LMFDB - Embedded newform 1620.3.b.a.809.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,3,Mod(809,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

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

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

chi := DirichletCharacter("1620.809");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1620.b (of order $2$, degree $1$, 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:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			809.12
Character	$\chi$	$=$	1620.809
Dual form			1620.3.b.a.809.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.557196 + 4.96886i) q^{5} -10.7259i q^{7} +O(q^{10})$	$q+(-0.557196 + 4.96886i) q^{5} -10.7259i q^{7} -12.1847i q^{11} +23.3979i q^{13} +1.67925 q^{17} +0.234357 q^{19} +19.8636 q^{23} +(-24.3791 - 5.53725i) q^{25} -34.0745i q^{29} -19.0688 q^{31} +(53.2956 + 5.97645i) q^{35} -32.7402i q^{37} -5.27816i q^{41} +52.9829i q^{43} +53.7146 q^{47} -66.0457 q^{49} -84.6464 q^{53} +(60.5441 + 6.78927i) q^{55} -88.3259i q^{59} -65.5965 q^{61} +(-116.261 - 13.0372i) q^{65} -99.4761i q^{67} -36.8639i q^{71} +79.0196i q^{73} -130.692 q^{77} +35.3962 q^{79} -27.8463 q^{83} +(-0.935673 + 8.34396i) q^{85} -152.513i q^{89} +250.965 q^{91} +(-0.130583 + 1.16449i) q^{95} -97.0911i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q - 48 q^{25} - 288 q^{49} - 36 q^{55} + 120 q^{61} + 480 q^{79} - 24 q^{85} + 48 q^{91}+O(q^{100}) $ 24 * q - 48 * q^25 - 288 * q^49 - 36 * q^55 + 120 * q^61 + 480 * q^79 - 24 * q^85 + 48 * q^91

Display 100 coefficients 10 coefficients

Character values

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

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$-1$	$-1$

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
$5$	−0.557196	+	4.96886i	−0.111439	+	0.993771i
$5$	−0.557196	+	4.96886i	−0.111439	+	0.993771i
$6$		0			0
$7$		−	10.7259i		−	1.53228i	−0.642676	−	0.766138i	$-0.722176\pi$
$7$		−	10.7259i		−	1.53228i	0.642676	−	0.766138i	$-0.277824\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	12.1847i		−	1.10770i	−0.832616	−	0.553850i	$-0.813158\pi$
$11$		−	12.1847i		−	1.10770i	0.832616	−	0.553850i	$-0.186842\pi$
$12$		0			0
$13$			23.3979i			1.79984i	0.436055	+	0.899920i	$0.356375\pi$
$13$			23.3979i			1.79984i	−0.436055	+	0.899920i	$0.643625\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.67925			0.0987796			0.0493898	−	0.998780i	$-0.484272\pi$
$17$	1.67925			0.0987796			0.0493898	+	0.998780i	$0.484272\pi$
$18$		0			0
$19$	0.234357			0.0123346			0.00616730	−	0.999981i	$-0.498037\pi$
$19$	0.234357			0.0123346			0.00616730	+	0.999981i	$0.498037\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	19.8636			0.863635			0.431818	−	0.901961i	$-0.357872\pi$
$23$	19.8636			0.863635			0.431818	+	0.901961i	$0.357872\pi$
$24$		0			0
$25$	−24.3791	−	5.53725i	−0.975163	−	0.221490i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	34.0745i		−	1.17498i	−0.809230	−	0.587492i	$-0.800116\pi$
$29$		−	34.0745i		−	1.17498i	0.809230	−	0.587492i	$-0.199884\pi$
$30$		0			0
$31$	−19.0688			−0.615122			−0.307561	−	0.951528i	$-0.599513\pi$
$31$	−19.0688			−0.615122			−0.307561	+	0.951528i	$0.599513\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	53.2956	+	5.97645i	1.52273	+	0.170756i
$36$		0			0
$37$		−	32.7402i		−	0.884871i	−0.896800	−	0.442436i	$-0.854114\pi$
$37$		−	32.7402i		−	0.884871i	0.896800	−	0.442436i	$-0.145886\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	5.27816i		−	0.128736i	−0.997926	−	0.0643679i	$-0.979497\pi$
$41$		−	5.27816i		−	0.128736i	0.997926	−	0.0643679i	$-0.0205031\pi$
$42$		0			0
$43$			52.9829i			1.23216i	0.787684	+	0.616080i	$0.211280\pi$
$43$			52.9829i			1.23216i	−0.787684	+	0.616080i	$0.788720\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	53.7146			1.14286			0.571432	−	0.820650i	$-0.306388\pi$
$47$	53.7146			1.14286			0.571432	+	0.820650i	$0.306388\pi$
$48$		0			0
$49$	−66.0457			−1.34787
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−84.6464			−1.59710			−0.798551	−	0.601927i	$-0.794400\pi$
$53$	−84.6464			−1.59710			−0.798551	+	0.601927i	$0.794400\pi$
$54$		0			0
$55$	60.5441	+	6.78927i	1.10080	+	0.123441i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	88.3259i		−	1.49705i	−0.663107	−	0.748525i	$-0.730763\pi$
$59$		−	88.3259i		−	1.49705i	0.663107	−	0.748525i	$-0.269237\pi$
$60$		0			0
$61$	−65.5965			−1.07535			−0.537676	−	0.843152i	$-0.680698\pi$
$61$	−65.5965			−1.07535			−0.537676	+	0.843152i	$0.680698\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−116.261	−	13.0372i	−1.78863	−	0.200573i
$66$		0			0
$67$		−	99.4761i		−	1.48472i	−0.670003	−	0.742359i	$-0.733707\pi$
$67$		−	99.4761i		−	1.48472i	0.670003	−	0.742359i	$-0.266293\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	36.8639i		−	0.519210i	−0.965715	−	0.259605i	$-0.916408\pi$
$71$		−	36.8639i		−	0.519210i	0.965715	−	0.259605i	$-0.0835924\pi$
$72$		0			0
$73$			79.0196i			1.08246i	0.840875	+	0.541230i	$0.182041\pi$
$73$			79.0196i			1.08246i	−0.840875	+	0.541230i	$0.817959\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−130.692			−1.69730
$78$		0			0
$79$	35.3962			0.448053			0.224027	−	0.974583i	$-0.428080\pi$
$79$	35.3962			0.448053			0.224027	+	0.974583i	$0.428080\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−27.8463			−0.335498			−0.167749	−	0.985830i	$-0.553650\pi$
$83$	−27.8463			−0.335498			−0.167749	+	0.985830i	$0.553650\pi$
$84$		0			0
$85$	−0.935673	+	8.34396i	−0.0110079	+	0.0981643i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	152.513i		−	1.71362i	−0.515628	−	0.856812i	$-0.672441\pi$
$89$		−	152.513i		−	1.71362i	0.515628	−	0.856812i	$-0.327559\pi$
$90$		0			0
$91$	250.965			2.75785
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−0.130583	+	1.16449i	−0.00137456	+	0.0122578i
$96$		0			0
$97$		−	97.0911i		−	1.00094i	−0.865754	−	0.500470i	$-0.833161\pi$
$97$		−	97.0911i		−	1.00094i	0.865754	−	0.500470i	$-0.166839\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	28.6313i		−	0.283478i	−0.989904	−	0.141739i	$-0.954731\pi$
$101$		−	28.6313i		−	0.283478i	0.989904	−	0.141739i	$-0.0452693\pi$
$102$		0			0
$103$			43.3806i			0.421171i	0.977575	+	0.210585i	$0.0675370\pi$
$103$			43.3806i			0.421171i	−0.977575	+	0.210585i	$0.932463\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−161.010			−1.50476			−0.752381	−	0.658728i	$-0.771095\pi$
$107$	−161.010			−1.50476			−0.752381	+	0.658728i	$0.771095\pi$
$108$		0			0
$109$	142.306			1.30556			0.652779	−	0.757548i	$-0.273603\pi$
$109$	142.306			1.30556			0.652779	+	0.757548i	$0.273603\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	84.0558			0.743856			0.371928	−	0.928262i	$-0.378697\pi$
$113$	84.0558			0.743856			0.371928	+	0.928262i	$0.378697\pi$
$114$		0			0
$115$	−11.0679	+	98.6994i	−0.0962428	+	0.858256i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	18.0116i		−	0.151358i
$120$		0			0
$121$	−27.4671			−0.227001
$122$		0			0
$123$		0			0
$124$		0			0
$125$	41.0977	−	118.051i	0.328782	−	0.944406i
$126$		0			0
$127$		−	188.651i		−	1.48544i	−0.669602	−	0.742720i	$-0.733535\pi$
$127$		−	188.651i		−	1.48544i	0.669602	−	0.742720i	$-0.266465\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	106.010i		−	0.809233i	−0.914486	−	0.404617i	$-0.867405\pi$
$131$		−	106.010i		−	0.809233i	0.914486	−	0.404617i	$-0.132595\pi$
$132$		0			0
$133$		−	2.51370i		−	0.0189000i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	256.133			1.86959			0.934793	−	0.355193i	$-0.115585\pi$
$137$	256.133			1.86959			0.934793	+	0.355193i	$0.115585\pi$
$138$		0			0
$139$	−181.688			−1.30711			−0.653555	−	0.756879i	$-0.726723\pi$
$139$	−181.688			−1.30711			−0.653555	+	0.756879i	$0.726723\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	285.097			1.99368
$144$		0			0
$145$	169.311	+	18.9862i	1.16766	+	0.130939i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	287.807i		−	1.93159i	−0.259301	−	0.965797i	$-0.583492\pi$
$149$		−	287.807i		−	1.93159i	0.259301	−	0.965797i	$-0.416508\pi$
$150$		0			0
$151$	−15.1408			−0.100271			−0.0501353	−	0.998742i	$-0.515965\pi$
$151$	−15.1408			−0.100271			−0.0501353	+	0.998742i	$0.515965\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	10.6250	−	94.7500i	0.0685487	−	0.611290i
$156$		0			0
$157$			90.1367i			0.574119i	0.957913	+	0.287059i	$0.0926777\pi$
$157$			90.1367i			0.574119i	−0.957913	+	0.287059i	$0.907322\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	213.056i		−	1.32333i
$162$		0			0
$163$		−	131.354i		−	0.805851i	−0.915233	−	0.402925i	$-0.867993\pi$
$163$		−	131.354i		−	0.805851i	0.915233	−	0.402925i	$-0.132007\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	155.940			0.933772			0.466886	−	0.884317i	$-0.345376\pi$
$167$	155.940			0.933772			0.466886	+	0.884317i	$0.345376\pi$
$168$		0			0
$169$	−378.463			−2.23943
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−134.319			−0.776409			−0.388205	−	0.921573i	$-0.626905\pi$
$173$	−134.319			−0.776409			−0.388205	+	0.921573i	$0.626905\pi$
$174$		0			0
$175$	−59.3922	+	261.488i	−0.339384	+	1.49422i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			201.999i			1.12849i	0.825608	+	0.564244i	$0.190832\pi$
$179$			201.999i			1.12849i	−0.825608	+	0.564244i	$0.809168\pi$
$180$		0			0
$181$	6.27313			0.0346582			0.0173291	−	0.999850i	$-0.494484\pi$
$181$	6.27313			0.0346582			0.0173291	+	0.999850i	$0.494484\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	162.682	+	18.2427i	0.879360	+	0.0986094i
$186$		0			0
$187$		−	20.4612i		−	0.109418i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	297.667i		−	1.55847i	−0.626734	−	0.779233i	$-0.715609\pi$
$191$		−	297.667i		−	1.55847i	0.626734	−	0.779233i	$-0.284391\pi$
$192$		0			0
$193$		−	55.6767i		−	0.288480i	−0.989543	−	0.144240i	$-0.953926\pi$
$193$		−	55.6767i		−	0.288480i	0.989543	−	0.144240i	$-0.0460738\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	22.9424			0.116459			0.0582296	−	0.998303i	$-0.481454\pi$
$197$	22.9424			0.116459			0.0582296	+	0.998303i	$0.481454\pi$
$198$		0			0
$199$	204.154			1.02590			0.512951	−	0.858418i	$-0.328552\pi$
$199$	204.154			1.02590			0.512951	+	0.858418i	$0.328552\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−365.481			−1.80040
$204$		0			0
$205$	26.2264	+	2.94097i	0.127934	+	0.0143462i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	2.85558i		−	0.0136630i
$210$		0			0
$211$	−357.794			−1.69571			−0.847854	−	0.530230i	$-0.822106\pi$
$211$	−357.794			−1.69571			−0.847854	+	0.530230i	$0.822106\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−263.264	−	29.5219i	−1.22449	−	0.137311i
$216$		0			0
$217$			204.531i			0.942537i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			39.2910i			0.177787i
$222$		0			0
$223$		−	244.092i		−	1.09458i	−0.836942	−	0.547292i	$-0.815659\pi$
$223$		−	244.092i		−	1.09458i	0.836942	−	0.547292i	$-0.184341\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−225.208			−0.992106			−0.496053	−	0.868292i	$-0.665218\pi$
$227$	−225.208			−0.992106			−0.496053	+	0.868292i	$0.665218\pi$
$228$		0			0
$229$	−45.1626			−0.197216			−0.0986082	−	0.995126i	$-0.531439\pi$
$229$	−45.1626			−0.197216			−0.0986082	+	0.995126i	$0.531439\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−55.4732			−0.238083			−0.119041	−	0.992889i	$-0.537982\pi$
$233$	−55.4732			−0.238083			−0.119041	+	0.992889i	$0.537982\pi$
$234$		0			0
$235$	−29.9296	+	266.900i	−0.127360	+	1.13575i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			12.1352i			0.0507750i	0.999678	+	0.0253875i	$0.00808196\pi$
$239$			12.1352i			0.0507750i	−0.999678	+	0.0253875i	$0.991918\pi$
$240$		0			0
$241$	52.8995			0.219500			0.109750	−	0.993959i	$-0.464995\pi$
$241$	52.8995			0.219500			0.109750	+	0.993959i	$0.464995\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	36.8004	−	328.172i	0.150206	−	1.33948i
$246$		0			0
$247$			5.48348i			0.0222003i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	44.3567i		−	0.176720i	−0.996089	−	0.0883599i	$-0.971837\pi$
$251$		−	44.3567i		−	0.176720i	0.996089	−	0.0883599i	$-0.0281626\pi$
$252$		0			0
$253$		−	242.032i		−	0.956649i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−171.300			−0.666538			−0.333269	−	0.942832i	$-0.608152\pi$
$257$	−171.300			−0.666538			−0.333269	+	0.942832i	$0.608152\pi$
$258$		0			0
$259$	−351.170			−1.35587
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−123.047			−0.467859			−0.233929	−	0.972254i	$-0.575158\pi$
$263$	−123.047			−0.467859			−0.233929	+	0.972254i	$0.575158\pi$
$264$		0			0
$265$	47.1646	−	420.596i	0.177980	−	1.58715i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			128.164i			0.476446i	0.971210	+	0.238223i	$0.0765649\pi$
$269$			128.164i			0.476446i	−0.971210	+	0.238223i	$0.923435\pi$
$270$		0			0
$271$	295.707			1.09117			0.545586	−	0.838055i	$-0.316307\pi$
$271$	295.707			1.09117			0.545586	+	0.838055i	$0.316307\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−67.4698	+	297.052i	−0.245345	+	1.08019i
$276$		0			0
$277$			485.006i			1.75092i	0.483287	+	0.875462i	$0.339443\pi$
$277$			485.006i			1.75092i	−0.483287	+	0.875462i	$0.660557\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	251.363i		−	0.894529i	−0.894402	−	0.447264i	$-0.852398\pi$
$281$		−	251.363i		−	0.894529i	0.894402	−	0.447264i	$-0.147602\pi$
$282$		0			0
$283$		−	276.885i		−	0.978394i	−0.872173	−	0.489197i	$-0.837290\pi$
$283$		−	276.885i		−	0.978394i	0.872173	−	0.489197i	$-0.162710\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−56.6133			−0.197259
$288$		0			0
$289$	−286.180			−0.990243
$290$		0			0
$291$		0			0
$292$		0			0
$293$	174.191			0.594510			0.297255	−	0.954798i	$-0.403929\pi$
$293$	174.191			0.594510			0.297255	+	0.954798i	$0.403929\pi$
$294$		0			0
$295$	438.879	+	49.2148i	1.48772	+	0.166830i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			464.767i			1.55441i
$300$		0			0
$301$	568.291			1.88801
$302$		0			0
$303$		0			0
$304$		0			0
$305$	36.5501	−	325.939i	0.119836	−	1.06865i
$306$		0			0
$307$			343.344i			1.11838i	0.829038	+	0.559192i	$0.188888\pi$
$307$			343.344i			1.11838i	−0.829038	+	0.559192i	$0.811112\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			230.352i			0.740681i	0.928896	+	0.370341i	$0.120759\pi$
$311$			230.352i			0.740681i	−0.928896	+	0.370341i	$0.879241\pi$
$312$		0			0
$313$			401.527i			1.28283i	0.767193	+	0.641416i	$0.221653\pi$
$313$			401.527i			1.28283i	−0.767193	+	0.641416i	$0.778347\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−416.574			−1.31411			−0.657057	−	0.753841i	$-0.728199\pi$
$317$	−416.574			−1.31411			−0.657057	+	0.753841i	$0.728199\pi$
$318$		0			0
$319$	−415.188			−1.30153
$320$		0			0
$321$		0			0
$322$		0			0
$323$	0.393545			0.00121841
$324$		0			0
$325$	129.560	−	570.420i	0.398647	−	1.75514i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	576.139i		−	1.75118i
$330$		0			0
$331$	201.348			0.608303			0.304151	−	0.952624i	$-0.401627\pi$
$331$	201.348			0.608303			0.304151	+	0.952624i	$0.401627\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	494.282	+	55.4277i	1.47547	+	0.165456i
$336$		0			0
$337$			44.8093i			0.132965i	0.997788	+	0.0664826i	$0.0211777\pi$
$337$			44.8093i			0.132965i	−0.997788	+	0.0664826i	$0.978822\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			232.347i			0.681371i
$342$		0			0
$343$			182.832i			0.533037i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	454.129			1.30873			0.654365	−	0.756179i	$-0.272936\pi$
$347$	454.129			1.30873			0.654365	+	0.756179i	$0.272936\pi$
$348$		0			0
$349$	−58.4464			−0.167468			−0.0837341	−	0.996488i	$-0.526685\pi$
$349$	−58.4464			−0.167468			−0.0837341	+	0.996488i	$0.526685\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	561.148			1.58965			0.794827	−	0.606836i	$-0.207562\pi$
$353$	561.148			1.58965			0.794827	+	0.606836i	$0.207562\pi$
$354$		0			0
$355$	183.172	+	20.5404i	0.515976	+	0.0578604i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			139.320i			0.388079i	0.980994	+	0.194040i	$0.0621590\pi$
$359$			139.320i			0.388079i	−0.980994	+	0.194040i	$0.937841\pi$
$360$		0			0
$361$	−360.945			−0.999848
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−392.637	−	44.0294i	−1.07572	−	0.120628i
$366$		0			0
$367$		−	340.199i		−	0.926973i	−0.886104	−	0.463486i	$-0.846598\pi$
$367$		−	340.199i		−	0.926973i	0.886104	−	0.463486i	$-0.153402\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			907.912i			2.44720i
$372$		0			0
$373$			387.588i			1.03911i	0.854437	+	0.519556i	$0.173902\pi$
$373$			387.588i			1.03911i	−0.854437	+	0.519556i	$0.826098\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	797.273			2.11478
$378$		0			0
$379$	−546.876			−1.44295			−0.721473	−	0.692443i	$-0.756535\pi$
$379$	−546.876			−1.44295			−0.721473	+	0.692443i	$0.756535\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	27.7134			0.0723588			0.0361794	−	0.999345i	$-0.488481\pi$
$383$	27.7134			0.0723588			0.0361794	+	0.999345i	$0.488481\pi$
$384$		0			0
$385$	72.8213	−	649.392i	0.189146	−	1.68673i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			483.655i			1.24333i	0.783284	+	0.621664i	$0.213543\pi$
$389$			483.655i			1.24333i	−0.783284	+	0.621664i	$0.786457\pi$
$390$		0			0
$391$	33.3560			0.0853095
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−19.7226	+	175.879i	−0.0499307	+	0.445262i
$396$		0			0
$397$		−	347.456i		−	0.875204i	−0.899169	−	0.437602i	$-0.855828\pi$
$397$		−	347.456i		−	0.875204i	0.899169	−	0.437602i	$-0.144172\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			277.268i			0.691442i	0.938337	+	0.345721i	$0.112366\pi$
$401$			277.268i			0.691442i	−0.938337	+	0.345721i	$0.887634\pi$
$402$		0			0
$403$		−	446.170i		−	1.10712i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−398.930			−0.980173
$408$		0			0
$409$	527.676			1.29016			0.645081	−	0.764115i	$-0.276824\pi$
$409$	527.676			1.29016			0.645081	+	0.764115i	$0.276824\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−947.378			−2.29389
$414$		0			0
$415$	15.5159	−	138.364i	0.0373876	−	0.333408i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	636.735i		−	1.51965i	−0.650125	−	0.759827i	$-0.725283\pi$
$419$		−	636.735i		−	1.51965i	0.650125	−	0.759827i	$-0.274717\pi$
$420$		0			0
$421$	30.6662			0.0728414			0.0364207	−	0.999337i	$-0.488404\pi$
$421$	30.6662			0.0728414			0.0364207	+	0.999337i	$0.488404\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−40.9386	−	9.29845i	−0.0963261	−	0.0218787i
$426$		0			0
$427$			703.584i			1.64774i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	381.687i		−	0.885584i	−0.896624	−	0.442792i	$-0.853988\pi$
$431$		−	381.687i		−	0.885584i	0.896624	−	0.442792i	$-0.146012\pi$
$432$		0			0
$433$			691.143i			1.59617i	0.602543	+	0.798086i	$0.294154\pi$
$433$			691.143i			1.59617i	−0.602543	+	0.798086i	$0.705846\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	4.65518			0.0106526
$438$		0			0
$439$	490.121			1.11645			0.558225	−	0.829690i	$-0.311483\pi$
$439$	490.121			1.11645			0.558225	+	0.829690i	$0.311483\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	575.877			1.29995			0.649974	−	0.759957i	$-0.274780\pi$
$443$	575.877			1.29995			0.649974	+	0.759957i	$0.274780\pi$
$444$		0			0
$445$	757.813	+	84.9794i	1.70295	+	0.190965i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			515.232i			1.14751i	0.819027	+	0.573755i	$0.194514\pi$
$449$			515.232i			1.14751i	−0.819027	+	0.573755i	$0.805486\pi$
$450$		0			0
$451$	−64.3129			−0.142601
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−139.837	+	1247.01i	−0.307333	+	2.74068i
$456$		0			0
$457$			96.8133i			0.211845i	0.994374	+	0.105923i	$0.0337796\pi$
$457$			96.8133i			0.211845i	−0.994374	+	0.105923i	$0.966220\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	622.795i		−	1.35097i	−0.737376	−	0.675483i	$-0.763935\pi$
$461$		−	622.795i		−	1.35097i	0.737376	−	0.675483i	$-0.236065\pi$
$462$		0			0
$463$			100.492i			0.217044i	0.994094	+	0.108522i	$0.0346119\pi$
$463$			100.492i			0.217044i	−0.994094	+	0.108522i	$0.965388\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	224.688			0.481131			0.240566	−	0.970633i	$-0.422667\pi$
$467$	224.688			0.481131			0.240566	+	0.970633i	$0.422667\pi$
$468$		0			0
$469$	−1066.97			−2.27500
$470$		0			0
$471$		0			0
$472$		0			0
$473$	645.581			1.36486
$474$		0			0
$475$	−5.71341	−	1.29770i	−0.0120282	−	0.00273199i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			699.123i			1.45955i	0.683689	+	0.729774i	$0.260375\pi$
$479$			699.123i			1.45955i	−0.683689	+	0.729774i	$0.739625\pi$
$480$		0			0
$481$	766.054			1.59263
$482$		0			0
$483$		0			0
$484$		0			0
$485$	482.432	+	54.0988i	0.994705	+	0.111544i
$486$		0			0
$487$			222.666i			0.457220i	0.973518	+	0.228610i	$0.0734181\pi$
$487$			222.666i			0.457220i	−0.973518	+	0.228610i	$0.926582\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			912.010i			1.85746i	0.370763	+	0.928728i	$0.379096\pi$
$491$			912.010i			1.85746i	−0.370763	+	0.928728i	$0.620904\pi$
$492$		0			0
$493$		−	57.2197i		−	0.116064i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−395.400			−0.795574
$498$		0			0
$499$	−785.750			−1.57465			−0.787324	−	0.616539i	$-0.788534\pi$
$499$	−785.750			−1.57465			−0.787324	+	0.616539i	$0.788534\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	288.196			0.572955			0.286477	−	0.958087i	$-0.407516\pi$
$503$	288.196			0.572955			0.286477	+	0.958087i	$0.407516\pi$
$504$		0			0
$505$	142.265	+	15.9532i	0.281712	+	0.0315905i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	476.690i		−	0.936522i	−0.883590	−	0.468261i	$-0.844881\pi$
$509$		−	476.690i		−	0.936522i	0.883590	−	0.468261i	$-0.155119\pi$
$510$		0			0
$511$	847.559			1.65863
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−215.552	−	24.1715i	−0.418548	−	0.0469350i
$516$		0			0
$517$		−	654.497i		−	1.26595i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			337.377i			0.647557i	0.946133	+	0.323778i	$0.104953\pi$
$521$			337.377i			0.647557i	−0.946133	+	0.323778i	$0.895047\pi$
$522$		0			0
$523$		−	820.192i		−	1.56824i	−0.620607	−	0.784122i	$-0.713114\pi$
$523$		−	820.192i		−	1.56824i	0.620607	−	0.784122i	$-0.286886\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−32.0213			−0.0607615
$528$		0			0
$529$	−134.437			−0.254134
$530$		0			0
$531$		0			0
$532$		0			0
$533$	123.498			0.231704
$534$		0			0
$535$	89.7139	−	800.034i	0.167690	−	1.49539i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			804.748i			1.49304i
$540$		0			0
$541$	−770.684			−1.42456			−0.712278	−	0.701898i	$-0.752336\pi$
$541$	−770.684			−1.42456			−0.712278	+	0.701898i	$0.752336\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−79.2923	+	707.098i	−0.145490	+	1.29743i
$546$		0			0
$547$			866.028i			1.58323i	0.611019	+	0.791616i	$0.290760\pi$
$547$			866.028i			1.58323i	−0.611019	+	0.791616i	$0.709240\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	7.98561i		−	0.0144929i
$552$		0			0
$553$		−	379.657i		−	0.686541i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−245.790			−0.441275			−0.220637	−	0.975356i	$-0.570814\pi$
$557$	−245.790			−0.441275			−0.220637	+	0.975356i	$0.570814\pi$
$558$		0			0
$559$	−1239.69			−2.21769
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−338.589			−0.601401			−0.300700	−	0.953719i	$-0.597220\pi$
$563$	−338.589			−0.601401			−0.300700	+	0.953719i	$0.597220\pi$
$564$		0			0
$565$	−46.8355	+	417.661i	−0.0828947	+	0.739223i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	306.670i		−	0.538963i	−0.963006	−	0.269481i	$-0.913148\pi$
$569$		−	306.670i		−	0.538963i	0.963006	−	0.269481i	$-0.0868523\pi$
$570$		0			0
$571$	18.1970			0.0318686			0.0159343	−	0.999873i	$-0.494928\pi$
$571$	18.1970			0.0318686			0.0159343	+	0.999873i	$0.494928\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−484.256	−	109.990i	−0.842185	−	0.191287i
$576$		0			0
$577$			766.613i			1.32862i	0.747458	+	0.664309i	$0.231274\pi$
$577$			766.613i			1.32862i	−0.747458	+	0.664309i	$0.768726\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			298.678i			0.514076i
$582$		0			0
$583$			1031.39i			1.76911i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	849.192			1.44666			0.723332	−	0.690500i	$-0.242609\pi$
$587$	849.192			1.44666			0.723332	+	0.690500i	$0.242609\pi$
$588$		0			0
$589$	−4.46891			−0.00758728
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−91.0522			−0.153545			−0.0767725	−	0.997049i	$-0.524462\pi$
$593$	−91.0522			−0.153545			−0.0767725	+	0.997049i	$0.524462\pi$
$594$		0			0
$595$	89.4968	+	10.0360i	0.150415	+	0.0168672i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	56.7169i		−	0.0946860i	−0.998879	−	0.0473430i	$-0.984925\pi$
$599$		−	56.7169i		−	0.0946860i	0.998879	−	0.0473430i	$-0.0150754\pi$
$600$		0			0
$601$	−214.647			−0.357150			−0.178575	−	0.983926i	$-0.557149\pi$
$601$	−214.647			−0.357150			−0.178575	+	0.983926i	$0.557149\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	15.3046	−	136.480i	0.0252968	−	0.225587i
$606$		0			0
$607$		−	271.542i		−	0.447351i	−0.974664	−	0.223675i	$-0.928194\pi$
$607$		−	271.542i		−	0.447351i	0.974664	−	0.223675i	$-0.0718055\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			1256.81i			2.05697i
$612$		0			0
$613$			576.529i			0.940504i	0.882532	+	0.470252i	$0.155837\pi$
$613$			576.529i			0.940504i	−0.882532	+	0.470252i	$0.844163\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−343.543			−0.556795			−0.278398	−	0.960466i	$-0.589803\pi$
$617$	−343.543			−0.556795			−0.278398	+	0.960466i	$0.589803\pi$
$618$		0			0
$619$	−393.346			−0.635454			−0.317727	−	0.948182i	$-0.602920\pi$
$619$	−393.346			−0.635454			−0.317727	+	0.948182i	$0.602920\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−1635.84			−2.62575
$624$		0			0
$625$	563.678	+	269.986i	0.901884	+	0.431978i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	54.9791i		−	0.0874072i
$630$		0			0
$631$	−224.300			−0.355467			−0.177734	−	0.984079i	$-0.556877\pi$
$631$	−224.300			−0.355467			−0.177734	+	0.984079i	$0.556877\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	937.379	+	105.116i	1.47619	+	0.165536i
$636$		0			0
$637$		−	1545.33i		−	2.42595i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			262.420i			0.409392i	0.978826	+	0.204696i	$0.0656205\pi$
$641$			262.420i			0.409392i	−0.978826	+	0.204696i	$0.934379\pi$
$642$		0			0
$643$		−	326.368i		−	0.507571i	−0.967261	−	0.253785i	$-0.918324\pi$
$643$		−	326.368i		−	0.507571i	0.967261	−	0.253785i	$-0.0816757\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−438.462			−0.677685			−0.338842	−	0.940843i	$-0.610035\pi$
$647$	−438.462			−0.677685			−0.338842	+	0.940843i	$0.610035\pi$
$648$		0			0
$649$	−1076.23			−1.65828
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−879.863			−1.34742			−0.673709	−	0.738997i	$-0.735300\pi$
$653$	−879.863			−1.34742			−0.673709	+	0.738997i	$0.735300\pi$
$654$		0			0
$655$	526.746	+	59.0681i	0.804193	+	0.0901803i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			1026.38i			1.55748i	0.627346	+	0.778741i	$0.284141\pi$
$659$			1026.38i			1.55748i	−0.627346	+	0.778741i	$0.715859\pi$
$660$		0			0
$661$	506.988			0.767002			0.383501	−	0.923540i	$-0.374718\pi$
$661$	506.988			0.767002			0.383501	+	0.923540i	$0.374718\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	12.4902	+	1.40063i	0.0187823	+	0.00210620i
$666$		0			0
$667$		−	676.843i		−	1.01476i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			799.274i			1.19117i
$672$		0			0
$673$		−	524.617i		−	0.779520i	−0.920917	−	0.389760i	$-0.872558\pi$
$673$		−	524.617i		−	0.779520i	0.920917	−	0.389760i	$-0.127442\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	44.2882			0.0654183			0.0327091	−	0.999465i	$-0.489587\pi$
$677$	44.2882			0.0654183			0.0327091	+	0.999465i	$0.489587\pi$
$678$		0			0
$679$	−1041.39			−1.53372
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1020.38			1.49396			0.746982	−	0.664845i	$-0.231502\pi$
$683$	1020.38			1.49396			0.746982	+	0.664845i	$0.231502\pi$
$684$		0			0
$685$	−142.716	+	1272.69i	−0.208345	+	1.85794i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	1980.55i		−	2.87453i
$690$		0			0
$691$	145.191			0.210117			0.105059	−	0.994466i	$-0.466497\pi$
$691$	145.191			0.210117			0.105059	+	0.994466i	$0.466497\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	101.236	−	902.784i	0.145663	−	1.29897i
$696$		0			0
$697$		−	8.86337i		−	0.0127165i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	1136.43i		−	1.62116i	−0.585629	−	0.810579i	$-0.699152\pi$
$701$		−	1136.43i		−	1.62116i	0.585629	−	0.810579i	$-0.300848\pi$
$702$		0			0
$703$		−	7.67292i		−	0.0109145i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−307.097			−0.434366
$708$		0			0
$709$	−805.357			−1.13591			−0.567953	−	0.823061i	$-0.692264\pi$
$709$	−805.357			−1.13591			−0.567953	+	0.823061i	$0.692264\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−378.775			−0.531241
$714$		0			0
$715$	−158.855	+	1416.61i	−0.222175	+	1.98127i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			1266.26i			1.76115i	0.473910	+	0.880573i	$0.342842\pi$
$719$			1266.26i			1.76115i	−0.473910	+	0.880573i	$0.657158\pi$
$720$		0			0
$721$	465.298			0.645351
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−188.679	+	830.705i	−0.260247	+	1.14580i
$726$		0			0
$727$			244.191i			0.335889i	0.985796	+	0.167944i	$0.0537129\pi$
$727$			244.191i			0.335889i	−0.985796	+	0.167944i	$0.946287\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			88.9716i			0.121712i
$732$		0			0
$733$			658.006i			0.897689i	0.893610	+	0.448844i	$0.148164\pi$
$733$			658.006i			0.897689i	−0.893610	+	0.448844i	$0.851836\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1212.09			−1.64462
$738$		0			0
$739$	178.113			0.241019			0.120509	−	0.992712i	$-0.461547\pi$
$739$	178.113			0.241019			0.120509	+	0.992712i	$0.461547\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	412.790			0.555572			0.277786	−	0.960643i	$-0.410399\pi$
$743$	412.790			0.555572			0.277786	+	0.960643i	$0.410399\pi$
$744$		0			0
$745$	1430.07	+	160.365i	1.91956	+	0.215255i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			1726.98i			2.30571i
$750$		0			0
$751$	399.613			0.532108			0.266054	−	0.963958i	$-0.414280\pi$
$751$	399.613			0.532108			0.266054	+	0.963958i	$0.414280\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	8.43642	−	75.2327i	0.0111741	−	0.0996460i
$756$		0			0
$757$		−	1032.23i		−	1.36358i	−0.731546	−	0.681792i	$-0.761201\pi$
$757$		−	1032.23i		−	1.36358i	0.731546	−	0.681792i	$-0.238799\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			286.616i			0.376630i	0.982109	+	0.188315i	$0.0603026\pi$
$761$			286.616i			0.376630i	−0.982109	+	0.188315i	$0.939697\pi$
$762$		0			0
$763$		−	1526.36i		−	2.00048i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	2066.64			2.69445
$768$		0			0
$769$	685.242			0.891082			0.445541	−	0.895262i	$-0.353011\pi$
$769$	685.242			0.891082			0.445541	+	0.895262i	$0.353011\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	1155.89			1.49533			0.747663	−	0.664079i	$-0.231176\pi$
$773$	1155.89			1.49533			0.747663	+	0.664079i	$0.231176\pi$
$774$		0			0
$775$	464.879	+	105.589i	0.599844	+	0.136243i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	1.23698i		−	0.00158790i
$780$		0			0
$781$	−449.176			−0.575129
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−447.876	−	50.2238i	−0.570543	−	0.0639794i
$786$		0			0
$787$			9.67882i			0.0122984i	0.999981	+	0.00614919i	$0.00195736\pi$
$787$			9.67882i			0.0122984i	−0.999981	+	0.00614919i	$0.998043\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	901.577i		−	1.13979i
$792$		0			0
$793$		−	1534.82i		−	1.93546i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1515.19			1.90112			0.950561	−	0.310539i	$-0.100510\pi$
$797$	1515.19			1.90112			0.950561	+	0.310539i	$0.100510\pi$
$798$		0			0
$799$	90.2004			0.112892
$800$		0			0
$801$		0			0
$802$		0			0
$803$	962.830			1.19904
$804$		0			0
$805$	1058.64	+	118.714i	1.31509	+	0.147471i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	628.159i		−	0.776464i	−0.921562	−	0.388232i	$-0.873086\pi$
$809$		−	628.159i		−	0.776464i	0.921562	−	0.388232i	$-0.126914\pi$
$810$		0			0
$811$	1027.82			1.26735			0.633675	−	0.773600i	$-0.281546\pi$
$811$	1027.82			1.26735			0.633675	+	0.773600i	$0.281546\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	652.677	+	73.1897i	0.800831	+	0.0898034i
$816$		0			0
$817$			12.4169i			0.0151982i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	1119.81i		−	1.36395i	−0.731374	−	0.681977i	$-0.761121\pi$
$821$		−	1119.81i		−	1.36395i	0.731374	−	0.681977i	$-0.238879\pi$
$822$		0			0
$823$		−	406.770i		−	0.494253i	−0.968983	−	0.247127i	$-0.920514\pi$
$823$		−	406.770i		−	0.494253i	0.968983	−	0.247127i	$-0.0794864\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	445.615			0.538833			0.269416	−	0.963024i	$-0.413169\pi$
$827$	445.615			0.538833			0.269416	+	0.963024i	$0.413169\pi$
$828$		0			0
$829$	763.809			0.921362			0.460681	−	0.887566i	$-0.347605\pi$
$829$	763.809			0.921362			0.460681	+	0.887566i	$0.347605\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−110.907			−0.133142
$834$		0			0
$835$	−86.8891	+	774.843i	−0.104059	+	0.927956i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	446.022i		−	0.531612i	−0.964027	−	0.265806i	$-0.914362\pi$
$839$		−	446.022i		−	0.531612i	0.964027	−	0.265806i	$-0.0856380\pi$
$840$		0			0
$841$	−320.073			−0.380586
$842$		0			0
$843$		0			0
$844$		0			0
$845$	210.878	−	1880.53i	0.249560	−	2.22548i
$846$		0			0
$847$			294.610i			0.347828i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	650.340i		−	0.764206i
$852$		0			0
$853$			1065.15i			1.24871i	0.781142	+	0.624354i	$0.214638\pi$
$853$			1065.15i			1.24871i	−0.781142	+	0.624354i	$0.785362\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	772.367			0.901245			0.450622	−	0.892715i	$-0.351202\pi$
$857$	772.367			0.901245			0.450622	+	0.892715i	$0.351202\pi$
$858$		0			0
$859$	68.1604			0.0793485			0.0396742	−	0.999213i	$-0.487368\pi$
$859$	68.1604			0.0793485			0.0396742	+	0.999213i	$0.487368\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	572.939			0.663892			0.331946	−	0.943298i	$-0.392295\pi$
$863$	572.939			0.663892			0.331946	+	0.943298i	$0.392295\pi$
$864$		0			0
$865$	74.8419	−	667.411i	0.0865224	−	0.771573i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	431.292i		−	0.496309i
$870$		0			0
$871$	2327.53			2.67225
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1266.20	−	440.812i	−1.44709	−	0.503785i
$876$		0			0
$877$		−	310.600i		−	0.354162i	−0.984196	−	0.177081i	$-0.943335\pi$
$877$		−	310.600i		−	0.354162i	0.984196	−	0.177081i	$-0.0566654\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			81.2933i			0.0922739i	0.998935	+	0.0461369i	$0.0146911\pi$
$881$			81.2933i			0.0922739i	−0.998935	+	0.0461369i	$0.985309\pi$
$882$		0			0
$883$			673.174i			0.762372i	0.924498	+	0.381186i	$0.124484\pi$
$883$			673.174i			0.762372i	−0.924498	+	0.381186i	$0.875516\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1743.23			−1.96531			−0.982654	−	0.185449i	$-0.940626\pi$
$887$	−1743.23			−1.96531			−0.982654	+	0.185449i	$0.940626\pi$
$888$		0			0
$889$	−2023.46			−2.27611
$890$		0			0
$891$		0			0
$892$		0			0
$893$	12.5884			0.0140968
$894$		0			0
$895$	−1003.71	−	112.553i	−1.12146	−	0.125758i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			649.759i			0.722758i
$900$		0			0
$901$	−142.143			−0.157761
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−3.49536	+	31.1703i	−0.00386228	+	0.0344423i
$906$		0			0
$907$		−	726.388i		−	0.800869i	−0.916325	−	0.400434i	$-0.868859\pi$
$907$		−	726.388i		−	0.800869i	0.916325	−	0.400434i	$-0.131141\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			1144.53i			1.25635i	0.778073	+	0.628174i	$0.216198\pi$
$911$			1144.53i			1.25635i	−0.778073	+	0.628174i	$0.783802\pi$
$912$		0			0
$913$			339.299i			0.371631i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1137.05			−1.23997
$918$		0			0
$919$	725.840			0.789815			0.394907	−	0.918721i	$-0.370777\pi$
$919$	725.840			0.789815			0.394907	+	0.918721i	$0.370777\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	862.539			0.934495
$924$		0			0
$925$	−181.291	+	798.177i	−0.195990	+	0.862894i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			720.396i			0.775453i	0.921774	+	0.387727i	$0.126740\pi$
$929$			720.396i			0.775453i	−0.921774	+	0.387727i	$0.873260\pi$
$930$		0			0
$931$	−15.4783			−0.0166255
$932$		0			0
$933$		0			0
$934$		0			0
$935$	101.669	+	11.4009i	0.108737	+	0.0121935i
$936$		0			0
$937$		−	821.308i		−	0.876529i	−0.898846	−	0.438265i	$-0.855593\pi$
$937$		−	821.308i		−	0.876529i	0.898846	−	0.438265i	$-0.144407\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	174.350i		−	0.185282i	−0.995700	−	0.0926410i	$-0.970469\pi$
$941$		−	174.350i		−	0.185282i	0.995700	−	0.0926410i	$-0.0295309\pi$
$942$		0			0
$943$		−	104.843i		−	0.111181i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1158.55			−1.22339			−0.611696	−	0.791093i	$-0.709513\pi$
$947$	−1158.55			−1.22339			−0.611696	+	0.791093i	$0.709513\pi$
$948$		0			0
$949$	−1848.89			−1.94825
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−734.811			−0.771051			−0.385525	−	0.922697i	$-0.625980\pi$
$953$	−734.811			−0.771051			−0.385525	+	0.922697i	$0.625980\pi$
$954$		0			0
$955$	1479.06	+	165.859i	1.54876	+	0.173674i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	2747.27i		−	2.86472i
$960$		0			0
$961$	−597.382			−0.621625
$962$		0			0
$963$		0			0
$964$		0			0
$965$	276.650	+	31.0229i	0.286684	+	0.0321480i
$966$		0			0
$967$		−	735.176i		−	0.760264i	−0.924932	−	0.380132i	$-0.875879\pi$
$967$		−	735.176i		−	0.760264i	0.924932	−	0.380132i	$-0.124121\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	630.994i		−	0.649839i	−0.945742	−	0.324920i	$-0.894663\pi$
$971$		−	630.994i		−	0.649839i	0.945742	−	0.324920i	$-0.105337\pi$
$972$		0			0
$973$			1948.78i			2.00286i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1127.99			−1.15454			−0.577271	−	0.816553i	$-0.695882\pi$
$977$	−1127.99			−1.15454			−0.577271	+	0.816553i	$0.695882\pi$
$978$		0			0
$979$	−1858.32			−1.89818
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−935.129			−0.951302			−0.475651	−	0.879634i	$-0.657787\pi$
$983$	−935.129			−0.951302			−0.475651	+	0.879634i	$0.657787\pi$
$984$		0			0
$985$	−12.7834	+	113.998i	−0.0129781	+	0.115734i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			1052.43i			1.06414i
$990$		0			0
$991$	839.459			0.847083			0.423541	−	0.905877i	$-0.360787\pi$
$991$	839.459			0.847083			0.423541	+	0.905877i	$0.360787\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−113.754	+	1014.41i	−0.114326	+	1.01951i
$996$		0			0
$997$			249.936i			0.250688i	0.992113	+	0.125344i	$0.0400035\pi$
$997$			249.936i			0.250688i	−0.992113	+	0.125344i	$0.959997\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.3.b.a.809.12	yes	24
3.2	odd	2	inner	1620.3.b.a.809.13	yes	24
5.4	even	2	inner	1620.3.b.a.809.14	yes	24
9.2	odd	6		1620.3.t.f.1349.19		48
9.4	even	3		1620.3.t.f.269.21		48
9.5	odd	6		1620.3.t.f.269.4		48
9.7	even	3		1620.3.t.f.1349.6		48
15.14	odd	2	inner	1620.3.b.a.809.11	✓	24
45.4	even	6		1620.3.t.f.269.19		48
45.14	odd	6		1620.3.t.f.269.6		48
45.29	odd	6		1620.3.t.f.1349.21		48
45.34	even	6		1620.3.t.f.1349.4		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1620.3.b.a.809.11	✓	24	15.14	odd	2	inner
1620.3.b.a.809.12	yes	24	1.1	even	1	trivial
1620.3.b.a.809.13	yes	24	3.2	odd	2	inner
1620.3.b.a.809.14	yes	24	5.4	even	2	inner
1620.3.t.f.269.4		48	9.5	odd	6
1620.3.t.f.269.6		48	45.14	odd	6
1620.3.t.f.269.19		48	45.4	even	6
1620.3.t.f.269.21		48	9.4	even	3
1620.3.t.f.1349.4		48	45.34	even	6
1620.3.t.f.1349.6		48	9.7	even	3
1620.3.t.f.1349.19		48	9.2	odd	6
1620.3.t.f.1349.21		48	45.29	odd	6

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.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.557196 + 4.96886i) q^{5} -10.7259i q^{7} +O(q^{10})\)	\(q+(-0.557196 + 4.96886i) q^{5} -10.7259i q^{7} -12.1847i q^{11} +23.3979i q^{13} +1.67925 q^{17} +0.234357 q^{19} +19.8636 q^{23} +(-24.3791 - 5.53725i) q^{25} -34.0745i q^{29} -19.0688 q^{31} +(53.2956 + 5.97645i) q^{35} -32.7402i q^{37} -5.27816i q^{41} +52.9829i q^{43} +53.7146 q^{47} -66.0457 q^{49} -84.6464 q^{53} +(60.5441 + 6.78927i) q^{55} -88.3259i q^{59} -65.5965 q^{61} +(-116.261 - 13.0372i) q^{65} -99.4761i q^{67} -36.8639i q^{71} +79.0196i q^{73} -130.692 q^{77} +35.3962 q^{79} -27.8463 q^{83} +(-0.935673 + 8.34396i) q^{85} -152.513i q^{89} +250.965 q^{91} +(-0.130583 + 1.16449i) q^{95} -97.0911i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q - 48 q^{25} - 288 q^{49} - 36 q^{55} + 120 q^{61} + 480 q^{79} - 24 q^{85} + 48 q^{91}+O(q^{100}) \) 24 * q - 48 * q^25 - 288 * q^49 - 36 * q^55 + 120 * q^61 + 480 * q^79 - 24 * q^85 + 48 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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