LMFDB - Embedded newform 9025.2.a.cb.1.3

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			1.3
Root			$-1.69217$ 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)

$f(q)$	$=$	$q-1.69217 q^{2} +1.52380 q^{3} +0.863428 q^{4} -2.57853 q^{6} +1.92327 q^{8} -0.678024 q^{9} +O(q^{10})$	$q-1.69217 q^{2} +1.52380 q^{3} +0.863428 q^{4} -2.57853 q^{6} +1.92327 q^{8} -0.678024 q^{9} -5.95117 q^{11} +1.31569 q^{12} -6.79166 q^{13} -4.98135 q^{16} +1.14733 q^{18} +10.0704 q^{22} +2.93068 q^{24} +11.4926 q^{26} -5.60458 q^{27} +4.58273 q^{32} -9.06841 q^{33} -0.585425 q^{36} +12.1390 q^{37} -10.3492 q^{39} -5.13841 q^{44} -7.59059 q^{48} -7.00000 q^{49} -5.86411 q^{52} -6.04380 q^{53} +9.48389 q^{54} -15.5804 q^{61} +2.20795 q^{64} +15.3453 q^{66} +3.36134 q^{67} -1.30402 q^{72} -20.5412 q^{74} +17.5125 q^{78} -6.50621 q^{81} -11.4457 q^{88} +6.98318 q^{96} +2.99619 q^{97} +11.8452 q^{98} +4.03504 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.69217	−1.19654	−0.598271	−	0.801294i	$-0.704145\pi$
$2$	−1.69217	−1.19654	−0.598271	+	0.801294i	$0.704145\pi$
$3$	1.52380	0.879768	0.439884	−	0.898055i	$-0.355020\pi$
$3$	1.52380	0.879768	0.439884	+	0.898055i	$0.355020\pi$
$4$	0.863428	0.431714
$5$	0	0
$5$	0	0
$6$	−2.57853	−1.05268
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	1.92327	0.679978
$9$	−0.678024	−0.226008
$10$	0	0
$11$	−5.95117	−1.79434	−0.897172	−	0.441680i	$-0.854382\pi$
$11$	−5.95117	−1.79434	−0.897172	+	0.441680i	$0.854382\pi$
$12$	1.31569	0.379808
$13$	−6.79166	−1.88367	−0.941834	−	0.336079i	$-0.890899\pi$
$13$	−6.79166	−1.88367	−0.941834	+	0.336079i	$0.890899\pi$
$14$	0	0
$15$	0	0
$16$	−4.98135	−1.24534
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	1.14733	0.270428
$19$	0	0
$19$	0	0
$20$	0	0
$21$	0	0
$22$	10.0704	2.14701
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	2.93068	0.598223
$25$	0	0
$26$	11.4926	2.25389
$27$	−5.60458	−1.07860
$28$	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	$0.5\pi$
$32$	4.58273	0.810120
$33$	−9.06841	−1.57861
$34$	0	0
$35$	0	0
$36$	−0.585425	−0.0975709
$37$	12.1390	1.99564	0.997820	−	0.0659893i	$-0.0210203\pi$
$37$	12.1390	1.99564	0.997820	+	0.0659893i	$0.0210203\pi$
$38$	0	0
$39$	−10.3492	−1.65719
$40$	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	$0.5\pi$
$44$	−5.13841	−0.774644
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	−7.59059	−1.09561
$49$	−7.00000	−1.00000
$50$	0	0
$51$	0	0
$52$	−5.86411	−0.813206
$53$	−6.04380	−0.830179	−0.415090	−	0.909781i	$-0.636250\pi$
$53$	−6.04380	−0.830179	−0.415090	+	0.909781i	$0.636250\pi$
$54$	9.48389	1.29059
$55$	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	$-0.977176\pi$
$61$	−15.5804	−1.99486	−0.997430	+	0.0716414i	$0.977176\pi$
$62$	0	0
$63$	0	0
$64$	2.20795	0.275994
$65$	0	0
$66$	15.3453	1.88887
$67$	3.36134	0.410653	0.205326	−	0.978694i	$-0.434174\pi$
$67$	3.36134	0.410653	0.205326	+	0.978694i	$0.434174\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	−1.30402	−0.153681
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	−20.5412	−2.38787
$75$	0	0
$76$	0	0
$77$	0	0
$78$	17.5125	1.98290
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	−6.50621	−0.722912
$82$	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
$88$	−11.4457	−1.22012
$89$	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
$96$	6.98318	0.712718
$97$	2.99619	0.304217	0.152109	−	0.988364i	$-0.451394\pi$
$97$	2.99619	0.304217	0.152109	+	0.988364i	$0.451394\pi$
$98$	11.8452	1.19654
$99$	4.03504	0.405537
$100$	0	0
$101$	−0.868264	−0.0863955	−0.0431977	−	0.999067i	$-0.513755\pi$
$101$	−0.868264	−0.0863955	−0.0431977	+	0.999067i	$0.513755\pi$
$102$	0	0
$103$	−16.9447	−1.66961	−0.834803	−	0.550548i	$-0.814419\pi$
$103$	−16.9447	−1.66961	−0.834803	+	0.550548i	$0.814419\pi$
$104$	−13.0622	−1.28085
$105$	0	0
$106$	10.2271	0.993345
$107$	−16.9906	−1.64255	−0.821274	−	0.570534i	$-0.806736\pi$
$107$	−16.9906	−1.64255	−0.821274	+	0.570534i	$0.806736\pi$
$108$	−4.83915	−0.465648
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	18.4975	1.75570
$112$	0	0
$113$	−13.6063	−1.27997	−0.639987	−	0.768386i	$-0.721060\pi$
$113$	−13.6063	−1.27997	−0.639987	+	0.768386i	$0.721060\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.60491	0.425724
$118$	0	0
$119$	0	0
$120$	0	0
$121$	24.4164	2.21967
$122$	26.3646	2.38694
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−19.7066	−1.74868	−0.874339	−	0.485315i	$-0.838705\pi$
$127$	−19.7066	−1.74868	−0.874339	+	0.485315i	$0.838705\pi$
$128$	−12.9017	−1.14036
$129$	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	$0.324356\pi$
$132$	−7.82992	−0.681507
$133$	0	0
$134$	−5.68795	−0.491364
$135$	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	$-0.378828\pi$
$139$	8.76093	0.743092	0.371546	+	0.928414i	$0.378828\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	40.4183	3.37995
$144$	3.37748	0.281456
$145$	0	0
$146$	0	0
$147$	−10.6666	−0.879768
$148$	10.4812	0.861546
$149$	22.9364	1.87902	0.939512	−	0.342516i	$-0.111279\pi$
$149$	22.9364	1.87902	0.939512	+	0.342516i	$0.111279\pi$
$150$	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
$156$	−8.93575	−0.715432
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	−9.20955	−0.730365
$160$	0	0
$161$	0	0
$162$	11.0096	0.864995
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	25.8018	1.99660	0.998302	−	0.0582442i	$-0.0185502\pi$
$167$	25.8018	1.99660	0.998302	+	0.0582442i	$0.0185502\pi$
$168$	0	0
$169$	33.1266	2.54820
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−6.83765	−0.519857	−0.259928	−	0.965628i	$-0.583699\pi$
$173$	−6.83765	−0.519857	−0.259928	+	0.965628i	$0.583699\pi$
$174$	0	0
$175$	0	0
$176$	29.6448	2.23456
$177$	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
$183$	−23.7414	−1.75501
$184$	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	$0.395109\pi$
$192$	3.36448	0.242810
$193$	21.2818	1.53190	0.765950	−	0.642901i	$-0.222269\pi$
$193$	21.2818	1.53190	0.765950	+	0.642901i	$0.222269\pi$
$194$	−5.07005	−0.364009
$195$	0	0
$196$	−6.04400	−0.431714
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	−6.82796	−0.485242
$199$	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
$201$	5.12202	0.361279
$202$	1.46925	0.103376
$203$	0	0
$204$	0	0
$205$	0	0
$206$	28.6732	1.99776
$207$	0	0
$208$	33.8316	2.34580
$209$	0	0
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	−5.21838	−0.358400
$213$	0	0
$214$	28.7510	1.96538
$215$	0	0
$216$	−10.7791	−0.733427
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	−31.3008	−2.10077
$223$	−28.8494	−1.93190	−0.965950	−	0.258728i	$-0.916697\pi$
$223$	−28.8494	−1.93190	−0.965950	+	0.258728i	$0.916697\pi$
$224$	0	0
$225$	0	0
$226$	23.0242	1.53154
$227$	−3.45331	−0.229205	−0.114602	−	0.993411i	$-0.536559\pi$
$227$	−3.45331	−0.229205	−0.114602	+	0.993411i	$0.536559\pi$
$228$	0	0
$229$	6.48779	0.428725	0.214362	−	0.976754i	$-0.431233\pi$
$229$	6.48779	0.428725	0.214362	+	0.976754i	$0.431233\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	−7.79228	−0.509397
$235$	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	$0.5\pi$
$242$	−41.3166	−2.65593
$243$	6.89957	0.442608
$244$	−13.4525	−0.861209
$245$	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
$254$	33.3469	2.09237
$255$	0	0
$256$	17.4159	1.08849
$257$	3.09902	0.193312	0.0966558	−	0.995318i	$-0.469185\pi$
$257$	3.09902	0.193312	0.0966558	+	0.995318i	$0.469185\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	−20.3060	−1.25451
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	−17.4410	−1.07342
$265$	0	0
$266$	0	0
$267$	0	0
$268$	2.90227	0.177285
$269$	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	$-0.261830\pi$
$271$	22.3998	1.36069	0.680345	+	0.732892i	$0.261830\pi$
$272$	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	$0.5\pi$
$278$	−14.8250	−0.889142
$279$	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
$286$	−68.3945	−4.04425
$287$	0	0
$288$	−3.10720	−0.183094
$289$	−17.0000	−1.00000
$290$	0	0
$291$	4.56560	0.267640
$292$	0	0
$293$	−24.3294	−1.42134	−0.710670	−	0.703525i	$-0.751608\pi$
$293$	−24.3294	−1.42134	−0.710670	+	0.703525i	$0.751608\pi$
$294$	18.0497	1.05268
$295$	0	0
$296$	23.3466	1.35699
$297$	33.3538	1.93539
$298$	−38.8122	−2.24833
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−1.32306	−0.0760080
$304$	0	0
$305$	0	0
$306$	0	0
$307$	23.8053	1.35864	0.679320	−	0.733842i	$-0.262275\pi$
$307$	23.8053	1.35864	0.679320	+	0.733842i	$0.262275\pi$
$308$	0	0
$309$	−25.8203	−1.46887
$310$	0	0
$311$	−16.1170	−0.913910	−0.456955	−	0.889490i	$-0.651060\pi$
$311$	−16.1170	−0.913910	−0.456955	+	0.889490i	$0.651060\pi$
$312$	−19.9042	−1.12685
$313$	0	0		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	33.9123	1.90471	0.952353	−	0.304999i	$-0.0986561\pi$
$317$	33.9123	1.90471	0.952353	+	0.304999i	$0.0986561\pi$
$318$	15.5841	0.873913
$319$	0	0
$320$	0	0
$321$	−25.8904	−1.44506
$322$	0	0
$323$	0	0
$324$	−5.61764	−0.312091
$325$	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
$333$	−8.23054	−0.451031
$334$	−43.6610	−2.38902
$335$	0	0
$336$	0	0
$337$	27.0977	1.47610	0.738052	−	0.674744i	$-0.235746\pi$
$337$	27.0977	1.47610	0.738052	+	0.674744i	$0.235746\pi$
$338$	−56.0558	−3.04903
$339$	−20.7333	−1.12608
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	11.5704	0.622031
$347$	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
$351$	38.0644	2.03173
$352$	−27.2726	−1.45364
$353$	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	$-0.235109\pi$
$359$	28.0193	1.47880	0.739401	+	0.673265i	$0.235109\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	37.2058	1.95280
$364$	0	0
$365$	0	0
$366$	40.1744	2.09995
$367$	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
$373$	6.14663	0.318260	0.159130	−	0.987258i	$-0.449131\pi$
$373$	6.14663	0.318260	0.159130	+	0.987258i	$0.449131\pi$
$374$	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
$381$	−30.0290	−1.53843
$382$	−15.1352	−0.774384
$383$	3.31535	0.169407	0.0847033	−	0.996406i	$-0.473006\pi$
$383$	3.31535	0.169407	0.0847033	+	0.996406i	$0.473006\pi$
$384$	−19.6596	−1.00325
$385$	0	0
$386$	−36.0124	−1.83298
$387$	0	0
$388$	2.58699	0.131335
$389$	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
$392$	−13.4629	−0.679978
$393$	18.2856	0.922388
$394$	0	0
$395$	0	0
$396$	3.48396	0.175076
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	−45.4056	−2.27598
$399$	0	0
$400$	0	0
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	−8.66731	−0.432286
$403$	0	0
$404$	−0.749683	−0.0372981
$405$	0	0
$406$	0	0
$407$	−72.2413	−3.58087
$408$	0	0
$409$	0	0		−	1.00000i	$-0.5\pi$
$409$	0	0			1.00000i	$0.5\pi$
$410$	0	0
$411$	0	0
$412$	−14.6305	−0.720793
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−31.1244	−1.52600
$417$	13.3499	0.653749
$418$	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
$424$	−11.6238	−0.564504
$425$	0	0
$426$	0	0
$427$	0	0
$428$	−14.6702	−0.709111
$429$	61.5896	2.97357
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	27.9184	1.34322
$433$	13.6523	0.656088	0.328044	−	0.944662i	$-0.393611\pi$
$433$	13.6523	0.656088	0.328044	+	0.944662i	$0.393611\pi$
$434$	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
$441$	4.74617	0.226008
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	15.9712	0.757961
$445$	0	0
$446$	48.8180	2.31160
$447$	34.9506	1.65311
$448$	0	0
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	0	0
$452$	−11.7481	−0.552583
$453$	0	0
$454$	5.84358	0.274253
$455$	0	0
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	−10.9784	−0.512988
$459$	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	$0.5\pi$
$468$	3.97601	0.183791
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	4.09784	0.187627
$478$	−40.6120	−1.85755
$479$	7.68770	0.351260	0.175630	−	0.984456i	$-0.443804\pi$
$479$	7.68770	0.351260	0.175630	+	0.984456i	$0.443804\pi$
$480$	0	0
$481$	−82.4440	−3.75912
$482$	0	0
$483$	0	0
$484$	21.0818	0.958264
$485$	0	0
$486$	−11.6752	−0.529599
$487$	−44.1113	−1.99887	−0.999437	−	0.0335531i	$-0.989318\pi$
$487$	−44.1113	−1.99887	−0.999437	+	0.0335531i	$0.989318\pi$
$488$	−29.9652	−1.35646
$489$	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	$-0.0307772\pi$
$499$	44.4679	1.99066	0.995329	+	0.0965389i	$0.0307772\pi$
$500$	0	0
$501$	39.3169	1.75655
$502$	30.2704	1.35103
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	50.4785	2.24183
$508$	−17.0152	−0.754929
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	−3.66724	−0.162071
$513$	0	0
$514$	−5.24406	−0.231306
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−10.4192	−0.457353
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	−43.9846	−1.92331	−0.961657	−	0.274256i	$-0.911568\pi$
$523$	−43.9846	−1.92331	−0.961657	+	0.274256i	$0.911568\pi$
$524$	10.3611	0.452628
$525$	0	0
$526$	0	0
$527$	0	0
$528$	45.1729	1.96590
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	6.46476	0.279235
$537$	0	0
$538$	0	0
$539$	41.6582	1.79434
$540$	0	0
$541$	−37.6485	−1.61864	−0.809318	−	0.587371i	$-0.800163\pi$
$541$	−37.6485	−1.61864	−0.809318	+	0.587371i	$0.800163\pi$
$542$	−37.9042	−1.62812
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−34.8418	−1.48973	−0.744864	−	0.667216i	$-0.767486\pi$
$547$	−34.8418	−1.48973	−0.744864	+	0.667216i	$0.767486\pi$
$548$	0	0
$549$	10.5639	0.450855
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	7.56443	0.320803
$557$	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
$563$	47.0322	1.98217	0.991086	−	0.133223i	$-0.0425327\pi$
$563$	47.0322	1.98217	0.991086	+	0.133223i	$0.0425327\pi$
$564$	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	$-0.309328\pi$
$571$	26.9461	1.12766	0.563829	+	0.825891i	$0.309328\pi$
$572$	34.8983	1.45917
$573$	13.6293	0.569373
$574$	0	0
$575$	0	0
$576$	−1.49704	−0.0623768
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	28.7668	1.19654
$579$	32.4293	1.34772
$580$	0	0
$581$	0	0
$582$	−7.72576	−0.320243
$583$	35.9677	1.48963
$584$	0	0
$585$	0	0
$586$	41.1695	1.70069
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	−9.20986	−0.379808
$589$	0	0
$590$	0	0
$591$	0	0
$592$	−60.4686	−2.48525
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	−56.4402	−2.31577
$595$	0	0
$596$	19.8039	0.811201
$597$	40.8879	1.67343
$598$	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
$603$	−2.27907	−0.0928109
$604$	0	0
$605$	0	0
$606$	2.23884	0.0909468
$607$	23.6673	0.960628	0.480314	−	0.877097i	$-0.340523\pi$
$607$	23.6673	0.960628	0.480314	+	0.877097i	$0.340523\pi$
$608$	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	$0.5\pi$
$614$	−40.2825	−1.62567
$615$	0	0
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	43.6923	1.75756
$619$	−35.3754	−1.42186	−0.710928	−	0.703265i	$-0.751725\pi$
$619$	−35.3754	−1.42186	−0.710928	+	0.703265i	$0.751725\pi$
$620$	0	0
$621$	0	0
$622$	27.2726	1.09353
$623$	0	0
$624$	51.5527	2.06376
$625$	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	$-0.929558\pi$
$631$	−49.0142	−1.95123	−0.975613	+	0.219499i	$0.929558\pi$
$632$	0	0
$633$	0	0
$634$	−57.3853	−2.27906
$635$	0	0
$636$	−7.95179	−0.315309
$637$	47.5416	1.88367
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	43.8109	1.72908
$643$	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	$0.5\pi$
$648$	−12.5132	−0.491565
$649$	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
$666$	13.9275	0.539678
$667$	0	0
$668$	22.2780	0.861962
$669$	−43.9608	−1.69962
$670$	0	0
$671$	92.7214	3.57947
$672$	0	0
$673$	27.2356	1.04986	0.524928	−	0.851147i	$-0.324092\pi$
$673$	27.2356	1.04986	0.524928	+	0.851147i	$0.324092\pi$
$674$	−45.8538	−1.76622
$675$	0	0
$676$	28.6025	1.10010
$677$	12.2418	0.470492	0.235246	−	0.971936i	$-0.424410\pi$
$677$	12.2418	0.470492	0.235246	+	0.971936i	$0.424410\pi$
$678$	35.0843	1.34740
$679$	0	0
$680$	0	0
$681$	−5.26217	−0.201647
$682$	0	0
$683$	28.7466	1.09996	0.549979	−	0.835179i	$-0.314636\pi$
$683$	28.7466	1.09996	0.549979	+	0.835179i	$0.314636\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	9.88611	0.377178
$688$	0	0
$689$	41.0474	1.56378
$690$	0	0
$691$	−0.331647	−0.0126165	−0.00630823	−	0.999980i	$-0.502008\pi$
$691$	−0.331647	−0.0126165	−0.00630823	+	0.999980i	$0.502008\pi$
$692$	−5.90382	−0.224429
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	22.7028	0.859314
$699$	0	0
$700$	0	0
$701$	21.1999	0.800709	0.400354	−	0.916360i	$-0.368887\pi$
$701$	21.1999	0.800709	0.400354	+	0.916360i	$0.368887\pi$
$702$	−64.4114	−2.43105
$703$	0	0
$704$	−13.1399	−0.495228
$705$	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
$717$	36.5713	1.36578
$718$	−47.4134	−1.76945
$719$	−51.8240	−1.93271	−0.966354	−	0.257214i	$-0.917195\pi$
$719$	−51.8240	−1.93271	−0.966354	+	0.257214i	$0.917195\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	−62.9584	−2.33661
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	30.0322	1.11230
$730$	0	0
$731$	0	0
$732$	−20.4990	−0.757664
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−20.0039	−0.736853
$738$	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
$743$	−1.62663	−0.0596754	−0.0298377	−	0.999555i	$-0.509499\pi$
$743$	−1.62663	−0.0596754	−0.0298377	+	0.999555i	$0.509499\pi$
$744$	0	0
$745$	0	0
$746$	−10.4011	−0.380812
$747$	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
$753$	−27.2586	−0.993359
$754$	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	$-0.446102\pi$
$761$	9.29755	0.337036	0.168518	+	0.985699i	$0.446102\pi$
$762$	50.8141	1.84080
$763$	0	0
$764$	7.72273	0.279399
$765$	0	0
$766$	−5.61013	−0.202702
$767$	0	0
$768$	26.5384	0.957622
$769$	−29.2192	−1.05367	−0.526836	−	0.849967i	$-0.676622\pi$
$769$	−29.2192	−1.05367	−0.526836	+	0.849967i	$0.676622\pi$
$770$	0	0
$771$	4.72230	0.170069
$772$	18.3753	0.661342
$773$	47.4496	1.70665	0.853323	−	0.521383i	$-0.174584\pi$
$773$	47.4496	1.70665	0.853323	+	0.521383i	$0.174584\pi$
$774$	0	0
$775$	0	0
$776$	5.76248	0.206861
$777$	0	0
$778$	−10.1530	−0.364003
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	34.8694	1.24534
$785$	0	0
$786$	−30.9423	−1.10368
$787$	25.6990	0.916070	0.458035	−	0.888934i	$-0.348553\pi$
$787$	25.6990	0.916070	0.458035	+	0.888934i	$0.348553\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	7.76046	0.275756
$793$	105.817	3.75765
$794$	0	0
$795$	0	0
$796$	23.1682	0.821175
$797$	−48.5046	−1.71812	−0.859061	−	0.511873i	$-0.828952\pi$
$797$	−48.5046	−1.71812	−0.859061	+	0.511873i	$0.828952\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	4.42249	0.155969
$805$	0	0
$806$	0	0
$807$	0	0
$808$	−1.66991	−0.0587471
$809$	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
$813$	34.1329	1.19709
$814$	122.244	4.28466
$815$	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	$0.5\pi$
$824$	−32.5891	−1.13530
$825$	0	0
$826$	0	0
$827$	−56.1750	−1.95340	−0.976699	−	0.214614i	$-0.931151\pi$
$827$	−56.1750	−1.95340	−0.976699	+	0.214614i	$0.931151\pi$
$828$	0	0
$829$	0	0		−	1.00000i	$-0.5\pi$
$829$	0	0			1.00000i	$0.5\pi$
$830$	0	0
$831$	0	0
$832$	−14.9956	−0.519880
$833$	0	0
$834$	−22.5903	−0.782238
$835$	0	0
$836$	0	0
$837$	0	0
$838$	−60.9180	−2.10438
$839$	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
$848$	30.1063	1.03385
$849$	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
$856$	−32.6776	−1.11690
$857$	−57.6474	−1.96920	−0.984599	−	0.174825i	$-0.944064\pi$
$857$	−57.6474	−1.96920	−0.984599	+	0.174825i	$0.944064\pi$
$858$	−104.220	−3.55800
$859$	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
$863$	−44.0875	−1.50075	−0.750377	−	0.661010i	$-0.770128\pi$
$863$	−44.0875	−1.50075	−0.750377	+	0.661010i	$0.770128\pi$
$864$	−25.6843	−0.873798
$865$	0	0
$866$	−23.1020	−0.785037
$867$	−25.9047	−0.879768
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−22.8291	−0.773534
$872$	0	0
$873$	−2.03149	−0.0687555
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−42.4094	−1.43206	−0.716032	−	0.698068i	$-0.754044\pi$
$877$	−42.4094	−1.43206	−0.716032	+	0.698068i	$0.754044\pi$
$878$	0	0
$879$	−37.0733	−1.25045
$880$	0	0
$881$	58.6434	1.97575	0.987874	−	0.155261i	$-0.0496217\pi$
$881$	58.6434	1.97575	0.987874	+	0.155261i	$0.0496217\pi$
$882$	−8.03131	−0.270428
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	16.8527	0.565858	0.282929	−	0.959141i	$-0.408694\pi$
$887$	16.8527	0.565858	0.282929	+	0.959141i	$0.408694\pi$
$888$	35.5756	1.19384
$889$	0	0
$890$	0	0
$891$	38.7195	1.29715
$892$	−24.9094	−0.834028
$893$	0	0
$894$	−59.1422	−1.97801
$895$	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
$904$	−26.1686	−0.870355
$905$	0	0
$906$	0	0
$907$	−17.0826	−0.567219	−0.283610	−	0.958940i	$-0.591532\pi$
$907$	−17.0826	−0.567219	−0.283610	+	0.958940i	$0.591532\pi$
$908$	−2.98169	−0.0989508
$909$	0.588704	0.0195261
$910$	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
$916$	5.60173	0.185087
$917$	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
$921$	36.2746	1.19529
$922$	−30.4590	−1.00311
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	11.4889	0.377345
$928$	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
$933$	−24.5591	−0.804029
$934$	0	0
$935$	0	0
$936$	8.85649	0.289483
$937$	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
$951$	51.6757	1.67570
$952$	0	0
$953$	−39.3618	−1.27505	−0.637527	−	0.770428i	$-0.720043\pi$
$953$	−39.3618	−1.27505	−0.637527	+	0.770428i	$0.720043\pi$
$954$	−6.93423	−0.224504
$955$	0	0
$956$	20.7223	0.670206
$957$	0	0
$958$	−13.0089	−0.420297
$959$	0	0
$960$	0	0
$961$	−31.0000	−1.00000
$962$	139.509	4.49795
$963$	11.5201	0.371229
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	46.9593	1.50933
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	5.95728	0.191080
$973$	0	0
$974$	74.6437	2.39174
$975$	0	0
$976$	77.6112	2.48427
$977$	−54.3563	−1.73901	−0.869506	−	0.493923i	$-0.835562\pi$
$977$	−54.3563	−1.73901	−0.869506	+	0.493923i	$0.835562\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	−60.5408	−1.93193
$983$	44.2033	1.40987	0.704933	−	0.709274i	$-0.250977\pi$
$983$	44.2033	1.40987	0.704933	+	0.709274i	$0.250977\pi$
$984$	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	$0.5\pi$
$998$	−75.2472	−2.38191
$999$	−68.0341	−2.15250

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9025.2.a.cb.1.3		8
5.2	odd	4		1805.2.b.h.1084.3	✓	8
5.3	odd	4		1805.2.b.h.1084.6	yes	8
5.4	even	2	inner	9025.2.a.cb.1.6		8
19.18	odd	2	inner	9025.2.a.cb.1.6		8
95.18	even	4		1805.2.b.h.1084.3	✓	8
95.37	even	4		1805.2.b.h.1084.6	yes	8
95.94	odd	2	CM	9025.2.a.cb.1.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1805.2.b.h.1084.3	✓	8	5.2	odd	4
1805.2.b.h.1084.3	✓	8	95.18	even	4
1805.2.b.h.1084.6	yes	8	5.3	odd	4
1805.2.b.h.1084.6	yes	8	95.37	even	4
9025.2.a.cb.1.3		8	1.1	even	1	trivial
9025.2.a.cb.1.3		8	95.94	odd	2	CM
9025.2.a.cb.1.6		8	5.4	even	2	inner
9025.2.a.cb.1.6		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)

\(f(q)\)	\(=\)	\(q-1.69217 q^{2} +1.52380 q^{3} +0.863428 q^{4} -2.57853 q^{6} +1.92327 q^{8} -0.678024 q^{9} +O(q^{10})\)	\(q-1.69217 q^{2} +1.52380 q^{3} +0.863428 q^{4} -2.57853 q^{6} +1.92327 q^{8} -0.678024 q^{9} -5.95117 q^{11} +1.31569 q^{12} -6.79166 q^{13} -4.98135 q^{16} +1.14733 q^{18} +10.0704 q^{22} +2.93068 q^{24} +11.4926 q^{26} -5.60458 q^{27} +4.58273 q^{32} -9.06841 q^{33} -0.585425 q^{36} +12.1390 q^{37} -10.3492 q^{39} -5.13841 q^{44} -7.59059 q^{48} -7.00000 q^{49} -5.86411 q^{52} -6.04380 q^{53} +9.48389 q^{54} -15.5804 q^{61} +2.20795 q^{64} +15.3453 q^{66} +3.36134 q^{67} -1.30402 q^{72} -20.5412 q^{74} +17.5125 q^{78} -6.50621 q^{81} -11.4457 q^{88} +6.98318 q^{96} +2.99619 q^{97} +11.8452 q^{98} +4.03504 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{4} + 24 q^{9}+O(q^{10}) \) 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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