LMFDB - Embedded newform 2340.2.fo.a.1601.9

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			1601.9
Character	$\chi$	$=$	2340.1601
Dual form			2340.2.fo.a.2321.9

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 + 0.707107i) q^{5} +(2.53048 - 0.678039i) q^{7} +(-3.89061 - 1.04249i) q^{11} +(-0.754020 - 3.52583i) q^{13} +(-0.545887 + 0.945504i) q^{17} +(-1.52046 - 5.67442i) q^{19} +(-3.15269 - 5.46062i) q^{23} +1.00000i q^{25} +(-5.81377 + 3.35658i) q^{29} +(-6.97310 + 6.97310i) q^{31} +(2.26876 + 1.30987i) q^{35} +(1.62994 - 6.08303i) q^{37} +(-0.202944 + 0.757396i) q^{41} +(-6.73137 - 3.88636i) q^{43} +(-2.91596 + 2.91596i) q^{47} +(-0.118605 + 0.0684765i) q^{49} -1.10890i q^{53} +(-2.01393 - 3.48823i) q^{55} +(-0.878261 - 3.27771i) q^{59} +(4.21393 - 7.29874i) q^{61} +(1.95996 - 3.02631i) q^{65} +(3.83062 + 1.02641i) q^{67} +(6.28238 - 1.68336i) q^{71} +(3.24844 + 3.24844i) q^{73} -10.5520 q^{77} +10.6689 q^{79} +(1.60450 + 1.60450i) q^{83} +(-1.05457 + 0.282572i) q^{85} +(-8.10685 - 2.17222i) q^{89} +(-4.29868 - 8.41077i) q^{91} +(2.93730 - 5.08755i) q^{95} +(-1.20898 - 4.51198i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	2.53048	−	0.678039i	0.956430	−	0.256275i	0.253341	−	0.967377i	$-0.418470\pi$
$7$	2.53048	−	0.678039i	0.956430	−	0.256275i	0.703089	+	0.711102i	$0.251804\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.89061	−	1.04249i	−1.17306	−	0.314322i	−0.380892	−	0.924620i	$-0.624383\pi$
$11$	−3.89061	−	1.04249i	−1.17306	−	0.314322i	−0.792172	+	0.610298i	$0.791050\pi$
$12$		0			0
$13$	−0.754020	−	3.52583i	−0.209128	−	0.977888i
$13$	−0.754020	−	3.52583i	−0.209128	−	0.977888i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.545887	+	0.945504i	−0.132397	+	0.229318i	−0.924600	−	0.380939i	$-0.875601\pi$
$17$	−0.545887	+	0.945504i	−0.132397	+	0.229318i	0.792203	+	0.610258i	$0.208934\pi$
$18$		0			0
$19$	−1.52046	−	5.67442i	−0.348817	−	1.30180i	−0.888089	−	0.459671i	$-0.847967\pi$
$19$	−1.52046	−	5.67442i	−0.348817	−	1.30180i	0.539273	−	0.842131i	$-0.318699\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.15269	−	5.46062i	−0.657382	−	1.13862i	−0.981291	−	0.192530i	$-0.938331\pi$
$23$	−3.15269	−	5.46062i	−0.657382	−	1.13862i	0.323909	−	0.946088i	$-0.395003\pi$
$24$		0			0
$25$			1.00000i			0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−5.81377	+	3.35658i	−1.07959	+	0.623301i	−0.930786	−	0.365565i	$-0.880876\pi$
$29$	−5.81377	+	3.35658i	−1.07959	+	0.623301i	−0.148804	+	0.988867i	$0.547542\pi$
$30$		0			0
$31$	−6.97310	+	6.97310i	−1.25241	+	1.25241i	−0.297767	+	0.954639i	$0.596242\pi$
$31$	−6.97310	+	6.97310i	−1.25241	+	1.25241i	−0.954639	+	0.297767i	$0.903758\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.26876	+	1.30987i	0.383491	+	0.221409i
$36$		0			0
$37$	1.62994	−	6.08303i	0.267961	−	1.00004i	−0.692452	−	0.721464i	$-0.743470\pi$
$37$	1.62994	−	6.08303i	0.267961	−	1.00004i	0.960413	−	0.278580i	$-0.0898637\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−0.202944	+	0.757396i	−0.0316945	+	0.118285i	−0.979961	−	0.199191i	$-0.936169\pi$
$41$	−0.202944	+	0.757396i	−0.0316945	+	0.118285i	0.948266	+	0.317476i	$0.102835\pi$
$42$		0			0
$43$	−6.73137	−	3.88636i	−1.02652	−	0.592664i	−0.110537	−	0.993872i	$-0.535257\pi$
$43$	−6.73137	−	3.88636i	−1.02652	−	0.592664i	−0.915987	+	0.401208i	$0.868591\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2.91596	+	2.91596i	−0.425336	+	0.425336i	−0.887036	−	0.461700i	$-0.847240\pi$
$47$	−2.91596	+	2.91596i	−0.425336	+	0.425336i	0.461700	+	0.887036i	$0.347240\pi$
$48$		0			0
$49$	−0.118605	+	0.0684765i	−0.0169435	+	0.00978236i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	1.10890i		−	0.152320i	−0.997096	−	0.0761599i	$-0.975734\pi$
$53$		−	1.10890i		−	0.152320i	0.997096	−	0.0761599i	$-0.0242659\pi$
$54$		0			0
$55$	−2.01393	−	3.48823i	−0.271558	−	0.470353i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−0.878261	−	3.27771i	−0.114340	−	0.426722i	0.884897	−	0.465787i	$-0.154229\pi$
$59$	−0.878261	−	3.27771i	−0.114340	−	0.426722i	−0.999237	+	0.0390651i	$0.987562\pi$
$60$		0			0
$61$	4.21393	−	7.29874i	0.539538	−	0.934507i	−0.459391	−	0.888234i	$-0.651932\pi$
$61$	4.21393	−	7.29874i	0.539538	−	0.934507i	0.998929	−	0.0462731i	$-0.0147344\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.95996	−	3.02631i	0.243104	−	0.375367i
$66$		0			0
$67$	3.83062	+	1.02641i	0.467985	+	0.125396i	0.485103	−	0.874457i	$-0.338782\pi$
$67$	3.83062	+	1.02641i	0.467985	+	0.125396i	−0.0171175	+	0.999853i	$0.505449\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	6.28238	−	1.68336i	0.745581	−	0.199778i	0.134024	−	0.990978i	$-0.457210\pi$
$71$	6.28238	−	1.68336i	0.745581	−	0.199778i	0.611557	+	0.791200i	$0.290543\pi$
$72$		0			0
$73$	3.24844	+	3.24844i	0.380201	+	0.380201i	0.871175	−	0.490973i	$-0.163359\pi$
$73$	3.24844	+	3.24844i	0.380201	+	0.380201i	−0.490973	+	0.871175i	$0.663359\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−10.5520			−1.20251
$78$		0			0
$79$	10.6689			1.20035			0.600174	−	0.799869i	$-0.295098\pi$
$79$	10.6689			1.20035			0.600174	+	0.799869i	$0.295098\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	1.60450	+	1.60450i	0.176117	+	0.176117i	0.789661	−	0.613544i	$-0.210257\pi$
$83$	1.60450	+	1.60450i	0.176117	+	0.176117i	−0.613544	+	0.789661i	$0.710257\pi$
$84$		0			0
$85$	−1.05457	+	0.282572i	−0.114384	+	0.0306492i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−8.10685	−	2.17222i	−0.859324	−	0.230255i	−0.197858	−	0.980231i	$-0.563399\pi$
$89$	−8.10685	−	2.17222i	−0.859324	−	0.230255i	−0.661466	+	0.749975i	$0.730065\pi$
$90$		0			0
$91$	−4.29868	−	8.41077i	−0.450624	−	0.881688i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.93730	−	5.08755i	0.301360	−	0.521972i
$96$		0			0
$97$	−1.20898	−	4.51198i	−0.122754	−	0.458123i	0.876996	−	0.480498i	$-0.159544\pi$
$97$	−1.20898	−	4.51198i	−0.122754	−	0.458123i	−0.999750	+	0.0223750i	$0.992877\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.22194	+	10.7767i	0.619107	+	1.07232i	0.989649	+	0.143509i	$0.0458384\pi$
$101$	6.22194	+	10.7767i	0.619107	+	1.07232i	−0.370543	+	0.928816i	$0.620828\pi$
$102$		0			0
$103$		−	6.95186i		−	0.684987i	−0.939520	−	0.342493i	$-0.888729\pi$
$103$		−	6.95186i		−	0.684987i	0.939520	−	0.342493i	$-0.111271\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−16.0738	+	9.28024i	−1.55392	+	0.897154i	−0.556100	+	0.831115i	$0.687703\pi$
$107$	−16.0738	+	9.28024i	−1.55392	+	0.897154i	−0.997817	+	0.0660389i	$0.978964\pi$
$108$		0			0
$109$	−5.24202	+	5.24202i	−0.502094	+	0.502094i	−0.912088	−	0.409994i	$-0.865531\pi$
$109$	−5.24202	+	5.24202i	−0.502094	+	0.502094i	0.409994	+	0.912088i	$0.365531\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	5.35897	+	3.09401i	0.504130	+	0.291060i	0.730417	−	0.683001i	$-0.239326\pi$
$113$	5.35897	+	3.09401i	0.504130	+	0.291060i	−0.226288	+	0.974061i	$0.572659\pi$
$114$		0			0
$115$	1.63195	−	6.09053i	0.152180	−	0.567945i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−0.740265	+	2.76271i	−0.0678600	+	0.253257i
$120$		0			0
$121$	4.52381	+	2.61183i	0.411256	+	0.237439i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.707107	+	0.707107i	−0.0632456	+	0.0632456i
$126$		0			0
$127$	3.84853	−	2.22195i	0.341502	−	0.197166i	−0.319434	−	0.947608i	$-0.603493\pi$
$127$	3.84853	−	2.22195i	0.341502	−	0.197166i	0.660936	+	0.750442i	$0.270159\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	8.01421i		−	0.700205i	−0.936711	−	0.350102i	$-0.886147\pi$
$131$		−	8.01421i		−	0.700205i	0.936711	−	0.350102i	$-0.113853\pi$
$132$		0			0
$133$	−7.69496	−	13.3281i	−0.667238	−	1.15569i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	4.41989	+	16.4953i	0.377617	+	1.40929i	0.849483	+	0.527615i	$0.176914\pi$
$137$	4.41989	+	16.4953i	0.377617	+	1.40929i	−0.471867	+	0.881670i	$0.656420\pi$
$138$		0			0
$139$	6.64826	−	11.5151i	0.563898	−	0.976700i	−0.433253	−	0.901272i	$-0.642634\pi$
$139$	6.64826	−	11.5151i	0.563898	−	0.976700i	0.997151	−	0.0754279i	$-0.0240323\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−0.742027	+	14.5037i	−0.0620514	+	1.21286i
$144$		0			0
$145$	−6.48442	−	1.73749i	−0.538501	−	0.144291i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−0.848793	+	0.227433i	−0.0695358	+	0.0186321i	−0.293419	−	0.955984i	$-0.594793\pi$
$149$	−0.848793	+	0.227433i	−0.0695358	+	0.0186321i	0.223883	+	0.974616i	$0.428127\pi$
$150$		0			0
$151$	−12.2620	−	12.2620i	−0.997870	−	0.997870i	0.00212755	−	0.999998i	$-0.499323\pi$
$151$	−12.2620	−	12.2620i	−0.997870	−	0.997870i	−0.999998	+	0.00212755i	$0.999323\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−9.86145			−0.792091
$156$		0			0
$157$	2.05163			0.163738			0.0818690	−	0.996643i	$-0.473911\pi$
$157$	2.05163			0.163738			0.0818690	+	0.996643i	$0.473911\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−11.6803	−	11.6803i	−0.920539	−	0.920539i
$162$		0			0
$163$	20.7802	−	5.56803i	1.62763	−	0.436122i	0.674399	−	0.738367i	$-0.264403\pi$
$163$	20.7802	−	5.56803i	1.62763	−	0.436122i	0.953230	+	0.302245i	$0.0977360\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−8.79950	−	2.35782i	−0.680926	−	0.182453i	−0.0982543	−	0.995161i	$-0.531326\pi$
$167$	−8.79950	−	2.35782i	−0.680926	−	0.182453i	−0.582671	+	0.812708i	$0.697993\pi$
$168$		0			0
$169$	−11.8629	+	5.31709i	−0.912531	+	0.409007i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	11.7536	−	20.3579i	0.893611	−	1.54778i	0.0580973	−	0.998311i	$-0.481497\pi$
$173$	11.7536	−	20.3579i	0.893611	−	1.54778i	0.835514	−	0.549469i	$-0.185170\pi$
$174$		0			0
$175$	0.678039	+	2.53048i	0.0512549	+	0.191286i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−4.23090	−	7.32813i	−0.316232	−	0.547730i	0.663467	−	0.748206i	$-0.269085\pi$
$179$	−4.23090	−	7.32813i	−0.316232	−	0.547730i	−0.979699	+	0.200476i	$0.935751\pi$
$180$		0			0
$181$		−	17.6932i		−	1.31513i	−0.753400	−	0.657563i	$-0.771587\pi$
$181$		−	17.6932i		−	1.31513i	0.753400	−	0.657563i	$-0.228413\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	5.45389	−	3.14881i	0.400978	−	0.231505i
$186$		0			0
$187$	3.10951	−	3.10951i	0.227390	−	0.227390i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−18.4412	−	10.6470i	−1.33436	−	0.770391i	−0.348392	−	0.937349i	$-0.613272\pi$
$191$	−18.4412	−	10.6470i	−1.33436	−	0.770391i	−0.985964	+	0.166958i	$0.946606\pi$
$192$		0			0
$193$	0.0881930	−	0.329141i	0.00634827	−	0.0236921i	−0.962679	−	0.270646i	$-0.912763\pi$
$193$	0.0881930	−	0.329141i	0.00634827	−	0.0236921i	0.969027	+	0.246954i	$0.0794295\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	6.61902	−	24.7025i	0.471586	−	1.75998i	−0.162490	−	0.986710i	$-0.551952\pi$
$197$	6.61902	−	24.7025i	0.471586	−	1.75998i	0.634075	−	0.773271i	$-0.281381\pi$
$198$		0			0
$199$	−7.20101	−	4.15750i	−0.510466	−	0.294718i	0.222559	−	0.974919i	$-0.428559\pi$
$199$	−7.20101	−	4.15750i	−0.510466	−	0.294718i	−0.733025	+	0.680202i	$0.761892\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−12.4357	+	12.4357i	−0.872816	+	0.872816i
$204$		0			0
$205$	−0.679062	+	0.392057i	−0.0474278	+	0.0273824i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			23.6620i			1.63674i
$210$		0			0
$211$	2.68836	+	4.65638i	0.185074	+	0.320558i	0.943602	−	0.331083i	$-0.107414\pi$
$211$	2.68836	+	4.65638i	0.185074	+	0.320558i	−0.758527	+	0.651641i	$0.774081\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−2.01173	−	7.50787i	−0.137199	−	0.512032i
$216$		0			0
$217$	−12.9172	+	22.3733i	−0.876879	+	1.51880i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	3.74529	+	1.21177i	0.251936	+	0.0815127i
$222$		0			0
$223$	−1.81321	−	0.485848i	−0.121421	−	0.0325348i	0.197597	−	0.980283i	$-0.436686\pi$
$223$	−1.81321	−	0.485848i	−0.121421	−	0.0325348i	−0.319018	+	0.947749i	$0.603353\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−10.7333	+	2.87597i	−0.712391	+	0.190885i	−0.596774	−	0.802409i	$-0.703551\pi$
$227$	−10.7333	+	2.87597i	−0.712391	+	0.190885i	−0.115617	+	0.993294i	$0.536885\pi$
$228$		0			0
$229$	7.85565	+	7.85565i	0.519116	+	0.519116i	0.917304	−	0.398188i	$-0.130361\pi$
$229$	7.85565	+	7.85565i	0.519116	+	0.519116i	−0.398188	+	0.917304i	$0.630361\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−21.0295			−1.37769			−0.688844	−	0.724910i	$-0.741881\pi$
$233$	−21.0295			−1.37769			−0.688844	+	0.724910i	$0.741881\pi$
$234$		0			0
$235$	−4.12379			−0.269006
$236$		0			0
$237$		0			0
$238$		0			0
$239$	4.56294	+	4.56294i	0.295152	+	0.295152i	0.839112	−	0.543959i	$-0.183075\pi$
$239$	4.56294	+	4.56294i	0.295152	+	0.295152i	−0.543959	+	0.839112i	$0.683075\pi$
$240$		0			0
$241$	−5.93422	+	1.59007i	−0.382257	+	0.102425i	−0.444830	−	0.895615i	$-0.646736\pi$
$241$	−5.93422	+	1.59007i	−0.382257	+	0.102425i	0.0625733	+	0.998040i	$0.480069\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−0.132286	−	0.0354460i	−0.00845147	−	0.00226456i
$246$		0			0
$247$	−18.8606	+	9.63950i	−1.20007	+	0.613347i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	10.9670	−	18.9955i	0.692234	−	1.19898i	−0.278871	−	0.960329i	$-0.589960\pi$
$251$	10.9670	−	18.9955i	0.692234	−	1.19898i	0.971104	−	0.238655i	$-0.0767066\pi$
$252$		0			0
$253$	6.57328	+	24.5318i	0.413258	+	1.54230i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	13.5734	+	23.5098i	0.846684	+	1.46650i	0.884150	+	0.467203i	$0.154738\pi$
$257$	13.5734	+	23.5098i	0.846684	+	1.46650i	−0.0374658	+	0.999298i	$0.511929\pi$
$258$		0			0
$259$		−	16.4981i		−	1.02514i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−1.83129	+	1.05730i	−0.112922	+	0.0651958i	−0.555397	−	0.831585i	$-0.687434\pi$
$263$	−1.83129	+	1.05730i	−0.112922	+	0.0651958i	0.442475	+	0.896781i	$0.354101\pi$
$264$		0			0
$265$	0.784114	−	0.784114i	0.0481677	−	0.0481677i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	14.6827	+	8.47708i	0.895222	+	0.516857i	0.875647	−	0.482952i	$-0.160435\pi$
$269$	14.6827	+	8.47708i	0.895222	+	0.516857i	0.0195750	+	0.999808i	$0.493769\pi$
$270$		0			0
$271$	6.28668	−	23.4622i	0.381888	−	1.42523i	−0.461125	−	0.887335i	$-0.652554\pi$
$271$	6.28668	−	23.4622i	0.381888	−	1.42523i	0.843014	−	0.537892i	$-0.180779\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.04249	−	3.89061i	0.0628643	−	0.234613i
$276$		0			0
$277$	24.7799	+	14.3067i	1.48888	+	0.859605i	0.999919	−	0.0127008i	$-0.00404291\pi$
$277$	24.7799	+	14.3067i	1.48888	+	0.859605i	0.488960	+	0.872306i	$0.337376\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	15.2677	−	15.2677i	0.910797	−	0.910797i	−0.0855381	−	0.996335i	$-0.527261\pi$
$281$	15.2677	−	15.2677i	0.910797	−	0.910797i	0.996335	+	0.0855381i	$0.0272609\pi$
$282$		0			0
$283$	−1.15381	+	0.666153i	−0.0685869	+	0.0395987i	−0.533901	−	0.845547i	$-0.679275\pi$
$283$	−1.15381	+	0.666153i	−0.0685869	+	0.0395987i	0.465314	+	0.885145i	$0.345941\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			2.05418i			0.121254i
$288$		0			0
$289$	7.90401	+	13.6902i	0.464942	+	0.805303i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−2.55199	−	9.52417i	−0.149089	−	0.556408i	−0.999539	−	0.0303524i	$-0.990337\pi$
$293$	−2.55199	−	9.52417i	−0.149089	−	0.556408i	0.850450	−	0.526056i	$-0.176330\pi$
$294$		0			0
$295$	1.69667	−	2.93872i	0.0987839	−	0.171099i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−16.8760	+	15.2333i	−0.975965	+	0.880962i
$300$		0			0
$301$	−19.6687	−	5.27021i	−1.13368	−	0.303770i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	8.14068	−	2.18129i	0.466134	−	0.124900i
$306$		0			0
$307$	14.7944	+	14.7944i	0.844360	+	0.844360i	0.989423	−	0.145062i	$-0.0463383\pi$
$307$	14.7944	+	14.7944i	0.844360	+	0.844360i	−0.145062	+	0.989423i	$0.546338\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	30.6541			1.73824			0.869118	−	0.494606i	$-0.164687\pi$
$311$	30.6541			1.73824			0.869118	+	0.494606i	$0.164687\pi$
$312$		0			0
$313$	−7.76880			−0.439119			−0.219559	−	0.975599i	$-0.570462\pi$
$313$	−7.76880			−0.439119			−0.219559	+	0.975599i	$0.570462\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1.62269	−	1.62269i	−0.0911396	−	0.0911396i	0.660067	−	0.751207i	$-0.270528\pi$
$317$	−1.62269	−	1.62269i	−0.0911396	−	0.0911396i	−0.751207	+	0.660067i	$0.770528\pi$
$318$		0			0
$319$	26.1183	−	6.99838i	1.46234	−	0.391834i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	6.19519	+	1.66000i	0.344710	+	0.0923646i
$324$		0			0
$325$	3.52583	−	0.754020i	0.195578	−	0.0418255i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−5.40163	+	9.35590i	−0.297801	+	0.515807i
$330$		0			0
$331$	−5.23761	−	19.5470i	−0.287885	−	1.07440i	−0.946705	−	0.322101i	$-0.895611\pi$
$331$	−5.23761	−	19.5470i	−0.287885	−	1.07440i	0.658820	−	0.752301i	$-0.271056\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	1.98288	+	3.43444i	0.108336	+	0.187644i
$336$		0			0
$337$			0.450414i			0.0245356i	0.999925	+	0.0122678i	$0.00390506\pi$
$337$			0.450414i			0.0245356i	−0.999925	+	0.0122678i	$0.996095\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	34.3990	−	19.8603i	1.86281	−	1.07549i
$342$		0			0
$343$	−13.2208	+	13.2208i	−0.713854	+	0.713854i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−6.09778	−	3.52056i	−0.327346	−	0.188993i	0.327316	−	0.944915i	$-0.393856\pi$
$347$	−6.09778	−	3.52056i	−0.327346	−	0.188993i	−0.654662	+	0.755921i	$0.727189\pi$
$348$		0			0
$349$	−9.14125	+	34.1156i	−0.489320	+	1.82617i	0.0704443	+	0.997516i	$0.477558\pi$
$349$	−9.14125	+	34.1156i	−0.489320	+	1.82617i	−0.559764	+	0.828652i	$0.689108\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−7.71144	+	28.7795i	−0.410439	+	1.53178i	0.383361	+	0.923599i	$0.374766\pi$
$353$	−7.71144	+	28.7795i	−0.410439	+	1.53178i	−0.793800	+	0.608179i	$0.791900\pi$
$354$		0			0
$355$	5.63263	+	3.25200i	0.298949	+	0.172598i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	16.5119	−	16.5119i	0.871463	−	0.871463i	−0.121169	−	0.992632i	$-0.538664\pi$
$359$	16.5119	−	16.5119i	0.871463	−	0.871463i	0.992632	+	0.121169i	$0.0386642\pi$
$360$		0			0
$361$	−13.4328	+	7.75544i	−0.706990	+	0.408181i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			4.59399i			0.240460i
$366$		0			0
$367$	12.7532	+	22.0892i	0.665713	+	1.15305i	0.979091	+	0.203421i	$0.0652059\pi$
$367$	12.7532	+	22.0892i	0.665713	+	1.15305i	−0.313378	+	0.949628i	$0.601461\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−0.751880	−	2.80606i	−0.0390357	−	0.145683i
$372$		0			0
$373$	−13.4770	+	23.3428i	−0.697811	+	1.20864i	0.271413	+	0.962463i	$0.412509\pi$
$373$	−13.4770	+	23.3428i	−0.697811	+	1.20864i	−0.969224	+	0.246181i	$0.920824\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	16.2184	+	17.9674i	0.835291	+	0.925369i
$378$		0			0
$379$	8.12361	+	2.17672i	0.417282	+	0.111810i	0.461350	−	0.887218i	$-0.347365\pi$
$379$	8.12361	+	2.17672i	0.417282	+	0.111810i	−0.0440682	+	0.999029i	$0.514032\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−10.7652	+	2.88453i	−0.550076	+	0.147392i	−0.523143	−	0.852245i	$-0.675241\pi$
$383$	−10.7652	+	2.88453i	−0.550076	+	0.147392i	−0.0269328	+	0.999637i	$0.508574\pi$
$384$		0			0
$385$	−7.46136	−	7.46136i	−0.380266	−	0.380266i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−17.8490			−0.904978			−0.452489	−	0.891770i	$-0.649464\pi$
$389$	−17.8490			−0.904978			−0.452489	+	0.891770i	$0.649464\pi$
$390$		0			0
$391$	6.88405			0.348142
$392$		0			0
$393$		0			0
$394$		0			0
$395$	7.54407	+	7.54407i	0.379584	+	0.379584i
$396$		0			0
$397$	0.492142	−	0.131869i	0.0246999	−	0.00661832i	−0.246448	−	0.969156i	$-0.579263\pi$
$397$	0.492142	−	0.131869i	0.0246999	−	0.00661832i	0.271148	+	0.962538i	$0.412597\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	10.8867	+	2.91709i	0.543657	+	0.145672i	0.520188	−	0.854052i	$-0.325862\pi$
$401$	10.8867	+	2.91709i	0.543657	+	0.145672i	0.0234696	+	0.999725i	$0.492529\pi$
$402$		0			0
$403$	29.8438	+	19.3281i	1.48663	+	0.962800i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−12.6830	+	21.9675i	−0.628671	+	1.08889i
$408$		0			0
$409$	−1.05818	−	3.94918i	−0.0523236	−	0.195275i	0.934817	−	0.355131i	$-0.115564\pi$
$409$	−1.05818	−	3.94918i	−0.0523236	−	0.195275i	−0.987140	+	0.159856i	$0.948897\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−4.44483	−	7.69868i	−0.218716	−	0.378827i
$414$		0			0
$415$			2.26911i			0.111386i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	27.5904	−	15.9293i	1.34788	−	0.778199i	0.359932	−	0.932979i	$-0.382800\pi$
$419$	27.5904	−	15.9293i	1.34788	−	0.778199i	0.987949	+	0.154780i	$0.0494667\pi$
$420$		0			0
$421$	−28.0296	+	28.0296i	−1.36608	+	1.36608i	−0.500122	+	0.865955i	$0.666711\pi$
$421$	−28.0296	+	28.0296i	−1.36608	+	1.36608i	−0.865955	+	0.500122i	$0.833289\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−0.945504	−	0.545887i	−0.0458637	−	0.0264794i
$426$		0			0
$427$	5.71441	−	21.3265i	0.276540	−	1.03206i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	6.50315	−	24.2701i	0.313246	−	1.16905i	−0.612366	−	0.790575i	$-0.709782\pi$
$431$	6.50315	−	24.2701i	0.313246	−	1.16905i	0.925612	−	0.378475i	$-0.123551\pi$
$432$		0			0
$433$	−3.93183	−	2.27004i	−0.188952	−	0.109091i	0.402540	−	0.915402i	$-0.368127\pi$
$433$	−3.93183	−	2.27004i	−0.188952	−	0.109091i	−0.591492	+	0.806311i	$0.701461\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−26.1923	+	26.1923i	−1.25295	+	1.25295i
$438$		0			0
$439$	−10.0206	+	5.78540i	−0.478258	+	0.276122i	−0.719690	−	0.694296i	$-0.755716\pi$
$439$	−10.0206	+	5.78540i	−0.478258	+	0.276122i	0.241433	+	0.970418i	$0.422383\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	16.7556i		−	0.796084i	−0.917367	−	0.398042i	$-0.869690\pi$
$443$		−	16.7556i		−	0.796084i	0.917367	−	0.398042i	$-0.130310\pi$
$444$		0			0
$445$	−4.19641	−	7.26840i	−0.198929	−	0.344555i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−3.85185	−	14.3753i	−0.181780	−	0.678413i	−0.995297	−	0.0968714i	$-0.969116\pi$
$449$	−3.85185	−	14.3753i	−0.181780	−	0.678413i	0.813517	−	0.581542i	$-0.197550\pi$
$450$		0			0
$451$	1.57915	−	2.73517i	0.0743593	−	0.128794i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	2.90769	−	8.98693i	0.136314	−	0.421314i
$456$		0			0
$457$	11.3942	+	3.05306i	0.532998	+	0.142816i	0.515272	−	0.857027i	$-0.327691\pi$
$457$	11.3942	+	3.05306i	0.532998	+	0.142816i	0.0177255	+	0.999843i	$0.494358\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	29.0081	−	7.77270i	1.35104	−	0.362011i	0.490523	−	0.871428i	$-0.336806\pi$
$461$	29.0081	−	7.77270i	1.35104	−	0.362011i	0.860519	+	0.509418i	$0.170139\pi$
$462$		0			0
$463$	−6.10145	−	6.10145i	−0.283559	−	0.283559i	0.550968	−	0.834526i	$-0.314259\pi$
$463$	−6.10145	−	6.10145i	−0.283559	−	0.283559i	−0.834526	+	0.550968i	$0.814259\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−10.7094			−0.495571			−0.247785	−	0.968815i	$-0.579703\pi$
$467$	−10.7094			−0.495571			−0.247785	+	0.968815i	$0.579703\pi$
$468$		0			0
$469$	10.3892			0.479731
$470$		0			0
$471$		0			0
$472$		0			0
$473$	22.1377	+	22.1377i	1.01789	+	1.01789i
$474$		0			0
$475$	5.67442	−	1.52046i	0.260360	−	0.0697634i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−16.4349	−	4.40371i	−0.750929	−	0.201211i	−0.136999	−	0.990571i	$-0.543746\pi$
$479$	−16.4349	−	4.40371i	−0.750929	−	0.201211i	−0.613930	+	0.789360i	$0.710412\pi$
$480$		0			0
$481$	−22.6767	−	1.16017i	−1.03397	−	0.0528992i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	2.33557	−	4.04533i	0.106053	−	0.183689i
$486$		0			0
$487$	−6.20483	−	23.1568i	−0.281168	−	1.04933i	−0.951594	−	0.307357i	$-0.900555\pi$
$487$	−6.20483	−	23.1568i	−0.281168	−	1.04933i	0.670426	−	0.741976i	$-0.266111\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−11.3582	−	19.6729i	−0.512586	−	0.887826i	−0.999893	−	0.0145951i	$-0.995354\pi$
$491$	−11.3582	−	19.6729i	−0.512586	−	0.887826i	0.487307	−	0.873231i	$-0.337979\pi$
$492$		0			0
$493$		−	7.32926i		−	0.330093i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	14.7560	−	8.51940i	0.661898	−	0.382147i
$498$		0			0
$499$	−5.58983	+	5.58983i	−0.250235	+	0.250235i	−0.821067	−	0.570832i	$-0.806621\pi$
$499$	−5.58983	+	5.58983i	−0.250235	+	0.250235i	0.570832	+	0.821067i	$0.306621\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	13.8310	+	7.98536i	0.616696	+	0.356050i	0.775582	−	0.631247i	$-0.217457\pi$
$503$	13.8310	+	7.98536i	0.616696	+	0.356050i	−0.158886	+	0.987297i	$0.550790\pi$
$504$		0			0
$505$	−3.22072	+	12.0199i	−0.143320	+	0.534877i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−5.21030	+	19.4451i	−0.230943	+	0.861890i	0.748993	+	0.662577i	$0.230537\pi$
$509$	−5.21030	+	19.4451i	−0.230943	+	0.861890i	−0.979936	+	0.199312i	$0.936129\pi$
$510$		0			0
$511$	10.4227	+	6.01753i	0.461072	+	0.266200i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	4.91571	−	4.91571i	0.216612	−	0.216612i
$516$		0			0
$517$	14.3847	−	8.30502i	0.632639	−	0.365254i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	16.3802i		−	0.717630i	−0.933409	−	0.358815i	$-0.883181\pi$
$521$		−	16.3802i		−	0.717630i	0.933409	−	0.358815i	$-0.116819\pi$
$522$		0			0
$523$	10.9018	+	18.8825i	0.476704	+	0.825675i	0.999644	−	0.0266945i	$-0.00849812\pi$
$523$	10.9018	+	18.8825i	0.476704	+	0.825675i	−0.522940	+	0.852370i	$0.675165\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−2.78657	−	10.3996i	−0.121385	−	0.453014i
$528$		0			0
$529$	−8.37892	+	14.5127i	−0.364301	+	0.630988i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	2.82347	+	0.144452i	0.122298	+	0.00625693i
$534$		0			0
$535$	−17.9280	−	4.80380i	−0.775097	−	0.207687i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	0.532831	−	0.142772i	0.0229507	−	0.00614961i
$540$		0			0
$541$	−9.72072	−	9.72072i	−0.417926	−	0.417926i	0.466562	−	0.884488i	$-0.345492\pi$
$541$	−9.72072	−	9.72072i	−0.417926	−	0.417926i	−0.884488	+	0.466562i	$0.845492\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−7.41333			−0.317552
$546$		0			0
$547$	29.8288			1.27539			0.637694	−	0.770290i	$-0.279888\pi$
$547$	29.8288			1.27539			0.637694	+	0.770290i	$0.279888\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	27.8862	+	27.8862i	1.18799	+	1.18799i
$552$		0			0
$553$	26.9975	−	7.23395i	1.14805	−	0.307619i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	29.7785	+	7.97912i	1.26175	+	0.338086i	0.826866	−	0.562399i	$-0.190121\pi$
$557$	29.7785	+	7.97912i	1.26175	+	0.338086i	0.434888	+	0.900485i	$0.356788\pi$
$558$		0			0
$559$	−8.62704	+	26.6640i	−0.364885	+	1.12777i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	1.12047	−	1.94071i	0.0472223	−	0.0817913i	−0.841448	−	0.540338i	$-0.818296\pi$
$563$	1.12047	−	1.94071i	0.0472223	−	0.0817913i	0.888670	+	0.458547i	$0.151630\pi$
$564$		0			0
$565$	1.60158	+	5.97716i	0.0673788	+	0.251461i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−10.2209	−	17.7032i	−0.428483	−	0.742155i	0.568255	−	0.822852i	$-0.307619\pi$
$569$	−10.2209	−	17.7032i	−0.428483	−	0.742155i	−0.996739	+	0.0806975i	$0.974285\pi$
$570$		0			0
$571$			8.90788i			0.372783i	0.982476	+	0.186392i	$0.0596793\pi$
$571$			8.90788i			0.372783i	−0.982476	+	0.186392i	$0.940321\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	5.46062	−	3.15269i	0.227724	−	0.131476i
$576$		0			0
$577$	−13.4827	+	13.4827i	−0.561293	+	0.561293i	−0.929675	−	0.368381i	$-0.879912\pi$
$577$	−13.4827	+	13.4827i	−0.561293	+	0.561293i	0.368381	+	0.929675i	$0.379912\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	5.14807	+	2.97224i	0.213578	+	0.123309i
$582$		0			0
$583$	−1.15602	+	4.31432i	−0.0478774	+	0.178681i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−5.95557	+	22.2265i	−0.245813	+	0.917385i	0.727161	+	0.686467i	$0.240839\pi$
$587$	−5.95557	+	22.2265i	−0.245813	+	0.917385i	−0.972973	+	0.230918i	$0.925827\pi$
$588$		0			0
$589$	50.1706	+	28.9660i	2.06724	+	1.19352i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	28.9170	−	28.9170i	1.18748	−	1.18748i	0.209716	−	0.977762i	$-0.432746\pi$
$593$	28.9170	−	28.9170i	1.18748	−	1.18748i	0.977762	−	0.209716i	$-0.0672540\pi$
$594$		0			0
$595$	−2.47698	+	1.43008i	−0.101546	+	0.0586277i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			12.5846i			0.514191i	0.966386	+	0.257095i	$0.0827655\pi$
$599$			12.5846i			0.514191i	−0.966386	+	0.257095i	$0.917235\pi$
$600$		0			0
$601$	19.7818	+	34.2631i	0.806916	+	1.39762i	0.914990	+	0.403476i	$0.132198\pi$
$601$	19.7818	+	34.2631i	0.806916	+	1.39762i	−0.108074	+	0.994143i	$0.534468\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.35198	+	5.04566i	0.0549658	+	0.205135i
$606$		0			0
$607$	−6.66054	+	11.5364i	−0.270343	+	0.468248i	−0.968950	−	0.247258i	$-0.920470\pi$
$607$	−6.66054	+	11.5364i	−0.270343	+	0.468248i	0.698607	+	0.715506i	$0.253804\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	12.4799	+	8.08247i	0.504881	+	0.326982i
$612$		0			0
$613$	31.0528	+	8.32058i	1.25421	+	0.336065i	0.823962	−	0.566645i	$-0.191759\pi$
$613$	31.0528	+	8.32058i	1.25421	+	0.336065i	0.430249	+	0.902710i	$0.358426\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−27.4433	+	7.35341i	−1.10483	+	0.296037i	−0.764728	−	0.644353i	$-0.777127\pi$
$617$	−27.4433	+	7.35341i	−1.10483	+	0.296037i	−0.340097	+	0.940390i	$0.610460\pi$
$618$		0			0
$619$	−29.4741	−	29.4741i	−1.18467	−	1.18467i	−0.978522	−	0.206144i	$-0.933909\pi$
$619$	−29.4741	−	29.4741i	−1.18467	−	1.18467i	−0.206144	−	0.978522i	$-0.566091\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−21.9870			−0.880892
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	4.86176	+	4.86176i	0.193851	+	0.193851i
$630$		0			0
$631$	−35.4849	+	9.50816i	−1.41263	+	0.378514i	−0.882864	−	0.469628i	$-0.844388\pi$
$631$	−35.4849	+	9.50816i	−1.41263	+	0.378514i	−0.529769	+	0.848142i	$0.677721\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	4.29248	+	1.15017i	0.170342	+	0.0456429i
$636$		0			0
$637$	0.330867	+	0.366547i	0.0131094	+	0.0145231i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	17.7288	−	30.7072i	0.700246	−	1.21286i	−0.268133	−	0.963382i	$-0.586407\pi$
$641$	17.7288	−	30.7072i	0.700246	−	1.21286i	0.968380	−	0.249481i	$-0.0802599\pi$
$642$		0			0
$643$	−5.62974	−	21.0105i	−0.222015	−	0.828573i	−0.983578	−	0.180482i	$-0.942234\pi$
$643$	−5.62974	−	21.0105i	−0.222015	−	0.828573i	0.761563	−	0.648091i	$-0.224432\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−14.6695	−	25.4084i	−0.576719	−	0.998906i	−0.995853	−	0.0909814i	$-0.971000\pi$
$647$	−14.6695	−	25.4084i	−0.576719	−	0.998906i	0.419134	−	0.907924i	$-0.362334\pi$
$648$		0			0
$649$			13.6679i			0.536512i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−43.5782	+	25.1599i	−1.70535	+	0.984582i	−0.765210	+	0.643781i	$0.777365\pi$
$653$	−43.5782	+	25.1599i	−1.70535	+	0.984582i	−0.940135	+	0.340801i	$0.889302\pi$
$654$		0			0
$655$	5.66690	−	5.66690i	0.221424	−	0.221424i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	9.65885	+	5.57654i	0.376255	+	0.217231i	0.676188	−	0.736729i	$-0.263631\pi$
$659$	9.65885	+	5.57654i	0.376255	+	0.217231i	−0.299932	+	0.953960i	$0.596964\pi$
$660$		0			0
$661$	2.69053	−	10.0412i	0.104649	−	0.390557i	−0.893656	−	0.448753i	$-0.851868\pi$
$661$	2.69053	−	10.0412i	0.104649	−	0.390557i	0.998305	+	0.0581962i	$0.0185349\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	3.98321	−	14.8655i	0.154462	−	0.576460i
$666$		0			0
$667$	36.6580	+	21.1645i	1.41940	+	0.819494i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−24.0036	+	24.0036i	−0.926649	+	0.926649i
$672$		0			0
$673$	28.0925	−	16.2192i	1.08289	−	0.625206i	0.151214	−	0.988501i	$-0.451682\pi$
$673$	28.0925	−	16.2192i	1.08289	−	0.625206i	0.931674	+	0.363296i	$0.118348\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			12.5092i			0.480768i	0.970678	+	0.240384i	$0.0772734\pi$
$677$			12.5092i			0.480768i	−0.970678	+	0.240384i	$0.922727\pi$
$678$		0			0
$679$	−6.11860	−	10.5977i	−0.234810	−	0.406704i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−6.74567	−	25.1752i	−0.258116	−	0.963302i	−0.966330	−	0.257305i	$-0.917166\pi$
$683$	−6.74567	−	25.1752i	−0.258116	−	0.963302i	0.708214	−	0.705998i	$-0.249501\pi$
$684$		0			0
$685$	−8.53857	+	14.7892i	−0.326242	+	0.565068i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−3.90980	+	0.836136i	−0.148952	+	0.0318542i
$690$		0			0
$691$	5.33260	+	1.42887i	0.202862	+	0.0543566i	0.358819	−	0.933407i	$-0.383180\pi$
$691$	5.33260	+	1.42887i	0.202862	+	0.0543566i	−0.155957	+	0.987764i	$0.549846\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	12.8435	−	3.44139i	0.487180	−	0.130539i
$696$		0			0
$697$	−0.605337	−	0.605337i	−0.0229288	−	0.0229288i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	31.7144			1.19784			0.598918	−	0.800810i	$-0.295598\pi$
$701$	31.7144			1.19784			0.598918	+	0.800810i	$0.295598\pi$
$702$		0			0
$703$	−36.9959			−1.39533
$704$		0			0
$705$		0			0
$706$		0			0
$707$	23.0515	+	23.0515i	0.866942	+	0.866942i
$708$		0			0
$709$	−29.3808	+	7.87255i	−1.10342	+	0.295660i	−0.764156	−	0.645032i	$-0.776844\pi$
$709$	−29.3808	+	7.87255i	−1.10342	+	0.295660i	−0.339262	+	0.940692i	$0.610177\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	60.0615	+	16.0934i	2.24932	+	0.602704i
$714$		0			0
$715$	−10.7803	+	9.73096i	−0.403162	+	0.363917i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−2.85595	+	4.94665i	−0.106509	+	0.184479i	−0.914354	−	0.404916i	$-0.867301\pi$
$719$	−2.85595	+	4.94665i	−0.106509	+	0.184479i	0.807845	+	0.589395i	$0.200634\pi$
$720$		0			0
$721$	−4.71363	−	17.5915i	−0.175545	−	0.655142i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−3.35658	−	5.81377i	−0.124660	−	0.215918i
$726$		0			0
$727$			28.3801i			1.05256i	0.850311	+	0.526280i	$0.176414\pi$
$727$			28.3801i			1.05256i	−0.850311	+	0.526280i	$0.823586\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	7.34914	−	4.24303i	0.271818	−	0.156934i
$732$		0			0
$733$	−36.8825	+	36.8825i	−1.36229	+	1.36229i	−0.491292	+	0.870995i	$0.663475\pi$
$733$	−36.8825	+	36.8825i	−1.36229	+	1.36229i	−0.870995	+	0.491292i	$0.836525\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−13.8334	−	7.98675i	−0.509562	−	0.294196i
$738$		0			0
$739$	4.11128	−	15.3435i	0.151236	−	0.564419i	−0.848163	−	0.529736i	$-0.822291\pi$
$739$	4.11128	−	15.3435i	0.151236	−	0.564419i	0.999398	−	0.0346835i	$-0.0110423\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−6.05887	+	22.6120i	−0.222278	+	0.829554i	0.761198	+	0.648519i	$0.224611\pi$
$743$	−6.05887	+	22.6120i	−0.222278	+	0.829554i	−0.983477	+	0.181035i	$0.942055\pi$
$744$		0			0
$745$	−0.761007	−	0.439368i	−0.0278811	−	0.0160972i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−34.3821	+	34.3821i	−1.25630	+	1.25630i
$750$		0			0
$751$	−8.07343	+	4.66119i	−0.294603	+	0.170089i	−0.640016	−	0.768362i	$-0.721072\pi$
$751$	−8.07343	+	4.66119i	−0.294603	+	0.170089i	0.345413	+	0.938451i	$0.387739\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	17.3411i		−	0.631109i
$756$		0			0
$757$	4.19011	+	7.25749i	0.152292	+	0.263778i	0.932070	−	0.362279i	$-0.118001\pi$
$757$	4.19011	+	7.25749i	0.152292	+	0.263778i	−0.779778	+	0.626057i	$0.784668\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−8.81096	−	32.8829i	−0.319397	−	1.19201i	−0.919826	−	0.392327i	$-0.871670\pi$
$761$	−8.81096	−	32.8829i	−0.319397	−	1.19201i	0.600429	−	0.799678i	$-0.294997\pi$
$762$		0			0
$763$	−9.71050	+	16.8191i	−0.351544	+	0.608892i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−10.8944	+	5.56806i	−0.393375	+	0.201051i
$768$		0			0
$769$	9.61170	+	2.57545i	0.346607	+	0.0928730i	0.427923	−	0.903815i	$-0.359246\pi$
$769$	9.61170	+	2.57545i	0.346607	+	0.0928730i	−0.0813159	+	0.996688i	$0.525912\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	35.8155	−	9.59673i	1.28819	−	0.345170i	0.451220	−	0.892413i	$-0.350989\pi$
$773$	35.8155	−	9.59673i	1.28819	−	0.345170i	0.836974	+	0.547242i	$0.184322\pi$
$774$		0			0
$775$	−6.97310	−	6.97310i	−0.250481	−	0.250481i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	4.60635			0.165040
$780$		0			0
$781$	−26.1972			−0.937409
$782$		0			0
$783$		0			0
$784$		0			0
$785$	1.45072	+	1.45072i	0.0517785	+	0.0517785i
$786$		0			0
$787$	−27.1668	+	7.27931i	−0.968391	+	0.259479i	−0.708148	−	0.706064i	$-0.750469\pi$
$787$	−27.1668	+	7.27931i	−0.968391	+	0.259479i	−0.260242	+	0.965543i	$0.583802\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	15.6586	+	4.19571i	0.556756	+	0.149182i
$792$		0			0
$793$	−28.9115	−	9.35418i	−1.02668	−	0.332177i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	3.30648	−	5.72699i	0.117122	−	0.202860i	−0.801504	−	0.597989i	$-0.795967\pi$
$797$	3.30648	−	5.72699i	0.117122	−	0.202860i	0.918626	+	0.395129i	$0.129300\pi$
$798$		0			0
$799$	−1.16527	−	4.34883i	−0.0412242	−	0.153851i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−9.25197	−	16.0249i	−0.326495	−	0.565506i
$804$		0			0
$805$		−	16.5185i		−	0.582200i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	30.7071	−	17.7288i	1.07960	−	0.623310i	0.148815	−	0.988865i	$-0.452454\pi$
$809$	30.7071	−	17.7288i	1.07960	−	0.623310i	0.930790	+	0.365555i	$0.119121\pi$
$810$		0			0
$811$	18.3653	−	18.3653i	0.644892	−	0.644892i	−0.306862	−	0.951754i	$-0.599279\pi$
$811$	18.3653	−	18.3653i	0.644892	−	0.644892i	0.951754	+	0.306862i	$0.0992790\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	18.6310	+	10.7566i	0.652616	+	0.376788i
$816$		0			0
$817$	−11.8181	+	44.1057i	−0.413462	+	1.54306i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1.27881	−	4.77257i	0.0446307	−	0.166564i	−0.940014	−	0.341137i	$-0.889188\pi$
$821$	1.27881	−	4.77257i	0.0446307	−	0.166564i	0.984644	+	0.174573i	$0.0558546\pi$
$822$		0			0
$823$	39.6375	+	22.8847i	1.38168	+	0.797712i	0.992358	−	0.123391i	$-0.0393770\pi$
$823$	39.6375	+	22.8847i	1.38168	+	0.797712i	0.389319	+	0.921103i	$0.372710\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−15.7153	+	15.7153i	−0.546476	+	0.546476i	−0.925420	−	0.378944i	$-0.876287\pi$
$827$	−15.7153	+	15.7153i	−0.546476	+	0.546476i	0.378944	+	0.925420i	$0.376287\pi$
$828$		0			0
$829$	33.6579	−	19.4324i	1.16899	−	0.674914i	0.215545	−	0.976494i	$-0.430847\pi$
$829$	33.6579	−	19.4324i	1.16899	−	0.674914i	0.953441	+	0.301580i	$0.0975140\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	0.149522i		−	0.00518062i
$834$		0			0
$835$	−4.55496	−	7.88941i	−0.157631	−	0.273024i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	10.9001	+	40.6798i	0.376314	+	1.40442i	0.851416	+	0.524491i	$0.175744\pi$
$839$	10.9001	+	40.6798i	0.376314	+	1.40442i	−0.475102	+	0.879931i	$0.657589\pi$
$840$		0			0
$841$	8.03327	−	13.9140i	0.277009	−	0.479794i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−12.1481	−	4.62859i	−0.417907	−	0.159228i
$846$		0			0
$847$	13.2183	+	3.54184i	0.454187	+	0.121699i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−38.3558	+	10.2774i	−1.31482	+	0.352305i
$852$		0			0
$853$	−38.4301	−	38.4301i	−1.31582	−	1.31582i	−0.917050	−	0.398772i	$-0.869437\pi$
$853$	−38.4301	−	38.4301i	−1.31582	−	1.31582i	−0.398772	−	0.917050i	$-0.630563\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−47.4349			−1.62035			−0.810173	−	0.586191i	$-0.800627\pi$
$857$	−47.4349			−1.62035			−0.810173	+	0.586191i	$0.800627\pi$
$858$		0			0
$859$	37.2436			1.27073			0.635367	−	0.772210i	$-0.280849\pi$
$859$	37.2436			1.27073			0.635367	+	0.772210i	$0.280849\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−28.6143	−	28.6143i	−0.974044	−	0.974044i	0.0256280	−	0.999672i	$-0.491841\pi$
$863$	−28.6143	−	28.6143i	−0.974044	−	0.974044i	−0.999672	+	0.0256280i	$0.991841\pi$
$864$		0			0
$865$	22.7063	−	6.08412i	0.772036	−	0.206866i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−41.5087	−	11.1222i	−1.40809	−	0.377295i
$870$		0			0
$871$	0.730586	−	14.2800i	0.0247550	−	0.483861i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1.30987	+	2.26876i	−0.0442817	+	0.0766982i
$876$		0			0
$877$	12.2798	+	45.8290i	0.414661	+	1.54754i	0.785514	+	0.618844i	$0.212399\pi$
$877$	12.2798	+	45.8290i	0.414661	+	1.54754i	−0.370853	+	0.928692i	$0.620935\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−20.7316	−	35.9082i	−0.698466	−	1.20978i	−0.968998	−	0.247068i	$-0.920533\pi$
$881$	−20.7316	−	35.9082i	−0.698466	−	1.20978i	0.270532	−	0.962711i	$-0.412800\pi$
$882$		0			0
$883$		−	41.3170i		−	1.39043i	−0.718803	−	0.695214i	$-0.755310\pi$
$883$		−	41.3170i		−	1.39043i	0.718803	−	0.695214i	$-0.244690\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1.25742	−	0.725973i	0.0422201	−	0.0243758i	−0.478741	−	0.877956i	$-0.658907\pi$
$887$	1.25742	−	0.725973i	0.0422201	−	0.0243758i	0.520961	+	0.853580i	$0.325574\pi$
$888$		0			0
$889$	8.23205	−	8.23205i	0.276094	−	0.276094i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	20.9800	+	12.1128i	0.702068	+	0.405339i
$894$		0			0
$895$	2.19007	−	8.17347i	0.0732061	−	0.273209i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	17.1342	−	63.9458i	0.571458	−	2.13271i
$900$		0			0
$901$	1.04847	+	0.605337i	0.0349297	+	0.0201667i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	12.5110	−	12.5110i	0.415879	−	0.415879i
$906$		0			0
$907$	33.8178	−	19.5247i	1.12290	−	0.648307i	0.180761	−	0.983527i	$-0.442144\pi$
$907$	33.8178	−	19.5247i	1.12290	−	0.648307i	0.942140	+	0.335220i	$0.108811\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	13.2628i		−	0.439417i	−0.975566	−	0.219709i	$-0.929489\pi$
$911$		−	13.2628i		−	0.439417i	0.975566	−	0.219709i	$-0.0705106\pi$
$912$		0			0
$913$	−4.56982	−	7.91517i	−0.151239	−	0.261954i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−5.43395	−	20.2798i	−0.179445	−	0.669697i
$918$		0			0
$919$	−10.0861	+	17.4697i	−0.332711	+	0.576273i	−0.983042	−	0.183378i	$-0.941297\pi$
$919$	−10.0861	+	17.4697i	−0.332711	+	0.576273i	0.650331	+	0.759651i	$0.274630\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−10.6723	−	20.8813i	−0.351282	−	0.687316i
$924$		0			0
$925$	6.08303	+	1.62994i	0.200009	+	0.0535922i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−38.4273	+	10.2966i	−1.26076	+	0.337819i	−0.826483	−	0.562962i	$-0.809662\pi$
$929$	−38.4273	+	10.2966i	−1.26076	+	0.337819i	−0.434275	+	0.900781i	$0.642995\pi$
$930$		0			0
$931$	0.568898	+	0.568898i	0.0186449	+	0.0186449i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	4.39751			0.143814
$936$		0			0
$937$	−24.4899			−0.800049			−0.400025	−	0.916504i	$-0.630998\pi$
$937$	−24.4899			−0.800049			−0.400025	+	0.916504i	$0.630998\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	10.8383	+	10.8383i	0.353318	+	0.353318i	0.861342	−	0.508025i	$-0.169624\pi$
$941$	10.8383	+	10.8383i	0.353318	+	0.353318i	−0.508025	+	0.861342i	$0.669624\pi$
$942$		0			0
$943$	4.77567	−	1.27964i	0.155517	−	0.0416707i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−14.6787	−	3.93314i	−0.476994	−	0.127810i	0.0123088	−	0.999924i	$-0.496082\pi$
$947$	−14.6787	−	3.93314i	−0.476994	−	0.127810i	−0.489302	+	0.872114i	$0.662749\pi$
$948$		0			0
$949$	9.00405	−	13.9028i	0.292284	−	0.451305i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	19.5246	−	33.8176i	0.632463	−	1.09546i	−0.354583	−	0.935025i	$-0.615377\pi$
$953$	19.5246	−	33.8176i	0.632463	−	1.09546i	0.987047	−	0.160434i	$-0.0512895\pi$
$954$		0			0
$955$	−5.51130	−	20.5685i	−0.178341	−	0.665579i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	22.3689	+	38.7440i	0.722328	+	1.25111i
$960$		0			0
$961$		−	66.2482i		−	2.13704i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	0.295100	−	0.170376i	0.00949959	−	0.00548459i
$966$		0			0
$967$	−24.3471	+	24.3471i	−0.782950	+	0.782950i	−0.980328	−	0.197378i	$-0.936757\pi$
$967$	−24.3471	+	24.3471i	−0.782950	+	0.782950i	0.197378	+	0.980328i	$0.436757\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−26.4150	−	15.2507i	−0.847697	−	0.489418i	0.0121759	−	0.999926i	$-0.496124\pi$
$971$	−26.4150	−	15.2507i	−0.847697	−	0.489418i	−0.859873	+	0.510508i	$0.829458\pi$
$972$		0			0
$973$	9.01556	−	33.6465i	0.289026	−	1.07866i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−3.72943	+	13.9184i	−0.119315	+	0.445290i	−0.999573	−	0.0292047i	$-0.990703\pi$
$977$	−3.72943	+	13.9184i	−0.119315	+	0.445290i	0.880258	+	0.474495i	$0.157369\pi$
$978$		0			0
$979$	29.2761	+	16.9026i	0.935668	+	0.540208i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−28.4967	+	28.4967i	−0.908905	+	0.908905i	−0.996184	−	0.0872793i	$-0.972183\pi$
$983$	−28.4967	+	28.4967i	−0.908905	+	0.908905i	0.0872793	+	0.996184i	$0.472183\pi$
$984$		0			0
$985$	22.1477	−	12.7870i	0.705684	−	0.407427i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			49.0100i			1.55843i
$990$		0			0
$991$	9.86429	+	17.0855i	0.313349	+	0.542737i	0.979085	−	0.203450i	$-0.0652155\pi$
$991$	9.86429	+	17.0855i	0.313349	+	0.542737i	−0.665736	+	0.746188i	$0.731882\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−2.15208	−	8.03168i	−0.0682256	−	0.254621i
$996$		0			0
$997$	21.1926	−	36.7066i	0.671175	−	1.16251i	−0.306396	−	0.951904i	$-0.599123\pi$
$997$	21.1926	−	36.7066i	0.671175	−	1.16251i	0.977571	−	0.210606i	$-0.0675436\pi$
$998$		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.1601.9	yes	40
3.2	odd	2	inner	2340.2.fo.a.1601.5	✓	40
13.7	odd	12	inner	2340.2.fo.a.2321.5	yes	40
39.20	even	12	inner	2340.2.fo.a.2321.9	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2340.2.fo.a.1601.5	✓	40	3.2	odd	2	inner
2340.2.fo.a.1601.9	yes	40	1.1	even	1	trivial
2340.2.fo.a.2321.5	yes	40	13.7	odd	12	inner
2340.2.fo.a.2321.9	yes	40	39.20	even	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)

\(f(q)\)	\(=\)	\(q+(0.707107 + 0.707107i) q^{5} +(2.53048 - 0.678039i) q^{7} +(-3.89061 - 1.04249i) q^{11} +(-0.754020 - 3.52583i) q^{13} +(-0.545887 + 0.945504i) q^{17} +(-1.52046 - 5.67442i) q^{19} +(-3.15269 - 5.46062i) q^{23} +1.00000i q^{25} +(-5.81377 + 3.35658i) q^{29} +(-6.97310 + 6.97310i) q^{31} +(2.26876 + 1.30987i) q^{35} +(1.62994 - 6.08303i) q^{37} +(-0.202944 + 0.757396i) q^{41} +(-6.73137 - 3.88636i) q^{43} +(-2.91596 + 2.91596i) q^{47} +(-0.118605 + 0.0684765i) q^{49} -1.10890i q^{53} +(-2.01393 - 3.48823i) q^{55} +(-0.878261 - 3.27771i) q^{59} +(4.21393 - 7.29874i) q^{61} +(1.95996 - 3.02631i) q^{65} +(3.83062 + 1.02641i) q^{67} +(6.28238 - 1.68336i) q^{71} +(3.24844 + 3.24844i) q^{73} -10.5520 q^{77} +10.6689 q^{79} +(1.60450 + 1.60450i) q^{83} +(-1.05457 + 0.282572i) q^{85} +(-8.10685 - 2.17222i) q^{89} +(-4.29868 - 8.41077i) q^{91} +(2.93730 - 5.08755i) q^{95} +(-1.20898 - 4.51198i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 12 q^{13} - 20 q^{19} - 28 q^{31} + 12 q^{49} + 16 q^{55} + 48 q^{61} + 16 q^{67} - 20 q^{73} + 80 q^{79} + 20 q^{85} + 4 q^{91} - 40 q^{97}+O(q^{100}) \) 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

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Database

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