LMFDB - Embedded newform 968.2.g.a.483.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [968,2,Mod(483,968)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(968, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1]))

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

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

chi := DirichletCharacter("968.483");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 968 = 2^{3} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	968.g (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$7.72951891566$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.64000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 $ x^8 - 2x^6 + 4x^4 - 8*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4}\cdot 11^{2} $
Twist minimal:	no (minimal twist has level 88)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			483.8
Root			$1.34500 + 0.437016i$ of defining polynomial
Character	$\chi$	$=$	968.483
Dual form			968.2.g.a.483.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} +3.30803 q^{3} -2.00000 q^{4} +4.67826i q^{6} -2.82843i q^{8} +7.94305 q^{9} -6.61606 q^{12} +4.00000 q^{16} -1.04978i q^{17} +11.2332i q^{18} +8.04031i q^{19} -9.35652i q^{24} +5.00000 q^{25} +16.3517 q^{27} +5.65685i q^{32} +1.48461 q^{34} -15.8861 q^{36} -11.3707 q^{38} -2.21021i q^{41} -6.88847i q^{43} +13.2321 q^{48} -7.00000 q^{49} +7.07107i q^{50} -3.47270i q^{51} +23.1249i q^{54} +26.5976i q^{57} -3.45844 q^{59} -8.00000 q^{64} -16.3138 q^{67} +2.09956i q^{68} -22.4663i q^{72} -7.14631i q^{73} +16.5401 q^{75} -16.0806i q^{76} +30.2629 q^{81} +3.12571 q^{82} -16.2450i q^{83} +9.74177 q^{86} +0.182318 q^{89} +18.7130i q^{96} -13.0498 q^{97} -9.89949i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{3} - 16 q^{4} + 28 q^{9} + 8 q^{12} + 32 q^{16} + 40 q^{25} + 8 q^{27} + 16 q^{34} - 56 q^{36} - 24 q^{38} - 16 q^{48} - 56 q^{49} + 12 q^{59} - 64 q^{64} - 28 q^{67} - 20 q^{75} + 112 q^{81}+ \cdots - 20 q^{97}+O(q^{100}) $ 8 * q - 4 * q^3 - 16 * q^4 + 28 * q^9 + 8 * q^12 + 32 * q^16 + 40 * q^25 + 8 * q^27 + 16 * q^34 - 56 * q^36 - 24 * q^38 - 16 * q^48 - 56 * q^49 + 12 * q^59 - 64 * q^64 - 28 * q^67 - 20 * q^75 + 112 * q^81 + 32 * q^82 + 24 * q^86 - 36 * q^89 - 20 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/968\mathbb{Z}\right)^\times$.

$n$	$485$	$727$	$849$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$	1.41421i	1.00000i
$2$	1.41421i	1.00000i
$3$	3.30803	1.90989	0.954945	−	0.296781i	$-0.0959133\pi$
$3$	3.30803	1.90989	0.954945	+	0.296781i	$0.0959133\pi$
$4$	−2.00000	−1.00000
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	4.67826i	1.90989i
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	− 2.82843i	− 1.00000i
$9$	7.94305	2.64768
$10$	0	0
$11$	0	0
$11$	0	0
$12$	−6.61606	−1.90989
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	1.00000
$17$	− 1.04978i	− 0.254609i	−0.991864	−	0.127304i	$-0.959367\pi$
$17$	− 1.04978i	− 0.254609i	0.991864	−	0.127304i	$-0.0406325\pi$
$18$	11.2332i	2.64768i
$19$	8.04031i	1.84457i	0.386507	+	0.922287i	$0.373682\pi$
$19$	8.04031i	1.84457i	−0.386507	+	0.922287i	$0.626318\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	− 9.35652i	− 1.90989i
$25$	5.00000	1.00000
$26$	0	0
$27$	16.3517	3.14690
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	5.65685i	1.00000i
$33$	0	0
$34$	1.48461	0.254609
$35$	0	0
$36$	−15.8861	−2.64768
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	−11.3707	−1.84457
$39$	0	0
$40$	0	0
$41$	− 2.21021i	− 0.345177i	−0.984994	−	0.172588i	$-0.944787\pi$
$41$	− 2.21021i	− 0.345177i	0.984994	−	0.172588i	$-0.0552131\pi$
$42$	0	0
$43$	− 6.88847i	− 1.05048i	−0.850954	−	0.525241i	$-0.823975\pi$
$43$	− 6.88847i	− 1.05048i	0.850954	−	0.525241i	$-0.176025\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	13.2321	1.90989
$49$	−7.00000	−1.00000
$50$	7.07107i	1.00000i
$51$	− 3.47270i	− 0.486275i
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	23.1249i	3.14690i
$55$	0	0
$56$	0	0
$57$	26.5976i	3.52293i
$58$	0	0
$59$	−3.45844	−0.450250	−0.225125	−	0.974330i	$-0.572279\pi$
$59$	−3.45844	−0.450250	−0.225125	+	0.974330i	$0.572279\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	0	0
$64$	−8.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	−16.3138	−1.99304	−0.996522	−	0.0833352i	$-0.973443\pi$
$67$	−16.3138	−1.99304	−0.996522	+	0.0833352i	$0.973443\pi$
$68$	2.09956i	0.254609i
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	− 22.4663i	− 2.64768i
$73$	− 7.14631i	− 0.836412i	−0.908352	−	0.418206i	$-0.862659\pi$
$73$	− 7.14631i	− 0.836412i	0.908352	−	0.418206i	$-0.137341\pi$
$74$	0	0
$75$	16.5401	1.90989
$76$	− 16.0806i	− 1.84457i
$77$	0	0
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	30.2629	3.36254
$82$	3.12571	0.345177
$83$	− 16.2450i	− 1.78312i	−0.452904	−	0.891559i	$-0.649612\pi$
$83$	− 16.2450i	− 1.78312i	0.452904	−	0.891559i	$-0.350388\pi$
$84$	0	0
$85$	0	0
$86$	9.74177	1.05048
$87$	0	0
$88$	0	0
$89$	0.182318	0.0193256	0.00966282	−	0.999953i	$-0.496924\pi$
$89$	0.182318	0.0193256	0.00966282	+	0.999953i	$0.496924\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	18.7130i	1.90989i
$97$	−13.0498	−1.32501	−0.662503	−	0.749059i	$-0.730506\pi$
$97$	−13.0498	−1.32501	−0.662503	+	0.749059i	$0.730506\pi$
$98$	− 9.89949i	− 1.00000i
$99$	0	0
$100$	−10.0000	−1.00000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	4.91114	0.486275
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 11.8246i	− 1.14312i	−0.820559	−	0.571562i	$-0.806338\pi$
$107$	− 11.8246i	− 1.14312i	0.820559	−	0.571562i	$-0.193662\pi$
$108$	−32.7035	−3.14690
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	7.91227	0.744324	0.372162	−	0.928168i	$-0.378617\pi$
$113$	7.91227	0.744324	0.372162	+	0.928168i	$0.378617\pi$
$114$	−37.6146	−3.52293
$115$	0	0
$116$	0	0
$117$	0	0
$118$	− 4.89097i	− 0.450250i
$119$	0	0
$120$	0	0
$121$	0	0
$122$	0	0
$123$	− 7.31144i	− 0.659250i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	− 11.3137i	− 1.00000i
$129$	− 22.7872i	− 2.00631i
$130$	0	0
$131$	22.0214i	1.92402i	0.273022	+	0.962008i	$0.411977\pi$
$131$	22.0214i	1.92402i	−0.273022	+	0.962008i	$0.588023\pi$
$132$	0	0
$133$	0	0
$134$	− 23.0711i	− 1.99304i
$135$	0	0
$136$	−2.96923	−0.254609
$137$	−19.6659	−1.68017	−0.840083	−	0.542457i	$-0.817494\pi$
$137$	−19.6659	−1.68017	−0.840083	+	0.542457i	$0.817494\pi$
$138$	0	0
$139$	8.48528i	0.719712i	0.933008	+	0.359856i	$0.117174\pi$
$139$	8.48528i	0.719712i	−0.933008	+	0.359856i	$0.882826\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	31.7722	2.64768
$145$	0	0
$146$	10.1064	0.836412
$147$	−23.1562	−1.90989
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	23.3913i	1.90989i
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	22.7414	1.84457
$153$	− 8.33845i	− 0.674124i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	42.7982i	3.36254i
$163$	−13.3445	−1.04522	−0.522612	−	0.852570i	$-0.675042\pi$
$163$	−13.3445	−1.04522	−0.522612	+	0.852570i	$0.675042\pi$
$164$	4.42042i	0.345177i
$165$	0	0
$166$	22.9739	1.78312
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	63.8646i	4.88384i
$172$	13.7769i	1.05048i
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−11.4406	−0.859929
$178$	0.257836i	0.0193256i
$179$	26.1999	1.95827	0.979135	−	0.203212i	$-0.0651381\pi$
$179$	26.1999	1.95827	0.979135	+	0.203212i	$0.0651381\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	−26.4642	−1.90989
$193$	− 16.9706i	− 1.22157i	−0.791797	−	0.610784i	$-0.790854\pi$
$193$	− 16.9706i	− 1.22157i	0.791797	−	0.610784i	$-0.209146\pi$
$194$	− 18.4552i	− 1.32501i
$195$	0	0
$196$	14.0000	1.00000
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	− 14.1421i	− 1.00000i
$201$	−53.9664	−3.80649
$202$	0	0
$203$	0	0
$204$	6.94540i	0.486275i
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	21.1811i	1.45817i	0.684425	+	0.729083i	$0.260053\pi$
$211$	21.1811i	1.45817i	−0.684425	+	0.729083i	$0.739947\pi$
$212$	0	0
$213$	0	0
$214$	16.7225	1.14312
$215$	0	0
$216$	− 46.2497i	− 3.14690i
$217$	0	0
$218$	0	0
$219$	− 23.6402i	− 1.59746i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	39.7152	2.64768
$226$	11.1896i	0.744324i
$227$	19.9218i	1.32226i	0.750273	+	0.661128i	$0.229922\pi$
$227$	19.9218i	1.32226i	−0.750273	+	0.661128i	$0.770078\pi$
$228$	− 53.1951i	− 3.52293i
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 30.2798i	− 1.98369i	−0.127438	−	0.991847i	$-0.540675\pi$
$233$	− 30.2798i	− 1.98369i	0.127438	−	0.991847i	$-0.459325\pi$
$234$	0	0
$235$	0	0
$236$	6.91687	0.450250
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	− 29.0119i	− 1.86882i	−0.356199	−	0.934410i	$-0.615928\pi$
$241$	− 29.0119i	− 1.86882i	0.356199	−	0.934410i	$-0.384072\pi$
$242$	0	0
$243$	51.0552	3.27520
$244$	0	0
$245$	0	0
$246$	10.3399	0.659250
$247$	0	0
$248$	0	0
$249$	− 53.7389i	− 3.40556i
$250$	0	0
$251$	6.00000	0.378717	0.189358	−	0.981908i	$-0.439359\pi$
$251$	6.00000	0.378717	0.189358	+	0.981908i	$0.439359\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	16.0000	1.00000
$257$	20.0305	1.24947	0.624734	−	0.780838i	$-0.285208\pi$
$257$	20.0305	1.24947	0.624734	+	0.780838i	$0.285208\pi$
$258$	32.2260	2.00631
$259$	0	0
$260$	0	0
$261$	0	0
$262$	−31.1429	−1.92402
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0.603112	0.0369099
$268$	32.6275	1.99304
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	− 4.19912i	− 0.254609i
$273$	0	0
$274$	− 27.8117i	− 1.68017i
$275$	0	0
$276$	0	0
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	−12.0000	−0.719712
$279$	0	0
$280$	0	0
$281$	− 25.8593i	− 1.54264i	−0.636448	−	0.771319i	$-0.719597\pi$
$281$	− 25.8593i	− 1.54264i	0.636448	−	0.771319i	$-0.280403\pi$
$282$	0	0
$283$	25.4558i	1.51319i	0.653882	+	0.756596i	$0.273139\pi$
$283$	25.4558i	1.51319i	−0.653882	+	0.756596i	$0.726861\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	44.9327i	2.64768i
$289$	15.8980	0.935174
$290$	0	0
$291$	−43.1691	−2.53062
$292$	14.2926i	0.836412i
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	− 32.7478i	− 1.90989i
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	−33.0803	−1.90989
$301$	0	0
$302$	0	0
$303$	0	0
$304$	32.1612i	1.84457i
$305$	0	0
$306$	11.7924	0.674124
$307$	34.9580i	1.99516i	0.0695319	+	0.997580i	$0.477849\pi$
$307$	34.9580i	1.99516i	−0.0695319	+	0.997580i	$0.522151\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	−11.8599	−0.670362	−0.335181	−	0.942154i	$-0.608798\pi$
$313$	−11.8599	−0.670362	−0.335181	+	0.942154i	$0.608798\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	− 39.1160i	− 2.18324i
$322$	0	0
$323$	8.44055	0.469645
$324$	−60.5258	−3.36254
$325$	0	0
$326$	− 18.8720i	− 1.04522i
$327$	0	0
$328$	−6.25142	−0.345177
$329$	0	0
$330$	0	0
$331$	16.1755	0.889086	0.444543	−	0.895757i	$-0.353366\pi$
$331$	16.1755	0.889086	0.444543	+	0.895757i	$0.353366\pi$
$332$	32.4900i	1.78312i
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	19.2300i	1.04752i	0.851865	+	0.523761i	$0.175472\pi$
$337$	19.2300i	1.04752i	−0.851865	+	0.523761i	$0.824528\pi$
$338$	− 18.3848i	− 1.00000i
$339$	26.1740	1.42158
$340$	0	0
$341$	0	0
$342$	−90.3181	−4.88384
$343$	0	0
$344$	−19.4835	−1.05048
$345$	0	0
$346$	0	0
$347$	− 26.2205i	− 1.40759i	−0.710404	−	0.703795i	$-0.751488\pi$
$347$	− 26.2205i	− 1.40759i	0.710404	−	0.703795i	$-0.248512\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	37.5706	1.99968	0.999840	−	0.0178943i	$-0.00569624\pi$
$353$	37.5706	1.99968	0.999840	+	0.0178943i	$0.00569624\pi$
$354$	− 16.1795i	− 0.859929i
$355$	0	0
$356$	−0.364636	−0.0193256
$357$	0	0
$358$	37.0522i	1.95827i
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−45.6465	−2.40245
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	− 17.5558i	− 0.913919i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0		−	1.00000i	$-0.5\pi$
$373$	0	0			1.00000i	$0.5\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−3.67266	−0.188652	−0.0943260	−	0.995541i	$-0.530070\pi$
$379$	−3.67266	−0.188652	−0.0943260	+	0.995541i	$0.530070\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	− 37.4261i	− 1.90989i
$385$	0	0
$386$	24.0000	1.22157
$387$	− 54.7155i	− 2.78134i
$388$	26.0996	1.32501
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	19.7990i	1.00000i
$393$	72.8473i	3.67466i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	20.0000	1.00000
$401$	39.5140	1.97324	0.986618	−	0.163049i	$-0.0521329\pi$
$401$	39.5140	1.97324	0.986618	+	0.163049i	$0.0521329\pi$
$402$	− 76.3200i	− 3.80649i
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	−9.82228	−0.486275
$409$	33.9411i	1.67828i	0.543915	+	0.839140i	$0.316941\pi$
$409$	33.9411i	1.67828i	−0.543915	+	0.839140i	$0.683059\pi$
$410$	0	0
$411$	−65.0552	−3.20894
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	28.0695i	1.37457i
$418$	0	0
$419$	−29.4076	−1.43666	−0.718328	−	0.695705i	$-0.755092\pi$
$419$	−29.4076	−1.43666	−0.718328	+	0.695705i	$0.755092\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	−29.9546	−1.45817
$423$	0	0
$424$	0	0
$425$	− 5.24890i	− 0.254609i
$426$	0	0
$427$	0	0
$428$	23.6491i	1.14312i
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	65.4070	3.14690
$433$	−20.7676	−0.998027	−0.499014	−	0.866594i	$-0.666304\pi$
$433$	−20.7676	−0.998027	−0.499014	+	0.866594i	$0.666304\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	33.4323	1.59746
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	−55.6013	−2.64768
$442$	0	0
$443$	−10.2887	−0.488832	−0.244416	−	0.969670i	$-0.578596\pi$
$443$	−10.2887	−0.488832	−0.244416	+	0.969670i	$0.578596\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−30.6537	−1.44664	−0.723319	−	0.690514i	$-0.757384\pi$
$449$	−30.6537	−1.44664	−0.723319	+	0.690514i	$0.757384\pi$
$450$	56.1658i	2.64768i
$451$	0	0
$452$	−15.8245	−0.744324
$453$	0	0
$454$	−28.1737	−1.32226
$455$	0	0
$456$	75.2293	3.52293
$457$	35.2159i	1.64733i	0.567078	+	0.823664i	$0.308074\pi$
$457$	35.2159i	1.64733i	−0.567078	+	0.823664i	$0.691926\pi$
$458$	0	0
$459$	− 17.1657i	− 0.801228i
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	42.8220	1.98369
$467$	30.0000	1.38823	0.694117	−	0.719862i	$-0.255795\pi$
$467$	30.0000	1.38823	0.694117	+	0.719862i	$0.255795\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	9.78194i	0.450250i
$473$	0	0
$474$	0	0
$475$	40.2015i	1.84457i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	0	0
$482$	41.0290	1.86882
$483$	0	0
$484$	0	0
$485$	0	0
$486$	72.2030i	3.27520i
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	−44.1441	−1.99627
$490$	0	0
$491$	44.3145i	1.99989i	0.0106338	+	0.999943i	$0.496615\pi$
$491$	44.3145i	1.99989i	−0.0106338	+	0.999943i	$0.503385\pi$
$492$	14.6229i	0.659250i
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	75.9982	3.40556
$499$	36.0237	1.61264	0.806321	−	0.591479i	$-0.201456\pi$
$499$	36.0237	1.61264	0.806321	+	0.591479i	$0.201456\pi$
$500$	0	0
$501$	0	0
$502$	8.48528i	0.378717i
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−43.0044	−1.90989
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	22.6274i	1.00000i
$513$	131.473i	5.80468i
$514$	28.3274i	1.24947i
$515$	0	0
$516$	45.5745i	2.00631i
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−21.7460	−0.952711	−0.476355	−	0.879253i	$-0.658042\pi$
$521$	−21.7460	−0.952711	−0.476355	+	0.879253i	$0.658042\pi$
$522$	0	0
$523$	− 1.74163i	− 0.0761561i	−0.999275	−	0.0380781i	$-0.987876\pi$
$523$	− 1.74163i	− 0.0761561i	0.999275	−	0.0380781i	$-0.0121236\pi$
$524$	− 44.0427i	− 1.92402i
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	23.0000	1.00000
$530$	0	0
$531$	−27.4705	−1.19212
$532$	0	0
$533$	0	0
$534$	0.852930i	0.0369099i
$535$	0	0
$536$	46.1423i	1.99304i
$537$	86.6699	3.74008
$538$	0	0
$539$	0	0
$540$	0	0
$541$	0	0		−	1.00000i	$-0.5\pi$
$541$	0	0			1.00000i	$0.5\pi$
$542$	0	0
$543$	0	0
$544$	5.93845	0.254609
$545$	0	0
$546$	0	0
$547$	33.9029i	1.44958i	0.688969	+	0.724791i	$0.258064\pi$
$547$	33.9029i	1.44958i	−0.688969	+	0.724791i	$0.741936\pi$
$548$	39.3317	1.68017
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	− 16.9706i	− 0.719712i
$557$	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
$562$	36.5706	1.54264
$563$	− 39.8941i	− 1.68134i	−0.541551	−	0.840668i	$-0.682163\pi$
$563$	− 39.8941i	− 1.68134i	0.541551	−	0.840668i	$-0.317837\pi$
$564$	0	0
$565$	0	0
$566$	−36.0000	−1.51319
$567$	0	0
$568$	0	0
$569$	− 42.9929i	− 1.80236i	−0.433447	−	0.901179i	$-0.642703\pi$
$569$	− 42.9929i	− 1.80236i	0.433447	−	0.901179i	$-0.357297\pi$
$570$	0	0
$571$	42.4264i	1.77549i	0.460336	+	0.887745i	$0.347729\pi$
$571$	42.4264i	1.77549i	−0.460336	+	0.887745i	$0.652271\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	−63.5444	−2.64768
$577$	−47.4567	−1.97565	−0.987824	−	0.155579i	$-0.950276\pi$
$577$	−47.4567	−1.97565	−0.987824	+	0.155579i	$0.950276\pi$
$578$	22.4831i	0.935174i
$579$	− 56.1391i	− 2.33306i
$580$	0	0
$581$	0	0
$582$	− 61.0503i	− 2.53062i
$583$	0	0
$584$	−20.2128	−0.836412
$585$	0	0
$586$	0	0
$587$	33.1167	1.36687	0.683437	−	0.730010i	$-0.260484\pi$
$587$	33.1167	1.36687	0.683437	+	0.730010i	$0.260484\pi$
$588$	46.3124	1.90989
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	47.1921i	1.93795i	0.247167	+	0.968973i	$0.420500\pi$
$593$	47.1921i	1.93795i	−0.247167	+	0.968973i	$0.579500\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	− 46.7826i	− 1.90989i
$601$	− 48.9928i	− 1.99846i	−0.0392649	−	0.999229i	$-0.512502\pi$
$601$	− 48.9928i	− 1.99846i	0.0392649	−	0.999229i	$-0.487498\pi$
$602$	0	0
$603$	−129.581	−5.27695
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	−45.4828	−1.84457
$609$	0	0
$610$	0	0
$611$	0	0
$612$	16.6769i	0.674124i
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	−49.4381	−1.99516
$615$	0	0
$616$	0	0
$617$	0.995401	0.0400733	0.0200367	−	0.999799i	$-0.493622\pi$
$617$	0.995401	0.0400733	0.0200367	+	0.999799i	$0.493622\pi$
$618$	0	0
$619$	−45.9721	−1.84777	−0.923887	−	0.382667i	$-0.875006\pi$
$619$	−45.9721	−1.84777	−0.923887	+	0.382667i	$0.875006\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	− 16.7725i	− 0.670362i
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	70.0676i	2.78494i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−39.8787	−1.57511	−0.787556	−	0.616243i	$-0.788654\pi$
$641$	−39.8787	−1.57511	−0.787556	+	0.616243i	$0.788654\pi$
$642$	55.3184	2.18324
$643$	23.5208	0.927571	0.463786	−	0.885948i	$-0.346491\pi$
$643$	23.5208	0.927571	0.463786	+	0.885948i	$0.346491\pi$
$644$	0	0
$645$	0	0
$646$	11.9367i	0.469645i
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	− 85.5964i	− 3.36254i
$649$	0	0
$650$	0	0
$651$	0	0
$652$	26.6891	1.04522
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	− 8.84084i	− 0.345177i
$657$	− 56.7635i	− 2.21455i
$658$	0	0
$659$	− 28.3200i	− 1.10319i	−0.834111	−	0.551596i	$-0.814019\pi$
$659$	− 28.3200i	− 1.10319i	0.834111	−	0.551596i	$-0.185981\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	22.8756i	0.889086i
$663$	0	0
$664$	−45.9478	−1.78312
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	− 35.3106i	− 1.36112i	−0.732691	−	0.680561i	$-0.761736\pi$
$673$	− 35.3106i	− 1.36112i	0.732691	−	0.680561i	$-0.238264\pi$
$674$	−27.1953	−1.04752
$675$	81.7587	3.14690
$676$	26.0000	1.00000
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	37.0157i	1.42158i
$679$	0	0
$680$	0	0
$681$	65.9019i	2.52537i
$682$	0	0
$683$	−42.0000	−1.60709	−0.803543	−	0.595247i	$-0.797054\pi$
$683$	−42.0000	−1.60709	−0.803543	+	0.595247i	$0.797054\pi$
$684$	− 127.729i	− 4.88384i
$685$	0	0
$686$	0	0
$687$	0	0
$688$	− 27.5539i	− 1.05048i
$689$	0	0
$690$	0	0
$691$	22.2522	0.846514	0.423257	−	0.906010i	$-0.360887\pi$
$691$	22.2522	0.846514	0.423257	+	0.906010i	$0.360887\pi$
$692$	0	0
$693$	0	0
$694$	37.0814	1.40759
$695$	0	0
$696$	0	0
$697$	−2.32023	−0.0878852
$698$	0	0
$699$	− 100.166i	− 3.78864i
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	53.1328i	1.99968i
$707$	0	0
$708$	22.8812	0.859929
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	− 0.515673i	− 0.0193256i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−52.3997	−1.95827
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	0	0
$722$	− 64.5540i	− 2.40245i
$723$	− 95.9721i	− 3.56924i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	78.1035	2.89272
$730$	0	0
$731$	−7.23137	−0.267462
$732$	0	0
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	24.8277	0.913919
$739$	14.3390i	0.527468i	0.964595	+	0.263734i	$0.0849541\pi$
$739$	14.3390i	0.527468i	−0.964595	+	0.263734i	$0.915046\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 129.035i	− 4.72113i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	19.8482	0.723307
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	− 5.19393i	− 0.188652i
$759$	0	0
$760$	0	0
$761$	40.8934i	1.48238i	0.671293	+	0.741192i	$0.265739\pi$
$761$	40.8934i	1.48238i	−0.671293	+	0.741192i	$0.734261\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	52.9284	1.90989
$769$	− 50.9117i	− 1.83592i	−0.396670	−	0.917961i	$-0.629834\pi$
$769$	− 50.9117i	− 1.83592i	0.396670	−	0.917961i	$-0.370166\pi$
$770$	0	0
$771$	66.2614	2.38635
$772$	33.9411i	1.22157i
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	77.3793	2.78134
$775$	0	0
$776$	36.9104i	1.32501i
$777$	0	0
$778$	0	0
$779$	17.7708	0.636704
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−28.0000	−1.00000
$785$	0	0
$786$	−103.022	−3.67466
$787$	49.9835i	1.78172i	0.454280	+	0.890859i	$0.349897\pi$
$787$	49.9835i	1.78172i	−0.454280	+	0.890859i	$0.650103\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	28.2843i	1.00000i
$801$	1.44816	0.0511682
$802$	55.8813i	1.97324i
$803$	0	0
$804$	107.933	3.80649
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	49.2916i	1.73300i	0.499176	+	0.866501i	$0.333636\pi$
$809$	49.2916i	1.73300i	−0.499176	+	0.866501i	$0.666364\pi$
$810$	0	0
$811$	49.2506i	1.72942i	0.502268	+	0.864712i	$0.332499\pi$
$811$	49.2506i	1.72942i	−0.502268	+	0.864712i	$0.667501\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	− 13.8908i	− 0.486275i
$817$	55.3854	1.93769
$818$	−48.0000	−1.67828
$819$	0	0
$820$	0	0
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	− 92.0019i	− 3.20894i
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	15.7227i	0.546731i	0.961910	+	0.273366i	$0.0881369\pi$
$827$	15.7227i	0.546731i	−0.961910	+	0.273366i	$0.911863\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	7.34846i	0.254609i
$834$	−39.6963	−1.37457
$835$	0	0
$836$	0	0
$837$	0	0
$838$	− 41.5887i	− 1.43666i
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	− 85.5434i	− 2.94627i
$844$	− 42.3622i	− 1.45817i
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	84.2086i	2.89003i
$850$	7.42306	0.254609
$851$	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
$856$	−33.4449	−1.14312
$857$	− 58.3493i	− 1.99317i	−0.0825467	−	0.996587i	$-0.526305\pi$
$857$	− 58.3493i	− 1.99317i	0.0825467	−	0.996587i	$-0.473695\pi$
$858$	0	0
$859$	−51.9105	−1.77116	−0.885582	−	0.464483i	$-0.846240\pi$
$859$	−51.9105	−1.77116	−0.885582	+	0.464483i	$0.846240\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	92.4994i	3.14690i
$865$	0	0
$866$	− 29.3698i	− 0.998027i
$867$	52.5909	1.78608
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−103.655	−3.50820
$874$	0	0
$875$	0	0
$876$	47.2804i	1.59746i
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	59.3622	1.99996	0.999981	−	0.00609171i	$-0.00193906\pi$
$881$	59.3622	1.99996	0.999981	+	0.00609171i	$0.00193906\pi$
$882$	− 78.6322i	− 2.64768i
$883$	−55.8718	−1.88024	−0.940119	−	0.340848i	$-0.889286\pi$
$883$	−55.8718	−1.88024	−0.940119	+	0.340848i	$0.889286\pi$
$884$	0	0
$885$	0	0
$886$	− 14.5504i	− 0.488832i
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	− 43.3509i	− 1.44664i
$899$	0	0
$900$	−79.4305	−2.64768
$901$	0	0
$902$	0	0
$903$	0	0
$904$	− 22.3793i	− 0.744324i
$905$	0	0
$906$	0	0
$907$	43.0028	1.42789	0.713943	−	0.700204i	$-0.246908\pi$
$907$	43.0028	1.42789	0.713943	+	0.700204i	$0.246908\pi$
$908$	− 39.8436i	− 1.32226i
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	106.390i	3.52293i
$913$	0	0
$914$	−49.8027	−1.64733
$915$	0	0
$916$	0	0
$917$	0	0
$918$	24.2760	0.801228
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	0	0
$921$	115.642i	3.81054i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	60.3120	1.97877	0.989386	−	0.145310i	$-0.0464179\pi$
$929$	60.3120	1.97877	0.989386	+	0.145310i	$0.0464179\pi$
$930$	0	0
$931$	− 56.2822i	− 1.84457i
$932$	60.5595i	1.98369i
$933$	0	0
$934$	42.4264i	1.38823i
$935$	0	0
$936$	0	0
$937$	− 61.1731i	− 1.99844i	−0.0395055	−	0.999219i	$-0.512578\pi$
$937$	− 61.1731i	− 1.99844i	0.0395055	−	0.999219i	$-0.487422\pi$
$938$	0	0
$939$	−39.2330	−1.28032
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	0	0
$944$	−13.8337	−0.450250
$945$	0	0
$946$	0	0
$947$	55.8582	1.81515	0.907573	−	0.419894i	$-0.137933\pi$
$947$	55.8582	1.81515	0.907573	+	0.419894i	$0.137933\pi$
$948$	0	0
$949$	0	0
$950$	−56.8536	−1.84457
$951$	0	0
$952$	0	0
$953$	53.9289i	1.74693i	0.486889	+	0.873464i	$0.338132\pi$
$953$	53.9289i	1.74693i	−0.486889	+	0.873464i	$0.661868\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	31.0000	1.00000
$962$	0	0
$963$	− 93.9231i	− 3.02663i
$964$	58.0238i	1.86882i
$965$	0	0
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	0	0
$969$	27.9216	0.896970
$970$	0	0
$971$	54.0000	1.73294	0.866471	−	0.499227i	$-0.166383\pi$
$971$	54.0000	1.73294	0.866471	+	0.499227i	$0.166383\pi$
$972$	−102.110	−3.27520
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	6.00000	0.191957	0.0959785	−	0.995383i	$-0.469402\pi$
$977$	6.00000	0.191957	0.0959785	+	0.995383i	$0.469402\pi$
$978$	− 62.4292i	− 1.99627i
$979$	0	0
$980$	0	0
$981$	0	0
$982$	−62.6702	−1.99989
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	−20.6799	−0.659250
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	53.5090	1.69806
$994$	0	0
$995$	0	0
$996$	107.478i	3.40556i
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	50.9452i	1.61264i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	968.2.g.a.483.8		8
4.3	odd	2		3872.2.g.b.1935.1		8
8.3	odd	2	CM	968.2.g.a.483.8		8
8.5	even	2		3872.2.g.b.1935.1		8
11.2	odd	10		968.2.k.d.403.1		8
11.3	even	5		88.2.k.a.35.2	✓	8
11.4	even	5		968.2.k.b.699.1		8
11.5	even	5		968.2.k.d.723.1		8
11.6	odd	10		968.2.k.c.723.2		8
11.7	odd	10		88.2.k.a.83.2	yes	8
11.8	odd	10		968.2.k.b.475.1		8
11.9	even	5		968.2.k.c.403.2		8
11.10	odd	2	inner	968.2.g.a.483.4		8
33.14	odd	10		792.2.bp.a.739.1		8
33.29	even	10		792.2.bp.a.523.1		8
44.3	odd	10		352.2.s.a.79.2		8
44.7	even	10		352.2.s.a.303.2		8
44.43	even	2		3872.2.g.b.1935.2		8
88.3	odd	10		88.2.k.a.35.2	✓	8
88.19	even	10		968.2.k.b.475.1		8
88.21	odd	2		3872.2.g.b.1935.2		8
88.27	odd	10		968.2.k.d.723.1		8
88.29	odd	10		352.2.s.a.303.2		8
88.35	even	10		968.2.k.d.403.1		8
88.43	even	2	inner	968.2.g.a.483.4		8
88.51	even	10		88.2.k.a.83.2	yes	8
88.59	odd	10		968.2.k.b.699.1		8
88.69	even	10		352.2.s.a.79.2		8
88.75	odd	10		968.2.k.c.403.2		8
88.83	even	10		968.2.k.c.723.2		8
264.179	even	10		792.2.bp.a.739.1		8
264.227	odd	10		792.2.bp.a.523.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.2.k.a.35.2	✓	8	11.3	even	5
88.2.k.a.35.2	✓	8	88.3	odd	10
88.2.k.a.83.2	yes	8	11.7	odd	10
88.2.k.a.83.2	yes	8	88.51	even	10
352.2.s.a.79.2		8	44.3	odd	10
352.2.s.a.79.2		8	88.69	even	10
352.2.s.a.303.2		8	44.7	even	10
352.2.s.a.303.2		8	88.29	odd	10
792.2.bp.a.523.1		8	33.29	even	10
792.2.bp.a.523.1		8	264.227	odd	10
792.2.bp.a.739.1		8	33.14	odd	10
792.2.bp.a.739.1		8	264.179	even	10
968.2.g.a.483.4		8	11.10	odd	2	inner
968.2.g.a.483.4		8	88.43	even	2	inner
968.2.g.a.483.8		8	1.1	even	1	trivial
968.2.g.a.483.8		8	8.3	odd	2	CM
968.2.k.b.475.1		8	11.8	odd	10
968.2.k.b.475.1		8	88.19	even	10
968.2.k.b.699.1		8	11.4	even	5
968.2.k.b.699.1		8	88.59	odd	10
968.2.k.c.403.2		8	11.9	even	5
968.2.k.c.403.2		8	88.75	odd	10
968.2.k.c.723.2		8	11.6	odd	10
968.2.k.c.723.2		8	88.83	even	10
968.2.k.d.403.1		8	11.2	odd	10
968.2.k.d.403.1		8	88.35	even	10
968.2.k.d.723.1		8	11.5	even	5
968.2.k.d.723.1		8	88.27	odd	10
3872.2.g.b.1935.1		8	4.3	odd	2
3872.2.g.b.1935.1		8	8.5	even	2
3872.2.g.b.1935.2		8	44.43	even	2
3872.2.g.b.1935.2		8	88.21	odd	2

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:	\( N \)	\(=\)	\( 968 = 2^{3} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	968.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} +3.30803 q^{3} -2.00000 q^{4} +4.67826i q^{6} -2.82843i q^{8} +7.94305 q^{9} -6.61606 q^{12} +4.00000 q^{16} -1.04978i q^{17} +11.2332i q^{18} +8.04031i q^{19} -9.35652i q^{24} +5.00000 q^{25} +16.3517 q^{27} +5.65685i q^{32} +1.48461 q^{34} -15.8861 q^{36} -11.3707 q^{38} -2.21021i q^{41} -6.88847i q^{43} +13.2321 q^{48} -7.00000 q^{49} +7.07107i q^{50} -3.47270i q^{51} +23.1249i q^{54} +26.5976i q^{57} -3.45844 q^{59} -8.00000 q^{64} -16.3138 q^{67} +2.09956i q^{68} -22.4663i q^{72} -7.14631i q^{73} +16.5401 q^{75} -16.0806i q^{76} +30.2629 q^{81} +3.12571 q^{82} -16.2450i q^{83} +9.74177 q^{86} +0.182318 q^{89} +18.7130i q^{96} -13.0498 q^{97} -9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{3} - 16 q^{4} + 28 q^{9} + 8 q^{12} + 32 q^{16} + 40 q^{25} + 8 q^{27} + 16 q^{34} - 56 q^{36} - 24 q^{38} - 16 q^{48} - 56 q^{49} + 12 q^{59} - 64 q^{64} - 28 q^{67} - 20 q^{75} + 112 q^{81}+ \cdots - 20 q^{97}+O(q^{100}) \) 8 * q - 4 * q^3 - 16 * q^4 + 28 * q^9 + 8 * q^12 + 32 * q^16 + 40 * q^25 + 8 * q^27 + 16 * q^34 - 56 * q^36 - 24 * q^38 - 16 * q^48 - 56 * q^49 + 12 * q^59 - 64 * q^64 - 28 * q^67 - 20 * q^75 + 112 * q^81 + 32 * q^82 + 24 * q^86 - 36 * q^89 - 20 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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