LMFDB - Embedded newform 784.2.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,2,Mod(1,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("784.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	784.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:	$6.26027151847$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 392)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	784.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843 q^{3} +2.82843 q^{5} +5.00000 q^{9} +4.00000 q^{11} -2.82843 q^{13} -8.00000 q^{15} -5.65685 q^{17} +2.82843 q^{19} +3.00000 q^{25} -5.65685 q^{27} +2.00000 q^{29} +5.65685 q^{31} -11.3137 q^{33} +10.0000 q^{37} +8.00000 q^{39} +5.65685 q^{41} +4.00000 q^{43} +14.1421 q^{45} -5.65685 q^{47} +16.0000 q^{51} +6.00000 q^{53} +11.3137 q^{55} -8.00000 q^{57} +2.82843 q^{59} +14.1421 q^{61} -8.00000 q^{65} -12.0000 q^{67} -8.48528 q^{75} -8.00000 q^{79} +1.00000 q^{81} +14.1421 q^{83} -16.0000 q^{85} -5.65685 q^{87} -16.0000 q^{93} +8.00000 q^{95} +5.65685 q^{97} +20.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 10 q^{9} + 8 q^{11} - 16 q^{15} + 6 q^{25} + 4 q^{29} + 20 q^{37} + 16 q^{39} + 8 q^{43} + 32 q^{51} + 12 q^{53} - 16 q^{57} - 16 q^{65} - 24 q^{67} - 16 q^{79} + 2 q^{81} - 32 q^{85} - 32 q^{93}+ \cdots + 40 q^{99}+O(q^{100}) $ 2 * q + 10 * q^9 + 8 * q^11 - 16 * q^15 + 6 * q^25 + 4 * q^29 + 20 * q^37 + 16 * q^39 + 8 * q^43 + 32 * q^51 + 12 * q^53 - 16 * q^57 - 16 * q^65 - 24 * q^67 - 16 * q^79 + 2 * q^81 - 32 * q^85 - 32 * q^93 + 16 * q^95 + 40 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−2.82843	−1.63299	−0.816497	−	0.577350i	$-0.804087\pi$
$3$	−2.82843	−1.63299	−0.816497	+	0.577350i	$0.804087\pi$
$4$	0	0
$5$	2.82843	1.26491	0.632456	−	0.774597i	$-0.282047\pi$
$5$	2.82843	1.26491	0.632456	+	0.774597i	$0.282047\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	5.00000	1.66667
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	−2.82843	−0.784465	−0.392232	−	0.919866i	$-0.628297\pi$
$13$	−2.82843	−0.784465	−0.392232	+	0.919866i	$0.628297\pi$
$14$	0	0
$15$	−8.00000	−2.06559
$16$	0	0
$17$	−5.65685	−1.37199	−0.685994	−	0.727607i	$-0.740633\pi$
$17$	−5.65685	−1.37199	−0.685994	+	0.727607i	$0.740633\pi$
$18$	0	0
$19$	2.82843	0.648886	0.324443	−	0.945905i	$-0.394823\pi$
$19$	2.82843	0.648886	0.324443	+	0.945905i	$0.394823\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	0	0
$27$	−5.65685	−1.08866
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	5.65685	1.01600	0.508001	−	0.861357i	$-0.330385\pi$
$31$	5.65685	1.01600	0.508001	+	0.861357i	$0.330385\pi$
$32$	0	0
$33$	−11.3137	−1.96946
$34$	0	0
$35$	0	0
$36$	0	0
$37$	10.0000	1.64399	0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000	1.64399	0.821995	+	0.569495i	$0.192861\pi$
$38$	0	0
$39$	8.00000	1.28103
$40$	0	0
$41$	5.65685	0.883452	0.441726	−	0.897150i	$-0.354366\pi$
$41$	5.65685	0.883452	0.441726	+	0.897150i	$0.354366\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	14.1421	2.10819
$46$	0	0
$47$	−5.65685	−0.825137	−0.412568	−	0.910927i	$-0.635368\pi$
$47$	−5.65685	−0.825137	−0.412568	+	0.910927i	$0.635368\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	16.0000	2.24045
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	11.3137	1.52554
$56$	0	0
$57$	−8.00000	−1.05963
$58$	0	0
$59$	2.82843	0.368230	0.184115	−	0.982905i	$-0.441058\pi$
$59$	2.82843	0.368230	0.184115	+	0.982905i	$0.441058\pi$
$60$	0	0
$61$	14.1421	1.81071	0.905357	−	0.424650i	$-0.139603\pi$
$61$	14.1421	1.81071	0.905357	+	0.424650i	$0.139603\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−8.00000	−0.992278
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	0	0
$75$	−8.48528	−0.979796
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	14.1421	1.55230	0.776151	−	0.630548i	$-0.217170\pi$
$83$	14.1421	1.55230	0.776151	+	0.630548i	$0.217170\pi$
$84$	0	0
$85$	−16.0000	−1.73544
$86$	0	0
$87$	−5.65685	−0.606478
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−16.0000	−1.65912
$94$	0	0
$95$	8.00000	0.820783
$96$	0	0
$97$	5.65685	0.574367	0.287183	−	0.957876i	$-0.407281\pi$
$97$	5.65685	0.574367	0.287183	+	0.957876i	$0.407281\pi$
$98$	0	0
$99$	20.0000	2.01008
$100$	0	0
$101$	−8.48528	−0.844317	−0.422159	−	0.906522i	$-0.638727\pi$
$101$	−8.48528	−0.844317	−0.422159	+	0.906522i	$0.638727\pi$
$102$	0	0
$103$	16.9706	1.67216	0.836080	−	0.548608i	$-0.184842\pi$
$103$	16.9706	1.67216	0.836080	+	0.548608i	$0.184842\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	−28.2843	−2.68462
$112$	0	0
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−14.1421	−1.30744
$118$	0	0
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	−16.0000	−1.44267
$124$	0	0
$125$	−5.65685	−0.505964
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	0	0
$129$	−11.3137	−0.996116
$130$	0	0
$131$	14.1421	1.23560	0.617802	−	0.786334i	$-0.288023\pi$
$131$	14.1421	1.23560	0.617802	+	0.786334i	$0.288023\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−16.0000	−1.37706
$136$	0	0
$137$	−10.0000	−0.854358	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000	−0.854358	−0.427179	+	0.904167i	$0.640493\pi$
$138$	0	0
$139$	−14.1421	−1.19952	−0.599760	−	0.800180i	$-0.704737\pi$
$139$	−14.1421	−1.19952	−0.599760	+	0.800180i	$0.704737\pi$
$140$	0	0
$141$	16.0000	1.34744
$142$	0	0
$143$	−11.3137	−0.946100
$144$	0	0
$145$	5.65685	0.469776
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	0	0
$153$	−28.2843	−2.28665
$154$	0	0
$155$	16.0000	1.28515
$156$	0	0
$157$	−14.1421	−1.12867	−0.564333	−	0.825547i	$-0.690866\pi$
$157$	−14.1421	−1.12867	−0.564333	+	0.825547i	$0.690866\pi$
$158$	0	0
$159$	−16.9706	−1.34585
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−20.0000	−1.56652	−0.783260	−	0.621694i	$-0.786445\pi$
$163$	−20.0000	−1.56652	−0.783260	+	0.621694i	$0.786445\pi$
$164$	0	0
$165$	−32.0000	−2.49120
$166$	0	0
$167$	5.65685	0.437741	0.218870	−	0.975754i	$-0.429763\pi$
$167$	5.65685	0.437741	0.218870	+	0.975754i	$0.429763\pi$
$168$	0	0
$169$	−5.00000	−0.384615
$170$	0	0
$171$	14.1421	1.08148
$172$	0	0
$173$	−2.82843	−0.215041	−0.107521	−	0.994203i	$-0.534291\pi$
$173$	−2.82843	−0.215041	−0.107521	+	0.994203i	$0.534291\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−8.00000	−0.601317
$178$	0	0
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	0	0
$181$	8.48528	0.630706	0.315353	−	0.948974i	$-0.397877\pi$
$181$	8.48528	0.630706	0.315353	+	0.948974i	$0.397877\pi$
$182$	0	0
$183$	−40.0000	−2.95689
$184$	0	0
$185$	28.2843	2.07950
$186$	0	0
$187$	−22.6274	−1.65468
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−24.0000	−1.73658	−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	0	0
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	0	0
$195$	22.6274	1.62038
$196$	0	0
$197$	22.0000	1.56744	0.783718	−	0.621117i	$-0.213321\pi$
$197$	22.0000	1.56744	0.783718	+	0.621117i	$0.213321\pi$
$198$	0	0
$199$	−16.9706	−1.20301	−0.601506	−	0.798869i	$-0.705432\pi$
$199$	−16.9706	−1.20301	−0.601506	+	0.798869i	$0.705432\pi$
$200$	0	0
$201$	33.9411	2.39402
$202$	0	0
$203$	0	0
$204$	0	0
$205$	16.0000	1.11749
$206$	0	0
$207$	0	0
$208$	0	0
$209$	11.3137	0.782586
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	11.3137	0.771589
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	16.0000	1.07628
$222$	0	0
$223$	11.3137	0.757622	0.378811	−	0.925474i	$-0.376333\pi$
$223$	11.3137	0.757622	0.378811	+	0.925474i	$0.376333\pi$
$224$	0	0
$225$	15.0000	1.00000
$226$	0	0
$227$	−8.48528	−0.563188	−0.281594	−	0.959534i	$-0.590863\pi$
$227$	−8.48528	−0.563188	−0.281594	+	0.959534i	$0.590863\pi$
$228$	0	0
$229$	−25.4558	−1.68217	−0.841085	−	0.540903i	$-0.818082\pi$
$229$	−25.4558	−1.68217	−0.841085	+	0.540903i	$0.818082\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	0	0
$235$	−16.0000	−1.04372
$236$	0	0
$237$	22.6274	1.46981
$238$	0	0
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	28.2843	1.82195	0.910975	−	0.412461i	$-0.135331\pi$
$241$	28.2843	1.82195	0.910975	+	0.412461i	$0.135331\pi$
$242$	0	0
$243$	14.1421	0.907218
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−8.00000	−0.509028
$248$	0	0
$249$	−40.0000	−2.53490
$250$	0	0
$251$	−14.1421	−0.892644	−0.446322	−	0.894873i	$-0.647266\pi$
$251$	−14.1421	−0.892644	−0.446322	+	0.894873i	$0.647266\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	45.2548	2.83397
$256$	0	0
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	10.0000	0.618984
$262$	0	0
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	0	0
$265$	16.9706	1.04249
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−14.1421	−0.862261	−0.431131	−	0.902290i	$-0.641885\pi$
$269$	−14.1421	−0.862261	−0.431131	+	0.902290i	$0.641885\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	12.0000	0.723627
$276$	0	0
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	0	0
$279$	28.2843	1.69334
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	2.82843	0.168133	0.0840663	−	0.996460i	$-0.473209\pi$
$283$	2.82843	0.168133	0.0840663	+	0.996460i	$0.473209\pi$
$284$	0	0
$285$	−22.6274	−1.34033
$286$	0	0
$287$	0	0
$288$	0	0
$289$	15.0000	0.882353
$290$	0	0
$291$	−16.0000	−0.937937
$292$	0	0
$293$	2.82843	0.165238	0.0826192	−	0.996581i	$-0.473671\pi$
$293$	2.82843	0.165238	0.0826192	+	0.996581i	$0.473671\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	0	0
$297$	−22.6274	−1.31298
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	24.0000	1.37876
$304$	0	0
$305$	40.0000	2.29039
$306$	0	0
$307$	−8.48528	−0.484281	−0.242140	−	0.970241i	$-0.577849\pi$
$307$	−8.48528	−0.484281	−0.242140	+	0.970241i	$0.577849\pi$
$308$	0	0
$309$	−48.0000	−2.73062
$310$	0	0
$311$	22.6274	1.28308	0.641542	−	0.767088i	$-0.278295\pi$
$311$	22.6274	1.28308	0.641542	+	0.767088i	$0.278295\pi$
$312$	0	0
$313$	−16.9706	−0.959233	−0.479616	−	0.877478i	$-0.659224\pi$
$313$	−16.9706	−0.959233	−0.479616	+	0.877478i	$0.659224\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	0	0
$319$	8.00000	0.447914
$320$	0	0
$321$	−33.9411	−1.89441
$322$	0	0
$323$	−16.0000	−0.890264
$324$	0	0
$325$	−8.48528	−0.470679
$326$	0	0
$327$	−5.65685	−0.312825
$328$	0	0
$329$	0	0
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	0	0
$333$	50.0000	2.73998
$334$	0	0
$335$	−33.9411	−1.85440
$336$	0	0
$337$	−10.0000	−0.544735	−0.272367	−	0.962193i	$-0.587807\pi$
$337$	−10.0000	−0.544735	−0.272367	+	0.962193i	$0.587807\pi$
$338$	0	0
$339$	−28.2843	−1.53619
$340$	0	0
$341$	22.6274	1.22534
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	20.0000	1.07366	0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000	1.07366	0.536828	+	0.843692i	$0.319622\pi$
$348$	0	0
$349$	−25.4558	−1.36262	−0.681310	−	0.731995i	$-0.738589\pi$
$349$	−25.4558	−1.36262	−0.681310	+	0.731995i	$0.738589\pi$
$350$	0	0
$351$	16.0000	0.854017
$352$	0	0
$353$	11.3137	0.602168	0.301084	−	0.953598i	$-0.402652\pi$
$353$	11.3137	0.602168	0.301084	+	0.953598i	$0.402652\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	0	0
$363$	−14.1421	−0.742270
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−33.9411	−1.77171	−0.885856	−	0.463960i	$-0.846428\pi$
$367$	−33.9411	−1.77171	−0.885856	+	0.463960i	$0.846428\pi$
$368$	0	0
$369$	28.2843	1.47242
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	0	0
$375$	16.0000	0.826236
$376$	0	0
$377$	−5.65685	−0.291343
$378$	0	0
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	0	0
$381$	22.6274	1.15924
$382$	0	0
$383$	−28.2843	−1.44526	−0.722629	−	0.691236i	$-0.757067\pi$
$383$	−28.2843	−1.44526	−0.722629	+	0.691236i	$0.757067\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	20.0000	1.01666
$388$	0	0
$389$	−22.0000	−1.11544	−0.557722	−	0.830028i	$-0.688325\pi$
$389$	−22.0000	−1.11544	−0.557722	+	0.830028i	$0.688325\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−40.0000	−2.01773
$394$	0	0
$395$	−22.6274	−1.13851
$396$	0	0
$397$	8.48528	0.425864	0.212932	−	0.977067i	$-0.431699\pi$
$397$	8.48528	0.425864	0.212932	+	0.977067i	$0.431699\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	0	0
$403$	−16.0000	−0.797017
$404$	0	0
$405$	2.82843	0.140546
$406$	0	0
$407$	40.0000	1.98273
$408$	0	0
$409$	−16.9706	−0.839140	−0.419570	−	0.907723i	$-0.637819\pi$
$409$	−16.9706	−0.839140	−0.419570	+	0.907723i	$0.637819\pi$
$410$	0	0
$411$	28.2843	1.39516
$412$	0	0
$413$	0	0
$414$	0	0
$415$	40.0000	1.96352
$416$	0	0
$417$	40.0000	1.95881
$418$	0	0
$419$	−2.82843	−0.138178	−0.0690889	−	0.997611i	$-0.522009\pi$
$419$	−2.82843	−0.138178	−0.0690889	+	0.997611i	$0.522009\pi$
$420$	0	0
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	0	0
$423$	−28.2843	−1.37523
$424$	0	0
$425$	−16.9706	−0.823193
$426$	0	0
$427$	0	0
$428$	0	0
$429$	32.0000	1.54497
$430$	0	0
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	0	0
$433$	5.65685	0.271851	0.135926	−	0.990719i	$-0.456599\pi$
$433$	5.65685	0.271851	0.135926	+	0.990719i	$0.456599\pi$
$434$	0	0
$435$	−16.0000	−0.767141
$436$	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
$441$	0	0
$442$	0	0
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	28.2843	1.33780
$448$	0	0
$449$	2.00000	0.0943858	0.0471929	−	0.998886i	$-0.484972\pi$
$449$	2.00000	0.0943858	0.0471929	+	0.998886i	$0.484972\pi$
$450$	0	0
$451$	22.6274	1.06548
$452$	0	0
$453$	45.2548	2.12626
$454$	0	0
$455$	0	0
$456$	0	0
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	0	0
$459$	32.0000	1.49363
$460$	0	0
$461$	−14.1421	−0.658665	−0.329332	−	0.944214i	$-0.606824\pi$
$461$	−14.1421	−0.658665	−0.329332	+	0.944214i	$0.606824\pi$
$462$	0	0
$463$	−40.0000	−1.85896	−0.929479	−	0.368875i	$-0.879743\pi$
$463$	−40.0000	−1.85896	−0.929479	+	0.368875i	$0.879743\pi$
$464$	0	0
$465$	−45.2548	−2.09864
$466$	0	0
$467$	−8.48528	−0.392652	−0.196326	−	0.980539i	$-0.562901\pi$
$467$	−8.48528	−0.392652	−0.196326	+	0.980539i	$0.562901\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	40.0000	1.84310
$472$	0	0
$473$	16.0000	0.735681
$474$	0	0
$475$	8.48528	0.389331
$476$	0	0
$477$	30.0000	1.37361
$478$	0	0
$479$	28.2843	1.29234	0.646171	−	0.763193i	$-0.276369\pi$
$479$	28.2843	1.29234	0.646171	+	0.763193i	$0.276369\pi$
$480$	0	0
$481$	−28.2843	−1.28965
$482$	0	0
$483$	0	0
$484$	0	0
$485$	16.0000	0.726523
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	0	0
$489$	56.5685	2.55812
$490$	0	0
$491$	−4.00000	−0.180517	−0.0902587	−	0.995918i	$-0.528769\pi$
$491$	−4.00000	−0.180517	−0.0902587	+	0.995918i	$0.528769\pi$
$492$	0	0
$493$	−11.3137	−0.509544
$494$	0	0
$495$	56.5685	2.54257
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−28.0000	−1.25345	−0.626726	−	0.779240i	$-0.715605\pi$
$499$	−28.0000	−1.25345	−0.626726	+	0.779240i	$0.715605\pi$
$500$	0	0
$501$	−16.0000	−0.714827
$502$	0	0
$503$	−11.3137	−0.504453	−0.252227	−	0.967668i	$-0.581163\pi$
$503$	−11.3137	−0.504453	−0.252227	+	0.967668i	$0.581163\pi$
$504$	0	0
$505$	−24.0000	−1.06799
$506$	0	0
$507$	14.1421	0.628074
$508$	0	0
$509$	−14.1421	−0.626839	−0.313420	−	0.949615i	$-0.601475\pi$
$509$	−14.1421	−0.626839	−0.313420	+	0.949615i	$0.601475\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−16.0000	−0.706417
$514$	0	0
$515$	48.0000	2.11513
$516$	0	0
$517$	−22.6274	−0.995153
$518$	0	0
$519$	8.00000	0.351161
$520$	0	0
$521$	−5.65685	−0.247831	−0.123916	−	0.992293i	$-0.539545\pi$
$521$	−5.65685	−0.247831	−0.123916	+	0.992293i	$0.539545\pi$
$522$	0	0
$523$	−19.7990	−0.865749	−0.432875	−	0.901454i	$-0.642501\pi$
$523$	−19.7990	−0.865749	−0.432875	+	0.901454i	$0.642501\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−32.0000	−1.39394
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	14.1421	0.613716
$532$	0	0
$533$	−16.0000	−0.693037
$534$	0	0
$535$	33.9411	1.46740
$536$	0	0
$537$	−56.5685	−2.44111
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	0	0
$543$	−24.0000	−1.02994
$544$	0	0
$545$	5.65685	0.242313
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	0	0
$549$	70.7107	3.01786
$550$	0	0
$551$	5.65685	0.240990
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−80.0000	−3.39581
$556$	0	0
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	0	0
$559$	−11.3137	−0.478519
$560$	0	0
$561$	64.0000	2.70208
$562$	0	0
$563$	42.4264	1.78806	0.894030	−	0.448007i	$-0.147866\pi$
$563$	42.4264	1.78806	0.894030	+	0.448007i	$0.147866\pi$
$564$	0	0
$565$	28.2843	1.18993
$566$	0	0
$567$	0	0
$568$	0	0
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	0	0
$573$	67.8823	2.83582
$574$	0	0
$575$	0	0
$576$	0	0
$577$	33.9411	1.41299	0.706494	−	0.707719i	$-0.250276\pi$
$577$	33.9411	1.41299	0.706494	+	0.707719i	$0.250276\pi$
$578$	0	0
$579$	−28.2843	−1.17545
$580$	0	0
$581$	0	0
$582$	0	0
$583$	24.0000	0.993978
$584$	0	0
$585$	−40.0000	−1.65380
$586$	0	0
$587$	42.4264	1.75113	0.875563	−	0.483105i	$-0.160491\pi$
$587$	42.4264	1.75113	0.875563	+	0.483105i	$0.160491\pi$
$588$	0	0
$589$	16.0000	0.659269
$590$	0	0
$591$	−62.2254	−2.55961
$592$	0	0
$593$	−11.3137	−0.464598	−0.232299	−	0.972644i	$-0.574625\pi$
$593$	−11.3137	−0.464598	−0.232299	+	0.972644i	$0.574625\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	48.0000	1.96451
$598$	0	0
$599$	−32.0000	−1.30748	−0.653742	−	0.756717i	$-0.726802\pi$
$599$	−32.0000	−1.30748	−0.653742	+	0.756717i	$0.726802\pi$
$600$	0	0
$601$	−22.6274	−0.922992	−0.461496	−	0.887142i	$-0.652687\pi$
$601$	−22.6274	−0.922992	−0.461496	+	0.887142i	$0.652687\pi$
$602$	0	0
$603$	−60.0000	−2.44339
$604$	0	0
$605$	14.1421	0.574960
$606$	0	0
$607$	−22.6274	−0.918419	−0.459209	−	0.888328i	$-0.651867\pi$
$607$	−22.6274	−0.918419	−0.459209	+	0.888328i	$0.651867\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	16.0000	0.647291
$612$	0	0
$613$	−6.00000	−0.242338	−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000	−0.242338	−0.121169	+	0.992632i	$0.538664\pi$
$614$	0	0
$615$	−45.2548	−1.82485
$616$	0	0
$617$	30.0000	1.20775	0.603877	−	0.797077i	$-0.293622\pi$
$617$	30.0000	1.20775	0.603877	+	0.797077i	$0.293622\pi$
$618$	0	0
$619$	14.1421	0.568420	0.284210	−	0.958762i	$-0.408269\pi$
$619$	14.1421	0.568420	0.284210	+	0.958762i	$0.408269\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−31.0000	−1.24000
$626$	0	0
$627$	−32.0000	−1.27796
$628$	0	0
$629$	−56.5685	−2.25554
$630$	0	0
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	0	0
$633$	−11.3137	−0.449680
$634$	0	0
$635$	−22.6274	−0.897942
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−10.0000	−0.394976	−0.197488	−	0.980305i	$-0.563278\pi$
$641$	−10.0000	−0.394976	−0.197488	+	0.980305i	$0.563278\pi$
$642$	0	0
$643$	−25.4558	−1.00388	−0.501940	−	0.864902i	$-0.667380\pi$
$643$	−25.4558	−1.00388	−0.501940	+	0.864902i	$0.667380\pi$
$644$	0	0
$645$	−32.0000	−1.26000
$646$	0	0
$647$	−16.9706	−0.667182	−0.333591	−	0.942718i	$-0.608260\pi$
$647$	−16.9706	−0.667182	−0.333591	+	0.942718i	$0.608260\pi$
$648$	0	0
$649$	11.3137	0.444102
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−14.0000	−0.547862	−0.273931	−	0.961749i	$-0.588324\pi$
$653$	−14.0000	−0.547862	−0.273931	+	0.961749i	$0.588324\pi$
$654$	0	0
$655$	40.0000	1.56293
$656$	0	0
$657$	0	0
$658$	0	0
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	0	0
$661$	14.1421	0.550065	0.275033	−	0.961435i	$-0.411311\pi$
$661$	14.1421	0.550065	0.275033	+	0.961435i	$0.411311\pi$
$662$	0	0
$663$	−45.2548	−1.75755
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−32.0000	−1.23719
$670$	0	0
$671$	56.5685	2.18380
$672$	0	0
$673$	30.0000	1.15642	0.578208	−	0.815890i	$-0.303752\pi$
$673$	30.0000	1.15642	0.578208	+	0.815890i	$0.303752\pi$
$674$	0	0
$675$	−16.9706	−0.653197
$676$	0	0
$677$	8.48528	0.326116	0.163058	−	0.986616i	$-0.447864\pi$
$677$	8.48528	0.326116	0.163058	+	0.986616i	$0.447864\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	24.0000	0.919682
$682$	0	0
$683$	−20.0000	−0.765279	−0.382639	−	0.923898i	$-0.624985\pi$
$683$	−20.0000	−0.765279	−0.382639	+	0.923898i	$0.624985\pi$
$684$	0	0
$685$	−28.2843	−1.08069
$686$	0	0
$687$	72.0000	2.74697
$688$	0	0
$689$	−16.9706	−0.646527
$690$	0	0
$691$	14.1421	0.537992	0.268996	−	0.963141i	$-0.413308\pi$
$691$	14.1421	0.537992	0.268996	+	0.963141i	$0.413308\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−40.0000	−1.51729
$696$	0	0
$697$	−32.0000	−1.21209
$698$	0	0
$699$	28.2843	1.06981
$700$	0	0
$701$	34.0000	1.28416	0.642081	−	0.766637i	$-0.278071\pi$
$701$	34.0000	1.28416	0.642081	+	0.766637i	$0.278071\pi$
$702$	0	0
$703$	28.2843	1.06676
$704$	0	0
$705$	45.2548	1.70440
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−22.0000	−0.826227	−0.413114	−	0.910679i	$-0.635559\pi$
$709$	−22.0000	−0.826227	−0.413114	+	0.910679i	$0.635559\pi$
$710$	0	0
$711$	−40.0000	−1.50012
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−32.0000	−1.19673
$716$	0	0
$717$	−22.6274	−0.845036
$718$	0	0
$719$	28.2843	1.05483	0.527413	−	0.849609i	$-0.323162\pi$
$719$	28.2843	1.05483	0.527413	+	0.849609i	$0.323162\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−80.0000	−2.97523
$724$	0	0
$725$	6.00000	0.222834
$726$	0	0
$727$	−28.2843	−1.04901	−0.524503	−	0.851409i	$-0.675749\pi$
$727$	−28.2843	−1.04901	−0.524503	+	0.851409i	$0.675749\pi$
$728$	0	0
$729$	−43.0000	−1.59259
$730$	0	0
$731$	−22.6274	−0.836905
$732$	0	0
$733$	25.4558	0.940233	0.470117	−	0.882604i	$-0.344212\pi$
$733$	25.4558	0.940233	0.470117	+	0.882604i	$0.344212\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−48.0000	−1.76810
$738$	0	0
$739$	12.0000	0.441427	0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000	0.441427	0.220714	+	0.975339i	$0.429161\pi$
$740$	0	0
$741$	22.6274	0.831239
$742$	0	0
$743$	16.0000	0.586983	0.293492	−	0.955962i	$-0.405183\pi$
$743$	16.0000	0.586983	0.293492	+	0.955962i	$0.405183\pi$
$744$	0	0
$745$	−28.2843	−1.03626
$746$	0	0
$747$	70.7107	2.58717
$748$	0	0
$749$	0	0
$750$	0	0
$751$	24.0000	0.875772	0.437886	−	0.899030i	$-0.355727\pi$
$751$	24.0000	0.875772	0.437886	+	0.899030i	$0.355727\pi$
$752$	0	0
$753$	40.0000	1.45768
$754$	0	0
$755$	−45.2548	−1.64699
$756$	0	0
$757$	10.0000	0.363456	0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000	0.363456	0.181728	+	0.983349i	$0.441831\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−28.2843	−1.02530	−0.512652	−	0.858596i	$-0.671337\pi$
$761$	−28.2843	−1.02530	−0.512652	+	0.858596i	$0.671337\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−80.0000	−2.89241
$766$	0	0
$767$	−8.00000	−0.288863
$768$	0	0
$769$	16.9706	0.611974	0.305987	−	0.952036i	$-0.401014\pi$
$769$	16.9706	0.611974	0.305987	+	0.952036i	$0.401014\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−53.7401	−1.93290	−0.966449	−	0.256859i	$-0.917312\pi$
$773$	−53.7401	−1.93290	−0.966449	+	0.256859i	$0.917312\pi$
$774$	0	0
$775$	16.9706	0.609601
$776$	0	0
$777$	0	0
$778$	0	0
$779$	16.0000	0.573259
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−11.3137	−0.404319
$784$	0	0
$785$	−40.0000	−1.42766
$786$	0	0
$787$	−8.48528	−0.302468	−0.151234	−	0.988498i	$-0.548325\pi$
$787$	−8.48528	−0.302468	−0.151234	+	0.988498i	$0.548325\pi$
$788$	0	0
$789$	−45.2548	−1.61111
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−40.0000	−1.42044
$794$	0	0
$795$	−48.0000	−1.70238
$796$	0	0
$797$	8.48528	0.300564	0.150282	−	0.988643i	$-0.451982\pi$
$797$	8.48528	0.300564	0.150282	+	0.988643i	$0.451982\pi$
$798$	0	0
$799$	32.0000	1.13208
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	40.0000	1.40807
$808$	0	0
$809$	18.0000	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$810$	0	0
$811$	14.1421	0.496598	0.248299	−	0.968683i	$-0.420129\pi$
$811$	14.1421	0.496598	0.248299	+	0.968683i	$0.420129\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−56.5685	−1.98151
$816$	0	0
$817$	11.3137	0.395817
$818$	0	0
$819$	0	0
$820$	0	0
$821$	54.0000	1.88461	0.942306	−	0.334751i	$-0.108652\pi$
$821$	54.0000	1.88461	0.942306	+	0.334751i	$0.108652\pi$
$822$	0	0
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	0	0
$825$	−33.9411	−1.18168
$826$	0	0
$827$	−20.0000	−0.695468	−0.347734	−	0.937593i	$-0.613049\pi$
$827$	−20.0000	−0.695468	−0.347734	+	0.937593i	$0.613049\pi$
$828$	0	0
$829$	−31.1127	−1.08059	−0.540294	−	0.841476i	$-0.681687\pi$
$829$	−31.1127	−1.08059	−0.540294	+	0.841476i	$0.681687\pi$
$830$	0	0
$831$	28.2843	0.981170
$832$	0	0
$833$	0	0
$834$	0	0
$835$	16.0000	0.553703
$836$	0	0
$837$	−32.0000	−1.10608
$838$	0	0
$839$	−28.2843	−0.976481	−0.488241	−	0.872709i	$-0.662361\pi$
$839$	−28.2843	−0.976481	−0.488241	+	0.872709i	$0.662361\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	−16.9706	−0.584497
$844$	0	0
$845$	−14.1421	−0.486504
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−8.00000	−0.274559
$850$	0	0
$851$	0	0
$852$	0	0
$853$	42.4264	1.45265	0.726326	−	0.687350i	$-0.241226\pi$
$853$	42.4264	1.45265	0.726326	+	0.687350i	$0.241226\pi$
$854$	0	0
$855$	40.0000	1.36797
$856$	0	0
$857$	28.2843	0.966172	0.483086	−	0.875573i	$-0.339516\pi$
$857$	28.2843	0.966172	0.483086	+	0.875573i	$0.339516\pi$
$858$	0	0
$859$	42.4264	1.44757	0.723785	−	0.690025i	$-0.242401\pi$
$859$	42.4264	1.44757	0.723785	+	0.690025i	$0.242401\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	0	0
$865$	−8.00000	−0.272008
$866$	0	0
$867$	−42.4264	−1.44088
$868$	0	0
$869$	−32.0000	−1.08553
$870$	0	0
$871$	33.9411	1.15005
$872$	0	0
$873$	28.2843	0.957278
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2.00000	0.0675352	0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000	0.0675352	0.0337676	+	0.999430i	$0.489249\pi$
$878$	0	0
$879$	−8.00000	−0.269833
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	0	0
$885$	−22.6274	−0.760612
$886$	0	0
$887$	−28.2843	−0.949693	−0.474846	−	0.880069i	$-0.657496\pi$
$887$	−28.2843	−0.949693	−0.474846	+	0.880069i	$0.657496\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	4.00000	0.134005
$892$	0	0
$893$	−16.0000	−0.535420
$894$	0	0
$895$	56.5685	1.89088
$896$	0	0
$897$	0	0
$898$	0	0
$899$	11.3137	0.377333
$900$	0	0
$901$	−33.9411	−1.13074
$902$	0	0
$903$	0	0
$904$	0	0
$905$	24.0000	0.797787
$906$	0	0
$907$	−20.0000	−0.664089	−0.332045	−	0.943264i	$-0.607738\pi$
$907$	−20.0000	−0.664089	−0.332045	+	0.943264i	$0.607738\pi$
$908$	0	0
$909$	−42.4264	−1.40720
$910$	0	0
$911$	40.0000	1.32526	0.662630	−	0.748947i	$-0.269440\pi$
$911$	40.0000	1.32526	0.662630	+	0.748947i	$0.269440\pi$
$912$	0	0
$913$	56.5685	1.87215
$914$	0	0
$915$	−113.137	−3.74020
$916$	0	0
$917$	0	0
$918$	0	0
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	0	0
$921$	24.0000	0.790827
$922$	0	0
$923$	0	0
$924$	0	0
$925$	30.0000	0.986394
$926$	0	0
$927$	84.8528	2.78693
$928$	0	0
$929$	39.5980	1.29917	0.649584	−	0.760290i	$-0.274943\pi$
$929$	39.5980	1.29917	0.649584	+	0.760290i	$0.274943\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−64.0000	−2.09527
$934$	0	0
$935$	−64.0000	−2.09302
$936$	0	0
$937$	−22.6274	−0.739205	−0.369603	−	0.929190i	$-0.620506\pi$
$937$	−22.6274	−0.739205	−0.369603	+	0.929190i	$0.620506\pi$
$938$	0	0
$939$	48.0000	1.56642
$940$	0	0
$941$	14.1421	0.461020	0.230510	−	0.973070i	$-0.425960\pi$
$941$	14.1421	0.461020	0.230510	+	0.973070i	$0.425960\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	50.9117	1.65092
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	0	0
$955$	−67.8823	−2.19662
$956$	0	0
$957$	−22.6274	−0.731441
$958$	0	0
$959$	0	0
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	60.0000	1.93347
$964$	0	0
$965$	28.2843	0.910503
$966$	0	0
$967$	−48.0000	−1.54358	−0.771788	−	0.635880i	$-0.780637\pi$
$967$	−48.0000	−1.54358	−0.771788	+	0.635880i	$0.780637\pi$
$968$	0	0
$969$	45.2548	1.45379
$970$	0	0
$971$	−19.7990	−0.635380	−0.317690	−	0.948195i	$-0.602907\pi$
$971$	−19.7990	−0.635380	−0.317690	+	0.948195i	$0.602907\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	24.0000	0.768615
$976$	0	0
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	10.0000	0.319275
$982$	0	0
$983$	39.5980	1.26298	0.631490	−	0.775384i	$-0.282444\pi$
$983$	39.5980	1.26298	0.631490	+	0.775384i	$0.282444\pi$
$984$	0	0
$985$	62.2254	1.98267
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	0	0
$993$	−56.5685	−1.79515
$994$	0	0
$995$	−48.0000	−1.52170
$996$	0	0
$997$	36.7696	1.16450	0.582252	−	0.813009i	$-0.302172\pi$
$997$	36.7696	1.16450	0.582252	+	0.813009i	$0.302172\pi$
$998$	0	0
$999$	−56.5685	−1.78975

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.2.a.n.1.1		2
3.2	odd	2		7056.2.a.cj.1.1		2
4.3	odd	2		392.2.a.h.1.2	yes	2
7.2	even	3		784.2.i.k.753.2		4
7.3	odd	6		784.2.i.k.177.1		4
7.4	even	3		784.2.i.k.177.2		4
7.5	odd	6		784.2.i.k.753.1		4
7.6	odd	2	inner	784.2.a.n.1.2		2
8.3	odd	2		3136.2.a.bt.1.1		2
8.5	even	2		3136.2.a.bq.1.2		2
12.11	even	2		3528.2.a.bj.1.1		2
20.19	odd	2		9800.2.a.bw.1.1		2
21.20	even	2		7056.2.a.cj.1.2		2
28.3	even	6		392.2.i.g.177.2		4
28.11	odd	6		392.2.i.g.177.1		4
28.19	even	6		392.2.i.g.361.2		4
28.23	odd	6		392.2.i.g.361.1		4
28.27	even	2		392.2.a.h.1.1	✓	2
56.13	odd	2		3136.2.a.bq.1.1		2
56.27	even	2		3136.2.a.bt.1.2		2
84.11	even	6		3528.2.s.be.3313.2		4
84.23	even	6		3528.2.s.be.361.2		4
84.47	odd	6		3528.2.s.be.361.1		4
84.59	odd	6		3528.2.s.be.3313.1		4
84.83	odd	2		3528.2.a.bj.1.2		2
140.139	even	2		9800.2.a.bw.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.a.h.1.1	✓	2	28.27	even	2
392.2.a.h.1.2	yes	2	4.3	odd	2
392.2.i.g.177.1		4	28.11	odd	6
392.2.i.g.177.2		4	28.3	even	6
392.2.i.g.361.1		4	28.23	odd	6
392.2.i.g.361.2		4	28.19	even	6
784.2.a.n.1.1		2	1.1	even	1	trivial
784.2.a.n.1.2		2	7.6	odd	2	inner
784.2.i.k.177.1		4	7.3	odd	6
784.2.i.k.177.2		4	7.4	even	3
784.2.i.k.753.1		4	7.5	odd	6
784.2.i.k.753.2		4	7.2	even	3
3136.2.a.bq.1.1		2	56.13	odd	2
3136.2.a.bq.1.2		2	8.5	even	2
3136.2.a.bt.1.1		2	8.3	odd	2
3136.2.a.bt.1.2		2	56.27	even	2
3528.2.a.bj.1.1		2	12.11	even	2
3528.2.a.bj.1.2		2	84.83	odd	2
3528.2.s.be.361.1		4	84.47	odd	6
3528.2.s.be.361.2		4	84.23	even	6
3528.2.s.be.3313.1		4	84.59	odd	6
3528.2.s.be.3313.2		4	84.11	even	6
7056.2.a.cj.1.1		2	3.2	odd	2
7056.2.a.cj.1.2		2	21.20	even	2
9800.2.a.bw.1.1		2	20.19	odd	2
9800.2.a.bw.1.2		2	140.139	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:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	784.a (trivial)

\(f(q)\)	\(=\)	\(q-2.82843 q^{3} +2.82843 q^{5} +5.00000 q^{9} +4.00000 q^{11} -2.82843 q^{13} -8.00000 q^{15} -5.65685 q^{17} +2.82843 q^{19} +3.00000 q^{25} -5.65685 q^{27} +2.00000 q^{29} +5.65685 q^{31} -11.3137 q^{33} +10.0000 q^{37} +8.00000 q^{39} +5.65685 q^{41} +4.00000 q^{43} +14.1421 q^{45} -5.65685 q^{47} +16.0000 q^{51} +6.00000 q^{53} +11.3137 q^{55} -8.00000 q^{57} +2.82843 q^{59} +14.1421 q^{61} -8.00000 q^{65} -12.0000 q^{67} -8.48528 q^{75} -8.00000 q^{79} +1.00000 q^{81} +14.1421 q^{83} -16.0000 q^{85} -5.65685 q^{87} -16.0000 q^{93} +8.00000 q^{95} +5.65685 q^{97} +20.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 10 q^{9} + 8 q^{11} - 16 q^{15} + 6 q^{25} + 4 q^{29} + 20 q^{37} + 16 q^{39} + 8 q^{43} + 32 q^{51} + 12 q^{53} - 16 q^{57} - 16 q^{65} - 24 q^{67} - 16 q^{79} + 2 q^{81} - 32 q^{85} - 32 q^{93}+ \cdots + 40 q^{99}+O(q^{100}) \) 2 * q + 10 * q^9 + 8 * q^11 - 16 * q^15 + 6 * q^25 + 4 * q^29 + 20 * q^37 + 16 * q^39 + 8 * q^43 + 32 * q^51 + 12 * q^53 - 16 * q^57 - 16 * q^65 - 24 * q^67 - 16 * q^79 + 2 * q^81 - 32 * q^85 - 32 * q^93 + 16 * q^95 + 40 * q^99

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