LMFDB - Embedded newform 32.7.c.a.31.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [32,7,Mod(31,32)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(32, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("32.31");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 32 = 2^{5} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	32.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$7.36173067584$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			31.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	32.31
Dual form			32.7.c.a.31.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000i q^{3} +50.0000 q^{5} -184.000i q^{7} +713.000 q^{9} +O(q^{10})$	$q-4.00000i q^{3} +50.0000 q^{5} -184.000i q^{7} +713.000 q^{9} -2108.00i q^{11} +2242.00 q^{13} -200.000i q^{15} +2898.00 q^{17} -8052.00i q^{19} -736.000 q^{21} +18584.0i q^{23} -13125.0 q^{25} -5768.00i q^{27} -20990.0 q^{29} +46880.0i q^{31} -8432.00 q^{33} -9200.00i q^{35} +402.000 q^{37} -8968.00i q^{39} +13330.0 q^{41} -5980.00i q^{43} +35650.0 q^{45} -144784. i q^{47} +83793.0 q^{49} -11592.0i q^{51} +171570. q^{53} -105400. i q^{55} -32208.0 q^{57} +147316. i q^{59} -270878. q^{61} -131192. i q^{63} +112100. q^{65} +431612. i q^{67} +74336.0 q^{69} -38648.0i q^{71} -696606. q^{73} +52500.0i q^{75} -387872. q^{77} +670096. i q^{79} +496705. q^{81} -301300. i q^{83} +144900. q^{85} +83960.0i q^{87} -161598. q^{89} -412528. i q^{91} +187520. q^{93} -402600. i q^{95} +520306. q^{97} -1.50300e6i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 100 q^{5} + 1426 q^{9}+O(q^{10}) $ 2 * q + 100 * q^5 + 1426 * q^9	$ 2 q + 100 q^{5} + 1426 q^{9} + 4484 q^{13} + 5796 q^{17} - 1472 q^{21} - 26250 q^{25} - 41980 q^{29} - 16864 q^{33} + 804 q^{37} + 26660 q^{41} + 71300 q^{45} + 167586 q^{49} + 343140 q^{53} - 64416 q^{57} - 541756 q^{61} + 224200 q^{65} + 148672 q^{69} - 1393212 q^{73} - 775744 q^{77} + 993410 q^{81} + 289800 q^{85} - 323196 q^{89} + 375040 q^{93} + 1040612 q^{97}+O(q^{100}) $ 2 * q + 100 * q^5 + 1426 * q^9 + 4484 * q^13 + 5796 * q^17 - 1472 * q^21 - 26250 * q^25 - 41980 * q^29 - 16864 * q^33 + 804 * q^37 + 26660 * q^41 + 71300 * q^45 + 167586 * q^49 + 343140 * q^53 - 64416 * q^57 - 541756 * q^61 + 224200 * q^65 + 148672 * q^69 - 1393212 * q^73 - 775744 * q^77 + 993410 * q^81 + 289800 * q^85 - 323196 * q^89 + 375040 * q^93 + 1040612 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/32\mathbb{Z}\right)^\times$.

$n$	$5$	$31$
$\chi(n)$	$1$	$-1$

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	− 0.148148i	−0.997253	−	0.0740741i	$-0.976400\pi$
$3$	− 4.00000i	− 0.148148i	0.997253	−	0.0740741i	$-0.0236001\pi$
$4$	0	0
$5$	50.0000	0.400000	0.200000	−	0.979796i	$-0.435906\pi$
$5$	50.0000	0.400000	0.200000	+	0.979796i	$0.435906\pi$
$6$	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
$11$	− 2108.00i	− 1.58377i	−0.610669	−	0.791886i	$-0.709099\pi$
$11$	− 2108.00i	− 1.58377i	0.610669	−	0.791886i	$-0.290901\pi$
$12$	0	0
$13$	2242.00	1.02048	0.510241	−	0.860031i	$-0.329556\pi$
$13$	2242.00	1.02048	0.510241	+	0.860031i	$0.329556\pi$
$14$	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
$19$	− 8052.00i	− 1.17393i	−0.809612	−	0.586966i	$-0.800322\pi$
$19$	− 8052.00i	− 1.17393i	0.809612	−	0.586966i	$-0.199678\pi$
$20$	0	0
$21$	−736.000	−0.0794731
$22$	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
$27$	− 5768.00i	− 0.293045i
$28$	0	0
$29$	−20990.0	−0.860634	−0.430317	−	0.902678i	$-0.641598\pi$
$29$	−20990.0	−0.860634	−0.430317	+	0.902678i	$0.641598\pi$
$30$	0	0
$31$	46880.0i	1.57363i	0.617189	+	0.786815i	$0.288271\pi$
$31$	46880.0i	1.57363i	−0.617189	+	0.786815i	$0.711729\pi$
$32$	0	0
$33$	−8432.00	−0.234633
$34$	0	0
$35$	− 9200.00i	− 0.214577i
$36$	0	0
$37$	402.000	0.00793635	0.00396818	−	0.999992i	$-0.498737\pi$
$37$	402.000	0.00793635	0.00396818	+	0.999992i	$0.498737\pi$
$38$	0	0
$39$	− 8968.00i	− 0.151183i
$40$	0	0
$41$	13330.0	0.193410	0.0967049	−	0.995313i	$-0.469170\pi$
$41$	13330.0	0.193410	0.0967049	+	0.995313i	$0.469170\pi$
$42$	0	0
$43$	− 5980.00i	− 0.0752135i	−0.999293	−	0.0376068i	$-0.988027\pi$
$43$	− 5980.00i	− 0.0752135i	0.999293	−	0.0376068i	$-0.0119734\pi$
$44$	0	0
$45$	35650.0	0.391221
$46$	0	0
$47$	− 144784.i	− 1.39453i	−0.716815	−	0.697264i	$-0.754401\pi$
$47$	− 144784.i	− 1.39453i	0.716815	−	0.697264i	$-0.245599\pi$
$48$	0	0
$49$	83793.0	0.712229
$50$	0	0
$51$	− 11592.0i	− 0.0873872i
$52$	0	0
$53$	171570.	1.15243	0.576214	−	0.817299i	$-0.304530\pi$
$53$	171570.	1.15243	0.576214	+	0.817299i	$0.304530\pi$
$54$	0	0
$55$	− 105400.i	− 0.633509i
$56$	0	0
$57$	−32208.0	−0.173916
$58$	0	0
$59$	147316.i	0.717289i	0.933474	+	0.358644i	$0.116761\pi$
$59$	147316.i	0.717289i	−0.933474	+	0.358644i	$0.883239\pi$
$60$	0	0
$61$	−270878.	−1.19340	−0.596698	−	0.802466i	$-0.703521\pi$
$61$	−270878.	−1.19340	−0.596698	+	0.802466i	$0.703521\pi$
$62$	0	0
$63$	− 131192.i	− 0.524669i
$64$	0	0
$65$	112100.	0.408193
$66$	0	0
$67$	431612.i	1.43506i	0.696529	+	0.717528i	$0.254727\pi$
$67$	431612.i	1.43506i	−0.696529	+	0.717528i	$0.745273\pi$
$68$	0	0
$69$	74336.0	0.226283
$70$	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	$-0.853068\pi$
$73$	−696606.	−1.79068	−0.895341	+	0.445381i	$0.853068\pi$
$74$	0	0
$75$	52500.0i	0.124444i
$76$	0	0
$77$	−387872.	−0.849603
$78$	0	0
$79$	670096.i	1.35911i	0.733623	+	0.679557i	$0.237828\pi$
$79$	670096.i	1.35911i	−0.733623	+	0.679557i	$0.762172\pi$
$80$	0	0
$81$	496705.	0.934638
$82$	0	0
$83$	− 301300.i	− 0.526944i	−0.964667	−	0.263472i	$-0.915132\pi$
$83$	− 301300.i	− 0.526944i	0.964667	−	0.263472i	$-0.0848677\pi$
$84$	0	0
$85$	144900.	0.235945
$86$	0	0
$87$	83960.0i	0.127501i
$88$	0	0
$89$	−161598.	−0.229227	−0.114614	−	0.993410i	$-0.536563\pi$
$89$	−161598.	−0.229227	−0.114614	+	0.993410i	$0.536563\pi$
$90$	0	0
$91$	− 412528.i	− 0.547431i
$92$	0	0
$93$	187520.	0.233130
$94$	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
$99$	− 1.50300e6i	− 1.54901i
$100$	0	0
$101$	1.59855e6	1.55153	0.775766	−	0.631020i	$-0.217363\pi$
$101$	1.59855e6	1.55153	0.775766	+	0.631020i	$0.217363\pi$
$102$	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.i	− 0.773586i	−0.922167	−	0.386793i	$-0.873583\pi$
$107$	− 947676.i	− 0.773586i	0.922167	−	0.386793i	$-0.126417\pi$
$108$	0	0
$109$	−1.96112e6	−1.51434	−0.757171	−	0.653216i	$-0.773419\pi$
$109$	−1.96112e6	−1.51434	−0.757171	+	0.653216i	$0.773419\pi$
$110$	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.i	0.610964i
$116$	0	0
$117$	1.59855e6	0.998085
$118$	0	0
$119$	− 533232.i	− 0.316428i
$120$	0	0
$121$	−2.67210e6	−1.50833
$122$	0	0
$123$	− 53320.0i	− 0.0286533i
$124$	0	0
$125$	−1.43750e6	−0.736000
$126$	0	0
$127$	1.97222e6i	0.962820i	0.876496	+	0.481410i	$0.159875\pi$
$127$	1.97222e6i	0.962820i	−0.876496	+	0.481410i	$0.840125\pi$
$128$	0	0
$129$	−23920.0	−0.0111427
$130$	0	0
$131$	2.25478e6i	1.00298i	0.865165	+	0.501488i	$0.167214\pi$
$131$	2.25478e6i	1.00298i	−0.865165	+	0.501488i	$0.832786\pi$
$132$	0	0
$133$	−1.48157e6	−0.629748
$134$	0	0
$135$	− 288400.i	− 0.117218i
$136$	0	0
$137$	−1.23064e6	−0.478596	−0.239298	−	0.970946i	$-0.576917\pi$
$137$	−1.23064e6	−0.478596	−0.239298	+	0.970946i	$0.576917\pi$
$138$	0	0
$139$	4.03776e6i	1.50348i	0.659462	+	0.751738i	$0.270784\pi$
$139$	4.03776e6i	1.50348i	−0.659462	+	0.751738i	$0.729216\pi$
$140$	0	0
$141$	−579136.	−0.206597
$142$	0	0
$143$	− 4.72614e6i	− 1.61621i
$144$	0	0
$145$	−1.04950e6	−0.344254
$146$	0	0
$147$	− 335172.i	− 0.105515i
$148$	0	0
$149$	1.84712e6	0.558389	0.279194	−	0.960235i	$-0.409933\pi$
$149$	1.84712e6	0.558389	0.279194	+	0.960235i	$0.409933\pi$
$150$	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
$155$	2.34400e6i	0.629452i
$156$	0	0
$157$	879842.	0.227356	0.113678	−	0.993518i	$-0.463737\pi$
$157$	879842.	0.227356	0.113678	+	0.993518i	$0.463737\pi$
$158$	0	0
$159$	− 686280.i	− 0.170730i
$160$	0	0
$161$	3.41946e6	0.819369
$162$	0	0
$163$	− 3.16784e6i	− 0.731478i	−0.930718	−	0.365739i	$-0.880816\pi$
$163$	− 3.16784e6i	− 0.731478i	0.930718	−	0.365739i	$-0.119184\pi$
$164$	0	0
$165$	−421600.	−0.0938531
$166$	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
$171$	− 5.74108e6i	− 1.14817i
$172$	0	0
$173$	3.81869e6	0.737524	0.368762	−	0.929524i	$-0.379782\pi$
$173$	3.81869e6	0.737524	0.368762	+	0.929524i	$0.379782\pi$
$174$	0	0
$175$	2.41500e6i	0.450612i
$176$	0	0
$177$	589264.	0.106265
$178$	0	0
$179$	4.02136e6i	0.701154i	0.936534	+	0.350577i	$0.114015\pi$
$179$	4.02136e6i	0.701154i	−0.936534	+	0.350577i	$0.885985\pi$
$180$	0	0
$181$	1.75432e6	0.295851	0.147926	−	0.988998i	$-0.452740\pi$
$181$	1.75432e6	0.295851	0.147926	+	0.988998i	$0.452740\pi$
$182$	0	0
$183$	1.08351e6i	0.176799i
$184$	0	0
$185$	20100.0	0.00317454
$186$	0	0
$187$	− 6.10898e6i	− 0.934209i
$188$	0	0
$189$	−1.06131e6	−0.157202
$190$	0	0
$191$	278336.i	0.0399456i	0.999801	+	0.0199728i	$0.00635797\pi$
$191$	278336.i	0.0399456i	−0.999801	+	0.0199728i	$0.993642\pi$
$192$	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.i	− 0.0604730i
$196$	0	0
$197$	−9.59425e6	−1.25491	−0.627455	−	0.778653i	$-0.715903\pi$
$197$	−9.59425e6	−1.25491	−0.627455	+	0.778653i	$0.715903\pi$
$198$	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
$203$	3.86216e6i	0.461681i
$204$	0	0
$205$	666500.	0.0773639
$206$	0	0
$207$	1.32504e7i	1.49389i
$208$	0	0
$209$	−1.69736e7	−1.85924
$210$	0	0
$211$	− 1.08290e7i	− 1.15277i	−0.817178	−	0.576385i	$-0.804463\pi$
$211$	− 1.08290e7i	− 1.15277i	0.817178	−	0.576385i	$-0.195537\pi$
$212$	0	0
$213$	−154592.	−0.0159974
$214$	0	0
$215$	− 299000.i	− 0.0300854i
$216$	0	0
$217$	8.62592e6	0.844163
$218$	0	0
$219$	2.78642e6i	0.265286i
$220$	0	0
$221$	6.49732e6	0.601945
$222$	0	0
$223$	− 5.41034e6i	− 0.487876i	−0.969791	−	0.243938i	$-0.921561\pi$
$223$	− 5.41034e6i	− 0.487876i	0.969791	−	0.243938i	$-0.0784394\pi$
$224$	0	0
$225$	−9.35812e6	−0.821564
$226$	0	0
$227$	9.45622e6i	0.808425i	0.914665	+	0.404213i	$0.132454\pi$
$227$	9.45622e6i	0.808425i	−0.914665	+	0.404213i	$0.867546\pi$
$228$	0	0
$229$	−1.47605e7	−1.22912	−0.614562	−	0.788869i	$-0.710667\pi$
$229$	−1.47605e7	−1.22912	−0.614562	+	0.788869i	$0.710667\pi$
$230$	0	0
$231$	1.55149e6i	0.125867i
$232$	0	0
$233$	1.53357e7	1.21237	0.606186	−	0.795323i	$-0.292699\pi$
$233$	1.53357e7	1.21237	0.606186	+	0.795323i	$0.292699\pi$
$234$	0	0
$235$	− 7.23920e6i	− 0.557811i
$236$	0	0
$237$	2.68038e6	0.201350
$238$	0	0
$239$	− 8.23509e6i	− 0.603218i	−0.953432	−	0.301609i	$-0.902476\pi$
$239$	− 8.23509e6i	− 0.603218i	0.953432	−	0.301609i	$-0.0975238\pi$
$240$	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
$243$	− 6.19169e6i	− 0.431510i
$244$	0	0
$245$	4.18965e6	0.284891
$246$	0	0
$247$	− 1.80526e7i	− 1.19798i
$248$	0	0
$249$	−1.20520e6	−0.0780658
$250$	0	0
$251$	1.19698e7i	0.756950i	0.925611	+	0.378475i	$0.123551\pi$
$251$	1.19698e7i	0.756950i	−0.925611	+	0.378475i	$0.876449\pi$
$252$	0	0
$253$	3.91751e7	2.41907
$254$	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
$259$	− 73968.0i	− 0.00425740i
$260$	0	0
$261$	−1.49659e7	−0.841745
$262$	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.i	0.0339596i
$268$	0	0
$269$	−2.08340e7	−1.07032	−0.535161	−	0.844750i	$-0.679749\pi$
$269$	−2.08340e7	−1.07032	−0.535161	+	0.844750i	$0.679749\pi$
$270$	0	0
$271$	− 1.56227e7i	− 0.784961i	−0.919760	−	0.392481i	$-0.871617\pi$
$271$	− 1.56227e7i	− 0.784961i	0.919760	−	0.392481i	$-0.128383\pi$
$272$	0	0
$273$	−1.65011e6	−0.0811009
$274$	0	0
$275$	2.76675e7i	1.33037i
$276$	0	0
$277$	1.58341e7	0.744996	0.372498	−	0.928033i	$-0.378501\pi$
$277$	1.58341e7	0.744996	0.372498	+	0.928033i	$0.378501\pi$
$278$	0	0
$279$	3.34254e7i	1.53909i
$280$	0	0
$281$	−2.96281e6	−0.133532	−0.0667660	−	0.997769i	$-0.521268\pi$
$281$	−2.96281e6	−0.133532	−0.0667660	+	0.997769i	$0.521268\pi$
$282$	0	0
$283$	− 7.15377e6i	− 0.315628i	−0.987469	−	0.157814i	$-0.949555\pi$
$283$	− 7.15377e6i	− 0.315628i	0.987469	−	0.157814i	$-0.0504447\pi$
$284$	0	0
$285$	−1.61040e6	−0.0695663
$286$	0	0
$287$	− 2.45272e6i	− 0.103753i
$288$	0	0
$289$	−1.57392e7	−0.652061
$290$	0	0
$291$	− 2.08122e6i	− 0.0844578i
$292$	0	0
$293$	−2.98632e7	−1.18723	−0.593614	−	0.804750i	$-0.702299\pi$
$293$	−2.98632e7	−1.18723	−0.593614	+	0.804750i	$0.702299\pi$
$294$	0	0
$295$	7.36580e6i	0.286915i
$296$	0	0
$297$	−1.21589e7	−0.464116
$298$	0	0
$299$	4.16653e7i	1.55870i
$300$	0	0
$301$	−1.10032e6	−0.0403478
$302$	0	0
$303$	− 6.39418e6i	− 0.229857i
$304$	0	0
$305$	−1.35439e7	−0.477358
$306$	0	0
$307$	− 9.47068e6i	− 0.327315i	−0.986517	−	0.163657i	$-0.947671\pi$
$307$	− 9.47068e6i	− 0.327315i	0.986517	−	0.163657i	$-0.0523292\pi$
$308$	0	0
$309$	3.54678e6	0.120215
$310$	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	$-0.361231\pi$
$313$	2.58976e7	0.844552	0.422276	+	0.906467i	$0.361231\pi$
$314$	0	0
$315$	− 6.55960e6i	− 0.209868i
$316$	0	0
$317$	3.29735e6	0.103511	0.0517555	−	0.998660i	$-0.483518\pi$
$317$	3.29735e6	0.103511	0.0517555	+	0.998660i	$0.483518\pi$
$318$	0	0
$319$	4.42469e7i	1.36305i
$320$	0	0
$321$	−3.79070e6	−0.114605
$322$	0	0
$323$	− 2.33347e7i	− 0.692460i
$324$	0	0
$325$	−2.94262e7	−0.857205
$326$	0	0
$327$	7.84447e6i	0.224347i
$328$	0	0
$329$	−2.66403e7	−0.748085
$330$	0	0
$331$	− 6.70721e7i	− 1.84952i	−0.380556	−	0.924758i	$-0.624267\pi$
$331$	− 6.70721e7i	− 1.84952i	0.380556	−	0.924758i	$-0.375733\pi$
$332$	0	0
$333$	286626.	0.00776217
$334$	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
$339$	− 7.06983e6i	− 0.181472i
$340$	0	0
$341$	9.88230e7	2.49227
$342$	0	0
$343$	− 3.70653e7i	− 0.918513i
$344$	0	0
$345$	3.71680e6	0.0905132
$346$	0	0
$347$	− 3.33170e7i	− 0.797402i	−0.917081	−	0.398701i	$-0.869461\pi$
$347$	− 3.33170e7i	− 0.797402i	0.917081	−	0.398701i	$-0.130539\pi$
$348$	0	0
$349$	−3.04510e7	−0.716350	−0.358175	−	0.933654i	$-0.616601\pi$
$349$	−3.04510e7	−0.716350	−0.358175	+	0.933654i	$0.616601\pi$
$350$	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
$355$	− 1.93240e6i	− 0.0431929i
$356$	0	0
$357$	−2.13293e6	−0.0468783
$358$	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
$363$	1.06884e7i	0.223457i
$364$	0	0
$365$	−3.48303e7	−0.716273
$366$	0	0
$367$	2.38924e7i	0.483350i	0.970357	+	0.241675i	$0.0776968\pi$
$367$	2.38924e7i	0.483350i	−0.970357	+	0.241675i	$0.922303\pi$
$368$	0	0
$369$	9.50429e6	0.189165
$370$	0	0
$371$	− 3.15689e7i	− 0.618212i
$372$	0	0
$373$	9.46760e7	1.82437	0.912186	−	0.409777i	$-0.134394\pi$
$373$	9.46760e7	1.82437	0.912186	+	0.409777i	$0.134394\pi$
$374$	0	0
$375$	5.75000e6i	0.109037i
$376$	0	0
$377$	−4.70596e7	−0.878262
$378$	0	0
$379$	5.17551e7i	0.950682i	0.879802	+	0.475341i	$0.157675\pi$
$379$	5.17551e7i	0.950682i	−0.879802	+	0.475341i	$0.842325\pi$
$380$	0	0
$381$	7.88890e6	0.142640
$382$	0	0
$383$	− 7.70121e7i	− 1.37076i	−0.728184	−	0.685382i	$-0.759635\pi$
$383$	− 7.70121e7i	− 1.37076i	0.728184	−	0.685382i	$-0.240365\pi$
$384$	0	0
$385$	−1.93936e7	−0.339841
$386$	0	0
$387$	− 4.26374e6i	− 0.0735627i
$388$	0	0
$389$	−8.92562e7	−1.51632	−0.758158	−	0.652071i	$-0.773900\pi$
$389$	−8.92562e7	−1.51632	−0.758158	+	0.652071i	$0.773900\pi$
$390$	0	0
$391$	5.38564e7i	0.900964i
$392$	0	0
$393$	9.01912e6	0.148589
$394$	0	0
$395$	3.35048e7i	0.543645i
$396$	0	0
$397$	−8.45175e7	−1.35075	−0.675375	−	0.737474i	$-0.736018\pi$
$397$	−8.45175e7	−1.35075	−0.675375	+	0.737474i	$0.736018\pi$
$398$	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
$403$	1.05105e8i	1.60586i
$404$	0	0
$405$	2.48352e7	0.373855
$406$	0	0
$407$	− 847416.i	− 0.0125694i
$408$	0	0
$409$	−1.16075e8	−1.69657	−0.848283	−	0.529544i	$-0.822363\pi$
$409$	−1.16075e8	−1.69657	−0.848283	+	0.529544i	$0.822363\pi$
$410$	0	0
$411$	4.92255e6i	0.0709030i
$412$	0	0
$413$	2.71061e7	0.384785
$414$	0	0
$415$	− 1.50650e7i	− 0.210778i
$416$	0	0
$417$	1.61511e7	0.222737
$418$	0	0
$419$	7.04530e7i	0.957761i	0.877880	+	0.478881i	$0.158957\pi$
$419$	7.04530e7i	0.957761i	−0.877880	+	0.478881i	$0.841043\pi$
$420$	0	0
$421$	−6.29023e7	−0.842986	−0.421493	−	0.906832i	$-0.638494\pi$
$421$	−6.29023e7	−0.842986	−0.421493	+	0.906832i	$0.638494\pi$
$422$	0	0
$423$	− 1.03231e8i	− 1.36392i
$424$	0	0
$425$	−3.80362e7	−0.495485
$426$	0	0
$427$	4.98416e7i	0.640189i
$428$	0	0
$429$	−1.89045e7	−0.239439
$430$	0	0
$431$	3.30262e7i	0.412502i	0.978499	+	0.206251i	$0.0661264\pi$
$431$	3.30262e7i	0.412502i	−0.978499	+	0.206251i	$0.933874\pi$
$432$	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
$435$	4.19800e6i	0.0510005i
$436$	0	0
$437$	1.49638e8	1.79308
$438$	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
$443$	− 5.74622e7i	− 0.660954i	−0.943814	−	0.330477i	$-0.892790\pi$
$443$	− 5.74622e7i	− 0.660954i	0.943814	−	0.330477i	$-0.107210\pi$
$444$	0	0
$445$	−8.07990e6	−0.0916908
$446$	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
$451$	− 2.80996e7i	− 0.306317i
$452$	0	0
$453$	−404384.	−0.00435010
$454$	0	0
$455$	− 2.06264e7i	− 0.218972i
$456$	0	0
$457$	3.89991e7	0.408607	0.204304	−	0.978908i	$-0.434507\pi$
$457$	3.89991e7	0.408607	0.204304	+	0.978908i	$0.434507\pi$
$458$	0	0
$459$	− 1.67157e7i	− 0.172856i
$460$	0	0
$461$	−9.51231e7	−0.970920	−0.485460	−	0.874259i	$-0.661348\pi$
$461$	−9.51231e7	−0.970920	−0.485460	+	0.874259i	$0.661348\pi$
$462$	0	0
$463$	− 1.26207e8i	− 1.27157i	−0.771867	−	0.635784i	$-0.780677\pi$
$463$	− 1.26207e8i	− 1.27157i	0.771867	−	0.635784i	$-0.219323\pi$
$464$	0	0
$465$	9.37600e6	0.0932521
$466$	0	0
$467$	− 1.94212e8i	− 1.90689i	−0.301562	−	0.953446i	$-0.597508\pi$
$467$	− 1.94212e8i	− 1.90689i	0.301562	−	0.953446i	$-0.402492\pi$
$468$	0	0
$469$	7.94166e7	0.769826
$470$	0	0
$471$	− 3.51937e6i	− 0.0336823i
$472$	0	0
$473$	−1.26058e7	−0.119121
$474$	0	0
$475$	1.05682e8i	0.986103i
$476$	0	0
$477$	1.22329e8	1.12713
$478$	0	0
$479$	1.46365e8i	1.33177i	0.746054	+	0.665886i	$0.231946\pi$
$479$	1.46365e8i	1.33177i	−0.746054	+	0.665886i	$0.768054\pi$
$480$	0	0
$481$	901284.	0.00809891
$482$	0	0
$483$	− 1.36778e7i	− 0.121388i
$484$	0	0
$485$	2.60153e7	0.228036
$486$	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
$491$	− 2.31258e8i	− 1.95368i	−0.213980	−	0.976838i	$-0.568643\pi$
$491$	− 2.31258e8i	− 1.95368i	0.213980	−	0.976838i	$-0.431357\pi$
$492$	0	0
$493$	−6.08290e7	−0.507657
$494$	0	0
$495$	− 7.51502e7i	− 0.619604i
$496$	0	0
$497$	−7.11123e6	−0.0579263
$498$	0	0
$499$	− 2.90337e7i	− 0.233669i	−0.993151	−	0.116834i	$-0.962725\pi$
$499$	− 2.90337e7i	− 0.233669i	0.993151	−	0.116834i	$-0.0372747\pi$
$500$	0	0
$501$	−2.10592e7	−0.167467
$502$	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.i	− 0.00613103i
$508$	0	0
$509$	5.02103e7	0.380750	0.190375	−	0.981711i	$-0.439030\pi$
$509$	5.02103e7	0.380750	0.190375	+	0.981711i	$0.439030\pi$
$510$	0	0
$511$	1.28176e8i	0.960599i
$512$	0	0
$513$	−4.64439e7	−0.344015
$514$	0	0
$515$	4.43348e7i	0.324581i
$516$	0	0
$517$	−3.05205e8	−2.20861
$518$	0	0
$519$	− 1.52748e7i	− 0.109263i
$520$	0	0
$521$	−4.30325e7	−0.304287	−0.152143	−	0.988358i	$-0.548618\pi$
$521$	−4.30325e7	−0.304287	−0.152143	+	0.988358i	$0.548618\pi$
$522$	0	0
$523$	1.72509e8i	1.20589i	0.797784	+	0.602943i	$0.206005\pi$
$523$	1.72509e8i	1.20589i	−0.797784	+	0.602943i	$0.793995\pi$
$524$	0	0
$525$	9.66000e6	0.0667574
$526$	0	0
$527$	1.35858e8i	0.928227i
$528$	0	0
$529$	−1.97329e8	−1.33298
$530$	0	0
$531$	1.05036e8i	0.701546i
$532$	0	0
$533$	2.98859e7	0.197371
$534$	0	0
$535$	− 4.73838e7i	− 0.309434i
$536$	0	0
$537$	1.60854e7	0.103875
$538$	0	0
$539$	− 1.76636e8i	− 1.12801i
$540$	0	0
$541$	1.59475e8	1.00716	0.503582	−	0.863947i	$-0.332015\pi$
$541$	1.59475e8	1.00716	0.503582	+	0.863947i	$0.332015\pi$
$542$	0	0
$543$	− 7.01729e6i	− 0.0438298i
$544$	0	0
$545$	−9.80559e7	−0.605737
$546$	0	0
$547$	− 3.06557e7i	− 0.187305i	−0.995605	−	0.0936523i	$-0.970146\pi$
$547$	− 3.06557e7i	− 0.187305i	0.995605	−	0.0936523i	$-0.0298542\pi$
$548$	0	0
$549$	−1.93136e8	−1.16720
$550$	0	0
$551$	1.69011e8i	1.01033i
$552$	0	0
$553$	1.23298e8	0.729087
$554$	0	0
$555$	− 80400.0i	0 0.000470302i
$556$	0	0
$557$	1.99783e6	0.0115609	0.00578046	−	0.999983i	$-0.498160\pi$
$557$	1.99783e6	0.0115609	0.00578046	+	0.999983i	$0.498160\pi$
$558$	0	0
$559$	− 1.34072e7i	− 0.0767541i
$560$	0	0
$561$	−2.44359e7	−0.138401
$562$	0	0
$563$	1.55895e8i	0.873588i	0.899562	+	0.436794i	$0.143886\pi$
$563$	1.55895e8i	0.873588i	−0.899562	+	0.436794i	$0.856114\pi$
$564$	0	0
$565$	8.83729e7	0.489975
$566$	0	0
$567$	− 9.13937e7i	− 0.501380i
$568$	0	0
$569$	1.06750e8	0.579468	0.289734	−	0.957107i	$-0.406433\pi$
$569$	1.06750e8	0.579468	0.289734	+	0.957107i	$0.406433\pi$
$570$	0	0
$571$	− 5.29306e7i	− 0.284314i	−0.989844	−	0.142157i	$-0.954596\pi$
$571$	− 5.29306e7i	− 0.284314i	0.989844	−	0.142157i	$-0.0454038\pi$
$572$	0	0
$573$	1.11334e6	0.00591787
$574$	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
$579$	7.36748e6i	0.0379562i
$580$	0	0
$581$	−5.54392e7	−0.282676
$582$	0	0
$583$	− 3.61670e8i	− 1.82518i
$584$	0	0
$585$	7.99273e7	0.399234
$586$	0	0
$587$	1.89126e8i	0.935057i	0.883978	+	0.467528i	$0.154855\pi$
$587$	1.89126e8i	0.935057i	−0.883978	+	0.467528i	$0.845145\pi$
$588$	0	0
$589$	3.77478e8	1.84733
$590$	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
$595$	− 2.66616e7i	− 0.126571i
$596$	0	0
$597$	5.47738e7	0.257424
$598$	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	$-0.439433\pi$
$601$	8.21137e7	0.378262	0.189131	+	0.981952i	$0.439433\pi$
$602$	0	0
$603$	3.07739e8i	1.40356i
$604$	0	0
$605$	−1.33605e8	−0.603333
$606$	0	0
$607$	− 2.93964e8i	− 1.31440i	−0.753715	−	0.657202i	$-0.771740\pi$
$607$	− 2.93964e8i	− 1.31440i	0.753715	−	0.657202i	$-0.228260\pi$
$608$	0	0
$609$	1.54486e7	0.0683972
$610$	0	0
$611$	− 3.24606e8i	− 1.42309i
$612$	0	0
$613$	−1.86200e8	−0.808348	−0.404174	−	0.914682i	$-0.632441\pi$
$613$	−1.86200e8	−0.808348	−0.404174	+	0.914682i	$0.632441\pi$
$614$	0	0
$615$	− 2.66600e6i	− 0.0114613i
$616$	0	0
$617$	2.92906e7	0.124702	0.0623509	−	0.998054i	$-0.480140\pi$
$617$	2.92906e7	0.124702	0.0623509	+	0.998054i	$0.480140\pi$
$618$	0	0
$619$	− 1.87910e8i	− 0.792277i	−0.918191	−	0.396139i	$-0.870350\pi$
$619$	− 1.87910e8i	− 0.792277i	0.918191	−	0.396139i	$-0.129650\pi$
$620$	0	0
$621$	1.07193e8	0.447600
$622$	0	0
$623$	2.97340e7i	0.122967i
$624$	0	0
$625$	1.33203e8	0.545600
$626$	0	0
$627$	6.78945e7i	0.275443i
$628$	0	0
$629$	1.16500e6	0.00468136
$630$	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
$635$	9.86112e7i	0.385128i
$636$	0	0
$637$	1.87864e8	0.726817
$638$	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
$643$	4.20041e8i	1.58000i	0.613104	+	0.790002i	$0.289921\pi$
$643$	4.20041e8i	1.58000i	−0.613104	+	0.790002i	$0.710079\pi$
$644$	0	0
$645$	−1.19600e6	−0.00445710
$646$	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
$651$	− 3.45037e7i	− 0.125061i
$652$	0	0
$653$	−2.23550e8	−0.802853	−0.401427	−	0.915891i	$-0.631485\pi$
$653$	−2.23550e8	−0.802853	−0.401427	+	0.915891i	$0.631485\pi$
$654$	0	0
$655$	1.12739e8i	0.401190i
$656$	0	0
$657$	−4.96680e8	−1.75138
$658$	0	0
$659$	− 1.13133e7i	− 0.0395307i	−0.999805	−	0.0197653i	$-0.993708\pi$
$659$	− 1.13133e7i	− 0.0395307i	0.999805	−	0.0197653i	$-0.00629191\pi$
$660$	0	0
$661$	−2.28431e8	−0.790955	−0.395477	−	0.918476i	$-0.629421\pi$
$661$	−2.28431e8	−0.790955	−0.395477	+	0.918476i	$0.629421\pi$
$662$	0	0
$663$	− 2.59893e7i	− 0.0891771i
$664$	0	0
$665$	−7.40784e7	−0.251899
$666$	0	0
$667$	− 3.90078e8i	− 1.31454i
$668$	0	0
$669$	−2.16413e7	−0.0722780
$670$	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
$675$	7.57050e7i	0.246158i
$676$	0	0
$677$	3.39715e7	0.109483	0.0547417	−	0.998501i	$-0.482566\pi$
$677$	3.39715e7	0.109483	0.0547417	+	0.998501i	$0.482566\pi$
$678$	0	0
$679$	− 9.57363e7i	− 0.305821i
$680$	0	0
$681$	3.78249e7	0.119767
$682$	0	0
$683$	2.56442e8i	0.804872i	0.915448	+	0.402436i	$0.131836\pi$
$683$	2.56442e8i	0.804872i	−0.915448	+	0.402436i	$0.868164\pi$
$684$	0	0
$685$	−6.15319e7	−0.191438
$686$	0	0
$687$	5.90421e7i	0.182092i
$688$	0	0
$689$	3.84660e8	1.17603
$690$	0	0
$691$	− 2.73283e8i	− 0.828282i	−0.910213	−	0.414141i	$-0.864082\pi$
$691$	− 2.73283e8i	− 0.828282i	0.910213	−	0.414141i	$-0.135918\pi$
$692$	0	0
$693$	−2.76553e8	−0.830956
$694$	0	0
$695$	2.01888e8i	0.601390i
$696$	0	0
$697$	3.86303e7	0.114085
$698$	0	0
$699$	− 6.13429e7i	− 0.179611i
$700$	0	0
$701$	−1.21405e8	−0.352438	−0.176219	−	0.984351i	$-0.556387\pi$
$701$	−1.21405e8	−0.352438	−0.176219	+	0.984351i	$0.556387\pi$
$702$	0	0
$703$	− 3.23690e6i	− 0.00931674i
$704$	0	0
$705$	−2.89568e7	−0.0826387
$706$	0	0
$707$	− 2.94132e8i	− 0.832309i
$708$	0	0
$709$	5.56778e8	1.56222	0.781112	−	0.624391i	$-0.214653\pi$
$709$	5.56778e8	1.56222	0.781112	+	0.624391i	$0.214653\pi$
$710$	0	0
$711$	4.77778e8i	1.32928i
$712$	0	0
$713$	−8.71218e8	−2.40358
$714$	0	0
$715$	− 2.36307e8i	− 0.646484i
$716$	0	0
$717$	−3.29404e7	−0.0893657
$718$	0	0
$719$	− 5.79277e8i	− 1.55847i	−0.626730	−	0.779236i	$-0.715607\pi$
$719$	− 5.79277e8i	− 1.55847i	0.626730	−	0.779236i	$-0.284393\pi$
$720$	0	0
$721$	1.63152e8	0.435298
$722$	0	0
$723$	− 3.23610e7i	− 0.0856264i
$724$	0	0
$725$	2.75494e8	0.722932
$726$	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
$731$	− 1.73300e7i	− 0.0443657i
$732$	0	0
$733$	1.35797e8	0.344808	0.172404	−	0.985026i	$-0.444847\pi$
$733$	1.35797e8	0.344808	0.172404	+	0.985026i	$0.444847\pi$
$734$	0	0
$735$	− 1.67586e7i	− 0.0422061i
$736$	0	0
$737$	9.09838e8	2.27280
$738$	0	0
$739$	− 3.97608e8i	− 0.985195i	−0.870257	−	0.492597i	$-0.836047\pi$
$739$	− 3.97608e8i	− 0.985195i	0.870257	−	0.492597i	$-0.163953\pi$
$740$	0	0
$741$	−7.22103e7	−0.177478
$742$	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
$747$	− 2.14827e8i	− 0.515379i
$748$	0	0
$749$	−1.74372e8	−0.414985
$750$	0	0
$751$	1.11382e8i	0.262964i	0.991319	+	0.131482i	$0.0419735\pi$
$751$	1.11382e8i	0.262964i	−0.991319	+	0.131482i	$0.958026\pi$
$752$	0	0
$753$	4.78794e7	0.112141
$754$	0	0
$755$	− 5.05480e6i	− 0.0117453i
$756$	0	0
$757$	−9.60559e7	−0.221430	−0.110715	−	0.993852i	$-0.535314\pi$
$757$	−9.60559e7	−0.221430	−0.110715	+	0.993852i	$0.535314\pi$
$758$	0	0
$759$	− 1.56700e8i	− 0.358381i
$760$	0	0
$761$	−2.27306e8	−0.515770	−0.257885	−	0.966176i	$-0.583026\pi$
$761$	−2.27306e8	−0.515770	−0.257885	+	0.966176i	$0.583026\pi$
$762$	0	0
$763$	3.60846e8i	0.812359i
$764$	0	0
$765$	1.03314e8	0.230767
$766$	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
$771$	7.87660e6i	0.0171860i
$772$	0	0
$773$	7.57894e8	1.64086	0.820428	−	0.571750i	$-0.193735\pi$
$773$	7.57894e8	1.64086	0.820428	+	0.571750i	$0.193735\pi$
$774$	0	0
$775$	− 6.15300e8i	− 1.32185i
$776$	0	0
$777$	−295872.	−0.000630726 0
$778$	0	0
$779$	− 1.07333e8i	− 0.227050i
$780$	0	0
$781$	−8.14700e7	−0.171019
$782$	0	0
$783$	1.21070e8i	0.252204i
$784$	0	0
$785$	4.39921e7	0.0909423
$786$	0	0
$787$	3.29425e8i	0.675823i	0.941178	+	0.337911i	$0.109720\pi$
$787$	3.29425e8i	0.675823i	−0.941178	+	0.337911i	$0.890280\pi$
$788$	0	0
$789$	−1.15249e8	−0.234643
$790$	0	0
$791$	− 3.25212e8i	− 0.657109i
$792$	0	0
$793$	−6.07308e8	−1.21784
$794$	0	0
$795$	− 3.43140e7i	− 0.0682920i
$796$	0	0
$797$	−8.12683e8	−1.60526	−0.802632	−	0.596475i	$-0.796568\pi$
$797$	−8.12683e8	−1.60526	−0.802632	+	0.596475i	$0.796568\pi$
$798$	0	0
$799$	− 4.19584e8i	− 0.822581i
$800$	0	0
$801$	−1.15219e8	−0.224196
$802$	0	0
$803$	1.46845e9i	2.83603i
$804$	0	0
$805$	1.70973e8	0.327747
$806$	0	0
$807$	8.33358e7i	0.158566i
$808$	0	0
$809$	−7.79881e7	−0.147293	−0.0736466	−	0.997284i	$-0.523464\pi$
$809$	−7.79881e7	−0.147293	−0.0736466	+	0.997284i	$0.523464\pi$
$810$	0	0
$811$	5.37023e8i	1.00677i	0.864062	+	0.503385i	$0.167912\pi$
$811$	5.37023e8i	1.00677i	−0.864062	+	0.503385i	$0.832088\pi$
$812$	0	0
$813$	−6.24908e7	−0.116291
$814$	0	0
$815$	− 1.58392e8i	− 0.292591i
$816$	0	0
$817$	−4.81510e7	−0.0882955
$818$	0	0
$819$	− 2.94132e8i	− 0.535416i
$820$	0	0
$821$	−3.74396e7	−0.0676553	−0.0338277	−	0.999428i	$-0.510770\pi$
$821$	−3.74396e7	−0.0676553	−0.0338277	+	0.999428i	$0.510770\pi$
$822$	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
$827$	2.21271e8i	0.391207i	0.980683	+	0.195604i	$0.0626666\pi$
$827$	2.21271e8i	0.391207i	−0.980683	+	0.195604i	$0.937333\pi$
$828$	0	0
$829$	−4.66636e8	−0.819058	−0.409529	−	0.912297i	$-0.634307\pi$
$829$	−4.66636e8	−0.819058	−0.409529	+	0.912297i	$0.634307\pi$
$830$	0	0
$831$	− 6.33364e7i	− 0.110370i
$832$	0	0
$833$	2.42832e8	0.420118
$834$	0	0
$835$	− 2.63240e8i	− 0.452160i
$836$	0	0
$837$	2.70404e8	0.461144
$838$	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
$843$	1.18513e7i	0.0197825i
$844$	0	0
$845$	9.98775e6	0.0165538
$846$	0	0
$847$	4.91667e8i	0.809135i
$848$	0	0
$849$	−2.86151e7	−0.0467597
$850$	0	0
$851$	7.47077e6i	0.0121221i
$852$	0	0
$853$	−1.02826e9	−1.65674	−0.828371	−	0.560181i	$-0.810732\pi$
$853$	−1.02826e9	−1.65674	−0.828371	+	0.560181i	$0.810732\pi$
$854$	0	0
$855$	− 2.87054e8i	− 0.459267i
$856$	0	0
$857$	−1.03110e9	−1.63816	−0.819081	−	0.573678i	$-0.805516\pi$
$857$	−1.03110e9	−1.63816	−0.819081	+	0.573678i	$0.805516\pi$
$858$	0	0
$859$	3.47524e8i	0.548284i	0.961689	+	0.274142i	$0.0883938\pi$
$859$	3.47524e8i	0.548284i	−0.961689	+	0.274142i	$0.911606\pi$
$860$	0	0
$861$	−9.81088e6	−0.0153709
$862$	0	0
$863$	− 1.05023e9i	− 1.63400i	−0.576641	−	0.816998i	$-0.695637\pi$
$863$	− 1.05023e9i	− 1.63400i	0.576641	−	0.816998i	$-0.304363\pi$
$864$	0	0
$865$	1.90934e8	0.295010
$866$	0	0
$867$	6.29567e7i	0.0966016i
$868$	0	0
$869$	1.41256e9	2.15253
$870$	0	0
$871$	9.67674e8i	1.46445i
$872$	0	0
$873$	3.70978e8	0.557578
$874$	0	0
$875$	2.64500e8i	0.394822i
$876$	0	0
$877$	1.68212e8	0.249379	0.124689	−	0.992196i	$-0.460207\pi$
$877$	1.68212e8	0.249379	0.124689	+	0.992196i	$0.460207\pi$
$878$	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
$883$	− 4.04332e7i	− 0.0587294i	−0.999569	−	0.0293647i	$-0.990652\pi$
$883$	− 4.04332e7i	− 0.0587294i	0.999569	−	0.0293647i	$-0.00934842\pi$
$884$	0	0
$885$	2.94632e7	0.0425060
$886$	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
$891$	− 1.04705e9i	− 1.48025i
$892$	0	0
$893$	−1.16580e9	−1.63708
$894$	0	0
$895$	2.01068e8i	0.280462i
$896$	0	0
$897$	1.66661e8	0.230918
$898$	0	0
$899$	− 9.84011e8i	− 1.35432i
$900$	0	0
$901$	4.97210e8	0.679775
$902$	0	0
$903$	4.40128e6i	0.00597745i
$904$	0	0
$905$	8.77161e7	0.118341
$906$	0	0
$907$	− 9.04088e8i	− 1.21168i	−0.795586	−	0.605841i	$-0.792837\pi$
$907$	− 9.04088e8i	− 1.21168i	0.795586	−	0.605841i	$-0.207163\pi$
$908$	0	0
$909$	1.13976e9	1.51748
$910$	0	0
$911$	1.94818e8i	0.257676i	0.991666	+	0.128838i	$0.0411248\pi$
$911$	1.94818e8i	0.257676i	−0.991666	+	0.128838i	$0.958875\pi$
$912$	0	0
$913$	−6.35140e8	−0.834560
$914$	0	0
$915$	5.41756e7i	0.0707197i
$916$	0	0
$917$	4.14880e8	0.538039
$918$	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
$923$	− 8.66488e7i	− 0.110194i
$924$	0	0
$925$	−5.27625e6	−0.00666654
$926$	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
$931$	− 6.74701e8i	− 0.836108i
$932$	0	0
$933$	5.70212e7	0.0702087
$934$	0	0
$935$	− 3.05449e8i	− 0.373684i
$936$	0	0
$937$	−5.61007e7	−0.0681945	−0.0340973	−	0.999419i	$-0.510856\pi$
$937$	−5.61007e7	−0.0681945	−0.0340973	+	0.999419i	$0.510856\pi$
$938$	0	0
$939$	− 1.03590e8i	− 0.125119i
$940$	0	0
$941$	3.14919e8	0.377947	0.188973	−	0.981982i	$-0.439484\pi$
$941$	3.14919e8	0.377947	0.188973	+	0.981982i	$0.439484\pi$
$942$	0	0
$943$	2.47725e8i	0.295416i
$944$	0	0
$945$	−5.30656e7	−0.0628807
$946$	0	0
$947$	2.73886e8i	0.322493i	0.986914	+	0.161246i	$0.0515513\pi$
$947$	2.73886e8i	0.322493i	−0.986914	+	0.161246i	$0.948449\pi$
$948$	0	0
$949$	−1.56179e9	−1.82736
$950$	0	0
$951$	− 1.31894e7i	− 0.0153350i
$952$	0	0
$953$	1.04389e9	1.20608	0.603042	−	0.797709i	$-0.293955\pi$
$953$	1.04389e9	1.20608	0.603042	+	0.797709i	$0.293955\pi$
$954$	0	0
$955$	1.39168e7i	0.0159783i
$956$	0	0
$957$	1.76988e8	0.201933
$958$	0	0
$959$	2.26437e8i	0.256739i
$960$	0	0
$961$	−1.31023e9	−1.47631
$962$	0	0
$963$	− 6.75693e8i	− 0.756607i
$964$	0	0
$965$	−9.20935e7	−0.102482
$966$	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
$971$	4.72983e8i	0.516640i	0.966059	+	0.258320i	$0.0831688\pi$
$971$	4.72983e8i	0.516640i	−0.966059	+	0.258320i	$0.916831\pi$
$972$	0	0
$973$	7.42949e8	0.806529
$974$	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
$979$	3.40649e8i	0.363043i
$980$	0	0
$981$	−1.39828e9	−1.48111
$982$	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
$987$	1.06561e8i	0.110827i
$988$	0	0
$989$	1.11132e8	0.114882
$990$	0	0
$991$	9.42496e8i	0.968408i	0.874955	+	0.484204i	$0.160891\pi$
$991$	9.42496e8i	0.968408i	−0.874955	+	0.484204i	$0.839109\pi$
$992$	0	0
$993$	−2.68288e8	−0.274002
$994$	0	0
$995$	6.84672e8i	0.695046i
$996$	0	0
$997$	6.48931e7	0.0654806	0.0327403	−	0.999464i	$-0.489577\pi$
$997$	6.48931e7	0.0654806	0.0327403	+	0.999464i	$0.489577\pi$
$998$	0	0
$999$	− 2.31874e6i	− 0.00232571i

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.7.c.a.31.1	✓	2	1.1	even	1	trivial
32.7.c.a.31.2	yes	2	4.3	odd	2	inner
64.7.c.b.63.1		2	8.3	odd	2
64.7.c.b.63.2		2	8.5	even	2
256.7.d.a.127.1		2	16.3	odd	4
256.7.d.a.127.2		2	16.5	even	4
256.7.d.c.127.1		2	16.13	even	4
256.7.d.c.127.2		2	16.11	odd	4
288.7.g.a.127.1		2	3.2	odd	2
288.7.g.a.127.2		2	12.11	even	2
576.7.g.i.127.1		2	24.5	odd	2
576.7.g.i.127.2		2	24.11	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:	\( N \)	\(=\)	\( 32 = 2^{5} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	32.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.00000i q^{3} +50.0000 q^{5} -184.000i q^{7} +713.000 q^{9} +O(q^{10})\)	\(q-4.00000i q^{3} +50.0000 q^{5} -184.000i q^{7} +713.000 q^{9} -2108.00i q^{11} +2242.00 q^{13} -200.000i q^{15} +2898.00 q^{17} -8052.00i q^{19} -736.000 q^{21} +18584.0i q^{23} -13125.0 q^{25} -5768.00i q^{27} -20990.0 q^{29} +46880.0i q^{31} -8432.00 q^{33} -9200.00i q^{35} +402.000 q^{37} -8968.00i q^{39} +13330.0 q^{41} -5980.00i q^{43} +35650.0 q^{45} -144784. i q^{47} +83793.0 q^{49} -11592.0i q^{51} +171570. q^{53} -105400. i q^{55} -32208.0 q^{57} +147316. i q^{59} -270878. q^{61} -131192. i q^{63} +112100. q^{65} +431612. i q^{67} +74336.0 q^{69} -38648.0i q^{71} -696606. q^{73} +52500.0i q^{75} -387872. q^{77} +670096. i q^{79} +496705. q^{81} -301300. i q^{83} +144900. q^{85} +83960.0i q^{87} -161598. q^{89} -412528. i q^{91} +187520. q^{93} -402600. i q^{95} +520306. q^{97} -1.50300e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 100 q^{5} + 1426 q^{9}+O(q^{10}) \) 2 * q + 100 * q^5 + 1426 * q^9	\( 2 q + 100 q^{5} + 1426 q^{9} + 4484 q^{13} + 5796 q^{17} - 1472 q^{21} - 26250 q^{25} - 41980 q^{29} - 16864 q^{33} + 804 q^{37} + 26660 q^{41} + 71300 q^{45} + 167586 q^{49} + 343140 q^{53} - 64416 q^{57} - 541756 q^{61} + 224200 q^{65} + 148672 q^{69} - 1393212 q^{73} - 775744 q^{77} + 993410 q^{81} + 289800 q^{85} - 323196 q^{89} + 375040 q^{93} + 1040612 q^{97}+O(q^{100}) \) 2 * q + 100 * q^5 + 1426 * q^9 + 4484 * q^13 + 5796 * q^17 - 1472 * q^21 - 26250 * q^25 - 41980 * q^29 - 16864 * q^33 + 804 * q^37 + 26660 * q^41 + 71300 * q^45 + 167586 * q^49 + 343140 * q^53 - 64416 * q^57 - 541756 * q^61 + 224200 * q^65 + 148672 * q^69 - 1393212 * q^73 - 775744 * q^77 + 993410 * q^81 + 289800 * q^85 - 323196 * q^89 + 375040 * q^93 + 1040612 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more