LMFDB - Embedded newform 1800.2.b.f.251.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1800,2,Mod(251,1800)]

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([1, 1, 1, 0]))

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

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

chi := DirichletCharacter("1800.251");

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.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$14.3730723638$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.40960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 1 $ x^8 + 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			251.1
Root			$0.437016 - 0.437016i$ of defining polynomial
Character	$\chi$	$=$	1800.251
Dual form			1800.2.b.f.251.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +2.00000 q^{4} -5.16228i q^{7} -2.82843 q^{8} +O(q^{10})$	$q-1.41421 q^{2} +2.00000 q^{4} -5.16228i q^{7} -2.82843 q^{8} -3.05792i q^{11} -0.837722i q^{13} +7.30056i q^{14} +4.00000 q^{16} +6.32456 q^{19} +4.32456i q^{22} +4.47214 q^{23} +1.18472i q^{26} -10.3246i q^{28} -5.65685 q^{32} -11.1623i q^{37} -8.94427 q^{38} +10.3585i q^{41} -6.11584i q^{44} -6.32456 q^{46} -2.82843 q^{47} -19.6491 q^{49} -1.67544i q^{52} -5.65685 q^{53} +14.6011i q^{56} -5.42736i q^{59} +8.00000 q^{64} +15.7858i q^{74} +12.6491 q^{76} -15.7858 q^{77} -14.6491i q^{82} +8.64911i q^{88} -18.8438i q^{89} -4.32456 q^{91} +8.94427 q^{92} +4.00000 q^{94} +27.7880 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{4}+O(q^{10}) $ 8 * q + 16 * q^4	$ 8 q + 16 q^{4} + 32 q^{16} - 56 q^{49} + 64 q^{64} + 16 q^{91} + 32 q^{94}+O(q^{100}) $ 8 * q + 16 * q^4 + 32 * q^16 - 56 * q^49 + 64 * q^64 + 16 * q^91 + 32 * 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 $

$2$	−1.41421	−1.00000
$2$	−1.41421	−1.00000
$3$	0	0
$3$	0	0
$4$	2.00000	1.00000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 5.16228i	− 1.95116i	−0.219650	−	0.975579i	$-0.570491\pi$
$7$	− 5.16228i	− 1.95116i	0.219650	−	0.975579i	$-0.429509\pi$
$8$	−2.82843	−1.00000
$9$	0	0
$10$	0	0
$11$	− 3.05792i	− 0.921998i	−0.887401	−	0.460999i	$-0.847491\pi$
$11$	− 3.05792i	− 0.921998i	0.887401	−	0.460999i	$-0.152509\pi$
$12$	0	0
$13$	− 0.837722i	− 0.232342i	−0.993229	−	0.116171i	$-0.962938\pi$
$13$	− 0.837722i	− 0.232342i	0.993229	−	0.116171i	$-0.0370621\pi$
$14$	7.30056i	1.95116i
$15$	0	0
$16$	4.00000	1.00000
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	6.32456	1.45095	0.725476	−	0.688247i	$-0.241620\pi$
$19$	6.32456	1.45095	0.725476	+	0.688247i	$0.241620\pi$
$20$	0	0
$21$	0	0
$22$	4.32456i	0.921998i
$23$	4.47214	0.932505	0.466252	−	0.884652i	$-0.345604\pi$
$23$	4.47214	0.932505	0.466252	+	0.884652i	$0.345604\pi$
$24$	0	0
$25$	0	0
$26$	1.18472i	0.232342i
$27$	0	0
$28$	− 10.3246i	− 1.95116i
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	−5.65685	−1.00000
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 11.1623i	− 1.83507i	−0.397658	−	0.917534i	$-0.630177\pi$
$37$	− 11.1623i	− 1.83507i	0.397658	−	0.917534i	$-0.369823\pi$
$38$	−8.94427	−1.45095
$39$	0	0
$40$	0	0
$41$	10.3585i	1.61772i	0.587999	+	0.808862i	$0.299916\pi$
$41$	10.3585i	1.61772i	−0.587999	+	0.808862i	$0.700084\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	− 6.11584i	− 0.921998i
$45$	0	0
$46$	−6.32456	−0.932505
$47$	−2.82843	−0.412568	−0.206284	−	0.978492i	$-0.566137\pi$
$47$	−2.82843	−0.412568	−0.206284	+	0.978492i	$0.566137\pi$
$48$	0	0
$49$	−19.6491	−2.80702
$50$	0	0
$51$	0	0
$52$	− 1.67544i	− 0.232342i
$53$	−5.65685	−0.777029	−0.388514	−	0.921443i	$-0.627012\pi$
$53$	−5.65685	−0.777029	−0.388514	+	0.921443i	$0.627012\pi$
$54$	0	0
$55$	0	0
$56$	14.6011i	1.95116i
$57$	0	0
$58$	0	0
$59$	− 5.42736i	− 0.706582i	−0.935513	−	0.353291i	$-0.885063\pi$
$59$	− 5.42736i	− 0.706582i	0.935513	−	0.353291i	$-0.114937\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	8.00000	1.00000
$65$	0	0
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	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	$0.5\pi$
$74$	15.7858i	1.83507i
$75$	0	0
$76$	12.6491	1.45095
$77$	−15.7858	−1.79896
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	0	0
$82$	− 14.6491i	− 1.61772i
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	8.64911i	0.921998i
$89$	− 18.8438i	− 1.99744i	−0.0506267	−	0.998718i	$-0.516122\pi$
$89$	− 18.8438i	− 1.99744i	0.0506267	−	0.998718i	$-0.483878\pi$
$90$	0	0
$91$	−4.32456	−0.453337
$92$	8.94427	0.932505
$93$	0	0
$94$	4.00000	0.412568
$95$	0	0
$96$	0	0
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	27.7880	2.80702
$99$	0	0
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	17.1623i	1.69105i	0.533936	+	0.845525i	$0.320712\pi$
$103$	17.1623i	1.69105i	−0.533936	+	0.845525i	$0.679288\pi$
$104$	2.36944i	0.232342i
$105$	0	0
$106$	8.00000	0.777029
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	− 20.6491i	− 1.95116i
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	7.67544i	0.706582i
$119$	0	0
$120$	0	0
$121$	1.64911	0.149919
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	13.8114i	1.22556i	0.790253	+	0.612781i	$0.209949\pi$
$127$	13.8114i	1.22556i	−0.790253	+	0.612781i	$0.790051\pi$
$128$	−11.3137	−1.00000
$129$	0	0
$130$	0	0
$131$	− 20.0285i	− 1.74990i	−0.484216	−	0.874948i	$-0.660895\pi$
$131$	− 20.0285i	− 1.74990i	0.484216	−	0.874948i	$-0.339105\pi$
$132$	0	0
$133$	− 32.6491i	− 2.83104i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2.56169	−0.214219
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	− 22.3246i	− 1.83507i
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	−17.8885	−1.45095
$153$	0	0
$154$	22.3246	1.79896
$155$	0	0
$156$	0	0
$157$	7.81139i	0.623417i	0.950178	+	0.311708i	$0.100901\pi$
$157$	7.81139i	0.623417i	−0.950178	+	0.311708i	$0.899099\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 23.0864i	− 1.81946i
$162$	0	0
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	20.7170i	1.61772i
$165$	0	0
$166$	0	0
$167$	−4.47214	−0.346064	−0.173032	−	0.984916i	$-0.555356\pi$
$167$	−4.47214	−0.346064	−0.173032	+	0.984916i	$0.555356\pi$
$168$	0	0
$169$	12.2982	0.946017
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−22.6274	−1.72033	−0.860165	−	0.510015i	$-0.829640\pi$
$173$	−22.6274	−1.72033	−0.860165	+	0.510015i	$0.829640\pi$
$174$	0	0
$175$	0	0
$176$	− 12.2317i	− 0.921998i
$177$	0	0
$178$	26.6491i	1.99744i
$179$	− 13.9126i	− 1.03988i	−0.854203	−	0.519940i	$-0.825954\pi$
$179$	− 13.9126i	− 1.03988i	0.854203	−	0.519940i	$-0.174046\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	6.11584	0.453337
$183$	0	0
$184$	−12.6491	−0.932505
$185$	0	0
$186$	0	0
$187$	0	0
$188$	−5.65685	−0.412568
$189$	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
$196$	−39.2982	−2.80702
$197$	22.3607	1.59313	0.796566	−	0.604551i	$-0.206648\pi$
$197$	22.3607	1.59313	0.796566	+	0.604551i	$0.206648\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	− 24.2711i	− 1.69105i
$207$	0	0
$208$	− 3.35089i	− 0.232342i
$209$	− 19.3400i	− 1.33778i
$210$	0	0
$211$	−22.0000	−1.51454	−0.757271	−	0.653101i	$-0.773468\pi$
$211$	−22.0000	−1.51454	−0.757271	+	0.653101i	$0.773468\pi$
$212$	−11.3137	−0.777029
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 1.81139i	− 0.121300i	−0.998159	−	0.0606498i	$-0.980683\pi$
$223$	− 1.81139i	− 0.121300i	0.998159	−	0.0606498i	$-0.0193173\pi$
$224$	29.2023i	1.95116i
$225$	0	0
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	0	0
$236$	− 10.8547i	− 0.706582i
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−25.2982	−1.62960	−0.814801	−	0.579741i	$-0.803154\pi$
$241$	−25.2982	−1.62960	−0.814801	+	0.579741i	$0.803154\pi$
$242$	−2.33219	−0.149919
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 5.29822i	− 0.337118i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	23.7749i	1.50066i	0.661065	+	0.750329i	$0.270105\pi$
$251$	23.7749i	1.50066i	−0.661065	+	0.750329i	$0.729895\pi$
$252$	0	0
$253$	− 13.6754i	− 0.859768i
$254$	− 19.5323i	− 1.22556i
$255$	0	0
$256$	16.0000	1.00000
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	−57.6228	−3.58051
$260$	0	0
$261$	0	0
$262$	28.3246i	1.74990i
$263$	31.3050	1.93035	0.965173	−	0.261612i	$-0.0842542\pi$
$263$	31.3050	1.93035	0.965173	+	0.261612i	$0.0842542\pi$
$264$	0	0
$265$	0	0
$266$	46.1728i	2.83104i
$267$	0	0
$268$	0	0
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 30.1359i	− 1.81069i	−0.424673	−	0.905347i	$-0.639611\pi$
$277$	− 30.1359i	− 1.81069i	0.424673	−	0.905347i	$-0.360389\pi$
$278$	19.7990	1.18746
$279$	0	0
$280$	0	0
$281$	− 33.4449i	− 1.99516i	−0.0695635	−	0.997578i	$-0.522161\pi$
$281$	− 33.4449i	− 1.99516i	0.0695635	−	0.997578i	$-0.477839\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	0	0
$285$	0	0
$286$	3.62278	0.214219
$287$	53.4734	3.15643
$288$	0	0
$289$	17.0000	1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	4.47214	0.261265	0.130632	−	0.991431i	$-0.458299\pi$
$293$	4.47214	0.261265	0.130632	+	0.991431i	$0.458299\pi$
$294$	0	0
$295$	0	0
$296$	31.5717i	1.83507i
$297$	0	0
$298$	0	0
$299$	− 3.74641i	− 0.216660i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	25.2982	1.45095
$305$	0	0
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	−31.5717	−1.79896
$309$	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	$0.5\pi$
$314$	− 11.0470i	− 0.623417i
$315$	0	0
$316$	0	0
$317$	−31.3050	−1.75826	−0.879131	−	0.476581i	$-0.841876\pi$
$317$	−31.3050	−1.75826	−0.879131	+	0.476581i	$0.841876\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	32.6491i	1.81946i
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	− 29.2982i	− 1.61772i
$329$	14.6011i	0.804986i
$330$	0	0
$331$	31.6228	1.73814	0.869072	−	0.494685i	$-0.164716\pi$
$331$	31.6228	1.73814	0.869072	+	0.494685i	$0.164716\pi$
$332$	0	0
$333$	0	0
$334$	6.32456	0.346064
$335$	0	0
$336$	0	0
$337$	0	0		−	1.00000i	$-0.5\pi$
$337$	0	0			1.00000i	$0.5\pi$
$338$	−17.3923	−0.946017
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	65.2982i	3.52577i
$344$	0	0
$345$	0	0
$346$	32.0000	1.72033
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	17.2982i	0.921998i
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	− 37.6875i	− 1.99744i
$357$	0	0
$358$	19.6754i	1.03988i
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	21.0000	1.10526
$362$	0	0
$363$	0	0
$364$	−8.64911	−0.453337
$365$	0	0
$366$	0	0
$367$	37.8114i	1.97374i	0.161521	+	0.986869i	$0.448360\pi$
$367$	37.8114i	1.97374i	−0.161521	+	0.986869i	$0.551640\pi$
$368$	17.8885	0.932505
$369$	0	0
$370$	0	0
$371$	29.2023i	1.51611i
$372$	0	0
$373$	6.13594i	0.317707i	0.987302	+	0.158854i	$0.0507798\pi$
$373$	6.13594i	0.317707i	−0.987302	+	0.158854i	$0.949220\pi$
$374$	0	0
$375$	0	0
$376$	8.00000	0.412568
$377$	0	0
$378$	0	0
$379$	34.0000	1.74646	0.873231	−	0.487306i	$-0.162020\pi$
$379$	34.0000	1.74646	0.873231	+	0.487306i	$0.162020\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	36.7696	1.87884	0.939418	−	0.342773i	$-0.111366\pi$
$383$	36.7696	1.87884	0.939418	+	0.342773i	$0.111366\pi$
$384$	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
$391$	0	0
$392$	55.5761	2.80702
$393$	0	0
$394$	−31.6228	−1.59313
$395$	0	0
$396$	0	0
$397$	24.8377i	1.24657i	0.781995	+	0.623285i	$0.214202\pi$
$397$	24.8377i	1.24657i	−0.781995	+	0.623285i	$0.785798\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	35.8143i	1.78848i	0.447586	+	0.894241i	$0.352284\pi$
$401$	35.8143i	1.78848i	−0.447586	+	0.894241i	$0.647716\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−34.1334	−1.69193
$408$	0	0
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	0	0
$412$	34.3246i	1.69105i
$413$	−28.0175	−1.37865
$414$	0	0
$415$	0	0
$416$	4.73887i	0.232342i
$417$	0	0
$418$	27.3509i	1.33778i
$419$	− 40.7455i	− 1.99055i	−0.0971178	−	0.995273i	$-0.530962\pi$
$419$	− 40.7455i	− 1.99055i	0.0971178	−	0.995273i	$-0.469038\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	31.1127	1.51454
$423$	0	0
$424$	16.0000	0.777029
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	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
$436$	0	0
$437$	28.2843	1.35302
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	0	0
$446$	2.56169i	0.121300i
$447$	0	0
$448$	− 41.2982i	− 1.95116i
$449$	41.9302i	1.97881i	0.145191	+	0.989404i	$0.453620\pi$
$449$	41.9302i	1.97881i	−0.145191	+	0.989404i	$0.546380\pi$
$450$	0	0
$451$	31.6754	1.49154
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	10.1886i	0.473505i	0.971570	+	0.236752i	$0.0760830\pi$
$463$	10.1886i	0.473505i	−0.971570	+	0.236752i	$0.923917\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	15.3509i	0.706582i
$473$	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
$481$	−9.35089	−0.426364
$482$	35.7771	1.62960
$483$	0	0
$484$	3.29822	0.149919
$485$	0	0
$486$	0	0
$487$	− 0.135944i	− 0.00616019i	−0.999995	−	0.00308010i	$-0.999020\pi$
$487$	− 0.135944i	− 0.00616019i	0.999995	−	0.00308010i	$-0.000980427\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 29.8907i	− 1.34895i	−0.738298	−	0.674475i	$-0.764370\pi$
$491$	− 29.8907i	− 1.34895i	0.738298	−	0.674475i	$-0.235630\pi$
$492$	0	0
$493$	0	0
$494$	7.49282i	0.337118i
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	44.2719	1.98188	0.990941	−	0.134298i	$-0.0428781\pi$
$499$	44.2719	1.98188	0.990941	+	0.134298i	$0.0428781\pi$
$500$	0	0
$501$	0	0
$502$	− 33.6228i	− 1.50066i
$503$	19.7990	0.882793	0.441397	−	0.897312i	$-0.354483\pi$
$503$	19.7990	0.882793	0.441397	+	0.897312i	$0.354483\pi$
$504$	0	0
$505$	0	0
$506$	19.3400i	0.859768i
$507$	0	0
$508$	27.6228i	1.22556i
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	−22.6274	−1.00000
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	8.64911i	0.380387i
$518$	81.4909	3.58051
$519$	0	0
$520$	0	0
$521$	8.98151i	0.393487i	0.980455	+	0.196744i	$0.0630367\pi$
$521$	8.98151i	0.393487i	−0.980455	+	0.196744i	$0.936963\pi$
$522$	0	0
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	− 40.0570i	− 1.74990i
$525$	0	0
$526$	−44.2719	−1.93035
$527$	0	0
$528$	0	0
$529$	−3.00000	−0.130435
$530$	0	0
$531$	0	0
$532$	− 65.2982i	− 2.83104i
$533$	8.67753	0.375866
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	60.0855i	2.58806i
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	42.6187i	1.81069i
$555$	0	0
$556$	−28.0000	−1.18746
$557$	−45.2548	−1.91751	−0.958754	−	0.284236i	$-0.908260\pi$
$557$	−45.2548	−1.91751	−0.958754	+	0.284236i	$0.908260\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	47.2982i	1.99516i
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	23.5826i	0.988636i	0.869281	+	0.494318i	$0.164582\pi$
$569$	23.5826i	0.988636i	−0.869281	+	0.494318i	$0.835418\pi$
$570$	0	0
$571$	−44.2719	−1.85272	−0.926360	−	0.376638i	$-0.877080\pi$
$571$	−44.2719	−1.85272	−0.926360	+	0.376638i	$0.877080\pi$
$572$	−5.12338	−0.214219
$573$	0	0
$574$	−75.6228	−3.15643
$575$	0	0
$576$	0	0
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	−24.0416	−1.00000
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	17.2982i	0.716419i
$584$	0	0
$585$	0	0
$586$	−6.32456	−0.261265
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	− 44.6491i	− 1.83507i
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	5.29822i	0.216660i
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−25.2982	−1.03194	−0.515968	−	0.856608i	$-0.672568\pi$
$601$	−25.2982	−1.03194	−0.515968	+	0.856608i	$0.672568\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	32.7851i	1.33070i	0.746530	+	0.665352i	$0.231719\pi$
$607$	32.7851i	1.33070i	−0.746530	+	0.665352i	$0.768281\pi$
$608$	−35.7771	−1.45095
$609$	0	0
$610$	0	0
$611$	2.36944i	0.0958571i
$612$	0	0
$613$	− 38.7851i	− 1.56651i	−0.621698	−	0.783257i	$-0.713557\pi$
$613$	− 38.7851i	− 1.56651i	0.621698	−	0.783257i	$-0.286443\pi$
$614$	0	0
$615$	0	0
$616$	44.6491	1.79896
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	46.0000	1.84890	0.924448	−	0.381308i	$-0.124526\pi$
$619$	46.0000	1.84890	0.924448	+	0.381308i	$0.124526\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−97.2768	−3.89731
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	15.6228i	0.623417i
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	44.2719	1.75826
$635$	0	0
$636$	0	0
$637$	16.4605i	0.652189i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	37.1913i	1.46897i	0.678626	+	0.734484i	$0.262576\pi$
$641$	37.1913i	1.46897i	−0.678626	+	0.734484i	$0.737424\pi$
$642$	0	0
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	− 46.1728i	− 1.81946i
$645$	0	0
$646$	0	0
$647$	31.1127	1.22317	0.611583	−	0.791180i	$-0.290533\pi$
$647$	31.1127	1.22317	0.611583	+	0.791180i	$0.290533\pi$
$648$	0	0
$649$	−16.5964	−0.651467
$650$	0	0
$651$	0	0
$652$	0	0
$653$	4.47214	0.175008	0.0875041	−	0.996164i	$-0.472111\pi$
$653$	4.47214	0.175008	0.0875041	+	0.996164i	$0.472111\pi$
$654$	0	0
$655$	0	0
$656$	41.4339i	1.61772i
$657$	0	0
$658$	− 20.6491i	− 0.804986i
$659$	12.9202i	0.503299i	0.967818	+	0.251649i	$0.0809729\pi$
$659$	12.9202i	0.503299i	−0.967818	+	0.251649i	$0.919027\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	−44.7214	−1.73814
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	−8.94427	−0.346064
$669$	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
$676$	24.5964	0.946017
$677$	49.1935	1.89066	0.945330	−	0.326116i	$-0.105740\pi$
$677$	49.1935	1.89066	0.945330	+	0.326116i	$0.105740\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	0	0
$686$	− 92.3456i	− 3.52577i
$687$	0	0
$688$	0	0
$689$	4.73887i	0.180537i
$690$	0	0
$691$	31.6228	1.20299	0.601494	−	0.798878i	$-0.294573\pi$
$691$	31.6228	1.20299	0.601494	+	0.798878i	$0.294573\pi$
$692$	−45.2548	−1.72033
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	− 70.5964i	− 2.66260i
$704$	− 24.4634i	− 0.921998i
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	53.2982i	1.99744i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	− 27.8253i	− 1.03988i
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	88.5964	3.29950
$722$	−29.6985	−1.10526
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 41.1623i	− 1.52662i	−0.646030	−	0.763312i	$-0.723572\pi$
$727$	− 41.1623i	− 1.52662i	0.646030	−	0.763312i	$-0.276428\pi$
$728$	12.2317	0.453337
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	− 50.7851i	− 1.87579i	−0.346921	−	0.937894i	$-0.612773\pi$
$733$	− 50.7851i	− 1.87579i	0.346921	−	0.937894i	$-0.387227\pi$
$734$	− 53.4734i	− 1.97374i
$735$	0	0
$736$	−25.2982	−0.932505
$737$	0	0
$738$	0	0
$739$	6.32456	0.232653	0.116326	−	0.993211i	$-0.462888\pi$
$739$	6.32456	0.232653	0.116326	+	0.993211i	$0.462888\pi$
$740$	0	0
$741$	0	0
$742$	− 41.2982i	− 1.51611i
$743$	36.7696	1.34894	0.674472	−	0.738300i	$-0.264371\pi$
$743$	36.7696	1.34894	0.674472	+	0.738300i	$0.264371\pi$
$744$	0	0
$745$	0	0
$746$	− 8.67753i	− 0.317707i
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	−11.3137	−0.412568
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 54.1359i	− 1.96760i	−0.179258	−	0.983802i	$-0.557370\pi$
$757$	− 54.1359i	− 1.96760i	0.179258	−	0.983802i	$-0.442630\pi$
$758$	−48.0833	−1.74646
$759$	0	0
$760$	0	0
$761$	20.2207i	0.733001i	0.930418	+	0.366501i	$0.119444\pi$
$761$	20.2207i	0.733001i	−0.930418	+	0.366501i	$0.880556\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	−52.0000	−1.87884
$767$	−4.54662	−0.164169
$768$	0	0
$769$	−12.6491	−0.456139	−0.228069	−	0.973645i	$-0.573241\pi$
$769$	−12.6491	−0.456139	−0.228069	+	0.973645i	$0.573241\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−22.3607	−0.804258	−0.402129	−	0.915583i	$-0.631730\pi$
$773$	−22.3607	−0.804258	−0.402129	+	0.915583i	$0.631730\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	65.5128i	2.34724i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−78.5964	−2.80702
$785$	0	0
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	44.7214	1.59313
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	− 35.1258i	− 1.24657i
$795$	0	0
$796$	0	0
$797$	39.5980	1.40263	0.701316	−	0.712850i	$-0.252596\pi$
$797$	39.5980	1.40263	0.701316	+	0.712850i	$0.252596\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	− 50.6491i	− 1.78848i
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 54.1619i	− 1.90423i	−0.305741	−	0.952115i	$-0.598904\pi$
$809$	− 54.1619i	− 1.90423i	0.305741	−	0.952115i	$-0.401096\pi$
$810$	0	0
$811$	2.00000	0.0702295	0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000	0.0702295	0.0351147	+	0.999383i	$0.488820\pi$
$812$	0	0
$813$	0	0
$814$	48.2719	1.69193
$815$	0	0
$816$	0	0
$817$	0	0
$818$	19.7990	0.692255
$819$	0	0
$820$	0	0
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	0	0
$823$	− 11.8641i	− 0.413555i	−0.978388	−	0.206778i	$-0.933702\pi$
$823$	− 11.8641i	− 0.413555i	0.978388	−	0.206778i	$-0.0662976\pi$
$824$	− 48.5423i	− 1.69105i
$825$	0	0
$826$	39.6228	1.37865
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	− 6.70178i	− 0.232342i
$833$	0	0
$834$	0	0
$835$	0	0
$836$	− 38.6800i	− 1.33778i
$837$	0	0
$838$	57.6228i	1.99055i
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	0	0
$844$	−44.0000	−1.51454
$845$	0	0
$846$	0	0
$847$	− 8.51317i	− 0.292516i
$848$	−22.6274	−0.777029
$849$	0	0
$850$	0	0
$851$	− 49.9192i	− 1.71121i
$852$	0	0
$853$	37.1096i	1.27061i	0.772262	+	0.635304i	$0.219125\pi$
$853$	37.1096i	1.27061i	−0.772262	+	0.635304i	$0.780875\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	−14.0000	−0.477674	−0.238837	−	0.971060i	$-0.576766\pi$
$859$	−14.0000	−0.477674	−0.238837	+	0.971060i	$0.576766\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	58.1378	1.97903	0.989516	−	0.144421i	$-0.0461320\pi$
$863$	58.1378	1.97903	0.989516	+	0.144421i	$0.0461320\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	−40.0000	−1.35302
$875$	0	0
$876$	0	0
$877$	− 49.1096i	− 1.65831i	−0.559016	−	0.829157i	$-0.688821\pi$
$877$	− 49.1096i	− 1.65831i	0.559016	−	0.829157i	$-0.311179\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 32.0679i	− 1.08040i	−0.841538	−	0.540198i	$-0.818349\pi$
$881$	− 32.0679i	− 1.08040i	0.841538	−	0.540198i	$-0.181651\pi$
$882$	0	0
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−4.47214	−0.150160	−0.0750798	−	0.997178i	$-0.523921\pi$
$887$	−4.47214	−0.150160	−0.0750798	+	0.997178i	$0.523921\pi$
$888$	0	0
$889$	71.2982	2.39127
$890$	0	0
$891$	0	0
$892$	− 3.62278i	− 0.121300i
$893$	−17.8885	−0.598617
$894$	0	0
$895$	0	0
$896$	58.4045i	1.95116i
$897$	0	0
$898$	− 59.2982i	− 1.97881i
$899$	0	0
$900$	0	0
$901$	0	0
$902$	−44.7958	−1.49154
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	0	0
$909$	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
$916$	0	0
$917$	−103.393	−3.41432
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	− 14.4089i	− 0.473505i
$927$	0	0
$928$	0	0
$929$	− 11.7355i	− 0.385028i	−0.981294	−	0.192514i	$-0.938336\pi$
$929$	− 11.7355i	− 0.385028i	0.981294	−	0.192514i	$-0.0616641\pi$
$930$	0	0
$931$	−124.272	−4.07285
$932$	0	0
$933$	0	0
$934$	0	0
$935$	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
$943$	46.3246i	1.50854i
$944$	− 21.7094i	− 0.706582i
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	31.0000	1.00000
$962$	13.2242	0.426364
$963$	0	0
$964$	−50.5964	−1.62960
$965$	0	0
$966$	0	0
$967$	− 60.1359i	− 1.93384i	−0.255077	−	0.966921i	$-0.582101\pi$
$967$	− 60.1359i	− 1.93384i	0.255077	−	0.966921i	$-0.417899\pi$
$968$	−4.66439	−0.149919
$969$	0	0
$970$	0	0
$971$	39.3685i	1.26339i	0.775215	+	0.631697i	$0.217641\pi$
$971$	39.3685i	1.26339i	−0.775215	+	0.631697i	$0.782359\pi$
$972$	0	0
$973$	72.2719i	2.31693i
$974$	0.192253i	0.00616019i
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	−57.6228	−1.84163
$980$	0	0
$981$	0	0
$982$	42.2719i	1.34895i
$983$	−48.0833	−1.53362	−0.766809	−	0.641875i	$-0.778157\pi$
$983$	−48.0833	−1.53362	−0.766809	+	0.641875i	$0.778157\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	− 10.5964i	− 0.337118i
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	62.7851i	1.98842i	0.107442	+	0.994211i	$0.465734\pi$
$997$	62.7851i	1.98842i	−0.107442	+	0.994211i	$0.534266\pi$
$998$	−62.6099	−1.98188
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1800.2.b.f.251.1		8
3.2	odd	2	inner	1800.2.b.f.251.5		8
4.3	odd	2		7200.2.b.f.4751.8		8
5.2	odd	4		360.2.m.b.179.1	yes	4
5.3	odd	4		360.2.m.a.179.4	yes	4
5.4	even	2	inner	1800.2.b.f.251.8		8
8.3	odd	2	inner	1800.2.b.f.251.8		8
8.5	even	2		7200.2.b.f.4751.2		8
12.11	even	2		7200.2.b.f.4751.7		8
15.2	even	4		360.2.m.b.179.4	yes	4
15.8	even	4		360.2.m.a.179.1	✓	4
15.14	odd	2	inner	1800.2.b.f.251.4		8
20.3	even	4		1440.2.m.b.719.4		4
20.7	even	4		1440.2.m.a.719.1		4
20.19	odd	2		7200.2.b.f.4751.2		8
24.5	odd	2		7200.2.b.f.4751.1		8
24.11	even	2	inner	1800.2.b.f.251.4		8
40.3	even	4		360.2.m.b.179.1	yes	4
40.13	odd	4		1440.2.m.a.719.1		4
40.19	odd	2	CM	1800.2.b.f.251.1		8
40.27	even	4		360.2.m.a.179.4	yes	4
40.29	even	2		7200.2.b.f.4751.8		8
40.37	odd	4		1440.2.m.b.719.4		4
60.23	odd	4		1440.2.m.b.719.2		4
60.47	odd	4		1440.2.m.a.719.3		4
60.59	even	2		7200.2.b.f.4751.1		8
120.29	odd	2		7200.2.b.f.4751.7		8
120.53	even	4		1440.2.m.a.719.3		4
120.59	even	2	inner	1800.2.b.f.251.5		8
120.77	even	4		1440.2.m.b.719.2		4
120.83	odd	4		360.2.m.b.179.4	yes	4
120.107	odd	4		360.2.m.a.179.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.m.a.179.1	✓	4	15.8	even	4
360.2.m.a.179.1	✓	4	120.107	odd	4
360.2.m.a.179.4	yes	4	5.3	odd	4
360.2.m.a.179.4	yes	4	40.27	even	4
360.2.m.b.179.1	yes	4	5.2	odd	4
360.2.m.b.179.1	yes	4	40.3	even	4
360.2.m.b.179.4	yes	4	15.2	even	4
360.2.m.b.179.4	yes	4	120.83	odd	4
1440.2.m.a.719.1		4	20.7	even	4
1440.2.m.a.719.1		4	40.13	odd	4
1440.2.m.a.719.3		4	60.47	odd	4
1440.2.m.a.719.3		4	120.53	even	4
1440.2.m.b.719.2		4	60.23	odd	4
1440.2.m.b.719.2		4	120.77	even	4
1440.2.m.b.719.4		4	20.3	even	4
1440.2.m.b.719.4		4	40.37	odd	4
1800.2.b.f.251.1		8	1.1	even	1	trivial
1800.2.b.f.251.1		8	40.19	odd	2	CM
1800.2.b.f.251.4		8	15.14	odd	2	inner
1800.2.b.f.251.4		8	24.11	even	2	inner
1800.2.b.f.251.5		8	3.2	odd	2	inner
1800.2.b.f.251.5		8	120.59	even	2	inner
1800.2.b.f.251.8		8	5.4	even	2	inner
1800.2.b.f.251.8		8	8.3	odd	2	inner
7200.2.b.f.4751.1		8	24.5	odd	2
7200.2.b.f.4751.1		8	60.59	even	2
7200.2.b.f.4751.2		8	8.5	even	2
7200.2.b.f.4751.2		8	20.19	odd	2
7200.2.b.f.4751.7		8	12.11	even	2
7200.2.b.f.4751.7		8	120.29	odd	2
7200.2.b.f.4751.8		8	4.3	odd	2
7200.2.b.f.4751.8		8	40.29	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.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +2.00000 q^{4} -5.16228i q^{7} -2.82843 q^{8} +O(q^{10})\)	\(q-1.41421 q^{2} +2.00000 q^{4} -5.16228i q^{7} -2.82843 q^{8} -3.05792i q^{11} -0.837722i q^{13} +7.30056i q^{14} +4.00000 q^{16} +6.32456 q^{19} +4.32456i q^{22} +4.47214 q^{23} +1.18472i q^{26} -10.3246i q^{28} -5.65685 q^{32} -11.1623i q^{37} -8.94427 q^{38} +10.3585i q^{41} -6.11584i q^{44} -6.32456 q^{46} -2.82843 q^{47} -19.6491 q^{49} -1.67544i q^{52} -5.65685 q^{53} +14.6011i q^{56} -5.42736i q^{59} +8.00000 q^{64} +15.7858i q^{74} +12.6491 q^{76} -15.7858 q^{77} -14.6491i q^{82} +8.64911i q^{88} -18.8438i q^{89} -4.32456 q^{91} +8.94427 q^{92} +4.00000 q^{94} +27.7880 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{4}+O(q^{10}) \) 8 * q + 16 * q^4	\( 8 q + 16 q^{4} + 32 q^{16} - 56 q^{49} + 64 q^{64} + 16 q^{91} + 32 q^{94}+O(q^{100}) \) 8 * q + 16 * q^4 + 32 * q^16 - 56 * q^49 + 64 * q^64 + 16 * q^91 + 32 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more