LMFDB - Embedded newform 256.7.d.c.127.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,7,Mod(127,256)]

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([1, 1]))

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("256.127");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	256.d (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:	$58.8938454067$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 32)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	256.127
Dual form			256.7.d.c.127.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000 q^{3} +50.0000i q^{5} -184.000i q^{7} -713.000 q^{9} +O(q^{10})$	$q+4.00000 q^{3} +50.0000i q^{5} -184.000i q^{7} -713.000 q^{9} -2108.00 q^{11} -2242.00i q^{13} +200.000i q^{15} +2898.00 q^{17} +8052.00 q^{19} -736.000i q^{21} +18584.0i q^{23} +13125.0 q^{25} -5768.00 q^{27} +20990.0i q^{29} -46880.0i q^{31} -8432.00 q^{33} +9200.00 q^{35} +402.000i q^{37} -8968.00i q^{39} -13330.0 q^{41} -5980.00 q^{43} -35650.0i q^{45} +144784. i q^{47} +83793.0 q^{49} +11592.0 q^{51} +171570. i q^{53} -105400. i q^{55} +32208.0 q^{57} +147316. q^{59} +270878. i q^{61} +131192. i q^{63} +112100. q^{65} -431612. q^{67} +74336.0i q^{69} -38648.0i q^{71} +696606. q^{73} +52500.0 q^{75} +387872. i q^{77} -670096. i q^{79} +496705. q^{81} +301300. q^{83} +144900. i q^{85} +83960.0i q^{87} +161598. q^{89} -412528. q^{91} -187520. i q^{93} +402600. i q^{95} +520306. q^{97} +1.50300e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{3} - 1426 q^{9}+O(q^{10}) $ 2 * q + 8 * q^3 - 1426 * q^9	$ 2 q + 8 q^{3} - 1426 q^{9} - 4216 q^{11} + 5796 q^{17} + 16104 q^{19} + 26250 q^{25} - 11536 q^{27} - 16864 q^{33} + 18400 q^{35} - 26660 q^{41} - 11960 q^{43} + 167586 q^{49} + 23184 q^{51} + 64416 q^{57} + 294632 q^{59} + 224200 q^{65} - 863224 q^{67} + 1393212 q^{73} + 105000 q^{75} + 993410 q^{81} + 602600 q^{83} + 323196 q^{89} - 825056 q^{91} + 1040612 q^{97} + 3006008 q^{99}+O(q^{100}) $ 2 * q + 8 * q^3 - 1426 * q^9 - 4216 * q^11 + 5796 * q^17 + 16104 * q^19 + 26250 * q^25 - 11536 * q^27 - 16864 * q^33 + 18400 * q^35 - 26660 * q^41 - 11960 * q^43 + 167586 * q^49 + 23184 * q^51 + 64416 * q^57 + 294632 * q^59 + 224200 * q^65 - 863224 * q^67 + 1393212 * q^73 + 105000 * q^75 + 993410 * q^81 + 602600 * q^83 + 323196 * q^89 - 825056 * q^91 + 1040612 * q^97 + 3006008 * q^99

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
$3$	4.00000	0.148148	0.0740741	−	0.997253i	$-0.476400\pi$
$3$	4.00000	0.148148	0.0740741	+	0.997253i	$0.476400\pi$
$4$	0	0
$5$	50.0000i	0.400000i	0.979796	+	0.200000i	$0.0640942\pi$
$5$	50.0000i	0.400000i	−0.979796	+	0.200000i	$0.935906\pi$
$6$	0	0
$7$	− 184.000i	− 0.536443i	−0.963357	−	0.268222i	$-0.913564\pi$
$7$	− 184.000i	− 0.536443i	0.963357	−	0.268222i	$-0.0864359\pi$
$8$	0	0
$9$	−713.000	−0.978052
$10$	0	0
$11$	−2108.00	−1.58377	−0.791886	−	0.610669i	$-0.790901\pi$
$11$	−2108.00	−1.58377	−0.791886	+	0.610669i	$0.790901\pi$
$12$	0	0
$13$	− 2242.00i	− 1.02048i	−0.860031	−	0.510241i	$-0.829556\pi$
$13$	− 2242.00i	− 1.02048i	0.860031	−	0.510241i	$-0.170444\pi$
$14$	0	0
$15$	200.000i	0.0592593i
$16$	0	0
$17$	2898.00	0.589864	0.294932	−	0.955518i	$-0.404703\pi$
$17$	2898.00	0.589864	0.294932	+	0.955518i	$0.404703\pi$
$18$	0	0
$19$	8052.00	1.17393	0.586966	−	0.809612i	$-0.300322\pi$
$19$	8052.00	1.17393	0.586966	+	0.809612i	$0.300322\pi$
$20$	0	0
$21$	− 736.000i	− 0.0794731i
$22$	0	0
$23$	18584.0i	1.52741i	0.645565	+	0.763705i	$0.276622\pi$
$23$	18584.0i	1.52741i	−0.645565	+	0.763705i	$0.723378\pi$
$24$	0	0
$25$	13125.0	0.840000
$26$	0	0
$27$	−5768.00	−0.293045
$28$	0	0
$29$	20990.0i	0.860634i	0.902678	+	0.430317i	$0.141598\pi$
$29$	20990.0i	0.860634i	−0.902678	+	0.430317i	$0.858402\pi$
$30$	0	0
$31$	− 46880.0i	− 1.57363i	−0.617189	−	0.786815i	$-0.711729\pi$
$31$	− 46880.0i	− 1.57363i	0.617189	−	0.786815i	$-0.288271\pi$
$32$	0	0
$33$	−8432.00	−0.234633
$34$	0	0
$35$	9200.00	0.214577
$36$	0	0
$37$	402.000i	0.00793635i	0.999992	+	0.00396818i	$0.00126311\pi$
$37$	402.000i	0.00793635i	−0.999992	+	0.00396818i	$0.998737\pi$
$38$	0	0
$39$	− 8968.00i	− 0.151183i
$40$	0	0
$41$	−13330.0	−0.193410	−0.0967049	−	0.995313i	$-0.530830\pi$
$41$	−13330.0	−0.193410	−0.0967049	+	0.995313i	$0.530830\pi$
$42$	0	0
$43$	−5980.00	−0.0752135	−0.0376068	−	0.999293i	$-0.511973\pi$
$43$	−5980.00	−0.0752135	−0.0376068	+	0.999293i	$0.511973\pi$
$44$	0	0
$45$	− 35650.0i	− 0.391221i
$46$	0	0
$47$	144784.i	1.39453i	0.716815	+	0.697264i	$0.245599\pi$
$47$	144784.i	1.39453i	−0.716815	+	0.697264i	$0.754401\pi$
$48$	0	0
$49$	83793.0	0.712229
$50$	0	0
$51$	11592.0	0.0873872
$52$	0	0
$53$	171570.i	1.15243i	0.817299	+	0.576214i	$0.195470\pi$
$53$	171570.i	1.15243i	−0.817299	+	0.576214i	$0.804530\pi$
$54$	0	0
$55$	− 105400.i	− 0.633509i
$56$	0	0
$57$	32208.0	0.173916
$58$	0	0
$59$	147316.	0.717289	0.358644	−	0.933474i	$-0.383239\pi$
$59$	147316.	0.717289	0.358644	+	0.933474i	$0.383239\pi$
$60$	0	0
$61$	270878.i	1.19340i	0.802466	+	0.596698i	$0.203521\pi$
$61$	270878.i	1.19340i	−0.802466	+	0.596698i	$0.796479\pi$
$62$	0	0
$63$	131192.i	0.524669i
$64$	0	0
$65$	112100.	0.408193
$66$	0	0
$67$	−431612.	−1.43506	−0.717528	−	0.696529i	$-0.754727\pi$
$67$	−431612.	−1.43506	−0.717528	+	0.696529i	$0.754727\pi$
$68$	0	0
$69$	74336.0i	0.226283i
$70$	0	0
$71$	− 38648.0i	− 0.107982i	−0.998541	−	0.0539911i	$-0.982806\pi$
$71$	− 38648.0i	− 0.107982i	0.998541	−	0.0539911i	$-0.0171943\pi$
$72$	0	0
$73$	696606.	1.79068	0.895341	−	0.445381i	$-0.146932\pi$
$73$	696606.	1.79068	0.895341	+	0.445381i	$0.146932\pi$
$74$	0	0
$75$	52500.0	0.124444
$76$	0	0
$77$	387872.i	0.849603i
$78$	0	0
$79$	− 670096.i	− 1.35911i	−0.733623	−	0.679557i	$-0.762172\pi$
$79$	− 670096.i	− 1.35911i	0.733623	−	0.679557i	$-0.237828\pi$
$80$	0	0
$81$	496705.	0.934638
$82$	0	0
$83$	301300.	0.526944	0.263472	−	0.964667i	$-0.415132\pi$
$83$	301300.	0.526944	0.263472	+	0.964667i	$0.415132\pi$
$84$	0	0
$85$	144900.i	0.235945i
$86$	0	0
$87$	83960.0i	0.127501i
$88$	0	0
$89$	161598.	0.229227	0.114614	−	0.993410i	$-0.463437\pi$
$89$	161598.	0.229227	0.114614	+	0.993410i	$0.463437\pi$
$90$	0	0
$91$	−412528.	−0.547431
$92$	0	0
$93$	− 187520.i	− 0.233130i
$94$	0	0
$95$	402600.i	0.469573i
$96$	0	0
$97$	520306.	0.570090	0.285045	−	0.958514i	$-0.407991\pi$
$97$	520306.	0.570090	0.285045	+	0.958514i	$0.407991\pi$
$98$	0	0
$99$	1.50300e6	1.54901
$100$	0	0
$101$	1.59855e6i	1.55153i	0.631020	+	0.775766i	$0.282637\pi$
$101$	1.59855e6i	1.55153i	−0.631020	+	0.775766i	$0.717363\pi$
$102$	0	0
$103$	886696.i	0.811452i	0.913995	+	0.405726i	$0.132981\pi$
$103$	886696.i	0.811452i	−0.913995	+	0.405726i	$0.867019\pi$
$104$	0	0
$105$	36800.0	0.0317892
$106$	0	0
$107$	−947676.	−0.773586	−0.386793	−	0.922167i	$-0.626417\pi$
$107$	−947676.	−0.773586	−0.386793	+	0.922167i	$0.626417\pi$
$108$	0	0
$109$	1.96112e6i	1.51434i	0.653216	+	0.757171i	$0.273419\pi$
$109$	1.96112e6i	1.51434i	−0.653216	+	0.757171i	$0.726581\pi$
$110$	0	0
$111$	1608.00i	0.00117576i
$112$	0	0
$113$	1.76746e6	1.22494	0.612469	−	0.790495i	$-0.290177\pi$
$113$	1.76746e6	1.22494	0.612469	+	0.790495i	$0.290177\pi$
$114$	0	0
$115$	−929200.	−0.610964
$116$	0	0
$117$	1.59855e6i	0.998085i
$118$	0	0
$119$	− 533232.i	− 0.316428i
$120$	0	0
$121$	2.67210e6	1.50833
$122$	0	0
$123$	−53320.0	−0.0286533
$124$	0	0
$125$	1.43750e6i	0.736000i
$126$	0	0
$127$	− 1.97222e6i	− 0.962820i	−0.876496	−	0.481410i	$-0.840125\pi$
$127$	− 1.97222e6i	− 0.962820i	0.876496	−	0.481410i	$-0.159875\pi$
$128$	0	0
$129$	−23920.0	−0.0111427
$130$	0	0
$131$	−2.25478e6	−1.00298	−0.501488	−	0.865165i	$-0.667214\pi$
$131$	−2.25478e6	−1.00298	−0.501488	+	0.865165i	$0.667214\pi$
$132$	0	0
$133$	− 1.48157e6i	− 0.629748i
$134$	0	0
$135$	− 288400.i	− 0.117218i
$136$	0	0
$137$	1.23064e6	0.478596	0.239298	−	0.970946i	$-0.423083\pi$
$137$	1.23064e6	0.478596	0.239298	+	0.970946i	$0.423083\pi$
$138$	0	0
$139$	4.03776e6	1.50348	0.751738	−	0.659462i	$-0.229216\pi$
$139$	4.03776e6	1.50348	0.751738	+	0.659462i	$0.229216\pi$
$140$	0	0
$141$	579136.i	0.206597i
$142$	0	0
$143$	4.72614e6i	1.61621i
$144$	0	0
$145$	−1.04950e6	−0.344254
$146$	0	0
$147$	335172.	0.105515
$148$	0	0
$149$	1.84712e6i	0.558389i	0.960235	+	0.279194i	$0.0900674\pi$
$149$	1.84712e6i	0.558389i	−0.960235	+	0.279194i	$0.909933\pi$
$150$	0	0
$151$	− 101096.i	− 0.0293632i	−0.999892	−	0.0146816i	$-0.995327\pi$
$151$	− 101096.i	− 0.0293632i	0.999892	−	0.0146816i	$-0.00467346\pi$
$152$	0	0
$153$	−2.06627e6	−0.576917
$154$	0	0
$155$	2.34400e6	0.629452
$156$	0	0
$157$	− 879842.i	− 0.227356i	−0.993518	−	0.113678i	$-0.963737\pi$
$157$	− 879842.i	− 0.227356i	0.993518	−	0.113678i	$-0.0362632\pi$
$158$	0	0
$159$	686280.i	0.170730i
$160$	0	0
$161$	3.41946e6	0.819369
$162$	0	0
$163$	3.16784e6	0.731478	0.365739	−	0.930718i	$-0.380816\pi$
$163$	3.16784e6	0.731478	0.365739	+	0.930718i	$0.380816\pi$
$164$	0	0
$165$	− 421600.i	− 0.0938531i
$166$	0	0
$167$	− 5.26479e6i	− 1.13040i	−0.824954	−	0.565200i	$-0.808799\pi$
$167$	− 5.26479e6i	− 1.13040i	0.824954	−	0.565200i	$-0.191201\pi$
$168$	0	0
$169$	−199755.	−0.0413845
$170$	0	0
$171$	−5.74108e6	−1.14817
$172$	0	0
$173$	− 3.81869e6i	− 0.737524i	−0.929524	−	0.368762i	$-0.879782\pi$
$173$	− 3.81869e6i	− 0.737524i	0.929524	−	0.368762i	$-0.120218\pi$
$174$	0	0
$175$	− 2.41500e6i	− 0.450612i
$176$	0	0
$177$	589264.	0.106265
$178$	0	0
$179$	−4.02136e6	−0.701154	−0.350577	−	0.936534i	$-0.614015\pi$
$179$	−4.02136e6	−0.701154	−0.350577	+	0.936534i	$0.614015\pi$
$180$	0	0
$181$	1.75432e6i	0.295851i	0.988998	+	0.147926i	$0.0472596\pi$
$181$	1.75432e6i	0.295851i	−0.988998	+	0.147926i	$0.952740\pi$
$182$	0	0
$183$	1.08351e6i	0.176799i
$184$	0	0
$185$	−20100.0	−0.00317454
$186$	0	0
$187$	−6.10898e6	−0.934209
$188$	0	0
$189$	1.06131e6i	0.157202i
$190$	0	0
$191$	− 278336.i	− 0.0399456i	−0.999801	−	0.0199728i	$-0.993642\pi$
$191$	− 278336.i	− 0.0399456i	0.999801	−	0.0199728i	$-0.00635797\pi$
$192$	0	0
$193$	−1.84187e6	−0.256205	−0.128102	−	0.991761i	$-0.540889\pi$
$193$	−1.84187e6	−0.256205	−0.128102	+	0.991761i	$0.540889\pi$
$194$	0	0
$195$	448400.	0.0604730
$196$	0	0
$197$	− 9.59425e6i	− 1.25491i	−0.778653	−	0.627455i	$-0.784097\pi$
$197$	− 9.59425e6i	− 1.25491i	0.778653	−	0.627455i	$-0.215903\pi$
$198$	0	0
$199$	1.36934e7i	1.73762i	0.495150	+	0.868808i	$0.335113\pi$
$199$	1.36934e7i	1.73762i	−0.495150	+	0.868808i	$0.664887\pi$
$200$	0	0
$201$	−1.72645e6	−0.212601
$202$	0	0
$203$	3.86216e6	0.461681
$204$	0	0
$205$	− 666500.i	− 0.0773639i
$206$	0	0
$207$	− 1.32504e7i	− 1.49389i
$208$	0	0
$209$	−1.69736e7	−1.85924
$210$	0	0
$211$	1.08290e7	1.15277	0.576385	−	0.817178i	$-0.304463\pi$
$211$	1.08290e7	1.15277	0.576385	+	0.817178i	$0.304463\pi$
$212$	0	0
$213$	− 154592.i	− 0.0159974i
$214$	0	0
$215$	− 299000.i	− 0.0300854i
$216$	0	0
$217$	−8.62592e6	−0.844163
$218$	0	0
$219$	2.78642e6	0.265286
$220$	0	0
$221$	− 6.49732e6i	− 0.601945i
$222$	0	0
$223$	5.41034e6i	0.487876i	0.969791	+	0.243938i	$0.0784394\pi$
$223$	5.41034e6i	0.487876i	−0.969791	+	0.243938i	$0.921561\pi$
$224$	0	0
$225$	−9.35812e6	−0.821564
$226$	0	0
$227$	−9.45622e6	−0.808425	−0.404213	−	0.914665i	$-0.632454\pi$
$227$	−9.45622e6	−0.808425	−0.404213	+	0.914665i	$0.632454\pi$
$228$	0	0
$229$	− 1.47605e7i	− 1.22912i	−0.788869	−	0.614562i	$-0.789333\pi$
$229$	− 1.47605e7i	− 1.22912i	0.788869	−	0.614562i	$-0.210667\pi$
$230$	0	0
$231$	1.55149e6i	0.125867i
$232$	0	0
$233$	−1.53357e7	−1.21237	−0.606186	−	0.795323i	$-0.707301\pi$
$233$	−1.53357e7	−1.21237	−0.606186	+	0.795323i	$0.707301\pi$
$234$	0	0
$235$	−7.23920e6	−0.557811
$236$	0	0
$237$	− 2.68038e6i	− 0.201350i
$238$	0	0
$239$	8.23509e6i	0.603218i	0.953432	+	0.301609i	$0.0975238\pi$
$239$	8.23509e6i	0.603218i	−0.953432	+	0.301609i	$0.902476\pi$
$240$	0	0
$241$	8.09026e6	0.577978	0.288989	−	0.957332i	$-0.406681\pi$
$241$	8.09026e6	0.577978	0.288989	+	0.957332i	$0.406681\pi$
$242$	0	0
$243$	6.19169e6	0.431510
$244$	0	0
$245$	4.18965e6i	0.284891i
$246$	0	0
$247$	− 1.80526e7i	− 1.19798i
$248$	0	0
$249$	1.20520e6	0.0780658
$250$	0	0
$251$	1.19698e7	0.756950	0.378475	−	0.925611i	$-0.376449\pi$
$251$	1.19698e7	0.756950	0.378475	+	0.925611i	$0.376449\pi$
$252$	0	0
$253$	− 3.91751e7i	− 2.41907i
$254$	0	0
$255$	579600.i	0.0349549i
$256$	0	0
$257$	−1.96915e6	−0.116006	−0.0580029	−	0.998316i	$-0.518473\pi$
$257$	−1.96915e6	−0.116006	−0.0580029	+	0.998316i	$0.518473\pi$
$258$	0	0
$259$	73968.0	0.00425740
$260$	0	0
$261$	− 1.49659e7i	− 0.841745i
$262$	0	0
$263$	− 2.88123e7i	− 1.58384i	−0.610625	−	0.791920i	$-0.709082\pi$
$263$	− 2.88123e7i	− 1.58384i	0.610625	−	0.791920i	$-0.290918\pi$
$264$	0	0
$265$	−8.57850e6	−0.460971
$266$	0	0
$267$	646392.	0.0339596
$268$	0	0
$269$	2.08340e7i	1.07032i	0.844750	+	0.535161i	$0.179749\pi$
$269$	2.08340e7i	1.07032i	−0.844750	+	0.535161i	$0.820251\pi$
$270$	0	0
$271$	1.56227e7i	0.784961i	0.919760	+	0.392481i	$0.128383\pi$
$271$	1.56227e7i	0.784961i	−0.919760	+	0.392481i	$0.871617\pi$
$272$	0	0
$273$	−1.65011e6	−0.0811009
$274$	0	0
$275$	−2.76675e7	−1.33037
$276$	0	0
$277$	1.58341e7i	0.744996i	0.928033	+	0.372498i	$0.121499\pi$
$277$	1.58341e7i	0.744996i	−0.928033	+	0.372498i	$0.878501\pi$
$278$	0	0
$279$	3.34254e7i	1.53909i
$280$	0	0
$281$	2.96281e6	0.133532	0.0667660	−	0.997769i	$-0.478732\pi$
$281$	2.96281e6	0.133532	0.0667660	+	0.997769i	$0.478732\pi$
$282$	0	0
$283$	−7.15377e6	−0.315628	−0.157814	−	0.987469i	$-0.550445\pi$
$283$	−7.15377e6	−0.315628	−0.157814	+	0.987469i	$0.550445\pi$
$284$	0	0
$285$	1.61040e6i	0.0695663i
$286$	0	0
$287$	2.45272e6i	0.103753i
$288$	0	0
$289$	−1.57392e7	−0.652061
$290$	0	0
$291$	2.08122e6	0.0844578
$292$	0	0
$293$	− 2.98632e7i	− 1.18723i	−0.804750	−	0.593614i	$-0.797701\pi$
$293$	− 2.98632e7i	− 1.18723i	0.804750	−	0.593614i	$-0.202299\pi$
$294$	0	0
$295$	7.36580e6i	0.286915i
$296$	0	0
$297$	1.21589e7	0.464116
$298$	0	0
$299$	4.16653e7	1.55870
$300$	0	0
$301$	1.10032e6i	0.0403478i
$302$	0	0
$303$	6.39418e6i	0.229857i
$304$	0	0
$305$	−1.35439e7	−0.477358
$306$	0	0
$307$	9.47068e6	0.327315	0.163657	−	0.986517i	$-0.447671\pi$
$307$	9.47068e6	0.327315	0.163657	+	0.986517i	$0.447671\pi$
$308$	0	0
$309$	3.54678e6i	0.120215i
$310$	0	0
$311$	1.42553e7i	0.473909i	0.971521	+	0.236954i	$0.0761492\pi$
$311$	1.42553e7i	0.473909i	−0.971521	+	0.236954i	$0.923851\pi$
$312$	0	0
$313$	−2.58976e7	−0.844552	−0.422276	−	0.906467i	$-0.638769\pi$
$313$	−2.58976e7	−0.844552	−0.422276	+	0.906467i	$0.638769\pi$
$314$	0	0
$315$	−6.55960e6	−0.209868
$316$	0	0
$317$	− 3.29735e6i	− 0.103511i	−0.998660	−	0.0517555i	$-0.983518\pi$
$317$	− 3.29735e6i	− 0.103511i	0.998660	−	0.0517555i	$-0.0164817\pi$
$318$	0	0
$319$	− 4.42469e7i	− 1.36305i
$320$	0	0
$321$	−3.79070e6	−0.114605
$322$	0	0
$323$	2.33347e7	0.692460
$324$	0	0
$325$	− 2.94262e7i	− 0.857205i
$326$	0	0
$327$	7.84447e6i	0.224347i
$328$	0	0
$329$	2.66403e7	0.748085
$330$	0	0
$331$	−6.70721e7	−1.84952	−0.924758	−	0.380556i	$-0.875733\pi$
$331$	−6.70721e7	−1.84952	−0.924758	+	0.380556i	$0.875733\pi$
$332$	0	0
$333$	− 286626.i	− 0.00776217i
$334$	0	0
$335$	− 2.15806e7i	− 0.574023i
$336$	0	0
$337$	3.61328e7	0.944086	0.472043	−	0.881576i	$-0.343517\pi$
$337$	3.61328e7	0.944086	0.472043	+	0.881576i	$0.343517\pi$
$338$	0	0
$339$	7.06983e6	0.181472
$340$	0	0
$341$	9.88230e7i	2.49227i
$342$	0	0
$343$	− 3.70653e7i	− 0.918513i
$344$	0	0
$345$	−3.71680e6	−0.0905132
$346$	0	0
$347$	−3.33170e7	−0.797402	−0.398701	−	0.917081i	$-0.630539\pi$
$347$	−3.33170e7	−0.797402	−0.398701	+	0.917081i	$0.630539\pi$
$348$	0	0
$349$	3.04510e7i	0.716350i	0.933654	+	0.358175i	$0.116601\pi$
$349$	3.04510e7i	0.716350i	−0.933654	+	0.358175i	$0.883399\pi$
$350$	0	0
$351$	1.29319e7i	0.299047i
$352$	0	0
$353$	5.43073e7	1.23462	0.617311	−	0.786719i	$-0.288222\pi$
$353$	5.43073e7	1.23462	0.617311	+	0.786719i	$0.288222\pi$
$354$	0	0
$355$	1.93240e6	0.0431929
$356$	0	0
$357$	− 2.13293e6i	− 0.0468783i
$358$	0	0
$359$	− 1.62646e6i	− 0.0351527i	−0.999846	−	0.0175764i	$-0.994405\pi$
$359$	− 1.62646e6i	− 0.0351527i	0.999846	−	0.0175764i	$-0.00559502\pi$
$360$	0	0
$361$	1.77888e7	0.378116
$362$	0	0
$363$	1.06884e7	0.223457
$364$	0	0
$365$	3.48303e7i	0.716273i
$366$	0	0
$367$	− 2.38924e7i	− 0.483350i	−0.970357	−	0.241675i	$-0.922303\pi$
$367$	− 2.38924e7i	− 0.483350i	0.970357	−	0.241675i	$-0.0776968\pi$
$368$	0	0
$369$	9.50429e6	0.189165
$370$	0	0
$371$	3.15689e7	0.618212
$372$	0	0
$373$	9.46760e7i	1.82437i	0.409777	+	0.912186i	$0.365606\pi$
$373$	9.46760e7i	1.82437i	−0.409777	+	0.912186i	$0.634394\pi$
$374$	0	0
$375$	5.75000e6i	0.109037i
$376$	0	0
$377$	4.70596e7	0.878262
$378$	0	0
$379$	5.17551e7	0.950682	0.475341	−	0.879802i	$-0.342325\pi$
$379$	5.17551e7	0.950682	0.475341	+	0.879802i	$0.342325\pi$
$380$	0	0
$381$	− 7.88890e6i	− 0.142640i
$382$	0	0
$383$	7.70121e7i	1.37076i	0.728184	+	0.685382i	$0.240365\pi$
$383$	7.70121e7i	1.37076i	−0.728184	+	0.685382i	$0.759635\pi$
$384$	0	0
$385$	−1.93936e7	−0.339841
$386$	0	0
$387$	4.26374e6	0.0735627
$388$	0	0
$389$	− 8.92562e7i	− 1.51632i	−0.652071	−	0.758158i	$-0.726100\pi$
$389$	− 8.92562e7i	− 1.51632i	0.652071	−	0.758158i	$-0.273900\pi$
$390$	0	0
$391$	5.38564e7i	0.900964i
$392$	0	0
$393$	−9.01912e6	−0.148589
$394$	0	0
$395$	3.35048e7	0.543645
$396$	0	0
$397$	8.45175e7i	1.35075i	0.737474	+	0.675375i	$0.236018\pi$
$397$	8.45175e7i	1.35075i	−0.737474	+	0.675375i	$0.763982\pi$
$398$	0	0
$399$	− 5.92627e6i	− 0.0932960i
$400$	0	0
$401$	6.78525e7	1.05228	0.526142	−	0.850397i	$-0.323638\pi$
$401$	6.78525e7	1.05228	0.526142	+	0.850397i	$0.323638\pi$
$402$	0	0
$403$	−1.05105e8	−1.60586
$404$	0	0
$405$	2.48352e7i	0.373855i
$406$	0	0
$407$	− 847416.i	− 0.0125694i
$408$	0	0
$409$	1.16075e8	1.69657	0.848283	−	0.529544i	$-0.177637\pi$
$409$	1.16075e8	1.69657	0.848283	+	0.529544i	$0.177637\pi$
$410$	0	0
$411$	4.92255e6	0.0709030
$412$	0	0
$413$	− 2.71061e7i	− 0.384785i
$414$	0	0
$415$	1.50650e7i	0.210778i
$416$	0	0
$417$	1.61511e7	0.222737
$418$	0	0
$419$	−7.04530e7	−0.957761	−0.478881	−	0.877880i	$-0.658957\pi$
$419$	−7.04530e7	−0.957761	−0.478881	+	0.877880i	$0.658957\pi$
$420$	0	0
$421$	− 6.29023e7i	− 0.842986i	−0.906832	−	0.421493i	$-0.861506\pi$
$421$	− 6.29023e7i	− 0.842986i	0.906832	−	0.421493i	$-0.138494\pi$
$422$	0	0
$423$	− 1.03231e8i	− 1.36392i
$424$	0	0
$425$	3.80362e7	0.495485
$426$	0	0
$427$	4.98416e7	0.640189
$428$	0	0
$429$	1.89045e7i	0.239439i
$430$	0	0
$431$	− 3.30262e7i	− 0.412502i	−0.978499	−	0.206251i	$-0.933874\pi$
$431$	− 3.30262e7i	− 0.412502i	0.978499	−	0.206251i	$-0.0661264\pi$
$432$	0	0
$433$	−4.83844e7	−0.595993	−0.297997	−	0.954567i	$-0.596318\pi$
$433$	−4.83844e7	−0.595993	−0.297997	+	0.954567i	$0.596318\pi$
$434$	0	0
$435$	−4.19800e6	−0.0510005
$436$	0	0
$437$	1.49638e8i	1.79308i
$438$	0	0
$439$	− 1.10794e7i	− 0.130955i	−0.997854	−	0.0654774i	$-0.979143\pi$
$439$	− 1.10794e7i	− 0.130955i	0.997854	−	0.0654774i	$-0.0208570\pi$
$440$	0	0
$441$	−5.97444e7	−0.696597
$442$	0	0
$443$	−5.74622e7	−0.660954	−0.330477	−	0.943814i	$-0.607210\pi$
$443$	−5.74622e7	−0.660954	−0.330477	+	0.943814i	$0.607210\pi$
$444$	0	0
$445$	8.07990e6i	0.0916908i
$446$	0	0
$447$	7.38849e6i	0.0827243i
$448$	0	0
$449$	9.59427e7	1.05992	0.529960	−	0.848023i	$-0.322207\pi$
$449$	9.59427e7	1.05992	0.529960	+	0.848023i	$0.322207\pi$
$450$	0	0
$451$	2.80996e7	0.306317
$452$	0	0
$453$	− 404384.i	− 0.00435010i
$454$	0	0
$455$	− 2.06264e7i	− 0.218972i
$456$	0	0
$457$	−3.89991e7	−0.408607	−0.204304	−	0.978908i	$-0.565493\pi$
$457$	−3.89991e7	−0.408607	−0.204304	+	0.978908i	$0.565493\pi$
$458$	0	0
$459$	−1.67157e7	−0.172856
$460$	0	0
$461$	9.51231e7i	0.970920i	0.874259	+	0.485460i	$0.161348\pi$
$461$	9.51231e7i	0.970920i	−0.874259	+	0.485460i	$0.838652\pi$
$462$	0	0
$463$	1.26207e8i	1.27157i	0.771867	+	0.635784i	$0.219323\pi$
$463$	1.26207e8i	1.27157i	−0.771867	+	0.635784i	$0.780677\pi$
$464$	0	0
$465$	9.37600e6	0.0932521
$466$	0	0
$467$	1.94212e8	1.90689	0.953446	−	0.301562i	$-0.0975081\pi$
$467$	1.94212e8	1.90689	0.953446	+	0.301562i	$0.0975081\pi$
$468$	0	0
$469$	7.94166e7i	0.769826i
$470$	0	0
$471$	− 3.51937e6i	− 0.0336823i
$472$	0	0
$473$	1.26058e7	0.119121
$474$	0	0
$475$	1.05682e8	0.986103
$476$	0	0
$477$	− 1.22329e8i	− 1.12713i
$478$	0	0
$479$	− 1.46365e8i	− 1.33177i	−0.746054	−	0.665886i	$-0.768054\pi$
$479$	− 1.46365e8i	− 1.33177i	0.746054	−	0.665886i	$-0.231946\pi$
$480$	0	0
$481$	901284.	0.00809891
$482$	0	0
$483$	1.36778e7	0.121388
$484$	0	0
$485$	2.60153e7i	0.228036i
$486$	0	0
$487$	1.27454e8i	1.10348i	0.834016	+	0.551741i	$0.186036\pi$
$487$	1.27454e8i	1.10348i	−0.834016	+	0.551741i	$0.813964\pi$
$488$	0	0
$489$	1.26714e7	0.108367
$490$	0	0
$491$	−2.31258e8	−1.95368	−0.976838	−	0.213980i	$-0.931357\pi$
$491$	−2.31258e8	−1.95368	−0.976838	+	0.213980i	$0.931357\pi$
$492$	0	0
$493$	6.08290e7i	0.507657i
$494$	0	0
$495$	7.51502e7i	0.619604i
$496$	0	0
$497$	−7.11123e6	−0.0579263
$498$	0	0
$499$	2.90337e7	0.233669	0.116834	−	0.993151i	$-0.462725\pi$
$499$	2.90337e7	0.233669	0.116834	+	0.993151i	$0.462725\pi$
$500$	0	0
$501$	− 2.10592e7i	− 0.167467i
$502$	0	0
$503$	9.61500e7i	0.755519i	0.925904	+	0.377759i	$0.123305\pi$
$503$	9.61500e7i	0.755519i	−0.925904	+	0.377759i	$0.876695\pi$
$504$	0	0
$505$	−7.99273e7	−0.620613
$506$	0	0
$507$	−799020.	−0.00613103
$508$	0	0
$509$	− 5.02103e7i	− 0.380750i	−0.981711	−	0.190375i	$-0.939030\pi$
$509$	− 5.02103e7i	− 0.380750i	0.981711	−	0.190375i	$-0.0609704\pi$
$510$	0	0
$511$	− 1.28176e8i	− 0.960599i
$512$	0	0
$513$	−4.64439e7	−0.344015
$514$	0	0
$515$	−4.43348e7	−0.324581
$516$	0	0
$517$	− 3.05205e8i	− 2.20861i
$518$	0	0
$519$	− 1.52748e7i	− 0.109263i
$520$	0	0
$521$	4.30325e7	0.304287	0.152143	−	0.988358i	$-0.451382\pi$
$521$	4.30325e7	0.304287	0.152143	+	0.988358i	$0.451382\pi$
$522$	0	0
$523$	1.72509e8	1.20589	0.602943	−	0.797784i	$-0.293995\pi$
$523$	1.72509e8	1.20589	0.602943	+	0.797784i	$0.293995\pi$
$524$	0	0
$525$	− 9.66000e6i	− 0.0667574i
$526$	0	0
$527$	− 1.35858e8i	− 0.928227i
$528$	0	0
$529$	−1.97329e8	−1.33298
$530$	0	0
$531$	−1.05036e8	−0.701546
$532$	0	0
$533$	2.98859e7i	0.197371i
$534$	0	0
$535$	− 4.73838e7i	− 0.309434i
$536$	0	0
$537$	−1.60854e7	−0.103875
$538$	0	0
$539$	−1.76636e8	−1.12801
$540$	0	0
$541$	− 1.59475e8i	− 1.00716i	−0.863947	−	0.503582i	$-0.832015\pi$
$541$	− 1.59475e8i	− 1.00716i	0.863947	−	0.503582i	$-0.167985\pi$
$542$	0	0
$543$	7.01729e6i	0.0438298i
$544$	0	0
$545$	−9.80559e7	−0.605737
$546$	0	0
$547$	3.06557e7	0.187305	0.0936523	−	0.995605i	$-0.470146\pi$
$547$	3.06557e7	0.187305	0.0936523	+	0.995605i	$0.470146\pi$
$548$	0	0
$549$	− 1.93136e8i	− 1.16720i
$550$	0	0
$551$	1.69011e8i	1.01033i
$552$	0	0
$553$	−1.23298e8	−0.729087
$554$	0	0
$555$	−80400.0	−0.000470302 0
$556$	0	0
$557$	− 1.99783e6i	− 0.0115609i	−0.999983	−	0.00578046i	$-0.998160\pi$
$557$	− 1.99783e6i	− 0.0115609i	0.999983	−	0.00578046i	$-0.00183999\pi$
$558$	0	0
$559$	1.34072e7i	0.0767541i
$560$	0	0
$561$	−2.44359e7	−0.138401
$562$	0	0
$563$	−1.55895e8	−0.873588	−0.436794	−	0.899562i	$-0.643886\pi$
$563$	−1.55895e8	−0.873588	−0.436794	+	0.899562i	$0.643886\pi$
$564$	0	0
$565$	8.83729e7i	0.489975i
$566$	0	0
$567$	− 9.13937e7i	− 0.501380i
$568$	0	0
$569$	−1.06750e8	−0.579468	−0.289734	−	0.957107i	$-0.593567\pi$
$569$	−1.06750e8	−0.579468	−0.289734	+	0.957107i	$0.593567\pi$
$570$	0	0
$571$	−5.29306e7	−0.284314	−0.142157	−	0.989844i	$-0.545404\pi$
$571$	−5.29306e7	−0.284314	−0.142157	+	0.989844i	$0.545404\pi$
$572$	0	0
$573$	− 1.11334e6i	− 0.00591787i
$574$	0	0
$575$	2.43915e8i	1.28302i
$576$	0	0
$577$	−5.19482e7	−0.270423	−0.135211	−	0.990817i	$-0.543171\pi$
$577$	−5.19482e7	−0.270423	−0.135211	+	0.990817i	$0.543171\pi$
$578$	0	0
$579$	−7.36748e6	−0.0379562
$580$	0	0
$581$	− 5.54392e7i	− 0.282676i
$582$	0	0
$583$	− 3.61670e8i	− 1.82518i
$584$	0	0
$585$	−7.99273e7	−0.399234
$586$	0	0
$587$	1.89126e8	0.935057	0.467528	−	0.883978i	$-0.345145\pi$
$587$	1.89126e8	0.935057	0.467528	+	0.883978i	$0.345145\pi$
$588$	0	0
$589$	− 3.77478e8i	− 1.84733i
$590$	0	0
$591$	− 3.83770e7i	− 0.185913i
$592$	0	0
$593$	1.09940e8	0.527221	0.263611	−	0.964629i	$-0.415087\pi$
$593$	1.09940e8	0.527221	0.263611	+	0.964629i	$0.415087\pi$
$594$	0	0
$595$	2.66616e7	0.126571
$596$	0	0
$597$	5.47738e7i	0.257424i
$598$	0	0
$599$	− 6.60256e7i	− 0.307208i	−0.988133	−	0.153604i	$-0.950912\pi$
$599$	− 6.60256e7i	− 0.307208i	0.988133	−	0.153604i	$-0.0490880\pi$
$600$	0	0
$601$	−8.21137e7	−0.378262	−0.189131	−	0.981952i	$-0.560567\pi$
$601$	−8.21137e7	−0.378262	−0.189131	+	0.981952i	$0.560567\pi$
$602$	0	0
$603$	3.07739e8	1.40356
$604$	0	0
$605$	1.33605e8i	0.603333i
$606$	0	0
$607$	2.93964e8i	1.31440i	0.753715	+	0.657202i	$0.228260\pi$
$607$	2.93964e8i	1.31440i	−0.753715	+	0.657202i	$0.771740\pi$
$608$	0	0
$609$	1.54486e7	0.0683972
$610$	0	0
$611$	3.24606e8	1.42309
$612$	0	0
$613$	− 1.86200e8i	− 0.808348i	−0.914682	−	0.404174i	$-0.867559\pi$
$613$	− 1.86200e8i	− 0.808348i	0.914682	−	0.404174i	$-0.132441\pi$
$614$	0	0
$615$	− 2.66600e6i	− 0.0114613i
$616$	0	0
$617$	−2.92906e7	−0.124702	−0.0623509	−	0.998054i	$-0.519860\pi$
$617$	−2.92906e7	−0.124702	−0.0623509	+	0.998054i	$0.519860\pi$
$618$	0	0
$619$	−1.87910e8	−0.792277	−0.396139	−	0.918191i	$-0.629650\pi$
$619$	−1.87910e8	−0.792277	−0.396139	+	0.918191i	$0.629650\pi$
$620$	0	0
$621$	− 1.07193e8i	− 0.447600i
$622$	0	0
$623$	− 2.97340e7i	− 0.122967i
$624$	0	0
$625$	1.33203e8	0.545600
$626$	0	0
$627$	−6.78945e7	−0.275443
$628$	0	0
$629$	1.16500e6i	0.00468136i
$630$	0	0
$631$	− 2.79075e7i	− 0.111079i	−0.998456	−	0.0555395i	$-0.982312\pi$
$631$	− 2.79075e7i	− 0.111079i	0.998456	−	0.0555395i	$-0.0176879\pi$
$632$	0	0
$633$	4.33162e7	0.170781
$634$	0	0
$635$	9.86112e7	0.385128
$636$	0	0
$637$	− 1.87864e8i	− 0.726817i
$638$	0	0
$639$	2.75560e7i	0.105612i
$640$	0	0
$641$	−3.88769e8	−1.47611	−0.738053	−	0.674742i	$-0.764255\pi$
$641$	−3.88769e8	−1.47611	−0.738053	+	0.674742i	$0.764255\pi$
$642$	0	0
$643$	−4.20041e8	−1.58000	−0.790002	−	0.613104i	$-0.789921\pi$
$643$	−4.20041e8	−1.58000	−0.790002	+	0.613104i	$0.789921\pi$
$644$	0	0
$645$	− 1.19600e6i	− 0.00445710i
$646$	0	0
$647$	3.30167e8i	1.21905i	0.792767	+	0.609524i	$0.208639\pi$
$647$	3.30167e8i	1.21905i	−0.792767	+	0.609524i	$0.791361\pi$
$648$	0	0
$649$	−3.10542e8	−1.13602
$650$	0	0
$651$	−3.45037e7	−0.125061
$652$	0	0
$653$	2.23550e8i	0.802853i	0.915891	+	0.401427i	$0.131485\pi$
$653$	2.23550e8i	0.802853i	−0.915891	+	0.401427i	$0.868515\pi$
$654$	0	0
$655$	− 1.12739e8i	− 0.401190i
$656$	0	0
$657$	−4.96680e8	−1.75138
$658$	0	0
$659$	1.13133e7	0.0395307	0.0197653	−	0.999805i	$-0.493708\pi$
$659$	1.13133e7	0.0395307	0.0197653	+	0.999805i	$0.493708\pi$
$660$	0	0
$661$	− 2.28431e8i	− 0.790955i	−0.918476	−	0.395477i	$-0.870579\pi$
$661$	− 2.28431e8i	− 0.790955i	0.918476	−	0.395477i	$-0.129421\pi$
$662$	0	0
$663$	− 2.59893e7i	− 0.0891771i
$664$	0	0
$665$	7.40784e7	0.251899
$666$	0	0
$667$	−3.90078e8	−1.31454
$668$	0	0
$669$	2.16413e7i	0.0722780i
$670$	0	0
$671$	− 5.71011e8i	− 1.89007i
$672$	0	0
$673$	−5.63585e7	−0.184890	−0.0924452	−	0.995718i	$-0.529468\pi$
$673$	−5.63585e7	−0.184890	−0.0924452	+	0.995718i	$0.529468\pi$
$674$	0	0
$675$	−7.57050e7	−0.246158
$676$	0	0
$677$	3.39715e7i	0.109483i	0.998501	+	0.0547417i	$0.0174335\pi$
$677$	3.39715e7i	0.109483i	−0.998501	+	0.0547417i	$0.982566\pi$
$678$	0	0
$679$	− 9.57363e7i	− 0.305821i
$680$	0	0
$681$	−3.78249e7	−0.119767
$682$	0	0
$683$	2.56442e8	0.804872	0.402436	−	0.915448i	$-0.368164\pi$
$683$	2.56442e8	0.804872	0.402436	+	0.915448i	$0.368164\pi$
$684$	0	0
$685$	6.15319e7i	0.191438i
$686$	0	0
$687$	− 5.90421e7i	− 0.182092i
$688$	0	0
$689$	3.84660e8	1.17603
$690$	0	0
$691$	2.73283e8	0.828282	0.414141	−	0.910213i	$-0.364082\pi$
$691$	2.73283e8	0.828282	0.414141	+	0.910213i	$0.364082\pi$
$692$	0	0
$693$	− 2.76553e8i	− 0.830956i
$694$	0	0
$695$	2.01888e8i	0.601390i
$696$	0	0
$697$	−3.86303e7	−0.114085
$698$	0	0
$699$	−6.13429e7	−0.179611
$700$	0	0
$701$	1.21405e8i	0.352438i	0.984351	+	0.176219i	$0.0563867\pi$
$701$	1.21405e8i	0.352438i	−0.984351	+	0.176219i	$0.943613\pi$
$702$	0	0
$703$	3.23690e6i	0.00931674i
$704$	0	0
$705$	−2.89568e7	−0.0826387
$706$	0	0
$707$	2.94132e8	0.832309
$708$	0	0
$709$	5.56778e8i	1.56222i	0.624391	+	0.781112i	$0.285347\pi$
$709$	5.56778e8i	1.56222i	−0.624391	+	0.781112i	$0.714653\pi$
$710$	0	0
$711$	4.77778e8i	1.32928i
$712$	0	0
$713$	8.71218e8	2.40358
$714$	0	0
$715$	−2.36307e8	−0.646484
$716$	0	0
$717$	3.29404e7i	0.0893657i
$718$	0	0
$719$	5.79277e8i	1.55847i	0.626730	+	0.779236i	$0.284393\pi$
$719$	5.79277e8i	1.55847i	−0.626730	+	0.779236i	$0.715607\pi$
$720$	0	0
$721$	1.63152e8	0.435298
$722$	0	0
$723$	3.23610e7	0.0856264
$724$	0	0
$725$	2.75494e8i	0.722932i
$726$	0	0
$727$	3.52065e8i	0.916263i	0.888884	+	0.458132i	$0.151481\pi$
$727$	3.52065e8i	0.916263i	−0.888884	+	0.458132i	$0.848519\pi$
$728$	0	0
$729$	−3.37331e8	−0.870711
$730$	0	0
$731$	−1.73300e7	−0.0443657
$732$	0	0
$733$	− 1.35797e8i	− 0.344808i	−0.985026	−	0.172404i	$-0.944847\pi$
$733$	− 1.35797e8i	− 0.344808i	0.985026	−	0.172404i	$-0.0551535\pi$
$734$	0	0
$735$	1.67586e7i	0.0422061i
$736$	0	0
$737$	9.09838e8	2.27280
$738$	0	0
$739$	3.97608e8	0.985195	0.492597	−	0.870257i	$-0.336047\pi$
$739$	3.97608e8	0.985195	0.492597	+	0.870257i	$0.336047\pi$
$740$	0	0
$741$	− 7.22103e7i	− 0.177478i
$742$	0	0
$743$	− 6.04512e8i	− 1.47380i	−0.676002	−	0.736899i	$-0.736289\pi$
$743$	− 6.04512e8i	− 1.47380i	0.676002	−	0.736899i	$-0.263711\pi$
$744$	0	0
$745$	−9.23561e7	−0.223356
$746$	0	0
$747$	−2.14827e8	−0.515379
$748$	0	0
$749$	1.74372e8i	0.414985i
$750$	0	0
$751$	− 1.11382e8i	− 0.262964i	−0.991319	−	0.131482i	$-0.958026\pi$
$751$	− 1.11382e8i	− 0.262964i	0.991319	−	0.131482i	$-0.0419735\pi$
$752$	0	0
$753$	4.78794e7	0.112141
$754$	0	0
$755$	5.05480e6	0.0117453
$756$	0	0
$757$	− 9.60559e7i	− 0.221430i	−0.993852	−	0.110715i	$-0.964686\pi$
$757$	− 9.60559e7i	− 0.221430i	0.993852	−	0.110715i	$-0.0353141\pi$
$758$	0	0
$759$	− 1.56700e8i	− 0.358381i
$760$	0	0
$761$	2.27306e8	0.515770	0.257885	−	0.966176i	$-0.416974\pi$
$761$	2.27306e8	0.515770	0.257885	+	0.966176i	$0.416974\pi$
$762$	0	0
$763$	3.60846e8	0.812359
$764$	0	0
$765$	− 1.03314e8i	− 0.230767i
$766$	0	0
$767$	− 3.30282e8i	− 0.731980i
$768$	0	0
$769$	−6.61973e7	−0.145566	−0.0727832	−	0.997348i	$-0.523188\pi$
$769$	−6.61973e7	−0.145566	−0.0727832	+	0.997348i	$0.523188\pi$
$770$	0	0
$771$	−7.87660e6	−0.0171860
$772$	0	0
$773$	7.57894e8i	1.64086i	0.571750	+	0.820428i	$0.306265\pi$
$773$	7.57894e8i	1.64086i	−0.571750	+	0.820428i	$0.693735\pi$
$774$	0	0
$775$	− 6.15300e8i	− 1.32185i
$776$	0	0
$777$	295872.	0.000630726 0
$778$	0	0
$779$	−1.07333e8	−0.227050
$780$	0	0
$781$	8.14700e7i	0.171019i
$782$	0	0
$783$	− 1.21070e8i	− 0.252204i
$784$	0	0
$785$	4.39921e7	0.0909423
$786$	0	0
$787$	−3.29425e8	−0.675823	−0.337911	−	0.941178i	$-0.609720\pi$
$787$	−3.29425e8	−0.675823	−0.337911	+	0.941178i	$0.609720\pi$
$788$	0	0
$789$	− 1.15249e8i	− 0.234643i
$790$	0	0
$791$	− 3.25212e8i	− 0.657109i
$792$	0	0
$793$	6.07308e8	1.21784
$794$	0	0
$795$	−3.43140e7	−0.0682920
$796$	0	0
$797$	8.12683e8i	1.60526i	0.596475	+	0.802632i	$0.296568\pi$
$797$	8.12683e8i	1.60526i	−0.596475	+	0.802632i	$0.703432\pi$
$798$	0	0
$799$	4.19584e8i	0.822581i
$800$	0	0
$801$	−1.15219e8	−0.224196
$802$	0	0
$803$	−1.46845e9	−2.83603
$804$	0	0
$805$	1.70973e8i	0.327747i
$806$	0	0
$807$	8.33358e7i	0.158566i
$808$	0	0
$809$	7.79881e7	0.147293	0.0736466	−	0.997284i	$-0.476536\pi$
$809$	7.79881e7	0.147293	0.0736466	+	0.997284i	$0.476536\pi$
$810$	0	0
$811$	5.37023e8	1.00677	0.503385	−	0.864062i	$-0.332088\pi$
$811$	5.37023e8	1.00677	0.503385	+	0.864062i	$0.332088\pi$
$812$	0	0
$813$	6.24908e7i	0.116291i
$814$	0	0
$815$	1.58392e8i	0.292591i
$816$	0	0
$817$	−4.81510e7	−0.0882955
$818$	0	0
$819$	2.94132e8	0.535416
$820$	0	0
$821$	− 3.74396e7i	− 0.0676553i	−0.999428	−	0.0338277i	$-0.989230\pi$
$821$	− 3.74396e7i	− 0.0676553i	0.999428	−	0.0338277i	$-0.0107697\pi$
$822$	0	0
$823$	5.95822e8i	1.06885i	0.845216	+	0.534425i	$0.179472\pi$
$823$	5.95822e8i	1.06885i	−0.845216	+	0.534425i	$0.820528\pi$
$824$	0	0
$825$	−1.10670e8	−0.197092
$826$	0	0
$827$	2.21271e8	0.391207	0.195604	−	0.980683i	$-0.437333\pi$
$827$	2.21271e8	0.391207	0.195604	+	0.980683i	$0.437333\pi$
$828$	0	0
$829$	4.66636e8i	0.819058i	0.912297	+	0.409529i	$0.134307\pi$
$829$	4.66636e8i	0.819058i	−0.912297	+	0.409529i	$0.865693\pi$
$830$	0	0
$831$	6.33364e7i	0.110370i
$832$	0	0
$833$	2.42832e8	0.420118
$834$	0	0
$835$	2.63240e8	0.452160
$836$	0	0
$837$	2.70404e8i	0.461144i
$838$	0	0
$839$	8.41732e7i	0.142524i	0.997458	+	0.0712620i	$0.0227026\pi$
$839$	8.41732e7i	0.142524i	−0.997458	+	0.0712620i	$0.977297\pi$
$840$	0	0
$841$	1.54243e8	0.259309
$842$	0	0
$843$	1.18513e7	0.0197825
$844$	0	0
$845$	− 9.98775e6i	− 0.0165538i
$846$	0	0
$847$	− 4.91667e8i	− 0.809135i
$848$	0	0
$849$	−2.86151e7	−0.0467597
$850$	0	0
$851$	−7.47077e6	−0.0121221
$852$	0	0
$853$	− 1.02826e9i	− 1.65674i	−0.560181	−	0.828371i	$-0.689268\pi$
$853$	− 1.02826e9i	− 1.65674i	0.560181	−	0.828371i	$-0.310732\pi$
$854$	0	0
$855$	− 2.87054e8i	− 0.459267i
$856$	0	0
$857$	1.03110e9	1.63816	0.819081	−	0.573678i	$-0.194484\pi$
$857$	1.03110e9	1.63816	0.819081	+	0.573678i	$0.194484\pi$
$858$	0	0
$859$	3.47524e8	0.548284	0.274142	−	0.961689i	$-0.411606\pi$
$859$	3.47524e8	0.548284	0.274142	+	0.961689i	$0.411606\pi$
$860$	0	0
$861$	9.81088e6i	0.0153709i
$862$	0	0
$863$	1.05023e9i	1.63400i	0.576641	+	0.816998i	$0.304363\pi$
$863$	1.05023e9i	1.63400i	−0.576641	+	0.816998i	$0.695637\pi$
$864$	0	0
$865$	1.90934e8	0.295010
$866$	0	0
$867$	−6.29567e7	−0.0966016
$868$	0	0
$869$	1.41256e9i	2.15253i
$870$	0	0
$871$	9.67674e8i	1.46445i
$872$	0	0
$873$	−3.70978e8	−0.557578
$874$	0	0
$875$	2.64500e8	0.394822
$876$	0	0
$877$	− 1.68212e8i	− 0.249379i	−0.992196	−	0.124689i	$-0.960207\pi$
$877$	− 1.68212e8i	− 0.249379i	0.992196	−	0.124689i	$-0.0397934\pi$
$878$	0	0
$879$	− 1.19453e8i	− 0.175886i
$880$	0	0
$881$	7.89298e8	1.15429	0.577143	−	0.816643i	$-0.304168\pi$
$881$	7.89298e8	1.15429	0.577143	+	0.816643i	$0.304168\pi$
$882$	0	0
$883$	4.04332e7	0.0587294	0.0293647	−	0.999569i	$-0.490652\pi$
$883$	4.04332e7	0.0587294	0.0293647	+	0.999569i	$0.490652\pi$
$884$	0	0
$885$	2.94632e7i	0.0425060i
$886$	0	0
$887$	− 1.21282e9i	− 1.73791i	−0.494895	−	0.868953i	$-0.664793\pi$
$887$	− 1.21282e9i	− 1.73791i	0.494895	−	0.868953i	$-0.335207\pi$
$888$	0	0
$889$	−3.62889e8	−0.516498
$890$	0	0
$891$	−1.04705e9	−1.48025
$892$	0	0
$893$	1.16580e9i	1.63708i
$894$	0	0
$895$	− 2.01068e8i	− 0.280462i
$896$	0	0
$897$	1.66661e8	0.230918
$898$	0	0
$899$	9.84011e8	1.35432
$900$	0	0
$901$	4.97210e8i	0.679775i
$902$	0	0
$903$	4.40128e6i	0.00597745i
$904$	0	0
$905$	−8.77161e7	−0.118341
$906$	0	0
$907$	−9.04088e8	−1.21168	−0.605841	−	0.795586i	$-0.707163\pi$
$907$	−9.04088e8	−1.21168	−0.605841	+	0.795586i	$0.707163\pi$
$908$	0	0
$909$	− 1.13976e9i	− 1.51748i
$910$	0	0
$911$	− 1.94818e8i	− 0.257676i	−0.991666	−	0.128838i	$-0.958875\pi$
$911$	− 1.94818e8i	− 0.257676i	0.991666	−	0.128838i	$-0.0411248\pi$
$912$	0	0
$913$	−6.35140e8	−0.834560
$914$	0	0
$915$	−5.41756e7	−0.0707197
$916$	0	0
$917$	4.14880e8i	0.538039i
$918$	0	0
$919$	9.23478e8i	1.18982i	0.803794	+	0.594908i	$0.202812\pi$
$919$	9.23478e8i	1.18982i	−0.803794	+	0.594908i	$0.797188\pi$
$920$	0	0
$921$	3.78827e7	0.0484911
$922$	0	0
$923$	−8.66488e7	−0.110194
$924$	0	0
$925$	5.27625e6i	0.00666654i
$926$	0	0
$927$	− 6.32214e8i	− 0.793643i
$928$	0	0
$929$	6.87235e8	0.857153	0.428576	−	0.903506i	$-0.359015\pi$
$929$	6.87235e8	0.857153	0.428576	+	0.903506i	$0.359015\pi$
$930$	0	0
$931$	6.74701e8	0.836108
$932$	0	0
$933$	5.70212e7i	0.0702087i
$934$	0	0
$935$	− 3.05449e8i	− 0.373684i
$936$	0	0
$937$	5.61007e7	0.0681945	0.0340973	−	0.999419i	$-0.489144\pi$
$937$	5.61007e7	0.0681945	0.0340973	+	0.999419i	$0.489144\pi$
$938$	0	0
$939$	−1.03590e8	−0.125119
$940$	0	0
$941$	− 3.14919e8i	− 0.377947i	−0.981982	−	0.188973i	$-0.939484\pi$
$941$	− 3.14919e8i	− 0.377947i	0.981982	−	0.188973i	$-0.0605160\pi$
$942$	0	0
$943$	− 2.47725e8i	− 0.295416i
$944$	0	0
$945$	−5.30656e7	−0.0628807
$946$	0	0
$947$	−2.73886e8	−0.322493	−0.161246	−	0.986914i	$-0.551551\pi$
$947$	−2.73886e8	−0.322493	−0.161246	+	0.986914i	$0.551551\pi$
$948$	0	0
$949$	− 1.56179e9i	− 1.82736i
$950$	0	0
$951$	− 1.31894e7i	− 0.0153350i
$952$	0	0
$953$	−1.04389e9	−1.20608	−0.603042	−	0.797709i	$-0.706045\pi$
$953$	−1.04389e9	−1.20608	−0.603042	+	0.797709i	$0.706045\pi$
$954$	0	0
$955$	1.39168e7	0.0159783
$956$	0	0
$957$	− 1.76988e8i	− 0.201933i
$958$	0	0
$959$	− 2.26437e8i	− 0.256739i
$960$	0	0
$961$	−1.31023e9	−1.47631
$962$	0	0
$963$	6.75693e8	0.756607
$964$	0	0
$965$	− 9.20935e7i	− 0.102482i
$966$	0	0
$967$	5.73542e7i	0.0634288i	0.999497	+	0.0317144i	$0.0100967\pi$
$967$	5.73542e7i	0.0634288i	−0.999497	+	0.0317144i	$0.989903\pi$
$968$	0	0
$969$	9.33388e7	0.102587
$970$	0	0
$971$	4.72983e8	0.516640	0.258320	−	0.966059i	$-0.416831\pi$
$971$	4.72983e8	0.516640	0.258320	+	0.966059i	$0.416831\pi$
$972$	0	0
$973$	− 7.42949e8i	− 0.806529i
$974$	0	0
$975$	− 1.17705e8i	− 0.126993i
$976$	0	0
$977$	−1.15739e8	−0.124107	−0.0620536	−	0.998073i	$-0.519765\pi$
$977$	−1.15739e8	−0.124107	−0.0620536	+	0.998073i	$0.519765\pi$
$978$	0	0
$979$	−3.40649e8	−0.363043
$980$	0	0
$981$	− 1.39828e9i	− 1.48111i
$982$	0	0
$983$	7.96459e8i	0.838500i	0.907871	+	0.419250i	$0.137707\pi$
$983$	7.96459e8i	0.838500i	−0.907871	+	0.419250i	$0.862293\pi$
$984$	0	0
$985$	4.79713e8	0.501964
$986$	0	0
$987$	1.06561e8	0.110827
$988$	0	0
$989$	− 1.11132e8i	− 0.114882i
$990$	0	0
$991$	− 9.42496e8i	− 0.968408i	−0.874955	−	0.484204i	$-0.839109\pi$
$991$	− 9.42496e8i	− 0.968408i	0.874955	−	0.484204i	$-0.160891\pi$
$992$	0	0
$993$	−2.68288e8	−0.274002
$994$	0	0
$995$	−6.84672e8	−0.695046
$996$	0	0
$997$	6.48931e7i	0.0654806i	0.999464	+	0.0327403i	$0.0104234\pi$
$997$	6.48931e7i	0.0654806i	−0.999464	+	0.0327403i	$0.989577\pi$
$998$	0	0
$999$	− 2.31874e6i	− 0.00232571i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.7.d.c.127.2		2
4.3	odd	2		256.7.d.a.127.2		2
8.3	odd	2	inner	256.7.d.c.127.1		2
8.5	even	2		256.7.d.a.127.1		2
16.3	odd	4		32.7.c.a.31.1	✓	2
16.5	even	4		64.7.c.b.63.1		2
16.11	odd	4		64.7.c.b.63.2		2
16.13	even	4		32.7.c.a.31.2	yes	2
48.5	odd	4		576.7.g.i.127.2		2
48.11	even	4		576.7.g.i.127.1		2
48.29	odd	4		288.7.g.a.127.2		2
48.35	even	4		288.7.g.a.127.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.7.c.a.31.1	✓	2	16.3	odd	4
32.7.c.a.31.2	yes	2	16.13	even	4
64.7.c.b.63.1		2	16.5	even	4
64.7.c.b.63.2		2	16.11	odd	4
256.7.d.a.127.1		2	8.5	even	2
256.7.d.a.127.2		2	4.3	odd	2
256.7.d.c.127.1		2	8.3	odd	2	inner
256.7.d.c.127.2		2	1.1	even	1	trivial
288.7.g.a.127.1		2	48.35	even	4
288.7.g.a.127.2		2	48.29	odd	4
576.7.g.i.127.1		2	48.11	even	4
576.7.g.i.127.2		2	48.5	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:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	256.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.00000 q^{3} +50.0000i q^{5} -184.000i q^{7} -713.000 q^{9} +O(q^{10})\)	\(q+4.00000 q^{3} +50.0000i q^{5} -184.000i q^{7} -713.000 q^{9} -2108.00 q^{11} -2242.00i q^{13} +200.000i q^{15} +2898.00 q^{17} +8052.00 q^{19} -736.000i q^{21} +18584.0i q^{23} +13125.0 q^{25} -5768.00 q^{27} +20990.0i q^{29} -46880.0i q^{31} -8432.00 q^{33} +9200.00 q^{35} +402.000i q^{37} -8968.00i q^{39} -13330.0 q^{41} -5980.00 q^{43} -35650.0i q^{45} +144784. i q^{47} +83793.0 q^{49} +11592.0 q^{51} +171570. i q^{53} -105400. i q^{55} +32208.0 q^{57} +147316. q^{59} +270878. i q^{61} +131192. i q^{63} +112100. q^{65} -431612. q^{67} +74336.0i q^{69} -38648.0i q^{71} +696606. q^{73} +52500.0 q^{75} +387872. i q^{77} -670096. i q^{79} +496705. q^{81} +301300. q^{83} +144900. i q^{85} +83960.0i q^{87} +161598. q^{89} -412528. q^{91} -187520. i q^{93} +402600. i q^{95} +520306. q^{97} +1.50300e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{3} - 1426 q^{9}+O(q^{10}) \) 2 * q + 8 * q^3 - 1426 * q^9	\( 2 q + 8 q^{3} - 1426 q^{9} - 4216 q^{11} + 5796 q^{17} + 16104 q^{19} + 26250 q^{25} - 11536 q^{27} - 16864 q^{33} + 18400 q^{35} - 26660 q^{41} - 11960 q^{43} + 167586 q^{49} + 23184 q^{51} + 64416 q^{57} + 294632 q^{59} + 224200 q^{65} - 863224 q^{67} + 1393212 q^{73} + 105000 q^{75} + 993410 q^{81} + 602600 q^{83} + 323196 q^{89} - 825056 q^{91} + 1040612 q^{97} + 3006008 q^{99}+O(q^{100}) \) 2 * q + 8 * q^3 - 1426 * q^9 - 4216 * q^11 + 5796 * q^17 + 16104 * q^19 + 26250 * q^25 - 11536 * q^27 - 16864 * q^33 + 18400 * q^35 - 26660 * q^41 - 11960 * q^43 + 167586 * q^49 + 23184 * q^51 + 64416 * q^57 + 294632 * q^59 + 224200 * q^65 - 863224 * q^67 + 1393212 * q^73 + 105000 * q^75 + 993410 * q^81 + 602600 * q^83 + 323196 * q^89 - 825056 * q^91 + 1040612 * q^97 + 3006008 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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