LMFDB - Embedded newform 200.3.g.c.51.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [200,3,Mod(51,200)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("200.51"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 200 = 2^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	200.g (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	no
Analytic conductor:	$5.44960528721$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			51.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	200.51
Dual form			200.3.g.c.51.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{2} -4.00000 q^{4} -6.00000i q^{7} -8.00000i q^{8} -9.00000 q^{9} -18.0000 q^{11} -6.00000i q^{13} +12.0000 q^{14} +16.0000 q^{16} -18.0000i q^{18} +2.00000 q^{19} -36.0000i q^{22} -26.0000i q^{23} +12.0000 q^{26} +24.0000i q^{28} +32.0000i q^{32} +36.0000 q^{36} +54.0000i q^{37} +4.00000i q^{38} -78.0000 q^{41} +72.0000 q^{44} +52.0000 q^{46} -86.0000i q^{47} +13.0000 q^{49} +24.0000i q^{52} +74.0000i q^{53} -48.0000 q^{56} -78.0000 q^{59} +54.0000i q^{63} -64.0000 q^{64} +72.0000i q^{72} -108.000 q^{74} -8.00000 q^{76} +108.000i q^{77} +81.0000 q^{81} -156.000i q^{82} +144.000i q^{88} -18.0000 q^{89} -36.0000 q^{91} +104.000i q^{92} +172.000 q^{94} +26.0000i q^{98} +162.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{4} - 18 q^{9} - 36 q^{11} + 24 q^{14} + 32 q^{16} + 4 q^{19} + 24 q^{26} + 72 q^{36} - 156 q^{41} + 144 q^{44} + 104 q^{46} + 26 q^{49} - 96 q^{56} - 156 q^{59} - 128 q^{64} - 216 q^{74} - 16 q^{76}+ \cdots + 324 q^{99}+O(q^{100}) $ 2 * q - 8 * q^4 - 18 * q^9 - 36 * q^11 + 24 * q^14 + 32 * q^16 + 4 * q^19 + 24 * q^26 + 72 * q^36 - 156 * q^41 + 144 * q^44 + 104 * q^46 + 26 * q^49 - 96 * q^56 - 156 * q^59 - 128 * q^64 - 216 * q^74 - 16 * q^76 + 162 * q^81 - 36 * q^89 - 72 * q^91 + 344 * q^94 + 324 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/200\mathbb{Z}\right)^\times$.

$n$	$101$	$151$	$177$
$\chi(n)$	$-1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.00000i	1.00000i
$2$	2.00000i	1.00000i
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	−4.00000	−1.00000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 6.00000i	− 0.857143i	−0.903508	−	0.428571i	$-0.859017\pi$
$7$	− 6.00000i	− 0.857143i	0.903508	−	0.428571i	$-0.140983\pi$
$8$	− 8.00000i	− 1.00000i
$9$	−9.00000	−1.00000
$10$	0	0
$11$	−18.0000	−1.63636	−0.818182	−	0.574960i	$-0.805018\pi$
$11$	−18.0000	−1.63636	−0.818182	+	0.574960i	$0.805018\pi$
$12$	0	0
$13$	− 6.00000i	− 0.461538i	−0.973009	−	0.230769i	$-0.925876\pi$
$13$	− 6.00000i	− 0.461538i	0.973009	−	0.230769i	$-0.0741242\pi$
$14$	12.0000	0.857143
$15$	0	0
$16$	16.0000	1.00000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	− 18.0000i	− 1.00000i
$19$	2.00000	0.105263	0.0526316	−	0.998614i	$-0.483239\pi$
$19$	2.00000	0.105263	0.0526316	+	0.998614i	$0.483239\pi$
$20$	0	0
$21$	0	0
$22$	− 36.0000i	− 1.63636i
$23$	− 26.0000i	− 1.13043i	−0.824942	−	0.565217i	$-0.808792\pi$
$23$	− 26.0000i	− 1.13043i	0.824942	−	0.565217i	$-0.191208\pi$
$24$	0	0
$25$	0	0
$26$	12.0000	0.461538
$27$	0	0
$28$	24.0000i	0.857143i
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	32.0000i	1.00000i
$33$	0	0
$34$	0	0
$35$	0	0
$36$	36.0000	1.00000
$37$	54.0000i	1.45946i	0.683736	+	0.729730i	$0.260354\pi$
$37$	54.0000i	1.45946i	−0.683736	+	0.729730i	$0.739646\pi$
$38$	4.00000i	0.105263i
$39$	0	0
$40$	0	0
$41$	−78.0000	−1.90244	−0.951220	−	0.308515i	$-0.900168\pi$
$41$	−78.0000	−1.90244	−0.951220	+	0.308515i	$0.900168\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	72.0000	1.63636
$45$	0	0
$46$	52.0000	1.13043
$47$	− 86.0000i	− 1.82979i	−0.403695	−	0.914894i	$-0.632274\pi$
$47$	− 86.0000i	− 1.82979i	0.403695	−	0.914894i	$-0.367726\pi$
$48$	0	0
$49$	13.0000	0.265306
$50$	0	0
$51$	0	0
$52$	24.0000i	0.461538i
$53$	74.0000i	1.39623i	0.715987	+	0.698113i	$0.245977\pi$
$53$	74.0000i	1.39623i	−0.715987	+	0.698113i	$0.754023\pi$
$54$	0	0
$55$	0	0
$56$	−48.0000	−0.857143
$57$	0	0
$58$	0	0
$59$	−78.0000	−1.32203	−0.661017	−	0.750371i	$-0.729875\pi$
$59$	−78.0000	−1.32203	−0.661017	+	0.750371i	$0.729875\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	54.0000i	0.857143i
$64$	−64.0000	−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	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	72.0000i	1.00000i
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	−108.000	−1.45946
$75$	0	0
$76$	−8.00000	−0.105263
$77$	108.000i	1.40260i
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	81.0000	1.00000
$82$	− 156.000i	− 1.90244i
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	144.000i	1.63636i
$89$	−18.0000	−0.202247	−0.101124	−	0.994874i	$-0.532244\pi$
$89$	−18.0000	−0.202247	−0.101124	+	0.994874i	$0.532244\pi$
$90$	0	0
$91$	−36.0000	−0.395604
$92$	104.000i	1.13043i
$93$	0	0
$94$	172.000	1.82979
$95$	0	0
$96$	0	0
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	26.0000i	0.265306i
$99$	162.000	1.63636
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	− 186.000i	− 1.80583i	−0.429824	−	0.902913i	$-0.641424\pi$
$103$	− 186.000i	− 1.80583i	0.429824	−	0.902913i	$-0.358576\pi$
$104$	−48.0000	−0.461538
$105$	0	0
$106$	−148.000	−1.39623
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	− 96.0000i	− 0.857143i
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	54.0000i	0.461538i
$118$	− 156.000i	− 1.32203i
$119$	0	0
$120$	0	0
$121$	203.000	1.67769
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	−108.000	−0.857143
$127$	− 246.000i	− 1.93701i	−0.248998	−	0.968504i	$-0.580101\pi$
$127$	− 246.000i	− 1.93701i	0.248998	−	0.968504i	$-0.419899\pi$
$128$	− 128.000i	− 1.00000i
$129$	0	0
$130$	0	0
$131$	222.000	1.69466	0.847328	−	0.531070i	$-0.178210\pi$
$131$	222.000	1.69466	0.847328	+	0.531070i	$0.178210\pi$
$132$	0	0
$133$	− 12.0000i	− 0.0902256i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	82.0000	0.589928	0.294964	−	0.955508i	$-0.404692\pi$
$139$	82.0000	0.589928	0.294964	+	0.955508i	$0.404692\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	108.000i	0.755245i
$144$	−144.000	−1.00000
$145$	0	0
$146$	0	0
$147$	0	0
$148$	− 216.000i	− 1.45946i
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	− 16.0000i	− 0.105263i
$153$	0	0
$154$	−216.000	−1.40260
$155$	0	0
$156$	0	0
$157$	− 186.000i	− 1.18471i	−0.805676	−	0.592357i	$-0.798198\pi$
$157$	− 186.000i	− 1.18471i	0.805676	−	0.592357i	$-0.201802\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−156.000	−0.968944
$162$	162.000i	1.00000i
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	312.000	1.90244
$165$	0	0
$166$	0	0
$167$	314.000i	1.88024i	0.340844	+	0.940120i	$0.389287\pi$
$167$	314.000i	1.88024i	−0.340844	+	0.940120i	$0.610713\pi$
$168$	0	0
$169$	133.000	0.786982
$170$	0	0
$171$	−18.0000	−0.105263
$172$	0	0
$173$	− 166.000i	− 0.959538i	−0.877395	−	0.479769i	$-0.840721\pi$
$173$	− 166.000i	− 0.959538i	0.877395	−	0.479769i	$-0.159279\pi$
$174$	0	0
$175$	0	0
$176$	−288.000	−1.63636
$177$	0	0
$178$	− 36.0000i	− 0.202247i
$179$	−318.000	−1.77654	−0.888268	−	0.459325i	$-0.848091\pi$
$179$	−318.000	−1.77654	−0.888268	+	0.459325i	$0.848091\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	− 72.0000i	− 0.395604i
$183$	0	0
$184$	−208.000	−1.13043
$185$	0	0
$186$	0	0
$187$	0	0
$188$	344.000i	1.82979i
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	0	0
$195$	0	0
$196$	−52.0000	−0.265306
$197$	− 106.000i	− 0.538071i	−0.963130	−	0.269036i	$-0.913295\pi$
$197$	− 106.000i	− 0.538071i	0.963130	−	0.269036i	$-0.0867049\pi$
$198$	324.000i	1.63636i
$199$	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
$206$	372.000	1.80583
$207$	234.000i	1.13043i
$208$	− 96.0000i	− 0.461538i
$209$	−36.0000	−0.172249
$210$	0	0
$211$	62.0000	0.293839	0.146919	−	0.989148i	$-0.453064\pi$
$211$	62.0000	0.293839	0.146919	+	0.989148i	$0.453064\pi$
$212$	− 296.000i	− 1.39623i
$213$	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
$223$	54.0000i	0.242152i	0.992643	+	0.121076i	$0.0386346\pi$
$223$	54.0000i	0.242152i	−0.992643	+	0.121076i	$0.961365\pi$
$224$	192.000	0.857143
$225$	0	0
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	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		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	−108.000	−0.461538
$235$	0	0
$236$	312.000	1.32203
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	−158.000	−0.655602	−0.327801	−	0.944747i	$-0.606307\pi$
$241$	−158.000	−0.655602	−0.327801	+	0.944747i	$0.606307\pi$
$242$	406.000i	1.67769i
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 12.0000i	− 0.0485830i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−498.000	−1.98406	−0.992032	−	0.125987i	$-0.959790\pi$
$251$	−498.000	−1.98406	−0.992032	+	0.125987i	$0.959790\pi$
$252$	− 216.000i	− 0.857143i
$253$	468.000i	1.84980i
$254$	492.000	1.93701
$255$	0	0
$256$	256.000	1.00000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	324.000	1.25097
$260$	0	0
$261$	0	0
$262$	444.000i	1.69466i
$263$	454.000i	1.72624i	0.505003	+	0.863118i	$0.331492\pi$
$263$	454.000i	1.72624i	−0.505003	+	0.863118i	$0.668508\pi$
$264$	0	0
$265$	0	0
$266$	24.0000	0.0902256
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	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
$277$	− 426.000i	− 1.53791i	−0.639305	−	0.768953i	$-0.720778\pi$
$277$	− 426.000i	− 1.53791i	0.639305	−	0.768953i	$-0.279222\pi$
$278$	164.000i	0.589928i
$279$	0	0
$280$	0	0
$281$	−78.0000	−0.277580	−0.138790	−	0.990322i	$-0.544321\pi$
$281$	−78.0000	−0.277580	−0.138790	+	0.990322i	$0.544321\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	0	0
$285$	0	0
$286$	−216.000	−0.755245
$287$	468.000i	1.63066i
$288$	− 288.000i	− 1.00000i
$289$	−289.000	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 566.000i	− 1.93174i	−0.259026	−	0.965870i	$-0.583402\pi$
$293$	− 566.000i	− 1.93174i	0.259026	−	0.965870i	$-0.416598\pi$
$294$	0	0
$295$	0	0
$296$	432.000	1.45946
$297$	0	0
$298$	0	0
$299$	−156.000	−0.521739
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	32.0000	0.105263
$305$	0	0
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	− 432.000i	− 1.40260i
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	0	0		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	372.000	1.18471
$315$	0	0
$316$	0	0
$317$	− 346.000i	− 1.09148i	−0.837954	−	0.545741i	$-0.816248\pi$
$317$	− 346.000i	− 1.09148i	0.837954	−	0.545741i	$-0.183752\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	− 312.000i	− 0.968944i
$323$	0	0
$324$	−324.000	−1.00000
$325$	0	0
$326$	0	0
$327$	0	0
$328$	624.000i	1.90244i
$329$	−516.000	−1.56839
$330$	0	0
$331$	−338.000	−1.02115	−0.510574	−	0.859834i	$-0.670567\pi$
$331$	−338.000	−1.02115	−0.510574	+	0.859834i	$0.670567\pi$
$332$	0	0
$333$	− 486.000i	− 1.45946i
$334$	−628.000	−1.88024
$335$	0	0
$336$	0	0
$337$	0	0		−	1.00000i	$-0.5\pi$
$337$	0	0			1.00000i	$0.5\pi$
$338$	266.000i	0.786982i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	− 36.0000i	− 0.105263i
$343$	− 372.000i	− 1.08455i
$344$	0	0
$345$	0	0
$346$	332.000	0.959538
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	− 576.000i	− 1.63636i
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	0	0
$356$	72.0000	0.202247
$357$	0	0
$358$	− 636.000i	− 1.77654i
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	−357.000	−0.988920
$362$	0	0
$363$	0	0
$364$	144.000	0.395604
$365$	0	0
$366$	0	0
$367$	234.000i	0.637602i	0.947822	+	0.318801i	$0.103280\pi$
$367$	234.000i	0.637602i	−0.947822	+	0.318801i	$0.896720\pi$
$368$	− 416.000i	− 1.13043i
$369$	702.000	1.90244
$370$	0	0
$371$	444.000	1.19677
$372$	0	0
$373$	234.000i	0.627346i	0.949531	+	0.313673i	$0.101560\pi$
$373$	234.000i	0.627346i	−0.949531	+	0.313673i	$0.898440\pi$
$374$	0	0
$375$	0	0
$376$	−688.000	−1.82979
$377$	0	0
$378$	0	0
$379$	−398.000	−1.05013	−0.525066	−	0.851062i	$-0.675959\pi$
$379$	−398.000	−1.05013	−0.525066	+	0.851062i	$0.675959\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 586.000i	− 1.53003i	−0.644015	−	0.765013i	$-0.722732\pi$
$383$	− 586.000i	− 1.53003i	0.644015	−	0.765013i	$-0.277268\pi$
$384$	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
$391$	0	0
$392$	− 104.000i	− 0.265306i
$393$	0	0
$394$	212.000	0.538071
$395$	0	0
$396$	−648.000	−1.63636
$397$	774.000i	1.94962i	0.223032	+	0.974811i	$0.428404\pi$
$397$	774.000i	1.94962i	−0.223032	+	0.974811i	$0.571596\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	642.000	1.60100	0.800499	−	0.599334i	$-0.204568\pi$
$401$	642.000	1.60100	0.800499	+	0.599334i	$0.204568\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 972.000i	− 2.38821i
$408$	0	0
$409$	622.000	1.52078	0.760391	−	0.649465i	$-0.225007\pi$
$409$	622.000	1.52078	0.760391	+	0.649465i	$0.225007\pi$
$410$	0	0
$411$	0	0
$412$	744.000i	1.80583i
$413$	468.000i	1.13317i
$414$	−468.000	−1.13043
$415$	0	0
$416$	192.000	0.461538
$417$	0	0
$418$	− 72.0000i	− 0.172249i
$419$	162.000	0.386635	0.193317	−	0.981136i	$-0.438075\pi$
$419$	162.000	0.386635	0.193317	+	0.981136i	$0.438075\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	124.000i	0.293839i
$423$	774.000i	1.82979i
$424$	592.000	1.39623
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 52.0000i	− 0.118993i
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	−117.000	−0.265306
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	0	0
$446$	−108.000	−0.242152
$447$	0	0
$448$	384.000i	0.857143i
$449$	−258.000	−0.574610	−0.287305	−	0.957839i	$-0.592759\pi$
$449$	−258.000	−0.574610	−0.287305	+	0.957839i	$0.592759\pi$
$450$	0	0
$451$	1404.00	3.11308
$452$	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	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	− 426.000i	− 0.920086i	−0.887897	−	0.460043i	$-0.847834\pi$
$463$	− 426.000i	− 0.920086i	0.887897	−	0.460043i	$-0.152166\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	− 216.000i	− 0.461538i
$469$	0	0
$470$	0	0
$471$	0	0
$472$	624.000i	1.32203i
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 666.000i	− 1.39623i
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	324.000	0.673597
$482$	− 316.000i	− 0.655602i
$483$	0	0
$484$	−812.000	−1.67769
$485$	0	0
$486$	0	0
$487$	− 6.00000i	− 0.0123203i	−0.999981	−	0.00616016i	$-0.998039\pi$
$487$	− 6.00000i	− 0.0123203i	0.999981	−	0.00616016i	$-0.00196085\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−978.000	−1.99185	−0.995927	−	0.0901668i	$-0.971260\pi$
$491$	−978.000	−1.99185	−0.995927	+	0.0901668i	$0.971260\pi$
$492$	0	0
$493$	0	0
$494$	24.0000	0.0485830
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	962.000	1.92786	0.963928	−	0.266164i	$-0.0857562\pi$
$499$	962.000	1.92786	0.963928	+	0.266164i	$0.0857562\pi$
$500$	0	0
$501$	0	0
$502$	− 996.000i	− 1.98406i
$503$	614.000i	1.22068i	0.792141	+	0.610338i	$0.208966\pi$
$503$	614.000i	1.22068i	−0.792141	+	0.610338i	$0.791034\pi$
$504$	432.000	0.857143
$505$	0	0
$506$	−936.000	−1.84980
$507$	0	0
$508$	984.000i	1.93701i
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	512.000i	1.00000i
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1548.00i	2.99420i
$518$	648.000i	1.25097i
$519$	0	0
$520$	0	0
$521$	402.000	0.771593	0.385797	−	0.922584i	$-0.373927\pi$
$521$	402.000	0.771593	0.385797	+	0.922584i	$0.373927\pi$
$522$	0	0
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	−888.000	−1.69466
$525$	0	0
$526$	−908.000	−1.72624
$527$	0	0
$528$	0	0
$529$	−147.000	−0.277883
$530$	0	0
$531$	702.000	1.32203
$532$	48.0000i	0.0902256i
$533$	468.000i	0.878049i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−234.000	−0.434137
$540$	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
$554$	852.000	1.53791
$555$	0	0
$556$	−328.000	−0.589928
$557$	934.000i	1.67684i	0.545025	+	0.838420i	$0.316520\pi$
$557$	934.000i	1.67684i	−0.545025	+	0.838420i	$0.683480\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	− 156.000i	− 0.277580i
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 486.000i	− 0.857143i
$568$	0	0
$569$	−978.000	−1.71880	−0.859402	−	0.511300i	$-0.829164\pi$
$569$	−978.000	−1.71880	−0.859402	+	0.511300i	$0.829164\pi$
$570$	0	0
$571$	−818.000	−1.43257	−0.716287	−	0.697806i	$-0.754160\pi$
$571$	−818.000	−1.43257	−0.716287	+	0.697806i	$0.754160\pi$
$572$	− 432.000i	− 0.755245i
$573$	0	0
$574$	−936.000	−1.63066
$575$	0	0
$576$	576.000	1.00000
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	− 578.000i	− 1.00000i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 1332.00i	− 2.28473i
$584$	0	0
$585$	0	0
$586$	1132.00	1.93174
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	864.000i	1.45946i
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	− 312.000i	− 0.521739i
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	562.000	0.935108	0.467554	−	0.883964i	$-0.345135\pi$
$601$	562.000	0.935108	0.467554	+	0.883964i	$0.345135\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 1206.00i	− 1.98682i	−0.114613	−	0.993410i	$-0.536563\pi$
$607$	− 1206.00i	− 1.98682i	0.114613	−	0.993410i	$-0.463437\pi$
$608$	64.0000i	0.105263i
$609$	0	0
$610$	0	0
$611$	−516.000	−0.844517
$612$	0	0
$613$	1194.00i	1.94780i	0.226982	+	0.973899i	$0.427114\pi$
$613$	1194.00i	1.94780i	−0.226982	+	0.973899i	$0.572886\pi$
$614$	0	0
$615$	0	0
$616$	864.000	1.40260
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	−878.000	−1.41842	−0.709208	−	0.704999i	$-0.750947\pi$
$619$	−878.000	−1.41842	−0.709208	+	0.704999i	$0.750947\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	108.000i	0.173355i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	744.000i	1.18471i
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	692.000	1.09148
$635$	0	0
$636$	0	0
$637$	− 78.0000i	− 0.122449i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−1278.00	−1.99376	−0.996880	−	0.0789336i	$-0.974848\pi$
$641$	−1278.00	−1.99376	−0.996880	+	0.0789336i	$0.974848\pi$
$642$	0	0
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	624.000	0.968944
$645$	0	0
$646$	0	0
$647$	− 326.000i	− 0.503864i	−0.967745	−	0.251932i	$-0.918934\pi$
$647$	− 326.000i	− 0.503864i	0.967745	−	0.251932i	$-0.0810659\pi$
$648$	− 648.000i	− 1.00000i
$649$	1404.00	2.16333
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 1286.00i	− 1.96937i	−0.174337	−	0.984686i	$-0.555778\pi$
$653$	− 1286.00i	− 1.96937i	0.174337	−	0.984686i	$-0.444222\pi$
$654$	0	0
$655$	0	0
$656$	−1248.00	−1.90244
$657$	0	0
$658$	− 1032.00i	− 1.56839i
$659$	642.000	0.974203	0.487102	−	0.873345i	$-0.338054\pi$
$659$	642.000	0.974203	0.487102	+	0.873345i	$0.338054\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	− 676.000i	− 1.02115i
$663$	0	0
$664$	0	0
$665$	0	0
$666$	972.000	1.45946
$667$	0	0
$668$	− 1256.00i	− 1.88024i
$669$	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
$676$	−532.000	−0.786982
$677$	− 1066.00i	− 1.57459i	−0.616574	−	0.787297i	$-0.711480\pi$
$677$	− 1066.00i	− 1.57459i	0.616574	−	0.787297i	$-0.288520\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	72.0000	0.105263
$685$	0	0
$686$	744.000	1.08455
$687$	0	0
$688$	0	0
$689$	444.000	0.644412
$690$	0	0
$691$	382.000	0.552822	0.276411	−	0.961040i	$-0.410855\pi$
$691$	382.000	0.552822	0.276411	+	0.961040i	$0.410855\pi$
$692$	664.000i	0.959538i
$693$	− 972.000i	− 1.40260i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	108.000i	0.153627i
$704$	1152.00	1.63636
$705$	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
$712$	144.000i	0.202247i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	1272.00	1.77654
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	−1116.00	−1.54785
$722$	− 714.000i	− 0.988920i
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1434.00i	1.97249i	0.165291	+	0.986245i	$0.447144\pi$
$727$	1434.00i	1.97249i	−0.165291	+	0.986245i	$0.552856\pi$
$728$	288.000i	0.395604i
$729$	−729.000	−1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	954.000i	1.30150i	0.759292	+	0.650750i	$0.225546\pi$
$733$	954.000i	1.30150i	−0.759292	+	0.650750i	$0.774454\pi$
$734$	−468.000	−0.637602
$735$	0	0
$736$	832.000	1.13043
$737$	0	0
$738$	1404.00i	1.90244i
$739$	−1438.00	−1.94587	−0.972936	−	0.231073i	$-0.925776\pi$
$739$	−1438.00	−1.94587	−0.972936	+	0.231073i	$0.925776\pi$
$740$	0	0
$741$	0	0
$742$	888.000i	1.19677i
$743$	134.000i	0.180350i	0.995926	+	0.0901750i	$0.0287426\pi$
$743$	134.000i	0.180350i	−0.995926	+	0.0901750i	$0.971257\pi$
$744$	0	0
$745$	0	0
$746$	−468.000	−0.627346
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	− 1376.00i	− 1.82979i
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	534.000i	0.705416i	0.935733	+	0.352708i	$0.114739\pi$
$757$	534.000i	0.705416i	−0.935733	+	0.352708i	$0.885261\pi$
$758$	− 796.000i	− 1.05013i
$759$	0	0
$760$	0	0
$761$	−1038.00	−1.36399	−0.681997	−	0.731355i	$-0.738888\pi$
$761$	−1038.00	−1.36399	−0.681997	+	0.731355i	$0.738888\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	1172.00	1.53003
$767$	468.000i	0.610169i
$768$	0	0
$769$	−1378.00	−1.79194	−0.895969	−	0.444117i	$-0.853517\pi$
$769$	−1378.00	−1.79194	−0.895969	+	0.444117i	$0.853517\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 1046.00i	− 1.35317i	−0.736365	−	0.676585i	$-0.763459\pi$
$773$	− 1046.00i	− 1.35317i	0.736365	−	0.676585i	$-0.236541\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−156.000	−0.200257
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	208.000	0.265306
$785$	0	0
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	424.000i	0.538071i
$789$	0	0
$790$	0	0
$791$	0	0
$792$	− 1296.00i	− 1.63636i
$793$	0	0
$794$	−1548.00	−1.94962
$795$	0	0
$796$	0	0
$797$	− 26.0000i	− 0.0326223i	−0.999867	−	0.0163112i	$-0.994808\pi$
$797$	− 26.0000i	− 0.0326223i	0.999867	−	0.0163112i	$-0.00519224\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	162.000	0.202247
$802$	1284.00i	1.60100i
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	942.000	1.16440	0.582200	−	0.813045i	$-0.302192\pi$
$809$	942.000	1.16440	0.582200	+	0.813045i	$0.302192\pi$
$810$	0	0
$811$	−1618.00	−1.99507	−0.997534	−	0.0701862i	$-0.977641\pi$
$811$	−1618.00	−1.99507	−0.997534	+	0.0701862i	$0.977641\pi$
$812$	0	0
$813$	0	0
$814$	1944.00	2.38821
$815$	0	0
$816$	0	0
$817$	0	0
$818$	1244.00i	1.52078i
$819$	324.000	0.395604
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	− 666.000i	− 0.809235i	−0.914486	−	0.404617i	$-0.867405\pi$
$823$	− 666.000i	− 0.809235i	0.914486	−	0.404617i	$-0.132595\pi$
$824$	−1488.00	−1.80583
$825$	0	0
$826$	−936.000	−1.13317
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	− 936.000i	− 1.13043i
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	384.000i	0.461538i
$833$	0	0
$834$	0	0
$835$	0	0
$836$	144.000	0.172249
$837$	0	0
$838$	324.000i	0.386635i
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	841.000	1.00000
$842$	0	0
$843$	0	0
$844$	−248.000	−0.293839
$845$	0	0
$846$	−1548.00	−1.82979
$847$	− 1218.00i	− 1.43802i
$848$	1184.00i	1.39623i
$849$	0	0
$850$	0	0
$851$	1404.00	1.64982
$852$	0	0
$853$	1674.00i	1.96249i	0.192777	+	0.981243i	$0.438251\pi$
$853$	1674.00i	1.96249i	−0.192777	+	0.981243i	$0.561749\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	1522.00	1.77183	0.885914	−	0.463850i	$-0.153532\pi$
$859$	1522.00	1.77183	0.885914	+	0.463850i	$0.153532\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1654.00i	1.91657i	0.285814	+	0.958285i	$0.407736\pi$
$863$	1654.00i	1.91657i	−0.285814	+	0.958285i	$0.592264\pi$
$864$	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
$874$	104.000	0.118993
$875$	0	0
$876$	0	0
$877$	− 1626.00i	− 1.85405i	−0.375002	−	0.927024i	$-0.622358\pi$
$877$	− 1626.00i	− 1.85405i	0.375002	−	0.927024i	$-0.377642\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1602.00	1.81839	0.909194	−	0.416373i	$-0.136699\pi$
$881$	1602.00	1.81839	0.909194	+	0.416373i	$0.136699\pi$
$882$	− 234.000i	− 0.265306i
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1754.00i	1.97745i	0.149736	+	0.988726i	$0.452158\pi$
$887$	1754.00i	1.97745i	−0.149736	+	0.988726i	$0.547842\pi$
$888$	0	0
$889$	−1476.00	−1.66029
$890$	0	0
$891$	−1458.00	−1.63636
$892$	− 216.000i	− 0.242152i
$893$	− 172.000i	− 0.192609i
$894$	0	0
$895$	0	0
$896$	−768.000	−0.857143
$897$	0	0
$898$	− 516.000i	− 0.574610i
$899$	0	0
$900$	0	0
$901$	0	0
$902$	2808.00i	3.11308i
$903$	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	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 1332.00i	− 1.45256i
$918$	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
$926$	852.000	0.920086
$927$	1674.00i	1.80583i
$928$	0	0
$929$	702.000	0.755651	0.377826	−	0.925877i	$-0.376672\pi$
$929$	702.000	0.755651	0.377826	+	0.925877i	$0.376672\pi$
$930$	0	0
$931$	26.0000	0.0279270
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	432.000	0.461538
$937$	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
$943$	2028.00i	2.15058i
$944$	−1248.00	−1.32203
$945$	0	0
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	1332.00	1.39623
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	961.000	1.00000
$962$	648.000i	0.673597i
$963$	0	0
$964$	632.000	0.655602
$965$	0	0
$966$	0	0
$967$	954.000i	0.986556i	0.869872	+	0.493278i	$0.164202\pi$
$967$	954.000i	0.986556i	−0.869872	+	0.493278i	$0.835798\pi$
$968$	− 1624.00i	− 1.67769i
$969$	0	0
$970$	0	0
$971$	1902.00	1.95881	0.979403	−	0.201917i	$-0.0647171\pi$
$971$	1902.00	1.95881	0.979403	+	0.201917i	$0.0647171\pi$
$972$	0	0
$973$	− 492.000i	− 0.505653i
$974$	12.0000	0.0123203
$975$	0	0
$976$	0	0
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	324.000	0.330950
$980$	0	0
$981$	0	0
$982$	− 1956.00i	− 1.99185i
$983$	− 346.000i	− 0.351984i	−0.984392	−	0.175992i	$-0.943687\pi$
$983$	− 346.000i	− 0.351984i	0.984392	−	0.175992i	$-0.0563132\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	48.0000i	0.0485830i
$989$	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
$997$	− 426.000i	− 0.427282i	−0.976912	−	0.213641i	$-0.931468\pi$
$997$	− 426.000i	− 0.427282i	0.976912	−	0.213641i	$-0.0685323\pi$
$998$	1924.00i	1.92786i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	200.3.g.c.51.2		2
4.3	odd	2		800.3.g.c.751.2		2
5.2	odd	4		40.3.e.a.19.1	✓	1
5.3	odd	4		40.3.e.b.19.1	yes	1
5.4	even	2	inner	200.3.g.c.51.1		2
8.3	odd	2	inner	200.3.g.c.51.1		2
8.5	even	2		800.3.g.c.751.1		2
15.2	even	4		360.3.p.b.19.1		1
15.8	even	4		360.3.p.a.19.1		1
20.3	even	4		160.3.e.a.79.1		1
20.7	even	4		160.3.e.b.79.1		1
20.19	odd	2		800.3.g.c.751.1		2
40.3	even	4		40.3.e.a.19.1	✓	1
40.13	odd	4		160.3.e.b.79.1		1
40.19	odd	2	CM	200.3.g.c.51.2		2
40.27	even	4		40.3.e.b.19.1	yes	1
40.29	even	2		800.3.g.c.751.2		2
40.37	odd	4		160.3.e.a.79.1		1
60.23	odd	4		1440.3.p.b.559.1		1
60.47	odd	4		1440.3.p.a.559.1		1
80.3	even	4		1280.3.h.b.1279.2		2
80.13	odd	4		1280.3.h.c.1279.2		2
80.27	even	4		1280.3.h.c.1279.2		2
80.37	odd	4		1280.3.h.b.1279.2		2
80.43	even	4		1280.3.h.b.1279.1		2
80.53	odd	4		1280.3.h.c.1279.1		2
80.67	even	4		1280.3.h.c.1279.1		2
80.77	odd	4		1280.3.h.b.1279.1		2
120.53	even	4		1440.3.p.a.559.1		1
120.77	even	4		1440.3.p.b.559.1		1
120.83	odd	4		360.3.p.b.19.1		1
120.107	odd	4		360.3.p.a.19.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.3.e.a.19.1	✓	1	5.2	odd	4
40.3.e.a.19.1	✓	1	40.3	even	4
40.3.e.b.19.1	yes	1	5.3	odd	4
40.3.e.b.19.1	yes	1	40.27	even	4
160.3.e.a.79.1		1	20.3	even	4
160.3.e.a.79.1		1	40.37	odd	4
160.3.e.b.79.1		1	20.7	even	4
160.3.e.b.79.1		1	40.13	odd	4
200.3.g.c.51.1		2	5.4	even	2	inner
200.3.g.c.51.1		2	8.3	odd	2	inner
200.3.g.c.51.2		2	1.1	even	1	trivial
200.3.g.c.51.2		2	40.19	odd	2	CM
360.3.p.a.19.1		1	15.8	even	4
360.3.p.a.19.1		1	120.107	odd	4
360.3.p.b.19.1		1	15.2	even	4
360.3.p.b.19.1		1	120.83	odd	4
800.3.g.c.751.1		2	8.5	even	2
800.3.g.c.751.1		2	20.19	odd	2
800.3.g.c.751.2		2	4.3	odd	2
800.3.g.c.751.2		2	40.29	even	2
1280.3.h.b.1279.1		2	80.43	even	4
1280.3.h.b.1279.1		2	80.77	odd	4
1280.3.h.b.1279.2		2	80.3	even	4
1280.3.h.b.1279.2		2	80.37	odd	4
1280.3.h.c.1279.1		2	80.53	odd	4
1280.3.h.c.1279.1		2	80.67	even	4
1280.3.h.c.1279.2		2	80.13	odd	4
1280.3.h.c.1279.2		2	80.27	even	4
1440.3.p.a.559.1		1	60.47	odd	4
1440.3.p.a.559.1		1	120.53	even	4
1440.3.p.b.559.1		1	60.23	odd	4
1440.3.p.b.559.1		1	120.77	even	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 200 = 2^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	200.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} -4.00000 q^{4} -6.00000i q^{7} -8.00000i q^{8} -9.00000 q^{9} -18.0000 q^{11} -6.00000i q^{13} +12.0000 q^{14} +16.0000 q^{16} -18.0000i q^{18} +2.00000 q^{19} -36.0000i q^{22} -26.0000i q^{23} +12.0000 q^{26} +24.0000i q^{28} +32.0000i q^{32} +36.0000 q^{36} +54.0000i q^{37} +4.00000i q^{38} -78.0000 q^{41} +72.0000 q^{44} +52.0000 q^{46} -86.0000i q^{47} +13.0000 q^{49} +24.0000i q^{52} +74.0000i q^{53} -48.0000 q^{56} -78.0000 q^{59} +54.0000i q^{63} -64.0000 q^{64} +72.0000i q^{72} -108.000 q^{74} -8.00000 q^{76} +108.000i q^{77} +81.0000 q^{81} -156.000i q^{82} +144.000i q^{88} -18.0000 q^{89} -36.0000 q^{91} +104.000i q^{92} +172.000 q^{94} +26.0000i q^{98} +162.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4} - 18 q^{9} - 36 q^{11} + 24 q^{14} + 32 q^{16} + 4 q^{19} + 24 q^{26} + 72 q^{36} - 156 q^{41} + 144 q^{44} + 104 q^{46} + 26 q^{49} - 96 q^{56} - 156 q^{59} - 128 q^{64} - 216 q^{74} - 16 q^{76}+ \cdots + 324 q^{99}+O(q^{100}) \) 2 * q - 8 * q^4 - 18 * q^9 - 36 * q^11 + 24 * q^14 + 32 * q^16 + 4 * q^19 + 24 * q^26 + 72 * q^36 - 156 * q^41 + 144 * q^44 + 104 * q^46 + 26 * q^49 - 96 * q^56 - 156 * q^59 - 128 * q^64 - 216 * q^74 - 16 * q^76 + 162 * q^81 - 36 * q^89 - 72 * q^91 + 344 * q^94 + 324 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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