LMFDB - Embedded newform 2340.2.fo.a.1601.7

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			1601.7
Character	$\chi$	$=$	2340.1601
Dual form			2340.2.fo.a.2321.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 + 0.707107i) q^{5} +(-3.16623 + 0.848389i) q^{7} +(-1.87883 - 0.503432i) q^{11} +(3.59250 - 0.306557i) q^{13} +(0.497031 - 0.860883i) q^{17} +(-0.513203 - 1.91530i) q^{19} +(2.92185 + 5.06079i) q^{23} +1.00000i q^{25} +(1.74433 - 1.00709i) q^{29} +(-3.03417 + 3.03417i) q^{31} +(-2.83876 - 1.63896i) q^{35} +(-1.27888 + 4.77285i) q^{37} +(1.06083 - 3.95907i) q^{41} +(-2.27024 - 1.31073i) q^{43} +(-7.28328 + 7.28328i) q^{47} +(3.24307 - 1.87239i) q^{49} +5.79649i q^{53} +(-0.972556 - 1.68452i) q^{55} +(3.40791 + 12.7185i) q^{59} +(-6.08039 + 10.5315i) q^{61} +(2.75705 + 2.32351i) q^{65} +(-11.2096 - 3.00361i) q^{67} +(-8.18898 + 2.19423i) q^{71} +(-9.72222 - 9.72222i) q^{73} +6.37593 q^{77} -15.3101 q^{79} +(4.87578 + 4.87578i) q^{83} +(0.960190 - 0.257282i) q^{85} +(3.14248 + 0.842026i) q^{89} +(-11.1146 + 4.01846i) q^{91} +(0.991432 - 1.71721i) q^{95} +(-1.66503 - 6.21398i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−3.16623	+	0.848389i	−1.19672	+	0.320661i	−0.801540	−	0.597942i	$-0.795985\pi$
$7$	−3.16623	+	0.848389i	−1.19672	+	0.320661i	−0.395183	+	0.918603i	$0.629319\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.87883	−	0.503432i	−0.566490	−	0.151790i	−0.0358036	−	0.999359i	$-0.511399\pi$
$11$	−1.87883	−	0.503432i	−0.566490	−	0.151790i	−0.530686	+	0.847568i	$0.678066\pi$
$12$		0			0
$13$	3.59250	−	0.306557i	0.996379	−	0.0850235i
$13$	3.59250	−	0.306557i	0.996379	−	0.0850235i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.497031	−	0.860883i	0.120548	−	0.208795i	−0.799436	−	0.600751i	$-0.794868\pi$
$17$	0.497031	−	0.860883i	0.120548	−	0.208795i	0.919984	+	0.391956i	$0.128202\pi$
$18$		0			0
$19$	−0.513203	−	1.91530i	−0.117737	−	0.439400i	0.881740	−	0.471735i	$-0.156372\pi$
$19$	−0.513203	−	1.91530i	−0.117737	−	0.439400i	−0.999477	+	0.0323354i	$0.989706\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.92185	+	5.06079i	0.609247	+	1.05525i	0.991365	+	0.131133i	$0.0418616\pi$
$23$	2.92185	+	5.06079i	0.609247	+	1.05525i	−0.382118	+	0.924114i	$0.624805\pi$
$24$		0			0
$25$			1.00000i			0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.74433	−	1.00709i	0.323914	−	0.187012i	−0.329222	−	0.944253i	$-0.606786\pi$
$29$	1.74433	−	1.00709i	0.323914	−	0.187012i	0.653136	+	0.757241i	$0.273453\pi$
$30$		0			0
$31$	−3.03417	+	3.03417i	−0.544953	+	0.544953i	−0.924977	−	0.380023i	$-0.875916\pi$
$31$	−3.03417	+	3.03417i	−0.544953	+	0.544953i	0.380023	+	0.924977i	$0.375916\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.83876	−	1.63896i	−0.479839	−	0.277035i
$36$		0			0
$37$	−1.27888	+	4.77285i	−0.210247	+	0.784652i	0.777539	+	0.628835i	$0.216468\pi$
$37$	−1.27888	+	4.77285i	−0.210247	+	0.784652i	−0.987786	+	0.155817i	$0.950199\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	1.06083	−	3.95907i	0.165674	−	0.618303i	−0.832279	−	0.554356i	$-0.812964\pi$
$41$	1.06083	−	3.95907i	0.165674	−	0.618303i	0.997953	−	0.0639470i	$-0.0203689\pi$
$42$		0			0
$43$	−2.27024	−	1.31073i	−0.346209	−	0.199884i	0.316805	−	0.948491i	$-0.397390\pi$
$43$	−2.27024	−	1.31073i	−0.346209	−	0.199884i	−0.663014	+	0.748607i	$0.730723\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−7.28328	+	7.28328i	−1.06238	+	1.06238i	−0.0644556	+	0.997921i	$0.520531\pi$
$47$	−7.28328	+	7.28328i	−1.06238	+	1.06238i	−0.997921	+	0.0644556i	$0.979469\pi$
$48$		0			0
$49$	3.24307	−	1.87239i	0.463295	−	0.267484i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			5.79649i			0.796209i	0.917340	+	0.398104i	$0.130332\pi$
$53$			5.79649i			0.796209i	−0.917340	+	0.398104i	$0.869668\pi$
$54$		0			0
$55$	−0.972556	−	1.68452i	−0.131139	−	0.227140i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.40791	+	12.7185i	0.443672	+	1.65581i	0.719418	+	0.694577i	$0.244408\pi$
$59$	3.40791	+	12.7185i	0.443672	+	1.65581i	−0.275746	+	0.961230i	$0.588925\pi$
$60$		0			0
$61$	−6.08039	+	10.5315i	−0.778514	+	1.34843i	0.154284	+	0.988027i	$0.450693\pi$
$61$	−6.08039	+	10.5315i	−0.778514	+	1.34843i	−0.932798	+	0.360399i	$0.882640\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.75705	+	2.32351i	0.341969	+	0.288196i
$66$		0			0
$67$	−11.2096	−	3.00361i	−1.36947	−	0.366949i	−0.502187	−	0.864759i	$-0.667471\pi$
$67$	−11.2096	−	3.00361i	−1.36947	−	0.366949i	−0.867286	+	0.497810i	$0.834138\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−8.18898	+	2.19423i	−0.971853	+	0.260407i	−0.709610	−	0.704594i	$-0.751129\pi$
$71$	−8.18898	+	2.19423i	−0.971853	+	0.260407i	−0.262243	+	0.965002i	$0.584462\pi$
$72$		0			0
$73$	−9.72222	−	9.72222i	−1.13790	−	1.13790i	−0.988826	−	0.149073i	$-0.952371\pi$
$73$	−9.72222	−	9.72222i	−1.13790	−	1.13790i	−0.149073	−	0.988826i	$-0.547629\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	6.37593			0.726604
$78$		0			0
$79$	−15.3101			−1.72252			−0.861259	−	0.508166i	$-0.830324\pi$
$79$	−15.3101			−1.72252			−0.861259	+	0.508166i	$0.830324\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	4.87578	+	4.87578i	0.535186	+	0.535186i	0.922111	−	0.386925i	$-0.126463\pi$
$83$	4.87578	+	4.87578i	0.535186	+	0.535186i	−0.386925	+	0.922111i	$0.626463\pi$
$84$		0			0
$85$	0.960190	−	0.257282i	0.104147	−	0.0279062i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	3.14248	+	0.842026i	0.333103	+	0.0892546i	0.421494	−	0.906831i	$-0.361506\pi$
$89$	3.14248	+	0.842026i	0.333103	+	0.0892546i	−0.0883913	+	0.996086i	$0.528173\pi$
$90$		0			0
$91$	−11.1146	+	4.01846i	−1.16513	+	0.421249i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	0.991432	−	1.71721i	0.101719	−	0.176182i
$96$		0			0
$97$	−1.66503	−	6.21398i	−0.169058	−	0.630934i	−0.997488	−	0.0708404i	$-0.977432\pi$
$97$	−1.66503	−	6.21398i	−0.169058	−	0.630934i	0.828429	−	0.560093i	$-0.189235\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−6.82034	−	11.8132i	−0.678649	−	1.17545i	−0.975388	−	0.220496i	$-0.929232\pi$
$101$	−6.82034	−	11.8132i	−0.678649	−	1.17545i	0.296739	−	0.954959i	$-0.404101\pi$
$102$		0			0
$103$			2.64257i			0.260381i	0.991489	+	0.130190i	$0.0415588\pi$
$103$			2.64257i			0.260381i	−0.991489	+	0.130190i	$0.958441\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	5.15639	−	2.97704i	0.498487	−	0.287802i	−0.229601	−	0.973285i	$-0.573742\pi$
$107$	5.15639	−	2.97704i	0.498487	−	0.287802i	0.728089	+	0.685483i	$0.240409\pi$
$108$		0			0
$109$	−9.55544	+	9.55544i	−0.915245	+	0.915245i	−0.996679	−	0.0814335i	$-0.974050\pi$
$109$	−9.55544	+	9.55544i	−0.915245	+	0.915245i	0.0814335	+	0.996679i	$0.474050\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	15.4719	+	8.93270i	1.45547	+	0.840318i	0.998784	−	0.0493082i	$-0.0157017\pi$
$113$	15.4719	+	8.93270i	1.45547	+	0.840318i	0.456690	+	0.889626i	$0.349035\pi$
$114$		0			0
$115$	−1.51246	+	5.64457i	−0.141038	+	0.526359i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−0.843351	+	3.14743i	−0.0773098	+	0.288524i
$120$		0			0
$121$	−6.24971	−	3.60827i	−0.568155	−	0.328024i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.707107	+	0.707107i	−0.0632456	+	0.0632456i
$126$		0			0
$127$	7.76627	−	4.48386i	0.689145	−	0.397878i	−0.114147	−	0.993464i	$-0.536413\pi$
$127$	7.76627	−	4.48386i	0.689145	−	0.397878i	0.803292	+	0.595586i	$0.203080\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			10.6699i			0.932235i	0.884723	+	0.466118i	$0.154348\pi$
$131$			10.6699i			0.932235i	−0.884723	+	0.466118i	$0.845652\pi$
$132$		0			0
$133$	3.24984	+	5.62888i	0.281797	+	0.488086i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	2.28456	+	8.52609i	0.195183	+	0.728433i	0.992219	+	0.124502i	$0.0397333\pi$
$137$	2.28456	+	8.52609i	0.195183	+	0.728433i	−0.797036	+	0.603931i	$0.793600\pi$
$138$		0			0
$139$	1.28785	−	2.23062i	0.109234	−	0.189199i	−0.806226	−	0.591608i	$-0.798494\pi$
$139$	1.28785	−	2.23062i	0.109234	−	0.189199i	0.915460	+	0.402409i	$0.131827\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−6.90403	−	1.23261i	−0.577344	−	0.103076i
$144$		0			0
$145$	1.94555	+	0.521308i	0.161569	+	0.0432923i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	0.268281	−	0.0718856i	0.0219784	−	0.00588910i	−0.247813	−	0.968808i	$-0.579712\pi$
$149$	0.268281	−	0.0718856i	0.0219784	−	0.00588910i	0.269792	+	0.962919i	$0.413045\pi$
$150$		0			0
$151$	−7.85814	−	7.85814i	−0.639486	−	0.639486i	0.310942	−	0.950429i	$-0.399355\pi$
$151$	−7.85814	−	7.85814i	−0.639486	−	0.639486i	−0.950429	+	0.310942i	$0.899355\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−4.29097			−0.344659
$156$		0			0
$157$	9.32922			0.744553			0.372276	−	0.928122i	$-0.378577\pi$
$157$	9.32922			0.744553			0.372276	+	0.928122i	$0.378577\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−13.5447	−	13.5447i	−1.06748	−	1.06748i
$162$		0			0
$163$	2.65879	−	0.712421i	0.208253	−	0.0558011i	−0.153184	−	0.988198i	$-0.548953\pi$
$163$	2.65879	−	0.712421i	0.208253	−	0.0558011i	0.361437	+	0.932396i	$0.382286\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	15.4704	+	4.14528i	1.19714	+	0.320772i	0.801702	−	0.597723i	$-0.203928\pi$
$167$	15.4704	+	4.14528i	1.19714	+	0.320772i	0.395433	+	0.918495i	$0.370595\pi$
$168$		0			0
$169$	12.8120	−	2.20261i	0.985542	−	0.169431i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	5.19819	−	9.00353i	0.395211	−	0.684526i	−0.597917	−	0.801558i	$-0.704005\pi$
$173$	5.19819	−	9.00353i	0.395211	−	0.684526i	0.993128	+	0.117032i	$0.0373380\pi$
$174$		0			0
$175$	−0.848389	−	3.16623i	−0.0641322	−	0.239344i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−4.98194	−	8.62897i	−0.372367	−	0.644959i	0.617562	−	0.786522i	$-0.288121\pi$
$179$	−4.98194	−	8.62897i	−0.372367	−	0.644959i	−0.989929	+	0.141563i	$0.954787\pi$
$180$		0			0
$181$			8.36779i			0.621973i	0.950414	+	0.310986i	$0.100659\pi$
$181$			8.36779i			0.621973i	−0.950414	+	0.310986i	$0.899341\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−4.27922	+	2.47061i	−0.314615	+	0.181643i
$186$		0			0
$187$	−1.36723	+	1.36723i	−0.0999821	+	0.0999821i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−2.02984	−	1.17193i	−0.146874	−	0.0847976i	0.424762	−	0.905305i	$-0.360358\pi$
$191$	−2.02984	−	1.17193i	−0.146874	−	0.0847976i	−0.571636	+	0.820507i	$0.693691\pi$
$192$		0			0
$193$	−3.61828	+	13.5036i	−0.260449	+	0.972011i	0.704528	+	0.709677i	$0.251159\pi$
$193$	−3.61828	+	13.5036i	−0.260449	+	0.972011i	−0.964977	+	0.262334i	$0.915508\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1.94998	+	7.27742i	−0.138930	+	0.518494i	0.861021	+	0.508570i	$0.169826\pi$
$197$	−1.94998	+	7.27742i	−0.138930	+	0.518494i	−0.999951	+	0.00992437i	$0.996841\pi$
$198$		0			0
$199$	5.04156	+	2.91074i	0.357387	+	0.206337i	0.667934	−	0.744221i	$-0.267179\pi$
$199$	5.04156	+	2.91074i	0.357387	+	0.206337i	−0.310547	+	0.950558i	$0.600512\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−4.66855	+	4.66855i	−0.327668	+	0.327668i
$204$		0			0
$205$	3.54961	−	2.04937i	0.247915	−	0.143134i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			3.85689i			0.266787i
$210$		0			0
$211$	8.60747	+	14.9086i	0.592563	+	1.02635i	0.993886	+	0.110412i	$0.0352171\pi$
$211$	8.60747	+	14.9086i	0.592563	+	1.02635i	−0.401323	+	0.915937i	$0.631450\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−0.678482	−	2.53213i	−0.0462721	−	0.172690i
$216$		0			0
$217$	7.03273	−	12.1810i	0.477413	−	0.826903i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1.52167	−	3.24509i	0.102359	−	0.218288i
$222$		0			0
$223$	−5.13910	−	1.37702i	−0.344140	−	0.0922120i	0.0826094	−	0.996582i	$-0.473675\pi$
$223$	−5.13910	−	1.37702i	−0.344140	−	0.0922120i	−0.426749	+	0.904370i	$0.640341\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	18.1412	−	4.86093i	1.20408	−	0.322631i	0.399641	−	0.916672i	$-0.369135\pi$
$227$	18.1412	−	4.86093i	1.20408	−	0.322631i	0.804435	+	0.594040i	$0.202468\pi$
$228$		0			0
$229$	−3.88768	−	3.88768i	−0.256905	−	0.256905i	0.566889	−	0.823794i	$-0.308147\pi$
$229$	−3.88768	−	3.88768i	−0.256905	−	0.256905i	−0.823794	+	0.566889i	$0.808147\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−26.1552			−1.71349			−0.856743	−	0.515743i	$-0.827516\pi$
$233$	−26.1552			−1.71349			−0.856743	+	0.515743i	$0.827516\pi$
$234$		0			0
$235$	−10.3001			−0.671906
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1.76891	+	1.76891i	0.114421	+	0.114421i	0.761999	−	0.647578i	$-0.224218\pi$
$239$	1.76891	+	1.76891i	0.114421	+	0.114421i	−0.647578	+	0.761999i	$0.724218\pi$
$240$		0			0
$241$	−13.2656	+	3.55450i	−0.854511	+	0.228966i	−0.659378	−	0.751811i	$-0.729180\pi$
$241$	−13.2656	+	3.55450i	−0.854511	+	0.228966i	−0.195133	+	0.980777i	$0.562514\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	3.61717	+	0.969218i	0.231093	+	0.0619211i
$246$		0			0
$247$	−2.43083	−	6.72338i	−0.154670	−	0.427798i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−5.71269	+	9.89467i	−0.360582	+	0.624546i	−0.988057	−	0.154091i	$-0.950755\pi$
$251$	−5.71269	+	9.89467i	−0.360582	+	0.624546i	0.627475	+	0.778637i	$0.284089\pi$
$252$		0			0
$253$	−2.94190	−	10.9793i	−0.184956	−	0.690265i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	7.20777	+	12.4842i	0.449608	+	0.778745i	0.998360	−	0.0572405i	$-0.0182302\pi$
$257$	7.20777	+	12.4842i	0.449608	+	0.778745i	−0.548752	+	0.835985i	$0.684897\pi$
$258$		0			0
$259$		−	16.1969i		−	1.00643i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	10.3567	−	5.97943i	0.638620	−	0.368707i	−0.145463	−	0.989364i	$-0.546467\pi$
$263$	10.3567	−	5.97943i	0.638620	−	0.368707i	0.784083	+	0.620656i	$0.213134\pi$
$264$		0			0
$265$	−4.09873	+	4.09873i	−0.251783	+	0.251783i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.62881	−	0.940396i	−0.0993105	−	0.0573369i	0.449522	−	0.893269i	$-0.351594\pi$
$269$	−1.62881	−	0.940396i	−0.0993105	−	0.0573369i	−0.548833	+	0.835932i	$0.684928\pi$
$270$		0			0
$271$	−5.38499	+	20.0971i	−0.327115	+	1.22081i	0.585053	+	0.810995i	$0.301074\pi$
$271$	−5.38499	+	20.0971i	−0.327115	+	1.22081i	−0.912169	+	0.409815i	$0.865593\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.503432	−	1.87883i	0.0303581	−	0.113298i
$276$		0			0
$277$	17.1315	+	9.89089i	1.02933	+	0.594286i	0.916794	−	0.399361i	$-0.130768\pi$
$277$	17.1315	+	9.89089i	1.02933	+	0.594286i	0.112540	+	0.993647i	$0.464101\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−10.5223	+	10.5223i	−0.627708	+	0.627708i	−0.947491	−	0.319783i	$-0.896390\pi$
$281$	−10.5223	+	10.5223i	−0.627708	+	0.627708i	0.319783	+	0.947491i	$0.396390\pi$
$282$		0			0
$283$	1.19558	−	0.690267i	0.0710697	−	0.0410321i	−0.464044	−	0.885812i	$-0.653602\pi$
$283$	1.19558	−	0.690267i	0.0710697	−	0.0410321i	0.535114	+	0.844780i	$0.320269\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			13.4353i			0.793063i
$288$		0			0
$289$	8.00592	+	13.8667i	0.470937	+	0.815686i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−3.96539	−	14.7990i	−0.231660	−	0.864568i	−0.979626	−	0.200831i	$-0.935636\pi$
$293$	−3.96539	−	14.7990i	−0.231660	−	0.864568i	0.747966	−	0.663737i	$-0.231031\pi$
$294$		0			0
$295$	−6.58358	+	11.4031i	−0.383311	+	0.663914i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	12.0481	+	17.2851i	0.696762	+	0.999626i
$300$		0			0
$301$	8.30012	+	2.22401i	0.478411	+	0.128190i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−11.7464	+	3.14744i	−0.672597	+	0.180222i
$306$		0			0
$307$	−16.9383	−	16.9383i	−0.966722	−	0.966722i	0.0327423	−	0.999464i	$-0.489576\pi$
$307$	−16.9383	−	16.9383i	−0.966722	−	0.966722i	−0.999464	+	0.0327423i	$0.989576\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−12.4858			−0.708004			−0.354002	−	0.935245i	$-0.615179\pi$
$311$	−12.4858			−0.708004			−0.354002	+	0.935245i	$0.615179\pi$
$312$		0			0
$313$	2.42172			0.136884			0.0684418	−	0.997655i	$-0.478197\pi$
$313$	2.42172			0.136884			0.0684418	+	0.997655i	$0.478197\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	18.3492	+	18.3492i	1.03059	+	1.03059i	0.999517	+	0.0310739i	$0.00989271\pi$
$317$	18.3492	+	18.3492i	1.03059	+	1.03059i	0.0310739	+	0.999517i	$0.490107\pi$
$318$		0			0
$319$	−3.78431	+	1.01400i	−0.211881	+	0.0567732i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−1.90393	−	0.510156i	−0.105937	−	0.0283858i
$324$		0			0
$325$	0.306557	+	3.59250i	0.0170047	+	0.199276i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	16.8815	−	29.2396i	0.930707	−	1.61203i
$330$		0			0
$331$	−6.76093	−	25.2321i	−0.371615	−	1.38688i	−0.858229	−	0.513267i	$-0.828435\pi$
$331$	−6.76093	−	25.2321i	−0.371615	−	1.38688i	0.486614	−	0.873617i	$-0.338232\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−5.80253	−	10.0503i	−0.317026	−	0.549105i
$336$		0			0
$337$		−	13.7215i		−	0.747457i	−0.927538	−	0.373729i	$-0.878079\pi$
$337$		−	13.7215i		−	0.747457i	0.927538	−	0.373729i	$-0.121921\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	7.22821	−	4.17321i	0.391429	−	0.225992i
$342$		0			0
$343$	7.54510	−	7.54510i	0.407397	−	0.407397i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1.34771	−	0.778100i	−0.0723488	−	0.0417706i	0.463389	−	0.886155i	$-0.346633\pi$
$347$	−1.34771	−	0.778100i	−0.0723488	−	0.0417706i	−0.535738	+	0.844384i	$0.679967\pi$
$348$		0			0
$349$	6.78967	−	25.3394i	0.363443	−	1.35639i	−0.506077	−	0.862488i	$-0.668905\pi$
$349$	6.78967	−	25.3394i	0.363443	−	1.35639i	0.869520	−	0.493898i	$-0.164428\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	0.917760	−	3.42513i	0.0488474	−	0.182301i	−0.937192	−	0.348815i	$-0.886584\pi$
$353$	0.917760	−	3.42513i	0.0488474	−	0.182301i	0.986039	+	0.166514i	$0.0532509\pi$
$354$		0			0
$355$	−7.34204	−	4.23893i	−0.389675	−	0.224979i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−14.5765	+	14.5765i	−0.769318	+	0.769318i	−0.977986	−	0.208669i	$-0.933087\pi$
$359$	−14.5765	+	14.5765i	−0.769318	+	0.769318i	0.208669	+	0.977986i	$0.433087\pi$
$360$		0			0
$361$	13.0495	−	7.53412i	0.686815	−	0.396533i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	13.7493i		−	0.719671i
$366$		0			0
$367$	15.2778	+	26.4619i	0.797495	+	1.38130i	0.921243	+	0.388988i	$0.127175\pi$
$367$	15.2778	+	26.4619i	0.797495	+	1.38130i	−0.123747	+	0.992314i	$0.539491\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−4.91767	−	18.3530i	−0.255313	−	0.952840i
$372$		0			0
$373$	−5.57744	+	9.66041i	−0.288789	+	0.500197i	−0.973521	−	0.228598i	$-0.926586\pi$
$373$	−5.57744	+	9.66041i	−0.288789	+	0.500197i	0.684732	+	0.728795i	$0.259919\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	5.95777	−	4.15270i	0.306841	−	0.213875i
$378$		0			0
$379$	11.2346	+	3.01032i	0.577085	+	0.154630i	0.535544	−	0.844507i	$-0.320107\pi$
$379$	11.2346	+	3.01032i	0.577085	+	0.154630i	0.0415411	+	0.999137i	$0.486773\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	3.43670	−	0.920861i	0.175607	−	0.0470538i	−0.169944	−	0.985454i	$-0.554359\pi$
$383$	3.43670	−	0.920861i	0.175607	−	0.0470538i	0.345551	+	0.938400i	$0.387692\pi$
$384$		0			0
$385$	4.50846	+	4.50846i	0.229772	+	0.229772i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	28.5970			1.44993			0.724963	−	0.688788i	$-0.241857\pi$
$389$	28.5970			1.44993			0.724963	+	0.688788i	$0.241857\pi$
$390$		0			0
$391$	5.80899			0.293773
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−10.8259	−	10.8259i	−0.544708	−	0.544708i
$396$		0			0
$397$	32.3646	−	8.67206i	1.62433	−	0.435238i	0.672061	−	0.740495i	$-0.265409\pi$
$397$	32.3646	−	8.67206i	1.62433	−	0.435238i	0.952270	+	0.305257i	$0.0987424\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−23.3703	−	6.26204i	−1.16706	−	0.312712i	−0.377275	−	0.926101i	$-0.623139\pi$
$401$	−23.3703	−	6.26204i	−1.16706	−	0.312712i	−0.789780	+	0.613390i	$0.789806\pi$
$402$		0			0
$403$	−9.97011	+	11.8304i	−0.496646	+	0.589314i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	4.80561	−	8.32357i	0.238205	−	0.412584i
$408$		0			0
$409$	−2.45860	−	9.17563i	−0.121570	−	0.453706i	0.878124	−	0.478433i	$-0.158795\pi$
$409$	−2.45860	−	9.17563i	−0.121570	−	0.453706i	−0.999694	+	0.0247270i	$0.992128\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−21.5805	−	37.3784i	−1.06190	−	1.83927i
$414$		0			0
$415$			6.89539i			0.338481i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−8.93265	+	5.15727i	−0.436388	+	0.251949i	−0.702064	−	0.712113i	$-0.747738\pi$
$419$	−8.93265	+	5.15727i	−0.436388	+	0.251949i	0.265676	+	0.964062i	$0.414405\pi$
$420$		0			0
$421$	−9.33390	+	9.33390i	−0.454906	+	0.454906i	−0.896979	−	0.442073i	$-0.854243\pi$
$421$	−9.33390	+	9.33390i	−0.454906	+	0.454906i	0.442073	+	0.896979i	$0.354243\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	0.860883	+	0.497031i	0.0417589	+	0.0241095i
$426$		0			0
$427$	10.3171	−	38.5038i	0.499278	−	1.86333i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	8.16959	−	30.4893i	0.393515	−	1.46862i	−0.430779	−	0.902457i	$-0.641761\pi$
$431$	8.16959	−	30.4893i	0.393515	−	1.46862i	0.824295	−	0.566161i	$-0.191572\pi$
$432$		0			0
$433$	23.8112	+	13.7474i	1.14429	+	0.660658i	0.947490	−	0.319785i	$-0.103611\pi$
$433$	23.8112	+	13.7474i	1.14429	+	0.660658i	0.196803	+	0.980443i	$0.436944\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	8.19342	−	8.19342i	0.391945	−	0.391945i
$438$		0			0
$439$	13.5061	−	7.79775i	0.644611	−	0.372166i	−0.141778	−	0.989899i	$-0.545282\pi$
$439$	13.5061	−	7.79775i	0.644611	−	0.372166i	0.786388	+	0.617732i	$0.211948\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			24.1783i			1.14875i	0.818593	+	0.574374i	$0.194754\pi$
$443$			24.1783i			1.14875i	−0.818593	+	0.574374i	$0.805246\pi$
$444$		0			0
$445$	1.62667	+	2.81747i	0.0771115	+	0.133561i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−5.00655	−	18.6847i	−0.236274	−	0.881786i	−0.977571	−	0.210608i	$-0.932456\pi$
$449$	−5.00655	−	18.6847i	−0.236274	−	0.881786i	0.741297	−	0.671177i	$-0.234211\pi$
$450$		0			0
$451$	−3.98625	+	6.90439i	−0.187705	+	0.325115i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−10.7007	−	5.01772i	−0.501656	−	0.235234i
$456$		0			0
$457$	6.28836	+	1.68496i	0.294157	+	0.0788192i	0.402880	−	0.915253i	$-0.368009\pi$
$457$	6.28836	+	1.68496i	0.294157	+	0.0788192i	−0.108722	+	0.994072i	$0.534676\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−6.39391	+	1.71324i	−0.297794	+	0.0797936i	−0.404622	−	0.914484i	$-0.632597\pi$
$461$	−6.39391	+	1.71324i	−0.297794	+	0.0797936i	0.106829	+	0.994277i	$0.465930\pi$
$462$		0			0
$463$	0.527603	+	0.527603i	0.0245198	+	0.0245198i	0.719260	−	0.694741i	$-0.244481\pi$
$463$	0.527603	+	0.527603i	0.0245198	+	0.0245198i	−0.694741	+	0.719260i	$0.744481\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−18.9948			−0.878973			−0.439487	−	0.898249i	$-0.644840\pi$
$467$	−18.9948			−0.878973			−0.439487	+	0.898249i	$0.644840\pi$
$468$		0			0
$469$	38.0405			1.75655
$470$		0			0
$471$		0			0
$472$		0			0
$473$	3.60555	+	3.60555i	0.165783	+	0.165783i
$474$		0			0
$475$	1.91530	−	0.513203i	0.0878800	−	0.0235474i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	34.6043	+	9.27218i	1.58111	+	0.423657i	0.939269	−	0.343182i	$-0.111505\pi$
$479$	34.6043	+	9.27218i	1.58111	+	0.423657i	0.641840	+	0.766839i	$0.278171\pi$
$480$		0			0
$481$	−3.13123	+	17.5385i	−0.142772	+	0.799687i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	3.21659	−	5.57130i	0.146058	−	0.252980i
$486$		0			0
$487$	−1.92975	−	7.20193i	−0.0874454	−	0.326351i	0.908321	−	0.418275i	$-0.137365\pi$
$487$	−1.92975	−	7.20193i	−0.0874454	−	0.326351i	−0.995766	+	0.0919240i	$0.970698\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−4.30435	−	7.45536i	−0.194253	−	0.336455i	0.752403	−	0.658703i	$-0.228895\pi$
$491$	−4.30435	−	7.45536i	−0.194253	−	0.336455i	−0.946655	+	0.322248i	$0.895561\pi$
$492$		0			0
$493$		−	2.00222i		−	0.0901754i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	24.0666	−	13.8949i	1.07954	−	0.623270i
$498$		0			0
$499$	−3.44687	+	3.44687i	−0.154303	+	0.154303i	−0.780037	−	0.625734i	$-0.784800\pi$
$499$	−3.44687	+	3.44687i	−0.154303	+	0.154303i	0.625734	+	0.780037i	$0.284800\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	4.92108	+	2.84119i	0.219420	+	0.126682i	0.605682	−	0.795707i	$-0.292900\pi$
$503$	4.92108	+	2.84119i	0.219420	+	0.126682i	−0.386262	+	0.922389i	$0.626234\pi$
$504$		0			0
$505$	3.53047	−	13.1759i	0.157104	−	0.586319i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−7.50006	+	27.9906i	−0.332434	+	1.24066i	0.574190	+	0.818722i	$0.305317\pi$
$509$	−7.50006	+	27.9906i	−0.332434	+	1.24066i	−0.906624	+	0.421939i	$0.861350\pi$
$510$		0			0
$511$	39.0310	+	22.5346i	1.72663	+	0.996870i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1.86858	+	1.86858i	−0.0823396	+	0.0823396i
$516$		0			0
$517$	17.3507	−	10.0174i	0.763084	−	0.440567i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	31.7509i		−	1.39103i	−0.718510	−	0.695517i	$-0.755176\pi$
$521$		−	31.7509i		−	1.39103i	0.718510	−	0.695517i	$-0.244824\pi$
$522$		0			0
$523$	−18.2947	−	31.6874i	−0.799972	−	1.38559i	−0.919634	−	0.392777i	$-0.871514\pi$
$523$	−18.2947	−	31.6874i	−0.799972	−	1.38559i	0.119662	−	0.992815i	$-0.461819\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1.10399	+	4.12014i	0.0480905	+	0.179476i
$528$		0			0
$529$	−5.57437	+	9.65510i	−0.242364	+	0.419787i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	2.59735	−	14.5482i	0.112504	−	0.630151i
$534$		0			0
$535$	5.75121	+	1.54103i	0.248646	+	0.0666246i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−7.03580	+	1.88524i	−0.303054	+	0.0812030i
$540$		0			0
$541$	19.9782	+	19.9782i	0.858930	+	0.858930i	0.991212	−	0.132282i	$-0.0422305\pi$
$541$	19.9782	+	19.9782i	0.858930	+	0.858930i	−0.132282	+	0.991212i	$0.542230\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−13.5134			−0.578852
$546$		0			0
$547$	−4.31118			−0.184333			−0.0921663	−	0.995744i	$-0.529379\pi$
$547$	−4.31118			−0.184333			−0.0921663	+	0.995744i	$0.529379\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−2.82407	−	2.82407i	−0.120310	−	0.120310i
$552$		0			0
$553$	48.4752	−	12.9889i	2.06138	−	0.552344i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−2.26713	−	0.607476i	−0.0960615	−	0.0257396i	0.210468	−	0.977601i	$-0.432501\pi$
$557$	−2.26713	−	0.607476i	−0.0960615	−	0.0257396i	−0.306530	+	0.951861i	$0.599168\pi$
$558$		0			0
$559$	−8.55765	−	4.01282i	−0.361950	−	0.169724i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−1.24261	+	2.15226i	−0.0523696	+	0.0907068i	−0.891022	−	0.453960i	$-0.850011\pi$
$563$	−1.24261	+	2.15226i	−0.0523696	+	0.0907068i	0.838652	+	0.544667i	$0.183344\pi$
$564$		0			0
$565$	4.62391	+	17.2567i	0.194529	+	0.725993i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	12.4872	+	21.6285i	0.523491	+	0.906713i	0.999626	+	0.0273409i	$0.00870395\pi$
$569$	12.4872	+	21.6285i	0.523491	+	0.906713i	−0.476135	+	0.879372i	$0.657963\pi$
$570$		0			0
$571$		−	29.5431i		−	1.23634i	−0.786044	−	0.618171i	$-0.787874\pi$
$571$		−	29.5431i		−	1.23634i	0.786044	−	0.618171i	$-0.212126\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−5.06079	+	2.92185i	−0.211049	+	0.121849i
$576$		0			0
$577$	−9.92049	+	9.92049i	−0.412996	+	0.412996i	−0.882781	−	0.469785i	$-0.844331\pi$
$577$	−9.92049	+	9.92049i	−0.412996	+	0.412996i	0.469785	+	0.882781i	$0.344331\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−19.5744	−	11.3013i	−0.812082	−	0.468856i
$582$		0			0
$583$	2.91814	−	10.8906i	0.120857	−	0.451044i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	5.25841	−	19.6246i	0.217038	−	0.809996i	−0.768402	−	0.639968i	$-0.778948\pi$
$587$	5.25841	−	19.6246i	0.217038	−	0.809996i	0.985439	−	0.170028i	$-0.0543857\pi$
$588$		0			0
$589$	7.36850	+	4.25420i	0.303614	+	0.175291i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	17.0108	−	17.0108i	0.698551	−	0.698551i	−0.265547	−	0.964098i	$-0.585553\pi$
$593$	17.0108	−	17.0108i	0.698551	−	0.698551i	0.964098	+	0.265547i	$0.0855525\pi$
$594$		0			0
$595$	−2.82191	+	1.62923i	−0.115687	+	0.0667919i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	45.4555i		−	1.85726i	−0.371005	−	0.928631i	$-0.620987\pi$
$599$		−	45.4555i		−	1.85726i	0.371005	−	0.928631i	$-0.379013\pi$
$600$		0			0
$601$	1.45074	+	2.51276i	0.0591771	+	0.102498i	0.894096	−	0.447875i	$-0.147819\pi$
$601$	1.45074	+	2.51276i	0.0591771	+	0.102498i	−0.834919	+	0.550373i	$0.814486\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.86778	−	6.97064i	−0.0759360	−	0.283397i
$606$		0			0
$607$	3.33380	−	5.77431i	0.135315	−	0.234372i	−0.790403	−	0.612587i	$-0.790129\pi$
$607$	3.33380	−	5.77431i	0.135315	−	0.234372i	0.925718	+	0.378215i	$0.123462\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−23.9324	+	28.3979i	−0.968202	+	1.14886i
$612$		0			0
$613$	−18.0197	−	4.82836i	−0.727808	−	0.195015i	−0.124155	−	0.992263i	$-0.539622\pi$
$613$	−18.0197	−	4.82836i	−0.727808	−	0.195015i	−0.603653	+	0.797247i	$0.706289\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−15.4598	+	4.14243i	−0.622386	+	0.166768i	−0.556212	−	0.831040i	$-0.687746\pi$
$617$	−15.4598	+	4.14243i	−0.622386	+	0.166768i	−0.0661739	+	0.997808i	$0.521079\pi$
$618$		0			0
$619$	−24.4411	−	24.4411i	−0.982372	−	0.982372i	0.0174755	−	0.999847i	$-0.494437\pi$
$619$	−24.4411	−	24.4411i	−0.982372	−	0.982372i	−0.999847	+	0.0174755i	$0.994437\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−10.6642			−0.427252
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	3.47322	+	3.47322i	0.138486	+	0.138486i
$630$		0			0
$631$	−13.2410	+	3.54793i	−0.527118	+	0.141241i	−0.512557	−	0.858653i	$-0.671302\pi$
$631$	−13.2410	+	3.54793i	−0.527118	+	0.141241i	−0.0145608	+	0.999894i	$0.504635\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	8.66214	+	2.32101i	0.343747	+	0.0921067i
$636$		0			0
$637$	11.0767	−	7.72072i	0.438875	−	0.305906i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	4.75984	−	8.24429i	0.188003	−	0.325630i	−0.756582	−	0.653899i	$-0.773132\pi$
$641$	4.75984	−	8.24429i	0.188003	−	0.325630i	0.944584	+	0.328269i	$0.106465\pi$
$642$		0			0
$643$	6.75344	+	25.2042i	0.266330	+	0.993956i	0.961432	+	0.275044i	$0.0886926\pi$
$643$	6.75344	+	25.2042i	0.266330	+	0.993956i	−0.695102	+	0.718911i	$0.744641\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1.29428	+	2.24176i	0.0508835	+	0.0881329i	0.890345	−	0.455286i	$-0.150463\pi$
$647$	1.29428	+	2.24176i	0.0508835	+	0.0881329i	−0.839462	+	0.543419i	$0.817130\pi$
$648$		0			0
$649$		−	25.6116i		−	1.00534i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	35.4836	−	20.4865i	1.38858	−	0.801697i	0.395425	−	0.918498i	$-0.370597\pi$
$653$	35.4836	−	20.4865i	1.38858	−	0.801697i	0.993155	+	0.116801i	$0.0372639\pi$
$654$		0			0
$655$	−7.54477	+	7.54477i	−0.294799	+	0.294799i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−19.6183	−	11.3267i	−0.764222	−	0.441224i	0.0665874	−	0.997781i	$-0.478789\pi$
$659$	−19.6183	−	11.3267i	−0.764222	−	0.441224i	−0.830810	+	0.556557i	$0.812122\pi$
$660$		0			0
$661$	−4.52193	+	16.8761i	−0.175883	+	0.656403i	0.820517	+	0.571622i	$0.193686\pi$
$661$	−4.52193	+	16.8761i	−0.175883	+	0.656403i	−0.996400	+	0.0847806i	$0.972981\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1.68224	+	6.27820i	−0.0652344	+	0.243458i
$666$		0			0
$667$	10.1933	+	5.88512i	0.394687	+	0.227873i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	16.7260	−	16.7260i	0.645699	−	0.645699i
$672$		0			0
$673$	23.2297	−	13.4117i	0.895441	−	0.516983i	0.0197226	−	0.999805i	$-0.493722\pi$
$673$	23.2297	−	13.4117i	0.895441	−	0.516983i	0.875718	+	0.482822i	$0.160388\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			11.5465i			0.443769i	0.975073	+	0.221884i	$0.0712207\pi$
$677$			11.5465i			0.443769i	−0.975073	+	0.221884i	$0.928779\pi$
$678$		0			0
$679$	10.5437	+	18.2623i	0.404631	+	0.700842i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	9.41674	+	35.1437i	0.360321	+	1.34474i	0.873654	+	0.486548i	$0.161744\pi$
$683$	9.41674	+	35.1437i	0.360321	+	1.34474i	−0.513332	+	0.858190i	$0.671589\pi$
$684$		0			0
$685$	−4.41343	+	7.64428i	−0.168628	+	0.292073i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1.77695	+	20.8238i	0.0676964	+	0.793325i
$690$		0			0
$691$	−6.72606	−	1.80224i	−0.255871	−	0.0685605i	0.128603	−	0.991696i	$-0.458951\pi$
$691$	−6.72606	−	1.80224i	−0.255871	−	0.0685605i	−0.384474	+	0.923136i	$0.625617\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	2.48794	−	0.666641i	0.0943728	−	0.0252871i
$696$		0			0
$697$	−2.88103	−	2.88103i	−0.109127	−	0.109127i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−38.9194			−1.46997			−0.734983	−	0.678086i	$-0.762810\pi$
$701$	−38.9194			−1.46997			−0.734983	+	0.678086i	$0.762810\pi$
$702$		0			0
$703$	9.79777			0.369530
$704$		0			0
$705$		0			0
$706$		0			0
$707$	31.6169	+	31.6169i	1.18908	+	1.18908i
$708$		0			0
$709$	41.0620	−	11.0025i	1.54212	−	0.413209i	0.615168	−	0.788396i	$-0.289088\pi$
$709$	41.0620	−	11.0025i	1.54212	−	0.413209i	0.926949	+	0.375187i	$0.122422\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−24.2207	−	6.48991i	−0.907072	−	0.243049i
$714$		0			0
$715$	−4.01030	−	5.75347i	−0.149977	−	0.215168i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	12.5747	−	21.7800i	0.468958	−	0.812258i	−0.530413	−	0.847739i	$-0.677963\pi$
$719$	12.5747	−	21.7800i	0.468958	−	0.812258i	0.999370	+	0.0354812i	$0.0112964\pi$
$720$		0			0
$721$	−2.24193	−	8.36700i	−0.0834939	−	0.311603i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1.00709	+	1.74433i	0.0374024	+	0.0647828i
$726$		0			0
$727$		−	17.8058i		−	0.660382i	−0.943914	−	0.330191i	$-0.892887\pi$
$727$		−	17.8058i		−	0.660382i	0.943914	−	0.330191i	$-0.107113\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−2.25676	+	1.30294i	−0.0834694	+	0.0481911i
$732$		0			0
$733$	−31.6431	+	31.6431i	−1.16876	+	1.16876i	−0.186264	+	0.982500i	$0.559638\pi$
$733$	−31.6431	+	31.6431i	−1.16876	+	1.16876i	−0.982500	+	0.186264i	$0.940362\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	19.5489	+	11.2866i	0.720093	+	0.415746i
$738$		0			0
$739$	0.725052	−	2.70593i	0.0266715	−	0.0995393i	−0.951307	−	0.308245i	$-0.900258\pi$
$739$	0.725052	−	2.70593i	0.0266715	−	0.0995393i	0.977979	+	0.208706i	$0.0669250\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−4.33319	+	16.1717i	−0.158969	+	0.593282i	0.839763	+	0.542953i	$0.182694\pi$
$743$	−4.33319	+	16.1717i	−0.158969	+	0.593282i	−0.998733	+	0.0503292i	$0.983973\pi$
$744$		0			0
$745$	0.240534	+	0.138872i	0.00881248	+	0.00508789i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−13.8006	+	13.8006i	−0.504264	+	0.504264i
$750$		0			0
$751$	−38.9158	+	22.4680i	−1.42006	+	0.819871i	−0.996303	−	0.0859067i	$-0.972621\pi$
$751$	−38.9158	+	22.4680i	−1.42006	+	0.819871i	−0.423754	+	0.905777i	$0.639288\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	11.1131i		−	0.404447i
$756$		0			0
$757$	−10.1405	−	17.5639i	−0.368563	−	0.638370i	0.620778	−	0.783986i	$-0.286817\pi$
$757$	−10.1405	−	17.5639i	−0.368563	−	0.638370i	−0.989341	+	0.145616i	$0.953483\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−10.6204	−	39.6358i	−0.384988	−	1.43680i	−0.838186	−	0.545385i	$-0.816383\pi$
$761$	−10.6204	−	39.6358i	−0.384988	−	1.43680i	0.453198	−	0.891410i	$-0.350283\pi$
$762$		0			0
$763$	22.1480	−	38.3614i	0.801811	−	1.38878i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	16.1418	+	44.6464i	0.582848	+	1.61209i
$768$		0			0
$769$	25.3183	+	6.78401i	0.913000	+	0.244638i	0.684591	−	0.728928i	$-0.259981\pi$
$769$	25.3183	+	6.78401i	0.913000	+	0.244638i	0.228409	+	0.973565i	$0.426648\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−8.47023	+	2.26959i	−0.304653	+	0.0816315i	−0.407907	−	0.913023i	$-0.633741\pi$
$773$	−8.47023	+	2.26959i	−0.304653	+	0.0816315i	0.103254	+	0.994655i	$0.467075\pi$
$774$		0			0
$775$	−3.03417	−	3.03417i	−0.108991	−	0.108991i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−8.12723			−0.291188
$780$		0			0
$781$	16.4904			0.590072
$782$		0			0
$783$		0			0
$784$		0			0
$785$	6.59675	+	6.59675i	0.235448	+	0.235448i
$786$		0			0
$787$	−17.6708	+	4.73488i	−0.629896	+	0.168780i	−0.559623	−	0.828748i	$-0.689054\pi$
$787$	−17.6708	+	4.73488i	−0.629896	+	0.168780i	−0.0702737	+	0.997528i	$0.522387\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−56.5660	−	15.1568i	−2.01125	−	0.538914i
$792$		0			0
$793$	−18.6153	+	39.6985i	−0.661047	+	1.40974i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−4.26464	+	7.38657i	−0.151061	+	0.261646i	−0.931618	−	0.363439i	$-0.881602\pi$
$797$	−4.26464	+	7.38657i	−0.151061	+	0.261646i	0.780557	+	0.625085i	$0.214936\pi$
$798$		0			0
$799$	2.65004	+	9.89007i	0.0937515	+	0.349886i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	13.3720	+	23.1609i	0.471886	+	0.817331i
$804$		0			0
$805$		−	19.1552i		−	0.675131i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−28.7439	+	16.5953i	−1.01058	+	0.583459i	−0.911362	−	0.411607i	$-0.864968\pi$
$809$	−28.7439	+	16.5953i	−1.01058	+	0.583459i	−0.0992189	+	0.995066i	$0.531634\pi$
$810$		0			0
$811$	−24.1650	+	24.1650i	−0.848547	+	0.848547i	−0.989952	−	0.141405i	$-0.954838\pi$
$811$	−24.1650	+	24.1650i	−0.848547	+	0.848547i	0.141405	+	0.989952i	$0.454838\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	2.38381	+	1.37629i	0.0835011	+	0.0482094i
$816$		0			0
$817$	−1.34534	+	5.02087i	−0.0470674	+	0.175658i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	7.36327	−	27.4801i	0.256980	−	0.959062i	−0.709998	−	0.704203i	$-0.751304\pi$
$821$	7.36327	−	27.4801i	0.256980	−	0.959062i	0.966978	−	0.254859i	$-0.0820290\pi$
$822$		0			0
$823$	40.3682	+	23.3066i	1.40715	+	0.812417i	0.995112	−	0.0987509i	$-0.0314847\pi$
$823$	40.3682	+	23.3066i	1.40715	+	0.812417i	0.412035	+	0.911168i	$0.364818\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−11.8641	+	11.8641i	−0.412554	+	0.412554i	−0.882627	−	0.470074i	$-0.844227\pi$
$827$	−11.8641	+	11.8641i	−0.412554	+	0.412554i	0.470074	+	0.882627i	$0.344227\pi$
$828$		0			0
$829$	−2.49255	+	1.43907i	−0.0865697	+	0.0499811i	−0.542660	−	0.839952i	$-0.682583\pi$
$829$	−2.49255	+	1.43907i	−0.0865697	+	0.0499811i	0.456090	+	0.889934i	$0.349249\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	3.72253i		−	0.128978i
$834$		0			0
$835$	8.00807	+	13.8704i	0.277131	+	0.480004i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−7.01685	−	26.1872i	−0.242249	−	0.904084i	−0.974746	−	0.223315i	$-0.928312\pi$
$839$	−7.01685	−	26.1872i	−0.242249	−	0.904084i	0.732498	−	0.680769i	$-0.238354\pi$
$840$		0			0
$841$	−12.4715	+	21.6013i	−0.430053	+	0.744874i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	10.6170	+	7.50201i	0.365235	+	0.258077i
$846$		0			0
$847$	22.8492	+	6.12243i	0.785108	+	0.210369i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−27.8911	+	7.47339i	−0.956094	+	0.256185i
$852$		0			0
$853$	−27.7512	−	27.7512i	−0.950184	−	0.950184i	0.0486329	−	0.998817i	$-0.484514\pi$
$853$	−27.7512	−	27.7512i	−0.950184	−	0.950184i	−0.998817	+	0.0486329i	$0.984514\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	35.6349			1.21727			0.608633	−	0.793452i	$-0.291718\pi$
$857$	35.6349			1.21727			0.608633	+	0.793452i	$0.291718\pi$
$858$		0			0
$859$	−36.8211			−1.25632			−0.628161	−	0.778084i	$-0.716192\pi$
$859$	−36.8211			−1.25632			−0.628161	+	0.778084i	$0.716192\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	40.0126	+	40.0126i	1.36204	+	1.36204i	0.871309	+	0.490735i	$0.163272\pi$
$863$	40.0126	+	40.0126i	1.36204	+	1.36204i	0.490735	+	0.871309i	$0.336728\pi$
$864$		0			0
$865$	10.0421	−	2.69078i	0.341443	−	0.0914894i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	28.7651	+	7.70758i	0.975789	+	0.261462i
$870$		0			0
$871$	−41.1913	−	7.35407i	−1.39571	−	0.249183i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.63896	−	2.83876i	0.0554070	−	0.0959677i
$876$		0			0
$877$	−3.41490	−	12.7446i	−0.115313	−	0.430353i	0.883997	−	0.467492i	$-0.154842\pi$
$877$	−3.41490	−	12.7446i	−0.115313	−	0.430353i	−0.999310	+	0.0371386i	$0.988176\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−14.9055	−	25.8171i	−0.502180	−	0.869801i	−0.999997	−	0.00251857i	$-0.999198\pi$
$881$	−14.9055	−	25.8171i	−0.502180	−	0.869801i	0.497817	−	0.867282i	$-0.334135\pi$
$882$		0			0
$883$		−	51.1759i		−	1.72221i	−0.508431	−	0.861103i	$-0.669774\pi$
$883$		−	51.1759i		−	1.72221i	0.508431	−	0.861103i	$-0.330226\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	17.2333	−	9.94965i	0.578637	−	0.334076i	−0.181954	−	0.983307i	$-0.558242\pi$
$887$	17.2333	−	9.94965i	0.578637	−	0.334076i	0.760592	+	0.649231i	$0.224909\pi$
$888$		0			0
$889$	−20.7857	+	20.7857i	−0.697131	+	0.697131i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	17.6875	+	10.2119i	0.591889	+	0.341727i
$894$		0			0
$895$	2.57884	−	9.62436i	0.0862011	−	0.321707i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−2.23692	+	8.34828i	−0.0746053	+	0.278431i
$900$		0			0
$901$	4.99009	+	2.88103i	0.166244	+	0.0959811i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−5.91692	+	5.91692i	−0.196685	+	0.196685i
$906$		0			0
$907$	−1.62081	+	0.935776i	−0.0538181	+	0.0310719i	−0.526668	−	0.850071i	$-0.676559\pi$
$907$	−1.62081	+	0.935776i	−0.0538181	+	0.0310719i	0.472850	+	0.881143i	$0.343225\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			33.2989i			1.10324i	0.834095	+	0.551620i	$0.185990\pi$
$911$			33.2989i			1.10324i	−0.834095	+	0.551620i	$0.814010\pi$
$912$		0			0
$913$	−6.70615	−	11.6154i	−0.221941	−	0.384414i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−9.05224	−	33.7834i	−0.298931	−	1.11563i
$918$		0			0
$919$	−13.4224	+	23.2483i	−0.442765	+	0.766892i	−0.997894	−	0.0648727i	$-0.979336\pi$
$919$	−13.4224	+	23.2483i	−0.442765	+	0.766892i	0.555128	+	0.831765i	$0.312669\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−28.7462	+	10.3932i	−0.946193	+	0.342095i
$924$		0			0
$925$	−4.77285	−	1.27888i	−0.156930	−	0.0420494i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	38.3256	−	10.2693i	1.25742	−	0.336925i	0.432223	−	0.901767i	$-0.357729\pi$
$929$	38.3256	−	10.2693i	1.25742	−	0.336925i	0.825199	+	0.564842i	$0.191063\pi$
$930$		0			0
$931$	−5.25053	−	5.25053i	−0.172079	−	0.172079i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−1.93356			−0.0632342
$936$		0			0
$937$	17.6999			0.578230			0.289115	−	0.957294i	$-0.406639\pi$
$937$	17.6999			0.578230			0.289115	+	0.957294i	$0.406639\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	39.1690	+	39.1690i	1.27687	+	1.27687i	0.942409	+	0.334464i	$0.108555\pi$
$941$	39.1690	+	39.1690i	1.27687	+	1.27687i	0.334464	+	0.942409i	$0.391445\pi$
$942$		0			0
$943$	23.1356	−	6.19917i	0.753399	−	0.201873i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−27.1761	−	7.28182i	−0.883105	−	0.236627i	−0.211359	−	0.977409i	$-0.567789\pi$
$947$	−27.1761	−	7.28182i	−0.883105	−	0.236627i	−0.671746	+	0.740781i	$0.734456\pi$
$948$		0			0
$949$	−37.9074	−	31.9466i	−1.23053	−	1.03703i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	0.534466	−	0.925722i	0.0173130	−	0.0299871i	−0.857239	−	0.514919i	$-0.827822\pi$
$953$	0.534466	−	0.925722i	0.0173130	−	0.0299871i	0.874552	+	0.484932i	$0.161155\pi$
$954$		0			0
$955$	−0.606634	−	2.26399i	−0.0196302	−	0.0732609i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−14.4669	−	25.0574i	−0.467160	−	0.809145i
$960$		0			0
$961$			12.5876i			0.406051i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−12.1070	+	6.98998i	−0.389738	+	0.225015i
$966$		0			0
$967$	8.93964	−	8.93964i	0.287479	−	0.287479i	−0.548603	−	0.836083i	$-0.684840\pi$
$967$	8.93964	−	8.93964i	0.287479	−	0.287479i	0.836083	+	0.548603i	$0.184840\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−34.6823	−	20.0238i	−1.11301	−	0.642595i	−0.173401	−	0.984851i	$-0.555476\pi$
$971$	−34.6823	−	20.0238i	−1.11301	−	0.642595i	−0.939607	+	0.342256i	$0.888809\pi$
$972$		0			0
$973$	−2.18520	+	8.15526i	−0.0700542	+	0.261446i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	7.04727	−	26.3008i	0.225462	−	0.841436i	−0.756757	−	0.653697i	$-0.773217\pi$
$977$	7.04727	−	26.3008i	0.225462	−	0.841436i	0.982219	−	0.187740i	$-0.0601161\pi$
$978$		0			0
$979$	−5.48030	−	3.16405i	−0.175151	−	0.101124i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	24.1627	−	24.1627i	0.770671	−	0.770671i	−0.207553	−	0.978224i	$-0.566550\pi$
$983$	24.1627	−	24.1627i	0.770671	−	0.770671i	0.978224	+	0.207553i	$0.0665498\pi$
$984$		0			0
$985$	−6.52475	+	3.76707i	−0.207896	+	0.120029i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	15.3190i		−	0.487115i
$990$		0			0
$991$	3.29089	+	5.69999i	0.104539	+	0.181066i	0.913550	−	0.406727i	$-0.133330\pi$
$991$	3.29089	+	5.69999i	0.104539	+	0.181066i	−0.809011	+	0.587794i	$0.799997\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1.50671	+	5.62313i	0.0477660	+	0.178265i
$996$		0			0
$997$	−7.97351	+	13.8105i	−0.252524	+	0.437384i	−0.964220	−	0.265103i	$-0.914594\pi$
$997$	−7.97351	+	13.8105i	−0.252524	+	0.437384i	0.711696	+	0.702487i	$0.247927\pi$
$998$		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.1601.7	yes	40
3.2	odd	2	inner	2340.2.fo.a.1601.2	✓	40
13.7	odd	12	inner	2340.2.fo.a.2321.2	yes	40
39.20	even	12	inner	2340.2.fo.a.2321.7	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2340.2.fo.a.1601.2	✓	40	3.2	odd	2	inner
2340.2.fo.a.1601.7	yes	40	1.1	even	1	trivial
2340.2.fo.a.2321.2	yes	40	13.7	odd	12	inner
2340.2.fo.a.2321.7	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)

\(f(q)\)	\(=\)	\(q+(0.707107 + 0.707107i) q^{5} +(-3.16623 + 0.848389i) q^{7} +(-1.87883 - 0.503432i) q^{11} +(3.59250 - 0.306557i) q^{13} +(0.497031 - 0.860883i) q^{17} +(-0.513203 - 1.91530i) q^{19} +(2.92185 + 5.06079i) q^{23} +1.00000i q^{25} +(1.74433 - 1.00709i) q^{29} +(-3.03417 + 3.03417i) q^{31} +(-2.83876 - 1.63896i) q^{35} +(-1.27888 + 4.77285i) q^{37} +(1.06083 - 3.95907i) q^{41} +(-2.27024 - 1.31073i) q^{43} +(-7.28328 + 7.28328i) q^{47} +(3.24307 - 1.87239i) q^{49} +5.79649i q^{53} +(-0.972556 - 1.68452i) q^{55} +(3.40791 + 12.7185i) q^{59} +(-6.08039 + 10.5315i) q^{61} +(2.75705 + 2.32351i) q^{65} +(-11.2096 - 3.00361i) q^{67} +(-8.18898 + 2.19423i) q^{71} +(-9.72222 - 9.72222i) q^{73} +6.37593 q^{77} -15.3101 q^{79} +(4.87578 + 4.87578i) q^{83} +(0.960190 - 0.257282i) q^{85} +(3.14248 + 0.842026i) q^{89} +(-11.1146 + 4.01846i) q^{91} +(0.991432 - 1.71721i) q^{95} +(-1.66503 - 6.21398i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 12 q^{13} - 20 q^{19} - 28 q^{31} + 12 q^{49} + 16 q^{55} + 48 q^{61} + 16 q^{67} - 20 q^{73} + 80 q^{79} + 20 q^{85} + 4 q^{91} - 40 q^{97}+O(q^{100}) \) 40 * q - 12 * q^13 - 20 * q^19 - 28 * q^31 + 12 * q^49 + 16 * q^55 + 48 * q^61 + 16 * q^67 - 20 * q^73 + 80 * q^79 + 20 * q^85 + 4 * q^91 - 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more