LMFDB - Embedded newform 1620.3.b.b.809.9

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:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			809.9
Character	$\chi$	$=$	1620.809
Dual form			1620.3.b.b.809.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.90866 - 4.62136i) q^{5} -0.764690i q^{7} +O(q^{10})$	$q+(-1.90866 - 4.62136i) q^{5} -0.764690i q^{7} +8.06808i q^{11} +2.16678i q^{13} -24.9128 q^{17} +20.7672 q^{19} +16.8403 q^{23} +(-17.7140 + 17.6412i) q^{25} +11.7377i q^{29} +0.669414 q^{31} +(-3.53391 + 1.45953i) q^{35} -48.2956i q^{37} +62.5657i q^{41} +71.3291i q^{43} +20.2838 q^{47} +48.4152 q^{49} -82.8011 q^{53} +(37.2856 - 15.3992i) q^{55} -5.39196i q^{59} +83.2195 q^{61} +(10.0135 - 4.13565i) q^{65} +10.6345i q^{67} -87.9667i q^{71} +15.5340i q^{73} +6.16958 q^{77} +33.7544 q^{79} +55.7163 q^{83} +(47.5502 + 115.131i) q^{85} +22.3583i q^{89} +1.65692 q^{91} +(-39.6375 - 95.9728i) q^{95} +68.9510i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q - 6 q^{25} - 60 q^{31} - 216 q^{49} + 42 q^{55} - 96 q^{61} - 228 q^{79} - 96 q^{85} - 84 q^{91}+O(q^{100}) $ 24 * q - 6 * q^25 - 60 * q^31 - 216 * q^49 + 42 * q^55 - 96 * q^61 - 228 * q^79 - 96 * q^85 - 84 * 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
$5$	−1.90866	−	4.62136i	−0.381732	−	0.924273i
$5$	−1.90866	−	4.62136i	−0.381732	−	0.924273i
$6$		0			0
$7$		−	0.764690i		−	0.109241i	−0.998507	−	0.0546207i	$-0.982605\pi$
$7$		−	0.764690i		−	0.109241i	0.998507	−	0.0546207i	$-0.0173950\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			8.06808i			0.733462i	0.930327	+	0.366731i	$0.119523\pi$
$11$			8.06808i			0.733462i	−0.930327	+	0.366731i	$0.880477\pi$
$12$		0			0
$13$			2.16678i			0.166676i	0.996521	+	0.0833378i	$0.0265581\pi$
$13$			2.16678i			0.166676i	−0.996521	+	0.0833378i	$0.973442\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−24.9128			−1.46546			−0.732731	−	0.680519i	$-0.761755\pi$
$17$	−24.9128			−1.46546			−0.732731	+	0.680519i	$0.761755\pi$
$18$		0			0
$19$	20.7672			1.09301			0.546505	−	0.837456i	$-0.315958\pi$
$19$	20.7672			1.09301			0.546505	+	0.837456i	$0.315958\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	16.8403			0.732186			0.366093	−	0.930578i	$-0.380695\pi$
$23$	16.8403			0.732186			0.366093	+	0.930578i	$0.380695\pi$
$24$		0			0
$25$	−17.7140	+	17.6412i	−0.708561	+	0.705649i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			11.7377i			0.404748i	0.979308	+	0.202374i	$0.0648657\pi$
$29$			11.7377i			0.404748i	−0.979308	+	0.202374i	$0.935134\pi$
$30$		0			0
$31$	0.669414			0.0215940			0.0107970	−	0.999942i	$-0.496563\pi$
$31$	0.669414			0.0215940			0.0107970	+	0.999942i	$0.496563\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−3.53391	+	1.45953i	−0.100969	+	0.0417010i
$36$		0			0
$37$		−	48.2956i		−	1.30529i	−0.757665	−	0.652644i	$-0.773660\pi$
$37$		−	48.2956i		−	1.30529i	0.757665	−	0.652644i	$-0.226340\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			62.5657i			1.52599i	0.646403	+	0.762997i	$0.276273\pi$
$41$			62.5657i			1.52599i	−0.646403	+	0.762997i	$0.723727\pi$
$42$		0			0
$43$			71.3291i			1.65882i	0.558644	+	0.829408i	$0.311322\pi$
$43$			71.3291i			1.65882i	−0.558644	+	0.829408i	$0.688678\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	20.2838			0.431570			0.215785	−	0.976441i	$-0.430769\pi$
$47$	20.2838			0.431570			0.215785	+	0.976441i	$0.430769\pi$
$48$		0			0
$49$	48.4152			0.988066
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−82.8011			−1.56228			−0.781142	−	0.624353i	$-0.785363\pi$
$53$	−82.8011			−1.56228			−0.781142	+	0.624353i	$0.785363\pi$
$54$		0			0
$55$	37.2856	−	15.3992i	0.677919	−	0.279986i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	5.39196i		−	0.0913892i	−0.998955	−	0.0456946i	$-0.985450\pi$
$59$		−	5.39196i		−	0.0913892i	0.998955	−	0.0456946i	$-0.0145501\pi$
$60$		0			0
$61$	83.2195			1.36425			0.682127	−	0.731234i	$-0.261055\pi$
$61$	83.2195			1.36425			0.682127	+	0.731234i	$0.261055\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	10.0135	−	4.13565i	0.154054	−	0.0636255i
$66$		0			0
$67$			10.6345i			0.158724i	0.996846	+	0.0793622i	$0.0252884\pi$
$67$			10.6345i			0.158724i	−0.996846	+	0.0793622i	$0.974712\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	87.9667i		−	1.23897i	−0.785010	−	0.619484i	$-0.787342\pi$
$71$		−	87.9667i		−	1.23897i	0.785010	−	0.619484i	$-0.212658\pi$
$72$		0			0
$73$			15.5340i			0.212794i	0.994324	+	0.106397i	$0.0339315\pi$
$73$			15.5340i			0.212794i	−0.994324	+	0.106397i	$0.966069\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	6.16958			0.0801245
$78$		0			0
$79$	33.7544			0.427271			0.213635	−	0.976913i	$-0.431470\pi$
$79$	33.7544			0.427271			0.213635	+	0.976913i	$0.431470\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	55.7163			0.671281			0.335640	−	0.941990i	$-0.391047\pi$
$83$	55.7163			0.671281			0.335640	+	0.941990i	$0.391047\pi$
$84$		0			0
$85$	47.5502	+	115.131i	0.559414	+	1.35449i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			22.3583i			0.251217i	0.992080	+	0.125608i	$0.0400883\pi$
$89$			22.3583i			0.251217i	−0.992080	+	0.125608i	$0.959912\pi$
$90$		0			0
$91$	1.65692			0.0182079
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−39.6375	−	95.9728i	−0.417237	−	1.01024i
$96$		0			0
$97$			68.9510i			0.710835i	0.934708	+	0.355418i	$0.115661\pi$
$97$			68.9510i			0.710835i	−0.934708	+	0.355418i	$0.884339\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	56.4228i		−	0.558642i	−0.960198	−	0.279321i	$-0.909891\pi$
$101$		−	56.4228i		−	0.558642i	0.960198	−	0.279321i	$-0.0901093\pi$
$102$		0			0
$103$		−	130.249i		−	1.26455i	−0.774744	−	0.632275i	$-0.782121\pi$
$103$		−	130.249i		−	1.26455i	0.774744	−	0.632275i	$-0.217879\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	109.062			1.01927			0.509635	−	0.860390i	$-0.329780\pi$
$107$	109.062			1.01927			0.509635	+	0.860390i	$0.329780\pi$
$108$		0			0
$109$	193.169			1.77219			0.886095	−	0.463503i	$-0.153408\pi$
$109$	193.169			1.77219			0.886095	+	0.463503i	$0.153408\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	182.208			1.61246			0.806230	−	0.591602i	$-0.201504\pi$
$113$	182.208			1.61246			0.806230	+	0.591602i	$0.201504\pi$
$114$		0			0
$115$	−32.1424	−	77.8251i	−0.279499	−	0.676740i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			19.0506i			0.160089i
$120$		0			0
$121$	55.9060			0.462033
$122$		0			0
$123$		0			0
$124$		0			0
$125$	115.337	+	48.1918i	0.922693	+	0.385535i
$126$		0			0
$127$			166.528i			1.31125i	0.755088	+	0.655623i	$0.227594\pi$
$127$			166.528i			1.31125i	−0.755088	+	0.655623i	$0.772406\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	73.2126i		−	0.558875i	−0.960164	−	0.279438i	$-0.909852\pi$
$131$		−	73.2126i		−	0.558875i	0.960164	−	0.279438i	$-0.0901480\pi$
$132$		0			0
$133$		−	15.8805i		−	0.119402i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−98.0784			−0.715901			−0.357950	−	0.933741i	$-0.616524\pi$
$137$	−98.0784			−0.715901			−0.357950	+	0.933741i	$0.616524\pi$
$138$		0			0
$139$	144.310			1.03820			0.519099	−	0.854714i	$-0.326267\pi$
$139$	144.310			1.03820			0.519099	+	0.854714i	$0.326267\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−17.4818			−0.122250
$144$		0			0
$145$	54.2442	−	22.4033i	0.374098	−	0.154505i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			7.25093i			0.0486640i	0.999704	+	0.0243320i	$0.00774587\pi$
$149$			7.25093i			0.0486640i	−0.999704	+	0.0243320i	$0.992254\pi$
$150$		0			0
$151$	187.581			1.24226			0.621130	−	0.783708i	$-0.286674\pi$
$151$	187.581			1.24226			0.621130	+	0.783708i	$0.286674\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1.27768	−	3.09361i	−0.00824313	−	0.0199588i
$156$		0			0
$157$			276.438i			1.76075i	0.474278	+	0.880375i	$0.342709\pi$
$157$			276.438i			1.76075i	−0.474278	+	0.880375i	$0.657291\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	12.8776i		−	0.0799851i
$162$		0			0
$163$			314.271i			1.92804i	0.265826	+	0.964021i	$0.414355\pi$
$163$			314.271i			1.92804i	−0.265826	+	0.964021i	$0.585645\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−269.113			−1.61146			−0.805728	−	0.592286i	$-0.798225\pi$
$167$	−269.113			−1.61146			−0.805728	+	0.592286i	$0.798225\pi$
$168$		0			0
$169$	164.305			0.972219
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−180.927			−1.04582			−0.522912	−	0.852387i	$-0.675154\pi$
$173$	−180.927			−1.04582			−0.522912	+	0.852387i	$0.675154\pi$
$174$		0			0
$175$	13.4901	+	13.5457i	0.0770862	+	0.0774042i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			234.728i			1.31133i	0.755053	+	0.655664i	$0.227611\pi$
$179$			234.728i			1.31133i	−0.755053	+	0.655664i	$0.772389\pi$
$180$		0			0
$181$	118.215			0.653119			0.326560	−	0.945177i	$-0.394111\pi$
$181$	118.215			0.653119			0.326560	+	0.945177i	$0.394111\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−223.192	+	92.1800i	−1.20644	+	0.498270i
$186$		0			0
$187$		−	200.999i		−	1.07486i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	77.5769i		−	0.406162i	−0.979162	−	0.203081i	$-0.934905\pi$
$191$		−	77.5769i		−	0.406162i	0.979162	−	0.203081i	$-0.0650955\pi$
$192$		0			0
$193$		−	194.161i		−	1.00602i	−0.864282	−	0.503008i	$-0.832227\pi$
$193$		−	194.161i		−	1.00602i	0.864282	−	0.503008i	$-0.167773\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	121.122			0.614835			0.307417	−	0.951575i	$-0.400535\pi$
$197$	121.122			0.614835			0.307417	+	0.951575i	$0.400535\pi$
$198$		0			0
$199$	−305.074			−1.53304			−0.766518	−	0.642223i	$-0.778012\pi$
$199$	−305.074			−1.53304			−0.766518	+	0.642223i	$0.778012\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	8.97569			0.0442152
$204$		0			0
$205$	289.139	−	119.417i	1.41043	−	0.582521i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			167.551i			0.801682i
$210$		0			0
$211$	198.226			0.939459			0.469730	−	0.882810i	$-0.344351\pi$
$211$	198.226			0.939459			0.469730	+	0.882810i	$0.344351\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	329.638	−	136.143i	1.53320	−	0.633223i
$216$		0			0
$217$		−	0.511894i		−	0.00235896i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	53.9807i		−	0.244257i
$222$		0			0
$223$			199.222i			0.893374i	0.894690	+	0.446687i	$0.147396\pi$
$223$			199.222i			0.893374i	−0.894690	+	0.446687i	$0.852604\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	126.435			0.556982			0.278491	−	0.960439i	$-0.410166\pi$
$227$	126.435			0.556982			0.278491	+	0.960439i	$0.410166\pi$
$228$		0			0
$229$	−138.672			−0.605555			−0.302777	−	0.953061i	$-0.597914\pi$
$229$	−138.672			−0.605555			−0.302777	+	0.953061i	$0.597914\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−306.314			−1.31465			−0.657326	−	0.753606i	$-0.728313\pi$
$233$	−306.314			−1.31465			−0.657326	+	0.753606i	$0.728313\pi$
$234$		0			0
$235$	−38.7149	−	93.7389i	−0.164744	−	0.398889i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			375.966i			1.57308i	0.617540	+	0.786539i	$0.288129\pi$
$239$			375.966i			1.57308i	−0.617540	+	0.786539i	$0.711871\pi$
$240$		0			0
$241$	−86.2884			−0.358043			−0.179022	−	0.983845i	$-0.557293\pi$
$241$	−86.2884			−0.358043			−0.179022	+	0.983845i	$0.557293\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−92.4083	−	223.745i	−0.377177	−	0.913243i
$246$		0			0
$247$			44.9980i			0.182178i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			280.285i			1.11667i	0.829615	+	0.558336i	$0.188560\pi$
$251$			280.285i			1.11667i	−0.829615	+	0.558336i	$0.811440\pi$
$252$		0			0
$253$			135.869i			0.537031i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	181.349			0.705640			0.352820	−	0.935691i	$-0.385223\pi$
$257$	181.349			0.705640			0.352820	+	0.935691i	$0.385223\pi$
$258$		0			0
$259$	−36.9312			−0.142591
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−14.2334			−0.0541193			−0.0270597	−	0.999634i	$-0.508614\pi$
$263$	−14.2334			−0.0541193			−0.0270597	+	0.999634i	$0.508614\pi$
$264$		0			0
$265$	158.039	+	382.654i	0.596374	+	1.44398i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	90.7714i		−	0.337440i	−0.985664	−	0.168720i	$-0.946037\pi$
$269$		−	90.7714i		−	0.337440i	0.985664	−	0.168720i	$-0.0539634\pi$
$270$		0			0
$271$	219.809			0.811105			0.405553	−	0.914072i	$-0.367079\pi$
$271$	219.809			0.811105			0.405553	+	0.914072i	$0.367079\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−142.331	−	142.918i	−0.517567	−	0.519703i
$276$		0			0
$277$			22.7044i			0.0819654i	0.999160	+	0.0409827i	$0.0130489\pi$
$277$			22.7044i			0.0819654i	−0.999160	+	0.0409827i	$0.986951\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			157.148i			0.559247i	0.960110	+	0.279623i	$0.0902096\pi$
$281$			157.148i			0.559247i	−0.960110	+	0.279623i	$0.909790\pi$
$282$		0			0
$283$		−	403.760i		−	1.42671i	−0.700801	−	0.713357i	$-0.747174\pi$
$283$		−	403.760i		−	1.42671i	0.700801	−	0.713357i	$-0.252826\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	47.8434			0.166702
$288$		0			0
$289$	331.650			1.14758
$290$		0			0
$291$		0			0
$292$		0			0
$293$	392.972			1.34120			0.670601	−	0.741818i	$-0.266036\pi$
$293$	392.972			1.34120			0.670601	+	0.741818i	$0.266036\pi$
$294$		0			0
$295$	−24.9182	+	10.2914i	−0.0844686	+	0.0348862i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			36.4892i			0.122038i
$300$		0			0
$301$	54.5446			0.181211
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−158.838	−	384.588i	−0.520780	−	1.26094i
$306$		0			0
$307$		−	349.575i		−	1.13868i	−0.822102	−	0.569340i	$-0.807199\pi$
$307$		−	349.575i		−	1.13868i	0.822102	−	0.569340i	$-0.192801\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	392.949i		−	1.26350i	−0.775172	−	0.631751i	$-0.782337\pi$
$311$		−	392.949i		−	1.26350i	0.775172	−	0.631751i	$-0.217663\pi$
$312$		0			0
$313$			380.055i			1.21423i	0.794613	+	0.607116i	$0.207674\pi$
$313$			380.055i			1.21423i	−0.794613	+	0.607116i	$0.792326\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−176.562			−0.556977			−0.278489	−	0.960440i	$-0.589833\pi$
$317$	−176.562			−0.556977			−0.278489	+	0.960440i	$0.589833\pi$
$318$		0			0
$319$	−94.7007			−0.296867
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−517.370			−1.60176
$324$		0			0
$325$	−38.2247	−	38.3825i	−0.117615	−	0.118100i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	15.5108i		−	0.0471454i
$330$		0			0
$331$	−651.103			−1.96708			−0.983540	−	0.180690i	$-0.942167\pi$
$331$	−651.103			−1.96708			−0.983540	+	0.180690i	$0.942167\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	49.1461	−	20.2977i	0.146705	−	0.0605902i
$336$		0			0
$337$		−	191.752i		−	0.568998i	−0.958676	−	0.284499i	$-0.908173\pi$
$337$		−	191.752i		−	0.568998i	0.958676	−	0.284499i	$-0.0918273\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			5.40089i			0.0158384i
$342$		0			0
$343$		−	74.4925i		−	0.217179i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−290.421			−0.836947			−0.418473	−	0.908229i	$-0.637435\pi$
$347$	−290.421			−0.836947			−0.418473	+	0.908229i	$0.637435\pi$
$348$		0			0
$349$	−52.1570			−0.149447			−0.0747235	−	0.997204i	$-0.523807\pi$
$349$	−52.1570			−0.149447			−0.0747235	+	0.997204i	$0.523807\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	587.530			1.66439			0.832195	−	0.554482i	$-0.187084\pi$
$353$	587.530			1.66439			0.832195	+	0.554482i	$0.187084\pi$
$354$		0			0
$355$	−406.526	+	167.899i	−1.14514	+	0.472954i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			112.104i			0.312268i	0.987736	+	0.156134i	$0.0499031\pi$
$359$			112.104i			0.312268i	−0.987736	+	0.156134i	$0.950097\pi$
$360$		0			0
$361$	70.2762			0.194671
$362$		0			0
$363$		0			0
$364$		0			0
$365$	71.7882	−	29.6491i	0.196680	−	0.0812304i
$366$		0			0
$367$		−	54.6520i		−	0.148915i	−0.997224	−	0.0744577i	$-0.976277\pi$
$367$		−	54.6520i		−	0.148915i	0.997224	−	0.0744577i	$-0.0237226\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			63.3171i			0.170666i
$372$		0			0
$373$			27.4898i			0.0736991i	0.999321	+	0.0368495i	$0.0117322\pi$
$373$			27.4898i			0.0736991i	−0.999321	+	0.0368495i	$0.988268\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−25.4330			−0.0674616
$378$		0			0
$379$	−396.721			−1.04676			−0.523379	−	0.852100i	$-0.675329\pi$
$379$	−396.721			−1.04676			−0.523379	+	0.852100i	$0.675329\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	652.030			1.70243			0.851214	−	0.524819i	$-0.175867\pi$
$383$	652.030			1.70243			0.851214	+	0.524819i	$0.175867\pi$
$384$		0			0
$385$	−11.7756	−	28.5119i	−0.0305861	−	0.0740569i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			537.093i			1.38070i	0.723475	+	0.690350i	$0.242544\pi$
$389$			537.093i			1.38070i	−0.723475	+	0.690350i	$0.757456\pi$
$390$		0			0
$391$	−419.539			−1.07299
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−64.4257	−	155.991i	−0.163103	−	0.394915i
$396$		0			0
$397$		−	428.508i		−	1.07937i	−0.841868	−	0.539683i	$-0.818544\pi$
$397$		−	428.508i		−	1.07937i	0.841868	−	0.539683i	$-0.181456\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			6.48702i			0.0161771i	0.999967	+	0.00808855i	$0.00257469\pi$
$401$			6.48702i			0.0161771i	−0.999967	+	0.00808855i	$0.997425\pi$
$402$		0			0
$403$			1.45048i			0.00359919i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	389.653			0.957379
$408$		0			0
$409$	134.748			0.329458			0.164729	−	0.986339i	$-0.447325\pi$
$409$	134.748			0.329458			0.164729	+	0.986339i	$0.447325\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−4.12318			−0.00998349
$414$		0			0
$415$	−106.344	−	257.485i	−0.256249	−	0.620447i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			511.367i			1.22045i	0.792230	+	0.610223i	$0.208920\pi$
$419$			511.367i			1.22045i	−0.792230	+	0.610223i	$0.791080\pi$
$420$		0			0
$421$	67.2579			0.159758			0.0798788	−	0.996805i	$-0.474547\pi$
$421$	67.2579			0.159758			0.0798788	+	0.996805i	$0.474547\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	441.307	−	439.493i	1.03837	−	1.03410i
$426$		0			0
$427$		−	63.6371i		−	0.149033i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	786.579i		−	1.82501i	−0.409066	−	0.912505i	$-0.634146\pi$
$431$		−	786.579i		−	1.82501i	0.409066	−	0.912505i	$-0.365854\pi$
$432$		0			0
$433$		−	279.636i		−	0.645810i	−0.946431	−	0.322905i	$-0.895341\pi$
$433$		−	279.636i		−	0.645810i	0.946431	−	0.322905i	$-0.104659\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	349.725			0.800287
$438$		0			0
$439$	−250.748			−0.571181			−0.285590	−	0.958352i	$-0.592190\pi$
$439$	−250.748			−0.571181			−0.285590	+	0.958352i	$0.592190\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−469.959			−1.06086			−0.530428	−	0.847730i	$-0.677969\pi$
$443$	−469.959			−1.06086			−0.530428	+	0.847730i	$0.677969\pi$
$444$		0			0
$445$	103.326	−	42.6744i	0.232193	−	0.0958975i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			471.154i			1.04934i	0.851306	+	0.524670i	$0.175811\pi$
$449$			471.154i			1.04934i	−0.851306	+	0.524670i	$0.824189\pi$
$450$		0			0
$451$	−504.786			−1.11926
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−3.16249	−	7.65722i	−0.00695054	−	0.0168291i
$456$		0			0
$457$			540.392i			1.18248i	0.806497	+	0.591239i	$0.201361\pi$
$457$			540.392i			1.18248i	−0.806497	+	0.591239i	$0.798639\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	879.590i		−	1.90800i	−0.299799	−	0.954002i	$-0.596920\pi$
$461$		−	879.590i		−	1.90800i	0.299799	−	0.954002i	$-0.403080\pi$
$462$		0			0
$463$		−	281.906i		−	0.608868i	−0.952533	−	0.304434i	$-0.901533\pi$
$463$		−	281.906i		−	0.608868i	0.952533	−	0.304434i	$-0.0984674\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−252.452			−0.540582			−0.270291	−	0.962779i	$-0.587120\pi$
$467$	−252.452			−0.540582			−0.270291	+	0.962779i	$0.587120\pi$
$468$		0			0
$469$	8.13212			0.0173393
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−575.489			−1.21668
$474$		0			0
$475$	−367.871	+	366.359i	−0.774464	+	0.771282i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			97.8182i			0.204213i	0.994773	+	0.102107i	$0.0325583\pi$
$479$			97.8182i			0.204213i	−0.994773	+	0.102107i	$0.967442\pi$
$480$		0			0
$481$	104.646			0.217560
$482$		0			0
$483$		0			0
$484$		0			0
$485$	318.648	−	131.604i	0.657006	−	0.271349i
$486$		0			0
$487$			235.388i			0.483344i	0.970358	+	0.241672i	$0.0776957\pi$
$487$			235.388i			0.483344i	−0.970358	+	0.241672i	$0.922304\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	36.6482i		−	0.0746400i	−0.999303	−	0.0373200i	$-0.988118\pi$
$491$		−	36.6482i		−	0.0746400i	0.999303	−	0.0373200i	$-0.0118821\pi$
$492$		0			0
$493$		−	292.419i		−	0.593143i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−67.2672			−0.135347
$498$		0			0
$499$	−524.574			−1.05125			−0.525625	−	0.850716i	$-0.676168\pi$
$499$	−524.574			−1.05125			−0.525625	+	0.850716i	$0.676168\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	37.5712			0.0746943			0.0373471	−	0.999302i	$-0.488109\pi$
$503$	37.5712			0.0746943			0.0373471	+	0.999302i	$0.488109\pi$
$504$		0			0
$505$	−260.750	+	107.692i	−0.516337	+	0.213252i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			624.654i			1.22722i	0.789610	+	0.613609i	$0.210283\pi$
$509$			624.654i			1.22722i	−0.789610	+	0.613609i	$0.789717\pi$
$510$		0			0
$511$	11.8787			0.0232459
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−601.926	+	248.600i	−1.16879	+	0.482719i
$516$		0			0
$517$			163.652i			0.316541i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			7.40805i			0.0142189i	0.999975	+	0.00710945i	$0.00226303\pi$
$521$			7.40805i			0.0142189i	−0.999975	+	0.00710945i	$0.997737\pi$
$522$		0			0
$523$		−	438.767i		−	0.838943i	−0.907768	−	0.419472i	$-0.862215\pi$
$523$		−	438.767i		−	0.838943i	0.907768	−	0.419472i	$-0.137785\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−16.6770			−0.0316452
$528$		0			0
$529$	−245.405			−0.463903
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−135.566			−0.254346
$534$		0			0
$535$	−208.162	−	504.015i	−0.389089	−	0.942085i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			390.618i			0.724709i
$540$		0			0
$541$	−566.949			−1.04797			−0.523983	−	0.851729i	$-0.675554\pi$
$541$	−566.949			−1.04797			−0.523983	+	0.851729i	$0.675554\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−368.694	−	892.704i	−0.676502	−	1.63799i
$546$		0			0
$547$		−	976.628i		−	1.78543i	−0.450626	−	0.892713i	$-0.648799\pi$
$547$		−	976.628i		−	1.78543i	0.450626	−	0.892713i	$-0.351201\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			243.759i			0.442394i
$552$		0			0
$553$		−	25.8116i		−	0.0466757i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	55.7939			0.100168			0.0500842	−	0.998745i	$-0.484051\pi$
$557$	55.7939			0.100168			0.0500842	+	0.998745i	$0.484051\pi$
$558$		0			0
$559$	−154.555			−0.276484
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−353.849			−0.628506			−0.314253	−	0.949339i	$-0.601754\pi$
$563$	−353.849			−0.628506			−0.314253	+	0.949339i	$0.601754\pi$
$564$		0			0
$565$	−347.773	−	842.049i	−0.615528	−	1.49035i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			484.145i			0.850870i	0.904989	+	0.425435i	$0.139879\pi$
$569$			484.145i			0.850870i	−0.904989	+	0.425435i	$0.860121\pi$
$570$		0			0
$571$	−939.774			−1.64584			−0.822920	−	0.568158i	$-0.807656\pi$
$571$	−939.774			−1.64584			−0.822920	+	0.568158i	$0.807656\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−298.309	+	297.083i	−0.518799	+	0.516667i
$576$		0			0
$577$			637.613i			1.10505i	0.833497	+	0.552524i	$0.186335\pi$
$577$			637.613i			1.10505i	−0.833497	+	0.552524i	$0.813665\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	42.6057i		−	0.0733317i
$582$		0			0
$583$		−	668.046i		−	1.14588i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−73.8273			−0.125771			−0.0628853	−	0.998021i	$-0.520030\pi$
$587$	−73.8273			−0.125771			−0.0628853	+	0.998021i	$0.520030\pi$
$588$		0			0
$589$	13.9019			0.0236025
$590$		0			0
$591$		0			0
$592$		0			0
$593$	804.091			1.35597			0.677986	−	0.735075i	$-0.262853\pi$
$593$	804.091			1.35597			0.677986	+	0.735075i	$0.262853\pi$
$594$		0			0
$595$	88.0398	−	36.3611i	0.147966	−	0.0611112i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	170.057i		−	0.283902i	−0.989874	−	0.141951i	$-0.954662\pi$
$599$		−	170.057i		−	0.283902i	0.989874	−	0.141951i	$-0.0453376\pi$
$600$		0			0
$601$	404.596			0.673204			0.336602	−	0.941647i	$-0.390722\pi$
$601$	404.596			0.673204			0.336602	+	0.941647i	$0.390722\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−106.706	−	258.362i	−0.176373	−	0.427045i
$606$		0			0
$607$			624.469i			1.02878i	0.857556	+	0.514390i	$0.171982\pi$
$607$			624.469i			1.02878i	−0.857556	+	0.514390i	$0.828018\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			43.9506i			0.0719323i
$612$		0			0
$613$			1080.05i			1.76191i	0.473199	+	0.880956i	$0.343099\pi$
$613$			1080.05i			1.76191i	−0.473199	+	0.880956i	$0.656901\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−296.107			−0.479914			−0.239957	−	0.970784i	$-0.577133\pi$
$617$	−296.107			−0.479914			−0.239957	+	0.970784i	$0.577133\pi$
$618$		0			0
$619$	−337.301			−0.544913			−0.272457	−	0.962168i	$-0.587836\pi$
$619$	−337.301			−0.544913			−0.272457	+	0.962168i	$0.587836\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	17.0972			0.0274433
$624$		0			0
$625$	2.57351	−	624.995i	0.00411762	−	0.999992i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			1203.18i			1.91285i
$630$		0			0
$631$	−449.564			−0.712463			−0.356232	−	0.934398i	$-0.615939\pi$
$631$	−449.564			−0.712463			−0.356232	+	0.934398i	$0.615939\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	769.588	−	317.846i	1.21195	−	0.500545i
$636$		0			0
$637$			104.905i			0.164687i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			379.891i			0.592654i	0.955086	+	0.296327i	$0.0957619\pi$
$641$			379.891i			0.592654i	−0.955086	+	0.296327i	$0.904238\pi$
$642$		0			0
$643$			428.717i			0.666744i	0.942795	+	0.333372i	$0.108187\pi$
$643$			428.717i			0.666744i	−0.942795	+	0.333372i	$0.891813\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	485.842			0.750916			0.375458	−	0.926840i	$-0.377486\pi$
$647$	485.842			0.750916			0.375458	+	0.926840i	$0.377486\pi$
$648$		0			0
$649$	43.5028			0.0670305
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−759.372			−1.16290			−0.581449	−	0.813583i	$-0.697514\pi$
$653$	−759.372			−1.16290			−0.581449	+	0.813583i	$0.697514\pi$
$654$		0			0
$655$	−338.342	+	139.738i	−0.516553	+	0.213341i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			1010.34i			1.53314i	0.642159	+	0.766572i	$0.278039\pi$
$659$			1010.34i			1.53314i	−0.642159	+	0.766572i	$0.721961\pi$
$660$		0			0
$661$	119.430			0.180681			0.0903404	−	0.995911i	$-0.471204\pi$
$661$	119.430			0.180681			0.0903404	+	0.995911i	$0.471204\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−73.3894	+	30.3104i	−0.110360	+	0.0455796i
$666$		0			0
$667$			197.666i			0.296351i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			671.422i			1.00063i
$672$		0			0
$673$		−	286.179i		−	0.425228i	−0.977136	−	0.212614i	$-0.931802\pi$
$673$		−	286.179i		−	0.425228i	0.977136	−	0.212614i	$-0.0681977\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−784.175			−1.15831			−0.579154	−	0.815218i	$-0.696617\pi$
$677$	−784.175			−1.15831			−0.579154	+	0.815218i	$0.696617\pi$
$678$		0			0
$679$	52.7261			0.0776526
$680$		0			0
$681$		0			0
$682$		0			0
$683$	829.221			1.21409			0.607043	−	0.794669i	$-0.292355\pi$
$683$	829.221			1.21409			0.607043	+	0.794669i	$0.292355\pi$
$684$		0			0
$685$	187.198	+	453.256i	0.273282	+	0.661688i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	179.412i		−	0.260395i
$690$		0			0
$691$	−170.062			−0.246110			−0.123055	−	0.992400i	$-0.539269\pi$
$691$	−170.062			−0.246110			−0.123055	+	0.992400i	$0.539269\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−275.438	−	666.907i	−0.396314	−	0.959579i
$696$		0			0
$697$		−	1558.69i		−	2.23628i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	780.669i		−	1.11365i	−0.830630	−	0.556825i	$-0.812019\pi$
$701$		−	780.669i		−	1.11365i	0.830630	−	0.556825i	$-0.187981\pi$
$702$		0			0
$703$		−	1002.96i		−	1.42669i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−43.1460			−0.0610268
$708$		0			0
$709$	484.792			0.683769			0.341884	−	0.939742i	$-0.388935\pi$
$709$	484.792			0.683769			0.341884	+	0.939742i	$0.388935\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	11.2731			0.0158108
$714$		0			0
$715$	33.3668	+	80.7897i	0.0466669	+	0.112993i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			772.608i			1.07456i	0.843404	+	0.537280i	$0.180548\pi$
$719$			772.608i			1.07456i	−0.843404	+	0.537280i	$0.819452\pi$
$720$		0			0
$721$	−99.5998			−0.138141
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−207.067	−	207.922i	−0.285610	−	0.286789i
$726$		0			0
$727$		−	342.384i		−	0.470955i	−0.971880	−	0.235477i	$-0.924335\pi$
$727$		−	342.384i		−	0.470955i	0.971880	−	0.235477i	$-0.0756654\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	1777.01i		−	2.43093i
$732$		0			0
$733$			1144.45i			1.56132i	0.624955	+	0.780661i	$0.285117\pi$
$733$			1144.45i			1.56132i	−0.624955	+	0.780661i	$0.714883\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−85.8003			−0.116418
$738$		0			0
$739$	−310.428			−0.420065			−0.210032	−	0.977694i	$-0.567357\pi$
$739$	−310.428			−0.420065			−0.210032	+	0.977694i	$0.567357\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−81.6372			−0.109875			−0.0549375	−	0.998490i	$-0.517496\pi$
$743$	−81.6372			−0.109875			−0.0549375	+	0.998490i	$0.517496\pi$
$744$		0			0
$745$	33.5092	−	13.8396i	0.0449788	−	0.0185766i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	83.3986i		−	0.111347i
$750$		0			0
$751$	−321.625			−0.428263			−0.214131	−	0.976805i	$-0.568692\pi$
$751$	−321.625			−0.428263			−0.214131	+	0.976805i	$0.568692\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−358.029	−	866.881i	−0.474210	−	1.14819i
$756$		0			0
$757$		−	215.279i		−	0.284384i	−0.989839	−	0.142192i	$-0.954585\pi$
$757$		−	215.279i		−	0.284384i	0.989839	−	0.142192i	$-0.0454151\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	148.015i		−	0.194500i	−0.995260	−	0.0972502i	$-0.968995\pi$
$761$		−	148.015i		−	0.194500i	0.995260	−	0.0972502i	$-0.0310047\pi$
$762$		0			0
$763$		−	147.714i		−	0.193597i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	11.6832			0.0152324
$768$		0			0
$769$	542.400			0.705332			0.352666	−	0.935749i	$-0.385275\pi$
$769$	542.400			0.705332			0.352666	+	0.935749i	$0.385275\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	441.852			0.571607			0.285803	−	0.958288i	$-0.407740\pi$
$773$	441.852			0.571607			0.285803	+	0.958288i	$0.407740\pi$
$774$		0			0
$775$	−11.8580	+	11.8093i	−0.0153007	+	0.0152378i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			1299.31i			1.66793i
$780$		0			0
$781$	709.723			0.908736
$782$		0			0
$783$		0			0
$784$		0			0
$785$	1277.52	−	527.626i	1.62741	−	0.672135i
$786$		0			0
$787$		−	479.118i		−	0.608790i	−0.952546	−	0.304395i	$-0.901546\pi$
$787$		−	479.118i		−	0.608790i	0.952546	−	0.304395i	$-0.0984543\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	139.333i		−	0.176147i
$792$		0			0
$793$			180.319i			0.227388i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	579.567			0.727186			0.363593	−	0.931558i	$-0.381550\pi$
$797$	579.567			0.727186			0.363593	+	0.931558i	$0.381550\pi$
$798$		0			0
$799$	−505.328			−0.632450
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−125.329			−0.156076
$804$		0			0
$805$	−59.5121	+	24.5790i	−0.0739280	+	0.0305329i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	906.264i		−	1.12023i	−0.828416	−	0.560114i	$-0.810757\pi$
$809$		−	906.264i		−	1.12023i	0.828416	−	0.560114i	$-0.189243\pi$
$810$		0			0
$811$	−593.696			−0.732055			−0.366027	−	0.930604i	$-0.619282\pi$
$811$	−593.696			−0.732055			−0.366027	+	0.930604i	$0.619282\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1452.36	−	599.836i	1.78204	−	0.735996i
$816$		0			0
$817$			1481.30i			1.81310i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			1216.40i			1.48161i	0.671721	+	0.740804i	$0.265555\pi$
$821$			1216.40i			1.48161i	−0.671721	+	0.740804i	$0.734445\pi$
$822$		0			0
$823$			1159.31i			1.40864i	0.709883	+	0.704320i	$0.248748\pi$
$823$			1159.31i			1.40864i	−0.709883	+	0.704320i	$0.751252\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−676.065			−0.817491			−0.408746	−	0.912648i	$-0.634034\pi$
$827$	−676.065			−0.817491			−0.408746	+	0.912648i	$0.634034\pi$
$828$		0			0
$829$	−528.868			−0.637959			−0.318979	−	0.947762i	$-0.603340\pi$
$829$	−528.868			−0.637959			−0.318979	+	0.947762i	$0.603340\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1206.16			−1.44797
$834$		0			0
$835$	513.646	+	1243.67i	0.615144	+	1.48942i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	523.108i		−	0.623490i	−0.950166	−	0.311745i	$-0.899087\pi$
$839$		−	523.108i		−	0.623490i	0.950166	−	0.311745i	$-0.100913\pi$
$840$		0			0
$841$	703.227			0.836179
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−313.603	−	759.314i	−0.371127	−	0.898596i
$846$		0			0
$847$		−	42.7508i		−	0.0504732i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	813.312i		−	0.955713i
$852$		0			0
$853$		−	1045.25i		−	1.22539i	−0.790321	−	0.612693i	$-0.790086\pi$
$853$		−	1045.25i		−	1.22539i	0.790321	−	0.612693i	$-0.209914\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	707.021			0.824995			0.412497	−	0.910959i	$-0.364657\pi$
$857$	707.021			0.824995			0.412497	+	0.910959i	$0.364657\pi$
$858$		0			0
$859$	−1322.89			−1.54004			−0.770020	−	0.638020i	$-0.779754\pi$
$859$	−1322.89			−1.54004			−0.770020	+	0.638020i	$0.779754\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	559.626			0.648466			0.324233	−	0.945977i	$-0.394894\pi$
$863$	559.626			0.648466			0.324233	+	0.945977i	$0.394894\pi$
$864$		0			0
$865$	345.329	+	836.132i	0.399224	+	0.966626i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			272.333i			0.313387i
$870$		0			0
$871$	−23.0427			−0.0264555
$872$		0			0
$873$		0			0
$874$		0			0
$875$	36.8518	−	88.1968i	0.0421164	−	0.100796i
$876$		0			0
$877$		−	493.866i		−	0.563131i	−0.959542	−	0.281565i	$-0.909146\pi$
$877$		−	493.866i		−	0.563131i	0.959542	−	0.281565i	$-0.0908536\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			765.716i			0.869145i	0.900637	+	0.434572i	$0.143100\pi$
$881$			765.716i			0.869145i	−0.900637	+	0.434572i	$0.856900\pi$
$882$		0			0
$883$		−	1647.52i		−	1.86582i	−0.360115	−	0.932908i	$-0.617263\pi$
$883$		−	1647.52i		−	1.86582i	0.360115	−	0.932908i	$-0.382737\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	306.218			0.345229			0.172615	−	0.984989i	$-0.444778\pi$
$887$	306.218			0.345229			0.172615	+	0.984989i	$0.444778\pi$
$888$		0			0
$889$	127.343			0.143242
$890$		0			0
$891$		0			0
$892$		0			0
$893$	421.238			0.471711
$894$		0			0
$895$	1084.76	−	448.016i	1.21203	−	0.500576i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			7.85738i			0.00874013i
$900$		0			0
$901$	2062.81			2.28947
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−225.632	−	546.313i	−0.249317	−	0.603661i
$906$		0			0
$907$		−	572.530i		−	0.631235i	−0.948887	−	0.315617i	$-0.897788\pi$
$907$		−	572.530i		−	0.631235i	0.948887	−	0.315617i	$-0.102212\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	257.597i		−	0.282763i	−0.989955	−	0.141382i	$-0.954846\pi$
$911$		−	257.597i		−	0.282763i	0.989955	−	0.141382i	$-0.0451544\pi$
$912$		0			0
$913$			449.524i			0.492359i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−55.9850			−0.0610523
$918$		0			0
$919$	935.435			1.01788			0.508942	−	0.860801i	$-0.330037\pi$
$919$	935.435			1.01788			0.508942	+	0.860801i	$0.330037\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	190.605			0.206506
$924$		0			0
$925$	851.995	+	855.510i	0.921075	+	0.924876i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			1202.59i			1.29450i	0.762277	+	0.647251i	$0.224081\pi$
$929$			1202.59i			1.29450i	−0.762277	+	0.647251i	$0.775919\pi$
$930$		0			0
$931$	1005.45			1.07997
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−928.890	+	383.639i	−0.993465	+	0.410309i
$936$		0			0
$937$		−	627.406i		−	0.669590i	−0.942291	−	0.334795i	$-0.891333\pi$
$937$		−	627.406i		−	0.669590i	0.942291	−	0.334795i	$-0.108667\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			76.0011i			0.0807663i	0.999184	+	0.0403831i	$0.0128579\pi$
$941$			76.0011i			0.0807663i	−0.999184	+	0.0403831i	$0.987142\pi$
$942$		0			0
$943$			1053.62i			1.11731i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1507.77			1.59215			0.796077	−	0.605195i	$-0.206905\pi$
$947$	1507.77			1.59215			0.796077	+	0.605195i	$0.206905\pi$
$948$		0			0
$949$	−33.6588			−0.0354676
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−355.068			−0.372580			−0.186290	−	0.982495i	$-0.559646\pi$
$953$	−355.068			−0.372580			−0.186290	+	0.982495i	$0.559646\pi$
$954$		0			0
$955$	−358.511	+	148.068i	−0.375405	+	0.155045i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			74.9996i			0.0782060i
$960$		0			0
$961$	−960.552			−0.999534
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−897.290	+	370.588i	−0.929834	+	0.384029i
$966$		0			0
$967$			1182.52i			1.22288i	0.791292	+	0.611439i	$0.209409\pi$
$967$			1182.52i			1.22288i	−0.791292	+	0.611439i	$0.790591\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	1230.36i		−	1.26711i	−0.773698	−	0.633555i	$-0.781595\pi$
$971$		−	1230.36i		−	1.26711i	0.773698	−	0.633555i	$-0.218405\pi$
$972$		0			0
$973$		−	110.352i		−	0.113414i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1383.47			1.41604			0.708018	−	0.706194i	$-0.249590\pi$
$977$	1383.47			1.41604			0.708018	+	0.706194i	$0.249590\pi$
$978$		0			0
$979$	−180.389			−0.184258
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−32.6140			−0.0331780			−0.0165890	−	0.999862i	$-0.505281\pi$
$983$	−32.6140			−0.0331780			−0.0165890	+	0.999862i	$0.505281\pi$
$984$		0			0
$985$	−231.182	−	559.751i	−0.234702	−	0.568275i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			1201.20i			1.21456i
$990$		0			0
$991$	1536.43			1.55039			0.775193	−	0.631724i	$-0.217653\pi$
$991$	1536.43			1.55039			0.775193	+	0.631724i	$0.217653\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	582.283	+	1409.86i	0.585209	+	1.41694i
$996$		0			0
$997$			552.983i			0.554647i	0.960777	+	0.277324i	$0.0894474\pi$
$997$			552.983i			0.554647i	−0.960777	+	0.277324i	$0.910553\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.3.b.b.809.9		24
3.2	odd	2	inner	1620.3.b.b.809.16		24
5.4	even	2	inner	1620.3.b.b.809.15		24
9.2	odd	6		540.3.t.a.449.1		24
9.4	even	3		540.3.t.a.89.4		24
9.5	odd	6		180.3.t.a.29.3	✓	24
9.7	even	3		180.3.t.a.149.10	yes	24
15.14	odd	2	inner	1620.3.b.b.809.10		24
45.2	even	12		2700.3.p.f.2501.5		24
45.4	even	6		540.3.t.a.89.1		24
45.7	odd	12		900.3.p.f.401.9		24
45.13	odd	12		2700.3.p.f.1601.8		24
45.14	odd	6		180.3.t.a.29.10	yes	24
45.22	odd	12		2700.3.p.f.1601.5		24
45.23	even	12		900.3.p.f.101.4		24
45.29	odd	6		540.3.t.a.449.4		24
45.32	even	12		900.3.p.f.101.9		24
45.34	even	6		180.3.t.a.149.3	yes	24
45.38	even	12		2700.3.p.f.2501.8		24
45.43	odd	12		900.3.p.f.401.4		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.3.t.a.29.3	✓	24	9.5	odd	6
180.3.t.a.29.10	yes	24	45.14	odd	6
180.3.t.a.149.3	yes	24	45.34	even	6
180.3.t.a.149.10	yes	24	9.7	even	3
540.3.t.a.89.1		24	45.4	even	6
540.3.t.a.89.4		24	9.4	even	3
540.3.t.a.449.1		24	9.2	odd	6
540.3.t.a.449.4		24	45.29	odd	6
900.3.p.f.101.4		24	45.23	even	12
900.3.p.f.101.9		24	45.32	even	12
900.3.p.f.401.4		24	45.43	odd	12
900.3.p.f.401.9		24	45.7	odd	12
1620.3.b.b.809.9		24	1.1	even	1	trivial
1620.3.b.b.809.10		24	15.14	odd	2	inner
1620.3.b.b.809.15		24	5.4	even	2	inner
1620.3.b.b.809.16		24	3.2	odd	2	inner
2700.3.p.f.1601.5		24	45.22	odd	12
2700.3.p.f.1601.8		24	45.13	odd	12
2700.3.p.f.2501.5		24	45.2	even	12
2700.3.p.f.2501.8		24	45.38	even	12

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)

\(f(q)\)	\(=\)	\(q+(-1.90866 - 4.62136i) q^{5} -0.764690i q^{7} +O(q^{10})\)	\(q+(-1.90866 - 4.62136i) q^{5} -0.764690i q^{7} +8.06808i q^{11} +2.16678i q^{13} -24.9128 q^{17} +20.7672 q^{19} +16.8403 q^{23} +(-17.7140 + 17.6412i) q^{25} +11.7377i q^{29} +0.669414 q^{31} +(-3.53391 + 1.45953i) q^{35} -48.2956i q^{37} +62.5657i q^{41} +71.3291i q^{43} +20.2838 q^{47} +48.4152 q^{49} -82.8011 q^{53} +(37.2856 - 15.3992i) q^{55} -5.39196i q^{59} +83.2195 q^{61} +(10.0135 - 4.13565i) q^{65} +10.6345i q^{67} -87.9667i q^{71} +15.5340i q^{73} +6.16958 q^{77} +33.7544 q^{79} +55.7163 q^{83} +(47.5502 + 115.131i) q^{85} +22.3583i q^{89} +1.65692 q^{91} +(-39.6375 - 95.9728i) q^{95} +68.9510i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q - 6 q^{25} - 60 q^{31} - 216 q^{49} + 42 q^{55} - 96 q^{61} - 228 q^{79} - 96 q^{85} - 84 q^{91}+O(q^{100}) \) 24 * q - 6 * q^25 - 60 * q^31 - 216 * q^49 + 42 * q^55 - 96 * q^61 - 228 * q^79 - 96 * q^85 - 84 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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