LMFDB - Embedded newform 1800.2.b.f.251.7

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			251.7
Root			$-0.437016 + 0.437016i$ of defining polynomial
Character	$\chi$	$=$	1800.251
Dual form			1800.2.b.f.251.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} +2.00000 q^{4} +1.16228i q^{7} +2.82843 q^{8} +O(q^{10})$	$q+1.41421 q^{2} +2.00000 q^{4} +1.16228i q^{7} +2.82843 q^{8} -5.88635i q^{11} -7.16228i q^{13} +1.64371i q^{14} +4.00000 q^{16} -6.32456 q^{19} -8.32456i q^{22} +4.47214 q^{23} -10.1290i q^{26} +2.32456i q^{28} +5.65685 q^{32} -4.83772i q^{37} -8.94427 q^{38} +7.53006i q^{41} -11.7727i q^{44} +6.32456 q^{46} +2.82843 q^{47} +5.64911 q^{49} -14.3246i q^{52} +5.65685 q^{53} +3.28742i q^{56} +14.3716i q^{59} +8.00000 q^{64} -6.84157i q^{74} -12.6491 q^{76} +6.84157 q^{77} +10.6491i q^{82} -16.6491i q^{88} +0.955223i q^{89} +8.32456 q^{91} +8.94427 q^{92} +4.00000 q^{94} +7.98905 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	1.16228i	0.439300i	0.975579	+	0.219650i	$0.0704915\pi$
$7$	1.16228i	0.439300i	−0.975579	+	0.219650i	$0.929509\pi$
$8$	2.82843	1.00000
$9$	0	0
$10$	0	0
$11$	− 5.88635i	− 1.77480i	−0.460999	−	0.887401i	$-0.652509\pi$
$11$	− 5.88635i	− 1.77480i	0.460999	−	0.887401i	$-0.347491\pi$
$12$	0	0
$13$	− 7.16228i	− 1.98646i	−0.116171	−	0.993229i	$-0.537062\pi$
$13$	− 7.16228i	− 1.98646i	0.116171	−	0.993229i	$-0.462938\pi$
$14$	1.64371i	0.439300i
$15$	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	$-0.758380\pi$
$19$	−6.32456	−1.45095	−0.725476	+	0.688247i	$0.758380\pi$
$20$	0	0
$21$	0	0
$22$	− 8.32456i	− 1.77480i
$23$	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
$26$	− 10.1290i	− 1.98646i
$27$	0	0
$28$	2.32456i	0.439300i
$29$	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
$37$	− 4.83772i	− 0.795317i	−0.917534	−	0.397658i	$-0.869823\pi$
$37$	− 4.83772i	− 0.795317i	0.917534	−	0.397658i	$-0.130177\pi$
$38$	−8.94427	−1.45095
$39$	0	0
$40$	0	0
$41$	7.53006i	1.17600i	0.808862	+	0.587999i	$0.200084\pi$
$41$	7.53006i	1.17600i	−0.808862	+	0.587999i	$0.799916\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	− 11.7727i	− 1.77480i
$45$	0	0
$46$	6.32456	0.932505
$47$	2.82843	0.412568	0.206284	−	0.978492i	$-0.433863\pi$
$47$	2.82843	0.412568	0.206284	+	0.978492i	$0.433863\pi$
$48$	0	0
$49$	5.64911	0.807016
$50$	0	0
$51$	0	0
$52$	− 14.3246i	− 1.98646i
$53$	5.65685	0.777029	0.388514	−	0.921443i	$-0.372988\pi$
$53$	5.65685	0.777029	0.388514	+	0.921443i	$0.372988\pi$
$54$	0	0
$55$	0	0
$56$	3.28742i	0.439300i
$57$	0	0
$58$	0	0
$59$	14.3716i	1.87103i	0.353291	+	0.935513i	$0.385063\pi$
$59$	14.3716i	1.87103i	−0.353291	+	0.935513i	$0.614937\pi$
$60$	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	$0.5\pi$
$74$	− 6.84157i	− 0.795317i
$75$	0	0
$76$	−12.6491	−1.45095
$77$	6.84157	0.779670
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	0	0
$82$	10.6491i	1.17600i
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	− 16.6491i	− 1.77480i
$89$	0.955223i	0.101253i	0.998718	+	0.0506267i	$0.0161219\pi$
$89$	0.955223i	0.101253i	−0.998718	+	0.0506267i	$0.983878\pi$
$90$	0	0
$91$	8.32456	0.872651
$92$	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	$0.5\pi$
$98$	7.98905	0.807016
$99$	0	0
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	10.8377i	1.06787i	0.845525	+	0.533936i	$0.179288\pi$
$103$	10.8377i	1.06787i	−0.845525	+	0.533936i	$0.820712\pi$
$104$	− 20.2580i	− 1.98646i
$105$	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
$112$	4.64911i	0.439300i
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	20.3246i	1.87103i
$119$	0	0
$120$	0	0
$121$	−23.6491	−2.14992
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 17.8114i	− 1.58051i	−0.612781	−	0.790253i	$-0.709949\pi$
$127$	− 17.8114i	− 1.58051i	0.612781	−	0.790253i	$-0.290051\pi$
$128$	11.3137	1.00000
$129$	0	0
$130$	0	0
$131$	11.0842i	0.968432i	0.874948	+	0.484216i	$0.160895\pi$
$131$	11.0842i	0.968432i	−0.874948	+	0.484216i	$0.839105\pi$
$132$	0	0
$133$	− 7.35089i	− 0.637403i
$134$	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
$143$	−42.1597	−3.52557
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	− 9.67544i	− 0.795317i
$149$	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
$154$	9.67544	0.779670
$155$	0	0
$156$	0	0
$157$	− 23.8114i	− 1.90036i	−0.311708	−	0.950178i	$-0.600901\pi$
$157$	− 23.8114i	− 1.90036i	0.311708	−	0.950178i	$-0.399099\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	5.19786i	0.409649i
$162$	0	0
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	15.0601i	1.17600i
$165$	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
$169$	−38.2982	−2.94602
$170$	0	0
$171$	0	0
$172$	0	0
$173$	22.6274	1.72033	0.860165	−	0.510015i	$-0.170360\pi$
$173$	22.6274	1.72033	0.860165	+	0.510015i	$0.170360\pi$
$174$	0	0
$175$	0	0
$176$	− 23.5454i	− 1.77480i
$177$	0	0
$178$	1.35089i	0.101253i
$179$	22.8569i	1.70841i	0.519940	+	0.854203i	$0.325954\pi$
$179$	22.8569i	1.70841i	−0.519940	+	0.854203i	$0.674046\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	11.7727	0.872651
$183$	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
$196$	11.2982	0.807016
$197$	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
$206$	15.3269i	1.06787i
$207$	0	0
$208$	− 28.6491i	− 1.98646i
$209$	37.2285i	2.57515i
$210$	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
$223$	29.8114i	1.99632i	0.0606498	+	0.998159i	$0.480683\pi$
$223$	29.8114i	1.99632i	−0.0606498	+	0.998159i	$0.519317\pi$
$224$	6.57484i	0.439300i
$225$	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
$236$	28.7433i	1.87103i
$237$	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	$-0.196846\pi$
$241$	25.2982	1.62960	0.814801	+	0.579741i	$0.196846\pi$
$242$	−33.4449	−2.14992
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	45.2982i	2.88226i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	20.9465i	1.32213i	0.750329	+	0.661065i	$0.229895\pi$
$251$	20.9465i	1.32213i	−0.750329	+	0.661065i	$0.770105\pi$
$252$	0	0
$253$	− 26.3246i	− 1.65501i
$254$	− 25.1891i	− 1.58051i
$255$	0	0
$256$	16.0000	1.00000
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	5.62278	0.349382
$260$	0	0
$261$	0	0
$262$	15.6754i	0.968432i
$263$	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
$266$	− 10.3957i	− 0.637403i
$267$	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
$277$	14.1359i	0.849347i	0.905347	+	0.424673i	$0.139611\pi$
$277$	14.1359i	0.849347i	−0.905347	+	0.424673i	$0.860389\pi$
$278$	−19.7990	−1.18746
$279$	0	0
$280$	0	0
$281$	− 2.33219i	− 0.139127i	−0.997578	−	0.0695635i	$-0.977839\pi$
$281$	− 2.33219i	− 0.139127i	0.997578	−	0.0695635i	$-0.0221607\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	0	0
$285$	0	0
$286$	−59.6228	−3.52557
$287$	−8.75202	−0.516615
$288$	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
$296$	− 13.6831i	− 0.795317i
$297$	0	0
$298$	0	0
$299$	− 32.0307i	− 1.85238i
$300$	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	$0.5\pi$
$308$	13.6831	0.779670
$309$	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	$0.5\pi$
$314$	− 33.6744i	− 1.90036i
$315$	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
$322$	7.35089i	0.409649i
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	21.2982i	1.17600i
$329$	3.28742i	0.181241i
$330$	0	0
$331$	−31.6228	−1.73814	−0.869072	−	0.494685i	$-0.835284\pi$
$331$	−31.6228	−1.73814	−0.869072	+	0.494685i	$0.835284\pi$
$332$	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	$0.5\pi$
$338$	−54.1619	−2.94602
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	14.7018i	0.793821i
$344$	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
$352$	− 33.2982i	− 1.77480i
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	1.91045i	0.101253i
$357$	0	0
$358$	32.3246i	1.70841i
$359$	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
$364$	16.6491	0.872651
$365$	0	0
$366$	0	0
$367$	6.18861i	0.323043i	0.986869	+	0.161521i	$0.0516401\pi$
$367$	6.18861i	0.323043i	−0.986869	+	0.161521i	$0.948360\pi$
$368$	17.8885	0.932505
$369$	0	0
$370$	0	0
$371$	6.57484i	0.341348i
$372$	0	0
$373$	− 38.1359i	− 1.97460i	−0.158854	−	0.987302i	$-0.550780\pi$
$373$	− 38.1359i	− 1.97460i	0.158854	−	0.987302i	$-0.449220\pi$
$374$	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	$-0.888634\pi$
$383$	−36.7696	−1.87884	−0.939418	+	0.342773i	$0.888634\pi$
$384$	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
$392$	15.9781	0.807016
$393$	0	0
$394$	31.6228	1.59313
$395$	0	0
$396$	0	0
$397$	31.1623i	1.56399i	0.623285	+	0.781995i	$0.285798\pi$
$397$	31.1623i	1.56399i	−0.623285	+	0.781995i	$0.714202\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 17.9258i	− 0.895171i	−0.894241	−	0.447586i	$-0.852284\pi$
$401$	− 17.9258i	− 0.895171i	0.894241	−	0.447586i	$-0.147716\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−28.4765	−1.41153
$408$	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
$412$	21.6754i	1.06787i
$413$	−16.7038	−0.821942
$414$	0	0
$415$	0	0
$416$	− 40.5160i	− 1.98646i
$417$	0	0
$418$	52.6491i	2.57515i
$419$	− 3.97590i	− 0.194236i	−0.995273	−	0.0971178i	$-0.969038\pi$
$419$	− 3.97590i	− 0.194236i	0.995273	−	0.0971178i	$-0.0309624\pi$
$420$	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
$446$	42.1597i	1.99632i
$447$	0	0
$448$	9.29822i	0.439300i
$449$	− 6.15309i	− 0.290382i	−0.989404	−	0.145191i	$-0.953620\pi$
$449$	− 6.15309i	− 0.290382i	0.989404	−	0.145191i	$-0.0463797\pi$
$450$	0	0
$451$	44.3246	2.08716
$452$	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
$463$	41.8114i	1.94314i	0.236752	+	0.971570i	$0.423917\pi$
$463$	41.8114i	1.94314i	−0.236752	+	0.971570i	$0.576083\pi$
$464$	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
$472$	40.6491i	1.87103i
$473$	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
$481$	−34.6491	−1.57986
$482$	35.7771	1.62960
$483$	0	0
$484$	−47.2982	−2.14992
$485$	0	0
$486$	0	0
$487$	44.1359i	1.99999i	0.00308010	+	0.999995i	$0.499020\pi$
$487$	44.1359i	1.99999i	−0.00308010	+	0.999995i	$0.500980\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 32.7192i	− 1.47660i	−0.674475	−	0.738298i	$-0.735630\pi$
$491$	− 32.7192i	− 1.47660i	0.674475	−	0.738298i	$-0.264370\pi$
$492$	0	0
$493$	0	0
$494$	64.0614i	2.88226i
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−44.2719	−1.98188	−0.990941	−	0.134298i	$-0.957122\pi$
$499$	−44.2719	−1.98188	−0.990941	+	0.134298i	$0.957122\pi$
$500$	0	0
$501$	0	0
$502$	29.6228i	1.32213i
$503$	−19.7990	−0.882793	−0.441397	−	0.897312i	$-0.645517\pi$
$503$	−19.7990	−0.882793	−0.441397	+	0.897312i	$0.645517\pi$
$504$	0	0
$505$	0	0
$506$	− 37.2285i	− 1.65501i
$507$	0	0
$508$	− 35.6228i	− 1.58051i
$509$	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
$517$	− 16.6491i	− 0.732227i
$518$	7.95181	0.349382
$519$	0	0
$520$	0	0
$521$	− 44.7586i	− 1.96091i	−0.196744	−	0.980455i	$-0.563037\pi$
$521$	− 44.7586i	− 1.96091i	0.196744	−	0.980455i	$-0.436963\pi$
$522$	0	0
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	22.1684i	0.968432i
$525$	0	0
$526$	44.2719	1.93035
$527$	0	0
$528$	0	0
$529$	−3.00000	−0.130435
$530$	0	0
$531$	0	0
$532$	− 14.7018i	− 0.637403i
$533$	53.9324	2.33607
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 33.2526i	− 1.43229i
$540$	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
$554$	19.9912i	0.849347i
$555$	0	0
$556$	−28.0000	−1.18746
$557$	45.2548	1.91751	0.958754	−	0.284236i	$-0.0917398\pi$
$557$	45.2548	1.91751	0.958754	+	0.284236i	$0.0917398\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	− 3.29822i	− 0.139127i
$563$	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
$569$	− 41.4712i	− 1.73856i	−0.494318	−	0.869281i	$-0.664582\pi$
$569$	− 41.4712i	− 1.73856i	0.494318	−	0.869281i	$-0.335418\pi$
$570$	0	0
$571$	44.2719	1.85272	0.926360	−	0.376638i	$-0.122920\pi$
$571$	44.2719	1.85272	0.926360	+	0.376638i	$0.122920\pi$
$572$	−84.3193	−3.52557
$573$	0	0
$574$	−12.3772	−0.516615
$575$	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
$583$	− 33.2982i	− 1.37907i
$584$	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
$592$	− 19.3509i	− 0.795317i
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	− 45.2982i	− 1.85238i
$599$	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	$-0.327432\pi$
$601$	25.2982	1.03194	0.515968	+	0.856608i	$0.327432\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 36.7851i	− 1.49306i	−0.665352	−	0.746530i	$-0.731719\pi$
$607$	− 36.7851i	− 1.49306i	0.665352	−	0.746530i	$-0.268281\pi$
$608$	−35.7771	−1.45095
$609$	0	0
$610$	0	0
$611$	− 20.2580i	− 0.819550i
$612$	0	0
$613$	30.7851i	1.24340i	0.783257	+	0.621698i	$0.213557\pi$
$613$	30.7851i	1.24340i	−0.783257	+	0.621698i	$0.786443\pi$
$614$	0	0
$615$	0	0
$616$	19.3509	0.779670
$617$	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
$623$	−1.11023	−0.0444806
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	− 47.6228i	− 1.90036i
$629$	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
$637$	− 40.4605i	− 1.60310i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	34.3629i	1.35725i	0.734484	+	0.678626i	$0.237424\pi$
$641$	34.3629i	1.35725i	−0.734484	+	0.678626i	$0.762576\pi$
$642$	0	0
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	10.3957i	0.409649i
$645$	0	0
$646$	0	0
$647$	−31.1127	−1.22317	−0.611583	−	0.791180i	$-0.709467\pi$
$647$	−31.1127	−1.22317	−0.611583	+	0.791180i	$0.709467\pi$
$648$	0	0
$649$	84.5964	3.32070
$650$	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
$656$	30.1202i	1.17600i
$657$	0	0
$658$	4.64911i	0.181241i
$659$	49.6897i	1.93564i	0.251649	+	0.967818i	$0.419027\pi$
$659$	49.6897i	1.93564i	−0.251649	+	0.967818i	$0.580973\pi$
$660$	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
$676$	−76.5964	−2.94602
$677$	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
$686$	20.7915i	0.793821i
$687$	0	0
$688$	0	0
$689$	− 40.5160i	− 1.54354i
$690$	0	0
$691$	−31.6228	−1.20299	−0.601494	−	0.798878i	$-0.705427\pi$
$691$	−31.6228	−1.20299	−0.601494	+	0.798878i	$0.705427\pi$
$692$	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
$703$	30.5964i	1.15397i
$704$	− 47.0908i	− 1.77480i
$705$	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
$712$	2.70178i	0.101253i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	45.7138i	1.70841i
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	−12.5964	−0.469116
$722$	29.6985	1.10526
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 34.8377i	− 1.29206i	−0.763312	−	0.646030i	$-0.776428\pi$
$727$	− 34.8377i	− 1.29206i	0.763312	−	0.646030i	$-0.223572\pi$
$728$	23.5454	0.872651
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	18.7851i	0.693842i	0.937894	+	0.346921i	$0.112773\pi$
$733$	18.7851i	0.693842i	−0.937894	+	0.346921i	$0.887227\pi$
$734$	8.75202i	0.323043i
$735$	0	0
$736$	25.2982	0.932505
$737$	0	0
$738$	0	0
$739$	−6.32456	−0.232653	−0.116326	−	0.993211i	$-0.537112\pi$
$739$	−6.32456	−0.232653	−0.116326	+	0.993211i	$0.537112\pi$
$740$	0	0
$741$	0	0
$742$	9.29822i	0.341348i
$743$	−36.7696	−1.34894	−0.674472	−	0.738300i	$-0.735629\pi$
$743$	−36.7696	−1.34894	−0.674472	+	0.738300i	$0.735629\pi$
$744$	0	0
$745$	0	0
$746$	− 53.9324i	− 1.97460i
$747$	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
$757$	− 9.86406i	− 0.358515i	−0.983802	−	0.179258i	$-0.942630\pi$
$757$	− 9.86406i	− 0.358515i	0.983802	−	0.179258i	$-0.0573696\pi$
$758$	48.0833	1.74646
$759$	0	0
$760$	0	0
$761$	51.3334i	1.86084i	0.366501	+	0.930418i	$0.380556\pi$
$761$	51.3334i	1.86084i	−0.366501	+	0.930418i	$0.619444\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	−52.0000	−1.87884
$767$	102.934	3.71672
$768$	0	0
$769$	12.6491	0.456139	0.228069	−	0.973645i	$-0.426759\pi$
$769$	12.6491	0.456139	0.228069	+	0.973645i	$0.426759\pi$
$770$	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
$779$	− 47.6243i	− 1.70632i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	22.5964	0.807016
$785$	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
$794$	44.0701i	1.56399i
$795$	0	0
$796$	0	0
$797$	−39.5980	−1.40263	−0.701316	−	0.712850i	$-0.747404\pi$
$797$	−39.5980	−1.40263	−0.701316	+	0.712850i	$0.747404\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	− 25.3509i	− 0.895171i
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 17.3923i	− 0.611481i	−0.952115	−	0.305741i	$-0.901096\pi$
$809$	− 17.3923i	− 0.611481i	0.952115	−	0.305741i	$-0.0989040\pi$
$810$	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
$814$	−40.2719	−1.41153
$815$	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
$823$	− 56.1359i	− 1.95678i	−0.206778	−	0.978388i	$-0.566298\pi$
$823$	− 56.1359i	− 1.95678i	0.206778	−	0.978388i	$-0.433702\pi$
$824$	30.6537i	1.06787i
$825$	0	0
$826$	−23.6228	−0.821942
$827$	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
$832$	− 57.2982i	− 1.98646i
$833$	0	0
$834$	0	0
$835$	0	0
$836$	74.4571i	2.57515i
$837$	0	0
$838$	− 5.62278i	− 0.194236i
$839$	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
$847$	− 27.4868i	− 0.944459i
$848$	22.6274	0.777029
$849$	0	0
$850$	0	0
$851$	− 21.6350i	− 0.741637i
$852$	0	0
$853$	− 45.1096i	− 1.54452i	−0.635304	−	0.772262i	$-0.719125\pi$
$853$	− 45.1096i	− 1.54452i	0.635304	−	0.772262i	$-0.280875\pi$
$854$	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
$877$	33.1096i	1.11803i	0.829157	+	0.559016i	$0.188821\pi$
$877$	33.1096i	1.11803i	−0.829157	+	0.559016i	$0.811179\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	49.9565i	1.68308i	0.540198	+	0.841538i	$0.318349\pi$
$881$	49.9565i	1.68308i	−0.540198	+	0.841538i	$0.681651\pi$
$882$	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
$889$	20.7018	0.694315
$890$	0	0
$891$	0	0
$892$	59.6228i	1.99632i
$893$	−17.8885	−0.598617
$894$	0	0
$895$	0	0
$896$	13.1497i	0.439300i
$897$	0	0
$898$	− 8.70178i	− 0.290382i
$899$	0	0
$900$	0	0
$901$	0	0
$902$	62.6844	2.08716
$903$	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
$917$	−12.8829	−0.425432
$918$	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
$926$	59.1302i	1.94314i
$927$	0	0
$928$	0	0
$929$	− 59.8187i	− 1.96259i	−0.192514	−	0.981294i	$-0.561664\pi$
$929$	− 59.8187i	− 1.96259i	0.192514	−	0.981294i	$-0.438336\pi$
$930$	0	0
$931$	−35.7281	−1.17094
$932$	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
$943$	33.6754i	1.09662i
$944$	57.4865i	1.87103i
$945$	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
$962$	−49.0012	−1.57986
$963$	0	0
$964$	50.5964	1.62960
$965$	0	0
$966$	0	0
$967$	− 15.8641i	− 0.510154i	−0.966921	−	0.255077i	$-0.917899\pi$
$967$	− 15.8641i	− 0.510154i	0.966921	−	0.255077i	$-0.0821008\pi$
$968$	−66.8898	−2.14992
$969$	0	0
$970$	0	0
$971$	− 48.3128i	− 1.55043i	−0.631697	−	0.775215i	$-0.717641\pi$
$971$	− 48.3128i	− 1.55043i	0.631697	−	0.775215i	$-0.282359\pi$
$972$	0	0
$973$	− 16.2719i	− 0.521653i
$974$	62.4177i	1.99999i
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	5.62278	0.179705
$980$	0	0
$981$	0	0
$982$	− 46.2719i	− 1.47660i
$983$	48.0833	1.53362	0.766809	−	0.641875i	$-0.221843\pi$
$983$	48.0833	1.53362	0.766809	+	0.641875i	$0.221843\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	90.5964i	2.88226i
$989$	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
$997$	− 6.78505i	− 0.214885i	−0.994211	−	0.107442i	$-0.965734\pi$
$997$	− 6.78505i	− 0.214885i	0.994211	−	0.107442i	$-0.0342661\pi$
$998$	−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.251.7		8
3.2	odd	2	inner	1800.2.b.f.251.3		8
4.3	odd	2		7200.2.b.f.4751.4		8
5.2	odd	4		360.2.m.b.179.3	yes	4
5.3	odd	4		360.2.m.a.179.2	✓	4
5.4	even	2	inner	1800.2.b.f.251.2		8
8.3	odd	2	inner	1800.2.b.f.251.2		8
8.5	even	2		7200.2.b.f.4751.6		8
12.11	even	2		7200.2.b.f.4751.3		8
15.2	even	4		360.2.m.b.179.2	yes	4
15.8	even	4		360.2.m.a.179.3	yes	4
15.14	odd	2	inner	1800.2.b.f.251.6		8
20.3	even	4		1440.2.m.b.719.3		4
20.7	even	4		1440.2.m.a.719.2		4
20.19	odd	2		7200.2.b.f.4751.6		8
24.5	odd	2		7200.2.b.f.4751.5		8
24.11	even	2	inner	1800.2.b.f.251.6		8
40.3	even	4		360.2.m.b.179.3	yes	4
40.13	odd	4		1440.2.m.a.719.2		4
40.19	odd	2	CM	1800.2.b.f.251.7		8
40.27	even	4		360.2.m.a.179.2	✓	4
40.29	even	2		7200.2.b.f.4751.4		8
40.37	odd	4		1440.2.m.b.719.3		4
60.23	odd	4		1440.2.m.b.719.1		4
60.47	odd	4		1440.2.m.a.719.4		4
60.59	even	2		7200.2.b.f.4751.5		8
120.29	odd	2		7200.2.b.f.4751.3		8
120.53	even	4		1440.2.m.a.719.4		4
120.59	even	2	inner	1800.2.b.f.251.3		8
120.77	even	4		1440.2.m.b.719.1		4
120.83	odd	4		360.2.m.b.179.2	yes	4
120.107	odd	4		360.2.m.a.179.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.m.a.179.2	✓	4	5.3	odd	4
360.2.m.a.179.2	✓	4	40.27	even	4
360.2.m.a.179.3	yes	4	15.8	even	4
360.2.m.a.179.3	yes	4	120.107	odd	4
360.2.m.b.179.2	yes	4	15.2	even	4
360.2.m.b.179.2	yes	4	120.83	odd	4
360.2.m.b.179.3	yes	4	5.2	odd	4
360.2.m.b.179.3	yes	4	40.3	even	4
1440.2.m.a.719.2		4	20.7	even	4
1440.2.m.a.719.2		4	40.13	odd	4
1440.2.m.a.719.4		4	60.47	odd	4
1440.2.m.a.719.4		4	120.53	even	4
1440.2.m.b.719.1		4	60.23	odd	4
1440.2.m.b.719.1		4	120.77	even	4
1440.2.m.b.719.3		4	20.3	even	4
1440.2.m.b.719.3		4	40.37	odd	4
1800.2.b.f.251.2		8	5.4	even	2	inner
1800.2.b.f.251.2		8	8.3	odd	2	inner
1800.2.b.f.251.3		8	3.2	odd	2	inner
1800.2.b.f.251.3		8	120.59	even	2	inner
1800.2.b.f.251.6		8	15.14	odd	2	inner
1800.2.b.f.251.6		8	24.11	even	2	inner
1800.2.b.f.251.7		8	1.1	even	1	trivial
1800.2.b.f.251.7		8	40.19	odd	2	CM
7200.2.b.f.4751.3		8	12.11	even	2
7200.2.b.f.4751.3		8	120.29	odd	2
7200.2.b.f.4751.4		8	4.3	odd	2
7200.2.b.f.4751.4		8	40.29	even	2
7200.2.b.f.4751.5		8	24.5	odd	2
7200.2.b.f.4751.5		8	60.59	even	2
7200.2.b.f.4751.6		8	8.5	even	2
7200.2.b.f.4751.6		8	20.19	odd	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)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +2.00000 q^{4} +1.16228i q^{7} +2.82843 q^{8} +O(q^{10})\)	\(q+1.41421 q^{2} +2.00000 q^{4} +1.16228i q^{7} +2.82843 q^{8} -5.88635i q^{11} -7.16228i q^{13} +1.64371i q^{14} +4.00000 q^{16} -6.32456 q^{19} -8.32456i q^{22} +4.47214 q^{23} -10.1290i q^{26} +2.32456i q^{28} +5.65685 q^{32} -4.83772i q^{37} -8.94427 q^{38} +7.53006i q^{41} -11.7727i q^{44} +6.32456 q^{46} +2.82843 q^{47} +5.64911 q^{49} -14.3246i q^{52} +5.65685 q^{53} +3.28742i q^{56} +14.3716i q^{59} +8.00000 q^{64} -6.84157i q^{74} -12.6491 q^{76} +6.84157 q^{77} +10.6491i q^{82} -16.6491i q^{88} +0.955223i q^{89} +8.32456 q^{91} +8.94427 q^{92} +4.00000 q^{94} +7.98905 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{4}+O(q^{10}) \) 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

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