LMFDB - Embedded newform 2340.2.fo.a.2321.3

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)

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);

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")

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			2321.3
Character	$\chi$	$=$	2340.2321
Dual form			2340.2.fo.a.1601.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 + 0.707107i) q^{5} +(1.13373 + 0.303782i) q^{7} +(-5.67905 + 1.52170i) q^{11} +(1.18813 - 3.40417i) q^{13} +(1.54614 + 2.67800i) q^{17} +(0.197693 - 0.737799i) q^{19} +(0.483360 - 0.837203i) q^{23} -1.00000i q^{25} +(-1.50966 - 0.871605i) q^{29} +(-6.73314 - 6.73314i) q^{31} +(-1.01647 + 0.586862i) q^{35} +(1.40322 + 5.23688i) q^{37} +(1.24298 + 4.63886i) q^{41} +(-0.193895 + 0.111945i) q^{43} +(-5.91426 - 5.91426i) q^{47} +(-4.86912 - 2.81119i) q^{49} -6.79176i q^{53} +(2.93969 - 5.09170i) q^{55} +(2.39404 - 8.93468i) q^{59} +(-5.59287 - 9.68714i) q^{61} +(1.56697 + 3.24724i) q^{65} +(1.68534 - 0.451587i) q^{67} +(0.138897 + 0.0372173i) q^{71} +(1.76023 - 1.76023i) q^{73} -6.90078 q^{77} +3.45768 q^{79} +(-10.0218 + 10.0218i) q^{83} +(-2.98692 - 0.800342i) q^{85} +(-8.76560 + 2.34874i) q^{89} +(2.38114 - 3.49847i) q^{91} +(0.381913 + 0.661493i) q^{95} +(0.444398 - 1.65852i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/2340\mathbb{Z}\right)^\times$.

$n$	$937$	$1081$	$1171$	$2081$
$\chi(n)$	$1$	$e\left(\frac{11}{12}\right)$	$1$	$-1$

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
$7$	1.13373	+	0.303782i	0.428510	+	0.114819i	0.466626	−	0.884454i	$-0.345469\pi$
$7$	1.13373	+	0.303782i	0.428510	+	0.114819i	−0.0381169	+	0.999273i	$0.512136\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−5.67905	+	1.52170i	−1.71230	+	0.458809i	−0.975986	−	0.217832i	$-0.930102\pi$
$11$	−5.67905	+	1.52170i	−1.71230	+	0.458809i	−0.736313	+	0.676641i	$0.763435\pi$
$12$		0			0
$13$	1.18813	−	3.40417i	0.329528	−	0.944146i
$13$	1.18813	−	3.40417i	0.329528	−	0.944146i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.54614	+	2.67800i	0.374994	+	0.649509i	0.990326	−	0.138759i	$-0.0443113\pi$
$17$	1.54614	+	2.67800i	0.374994	+	0.649509i	−0.615332	+	0.788268i	$0.710978\pi$
$18$		0			0
$19$	0.197693	−	0.737799i	0.0453538	−	0.169263i	−0.939534	−	0.342455i	$-0.888742\pi$
$19$	0.197693	−	0.737799i	0.0453538	−	0.169263i	0.984888	+	0.173192i	$0.0554082\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.483360	−	0.837203i	0.100787	−	0.174569i	−0.811222	−	0.584738i	$-0.801197\pi$
$23$	0.483360	−	0.837203i	0.100787	−	0.174569i	0.912009	+	0.410170i	$0.134530\pi$
$24$		0			0
$25$		−	1.00000i		−	0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1.50966	−	0.871605i	−0.280338	−	0.161853i	0.353239	−	0.935533i	$-0.385080\pi$
$29$	−1.50966	−	0.871605i	−0.280338	−	0.161853i	−0.633576	+	0.773680i	$0.718414\pi$
$30$		0			0
$31$	−6.73314	−	6.73314i	−1.20931	−	1.20931i	−0.971251	−	0.238056i	$-0.923490\pi$
$31$	−6.73314	−	6.73314i	−1.20931	−	1.20931i	−0.238056	−	0.971251i	$-0.576510\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.01647	+	0.586862i	−0.171816	+	0.0991977i
$36$		0			0
$37$	1.40322	+	5.23688i	0.230687	+	0.860937i	0.980046	+	0.198772i	$0.0636955\pi$
$37$	1.40322	+	5.23688i	0.230687	+	0.860937i	−0.749358	+	0.662165i	$0.769638\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	1.24298	+	4.63886i	0.194121	+	0.724468i	0.992493	+	0.122304i	$0.0390282\pi$
$41$	1.24298	+	4.63886i	0.194121	+	0.724468i	−0.798372	+	0.602165i	$0.794305\pi$
$42$		0			0
$43$	−0.193895	+	0.111945i	−0.0295686	+	0.0170715i	−0.514711	−	0.857363i	$-0.672101\pi$
$43$	−0.193895	+	0.111945i	−0.0295686	+	0.0170715i	0.485143	+	0.874435i	$0.338768\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−5.91426	−	5.91426i	−0.862683	−	0.862683i	0.128966	−	0.991649i	$-0.458834\pi$
$47$	−5.91426	−	5.91426i	−0.862683	−	0.862683i	−0.991649	+	0.128966i	$0.958834\pi$
$48$		0			0
$49$	−4.86912	−	2.81119i	−0.695588	−	0.401598i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	6.79176i		−	0.932921i	−0.884542	−	0.466460i	$-0.845529\pi$
$53$		−	6.79176i		−	0.932921i	0.884542	−	0.466460i	$-0.154471\pi$
$54$		0			0
$55$	2.93969	−	5.09170i	0.396388	−	0.686565i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	2.39404	−	8.93468i	0.311678	−	1.16320i	−0.615366	−	0.788242i	$-0.710992\pi$
$59$	2.39404	−	8.93468i	0.311678	−	1.16320i	0.927043	−	0.374955i	$-0.122342\pi$
$60$		0			0
$61$	−5.59287	−	9.68714i	−0.716094	−	1.24031i	−0.962536	−	0.271154i	$-0.912595\pi$
$61$	−5.59287	−	9.68714i	−0.716094	−	1.24031i	0.246442	−	0.969158i	$-0.420739\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.56697	+	3.24724i	0.194359	+	0.402771i
$66$		0			0
$67$	1.68534	−	0.451587i	0.205898	−	0.0551701i	−0.154396	−	0.988009i	$-0.549343\pi$
$67$	1.68534	−	0.451587i	0.205898	−	0.0551701i	0.360294	+	0.932839i	$0.382676\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	0.138897	+	0.0372173i	0.0164840	+	0.00441689i	0.267052	−	0.963682i	$-0.413951\pi$
$71$	0.138897	+	0.0372173i	0.0164840	+	0.00441689i	−0.250568	+	0.968099i	$0.580617\pi$
$72$		0			0
$73$	1.76023	−	1.76023i	0.206019	−	0.206019i	−0.596554	−	0.802573i	$-0.703464\pi$
$73$	1.76023	−	1.76023i	0.206019	−	0.206019i	0.802573	+	0.596554i	$0.203464\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−6.90078			−0.786417
$78$		0			0
$79$	3.45768			0.389020			0.194510	−	0.980901i	$-0.437688\pi$
$79$	3.45768			0.389020			0.194510	+	0.980901i	$0.437688\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−10.0218	+	10.0218i	−1.10003	+	1.10003i	−0.105625	+	0.994406i	$0.533684\pi$
$83$	−10.0218	+	10.0218i	−1.10003	+	1.10003i	−0.994406	+	0.105625i	$0.966316\pi$
$84$		0			0
$85$	−2.98692	−	0.800342i	−0.323977	−	0.0868093i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−8.76560	+	2.34874i	−0.929152	+	0.248965i	−0.691493	−	0.722383i	$-0.743047\pi$
$89$	−8.76560	+	2.34874i	−0.929152	+	0.248965i	−0.237659	+	0.971349i	$0.576380\pi$
$90$		0			0
$91$	2.38114	−	3.49847i	0.249612	−	0.366740i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	0.381913	+	0.661493i	0.0391834	+	0.0678677i
$96$		0			0
$97$	0.444398	−	1.65852i	0.0451218	−	0.168397i	−0.939688	−	0.342032i	$-0.888885\pi$
$97$	0.444398	−	1.65852i	0.0451218	−	0.168397i	0.984810	+	0.173636i	$0.0555515\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	0.271625	−	0.470468i	0.0270277	−	0.0468133i	−0.852195	−	0.523224i	$-0.824729\pi$
$101$	0.271625	−	0.470468i	0.0270277	−	0.0468133i	0.879223	+	0.476411i	$0.158062\pi$
$102$		0			0
$103$		−	11.2343i		−	1.10695i	−0.832866	−	0.553474i	$-0.813302\pi$
$103$		−	11.2343i		−	1.10695i	0.832866	−	0.553474i	$-0.186698\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−4.65278	−	2.68629i	−0.449802	−	0.259693i	0.257945	−	0.966160i	$-0.416955\pi$
$107$	−4.65278	−	2.68629i	−0.449802	−	0.259693i	−0.707746	+	0.706467i	$0.750288\pi$
$108$		0			0
$109$	5.32031	+	5.32031i	0.509594	+	0.509594i	0.914402	−	0.404808i	$-0.132662\pi$
$109$	5.32031	+	5.32031i	0.509594	+	0.509594i	−0.404808	+	0.914402i	$0.632662\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	9.40694	−	5.43110i	0.884931	−	0.510915i	0.0126497	−	0.999920i	$-0.495973\pi$
$113$	9.40694	−	5.43110i	0.884931	−	0.510915i	0.872281	+	0.489005i	$0.162640\pi$
$114$		0			0
$115$	0.250205	+	0.933779i	0.0233318	+	0.0870753i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	0.939380	+	3.50581i	0.0861128	+	0.321377i
$120$		0			0
$121$	20.4098	−	11.7836i	1.85544	−	1.07124i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	0.707107	+	0.707107i	0.0632456	+	0.0632456i
$126$		0			0
$127$	−5.16094	−	2.97967i	−0.457960	−	0.264403i	0.253226	−	0.967407i	$-0.418508\pi$
$127$	−5.16094	−	2.97967i	−0.457960	−	0.264403i	−0.711186	+	0.703004i	$0.751842\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			8.88342i			0.776148i	0.921628	+	0.388074i	$0.126859\pi$
$131$			8.88342i			0.776148i	−0.921628	+	0.388074i	$0.873141\pi$
$132$		0			0
$133$	0.448260	−	0.776410i	0.0388691	−	0.0673233i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	2.45265	−	9.15341i	0.209544	−	0.782029i	−0.778472	−	0.627679i	$-0.784005\pi$
$137$	2.45265	−	9.15341i	0.209544	−	0.782029i	0.988016	−	0.154350i	$-0.0493282\pi$
$138$		0			0
$139$	−5.68647	−	9.84925i	−0.482320	−	0.835402i	0.517474	−	0.855699i	$-0.326872\pi$
$139$	−5.68647	−	9.84925i	−0.482320	−	0.835402i	−0.999794	+	0.0202965i	$0.993539\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.56734	+	21.1404i	−0.131067	+	1.76785i
$144$		0			0
$145$	1.68381	−	0.451176i	0.139833	−	0.0374681i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−15.1009	−	4.04627i	−1.23711	−	0.331483i	−0.419768	−	0.907632i	$-0.637888\pi$
$149$	−15.1009	−	4.04627i	−1.23711	−	0.331483i	−0.817345	+	0.576148i	$0.804555\pi$
$150$		0			0
$151$	5.06855	−	5.06855i	0.412473	−	0.412473i	−0.470126	−	0.882599i	$-0.655792\pi$
$151$	5.06855	−	5.06855i	0.412473	−	0.412473i	0.882599	+	0.470126i	$0.155792\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	9.52210			0.764833
$156$		0			0
$157$	−10.4700			−0.835594			−0.417797	−	0.908540i	$-0.637198\pi$
$157$	−10.4700			−0.835594			−0.417797	+	0.908540i	$0.637198\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	0.802327	−	0.802327i	0.0632322	−	0.0632322i
$162$		0			0
$163$	−21.1260	−	5.66069i	−1.65471	−	0.443379i	−0.693786	−	0.720181i	$-0.744059\pi$
$163$	−21.1260	−	5.66069i	−1.65471	−	0.443379i	−0.960927	+	0.276802i	$0.910725\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−15.0023	+	4.01985i	−1.16091	+	0.311066i	−0.787330	−	0.616532i	$-0.788537\pi$
$167$	−15.0023	+	4.01985i	−1.16091	+	0.311066i	−0.373582	+	0.927597i	$0.621871\pi$
$168$		0			0
$169$	−10.1767	−	8.08918i	−0.782823	−	0.622245i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−8.09287	−	14.0173i	−0.615289	−	1.06571i	−0.990334	−	0.138705i	$-0.955706\pi$
$173$	−8.09287	−	14.0173i	−0.615289	−	1.06571i	0.375044	−	0.927007i	$-0.377627\pi$
$174$		0			0
$175$	0.303782	−	1.13373i	0.0229638	−	0.0857019i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−2.31122	+	4.00315i	−0.172749	+	0.299210i	−0.939380	−	0.342878i	$-0.888598\pi$
$179$	−2.31122	+	4.00315i	−0.172749	+	0.299210i	0.766631	+	0.642088i	$0.221932\pi$
$180$		0			0
$181$			16.5937i			1.23340i	0.787198	+	0.616700i	$0.211531\pi$
$181$			16.5937i			1.23340i	−0.787198	+	0.616700i	$0.788469\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−4.69526	−	2.71081i	−0.345202	−	0.199302i
$186$		0			0
$187$	−12.8557	−	12.8557i	−0.940103	−	0.940103i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	6.57704	−	3.79725i	0.475898	−	0.274760i	−0.242808	−	0.970074i	$-0.578068\pi$
$191$	6.57704	−	3.79725i	0.475898	−	0.274760i	0.718705	+	0.695315i	$0.244735\pi$
$192$		0			0
$193$	3.22758	+	12.0455i	0.232327	+	0.867054i	0.979336	+	0.202241i	$0.0648224\pi$
$193$	3.22758	+	12.0455i	0.232327	+	0.867054i	−0.747009	+	0.664814i	$0.768511\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−3.61312	−	13.4843i	−0.257424	−	0.960720i	−0.966726	−	0.255815i	$-0.917656\pi$
$197$	−3.61312	−	13.4843i	−0.257424	−	0.960720i	0.709302	−	0.704905i	$-0.249010\pi$
$198$		0			0
$199$	8.63320	−	4.98438i	0.611991	−	0.353333i	−0.161753	−	0.986831i	$-0.551715\pi$
$199$	8.63320	−	4.98438i	0.611991	−	0.353333i	0.773744	+	0.633498i	$0.218381\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1.44677	−	1.44677i	−0.101544	−	0.101544i
$204$		0			0
$205$	−4.15909	−	2.40125i	−0.290483	−	0.167711i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			4.49083i			0.310637i
$210$		0			0
$211$	−2.53786	+	4.39571i	−0.174714	+	0.302613i	−0.940062	−	0.341003i	$-0.889233\pi$
$211$	−2.53786	+	4.39571i	−0.174714	+	0.302613i	0.765348	+	0.643616i	$0.222567\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	0.0579470	−	0.216261i	0.00395195	−	0.0147489i
$216$		0			0
$217$	−5.58816	−	9.67897i	−0.379349	−	0.657051i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	10.9534	−	2.08152i	0.736803	−	0.140018i
$222$		0			0
$223$	14.3790	−	3.85285i	0.962891	−	0.258006i	0.257067	−	0.966394i	$-0.417244\pi$
$223$	14.3790	−	3.85285i	0.962891	−	0.258006i	0.705823	+	0.708388i	$0.250577\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−15.6176	−	4.18473i	−1.03658	−	0.277750i	−0.299883	−	0.953976i	$-0.596948\pi$
$227$	−15.6176	−	4.18473i	−1.03658	−	0.277750i	−0.736694	+	0.676226i	$0.763614\pi$
$228$		0			0
$229$	−10.0529	+	10.0529i	−0.664313	+	0.664313i	−0.956394	−	0.292080i	$-0.905653\pi$
$229$	−10.0529	+	10.0529i	−0.664313	+	0.664313i	0.292080	+	0.956394i	$0.405653\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	12.1088			0.793272			0.396636	−	0.917976i	$-0.370178\pi$
$233$	12.1088			0.793272			0.396636	+	0.917976i	$0.370178\pi$
$234$		0			0
$235$	8.36402			0.545609
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1.03894	−	1.03894i	0.0672037	−	0.0672037i	−0.672706	−	0.739910i	$-0.734868\pi$
$239$	1.03894	−	1.03894i	0.0672037	−	0.0672037i	0.739910	+	0.672706i	$0.234868\pi$
$240$		0			0
$241$	26.7924	+	7.17901i	1.72585	+	0.462441i	0.979221	−	0.202796i	$-0.0650028\pi$
$241$	26.7924	+	7.17901i	1.72585	+	0.462441i	0.746632	+	0.665237i	$0.231669\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	5.43080	−	1.45518i	0.346961	−	0.0929679i
$246$		0			0
$247$	−2.27671	−	1.54958i	−0.144863	−	0.0985974i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	10.8938	+	18.8687i	0.687614	+	1.19098i	0.972608	+	0.232452i	$0.0746750\pi$
$251$	10.8938	+	18.8687i	0.687614	+	1.19098i	−0.284994	+	0.958529i	$0.591992\pi$
$252$		0			0
$253$	−1.47105	+	5.49005i	−0.0924844	+	0.345157i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	6.86167	−	11.8848i	0.428019	−	0.741351i	−0.568678	−	0.822560i	$-0.692545\pi$
$257$	6.86167	−	11.8848i	0.428019	−	0.741351i	0.996697	+	0.0812092i	$0.0258782\pi$
$258$		0			0
$259$			6.36348i			0.395407i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−27.9418	−	16.1322i	−1.72296	−	0.994753i	−0.912636	−	0.408774i	$-0.865956\pi$
$263$	−27.9418	−	16.1322i	−1.72296	−	0.994753i	−0.810327	−	0.585978i	$-0.800710\pi$
$264$		0			0
$265$	4.80250	+	4.80250i	0.295015	+	0.295015i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−8.48371	+	4.89807i	−0.517261	+	0.298641i	−0.735813	−	0.677185i	$-0.763200\pi$
$269$	−8.48371	+	4.89807i	−0.517261	+	0.298641i	0.218552	+	0.975825i	$0.429867\pi$
$270$		0			0
$271$	−1.73449	−	6.47322i	−0.105363	−	0.393220i	0.893023	−	0.450011i	$-0.148580\pi$
$271$	−1.73449	−	6.47322i	−0.105363	−	0.393220i	−0.998386	+	0.0567907i	$0.981913\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.52170	+	5.67905i	0.0917618	+	0.342460i
$276$		0			0
$277$	2.04724	−	1.18198i	0.123007	−	0.0710180i	−0.437234	−	0.899348i	$-0.644042\pi$
$277$	2.04724	−	1.18198i	0.123007	−	0.0710180i	0.560241	+	0.828330i	$0.310709\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	9.52176	+	9.52176i	0.568021	+	0.568021i	0.931574	−	0.363553i	$-0.118436\pi$
$281$	9.52176	+	9.52176i	0.568021	+	0.568021i	−0.363553	+	0.931574i	$0.618436\pi$
$282$		0			0
$283$	3.62127	+	2.09074i	0.215262	+	0.124282i	0.603754	−	0.797170i	$-0.293671\pi$
$283$	3.62127	+	2.09074i	0.215262	+	0.124282i	−0.388493	+	0.921452i	$0.627004\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			5.63681i			0.332730i
$288$		0			0
$289$	3.71889	−	6.44131i	0.218758	−	0.378901i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−5.64517	+	21.0681i	−0.329794	+	1.23081i	0.579609	+	0.814894i	$0.303205\pi$
$293$	−5.64517	+	21.0681i	−0.329794	+	1.23081i	−0.909404	+	0.415915i	$0.863462\pi$
$294$		0			0
$295$	4.62493	+	8.01062i	0.269274	+	0.466396i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2.27569	−	2.64014i	−0.131606	−	0.152683i
$300$		0			0
$301$	−0.253831	+	0.0680138i	−0.0146306	+	0.00392025i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	10.8046	+	2.89508i	0.618670	+	0.165772i
$306$		0			0
$307$	−14.4907	+	14.4907i	−0.827031	+	0.827031i	−0.987105	−	0.160074i	$-0.948827\pi$
$307$	−14.4907	+	14.4907i	−0.827031	+	0.827031i	0.160074	+	0.987105i	$0.448827\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	10.6627			0.604624			0.302312	−	0.953209i	$-0.402242\pi$
$311$	10.6627			0.604624			0.302312	+	0.953209i	$0.402242\pi$
$312$		0			0
$313$	11.6939			0.660976			0.330488	−	0.943810i	$-0.392787\pi$
$313$	11.6939			0.660976			0.330488	+	0.943810i	$0.392787\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−3.36308	+	3.36308i	−0.188889	+	0.188889i	−0.795216	−	0.606327i	$-0.792642\pi$
$317$	−3.36308	+	3.36308i	−0.188889	+	0.188889i	0.606327	+	0.795216i	$0.292642\pi$
$318$		0			0
$319$	9.89979	+	2.65264i	0.554282	+	0.148519i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	2.28148	−	0.611322i	0.126945	−	0.0340149i
$324$		0			0
$325$	−3.40417	−	1.18813i	−0.188829	−	0.0659055i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−4.90853	−	8.50182i	−0.270616	−	0.468720i
$330$		0			0
$331$	6.15844	−	22.9836i	0.338498	−	1.26329i	−0.561528	−	0.827457i	$-0.689786\pi$
$331$	6.15844	−	22.9836i	0.338498	−	1.26329i	0.900027	−	0.435835i	$-0.143547\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−0.872399	+	1.51104i	−0.0476642	+	0.0825569i
$336$		0			0
$337$			33.7290i			1.83733i	0.395033	+	0.918667i	$0.370733\pi$
$337$			33.7290i			1.83733i	−0.395033	+	0.918667i	$0.629267\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	48.4837	+	27.9921i	2.62554	+	1.51586i
$342$		0			0
$343$	−10.4759	−	10.4759i	−0.565646	−	0.565646i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−11.5063	+	6.64314i	−0.617688	+	0.356622i	−0.775968	−	0.630772i	$-0.782738\pi$
$347$	−11.5063	+	6.64314i	−0.617688	+	0.356622i	0.158280	+	0.987394i	$0.449405\pi$
$348$		0			0
$349$	2.75606	+	10.2858i	0.147529	+	0.550584i	0.999630	+	0.0272072i	$0.00866140\pi$
$349$	2.75606	+	10.2858i	0.147529	+	0.550584i	−0.852101	+	0.523377i	$0.824672\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−4.06465	−	15.1695i	−0.216340	−	0.807391i	−0.985691	−	0.168565i	$-0.946087\pi$
$353$	−4.06465	−	15.1695i	−0.216340	−	0.807391i	0.769351	−	0.638826i	$-0.220580\pi$
$354$		0			0
$355$	−0.124532	+	0.0718984i	−0.00660945	+	0.00381597i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−3.93861	−	3.93861i	−0.207872	−	0.207872i	0.595491	−	0.803362i	$-0.296958\pi$
$359$	−3.93861	−	3.93861i	−0.207872	−	0.207872i	−0.803362	+	0.595491i	$0.796958\pi$
$360$		0			0
$361$	15.9492	+	9.20828i	0.839432	+	0.484647i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			2.48934i			0.130298i
$366$		0			0
$367$	−10.0127	+	17.3426i	−0.522661	+	0.905276i	0.476991	+	0.878908i	$0.341727\pi$
$367$	−10.0127	+	17.3426i	−0.522661	+	0.905276i	−0.999652	+	0.0263676i	$0.991606\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	2.06322	−	7.70003i	0.107117	−	0.399765i
$372$		0			0
$373$	15.8687	+	27.4854i	0.821650	+	1.42314i	0.904453	+	0.426573i	$0.140279\pi$
$373$	15.8687	+	27.4854i	0.821650	+	1.42314i	−0.0828034	+	0.996566i	$0.526387\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−4.76077	+	4.10357i	−0.245192	+	0.211345i
$378$		0			0
$379$	−21.9570	+	5.88337i	−1.12786	+	0.302208i	−0.774059	−	0.633113i	$-0.781777\pi$
$379$	−21.9570	+	5.88337i	−1.12786	+	0.302208i	−0.353798	+	0.935322i	$0.615110\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	25.5676	+	6.85081i	1.30644	+	0.350060i	0.843883	−	0.536528i	$-0.180264\pi$
$383$	25.5676	+	6.85081i	1.30644	+	0.350060i	0.462560	+	0.886588i	$0.346931\pi$
$384$		0			0
$385$	4.87959	−	4.87959i	0.248687	−	0.248687i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	23.8866			1.21110			0.605550	−	0.795807i	$-0.292953\pi$
$389$	23.8866			1.21110			0.605550	+	0.795807i	$0.292953\pi$
$390$		0			0
$391$	2.98937			0.151179
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−2.44495	+	2.44495i	−0.123019	+	0.123019i
$396$		0			0
$397$	29.9528	+	8.02584i	1.50329	+	0.402805i	0.914200	−	0.405262i	$-0.132820\pi$
$397$	29.9528	+	8.02584i	1.50329	+	0.402805i	0.589090	+	0.808068i	$0.299487\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	21.2106	−	5.68338i	1.05921	−	0.283814i	0.313157	−	0.949701i	$-0.398613\pi$
$401$	21.2106	−	5.68338i	1.05921	−	0.283814i	0.746052	+	0.665887i	$0.231947\pi$
$402$		0			0
$403$	−30.9206	+	14.9209i	−1.54026	+	0.743262i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−15.9379	−	27.6052i	−0.790012	−	1.36834i
$408$		0			0
$409$	−10.3011	+	38.4442i	−0.509357	+	1.90094i	−0.0825886	+	0.996584i	$0.526319\pi$
$409$	−10.3011	+	38.4442i	−0.509357	+	1.90094i	−0.426768	+	0.904361i	$0.640348\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	5.42839	−	9.40225i	0.267114	−	0.462654i
$414$		0			0
$415$		−	14.1729i		−	0.695721i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−23.9716	−	13.8400i	−1.17109	−	0.676129i	−0.217153	−	0.976137i	$-0.569677\pi$
$419$	−23.9716	−	13.8400i	−1.17109	−	0.676129i	−0.953937	+	0.300008i	$0.903011\pi$
$420$		0			0
$421$	−17.5972	−	17.5972i	−0.857636	−	0.857636i	0.133423	−	0.991059i	$-0.457403\pi$
$421$	−17.5972	−	17.5972i	−0.857636	−	0.857636i	−0.991059	+	0.133423i	$0.957403\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	2.67800	−	1.54614i	0.129902	−	0.0749989i
$426$		0			0
$427$	−3.39803	−	12.6816i	−0.164442	−	0.613706i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	4.94492	+	18.4547i	0.238189	+	0.888932i	0.976686	+	0.214675i	$0.0688693\pi$
$431$	4.94492	+	18.4547i	0.238189	+	0.888932i	−0.738497	+	0.674257i	$0.764464\pi$
$432$		0			0
$433$	−7.43829	+	4.29450i	−0.357462	+	0.206381i	−0.667967	−	0.744191i	$-0.732835\pi$
$433$	−7.43829	+	4.29450i	−0.357462	+	0.206381i	0.310505	+	0.950572i	$0.399502\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−0.522131	−	0.522131i	−0.0249769	−	0.0249769i
$438$		0			0
$439$	2.94438	+	1.69994i	0.140528	+	0.0811336i	0.568615	−	0.822603i	$-0.307479\pi$
$439$	2.94438	+	1.69994i	0.140528	+	0.0811336i	−0.428088	+	0.903737i	$0.640813\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	23.4089i		−	1.11219i	−0.831118	−	0.556096i	$-0.812299\pi$
$443$		−	23.4089i		−	1.11219i	0.831118	−	0.556096i	$-0.187701\pi$
$444$		0			0
$445$	4.53741	−	7.85902i	0.215094	−	0.372553i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	2.08592	−	7.78476i	0.0984407	−	0.367386i	−0.899078	−	0.437789i	$-0.855762\pi$
$449$	2.08592	−	7.78476i	0.0984407	−	0.367386i	0.997519	+	0.0704029i	$0.0224285\pi$
$450$		0			0
$451$	−14.1179	−	24.4529i	−0.664786	−	1.15144i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	0.790072	+	4.15752i	0.0370392	+	0.194907i
$456$		0			0
$457$	−37.3815	+	10.0164i	−1.74863	+	0.468545i	−0.984334	−	0.176315i	$-0.943582\pi$
$457$	−37.3815	+	10.0164i	−1.74863	+	0.468545i	−0.764301	+	0.644860i	$0.776916\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−2.24005	−	0.600218i	−0.104329	−	0.0279550i	0.206277	−	0.978494i	$-0.433865\pi$
$461$	−2.24005	−	0.600218i	−0.104329	−	0.0279550i	−0.310606	+	0.950539i	$0.600532\pi$
$462$		0			0
$463$	−15.1607	+	15.1607i	−0.704578	+	0.704578i	−0.965390	−	0.260812i	$-0.916010\pi$
$463$	−15.1607	+	15.1607i	−0.704578	+	0.704578i	0.260812	+	0.965390i	$0.416010\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	12.0039			0.555474			0.277737	−	0.960657i	$-0.410416\pi$
$467$	12.0039			0.555474			0.277737	+	0.960657i	$0.410416\pi$
$468$		0			0
$469$	2.04791			0.0945637
$470$		0			0
$471$		0			0
$472$		0			0
$473$	0.930791	−	0.930791i	0.0427978	−	0.0427978i
$474$		0			0
$475$	−0.737799	−	0.197693i	−0.0338526	−	0.00907076i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	36.3400	−	9.73727i	1.66042	−	0.444907i	0.697914	−	0.716181i	$-0.254112\pi$
$479$	36.3400	−	9.73727i	1.66042	−	0.444907i	0.962502	+	0.271274i	$0.0874449\pi$
$480$		0			0
$481$	19.4944	+	1.44530i	0.888868	+	0.0659001i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	0.858511	+	1.48698i	0.0389830	+	0.0675205i
$486$		0			0
$487$	−10.3835	+	38.7519i	−0.470523	+	1.75601i	0.167376	+	0.985893i	$0.446471\pi$
$487$	−10.3835	+	38.7519i	−0.470523	+	1.75601i	−0.637898	+	0.770121i	$0.720196\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−3.39910	+	5.88742i	−0.153399	+	0.265696i	−0.932475	−	0.361234i	$-0.882355\pi$
$491$	−3.39910	+	5.88742i	−0.153399	+	0.265696i	0.779076	+	0.626930i	$0.215689\pi$
$492$		0			0
$493$		−	5.39050i		−	0.242776i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	0.146166	+	0.0843888i	0.00655643	+	0.00378536i
$498$		0			0
$499$	−18.8104	−	18.8104i	−0.842071	−	0.842071i	0.147057	−	0.989128i	$-0.453020\pi$
$499$	−18.8104	−	18.8104i	−0.842071	−	0.842071i	−0.989128	+	0.147057i	$0.953020\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−4.30951	+	2.48810i	−0.192152	+	0.110939i	−0.592989	−	0.805210i	$-0.702052\pi$
$503$	−4.30951	+	2.48810i	−0.192152	+	0.110939i	0.400838	+	0.916149i	$0.368719\pi$
$504$		0			0
$505$	0.140603	+	0.524739i	0.00625677	+	0.0233506i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−8.63306	−	32.2190i	−0.382654	−	1.42808i	−0.841832	−	0.539739i	$-0.818523\pi$
$509$	−8.63306	−	32.2190i	−0.382654	−	1.42808i	0.459179	−	0.888344i	$-0.348144\pi$
$510$		0			0
$511$	2.53035	−	1.46090i	0.111936	−	0.0646263i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	7.94385	+	7.94385i	0.350048	+	0.350048i
$516$		0			0
$517$	42.5871	+	24.5877i	1.87298	+	1.08136i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			34.2554i			1.50076i	0.661008	+	0.750378i	$0.270129\pi$
$521$			34.2554i			1.50076i	−0.661008	+	0.750378i	$0.729871\pi$
$522$		0			0
$523$	16.7587	−	29.0269i	0.732807	−	1.26926i	−0.222872	−	0.974848i	$-0.571543\pi$
$523$	16.7587	−	29.0269i	0.732807	−	1.26926i	0.955679	−	0.294411i	$-0.0951234\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	7.62093	−	28.4417i	0.331973	−	1.23894i
$528$		0			0
$529$	11.0327	+	19.1092i	0.479684	+	0.830837i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	17.2683	+	1.28026i	0.747972	+	0.0554541i
$534$		0			0
$535$	5.18951	−	1.39052i	0.224362	−	0.0601176i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	31.9298	+	8.55555i	1.37531	+	0.368514i
$540$		0			0
$541$	−8.41949	+	8.41949i	−0.361982	+	0.361982i	−0.864542	−	0.502560i	$-0.832392\pi$
$541$	−8.41949	+	8.41949i	−0.361982	+	0.361982i	0.502560	+	0.864542i	$0.332392\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−7.52406			−0.322295
$546$		0			0
$547$	24.7118			1.05660			0.528301	−	0.849057i	$-0.322829\pi$
$547$	24.7118			1.05660			0.528301	+	0.849057i	$0.322829\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−0.941519	+	0.941519i	−0.0401101	+	0.0401101i
$552$		0			0
$553$	3.92008	+	1.05038i	0.166699	+	0.0446668i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	25.3132	−	6.78265i	1.07255	−	0.287390i	0.321012	−	0.947075i	$-0.395977\pi$
$557$	25.3132	−	6.78265i	1.07255	−	0.287390i	0.751542	+	0.659685i	$0.229310\pi$
$558$		0			0
$559$	0.150708	+	0.793055i	0.00637426	+	0.0335426i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−19.2227	−	33.2947i	−0.810141	−	1.40320i	−0.912765	−	0.408485i	$-0.866057\pi$
$563$	−19.2227	−	33.2947i	−0.810141	−	1.40320i	0.102624	−	0.994720i	$-0.467276\pi$
$564$		0			0
$565$	−2.81134	+	10.4921i	−0.118274	+	0.441405i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	5.91258	−	10.2409i	0.247868	−	0.429321i	−0.715066	−	0.699057i	$-0.753603\pi$
$569$	5.91258	−	10.2409i	0.247868	−	0.429321i	0.962934	+	0.269737i	$0.0869366\pi$
$570$		0			0
$571$			2.11878i			0.0886680i	0.999017	+	0.0443340i	$0.0141166\pi$
$571$			2.11878i			0.0886680i	−0.999017	+	0.0443340i	$0.985883\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.837203	−	0.483360i	−0.0349138	−	0.0201575i
$576$		0			0
$577$	−3.65033	−	3.65033i	−0.151965	−	0.151965i	0.627030	−	0.778995i	$-0.284270\pi$
$577$	−3.65033	−	3.65033i	−0.151965	−	0.151965i	−0.778995	+	0.627030i	$0.784270\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−14.4064	+	8.31754i	−0.597678	+	0.345070i
$582$		0			0
$583$	10.3350	+	38.5708i	0.428033	+	1.59744i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	2.42609	+	9.05429i	0.100135	+	0.373710i	0.997748	−	0.0670750i	$-0.0213667\pi$
$587$	2.42609	+	9.05429i	0.100135	+	0.373710i	−0.897613	+	0.440785i	$0.854700\pi$
$588$		0			0
$589$	−6.29880	+	3.63661i	−0.259537	+	0.149844i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	28.5139	+	28.5139i	1.17093	+	1.17093i	0.981989	+	0.188937i	$0.0605041\pi$
$593$	28.5139	+	28.5139i	1.17093	+	1.17093i	0.188937	+	0.981989i	$0.439496\pi$
$594$		0			0
$595$	−3.14323	−	1.81474i	−0.128860	−	0.0743972i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			16.8078i			0.686749i	0.939199	+	0.343374i	$0.111570\pi$
$599$			16.8078i			0.686749i	−0.939199	+	0.343374i	$0.888430\pi$
$600$		0			0
$601$	4.48665	−	7.77111i	0.183014	−	0.316990i	−0.759891	−	0.650050i	$-0.774748\pi$
$601$	4.48665	−	7.77111i	0.183014	−	0.316990i	0.942906	+	0.333060i	$0.108081\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−6.09965	+	22.7642i	−0.247986	+	0.925496i
$606$		0			0
$607$	0.117780	+	0.204001i	0.00478054	+	0.00828014i	0.868406	−	0.495854i	$-0.165145\pi$
$607$	0.117780	+	0.204001i	0.00478054	+	0.00828014i	−0.863625	+	0.504134i	$0.831812\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−27.1600	+	13.1062i	−1.09878	+	0.530221i
$612$		0			0
$613$	27.9862	−	7.49887i	1.13035	−	0.302877i	0.355285	−	0.934758i	$-0.384384\pi$
$613$	27.9862	−	7.49887i	1.13035	−	0.302877i	0.775065	+	0.631881i	$0.217717\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−13.7886	−	3.69465i	−0.555109	−	0.148741i	−0.0296504	−	0.999560i	$-0.509439\pi$
$617$	−13.7886	−	3.69465i	−0.555109	−	0.148741i	−0.525458	+	0.850819i	$0.676106\pi$
$618$		0			0
$619$	4.14846	−	4.14846i	0.166740	−	0.166740i	−0.618805	−	0.785545i	$-0.712383\pi$
$619$	4.14846	−	4.14846i	0.166740	−	0.166740i	0.785545	+	0.618805i	$0.212383\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−10.6513			−0.426736
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−11.8548	+	11.8548i	−0.472680	+	0.472680i
$630$		0			0
$631$	26.3618	+	7.06362i	1.04945	+	0.281198i	0.742023	−	0.670375i	$-0.233867\pi$
$631$	26.3618	+	7.06362i	1.04945	+	0.281198i	0.307423	+	0.951573i	$0.400533\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	5.75628	−	1.54239i	0.228431	−	0.0612079i
$636$		0			0
$637$	−15.3549	+	13.2352i	−0.608383	+	0.524399i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−7.82715	−	13.5570i	−0.309154	−	0.535470i	0.669024	−	0.743241i	$-0.266712\pi$
$641$	−7.82715	−	13.5570i	−0.309154	−	0.535470i	−0.978177	+	0.207771i	$0.933379\pi$
$642$		0			0
$643$	−11.8967	+	44.3990i	−0.469159	+	1.75093i	0.173560	+	0.984823i	$0.444473\pi$
$643$	−11.8967	+	44.3990i	−0.469159	+	1.75093i	−0.642719	+	0.766102i	$0.722194\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−3.40387	+	5.89567i	−0.133820	+	0.231783i	−0.925146	−	0.379611i	$-0.876058\pi$
$647$	−3.40387	+	5.89567i	−0.133820	+	0.231783i	0.791326	+	0.611394i	$0.209391\pi$
$648$		0			0
$649$			54.3835i			2.13474i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	14.1249	+	8.15499i	0.552748	+	0.319129i	0.750230	−	0.661177i	$-0.229943\pi$
$653$	14.1249	+	8.15499i	0.552748	+	0.319129i	−0.197481	+	0.980307i	$0.563276\pi$
$654$		0			0
$655$	−6.28153	−	6.28153i	−0.245440	−	0.245440i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−12.0483	+	6.95612i	−0.469337	+	0.270972i	−0.715962	−	0.698139i	$-0.754012\pi$
$659$	−12.0483	+	6.95612i	−0.469337	+	0.270972i	0.246625	+	0.969111i	$0.420678\pi$
$660$		0			0
$661$	−9.29375	−	34.6847i	−0.361485	−	1.34908i	−0.872124	−	0.489285i	$-0.837258\pi$
$661$	−9.29375	−	34.6847i	−0.361485	−	1.34908i	0.510639	−	0.859795i	$-0.329409\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	0.232037	+	0.865972i	0.00899799	+	0.0335810i
$666$		0			0
$667$	−1.45942	+	0.842597i	−0.0565090	+	0.0326255i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	46.5031	+	46.5031i	1.79523	+	1.79523i
$672$		0			0
$673$	−21.4063	−	12.3589i	−0.825151	−	0.476401i	0.0270385	−	0.999634i	$-0.491392\pi$
$673$	−21.4063	−	12.3589i	−0.825151	−	0.476401i	−0.852190	+	0.523233i	$0.824726\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	31.3290i		−	1.20407i	−0.798469	−	0.602036i	$-0.794356\pi$
$677$		−	31.3290i		−	1.20407i	0.798469	−	0.602036i	$-0.205644\pi$
$678$		0			0
$679$	1.00765	−	1.74531i	0.0386702	−	0.0669788i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−11.3532	+	42.3707i	−0.434418	+	1.62127i	0.308037	+	0.951374i	$0.400328\pi$
$683$	−11.3532	+	42.3707i	−0.434418	+	1.62127i	−0.742455	+	0.669896i	$0.766339\pi$
$684$		0			0
$685$	4.73815	+	8.20672i	0.181036	+	0.313563i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−23.1203	−	8.06949i	−0.880813	−	0.307423i
$690$		0			0
$691$	−3.53837	+	0.948103i	−0.134606	+	0.0360676i	−0.325493	−	0.945544i	$-0.605530\pi$
$691$	−3.53837	+	0.948103i	−0.134606	+	0.0360676i	0.190887	+	0.981612i	$0.438864\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	10.9854	+	2.94353i	0.416700	+	0.111654i
$696$		0			0
$697$	−10.5010	+	10.5010i	−0.397755	+	0.397755i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	45.3964			1.71460			0.857298	−	0.514820i	$-0.172141\pi$
$701$	45.3964			1.71460			0.857298	+	0.514820i	$0.172141\pi$
$702$		0			0
$703$	4.14117			0.156187
$704$		0			0
$705$		0			0
$706$		0			0
$707$	0.450869	−	0.450869i	0.0169567	−	0.0169567i
$708$		0			0
$709$	25.7328	+	6.89508i	0.966415	+	0.258950i	0.707313	−	0.706900i	$-0.249907\pi$
$709$	25.7328	+	6.89508i	0.966415	+	0.258950i	0.259101	+	0.965850i	$0.416574\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−8.89154	+	2.38248i	−0.332991	+	0.0892246i
$714$		0			0
$715$	−13.8403	−	16.0568i	−0.517596	−	0.600491i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−6.72876	−	11.6546i	−0.250940	−	0.434642i	0.712845	−	0.701322i	$-0.247406\pi$
$719$	−6.72876	−	11.6546i	−0.250940	−	0.434642i	−0.963785	+	0.266681i	$0.914073\pi$
$720$		0			0
$721$	3.41278	−	12.7367i	0.127099	−	0.474338i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−0.871605	+	1.50966i	−0.0323706	+	0.0560675i
$726$		0			0
$727$			44.7131i			1.65832i	0.559014	+	0.829158i	$0.311180\pi$
$727$			44.7131i			1.65832i	−0.559014	+	0.829158i	$0.688820\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−0.599577	−	0.346166i	−0.0221761	−	0.0128034i
$732$		0			0
$733$	−13.0888	−	13.0888i	−0.483444	−	0.483444i	0.422785	−	0.906230i	$-0.361052\pi$
$733$	−13.0888	−	13.0888i	−0.483444	−	0.483444i	−0.906230	+	0.422785i	$0.861052\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−8.88398	+	5.12917i	−0.327246	+	0.188935i
$738$		0			0
$739$	−2.75261	−	10.2729i	−0.101256	−	0.377894i	0.896637	−	0.442766i	$-0.146003\pi$
$739$	−2.75261	−	10.2729i	−0.101256	−	0.377894i	−0.997894	+	0.0648721i	$0.979336\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−2.54705	−	9.50571i	−0.0934421	−	0.348731i	0.903337	−	0.428932i	$-0.141110\pi$
$743$	−2.54705	−	9.50571i	−0.0934421	−	0.348731i	−0.996779	+	0.0802018i	$0.974444\pi$
$744$		0			0
$745$	13.5391	−	7.81679i	0.496034	−	0.286385i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−4.45896	−	4.45896i	−0.162927	−	0.162927i
$750$		0			0
$751$	43.2940	+	24.9958i	1.57982	+	0.912109i	0.994883	+	0.101029i	$0.0322134\pi$
$751$	43.2940	+	24.9958i	1.57982	+	0.912109i	0.584935	+	0.811080i	$0.301120\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			7.16801i			0.260871i
$756$		0			0
$757$	−5.05039	+	8.74753i	−0.183560	+	0.317934i	−0.943090	−	0.332537i	$-0.892095\pi$
$757$	−5.05039	+	8.74753i	−0.183560	+	0.317934i	0.759531	+	0.650472i	$0.225429\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	10.7707	−	40.1967i	0.390437	−	1.45713i	−0.438978	−	0.898498i	$-0.644659\pi$
$761$	10.7707	−	40.1967i	0.390437	−	1.45713i	0.829415	−	0.558633i	$-0.188674\pi$
$762$		0			0
$763$	4.41558	+	7.64801i	0.159855	+	0.276877i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−27.5707	−	18.7653i	−0.995521	−	0.677575i
$768$		0			0
$769$	−17.3911	+	4.65992i	−0.627137	+	0.168041i	−0.558370	−	0.829592i	$-0.688573\pi$
$769$	−17.3911	+	4.65992i	−0.627137	+	0.168041i	−0.0687670	+	0.997633i	$0.521907\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	18.4344	+	4.93949i	0.663040	+	0.177661i	0.574617	−	0.818422i	$-0.305151\pi$
$773$	18.4344	+	4.93949i	0.663040	+	0.177661i	0.0884223	+	0.996083i	$0.471818\pi$
$774$		0			0
$775$	−6.73314	+	6.73314i	−0.241862	+	0.241862i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	3.66828			0.131430
$780$		0			0
$781$	−0.845437			−0.0302521
$782$		0			0
$783$		0			0
$784$		0			0
$785$	7.40338	−	7.40338i	0.264238	−	0.264238i
$786$		0			0
$787$	−38.2403	−	10.2465i	−1.36312	−	0.365247i	−0.498159	−	0.867086i	$-0.665990\pi$
$787$	−38.2403	−	10.2465i	−1.36312	−	0.365247i	−0.864961	+	0.501839i	$0.832657\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	12.3148	−	3.29974i	0.437864	−	0.117325i
$792$		0			0
$793$	−39.6217	+	7.52950i	−1.40701	+	0.267380i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−24.6196	−	42.6424i	−0.872071	−	1.51047i	−0.859851	−	0.510545i	$-0.829444\pi$
$797$	−24.6196	−	42.6424i	−0.872071	−	1.51047i	−0.0122200	−	0.999925i	$-0.503890\pi$
$798$		0			0
$799$	6.69408	−	24.9826i	0.236819	−	0.883822i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−7.31789	+	12.6750i	−0.258243	+	0.447290i
$804$		0			0
$805$			1.13466i			0.0399916i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−4.92177	−	2.84158i	−0.173040	−	0.0999048i	0.410978	−	0.911645i	$-0.365187\pi$
$809$	−4.92177	−	2.84158i	−0.173040	−	0.0999048i	−0.584019	+	0.811740i	$0.698520\pi$
$810$		0			0
$811$	−3.73383	−	3.73383i	−0.131113	−	0.131113i	0.638505	−	0.769618i	$-0.279553\pi$
$811$	−3.73383	−	3.73383i	−0.131113	−	0.131113i	−0.769618	+	0.638505i	$0.779553\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	18.9410	−	10.9356i	0.663475	−	0.383058i
$816$		0			0
$817$	0.0442615	+	0.165186i	0.00154851	+	0.00577913i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	2.07964	+	7.76133i	0.0725800	+	0.270872i	0.992674	−	0.120826i	$-0.0385543\pi$
$821$	2.07964	+	7.76133i	0.0725800	+	0.270872i	−0.920094	+	0.391698i	$0.871888\pi$
$822$		0			0
$823$	30.5320	−	17.6277i	1.06428	−	0.614462i	0.137667	−	0.990479i	$-0.456040\pi$
$823$	30.5320	−	17.6277i	1.06428	−	0.614462i	0.926613	+	0.376016i	$0.122706\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	12.3633	+	12.3633i	0.429914	+	0.429914i	0.888599	−	0.458685i	$-0.151679\pi$
$827$	12.3633	+	12.3633i	0.429914	+	0.429914i	−0.458685	+	0.888599i	$0.651679\pi$
$828$		0			0
$829$	33.7255	+	19.4714i	1.17134	+	0.676271i	0.953994	−	0.299825i	$-0.0969283\pi$
$829$	33.7255	+	19.4714i	1.17134	+	0.676271i	0.217341	+	0.976096i	$0.430262\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	17.3860i		−	0.602388i
$834$		0			0
$835$	7.76576	−	13.4507i	0.268745	−	0.465480i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−3.92058	+	14.6318i	−0.135354	+	0.505146i	0.864643	+	0.502387i	$0.167545\pi$
$839$	−3.92058	+	14.6318i	−0.135354	+	0.505146i	−0.999996	+	0.00275889i	$0.999122\pi$
$840$		0			0
$841$	−12.9806	−	22.4831i	−0.447607	−	0.775278i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	12.9159	−	1.47610i	0.444321	−	0.0507794i
$846$		0			0
$847$	26.7189	−	7.15930i	0.918071	−	0.245996i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	5.06259	+	1.35652i	0.173543	+	0.0465008i
$852$		0			0
$853$	−19.8670	+	19.8670i	−0.680234	+	0.680234i	−0.960053	−	0.279819i	$-0.909726\pi$
$853$	−19.8670	+	19.8670i	−0.680234	+	0.680234i	0.279819	+	0.960053i	$0.409726\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−11.1391			−0.380503			−0.190251	−	0.981735i	$-0.560930\pi$
$857$	−11.1391			−0.380503			−0.190251	+	0.981735i	$0.560930\pi$
$858$		0			0
$859$	−46.6796			−1.59269			−0.796343	−	0.604845i	$-0.793235\pi$
$859$	−46.6796			−1.59269			−0.796343	+	0.604845i	$0.793235\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	12.7535	−	12.7535i	0.434134	−	0.434134i	−0.455898	−	0.890032i	$-0.650682\pi$
$863$	12.7535	−	12.7535i	0.434134	−	0.434134i	0.890032	+	0.455898i	$0.150682\pi$
$864$		0			0
$865$	15.6342	+	4.18918i	0.531579	+	0.142436i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−19.6364	+	5.26155i	−0.666118	+	0.178486i
$870$		0			0
$871$	0.465131	−	6.27374i	0.0157604	−	0.212577i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	0.586862	+	1.01647i	0.0198395	+	0.0343631i
$876$		0			0
$877$	9.37679	−	34.9947i	0.316632	−	1.18169i	−0.605829	−	0.795595i	$-0.707158\pi$
$877$	9.37679	−	34.9947i	0.316632	−	1.18169i	0.922461	−	0.386091i	$-0.126175\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	20.9715	−	36.3237i	0.706547	−	1.22378i	−0.259583	−	0.965721i	$-0.583585\pi$
$881$	20.9715	−	36.3237i	0.706547	−	1.22378i	0.966130	−	0.258055i	$-0.0830816\pi$
$882$		0			0
$883$		−	8.28843i		−	0.278928i	−0.990227	−	0.139464i	$-0.955462\pi$
$883$		−	8.28843i		−	0.278928i	0.990227	−	0.139464i	$-0.0445379\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−9.21496	−	5.32026i	−0.309408	−	0.178637i	0.337253	−	0.941414i	$-0.390502\pi$
$887$	−9.21496	−	5.32026i	−0.309408	−	0.178637i	−0.646662	+	0.762777i	$0.723835\pi$
$888$		0			0
$889$	−4.94594	−	4.94594i	−0.165882	−	0.165882i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−5.53274	+	3.19433i	−0.185146	+	0.106894i
$894$		0			0
$895$	−1.19638	−	4.46494i	−0.0399905	−	0.149246i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	4.29614	+	16.0334i	0.143284	+	0.534745i
$900$		0			0
$901$	18.1883	−	10.5010i	0.605941	−	0.349840i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−11.7335	−	11.7335i	−0.390035	−	0.390035i
$906$		0			0
$907$	−35.0965	−	20.2630i	−1.16536	−	0.672821i	−0.212777	−	0.977101i	$-0.568251\pi$
$907$	−35.0965	−	20.2630i	−1.16536	−	0.672821i	−0.952583	+	0.304280i	$0.901584\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			7.99976i			0.265044i	0.991180	+	0.132522i	$0.0423075\pi$
$911$			7.99976i			0.265044i	−0.991180	+	0.132522i	$0.957693\pi$
$912$		0			0
$913$	41.6640	−	72.1642i	1.37888	−	2.38829i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−2.69862	+	10.0714i	−0.0891164	+	0.332587i
$918$		0			0
$919$	−10.4768	−	18.1464i	−0.345598	−	0.598593i	0.639864	−	0.768488i	$-0.278990\pi$
$919$	−10.4768	−	18.1464i	−0.345598	−	0.598593i	−0.985462	+	0.169895i	$0.945657\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	0.291722	−	0.428610i	0.00960214	−	0.0141079i
$924$		0			0
$925$	5.23688	−	1.40322i	0.172187	−	0.0461375i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−25.4791	−	6.82710i	−0.835942	−	0.223990i	−0.184638	−	0.982807i	$-0.559111\pi$
$929$	−25.4791	−	6.82710i	−0.835942	−	0.223990i	−0.651304	+	0.758817i	$0.725778\pi$
$930$		0			0
$931$	−3.03668	+	3.03668i	−0.0995232	+	0.0995232i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	18.1807			0.594574
$936$		0			0
$937$	−55.5630			−1.81516			−0.907582	−	0.419875i	$-0.862074\pi$
$937$	−55.5630			−1.81516			−0.907582	+	0.419875i	$0.862074\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−25.7216	+	25.7216i	−0.838499	+	0.838499i	−0.988661	−	0.150162i	$-0.952020\pi$
$941$	−25.7216	+	25.7216i	−0.838499	+	0.838499i	0.150162	+	0.988661i	$0.452020\pi$
$942$		0			0
$943$	4.48448	+	1.20161i	0.146035	+	0.0391299i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−38.9094	+	10.4257i	−1.26439	+	0.338791i	−0.827877	−	0.560909i	$-0.810452\pi$
$947$	−38.9094	+	10.4257i	−1.26439	+	0.338791i	−0.436508	+	0.899700i	$0.643785\pi$
$948$		0			0
$949$	−3.90073	−	8.08349i	−0.126623	−	0.262401i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	7.08429	+	12.2703i	0.229483	+	0.397475i	0.957655	−	0.287919i	$-0.0929633\pi$
$953$	7.08429	+	12.2703i	0.229483	+	0.397475i	−0.728172	+	0.685394i	$0.759630\pi$
$954$		0			0
$955$	−1.96560	+	7.33573i	−0.0636054	+	0.237379i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	5.56128	−	9.63242i	0.179583	−	0.311047i
$960$		0			0
$961$			59.6704i			1.92485i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−10.7997	−	6.23521i	−0.347655	−	0.200719i
$966$		0			0
$967$	−10.7941	−	10.7941i	−0.347115	−	0.347115i	0.511919	−	0.859034i	$-0.328935\pi$
$967$	−10.7941	−	10.7941i	−0.347115	−	0.347115i	−0.859034	+	0.511919i	$0.828935\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−24.9093	+	14.3814i	−0.799379	+	0.461522i	−0.843254	−	0.537515i	$-0.819363\pi$
$971$	−24.9093	+	14.3814i	−0.799379	+	0.461522i	0.0438751	+	0.999037i	$0.486030\pi$
$972$		0			0
$973$	−3.45489	−	12.8938i	−0.110759	−	0.413357i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−2.23226	−	8.33091i	−0.0714164	−	0.266530i	0.920980	−	0.389609i	$-0.127390\pi$
$977$	−2.23226	−	8.33091i	−0.0714164	−	0.266530i	−0.992397	+	0.123079i	$0.960723\pi$
$978$		0			0
$979$	46.2063	−	26.6772i	1.47676	−	0.852607i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−43.9278	−	43.9278i	−1.40108	−	1.40108i	−0.796687	−	0.604392i	$-0.793416\pi$
$983$	−43.9278	−	43.9278i	−1.40108	−	1.40108i	−0.604392	−	0.796687i	$-0.706584\pi$
$984$		0			0
$985$	12.0897	+	6.98001i	0.385211	+	0.222402i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			0.216439i			0.00688236i
$990$		0			0
$991$	8.85788	−	15.3423i	0.281380	−	0.487364i	−0.690345	−	0.723480i	$-0.742541\pi$
$991$	8.85788	−	15.3423i	0.281380	−	0.487364i	0.971725	+	0.236116i	$0.0758746\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−2.58010	+	9.62908i	−0.0817948	+	0.305262i
$996$		0			0
$997$	−19.0469	−	32.9901i	−0.603220	−	1.04481i	−0.992330	−	0.123617i	$-0.960551\pi$
$997$	−19.0469	−	32.9901i	−0.603220	−	1.04481i	0.389110	−	0.921191i	$-0.372783\pi$
$998$		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.2321.3	yes	40
3.2	odd	2	inner	2340.2.fo.a.2321.6	yes	40
13.2	odd	12	inner	2340.2.fo.a.1601.6	yes	40
39.2	even	12	inner	2340.2.fo.a.1601.3	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2340.2.fo.a.1601.3	✓	40	39.2	even	12	inner
2340.2.fo.a.1601.6	yes	40	13.2	odd	12	inner
2340.2.fo.a.2321.3	yes	40	1.1	even	1	trivial
2340.2.fo.a.2321.6	yes	40	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-0.707107 + 0.707107i) q^{5} +(1.13373 + 0.303782i) q^{7} +(-5.67905 + 1.52170i) q^{11} +(1.18813 - 3.40417i) q^{13} +(1.54614 + 2.67800i) q^{17} +(0.197693 - 0.737799i) q^{19} +(0.483360 - 0.837203i) q^{23} -1.00000i q^{25} +(-1.50966 - 0.871605i) q^{29} +(-6.73314 - 6.73314i) q^{31} +(-1.01647 + 0.586862i) q^{35} +(1.40322 + 5.23688i) q^{37} +(1.24298 + 4.63886i) q^{41} +(-0.193895 + 0.111945i) q^{43} +(-5.91426 - 5.91426i) q^{47} +(-4.86912 - 2.81119i) q^{49} -6.79176i q^{53} +(2.93969 - 5.09170i) q^{55} +(2.39404 - 8.93468i) q^{59} +(-5.59287 - 9.68714i) q^{61} +(1.56697 + 3.24724i) q^{65} +(1.68534 - 0.451587i) q^{67} +(0.138897 + 0.0372173i) q^{71} +(1.76023 - 1.76023i) q^{73} -6.90078 q^{77} +3.45768 q^{79} +(-10.0218 + 10.0218i) q^{83} +(-2.98692 - 0.800342i) q^{85} +(-8.76560 + 2.34874i) q^{89} +(2.38114 - 3.49847i) q^{91} +(0.381913 + 0.661493i) q^{95} +(0.444398 - 1.65852i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 12 q^{13} - 20 q^{19} - 28 q^{31} + 12 q^{49} + 16 q^{55} + 48 q^{61} + 16 q^{67} - 20 q^{73} + 80 q^{79} + 20 q^{85} + 4 q^{91} - 40 q^{97}+O(q^{100}) \) 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

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