LMFDB - Embedded newform 1024.2.e.a.257.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1024,2,Mod(257,1024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1024, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3]))

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

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

chi := DirichletCharacter("1024.257");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1024 = 2^{10} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1024.e (of order $4$, degree $2$, not 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:	$8.17668116698$
Analytic rank:	$1$
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:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 128)
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			257.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1024.257
Dual form			1024.2.e.a.769.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.00000 - 2.00000i) q^{3} +5.00000i q^{9} +O(q^{10})$	$q+(-2.00000 - 2.00000i) q^{3} +5.00000i q^{9} +(2.00000 - 2.00000i) q^{11} -6.00000 q^{17} +(-6.00000 - 6.00000i) q^{19} +5.00000i q^{25} +(4.00000 - 4.00000i) q^{27} -8.00000 q^{33} +6.00000i q^{41} +(-6.00000 + 6.00000i) q^{43} +7.00000 q^{49} +(12.0000 + 12.0000i) q^{51} +24.0000i q^{57} +(-10.0000 + 10.0000i) q^{59} +(6.00000 + 6.00000i) q^{67} +2.00000i q^{73} +(10.0000 - 10.0000i) q^{75} -1.00000 q^{81} +(2.00000 + 2.00000i) q^{83} -18.0000i q^{89} -10.0000 q^{97} +(10.0000 + 10.0000i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{3}+O(q^{10}) $ 2 * q - 4 * q^3	$ 2 q - 4 q^{3} + 4 q^{11} - 12 q^{17} - 12 q^{19} + 8 q^{27} - 16 q^{33} - 12 q^{43} + 14 q^{49} + 24 q^{51} - 20 q^{59} + 12 q^{67} + 20 q^{75} - 2 q^{81} + 4 q^{83} - 20 q^{97} + 20 q^{99}+O(q^{100}) $ 2 * q - 4 * q^3 + 4 * q^11 - 12 * q^17 - 12 * q^19 + 8 * q^27 - 16 * q^33 - 12 * q^43 + 14 * q^49 + 24 * q^51 - 20 * q^59 + 12 * q^67 + 20 * q^75 - 2 * q^81 + 4 * q^83 - 20 * q^97 + 20 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1024\mathbb{Z}\right)^\times$.

$n$	$5$	$1023$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$

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
$3$	−2.00000	−	2.00000i	−1.15470	−	1.15470i	−0.985599	−	0.169102i	$-0.945913\pi$
$3$	−2.00000	−	2.00000i	−1.15470	−	1.15470i	−0.169102	−	0.985599i	$-0.554087\pi$
$4$		0			0
$5$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$5$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$			5.00000i			1.66667i
$10$		0			0
$11$	2.00000	−	2.00000i	0.603023	−	0.603023i	−0.338091	−	0.941113i	$-0.609781\pi$
$11$	2.00000	−	2.00000i	0.603023	−	0.603023i	0.941113	+	0.338091i	$0.109781\pi$
$12$		0			0
$13$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$13$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−6.00000			−1.45521			−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000			−1.45521			−0.727607	+	0.685994i	$0.759367\pi$
$18$		0			0
$19$	−6.00000	−	6.00000i	−1.37649	−	1.37649i	−0.850469	−	0.526026i	$-0.823682\pi$
$19$	−6.00000	−	6.00000i	−1.37649	−	1.37649i	−0.526026	−	0.850469i	$-0.676318\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$			5.00000i			1.00000i
$26$		0			0
$27$	4.00000	−	4.00000i	0.769800	−	0.769800i
$28$		0			0
$29$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$29$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$		0			0
$33$	−8.00000			−1.39262
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$37$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			6.00000i			0.937043i	0.883452	+	0.468521i	$0.155213\pi$
$41$			6.00000i			0.937043i	−0.883452	+	0.468521i	$0.844787\pi$
$42$		0			0
$43$	−6.00000	+	6.00000i	−0.914991	+	0.914991i	−0.996660	−	0.0816682i	$-0.973975\pi$
$43$	−6.00000	+	6.00000i	−0.914991	+	0.914991i	0.0816682	+	0.996660i	$0.473975\pi$
$44$		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
$51$	12.0000	+	12.0000i	1.68034	+	1.68034i
$52$		0			0
$53$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$53$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$			24.0000i			3.17888i
$58$		0			0
$59$	−10.0000	+	10.0000i	−1.30189	+	1.30189i	−0.374772	+	0.927117i	$0.622279\pi$
$59$	−10.0000	+	10.0000i	−1.30189	+	1.30189i	−0.927117	+	0.374772i	$0.877721\pi$
$60$		0			0
$61$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$61$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	6.00000	+	6.00000i	0.733017	+	0.733017i	0.971216	−	0.238200i	$-0.0765572\pi$
$67$	6.00000	+	6.00000i	0.733017	+	0.733017i	−0.238200	+	0.971216i	$0.576557\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$			2.00000i			0.234082i	0.993127	+	0.117041i	$0.0373409\pi$
$73$			2.00000i			0.234082i	−0.993127	+	0.117041i	$0.962659\pi$
$74$		0			0
$75$	10.0000	−	10.0000i	1.15470	−	1.15470i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	−1.00000			−0.111111
$82$		0			0
$83$	2.00000	+	2.00000i	0.219529	+	0.219529i	0.808300	−	0.588771i	$-0.200388\pi$
$83$	2.00000	+	2.00000i	0.219529	+	0.219529i	−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	$-0.596924\pi$
$89$		−	18.0000i		−	1.90800i	0.299813	−	0.953998i	$-0.403076\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−10.0000			−1.01535			−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000			−1.01535			−0.507673	+	0.861550i	$0.669494\pi$
$98$		0			0
$99$	10.0000	+	10.0000i	1.00504	+	1.00504i
$100$		0			0
$101$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$101$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−14.0000	+	14.0000i	−1.35343	+	1.35343i	−0.471640	+	0.881791i	$0.656338\pi$
$107$	−14.0000	+	14.0000i	−1.35343	+	1.35343i	−0.881791	+	0.471640i	$0.843662\pi$
$108$		0			0
$109$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$109$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−18.0000			−1.69330			−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000			−1.69330			−0.846649	+	0.532152i	$0.821383\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$			3.00000i			0.272727i
$122$		0			0
$123$	12.0000	−	12.0000i	1.08200	−	1.08200i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$		0			0
$129$	24.0000			2.11308
$130$		0			0
$131$	−10.0000	−	10.0000i	−0.873704	−	0.873704i	0.119170	−	0.992874i	$-0.461977\pi$
$131$	−10.0000	−	10.0000i	−0.873704	−	0.873704i	−0.992874	+	0.119170i	$0.961977\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$			6.00000i			0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$			6.00000i			0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$		0			0
$139$	−6.00000	+	6.00000i	−0.508913	+	0.508913i	−0.914193	−	0.405279i	$-0.867174\pi$
$139$	−6.00000	+	6.00000i	−0.508913	+	0.508913i	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
$147$	−14.0000	−	14.0000i	−1.15470	−	1.15470i
$148$		0			0
$149$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$149$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$		−	30.0000i		−	2.42536i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$157$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−18.0000	−	18.0000i	−1.40987	−	1.40987i	−0.760319	−	0.649550i	$-0.774958\pi$
$163$	−18.0000	−	18.0000i	−1.40987	−	1.40987i	−0.649550	−	0.760319i	$-0.725042\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$		−	13.0000i		−	1.00000i
$170$		0			0
$171$	30.0000	−	30.0000i	2.29416	−	2.29416i
$172$		0			0
$173$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$173$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	40.0000			3.00658
$178$		0			0
$179$	−14.0000	−	14.0000i	−1.04641	−	1.04641i	−0.998869	−	0.0475398i	$-0.984862\pi$
$179$	−14.0000	−	14.0000i	−1.04641	−	1.04641i	−0.0475398	−	0.998869i	$-0.515138\pi$
$180$		0			0
$181$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$181$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−12.0000	+	12.0000i	−0.877527	+	0.877527i
$188$		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
$193$	22.0000			1.58359			0.791797	−	0.610784i	$-0.209146\pi$
$193$	22.0000			1.58359			0.791797	+	0.610784i	$0.209146\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$197$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$		−	24.0000i		−	1.69283i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−24.0000			−1.66011
$210$		0			0
$211$	18.0000	+	18.0000i	1.23917	+	1.23917i	0.960340	+	0.278831i	$0.0899469\pi$
$211$	18.0000	+	18.0000i	1.23917	+	1.23917i	0.278831	+	0.960340i	$0.410053\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	4.00000	−	4.00000i	0.270295	−	0.270295i
$220$		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
$225$	−25.0000			−1.66667
$226$		0			0
$227$	−2.00000	−	2.00000i	−0.132745	−	0.132745i	0.637613	−	0.770357i	$-0.279922\pi$
$227$	−2.00000	−	2.00000i	−0.132745	−	0.132745i	−0.770357	+	0.637613i	$0.779922\pi$
$228$		0			0
$229$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$229$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$230$		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
$241$	26.0000			1.67481			0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000			1.67481			0.837404	+	0.546585i	$0.184072\pi$
$242$		0			0
$243$	−10.0000	−	10.0000i	−0.641500	−	0.641500i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		−	8.00000i		−	0.506979i
$250$		0			0
$251$	22.0000	−	22.0000i	1.38863	−	1.38863i	0.560417	−	0.828210i	$-0.310641\pi$
$251$	22.0000	−	22.0000i	1.38863	−	1.38863i	0.828210	−	0.560417i	$-0.189359\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−30.0000			−1.87135			−0.935674	−	0.352865i	$-0.885208\pi$
$257$	−30.0000			−1.87135			−0.935674	+	0.352865i	$0.885208\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−36.0000	+	36.0000i	−2.20316	+	2.20316i
$268$		0			0
$269$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$269$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	10.0000	+	10.0000i	0.603023	+	0.603023i
$276$		0			0
$277$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$277$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$278$		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
$283$	−18.0000	+	18.0000i	−1.06999	+	1.06999i	−0.0726300	+	0.997359i	$0.523139\pi$
$283$	−18.0000	+	18.0000i	−1.06999	+	1.06999i	−0.997359	+	0.0726300i	$0.976861\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	19.0000			1.11765
$290$		0			0
$291$	20.0000	+	20.0000i	1.17242	+	1.17242i
$292$		0			0
$293$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$293$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		−	16.0000i		−	0.928414i
$298$		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
$307$	−6.00000	−	6.00000i	−0.342438	−	0.342438i	0.514845	−	0.857283i	$-0.327849\pi$
$307$	−6.00000	−	6.00000i	−0.342438	−	0.342438i	−0.857283	+	0.514845i	$0.827849\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$			10.0000i			0.565233i	0.959233	+	0.282617i	$0.0912024\pi$
$313$			10.0000i			0.565233i	−0.959233	+	0.282617i	$0.908798\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$317$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	56.0000			3.12562
$322$		0			0
$323$	36.0000	+	36.0000i	2.00309	+	2.00309i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	18.0000	−	18.0000i	0.989369	−	0.989369i	−0.0105746	−	0.999944i	$-0.503366\pi$
$331$	18.0000	−	18.0000i	0.989369	−	0.989369i	0.999944	+	0.0105746i	$0.00336607\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	14.0000			0.762629			0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000			0.762629			0.381314	+	0.924445i	$0.375472\pi$
$338$		0			0
$339$	36.0000	+	36.0000i	1.95525	+	1.95525i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−26.0000	+	26.0000i	−1.39575	+	1.39575i	−0.583998	+	0.811755i	$0.698512\pi$
$347$	−26.0000	+	26.0000i	−1.39575	+	1.39575i	−0.811755	+	0.583998i	$0.801488\pi$
$348$		0			0
$349$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$349$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−30.0000			−1.59674			−0.798369	−	0.602168i	$-0.794304\pi$
$353$	−30.0000			−1.59674			−0.798369	+	0.602168i	$0.794304\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$			53.0000i			2.78947i
$362$		0			0
$363$	6.00000	−	6.00000i	0.314918	−	0.314918i
$364$		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
$369$	−30.0000			−1.56174
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$373$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	6.00000	−	6.00000i	0.308199	−	0.308199i	−0.536011	−	0.844211i	$-0.680070\pi$
$379$	6.00000	−	6.00000i	0.308199	−	0.308199i	0.844211	+	0.536011i	$0.180070\pi$
$380$		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
$387$	−30.0000	−	30.0000i	−1.52499	−	1.52499i
$388$		0			0
$389$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$389$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$			40.0000i			2.01773i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$397$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−6.00000			−0.299626			−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000			−0.299626			−0.149813	+	0.988714i	$0.547867\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	22.0000i		−	1.08783i	−0.839140	−	0.543915i	$-0.816941\pi$
$409$		−	22.0000i		−	1.08783i	0.839140	−	0.543915i	$-0.183059\pi$
$410$		0			0
$411$	12.0000	−	12.0000i	0.591916	−	0.591916i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	24.0000			1.17529
$418$		0			0
$419$	−26.0000	−	26.0000i	−1.27018	−	1.27018i	−0.945991	−	0.324192i	$-0.894908\pi$
$419$	−26.0000	−	26.0000i	−1.27018	−	1.27018i	−0.324192	−	0.945991i	$-0.605092\pi$
$420$		0			0
$421$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$421$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$422$		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
$433$	−38.0000			−1.82616			−0.913082	−	0.407777i	$-0.866304\pi$
$433$	−38.0000			−1.82616			−0.913082	+	0.407777i	$0.866304\pi$
$434$		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
$441$			35.0000i			1.66667i
$442$		0			0
$443$	−2.00000	+	2.00000i	−0.0950229	+	0.0950229i	−0.753020	−	0.657997i	$-0.771404\pi$
$443$	−2.00000	+	2.00000i	−0.0950229	+	0.0950229i	0.657997	+	0.753020i	$0.271404\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−42.0000			−1.98210			−0.991051	−	0.133482i	$-0.957384\pi$
$449$	−42.0000			−1.98210			−0.991051	+	0.133482i	$0.957384\pi$
$450$		0			0
$451$	12.0000	+	12.0000i	0.565058	+	0.565058i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	26.0000i		−	1.21623i	−0.793849	−	0.608114i	$-0.791926\pi$
$457$		−	26.0000i		−	1.21623i	0.793849	−	0.608114i	$-0.208074\pi$
$458$		0			0
$459$	−24.0000	+	24.0000i	−1.12022	+	1.12022i
$460$		0			0
$461$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$461$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−22.0000	−	22.0000i	−1.01804	−	1.01804i	−0.999834	−	0.0182043i	$-0.994205\pi$
$467$	−22.0000	−	22.0000i	−1.01804	−	1.01804i	−0.0182043	−	0.999834i	$-0.505795\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$			24.0000i			1.10352i
$474$		0			0
$475$	30.0000	−	30.0000i	1.37649	−	1.37649i
$476$		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
$489$			72.0000i			3.25595i
$490$		0			0
$491$	10.0000	−	10.0000i	0.451294	−	0.451294i	−0.444490	−	0.895784i	$-0.646615\pi$
$491$	10.0000	−	10.0000i	0.451294	−	0.451294i	0.895784	+	0.444490i	$0.146615\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−30.0000	−	30.0000i	−1.34298	−	1.34298i	−0.893073	−	0.449911i	$-0.851456\pi$
$499$	−30.0000	−	30.0000i	−1.34298	−	1.34298i	−0.449911	−	0.893073i	$-0.648544\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−26.0000	+	26.0000i	−1.15470	+	1.15470i
$508$		0			0
$509$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$509$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−48.0000			−2.11925
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$			6.00000i			0.262865i	0.991325	+	0.131432i	$0.0419576\pi$
$521$			6.00000i			0.262865i	−0.991325	+	0.131432i	$0.958042\pi$
$522$		0			0
$523$	18.0000	−	18.0000i	0.787085	−	0.787085i	−0.193930	−	0.981015i	$-0.562124\pi$
$523$	18.0000	−	18.0000i	0.787085	−	0.787085i	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
$531$	−50.0000	−	50.0000i	−2.16982	−	2.16982i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$			56.0000i			2.41658i
$538$		0			0
$539$	14.0000	−	14.0000i	0.603023	−	0.603023i
$540$		0			0
$541$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$541$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	6.00000	+	6.00000i	0.256541	+	0.256541i	0.823646	−	0.567104i	$-0.191936\pi$
$547$	6.00000	+	6.00000i	0.256541	+	0.256541i	−0.567104	+	0.823646i	$0.691936\pi$
$548$		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		0.707107	−	0.707107i	$-0.250000\pi$
$557$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	48.0000			2.02656
$562$		0			0
$563$	26.0000	+	26.0000i	1.09577	+	1.09577i	0.994900	+	0.100870i	$0.0321625\pi$
$563$	26.0000	+	26.0000i	1.09577	+	1.09577i	0.100870	+	0.994900i	$0.467837\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$			42.0000i			1.76073i	0.474295	+	0.880366i	$0.342703\pi$
$569$			42.0000i			1.76073i	−0.474295	+	0.880366i	$0.657297\pi$
$570$		0			0
$571$	30.0000	−	30.0000i	1.25546	−	1.25546i	0.302224	−	0.953237i	$-0.402271\pi$
$571$	30.0000	−	30.0000i	1.25546	−	1.25546i	0.953237	−	0.302224i	$-0.0977291\pi$
$572$		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
$579$	−44.0000	−	44.0000i	−1.82858	−	1.82858i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	34.0000	−	34.0000i	1.40333	−	1.40333i	0.614109	−	0.789221i	$-0.289516\pi$
$587$	34.0000	−	34.0000i	1.40333	−	1.40333i	0.789221	−	0.614109i	$-0.210484\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−18.0000			−0.739171			−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000			−0.739171			−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		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$			46.0000i			1.87638i	0.346122	+	0.938190i	$0.387498\pi$
$601$			46.0000i			1.87638i	−0.346122	+	0.938190i	$0.612502\pi$
$602$		0			0
$603$	−30.0000	+	30.0000i	−1.22169	+	1.22169i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$613$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	30.0000i		−	1.20775i	−0.797077	−	0.603877i	$-0.793622\pi$
$617$		−	30.0000i		−	1.20775i	0.797077	−	0.603877i	$-0.206378\pi$
$618$		0			0
$619$	−30.0000	+	30.0000i	−1.20580	+	1.20580i	−0.233428	+	0.972374i	$0.574994\pi$
$619$	−30.0000	+	30.0000i	−1.20580	+	1.20580i	−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
$627$	48.0000	+	48.0000i	1.91694	+	1.91694i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		−	72.0000i		−	2.86174i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−42.0000			−1.65890			−0.829450	−	0.558581i	$-0.811346\pi$
$641$	−42.0000			−1.65890			−0.829450	+	0.558581i	$0.811346\pi$
$642$		0			0
$643$	6.00000	+	6.00000i	0.236617	+	0.236617i	0.815448	−	0.578831i	$-0.196491\pi$
$643$	6.00000	+	6.00000i	0.236617	+	0.236617i	−0.578831	+	0.815448i	$0.696491\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$			40.0000i			1.57014i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$653$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−10.0000			−0.390137
$658$		0			0
$659$	34.0000	+	34.0000i	1.32445	+	1.32445i	0.910131	+	0.414321i	$0.135981\pi$
$659$	34.0000	+	34.0000i	1.32445	+	1.32445i	0.414321	+	0.910131i	$0.364019\pi$
$660$		0			0
$661$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$661$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$662$		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
$673$	−10.0000			−0.385472			−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000			−0.385472			−0.192736	+	0.981251i	$0.561736\pi$
$674$		0			0
$675$	20.0000	+	20.0000i	0.769800	+	0.769800i
$676$		0			0
$677$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$677$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$			8.00000i			0.306561i
$682$		0			0
$683$	−22.0000	+	22.0000i	−0.841807	+	0.841807i	−0.989094	−	0.147287i	$-0.952946\pi$
$683$	−22.0000	+	22.0000i	−0.841807	+	0.841807i	0.147287	+	0.989094i	$0.452946\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	18.0000	+	18.0000i	0.684752	+	0.684752i	0.961067	−	0.276315i	$-0.0891133\pi$
$691$	18.0000	+	18.0000i	0.684752	+	0.684752i	−0.276315	+	0.961067i	$0.589113\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	36.0000i		−	1.36360i
$698$		0			0
$699$	−60.0000	+	60.0000i	−2.26941	+	2.26941i
$700$		0			0
$701$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$701$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$709$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$710$		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
$723$	−52.0000	−	52.0000i	−1.93390	−	1.93390i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$			43.0000i			1.59259i
$730$		0			0
$731$	36.0000	−	36.0000i	1.33151	−	1.33151i
$732$		0			0
$733$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$733$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	24.0000			0.884051
$738$		0			0
$739$	30.0000	+	30.0000i	1.10357	+	1.10357i	0.993977	+	0.109591i	$0.0349541\pi$
$739$	30.0000	+	30.0000i	1.10357	+	1.10357i	0.109591	+	0.993977i	$0.465046\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−10.0000	+	10.0000i	−0.365881	+	0.365881i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$		0			0
$753$	−88.0000			−3.20690
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$757$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	54.0000i		−	1.95750i	−0.205061	−	0.978749i	$-0.565739\pi$
$761$		−	54.0000i		−	1.95750i	0.205061	−	0.978749i	$-0.434261\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	22.0000			0.793340			0.396670	−	0.917961i	$-0.370166\pi$
$769$	22.0000			0.793340			0.396670	+	0.917961i	$0.370166\pi$
$770$		0			0
$771$	60.0000	+	60.0000i	2.16085	+	2.16085i
$772$		0			0
$773$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$773$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	36.0000	−	36.0000i	1.28983	−	1.28983i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	18.0000	+	18.0000i	0.641631	+	0.641631i	0.950956	−	0.309326i	$-0.100103\pi$
$787$	18.0000	+	18.0000i	0.641631	+	0.641631i	−0.309326	+	0.950956i	$0.600103\pi$
$788$		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		0.707107	−	0.707107i	$-0.250000\pi$
$797$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	90.0000			3.17999
$802$		0			0
$803$	4.00000	+	4.00000i	0.141157	+	0.141157i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			6.00000i			0.210949i	0.994422	+	0.105474i	$0.0336361\pi$
$809$			6.00000i			0.210949i	−0.994422	+	0.105474i	$0.966364\pi$
$810$		0			0
$811$	−30.0000	+	30.0000i	−1.05344	+	1.05344i	−0.0549536	+	0.998489i	$0.517501\pi$
$811$	−30.0000	+	30.0000i	−1.05344	+	1.05344i	−0.998489	+	0.0549536i	$0.982499\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	72.0000			2.51896
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$821$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$822$		0			0
$823$		0			0		1.00000			$0$
$823$		0			0		−1.00000			$\pi$
$824$		0			0
$825$		−	40.0000i		−	1.39262i
$826$		0			0
$827$	14.0000	−	14.0000i	0.486828	−	0.486828i	−0.420476	−	0.907304i	$-0.638137\pi$
$827$	14.0000	−	14.0000i	0.486828	−	0.486828i	0.907304	+	0.420476i	$0.138137\pi$
$828$		0			0
$829$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$829$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−42.0000			−1.45521
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$		−	29.0000i		−	1.00000i
$842$		0			0
$843$	−36.0000	+	36.0000i	−1.23991	+	1.23991i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	72.0000			2.47103
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$853$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	54.0000i		−	1.84460i	−0.386469	−	0.922302i	$-0.626305\pi$
$857$		−	54.0000i		−	1.84460i	0.386469	−	0.922302i	$-0.373695\pi$
$858$		0			0
$859$	6.00000	−	6.00000i	0.204717	−	0.204717i	−0.597300	−	0.802018i	$-0.703760\pi$
$859$	6.00000	−	6.00000i	0.204717	−	0.204717i	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
$867$	−38.0000	−	38.0000i	−1.29055	−	1.29055i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		−	50.0000i		−	1.69224i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$877$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−18.0000			−0.606435			−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000			−0.606435			−0.303218	+	0.952921i	$0.598061\pi$
$882$		0			0
$883$	42.0000	+	42.0000i	1.41341	+	1.41341i	0.730502	+	0.682910i	$0.239286\pi$
$883$	42.0000	+	42.0000i	1.41341	+	1.41341i	0.682910	+	0.730502i	$0.260714\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−2.00000	+	2.00000i	−0.0670025	+	0.0670025i
$892$		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
$907$	42.0000	−	42.0000i	1.39459	−	1.39459i	0.579898	−	0.814689i	$-0.303092\pi$
$907$	42.0000	−	42.0000i	1.39459	−	1.39459i	0.814689	−	0.579898i	$-0.196908\pi$
$908$		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
$913$	8.00000			0.264761
$914$		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
$921$			24.0000i			0.790827i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	54.0000			1.77168			0.885841	−	0.463988i	$-0.153582\pi$
$929$	54.0000			1.77168			0.885841	+	0.463988i	$0.153582\pi$
$930$		0			0
$931$	−42.0000	−	42.0000i	−1.37649	−	1.37649i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			34.0000i			1.11073i	0.831606	+	0.555366i	$0.187422\pi$
$937$			34.0000i			1.11073i	−0.831606	+	0.555366i	$0.812578\pi$
$938$		0			0
$939$	20.0000	−	20.0000i	0.652675	−	0.652675i
$940$		0			0
$941$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$941$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−38.0000	−	38.0000i	−1.23483	−	1.23483i	−0.962085	−	0.272749i	$-0.912067\pi$
$947$	−38.0000	−	38.0000i	−1.23483	−	1.23483i	−0.272749	−	0.962085i	$-0.587933\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$			42.0000i			1.36051i	0.732974	+	0.680257i	$0.238132\pi$
$953$			42.0000i			1.36051i	−0.732974	+	0.680257i	$0.761868\pi$
$954$		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
$963$	−70.0000	−	70.0000i	−2.25572	−	2.25572i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		0			0
$969$		−	144.000i		−	4.62595i
$970$		0			0
$971$	−22.0000	+	22.0000i	−0.706014	+	0.706014i	−0.965694	−	0.259681i	$-0.916383\pi$
$971$	−22.0000	+	22.0000i	−0.706014	+	0.706014i	0.259681	+	0.965694i	$0.416383\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−6.00000			−0.191957			−0.0959785	−	0.995383i	$-0.530598\pi$
$977$	−6.00000			−0.191957			−0.0959785	+	0.995383i	$0.530598\pi$
$978$		0			0
$979$	−36.0000	−	36.0000i	−1.15056	−	1.15056i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$		0			0
$993$	−72.0000			−2.28485
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$997$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1024.2.e.a.257.1		2
4.3	odd	2		1024.2.e.f.257.1		2
8.3	odd	2	CM	1024.2.e.a.257.1		2
8.5	even	2		1024.2.e.f.257.1		2
16.3	odd	4	inner	1024.2.e.a.769.1		2
16.5	even	4	inner	1024.2.e.a.769.1		2
16.11	odd	4		1024.2.e.f.769.1		2
16.13	even	4		1024.2.e.f.769.1		2
32.3	odd	8		128.2.b.a.65.2	yes	2
32.5	even	8		256.2.a.e.1.2		2
32.11	odd	8		256.2.a.e.1.2		2
32.13	even	8		128.2.b.a.65.2	yes	2
32.19	odd	8		128.2.b.a.65.1	✓	2
32.21	even	8		256.2.a.e.1.1		2
32.27	odd	8		256.2.a.e.1.1		2
32.29	even	8		128.2.b.a.65.1	✓	2
96.5	odd	8		2304.2.a.t.1.2		2
96.11	even	8		2304.2.a.t.1.2		2
96.29	odd	8		1152.2.d.c.577.2		2
96.35	even	8		1152.2.d.c.577.1		2
96.53	odd	8		2304.2.a.t.1.1		2
96.59	even	8		2304.2.a.t.1.1		2
96.77	odd	8		1152.2.d.c.577.1		2
96.83	even	8		1152.2.d.c.577.2		2
160.3	even	8		3200.2.f.o.449.2		4
160.13	odd	8		3200.2.f.o.449.2		4
160.19	odd	8		3200.2.d.c.1601.2		2
160.29	even	8		3200.2.d.c.1601.2		2
160.59	odd	8		6400.2.a.by.1.2		2
160.67	even	8		3200.2.f.o.449.4		4
160.69	even	8		6400.2.a.by.1.1		2
160.77	odd	8		3200.2.f.o.449.4		4
160.83	even	8		3200.2.f.o.449.3		4
160.93	odd	8		3200.2.f.o.449.3		4
160.99	odd	8		3200.2.d.c.1601.1		2
160.109	even	8		3200.2.d.c.1601.1		2
160.139	odd	8		6400.2.a.by.1.1		2
160.147	even	8		3200.2.f.o.449.1		4
160.149	even	8		6400.2.a.by.1.2		2
160.157	odd	8		3200.2.f.o.449.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.2.b.a.65.1	✓	2	32.19	odd	8
128.2.b.a.65.1	✓	2	32.29	even	8
128.2.b.a.65.2	yes	2	32.3	odd	8
128.2.b.a.65.2	yes	2	32.13	even	8
256.2.a.e.1.1		2	32.21	even	8
256.2.a.e.1.1		2	32.27	odd	8
256.2.a.e.1.2		2	32.5	even	8
256.2.a.e.1.2		2	32.11	odd	8
1024.2.e.a.257.1		2	1.1	even	1	trivial
1024.2.e.a.257.1		2	8.3	odd	2	CM
1024.2.e.a.769.1		2	16.3	odd	4	inner
1024.2.e.a.769.1		2	16.5	even	4	inner
1024.2.e.f.257.1		2	4.3	odd	2
1024.2.e.f.257.1		2	8.5	even	2
1024.2.e.f.769.1		2	16.11	odd	4
1024.2.e.f.769.1		2	16.13	even	4
1152.2.d.c.577.1		2	96.35	even	8
1152.2.d.c.577.1		2	96.77	odd	8
1152.2.d.c.577.2		2	96.29	odd	8
1152.2.d.c.577.2		2	96.83	even	8
2304.2.a.t.1.1		2	96.53	odd	8
2304.2.a.t.1.1		2	96.59	even	8
2304.2.a.t.1.2		2	96.5	odd	8
2304.2.a.t.1.2		2	96.11	even	8
3200.2.d.c.1601.1		2	160.99	odd	8
3200.2.d.c.1601.1		2	160.109	even	8
3200.2.d.c.1601.2		2	160.19	odd	8
3200.2.d.c.1601.2		2	160.29	even	8
3200.2.f.o.449.1		4	160.147	even	8
3200.2.f.o.449.1		4	160.157	odd	8
3200.2.f.o.449.2		4	160.3	even	8
3200.2.f.o.449.2		4	160.13	odd	8
3200.2.f.o.449.3		4	160.83	even	8
3200.2.f.o.449.3		4	160.93	odd	8
3200.2.f.o.449.4		4	160.67	even	8
3200.2.f.o.449.4		4	160.77	odd	8
6400.2.a.by.1.1		2	160.69	even	8
6400.2.a.by.1.1		2	160.139	odd	8
6400.2.a.by.1.2		2	160.59	odd	8
6400.2.a.by.1.2		2	160.149	even	8

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:	\( N \)	\(=\)	\( 1024 = 2^{10} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1024.e (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.00000 - 2.00000i) q^{3} +5.00000i q^{9} +O(q^{10})\)	\(q+(-2.00000 - 2.00000i) q^{3} +5.00000i q^{9} +(2.00000 - 2.00000i) q^{11} -6.00000 q^{17} +(-6.00000 - 6.00000i) q^{19} +5.00000i q^{25} +(4.00000 - 4.00000i) q^{27} -8.00000 q^{33} +6.00000i q^{41} +(-6.00000 + 6.00000i) q^{43} +7.00000 q^{49} +(12.0000 + 12.0000i) q^{51} +24.0000i q^{57} +(-10.0000 + 10.0000i) q^{59} +(6.00000 + 6.00000i) q^{67} +2.00000i q^{73} +(10.0000 - 10.0000i) q^{75} -1.00000 q^{81} +(2.00000 + 2.00000i) q^{83} -18.0000i q^{89} -10.0000 q^{97} +(10.0000 + 10.0000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{3}+O(q^{10}) \) 2 * q - 4 * q^3	\( 2 q - 4 q^{3} + 4 q^{11} - 12 q^{17} - 12 q^{19} + 8 q^{27} - 16 q^{33} - 12 q^{43} + 14 q^{49} + 24 q^{51} - 20 q^{59} + 12 q^{67} + 20 q^{75} - 2 q^{81} + 4 q^{83} - 20 q^{97} + 20 q^{99}+O(q^{100}) \) 2 * q - 4 * q^3 + 4 * q^11 - 12 * q^17 - 12 * q^19 + 8 * q^27 - 16 * q^33 - 12 * q^43 + 14 * q^49 + 24 * q^51 - 20 * q^59 + 12 * q^67 + 20 * q^75 - 2 * q^81 + 4 * q^83 - 20 * q^97 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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