LMFDB - Embedded newform 9025.2.a.cb.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9025,2,Mod(1,9025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9025, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("9025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9025 = 5^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9025.a (trivial)

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:	yes
Analytic conductor:	$72.0649878242$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.280944640000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 16x^{6} + 80x^{4} - 128x^{2} + 19 $ x^8 - 16x^6 + 80x^4 - 128*x^2 + 19
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 1805)
Fricke sign:	$-1$
Sato-Tate group:	$N(\mathrm{U}(1))$

Embedding invariants

Embedding label			1.5
Root			$0.406045$ of defining polynomial
Character	$\chi$	$=$	9025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.406045 q^{2} -3.27727 q^{3} -1.83513 q^{4} -1.33072 q^{6} -1.55723 q^{8} +7.74048 q^{9} +O(q^{10})$	$q+0.406045 q^{2} -3.27727 q^{3} -1.83513 q^{4} -1.33072 q^{6} -1.55723 q^{8} +7.74048 q^{9} -2.92978 q^{11} +6.01420 q^{12} -6.51610 q^{13} +3.03795 q^{16} +3.14298 q^{18} -1.18962 q^{22} +5.10347 q^{24} -2.64583 q^{26} -15.5358 q^{27} +4.34801 q^{32} +9.60166 q^{33} -14.2048 q^{36} -8.01591 q^{37} +21.3550 q^{39} +5.37651 q^{44} -9.95617 q^{48} -7.00000 q^{49} +11.9579 q^{52} -5.09315 q^{53} -6.30824 q^{54} -1.11908 q^{61} -4.31041 q^{64} +3.89870 q^{66} -8.95237 q^{67} -12.0537 q^{72} -3.25482 q^{74} +8.67109 q^{78} +27.6936 q^{81} +4.56235 q^{88} -14.2496 q^{96} +11.6477 q^{97} -2.84231 q^{98} -22.6779 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{4} + 24 q^{9}+O(q^{10}) $ 8 * q + 16 * q^4 + 24 * q^9	$ 8 q + 16 q^{4} + 24 q^{9} + 32 q^{16} + 8 q^{24} + 24 q^{26} - 8 q^{36} + 72 q^{44} - 56 q^{49} + 88 q^{54} + 64 q^{64} + 104 q^{66} + 72 q^{81} - 120 q^{96} + 32 q^{99}+O(q^{100}) $ 8 * q + 16 * q^4 + 24 * q^9 + 32 * q^16 + 8 * q^24 + 24 * q^26 - 8 * q^36 + 72 * q^44 - 56 * q^49 + 88 * q^54 + 64 * q^64 + 104 * q^66 + 72 * q^81 - 120 * q^96 + 32 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	0.406045	0.287117	0.143559	−	0.989642i	$-0.454145\pi$
$2$	0.406045	0.287117	0.143559	+	0.989642i	$0.454145\pi$
$3$	−3.27727	−1.89213	−0.946065	−	0.323976i	$-0.894980\pi$
$3$	−3.27727	−1.89213	−0.946065	+	0.323976i	$0.894980\pi$
$4$	−1.83513	−0.917564
$5$	0	0
$5$	0	0
$6$	−1.33072	−0.543263
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	−1.55723	−0.550565
$9$	7.74048	2.58016
$10$	0	0
$11$	−2.92978	−0.883361	−0.441680	−	0.897172i	$-0.645618\pi$
$11$	−2.92978	−0.883361	−0.441680	+	0.897172i	$0.645618\pi$
$12$	6.01420	1.73615
$13$	−6.51610	−1.80724	−0.903621	−	0.428333i	$-0.859101\pi$
$13$	−6.51610	−1.80724	−0.903621	+	0.428333i	$0.859101\pi$
$14$	0	0
$15$	0	0
$16$	3.03795	0.759487
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	3.14298	0.740808
$19$	0	0
$19$	0	0
$20$	0	0
$21$	0	0
$22$	−1.18962	−0.253628
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	5.10347	1.04174
$25$	0	0
$26$	−2.64583	−0.518890
$27$	−15.5358	−2.98987
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	4.34801	0.768627
$33$	9.60166	1.67143
$34$	0	0
$35$	0	0
$36$	−14.2048	−2.36746
$37$	−8.01591	−1.31781	−0.658904	−	0.752227i	$-0.728980\pi$
$37$	−8.01591	−1.31781	−0.658904	+	0.752227i	$0.728980\pi$
$38$	0	0
$39$	21.3550	3.41954
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	5.37651	0.810540
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	−9.95617	−1.43705
$49$	−7.00000	−1.00000
$50$	0	0
$51$	0	0
$52$	11.9579	1.65826
$53$	−5.09315	−0.699599	−0.349799	−	0.936825i	$-0.613750\pi$
$53$	−5.09315	−0.699599	−0.349799	+	0.936825i	$0.613750\pi$
$54$	−6.30824	−0.858442
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−1.11908	−0.143283	−0.0716414	−	0.997430i	$-0.522824\pi$
$61$	−1.11908	−0.143283	−0.0716414	+	0.997430i	$0.522824\pi$
$62$	0	0
$63$	0	0
$64$	−4.31041	−0.538801
$65$	0	0
$66$	3.89870	0.479897
$67$	−8.95237	−1.09371	−0.546853	−	0.837229i	$-0.684174\pi$
$67$	−8.95237	−1.09371	−0.546853	+	0.837229i	$0.684174\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	−12.0537	−1.42055
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	−3.25482	−0.378365
$75$	0	0
$76$	0	0
$77$	0	0
$78$	8.67109	0.981807
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	27.6936	3.07706
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	4.56235	0.486348
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	−14.2496	−1.45434
$97$	11.6477	1.18264	0.591322	−	0.806436i	$-0.298606\pi$
$97$	11.6477	1.18264	0.591322	+	0.806436i	$0.298606\pi$
$98$	−2.84231	−0.287117
$99$	−22.6779	−2.27921
$100$	0	0
$101$	−20.0810	−1.99813	−0.999067	−	0.0431977i	$-0.986245\pi$
$101$	−20.0810	−1.99813	−0.999067	+	0.0431977i	$0.986245\pi$
$102$	0	0
$103$	−4.07983	−0.401998	−0.200999	−	0.979591i	$-0.564419\pi$
$103$	−4.07983	−0.401998	−0.200999	+	0.979591i	$0.564419\pi$
$104$	10.1471	0.995004
$105$	0	0
$106$	−2.06805	−0.200867
$107$	−20.3604	−1.96831	−0.984157	−	0.177300i	$-0.943264\pi$
$107$	−20.3604	−1.96831	−0.984157	+	0.177300i	$0.943264\pi$
$108$	28.5102	2.74340
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	26.2703	2.49347
$112$	0	0
$113$	−21.1725	−1.99174	−0.995870	−	0.0907914i	$-0.971060\pi$
$113$	−21.1725	−1.99174	−0.995870	+	0.0907914i	$0.971060\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−50.4377	−4.66297
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−2.41641	−0.219673
$122$	−0.454395	−0.0411390
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	6.20003	0.550163	0.275082	−	0.961421i	$-0.411295\pi$
$127$	6.20003	0.550163	0.275082	+	0.961421i	$0.411295\pi$
$128$	−10.4462	−0.923326
$129$	0	0
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	−17.6203	−1.53365
$133$	0	0
$134$	−3.63506	−0.314022
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	−21.8917	−1.85683	−0.928414	−	0.371546i	$-0.878828\pi$
$139$	−21.8917	−1.85683	−0.928414	+	0.371546i	$0.878828\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	19.0907	1.59645
$144$	23.5152	1.95960
$145$	0	0
$146$	0	0
$147$	22.9409	1.89213
$148$	14.7102	1.20917
$149$	−8.36188	−0.685032	−0.342516	−	0.939512i	$-0.611279\pi$
$149$	−8.36188	−0.685032	−0.342516	+	0.939512i	$0.611279\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	−39.1892	−3.13764
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	16.6916	1.32373
$160$	0	0
$161$	0	0
$162$	11.2448	0.883477
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−19.3091	−1.49418	−0.747091	−	0.664721i	$-0.768550\pi$
$167$	−19.3091	−1.49418	−0.747091	+	0.664721i	$0.768550\pi$
$168$	0	0
$169$	29.4596	2.26612
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−22.7967	−1.73320	−0.866599	−	0.499005i	$-0.833699\pi$
$173$	−22.7967	−1.73320	−0.866599	+	0.499005i	$0.833699\pi$
$174$	0	0
$175$	0	0
$176$	−8.90051	−0.670901
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	3.66751	0.271110
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−8.94427	−0.647185	−0.323592	−	0.946197i	$-0.604891\pi$
$191$	−8.94427	−0.647185	−0.323592	+	0.946197i	$0.604891\pi$
$192$	14.1264	1.01948
$193$	−27.6795	−1.99242	−0.996208	−	0.0870089i	$-0.972269\pi$
$193$	−27.6795	−1.99242	−0.996208	+	0.0870089i	$0.972269\pi$
$194$	4.72948	0.339557
$195$	0	0
$196$	12.8459	0.917564
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	−9.20823	−0.654401
$199$	−26.8328	−1.90213	−0.951064	−	0.308994i	$-0.900008\pi$
$199$	−26.8328	−1.90213	−0.951064	+	0.308994i	$0.900008\pi$
$200$	0	0
$201$	29.3393	2.06944
$202$	−8.15378	−0.573698
$203$	0	0
$204$	0	0
$205$	0	0
$206$	−1.65660	−0.115420
$207$	0	0
$208$	−19.7956	−1.37258
$209$	0	0
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	9.34659	0.641926
$213$	0	0
$214$	−8.26723	−0.565136
$215$	0	0
$216$	24.1929	1.64612
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	10.6669	0.715916
$223$	25.8636	1.73196	0.865978	−	0.500082i	$-0.166697\pi$
$223$	25.8636	1.73196	0.865978	+	0.500082i	$0.166697\pi$
$224$	0	0
$225$	0	0
$226$	−8.59698	−0.571862
$227$	−23.6088	−1.56697	−0.783484	−	0.621412i	$-0.786559\pi$
$227$	−23.6088	−1.56697	−0.783484	+	0.621412i	$0.786559\pi$
$228$	0	0
$229$	−29.5619	−1.95351	−0.976754	−	0.214362i	$-0.931233\pi$
$229$	−29.5619	−1.95351	−0.976754	+	0.214362i	$0.931233\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	−20.4800	−1.33882
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	−0.981170	−0.0630720
$243$	−44.1518	−2.83234
$244$	2.05365	0.131471
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	17.8885	1.12911	0.564557	−	0.825394i	$-0.309047\pi$
$251$	17.8885	1.12911	0.564557	+	0.825394i	$0.309047\pi$
$252$	0	0
$253$	0	0
$254$	2.51749	0.157961
$255$	0	0
$256$	4.37918	0.273699
$257$	−24.7568	−1.54428	−0.772142	−	0.635450i	$-0.780815\pi$
$257$	−24.7568	−1.54428	−0.772142	+	0.635450i	$0.780815\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	4.87254	0.301026
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	−14.9520	−0.920234
$265$	0	0
$266$	0	0
$267$	0	0
$268$	16.4287	1.00355
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	24.1298	1.46578	0.732892	−	0.680345i	$-0.238170\pi$
$271$	24.1298	1.46578	0.732892	+	0.680345i	$0.238170\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	−8.88901	−0.533127
$279$	0	0
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	0	0
$285$	0	0
$286$	7.75169	0.458367
$287$	0	0
$288$	33.6557	1.98318
$289$	−17.0000	−1.00000
$290$	0	0
$291$	−38.1726	−2.23772
$292$	0	0
$293$	34.2340	1.99997	0.999987	−	0.00505234i	$-0.00160822\pi$
$293$	34.2340	1.99997	0.999987	+	0.00505234i	$0.00160822\pi$
$294$	9.31502	0.543263
$295$	0	0
$296$	12.4826	0.725539
$297$	45.5165	2.64113
$298$	−3.39530	−0.196684
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	65.8108	3.78073
$304$	0	0
$305$	0	0
$306$	0	0
$307$	35.0168	1.99851	0.999257	−	0.0385528i	$-0.0122748\pi$
$307$	35.0168	1.99851	0.999257	+	0.0385528i	$0.0122748\pi$
$308$	0	0
$309$	13.3707	0.760633
$310$	0	0
$311$	31.3726	1.77898	0.889490	−	0.456955i	$-0.151060\pi$
$311$	31.3726	1.77898	0.889490	+	0.456955i	$0.151060\pi$
$312$	−33.2547	−1.88268
$313$	0	0		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	16.2999	0.915496	0.457748	−	0.889082i	$-0.348656\pi$
$317$	16.2999	0.915496	0.457748	+	0.889082i	$0.348656\pi$
$318$	6.77755	0.380066
$319$	0	0
$320$	0	0
$321$	66.7265	3.72431
$322$	0	0
$323$	0	0
$324$	−50.8212	−2.82340
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	0	0
$333$	−62.0470	−3.40015
$334$	−7.84036	−0.429005
$335$	0	0
$336$	0	0
$337$	1.64356	0.0895307	0.0447653	−	0.998998i	$-0.485746\pi$
$337$	1.64356	0.0895307	0.0447653	+	0.998998i	$0.485746\pi$
$338$	11.9619	0.650642
$339$	69.3879	3.76863
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	−9.25647	−0.497631
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	13.4164	0.718164	0.359082	−	0.933306i	$-0.383090\pi$
$349$	13.4164	0.718164	0.359082	+	0.933306i	$0.383090\pi$
$350$	0	0
$351$	101.233	5.40341
$352$	−12.7387	−0.678975
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−25.5131	−1.34653	−0.673265	−	0.739401i	$-0.735109\pi$
$359$	−25.5131	−1.34653	−0.673265	+	0.739401i	$0.735109\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	7.91921	0.415651
$364$	0	0
$365$	0	0
$366$	1.48917	0.0778403
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−31.3113	−1.62124	−0.810619	−	0.585575i	$-0.800869\pi$
$373$	−31.3113	−1.62124	−0.810619	+	0.585575i	$0.800869\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	0	0
$381$	−20.3191	−1.04098
$382$	−3.63178	−0.185818
$383$	−25.2329	−1.28934	−0.644671	−	0.764460i	$-0.723006\pi$
$383$	−25.2329	−1.28934	−0.644671	+	0.764460i	$0.723006\pi$
$384$	34.2351	1.74705
$385$	0	0
$386$	−11.2391	−0.572056
$387$	0	0
$388$	−21.3750	−1.08515
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	0	0
$392$	10.9006	0.550565
$393$	−39.3272	−1.98379
$394$	0	0
$395$	0	0
$396$	41.6168	2.09132
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	−10.8953	−0.546133
$399$	0	0
$400$	0	0
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	11.9131	0.594170
$403$	0	0
$404$	36.8512	1.83341
$405$	0	0
$406$	0	0
$407$	23.4848	1.16410
$408$	0	0
$409$	0	0		−	1.00000i	$-0.5\pi$
$409$	0	0			1.00000i	$0.5\pi$
$410$	0	0
$411$	0	0
$412$	7.48701	0.368859
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−28.3321	−1.38909
$417$	71.7449	3.51336
$418$	0	0
$419$	36.0000	1.75872	0.879358	−	0.476162i	$-0.157972\pi$
$419$	36.0000	1.75872	0.879358	+	0.476162i	$0.157972\pi$
$420$	0	0
$421$	0	0		−	1.00000i	$-0.5\pi$
$421$	0	0			1.00000i	$0.5\pi$
$422$	0	0
$423$	0	0
$424$	7.93123	0.385175
$425$	0	0
$426$	0	0
$427$	0	0
$428$	37.3639	1.80605
$429$	−62.5654	−3.02069
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	−47.1970	−2.27077
$433$	37.4530	1.79988	0.899939	−	0.436015i	$-0.143611\pi$
$433$	37.4530	1.79988	0.899939	+	0.436015i	$0.143611\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	−54.1833	−2.58016
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	−48.2093	−2.28791
$445$	0	0
$446$	10.5018	0.497274
$447$	27.4041	1.29617
$448$	0	0
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	0	0
$452$	38.8542	1.82755
$453$	0	0
$454$	−9.58621	−0.449903
$455$	0	0
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	−12.0035	−0.560886
$459$	0	0
$460$	0	0
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	92.5597	4.27857
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−39.4235	−1.80508
$478$	9.74507	0.445729
$479$	43.0918	1.96891	0.984456	−	0.175630i	$-0.0561962\pi$
$479$	43.0918	1.96891	0.984456	+	0.175630i	$0.0561962\pi$
$480$	0	0
$481$	52.2325	2.38160
$482$	0	0
$483$	0	0
$484$	4.43442	0.201564
$485$	0	0
$486$	−17.9276	−0.813213
$487$	−30.1442	−1.36597	−0.682983	−	0.730434i	$-0.739318\pi$
$487$	−30.1442	−1.36597	−0.682983	+	0.730434i	$0.739318\pi$
$488$	1.74266	0.0788866
$489$	0	0
$490$	0	0
$491$	−35.7771	−1.61460	−0.807299	−	0.590143i	$-0.799071\pi$
$491$	−35.7771	−1.61460	−0.807299	+	0.590143i	$0.799071\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−4.31303	−0.193078	−0.0965389	−	0.995329i	$-0.530777\pi$
$499$	−4.31303	−0.193078	−0.0965389	+	0.995329i	$0.530777\pi$
$500$	0	0
$501$	63.2811	2.82719
$502$	7.26355	0.324188
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−96.5469	−4.28780
$508$	−11.3778	−0.504810
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	22.6706	1.00191
$513$	0	0
$514$	−10.0524	−0.443390
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	74.7108	3.27944
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	22.2319	0.972131	0.486066	−	0.873922i	$-0.338432\pi$
$523$	22.2319	0.972131	0.486066	+	0.873922i	$0.338432\pi$
$524$	−22.0215	−0.962015
$525$	0	0
$526$	0	0
$527$	0	0
$528$	29.1694	1.26943
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	13.9409	0.602157
$537$	0	0
$538$	0	0
$539$	20.5084	0.883361
$540$	0	0
$541$	27.3238	1.17474	0.587371	−	0.809318i	$-0.300163\pi$
$541$	27.3238	1.17474	0.587371	+	0.809318i	$0.300163\pi$
$542$	9.79780	0.420851
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	2.56825	0.109810	0.0549052	−	0.998492i	$-0.482514\pi$
$547$	2.56825	0.109810	0.0549052	+	0.998492i	$0.482514\pi$
$548$	0	0
$549$	−8.66218	−0.369693
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	40.1740	1.70376
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−28.7864	−1.21320	−0.606601	−	0.795007i	$-0.707467\pi$
$563$	−28.7864	−1.21320	−0.606601	+	0.795007i	$0.707467\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	39.4704	1.65178	0.825891	−	0.563829i	$-0.190672\pi$
$571$	39.4704	1.65178	0.825891	+	0.563829i	$0.190672\pi$
$572$	−35.0339	−1.46484
$573$	29.3128	1.22456
$574$	0	0
$575$	0	0
$576$	−33.3646	−1.39019
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	−6.90276	−0.287117
$579$	90.7132	3.76991
$580$	0	0
$581$	0	0
$582$	−15.4998	−0.642486
$583$	14.9218	0.617998
$584$	0	0
$585$	0	0
$586$	13.9006	0.574227
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	−42.0994	−1.73615
$589$	0	0
$590$	0	0
$591$	0	0
$592$	−24.3519	−1.00086
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	18.4817	0.758314
$595$	0	0
$596$	15.3451	0.628561
$597$	87.9383	3.59907
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	0	0		−	1.00000i	$-0.5\pi$
$601$	0	0			1.00000i	$0.5\pi$
$602$	0	0
$603$	−69.2956	−2.82194
$604$	0	0
$605$	0	0
$606$	26.7221	1.08551
$607$	−13.8249	−0.561136	−0.280568	−	0.959834i	$-0.590523\pi$
$607$	−13.8249	−0.561136	−0.280568	+	0.959834i	$0.590523\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	14.2184	0.573807
$615$	0	0
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	5.42910	0.218391
$619$	34.9941	1.40653	0.703265	−	0.710928i	$-0.251725\pi$
$619$	34.9941	1.40653	0.703265	+	0.710928i	$0.251725\pi$
$620$	0	0
$621$	0	0
$622$	12.7387	0.510775
$623$	0	0
$624$	64.8754	2.59709
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−11.0275	−0.438997	−0.219499	−	0.975613i	$-0.570442\pi$
$631$	−11.0275	−0.438997	−0.219499	+	0.975613i	$0.570442\pi$
$632$	0	0
$633$	0	0
$634$	6.61851	0.262855
$635$	0	0
$636$	−30.6313	−1.21461
$637$	45.6127	1.80724
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	27.0939	1.06931
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	−43.1254	−1.69412
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	−25.1939	−0.976242
$667$	0	0
$668$	35.4347	1.37101
$669$	−84.7620	−3.27709
$670$	0	0
$671$	3.27864	0.126571
$672$	0	0
$673$	50.4853	1.94606	0.973032	−	0.230671i	$-0.0740922\pi$
$673$	50.4853	1.94606	0.973032	+	0.230671i	$0.0740922\pi$
$674$	0.667361	0.0257058
$675$	0	0
$676$	−54.0621	−2.07931
$677$	−44.4204	−1.70721	−0.853607	−	0.520918i	$-0.825590\pi$
$677$	−44.4204	−1.70721	−0.853607	+	0.520918i	$0.825590\pi$
$678$	28.1746	1.08204
$679$	0	0
$680$	0	0
$681$	77.3722	2.96491
$682$	0	0
$683$	10.5408	0.403333	0.201667	−	0.979454i	$-0.435364\pi$
$683$	10.5408	0.403333	0.201667	+	0.979454i	$0.435364\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	96.8824	3.69629
$688$	0	0
$689$	33.1875	1.26434
$690$	0	0
$691$	−52.5727	−1.99996	−0.999980	−	0.00630823i	$-0.997992\pi$
$691$	−52.5727	−1.99996	−0.999980	+	0.00630823i	$0.997992\pi$
$692$	41.8348	1.59032
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	5.44766	0.206197
$699$	0	0
$700$	0	0
$701$	−48.5239	−1.83272	−0.916360	−	0.400354i	$-0.868887\pi$
$701$	−48.5239	−1.83272	−0.916360	+	0.400354i	$0.868887\pi$
$702$	41.1051	1.55141
$703$	0	0
$704$	12.6285	0.475956
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−40.2492	−1.51159	−0.755796	−	0.654808i	$-0.772750\pi$
$709$	−40.2492	−1.51159	−0.755796	+	0.654808i	$0.772750\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−78.6544	−2.93740
$718$	−10.3595	−0.386612
$719$	13.7940	0.514429	0.257214	−	0.966354i	$-0.417195\pi$
$719$	13.7940	0.514429	0.257214	+	0.966354i	$0.417195\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	3.21556	0.119340
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	61.6165	2.28209
$730$	0	0
$731$	0	0
$732$	−6.73035	−0.248761
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	26.2284	0.966137
$738$	0	0
$739$	53.6656	1.97412	0.987061	−	0.160345i	$-0.0512606\pi$
$739$	53.6656	1.97412	0.987061	+	0.160345i	$0.0512606\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	39.6817	1.45578	0.727890	−	0.685693i	$-0.240501\pi$
$743$	39.6817	1.45578	0.727890	+	0.685693i	$0.240501\pi$
$744$	0	0
$745$	0	0
$746$	−12.7138	−0.465485
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	0	0
$753$	−58.6255	−2.13643
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−54.3834	−1.97140	−0.985699	−	0.168518i	$-0.946102\pi$
$761$	−54.3834	−1.97140	−0.985699	+	0.168518i	$0.946102\pi$
$762$	−8.25048	−0.298883
$763$	0	0
$764$	16.4139	0.593833
$765$	0	0
$766$	−10.2457	−0.370192
$767$	0	0
$768$	−14.3517	−0.517874
$769$	−47.1406	−1.69993	−0.849967	−	0.526836i	$-0.823378\pi$
$769$	−47.1406	−1.69993	−0.849967	+	0.526836i	$0.823378\pi$
$770$	0	0
$771$	81.1345	2.92199
$772$	50.7954	1.82817
$773$	13.0516	0.469433	0.234717	−	0.972064i	$-0.424584\pi$
$773$	13.0516	0.469433	0.234717	+	0.972064i	$0.424584\pi$
$774$	0	0
$775$	0	0
$776$	−18.1382	−0.651122
$777$	0	0
$778$	2.43627	0.0873445
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−21.2656	−0.759487
$785$	0	0
$786$	−15.9686	−0.569581
$787$	17.0954	0.609383	0.304692	−	0.952451i	$-0.401447\pi$
$787$	17.0954	0.609383	0.304692	+	0.952451i	$0.401447\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	35.3147	1.25485
$793$	7.29201	0.258947
$794$	0	0
$795$	0	0
$796$	49.2416	1.74532
$797$	13.8614	0.490997	0.245499	−	0.969397i	$-0.421048\pi$
$797$	13.8614	0.490997	0.245499	+	0.969397i	$0.421048\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	−53.8414	−1.89884
$805$	0	0
$806$	0	0
$807$	0	0
$808$	31.2708	1.10010
$809$	−22.3607	−0.786160	−0.393080	−	0.919504i	$-0.628590\pi$
$809$	−22.3607	−0.786160	−0.393080	+	0.919504i	$0.628590\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	−79.0799	−2.77345
$814$	9.53590	0.334233
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−42.0000	−1.46581	−0.732905	−	0.680331i	$-0.761836\pi$
$821$	−42.0000	−1.46581	−0.732905	+	0.680331i	$0.761836\pi$
$822$	0	0
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	6.35325	0.221326
$825$	0	0
$826$	0	0
$827$	48.4500	1.68477	0.842386	−	0.538875i	$-0.181151\pi$
$827$	48.4500	1.68477	0.842386	+	0.538875i	$0.181151\pi$
$828$	0	0
$829$	0	0		−	1.00000i	$-0.5\pi$
$829$	0	0			1.00000i	$0.5\pi$
$830$	0	0
$831$	0	0
$832$	28.0871	0.973744
$833$	0	0
$834$	29.1316	1.00875
$835$	0	0
$836$	0	0
$837$	0	0
$838$	14.6176	0.504957
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	−15.4727	−0.531336
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	0	0
$856$	31.7059	1.08369
$857$	33.5250	1.14519	0.572597	−	0.819837i	$-0.305936\pi$
$857$	33.5250	1.14519	0.572597	+	0.819837i	$0.305936\pi$
$858$	−25.4044	−0.867290
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	58.6363	1.99600	0.998001	−	0.0631923i	$-0.0201281\pi$
$863$	58.6363	1.99600	0.998001	+	0.0631923i	$0.0201281\pi$
$864$	−67.5499	−2.29809
$865$	0	0
$866$	15.2076	0.516776
$867$	55.7135	1.89213
$868$	0	0
$869$	0	0
$870$	0	0
$871$	58.3346	1.97659
$872$	0	0
$873$	90.1587	3.05141
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0.752363	0.0254055	0.0127028	−	0.999919i	$-0.495956\pi$
$877$	0.752363	0.0254055	0.0127028	+	0.999919i	$0.495956\pi$
$878$	0	0
$879$	−112.194	−3.78421
$880$	0	0
$881$	9.21678	0.310521	0.155261	−	0.987874i	$-0.450378\pi$
$881$	9.21678	0.310521	0.155261	+	0.987874i	$0.450378\pi$
$882$	−22.0009	−0.740808
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−28.4813	−0.956308	−0.478154	−	0.878276i	$-0.658694\pi$
$887$	−28.4813	−0.956308	−0.478154	+	0.878276i	$0.658694\pi$
$888$	−40.9090	−1.37282
$889$	0	0
$890$	0	0
$891$	−81.1360	−2.71816
$892$	−47.4631	−1.58918
$893$	0	0
$894$	11.1273	0.372153
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	32.9705	1.09658
$905$	0	0
$906$	0	0
$907$	−52.9215	−1.75723	−0.878615	−	0.477531i	$-0.841532\pi$
$907$	−52.9215	−1.75723	−0.878615	+	0.477531i	$0.841532\pi$
$908$	43.3251	1.43779
$909$	−155.436	−5.15550
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	54.2499	1.79247
$917$	0	0
$918$	0	0
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	−114.759	−3.78145
$922$	7.30881	0.240703
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−31.5799	−1.03722
$928$	0	0
$929$	−31.3050	−1.02708	−0.513541	−	0.858065i	$-0.671667\pi$
$929$	−31.3050	−1.02708	−0.513541	+	0.858065i	$0.671667\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−102.817	−3.36606
$934$	0	0
$935$	0	0
$936$	78.5434	2.56727
$937$	0	0		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−53.4193	−1.73224
$952$	0	0
$953$	−5.80217	−0.187951	−0.0939754	−	0.995575i	$-0.529957\pi$
$953$	−5.80217	−0.187951	−0.0939754	+	0.995575i	$0.529957\pi$
$954$	−16.0077	−0.518268
$955$	0	0
$956$	−44.0431	−1.42445
$957$	0	0
$958$	17.4972	0.565308
$959$	0	0
$960$	0	0
$961$	−31.0000	−1.00000
$962$	21.2087	0.683797
$963$	−157.599	−5.07856
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	3.76291	0.120945
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	81.0242	2.59885
$973$	0	0
$974$	−12.2399	−0.392192
$975$	0	0
$976$	−3.39969	−0.108822
$977$	−60.2691	−1.92818	−0.964090	−	0.265577i	$-0.914438\pi$
$977$	−60.2691	−1.92818	−0.964090	+	0.265577i	$0.914438\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	−14.5271	−0.463578
$983$	62.7054	1.99999	0.999995	−	0.00306979i	$-0.000977145\pi$
$983$	62.7054	1.99999	0.999995	+	0.00306979i	$0.000977145\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	−1.75128	−0.0554359
$999$	124.534	3.94007

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9025.2.a.cb.1.5		8
5.2	odd	4		1805.2.b.h.1084.5	yes	8
5.3	odd	4		1805.2.b.h.1084.4	✓	8
5.4	even	2	inner	9025.2.a.cb.1.4		8
19.18	odd	2	inner	9025.2.a.cb.1.4		8
95.18	even	4		1805.2.b.h.1084.5	yes	8
95.37	even	4		1805.2.b.h.1084.4	✓	8
95.94	odd	2	CM	9025.2.a.cb.1.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1805.2.b.h.1084.4	✓	8	5.3	odd	4
1805.2.b.h.1084.4	✓	8	95.37	even	4
1805.2.b.h.1084.5	yes	8	5.2	odd	4
1805.2.b.h.1084.5	yes	8	95.18	even	4
9025.2.a.cb.1.4		8	5.4	even	2	inner
9025.2.a.cb.1.4		8	19.18	odd	2	inner
9025.2.a.cb.1.5		8	1.1	even	1	trivial
9025.2.a.cb.1.5		8	95.94	odd	2	CM

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:	\( N \)	\(=\)	\( 9025 = 5^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9025.a (trivial)

\(f(q)\)	\(=\)	\(q+0.406045 q^{2} -3.27727 q^{3} -1.83513 q^{4} -1.33072 q^{6} -1.55723 q^{8} +7.74048 q^{9} +O(q^{10})\)	\(q+0.406045 q^{2} -3.27727 q^{3} -1.83513 q^{4} -1.33072 q^{6} -1.55723 q^{8} +7.74048 q^{9} -2.92978 q^{11} +6.01420 q^{12} -6.51610 q^{13} +3.03795 q^{16} +3.14298 q^{18} -1.18962 q^{22} +5.10347 q^{24} -2.64583 q^{26} -15.5358 q^{27} +4.34801 q^{32} +9.60166 q^{33} -14.2048 q^{36} -8.01591 q^{37} +21.3550 q^{39} +5.37651 q^{44} -9.95617 q^{48} -7.00000 q^{49} +11.9579 q^{52} -5.09315 q^{53} -6.30824 q^{54} -1.11908 q^{61} -4.31041 q^{64} +3.89870 q^{66} -8.95237 q^{67} -12.0537 q^{72} -3.25482 q^{74} +8.67109 q^{78} +27.6936 q^{81} +4.56235 q^{88} -14.2496 q^{96} +11.6477 q^{97} -2.84231 q^{98} -22.6779 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{4} + 24 q^{9}+O(q^{10}) \) 8 * q + 16 * q^4 + 24 * q^9	\( 8 q + 16 q^{4} + 24 q^{9} + 32 q^{16} + 8 q^{24} + 24 q^{26} - 8 q^{36} + 72 q^{44} - 56 q^{49} + 88 q^{54} + 64 q^{64} + 104 q^{66} + 72 q^{81} - 120 q^{96} + 32 q^{99}+O(q^{100}) \) 8 * q + 16 * q^4 + 24 * q^9 + 32 * q^16 + 8 * q^24 + 24 * q^26 - 8 * q^36 + 72 * q^44 - 56 * q^49 + 88 * q^54 + 64 * q^64 + 104 * q^66 + 72 * q^81 - 120 * q^96 + 32 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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