LMFDB - Embedded newform 2340.2.fo.b.1961.1

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 = [40,0,0,0,0,0,8] 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			1961.1
Character	$\chi$	$=$	2340.1961
Dual form			2340.2.fo.b.1241.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 + 0.707107i) q^{5} +(0.0943773 + 0.352221i) q^{7} +(0.741884 - 2.76875i) q^{11} +(2.55727 - 2.54173i) q^{13} +(-2.67668 + 4.63615i) q^{17} +(-0.440635 + 0.118068i) q^{19} +(4.23522 + 7.33561i) q^{23} -1.00000i q^{25} +(3.94905 - 2.27999i) q^{29} +(-5.97321 - 5.97321i) q^{31} +(-0.315793 - 0.182323i) q^{35} +(8.63123 + 2.31273i) q^{37} +(-4.92527 - 1.31972i) q^{41} +(3.85361 + 2.22488i) q^{43} +(-1.40381 - 1.40381i) q^{47} +(5.94703 - 3.43352i) q^{49} -9.32366i q^{53} +(1.43321 + 2.48239i) q^{55} +(5.14683 - 1.37909i) q^{59} +(-2.66247 + 4.61153i) q^{61} +(-0.0109895 + 3.60553i) q^{65} +(3.61721 - 13.4996i) q^{67} +(3.70429 + 13.8246i) q^{71} +(6.70176 - 6.70176i) q^{73} +1.04523 q^{77} +8.79220 q^{79} +(-5.75093 + 5.75093i) q^{83} +(-1.38555 - 5.17096i) q^{85} +(-3.42864 + 12.7959i) q^{89} +(1.13660 + 0.660842i) q^{91} +(0.228090 - 0.395063i) q^{95} +(6.07736 - 1.62842i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 8 q^{7} + 12 q^{13} + 28 q^{19} + 36 q^{31} - 40 q^{37} - 72 q^{43} - 36 q^{49} - 32 q^{61} + 16 q^{67} + 28 q^{73} + 48 q^{79} - 20 q^{85} + 132 q^{91} + 48 q^{97}+O(q^{100}) $ 40 * q + 8 * q^7 + 12 * q^13 + 28 * q^19 + 36 * q^31 - 40 * q^37 - 72 * q^43 - 36 * q^49 - 32 * q^61 + 16 * q^67 + 28 * q^73 + 48 * q^79 - 20 * q^85 + 132 * q^91 + 48 * q^97

Character values

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

$n$	$937$	$1081$	$1171$	$2081$
$\chi(n)$	$1$	$e\left(\frac{7}{12}\right)$	$1$	$-1$

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
$7$	0.0943773	+	0.352221i	0.0356713	+	0.133127i	0.981465	−	0.191640i	$-0.0613807\pi$
$7$	0.0943773	+	0.352221i	0.0356713	+	0.133127i	−0.945794	+	0.324767i	$0.894714\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.741884	−	2.76875i	0.223686	−	0.834809i	−0.759240	−	0.650811i	$-0.774429\pi$
$11$	0.741884	−	2.76875i	0.223686	−	0.834809i	0.982927	−	0.183999i	$-0.0589042\pi$
$12$		0			0
$13$	2.55727	−	2.54173i	0.709259	−	0.704948i
$13$	2.55727	−	2.54173i	0.709259	−	0.704948i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.67668	+	4.63615i	−0.649191	+	1.12443i	0.334125	+	0.942529i	$0.391559\pi$
$17$	−2.67668	+	4.63615i	−0.649191	+	1.12443i	−0.983316	+	0.181903i	$0.941774\pi$
$18$		0			0
$19$	−0.440635	+	0.118068i	−0.101089	+	0.0270866i	−0.309009	−	0.951059i	$-0.599997\pi$
$19$	−0.440635	+	0.118068i	−0.101089	+	0.0270866i	0.207920	+	0.978146i	$0.433331\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	4.23522	+	7.33561i	0.883104	+	1.52958i	0.847872	+	0.530202i	$0.177884\pi$
$23$	4.23522	+	7.33561i	0.883104	+	1.52958i	0.0352322	+	0.999379i	$0.488783\pi$
$24$		0			0
$25$		−	1.00000i		−	0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.94905	−	2.27999i	0.733320	−	0.423383i	−0.0863154	−	0.996268i	$-0.527509\pi$
$29$	3.94905	−	2.27999i	0.733320	−	0.423383i	0.819636	+	0.572885i	$0.194176\pi$
$30$		0			0
$31$	−5.97321	−	5.97321i	−1.07282	−	1.07282i	−0.997132	−	0.0756883i	$-0.975885\pi$
$31$	−5.97321	−	5.97321i	−1.07282	−	1.07282i	−0.0756883	−	0.997132i	$-0.524115\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.315793	−	0.182323i	−0.0533787	−	0.0308182i
$36$		0			0
$37$	8.63123	+	2.31273i	1.41897	+	0.380211i	0.885117	−	0.465369i	$-0.154078\pi$
$37$	8.63123	+	2.31273i	1.41897	+	0.380211i	0.533849	+	0.845580i	$0.320745\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−4.92527	−	1.31972i	−0.769199	−	0.206106i	−0.147181	−	0.989110i	$-0.547020\pi$
$41$	−4.92527	−	1.31972i	−0.769199	−	0.206106i	−0.622017	+	0.783003i	$0.713687\pi$
$42$		0			0
$43$	3.85361	+	2.22488i	0.587670	+	0.339292i	0.764176	−	0.645008i	$-0.223146\pi$
$43$	3.85361	+	2.22488i	0.587670	+	0.339292i	−0.176505	+	0.984300i	$0.556479\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.40381	−	1.40381i	−0.204766	−	0.204766i	0.597272	−	0.802039i	$-0.296251\pi$
$47$	−1.40381	−	1.40381i	−0.204766	−	0.204766i	−0.802039	+	0.597272i	$0.796251\pi$
$48$		0			0
$49$	5.94703	−	3.43352i	0.849575	−	0.490502i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	9.32366i		−	1.28070i	−0.768082	−	0.640352i	$-0.778789\pi$
$53$		−	9.32366i		−	1.28070i	0.768082	−	0.640352i	$-0.221211\pi$
$54$		0			0
$55$	1.43321	+	2.48239i	0.193254	+	0.334726i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	5.14683	−	1.37909i	0.670060	−	0.179542i	0.0922781	−	0.995733i	$-0.470585\pi$
$59$	5.14683	−	1.37909i	0.670060	−	0.179542i	0.577782	+	0.816191i	$0.303918\pi$
$60$		0			0
$61$	−2.66247	+	4.61153i	−0.340894	+	0.590445i	−0.984599	−	0.174828i	$-0.944063\pi$
$61$	−2.66247	+	4.61153i	−0.340894	+	0.590445i	0.643705	+	0.765274i	$0.277396\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.0109895	+	3.60553i	−0.00136308	+	0.447212i
$66$		0			0
$67$	3.61721	−	13.4996i	0.441912	−	1.64924i	−0.282050	−	0.959400i	$-0.591014\pi$
$67$	3.61721	−	13.4996i	0.441912	−	1.64924i	0.723962	−	0.689839i	$-0.242319\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	3.70429	+	13.8246i	0.439619	+	1.64068i	0.729766	+	0.683697i	$0.239629\pi$
$71$	3.70429	+	13.8246i	0.439619	+	1.64068i	−0.290148	+	0.956982i	$0.593704\pi$
$72$		0			0
$73$	6.70176	−	6.70176i	0.784382	−	0.784382i	−0.196185	−	0.980567i	$-0.562855\pi$
$73$	6.70176	−	6.70176i	0.784382	−	0.784382i	0.980567	+	0.196185i	$0.0628553\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.04523			0.119115
$78$		0			0
$79$	8.79220			0.989200			0.494600	−	0.869121i	$-0.335315\pi$
$79$	8.79220			0.989200			0.494600	+	0.869121i	$0.335315\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−5.75093	+	5.75093i	−0.631246	+	0.631246i	−0.948381	−	0.317134i	$-0.897279\pi$
$83$	−5.75093	+	5.75093i	−0.631246	+	0.631246i	0.317134	+	0.948381i	$0.397279\pi$
$84$		0			0
$85$	−1.38555	−	5.17096i	−0.150284	−	0.560869i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−3.42864	+	12.7959i	−0.363435	+	1.35636i	0.506094	+	0.862478i	$0.331089\pi$
$89$	−3.42864	+	12.7959i	−0.363435	+	1.35636i	−0.869530	+	0.493881i	$0.835578\pi$
$90$		0			0
$91$	1.13660	+	0.660842i	0.119148	+	0.0692751i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	0.228090	−	0.395063i	0.0234015	−	0.0405326i
$96$		0			0
$97$	6.07736	−	1.62842i	0.617063	−	0.165341i	0.0632707	−	0.997996i	$-0.479847\pi$
$97$	6.07736	−	1.62842i	0.617063	−	0.165341i	0.553792	+	0.832655i	$0.313180\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.37827	+	11.0475i	0.634661	+	1.09927i	0.986587	+	0.163237i	$0.0521936\pi$
$101$	6.37827	+	11.0475i	0.634661	+	1.09927i	−0.351926	+	0.936028i	$0.614473\pi$
$102$		0			0
$103$			11.2042i			1.10398i	0.833850	+	0.551991i	$0.186132\pi$
$103$			11.2042i			1.10398i	−0.833850	+	0.551991i	$0.813868\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	11.0311	−	6.36880i	1.06642	−	0.615696i	0.139216	−	0.990262i	$-0.455542\pi$
$107$	11.0311	−	6.36880i	1.06642	−	0.615696i	0.927200	+	0.374567i	$0.122208\pi$
$108$		0			0
$109$	9.42497	+	9.42497i	0.902749	+	0.902749i	0.995673	−	0.0929245i	$-0.0296215\pi$
$109$	9.42497	+	9.42497i	0.902749	+	0.902749i	−0.0929245	+	0.995673i	$0.529622\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	13.9331	+	8.04429i	1.31072	+	0.756743i	0.982215	−	0.187760i	$-0.0601228\pi$
$113$	13.9331	+	8.04429i	1.31072	+	0.756743i	0.328502	+	0.944503i	$0.393456\pi$
$114$		0			0
$115$	−8.18181	−	2.19231i	−0.762958	−	0.204434i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1.88557	−	0.505236i	−0.172850	−	0.0463149i
$120$		0			0
$121$	2.41070	+	1.39182i	0.219155	+	0.126529i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	0.707107	+	0.707107i	0.0632456	+	0.0632456i
$126$		0			0
$127$	−11.4078	+	6.58632i	−1.01228	+	0.584441i	−0.911859	−	0.410504i	$-0.865353\pi$
$127$	−11.4078	+	6.58632i	−1.01228	+	0.584441i	−0.100423	+	0.994945i	$0.532020\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			10.4866i			0.916217i	0.888896	+	0.458109i	$0.151473\pi$
$131$			10.4866i			0.916217i	−0.888896	+	0.458109i	$0.848527\pi$
$132$		0			0
$133$	−0.0831720	−	0.144058i	−0.00721192	−	0.0124914i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	7.61194	−	2.03961i	0.650332	−	0.174256i	0.0814531	−	0.996677i	$-0.474044\pi$
$137$	7.61194	−	2.03961i	0.650332	−	0.174256i	0.568879	+	0.822421i	$0.307377\pi$
$138$		0			0
$139$	2.34623	−	4.06379i	0.199004	−	0.344686i	−0.749201	−	0.662342i	$-0.769563\pi$
$139$	2.34623	−	4.06379i	0.199004	−	0.344686i	0.948206	+	0.317656i	$0.102896\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−5.14021	−	8.96610i	−0.429846	−	0.749783i
$144$		0			0
$145$	−1.18021	+	4.40459i	−0.0980109	+	0.365782i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	0.968416	+	3.61418i	0.0793358	+	0.296085i	0.994181	−	0.107719i	$-0.0343545\pi$
$149$	0.968416	+	3.61418i	0.0793358	+	0.296085i	−0.914846	+	0.403804i	$0.867688\pi$
$150$		0			0
$151$	13.2072	−	13.2072i	1.07478	−	1.07478i	0.0778166	−	0.996968i	$-0.475205\pi$
$151$	13.2072	−	13.2072i	1.07478	−	1.07478i	0.996968	−	0.0778166i	$-0.0247949\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	8.44739			0.678511
$156$		0			0
$157$	19.1626			1.52935			0.764673	−	0.644419i	$-0.222901\pi$
$157$	19.1626			1.52935			0.764673	+	0.644419i	$0.222901\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−2.18405	+	2.18405i	−0.172127	+	0.172127i
$162$		0			0
$163$	−1.59062	−	5.93628i	−0.124587	−	0.464966i	0.875237	−	0.483694i	$-0.160705\pi$
$163$	−1.59062	−	5.93628i	−0.124587	−	0.464966i	−0.999825	+	0.0187278i	$0.994038\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	3.04153	−	11.3511i	0.235361	−	0.878378i	−0.742625	−	0.669707i	$-0.766420\pi$
$167$	3.04153	−	11.3511i	0.235361	−	0.878378i	0.977986	−	0.208671i	$-0.0669136\pi$
$168$		0			0
$169$	0.0792459	−	12.9998i	0.00609584	−	0.999981i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.45218	+	2.51525i	−0.110407	+	0.191231i	−0.915935	−	0.401328i	$-0.868549\pi$
$173$	−1.45218	+	2.51525i	−0.110407	+	0.191231i	0.805527	+	0.592559i	$0.201882\pi$
$174$		0			0
$175$	0.352221	−	0.0943773i	0.0266254	−	0.00713425i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−6.59526	−	11.4233i	−0.492953	−	0.853820i	0.507014	−	0.861938i	$-0.330749\pi$
$179$	−6.59526	−	11.4233i	−0.492953	−	0.853820i	−0.999967	+	0.00811814i	$0.997416\pi$
$180$		0			0
$181$		−	23.1725i		−	1.72240i	−0.508266	−	0.861200i	$-0.669713\pi$
$181$		−	23.1725i		−	1.72240i	0.508266	−	0.861200i	$-0.330287\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−7.73855	+	4.46786i	−0.568950	+	0.328483i
$186$		0			0
$187$	10.8506	+	10.8506i	0.793471	+	0.793471i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−19.1414	−	11.0513i	−1.38502	−	0.799642i	−0.392271	−	0.919850i	$-0.628311\pi$
$191$	−19.1414	−	11.0513i	−1.38502	−	0.799642i	−0.992749	+	0.120208i	$0.961644\pi$
$192$		0			0
$193$	−19.8294	−	5.31327i	−1.42735	−	0.382458i	−0.539266	−	0.842136i	$-0.681298\pi$
$193$	−19.8294	−	5.31327i	−1.42735	−	0.382458i	−0.888086	+	0.459678i	$0.847965\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	7.88280	+	2.11219i	0.561626	+	0.150487i	0.528453	−	0.848962i	$-0.322772\pi$
$197$	7.88280	+	2.11219i	0.561626	+	0.150487i	0.0331729	+	0.999450i	$0.489439\pi$
$198$		0			0
$199$	−6.89149	−	3.97881i	−0.488525	−	0.282050i	0.235437	−	0.971890i	$-0.424348\pi$
$199$	−6.89149	−	3.97881i	−0.488525	−	0.282050i	−0.723962	+	0.689839i	$0.757681\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1.17576	+	1.17576i	0.0825221	+	0.0825221i
$204$		0			0
$205$	4.41588	−	2.54951i	0.308418	−	0.178065i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			1.30760i			0.0904487i
$210$		0			0
$211$	3.48632	+	6.03848i	0.240008	+	0.415706i	0.960716	−	0.277532i	$-0.0895166\pi$
$211$	3.48632	+	6.03848i	0.240008	+	0.415706i	−0.720708	+	0.693239i	$0.756183\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−4.29815	+	1.15168i	−0.293131	+	0.0785443i
$216$		0			0
$217$	1.54015	−	2.66762i	0.104552	−	0.181090i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	4.93883	+	18.6593i	0.332222	+	1.25516i
$222$		0			0
$223$	0.596124	−	2.22477i	0.0399194	−	0.148981i	−0.943090	−	0.332539i	$-0.892095\pi$
$223$	0.596124	−	2.22477i	0.0399194	−	0.148981i	0.983009	+	0.183558i	$0.0587614\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−0.858436	−	3.20373i	−0.0569764	−	0.212639i	0.931569	−	0.363566i	$-0.118441\pi$
$227$	−0.858436	−	3.20373i	−0.0569764	−	0.212639i	−0.988545	+	0.150927i	$0.951774\pi$
$228$		0			0
$229$	−9.86704	+	9.86704i	−0.652032	+	0.652032i	−0.953482	−	0.301450i	$-0.902529\pi$
$229$	−9.86704	+	9.86704i	−0.652032	+	0.652032i	0.301450	+	0.953482i	$0.402529\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−16.6009			−1.08756			−0.543780	−	0.839228i	$-0.683007\pi$
$233$	−16.6009			−1.08756			−0.543780	+	0.839228i	$0.683007\pi$
$234$		0			0
$235$	1.98528			0.129506
$236$		0			0
$237$		0			0
$238$		0			0
$239$	13.9444	−	13.9444i	0.901991	−	0.901991i	−0.0936174	−	0.995608i	$-0.529843\pi$
$239$	13.9444	−	13.9444i	0.901991	−	0.901991i	0.995608	+	0.0936174i	$0.0298430\pi$
$240$		0			0
$241$	4.81769	+	17.9799i	0.310335	+	1.15818i	0.928255	+	0.371944i	$0.121309\pi$
$241$	4.81769	+	17.9799i	0.310335	+	1.15818i	−0.617920	+	0.786241i	$0.712025\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.77732	+	6.63304i	−0.113549	+	0.423770i
$246$		0			0
$247$	−0.826727	+	1.42191i	−0.0526034	+	0.0904737i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−14.3742	+	24.8968i	−0.907288	+	1.57147i	−0.0894727	+	0.995989i	$0.528518\pi$
$251$	−14.3742	+	24.8968i	−0.907288	+	1.57147i	−0.817816	+	0.575480i	$0.804815\pi$
$252$		0			0
$253$	23.4525	−	6.28408i	1.47445	−	0.395077i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−9.48872	−	16.4349i	−0.591890	−	1.02518i	−0.993978	−	0.109583i	$-0.965049\pi$
$257$	−9.48872	−	16.4349i	−0.591890	−	1.02518i	0.402088	−	0.915601i	$-0.368285\pi$
$258$		0			0
$259$			3.25837i			0.202465i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	13.9273	−	8.04093i	0.858794	−	0.495825i	−0.00481415	−	0.999988i	$-0.501532\pi$
$263$	13.9273	−	8.04093i	0.858794	−	0.495825i	0.863608	+	0.504163i	$0.168199\pi$
$264$		0			0
$265$	6.59282	+	6.59282i	0.404994	+	0.404994i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−19.4762	−	11.2446i	−1.18748	−	0.685593i	−0.229748	−	0.973250i	$-0.573790\pi$
$269$	−19.4762	−	11.2446i	−1.18748	−	0.685593i	−0.957733	+	0.287657i	$0.907124\pi$
$270$		0			0
$271$	−31.1063	−	8.33492i	−1.88958	−	0.506310i	−0.998637	−	0.0521842i	$-0.983382\pi$
$271$	−31.1063	−	8.33492i	−1.88958	−	0.506310i	−0.890938	−	0.454126i	$-0.849952\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2.76875	−	0.741884i	−0.166962	−	0.0447373i
$276$		0			0
$277$	10.0765	+	5.81765i	0.605437	+	0.349549i	0.771177	−	0.636620i	$-0.219668\pi$
$277$	10.0765	+	5.81765i	0.605437	+	0.349549i	−0.165741	+	0.986169i	$0.553002\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	17.0909	+	17.0909i	1.01956	+	1.01956i	0.999805	+	0.0197551i	$0.00628864\pi$
$281$	17.0909	+	17.0909i	1.01956	+	1.01956i	0.0197551	+	0.999805i	$0.493711\pi$
$282$		0			0
$283$	3.86509	−	2.23151i	0.229755	−	0.132649i	−0.380704	−	0.924697i	$-0.624318\pi$
$283$	3.86509	−	2.23151i	0.229755	−	0.132649i	0.610459	+	0.792048i	$0.290985\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	1.85934i		−	0.109753i
$288$		0			0
$289$	−5.82927	−	10.0966i	−0.342898	−	0.593917i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−2.88583	+	0.773256i	−0.168592	+	0.0451741i	−0.342128	−	0.939654i	$-0.611148\pi$
$293$	−2.88583	+	0.773256i	−0.168592	+	0.0451741i	0.173536	+	0.984828i	$0.444481\pi$
$294$		0			0
$295$	−2.66419	+	4.61452i	−0.155115	+	0.268668i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	29.4757	+	7.99436i	1.70462	+	0.462326i
$300$		0			0
$301$	−0.419957	+	1.56730i	−0.0242059	+	0.0903377i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.37819	−	5.14349i	−0.0789151	−	0.294515i
$306$		0			0
$307$	−3.16535	+	3.16535i	−0.180656	+	0.180656i	−0.791642	−	0.610986i	$-0.790773\pi$
$307$	−3.16535	+	3.16535i	−0.180656	+	0.180656i	0.610986	+	0.791642i	$0.290773\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	4.00398			0.227045			0.113522	−	0.993535i	$-0.463787\pi$
$311$	4.00398			0.227045			0.113522	+	0.993535i	$0.463787\pi$
$312$		0			0
$313$	5.88412			0.332590			0.166295	−	0.986076i	$-0.446820\pi$
$313$	5.88412			0.332590			0.166295	+	0.986076i	$0.446820\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	14.7128	−	14.7128i	0.826356	−	0.826356i	−0.160655	−	0.987011i	$-0.551361\pi$
$317$	14.7128	−	14.7128i	0.826356	−	0.826356i	0.987011	+	0.160655i	$0.0513607\pi$
$318$		0			0
$319$	−3.38297	−	12.6254i	−0.189410	−	0.706888i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	0.632061	−	2.35888i	0.0351688	−	0.131252i
$324$		0			0
$325$	−2.54173	−	2.55727i	−0.140990	−	0.141852i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	0.361963	−	0.626938i	0.0199557	−	0.0345642i
$330$		0			0
$331$	−22.5912	+	6.05330i	−1.24173	+	0.332720i	−0.819135	−	0.573601i	$-0.805546\pi$
$331$	−22.5912	+	6.05330i	−1.24173	+	0.332720i	−0.422592	+	0.906320i	$0.638880\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	6.98791	+	12.1034i	0.381790	+	0.661280i
$336$		0			0
$337$			30.4217i			1.65717i	0.559860	+	0.828587i	$0.310855\pi$
$337$			30.4217i			1.65717i	−0.559860	+	0.828587i	$0.689145\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−20.9697	+	12.1069i	−1.13558	+	0.655625i
$342$		0			0
$343$	3.57552	+	3.57552i	0.193060	+	0.193060i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−21.0900	−	12.1763i	−1.13217	−	0.653660i	−0.187691	−	0.982228i	$-0.560101\pi$
$347$	−21.0900	−	12.1763i	−1.13217	−	0.653660i	−0.944480	+	0.328568i	$0.893434\pi$
$348$		0			0
$349$	20.3200	+	5.44474i	1.08771	+	0.291450i	0.757750	−	0.652545i	$-0.226299\pi$
$349$	20.3200	+	5.44474i	1.08771	+	0.291450i	0.329958	+	0.943996i	$0.392966\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	6.01614	+	1.61202i	0.320207	+	0.0857991i	0.415342	−	0.909665i	$-0.363662\pi$
$353$	6.01614	+	1.61202i	0.320207	+	0.0857991i	−0.0951356	+	0.995464i	$0.530328\pi$
$354$		0			0
$355$	−12.3948	−	7.15614i	−0.657848	−	0.379809i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−21.9258	−	21.9258i	−1.15720	−	1.15720i	−0.985076	−	0.172121i	$-0.944938\pi$
$359$	−21.9258	−	21.9258i	−1.15720	−	1.15720i	−0.172121	−	0.985076i	$-0.555062\pi$
$360$		0			0
$361$	−16.2743	+	9.39595i	−0.856540	+	0.494524i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			9.47772i			0.496087i
$366$		0			0
$367$	−5.59558	−	9.69183i	−0.292087	−	0.505909i	0.682216	−	0.731151i	$-0.261016\pi$
$367$	−5.59558	−	9.69183i	−0.292087	−	0.505909i	−0.974303	+	0.225241i	$0.927683\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	3.28399	−	0.879942i	0.170496	−	0.0456843i
$372$		0			0
$373$	−10.2637	+	17.7772i	−0.531433	+	0.920468i	0.467894	+	0.883784i	$0.345013\pi$
$373$	−10.2637	+	17.7772i	−0.531433	+	0.920468i	−0.999327	+	0.0366837i	$0.988321\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	4.30368	−	15.8679i	0.221651	−	0.817241i
$378$		0			0
$379$	−9.06352	+	33.8255i	−0.465562	+	1.73750i	0.189458	+	0.981889i	$0.439327\pi$
$379$	−9.06352	+	33.8255i	−0.465562	+	1.73750i	−0.655020	+	0.755611i	$0.727340\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−4.84644	−	18.0872i	−0.247641	−	0.924210i	−0.972037	−	0.234826i	$-0.924548\pi$
$383$	−4.84644	−	18.0872i	−0.247641	−	0.924210i	0.724396	−	0.689384i	$-0.242119\pi$
$384$		0			0
$385$	−0.739088	+	0.739088i	−0.0376674	+	0.0376674i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−3.92290			−0.198899			−0.0994495	−	0.995043i	$-0.531708\pi$
$389$	−3.92290			−0.198899			−0.0994495	+	0.995043i	$0.531708\pi$
$390$		0			0
$391$	−45.3453			−2.29321
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−6.21703	+	6.21703i	−0.312813	+	0.312813i
$396$		0			0
$397$	−6.00536	−	22.4123i	−0.301400	−	1.12484i	−0.936000	−	0.352001i	$-0.885502\pi$
$397$	−6.00536	−	22.4123i	−0.301400	−	1.12484i	0.634599	−	0.772841i	$-0.281165\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	0.136017	−	0.507621i	0.00679234	−	0.0253494i	−0.962446	−	0.271471i	$-0.912490\pi$
$401$	0.136017	−	0.507621i	0.00679234	−	0.0253494i	0.969239	+	0.246122i	$0.0791564\pi$
$402$		0			0
$403$	−30.4574	−	0.0928324i	−1.51719	−	0.00462431i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	12.8067	−	22.1819i	0.634807	−	1.09952i
$408$		0			0
$409$	26.8622	−	7.19771i	1.32825	−	0.355904i	0.476188	−	0.879343i	$-0.342018\pi$
$409$	26.8622	−	7.19771i	1.32825	−	0.355904i	0.852063	+	0.523440i	$0.175351\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	0.971487	+	1.68267i	0.0478038	+	0.0827986i
$414$		0			0
$415$		−	8.13304i		−	0.399235i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	0.200186	−	0.115578i	0.00977975	−	0.00564634i	−0.495102	−	0.868835i	$-0.664869\pi$
$419$	0.200186	−	0.115578i	0.00977975	−	0.00564634i	0.504882	+	0.863188i	$0.331536\pi$
$420$		0			0
$421$	−18.3716	−	18.3716i	−0.895378	−	0.895378i	0.0996449	−	0.995023i	$-0.468229\pi$
$421$	−18.3716	−	18.3716i	−0.895378	−	0.895378i	−0.995023	+	0.0996449i	$0.968229\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	4.63615	+	2.67668i	0.224886	+	0.129838i
$426$		0			0
$427$	−1.87555	−	0.502553i	−0.0907643	−	0.0243202i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−16.2167	−	4.34525i	−0.781132	−	0.209304i	−0.153848	−	0.988095i	$-0.549167\pi$
$431$	−16.2167	−	4.34525i	−0.781132	−	0.209304i	−0.627284	+	0.778791i	$0.715833\pi$
$432$		0			0
$433$	25.8889	+	14.9470i	1.24414	+	0.718305i	0.969935	−	0.243365i	$-0.0782514\pi$
$433$	25.8889	+	14.9470i	1.24414	+	0.718305i	0.274207	+	0.961671i	$0.411585\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−2.73229	−	2.73229i	−0.130703	−	0.130703i
$438$		0			0
$439$	−13.4168	+	7.74619i	−0.640348	+	0.369705i	−0.784749	−	0.619814i	$-0.787208\pi$
$439$	−13.4168	+	7.74619i	−0.640348	+	0.369705i	0.144400	+	0.989519i	$0.453875\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			5.99029i			0.284607i	0.989823	+	0.142304i	$0.0454509\pi$
$443$			5.99029i			0.284607i	−0.989823	+	0.142304i	$0.954549\pi$
$444$		0			0
$445$	−6.62363	−	11.4725i	−0.313990	−	0.543847i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	18.6369	−	4.99374i	0.879530	−	0.235669i	0.209325	−	0.977846i	$-0.432873\pi$
$449$	18.6369	−	4.99374i	0.879530	−	0.235669i	0.670204	+	0.742177i	$0.266207\pi$
$450$		0			0
$451$	−7.30797	+	12.6578i	−0.344119	+	0.596031i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1.27098	+	0.336410i	−0.0595845	+	0.0157711i
$456$		0			0
$457$	−1.90753	+	7.11901i	−0.0892306	+	0.333013i	−0.996082	−	0.0884380i	$-0.971812\pi$
$457$	−1.90753	+	7.11901i	−0.0892306	+	0.333013i	0.906851	+	0.421451i	$0.138479\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−1.55335	−	5.79719i	−0.0723469	−	0.270002i	0.920272	−	0.391280i	$-0.127968\pi$
$461$	−1.55335	−	5.79719i	−0.0723469	−	0.270002i	−0.992619	+	0.121278i	$0.961301\pi$
$462$		0			0
$463$	3.47393	−	3.47393i	0.161447	−	0.161447i	−0.621760	−	0.783208i	$-0.713582\pi$
$463$	3.47393	−	3.47393i	0.161447	−	0.161447i	0.783208	+	0.621760i	$0.213582\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−33.9452			−1.57080			−0.785398	−	0.618991i	$-0.787542\pi$
$467$	−33.9452			−1.57080			−0.785398	+	0.618991i	$0.787542\pi$
$468$		0			0
$469$	5.09622			0.235322
$470$		0			0
$471$		0			0
$472$		0			0
$473$	9.01908	−	9.01908i	0.414698	−	0.414698i
$474$		0			0
$475$	0.118068	+	0.440635i	0.00541733	+	0.0202177i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	7.98717	−	29.8085i	0.364943	−	1.36199i	−0.502555	−	0.864545i	$-0.667607\pi$
$479$	7.98717	−	29.8085i	0.364943	−	1.36199i	0.867498	−	0.497440i	$-0.165727\pi$
$480$		0			0
$481$	27.9507	−	16.0240i	1.27444	−	0.730630i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−3.14588	+	5.44882i	−0.142847	+	0.247418i
$486$		0			0
$487$	16.2642	−	4.35798i	0.737001	−	0.197479i	0.129256	−	0.991611i	$-0.458741\pi$
$487$	16.2642	−	4.35798i	0.737001	−	0.197479i	0.607745	+	0.794132i	$0.292074\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	1.26215	+	2.18611i	0.0569601	+	0.0986578i	0.893099	−	0.449859i	$-0.148526\pi$
$491$	1.26215	+	2.18611i	0.0569601	+	0.0986578i	−0.836139	+	0.548517i	$0.815193\pi$
$492$		0			0
$493$			24.4112i			1.09943i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−4.51971	+	2.60946i	−0.202737	+	0.117050i
$498$		0			0
$499$	7.14857	+	7.14857i	0.320014	+	0.320014i	0.848772	−	0.528759i	$-0.177342\pi$
$499$	7.14857	+	7.14857i	0.320014	+	0.320014i	−0.528759	+	0.848772i	$0.677342\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−15.0961	−	8.71572i	−0.673100	−	0.388615i	0.124150	−	0.992263i	$-0.460380\pi$
$503$	−15.0961	−	8.71572i	−0.673100	−	0.388615i	−0.797250	+	0.603649i	$0.793713\pi$
$504$		0			0
$505$	−12.3219	−	3.30163i	−0.548316	−	0.146921i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	6.08608	+	1.63076i	0.269761	+	0.0722821i	0.391164	−	0.920321i	$-0.372073\pi$
$509$	6.08608	+	1.63076i	0.269761	+	0.0722821i	−0.121403	+	0.992603i	$0.538739\pi$
$510$		0			0
$511$	2.99299	+	1.72801i	0.132402	+	0.0764425i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−7.92255	−	7.92255i	−0.349109	−	0.349109i
$516$		0			0
$517$	−4.92825	+	2.84533i	−0.216744	+	0.125137i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			11.1078i			0.486643i	0.969946	+	0.243321i	$0.0782369\pi$
$521$			11.1078i			0.486643i	−0.969946	+	0.243321i	$0.921763\pi$
$522$		0			0
$523$	−17.1039	−	29.6248i	−0.747900	−	1.29540i	−0.948828	−	0.315795i	$-0.897729\pi$
$523$	−17.1039	−	29.6248i	−0.747900	−	1.29540i	0.200927	−	0.979606i	$-0.435604\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	43.6811	−	11.7043i	1.90278	−	0.509848i
$528$		0			0
$529$	−24.3741	+	42.2172i	−1.05974	+	1.83553i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−15.9496	+	9.14382i	−0.690855	+	0.396063i
$534$		0			0
$535$	−3.29674	+	12.3036i	−0.142530	+	0.531930i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−5.09454	−	19.0131i	−0.219437	−	0.818952i
$540$		0			0
$541$	−26.9297	+	26.9297i	−1.15780	+	1.15780i	−0.172853	+	0.984948i	$0.555298\pi$
$541$	−26.9297	+	26.9297i	−1.15780	+	1.15780i	−0.984948	+	0.172853i	$0.944702\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−13.3289			−0.570948
$546$		0			0
$547$	5.96163			0.254901			0.127450	−	0.991845i	$-0.459321\pi$
$547$	5.96163			0.254901			0.127450	+	0.991845i	$0.459321\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−1.47090	+	1.47090i	−0.0626624	+	0.0626624i
$552$		0			0
$553$	0.829784	+	3.09680i	0.0352860	+	0.131689i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	3.99877	−	14.9236i	0.169433	−	0.632333i	−0.828000	−	0.560728i	$-0.810521\pi$
$557$	3.99877	−	14.9236i	0.169433	−	0.632333i	0.997433	−	0.0716048i	$-0.0228121\pi$
$558$		0			0
$559$	15.5098	−	4.10521i	0.655993	−	0.173632i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−3.55059	+	6.14980i	−0.149639	+	0.259183i	−0.931094	−	0.364779i	$-0.881145\pi$
$563$	−3.55059	+	6.14980i	−0.149639	+	0.259183i	0.781455	+	0.623962i	$0.214478\pi$
$564$		0			0
$565$	−15.5404	+	4.16403i	−0.653788	+	0.175182i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	8.42204	+	14.5874i	0.353070	+	0.611536i	0.986786	−	0.162030i	$-0.0518043\pi$
$569$	8.42204	+	14.5874i	0.353070	+	0.611536i	−0.633715	+	0.773566i	$0.718471\pi$
$570$		0			0
$571$			19.8996i			0.832774i	0.909187	+	0.416387i	$0.136704\pi$
$571$			19.8996i			0.832774i	−0.909187	+	0.416387i	$0.863296\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	7.33561	−	4.23522i	0.305916	−	0.176621i
$576$		0			0
$577$	−0.530957	−	0.530957i	−0.0221040	−	0.0221040i	0.695968	−	0.718072i	$-0.254975\pi$
$577$	−0.530957	−	0.530957i	−0.0221040	−	0.0221040i	−0.718072	+	0.695968i	$0.754975\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−2.56835	−	1.48284i	−0.106553	−	0.0615185i
$582$		0			0
$583$	−25.8149	−	6.91707i	−1.06914	−	0.286476i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	3.92004	+	1.05037i	0.161798	+	0.0433535i	0.338808	−	0.940855i	$-0.389976\pi$
$587$	3.92004	+	1.05037i	0.161798	+	0.0433535i	−0.177011	+	0.984209i	$0.556643\pi$
$588$		0			0
$589$	3.33725	+	1.92676i	0.137509	+	0.0793909i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	1.59697	+	1.59697i	0.0655796	+	0.0655796i	0.739136	−	0.673556i	$-0.235234\pi$
$593$	1.59697	+	1.59697i	0.0655796	+	0.0655796i	−0.673556	+	0.739136i	$0.735234\pi$
$594$		0			0
$595$	1.69055	−	0.976041i	0.0693059	−	0.0400138i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	4.71920i		−	0.192821i	−0.995342	−	0.0964107i	$-0.969264\pi$
$599$		−	4.71920i		−	0.192821i	0.995342	−	0.0964107i	$-0.0307362\pi$
$600$		0			0
$601$	−13.8054	−	23.9116i	−0.563132	−	0.975374i	−0.997221	−	0.0745035i	$-0.976263\pi$
$601$	−13.8054	−	23.9116i	−0.563132	−	0.975374i	0.434088	−	0.900870i	$-0.357071\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−2.68879	+	0.720458i	−0.109315	+	0.0292908i
$606$		0			0
$607$	−15.2581	+	26.4278i	−0.619307	+	1.07267i	0.370306	+	0.928910i	$0.379253\pi$
$607$	−15.2581	+	26.4278i	−0.619307	+	1.07267i	−0.989612	+	0.143761i	$0.954080\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−7.15801	−	0.0218172i	−0.289582	−	0.000882631i
$612$		0			0
$613$	−8.23877	+	30.7475i	−0.332761	+	1.24188i	0.573516	+	0.819195i	$0.305579\pi$
$613$	−8.23877	+	30.7475i	−0.332761	+	1.24188i	−0.906276	+	0.422686i	$0.861087\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	2.11071	+	7.87728i	0.0849741	+	0.317127i	0.995309	−	0.0967442i	$-0.0308429\pi$
$617$	2.11071	+	7.87728i	0.0849741	+	0.317127i	−0.910335	+	0.413872i	$0.864176\pi$
$618$		0			0
$619$	−9.45659	+	9.45659i	−0.380093	+	0.380093i	−0.871135	−	0.491043i	$-0.836616\pi$
$619$	−9.45659	+	9.45659i	−0.380093	+	0.380093i	0.491043	+	0.871135i	$0.336616\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−4.83056			−0.193532
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−33.8253	+	33.8253i	−1.34870	+	1.34870i
$630$		0			0
$631$	−6.88603	−	25.6990i	−0.274129	−	1.02306i	−0.956423	−	0.291984i	$-0.905685\pi$
$631$	−6.88603	−	25.6990i	−0.274129	−	1.02306i	0.682295	−	0.731077i	$-0.260982\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	3.40933	−	12.7238i	0.135295	−	0.504928i
$636$		0			0
$637$	6.48108	−	23.8961i	0.256790	−	0.946800i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−20.1230	+	34.8540i	−0.794809	+	1.37665i	0.128152	+	0.991755i	$0.459096\pi$
$641$	−20.1230	+	34.8540i	−0.794809	+	1.37665i	−0.922960	+	0.384895i	$0.874238\pi$
$642$		0			0
$643$	−0.868262	+	0.232650i	−0.0342409	+	0.00917482i	−0.275899	−	0.961187i	$-0.588975\pi$
$643$	−0.868262	+	0.232650i	−0.0342409	+	0.00917482i	0.241658	+	0.970362i	$0.422309\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−3.47623	−	6.02100i	−0.136665	−	0.236710i	0.789568	−	0.613664i	$-0.210305\pi$
$647$	−3.47623	−	6.02100i	−0.136665	−	0.236710i	−0.926232	+	0.376954i	$0.876972\pi$
$648$		0			0
$649$		−	15.2734i		−	0.599533i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	41.8416	−	24.1572i	1.63739	−	0.945345i	0.655658	−	0.755058i	$-0.272391\pi$
$653$	41.8416	−	24.1572i	1.63739	−	0.945345i	0.981728	−	0.190287i	$-0.0609420\pi$
$654$		0			0
$655$	−7.41514	−	7.41514i	−0.289733	−	0.289733i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	3.87776	+	2.23882i	0.151056	+	0.0872122i	0.573623	−	0.819120i	$-0.305538\pi$
$659$	3.87776	+	2.23882i	0.151056	+	0.0872122i	−0.422567	+	0.906332i	$0.638871\pi$
$660$		0			0
$661$	−38.0839	−	10.2046i	−1.48129	−	0.396911i	−0.574506	−	0.818501i	$-0.694806\pi$
$661$	−38.0839	−	10.2046i	−1.48129	−	0.396911i	−0.906787	+	0.421589i	$0.861472\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	0.160676	+	0.0430530i	0.00623074	+	0.00166952i
$666$		0			0
$667$	33.4502	+	19.3125i	1.29520	+	0.747782i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	10.7929	+	10.7929i	0.416656	+	0.416656i
$672$		0			0
$673$	15.4580	−	8.92469i	0.595863	−	0.344022i	−0.171549	−	0.985176i	$-0.554877\pi$
$673$	15.4580	−	8.92469i	0.595863	−	0.344022i	0.767412	+	0.641154i	$0.221544\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	6.42183i		−	0.246811i	−0.992356	−	0.123406i	$-0.960618\pi$
$677$		−	6.42183i		−	0.246811i	0.992356	−	0.123406i	$-0.0393816\pi$
$678$		0			0
$679$	1.14713	+	1.98689i	0.0440228	+	0.0762498i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−6.84317	+	1.83362i	−0.261847	+	0.0701616i	−0.387354	−	0.921931i	$-0.626611\pi$
$683$	−6.84317	+	1.83362i	−0.261847	+	0.0701616i	0.125507	+	0.992093i	$0.459944\pi$
$684$		0			0
$685$	−3.94023	+	6.82468i	−0.150548	+	0.260758i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−23.6982	−	23.8431i	−0.902829	−	0.908350i
$690$		0			0
$691$	2.10851	−	7.86908i	0.0802117	−	0.299354i	−0.914153	−	0.405370i	$-0.867143\pi$
$691$	2.10851	−	7.86908i	0.0802117	−	0.299354i	0.994364	+	0.106016i	$0.0338094\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1.21450	+	4.53256i	0.0460685	+	0.171930i
$696$		0			0
$697$	19.3018	−	19.3018i	0.731109	−	0.731109i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−11.4270			−0.431593			−0.215796	−	0.976438i	$-0.569235\pi$
$701$	−11.4270			−0.431593			−0.215796	+	0.976438i	$0.569235\pi$
$702$		0			0
$703$	−4.07629			−0.153740
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−3.28919	+	3.28919i	−0.123703	+	0.123703i
$708$		0			0
$709$	−5.12896	−	19.1415i	−0.192622	−	0.718876i	−0.992870	−	0.119206i	$-0.961965\pi$
$709$	−5.12896	−	19.1415i	−0.192622	−	0.718876i	0.800247	−	0.599670i	$-0.204701\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	18.5193	−	69.1150i	0.693553	−	2.58838i
$714$		0			0
$715$	9.97467	+	2.70532i	0.373031	+	0.101173i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	20.0442	−	34.7175i	0.747521	−	1.29475i	−0.201486	−	0.979491i	$-0.564577\pi$
$719$	20.0442	−	34.7175i	0.747521	−	1.29475i	0.949007	−	0.315254i	$-0.102090\pi$
$720$		0			0
$721$	−3.94635	+	1.05742i	−0.146970	+	0.0393804i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−2.27999	−	3.94905i	−0.0846765	−	0.146664i
$726$		0			0
$727$		−	4.19951i		−	0.155751i	−0.996963	−	0.0778755i	$-0.975186\pi$
$727$		−	4.19951i		−	0.155751i	0.996963	−	0.0778755i	$-0.0248137\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−20.6298	+	11.9106i	−0.763021	+	0.440530i
$732$		0			0
$733$	−3.94126	−	3.94126i	−0.145574	−	0.145574i	0.630564	−	0.776137i	$-0.282824\pi$
$733$	−3.94126	−	3.94126i	−0.145574	−	0.145574i	−0.776137	+	0.630564i	$0.782824\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−34.6935	−	20.0303i	−1.27795	−	0.737825i
$738$		0			0
$739$	−26.8799	−	7.20245i	−0.988794	−	0.264946i	−0.272050	−	0.962283i	$-0.587702\pi$
$739$	−26.8799	−	7.20245i	−0.988794	−	0.264946i	−0.716744	+	0.697337i	$0.754368\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−25.0468	−	6.71128i	−0.918879	−	0.246213i	−0.231773	−	0.972770i	$-0.574453\pi$
$743$	−25.0468	−	6.71128i	−0.918879	−	0.246213i	−0.687106	+	0.726557i	$0.741119\pi$
$744$		0			0
$745$	−3.24038	−	1.87084i	−0.118718	−	0.0685422i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	3.28431	+	3.28431i	0.120006	+	0.120006i
$750$		0			0
$751$	−24.7819	+	14.3079i	−0.904306	+	0.522101i	−0.878595	−	0.477568i	$-0.841518\pi$
$751$	−24.7819	+	14.3079i	−0.904306	+	0.522101i	−0.0257111	+	0.999669i	$0.508185\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			18.6778i			0.679753i
$756$		0			0
$757$	23.4332	+	40.5875i	0.851694	+	1.47518i	0.879678	+	0.475569i	$0.157758\pi$
$757$	23.4332	+	40.5875i	0.851694	+	1.47518i	−0.0279844	+	0.999608i	$0.508909\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−10.7826	+	2.88919i	−0.390869	+	0.104733i	−0.448901	−	0.893582i	$-0.648184\pi$
$761$	−10.7826	+	2.88919i	−0.390869	+	0.104733i	0.0580315	+	0.998315i	$0.481518\pi$
$762$		0			0
$763$	−2.43017	+	4.20917i	−0.0879780	+	0.152382i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	9.65656	−	16.6085i	0.348678	−	0.599699i
$768$		0			0
$769$	11.3385	−	42.3157i	0.408876	−	1.52594i	−0.387919	−	0.921694i	$-0.626806\pi$
$769$	11.3385	−	42.3157i	0.408876	−	1.52594i	0.796794	−	0.604251i	$-0.206527\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	1.44957	+	5.40985i	0.0521373	+	0.194579i	0.987082	−	0.160213i	$-0.0512182\pi$
$773$	1.44957	+	5.40985i	0.0521373	+	0.194579i	−0.934945	+	0.354792i	$0.884552\pi$
$774$		0			0
$775$	−5.97321	+	5.97321i	−0.214564	+	0.214564i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	2.32607			0.0833400
$780$		0			0
$781$	41.0250			1.46799
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−13.5500	+	13.5500i	−0.483622	+	0.483622i
$786$		0			0
$787$	10.4823	+	39.1205i	0.373654	+	1.39450i	0.855301	+	0.518131i	$0.173372\pi$
$787$	10.4823	+	39.1205i	0.373654	+	1.39450i	−0.481647	+	0.876365i	$0.659961\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−1.51840	+	5.66673i	−0.0539879	+	0.201486i
$792$		0			0
$793$	4.91260	+	18.5602i	0.174452	+	0.659091i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−0.465021	+	0.805441i	−0.0164719	+	0.0285302i	−0.874144	−	0.485667i	$-0.838577\pi$
$797$	−0.465021	+	0.805441i	−0.0164719	+	0.0285302i	0.857672	+	0.514197i	$0.171910\pi$
$798$		0			0
$799$	10.2658	−	2.75072i	0.363178	−	0.0973134i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−13.5836	−	23.5274i	−0.479354	−	0.830265i
$804$		0			0
$805$		−	3.08871i		−	0.108863i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−6.15315	+	3.55252i	−0.216333	+	0.124900i	−0.604251	−	0.796794i	$-0.706528\pi$
$809$	−6.15315	+	3.55252i	−0.216333	+	0.124900i	0.387918	+	0.921694i	$0.373194\pi$
$810$		0			0
$811$	2.95262	+	2.95262i	0.103681	+	0.103681i	0.757044	−	0.653364i	$-0.226643\pi$
$811$	2.95262	+	2.95262i	0.103681	+	0.103681i	−0.653364	+	0.757044i	$0.726643\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	5.32233	+	3.07285i	0.186433	+	0.107637i
$816$		0			0
$817$	−1.96073	−	0.525375i	−0.0685971	−	0.0183805i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	39.4238	+	10.5636i	1.37590	+	0.368671i	0.869630	−	0.493704i	$-0.164357\pi$
$821$	39.4238	+	10.5636i	1.37590	+	0.368671i	0.506270	+	0.862375i	$0.331024\pi$
$822$		0			0
$823$	−39.2644	−	22.6693i	−1.36867	−	0.790202i	−0.377912	−	0.925842i	$-0.623358\pi$
$823$	−39.2644	−	22.6693i	−1.36867	−	0.790202i	−0.990758	+	0.135640i	$0.956691\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−14.6636	−	14.6636i	−0.509903	−	0.509903i	0.404593	−	0.914497i	$-0.367413\pi$
$827$	−14.6636	−	14.6636i	−0.509903	−	0.509903i	−0.914497	+	0.404593i	$0.867413\pi$
$828$		0			0
$829$	−27.3885	+	15.8128i	−0.951242	+	0.549200i	−0.893467	−	0.449130i	$-0.851734\pi$
$829$	−27.3885	+	15.8128i	−0.951242	+	0.549200i	−0.0577756	+	0.998330i	$0.518401\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			36.7618i			1.27372i
$834$		0			0
$835$	5.87578	+	10.1772i	0.203340	+	0.352195i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	32.2463	−	8.64036i	1.11326	−	0.298298i	0.345110	−	0.938562i	$-0.387842\pi$
$839$	32.2463	−	8.64036i	1.11326	−	0.298298i	0.768155	+	0.640264i	$0.221175\pi$
$840$		0			0
$841$	−4.10333	+	7.10718i	−0.141494	+	0.245075i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	9.13618	+	9.24825i	0.314294	+	0.318150i
$846$		0			0
$847$	−0.262712	+	0.980455i	−0.00902689	+	0.0336888i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	19.5898	+	73.1103i	0.671531	+	2.50619i
$852$		0			0
$853$	32.7512	−	32.7512i	1.12138	−	1.12138i	0.129845	−	0.991534i	$-0.458552\pi$
$853$	32.7512	−	32.7512i	1.12138	−	1.12138i	0.991534	−	0.129845i	$-0.0414479\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	47.2081			1.61260			0.806299	−	0.591509i	$-0.201468\pi$
$857$	47.2081			1.61260			0.806299	+	0.591509i	$0.201468\pi$
$858$		0			0
$859$	45.6328			1.55697			0.778486	−	0.627662i	$-0.215988\pi$
$859$	45.6328			1.55697			0.778486	+	0.627662i	$0.215988\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−36.5583	+	36.5583i	−1.24446	+	1.24446i	−0.286326	+	0.958132i	$0.592434\pi$
$863$	−36.5583	+	36.5583i	−1.24446	+	1.24446i	−0.958132	+	0.286326i	$0.907566\pi$
$864$		0			0
$865$	−0.751705	−	2.80540i	−0.0255587	−	0.0953865i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	6.52280	−	24.3434i	0.221271	−	0.825793i
$870$		0			0
$871$	−25.0621	−	43.7161i	−0.849198	−	1.48126i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−0.182323	+	0.315793i	−0.00616364	+	0.0106757i
$876$		0			0
$877$	19.2133	−	5.14819i	0.648787	−	0.173842i	0.0806066	−	0.996746i	$-0.474314\pi$
$877$	19.2133	−	5.14819i	0.648787	−	0.173842i	0.568180	+	0.822904i	$0.307648\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−16.6875	−	28.9035i	−0.562215	−	0.973785i	−0.997303	−	0.0733970i	$-0.976616\pi$
$881$	−16.6875	−	28.9035i	−0.562215	−	0.973785i	0.435088	−	0.900388i	$-0.356717\pi$
$882$		0			0
$883$			12.8119i			0.431155i	0.976487	+	0.215577i	$0.0691633\pi$
$883$			12.8119i			0.431155i	−0.976487	+	0.215577i	$0.930837\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	27.5297	−	15.8943i	0.924358	−	0.533678i	0.0393350	−	0.999226i	$-0.487476\pi$
$887$	27.5297	−	15.8943i	0.924358	−	0.533678i	0.885023	+	0.465548i	$0.154143\pi$
$888$		0			0
$889$	−3.39648	−	3.39648i	−0.113914	−	0.113914i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	0.784312	+	0.452823i	0.0262460	+	0.0151531i
$894$		0			0
$895$	12.7411	+	3.41396i	0.425887	+	0.114116i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−37.2073	−	9.96967i	−1.24093	−	0.332507i
$900$		0			0
$901$	43.2259	+	24.9565i	1.44006	+	0.831421i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	16.3854	+	16.3854i	0.544671	+	0.544671i
$906$		0			0
$907$	11.9402	−	6.89370i	0.396469	−	0.228902i	−0.288490	−	0.957483i	$-0.593153\pi$
$907$	11.9402	−	6.89370i	0.396469	−	0.228902i	0.684959	+	0.728581i	$0.259820\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			11.4607i			0.379709i	0.981812	+	0.189855i	$0.0608016\pi$
$911$			11.4607i			0.379709i	−0.981812	+	0.189855i	$0.939198\pi$
$912$		0			0
$913$	11.6564	+	20.1894i	0.385769	+	0.668171i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−3.69359	+	0.989695i	−0.121973	+	0.0326826i
$918$		0			0
$919$	0.380145	−	0.658430i	0.0125398	−	0.0217196i	−0.859687	−	0.510820i	$-0.829342\pi$
$919$	0.380145	−	0.658430i	0.0125398	−	0.0217196i	0.872227	+	0.489101i	$0.162675\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	44.6112	+	25.9379i	1.46840	+	0.853757i
$924$		0			0
$925$	2.31273	−	8.63123i	0.0760422	−	0.283793i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	6.84747	+	25.5551i	0.224658	+	0.838436i	0.982541	+	0.186045i	$0.0595671\pi$
$929$	6.84747	+	25.5551i	0.224658	+	0.838436i	−0.757883	+	0.652391i	$0.773766\pi$
$930$		0			0
$931$	−2.21508	+	2.21508i	−0.0725964	+	0.0725964i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−15.3450			−0.501835
$936$		0			0
$937$	−11.1139			−0.363076			−0.181538	−	0.983384i	$-0.558108\pi$
$937$	−11.1139			−0.363076			−0.181538	+	0.983384i	$0.558108\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1.32739	+	1.32739i	−0.0432716	+	0.0432716i	−0.728412	−	0.685140i	$-0.759741\pi$
$941$	−1.32739	+	1.32739i	−0.0432716	+	0.0432716i	0.685140	+	0.728412i	$0.259741\pi$
$942$		0			0
$943$	−11.1786	−	41.7192i	−0.364026	−	1.35856i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1.64195	−	6.12785i	0.0533562	−	0.199128i	−0.934103	−	0.357005i	$-0.883798\pi$
$947$	1.64195	−	6.12785i	0.0533562	−	0.199128i	0.987459	+	0.157876i	$0.0504647\pi$
$948$		0			0
$949$	0.104155	−	34.1723i	0.00338102	−	1.10928i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−24.7384	+	42.8482i	−0.801355	+	1.38799i	0.117369	+	0.993088i	$0.462554\pi$
$953$	−24.7384	+	42.8482i	−0.801355	+	1.38799i	−0.918724	+	0.394900i	$0.870779\pi$
$954$		0			0
$955$	21.3494	−	5.72056i	0.690851	−	0.185113i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1.43679	+	2.48859i	0.0463963	+	0.0803608i
$960$		0			0
$961$			40.3584i			1.30188i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	17.7786	−	10.2645i	0.572312	−	0.330424i
$966$		0			0
$967$	−23.3987	−	23.3987i	−0.752452	−	0.752452i	0.222484	−	0.974936i	$-0.428583\pi$
$967$	−23.3987	−	23.3987i	−0.752452	−	0.752452i	−0.974936	+	0.222484i	$0.928583\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−7.38401	−	4.26316i	−0.236964	−	0.136811i	0.376816	−	0.926288i	$-0.377019\pi$
$971$	−7.38401	−	4.26316i	−0.236964	−	0.136811i	−0.613781	+	0.789477i	$0.710352\pi$
$972$		0			0
$973$	1.65278	+	0.442861i	0.0529857	+	0.0141975i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−9.05268	−	2.42566i	−0.289621	−	0.0776037i	0.111084	−	0.993811i	$-0.464568\pi$
$977$	−9.05268	−	2.42566i	−0.289621	−	0.0776037i	−0.400704	+	0.916207i	$0.631235\pi$
$978$		0			0
$979$	32.8849	+	18.9861i	1.05101	+	0.606798i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	8.80808	+	8.80808i	0.280934	+	0.280934i	0.833481	−	0.552547i	$-0.186344\pi$
$983$	8.80808	+	8.80808i	0.280934	+	0.280934i	−0.552547	+	0.833481i	$0.686344\pi$
$984$		0			0
$985$	−7.06752	+	4.08044i	−0.225190	+	0.130014i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			37.6915i			1.19852i
$990$		0			0
$991$	−21.2481	−	36.8028i	−0.674968	−	1.16908i	−0.976478	−	0.215616i	$-0.930824\pi$
$991$	−21.2481	−	36.8028i	−0.674968	−	1.16908i	0.301510	−	0.953463i	$-0.402509\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	7.68646	−	2.05958i	0.243677	−	0.0652931i
$996$		0			0
$997$	−9.11451	+	15.7868i	−0.288660	+	0.499973i	−0.973490	−	0.228729i	$-0.926543\pi$
$997$	−9.11451	+	15.7868i	−0.288660	+	0.499973i	0.684831	+	0.728702i	$0.259876\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2340.2.fo.b.1961.1	yes	40
3.2	odd	2	inner	2340.2.fo.b.1961.9	yes	40
13.6	odd	12	inner	2340.2.fo.b.1241.9	yes	40
39.32	even	12	inner	2340.2.fo.b.1241.1	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2340.2.fo.b.1241.1	✓	40	39.32	even	12	inner
2340.2.fo.b.1241.9	yes	40	13.6	odd	12	inner
2340.2.fo.b.1961.1	yes	40	1.1	even	1	trivial
2340.2.fo.b.1961.9	yes	40	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
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)

\(f(q)\)	\(=\)	\(q+(-0.707107 + 0.707107i) q^{5} +(0.0943773 + 0.352221i) q^{7} +(0.741884 - 2.76875i) q^{11} +(2.55727 - 2.54173i) q^{13} +(-2.67668 + 4.63615i) q^{17} +(-0.440635 + 0.118068i) q^{19} +(4.23522 + 7.33561i) q^{23} -1.00000i q^{25} +(3.94905 - 2.27999i) q^{29} +(-5.97321 - 5.97321i) q^{31} +(-0.315793 - 0.182323i) q^{35} +(8.63123 + 2.31273i) q^{37} +(-4.92527 - 1.31972i) q^{41} +(3.85361 + 2.22488i) q^{43} +(-1.40381 - 1.40381i) q^{47} +(5.94703 - 3.43352i) q^{49} -9.32366i q^{53} +(1.43321 + 2.48239i) q^{55} +(5.14683 - 1.37909i) q^{59} +(-2.66247 + 4.61153i) q^{61} +(-0.0109895 + 3.60553i) q^{65} +(3.61721 - 13.4996i) q^{67} +(3.70429 + 13.8246i) q^{71} +(6.70176 - 6.70176i) q^{73} +1.04523 q^{77} +8.79220 q^{79} +(-5.75093 + 5.75093i) q^{83} +(-1.38555 - 5.17096i) q^{85} +(-3.42864 + 12.7959i) q^{89} +(1.13660 + 0.660842i) q^{91} +(0.228090 - 0.395063i) q^{95} +(6.07736 - 1.62842i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 8 q^{7} + 12 q^{13} + 28 q^{19} + 36 q^{31} - 40 q^{37} - 72 q^{43} - 36 q^{49} - 32 q^{61} + 16 q^{67} + 28 q^{73} + 48 q^{79} - 20 q^{85} + 132 q^{91} + 48 q^{97}+O(q^{100}) \) 40 * q + 8 * q^7 + 12 * q^13 + 28 * q^19 + 36 * q^31 - 40 * q^37 - 72 * q^43 - 36 * q^49 - 32 * q^61 + 16 * q^67 + 28 * q^73 + 48 * q^79 - 20 * q^85 + 132 * q^91 + 48 * q^97

Introduction

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