LMFDB - Embedded newform 1620.3.o.g.701.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,3,Mod(701,1620)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 5, 0]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1620.701");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1620.o (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$44.1418028264$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			701.4
Character	$\chi$	$=$	1620.701
Dual form			1620.3.o.g.1241.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.93649 - 1.11803i) q^{5} +(3.57283 + 6.18832i) q^{7} +O(q^{10})$	$q+(-1.93649 - 1.11803i) q^{5} +(3.57283 + 6.18832i) q^{7} +(2.20431 - 1.27266i) q^{11} +(-5.57892 + 9.66297i) q^{13} -2.05814i q^{17} -19.3781 q^{19} +(10.4254 + 6.01908i) q^{23} +(2.50000 + 4.33013i) q^{25} +(19.3077 - 11.1473i) q^{29} +(-7.08342 + 12.2689i) q^{31} -15.9782i q^{35} -46.3319 q^{37} +(-15.0752 - 8.70366i) q^{41} +(-13.5098 - 23.3997i) q^{43} +(-42.2514 + 24.3938i) q^{47} +(-1.03020 + 1.78437i) q^{49} +9.84048i q^{53} -5.69151 q^{55} +(76.1103 + 43.9423i) q^{59} +(-29.9240 - 51.8300i) q^{61} +(21.6071 - 12.4748i) q^{65} +(3.85720 - 6.68087i) q^{67} -79.5282i q^{71} -6.01802 q^{73} +(15.7513 + 9.09400i) q^{77} +(8.46386 + 14.6598i) q^{79} +(-127.148 + 73.4091i) q^{83} +(-2.30107 + 3.98557i) q^{85} -100.304i q^{89} -79.7301 q^{91} +(37.5255 + 21.6653i) q^{95} +(-36.3075 - 62.8865i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 8 q^{7}+O(q^{10}) $ 32 * q - 8 * q^7	$ 32 q - 8 q^{7} + 40 q^{13} + 112 q^{19} + 80 q^{25} + 64 q^{31} - 176 q^{37} - 128 q^{43} - 216 q^{49} - 8 q^{61} + 40 q^{67} + 112 q^{73} + 136 q^{79} - 784 q^{91} - 128 q^{97}+O(q^{100}) $ 32 * q - 8 * q^7 + 40 * q^13 + 112 * q^19 + 80 * q^25 + 64 * q^31 - 176 * q^37 - 128 * q^43 - 216 * q^49 - 8 * q^61 + 40 * q^67 + 112 * q^73 + 136 * q^79 - 784 * q^91 - 128 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$.

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$1$	$e\left(\frac{5}{6}\right)$

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
$5$	−1.93649	−	1.11803i	−0.387298	−	0.223607i
$5$	−1.93649	−	1.11803i	−0.387298	−	0.223607i
$6$		0			0
$7$	3.57283	+	6.18832i	0.510404	+	0.884046i	0.999927	+	0.0120555i	$0.00383748\pi$
$7$	3.57283	+	6.18832i	0.510404	+	0.884046i	−0.489523	+	0.871990i	$0.662829\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.20431	−	1.27266i	0.200392	−	0.115696i	−0.396446	−	0.918058i	$-0.629757\pi$
$11$	2.20431	−	1.27266i	0.200392	−	0.115696i	0.596838	+	0.802361i	$0.296423\pi$
$12$		0			0
$13$	−5.57892	+	9.66297i	−0.429147	+	0.743305i	−0.996798	−	0.0799646i	$-0.974519\pi$
$13$	−5.57892	+	9.66297i	−0.429147	+	0.743305i	0.567650	+	0.823270i	$0.307853\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	2.05814i		−	0.121067i	−0.998166	−	0.0605335i	$-0.980720\pi$
$17$		−	2.05814i		−	0.121067i	0.998166	−	0.0605335i	$-0.0192802\pi$
$18$		0			0
$19$	−19.3781			−1.01990			−0.509949	−	0.860204i	$-0.670336\pi$
$19$	−19.3781			−1.01990			−0.509949	+	0.860204i	$0.670336\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	10.4254	+	6.01908i	0.453276	+	0.261699i	0.709213	−	0.704994i	$-0.249051\pi$
$23$	10.4254	+	6.01908i	0.453276	+	0.261699i	−0.255937	+	0.966694i	$0.582384\pi$
$24$		0			0
$25$	2.50000	+	4.33013i	0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	19.3077	−	11.1473i	0.665782	−	0.384390i	−0.128694	−	0.991684i	$-0.541079\pi$
$29$	19.3077	−	11.1473i	0.665782	−	0.384390i	0.794477	+	0.607295i	$0.207745\pi$
$30$		0			0
$31$	−7.08342	+	12.2689i	−0.228498	+	0.395769i	−0.957363	−	0.288888i	$-0.906715\pi$
$31$	−7.08342	+	12.2689i	−0.228498	+	0.395769i	0.728865	+	0.684657i	$0.240048\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	15.9782i		−	0.456519i
$36$		0			0
$37$	−46.3319			−1.25221			−0.626107	−	0.779737i	$-0.715353\pi$
$37$	−46.3319			−1.25221			−0.626107	+	0.779737i	$0.715353\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−15.0752	−	8.70366i	−0.367687	−	0.212284i	0.304760	−	0.952429i	$-0.401424\pi$
$41$	−15.0752	−	8.70366i	−0.367687	−	0.212284i	−0.672448	+	0.740145i	$0.734757\pi$
$42$		0			0
$43$	−13.5098	−	23.3997i	−0.314183	−	0.544180i	0.665081	−	0.746771i	$-0.268397\pi$
$43$	−13.5098	−	23.3997i	−0.314183	−	0.544180i	−0.979263	+	0.202591i	$0.935064\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−42.2514	+	24.3938i	−0.898965	+	0.519018i	−0.876864	−	0.480738i	$-0.840369\pi$
$47$	−42.2514	+	24.3938i	−0.898965	+	0.519018i	−0.0221009	+	0.999756i	$0.507036\pi$
$48$		0			0
$49$	−1.03020	+	1.78437i	−0.0210246	+	0.0364157i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			9.84048i			0.185670i	0.995682	+	0.0928348i	$0.0295928\pi$
$53$			9.84048i			0.185670i	−0.995682	+	0.0928348i	$0.970407\pi$
$54$		0			0
$55$	−5.69151			−0.103482
$56$		0			0
$57$		0			0
$58$		0			0
$59$	76.1103	+	43.9423i	1.29001	+	0.744785i	0.978655	−	0.205510i	$-0.0658854\pi$
$59$	76.1103	+	43.9423i	1.29001	+	0.744785i	0.311350	+	0.950295i	$0.399219\pi$
$60$		0			0
$61$	−29.9240	−	51.8300i	−0.490558	−	0.849672i	0.509383	−	0.860540i	$-0.329874\pi$
$61$	−29.9240	−	51.8300i	−0.490558	−	0.849672i	−0.999941	+	0.0108684i	$0.996540\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	21.6071	−	12.4748i	0.332416	−	0.191921i
$66$		0			0
$67$	3.85720	−	6.68087i	0.0575701	−	0.0997144i	−0.835804	−	0.549028i	$-0.814998\pi$
$67$	3.85720	−	6.68087i	0.0575701	−	0.0997144i	0.893374	+	0.449314i	$0.148331\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	79.5282i		−	1.12012i	−0.828453	−	0.560058i	$-0.810779\pi$
$71$		−	79.5282i		−	1.12012i	0.828453	−	0.560058i	$-0.189221\pi$
$72$		0			0
$73$	−6.01802			−0.0824387			−0.0412193	−	0.999150i	$-0.513124\pi$
$73$	−6.01802			−0.0824387			−0.0412193	+	0.999150i	$0.513124\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	15.7513	+	9.09400i	0.204562	+	0.118104i
$78$		0			0
$79$	8.46386	+	14.6598i	0.107137	+	0.185568i	0.914609	−	0.404338i	$-0.132498\pi$
$79$	8.46386	+	14.6598i	0.107137	+	0.185568i	−0.807472	+	0.589906i	$0.799165\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−127.148	+	73.4091i	−1.53191	+	0.884447i	−0.532633	+	0.846347i	$0.678797\pi$
$83$	−127.148	+	73.4091i	−1.53191	+	0.884447i	−0.999274	+	0.0381001i	$0.987869\pi$
$84$		0			0
$85$	−2.30107	+	3.98557i	−0.0270714	+	0.0468890i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	100.304i		−	1.12701i	−0.826113	−	0.563504i	$-0.809453\pi$
$89$		−	100.304i		−	1.12701i	0.826113	−	0.563504i	$-0.190547\pi$
$90$		0			0
$91$	−79.7301			−0.876154
$92$		0			0
$93$		0			0
$94$		0			0
$95$	37.5255	+	21.6653i	0.395005	+	0.228056i
$96$		0			0
$97$	−36.3075	−	62.8865i	−0.374304	−	0.648314i	0.615918	−	0.787810i	$-0.288785\pi$
$97$	−36.3075	−	62.8865i	−0.374304	−	0.648314i	−0.990223	+	0.139496i	$0.955452\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−128.595	+	74.2442i	−1.27321	+	0.735091i	−0.975592	−	0.219593i	$-0.929527\pi$
$101$	−128.595	+	74.2442i	−1.27321	+	0.735091i	−0.297623	+	0.954683i	$0.596194\pi$
$102$		0			0
$103$	−70.9917	+	122.961i	−0.689240	+	1.19380i	0.282844	+	0.959166i	$0.408722\pi$
$103$	−70.9917	+	122.961i	−0.689240	+	1.19380i	−0.972084	+	0.234633i	$0.924611\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			3.62138i			0.0338447i	0.999857	+	0.0169224i	$0.00538681\pi$
$107$			3.62138i			0.0338447i	−0.999857	+	0.0169224i	$0.994613\pi$
$108$		0			0
$109$	−193.000			−1.77064			−0.885320	−	0.464983i	$-0.846060\pi$
$109$	−193.000			−1.77064			−0.885320	+	0.464983i	$0.846060\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	75.2851	+	43.4659i	0.666240	+	0.384654i	0.794650	−	0.607067i	$-0.207654\pi$
$113$	75.2851	+	43.4659i	0.666240	+	0.384654i	−0.128411	+	0.991721i	$0.540988\pi$
$114$		0			0
$115$	−13.4591	−	23.3118i	−0.117035	−	0.202711i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	12.7364	−	7.35337i	0.107029	−	0.0617931i
$120$		0			0
$121$	−57.2607	+	99.1784i	−0.473229	+	0.819656i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	11.1803i		−	0.0894427i
$126$		0			0
$127$	−83.5634			−0.657979			−0.328990	−	0.944334i	$-0.606708\pi$
$127$	−83.5634			−0.657979			−0.328990	+	0.944334i	$0.606708\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−19.2345	−	11.1050i	−0.146828	−	0.0847712i	0.424786	−	0.905294i	$-0.360349\pi$
$131$	−19.2345	−	11.1050i	−0.146828	−	0.0847712i	−0.571614	+	0.820522i	$0.693683\pi$
$132$		0			0
$133$	−69.2345	−	119.918i	−0.520560	−	0.901637i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−9.28159	+	5.35873i	−0.0677489	+	0.0391148i	−0.533492	−	0.845805i	$-0.679120\pi$
$137$	−9.28159	+	5.35873i	−0.0677489	+	0.0391148i	0.465743	+	0.884920i	$0.345787\pi$
$138$		0			0
$139$	21.2704	−	36.8414i	0.153025	−	0.265046i	−0.779313	−	0.626634i	$-0.784432\pi$
$139$	21.2704	−	36.8414i	0.153025	−	0.265046i	0.932338	+	0.361588i	$0.117765\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			28.4003i			0.198603i
$144$		0			0
$145$	−49.8522			−0.343808
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−84.7559	−	48.9338i	−0.568831	−	0.328415i	0.187851	−	0.982198i	$-0.439848\pi$
$149$	−84.7559	−	48.9338i	−0.568831	−	0.328415i	−0.756682	+	0.653783i	$0.773181\pi$
$150$		0			0
$151$	2.06713	+	3.58037i	0.0136896	+	0.0237111i	0.872789	−	0.488098i	$-0.162309\pi$
$151$	2.06713	+	3.58037i	0.0136896	+	0.0237111i	−0.859099	+	0.511809i	$0.828976\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	27.4340	−	15.8390i	0.176993	−	0.102187i
$156$		0			0
$157$	−115.393	+	199.866i	−0.734986	+	1.27303i	0.219744	+	0.975558i	$0.429478\pi$
$157$	−115.393	+	199.866i	−0.734986	+	1.27303i	−0.954730	+	0.297475i	$0.903855\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			86.0205i			0.534289i
$162$		0			0
$163$	−76.8009			−0.471171			−0.235586	−	0.971854i	$-0.575701\pi$
$163$	−76.8009			−0.471171			−0.235586	+	0.971854i	$0.575701\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	83.0831	+	47.9680i	0.497503	+	0.287234i	0.727682	−	0.685915i	$-0.240598\pi$
$167$	83.0831	+	47.9680i	0.497503	+	0.287234i	−0.230179	+	0.973148i	$0.573931\pi$
$168$		0			0
$169$	22.2514	+	38.5405i	0.131665	+	0.228050i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	88.2873	−	50.9727i	0.510331	−	0.294640i	−0.222639	−	0.974901i	$-0.571467\pi$
$173$	88.2873	−	50.9727i	0.510331	−	0.294640i	0.732970	+	0.680261i	$0.238134\pi$
$174$		0			0
$175$	−17.8641	+	30.9416i	−0.102081	+	0.176809i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	174.549i		−	0.975131i	−0.873086	−	0.487566i	$-0.837885\pi$
$179$		−	174.549i		−	0.975131i	0.873086	−	0.487566i	$-0.162115\pi$
$180$		0			0
$181$	−299.220			−1.65315			−0.826575	−	0.562827i	$-0.809714\pi$
$181$	−299.220			−1.65315			−0.826575	+	0.562827i	$0.809714\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	89.7214	+	51.8006i	0.484980	+	0.280003i
$186$		0			0
$187$	−2.61931	−	4.53678i	−0.0140070	−	0.0242609i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−123.233	+	71.1486i	−0.645199	+	0.372506i	−0.786615	−	0.617444i	$-0.788168\pi$
$191$	−123.233	+	71.1486i	−0.645199	+	0.372506i	0.141415	+	0.989950i	$0.454835\pi$
$192$		0			0
$193$	−111.425	+	192.995i	−0.577334	+	0.999972i	0.418450	+	0.908240i	$0.362574\pi$
$193$	−111.425	+	192.995i	−0.577334	+	0.999972i	−0.995784	+	0.0917319i	$0.970760\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	270.569i		−	1.37345i	−0.726919	−	0.686723i	$-0.759048\pi$
$197$		−	270.569i		−	1.37345i	0.726919	−	0.686723i	$-0.240952\pi$
$198$		0			0
$199$	−140.476			−0.705912			−0.352956	−	0.935640i	$-0.614823\pi$
$199$	−140.476			−0.705912			−0.352956	+	0.935640i	$0.614823\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	137.966	+	79.6548i	0.679636	+	0.392388i
$204$		0			0
$205$	19.4620	+	33.7091i	0.0949365	+	0.164435i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−42.7154	+	24.6617i	−0.204380	+	0.117999i
$210$		0			0
$211$	103.996	−	180.127i	0.492874	−	0.853682i	−0.507093	−	0.861892i	$-0.669280\pi$
$211$	103.996	−	180.127i	0.492874	−	0.853682i	0.999966	+	0.00820932i	$0.00261314\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			60.4179i			0.281013i
$216$		0			0
$217$	−101.231			−0.466504
$218$		0			0
$219$		0			0
$220$		0			0
$221$	19.8877	+	11.4822i	0.0899897	+	0.0519556i
$222$		0			0
$223$	−110.548	−	191.476i	−0.495733	−	0.858635i	0.504255	−	0.863555i	$-0.331767\pi$
$223$	−110.548	−	191.476i	−0.495733	−	0.858635i	−0.999988	+	0.00491996i	$0.998434\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−151.949	+	87.7280i	−0.669380	+	0.386467i	−0.795842	−	0.605505i	$-0.792971\pi$
$227$	−151.949	+	87.7280i	−0.669380	+	0.386467i	0.126461	+	0.991972i	$0.459638\pi$
$228$		0			0
$229$	−107.009	+	185.344i	−0.467287	+	0.809364i	−0.999301	−	0.0373707i	$-0.988102\pi$
$229$	−107.009	+	185.344i	−0.467287	+	0.809364i	0.532015	+	0.846735i	$0.321435\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			9.77506i			0.0419530i	0.999780	+	0.0209765i	$0.00667752\pi$
$233$			9.77506i			0.0419530i	−0.999780	+	0.0209765i	$0.993322\pi$
$234$		0			0
$235$	109.093			0.464224
$236$		0			0
$237$		0			0
$238$		0			0
$239$	248.842	+	143.669i	1.04118	+	0.601125i	0.920167	−	0.391526i	$-0.128053\pi$
$239$	248.842	+	143.669i	1.04118	+	0.601125i	0.121012	+	0.992651i	$0.461386\pi$
$240$		0			0
$241$	163.672	+	283.488i	0.679137	+	1.17630i	0.975241	+	0.221143i	$0.0709789\pi$
$241$	163.672	+	283.488i	0.679137	+	1.17630i	−0.296105	+	0.955155i	$0.595688\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	3.98997	−	2.30361i	0.0162856	−	0.00940248i
$246$		0			0
$247$	108.109	−	187.250i	0.437687	−	0.758096i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	290.903i		−	1.15897i	−0.814981	−	0.579487i	$-0.803253\pi$
$251$		−	290.903i		−	1.15897i	0.814981	−	0.579487i	$-0.196747\pi$
$252$		0			0
$253$	30.6410			0.121111
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−276.077	−	159.393i	−1.07423	−	0.620207i	−0.144896	−	0.989447i	$-0.546285\pi$
$257$	−276.077	−	159.393i	−1.07423	−	0.620207i	−0.929334	+	0.369240i	$0.879618\pi$
$258$		0			0
$259$	−165.536	−	286.717i	−0.639135	−	1.10701i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	145.630	−	84.0798i	0.553728	−	0.319695i	−0.196896	−	0.980424i	$-0.563086\pi$
$263$	145.630	−	84.0798i	0.553728	−	0.319695i	0.750624	+	0.660729i	$0.229753\pi$
$264$		0			0
$265$	11.0020	−	19.0560i	0.0415170	−	0.0719095i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			129.572i			0.481679i	0.970565	+	0.240839i	$0.0774227\pi$
$269$			129.572i			0.481679i	−0.970565	+	0.240839i	$0.922577\pi$
$270$		0			0
$271$	−15.6638			−0.0578000			−0.0289000	−	0.999582i	$-0.509200\pi$
$271$	−15.6638			−0.0578000			−0.0289000	+	0.999582i	$0.509200\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	11.0216	+	6.36331i	0.0400784	+	0.0231393i
$276$		0			0
$277$	46.1348	+	79.9078i	0.166552	+	0.288476i	0.937205	−	0.348778i	$-0.113403\pi$
$277$	46.1348	+	79.9078i	0.166552	+	0.288476i	−0.770654	+	0.637254i	$0.780070\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−72.8094	+	42.0365i	−0.259108	+	0.149596i	−0.623928	−	0.781482i	$-0.714464\pi$
$281$	−72.8094	+	42.0365i	−0.259108	+	0.149596i	0.364819	+	0.931078i	$0.381131\pi$
$282$		0			0
$283$	131.855	−	228.380i	0.465920	−	0.806996i	−0.533323	−	0.845912i	$-0.679057\pi$
$283$	131.855	−	228.380i	0.465920	−	0.806996i	0.999243	+	0.0389154i	$0.0123903\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	124.387i		−	0.433403i
$288$		0			0
$289$	284.764			0.985343
$290$		0			0
$291$		0			0
$292$		0			0
$293$	475.043	+	274.266i	1.62131	+	0.936061i	0.986571	+	0.163331i	$0.0522239\pi$
$293$	475.043	+	274.266i	1.62131	+	0.936061i	0.634735	+	0.772730i	$0.281109\pi$
$294$		0			0
$295$	−98.2580	−	170.188i	−0.333078	−	0.576908i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−116.324	+	67.1599i	−0.389045	+	0.224615i
$300$		0			0
$301$	96.5367	−	167.207i	0.320720	−	0.555503i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			133.824i			0.438769i
$306$		0			0
$307$	−192.466			−0.626926			−0.313463	−	0.949600i	$-0.601489\pi$
$307$	−192.466			−0.626926			−0.313463	+	0.949600i	$0.601489\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−392.392	−	226.548i	−1.26171	−	0.728449i	−0.288305	−	0.957539i	$-0.593092\pi$
$311$	−392.392	−	226.548i	−1.26171	−	0.728449i	−0.973405	+	0.229090i	$0.926425\pi$
$312$		0			0
$313$	32.8644	+	56.9228i	0.104998	+	0.181862i	0.913737	−	0.406305i	$-0.133183\pi$
$313$	32.8644	+	56.9228i	0.104998	+	0.181862i	−0.808739	+	0.588167i	$0.799850\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	4.16081	−	2.40225i	0.0131256	−	0.00757806i	−0.493423	−	0.869790i	$-0.664254\pi$
$317$	4.16081	−	2.40225i	0.0131256	−	0.00757806i	0.506548	+	0.862212i	$0.330921\pi$
$318$		0			0
$319$	28.3735	−	49.1443i	0.0889450	−	0.154057i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			39.8827i			0.123476i
$324$		0			0
$325$	−55.7892			−0.171659
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−301.914	−	174.310i	−0.917671	−	0.529818i
$330$		0			0
$331$	−72.6006	−	125.748i	−0.219337	−	0.379903i	0.735268	−	0.677776i	$-0.237056\pi$
$331$	−72.6006	−	125.748i	−0.219337	−	0.379903i	−0.954605	+	0.297873i	$0.903723\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−14.9389	+	8.62496i	−0.0445936	+	0.0257462i
$336$		0			0
$337$	72.8538	−	126.187i	0.216183	−	0.374441i	−0.737455	−	0.675397i	$-0.763972\pi$
$337$	72.8538	−	126.187i	0.216183	−	0.374441i	0.953638	+	0.300956i	$0.0973057\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			36.0592i			0.105745i
$342$		0			0
$343$	335.414			0.977884
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−161.823	−	93.4285i	−0.466349	−	0.269246i	0.248361	−	0.968667i	$-0.420108\pi$
$347$	−161.823	−	93.4285i	−0.466349	−	0.269246i	−0.714710	+	0.699421i	$0.753441\pi$
$348$		0			0
$349$	90.7738	+	157.225i	0.260097	+	0.450501i	0.966267	−	0.257541i	$-0.0829122\pi$
$349$	90.7738	+	157.225i	0.260097	+	0.450501i	−0.706171	+	0.708042i	$0.749579\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	584.949	−	337.721i	1.65708	−	0.956715i	0.683027	−	0.730393i	$-0.260663\pi$
$353$	584.949	−	337.721i	1.65708	−	0.956715i	0.974052	−	0.226322i	$-0.0726703\pi$
$354$		0			0
$355$	−88.9153	+	154.006i	−0.250466	+	0.433819i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			361.492i			1.00694i	0.864012	+	0.503471i	$0.167944\pi$
$359$			361.492i			1.00694i	−0.864012	+	0.503471i	$0.832056\pi$
$360$		0			0
$361$	14.5096			0.0401929
$362$		0			0
$363$		0			0
$364$		0			0
$365$	11.6539	+	6.72836i	0.0319284	+	0.0184339i
$366$		0			0
$367$	47.6816	+	82.5869i	0.129923	+	0.225033i	0.923646	−	0.383246i	$-0.125194\pi$
$367$	47.6816	+	82.5869i	0.129923	+	0.225033i	−0.793724	+	0.608278i	$0.791860\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−60.8961	+	35.1584i	−0.164140	+	0.0947665i
$372$		0			0
$373$	−296.189	+	513.014i	−0.794071	+	1.37537i	0.129356	+	0.991598i	$0.458709\pi$
$373$	−296.189	+	513.014i	−0.794071	+	1.37537i	−0.923427	+	0.383774i	$0.874624\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			248.759i			0.659839i
$378$		0			0
$379$	289.491			0.763830			0.381915	−	0.924198i	$-0.375265\pi$
$379$	289.491			0.763830			0.381915	+	0.924198i	$0.375265\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	543.273	+	313.659i	1.41847	+	0.818953i	0.996164	−	0.0875011i	$-0.0278881\pi$
$383$	543.273	+	313.659i	1.41847	+	0.818953i	0.422304	+	0.906454i	$0.361221\pi$
$384$		0			0
$385$	−20.3348	−	35.2209i	−0.0528177	−	0.0914829i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−345.046	+	199.212i	−0.887008	+	0.512114i	−0.872963	−	0.487787i	$-0.837804\pi$
$389$	−345.046	+	199.212i	−0.887008	+	0.512114i	−0.0140453	+	0.999901i	$0.504471\pi$
$390$		0			0
$391$	12.3881	−	21.4568i	0.0316831	−	0.0548767i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	37.8515i		−	0.0958267i
$396$		0			0
$397$	101.894			0.256660			0.128330	−	0.991731i	$-0.459038\pi$
$397$	101.894			0.256660			0.128330	+	0.991731i	$0.459038\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−445.055	−	256.953i	−1.10986	−	0.640779i	−0.171069	−	0.985259i	$-0.554722\pi$
$401$	−445.055	−	256.953i	−1.10986	−	0.640779i	−0.938794	+	0.344480i	$0.888055\pi$
$402$		0			0
$403$	−79.0357	−	136.894i	−0.196118	−	0.339687i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−102.130	+	58.9648i	−0.250934	+	0.144877i
$408$		0			0
$409$	−87.3358	+	151.270i	−0.213535	+	0.369854i	−0.952818	−	0.303541i	$-0.901831\pi$
$409$	−87.3358	+	151.270i	−0.213535	+	0.369854i	0.739283	+	0.673394i	$0.235164\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			627.993i			1.52056i
$414$		0			0
$415$	328.295			0.791073
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−404.169	−	233.347i	−0.964604	−	0.556914i	−0.0670171	−	0.997752i	$-0.521348\pi$
$419$	−404.169	−	233.347i	−0.964604	−	0.556914i	−0.897587	+	0.440837i	$0.854682\pi$
$420$		0			0
$421$	335.886	+	581.771i	0.797828	+	1.38188i	0.921028	+	0.389497i	$0.127351\pi$
$421$	335.886	+	581.771i	0.797828	+	1.38188i	−0.123200	+	0.992382i	$0.539316\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	8.91200	−	5.14534i	0.0209694	−	0.0121067i
$426$		0			0
$427$	213.827	−	370.359i	0.500766	−	0.867352i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	690.758i		−	1.60269i	−0.598204	−	0.801344i	$-0.704119\pi$
$431$		−	690.758i		−	1.60269i	0.598204	−	0.801344i	$-0.295881\pi$
$432$		0			0
$433$	418.521			0.966562			0.483281	−	0.875465i	$-0.339445\pi$
$433$	418.521			0.966562			0.483281	+	0.875465i	$0.339445\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−202.023	−	116.638i	−0.462296	−	0.266906i
$438$		0			0
$439$	180.726	+	313.027i	0.411677	+	0.713045i	0.995073	−	0.0991427i	$-0.0316100\pi$
$439$	180.726	+	313.027i	0.411677	+	0.713045i	−0.583397	+	0.812187i	$0.698277\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	579.959	−	334.839i	1.30916	−	0.755845i	0.327206	−	0.944953i	$-0.393893\pi$
$443$	579.959	−	334.839i	1.30916	−	0.755845i	0.981956	+	0.189108i	$0.0605595\pi$
$444$		0			0
$445$	−112.143	+	194.237i	−0.252007	+	0.436489i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			236.187i			0.526028i	0.964792	+	0.263014i	$0.0847166\pi$
$449$			236.187i			0.526028i	−0.964792	+	0.263014i	$0.915283\pi$
$450$		0			0
$451$	−44.3072			−0.0982422
$452$		0			0
$453$		0			0
$454$		0			0
$455$	154.397	+	89.1409i	0.339333	+	0.195914i
$456$		0			0
$457$	346.900	+	600.848i	0.759080	+	1.31477i	0.943320	+	0.331884i	$0.107684\pi$
$457$	346.900	+	600.848i	0.759080	+	1.31477i	−0.184240	+	0.982881i	$0.558982\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	290.417	−	167.672i	0.629971	−	0.363714i	−0.150770	−	0.988569i	$-0.548175\pi$
$461$	290.417	−	167.672i	0.629971	−	0.363714i	0.780741	+	0.624855i	$0.214842\pi$
$462$		0			0
$463$	180.088	−	311.922i	0.388960	−	0.673698i	−0.603350	−	0.797476i	$-0.706168\pi$
$463$	180.088	−	311.922i	0.388960	−	0.673698i	0.992310	+	0.123778i	$0.0395012\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			679.050i			1.45407i	0.686602	+	0.727034i	$0.259102\pi$
$467$			679.050i			1.45407i	−0.686602	+	0.727034i	$0.740898\pi$
$468$		0			0
$469$	55.1245			0.117536
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−59.5599	−	34.3869i	−0.125919	−	0.0726996i
$474$		0			0
$475$	−48.4452	−	83.9095i	−0.101990	−	0.176652i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−188.160	+	108.634i	−0.392818	+	0.226794i	−0.683381	−	0.730062i	$-0.739491\pi$
$479$	−188.160	+	108.634i	−0.392818	+	0.226794i	0.290562	+	0.956856i	$0.406158\pi$
$480$		0			0
$481$	258.482	−	447.704i	0.537384	−	0.930777i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			162.372i			0.334788i
$486$		0			0
$487$	737.594			1.51457			0.757283	−	0.653086i	$-0.226526\pi$
$487$	737.594			1.51457			0.757283	+	0.653086i	$0.226526\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−657.536	−	379.629i	−1.33918	−	0.773175i	−0.352492	−	0.935815i	$-0.614666\pi$
$491$	−657.536	−	379.629i	−1.33918	−	0.773175i	−0.986685	+	0.162640i	$0.947999\pi$
$492$		0			0
$493$	−22.9427	−	39.7379i	−0.0465369	−	0.0806042i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	492.146	−	284.141i	0.990234	−	0.571712i
$498$		0			0
$499$	333.289	−	577.274i	0.667915	−	1.15686i	−0.310572	−	0.950550i	$-0.600520\pi$
$499$	333.289	−	577.274i	0.667915	−	1.15686i	0.978486	−	0.206312i	$-0.0661462\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	180.800i		−	0.359444i	−0.983717	−	0.179722i	$-0.942480\pi$
$503$		−	180.800i		−	0.359444i	0.983717	−	0.179722i	$-0.0575199\pi$
$504$		0			0
$505$	332.030			0.657485
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−318.561	−	183.921i	−0.625856	−	0.361338i	0.153289	−	0.988181i	$-0.451013\pi$
$509$	−318.561	−	183.921i	−0.625856	−	0.361338i	−0.779146	+	0.626843i	$0.784347\pi$
$510$		0			0
$511$	−21.5014	−	37.2415i	−0.0420770	−	0.0728796i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	274.950	−	158.742i	0.533883	−	0.308238i
$516$		0			0
$517$	−62.0902	+	107.543i	−0.120097	+	0.208014i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			428.171i			0.821825i	0.911675	+	0.410913i	$0.134790\pi$
$521$			428.171i			0.821825i	−0.911675	+	0.410913i	$0.865210\pi$
$522$		0			0
$523$	742.878			1.42042			0.710208	−	0.703992i	$-0.248601\pi$
$523$	742.878			1.42042			0.710208	+	0.703992i	$0.248601\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	25.2510	+	14.5787i	0.0479146	+	0.0276635i
$528$		0			0
$529$	−192.041	−	332.625i	−0.363027	−	0.628782i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	168.206	−	97.1140i	0.315584	−	0.182203i
$534$		0			0
$535$	4.04883	−	7.01278i	0.00756791	−	0.0131080i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			5.24441i			0.00972989i
$540$		0			0
$541$	−431.532			−0.797656			−0.398828	−	0.917026i	$-0.630583\pi$
$541$	−431.532			−0.797656			−0.398828	+	0.917026i	$0.630583\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	373.742	+	215.780i	0.685766	+	0.395927i
$546$		0			0
$547$	−72.9216	−	126.304i	−0.133312	−	0.230903i	0.791639	−	0.610989i	$-0.209228\pi$
$547$	−72.9216	−	126.304i	−0.133312	−	0.230903i	−0.924951	+	0.380086i	$0.875895\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−374.146	+	216.013i	−0.679030	+	0.392038i
$552$		0			0
$553$	−60.4799	+	104.754i	−0.109367	+	0.189429i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	680.396i		−	1.22154i	−0.791809	−	0.610769i	$-0.790860\pi$
$557$		−	680.396i		−	1.22154i	0.791809	−	0.610769i	$-0.209140\pi$
$558$		0			0
$559$	301.481			0.539323
$560$		0			0
$561$		0			0
$562$		0			0
$563$	797.888	+	460.661i	1.41721	+	0.818226i	0.996053	−	0.0887629i	$-0.0282914\pi$
$563$	797.888	+	460.661i	1.41721	+	0.818226i	0.421155	+	0.906988i	$0.361625\pi$
$564$		0			0
$565$	−97.1926	−	168.343i	−0.172022	−	0.297951i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	685.403	−	395.717i	1.20457	−	0.695461i	0.243005	−	0.970025i	$-0.421867\pi$
$569$	685.403	−	395.717i	1.20457	−	0.695461i	0.961569	+	0.274564i	$0.0885335\pi$
$570$		0			0
$571$	529.700	−	917.468i	0.927672	−	1.60677i	0.140464	−	0.990086i	$-0.455141\pi$
$571$	529.700	−	917.468i	0.927672	−	1.60677i	0.787207	−	0.616688i	$-0.211526\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			60.1908i			0.104680i
$576$		0			0
$577$	−526.293			−0.912119			−0.456060	−	0.889949i	$-0.650740\pi$
$577$	−526.293			−0.912119			−0.456060	+	0.889949i	$0.650740\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−908.558	−	524.556i	−1.56378	−	0.902850i
$582$		0			0
$583$	12.5236	+	21.6915i	0.0214813	+	0.0372067i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−808.682	+	466.893i	−1.37765	+	0.795388i	−0.991877	−	0.127204i	$-0.959400\pi$
$587$	−808.682	+	466.893i	−1.37765	+	0.795388i	−0.385776	+	0.922592i	$0.626066\pi$
$588$		0			0
$589$	137.263	−	237.747i	0.233044	−	0.403645i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			686.623i			1.15788i	0.815370	+	0.578940i	$0.196533\pi$
$593$			686.623i			1.15788i	−0.815370	+	0.578940i	$0.803467\pi$
$594$		0			0
$595$	−32.8853			−0.0552694
$596$		0			0
$597$		0			0
$598$		0			0
$599$	217.781	+	125.736i	0.363574	+	0.209910i	0.670647	−	0.741776i	$-0.266016\pi$
$599$	217.781	+	125.736i	0.363574	+	0.209910i	−0.307073	+	0.951686i	$0.599350\pi$
$600$		0			0
$601$	304.733	+	527.812i	0.507043	+	0.878223i	0.999967	+	0.00815116i	$0.00259462\pi$
$601$	304.733	+	527.812i	0.507043	+	0.878223i	−0.492924	+	0.870072i	$0.664072\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	221.770	−	128.039i	0.366561	−	0.211634i
$606$		0			0
$607$	414.080	−	717.208i	0.682175	−	1.18156i	−0.292140	−	0.956375i	$-0.594367\pi$
$607$	414.080	−	717.208i	0.682175	−	1.18156i	0.974316	−	0.225187i	$-0.0722993\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	544.365i		−	0.890941i
$612$		0			0
$613$	534.809			0.872445			0.436223	−	0.899839i	$-0.356316\pi$
$613$	534.809			0.872445			0.436223	+	0.899839i	$0.356316\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−384.859	−	222.199i	−0.623759	−	0.360127i	0.154572	−	0.987982i	$-0.450600\pi$
$617$	−384.859	−	222.199i	−0.623759	−	0.360127i	−0.778331	+	0.627854i	$0.783933\pi$
$618$		0			0
$619$	149.322	+	258.633i	0.241231	+	0.417824i	0.961065	−	0.276322i	$-0.0891156\pi$
$619$	149.322	+	258.633i	0.241231	+	0.417824i	−0.719834	+	0.694146i	$0.755782\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	620.712	−	358.368i	0.996327	−	0.575230i
$624$		0			0
$625$	−12.5000	+	21.6506i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			95.3574i			0.151602i
$630$		0			0
$631$	1154.93			1.83031			0.915155	−	0.403103i	$-0.132068\pi$
$631$	1154.93			1.83031			0.915155	+	0.403103i	$0.132068\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	161.820	+	93.4267i	0.254834	+	0.147129i
$636$		0			0
$637$	−11.4949	−	19.9097i	−0.0180453	−	0.0312554i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−588.759	+	339.920i	−0.918502	+	0.530297i	−0.883157	−	0.469078i	$-0.844586\pi$
$641$	−588.759	+	339.920i	−0.918502	+	0.530297i	−0.0353448	+	0.999375i	$0.511253\pi$
$642$		0			0
$643$	−217.288	+	376.354i	−0.337929	+	0.585310i	−0.984043	−	0.177931i	$-0.943060\pi$
$643$	−217.288	+	376.354i	−0.337929	+	0.585310i	0.646114	+	0.763241i	$0.276393\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	222.972i		−	0.344625i	−0.985042	−	0.172312i	$-0.944876\pi$
$647$		−	222.972i		−	0.344625i	0.985042	−	0.172312i	$-0.0551239\pi$
$648$		0			0
$649$	223.695			0.344676
$650$		0			0
$651$		0			0
$652$		0			0
$653$	453.763	+	261.980i	0.694889	+	0.401194i	0.805441	−	0.592676i	$-0.201929\pi$
$653$	453.763	+	261.980i	0.694889	+	0.401194i	−0.110552	+	0.993870i	$0.535262\pi$
$654$		0			0
$655$	24.8316	+	43.0096i	0.0379109	+	0.0656635i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	249.789	−	144.216i	0.379043	−	0.218841i	−0.298359	−	0.954454i	$-0.596439\pi$
$659$	249.789	−	144.216i	0.379043	−	0.218841i	0.677402	+	0.735613i	$0.263106\pi$
$660$		0			0
$661$	−24.7991	+	42.9533i	−0.0375176	+	0.0649823i	−0.884174	−	0.467157i	$-0.845278\pi$
$661$	−24.7991	+	42.9533i	−0.0375176	+	0.0649823i	0.846657	+	0.532139i	$0.178612\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			309.626i			0.465603i
$666$		0			0
$667$	268.386			0.402378
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−131.924	−	76.1664i	−0.196608	−	0.113512i
$672$		0			0
$673$	−306.034	−	530.066i	−0.454731	−	0.787617i	0.543942	−	0.839123i	$-0.316931\pi$
$673$	−306.034	−	530.066i	−0.454731	−	0.787617i	−0.998673	+	0.0515058i	$0.983598\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	931.146	−	537.597i	1.37540	−	0.794087i	0.383798	−	0.923417i	$-0.374616\pi$
$677$	931.146	−	537.597i	1.37540	−	0.794087i	0.991602	+	0.129330i	$0.0412825\pi$
$678$		0			0
$679$	259.441	−	449.365i	0.382093	−	0.661804i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			718.621i			1.05215i	0.850437	+	0.526077i	$0.176337\pi$
$683$			718.621i			1.05215i	−0.850437	+	0.526077i	$0.823663\pi$
$684$		0			0
$685$	23.9650			0.0349854
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−95.0883	−	54.8992i	−0.138009	−	0.0796796i
$690$		0			0
$691$	420.368	+	728.099i	0.608347	+	1.05369i	0.991513	+	0.130009i	$0.0415006\pi$
$691$	420.368	+	728.099i	0.608347	+	1.05369i	−0.383165	+	0.923680i	$0.625166\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−82.3800	+	47.5621i	−0.118532	+	0.0684347i
$696$		0			0
$697$	−17.9133	+	31.0268i	−0.0257006	+	0.0445148i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			614.419i			0.876490i	0.898856	+	0.438245i	$0.144400\pi$
$701$			614.419i			0.876490i	−0.898856	+	0.438245i	$0.855600\pi$
$702$		0			0
$703$	897.823			1.27713
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−918.893	−	530.523i	−1.29971	−	0.750387i
$708$		0			0
$709$	434.681	+	752.890i	0.613091	+	1.06190i	0.990716	+	0.135946i	$0.0434074\pi$
$709$	434.681	+	752.890i	0.613091	+	1.06190i	−0.377625	+	0.925958i	$0.623259\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−147.694	+	85.2714i	−0.207145	+	0.119595i
$714$		0			0
$715$	31.7525	−	54.9969i	0.0444091	−	0.0769188i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	1307.10i		−	1.81794i	−0.416859	−	0.908971i	$-0.636869\pi$
$719$		−	1307.10i		−	1.81794i	0.416859	−	0.908971i	$-0.363131\pi$
$720$		0			0
$721$	−1014.57			−1.40716
$722$		0			0
$723$		0			0
$724$		0			0
$725$	96.5384	+	55.7365i	0.133156	+	0.0768779i
$726$		0			0
$727$	112.223	+	194.376i	0.154365	+	0.267367i	0.932827	−	0.360323i	$-0.117334\pi$
$727$	112.223	+	194.376i	0.154365	+	0.267367i	−0.778463	+	0.627691i	$0.784000\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−48.1599	+	27.8051i	−0.0658822	+	0.0380371i
$732$		0			0
$733$	−571.310	+	989.538i	−0.779413	+	1.34998i	0.152867	+	0.988247i	$0.451149\pi$
$733$	−571.310	+	989.538i	−0.779413	+	1.34998i	−0.932280	+	0.361737i	$0.882184\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	19.6356i		−	0.0266427i
$738$		0			0
$739$	−450.991			−0.610272			−0.305136	−	0.952309i	$-0.598702\pi$
$739$	−450.991			−0.610272			−0.305136	+	0.952309i	$0.598702\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1074.55	+	620.394i	1.44624	+	0.834985i	0.998255	−	0.0590568i	$-0.0188093\pi$
$743$	1074.55	+	620.394i	1.44624	+	0.834985i	0.447983	+	0.894042i	$0.352143\pi$
$744$		0			0
$745$	109.419	+	189.520i	0.146872	+	0.254389i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−22.4103	+	12.9386i	−0.0299203	+	0.0172745i
$750$		0			0
$751$	226.995	−	393.168i	0.302257	−	0.523525i	−0.674389	−	0.738376i	$-0.735593\pi$
$751$	226.995	−	393.168i	0.302257	−	0.523525i	0.976647	+	0.214850i	$0.0689264\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	9.24448i		−	0.0122443i
$756$		0			0
$757$	−18.7572			−0.0247784			−0.0123892	−	0.999923i	$-0.503944\pi$
$757$	−18.7572			−0.0247784			−0.0123892	+	0.999923i	$0.503944\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−655.215	−	378.289i	−0.860992	−	0.497094i	0.00335213	−	0.999994i	$-0.498933\pi$
$761$	−655.215	−	378.289i	−0.860992	−	0.497094i	−0.864344	+	0.502900i	$0.832266\pi$
$762$		0			0
$763$	−689.555	−	1194.34i	−0.903742	−	1.56533i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−849.226	+	490.301i	−1.10721	+	0.639245i
$768$		0			0
$769$	−583.868	+	1011.29i	−0.759257	+	1.31507i	0.183974	+	0.982931i	$0.441104\pi$
$769$	−583.868	+	1011.29i	−0.759257	+	1.31507i	−0.943230	+	0.332140i	$0.892229\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			763.809i			0.988110i	0.869431	+	0.494055i	$0.164486\pi$
$773$			763.809i			0.988110i	−0.869431	+	0.494055i	$0.835514\pi$
$774$		0			0
$775$	−70.8342			−0.0913990
$776$		0			0
$777$		0			0
$778$		0			0
$779$	292.128	+	168.660i	0.375004	+	0.216509i
$780$		0			0
$781$	−101.213	−	175.305i	−0.129594	−	0.224463i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	446.914	−	258.026i	0.569318	−	0.328696i
$786$		0			0
$787$	−248.732	+	430.817i	−0.316051	+	0.547416i	−0.979660	−	0.200663i	$-0.935690\pi$
$787$	−248.732	+	430.817i	−0.316051	+	0.547416i	0.663609	+	0.748079i	$0.269024\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			621.184i			0.785315i
$792$		0			0
$793$	667.775			0.842087
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1163.28	−	671.621i	−1.45958	−	0.842686i	−0.460585	−	0.887616i	$-0.652360\pi$
$797$	−1163.28	−	671.621i	−1.45958	−	0.842686i	−0.998990	+	0.0449296i	$0.985694\pi$
$798$		0			0
$799$	50.2059	+	86.9591i	0.0628359	+	0.108835i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−13.2656	+	7.65891i	−0.0165201	+	0.00953787i
$804$		0			0
$805$	96.1739	−	166.578i	0.119471	−	0.206929i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			244.633i			0.302389i	0.988504	+	0.151195i	$0.0483120\pi$
$809$			244.633i			0.302389i	−0.988504	+	0.151195i	$0.951688\pi$
$810$		0			0
$811$	−1182.65			−1.45827			−0.729133	−	0.684372i	$-0.760076\pi$
$811$	−1182.65			−1.45827			−0.729133	+	0.684372i	$0.760076\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	148.724	+	85.8660i	0.182484	+	0.105357i
$816$		0			0
$817$	261.795	+	453.442i	0.320434	+	0.555008i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−795.256	+	459.141i	−0.968643	+	0.559246i	−0.898822	−	0.438313i	$-0.855576\pi$
$821$	−795.256	+	459.141i	−0.968643	+	0.559246i	−0.0698205	+	0.997560i	$0.522243\pi$
$822$		0			0
$823$	−406.279	+	703.695i	−0.493656	+	0.855037i	−0.999973	−	0.00731049i	$-0.997673\pi$
$823$	−406.279	+	703.695i	−0.493656	+	0.855037i	0.506318	+	0.862347i	$0.331006\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	476.892i		−	0.576653i	−0.957532	−	0.288327i	$-0.906901\pi$
$827$		−	476.892i		−	0.576653i	0.957532	−	0.288327i	$-0.0930989\pi$
$828$		0			0
$829$	−369.534			−0.445758			−0.222879	−	0.974846i	$-0.571546\pi$
$829$	−369.534			−0.445758			−0.222879	+	0.974846i	$0.571546\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	3.67247	+	2.12030i	0.00440873	+	0.00254538i
$834$		0			0
$835$	−107.260	−	185.779i	−0.128455	−	0.222490i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	728.282	−	420.474i	0.868036	−	0.501161i	0.00134090	−	0.999999i	$-0.499573\pi$
$839$	728.282	−	420.474i	0.868036	−	0.501161i	0.866695	+	0.498838i	$0.166240\pi$
$840$		0			0
$841$	−171.976	+	297.870i	−0.204489	+	0.354186i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	99.5111i		−	0.117765i
$846$		0			0
$847$	−818.330			−0.966151
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−483.026	−	278.875i	−0.567599	−	0.327703i
$852$		0			0
$853$	−362.604	−	628.048i	−0.425092	−	0.736282i	0.571337	−	0.820716i	$-0.306425\pi$
$853$	−362.604	−	628.048i	−0.425092	−	0.736282i	−0.996429	+	0.0844341i	$0.973092\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−732.868	+	423.121i	−0.855155	+	0.493724i	−0.862387	−	0.506250i	$-0.831031\pi$
$857$	−732.868	+	423.121i	−0.855155	+	0.493724i	0.00723204	+	0.999974i	$0.497698\pi$
$858$		0			0
$859$	352.776	−	611.025i	0.410682	−	0.711321i	−0.584283	−	0.811550i	$-0.698624\pi$
$859$	352.776	−	611.025i	0.410682	−	0.711321i	0.994964	+	0.100229i	$0.0319574\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			626.198i			0.725606i	0.931866	+	0.362803i	$0.118180\pi$
$863$			626.198i			0.725606i	−0.931866	+	0.362803i	$0.881820\pi$
$864$		0			0
$865$	−227.957			−0.263534
$866$		0			0
$867$		0			0
$868$		0			0
$869$	37.3140	+	21.5433i	0.0429390	+	0.0247909i
$870$		0			0
$871$	43.0380	+	74.5440i	0.0494122	+	0.0855844i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	69.1875	−	39.9454i	0.0790715	−	0.0456519i
$876$		0			0
$877$	288.812	−	500.237i	0.329318	−	0.570395i	−0.653059	−	0.757307i	$-0.726515\pi$
$877$	288.812	−	500.237i	0.329318	−	0.570395i	0.982377	+	0.186912i	$0.0598479\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	1719.11i		−	1.95131i	−0.219306	−	0.975656i	$-0.570379\pi$
$881$		−	1719.11i		−	1.95131i	0.219306	−	0.975656i	$-0.429621\pi$
$882$		0			0
$883$	−356.656			−0.403914			−0.201957	−	0.979394i	$-0.564730\pi$
$883$	−356.656			−0.403914			−0.201957	+	0.979394i	$0.564730\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	428.591	+	247.447i	0.483192	+	0.278971i	0.721746	−	0.692158i	$-0.243340\pi$
$887$	428.591	+	247.447i	0.483192	+	0.278971i	−0.238554	+	0.971129i	$0.576673\pi$
$888$		0			0
$889$	−298.558	−	517.117i	−0.335835	−	0.581684i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	818.750	−	472.705i	0.916853	−	0.529345i
$894$		0			0
$895$	−195.151	+	338.012i	−0.218046	+	0.377667i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			315.844i			0.351328i
$900$		0			0
$901$	20.2531			0.0224784
$902$		0			0
$903$		0			0
$904$		0			0
$905$	579.437	+	334.538i	0.640262	+	0.369655i
$906$		0			0
$907$	−311.723	−	539.919i	−0.343685	−	0.595280i	0.641429	−	0.767183i	$-0.278342\pi$
$907$	−311.723	−	539.919i	−0.343685	−	0.595280i	−0.985114	+	0.171902i	$0.945009\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1527.46	+	881.879i	−1.67668	+	0.968034i	−0.712931	+	0.701234i	$0.752633\pi$
$911$	−1527.46	+	881.879i	−1.67668	+	0.968034i	−0.963752	+	0.266800i	$0.914034\pi$
$912$		0			0
$913$	−186.850	+	323.633i	−0.204655	+	0.354472i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	158.705i		−	0.173070i
$918$		0			0
$919$	−540.616			−0.588266			−0.294133	−	0.955765i	$-0.595031\pi$
$919$	−540.616			−0.588266			−0.294133	+	0.955765i	$0.595031\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	768.479	+	443.682i	0.832588	+	0.480695i
$924$		0			0
$925$	−115.830	−	200.623i	−0.125221	−	0.216890i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	206.950	−	119.482i	0.222766	−	0.128614i	−0.384464	−	0.923140i	$-0.625614\pi$
$929$	206.950	−	119.482i	0.222766	−	0.128614i	0.607230	+	0.794526i	$0.292281\pi$
$930$		0			0
$931$	19.9634	−	34.5776i	0.0214429	−	0.0371403i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			11.7139i			0.0125283i
$936$		0			0
$937$	−75.1773			−0.0802320			−0.0401160	−	0.999195i	$-0.512773\pi$
$937$	−75.1773			−0.0802320			−0.0401160	+	0.999195i	$0.512773\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	57.3372	+	33.1037i	0.0609322	+	0.0351792i	0.530157	−	0.847900i	$-0.322133\pi$
$941$	57.3372	+	33.1037i	0.0609322	+	0.0351792i	−0.469224	+	0.883079i	$0.655466\pi$
$942$		0			0
$943$	−104.776	−	181.477i	−0.111109	−	0.192447i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−225.603	+	130.252i	−0.238230	+	0.137542i	−0.614363	−	0.789024i	$-0.710587\pi$
$947$	−225.603	+	130.252i	−0.238230	+	0.137542i	0.376133	+	0.926566i	$0.377254\pi$
$948$		0			0
$949$	33.5741	−	58.1520i	0.0353784	−	0.0612771i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	502.854i		−	0.527654i	−0.964570	−	0.263827i	$-0.915015\pi$
$953$		−	502.854i		−	0.527654i	0.964570	−	0.263827i	$-0.0849847\pi$
$954$		0			0
$955$	318.186			0.333179
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−66.3231	−	38.2917i	−0.0691586	−	0.0399287i
$960$		0			0
$961$	380.150	+	658.439i	0.395578	+	0.685161i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	431.549	−	249.155i	0.447201	−	0.258192i
$966$		0			0
$967$	−308.935	+	535.091i	−0.319478	+	0.553352i	−0.980379	−	0.197121i	$-0.936841\pi$
$967$	−308.935	+	535.091i	−0.319478	+	0.553352i	0.660901	+	0.750473i	$0.270174\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	240.669i		−	0.247856i	−0.992291	−	0.123928i	$-0.960451\pi$
$971$		−	240.669i		−	0.247856i	0.992291	−	0.123928i	$-0.0395492\pi$
$972$		0			0
$973$	303.982			0.312417
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1293.59	+	746.857i	1.32405	+	0.764439i	0.984372	−	0.176104i	$-0.0563494\pi$
$977$	1293.59	+	746.857i	1.32405	+	0.764439i	0.339675	+	0.940543i	$0.389683\pi$
$978$		0			0
$979$	−127.653	−	221.101i	−0.130391	−	0.225844i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−674.535	+	389.443i	−0.686200	+	0.396178i	−0.802187	−	0.597073i	$-0.796330\pi$
$983$	−674.535	+	389.443i	−0.686200	+	0.396178i	0.115987	+	0.993251i	$0.462997\pi$
$984$		0			0
$985$	−302.505	+	523.955i	−0.307112	+	0.531934i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	325.267i		−	0.328885i
$990$		0			0
$991$	−703.289			−0.709676			−0.354838	−	0.934928i	$-0.615464\pi$
$991$	−703.289			−0.709676			−0.354838	+	0.934928i	$0.615464\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	272.032	+	157.057i	0.273399	+	0.157847i
$996$		0			0
$997$	293.948	+	509.133i	0.294833	+	0.510665i	0.974946	−	0.222442i	$-0.0714027\pi$
$997$	293.948	+	509.133i	0.294833	+	0.510665i	−0.680113	+	0.733107i	$0.738069\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.3.o.g.701.4		32
3.2	odd	2	inner	1620.3.o.g.701.12		32
9.2	odd	6	inner	1620.3.o.g.1241.4		32
9.4	even	3		1620.3.g.c.161.10	yes	16
9.5	odd	6		1620.3.g.c.161.2	✓	16
9.7	even	3	inner	1620.3.o.g.1241.12		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1620.3.g.c.161.2	✓	16	9.5	odd	6
1620.3.g.c.161.10	yes	16	9.4	even	3
1620.3.o.g.701.4		32	1.1	even	1	trivial
1620.3.o.g.701.12		32	3.2	odd	2	inner
1620.3.o.g.1241.4		32	9.2	odd	6	inner
1620.3.o.g.1241.12		32	9.7	even	3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1620.o (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.93649 - 1.11803i) q^{5} +(3.57283 + 6.18832i) q^{7} +O(q^{10})\)	\(q+(-1.93649 - 1.11803i) q^{5} +(3.57283 + 6.18832i) q^{7} +(2.20431 - 1.27266i) q^{11} +(-5.57892 + 9.66297i) q^{13} -2.05814i q^{17} -19.3781 q^{19} +(10.4254 + 6.01908i) q^{23} +(2.50000 + 4.33013i) q^{25} +(19.3077 - 11.1473i) q^{29} +(-7.08342 + 12.2689i) q^{31} -15.9782i q^{35} -46.3319 q^{37} +(-15.0752 - 8.70366i) q^{41} +(-13.5098 - 23.3997i) q^{43} +(-42.2514 + 24.3938i) q^{47} +(-1.03020 + 1.78437i) q^{49} +9.84048i q^{53} -5.69151 q^{55} +(76.1103 + 43.9423i) q^{59} +(-29.9240 - 51.8300i) q^{61} +(21.6071 - 12.4748i) q^{65} +(3.85720 - 6.68087i) q^{67} -79.5282i q^{71} -6.01802 q^{73} +(15.7513 + 9.09400i) q^{77} +(8.46386 + 14.6598i) q^{79} +(-127.148 + 73.4091i) q^{83} +(-2.30107 + 3.98557i) q^{85} -100.304i q^{89} -79.7301 q^{91} +(37.5255 + 21.6653i) q^{95} +(-36.3075 - 62.8865i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 8 q^{7}+O(q^{10}) \) 32 * q - 8 * q^7	\( 32 q - 8 q^{7} + 40 q^{13} + 112 q^{19} + 80 q^{25} + 64 q^{31} - 176 q^{37} - 128 q^{43} - 216 q^{49} - 8 q^{61} + 40 q^{67} + 112 q^{73} + 136 q^{79} - 784 q^{91} - 128 q^{97}+O(q^{100}) \) 32 * q - 8 * q^7 + 40 * q^13 + 112 * q^19 + 80 * q^25 + 64 * q^31 - 176 * q^37 - 128 * q^43 - 216 * q^49 - 8 * q^61 + 40 * q^67 + 112 * q^73 + 136 * q^79 - 784 * q^91 - 128 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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