LMFDB - Embedded newform 900.3.c.i.451.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [900,3,Mod(451,900)]

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([1, 0, 0]))

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

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

chi := DirichletCharacter("900.451");

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.c (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:	$2$
Coefficient field:	$\Q(\sqrt{-15}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 4 $ x^2 - x + 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			451.2
Root			$0.500000 - 1.93649i$ of defining polynomial
Character	$\chi$	$=$	900.451
Dual form			900.3.c.i.451.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 + 1.93649i) q^{2} +(-3.50000 + 1.93649i) q^{4} +(-5.50000 - 5.80948i) q^{8} +O(q^{10})$	$q+(0.500000 + 1.93649i) q^{2} +(-3.50000 + 1.93649i) q^{4} +(-5.50000 - 5.80948i) q^{8} +(8.50000 - 13.5554i) q^{16} +14.0000 q^{17} -30.9839i q^{19} -30.9839i q^{23} +61.9677i q^{31} +(30.5000 + 9.68246i) q^{32} +(7.00000 + 27.1109i) q^{34} +(60.0000 - 15.4919i) q^{38} +(60.0000 - 15.4919i) q^{46} -92.9516i q^{47} +49.0000 q^{49} +86.0000 q^{53} +118.000 q^{61} +(-120.000 + 30.9839i) q^{62} +(-3.50000 + 63.9042i) q^{64} +(-49.0000 + 27.1109i) q^{68} +(60.0000 + 108.444i) q^{76} +123.935i q^{79} -61.9677i q^{83} +(60.0000 + 108.444i) q^{92} +(180.000 - 46.4758i) q^{94} +(24.5000 + 94.8881i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - 7 q^{4} - 11 q^{8}+O(q^{10}) $ 2 * q + q^2 - 7 * q^4 - 11 * q^8	$ 2 q + q^{2} - 7 q^{4} - 11 q^{8} + 17 q^{16} + 28 q^{17} + 61 q^{32} + 14 q^{34} + 120 q^{38} + 120 q^{46} + 98 q^{49} + 172 q^{53} + 236 q^{61} - 240 q^{62} - 7 q^{64} - 98 q^{68} + 120 q^{76} + 120 q^{92} + 360 q^{94} + 49 q^{98}+O(q^{100}) $ 2 * q + q^2 - 7 * q^4 - 11 * q^8 + 17 * q^16 + 28 * q^17 + 61 * q^32 + 14 * q^34 + 120 * q^38 + 120 * q^46 + 98 * q^49 + 172 * q^53 + 236 * q^61 - 240 * q^62 - 7 * q^64 - 98 * q^68 + 120 * q^76 + 120 * q^92 + 360 * q^94 + 49 * q^98

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 $

$2$	0.500000	+	1.93649i	0.250000	+	0.968246i
$2$	0.500000	+	1.93649i	0.250000	+	0.968246i
$3$		0			0
$3$		0			0
$4$	−3.50000	+	1.93649i	−0.875000	+	0.484123i
$5$		0			0
$5$		0			0
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$	−5.50000	−	5.80948i	−0.687500	−	0.726184i
$9$		0			0
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$		0			0
$16$	8.50000	−	13.5554i	0.531250	−	0.847215i
$17$	14.0000			0.823529			0.411765	−	0.911290i	$-0.364913\pi$
$17$	14.0000			0.823529			0.411765	+	0.911290i	$0.364913\pi$
$18$		0			0
$19$		−	30.9839i		−	1.63073i	−0.578947	−	0.815365i	$-0.696536\pi$
$19$		−	30.9839i		−	1.63073i	0.578947	−	0.815365i	$-0.303464\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	30.9839i		−	1.34712i	−0.739130	−	0.673562i	$-0.764763\pi$
$23$		−	30.9839i		−	1.34712i	0.739130	−	0.673562i	$-0.235237\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$			61.9677i			1.99896i	0.0322581	+	0.999480i	$0.489730\pi$
$31$			61.9677i			1.99896i	−0.0322581	+	0.999480i	$0.510270\pi$
$32$	30.5000	+	9.68246i	0.953125	+	0.302577i
$33$		0			0
$34$	7.00000	+	27.1109i	0.205882	+	0.797379i
$35$		0			0
$36$		0			0
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$	60.0000	−	15.4919i	1.57895	−	0.407682i
$39$		0			0
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$		0			0
$46$	60.0000	−	15.4919i	1.30435	−	0.336781i
$47$		−	92.9516i		−	1.97769i	−0.148936	−	0.988847i	$-0.547585\pi$
$47$		−	92.9516i		−	1.97769i	0.148936	−	0.988847i	$-0.452415\pi$
$48$		0			0
$49$	49.0000			1.00000
$50$		0			0
$51$		0			0
$52$		0			0
$53$	86.0000			1.62264			0.811321	−	0.584601i	$-0.198749\pi$
$53$	86.0000			1.62264			0.811321	+	0.584601i	$0.198749\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$		0			0
$61$	118.000			1.93443			0.967213	−	0.253966i	$-0.0817352\pi$
$61$	118.000			1.93443			0.967213	+	0.253966i	$0.0817352\pi$
$62$	−120.000	+	30.9839i	−1.93548	+	0.499740i
$63$		0			0
$64$	−3.50000	+	63.9042i	−0.0546875	+	0.998504i
$65$		0			0
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$	−49.0000	+	27.1109i	−0.720588	+	0.398689i
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$		0			0
$75$		0			0
$76$	60.0000	+	108.444i	0.789474	+	1.42689i
$77$		0			0
$78$		0			0
$79$			123.935i			1.56880i	0.620253	+	0.784402i	$0.287030\pi$
$79$			123.935i			1.56880i	−0.620253	+	0.784402i	$0.712970\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	61.9677i		−	0.746599i	−0.927711	−	0.373300i	$-0.878226\pi$
$83$		−	61.9677i		−	0.746599i	0.927711	−	0.373300i	$-0.121774\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$	60.0000	+	108.444i	0.652174	+	1.17873i
$93$		0			0
$94$	180.000	−	46.4758i	1.91489	−	0.494423i
$95$		0			0
$96$		0			0
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$	24.5000	+	94.8881i	0.250000	+	0.968246i
$99$		0			0
$100$		0			0
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$	43.0000	+	166.538i	0.405660	+	1.57112i
$107$		−	185.903i		−	1.73741i	−0.495327	−	0.868707i	$-0.664952\pi$
$107$		−	185.903i		−	1.73741i	0.495327	−	0.868707i	$-0.335048\pi$
$108$		0			0
$109$	22.0000			0.201835			0.100917	−	0.994895i	$-0.467822\pi$
$109$	22.0000			0.201835			0.100917	+	0.994895i	$0.467822\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−206.000			−1.82301			−0.911504	−	0.411290i	$-0.865078\pi$
$113$	−206.000			−1.82301			−0.911504	+	0.411290i	$0.865078\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	121.000			1.00000
$122$	59.0000	+	228.506i	0.483607	+	1.87300i
$123$		0			0
$124$	−120.000	−	216.887i	−0.967742	−	1.74909i
$125$		0			0
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$	−125.500	+	25.1744i	−0.980469	+	0.196675i
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−77.0000	−	81.3327i	−0.566176	−	0.598034i
$137$	226.000			1.64964			0.824818	−	0.565399i	$-0.191278\pi$
$137$	226.000			1.64964			0.824818	+	0.565399i	$0.191278\pi$
$138$		0			0
$139$			92.9516i			0.668717i	0.942446	+	0.334358i	$0.108520\pi$
$139$			92.9516i			0.668717i	−0.942446	+	0.334358i	$0.891480\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$		−	185.903i		−	1.23115i	−0.788079	−	0.615574i	$-0.788924\pi$
$151$		−	185.903i		−	1.23115i	0.788079	−	0.615574i	$-0.211076\pi$
$152$	−180.000	+	170.411i	−1.18421	+	1.12113i
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$	−240.000	+	61.9677i	−1.51899	+	0.392201i
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$		0			0
$166$	120.000	−	30.9839i	0.722892	−	0.186650i
$167$		−	216.887i		−	1.29872i	−0.760479	−	0.649362i	$-0.775036\pi$
$167$		−	216.887i		−	1.29872i	0.760479	−	0.649362i	$-0.224964\pi$
$168$		0			0
$169$	−169.000			−1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$	154.000			0.890173			0.445087	−	0.895487i	$-0.353173\pi$
$173$	154.000			0.890173			0.445087	+	0.895487i	$0.353173\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$	−122.000			−0.674033			−0.337017	−	0.941499i	$-0.609418\pi$
$181$	−122.000			−0.674033			−0.337017	+	0.941499i	$0.609418\pi$
$182$		0			0
$183$		0			0
$184$	−180.000	+	170.411i	−0.978261	+	0.926148i
$185$		0			0
$186$		0			0
$187$		0			0
$188$	180.000	+	325.331i	0.957447	+	1.73048i
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$		0			0			−	1.00000i	$-0.5\pi$
$193$		0			0				1.00000i	$0.5\pi$
$194$		0			0
$195$		0			0
$196$	−171.500	+	94.8881i	−0.875000	+	0.484123i
$197$	−374.000			−1.89848			−0.949239	−	0.314557i	$-0.898144\pi$
$197$	−374.000			−1.89848			−0.949239	+	0.314557i	$0.898144\pi$
$198$		0			0
$199$		−	371.806i		−	1.86837i	−0.356784	−	0.934187i	$-0.616127\pi$
$199$		−	371.806i		−	1.86837i	0.356784	−	0.934187i	$-0.383873\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$			216.887i			1.02790i	0.857820	+	0.513950i	$0.171818\pi$
$211$			216.887i			1.02790i	−0.857820	+	0.513950i	$0.828182\pi$
$212$	−301.000	+	166.538i	−1.41981	+	0.785558i
$213$		0			0
$214$	360.000	−	92.9516i	1.68224	−	0.434353i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	11.0000	+	42.6028i	0.0504587	+	0.195426i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$		0			0
$226$	−103.000	−	398.917i	−0.455752	−	1.76512i
$227$		−	433.774i		−	1.91090i	−0.295154	−	0.955450i	$-0.595371\pi$
$227$		−	433.774i		−	1.91090i	0.295154	−	0.955450i	$-0.404629\pi$
$228$		0			0
$229$	218.000			0.951965			0.475983	−	0.879455i	$-0.342093\pi$
$229$	218.000			0.951965			0.475983	+	0.879455i	$0.342093\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−34.0000			−0.145923			−0.0729614	−	0.997335i	$-0.523245\pi$
$233$	−34.0000			−0.145923			−0.0729614	+	0.997335i	$0.523245\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	−478.000			−1.98340			−0.991701	−	0.128564i	$-0.958963\pi$
$241$	−478.000			−1.98340			−0.991701	+	0.128564i	$0.958963\pi$
$242$	60.5000	+	234.315i	0.250000	+	0.968246i
$243$		0			0
$244$	−413.000	+	228.506i	−1.69262	+	0.936500i
$245$		0			0
$246$		0			0
$247$		0			0
$248$	360.000	−	340.823i	1.45161	−	1.37428i
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−111.500	−	230.443i	−0.435547	−	0.900166i
$257$	466.000			1.81323			0.906615	−	0.421959i	$-0.138657\pi$
$257$	466.000			1.81323			0.906615	+	0.421959i	$0.138657\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	278.855i		−	1.06028i	−0.847909	−	0.530142i	$-0.822139\pi$
$263$		−	278.855i		−	1.06028i	0.847909	−	0.530142i	$-0.177861\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$			247.871i			0.914653i	0.889299	+	0.457326i	$0.151193\pi$
$271$			247.871i			0.914653i	−0.889299	+	0.457326i	$0.848807\pi$
$272$	119.000	−	189.776i	0.437500	−	0.697707i
$273$		0			0
$274$	113.000	+	437.647i	0.412409	+	1.59725i
$275$		0			0
$276$		0			0
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$	−180.000	+	46.4758i	−0.647482	+	0.167179i
$279$		0			0
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$		0			0
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−93.0000			−0.321799
$290$		0			0
$291$		0			0
$292$		0			0
$293$	394.000			1.34471			0.672355	−	0.740229i	$-0.265283\pi$
$293$	394.000			1.34471			0.672355	+	0.740229i	$0.265283\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$	360.000	−	92.9516i	1.19205	−	0.307787i
$303$		0			0
$304$	−420.000	−	263.363i	−1.38158	−	0.866325i
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$		0			0			−	1.00000i	$-0.5\pi$
$313$		0			0				1.00000i	$0.5\pi$
$314$		0			0
$315$		0			0
$316$	−240.000	−	433.774i	−0.759494	−	1.37270i
$317$	134.000			0.422713			0.211356	−	0.977409i	$-0.432212\pi$
$317$	134.000			0.422713			0.211356	+	0.977409i	$0.432212\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	433.774i		−	1.34295i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		−	650.661i		−	1.96574i	−0.184290	−	0.982872i	$-0.558999\pi$
$331$		−	650.661i		−	1.96574i	0.184290	−	0.982872i	$-0.441001\pi$
$332$	120.000	+	216.887i	0.361446	+	0.653274i
$333$		0			0
$334$	420.000	−	108.444i	1.25749	−	0.324681i
$335$		0			0
$336$		0			0
$337$		0			0			−	1.00000i	$-0.5\pi$
$337$		0			0				1.00000i	$0.5\pi$
$338$	−84.5000	−	327.267i	−0.250000	−	0.968246i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	77.0000	+	298.220i	0.222543	+	0.861907i
$347$			371.806i			1.07149i	0.844380	+	0.535744i	$0.179969\pi$
$347$			371.806i			1.07149i	−0.844380	+	0.535744i	$0.820031\pi$
$348$		0			0
$349$	−458.000			−1.31232			−0.656160	−	0.754621i	$-0.727821\pi$
$349$	−458.000			−1.31232			−0.656160	+	0.754621i	$0.727821\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−274.000			−0.776204			−0.388102	−	0.921616i	$-0.626869\pi$
$353$	−274.000			−0.776204			−0.388102	+	0.921616i	$0.626869\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	−599.000			−1.65928
$362$	−61.0000	−	236.252i	−0.168508	−	0.652630i
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$	−420.000	−	263.363i	−1.14130	−	0.715660i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$		0			0
$376$	−540.000	+	511.234i	−1.43617	+	1.35966i
$377$		0			0
$378$		0			0
$379$		−	154.919i		−	0.408758i	−0.978892	−	0.204379i	$-0.934482\pi$
$379$		−	154.919i		−	0.408758i	0.978892	−	0.204379i	$-0.0655175\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			340.823i			0.889876i	0.895561	+	0.444938i	$0.146774\pi$
$383$			340.823i			0.889876i	−0.895561	+	0.444938i	$0.853226\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		−	433.774i		−	1.10940i
$392$	−269.500	−	284.664i	−0.687500	−	0.726184i
$393$		0			0
$394$	−187.000	−	724.248i	−0.474619	−	1.83819i
$395$		0			0
$396$		0			0
$397$		0			0			−	1.00000i	$-0.5\pi$
$397$		0			0				1.00000i	$0.5\pi$
$398$	720.000	−	185.903i	1.80905	−	0.467093i
$399$		0			0
$400$		0			0
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	142.000			0.347188			0.173594	−	0.984817i	$-0.444462\pi$
$409$	142.000			0.347188			0.173594	+	0.984817i	$0.444462\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	602.000			1.42993			0.714964	−	0.699161i	$-0.246443\pi$
$421$	602.000			1.42993			0.714964	+	0.699161i	$0.246443\pi$
$422$	−420.000	+	108.444i	−0.995261	+	0.256975i
$423$		0			0
$424$	−473.000	−	499.615i	−1.11557	−	1.17834i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	360.000	+	650.661i	0.841121	+	1.52024i
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$	−77.0000	+	42.6028i	−0.176606	+	0.0977129i
$437$	−960.000			−2.19680
$438$		0			0
$439$			619.677i			1.41157i	0.708428	+	0.705783i	$0.249405\pi$
$439$			619.677i			1.41157i	−0.708428	+	0.705783i	$0.750595\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			681.645i			1.53870i	0.638826	+	0.769351i	$0.279420\pi$
$443$			681.645i			1.53870i	−0.638826	+	0.769351i	$0.720580\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$	721.000	−	398.917i	1.59513	−	0.882560i
$453$		0			0
$454$	840.000	−	216.887i	1.85022	−	0.477725i
$455$		0			0
$456$		0			0
$457$		0			0			−	1.00000i	$-0.5\pi$
$457$		0			0				1.00000i	$0.5\pi$
$458$	109.000	+	422.155i	0.237991	+	0.921736i
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$	−17.0000	−	65.8407i	−0.0364807	−	0.141289i
$467$			867.548i			1.85771i	0.370450	+	0.928853i	$0.379204\pi$
$467$			867.548i			1.85771i	−0.370450	+	0.928853i	$0.620796\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		1.00000			$0$
$479$		0			0		−1.00000			$\pi$
$480$		0			0
$481$		0			0
$482$	−239.000	−	925.643i	−0.495851	−	1.92042i
$483$		0			0
$484$	−423.500	+	234.315i	−0.875000	+	0.484123i
$485$		0			0
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$	−649.000	−	685.518i	−1.32992	−	1.40475i
$489$		0			0
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	840.000	+	526.726i	1.69355	+	1.06195i
$497$		0			0
$498$		0			0
$499$			340.823i			0.683011i	0.939880	+	0.341506i	$0.110937\pi$
$499$			340.823i			0.683011i	−0.939880	+	0.341506i	$0.889063\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			154.919i			0.307991i	0.988072	+	0.153995i	$0.0492141\pi$
$503$			154.919i			0.307991i	−0.988072	+	0.153995i	$0.950786\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$	390.500	−	331.140i	0.762695	−	0.646758i
$513$		0			0
$514$	233.000	+	902.405i	0.453307	+	1.75565i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$		0			0
$525$		0			0
$526$	540.000	−	139.427i	1.02662	−	0.265071i
$527$			867.548i			1.64620i
$528$		0			0
$529$	−431.000			−0.814745
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	1078.00			1.99261			0.996303	−	0.0859072i	$-0.0273789\pi$
$541$	1078.00			1.99261			0.996303	+	0.0859072i	$0.0273789\pi$
$542$	−480.000	+	123.935i	−0.885609	+	0.228663i
$543$		0			0
$544$	427.000	+	135.554i	0.784926	+	0.249181i
$545$		0			0
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$	−791.000	+	437.647i	−1.44343	+	0.798626i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−180.000	−	325.331i	−0.323741	−	0.585127i
$557$	614.000			1.10233			0.551167	−	0.834395i	$-0.314183\pi$
$557$	614.000			1.10233			0.551167	+	0.834395i	$0.314183\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$			1115.42i			1.98121i	0.136767	+	0.990603i	$0.456329\pi$
$563$			1115.42i			1.98121i	−0.136767	+	0.990603i	$0.543671\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$			1084.44i			1.89919i	0.313485	+	0.949593i	$0.398503\pi$
$571$			1084.44i			1.89919i	−0.313485	+	0.949593i	$0.601497\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$		0			0			−	1.00000i	$-0.5\pi$
$577$		0			0				1.00000i	$0.5\pi$
$578$	−46.5000	−	180.094i	−0.0804498	−	0.311581i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	197.000	+	762.978i	0.336177	+	1.30201i
$587$			805.581i			1.37237i	0.727428	+	0.686184i	$0.240716\pi$
$587$			805.581i			1.37237i	−0.727428	+	0.686184i	$0.759284\pi$
$588$		0			0
$589$	1920.00			3.25976
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1166.00			−1.96627			−0.983137	−	0.182873i	$-0.941460\pi$
$593$	−1166.00			−1.96627			−0.983137	+	0.182873i	$0.941460\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$	−242.000			−0.402662			−0.201331	−	0.979523i	$-0.564527\pi$
$601$	−242.000			−0.402662			−0.201331	+	0.979523i	$0.564527\pi$
$602$		0			0
$603$		0			0
$604$	360.000	+	650.661i	0.596026	+	1.07725i
$605$		0			0
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$	300.000	−	945.008i	0.493421	−	1.55429i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1186.00			−1.92220			−0.961102	−	0.276193i	$-0.910927\pi$
$617$	−1186.00			−1.92220			−0.961102	+	0.276193i	$0.910927\pi$
$618$		0			0
$619$		−	1022.47i		−	1.65181i	−0.563813	−	0.825903i	$-0.690666\pi$
$619$		−	1022.47i		−	1.65181i	0.563813	−	0.825903i	$-0.309334\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$			1239.35i			1.96411i	0.188590	+	0.982056i	$0.439608\pi$
$631$			1239.35i			1.96411i	−0.188590	+	0.982056i	$0.560392\pi$
$632$	720.000	−	681.645i	1.13924	−	1.07855i
$633$		0			0
$634$	67.0000	+	259.490i	0.105678	+	0.409290i
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$	840.000	−	216.887i	1.30031	−	0.335738i
$647$			1084.44i			1.67610i	0.545595	+	0.838049i	$0.316304\pi$
$647$			1084.44i			1.67610i	−0.545595	+	0.838049i	$0.683696\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1114.00			−1.70597			−0.852986	−	0.521933i	$-0.825211\pi$
$653$	−1114.00			−1.70597			−0.852986	+	0.521933i	$0.825211\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	838.000			1.26778			0.633888	−	0.773425i	$-0.281458\pi$
$661$	838.000			1.26778			0.633888	+	0.773425i	$0.281458\pi$
$662$	1260.00	−	325.331i	1.90332	−	0.491436i
$663$		0			0
$664$	−360.000	+	340.823i	−0.542169	+	0.513287i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	420.000	+	759.105i	0.628743	+	1.13638i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0			−	1.00000i	$-0.5\pi$
$673$		0			0				1.00000i	$0.5\pi$
$674$		0			0
$675$		0			0
$676$	591.500	−	327.267i	0.875000	−	0.484123i
$677$	374.000			0.552437			0.276219	−	0.961095i	$-0.410919\pi$
$677$	374.000			0.552437			0.276219	+	0.961095i	$0.410919\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	1363.29i		−	1.99603i	−0.0629575	−	0.998016i	$-0.520053\pi$
$683$		−	1363.29i		−	1.99603i	0.0629575	−	0.998016i	$-0.479947\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$			402.790i			0.582909i	0.956585	+	0.291455i	$0.0941392\pi$
$691$			402.790i			0.582909i	−0.956585	+	0.291455i	$0.905861\pi$
$692$	−539.000	+	298.220i	−0.778902	+	0.430953i
$693$		0			0
$694$	−720.000	+	185.903i	−1.03746	+	0.267872i
$695$		0			0
$696$		0			0
$697$		0			0
$698$	−229.000	−	886.913i	−0.328080	−	1.27065i
$699$		0			0
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−137.000	−	530.599i	−0.194051	−	0.751556i
$707$		0			0
$708$		0			0
$709$	−742.000			−1.04654			−0.523272	−	0.852166i	$-0.675289\pi$
$709$	−742.000			−1.04654			−0.523272	+	0.852166i	$0.675289\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1920.00			2.69285
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$		0			0
$722$	−299.500	−	1159.96i	−0.414820	−	1.60659i
$723$		0			0
$724$	427.000	−	236.252i	0.589779	−	0.326315i
$725$		0			0
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$		0			0
$735$		0			0
$736$	300.000	−	945.008i	0.407609	−	1.28398i
$737$		0			0
$738$		0			0
$739$			216.887i			0.293487i	0.989175	+	0.146744i	$0.0468792\pi$
$739$			216.887i			0.293487i	−0.989175	+	0.146744i	$0.953121\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			1394.27i			1.87655i	0.345895	+	0.938273i	$0.387575\pi$
$743$			1394.27i			1.87655i	−0.345895	+	0.938273i	$0.612425\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		−	433.774i		−	0.577595i	−0.957390	−	0.288798i	$-0.906745\pi$
$751$		−	433.774i		−	0.577595i	0.957390	−	0.288798i	$-0.0932555\pi$
$752$	−1260.00	−	790.089i	−1.67553	−	1.05065i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$	300.000	−	77.4597i	0.395778	−	0.102190i
$759$		0			0
$760$		0			0
$761$		0			0			−	1.00000i	$-0.5\pi$
$761$		0			0				1.00000i	$0.5\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	−660.000	+	170.411i	−0.861619	+	0.222469i
$767$		0			0
$768$		0			0
$769$	−578.000			−0.751625			−0.375813	−	0.926696i	$-0.622636\pi$
$769$	−578.000			−0.751625			−0.375813	+	0.926696i	$0.622636\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	1526.00			1.97413			0.987063	−	0.160330i	$-0.0512560\pi$
$773$	1526.00			1.97413			0.987063	+	0.160330i	$0.0512560\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$	840.000	−	216.887i	1.07417	−	0.277349i
$783$		0			0
$784$	416.500	−	664.217i	0.531250	−	0.847215i
$785$		0			0
$786$		0			0
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$	1309.00	−	724.248i	1.66117	−	0.919096i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	720.000	+	1301.32i	0.904523	+	1.63483i
$797$	826.000			1.03639			0.518193	−	0.855264i	$-0.326605\pi$
$797$	826.000			1.03639			0.518193	+	0.855264i	$0.326605\pi$
$798$		0			0
$799$		−	1301.32i		−	1.62869i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$		−	1208.37i		−	1.48998i	−0.667078	−	0.744988i	$-0.732455\pi$
$811$		−	1208.37i		−	1.48998i	0.667078	−	0.744988i	$-0.267545\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	71.0000	+	274.982i	0.0867971	+	0.336164i
$819$		0			0
$820$		0			0
$821$		0			0			−	1.00000i	$-0.5\pi$
$821$		0			0				1.00000i	$0.5\pi$
$822$		0			0
$823$		0			0		1.00000			$0$
$823$		0			0		−1.00000			$\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	1611.16i		−	1.94820i	−0.226119	−	0.974100i	$-0.572604\pi$
$827$		−	1611.16i		−	1.94820i	0.226119	−	0.974100i	$-0.427396\pi$
$828$		0			0
$829$	502.000			0.605549			0.302774	−	0.953062i	$-0.402087\pi$
$829$	502.000			0.605549			0.302774	+	0.953062i	$0.402087\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	686.000			0.823529
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	−841.000			−1.00000
$842$	301.000	+	1165.77i	0.357482	+	1.38452i
$843$		0			0
$844$	−420.000	−	759.105i	−0.497630	−	0.899413i
$845$		0			0
$846$		0			0
$847$		0			0
$848$	731.000	−	1165.77i	0.862028	−	1.37473i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$	−1080.00	+	1022.47i	−1.26168	+	1.19447i
$857$	−1666.00			−1.94399			−0.971995	−	0.235000i	$-0.924491\pi$
$857$	−1666.00			−1.94399			−0.971995	+	0.235000i	$0.924491\pi$
$858$		0			0
$859$			1704.11i			1.98383i	0.126892	+	0.991917i	$0.459500\pi$
$859$			1704.11i			1.98383i	−0.126892	+	0.991917i	$0.540500\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	1704.11i		−	1.97464i	−0.158749	−	0.987319i	$-0.550746\pi$
$863$		−	1704.11i		−	1.97464i	0.158749	−	0.987319i	$-0.449254\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−121.000	−	127.808i	−0.138761	−	0.146569i
$873$		0			0
$874$	−480.000	−	1859.03i	−0.549199	−	2.12704i
$875$		0			0
$876$		0			0
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$	−1200.00	+	309.839i	−1.36674	+	0.352891i
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$	−1320.00	+	340.823i	−1.48984	+	0.384676i
$887$			526.726i			0.593828i	0.954904	+	0.296914i	$0.0959575\pi$
$887$			526.726i			0.593828i	−0.954904	+	0.296914i	$0.904042\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−2880.00			−3.22508
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	1204.00			1.33629
$902$		0			0
$903$		0			0
$904$	1133.00	+	1196.75i	1.25332	+	1.32384i
$905$		0			0
$906$		0			0
$907$		0			0		1.00000			$0$
$907$		0			0		−1.00000			$\pi$
$908$	840.000	+	1518.21i	0.925110	+	1.67204i
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−763.000	+	422.155i	−0.832969	+	0.460868i
$917$		0			0
$918$		0			0
$919$			1301.32i			1.41602i	0.706202	+	0.708010i	$0.250407\pi$
$919$			1301.32i			1.41602i	−0.706202	+	0.708010i	$0.749593\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$		−	1518.21i		−	1.63073i
$932$	119.000	−	65.8407i	0.127682	−	0.0706445i
$933$		0			0
$934$	−1680.00	+	433.774i	−1.79872	+	0.464426i
$935$		0			0
$936$		0			0
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$			1053.45i			1.11241i	0.831045	+	0.556205i	$0.187743\pi$
$947$			1053.45i			1.11241i	−0.831045	+	0.556205i	$0.812257\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1474.00			1.54669			0.773347	−	0.633983i	$-0.218581\pi$
$953$	1474.00			1.54669			0.773347	+	0.633983i	$0.218581\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−2879.00			−2.99584
$962$		0			0
$963$		0			0
$964$	1673.00	−	925.643i	1.73548	−	0.960211i
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$	−665.500	−	702.946i	−0.687500	−	0.726184i
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	1003.00	−	1599.54i	1.02766	−	1.63888i
$977$	1934.00			1.97953			0.989765	−	0.142710i	$-0.0455815\pi$
$977$	1934.00			1.97953			0.989765	+	0.142710i	$0.0455815\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	216.887i		−	0.220638i	−0.993896	−	0.110319i	$-0.964813\pi$
$983$		−	216.887i		−	0.220638i	0.993896	−	0.110319i	$-0.0351872\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		−	1735.10i		−	1.75085i	−0.483350	−	0.875427i	$-0.660580\pi$
$991$		−	1735.10i		−	1.75085i	0.483350	−	0.875427i	$-0.339420\pi$
$992$	−600.000	+	1890.02i	−0.604839	+	1.90526i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0			−	1.00000i	$-0.5\pi$
$997$		0			0				1.00000i	$0.5\pi$
$998$	−660.000	+	170.411i	−0.661323	+	0.170753i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.3.c.i.451.2		2
3.2	odd	2		900.3.c.g.451.1		2
4.3	odd	2	inner	900.3.c.i.451.1		2
5.2	odd	4		180.3.f.f.19.2	yes	4
5.3	odd	4		180.3.f.f.19.3	yes	4
5.4	even	2		900.3.c.g.451.1		2
12.11	even	2		900.3.c.g.451.2		2
15.2	even	4		180.3.f.f.19.3	yes	4
15.8	even	4		180.3.f.f.19.2	yes	4
15.14	odd	2	CM	900.3.c.i.451.2		2
20.3	even	4		180.3.f.f.19.1	✓	4
20.7	even	4		180.3.f.f.19.4	yes	4
20.19	odd	2		900.3.c.g.451.2		2
60.23	odd	4		180.3.f.f.19.4	yes	4
60.47	odd	4		180.3.f.f.19.1	✓	4
60.59	even	2	inner	900.3.c.i.451.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.3.f.f.19.1	✓	4	20.3	even	4
180.3.f.f.19.1	✓	4	60.47	odd	4
180.3.f.f.19.2	yes	4	5.2	odd	4
180.3.f.f.19.2	yes	4	15.8	even	4
180.3.f.f.19.3	yes	4	5.3	odd	4
180.3.f.f.19.3	yes	4	15.2	even	4
180.3.f.f.19.4	yes	4	20.7	even	4
180.3.f.f.19.4	yes	4	60.23	odd	4
900.3.c.g.451.1		2	3.2	odd	2
900.3.c.g.451.1		2	5.4	even	2
900.3.c.g.451.2		2	12.11	even	2
900.3.c.g.451.2		2	20.19	odd	2
900.3.c.i.451.1		2	4.3	odd	2	inner
900.3.c.i.451.1		2	60.59	even	2	inner
900.3.c.i.451.2		2	1.1	even	1	trivial
900.3.c.i.451.2		2	15.14	odd	2	CM

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

\(f(q)\)	\(=\)	\(q+(0.500000 + 1.93649i) q^{2} +(-3.50000 + 1.93649i) q^{4} +(-5.50000 - 5.80948i) q^{8} +O(q^{10})\)	\(q+(0.500000 + 1.93649i) q^{2} +(-3.50000 + 1.93649i) q^{4} +(-5.50000 - 5.80948i) q^{8} +(8.50000 - 13.5554i) q^{16} +14.0000 q^{17} -30.9839i q^{19} -30.9839i q^{23} +61.9677i q^{31} +(30.5000 + 9.68246i) q^{32} +(7.00000 + 27.1109i) q^{34} +(60.0000 - 15.4919i) q^{38} +(60.0000 - 15.4919i) q^{46} -92.9516i q^{47} +49.0000 q^{49} +86.0000 q^{53} +118.000 q^{61} +(-120.000 + 30.9839i) q^{62} +(-3.50000 + 63.9042i) q^{64} +(-49.0000 + 27.1109i) q^{68} +(60.0000 + 108.444i) q^{76} +123.935i q^{79} -61.9677i q^{83} +(60.0000 + 108.444i) q^{92} +(180.000 - 46.4758i) q^{94} +(24.5000 + 94.8881i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - 7 q^{4} - 11 q^{8}+O(q^{10}) \) 2 * q + q^2 - 7 * q^4 - 11 * q^8	\( 2 q + q^{2} - 7 q^{4} - 11 q^{8} + 17 q^{16} + 28 q^{17} + 61 q^{32} + 14 q^{34} + 120 q^{38} + 120 q^{46} + 98 q^{49} + 172 q^{53} + 236 q^{61} - 240 q^{62} - 7 q^{64} - 98 q^{68} + 120 q^{76} + 120 q^{92} + 360 q^{94} + 49 q^{98}+O(q^{100}) \) 2 * q + q^2 - 7 * q^4 - 11 * q^8 + 17 * q^16 + 28 * q^17 + 61 * q^32 + 14 * q^34 + 120 * q^38 + 120 * q^46 + 98 * q^49 + 172 * q^53 + 236 * q^61 - 240 * q^62 - 7 * q^64 - 98 * q^68 + 120 * q^76 + 120 * q^92 + 360 * q^94 + 49 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more