LMFDB - Embedded newform 775.2.b.c.249.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [775,2,Mod(249,775)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(775, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("775.249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 775 = 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	775.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:	$6.18840615665$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 155)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			249.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	775.249
Dual form			775.2.b.c.249.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +2.00000 q^{4} +2.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +2.00000 q^{4} +2.00000 q^{9} -4.00000 q^{11} -2.00000i q^{12} -6.00000i q^{13} +4.00000 q^{16} -5.00000i q^{17} +1.00000 q^{19} +8.00000i q^{23} -5.00000i q^{27} +10.0000 q^{29} -1.00000 q^{31} +4.00000i q^{33} +4.00000 q^{36} -1.00000i q^{37} -6.00000 q^{39} -3.00000 q^{41} -7.00000i q^{43} -8.00000 q^{44} +6.00000i q^{47} -4.00000i q^{48} +7.00000 q^{49} -5.00000 q^{51} -12.0000i q^{52} +5.00000i q^{53} -1.00000i q^{57} -11.0000 q^{59} -12.0000 q^{61} +8.00000 q^{64} +2.00000i q^{67} -10.0000i q^{68} +8.00000 q^{69} +9.00000 q^{71} -9.00000i q^{73} +2.00000 q^{76} +10.0000 q^{79} +1.00000 q^{81} +9.00000i q^{83} -10.0000i q^{87} +16.0000i q^{92} +1.00000i q^{93} +14.0000i q^{97} -8.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{4} + 4 q^{9}+O(q^{10}) $ 2 * q + 4 * q^4 + 4 * q^9	$ 2 q + 4 q^{4} + 4 q^{9} - 8 q^{11} + 8 q^{16} + 2 q^{19} + 20 q^{29} - 2 q^{31} + 8 q^{36} - 12 q^{39} - 6 q^{41} - 16 q^{44} + 14 q^{49} - 10 q^{51} - 22 q^{59} - 24 q^{61} + 16 q^{64} + 16 q^{69} + 18 q^{71} + 4 q^{76} + 20 q^{79} + 2 q^{81} - 16 q^{99}+O(q^{100}) $ 2 * q + 4 * q^4 + 4 * q^9 - 8 * q^11 + 8 * q^16 + 2 * q^19 + 20 * q^29 - 2 * q^31 + 8 * q^36 - 12 * q^39 - 6 * q^41 - 16 * q^44 + 14 * q^49 - 10 * q^51 - 22 * q^59 - 24 * q^61 + 16 * q^64 + 16 * q^69 + 18 * q^71 + 4 * q^76 + 20 * q^79 + 2 * q^81 - 16 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/775\mathbb{Z}\right)^\times$.

$n$	$251$	$652$
$\chi(n)$	$1$	$-1$

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	1.00000			$0$
$2$	0	0	−1.00000			$\pi$
$3$	− 1.00000i	− 0.577350i	−0.957427	−	0.288675i	$-0.906785\pi$
$3$	− 1.00000i	− 0.577350i	0.957427	−	0.288675i	$-0.0932147\pi$
$4$	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
$9$	2.00000	0.666667
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	− 2.00000i	− 0.577350i
$13$	− 6.00000i	− 1.66410i	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	− 6.00000i	− 1.66410i	0.554700	−	0.832050i	$-0.312833\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	1.00000
$17$	− 5.00000i	− 1.21268i	−0.795206	−	0.606339i	$-0.792637\pi$
$17$	− 5.00000i	− 1.21268i	0.795206	−	0.606339i	$-0.207363\pi$
$18$	0	0
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.00000i	1.66812i	0.551677	+	0.834058i	$0.313988\pi$
$23$	8.00000i	1.66812i	−0.551677	+	0.834058i	$0.686012\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 5.00000i	− 0.962250i
$28$	0	0
$29$	10.0000	1.85695	0.928477	−	0.371391i	$-0.121119\pi$
$29$	10.0000	1.85695	0.928477	+	0.371391i	$0.121119\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	4.00000i	0.696311i
$34$	0	0
$35$	0	0
$36$	4.00000	0.666667
$37$	− 1.00000i	− 0.164399i	−0.996616	−	0.0821995i	$-0.973806\pi$
$37$	− 1.00000i	− 0.164399i	0.996616	−	0.0821995i	$-0.0261945\pi$
$38$	0	0
$39$	−6.00000	−0.960769
$40$	0	0
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	0	0
$43$	− 7.00000i	− 1.06749i	−0.845645	−	0.533745i	$-0.820784\pi$
$43$	− 7.00000i	− 1.06749i	0.845645	−	0.533745i	$-0.179216\pi$
$44$	−8.00000	−1.20605
$45$	0	0
$46$	0	0
$47$	6.00000i	0.875190i	0.899172	+	0.437595i	$0.144170\pi$
$47$	6.00000i	0.875190i	−0.899172	+	0.437595i	$0.855830\pi$
$48$	− 4.00000i	− 0.577350i
$49$	7.00000	1.00000
$50$	0	0
$51$	−5.00000	−0.700140
$52$	− 12.0000i	− 1.66410i
$53$	5.00000i	0.686803i	0.939189	+	0.343401i	$0.111579\pi$
$53$	5.00000i	0.686803i	−0.939189	+	0.343401i	$0.888421\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 1.00000i	− 0.132453i
$58$	0	0
$59$	−11.0000	−1.43208	−0.716039	−	0.698060i	$-0.754047\pi$
$59$	−11.0000	−1.43208	−0.716039	+	0.698060i	$0.754047\pi$
$60$	0	0
$61$	−12.0000	−1.53644	−0.768221	−	0.640184i	$-0.778858\pi$
$61$	−12.0000	−1.53644	−0.768221	+	0.640184i	$0.778858\pi$
$62$	0	0
$63$	0	0
$64$	8.00000	1.00000
$65$	0	0
$66$	0	0
$67$	2.00000i	0.244339i	0.992509	+	0.122169i	$0.0389851\pi$
$67$	2.00000i	0.244339i	−0.992509	+	0.122169i	$0.961015\pi$
$68$	− 10.0000i	− 1.21268i
$69$	8.00000	0.963087
$70$	0	0
$71$	9.00000	1.06810	0.534052	−	0.845452i	$-0.320669\pi$
$71$	9.00000	1.06810	0.534052	+	0.845452i	$0.320669\pi$
$72$	0	0
$73$	− 9.00000i	− 1.05337i	−0.850060	−	0.526685i	$-0.823435\pi$
$73$	− 9.00000i	− 1.05337i	0.850060	−	0.526685i	$-0.176565\pi$
$74$	0	0
$75$	0	0
$76$	2.00000	0.229416
$77$	0	0
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	9.00000i	0.987878i	0.869496	+	0.493939i	$0.164443\pi$
$83$	9.00000i	0.987878i	−0.869496	+	0.493939i	$0.835557\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 10.0000i	− 1.07211i
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	16.0000i	1.66812i
$93$	1.00000i	0.103695i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	14.0000i	1.42148i	0.703452	+	0.710742i	$0.251641\pi$
$97$	14.0000i	1.42148i	−0.703452	+	0.710742i	$0.748359\pi$
$98$	0	0
$99$	−8.00000	−0.804030
$100$	0	0
$101$	−7.00000	−0.696526	−0.348263	−	0.937397i	$-0.613228\pi$
$101$	−7.00000	−0.696526	−0.348263	+	0.937397i	$0.613228\pi$
$102$	0	0
$103$	8.00000i	0.788263i	0.919054	+	0.394132i	$0.128955\pi$
$103$	8.00000i	0.788263i	−0.919054	+	0.394132i	$0.871045\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.00000i	0.193347i	0.995316	+	0.0966736i	$0.0308203\pi$
$107$	2.00000i	0.193347i	−0.995316	+	0.0966736i	$0.969180\pi$
$108$	− 10.0000i	− 0.962250i
$109$	−15.0000	−1.43674	−0.718370	−	0.695662i	$-0.755111\pi$
$109$	−15.0000	−1.43674	−0.718370	+	0.695662i	$0.755111\pi$
$110$	0	0
$111$	−1.00000	−0.0949158
$112$	0	0
$113$	− 4.00000i	− 0.376288i	−0.982141	−	0.188144i	$-0.939753\pi$
$113$	− 4.00000i	− 0.376288i	0.982141	−	0.188144i	$-0.0602472\pi$
$114$	0	0
$115$	0	0
$116$	20.0000	1.85695
$117$	− 12.0000i	− 1.10940i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	3.00000i	0.270501i
$124$	−2.00000	−0.179605
$125$	0	0
$126$	0	0
$127$	8.00000i	0.709885i	0.934888	+	0.354943i	$0.115500\pi$
$127$	8.00000i	0.709885i	−0.934888	+	0.354943i	$0.884500\pi$
$128$	0	0
$129$	−7.00000	−0.616316
$130$	0	0
$131$	−1.00000	−0.0873704	−0.0436852	−	0.999045i	$-0.513910\pi$
$131$	−1.00000	−0.0873704	−0.0436852	+	0.999045i	$0.513910\pi$
$132$	8.00000i	0.696311i
$133$	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
$139$	−2.00000	−0.169638	−0.0848189	−	0.996396i	$-0.527031\pi$
$139$	−2.00000	−0.169638	−0.0848189	+	0.996396i	$0.527031\pi$
$140$	0	0
$141$	6.00000	0.505291
$142$	0	0
$143$	24.0000i	2.00698i
$144$	8.00000	0.666667
$145$	0	0
$146$	0	0
$147$	− 7.00000i	− 0.577350i
$148$	− 2.00000i	− 0.164399i
$149$	−1.00000	−0.0819232	−0.0409616	−	0.999161i	$-0.513042\pi$
$149$	−1.00000	−0.0819232	−0.0409616	+	0.999161i	$0.513042\pi$
$150$	0	0
$151$	10.0000	0.813788	0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000	0.813788	0.406894	+	0.913475i	$0.366612\pi$
$152$	0	0
$153$	− 10.0000i	− 0.808452i
$154$	0	0
$155$	0	0
$156$	−12.0000	−0.960769
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	5.00000	0.396526
$160$	0	0
$161$	0	0
$162$	0	0
$163$	24.0000i	1.87983i	0.341415	+	0.939913i	$0.389094\pi$
$163$	24.0000i	1.87983i	−0.341415	+	0.939913i	$0.610906\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	0	0
$167$	15.0000i	1.16073i	0.814355	+	0.580367i	$0.197091\pi$
$167$	15.0000i	1.16073i	−0.814355	+	0.580367i	$0.802909\pi$
$168$	0	0
$169$	−23.0000	−1.76923
$170$	0	0
$171$	2.00000	0.152944
$172$	− 14.0000i	− 1.06749i
$173$	4.00000i	0.304114i	0.988372	+	0.152057i	$0.0485898\pi$
$173$	4.00000i	0.304114i	−0.988372	+	0.152057i	$0.951410\pi$
$174$	0	0
$175$	0	0
$176$	−16.0000	−1.20605
$177$	11.0000i	0.826811i
$178$	0	0
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	0	0
$181$	−18.0000	−1.33793	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000	−1.33793	−0.668965	+	0.743294i	$0.733262\pi$
$182$	0	0
$183$	12.0000i	0.887066i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	20.0000i	1.46254i
$188$	12.0000i	0.875190i
$189$	0	0
$190$	0	0
$191$	4.00000	0.289430	0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000	0.289430	0.144715	+	0.989473i	$0.453773\pi$
$192$	− 8.00000i	− 0.577350i
$193$	14.0000i	1.00774i	0.863779	+	0.503871i	$0.168091\pi$
$193$	14.0000i	1.00774i	−0.863779	+	0.503871i	$0.831909\pi$
$194$	0	0
$195$	0	0
$196$	14.0000	1.00000
$197$	22.0000i	1.56744i	0.621117	+	0.783718i	$0.286679\pi$
$197$	22.0000i	1.56744i	−0.621117	+	0.783718i	$0.713321\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	2.00000	0.141069
$202$	0	0
$203$	0	0
$204$	−10.0000	−0.700140
$205$	0	0
$206$	0	0
$207$	16.0000i	1.11208i
$208$	− 24.0000i	− 1.66410i
$209$	−4.00000	−0.276686
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	10.0000i	0.686803i
$213$	− 9.00000i	− 0.616670i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−9.00000	−0.608164
$220$	0	0
$221$	−30.0000	−2.01802
$222$	0	0
$223$	− 17.0000i	− 1.13840i	−0.822198	−	0.569202i	$-0.807252\pi$
$223$	− 17.0000i	− 1.13840i	0.822198	−	0.569202i	$-0.192748\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	10.0000i	0.663723i	0.943328	+	0.331862i	$0.107677\pi$
$227$	10.0000i	0.663723i	−0.943328	+	0.331862i	$0.892323\pi$
$228$	− 2.00000i	− 0.132453i
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	16.0000i	1.04819i	0.851658	+	0.524097i	$0.175597\pi$
$233$	16.0000i	1.04819i	−0.851658	+	0.524097i	$0.824403\pi$
$234$	0	0
$235$	0	0
$236$	−22.0000	−1.43208
$237$	− 10.0000i	− 0.649570i
$238$	0	0
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	4.00000	0.257663	0.128831	−	0.991667i	$-0.458877\pi$
$241$	4.00000	0.257663	0.128831	+	0.991667i	$0.458877\pi$
$242$	0	0
$243$	− 16.0000i	− 1.02640i
$244$	−24.0000	−1.53644
$245$	0	0
$246$	0	0
$247$	− 6.00000i	− 0.381771i
$248$	0	0
$249$	9.00000	0.570352
$250$	0	0
$251$	−22.0000	−1.38863	−0.694314	−	0.719672i	$-0.744292\pi$
$251$	−22.0000	−1.38863	−0.694314	+	0.719672i	$0.744292\pi$
$252$	0	0
$253$	− 32.0000i	− 2.01182i
$254$	0	0
$255$	0	0
$256$	16.0000	1.00000
$257$	− 10.0000i	− 0.623783i	−0.950118	−	0.311891i	$-0.899037\pi$
$257$	− 10.0000i	− 0.623783i	0.950118	−	0.311891i	$-0.100963\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	20.0000	1.23797
$262$	0	0
$263$	− 9.00000i	− 0.554964i	−0.960731	−	0.277482i	$-0.910500\pi$
$263$	− 9.00000i	− 0.554964i	0.960731	−	0.277482i	$-0.0894999\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	4.00000i	0.244339i
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	−14.0000	−0.850439	−0.425220	−	0.905090i	$-0.639803\pi$
$271$	−14.0000	−0.850439	−0.425220	+	0.905090i	$0.639803\pi$
$272$	− 20.0000i	− 1.21268i
$273$	0	0
$274$	0	0
$275$	0	0
$276$	16.0000	0.963087
$277$	− 9.00000i	− 0.540758i	−0.962754	−	0.270379i	$-0.912851\pi$
$277$	− 9.00000i	− 0.540758i	0.962754	−	0.270379i	$-0.0871489\pi$
$278$	0	0
$279$	−2.00000	−0.119737
$280$	0	0
$281$	7.00000	0.417585	0.208792	−	0.977960i	$-0.433047\pi$
$281$	7.00000	0.417585	0.208792	+	0.977960i	$0.433047\pi$
$282$	0	0
$283$	− 6.00000i	− 0.356663i	−0.983970	−	0.178331i	$-0.942930\pi$
$283$	− 6.00000i	− 0.356663i	0.983970	−	0.178331i	$-0.0570699\pi$
$284$	18.0000	1.06810
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	14.0000	0.820695
$292$	− 18.0000i	− 1.05337i
$293$	6.00000i	0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$	6.00000i	0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	20.0000i	1.16052i
$298$	0	0
$299$	48.0000	2.77591
$300$	0	0
$301$	0	0
$302$	0	0
$303$	7.00000i	0.402139i
$304$	4.00000	0.229416
$305$	0	0
$306$	0	0
$307$	− 20.0000i	− 1.14146i	−0.821138	−	0.570730i	$-0.806660\pi$
$307$	− 20.0000i	− 1.14146i	0.821138	−	0.570730i	$-0.193340\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	−7.00000	−0.396934	−0.198467	−	0.980108i	$-0.563596\pi$
$311$	−7.00000	−0.396934	−0.198467	+	0.980108i	$0.563596\pi$
$312$	0	0
$313$	29.0000i	1.63918i	0.572953	+	0.819588i	$0.305798\pi$
$313$	29.0000i	1.63918i	−0.572953	+	0.819588i	$0.694202\pi$
$314$	0	0
$315$	0	0
$316$	20.0000	1.12509
$317$	− 32.0000i	− 1.79730i	−0.438667	−	0.898650i	$-0.644549\pi$
$317$	− 32.0000i	− 1.79730i	0.438667	−	0.898650i	$-0.355451\pi$
$318$	0	0
$319$	−40.0000	−2.23957
$320$	0	0
$321$	2.00000	0.111629
$322$	0	0
$323$	− 5.00000i	− 0.278207i
$324$	2.00000	0.111111
$325$	0	0
$326$	0	0
$327$	15.0000i	0.829502i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	18.0000i	0.987878i
$333$	− 2.00000i	− 0.109599i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	19.0000i	1.03500i	0.855684	+	0.517498i	$0.173136\pi$
$337$	19.0000i	1.03500i	−0.855684	+	0.517498i	$0.826864\pi$
$338$	0	0
$339$	−4.00000	−0.217250
$340$	0	0
$341$	4.00000	0.216612
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 8.00000i	− 0.429463i	−0.976673	−	0.214731i	$-0.931112\pi$
$347$	− 8.00000i	− 0.429463i	0.976673	−	0.214731i	$-0.0688876\pi$
$348$	− 20.0000i	− 1.07211i
$349$	29.0000	1.55233	0.776167	−	0.630527i	$-0.217161\pi$
$349$	29.0000	1.55233	0.776167	+	0.630527i	$0.217161\pi$
$350$	0	0
$351$	−30.0000	−1.60128
$352$	0	0
$353$	− 6.00000i	− 0.319348i	−0.987170	−	0.159674i	$-0.948956\pi$
$353$	− 6.00000i	− 0.319348i	0.987170	−	0.159674i	$-0.0510443\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−16.0000	−0.844448	−0.422224	−	0.906492i	$-0.638750\pi$
$359$	−16.0000	−0.844448	−0.422224	+	0.906492i	$0.638750\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	− 5.00000i	− 0.262432i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 25.0000i	− 1.30499i	−0.757793	−	0.652495i	$-0.773722\pi$
$367$	− 25.0000i	− 1.30499i	0.757793	−	0.652495i	$-0.226278\pi$
$368$	32.0000i	1.66812i
$369$	−6.00000	−0.312348
$370$	0	0
$371$	0	0
$372$	2.00000i	0.103695i
$373$	− 26.0000i	− 1.34623i	−0.739538	−	0.673114i	$-0.764956\pi$
$373$	− 26.0000i	− 1.34623i	0.739538	−	0.673114i	$-0.235044\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 60.0000i	− 3.09016i
$378$	0	0
$379$	5.00000	0.256833	0.128416	−	0.991720i	$-0.459011\pi$
$379$	5.00000	0.256833	0.128416	+	0.991720i	$0.459011\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	0	0
$383$	1.00000i	0.0510976i	0.999674	+	0.0255488i	$0.00813332\pi$
$383$	1.00000i	0.0510976i	−0.999674	+	0.0255488i	$0.991867\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 14.0000i	− 0.711660i
$388$	28.0000i	1.42148i
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	0	0
$391$	40.0000	2.02289
$392$	0	0
$393$	1.00000i	0.0504433i
$394$	0	0
$395$	0	0
$396$	−16.0000	−0.804030
$397$	2.00000i	0.100377i	0.998740	+	0.0501886i	$0.0159822\pi$
$397$	2.00000i	0.100377i	−0.998740	+	0.0501886i	$0.984018\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−16.0000	−0.799002	−0.399501	−	0.916733i	$-0.630817\pi$
$401$	−16.0000	−0.799002	−0.399501	+	0.916733i	$0.630817\pi$
$402$	0	0
$403$	6.00000i	0.298881i
$404$	−14.0000	−0.696526
$405$	0	0
$406$	0	0
$407$	4.00000i	0.198273i
$408$	0	0
$409$	30.0000	1.48340	0.741702	−	0.670729i	$-0.234019\pi$
$409$	30.0000	1.48340	0.741702	+	0.670729i	$0.234019\pi$
$410$	0	0
$411$	−3.00000	−0.147979
$412$	16.0000i	0.788263i
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	2.00000i	0.0979404i
$418$	0	0
$419$	15.0000	0.732798	0.366399	−	0.930458i	$-0.380591\pi$
$419$	15.0000	0.732798	0.366399	+	0.930458i	$0.380591\pi$
$420$	0	0
$421$	−1.00000	−0.0487370	−0.0243685	−	0.999703i	$-0.507758\pi$
$421$	−1.00000	−0.0487370	−0.0243685	+	0.999703i	$0.507758\pi$
$422$	0	0
$423$	12.0000i	0.583460i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	4.00000i	0.193347i
$429$	24.0000	1.15873
$430$	0	0
$431$	−33.0000	−1.58955	−0.794777	−	0.606902i	$-0.792412\pi$
$431$	−33.0000	−1.58955	−0.794777	+	0.606902i	$0.792412\pi$
$432$	− 20.0000i	− 0.962250i
$433$	14.0000i	0.672797i	0.941720	+	0.336399i	$0.109209\pi$
$433$	14.0000i	0.672797i	−0.941720	+	0.336399i	$0.890791\pi$
$434$	0	0
$435$	0	0
$436$	−30.0000	−1.43674
$437$	8.00000i	0.382692i
$438$	0	0
$439$	35.0000	1.67046	0.835229	−	0.549902i	$-0.185335\pi$
$439$	35.0000	1.67046	0.835229	+	0.549902i	$0.185335\pi$
$440$	0	0
$441$	14.0000	0.666667
$442$	0	0
$443$	− 34.0000i	− 1.61539i	−0.589601	−	0.807694i	$-0.700715\pi$
$443$	− 34.0000i	− 1.61539i	0.589601	−	0.807694i	$-0.299285\pi$
$444$	−2.00000	−0.0949158
$445$	0	0
$446$	0	0
$447$	1.00000i	0.0472984i
$448$	0	0
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	12.0000	0.565058
$452$	− 8.00000i	− 0.376288i
$453$	− 10.0000i	− 0.469841i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 26.0000i	− 1.21623i	−0.793849	−	0.608114i	$-0.791926\pi$
$457$	− 26.0000i	− 1.21623i	0.793849	−	0.608114i	$-0.208074\pi$
$458$	0	0
$459$	−25.0000	−1.16690
$460$	0	0
$461$	4.00000	0.186299	0.0931493	−	0.995652i	$-0.470307\pi$
$461$	4.00000	0.186299	0.0931493	+	0.995652i	$0.470307\pi$
$462$	0	0
$463$	− 12.0000i	− 0.557687i	−0.960337	−	0.278844i	$-0.910049\pi$
$463$	− 12.0000i	− 0.557687i	0.960337	−	0.278844i	$-0.0899511\pi$
$464$	40.0000	1.85695
$465$	0	0
$466$	0	0
$467$	28.0000i	1.29569i	0.761774	+	0.647843i	$0.224329\pi$
$467$	28.0000i	1.29569i	−0.761774	+	0.647843i	$0.775671\pi$
$468$	− 24.0000i	− 1.10940i
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	28.0000i	1.28744i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	10.0000i	0.457869i
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	0	0
$483$	0	0
$484$	10.0000	0.454545
$485$	0	0
$486$	0	0
$487$	19.0000i	0.860972i	0.902597	+	0.430486i	$0.141658\pi$
$487$	19.0000i	0.860972i	−0.902597	+	0.430486i	$0.858342\pi$
$488$	0	0
$489$	24.0000	1.08532
$490$	0	0
$491$	−4.00000	−0.180517	−0.0902587	−	0.995918i	$-0.528769\pi$
$491$	−4.00000	−0.180517	−0.0902587	+	0.995918i	$0.528769\pi$
$492$	6.00000i	0.270501i
$493$	− 50.0000i	− 2.25189i
$494$	0	0
$495$	0	0
$496$	−4.00000	−0.179605
$497$	0	0
$498$	0	0
$499$	−34.0000	−1.52205	−0.761025	−	0.648723i	$-0.775303\pi$
$499$	−34.0000	−1.52205	−0.761025	+	0.648723i	$0.775303\pi$
$500$	0	0
$501$	15.0000	0.670151
$502$	0	0
$503$	− 22.0000i	− 0.980932i	−0.871460	−	0.490466i	$-0.836827\pi$
$503$	− 22.0000i	− 0.980932i	0.871460	−	0.490466i	$-0.163173\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	23.0000i	1.02147i
$508$	16.0000i	0.709885i
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	− 5.00000i	− 0.220755i
$514$	0	0
$515$	0	0
$516$	−14.0000	−0.616316
$517$	− 24.0000i	− 1.05552i
$518$	0	0
$519$	4.00000	0.175581
$520$	0	0
$521$	19.0000	0.832405	0.416203	−	0.909272i	$-0.363361\pi$
$521$	19.0000	0.832405	0.416203	+	0.909272i	$0.363361\pi$
$522$	0	0
$523$	11.0000i	0.480996i	0.970650	+	0.240498i	$0.0773108\pi$
$523$	11.0000i	0.480996i	−0.970650	+	0.240498i	$0.922689\pi$
$524$	−2.00000	−0.0873704
$525$	0	0
$526$	0	0
$527$	5.00000i	0.217803i
$528$	16.0000i	0.696311i
$529$	−41.0000	−1.78261
$530$	0	0
$531$	−22.0000	−0.954719
$532$	0	0
$533$	18.0000i	0.779667i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 4.00000i	− 0.172613i
$538$	0	0
$539$	−28.0000	−1.20605
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	0	0
$543$	18.0000i	0.772454i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 40.0000i	− 1.71028i	−0.518400	−	0.855138i	$-0.673472\pi$
$547$	− 40.0000i	− 1.71028i	0.518400	−	0.855138i	$-0.326528\pi$
$548$	− 6.00000i	− 0.256307i
$549$	−24.0000	−1.02430
$550$	0	0
$551$	10.0000	0.426014
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	−4.00000	−0.169638
$557$	− 30.0000i	− 1.27114i	−0.772043	−	0.635570i	$-0.780765\pi$
$557$	− 30.0000i	− 1.27114i	0.772043	−	0.635570i	$-0.219235\pi$
$558$	0	0
$559$	−42.0000	−1.77641
$560$	0	0
$561$	20.0000	0.844401
$562$	0	0
$563$	− 6.00000i	− 0.252870i	−0.991975	−	0.126435i	$-0.959647\pi$
$563$	− 6.00000i	− 0.252870i	0.991975	−	0.126435i	$-0.0403535\pi$
$564$	12.0000	0.505291
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−24.0000	−1.00613	−0.503066	−	0.864248i	$-0.667795\pi$
$569$	−24.0000	−1.00613	−0.503066	+	0.864248i	$0.667795\pi$
$570$	0	0
$571$	−26.0000	−1.08807	−0.544033	−	0.839064i	$-0.683103\pi$
$571$	−26.0000	−1.08807	−0.544033	+	0.839064i	$0.683103\pi$
$572$	48.0000i	2.00698i
$573$	− 4.00000i	− 0.167102i
$574$	0	0
$575$	0	0
$576$	16.0000	0.666667
$577$	− 6.00000i	− 0.249783i	−0.992170	−	0.124892i	$-0.960142\pi$
$577$	− 6.00000i	− 0.249783i	0.992170	−	0.124892i	$-0.0398583\pi$
$578$	0	0
$579$	14.0000	0.581820
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 20.0000i	− 0.828315i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8.00000i	0.330195i	0.986277	+	0.165098i	$0.0527939\pi$
$587$	8.00000i	0.330195i	−0.986277	+	0.165098i	$0.947206\pi$
$588$	− 14.0000i	− 0.577350i
$589$	−1.00000	−0.0412043
$590$	0	0
$591$	22.0000	0.904959
$592$	− 4.00000i	− 0.164399i
$593$	− 4.00000i	− 0.164260i	−0.996622	−	0.0821302i	$-0.973828\pi$
$593$	− 4.00000i	− 0.164260i	0.996622	−	0.0821302i	$-0.0261723\pi$
$594$	0	0
$595$	0	0
$596$	−2.00000	−0.0819232
$597$	− 8.00000i	− 0.327418i
$598$	0	0
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	0	0
$601$	−8.00000	−0.326327	−0.163163	−	0.986599i	$-0.552170\pi$
$601$	−8.00000	−0.326327	−0.163163	+	0.986599i	$0.552170\pi$
$602$	0	0
$603$	4.00000i	0.162893i
$604$	20.0000	0.813788
$605$	0	0
$606$	0	0
$607$	42.0000i	1.70473i	0.522949	+	0.852364i	$0.324832\pi$
$607$	42.0000i	1.70473i	−0.522949	+	0.852364i	$0.675168\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	36.0000	1.45640
$612$	− 20.0000i	− 0.808452i
$613$	− 5.00000i	− 0.201948i	−0.994889	−	0.100974i	$-0.967804\pi$
$613$	− 5.00000i	− 0.201948i	0.994889	−	0.100974i	$-0.0321959\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 16.0000i	− 0.644136i	−0.946717	−	0.322068i	$-0.895622\pi$
$617$	− 16.0000i	− 0.644136i	0.946717	−	0.322068i	$-0.104378\pi$
$618$	0	0
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	0	0
$621$	40.0000	1.60514
$622$	0	0
$623$	0	0
$624$	−24.0000	−0.960769
$625$	0	0
$626$	0	0
$627$	4.00000i	0.159745i
$628$	0	0
$629$	−5.00000	−0.199363
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	0	0
$633$	− 20.0000i	− 0.794929i
$634$	0	0
$635$	0	0
$636$	10.0000	0.396526
$637$	− 42.0000i	− 1.66410i
$638$	0	0
$639$	18.0000	0.712069
$640$	0	0
$641$	−24.0000	−0.947943	−0.473972	−	0.880540i	$-0.657180\pi$
$641$	−24.0000	−0.947943	−0.473972	+	0.880540i	$0.657180\pi$
$642$	0	0
$643$	7.00000i	0.276053i	0.990429	+	0.138027i	$0.0440759\pi$
$643$	7.00000i	0.276053i	−0.990429	+	0.138027i	$0.955924\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	21.0000i	0.825595i	0.910823	+	0.412798i	$0.135448\pi$
$647$	21.0000i	0.825595i	−0.910823	+	0.412798i	$0.864552\pi$
$648$	0	0
$649$	44.0000	1.72715
$650$	0	0
$651$	0	0
$652$	48.0000i	1.87983i
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	−12.0000	−0.468521
$657$	− 18.0000i	− 0.702247i
$658$	0	0
$659$	−39.0000	−1.51922	−0.759612	−	0.650376i	$-0.774611\pi$
$659$	−39.0000	−1.51922	−0.759612	+	0.650376i	$0.774611\pi$
$660$	0	0
$661$	29.0000	1.12797	0.563985	−	0.825785i	$-0.309268\pi$
$661$	29.0000	1.12797	0.563985	+	0.825785i	$0.309268\pi$
$662$	0	0
$663$	30.0000i	1.16510i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	80.0000i	3.09761i
$668$	30.0000i	1.16073i
$669$	−17.0000	−0.657258
$670$	0	0
$671$	48.0000	1.85302
$672$	0	0
$673$	19.0000i	0.732396i	0.930537	+	0.366198i	$0.119341\pi$
$673$	19.0000i	0.732396i	−0.930537	+	0.366198i	$0.880659\pi$
$674$	0	0
$675$	0	0
$676$	−46.0000	−1.76923
$677$	− 27.0000i	− 1.03769i	−0.854867	−	0.518847i	$-0.826361\pi$
$677$	− 27.0000i	− 1.03769i	0.854867	−	0.518847i	$-0.173639\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	10.0000	0.383201
$682$	0	0
$683$	48.0000i	1.83667i	0.395805	+	0.918334i	$0.370466\pi$
$683$	48.0000i	1.83667i	−0.395805	+	0.918334i	$0.629534\pi$
$684$	4.00000	0.152944
$685$	0	0
$686$	0	0
$687$	− 14.0000i	− 0.534133i
$688$	− 28.0000i	− 1.06749i
$689$	30.0000	1.14291
$690$	0	0
$691$	−17.0000	−0.646710	−0.323355	−	0.946278i	$-0.604811\pi$
$691$	−17.0000	−0.646710	−0.323355	+	0.946278i	$0.604811\pi$
$692$	8.00000i	0.304114i
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	15.0000i	0.568166i
$698$	0	0
$699$	16.0000	0.605176
$700$	0	0
$701$	−26.0000	−0.982006	−0.491003	−	0.871158i	$-0.663370\pi$
$701$	−26.0000	−0.982006	−0.491003	+	0.871158i	$0.663370\pi$
$702$	0	0
$703$	− 1.00000i	− 0.0377157i
$704$	−32.0000	−1.20605
$705$	0	0
$706$	0	0
$707$	0	0
$708$	22.0000i	0.826811i
$709$	−14.0000	−0.525781	−0.262891	−	0.964826i	$-0.584676\pi$
$709$	−14.0000	−0.525781	−0.262891	+	0.964826i	$0.584676\pi$
$710$	0	0
$711$	20.0000	0.750059
$712$	0	0
$713$	− 8.00000i	− 0.299602i
$714$	0	0
$715$	0	0
$716$	8.00000	0.298974
$717$	6.00000i	0.224074i
$718$	0	0
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 4.00000i	− 0.148762i
$724$	−36.0000	−1.33793
$725$	0	0
$726$	0	0
$727$	− 28.0000i	− 1.03846i	−0.854634	−	0.519231i	$-0.826218\pi$
$727$	− 28.0000i	− 1.03846i	0.854634	−	0.519231i	$-0.173782\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	−35.0000	−1.29452
$732$	24.0000i	0.887066i
$733$	− 40.0000i	− 1.47743i	−0.674016	−	0.738717i	$-0.735432\pi$
$733$	− 40.0000i	− 1.47743i	0.674016	−	0.738717i	$-0.264568\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 8.00000i	− 0.294684i
$738$	0	0
$739$	−2.00000	−0.0735712	−0.0367856	−	0.999323i	$-0.511712\pi$
$739$	−2.00000	−0.0735712	−0.0367856	+	0.999323i	$0.511712\pi$
$740$	0	0
$741$	−6.00000	−0.220416
$742$	0	0
$743$	− 15.0000i	− 0.550297i	−0.961402	−	0.275148i	$-0.911273\pi$
$743$	− 15.0000i	− 0.550297i	0.961402	−	0.275148i	$-0.0887270\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	18.0000i	0.658586i
$748$	40.0000i	1.46254i
$749$	0	0
$750$	0	0
$751$	−12.0000	−0.437886	−0.218943	−	0.975738i	$-0.570261\pi$
$751$	−12.0000	−0.437886	−0.218943	+	0.975738i	$0.570261\pi$
$752$	24.0000i	0.875190i
$753$	22.0000i	0.801725i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	45.0000i	1.63555i	0.575536	+	0.817776i	$0.304793\pi$
$757$	45.0000i	1.63555i	−0.575536	+	0.817776i	$0.695207\pi$
$758$	0	0
$759$	−32.0000	−1.16153
$760$	0	0
$761$	−44.0000	−1.59500	−0.797499	−	0.603320i	$-0.793844\pi$
$761$	−44.0000	−1.59500	−0.797499	+	0.603320i	$0.793844\pi$
$762$	0	0
$763$	0	0
$764$	8.00000	0.289430
$765$	0	0
$766$	0	0
$767$	66.0000i	2.38312i
$768$	− 16.0000i	− 0.577350i
$769$	29.0000	1.04577	0.522883	−	0.852404i	$-0.324856\pi$
$769$	29.0000	1.04577	0.522883	+	0.852404i	$0.324856\pi$
$770$	0	0
$771$	−10.0000	−0.360141
$772$	28.0000i	1.00774i
$773$	2.00000i	0.0719350i	0.999353	+	0.0359675i	$0.0114513\pi$
$773$	2.00000i	0.0719350i	−0.999353	+	0.0359675i	$0.988549\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3.00000	−0.107486
$780$	0	0
$781$	−36.0000	−1.28818
$782$	0	0
$783$	− 50.0000i	− 1.78685i
$784$	28.0000	1.00000
$785$	0	0
$786$	0	0
$787$	20.0000i	0.712923i	0.934310	+	0.356462i	$0.116017\pi$
$787$	20.0000i	0.712923i	−0.934310	+	0.356462i	$0.883983\pi$
$788$	44.0000i	1.56744i
$789$	−9.00000	−0.320408
$790$	0	0
$791$	0	0
$792$	0	0
$793$	72.0000i	2.55679i
$794$	0	0
$795$	0	0
$796$	16.0000	0.567105
$797$	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
$799$	30.0000	1.06132
$800$	0	0
$801$	0	0
$802$	0	0
$803$	36.0000i	1.27041i
$804$	4.00000	0.141069
$805$	0	0
$806$	0	0
$807$	6.00000i	0.211210i
$808$	0	0
$809$	22.0000	0.773479	0.386739	−	0.922189i	$-0.373601\pi$
$809$	22.0000	0.773479	0.386739	+	0.922189i	$0.373601\pi$
$810$	0	0
$811$	−3.00000	−0.105344	−0.0526721	−	0.998612i	$-0.516774\pi$
$811$	−3.00000	−0.105344	−0.0526721	+	0.998612i	$0.516774\pi$
$812$	0	0
$813$	14.0000i	0.491001i
$814$	0	0
$815$	0	0
$816$	−20.0000	−0.700140
$817$	− 7.00000i	− 0.244899i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	56.0000	1.95441	0.977207	−	0.212290i	$-0.0680921\pi$
$821$	56.0000	1.95441	0.977207	+	0.212290i	$0.0680921\pi$
$822$	0	0
$823$	− 23.0000i	− 0.801730i	−0.916137	−	0.400865i	$-0.868710\pi$
$823$	− 23.0000i	− 0.801730i	0.916137	−	0.400865i	$-0.131290\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 37.0000i	− 1.28662i	−0.765607	−	0.643308i	$-0.777561\pi$
$827$	− 37.0000i	− 1.28662i	0.765607	−	0.643308i	$-0.222439\pi$
$828$	32.0000i	1.11208i
$829$	46.0000	1.59765	0.798823	−	0.601566i	$-0.205456\pi$
$829$	46.0000	1.59765	0.798823	+	0.601566i	$0.205456\pi$
$830$	0	0
$831$	−9.00000	−0.312207
$832$	− 48.0000i	− 1.66410i
$833$	− 35.0000i	− 1.21268i
$834$	0	0
$835$	0	0
$836$	−8.00000	−0.276686
$837$	5.00000i	0.172825i
$838$	0	0
$839$	35.0000	1.20833	0.604167	−	0.796858i	$-0.293506\pi$
$839$	35.0000	1.20833	0.604167	+	0.796858i	$0.293506\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	0	0
$843$	− 7.00000i	− 0.241093i
$844$	40.0000	1.37686
$845$	0	0
$846$	0	0
$847$	0	0
$848$	20.0000i	0.686803i
$849$	−6.00000	−0.205919
$850$	0	0
$851$	8.00000	0.274236
$852$	− 18.0000i	− 0.616670i
$853$	− 32.0000i	− 1.09566i	−0.836590	−	0.547830i	$-0.815454\pi$
$853$	− 32.0000i	− 1.09566i	0.836590	−	0.547830i	$-0.184546\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 36.0000i	− 1.22974i	−0.788630	−	0.614868i	$-0.789209\pi$
$857$	− 36.0000i	− 1.22974i	0.788630	−	0.614868i	$-0.210791\pi$
$858$	0	0
$859$	−36.0000	−1.22830	−0.614152	−	0.789188i	$-0.710502\pi$
$859$	−36.0000	−1.22830	−0.614152	+	0.789188i	$0.710502\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 45.0000i	− 1.53182i	−0.642949	−	0.765909i	$-0.722289\pi$
$863$	− 45.0000i	− 1.53182i	0.642949	−	0.765909i	$-0.277711\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	8.00000i	0.271694i
$868$	0	0
$869$	−40.0000	−1.35691
$870$	0	0
$871$	12.0000	0.406604
$872$	0	0
$873$	28.0000i	0.947656i
$874$	0	0
$875$	0	0
$876$	−18.0000	−0.608164
$877$	44.0000i	1.48577i	0.669417	+	0.742887i	$0.266544\pi$
$877$	44.0000i	1.48577i	−0.669417	+	0.742887i	$0.733456\pi$
$878$	0	0
$879$	6.00000	0.202375
$880$	0	0
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	0	0
$883$	− 11.0000i	− 0.370179i	−0.982722	−	0.185090i	$-0.940742\pi$
$883$	− 11.0000i	− 0.370179i	0.982722	−	0.185090i	$-0.0592576\pi$
$884$	−60.0000	−2.01802
$885$	0	0
$886$	0	0
$887$	− 18.0000i	− 0.604381i	−0.953248	−	0.302190i	$-0.902282\pi$
$887$	− 18.0000i	− 0.604381i	0.953248	−	0.302190i	$-0.0977178\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−4.00000	−0.134005
$892$	− 34.0000i	− 1.13840i
$893$	6.00000i	0.200782i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 48.0000i	− 1.60267i
$898$	0	0
$899$	−10.0000	−0.333519
$900$	0	0
$901$	25.0000	0.832871
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	20.0000i	0.664089i	0.943264	+	0.332045i	$0.107738\pi$
$907$	20.0000i	0.664089i	−0.943264	+	0.332045i	$0.892262\pi$
$908$	20.0000i	0.663723i
$909$	−14.0000	−0.464351
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	− 4.00000i	− 0.132453i
$913$	− 36.0000i	− 1.19143i
$914$	0	0
$915$	0	0
$916$	28.0000	0.925146
$917$	0	0
$918$	0	0
$919$	17.0000	0.560778	0.280389	−	0.959886i	$-0.409536\pi$
$919$	17.0000	0.560778	0.280389	+	0.959886i	$0.409536\pi$
$920$	0	0
$921$	−20.0000	−0.659022
$922$	0	0
$923$	− 54.0000i	− 1.77743i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	16.0000i	0.525509i
$928$	0	0
$929$	−48.0000	−1.57483	−0.787414	−	0.616424i	$-0.788581\pi$
$929$	−48.0000	−1.57483	−0.787414	+	0.616424i	$0.788581\pi$
$930$	0	0
$931$	7.00000	0.229416
$932$	32.0000i	1.04819i
$933$	7.00000i	0.229170i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 28.0000i	− 0.914720i	−0.889282	−	0.457360i	$-0.848795\pi$
$937$	− 28.0000i	− 0.914720i	0.889282	−	0.457360i	$-0.151205\pi$
$938$	0	0
$939$	29.0000	0.946379
$940$	0	0
$941$	54.0000	1.76035	0.880175	−	0.474650i	$-0.157425\pi$
$941$	54.0000	1.76035	0.880175	+	0.474650i	$0.157425\pi$
$942$	0	0
$943$	− 24.0000i	− 0.781548i
$944$	−44.0000	−1.43208
$945$	0	0
$946$	0	0
$947$	− 55.0000i	− 1.78726i	−0.448805	−	0.893630i	$-0.648150\pi$
$947$	− 55.0000i	− 1.78726i	0.448805	−	0.893630i	$-0.351850\pi$
$948$	− 20.0000i	− 0.649570i
$949$	−54.0000	−1.75291
$950$	0	0
$951$	−32.0000	−1.03767
$952$	0	0
$953$	− 9.00000i	− 0.291539i	−0.989319	−	0.145769i	$-0.953434\pi$
$953$	− 9.00000i	− 0.291539i	0.989319	−	0.145769i	$-0.0465657\pi$
$954$	0	0
$955$	0	0
$956$	−12.0000	−0.388108
$957$	40.0000i	1.29302i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	4.00000i	0.128898i
$964$	8.00000	0.257663
$965$	0	0
$966$	0	0
$967$	8.00000i	0.257263i	0.991692	+	0.128631i	$0.0410584\pi$
$967$	8.00000i	0.257263i	−0.991692	+	0.128631i	$0.958942\pi$
$968$	0	0
$969$	−5.00000	−0.160623
$970$	0	0
$971$	33.0000	1.05902	0.529510	−	0.848304i	$-0.322376\pi$
$971$	33.0000	1.05902	0.529510	+	0.848304i	$0.322376\pi$
$972$	− 32.0000i	− 1.02640i
$973$	0	0
$974$	0	0
$975$	0	0
$976$	−48.0000	−1.53644
$977$	44.0000i	1.40768i	0.710356	+	0.703842i	$0.248534\pi$
$977$	44.0000i	1.40768i	−0.710356	+	0.703842i	$0.751466\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−30.0000	−0.957826
$982$	0	0
$983$	56.0000i	1.78612i	0.449935	+	0.893061i	$0.351447\pi$
$983$	56.0000i	1.78612i	−0.449935	+	0.893061i	$0.648553\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	− 12.0000i	− 0.381771i
$989$	56.0000	1.78070
$990$	0	0
$991$	20.0000	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$992$	0	0
$993$	− 12.0000i	− 0.380808i
$994$	0	0
$995$	0	0
$996$	18.0000	0.570352
$997$	28.0000i	0.886769i	0.896332	+	0.443384i	$0.146222\pi$
$997$	28.0000i	0.886769i	−0.896332	+	0.443384i	$0.853778\pi$
$998$	0	0
$999$	−5.00000	−0.158193

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	775.2.b.c.249.1		2
5.2	odd	4		155.2.a.c.1.1	✓	1
5.3	odd	4		775.2.a.a.1.1		1
5.4	even	2	inner	775.2.b.c.249.2		2
15.2	even	4		1395.2.a.b.1.1		1
15.8	even	4		6975.2.a.l.1.1		1
20.7	even	4		2480.2.a.k.1.1		1
35.27	even	4		7595.2.a.h.1.1		1
40.27	even	4		9920.2.a.l.1.1		1
40.37	odd	4		9920.2.a.ba.1.1		1
155.92	even	4		4805.2.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
155.2.a.c.1.1	✓	1	5.2	odd	4
775.2.a.a.1.1		1	5.3	odd	4
775.2.b.c.249.1		2	1.1	even	1	trivial
775.2.b.c.249.2		2	5.4	even	2	inner
1395.2.a.b.1.1		1	15.2	even	4
2480.2.a.k.1.1		1	20.7	even	4
4805.2.a.e.1.1		1	155.92	even	4
6975.2.a.l.1.1		1	15.8	even	4
7595.2.a.h.1.1		1	35.27	even	4
9920.2.a.l.1.1		1	40.27	even	4
9920.2.a.ba.1.1		1	40.37	odd	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:	\( N \)	\(=\)	\( 775 = 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	775.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +2.00000 q^{4} +2.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +2.00000 q^{4} +2.00000 q^{9} -4.00000 q^{11} -2.00000i q^{12} -6.00000i q^{13} +4.00000 q^{16} -5.00000i q^{17} +1.00000 q^{19} +8.00000i q^{23} -5.00000i q^{27} +10.0000 q^{29} -1.00000 q^{31} +4.00000i q^{33} +4.00000 q^{36} -1.00000i q^{37} -6.00000 q^{39} -3.00000 q^{41} -7.00000i q^{43} -8.00000 q^{44} +6.00000i q^{47} -4.00000i q^{48} +7.00000 q^{49} -5.00000 q^{51} -12.0000i q^{52} +5.00000i q^{53} -1.00000i q^{57} -11.0000 q^{59} -12.0000 q^{61} +8.00000 q^{64} +2.00000i q^{67} -10.0000i q^{68} +8.00000 q^{69} +9.00000 q^{71} -9.00000i q^{73} +2.00000 q^{76} +10.0000 q^{79} +1.00000 q^{81} +9.00000i q^{83} -10.0000i q^{87} +16.0000i q^{92} +1.00000i q^{93} +14.0000i q^{97} -8.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{4} + 4 q^{9}+O(q^{10}) \) 2 * q + 4 * q^4 + 4 * q^9	\( 2 q + 4 q^{4} + 4 q^{9} - 8 q^{11} + 8 q^{16} + 2 q^{19} + 20 q^{29} - 2 q^{31} + 8 q^{36} - 12 q^{39} - 6 q^{41} - 16 q^{44} + 14 q^{49} - 10 q^{51} - 22 q^{59} - 24 q^{61} + 16 q^{64} + 16 q^{69} + 18 q^{71} + 4 q^{76} + 20 q^{79} + 2 q^{81} - 16 q^{99}+O(q^{100}) \) 2 * q + 4 * q^4 + 4 * q^9 - 8 * q^11 + 8 * q^16 + 2 * q^19 + 20 * q^29 - 2 * q^31 + 8 * q^36 - 12 * q^39 - 6 * q^41 - 16 * q^44 + 14 * q^49 - 10 * q^51 - 22 * q^59 - 24 * q^61 + 16 * q^64 + 16 * q^69 + 18 * q^71 + 4 * q^76 + 20 * q^79 + 2 * q^81 - 16 * q^99

Introduction

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