LMFDB - Embedded newform 1620.3.b.b.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:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.21895 + 4.84914i) q^{5} -11.5030i q^{7} +O(q^{10})$	$q+(-1.21895 + 4.84914i) q^{5} -11.5030i q^{7} -15.2470i q^{11} -17.0417i q^{13} +10.4260 q^{17} -23.1507 q^{19} -13.9612 q^{23} +(-22.0283 - 11.8217i) q^{25} +44.8342i q^{29} +5.44783 q^{31} +(55.7795 + 14.0215i) q^{35} +50.7314i q^{37} +42.6658i q^{41} -6.14899i q^{43} -32.0932 q^{47} -83.3183 q^{49} +16.7365 q^{53} +(73.9350 + 18.5854i) q^{55} -19.5475i q^{59} +6.59937 q^{61} +(82.6376 + 20.7730i) q^{65} +35.5626i q^{67} -65.5016i q^{71} +104.118i q^{73} -175.386 q^{77} -150.526 q^{79} -81.4589 q^{83} +(-12.7088 + 50.5571i) q^{85} +14.3858i q^{89} -196.030 q^{91} +(28.2195 - 112.261i) q^{95} +26.8666i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−1.21895	+	4.84914i	−0.243790	+	0.969828i
$5$	−1.21895	+	4.84914i	−0.243790	+	0.969828i
$6$		0			0
$7$		−	11.5030i		−	1.64328i	−0.570006	−	0.821641i	$-0.693059\pi$
$7$		−	11.5030i		−	1.64328i	0.570006	−	0.821641i	$-0.306941\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	15.2470i		−	1.38609i	−0.720893	−	0.693047i	$-0.756268\pi$
$11$		−	15.2470i		−	1.38609i	0.720893	−	0.693047i	$-0.243732\pi$
$12$		0			0
$13$		−	17.0417i		−	1.31090i	−0.755239	−	0.655450i	$-0.772479\pi$
$13$		−	17.0417i		−	1.31090i	0.755239	−	0.655450i	$-0.227521\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	10.4260			0.613294			0.306647	−	0.951823i	$-0.400793\pi$
$17$	10.4260			0.613294			0.306647	+	0.951823i	$0.400793\pi$
$18$		0			0
$19$	−23.1507			−1.21846			−0.609228	−	0.792995i	$-0.708521\pi$
$19$	−23.1507			−1.21846			−0.609228	+	0.792995i	$0.708521\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−13.9612			−0.607009			−0.303504	−	0.952830i	$-0.598157\pi$
$23$	−13.9612			−0.607009			−0.303504	+	0.952830i	$0.598157\pi$
$24$		0			0
$25$	−22.0283	−	11.8217i	−0.881133	−	0.472869i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			44.8342i			1.54601i	0.634401	+	0.773004i	$0.281247\pi$
$29$			44.8342i			1.54601i	−0.634401	+	0.773004i	$0.718753\pi$
$30$		0			0
$31$	5.44783			0.175736			0.0878682	−	0.996132i	$-0.471995\pi$
$31$	5.44783			0.175736			0.0878682	+	0.996132i	$0.471995\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	55.7795	+	14.0215i	1.59370	+	0.400615i
$36$		0			0
$37$			50.7314i			1.37112i	0.728017	+	0.685559i	$0.240442\pi$
$37$			50.7314i			1.37112i	−0.728017	+	0.685559i	$0.759558\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			42.6658i			1.04063i	0.853975	+	0.520315i	$0.174185\pi$
$41$			42.6658i			1.04063i	−0.853975	+	0.520315i	$0.825815\pi$
$42$		0			0
$43$		−	6.14899i		−	0.143000i	−0.997441	−	0.0714998i	$-0.977221\pi$
$43$		−	6.14899i		−	0.143000i	0.997441	−	0.0714998i	$-0.0227785\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−32.0932			−0.682833			−0.341417	−	0.939912i	$-0.610907\pi$
$47$	−32.0932			−0.682833			−0.341417	+	0.939912i	$0.610907\pi$
$48$		0			0
$49$	−83.3183			−1.70037
$50$		0			0
$51$		0			0
$52$		0			0
$53$	16.7365			0.315783			0.157892	−	0.987456i	$-0.449530\pi$
$53$	16.7365			0.315783			0.157892	+	0.987456i	$0.449530\pi$
$54$		0			0
$55$	73.9350	+	18.5854i	1.34427	+	0.337916i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	19.5475i		−	0.331313i	−0.986183	−	0.165657i	$-0.947026\pi$
$59$		−	19.5475i		−	0.331313i	0.986183	−	0.165657i	$-0.0529743\pi$
$60$		0			0
$61$	6.59937			0.108186			0.0540932	−	0.998536i	$-0.482773\pi$
$61$	6.59937			0.108186			0.0540932	+	0.998536i	$0.482773\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	82.6376	+	20.7730i	1.27135	+	0.319584i
$66$		0			0
$67$			35.5626i			0.530785i	0.964140	+	0.265392i	$0.0855015\pi$
$67$			35.5626i			0.530785i	−0.964140	+	0.265392i	$0.914498\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	65.5016i		−	0.922557i	−0.887255	−	0.461279i	$-0.847391\pi$
$71$		−	65.5016i		−	0.922557i	0.887255	−	0.461279i	$-0.152609\pi$
$72$		0			0
$73$			104.118i			1.42628i	0.701023	+	0.713139i	$0.252727\pi$
$73$			104.118i			1.42628i	−0.701023	+	0.713139i	$0.747273\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−175.386			−2.27774
$78$		0			0
$79$	−150.526			−1.90539			−0.952694	−	0.303933i	$-0.901700\pi$
$79$	−150.526			−1.90539			−0.952694	+	0.303933i	$0.901700\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−81.4589			−0.981433			−0.490716	−	0.871319i	$-0.663265\pi$
$83$	−81.4589			−0.981433			−0.490716	+	0.871319i	$0.663265\pi$
$84$		0			0
$85$	−12.7088	+	50.5571i	−0.149515	+	0.594790i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			14.3858i			0.161638i	0.996729	+	0.0808189i	$0.0257536\pi$
$89$			14.3858i			0.161638i	−0.996729	+	0.0808189i	$0.974246\pi$
$90$		0			0
$91$	−196.030			−2.15418
$92$		0			0
$93$		0			0
$94$		0			0
$95$	28.2195	−	112.261i	0.297048	−	1.18169i
$96$		0			0
$97$			26.8666i			0.276975i	0.990364	+	0.138488i	$0.0442241\pi$
$97$			26.8666i			0.276975i	−0.990364	+	0.138488i	$0.955776\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	95.4997i		−	0.945542i	−0.881185	−	0.472771i	$-0.843254\pi$
$101$		−	95.4997i		−	0.945542i	0.881185	−	0.472771i	$-0.156746\pi$
$102$		0			0
$103$		−	57.7844i		−	0.561014i	−0.959852	−	0.280507i	$-0.909497\pi$
$103$		−	57.7844i		−	0.561014i	0.959852	−	0.280507i	$-0.0905025\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−64.8388			−0.605970			−0.302985	−	0.952995i	$-0.597983\pi$
$107$	−64.8388			−0.605970			−0.302985	+	0.952995i	$0.597983\pi$
$108$		0			0
$109$	−40.9173			−0.375388			−0.187694	−	0.982228i	$-0.560101\pi$
$109$	−40.9173			−0.375388			−0.187694	+	0.982228i	$0.560101\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	216.442			1.91542			0.957709	−	0.287737i	$-0.0929030\pi$
$113$	216.442			1.91542			0.957709	+	0.287737i	$0.0929030\pi$
$114$		0			0
$115$	17.0180	−	67.6998i	0.147983	−	0.588694i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	119.930i		−	1.00781i
$120$		0			0
$121$	−111.472			−0.921254
$122$		0			0
$123$		0			0
$124$		0			0
$125$	84.1766	−	92.4083i	0.673413	−	0.739267i
$126$		0			0
$127$		−	128.534i		−	1.01208i	−0.862510	−	0.506041i	$-0.831109\pi$
$127$		−	128.534i		−	1.01208i	0.862510	−	0.506041i	$-0.168891\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			26.7783i			0.204415i	0.994763	+	0.102207i	$0.0325905\pi$
$131$			26.7783i			0.204415i	−0.994763	+	0.102207i	$0.967409\pi$
$132$		0			0
$133$			266.301i			2.00227i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−99.8473			−0.728812			−0.364406	−	0.931240i	$-0.618728\pi$
$137$	−99.8473			−0.728812			−0.364406	+	0.931240i	$0.618728\pi$
$138$		0			0
$139$	−15.3879			−0.110704			−0.0553522	−	0.998467i	$-0.517628\pi$
$139$	−15.3879			−0.110704			−0.0553522	+	0.998467i	$0.517628\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−259.835			−1.81703
$144$		0			0
$145$	−217.407	−	54.6507i	−1.49936	−	0.376901i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			120.257i			0.807097i	0.914958	+	0.403548i	$0.132223\pi$
$149$			120.257i			0.807097i	−0.914958	+	0.403548i	$0.867777\pi$
$150$		0			0
$151$	−44.3135			−0.293467			−0.146733	−	0.989176i	$-0.546876\pi$
$151$	−44.3135			−0.293467			−0.146733	+	0.989176i	$0.546876\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−6.64063	+	26.4173i	−0.0428428	+	0.170434i
$156$		0			0
$157$			59.5447i			0.379266i	0.981855	+	0.189633i	$0.0607298\pi$
$157$			59.5447i			0.379266i	−0.981855	+	0.189633i	$0.939270\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			160.595i			0.997486i
$162$		0			0
$163$		−	200.140i		−	1.22785i	−0.789363	−	0.613927i	$-0.789589\pi$
$163$		−	200.140i		−	1.22785i	0.789363	−	0.613927i	$-0.210411\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	77.1260			0.461832			0.230916	−	0.972974i	$-0.425828\pi$
$167$	77.1260			0.461832			0.230916	+	0.972974i	$0.425828\pi$
$168$		0			0
$169$	−121.419			−0.718458
$170$		0			0
$171$		0			0
$172$		0			0
$173$	258.711			1.49544			0.747718	−	0.664016i	$-0.231149\pi$
$173$	258.711			1.49544			0.747718	+	0.664016i	$0.231149\pi$
$174$		0			0
$175$	−135.985	+	253.391i	−0.777056	+	1.44795i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			174.572i			0.975261i	0.873050	+	0.487630i	$0.162139\pi$
$179$			174.572i			0.975261i	−0.873050	+	0.487630i	$0.837861\pi$
$180$		0			0
$181$	−22.3291			−0.123365			−0.0616825	−	0.998096i	$-0.519647\pi$
$181$	−22.3291			−0.123365			−0.0616825	+	0.998096i	$0.519647\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−246.003	−	61.8390i	−1.32975	−	0.334265i
$186$		0			0
$187$		−	158.965i		−	0.850083i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			219.639i			1.14994i	0.818174	+	0.574971i	$0.194987\pi$
$191$			219.639i			1.14994i	−0.818174	+	0.574971i	$0.805013\pi$
$192$		0			0
$193$		−	341.447i		−	1.76916i	−0.466390	−	0.884579i	$-0.654446\pi$
$193$		−	341.447i		−	1.76916i	0.466390	−	0.884579i	$-0.345554\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−200.719			−1.01888			−0.509439	−	0.860507i	$-0.670147\pi$
$197$	−200.719			−1.01888			−0.509439	+	0.860507i	$0.670147\pi$
$198$		0			0
$199$	−79.9313			−0.401665			−0.200833	−	0.979626i	$-0.564365\pi$
$199$	−79.9313			−0.401665			−0.200833	+	0.979626i	$0.564365\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	515.727			2.54053
$204$		0			0
$205$	−206.892	−	52.0075i	−1.00923	−	0.253695i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			352.979i			1.68889i
$210$		0			0
$211$	−71.8961			−0.340740			−0.170370	−	0.985380i	$-0.554496\pi$
$211$	−71.8961			−0.340740			−0.170370	+	0.985380i	$0.554496\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	29.8173	+	7.49531i	0.138685	+	0.0348619i
$216$		0			0
$217$		−	62.6662i		−	0.288784i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	177.677i		−	0.803967i
$222$		0			0
$223$			353.131i			1.58355i	0.610816	+	0.791773i	$0.290842\pi$
$223$			353.131i			1.58355i	−0.610816	+	0.791773i	$0.709158\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	171.402			0.755075			0.377538	−	0.925994i	$-0.376771\pi$
$227$	171.402			0.755075			0.377538	+	0.925994i	$0.376771\pi$
$228$		0			0
$229$	−165.801			−0.724021			−0.362011	−	0.932174i	$-0.617910\pi$
$229$	−165.801			−0.724021			−0.362011	+	0.932174i	$0.617910\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	6.09276			0.0261492			0.0130746	−	0.999915i	$-0.495838\pi$
$233$	6.09276			0.0261492			0.0130746	+	0.999915i	$0.495838\pi$
$234$		0			0
$235$	39.1200	−	155.624i	0.166468	−	0.662231i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			35.5326i			0.148672i	0.997233	+	0.0743360i	$0.0236837\pi$
$239$			35.5326i			0.148672i	−0.997233	+	0.0743360i	$0.976316\pi$
$240$		0			0
$241$	−300.304			−1.24608			−0.623038	−	0.782192i	$-0.714102\pi$
$241$	−300.304			−1.24608			−0.623038	+	0.782192i	$0.714102\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	101.561	−	404.022i	0.414534	−	1.64907i
$246$		0			0
$247$			394.527i			1.59727i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	240.776i		−	0.959268i	−0.877469	−	0.479634i	$-0.840770\pi$
$251$		−	240.776i		−	0.959268i	0.877469	−	0.479634i	$-0.159230\pi$
$252$		0			0
$253$			212.867i			0.841371i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−134.051			−0.521598			−0.260799	−	0.965393i	$-0.583986\pi$
$257$	−134.051			−0.521598			−0.260799	+	0.965393i	$0.583986\pi$
$258$		0			0
$259$	583.561			2.25313
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−437.151			−1.66217			−0.831085	−	0.556145i	$-0.812280\pi$
$263$	−437.151			−1.66217			−0.831085	+	0.556145i	$0.812280\pi$
$264$		0			0
$265$	−20.4010	+	81.1576i	−0.0769847	+	0.306255i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			39.0128i			0.145029i	0.997367	+	0.0725146i	$0.0231024\pi$
$269$			39.0128i			0.145029i	−0.997367	+	0.0725146i	$0.976898\pi$
$270$		0			0
$271$	225.452			0.831925			0.415963	−	0.909382i	$-0.363445\pi$
$271$	225.452			0.831925			0.415963	+	0.909382i	$0.363445\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−180.246	+	335.866i	−0.655440	+	1.22133i
$276$		0			0
$277$			228.089i			0.823424i	0.911314	+	0.411712i	$0.135069\pi$
$277$			228.089i			0.823424i	−0.911314	+	0.411712i	$0.864931\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	240.786i		−	0.856890i	−0.903568	−	0.428445i	$-0.859062\pi$
$281$		−	240.786i		−	0.856890i	0.903568	−	0.428445i	$-0.140938\pi$
$282$		0			0
$283$		−	46.4756i		−	0.164225i	−0.996623	−	0.0821123i	$-0.973833\pi$
$283$		−	46.4756i		−	0.164225i	0.996623	−	0.0821123i	$-0.0261666\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	490.783			1.71005
$288$		0			0
$289$	−180.299			−0.623871
$290$		0			0
$291$		0			0
$292$		0			0
$293$	96.5401			0.329489			0.164744	−	0.986336i	$-0.447320\pi$
$293$	96.5401			0.329489			0.164744	+	0.986336i	$0.447320\pi$
$294$		0			0
$295$	94.7885	+	23.8274i	0.321317	+	0.0807709i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			237.923i			0.795728i
$300$		0			0
$301$	−70.7316			−0.234989
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−8.04430	+	32.0013i	−0.0263748	+	0.104922i
$306$		0			0
$307$			398.954i			1.29952i	0.760138	+	0.649762i	$0.225131\pi$
$307$			398.954i			1.29952i	−0.760138	+	0.649762i	$0.774869\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	400.220i		−	1.28688i	−0.765497	−	0.643440i	$-0.777507\pi$
$311$		−	400.220i		−	1.28688i	0.765497	−	0.643440i	$-0.222493\pi$
$312$		0			0
$313$		−	206.799i		−	0.660700i	−0.943859	−	0.330350i	$-0.892833\pi$
$313$		−	206.799i		−	0.660700i	0.943859	−	0.330350i	$-0.107167\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	62.7971			0.198098			0.0990491	−	0.995083i	$-0.468420\pi$
$317$	62.7971			0.198098			0.0990491	+	0.995083i	$0.468420\pi$
$318$		0			0
$319$	683.589			2.14291
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−241.369			−0.747272
$324$		0			0
$325$	−201.462	+	375.400i	−0.619883	+	1.15508i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			369.167i			1.12209i
$330$		0			0
$331$	−240.765			−0.727387			−0.363693	−	0.931519i	$-0.618484\pi$
$331$	−240.765			−0.727387			−0.363693	+	0.931519i	$0.618484\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−172.448	−	43.3490i	−0.514770	−	0.129400i
$336$		0			0
$337$		−	342.878i		−	1.01744i	−0.860932	−	0.508720i	$-0.830119\pi$
$337$		−	342.878i		−	1.01744i	0.860932	−	0.508720i	$-0.169881\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	83.0632i		−	0.243587i
$342$		0			0
$343$			394.762i			1.15091i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−91.3625			−0.263292			−0.131646	−	0.991297i	$-0.542026\pi$
$347$	−91.3625			−0.263292			−0.131646	+	0.991297i	$0.542026\pi$
$348$		0			0
$349$	−62.0889			−0.177905			−0.0889526	−	0.996036i	$-0.528352\pi$
$349$	−62.0889			−0.177905			−0.0889526	+	0.996036i	$0.528352\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−132.188			−0.374470			−0.187235	−	0.982315i	$-0.559953\pi$
$353$	−132.188			−0.374470			−0.187235	+	0.982315i	$0.559953\pi$
$354$		0			0
$355$	317.626	+	79.8431i	0.894722	+	0.224910i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			299.807i			0.835118i	0.908650	+	0.417559i	$0.137114\pi$
$359$			299.807i			0.835118i	−0.908650	+	0.417559i	$0.862886\pi$
$360$		0			0
$361$	174.954			0.484637
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−504.884	−	126.915i	−1.38324	−	0.347712i
$366$		0			0
$367$		−	428.470i		−	1.16749i	−0.811935	−	0.583747i	$-0.801586\pi$
$367$		−	428.470i		−	1.16749i	0.811935	−	0.583747i	$-0.198414\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	192.519i		−	0.518920i
$372$		0			0
$373$		−	419.109i		−	1.12362i	−0.827267	−	0.561809i	$-0.810106\pi$
$373$		−	419.109i		−	1.12362i	0.827267	−	0.561809i	$-0.189894\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	764.051			2.02666
$378$		0			0
$379$	394.789			1.04166			0.520830	−	0.853660i	$-0.325622\pi$
$379$	394.789			1.04166			0.520830	+	0.853660i	$0.325622\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−515.753			−1.34661			−0.673307	−	0.739363i	$-0.735127\pi$
$383$	−515.753			−1.34661			−0.673307	+	0.739363i	$0.735127\pi$
$384$		0			0
$385$	213.787	−	850.472i	0.555290	−	2.20902i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	108.711i		−	0.279462i	−0.990189	−	0.139731i	$-0.955376\pi$
$389$		−	108.711i		−	0.279462i	0.990189	−	0.139731i	$-0.0446238\pi$
$390$		0			0
$391$	−145.559			−0.372275
$392$		0			0
$393$		0			0
$394$		0			0
$395$	183.483	−	729.920i	0.464514	−	1.84790i
$396$		0			0
$397$		−	319.633i		−	0.805121i	−0.915393	−	0.402560i	$-0.868120\pi$
$397$		−	319.633i		−	0.805121i	0.915393	−	0.402560i	$-0.131880\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	86.2638i		−	0.215122i	−0.994199	−	0.107561i	$-0.965696\pi$
$401$		−	86.2638i		−	0.215122i	0.994199	−	0.107561i	$-0.0343040\pi$
$402$		0			0
$403$		−	92.8402i		−	0.230373i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	773.502			1.90050
$408$		0			0
$409$	−670.973			−1.64052			−0.820260	−	0.571991i	$-0.806171\pi$
$409$	−670.973			−1.64052			−0.820260	+	0.571991i	$0.806171\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−224.854			−0.544441
$414$		0			0
$415$	99.2943	−	395.006i	0.239263	−	0.951821i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			712.906i			1.70145i	0.525615	+	0.850723i	$0.323835\pi$
$419$			712.906i			1.70145i	−0.525615	+	0.850723i	$0.676165\pi$
$420$		0			0
$421$	411.724			0.977967			0.488983	−	0.872293i	$-0.337368\pi$
$421$	411.724			0.977967			0.488983	+	0.872293i	$0.337368\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−229.667	−	123.253i	−0.540393	−	0.290008i
$426$		0			0
$427$		−	75.9124i		−	0.177781i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			165.609i			0.384243i	0.981371	+	0.192122i	$0.0615368\pi$
$431$			165.609i			0.384243i	−0.981371	+	0.192122i	$0.938463\pi$
$432$		0			0
$433$		−	227.973i		−	0.526497i	−0.964728	−	0.263248i	$-0.915206\pi$
$433$		−	227.973i		−	0.526497i	0.964728	−	0.263248i	$-0.0847938\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	323.211			0.739614
$438$		0			0
$439$	−476.210			−1.08476			−0.542380	−	0.840133i	$-0.682477\pi$
$439$	−476.210			−1.08476			−0.542380	+	0.840133i	$0.682477\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−541.053			−1.22134			−0.610669	−	0.791886i	$-0.709099\pi$
$443$	−541.053			−1.22134			−0.610669	+	0.791886i	$0.709099\pi$
$444$		0			0
$445$	−69.7586	−	17.5355i	−0.156761	−	0.0394057i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	98.6418i		−	0.219692i	−0.993949	−	0.109846i	$-0.964964\pi$
$449$		−	98.6418i		−	0.219692i	0.993949	−	0.109846i	$-0.0350358\pi$
$450$		0			0
$451$	650.527			1.44241
$452$		0			0
$453$		0			0
$454$		0			0
$455$	238.951	−	950.577i	0.525167	−	2.08918i
$456$		0			0
$457$		−	275.913i		−	0.603749i	−0.953348	−	0.301874i	$-0.902388\pi$
$457$		−	275.913i		−	0.603749i	0.953348	−	0.301874i	$-0.0976123\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			793.261i			1.72074i	0.509670	+	0.860370i	$0.329767\pi$
$461$			793.261i			1.72074i	−0.509670	+	0.860370i	$0.670233\pi$
$462$		0			0
$463$		−	779.199i		−	1.68294i	−0.540308	−	0.841468i	$-0.681692\pi$
$463$		−	779.199i		−	1.68294i	0.540308	−	0.841468i	$-0.318308\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	233.839			0.500726			0.250363	−	0.968152i	$-0.419450\pi$
$467$	233.839			0.500726			0.250363	+	0.968152i	$0.419450\pi$
$468$		0			0
$469$	409.075			0.872229
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−93.7538			−0.198211
$474$		0			0
$475$	509.971	+	273.681i	1.07362	+	0.576170i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			96.4359i			0.201328i	0.994921	+	0.100664i	$0.0320966\pi$
$479$			96.4359i			0.201328i	−0.994921	+	0.100664i	$0.967903\pi$
$480$		0			0
$481$	864.548			1.79740
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−130.280	−	32.7491i	−0.268618	−	0.0675238i
$486$		0			0
$487$			171.674i			0.352513i	0.984344	+	0.176256i	$0.0563988\pi$
$487$			171.674i			0.352513i	−0.984344	+	0.176256i	$0.943601\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	21.6217i		−	0.0440361i	−0.999758	−	0.0220181i	$-0.992991\pi$
$491$		−	21.6217i		−	0.0440361i	0.999758	−	0.0220181i	$-0.00700913\pi$
$492$		0			0
$493$			467.441i			0.948157i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−753.462			−1.51602
$498$		0			0
$499$	519.959			1.04200			0.521001	−	0.853556i	$-0.325559\pi$
$499$	519.959			1.04200			0.521001	+	0.853556i	$0.325559\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−362.526			−0.720727			−0.360363	−	0.932812i	$-0.617347\pi$
$503$	−362.526			−0.720727			−0.360363	+	0.932812i	$0.617347\pi$
$504$		0			0
$505$	463.092	+	116.409i	0.917013	+	0.230514i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	451.564i		−	0.887159i	−0.896235	−	0.443580i	$-0.853708\pi$
$509$		−	451.564i		−	0.887159i	0.896235	−	0.443580i	$-0.146292\pi$
$510$		0			0
$511$	1197.67			2.34378
$512$		0			0
$513$		0			0
$514$		0			0
$515$	280.205	+	70.4363i	0.544087	+	0.136770i
$516$		0			0
$517$			489.325i			0.946471i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	901.613i		−	1.73054i	−0.501303	−	0.865272i	$-0.667146\pi$
$521$		−	901.613i		−	1.73054i	0.501303	−	0.865272i	$-0.332854\pi$
$522$		0			0
$523$		−	292.527i		−	0.559326i	−0.960098	−	0.279663i	$-0.909777\pi$
$523$		−	292.527i		−	0.559326i	0.960098	−	0.279663i	$-0.0902227\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	56.7990			0.107778
$528$		0			0
$529$	−334.085			−0.631540
$530$		0			0
$531$		0			0
$532$		0			0
$533$	727.098			1.36416
$534$		0			0
$535$	79.0353	−	314.413i	0.147729	−	0.587687i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			1270.36i			2.35688i
$540$		0			0
$541$	−790.789			−1.46172			−0.730858	−	0.682529i	$-0.760880\pi$
$541$	−790.789			−1.46172			−0.730858	+	0.682529i	$0.760880\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	49.8761	−	198.414i	0.0915158	−	0.364062i
$546$		0			0
$547$		−	158.225i		−	0.289259i	−0.989486	−	0.144629i	$-0.953801\pi$
$547$		−	158.225i		−	0.289259i	0.989486	−	0.144629i	$-0.0461990\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	1037.94i		−	1.88374i
$552$		0			0
$553$			1731.49i			3.13109i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−1011.82			−1.81655			−0.908273	−	0.418378i	$-0.862599\pi$
$557$	−1011.82			−1.81655			−0.908273	+	0.418378i	$0.862599\pi$
$558$		0			0
$559$	−104.789			−0.187458
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−80.9702			−0.143819			−0.0719096	−	0.997411i	$-0.522909\pi$
$563$	−80.9702			−0.143819			−0.0719096	+	0.997411i	$0.522909\pi$
$564$		0			0
$565$	−263.832	+	1049.56i	−0.466960	+	1.85763i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			772.474i			1.35760i	0.734323	+	0.678800i	$0.237500\pi$
$569$			772.474i			1.35760i	−0.734323	+	0.678800i	$0.762500\pi$
$570$		0			0
$571$	768.777			1.34637			0.673185	−	0.739474i	$-0.264926\pi$
$571$	768.777			1.34637			0.673185	+	0.739474i	$0.264926\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	307.542	+	165.045i	0.534856	+	0.287036i
$576$		0			0
$577$		−	188.653i		−	0.326955i	−0.986547	−	0.163477i	$-0.947729\pi$
$577$		−	188.653i		−	0.326955i	0.986547	−	0.163477i	$-0.0522710\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			937.019i			1.61277i
$582$		0			0
$583$		−	255.182i		−	0.437705i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	556.656			0.948307			0.474153	−	0.880442i	$-0.342754\pi$
$587$	556.656			0.948307			0.474153	+	0.880442i	$0.342754\pi$
$588$		0			0
$589$	−126.121			−0.214127
$590$		0			0
$591$		0			0
$592$		0			0
$593$	715.570			1.20670			0.603348	−	0.797478i	$-0.293833\pi$
$593$	715.570			1.20670			0.603348	+	0.797478i	$0.293833\pi$
$594$		0			0
$595$	581.557	+	146.189i	0.977407	+	0.245695i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			1171.15i			1.95517i	0.210541	+	0.977585i	$0.432477\pi$
$599$			1171.15i			1.95517i	−0.210541	+	0.977585i	$0.567523\pi$
$600$		0			0
$601$	229.820			0.382396			0.191198	−	0.981552i	$-0.438763\pi$
$601$	229.820			0.382396			0.191198	+	0.981552i	$0.438763\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	135.879	−	540.542i	0.224593	−	0.893458i
$606$		0			0
$607$		−	1034.42i		−	1.70415i	−0.523416	−	0.852077i	$-0.675343\pi$
$607$		−	1034.42i		−	1.70415i	0.523416	−	0.852077i	$-0.324657\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			546.922i			0.895126i
$612$		0			0
$613$			69.5129i			0.113398i	0.998391	+	0.0566989i	$0.0180575\pi$
$613$			69.5129i			0.113398i	−0.998391	+	0.0566989i	$0.981942\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1034.67			1.67694			0.838468	−	0.544951i	$-0.183452\pi$
$617$	1034.67			1.67694			0.838468	+	0.544951i	$0.183452\pi$
$618$		0			0
$619$	976.692			1.57785			0.788927	−	0.614487i	$-0.210637\pi$
$619$	976.692			1.57785			0.788927	+	0.614487i	$0.210637\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	165.479			0.265616
$624$		0			0
$625$	345.494	+	520.825i	0.552790	+	0.833320i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			528.925i			0.840898i
$630$		0			0
$631$	257.928			0.408761			0.204380	−	0.978892i	$-0.434482\pi$
$631$	257.928			0.408761			0.204380	+	0.978892i	$0.434482\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	623.281	+	156.677i	0.981545	+	0.246735i
$636$		0			0
$637$			1419.88i			2.22902i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	756.232i		−	1.17977i	−0.807487	−	0.589885i	$-0.799173\pi$
$641$		−	756.232i		−	1.17977i	0.807487	−	0.589885i	$-0.200827\pi$
$642$		0			0
$643$			249.569i			0.388132i	0.980988	+	0.194066i	$0.0621676\pi$
$643$			249.569i			0.388132i	−0.980988	+	0.194066i	$0.937832\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1210.85			−1.87149			−0.935743	−	0.352682i	$-0.885270\pi$
$647$	−1210.85			−1.87149			−0.935743	+	0.352682i	$0.885270\pi$
$648$		0			0
$649$	−298.041			−0.459231
$650$		0			0
$651$		0			0
$652$		0			0
$653$	829.225			1.26987			0.634935	−	0.772566i	$-0.281027\pi$
$653$	829.225			1.26987			0.634935	+	0.772566i	$0.281027\pi$
$654$		0			0
$655$	−129.852	−	32.6414i	−0.198247	−	0.0498342i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			1210.69i			1.83717i	0.395224	+	0.918585i	$0.370667\pi$
$659$			1210.69i			1.83717i	−0.395224	+	0.918585i	$0.629333\pi$
$660$		0			0
$661$	−245.005			−0.370658			−0.185329	−	0.982677i	$-0.559335\pi$
$661$	−245.005			−0.370658			−0.185329	+	0.982677i	$0.559335\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1291.33	−	324.608i	−1.94185	−	0.488133i
$666$		0			0
$667$		−	625.940i		−	0.938440i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	100.621i		−	0.149956i
$672$		0			0
$673$		−	1176.79i		−	1.74857i	−0.485409	−	0.874287i	$-0.661329\pi$
$673$		−	1176.79i		−	1.74857i	0.485409	−	0.874287i	$-0.338671\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−733.165			−1.08296			−0.541481	−	0.840713i	$-0.682136\pi$
$677$	−733.165			−1.08296			−0.541481	+	0.840713i	$0.682136\pi$
$678$		0			0
$679$	309.046			0.455148
$680$		0			0
$681$		0			0
$682$		0			0
$683$	770.725			1.12844			0.564221	−	0.825624i	$-0.309177\pi$
$683$	770.725			1.12844			0.564221	+	0.825624i	$0.309177\pi$
$684$		0			0
$685$	121.709	−	484.174i	0.177677	−	0.706823i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	285.218i		−	0.413960i
$690$		0			0
$691$	−166.635			−0.241151			−0.120575	−	0.992704i	$-0.538474\pi$
$691$	−166.635			−0.241151			−0.120575	+	0.992704i	$0.538474\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	18.7571	−	74.6182i	0.0269886	−	0.107364i
$696$		0			0
$697$			444.834i			0.638212i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			857.331i			1.22301i	0.791240	+	0.611506i	$0.209436\pi$
$701$			857.331i			1.22301i	−0.791240	+	0.611506i	$0.790564\pi$
$702$		0			0
$703$		−	1174.47i		−	1.67065i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1098.53			−1.55379
$708$		0			0
$709$	1124.15			1.58554			0.792770	−	0.609521i	$-0.208638\pi$
$709$	1124.15			1.58554			0.792770	+	0.609521i	$0.208638\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−76.0582			−0.106674
$714$		0			0
$715$	316.726	−	1259.98i	0.442973	−	1.76221i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	772.421i		−	1.07430i	−0.843487	−	0.537150i	$-0.819501\pi$
$719$		−	772.421i		−	1.07430i	0.843487	−	0.537150i	$-0.180499\pi$
$720$		0			0
$721$	−664.692			−0.921903
$722$		0			0
$723$		0			0
$724$		0			0
$725$	530.017	−	987.623i	0.731059	−	1.36224i
$726$		0			0
$727$		−	261.129i		−	0.359186i	−0.983741	−	0.179593i	$-0.942522\pi$
$727$		−	261.129i		−	0.359186i	0.983741	−	0.179593i	$-0.0574782\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	64.1093i		−	0.0877008i
$732$		0			0
$733$		−	517.385i		−	0.705845i	−0.935652	−	0.352923i	$-0.885188\pi$
$733$		−	517.385i		−	0.705845i	0.935652	−	0.352923i	$-0.114812\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	542.224			0.735717
$738$		0			0
$739$	396.875			0.537044			0.268522	−	0.963274i	$-0.413465\pi$
$739$	396.875			0.537044			0.268522	+	0.963274i	$0.413465\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−777.037			−1.04581			−0.522905	−	0.852391i	$-0.675152\pi$
$743$	−777.037			−1.04581			−0.522905	+	0.852391i	$0.675152\pi$
$744$		0			0
$745$	−583.145	−	146.588i	−0.782745	−	0.196762i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			745.839i			0.995780i
$750$		0			0
$751$	−469.264			−0.624852			−0.312426	−	0.949942i	$-0.601142\pi$
$751$	−469.264			−0.624852			−0.312426	+	0.949942i	$0.601142\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	54.0160	−	214.882i	0.0715443	−	0.284613i
$756$		0			0
$757$		−	578.062i		−	0.763622i	−0.924240	−	0.381811i	$-0.875301\pi$
$757$		−	578.062i		−	0.763622i	0.924240	−	0.381811i	$-0.124699\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	801.033i		−	1.05261i	−0.850297	−	0.526303i	$-0.823578\pi$
$761$		−	801.033i		−	1.05261i	0.850297	−	0.526303i	$-0.176422\pi$
$762$		0			0
$763$			470.670i			0.616868i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−333.122			−0.434318
$768$		0			0
$769$	620.247			0.806563			0.403282	−	0.915076i	$-0.367869\pi$
$769$	620.247			0.806563			0.403282	+	0.915076i	$0.367869\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−261.647			−0.338483			−0.169241	−	0.985575i	$-0.554132\pi$
$773$	−261.647			−0.338483			−0.169241	+	0.985575i	$0.554132\pi$
$774$		0			0
$775$	−120.007	−	64.4027i	−0.154847	−	0.0831002i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	987.742i		−	1.26796i
$780$		0			0
$781$	−998.704			−1.27875
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−288.741	−	72.5820i	−0.367823	−	0.0924612i
$786$		0			0
$787$		−	7.52301i		−	0.00955910i	−0.999989	−	0.00477955i	$-0.998479\pi$
$787$		−	7.52301i		−	0.00955910i	0.999989	−	0.00477955i	$-0.00152138\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	2489.73i		−	3.14757i
$792$		0			0
$793$		−	112.464i		−	0.141822i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−338.711			−0.424982			−0.212491	−	0.977163i	$-0.568158\pi$
$797$	−338.711			−0.424982			−0.212491	+	0.977163i	$0.568158\pi$
$798$		0			0
$799$	−334.603			−0.418778
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1587.49			1.97695
$804$		0			0
$805$	−778.749	−	195.758i	−0.967390	−	0.243177i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			1370.18i			1.69367i	0.531857	+	0.846834i	$0.321494\pi$
$809$			1370.18i			1.69367i	−0.531857	+	0.846834i	$0.678506\pi$
$810$		0			0
$811$	395.829			0.488075			0.244037	−	0.969766i	$-0.421528\pi$
$811$	395.829			0.488075			0.244037	+	0.969766i	$0.421528\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	970.508	+	243.961i	1.19081	+	0.299339i
$816$		0			0
$817$			142.353i			0.174239i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	957.135i		−	1.16582i	−0.812538	−	0.582908i	$-0.801915\pi$
$821$		−	957.135i		−	1.16582i	0.812538	−	0.582908i	$-0.198085\pi$
$822$		0			0
$823$			1296.92i			1.57585i	0.615773	+	0.787923i	$0.288844\pi$
$823$			1296.92i			1.57585i	−0.615773	+	0.787923i	$0.711156\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1097.08			−1.32658			−0.663291	−	0.748361i	$-0.730841\pi$
$827$	−1097.08			−1.32658			−0.663291	+	0.748361i	$0.730841\pi$
$828$		0			0
$829$	286.317			0.345376			0.172688	−	0.984977i	$-0.444755\pi$
$829$	286.317			0.345376			0.172688	+	0.984977i	$0.444755\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−868.676			−1.04283
$834$		0			0
$835$	−94.0127	+	373.995i	−0.112590	+	0.447898i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			336.627i			0.401224i	0.979671	+	0.200612i	$0.0642931\pi$
$839$			336.627i			0.401224i	−0.979671	+	0.200612i	$0.935707\pi$
$840$		0			0
$841$	−1169.11			−1.39014
$842$		0			0
$843$		0			0
$844$		0			0
$845$	148.004	−	588.779i	0.175153	−	0.696780i
$846$		0			0
$847$			1282.26i			1.51388i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	708.271i		−	0.832281i
$852$		0			0
$853$		−	1218.01i		−	1.42792i	−0.700188	−	0.713958i	$-0.746901\pi$
$853$		−	1218.01i		−	1.42792i	0.700188	−	0.713958i	$-0.253099\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1206.24			−1.40752			−0.703758	−	0.710440i	$-0.748496\pi$
$857$	−1206.24			−1.40752			−0.703758	+	0.710440i	$0.748496\pi$
$858$		0			0
$859$	−1559.34			−1.81529			−0.907647	−	0.419734i	$-0.862123\pi$
$859$	−1559.34			−1.81529			−0.907647	+	0.419734i	$0.862123\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1124.20			1.30266			0.651332	−	0.758793i	$-0.274211\pi$
$863$	1124.20			1.30266			0.651332	+	0.758793i	$0.274211\pi$
$864$		0			0
$865$	−315.355	+	1254.52i	−0.364573	+	1.45032i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			2295.07i			2.64104i
$870$		0			0
$871$	606.047			0.695806
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1062.97	−	968.281i	−1.21482	−	1.10661i
$876$		0			0
$877$		−	1150.78i		−	1.31218i	−0.754684	−	0.656088i	$-0.772210\pi$
$877$		−	1150.78i		−	1.31218i	0.754684	−	0.656088i	$-0.227790\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	234.506i		−	0.266181i	−0.991104	−	0.133091i	$-0.957510\pi$
$881$		−	234.506i		−	0.266181i	0.991104	−	0.133091i	$-0.0424902\pi$
$882$		0			0
$883$			1211.86i			1.37243i	0.727399	+	0.686215i	$0.240729\pi$
$883$			1211.86i			1.37243i	−0.727399	+	0.686215i	$0.759271\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1323.03			−1.49158			−0.745790	−	0.666181i	$-0.767928\pi$
$887$	−1323.03			−1.49158			−0.745790	+	0.666181i	$0.767928\pi$
$888$		0			0
$889$	−1478.53			−1.66313
$890$		0			0
$891$		0			0
$892$		0			0
$893$	742.979			0.832003
$894$		0			0
$895$	−846.522	−	212.794i	−0.945835	−	0.237759i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			244.249i			0.271690i
$900$		0			0
$901$	174.495			0.193668
$902$		0			0
$903$		0			0
$904$		0			0
$905$	27.2180	−	108.277i	0.0300752	−	0.119643i
$906$		0			0
$907$		−	494.733i		−	0.545461i	−0.962090	−	0.272730i	$-0.912073\pi$
$907$		−	494.733i		−	0.545461i	0.962090	−	0.272730i	$-0.0879267\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			172.751i			0.189627i	0.995495	+	0.0948137i	$0.0302255\pi$
$911$			172.751i			0.189627i	−0.995495	+	0.0948137i	$0.969774\pi$
$912$		0			0
$913$			1242.01i			1.36036i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	308.030			0.335911
$918$		0			0
$919$	414.019			0.450510			0.225255	−	0.974300i	$-0.427679\pi$
$919$	414.019			0.450510			0.225255	+	0.974300i	$0.427679\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−1116.26			−1.20938
$924$		0			0
$925$	599.732	−	1117.53i	0.648359	−	1.20814i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	1326.17i		−	1.42752i	−0.700390	−	0.713760i	$-0.746991\pi$
$929$		−	1326.17i		−	1.42752i	0.700390	−	0.713760i	$-0.253009\pi$
$930$		0			0
$931$	1928.87			2.07183
$932$		0			0
$933$		0			0
$934$		0			0
$935$	770.846	+	193.771i	0.824434	+	0.207242i
$936$		0			0
$937$			419.126i			0.447306i	0.974669	+	0.223653i	$0.0717982\pi$
$937$			419.126i			0.447306i	−0.974669	+	0.223653i	$0.928202\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	669.413i		−	0.711385i	−0.934603	−	0.355692i	$-0.884245\pi$
$941$		−	669.413i		−	0.711385i	0.934603	−	0.355692i	$-0.115755\pi$
$942$		0			0
$943$		−	595.666i		−	0.631671i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−303.580			−0.320570			−0.160285	−	0.987071i	$-0.551241\pi$
$947$	−303.580			−0.320570			−0.160285	+	0.987071i	$0.551241\pi$
$948$		0			0
$949$	1774.35			1.86971
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1452.57			1.52421			0.762103	−	0.647455i	$-0.224167\pi$
$953$	1452.57			1.52421			0.762103	+	0.647455i	$0.224167\pi$
$954$		0			0
$955$	−1065.06	−	267.729i	−1.11525	−	0.280344i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			1148.54i			1.19764i
$960$		0			0
$961$	−931.321			−0.969117
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1655.73	+	416.207i	1.71578	+	0.431303i
$966$		0			0
$967$			41.8858i			0.0433152i	0.999765	+	0.0216576i	$0.00689437\pi$
$967$			41.8858i			0.0433152i	−0.999765	+	0.0216576i	$0.993106\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			513.822i			0.529168i	0.964363	+	0.264584i	$0.0852346\pi$
$971$			513.822i			0.529168i	−0.964363	+	0.264584i	$0.914765\pi$
$972$		0			0
$973$			177.007i			0.181919i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1723.36			1.76393			0.881965	−	0.471314i	$-0.156220\pi$
$977$	1723.36			1.76393			0.881965	+	0.471314i	$0.156220\pi$
$978$		0			0
$979$	219.340			0.224045
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−797.891			−0.811690			−0.405845	−	0.913942i	$-0.633023\pi$
$983$	−797.891			−0.811690			−0.405845	+	0.913942i	$0.633023\pi$
$984$		0			0
$985$	244.666	−	973.315i	0.248392	−	0.988137i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			85.8473i			0.0868021i
$990$		0			0
$991$	−920.998			−0.929363			−0.464681	−	0.885478i	$-0.653831\pi$
$991$	−920.998			−0.929363			−0.464681	+	0.885478i	$0.653831\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	97.4323	−	387.598i	0.0979219	−	0.389546i
$996$		0			0
$997$			896.823i			0.899521i	0.893149	+	0.449761i	$0.148491\pi$
$997$			896.823i			0.899521i	−0.893149	+	0.449761i	$0.851509\pi$
$998$		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.809.12		24
3.2	odd	2	inner	1620.3.b.b.809.13		24
5.4	even	2	inner	1620.3.b.b.809.14		24
9.2	odd	6		540.3.t.a.449.9		24
9.4	even	3		540.3.t.a.89.11		24
9.5	odd	6		180.3.t.a.29.4	✓	24
9.7	even	3		180.3.t.a.149.9	yes	24
15.14	odd	2	inner	1620.3.b.b.809.11		24
45.2	even	12		2700.3.p.f.2501.2		24
45.4	even	6		540.3.t.a.89.9		24
45.7	odd	12		900.3.p.f.401.10		24
45.13	odd	12		2700.3.p.f.1601.11		24
45.14	odd	6		180.3.t.a.29.9	yes	24
45.22	odd	12		2700.3.p.f.1601.2		24
45.23	even	12		900.3.p.f.101.3		24
45.29	odd	6		540.3.t.a.449.11		24
45.32	even	12		900.3.p.f.101.10		24
45.34	even	6		180.3.t.a.149.4	yes	24
45.38	even	12		2700.3.p.f.2501.11		24
45.43	odd	12		900.3.p.f.401.3		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.3.t.a.29.4	✓	24	9.5	odd	6
180.3.t.a.29.9	yes	24	45.14	odd	6
180.3.t.a.149.4	yes	24	45.34	even	6
180.3.t.a.149.9	yes	24	9.7	even	3
540.3.t.a.89.9		24	45.4	even	6
540.3.t.a.89.11		24	9.4	even	3
540.3.t.a.449.9		24	9.2	odd	6
540.3.t.a.449.11		24	45.29	odd	6
900.3.p.f.101.3		24	45.23	even	12
900.3.p.f.101.10		24	45.32	even	12
900.3.p.f.401.3		24	45.43	odd	12
900.3.p.f.401.10		24	45.7	odd	12
1620.3.b.b.809.11		24	15.14	odd	2	inner
1620.3.b.b.809.12		24	1.1	even	1	trivial
1620.3.b.b.809.13		24	3.2	odd	2	inner
1620.3.b.b.809.14		24	5.4	even	2	inner
2700.3.p.f.1601.2		24	45.22	odd	12
2700.3.p.f.1601.11		24	45.13	odd	12
2700.3.p.f.2501.2		24	45.2	even	12
2700.3.p.f.2501.11		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)

\(f(q)\)	\(=\)	\(q+(-1.21895 + 4.84914i) q^{5} -11.5030i q^{7} +O(q^{10})\)	\(q+(-1.21895 + 4.84914i) q^{5} -11.5030i q^{7} -15.2470i q^{11} -17.0417i q^{13} +10.4260 q^{17} -23.1507 q^{19} -13.9612 q^{23} +(-22.0283 - 11.8217i) q^{25} +44.8342i q^{29} +5.44783 q^{31} +(55.7795 + 14.0215i) q^{35} +50.7314i q^{37} +42.6658i q^{41} -6.14899i q^{43} -32.0932 q^{47} -83.3183 q^{49} +16.7365 q^{53} +(73.9350 + 18.5854i) q^{55} -19.5475i q^{59} +6.59937 q^{61} +(82.6376 + 20.7730i) q^{65} +35.5626i q^{67} -65.5016i q^{71} +104.118i q^{73} -175.386 q^{77} -150.526 q^{79} -81.4589 q^{83} +(-12.7088 + 50.5571i) q^{85} +14.3858i q^{89} -196.030 q^{91} +(28.2195 - 112.261i) q^{95} +26.8666i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 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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