LMFDB - Embedded newform 504.2.p.c.307.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [504,2,Mod(307,504)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(504, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("504.307");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 504 = 2^{3} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	504.p (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:	$4.02446026187$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 8x^{2} + 9 $ x^4 + 8*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			307.4
Root			$-2.57794i$ of defining polynomial
Character	$\chi$	$=$	504.307
Dual form			504.2.p.c.307.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} -2.00000 q^{4} +2.64575 q^{7} -2.82843i q^{8} +O(q^{10})$	$q+1.41421i q^{2} -2.00000 q^{4} +2.64575 q^{7} -2.82843i q^{8} +5.29150 q^{13} +3.74166i q^{14} +4.00000 q^{16} +7.48331i q^{17} +2.82843i q^{23} -5.00000 q^{25} +7.48331i q^{26} -5.29150 q^{28} -5.65685i q^{29} +5.29150 q^{31} +5.65685i q^{32} -10.5830 q^{34} +7.48331i q^{41} +2.00000 q^{43} -4.00000 q^{46} +7.00000 q^{49} -7.07107i q^{50} -10.5830 q^{52} +11.3137i q^{53} -7.48331i q^{56} +8.00000 q^{58} -14.9666i q^{59} +5.29150 q^{61} +7.48331i q^{62} -8.00000 q^{64} -10.0000 q^{67} -14.9666i q^{68} -14.1421i q^{71} -10.5830 q^{82} -14.9666i q^{83} +2.82843i q^{86} +7.48331i q^{89} +14.0000 q^{91} -5.65685i q^{92} +9.89949i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4}+O(q^{10}) $ 4 * q - 8 * q^4	$ 4 q - 8 q^{4} + 16 q^{16} - 20 q^{25} + 8 q^{43} - 16 q^{46} + 28 q^{49} + 32 q^{58} - 32 q^{64} - 40 q^{67} + 56 q^{91}+O(q^{100}) $ 4 * q - 8 * q^4 + 16 * q^16 - 20 * q^25 + 8 * q^43 - 16 * q^46 + 28 * q^49 + 32 * q^58 - 32 * q^64 - 40 * q^67 + 56 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/504\mathbb{Z}\right)^\times$.

$n$	$73$	$127$	$253$	$281$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

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 $

$2$	1.41421i	1.00000i
$2$	1.41421i	1.00000i
$3$	0	0
$3$	0	0
$4$	−2.00000	−1.00000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	2.64575	1.00000
$7$	2.64575	1.00000
$8$	− 2.82843i	− 1.00000i
$9$	0	0
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	5.29150	1.46760	0.733799	−	0.679366i	$-0.237745\pi$
$13$	5.29150	1.46760	0.733799	+	0.679366i	$0.237745\pi$
$14$	3.74166i	1.00000i
$15$	0	0
$16$	4.00000	1.00000
$17$	7.48331i	1.81497i	0.420084	+	0.907485i	$0.362001\pi$
$17$	7.48331i	1.81497i	−0.420084	+	0.907485i	$0.637999\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.82843i	0.589768i	0.955533	+	0.294884i	$0.0952810\pi$
$23$	2.82843i	0.589768i	−0.955533	+	0.294884i	$0.904719\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	7.48331i	1.46760i
$27$	0	0
$28$	−5.29150	−1.00000
$29$	− 5.65685i	− 1.05045i	−0.850963	−	0.525226i	$-0.823981\pi$
$29$	− 5.65685i	− 1.05045i	0.850963	−	0.525226i	$-0.176019\pi$
$30$	0	0
$31$	5.29150	0.950382	0.475191	−	0.879883i	$-0.342379\pi$
$31$	5.29150	0.950382	0.475191	+	0.879883i	$0.342379\pi$
$32$	5.65685i	1.00000i
$33$	0	0
$34$	−10.5830	−1.81497
$35$	0	0
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.48331i	1.16870i	0.811503	+	0.584349i	$0.198650\pi$
$41$	7.48331i	1.16870i	−0.811503	+	0.584349i	$0.801350\pi$
$42$	0	0
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	0	0
$45$	0	0
$46$	−4.00000	−0.589768
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	7.00000	1.00000
$50$	− 7.07107i	− 1.00000i
$51$	0	0
$52$	−10.5830	−1.46760
$53$	11.3137i	1.55406i	0.629465	+	0.777029i	$0.283274\pi$
$53$	11.3137i	1.55406i	−0.629465	+	0.777029i	$0.716726\pi$
$54$	0	0
$55$	0	0
$56$	− 7.48331i	− 1.00000i
$57$	0	0
$58$	8.00000	1.05045
$59$	− 14.9666i	− 1.94849i	−0.225494	−	0.974245i	$-0.572400\pi$
$59$	− 14.9666i	− 1.94849i	0.225494	−	0.974245i	$-0.427600\pi$
$60$	0	0
$61$	5.29150	0.677507	0.338754	−	0.940875i	$-0.389995\pi$
$61$	5.29150	0.677507	0.338754	+	0.940875i	$0.389995\pi$
$62$	7.48331i	0.950382i
$63$	0	0
$64$	−8.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	− 14.9666i	− 1.81497i
$69$	0	0
$70$	0	0
$71$	− 14.1421i	− 1.67836i	−0.543852	−	0.839181i	$-0.683035\pi$
$71$	− 14.1421i	− 1.67836i	0.543852	−	0.839181i	$-0.316965\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	0	0
$82$	−10.5830	−1.16870
$83$	− 14.9666i	− 1.64280i	−0.570352	−	0.821401i	$-0.693193\pi$
$83$	− 14.9666i	− 1.64280i	0.570352	−	0.821401i	$-0.306807\pi$
$84$	0	0
$85$	0	0
$86$	2.82843i	0.304997i
$87$	0	0
$88$	0	0
$89$	7.48331i	0.793230i	0.917985	+	0.396615i	$0.129815\pi$
$89$	7.48331i	0.793230i	−0.917985	+	0.396615i	$0.870185\pi$
$90$	0	0
$91$	14.0000	1.46760
$92$	− 5.65685i	− 0.589768i
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	9.89949i	1.00000i
$99$	0	0
$100$	10.0000	1.00000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	5.29150	0.521387	0.260694	−	0.965422i	$-0.416049\pi$
$103$	5.29150	0.521387	0.260694	+	0.965422i	$0.416049\pi$
$104$	− 14.9666i	− 1.46760i
$105$	0	0
$106$	−16.0000	−1.55406
$107$	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
$112$	10.5830	1.00000
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	0	0
$116$	11.3137i	1.05045i
$117$	0	0
$118$	21.1660	1.94849
$119$	19.7990i	1.81497i
$120$	0	0
$121$	−11.0000	−1.00000
$122$	7.48331i	0.677507i
$123$	0	0
$124$	−10.5830	−0.950382
$125$	0	0
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	− 11.3137i	− 1.00000i
$129$	0	0
$130$	0	0
$131$	− 14.9666i	− 1.30764i	−0.756650	−	0.653820i	$-0.773165\pi$
$131$	− 14.9666i	− 1.30764i	0.756650	−	0.653820i	$-0.226835\pi$
$132$	0	0
$133$	0	0
$134$	− 14.1421i	− 1.22169i
$135$	0	0
$136$	21.1660	1.81497
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	0	0
$142$	20.0000	1.67836
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 22.6274i	− 1.85371i	−0.375419	−	0.926855i	$-0.622501\pi$
$149$	− 22.6274i	− 1.85371i	0.375419	−	0.926855i	$-0.377499\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	5.29150	0.422308	0.211154	−	0.977453i	$-0.432278\pi$
$157$	5.29150	0.422308	0.211154	+	0.977453i	$0.432278\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	7.48331i	0.589768i
$162$	0	0
$163$	−22.0000	−1.72317	−0.861586	−	0.507611i	$-0.830529\pi$
$163$	−22.0000	−1.72317	−0.861586	+	0.507611i	$0.830529\pi$
$164$	− 14.9666i	− 1.16870i
$165$	0	0
$166$	21.1660	1.64280
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	15.0000	1.15385
$170$	0	0
$171$	0	0
$172$	−4.00000	−0.304997
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	−13.2288	−1.00000
$176$	0	0
$177$	0	0
$178$	−10.5830	−0.793230
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−26.4575	−1.96657	−0.983286	−	0.182069i	$-0.941721\pi$
$181$	−26.4575	−1.96657	−0.983286	+	0.182069i	$0.941721\pi$
$182$	19.7990i	1.46760i
$183$	0	0
$184$	8.00000	0.589768
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2.82843i	0.204658i	0.994751	+	0.102329i	$0.0326294\pi$
$191$	2.82843i	0.204658i	−0.994751	+	0.102329i	$0.967371\pi$
$192$	0	0
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	0	0
$195$	0	0
$196$	−14.0000	−1.00000
$197$	− 5.65685i	− 0.403034i	−0.979485	−	0.201517i	$-0.935413\pi$
$197$	− 5.65685i	− 0.403034i	0.979485	−	0.201517i	$-0.0645872\pi$
$198$	0	0
$199$	−26.4575	−1.87552	−0.937762	−	0.347279i	$-0.887106\pi$
$199$	−26.4575	−1.87552	−0.937762	+	0.347279i	$0.887106\pi$
$200$	14.1421i	1.00000i
$201$	0	0
$202$	0	0
$203$	− 14.9666i	− 1.05045i
$204$	0	0
$205$	0	0
$206$	7.48331i	0.521387i
$207$	0	0
$208$	21.1660	1.46760
$209$	0	0
$210$	0	0
$211$	26.0000	1.78991	0.894957	−	0.446153i	$-0.147206\pi$
$211$	26.0000	1.78991	0.894957	+	0.446153i	$0.147206\pi$
$212$	− 22.6274i	− 1.55406i
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	14.0000	0.950382
$218$	0	0
$219$	0	0
$220$	0	0
$221$	39.5980i	2.66365i
$222$	0	0
$223$	5.29150	0.354345	0.177173	−	0.984180i	$-0.443305\pi$
$223$	5.29150	0.354345	0.177173	+	0.984180i	$0.443305\pi$
$224$	14.9666i	1.00000i
$225$	0	0
$226$	0	0
$227$	29.9333i	1.98674i	0.114960	+	0.993370i	$0.463326\pi$
$227$	29.9333i	1.98674i	−0.114960	+	0.993370i	$0.536674\pi$
$228$	0	0
$229$	−26.4575	−1.74836	−0.874181	−	0.485601i	$-0.838601\pi$
$229$	−26.4575	−1.74836	−0.874181	+	0.485601i	$0.838601\pi$
$230$	0	0
$231$	0	0
$232$	−16.0000	−1.05045
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	0	0
$236$	29.9333i	1.94849i
$237$	0	0
$238$	−28.0000	−1.81497
$239$	− 14.1421i	− 0.914779i	−0.889267	−	0.457389i	$-0.848785\pi$
$239$	− 14.1421i	− 0.914779i	0.889267	−	0.457389i	$-0.151215\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	− 15.5563i	− 1.00000i
$243$	0	0
$244$	−10.5830	−0.677507
$245$	0	0
$246$	0	0
$247$	0	0
$248$	− 14.9666i	− 0.950382i
$249$	0	0
$250$	0	0
$251$	29.9333i	1.88937i	0.327978	+	0.944685i	$0.393633\pi$
$251$	29.9333i	1.88937i	−0.327978	+	0.944685i	$0.606367\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	16.0000	1.00000
$257$	7.48331i	0.466796i	0.972381	+	0.233398i	$0.0749846\pi$
$257$	7.48331i	0.466796i	−0.972381	+	0.233398i	$0.925015\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	21.1660	1.30764
$263$	− 31.1127i	− 1.91849i	−0.282574	−	0.959246i	$-0.591188\pi$
$263$	− 31.1127i	− 1.91849i	0.282574	−	0.959246i	$-0.408812\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	20.0000	1.22169
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	−26.4575	−1.60718	−0.803590	−	0.595184i	$-0.797079\pi$
$271$	−26.4575	−1.60718	−0.803590	+	0.595184i	$0.797079\pi$
$272$	29.9333i	1.81497i
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	28.2843i	1.67836i
$285$	0	0
$286$	0	0
$287$	19.7990i	1.16870i
$288$	0	0
$289$	−39.0000	−2.29412
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	32.0000	1.85371
$299$	14.9666i	0.865543i
$300$	0	0
$301$	5.29150	0.304997
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	7.48331i	0.422308i
$315$	0	0
$316$	0	0
$317$	− 22.6274i	− 1.27088i	−0.772149	−	0.635441i	$-0.780818\pi$
$317$	− 22.6274i	− 1.27088i	0.772149	−	0.635441i	$-0.219182\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	−10.5830	−0.589768
$323$	0	0
$324$	0	0
$325$	−26.4575	−1.46760
$326$	− 31.1127i	− 1.72317i
$327$	0	0
$328$	21.1660	1.16870
$329$	0	0
$330$	0	0
$331$	−34.0000	−1.86881	−0.934405	−	0.356214i	$-0.884068\pi$
$331$	−34.0000	−1.86881	−0.934405	+	0.356214i	$0.884068\pi$
$332$	29.9333i	1.64280i
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	26.0000	1.41631	0.708155	−	0.706057i	$-0.249528\pi$
$337$	26.0000	1.41631	0.708155	+	0.706057i	$0.249528\pi$
$338$	21.2132i	1.15385i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	18.5203	1.00000
$344$	− 5.65685i	− 0.304997i
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	37.0405	1.98273	0.991367	−	0.131118i	$-0.0418567\pi$
$349$	37.0405	1.98273	0.991367	+	0.131118i	$0.0418567\pi$
$350$	− 18.7083i	− 1.00000i
$351$	0	0
$352$	0	0
$353$	− 37.4166i	− 1.99148i	−0.0921878	−	0.995742i	$-0.529386\pi$
$353$	− 37.4166i	− 1.99148i	0.0921878	−	0.995742i	$-0.470614\pi$
$354$	0	0
$355$	0	0
$356$	− 14.9666i	− 0.793230i
$357$	0	0
$358$	0	0
$359$	36.7696i	1.94062i	0.241859	+	0.970311i	$0.422243\pi$
$359$	36.7696i	1.94062i	−0.241859	+	0.970311i	$0.577757\pi$
$360$	0	0
$361$	19.0000	1.00000
$362$	− 37.4166i	− 1.96657i
$363$	0	0
$364$	−28.0000	−1.46760
$365$	0	0
$366$	0	0
$367$	37.0405	1.93350	0.966750	−	0.255725i	$-0.0823140\pi$
$367$	37.0405	1.93350	0.966750	+	0.255725i	$0.0823140\pi$
$368$	11.3137i	0.589768i
$369$	0	0
$370$	0	0
$371$	29.9333i	1.55406i
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 29.9333i	− 1.54164i
$378$	0	0
$379$	2.00000	0.102733	0.0513665	−	0.998680i	$-0.483642\pi$
$379$	2.00000	0.102733	0.0513665	+	0.998680i	$0.483642\pi$
$380$	0	0
$381$	0	0
$382$	−4.00000	−0.204658
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	− 14.1421i	− 0.719816i
$387$	0	0
$388$	0	0
$389$	28.2843i	1.43407i	0.697037	+	0.717035i	$0.254501\pi$
$389$	28.2843i	1.43407i	−0.697037	+	0.717035i	$0.745499\pi$
$390$	0	0
$391$	−21.1660	−1.07041
$392$	− 19.7990i	− 1.00000i
$393$	0	0
$394$	8.00000	0.403034
$395$	0	0
$396$	0	0
$397$	37.0405	1.85901	0.929505	−	0.368809i	$-0.120234\pi$
$397$	37.0405	1.85901	0.929505	+	0.368809i	$0.120234\pi$
$398$	− 37.4166i	− 1.87552i
$399$	0	0
$400$	−20.0000	−1.00000
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	28.0000	1.39478
$404$	0	0
$405$	0	0
$406$	21.1660	1.05045
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	−10.5830	−0.521387
$413$	− 39.5980i	− 1.94849i
$414$	0	0
$415$	0	0
$416$	29.9333i	1.46760i
$417$	0	0
$418$	0	0
$419$	− 14.9666i	− 0.731168i	−0.930778	−	0.365584i	$-0.880869\pi$
$419$	− 14.9666i	− 0.731168i	0.930778	−	0.365584i	$-0.119131\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	36.7696i	1.78991i
$423$	0	0
$424$	32.0000	1.55406
$425$	− 37.4166i	− 1.81497i
$426$	0	0
$427$	14.0000	0.677507
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 31.1127i	− 1.49865i	−0.662205	−	0.749323i	$-0.730379\pi$
$431$	− 31.1127i	− 1.49865i	0.662205	−	0.749323i	$-0.269621\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	19.7990i	0.950382i
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	37.0405	1.76785	0.883924	−	0.467631i	$-0.154892\pi$
$439$	37.0405	1.76785	0.883924	+	0.467631i	$0.154892\pi$
$440$	0	0
$441$	0	0
$442$	−56.0000	−2.66365
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	0	0
$446$	7.48331i	0.354345i
$447$	0	0
$448$	−21.1660	−1.00000
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	−42.3320	−1.98674
$455$	0	0
$456$	0	0
$457$	−34.0000	−1.59045	−0.795226	−	0.606313i	$-0.792648\pi$
$457$	−34.0000	−1.59045	−0.795226	+	0.606313i	$0.792648\pi$
$458$	− 37.4166i	− 1.74836i
$459$	0	0
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	− 22.6274i	− 1.05045i
$465$	0	0
$466$	0	0
$467$	29.9333i	1.38515i	0.721348	+	0.692573i	$0.243523\pi$
$467$	29.9333i	1.38515i	−0.721348	+	0.692573i	$0.756477\pi$
$468$	0	0
$469$	−26.4575	−1.22169
$470$	0	0
$471$	0	0
$472$	−42.3320	−1.94849
$473$	0	0
$474$	0	0
$475$	0	0
$476$	− 39.5980i	− 1.81497i
$477$	0	0
$478$	20.0000	0.914779
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	22.0000	1.00000
$485$	0	0
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	− 14.9666i	− 0.677507i
$489$	0	0
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	42.3320	1.90654
$494$	0	0
$495$	0	0
$496$	21.1660	0.950382
$497$	− 37.4166i	− 1.67836i
$498$	0	0
$499$	−22.0000	−0.984855	−0.492428	−	0.870353i	$-0.663890\pi$
$499$	−22.0000	−0.984855	−0.492428	+	0.870353i	$0.663890\pi$
$500$	0	0
$501$	0	0
$502$	−42.3320	−1.88937
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	22.6274i	1.00000i
$513$	0	0
$514$	−10.5830	−0.466796
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	7.48331i	0.327850i	0.986473	+	0.163925i	$0.0524155\pi$
$521$	7.48331i	0.327850i	−0.986473	+	0.163925i	$0.947584\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	29.9333i	1.30764i
$525$	0	0
$526$	44.0000	1.91849
$527$	39.5980i	1.72492i
$528$	0	0
$529$	15.0000	0.652174
$530$	0	0
$531$	0	0
$532$	0	0
$533$	39.5980i	1.71518i
$534$	0	0
$535$	0	0
$536$	28.2843i	1.22169i
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	− 37.4166i	− 1.60718i
$543$	0	0
$544$	−42.3320	−1.81497
$545$	0	0
$546$	0	0
$547$	26.0000	1.11168	0.555840	−	0.831289i	$-0.312397\pi$
$547$	26.0000	1.11168	0.555840	+	0.831289i	$0.312397\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	11.3137i	0.479377i	0.970850	+	0.239689i	$0.0770453\pi$
$557$	11.3137i	0.479377i	−0.970850	+	0.239689i	$0.922955\pi$
$558$	0	0
$559$	10.5830	0.447613
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 14.9666i	− 0.630768i	−0.948964	−	0.315384i	$-0.897867\pi$
$563$	− 14.9666i	− 0.630768i	0.948964	−	0.315384i	$-0.102133\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	−40.0000	−1.67836
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	−46.0000	−1.92504	−0.962520	−	0.271211i	$-0.912576\pi$
$571$	−46.0000	−1.92504	−0.962520	+	0.271211i	$0.912576\pi$
$572$	0	0
$573$	0	0
$574$	−28.0000	−1.16870
$575$	− 14.1421i	− 0.589768i
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	− 55.1543i	− 2.29412i
$579$	0	0
$580$	0	0
$581$	− 39.5980i	− 1.64280i
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	29.9333i	1.23548i	0.786383	+	0.617739i	$0.211951\pi$
$587$	29.9333i	1.23548i	−0.786383	+	0.617739i	$0.788049\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 37.4166i	− 1.53651i	−0.640141	−	0.768257i	$-0.721124\pi$
$593$	− 37.4166i	− 1.53651i	0.640141	−	0.768257i	$-0.278876\pi$
$594$	0	0
$595$	0	0
$596$	45.2548i	1.85371i
$597$	0	0
$598$	−21.1660	−0.865543
$599$	− 48.0833i	− 1.96463i	−0.187239	−	0.982314i	$-0.559954\pi$
$599$	− 48.0833i	− 1.96463i	0.187239	−	0.982314i	$-0.440046\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	7.48331i	0.304997i
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	5.29150	0.214775	0.107388	−	0.994217i	$-0.465751\pi$
$607$	5.29150	0.214775	0.107388	+	0.994217i	$0.465751\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	19.7990i	0.793230i
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	−10.5830	−0.422308
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	32.0000	1.27088
$635$	0	0
$636$	0	0
$637$	37.0405	1.46760
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	− 14.9666i	− 0.589768i
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	0	0
$650$	− 37.4166i	− 1.46760i
$651$	0	0
$652$	44.0000	1.72317
$653$	− 22.6274i	− 0.885479i	−0.896650	−	0.442740i	$-0.854007\pi$
$653$	− 22.6274i	− 0.885479i	0.896650	−	0.442740i	$-0.145993\pi$
$654$	0	0
$655$	0	0
$656$	29.9333i	1.16870i
$657$	0	0
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	−26.4575	−1.02908	−0.514539	−	0.857467i	$-0.672037\pi$
$661$	−26.4575	−1.02908	−0.514539	+	0.857467i	$0.672037\pi$
$662$	− 48.0833i	− 1.86881i
$663$	0	0
$664$	−42.3320	−1.64280
$665$	0	0
$666$	0	0
$667$	16.0000	0.619522
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	2.00000	0.0770943	0.0385472	−	0.999257i	$-0.487727\pi$
$673$	2.00000	0.0770943	0.0385472	+	0.999257i	$0.487727\pi$
$674$	36.7696i	1.41631i
$675$	0	0
$676$	−30.0000	−1.15385
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	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	$0.5\pi$
$684$	0	0
$685$	0	0
$686$	26.1916i	1.00000i
$687$	0	0
$688$	8.00000	0.304997
$689$	59.8665i	2.28073i
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−56.0000	−2.12115
$698$	52.3832i	1.98273i
$699$	0	0
$700$	26.4575	1.00000
$701$	45.2548i	1.70925i	0.519244	+	0.854626i	$0.326213\pi$
$701$	45.2548i	1.70925i	−0.519244	+	0.854626i	$0.673787\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	52.9150	1.99148
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	21.1660	0.793230
$713$	14.9666i	0.560505i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	−52.0000	−1.94062
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	14.0000	0.521387
$722$	26.8701i	1.00000i
$723$	0	0
$724$	52.9150	1.96657
$725$	28.2843i	1.05045i
$726$	0	0
$727$	37.0405	1.37376	0.686878	−	0.726772i	$-0.258980\pi$
$727$	37.0405	1.37376	0.686878	+	0.726772i	$0.258980\pi$
$728$	− 39.5980i	− 1.46760i
$729$	0	0
$730$	0	0
$731$	14.9666i	0.553561i
$732$	0	0
$733$	5.29150	0.195446	0.0977231	−	0.995214i	$-0.468844\pi$
$733$	5.29150	0.195446	0.0977231	+	0.995214i	$0.468844\pi$
$734$	52.3832i	1.93350i
$735$	0	0
$736$	−16.0000	−0.589768
$737$	0	0
$738$	0	0
$739$	38.0000	1.39785	0.698926	−	0.715194i	$-0.253662\pi$
$739$	38.0000	1.39785	0.698926	+	0.715194i	$0.253662\pi$
$740$	0	0
$741$	0	0
$742$	−42.3320	−1.55406
$743$	53.7401i	1.97153i	0.168118	+	0.985767i	$0.446231\pi$
$743$	53.7401i	1.97153i	−0.168118	+	0.985767i	$0.553769\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0	0
$754$	42.3320	1.54164
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	2.82843i	0.102733i
$759$	0	0
$760$	0	0
$761$	52.3832i	1.89889i	0.313934	+	0.949445i	$0.398353\pi$
$761$	52.3832i	1.89889i	−0.313934	+	0.949445i	$0.601647\pi$
$762$	0	0
$763$	0	0
$764$	− 5.65685i	− 0.204658i
$765$	0	0
$766$	0	0
$767$	− 79.1960i	− 2.85960i
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	20.0000	0.719816
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	−26.4575	−0.950382
$776$	0	0
$777$	0	0
$778$	−40.0000	−1.43407
$779$	0	0
$780$	0	0
$781$	0	0
$782$	− 29.9333i	− 1.07041i
$783$	0	0
$784$	28.0000	1.00000
$785$	0	0
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	11.3137i	0.403034i
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	28.0000	0.994309
$794$	52.3832i	1.85901i
$795$	0	0
$796$	52.9150	1.87552
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	− 28.2843i	− 1.00000i
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	39.5980i	1.39478i
$807$	0	0
$808$	0	0
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	29.9333i	1.05045i
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 56.5685i	− 1.97426i	−0.159933	−	0.987128i	$-0.551128\pi$
$821$	− 56.5685i	− 1.97426i	0.159933	−	0.987128i	$-0.448872\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	− 14.9666i	− 0.521387i
$825$	0	0
$826$	56.0000	1.94849
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	37.0405	1.28647	0.643235	−	0.765669i	$-0.277592\pi$
$829$	37.0405	1.28647	0.643235	+	0.765669i	$0.277592\pi$
$830$	0	0
$831$	0	0
$832$	−42.3320	−1.46760
$833$	52.3832i	1.81497i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	21.1660	0.731168
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−3.00000	−0.103448
$842$	0	0
$843$	0	0
$844$	−52.0000	−1.78991
$845$	0	0
$846$	0	0
$847$	−29.1033	−1.00000
$848$	45.2548i	1.55406i
$849$	0	0
$850$	52.9150	1.81497
$851$	0	0
$852$	0	0
$853$	−58.2065	−1.99295	−0.996477	−	0.0838689i	$-0.973272\pi$
$853$	−58.2065	−1.99295	−0.996477	+	0.0838689i	$0.973272\pi$
$854$	19.7990i	0.677507i
$855$	0	0
$856$	0	0
$857$	− 37.4166i	− 1.27813i	−0.769154	−	0.639063i	$-0.779322\pi$
$857$	− 37.4166i	− 1.27813i	0.769154	−	0.639063i	$-0.220678\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	44.0000	1.49865
$863$	36.7696i	1.25165i	0.779963	+	0.625825i	$0.215238\pi$
$863$	36.7696i	1.25165i	−0.779963	+	0.625825i	$0.784762\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	−28.0000	−0.950382
$869$	0	0
$870$	0	0
$871$	−52.9150	−1.79296
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	52.3832i	1.76785i
$879$	0	0
$880$	0	0
$881$	7.48331i	0.252119i	0.992023	+	0.126060i	$0.0402331\pi$
$881$	7.48331i	0.252119i	−0.992023	+	0.126060i	$0.959767\pi$
$882$	0	0
$883$	−58.0000	−1.95186	−0.975928	−	0.218094i	$-0.930016\pi$
$883$	−58.0000	−1.95186	−0.975928	+	0.218094i	$0.930016\pi$
$884$	− 79.1960i	− 2.66365i
$885$	0	0
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	−10.5830	−0.354345
$893$	0	0
$894$	0	0
$895$	0	0
$896$	− 29.9333i	− 1.00000i
$897$	0	0
$898$	0	0
$899$	− 29.9333i	− 0.998330i
$900$	0	0
$901$	−84.6640	−2.82057
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−46.0000	−1.52740	−0.763702	−	0.645568i	$-0.776621\pi$
$907$	−46.0000	−1.52740	−0.763702	+	0.645568i	$0.776621\pi$
$908$	− 59.8665i	− 1.98674i
$909$	0	0
$910$	0	0
$911$	53.7401i	1.78049i	0.455483	+	0.890245i	$0.349467\pi$
$911$	53.7401i	1.78049i	−0.455483	+	0.890245i	$0.650533\pi$
$912$	0	0
$913$	0	0
$914$	− 48.0833i	− 1.59045i
$915$	0	0
$916$	52.9150	1.74836
$917$	− 39.5980i	− 1.30764i
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 74.8331i	− 2.46316i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	32.0000	1.05045
$929$	52.3832i	1.71864i	0.511441	+	0.859319i	$0.329112\pi$
$929$	52.3832i	1.71864i	−0.511441	+	0.859319i	$0.670888\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	−42.3320	−1.38515
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	− 37.4166i	− 1.22169i
$939$	0	0
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	−21.1660	−0.689260
$944$	− 59.8665i	− 1.94849i
$945$	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
$952$	56.0000	1.81497
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	28.2843i	0.914779i
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−3.00000	−0.0967742
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	31.1127i	1.00000i
$969$	0	0
$970$	0	0
$971$	− 59.8665i	− 1.92121i	−0.277921	−	0.960604i	$-0.589645\pi$
$971$	− 59.8665i	− 1.92121i	0.277921	−	0.960604i	$-0.410355\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	21.1660	0.677507
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	0	0
$986$	59.8665i	1.90654i
$987$	0	0
$988$	0	0
$989$	5.65685i	0.179878i
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	29.9333i	0.950382i
$993$	0	0
$994$	52.9150	1.67836
$995$	0	0
$996$	0	0
$997$	−58.2065	−1.84342	−0.921710	−	0.387881i	$-0.873207\pi$
$997$	−58.2065	−1.84342	−0.921710	+	0.387881i	$0.873207\pi$
$998$	− 31.1127i	− 0.984855i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	504.2.p.c.307.4	yes	4
3.2	odd	2	inner	504.2.p.c.307.2	yes	4
4.3	odd	2		2016.2.p.c.559.2		4
7.6	odd	2	inner	504.2.p.c.307.3	yes	4
8.3	odd	2	inner	504.2.p.c.307.1	✓	4
8.5	even	2		2016.2.p.c.559.4		4
12.11	even	2		2016.2.p.c.559.1		4
21.20	even	2	inner	504.2.p.c.307.1	✓	4
24.5	odd	2		2016.2.p.c.559.3		4
24.11	even	2	inner	504.2.p.c.307.3	yes	4
28.27	even	2		2016.2.p.c.559.3		4
56.13	odd	2		2016.2.p.c.559.1		4
56.27	even	2	inner	504.2.p.c.307.2	yes	4
84.83	odd	2		2016.2.p.c.559.4		4
168.83	odd	2	CM	504.2.p.c.307.4	yes	4
168.125	even	2		2016.2.p.c.559.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.p.c.307.1	✓	4	8.3	odd	2	inner
504.2.p.c.307.1	✓	4	21.20	even	2	inner
504.2.p.c.307.2	yes	4	3.2	odd	2	inner
504.2.p.c.307.2	yes	4	56.27	even	2	inner
504.2.p.c.307.3	yes	4	7.6	odd	2	inner
504.2.p.c.307.3	yes	4	24.11	even	2	inner
504.2.p.c.307.4	yes	4	1.1	even	1	trivial
504.2.p.c.307.4	yes	4	168.83	odd	2	CM
2016.2.p.c.559.1		4	12.11	even	2
2016.2.p.c.559.1		4	56.13	odd	2
2016.2.p.c.559.2		4	4.3	odd	2
2016.2.p.c.559.2		4	168.125	even	2
2016.2.p.c.559.3		4	24.5	odd	2
2016.2.p.c.559.3		4	28.27	even	2
2016.2.p.c.559.4		4	8.5	even	2
2016.2.p.c.559.4		4	84.83	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:	\( N \)	\(=\)	\( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	504.p (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -2.00000 q^{4} +2.64575 q^{7} -2.82843i q^{8} +O(q^{10})\)	\(q+1.41421i q^{2} -2.00000 q^{4} +2.64575 q^{7} -2.82843i q^{8} +5.29150 q^{13} +3.74166i q^{14} +4.00000 q^{16} +7.48331i q^{17} +2.82843i q^{23} -5.00000 q^{25} +7.48331i q^{26} -5.29150 q^{28} -5.65685i q^{29} +5.29150 q^{31} +5.65685i q^{32} -10.5830 q^{34} +7.48331i q^{41} +2.00000 q^{43} -4.00000 q^{46} +7.00000 q^{49} -7.07107i q^{50} -10.5830 q^{52} +11.3137i q^{53} -7.48331i q^{56} +8.00000 q^{58} -14.9666i q^{59} +5.29150 q^{61} +7.48331i q^{62} -8.00000 q^{64} -10.0000 q^{67} -14.9666i q^{68} -14.1421i q^{71} -10.5830 q^{82} -14.9666i q^{83} +2.82843i q^{86} +7.48331i q^{89} +14.0000 q^{91} -5.65685i q^{92} +9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4}+O(q^{10}) \) 4 * q - 8 * q^4	\( 4 q - 8 q^{4} + 16 q^{16} - 20 q^{25} + 8 q^{43} - 16 q^{46} + 28 q^{49} + 32 q^{58} - 32 q^{64} - 40 q^{67} + 56 q^{91}+O(q^{100}) \) 4 * q - 8 * q^4 + 16 * q^16 - 20 * q^25 + 8 * q^43 - 16 * q^46 + 28 * q^49 + 32 * q^58 - 32 * q^64 - 40 * q^67 + 56 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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