LMFDB - Embedded newform 200.3.e.a.99.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [200,3,Mod(99,200)] 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("200.99"); 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, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 200 = 2^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	200.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == 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 8)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			99.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	200.99
Dual form			200.3.e.a.99.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{2} -2.00000i q^{3} -4.00000 q^{4} +4.00000 q^{6} -8.00000i q^{8} +5.00000 q^{9} +14.0000 q^{11} +8.00000i q^{12} +16.0000 q^{16} -2.00000i q^{17} +10.0000i q^{18} +34.0000 q^{19} +28.0000i q^{22} -16.0000 q^{24} -28.0000i q^{27} +32.0000i q^{32} -28.0000i q^{33} +4.00000 q^{34} -20.0000 q^{36} +68.0000i q^{38} -46.0000 q^{41} +14.0000i q^{43} -56.0000 q^{44} -32.0000i q^{48} -49.0000 q^{49} -4.00000 q^{51} +56.0000 q^{54} -68.0000i q^{57} +82.0000 q^{59} -64.0000 q^{64} +56.0000 q^{66} -62.0000i q^{67} +8.00000i q^{68} -40.0000i q^{72} -142.000i q^{73} -136.000 q^{76} -11.0000 q^{81} -92.0000i q^{82} +158.000i q^{83} -28.0000 q^{86} -112.000i q^{88} -146.000 q^{89} +64.0000 q^{96} +94.0000i q^{97} -98.0000i q^{98} +70.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{4} + 8 q^{6} + 10 q^{9} + 28 q^{11} + 32 q^{16} + 68 q^{19} - 32 q^{24} + 8 q^{34} - 40 q^{36} - 92 q^{41} - 112 q^{44} - 98 q^{49} - 8 q^{51} + 112 q^{54} + 164 q^{59} - 128 q^{64} + 112 q^{66}+ \cdots + 140 q^{99}+O(q^{100}) $ 2 * q - 8 * q^4 + 8 * q^6 + 10 * q^9 + 28 * q^11 + 32 * q^16 + 68 * q^19 - 32 * q^24 + 8 * q^34 - 40 * q^36 - 92 * q^41 - 112 * q^44 - 98 * q^49 - 8 * q^51 + 112 * q^54 + 164 * q^59 - 128 * q^64 + 112 * q^66 - 272 * q^76 - 22 * q^81 - 56 * q^86 - 292 * q^89 + 128 * q^96 + 140 * 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
$3$	− 2.00000i	− 0.666667i	−0.942809	−	0.333333i	$-0.891827\pi$
$3$	− 2.00000i	− 0.666667i	0.942809	−	0.333333i	$-0.108173\pi$
$4$	−4.00000	−1.00000
$5$	0	0
$5$	0	0
$6$	4.00000	0.666667
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	− 8.00000i	− 1.00000i
$9$	5.00000	0.555556
$10$	0	0
$11$	14.0000	1.27273	0.636364	−	0.771389i	$-0.280438\pi$
$11$	14.0000	1.27273	0.636364	+	0.771389i	$0.280438\pi$
$12$	8.00000i	0.666667i
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	16.0000	1.00000
$17$	− 2.00000i	− 0.117647i	−0.998268	−	0.0588235i	$-0.981265\pi$
$17$	− 2.00000i	− 0.117647i	0.998268	−	0.0588235i	$-0.0187349\pi$
$18$	10.0000i	0.555556i
$19$	34.0000	1.78947	0.894737	−	0.446594i	$-0.147363\pi$
$19$	34.0000	1.78947	0.894737	+	0.446594i	$0.147363\pi$
$20$	0	0
$21$	0	0
$22$	28.0000i	1.27273i
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	−16.0000	−0.666667
$25$	0	0
$26$	0	0
$27$	− 28.0000i	− 1.03704i
$28$	0	0
$29$	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
$33$	− 28.0000i	− 0.848485i
$34$	4.00000	0.117647
$35$	0	0
$36$	−20.0000	−0.555556
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	68.0000i	1.78947i
$39$	0	0
$40$	0	0
$41$	−46.0000	−1.12195	−0.560976	−	0.827832i	$-0.689574\pi$
$41$	−46.0000	−1.12195	−0.560976	+	0.827832i	$0.689574\pi$
$42$	0	0
$43$	14.0000i	0.325581i	0.986661	+	0.162791i	$0.0520495\pi$
$43$	14.0000i	0.325581i	−0.986661	+	0.162791i	$0.947950\pi$
$44$	−56.0000	−1.27273
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	− 32.0000i	− 0.666667i
$49$	−49.0000	−1.00000
$50$	0	0
$51$	−4.00000	−0.0784314
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	56.0000	1.03704
$55$	0	0
$56$	0	0
$57$	− 68.0000i	− 1.19298i
$58$	0	0
$59$	82.0000	1.38983	0.694915	−	0.719092i	$-0.255442\pi$
$59$	82.0000	1.38983	0.694915	+	0.719092i	$0.255442\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	−64.0000	−1.00000
$65$	0	0
$66$	56.0000	0.848485
$67$	− 62.0000i	− 0.925373i	−0.886522	−	0.462687i	$-0.846886\pi$
$67$	− 62.0000i	− 0.925373i	0.886522	−	0.462687i	$-0.153114\pi$
$68$	8.00000i	0.117647i
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	− 40.0000i	− 0.555556i
$73$	− 142.000i	− 1.94521i	−0.232473	−	0.972603i	$-0.574682\pi$
$73$	− 142.000i	− 1.94521i	0.232473	−	0.972603i	$-0.425318\pi$
$74$	0	0
$75$	0	0
$76$	−136.000	−1.78947
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	−11.0000	−0.135802
$82$	− 92.0000i	− 1.12195i
$83$	158.000i	1.90361i	0.306697	+	0.951807i	$0.400776\pi$
$83$	158.000i	1.90361i	−0.306697	+	0.951807i	$0.599224\pi$
$84$	0	0
$85$	0	0
$86$	−28.0000	−0.325581
$87$	0	0
$88$	− 112.000i	− 1.27273i
$89$	−146.000	−1.64045	−0.820225	−	0.572041i	$-0.806152\pi$
$89$	−146.000	−1.64045	−0.820225	+	0.572041i	$0.806152\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	64.0000	0.666667
$97$	94.0000i	0.969072i	0.874771	+	0.484536i	$0.161012\pi$
$97$	94.0000i	0.969072i	−0.874771	+	0.484536i	$0.838988\pi$
$98$	− 98.0000i	− 1.00000i
$99$	70.0000	0.707071
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	− 8.00000i	− 0.0784314i
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	178.000i	1.66355i	0.555112	+	0.831776i	$0.312675\pi$
$107$	178.000i	1.66355i	−0.555112	+	0.831776i	$0.687325\pi$
$108$	112.000i	1.03704i
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	98.0000i	0.867257i	0.901092	+	0.433628i	$0.142767\pi$
$113$	98.0000i	0.867257i	−0.901092	+	0.433628i	$0.857233\pi$
$114$	136.000	1.19298
$115$	0	0
$116$	0	0
$117$	0	0
$118$	164.000i	1.38983i
$119$	0	0
$120$	0	0
$121$	75.0000	0.619835
$122$	0	0
$123$	92.0000i	0.747967i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	− 128.000i	− 1.00000i
$129$	28.0000	0.217054
$130$	0	0
$131$	62.0000	0.473282	0.236641	−	0.971597i	$-0.423953\pi$
$131$	62.0000	0.473282	0.236641	+	0.971597i	$0.423953\pi$
$132$	112.000i	0.848485i
$133$	0	0
$134$	124.000	0.925373
$135$	0	0
$136$	−16.0000	−0.117647
$137$	238.000i	1.73723i	0.495491	+	0.868613i	$0.334988\pi$
$137$	238.000i	1.73723i	−0.495491	+	0.868613i	$0.665012\pi$
$138$	0	0
$139$	−206.000	−1.48201	−0.741007	−	0.671497i	$-0.765652\pi$
$139$	−206.000	−1.48201	−0.741007	+	0.671497i	$0.765652\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	80.0000	0.555556
$145$	0	0
$146$	284.000	1.94521
$147$	98.0000i	0.666667i
$148$	0	0
$149$	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			$\pi$
$152$	− 272.000i	− 1.78947i
$153$	− 10.0000i	− 0.0653595i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	− 22.0000i	− 0.135802i
$163$	− 322.000i	− 1.97546i	−0.156171	−	0.987730i	$-0.549915\pi$
$163$	− 322.000i	− 1.97546i	0.156171	−	0.987730i	$-0.450085\pi$
$164$	184.000	1.12195
$165$	0	0
$166$	−316.000	−1.90361
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−169.000	−1.00000
$170$	0	0
$171$	170.000	0.994152
$172$	− 56.0000i	− 0.325581i
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	224.000	1.27273
$177$	− 164.000i	− 0.926554i
$178$	− 292.000i	− 1.64045i
$179$	34.0000	0.189944	0.0949721	−	0.995480i	$-0.469724\pi$
$179$	34.0000	0.189944	0.0949721	+	0.995480i	$0.469724\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 28.0000i	− 0.149733i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	128.000i	0.666667i
$193$	98.0000i	0.507772i	0.967234	+	0.253886i	$0.0817088\pi$
$193$	98.0000i	0.507772i	−0.967234	+	0.253886i	$0.918291\pi$
$194$	−188.000	−0.969072
$195$	0	0
$196$	196.000	1.00000
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	140.000i	0.707071i
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0	0
$201$	−124.000	−0.616915
$202$	0	0
$203$	0	0
$204$	16.0000	0.0784314
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	476.000	2.27751
$210$	0	0
$211$	−226.000	−1.07109	−0.535545	−	0.844507i	$-0.679894\pi$
$211$	−226.000	−1.07109	−0.535545	+	0.844507i	$0.679894\pi$
$212$	0	0
$213$	0	0
$214$	−356.000	−1.66355
$215$	0	0
$216$	−224.000	−1.03704
$217$	0	0
$218$	0	0
$219$	−284.000	−1.29680
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	0	0
$226$	−196.000	−0.867257
$227$	− 446.000i	− 1.96476i	−0.186900	−	0.982379i	$-0.559844\pi$
$227$	− 446.000i	− 1.96476i	0.186900	−	0.982379i	$-0.440156\pi$
$228$	272.000i	1.19298i
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	434.000i	1.86266i	0.364175	+	0.931330i	$0.381351\pi$
$233$	434.000i	1.86266i	−0.364175	+	0.931330i	$0.618649\pi$
$234$	0	0
$235$	0	0
$236$	−328.000	−1.38983
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	194.000	0.804979	0.402490	−	0.915425i	$-0.368145\pi$
$241$	194.000	0.804979	0.402490	+	0.915425i	$0.368145\pi$
$242$	150.000i	0.619835i
$243$	− 230.000i	− 0.946502i
$244$	0	0
$245$	0	0
$246$	−184.000	−0.747967
$247$	0	0
$248$	0	0
$249$	316.000	1.26908
$250$	0	0
$251$	−466.000	−1.85657	−0.928287	−	0.371865i	$-0.878718\pi$
$251$	−466.000	−1.85657	−0.928287	+	0.371865i	$0.878718\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	256.000	1.00000
$257$	− 386.000i	− 1.50195i	−0.660333	−	0.750973i	$-0.729585\pi$
$257$	− 386.000i	− 1.50195i	0.660333	−	0.750973i	$-0.270415\pi$
$258$	56.0000i	0.217054i
$259$	0	0
$260$	0	0
$261$	0	0
$262$	124.000i	0.473282i
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	−224.000	−0.848485
$265$	0	0
$266$	0	0
$267$	292.000i	1.09363i
$268$	248.000i	0.925373i
$269$	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			$\pi$
$272$	− 32.0000i	− 0.117647i
$273$	0	0
$274$	−476.000	−1.73723
$275$	0	0
$276$	0	0
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	− 412.000i	− 1.48201i
$279$	0	0
$280$	0	0
$281$	−238.000	−0.846975	−0.423488	−	0.905902i	$-0.639194\pi$
$281$	−238.000	−0.846975	−0.423488	+	0.905902i	$0.639194\pi$
$282$	0	0
$283$	− 82.0000i	− 0.289753i	−0.989450	−	0.144876i	$-0.953722\pi$
$283$	− 82.0000i	− 0.289753i	0.989450	−	0.144876i	$-0.0462784\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	160.000i	0.555556i
$289$	285.000	0.986159
$290$	0	0
$291$	188.000	0.646048
$292$	568.000i	1.94521i
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	−196.000	−0.666667
$295$	0	0
$296$	0	0
$297$	− 392.000i	− 1.31987i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	544.000	1.78947
$305$	0	0
$306$	20.0000	0.0653595
$307$	− 542.000i	− 1.76547i	−0.469869	−	0.882736i	$-0.655699\pi$
$307$	− 542.000i	− 1.76547i	0.469869	−	0.882736i	$-0.344301\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	− 526.000i	− 1.68051i	−0.542191	−	0.840256i	$-0.682405\pi$
$313$	− 526.000i	− 1.68051i	0.542191	−	0.840256i	$-0.317595\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	356.000	1.10903
$322$	0	0
$323$	− 68.0000i	− 0.210526i
$324$	44.0000	0.135802
$325$	0	0
$326$	644.000	1.97546
$327$	0	0
$328$	368.000i	1.12195i
$329$	0	0
$330$	0	0
$331$	14.0000	0.0422961	0.0211480	−	0.999776i	$-0.493268\pi$
$331$	14.0000	0.0422961	0.0211480	+	0.999776i	$0.493268\pi$
$332$	− 632.000i	− 1.90361i
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	478.000i	1.41840i	0.705009	+	0.709199i	$0.250943\pi$
$337$	478.000i	1.41840i	−0.705009	+	0.709199i	$0.749057\pi$
$338$	− 338.000i	− 1.00000i
$339$	196.000	0.578171
$340$	0	0
$341$	0	0
$342$	340.000i	0.994152i
$343$	0	0
$344$	112.000	0.325581
$345$	0	0
$346$	0	0
$347$	658.000i	1.89625i	0.317892	+	0.948127i	$0.397025\pi$
$347$	658.000i	1.89625i	−0.317892	+	0.948127i	$0.602975\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	448.000i	1.27273i
$353$	194.000i	0.549575i	0.961505	+	0.274788i	$0.0886075\pi$
$353$	194.000i	0.549575i	−0.961505	+	0.274788i	$0.911392\pi$
$354$	328.000	0.926554
$355$	0	0
$356$	584.000	1.64045
$357$	0	0
$358$	68.0000i	0.189944i
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	795.000	2.20222
$362$	0	0
$363$	− 150.000i	− 0.413223i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	−230.000	−0.623306
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0		−	1.00000i	$-0.5\pi$
$373$	0	0			1.00000i	$0.5\pi$
$374$	56.0000	0.149733
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−686.000	−1.81003	−0.905013	−	0.425383i	$-0.860139\pi$
$379$	−686.000	−1.81003	−0.905013	+	0.425383i	$0.860139\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	−256.000	−0.666667
$385$	0	0
$386$	−196.000	−0.507772
$387$	70.0000i	0.180879i
$388$	− 376.000i	− 0.969072i
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	392.000i	1.00000i
$393$	− 124.000i	− 0.315522i
$394$	0	0
$395$	0	0
$396$	−280.000	−0.707071
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−766.000	−1.91022	−0.955112	−	0.296244i	$-0.904266\pi$
$401$	−766.000	−1.91022	−0.955112	+	0.296244i	$0.904266\pi$
$402$	− 248.000i	− 0.616915i
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	32.0000i	0.0784314i
$409$	334.000	0.816626	0.408313	−	0.912842i	$-0.366117\pi$
$409$	334.000	0.816626	0.408313	+	0.912842i	$0.366117\pi$
$410$	0	0
$411$	476.000	1.15815
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	412.000i	0.988010i
$418$	952.000i	2.27751i
$419$	514.000	1.22673	0.613365	−	0.789799i	$-0.289815\pi$
$419$	514.000	1.22673	0.613365	+	0.789799i	$0.289815\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	− 452.000i	− 1.07109i
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	− 712.000i	− 1.66355i
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	− 448.000i	− 1.03704i
$433$	578.000i	1.33487i	0.744667	+	0.667436i	$0.232608\pi$
$433$	578.000i	1.33487i	−0.744667	+	0.667436i	$0.767392\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	− 568.000i	− 1.29680i
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	−245.000	−0.555556
$442$	0	0
$443$	878.000i	1.98194i	0.134079	+	0.990971i	$0.457192\pi$
$443$	878.000i	1.98194i	−0.134079	+	0.990971i	$0.542808\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−866.000	−1.92873	−0.964365	−	0.264574i	$-0.914769\pi$
$449$	−866.000	−1.92873	−0.964365	+	0.264574i	$0.914769\pi$
$450$	0	0
$451$	−644.000	−1.42794
$452$	− 392.000i	− 0.867257i
$453$	0	0
$454$	892.000	1.96476
$455$	0	0
$456$	−544.000	−1.19298
$457$	238.000i	0.520788i	0.965502	+	0.260394i	$0.0838524\pi$
$457$	238.000i	0.520788i	−0.965502	+	0.260394i	$0.916148\pi$
$458$	0	0
$459$	−56.0000	−0.122004
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	0	0
$465$	0	0
$466$	−868.000	−1.86266
$467$	34.0000i	0.0728051i	0.999337	+	0.0364026i	$0.0115899\pi$
$467$	34.0000i	0.0728051i	−0.999337	+	0.0364026i	$0.988410\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	− 656.000i	− 1.38983i
$473$	196.000i	0.414376i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	388.000i	0.804979i
$483$	0	0
$484$	−300.000	−0.619835
$485$	0	0
$486$	460.000	0.946502
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	0	0
$489$	−644.000	−1.31697
$490$	0	0
$491$	782.000	1.59267	0.796334	−	0.604857i	$-0.206770\pi$
$491$	782.000	1.59267	0.796334	+	0.604857i	$0.206770\pi$
$492$	− 368.000i	− 0.747967i
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	632.000i	1.26908i
$499$	802.000	1.60721	0.803607	−	0.595160i	$-0.202911\pi$
$499$	802.000	1.60721	0.803607	+	0.595160i	$0.202911\pi$
$500$	0	0
$501$	0	0
$502$	− 932.000i	− 1.85657i
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	338.000i	0.666667i
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	512.000i	1.00000i
$513$	− 952.000i	− 1.85575i
$514$	772.000	1.50195
$515$	0	0
$516$	−112.000	−0.217054
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1006.00	−1.93090	−0.965451	−	0.260584i	$-0.916085\pi$
$521$	−1006.00	−1.93090	−0.965451	+	0.260584i	$0.916085\pi$
$522$	0	0
$523$	398.000i	0.760994i	0.924782	+	0.380497i	$0.124247\pi$
$523$	398.000i	0.760994i	−0.924782	+	0.380497i	$0.875753\pi$
$524$	−248.000	−0.473282
$525$	0	0
$526$	0	0
$527$	0	0
$528$	− 448.000i	− 0.848485i
$529$	−529.000	−1.00000
$530$	0	0
$531$	410.000	0.772128
$532$	0	0
$533$	0	0
$534$	−584.000	−1.09363
$535$	0	0
$536$	−496.000	−0.925373
$537$	− 68.0000i	− 0.126629i
$538$	0	0
$539$	−686.000	−1.27273
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	64.0000	0.117647
$545$	0	0
$546$	0	0
$547$	− 1022.00i	− 1.86837i	−0.356785	−	0.934186i	$-0.616127\pi$
$547$	− 1022.00i	− 1.86837i	0.356785	−	0.934186i	$-0.383873\pi$
$548$	− 952.000i	− 1.73723i
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	824.000	1.48201
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−56.0000	−0.0998217
$562$	− 476.000i	− 0.846975i
$563$	− 226.000i	− 0.401421i	−0.979651	−	0.200710i	$-0.935675\pi$
$563$	− 226.000i	− 0.401421i	0.979651	−	0.200710i	$-0.0643251\pi$
$564$	0	0
$565$	0	0
$566$	164.000	0.289753
$567$	0	0
$568$	0	0
$569$	−626.000	−1.10018	−0.550088	−	0.835107i	$-0.685406\pi$
$569$	−626.000	−1.10018	−0.550088	+	0.835107i	$0.685406\pi$
$570$	0	0
$571$	−658.000	−1.15236	−0.576182	−	0.817321i	$-0.695458\pi$
$571$	−658.000	−1.15236	−0.576182	+	0.817321i	$0.695458\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	−320.000	−0.555556
$577$	− 2.00000i	− 0.00346620i	−0.999998	−	0.00173310i	$-0.999448\pi$
$577$	− 2.00000i	− 0.00346620i	0.999998	−	0.00173310i	$-0.000551664\pi$
$578$	570.000i	0.986159i
$579$	196.000	0.338515
$580$	0	0
$581$	0	0
$582$	376.000i	0.646048i
$583$	0	0
$584$	−1136.00	−1.94521
$585$	0	0
$586$	0	0
$587$	1138.00i	1.93867i	0.245741	+	0.969336i	$0.420969\pi$
$587$	1138.00i	1.93867i	−0.245741	+	0.969336i	$0.579031\pi$
$588$	− 392.000i	− 0.666667i
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 862.000i	− 1.45363i	−0.686836	−	0.726813i	$-0.741001\pi$
$593$	− 862.000i	− 1.45363i	0.686836	−	0.726813i	$-0.258999\pi$
$594$	784.000	1.31987
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	914.000	1.52080	0.760399	−	0.649456i	$-0.225003\pi$
$601$	914.000	1.52080	0.760399	+	0.649456i	$0.225003\pi$
$602$	0	0
$603$	− 310.000i	− 0.514096i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	1088.00i	1.78947i
$609$	0	0
$610$	0	0
$611$	0	0
$612$	40.0000i	0.0653595i
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	1084.00	1.76547
$615$	0	0
$616$	0	0
$617$	334.000i	0.541329i	0.962674	+	0.270665i	$0.0872434\pi$
$617$	334.000i	0.541329i	−0.962674	+	0.270665i	$0.912757\pi$
$618$	0	0
$619$	562.000	0.907916	0.453958	−	0.891023i	$-0.350012\pi$
$619$	562.000	0.907916	0.453958	+	0.891023i	$0.350012\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	1052.00	1.68051
$627$	− 952.000i	− 1.51834i
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	452.000i	0.714060i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	482.000	0.751950	0.375975	−	0.926630i	$-0.377308\pi$
$641$	482.000	0.751950	0.375975	+	0.926630i	$0.377308\pi$
$642$	712.000i	1.10903i
$643$	1214.00i	1.88802i	0.329910	+	0.944012i	$0.392982\pi$
$643$	1214.00i	1.88802i	−0.329910	+	0.944012i	$0.607018\pi$
$644$	0	0
$645$	0	0
$646$	136.000	0.210526
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	88.0000i	0.135802i
$649$	1148.00	1.76888
$650$	0	0
$651$	0	0
$652$	1288.00i	1.97546i
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	0	0
$656$	−736.000	−1.12195
$657$	− 710.000i	− 1.08067i
$658$	0	0
$659$	994.000	1.50835	0.754173	−	0.656676i	$-0.228038\pi$
$659$	994.000	1.50835	0.754173	+	0.656676i	$0.228038\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	28.0000i	0.0422961i
$663$	0	0
$664$	1264.00	1.90361
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	− 1246.00i	− 1.85141i	−0.378244	−	0.925706i	$-0.623472\pi$
$673$	− 1246.00i	− 1.85141i	0.378244	−	0.925706i	$-0.376528\pi$
$674$	−956.000	−1.41840
$675$	0	0
$676$	676.000	1.00000
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	392.000i	0.578171i
$679$	0	0
$680$	0	0
$681$	−892.000	−1.30984
$682$	0	0
$683$	398.000i	0.582723i	0.956613	+	0.291362i	$0.0941083\pi$
$683$	398.000i	0.582723i	−0.956613	+	0.291362i	$0.905892\pi$
$684$	−680.000	−0.994152
$685$	0	0
$686$	0	0
$687$	0	0
$688$	224.000i	0.325581i
$689$	0	0
$690$	0	0
$691$	734.000	1.06223	0.531114	−	0.847300i	$-0.321773\pi$
$691$	734.000	1.06223	0.531114	+	0.847300i	$0.321773\pi$
$692$	0	0
$693$	0	0
$694$	−1316.00	−1.89625
$695$	0	0
$696$	0	0
$697$	92.0000i	0.131994i
$698$	0	0
$699$	868.000	1.24177
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	−896.000	−1.27273
$705$	0	0
$706$	−388.000	−0.549575
$707$	0	0
$708$	656.000i	0.926554i
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	1168.00i	1.64045i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−136.000	−0.189944
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	0	0
$722$	1590.00i	2.20222i
$723$	− 388.000i	− 0.536653i
$724$	0	0
$725$	0	0
$726$	300.000	0.413223
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	−559.000	−0.766804
$730$	0	0
$731$	28.0000	0.0383037
$732$	0	0
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 868.000i	− 1.17775i
$738$	− 460.000i	− 0.623306i
$739$	322.000	0.435724	0.217862	−	0.975980i	$-0.430092\pi$
$739$	322.000	0.435724	0.217862	+	0.975980i	$0.430092\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	790.000i	1.05756i
$748$	112.000i	0.149733i
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	932.000i	1.23772i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	− 1372.00i	− 1.81003i
$759$	0	0
$760$	0	0
$761$	1394.00	1.83180	0.915900	−	0.401406i	$-0.131478\pi$
$761$	1394.00	1.83180	0.915900	+	0.401406i	$0.131478\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	− 512.000i	− 0.666667i
$769$	1054.00	1.37061	0.685306	−	0.728256i	$-0.259669\pi$
$769$	1054.00	1.37061	0.685306	+	0.728256i	$0.259669\pi$
$770$	0	0
$771$	−772.000	−1.00130
$772$	− 392.000i	− 0.507772i
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	−140.000	−0.180879
$775$	0	0
$776$	752.000	0.969072
$777$	0	0
$778$	0	0
$779$	−1564.00	−2.00770
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−784.000	−1.00000
$785$	0	0
$786$	248.000	0.315522
$787$	− 926.000i	− 1.17662i	−0.808635	−	0.588310i	$-0.799793\pi$
$787$	− 926.000i	− 1.17662i	0.808635	−	0.588310i	$-0.200207\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	− 560.000i	− 0.707071i
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−730.000	−0.911361
$802$	− 1532.00i	− 1.91022i
$803$	− 1988.00i	− 2.47572i
$804$	496.000	0.616915
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1582.00	1.95550	0.977750	−	0.209772i	$-0.0672722\pi$
$809$	1582.00	1.95550	0.977750	+	0.209772i	$0.0672722\pi$
$810$	0	0
$811$	−178.000	−0.219482	−0.109741	−	0.993960i	$-0.535002\pi$
$811$	−178.000	−0.219482	−0.109741	+	0.993960i	$0.535002\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	−64.0000	−0.0784314
$817$	476.000i	0.582619i
$818$	668.000i	0.816626i
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	952.000i	1.15815i
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 1262.00i	− 1.52600i	−0.646400	−	0.762999i	$-0.723726\pi$
$827$	− 1262.00i	− 1.52600i	0.646400	−	0.762999i	$-0.276274\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	98.0000i	0.117647i
$834$	−824.000	−0.988010
$835$	0	0
$836$	−1904.00	−2.27751
$837$	0	0
$838$	1028.00i	1.22673i
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	841.000	1.00000
$842$	0	0
$843$	476.000i	0.564650i
$844$	904.000	1.07109
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−164.000	−0.193168
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	0	0
$856$	1424.00	1.66355
$857$	− 1202.00i	− 1.40257i	−0.712882	−	0.701284i	$-0.752611\pi$
$857$	− 1202.00i	− 1.40257i	0.712882	−	0.701284i	$-0.247389\pi$
$858$	0	0
$859$	−1646.00	−1.91618	−0.958091	−	0.286465i	$-0.907520\pi$
$859$	−1646.00	−1.91618	−0.958091	+	0.286465i	$0.907520\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	896.000	1.03704
$865$	0	0
$866$	−1156.00	−1.33487
$867$	− 570.000i	− 0.657439i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	470.000i	0.538373i
$874$	0	0
$875$	0	0
$876$	1136.00	1.29680
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1438.00	−1.63224	−0.816118	−	0.577885i	$-0.803878\pi$
$881$	−1438.00	−1.63224	−0.816118	+	0.577885i	$0.803878\pi$
$882$	− 490.000i	− 0.555556i
$883$	− 1762.00i	− 1.99547i	−0.0672672	−	0.997735i	$-0.521428\pi$
$883$	− 1762.00i	− 1.99547i	0.0672672	−	0.997735i	$-0.478572\pi$
$884$	0	0
$885$	0	0
$886$	−1756.00	−1.98194
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−154.000	−0.172840
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	− 1732.00i	− 1.92873i
$899$	0	0
$900$	0	0
$901$	0	0
$902$	− 1288.00i	− 1.42794i
$903$	0	0
$904$	784.000	0.867257
$905$	0	0
$906$	0	0
$907$	1714.00i	1.88975i	0.327436	+	0.944873i	$0.393815\pi$
$907$	1714.00i	1.88975i	−0.327436	+	0.944873i	$0.606185\pi$
$908$	1784.00i	1.96476i
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	− 1088.00i	− 1.19298i
$913$	2212.00i	2.42278i
$914$	−476.000	−0.520788
$915$	0	0
$916$	0	0
$917$	0	0
$918$	− 112.000i	− 0.122004i
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	−1084.00	−1.17698
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1058.00	−1.13886	−0.569429	−	0.822040i	$-0.692836\pi$
$929$	−1058.00	−1.13886	−0.569429	+	0.822040i	$0.692836\pi$
$930$	0	0
$931$	−1666.00	−1.78947
$932$	− 1736.00i	− 1.86266i
$933$	0	0
$934$	−68.0000	−0.0728051
$935$	0	0
$936$	0	0
$937$	718.000i	0.766275i	0.923691	+	0.383138i	$0.125157\pi$
$937$	718.000i	0.766275i	−0.923691	+	0.383138i	$0.874843\pi$
$938$	0	0
$939$	−1052.00	−1.12034
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	1312.00	1.38983
$945$	0	0
$946$	−392.000	−0.414376
$947$	994.000i	1.04963i	0.851216	+	0.524815i	$0.175866\pi$
$947$	994.000i	1.04963i	−0.851216	+	0.524815i	$0.824134\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 142.000i	− 0.149003i	−0.997221	−	0.0745016i	$-0.976263\pi$
$953$	− 142.000i	− 0.149003i	0.997221	−	0.0745016i	$-0.0237366\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	961.000	1.00000
$962$	0	0
$963$	890.000i	0.924195i
$964$	−776.000	−0.804979
$965$	0	0
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	− 600.000i	− 0.619835i
$969$	−136.000	−0.140351
$970$	0	0
$971$	974.000	1.00309	0.501545	−	0.865132i	$-0.332765\pi$
$971$	974.000	1.00309	0.501545	+	0.865132i	$0.332765\pi$
$972$	920.000i	0.946502i
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1918.00i	1.96315i	0.191071	+	0.981576i	$0.438804\pi$
$977$	1918.00i	1.96315i	−0.191071	+	0.981576i	$0.561196\pi$
$978$	− 1288.00i	− 1.31697i
$979$	−2044.00	−2.08784
$980$	0	0
$981$	0	0
$982$	1564.00i	1.59267i
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	736.000	0.747967
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	− 28.0000i	− 0.0281974i
$994$	0	0
$995$	0	0
$996$	−1264.00	−1.26908
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	1604.00i	1.60721i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	200.3.e.a.99.2		2
4.3	odd	2		800.3.e.a.399.2		2
5.2	odd	4		8.3.d.a.3.1	✓	1
5.3	odd	4		200.3.g.a.51.1		1
5.4	even	2	inner	200.3.e.a.99.1		2
8.3	odd	2	CM	200.3.e.a.99.2		2
8.5	even	2		800.3.e.a.399.2		2
15.2	even	4		72.3.b.a.19.1		1
20.3	even	4		800.3.g.a.751.1		1
20.7	even	4		32.3.d.a.15.1		1
20.19	odd	2		800.3.e.a.399.1		2
35.2	odd	12		392.3.k.d.67.1		2
35.12	even	12		392.3.k.b.67.1		2
35.17	even	12		392.3.k.b.275.1		2
35.27	even	4		392.3.g.a.99.1		1
35.32	odd	12		392.3.k.d.275.1		2
40.3	even	4		200.3.g.a.51.1		1
40.13	odd	4		800.3.g.a.751.1		1
40.19	odd	2	inner	200.3.e.a.99.1		2
40.27	even	4		8.3.d.a.3.1	✓	1
40.29	even	2		800.3.e.a.399.1		2
40.37	odd	4		32.3.d.a.15.1		1
60.47	odd	4		288.3.b.a.271.1		1
80.27	even	4		256.3.c.e.255.1		2
80.37	odd	4		256.3.c.e.255.2		2
80.67	even	4		256.3.c.e.255.2		2
80.77	odd	4		256.3.c.e.255.1		2
120.77	even	4		288.3.b.a.271.1		1
120.107	odd	4		72.3.b.a.19.1		1
140.27	odd	4		1568.3.g.a.687.1		1
240.77	even	4		2304.3.g.j.1279.2		2
240.107	odd	4		2304.3.g.j.1279.2		2
240.197	even	4		2304.3.g.j.1279.1		2
240.227	odd	4		2304.3.g.j.1279.1		2
280.27	odd	4		392.3.g.a.99.1		1
280.67	even	12		392.3.k.d.275.1		2
280.107	even	12		392.3.k.d.67.1		2
280.187	odd	12		392.3.k.b.67.1		2
280.227	odd	12		392.3.k.b.275.1		2
280.237	even	4		1568.3.g.a.687.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.3.d.a.3.1	✓	1	5.2	odd	4
8.3.d.a.3.1	✓	1	40.27	even	4
32.3.d.a.15.1		1	20.7	even	4
32.3.d.a.15.1		1	40.37	odd	4
72.3.b.a.19.1		1	15.2	even	4
72.3.b.a.19.1		1	120.107	odd	4
200.3.e.a.99.1		2	5.4	even	2	inner
200.3.e.a.99.1		2	40.19	odd	2	inner
200.3.e.a.99.2		2	1.1	even	1	trivial
200.3.e.a.99.2		2	8.3	odd	2	CM
200.3.g.a.51.1		1	5.3	odd	4
200.3.g.a.51.1		1	40.3	even	4
256.3.c.e.255.1		2	80.27	even	4
256.3.c.e.255.1		2	80.77	odd	4
256.3.c.e.255.2		2	80.37	odd	4
256.3.c.e.255.2		2	80.67	even	4
288.3.b.a.271.1		1	60.47	odd	4
288.3.b.a.271.1		1	120.77	even	4
392.3.g.a.99.1		1	35.27	even	4
392.3.g.a.99.1		1	280.27	odd	4
392.3.k.b.67.1		2	35.12	even	12
392.3.k.b.67.1		2	280.187	odd	12
392.3.k.b.275.1		2	35.17	even	12
392.3.k.b.275.1		2	280.227	odd	12
392.3.k.d.67.1		2	35.2	odd	12
392.3.k.d.67.1		2	280.107	even	12
392.3.k.d.275.1		2	35.32	odd	12
392.3.k.d.275.1		2	280.67	even	12
800.3.e.a.399.1		2	20.19	odd	2
800.3.e.a.399.1		2	40.29	even	2
800.3.e.a.399.2		2	4.3	odd	2
800.3.e.a.399.2		2	8.5	even	2
800.3.g.a.751.1		1	20.3	even	4
800.3.g.a.751.1		1	40.13	odd	4
1568.3.g.a.687.1		1	140.27	odd	4
1568.3.g.a.687.1		1	280.237	even	4
2304.3.g.j.1279.1		2	240.197	even	4
2304.3.g.j.1279.1		2	240.227	odd	4
2304.3.g.j.1279.2		2	240.77	even	4
2304.3.g.j.1279.2		2	240.107	odd	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.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} -2.00000i q^{3} -4.00000 q^{4} +4.00000 q^{6} -8.00000i q^{8} +5.00000 q^{9} +14.0000 q^{11} +8.00000i q^{12} +16.0000 q^{16} -2.00000i q^{17} +10.0000i q^{18} +34.0000 q^{19} +28.0000i q^{22} -16.0000 q^{24} -28.0000i q^{27} +32.0000i q^{32} -28.0000i q^{33} +4.00000 q^{34} -20.0000 q^{36} +68.0000i q^{38} -46.0000 q^{41} +14.0000i q^{43} -56.0000 q^{44} -32.0000i q^{48} -49.0000 q^{49} -4.00000 q^{51} +56.0000 q^{54} -68.0000i q^{57} +82.0000 q^{59} -64.0000 q^{64} +56.0000 q^{66} -62.0000i q^{67} +8.00000i q^{68} -40.0000i q^{72} -142.000i q^{73} -136.000 q^{76} -11.0000 q^{81} -92.0000i q^{82} +158.000i q^{83} -28.0000 q^{86} -112.000i q^{88} -146.000 q^{89} +64.0000 q^{96} +94.0000i q^{97} -98.0000i q^{98} +70.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4} + 8 q^{6} + 10 q^{9} + 28 q^{11} + 32 q^{16} + 68 q^{19} - 32 q^{24} + 8 q^{34} - 40 q^{36} - 92 q^{41} - 112 q^{44} - 98 q^{49} - 8 q^{51} + 112 q^{54} + 164 q^{59} - 128 q^{64} + 112 q^{66}+ \cdots + 140 q^{99}+O(q^{100}) \) 2 * q - 8 * q^4 + 8 * q^6 + 10 * q^9 + 28 * q^11 + 32 * q^16 + 68 * q^19 - 32 * q^24 + 8 * q^34 - 40 * q^36 - 92 * q^41 - 112 * q^44 - 98 * q^49 - 8 * q^51 + 112 * q^54 + 164 * q^59 - 128 * q^64 + 112 * q^66 - 272 * q^76 - 22 * q^81 - 56 * q^86 - 292 * q^89 + 128 * q^96 + 140 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

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