LMFDB - Embedded newform 2340.2.fo.a.1601.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2340,2,Mod(1241,2340)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2340, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 6, 0, 5])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2340.1241"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2340.fo (of order $12$, degree $4$, minimal)

Newform invariants

comment:select newform

sage:traces = [40,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$18.6849940730$
Analytic rank:	$0$
Dimension:	$40$
Relative dimension:	$10$ over $\Q(\zeta_{12})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			1601.1
Character	$\chi$	$=$	2340.1601
Dual form			2340.2.fo.a.2321.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 - 0.707107i) q^{5} +(-4.01056 + 1.07463i) q^{7} +(-3.52935 - 0.945687i) q^{11} +(-3.53472 - 0.711154i) q^{13} +(-0.250531 + 0.433933i) q^{17} +(-0.655804 - 2.44749i) q^{19} +(4.31373 + 7.47161i) q^{23} +1.00000i q^{25} +(1.88803 - 1.09005i) q^{29} +(7.27566 - 7.27566i) q^{31} +(3.59577 + 2.07602i) q^{35} +(-0.373943 + 1.39557i) q^{37} +(-0.600166 + 2.23985i) q^{41} +(10.4702 + 6.04499i) q^{43} +(3.63444 - 3.63444i) q^{47} +(8.86761 - 5.11972i) q^{49} -3.27937i q^{53} +(1.82693 + 3.16433i) q^{55} +(2.14790 + 8.01606i) q^{59} +(4.60699 - 7.97954i) q^{61} +(1.99656 + 3.00229i) q^{65} +(12.6951 + 3.40165i) q^{67} +(-12.8153 + 3.43386i) q^{71} +(-7.04050 - 7.04050i) q^{73} +15.1709 q^{77} +2.00972 q^{79} +(10.9028 + 10.9028i) q^{83} +(0.483989 - 0.129685i) q^{85} +(-12.3381 - 3.30598i) q^{89} +(14.9405 - 0.946382i) q^{91} +(-1.26692 + 2.19436i) q^{95} +(-2.79497 - 10.4310i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 12 q^{13} - 20 q^{19} - 28 q^{31} + 12 q^{49} + 16 q^{55} + 48 q^{61} + 16 q^{67} - 20 q^{73} + 80 q^{79} + 20 q^{85} + 4 q^{91} - 40 q^{97}+O(q^{100}) $ 40 * q - 12 * q^13 - 20 * q^19 - 28 * q^31 + 12 * q^49 + 16 * q^55 + 48 * q^61 + 16 * q^67 - 20 * q^73 + 80 * q^79 + 20 * q^85 + 4 * q^91 - 40 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2340\mathbb{Z}\right)^\times$.

$n$	$937$	$1081$	$1171$	$2081$
$\chi(n)$	$1$	$e\left(\frac{1}{12}\right)$	$1$	$-1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−0.707107	−	0.707107i	−0.316228	−	0.316228i
$5$	−0.707107	−	0.707107i	−0.316228	−	0.316228i
$6$		0			0
$7$	−4.01056	+	1.07463i	−1.51585	+	0.406171i	−0.918373	−	0.395715i	$-0.870497\pi$
$7$	−4.01056	+	1.07463i	−1.51585	+	0.406171i	−0.597477	+	0.801886i	$0.703830\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.52935	−	0.945687i	−1.06414	−	0.285135i	−0.316056	−	0.948740i	$-0.602359\pi$
$11$	−3.52935	−	0.945687i	−1.06414	−	0.285135i	−0.748083	+	0.663605i	$0.769026\pi$
$12$		0			0
$13$	−3.53472	−	0.711154i	−0.980356	−	0.197239i
$13$	−3.53472	−	0.711154i	−0.980356	−	0.197239i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.250531	+	0.433933i	−0.0607628	+	0.105244i	−0.894807	−	0.446454i	$-0.852687\pi$
$17$	−0.250531	+	0.433933i	−0.0607628	+	0.105244i	0.834044	+	0.551698i	$0.186020\pi$
$18$		0			0
$19$	−0.655804	−	2.44749i	−0.150452	−	0.561494i	−0.999452	−	0.0331012i	$-0.989462\pi$
$19$	−0.655804	−	2.44749i	−0.150452	−	0.561494i	0.849000	−	0.528393i	$-0.177205\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	4.31373	+	7.47161i	0.899476	+	1.55794i	0.828166	+	0.560483i	$0.189385\pi$
$23$	4.31373	+	7.47161i	0.899476	+	1.55794i	0.0713101	+	0.997454i	$0.477282\pi$
$24$		0			0
$25$			1.00000i			0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.88803	−	1.09005i	0.350598	−	0.202418i	−0.314351	−	0.949307i	$-0.601787\pi$
$29$	1.88803	−	1.09005i	0.350598	−	0.202418i	0.664949	+	0.746889i	$0.268453\pi$
$30$		0			0
$31$	7.27566	−	7.27566i	1.30675	−	1.30675i	0.382999	−	0.923749i	$-0.374891\pi$
$31$	7.27566	−	7.27566i	1.30675	−	1.30675i	0.923749	−	0.382999i	$-0.125109\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	3.59577	+	2.07602i	0.607796	+	0.350911i
$36$		0			0
$37$	−0.373943	+	1.39557i	−0.0614758	+	0.229431i	−0.989828	−	0.142272i	$-0.954559\pi$
$37$	−0.373943	+	1.39557i	−0.0614758	+	0.229431i	0.928352	+	0.371703i	$0.121226\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−0.600166	+	2.23985i	−0.0937302	+	0.349806i	−0.996824	−	0.0796378i	$-0.974624\pi$
$41$	−0.600166	+	2.23985i	−0.0937302	+	0.349806i	0.903094	+	0.429444i	$0.141290\pi$
$42$		0			0
$43$	10.4702	+	6.04499i	1.59670	+	0.921853i	0.992118	+	0.125307i	$0.0399915\pi$
$43$	10.4702	+	6.04499i	1.59670	+	0.921853i	0.604578	+	0.796546i	$0.293342\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.63444	−	3.63444i	0.530138	−	0.530138i	−0.390475	−	0.920613i	$-0.627689\pi$
$47$	3.63444	−	3.63444i	0.530138	−	0.530138i	0.920613	+	0.390475i	$0.127689\pi$
$48$		0			0
$49$	8.86761	−	5.11972i	1.26680	−	0.731388i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	3.27937i		−	0.450456i	−0.974306	−	0.225228i	$-0.927687\pi$
$53$		−	3.27937i		−	0.450456i	0.974306	−	0.225228i	$-0.0723127\pi$
$54$		0			0
$55$	1.82693	+	3.16433i	0.246343	+	0.426678i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	2.14790	+	8.01606i	0.279632	+	1.04360i	0.952673	+	0.303997i	$0.0983213\pi$
$59$	2.14790	+	8.01606i	0.279632	+	1.04360i	−0.673041	+	0.739606i	$0.735012\pi$
$60$		0			0
$61$	4.60699	−	7.97954i	0.589865	−	1.02168i	−0.404385	−	0.914589i	$-0.632514\pi$
$61$	4.60699	−	7.97954i	0.589865	−	1.02168i	0.994250	−	0.107087i	$-0.0341524\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.99656	+	3.00229i	0.247643	+	0.372388i
$66$		0			0
$67$	12.6951	+	3.40165i	1.55096	+	0.415578i	0.929787	−	0.368097i	$-0.119991\pi$
$67$	12.6951	+	3.40165i	1.55096	+	0.415578i	0.621171	+	0.783675i	$0.286657\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−12.8153	+	3.43386i	−1.52090	+	0.407524i	−0.920038	−	0.391829i	$-0.871842\pi$
$71$	−12.8153	+	3.43386i	−1.52090	+	0.407524i	−0.600861	+	0.799353i	$0.705176\pi$
$72$		0			0
$73$	−7.04050	−	7.04050i	−0.824028	−	0.824028i	0.162655	−	0.986683i	$-0.447994\pi$
$73$	−7.04050	−	7.04050i	−0.824028	−	0.824028i	−0.986683	+	0.162655i	$0.947994\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	15.1709			1.72889
$78$		0			0
$79$	2.00972			0.226111			0.113056	−	0.993589i	$-0.463936\pi$
$79$	2.00972			0.226111			0.113056	+	0.993589i	$0.463936\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	10.9028	+	10.9028i	1.19674	+	1.19674i	0.975138	+	0.221597i	$0.0711271\pi$
$83$	10.9028	+	10.9028i	1.19674	+	1.19674i	0.221597	+	0.975138i	$0.428873\pi$
$84$		0			0
$85$	0.483989	−	0.129685i	0.0524960	−	0.0140663i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−12.3381	−	3.30598i	−1.30783	−	0.350433i	−0.463427	−	0.886135i	$-0.653380\pi$
$89$	−12.3381	−	3.30598i	−1.30783	−	0.350433i	−0.844407	+	0.535702i	$0.820047\pi$
$90$		0			0
$91$	14.9405	−	0.946382i	1.56618	−	0.0992078i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.26692	+	2.19436i	−0.129983	+	0.225137i
$96$		0			0
$97$	−2.79497	−	10.4310i	−0.283786	−	1.05911i	−0.949722	−	0.313096i	$-0.898634\pi$
$97$	−2.79497	−	10.4310i	−0.283786	−	1.05911i	0.665935	−	0.746010i	$-0.268033\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	4.16671	+	7.21696i	0.414603	+	0.718114i	0.995387	−	0.0959440i	$-0.0305870\pi$
$101$	4.16671	+	7.21696i	0.414603	+	0.718114i	−0.580783	+	0.814058i	$0.697254\pi$
$102$		0			0
$103$			3.43554i			0.338514i	0.985572	+	0.169257i	$0.0541368\pi$
$103$			3.43554i			0.338514i	−0.985572	+	0.169257i	$0.945863\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−11.2736	+	6.50880i	−1.08986	+	0.629230i	−0.933539	−	0.358477i	$-0.883296\pi$
$107$	−11.2736	+	6.50880i	−1.08986	+	0.629230i	−0.156319	+	0.987707i	$0.549963\pi$
$108$		0			0
$109$	−7.42028	+	7.42028i	−0.710734	+	0.710734i	−0.966689	−	0.255955i	$-0.917610\pi$
$109$	−7.42028	+	7.42028i	−0.710734	+	0.710734i	0.255955	+	0.966689i	$0.417610\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.29483	+	0.747569i	0.121807	+	0.0703254i	0.559666	−	0.828719i	$-0.310930\pi$
$113$	1.29483	+	0.747569i	0.121807	+	0.0703254i	−0.437858	+	0.899044i	$0.644263\pi$
$114$		0			0
$115$	2.23295	−	8.33349i	0.208224	−	0.777102i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	0.538455	−	2.00954i	0.0493601	−	0.184214i
$120$		0			0
$121$	2.03571	+	1.17532i	0.185065	+	0.106847i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	0.707107	−	0.707107i	0.0632456	−	0.0632456i
$126$		0			0
$127$	16.3998	−	9.46843i	1.45525	−	0.840187i	0.456475	−	0.889736i	$-0.349112\pi$
$127$	16.3998	−	9.46843i	1.45525	−	0.840187i	0.998772	+	0.0495491i	$0.0157784\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	7.71300i		−	0.673888i	−0.941525	−	0.336944i	$-0.890607\pi$
$131$		−	7.71300i		−	0.673888i	0.941525	−	0.336944i	$-0.109393\pi$
$132$		0			0
$133$	5.26029	+	9.11109i	0.456125	+	0.790031i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	3.17079	+	11.8336i	0.270899	+	1.01101i	0.958540	+	0.284958i	$0.0919797\pi$
$137$	3.17079	+	11.8336i	0.270899	+	1.01101i	−0.687641	+	0.726051i	$0.741354\pi$
$138$		0			0
$139$	−1.74701	+	3.02591i	−0.148179	+	0.256654i	−0.930555	−	0.366153i	$-0.880675\pi$
$139$	−1.74701	+	3.02591i	−0.148179	+	0.256654i	0.782375	+	0.622807i	$0.214008\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	11.8027	+	5.85265i	0.986995	+	0.489423i
$144$		0			0
$145$	−2.10582	−	0.564253i	−0.174879	−	0.0468587i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	21.8376	−	5.85137i	1.78901	−	0.479363i	0.796831	−	0.604202i	$-0.206508\pi$
$149$	21.8376	−	5.85137i	1.78901	−	0.479363i	0.992177	+	0.124839i	$0.0398414\pi$
$150$		0			0
$151$	−15.3695	−	15.3695i	−1.25075	−	1.25075i	−0.955383	−	0.295371i	$-0.904557\pi$
$151$	−15.3695	−	15.3695i	−1.25075	−	1.25075i	−0.295371	−	0.955383i	$-0.595443\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−10.2893			−0.826460
$156$		0			0
$157$	−2.01108			−0.160501			−0.0802507	−	0.996775i	$-0.525572\pi$
$157$	−2.01108			−0.160501			−0.0802507	+	0.996775i	$0.525572\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−25.3297	−	25.3297i	−1.99626	−	1.99626i
$162$		0			0
$163$	1.46133	−	0.391562i	0.114460	−	0.0306695i	−0.201134	−	0.979564i	$-0.564463\pi$
$163$	1.46133	−	0.391562i	0.114460	−	0.0306695i	0.315594	+	0.948894i	$0.397796\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	2.50924	+	0.672350i	0.194171	+	0.0520280i	0.354594	−	0.935020i	$-0.384619\pi$
$167$	2.50924	+	0.672350i	0.194171	+	0.0520280i	−0.160423	+	0.987048i	$0.551286\pi$
$168$		0			0
$169$	11.9885	+	5.02746i	0.922194	+	0.386728i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	8.84184	−	15.3145i	0.672233	−	1.16434i	−0.305037	−	0.952340i	$-0.598669\pi$
$173$	8.84184	−	15.3145i	0.672233	−	1.16434i	0.977270	−	0.212000i	$-0.0679978\pi$
$174$		0			0
$175$	−1.07463	−	4.01056i	−0.0812342	−	0.303170i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	9.44475	+	16.3588i	0.705934	+	1.22271i	0.966353	+	0.257218i	$0.0828058\pi$
$179$	9.44475	+	16.3588i	0.705934	+	1.22271i	−0.260419	+	0.965496i	$0.583861\pi$
$180$		0			0
$181$		−	8.53654i		−	0.634516i	−0.948339	−	0.317258i	$-0.897238\pi$
$181$		−	8.53654i		−	0.634516i	0.948339	−	0.317258i	$-0.102762\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.25124	−	0.722402i	0.0919928	−	0.0531121i
$186$		0			0
$187$	1.29458	−	1.29458i	0.0946689	−	0.0946689i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−3.46898	−	2.00282i	−0.251007	−	0.144919i	0.369218	−	0.929343i	$-0.379625\pi$
$191$	−3.46898	−	2.00282i	−0.251007	−	0.144919i	−0.620225	+	0.784424i	$0.712959\pi$
$192$		0			0
$193$	−1.73011	+	6.45685i	−0.124536	+	0.464774i	−0.999823	−	0.0188299i	$-0.994006\pi$
$193$	−1.73011	+	6.45685i	−0.124536	+	0.464774i	0.875287	+	0.483604i	$0.160673\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1.23033	+	4.59166i	−0.0876575	+	0.327142i	−0.995804	−	0.0915100i	$-0.970831\pi$
$197$	−1.23033	+	4.59166i	−0.0876575	+	0.327142i	0.908147	+	0.418652i	$0.137497\pi$
$198$		0			0
$199$	−0.876215	−	0.505883i	−0.0621132	−	0.0358611i	0.468622	−	0.883399i	$-0.344751\pi$
$199$	−0.876215	−	0.505883i	−0.0621132	−	0.0358611i	−0.530735	+	0.847538i	$0.678084\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−6.40065	+	6.40065i	−0.449238	+	0.449238i
$204$		0			0
$205$	2.00819	−	1.15943i	0.140258	−	0.0809782i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			9.25825i			0.640407i
$210$		0			0
$211$	2.98181	+	5.16465i	0.205276	+	0.355549i	0.950221	−	0.311577i	$-0.100857\pi$
$211$	2.98181	+	5.16465i	0.205276	+	0.355549i	−0.744944	+	0.667127i	$0.767524\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−3.12912	−	11.6780i	−0.213404	−	0.796435i
$216$		0			0
$217$	−21.3609	+	36.9981i	−1.45007	+	2.51160i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1.19415	−	1.35567i	0.0803273	−	0.0911920i
$222$		0			0
$223$	4.41425	+	1.18279i	0.295600	+	0.0792057i	0.403571	−	0.914948i	$-0.367769\pi$
$223$	4.41425	+	1.18279i	0.295600	+	0.0792057i	−0.107971	+	0.994154i	$0.534435\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	18.0088	−	4.82545i	1.19529	−	0.320277i	0.394315	−	0.918976i	$-0.370982\pi$
$227$	18.0088	−	4.82545i	1.19529	−	0.320277i	0.800974	+	0.598699i	$0.204315\pi$
$228$		0			0
$229$	−5.88703	−	5.88703i	−0.389026	−	0.389026i	0.485314	−	0.874340i	$-0.338705\pi$
$229$	−5.88703	−	5.88703i	−0.389026	−	0.389026i	−0.874340	+	0.485314i	$0.838705\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−6.90758			−0.452531			−0.226265	−	0.974066i	$-0.572652\pi$
$233$	−6.90758			−0.452531			−0.226265	+	0.974066i	$0.572652\pi$
$234$		0			0
$235$	−5.13988			−0.335289
$236$		0			0
$237$		0			0
$238$		0			0
$239$	14.2978	+	14.2978i	0.924848	+	0.924848i	0.997367	−	0.0725191i	$-0.0231038\pi$
$239$	14.2978	+	14.2978i	0.924848	+	0.924848i	−0.0725191	+	0.997367i	$0.523104\pi$
$240$		0			0
$241$	15.5944	−	4.17850i	1.00452	−	0.269161i	0.281183	−	0.959654i	$-0.409273\pi$
$241$	15.5944	−	4.17850i	1.00452	−	0.269161i	0.723339	+	0.690493i	$0.242607\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−9.89054	−	2.65016i	−0.631883	−	0.169313i
$246$		0			0
$247$	0.577541	+	9.11759i	0.0367481	+	0.580138i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−12.8248	+	22.2133i	−0.809497	+	1.40209i	0.103715	+	0.994607i	$0.466927\pi$
$251$	−12.8248	+	22.2133i	−0.809497	+	1.40209i	−0.913213	+	0.407483i	$0.866406\pi$
$252$		0			0
$253$	−8.15888	−	30.4494i	−0.512944	−	1.91433i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	8.08608	+	14.0055i	0.504396	+	0.873639i	0.999987	+	0.00508346i	$0.00161812\pi$
$257$	8.08608	+	14.0055i	0.504396	+	0.873639i	−0.495591	+	0.868556i	$0.665049\pi$
$258$		0			0
$259$		−	5.99888i		−	0.372753i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−26.3248	+	15.1986i	−1.62326	+	0.937188i	−0.637215	+	0.770686i	$0.719914\pi$
$263$	−26.3248	+	15.1986i	−1.62326	+	0.937188i	−0.986041	+	0.166502i	$0.946753\pi$
$264$		0			0
$265$	−2.31886	+	2.31886i	−0.142447	+	0.142447i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	6.15094	+	3.55125i	0.375030	+	0.216523i	0.675653	−	0.737219i	$-0.263862\pi$
$269$	6.15094	+	3.55125i	0.375030	+	0.216523i	−0.300624	+	0.953743i	$0.597195\pi$
$270$		0			0
$271$	0.301356	−	1.12468i	0.0183061	−	0.0683192i	−0.956169	−	0.292817i	$-0.905407\pi$
$271$	0.301356	−	1.12468i	0.0183061	−	0.0683192i	0.974475	+	0.224498i	$0.0720741\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.945687	−	3.52935i	0.0570270	−	0.212828i
$276$		0			0
$277$	−0.107751	−	0.0622103i	−0.00647415	−	0.00373786i	0.496759	−	0.867888i	$-0.334523\pi$
$277$	−0.107751	−	0.0622103i	−0.00647415	−	0.00373786i	−0.503234	+	0.864150i	$0.667856\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	23.1030	−	23.1030i	1.37821	−	1.37821i	0.530570	−	0.847641i	$-0.321978\pi$
$281$	23.1030	−	23.1030i	1.37821	−	1.37821i	0.847641	−	0.530570i	$-0.178022\pi$
$282$		0			0
$283$	11.0483	−	6.37873i	0.656753	−	0.379176i	−0.134286	−	0.990943i	$-0.542874\pi$
$283$	11.0483	−	6.37873i	0.656753	−	0.379176i	0.791039	+	0.611766i	$0.209541\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	9.62801i		−	0.568324i
$288$		0			0
$289$	8.37447	+	14.5050i	0.492616	+	0.853236i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−6.38558	−	23.8313i	−0.373050	−	1.39224i	−0.856174	−	0.516688i	$-0.827165\pi$
$293$	−6.38558	−	23.8313i	−0.373050	−	1.39224i	0.483124	−	0.875552i	$-0.339502\pi$
$294$		0			0
$295$	4.14942	−	7.18701i	0.241589	−	0.418444i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−9.93439	−	29.4778i	−0.574521	−	1.70474i
$300$		0			0
$301$	−48.4876	−	12.9922i	−2.79478	−	0.748859i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−8.90003	+	2.38475i	−0.509614	+	0.136551i
$306$		0			0
$307$	3.08427	+	3.08427i	0.176028	+	0.176028i	0.789622	−	0.613594i	$-0.210277\pi$
$307$	3.08427	+	3.08427i	0.176028	+	0.176028i	−0.613594	+	0.789622i	$0.710277\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−1.46711			−0.0831922			−0.0415961	−	0.999135i	$-0.513244\pi$
$311$	−1.46711			−0.0831922			−0.0415961	+	0.999135i	$0.513244\pi$
$312$		0			0
$313$	23.6515			1.33686			0.668430	−	0.743775i	$-0.266967\pi$
$313$	23.6515			1.33686			0.668430	+	0.743775i	$0.266967\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	13.8476	+	13.8476i	0.777756	+	0.777756i	0.979449	−	0.201692i	$-0.0646441\pi$
$317$	13.8476	+	13.8476i	0.777756	+	0.777756i	−0.201692	+	0.979449i	$0.564644\pi$
$318$		0			0
$319$	−7.69436	+	2.06170i	−0.430801	+	0.115433i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	1.22635	+	0.328599i	0.0682358	+	0.0182837i
$324$		0			0
$325$	0.711154	−	3.53472i	0.0394477	−	0.196071i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−10.6705	+	18.4818i	−0.588283	+	1.01894i
$330$		0			0
$331$	1.63770	+	6.11198i	0.0900161	+	0.335945i	0.996217	−	0.0869033i	$-0.0276971\pi$
$331$	1.63770	+	6.11198i	0.0900161	+	0.335945i	−0.906201	+	0.422848i	$0.861030\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−6.57149	−	11.3822i	−0.359039	−	0.621873i
$336$		0			0
$337$			14.7494i			0.803452i	0.915760	+	0.401726i	$0.131590\pi$
$337$			14.7494i			0.803452i	−0.915760	+	0.401726i	$0.868410\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−32.5589	+	18.7979i	−1.76316	+	1.01796i
$342$		0			0
$343$	−9.51078	+	9.51078i	−0.513534	+	0.513534i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	9.41926	+	5.43821i	0.505653	+	0.291939i	0.731045	−	0.682329i	$-0.239033\pi$
$347$	9.41926	+	5.43821i	0.505653	+	0.291939i	−0.225392	+	0.974268i	$0.572366\pi$
$348$		0			0
$349$	−2.31436	+	8.63731i	−0.123885	+	0.462345i	−0.999797	−	0.0201251i	$-0.993594\pi$
$349$	−2.31436	+	8.63731i	−0.123885	+	0.462345i	0.875913	+	0.482470i	$0.160260\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−5.05346	+	18.8598i	−0.268968	+	1.00380i	0.690808	+	0.723038i	$0.257255\pi$
$353$	−5.05346	+	18.8598i	−0.268968	+	1.00380i	−0.959776	+	0.280765i	$0.909412\pi$
$354$		0			0
$355$	11.4899	+	6.63370i	0.609821	+	0.352080i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	10.5133	−	10.5133i	0.554872	−	0.554872i	−0.372971	−	0.927843i	$-0.621661\pi$
$359$	10.5133	−	10.5133i	0.554872	−	0.554872i	0.927843	+	0.372971i	$0.121661\pi$
$360$		0			0
$361$	10.8943	−	6.28985i	0.573386	−	0.331045i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			9.95677i			0.521161i
$366$		0			0
$367$	−4.72665	−	8.18679i	−0.246729	−	0.427347i	0.715887	−	0.698216i	$-0.246022\pi$
$367$	−4.72665	−	8.18679i	−0.246729	−	0.427347i	−0.962616	+	0.270869i	$0.912689\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	3.52410	+	13.1521i	0.182962	+	0.682823i
$372$		0			0
$373$	6.19481	−	10.7297i	0.320755	−	0.555564i	−0.659889	−	0.751363i	$-0.729397\pi$
$373$	6.19481	−	10.7297i	0.320755	−	0.555564i	0.980644	+	0.195799i	$0.0627300\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−7.44885	+	2.51036i	−0.383635	+	0.129290i
$378$		0			0
$379$	31.2262	+	8.36703i	1.60398	+	0.429785i	0.946242	−	0.323461i	$-0.104846\pi$
$379$	31.2262	+	8.36703i	1.60398	+	0.429785i	0.657739	+	0.753246i	$0.271513\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	19.4649	−	5.21560i	0.994610	−	0.266505i	0.275424	−	0.961323i	$-0.411182\pi$
$383$	19.4649	−	5.21560i	0.994610	−	0.266505i	0.719186	+	0.694818i	$0.244515\pi$
$384$		0			0
$385$	−10.7275	−	10.7275i	−0.546723	−	0.546723i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	1.48657			0.0753720			0.0376860	−	0.999290i	$-0.488001\pi$
$389$	1.48657			0.0753720			0.0376860	+	0.999290i	$0.488001\pi$
$390$		0			0
$391$	−4.32290			−0.218618
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−1.42109	−	1.42109i	−0.0715026	−	0.0715026i
$396$		0			0
$397$	2.71632	−	0.727836i	0.136328	−	0.0365290i	−0.190010	−	0.981782i	$-0.560852\pi$
$397$	2.71632	−	0.727836i	0.136328	−	0.0365290i	0.326338	+	0.945253i	$0.394185\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−2.05571	−	0.550825i	−0.102657	−	0.0275069i	0.207125	−	0.978315i	$-0.433589\pi$
$401$	−2.05571	−	0.550825i	−0.102657	−	0.0275069i	−0.309782	+	0.950808i	$0.600256\pi$
$402$		0			0
$403$	−30.8916	+	20.5433i	−1.53882	+	1.02334i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	2.63955	−	4.57184i	0.130838	−	0.226617i
$408$		0			0
$409$	−1.17797	−	4.39625i	−0.0582470	−	0.217381i	0.930668	−	0.365866i	$-0.119227\pi$
$409$	−1.17797	−	4.39625i	−0.0582470	−	0.217381i	−0.988915	+	0.148485i	$0.952560\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−17.2286	−	29.8407i	−0.847762	−	1.46837i
$414$		0			0
$415$		−	15.4189i		−	0.756882i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	14.0440	−	8.10829i	0.686093	−	0.396116i	−0.116054	−	0.993243i	$-0.537025\pi$
$419$	14.0440	−	8.10829i	0.686093	−	0.396116i	0.802147	+	0.597127i	$0.203691\pi$
$420$		0			0
$421$	−16.4992	+	16.4992i	−0.804122	+	0.804122i	−0.983737	−	0.179615i	$-0.942515\pi$
$421$	−16.4992	+	16.4992i	−0.804122	+	0.804122i	0.179615	+	0.983737i	$0.442515\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−0.433933	−	0.250531i	−0.0210488	−	0.0121526i
$426$		0			0
$427$	−9.90160	+	36.9533i	−0.479172	+	1.78829i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	3.63236	−	13.5561i	0.174965	−	0.652977i	−0.821593	−	0.570074i	$-0.806914\pi$
$431$	3.63236	−	13.5561i	0.174965	−	0.652977i	0.996558	−	0.0829022i	$-0.0264189\pi$
$432$		0			0
$433$	−0.186315	−	0.107569i	−0.00895373	−	0.00516944i	0.495516	−	0.868599i	$-0.334979\pi$
$433$	−0.186315	−	0.107569i	−0.00895373	−	0.00516944i	−0.504470	+	0.863429i	$0.668312\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	15.4578	−	15.4578i	0.739445	−	0.739445i
$438$		0			0
$439$	−9.59192	+	5.53790i	−0.457797	+	0.264309i	−0.711118	−	0.703073i	$-0.751811\pi$
$439$	−9.59192	+	5.53790i	−0.457797	+	0.264309i	0.253320	+	0.967382i	$0.418477\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	19.6249i		−	0.932408i	−0.884677	−	0.466204i	$-0.845621\pi$
$443$		−	19.6249i		−	0.932408i	0.884677	−	0.466204i	$-0.154379\pi$
$444$		0			0
$445$	6.38666	+	11.0620i	0.302757	+	0.524390i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−4.83837	−	18.0571i	−0.228337	−	0.852165i	−0.981040	−	0.193805i	$-0.937917\pi$
$449$	−4.83837	−	18.0571i	−0.228337	−	0.852165i	0.752703	−	0.658360i	$-0.228750\pi$
$450$		0			0
$451$	4.23639	−	7.33765i	0.199484	−	0.345516i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−11.2337	−	9.89530i	−0.526643	−	0.463899i
$456$		0			0
$457$	−30.5132	−	8.17597i	−1.42734	−	0.382456i	−0.539262	−	0.842138i	$-0.681297\pi$
$457$	−30.5132	−	8.17597i	−1.42734	−	0.382456i	−0.888083	+	0.459682i	$0.847963\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−23.8645	+	6.39448i	−1.11148	+	0.297821i	−0.767432	−	0.641131i	$-0.778466\pi$
$461$	−23.8645	+	6.39448i	−1.11148	+	0.297821i	−0.344050	+	0.938951i	$0.611799\pi$
$462$		0			0
$463$	−28.7285	−	28.7285i	−1.33513	−	1.33513i	−0.900714	−	0.434413i	$-0.856956\pi$
$463$	−28.7285	−	28.7285i	−1.33513	−	1.33513i	−0.434413	−	0.900714i	$-0.643044\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	34.9601			1.61776			0.808880	−	0.587974i	$-0.200074\pi$
$467$	34.9601			1.61776			0.808880	+	0.587974i	$0.200074\pi$
$468$		0			0
$469$	−54.5702			−2.51982
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−31.2365	−	31.2365i	−1.43625	−	1.43625i
$474$		0			0
$475$	2.44749	−	0.655804i	0.112299	−	0.0300904i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−7.34671	−	1.96854i	−0.335680	−	0.0899451i	0.0870421	−	0.996205i	$-0.472259\pi$
$479$	−7.34671	−	1.96854i	−0.335680	−	0.0899451i	−0.422722	+	0.906260i	$0.638925\pi$
$480$		0			0
$481$	2.31425	−	4.66703i	0.105521	−	0.212798i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−5.39947	+	9.35216i	−0.245177	+	0.424660i
$486$		0			0
$487$	5.20417	+	19.4222i	0.235824	+	0.880106i	0.977776	+	0.209653i	$0.0672334\pi$
$487$	5.20417	+	19.4222i	0.235824	+	0.880106i	−0.741952	+	0.670453i	$0.766100\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−8.72223	−	15.1073i	−0.393629	−	0.681785i	0.599296	−	0.800527i	$-0.295447\pi$
$491$	−8.72223	−	15.1073i	−0.393629	−	0.681785i	−0.992925	+	0.118742i	$0.962114\pi$
$492$		0			0
$493$			1.09237i			0.0491978i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	47.7066	−	27.5434i	2.13993	−	1.23549i
$498$		0			0
$499$	−14.7662	+	14.7662i	−0.661028	+	0.661028i	−0.955622	−	0.294594i	$-0.904815\pi$
$499$	−14.7662	+	14.7662i	−0.661028	+	0.661028i	0.294594	+	0.955622i	$0.404815\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−12.3907	−	7.15375i	−0.552472	−	0.318970i	0.197646	−	0.980273i	$-0.436670\pi$
$503$	−12.3907	−	7.15375i	−0.552472	−	0.318970i	−0.750119	+	0.661303i	$0.770004\pi$
$504$		0			0
$505$	2.15685	−	8.04947i	0.0959785	−	0.358197i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	8.69157	−	32.4374i	0.385247	−	1.43776i	−0.452529	−	0.891749i	$-0.649478\pi$
$509$	8.69157	−	32.4374i	0.385247	−	1.43776i	0.837777	−	0.546013i	$-0.183855\pi$
$510$		0			0
$511$	35.8023	+	20.6705i	1.58380	+	0.914407i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	2.42929	−	2.42929i	0.107047	−	0.107047i
$516$		0			0
$517$	−16.2643	+	9.39018i	−0.715302	+	0.412980i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	40.0463i		−	1.75446i	−0.480071	−	0.877229i	$-0.659389\pi$
$521$		−	40.0463i		−	1.75446i	0.480071	−	0.877229i	$-0.340611\pi$
$522$		0			0
$523$	17.1739	+	29.7461i	0.750962	+	1.30070i	0.947357	+	0.320180i	$0.103743\pi$
$523$	17.1739	+	29.7461i	0.750962	+	1.30070i	−0.196395	+	0.980525i	$0.562923\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1.33437	+	4.97993i	0.0581260	+	0.216929i
$528$		0			0
$529$	−25.7166	+	44.5425i	−1.11811	+	1.93663i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	3.71430	−	7.49044i	0.160884	−	0.324447i
$534$		0			0
$535$	12.5740	+	3.36920i	0.543623	+	0.145663i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−36.1386	+	9.68330i	−1.55660	+	0.417089i
$540$		0			0
$541$	4.91604	+	4.91604i	0.211357	+	0.211357i	0.804844	−	0.593487i	$-0.202249\pi$
$541$	4.91604	+	4.91604i	0.211357	+	0.211357i	−0.593487	+	0.804844i	$0.702249\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	10.4939			0.449507
$546$		0			0
$547$	−9.71191			−0.415251			−0.207626	−	0.978208i	$-0.566574\pi$
$547$	−9.71191			−0.415251			−0.207626	+	0.978208i	$0.566574\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−3.90607	−	3.90607i	−0.166404	−	0.166404i
$552$		0			0
$553$	−8.06011	+	2.15970i	−0.342751	+	0.0918397i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	6.54304	+	1.75320i	0.277237	+	0.0742855i	0.394759	−	0.918785i	$-0.370828\pi$
$557$	6.54304	+	1.75320i	0.277237	+	0.0742855i	−0.117521	+	0.993070i	$0.537495\pi$
$558$		0			0
$559$	−32.7105	−	28.8133i	−1.38350	−	1.21867i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−12.2808	+	21.2710i	−0.517574	+	0.896464i	0.482218	+	0.876051i	$0.339831\pi$
$563$	−12.2808	+	21.2710i	−0.517574	+	0.896464i	−0.999792	+	0.0204126i	$0.993502\pi$
$564$		0			0
$565$	−0.386970	−	1.44419i	−0.0162800	−	0.0607577i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	10.2349	+	17.7274i	0.429071	+	0.743172i	0.996791	−	0.0800495i	$-0.0255078\pi$
$569$	10.2349	+	17.7274i	0.429071	+	0.743172i	−0.567720	+	0.823222i	$0.692174\pi$
$570$		0			0
$571$		−	37.1325i		−	1.55395i	−0.629534	−	0.776973i	$-0.716754\pi$
$571$		−	37.1325i		−	1.55395i	0.629534	−	0.776973i	$-0.283246\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−7.47161	+	4.31373i	−0.311588	+	0.179895i
$576$		0			0
$577$	−23.3196	+	23.3196i	−0.970806	+	0.970806i	−0.999586	−	0.0287801i	$-0.990838\pi$
$577$	−23.3196	+	23.3196i	−0.970806	+	0.970806i	0.0287801	+	0.999586i	$0.490838\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−55.4427	−	32.0099i	−2.30015	−	1.32799i
$582$		0			0
$583$	−3.10125	+	11.5740i	−0.128441	+	0.479348i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−10.6357	+	39.6930i	−0.438983	+	1.63831i	0.292369	+	0.956306i	$0.405557\pi$
$587$	−10.6357	+	39.6930i	−0.438983	+	1.63831i	−0.731352	+	0.682001i	$0.761110\pi$
$588$		0			0
$589$	−22.5786	−	13.0357i	−0.930333	−	0.537128i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−19.4085	+	19.4085i	−0.797013	+	0.797013i	−0.982623	−	0.185611i	$-0.940574\pi$
$593$	−19.4085	+	19.4085i	−0.797013	+	0.797013i	0.185611	+	0.982623i	$0.440574\pi$
$594$		0			0
$595$	−1.80171	+	1.04022i	−0.0738628	+	0.0426447i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			12.6710i			0.517725i	0.965914	+	0.258862i	$0.0833476\pi$
$599$			12.6710i			0.517725i	−0.965914	+	0.258862i	$0.916652\pi$
$600$		0			0
$601$	−22.0949	−	38.2695i	−0.901270	−	1.56104i	−0.825848	−	0.563893i	$-0.809303\pi$
$601$	−22.0949	−	38.2695i	−0.901270	−	1.56104i	−0.0754220	−	0.997152i	$-0.524030\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−0.608390	−	2.27054i	−0.0247346	−	0.0923107i
$606$		0			0
$607$	−3.28574	+	5.69107i	−0.133364	+	0.230993i	−0.924971	−	0.380037i	$-0.875911\pi$
$607$	−3.28574	+	5.69107i	−0.133364	+	0.230993i	0.791607	+	0.611030i	$0.209245\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−15.4314	+	10.2621i	−0.624288	+	0.415160i
$612$		0			0
$613$	0.885895	+	0.237375i	0.0357810	+	0.00958748i	0.276665	−	0.960966i	$-0.410771\pi$
$613$	0.885895	+	0.237375i	0.0357810	+	0.00958748i	−0.240884	+	0.970554i	$0.577437\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	13.5849	−	3.64006i	0.546908	−	0.146543i	0.0252230	−	0.999682i	$-0.491970\pi$
$617$	13.5849	−	3.64006i	0.546908	−	0.146543i	0.521685	+	0.853138i	$0.325304\pi$
$618$		0			0
$619$	23.9475	+	23.9475i	0.962532	+	0.962532i	0.999323	−	0.0367914i	$-0.0117137\pi$
$619$	23.9475	+	23.9475i	0.962532	+	0.962532i	−0.0367914	+	0.999323i	$0.511714\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	53.0353			2.12482
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−0.511901	−	0.511901i	−0.0204108	−	0.0204108i
$630$		0			0
$631$	8.08096	−	2.16529i	0.321698	−	0.0861987i	−0.0943562	−	0.995538i	$-0.530079\pi$
$631$	8.08096	−	2.16529i	0.321698	−	0.0861987i	0.416054	+	0.909340i	$0.363413\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−18.2916	−	4.90122i	−0.725880	−	0.194499i
$636$		0			0
$637$	−34.9855	+	11.7905i	−1.38617	+	0.467159i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−2.01509	+	3.49023i	−0.0795911	+	0.137856i	−0.903074	−	0.429486i	$-0.858695\pi$
$641$	−2.01509	+	3.49023i	−0.0795911	+	0.137856i	0.823482	+	0.567342i	$0.192028\pi$
$642$		0			0
$643$	−0.771371	−	2.87880i	−0.0304199	−	0.113529i	0.949047	−	0.315136i	$-0.102050\pi$
$643$	−0.771371	−	2.87880i	−0.0304199	−	0.113529i	−0.979466	+	0.201607i	$0.935383\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	10.5827	+	18.3297i	0.416048	+	0.720616i	0.995538	−	0.0943632i	$-0.0300815\pi$
$647$	10.5827	+	18.3297i	0.416048	+	0.720616i	−0.579490	+	0.814979i	$0.696748\pi$
$648$		0			0
$649$		−	30.3227i		−	1.19027i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−5.03699	+	2.90811i	−0.197113	+	0.113803i	−0.595308	−	0.803498i	$-0.702970\pi$
$653$	−5.03699	+	2.90811i	−0.197113	+	0.113803i	0.398195	+	0.917301i	$0.369637\pi$
$654$		0			0
$655$	−5.45391	+	5.45391i	−0.213102	+	0.213102i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	8.33083	+	4.80981i	0.324523	+	0.187363i	0.653407	−	0.757007i	$-0.273339\pi$
$659$	8.33083	+	4.80981i	0.324523	+	0.187363i	−0.328884	+	0.944370i	$0.606672\pi$
$660$		0			0
$661$	2.91475	−	10.8780i	0.113370	−	0.423104i	−0.885789	−	0.464088i	$-0.846382\pi$
$661$	2.91475	−	10.8780i	0.113370	−	0.423104i	0.999160	+	0.0409831i	$0.0130490\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	2.72293	−	10.1621i	0.105591	−	0.394069i
$666$		0			0
$667$	16.2889	+	9.40440i	0.630708	+	0.364140i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−23.8058	+	23.8058i	−0.919014	+	0.919014i
$672$		0			0
$673$	−20.4676	+	11.8170i	−0.788970	+	0.455512i	−0.839600	−	0.543206i	$-0.817210\pi$
$673$	−20.4676	+	11.8170i	−0.788970	+	0.455512i	0.0506299	+	0.998717i	$0.483877\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	1.45196i		−	0.0558033i	−0.999611	−	0.0279017i	$-0.991117\pi$
$677$		−	1.45196i		−	0.0558033i	0.999611	−	0.0279017i	$-0.00888253\pi$
$678$		0			0
$679$	22.4188	+	38.8305i	0.860355	+	1.49018i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1.94637	−	7.26395i	−0.0744757	−	0.277947i	0.918638	−	0.395100i	$-0.129290\pi$
$683$	−1.94637	−	7.26395i	−0.0744757	−	0.277947i	−0.993114	+	0.117153i	$0.962623\pi$
$684$		0			0
$685$	6.12550	−	10.6097i	0.234043	−	0.405375i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−2.33213	+	11.5917i	−0.0888472	+	0.441607i
$690$		0			0
$691$	11.7691	+	3.15352i	0.447718	+	0.119966i	0.475631	−	0.879645i	$-0.342220\pi$
$691$	11.7691	+	3.15352i	0.447718	+	0.119966i	−0.0279135	+	0.999610i	$0.508886\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	3.37496	−	0.904318i	0.128020	−	0.0343027i
$696$		0			0
$697$	−0.821584	−	0.821584i	−0.0311197	−	0.0311197i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−9.91579			−0.374514			−0.187257	−	0.982311i	$-0.559960\pi$
$701$	−9.91579			−0.374514			−0.187257	+	0.982311i	$0.559960\pi$
$702$		0			0
$703$	3.66089			0.138073
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−24.4664	−	24.4664i	−0.920154	−	0.920154i
$708$		0			0
$709$	39.2772	−	10.5243i	1.47509	−	0.395248i	0.570415	−	0.821357i	$-0.306782\pi$
$709$	39.2772	−	10.5243i	1.47509	−	0.395248i	0.904672	+	0.426108i	$0.140116\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	85.7462	+	22.9756i	3.21122	+	0.860444i
$714$		0			0
$715$	−4.20735	−	12.4842i	−0.157346	−	0.466884i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−15.5501	+	26.9335i	−0.579920	+	1.00445i	0.415567	+	0.909562i	$0.363583\pi$
$719$	−15.5501	+	26.9335i	−0.579920	+	1.00445i	−0.995488	+	0.0948892i	$0.969750\pi$
$720$		0			0
$721$	−3.69193	−	13.7785i	−0.137494	−	0.513136i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1.09005	+	1.88803i	0.0404835	+	0.0701196i
$726$		0			0
$727$		−	29.5381i		−	1.09551i	−0.836640	−	0.547753i	$-0.815483\pi$
$727$		−	29.5381i		−	1.09551i	0.836640	−	0.547753i	$-0.184517\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−5.24624	+	3.02892i	−0.194039	+	0.112029i
$732$		0			0
$733$	−15.4667	+	15.4667i	−0.571275	+	0.571275i	−0.932485	−	0.361209i	$-0.882364\pi$
$733$	−15.4667	+	15.4667i	−0.571275	+	0.571275i	0.361209	+	0.932485i	$0.382364\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−41.5887	−	24.0113i	−1.53194	−	0.884466i
$738$		0			0
$739$	12.7326	−	47.5186i	0.468375	−	1.74800i	−0.177076	−	0.984197i	$-0.556664\pi$
$739$	12.7326	−	47.5186i	0.468375	−	1.74800i	0.645451	−	0.763802i	$-0.276669\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−3.63939	+	13.5824i	−0.133516	+	0.498289i	−1.00000		0.000916616i	$-0.999708\pi$
$743$	−3.63939	+	13.5824i	−0.133516	+	0.498289i	0.866483	+	0.499206i	$0.166375\pi$
$744$		0			0
$745$	−19.5791	−	11.3040i	−0.717322	−	0.414146i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	38.2189	−	38.2189i	1.39649	−	1.39649i
$750$		0			0
$751$	28.4252	−	16.4113i	1.03725	−	0.598856i	0.118196	−	0.992990i	$-0.462289\pi$
$751$	28.4252	−	16.4113i	1.03725	−	0.598856i	0.919053	+	0.394135i	$0.128956\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			21.7358i			0.791046i
$756$		0			0
$757$	18.6587	+	32.3179i	0.678163	+	1.17461i	0.975534	+	0.219850i	$0.0705569\pi$
$757$	18.6587	+	32.3179i	0.678163	+	1.17461i	−0.297371	+	0.954762i	$0.596110\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	5.92952	+	22.1293i	0.214945	+	0.802185i	0.986186	+	0.165642i	$0.0529696\pi$
$761$	5.92952	+	22.1293i	0.214945	+	0.802185i	−0.771241	+	0.636543i	$0.780364\pi$
$762$		0			0
$763$	21.7855	−	37.7335i	0.788686	−	1.36605i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1.89157	−	29.8620i	−0.0683006	−	1.07826i
$768$		0			0
$769$	12.0913	+	3.23984i	0.436022	+	0.116832i	0.470152	−	0.882586i	$-0.344199\pi$
$769$	12.0913	+	3.23984i	0.436022	+	0.116832i	−0.0341296	+	0.999417i	$0.510866\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	2.38066	−	0.637896i	0.0856264	−	0.0229435i	−0.215752	−	0.976448i	$-0.569220\pi$
$773$	2.38066	−	0.637896i	0.0856264	−	0.0229435i	0.301378	+	0.953505i	$0.402553\pi$
$774$		0			0
$775$	7.27566	+	7.27566i	0.261350	+	0.261350i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	5.87561			0.210516
$780$		0			0
$781$	48.4771			1.73465
$782$		0			0
$783$		0			0
$784$		0			0
$785$	1.42205	+	1.42205i	0.0507550	+	0.0507550i
$786$		0			0
$787$	27.1713	−	7.28053i	0.968552	−	0.259523i	0.260336	−	0.965518i	$-0.416167\pi$
$787$	27.1713	−	7.28053i	0.968552	−	0.259523i	0.708217	+	0.705995i	$0.249500\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−5.99635	−	1.60672i	−0.213206	−	0.0571283i
$792$		0			0
$793$	−21.9591	+	24.9292i	−0.779791	+	0.885262i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−4.10345	+	7.10738i	−0.145352	+	0.251756i	−0.929504	−	0.368812i	$-0.879765\pi$
$797$	−4.10345	+	7.10738i	−0.145352	+	0.251756i	0.784152	+	0.620568i	$0.213098\pi$
$798$		0			0
$799$	0.666563	+	2.48765i	0.0235813	+	0.0880066i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	18.1903	+	31.5065i	0.641921	+	1.11184i
$804$		0			0
$805$			35.8216i			1.26255i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	46.0797	−	26.6041i	1.62008	−	0.935351i	0.633178	−	0.774006i	$-0.281750\pi$
$809$	46.0797	−	26.6041i	1.62008	−	0.935351i	0.986898	−	0.161346i	$-0.0515834\pi$
$810$		0			0
$811$	−26.9869	+	26.9869i	−0.947638	+	0.947638i	−0.998696	−	0.0510577i	$-0.983741\pi$
$811$	−26.9869	+	26.9869i	−0.947638	+	0.947638i	0.0510577	+	0.998696i	$0.483741\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−1.31019	−	0.756440i	−0.0458940	−	0.0264969i
$816$		0			0
$817$	7.92866	−	29.5902i	0.277389	−	1.03523i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−0.0878520	+	0.327868i	−0.00306606	+	0.0114427i	−0.967442	−	0.253093i	$-0.918552\pi$
$821$	−0.0878520	+	0.327868i	−0.00306606	+	0.0114427i	0.964376	+	0.264536i	$0.0852187\pi$
$822$		0			0
$823$	19.3093	+	11.1482i	0.673080	+	0.388603i	0.797243	−	0.603659i	$-0.206291\pi$
$823$	19.3093	+	11.1482i	0.673080	+	0.388603i	−0.124162	+	0.992262i	$0.539624\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	3.10744	−	3.10744i	0.108056	−	0.108056i	−0.651012	−	0.759068i	$-0.725655\pi$
$827$	3.10744	−	3.10744i	0.108056	−	0.108056i	0.759068	+	0.651012i	$0.225655\pi$
$828$		0			0
$829$	−5.78649	+	3.34083i	−0.200973	+	0.116032i	−0.597109	−	0.802160i	$-0.703684\pi$
$829$	−5.78649	+	3.34083i	−0.200973	+	0.116032i	0.396136	+	0.918192i	$0.370351\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			5.13060i			0.177765i
$834$		0			0
$835$	−1.29888	−	2.24973i	−0.0449496	−	0.0778550i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	14.2210	+	53.0734i	0.490962	+	1.83230i	0.551562	+	0.834134i	$0.314032\pi$
$839$	14.2210	+	53.0734i	0.490962	+	1.83230i	−0.0605993	+	0.998162i	$0.519301\pi$
$840$		0			0
$841$	−12.1236	+	20.9986i	−0.418054	+	0.724091i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−4.92221	−	12.0321i	−0.169329	−	0.413917i
$846$		0			0
$847$	−9.42738	−	2.52606i	−0.323929	−	0.0867964i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−12.0403	+	3.22618i	−0.412735	+	0.110592i
$852$		0			0
$853$	20.3851	+	20.3851i	0.697972	+	0.697972i	0.963973	−	0.266001i	$-0.0857024\pi$
$853$	20.3851	+	20.3851i	0.697972	+	0.697972i	−0.266001	+	0.963973i	$0.585702\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−23.8038			−0.813122			−0.406561	−	0.913624i	$-0.633272\pi$
$857$	−23.8038			−0.813122			−0.406561	+	0.913624i	$0.633272\pi$
$858$		0			0
$859$	35.6823			1.21746			0.608732	−	0.793376i	$-0.291679\pi$
$859$	35.6823			1.21746			0.608732	+	0.793376i	$0.291679\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−10.9082	−	10.9082i	−0.371320	−	0.371320i	0.496638	−	0.867958i	$-0.334568\pi$
$863$	−10.9082	−	10.9082i	−0.371320	−	0.371320i	−0.867958	+	0.496638i	$0.834568\pi$
$864$		0			0
$865$	−17.0811	+	4.57687i	−0.580776	+	0.155618i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−7.09300	−	1.90056i	−0.240614	−	0.0644722i
$870$		0			0
$871$	−42.4547	−	21.0521i	−1.43852	−	0.713323i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−2.07602	+	3.59577i	−0.0701823	+	0.121559i
$876$		0			0
$877$	−5.41299	−	20.2016i	−0.182784	−	0.682158i	−0.995094	−	0.0989328i	$-0.968457\pi$
$877$	−5.41299	−	20.2016i	−0.182784	−	0.682158i	0.812310	−	0.583225i	$-0.198210\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−20.5597	−	35.6104i	−0.692673	−	1.19974i	−0.970959	−	0.239246i	$-0.923100\pi$
$881$	−20.5597	−	35.6104i	−0.692673	−	1.19974i	0.278286	−	0.960498i	$-0.410234\pi$
$882$		0			0
$883$			45.2514i			1.52283i	0.648264	+	0.761415i	$0.275495\pi$
$883$			45.2514i			1.52283i	−0.648264	+	0.761415i	$0.724505\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−14.7677	+	8.52612i	−0.495850	+	0.286279i	−0.726998	−	0.686640i	$-0.759085\pi$
$887$	−14.7677	+	8.52612i	−0.495850	+	0.286279i	0.231148	+	0.972919i	$0.425752\pi$
$888$		0			0
$889$	−55.5974	+	55.5974i	−1.86468	+	1.86468i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−11.2788	−	6.51180i	−0.377430	−	0.217909i
$894$		0			0
$895$	4.88896	−	18.2459i	0.163420	−	0.609892i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	5.80579	−	21.6675i	0.193634	−	0.722652i
$900$		0			0
$901$	1.42303	+	0.821584i	0.0474078	+	0.0273709i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−6.03624	+	6.03624i	−0.200652	+	0.200652i
$906$		0			0
$907$	−9.11939	+	5.26508i	−0.302804	+	0.174824i	−0.643702	−	0.765276i	$-0.722602\pi$
$907$	−9.11939	+	5.26508i	−0.302804	+	0.174824i	0.340898	+	0.940100i	$0.389269\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	46.1066i		−	1.52758i	−0.645465	−	0.763790i	$-0.723336\pi$
$911$		−	46.1066i		−	1.52758i	0.645465	−	0.763790i	$-0.276664\pi$
$912$		0			0
$913$	−28.1691	−	48.7903i	−0.932262	−	1.61472i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	8.28860	+	30.9335i	0.273714	+	1.02151i
$918$		0			0
$919$	−2.25883	+	3.91240i	−0.0745118	+	0.129058i	−0.900874	−	0.434081i	$-0.857073\pi$
$919$	−2.25883	+	3.91240i	−0.0745118	+	0.129058i	0.826362	+	0.563139i	$0.190407\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	47.7406	−	3.02406i	1.57140	−	0.0995382i
$924$		0			0
$925$	−1.39557	−	0.373943i	−0.0458862	−	0.0122952i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	31.5770	−	8.46103i	1.03601	−	0.277597i	0.299550	−	0.954081i	$-0.403164\pi$
$929$	31.5770	−	8.46103i	1.03601	−	0.277597i	0.736458	+	0.676483i	$0.236497\pi$
$930$		0			0
$931$	−18.3459	−	18.3459i	−0.601263	−	0.601263i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−1.83081			−0.0598738
$936$		0			0
$937$	36.8605			1.20418			0.602090	−	0.798428i	$-0.294335\pi$
$937$	36.8605			1.20418			0.602090	+	0.798428i	$0.294335\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−17.5417	−	17.5417i	−0.571842	−	0.571842i	0.360801	−	0.932643i	$-0.382504\pi$
$941$	−17.5417	−	17.5417i	−0.571842	−	0.571842i	−0.932643	+	0.360801i	$0.882504\pi$
$942$		0			0
$943$	−19.3242	+	5.17791i	−0.629284	+	0.168616i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	40.7546	+	10.9202i	1.32435	+	0.354857i	0.850604	−	0.525807i	$-0.176237\pi$
$947$	40.7546	+	10.9202i	1.32435	+	0.354857i	0.473741	+	0.880664i	$0.342903\pi$
$948$		0			0
$949$	19.8793	+	29.8931i	0.645311	+	0.970371i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−17.2677	+	29.9085i	−0.559355	+	0.968832i	0.438195	+	0.898880i	$0.355618\pi$
$953$	−17.2677	+	29.9085i	−0.559355	+	0.968832i	−0.997550	+	0.0699518i	$0.977715\pi$
$954$		0			0
$955$	1.03673	+	3.86915i	0.0335479	+	0.125203i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−25.4333	−	44.0518i	−0.821285	−	1.42251i
$960$		0			0
$961$		−	74.8706i		−	2.41518i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	5.78905	−	3.34231i	0.186356	−	0.107593i
$966$		0			0
$967$	−20.7723	+	20.7723i	−0.667992	+	0.667992i	−0.957251	−	0.289259i	$-0.906591\pi$
$967$	−20.7723	+	20.7723i	−0.667992	+	0.667992i	0.289259	+	0.957251i	$0.406591\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	17.4856	+	10.0953i	0.561140	+	0.323974i	0.753603	−	0.657330i	$-0.228314\pi$
$971$	17.4856	+	10.0953i	0.561140	+	0.323974i	−0.192463	+	0.981304i	$0.561648\pi$
$972$		0			0
$973$	3.75476	−	14.0130i	0.120372	−	0.449235i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	3.38872	−	12.6469i	0.108415	−	0.404609i	−0.890295	−	0.455383i	$-0.849502\pi$
$977$	3.38872	−	12.6469i	0.108415	−	0.404609i	0.998710	+	0.0507742i	$0.0161689\pi$
$978$		0			0
$979$	40.4190	+	23.3359i	1.29180	+	0.745819i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	18.2463	−	18.2463i	0.581966	−	0.581966i	−0.353477	−	0.935443i	$-0.615001\pi$
$983$	18.2463	−	18.2463i	0.581966	−	0.581966i	0.935443	+	0.353477i	$0.115001\pi$
$984$		0			0
$985$	4.11677	−	2.37682i	0.131171	−	0.0757317i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			104.306i			3.31674i
$990$		0			0
$991$	−16.3995	−	28.4047i	−0.520947	−	0.902306i	−0.999703	−	0.0243585i	$-0.992246\pi$
$991$	−16.3995	−	28.4047i	−0.520947	−	0.902306i	0.478757	−	0.877948i	$-0.341088\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	0.261864	+	0.977291i	0.00830166	+	0.0309822i
$996$		0			0
$997$	18.4237	−	31.9108i	0.583485	−	1.01063i	−0.411577	−	0.911375i	$-0.635022\pi$
$997$	18.4237	−	31.9108i	0.583485	−	1.01063i	0.995062	−	0.0992512i	$-0.0316447\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2340.2.fo.a.1601.1	✓	40
3.2	odd	2	inner	2340.2.fo.a.1601.8	yes	40
13.7	odd	12	inner	2340.2.fo.a.2321.8	yes	40
39.20	even	12	inner	2340.2.fo.a.2321.1	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2340.2.fo.a.1601.1	✓	40	1.1	even	1	trivial
2340.2.fo.a.1601.8	yes	40	3.2	odd	2	inner
2340.2.fo.a.2321.1	yes	40	39.20	even	12	inner
2340.2.fo.a.2321.8	yes	40	13.7	odd	12	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2340.fo (of order \(12\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 - 0.707107i) q^{5} +(-4.01056 + 1.07463i) q^{7} +(-3.52935 - 0.945687i) q^{11} +(-3.53472 - 0.711154i) q^{13} +(-0.250531 + 0.433933i) q^{17} +(-0.655804 - 2.44749i) q^{19} +(4.31373 + 7.47161i) q^{23} +1.00000i q^{25} +(1.88803 - 1.09005i) q^{29} +(7.27566 - 7.27566i) q^{31} +(3.59577 + 2.07602i) q^{35} +(-0.373943 + 1.39557i) q^{37} +(-0.600166 + 2.23985i) q^{41} +(10.4702 + 6.04499i) q^{43} +(3.63444 - 3.63444i) q^{47} +(8.86761 - 5.11972i) q^{49} -3.27937i q^{53} +(1.82693 + 3.16433i) q^{55} +(2.14790 + 8.01606i) q^{59} +(4.60699 - 7.97954i) q^{61} +(1.99656 + 3.00229i) q^{65} +(12.6951 + 3.40165i) q^{67} +(-12.8153 + 3.43386i) q^{71} +(-7.04050 - 7.04050i) q^{73} +15.1709 q^{77} +2.00972 q^{79} +(10.9028 + 10.9028i) q^{83} +(0.483989 - 0.129685i) q^{85} +(-12.3381 - 3.30598i) q^{89} +(14.9405 - 0.946382i) q^{91} +(-1.26692 + 2.19436i) q^{95} +(-2.79497 - 10.4310i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 12 q^{13} - 20 q^{19} - 28 q^{31} + 12 q^{49} + 16 q^{55} + 48 q^{61} + 16 q^{67} - 20 q^{73} + 80 q^{79} + 20 q^{85} + 4 q^{91} - 40 q^{97}+O(q^{100}) \) 40 * q - 12 * q^13 - 20 * q^19 - 28 * q^31 + 12 * q^49 + 16 * q^55 + 48 * q^61 + 16 * q^67 - 20 * q^73 + 80 * q^79 + 20 * q^85 + 4 * q^91 - 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

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