LMFDB - Embedded newform 700.4.e.a.449.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [700,4,Mod(449,700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(700, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("700.449");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 700 = 2^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	700.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:	$41.3013370040$
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:	$ 1 $
Twist minimal:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	700.449
Dual form			700.4.e.a.449.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-10.0000i q^{3} +7.00000i q^{7} -73.0000 q^{9} +O(q^{10})$	$q-10.0000i q^{3} +7.00000i q^{7} -73.0000 q^{9} -40.0000 q^{11} -12.0000i q^{13} +58.0000i q^{17} -26.0000 q^{19} +70.0000 q^{21} -64.0000i q^{23} +460.000i q^{27} +62.0000 q^{29} +252.000 q^{31} +400.000i q^{33} -26.0000i q^{37} -120.000 q^{39} +6.00000 q^{41} +416.000i q^{43} +396.000i q^{47} -49.0000 q^{49} +580.000 q^{51} -450.000i q^{53} +260.000i q^{57} -274.000 q^{59} -576.000 q^{61} -511.000i q^{63} +476.000i q^{67} -640.000 q^{69} -448.000 q^{71} -158.000i q^{73} -280.000i q^{77} +936.000 q^{79} +2629.00 q^{81} +530.000i q^{83} -620.000i q^{87} +390.000 q^{89} +84.0000 q^{91} -2520.00i q^{93} -214.000i q^{97} +2920.00 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 146 q^{9}+O(q^{10}) $ 2 * q - 146 * q^9	$ 2 q - 146 q^{9} - 80 q^{11} - 52 q^{19} + 140 q^{21} + 124 q^{29} + 504 q^{31} - 240 q^{39} + 12 q^{41} - 98 q^{49} + 1160 q^{51} - 548 q^{59} - 1152 q^{61} - 1280 q^{69} - 896 q^{71} + 1872 q^{79} + 5258 q^{81} + 780 q^{89} + 168 q^{91} + 5840 q^{99}+O(q^{100}) $ 2 * q - 146 * q^9 - 80 * q^11 - 52 * q^19 + 140 * q^21 + 124 * q^29 + 504 * q^31 - 240 * q^39 + 12 * q^41 - 98 * q^49 + 1160 * q^51 - 548 * q^59 - 1152 * q^61 - 1280 * q^69 - 896 * q^71 + 1872 * q^79 + 5258 * q^81 + 780 * q^89 + 168 * q^91 + 5840 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/700\mathbb{Z}\right)^\times$.

$n$	$101$	$351$	$477$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 10.0000i	− 1.92450i	−0.272166	−	0.962250i	$-0.587740\pi$
$3$	− 10.0000i	− 1.92450i	0.272166	−	0.962250i	$-0.412260\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	7.00000i	0.377964i
$7$	7.00000i	0.377964i
$8$	0	0
$9$	−73.0000	−2.70370
$10$	0	0
$11$	−40.0000	−1.09640	−0.548202	−	0.836346i	$-0.684688\pi$
$11$	−40.0000	−1.09640	−0.548202	+	0.836346i	$0.684688\pi$
$12$	0	0
$13$	− 12.0000i	− 0.256015i	−0.991773	−	0.128008i	$-0.959142\pi$
$13$	− 12.0000i	− 0.256015i	0.991773	−	0.128008i	$-0.0408582\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	58.0000i	0.827474i	0.910396	+	0.413737i	$0.135777\pi$
$17$	58.0000i	0.827474i	−0.910396	+	0.413737i	$0.864223\pi$
$18$	0	0
$19$	−26.0000	−0.313937	−0.156969	−	0.987604i	$-0.550172\pi$
$19$	−26.0000	−0.313937	−0.156969	+	0.987604i	$0.550172\pi$
$20$	0	0
$21$	70.0000	0.727393
$22$	0	0
$23$	− 64.0000i	− 0.580214i	−0.956994	−	0.290107i	$-0.906309\pi$
$23$	− 64.0000i	− 0.580214i	0.956994	−	0.290107i	$-0.0936909\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	460.000i	3.27878i
$28$	0	0
$29$	62.0000	0.397004	0.198502	−	0.980101i	$-0.436392\pi$
$29$	62.0000	0.397004	0.198502	+	0.980101i	$0.436392\pi$
$30$	0	0
$31$	252.000	1.46002	0.730009	−	0.683438i	$-0.239516\pi$
$31$	252.000	1.46002	0.730009	+	0.683438i	$0.239516\pi$
$32$	0	0
$33$	400.000i	2.11003i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 26.0000i	− 0.115524i	−0.998330	−	0.0577618i	$-0.981604\pi$
$37$	− 26.0000i	− 0.115524i	0.998330	−	0.0577618i	$-0.0183964\pi$
$38$	0	0
$39$	−120.000	−0.492702
$40$	0	0
$41$	6.00000	0.0228547	0.0114273	−	0.999935i	$-0.496362\pi$
$41$	6.00000	0.0228547	0.0114273	+	0.999935i	$0.496362\pi$
$42$	0	0
$43$	416.000i	1.47534i	0.675164	+	0.737668i	$0.264073\pi$
$43$	416.000i	1.47534i	−0.675164	+	0.737668i	$0.735927\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	396.000i	1.22899i	0.788921	+	0.614495i	$0.210640\pi$
$47$	396.000i	1.22899i	−0.788921	+	0.614495i	$0.789360\pi$
$48$	0	0
$49$	−49.0000	−0.142857
$50$	0	0
$51$	580.000	1.59248
$52$	0	0
$53$	− 450.000i	− 1.16627i	−0.812376	−	0.583134i	$-0.801826\pi$
$53$	− 450.000i	− 1.16627i	0.812376	−	0.583134i	$-0.198174\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	260.000i	0.604173i
$58$	0	0
$59$	−274.000	−0.604606	−0.302303	−	0.953212i	$-0.597755\pi$
$59$	−274.000	−0.604606	−0.302303	+	0.953212i	$0.597755\pi$
$60$	0	0
$61$	−576.000	−1.20900	−0.604502	−	0.796604i	$-0.706628\pi$
$61$	−576.000	−1.20900	−0.604502	+	0.796604i	$0.706628\pi$
$62$	0	0
$63$	− 511.000i	− 1.02190i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	476.000i	0.867950i	0.900925	+	0.433975i	$0.142889\pi$
$67$	476.000i	0.867950i	−0.900925	+	0.433975i	$0.857111\pi$
$68$	0	0
$69$	−640.000	−1.11662
$70$	0	0
$71$	−448.000	−0.748843	−0.374421	−	0.927259i	$-0.622159\pi$
$71$	−448.000	−0.748843	−0.374421	+	0.927259i	$0.622159\pi$
$72$	0	0
$73$	− 158.000i	− 0.253322i	−0.991946	−	0.126661i	$-0.959574\pi$
$73$	− 158.000i	− 0.253322i	0.991946	−	0.126661i	$-0.0404260\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 280.000i	− 0.414402i
$78$	0	0
$79$	936.000	1.33302	0.666508	−	0.745498i	$-0.267788\pi$
$79$	936.000	1.33302	0.666508	+	0.745498i	$0.267788\pi$
$80$	0	0
$81$	2629.00	3.60631
$82$	0	0
$83$	530.000i	0.700904i	0.936581	+	0.350452i	$0.113972\pi$
$83$	530.000i	0.700904i	−0.936581	+	0.350452i	$0.886028\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 620.000i	− 0.764034i
$88$	0	0
$89$	390.000	0.464493	0.232247	−	0.972657i	$-0.425392\pi$
$89$	390.000	0.464493	0.232247	+	0.972657i	$0.425392\pi$
$90$	0	0
$91$	84.0000	0.0967648
$92$	0	0
$93$	− 2520.00i	− 2.80980i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 214.000i	− 0.224004i	−0.993708	−	0.112002i	$-0.964274\pi$
$97$	− 214.000i	− 0.224004i	0.993708	−	0.112002i	$-0.0357263\pi$
$98$	0	0
$99$	2920.00	2.96435
$100$	0	0
$101$	1432.00	1.41079	0.705393	−	0.708817i	$-0.250771\pi$
$101$	1432.00	1.41079	0.705393	+	0.708817i	$0.250771\pi$
$102$	0	0
$103$	764.000i	0.730866i	0.930838	+	0.365433i	$0.119079\pi$
$103$	764.000i	0.730866i	−0.930838	+	0.365433i	$0.880921\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 324.000i	− 0.292731i	−0.989231	−	0.146366i	$-0.953242\pi$
$107$	− 324.000i	− 0.292731i	0.989231	−	0.146366i	$-0.0467576\pi$
$108$	0	0
$109$	1334.00	1.17224	0.586119	−	0.810225i	$-0.300655\pi$
$109$	1334.00	1.17224	0.586119	+	0.810225i	$0.300655\pi$
$110$	0	0
$111$	−260.000	−0.222325
$112$	0	0
$113$	1798.00i	1.49683i	0.663232	+	0.748414i	$0.269184\pi$
$113$	1798.00i	1.49683i	−0.663232	+	0.748414i	$0.730816\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	876.000i	0.692190i
$118$	0	0
$119$	−406.000	−0.312756
$120$	0	0
$121$	269.000	0.202104
$122$	0	0
$123$	− 60.0000i	− 0.0439839i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	384.000i	0.268303i	0.990961	+	0.134152i	$0.0428309\pi$
$127$	384.000i	0.268303i	−0.990961	+	0.134152i	$0.957169\pi$
$128$	0	0
$129$	4160.00	2.83928
$130$	0	0
$131$	−1814.00	−1.20985	−0.604923	−	0.796284i	$-0.706796\pi$
$131$	−1814.00	−1.20985	−0.604923	+	0.796284i	$0.706796\pi$
$132$	0	0
$133$	− 182.000i	− 0.118657i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1666.00i	− 1.03895i	−0.854486	−	0.519474i	$-0.826128\pi$
$137$	− 1666.00i	− 1.03895i	0.854486	−	0.519474i	$-0.173872\pi$
$138$	0	0
$139$	−1126.00	−0.687094	−0.343547	−	0.939135i	$-0.611628\pi$
$139$	−1126.00	−0.687094	−0.343547	+	0.939135i	$0.611628\pi$
$140$	0	0
$141$	3960.00	2.36519
$142$	0	0
$143$	480.000i	0.280697i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	490.000i	0.274929i
$148$	0	0
$149$	−2694.00	−1.48122	−0.740608	−	0.671938i	$-0.765462\pi$
$149$	−2694.00	−1.48122	−0.740608	+	0.671938i	$0.765462\pi$
$150$	0	0
$151$	−2648.00	−1.42709	−0.713547	−	0.700607i	$-0.752912\pi$
$151$	−2648.00	−1.42709	−0.713547	+	0.700607i	$0.752912\pi$
$152$	0	0
$153$	− 4234.00i	− 2.23725i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	556.000i	0.282635i	0.989964	+	0.141317i	$0.0451338\pi$
$157$	556.000i	0.282635i	−0.989964	+	0.141317i	$0.954866\pi$
$158$	0	0
$159$	−4500.00	−2.24449
$160$	0	0
$161$	448.000	0.219300
$162$	0	0
$163$	− 328.000i	− 0.157613i	−0.996890	−	0.0788066i	$-0.974889\pi$
$163$	− 328.000i	− 0.157613i	0.996890	−	0.0788066i	$-0.0251109\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4268.00i	1.97765i	0.149077	+	0.988826i	$0.452370\pi$
$167$	4268.00i	1.97765i	−0.149077	+	0.988826i	$0.547630\pi$
$168$	0	0
$169$	2053.00	0.934456
$170$	0	0
$171$	1898.00	0.848793
$172$	0	0
$173$	− 3476.00i	− 1.52760i	−0.645451	−	0.763802i	$-0.723331\pi$
$173$	− 3476.00i	− 1.52760i	0.645451	−	0.763802i	$-0.276669\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2740.00i	1.16357i
$178$	0	0
$179$	−2268.00	−0.947029	−0.473515	−	0.880786i	$-0.657015\pi$
$179$	−2268.00	−0.947029	−0.473515	+	0.880786i	$0.657015\pi$
$180$	0	0
$181$	−276.000	−0.113342	−0.0566710	−	0.998393i	$-0.518049\pi$
$181$	−276.000	−0.113342	−0.0566710	+	0.998393i	$0.518049\pi$
$182$	0	0
$183$	5760.00i	2.32673i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 2320.00i	− 0.907247i
$188$	0	0
$189$	−3220.00	−1.23926
$190$	0	0
$191$	−3000.00	−1.13650	−0.568252	−	0.822854i	$-0.692380\pi$
$191$	−3000.00	−1.13650	−0.568252	+	0.822854i	$0.692380\pi$
$192$	0	0
$193$	3278.00i	1.22257i	0.791411	+	0.611284i	$0.209347\pi$
$193$	3278.00i	1.22257i	−0.791411	+	0.611284i	$0.790653\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2362.00i	0.854241i	0.904195	+	0.427121i	$0.140472\pi$
$197$	2362.00i	0.854241i	−0.904195	+	0.427121i	$0.859528\pi$
$198$	0	0
$199$	1036.00	0.369046	0.184523	−	0.982828i	$-0.440926\pi$
$199$	1036.00	0.369046	0.184523	+	0.982828i	$0.440926\pi$
$200$	0	0
$201$	4760.00	1.67037
$202$	0	0
$203$	434.000i	0.150053i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	4672.00i	1.56873i
$208$	0	0
$209$	1040.00	0.344202
$210$	0	0
$211$	3524.00	1.14977	0.574887	−	0.818233i	$-0.305046\pi$
$211$	3524.00	1.14977	0.574887	+	0.818233i	$0.305046\pi$
$212$	0	0
$213$	4480.00i	1.44115i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1764.00i	0.551835i
$218$	0	0
$219$	−1580.00	−0.487518
$220$	0	0
$221$	696.000	0.211846
$222$	0	0
$223$	− 1336.00i	− 0.401189i	−0.979674	−	0.200595i	$-0.935713\pi$
$223$	− 1336.00i	− 0.401189i	0.979674	−	0.200595i	$-0.0642874\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 1290.00i	− 0.377182i	−0.982056	−	0.188591i	$-0.939608\pi$
$227$	− 1290.00i	− 0.377182i	0.982056	−	0.188591i	$-0.0603920\pi$
$228$	0	0
$229$	−5524.00	−1.59404	−0.797022	−	0.603950i	$-0.793593\pi$
$229$	−5524.00	−1.59404	−0.797022	+	0.603950i	$0.793593\pi$
$230$	0	0
$231$	−2800.00	−0.797517
$232$	0	0
$233$	6314.00i	1.77530i	0.460523	+	0.887648i	$0.347662\pi$
$233$	6314.00i	1.77530i	−0.460523	+	0.887648i	$0.652338\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 9360.00i	− 2.56539i
$238$	0	0
$239$	3960.00	1.07176	0.535881	−	0.844294i	$-0.319980\pi$
$239$	3960.00	1.07176	0.535881	+	0.844294i	$0.319980\pi$
$240$	0	0
$241$	−7018.00	−1.87581	−0.937903	−	0.346898i	$-0.887235\pi$
$241$	−7018.00	−1.87581	−0.937903	+	0.346898i	$0.887235\pi$
$242$	0	0
$243$	− 13870.0i	− 3.66157i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	312.000i	0.0803728i
$248$	0	0
$249$	5300.00	1.34889
$250$	0	0
$251$	−2394.00	−0.602024	−0.301012	−	0.953620i	$-0.597324\pi$
$251$	−2394.00	−0.602024	−0.301012	+	0.953620i	$0.597324\pi$
$252$	0	0
$253$	2560.00i	0.636149i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2766.00i	0.671355i	0.941977	+	0.335678i	$0.108965\pi$
$257$	2766.00i	0.671355i	−0.941977	+	0.335678i	$0.891035\pi$
$258$	0	0
$259$	182.000	0.0436638
$260$	0	0
$261$	−4526.00	−1.07338
$262$	0	0
$263$	7968.00i	1.86817i	0.357055	+	0.934084i	$0.383781\pi$
$263$	7968.00i	1.86817i	−0.357055	+	0.934084i	$0.616219\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 3900.00i	− 0.893918i
$268$	0	0
$269$	2900.00	0.657309	0.328654	−	0.944450i	$-0.393405\pi$
$269$	2900.00	0.657309	0.328654	+	0.944450i	$0.393405\pi$
$270$	0	0
$271$	2640.00	0.591766	0.295883	−	0.955224i	$-0.404386\pi$
$271$	2640.00	0.591766	0.295883	+	0.955224i	$0.404386\pi$
$272$	0	0
$273$	− 840.000i	− 0.186224i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1522.00i	0.330138i	0.986282	+	0.165069i	$0.0527846\pi$
$277$	1522.00i	0.330138i	−0.986282	+	0.165069i	$0.947215\pi$
$278$	0	0
$279$	−18396.0	−3.94745
$280$	0	0
$281$	−4534.00	−0.962547	−0.481274	−	0.876570i	$-0.659826\pi$
$281$	−4534.00	−0.962547	−0.481274	+	0.876570i	$0.659826\pi$
$282$	0	0
$283$	4834.00i	1.01538i	0.861541	+	0.507688i	$0.169500\pi$
$283$	4834.00i	1.01538i	−0.861541	+	0.507688i	$0.830500\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	42.0000i	0.00863826i
$288$	0	0
$289$	1549.00	0.315286
$290$	0	0
$291$	−2140.00	−0.431096
$292$	0	0
$293$	− 4656.00i	− 0.928350i	−0.885744	−	0.464175i	$-0.846351\pi$
$293$	− 4656.00i	− 0.928350i	0.885744	−	0.464175i	$-0.153649\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 18400.0i	− 3.59487i
$298$	0	0
$299$	−768.000	−0.148544
$300$	0	0
$301$	−2912.00	−0.557624
$302$	0	0
$303$	− 14320.0i	− 2.71506i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	7238.00i	1.34558i	0.739831	+	0.672792i	$0.234905\pi$
$307$	7238.00i	1.34558i	−0.739831	+	0.672792i	$0.765095\pi$
$308$	0	0
$309$	7640.00	1.40655
$310$	0	0
$311$	1096.00	0.199834	0.0999171	−	0.994996i	$-0.468142\pi$
$311$	1096.00	0.199834	0.0999171	+	0.994996i	$0.468142\pi$
$312$	0	0
$313$	− 3818.00i	− 0.689476i	−0.938699	−	0.344738i	$-0.887968\pi$
$313$	− 3818.00i	− 0.689476i	0.938699	−	0.344738i	$-0.112032\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 1998.00i	− 0.354003i	−0.984211	−	0.177001i	$-0.943360\pi$
$317$	− 1998.00i	− 0.354003i	0.984211	−	0.177001i	$-0.0566397\pi$
$318$	0	0
$319$	−2480.00	−0.435277
$320$	0	0
$321$	−3240.00	−0.563362
$322$	0	0
$323$	− 1508.00i	− 0.259775i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 13340.0i	− 2.25597i
$328$	0	0
$329$	−2772.00	−0.464515
$330$	0	0
$331$	−7936.00	−1.31783	−0.658915	−	0.752217i	$-0.728984\pi$
$331$	−7936.00	−1.31783	−0.658915	+	0.752217i	$0.728984\pi$
$332$	0	0
$333$	1898.00i	0.312342i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 2766.00i	− 0.447103i	−0.974692	−	0.223551i	$-0.928235\pi$
$337$	− 2766.00i	− 0.447103i	0.974692	−	0.223551i	$-0.0717650\pi$
$338$	0	0
$339$	17980.0	2.88065
$340$	0	0
$341$	−10080.0	−1.60077
$342$	0	0
$343$	− 343.000i	− 0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 8352.00i	− 1.29210i	−0.763295	−	0.646050i	$-0.776420\pi$
$347$	− 8352.00i	− 1.29210i	0.763295	−	0.646050i	$-0.223580\pi$
$348$	0	0
$349$	5924.00	0.908609	0.454304	−	0.890847i	$-0.349888\pi$
$349$	5924.00	0.908609	0.454304	+	0.890847i	$0.349888\pi$
$350$	0	0
$351$	5520.00	0.839418
$352$	0	0
$353$	2226.00i	0.335632i	0.985818	+	0.167816i	$0.0536714\pi$
$353$	2226.00i	0.335632i	−0.985818	+	0.167816i	$0.946329\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4060.00i	0.601899i
$358$	0	0
$359$	−3880.00	−0.570414	−0.285207	−	0.958466i	$-0.592062\pi$
$359$	−3880.00	−0.570414	−0.285207	+	0.958466i	$0.592062\pi$
$360$	0	0
$361$	−6183.00	−0.901443
$362$	0	0
$363$	− 2690.00i	− 0.388949i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	2584.00i	0.367531i	0.982970	+	0.183765i	$0.0588286\pi$
$367$	2584.00i	0.367531i	−0.982970	+	0.183765i	$0.941171\pi$
$368$	0	0
$369$	−438.000	−0.0617923
$370$	0	0
$371$	3150.00	0.440808
$372$	0	0
$373$	10534.0i	1.46228i	0.682228	+	0.731139i	$0.261011\pi$
$373$	10534.0i	1.46228i	−0.682228	+	0.731139i	$0.738989\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 744.000i	− 0.101639i
$378$	0	0
$379$	4472.00	0.606098	0.303049	−	0.952975i	$-0.401995\pi$
$379$	4472.00	0.606098	0.303049	+	0.952975i	$0.401995\pi$
$380$	0	0
$381$	3840.00	0.516350
$382$	0	0
$383$	2468.00i	0.329266i	0.986355	+	0.164633i	$0.0526440\pi$
$383$	2468.00i	0.329266i	−0.986355	+	0.164633i	$0.947356\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 30368.0i	− 3.98887i
$388$	0	0
$389$	1046.00	0.136335	0.0681675	−	0.997674i	$-0.478285\pi$
$389$	1046.00	0.136335	0.0681675	+	0.997674i	$0.478285\pi$
$390$	0	0
$391$	3712.00	0.480112
$392$	0	0
$393$	18140.0i	2.32835i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 2124.00i	− 0.268515i	−0.990946	−	0.134258i	$-0.957135\pi$
$397$	− 2124.00i	− 0.268515i	0.990946	−	0.134258i	$-0.0428649\pi$
$398$	0	0
$399$	−1820.00	−0.228356
$400$	0	0
$401$	11598.0	1.44433	0.722165	−	0.691721i	$-0.243147\pi$
$401$	11598.0	1.44433	0.722165	+	0.691721i	$0.243147\pi$
$402$	0	0
$403$	− 3024.00i	− 0.373787i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1040.00i	0.126661i
$408$	0	0
$409$	2770.00	0.334884	0.167442	−	0.985882i	$-0.446449\pi$
$409$	2770.00	0.334884	0.167442	+	0.985882i	$0.446449\pi$
$410$	0	0
$411$	−16660.0	−1.99946
$412$	0	0
$413$	− 1918.00i	− 0.228520i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	11260.0i	1.32231i
$418$	0	0
$419$	−9438.00	−1.10042	−0.550211	−	0.835026i	$-0.685453\pi$
$419$	−9438.00	−1.10042	−0.550211	+	0.835026i	$0.685453\pi$
$420$	0	0
$421$	5550.00	0.642495	0.321248	−	0.946995i	$-0.395898\pi$
$421$	5550.00	0.642495	0.321248	+	0.946995i	$0.395898\pi$
$422$	0	0
$423$	− 28908.0i	− 3.32283i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 4032.00i	− 0.456961i
$428$	0	0
$429$	4800.00	0.540201
$430$	0	0
$431$	−3000.00	−0.335278	−0.167639	−	0.985848i	$-0.553614\pi$
$431$	−3000.00	−0.335278	−0.167639	+	0.985848i	$0.553614\pi$
$432$	0	0
$433$	12926.0i	1.43460i	0.696762	+	0.717302i	$0.254623\pi$
$433$	12926.0i	1.43460i	−0.696762	+	0.717302i	$0.745377\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1664.00i	0.182151i
$438$	0	0
$439$	408.000	0.0443571	0.0221786	−	0.999754i	$-0.492940\pi$
$439$	408.000	0.0443571	0.0221786	+	0.999754i	$0.492940\pi$
$440$	0	0
$441$	3577.00	0.386243
$442$	0	0
$443$	− 14452.0i	− 1.54997i	−0.631982	−	0.774983i	$-0.717758\pi$
$443$	− 14452.0i	− 1.54997i	0.631982	−	0.774983i	$-0.282242\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	26940.0i	2.85060i
$448$	0	0
$449$	−10258.0	−1.07818	−0.539092	−	0.842247i	$-0.681233\pi$
$449$	−10258.0	−1.07818	−0.539092	+	0.842247i	$0.681233\pi$
$450$	0	0
$451$	−240.000	−0.0250580
$452$	0	0
$453$	26480.0i	2.74644i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5498.00i	0.562769i	0.959595	+	0.281385i	$0.0907937\pi$
$457$	5498.00i	0.562769i	−0.959595	+	0.281385i	$0.909206\pi$
$458$	0	0
$459$	−26680.0	−2.71311
$460$	0	0
$461$	−16316.0	−1.64840	−0.824199	−	0.566300i	$-0.808375\pi$
$461$	−16316.0	−1.64840	−0.824199	+	0.566300i	$0.808375\pi$
$462$	0	0
$463$	8944.00i	0.897760i	0.893592	+	0.448880i	$0.148177\pi$
$463$	8944.00i	0.897760i	−0.893592	+	0.448880i	$0.851823\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	9422.00i	0.933615i	0.884359	+	0.466807i	$0.154596\pi$
$467$	9422.00i	0.933615i	−0.884359	+	0.466807i	$0.845404\pi$
$468$	0	0
$469$	−3332.00	−0.328054
$470$	0	0
$471$	5560.00	0.543931
$472$	0	0
$473$	− 16640.0i	− 1.61756i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	32850.0i	3.15325i
$478$	0	0
$479$	−13820.0	−1.31827	−0.659136	−	0.752024i	$-0.729078\pi$
$479$	−13820.0	−1.31827	−0.659136	+	0.752024i	$0.729078\pi$
$480$	0	0
$481$	−312.000	−0.0295758
$482$	0	0
$483$	− 4480.00i	− 0.422044i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	13264.0i	1.23419i	0.786890	+	0.617094i	$0.211690\pi$
$487$	13264.0i	1.23419i	−0.786890	+	0.617094i	$0.788310\pi$
$488$	0	0
$489$	−3280.00	−0.303327
$490$	0	0
$491$	−5940.00	−0.545964	−0.272982	−	0.962019i	$-0.588010\pi$
$491$	−5940.00	−0.545964	−0.272982	+	0.962019i	$0.588010\pi$
$492$	0	0
$493$	3596.00i	0.328511i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 3136.00i	− 0.283036i
$498$	0	0
$499$	8252.00	0.740301	0.370151	−	0.928972i	$-0.379306\pi$
$499$	8252.00	0.740301	0.370151	+	0.928972i	$0.379306\pi$
$500$	0	0
$501$	42680.0	3.80599
$502$	0	0
$503$	4704.00i	0.416980i	0.978024	+	0.208490i	$0.0668549\pi$
$503$	4704.00i	0.416980i	−0.978024	+	0.208490i	$0.933145\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 20530.0i	− 1.79836i
$508$	0	0
$509$	10788.0	0.939430	0.469715	−	0.882818i	$-0.344357\pi$
$509$	10788.0	0.939430	0.469715	+	0.882818i	$0.344357\pi$
$510$	0	0
$511$	1106.00	0.0957467
$512$	0	0
$513$	− 11960.0i	− 1.02933i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 15840.0i	− 1.34747i
$518$	0	0
$519$	−34760.0	−2.93987
$520$	0	0
$521$	−14586.0	−1.22653	−0.613267	−	0.789876i	$-0.710145\pi$
$521$	−14586.0	−1.22653	−0.613267	+	0.789876i	$0.710145\pi$
$522$	0	0
$523$	26.0000i	0.00217381i	0.999999	+	0.00108690i	$0.000345972\pi$
$523$	26.0000i	0.00217381i	−0.999999	+	0.00108690i	$0.999654\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	14616.0i	1.20813i
$528$	0	0
$529$	8071.00	0.663352
$530$	0	0
$531$	20002.0	1.63468
$532$	0	0
$533$	− 72.0000i	− 0.00585116i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	22680.0i	1.82256i
$538$	0	0
$539$	1960.00	0.156629
$540$	0	0
$541$	11214.0	0.891178	0.445589	−	0.895238i	$-0.352994\pi$
$541$	11214.0	0.891178	0.445589	+	0.895238i	$0.352994\pi$
$542$	0	0
$543$	2760.00i	0.218127i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	5424.00i	0.423973i	0.977273	+	0.211987i	$0.0679934\pi$
$547$	5424.00i	0.423973i	−0.977273	+	0.211987i	$0.932007\pi$
$548$	0	0
$549$	42048.0	3.26879
$550$	0	0
$551$	−1612.00	−0.124634
$552$	0	0
$553$	6552.00i	0.503833i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	17618.0i	1.34021i	0.742265	+	0.670106i	$0.233752\pi$
$557$	17618.0i	1.34021i	−0.742265	+	0.670106i	$0.766248\pi$
$558$	0	0
$559$	4992.00	0.377709
$560$	0	0
$561$	−23200.0	−1.74600
$562$	0	0
$563$	− 3562.00i	− 0.266644i	−0.991073	−	0.133322i	$-0.957436\pi$
$563$	− 3562.00i	− 0.266644i	0.991073	−	0.133322i	$-0.0425644\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	18403.0i	1.36306i
$568$	0	0
$569$	−2838.00	−0.209095	−0.104548	−	0.994520i	$-0.533339\pi$
$569$	−2838.00	−0.209095	−0.104548	+	0.994520i	$0.533339\pi$
$570$	0	0
$571$	−360.000	−0.0263845	−0.0131922	−	0.999913i	$-0.504199\pi$
$571$	−360.000	−0.0263845	−0.0131922	+	0.999913i	$0.504199\pi$
$572$	0	0
$573$	30000.0i	2.18720i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 22018.0i	− 1.58860i	−0.607527	−	0.794299i	$-0.707838\pi$
$577$	− 22018.0i	− 1.58860i	0.607527	−	0.794299i	$-0.292162\pi$
$578$	0	0
$579$	32780.0	2.35283
$580$	0	0
$581$	−3710.00	−0.264917
$582$	0	0
$583$	18000.0i	1.27870i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 1454.00i	− 0.102237i	−0.998693	−	0.0511184i	$-0.983721\pi$
$587$	− 1454.00i	− 0.102237i	0.998693	−	0.0511184i	$-0.0162786\pi$
$588$	0	0
$589$	−6552.00	−0.458354
$590$	0	0
$591$	23620.0	1.64399
$592$	0	0
$593$	13818.0i	0.956892i	0.878117	+	0.478446i	$0.158800\pi$
$593$	13818.0i	0.956892i	−0.878117	+	0.478446i	$0.841200\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 10360.0i	− 0.710229i
$598$	0	0
$599$	−6696.00	−0.456746	−0.228373	−	0.973574i	$-0.573341\pi$
$599$	−6696.00	−0.456746	−0.228373	+	0.973574i	$0.573341\pi$
$600$	0	0
$601$	10010.0	0.679395	0.339698	−	0.940535i	$-0.389675\pi$
$601$	10010.0	0.679395	0.339698	+	0.940535i	$0.389675\pi$
$602$	0	0
$603$	− 34748.0i	− 2.34668i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 2880.00i	− 0.192579i	−0.995353	−	0.0962896i	$-0.969303\pi$
$607$	− 2880.00i	− 0.192579i	0.995353	−	0.0962896i	$-0.0306975\pi$
$608$	0	0
$609$	4340.00	0.288778
$610$	0	0
$611$	4752.00	0.314640
$612$	0	0
$613$	6522.00i	0.429724i	0.976644	+	0.214862i	$0.0689303\pi$
$613$	6522.00i	0.429724i	−0.976644	+	0.214862i	$0.931070\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 6614.00i	− 0.431555i	−0.976443	−	0.215778i	$-0.930771\pi$
$617$	− 6614.00i	− 0.431555i	0.976443	−	0.215778i	$-0.0692286\pi$
$618$	0	0
$619$	−5266.00	−0.341936	−0.170968	−	0.985277i	$-0.554689\pi$
$619$	−5266.00	−0.341936	−0.170968	+	0.985277i	$0.554689\pi$
$620$	0	0
$621$	29440.0	1.90239
$622$	0	0
$623$	2730.00i	0.175562i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 10400.0i	− 0.662418i
$628$	0	0
$629$	1508.00	0.0955928
$630$	0	0
$631$	3344.00	0.210971	0.105485	−	0.994421i	$-0.466360\pi$
$631$	3344.00	0.210971	0.105485	+	0.994421i	$0.466360\pi$
$632$	0	0
$633$	− 35240.0i	− 2.21274i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	588.000i	0.0365736i
$638$	0	0
$639$	32704.0	2.02465
$640$	0	0
$641$	−4882.00	−0.300823	−0.150411	−	0.988623i	$-0.548060\pi$
$641$	−4882.00	−0.300823	−0.150411	+	0.988623i	$0.548060\pi$
$642$	0	0
$643$	− 15898.0i	− 0.975048i	−0.873110	−	0.487524i	$-0.837900\pi$
$643$	− 15898.0i	− 0.975048i	0.873110	−	0.487524i	$-0.162100\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 6132.00i	− 0.372602i	−0.982493	−	0.186301i	$-0.940350\pi$
$647$	− 6132.00i	− 0.372602i	0.982493	−	0.186301i	$-0.0596500\pi$
$648$	0	0
$649$	10960.0	0.662893
$650$	0	0
$651$	17640.0	1.06201
$652$	0	0
$653$	− 24198.0i	− 1.45014i	−0.688676	−	0.725070i	$-0.741808\pi$
$653$	− 24198.0i	− 1.45014i	0.688676	−	0.725070i	$-0.258192\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	11534.0i	0.684907i
$658$	0	0
$659$	−17456.0	−1.03185	−0.515925	−	0.856634i	$-0.672552\pi$
$659$	−17456.0	−1.03185	−0.515925	+	0.856634i	$0.672552\pi$
$660$	0	0
$661$	−656.000	−0.0386013	−0.0193006	−	0.999814i	$-0.506144\pi$
$661$	−656.000	−0.0386013	−0.0193006	+	0.999814i	$0.506144\pi$
$662$	0	0
$663$	− 6960.00i	− 0.407698i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 3968.00i	− 0.230347i
$668$	0	0
$669$	−13360.0	−0.772089
$670$	0	0
$671$	23040.0	1.32556
$672$	0	0
$673$	− 18214.0i	− 1.04324i	−0.853179	−	0.521618i	$-0.825329\pi$
$673$	− 18214.0i	− 1.04324i	0.853179	−	0.521618i	$-0.174671\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 30252.0i	− 1.71740i	−0.512480	−	0.858699i	$-0.671273\pi$
$677$	− 30252.0i	− 1.71740i	0.512480	−	0.858699i	$-0.328727\pi$
$678$	0	0
$679$	1498.00	0.0846656
$680$	0	0
$681$	−12900.0	−0.725887
$682$	0	0
$683$	− 10836.0i	− 0.607069i	−0.952820	−	0.303534i	$-0.901833\pi$
$683$	− 10836.0i	− 0.607069i	0.952820	−	0.303534i	$-0.0981667\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	55240.0i	3.06774i
$688$	0	0
$689$	−5400.00	−0.298583
$690$	0	0
$691$	9578.00	0.527300	0.263650	−	0.964618i	$-0.415074\pi$
$691$	9578.00	0.527300	0.263650	+	0.964618i	$0.415074\pi$
$692$	0	0
$693$	20440.0i	1.12042i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	348.000i	0.0189117i
$698$	0	0
$699$	63140.0	3.41656
$700$	0	0
$701$	12442.0	0.670368	0.335184	−	0.942153i	$-0.391202\pi$
$701$	12442.0	0.670368	0.335184	+	0.942153i	$0.391202\pi$
$702$	0	0
$703$	676.000i	0.0362672i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	10024.0i	0.533227i
$708$	0	0
$709$	25174.0	1.33347	0.666734	−	0.745295i	$-0.267692\pi$
$709$	25174.0	1.33347	0.666734	+	0.745295i	$0.267692\pi$
$710$	0	0
$711$	−68328.0	−3.60408
$712$	0	0
$713$	− 16128.0i	− 0.847123i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 39600.0i	− 2.06261i
$718$	0	0
$719$	−34188.0	−1.77329	−0.886646	−	0.462448i	$-0.846971\pi$
$719$	−34188.0	−1.77329	−0.886646	+	0.462448i	$0.846971\pi$
$720$	0	0
$721$	−5348.00	−0.276241
$722$	0	0
$723$	70180.0i	3.60999i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	5204.00i	0.265482i	0.991151	+	0.132741i	$0.0423779\pi$
$727$	5204.00i	0.265482i	−0.991151	+	0.132741i	$0.957622\pi$
$728$	0	0
$729$	−67717.0	−3.44038
$730$	0	0
$731$	−24128.0	−1.22080
$732$	0	0
$733$	− 32880.0i	− 1.65682i	−0.560121	−	0.828411i	$-0.689245\pi$
$733$	− 32880.0i	− 1.65682i	0.560121	−	0.828411i	$-0.310755\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 19040.0i	− 0.951625i
$738$	0	0
$739$	3912.00	0.194730	0.0973648	−	0.995249i	$-0.468959\pi$
$739$	3912.00	0.194730	0.0973648	+	0.995249i	$0.468959\pi$
$740$	0	0
$741$	3120.00	0.154678
$742$	0	0
$743$	− 16008.0i	− 0.790413i	−0.918592	−	0.395206i	$-0.870673\pi$
$743$	− 16008.0i	− 0.790413i	0.918592	−	0.395206i	$-0.129327\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 38690.0i	− 1.89504i
$748$	0	0
$749$	2268.00	0.110642
$750$	0	0
$751$	9960.00	0.483949	0.241974	−	0.970283i	$-0.422205\pi$
$751$	9960.00	0.483949	0.241974	+	0.970283i	$0.422205\pi$
$752$	0	0
$753$	23940.0i	1.15860i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 12378.0i	− 0.594301i	−0.954831	−	0.297151i	$-0.903964\pi$
$757$	− 12378.0i	− 0.594301i	0.954831	−	0.297151i	$-0.0960364\pi$
$758$	0	0
$759$	25600.0	1.22427
$760$	0	0
$761$	34670.0	1.65149	0.825747	−	0.564041i	$-0.190754\pi$
$761$	34670.0	1.65149	0.825747	+	0.564041i	$0.190754\pi$
$762$	0	0
$763$	9338.00i	0.443065i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3288.00i	0.154789i
$768$	0	0
$769$	10898.0	0.511043	0.255521	−	0.966803i	$-0.417753\pi$
$769$	10898.0	0.511043	0.255521	+	0.966803i	$0.417753\pi$
$770$	0	0
$771$	27660.0	1.29202
$772$	0	0
$773$	− 25808.0i	− 1.20084i	−0.799685	−	0.600420i	$-0.795000\pi$
$773$	− 25808.0i	− 1.20084i	0.799685	−	0.600420i	$-0.205000\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 1820.00i	− 0.0840311i
$778$	0	0
$779$	−156.000	−0.00717494
$780$	0	0
$781$	17920.0	0.821035
$782$	0	0
$783$	28520.0i	1.30169i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	21054.0i	0.953614i	0.879008	+	0.476807i	$0.158206\pi$
$787$	21054.0i	0.953614i	−0.879008	+	0.476807i	$0.841794\pi$
$788$	0	0
$789$	79680.0	3.59529
$790$	0	0
$791$	−12586.0	−0.565748
$792$	0	0
$793$	6912.00i	0.309524i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	24276.0i	1.07892i	0.842011	+	0.539461i	$0.181372\pi$
$797$	24276.0i	1.07892i	−0.842011	+	0.539461i	$0.818628\pi$
$798$	0	0
$799$	−22968.0	−1.01696
$800$	0	0
$801$	−28470.0	−1.25585
$802$	0	0
$803$	6320.00i	0.277743i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 29000.0i	− 1.26499i
$808$	0	0
$809$	−21526.0	−0.935493	−0.467747	−	0.883863i	$-0.654934\pi$
$809$	−21526.0	−0.935493	−0.467747	+	0.883863i	$0.654934\pi$
$810$	0	0
$811$	−12806.0	−0.554475	−0.277238	−	0.960801i	$-0.589419\pi$
$811$	−12806.0	−0.554475	−0.277238	+	0.960801i	$0.589419\pi$
$812$	0	0
$813$	− 26400.0i	− 1.13885i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 10816.0i	− 0.463163i
$818$	0	0
$819$	−6132.00	−0.261623
$820$	0	0
$821$	13214.0	0.561720	0.280860	−	0.959749i	$-0.409380\pi$
$821$	13214.0	0.561720	0.280860	+	0.959749i	$0.409380\pi$
$822$	0	0
$823$	32248.0i	1.36585i	0.730488	+	0.682925i	$0.239292\pi$
$823$	32248.0i	1.36585i	−0.730488	+	0.682925i	$0.760708\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	14316.0i	0.601954i	0.953631	+	0.300977i	$0.0973128\pi$
$827$	14316.0i	0.601954i	−0.953631	+	0.300977i	$0.902687\pi$
$828$	0	0
$829$	−25168.0	−1.05443	−0.527214	−	0.849733i	$-0.676763\pi$
$829$	−25168.0	−1.05443	−0.527214	+	0.849733i	$0.676763\pi$
$830$	0	0
$831$	15220.0	0.635350
$832$	0	0
$833$	− 2842.00i	− 0.118211i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	115920.i	4.78707i
$838$	0	0
$839$	−9356.00	−0.384988	−0.192494	−	0.981298i	$-0.561658\pi$
$839$	−9356.00	−0.384988	−0.192494	+	0.981298i	$0.561658\pi$
$840$	0	0
$841$	−20545.0	−0.842388
$842$	0	0
$843$	45340.0i	1.85242i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1883.00i	0.0763880i
$848$	0	0
$849$	48340.0	1.95409
$850$	0	0
$851$	−1664.00	−0.0670284
$852$	0	0
$853$	2372.00i	0.0952119i	0.998866	+	0.0476059i	$0.0151592\pi$
$853$	2372.00i	0.0952119i	−0.998866	+	0.0476059i	$0.984841\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 11694.0i	− 0.466114i	−0.972463	−	0.233057i	$-0.925127\pi$
$857$	− 11694.0i	− 0.466114i	0.972463	−	0.233057i	$-0.0748728\pi$
$858$	0	0
$859$	20506.0	0.814500	0.407250	−	0.913317i	$-0.366488\pi$
$859$	20506.0	0.814500	0.407250	+	0.913317i	$0.366488\pi$
$860$	0	0
$861$	420.000	0.0166243
$862$	0	0
$863$	− 28136.0i	− 1.10980i	−0.831916	−	0.554902i	$-0.812756\pi$
$863$	− 28136.0i	− 1.10980i	0.831916	−	0.554902i	$-0.187244\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 15490.0i	− 0.606768i
$868$	0	0
$869$	−37440.0	−1.46152
$870$	0	0
$871$	5712.00	0.222209
$872$	0	0
$873$	15622.0i	0.605641i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	37070.0i	1.42733i	0.700489	+	0.713663i	$0.252965\pi$
$877$	37070.0i	1.42733i	−0.700489	+	0.713663i	$0.747035\pi$
$878$	0	0
$879$	−46560.0	−1.78661
$880$	0	0
$881$	−6198.00	−0.237021	−0.118511	−	0.992953i	$-0.537812\pi$
$881$	−6198.00	−0.237021	−0.118511	+	0.992953i	$0.537812\pi$
$882$	0	0
$883$	− 31876.0i	− 1.21485i	−0.794377	−	0.607425i	$-0.792202\pi$
$883$	− 31876.0i	− 1.21485i	0.794377	−	0.607425i	$-0.207798\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	132.000i	0.00499676i	0.999997	+	0.00249838i	$0.000795260\pi$
$887$	132.000i	0.00499676i	−0.999997	+	0.00249838i	$0.999205\pi$
$888$	0	0
$889$	−2688.00	−0.101409
$890$	0	0
$891$	−105160.	−3.95398
$892$	0	0
$893$	− 10296.0i	− 0.385826i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	7680.00i	0.285873i
$898$	0	0
$899$	15624.0	0.579632
$900$	0	0
$901$	26100.0	0.965058
$902$	0	0
$903$	29120.0i	1.07315i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 38244.0i	− 1.40008i	−0.714104	−	0.700039i	$-0.753166\pi$
$907$	− 38244.0i	− 1.40008i	0.714104	−	0.700039i	$-0.246834\pi$
$908$	0	0
$909$	−104536.	−3.81435
$910$	0	0
$911$	−7008.00	−0.254869	−0.127434	−	0.991847i	$-0.540674\pi$
$911$	−7008.00	−0.254869	−0.127434	+	0.991847i	$0.540674\pi$
$912$	0	0
$913$	− 21200.0i	− 0.768475i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 12698.0i	− 0.457279i
$918$	0	0
$919$	−36664.0	−1.31603	−0.658016	−	0.753004i	$-0.728604\pi$
$919$	−36664.0	−1.31603	−0.658016	+	0.753004i	$0.728604\pi$
$920$	0	0
$921$	72380.0	2.58958
$922$	0	0
$923$	5376.00i	0.191715i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 55772.0i	− 1.97604i
$928$	0	0
$929$	−45510.0	−1.60725	−0.803625	−	0.595136i	$-0.797098\pi$
$929$	−45510.0	−1.60725	−0.803625	+	0.595136i	$0.797098\pi$
$930$	0	0
$931$	1274.00	0.0448482
$932$	0	0
$933$	− 10960.0i	− 0.384581i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	3838.00i	0.133812i	0.997759	+	0.0669061i	$0.0213128\pi$
$937$	3838.00i	0.133812i	−0.997759	+	0.0669061i	$0.978687\pi$
$938$	0	0
$939$	−38180.0	−1.32690
$940$	0	0
$941$	−16832.0	−0.583111	−0.291556	−	0.956554i	$-0.594173\pi$
$941$	−16832.0	−0.583111	−0.291556	+	0.956554i	$0.594173\pi$
$942$	0	0
$943$	− 384.000i	− 0.0132606i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	40928.0i	1.40442i	0.711972	+	0.702208i	$0.247802\pi$
$947$	40928.0i	1.40442i	−0.711972	+	0.702208i	$0.752198\pi$
$948$	0	0
$949$	−1896.00	−0.0648543
$950$	0	0
$951$	−19980.0	−0.681279
$952$	0	0
$953$	− 24070.0i	− 0.818157i	−0.912499	−	0.409079i	$-0.865850\pi$
$953$	− 24070.0i	− 0.818157i	0.912499	−	0.409079i	$-0.134150\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	24800.0i	0.837691i
$958$	0	0
$959$	11662.0	0.392686
$960$	0	0
$961$	33713.0	1.13165
$962$	0	0
$963$	23652.0i	0.791459i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	17152.0i	0.570394i	0.958469	+	0.285197i	$0.0920590\pi$
$967$	17152.0i	0.570394i	−0.958469	+	0.285197i	$0.907941\pi$
$968$	0	0
$969$	−15080.0	−0.499937
$970$	0	0
$971$	−32910.0	−1.08767	−0.543837	−	0.839191i	$-0.683029\pi$
$971$	−32910.0	−1.08767	−0.543837	+	0.839191i	$0.683029\pi$
$972$	0	0
$973$	− 7882.00i	− 0.259697i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	6822.00i	0.223393i	0.993742	+	0.111697i	$0.0356285\pi$
$977$	6822.00i	0.223393i	−0.993742	+	0.111697i	$0.964372\pi$
$978$	0	0
$979$	−15600.0	−0.509273
$980$	0	0
$981$	−97382.0	−3.16939
$982$	0	0
$983$	− 48420.0i	− 1.57107i	−0.618820	−	0.785533i	$-0.712389\pi$
$983$	− 48420.0i	− 1.57107i	0.618820	−	0.785533i	$-0.287611\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	27720.0i	0.893959i
$988$	0	0
$989$	26624.0	0.856010
$990$	0	0
$991$	−49216.0	−1.57760	−0.788798	−	0.614652i	$-0.789296\pi$
$991$	−49216.0	−1.57760	−0.788798	+	0.614652i	$0.789296\pi$
$992$	0	0
$993$	79360.0i	2.53617i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 35264.0i	− 1.12018i	−0.828431	−	0.560091i	$-0.810766\pi$
$997$	− 35264.0i	− 1.12018i	0.828431	−	0.560091i	$-0.189234\pi$
$998$	0	0
$999$	11960.0	0.378776

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	700.4.e.a.449.1		2
5.2	odd	4		28.4.a.a.1.1	✓	1
5.3	odd	4		700.4.a.n.1.1		1
5.4	even	2	inner	700.4.e.a.449.2		2
15.2	even	4		252.4.a.d.1.1		1
20.7	even	4		112.4.a.g.1.1		1
35.2	odd	12		196.4.e.f.165.1		2
35.12	even	12		196.4.e.a.165.1		2
35.17	even	12		196.4.e.a.177.1		2
35.27	even	4		196.4.a.d.1.1		1
35.32	odd	12		196.4.e.f.177.1		2
40.27	even	4		448.4.a.a.1.1		1
40.37	odd	4		448.4.a.p.1.1		1
60.47	odd	4		1008.4.a.o.1.1		1
105.2	even	12		1764.4.k.d.361.1		2
105.17	odd	12		1764.4.k.m.1549.1		2
105.32	even	12		1764.4.k.d.1549.1		2
105.47	odd	12		1764.4.k.m.361.1		2
105.62	odd	4		1764.4.a.c.1.1		1
140.27	odd	4		784.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.4.a.a.1.1	✓	1	5.2	odd	4
112.4.a.g.1.1		1	20.7	even	4
196.4.a.d.1.1		1	35.27	even	4
196.4.e.a.165.1		2	35.12	even	12
196.4.e.a.177.1		2	35.17	even	12
196.4.e.f.165.1		2	35.2	odd	12
196.4.e.f.177.1		2	35.32	odd	12
252.4.a.d.1.1		1	15.2	even	4
448.4.a.a.1.1		1	40.27	even	4
448.4.a.p.1.1		1	40.37	odd	4
700.4.a.n.1.1		1	5.3	odd	4
700.4.e.a.449.1		2	1.1	even	1	trivial
700.4.e.a.449.2		2	5.4	even	2	inner
784.4.a.a.1.1		1	140.27	odd	4
1008.4.a.o.1.1		1	60.47	odd	4
1764.4.a.c.1.1		1	105.62	odd	4
1764.4.k.d.361.1		2	105.2	even	12
1764.4.k.d.1549.1		2	105.32	even	12
1764.4.k.m.361.1		2	105.47	odd	12
1764.4.k.m.1549.1		2	105.17	odd	12

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:	\( N \)	\(=\)	\( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	700.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-10.0000i q^{3} +7.00000i q^{7} -73.0000 q^{9} +O(q^{10})\)	\(q-10.0000i q^{3} +7.00000i q^{7} -73.0000 q^{9} -40.0000 q^{11} -12.0000i q^{13} +58.0000i q^{17} -26.0000 q^{19} +70.0000 q^{21} -64.0000i q^{23} +460.000i q^{27} +62.0000 q^{29} +252.000 q^{31} +400.000i q^{33} -26.0000i q^{37} -120.000 q^{39} +6.00000 q^{41} +416.000i q^{43} +396.000i q^{47} -49.0000 q^{49} +580.000 q^{51} -450.000i q^{53} +260.000i q^{57} -274.000 q^{59} -576.000 q^{61} -511.000i q^{63} +476.000i q^{67} -640.000 q^{69} -448.000 q^{71} -158.000i q^{73} -280.000i q^{77} +936.000 q^{79} +2629.00 q^{81} +530.000i q^{83} -620.000i q^{87} +390.000 q^{89} +84.0000 q^{91} -2520.00i q^{93} -214.000i q^{97} +2920.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 146 q^{9}+O(q^{10}) \) 2 * q - 146 * q^9	\( 2 q - 146 q^{9} - 80 q^{11} - 52 q^{19} + 140 q^{21} + 124 q^{29} + 504 q^{31} - 240 q^{39} + 12 q^{41} - 98 q^{49} + 1160 q^{51} - 548 q^{59} - 1152 q^{61} - 1280 q^{69} - 896 q^{71} + 1872 q^{79} + 5258 q^{81} + 780 q^{89} + 168 q^{91} + 5840 q^{99}+O(q^{100}) \) 2 * q - 146 * q^9 - 80 * q^11 - 52 * q^19 + 140 * q^21 + 124 * q^29 + 504 * q^31 - 240 * q^39 + 12 * q^41 - 98 * q^49 + 1160 * q^51 - 548 * q^59 - 1152 * q^61 - 1280 * q^69 - 896 * q^71 + 1872 * q^79 + 5258 * q^81 + 780 * q^89 + 168 * q^91 + 5840 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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