LMFDB - Embedded newform 9025.2.a.cb.1.2

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			1.2
Root			$-2.26640$ 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)

$f(q)$	$=$	$q-2.26640 q^{2} -3.11095 q^{3} +3.13657 q^{4} +7.05066 q^{6} -2.57593 q^{8} +6.67802 q^{9} +O(q^{10})$	$q-2.26640 q^{2} -3.11095 q^{3} +3.13657 q^{4} +7.05066 q^{6} -2.57593 q^{8} +6.67802 q^{9} +5.95117 q^{11} -9.75773 q^{12} +2.42350 q^{13} -0.435060 q^{16} -15.1351 q^{18} -13.4877 q^{22} +8.01359 q^{24} -5.49262 q^{26} -11.4422 q^{27} +6.13788 q^{32} -18.5138 q^{33} +20.9461 q^{36} +0.802795 q^{37} -7.53939 q^{39} +18.6663 q^{44} +1.35345 q^{48} -7.00000 q^{49} +7.60148 q^{52} -13.2466 q^{53} +25.9325 q^{54} +15.5804 q^{61} -13.0408 q^{64} +41.9597 q^{66} +16.0219 q^{67} -17.2021 q^{72} -1.81946 q^{74} +17.0873 q^{78} +15.5619 q^{81} -15.3298 q^{88} -19.0946 q^{96} +19.4685 q^{97} +15.8648 q^{98} +39.7420 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.26640	−1.60259	−0.801294	−	0.598271i	$-0.795855\pi$
$2$	−2.26640	−1.60259	−0.801294	+	0.598271i	$0.795855\pi$
$3$	−3.11095	−1.79611	−0.898055	−	0.439884i	$-0.855020\pi$
$3$	−3.11095	−1.79611	−0.898055	+	0.439884i	$0.855020\pi$
$4$	3.13657	1.56829
$5$	0	0
$5$	0	0
$6$	7.05066	2.87842
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	−2.57593	−0.910728
$9$	6.67802	2.22601
$10$	0	0
$11$	5.95117	1.79434	0.897172	−	0.441680i	$-0.145618\pi$
$11$	5.95117	1.79434	0.897172	+	0.441680i	$0.145618\pi$
$12$	−9.75773	−2.81681
$13$	2.42350	0.672158	0.336079	−	0.941834i	$-0.390899\pi$
$13$	2.42350	0.672158	0.336079	+	0.941834i	$0.390899\pi$
$14$	0	0
$15$	0	0
$16$	−0.435060	−0.108765
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	−15.1351	−3.56737
$19$	0	0
$19$	0	0
$20$	0	0
$21$	0	0
$22$	−13.4877	−2.87559
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	8.01359	1.63577
$25$	0	0
$26$	−5.49262	−1.07719
$27$	−11.4422	−2.20204
$28$	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	$0.5\pi$
$32$	6.13788	1.08503
$33$	−18.5138	−3.22284
$34$	0	0
$35$	0	0
$36$	20.9461	3.49102
$37$	0.802795	0.131979	0.0659893	−	0.997820i	$-0.478980\pi$
$37$	0.802795	0.131979	0.0659893	+	0.997820i	$0.478980\pi$
$38$	0	0
$39$	−7.53939	−1.20727
$40$	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	$0.5\pi$
$44$	18.6663	2.81405
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	1.35345	0.195354
$49$	−7.00000	−1.00000
$50$	0	0
$51$	0	0
$52$	7.60148	1.05414
$53$	−13.2466	−1.81956	−0.909781	−	0.415090i	$-0.863750\pi$
$53$	−13.2466	−1.81956	−0.909781	+	0.415090i	$0.863750\pi$
$54$	25.9325	3.52897
$55$	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
$61$	15.5804	1.99486	0.997430	−	0.0716414i	$-0.0228237\pi$
$61$	15.5804	1.99486	0.997430	+	0.0716414i	$0.0228237\pi$
$62$	0	0
$63$	0	0
$64$	−13.0408	−1.63010
$65$	0	0
$66$	41.9597	5.16488
$67$	16.0219	1.95739	0.978694	−	0.205326i	$-0.0658256\pi$
$67$	16.0219	1.95739	0.978694	+	0.205326i	$0.0658256\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	−17.2021	−2.02729
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	−1.81946	−0.211507
$75$	0	0
$76$	0	0
$77$	0	0
$78$	17.0873	1.93475
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	15.5619	1.72910
$82$	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
$88$	−15.3298	−1.63416
$89$	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
$96$	−19.0946	−1.94884
$97$	19.4685	1.97673	0.988364	−	0.152109i	$-0.0486063\pi$
$97$	19.4685	1.97673	0.988364	+	0.152109i	$0.0486063\pi$
$98$	15.8648	1.60259
$99$	39.7420	3.99423
$100$	0	0
$101$	0.868264	0.0863955	0.0431977	−	0.999067i	$-0.486245\pi$
$101$	0.868264	0.0863955	0.0431977	+	0.999067i	$0.486245\pi$
$102$	0	0
$103$	−11.1749	−1.10110	−0.550548	−	0.834803i	$-0.685581\pi$
$103$	−11.1749	−1.10110	−0.550548	+	0.834803i	$0.685581\pi$
$104$	−6.24276	−0.612153
$105$	0	0
$106$	30.0221	2.91601
$107$	11.8033	1.14107	0.570534	−	0.821274i	$-0.306736\pi$
$107$	11.8033	1.14107	0.570534	+	0.821274i	$0.306736\pi$
$108$	−35.8892	−3.45344
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	−2.49746	−0.237048
$112$	0	0
$113$	16.3361	1.53677	0.768386	−	0.639987i	$-0.221060\pi$
$113$	16.3361	1.53677	0.768386	+	0.639987i	$0.221060\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	16.1842	1.49623
$118$	0	0
$119$	0	0
$120$	0	0
$121$	24.4164	2.21967
$122$	−35.3113	−3.19694
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−10.9384	−0.970630	−0.485315	−	0.874339i	$-0.661295\pi$
$127$	−10.9384	−0.970630	−0.485315	+	0.874339i	$0.661295\pi$
$128$	17.2798	1.52734
$129$	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	$0.324356\pi$
$132$	−58.0699	−5.05433
$133$	0	0
$134$	−36.3121	−3.13688
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	−8.76093	−0.743092	−0.371546	−	0.928414i	$-0.621172\pi$
$139$	−8.76093	−0.743092	−0.371546	+	0.928414i	$0.621172\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	14.4227	1.20608
$144$	−2.90534	−0.242112
$145$	0	0
$146$	0	0
$147$	21.7767	1.79611
$148$	2.51802	0.206980
$149$	−22.9364	−1.87902	−0.939512	−	0.342516i	$-0.888721\pi$
$149$	−22.9364	−1.87902	−0.939512	+	0.342516i	$0.888721\pi$
$150$	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
$156$	−23.6478	−1.89334
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	41.2096	3.26813
$160$	0	0
$161$	0	0
$162$	−35.2696	−2.77104
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.50536	−0.116488	−0.0582442	−	0.998302i	$-0.518550\pi$
$167$	−1.50536	−0.116488	−0.0582442	+	0.998302i	$0.518550\pi$
$168$	0	0
$169$	−7.12665	−0.548204
$170$	0	0
$171$	0	0
$172$	0	0
$173$	25.4017	1.93126	0.965628	−	0.259928i	$-0.0836990\pi$
$173$	25.4017	1.93126	0.965628	+	0.259928i	$0.0836990\pi$
$174$	0	0
$175$	0	0
$176$	−2.58911	−0.195162
$177$	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
$183$	−48.4698	−3.58299
$184$	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	$-0.395109\pi$
$191$	8.94427	0.647185	0.323592	+	0.946197i	$0.395109\pi$
$192$	40.5692	2.92783
$193$	−17.8629	−1.28580	−0.642901	−	0.765950i	$-0.722269\pi$
$193$	−17.8629	−1.28580	−0.642901	+	0.765950i	$0.722269\pi$
$194$	−44.1234	−3.16788
$195$	0	0
$196$	−21.9560	−1.56829
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	−90.0714	−6.40110
$199$	26.8328	1.90213	0.951064	−	0.308994i	$-0.0999924\pi$
$199$	26.8328	1.90213	0.951064	+	0.308994i	$0.0999924\pi$
$200$	0	0
$201$	−49.8434	−3.51568
$202$	−1.96783	−0.138456
$203$	0	0
$204$	0	0
$205$	0	0
$206$	25.3268	1.76460
$207$	0	0
$208$	−1.05437	−0.0731072
$209$	0	0
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	−41.5489	−2.85359
$213$	0	0
$214$	−26.7510	−1.82866
$215$	0	0
$216$	29.4742	2.00546
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	5.66024	0.379890
$223$	7.72727	0.517456	0.258728	−	0.965950i	$-0.416697\pi$
$223$	7.72727	0.517456	0.258728	+	0.965950i	$0.416697\pi$
$224$	0	0
$225$	0	0
$226$	−37.0242	−2.46281
$227$	29.9345	1.98682	0.993411	−	0.114602i	$-0.0365594\pi$
$227$	29.9345	1.98682	0.993411	+	0.114602i	$0.0365594\pi$
$228$	0	0
$229$	−6.48779	−0.428725	−0.214362	−	0.976754i	$-0.568767\pi$
$229$	−6.48779	−0.428725	−0.214362	+	0.976754i	$0.568767\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	−36.6799	−2.39784
$235$	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	$0.5\pi$
$242$	−55.3374	−3.55722
$243$	−14.0860	−0.903615
$244$	48.8689	3.12851
$245$	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	$-0.690953\pi$
$251$	−17.8885	−1.12911	−0.564557	+	0.825394i	$0.690953\pi$
$252$	0	0
$253$	0	0
$254$	24.7909	1.55552
$255$	0	0
$256$	−13.0815	−0.817596
$257$	−31.9123	−1.99064	−0.995318	−	0.0966558i	$-0.969185\pi$
$257$	−31.9123	−1.99064	−0.995318	+	0.0966558i	$0.969185\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	−27.1968	−1.68022
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	47.6902	2.93513
$265$	0	0
$266$	0	0
$267$	0	0
$268$	50.2539	3.06974
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	−22.3998	−1.36069	−0.680345	−	0.732892i	$-0.738170\pi$
$271$	−22.3998	−1.36069	−0.680345	+	0.732892i	$0.738170\pi$
$272$	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	$0.5\pi$
$278$	19.8558	1.19087
$279$	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
$286$	−32.6875	−1.93285
$287$	0	0
$288$	40.9889	2.41529
$289$	−17.0000	−1.00000
$290$	0	0
$291$	−60.5656	−3.55042
$292$	0	0
$293$	24.0848	1.40705	0.703525	−	0.710670i	$-0.251608\pi$
$293$	24.0848	1.40705	0.703525	+	0.710670i	$0.251608\pi$
$294$	−49.3547	−2.87842
$295$	0	0
$296$	−2.06794	−0.120197
$297$	−68.0942	−3.95123
$298$	51.9831	3.01130
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−2.70113	−0.155176
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−25.7159	−1.46768	−0.733842	−	0.679320i	$-0.762275\pi$
$307$	−25.7159	−1.46768	−0.733842	+	0.679320i	$0.762275\pi$
$308$	0	0
$309$	34.7646	1.97769
$310$	0	0
$311$	16.1170	0.913910	0.456955	−	0.889490i	$-0.348940\pi$
$311$	16.1170	0.913910	0.456955	+	0.889490i	$0.348940\pi$
$312$	19.4209	1.09949
$313$	0	0		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	10.8607	0.609998	0.304999	−	0.952353i	$-0.401344\pi$
$317$	10.8607	0.609998	0.304999	+	0.952353i	$0.401344\pi$
$318$	−93.3974	−5.23746
$319$	0	0
$320$	0	0
$321$	−36.7195	−2.04948
$322$	0	0
$323$	0	0
$324$	48.8111	2.71173
$325$	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
$333$	5.36108	0.293786
$334$	3.41175	0.186683
$335$	0	0
$336$	0	0
$337$	24.7733	1.34949	0.674744	−	0.738052i	$-0.264254\pi$
$337$	24.7733	1.34949	0.674744	+	0.738052i	$0.264254\pi$
$338$	16.1518	0.878545
$339$	−50.8208	−2.76021
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	−57.5704	−3.09501
$347$	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	$-0.616910\pi$
$349$	−13.4164	−0.718164	−0.359082	+	0.933306i	$0.616910\pi$
$350$	0	0
$351$	−27.7301	−1.48012
$352$	36.5275	1.94692
$353$	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
$359$	−28.0193	−1.47880	−0.739401	−	0.673265i	$-0.764891\pi$
$359$	−28.0193	−1.47880	−0.739401	+	0.673265i	$0.764891\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	−75.9583	−3.98678
$364$	0	0
$365$	0	0
$366$	109.852	5.74205
$367$	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
$373$	−38.1342	−1.97452	−0.987258	−	0.159130i	$-0.949131\pi$
$373$	−38.1342	−1.97452	−0.987258	+	0.159130i	$0.949131\pi$
$374$	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
$381$	34.0290	1.74336
$382$	−20.2713	−1.03717
$383$	39.0001	1.99281	0.996406	−	0.0847033i	$-0.0269942\pi$
$383$	39.0001	1.99281	0.996406	+	0.0847033i	$0.0269942\pi$
$384$	−53.7568	−2.74326
$385$	0	0
$386$	40.4845	2.06061
$387$	0	0
$388$	61.0644	3.10007
$389$	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
$392$	18.0315	0.910728
$393$	−37.3314	−1.88312
$394$	0	0
$395$	0	0
$396$	124.654	6.26409
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	−60.8139	−3.04833
$399$	0	0
$400$	0	0
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	112.965	5.63419
$403$	0	0
$404$	2.72337	0.135493
$405$	0	0
$406$	0	0
$407$	4.77757	0.236815
$408$	0	0
$409$	0	0		−	1.00000i	$-0.5\pi$
$409$	0	0			1.00000i	$0.5\pi$
$410$	0	0
$411$	0	0
$412$	−35.0509	−1.72683
$413$	0	0
$414$	0	0
$415$	0	0
$416$	14.8751	0.729314
$417$	27.2548	1.33467
$418$	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
$424$	34.1223	1.65713
$425$	0	0
$426$	0	0
$427$	0	0
$428$	37.0219	1.78952
$429$	−44.8682	−2.16626
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	4.97802	0.239505
$433$	−39.3143	−1.88932	−0.944662	−	0.328044i	$-0.893611\pi$
$433$	−39.3143	−1.88932	−0.944662	+	0.328044i	$0.893611\pi$
$434$	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
$441$	−46.7462	−2.22601
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	−7.83345	−0.371759
$445$	0	0
$446$	−17.5131	−0.829269
$447$	71.3541	3.37493
$448$	0	0
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	0	0
$452$	51.2394	2.41010
$453$	0	0
$454$	−67.8436	−3.18406
$455$	0	0
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	14.7039	0.687069
$459$	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	$0.5\pi$
$468$	50.7629	2.34651
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−88.4611	−4.05036
$478$	−54.3936	−2.48791
$479$	−7.68770	−0.351260	−0.175630	−	0.984456i	$-0.556196\pi$
$479$	−7.68770	−0.351260	−0.175630	+	0.984456i	$0.556196\pi$
$480$	0	0
$481$	1.94557	0.0887105
$482$	0	0
$483$	0	0
$484$	76.5838	3.48108
$485$	0	0
$486$	31.9245	1.44812
$487$	−1.48091	−0.0671063	−0.0335531	−	0.999437i	$-0.510682\pi$
$487$	−1.48091	−0.0671063	−0.0335531	+	0.999437i	$0.510682\pi$
$488$	−40.1339	−1.81678
$489$	0	0
$490$	0	0
$491$	35.7771	1.61460	0.807299	−	0.590143i	$-0.200929\pi$
$491$	35.7771	1.61460	0.807299	+	0.590143i	$0.200929\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−44.4679	−1.99066	−0.995329	−	0.0965389i	$-0.969223\pi$
$499$	−44.4679	−1.99066	−0.995329	+	0.0965389i	$0.969223\pi$
$500$	0	0
$501$	4.68311	0.209226
$502$	40.5426	1.80951
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	22.1707	0.984634
$508$	−34.3092	−1.52223
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	−4.91171	−0.217069
$513$	0	0
$514$	72.3261	3.19017
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−79.0235	−3.46875
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	−12.5440	−0.548512	−0.274256	−	0.961657i	$-0.588432\pi$
$523$	−12.5440	−0.548512	−0.274256	+	0.961657i	$0.588432\pi$
$524$	37.6389	1.64426
$525$	0	0
$526$	0	0
$527$	0	0
$528$	8.05461	0.350532
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	−41.2713	−1.78265
$537$	0	0
$538$	0	0
$539$	−41.6582	−1.79434
$540$	0	0
$541$	37.6485	1.61864	0.809318	−	0.587371i	$-0.199837\pi$
$541$	37.6485	1.61864	0.809318	+	0.587371i	$0.199837\pi$
$542$	50.7669	2.18063
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−31.2098	−1.33443	−0.667216	−	0.744864i	$-0.732514\pi$
$547$	−31.2098	−1.33443	−0.667216	+	0.744864i	$0.732514\pi$
$548$	0	0
$549$	104.046	4.44058
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	−27.4793	−1.16538
$557$	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
$563$	6.32213	0.266446	0.133223	−	0.991086i	$-0.457467\pi$
$563$	6.32213	0.266446	0.133223	+	0.991086i	$0.457467\pi$
$564$	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
$571$	−26.9461	−1.12766	−0.563829	−	0.825891i	$-0.690672\pi$
$571$	−26.9461	−1.12766	−0.563829	+	0.825891i	$0.690672\pi$
$572$	45.2377	1.89148
$573$	−27.8252	−1.16241
$574$	0	0
$575$	0	0
$576$	−87.0865	−3.62861
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	38.5288	1.60259
$579$	55.5707	2.30944
$580$	0	0
$581$	0	0
$582$	137.266	5.68986
$583$	−78.8328	−3.26492
$584$	0	0
$585$	0	0
$586$	−54.5859	−2.25492
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	68.3041	2.81681
$589$	0	0
$590$	0	0
$591$	0	0
$592$	−0.349264	−0.0143547
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	154.329	6.33219
$595$	0	0
$596$	−71.9417	−2.94685
$597$	−83.4756	−3.41643
$598$	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
$603$	106.995	4.35716
$604$	0	0
$605$	0	0
$606$	6.12184	0.248683
$607$	43.2187	1.75419	0.877097	−	0.480314i	$-0.159477\pi$
$607$	43.2187	1.75419	0.877097	+	0.480314i	$0.159477\pi$
$608$	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	$0.5\pi$
$614$	58.2825	2.35209
$615$	0	0
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	−78.7905	−3.16942
$619$	35.3754	1.42186	0.710928	−	0.703265i	$-0.248275\pi$
$619$	35.3754	1.42186	0.710928	+	0.703265i	$0.248275\pi$
$620$	0	0
$621$	0	0
$622$	−36.5275	−1.46462
$623$	0	0
$624$	3.28009	0.131309
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	49.0142	1.95123	0.975613	−	0.219499i	$-0.0704421\pi$
$631$	49.0142	1.95123	0.975613	+	0.219499i	$0.0704421\pi$
$632$	0	0
$633$	0	0
$634$	−24.6147	−0.977575
$635$	0	0
$636$	129.257	5.12536
$637$	−16.9645	−0.672158
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	83.2211	3.28448
$643$	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	$0.5\pi$
$648$	−40.0864	−1.57474
$649$	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
$666$	−12.1504	−0.470817
$667$	0	0
$668$	−4.72168	−0.182687
$669$	−24.0392	−0.929408
$670$	0	0
$671$	92.7214	3.57947
$672$	0	0
$673$	−44.1613	−1.70229	−0.851147	−	0.524928i	$-0.824092\pi$
$673$	−44.1613	−1.70229	−0.851147	+	0.524928i	$0.824092\pi$
$674$	−56.1462	−2.16267
$675$	0	0
$676$	−22.3533	−0.859740
$677$	−50.5780	−1.94387	−0.971936	−	0.235246i	$-0.924410\pi$
$677$	−50.5780	−1.94387	−0.971936	+	0.235246i	$0.924410\pi$
$678$	115.180	4.42348
$679$	0	0
$680$	0	0
$681$	−93.1248	−3.56855
$682$	0	0
$683$	43.6536	1.67036	0.835179	−	0.549979i	$-0.185364\pi$
$683$	43.6536	1.67036	0.835179	+	0.549979i	$0.185364\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	20.1832	0.770037
$688$	0	0
$689$	−32.1031	−1.22303
$690$	0	0
$691$	0.331647	0.0126165	0.00630823	−	0.999980i	$-0.497992\pi$
$691$	0.331647	0.0126165	0.00630823	+	0.999980i	$0.497992\pi$
$692$	79.6743	3.02876
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	30.4070	1.15092
$699$	0	0
$700$	0	0
$701$	−21.1999	−0.800709	−0.400354	−	0.916360i	$-0.631113\pi$
$701$	−21.1999	−0.800709	−0.400354	+	0.916360i	$0.631113\pi$
$702$	62.8474	2.37202
$703$	0	0
$704$	−77.6078	−2.92495
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	40.2492	1.51159	0.755796	−	0.654808i	$-0.227250\pi$
$709$	40.2492	1.51159	0.755796	+	0.654808i	$0.227250\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−74.6629	−2.78834
$718$	63.5030	2.36991
$719$	51.8240	1.93271	0.966354	−	0.257214i	$-0.0828047\pi$
$719$	51.8240	1.93271	0.966354	+	0.257214i	$0.0828047\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	172.152	6.38916
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	−2.86503	−0.106112
$730$	0	0
$731$	0	0
$732$	−152.029	−5.61915
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	95.3491	3.51223
$738$	0	0
$739$	−53.6656	−1.97412	−0.987061	−	0.160345i	$-0.948739\pi$
$739$	−53.6656	−1.97412	−0.987061	+	0.160345i	$0.948739\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	54.4918	1.99911	0.999555	−	0.0298377i	$-0.00949905\pi$
$743$	54.4918	1.99911	0.999555	+	0.0298377i	$0.00949905\pi$
$744$	0	0
$745$	0	0
$746$	86.4274	3.16433
$747$	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
$753$	55.6504	2.02801
$754$	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
$761$	−9.29755	−0.337036	−0.168518	−	0.985699i	$-0.553898\pi$
$761$	−9.29755	−0.337036	−0.168518	+	0.985699i	$0.553898\pi$
$762$	−77.1233	−2.79388
$763$	0	0
$764$	28.0544	1.01497
$765$	0	0
$766$	−88.3899	−3.19366
$767$	0	0
$768$	40.6960	1.46849
$769$	29.2192	1.05367	0.526836	−	0.849967i	$-0.323378\pi$
$769$	29.2192	1.05367	0.526836	+	0.849967i	$0.323378\pi$
$770$	0	0
$771$	99.2777	3.57540
$772$	−56.0283	−2.01650
$773$	28.9919	1.04277	0.521383	−	0.853323i	$-0.325416\pi$
$773$	28.9919	1.04277	0.521383	+	0.853323i	$0.325416\pi$
$774$	0	0
$775$	0	0
$776$	−50.1495	−1.80026
$777$	0	0
$778$	−13.5984	−0.487526
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	3.04542	0.108765
$785$	0	0
$786$	84.6080	3.01787
$787$	49.8755	1.77787	0.888934	−	0.458035i	$-0.151447\pi$
$787$	49.8755	1.77787	0.888934	+	0.458035i	$0.151447\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	−102.373	−3.63765
$793$	37.7590	1.34086
$794$	0	0
$795$	0	0
$796$	84.1631	2.98308
$797$	−28.9016	−1.02375	−0.511873	−	0.859061i	$-0.671048\pi$
$797$	−28.9016	−1.02375	−0.511873	+	0.859061i	$0.671048\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	−156.337	−5.51359
$805$	0	0
$806$	0	0
$807$	0	0
$808$	−2.23658	−0.0786828
$809$	22.3607	0.786160	0.393080	−	0.919504i	$-0.371410\pi$
$809$	22.3607	0.786160	0.393080	+	0.919504i	$0.371410\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	69.6847	2.44395
$814$	−10.8279	−0.379517
$815$	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	$0.5\pi$
$824$	28.7857	1.00280
$825$	0	0
$826$	0	0
$827$	12.3436	0.429228	0.214614	−	0.976699i	$-0.431151\pi$
$827$	12.3436	0.429228	0.214614	+	0.976699i	$0.431151\pi$
$828$	0	0
$829$	0	0		−	1.00000i	$-0.5\pi$
$829$	0	0			1.00000i	$0.5\pi$
$830$	0	0
$831$	0	0
$832$	−31.6043	−1.09568
$833$	0	0
$834$	−61.7704	−2.13893
$835$	0	0
$836$	0	0
$837$	0	0
$838$	−81.5904	−2.81849
$839$	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
$848$	5.76306	0.197904
$849$	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
$856$	−30.4044	−1.03920
$857$	−10.2359	−0.349651	−0.174825	−	0.984599i	$-0.555936\pi$
$857$	−10.2359	−0.349651	−0.174825	+	0.984599i	$0.555936\pi$
$858$	101.689	3.47162
$859$	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
$863$	38.8368	1.32202	0.661010	−	0.750377i	$-0.270128\pi$
$863$	38.8368	1.32202	0.661010	+	0.750377i	$0.270128\pi$
$864$	−70.2305	−2.38929
$865$	0	0
$866$	89.1020	3.02781
$867$	52.8862	1.79611
$868$	0	0
$869$	0	0
$870$	0	0
$871$	38.8291	1.31567
$872$	0	0
$873$	130.011	4.40021
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−41.3454	−1.39614	−0.698068	−	0.716032i	$-0.745956\pi$
$877$	−41.3454	−1.39614	−0.698068	+	0.716032i	$0.745956\pi$
$878$	0	0
$879$	−74.9267	−2.52722
$880$	0	0
$881$	−58.6434	−1.97575	−0.987874	−	0.155261i	$-0.950378\pi$
$881$	−58.6434	−1.97575	−0.987874	+	0.155261i	$0.950378\pi$
$882$	105.946	3.56737
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	57.1313	1.91828	0.959141	−	0.282929i	$-0.0913060\pi$
$887$	57.1313	1.91828	0.959141	+	0.282929i	$0.0913060\pi$
$888$	6.43327	0.215886
$889$	0	0
$890$	0	0
$891$	92.6117	3.10261
$892$	24.2371	0.811519
$893$	0	0
$894$	−161.717	−5.40862
$895$	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
$904$	−42.0806	−1.39958
$905$	0	0
$906$	0	0
$907$	57.7597	1.91788	0.958940	−	0.283610i	$-0.0915320\pi$
$907$	57.7597	1.91788	0.958940	+	0.283610i	$0.0915320\pi$
$908$	93.8917	3.11591
$909$	5.79829	0.192317
$910$	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
$916$	−20.3494	−0.672363
$917$	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
$921$	80.0009	2.63612
$922$	−40.7952	−1.34352
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−74.6263	−2.45105
$928$	0	0
$929$	31.3050	1.02708	0.513541	−	0.858065i	$-0.328333\pi$
$929$	31.3050	1.02708	0.513541	+	0.858065i	$0.328333\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−50.1392	−1.64148
$934$	0	0
$935$	0	0
$936$	−41.6893	−1.36266
$937$	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
$951$	−33.7871	−1.09562
$952$	0	0
$953$	−47.5673	−1.54086	−0.770428	−	0.637527i	$-0.779957\pi$
$953$	−47.5673	−1.54086	−0.770428	+	0.637527i	$0.779957\pi$
$954$	200.488	6.49105
$955$	0	0
$956$	75.2777	2.43466
$957$	0	0
$958$	17.4234	0.562925
$959$	0	0
$960$	0	0
$961$	−31.0000	−1.00000
$962$	−4.40945	−0.142166
$963$	78.8227	2.54003
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	−62.8949	−2.02152
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	−44.1817	−1.41713
$973$	0	0
$974$	3.35633	0.107544
$975$	0	0
$976$	−6.77839	−0.216971
$977$	30.8771	0.987846	0.493923	−	0.869506i	$-0.335562\pi$
$977$	30.8771	0.987846	0.493923	+	0.869506i	$0.335562\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	−81.0852	−2.58753
$983$	−44.4755	−1.41855	−0.709274	−	0.704933i	$-0.750977\pi$
$983$	−44.4755	−1.41855	−0.709274	+	0.704933i	$0.750977\pi$
$984$	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	$0.5\pi$
$998$	100.782	3.19020
$999$	−9.18571	−0.290623

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9025.2.a.cb.1.2		8
5.2	odd	4		1805.2.b.h.1084.2	✓	8
5.3	odd	4		1805.2.b.h.1084.7	yes	8
5.4	even	2	inner	9025.2.a.cb.1.7		8
19.18	odd	2	inner	9025.2.a.cb.1.7		8
95.18	even	4		1805.2.b.h.1084.2	✓	8
95.37	even	4		1805.2.b.h.1084.7	yes	8
95.94	odd	2	CM	9025.2.a.cb.1.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1805.2.b.h.1084.2	✓	8	5.2	odd	4
1805.2.b.h.1084.2	✓	8	95.18	even	4
1805.2.b.h.1084.7	yes	8	5.3	odd	4
1805.2.b.h.1084.7	yes	8	95.37	even	4
9025.2.a.cb.1.2		8	1.1	even	1	trivial
9025.2.a.cb.1.2		8	95.94	odd	2	CM
9025.2.a.cb.1.7		8	5.4	even	2	inner
9025.2.a.cb.1.7		8	19.18	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-2.26640 q^{2} -3.11095 q^{3} +3.13657 q^{4} +7.05066 q^{6} -2.57593 q^{8} +6.67802 q^{9} +O(q^{10})\)	\(q-2.26640 q^{2} -3.11095 q^{3} +3.13657 q^{4} +7.05066 q^{6} -2.57593 q^{8} +6.67802 q^{9} +5.95117 q^{11} -9.75773 q^{12} +2.42350 q^{13} -0.435060 q^{16} -15.1351 q^{18} -13.4877 q^{22} +8.01359 q^{24} -5.49262 q^{26} -11.4422 q^{27} +6.13788 q^{32} -18.5138 q^{33} +20.9461 q^{36} +0.802795 q^{37} -7.53939 q^{39} +18.6663 q^{44} +1.35345 q^{48} -7.00000 q^{49} +7.60148 q^{52} -13.2466 q^{53} +25.9325 q^{54} +15.5804 q^{61} -13.0408 q^{64} +41.9597 q^{66} +16.0219 q^{67} -17.2021 q^{72} -1.81946 q^{74} +17.0873 q^{78} +15.5619 q^{81} -15.3298 q^{88} -19.0946 q^{96} +19.4685 q^{97} +15.8648 q^{98} +39.7420 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{4} + 24 q^{9}+O(q^{10}) \) 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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