LMFDB - Embedded newform 1600.2.n.q.1407.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,2,Mod(1343,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 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("1600.1343");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1600.n (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:	$12.7760643234$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 400)
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			1407.1
Root			$1.61803i$ of defining polynomial
Character	$\chi$	$=$	1600.1407
Dual form			1600.2.n.q.1343.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.23607 - 2.23607i) q^{3} +(2.23607 - 2.23607i) q^{7} +7.00000i q^{9} +O(q^{10})$	$q+(-2.23607 - 2.23607i) q^{3} +(2.23607 - 2.23607i) q^{7} +7.00000i q^{9} -10.0000 q^{21} +(-6.70820 - 6.70820i) q^{23} +(8.94427 - 8.94427i) q^{27} -6.00000i q^{29} -12.0000 q^{41} +(2.23607 + 2.23607i) q^{43} +(6.70820 - 6.70820i) q^{47} -3.00000i q^{49} -8.00000 q^{61} +(15.6525 + 15.6525i) q^{63} +(-11.1803 + 11.1803i) q^{67} +30.0000i q^{69} -19.0000 q^{81} +(-6.70820 - 6.70820i) q^{83} +(-13.4164 + 13.4164i) q^{87} +6.00000i q^{89} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 40 q^{21} - 48 q^{41} - 32 q^{61} - 76 q^{81}+O(q^{100}) $ 4 * q - 40 * q^21 - 48 * q^41 - 32 * q^61 - 76 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1151$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$-1$

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
$3$	−2.23607	−	2.23607i	−1.29099	−	1.29099i	−0.934172	−	0.356822i	$-0.883860\pi$
$3$	−2.23607	−	2.23607i	−1.29099	−	1.29099i	−0.356822	−	0.934172i	$-0.616140\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	2.23607	−	2.23607i	0.845154	−	0.845154i	−0.144370	−	0.989524i	$-0.546115\pi$
$7$	2.23607	−	2.23607i	0.845154	−	0.845154i	0.989524	+	0.144370i	$0.0461154\pi$
$8$		0			0
$9$			7.00000i			2.33333i
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$13$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$17$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$	−10.0000			−2.18218
$22$		0			0
$23$	−6.70820	−	6.70820i	−1.39876	−	1.39876i	−0.803636	−	0.595121i	$-0.797104\pi$
$23$	−6.70820	−	6.70820i	−1.39876	−	1.39876i	−0.595121	−	0.803636i	$-0.702896\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	8.94427	−	8.94427i	1.72133	−	1.72133i
$28$		0			0
$29$		−	6.00000i		−	1.11417i	−0.830455	−	0.557086i	$-0.811919\pi$
$29$		−	6.00000i		−	1.11417i	0.830455	−	0.557086i	$-0.188081\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$37$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−12.0000			−1.87409			−0.937043	−	0.349215i	$-0.886448\pi$
$41$	−12.0000			−1.87409			−0.937043	+	0.349215i	$0.886448\pi$
$42$		0			0
$43$	2.23607	+	2.23607i	0.340997	+	0.340997i	0.856742	−	0.515745i	$-0.172485\pi$
$43$	2.23607	+	2.23607i	0.340997	+	0.340997i	−0.515745	+	0.856742i	$0.672485\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	6.70820	−	6.70820i	0.978492	−	0.978492i	−0.0212814	−	0.999774i	$-0.506775\pi$
$47$	6.70820	−	6.70820i	0.978492	−	0.978492i	0.999774	+	0.0212814i	$0.00677460\pi$
$48$		0			0
$49$		−	3.00000i		−	0.428571i
$50$		0			0
$51$		0			0
$52$		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
$57$		0			0
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	−8.00000			−1.02430			−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000			−1.02430			−0.512148	+	0.858898i	$0.671150\pi$
$62$		0			0
$63$	15.6525	+	15.6525i	1.97203	+	1.97203i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−11.1803	+	11.1803i	−1.36590	+	1.36590i	−0.499694	+	0.866202i	$0.666554\pi$
$67$	−11.1803	+	11.1803i	−1.36590	+	1.36590i	−0.866202	+	0.499694i	$0.833446\pi$
$68$		0			0
$69$			30.0000i			3.61158i
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$73$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$74$		0			0
$75$		0			0
$76$		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
$81$	−19.0000			−2.11111
$82$		0			0
$83$	−6.70820	−	6.70820i	−0.736321	−	0.736321i	0.235543	−	0.971864i	$-0.424313\pi$
$83$	−6.70820	−	6.70820i	−0.736321	−	0.736321i	−0.971864	+	0.235543i	$0.924313\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−13.4164	+	13.4164i	−1.43839	+	1.43839i
$88$		0			0
$89$			6.00000i			0.635999i	0.948091	+	0.317999i	$0.103011\pi$
$89$			6.00000i			0.635999i	−0.948091	+	0.317999i	$0.896989\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$97$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	18.0000			1.79107			0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000			1.79107			0.895533	+	0.444994i	$0.146794\pi$
$102$		0			0
$103$	−11.1803	−	11.1803i	−1.10163	−	1.10163i	−0.994214	−	0.107418i	$-0.965742\pi$
$103$	−11.1803	−	11.1803i	−1.10163	−	1.10163i	−0.107418	−	0.994214i	$-0.534258\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−6.70820	+	6.70820i	−0.648507	+	0.648507i	−0.952632	−	0.304125i	$-0.901636\pi$
$107$	−6.70820	+	6.70820i	−0.648507	+	0.648507i	0.304125	+	0.952632i	$0.401636\pi$
$108$		0			0
$109$			16.0000i			1.53252i	0.642529	+	0.766261i	$0.277885\pi$
$109$			16.0000i			1.53252i	−0.642529	+	0.766261i	$0.722115\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$113$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$	26.8328	+	26.8328i	2.41943	+	2.41943i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−15.6525	+	15.6525i	−1.38893	+	1.38893i	−0.561363	+	0.827570i	$0.689723\pi$
$127$	−15.6525	+	15.6525i	−1.38893	+	1.38893i	−0.827570	+	0.561363i	$0.810277\pi$
$128$		0			0
$129$		−	10.0000i		−	0.880451i
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$137$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$	−30.0000			−2.52646
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−6.70820	+	6.70820i	−0.553283	+	0.553283i
$148$		0			0
$149$		−	24.0000i		−	1.96616i	−0.183186	−	0.983078i	$-0.558641\pi$
$149$		−	24.0000i		−	1.96616i	0.183186	−	0.983078i	$-0.441359\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		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
$161$	−30.0000			−2.36433
$162$		0			0
$163$	−15.6525	−	15.6525i	−1.22600	−	1.22600i	−0.965465	−	0.260531i	$-0.916102\pi$
$163$	−15.6525	−	15.6525i	−1.22600	−	1.22600i	−0.260531	−	0.965465i	$-0.583898\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−6.70820	+	6.70820i	−0.519096	+	0.519096i	−0.917298	−	0.398202i	$-0.869634\pi$
$167$	−6.70820	+	6.70820i	−0.519096	+	0.519096i	0.398202	+	0.917298i	$0.369634\pi$
$168$		0			0
$169$			13.0000i			1.00000i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$173$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	−2.00000			−0.148659			−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000			−0.148659			−0.0743294	+	0.997234i	$0.523682\pi$
$182$		0			0
$183$	17.8885	+	17.8885i	1.32236	+	1.32236i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		−	40.0000i		−	2.90957i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$193$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$197$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$	50.0000			3.52673
$202$		0			0
$203$	−13.4164	−	13.4164i	−0.941647	−	0.941647i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	46.9574	−	46.9574i	3.26377	−	3.26377i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		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
$223$	2.23607	+	2.23607i	0.149738	+	0.149738i	0.778001	−	0.628263i	$-0.216234\pi$
$223$	2.23607	+	2.23607i	0.149738	+	0.149738i	−0.628263	+	0.778001i	$0.716234\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	20.1246	−	20.1246i	1.33572	−	1.33572i	0.435556	−	0.900162i	$-0.356552\pi$
$227$	20.1246	−	20.1246i	1.33572	−	1.33572i	0.900162	−	0.435556i	$-0.143448\pi$
$228$		0			0
$229$			14.0000i			0.925146i	0.886581	+	0.462573i	$0.153074\pi$
$229$			14.0000i			0.925146i	−0.886581	+	0.462573i	$0.846926\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$233$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$234$		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
$241$	28.0000			1.80364			0.901819	−	0.432113i	$-0.142232\pi$
$241$	28.0000			1.80364			0.901819	+	0.432113i	$0.142232\pi$
$242$		0			0
$243$	15.6525	+	15.6525i	1.00411	+	1.00411i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$			30.0000i			1.90117i
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$257$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	42.0000			2.59973
$262$		0			0
$263$	−20.1246	−	20.1246i	−1.24094	−	1.24094i	−0.959613	−	0.281324i	$-0.909226\pi$
$263$	−20.1246	−	20.1246i	−1.24094	−	1.24094i	−0.281324	−	0.959613i	$-0.590774\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	13.4164	−	13.4164i	0.821071	−	0.821071i
$268$		0			0
$269$		−	24.0000i		−	1.46331i	−0.681677	−	0.731653i	$-0.738749\pi$
$269$		−	24.0000i		−	1.46331i	0.681677	−	0.731653i	$-0.261251\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$277$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−12.0000			−0.715860			−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000			−0.715860			−0.357930	+	0.933748i	$0.616517\pi$
$282$		0			0
$283$	−11.1803	−	11.1803i	−0.664602	−	0.664602i	0.291859	−	0.956461i	$-0.405726\pi$
$283$	−11.1803	−	11.1803i	−0.664602	−	0.664602i	−0.956461	+	0.291859i	$0.905726\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−26.8328	+	26.8328i	−1.58389	+	1.58389i
$288$		0			0
$289$		−	17.0000i		−	1.00000i
$290$		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
$301$	10.0000			0.576390
$302$		0			0
$303$	−40.2492	−	40.2492i	−2.31226	−	2.31226i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	24.5967	−	24.5967i	1.40381	−	1.40381i	0.616296	−	0.787515i	$-0.288633\pi$
$307$	24.5967	−	24.5967i	1.40381	−	1.40381i	0.787515	−	0.616296i	$-0.211367\pi$
$308$		0			0
$309$			50.0000i			2.84440i
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$313$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$314$		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
$321$	30.0000			1.67444
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$	35.7771	−	35.7771i	1.97848	−	1.97848i
$328$		0			0
$329$		−	30.0000i		−	1.65395i
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$337$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	8.94427	+	8.94427i	0.482945	+	0.482945i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−20.1246	+	20.1246i	−1.08035	+	1.08035i	−0.0838690	+	0.996477i	$0.526728\pi$
$347$	−20.1246	+	20.1246i	−1.08035	+	1.08035i	−0.996477	+	0.0838690i	$0.973272\pi$
$348$		0			0
$349$		−	26.0000i		−	1.39175i	−0.718164	−	0.695874i	$-0.755017\pi$
$349$		−	26.0000i		−	1.39175i	0.718164	−	0.695874i	$-0.244983\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$353$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$354$		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
$361$	−19.0000			−1.00000
$362$		0			0
$363$	−24.5967	−	24.5967i	−1.29099	−	1.29099i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−2.23607	+	2.23607i	−0.116722	+	0.116722i	−0.763055	−	0.646333i	$-0.776302\pi$
$367$	−2.23607	+	2.23607i	−0.116722	+	0.116722i	0.646333	+	0.763055i	$0.276302\pi$
$368$		0			0
$369$		−	84.0000i		−	4.37287i
$370$		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
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$	70.0000			3.58621
$382$		0			0
$383$	20.1246	+	20.1246i	1.02832	+	1.02832i	0.999587	+	0.0287325i	$0.00914709\pi$
$383$	20.1246	+	20.1246i	1.02832	+	1.02832i	0.0287325	+	0.999587i	$0.490853\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−15.6525	+	15.6525i	−0.795660	+	0.795660i
$388$		0			0
$389$		−	24.0000i		−	1.21685i	−0.793612	−	0.608424i	$-0.791802\pi$
$389$		−	24.0000i		−	1.21685i	0.793612	−	0.608424i	$-0.208198\pi$
$390$		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
$401$	−18.0000			−0.898877			−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000			−0.898877			−0.449439	+	0.893311i	$0.648376\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$			4.00000i			0.197787i	0.995098	+	0.0988936i	$0.0315304\pi$
$409$			4.00000i			0.197787i	−0.995098	+	0.0988936i	$0.968470\pi$
$410$		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			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−8.00000			−0.389896			−0.194948	−	0.980814i	$-0.562454\pi$
$421$	−8.00000			−0.389896			−0.194948	+	0.980814i	$0.562454\pi$
$422$		0			0
$423$	46.9574	+	46.9574i	2.28315	+	2.28315i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−17.8885	+	17.8885i	−0.865687	+	0.865687i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$433$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	21.0000			1.00000
$442$		0			0
$443$	−6.70820	−	6.70820i	−0.318716	−	0.318716i	0.529558	−	0.848274i	$-0.322358\pi$
$443$	−6.70820	−	6.70820i	−0.318716	−	0.318716i	−0.848274	+	0.529558i	$0.822358\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−53.6656	+	53.6656i	−2.53830	+	2.53830i
$448$		0			0
$449$		−	36.0000i		−	1.69895i	−0.527633	−	0.849473i	$-0.676920\pi$
$449$		−	36.0000i		−	1.69895i	0.527633	−	0.849473i	$-0.323080\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$457$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−42.0000			−1.95614			−0.978068	−	0.208288i	$-0.933211\pi$
$461$	−42.0000			−1.95614			−0.978068	+	0.208288i	$0.933211\pi$
$462$		0			0
$463$	−29.0689	−	29.0689i	−1.35095	−	1.35095i	−0.884606	−	0.466340i	$-0.845572\pi$
$463$	−29.0689	−	29.0689i	−1.35095	−	1.35095i	−0.466340	−	0.884606i	$-0.654428\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−20.1246	+	20.1246i	−0.931256	+	0.931256i	−0.997785	−	0.0665285i	$-0.978808\pi$
$467$	−20.1246	+	20.1246i	−0.931256	+	0.931256i	0.0665285	+	0.997785i	$0.478808\pi$
$468$		0			0
$469$			50.0000i			2.30879i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		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
$483$	67.0820	+	67.0820i	3.05234	+	3.05234i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	15.6525	−	15.6525i	0.709281	−	0.709281i	−0.257103	−	0.966384i	$-0.582768\pi$
$487$	15.6525	−	15.6525i	0.709281	−	0.709281i	0.966384	+	0.257103i	$0.0827679\pi$
$488$		0			0
$489$			70.0000i			3.16551i
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$		0			0
$501$	30.0000			1.34030
$502$		0			0
$503$	6.70820	+	6.70820i	0.299104	+	0.299104i	0.840663	−	0.541559i	$-0.182166\pi$
$503$	6.70820	+	6.70820i	0.299104	+	0.299104i	−0.541559	+	0.840663i	$0.682166\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	29.0689	−	29.0689i	1.29099	−	1.29099i
$508$		0			0
$509$		−	6.00000i		−	0.265945i	−0.991120	−	0.132973i	$-0.957548\pi$
$509$		−	6.00000i		−	0.265945i	0.991120	−	0.132973i	$-0.0424523\pi$
$510$		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
$521$	42.0000			1.84005			0.920027	−	0.391856i	$-0.128167\pi$
$521$	42.0000			1.84005			0.920027	+	0.391856i	$0.128167\pi$
$522$		0			0
$523$	24.5967	+	24.5967i	1.07554	+	1.07554i	0.996903	+	0.0786374i	$0.0250569\pi$
$523$	24.5967	+	24.5967i	1.07554	+	1.07554i	0.0786374	+	0.996903i	$0.474943\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$			67.0000i			2.91304i
$530$		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
$539$		0			0
$540$		0			0
$541$	38.0000			1.63375			0.816874	−	0.576816i	$-0.195705\pi$
$541$	38.0000			1.63375			0.816874	+	0.576816i	$0.195705\pi$
$542$		0			0
$543$	4.47214	+	4.47214i	0.191918	+	0.191918i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	2.23607	−	2.23607i	0.0956074	−	0.0956074i	−0.657685	−	0.753293i	$-0.728464\pi$
$547$	2.23607	−	2.23607i	0.0956074	−	0.0956074i	0.753293	+	0.657685i	$0.228464\pi$
$548$		0			0
$549$		−	56.0000i		−	2.39002i
$550$		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
$561$		0			0
$562$		0			0
$563$	33.5410	+	33.5410i	1.41359	+	1.41359i	0.727865	+	0.685720i	$0.240513\pi$
$563$	33.5410	+	33.5410i	1.41359	+	1.41359i	0.685720	+	0.727865i	$0.259487\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−42.4853	+	42.4853i	−1.78421	+	1.78421i
$568$		0			0
$569$		−	36.0000i		−	1.50920i	−0.656186	−	0.754599i	$-0.727831\pi$
$569$		−	36.0000i		−	1.50920i	0.656186	−	0.754599i	$-0.272169\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$577$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−30.0000			−1.24461
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−33.5410	+	33.5410i	−1.38439	+	1.38439i	−0.547733	+	0.836653i	$0.684509\pi$
$587$	−33.5410	+	33.5410i	−1.38439	+	1.38439i	−0.836653	+	0.547733i	$0.815491\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$593$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$594$		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
$601$	28.0000			1.14214			0.571072	−	0.820900i	$-0.306528\pi$
$601$	28.0000			1.14214			0.571072	+	0.820900i	$0.306528\pi$
$602$		0			0
$603$	−78.2624	−	78.2624i	−3.18709	−	3.18709i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−11.1803	+	11.1803i	−0.453796	+	0.453796i	−0.896612	−	0.442816i	$-0.853979\pi$
$607$	−11.1803	+	11.1803i	−0.453796	+	0.453796i	0.442816	+	0.896612i	$0.353979\pi$
$608$		0			0
$609$			60.0000i			2.43132i
$610$		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
$617$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$617$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$	−120.000			−4.81543
$622$		0			0
$623$	13.4164	+	13.4164i	0.537517	+	0.537517i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−12.0000			−0.473972			−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000			−0.473972			−0.236986	+	0.971513i	$0.576159\pi$
$642$		0			0
$643$	−29.0689	−	29.0689i	−1.14636	−	1.14636i	−0.987262	−	0.159103i	$-0.949140\pi$
$643$	−29.0689	−	29.0689i	−1.14636	−	1.14636i	−0.159103	−	0.987262i	$-0.550860\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	33.5410	−	33.5410i	1.31863	−	1.31863i	0.403775	−	0.914858i	$-0.367698\pi$
$647$	33.5410	−	33.5410i	1.31863	−	1.31863i	0.914858	−	0.403775i	$-0.132302\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$653$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	32.0000			1.24466			0.622328	−	0.782757i	$-0.286187\pi$
$661$	32.0000			1.24466			0.622328	+	0.782757i	$0.286187\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−40.2492	+	40.2492i	−1.55846	+	1.55846i
$668$		0			0
$669$		−	10.0000i		−	0.386622i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$673$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$677$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−90.0000			−3.44881
$682$		0			0
$683$	20.1246	+	20.1246i	0.770047	+	0.770047i	0.978114	−	0.208068i	$-0.0667174\pi$
$683$	20.1246	+	20.1246i	0.770047	+	0.770047i	−0.208068	+	0.978114i	$0.566717\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	31.3050	−	31.3050i	1.19436	−	1.19436i
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−48.0000			−1.81293			−0.906467	−	0.422276i	$-0.861231\pi$
$701$	−48.0000			−1.81293			−0.906467	+	0.422276i	$0.861231\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	40.2492	−	40.2492i	1.51373	−	1.51373i
$708$		0			0
$709$		−	46.0000i		−	1.72757i	−0.503864	−	0.863783i	$-0.668089\pi$
$709$		−	46.0000i		−	1.72757i	0.503864	−	0.863783i	$-0.331911\pi$
$710$		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
$721$	−50.0000			−1.86210
$722$		0			0
$723$	−62.6099	−	62.6099i	−2.32849	−	2.32849i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	38.0132	−	38.0132i	1.40983	−	1.40983i	0.649283	−	0.760547i	$-0.275069\pi$
$727$	38.0132	−	38.0132i	1.40983	−	1.40983i	0.760547	−	0.649283i	$-0.224931\pi$
$728$		0			0
$729$		−	13.0000i		−	0.481481i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$733$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−33.5410	−	33.5410i	−1.23050	−	1.23050i	−0.963772	−	0.266729i	$-0.914057\pi$
$743$	−33.5410	−	33.5410i	−1.23050	−	1.23050i	−0.266729	−	0.963772i	$-0.585943\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	46.9574	−	46.9574i	1.71808	−	1.71808i
$748$		0			0
$749$			30.0000i			1.09618i
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$757$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	42.0000			1.52250			0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000			1.52250			0.761249	+	0.648459i	$0.224586\pi$
$762$		0			0
$763$	35.7771	+	35.7771i	1.29522	+	1.29522i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		−	14.0000i		−	0.504853i	−0.967616	−	0.252426i	$-0.918771\pi$
$769$		−	14.0000i		−	0.504853i	0.967616	−	0.252426i	$-0.0812286\pi$
$770$		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
$781$		0			0
$782$		0			0
$783$	−53.6656	−	53.6656i	−1.91785	−	1.91785i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	29.0689	−	29.0689i	1.03619	−	1.03619i	0.0368739	−	0.999320i	$-0.488260\pi$
$787$	29.0689	−	29.0689i	1.03619	−	1.03619i	0.999320	−	0.0368739i	$-0.0117400\pi$
$788$		0			0
$789$			90.0000i			3.20408i
$790$		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
$801$	−42.0000			−1.48400
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−53.6656	+	53.6656i	−1.88912	+	1.88912i
$808$		0			0
$809$		−	54.0000i		−	1.89854i	−0.314464	−	0.949269i	$-0.601825\pi$
$809$		−	54.0000i		−	1.89854i	0.314464	−	0.949269i	$-0.398175\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		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
$821$	−48.0000			−1.67521			−0.837606	−	0.546275i	$-0.816045\pi$
$821$	−48.0000			−1.67521			−0.837606	+	0.546275i	$0.816045\pi$
$822$		0			0
$823$	11.1803	+	11.1803i	0.389722	+	0.389722i	0.874588	−	0.484866i	$-0.161132\pi$
$823$	11.1803	+	11.1803i	0.389722	+	0.389722i	−0.484866	+	0.874588i	$0.661132\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	33.5410	−	33.5410i	1.16634	−	1.16634i	0.183274	−	0.983062i	$-0.441331\pi$
$827$	33.5410	−	33.5410i	1.16634	−	1.16634i	0.983062	−	0.183274i	$-0.0586694\pi$
$828$		0			0
$829$			56.0000i			1.94496i	0.232986	+	0.972480i	$0.425151\pi$
$829$			56.0000i			1.94496i	−0.232986	+	0.972480i	$0.574849\pi$
$830$		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			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−7.00000			−0.241379
$842$		0			0
$843$	26.8328	+	26.8328i	0.924171	+	0.924171i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	24.5967	−	24.5967i	0.845154	−	0.845154i
$848$		0			0
$849$			50.0000i			1.71600i
$850$		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
$857$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$857$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$858$		0			0
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$		0			0
$861$	120.000			4.08959
$862$		0			0
$863$	−6.70820	−	6.70820i	−0.228350	−	0.228350i	0.583653	−	0.812003i	$-0.301623\pi$
$863$	−6.70820	−	6.70820i	−0.228350	−	0.228350i	−0.812003	+	0.583653i	$0.801623\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−38.0132	+	38.0132i	−1.29099	+	1.29099i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		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
$881$	−12.0000			−0.404290			−0.202145	−	0.979356i	$-0.564791\pi$
$881$	−12.0000			−0.404290			−0.202145	+	0.979356i	$0.564791\pi$
$882$		0			0
$883$	15.6525	+	15.6525i	0.526748	+	0.526748i	0.919601	−	0.392853i	$-0.128512\pi$
$883$	15.6525	+	15.6525i	0.526748	+	0.526748i	−0.392853	+	0.919601i	$0.628512\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	20.1246	−	20.1246i	0.675718	−	0.675718i	−0.283310	−	0.959028i	$-0.591433\pi$
$887$	20.1246	−	20.1246i	0.675718	−	0.675718i	0.959028	+	0.283310i	$0.0914325\pi$
$888$		0			0
$889$			70.0000i			2.34772i
$890$		0			0
$891$		0			0
$892$		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
$903$	−22.3607	−	22.3607i	−0.744117	−	0.744117i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	42.4853	−	42.4853i	1.41070	−	1.41070i	0.655544	−	0.755157i	$-0.272439\pi$
$907$	42.4853	−	42.4853i	1.41070	−	1.41070i	0.755157	−	0.655544i	$-0.227561\pi$
$908$		0			0
$909$			126.000i			4.17916i
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$	−110.000			−3.62462
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	78.2624	−	78.2624i	2.57047	−	2.57047i
$928$		0			0
$929$		−	36.0000i		−	1.18112i	−0.806993	−	0.590561i	$-0.798907\pi$
$929$		−	36.0000i		−	1.18112i	0.806993	−	0.590561i	$-0.201093\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$937$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−42.0000			−1.36916			−0.684580	−	0.728937i	$-0.740015\pi$
$941$	−42.0000			−1.36916			−0.684580	+	0.728937i	$0.740015\pi$
$942$		0			0
$943$	80.4984	+	80.4984i	2.62139	+	2.62139i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	6.70820	−	6.70820i	0.217987	−	0.217987i	−0.589662	−	0.807650i	$-0.700739\pi$
$947$	6.70820	−	6.70820i	0.217987	−	0.217987i	0.807650	+	0.589662i	$0.200739\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$953$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$954$		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
$963$	−46.9574	−	46.9574i	−1.51318	−	1.51318i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−29.0689	+	29.0689i	−0.934792	+	0.934792i	−0.998000	−	0.0632081i	$-0.979867\pi$
$967$	−29.0689	+	29.0689i	−0.934792	+	0.934792i	0.0632081	+	0.998000i	$0.479867\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$977$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−112.000			−3.57588
$982$		0			0
$983$	33.5410	+	33.5410i	1.06979	+	1.06979i	0.997374	+	0.0724180i	$0.0230716\pi$
$983$	33.5410	+	33.5410i	1.06979	+	1.06979i	0.0724180	+	0.997374i	$0.476928\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−67.0820	+	67.0820i	−2.13524	+	2.13524i
$988$		0			0
$989$		−	30.0000i		−	0.953945i
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$997$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.2.n.q.1407.1		4
4.3	odd	2	inner	1600.2.n.q.1407.2		4
5.2	odd	4	inner	1600.2.n.q.1343.1		4
5.3	odd	4	inner	1600.2.n.q.1343.2		4
5.4	even	2	inner	1600.2.n.q.1407.2		4
8.3	odd	2		400.2.n.c.207.1	yes	4
8.5	even	2		400.2.n.c.207.2	yes	4
20.3	even	4	inner	1600.2.n.q.1343.1		4
20.7	even	4	inner	1600.2.n.q.1343.2		4
20.19	odd	2	CM	1600.2.n.q.1407.1		4
24.5	odd	2		3600.2.x.f.3007.2		4
24.11	even	2		3600.2.x.f.3007.1		4
40.3	even	4		400.2.n.c.143.2	yes	4
40.13	odd	4		400.2.n.c.143.1	✓	4
40.19	odd	2		400.2.n.c.207.2	yes	4
40.27	even	4		400.2.n.c.143.1	✓	4
40.29	even	2		400.2.n.c.207.1	yes	4
40.37	odd	4		400.2.n.c.143.2	yes	4
120.29	odd	2		3600.2.x.f.3007.1		4
120.53	even	4		3600.2.x.f.2143.1		4
120.59	even	2		3600.2.x.f.3007.2		4
120.77	even	4		3600.2.x.f.2143.2		4
120.83	odd	4		3600.2.x.f.2143.2		4
120.107	odd	4		3600.2.x.f.2143.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.n.c.143.1	✓	4	40.13	odd	4
400.2.n.c.143.1	✓	4	40.27	even	4
400.2.n.c.143.2	yes	4	40.3	even	4
400.2.n.c.143.2	yes	4	40.37	odd	4
400.2.n.c.207.1	yes	4	8.3	odd	2
400.2.n.c.207.1	yes	4	40.29	even	2
400.2.n.c.207.2	yes	4	8.5	even	2
400.2.n.c.207.2	yes	4	40.19	odd	2
1600.2.n.q.1343.1		4	5.2	odd	4	inner
1600.2.n.q.1343.1		4	20.3	even	4	inner
1600.2.n.q.1343.2		4	5.3	odd	4	inner
1600.2.n.q.1343.2		4	20.7	even	4	inner
1600.2.n.q.1407.1		4	1.1	even	1	trivial
1600.2.n.q.1407.1		4	20.19	odd	2	CM
1600.2.n.q.1407.2		4	4.3	odd	2	inner
1600.2.n.q.1407.2		4	5.4	even	2	inner
3600.2.x.f.2143.1		4	120.53	even	4
3600.2.x.f.2143.1		4	120.107	odd	4
3600.2.x.f.2143.2		4	120.77	even	4
3600.2.x.f.2143.2		4	120.83	odd	4
3600.2.x.f.3007.1		4	24.11	even	2
3600.2.x.f.3007.1		4	120.29	odd	2
3600.2.x.f.3007.2		4	24.5	odd	2
3600.2.x.f.3007.2		4	120.59	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1600.n (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.23607 - 2.23607i) q^{3} +(2.23607 - 2.23607i) q^{7} +7.00000i q^{9} +O(q^{10})\)	\(q+(-2.23607 - 2.23607i) q^{3} +(2.23607 - 2.23607i) q^{7} +7.00000i q^{9} -10.0000 q^{21} +(-6.70820 - 6.70820i) q^{23} +(8.94427 - 8.94427i) q^{27} -6.00000i q^{29} -12.0000 q^{41} +(2.23607 + 2.23607i) q^{43} +(6.70820 - 6.70820i) q^{47} -3.00000i q^{49} -8.00000 q^{61} +(15.6525 + 15.6525i) q^{63} +(-11.1803 + 11.1803i) q^{67} +30.0000i q^{69} -19.0000 q^{81} +(-6.70820 - 6.70820i) q^{83} +(-13.4164 + 13.4164i) q^{87} +6.00000i q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 40 q^{21} - 48 q^{41} - 32 q^{61} - 76 q^{81}+O(q^{100}) \) 4 * q - 40 * q^21 - 48 * q^41 - 32 * q^61 - 76 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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