LMFDB - Embedded newform 704.4.e.g.703.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [704,4,Mod(703,704)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(704, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("704.703");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 704 = 2^{6} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	704.e (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:	$41.5373446440$
Analytic rank:	$0$
Dimension:	$36$
Twist minimal:	no (minimal twist has level 352)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			703.19
Character	$\chi$	$=$	704.703
Dual form			704.4.e.g.703.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.08315i q^{3} +19.1950 q^{5} -20.3180 q^{7} +17.4942 q^{9} +(27.3461 + 24.1493i) q^{11} +42.7879i q^{13} -59.1812i q^{15} +92.2301i q^{17} -89.6650 q^{19} +62.6434i q^{21} +31.4531i q^{23} +243.449 q^{25} -137.182i q^{27} +226.048i q^{29} +300.966i q^{31} +(74.4559 - 84.3123i) q^{33} -390.004 q^{35} -291.432 q^{37} +131.921 q^{39} -297.331i q^{41} +43.4091 q^{43} +335.801 q^{45} +226.344i q^{47} +69.8206 q^{49} +284.359 q^{51} -63.5040 q^{53} +(524.910 + 463.547i) q^{55} +276.451i q^{57} +12.9192i q^{59} +173.175i q^{61} -355.447 q^{63} +821.315i q^{65} +155.683i q^{67} +96.9748 q^{69} -912.204i q^{71} -849.892i q^{73} -750.591i q^{75} +(-555.619 - 490.665i) q^{77} +419.424 q^{79} +49.3895 q^{81} -1354.67 q^{83} +1770.36i q^{85} +696.940 q^{87} +759.067 q^{89} -869.364i q^{91} +927.923 q^{93} -1721.12 q^{95} +1812.03 q^{97} +(478.398 + 422.472i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q - 364 q^{9} + 1236 q^{25} + 592 q^{33} - 984 q^{45} + 2372 q^{49} - 376 q^{53} + 2824 q^{69} + 2592 q^{77} + 6756 q^{81} + 512 q^{89} + 872 q^{93} + 3904 q^{97}+O(q^{100}) $ 36 * q - 364 * q^9 + 1236 * q^25 + 592 * q^33 - 984 * q^45 + 2372 * q^49 - 376 * q^53 + 2824 * q^69 + 2592 * q^77 + 6756 * q^81 + 512 * q^89 + 872 * q^93 + 3904 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/704\mathbb{Z}\right)^\times$.

$n$	$133$	$321$	$639$
$\chi(n)$	$1$	$-1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		−	3.08315i		−	0.593353i	−0.954978	−	0.296676i	$-0.904122\pi$
$3$		−	3.08315i		−	0.593353i	0.954978	−	0.296676i	$-0.0958782\pi$
$4$		0			0
$5$	19.1950			1.71686			0.858428	−	0.512934i	$-0.171441\pi$
$5$	19.1950			1.71686			0.858428	+	0.512934i	$0.171441\pi$
$6$		0			0
$7$	−20.3180			−1.09707			−0.548534	−	0.836128i	$-0.684814\pi$
$7$	−20.3180			−1.09707			−0.548534	+	0.836128i	$0.684814\pi$
$8$		0			0
$9$	17.4942			0.647933
$10$		0			0
$11$	27.3461	+	24.1493i	0.749561	+	0.661935i
$11$	27.3461	+	24.1493i	0.749561	+	0.661935i
$12$		0			0
$13$			42.7879i			0.912863i	0.889758	+	0.456432i	$0.150873\pi$
$13$			42.7879i			0.912863i	−0.889758	+	0.456432i	$0.849127\pi$
$14$		0			0
$15$		−	59.1812i		−	1.01870i
$16$		0			0
$17$			92.2301i			1.31583i	0.753093	+	0.657914i	$0.228561\pi$
$17$			92.2301i			1.31583i	−0.753093	+	0.657914i	$0.771439\pi$
$18$		0			0
$19$	−89.6650			−1.08266			−0.541331	−	0.840810i	$-0.682079\pi$
$19$	−89.6650			−1.08266			−0.541331	+	0.840810i	$0.682079\pi$
$20$		0			0
$21$			62.6434i			0.650948i
$22$		0			0
$23$			31.4531i			0.285149i	0.989784	+	0.142575i	$0.0455381\pi$
$23$			31.4531i			0.285149i	−0.989784	+	0.142575i	$0.954462\pi$
$24$		0			0
$25$	243.449			1.94759
$26$		0			0
$27$		−	137.182i		−	0.977805i
$28$		0			0
$29$			226.048i			1.44745i	0.690088	+	0.723725i	$0.257572\pi$
$29$			226.048i			1.44745i	−0.690088	+	0.723725i	$0.742428\pi$
$30$		0			0
$31$			300.966i			1.74371i	0.489763	+	0.871856i	$0.337083\pi$
$31$			300.966i			1.74371i	−0.489763	+	0.871856i	$0.662917\pi$
$32$		0			0
$33$	74.4559	−	84.3123i	0.392761	−	0.444754i
$34$		0			0
$35$	−390.004			−1.88351
$36$		0			0
$37$	−291.432			−1.29489			−0.647447	−	0.762111i	$-0.724163\pi$
$37$	−291.432			−1.29489			−0.647447	+	0.762111i	$0.724163\pi$
$38$		0			0
$39$	131.921			0.541650
$40$		0			0
$41$		−	297.331i		−	1.13257i	−0.824209	−	0.566285i	$-0.808380\pi$
$41$		−	297.331i		−	1.13257i	0.824209	−	0.566285i	$-0.191620\pi$
$42$		0			0
$43$	43.4091			0.153950			0.0769748	−	0.997033i	$-0.475474\pi$
$43$	43.4091			0.153950			0.0769748	+	0.997033i	$0.475474\pi$
$44$		0			0
$45$	335.801			1.11241
$46$		0			0
$47$			226.344i			0.702461i	0.936289	+	0.351231i	$0.114237\pi$
$47$			226.344i			0.702461i	−0.936289	+	0.351231i	$0.885763\pi$
$48$		0			0
$49$	69.8206			0.203559
$50$		0			0
$51$	284.359			0.780750
$52$		0			0
$53$	−63.5040			−0.164584			−0.0822919	−	0.996608i	$-0.526224\pi$
$53$	−63.5040			−0.164584			−0.0822919	+	0.996608i	$0.526224\pi$
$54$		0			0
$55$	524.910	+	463.547i	1.28689	+	1.13645i
$56$		0			0
$57$			276.451i			0.642400i
$58$		0			0
$59$			12.9192i			0.0285073i	0.999898	+	0.0142537i	$0.00453724\pi$
$59$			12.9192i			0.0285073i	−0.999898	+	0.0142537i	$0.995463\pi$
$60$		0			0
$61$			173.175i			0.363488i	0.983346	+	0.181744i	$0.0581742\pi$
$61$			173.175i			0.363488i	−0.983346	+	0.181744i	$0.941826\pi$
$62$		0			0
$63$	−355.447			−0.710826
$64$		0			0
$65$			821.315i			1.56726i
$66$		0			0
$67$			155.683i			0.283877i	0.989876	+	0.141938i	$0.0453334\pi$
$67$			155.683i			0.283877i	−0.989876	+	0.141938i	$0.954667\pi$
$68$		0			0
$69$	96.9748			0.169194
$70$		0			0
$71$		−	912.204i		−	1.52477i	−0.647123	−	0.762385i	$-0.724028\pi$
$71$		−	912.204i		−	1.52477i	0.647123	−	0.762385i	$-0.275972\pi$
$72$		0			0
$73$		−	849.892i		−	1.36263i	−0.731988	−	0.681317i	$-0.761407\pi$
$73$		−	849.892i		−	1.36263i	0.731988	−	0.681317i	$-0.238593\pi$
$74$		0			0
$75$		−	750.591i		−	1.15561i
$76$		0			0
$77$	−555.619	−	490.665i	−0.822320	−	0.726188i
$78$		0			0
$79$	419.424			0.597327			0.298664	−	0.954358i	$-0.403459\pi$
$79$	419.424			0.597327			0.298664	+	0.954358i	$0.403459\pi$
$80$		0			0
$81$	49.3895			0.0677497
$82$		0			0
$83$	−1354.67			−1.79150			−0.895752	−	0.444555i	$-0.853362\pi$
$83$	−1354.67			−1.79150			−0.895752	+	0.444555i	$0.853362\pi$
$84$		0			0
$85$			1770.36i			2.25909i
$86$		0			0
$87$	696.940			0.858849
$88$		0			0
$89$	759.067			0.904055			0.452027	−	0.892004i	$-0.350701\pi$
$89$	759.067			0.904055			0.452027	+	0.892004i	$0.350701\pi$
$90$		0			0
$91$		−	869.364i		−	1.00147i
$92$		0			0
$93$	927.923			1.03464
$94$		0			0
$95$	−1721.12			−1.85877
$96$		0			0
$97$	1812.03			1.89674			0.948372	−	0.317161i	$-0.102730\pi$
$97$	1812.03			1.89674			0.948372	+	0.317161i	$0.102730\pi$
$98$		0			0
$99$	478.398	+	422.472i	0.485665	+	0.428890i
$100$		0			0
$101$		−	446.439i		−	0.439825i	−0.975520	−	0.219912i	$-0.929423\pi$
$101$		−	446.439i		−	0.439825i	0.975520	−	0.219912i	$-0.0705772\pi$
$102$		0			0
$103$			572.642i			0.547806i	0.961757	+	0.273903i	$0.0883148\pi$
$103$			572.642i			0.547806i	−0.961757	+	0.273903i	$0.911685\pi$
$104$		0			0
$105$			1202.44i			1.11758i
$106$		0			0
$107$	1080.96			0.976636			0.488318	−	0.872666i	$-0.337611\pi$
$107$	1080.96			0.976636			0.488318	+	0.872666i	$0.337611\pi$
$108$		0			0
$109$			1959.95i			1.72228i	0.508365	+	0.861142i	$0.330250\pi$
$109$			1959.95i			1.72228i	−0.508365	+	0.861142i	$0.669750\pi$
$110$		0			0
$111$			898.527i			0.768328i
$112$		0			0
$113$	−322.085			−0.268135			−0.134067	−	0.990972i	$-0.542804\pi$
$113$	−322.085			−0.268135			−0.134067	+	0.990972i	$0.542804\pi$
$114$		0			0
$115$			603.744i			0.489560i
$116$		0			0
$117$			748.539i			0.591474i
$118$		0			0
$119$		−	1873.93i		−	1.44355i
$120$		0			0
$121$	164.623	+	1320.78i	0.123684	+	0.992322i
$122$		0			0
$123$	−916.718			−0.672013
$124$		0			0
$125$	2273.64			1.62688
$126$		0			0
$127$	1824.58			1.27485			0.637423	−	0.770514i	$-0.280000\pi$
$127$	1824.58			1.27485			0.637423	+	0.770514i	$0.280000\pi$
$128$		0			0
$129$		−	133.837i		−	0.0913463i
$130$		0			0
$131$	2596.42			1.73168			0.865842	−	0.500317i	$-0.166783\pi$
$131$	2596.42			1.73168			0.865842	+	0.500317i	$0.166783\pi$
$132$		0			0
$133$	1821.81			1.18775
$134$		0			0
$135$		−	2633.22i		−	1.67875i
$136$		0			0
$137$	−340.359			−0.212254			−0.106127	−	0.994353i	$-0.533845\pi$
$137$	−340.359			−0.212254			−0.106127	+	0.994353i	$0.533845\pi$
$138$		0			0
$139$	1472.75			0.898686			0.449343	−	0.893359i	$-0.351658\pi$
$139$	1472.75			0.898686			0.449343	+	0.893359i	$0.351658\pi$
$140$		0			0
$141$	697.853			0.416807
$142$		0			0
$143$	−1033.30	+	1170.08i	−0.604256	+	0.684247i
$144$		0			0
$145$			4339.00i			2.48506i
$146$		0			0
$147$		−	215.267i		−	0.120782i
$148$		0			0
$149$		−	2312.02i		−	1.27119i	−0.772021	−	0.635597i	$-0.780754\pi$
$149$		−	2312.02i		−	1.27119i	0.772021	−	0.635597i	$-0.219246\pi$
$150$		0			0
$151$	200.926			0.108286			0.0541429	−	0.998533i	$-0.482757\pi$
$151$	200.926			0.108286			0.0541429	+	0.998533i	$0.482757\pi$
$152$		0			0
$153$			1613.49i			0.852569i
$154$		0			0
$155$			5777.05i			2.99370i
$156$		0			0
$157$	−320.265			−0.162802			−0.0814011	−	0.996681i	$-0.525939\pi$
$157$	−320.265			−0.162802			−0.0814011	+	0.996681i	$0.525939\pi$
$158$		0			0
$159$			195.792i			0.0976562i
$160$		0			0
$161$		−	639.065i		−	0.312828i
$162$		0			0
$163$		−	829.888i		−	0.398784i	−0.979920	−	0.199392i	$-0.936103\pi$
$163$		−	829.888i		−	0.398784i	0.979920	−	0.199392i	$-0.0638967\pi$
$164$		0			0
$165$	1429.18	−	1618.38i	0.674314	−	0.763578i
$166$		0			0
$167$	122.431			0.0567306			0.0283653	−	0.999598i	$-0.490970\pi$
$167$	122.431			0.0567306			0.0283653	+	0.999598i	$0.490970\pi$
$168$		0			0
$169$	366.197			0.166680
$170$		0			0
$171$	−1568.62			−0.701492
$172$		0			0
$173$			1701.79i			0.747889i	0.927451	+	0.373945i	$0.121995\pi$
$173$			1701.79i			0.747889i	−0.927451	+	0.373945i	$0.878005\pi$
$174$		0			0
$175$	−4946.40			−2.13664
$176$		0			0
$177$	39.8318			0.0169149
$178$		0			0
$179$		−	2653.51i		−	1.10800i	−0.832516	−	0.554002i	$-0.813100\pi$
$179$		−	2653.51i		−	1.10800i	0.832516	−	0.554002i	$-0.186900\pi$
$180$		0			0
$181$	658.860			0.270567			0.135284	−	0.990807i	$-0.456805\pi$
$181$	658.860			0.270567			0.135284	+	0.990807i	$0.456805\pi$
$182$		0			0
$183$	533.924			0.215676
$184$		0			0
$185$	−5594.04			−2.22315
$186$		0			0
$187$	−2227.29	+	2522.14i	−0.870993	+	0.986294i
$188$		0			0
$189$			2787.27i			1.07272i
$190$		0			0
$191$		−	1840.05i		−	0.697076i	−0.937295	−	0.348538i	$-0.886678\pi$
$191$		−	1840.05i		−	0.697076i	0.937295	−	0.348538i	$-0.113322\pi$
$192$		0			0
$193$		−	2162.04i		−	0.806356i	−0.915122	−	0.403178i	$-0.867906\pi$
$193$		−	2162.04i		−	0.806356i	0.915122	−	0.403178i	$-0.132094\pi$
$194$		0			0
$195$	2532.24			0.929935
$196$		0			0
$197$		−	2078.87i		−	0.751845i	−0.926651	−	0.375923i	$-0.877326\pi$
$197$		−	2078.87i		−	0.751845i	0.926651	−	0.375923i	$-0.122674\pi$
$198$		0			0
$199$			4059.29i			1.44601i	0.690844	+	0.723004i	$0.257239\pi$
$199$			4059.29i			1.44601i	−0.690844	+	0.723004i	$0.742761\pi$
$200$		0			0
$201$	479.995			0.168439
$202$		0			0
$203$		−	4592.84i		−	1.58795i
$204$		0			0
$205$		−	5707.29i		−	1.94446i
$206$		0			0
$207$			550.247i			0.184758i
$208$		0			0
$209$	−2451.99	−	2165.35i	−0.811521	−	0.716652i
$210$		0			0
$211$	−1057.19			−0.344928			−0.172464	−	0.985016i	$-0.555173\pi$
$211$	−1057.19			−0.344928			−0.172464	+	0.985016i	$0.555173\pi$
$212$		0			0
$213$	−2812.46			−0.904727
$214$		0			0
$215$	833.240			0.264309
$216$		0			0
$217$		−	6115.02i		−	1.91297i
$218$		0			0
$219$	−2620.34			−0.808522
$220$		0			0
$221$	−3946.33			−1.20117
$222$		0			0
$223$		−	3910.29i		−	1.17423i	−0.809505	−	0.587113i	$-0.800264\pi$
$223$		−	3910.29i		−	1.17423i	0.809505	−	0.587113i	$-0.199736\pi$
$224$		0			0
$225$	4258.95			1.26191
$226$		0			0
$227$	−1826.02			−0.533907			−0.266954	−	0.963709i	$-0.586017\pi$
$227$	−1826.02			−0.533907			−0.266954	+	0.963709i	$0.586017\pi$
$228$		0			0
$229$	−3776.33			−1.08972			−0.544862	−	0.838526i	$-0.683418\pi$
$229$	−3776.33			−1.08972			−0.544862	+	0.838526i	$0.683418\pi$
$230$		0			0
$231$	−1512.79	+	1713.06i	−0.430886	+	0.487925i
$232$		0			0
$233$		−	629.948i		−	0.177121i	−0.996071	−	0.0885607i	$-0.971773\pi$
$233$		−	629.948i		−	0.177121i	0.996071	−	0.0885607i	$-0.0282267\pi$
$234$		0			0
$235$			4344.68i			1.20602i
$236$		0			0
$237$		−	1293.15i		−	0.354426i
$238$		0			0
$239$	−1858.77			−0.503069			−0.251535	−	0.967848i	$-0.580935\pi$
$239$	−1858.77			−0.503069			−0.251535	+	0.967848i	$0.580935\pi$
$240$		0			0
$241$		−	76.5266i		−	0.0204544i	−0.999948	−	0.0102272i	$-0.996745\pi$
$241$		−	76.5266i		−	0.0204544i	0.999948	−	0.0102272i	$-0.00325548\pi$
$242$		0			0
$243$		−	3856.20i		−	1.01800i
$244$		0			0
$245$	1340.21			0.349481
$246$		0			0
$247$		−	3836.58i		−	0.988322i
$248$		0			0
$249$			4176.66i			1.06299i
$250$		0			0
$251$			3895.72i			0.979663i	0.871817	+	0.489831i	$0.162942\pi$
$251$			3895.72i			0.979663i	−0.871817	+	0.489831i	$0.837058\pi$
$252$		0			0
$253$	−759.571	+	860.122i	−0.188750	+	0.213737i
$254$		0			0
$255$	5458.29			1.34044
$256$		0			0
$257$	1153.23			0.279908			0.139954	−	0.990158i	$-0.455305\pi$
$257$	1153.23			0.279908			0.139954	+	0.990158i	$0.455305\pi$
$258$		0			0
$259$	5921.30			1.42059
$260$		0			0
$261$			3954.53i			0.937851i
$262$		0			0
$263$	−3055.93			−0.716489			−0.358245	−	0.933628i	$-0.616625\pi$
$263$	−3055.93			−0.716489			−0.358245	+	0.933628i	$0.616625\pi$
$264$		0			0
$265$	−1218.96			−0.282567
$266$		0			0
$267$		−	2340.32i		−	0.536423i
$268$		0			0
$269$	725.783			0.164505			0.0822524	−	0.996612i	$-0.473789\pi$
$269$	725.783			0.164505			0.0822524	+	0.996612i	$0.473789\pi$
$270$		0			0
$271$	5876.46			1.31723			0.658615	−	0.752480i	$-0.271142\pi$
$271$	5876.46			1.31723			0.658615	+	0.752480i	$0.271142\pi$
$272$		0			0
$273$	−2680.38			−0.594227
$274$		0			0
$275$	6657.40	+	5879.13i	1.45984	+	1.28918i
$276$		0			0
$277$		−	4022.34i		−	0.872487i	−0.899829	−	0.436244i	$-0.856309\pi$
$277$		−	4022.34i		−	0.872487i	0.899829	−	0.436244i	$-0.143691\pi$
$278$		0			0
$279$			5265.15i			1.12981i
$280$		0			0
$281$		−	3390.84i		−	0.719859i	−0.932979	−	0.359930i	$-0.882801\pi$
$281$		−	3390.84i		−	0.719859i	0.932979	−	0.359930i	$-0.117199\pi$
$282$		0			0
$283$	−6948.19			−1.45946			−0.729729	−	0.683736i	$-0.760354\pi$
$283$	−6948.19			−1.45946			−0.729729	+	0.683736i	$0.760354\pi$
$284$		0			0
$285$			5306.48i			1.10291i
$286$		0			0
$287$			6041.18i			1.24251i
$288$		0			0
$289$	−3593.40			−0.731406
$290$		0			0
$291$		−	5586.77i		−	1.12544i
$292$		0			0
$293$		−	5267.68i		−	1.05031i	−0.851006	−	0.525156i	$-0.824007\pi$
$293$		−	5267.68i		−	1.05031i	0.851006	−	0.525156i	$-0.175993\pi$
$294$		0			0
$295$			247.984i			0.0489430i
$296$		0			0
$297$	3312.86	−	3751.41i	0.647244	−	0.732925i
$298$		0			0
$299$	−1345.81			−0.260302
$300$		0			0
$301$	−881.986			−0.168893
$302$		0			0
$303$	−1376.44			−0.260971
$304$		0			0
$305$			3324.10i			0.624056i
$306$		0			0
$307$	1644.72			0.305762			0.152881	−	0.988245i	$-0.451145\pi$
$307$	1644.72			0.305762			0.152881	+	0.988245i	$0.451145\pi$
$308$		0			0
$309$	1765.54			0.325042
$310$		0			0
$311$		−	6181.97i		−	1.12716i	−0.826061	−	0.563581i	$-0.809423\pi$
$311$		−	6181.97i		−	1.12716i	0.826061	−	0.563581i	$-0.190577\pi$
$312$		0			0
$313$	5778.02			1.04343			0.521714	−	0.853120i	$-0.325293\pi$
$313$	5778.02			1.04343			0.521714	+	0.853120i	$0.325293\pi$
$314$		0			0
$315$	−6822.81			−1.22039
$316$		0			0
$317$	−1817.51			−0.322023			−0.161011	−	0.986953i	$-0.551476\pi$
$317$	−1817.51			−0.322023			−0.161011	+	0.986953i	$0.551476\pi$
$318$		0			0
$319$	−5458.90	+	6181.54i	−0.958119	+	1.08495i
$320$		0			0
$321$		−	3332.75i		−	0.579490i
$322$		0			0
$323$		−	8269.82i		−	1.42460i
$324$		0			0
$325$			10416.7i			1.77789i
$326$		0			0
$327$	6042.81			1.02192
$328$		0			0
$329$		−	4598.86i		−	0.770648i
$330$		0			0
$331$		−	2004.29i		−	0.332828i	−0.986056	−	0.166414i	$-0.946781\pi$
$331$		−	2004.29i		−	0.332828i	0.986056	−	0.166414i	$-0.0532187\pi$
$332$		0			0
$333$	−5098.36			−0.839004
$334$		0			0
$335$			2988.34i			0.487375i
$336$		0			0
$337$		−	2087.92i		−	0.337497i	−0.985659	−	0.168749i	$-0.946027\pi$
$337$		−	2087.92i		−	0.337497i	0.985659	−	0.168749i	$-0.0539726\pi$
$338$		0			0
$339$			993.037i			0.159098i
$340$		0			0
$341$	−7268.11	+	8230.25i	−1.15422	+	1.30702i
$342$		0			0
$343$	5550.46			0.873751
$344$		0			0
$345$	1861.43			0.290482
$346$		0			0
$347$	−7344.94			−1.13630			−0.568151	−	0.822924i	$-0.692341\pi$
$347$	−7344.94			−1.13630			−0.568151	+	0.822924i	$0.692341\pi$
$348$		0			0
$349$		−	5535.38i		−	0.849003i	−0.905427	−	0.424501i	$-0.860449\pi$
$349$		−	5535.38i		−	0.849003i	0.905427	−	0.424501i	$-0.139551\pi$
$350$		0			0
$351$	5869.74			0.892602
$352$		0			0
$353$	4595.47			0.692896			0.346448	−	0.938069i	$-0.387388\pi$
$353$	4595.47			0.692896			0.346448	+	0.938069i	$0.387388\pi$
$354$		0			0
$355$		−	17509.8i		−	2.61781i
$356$		0			0
$357$	−5777.61			−0.856536
$358$		0			0
$359$	11954.3			1.75745			0.878725	−	0.477328i	$-0.158395\pi$
$359$	11954.3			1.75745			0.878725	+	0.477328i	$0.158395\pi$
$360$		0			0
$361$	1180.82			0.172156
$362$		0			0
$363$	4072.16	−	507.557i	0.588797	−	0.0733880i
$364$		0			0
$365$		−	16313.7i		−	2.33945i
$366$		0			0
$367$			36.1633i			0.00514362i	0.999997	+	0.00257181i	$0.000818633\pi$
$367$			36.1633i			0.00514362i	−0.999997	+	0.00257181i	$0.999181\pi$
$368$		0			0
$369$		−	5201.57i		−	0.733829i
$370$		0			0
$371$	1290.27			0.180560
$372$		0			0
$373$		−	8053.88i		−	1.11800i	−0.829168	−	0.559000i	$-0.811185\pi$
$373$		−	8053.88i		−	1.11800i	0.829168	−	0.559000i	$-0.188815\pi$
$374$		0			0
$375$		−	7009.97i		−	0.965316i
$376$		0			0
$377$	−9672.12			−1.32132
$378$		0			0
$379$			10639.5i			1.44200i	0.692938	+	0.720998i	$0.256316\pi$
$379$			10639.5i			1.44200i	−0.692938	+	0.720998i	$0.743684\pi$
$380$		0			0
$381$		−	5625.46i		−	0.756433i
$382$		0			0
$383$			6677.81i			0.890914i	0.895303	+	0.445457i	$0.146959\pi$
$383$			6677.81i			0.890914i	−0.895303	+	0.445457i	$0.853041\pi$
$384$		0			0
$385$	−10665.1	−	9418.33i	−1.41180	−	1.24676i
$386$		0			0
$387$	759.407			0.0997490
$388$		0			0
$389$	3002.22			0.391308			0.195654	−	0.980673i	$-0.437317\pi$
$389$	3002.22			0.391308			0.195654	+	0.980673i	$0.437317\pi$
$390$		0			0
$391$	−2900.93			−0.375208
$392$		0			0
$393$		−	8005.17i		−	1.02750i
$394$		0			0
$395$	8050.85			1.02553
$396$		0			0
$397$	−5728.92			−0.724248			−0.362124	−	0.932130i	$-0.617948\pi$
$397$	−5728.92			−0.724248			−0.362124	+	0.932130i	$0.617948\pi$
$398$		0			0
$399$		−	5616.92i		−	0.704757i
$400$		0			0
$401$	−8272.92			−1.03025			−0.515124	−	0.857115i	$-0.672254\pi$
$401$	−8272.92			−1.03025			−0.515124	+	0.857115i	$0.672254\pi$
$402$		0			0
$403$	−12877.7			−1.59177
$404$		0			0
$405$	948.034			0.116317
$406$		0			0
$407$	−7969.53	−	7037.87i	−0.970602	−	0.857135i
$408$		0			0
$409$			2161.72i			0.261345i	0.991426	+	0.130673i	$0.0417137\pi$
$409$			2161.72i			0.261345i	−0.991426	+	0.130673i	$0.958286\pi$
$410$		0			0
$411$			1049.38i			0.125942i
$412$		0			0
$413$		−	262.492i		−	0.0312745i
$414$		0			0
$415$	−26003.0			−3.07575
$416$		0			0
$417$		−	4540.72i		−	0.533237i
$418$		0			0
$419$			15653.3i			1.82509i	0.408975	+	0.912545i	$0.365886\pi$
$419$			15653.3i			1.82509i	−0.408975	+	0.912545i	$0.634114\pi$
$420$		0			0
$421$	−13627.3			−1.57756			−0.788782	−	0.614672i	$-0.789288\pi$
$421$	−13627.3			−1.57756			−0.788782	+	0.614672i	$0.789288\pi$
$422$		0			0
$423$			3959.71i			0.455148i
$424$		0			0
$425$			22453.4i			2.56270i
$426$		0			0
$427$		−	3518.56i		−	0.398771i
$428$		0			0
$429$	3607.54	+	3185.81i	0.406000	+	0.358537i
$430$		0			0
$431$	−1499.14			−0.167544			−0.0837718	−	0.996485i	$-0.526697\pi$
$431$	−1499.14			−0.167544			−0.0837718	+	0.996485i	$0.526697\pi$
$432$		0			0
$433$	−16133.2			−1.79055			−0.895277	−	0.445510i	$-0.853022\pi$
$433$	−16133.2			−1.79055			−0.895277	+	0.445510i	$0.853022\pi$
$434$		0			0
$435$	13377.8			1.47452
$436$		0			0
$437$		−	2820.25i		−	0.308720i
$438$		0			0
$439$	13311.3			1.44719			0.723593	−	0.690227i	$-0.242489\pi$
$439$	13311.3			1.44719			0.723593	+	0.690227i	$0.242489\pi$
$440$		0			0
$441$	1221.45			0.131892
$442$		0			0
$443$			8498.88i			0.911499i	0.890108	+	0.455749i	$0.150629\pi$
$443$			8498.88i			0.911499i	−0.890108	+	0.455749i	$0.849371\pi$
$444$		0			0
$445$	14570.3			1.55213
$446$		0			0
$447$	−7128.30			−0.754266
$448$		0			0
$449$	1143.89			0.120230			0.0601152	−	0.998191i	$-0.480853\pi$
$449$	1143.89			0.120230			0.0601152	+	0.998191i	$0.480853\pi$
$450$		0			0
$451$	7180.35	−	8130.87i	0.749688	−	0.848930i
$452$		0			0
$453$		−	619.486i		−	0.0642516i
$454$		0			0
$455$		−	16687.5i		−	1.71939i
$456$		0			0
$457$			415.516i			0.0425318i	0.999774	+	0.0212659i	$0.00676966\pi$
$457$			415.516i			0.0425318i	−0.999774	+	0.0212659i	$0.993230\pi$
$458$		0			0
$459$	12652.3			1.28662
$460$		0			0
$461$			2496.11i			0.252181i	0.992019	+	0.126090i	$0.0402430\pi$
$461$			2496.11i			0.252181i	−0.992019	+	0.126090i	$0.959757\pi$
$462$		0			0
$463$			1159.10i			0.116346i	0.998307	+	0.0581729i	$0.0185275\pi$
$463$			1159.10i			0.116346i	−0.998307	+	0.0581729i	$0.981473\pi$
$464$		0			0
$465$	17811.5			1.77632
$466$		0			0
$467$			11650.9i			1.15447i	0.816577	+	0.577236i	$0.195869\pi$
$467$			11650.9i			1.15447i	−0.816577	+	0.577236i	$0.804131\pi$
$468$		0			0
$469$		−	3163.17i		−	0.311432i
$470$		0			0
$471$			987.426i			0.0965991i
$472$		0			0
$473$	1187.07	+	1048.30i	0.115395	+	0.101905i
$474$		0			0
$475$	−21828.9			−2.10859
$476$		0			0
$477$	−1110.95			−0.106639
$478$		0			0
$479$	−13319.4			−1.27052			−0.635258	−	0.772300i	$-0.719106\pi$
$479$	−13319.4			−1.27052			−0.635258	+	0.772300i	$0.719106\pi$
$480$		0			0
$481$		−	12469.7i		−	1.18206i
$482$		0			0
$483$	−1970.33			−0.185617
$484$		0			0
$485$	34782.0			3.25644
$486$		0			0
$487$		−	699.293i		−	0.0650678i	−0.999471	−	0.0325339i	$-0.989642\pi$
$487$		−	699.293i		−	0.0650678i	0.999471	−	0.0325339i	$-0.0103577\pi$
$488$		0			0
$489$	−2558.67			−0.236620
$490$		0			0
$491$	5152.60			0.473592			0.236796	−	0.971559i	$-0.423903\pi$
$491$	5152.60			0.473592			0.236796	+	0.971559i	$0.423903\pi$
$492$		0			0
$493$	−20848.4			−1.90460
$494$		0			0
$495$	9182.87	+	8109.37i	0.833817	+	0.736342i
$496$		0			0
$497$			18534.2i			1.67278i
$498$		0			0
$499$			622.541i			0.0558492i	0.999610	+	0.0279246i	$0.00888984\pi$
$499$			622.541i			0.0558492i	−0.999610	+	0.0279246i	$0.991110\pi$
$500$		0			0
$501$		−	377.474i		−	0.0336612i
$502$		0			0
$503$	7478.46			0.662918			0.331459	−	0.943470i	$-0.392459\pi$
$503$	7478.46			0.662918			0.331459	+	0.943470i	$0.392459\pi$
$504$		0			0
$505$		−	8569.41i		−	0.755116i
$506$		0			0
$507$		−	1129.04i		−	0.0989002i
$508$		0			0
$509$	18238.5			1.58822			0.794111	−	0.607772i	$-0.207937\pi$
$509$	18238.5			1.58822			0.794111	+	0.607772i	$0.207937\pi$
$510$		0			0
$511$			17268.1i			1.49490i
$512$		0			0
$513$			12300.5i			1.05863i
$514$		0			0
$515$			10991.9i			0.940505i
$516$		0			0
$517$	−5466.05	+	6189.64i	−0.464984	+	0.526538i
$518$		0			0
$519$	5246.88			0.443762
$520$		0			0
$521$	19257.2			1.61933			0.809667	−	0.586890i	$-0.199648\pi$
$521$	19257.2			1.61933			0.809667	+	0.586890i	$0.199648\pi$
$522$		0			0
$523$	−8373.24			−0.700069			−0.350035	−	0.936737i	$-0.613830\pi$
$523$	−8373.24			−0.700069			−0.350035	+	0.936737i	$0.613830\pi$
$524$		0			0
$525$			15250.5i			1.26778i
$526$		0			0
$527$	−27758.1			−2.29443
$528$		0			0
$529$	11177.7			0.918690
$530$		0			0
$531$			226.010i			0.0184708i
$532$		0			0
$533$	12722.2			1.03388
$534$		0			0
$535$	20749.0			1.67674
$536$		0			0
$537$	−8181.17			−0.657437
$538$		0			0
$539$	1909.32	+	1686.12i	0.152580	+	0.134743i
$540$		0			0
$541$		−	5729.06i		−	0.455289i	−0.973744	−	0.227645i	$-0.926898\pi$
$541$		−	5729.06i		−	0.455289i	0.973744	−	0.227645i	$-0.0731024\pi$
$542$		0			0
$543$		−	2031.36i		−	0.160542i
$544$		0			0
$545$			37621.3i			2.95691i
$546$		0			0
$547$	21986.8			1.71863			0.859313	−	0.511450i	$-0.170891\pi$
$547$	21986.8			1.71863			0.859313	+	0.511450i	$0.170891\pi$
$548$		0			0
$549$			3029.55i			0.235516i
$550$		0			0
$551$		−	20268.6i		−	1.56710i
$552$		0			0
$553$	−8521.85			−0.655309
$554$		0			0
$555$			17247.3i			1.31911i
$556$		0			0
$557$			19691.1i			1.49792i	0.662617	+	0.748958i	$0.269446\pi$
$557$			19691.1i			1.49792i	−0.662617	+	0.748958i	$0.730554\pi$
$558$		0			0
$559$			1857.38i			0.140535i
$560$		0			0
$561$	7776.13	+	6867.08i	0.585220	+	0.516806i
$562$		0			0
$563$	−8495.34			−0.635943			−0.317971	−	0.948100i	$-0.603002\pi$
$563$	−8495.34			−0.635943			−0.317971	+	0.948100i	$0.603002\pi$
$564$		0			0
$565$	−6182.44			−0.460349
$566$		0			0
$567$	−1003.50			−0.0743261
$568$		0			0
$569$			20364.8i			1.50042i	0.661200	+	0.750210i	$0.270048\pi$
$569$			20364.8i			1.50042i	−0.661200	+	0.750210i	$0.729952\pi$
$570$		0			0
$571$	956.631			0.0701117			0.0350559	−	0.999385i	$-0.488839\pi$
$571$	956.631			0.0701117			0.0350559	+	0.999385i	$0.488839\pi$
$572$		0			0
$573$	−5673.16			−0.413612
$574$		0			0
$575$			7657.25i			0.555355i
$576$		0			0
$577$	−296.808			−0.0214147			−0.0107073	−	0.999943i	$-0.503408\pi$
$577$	−296.808			−0.0214147			−0.0107073	+	0.999943i	$0.503408\pi$
$578$		0			0
$579$	−6665.88			−0.478454
$580$		0			0
$581$	27524.2			1.96540
$582$		0			0
$583$	−1736.59	−	1533.58i	−0.123366	−	0.108944i
$584$		0			0
$585$			14368.2i			1.01548i
$586$		0			0
$587$		−	19249.0i		−	1.35347i	−0.736225	−	0.676737i	$-0.763393\pi$
$587$		−	19249.0i		−	1.35347i	0.736225	−	0.676737i	$-0.236607\pi$
$588$		0			0
$589$		−	26986.1i		−	1.88785i
$590$		0			0
$591$	−6409.48			−0.446109
$592$		0			0
$593$		−	25489.7i		−	1.76516i	−0.470166	−	0.882578i	$-0.655806\pi$
$593$		−	25489.7i		−	1.76516i	0.470166	−	0.882578i	$-0.344194\pi$
$594$		0			0
$595$		−	35970.2i		−	2.47837i
$596$		0			0
$597$	12515.4			0.857992
$598$		0			0
$599$		−	4642.87i		−	0.316699i	−0.987383	−	0.158349i	$-0.949383\pi$
$599$		−	4642.87i		−	0.316699i	0.987383	−	0.158349i	$-0.0506172\pi$
$600$		0			0
$601$		−	7278.12i		−	0.493978i	−0.969018	−	0.246989i	$-0.920559\pi$
$601$		−	7278.12i		−	0.493978i	0.969018	−	0.246989i	$-0.0794411\pi$
$602$		0			0
$603$			2723.55i			0.183933i
$604$		0			0
$605$	3159.94	+	25352.4i	0.212347	+	1.70367i
$606$		0			0
$607$	6122.64			0.409407			0.204704	−	0.978824i	$-0.434377\pi$
$607$	6122.64			0.409407			0.204704	+	0.978824i	$0.434377\pi$
$608$		0			0
$609$	−14160.4			−0.942215
$610$		0			0
$611$	−9684.79			−0.641251
$612$		0			0
$613$			24978.5i			1.64579i	0.568192	+	0.822896i	$0.307643\pi$
$613$			24978.5i			1.64579i	−0.568192	+	0.822896i	$0.692357\pi$
$614$		0			0
$615$	−17596.4			−1.15375
$616$		0			0
$617$	−18385.9			−1.19966			−0.599828	−	0.800129i	$-0.704764\pi$
$617$	−18385.9			−1.19966			−0.599828	+	0.800129i	$0.704764\pi$
$618$		0			0
$619$			8889.98i			0.577251i	0.957442	+	0.288625i	$0.0931982\pi$
$619$			8889.98i			0.577251i	−0.957442	+	0.288625i	$0.906802\pi$
$620$		0			0
$621$	4314.81			0.278820
$622$		0			0
$623$	−15422.7			−0.991810
$624$		0			0
$625$	13211.4			0.845530
$626$		0			0
$627$	−6676.09	+	7559.86i	−0.425227	+	0.481518i
$628$		0			0
$629$		−	26878.8i		−	1.70386i
$630$		0			0
$631$		−	28313.1i		−	1.78625i	−0.449807	−	0.893126i	$-0.648507\pi$
$631$		−	28313.1i		−	1.78625i	0.449807	−	0.893126i	$-0.351493\pi$
$632$		0			0
$633$			3259.46i			0.204664i
$634$		0			0
$635$	35022.9			2.18873
$636$		0			0
$637$			2987.48i			0.185821i
$638$		0			0
$639$		−	15958.3i		−	0.987949i
$640$		0			0
$641$	8638.10			0.532269			0.266134	−	0.963936i	$-0.414253\pi$
$641$	8638.10			0.532269			0.266134	+	0.963936i	$0.414253\pi$
$642$		0			0
$643$			22129.9i			1.35726i	0.734480	+	0.678630i	$0.237426\pi$
$643$			22129.9i			1.35726i	−0.734480	+	0.678630i	$0.762574\pi$
$644$		0			0
$645$		−	2569.00i		−	0.156829i
$646$		0			0
$647$			23922.1i			1.45359i	0.686854	+	0.726796i	$0.258991\pi$
$647$			23922.1i			1.45359i	−0.686854	+	0.726796i	$0.741009\pi$
$648$		0			0
$649$	−311.989	+	353.290i	−0.0188700	+	0.0213680i
$650$		0			0
$651$	−18853.5			−1.13507
$652$		0			0
$653$	21734.4			1.30250			0.651251	−	0.758862i	$-0.274244\pi$
$653$	21734.4			1.30250			0.651251	+	0.758862i	$0.274244\pi$
$654$		0			0
$655$	49838.4			2.97305
$656$		0			0
$657$		−	14868.2i		−	0.882895i
$658$		0			0
$659$	13202.5			0.780422			0.390211	−	0.920725i	$-0.372402\pi$
$659$	13202.5			0.780422			0.390211	+	0.920725i	$0.372402\pi$
$660$		0			0
$661$	−18389.0			−1.08207			−0.541036	−	0.840999i	$-0.681968\pi$
$661$	−18389.0			−1.08207			−0.541036	+	0.840999i	$0.681968\pi$
$662$		0			0
$663$			12167.1i			0.712718i
$664$		0			0
$665$	34969.8			2.03920
$666$		0			0
$667$	−7109.92			−0.412740
$668$		0			0
$669$	−12056.0			−0.696730
$670$		0			0
$671$	−4182.05	+	4735.66i	−0.240605	+	0.272456i
$672$		0			0
$673$			577.342i			0.0330682i	0.999863	+	0.0165341i	$0.00526321\pi$
$673$			577.342i			0.0330682i	−0.999863	+	0.0165341i	$0.994737\pi$
$674$		0			0
$675$		−	33396.9i		−	1.90437i
$676$		0			0
$677$		−	7296.39i		−	0.414214i	−0.978318	−	0.207107i	$-0.933595\pi$
$677$		−	7296.39i		−	0.414214i	0.978318	−	0.207107i	$-0.0664049\pi$
$678$		0			0
$679$	−36816.9			−2.08086
$680$		0			0
$681$			5629.88i			0.316795i
$682$		0			0
$683$		−	27059.0i		−	1.51593i	−0.652293	−	0.757967i	$-0.726193\pi$
$683$		−	27059.0i		−	1.51593i	0.652293	−	0.757967i	$-0.273807\pi$
$684$		0			0
$685$	−6533.20			−0.364410
$686$		0			0
$687$			11643.0i			0.646590i
$688$		0			0
$689$		−	2717.20i		−	0.150243i
$690$		0			0
$691$		−	2339.57i		−	0.128801i	−0.997924	−	0.0644005i	$-0.979486\pi$
$691$		−	2339.57i		−	0.128801i	0.997924	−	0.0644005i	$-0.0205135\pi$
$692$		0			0
$693$	−9720.09	−	8583.79i	−0.532808	−	0.470521i
$694$		0			0
$695$	28269.5			1.54291
$696$		0			0
$697$	27422.9			1.49027
$698$		0			0
$699$	−1942.23			−0.105095
$700$		0			0
$701$			12256.2i			0.660358i	0.943918	+	0.330179i	$0.107109\pi$
$701$			12256.2i			0.660358i	−0.943918	+	0.330179i	$0.892891\pi$
$702$		0			0
$703$	26131.2			1.40193
$704$		0			0
$705$	13395.3			0.715598
$706$		0			0
$707$			9070.74i			0.482518i
$708$		0			0
$709$	−26312.8			−1.39379			−0.696896	−	0.717172i	$-0.745436\pi$
$709$	−26312.8			−1.39379			−0.696896	+	0.717172i	$0.745436\pi$
$710$		0			0
$711$	7337.48			0.387028
$712$		0			0
$713$	−9466.32			−0.497218
$714$		0			0
$715$	−19834.2	+	22459.8i	−1.03742	+	1.17475i
$716$		0			0
$717$			5730.86i			0.298498i
$718$		0			0
$719$		−	25795.2i		−	1.33797i	−0.743276	−	0.668985i	$-0.766729\pi$
$719$		−	25795.2i		−	1.33797i	0.743276	−	0.668985i	$-0.233271\pi$
$720$		0			0
$721$		−	11634.9i		−	0.600981i
$722$		0			0
$723$	−235.943			−0.0121367
$724$		0			0
$725$			55031.2i			2.81905i
$726$		0			0
$727$			7230.20i			0.368849i	0.982847	+	0.184425i	$0.0590421\pi$
$727$			7230.20i			0.368849i	−0.982847	+	0.184425i	$0.940958\pi$
$728$		0			0
$729$	−10555.7			−0.536286
$730$		0			0
$731$			4003.63i			0.202571i
$732$		0			0
$733$		−	391.505i		−	0.0197279i	−0.999951	−	0.00986397i	$-0.996860\pi$
$733$		−	391.505i		−	0.0197279i	0.999951	−	0.00986397i	$-0.00313985\pi$
$734$		0			0
$735$		−	4132.07i		−	0.207365i
$736$		0			0
$737$	−3759.64	+	4257.33i	−0.187908	+	0.212783i
$738$		0			0
$739$	8067.58			0.401584			0.200792	−	0.979634i	$-0.435648\pi$
$739$	8067.58			0.401584			0.200792	+	0.979634i	$0.435648\pi$
$740$		0			0
$741$	−11828.7			−0.586423
$742$		0			0
$743$	5919.90			0.292302			0.146151	−	0.989262i	$-0.453312\pi$
$743$	5919.90			0.292302			0.146151	+	0.989262i	$0.453312\pi$
$744$		0			0
$745$		−	44379.3i		−	2.18246i
$746$		0			0
$747$	−23698.9			−1.16077
$748$		0			0
$749$	−21962.9			−1.07144
$750$		0			0
$751$			23795.9i			1.15622i	0.815958	+	0.578112i	$0.196210\pi$
$751$			23795.9i			1.15622i	−0.815958	+	0.578112i	$0.803790\pi$
$752$		0			0
$753$	12011.1			0.581285
$754$		0			0
$755$	3856.79			0.185911
$756$		0			0
$757$	8772.63			0.421198			0.210599	−	0.977573i	$-0.432459\pi$
$757$	8772.63			0.421198			0.210599	+	0.977573i	$0.432459\pi$
$758$		0			0
$759$	2651.89	+	2341.87i	0.126821	+	0.111996i
$760$		0			0
$761$			24086.4i			1.14735i	0.819084	+	0.573673i	$0.194482\pi$
$761$			24086.4i			1.14735i	−0.819084	+	0.573673i	$0.805518\pi$
$762$		0			0
$763$		−	39822.2i		−	1.88946i
$764$		0			0
$765$			30971.0i			1.46374i
$766$		0			0
$767$	−552.784			−0.0260233
$768$		0			0
$769$		−	3920.45i		−	0.183843i	−0.995766	−	0.0919214i	$-0.970699\pi$
$769$		−	3920.45i		−	0.183843i	0.995766	−	0.0919214i	$-0.0293009\pi$
$770$		0			0
$771$		−	3555.57i		−	0.166084i
$772$		0			0
$773$	−37620.2			−1.75046			−0.875229	−	0.483709i	$-0.839289\pi$
$773$	−37620.2			−1.75046			−0.875229	+	0.483709i	$0.839289\pi$
$774$		0			0
$775$			73269.9i			3.39604i
$776$		0			0
$777$		−	18256.3i		−	0.842908i
$778$		0			0
$779$			26660.2i			1.22619i
$780$		0			0
$781$	22029.1	−	24945.3i	1.00930	−	1.14291i
$782$		0			0
$783$	31009.8			1.41532
$784$		0			0
$785$	−6147.50			−0.279508
$786$		0			0
$787$	9586.49			0.434208			0.217104	−	0.976149i	$-0.430339\pi$
$787$	9586.49			0.434208			0.217104	+	0.976149i	$0.430339\pi$
$788$		0			0
$789$			9421.89i			0.425131i
$790$		0			0
$791$	6544.12			0.294162
$792$		0			0
$793$	−7409.78			−0.331815
$794$		0			0
$795$			3758.24i			0.167662i
$796$		0			0
$797$	2797.38			0.124327			0.0621633	−	0.998066i	$-0.480200\pi$
$797$	2797.38			0.124327			0.0621633	+	0.998066i	$0.480200\pi$
$798$		0			0
$799$	−20875.7			−0.924319
$800$		0			0
$801$	13279.3			0.585767
$802$		0			0
$803$	20524.3	−	23241.3i	0.901976	−	1.02138i
$804$		0			0
$805$		−	12266.9i		−	0.537081i
$806$		0			0
$807$		−	2237.70i		−	0.0976093i
$808$		0			0
$809$			18308.0i			0.795644i	0.917463	+	0.397822i	$0.130234\pi$
$809$			18308.0i			0.795644i	−0.917463	+	0.397822i	$0.869766\pi$
$810$		0			0
$811$	23483.2			1.01678			0.508388	−	0.861128i	$-0.330242\pi$
$811$	23483.2			1.01678			0.508388	+	0.861128i	$0.330242\pi$
$812$		0			0
$813$		−	18118.0i		−	0.781582i
$814$		0			0
$815$		−	15929.7i		−	0.684655i
$816$		0			0
$817$	−3892.28			−0.166675
$818$		0			0
$819$		−	15208.8i		−	0.648887i
$820$		0			0
$821$			4309.63i			0.183200i	0.995796	+	0.0916000i	$0.0291981\pi$
$821$			4309.63i			0.183200i	−0.995796	+	0.0916000i	$0.970802\pi$
$822$		0			0
$823$		−	9337.21i		−	0.395473i	−0.980255	−	0.197737i	$-0.936641\pi$
$823$		−	9337.21i		−	0.395473i	0.980255	−	0.197737i	$-0.0633591\pi$
$824$		0			0
$825$	18126.2	−	20525.8i	0.764939	−	0.866200i
$826$		0			0
$827$	−15376.3			−0.646535			−0.323268	−	0.946308i	$-0.604781\pi$
$827$	−15376.3			−0.646535			−0.323268	+	0.946308i	$0.604781\pi$
$828$		0			0
$829$	29696.0			1.24413			0.622065	−	0.782966i	$-0.286294\pi$
$829$	29696.0			1.24413			0.622065	+	0.782966i	$0.286294\pi$
$830$		0			0
$831$	−12401.5			−0.517692
$832$		0			0
$833$			6439.56i			0.267848i
$834$		0			0
$835$	2350.07			0.0973982
$836$		0			0
$837$	41287.2			1.70501
$838$		0			0
$839$		−	29560.4i		−	1.21638i	−0.793793	−	0.608188i	$-0.791897\pi$
$839$		−	29560.4i		−	1.21638i	0.793793	−	0.608188i	$-0.208103\pi$
$840$		0			0
$841$	−26708.7			−1.09511
$842$		0			0
$843$	−10454.5			−0.427130
$844$		0			0
$845$	7029.16			0.286166
$846$		0			0
$847$	−3344.80	−	26835.6i	−0.135689	−	1.08864i
$848$		0			0
$849$			21422.3i			0.865973i
$850$		0			0
$851$		−	9166.44i		−	0.369238i
$852$		0			0
$853$		−	24061.3i		−	0.965819i	−0.875670	−	0.482910i	$-0.839580\pi$
$853$		−	24061.3i		−	0.965819i	0.875670	−	0.482910i	$-0.160420\pi$
$854$		0			0
$855$	−30109.7			−1.20436
$856$		0			0
$857$			2247.42i			0.0895803i	0.998996	+	0.0447901i	$0.0142619\pi$
$857$			2247.42i			0.0895803i	−0.998996	+	0.0447901i	$0.985738\pi$
$858$		0			0
$859$			27776.7i			1.10329i	0.834078	+	0.551647i	$0.186000\pi$
$859$			27776.7i			1.10329i	−0.834078	+	0.551647i	$0.814000\pi$
$860$		0			0
$861$	18625.9			0.737244
$862$		0			0
$863$			37405.1i			1.47542i	0.675119	+	0.737709i	$0.264092\pi$
$863$			37405.1i			1.47542i	−0.675119	+	0.737709i	$0.735908\pi$
$864$		0			0
$865$			32666.0i			1.28402i
$866$		0			0
$867$			11079.0i			0.433981i
$868$		0			0
$869$	11469.6	+	10128.8i	0.447733	+	0.395392i
$870$		0			0
$871$	−6661.35			−0.259140
$872$		0			0
$873$	31700.0			1.22896
$874$		0			0
$875$	−46195.8			−1.78480
$876$		0			0
$877$			1996.92i			0.0768885i	0.999261	+	0.0384443i	$0.0122402\pi$
$877$			1996.92i			0.0768885i	−0.999261	+	0.0384443i	$0.987760\pi$
$878$		0			0
$879$	−16241.1			−0.623205
$880$		0			0
$881$	21494.7			0.821990			0.410995	−	0.911638i	$-0.365181\pi$
$881$	21494.7			0.821990			0.410995	+	0.911638i	$0.365181\pi$
$882$		0			0
$883$			4664.50i			0.177772i	0.996042	+	0.0888862i	$0.0283307\pi$
$883$			4664.50i			0.177772i	−0.996042	+	0.0888862i	$0.971669\pi$
$884$		0			0
$885$	764.572			0.0290405
$886$		0			0
$887$	30507.3			1.15483			0.577415	−	0.816451i	$-0.304061\pi$
$887$	30507.3			1.15483			0.577415	+	0.816451i	$0.304061\pi$
$888$		0			0
$889$	−37071.8			−1.39859
$890$		0			0
$891$	1350.61	+	1192.72i	0.0507825	+	0.0448459i
$892$		0			0
$893$		−	20295.2i		−	0.760528i
$894$		0			0
$895$		−	50934.2i		−	1.90228i
$896$		0			0
$897$			4149.35i			0.154451i
$898$		0			0
$899$	−68032.7			−2.52394
$900$		0			0
$901$		−	5856.98i		−	0.216564i
$902$		0			0
$903$			2719.30i			0.100213i
$904$		0			0
$905$	12646.8			0.464525
$906$		0			0
$907$			52828.4i			1.93400i	0.254777	+	0.967000i	$0.417998\pi$
$907$			52828.4i			1.93400i	−0.254777	+	0.967000i	$0.582002\pi$
$908$		0			0
$909$		−	7810.08i		−	0.284977i
$910$		0			0
$911$		−	22840.9i		−	0.830685i	−0.909665	−	0.415343i	$-0.863662\pi$
$911$		−	22840.9i		−	0.830685i	0.909665	−	0.415343i	$-0.136338\pi$
$912$		0			0
$913$	−37045.1	−	32714.4i	−1.34284	−	1.18586i
$914$		0			0
$915$	10248.7			0.370285
$916$		0			0
$917$	−52754.1			−1.89978
$918$		0			0
$919$	−31840.9			−1.14291			−0.571456	−	0.820633i	$-0.693621\pi$
$919$	−31840.9			−1.14291			−0.571456	+	0.820633i	$0.693621\pi$
$920$		0			0
$921$		−	5070.91i		−	0.181425i
$922$		0			0
$923$	39031.3			1.39191
$924$		0			0
$925$	−70948.8			−2.52193
$926$		0			0
$927$			10017.9i			0.354942i
$928$		0			0
$929$	−32093.2			−1.13342			−0.566709	−	0.823918i	$-0.691784\pi$
$929$	−32093.2			−1.13342			−0.566709	+	0.823918i	$0.691784\pi$
$930$		0			0
$931$	−6260.47			−0.220385
$932$		0			0
$933$	−19060.0			−0.668805
$934$		0			0
$935$	−42753.0	+	48412.5i	−1.49537	+	1.69333i
$936$		0			0
$937$			13617.4i			0.474773i	0.971415	+	0.237387i	$0.0762908\pi$
$937$			13617.4i			0.474773i	−0.971415	+	0.237387i	$0.923709\pi$
$938$		0			0
$939$		−	17814.5i		−	0.619121i
$940$		0			0
$941$		−	91.4083i		−	0.00316666i	−0.999999	−	0.00158333i	$-0.999496\pi$
$941$		−	91.4083i		−	0.00316666i	0.999999	−	0.00158333i	$-0.000503990\pi$
$942$		0			0
$943$	9352.01			0.322952
$944$		0			0
$945$			53501.7i			1.84170i
$946$		0			0
$947$		−	30951.5i		−	1.06208i	−0.847347	−	0.531039i	$-0.821802\pi$
$947$		−	30951.5i		−	1.06208i	0.847347	−	0.531039i	$-0.178198\pi$
$948$		0			0
$949$	36365.1			1.24390
$950$		0			0
$951$			5603.64i			0.191073i
$952$		0			0
$953$		−	51183.2i		−	1.73975i	−0.493269	−	0.869877i	$-0.664198\pi$
$953$		−	51183.2i		−	1.73975i	0.493269	−	0.869877i	$-0.335802\pi$
$954$		0			0
$955$		−	35319.9i		−	1.19678i
$956$		0			0
$957$	19058.6	+	16830.6i	0.643759	+	0.568502i
$958$		0			0
$959$	6915.41			0.232857
$960$		0			0
$961$	−60789.4			−2.04053
$962$		0			0
$963$	18910.5			0.632795
$964$		0			0
$965$		−	41500.4i		−	1.38440i
$966$		0			0
$967$	−18209.8			−0.605571			−0.302785	−	0.953059i	$-0.597917\pi$
$967$	−18209.8			−0.605571			−0.302785	+	0.953059i	$0.597917\pi$
$968$		0			0
$969$	−25497.1			−0.845288
$970$		0			0
$971$		−	33508.5i		−	1.10746i	−0.832698	−	0.553728i	$-0.813205\pi$
$971$		−	33508.5i		−	1.10746i	0.832698	−	0.553728i	$-0.186795\pi$
$972$		0			0
$973$	−29923.4			−0.985919
$974$		0			0
$975$	32116.2			1.05491
$976$		0			0
$977$	40753.4			1.33451			0.667256	−	0.744829i	$-0.267469\pi$
$977$	40753.4			1.33451			0.667256	+	0.744829i	$0.267469\pi$
$978$		0			0
$979$	20757.5	+	18330.9i	0.677644	+	0.598426i
$980$		0			0
$981$			34287.7i			1.11592i
$982$		0			0
$983$			675.886i			0.0219302i	0.999940	+	0.0109651i	$0.00349038\pi$
$983$			675.886i			0.0219302i	−0.999940	+	0.0109651i	$0.996510\pi$
$984$		0			0
$985$		−	39904.0i		−	1.29081i
$986$		0			0
$987$	−14179.0			−0.457266
$988$		0			0
$989$			1365.35i			0.0438986i
$990$		0			0
$991$			22802.9i			0.730936i	0.930824	+	0.365468i	$0.119091\pi$
$991$			22802.9i			0.730936i	−0.930824	+	0.365468i	$0.880909\pi$
$992$		0			0
$993$	−6179.54			−0.197484
$994$		0			0
$995$			77918.2i			2.48259i
$996$		0			0
$997$		−	59937.6i		−	1.90396i	−0.306166	−	0.951978i	$-0.599046\pi$
$997$		−	59937.6i		−	1.90396i	0.306166	−	0.951978i	$-0.400954\pi$
$998$		0			0
$999$			39979.2i			1.26615i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	704.4.e.g.703.19		36
4.3	odd	2	inner	704.4.e.g.703.4		36
8.3	odd	2		352.4.e.a.351.33	yes	36
8.5	even	2		352.4.e.a.351.18	yes	36
11.10	odd	2	inner	704.4.e.g.703.3		36
44.43	even	2	inner	704.4.e.g.703.20		36
88.21	odd	2		352.4.e.a.351.34	yes	36
88.43	even	2		352.4.e.a.351.17	✓	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
352.4.e.a.351.17	✓	36	88.43	even	2
352.4.e.a.351.18	yes	36	8.5	even	2
352.4.e.a.351.33	yes	36	8.3	odd	2
352.4.e.a.351.34	yes	36	88.21	odd	2
704.4.e.g.703.3		36	11.10	odd	2	inner
704.4.e.g.703.4		36	4.3	odd	2	inner
704.4.e.g.703.19		36	1.1	even	1	trivial
704.4.e.g.703.20		36	44.43	even	2	inner

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:	\( N \)	\(=\)	\( 704 = 2^{6} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	704.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.08315i q^{3} +19.1950 q^{5} -20.3180 q^{7} +17.4942 q^{9} +(27.3461 + 24.1493i) q^{11} +42.7879i q^{13} -59.1812i q^{15} +92.2301i q^{17} -89.6650 q^{19} +62.6434i q^{21} +31.4531i q^{23} +243.449 q^{25} -137.182i q^{27} +226.048i q^{29} +300.966i q^{31} +(74.4559 - 84.3123i) q^{33} -390.004 q^{35} -291.432 q^{37} +131.921 q^{39} -297.331i q^{41} +43.4091 q^{43} +335.801 q^{45} +226.344i q^{47} +69.8206 q^{49} +284.359 q^{51} -63.5040 q^{53} +(524.910 + 463.547i) q^{55} +276.451i q^{57} +12.9192i q^{59} +173.175i q^{61} -355.447 q^{63} +821.315i q^{65} +155.683i q^{67} +96.9748 q^{69} -912.204i q^{71} -849.892i q^{73} -750.591i q^{75} +(-555.619 - 490.665i) q^{77} +419.424 q^{79} +49.3895 q^{81} -1354.67 q^{83} +1770.36i q^{85} +696.940 q^{87} +759.067 q^{89} -869.364i q^{91} +927.923 q^{93} -1721.12 q^{95} +1812.03 q^{97} +(478.398 + 422.472i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 364 q^{9} + 1236 q^{25} + 592 q^{33} - 984 q^{45} + 2372 q^{49} - 376 q^{53} + 2824 q^{69} + 2592 q^{77} + 6756 q^{81} + 512 q^{89} + 872 q^{93} + 3904 q^{97}+O(q^{100}) \) 36 * q - 364 * q^9 + 1236 * q^25 + 592 * q^33 - 984 * q^45 + 2372 * q^49 - 376 * q^53 + 2824 * q^69 + 2592 * q^77 + 6756 * q^81 + 512 * q^89 + 872 * q^93 + 3904 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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