LMFDB - Embedded newform 1800.2.k.l.901.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1800,2,Mod(901,1800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1800, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1800.901");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1800.k (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:	$14.3730723638$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{-5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + x^{2} + 4 $ x^4 + x^2 + 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			901.2
Root			$0.866025 - 1.11803i$ of defining polynomial
Character	$\chi$	$=$	1800.901
Dual form			1800.2.k.l.901.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.866025 + 1.11803i) q^{2} +(-0.500000 - 1.93649i) q^{4} +(2.59808 + 1.11803i) q^{8} +O(q^{10})$	$q+(-0.866025 + 1.11803i) q^{2} +(-0.500000 - 1.93649i) q^{4} +(2.59808 + 1.11803i) q^{8} +(-3.50000 + 1.93649i) q^{16} +6.92820 q^{17} -7.74597i q^{19} -3.46410 q^{23} -8.00000 q^{31} +(0.866025 - 5.59017i) q^{32} +(-6.00000 + 7.74597i) q^{34} +(8.66025 + 6.70820i) q^{38} +(3.00000 - 3.87298i) q^{46} +10.3923 q^{47} -7.00000 q^{49} +4.47214i q^{53} -15.4919i q^{61} +(6.92820 - 8.94427i) q^{62} +(5.50000 + 5.80948i) q^{64} +(-3.46410 - 13.4164i) q^{68} +(-15.0000 + 3.87298i) q^{76} +16.0000 q^{79} -17.8885i q^{83} +(1.73205 + 6.70820i) q^{92} +(-9.00000 + 11.6190i) q^{94} +(6.06218 - 7.82624i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{4}+O(q^{10}) $ 4 * q - 2 * q^4	$ 4 q - 2 q^{4} - 14 q^{16} - 32 q^{31} - 24 q^{34} + 12 q^{46} - 28 q^{49} + 22 q^{64} - 60 q^{76} + 64 q^{79} - 36 q^{94}+O(q^{100}) $ 4 * q - 2 * q^4 - 14 * q^16 - 32 * q^31 - 24 * q^34 + 12 * q^46 - 28 * q^49 + 22 * q^64 - 60 * q^76 + 64 * q^79 - 36 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1001$	$1351$
$\chi(n)$	$1$	$-1$	$1$	$1$

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 $

$2$	−0.866025	+	1.11803i	−0.612372	+	0.790569i
$2$	−0.866025	+	1.11803i	−0.612372	+	0.790569i
$3$		0			0
$3$		0			0
$4$	−0.500000	−	1.93649i	−0.250000	−	0.968246i
$5$		0			0
$5$		0			0
$6$		0			0
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$	2.59808	+	1.11803i	0.918559	+	0.395285i
$9$		0			0
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$		0			0
$16$	−3.50000	+	1.93649i	−0.875000	+	0.484123i
$17$	6.92820			1.68034			0.840168	−	0.542326i	$-0.182456\pi$
$17$	6.92820			1.68034			0.840168	+	0.542326i	$0.182456\pi$
$18$		0			0
$19$		−	7.74597i		−	1.77705i	−0.458831	−	0.888523i	$-0.651732\pi$
$19$		−	7.74597i		−	1.77705i	0.458831	−	0.888523i	$-0.348268\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.46410			−0.722315			−0.361158	−	0.932505i	$-0.617618\pi$
$23$	−3.46410			−0.722315			−0.361158	+	0.932505i	$0.617618\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	−8.00000			−1.43684			−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000			−1.43684			−0.718421	+	0.695608i	$0.755135\pi$
$32$	0.866025	−	5.59017i	0.153093	−	0.988212i
$33$		0			0
$34$	−6.00000	+	7.74597i	−1.02899	+	1.32842i
$35$		0			0
$36$		0			0
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	8.66025	+	6.70820i	1.40488	+	1.08821i
$39$		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
$46$	3.00000	−	3.87298i	0.442326	−	0.571040i
$47$	10.3923			1.51587			0.757937	−	0.652328i	$-0.226208\pi$
$47$	10.3923			1.51587			0.757937	+	0.652328i	$0.226208\pi$
$48$		0			0
$49$	−7.00000			−1.00000
$50$		0			0
$51$		0			0
$52$		0			0
$53$			4.47214i			0.614295i	0.951662	+	0.307148i	$0.0993745\pi$
$53$			4.47214i			0.614295i	−0.951662	+	0.307148i	$0.900625\pi$
$54$		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
$61$		−	15.4919i		−	1.98354i	−0.128037	−	0.991769i	$-0.540868\pi$
$61$		−	15.4919i		−	1.98354i	0.128037	−	0.991769i	$-0.459132\pi$
$62$	6.92820	−	8.94427i	0.879883	−	1.13592i
$63$		0			0
$64$	5.50000	+	5.80948i	0.687500	+	0.726184i
$65$		0			0
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$	−3.46410	−	13.4164i	−0.420084	−	1.62698i
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$		0			0
$75$		0			0
$76$	−15.0000	+	3.87298i	−1.72062	+	0.444262i
$77$		0			0
$78$		0			0
$79$	16.0000			1.80014			0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000			1.80014			0.900070	+	0.435745i	$0.143515\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	17.8885i		−	1.96352i	−0.190117	−	0.981761i	$-0.560887\pi$
$83$		−	17.8885i		−	1.96352i	0.190117	−	0.981761i	$-0.439113\pi$
$84$		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
$92$	1.73205	+	6.70820i	0.180579	+	0.699379i
$93$		0			0
$94$	−9.00000	+	11.6190i	−0.928279	+	1.19840i
$95$		0			0
$96$		0			0
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$	6.06218	−	7.82624i	0.612372	−	0.790569i
$99$		0			0
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$		0			0
$106$	−5.00000	−	3.87298i	−0.485643	−	0.376177i
$107$		−	17.8885i		−	1.72935i	−0.502331	−	0.864675i	$-0.667524\pi$
$107$		−	17.8885i		−	1.72935i	0.502331	−	0.864675i	$-0.332476\pi$
$108$		0			0
$109$		−	15.4919i		−	1.48386i	−0.670478	−	0.741929i	$-0.733911\pi$
$109$		−	15.4919i		−	1.48386i	0.670478	−	0.741929i	$-0.266089\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	20.7846			1.95525			0.977626	−	0.210352i	$-0.0674609\pi$
$113$	20.7846			1.95525			0.977626	+	0.210352i	$0.0674609\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	11.0000			1.00000
$122$	17.3205	+	13.4164i	1.56813	+	1.21466i
$123$		0			0
$124$	4.00000	+	15.4919i	0.359211	+	1.39122i
$125$		0			0
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$	−11.2583	+	1.11803i	−0.995105	+	0.0988212i
$129$		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
$136$	18.0000	+	7.74597i	1.54349	+	0.664211i
$137$	6.92820			0.591916			0.295958	−	0.955201i	$-0.404361\pi$
$137$	6.92820			0.591916			0.295958	+	0.955201i	$0.404361\pi$
$138$		0			0
$139$		−	23.2379i		−	1.97101i	−0.169638	−	0.985506i	$-0.554260\pi$
$139$		−	23.2379i		−	1.97101i	0.169638	−	0.985506i	$-0.445740\pi$
$140$		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		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$	8.00000			0.651031			0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000			0.651031			0.325515	+	0.945537i	$0.394462\pi$
$152$	8.66025	−	20.1246i	0.702439	−	1.63232i
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$	−13.8564	+	17.8885i	−1.10236	+	1.42314i
$159$		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
$166$	20.0000	+	15.4919i	1.55230	+	1.20241i
$167$	−24.2487			−1.87642			−0.938211	−	0.346064i	$-0.887518\pi$
$167$	−24.2487			−1.87642			−0.938211	+	0.346064i	$0.887518\pi$
$168$		0			0
$169$	13.0000			1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$			22.3607i			1.70005i	0.526742	+	0.850026i	$0.323414\pi$
$173$			22.3607i			1.70005i	−0.526742	+	0.850026i	$0.676586\pi$
$174$		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
$181$			15.4919i			1.15151i	0.817624	+	0.575753i	$0.195291\pi$
$181$			15.4919i			1.15151i	−0.817624	+	0.575753i	$0.804709\pi$
$182$		0			0
$183$		0			0
$184$	−9.00000	−	3.87298i	−0.663489	−	0.285520i
$185$		0			0
$186$		0			0
$187$		0			0
$188$	−5.19615	−	20.1246i	−0.378968	−	1.46774i
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$		0			0			−	1.00000i	$-0.5\pi$
$193$		0			0				1.00000i	$0.5\pi$
$194$		0			0
$195$		0			0
$196$	3.50000	+	13.5554i	0.250000	+	0.968246i
$197$			4.47214i			0.318626i	0.987228	+	0.159313i	$0.0509280\pi$
$197$			4.47214i			0.318626i	−0.987228	+	0.159313i	$0.949072\pi$
$198$		0			0
$199$	−16.0000			−1.13421			−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000			−1.13421			−0.567105	+	0.823646i	$0.691937\pi$
$200$		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
$211$		−	7.74597i		−	0.533254i	−0.963800	−	0.266627i	$-0.914091\pi$
$211$		−	7.74597i		−	0.533254i	0.963800	−	0.266627i	$-0.0859092\pi$
$212$	8.66025	−	2.23607i	0.594789	−	0.153574i
$213$		0			0
$214$	20.0000	+	15.4919i	1.36717	+	1.05901i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	17.3205	+	13.4164i	1.17309	+	0.908674i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$		0			0
$226$	−18.0000	+	23.2379i	−1.19734	+	1.54576i
$227$			17.8885i			1.18730i	0.804722	+	0.593652i	$0.202314\pi$
$227$			17.8885i			1.18730i	−0.804722	+	0.593652i	$0.797686\pi$
$228$		0			0
$229$		−	15.4919i		−	1.02374i	−0.859064	−	0.511868i	$-0.828954\pi$
$229$		−	15.4919i		−	1.02374i	0.859064	−	0.511868i	$-0.171046\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	20.7846			1.36165			0.680823	−	0.732448i	$-0.261622\pi$
$233$	20.7846			1.36165			0.680823	+	0.732448i	$0.261622\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−2.00000			−0.128831			−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000			−0.128831			−0.0644157	+	0.997923i	$0.520518\pi$
$242$	−9.52628	+	12.2984i	−0.612372	+	0.790569i
$243$		0			0
$244$	−30.0000	+	7.74597i	−1.92055	+	0.495885i
$245$		0			0
$246$		0			0
$247$		0			0
$248$	−20.7846	−	8.94427i	−1.31982	−	0.567962i
$249$		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
$256$	8.50000	−	13.5554i	0.531250	−	0.847215i
$257$	−6.92820			−0.432169			−0.216085	−	0.976375i	$-0.569329\pi$
$257$	−6.92820			−0.432169			−0.216085	+	0.976375i	$0.569329\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−31.1769			−1.92245			−0.961225	−	0.275764i	$-0.911069\pi$
$263$	−31.1769			−1.92245			−0.961225	+	0.275764i	$0.911069\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	32.0000			1.94386			0.971931	−	0.235267i	$-0.0755965\pi$
$271$	32.0000			1.94386			0.971931	+	0.235267i	$0.0755965\pi$
$272$	−24.2487	+	13.4164i	−1.47029	+	0.813489i
$273$		0			0
$274$	−6.00000	+	7.74597i	−0.362473	+	0.467951i
$275$		0			0
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$	25.9808	+	20.1246i	1.55822	+	1.20699i
$279$		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
$289$	31.0000			1.82353
$290$		0			0
$291$		0			0
$292$		0			0
$293$			31.3050i			1.82885i	0.404750	+	0.914427i	$0.367359\pi$
$293$			31.3050i			1.82885i	−0.404750	+	0.914427i	$0.632641\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$	−6.92820	+	8.94427i	−0.398673	+	0.514685i
$303$		0			0
$304$	15.0000	+	27.1109i	0.860309	+	1.55492i
$305$		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			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		0			0			−	1.00000i	$-0.5\pi$
$313$		0			0				1.00000i	$0.5\pi$
$314$		0			0
$315$		0			0
$316$	−8.00000	−	30.9839i	−0.450035	−	1.74298i
$317$			22.3607i			1.25590i	0.778253	+	0.627950i	$0.216106\pi$
$317$			22.3607i			1.25590i	−0.778253	+	0.627950i	$0.783894\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	53.6656i		−	2.98604i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		−	23.2379i		−	1.27727i	−0.769510	−	0.638635i	$-0.779499\pi$
$331$		−	23.2379i		−	1.27727i	0.769510	−	0.638635i	$-0.220501\pi$
$332$	−34.6410	+	8.94427i	−1.90117	+	0.490881i
$333$		0			0
$334$	21.0000	−	27.1109i	1.14907	−	1.48344i
$335$		0			0
$336$		0			0
$337$		0			0			−	1.00000i	$-0.5\pi$
$337$		0			0				1.00000i	$0.5\pi$
$338$	−11.2583	+	14.5344i	−0.612372	+	0.790569i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−25.0000	−	19.3649i	−1.34401	−	1.04106i
$347$		−	35.7771i		−	1.92061i	−0.278944	−	0.960307i	$-0.589984\pi$
$347$		−	35.7771i		−	1.92061i	0.278944	−	0.960307i	$-0.410016\pi$
$348$		0			0
$349$		−	15.4919i		−	0.829264i	−0.909989	−	0.414632i	$-0.863910\pi$
$349$		−	15.4919i		−	0.829264i	0.909989	−	0.414632i	$-0.136090\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−20.7846			−1.10625			−0.553127	−	0.833097i	$-0.686565\pi$
$353$	−20.7846			−1.10625			−0.553127	+	0.833097i	$0.686565\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−41.0000			−2.15789
$362$	−17.3205	−	13.4164i	−0.910346	−	0.705151i
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$	12.1244	−	6.70820i	0.632026	−	0.349689i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$	27.0000	+	11.6190i	1.39242	+	0.599202i
$377$		0			0
$378$		0			0
$379$			38.7298i			1.98942i	0.102733	+	0.994709i	$0.467241\pi$
$379$			38.7298i			1.98942i	−0.102733	+	0.994709i	$0.532759\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−38.1051			−1.94708			−0.973540	−	0.228515i	$-0.926613\pi$
$383$	−38.1051			−1.94708			−0.973540	+	0.228515i	$0.926613\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$	−24.0000			−1.21373
$392$	−18.1865	−	7.82624i	−0.918559	−	0.395285i
$393$		0			0
$394$	−5.00000	−	3.87298i	−0.251896	−	0.195118i
$395$		0			0
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$	13.8564	−	17.8885i	0.694559	−	0.896672i
$399$		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
$409$	−26.0000			−1.28562			−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000			−1.28562			−0.642809	+	0.766027i	$0.722231\pi$
$410$		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
$421$		−	15.4919i		−	0.755031i	−0.926003	−	0.377515i	$-0.876779\pi$
$421$		−	15.4919i		−	0.755031i	0.926003	−	0.377515i	$-0.123221\pi$
$422$	8.66025	+	6.70820i	0.421575	+	0.326550i
$423$		0			0
$424$	−5.00000	+	11.6190i	−0.242821	+	0.564266i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−34.6410	+	8.94427i	−1.67444	+	0.432338i
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$	−30.0000	+	7.74597i	−1.43674	+	0.370965i
$437$			26.8328i			1.28359i
$438$		0			0
$439$	−16.0000			−0.763638			−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000			−0.763638			−0.381819	+	0.924237i	$0.624702\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			17.8885i			0.849910i	0.905214	+	0.424955i	$0.139710\pi$
$443$			17.8885i			0.849910i	−0.905214	+	0.424955i	$0.860290\pi$
$444$		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
$452$	−10.3923	−	40.2492i	−0.488813	−	1.89316i
$453$		0			0
$454$	−20.0000	−	15.4919i	−0.938647	−	0.727072i
$455$		0			0
$456$		0			0
$457$		0			0			−	1.00000i	$-0.5\pi$
$457$		0			0				1.00000i	$0.5\pi$
$458$	17.3205	+	13.4164i	0.809334	+	0.626908i
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		0			0
$465$		0			0
$466$	−18.0000	+	23.2379i	−0.833834	+	1.07647i
$467$			35.7771i			1.65557i	0.561048	+	0.827783i	$0.310398\pi$
$467$			35.7771i			1.65557i	−0.561048	+	0.827783i	$0.689602\pi$
$468$		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			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$	1.73205	−	2.23607i	0.0788928	−	0.101850i
$483$		0			0
$484$	−5.50000	−	21.3014i	−0.250000	−	0.968246i
$485$		0			0
$486$		0			0
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$	17.3205	−	40.2492i	0.784063	−	1.82200i
$489$		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
$496$	28.0000	−	15.4919i	1.25724	−	0.695608i
$497$		0			0
$498$		0			0
$499$		−	7.74597i		−	0.346757i	−0.984855	−	0.173379i	$-0.944532\pi$
$499$		−	7.74597i		−	0.346757i	0.984855	−	0.173379i	$-0.0554684\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	3.46410			0.154457			0.0772283	−	0.997013i	$-0.475393\pi$
$503$	3.46410			0.154457			0.0772283	+	0.997013i	$0.475393\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$		0			0
$512$	7.79423	+	21.2426i	0.344459	+	0.938801i
$513$		0			0
$514$	6.00000	−	7.74597i	0.264649	−	0.341660i
$515$		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
$526$	27.0000	−	34.8569i	1.17726	−	1.51983i
$527$	−55.4256			−2.41438
$528$		0			0
$529$	−11.0000			−0.478261
$530$		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
$541$			46.4758i			1.99815i	0.0429934	+	0.999075i	$0.486311\pi$
$541$			46.4758i			1.99815i	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	−27.7128	+	35.7771i	−1.19037	+	1.53676i
$543$		0			0
$544$	6.00000	−	38.7298i	0.257248	−	1.66053i
$545$		0			0
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$	−3.46410	−	13.4164i	−0.147979	−	0.573121i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−45.0000	+	11.6190i	−1.90843	+	0.492753i
$557$		−	22.3607i		−	0.947452i	−0.880672	−	0.473726i	$-0.842909\pi$
$557$		−	22.3607i		−	0.947452i	0.880672	−	0.473726i	$-0.157091\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	35.7771i		−	1.50782i	−0.656975	−	0.753912i	$-0.728164\pi$
$563$		−	35.7771i		−	1.50782i	0.656975	−	0.753912i	$-0.271836\pi$
$564$		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
$571$			38.7298i			1.62079i	0.585882	+	0.810397i	$0.300748\pi$
$571$			38.7298i			1.62079i	−0.585882	+	0.810397i	$0.699252\pi$
$572$		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	$0.5\pi$
$578$	−26.8468	+	34.6591i	−1.11668	+	1.44163i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	−35.0000	−	27.1109i	−1.44584	−	1.11994i
$587$		−	17.8885i		−	0.738339i	−0.929362	−	0.369170i	$-0.879642\pi$
$587$		−	17.8885i		−	0.738339i	0.929362	−	0.369170i	$-0.120358\pi$
$588$		0			0
$589$			61.9677i			2.55334i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	48.4974			1.99155			0.995775	−	0.0918243i	$-0.0292698\pi$
$593$	48.4974			1.99155			0.995775	+	0.0918243i	$0.0292698\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	38.0000			1.55005			0.775026	−	0.631929i	$-0.217737\pi$
$601$	38.0000			1.55005			0.775026	+	0.631929i	$0.217737\pi$
$602$		0			0
$603$		0			0
$604$	−4.00000	−	15.4919i	−0.162758	−	0.630358i
$605$		0			0
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$	−43.3013	−	6.70820i	−1.75610	−	0.272054i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−6.92820			−0.278919			−0.139459	−	0.990228i	$-0.544536\pi$
$617$	−6.92820			−0.278919			−0.139459	+	0.990228i	$0.544536\pi$
$618$		0			0
$619$		−	23.2379i		−	0.934010i	−0.884255	−	0.467005i	$-0.845333\pi$
$619$		−	23.2379i		−	0.934010i	0.884255	−	0.467005i	$-0.154667\pi$
$620$		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
$631$	−32.0000			−1.27390			−0.636950	−	0.770905i	$-0.719804\pi$
$631$	−32.0000			−1.27390			−0.636950	+	0.770905i	$0.719804\pi$
$632$	41.5692	+	17.8885i	1.65353	+	0.711568i
$633$		0			0
$634$	−25.0000	−	19.3649i	−0.992877	−	0.769079i
$635$		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
$646$	60.0000	+	46.4758i	2.36067	+	1.82857i
$647$	24.2487			0.953315			0.476658	−	0.879089i	$-0.341848\pi$
$647$	24.2487			0.953315			0.476658	+	0.879089i	$0.341848\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$			49.1935i			1.92509i	0.271122	+	0.962545i	$0.412605\pi$
$653$			49.1935i			1.92509i	−0.271122	+	0.962545i	$0.587395\pi$
$654$		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
$661$		−	46.4758i		−	1.80770i	−0.427850	−	0.903850i	$-0.640729\pi$
$661$		−	46.4758i		−	1.80770i	0.427850	−	0.903850i	$-0.359271\pi$
$662$	25.9808	+	20.1246i	1.00977	+	0.782165i
$663$		0			0
$664$	20.0000	−	46.4758i	0.776151	−	1.80361i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	12.1244	+	46.9574i	0.469105	+	1.81684i
$669$		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
$676$	−6.50000	−	25.1744i	−0.250000	−	0.968246i
$677$		−	31.3050i		−	1.20315i	−0.798817	−	0.601574i	$-0.794541\pi$
$677$		−	31.3050i		−	1.20315i	0.798817	−	0.601574i	$-0.205459\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$			35.7771i			1.36897i	0.729026	+	0.684486i	$0.239973\pi$
$683$			35.7771i			1.36897i	−0.729026	+	0.684486i	$0.760027\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$			7.74597i			0.294670i	0.989087	+	0.147335i	$0.0470696\pi$
$691$			7.74597i			0.294670i	−0.989087	+	0.147335i	$0.952930\pi$
$692$	43.3013	−	11.1803i	1.64607	−	0.425013i
$693$		0			0
$694$	40.0000	+	30.9839i	1.51838	+	1.17613i
$695$		0			0
$696$		0			0
$697$		0			0
$698$	17.3205	+	13.4164i	0.655591	+	0.507819i
$699$		0			0
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	18.0000	−	23.2379i	0.677439	−	0.874570i
$707$		0			0
$708$		0			0
$709$			46.4758i			1.74544i	0.488225	+	0.872718i	$0.337644\pi$
$709$			46.4758i			1.74544i	−0.488225	+	0.872718i	$0.662356\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	27.7128			1.03785
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$	35.5070	−	45.8394i	1.32144	−	1.70597i
$723$		0			0
$724$	30.0000	−	7.74597i	1.11494	−	0.287877i
$725$		0			0
$726$		0			0
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$		0			0
$736$	−3.00000	+	19.3649i	−0.110581	+	0.713800i
$737$		0			0
$738$		0			0
$739$			54.2218i			1.99458i	0.0735712	+	0.997290i	$0.476560\pi$
$739$			54.2218i			1.99458i	−0.0735712	+	0.997290i	$0.523440\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	31.1769			1.14377			0.571885	−	0.820334i	$-0.306212\pi$
$743$	31.1769			1.14377			0.571885	+	0.820334i	$0.306212\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−8.00000			−0.291924			−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000			−0.291924			−0.145962	+	0.989290i	$0.546628\pi$
$752$	−36.3731	+	20.1246i	−1.32639	+	0.733869i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	−43.3013	−	33.5410i	−1.57277	−	1.21826i
$759$		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
$766$	33.0000	−	42.6028i	1.19234	−	1.53930i
$767$		0			0
$768$		0			0
$769$	−46.0000			−1.65880			−0.829401	−	0.558653i	$-0.811318\pi$
$769$	−46.0000			−1.65880			−0.829401	+	0.558653i	$0.811318\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	4.47214i		−	0.160852i	−0.996761	−	0.0804258i	$-0.974372\pi$
$773$		−	4.47214i		−	0.160852i	0.996761	−	0.0804258i	$-0.0256280\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$	20.7846	−	26.8328i	0.743256	−	0.959540i
$783$		0			0
$784$	24.5000	−	13.5554i	0.875000	−	0.484123i
$785$		0			0
$786$		0			0
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$	8.66025	−	2.23607i	0.308509	−	0.0796566i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	8.00000	+	30.9839i	0.283552	+	1.09819i
$797$			49.1935i			1.74252i	0.490819	+	0.871262i	$0.336698\pi$
$797$			49.1935i			1.74252i	−0.490819	+	0.871262i	$0.663302\pi$
$798$		0			0
$799$	72.0000			2.54718
$800$		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
$811$			23.2379i			0.815993i	0.912983	+	0.407997i	$0.133772\pi$
$811$			23.2379i			0.815993i	−0.912983	+	0.407997i	$0.866228\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	22.5167	−	29.0689i	0.787277	−	1.01637i
$819$		0			0
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$		0			0			−	1.00000i	$-0.5\pi$
$823$		0			0				1.00000i	$0.5\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	35.7771i		−	1.24409i	−0.782981	−	0.622046i	$-0.786302\pi$
$827$		−	35.7771i		−	1.24409i	0.782981	−	0.622046i	$-0.213698\pi$
$828$		0			0
$829$			46.4758i			1.61417i	0.590434	+	0.807086i	$0.298956\pi$
$829$			46.4758i			1.61417i	−0.590434	+	0.807086i	$0.701044\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−48.4974			−1.68034
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$	17.3205	+	13.4164i	0.596904	+	0.462360i
$843$		0			0
$844$	−15.0000	+	3.87298i	−0.516321	+	0.133314i
$845$		0			0
$846$		0			0
$847$		0			0
$848$	−8.66025	−	15.6525i	−0.297394	−	0.537508i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		0			0
$855$		0			0
$856$	20.0000	−	46.4758i	0.683586	−	1.58851i
$857$	−6.92820			−0.236663			−0.118331	−	0.992974i	$-0.537755\pi$
$857$	−6.92820			−0.236663			−0.118331	+	0.992974i	$0.537755\pi$
$858$		0			0
$859$			38.7298i			1.32144i	0.750630	+	0.660722i	$0.229750\pi$
$859$			38.7298i			1.32144i	−0.750630	+	0.660722i	$0.770250\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	38.1051			1.29711			0.648557	−	0.761166i	$-0.275373\pi$
$863$	38.1051			1.29711			0.648557	+	0.761166i	$0.275373\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	17.3205	−	40.2492i	0.586546	−	1.36301i
$873$		0			0
$874$	−30.0000	−	23.2379i	−1.01477	−	0.786034i
$875$		0			0
$876$		0			0
$877$		0			0		1.00000			$0$
$877$		0			0		−1.00000			$\pi$
$878$	13.8564	−	17.8885i	0.467631	−	0.603709i
$879$		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
$886$	−20.0000	−	15.4919i	−0.671913	−	0.520462i
$887$	58.8897			1.97732			0.988662	−	0.150160i	$-0.0479788\pi$
$887$	58.8897			1.97732			0.988662	+	0.150160i	$0.0479788\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	80.4984i		−	2.69378i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$			30.9839i			1.03222i
$902$		0			0
$903$		0			0
$904$	54.0000	+	23.2379i	1.79601	+	0.772881i
$905$		0			0
$906$		0			0
$907$		0			0		1.00000			$0$
$907$		0			0		−1.00000			$\pi$
$908$	34.6410	−	8.94427i	1.14960	−	0.296826i
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−30.0000	+	7.74597i	−0.991228	+	0.255934i
$917$		0			0
$918$		0			0
$919$	−56.0000			−1.84727			−0.923635	−	0.383274i	$-0.874797\pi$
$919$	−56.0000			−1.84727			−0.923635	+	0.383274i	$0.874797\pi$
$920$		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
$931$			54.2218i			1.77705i
$932$	−10.3923	−	40.2492i	−0.340411	−	1.31841i
$933$		0			0
$934$	−40.0000	−	30.9839i	−1.30884	−	1.01382i
$935$		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		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$			17.8885i			0.581300i	0.956830	+	0.290650i	$0.0938715\pi$
$947$			17.8885i			0.581300i	−0.956830	+	0.290650i	$0.906129\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−20.7846			−0.673280			−0.336640	−	0.941634i	$-0.609290\pi$
$953$	−20.7846			−0.673280			−0.336640	+	0.941634i	$0.609290\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	33.0000			1.06452
$962$		0			0
$963$		0			0
$964$	1.00000	+	3.87298i	0.0322078	+	0.124740i
$965$		0			0
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$	28.5788	+	12.2984i	0.918559	+	0.395285i
$969$		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
$976$	30.0000	+	54.2218i	0.960277	+	1.73560i
$977$	62.3538			1.99488			0.997438	−	0.0715382i	$-0.0227908\pi$
$977$	62.3538			1.99488			0.997438	+	0.0715382i	$0.0227908\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	3.46410			0.110488			0.0552438	−	0.998473i	$-0.482406\pi$
$983$	3.46410			0.110488			0.0552438	+	0.998473i	$0.482406\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	32.0000			1.01651			0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000			1.01651			0.508257	+	0.861206i	$0.330290\pi$
$992$	−6.92820	+	44.7214i	−0.219971	+	1.41990i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		1.00000			$0$
$997$		0			0		−1.00000			$\pi$
$998$	8.66025	+	6.70820i	0.274136	+	0.212344i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1800.2.k.l.901.2		4
3.2	odd	2	inner	1800.2.k.l.901.3		4
4.3	odd	2		7200.2.k.n.3601.4		4
5.2	odd	4		360.2.d.c.109.1	✓	4
5.3	odd	4		360.2.d.c.109.4	yes	4
5.4	even	2	inner	1800.2.k.l.901.3		4
8.3	odd	2		7200.2.k.n.3601.3		4
8.5	even	2	inner	1800.2.k.l.901.1		4
12.11	even	2		7200.2.k.n.3601.2		4
15.2	even	4		360.2.d.c.109.4	yes	4
15.8	even	4		360.2.d.c.109.1	✓	4
15.14	odd	2	CM	1800.2.k.l.901.2		4
20.3	even	4		1440.2.d.d.1009.3		4
20.7	even	4		1440.2.d.d.1009.2		4
20.19	odd	2		7200.2.k.n.3601.2		4
24.5	odd	2	inner	1800.2.k.l.901.4		4
24.11	even	2		7200.2.k.n.3601.1		4
40.3	even	4		1440.2.d.d.1009.1		4
40.13	odd	4		360.2.d.c.109.2	yes	4
40.19	odd	2		7200.2.k.n.3601.1		4
40.27	even	4		1440.2.d.d.1009.4		4
40.29	even	2	inner	1800.2.k.l.901.4		4
40.37	odd	4		360.2.d.c.109.3	yes	4
60.23	odd	4		1440.2.d.d.1009.2		4
60.47	odd	4		1440.2.d.d.1009.3		4
60.59	even	2		7200.2.k.n.3601.4		4
120.29	odd	2	inner	1800.2.k.l.901.1		4
120.53	even	4		360.2.d.c.109.3	yes	4
120.59	even	2		7200.2.k.n.3601.3		4
120.77	even	4		360.2.d.c.109.2	yes	4
120.83	odd	4		1440.2.d.d.1009.4		4
120.107	odd	4		1440.2.d.d.1009.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.d.c.109.1	✓	4	5.2	odd	4
360.2.d.c.109.1	✓	4	15.8	even	4
360.2.d.c.109.2	yes	4	40.13	odd	4
360.2.d.c.109.2	yes	4	120.77	even	4
360.2.d.c.109.3	yes	4	40.37	odd	4
360.2.d.c.109.3	yes	4	120.53	even	4
360.2.d.c.109.4	yes	4	5.3	odd	4
360.2.d.c.109.4	yes	4	15.2	even	4
1440.2.d.d.1009.1		4	40.3	even	4
1440.2.d.d.1009.1		4	120.107	odd	4
1440.2.d.d.1009.2		4	20.7	even	4
1440.2.d.d.1009.2		4	60.23	odd	4
1440.2.d.d.1009.3		4	20.3	even	4
1440.2.d.d.1009.3		4	60.47	odd	4
1440.2.d.d.1009.4		4	40.27	even	4
1440.2.d.d.1009.4		4	120.83	odd	4
1800.2.k.l.901.1		4	8.5	even	2	inner
1800.2.k.l.901.1		4	120.29	odd	2	inner
1800.2.k.l.901.2		4	1.1	even	1	trivial
1800.2.k.l.901.2		4	15.14	odd	2	CM
1800.2.k.l.901.3		4	3.2	odd	2	inner
1800.2.k.l.901.3		4	5.4	even	2	inner
1800.2.k.l.901.4		4	24.5	odd	2	inner
1800.2.k.l.901.4		4	40.29	even	2	inner
7200.2.k.n.3601.1		4	24.11	even	2
7200.2.k.n.3601.1		4	40.19	odd	2
7200.2.k.n.3601.2		4	12.11	even	2
7200.2.k.n.3601.2		4	20.19	odd	2
7200.2.k.n.3601.3		4	8.3	odd	2
7200.2.k.n.3601.3		4	120.59	even	2
7200.2.k.n.3601.4		4	4.3	odd	2
7200.2.k.n.3601.4		4	60.59	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:	\( N \)	\(=\)	\( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1800.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.866025 + 1.11803i) q^{2} +(-0.500000 - 1.93649i) q^{4} +(2.59808 + 1.11803i) q^{8} +O(q^{10})\)	\(q+(-0.866025 + 1.11803i) q^{2} +(-0.500000 - 1.93649i) q^{4} +(2.59808 + 1.11803i) q^{8} +(-3.50000 + 1.93649i) q^{16} +6.92820 q^{17} -7.74597i q^{19} -3.46410 q^{23} -8.00000 q^{31} +(0.866025 - 5.59017i) q^{32} +(-6.00000 + 7.74597i) q^{34} +(8.66025 + 6.70820i) q^{38} +(3.00000 - 3.87298i) q^{46} +10.3923 q^{47} -7.00000 q^{49} +4.47214i q^{53} -15.4919i q^{61} +(6.92820 - 8.94427i) q^{62} +(5.50000 + 5.80948i) q^{64} +(-3.46410 - 13.4164i) q^{68} +(-15.0000 + 3.87298i) q^{76} +16.0000 q^{79} -17.8885i q^{83} +(1.73205 + 6.70820i) q^{92} +(-9.00000 + 11.6190i) q^{94} +(6.06218 - 7.82624i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{4}+O(q^{10}) \) 4 * q - 2 * q^4	\( 4 q - 2 q^{4} - 14 q^{16} - 32 q^{31} - 24 q^{34} + 12 q^{46} - 28 q^{49} + 22 q^{64} - 60 q^{76} + 64 q^{79} - 36 q^{94}+O(q^{100}) \) 4 * q - 2 * q^4 - 14 * q^16 - 32 * q^31 - 24 * q^34 + 12 * q^46 - 28 * q^49 + 22 * q^64 - 60 * q^76 + 64 * q^79 - 36 * q^94

Introduction

L-functions

Modular forms

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Representations

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Database

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