LMFDB - Embedded newform 1536.2.c.f.1535.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1536,2,Mod(1535,1536)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1536, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1536.1535");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1536 = 2^{9} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1536.c (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:	$12.2650217505$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			1535.4
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1536.1535
Dual form			1536.2.c.f.1535.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.292893 + 1.70711i) q^{3} +(-2.82843 - 1.00000i) q^{9} +O(q^{10})$	$q+(-0.292893 + 1.70711i) q^{3} +(-2.82843 - 1.00000i) q^{9} +6.24264 q^{11} +6.00000i q^{17} -4.58579i q^{19} +5.00000 q^{25} +(2.53553 - 4.53553i) q^{27} +(-1.82843 + 10.6569i) q^{33} +11.3137i q^{41} -1.07107i q^{43} +7.00000 q^{49} +(-10.2426 - 1.75736i) q^{51} +(7.82843 + 1.34315i) q^{57} +14.2426 q^{59} +15.8995i q^{67} -16.9706 q^{73} +(-1.46447 + 8.53553i) q^{75} +(7.00000 + 5.65685i) q^{81} -10.7279 q^{83} +18.0000i q^{89} -16.9706 q^{97} +(-17.6569 - 6.24264i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3}+O(q^{10}) $ 4 * q - 4 * q^3	$ 4 q - 4 q^{3} + 8 q^{11} + 20 q^{25} - 4 q^{27} + 4 q^{33} + 28 q^{49} - 24 q^{51} + 20 q^{57} + 40 q^{59} - 20 q^{75} + 28 q^{81} + 8 q^{83} - 48 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + 8 * q^11 + 20 * q^25 - 4 * q^27 + 4 * q^33 + 28 * q^49 - 24 * q^51 + 20 * q^57 + 40 * q^59 - 20 * q^75 + 28 * q^81 + 8 * q^83 - 48 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1536\mathbb{Z}\right)^\times$.

$n$	$511$	$517$	$1025$
$\chi(n)$	$-1$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$	−0.292893	+	1.70711i	−0.169102	+	0.985599i
$3$	−0.292893	+	1.70711i	−0.169102	+	0.985599i
$4$		0			0
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$	−2.82843	−	1.00000i	−0.942809	−	0.333333i
$10$		0			0
$11$	6.24264			1.88223			0.941113	−	0.338091i	$-0.109781\pi$
$11$	6.24264			1.88223			0.941113	+	0.338091i	$0.109781\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			6.00000i			1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$			6.00000i			1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$		0			0
$19$		−	4.58579i		−	1.05205i	−0.850469	−	0.526026i	$-0.823682\pi$
$19$		−	4.58579i		−	1.05205i	0.850469	−	0.526026i	$-0.176318\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$	5.00000			1.00000
$26$		0			0
$27$	2.53553	−	4.53553i	0.487964	−	0.872864i
$28$		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			$\pi$
$32$		0			0
$33$	−1.82843	+	10.6569i	−0.318288	+	1.85512i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			11.3137i			1.76690i	0.468521	+	0.883452i	$0.344787\pi$
$41$			11.3137i			1.76690i	−0.468521	+	0.883452i	$0.655213\pi$
$42$		0			0
$43$		−	1.07107i		−	0.163336i	−0.996660	−	0.0816682i	$-0.973975\pi$
$43$		−	1.07107i		−	0.163336i	0.996660	−	0.0816682i	$-0.0260248\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$	−10.2426	−	1.75736i	−1.43426	−	0.246080i
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	7.82843	+	1.34315i	1.03690	+	0.177904i
$58$		0			0
$59$	14.2426			1.85423			0.927117	−	0.374772i	$-0.122279\pi$
$59$	14.2426			1.85423			0.927117	+	0.374772i	$0.122279\pi$
$60$		0			0
$61$		0			0			−	1.00000i	$-0.5\pi$
$61$		0			0				1.00000i	$0.5\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$			15.8995i			1.94243i	0.238200	+	0.971216i	$0.423443\pi$
$67$			15.8995i			1.94243i	−0.238200	+	0.971216i	$0.576557\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	−16.9706			−1.98625			−0.993127	−	0.117041i	$-0.962659\pi$
$73$	−16.9706			−1.98625			−0.993127	+	0.117041i	$0.962659\pi$
$74$		0			0
$75$	−1.46447	+	8.53553i	−0.169102	+	0.985599i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	7.00000	+	5.65685i	0.777778	+	0.628539i
$82$		0			0
$83$	−10.7279			−1.17754			−0.588771	−	0.808300i	$-0.700388\pi$
$83$	−10.7279			−1.17754			−0.588771	+	0.808300i	$0.700388\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$			18.0000i			1.90800i	0.299813	+	0.953998i	$0.403076\pi$
$89$			18.0000i			1.90800i	−0.299813	+	0.953998i	$0.596924\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−16.9706			−1.72310			−0.861550	−	0.507673i	$-0.830506\pi$
$97$	−16.9706			−1.72310			−0.861550	+	0.507673i	$0.830506\pi$
$98$		0			0
$99$	−17.6569	−	6.24264i	−1.77458	−	0.627409i
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	9.75736			0.943280			0.471640	−	0.881791i	$-0.343662\pi$
$107$	9.75736			0.943280			0.471640	+	0.881791i	$0.343662\pi$
$108$		0			0
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	11.3137i		−	1.06430i	−0.846649	−	0.532152i	$-0.821383\pi$
$113$		−	11.3137i		−	1.06430i	0.846649	−	0.532152i	$-0.178617\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	27.9706			2.54278
$122$		0			0
$123$	−19.3137	−	3.31371i	−1.74146	−	0.298787i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		0			0
$129$	1.82843	+	0.313708i	0.160984	+	0.0276205i
$130$		0			0
$131$	−2.72792			−0.238340			−0.119170	−	0.992874i	$-0.538023\pi$
$131$	−2.72792			−0.238340			−0.119170	+	0.992874i	$0.538023\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$			22.6274i			1.93319i	0.256307	+	0.966595i	$0.417494\pi$
$137$			22.6274i			1.93319i	−0.256307	+	0.966595i	$0.582506\pi$
$138$		0			0
$139$		−	21.5563i		−	1.82839i	−0.405279	−	0.914193i	$-0.632826\pi$
$139$		−	21.5563i		−	1.82839i	0.405279	−	0.914193i	$-0.367174\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−2.05025	+	11.9497i	−0.169102	+	0.985599i
$148$		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			$\pi$
$152$		0			0
$153$	6.00000	−	16.9706i	0.485071	−	1.37199i
$154$		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
$162$		0			0
$163$		−	19.4142i		−	1.52064i	−0.649550	−	0.760319i	$-0.725042\pi$
$163$		−	19.4142i		−	1.52064i	0.649550	−	0.760319i	$-0.274958\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	−13.0000			−1.00000
$170$		0			0
$171$	−4.58579	+	12.9706i	−0.350684	+	0.991884i
$172$		0			0
$173$		0			0		1.00000			$0$
$173$		0			0		−1.00000			$\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−4.17157	+	24.3137i	−0.313555	+	1.82753i
$178$		0			0
$179$	26.7279			1.99774			0.998869	−	0.0475398i	$-0.0151381\pi$
$179$	26.7279			1.99774			0.998869	+	0.0475398i	$0.0151381\pi$
$180$		0			0
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$			37.4558i			2.73904i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$	16.9706			1.22157			0.610784	−	0.791797i	$-0.290854\pi$
$193$	16.9706			1.22157			0.610784	+	0.791797i	$0.290854\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$	−27.1421	−	4.65685i	−1.91446	−	0.328469i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	28.6274i		−	1.98020i
$210$		0			0
$211$			8.10051i			0.557662i	0.960340	+	0.278831i	$0.0899469\pi$
$211$			8.10051i			0.557662i	−0.960340	+	0.278831i	$0.910053\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	4.97056	−	28.9706i	0.335880	−	1.95765i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	−14.1421	−	5.00000i	−0.942809	−	0.333333i
$226$		0			0
$227$	23.2132			1.54071			0.770357	−	0.637613i	$-0.220078\pi$
$227$	23.2132			1.54071			0.770357	+	0.637613i	$0.220078\pi$
$228$		0			0
$229$		0			0			−	1.00000i	$-0.5\pi$
$229$		0			0				1.00000i	$0.5\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	30.0000i		−	1.96537i	−0.185296	−	0.982683i	$-0.559325\pi$
$233$		−	30.0000i		−	1.96537i	0.185296	−	0.982683i	$-0.440675\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	16.9706			1.09317			0.546585	−	0.837404i	$-0.315928\pi$
$241$	16.9706			1.09317			0.546585	+	0.837404i	$0.315928\pi$
$242$		0			0
$243$	−11.7071	+	10.2929i	−0.751011	+	0.660289i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	3.14214	−	18.3137i	0.199125	−	1.16058i
$250$		0			0
$251$	17.7574			1.12083			0.560417	−	0.828210i	$-0.310641\pi$
$251$	17.7574			1.12083			0.560417	+	0.828210i	$0.310641\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	11.3137i		−	0.705730i	−0.935674	−	0.352865i	$-0.885208\pi$
$257$		−	11.3137i		−	0.705730i	0.935674	−	0.352865i	$-0.114792\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−30.7279	−	5.27208i	−1.88052	−	0.322646i
$268$		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
$275$	31.2132			1.88223
$276$		0			0
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	18.0000i		−	1.07379i	−0.843649	−	0.536895i	$-0.819597\pi$
$281$		−	18.0000i		−	1.07379i	0.843649	−	0.536895i	$-0.180403\pi$
$282$		0			0
$283$		−	2.44365i		−	0.145260i	−0.997359	−	0.0726300i	$-0.976861\pi$
$283$		−	2.44365i		−	0.145260i	0.997359	−	0.0726300i	$-0.0231392\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−19.0000			−1.11765
$290$		0			0
$291$	4.97056	−	28.9706i	0.291380	−	1.69828i
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	15.8284	−	28.3137i	0.918458	−	1.64293i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	18.0416i		−	1.02969i	−0.857283	−	0.514845i	$-0.827849\pi$
$307$		−	18.0416i		−	1.02969i	0.857283	−	0.514845i	$-0.172151\pi$
$308$		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
$313$	−10.0000			−0.565233			−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000			−0.565233			−0.282617	+	0.959233i	$0.591202\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		1.00000			$0$
$317$		0			0		−1.00000			$\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−2.85786	+	16.6569i	−0.159510	+	0.929695i
$322$		0			0
$323$	27.5147			1.53096
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		−	36.3848i		−	1.99989i	−0.0105746	−	0.999944i	$-0.503366\pi$
$331$		−	36.3848i		−	1.99989i	0.0105746	−	0.999944i	$-0.496634\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−14.0000			−0.762629			−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000			−0.762629			−0.381314	+	0.924445i	$0.624528\pi$
$338$		0			0
$339$	19.3137	+	3.31371i	1.04898	+	0.179976i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−30.2426			−1.62351			−0.811755	−	0.583998i	$-0.801488\pi$
$347$	−30.2426			−1.62351			−0.811755	+	0.583998i	$0.801488\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	22.6274i		−	1.20434i	−0.798369	−	0.602168i	$-0.794304\pi$
$353$		−	22.6274i		−	1.20434i	0.798369	−	0.602168i	$-0.205696\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−2.02944			−0.106812
$362$		0			0
$363$	−8.19239	+	47.7487i	−0.429989	+	2.50616i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		0			0
$369$	11.3137	−	32.0000i	0.588968	−	1.66585i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$			32.8701i			1.68842i	0.536011	+	0.844211i	$0.319930\pi$
$379$			32.8701i			1.68842i	−0.536011	+	0.844211i	$0.680070\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−1.07107	+	3.02944i	−0.0544454	+	0.153995i
$388$		0			0
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	0.798990	−	4.65685i	0.0403037	−	0.234907i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0			−	1.00000i	$-0.5\pi$
$397$		0			0				1.00000i	$0.5\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	6.00000i		−	0.299626i	−0.988714	−	0.149813i	$-0.952133\pi$
$401$		−	6.00000i		−	0.299626i	0.988714	−	0.149813i	$-0.0478671\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−22.0000			−1.08783			−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000			−1.08783			−0.543915	+	0.839140i	$0.683059\pi$
$410$		0			0
$411$	−38.6274	−	6.62742i	−1.90535	−	0.326906i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	36.7990	+	6.31371i	1.80205	+	0.309184i
$418$		0			0
$419$	−13.2721			−0.648383			−0.324192	−	0.945991i	$-0.605092\pi$
$419$	−13.2721			−0.648383			−0.324192	+	0.945991i	$0.605092\pi$
$420$		0			0
$421$		0			0			−	1.00000i	$-0.5\pi$
$421$		0			0				1.00000i	$0.5\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			30.0000i			1.45521i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$	−16.9706			−0.815553			−0.407777	−	0.913082i	$-0.633696\pi$
$433$	−16.9706			−0.815553			−0.407777	+	0.913082i	$0.633696\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	−19.7990	−	7.00000i	−0.942809	−	0.333333i
$442$		0			0
$443$	−27.6985			−1.31599			−0.657997	−	0.753020i	$-0.728596\pi$
$443$	−27.6985			−1.31599			−0.657997	+	0.753020i	$0.728596\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	42.0000i		−	1.98210i	−0.133482	−	0.991051i	$-0.542616\pi$
$449$		−	42.0000i		−	1.98210i	0.133482	−	0.991051i	$-0.457384\pi$
$450$		0			0
$451$			70.6274i			3.32572i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	26.0000			1.21623			0.608114	−	0.793849i	$-0.291926\pi$
$457$	26.0000			1.21623			0.608114	+	0.793849i	$0.291926\pi$
$458$		0			0
$459$	27.2132	+	15.2132i	1.27020	+	0.710092i
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	0.786797			0.0364086			0.0182043	−	0.999834i	$-0.494205\pi$
$467$	0.786797			0.0364086			0.0182043	+	0.999834i	$0.494205\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	6.68629i		−	0.307436i
$474$		0			0
$475$		−	22.9289i		−	1.05205i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$	33.1421	+	5.68629i	1.49874	+	0.257143i
$490$		0			0
$491$	−19.6985			−0.888980			−0.444490	−	0.895784i	$-0.646615\pi$
$491$	−19.6985			−0.888980			−0.444490	+	0.895784i	$0.646615\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		−	39.8995i		−	1.78615i	−0.449911	−	0.893073i	$-0.648544\pi$
$499$		−	39.8995i		−	1.78615i	0.449911	−	0.893073i	$-0.351456\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	3.80761	−	22.1924i	0.169102	−	0.985599i
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−20.7990	−	11.6274i	−0.918298	−	0.513363i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	45.2548i		−	1.98265i	−0.131432	−	0.991325i	$-0.541958\pi$
$521$		−	45.2548i		−	1.98265i	0.131432	−	0.991325i	$-0.458042\pi$
$522$		0			0
$523$		−	8.87006i		−	0.387861i	−0.981015	−	0.193930i	$-0.937876\pi$
$523$		−	8.87006i		−	0.387861i	0.981015	−	0.193930i	$-0.0621236\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−23.0000			−1.00000
$530$		0			0
$531$	−40.2843	−	14.2426i	−1.74819	−	0.618078i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−7.82843	+	45.6274i	−0.337822	+	1.96897i
$538$		0			0
$539$	43.6985			1.88223
$540$		0			0
$541$		0			0			−	1.00000i	$-0.5\pi$
$541$		0			0				1.00000i	$0.5\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	38.5269i		−	1.64729i	−0.567104	−	0.823646i	$-0.691936\pi$
$547$		−	38.5269i		−	1.64729i	0.567104	−	0.823646i	$-0.308064\pi$
$548$		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
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−63.9411	−	10.9706i	−2.69960	−	0.463178i
$562$		0			0
$563$	−47.2132			−1.98980			−0.994900	−	0.100870i	$-0.967837\pi$
$563$	−47.2132			−1.98980			−0.994900	+	0.100870i	$0.967837\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$			22.6274i			0.948591i	0.880366	+	0.474295i	$0.157297\pi$
$569$			22.6274i			0.948591i	−0.880366	+	0.474295i	$0.842703\pi$
$570$		0			0
$571$			45.5563i			1.90647i	0.302224	+	0.953237i	$0.402271\pi$
$571$			45.5563i			1.90647i	−0.302224	+	0.953237i	$0.597729\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	34.0000			1.41544			0.707719	−	0.706494i	$-0.249724\pi$
$577$	34.0000			1.41544			0.707719	+	0.706494i	$0.249724\pi$
$578$		0			0
$579$	−4.97056	+	28.9706i	−0.206570	+	1.20398i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−38.2426			−1.57844			−0.789221	−	0.614109i	$-0.789516\pi$
$587$	−38.2426			−1.57844			−0.789221	+	0.614109i	$0.789516\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$			45.2548i			1.85839i	0.369586	+	0.929197i	$0.379500\pi$
$593$			45.2548i			1.85839i	−0.369586	+	0.929197i	$0.620500\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	−16.9706			−0.692244			−0.346122	−	0.938190i	$-0.612502\pi$
$601$	−16.9706			−0.692244			−0.346122	+	0.938190i	$0.612502\pi$
$602$		0			0
$603$	15.8995	−	44.9706i	0.647477	−	1.83134i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			30.0000i			1.20775i	0.797077	+	0.603877i	$0.206378\pi$
$617$			30.0000i			1.20775i	−0.797077	+	0.603877i	$0.793622\pi$
$618$		0			0
$619$			11.6152i			0.466855i	0.972374	+	0.233428i	$0.0749942\pi$
$619$			11.6152i			0.466855i	−0.972374	+	0.233428i	$0.925006\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$	48.8701	+	8.38478i	1.95168	+	0.334856i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$	−13.8284	−	2.37258i	−0.549631	−	0.0943017i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$			42.0000i			1.65890i	0.558581	+	0.829450i	$0.311346\pi$
$641$			42.0000i			1.65890i	−0.558581	+	0.829450i	$0.688654\pi$
$642$		0			0
$643$			29.3553i			1.15766i	0.815448	+	0.578831i	$0.196491\pi$
$643$			29.3553i			1.15766i	−0.815448	+	0.578831i	$0.803509\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$	88.9117			3.49009
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	48.0000	+	16.9706i	1.87266	+	0.662085i
$658$		0			0
$659$	−21.2721			−0.828643			−0.414321	−	0.910131i	$-0.635981\pi$
$659$	−21.2721			−0.828643			−0.414321	+	0.910131i	$0.635981\pi$
$660$		0			0
$661$		0			0			−	1.00000i	$-0.5\pi$
$661$		0			0				1.00000i	$0.5\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	50.9117			1.96250			0.981251	−	0.192736i	$-0.0617360\pi$
$673$	50.9117			1.96250			0.981251	+	0.192736i	$0.0617360\pi$
$674$		0			0
$675$	12.6777	−	22.6777i	0.487964	−	0.872864i
$676$		0			0
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−6.79899	+	39.6274i	−0.260538	+	1.51853i
$682$		0			0
$683$	51.6985			1.97819			0.989094	−	0.147287i	$-0.0470541\pi$
$683$	51.6985			1.97819			0.989094	+	0.147287i	$0.0470541\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$			14.5269i			0.552630i	0.961067	+	0.276315i	$0.0891133\pi$
$691$			14.5269i			0.552630i	−0.961067	+	0.276315i	$0.910887\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−67.8823			−2.57122
$698$		0			0
$699$	51.2132	+	8.78680i	1.93706	+	0.332347i
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0			−	1.00000i	$-0.5\pi$
$709$		0			0				1.00000i	$0.5\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−4.97056	+	28.9706i	−0.184857	+	1.07743i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	−14.1421	−	23.0000i	−0.523783	−	0.851852i
$730$		0			0
$731$	6.42641			0.237689
$732$		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
$737$			99.2548i			3.65610i
$738$		0			0
$739$		−	5.95837i		−	0.219182i	−0.993977	−	0.109591i	$-0.965046\pi$
$739$		−	5.95837i		−	0.219182i	0.993977	−	0.109591i	$-0.0349541\pi$
$740$		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
$747$	30.3431	+	10.7279i	1.11020	+	0.392514i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$	−5.20101	+	30.3137i	−0.189535	+	1.10469i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			11.3137i			0.410122i	0.978749	+	0.205061i	$0.0657392\pi$
$761$			11.3137i			0.410122i	−0.978749	+	0.205061i	$0.934261\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−50.9117			−1.83592			−0.917961	−	0.396670i	$-0.870166\pi$
$769$	−50.9117			−1.83592			−0.917961	+	0.396670i	$0.870166\pi$
$770$		0			0
$771$	19.3137	+	3.31371i	0.695566	+	0.119340i
$772$		0			0
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	51.8823			1.85887
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	53.3553i		−	1.90191i	−0.309326	−	0.950956i	$-0.600103\pi$
$787$		−	53.3553i		−	1.90191i	0.309326	−	0.950956i	$-0.399897\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	18.0000	−	50.9117i	0.635999	−	1.79888i
$802$		0			0
$803$	−105.941			−3.73858
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	56.5685i		−	1.98884i	−0.105474	−	0.994422i	$-0.533636\pi$
$809$		−	56.5685i		−	1.98884i	0.105474	−	0.994422i	$-0.466364\pi$
$810$		0			0
$811$		−	56.8701i		−	1.99698i	−0.0549536	−	0.998489i	$-0.517501\pi$
$811$		−	56.8701i		−	1.99698i	0.0549536	−	0.998489i	$-0.482499\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−4.91169			−0.171838
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$		0			0		1.00000			$0$
$823$		0			0		−1.00000			$\pi$
$824$		0			0
$825$	−9.14214	+	53.2843i	−0.318288	+	1.85512i
$826$		0			0
$827$	−24.1838			−0.840952			−0.420476	−	0.907304i	$-0.638137\pi$
$827$	−24.1838			−0.840952			−0.420476	+	0.907304i	$0.638137\pi$
$828$		0			0
$829$		0			0			−	1.00000i	$-0.5\pi$
$829$		0			0				1.00000i	$0.5\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			42.0000i			1.45521i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$		0			0
$843$	30.7279	+	5.27208i	1.05833	+	0.181580i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	4.17157	+	0.715729i	0.143168	+	0.0245637i
$850$		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
$856$		0			0
$857$			22.6274i			0.772938i	0.922302	+	0.386469i	$0.126305\pi$
$857$			22.6274i			0.772938i	−0.922302	+	0.386469i	$0.873695\pi$
$858$		0			0
$859$		−	35.0122i		−	1.19460i	−0.802018	−	0.597300i	$-0.796240\pi$
$859$		−	35.0122i		−	1.19460i	0.802018	−	0.597300i	$-0.203760\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	5.56497	−	32.4350i	0.188996	−	1.10155i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	48.0000	+	16.9706i	1.62455	+	0.574367i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			56.5685i			1.90584i	0.303218	+	0.952921i	$0.401939\pi$
$881$			56.5685i			1.90584i	−0.303218	+	0.952921i	$0.598061\pi$
$882$		0			0
$883$			43.4142i			1.46100i	0.682910	+	0.730502i	$0.260714\pi$
$883$			43.4142i			1.46100i	−0.682910	+	0.730502i	$0.739286\pi$
$884$		0			0
$885$		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
$891$	43.6985	+	35.3137i	1.46395	+	1.18305i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	49.0711i		−	1.62938i	−0.579898	−	0.814689i	$-0.696908\pi$
$907$		−	49.0711i		−	1.62938i	0.579898	−	0.814689i	$-0.303092\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$	−66.9706			−2.21640
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	30.7990	+	5.28427i	1.01486	+	0.174123i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	54.0000i		−	1.77168i	−0.463988	−	0.885841i	$-0.653582\pi$
$929$		−	54.0000i		−	1.77168i	0.463988	−	0.885841i	$-0.346418\pi$
$930$		0			0
$931$		−	32.1005i		−	1.05205i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−50.9117			−1.66321			−0.831606	−	0.555366i	$-0.812578\pi$
$937$	−50.9117			−1.66321			−0.831606	+	0.555366i	$0.812578\pi$
$938$		0			0
$939$	2.92893	−	17.0711i	0.0955821	−	0.557093i
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−16.7868			−0.545498			−0.272749	−	0.962085i	$-0.587933\pi$
$947$	−16.7868			−0.545498			−0.272749	+	0.962085i	$0.587933\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	45.2548i		−	1.46595i	−0.680257	−	0.732974i	$-0.738132\pi$
$953$		−	45.2548i		−	1.46595i	0.680257	−	0.732974i	$-0.261868\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$	−27.5980	−	9.75736i	−0.889332	−	0.314427i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		0			0
$969$	−8.05887	+	46.9706i	−0.258888	+	1.50891i
$970$		0			0
$971$	−16.1838			−0.519362			−0.259681	−	0.965694i	$-0.583617\pi$
$971$	−16.1838			−0.519362			−0.259681	+	0.965694i	$0.583617\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$			6.00000i			0.191957i	0.995383	+	0.0959785i	$0.0305980\pi$
$977$			6.00000i			0.191957i	−0.995383	+	0.0959785i	$0.969402\pi$
$978$		0			0
$979$			112.368i			3.59128i
$980$		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
$986$		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
$993$	62.1127	+	10.6569i	1.97109	+	0.338185i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0			−	1.00000i	$-0.5\pi$
$997$		0			0				1.00000i	$0.5\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1536.2.c.f.1535.4	yes	4
3.2	odd	2		1536.2.c.j.1535.2	yes	4
4.3	odd	2		1536.2.c.j.1535.1	yes	4
8.3	odd	2	CM	1536.2.c.f.1535.4	yes	4
8.5	even	2		1536.2.c.j.1535.1	yes	4
12.11	even	2	inner	1536.2.c.f.1535.3	✓	4
16.3	odd	4		1536.2.f.j.767.4		4
16.5	even	4		1536.2.f.j.767.4		4
16.11	odd	4		1536.2.f.a.767.1		4
16.13	even	4		1536.2.f.a.767.1		4
24.5	odd	2	inner	1536.2.c.f.1535.3	✓	4
24.11	even	2		1536.2.c.j.1535.2	yes	4
48.5	odd	4		1536.2.f.j.767.3		4
48.11	even	4		1536.2.f.a.767.2		4
48.29	odd	4		1536.2.f.a.767.2		4
48.35	even	4		1536.2.f.j.767.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1536.2.c.f.1535.3	✓	4	12.11	even	2	inner
1536.2.c.f.1535.3	✓	4	24.5	odd	2	inner
1536.2.c.f.1535.4	yes	4	1.1	even	1	trivial
1536.2.c.f.1535.4	yes	4	8.3	odd	2	CM
1536.2.c.j.1535.1	yes	4	4.3	odd	2
1536.2.c.j.1535.1	yes	4	8.5	even	2
1536.2.c.j.1535.2	yes	4	3.2	odd	2
1536.2.c.j.1535.2	yes	4	24.11	even	2
1536.2.f.a.767.1		4	16.11	odd	4
1536.2.f.a.767.1		4	16.13	even	4
1536.2.f.a.767.2		4	48.11	even	4
1536.2.f.a.767.2		4	48.29	odd	4
1536.2.f.j.767.3		4	48.5	odd	4
1536.2.f.j.767.3		4	48.35	even	4
1536.2.f.j.767.4		4	16.3	odd	4
1536.2.f.j.767.4		4	16.5	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1536 = 2^{9} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1536.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.292893 + 1.70711i) q^{3} +(-2.82843 - 1.00000i) q^{9} +O(q^{10})\)	\(q+(-0.292893 + 1.70711i) q^{3} +(-2.82843 - 1.00000i) q^{9} +6.24264 q^{11} +6.00000i q^{17} -4.58579i q^{19} +5.00000 q^{25} +(2.53553 - 4.53553i) q^{27} +(-1.82843 + 10.6569i) q^{33} +11.3137i q^{41} -1.07107i q^{43} +7.00000 q^{49} +(-10.2426 - 1.75736i) q^{51} +(7.82843 + 1.34315i) q^{57} +14.2426 q^{59} +15.8995i q^{67} -16.9706 q^{73} +(-1.46447 + 8.53553i) q^{75} +(7.00000 + 5.65685i) q^{81} -10.7279 q^{83} +18.0000i q^{89} -16.9706 q^{97} +(-17.6569 - 6.24264i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3}+O(q^{10}) \) 4 * q - 4 * q^3	\( 4 q - 4 q^{3} + 8 q^{11} + 20 q^{25} - 4 q^{27} + 4 q^{33} + 28 q^{49} - 24 q^{51} + 20 q^{57} + 40 q^{59} - 20 q^{75} + 28 q^{81} + 8 q^{83} - 48 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + 8 * q^11 + 20 * q^25 - 4 * q^27 + 4 * q^33 + 28 * q^49 - 24 * q^51 + 20 * q^57 + 40 * q^59 - 20 * q^75 + 28 * q^81 + 8 * q^83 - 48 * q^99

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