LMFDB - Embedded newform 900.3.g.c.701.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [900,3,Mod(701,900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(900, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("900.701");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	900.g (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:	$24.5232237924$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-23})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} + 17x^{2} - 16x + 18 $ x^4 - 2x^3 + 17x^2 - 16*x + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			701.2
Root			$0.500000 + 0.983702i$ of defining polynomial
Character	$\chi$	$=$	900.701
Dual form			900.3.g.c.701.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.78233 q^{7} +O(q^{10})$	$q-6.78233 q^{7} +9.89949i q^{11} +20.3470 q^{13} -19.1833i q^{17} -12.0000 q^{19} +9.59166i q^{23} +8.48528i q^{29} -38.0000 q^{31} -6.78233 q^{37} +69.2965i q^{41} -67.8233 q^{43} +76.7333i q^{47} -3.00000 q^{49} +83.4386i q^{59} -70.0000 q^{61} +108.517 q^{67} +118.794i q^{71} +13.5647 q^{73} -67.1416i q^{77} -30.0000 q^{79} -134.283i q^{83} -32.5269i q^{89} -138.000 q^{91} -94.9526 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 48 q^{19} - 152 q^{31} - 12 q^{49} - 280 q^{61} - 120 q^{79} - 552 q^{91}+O(q^{100}) $ 4 * q - 48 * q^19 - 152 * q^31 - 12 * q^49 - 280 * q^61 - 120 * q^79 - 552 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/900\mathbb{Z}\right)^\times$.

$n$	$101$	$451$	$577$
$\chi(n)$	$-1$	$1$	$1$

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	0
$5$	0	0
$6$	0	0
$7$	−6.78233	−0.968904	−0.484452	−	0.874818i	$-0.660981\pi$
$7$	−6.78233	−0.968904	−0.484452	+	0.874818i	$0.660981\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	9.89949i	0.899954i	0.893040	+	0.449977i	$0.148568\pi$
$11$	9.89949i	0.899954i	−0.893040	+	0.449977i	$0.851432\pi$
$12$	0	0
$13$	20.3470	1.56515	0.782577	−	0.622554i	$-0.213905\pi$
$13$	20.3470	1.56515	0.782577	+	0.622554i	$0.213905\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 19.1833i	− 1.12843i	−0.825628	−	0.564215i	$-0.809179\pi$
$17$	− 19.1833i	− 1.12843i	0.825628	−	0.564215i	$-0.190821\pi$
$18$	0	0
$19$	−12.0000	−0.631579	−0.315789	−	0.948829i	$-0.602269\pi$
$19$	−12.0000	−0.631579	−0.315789	+	0.948829i	$0.602269\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	9.59166i	0.417029i	0.978019	+	0.208514i	$0.0668628\pi$
$23$	9.59166i	0.417029i	−0.978019	+	0.208514i	$0.933137\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	8.48528i	0.292596i	0.989241	+	0.146298i	$0.0467358\pi$
$29$	8.48528i	0.292596i	−0.989241	+	0.146298i	$0.953264\pi$
$30$	0	0
$31$	−38.0000	−1.22581	−0.612903	−	0.790158i	$-0.709998\pi$
$31$	−38.0000	−1.22581	−0.612903	+	0.790158i	$0.709998\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−6.78233	−0.183306	−0.0916531	−	0.995791i	$-0.529215\pi$
$37$	−6.78233	−0.183306	−0.0916531	+	0.995791i	$0.529215\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	69.2965i	1.69016i	0.534642	+	0.845079i	$0.320447\pi$
$41$	69.2965i	1.69016i	−0.534642	+	0.845079i	$0.679553\pi$
$42$	0	0
$43$	−67.8233	−1.57729	−0.788643	−	0.614851i	$-0.789216\pi$
$43$	−67.8233	−1.57729	−0.788643	+	0.614851i	$0.789216\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	76.7333i	1.63262i	0.577612	+	0.816312i	$0.303985\pi$
$47$	76.7333i	1.63262i	−0.577612	+	0.816312i	$0.696015\pi$
$48$	0	0
$49$	−3.00000	−0.0612245
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	83.4386i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$59$	83.4386i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$60$	0	0
$61$	−70.0000	−1.14754	−0.573770	−	0.819016i	$-0.694520\pi$
$61$	−70.0000	−1.14754	−0.573770	+	0.819016i	$0.694520\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	108.517	1.61966	0.809830	−	0.586664i	$-0.199559\pi$
$67$	108.517	1.61966	0.809830	+	0.586664i	$0.199559\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	118.794i	1.67315i	0.547849	+	0.836577i	$0.315447\pi$
$71$	118.794i	1.67315i	−0.547849	+	0.836577i	$0.684553\pi$
$72$	0	0
$73$	13.5647	0.185817	0.0929086	−	0.995675i	$-0.470384\pi$
$73$	13.5647	0.185817	0.0929086	+	0.995675i	$0.470384\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 67.1416i	− 0.871969i
$78$	0	0
$79$	−30.0000	−0.379747	−0.189873	−	0.981809i	$-0.560808\pi$
$79$	−30.0000	−0.379747	−0.189873	+	0.981809i	$0.560808\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 134.283i	− 1.61787i	−0.587897	−	0.808935i	$-0.700044\pi$
$83$	− 134.283i	− 1.61787i	0.587897	−	0.808935i	$-0.299956\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 32.5269i	− 0.365471i	−0.983162	−	0.182735i	$-0.941505\pi$
$89$	− 32.5269i	− 0.365471i	0.983162	−	0.182735i	$-0.0584952\pi$
$90$	0	0
$91$	−138.000	−1.51648
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−94.9526	−0.978893	−0.489446	−	0.872033i	$-0.662801\pi$
$97$	−94.9526	−0.978893	−0.489446	+	0.872033i	$0.662801\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	79.1960i	0.784118i	0.919940	+	0.392059i	$0.128237\pi$
$101$	79.1960i	0.784118i	−0.919940	+	0.392059i	$0.871763\pi$
$102$	0	0
$103$	88.1703	0.856022	0.428011	−	0.903773i	$-0.359214\pi$
$103$	88.1703	0.856022	0.428011	+	0.903773i	$0.359214\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 57.5500i	− 0.537850i	−0.963161	−	0.268925i	$-0.913332\pi$
$107$	− 57.5500i	− 0.537850i	0.963161	−	0.268925i	$-0.0866684\pi$
$108$	0	0
$109$	−74.0000	−0.678899	−0.339450	−	0.940624i	$-0.610241\pi$
$109$	−74.0000	−0.678899	−0.339450	+	0.940624i	$0.610241\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	172.650i	1.52788i	0.645290	+	0.763938i	$0.276737\pi$
$113$	172.650i	1.52788i	−0.645290	+	0.763938i	$0.723263\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	130.108i	1.09334i
$120$	0	0
$121$	23.0000	0.190083
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−169.558	−1.33510	−0.667552	−	0.744563i	$-0.732658\pi$
$127$	−169.558	−1.33510	−0.667552	+	0.744563i	$0.732658\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	165.463i	1.26308i	0.775345	+	0.631538i	$0.217576\pi$
$131$	165.463i	1.26308i	−0.775345	+	0.631538i	$0.782424\pi$
$132$	0	0
$133$	81.3880	0.611940
$134$	0	0
$135$	0	0
$136$	0	0
$137$	115.100i	0.840146i	0.907490	+	0.420073i	$0.137995\pi$
$137$	115.100i	0.840146i	−0.907490	+	0.420073i	$0.862005\pi$
$138$	0	0
$139$	−62.0000	−0.446043	−0.223022	−	0.974814i	$-0.571592\pi$
$139$	−62.0000	−0.446043	−0.223022	+	0.974814i	$0.571592\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	201.425i	1.40857i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 158.392i	− 1.06303i	−0.847048	−	0.531517i	$-0.821622\pi$
$149$	− 158.392i	− 1.06303i	0.847048	−	0.531517i	$-0.178378\pi$
$150$	0	0
$151$	70.0000	0.463576	0.231788	−	0.972766i	$-0.425542\pi$
$151$	70.0000	0.463576	0.231788	+	0.972766i	$0.425542\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	47.4763	0.302397	0.151198	−	0.988503i	$-0.451687\pi$
$157$	47.4763	0.302397	0.151198	+	0.988503i	$0.451687\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 65.0538i	− 0.404061i
$162$	0	0
$163$	−94.9526	−0.582531	−0.291266	−	0.956642i	$-0.594076\pi$
$163$	−94.9526	−0.582531	−0.291266	+	0.956642i	$0.594076\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 201.425i	− 1.20614i	−0.797689	−	0.603069i	$-0.793945\pi$
$167$	− 201.425i	− 1.20614i	0.797689	−	0.603069i	$-0.206055\pi$
$168$	0	0
$169$	245.000	1.44970
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 268.567i	− 1.55241i	−0.630482	−	0.776204i	$-0.717143\pi$
$173$	− 268.567i	− 1.55241i	0.630482	−	0.776204i	$-0.282857\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	134.350i	0.750560i	0.926911	+	0.375280i	$0.122453\pi$
$179$	134.350i	0.750560i	−0.926911	+	0.375280i	$0.877547\pi$
$180$	0	0
$181$	−22.0000	−0.121547	−0.0607735	−	0.998152i	$-0.519357\pi$
$181$	−22.0000	−0.121547	−0.0607735	+	0.998152i	$0.519357\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	189.905	1.01554
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 195.161i	− 1.02179i	−0.859644	−	0.510894i	$-0.829314\pi$
$191$	− 195.161i	− 1.02179i	0.859644	−	0.510894i	$-0.170686\pi$
$192$	0	0
$193$	−271.293	−1.40566	−0.702832	−	0.711356i	$-0.748081\pi$
$193$	−271.293	−1.40566	−0.702832	+	0.711356i	$0.748081\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	163.058i	0.827707i	0.910344	+	0.413853i	$0.135817\pi$
$197$	163.058i	0.827707i	−0.910344	+	0.413853i	$0.864183\pi$
$198$	0	0
$199$	−294.000	−1.47739	−0.738693	−	0.674042i	$-0.764557\pi$
$199$	−294.000	−1.47739	−0.738693	+	0.674042i	$0.764557\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 57.5500i	− 0.283497i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 118.794i	− 0.568392i
$210$	0	0
$211$	−42.0000	−0.199052	−0.0995261	−	0.995035i	$-0.531733\pi$
$211$	−42.0000	−0.199052	−0.0995261	+	0.995035i	$0.531733\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	257.729	1.18769
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 390.323i	− 1.76617i
$222$	0	0
$223$	196.688	0.882007	0.441004	−	0.897505i	$-0.354623\pi$
$223$	196.688	0.882007	0.441004	+	0.897505i	$0.354623\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	268.567i	1.18311i	0.806264	+	0.591556i	$0.201486\pi$
$227$	268.567i	1.18311i	−0.806264	+	0.591556i	$0.798514\pi$
$228$	0	0
$229$	422.000	1.84279	0.921397	−	0.388622i	$-0.127049\pi$
$229$	422.000	1.84279	0.921397	+	0.388622i	$0.127049\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 211.017i	− 0.905651i	−0.891599	−	0.452825i	$-0.850416\pi$
$233$	− 211.017i	− 0.905651i	0.891599	−	0.452825i	$-0.149584\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 70.7107i	− 0.295861i	−0.988998	−	0.147930i	$-0.952739\pi$
$239$	− 70.7107i	− 0.295861i	0.988998	−	0.147930i	$-0.0472611\pi$
$240$	0	0
$241$	280.000	1.16183	0.580913	−	0.813966i	$-0.302696\pi$
$241$	280.000	1.16183	0.580913	+	0.813966i	$0.302696\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−244.164	−0.988518
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 80.6102i	− 0.321156i	−0.987023	−	0.160578i	$-0.948664\pi$
$251$	− 80.6102i	− 0.321156i	0.987023	−	0.160578i	$-0.0513358\pi$
$252$	0	0
$253$	−94.9526	−0.375307
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	46.0000	0.177606
$260$	0	0
$261$	0	0
$262$	0	0
$263$	67.1416i	0.255291i	0.991820	+	0.127646i	$0.0407420\pi$
$263$	67.1416i	0.255291i	−0.991820	+	0.127646i	$0.959258\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 263.044i	− 0.977858i	−0.872324	−	0.488929i	$-0.837388\pi$
$269$	− 263.044i	− 0.977858i	0.872324	−	0.488929i	$-0.162612\pi$
$270$	0	0
$271$	322.000	1.18819	0.594096	−	0.804394i	$-0.297510\pi$
$271$	322.000	1.18819	0.594096	+	0.804394i	$0.297510\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	33.9116	0.122425	0.0612124	−	0.998125i	$-0.480503\pi$
$277$	33.9116	0.122425	0.0612124	+	0.998125i	$0.480503\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 326.683i	− 1.16257i	−0.813699	−	0.581287i	$-0.802549\pi$
$281$	− 326.683i	− 1.16257i	0.813699	−	0.581287i	$-0.197451\pi$
$282$	0	0
$283$	108.517	0.383453	0.191727	−	0.981448i	$-0.438591\pi$
$283$	108.517	0.383453	0.191727	+	0.981448i	$0.438591\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 469.991i	− 1.63760i
$288$	0	0
$289$	−79.0000	−0.273356
$290$	0	0
$291$	0	0
$292$	0	0
$293$	201.425i	0.687457i	0.939069	+	0.343729i	$0.111690\pi$
$293$	201.425i	0.687457i	−0.939069	+	0.343729i	$0.888310\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	195.161i	0.652714i
$300$	0	0
$301$	460.000	1.52824
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−162.776	−0.530215	−0.265107	−	0.964219i	$-0.585407\pi$
$307$	−162.776	−0.530215	−0.265107	+	0.964219i	$0.585407\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 178.191i	− 0.572961i	−0.958086	−	0.286481i	$-0.907515\pi$
$311$	− 178.191i	− 0.572961i	0.958086	−	0.286481i	$-0.0924854\pi$
$312$	0	0
$313$	54.2586	0.173350	0.0866751	−	0.996237i	$-0.472376\pi$
$313$	54.2586	0.173350	0.0866751	+	0.996237i	$0.472376\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 278.158i	− 0.877471i	−0.898616	−	0.438735i	$-0.855427\pi$
$317$	− 278.158i	− 0.877471i	0.898616	−	0.438735i	$-0.144573\pi$
$318$	0	0
$319$	−84.0000	−0.263323
$320$	0	0
$321$	0	0
$322$	0	0
$323$	230.200i	0.712693i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 520.431i	− 1.58186i
$330$	0	0
$331$	40.0000	0.120846	0.0604230	−	0.998173i	$-0.480755\pi$
$331$	40.0000	0.120846	0.0604230	+	0.998173i	$0.480755\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−311.987	−0.925778	−0.462889	−	0.886416i	$-0.653187\pi$
$337$	−311.987	−0.925778	−0.462889	+	0.886416i	$0.653187\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 376.181i	− 1.10317i
$342$	0	0
$343$	352.681	1.02822
$344$	0	0
$345$	0	0
$346$	0	0
$347$	402.850i	1.16095i	0.814278	+	0.580475i	$0.197133\pi$
$347$	402.850i	1.16095i	−0.814278	+	0.580475i	$0.802867\pi$
$348$	0	0
$349$	406.000	1.16332	0.581662	−	0.813431i	$-0.302403\pi$
$349$	406.000	1.16332	0.581662	+	0.813431i	$0.302403\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	537.133i	1.52162i	0.648973	+	0.760812i	$0.275199\pi$
$353$	537.133i	1.52162i	−0.648973	+	0.760812i	$0.724801\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	161.220i	0.449082i	0.974465	+	0.224541i	$0.0720882\pi$
$359$	161.220i	0.449082i	−0.974465	+	0.224541i	$0.927912\pi$
$360$	0	0
$361$	−217.000	−0.601108
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−250.946	−0.683777	−0.341889	−	0.939740i	$-0.611067\pi$
$367$	−250.946	−0.683777	−0.341889	+	0.939740i	$0.611067\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	101.735	0.272748	0.136374	−	0.990657i	$-0.456455\pi$
$373$	101.735	0.272748	0.136374	+	0.990657i	$0.456455\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	172.650i	0.457957i
$378$	0	0
$379$	−538.000	−1.41953	−0.709763	−	0.704441i	$-0.751198\pi$
$379$	−538.000	−1.41953	−0.709763	+	0.704441i	$0.751198\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 268.567i	− 0.701218i	−0.936522	−	0.350609i	$-0.885975\pi$
$383$	− 268.567i	− 0.701218i	0.936522	−	0.350609i	$-0.114025\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 494.975i	− 1.27243i	−0.771512	−	0.636214i	$-0.780499\pi$
$389$	− 494.975i	− 1.27243i	0.771512	−	0.636214i	$-0.219501\pi$
$390$	0	0
$391$	184.000	0.470588
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	278.076	0.700442	0.350221	−	0.936667i	$-0.386106\pi$
$397$	278.076	0.700442	0.350221	+	0.936667i	$0.386106\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 247.487i	− 0.617175i	−0.951196	−	0.308588i	$-0.900144\pi$
$401$	− 247.487i	− 0.617175i	0.951196	−	0.308588i	$-0.0998563\pi$
$402$	0	0
$403$	−773.186	−1.91857
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 67.1416i	− 0.164967i
$408$	0	0
$409$	242.000	0.591687	0.295844	−	0.955236i	$-0.404399\pi$
$409$	242.000	0.591687	0.295844	+	0.955236i	$0.404399\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 565.908i	− 1.37024i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	790.545i	1.88674i	0.331738	+	0.943372i	$0.392365\pi$
$419$	790.545i	1.88674i	−0.331738	+	0.943372i	$0.607635\pi$
$420$	0	0
$421$	358.000	0.850356	0.425178	−	0.905110i	$-0.360211\pi$
$421$	358.000	0.850356	0.425178	+	0.905110i	$0.360211\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	474.763	1.11186
$428$	0	0
$429$	0	0
$430$	0	0
$431$	14.1421i	0.0328124i	0.999865	+	0.0164062i	$0.00522249\pi$
$431$	14.1421i	0.0328124i	−0.999865	+	0.0164062i	$0.994778\pi$
$432$	0	0
$433$	257.729	0.595216	0.297608	−	0.954688i	$-0.403811\pi$
$433$	257.729	0.595216	0.297608	+	0.954688i	$0.403811\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 115.100i	− 0.263387i
$438$	0	0
$439$	690.000	1.57175	0.785877	−	0.618383i	$-0.212212\pi$
$439$	690.000	1.57175	0.785877	+	0.618383i	$0.212212\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 402.850i	− 0.909368i	−0.890653	−	0.454684i	$-0.849752\pi$
$443$	− 402.850i	− 0.909368i	0.890653	−	0.454684i	$-0.150248\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 80.6102i	− 0.179533i	−0.995963	−	0.0897663i	$-0.971388\pi$
$449$	− 80.6102i	− 0.179533i	0.995963	−	0.0897663i	$-0.0286120\pi$
$450$	0	0
$451$	−686.000	−1.52106
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	339.116	0.742049	0.371025	−	0.928623i	$-0.379007\pi$
$457$	339.116	0.742049	0.371025	+	0.928623i	$0.379007\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	644.881i	1.39888i	0.714694	+	0.699438i	$0.246566\pi$
$461$	644.881i	1.39888i	−0.714694	+	0.699438i	$0.753434\pi$
$462$	0	0
$463$	−264.511	−0.571298	−0.285649	−	0.958334i	$-0.592209\pi$
$463$	−264.511	−0.571298	−0.285649	+	0.958334i	$0.592209\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	460.400i	0.985867i	0.870067	+	0.492933i	$0.164075\pi$
$467$	460.400i	0.985867i	−0.870067	+	0.492933i	$0.835925\pi$
$468$	0	0
$469$	−736.000	−1.56930
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 671.416i	− 1.41949i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 79.1960i	− 0.165336i	−0.996577	−	0.0826680i	$-0.973656\pi$
$479$	− 79.1960i	− 0.165336i	0.996577	−	0.0826680i	$-0.0263441\pi$
$480$	0	0
$481$	−138.000	−0.286902
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	793.533	1.62943	0.814715	−	0.579861i	$-0.196893\pi$
$487$	793.533	1.62943	0.814715	+	0.579861i	$0.196893\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	564.271i	1.14923i	0.818424	+	0.574614i	$0.194848\pi$
$491$	564.271i	1.14923i	−0.818424	+	0.574614i	$0.805152\pi$
$492$	0	0
$493$	162.776	0.330174
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 805.700i	− 1.62113i
$498$	0	0
$499$	72.0000	0.144289	0.0721443	−	0.997394i	$-0.477016\pi$
$499$	72.0000	0.144289	0.0721443	+	0.997394i	$0.477016\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	230.200i	0.457654i	0.973467	+	0.228827i	$0.0734890\pi$
$503$	230.200i	0.457654i	−0.973467	+	0.228827i	$0.926511\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	492.146i	0.966889i	0.875375	+	0.483444i	$0.160614\pi$
$509$	492.146i	0.966889i	−0.875375	+	0.483444i	$0.839386\pi$
$510$	0	0
$511$	−92.0000	−0.180039
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−759.621	−1.46929
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 69.2965i	− 0.133007i	−0.997786	−	0.0665033i	$-0.978816\pi$
$521$	− 69.2965i	− 0.133007i	0.997786	−	0.0665033i	$-0.0211843\pi$
$522$	0	0
$523$	−339.116	−0.648406	−0.324203	−	0.945987i	$-0.605096\pi$
$523$	−339.116	−0.648406	−0.324203	+	0.945987i	$0.605096\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	728.966i	1.38324i
$528$	0	0
$529$	437.000	0.826087
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1409.97i	2.64536i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 29.6985i	− 0.0550992i
$540$	0	0
$541$	−590.000	−1.09057	−0.545287	−	0.838250i	$-0.683579\pi$
$541$	−590.000	−1.09057	−0.545287	+	0.838250i	$0.683579\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−230.599	−0.421571	−0.210785	−	0.977532i	$-0.567602\pi$
$547$	−230.599	−0.421571	−0.210785	+	0.977532i	$0.567602\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 101.823i	− 0.184797i
$552$	0	0
$553$	203.470	0.367938
$554$	0	0
$555$	0	0
$556$	0	0
$557$	690.600i	1.23986i	0.784659	+	0.619928i	$0.212838\pi$
$557$	690.600i	1.23986i	−0.784659	+	0.619928i	$0.787162\pi$
$558$	0	0
$559$	−1380.00	−2.46869
$560$	0	0
$561$	0	0
$562$	0	0
$563$	19.1833i	0.0340734i	0.999855	+	0.0170367i	$0.00542321\pi$
$563$	19.1833i	0.0340734i	−0.999855	+	0.0170367i	$0.994577\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 108.894i	− 0.191379i	−0.995411	−	0.0956893i	$-0.969494\pi$
$569$	− 108.894i	− 0.191379i	0.995411	−	0.0956893i	$-0.0305055\pi$
$570$	0	0
$571$	−704.000	−1.23292	−0.616462	−	0.787384i	$-0.711435\pi$
$571$	−704.000	−1.23292	−0.616462	+	0.787384i	$0.711435\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−1125.87	−1.95124	−0.975621	−	0.219462i	$-0.929570\pi$
$577$	−1125.87	−1.95124	−0.975621	+	0.219462i	$0.929570\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	910.754i	1.56756i
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 978.350i	− 1.66669i	−0.552750	−	0.833347i	$-0.686422\pi$
$587$	− 978.350i	− 1.66669i	0.552750	−	0.833347i	$-0.313578\pi$
$588$	0	0
$589$	456.000	0.774194
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	296.985i	0.495801i	0.968785	+	0.247901i	$0.0797406\pi$
$599$	296.985i	0.495801i	−0.968785	+	0.247901i	$0.920259\pi$
$600$	0	0
$601$	20.0000	0.0332779	0.0166389	−	0.999862i	$-0.494703\pi$
$601$	20.0000	0.0332779	0.0166389	+	0.999862i	$0.494703\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	6.78233	0.0111735	0.00558676	−	0.999984i	$-0.498222\pi$
$607$	6.78233	0.0111735	0.00558676	+	0.999984i	$0.498222\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1561.29i	2.55531i
$612$	0	0
$613$	−47.4763	−0.0774491	−0.0387246	−	0.999250i	$-0.512329\pi$
$613$	−47.4763	−0.0774491	−0.0387246	+	0.999250i	$0.512329\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	38.3667i	0.0621826i	0.999517	+	0.0310913i	$0.00989826\pi$
$617$	38.3667i	0.0621826i	−0.999517	+	0.0310913i	$0.990102\pi$
$618$	0	0
$619$	−178.000	−0.287561	−0.143780	−	0.989610i	$-0.545926\pi$
$619$	−178.000	−0.287561	−0.143780	+	0.989610i	$0.545926\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	220.608i	0.354106i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	130.108i	0.206848i
$630$	0	0
$631$	−154.000	−0.244057	−0.122029	−	0.992527i	$-0.538940\pi$
$631$	−154.000	−0.244057	−0.122029	+	0.992527i	$0.538940\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−61.0410	−0.0958257
$638$	0	0
$639$	0	0
$640$	0	0
$641$	807.516i	1.25978i	0.776686	+	0.629888i	$0.216899\pi$
$641$	807.516i	1.25978i	−0.776686	+	0.629888i	$0.783101\pi$
$642$	0	0
$643$	1017.35	1.58219	0.791096	−	0.611692i	$-0.209511\pi$
$643$	1017.35	1.58219	0.791096	+	0.611692i	$0.209511\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 805.700i	− 1.24529i	−0.782506	−	0.622643i	$-0.786059\pi$
$647$	− 805.700i	− 1.24529i	0.782506	−	0.622643i	$-0.213941\pi$
$648$	0	0
$649$	−826.000	−1.27273
$650$	0	0
$651$	0	0
$652$	0	0
$653$	67.1416i	0.102820i	0.998678	+	0.0514101i	$0.0163716\pi$
$653$	67.1416i	0.102820i	−0.998678	+	0.0514101i	$0.983628\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	524.673i	0.796166i	0.917349	+	0.398083i	$0.130324\pi$
$659$	524.673i	0.796166i	−0.917349	+	0.398083i	$0.869676\pi$
$660$	0	0
$661$	74.0000	0.111952	0.0559758	−	0.998432i	$-0.482173\pi$
$661$	74.0000	0.111952	0.0559758	+	0.998432i	$0.482173\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−81.3880	−0.122021
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 692.965i	− 1.03273i
$672$	0	0
$673$	691.798	1.02793	0.513966	−	0.857811i	$-0.328176\pi$
$673$	691.798	1.02793	0.513966	+	0.857811i	$0.328176\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 278.158i	− 0.410869i	−0.978671	−	0.205434i	$-0.934139\pi$
$677$	− 278.158i	− 0.410869i	0.978671	−	0.205434i	$-0.0658607\pi$
$678$	0	0
$679$	644.000	0.948454
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 422.033i	− 0.617911i	−0.951077	−	0.308955i	$-0.900021\pi$
$683$	− 422.033i	− 0.617911i	0.951077	−	0.308955i	$-0.0999794\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−68.0000	−0.0984081	−0.0492041	−	0.998789i	$-0.515668\pi$
$691$	−68.0000	−0.0984081	−0.0492041	+	0.998789i	$0.515668\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1329.34	1.90723
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 130.108i	− 0.185603i	−0.995685	−	0.0928015i	$-0.970418\pi$
$701$	− 130.108i	− 0.185603i	0.995685	−	0.0928015i	$-0.0295822\pi$
$702$	0	0
$703$	81.3880	0.115772
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 537.133i	− 0.759736i
$708$	0	0
$709$	210.000	0.296192	0.148096	−	0.988973i	$-0.452686\pi$
$709$	210.000	0.296192	0.148096	+	0.988973i	$0.452686\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 364.483i	− 0.511197i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 121.622i	− 0.169155i	−0.996417	−	0.0845774i	$-0.973046\pi$
$719$	− 121.622i	− 0.169155i	0.996417	−	0.0845774i	$-0.0269540\pi$
$720$	0	0
$721$	−598.000	−0.829404
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	20.3470	0.0279876	0.0139938	−	0.999902i	$-0.495545\pi$
$727$	20.3470	0.0279876	0.0139938	+	0.999902i	$0.495545\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1301.08i	1.77986i
$732$	0	0
$733$	−1417.51	−1.93384	−0.966922	−	0.255074i	$-0.917900\pi$
$733$	−1417.51	−1.93384	−0.966922	+	0.255074i	$0.917900\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1074.27i	1.45762i
$738$	0	0
$739$	−416.000	−0.562923	−0.281461	−	0.959573i	$-0.590819\pi$
$739$	−416.000	−0.562923	−0.281461	+	0.959573i	$0.590819\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 805.700i	− 1.08439i	−0.840254	−	0.542194i	$-0.817594\pi$
$743$	− 805.700i	− 1.08439i	0.840254	−	0.542194i	$-0.182406\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	390.323i	0.521125i
$750$	0	0
$751$	86.0000	0.114514	0.0572570	−	0.998359i	$-0.481765\pi$
$751$	86.0000	0.114514	0.0572570	+	0.998359i	$0.481765\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−47.4763	−0.0627164	−0.0313582	−	0.999508i	$-0.509983\pi$
$757$	−47.4763	−0.0627164	−0.0313582	+	0.999508i	$0.509983\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	369.110i	0.485033i	0.970147	+	0.242516i	$0.0779728\pi$
$761$	369.110i	0.485033i	−0.970147	+	0.242516i	$0.922027\pi$
$762$	0	0
$763$	501.892	0.657788
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1697.72i	2.21346i
$768$	0	0
$769$	−384.000	−0.499350	−0.249675	−	0.968330i	$-0.580324\pi$
$769$	−384.000	−0.499350	−0.249675	+	0.968330i	$0.580324\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 201.425i	− 0.260576i	−0.991476	−	0.130288i	$-0.958410\pi$
$773$	− 201.425i	− 0.260576i	0.991476	−	0.130288i	$-0.0415901\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 831.558i	− 1.06747i
$780$	0	0
$781$	−1176.00	−1.50576
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−474.763	−0.603257	−0.301628	−	0.953426i	$-0.597530\pi$
$787$	−474.763	−0.603257	−0.301628	+	0.953426i	$0.597530\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 1170.97i	− 1.48037i
$792$	0	0
$793$	−1424.29	−1.79608
$794$	0	0
$795$	0	0
$796$	0	0
$797$	498.766i	0.625805i	0.949785	+	0.312902i	$0.101301\pi$
$797$	498.766i	0.625805i	−0.949785	+	0.312902i	$0.898699\pi$
$798$	0	0
$799$	1472.00	1.84230
$800$	0	0
$801$	0	0
$802$	0	0
$803$	134.283i	0.167227i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1480.68i	1.83026i	0.403157	+	0.915131i	$0.367913\pi$
$809$	1480.68i	1.83026i	−0.403157	+	0.915131i	$0.632087\pi$
$810$	0	0
$811$	−602.000	−0.742293	−0.371147	−	0.928574i	$-0.621035\pi$
$811$	−602.000	−0.742293	−0.371147	+	0.928574i	$0.621035\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	813.880	0.996181
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 1405.73i	− 1.71221i	−0.516798	−	0.856107i	$-0.672876\pi$
$821$	− 1405.73i	− 1.71221i	0.516798	−	0.856107i	$-0.327124\pi$
$822$	0	0
$823$	−1281.86	−1.55755	−0.778773	−	0.627306i	$-0.784158\pi$
$823$	−1281.86	−1.55755	−0.778773	+	0.627306i	$0.784158\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 1170.18i	− 1.41497i	−0.706727	−	0.707487i	$-0.749829\pi$
$827$	− 1170.18i	− 1.41497i	0.706727	−	0.707487i	$-0.250171\pi$
$828$	0	0
$829$	−1074.00	−1.29554	−0.647768	−	0.761837i	$-0.724298\pi$
$829$	−1074.00	−1.29554	−0.647768	+	0.761837i	$0.724298\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	57.5500i	0.0690876i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 743.876i	− 0.886623i	−0.896368	−	0.443311i	$-0.853804\pi$
$839$	− 743.876i	− 0.886623i	0.896368	−	0.443311i	$-0.146196\pi$
$840$	0	0
$841$	769.000	0.914388
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−155.994	−0.184172
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 65.0538i	− 0.0764440i
$852$	0	0
$853$	169.558	0.198779	0.0993894	−	0.995049i	$-0.468311\pi$
$853$	169.558	0.198779	0.0993894	+	0.995049i	$0.468311\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 115.100i	− 0.134306i	−0.997743	−	0.0671528i	$-0.978608\pi$
$857$	− 115.100i	− 0.134306i	0.997743	−	0.0671528i	$-0.0213915\pi$
$858$	0	0
$859$	−154.000	−0.179278	−0.0896391	−	0.995974i	$-0.528571\pi$
$859$	−154.000	−0.179278	−0.0896391	+	0.995974i	$0.528571\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1064.67i	1.23369i	0.787085	+	0.616845i	$0.211589\pi$
$863$	1064.67i	1.23369i	−0.787085	+	0.616845i	$0.788411\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 296.985i	− 0.341755i
$870$	0	0
$871$	2208.00	2.53502
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1390.38	1.58538	0.792690	−	0.609625i	$-0.208680\pi$
$877$	1390.38	1.58538	0.792690	+	0.609625i	$0.208680\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	861.256i	0.977589i	0.872399	+	0.488795i	$0.162563\pi$
$881$	861.256i	0.977589i	−0.872399	+	0.488795i	$0.837437\pi$
$882$	0	0
$883$	1017.35	1.15215	0.576076	−	0.817396i	$-0.304583\pi$
$883$	1017.35	1.15215	0.576076	+	0.817396i	$0.304583\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	604.275i	0.681257i	0.940198	+	0.340628i	$0.110640\pi$
$887$	604.275i	0.681257i	−0.940198	+	0.340628i	$0.889360\pi$
$888$	0	0
$889$	1150.00	1.29359
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 920.800i	− 1.03113i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 322.441i	− 0.358666i
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−935.962	−1.03193	−0.515966	−	0.856609i	$-0.672567\pi$
$907$	−935.962	−1.03193	−0.515966	+	0.856609i	$0.672567\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 763.675i	− 0.838282i	−0.907921	−	0.419141i	$-0.862331\pi$
$911$	− 763.675i	− 0.838282i	0.907921	−	0.419141i	$-0.137669\pi$
$912$	0	0
$913$	1329.34	1.45601
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 1122.22i	− 1.22380i
$918$	0	0
$919$	54.0000	0.0587595	0.0293798	−	0.999568i	$-0.490647\pi$
$919$	54.0000	0.0587595	0.0293798	+	0.999568i	$0.490647\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	2417.10i	2.61874i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 148.492i	− 0.159841i	−0.996801	−	0.0799206i	$-0.974533\pi$
$929$	− 148.492i	− 0.159841i	0.996801	−	0.0799206i	$-0.0254667\pi$
$930$	0	0
$931$	36.0000	0.0386681
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−786.750	−0.839648	−0.419824	−	0.907606i	$-0.637908\pi$
$937$	−786.750	−0.839648	−0.419824	+	0.907606i	$0.637908\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 342.240i	− 0.363698i	−0.983326	−	0.181849i	$-0.941792\pi$
$941$	− 342.240i	− 0.363698i	0.983326	−	0.181849i	$-0.0582082\pi$
$942$	0	0
$943$	−664.668	−0.704844
$944$	0	0
$945$	0	0
$946$	0	0
$947$	402.850i	0.425396i	0.977118	+	0.212698i	$0.0682250\pi$
$947$	402.850i	0.425396i	−0.977118	+	0.212698i	$0.931775\pi$
$948$	0	0
$949$	276.000	0.290832
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 1074.27i	− 1.12725i	−0.826032	−	0.563623i	$-0.809407\pi$
$953$	− 1074.27i	− 1.12725i	0.826032	−	0.563623i	$-0.190593\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 780.646i	− 0.814021i
$960$	0	0
$961$	483.000	0.502601
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−400.157	−0.413813	−0.206907	−	0.978361i	$-0.566340\pi$
$967$	−400.157	−0.413813	−0.206907	+	0.978361i	$0.566340\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 683.065i	− 0.703466i	−0.936100	−	0.351733i	$-0.885593\pi$
$971$	− 683.065i	− 0.703466i	0.936100	−	0.351733i	$-0.114407\pi$
$972$	0	0
$973$	420.504	0.432173
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 1611.40i	− 1.64933i	−0.565618	−	0.824667i	$-0.691363\pi$
$977$	− 1611.40i	− 1.64933i	0.565618	−	0.824667i	$-0.308637\pi$
$978$	0	0
$979$	322.000	0.328907
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 959.166i	− 0.975754i	−0.872912	−	0.487877i	$-0.837771\pi$
$983$	− 959.166i	− 0.975754i	0.872912	−	0.487877i	$-0.162229\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 650.538i	− 0.657774i
$990$	0	0
$991$	1054.00	1.06357	0.531786	−	0.846879i	$-0.321521\pi$
$991$	1054.00	1.06357	0.531786	+	0.846879i	$0.321521\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−1200.47	−1.20408	−0.602042	−	0.798464i	$-0.705646\pi$
$997$	−1200.47	−1.20408	−0.602042	+	0.798464i	$0.705646\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.3.g.c.701.2		4
3.2	odd	2	inner	900.3.g.c.701.1		4
4.3	odd	2		3600.3.l.r.1601.3		4
5.2	odd	4		180.3.b.a.89.3	yes	4
5.3	odd	4		180.3.b.a.89.1	✓	4
5.4	even	2	inner	900.3.g.c.701.4		4
12.11	even	2		3600.3.l.r.1601.4		4
15.2	even	4		180.3.b.a.89.2	yes	4
15.8	even	4		180.3.b.a.89.4	yes	4
15.14	odd	2	inner	900.3.g.c.701.3		4
20.3	even	4		720.3.c.b.449.1		4
20.7	even	4		720.3.c.b.449.3		4
20.19	odd	2		3600.3.l.r.1601.1		4
40.3	even	4		2880.3.c.f.449.4		4
40.13	odd	4		2880.3.c.c.449.4		4
40.27	even	4		2880.3.c.f.449.2		4
40.37	odd	4		2880.3.c.c.449.2		4
45.2	even	12		1620.3.t.c.1349.3		8
45.7	odd	12		1620.3.t.c.1349.2		8
45.13	odd	12		1620.3.t.c.269.3		8
45.22	odd	12		1620.3.t.c.269.1		8
45.23	even	12		1620.3.t.c.269.2		8
45.32	even	12		1620.3.t.c.269.4		8
45.38	even	12		1620.3.t.c.1349.1		8
45.43	odd	12		1620.3.t.c.1349.4		8
60.23	odd	4		720.3.c.b.449.4		4
60.47	odd	4		720.3.c.b.449.2		4
60.59	even	2		3600.3.l.r.1601.2		4
120.53	even	4		2880.3.c.c.449.1		4
120.77	even	4		2880.3.c.c.449.3		4
120.83	odd	4		2880.3.c.f.449.1		4
120.107	odd	4		2880.3.c.f.449.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.3.b.a.89.1	✓	4	5.3	odd	4
180.3.b.a.89.2	yes	4	15.2	even	4
180.3.b.a.89.3	yes	4	5.2	odd	4
180.3.b.a.89.4	yes	4	15.8	even	4
720.3.c.b.449.1		4	20.3	even	4
720.3.c.b.449.2		4	60.47	odd	4
720.3.c.b.449.3		4	20.7	even	4
720.3.c.b.449.4		4	60.23	odd	4
900.3.g.c.701.1		4	3.2	odd	2	inner
900.3.g.c.701.2		4	1.1	even	1	trivial
900.3.g.c.701.3		4	15.14	odd	2	inner
900.3.g.c.701.4		4	5.4	even	2	inner
1620.3.t.c.269.1		8	45.22	odd	12
1620.3.t.c.269.2		8	45.23	even	12
1620.3.t.c.269.3		8	45.13	odd	12
1620.3.t.c.269.4		8	45.32	even	12
1620.3.t.c.1349.1		8	45.38	even	12
1620.3.t.c.1349.2		8	45.7	odd	12
1620.3.t.c.1349.3		8	45.2	even	12
1620.3.t.c.1349.4		8	45.43	odd	12
2880.3.c.c.449.1		4	120.53	even	4
2880.3.c.c.449.2		4	40.37	odd	4
2880.3.c.c.449.3		4	120.77	even	4
2880.3.c.c.449.4		4	40.13	odd	4
2880.3.c.f.449.1		4	120.83	odd	4
2880.3.c.f.449.2		4	40.27	even	4
2880.3.c.f.449.3		4	120.107	odd	4
2880.3.c.f.449.4		4	40.3	even	4
3600.3.l.r.1601.1		4	20.19	odd	2
3600.3.l.r.1601.2		4	60.59	even	2
3600.3.l.r.1601.3		4	4.3	odd	2
3600.3.l.r.1601.4		4	12.11	even	2

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:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	900.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-6.78233 q^{7} +O(q^{10})\)	\(q-6.78233 q^{7} +9.89949i q^{11} +20.3470 q^{13} -19.1833i q^{17} -12.0000 q^{19} +9.59166i q^{23} +8.48528i q^{29} -38.0000 q^{31} -6.78233 q^{37} +69.2965i q^{41} -67.8233 q^{43} +76.7333i q^{47} -3.00000 q^{49} +83.4386i q^{59} -70.0000 q^{61} +108.517 q^{67} +118.794i q^{71} +13.5647 q^{73} -67.1416i q^{77} -30.0000 q^{79} -134.283i q^{83} -32.5269i q^{89} -138.000 q^{91} -94.9526 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 48 q^{19} - 152 q^{31} - 12 q^{49} - 280 q^{61} - 120 q^{79} - 552 q^{91}+O(q^{100}) \) 4 * q - 48 * q^19 - 152 * q^31 - 12 * q^49 - 280 * q^61 - 120 * q^79 - 552 * q^91

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