LMFDB - Embedded newform 6975.2.a.ck.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6975,2,Mod(1,6975)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6975.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6975, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6975 = 3^{2} \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6975.a (trivial)

Newform invariants

comment:select newform

sage:traces = [16,0,0,12,0,0,0,0,0,0,0,0,0,0,0,44,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$55.6956554098$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 20x^{14} + 152x^{12} - 571x^{10} + 1130x^{8} - 1138x^{6} + 492x^{4} - 43x^{2} + 1 $ x^16 - 20x^14 + 152x^12 - 571x^10 + 1130x^8 - 1138x^6 + 492x^4 - 43*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 5 $
Twist minimal:	no (minimal twist has level 1395)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.80230$ of defining polynomial
Character	$\chi$	$=$	6975.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.72869 q^{2} +5.44577 q^{4} -0.363635 q^{7} -9.40246 q^{8} -5.69456 q^{11} -2.16431 q^{13} +0.992249 q^{14} +14.7649 q^{16} -1.91527 q^{17} +1.10511 q^{19} +15.5387 q^{22} -7.73467 q^{23} +5.90573 q^{26} -1.98027 q^{28} +6.52907 q^{29} -1.00000 q^{31} -21.4840 q^{32} +5.22619 q^{34} +5.51041 q^{37} -3.01552 q^{38} +8.62833 q^{41} -9.99008 q^{43} -31.0113 q^{44} +21.1056 q^{46} +4.18224 q^{47} -6.86777 q^{49} -11.7863 q^{52} +5.03921 q^{53} +3.41906 q^{56} -17.8158 q^{58} -7.34671 q^{59} +1.24112 q^{61} +2.72869 q^{62} +29.0934 q^{64} +1.06897 q^{67} -10.4301 q^{68} -11.7215 q^{71} -11.7863 q^{73} -15.0362 q^{74} +6.01820 q^{76} +2.07074 q^{77} +0.773812 q^{79} -23.5441 q^{82} -8.37479 q^{83} +27.2599 q^{86} +53.5429 q^{88} +5.48339 q^{89} +0.787018 q^{91} -42.1213 q^{92} -11.4121 q^{94} -7.34931 q^{97} +18.7400 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 12 q^{4} + 44 q^{16} - 16 q^{31} + 24 q^{34} + 88 q^{46} + 16 q^{49} + 64 q^{61} + 176 q^{64} - 12 q^{76} + 72 q^{79} - 16 q^{91} - 8 q^{94}+O(q^{100}) $ 16 * q + 12 * q^4 + 44 * q^16 - 16 * q^31 + 24 * q^34 + 88 * q^46 + 16 * q^49 + 64 * q^61 + 176 * q^64 - 12 * q^76 + 72 * q^79 - 16 * q^91 - 8 * q^94

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 $

$2$	−2.72869	−1.92948	−0.964739	−	0.263208i	$-0.915219\pi$
$2$	−2.72869	−1.92948	−0.964739	+	0.263208i	$0.915219\pi$
$3$	0	0
$3$	0	0
$4$	5.44577	2.72289
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.363635	−0.137441	−0.0687206	−	0.997636i	$-0.521892\pi$
$7$	−0.363635	−0.137441	−0.0687206	+	0.997636i	$0.521892\pi$
$8$	−9.40246	−3.32427
$9$	0	0
$10$	0	0
$11$	−5.69456	−1.71697	−0.858487	−	0.512835i	$-0.828595\pi$
$11$	−5.69456	−1.71697	−0.858487	+	0.512835i	$0.828595\pi$
$12$	0	0
$13$	−2.16431	−0.600271	−0.300135	−	0.953897i	$-0.597032\pi$
$13$	−2.16431	−0.600271	−0.300135	+	0.953897i	$0.597032\pi$
$14$	0.992249	0.265190
$15$	0	0
$16$	14.7649	3.69122
$17$	−1.91527	−0.464521	−0.232261	−	0.972654i	$-0.574612\pi$
$17$	−1.91527	−0.464521	−0.232261	+	0.972654i	$0.574612\pi$
$18$	0	0
$19$	1.10511	0.253530	0.126765	−	0.991933i	$-0.459541\pi$
$19$	1.10511	0.253530	0.126765	+	0.991933i	$0.459541\pi$
$20$	0	0
$21$	0	0
$22$	15.5387	3.31286
$23$	−7.73467	−1.61279	−0.806395	−	0.591377i	$-0.798584\pi$
$23$	−7.73467	−1.61279	−0.806395	+	0.591377i	$0.798584\pi$
$24$	0	0
$25$	0	0
$26$	5.90573	1.15821
$27$	0	0
$28$	−1.98027	−0.374237
$29$	6.52907	1.21242	0.606209	−	0.795305i	$-0.292689\pi$
$29$	6.52907	1.21242	0.606209	+	0.795305i	$0.292689\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−21.4840	−3.79786
$33$	0	0
$34$	5.22619	0.896284
$35$	0	0
$36$	0	0
$37$	5.51041	0.905906	0.452953	−	0.891534i	$-0.350370\pi$
$37$	5.51041	0.905906	0.452953	+	0.891534i	$0.350370\pi$
$38$	−3.01552	−0.489181
$39$	0	0
$40$	0	0
$41$	8.62833	1.34752	0.673760	−	0.738950i	$-0.264678\pi$
$41$	8.62833	1.34752	0.673760	+	0.738950i	$0.264678\pi$
$42$	0	0
$43$	−9.99008	−1.52347	−0.761736	−	0.647887i	$-0.775653\pi$
$43$	−9.99008	−1.52347	−0.761736	+	0.647887i	$0.775653\pi$
$44$	−31.0113	−4.67513
$45$	0	0
$46$	21.1056	3.11184
$47$	4.18224	0.610042	0.305021	−	0.952346i	$-0.401336\pi$
$47$	4.18224	0.610042	0.305021	+	0.952346i	$0.401336\pi$
$48$	0	0
$49$	−6.86777	−0.981110
$50$	0	0
$51$	0	0
$52$	−11.7863	−1.63447
$53$	5.03921	0.692189	0.346095	−	0.938200i	$-0.387508\pi$
$53$	5.03921	0.692189	0.346095	+	0.938200i	$0.387508\pi$
$54$	0	0
$55$	0	0
$56$	3.41906	0.456892
$57$	0	0
$58$	−17.8158	−2.33934
$59$	−7.34671	−0.956460	−0.478230	−	0.878235i	$-0.658722\pi$
$59$	−7.34671	−0.956460	−0.478230	+	0.878235i	$0.658722\pi$
$60$	0	0
$61$	1.24112	0.158910	0.0794548	−	0.996838i	$-0.474682\pi$
$61$	1.24112	0.158910	0.0794548	+	0.996838i	$0.474682\pi$
$62$	2.72869	0.346545
$63$	0	0
$64$	29.0934	3.63667
$65$	0	0
$66$	0	0
$67$	1.06897	0.130596	0.0652978	−	0.997866i	$-0.479200\pi$
$67$	1.06897	0.130596	0.0652978	+	0.997866i	$0.479200\pi$
$68$	−10.4301	−1.26484
$69$	0	0
$70$	0	0
$71$	−11.7215	−1.39108	−0.695541	−	0.718486i	$-0.744835\pi$
$71$	−11.7215	−1.39108	−0.695541	+	0.718486i	$0.744835\pi$
$72$	0	0
$73$	−11.7863	−1.37948	−0.689742	−	0.724055i	$-0.742276\pi$
$73$	−11.7863	−1.37948	−0.689742	+	0.724055i	$0.742276\pi$
$74$	−15.0362	−1.74793
$75$	0	0
$76$	6.01820	0.690335
$77$	2.07074	0.235983
$78$	0	0
$79$	0.773812	0.0870607	0.0435304	−	0.999052i	$-0.486139\pi$
$79$	0.773812	0.0870607	0.0435304	+	0.999052i	$0.486139\pi$
$80$	0	0
$81$	0	0
$82$	−23.5441	−2.60001
$83$	−8.37479	−0.919253	−0.459626	−	0.888112i	$-0.652017\pi$
$83$	−8.37479	−0.919253	−0.459626	+	0.888112i	$0.652017\pi$
$84$	0	0
$85$	0	0
$86$	27.2599	2.93951
$87$	0	0
$88$	53.5429	5.70769
$89$	5.48339	0.581238	0.290619	−	0.956839i	$-0.406139\pi$
$89$	5.48339	0.581238	0.290619	+	0.956839i	$0.406139\pi$
$90$	0	0
$91$	0.787018	0.0825019
$92$	−42.1213	−4.39144
$93$	0	0
$94$	−11.4121	−1.17706
$95$	0	0
$96$	0	0
$97$	−7.34931	−0.746209	−0.373105	−	0.927789i	$-0.621707\pi$
$97$	−7.34931	−0.746209	−0.373105	+	0.927789i	$0.621707\pi$
$98$	18.7400	1.89303
$99$	0	0
$100$	0	0
$101$	−15.3686	−1.52923	−0.764615	−	0.644487i	$-0.777071\pi$
$101$	−15.3686	−1.52923	−0.764615	+	0.644487i	$0.777071\pi$
$102$	0	0
$103$	−9.98565	−0.983915	−0.491958	−	0.870619i	$-0.663718\pi$
$103$	−9.98565	−0.983915	−0.491958	+	0.870619i	$0.663718\pi$
$104$	20.3498	1.99546
$105$	0	0
$106$	−13.7505	−1.33556
$107$	−17.7428	−1.71526	−0.857632	−	0.514265i	$-0.828065\pi$
$107$	−17.7428	−1.71526	−0.857632	+	0.514265i	$0.828065\pi$
$108$	0	0
$109$	−0.0115945	−0.00111055	−0.000555277	−	1.00000i	$-0.500177\pi$
$109$	−0.0115945	−0.00111055	−0.000555277		1.00000i	$0.500177\pi$
$110$	0	0
$111$	0	0
$112$	−5.36903	−0.507326
$113$	9.89582	0.930921	0.465460	−	0.885069i	$-0.345889\pi$
$113$	9.89582	0.930921	0.465460	+	0.885069i	$0.345889\pi$
$114$	0	0
$115$	0	0
$116$	35.5559	3.30128
$117$	0	0
$118$	20.0469	1.84547
$119$	0.696460	0.0638443
$120$	0	0
$121$	21.4280	1.94800
$122$	−3.38665	−0.306613
$123$	0	0
$124$	−5.44577	−0.489045
$125$	0	0
$126$	0	0
$127$	5.91670	0.525022	0.262511	−	0.964929i	$-0.415449\pi$
$127$	5.91670	0.525022	0.262511	+	0.964929i	$0.415449\pi$
$128$	−36.4190	−3.21901
$129$	0	0
$130$	0	0
$131$	−18.7876	−1.64148	−0.820742	−	0.571299i	$-0.806440\pi$
$131$	−18.7876	−1.64148	−0.820742	+	0.571299i	$0.806440\pi$
$132$	0	0
$133$	−0.401858	−0.0348455
$134$	−2.91690	−0.251981
$135$	0	0
$136$	18.0083	1.54419
$137$	−13.8322	−1.18176	−0.590881	−	0.806758i	$-0.701220\pi$
$137$	−13.8322	−1.18176	−0.590881	+	0.806758i	$0.701220\pi$
$138$	0	0
$139$	15.4402	1.30962	0.654810	−	0.755793i	$-0.272749\pi$
$139$	15.4402	1.30962	0.654810	+	0.755793i	$0.272749\pi$
$140$	0	0
$141$	0	0
$142$	31.9843	2.68406
$143$	12.3248	1.03065
$144$	0	0
$145$	0	0
$146$	32.1613	2.66168
$147$	0	0
$148$	30.0085	2.46668
$149$	13.7546	1.12682	0.563411	−	0.826177i	$-0.309489\pi$
$149$	13.7546	1.12682	0.563411	+	0.826177i	$0.309489\pi$
$150$	0	0
$151$	7.97186	0.648741	0.324370	−	0.945930i	$-0.394848\pi$
$151$	7.97186	0.648741	0.324370	+	0.945930i	$0.394848\pi$
$152$	−10.3908	−0.842804
$153$	0	0
$154$	−5.65042	−0.455324
$155$	0	0
$156$	0	0
$157$	11.9800	0.956110	0.478055	−	0.878330i	$-0.341342\pi$
$157$	11.9800	0.956110	0.478055	+	0.878330i	$0.341342\pi$
$158$	−2.11150	−0.167982
$159$	0	0
$160$	0	0
$161$	2.81260	0.221664
$162$	0	0
$163$	−12.6691	−0.992318	−0.496159	−	0.868232i	$-0.665257\pi$
$163$	−12.6691	−0.992318	−0.496159	+	0.868232i	$0.665257\pi$
$164$	46.9879	3.66914
$165$	0	0
$166$	22.8522	1.77368
$167$	20.3020	1.57102	0.785509	−	0.618851i	$-0.212401\pi$
$167$	20.3020	1.57102	0.785509	+	0.618851i	$0.212401\pi$
$168$	0	0
$169$	−8.31578	−0.639675
$170$	0	0
$171$	0	0
$172$	−54.4037	−4.14824
$173$	20.2355	1.53848	0.769240	−	0.638960i	$-0.220635\pi$
$173$	20.2355	1.53848	0.769240	+	0.638960i	$0.220635\pi$
$174$	0	0
$175$	0	0
$176$	−84.0796	−6.33774
$177$	0	0
$178$	−14.9625	−1.12149
$179$	−9.62221	−0.719198	−0.359599	−	0.933107i	$-0.617086\pi$
$179$	−9.62221	−0.719198	−0.359599	+	0.933107i	$0.617086\pi$
$180$	0	0
$181$	5.66158	0.420822	0.210411	−	0.977613i	$-0.432520\pi$
$181$	5.66158	0.420822	0.210411	+	0.977613i	$0.432520\pi$
$182$	−2.14753	−0.159186
$183$	0	0
$184$	72.7249	5.36135
$185$	0	0
$186$	0	0
$187$	10.9066	0.797571
$188$	22.7755	1.66108
$189$	0	0
$190$	0	0
$191$	−10.7827	−0.780205	−0.390103	−	0.920771i	$-0.627561\pi$
$191$	−10.7827	−0.780205	−0.390103	+	0.920771i	$0.627561\pi$
$192$	0	0
$193$	10.0239	0.721534	0.360767	−	0.932656i	$-0.382515\pi$
$193$	10.0239	0.721534	0.360767	+	0.932656i	$0.382515\pi$
$194$	20.0540	1.43979
$195$	0	0
$196$	−37.4003	−2.67145
$197$	14.5562	1.03709	0.518543	−	0.855051i	$-0.326475\pi$
$197$	14.5562	1.03709	0.518543	+	0.855051i	$0.326475\pi$
$198$	0	0
$199$	−3.35886	−0.238103	−0.119051	−	0.992888i	$-0.537985\pi$
$199$	−3.35886	−0.238103	−0.119051	+	0.992888i	$0.537985\pi$
$200$	0	0
$201$	0	0
$202$	41.9362	2.95062
$203$	−2.37420	−0.166636
$204$	0	0
$205$	0	0
$206$	27.2478	1.89844
$207$	0	0
$208$	−31.9557	−2.21573
$209$	−6.29314	−0.435305
$210$	0	0
$211$	−10.9845	−0.756203	−0.378101	−	0.925764i	$-0.623423\pi$
$211$	−10.9845	−0.756203	−0.378101	+	0.925764i	$0.623423\pi$
$212$	27.4424	1.88475
$213$	0	0
$214$	48.4147	3.30956
$215$	0	0
$216$	0	0
$217$	0.363635	0.0246852
$218$	0.0316379	0.00214279
$219$	0	0
$220$	0	0
$221$	4.14523	0.278838
$222$	0	0
$223$	20.1694	1.35064	0.675322	−	0.737523i	$-0.264004\pi$
$223$	20.1694	1.35064	0.675322	+	0.737523i	$0.264004\pi$
$224$	7.81232	0.521983
$225$	0	0
$226$	−27.0027	−1.79619
$227$	20.6209	1.36866	0.684328	−	0.729174i	$-0.260096\pi$
$227$	20.6209	1.36866	0.684328	+	0.729174i	$0.260096\pi$
$228$	0	0
$229$	−1.19805	−0.0791691	−0.0395846	−	0.999216i	$-0.512603\pi$
$229$	−1.19805	−0.0791691	−0.0395846	+	0.999216i	$0.512603\pi$
$230$	0	0
$231$	0	0
$232$	−61.3894	−4.03041
$233$	−12.2854	−0.804845	−0.402423	−	0.915454i	$-0.631832\pi$
$233$	−12.2854	−0.804845	−0.402423	+	0.915454i	$0.631832\pi$
$234$	0	0
$235$	0	0
$236$	−40.0085	−2.60433
$237$	0	0
$238$	−1.90043	−0.123186
$239$	−6.59202	−0.426403	−0.213201	−	0.977008i	$-0.568389\pi$
$239$	−6.59202	−0.426403	−0.213201	+	0.977008i	$0.568389\pi$
$240$	0	0
$241$	−3.65042	−0.235144	−0.117572	−	0.993064i	$-0.537511\pi$
$241$	−3.65042	−0.235144	−0.117572	+	0.993064i	$0.537511\pi$
$242$	−58.4705	−3.75863
$243$	0	0
$244$	6.75888	0.432693
$245$	0	0
$246$	0	0
$247$	−2.39180	−0.152187
$248$	9.40246	0.597057
$249$	0	0
$250$	0	0
$251$	−9.82696	−0.620272	−0.310136	−	0.950692i	$-0.600375\pi$
$251$	−9.82696	−0.620272	−0.310136	+	0.950692i	$0.600375\pi$
$252$	0	0
$253$	44.0455	2.76912
$254$	−16.1449	−1.01302
$255$	0	0
$256$	41.1896	2.57435
$257$	28.4788	1.77646	0.888229	−	0.459401i	$-0.151936\pi$
$257$	28.4788	1.77646	0.888229	+	0.459401i	$0.151936\pi$
$258$	0	0
$259$	−2.00378	−0.124509
$260$	0	0
$261$	0	0
$262$	51.2657	3.16721
$263$	23.2395	1.43301	0.716506	−	0.697581i	$-0.245740\pi$
$263$	23.2395	1.43301	0.716506	+	0.697581i	$0.245740\pi$
$264$	0	0
$265$	0	0
$266$	1.09655	0.0672337
$267$	0	0
$268$	5.82138	0.355597
$269$	5.41403	0.330099	0.165049	−	0.986285i	$-0.447222\pi$
$269$	5.41403	0.330099	0.165049	+	0.986285i	$0.447222\pi$
$270$	0	0
$271$	2.30650	0.140110	0.0700550	−	0.997543i	$-0.477683\pi$
$271$	2.30650	0.140110	0.0700550	+	0.997543i	$0.477683\pi$
$272$	−28.2788	−1.71465
$273$	0	0
$274$	37.7438	2.28019
$275$	0	0
$276$	0	0
$277$	−16.7851	−1.00852	−0.504259	−	0.863552i	$-0.668234\pi$
$277$	−16.7851	−1.00852	−0.504259	+	0.863552i	$0.668234\pi$
$278$	−42.1316	−2.52688
$279$	0	0
$280$	0	0
$281$	28.7008	1.71215	0.856075	−	0.516852i	$-0.172896\pi$
$281$	28.7008	1.71215	0.856075	+	0.516852i	$0.172896\pi$
$282$	0	0
$283$	−7.82577	−0.465194	−0.232597	−	0.972573i	$-0.574722\pi$
$283$	−7.82577	−0.465194	−0.232597	+	0.972573i	$0.574722\pi$
$284$	−63.8324	−3.78776
$285$	0	0
$286$	−33.6305	−1.98862
$287$	−3.13757	−0.185205
$288$	0	0
$289$	−13.3317	−0.784220
$290$	0	0
$291$	0	0
$292$	−64.1856	−3.75618
$293$	20.2763	1.18455	0.592277	−	0.805734i	$-0.298229\pi$
$293$	20.2763	1.18455	0.592277	+	0.805734i	$0.298229\pi$
$294$	0	0
$295$	0	0
$296$	−51.8114	−3.01148
$297$	0	0
$298$	−37.5321	−2.17418
$299$	16.7402	0.968111
$300$	0	0
$301$	3.63274	0.209388
$302$	−21.7528	−1.25173
$303$	0	0
$304$	16.3169	0.935837
$305$	0	0
$306$	0	0
$307$	5.83583	0.333068	0.166534	−	0.986036i	$-0.446742\pi$
$307$	5.83583	0.333068	0.166534	+	0.986036i	$0.446742\pi$
$308$	11.2768	0.642555
$309$	0	0
$310$	0	0
$311$	19.5768	1.11010	0.555048	−	0.831818i	$-0.312700\pi$
$311$	19.5768	1.11010	0.555048	+	0.831818i	$0.312700\pi$
$312$	0	0
$313$	−30.9264	−1.74806	−0.874031	−	0.485870i	$-0.838503\pi$
$313$	−30.9264	−1.74806	−0.874031	+	0.485870i	$0.838503\pi$
$314$	−32.6898	−1.84479
$315$	0	0
$316$	4.21401	0.237056
$317$	−11.3161	−0.635577	−0.317789	−	0.948162i	$-0.602940\pi$
$317$	−11.3161	−0.635577	−0.317789	+	0.948162i	$0.602940\pi$
$318$	0	0
$319$	−37.1802	−2.08169
$320$	0	0
$321$	0	0
$322$	−7.67472	−0.427695
$323$	−2.11659	−0.117770
$324$	0	0
$325$	0	0
$326$	34.5700	1.91466
$327$	0	0
$328$	−81.1276	−4.47952
$329$	−1.52081	−0.0838449
$330$	0	0
$331$	9.87111	0.542565	0.271283	−	0.962500i	$-0.412552\pi$
$331$	9.87111	0.542565	0.271283	+	0.962500i	$0.412552\pi$
$332$	−45.6072	−2.50302
$333$	0	0
$334$	−55.3980	−3.03124
$335$	0	0
$336$	0	0
$337$	23.2517	1.26660	0.633299	−	0.773907i	$-0.281700\pi$
$337$	23.2517	1.26660	0.633299	+	0.773907i	$0.281700\pi$
$338$	22.6912	1.23424
$339$	0	0
$340$	0	0
$341$	5.69456	0.308378
$342$	0	0
$343$	5.04281	0.272286
$344$	93.9313	5.06444
$345$	0	0
$346$	−55.2166	−2.96846
$347$	22.1077	1.18680	0.593401	−	0.804907i	$-0.297785\pi$
$347$	22.1077	1.18680	0.593401	+	0.804907i	$0.297785\pi$
$348$	0	0
$349$	−4.31475	−0.230964	−0.115482	−	0.993310i	$-0.536841\pi$
$349$	−4.31475	−0.230964	−0.115482	+	0.993310i	$0.536841\pi$
$350$	0	0
$351$	0	0
$352$	122.342	6.52083
$353$	−3.37872	−0.179831	−0.0899157	−	0.995949i	$-0.528660\pi$
$353$	−3.37872	−0.179831	−0.0899157	+	0.995949i	$0.528660\pi$
$354$	0	0
$355$	0	0
$356$	29.8613	1.58265
$357$	0	0
$358$	26.2561	1.38768
$359$	2.37338	0.125262	0.0626311	−	0.998037i	$-0.480051\pi$
$359$	2.37338	0.125262	0.0626311	+	0.998037i	$0.480051\pi$
$360$	0	0
$361$	−17.7787	−0.935722
$362$	−15.4487	−0.811966
$363$	0	0
$364$	4.28592	0.224643
$365$	0	0
$366$	0	0
$367$	−11.7481	−0.613246	−0.306623	−	0.951831i	$-0.599199\pi$
$367$	−11.7481	−0.613246	−0.306623	+	0.951831i	$0.599199\pi$
$368$	−114.202	−5.95317
$369$	0	0
$370$	0	0
$371$	−1.83244	−0.0951353
$372$	0	0
$373$	7.01763	0.363359	0.181679	−	0.983358i	$-0.441847\pi$
$373$	7.01763	0.363359	0.181679	+	0.983358i	$0.441847\pi$
$374$	−29.7608	−1.53890
$375$	0	0
$376$	−39.3233	−2.02795
$377$	−14.1309	−0.727779
$378$	0	0
$379$	−10.3561	−0.531957	−0.265979	−	0.963979i	$-0.585695\pi$
$379$	−10.3561	−0.531957	−0.265979	+	0.963979i	$0.585695\pi$
$380$	0	0
$381$	0	0
$382$	29.4226	1.50539
$383$	13.7657	0.703395	0.351697	−	0.936114i	$-0.385605\pi$
$383$	13.7657	0.703395	0.351697	+	0.936114i	$0.385605\pi$
$384$	0	0
$385$	0	0
$386$	−27.3521	−1.39218
$387$	0	0
$388$	−40.0227	−2.03184
$389$	11.7765	0.597094	0.298547	−	0.954395i	$-0.403498\pi$
$389$	11.7765	0.597094	0.298547	+	0.954395i	$0.403498\pi$
$390$	0	0
$391$	14.8140	0.749175
$392$	64.5739	3.26148
$393$	0	0
$394$	−39.7194	−2.00104
$395$	0	0
$396$	0	0
$397$	−14.7131	−0.738427	−0.369214	−	0.929345i	$-0.620373\pi$
$397$	−14.7131	−0.738427	−0.369214	+	0.929345i	$0.620373\pi$
$398$	9.16529	0.459415
$399$	0	0
$400$	0	0
$401$	14.2081	0.709520	0.354760	−	0.934957i	$-0.384563\pi$
$401$	14.2081	0.709520	0.354760	+	0.934957i	$0.384563\pi$
$402$	0	0
$403$	2.16431	0.107812
$404$	−83.6938	−4.16392
$405$	0	0
$406$	6.47847	0.321521
$407$	−31.3794	−1.55542
$408$	0	0
$409$	33.1914	1.64121	0.820604	−	0.571498i	$-0.193637\pi$
$409$	33.1914	1.64121	0.820604	+	0.571498i	$0.193637\pi$
$410$	0	0
$411$	0	0
$412$	−54.3796	−2.67909
$413$	2.67152	0.131457
$414$	0	0
$415$	0	0
$416$	46.4979	2.27975
$417$	0	0
$418$	17.1720	0.839912
$419$	−17.0350	−0.832216	−0.416108	−	0.909315i	$-0.636606\pi$
$419$	−17.0350	−0.832216	−0.416108	+	0.909315i	$0.636606\pi$
$420$	0	0
$421$	−21.6984	−1.05752	−0.528758	−	0.848773i	$-0.677342\pi$
$421$	−21.6984	−1.05752	−0.528758	+	0.848773i	$0.677342\pi$
$422$	29.9733	1.45908
$423$	0	0
$424$	−47.3810	−2.30103
$425$	0	0
$426$	0	0
$427$	−0.451316	−0.0218407
$428$	−96.6234	−4.67047
$429$	0	0
$430$	0	0
$431$	−11.5438	−0.556043	−0.278022	−	0.960575i	$-0.589679\pi$
$431$	−11.5438	−0.556043	−0.278022	+	0.960575i	$0.589679\pi$
$432$	0	0
$433$	−5.14235	−0.247125	−0.123563	−	0.992337i	$-0.539432\pi$
$433$	−5.14235	−0.247125	−0.123563	+	0.992337i	$0.539432\pi$
$434$	−0.992249	−0.0476295
$435$	0	0
$436$	−0.0631411	−0.00302391
$437$	−8.54769	−0.408891
$438$	0	0
$439$	−18.5498	−0.885333	−0.442666	−	0.896686i	$-0.645967\pi$
$439$	−18.5498	−0.885333	−0.442666	+	0.896686i	$0.645967\pi$
$440$	0	0
$441$	0	0
$442$	−11.3111	−0.538013
$443$	−0.829840	−0.0394269	−0.0197135	−	0.999806i	$-0.506275\pi$
$443$	−0.829840	−0.0394269	−0.0197135	+	0.999806i	$0.506275\pi$
$444$	0	0
$445$	0	0
$446$	−55.0362	−2.60604
$447$	0	0
$448$	−10.5794	−0.499828
$449$	19.9721	0.942540	0.471270	−	0.881989i	$-0.343796\pi$
$449$	19.9721	0.942540	0.471270	+	0.881989i	$0.343796\pi$
$450$	0	0
$451$	−49.1346	−2.31366
$452$	53.8904	2.53479
$453$	0	0
$454$	−56.2681	−2.64079
$455$	0	0
$456$	0	0
$457$	14.8949	0.696754	0.348377	−	0.937354i	$-0.386733\pi$
$457$	14.8949	0.696754	0.348377	+	0.937354i	$0.386733\pi$
$458$	3.26910	0.152755
$459$	0	0
$460$	0	0
$461$	−10.0510	−0.468120	−0.234060	−	0.972222i	$-0.575201\pi$
$461$	−10.0510	−0.468120	−0.234060	+	0.972222i	$0.575201\pi$
$462$	0	0
$463$	34.0072	1.58045	0.790225	−	0.612817i	$-0.209964\pi$
$463$	34.0072	1.58045	0.790225	+	0.612817i	$0.209964\pi$
$464$	96.4011	4.47531
$465$	0	0
$466$	33.5232	1.55293
$467$	36.3361	1.68143	0.840717	−	0.541474i	$-0.182134\pi$
$467$	36.3361	1.68143	0.840717	+	0.541474i	$0.182134\pi$
$468$	0	0
$469$	−0.388716	−0.0179492
$470$	0	0
$471$	0	0
$472$	69.0772	3.17953
$473$	56.8891	2.61576
$474$	0	0
$475$	0	0
$476$	3.79276	0.173841
$477$	0	0
$478$	17.9876	0.822735
$479$	37.1451	1.69720	0.848600	−	0.529034i	$-0.177446\pi$
$479$	37.1451	1.69720	0.848600	+	0.529034i	$0.177446\pi$
$480$	0	0
$481$	−11.9262	−0.543789
$482$	9.96088	0.453706
$483$	0	0
$484$	116.692	5.30419
$485$	0	0
$486$	0	0
$487$	−10.0458	−0.455219	−0.227609	−	0.973753i	$-0.573091\pi$
$487$	−10.0458	−0.455219	−0.227609	+	0.973753i	$0.573091\pi$
$488$	−11.6696	−0.528259
$489$	0	0
$490$	0	0
$491$	−21.7701	−0.982469	−0.491235	−	0.871027i	$-0.663454\pi$
$491$	−21.7701	−0.982469	−0.491235	+	0.871027i	$0.663454\pi$
$492$	0	0
$493$	−12.5049	−0.563194
$494$	6.52650	0.293641
$495$	0	0
$496$	−14.7649	−0.662963
$497$	4.26234	0.191192
$498$	0	0
$499$	26.8960	1.20403	0.602016	−	0.798484i	$-0.294365\pi$
$499$	26.8960	1.20403	0.602016	+	0.798484i	$0.294365\pi$
$500$	0	0
$501$	0	0
$502$	26.8148	1.19680
$503$	−23.8273	−1.06241	−0.531204	−	0.847244i	$-0.678260\pi$
$503$	−23.8273	−1.06241	−0.531204	+	0.847244i	$0.678260\pi$
$504$	0	0
$505$	0	0
$506$	−120.187	−5.34296
$507$	0	0
$508$	32.2210	1.42958
$509$	−14.6305	−0.648484	−0.324242	−	0.945974i	$-0.605109\pi$
$509$	−14.6305	−0.648484	−0.324242	+	0.945974i	$0.605109\pi$
$510$	0	0
$511$	4.28592	0.189598
$512$	−39.5557	−1.74813
$513$	0	0
$514$	−77.7099	−3.42764
$515$	0	0
$516$	0	0
$517$	−23.8160	−1.04743
$518$	5.46770	0.240237
$519$	0	0
$520$	0	0
$521$	−3.72289	−0.163103	−0.0815514	−	0.996669i	$-0.525987\pi$
$521$	−3.72289	−0.163103	−0.0815514	+	0.996669i	$0.525987\pi$
$522$	0	0
$523$	39.2901	1.71804	0.859019	−	0.511944i	$-0.171074\pi$
$523$	39.2901	1.71804	0.859019	+	0.511944i	$0.171074\pi$
$524$	−102.313	−4.46957
$525$	0	0
$526$	−63.4136	−2.76497
$527$	1.91527	0.0834305
$528$	0	0
$529$	36.8251	1.60109
$530$	0	0
$531$	0	0
$532$	−2.18843	−0.0948804
$533$	−18.6744	−0.808876
$534$	0	0
$535$	0	0
$536$	−10.0510	−0.434135
$537$	0	0
$538$	−14.7732	−0.636919
$539$	39.1089	1.68454
$540$	0	0
$541$	40.5855	1.74491	0.872453	−	0.488698i	$-0.162528\pi$
$541$	40.5855	1.74491	0.872453	+	0.488698i	$0.162528\pi$
$542$	−6.29374	−0.270339
$543$	0	0
$544$	41.1476	1.76419
$545$	0	0
$546$	0	0
$547$	−22.1212	−0.945832	−0.472916	−	0.881107i	$-0.656799\pi$
$547$	−22.1212	−0.945832	−0.472916	+	0.881107i	$0.656799\pi$
$548$	−75.3269	−3.21781
$549$	0	0
$550$	0	0
$551$	7.21537	0.307385
$552$	0	0
$553$	−0.281385	−0.0119657
$554$	45.8014	1.94592
$555$	0	0
$556$	84.0838	3.56595
$557$	4.74048	0.200861	0.100430	−	0.994944i	$-0.467978\pi$
$557$	4.74048	0.200861	0.100430	+	0.994944i	$0.467978\pi$
$558$	0	0
$559$	21.6216	0.914496
$560$	0	0
$561$	0	0
$562$	−78.3158	−3.30355
$563$	34.7932	1.46636	0.733178	−	0.680036i	$-0.238036\pi$
$563$	34.7932	1.46636	0.733178	+	0.680036i	$0.238036\pi$
$564$	0	0
$565$	0	0
$566$	21.3541	0.897581
$567$	0	0
$568$	110.211	4.62433
$569$	−30.7307	−1.28830	−0.644150	−	0.764899i	$-0.722789\pi$
$569$	−30.7307	−1.28830	−0.644150	+	0.764899i	$0.722789\pi$
$570$	0	0
$571$	38.4989	1.61113	0.805564	−	0.592509i	$-0.201862\pi$
$571$	38.4989	1.61113	0.805564	+	0.592509i	$0.201862\pi$
$572$	67.1179	2.80634
$573$	0	0
$574$	8.56146	0.357348
$575$	0	0
$576$	0	0
$577$	−20.7281	−0.862924	−0.431462	−	0.902131i	$-0.642002\pi$
$577$	−20.7281	−0.862924	−0.431462	+	0.902131i	$0.642002\pi$
$578$	36.3782	1.51314
$579$	0	0
$580$	0	0
$581$	3.04537	0.126343
$582$	0	0
$583$	−28.6961	−1.18847
$584$	110.820	4.58578
$585$	0	0
$586$	−55.3278	−2.28557
$587$	−15.6299	−0.645115	−0.322558	−	0.946550i	$-0.604543\pi$
$587$	−15.6299	−0.645115	−0.322558	+	0.946550i	$0.604543\pi$
$588$	0	0
$589$	−1.10511	−0.0455354
$590$	0	0
$591$	0	0
$592$	81.3607	3.34390
$593$	−9.50337	−0.390257	−0.195128	−	0.980778i	$-0.562512\pi$
$593$	−9.50337	−0.390257	−0.195128	+	0.980778i	$0.562512\pi$
$594$	0	0
$595$	0	0
$596$	74.9045	3.06821
$597$	0	0
$598$	−45.6789	−1.86795
$599$	27.8631	1.13845	0.569227	−	0.822180i	$-0.307242\pi$
$599$	27.8631	1.13845	0.569227	+	0.822180i	$0.307242\pi$
$600$	0	0
$601$	−12.6250	−0.514986	−0.257493	−	0.966280i	$-0.582896\pi$
$601$	−12.6250	−0.514986	−0.257493	+	0.966280i	$0.582896\pi$
$602$	−9.91264	−0.404009
$603$	0	0
$604$	43.4129	1.76645
$605$	0	0
$606$	0	0
$607$	39.0820	1.58629	0.793144	−	0.609034i	$-0.208443\pi$
$607$	39.0820	1.58629	0.793144	+	0.609034i	$0.208443\pi$
$608$	−23.7422	−0.962874
$609$	0	0
$610$	0	0
$611$	−9.05165	−0.366190
$612$	0	0
$613$	−19.4039	−0.783718	−0.391859	−	0.920025i	$-0.628168\pi$
$613$	−19.4039	−0.783718	−0.391859	+	0.920025i	$0.628168\pi$
$614$	−15.9242	−0.642648
$615$	0	0
$616$	−19.4701	−0.784471
$617$	14.9950	0.603676	0.301838	−	0.953359i	$-0.402400\pi$
$617$	14.9950	0.603676	0.301838	+	0.953359i	$0.402400\pi$
$618$	0	0
$619$	−40.7680	−1.63860	−0.819302	−	0.573362i	$-0.805639\pi$
$619$	−40.7680	−1.63860	−0.819302	+	0.573362i	$0.805639\pi$
$620$	0	0
$621$	0	0
$622$	−53.4190	−2.14191
$623$	−1.99395	−0.0798860
$624$	0	0
$625$	0	0
$626$	84.3886	3.37285
$627$	0	0
$628$	65.2405	2.60338
$629$	−10.5539	−0.420813
$630$	0	0
$631$	15.1448	0.602907	0.301453	−	0.953481i	$-0.402528\pi$
$631$	15.1448	0.602907	0.301453	+	0.953481i	$0.402528\pi$
$632$	−7.27574	−0.289413
$633$	0	0
$634$	30.8783	1.22633
$635$	0	0
$636$	0	0
$637$	14.8640	0.588931
$638$	101.453	4.01658
$639$	0	0
$640$	0	0
$641$	14.1668	0.559554	0.279777	−	0.960065i	$-0.409739\pi$
$641$	14.1668	0.559554	0.279777	+	0.960065i	$0.409739\pi$
$642$	0	0
$643$	27.7195	1.09315	0.546574	−	0.837411i	$-0.315932\pi$
$643$	27.7195	1.09315	0.546574	+	0.837411i	$0.315932\pi$
$644$	15.3168	0.603565
$645$	0	0
$646$	5.77553	0.227235
$647$	−29.0621	−1.14255	−0.571275	−	0.820758i	$-0.693551\pi$
$647$	−29.0621	−1.14255	−0.571275	+	0.820758i	$0.693551\pi$
$648$	0	0
$649$	41.8363	1.64222
$650$	0	0
$651$	0	0
$652$	−68.9929	−2.70197
$653$	9.36152	0.366345	0.183172	−	0.983081i	$-0.441363\pi$
$653$	9.36152	0.366345	0.183172	+	0.983081i	$0.441363\pi$
$654$	0	0
$655$	0	0
$656$	127.396	4.97400
$657$	0	0
$658$	4.14982	0.161777
$659$	−30.5155	−1.18872	−0.594358	−	0.804201i	$-0.702594\pi$
$659$	−30.5155	−1.18872	−0.594358	+	0.804201i	$0.702594\pi$
$660$	0	0
$661$	21.9723	0.854623	0.427312	−	0.904104i	$-0.359461\pi$
$661$	21.9723	0.854623	0.427312	+	0.904104i	$0.359461\pi$
$662$	−26.9352	−1.04687
$663$	0	0
$664$	78.7436	3.05585
$665$	0	0
$666$	0	0
$667$	−50.5002	−1.95538
$668$	110.560	4.27770
$669$	0	0
$670$	0	0
$671$	−7.06765	−0.272844
$672$	0	0
$673$	−20.6046	−0.794250	−0.397125	−	0.917765i	$-0.629992\pi$
$673$	−20.6046	−0.794250	−0.397125	+	0.917765i	$0.629992\pi$
$674$	−63.4467	−2.44387
$675$	0	0
$676$	−45.2858	−1.74176
$677$	17.4511	0.670700	0.335350	−	0.942094i	$-0.391145\pi$
$677$	17.4511	0.670700	0.335350	+	0.942094i	$0.391145\pi$
$678$	0	0
$679$	2.67247	0.102560
$680$	0	0
$681$	0	0
$682$	−15.5387	−0.595008
$683$	6.69829	0.256303	0.128151	−	0.991755i	$-0.459096\pi$
$683$	6.69829	0.256303	0.128151	+	0.991755i	$0.459096\pi$
$684$	0	0
$685$	0	0
$686$	−13.7603	−0.525370
$687$	0	0
$688$	−147.502	−5.62348
$689$	−10.9064	−0.415501
$690$	0	0
$691$	20.2732	0.771227	0.385614	−	0.922660i	$-0.373990\pi$
$691$	20.2732	0.771227	0.385614	+	0.922660i	$0.373990\pi$
$692$	110.198	4.18910
$693$	0	0
$694$	−60.3251	−2.28991
$695$	0	0
$696$	0	0
$697$	−16.5256	−0.625952
$698$	11.7736	0.445639
$699$	0	0
$700$	0	0
$701$	9.05067	0.341839	0.170920	−	0.985285i	$-0.445326\pi$
$701$	9.05067	0.341839	0.170920	+	0.985285i	$0.445326\pi$
$702$	0	0
$703$	6.08963	0.229675
$704$	−165.674	−6.24407
$705$	0	0
$706$	9.21950	0.346981
$707$	5.58855	0.210179
$708$	0	0
$709$	37.3240	1.40173	0.700867	−	0.713292i	$-0.252797\pi$
$709$	37.3240	1.40173	0.700867	+	0.713292i	$0.252797\pi$
$710$	0	0
$711$	0	0
$712$	−51.5574	−1.93219
$713$	7.73467	0.289666
$714$	0	0
$715$	0	0
$716$	−52.4003	−1.95829
$717$	0	0
$718$	−6.47623	−0.241691
$719$	−11.7669	−0.438830	−0.219415	−	0.975632i	$-0.570415\pi$
$719$	−11.7669	−0.438830	−0.219415	+	0.975632i	$0.570415\pi$
$720$	0	0
$721$	3.63113	0.135230
$722$	48.5127	1.80546
$723$	0	0
$724$	30.8317	1.14585
$725$	0	0
$726$	0	0
$727$	13.1618	0.488145	0.244073	−	0.969757i	$-0.421517\pi$
$727$	13.1618	0.488145	0.244073	+	0.969757i	$0.421517\pi$
$728$	−7.39990	−0.274259
$729$	0	0
$730$	0	0
$731$	19.1337	0.707685
$732$	0	0
$733$	−49.0781	−1.81274	−0.906371	−	0.422483i	$-0.861159\pi$
$733$	−49.0781	−1.81274	−0.906371	+	0.422483i	$0.861159\pi$
$734$	32.0570	1.18324
$735$	0	0
$736$	166.171	6.12516
$737$	−6.08732	−0.224229
$738$	0	0
$739$	30.4019	1.11835	0.559176	−	0.829049i	$-0.311117\pi$
$739$	30.4019	1.11835	0.559176	+	0.829049i	$0.311117\pi$
$740$	0	0
$741$	0	0
$742$	5.00016	0.183561
$743$	−18.4888	−0.678287	−0.339143	−	0.940735i	$-0.610137\pi$
$743$	−18.4888	−0.678287	−0.339143	+	0.940735i	$0.610137\pi$
$744$	0	0
$745$	0	0
$746$	−19.1490	−0.701093
$747$	0	0
$748$	59.3950	2.17170
$749$	6.45191	0.235748
$750$	0	0
$751$	−28.8407	−1.05241	−0.526205	−	0.850358i	$-0.676386\pi$
$751$	−28.8407	−1.05241	−0.526205	+	0.850358i	$0.676386\pi$
$752$	61.7503	2.25180
$753$	0	0
$754$	38.5590	1.40423
$755$	0	0
$756$	0	0
$757$	−4.12046	−0.149761	−0.0748803	−	0.997193i	$-0.523857\pi$
$757$	−4.12046	−0.149761	−0.0748803	+	0.997193i	$0.523857\pi$
$758$	28.2586	1.02640
$759$	0	0
$760$	0	0
$761$	−8.64846	−0.313506	−0.156753	−	0.987638i	$-0.550103\pi$
$761$	−8.64846	−0.313506	−0.156753	+	0.987638i	$0.550103\pi$
$762$	0	0
$763$	0.00421617	0.000152636 0
$764$	−58.7199	−2.12441
$765$	0	0
$766$	−37.5624	−1.35718
$767$	15.9005	0.574135
$768$	0	0
$769$	−15.9750	−0.576075	−0.288037	−	0.957619i	$-0.593003\pi$
$769$	−15.9750	−0.576075	−0.288037	+	0.957619i	$0.593003\pi$
$770$	0	0
$771$	0	0
$772$	54.5877	1.96466
$773$	42.3024	1.52151	0.760757	−	0.649037i	$-0.224828\pi$
$773$	42.3024	1.52151	0.760757	+	0.649037i	$0.224828\pi$
$774$	0	0
$775$	0	0
$776$	69.1016	2.48060
$777$	0	0
$778$	−32.1345	−1.15208
$779$	9.53529	0.341637
$780$	0	0
$781$	66.7486	2.38845
$782$	−40.4228	−1.44552
$783$	0	0
$784$	−101.402	−3.62150
$785$	0	0
$786$	0	0
$787$	31.5495	1.12462	0.562308	−	0.826928i	$-0.309913\pi$
$787$	31.5495	1.12462	0.562308	+	0.826928i	$0.309913\pi$
$788$	79.2698	2.82387
$789$	0	0
$790$	0	0
$791$	−3.59847	−0.127947
$792$	0	0
$793$	−2.68617	−0.0953887
$794$	40.1475	1.42478
$795$	0	0
$796$	−18.2916	−0.648327
$797$	24.5171	0.868440	0.434220	−	0.900807i	$-0.357024\pi$
$797$	24.5171	0.868440	0.434220	+	0.900807i	$0.357024\pi$
$798$	0	0
$799$	−8.01012	−0.283378
$800$	0	0
$801$	0	0
$802$	−38.7696	−1.36900
$803$	67.1179	2.36854
$804$	0	0
$805$	0	0
$806$	−5.90573	−0.208020
$807$	0	0
$808$	144.502	5.08358
$809$	−10.5514	−0.370966	−0.185483	−	0.982647i	$-0.559385\pi$
$809$	−10.5514	−0.370966	−0.185483	+	0.982647i	$0.559385\pi$
$810$	0	0
$811$	−2.45238	−0.0861145	−0.0430573	−	0.999073i	$-0.513710\pi$
$811$	−2.45238	−0.0861145	−0.0430573	+	0.999073i	$0.513710\pi$
$812$	−12.9294	−0.453731
$813$	0	0
$814$	85.6247	3.00115
$815$	0	0
$816$	0	0
$817$	−11.0402	−0.386247
$818$	−90.5691	−3.16667
$819$	0	0
$820$	0	0
$821$	32.9646	1.15047	0.575236	−	0.817987i	$-0.304910\pi$
$821$	32.9646	1.15047	0.575236	+	0.817987i	$0.304910\pi$
$822$	0	0
$823$	21.3701	0.744915	0.372458	−	0.928049i	$-0.378515\pi$
$823$	21.3701	0.744915	0.372458	+	0.928049i	$0.378515\pi$
$824$	93.8897	3.27080
$825$	0	0
$826$	−7.28977	−0.253643
$827$	25.5351	0.887941	0.443971	−	0.896041i	$-0.353569\pi$
$827$	25.5351	0.887941	0.443971	+	0.896041i	$0.353569\pi$
$828$	0	0
$829$	−50.1867	−1.74306	−0.871529	−	0.490344i	$-0.836871\pi$
$829$	−50.1867	−1.74306	−0.871529	+	0.490344i	$0.836871\pi$
$830$	0	0
$831$	0	0
$832$	−62.9670	−2.18299
$833$	13.1536	0.455746
$834$	0	0
$835$	0	0
$836$	−34.2710	−1.18529
$837$	0	0
$838$	46.4834	1.60574
$839$	−32.7185	−1.12957	−0.564784	−	0.825239i	$-0.691040\pi$
$839$	−32.7185	−1.12957	−0.564784	+	0.825239i	$0.691040\pi$
$840$	0	0
$841$	13.6288	0.469959
$842$	59.2083	2.04045
$843$	0	0
$844$	−59.8190	−2.05905
$845$	0	0
$846$	0	0
$847$	−7.79198	−0.267736
$848$	74.4035	2.55503
$849$	0	0
$850$	0	0
$851$	−42.6212	−1.46104
$852$	0	0
$853$	56.5855	1.93745	0.968725	−	0.248136i	$-0.0798179\pi$
$853$	56.5855	1.93745	0.968725	+	0.248136i	$0.0798179\pi$
$854$	1.23150	0.0421412
$855$	0	0
$856$	166.826	5.70200
$857$	−9.30014	−0.317687	−0.158843	−	0.987304i	$-0.550776\pi$
$857$	−9.30014	−0.317687	−0.158843	+	0.987304i	$0.550776\pi$
$858$	0	0
$859$	29.5054	1.00671	0.503356	−	0.864079i	$-0.332098\pi$
$859$	29.5054	1.00671	0.503356	+	0.864079i	$0.332098\pi$
$860$	0	0
$861$	0	0
$862$	31.4994	1.07287
$863$	12.2429	0.416753	0.208376	−	0.978049i	$-0.433182\pi$
$863$	12.2429	0.416753	0.208376	+	0.978049i	$0.433182\pi$
$864$	0	0
$865$	0	0
$866$	14.0319	0.476823
$867$	0	0
$868$	1.98027	0.0672149
$869$	−4.40652	−0.149481
$870$	0	0
$871$	−2.31358	−0.0783927
$872$	0.109017	0.00369178
$873$	0	0
$874$	23.3240	0.788947
$875$	0	0
$876$	0	0
$877$	−25.6754	−0.866997	−0.433499	−	0.901154i	$-0.642721\pi$
$877$	−25.6754	−0.866997	−0.433499	+	0.901154i	$0.642721\pi$
$878$	50.6167	1.70823
$879$	0	0
$880$	0	0
$881$	51.6637	1.74059	0.870297	−	0.492527i	$-0.163927\pi$
$881$	51.6637	1.74059	0.870297	+	0.492527i	$0.163927\pi$
$882$	0	0
$883$	1.40767	0.0473719	0.0236860	−	0.999719i	$-0.492460\pi$
$883$	1.40767	0.0473719	0.0236860	+	0.999719i	$0.492460\pi$
$884$	22.5740	0.759245
$885$	0	0
$886$	2.26438	0.0760734
$887$	9.06740	0.304453	0.152227	−	0.988346i	$-0.451356\pi$
$887$	9.06740	0.304453	0.152227	+	0.988346i	$0.451356\pi$
$888$	0	0
$889$	−2.15152	−0.0721597
$890$	0	0
$891$	0	0
$892$	109.838	3.67765
$893$	4.62185	0.154664
$894$	0	0
$895$	0	0
$896$	13.2432	0.442425
$897$	0	0
$898$	−54.4976	−1.81861
$899$	−6.52907	−0.217757
$900$	0	0
$901$	−9.65146	−0.321537
$902$	134.073	4.46415
$903$	0	0
$904$	−93.0451	−3.09463
$905$	0	0
$906$	0	0
$907$	−34.2737	−1.13804	−0.569020	−	0.822324i	$-0.692677\pi$
$907$	−34.2737	−1.13804	−0.569020	+	0.822324i	$0.692677\pi$
$908$	112.297	3.72669
$909$	0	0
$910$	0	0
$911$	44.0662	1.45998	0.729989	−	0.683459i	$-0.239525\pi$
$911$	44.0662	1.45998	0.729989	+	0.683459i	$0.239525\pi$
$912$	0	0
$913$	47.6908	1.57833
$914$	−40.6436	−1.34437
$915$	0	0
$916$	−6.52429	−0.215569
$917$	6.83185	0.225607
$918$	0	0
$919$	−3.21793	−0.106150	−0.0530749	−	0.998591i	$-0.516902\pi$
$919$	−3.21793	−0.106150	−0.0530749	+	0.998591i	$0.516902\pi$
$920$	0	0
$921$	0	0
$922$	27.4260	0.903227
$923$	25.3688	0.835026
$924$	0	0
$925$	0	0
$926$	−92.7953	−3.04944
$927$	0	0
$928$	−140.270	−4.60460
$929$	−40.7022	−1.33540	−0.667698	−	0.744433i	$-0.732720\pi$
$929$	−40.7022	−1.33540	−0.667698	+	0.744433i	$0.732720\pi$
$930$	0	0
$931$	−7.58967	−0.248741
$932$	−66.9037	−2.19150
$933$	0	0
$934$	−99.1501	−3.24429
$935$	0	0
$936$	0	0
$937$	−5.49291	−0.179446	−0.0897228	−	0.995967i	$-0.528598\pi$
$937$	−5.49291	−0.179446	−0.0897228	+	0.995967i	$0.528598\pi$
$938$	1.06069	0.0346326
$939$	0	0
$940$	0	0
$941$	−25.9336	−0.845413	−0.422706	−	0.906267i	$-0.638920\pi$
$941$	−25.9336	−0.845413	−0.422706	+	0.906267i	$0.638920\pi$
$942$	0	0
$943$	−66.7373	−2.17327
$944$	−108.473	−3.53051
$945$	0	0
$946$	−155.233	−5.04706
$947$	32.9687	1.07134	0.535669	−	0.844428i	$-0.320060\pi$
$947$	32.9687	1.07134	0.535669	+	0.844428i	$0.320060\pi$
$948$	0	0
$949$	25.5092	0.828064
$950$	0	0
$951$	0	0
$952$	−6.54843	−0.212236
$953$	−6.46663	−0.209475	−0.104737	−	0.994500i	$-0.533400\pi$
$953$	−6.46663	−0.209475	−0.104737	+	0.994500i	$0.533400\pi$
$954$	0	0
$955$	0	0
$956$	−35.8987	−1.16105
$957$	0	0
$958$	−101.358	−3.27471
$959$	5.02987	0.162423
$960$	0	0
$961$	1.00000	0.0322581
$962$	32.5430	1.04923
$963$	0	0
$964$	−19.8794	−0.640271
$965$	0	0
$966$	0	0
$967$	20.1694	0.648605	0.324303	−	0.945953i	$-0.394870\pi$
$967$	20.1694	0.648605	0.324303	+	0.945953i	$0.394870\pi$
$968$	−201.476	−6.47568
$969$	0	0
$970$	0	0
$971$	27.2911	0.875811	0.437906	−	0.899021i	$-0.355720\pi$
$971$	27.2911	0.875811	0.437906	+	0.899021i	$0.355720\pi$
$972$	0	0
$973$	−5.61460	−0.179996
$974$	27.4119	0.878335
$975$	0	0
$976$	18.3251	0.586571
$977$	−30.9195	−0.989202	−0.494601	−	0.869120i	$-0.664686\pi$
$977$	−30.9195	−0.989202	−0.494601	+	0.869120i	$0.664686\pi$
$978$	0	0
$979$	−31.2255	−0.997971
$980$	0	0
$981$	0	0
$982$	59.4038	1.89565
$983$	−10.0088	−0.319230	−0.159615	−	0.987179i	$-0.551025\pi$
$983$	−10.0088	−0.319230	−0.159615	+	0.987179i	$0.551025\pi$
$984$	0	0
$985$	0	0
$986$	34.1222	1.08667
$987$	0	0
$988$	−13.0252	−0.414387
$989$	77.2700	2.45704
$990$	0	0
$991$	11.6459	0.369946	0.184973	−	0.982744i	$-0.440780\pi$
$991$	11.6459	0.369946	0.184973	+	0.982744i	$0.440780\pi$
$992$	21.4840	0.682116
$993$	0	0
$994$	−11.6306	−0.368901
$995$	0	0
$996$	0	0
$997$	8.93602	0.283007	0.141503	−	0.989938i	$-0.454806\pi$
$997$	8.93602	0.283007	0.141503	+	0.989938i	$0.454806\pi$
$998$	−73.3910	−2.32315
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6975.2.a.ck.1.1		16
3.2	odd	2	inner	6975.2.a.ck.1.15		16
5.2	odd	4		1395.2.c.g.559.2	yes	16
5.3	odd	4		1395.2.c.g.559.16	yes	16
5.4	even	2	inner	6975.2.a.ck.1.16		16
15.2	even	4		1395.2.c.g.559.15	yes	16
15.8	even	4		1395.2.c.g.559.1	✓	16
15.14	odd	2	inner	6975.2.a.ck.1.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1395.2.c.g.559.1	✓	16	15.8	even	4
1395.2.c.g.559.2	yes	16	5.2	odd	4
1395.2.c.g.559.15	yes	16	15.2	even	4
1395.2.c.g.559.16	yes	16	5.3	odd	4
6975.2.a.ck.1.1		16	1.1	even	1	trivial
6975.2.a.ck.1.2		16	15.14	odd	2	inner
6975.2.a.ck.1.15		16	3.2	odd	2	inner
6975.2.a.ck.1.16		16	5.4	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6975 = 3^{2} \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6975.a (trivial)

\(f(q)\)	\(=\)	\(q-2.72869 q^{2} +5.44577 q^{4} -0.363635 q^{7} -9.40246 q^{8} -5.69456 q^{11} -2.16431 q^{13} +0.992249 q^{14} +14.7649 q^{16} -1.91527 q^{17} +1.10511 q^{19} +15.5387 q^{22} -7.73467 q^{23} +5.90573 q^{26} -1.98027 q^{28} +6.52907 q^{29} -1.00000 q^{31} -21.4840 q^{32} +5.22619 q^{34} +5.51041 q^{37} -3.01552 q^{38} +8.62833 q^{41} -9.99008 q^{43} -31.0113 q^{44} +21.1056 q^{46} +4.18224 q^{47} -6.86777 q^{49} -11.7863 q^{52} +5.03921 q^{53} +3.41906 q^{56} -17.8158 q^{58} -7.34671 q^{59} +1.24112 q^{61} +2.72869 q^{62} +29.0934 q^{64} +1.06897 q^{67} -10.4301 q^{68} -11.7215 q^{71} -11.7863 q^{73} -15.0362 q^{74} +6.01820 q^{76} +2.07074 q^{77} +0.773812 q^{79} -23.5441 q^{82} -8.37479 q^{83} +27.2599 q^{86} +53.5429 q^{88} +5.48339 q^{89} +0.787018 q^{91} -42.1213 q^{92} -11.4121 q^{94} -7.34931 q^{97} +18.7400 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 12 q^{4} + 44 q^{16} - 16 q^{31} + 24 q^{34} + 88 q^{46} + 16 q^{49} + 64 q^{61} + 176 q^{64} - 12 q^{76} + 72 q^{79} - 16 q^{91} - 8 q^{94}+O(q^{100}) \) 16 * q + 12 * q^4 + 44 * q^16 - 16 * q^31 + 24 * q^34 + 88 * q^46 + 16 * q^49 + 64 * q^61 + 176 * q^64 - 12 * q^76 + 72 * q^79 - 16 * q^91 - 8 * q^94

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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