LMFDB - Embedded newform 3328.2.b.x.1665.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3328,2,Mod(1665,3328)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3328, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3328.1665");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3328 = 2^{8} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3328.b (of order $2$, degree $1$, 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:	$26.5742137927$
Analytic rank:	$0$
Dimension:	$4$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 416)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1665.4
Root			$0.618034i$ of defining polynomial
Character	$\chi$	$=$	3328.1665
Dual form			3328.2.b.x.1665.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.23607i q^{3} +3.00000i q^{5} +2.23607 q^{7} -2.00000 q^{9} +O(q^{10})$	$q+2.23607i q^{3} +3.00000i q^{5} +2.23607 q^{7} -2.00000 q^{9} +4.47214i q^{11} +1.00000i q^{13} -6.70820 q^{15} -3.00000 q^{17} -4.47214i q^{19} +5.00000i q^{21} -8.94427 q^{23} -4.00000 q^{25} +2.23607i q^{27} -10.0000i q^{29} -10.0000 q^{33} +6.70820i q^{35} +3.00000i q^{37} -2.23607 q^{39} -6.70820i q^{43} -6.00000i q^{45} -2.23607 q^{47} -2.00000 q^{49} -6.70820i q^{51} +4.00000i q^{53} -13.4164 q^{55} +10.0000 q^{57} +4.47214i q^{59} -4.47214 q^{63} -3.00000 q^{65} +13.4164i q^{67} -20.0000i q^{69} +6.70820 q^{71} -14.0000 q^{73} -8.94427i q^{75} +10.0000i q^{77} +8.94427 q^{79} -11.0000 q^{81} +17.8885i q^{83} -9.00000i q^{85} +22.3607 q^{87} +10.0000 q^{89} +2.23607i q^{91} +13.4164 q^{95} -2.00000 q^{97} -8.94427i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{9}+O(q^{10}) $ 4 * q - 8 * q^9	$ 4 q - 8 q^{9} - 12 q^{17} - 16 q^{25} - 40 q^{33} - 8 q^{49} + 40 q^{57} - 12 q^{65} - 56 q^{73} - 44 q^{81} + 40 q^{89} - 8 q^{97}+O(q^{100}) $ 4 * q - 8 * q^9 - 12 * q^17 - 16 * q^25 - 40 * q^33 - 8 * q^49 + 40 * q^57 - 12 * q^65 - 56 * q^73 - 44 * q^81 + 40 * q^89 - 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3328\mathbb{Z}\right)^\times$.

$n$	$261$	$769$	$1535$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	2.23607i	1.29099i	0.763763	+	0.645497i	$0.223350\pi$
$3$	2.23607i	1.29099i	−0.763763	+	0.645497i	$0.776650\pi$
$4$	0	0
$5$	3.00000i	1.34164i	0.741620	+	0.670820i	$0.234058\pi$
$5$	3.00000i	1.34164i	−0.741620	+	0.670820i	$0.765942\pi$
$6$	0	0
$7$	2.23607	0.845154	0.422577	−	0.906327i	$-0.361126\pi$
$7$	2.23607	0.845154	0.422577	+	0.906327i	$0.361126\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	4.47214i	1.34840i	0.738549	+	0.674200i	$0.235511\pi$
$11$	4.47214i	1.34840i	−0.738549	+	0.674200i	$0.764489\pi$
$12$	0	0
$13$	1.00000i	0.277350i
$13$	1.00000i	0.277350i
$14$	0	0
$15$	−6.70820	−1.73205
$16$	0	0
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	− 4.47214i	− 1.02598i	−0.858395	−	0.512989i	$-0.828538\pi$
$19$	− 4.47214i	− 1.02598i	0.858395	−	0.512989i	$-0.171462\pi$
$20$	0	0
$21$	5.00000i	1.09109i
$22$	0	0
$23$	−8.94427	−1.86501	−0.932505	−	0.361158i	$-0.882382\pi$
$23$	−8.94427	−1.86501	−0.932505	+	0.361158i	$0.882382\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	2.23607i	0.430331i
$28$	0	0
$29$	− 10.0000i	− 1.85695i	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	− 10.0000i	− 1.85695i	0.371391	−	0.928477i	$-0.378881\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	−10.0000	−1.74078
$34$	0	0
$35$	6.70820i	1.13389i
$36$	0	0
$37$	3.00000i	0.493197i	0.969118	+	0.246598i	$0.0793129\pi$
$37$	3.00000i	0.493197i	−0.969118	+	0.246598i	$0.920687\pi$
$38$	0	0
$39$	−2.23607	−0.358057
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	− 6.70820i	− 1.02299i	−0.859286	−	0.511496i	$-0.829092\pi$
$43$	− 6.70820i	− 1.02299i	0.859286	−	0.511496i	$-0.170908\pi$
$44$	0	0
$45$	− 6.00000i	− 0.894427i
$46$	0	0
$47$	−2.23607	−0.326164	−0.163082	−	0.986613i	$-0.552144\pi$
$47$	−2.23607	−0.326164	−0.163082	+	0.986613i	$0.552144\pi$
$48$	0	0
$49$	−2.00000	−0.285714
$50$	0	0
$51$	− 6.70820i	− 0.939336i
$52$	0	0
$53$	4.00000i	0.549442i	0.961524	+	0.274721i	$0.0885855\pi$
$53$	4.00000i	0.549442i	−0.961524	+	0.274721i	$0.911414\pi$
$54$	0	0
$55$	−13.4164	−1.80907
$56$	0	0
$57$	10.0000	1.32453
$58$	0	0
$59$	4.47214i	0.582223i	0.956689	+	0.291111i	$0.0940250\pi$
$59$	4.47214i	0.582223i	−0.956689	+	0.291111i	$0.905975\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	−4.47214	−0.563436
$64$	0	0
$65$	−3.00000	−0.372104
$66$	0	0
$67$	13.4164i	1.63908i	0.573025	+	0.819538i	$0.305770\pi$
$67$	13.4164i	1.63908i	−0.573025	+	0.819538i	$0.694230\pi$
$68$	0	0
$69$	− 20.0000i	− 2.40772i
$70$	0	0
$71$	6.70820	0.796117	0.398059	−	0.917360i	$-0.369684\pi$
$71$	6.70820	0.796117	0.398059	+	0.917360i	$0.369684\pi$
$72$	0	0
$73$	−14.0000	−1.63858	−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000	−1.63858	−0.819288	+	0.573382i	$0.805631\pi$
$74$	0	0
$75$	− 8.94427i	− 1.03280i
$76$	0	0
$77$	10.0000i	1.13961i
$78$	0	0
$79$	8.94427	1.00631	0.503155	−	0.864196i	$-0.332173\pi$
$79$	8.94427	1.00631	0.503155	+	0.864196i	$0.332173\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	17.8885i	1.96352i	0.190117	+	0.981761i	$0.439113\pi$
$83$	17.8885i	1.96352i	−0.190117	+	0.981761i	$0.560887\pi$
$84$	0	0
$85$	− 9.00000i	− 0.976187i
$86$	0	0
$87$	22.3607	2.39732
$88$	0	0
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	2.23607i	0.234404i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	13.4164	1.37649
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	− 8.94427i	− 0.898933i
$100$	0	0
$101$	− 12.0000i	− 1.19404i	−0.802225	−	0.597022i	$-0.796350\pi$
$101$	− 12.0000i	− 1.19404i	0.802225	−	0.597022i	$-0.203650\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	−15.0000	−1.46385
$106$	0	0
$107$	17.8885i	1.72935i	0.502331	+	0.864675i	$0.332476\pi$
$107$	17.8885i	1.72935i	−0.502331	+	0.864675i	$0.667524\pi$
$108$	0	0
$109$	− 11.0000i	− 1.05361i	−0.849987	−	0.526804i	$-0.823390\pi$
$109$	− 11.0000i	− 1.05361i	0.849987	−	0.526804i	$-0.176610\pi$
$110$	0	0
$111$	−6.70820	−0.636715
$112$	0	0
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	0	0
$115$	− 26.8328i	− 2.50217i
$116$	0	0
$117$	− 2.00000i	− 0.184900i
$118$	0	0
$119$	−6.70820	−0.614940
$120$	0	0
$121$	−9.00000	−0.818182
$122$	0	0
$123$	0	0
$124$	0	0
$125$	3.00000i	0.268328i
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0	0
$129$	15.0000	1.32068
$130$	0	0
$131$	15.6525i	1.36756i	0.729687	+	0.683782i	$0.239666\pi$
$131$	15.6525i	1.36756i	−0.729687	+	0.683782i	$0.760334\pi$
$132$	0	0
$133$	− 10.0000i	− 0.867110i
$134$	0	0
$135$	−6.70820	−0.577350
$136$	0	0
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	0	0
$139$	11.1803i	0.948304i	0.880443	+	0.474152i	$0.157245\pi$
$139$	11.1803i	0.948304i	−0.880443	+	0.474152i	$0.842755\pi$
$140$	0	0
$141$	− 5.00000i	− 0.421076i
$142$	0	0
$143$	−4.47214	−0.373979
$144$	0	0
$145$	30.0000	2.49136
$146$	0	0
$147$	− 4.47214i	− 0.368856i
$148$	0	0
$149$	14.0000i	1.14692i	0.819232	+	0.573462i	$0.194400\pi$
$149$	14.0000i	1.14692i	−0.819232	+	0.573462i	$0.805600\pi$
$150$	0	0
$151$	6.70820	0.545906	0.272953	−	0.962027i	$-0.412000\pi$
$151$	6.70820	0.545906	0.272953	+	0.962027i	$0.412000\pi$
$152$	0	0
$153$	6.00000	0.485071
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 22.0000i	− 1.75579i	−0.478852	−	0.877896i	$-0.658947\pi$
$157$	− 22.0000i	− 1.75579i	0.478852	−	0.877896i	$-0.341053\pi$
$158$	0	0
$159$	−8.94427	−0.709327
$160$	0	0
$161$	−20.0000	−1.57622
$162$	0	0
$163$	8.94427i	0.700569i	0.936643	+	0.350285i	$0.113915\pi$
$163$	8.94427i	0.700569i	−0.936643	+	0.350285i	$0.886085\pi$
$164$	0	0
$165$	− 30.0000i	− 2.33550i
$166$	0	0
$167$	−8.94427	−0.692129	−0.346064	−	0.938211i	$-0.612482\pi$
$167$	−8.94427	−0.692129	−0.346064	+	0.938211i	$0.612482\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	8.94427i	0.683986i
$172$	0	0
$173$	− 4.00000i	− 0.304114i	−0.988372	−	0.152057i	$-0.951410\pi$
$173$	− 4.00000i	− 0.304114i	0.988372	−	0.152057i	$-0.0485898\pi$
$174$	0	0
$175$	−8.94427	−0.676123
$176$	0	0
$177$	−10.0000	−0.751646
$178$	0	0
$179$	− 11.1803i	− 0.835658i	−0.908526	−	0.417829i	$-0.862791\pi$
$179$	− 11.1803i	− 0.835658i	0.908526	−	0.417829i	$-0.137209\pi$
$180$	0	0
$181$	− 8.00000i	− 0.594635i	−0.954779	−	0.297318i	$-0.903908\pi$
$181$	− 8.00000i	− 0.594635i	0.954779	−	0.297318i	$-0.0960920\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−9.00000	−0.661693
$186$	0	0
$187$	− 13.4164i	− 0.981105i
$188$	0	0
$189$	5.00000i	0.363696i
$190$	0	0
$191$	−4.47214	−0.323592	−0.161796	−	0.986824i	$-0.551729\pi$
$191$	−4.47214	−0.323592	−0.161796	+	0.986824i	$0.551729\pi$
$192$	0	0
$193$	16.0000	1.15171	0.575853	−	0.817554i	$-0.304670\pi$
$193$	16.0000	1.15171	0.575853	+	0.817554i	$0.304670\pi$
$194$	0	0
$195$	− 6.70820i	− 0.480384i
$196$	0	0
$197$	− 7.00000i	− 0.498729i	−0.968410	−	0.249365i	$-0.919778\pi$
$197$	− 7.00000i	− 0.498729i	0.968410	−	0.249365i	$-0.0802218\pi$
$198$	0	0
$199$	13.4164	0.951064	0.475532	−	0.879698i	$-0.342256\pi$
$199$	13.4164	0.951064	0.475532	+	0.879698i	$0.342256\pi$
$200$	0	0
$201$	−30.0000	−2.11604
$202$	0	0
$203$	− 22.3607i	− 1.56941i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	17.8885	1.24334
$208$	0	0
$209$	20.0000	1.38343
$210$	0	0
$211$	15.6525i	1.07756i	0.842446	+	0.538780i	$0.181115\pi$
$211$	15.6525i	1.07756i	−0.842446	+	0.538780i	$0.818885\pi$
$212$	0	0
$213$	15.0000i	1.02778i
$214$	0	0
$215$	20.1246	1.37249
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 31.3050i	− 2.11539i
$220$	0	0
$221$	− 3.00000i	− 0.201802i
$222$	0	0
$223$	2.23607	0.149738	0.0748691	−	0.997193i	$-0.476146\pi$
$223$	2.23607	0.149738	0.0748691	+	0.997193i	$0.476146\pi$
$224$	0	0
$225$	8.00000	0.533333
$226$	0	0
$227$	− 17.8885i	− 1.18730i	−0.804722	−	0.593652i	$-0.797686\pi$
$227$	− 17.8885i	− 1.18730i	0.804722	−	0.593652i	$-0.202314\pi$
$228$	0	0
$229$	5.00000i	0.330409i	0.986259	+	0.165205i	$0.0528285\pi$
$229$	5.00000i	0.330409i	−0.986259	+	0.165205i	$0.947172\pi$
$230$	0	0
$231$	−22.3607	−1.47122
$232$	0	0
$233$	−1.00000	−0.0655122	−0.0327561	−	0.999463i	$-0.510428\pi$
$233$	−1.00000	−0.0655122	−0.0327561	+	0.999463i	$0.510428\pi$
$234$	0	0
$235$	− 6.70820i	− 0.437595i
$236$	0	0
$237$	20.0000i	1.29914i
$238$	0	0
$239$	15.6525	1.01247	0.506237	−	0.862394i	$-0.331036\pi$
$239$	15.6525	1.01247	0.506237	+	0.862394i	$0.331036\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	0	0
$243$	− 17.8885i	− 1.14755i
$244$	0	0
$245$	− 6.00000i	− 0.383326i
$246$	0	0
$247$	4.47214	0.284555
$248$	0	0
$249$	−40.0000	−2.53490
$250$	0	0
$251$	− 8.94427i	− 0.564557i	−0.959332	−	0.282279i	$-0.908910\pi$
$251$	− 8.94427i	− 0.564557i	0.959332	−	0.282279i	$-0.0910903\pi$
$252$	0	0
$253$	− 40.0000i	− 2.51478i
$254$	0	0
$255$	20.1246	1.26025
$256$	0	0
$257$	−23.0000	−1.43470	−0.717350	−	0.696713i	$-0.754645\pi$
$257$	−23.0000	−1.43470	−0.717350	+	0.696713i	$0.754645\pi$
$258$	0	0
$259$	6.70820i	0.416828i
$260$	0	0
$261$	20.0000i	1.23797i
$262$	0	0
$263$	8.94427	0.551527	0.275764	−	0.961225i	$-0.411069\pi$
$263$	8.94427	0.551527	0.275764	+	0.961225i	$0.411069\pi$
$264$	0	0
$265$	−12.0000	−0.737154
$266$	0	0
$267$	22.3607i	1.36845i
$268$	0	0
$269$	16.0000i	0.975537i	0.872973	+	0.487769i	$0.162189\pi$
$269$	16.0000i	0.975537i	−0.872973	+	0.487769i	$0.837811\pi$
$270$	0	0
$271$	−20.1246	−1.22248	−0.611242	−	0.791444i	$-0.709330\pi$
$271$	−20.1246	−1.22248	−0.611242	+	0.791444i	$0.709330\pi$
$272$	0	0
$273$	−5.00000	−0.302614
$274$	0	0
$275$	− 17.8885i	− 1.07872i
$276$	0	0
$277$	− 28.0000i	− 1.68236i	−0.540758	−	0.841178i	$-0.681862\pi$
$277$	− 28.0000i	− 1.68236i	0.540758	−	0.841178i	$-0.318138\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	0	0
$283$	− 17.8885i	− 1.06336i	−0.846944	−	0.531682i	$-0.821560\pi$
$283$	− 17.8885i	− 1.06336i	0.846944	−	0.531682i	$-0.178440\pi$
$284$	0	0
$285$	30.0000i	1.77705i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	− 4.47214i	− 0.262161i
$292$	0	0
$293$	31.0000i	1.81104i	0.424304	+	0.905520i	$0.360519\pi$
$293$	31.0000i	1.81104i	−0.424304	+	0.905520i	$0.639481\pi$
$294$	0	0
$295$	−13.4164	−0.781133
$296$	0	0
$297$	−10.0000	−0.580259
$298$	0	0
$299$	− 8.94427i	− 0.517261i
$300$	0	0
$301$	− 15.0000i	− 0.864586i
$302$	0	0
$303$	26.8328	1.54150
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 4.47214i	− 0.255238i	−0.991823	−	0.127619i	$-0.959266\pi$
$307$	− 4.47214i	− 0.255238i	0.991823	−	0.127619i	$-0.0407335\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	4.47214	0.253592	0.126796	−	0.991929i	$-0.459531\pi$
$311$	4.47214	0.253592	0.126796	+	0.991929i	$0.459531\pi$
$312$	0	0
$313$	29.0000	1.63918	0.819588	−	0.572953i	$-0.194202\pi$
$313$	29.0000	1.63918	0.819588	+	0.572953i	$0.194202\pi$
$314$	0	0
$315$	− 13.4164i	− 0.755929i
$316$	0	0
$317$	18.0000i	1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$	18.0000i	1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$	0	0
$319$	44.7214	2.50392
$320$	0	0
$321$	−40.0000	−2.23258
$322$	0	0
$323$	13.4164i	0.746509i
$324$	0	0
$325$	− 4.00000i	− 0.221880i
$326$	0	0
$327$	24.5967	1.36020
$328$	0	0
$329$	−5.00000	−0.275659
$330$	0	0
$331$	26.8328i	1.47486i	0.675421	+	0.737432i	$0.263962\pi$
$331$	26.8328i	1.47486i	−0.675421	+	0.737432i	$0.736038\pi$
$332$	0	0
$333$	− 6.00000i	− 0.328798i
$334$	0	0
$335$	−40.2492	−2.19905
$336$	0	0
$337$	3.00000	0.163420	0.0817102	−	0.996656i	$-0.473962\pi$
$337$	3.00000	0.163420	0.0817102	+	0.996656i	$0.473962\pi$
$338$	0	0
$339$	− 31.3050i	− 1.70025i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−20.1246	−1.08663
$344$	0	0
$345$	60.0000	3.23029
$346$	0	0
$347$	11.1803i	0.600192i	0.953909	+	0.300096i	$0.0970187\pi$
$347$	11.1803i	0.600192i	−0.953909	+	0.300096i	$0.902981\pi$
$348$	0	0
$349$	5.00000i	0.267644i	0.991005	+	0.133822i	$0.0427250\pi$
$349$	5.00000i	0.267644i	−0.991005	+	0.133822i	$0.957275\pi$
$350$	0	0
$351$	−2.23607	−0.119352
$352$	0	0
$353$	4.00000	0.212899	0.106449	−	0.994318i	$-0.466052\pi$
$353$	4.00000	0.212899	0.106449	+	0.994318i	$0.466052\pi$
$354$	0	0
$355$	20.1246i	1.06810i
$356$	0	0
$357$	− 15.0000i	− 0.793884i
$358$	0	0
$359$	8.94427	0.472061	0.236030	−	0.971746i	$-0.424154\pi$
$359$	8.94427	0.472061	0.236030	+	0.971746i	$0.424154\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	− 20.1246i	− 1.05627i
$364$	0	0
$365$	− 42.0000i	− 2.19838i
$366$	0	0
$367$	4.47214	0.233444	0.116722	−	0.993165i	$-0.462761\pi$
$367$	4.47214	0.233444	0.116722	+	0.993165i	$0.462761\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	8.94427i	0.464363i
$372$	0	0
$373$	4.00000i	0.207112i	0.994624	+	0.103556i	$0.0330221\pi$
$373$	4.00000i	0.207112i	−0.994624	+	0.103556i	$0.966978\pi$
$374$	0	0
$375$	−6.70820	−0.346410
$376$	0	0
$377$	10.0000	0.515026
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−15.6525	−0.799804	−0.399902	−	0.916558i	$-0.630956\pi$
$383$	−15.6525	−0.799804	−0.399902	+	0.916558i	$0.630956\pi$
$384$	0	0
$385$	−30.0000	−1.52894
$386$	0	0
$387$	13.4164i	0.681994i
$388$	0	0
$389$	26.0000i	1.31825i	0.752032	+	0.659126i	$0.229074\pi$
$389$	26.0000i	1.31825i	−0.752032	+	0.659126i	$0.770926\pi$
$390$	0	0
$391$	26.8328	1.35699
$392$	0	0
$393$	−35.0000	−1.76552
$394$	0	0
$395$	26.8328i	1.35011i
$396$	0	0
$397$	22.0000i	1.10415i	0.833795	+	0.552074i	$0.186163\pi$
$397$	22.0000i	1.10415i	−0.833795	+	0.552074i	$0.813837\pi$
$398$	0	0
$399$	22.3607	1.11943
$400$	0	0
$401$	−8.00000	−0.399501	−0.199750	−	0.979847i	$-0.564013\pi$
$401$	−8.00000	−0.399501	−0.199750	+	0.979847i	$0.564013\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 33.0000i	− 1.63978i
$406$	0	0
$407$	−13.4164	−0.665027
$408$	0	0
$409$	−4.00000	−0.197787	−0.0988936	−	0.995098i	$-0.531530\pi$
$409$	−4.00000	−0.197787	−0.0988936	+	0.995098i	$0.531530\pi$
$410$	0	0
$411$	26.8328i	1.32357i
$412$	0	0
$413$	10.0000i	0.492068i
$414$	0	0
$415$	−53.6656	−2.63434
$416$	0	0
$417$	−25.0000	−1.22426
$418$	0	0
$419$	2.23607i	0.109239i	0.998507	+	0.0546195i	$0.0173946\pi$
$419$	2.23607i	0.109239i	−0.998507	+	0.0546195i	$0.982605\pi$
$420$	0	0
$421$	15.0000i	0.731055i	0.930800	+	0.365528i	$0.119111\pi$
$421$	15.0000i	0.731055i	−0.930800	+	0.365528i	$0.880889\pi$
$422$	0	0
$423$	4.47214	0.217443
$424$	0	0
$425$	12.0000	0.582086
$426$	0	0
$427$	0	0
$428$	0	0
$429$	− 10.0000i	− 0.482805i
$430$	0	0
$431$	15.6525	0.753953	0.376977	−	0.926223i	$-0.376964\pi$
$431$	15.6525	0.753953	0.376977	+	0.926223i	$0.376964\pi$
$432$	0	0
$433$	−21.0000	−1.00920	−0.504598	−	0.863355i	$-0.668359\pi$
$433$	−21.0000	−1.00920	−0.504598	+	0.863355i	$0.668359\pi$
$434$	0	0
$435$	67.0820i	3.21634i
$436$	0	0
$437$	40.0000i	1.91346i
$438$	0	0
$439$	4.47214	0.213443	0.106722	−	0.994289i	$-0.465965\pi$
$439$	4.47214	0.213443	0.106722	+	0.994289i	$0.465965\pi$
$440$	0	0
$441$	4.00000	0.190476
$442$	0	0
$443$	15.6525i	0.743672i	0.928299	+	0.371836i	$0.121272\pi$
$443$	15.6525i	0.743672i	−0.928299	+	0.371836i	$0.878728\pi$
$444$	0	0
$445$	30.0000i	1.42214i
$446$	0	0
$447$	−31.3050	−1.48067
$448$	0	0
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	15.0000i	0.704761i
$454$	0	0
$455$	−6.70820	−0.314485
$456$	0	0
$457$	−2.00000	−0.0935561	−0.0467780	−	0.998905i	$-0.514895\pi$
$457$	−2.00000	−0.0935561	−0.0467780	+	0.998905i	$0.514895\pi$
$458$	0	0
$459$	− 6.70820i	− 0.313112i
$460$	0	0
$461$	− 3.00000i	− 0.139724i	−0.997557	−	0.0698620i	$-0.977744\pi$
$461$	− 3.00000i	− 0.139724i	0.997557	−	0.0698620i	$-0.0222559\pi$
$462$	0	0
$463$	−8.94427	−0.415676	−0.207838	−	0.978163i	$-0.566643\pi$
$463$	−8.94427	−0.415676	−0.207838	+	0.978163i	$0.566643\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	30.0000i	1.38527i
$470$	0	0
$471$	49.1935	2.26672
$472$	0	0
$473$	30.0000	1.37940
$474$	0	0
$475$	17.8885i	0.820783i
$476$	0	0
$477$	− 8.00000i	− 0.366295i
$478$	0	0
$479$	−20.1246	−0.919517	−0.459758	−	0.888044i	$-0.652064\pi$
$479$	−20.1246	−0.919517	−0.459758	+	0.888044i	$0.652064\pi$
$480$	0	0
$481$	−3.00000	−0.136788
$482$	0	0
$483$	− 44.7214i	− 2.03489i
$484$	0	0
$485$	− 6.00000i	− 0.272446i
$486$	0	0
$487$	−26.8328	−1.21591	−0.607955	−	0.793971i	$-0.708010\pi$
$487$	−26.8328	−1.21591	−0.607955	+	0.793971i	$0.708010\pi$
$488$	0	0
$489$	−20.0000	−0.904431
$490$	0	0
$491$	11.1803i	0.504562i	0.967654	+	0.252281i	$0.0811807\pi$
$491$	11.1803i	0.504562i	−0.967654	+	0.252281i	$0.918819\pi$
$492$	0	0
$493$	30.0000i	1.35113i
$494$	0	0
$495$	26.8328	1.20605
$496$	0	0
$497$	15.0000	0.672842
$498$	0	0
$499$	− 35.7771i	− 1.60160i	−0.598930	−	0.800801i	$-0.704407\pi$
$499$	− 35.7771i	− 1.60160i	0.598930	−	0.800801i	$-0.295593\pi$
$500$	0	0
$501$	− 20.0000i	− 0.893534i
$502$	0	0
$503$	−31.3050	−1.39582	−0.697909	−	0.716186i	$-0.745886\pi$
$503$	−31.3050	−1.39582	−0.697909	+	0.716186i	$0.745886\pi$
$504$	0	0
$505$	36.0000	1.60198
$506$	0	0
$507$	− 2.23607i	− 0.0993073i
$508$	0	0
$509$	30.0000i	1.32973i	0.746965	+	0.664863i	$0.231510\pi$
$509$	30.0000i	1.32973i	−0.746965	+	0.664863i	$0.768490\pi$
$510$	0	0
$511$	−31.3050	−1.38485
$512$	0	0
$513$	10.0000	0.441511
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 10.0000i	− 0.439799i
$518$	0	0
$519$	8.94427	0.392610
$520$	0	0
$521$	−3.00000	−0.131432	−0.0657162	−	0.997838i	$-0.520933\pi$
$521$	−3.00000	−0.131432	−0.0657162	+	0.997838i	$0.520933\pi$
$522$	0	0
$523$	17.8885i	0.782211i	0.920346	+	0.391106i	$0.127907\pi$
$523$	17.8885i	0.782211i	−0.920346	+	0.391106i	$0.872093\pi$
$524$	0	0
$525$	− 20.0000i	− 0.872872i
$526$	0	0
$527$	0	0
$528$	0	0
$529$	57.0000	2.47826
$530$	0	0
$531$	− 8.94427i	− 0.388148i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−53.6656	−2.32017
$536$	0	0
$537$	25.0000	1.07883
$538$	0	0
$539$	− 8.94427i	− 0.385257i
$540$	0	0
$541$	− 33.0000i	− 1.41878i	−0.704816	−	0.709390i	$-0.748970\pi$
$541$	− 33.0000i	− 1.41878i	0.704816	−	0.709390i	$-0.251030\pi$
$542$	0	0
$543$	17.8885	0.767671
$544$	0	0
$545$	33.0000	1.41356
$546$	0	0
$547$	38.0132i	1.62533i	0.582735	+	0.812663i	$0.301983\pi$
$547$	38.0132i	1.62533i	−0.582735	+	0.812663i	$0.698017\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−44.7214	−1.90519
$552$	0	0
$553$	20.0000	0.850487
$554$	0	0
$555$	− 20.1246i	− 0.854242i
$556$	0	0
$557$	7.00000i	0.296600i	0.988942	+	0.148300i	$0.0473800\pi$
$557$	7.00000i	0.296600i	−0.988942	+	0.148300i	$0.952620\pi$
$558$	0	0
$559$	6.70820	0.283727
$560$	0	0
$561$	30.0000	1.26660
$562$	0	0
$563$	15.6525i	0.659673i	0.944038	+	0.329837i	$0.106994\pi$
$563$	15.6525i	0.659673i	−0.944038	+	0.329837i	$0.893006\pi$
$564$	0	0
$565$	− 42.0000i	− 1.76695i
$566$	0	0
$567$	−24.5967	−1.03297
$568$	0	0
$569$	9.00000	0.377300	0.188650	−	0.982044i	$-0.439589\pi$
$569$	9.00000	0.377300	0.188650	+	0.982044i	$0.439589\pi$
$570$	0	0
$571$	24.5967i	1.02934i	0.857388	+	0.514671i	$0.172086\pi$
$571$	24.5967i	1.02934i	−0.857388	+	0.514671i	$0.827914\pi$
$572$	0	0
$573$	− 10.0000i	− 0.417756i
$574$	0	0
$575$	35.7771	1.49201
$576$	0	0
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	0	0
$579$	35.7771i	1.48685i
$580$	0	0
$581$	40.0000i	1.65948i
$582$	0	0
$583$	−17.8885	−0.740868
$584$	0	0
$585$	6.00000	0.248069
$586$	0	0
$587$	− 44.7214i	− 1.84585i	−0.384982	−	0.922924i	$-0.625792\pi$
$587$	− 44.7214i	− 1.84585i	0.384982	−	0.922924i	$-0.374208\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	15.6525	0.643857
$592$	0	0
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	0	0
$595$	− 20.1246i	− 0.825029i
$596$	0	0
$597$	30.0000i	1.22782i
$598$	0	0
$599$	31.3050	1.27909	0.639543	−	0.768755i	$-0.279124\pi$
$599$	31.3050	1.27909	0.639543	+	0.768755i	$0.279124\pi$
$600$	0	0
$601$	−5.00000	−0.203954	−0.101977	−	0.994787i	$-0.532517\pi$
$601$	−5.00000	−0.203954	−0.101977	+	0.994787i	$0.532517\pi$
$602$	0	0
$603$	− 26.8328i	− 1.09272i
$604$	0	0
$605$	− 27.0000i	− 1.09771i
$606$	0	0
$607$	−13.4164	−0.544555	−0.272278	−	0.962219i	$-0.587777\pi$
$607$	−13.4164	−0.544555	−0.272278	+	0.962219i	$0.587777\pi$
$608$	0	0
$609$	50.0000	2.02610
$610$	0	0
$611$	− 2.23607i	− 0.0904616i
$612$	0	0
$613$	26.0000i	1.05013i	0.851062	+	0.525065i	$0.175959\pi$
$613$	26.0000i	1.05013i	−0.851062	+	0.525065i	$0.824041\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	8.00000	0.322068	0.161034	−	0.986949i	$-0.448517\pi$
$617$	8.00000	0.322068	0.161034	+	0.986949i	$0.448517\pi$
$618$	0	0
$619$	− 26.8328i	− 1.07850i	−0.842145	−	0.539251i	$-0.818707\pi$
$619$	− 26.8328i	− 1.07850i	0.842145	−	0.539251i	$-0.181293\pi$
$620$	0	0
$621$	− 20.0000i	− 0.802572i
$622$	0	0
$623$	22.3607	0.895862
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	44.7214i	1.78600i
$628$	0	0
$629$	− 9.00000i	− 0.358854i
$630$	0	0
$631$	33.5410	1.33525	0.667623	−	0.744499i	$-0.267312\pi$
$631$	33.5410	1.33525	0.667623	+	0.744499i	$0.267312\pi$
$632$	0	0
$633$	−35.0000	−1.39113
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 2.00000i	− 0.0792429i
$638$	0	0
$639$	−13.4164	−0.530745
$640$	0	0
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	0	0
$643$	− 22.3607i	− 0.881819i	−0.897552	−	0.440910i	$-0.854656\pi$
$643$	− 22.3607i	− 0.881819i	0.897552	−	0.440910i	$-0.145344\pi$
$644$	0	0
$645$	45.0000i	1.77187i
$646$	0	0
$647$	40.2492	1.58236	0.791180	−	0.611583i	$-0.209467\pi$
$647$	40.2492	1.58236	0.791180	+	0.611583i	$0.209467\pi$
$648$	0	0
$649$	−20.0000	−0.785069
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 24.0000i	− 0.939193i	−0.882881	−	0.469596i	$-0.844399\pi$
$653$	− 24.0000i	− 0.939193i	0.882881	−	0.469596i	$-0.155601\pi$
$654$	0	0
$655$	−46.9574	−1.83478
$656$	0	0
$657$	28.0000	1.09238
$658$	0	0
$659$	17.8885i	0.696839i	0.937339	+	0.348419i	$0.113281\pi$
$659$	17.8885i	0.696839i	−0.937339	+	0.348419i	$0.886719\pi$
$660$	0	0
$661$	− 10.0000i	− 0.388955i	−0.980907	−	0.194477i	$-0.937699\pi$
$661$	− 10.0000i	− 0.388955i	0.980907	−	0.194477i	$-0.0623011\pi$
$662$	0	0
$663$	6.70820	0.260525
$664$	0	0
$665$	30.0000	1.16335
$666$	0	0
$667$	89.4427i	3.46324i
$668$	0	0
$669$	5.00000i	0.193311i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	29.0000	1.11787	0.558934	−	0.829212i	$-0.311211\pi$
$673$	29.0000	1.11787	0.558934	+	0.829212i	$0.311211\pi$
$674$	0	0
$675$	− 8.94427i	− 0.344265i
$676$	0	0
$677$	− 12.0000i	− 0.461197i	−0.973049	−	0.230599i	$-0.925932\pi$
$677$	− 12.0000i	− 0.461197i	0.973049	−	0.230599i	$-0.0740685\pi$
$678$	0	0
$679$	−4.47214	−0.171625
$680$	0	0
$681$	40.0000	1.53280
$682$	0	0
$683$	26.8328i	1.02673i	0.858171	+	0.513365i	$0.171601\pi$
$683$	26.8328i	1.02673i	−0.858171	+	0.513365i	$0.828399\pi$
$684$	0	0
$685$	36.0000i	1.37549i
$686$	0	0
$687$	−11.1803	−0.426557
$688$	0	0
$689$	−4.00000	−0.152388
$690$	0	0
$691$	17.8885i	0.680512i	0.940333	+	0.340256i	$0.110514\pi$
$691$	17.8885i	0.680512i	−0.940333	+	0.340256i	$0.889486\pi$
$692$	0	0
$693$	− 20.0000i	− 0.759737i
$694$	0	0
$695$	−33.5410	−1.27228
$696$	0	0
$697$	0	0
$698$	0	0
$699$	− 2.23607i	− 0.0845759i
$700$	0	0
$701$	20.0000i	0.755390i	0.925930	+	0.377695i	$0.123283\pi$
$701$	20.0000i	0.755390i	−0.925930	+	0.377695i	$0.876717\pi$
$702$	0	0
$703$	13.4164	0.506009
$704$	0	0
$705$	15.0000	0.564933
$706$	0	0
$707$	− 26.8328i	− 1.00915i
$708$	0	0
$709$	− 10.0000i	− 0.375558i	−0.982211	−	0.187779i	$-0.939871\pi$
$709$	− 10.0000i	− 0.375558i	0.982211	−	0.187779i	$-0.0601289\pi$
$710$	0	0
$711$	−17.8885	−0.670873
$712$	0	0
$713$	0	0
$714$	0	0
$715$	− 13.4164i	− 0.501745i
$716$	0	0
$717$	35.0000i	1.30710i
$718$	0	0
$719$	−40.2492	−1.50104	−0.750521	−	0.660846i	$-0.770198\pi$
$719$	−40.2492	−1.50104	−0.750521	+	0.660846i	$0.770198\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	22.3607i	0.831603i
$724$	0	0
$725$	40.0000i	1.48556i
$726$	0	0
$727$	−31.3050	−1.16104	−0.580518	−	0.814247i	$-0.697150\pi$
$727$	−31.3050	−1.16104	−0.580518	+	0.814247i	$0.697150\pi$
$728$	0	0
$729$	7.00000	0.259259
$730$	0	0
$731$	20.1246i	0.744336i
$732$	0	0
$733$	− 9.00000i	− 0.332423i	−0.986090	−	0.166211i	$-0.946847\pi$
$733$	− 9.00000i	− 0.332423i	0.986090	−	0.166211i	$-0.0531534\pi$
$734$	0	0
$735$	13.4164	0.494872
$736$	0	0
$737$	−60.0000	−2.21013
$738$	0	0
$739$	26.8328i	0.987061i	0.869728	+	0.493531i	$0.164294\pi$
$739$	26.8328i	0.987061i	−0.869728	+	0.493531i	$0.835706\pi$
$740$	0	0
$741$	10.0000i	0.367359i
$742$	0	0
$743$	29.0689	1.06643	0.533217	−	0.845979i	$-0.320983\pi$
$743$	29.0689	1.06643	0.533217	+	0.845979i	$0.320983\pi$
$744$	0	0
$745$	−42.0000	−1.53876
$746$	0	0
$747$	− 35.7771i	− 1.30902i
$748$	0	0
$749$	40.0000i	1.46157i
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	0	0
$753$	20.0000	0.728841
$754$	0	0
$755$	20.1246i	0.732410i
$756$	0	0
$757$	− 28.0000i	− 1.01768i	−0.860862	−	0.508839i	$-0.830075\pi$
$757$	− 28.0000i	− 1.01768i	0.860862	−	0.508839i	$-0.169925\pi$
$758$	0	0
$759$	89.4427	3.24657
$760$	0	0
$761$	18.0000	0.652499	0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000	0.652499	0.326250	+	0.945284i	$0.394215\pi$
$762$	0	0
$763$	− 24.5967i	− 0.890462i
$764$	0	0
$765$	18.0000i	0.650791i
$766$	0	0
$767$	−4.47214	−0.161479
$768$	0	0
$769$	0	0		−	1.00000i	$-0.5\pi$
$769$	0	0			1.00000i	$0.5\pi$
$770$	0	0
$771$	− 51.4296i	− 1.85219i
$772$	0	0
$773$	− 9.00000i	− 0.323708i	−0.986815	−	0.161854i	$-0.948253\pi$
$773$	− 9.00000i	− 0.323708i	0.986815	−	0.161854i	$-0.0517473\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−15.0000	−0.538122
$778$	0	0
$779$	0	0
$780$	0	0
$781$	30.0000i	1.07348i
$782$	0	0
$783$	22.3607	0.799106
$784$	0	0
$785$	66.0000	2.35564
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	0	0
$789$	20.0000i	0.712019i
$790$	0	0
$791$	−31.3050	−1.11308
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 26.8328i	− 0.951662i
$796$	0	0
$797$	18.0000i	0.637593i	0.947823	+	0.318796i	$0.103279\pi$
$797$	18.0000i	0.637593i	−0.947823	+	0.318796i	$0.896721\pi$
$798$	0	0
$799$	6.70820	0.237319
$800$	0	0
$801$	−20.0000	−0.706665
$802$	0	0
$803$	− 62.6099i	− 2.20946i
$804$	0	0
$805$	− 60.0000i	− 2.11472i
$806$	0	0
$807$	−35.7771	−1.25941
$808$	0	0
$809$	−35.0000	−1.23053	−0.615267	−	0.788319i	$-0.710952\pi$
$809$	−35.0000	−1.23053	−0.615267	+	0.788319i	$0.710952\pi$
$810$	0	0
$811$	− 8.94427i	− 0.314076i	−0.987593	−	0.157038i	$-0.949806\pi$
$811$	− 8.94427i	− 0.314076i	0.987593	−	0.157038i	$-0.0501945\pi$
$812$	0	0
$813$	− 45.0000i	− 1.57822i
$814$	0	0
$815$	−26.8328	−0.939913
$816$	0	0
$817$	−30.0000	−1.04957
$818$	0	0
$819$	− 4.47214i	− 0.156269i
$820$	0	0
$821$	− 45.0000i	− 1.57051i	−0.619172	−	0.785255i	$-0.712532\pi$
$821$	− 45.0000i	− 1.57051i	0.619172	−	0.785255i	$-0.287468\pi$
$822$	0	0
$823$	31.3050	1.09122	0.545611	−	0.838039i	$-0.316298\pi$
$823$	31.3050	1.09122	0.545611	+	0.838039i	$0.316298\pi$
$824$	0	0
$825$	40.0000	1.39262
$826$	0	0
$827$	22.3607i	0.777557i	0.921331	+	0.388779i	$0.127103\pi$
$827$	22.3607i	0.777557i	−0.921331	+	0.388779i	$0.872897\pi$
$828$	0	0
$829$	46.0000i	1.59765i	0.601566	+	0.798823i	$0.294544\pi$
$829$	46.0000i	1.59765i	−0.601566	+	0.798823i	$0.705456\pi$
$830$	0	0
$831$	62.6099	2.17191
$832$	0	0
$833$	6.00000	0.207888
$834$	0	0
$835$	− 26.8328i	− 0.928588i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−26.8328	−0.926372	−0.463186	−	0.886261i	$-0.653294\pi$
$839$	−26.8328	−0.926372	−0.463186	+	0.886261i	$0.653294\pi$
$840$	0	0
$841$	−71.0000	−2.44828
$842$	0	0
$843$	22.3607i	0.770143i
$844$	0	0
$845$	− 3.00000i	− 0.103203i
$846$	0	0
$847$	−20.1246	−0.691490
$848$	0	0
$849$	40.0000	1.37280
$850$	0	0
$851$	− 26.8328i	− 0.919817i
$852$	0	0
$853$	− 19.0000i	− 0.650548i	−0.945620	−	0.325274i	$-0.894544\pi$
$853$	− 19.0000i	− 0.650548i	0.945620	−	0.325274i	$-0.105456\pi$
$854$	0	0
$855$	−26.8328	−0.917663
$856$	0	0
$857$	2.00000	0.0683187	0.0341593	−	0.999416i	$-0.489125\pi$
$857$	2.00000	0.0683187	0.0341593	+	0.999416i	$0.489125\pi$
$858$	0	0
$859$	35.7771i	1.22070i	0.792132	+	0.610349i	$0.208971\pi$
$859$	35.7771i	1.22070i	−0.792132	+	0.610349i	$0.791029\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−29.0689	−0.989516	−0.494758	−	0.869031i	$-0.664743\pi$
$863$	−29.0689	−0.989516	−0.494758	+	0.869031i	$0.664743\pi$
$864$	0	0
$865$	12.0000	0.408012
$866$	0	0
$867$	− 17.8885i	− 0.607527i
$868$	0	0
$869$	40.0000i	1.35691i
$870$	0	0
$871$	−13.4164	−0.454598
$872$	0	0
$873$	4.00000	0.135379
$874$	0	0
$875$	6.70820i	0.226779i
$876$	0	0
$877$	3.00000i	0.101303i	0.998716	+	0.0506514i	$0.0161297\pi$
$877$	3.00000i	0.101303i	−0.998716	+	0.0506514i	$0.983870\pi$
$878$	0	0
$879$	−69.3181	−2.33804
$880$	0	0
$881$	5.00000	0.168454	0.0842271	−	0.996447i	$-0.473158\pi$
$881$	5.00000	0.168454	0.0842271	+	0.996447i	$0.473158\pi$
$882$	0	0
$883$	2.23607i	0.0752497i	0.999292	+	0.0376248i	$0.0119792\pi$
$883$	2.23607i	0.0752497i	−0.999292	+	0.0376248i	$0.988021\pi$
$884$	0	0
$885$	− 30.0000i	− 1.00844i
$886$	0	0
$887$	−17.8885	−0.600639	−0.300319	−	0.953839i	$-0.597093\pi$
$887$	−17.8885	−0.600639	−0.300319	+	0.953839i	$0.597093\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	− 49.1935i	− 1.64804i
$892$	0	0
$893$	10.0000i	0.334637i
$894$	0	0
$895$	33.5410	1.12115
$896$	0	0
$897$	20.0000	0.667781
$898$	0	0
$899$	0	0
$900$	0	0
$901$	− 12.0000i	− 0.399778i
$902$	0	0
$903$	33.5410	1.11618
$904$	0	0
$905$	24.0000	0.797787
$906$	0	0
$907$	− 33.5410i	− 1.11371i	−0.830609	−	0.556856i	$-0.812008\pi$
$907$	− 33.5410i	− 1.11371i	0.830609	−	0.556856i	$-0.187992\pi$
$908$	0	0
$909$	24.0000i	0.796030i
$910$	0	0
$911$	−4.47214	−0.148168	−0.0740842	−	0.997252i	$-0.523603\pi$
$911$	−4.47214	−0.148168	−0.0740842	+	0.997252i	$0.523603\pi$
$912$	0	0
$913$	−80.0000	−2.64761
$914$	0	0
$915$	0	0
$916$	0	0
$917$	35.0000i	1.15580i
$918$	0	0
$919$	35.7771	1.18018	0.590089	−	0.807338i	$-0.299093\pi$
$919$	35.7771	1.18018	0.590089	+	0.807338i	$0.299093\pi$
$920$	0	0
$921$	10.0000	0.329511
$922$	0	0
$923$	6.70820i	0.220803i
$924$	0	0
$925$	− 12.0000i	− 0.394558i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	4.00000	0.131236	0.0656179	−	0.997845i	$-0.479098\pi$
$929$	4.00000	0.131236	0.0656179	+	0.997845i	$0.479098\pi$
$930$	0	0
$931$	8.94427i	0.293137i
$932$	0	0
$933$	10.0000i	0.327385i
$934$	0	0
$935$	40.2492	1.31629
$936$	0	0
$937$	58.0000	1.89478	0.947389	−	0.320085i	$-0.103712\pi$
$937$	58.0000	1.89478	0.947389	+	0.320085i	$0.103712\pi$
$938$	0	0
$939$	64.8460i	2.11617i
$940$	0	0
$941$	3.00000i	0.0977972i	0.998804	+	0.0488986i	$0.0155711\pi$
$941$	3.00000i	0.0977972i	−0.998804	+	0.0488986i	$0.984429\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−15.0000	−0.487950
$946$	0	0
$947$	− 4.47214i	− 0.145325i	−0.997357	−	0.0726624i	$-0.976850\pi$
$947$	− 4.47214i	− 0.145325i	0.997357	−	0.0726624i	$-0.0231496\pi$
$948$	0	0
$949$	− 14.0000i	− 0.454459i
$950$	0	0
$951$	−40.2492	−1.30517
$952$	0	0
$953$	9.00000	0.291539	0.145769	−	0.989319i	$-0.453434\pi$
$953$	9.00000	0.291539	0.145769	+	0.989319i	$0.453434\pi$
$954$	0	0
$955$	− 13.4164i	− 0.434145i
$956$	0	0
$957$	100.000i	3.23254i
$958$	0	0
$959$	26.8328	0.866477
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	− 35.7771i	− 1.15290i
$964$	0	0
$965$	48.0000i	1.54517i
$966$	0	0
$967$	−38.0132	−1.22242	−0.611210	−	0.791468i	$-0.709317\pi$
$967$	−38.0132	−1.22242	−0.611210	+	0.791468i	$0.709317\pi$
$968$	0	0
$969$	−30.0000	−0.963739
$970$	0	0
$971$	− 55.9017i	− 1.79397i	−0.442060	−	0.896985i	$-0.645752\pi$
$971$	− 55.9017i	− 1.79397i	0.442060	−	0.896985i	$-0.354248\pi$
$972$	0	0
$973$	25.0000i	0.801463i
$974$	0	0
$975$	8.94427	0.286446
$976$	0	0
$977$	58.0000	1.85558	0.927792	−	0.373097i	$-0.121704\pi$
$977$	58.0000	1.85558	0.927792	+	0.373097i	$0.121704\pi$
$978$	0	0
$979$	44.7214i	1.42930i
$980$	0	0
$981$	22.0000i	0.702406i
$982$	0	0
$983$	2.23607	0.0713195	0.0356597	−	0.999364i	$-0.488647\pi$
$983$	2.23607	0.0713195	0.0356597	+	0.999364i	$0.488647\pi$
$984$	0	0
$985$	21.0000	0.669116
$986$	0	0
$987$	− 11.1803i	− 0.355874i
$988$	0	0
$989$	60.0000i	1.90789i
$990$	0	0
$991$	31.3050	0.994435	0.497217	−	0.867626i	$-0.334355\pi$
$991$	31.3050	0.994435	0.497217	+	0.867626i	$0.334355\pi$
$992$	0	0
$993$	−60.0000	−1.90404
$994$	0	0
$995$	40.2492i	1.27599i
$996$	0	0
$997$	22.0000i	0.696747i	0.937356	+	0.348373i	$0.113266\pi$
$997$	22.0000i	0.696747i	−0.937356	+	0.348373i	$0.886734\pi$
$998$	0	0
$999$	−6.70820	−0.212238

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3328.2.b.x.1665.4		4
4.3	odd	2	inner	3328.2.b.x.1665.2		4
8.3	odd	2	inner	3328.2.b.x.1665.3		4
8.5	even	2	inner	3328.2.b.x.1665.1		4
16.3	odd	4		416.2.a.d.1.2	yes	2
16.5	even	4		832.2.a.m.1.2		2
16.11	odd	4		832.2.a.m.1.1		2
16.13	even	4		416.2.a.d.1.1	✓	2
48.5	odd	4		7488.2.a.cw.1.1		2
48.11	even	4		7488.2.a.cw.1.2		2
48.29	odd	4		3744.2.a.q.1.1		2
48.35	even	4		3744.2.a.q.1.2		2
208.51	odd	4		5408.2.a.q.1.2		2
208.77	even	4		5408.2.a.q.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
416.2.a.d.1.1	✓	2	16.13	even	4
416.2.a.d.1.2	yes	2	16.3	odd	4
832.2.a.m.1.1		2	16.11	odd	4
832.2.a.m.1.2		2	16.5	even	4
3328.2.b.x.1665.1		4	8.5	even	2	inner
3328.2.b.x.1665.2		4	4.3	odd	2	inner
3328.2.b.x.1665.3		4	8.3	odd	2	inner
3328.2.b.x.1665.4		4	1.1	even	1	trivial
3744.2.a.q.1.1		2	48.29	odd	4
3744.2.a.q.1.2		2	48.35	even	4
5408.2.a.q.1.1		2	208.77	even	4
5408.2.a.q.1.2		2	208.51	odd	4
7488.2.a.cw.1.1		2	48.5	odd	4
7488.2.a.cw.1.2		2	48.11	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3328 = 2^{8} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3328.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.23607i q^{3} +3.00000i q^{5} +2.23607 q^{7} -2.00000 q^{9} +O(q^{10})\)	\(q+2.23607i q^{3} +3.00000i q^{5} +2.23607 q^{7} -2.00000 q^{9} +4.47214i q^{11} +1.00000i q^{13} -6.70820 q^{15} -3.00000 q^{17} -4.47214i q^{19} +5.00000i q^{21} -8.94427 q^{23} -4.00000 q^{25} +2.23607i q^{27} -10.0000i q^{29} -10.0000 q^{33} +6.70820i q^{35} +3.00000i q^{37} -2.23607 q^{39} -6.70820i q^{43} -6.00000i q^{45} -2.23607 q^{47} -2.00000 q^{49} -6.70820i q^{51} +4.00000i q^{53} -13.4164 q^{55} +10.0000 q^{57} +4.47214i q^{59} -4.47214 q^{63} -3.00000 q^{65} +13.4164i q^{67} -20.0000i q^{69} +6.70820 q^{71} -14.0000 q^{73} -8.94427i q^{75} +10.0000i q^{77} +8.94427 q^{79} -11.0000 q^{81} +17.8885i q^{83} -9.00000i q^{85} +22.3607 q^{87} +10.0000 q^{89} +2.23607i q^{91} +13.4164 q^{95} -2.00000 q^{97} -8.94427i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{9}+O(q^{10}) \) 4 * q - 8 * q^9	\( 4 q - 8 q^{9} - 12 q^{17} - 16 q^{25} - 40 q^{33} - 8 q^{49} + 40 q^{57} - 12 q^{65} - 56 q^{73} - 44 q^{81} + 40 q^{89} - 8 q^{97}+O(q^{100}) \) 4 * q - 8 * q^9 - 12 * q^17 - 16 * q^25 - 40 * q^33 - 8 * q^49 + 40 * q^57 - 12 * q^65 - 56 * q^73 - 44 * q^81 + 40 * q^89 - 8 * q^97

Introduction

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Modular forms

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Database

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