LMFDB - Embedded newform 448.4.j.a.335.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,4,Mod(111,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3, 2]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("448.111");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	448.j (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:	$26.4328556826$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 3x^{2} + 4 $ x^4 - 3*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			335.1
Root			$-1.32288 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	448.335
Dual form			448.4.j.a.111.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-18.5203 q^{7} +27.0000i q^{9} +O(q^{10})$	$q-18.5203 q^{7} +27.0000i q^{9} +(47.2288 - 47.2288i) q^{11} -40.0000 q^{23} +125.000i q^{25} +(-215.288 + 215.288i) q^{29} +(219.708 + 219.708i) q^{37} +(-357.221 + 357.221i) q^{43} +343.000 q^{49} +(46.2994 + 46.2994i) q^{53} -500.047i q^{63} +(774.800 + 774.800i) q^{67} -978.928 q^{71} +(-874.689 + 874.689i) q^{77} +1384.00i q^{79} -729.000 q^{81} +(1275.18 + 1275.18i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 136 q^{11} - 160 q^{23} - 332 q^{29} + 900 q^{37} - 360 q^{43} + 1372 q^{49} + 1180 q^{53} + 1480 q^{67} - 980 q^{77} - 2916 q^{81} + 3672 q^{99}+O(q^{100}) $ 4 * q + 136 * q^11 - 160 * q^23 - 332 * q^29 + 900 * q^37 - 360 * q^43 + 1372 * q^49 + 1180 * q^53 + 1480 * q^67 - 980 * q^77 - 2916 * q^81 + 3672 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/448\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{1}{4}\right)$

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			0		−0.707107	−	0.707107i	$-0.750000\pi$
$3$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$4$		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
$7$	−18.5203			−1.00000
$7$	−18.5203			−1.00000
$8$		0			0
$9$			27.0000i			1.00000i
$10$		0			0
$11$	47.2288	−	47.2288i	1.29455	−	1.29455i	0.362602	−	0.931944i	$-0.381889\pi$
$11$	47.2288	−	47.2288i	1.29455	−	1.29455i	0.931944	−	0.362602i	$-0.118111\pi$
$12$		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
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		0			0
$19$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$19$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−40.0000			−0.362634			−0.181317	−	0.983425i	$-0.558036\pi$
$23$	−40.0000			−0.362634			−0.181317	+	0.983425i	$0.558036\pi$
$24$		0			0
$25$			125.000i			1.00000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−215.288	+	215.288i	−1.37855	+	1.37855i	−0.531473	+	0.847075i	$0.678361\pi$
$29$	−215.288	+	215.288i	−1.37855	+	1.37855i	−0.847075	+	0.531473i	$0.821639\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	219.708	+	219.708i	0.976212	+	0.976212i	0.999724	−	0.0235113i	$-0.00748457\pi$
$37$	219.708	+	219.708i	0.976212	+	0.976212i	−0.0235113	+	0.999724i	$0.507485\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	−357.221	+	357.221i	−1.26688	+	1.26688i	−0.319183	+	0.947693i	$0.603408\pi$
$43$	−357.221	+	357.221i	−1.26688	+	1.26688i	−0.947693	+	0.319183i	$0.896592\pi$
$44$		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
$49$	343.000			1.00000
$50$		0			0
$51$		0			0
$52$		0			0
$53$	46.2994	+	46.2994i	0.119995	+	0.119995i	0.764554	−	0.644560i	$-0.222959\pi$
$53$	46.2994	+	46.2994i	0.119995	+	0.119995i	−0.644560	+	0.764554i	$0.722959\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$59$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$60$		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
$63$		−	500.047i		−	1.00000i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	774.800	+	774.800i	1.41279	+	1.41279i	0.738122	+	0.674667i	$0.235713\pi$
$67$	774.800	+	774.800i	1.41279	+	1.41279i	0.674667	+	0.738122i	$0.264287\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−978.928			−1.63630			−0.818151	−	0.575004i	$-0.805000\pi$
$71$	−978.928			−1.63630			−0.818151	+	0.575004i	$0.805000\pi$
$72$		0			0
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−874.689	+	874.689i	−1.29455	+	1.29455i
$78$		0			0
$79$			1384.00i			1.97104i	0.169559	+	0.985520i	$0.445766\pi$
$79$			1384.00i			1.97104i	−0.169559	+	0.985520i	$0.554234\pi$
$80$		0			0
$81$	−729.000			−1.00000
$82$		0			0
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		0			0
$99$	1275.18	+	1275.18i	1.29455	+	1.29455i
$100$		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
$107$	14.7949	−	14.7949i	0.0133670	−	0.0133670i	−0.700392	−	0.713759i	$-0.746991\pi$
$107$	14.7949	−	14.7949i	0.0133670	−	0.0133670i	0.713759	+	0.700392i	$0.246991\pi$
$108$		0			0
$109$	−1110.67	+	1110.67i	−0.975993	+	0.975993i	−0.999718	−	0.0237260i	$-0.992447\pi$
$109$	−1110.67	+	1110.67i	−0.975993	+	0.975993i	0.0237260	+	0.999718i	$0.492447\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	2307.10			1.92065			0.960324	−	0.278886i	$-0.0899653\pi$
$113$	2307.10			1.92065			0.960324	+	0.278886i	$0.0899653\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$		−	3130.11i		−	2.35170i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	2047.81i		−	1.43082i	−0.698706	−	0.715409i	$-0.746240\pi$
$127$		−	2047.81i		−	1.43082i	0.698706	−	0.715409i	$-0.253760\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$131$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$			783.142i			0.488382i	0.969727	+	0.244191i	$0.0785224\pi$
$137$			783.142i			0.488382i	−0.969727	+	0.244191i	$0.921478\pi$
$138$		0			0
$139$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$139$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1365.65	−	1365.65i	−0.750864	−	0.750864i	0.223777	−	0.974640i	$-0.428161\pi$
$149$	−1365.65	−	1365.65i	−0.750864	−	0.750864i	−0.974640	+	0.223777i	$0.928161\pi$
$150$		0			0
$151$	−2952.00			−1.59093			−0.795465	−	0.606000i	$-0.792773\pi$
$151$	−2952.00			−1.59093			−0.795465	+	0.606000i	$0.792773\pi$
$152$		0			0
$153$		0			0
$154$		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
$161$	740.810			0.362634
$162$		0			0
$163$	991.129	+	991.129i	0.476265	+	0.476265i	0.903935	−	0.427670i	$-0.140665\pi$
$163$	991.129	+	991.129i	0.476265	+	0.476265i	−0.427670	+	0.903935i	$0.640665\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$		−	2197.00i		−	1.00000i
$170$		0			0
$171$		0			0
$172$		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
$175$		−	2315.03i		−	1.00000i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	1114.29	+	1114.29i	0.465283	+	0.465283i	0.900383	−	0.435099i	$-0.143287\pi$
$179$	1114.29	+	1114.29i	0.465283	+	0.465283i	−0.435099	+	0.900383i	$0.643287\pi$
$180$		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
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	4072.00i		−	1.54262i	−0.636462	−	0.771308i	$-0.719603\pi$
$191$		−	4072.00i		−	1.54262i	0.636462	−	0.771308i	$-0.280397\pi$
$192$		0			0
$193$	2772.75			1.03413			0.517064	−	0.855947i	$-0.327025\pi$
$193$	2772.75			1.03413			0.517064	+	0.855947i	$0.327025\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	3639.63	+	3639.63i	1.31631	+	1.31631i	0.916675	+	0.399634i	$0.130863\pi$
$197$	3639.63	+	3639.63i	1.31631	+	1.31631i	0.399634	+	0.916675i	$0.369137\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	3987.18	−	3987.18i	1.37855	−	1.37855i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		−	1080.00i		−	0.362634i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−3820.33	−	3820.33i	−1.24646	−	1.24646i	−0.957274	−	0.289181i	$-0.906617\pi$
$211$	−3820.33	−	3820.33i	−1.24646	−	1.24646i	−0.289181	−	0.957274i	$-0.593383\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$	−3375.00			−1.00000
$226$		0			0
$227$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$227$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$228$		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
$233$		−	5312.67i		−	1.49375i	−0.664963	−	0.746877i	$-0.731553\pi$
$233$		−	5312.67i		−	1.49375i	0.664963	−	0.746877i	$-0.268447\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	449.778i		−	0.121731i	−0.998146	−	0.0608655i	$-0.980614\pi$
$239$		−	449.778i		−	0.121731i	0.998146	−	0.0608655i	$-0.0193861\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$251$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$252$		0			0
$253$	−1889.15	+	1889.15i	−0.469446	+	0.469446i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$	−4069.06	−	4069.06i	−0.976212	−	0.976212i
$260$		0			0
$261$	−5812.76	−	5812.76i	−1.37855	−	1.37855i
$262$		0			0
$263$	4026.83			0.944126			0.472063	−	0.881565i	$-0.343509\pi$
$263$	4026.83			0.944126			0.472063	+	0.881565i	$0.343509\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		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
$275$	5903.59	+	5903.59i	1.29455	+	1.29455i
$276$		0			0
$277$	−845.212	−	845.212i	−0.183335	−	0.183335i	0.609472	−	0.792807i	$-0.291381\pi$
$277$	−845.212	−	845.212i	−0.183335	−	0.183335i	−0.792807	+	0.609472i	$0.791381\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			8360.57i			1.77491i	0.460893	+	0.887456i	$0.347529\pi$
$281$			8360.57i			1.77491i	−0.460893	+	0.887456i	$0.652471\pi$
$282$		0			0
$283$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$283$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	4913.00			1.00000
$290$		0			0
$291$		0			0
$292$		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
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	6615.82	−	6615.82i	1.26688	−	1.26688i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$307$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$		0			0			−	1.00000i	$-0.5\pi$
$313$		0			0				1.00000i	$0.5\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	7924.57	−	7924.57i	1.40406	−	1.40406i	0.617467	−	0.786597i	$-0.288159\pi$
$317$	7924.57	−	7924.57i	1.40406	−	1.40406i	0.786597	−	0.617467i	$-0.211841\pi$
$318$		0			0
$319$			20335.5i			3.56919i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−2900.85	+	2900.85i	−0.481707	+	0.481707i	−0.905677	−	0.423969i	$-0.860636\pi$
$331$	−2900.85	+	2900.85i	−0.481707	+	0.481707i	0.423969	+	0.905677i	$0.360636\pi$
$332$		0			0
$333$	−5932.13	+	5932.13i	−0.976212	+	0.976212i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−11916.5			−1.92621			−0.963103	−	0.269135i	$-0.913262\pi$
$337$	−11916.5			−1.92621			−0.963103	+	0.269135i	$0.913262\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−6352.45			−1.00000
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−8180.21	+	8180.21i	−1.26552	+	1.26552i	−0.317146	+	0.948377i	$0.602725\pi$
$347$	−8180.21	+	8180.21i	−1.26552	+	1.26552i	−0.948377	+	0.317146i	$0.897275\pi$
$348$		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
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	8104.00			1.19140			0.595700	−	0.803207i	$-0.296875\pi$
$359$	8104.00			1.19140			0.595700	+	0.803207i	$0.296875\pi$
$360$		0			0
$361$			6859.00i			1.00000i
$362$		0			0
$363$		0			0
$364$		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
$369$		0			0
$370$		0			0
$371$	−857.476	−	857.476i	−0.119995	−	0.119995i
$372$		0			0
$373$	5222.93	+	5222.93i	0.725022	+	0.725022i	0.969624	−	0.244602i	$-0.0786573\pi$
$373$	5222.93	+	5222.93i	0.725022	+	0.725022i	−0.244602	+	0.969624i	$0.578657\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−1605.74	+	1605.74i	−0.217629	+	0.217629i	−0.807498	−	0.589870i	$-0.799179\pi$
$379$	−1605.74	+	1605.74i	−0.217629	+	0.217629i	0.589870	+	0.807498i	$0.299179\pi$
$380$		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
$387$	−9644.96	−	9644.96i	−1.26688	−	1.26688i
$388$		0			0
$389$	319.535	+	319.535i	0.0416480	+	0.0416480i	0.727624	−	0.685976i	$-0.240625\pi$
$389$	319.535	+	319.535i	0.0416480	+	0.0416480i	−0.685976	+	0.727624i	$0.740625\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		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
$401$	−15980.3			−1.99007			−0.995037	−	0.0995016i	$-0.968275\pi$
$401$	−15980.3			−1.99007			−0.995037	+	0.0995016i	$0.968275\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	20753.1			2.52750
$408$		0			0
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$419$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$420$		0			0
$421$	11679.0	+	11679.0i	1.35202	+	1.35202i	0.883402	+	0.468616i	$0.155247\pi$
$421$	11679.0	+	11679.0i	1.35202	+	1.35202i	0.468616	+	0.883402i	$0.344753\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$			15689.3i			1.75343i	0.481012	+	0.876714i	$0.340269\pi$
$431$			15689.3i			1.75343i	−0.481012	+	0.876714i	$0.659731\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		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
$441$			9261.00i			1.00000i
$442$		0			0
$443$	10086.4	−	10086.4i	1.08176	−	1.08176i	0.0854102	−	0.996346i	$-0.472780\pi$
$443$	10086.4	−	10086.4i	1.08176	−	1.08176i	0.996346	−	0.0854102i	$-0.0272201\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	2686.00			0.282317			0.141158	−	0.989987i	$-0.454917\pi$
$449$	2686.00			0.282317			0.141158	+	0.989987i	$0.454917\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$			8010.00i			0.819895i	0.912109	+	0.409947i	$0.134453\pi$
$457$			8010.00i			0.819895i	−0.912109	+	0.409947i	$0.865547\pi$
$458$		0			0
$459$		0			0
$460$		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
$463$		−	8440.00i		−	0.847171i	−0.905856	−	0.423585i	$-0.860771\pi$
$463$		−	8440.00i		−	0.847171i	0.905856	−	0.423585i	$-0.139229\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$467$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$468$		0			0
$469$	−14349.5	−	14349.5i	−1.41279	−	1.41279i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			33742.2i			3.28006i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−1250.08	+	1250.08i	−0.119995	+	0.119995i
$478$		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
$487$	−21240.0			−1.97634			−0.988169	−	0.153371i	$-0.950987\pi$
$487$	−21240.0			−1.97634			−0.988169	+	0.153371i	$0.950987\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−14009.1	+	14009.1i	−1.28762	+	1.28762i	−0.351394	+	0.936228i	$0.614292\pi$
$491$	−14009.1	+	14009.1i	−1.28762	+	1.28762i	−0.936228	+	0.351394i	$0.885708\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	18130.0			1.63630
$498$		0			0
$499$	−14161.3	−	14161.3i	−1.27044	−	1.27044i	−0.945859	−	0.324577i	$-0.894778\pi$
$499$	−14161.3	−	14161.3i	−1.27044	−	1.27044i	−0.324577	−	0.945859i	$-0.605222\pi$
$500$		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
$507$		0			0
$508$		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
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$523$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−10567.0			−0.868497
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	16199.5	−	16199.5i	1.29455	−	1.29455i
$540$		0			0
$541$	17701.8	−	17701.8i	1.40677	−	1.40677i	0.630914	−	0.775853i	$-0.282680\pi$
$541$	17701.8	−	17701.8i	1.40677	−	1.40677i	0.775853	−	0.630914i	$-0.217320\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	4534.85	+	4534.85i	0.354472	+	0.354472i	0.861770	−	0.507299i	$-0.169356\pi$
$547$	4534.85	+	4534.85i	0.354472	+	0.354472i	−0.507299	+	0.861770i	$0.669356\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		−	25632.0i		−	1.97104i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	1985.55	−	1985.55i	0.151042	−	0.151042i	−0.627541	−	0.778583i	$-0.715939\pi$
$557$	1985.55	−	1985.55i	0.151042	−	0.151042i	0.778583	+	0.627541i	$0.215939\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$563$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	13501.3			1.00000
$568$		0			0
$569$		−	26906.0i		−	1.98235i	−0.132553	−	0.991176i	$-0.542317\pi$
$569$		−	26906.0i		−	1.98235i	0.132553	−	0.991176i	$-0.457683\pi$
$570$		0			0
$571$	9821.53	−	9821.53i	0.719822	−	0.719822i	−0.248747	−	0.968569i	$-0.580019\pi$
$571$	9821.53	−	9821.53i	0.719822	−	0.719822i	0.968569	+	0.248747i	$0.0800187\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	5000.00i		−	0.362634i
$576$		0			0
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	4373.32			0.310677
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$587$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	15742.2			1.07381			0.536903	−	0.843644i	$-0.319594\pi$
$599$	15742.2			1.07381			0.536903	+	0.843644i	$0.319594\pi$
$600$		0			0
$601$		0			0			−	1.00000i	$-0.5\pi$
$601$		0			0				1.00000i	$0.5\pi$
$602$		0			0
$603$	−20919.6	+	20919.6i	−1.41279	+	1.41279i
$604$		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
$613$	5686.72	+	5686.72i	0.374689	+	0.374689i	0.869182	−	0.494493i	$-0.164646\pi$
$613$	5686.72	+	5686.72i	0.374689	+	0.374689i	−0.494493	+	0.869182i	$0.664646\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	30550.0i		−	1.99335i	−0.0814823	−	0.996675i	$-0.525965\pi$
$617$		−	30550.0i		−	1.99335i	0.0814823	−	0.996675i	$-0.474035\pi$
$618$		0			0
$619$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$619$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−15625.0			−1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−17858.8			−1.12670			−0.563351	−	0.826218i	$-0.690488\pi$
$631$	−17858.8			−1.12670			−0.563351	+	0.826218i	$0.690488\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		−	26431.1i		−	1.63630i
$640$		0			0
$641$	31219.9			1.92373			0.961865	−	0.273526i	$-0.0881899\pi$
$641$	31219.9			1.92373			0.961865	+	0.273526i	$0.0881899\pi$
$642$		0			0
$643$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$643$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−23298.4	+	23298.4i	−1.39623	+	1.39623i	−0.585701	+	0.810527i	$0.699181\pi$
$653$	−23298.4	+	23298.4i	−1.39623	+	1.39623i	−0.810527	+	0.585701i	$0.800819\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	15991.1	+	15991.1i	0.945259	+	0.945259i	0.998578	−	0.0533186i	$-0.0169799\pi$
$659$	15991.1	+	15991.1i	0.945259	+	0.945259i	−0.0533186	+	0.998578i	$0.516980\pi$
$660$		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
$667$	8611.50	−	8611.50i	0.499908	−	0.499908i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−33570.0			−1.92278			−0.961388	−	0.275196i	$-0.911257\pi$
$673$	−33570.0			−1.92278			−0.961388	+	0.275196i	$0.911257\pi$
$674$		0			0
$675$		0			0
$676$		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
$681$		0			0
$682$		0			0
$683$	11682.9	−	11682.9i	0.654517	−	0.654517i	−0.299560	−	0.954077i	$-0.596840\pi$
$683$	11682.9	−	11682.9i	0.654517	−	0.654517i	0.954077	+	0.299560i	$0.0968400\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$691$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$692$		0			0
$693$	−23616.6	−	23616.6i	−1.29455	−	1.29455i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	20539.9	−	20539.9i	1.10668	−	1.10668i	0.113093	−	0.993584i	$-0.463924\pi$
$701$	20539.9	−	20539.9i	1.10668	−	1.10668i	0.993584	−	0.113093i	$-0.0360758\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	11532.9	+	11532.9i	0.610899	+	0.610899i	0.943180	−	0.332281i	$-0.107818\pi$
$709$	11532.9	+	11532.9i	0.610899	+	0.610899i	−0.332281	+	0.943180i	$0.607818\pi$
$710$		0			0
$711$	−37368.0			−1.97104
$712$		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
$723$		0			0
$724$		0			0
$725$	−26910.9	−	26910.9i	−1.37855	−	1.37855i
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$		−	19683.0i		−	1.00000i
$730$		0			0
$731$		0			0
$732$		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
$737$	73185.7			3.65784
$738$		0			0
$739$	28258.7	+	28258.7i	1.40665	+	1.40665i	0.776365	+	0.630283i	$0.217061\pi$
$739$	28258.7	+	28258.7i	1.40665	+	1.40665i	0.630283	+	0.776365i	$0.282939\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	25160.0			1.24230			0.621151	−	0.783691i	$-0.286665\pi$
$743$	25160.0			1.24230			0.621151	+	0.783691i	$0.286665\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−274.005	+	274.005i	−0.0133670	+	0.0133670i
$750$		0			0
$751$		−	41088.5i		−	1.99646i	−0.0594732	−	0.998230i	$-0.518942\pi$
$751$		−	41088.5i		−	1.99646i	0.0594732	−	0.998230i	$-0.481058\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	5990.65	+	5990.65i	0.287627	+	0.287627i	0.836141	−	0.548514i	$-0.184806\pi$
$757$	5990.65	+	5990.65i	0.287627	+	0.287627i	−0.548514	+	0.836141i	$0.684806\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0			−	1.00000i	$-0.5\pi$
$761$		0			0				1.00000i	$0.5\pi$
$762$		0			0
$763$	20570.0	−	20570.0i	0.975993	−	0.975993i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$		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
$779$		0			0
$780$		0			0
$781$	−46233.6	+	46233.6i	−2.11827	+	2.11827i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$787$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−42728.0			−1.92065
$792$		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
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			37354.0i			1.62336i	0.584104	+	0.811679i	$0.301446\pi$
$809$			37354.0i			1.62336i	−0.584104	+	0.811679i	$0.698554\pi$
$810$		0			0
$811$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$811$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	12852.8	+	12852.8i	0.546366	+	0.546366i	0.925388	−	0.379022i	$-0.123739\pi$
$821$	12852.8	+	12852.8i	0.546366	+	0.546366i	−0.379022	+	0.925388i	$0.623739\pi$
$822$		0			0
$823$	−9572.33			−0.405432			−0.202716	−	0.979238i	$-0.564977\pi$
$823$	−9572.33			−0.405432			−0.202716	+	0.979238i	$0.564977\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−32529.0	+	32529.0i	−1.36777	+	1.36777i	−0.504151	+	0.863615i	$0.668195\pi$
$827$	−32529.0	+	32529.0i	−1.36777	+	1.36777i	−0.863615	+	0.504151i	$0.831805\pi$
$828$		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
$833$		0			0
$834$		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
$841$		−	68308.5i		−	2.80079i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$			57970.5i			2.35170i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−8788.34	−	8788.34i	−0.354008	−	0.354008i
$852$		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
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$859$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	20200.0i		−	0.796774i	−0.917217	−	0.398387i	$-0.869570\pi$
$863$		−	20200.0i		−	0.796774i	0.917217	−	0.398387i	$-0.130430\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	65364.6	+	65364.6i	2.55160	+	2.55160i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−29039.3	+	29039.3i	−1.11812	+	1.11812i	−0.126099	+	0.992018i	$0.540246\pi$
$877$	−29039.3	+	29039.3i	−1.11812	+	1.11812i	−0.992018	+	0.126099i	$0.959754\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	6477.31	+	6477.31i	0.246862	+	0.246862i	0.819681	−	0.572820i	$-0.194150\pi$
$883$	6477.31	+	6477.31i	0.246862	+	0.246862i	−0.572820	+	0.819681i	$0.694150\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$			37926.0i			1.43082i
$890$		0			0
$891$	−34429.8	+	34429.8i	−1.29455	+	1.29455i
$892$		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
$907$	33495.0	−	33495.0i	1.22622	−	1.22622i	0.260840	−	0.965382i	$-0.416000\pi$
$907$	33495.0	−	33495.0i	1.22622	−	1.22622i	0.965382	−	0.260840i	$-0.0839996\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	38125.3i		−	1.38655i	−0.720673	−	0.693275i	$-0.756167\pi$
$911$		−	38125.3i		−	1.38655i	0.720673	−	0.693275i	$-0.243833\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−51301.1			−1.84142			−0.920711	−	0.390244i	$-0.872391\pi$
$919$	−51301.1			−1.84142			−0.920711	+	0.390244i	$0.872391\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−27463.6	+	27463.6i	−0.976212	+	0.976212i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		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
$947$	40329.5	+	40329.5i	1.38388	+	1.38388i	0.837612	+	0.546266i	$0.183951\pi$
$947$	40329.5	+	40329.5i	1.38388	+	1.38388i	0.546266	+	0.837612i	$0.316049\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$			51031.3i			1.73459i	0.497794	+	0.867295i	$0.334143\pi$
$953$			51031.3i			1.73459i	−0.497794	+	0.867295i	$0.665857\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	14504.0i		−	0.488382i
$960$		0			0
$961$	−29791.0			−1.00000
$962$		0			0
$963$	399.461	+	399.461i	0.0133670	+	0.0133670i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	52040.0			1.73060			0.865302	−	0.501251i	$-0.167127\pi$
$967$	52040.0			1.73060			0.865302	+	0.501251i	$0.167127\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$971$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	37490.0			1.22765			0.613824	−	0.789443i	$-0.289631\pi$
$977$	37490.0			1.22765			0.613824	+	0.789443i	$0.289631\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−29988.2	−	29988.2i	−0.975993	−	0.975993i
$982$		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
$989$	14288.8	−	14288.8i	0.459412	−	0.459412i
$990$		0			0
$991$			57528.0i			1.84403i	0.387150	+	0.922017i	$0.373460\pi$
$991$			57528.0i			1.84403i	−0.387150	+	0.922017i	$0.626540\pi$
$992$		0			0
$993$		0			0
$994$		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	448.4.j.a.335.1		4
4.3	odd	2		112.4.j.a.27.1	✓	4
7.6	odd	2	CM	448.4.j.a.335.1		4
16.3	odd	4	inner	448.4.j.a.111.1		4
16.13	even	4		112.4.j.a.83.1	yes	4
28.27	even	2		112.4.j.a.27.1	✓	4
112.13	odd	4		112.4.j.a.83.1	yes	4
112.83	even	4	inner	448.4.j.a.111.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.4.j.a.27.1	✓	4	4.3	odd	2
112.4.j.a.27.1	✓	4	28.27	even	2
112.4.j.a.83.1	yes	4	16.13	even	4
112.4.j.a.83.1	yes	4	112.13	odd	4
448.4.j.a.111.1		4	16.3	odd	4	inner
448.4.j.a.111.1		4	112.83	even	4	inner
448.4.j.a.335.1		4	1.1	even	1	trivial
448.4.j.a.335.1		4	7.6	odd	2	CM

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:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	448.j (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q-18.5203 q^{7} +27.0000i q^{9} +O(q^{10})\)	\(q-18.5203 q^{7} +27.0000i q^{9} +(47.2288 - 47.2288i) q^{11} -40.0000 q^{23} +125.000i q^{25} +(-215.288 + 215.288i) q^{29} +(219.708 + 219.708i) q^{37} +(-357.221 + 357.221i) q^{43} +343.000 q^{49} +(46.2994 + 46.2994i) q^{53} -500.047i q^{63} +(774.800 + 774.800i) q^{67} -978.928 q^{71} +(-874.689 + 874.689i) q^{77} +1384.00i q^{79} -729.000 q^{81} +(1275.18 + 1275.18i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 136 q^{11} - 160 q^{23} - 332 q^{29} + 900 q^{37} - 360 q^{43} + 1372 q^{49} + 1180 q^{53} + 1480 q^{67} - 980 q^{77} - 2916 q^{81} + 3672 q^{99}+O(q^{100}) \) 4 * q + 136 * q^11 - 160 * q^23 - 332 * q^29 + 900 * q^37 - 360 * q^43 + 1372 * q^49 + 1180 * q^53 + 1480 * q^67 - 980 * q^77 - 2916 * q^81 + 3672 * q^99

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