LMFDB - Embedded newform 968.2.g.a.483.3

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			483.3
Root			$0.831254 + 1.14412i$ of defining polynomial
Character	$\chi$	$=$	968.483
Dual form			968.2.g.a.483.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} +0.0444738 q^{3} -2.00000 q^{4} -0.0628954i q^{6} +2.82843i q^{8} -2.99802 q^{9} -0.0889475 q^{12} +4.00000 q^{16} +3.95827i q^{17} +4.23984i q^{18} +4.52421i q^{19} +0.125791i q^{24} +5.00000 q^{25} -0.266755 q^{27} -5.65685i q^{32} +5.59785 q^{34} +5.99604 q^{36} +6.39820 q^{38} +12.6797i q^{41} +12.7426i q^{43} +0.177895 q^{48} -7.00000 q^{49} -7.07107i q^{50} +0.176039i q^{51} +0.377248i q^{54} +0.201209i q^{57} +11.5959 q^{59} -8.00000 q^{64} +12.3962 q^{67} -7.91655i q^{68} -8.47969i q^{72} -12.5539i q^{73} +0.222369 q^{75} -9.04842i q^{76} +8.98220 q^{81} +17.9318 q^{82} +12.8684i q^{83} +18.0207 q^{86} -17.8873 q^{89} -0.251582i q^{96} -18.0652 q^{97} +9.89949i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.0444738	0.0256769	0.0128385	−	0.999918i	$-0.495913\pi$
$3$	0.0444738	0.0256769	0.0128385	+	0.999918i	$0.495913\pi$
$4$	−2.00000	−1.00000
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	− 0.0628954i	− 0.0256769i
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	2.82843i	1.00000i
$9$	−2.99802	−0.999341
$10$	0	0
$11$	0	0
$11$	0	0
$12$	−0.0889475	−0.0256769
$13$	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
$17$	3.95827i	0.960023i	0.877262	+	0.480011i	$0.159367\pi$
$17$	3.95827i	0.960023i	−0.877262	+	0.480011i	$0.840633\pi$
$18$	4.23984i	0.999341i
$19$	4.52421i	1.03792i	0.854797	+	0.518962i	$0.173682\pi$
$19$	4.52421i	1.03792i	−0.854797	+	0.518962i	$0.826318\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0.125791i	0.0256769i
$25$	5.00000	1.00000
$26$	0	0
$27$	−0.266755	−0.0513369
$28$	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
$34$	5.59785	0.960023
$35$	0	0
$36$	5.99604	0.999341
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	6.39820	1.03792
$39$	0	0
$40$	0	0
$41$	12.6797i	1.98024i	0.140238	+	0.990118i	$0.455213\pi$
$41$	12.6797i	1.98024i	−0.140238	+	0.990118i	$0.544787\pi$
$42$	0	0
$43$	12.7426i	1.94323i	0.236575	+	0.971613i	$0.423975\pi$
$43$	12.7426i	1.94323i	−0.236575	+	0.971613i	$0.576025\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0.177895	0.0256769
$49$	−7.00000	−1.00000
$50$	− 7.07107i	− 1.00000i
$51$	0.176039i	0.0246504i
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0.377248i	0.0513369i
$55$	0	0
$56$	0	0
$57$	0.201209i	0.0266507i
$58$	0	0
$59$	11.5959	1.50965	0.754827	−	0.655924i	$-0.227721\pi$
$59$	11.5959	1.50965	0.754827	+	0.655924i	$0.227721\pi$
$60$	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
$67$	12.3962	1.51444	0.757220	−	0.653160i	$-0.226557\pi$
$67$	12.3962	1.51444	0.757220	+	0.653160i	$0.226557\pi$
$68$	− 7.91655i	− 0.960023i
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	− 8.47969i	− 0.999341i
$73$	− 12.5539i	− 1.46932i	−0.678434	−	0.734662i	$-0.737341\pi$
$73$	− 12.5539i	− 1.46932i	0.678434	−	0.734662i	$-0.262659\pi$
$74$	0	0
$75$	0.222369	0.0256769
$76$	− 9.04842i	− 1.03792i
$77$	0	0
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	8.98220	0.998023
$82$	17.9318	1.98024
$83$	12.8684i	1.41249i	0.707968	+	0.706244i	$0.249612\pi$
$83$	12.8684i	1.41249i	−0.707968	+	0.706244i	$0.750388\pi$
$84$	0	0
$85$	0	0
$86$	18.0207	1.94323
$87$	0	0
$88$	0	0
$89$	−17.8873	−1.89605	−0.948026	−	0.318192i	$-0.896924\pi$
$89$	−17.8873	−1.89605	−0.948026	+	0.318192i	$0.896924\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	− 0.251582i	− 0.0256769i
$97$	−18.0652	−1.83424	−0.917122	−	0.398606i	$-0.869494\pi$
$97$	−18.0652	−1.83424	−0.917122	+	0.398606i	$0.869494\pi$
$98$	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	$0.5\pi$
$102$	0.248957	0.0246504
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 12.4910i	− 1.20755i	−0.797154	−	0.603776i	$-0.793662\pi$
$107$	− 12.4910i	− 1.20755i	0.797154	−	0.603776i	$-0.206338\pi$
$108$	0.533509	0.0513369
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	5.19767	0.488956	0.244478	−	0.969655i	$-0.421383\pi$
$113$	5.19767	0.488956	0.244478	+	0.969655i	$0.421383\pi$
$114$	0.284552	0.0266507
$115$	0	0
$116$	0	0
$117$	0	0
$118$	− 16.3990i	− 1.50965i
$119$	0	0
$120$	0	0
$121$	0	0
$122$	0	0
$123$	0.563914i	0.0508464i
$124$	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
$129$	0.566711i	0.0498961i
$130$	0	0
$131$	21.4892i	1.87752i	0.344574	+	0.938759i	$0.388023\pi$
$131$	21.4892i	1.87752i	−0.344574	+	0.938759i	$0.611977\pi$
$132$	0	0
$133$	0	0
$134$	− 17.5309i	− 1.51444i
$135$	0	0
$136$	−11.1957	−0.960023
$137$	−18.1542	−1.55101	−0.775507	−	0.631338i	$-0.782506\pi$
$137$	−18.1542	−1.55101	−0.775507	+	0.631338i	$0.782506\pi$
$138$	0	0
$139$	− 8.48528i	− 0.719712i	−0.933008	−	0.359856i	$-0.882826\pi$
$139$	− 8.48528i	− 0.719712i	0.933008	−	0.359856i	$-0.117174\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	−11.9921	−0.999341
$145$	0	0
$146$	−17.7539	−1.46932
$147$	−0.311316	−0.0256769
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	− 0.314477i	− 0.0256769i
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	−12.7964	−1.03792
$153$	− 11.8670i	− 0.959390i
$154$	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
$162$	− 12.7028i	− 0.998023i
$163$	23.5919	1.84786	0.923931	−	0.382560i	$-0.124958\pi$
$163$	23.5919	1.84786	0.923931	+	0.382560i	$0.124958\pi$
$164$	− 25.3594i	− 1.98024i
$165$	0	0
$166$	18.1986	1.41249
$167$	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
$171$	− 13.5637i	− 1.03724i
$172$	− 25.4852i	− 1.94323i
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0.515712	0.0387633
$178$	25.2965i	1.89605i
$179$	−24.3923	−1.82316	−0.911582	−	0.411119i	$-0.865138\pi$
$179$	−24.3923	−1.82316	−0.911582	+	0.411119i	$0.865138\pi$
$180$	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			$\pi$
$192$	−0.355790	−0.0256769
$193$	16.9706i	1.22157i	0.791797	+	0.610784i	$0.209146\pi$
$193$	16.9706i	1.22157i	−0.791797	+	0.610784i	$0.790854\pi$
$194$	25.5481i	1.83424i
$195$	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
$201$	0.551307	0.0388862
$202$	0	0
$203$	0	0
$204$	− 0.352079i	− 0.0246504i
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	12.3652i	0.851257i	0.904898	+	0.425628i	$0.139947\pi$
$211$	12.3652i	0.851257i	−0.904898	+	0.425628i	$0.860053\pi$
$212$	0	0
$213$	0	0
$214$	−17.6650	−1.20755
$215$	0	0
$216$	− 0.754496i	− 0.0513369i
$217$	0	0
$218$	0	0
$219$	− 0.558319i	− 0.0377277i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	−14.9901	−0.999341
$226$	− 7.35062i	− 0.488956i
$227$	29.4057i	1.95173i	0.218382	+	0.975863i	$0.429922\pi$
$227$	29.4057i	1.95173i	−0.218382	+	0.975863i	$0.570078\pi$
$228$	− 0.402417i	− 0.0266507i
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	13.0571i	0.855397i	0.903921	+	0.427698i	$0.140675\pi$
$233$	13.0571i	0.855397i	−0.903921	+	0.427698i	$0.859325\pi$
$234$	0	0
$235$	0	0
$236$	−23.1917	−1.50965
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	− 29.9717i	− 1.93064i	−0.261061	−	0.965322i	$-0.584072\pi$
$241$	− 29.9717i	− 1.93064i	0.261061	−	0.965322i	$-0.415928\pi$
$242$	0	0
$243$	1.19974	0.0769631
$244$	0	0
$245$	0	0
$246$	0.797494	0.0508464
$247$	0	0
$248$	0	0
$249$	0.572305i	0.0362684i
$250$	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
$257$	−17.6205	−1.09914	−0.549568	−	0.835449i	$-0.685208\pi$
$257$	−17.6205	−1.09914	−0.549568	+	0.835449i	$0.685208\pi$
$258$	0.801450	0.0498961
$259$	0	0
$260$	0	0
$261$	0	0
$262$	30.3903	1.87752
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−0.795516	−0.0486848
$268$	−24.7924	−1.51444
$269$	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	$0.5\pi$
$272$	15.8331i	0.960023i
$273$	0	0
$274$	25.6739i	1.55101i
$275$	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
$281$	− 12.3023i	− 0.733895i	−0.930242	−	0.366947i	$-0.880403\pi$
$281$	− 12.3023i	− 0.733895i	0.930242	−	0.366947i	$-0.119597\pi$
$282$	0	0
$283$	− 25.4558i	− 1.51319i	−0.653882	−	0.756596i	$-0.726861\pi$
$283$	− 25.4558i	− 1.51319i	0.653882	−	0.756596i	$-0.273139\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	16.9594i	0.999341i
$289$	1.33206	0.0783564
$290$	0	0
$291$	−0.803428	−0.0470978
$292$	25.1078i	1.46932i
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0.440268i	0.0256769i
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	−0.444738	−0.0256769
$301$	0	0
$302$	0	0
$303$	0	0
$304$	18.0968i	1.03792i
$305$	0	0
$306$	−16.7825	−0.959390
$307$	− 13.1200i	− 0.748796i	−0.927268	−	0.374398i	$-0.877849\pi$
$307$	− 13.1200i	− 0.748796i	0.927268	−	0.374398i	$-0.122151\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	29.1898	1.64990	0.824951	−	0.565204i	$-0.191202\pi$
$313$	29.1898	1.64990	0.824951	+	0.565204i	$0.191202\pi$
$314$	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
$321$	− 0.555522i	− 0.0310062i
$322$	0	0
$323$	−17.9081	−0.996431
$324$	−17.9644	−0.998023
$325$	0	0
$326$	− 33.3640i	− 1.84786i
$327$	0	0
$328$	−35.8636	−1.98024
$329$	0	0
$330$	0	0
$331$	35.9970	1.97857	0.989287	−	0.145981i	$-0.0466339\pi$
$331$	35.9970	1.97857	0.989287	+	0.145981i	$0.0466339\pi$
$332$	− 25.7368i	− 1.41249i
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 2.82641i	− 0.153964i	−0.997032	−	0.0769821i	$-0.975472\pi$
$337$	− 2.82641i	− 0.153964i	0.997032	−	0.0769821i	$-0.0245284\pi$
$338$	18.3848i	1.00000i
$339$	0.231160	0.0125549
$340$	0	0
$341$	0	0
$342$	−19.1819	−1.03724
$343$	0	0
$344$	−36.0415	−1.94323
$345$	0	0
$346$	0	0
$347$	− 5.65608i	− 0.303634i	−0.988409	−	0.151817i	$-0.951488\pi$
$347$	− 5.65608i	− 0.303634i	0.988409	−	0.151817i	$-0.0485125\pi$
$348$	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
$353$	−30.7905	−1.63881	−0.819405	−	0.573214i	$-0.805696\pi$
$353$	−30.7905	−1.63881	−0.819405	+	0.573214i	$0.805696\pi$
$354$	− 0.729327i	− 0.0387633i
$355$	0	0
$356$	35.7746	1.89605
$357$	0	0
$358$	34.4959i	1.82316i
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−1.46847	−0.0772878
$362$	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
$369$	− 38.0140i	− 1.97893i
$370$	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
$379$	35.7302	1.83534	0.917668	−	0.397349i	$-0.130070\pi$
$379$	35.7302	1.83534	0.917668	+	0.397349i	$0.130070\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0.503163i	0.0256769i
$385$	0	0
$386$	24.0000	1.22157
$387$	− 38.2026i	− 1.94195i
$388$	36.1304	1.83424
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	− 19.7990i	− 1.00000i
$393$	0.955705i	0.0482089i
$394$	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
$401$	18.4210	0.919901	0.459951	−	0.887945i	$-0.347867\pi$
$401$	18.4210	0.919901	0.459951	+	0.887945i	$0.347867\pi$
$402$	− 0.779665i	− 0.0388862i
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	−0.497915	−0.0246504
$409$	− 33.9411i	− 1.67828i	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	− 33.9411i	− 1.67828i	0.543915	−	0.839140i	$-0.316941\pi$
$410$	0	0
$411$	−0.807384	−0.0398253
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 0.377372i	− 0.0184800i
$418$	0	0
$419$	−36.1749	−1.76726	−0.883630	−	0.468186i	$-0.844908\pi$
$419$	−36.1749	−1.76726	−0.883630	+	0.468186i	$0.844908\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	17.4871	0.851257
$423$	0	0
$424$	0	0
$425$	19.7914i	0.960023i
$426$	0	0
$427$	0	0
$428$	24.9820i	1.20755i
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	−1.06702	−0.0513369
$433$	−4.39732	−0.211322	−0.105661	−	0.994402i	$-0.533696\pi$
$433$	−4.39732	−0.211322	−0.105661	+	0.994402i	$0.533696\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	−0.789583	−0.0377277
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	20.9862	0.999341
$442$	0	0
$443$	35.6412	1.69337	0.846683	−	0.532098i	$-0.178596\pi$
$443$	35.6412	1.69337	0.846683	+	0.532098i	$0.178596\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	7.59873	0.358606	0.179303	−	0.983794i	$-0.442616\pi$
$449$	7.59873	0.358606	0.179303	+	0.983794i	$0.442616\pi$
$450$	21.1992i	0.999341i
$451$	0	0
$452$	−10.3953	−0.488956
$453$	0	0
$454$	41.5860	1.95173
$455$	0	0
$456$	−0.569104	−0.0266507
$457$	12.1765i	0.569594i	0.958588	+	0.284797i	$0.0919262\pi$
$457$	12.1765i	0.569594i	−0.958588	+	0.284797i	$0.908074\pi$
$458$	0	0
$459$	− 1.05589i	− 0.0492846i
$460$	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
$466$	18.4655	0.855397
$467$	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
$472$	32.7981i	1.50965i
$473$	0	0
$474$	0	0
$475$	22.6210i	1.03792i
$476$	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
$482$	−42.3863	−1.93064
$483$	0	0
$484$	0	0
$485$	0	0
$486$	− 1.69668i	− 0.0769631i
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	1.04922	0.0474474
$490$	0	0
$491$	− 13.2458i	− 0.597772i	−0.954289	−	0.298886i	$-0.903385\pi$
$491$	− 13.2458i	− 0.597772i	0.954289	−	0.298886i	$-0.0966151\pi$
$492$	− 1.12783i	− 0.0508464i
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0.809362	0.0362684
$499$	36.2639	1.62339	0.811697	−	0.584079i	$-0.198544\pi$
$499$	36.2639	1.62339	0.811697	+	0.584079i	$0.198544\pi$
$500$	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
$507$	−0.578159	−0.0256769
$508$	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
$513$	− 1.20685i	− 0.0532839i
$514$	24.9191i	1.09914i
$515$	0	0
$516$	− 1.13342i	− 0.0498961i
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	41.1858	1.80438	0.902191	−	0.431336i	$-0.141958\pi$
$521$	41.1858	1.80438	0.902191	+	0.431336i	$0.141958\pi$
$522$	0	0
$523$	− 28.2739i	− 1.23633i	−0.786049	−	0.618165i	$-0.787876\pi$
$523$	− 28.2739i	− 1.23633i	0.786049	−	0.618165i	$-0.212124\pi$
$524$	− 42.9784i	− 1.87752i
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	23.0000	1.00000
$530$	0	0
$531$	−34.7647	−1.50866
$532$	0	0
$533$	0	0
$534$	1.12503i	0.0486848i
$535$	0	0
$536$	35.0618i	1.51444i
$537$	−1.08482	−0.0468133
$538$	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
$544$	22.3914	0.960023
$545$	0	0
$546$	0	0
$547$	46.3707i	1.98267i	0.131366	+	0.991334i	$0.458064\pi$
$547$	46.3707i	1.98267i	−0.131366	+	0.991334i	$0.541936\pi$
$548$	36.3083	1.55101
$549$	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
$562$	−17.3981	−0.733895
$563$	− 12.1136i	− 0.510529i	−0.966871	−	0.255264i	$-0.917837\pi$
$563$	− 12.1136i	− 0.510529i	0.966871	−	0.255264i	$-0.0821625\pi$
$564$	0	0
$565$	0	0
$566$	−36.0000	−1.51319
$567$	0	0
$568$	0	0
$569$	− 46.9366i	− 1.96769i	−0.179034	−	0.983843i	$-0.557297\pi$
$569$	− 46.9366i	− 1.96769i	0.179034	−	0.983843i	$-0.442703\pi$
$570$	0	0
$571$	− 42.4264i	− 1.77549i	−0.460336	−	0.887745i	$-0.652271\pi$
$571$	− 42.4264i	− 1.77549i	0.460336	−	0.887745i	$-0.347729\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	23.9842	0.999341
$577$	42.7865	1.78123	0.890613	−	0.454762i	$-0.150276\pi$
$577$	42.7865	1.78123	0.890613	+	0.454762i	$0.150276\pi$
$578$	− 1.88382i	− 0.0783564i
$579$	0.754745i	0.0313661i
$580$	0	0
$581$	0	0
$582$	1.13622i	0.0470978i
$583$	0	0
$584$	35.5078	1.46932
$585$	0	0
$586$	0	0
$587$	−47.5840	−1.96400	−0.982001	−	0.188876i	$-0.939516\pi$
$587$	−47.5840	−1.96400	−0.982001	+	0.188876i	$0.939516\pi$
$588$	0.622633	0.0256769
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	31.1035i	1.27727i	0.769510	+	0.638634i	$0.220500\pi$
$593$	31.1035i	1.27727i	−0.769510	+	0.638634i	$0.779500\pi$
$594$	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			$\pi$
$600$	0.628954i	0.0256769i
$601$	13.3086i	0.542871i	0.962457	+	0.271436i	$0.0874984\pi$
$601$	13.3086i	0.542871i	−0.962457	+	0.271436i	$0.912502\pi$
$602$	0	0
$603$	−37.1641	−1.51344
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	25.5928	1.03792
$609$	0	0
$610$	0	0
$611$	0	0
$612$	23.7340i	0.959390i
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	−18.5544	−0.748796
$615$	0	0
$616$	0	0
$617$	28.3894	1.14291	0.571457	−	0.820632i	$-0.306378\pi$
$617$	28.3894	1.14291	0.571457	+	0.820632i	$0.306378\pi$
$618$	0	0
$619$	48.3844	1.94473	0.972366	−	0.233463i	$-0.0750058\pi$
$619$	48.3844	1.94473	0.972366	+	0.233463i	$0.0750058\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	− 41.2806i	− 1.64990i
$627$	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
$633$	0.549928i	0.0218577i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	17.3536	0.685427	0.342714	−	0.939440i	$-0.388654\pi$
$641$	17.3536	0.685427	0.342714	+	0.939440i	$0.388654\pi$
$642$	−0.785627	−0.0310062
$643$	−35.4633	−1.39854	−0.699269	−	0.714859i	$-0.746491\pi$
$643$	−35.4633	−1.39854	−0.699269	+	0.714859i	$0.746491\pi$
$644$	0	0
$645$	0	0
$646$	25.3258i	0.996431i
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	25.4055i	0.998023i
$649$	0	0
$650$	0	0
$651$	0	0
$652$	−47.1838	−1.84786
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	50.7188i	1.98024i
$657$	37.6369i	1.46835i
$658$	0	0
$659$	2.26047i	0.0880555i	0.999030	+	0.0440278i	$0.0140190\pi$
$659$	2.26047i	0.0880555i	−0.999030	+	0.0440278i	$0.985981\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	− 50.9075i	− 1.97857i
$663$	0	0
$664$	−36.3973	−1.41249
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	− 6.22201i	− 0.239841i	−0.992784	−	0.119920i	$-0.961736\pi$
$673$	− 6.22201i	− 0.239841i	0.992784	−	0.119920i	$-0.0382640\pi$
$674$	−3.99714	−0.153964
$675$	−1.33377	−0.0513369
$676$	26.0000	1.00000
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	− 0.326910i	− 0.0125549i
$679$	0	0
$680$	0	0
$681$	1.30778i	0.0501144i
$682$	0	0
$683$	−42.0000	−1.60709	−0.803543	−	0.595247i	$-0.797054\pi$
$683$	−42.0000	−1.60709	−0.803543	+	0.595247i	$0.797054\pi$
$684$	27.1274i	1.03724i
$685$	0	0
$686$	0	0
$687$	0	0
$688$	50.9704i	1.94323i
$689$	0	0
$690$	0	0
$691$	9.99516	0.380234	0.190117	−	0.981761i	$-0.439113\pi$
$691$	9.99516	0.380234	0.190117	+	0.981761i	$0.439113\pi$
$692$	0	0
$693$	0	0
$694$	−7.99890	−0.303634
$695$	0	0
$696$	0	0
$697$	−50.1897	−1.90107
$698$	0	0
$699$	0.580697i	0.0219640i
$700$	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
$706$	43.5443i	1.63881i
$707$	0	0
$708$	−1.03142	−0.0387633
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	− 50.5930i	− 1.89605i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	48.7845	1.82316
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	0	0
$722$	2.07673i	0.0772878i
$723$	− 1.33295i	− 0.0495730i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	−26.8933	−0.996046
$730$	0	0
$731$	−50.4387	−1.86554
$732$	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
$738$	−53.7599	−1.97893
$739$	− 19.2254i	− 0.707219i	−0.935393	−	0.353610i	$-0.884954\pi$
$739$	− 19.2254i	− 0.707219i	0.935393	−	0.353610i	$-0.115046\pi$
$740$	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
$747$	− 38.5797i	− 1.41156i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0.266843	0.00972428
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	− 50.5301i	− 1.83534i
$759$	0	0
$760$	0	0
$761$	54.8532i	1.98843i	0.107426	+	0.994213i	$0.465739\pi$
$761$	54.8532i	1.98843i	−0.107426	+	0.994213i	$0.534261\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0.711580	0.0256769
$769$	50.9117i	1.83592i	0.396670	+	0.917961i	$0.370166\pi$
$769$	50.9117i	1.83592i	−0.396670	+	0.917961i	$0.629834\pi$
$770$	0	0
$771$	−0.783649	−0.0282224
$772$	− 33.9411i	− 1.22157i
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	−54.0266	−1.94195
$775$	0	0
$776$	− 51.0961i	− 1.83424i
$777$	0	0
$778$	0	0
$779$	−57.3656	−2.05534
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−28.0000	−1.00000
$785$	0	0
$786$	1.35157	0.0482089
$787$	55.4191i	1.97548i	0.156114	+	0.987739i	$0.450103\pi$
$787$	55.4191i	1.97548i	−0.156114	+	0.987739i	$0.549897\pi$
$788$	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
$801$	53.6266	1.89480
$802$	− 26.0512i	− 0.919901i
$803$	0	0
$804$	−1.10261	−0.0388862
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	23.1870i	0.815211i	0.913158	+	0.407605i	$0.133636\pi$
$809$	23.1870i	0.815211i	−0.913158	+	0.407605i	$0.866364\pi$
$810$	0	0
$811$	11.9878i	0.420950i	0.977599	+	0.210475i	$0.0675011\pi$
$811$	11.9878i	0.420950i	−0.977599	+	0.210475i	$0.932499\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0.704158i	0.0246504i
$817$	−57.6501	−2.01692
$818$	−48.0000	−1.67828
$819$	0	0
$820$	0	0
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	1.14181i	0.0398253i
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	45.2388i	1.57311i	0.617521	+	0.786554i	$0.288137\pi$
$827$	45.2388i	1.57311i	−0.617521	+	0.786554i	$0.711863\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 27.7079i	− 0.960023i
$834$	−0.533685	−0.0184800
$835$	0	0
$836$	0	0
$837$	0	0
$838$	51.1590i	1.76726i
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	− 0.547131i	− 0.0188442i
$844$	− 24.7304i	− 0.851257i
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	− 1.13212i	− 0.0388542i
$850$	27.9892	0.960023
$851$	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
$856$	35.3299	1.20755
$857$	13.4344i	0.458912i	0.973319	+	0.229456i	$0.0736947\pi$
$857$	13.4344i	0.458912i	−0.973319	+	0.229456i	$0.926305\pi$
$858$	0	0
$859$	25.9930	0.886869	0.443434	−	0.896307i	$-0.353760\pi$
$859$	25.9930	0.886869	0.443434	+	0.896307i	$0.353760\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	1.50899i	0.0513369i
$865$	0	0
$866$	6.21875i	0.211322i
$867$	0.0592417	0.00201195
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	54.1599	1.83304
$874$	0	0
$875$	0	0
$876$	1.11664i	0.0377277i
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	18.6878	0.629610	0.314805	−	0.949156i	$-0.398061\pi$
$881$	18.6878	0.629610	0.314805	+	0.949156i	$0.398061\pi$
$882$	− 29.6789i	− 0.999341i
$883$	−36.5307	−1.22936	−0.614678	−	0.788778i	$-0.710714\pi$
$883$	−36.5307	−1.22936	−0.614678	+	0.788778i	$0.710714\pi$
$884$	0	0
$885$	0	0
$886$	− 50.4043i	− 1.69337i
$887$	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
$898$	− 10.7462i	− 0.358606i
$899$	0	0
$900$	29.9802	0.999341
$901$	0	0
$902$	0	0
$903$	0	0
$904$	14.7012i	0.488956i
$905$	0	0
$906$	0	0
$907$	−59.5800	−1.97832	−0.989161	−	0.146832i	$-0.953092\pi$
$907$	−59.5800	−1.97832	−0.989161	+	0.146832i	$0.953092\pi$
$908$	− 58.8115i	− 1.95173i
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0.804834i	0.0266507i
$913$	0	0
$914$	17.2202	0.569594
$915$	0	0
$916$	0	0
$917$	0	0
$918$	−1.49325	−0.0492846
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	0	0
$921$	− 0.583494i	− 0.0192268i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−43.5869	−1.43004	−0.715019	−	0.699105i	$-0.753582\pi$
$929$	−43.5869	−1.43004	−0.715019	+	0.699105i	$0.753582\pi$
$930$	0	0
$931$	− 31.6695i	− 1.03792i
$932$	− 26.1141i	− 0.855397i
$933$	0	0
$934$	− 42.4264i	− 1.38823i
$935$	0	0
$936$	0	0
$937$	− 48.0685i	− 1.57033i	−0.619287	−	0.785165i	$-0.712578\pi$
$937$	− 48.0685i	− 1.57033i	0.619287	−	0.785165i	$-0.287422\pi$
$938$	0	0
$939$	1.29818	0.0423645
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	0	0
$944$	46.3835	1.50965
$945$	0	0
$946$	0	0
$947$	−60.3804	−1.96210	−0.981050	−	0.193757i	$-0.937933\pi$
$947$	−60.3804	−1.96210	−0.981050	+	0.193757i	$0.937933\pi$
$948$	0	0
$949$	0	0
$950$	31.9910	1.03792
$951$	0	0
$952$	0	0
$953$	11.9249i	0.386287i	0.981171	+	0.193143i	$0.0618683\pi$
$953$	11.9249i	0.386287i	−0.981171	+	0.193143i	$0.938132\pi$
$954$	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
$963$	37.4483i	1.20676i
$964$	59.9433i	1.93064i
$965$	0	0
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	0	0
$969$	−0.796439	−0.0255853
$970$	0	0
$971$	54.0000	1.73294	0.866471	−	0.499227i	$-0.166383\pi$
$971$	54.0000	1.73294	0.866471	+	0.499227i	$0.166383\pi$
$972$	−2.39947	−0.0769631
$973$	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	$0.469402\pi$
$978$	− 1.48382i	− 0.0474474i
$979$	0	0
$980$	0	0
$981$	0	0
$982$	−18.7323	−0.597772
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	−1.59499	−0.0508464
$985$	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
$993$	1.60092	0.0508037
$994$	0	0
$995$	0	0
$996$	− 1.14461i	− 0.0362684i
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	− 51.2848i	− 1.62339i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	968.2.g.a.483.3		8
4.3	odd	2		3872.2.g.b.1935.4		8
8.3	odd	2	CM	968.2.g.a.483.3		8
8.5	even	2		3872.2.g.b.1935.4		8
11.2	odd	10		88.2.k.a.51.2	yes	8
11.3	even	5		968.2.k.c.475.1		8
11.4	even	5		968.2.k.d.699.2		8
11.5	even	5		88.2.k.a.19.2	✓	8
11.6	odd	10		968.2.k.b.723.1		8
11.7	odd	10		968.2.k.c.699.1		8
11.8	odd	10		968.2.k.d.475.2		8
11.9	even	5		968.2.k.b.403.1		8
11.10	odd	2	inner	968.2.g.a.483.7		8
33.2	even	10		792.2.bp.a.667.1		8
33.5	odd	10		792.2.bp.a.19.1		8
44.27	odd	10		352.2.s.a.239.1		8
44.35	even	10		352.2.s.a.271.1		8
44.43	even	2		3872.2.g.b.1935.3		8
88.3	odd	10		968.2.k.c.475.1		8
88.5	even	10		352.2.s.a.239.1		8
88.13	odd	10		352.2.s.a.271.1		8
88.19	even	10		968.2.k.d.475.2		8
88.21	odd	2		3872.2.g.b.1935.3		8
88.27	odd	10		88.2.k.a.19.2	✓	8
88.35	even	10		88.2.k.a.51.2	yes	8
88.43	even	2	inner	968.2.g.a.483.7		8
88.51	even	10		968.2.k.c.699.1		8
88.59	odd	10		968.2.k.d.699.2		8
88.75	odd	10		968.2.k.b.403.1		8
88.83	even	10		968.2.k.b.723.1		8
264.35	odd	10		792.2.bp.a.667.1		8
264.203	even	10		792.2.bp.a.19.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.2.k.a.19.2	✓	8	11.5	even	5
88.2.k.a.19.2	✓	8	88.27	odd	10
88.2.k.a.51.2	yes	8	11.2	odd	10
88.2.k.a.51.2	yes	8	88.35	even	10
352.2.s.a.239.1		8	44.27	odd	10
352.2.s.a.239.1		8	88.5	even	10
352.2.s.a.271.1		8	44.35	even	10
352.2.s.a.271.1		8	88.13	odd	10
792.2.bp.a.19.1		8	33.5	odd	10
792.2.bp.a.19.1		8	264.203	even	10
792.2.bp.a.667.1		8	33.2	even	10
792.2.bp.a.667.1		8	264.35	odd	10
968.2.g.a.483.3		8	1.1	even	1	trivial
968.2.g.a.483.3		8	8.3	odd	2	CM
968.2.g.a.483.7		8	11.10	odd	2	inner
968.2.g.a.483.7		8	88.43	even	2	inner
968.2.k.b.403.1		8	11.9	even	5
968.2.k.b.403.1		8	88.75	odd	10
968.2.k.b.723.1		8	11.6	odd	10
968.2.k.b.723.1		8	88.83	even	10
968.2.k.c.475.1		8	11.3	even	5
968.2.k.c.475.1		8	88.3	odd	10
968.2.k.c.699.1		8	11.7	odd	10
968.2.k.c.699.1		8	88.51	even	10
968.2.k.d.475.2		8	11.8	odd	10
968.2.k.d.475.2		8	88.19	even	10
968.2.k.d.699.2		8	11.4	even	5
968.2.k.d.699.2		8	88.59	odd	10
3872.2.g.b.1935.3		8	44.43	even	2
3872.2.g.b.1935.3		8	88.21	odd	2
3872.2.g.b.1935.4		8	4.3	odd	2
3872.2.g.b.1935.4		8	8.5	even	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)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} +0.0444738 q^{3} -2.00000 q^{4} -0.0628954i q^{6} +2.82843i q^{8} -2.99802 q^{9} -0.0889475 q^{12} +4.00000 q^{16} +3.95827i q^{17} +4.23984i q^{18} +4.52421i q^{19} +0.125791i q^{24} +5.00000 q^{25} -0.266755 q^{27} -5.65685i q^{32} +5.59785 q^{34} +5.99604 q^{36} +6.39820 q^{38} +12.6797i q^{41} +12.7426i q^{43} +0.177895 q^{48} -7.00000 q^{49} -7.07107i q^{50} +0.176039i q^{51} +0.377248i q^{54} +0.201209i q^{57} +11.5959 q^{59} -8.00000 q^{64} +12.3962 q^{67} -7.91655i q^{68} -8.47969i q^{72} -12.5539i q^{73} +0.222369 q^{75} -9.04842i q^{76} +8.98220 q^{81} +17.9318 q^{82} +12.8684i q^{83} +18.0207 q^{86} -17.8873 q^{89} -0.251582i q^{96} -18.0652 q^{97} +9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{3} - 16 q^{4} + 28 q^{9} + 8 q^{12} + 32 q^{16} + 40 q^{25} + 8 q^{27} + 16 q^{34} - 56 q^{36} - 24 q^{38} - 16 q^{48} - 56 q^{49} + 12 q^{59} - 64 q^{64} - 28 q^{67} - 20 q^{75} + 112 q^{81}+ \cdots - 20 q^{97}+O(q^{100}) \) 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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Groups

Database

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