q^{73} +(18991.9 + 18991.9i) q^{77} +6891.24i q^{79} +(41987.7 - 41987.7i) q^{83} -104880. q^{89} +149338. q^{91} +(16778.7 - 16778.7i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q + 13848 q^{31} - 28200 q^{61} + 908328 q^{91}+O(q^{100}) \) 24 * q + 13848 * q^31 - 28200 * q^61 + 908328 * q^91

Introduction

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Groups

Database

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