LMFDB - Embedded newform 825.2.f.b.626.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,2,Mod(626,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("825.626");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	825.f (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$6.58765816676$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 9 $ x^4 - 5*x^2 + 9
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 165)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			626.1
Root			$1.65831 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	825.626
Dual form			825.2.f.b.626.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.65831 - 0.500000i) q^{3} -2.00000 q^{4} +(2.50000 + 1.65831i) q^{9} +O(q^{10})$	$q+(-1.65831 - 0.500000i) q^{3} -2.00000 q^{4} +(2.50000 + 1.65831i) q^{9} +3.31662i q^{11} +(3.31662 + 1.00000i) q^{12} +4.00000 q^{16} -9.00000i q^{23} +(-3.31662 - 4.00000i) q^{27} -5.00000 q^{31} +(1.65831 - 5.50000i) q^{33} +(-5.00000 - 3.31662i) q^{36} -9.94987 q^{37} -6.63325i q^{44} -12.0000i q^{47} +(-6.63325 - 2.00000i) q^{48} +7.00000 q^{49} -6.00000i q^{53} -3.31662i q^{59} -8.00000 q^{64} +9.94987 q^{67} +(-4.50000 + 14.9248i) q^{69} -16.5831i q^{71} +(3.50000 + 8.29156i) q^{81} -16.5831i q^{89} +18.0000i q^{92} +(8.29156 + 2.50000i) q^{93} +9.94987 q^{97} +(-5.50000 + 8.29156i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4} + 10 q^{9}+O(q^{10}) $ 4 * q - 8 * q^4 + 10 * q^9	$ 4 q - 8 q^{4} + 10 q^{9} + 16 q^{16} - 20 q^{31} - 20 q^{36} + 28 q^{49} - 32 q^{64} - 18 q^{69} + 14 q^{81} - 22 q^{99}+O(q^{100}) $ 4 * q - 8 * q^4 + 10 * q^9 + 16 * q^16 - 20 * q^31 - 20 * q^36 + 28 * q^49 - 32 * q^64 - 18 * q^69 + 14 * q^81 - 22 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$.

$n$	$376$	$551$	$727$
$\chi(n)$	$-1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0			−	1.00000i	$-0.5\pi$
$2$		0			0				1.00000i	$0.5\pi$
$3$	−1.65831	−	0.500000i	−0.957427	−	0.288675i
$3$	−1.65831	−	0.500000i	−0.957427	−	0.288675i
$4$	−2.00000			−1.00000
$5$		0			0
$5$		0			0
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$	2.50000	+	1.65831i	0.833333	+	0.552771i
$10$		0			0
$11$			3.31662i			1.00000i
$11$			3.31662i			1.00000i
$12$	3.31662	+	1.00000i	0.957427	+	0.288675i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$		0			0
$16$	4.00000			1.00000
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	9.00000i		−	1.87663i	−0.345782	−	0.938315i	$-0.612386\pi$
$23$		−	9.00000i		−	1.87663i	0.345782	−	0.938315i	$-0.387614\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−3.31662	−	4.00000i	−0.638285	−	0.769800i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	−5.00000			−0.898027			−0.449013	−	0.893525i	$-0.648224\pi$
$31$	−5.00000			−0.898027			−0.449013	+	0.893525i	$0.648224\pi$
$32$		0			0
$33$	1.65831	−	5.50000i	0.288675	−	0.957427i
$34$		0			0
$35$		0			0
$36$	−5.00000	−	3.31662i	−0.833333	−	0.552771i
$37$	−9.94987			−1.63575			−0.817875	−	0.575396i	$-0.804848\pi$
$37$	−9.94987			−1.63575			−0.817875	+	0.575396i	$0.804848\pi$
$38$		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
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		−	6.63325i		−	1.00000i
$45$		0			0
$46$		0			0
$47$		−	12.0000i		−	1.75038i	−0.483779	−	0.875190i	$-0.660736\pi$
$47$		−	12.0000i		−	1.75038i	0.483779	−	0.875190i	$-0.339264\pi$
$48$	−6.63325	−	2.00000i	−0.957427	−	0.288675i
$49$	7.00000			1.00000
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	6.00000i		−	0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$		−	6.00000i		−	0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	3.31662i		−	0.431788i	−0.976417	−	0.215894i	$-0.930733\pi$
$59$		−	3.31662i		−	0.431788i	0.976417	−	0.215894i	$-0.0692665\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$		0			0
$64$	−8.00000			−1.00000
$65$		0			0
$66$		0			0
$67$	9.94987			1.21557			0.607785	−	0.794101i	$-0.292058\pi$
$67$	9.94987			1.21557			0.607785	+	0.794101i	$0.292058\pi$
$68$		0			0
$69$	−4.50000	+	14.9248i	−0.541736	+	1.79674i
$70$		0			0
$71$		−	16.5831i		−	1.96805i	−0.178017	−	0.984027i	$-0.556968\pi$
$71$		−	16.5831i		−	1.96805i	0.178017	−	0.984027i	$-0.443032\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	3.50000	+	8.29156i	0.388889	+	0.921285i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	16.5831i		−	1.75781i	−0.476999	−	0.878904i	$-0.658275\pi$
$89$		−	16.5831i		−	1.75781i	0.476999	−	0.878904i	$-0.341725\pi$
$90$		0			0
$91$		0			0
$92$			18.0000i			1.87663i
$93$	8.29156	+	2.50000i	0.859795	+	0.259238i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	9.94987			1.01026			0.505128	−	0.863044i	$-0.331445\pi$
$97$	9.94987			1.01026			0.505128	+	0.863044i	$0.331445\pi$
$98$		0			0
$99$	−5.50000	+	8.29156i	−0.552771	+	0.833333i
$100$		0			0
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$	−19.8997			−1.96078			−0.980390	−	0.197066i	$-0.936859\pi$
$103$	−19.8997			−1.96078			−0.980390	+	0.197066i	$0.936859\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$	6.63325	+	8.00000i	0.638285	+	0.769800i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$	16.5000	+	4.97494i	1.56611	+	0.472200i
$112$		0			0
$113$			21.0000i			1.97551i	0.156001	+	0.987757i	$0.450140\pi$
$113$			21.0000i			1.97551i	−0.156001	+	0.987757i	$0.549860\pi$
$114$		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
$123$		0			0
$124$	10.0000			0.898027
$125$		0			0
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$	−3.31662	+	11.0000i	−0.288675	+	0.957427i
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	3.00000i		−	0.256307i	−0.991754	−	0.128154i	$-0.959095\pi$
$137$		−	3.00000i		−	0.256307i	0.991754	−	0.128154i	$-0.0409051\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$	−6.00000	+	19.8997i	−0.505291	+	1.67586i
$142$		0			0
$143$		0			0
$144$	10.0000	+	6.63325i	0.833333	+	0.552771i
$145$		0			0
$146$		0			0
$147$	−11.6082	−	3.50000i	−0.957427	−	0.288675i
$148$	19.8997			1.63575
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		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
$157$	−9.94987			−0.794086			−0.397043	−	0.917800i	$-0.629964\pi$
$157$	−9.94987			−0.794086			−0.397043	+	0.917800i	$0.629964\pi$
$158$		0			0
$159$	−3.00000	+	9.94987i	−0.237915	+	0.789076i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	19.8997			1.55867			0.779334	−	0.626608i	$-0.215557\pi$
$163$	19.8997			1.55867			0.779334	+	0.626608i	$0.215557\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	13.0000			1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$		0			0
$176$			13.2665i			1.00000i
$177$	−1.65831	+	5.50000i	−0.124646	+	0.413405i
$178$		0			0
$179$		−	16.5831i		−	1.23948i	−0.784807	−	0.619740i	$-0.787238\pi$
$179$		−	16.5831i		−	1.23948i	0.784807	−	0.619740i	$-0.212762\pi$
$180$		0			0
$181$	25.0000			1.85824			0.929118	−	0.369784i	$-0.120568\pi$
$181$	25.0000			1.85824			0.929118	+	0.369784i	$0.120568\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$			24.0000i			1.75038i
$189$		0			0
$190$		0			0
$191$		−	23.2164i		−	1.67988i	−0.542681	−	0.839939i	$-0.682591\pi$
$191$		−	23.2164i		−	1.67988i	0.542681	−	0.839939i	$-0.317409\pi$
$192$	13.2665	+	4.00000i	0.957427	+	0.288675i
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$		0			0
$196$	−14.0000			−1.00000
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$	−20.0000			−1.41776			−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000			−1.41776			−0.708881	+	0.705328i	$0.750800\pi$
$200$		0			0
$201$	−16.5000	−	4.97494i	−1.16382	−	0.350905i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	14.9248	−	22.5000i	1.03735	−	1.56386i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$			12.0000i			0.824163i
$213$	−8.29156	+	27.5000i	−0.568128	+	1.88427i
$214$		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
$223$	−29.8496			−1.99888			−0.999439	−	0.0334825i	$-0.989340\pi$
$223$	−29.8496			−1.99888			−0.999439	+	0.0334825i	$0.989340\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$	−5.00000			−0.330409			−0.165205	−	0.986259i	$-0.552828\pi$
$229$	−5.00000			−0.330409			−0.165205	+	0.986259i	$0.552828\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$		0			0
$236$			6.63325i			0.431788i
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$	−1.65831	−	15.5000i	−0.106381	−	0.994325i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	16.5831i		−	1.04672i	−0.852112	−	0.523359i	$-0.824679\pi$
$251$		−	16.5831i		−	1.04672i	0.852112	−	0.523359i	$-0.175321\pi$
$252$		0			0
$253$	29.8496			1.87663
$254$		0			0
$255$		0			0
$256$	16.0000			1.00000
$257$		−	18.0000i		−	1.12281i	−0.827541	−	0.561405i	$-0.810261\pi$
$257$		−	18.0000i		−	1.12281i	0.827541	−	0.561405i	$-0.189739\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−8.29156	+	27.5000i	−0.507435	+	1.68297i
$268$	−19.8997			−1.21557
$269$		−	13.2665i		−	0.808873i	−0.914566	−	0.404436i	$-0.867468\pi$
$269$		−	13.2665i		−	0.808873i	0.914566	−	0.404436i	$-0.132532\pi$
$270$		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
$276$	9.00000	−	29.8496i	0.541736	−	1.79674i
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$	−12.5000	−	8.29156i	−0.748355	−	0.496403i
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$		0			0
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$			33.1662i			1.96805i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−17.0000			−1.00000
$290$		0			0
$291$	−16.5000	−	4.97494i	−0.967247	−	0.291636i
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	13.2665	−	11.0000i	0.769800	−	0.638285i
$298$		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
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$	33.0000	+	9.94987i	1.87730	+	0.566029i
$310$		0			0
$311$			33.1662i			1.88069i	0.340229	+	0.940343i	$0.389495\pi$
$311$			33.1662i			1.88069i	−0.340229	+	0.940343i	$0.610505\pi$
$312$		0			0
$313$	29.8496			1.68720			0.843600	−	0.536972i	$-0.180432\pi$
$313$	29.8496			1.68720			0.843600	+	0.536972i	$0.180432\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			27.0000i			1.51647i	0.651981	+	0.758236i	$0.273938\pi$
$317$			27.0000i			1.51647i	−0.651981	+	0.758236i	$0.726062\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−7.00000	−	16.5831i	−0.388889	−	0.921285i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−35.0000			−1.92377			−0.961887	−	0.273447i	$-0.911836\pi$
$331$	−35.0000			−1.92377			−0.961887	+	0.273447i	$0.911836\pi$
$332$		0			0
$333$	−24.8747	−	16.5000i	−1.36312	−	0.904194i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		0			0
$339$	10.5000	−	34.8246i	0.570282	−	1.89141i
$340$		0			0
$341$		−	16.5831i		−	0.898027i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			9.00000i			0.479022i	0.970894	+	0.239511i	$0.0769871\pi$
$353$			9.00000i			0.479022i	−0.970894	+	0.239511i	$0.923013\pi$
$354$		0			0
$355$		0			0
$356$			33.1662i			1.75781i
$357$		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
$363$	18.2414	+	5.50000i	0.957427	+	0.288675i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−9.94987			−0.519379			−0.259690	−	0.965692i	$-0.583620\pi$
$367$	−9.94987			−0.519379			−0.259690	+	0.965692i	$0.583620\pi$
$368$		−	36.0000i		−	1.87663i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−16.5831	−	5.00000i	−0.859795	−	0.259238i
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	25.0000			1.28416			0.642082	−	0.766636i	$-0.278071\pi$
$379$	25.0000			1.28416			0.642082	+	0.766636i	$0.278071\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	39.0000i		−	1.99281i	−0.0847358	−	0.996403i	$-0.527005\pi$
$383$		−	39.0000i		−	1.99281i	0.0847358	−	0.996403i	$-0.472995\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−19.8997			−1.01026
$389$			36.4829i			1.84976i	0.380265	+	0.924878i	$0.375833\pi$
$389$			36.4829i			1.84976i	−0.380265	+	0.924878i	$0.624167\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$	11.0000	−	16.5831i	0.552771	−	0.833333i
$397$	−39.7995			−1.99748			−0.998740	−	0.0501886i	$-0.984018\pi$
$397$	−39.7995			−1.99748			−0.998740	+	0.0501886i	$0.984018\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			26.5330i			1.32499i	0.749064	+	0.662497i	$0.230503\pi$
$401$			26.5330i			1.32499i	−0.749064	+	0.662497i	$0.769497\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	33.0000i		−	1.63575i
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$	−1.50000	+	4.97494i	−0.0739895	+	0.245396i
$412$	39.7995			1.96078
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$			33.1662i			1.62028i	0.586238	+	0.810139i	$0.300608\pi$
$419$			33.1662i			1.62028i	−0.586238	+	0.810139i	$0.699392\pi$
$420$		0			0
$421$	10.0000			0.487370			0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000			0.487370			0.243685	+	0.969854i	$0.421644\pi$
$422$		0			0
$423$	19.8997	−	30.0000i	0.967559	−	1.45865i
$424$		0			0
$425$		0			0
$426$		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	$0.5\pi$
$432$	−13.2665	−	16.0000i	−0.638285	−	0.769800i
$433$	−29.8496			−1.43448			−0.717241	−	0.696826i	$-0.754595\pi$
$433$	−29.8496			−1.43448			−0.717241	+	0.696826i	$0.754595\pi$
$434$		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
$441$	17.5000	+	11.6082i	0.833333	+	0.552771i
$442$		0			0
$443$		−	21.0000i		−	0.997740i	−0.866677	−	0.498870i	$-0.833748\pi$
$443$		−	21.0000i		−	0.997740i	0.866677	−	0.498870i	$-0.166252\pi$
$444$	−33.0000	−	9.94987i	−1.56611	−	0.472200i
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	16.5831i		−	0.782606i	−0.920262	−	0.391303i	$-0.872024\pi$
$449$		−	16.5831i		−	0.782606i	0.920262	−	0.391303i	$-0.127976\pi$
$450$		0			0
$451$		0			0
$452$		−	42.0000i		−	1.97551i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$	−29.8496			−1.38723			−0.693615	−	0.720346i	$-0.743983\pi$
$463$	−29.8496			−1.38723			−0.693615	+	0.720346i	$0.743983\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			3.00000i			0.138823i	0.997588	+	0.0694117i	$0.0221122\pi$
$467$			3.00000i			0.138823i	−0.997588	+	0.0694117i	$0.977888\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	16.5000	+	4.97494i	0.760280	+	0.229233i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	9.94987	−	15.0000i	0.455573	−	0.686803i
$478$		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
$484$	22.0000			1.00000
$485$		0			0
$486$		0			0
$487$	9.94987			0.450872			0.225436	−	0.974258i	$-0.427619\pi$
$487$	9.94987			0.450872			0.225436	+	0.974258i	$0.427619\pi$
$488$		0			0
$489$	−33.0000	−	9.94987i	−1.49231	−	0.449949i
$490$		0			0
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−20.0000			−0.898027
$497$		0			0
$498$		0			0
$499$	40.0000			1.79065			0.895323	−	0.445418i	$-0.146945\pi$
$499$	40.0000			1.79065			0.895323	+	0.445418i	$0.146945\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−21.5581	−	6.50000i	−0.957427	−	0.288675i
$508$		0			0
$509$		−	3.31662i		−	0.147007i	−0.997295	−	0.0735034i	$-0.976582\pi$
$509$		−	3.31662i		−	0.147007i	0.997295	−	0.0735034i	$-0.0234180\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	39.7995			1.75038
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	43.1161i		−	1.88895i	−0.328581	−	0.944476i	$-0.606570\pi$
$521$		−	43.1161i		−	1.88895i	0.328581	−	0.944476i	$-0.393430\pi$
$522$		0			0
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$	6.63325	−	22.0000i	0.288675	−	0.957427i
$529$	−58.0000			−2.52174
$530$		0			0
$531$	5.50000	−	8.29156i	0.238680	−	0.359823i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−8.29156	+	27.5000i	−0.357807	+	1.18671i
$538$		0			0
$539$			23.2164i			1.00000i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$	−41.4578	−	12.5000i	−1.77912	−	0.536426i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$			6.00000i			0.256307i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$	12.0000	−	39.7995i	0.505291	−	1.67586i
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$	−11.6082	+	38.5000i	−0.484939	+	1.60836i
$574$		0			0
$575$		0			0
$576$	−20.0000	−	13.2665i	−0.833333	−	0.552771i
$577$	9.94987			0.414219			0.207109	−	0.978318i	$-0.433594\pi$
$577$	9.94987			0.414219			0.207109	+	0.978318i	$0.433594\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	19.8997			0.824163
$584$		0			0
$585$		0			0
$586$		0			0
$587$			48.0000i			1.98117i	0.136892	+	0.990586i	$0.456289\pi$
$587$			48.0000i			1.98117i	−0.136892	+	0.990586i	$0.543711\pi$
$588$	23.2164	+	7.00000i	0.957427	+	0.288675i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−39.7995			−1.63575
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	33.1662	+	10.0000i	1.35740	+	0.409273i
$598$		0			0
$599$			33.1662i			1.35514i	0.735460	+	0.677568i	$0.236966\pi$
$599$			33.1662i			1.35514i	−0.735460	+	0.677568i	$0.763034\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$	24.8747	+	16.5000i	1.01298	+	0.671932i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			42.0000i			1.69086i	0.534089	+	0.845428i	$0.320655\pi$
$617$			42.0000i			1.69086i	−0.534089	+	0.845428i	$0.679345\pi$
$618$		0			0
$619$	−1.00000			−0.0401934			−0.0200967	−	0.999798i	$-0.506397\pi$
$619$	−1.00000			−0.0401934			−0.0200967	+	0.999798i	$0.506397\pi$
$620$		0			0
$621$	−36.0000	+	29.8496i	−1.44463	+	1.19782i
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$	19.8997			0.794086
$629$		0			0
$630$		0			0
$631$	−7.00000			−0.278666			−0.139333	−	0.990246i	$-0.544496\pi$
$631$	−7.00000			−0.278666			−0.139333	+	0.990246i	$0.544496\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$	6.00000	−	19.8997i	0.237915	−	0.789076i
$637$		0			0
$638$		0			0
$639$	27.5000	−	41.4578i	1.08788	−	1.64005i
$640$		0			0
$641$		−	23.2164i		−	0.916992i	−0.888697	−	0.458496i	$-0.848388\pi$
$641$		−	23.2164i		−	0.916992i	0.888697	−	0.458496i	$-0.151612\pi$
$642$		0			0
$643$	29.8496			1.17715			0.588577	−	0.808441i	$-0.299688\pi$
$643$	29.8496			1.17715			0.588577	+	0.808441i	$0.299688\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	27.0000i		−	1.06148i	−0.847535	−	0.530740i	$-0.821914\pi$
$647$		−	27.0000i		−	1.06148i	0.847535	−	0.530740i	$-0.178086\pi$
$648$		0			0
$649$	11.0000			0.431788
$650$		0			0
$651$		0			0
$652$	−39.7995			−1.55867
$653$			51.0000i			1.99578i	0.0648948	+	0.997892i	$0.479329\pi$
$653$			51.0000i			1.99578i	−0.0648948	+	0.997892i	$0.520671\pi$
$654$		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
$661$	−13.0000			−0.505641			−0.252821	−	0.967513i	$-0.581358\pi$
$661$	−13.0000			−0.505641			−0.252821	+	0.967513i	$0.581358\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	49.5000	+	14.9248i	1.91378	+	0.577027i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$		0			0
$676$	−26.0000			−1.00000
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	24.0000i		−	0.918334i	−0.888350	−	0.459167i	$-0.848148\pi$
$683$		−	24.0000i		−	0.918334i	0.888350	−	0.459167i	$-0.151852\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	8.29156	+	2.50000i	0.316343	+	0.0953809i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	17.0000			0.646710			0.323355	−	0.946278i	$-0.395189\pi$
$691$	17.0000			0.646710			0.323355	+	0.946278i	$0.395189\pi$
$692$		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
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$		−	26.5330i		−	1.00000i
$705$		0			0
$706$		0			0
$707$		0			0
$708$	3.31662	−	11.0000i	0.124646	−	0.413405i
$709$	−19.0000			−0.713560			−0.356780	−	0.934188i	$-0.616125\pi$
$709$	−19.0000			−0.713560			−0.356780	+	0.934188i	$0.616125\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			45.0000i			1.68526i
$714$		0			0
$715$		0			0
$716$			33.1662i			1.23948i
$717$		0			0
$718$		0			0
$719$		−	16.5831i		−	0.618446i	−0.950990	−	0.309223i	$-0.899931\pi$
$719$		−	16.5831i		−	0.618446i	0.950990	−	0.309223i	$-0.100069\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$	−50.0000			−1.85824
$725$		0			0
$726$		0			0
$727$	−9.94987			−0.369020			−0.184510	−	0.982831i	$-0.559070\pi$
$727$	−9.94987			−0.369020			−0.184510	+	0.982831i	$0.559070\pi$
$728$		0			0
$729$	−5.00000	+	26.5330i	−0.185185	+	0.982704i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			33.0000i			1.21557i
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	23.0000			0.839282			0.419641	−	0.907690i	$-0.362156\pi$
$751$	23.0000			0.839282			0.419641	+	0.907690i	$0.362156\pi$
$752$		−	48.0000i		−	1.75038i
$753$	−8.29156	+	27.5000i	−0.302161	+	1.00216i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	39.7995			1.44654			0.723269	−	0.690567i	$-0.242639\pi$
$757$	39.7995			1.44654			0.723269	+	0.690567i	$0.242639\pi$
$758$		0			0
$759$	−49.5000	−	14.9248i	−1.79674	−	0.541736i
$760$		0			0
$761$		0			0			−	1.00000i	$-0.5\pi$
$761$		0			0				1.00000i	$0.5\pi$
$762$		0			0
$763$		0			0
$764$			46.4327i			1.67988i
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−26.5330	−	8.00000i	−0.957427	−	0.288675i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	−9.00000	+	29.8496i	−0.324127	+	1.07501i
$772$		0			0
$773$			54.0000i			1.94225i	0.238581	+	0.971123i	$0.423318\pi$
$773$			54.0000i			1.94225i	−0.238581	+	0.971123i	$0.576682\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	55.0000			1.96805
$782$		0			0
$783$		0			0
$784$	28.0000			1.00000
$785$		0			0
$786$		0			0
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	40.0000			1.41776
$797$			3.00000i			0.106265i	0.998587	+	0.0531327i	$0.0169206\pi$
$797$			3.00000i			0.106265i	−0.998587	+	0.0531327i	$0.983079\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	27.5000	−	41.4578i	0.971665	−	1.46484i
$802$		0			0
$803$		0			0
$804$	33.0000	+	9.94987i	1.16382	+	0.350905i
$805$		0			0
$806$		0			0
$807$	−6.63325	+	22.0000i	−0.233501	+	0.774437i
$808$		0			0
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		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
$821$		0			0			−	1.00000i	$-0.5\pi$
$821$		0			0				1.00000i	$0.5\pi$
$822$		0			0
$823$	29.8496			1.04049			0.520246	−	0.854016i	$-0.325840\pi$
$823$	29.8496			1.04049			0.520246	+	0.854016i	$0.325840\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$	−29.8496	+	45.0000i	−1.03735	+	1.56386i
$829$	29.0000			1.00721			0.503606	−	0.863934i	$-0.332006\pi$
$829$	29.0000			1.00721			0.503606	+	0.863934i	$0.332006\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	16.5831	+	20.0000i	0.573197	+	0.691301i
$838$		0			0
$839$			36.4829i			1.25953i	0.776786	+	0.629764i	$0.216849\pi$
$839$			36.4829i			1.25953i	−0.776786	+	0.629764i	$0.783151\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		−	24.0000i		−	0.824163i
$849$		0			0
$850$		0			0
$851$			89.5489i			3.06970i
$852$	16.5831	−	55.0000i	0.568128	−	1.88427i
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		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
$859$	−31.0000			−1.05771			−0.528853	−	0.848713i	$-0.677378\pi$
$859$	−31.0000			−1.05771			−0.528853	+	0.848713i	$0.677378\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			36.0000i			1.22545i	0.790295	+	0.612727i	$0.209928\pi$
$863$			36.0000i			1.22545i	−0.790295	+	0.612727i	$0.790072\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	28.1913	+	8.50000i	0.957427	+	0.288675i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	24.8747	+	16.5000i	0.841881	+	0.558440i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		1.00000			$0$
$877$		0			0		−1.00000			$\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	16.5831i		−	0.558700i	−0.960189	−	0.279350i	$-0.909881\pi$
$881$		−	16.5831i		−	0.558700i	0.960189	−	0.279350i	$-0.0901189\pi$
$882$		0			0
$883$	−19.8997			−0.669680			−0.334840	−	0.942275i	$-0.608682\pi$
$883$	−19.8997			−0.669680			−0.334840	+	0.942275i	$0.608682\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−27.5000	+	11.6082i	−0.921285	+	0.388889i
$892$	59.6992			1.99888
$893$		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
$907$	−59.6992			−1.98228			−0.991140	−	0.132818i	$-0.957597\pi$
$907$	−59.6992			−1.98228			−0.991140	+	0.132818i	$0.957597\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			6.63325i			0.219769i	0.993944	+	0.109885i	$0.0350482\pi$
$911$			6.63325i			0.219769i	−0.993944	+	0.109885i	$0.964952\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	10.0000			0.330409
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−49.7494	−	33.0000i	−1.63398	−	1.08386i
$928$		0			0
$929$		−	53.0660i		−	1.74104i	−0.492134	−	0.870519i	$-0.663783\pi$
$929$		−	53.0660i		−	1.74104i	0.492134	−	0.870519i	$-0.336217\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	16.5831	−	55.0000i	0.542907	−	1.80062i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$	−49.5000	−	14.9248i	−1.61537	−	0.487053i
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$		0			0
$944$		−	13.2665i		−	0.431788i
$945$		0			0
$946$		0			0
$947$			57.0000i			1.85225i	0.377215	+	0.926126i	$0.376882\pi$
$947$			57.0000i			1.85225i	−0.377215	+	0.926126i	$0.623118\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	13.5000	−	44.7744i	0.437767	−	1.45191i
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−6.00000			−0.193548
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	43.1161i		−	1.38366i	−0.722059	−	0.691831i	$-0.756804\pi$
$971$		−	43.1161i		−	1.38366i	0.722059	−	0.691831i	$-0.243196\pi$
$972$	3.31662	+	31.0000i	0.106381	+	0.994325i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	27.0000i		−	0.863807i	−0.901920	−	0.431903i	$-0.857842\pi$
$977$		−	27.0000i		−	0.863807i	0.901920	−	0.431903i	$-0.142158\pi$
$978$		0			0
$979$	55.0000			1.75781
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	51.0000i		−	1.62665i	−0.581811	−	0.813324i	$-0.697656\pi$
$983$		−	51.0000i		−	1.62665i	0.581811	−	0.813324i	$-0.302344\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−20.0000			−0.635321			−0.317660	−	0.948205i	$-0.602897\pi$
$991$	−20.0000			−0.635321			−0.317660	+	0.948205i	$0.602897\pi$
$992$		0			0
$993$	58.0409	+	17.5000i	1.84187	+	0.555346i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		1.00000			$0$
$997$		0			0		−1.00000			$\pi$
$998$		0			0
$999$	33.0000	+	39.7995i	1.04407	+	1.25920i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.2.f.b.626.1		4
3.2	odd	2	inner	825.2.f.b.626.2		4
5.2	odd	4		165.2.d.a.164.2	yes	2
5.3	odd	4		165.2.d.b.164.1	yes	2
5.4	even	2	inner	825.2.f.b.626.4		4
11.10	odd	2	CM	825.2.f.b.626.1		4
15.2	even	4		165.2.d.b.164.2	yes	2
15.8	even	4		165.2.d.a.164.1	✓	2
15.14	odd	2	inner	825.2.f.b.626.3		4
33.32	even	2	inner	825.2.f.b.626.2		4
55.32	even	4		165.2.d.a.164.2	yes	2
55.43	even	4		165.2.d.b.164.1	yes	2
55.54	odd	2	inner	825.2.f.b.626.4		4
165.32	odd	4		165.2.d.b.164.2	yes	2
165.98	odd	4		165.2.d.a.164.1	✓	2
165.164	even	2	inner	825.2.f.b.626.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.d.a.164.1	✓	2	15.8	even	4
165.2.d.a.164.1	✓	2	165.98	odd	4
165.2.d.a.164.2	yes	2	5.2	odd	4
165.2.d.a.164.2	yes	2	55.32	even	4
165.2.d.b.164.1	yes	2	5.3	odd	4
165.2.d.b.164.1	yes	2	55.43	even	4
165.2.d.b.164.2	yes	2	15.2	even	4
165.2.d.b.164.2	yes	2	165.32	odd	4
825.2.f.b.626.1		4	1.1	even	1	trivial
825.2.f.b.626.1		4	11.10	odd	2	CM
825.2.f.b.626.2		4	3.2	odd	2	inner
825.2.f.b.626.2		4	33.32	even	2	inner
825.2.f.b.626.3		4	15.14	odd	2	inner
825.2.f.b.626.3		4	165.164	even	2	inner
825.2.f.b.626.4		4	5.4	even	2	inner
825.2.f.b.626.4		4	55.54	odd	2	inner

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:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	825.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.65831 - 0.500000i) q^{3} -2.00000 q^{4} +(2.50000 + 1.65831i) q^{9} +O(q^{10})\)	\(q+(-1.65831 - 0.500000i) q^{3} -2.00000 q^{4} +(2.50000 + 1.65831i) q^{9} +3.31662i q^{11} +(3.31662 + 1.00000i) q^{12} +4.00000 q^{16} -9.00000i q^{23} +(-3.31662 - 4.00000i) q^{27} -5.00000 q^{31} +(1.65831 - 5.50000i) q^{33} +(-5.00000 - 3.31662i) q^{36} -9.94987 q^{37} -6.63325i q^{44} -12.0000i q^{47} +(-6.63325 - 2.00000i) q^{48} +7.00000 q^{49} -6.00000i q^{53} -3.31662i q^{59} -8.00000 q^{64} +9.94987 q^{67} +(-4.50000 + 14.9248i) q^{69} -16.5831i q^{71} +(3.50000 + 8.29156i) q^{81} -16.5831i q^{89} +18.0000i q^{92} +(8.29156 + 2.50000i) q^{93} +9.94987 q^{97} +(-5.50000 + 8.29156i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4} + 10 q^{9}+O(q^{10}) \) 4 * q - 8 * q^4 + 10 * q^9	\( 4 q - 8 q^{4} + 10 q^{9} + 16 q^{16} - 20 q^{31} - 20 q^{36} + 28 q^{49} - 32 q^{64} - 18 q^{69} + 14 q^{81} - 22 q^{99}+O(q^{100}) \) 4 * q - 8 * q^4 + 10 * q^9 + 16 * q^16 - 20 * q^31 - 20 * q^36 + 28 * q^49 - 32 * q^64 - 18 * q^69 + 14 * q^81 - 22 * q^99

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