LMFDB - Embedded newform 3360.2.j.e.1009.13

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3360,2,Mod(1009,3360)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3360, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3360.1009"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3360.j (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [32,0,-32,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$26.8297350792$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 840)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1009.13
Character	$\chi$	$=$	3360.1009
Dual form			3360.2.j.e.1009.14

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +(-0.725275 - 2.11518i) q^{5} +1.00000i q^{7} +1.00000 q^{9} +1.98457i q^{11} +4.46045 q^{13} +(0.725275 + 2.11518i) q^{15} -6.83320i q^{17} +6.40466i q^{19} -1.00000i q^{21} -8.27003i q^{23} +(-3.94795 + 3.06817i) q^{25} -1.00000 q^{27} +2.08400i q^{29} -1.28335 q^{31} -1.98457i q^{33} +(2.11518 - 0.725275i) q^{35} +10.5278 q^{37} -4.46045 q^{39} -5.84384 q^{41} +0.807243 q^{43} +(-0.725275 - 2.11518i) q^{45} +9.01427i q^{47} -1.00000 q^{49} +6.83320i q^{51} +4.80213 q^{53} +(4.19773 - 1.43936i) q^{55} -6.40466i q^{57} +1.35143i q^{59} +11.1318i q^{61} +1.00000i q^{63} +(-3.23505 - 9.43463i) q^{65} -6.01414 q^{67} +8.27003i q^{69} +7.23080 q^{71} -0.829004i q^{73} +(3.94795 - 3.06817i) q^{75} -1.98457 q^{77} +11.0281 q^{79} +1.00000 q^{81} -2.69790 q^{83} +(-14.4534 + 4.95594i) q^{85} -2.08400i q^{87} +2.66995 q^{89} +4.46045i q^{91} +1.28335 q^{93} +(13.5470 - 4.64513i) q^{95} -0.641074i q^{97} +1.98457i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 32 q^{3} + 32 q^{9} + 32 q^{13} - 16 q^{25} - 32 q^{27} - 32 q^{31} - 4 q^{35} - 8 q^{37} - 32 q^{39} + 24 q^{41} + 8 q^{43} - 32 q^{49} - 24 q^{53} - 24 q^{55} + 8 q^{65} - 24 q^{67} - 40 q^{71}+ \cdots + 48 q^{95}+O(q^{100}) $ 32 * q - 32 * q^3 + 32 * q^9 + 32 * q^13 - 16 * q^25 - 32 * q^27 - 32 * q^31 - 4 * q^35 - 8 * q^37 - 32 * q^39 + 24 * q^41 + 8 * q^43 - 32 * q^49 - 24 * q^53 - 24 * q^55 + 8 * q^65 - 24 * q^67 - 40 * q^71 + 16 * q^75 - 16 * q^77 + 24 * q^79 + 32 * q^81 - 8 * q^85 + 24 * q^89 + 32 * q^93 + 48 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3360\mathbb{Z}\right)^\times$.

$n$	$421$	$1121$	$1471$	$1921$	$2017$
$\chi(n)$	$-1$	$1$	$1$	$1$	$-1$

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
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$		0			0
$5$	−0.725275	−	2.11518i	−0.324353	−	0.945936i
$5$	−0.725275	−	2.11518i	−0.324353	−	0.945936i
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			1.98457i			0.598372i	0.954195	+	0.299186i	$0.0967150\pi$
$11$			1.98457i			0.598372i	−0.954195	+	0.299186i	$0.903285\pi$
$12$		0			0
$13$	4.46045			1.23710			0.618552	−	0.785744i	$-0.287719\pi$
$13$	4.46045			1.23710			0.618552	+	0.785744i	$0.287719\pi$
$14$		0			0
$15$	0.725275	+	2.11518i	0.187265	+	0.546137i
$16$		0			0
$17$		−	6.83320i		−	1.65729i	−0.559772	−	0.828647i	$-0.689111\pi$
$17$		−	6.83320i		−	1.65729i	0.559772	−	0.828647i	$-0.310889\pi$
$18$		0			0
$19$			6.40466i			1.46933i	0.678431	+	0.734664i	$0.262660\pi$
$19$			6.40466i			1.46933i	−0.678431	+	0.734664i	$0.737340\pi$
$20$		0			0
$21$		−	1.00000i		−	0.218218i
$22$		0			0
$23$		−	8.27003i		−	1.72442i	−0.506551	−	0.862210i	$-0.669080\pi$
$23$		−	8.27003i		−	1.72442i	0.506551	−	0.862210i	$-0.330920\pi$
$24$		0			0
$25$	−3.94795	+	3.06817i	−0.789591	+	0.613634i
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$			2.08400i			0.386989i	0.981101	+	0.193494i	$0.0619821\pi$
$29$			2.08400i			0.386989i	−0.981101	+	0.193494i	$0.938018\pi$
$30$		0			0
$31$	−1.28335			−0.230497			−0.115249	−	0.993337i	$-0.536766\pi$
$31$	−1.28335			−0.230497			−0.115249	+	0.993337i	$0.536766\pi$
$32$		0			0
$33$		−	1.98457i		−	0.345470i
$34$		0			0
$35$	2.11518	−	0.725275i	0.357530	−	0.122594i
$36$		0			0
$37$	10.5278			1.73077			0.865384	−	0.501110i	$-0.167075\pi$
$37$	10.5278			1.73077			0.865384	+	0.501110i	$0.167075\pi$
$38$		0			0
$39$	−4.46045			−0.714243
$40$		0			0
$41$	−5.84384			−0.912654			−0.456327	−	0.889812i	$-0.650835\pi$
$41$	−5.84384			−0.912654			−0.456327	+	0.889812i	$0.650835\pi$
$42$		0			0
$43$	0.807243			0.123103			0.0615517	−	0.998104i	$-0.480395\pi$
$43$	0.807243			0.123103			0.0615517	+	0.998104i	$0.480395\pi$
$44$		0			0
$45$	−0.725275	−	2.11518i	−0.108118	−	0.315312i
$46$		0			0
$47$			9.01427i			1.31487i	0.753513	+	0.657433i	$0.228358\pi$
$47$			9.01427i			1.31487i	−0.753513	+	0.657433i	$0.771642\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$			6.83320i			0.956839i
$52$		0			0
$53$	4.80213			0.659623			0.329811	−	0.944047i	$-0.393015\pi$
$53$	4.80213			0.659623			0.329811	+	0.944047i	$0.393015\pi$
$54$		0			0
$55$	4.19773	−	1.43936i	0.566021	−	0.194083i
$56$		0			0
$57$		−	6.40466i		−	0.848317i
$58$		0			0
$59$			1.35143i			0.175942i	0.996123	+	0.0879708i	$0.0280382\pi$
$59$			1.35143i			0.175942i	−0.996123	+	0.0879708i	$0.971962\pi$
$60$		0			0
$61$			11.1318i			1.42527i	0.701532	+	0.712637i	$0.252500\pi$
$61$			11.1318i			1.42527i	−0.701532	+	0.712637i	$0.747500\pi$
$62$		0			0
$63$			1.00000i			0.125988i
$64$		0			0
$65$	−3.23505	−	9.43463i	−0.401258	−	1.17022i
$66$		0			0
$67$	−6.01414			−0.734744			−0.367372	−	0.930074i	$-0.619742\pi$
$67$	−6.01414			−0.734744			−0.367372	+	0.930074i	$0.619742\pi$
$68$		0			0
$69$			8.27003i			0.995594i
$70$		0			0
$71$	7.23080			0.858138			0.429069	−	0.903272i	$-0.358842\pi$
$71$	7.23080			0.858138			0.429069	+	0.903272i	$0.358842\pi$
$72$		0			0
$73$		−	0.829004i		−	0.0970276i	−0.998823	−	0.0485138i	$-0.984552\pi$
$73$		−	0.829004i		−	0.0970276i	0.998823	−	0.0485138i	$-0.0154485\pi$
$74$		0			0
$75$	3.94795	−	3.06817i	0.455870	−	0.354282i
$76$		0			0
$77$	−1.98457			−0.226163
$78$		0			0
$79$	11.0281			1.24076			0.620382	−	0.784300i	$-0.286978\pi$
$79$	11.0281			1.24076			0.620382	+	0.784300i	$0.286978\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	−2.69790			−0.296133			−0.148066	−	0.988977i	$-0.547305\pi$
$83$	−2.69790			−0.296133			−0.148066	+	0.988977i	$0.547305\pi$
$84$		0			0
$85$	−14.4534	+	4.95594i	−1.56769	+	0.537548i
$86$		0			0
$87$		−	2.08400i		−	0.223428i
$88$		0			0
$89$	2.66995			0.283014			0.141507	−	0.989937i	$-0.454805\pi$
$89$	2.66995			0.283014			0.141507	+	0.989937i	$0.454805\pi$
$90$		0			0
$91$			4.46045i			0.467582i
$92$		0			0
$93$	1.28335			0.133078
$94$		0			0
$95$	13.5470	−	4.64513i	1.38989	−	0.476581i
$96$		0			0
$97$		−	0.641074i		−	0.0650912i	−0.999470	−	0.0325456i	$-0.989639\pi$
$97$		−	0.641074i		−	0.0650912i	0.999470	−	0.0325456i	$-0.0103614\pi$
$98$		0			0
$99$			1.98457i			0.199457i
$100$		0			0
$101$		−	12.3440i		−	1.22828i	−0.789198	−	0.614139i	$-0.789503\pi$
$101$		−	12.3440i		−	1.22828i	0.789198	−	0.614139i	$-0.210497\pi$
$102$		0			0
$103$		−	15.6671i		−	1.54373i	−0.635787	−	0.771865i	$-0.719324\pi$
$103$		−	15.6671i		−	1.54373i	0.635787	−	0.771865i	$-0.280676\pi$
$104$		0			0
$105$	−2.11518	+	0.725275i	−0.206420	+	0.0707795i
$106$		0			0
$107$	16.6552			1.61011			0.805057	−	0.593197i	$-0.202134\pi$
$107$	16.6552			1.61011			0.805057	+	0.593197i	$0.202134\pi$
$108$		0			0
$109$		−	14.9152i		−	1.42861i	−0.699833	−	0.714306i	$-0.746742\pi$
$109$		−	14.9152i		−	1.42861i	0.699833	−	0.714306i	$-0.253258\pi$
$110$		0			0
$111$	−10.5278			−0.999259
$112$		0			0
$113$			2.41294i			0.226990i	0.993539	+	0.113495i	$0.0362046\pi$
$113$			2.41294i			0.226990i	−0.993539	+	0.113495i	$0.963795\pi$
$114$		0			0
$115$	−17.4926	+	5.99804i	−1.63119	+	0.559320i
$116$		0			0
$117$	4.46045			0.412368
$118$		0			0
$119$	6.83320			0.626398
$120$		0			0
$121$	7.06147			0.641952
$122$		0			0
$123$	5.84384			0.526921
$124$		0			0
$125$	9.35307	+	6.12536i	0.836564	+	0.547869i
$126$		0			0
$127$		−	18.6981i		−	1.65919i	−0.558366	−	0.829595i	$-0.688571\pi$
$127$		−	18.6981i		−	1.65919i	0.558366	−	0.829595i	$-0.311429\pi$
$128$		0			0
$129$	−0.807243			−0.0710738
$130$		0			0
$131$		−	14.7118i		−	1.28537i	−0.766129	−	0.642687i	$-0.777820\pi$
$131$		−	14.7118i		−	1.28537i	0.766129	−	0.642687i	$-0.222180\pi$
$132$		0			0
$133$	−6.40466			−0.555354
$134$		0			0
$135$	0.725275	+	2.11518i	0.0624217	+	0.182046i
$136$		0			0
$137$		−	10.8323i		−	0.925464i	−0.886498	−	0.462732i	$-0.846869\pi$
$137$		−	10.8323i		−	0.925464i	0.886498	−	0.462732i	$-0.153131\pi$
$138$		0			0
$139$		−	10.7222i		−	0.909441i	−0.890634	−	0.454721i	$-0.849739\pi$
$139$		−	10.7222i		−	0.909441i	0.890634	−	0.454721i	$-0.150261\pi$
$140$		0			0
$141$		−	9.01427i		−	0.759138i
$142$		0			0
$143$			8.85208i			0.740248i
$144$		0			0
$145$	4.40803	−	1.51147i	0.366067	−	0.125521i
$146$		0			0
$147$	1.00000			0.0824786
$148$		0			0
$149$		−	10.6182i		−	0.869876i	−0.900460	−	0.434938i	$-0.856770\pi$
$149$		−	10.6182i		−	0.869876i	0.900460	−	0.434938i	$-0.143230\pi$
$150$		0			0
$151$	5.53645			0.450550			0.225275	−	0.974295i	$-0.427672\pi$
$151$	5.53645			0.450550			0.225275	+	0.974295i	$0.427672\pi$
$152$		0			0
$153$		−	6.83320i		−	0.552431i
$154$		0			0
$155$	0.930784	+	2.71452i	0.0747624	+	0.218036i
$156$		0			0
$157$	15.0491			1.20105			0.600523	−	0.799607i	$-0.294959\pi$
$157$	15.0491			1.20105			0.600523	+	0.799607i	$0.294959\pi$
$158$		0			0
$159$	−4.80213			−0.380834
$160$		0			0
$161$	8.27003			0.651769
$162$		0			0
$163$	−6.23195			−0.488124			−0.244062	−	0.969760i	$-0.578480\pi$
$163$	−6.23195			−0.488124			−0.244062	+	0.969760i	$0.578480\pi$
$164$		0			0
$165$	−4.19773	+	1.43936i	−0.326793	+	0.112054i
$166$		0			0
$167$		−	6.54404i		−	0.506393i	−0.967415	−	0.253196i	$-0.918518\pi$
$167$		−	6.54404i		−	0.506393i	0.967415	−	0.253196i	$-0.0814819\pi$
$168$		0			0
$169$	6.89557			0.530428
$170$		0			0
$171$			6.40466i			0.489776i
$172$		0			0
$173$	−7.99047			−0.607504			−0.303752	−	0.952751i	$-0.598240\pi$
$173$	−7.99047			−0.607504			−0.303752	+	0.952751i	$0.598240\pi$
$174$		0			0
$175$	−3.06817	−	3.94795i	−0.231932	−	0.298437i
$176$		0			0
$177$		−	1.35143i		−	0.101580i
$178$		0			0
$179$		−	8.56129i		−	0.639901i	−0.947434	−	0.319950i	$-0.896334\pi$
$179$		−	8.56129i		−	0.639901i	0.947434	−	0.319950i	$-0.103666\pi$
$180$		0			0
$181$		−	6.14276i		−	0.456588i	−0.973592	−	0.228294i	$-0.926685\pi$
$181$		−	6.14276i		−	0.456588i	0.973592	−	0.228294i	$-0.0733147\pi$
$182$		0			0
$183$		−	11.1318i		−	0.822883i
$184$		0			0
$185$	−7.63558	−	22.2683i	−0.561379	−	1.63720i
$186$		0			0
$187$	13.5610			0.991678
$188$		0			0
$189$		−	1.00000i		−	0.0727393i
$190$		0			0
$191$	8.89586			0.643682			0.321841	−	0.946794i	$-0.395698\pi$
$191$	8.89586			0.643682			0.321841	+	0.946794i	$0.395698\pi$
$192$		0			0
$193$			9.05722i			0.651953i	0.945378	+	0.325977i	$0.105693\pi$
$193$			9.05722i			0.651953i	−0.945378	+	0.325977i	$0.894307\pi$
$194$		0			0
$195$	3.23505	+	9.43463i	0.231667	+	0.675628i
$196$		0			0
$197$	−7.65946			−0.545714			−0.272857	−	0.962055i	$-0.587969\pi$
$197$	−7.65946			−0.545714			−0.272857	+	0.962055i	$0.587969\pi$
$198$		0			0
$199$	−16.0795			−1.13985			−0.569923	−	0.821698i	$-0.693027\pi$
$199$	−16.0795			−1.13985			−0.569923	+	0.821698i	$0.693027\pi$
$200$		0			0
$201$	6.01414			0.424205
$202$		0			0
$203$	−2.08400			−0.146268
$204$		0			0
$205$	4.23839	+	12.3608i	0.296022	+	0.863313i
$206$		0			0
$207$		−	8.27003i		−	0.574807i
$208$		0			0
$209$	−12.7105			−0.879204
$210$		0			0
$211$			18.6220i			1.28199i	0.767545	+	0.640995i	$0.221478\pi$
$211$			18.6220i			1.28199i	−0.767545	+	0.640995i	$0.778522\pi$
$212$		0			0
$213$	−7.23080			−0.495446
$214$		0			0
$215$	−0.585473	−	1.70746i	−0.0399289	−	0.116448i
$216$		0			0
$217$		−	1.28335i		−	0.0871197i
$218$		0			0
$219$			0.829004i			0.0560189i
$220$		0			0
$221$		−	30.4791i		−	2.05025i
$222$		0			0
$223$		−	24.8064i		−	1.66116i	−0.556900	−	0.830579i	$-0.688009\pi$
$223$		−	24.8064i		−	1.66116i	0.556900	−	0.830579i	$-0.311991\pi$
$224$		0			0
$225$	−3.94795	+	3.06817i	−0.263197	+	0.204545i
$226$		0			0
$227$	6.10249			0.405037			0.202518	−	0.979278i	$-0.435087\pi$
$227$	6.10249			0.405037			0.202518	+	0.979278i	$0.435087\pi$
$228$		0			0
$229$			7.13626i			0.471577i	0.971804	+	0.235789i	$0.0757673\pi$
$229$			7.13626i			0.471577i	−0.971804	+	0.235789i	$0.924233\pi$
$230$		0			0
$231$	1.98457			0.130575
$232$		0			0
$233$			26.3887i			1.72878i	0.502819	+	0.864392i	$0.332296\pi$
$233$			26.3887i			1.72878i	−0.502819	+	0.864392i	$0.667704\pi$
$234$		0			0
$235$	19.0668	−	6.53782i	1.24378	−	0.426480i
$236$		0			0
$237$	−11.0281			−0.716355
$238$		0			0
$239$	−11.8328			−0.765399			−0.382699	−	0.923873i	$-0.625006\pi$
$239$	−11.8328			−0.765399			−0.382699	+	0.923873i	$0.625006\pi$
$240$		0			0
$241$	19.1963			1.23655			0.618273	−	0.785964i	$-0.287833\pi$
$241$	19.1963			1.23655			0.618273	+	0.785964i	$0.287833\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$	0.725275	+	2.11518i	0.0463361	+	0.135134i
$246$		0			0
$247$			28.5676i			1.81771i
$248$		0			0
$249$	2.69790			0.170972
$250$		0			0
$251$			1.62899i			0.102821i	0.998678	+	0.0514106i	$0.0163717\pi$
$251$			1.62899i			0.102821i	−0.998678	+	0.0514106i	$0.983628\pi$
$252$		0			0
$253$	16.4125			1.03184
$254$		0			0
$255$	14.4534	−	4.95594i	0.905109	−	0.310353i
$256$		0			0
$257$			1.89332i			0.118102i	0.998255	+	0.0590510i	$0.0188075\pi$
$257$			1.89332i			0.118102i	−0.998255	+	0.0590510i	$0.981193\pi$
$258$		0			0
$259$			10.5278i			0.654169i
$260$		0			0
$261$			2.08400i			0.128996i
$262$		0			0
$263$		−	4.66210i		−	0.287477i	−0.989616	−	0.143739i	$-0.954088\pi$
$263$		−	4.66210i		−	0.287477i	0.989616	−	0.143739i	$-0.0459125\pi$
$264$		0			0
$265$	−3.48286	−	10.1574i	−0.213950	−	0.623961i
$266$		0			0
$267$	−2.66995			−0.163398
$268$		0			0
$269$			12.3189i			0.751099i	0.926802	+	0.375549i	$0.122546\pi$
$269$			12.3189i			0.751099i	−0.926802	+	0.375549i	$0.877454\pi$
$270$		0			0
$271$	25.4743			1.54745			0.773726	−	0.633520i	$-0.218391\pi$
$271$	25.4743			1.54745			0.773726	+	0.633520i	$0.218391\pi$
$272$		0			0
$273$		−	4.46045i		−	0.269958i
$274$		0			0
$275$	−6.08901	−	7.83501i	−0.367181	−	0.472469i
$276$		0			0
$277$	15.5659			0.935267			0.467633	−	0.883923i	$-0.345107\pi$
$277$	15.5659			0.935267			0.467633	+	0.883923i	$0.345107\pi$
$278$		0			0
$279$	−1.28335			−0.0768324
$280$		0			0
$281$	31.7854			1.89616			0.948078	−	0.318039i	$-0.103024\pi$
$281$	31.7854			1.89616			0.948078	+	0.318039i	$0.103024\pi$
$282$		0			0
$283$	−24.6006			−1.46235			−0.731176	−	0.682189i	$-0.761028\pi$
$283$	−24.6006			−1.46235			−0.731176	+	0.682189i	$0.761028\pi$
$284$		0			0
$285$	−13.5470	+	4.64513i	−0.802454	+	0.275154i
$286$		0			0
$287$		−	5.84384i		−	0.344951i
$288$		0			0
$289$	−29.6926			−1.74662
$290$		0			0
$291$			0.641074i			0.0375804i
$292$		0			0
$293$	−3.65931			−0.213779			−0.106890	−	0.994271i	$-0.534089\pi$
$293$	−3.65931			−0.213779			−0.106890	+	0.994271i	$0.534089\pi$
$294$		0			0
$295$	2.85852	−	0.980160i	0.166430	−	0.0570671i
$296$		0			0
$297$		−	1.98457i		−	0.115157i
$298$		0			0
$299$		−	36.8880i		−	2.13329i
$300$		0			0
$301$			0.807243i			0.0465287i
$302$		0			0
$303$			12.3440i			0.709147i
$304$		0			0
$305$	23.5456	−	8.07358i	1.34822	−	0.462292i
$306$		0			0
$307$	23.6180			1.34795			0.673974	−	0.738755i	$-0.264586\pi$
$307$	23.6180			1.34795			0.673974	+	0.738755i	$0.264586\pi$
$308$		0			0
$309$			15.6671i			0.891272i
$310$		0			0
$311$	−29.5891			−1.67785			−0.838923	−	0.544250i	$-0.816814\pi$
$311$	−29.5891			−1.67785			−0.838923	+	0.544250i	$0.816814\pi$
$312$		0			0
$313$		−	33.1631i		−	1.87449i	−0.348671	−	0.937245i	$-0.613367\pi$
$313$		−	33.1631i		−	1.87449i	0.348671	−	0.937245i	$-0.386633\pi$
$314$		0			0
$315$	2.11518	−	0.725275i	0.119177	−	0.0408646i
$316$		0			0
$317$	20.3788			1.14459			0.572295	−	0.820048i	$-0.306053\pi$
$317$	20.3788			1.14459			0.572295	+	0.820048i	$0.306053\pi$
$318$		0			0
$319$	−4.13585			−0.231563
$320$		0			0
$321$	−16.6552			−0.929600
$322$		0			0
$323$	43.7643			2.43511
$324$		0			0
$325$	−17.6096	+	13.6854i	−0.976807	+	0.759129i
$326$		0			0
$327$			14.9152i			0.824810i
$328$		0			0
$329$	−9.01427			−0.496973
$330$		0			0
$331$		−	1.14980i		−	0.0631988i	−0.999501	−	0.0315994i	$-0.989940\pi$
$331$		−	1.14980i		−	0.0631988i	0.999501	−	0.0315994i	$-0.0100601\pi$
$332$		0			0
$333$	10.5278			0.576922
$334$		0			0
$335$	4.36190	+	12.7210i	0.238316	+	0.695021i
$336$		0			0
$337$			32.2193i			1.75510i	0.479488	+	0.877549i	$0.340822\pi$
$337$			32.2193i			1.75510i	−0.479488	+	0.877549i	$0.659178\pi$
$338$		0			0
$339$		−	2.41294i		−	0.131053i
$340$		0			0
$341$		−	2.54691i		−	0.137923i
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$	17.4926	−	5.99804i	0.941769	−	0.322924i
$346$		0			0
$347$	22.3813			1.20149			0.600746	−	0.799440i	$-0.294870\pi$
$347$	22.3813			1.20149			0.600746	+	0.799440i	$0.294870\pi$
$348$		0			0
$349$			16.9131i			0.905339i	0.891678	+	0.452670i	$0.149528\pi$
$349$			16.9131i			0.905339i	−0.891678	+	0.452670i	$0.850472\pi$
$350$		0			0
$351$	−4.46045			−0.238081
$352$		0			0
$353$		−	15.6792i		−	0.834521i	−0.908787	−	0.417261i	$-0.862990\pi$
$353$		−	15.6792i		−	0.834521i	0.908787	−	0.417261i	$-0.137010\pi$
$354$		0			0
$355$	−5.24431	−	15.2944i	−0.278339	−	0.811744i
$356$		0			0
$357$	−6.83320			−0.361651
$358$		0			0
$359$	3.20198			0.168994			0.0844972	−	0.996424i	$-0.473072\pi$
$359$	3.20198			0.168994			0.0844972	+	0.996424i	$0.473072\pi$
$360$		0			0
$361$	−22.0196			−1.15893
$362$		0			0
$363$	−7.06147			−0.370631
$364$		0			0
$365$	−1.75349	+	0.601256i	−0.0917819	+	0.0314712i
$366$		0			0
$367$			5.06988i			0.264645i	0.991207	+	0.132323i	$0.0422435\pi$
$367$			5.06988i			0.264645i	−0.991207	+	0.132323i	$0.957756\pi$
$368$		0			0
$369$	−5.84384			−0.304218
$370$		0			0
$371$			4.80213i			0.249314i
$372$		0			0
$373$	−10.6334			−0.550578			−0.275289	−	0.961361i	$-0.588774\pi$
$373$	−10.6334			−0.550578			−0.275289	+	0.961361i	$0.588774\pi$
$374$		0			0
$375$	−9.35307	−	6.12536i	−0.482991	−	0.316312i
$376$		0			0
$377$			9.29556i			0.478746i
$378$		0			0
$379$		−	6.22848i		−	0.319935i	−0.987122	−	0.159968i	$-0.948861\pi$
$379$		−	6.22848i		−	0.319935i	0.987122	−	0.159968i	$-0.0511390\pi$
$380$		0			0
$381$			18.6981i			0.957933i
$382$		0			0
$383$		−	8.33007i		−	0.425647i	−0.977091	−	0.212823i	$-0.931734\pi$
$383$		−	8.33007i		−	0.425647i	0.977091	−	0.212823i	$-0.0682659\pi$
$384$		0			0
$385$	1.43936	+	4.19773i	0.0733566	+	0.213936i
$386$		0			0
$387$	0.807243			0.0410345
$388$		0			0
$389$		−	5.78317i		−	0.293218i	−0.989195	−	0.146609i	$-0.953164\pi$
$389$		−	5.78317i		−	0.293218i	0.989195	−	0.146609i	$-0.0468359\pi$
$390$		0			0
$391$	−56.5107			−2.85787
$392$		0			0
$393$			14.7118i			0.742111i
$394$		0			0
$395$	−7.99843	−	23.3265i	−0.402445	−	1.17368i
$396$		0			0
$397$	9.77342			0.490514			0.245257	−	0.969458i	$-0.421128\pi$
$397$	9.77342			0.490514			0.245257	+	0.969458i	$0.421128\pi$
$398$		0			0
$399$	6.40466			0.320634
$400$		0			0
$401$	34.6449			1.73008			0.865041	−	0.501701i	$-0.167292\pi$
$401$	34.6449			1.73008			0.865041	+	0.501701i	$0.167292\pi$
$402$		0			0
$403$	−5.72433			−0.285149
$404$		0			0
$405$	−0.725275	−	2.11518i	−0.0360392	−	0.105104i
$406$		0			0
$407$			20.8933i			1.03564i
$408$		0			0
$409$	−7.32443			−0.362170			−0.181085	−	0.983467i	$-0.557961\pi$
$409$	−7.32443			−0.362170			−0.181085	+	0.983467i	$0.557961\pi$
$410$		0			0
$411$			10.8323i			0.534317i
$412$		0			0
$413$	−1.35143			−0.0664997
$414$		0			0
$415$	1.95672	+	5.70654i	0.0960515	+	0.280123i
$416$		0			0
$417$			10.7222i			0.525066i
$418$		0			0
$419$			2.85294i			0.139375i	0.997569	+	0.0696876i	$0.0222003\pi$
$419$			2.85294i			0.139375i	−0.997569	+	0.0696876i	$0.977800\pi$
$420$		0			0
$421$			21.3514i			1.04060i	0.853982	+	0.520302i	$0.174181\pi$
$421$			21.3514i			1.04060i	−0.853982	+	0.520302i	$0.825819\pi$
$422$		0			0
$423$			9.01427i			0.438289i
$424$		0			0
$425$	20.9654	+	26.9772i	1.01697	+	1.30858i
$426$		0			0
$427$	−11.1318			−0.538703
$428$		0			0
$429$		−	8.85208i		−	0.427383i
$430$		0			0
$431$	−6.78558			−0.326850			−0.163425	−	0.986556i	$-0.552254\pi$
$431$	−6.78558			−0.326850			−0.163425	+	0.986556i	$0.552254\pi$
$432$		0			0
$433$			3.90527i			0.187675i	0.995588	+	0.0938377i	$0.0299135\pi$
$433$			3.90527i			0.187675i	−0.995588	+	0.0938377i	$0.970087\pi$
$434$		0			0
$435$	−4.40803	+	1.51147i	−0.211349	+	0.0724695i
$436$		0			0
$437$	52.9667			2.53374
$438$		0			0
$439$	−10.1113			−0.482586			−0.241293	−	0.970452i	$-0.577571\pi$
$439$	−10.1113			−0.482586			−0.241293	+	0.970452i	$0.577571\pi$
$440$		0			0
$441$	−1.00000			−0.0476190
$442$		0			0
$443$	−16.6831			−0.792638			−0.396319	−	0.918113i	$-0.629713\pi$
$443$	−16.6831			−0.792638			−0.396319	+	0.918113i	$0.629713\pi$
$444$		0			0
$445$	−1.93644	−	5.64741i	−0.0917963	−	0.267713i
$446$		0			0
$447$			10.6182i			0.502223i
$448$		0			0
$449$	13.0239			0.614637			0.307319	−	0.951607i	$-0.400568\pi$
$449$	13.0239			0.614637			0.307319	+	0.951607i	$0.400568\pi$
$450$		0			0
$451$		−	11.5975i		−	0.546106i
$452$		0			0
$453$	−5.53645			−0.260125
$454$		0			0
$455$	9.43463	−	3.23505i	0.442302	−	0.151661i
$456$		0			0
$457$		−	10.8925i		−	0.509531i	−0.967003	−	0.254765i	$-0.918002\pi$
$457$		−	10.8925i		−	0.509531i	0.967003	−	0.254765i	$-0.0819982\pi$
$458$		0			0
$459$			6.83320i			0.318946i
$460$		0			0
$461$		−	10.5011i		−	0.489086i	−0.969638	−	0.244543i	$-0.921362\pi$
$461$		−	10.5011i		−	0.489086i	0.969638	−	0.244543i	$-0.0786380\pi$
$462$		0			0
$463$		−	2.39493i		−	0.111302i	−0.998450	−	0.0556509i	$-0.982277\pi$
$463$		−	2.39493i		−	0.111302i	0.998450	−	0.0556509i	$-0.0177234\pi$
$464$		0			0
$465$	−0.930784	−	2.71452i	−0.0431641	−	0.125883i
$466$		0			0
$467$	−3.22855			−0.149400			−0.0746998	−	0.997206i	$-0.523800\pi$
$467$	−3.22855			−0.149400			−0.0746998	+	0.997206i	$0.523800\pi$
$468$		0			0
$469$		−	6.01414i		−	0.277707i
$470$		0			0
$471$	−15.0491			−0.693425
$472$		0			0
$473$			1.60203i			0.0736616i
$474$		0			0
$475$	−19.6506	−	25.2853i	−0.901630	−	1.16017i
$476$		0			0
$477$	4.80213			0.219874
$478$		0			0
$479$	30.7731			1.40606			0.703029	−	0.711161i	$-0.251830\pi$
$479$	30.7731			1.40606			0.703029	+	0.711161i	$0.251830\pi$
$480$		0			0
$481$	46.9589			2.14114
$482$		0			0
$483$	−8.27003			−0.376299
$484$		0			0
$485$	−1.35598	+	0.464954i	−0.0615721	+	0.0211125i
$486$		0			0
$487$		−	15.8470i		−	0.718096i	−0.933319	−	0.359048i	$-0.883102\pi$
$487$		−	15.8470i		−	0.718096i	0.933319	−	0.359048i	$-0.116898\pi$
$488$		0			0
$489$	6.23195			0.281819
$490$		0			0
$491$		−	1.66042i		−	0.0749335i	−0.999298	−	0.0374668i	$-0.988071\pi$
$491$		−	1.66042i		−	0.0749335i	0.999298	−	0.0374668i	$-0.0119288\pi$
$492$		0			0
$493$	14.2404			0.641354
$494$		0			0
$495$	4.19773	−	1.43936i	0.188674	−	0.0646945i
$496$		0			0
$497$			7.23080i			0.324346i
$498$		0			0
$499$			13.1726i			0.589687i	0.955546	+	0.294843i	$0.0952675\pi$
$499$			13.1726i			0.589687i	−0.955546	+	0.294843i	$0.904733\pi$
$500$		0			0
$501$			6.54404i			0.292366i
$502$		0			0
$503$		−	23.5632i		−	1.05063i	−0.850907	−	0.525316i	$-0.823947\pi$
$503$		−	23.5632i		−	1.05063i	0.850907	−	0.525316i	$-0.176053\pi$
$504$		0			0
$505$	−26.1099	+	8.95282i	−1.16187	+	0.398395i
$506$		0			0
$507$	−6.89557			−0.306243
$508$		0			0
$509$			18.3350i			0.812683i	0.913721	+	0.406341i	$0.133196\pi$
$509$			18.3350i			0.812683i	−0.913721	+	0.406341i	$0.866804\pi$
$510$		0			0
$511$	0.829004			0.0366730
$512$		0			0
$513$		−	6.40466i		−	0.282772i
$514$		0			0
$515$	−33.1388	+	11.3630i	−1.46027	+	0.500713i
$516$		0			0
$517$	−17.8895			−0.786779
$518$		0			0
$519$	7.99047			0.350743
$520$		0			0
$521$	−19.9787			−0.875281			−0.437640	−	0.899150i	$-0.644186\pi$
$521$	−19.9787			−0.875281			−0.437640	+	0.899150i	$0.644186\pi$
$522$		0			0
$523$	−30.3642			−1.32773			−0.663867	−	0.747850i	$-0.731086\pi$
$523$	−30.3642			−1.32773			−0.663867	+	0.747850i	$0.731086\pi$
$524$		0			0
$525$	3.06817	+	3.94795i	0.133906	+	0.172303i
$526$		0			0
$527$			8.76941i			0.382002i
$528$		0			0
$529$	−45.3934			−1.97362
$530$		0			0
$531$			1.35143i			0.0586472i
$532$		0			0
$533$	−26.0661			−1.12905
$534$		0			0
$535$	−12.0796	−	35.2286i	−0.522245	−	1.52307i
$536$		0			0
$537$			8.56129i			0.369447i
$538$		0			0
$539$		−	1.98457i		−	0.0854816i
$540$		0			0
$541$			29.7815i			1.28041i	0.768206	+	0.640203i	$0.221150\pi$
$541$			29.7815i			1.28041i	−0.768206	+	0.640203i	$0.778850\pi$
$542$		0			0
$543$			6.14276i			0.263611i
$544$		0			0
$545$	−31.5482	+	10.8176i	−1.35138	+	0.463374i
$546$		0			0
$547$	−12.9079			−0.551901			−0.275951	−	0.961172i	$-0.588993\pi$
$547$	−12.9079			−0.551901			−0.275951	+	0.961172i	$0.588993\pi$
$548$		0			0
$549$			11.1318i			0.475092i
$550$		0			0
$551$	−13.3473			−0.568614
$552$		0			0
$553$			11.0281i			0.468964i
$554$		0			0
$555$	7.63558	+	22.2683i	0.324112	+	0.945235i
$556$		0			0
$557$	−39.3089			−1.66557			−0.832787	−	0.553594i	$-0.813256\pi$
$557$	−39.3089			−1.66557			−0.832787	+	0.553594i	$0.813256\pi$
$558$		0			0
$559$	3.60066			0.152292
$560$		0			0
$561$	−13.5610			−0.572545
$562$		0			0
$563$	−16.5242			−0.696411			−0.348205	−	0.937418i	$-0.613209\pi$
$563$	−16.5242			−0.696411			−0.348205	+	0.937418i	$0.613209\pi$
$564$		0			0
$565$	5.10379	−	1.75004i	0.214718	−	0.0736248i
$566$		0			0
$567$			1.00000i			0.0419961i
$568$		0			0
$569$	43.3602			1.81775			0.908876	−	0.417065i	$-0.136941\pi$
$569$	43.3602			1.81775			0.908876	+	0.417065i	$0.136941\pi$
$570$		0			0
$571$			13.8810i			0.580903i	0.956890	+	0.290452i	$0.0938055\pi$
$571$			13.8810i			0.580903i	−0.956890	+	0.290452i	$0.906194\pi$
$572$		0			0
$573$	−8.89586			−0.371630
$574$		0			0
$575$	25.3738	+	32.6497i	1.05816	+	1.36159i
$576$		0			0
$577$			5.64868i			0.235158i	0.993064	+	0.117579i	$0.0375133\pi$
$577$			5.64868i			0.235158i	−0.993064	+	0.117579i	$0.962487\pi$
$578$		0			0
$579$		−	9.05722i		−	0.376405i
$580$		0			0
$581$		−	2.69790i		−	0.111928i
$582$		0			0
$583$			9.53018i			0.394700i
$584$		0			0
$585$	−3.23505	−	9.43463i	−0.133753	−	0.390074i
$586$		0			0
$587$	42.8700			1.76944			0.884718	−	0.466127i	$-0.154351\pi$
$587$	42.8700			1.76944			0.884718	+	0.466127i	$0.154351\pi$
$588$		0			0
$589$		−	8.21944i		−	0.338676i
$590$		0			0
$591$	7.65946			0.315068
$592$		0			0
$593$			13.2361i			0.543540i	0.962362	+	0.271770i	$0.0876091\pi$
$593$			13.2361i			0.543540i	−0.962362	+	0.271770i	$0.912391\pi$
$594$		0			0
$595$	−4.95594	−	14.4534i	−0.203174	−	0.592533i
$596$		0			0
$597$	16.0795			0.658091
$598$		0			0
$599$	−17.7652			−0.725866			−0.362933	−	0.931815i	$-0.618225\pi$
$599$	−17.7652			−0.725866			−0.362933	+	0.931815i	$0.618225\pi$
$600$		0			0
$601$	−7.38082			−0.301070			−0.150535	−	0.988605i	$-0.548100\pi$
$601$	−7.38082			−0.301070			−0.150535	+	0.988605i	$0.548100\pi$
$602$		0			0
$603$	−6.01414			−0.244915
$604$		0			0
$605$	−5.12150	−	14.9363i	−0.208219	−	0.607245i
$606$		0			0
$607$			7.45733i			0.302683i	0.988481	+	0.151342i	$0.0483594\pi$
$607$			7.45733i			0.302683i	−0.988481	+	0.151342i	$0.951641\pi$
$608$		0			0
$609$	2.08400			0.0844479
$610$		0			0
$611$			40.2077i			1.62663i
$612$		0			0
$613$	−35.3238			−1.42672			−0.713358	−	0.700800i	$-0.752827\pi$
$613$	−35.3238			−1.42672			−0.713358	+	0.700800i	$0.752827\pi$
$614$		0			0
$615$	−4.23839	−	12.3608i	−0.170908	−	0.498434i
$616$		0			0
$617$			24.2477i			0.976176i	0.872794	+	0.488088i	$0.162306\pi$
$617$			24.2477i			0.976176i	−0.872794	+	0.488088i	$0.837694\pi$
$618$		0			0
$619$			16.9940i			0.683045i	0.939874	+	0.341522i	$0.110943\pi$
$619$			16.9940i			0.683045i	−0.939874	+	0.341522i	$0.889057\pi$
$620$		0			0
$621$			8.27003i			0.331865i
$622$		0			0
$623$			2.66995i			0.106969i
$624$		0			0
$625$	6.17268	−	24.2260i	0.246907	−	0.969039i
$626$		0			0
$627$	12.7105			0.507609
$628$		0			0
$629$		−	71.9389i		−	2.86839i
$630$		0			0
$631$	9.21604			0.366885			0.183442	−	0.983030i	$-0.441276\pi$
$631$	9.21604			0.366885			0.183442	+	0.983030i	$0.441276\pi$
$632$		0			0
$633$		−	18.6220i		−	0.740157i
$634$		0			0
$635$	−39.5498	+	13.5613i	−1.56949	+	0.538162i
$636$		0			0
$637$	−4.46045			−0.176729
$638$		0			0
$639$	7.23080			0.286046
$640$		0			0
$641$	−22.2379			−0.878344			−0.439172	−	0.898403i	$-0.644728\pi$
$641$	−22.2379			−0.878344			−0.439172	+	0.898403i	$0.644728\pi$
$642$		0			0
$643$	24.8103			0.978425			0.489212	−	0.872165i	$-0.337284\pi$
$643$	24.8103			0.978425			0.489212	+	0.872165i	$0.337284\pi$
$644$		0			0
$645$	0.585473	+	1.70746i	0.0230530	+	0.0672313i
$646$		0			0
$647$			27.1413i			1.06703i	0.845790	+	0.533516i	$0.179130\pi$
$647$			27.1413i			1.06703i	−0.845790	+	0.533516i	$0.820870\pi$
$648$		0			0
$649$	−2.68202			−0.105278
$650$		0			0
$651$			1.28335i			0.0502986i
$652$		0			0
$653$	12.6387			0.494589			0.247294	−	0.968940i	$-0.420458\pi$
$653$	12.6387			0.494589			0.247294	+	0.968940i	$0.420458\pi$
$654$		0			0
$655$	−31.1180	+	10.6701i	−1.21588	+	0.416914i
$656$		0			0
$657$		−	0.829004i		−	0.0323425i
$658$		0			0
$659$		−	20.7416i		−	0.807979i	−0.914764	−	0.403990i	$-0.867623\pi$
$659$		−	20.7416i		−	0.807979i	0.914764	−	0.403990i	$-0.132377\pi$
$660$		0			0
$661$		−	31.3436i		−	1.21913i	−0.792737	−	0.609563i	$-0.791345\pi$
$661$		−	31.3436i		−	1.21913i	0.792737	−	0.609563i	$-0.208655\pi$
$662$		0			0
$663$			30.4791i			1.18371i
$664$		0			0
$665$	4.64513	+	13.5470i	0.180131	+	0.525330i
$666$		0			0
$667$	17.2347			0.667331
$668$		0			0
$669$			24.8064i			0.959071i
$670$		0			0
$671$	−22.0918			−0.852844
$672$		0			0
$673$		−	24.9548i		−	0.961937i	−0.876738	−	0.480969i	$-0.840285\pi$
$673$		−	24.9548i		−	0.961937i	0.876738	−	0.480969i	$-0.159715\pi$
$674$		0			0
$675$	3.94795	−	3.06817i	0.151957	−	0.118094i
$676$		0			0
$677$	−5.72843			−0.220161			−0.110081	−	0.993923i	$-0.535111\pi$
$677$	−5.72843			−0.220161			−0.110081	+	0.993923i	$0.535111\pi$
$678$		0			0
$679$	0.641074			0.0246021
$680$		0			0
$681$	−6.10249			−0.233848
$682$		0			0
$683$	−5.51566			−0.211051			−0.105525	−	0.994417i	$-0.533652\pi$
$683$	−5.51566			−0.211051			−0.105525	+	0.994417i	$0.533652\pi$
$684$		0			0
$685$	−22.9122	+	7.85638i	−0.875430	+	0.300177i
$686$		0			0
$687$		−	7.13626i		−	0.272265i
$688$		0			0
$689$	21.4196			0.816023
$690$		0			0
$691$			22.2544i			0.846599i	0.905990	+	0.423299i	$0.139128\pi$
$691$			22.2544i			0.846599i	−0.905990	+	0.423299i	$0.860872\pi$
$692$		0			0
$693$	−1.98457			−0.0753877
$694$		0			0
$695$	−22.6793	+	7.77650i	−0.860273	+	0.294980i
$696$		0			0
$697$			39.9321i			1.51254i
$698$		0			0
$699$		−	26.3887i		−	0.998113i
$700$		0			0
$701$		−	34.2655i		−	1.29419i	−0.762410	−	0.647095i	$-0.775984\pi$
$701$		−	34.2655i		−	1.29419i	0.762410	−	0.647095i	$-0.224016\pi$
$702$		0			0
$703$			67.4272i			2.54307i
$704$		0			0
$705$	−19.0668	+	6.53782i	−0.718097	+	0.246229i
$706$		0			0
$707$	12.3440			0.464246
$708$		0			0
$709$			3.36438i			0.126352i	0.998002	+	0.0631759i	$0.0201229\pi$
$709$			3.36438i			0.126352i	−0.998002	+	0.0631759i	$0.979877\pi$
$710$		0			0
$711$	11.0281			0.413588
$712$		0			0
$713$			10.6134i			0.397474i
$714$		0			0
$715$	18.7237	−	6.42019i	0.700228	−	0.240101i
$716$		0			0
$717$	11.8328			0.441903
$718$		0			0
$719$	23.7926			0.887315			0.443657	−	0.896196i	$-0.353681\pi$
$719$	23.7926			0.887315			0.443657	+	0.896196i	$0.353681\pi$
$720$		0			0
$721$	15.6671			0.583475
$722$		0			0
$723$	−19.1963			−0.713920
$724$		0			0
$725$	−6.39406	−	8.22753i	−0.237469	−	0.305563i
$726$		0			0
$727$		−	10.4614i		−	0.387992i	−0.981002	−	0.193996i	$-0.937855\pi$
$727$		−	10.4614i		−	0.387992i	0.981002	−	0.193996i	$-0.0621448\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$		−	5.51605i		−	0.204019i
$732$		0			0
$733$	14.3975			0.531783			0.265891	−	0.964003i	$-0.414334\pi$
$733$	14.3975			0.531783			0.265891	+	0.964003i	$0.414334\pi$
$734$		0			0
$735$	−0.725275	−	2.11518i	−0.0267522	−	0.0780195i
$736$		0			0
$737$		−	11.9355i		−	0.439650i
$738$		0			0
$739$			19.5159i			0.717905i	0.933356	+	0.358952i	$0.116866\pi$
$739$			19.5159i			0.717905i	−0.933356	+	0.358952i	$0.883134\pi$
$740$		0			0
$741$		−	28.5676i		−	1.04946i
$742$		0			0
$743$		−	21.0471i		−	0.772143i	−0.922469	−	0.386072i	$-0.873832\pi$
$743$		−	21.0471i		−	0.772143i	0.922469	−	0.386072i	$-0.126168\pi$
$744$		0			0
$745$	−22.4594	+	7.70110i	−0.822847	+	0.282146i
$746$		0			0
$747$	−2.69790			−0.0987110
$748$		0			0
$749$			16.6552i			0.608566i
$750$		0			0
$751$	40.0984			1.46321			0.731606	−	0.681728i	$-0.238771\pi$
$751$	40.0984			1.46321			0.731606	+	0.681728i	$0.238771\pi$
$752$		0			0
$753$		−	1.62899i		−	0.0593638i
$754$		0			0
$755$	−4.01545	−	11.7106i	−0.146137	−	0.426192i
$756$		0			0
$757$	11.1584			0.405560			0.202780	−	0.979224i	$-0.435002\pi$
$757$	11.1584			0.405560			0.202780	+	0.979224i	$0.435002\pi$
$758$		0			0
$759$	−16.4125			−0.595735
$760$		0			0
$761$	−49.3013			−1.78717			−0.893586	−	0.448893i	$-0.851819\pi$
$761$	−49.3013			−1.78717			−0.893586	+	0.448893i	$0.851819\pi$
$762$		0			0
$763$	14.9152			0.539965
$764$		0			0
$765$	−14.4534	+	4.95594i	−0.522565	+	0.179183i
$766$		0			0
$767$			6.02799i			0.217658i
$768$		0			0
$769$	−45.0982			−1.62628			−0.813141	−	0.582066i	$-0.802244\pi$
$769$	−45.0982			−1.62628			−0.813141	+	0.582066i	$0.802244\pi$
$770$		0			0
$771$		−	1.89332i		−	0.0681862i
$772$		0			0
$773$	−28.0013			−1.00714			−0.503568	−	0.863956i	$-0.667980\pi$
$773$	−28.0013			−1.00714			−0.503568	+	0.863956i	$0.667980\pi$
$774$		0			0
$775$	5.06662	−	3.93755i	0.181998	−	0.141441i
$776$		0			0
$777$		−	10.5278i		−	0.377684i
$778$		0			0
$779$		−	37.4278i		−	1.34099i
$780$		0			0
$781$			14.3501i			0.513485i
$782$		0			0
$783$		−	2.08400i		−	0.0744760i
$784$		0			0
$785$	−10.9147	−	31.8315i	−0.389563	−	1.13611i
$786$		0			0
$787$	5.29498			0.188746			0.0943728	−	0.995537i	$-0.469915\pi$
$787$	5.29498			0.188746			0.0943728	+	0.995537i	$0.469915\pi$
$788$		0			0
$789$			4.66210i			0.165975i
$790$		0			0
$791$	−2.41294			−0.0857942
$792$		0			0
$793$			49.6526i			1.76321i
$794$		0			0
$795$	3.48286	+	10.1574i	0.123524	+	0.360244i
$796$		0			0
$797$	41.9018			1.48424			0.742119	−	0.670268i	$-0.233821\pi$
$797$	41.9018			1.48424			0.742119	+	0.670268i	$0.233821\pi$
$798$		0			0
$799$	61.5963			2.17912
$800$		0			0
$801$	2.66995			0.0943379
$802$		0			0
$803$	1.64522			0.0580585
$804$		0			0
$805$	−5.99804	−	17.4926i	−0.211403	−	0.616532i
$806$		0			0
$807$		−	12.3189i		−	0.433647i
$808$		0			0
$809$	−33.9710			−1.19436			−0.597178	−	0.802108i	$-0.703712\pi$
$809$	−33.9710			−1.19436			−0.597178	+	0.802108i	$0.703712\pi$
$810$		0			0
$811$			35.2089i			1.23635i	0.786040	+	0.618176i	$0.212128\pi$
$811$			35.2089i			1.23635i	−0.786040	+	0.618176i	$0.787872\pi$
$812$		0			0
$813$	−25.4743			−0.893422
$814$		0			0
$815$	4.51988	+	13.1817i	0.158324	+	0.461734i
$816$		0			0
$817$			5.17012i			0.180879i
$818$		0			0
$819$			4.46045i			0.155861i
$820$		0			0
$821$			3.87307i			0.135171i	0.997713	+	0.0675855i	$0.0215296\pi$
$821$			3.87307i			0.135171i	−0.997713	+	0.0675855i	$0.978470\pi$
$822$		0			0
$823$			11.7389i			0.409194i	0.978846	+	0.204597i	$0.0655884\pi$
$823$			11.7389i			0.409194i	−0.978846	+	0.204597i	$0.934412\pi$
$824$		0			0
$825$	6.08901	+	7.83501i	0.211992	+	0.272780i
$826$		0			0
$827$	−11.6682			−0.405744			−0.202872	−	0.979205i	$-0.565028\pi$
$827$	−11.6682			−0.405744			−0.202872	+	0.979205i	$0.565028\pi$
$828$		0			0
$829$		−	31.3408i		−	1.08851i	−0.838920	−	0.544255i	$-0.816813\pi$
$829$		−	31.3408i		−	1.08851i	0.838920	−	0.544255i	$-0.183187\pi$
$830$		0			0
$831$	−15.5659			−0.539976
$832$		0			0
$833$			6.83320i			0.236756i
$834$		0			0
$835$	−13.8418	+	4.74623i	−0.479015	+	0.164250i
$836$		0			0
$837$	1.28335			0.0443592
$838$		0			0
$839$	−36.2751			−1.25236			−0.626178	−	0.779680i	$-0.715382\pi$
$839$	−36.2751			−1.25236			−0.626178	+	0.779680i	$0.715382\pi$
$840$		0			0
$841$	24.6570			0.850240
$842$		0			0
$843$	−31.7854			−1.09475
$844$		0			0
$845$	−5.00118	−	14.5854i	−0.172046	−	0.501751i
$846$		0			0
$847$			7.06147i			0.242635i
$848$		0			0
$849$	24.6006			0.844290
$850$		0			0
$851$		−	87.0656i		−	2.98457i
$852$		0			0
$853$	−7.35109			−0.251696			−0.125848	−	0.992050i	$-0.540165\pi$
$853$	−7.35109			−0.251696			−0.125848	+	0.992050i	$0.540165\pi$
$854$		0			0
$855$	13.5470	−	4.64513i	0.463297	−	0.158860i
$856$		0			0
$857$			32.9054i			1.12403i	0.827128	+	0.562014i	$0.189973\pi$
$857$			32.9054i			1.12403i	−0.827128	+	0.562014i	$0.810027\pi$
$858$		0			0
$859$			47.6274i			1.62502i	0.582944	+	0.812512i	$0.301901\pi$
$859$			47.6274i			1.62502i	−0.582944	+	0.812512i	$0.698099\pi$
$860$		0			0
$861$			5.84384i			0.199157i
$862$		0			0
$863$			9.15980i			0.311803i	0.987773	+	0.155902i	$0.0498283\pi$
$863$			9.15980i			0.311803i	−0.987773	+	0.155902i	$0.950172\pi$
$864$		0			0
$865$	5.79529	+	16.9013i	0.197046	+	0.574661i
$866$		0			0
$867$	29.6926			1.00841
$868$		0			0
$869$			21.8862i			0.742437i
$870$		0			0
$871$	−26.8257			−0.908956
$872$		0			0
$873$		−	0.641074i		−	0.0216971i
$874$		0			0
$875$	−6.12536	+	9.35307i	−0.207075	+	0.316192i
$876$		0			0
$877$	−11.9012			−0.401876			−0.200938	−	0.979604i	$-0.564399\pi$
$877$	−11.9012			−0.401876			−0.200938	+	0.979604i	$0.564399\pi$
$878$		0			0
$879$	3.65931			0.123425
$880$		0			0
$881$	0.573821			0.0193325			0.00966626	−	0.999953i	$-0.496923\pi$
$881$	0.573821			0.0193325			0.00966626	+	0.999953i	$0.496923\pi$
$882$		0			0
$883$	−15.2575			−0.513456			−0.256728	−	0.966484i	$-0.582644\pi$
$883$	−15.2575			−0.513456			−0.256728	+	0.966484i	$0.582644\pi$
$884$		0			0
$885$	−2.85852	+	0.980160i	−0.0960881	+	0.0329477i
$886$		0			0
$887$		−	12.3853i		−	0.415858i	−0.978144	−	0.207929i	$-0.933328\pi$
$887$		−	12.3853i		−	0.415858i	0.978144	−	0.207929i	$-0.0666723\pi$
$888$		0			0
$889$	18.6981			0.627115
$890$		0			0
$891$			1.98457i			0.0664857i
$892$		0			0
$893$	−57.7333			−1.93197
$894$		0			0
$895$	−18.1086	+	6.20928i	−0.605305	+	0.207553i
$896$		0			0
$897$			36.8880i			1.23165i
$898$		0			0
$899$		−	2.67451i		−	0.0891998i
$900$		0			0
$901$		−	32.8139i		−	1.09319i
$902$		0			0
$903$		−	0.807243i		−	0.0268634i
$904$		0			0
$905$	−12.9930	+	4.45519i	−0.431903	+	0.148095i
$906$		0			0
$907$	17.4322			0.578826			0.289413	−	0.957204i	$-0.406540\pi$
$907$	17.4322			0.578826			0.289413	+	0.957204i	$0.406540\pi$
$908$		0			0
$909$		−	12.3440i		−	0.409426i
$910$		0			0
$911$	9.42203			0.312166			0.156083	−	0.987744i	$-0.450113\pi$
$911$	9.42203			0.312166			0.156083	+	0.987744i	$0.450113\pi$
$912$		0			0
$913$		−	5.35418i		−	0.177198i
$914$		0			0
$915$	−23.5456	+	8.07358i	−0.778395	+	0.266904i
$916$		0			0
$917$	14.7118			0.485825
$918$		0			0
$919$	20.0200			0.660398			0.330199	−	0.943911i	$-0.392884\pi$
$919$	20.0200			0.660398			0.330199	+	0.943911i	$0.392884\pi$
$920$		0			0
$921$	−23.6180			−0.778238
$922$		0			0
$923$	32.2526			1.06161
$924$		0			0
$925$	−41.5634	+	32.3012i	−1.36660	+	1.06206i
$926$		0			0
$927$		−	15.6671i		−	0.514576i
$928$		0			0
$929$	46.9433			1.54016			0.770081	−	0.637947i	$-0.220216\pi$
$929$	46.9433			1.54016			0.770081	+	0.637947i	$0.220216\pi$
$930$		0			0
$931$		−	6.40466i		−	0.209904i
$932$		0			0
$933$	29.5891			0.968705
$934$		0			0
$935$	−9.83544	−	28.6839i	−0.321653	−	0.938064i
$936$		0			0
$937$		−	48.8178i		−	1.59481i	−0.603446	−	0.797404i	$-0.706206\pi$
$937$		−	48.8178i		−	1.59481i	0.603446	−	0.797404i	$-0.293794\pi$
$938$		0			0
$939$			33.1631i			1.08224i
$940$		0			0
$941$		−	18.9062i		−	0.616324i	−0.951334	−	0.308162i	$-0.900286\pi$
$941$		−	18.9062i		−	0.616324i	0.951334	−	0.308162i	$-0.0997139\pi$
$942$		0			0
$943$			48.3287i			1.57380i
$944$		0			0
$945$	−2.11518	+	0.725275i	−0.0688067	+	0.0235932i
$946$		0			0
$947$	−23.8515			−0.775068			−0.387534	−	0.921855i	$-0.626673\pi$
$947$	−23.8515			−0.775068			−0.387534	+	0.921855i	$0.626673\pi$
$948$		0			0
$949$		−	3.69773i		−	0.120033i
$950$		0			0
$951$	−20.3788			−0.660829
$952$		0			0
$953$		−	8.36369i		−	0.270927i	−0.990782	−	0.135463i	$-0.956748\pi$
$953$		−	8.36369i		−	0.270927i	0.990782	−	0.135463i	$-0.0432523\pi$
$954$		0			0
$955$	−6.45194	−	18.8163i	−0.208780	−	0.608882i
$956$		0			0
$957$	4.13585			0.133693
$958$		0			0
$959$	10.8323			0.349793
$960$		0			0
$961$	−29.3530			−0.946871
$962$		0			0
$963$	16.6552			0.536705
$964$		0			0
$965$	19.1576	−	6.56897i	0.616706	−	0.211463i
$966$		0			0
$967$			12.3031i			0.395642i	0.980238	+	0.197821i	$0.0633866\pi$
$967$			12.3031i			0.395642i	−0.980238	+	0.197821i	$0.936613\pi$
$968$		0			0
$969$	−43.7643			−1.40591
$970$		0			0
$971$			22.8211i			0.732362i	0.930544	+	0.366181i	$0.119335\pi$
$971$			22.8211i			0.732362i	−0.930544	+	0.366181i	$0.880665\pi$
$972$		0			0
$973$	10.7222			0.343736
$974$		0			0
$975$	17.6096	−	13.6854i	0.563960	−	0.438284i
$976$		0			0
$977$			26.6256i			0.851828i	0.904764	+	0.425914i	$0.140047\pi$
$977$			26.6256i			0.851828i	−0.904764	+	0.425914i	$0.859953\pi$
$978$		0			0
$979$			5.29871i			0.169347i
$980$		0			0
$981$		−	14.9152i		−	0.476204i
$982$		0			0
$983$			52.9053i			1.68742i	0.536801	+	0.843709i	$0.319633\pi$
$983$			52.9053i			1.68742i	−0.536801	+	0.843709i	$0.680367\pi$
$984$		0			0
$985$	5.55521	+	16.2011i	0.177004	+	0.516210i
$986$		0			0
$987$	9.01427			0.286927
$988$		0			0
$989$		−	6.67593i		−	0.212282i
$990$		0			0
$991$	31.6648			1.00586			0.502932	−	0.864326i	$-0.332255\pi$
$991$	31.6648			1.00586			0.502932	+	0.864326i	$0.332255\pi$
$992$		0			0
$993$			1.14980i			0.0364878i
$994$		0			0
$995$	11.6621	+	34.0110i	0.369712	+	1.07822i
$996$		0			0
$997$	−41.9637			−1.32900			−0.664502	−	0.747286i	$-0.731356\pi$
$997$	−41.9637			−1.32900			−0.664502	+	0.747286i	$0.731356\pi$
$998$		0			0
$999$	−10.5278			−0.333086

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3360.2.j.e.1009.13		32
4.3	odd	2		840.2.j.f.589.22	yes	32
5.4	even	2		3360.2.j.f.1009.19		32
8.3	odd	2		840.2.j.e.589.12	yes	32
8.5	even	2		3360.2.j.f.1009.20		32
20.19	odd	2		840.2.j.e.589.11	✓	32
40.19	odd	2		840.2.j.f.589.21	yes	32
40.29	even	2	inner	3360.2.j.e.1009.14		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.j.e.589.11	✓	32	20.19	odd	2
840.2.j.e.589.12	yes	32	8.3	odd	2
840.2.j.f.589.21	yes	32	40.19	odd	2
840.2.j.f.589.22	yes	32	4.3	odd	2
3360.2.j.e.1009.13		32	1.1	even	1	trivial
3360.2.j.e.1009.14		32	40.29	even	2	inner
3360.2.j.f.1009.19		32	5.4	even	2
3360.2.j.f.1009.20		32	8.5	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3360.j (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +(-0.725275 - 2.11518i) q^{5} +1.00000i q^{7} +1.00000 q^{9} +1.98457i q^{11} +4.46045 q^{13} +(0.725275 + 2.11518i) q^{15} -6.83320i q^{17} +6.40466i q^{19} -1.00000i q^{21} -8.27003i q^{23} +(-3.94795 + 3.06817i) q^{25} -1.00000 q^{27} +2.08400i q^{29} -1.28335 q^{31} -1.98457i q^{33} +(2.11518 - 0.725275i) q^{35} +10.5278 q^{37} -4.46045 q^{39} -5.84384 q^{41} +0.807243 q^{43} +(-0.725275 - 2.11518i) q^{45} +9.01427i q^{47} -1.00000 q^{49} +6.83320i q^{51} +4.80213 q^{53} +(4.19773 - 1.43936i) q^{55} -6.40466i q^{57} +1.35143i q^{59} +11.1318i q^{61} +1.00000i q^{63} +(-3.23505 - 9.43463i) q^{65} -6.01414 q^{67} +8.27003i q^{69} +7.23080 q^{71} -0.829004i q^{73} +(3.94795 - 3.06817i) q^{75} -1.98457 q^{77} +11.0281 q^{79} +1.00000 q^{81} -2.69790 q^{83} +(-14.4534 + 4.95594i) q^{85} -2.08400i q^{87} +2.66995 q^{89} +4.46045i q^{91} +1.28335 q^{93} +(13.5470 - 4.64513i) q^{95} -0.641074i q^{97} +1.98457i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 32 q^{3} + 32 q^{9} + 32 q^{13} - 16 q^{25} - 32 q^{27} - 32 q^{31} - 4 q^{35} - 8 q^{37} - 32 q^{39} + 24 q^{41} + 8 q^{43} - 32 q^{49} - 24 q^{53} - 24 q^{55} + 8 q^{65} - 24 q^{67} - 40 q^{71}+ \cdots + 48 q^{95}+O(q^{100}) \) 32 * q - 32 * q^3 + 32 * q^9 + 32 * q^13 - 16 * q^25 - 32 * q^27 - 32 * q^31 - 4 * q^35 - 8 * q^37 - 32 * q^39 + 24 * q^41 + 8 * q^43 - 32 * q^49 - 24 * q^53 - 24 * q^55 + 8 * q^65 - 24 * q^67 - 40 * q^71 + 16 * q^75 - 16 * q^77 + 24 * q^79 + 32 * q^81 - 8 * q^85 + 24 * q^89 + 32 * q^93 + 48 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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