LMFDB - Embedded newform 256.4.b.g.129.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,4,Mod(129,256)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(256, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("256.129");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	256.b (of order $2$, degree $1$, not 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:	$15.1044889615$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 8)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	256.129
Dual form			256.4.b.g.129.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000i q^{3} -2.00000i q^{5} +24.0000 q^{7} +11.0000 q^{9} +O(q^{10})$	$q-4.00000i q^{3} -2.00000i q^{5} +24.0000 q^{7} +11.0000 q^{9} +44.0000i q^{11} -22.0000i q^{13} -8.00000 q^{15} +50.0000 q^{17} +44.0000i q^{19} -96.0000i q^{21} -56.0000 q^{23} +121.000 q^{25} -152.000i q^{27} -198.000i q^{29} +160.000 q^{31} +176.000 q^{33} -48.0000i q^{35} -162.000i q^{37} -88.0000 q^{39} +198.000 q^{41} -52.0000i q^{43} -22.0000i q^{45} -528.000 q^{47} +233.000 q^{49} -200.000i q^{51} -242.000i q^{53} +88.0000 q^{55} +176.000 q^{57} +668.000i q^{59} -550.000i q^{61} +264.000 q^{63} -44.0000 q^{65} +188.000i q^{67} +224.000i q^{69} +728.000 q^{71} -154.000 q^{73} -484.000i q^{75} +1056.00i q^{77} +656.000 q^{79} -311.000 q^{81} +236.000i q^{83} -100.000i q^{85} -792.000 q^{87} -714.000 q^{89} -528.000i q^{91} -640.000i q^{93} +88.0000 q^{95} -478.000 q^{97} +484.000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 48 q^{7} + 22 q^{9}+O(q^{10}) $ 2 * q + 48 * q^7 + 22 * q^9	$ 2 q + 48 q^{7} + 22 q^{9} - 16 q^{15} + 100 q^{17} - 112 q^{23} + 242 q^{25} + 320 q^{31} + 352 q^{33} - 176 q^{39} + 396 q^{41} - 1056 q^{47} + 466 q^{49} + 176 q^{55} + 352 q^{57} + 528 q^{63} - 88 q^{65} + 1456 q^{71} - 308 q^{73} + 1312 q^{79} - 622 q^{81} - 1584 q^{87} - 1428 q^{89} + 176 q^{95} - 956 q^{97}+O(q^{100}) $ 2 * q + 48 * q^7 + 22 * q^9 - 16 * q^15 + 100 * q^17 - 112 * q^23 + 242 * q^25 + 320 * q^31 + 352 * q^33 - 176 * q^39 + 396 * q^41 - 1056 * q^47 + 466 * q^49 + 176 * q^55 + 352 * q^57 + 528 * q^63 - 88 * q^65 + 1456 * q^71 - 308 * q^73 + 1312 * q^79 - 622 * q^81 - 1584 * q^87 - 1428 * q^89 + 176 * q^95 - 956 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/256\mathbb{Z}\right)^\times$.

$n$	$5$	$255$
$\chi(n)$	$-1$	$1$

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 $

$2$	0	0
$2$	0	0
$3$	− 4.00000i	− 0.769800i	−0.922958	−	0.384900i	$-0.874236\pi$
$3$	− 4.00000i	− 0.769800i	0.922958	−	0.384900i	$-0.125764\pi$
$4$	0	0
$5$	− 2.00000i	− 0.178885i	−0.995992	−	0.0894427i	$-0.971491\pi$
$5$	− 2.00000i	− 0.178885i	0.995992	−	0.0894427i	$-0.0285086\pi$
$6$	0	0
$7$	24.0000	1.29588	0.647939	−	0.761692i	$-0.275631\pi$
$7$	24.0000	1.29588	0.647939	+	0.761692i	$0.275631\pi$
$8$	0	0
$9$	11.0000	0.407407
$10$	0	0
$11$	44.0000i	1.20605i	0.797724	+	0.603023i	$0.206037\pi$
$11$	44.0000i	1.20605i	−0.797724	+	0.603023i	$0.793963\pi$
$12$	0	0
$13$	− 22.0000i	− 0.469362i	−0.972072	−	0.234681i	$-0.924595\pi$
$13$	− 22.0000i	− 0.469362i	0.972072	−	0.234681i	$-0.0754045\pi$
$14$	0	0
$15$	−8.00000	−0.137706
$16$	0	0
$17$	50.0000	0.713340	0.356670	−	0.934230i	$-0.383912\pi$
$17$	50.0000	0.713340	0.356670	+	0.934230i	$0.383912\pi$
$18$	0	0
$19$	44.0000i	0.531279i	0.964072	+	0.265639i	$0.0855830\pi$
$19$	44.0000i	0.531279i	−0.964072	+	0.265639i	$0.914417\pi$
$20$	0	0
$21$	− 96.0000i	− 0.997567i
$22$	0	0
$23$	−56.0000	−0.507687	−0.253844	−	0.967245i	$-0.581695\pi$
$23$	−56.0000	−0.507687	−0.253844	+	0.967245i	$0.581695\pi$
$24$	0	0
$25$	121.000	0.968000
$26$	0	0
$27$	− 152.000i	− 1.08342i
$28$	0	0
$29$	− 198.000i	− 1.26785i	−0.773394	−	0.633925i	$-0.781443\pi$
$29$	− 198.000i	− 1.26785i	0.773394	−	0.633925i	$-0.218557\pi$
$30$	0	0
$31$	160.000	0.926995	0.463498	−	0.886098i	$-0.346594\pi$
$31$	160.000	0.926995	0.463498	+	0.886098i	$0.346594\pi$
$32$	0	0
$33$	176.000	0.928414
$34$	0	0
$35$	− 48.0000i	− 0.231814i
$36$	0	0
$37$	− 162.000i	− 0.719801i	−0.932991	−	0.359900i	$-0.882811\pi$
$37$	− 162.000i	− 0.719801i	0.932991	−	0.359900i	$-0.117189\pi$
$38$	0	0
$39$	−88.0000	−0.361315
$40$	0	0
$41$	198.000	0.754205	0.377102	−	0.926172i	$-0.376920\pi$
$41$	198.000	0.754205	0.377102	+	0.926172i	$0.376920\pi$
$42$	0	0
$43$	− 52.0000i	− 0.184417i	−0.995740	−	0.0922084i	$-0.970607\pi$
$43$	− 52.0000i	− 0.184417i	0.995740	−	0.0922084i	$-0.0293926\pi$
$44$	0	0
$45$	− 22.0000i	− 0.0728793i
$46$	0	0
$47$	−528.000	−1.63865	−0.819327	−	0.573327i	$-0.805653\pi$
$47$	−528.000	−1.63865	−0.819327	+	0.573327i	$0.805653\pi$
$48$	0	0
$49$	233.000	0.679300
$50$	0	0
$51$	− 200.000i	− 0.549129i
$52$	0	0
$53$	− 242.000i	− 0.627194i	−0.949556	−	0.313597i	$-0.898466\pi$
$53$	− 242.000i	− 0.627194i	0.949556	−	0.313597i	$-0.101534\pi$
$54$	0	0
$55$	88.0000	0.215744
$56$	0	0
$57$	176.000	0.408978
$58$	0	0
$59$	668.000i	1.47400i	0.675891	+	0.737002i	$0.263759\pi$
$59$	668.000i	1.47400i	−0.675891	+	0.737002i	$0.736241\pi$
$60$	0	0
$61$	− 550.000i	− 1.15443i	−0.816592	−	0.577215i	$-0.804139\pi$
$61$	− 550.000i	− 1.15443i	0.816592	−	0.577215i	$-0.195861\pi$
$62$	0	0
$63$	264.000	0.527950
$64$	0	0
$65$	−44.0000	−0.0839620
$66$	0	0
$67$	188.000i	0.342804i	0.985201	+	0.171402i	$0.0548297\pi$
$67$	188.000i	0.342804i	−0.985201	+	0.171402i	$0.945170\pi$
$68$	0	0
$69$	224.000i	0.390818i
$70$	0	0
$71$	728.000	1.21687	0.608435	−	0.793604i	$-0.291798\pi$
$71$	728.000	1.21687	0.608435	+	0.793604i	$0.291798\pi$
$72$	0	0
$73$	−154.000	−0.246909	−0.123454	−	0.992350i	$-0.539397\pi$
$73$	−154.000	−0.246909	−0.123454	+	0.992350i	$0.539397\pi$
$74$	0	0
$75$	− 484.000i	− 0.745167i
$76$	0	0
$77$	1056.00i	1.56289i
$78$	0	0
$79$	656.000	0.934250	0.467125	−	0.884191i	$-0.345290\pi$
$79$	656.000	0.934250	0.467125	+	0.884191i	$0.345290\pi$
$80$	0	0
$81$	−311.000	−0.426612
$82$	0	0
$83$	236.000i	0.312101i	0.987749	+	0.156050i	$0.0498762\pi$
$83$	236.000i	0.312101i	−0.987749	+	0.156050i	$0.950124\pi$
$84$	0	0
$85$	− 100.000i	− 0.127606i
$86$	0	0
$87$	−792.000	−0.975992
$88$	0	0
$89$	−714.000	−0.850380	−0.425190	−	0.905104i	$-0.639793\pi$
$89$	−714.000	−0.850380	−0.425190	+	0.905104i	$0.639793\pi$
$90$	0	0
$91$	− 528.000i	− 0.608236i
$92$	0	0
$93$	− 640.000i	− 0.713601i
$94$	0	0
$95$	88.0000	0.0950380
$96$	0	0
$97$	−478.000	−0.500346	−0.250173	−	0.968201i	$-0.580487\pi$
$97$	−478.000	−0.500346	−0.250173	+	0.968201i	$0.580487\pi$
$98$	0	0
$99$	484.000i	0.491352i
$100$	0	0
$101$	1566.00i	1.54280i	0.636350	+	0.771400i	$0.280443\pi$
$101$	1566.00i	1.54280i	−0.636350	+	0.771400i	$0.719557\pi$
$102$	0	0
$103$	−968.000	−0.926018	−0.463009	−	0.886354i	$-0.653230\pi$
$103$	−968.000	−0.926018	−0.463009	+	0.886354i	$0.653230\pi$
$104$	0	0
$105$	−192.000	−0.178450
$106$	0	0
$107$	780.000i	0.704724i	0.935864	+	0.352362i	$0.114621\pi$
$107$	780.000i	0.704724i	−0.935864	+	0.352362i	$0.885379\pi$
$108$	0	0
$109$	1994.00i	1.75221i	0.482123	+	0.876103i	$0.339866\pi$
$109$	1994.00i	1.75221i	−0.482123	+	0.876103i	$0.660134\pi$
$110$	0	0
$111$	−648.000	−0.554103
$112$	0	0
$113$	−942.000	−0.784212	−0.392106	−	0.919920i	$-0.628253\pi$
$113$	−942.000	−0.784212	−0.392106	+	0.919920i	$0.628253\pi$
$114$	0	0
$115$	112.000i	0.0908179i
$116$	0	0
$117$	− 242.000i	− 0.191221i
$118$	0	0
$119$	1200.00	0.924402
$120$	0	0
$121$	−605.000	−0.454545
$122$	0	0
$123$	− 792.000i	− 0.580587i
$124$	0	0
$125$	− 492.000i	− 0.352047i
$126$	0	0
$127$	−1408.00	−0.983778	−0.491889	−	0.870658i	$-0.663693\pi$
$127$	−1408.00	−0.983778	−0.491889	+	0.870658i	$0.663693\pi$
$128$	0	0
$129$	−208.000	−0.141964
$130$	0	0
$131$	− 2692.00i	− 1.79543i	−0.440578	−	0.897714i	$-0.645227\pi$
$131$	− 2692.00i	− 1.79543i	0.440578	−	0.897714i	$-0.354773\pi$
$132$	0	0
$133$	1056.00i	0.688472i
$134$	0	0
$135$	−304.000	−0.193809
$136$	0	0
$137$	−1626.00	−1.01400	−0.507002	−	0.861945i	$-0.669246\pi$
$137$	−1626.00	−1.01400	−0.507002	+	0.861945i	$0.669246\pi$
$138$	0	0
$139$	684.000i	0.417382i	0.977982	+	0.208691i	$0.0669203\pi$
$139$	684.000i	0.417382i	−0.977982	+	0.208691i	$0.933080\pi$
$140$	0	0
$141$	2112.00i	1.26144i
$142$	0	0
$143$	968.000	0.566072
$144$	0	0
$145$	−396.000	−0.226800
$146$	0	0
$147$	− 932.000i	− 0.522926i
$148$	0	0
$149$	302.000i	0.166046i	0.996548	+	0.0830228i	$0.0264574\pi$
$149$	302.000i	0.166046i	−0.996548	+	0.0830228i	$0.973543\pi$
$150$	0	0
$151$	1352.00	0.728637	0.364319	−	0.931274i	$-0.381302\pi$
$151$	1352.00	0.728637	0.364319	+	0.931274i	$0.381302\pi$
$152$	0	0
$153$	550.000	0.290620
$154$	0	0
$155$	− 320.000i	− 0.165826i
$156$	0	0
$157$	− 3142.00i	− 1.59719i	−0.601868	−	0.798595i	$-0.705577\pi$
$157$	− 3142.00i	− 1.59719i	0.601868	−	0.798595i	$-0.294423\pi$
$158$	0	0
$159$	−968.000	−0.482814
$160$	0	0
$161$	−1344.00	−0.657901
$162$	0	0
$163$	3036.00i	1.45888i	0.684043	+	0.729441i	$0.260220\pi$
$163$	3036.00i	1.45888i	−0.684043	+	0.729441i	$0.739780\pi$
$164$	0	0
$165$	− 352.000i	− 0.166080i
$166$	0	0
$167$	−264.000	−0.122329	−0.0611645	−	0.998128i	$-0.519481\pi$
$167$	−264.000	−0.122329	−0.0611645	+	0.998128i	$0.519481\pi$
$168$	0	0
$169$	1713.00	0.779700
$170$	0	0
$171$	484.000i	0.216447i
$172$	0	0
$173$	2826.00i	1.24195i	0.783832	+	0.620973i	$0.213263\pi$
$173$	2826.00i	1.24195i	−0.783832	+	0.620973i	$0.786737\pi$
$174$	0	0
$175$	2904.00	1.25441
$176$	0	0
$177$	2672.00	1.13469
$178$	0	0
$179$	3084.00i	1.28776i	0.765127	+	0.643880i	$0.222676\pi$
$179$	3084.00i	1.28776i	−0.765127	+	0.643880i	$0.777324\pi$
$180$	0	0
$181$	− 2418.00i	− 0.992975i	−0.868044	−	0.496488i	$-0.834623\pi$
$181$	− 2418.00i	− 0.992975i	0.868044	−	0.496488i	$-0.165377\pi$
$182$	0	0
$183$	−2200.00	−0.888681
$184$	0	0
$185$	−324.000	−0.128762
$186$	0	0
$187$	2200.00i	0.860320i
$188$	0	0
$189$	− 3648.00i	− 1.40398i
$190$	0	0
$191$	960.000	0.363681	0.181841	−	0.983328i	$-0.441794\pi$
$191$	960.000	0.363681	0.181841	+	0.983328i	$0.441794\pi$
$192$	0	0
$193$	2882.00	1.07488	0.537438	−	0.843304i	$-0.319392\pi$
$193$	2882.00	1.07488	0.537438	+	0.843304i	$0.319392\pi$
$194$	0	0
$195$	176.000i	0.0646340i
$196$	0	0
$197$	1086.00i	0.392763i	0.980528	+	0.196381i	$0.0629191\pi$
$197$	1086.00i	0.392763i	−0.980528	+	0.196381i	$0.937081\pi$
$198$	0	0
$199$	88.0000	0.0313475	0.0156738	−	0.999877i	$-0.495011\pi$
$199$	88.0000	0.0313475	0.0156738	+	0.999877i	$0.495011\pi$
$200$	0	0
$201$	752.000	0.263890
$202$	0	0
$203$	− 4752.00i	− 1.64298i
$204$	0	0
$205$	− 396.000i	− 0.134916i
$206$	0	0
$207$	−616.000	−0.206836
$208$	0	0
$209$	−1936.00	−0.640746
$210$	0	0
$211$	− 3476.00i	− 1.13411i	−0.823679	−	0.567056i	$-0.808082\pi$
$211$	− 3476.00i	− 1.13411i	0.823679	−	0.567056i	$-0.191918\pi$
$212$	0	0
$213$	− 2912.00i	− 0.936746i
$214$	0	0
$215$	−104.000	−0.0329895
$216$	0	0
$217$	3840.00	1.20127
$218$	0	0
$219$	616.000i	0.190070i
$220$	0	0
$221$	− 1100.00i	− 0.334815i
$222$	0	0
$223$	−928.000	−0.278670	−0.139335	−	0.990245i	$-0.544497\pi$
$223$	−928.000	−0.278670	−0.139335	+	0.990245i	$0.544497\pi$
$224$	0	0
$225$	1331.00	0.394370
$226$	0	0
$227$	156.000i	0.0456127i	0.999740	+	0.0228064i	$0.00726012\pi$
$227$	156.000i	0.0456127i	−0.999740	+	0.0228064i	$0.992740\pi$
$228$	0	0
$229$	− 1634.00i	− 0.471519i	−0.971811	−	0.235759i	$-0.924242\pi$
$229$	− 1634.00i	− 0.471519i	0.971811	−	0.235759i	$-0.0757577\pi$
$230$	0	0
$231$	4224.00	1.20311
$232$	0	0
$233$	902.000	0.253614	0.126807	−	0.991927i	$-0.459527\pi$
$233$	902.000	0.253614	0.126807	+	0.991927i	$0.459527\pi$
$234$	0	0
$235$	1056.00i	0.293131i
$236$	0	0
$237$	− 2624.00i	− 0.719186i
$238$	0	0
$239$	−1616.00	−0.437365	−0.218683	−	0.975796i	$-0.570176\pi$
$239$	−1616.00	−0.437365	−0.218683	+	0.975796i	$0.570176\pi$
$240$	0	0
$241$	4818.00	1.28778	0.643889	−	0.765119i	$-0.277320\pi$
$241$	4818.00	1.28778	0.643889	+	0.765119i	$0.277320\pi$
$242$	0	0
$243$	− 2860.00i	− 0.755017i
$244$	0	0
$245$	− 466.000i	− 0.121517i
$246$	0	0
$247$	968.000	0.249362
$248$	0	0
$249$	944.000	0.240255
$250$	0	0
$251$	2140.00i	0.538150i	0.963119	+	0.269075i	$0.0867179\pi$
$251$	2140.00i	0.538150i	−0.963119	+	0.269075i	$0.913282\pi$
$252$	0	0
$253$	− 2464.00i	− 0.612294i
$254$	0	0
$255$	−400.000	−0.0982313
$256$	0	0
$257$	770.000	0.186892	0.0934461	−	0.995624i	$-0.470212\pi$
$257$	770.000	0.186892	0.0934461	+	0.995624i	$0.470212\pi$
$258$	0	0
$259$	− 3888.00i	− 0.932774i
$260$	0	0
$261$	− 2178.00i	− 0.516532i
$262$	0	0
$263$	−7400.00	−1.73499	−0.867497	−	0.497442i	$-0.834273\pi$
$263$	−7400.00	−1.73499	−0.867497	+	0.497442i	$0.834273\pi$
$264$	0	0
$265$	−484.000	−0.112196
$266$	0	0
$267$	2856.00i	0.654623i
$268$	0	0
$269$	2794.00i	0.633283i	0.948545	+	0.316642i	$0.102555\pi$
$269$	2794.00i	0.633283i	−0.948545	+	0.316642i	$0.897445\pi$
$270$	0	0
$271$	−8624.00	−1.93310	−0.966551	−	0.256474i	$-0.917439\pi$
$271$	−8624.00	−1.93310	−0.966551	+	0.256474i	$0.917439\pi$
$272$	0	0
$273$	−2112.00	−0.468220
$274$	0	0
$275$	5324.00i	1.16745i
$276$	0	0
$277$	− 1874.00i	− 0.406490i	−0.979128	−	0.203245i	$-0.934851\pi$
$277$	− 1874.00i	− 0.406490i	0.979128	−	0.203245i	$-0.0651488\pi$
$278$	0	0
$279$	1760.00	0.377665
$280$	0	0
$281$	−3338.00	−0.708642	−0.354321	−	0.935124i	$-0.615288\pi$
$281$	−3338.00	−0.708642	−0.354321	+	0.935124i	$0.615288\pi$
$282$	0	0
$283$	− 7172.00i	− 1.50647i	−0.657751	−	0.753235i	$-0.728492\pi$
$283$	− 7172.00i	− 1.50647i	0.657751	−	0.753235i	$-0.271508\pi$
$284$	0	0
$285$	− 352.000i	− 0.0731603i
$286$	0	0
$287$	4752.00	0.977358
$288$	0	0
$289$	−2413.00	−0.491146
$290$	0	0
$291$	1912.00i	0.385166i
$292$	0	0
$293$	5214.00i	1.03961i	0.854286	+	0.519804i	$0.173995\pi$
$293$	5214.00i	1.03961i	−0.854286	+	0.519804i	$0.826005\pi$
$294$	0	0
$295$	1336.00	0.263678
$296$	0	0
$297$	6688.00	1.30666
$298$	0	0
$299$	1232.00i	0.238289i
$300$	0	0
$301$	− 1248.00i	− 0.238982i
$302$	0	0
$303$	6264.00	1.18765
$304$	0	0
$305$	−1100.00	−0.206511
$306$	0	0
$307$	396.000i	0.0736186i	0.999322	+	0.0368093i	$0.0117194\pi$
$307$	396.000i	0.0736186i	−0.999322	+	0.0368093i	$0.988281\pi$
$308$	0	0
$309$	3872.00i	0.712849i
$310$	0	0
$311$	−4056.00	−0.739533	−0.369766	−	0.929125i	$-0.620562\pi$
$311$	−4056.00	−0.739533	−0.369766	+	0.929125i	$0.620562\pi$
$312$	0	0
$313$	−2154.00	−0.388982	−0.194491	−	0.980904i	$-0.562305\pi$
$313$	−2154.00	−0.388982	−0.194491	+	0.980904i	$0.562305\pi$
$314$	0	0
$315$	− 528.000i	− 0.0944426i
$316$	0	0
$317$	7386.00i	1.30864i	0.756217	+	0.654320i	$0.227045\pi$
$317$	7386.00i	1.30864i	−0.756217	+	0.654320i	$0.772955\pi$
$318$	0	0
$319$	8712.00	1.52909
$320$	0	0
$321$	3120.00	0.542497
$322$	0	0
$323$	2200.00i	0.378982i
$324$	0	0
$325$	− 2662.00i	− 0.454342i
$326$	0	0
$327$	7976.00	1.34885
$328$	0	0
$329$	−12672.0	−2.12350
$330$	0	0
$331$	1132.00i	0.187977i	0.995573	+	0.0939884i	$0.0299617\pi$
$331$	1132.00i	0.187977i	−0.995573	+	0.0939884i	$0.970038\pi$
$332$	0	0
$333$	− 1782.00i	− 0.293252i
$334$	0	0
$335$	376.000	0.0613226
$336$	0	0
$337$	−3342.00	−0.540209	−0.270104	−	0.962831i	$-0.587058\pi$
$337$	−3342.00	−0.540209	−0.270104	+	0.962831i	$0.587058\pi$
$338$	0	0
$339$	3768.00i	0.603686i
$340$	0	0
$341$	7040.00i	1.11800i
$342$	0	0
$343$	−2640.00	−0.415588
$344$	0	0
$345$	448.000	0.0699116
$346$	0	0
$347$	− 2244.00i	− 0.347159i	−0.984820	−	0.173580i	$-0.944467\pi$
$347$	− 2244.00i	− 0.347159i	0.984820	−	0.173580i	$-0.0555334\pi$
$348$	0	0
$349$	6522.00i	1.00033i	0.865931	+	0.500164i	$0.166727\pi$
$349$	6522.00i	1.00033i	−0.865931	+	0.500164i	$0.833273\pi$
$350$	0	0
$351$	−3344.00	−0.508517
$352$	0	0
$353$	−11230.0	−1.69324	−0.846618	−	0.532200i	$-0.821365\pi$
$353$	−11230.0	−1.69324	−0.846618	+	0.532200i	$0.821365\pi$
$354$	0	0
$355$	− 1456.00i	− 0.217680i
$356$	0	0
$357$	− 4800.00i	− 0.711605i
$358$	0	0
$359$	1848.00	0.271682	0.135841	−	0.990731i	$-0.456626\pi$
$359$	1848.00	0.271682	0.135841	+	0.990731i	$0.456626\pi$
$360$	0	0
$361$	4923.00	0.717743
$362$	0	0
$363$	2420.00i	0.349909i
$364$	0	0
$365$	308.000i	0.0441684i
$366$	0	0
$367$	−7120.00	−1.01270	−0.506350	−	0.862328i	$-0.669006\pi$
$367$	−7120.00	−1.01270	−0.506350	+	0.862328i	$0.669006\pi$
$368$	0	0
$369$	2178.00	0.307269
$370$	0	0
$371$	− 5808.00i	− 0.812766i
$372$	0	0
$373$	6350.00i	0.881476i	0.897636	+	0.440738i	$0.145283\pi$
$373$	6350.00i	0.881476i	−0.897636	+	0.440738i	$0.854717\pi$
$374$	0	0
$375$	−1968.00	−0.271006
$376$	0	0
$377$	−4356.00	−0.595081
$378$	0	0
$379$	7900.00i	1.07070i	0.844630	+	0.535351i	$0.179821\pi$
$379$	7900.00i	1.07070i	−0.844630	+	0.535351i	$0.820179\pi$
$380$	0	0
$381$	5632.00i	0.757313i
$382$	0	0
$383$	−10368.0	−1.38324	−0.691619	−	0.722263i	$-0.743102\pi$
$383$	−10368.0	−1.38324	−0.691619	+	0.722263i	$0.743102\pi$
$384$	0	0
$385$	2112.00	0.279578
$386$	0	0
$387$	− 572.000i	− 0.0751328i
$388$	0	0
$389$	8830.00i	1.15090i	0.817838	+	0.575448i	$0.195172\pi$
$389$	8830.00i	1.15090i	−0.817838	+	0.575448i	$0.804828\pi$
$390$	0	0
$391$	−2800.00	−0.362154
$392$	0	0
$393$	−10768.0	−1.38212
$394$	0	0
$395$	− 1312.00i	− 0.167124i
$396$	0	0
$397$	− 9878.00i	− 1.24877i	−0.781116	−	0.624386i	$-0.785349\pi$
$397$	− 9878.00i	− 1.24877i	0.781116	−	0.624386i	$-0.214651\pi$
$398$	0	0
$399$	4224.00	0.529986
$400$	0	0
$401$	−13134.0	−1.63561	−0.817806	−	0.575494i	$-0.804810\pi$
$401$	−13134.0	−1.63561	−0.817806	+	0.575494i	$0.804810\pi$
$402$	0	0
$403$	− 3520.00i	− 0.435096i
$404$	0	0
$405$	622.000i	0.0763146i
$406$	0	0
$407$	7128.00	0.868113
$408$	0	0
$409$	−906.000	−0.109533	−0.0547663	−	0.998499i	$-0.517441\pi$
$409$	−906.000	−0.109533	−0.0547663	+	0.998499i	$0.517441\pi$
$410$	0	0
$411$	6504.00i	0.780581i
$412$	0	0
$413$	16032.0i	1.91013i
$414$	0	0
$415$	472.000	0.0558303
$416$	0	0
$417$	2736.00	0.321301
$418$	0	0
$419$	− 5412.00i	− 0.631011i	−0.948924	−	0.315505i	$-0.897826\pi$
$419$	− 5412.00i	− 0.631011i	0.948924	−	0.315505i	$-0.102174\pi$
$420$	0	0
$421$	− 4642.00i	− 0.537381i	−0.963227	−	0.268690i	$-0.913409\pi$
$421$	− 4642.00i	− 0.537381i	0.963227	−	0.268690i	$-0.0865908\pi$
$422$	0	0
$423$	−5808.00	−0.667600
$424$	0	0
$425$	6050.00	0.690513
$426$	0	0
$427$	− 13200.0i	− 1.49600i
$428$	0	0
$429$	− 3872.00i	− 0.435762i
$430$	0	0
$431$	−656.000	−0.0733142	−0.0366571	−	0.999328i	$-0.511671\pi$
$431$	−656.000	−0.0733142	−0.0366571	+	0.999328i	$0.511671\pi$
$432$	0	0
$433$	9490.00	1.05326	0.526629	−	0.850096i	$-0.323456\pi$
$433$	9490.00	1.05326	0.526629	+	0.850096i	$0.323456\pi$
$434$	0	0
$435$	1584.00i	0.174591i
$436$	0	0
$437$	− 2464.00i	− 0.269723i
$438$	0	0
$439$	5544.00	0.602735	0.301368	−	0.953508i	$-0.402557\pi$
$439$	5544.00	0.602735	0.301368	+	0.953508i	$0.402557\pi$
$440$	0	0
$441$	2563.00	0.276752
$442$	0	0
$443$	− 7652.00i	− 0.820672i	−0.911935	−	0.410336i	$-0.865412\pi$
$443$	− 7652.00i	− 0.820672i	0.911935	−	0.410336i	$-0.134588\pi$
$444$	0	0
$445$	1428.00i	0.152121i
$446$	0	0
$447$	1208.00	0.127822
$448$	0	0
$449$	−446.000	−0.0468776	−0.0234388	−	0.999725i	$-0.507461\pi$
$449$	−446.000	−0.0468776	−0.0234388	+	0.999725i	$0.507461\pi$
$450$	0	0
$451$	8712.00i	0.909605i
$452$	0	0
$453$	− 5408.00i	− 0.560905i
$454$	0	0
$455$	−1056.00	−0.108804
$456$	0	0
$457$	−1562.00	−0.159885	−0.0799423	−	0.996799i	$-0.525474\pi$
$457$	−1562.00	−0.159885	−0.0799423	+	0.996799i	$0.525474\pi$
$458$	0	0
$459$	− 7600.00i	− 0.772849i
$460$	0	0
$461$	− 10582.0i	− 1.06910i	−0.845138	−	0.534548i	$-0.820482\pi$
$461$	− 10582.0i	− 1.06910i	0.845138	−	0.534548i	$-0.179518\pi$
$462$	0	0
$463$	10768.0	1.08085	0.540423	−	0.841394i	$-0.318264\pi$
$463$	10768.0	1.08085	0.540423	+	0.841394i	$0.318264\pi$
$464$	0	0
$465$	−1280.00	−0.127653
$466$	0	0
$467$	− 9876.00i	− 0.978601i	−0.872115	−	0.489301i	$-0.837252\pi$
$467$	− 9876.00i	− 0.978601i	0.872115	−	0.489301i	$-0.162748\pi$
$468$	0	0
$469$	4512.00i	0.444232i
$470$	0	0
$471$	−12568.0	−1.22952
$472$	0	0
$473$	2288.00	0.222415
$474$	0	0
$475$	5324.00i	0.514278i
$476$	0	0
$477$	− 2662.00i	− 0.255523i
$478$	0	0
$479$	352.000	0.0335768	0.0167884	−	0.999859i	$-0.494656\pi$
$479$	352.000	0.0335768	0.0167884	+	0.999859i	$0.494656\pi$
$480$	0	0
$481$	−3564.00	−0.337847
$482$	0	0
$483$	5376.00i	0.506452i
$484$	0	0
$485$	956.000i	0.0895046i
$486$	0	0
$487$	−15176.0	−1.41209	−0.706047	−	0.708165i	$-0.749523\pi$
$487$	−15176.0	−1.41209	−0.706047	+	0.708165i	$0.749523\pi$
$488$	0	0
$489$	12144.0	1.12305
$490$	0	0
$491$	8844.00i	0.812880i	0.913677	+	0.406440i	$0.133230\pi$
$491$	8844.00i	0.812880i	−0.913677	+	0.406440i	$0.866770\pi$
$492$	0	0
$493$	− 9900.00i	− 0.904409i
$494$	0	0
$495$	968.000	0.0878957
$496$	0	0
$497$	17472.0	1.57691
$498$	0	0
$499$	19404.0i	1.74077i	0.492375	+	0.870383i	$0.336129\pi$
$499$	19404.0i	1.74077i	−0.492375	+	0.870383i	$0.663871\pi$
$500$	0	0
$501$	1056.00i	0.0941689i
$502$	0	0
$503$	16488.0	1.46156	0.730779	−	0.682614i	$-0.239157\pi$
$503$	16488.0	1.46156	0.730779	+	0.682614i	$0.239157\pi$
$504$	0	0
$505$	3132.00	0.275984
$506$	0	0
$507$	− 6852.00i	− 0.600213i
$508$	0	0
$509$	12954.0i	1.12805i	0.825759	+	0.564024i	$0.190747\pi$
$509$	12954.0i	1.12805i	−0.825759	+	0.564024i	$0.809253\pi$
$510$	0	0
$511$	−3696.00	−0.319964
$512$	0	0
$513$	6688.00	0.575599
$514$	0	0
$515$	1936.00i	0.165651i
$516$	0	0
$517$	− 23232.0i	− 1.97629i
$518$	0	0
$519$	11304.0	0.956051
$520$	0	0
$521$	−10970.0	−0.922465	−0.461233	−	0.887279i	$-0.652593\pi$
$521$	−10970.0	−0.922465	−0.461233	+	0.887279i	$0.652593\pi$
$522$	0	0
$523$	16940.0i	1.41632i	0.706053	+	0.708159i	$0.250474\pi$
$523$	16940.0i	1.41632i	−0.706053	+	0.708159i	$0.749526\pi$
$524$	0	0
$525$	− 11616.0i	− 0.965645i
$526$	0	0
$527$	8000.00	0.661263
$528$	0	0
$529$	−9031.00	−0.742254
$530$	0	0
$531$	7348.00i	0.600520i
$532$	0	0
$533$	− 4356.00i	− 0.353995i
$534$	0	0
$535$	1560.00	0.126065
$536$	0	0
$537$	12336.0	0.991318
$538$	0	0
$539$	10252.0i	0.819267i
$540$	0	0
$541$	− 198.000i	− 0.0157351i	−0.999969	−	0.00786755i	$-0.997496\pi$
$541$	− 198.000i	− 0.0157351i	0.999969	−	0.00786755i	$-0.00250434\pi$
$542$	0	0
$543$	−9672.00	−0.764393
$544$	0	0
$545$	3988.00	0.313444
$546$	0	0
$547$	− 15268.0i	− 1.19344i	−0.802449	−	0.596721i	$-0.796470\pi$
$547$	− 15268.0i	− 1.19344i	0.802449	−	0.596721i	$-0.203530\pi$
$548$	0	0
$549$	− 6050.00i	− 0.470324i
$550$	0	0
$551$	8712.00	0.673582
$552$	0	0
$553$	15744.0	1.21067
$554$	0	0
$555$	1296.00i	0.0991210i
$556$	0	0
$557$	− 20854.0i	− 1.58638i	−0.608976	−	0.793189i	$-0.708419\pi$
$557$	− 20854.0i	− 1.58638i	0.608976	−	0.793189i	$-0.291581\pi$
$558$	0	0
$559$	−1144.00	−0.0865582
$560$	0	0
$561$	8800.00	0.662275
$562$	0	0
$563$	− 19316.0i	− 1.44595i	−0.690872	−	0.722977i	$-0.742773\pi$
$563$	− 19316.0i	− 1.44595i	0.690872	−	0.722977i	$-0.257227\pi$
$564$	0	0
$565$	1884.00i	0.140284i
$566$	0	0
$567$	−7464.00	−0.552837
$568$	0	0
$569$	−7018.00	−0.517065	−0.258532	−	0.966003i	$-0.583239\pi$
$569$	−7018.00	−0.517065	−0.258532	+	0.966003i	$0.583239\pi$
$570$	0	0
$571$	− 24420.0i	− 1.78975i	−0.446320	−	0.894873i	$-0.647266\pi$
$571$	− 24420.0i	− 1.78975i	0.446320	−	0.894873i	$-0.352734\pi$
$572$	0	0
$573$	− 3840.00i	− 0.279962i
$574$	0	0
$575$	−6776.00	−0.491441
$576$	0	0
$577$	23234.0	1.67633	0.838166	−	0.545415i	$-0.183628\pi$
$577$	23234.0	1.67633	0.838166	+	0.545415i	$0.183628\pi$
$578$	0	0
$579$	− 11528.0i	− 0.827439i
$580$	0	0
$581$	5664.00i	0.404445i
$582$	0	0
$583$	10648.0	0.756424
$584$	0	0
$585$	−484.000	−0.0342067
$586$	0	0
$587$	10604.0i	0.745611i	0.927909	+	0.372806i	$0.121604\pi$
$587$	10604.0i	0.745611i	−0.927909	+	0.372806i	$0.878396\pi$
$588$	0	0
$589$	7040.00i	0.492493i
$590$	0	0
$591$	4344.00	0.302349
$592$	0	0
$593$	−13838.0	−0.958277	−0.479139	−	0.877739i	$-0.659051\pi$
$593$	−13838.0	−0.958277	−0.479139	+	0.877739i	$0.659051\pi$
$594$	0	0
$595$	− 2400.00i	− 0.165362i
$596$	0	0
$597$	− 352.000i	− 0.0241313i
$598$	0	0
$599$	−3960.00	−0.270119	−0.135059	−	0.990837i	$-0.543123\pi$
$599$	−3960.00	−0.270119	−0.135059	+	0.990837i	$0.543123\pi$
$600$	0	0
$601$	5942.00	0.403293	0.201647	−	0.979458i	$-0.435371\pi$
$601$	5942.00	0.403293	0.201647	+	0.979458i	$0.435371\pi$
$602$	0	0
$603$	2068.00i	0.139661i
$604$	0	0
$605$	1210.00i	0.0813116i
$606$	0	0
$607$	3040.00	0.203278	0.101639	−	0.994821i	$-0.467591\pi$
$607$	3040.00	0.203278	0.101639	+	0.994821i	$0.467591\pi$
$608$	0	0
$609$	−19008.0	−1.26477
$610$	0	0
$611$	11616.0i	0.769121i
$612$	0	0
$613$	− 2530.00i	− 0.166698i	−0.996520	−	0.0833489i	$-0.973438\pi$
$613$	− 2530.00i	− 0.166698i	0.996520	−	0.0833489i	$-0.0265616\pi$
$614$	0	0
$615$	−1584.00	−0.103859
$616$	0	0
$617$	19206.0	1.25317	0.626584	−	0.779354i	$-0.284453\pi$
$617$	19206.0	1.25317	0.626584	+	0.779354i	$0.284453\pi$
$618$	0	0
$619$	− 10996.0i	− 0.714001i	−0.934104	−	0.357000i	$-0.883799\pi$
$619$	− 10996.0i	− 0.714001i	0.934104	−	0.357000i	$-0.116201\pi$
$620$	0	0
$621$	8512.00i	0.550040i
$622$	0	0
$623$	−17136.0	−1.10199
$624$	0	0
$625$	14141.0	0.905024
$626$	0	0
$627$	7744.00i	0.493247i
$628$	0	0
$629$	− 8100.00i	− 0.513463i
$630$	0	0
$631$	−6680.00	−0.421437	−0.210718	−	0.977547i	$-0.567580\pi$
$631$	−6680.00	−0.421437	−0.210718	+	0.977547i	$0.567580\pi$
$632$	0	0
$633$	−13904.0	−0.873040
$634$	0	0
$635$	2816.00i	0.175984i
$636$	0	0
$637$	− 5126.00i	− 0.318838i
$638$	0	0
$639$	8008.00	0.495761
$640$	0	0
$641$	6274.00	0.386596	0.193298	−	0.981140i	$-0.438082\pi$
$641$	6274.00	0.386596	0.193298	+	0.981140i	$0.438082\pi$
$642$	0	0
$643$	9084.00i	0.557135i	0.960417	+	0.278568i	$0.0898596\pi$
$643$	9084.00i	0.557135i	−0.960417	+	0.278568i	$0.910140\pi$
$644$	0	0
$645$	416.000i	0.0253953i
$646$	0	0
$647$	−23656.0	−1.43742	−0.718712	−	0.695308i	$-0.755268\pi$
$647$	−23656.0	−1.43742	−0.718712	+	0.695308i	$0.755268\pi$
$648$	0	0
$649$	−29392.0	−1.77771
$650$	0	0
$651$	− 15360.0i	− 0.924740i
$652$	0	0
$653$	6762.00i	0.405234i	0.979258	+	0.202617i	$0.0649446\pi$
$653$	6762.00i	0.405234i	−0.979258	+	0.202617i	$0.935055\pi$
$654$	0	0
$655$	−5384.00	−0.321176
$656$	0	0
$657$	−1694.00	−0.100592
$658$	0	0
$659$	15276.0i	0.902987i	0.892274	+	0.451494i	$0.149109\pi$
$659$	15276.0i	0.902987i	−0.892274	+	0.451494i	$0.850891\pi$
$660$	0	0
$661$	11054.0i	0.650455i	0.945636	+	0.325228i	$0.105441\pi$
$661$	11054.0i	0.650455i	−0.945636	+	0.325228i	$0.894559\pi$
$662$	0	0
$663$	−4400.00	−0.257740
$664$	0	0
$665$	2112.00	0.123158
$666$	0	0
$667$	11088.0i	0.643672i
$668$	0	0
$669$	3712.00i	0.214520i
$670$	0	0
$671$	24200.0	1.39230
$672$	0	0
$673$	−21278.0	−1.21873	−0.609366	−	0.792889i	$-0.708576\pi$
$673$	−21278.0	−1.21873	−0.609366	+	0.792889i	$0.708576\pi$
$674$	0	0
$675$	− 18392.0i	− 1.04875i
$676$	0	0
$677$	8926.00i	0.506727i	0.967371	+	0.253363i	$0.0815368\pi$
$677$	8926.00i	0.506727i	−0.967371	+	0.253363i	$0.918463\pi$
$678$	0	0
$679$	−11472.0	−0.648387
$680$	0	0
$681$	624.000	0.0351127
$682$	0	0
$683$	− 8116.00i	− 0.454685i	−0.973815	−	0.227343i	$-0.926996\pi$
$683$	− 8116.00i	− 0.454685i	0.973815	−	0.227343i	$-0.0730037\pi$
$684$	0	0
$685$	3252.00i	0.181391i
$686$	0	0
$687$	−6536.00	−0.362975
$688$	0	0
$689$	−5324.00	−0.294381
$690$	0	0
$691$	− 11764.0i	− 0.647646i	−0.946118	−	0.323823i	$-0.895032\pi$
$691$	− 11764.0i	− 0.647646i	0.946118	−	0.323823i	$-0.104968\pi$
$692$	0	0
$693$	11616.0i	0.636732i
$694$	0	0
$695$	1368.00	0.0746636
$696$	0	0
$697$	9900.00	0.538005
$698$	0	0
$699$	− 3608.00i	− 0.195232i
$700$	0	0
$701$	4698.00i	0.253126i	0.991959	+	0.126563i	$0.0403945\pi$
$701$	4698.00i	0.253126i	−0.991959	+	0.126563i	$0.959605\pi$
$702$	0	0
$703$	7128.00	0.382415
$704$	0	0
$705$	4224.00	0.225653
$706$	0	0
$707$	37584.0i	1.99928i
$708$	0	0
$709$	24638.0i	1.30508i	0.757756	+	0.652538i	$0.226296\pi$
$709$	24638.0i	1.30508i	−0.757756	+	0.652538i	$0.773704\pi$
$710$	0	0
$711$	7216.00	0.380620
$712$	0	0
$713$	−8960.00	−0.470624
$714$	0	0
$715$	− 1936.00i	− 0.101262i
$716$	0	0
$717$	6464.00i	0.336684i
$718$	0	0
$719$	−16624.0	−0.862268	−0.431134	−	0.902288i	$-0.641886\pi$
$719$	−16624.0	−0.862268	−0.431134	+	0.902288i	$0.641886\pi$
$720$	0	0
$721$	−23232.0	−1.20001
$722$	0	0
$723$	− 19272.0i	− 0.991332i
$724$	0	0
$725$	− 23958.0i	− 1.22728i
$726$	0	0
$727$	30216.0	1.54147	0.770735	−	0.637155i	$-0.219889\pi$
$727$	30216.0	1.54147	0.770735	+	0.637155i	$0.219889\pi$
$728$	0	0
$729$	−19837.0	−1.00782
$730$	0	0
$731$	− 2600.00i	− 0.131552i
$732$	0	0
$733$	3322.00i	0.167395i	0.996491	+	0.0836977i	$0.0266730\pi$
$733$	3322.00i	0.167395i	−0.996491	+	0.0836977i	$0.973327\pi$
$734$	0	0
$735$	−1864.00	−0.0935438
$736$	0	0
$737$	−8272.00	−0.413437
$738$	0	0
$739$	− 14692.0i	− 0.731331i	−0.930746	−	0.365666i	$-0.880841\pi$
$739$	− 14692.0i	− 0.731331i	0.930746	−	0.365666i	$-0.119159\pi$
$740$	0	0
$741$	− 3872.00i	− 0.191959i
$742$	0	0
$743$	28600.0	1.41216	0.706078	−	0.708134i	$-0.250463\pi$
$743$	28600.0	1.41216	0.706078	+	0.708134i	$0.250463\pi$
$744$	0	0
$745$	604.000	0.0297032
$746$	0	0
$747$	2596.00i	0.127152i
$748$	0	0
$749$	18720.0i	0.913236i
$750$	0	0
$751$	29616.0	1.43902	0.719509	−	0.694483i	$-0.244367\pi$
$751$	29616.0	1.43902	0.719509	+	0.694483i	$0.244367\pi$
$752$	0	0
$753$	8560.00	0.414268
$754$	0	0
$755$	− 2704.00i	− 0.130343i
$756$	0	0
$757$	2894.00i	0.138949i	0.997584	+	0.0694744i	$0.0221322\pi$
$757$	2894.00i	0.138949i	−0.997584	+	0.0694744i	$0.977868\pi$
$758$	0	0
$759$	−9856.00	−0.471344
$760$	0	0
$761$	−14762.0	−0.703183	−0.351591	−	0.936154i	$-0.614359\pi$
$761$	−14762.0	−0.703183	−0.351591	+	0.936154i	$0.614359\pi$
$762$	0	0
$763$	47856.0i	2.27065i
$764$	0	0
$765$	− 1100.00i	− 0.0519877i
$766$	0	0
$767$	14696.0	0.691841
$768$	0	0
$769$	−7678.00	−0.360047	−0.180023	−	0.983662i	$-0.557617\pi$
$769$	−7678.00	−0.360047	−0.180023	+	0.983662i	$0.557617\pi$
$770$	0	0
$771$	− 3080.00i	− 0.143870i
$772$	0	0
$773$	27390.0i	1.27445i	0.770678	+	0.637225i	$0.219918\pi$
$773$	27390.0i	1.27445i	−0.770678	+	0.637225i	$0.780082\pi$
$774$	0	0
$775$	19360.0	0.897331
$776$	0	0
$777$	−15552.0	−0.718050
$778$	0	0
$779$	8712.00i	0.400693i
$780$	0	0
$781$	32032.0i	1.46760i
$782$	0	0
$783$	−30096.0	−1.37362
$784$	0	0
$785$	−6284.00	−0.285714
$786$	0	0
$787$	19756.0i	0.894823i	0.894328	+	0.447411i	$0.147654\pi$
$787$	19756.0i	0.894823i	−0.894328	+	0.447411i	$0.852346\pi$
$788$	0	0
$789$	29600.0i	1.33560i
$790$	0	0
$791$	−22608.0	−1.01624
$792$	0	0
$793$	−12100.0	−0.541846
$794$	0	0
$795$	1936.00i	0.0863684i
$796$	0	0
$797$	− 38854.0i	− 1.72682i	−0.504499	−	0.863412i	$-0.668323\pi$
$797$	− 38854.0i	− 1.72682i	0.504499	−	0.863412i	$-0.331677\pi$
$798$	0	0
$799$	−26400.0	−1.16892
$800$	0	0
$801$	−7854.00	−0.346451
$802$	0	0
$803$	− 6776.00i	− 0.297783i
$804$	0	0
$805$	2688.00i	0.117689i
$806$	0	0
$807$	11176.0	0.487502
$808$	0	0
$809$	14278.0	0.620504	0.310252	−	0.950654i	$-0.399587\pi$
$809$	14278.0	0.620504	0.310252	+	0.950654i	$0.399587\pi$
$810$	0	0
$811$	716.000i	0.0310014i	0.999880	+	0.0155007i	$0.00493423\pi$
$811$	716.000i	0.0310014i	−0.999880	+	0.0155007i	$0.995066\pi$
$812$	0	0
$813$	34496.0i	1.48810i
$814$	0	0
$815$	6072.00	0.260973
$816$	0	0
$817$	2288.00	0.0979767
$818$	0	0
$819$	− 5808.00i	− 0.247800i
$820$	0	0
$821$	− 23538.0i	− 1.00059i	−0.865856	−	0.500293i	$-0.833225\pi$
$821$	− 23538.0i	− 1.00059i	0.865856	−	0.500293i	$-0.166775\pi$
$822$	0	0
$823$	−6616.00	−0.280218	−0.140109	−	0.990136i	$-0.544745\pi$
$823$	−6616.00	−0.280218	−0.140109	+	0.990136i	$0.544745\pi$
$824$	0	0
$825$	21296.0	0.898705
$826$	0	0
$827$	− 27236.0i	− 1.14521i	−0.819831	−	0.572605i	$-0.805933\pi$
$827$	− 27236.0i	− 1.14521i	0.819831	−	0.572605i	$-0.194067\pi$
$828$	0	0
$829$	− 12070.0i	− 0.505680i	−0.967508	−	0.252840i	$-0.918635\pi$
$829$	− 12070.0i	− 0.505680i	0.967508	−	0.252840i	$-0.0813646\pi$
$830$	0	0
$831$	−7496.00	−0.312916
$832$	0	0
$833$	11650.0	0.484572
$834$	0	0
$835$	528.000i	0.0218829i
$836$	0	0
$837$	− 24320.0i	− 1.00433i
$838$	0	0
$839$	−42024.0	−1.72924	−0.864618	−	0.502429i	$-0.832440\pi$
$839$	−42024.0	−1.72924	−0.864618	+	0.502429i	$0.832440\pi$
$840$	0	0
$841$	−14815.0	−0.607446
$842$	0	0
$843$	13352.0i	0.545513i
$844$	0	0
$845$	− 3426.00i	− 0.139477i
$846$	0	0
$847$	−14520.0	−0.589036
$848$	0	0
$849$	−28688.0	−1.15968
$850$	0	0
$851$	9072.00i	0.365434i
$852$	0	0
$853$	2414.00i	0.0968978i	0.998826	+	0.0484489i	$0.0154278\pi$
$853$	2414.00i	0.0968978i	−0.998826	+	0.0484489i	$0.984572\pi$
$854$	0	0
$855$	968.000	0.0387192
$856$	0	0
$857$	37686.0	1.50213	0.751067	−	0.660226i	$-0.229539\pi$
$857$	37686.0	1.50213	0.751067	+	0.660226i	$0.229539\pi$
$858$	0	0
$859$	− 40644.0i	− 1.61438i	−0.590289	−	0.807192i	$-0.700986\pi$
$859$	− 40644.0i	− 1.61438i	0.590289	−	0.807192i	$-0.299014\pi$
$860$	0	0
$861$	− 19008.0i	− 0.752370i
$862$	0	0
$863$	18656.0	0.735872	0.367936	−	0.929851i	$-0.380065\pi$
$863$	18656.0	0.735872	0.367936	+	0.929851i	$0.380065\pi$
$864$	0	0
$865$	5652.00	0.222166
$866$	0	0
$867$	9652.00i	0.378084i
$868$	0	0
$869$	28864.0i	1.12675i
$870$	0	0
$871$	4136.00	0.160899
$872$	0	0
$873$	−5258.00	−0.203845
$874$	0	0
$875$	− 11808.0i	− 0.456209i
$876$	0	0
$877$	13002.0i	0.500623i	0.968165	+	0.250311i	$0.0805330\pi$
$877$	13002.0i	0.500623i	−0.968165	+	0.250311i	$0.919467\pi$
$878$	0	0
$879$	20856.0	0.800291
$880$	0	0
$881$	49490.0	1.89258	0.946289	−	0.323323i	$-0.104800\pi$
$881$	49490.0	1.89258	0.946289	+	0.323323i	$0.104800\pi$
$882$	0	0
$883$	1100.00i	0.0419229i	0.999780	+	0.0209615i	$0.00667273\pi$
$883$	1100.00i	0.0419229i	−0.999780	+	0.0209615i	$0.993327\pi$
$884$	0	0
$885$	− 5344.00i	− 0.202979i
$886$	0	0
$887$	−14104.0	−0.533896	−0.266948	−	0.963711i	$-0.586015\pi$
$887$	−14104.0	−0.533896	−0.266948	+	0.963711i	$0.586015\pi$
$888$	0	0
$889$	−33792.0	−1.27486
$890$	0	0
$891$	− 13684.0i	− 0.514513i
$892$	0	0
$893$	− 23232.0i	− 0.870581i
$894$	0	0
$895$	6168.00	0.230361
$896$	0	0
$897$	4928.00	0.183435
$898$	0	0
$899$	− 31680.0i	− 1.17529i
$900$	0	0
$901$	− 12100.0i	− 0.447402i
$902$	0	0
$903$	−4992.00	−0.183968
$904$	0	0
$905$	−4836.00	−0.177629
$906$	0	0
$907$	12716.0i	0.465521i	0.972534	+	0.232761i	$0.0747759\pi$
$907$	12716.0i	0.465521i	−0.972534	+	0.232761i	$0.925224\pi$
$908$	0	0
$909$	17226.0i	0.628548i
$910$	0	0
$911$	39632.0	1.44135	0.720673	−	0.693275i	$-0.243833\pi$
$911$	39632.0	1.44135	0.720673	+	0.693275i	$0.243833\pi$
$912$	0	0
$913$	−10384.0	−0.376408
$914$	0	0
$915$	4400.00i	0.158972i
$916$	0	0
$917$	− 64608.0i	− 2.32666i
$918$	0	0
$919$	5704.00	0.204742	0.102371	−	0.994746i	$-0.467357\pi$
$919$	5704.00	0.204742	0.102371	+	0.994746i	$0.467357\pi$
$920$	0	0
$921$	1584.00	0.0566716
$922$	0	0
$923$	− 16016.0i	− 0.571152i
$924$	0	0
$925$	− 19602.0i	− 0.696767i
$926$	0	0
$927$	−10648.0	−0.377267
$928$	0	0
$929$	8162.00	0.288252	0.144126	−	0.989559i	$-0.453963\pi$
$929$	8162.00	0.288252	0.144126	+	0.989559i	$0.453963\pi$
$930$	0	0
$931$	10252.0i	0.360898i
$932$	0	0
$933$	16224.0i	0.569293i
$934$	0	0
$935$	4400.00	0.153899
$936$	0	0
$937$	55110.0	1.92141	0.960707	−	0.277564i	$-0.0895270\pi$
$937$	55110.0	1.92141	0.960707	+	0.277564i	$0.0895270\pi$
$938$	0	0
$939$	8616.00i	0.299438i
$940$	0	0
$941$	− 16374.0i	− 0.567245i	−0.958936	−	0.283622i	$-0.908464\pi$
$941$	− 16374.0i	− 0.567245i	0.958936	−	0.283622i	$-0.0915362\pi$
$942$	0	0
$943$	−11088.0	−0.382900
$944$	0	0
$945$	−7296.00	−0.251152
$946$	0	0
$947$	8460.00i	0.290299i	0.989410	+	0.145149i	$0.0463663\pi$
$947$	8460.00i	0.290299i	−0.989410	+	0.145149i	$0.953634\pi$
$948$	0	0
$949$	3388.00i	0.115889i
$950$	0	0
$951$	29544.0	1.00739
$952$	0	0
$953$	20502.0	0.696878	0.348439	−	0.937331i	$-0.386712\pi$
$953$	20502.0	0.696878	0.348439	+	0.937331i	$0.386712\pi$
$954$	0	0
$955$	− 1920.00i	− 0.0650573i
$956$	0	0
$957$	− 34848.0i	− 1.17709i
$958$	0	0
$959$	−39024.0	−1.31403
$960$	0	0
$961$	−4191.00	−0.140680
$962$	0	0
$963$	8580.00i	0.287110i
$964$	0	0
$965$	− 5764.00i	− 0.192280i
$966$	0	0
$967$	−36520.0	−1.21448	−0.607241	−	0.794518i	$-0.707724\pi$
$967$	−36520.0	−1.21448	−0.607241	+	0.794518i	$0.707724\pi$
$968$	0	0
$969$	8800.00	0.291741
$970$	0	0
$971$	− 20244.0i	− 0.669064i	−0.942384	−	0.334532i	$-0.891422\pi$
$971$	− 20244.0i	− 0.669064i	0.942384	−	0.334532i	$-0.108578\pi$
$972$	0	0
$973$	16416.0i	0.540876i
$974$	0	0
$975$	−10648.0	−0.349753
$976$	0	0
$977$	50034.0	1.63841	0.819206	−	0.573499i	$-0.194414\pi$
$977$	50034.0	1.63841	0.819206	+	0.573499i	$0.194414\pi$
$978$	0	0
$979$	− 31416.0i	− 1.02560i
$980$	0	0
$981$	21934.0i	0.713862i
$982$	0	0
$983$	37128.0	1.20468	0.602339	−	0.798240i	$-0.294235\pi$
$983$	37128.0	1.20468	0.602339	+	0.798240i	$0.294235\pi$
$984$	0	0
$985$	2172.00	0.0702596
$986$	0	0
$987$	50688.0i	1.63467i
$988$	0	0
$989$	2912.00i	0.0936261i
$990$	0	0
$991$	−27808.0	−0.891373	−0.445686	−	0.895189i	$-0.647040\pi$
$991$	−27808.0	−0.891373	−0.445686	+	0.895189i	$0.647040\pi$
$992$	0	0
$993$	4528.00	0.144705
$994$	0	0
$995$	− 176.000i	− 0.00560761i
$996$	0	0
$997$	− 28514.0i	− 0.905765i	−0.891570	−	0.452882i	$-0.850396\pi$
$997$	− 28514.0i	− 0.905765i	0.891570	−	0.452882i	$-0.149604\pi$
$998$	0	0
$999$	−24624.0	−0.779849

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.4.b.g.129.1		2
4.3	odd	2		256.4.b.a.129.2		2
8.3	odd	2		256.4.b.a.129.1		2
8.5	even	2	inner	256.4.b.g.129.2		2
16.3	odd	4		8.4.a.a.1.1	✓	1
16.5	even	4		64.4.a.b.1.1		1
16.11	odd	4		64.4.a.d.1.1		1
16.13	even	4		16.4.a.a.1.1		1
48.5	odd	4		576.4.a.j.1.1		1
48.11	even	4		576.4.a.k.1.1		1
48.29	odd	4		144.4.a.e.1.1		1
48.35	even	4		72.4.a.c.1.1		1
80.3	even	4		200.4.c.e.49.1		2
80.13	odd	4		400.4.c.i.49.2		2
80.19	odd	4		200.4.a.g.1.1		1
80.29	even	4		400.4.a.g.1.1		1
80.59	odd	4		1600.4.a.o.1.1		1
80.67	even	4		200.4.c.e.49.2		2
80.69	even	4		1600.4.a.bm.1.1		1
80.77	odd	4		400.4.c.i.49.1		2
112.3	even	12		392.4.i.b.177.1		2
112.13	odd	4		784.4.a.e.1.1		1
112.19	even	12		392.4.i.b.361.1		2
112.51	odd	12		392.4.i.g.361.1		2
112.67	odd	12		392.4.i.g.177.1		2
112.83	even	4		392.4.a.e.1.1		1
144.67	odd	12		648.4.i.h.217.1		2
144.83	even	12		648.4.i.e.433.1		2
144.115	odd	12		648.4.i.h.433.1		2
144.131	even	12		648.4.i.e.217.1		2
176.109	odd	4		1936.4.a.l.1.1		1
176.131	even	4		968.4.a.a.1.1		1
208.51	odd	4		1352.4.a.a.1.1		1
240.83	odd	4		1800.4.f.u.649.1		2
240.179	even	4		1800.4.a.d.1.1		1
240.227	odd	4		1800.4.f.u.649.2		2
272.67	odd	4		2312.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.4.a.a.1.1	✓	1	16.3	odd	4
16.4.a.a.1.1		1	16.13	even	4
64.4.a.b.1.1		1	16.5	even	4
64.4.a.d.1.1		1	16.11	odd	4
72.4.a.c.1.1		1	48.35	even	4
144.4.a.e.1.1		1	48.29	odd	4
200.4.a.g.1.1		1	80.19	odd	4
200.4.c.e.49.1		2	80.3	even	4
200.4.c.e.49.2		2	80.67	even	4
256.4.b.a.129.1		2	8.3	odd	2
256.4.b.a.129.2		2	4.3	odd	2
256.4.b.g.129.1		2	1.1	even	1	trivial
256.4.b.g.129.2		2	8.5	even	2	inner
392.4.a.e.1.1		1	112.83	even	4
392.4.i.b.177.1		2	112.3	even	12
392.4.i.b.361.1		2	112.19	even	12
392.4.i.g.177.1		2	112.67	odd	12
392.4.i.g.361.1		2	112.51	odd	12
400.4.a.g.1.1		1	80.29	even	4
400.4.c.i.49.1		2	80.77	odd	4
400.4.c.i.49.2		2	80.13	odd	4
576.4.a.j.1.1		1	48.5	odd	4
576.4.a.k.1.1		1	48.11	even	4
648.4.i.e.217.1		2	144.131	even	12
648.4.i.e.433.1		2	144.83	even	12
648.4.i.h.217.1		2	144.67	odd	12
648.4.i.h.433.1		2	144.115	odd	12
784.4.a.e.1.1		1	112.13	odd	4
968.4.a.a.1.1		1	176.131	even	4
1352.4.a.a.1.1		1	208.51	odd	4
1600.4.a.o.1.1		1	80.59	odd	4
1600.4.a.bm.1.1		1	80.69	even	4
1800.4.a.d.1.1		1	240.179	even	4
1800.4.f.u.649.1		2	240.83	odd	4
1800.4.f.u.649.2		2	240.227	odd	4
1936.4.a.l.1.1		1	176.109	odd	4
2312.4.a.a.1.1		1	272.67	odd	4

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:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	256.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.00000i q^{3} -2.00000i q^{5} +24.0000 q^{7} +11.0000 q^{9} +O(q^{10})\)	\(q-4.00000i q^{3} -2.00000i q^{5} +24.0000 q^{7} +11.0000 q^{9} +44.0000i q^{11} -22.0000i q^{13} -8.00000 q^{15} +50.0000 q^{17} +44.0000i q^{19} -96.0000i q^{21} -56.0000 q^{23} +121.000 q^{25} -152.000i q^{27} -198.000i q^{29} +160.000 q^{31} +176.000 q^{33} -48.0000i q^{35} -162.000i q^{37} -88.0000 q^{39} +198.000 q^{41} -52.0000i q^{43} -22.0000i q^{45} -528.000 q^{47} +233.000 q^{49} -200.000i q^{51} -242.000i q^{53} +88.0000 q^{55} +176.000 q^{57} +668.000i q^{59} -550.000i q^{61} +264.000 q^{63} -44.0000 q^{65} +188.000i q^{67} +224.000i q^{69} +728.000 q^{71} -154.000 q^{73} -484.000i q^{75} +1056.00i q^{77} +656.000 q^{79} -311.000 q^{81} +236.000i q^{83} -100.000i q^{85} -792.000 q^{87} -714.000 q^{89} -528.000i q^{91} -640.000i q^{93} +88.0000 q^{95} -478.000 q^{97} +484.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 48 q^{7} + 22 q^{9}+O(q^{10}) \) 2 * q + 48 * q^7 + 22 * q^9	\( 2 q + 48 q^{7} + 22 q^{9} - 16 q^{15} + 100 q^{17} - 112 q^{23} + 242 q^{25} + 320 q^{31} + 352 q^{33} - 176 q^{39} + 396 q^{41} - 1056 q^{47} + 466 q^{49} + 176 q^{55} + 352 q^{57} + 528 q^{63} - 88 q^{65} + 1456 q^{71} - 308 q^{73} + 1312 q^{79} - 622 q^{81} - 1584 q^{87} - 1428 q^{89} + 176 q^{95} - 956 q^{97}+O(q^{100}) \) 2 * q + 48 * q^7 + 22 * q^9 - 16 * q^15 + 100 * q^17 - 112 * q^23 + 242 * q^25 + 320 * q^31 + 352 * q^33 - 176 * q^39 + 396 * q^41 - 1056 * q^47 + 466 * q^49 + 176 * q^55 + 352 * q^57 + 528 * q^63 - 88 * q^65 + 1456 * q^71 - 308 * q^73 + 1312 * q^79 - 622 * q^81 - 1584 * q^87 - 1428 * q^89 + 176 * q^95 - 956 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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