LMFDB - Embedded newform 3364.2.c.b.1681.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3364,2,Mod(1681,3364)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3364, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3364.1681");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3364 = 2^{2} \cdot 29^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3364.c (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:	$26.8616752400$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 116)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1681.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	3364.1681
Dual form			3364.2.c.b.1681.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{3} -3.00000 q^{5} +4.00000 q^{7} -6.00000 q^{9} +O(q^{10})$	$q+3.00000i q^{3} -3.00000 q^{5} +4.00000 q^{7} -6.00000 q^{9} +1.00000i q^{11} +3.00000 q^{13} -9.00000i q^{15} -2.00000i q^{17} -4.00000i q^{19} +12.0000i q^{21} -6.00000 q^{23} +4.00000 q^{25} -9.00000i q^{27} -9.00000i q^{31} -3.00000 q^{33} -12.0000 q^{35} -8.00000i q^{37} +9.00000i q^{39} -8.00000i q^{41} +5.00000i q^{43} +18.0000 q^{45} -7.00000i q^{47} +9.00000 q^{49} +6.00000 q^{51} -5.00000 q^{53} -3.00000i q^{55} +12.0000 q^{57} -10.0000 q^{59} -10.0000i q^{61} -24.0000 q^{63} -9.00000 q^{65} -8.00000 q^{67} -18.0000i q^{69} +2.00000 q^{71} +12.0000i q^{75} +4.00000i q^{77} +1.00000i q^{79} +9.00000 q^{81} +6.00000 q^{83} +6.00000i q^{85} -12.0000i q^{89} +12.0000 q^{91} +27.0000 q^{93} +12.0000i q^{95} -6.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{5} + 8 q^{7} - 12 q^{9}+O(q^{10}) $ 2 * q - 6 * q^5 + 8 * q^7 - 12 * q^9	$ 2 q - 6 q^{5} + 8 q^{7} - 12 q^{9} + 6 q^{13} - 12 q^{23} + 8 q^{25} - 6 q^{33} - 24 q^{35} + 36 q^{45} + 18 q^{49} + 12 q^{51} - 10 q^{53} + 24 q^{57} - 20 q^{59} - 48 q^{63} - 18 q^{65} - 16 q^{67} + 4 q^{71} + 18 q^{81} + 12 q^{83} + 24 q^{91} + 54 q^{93}+O(q^{100}) $ 2 * q - 6 * q^5 + 8 * q^7 - 12 * q^9 + 6 * q^13 - 12 * q^23 + 8 * q^25 - 6 * q^33 - 24 * q^35 + 36 * q^45 + 18 * q^49 + 12 * q^51 - 10 * q^53 + 24 * q^57 - 20 * q^59 - 48 * q^63 - 18 * q^65 - 16 * q^67 + 4 * q^71 + 18 * q^81 + 12 * q^83 + 24 * q^91 + 54 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3364\mathbb{Z}\right)^\times$.

$n$	$1683$	$2525$
$\chi(n)$	$1$	$-1$

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
$2$	0	0
$3$	3.00000i	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$3$	3.00000i	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$4$	0	0
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	0	0
$7$	4.00000	1.51186	0.755929	−	0.654654i	$-0.227186\pi$
$7$	4.00000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	0	0
$9$	−6.00000	−2.00000
$10$	0	0
$11$	1.00000i	0.301511i	0.988571	+	0.150756i	$0.0481707\pi$
$11$	1.00000i	0.301511i	−0.988571	+	0.150756i	$0.951829\pi$
$12$	0	0
$13$	3.00000	0.832050	0.416025	−	0.909353i	$-0.363423\pi$
$13$	3.00000	0.832050	0.416025	+	0.909353i	$0.363423\pi$
$14$	0	0
$15$	− 9.00000i	− 2.32379i
$16$	0	0
$17$	− 2.00000i	− 0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	− 2.00000i	− 0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$	0	0
$19$	− 4.00000i	− 0.917663i	−0.888523	−	0.458831i	$-0.848268\pi$
$19$	− 4.00000i	− 0.917663i	0.888523	−	0.458831i	$-0.151732\pi$
$20$	0	0
$21$	12.0000i	2.61861i
$22$	0	0
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	− 9.00000i	− 1.73205i
$28$	0	0
$29$	0	0
$29$	0	0
$30$	0	0
$31$	− 9.00000i	− 1.61645i	−0.588875	−	0.808224i	$-0.700429\pi$
$31$	− 9.00000i	− 1.61645i	0.588875	−	0.808224i	$-0.299571\pi$
$32$	0	0
$33$	−3.00000	−0.522233
$34$	0	0
$35$	−12.0000	−2.02837
$36$	0	0
$37$	− 8.00000i	− 1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$	− 8.00000i	− 1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$	0	0
$39$	9.00000i	1.44115i
$40$	0	0
$41$	− 8.00000i	− 1.24939i	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	− 8.00000i	− 1.24939i	0.780869	−	0.624695i	$-0.214777\pi$
$42$	0	0
$43$	5.00000i	0.762493i	0.924473	+	0.381246i	$0.124505\pi$
$43$	5.00000i	0.762493i	−0.924473	+	0.381246i	$0.875495\pi$
$44$	0	0
$45$	18.0000	2.68328
$46$	0	0
$47$	− 7.00000i	− 1.02105i	−0.859861	−	0.510527i	$-0.829450\pi$
$47$	− 7.00000i	− 1.02105i	0.859861	−	0.510527i	$-0.170550\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	0	0
$51$	6.00000	0.840168
$52$	0	0
$53$	−5.00000	−0.686803	−0.343401	−	0.939189i	$-0.611579\pi$
$53$	−5.00000	−0.686803	−0.343401	+	0.939189i	$0.611579\pi$
$54$	0	0
$55$	− 3.00000i	− 0.404520i
$56$	0	0
$57$	12.0000	1.58944
$58$	0	0
$59$	−10.0000	−1.30189	−0.650945	−	0.759125i	$-0.725627\pi$
$59$	−10.0000	−1.30189	−0.650945	+	0.759125i	$0.725627\pi$
$60$	0	0
$61$	− 10.0000i	− 1.28037i	−0.768221	−	0.640184i	$-0.778858\pi$
$61$	− 10.0000i	− 1.28037i	0.768221	−	0.640184i	$-0.221142\pi$
$62$	0	0
$63$	−24.0000	−3.02372
$64$	0	0
$65$	−9.00000	−1.11631
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	0	0
$69$	− 18.0000i	− 2.16695i
$70$	0	0
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	12.0000i	1.38564i
$76$	0	0
$77$	4.00000i	0.455842i
$78$	0	0
$79$	1.00000i	0.112509i	0.998416	+	0.0562544i	$0.0179158\pi$
$79$	1.00000i	0.112509i	−0.998416	+	0.0562544i	$0.982084\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	6.00000i	0.650791i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 12.0000i	− 1.27200i	−0.771690	−	0.635999i	$-0.780588\pi$
$89$	− 12.0000i	− 1.27200i	0.771690	−	0.635999i	$-0.219412\pi$
$90$	0	0
$91$	12.0000	1.25794
$92$	0	0
$93$	27.0000	2.79977
$94$	0	0
$95$	12.0000i	1.23117i
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0	0
$99$	− 6.00000i	− 0.603023i
$100$	0	0
$101$	4.00000i	0.398015i	0.979998	+	0.199007i	$0.0637718\pi$
$101$	4.00000i	0.398015i	−0.979998	+	0.199007i	$0.936228\pi$
$102$	0	0
$103$	6.00000	0.591198	0.295599	−	0.955312i	$-0.404481\pi$
$103$	6.00000	0.591198	0.295599	+	0.955312i	$0.404481\pi$
$104$	0	0
$105$	− 36.0000i	− 3.51324i
$106$	0	0
$107$	−8.00000	−0.773389	−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000	−0.773389	−0.386695	+	0.922208i	$0.626383\pi$
$108$	0	0
$109$	5.00000	0.478913	0.239457	−	0.970907i	$-0.423031\pi$
$109$	5.00000	0.478913	0.239457	+	0.970907i	$0.423031\pi$
$110$	0	0
$111$	24.0000	2.27798
$112$	0	0
$113$	− 6.00000i	− 0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$	− 6.00000i	− 0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	0	0
$115$	18.0000	1.67851
$116$	0	0
$117$	−18.0000	−1.66410
$118$	0	0
$119$	− 8.00000i	− 0.733359i
$120$	0	0
$121$	10.0000	0.909091
$122$	0	0
$123$	24.0000	2.16401
$124$	0	0
$125$	3.00000	0.268328
$126$	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
$129$	−15.0000	−1.32068
$130$	0	0
$131$	12.0000i	1.04844i	0.851581	+	0.524222i	$0.175644\pi$
$131$	12.0000i	1.04844i	−0.851581	+	0.524222i	$0.824356\pi$
$132$	0	0
$133$	− 16.0000i	− 1.38738i
$134$	0	0
$135$	27.0000i	2.32379i
$136$	0	0
$137$	− 4.00000i	− 0.341743i	−0.985293	−	0.170872i	$-0.945342\pi$
$137$	− 4.00000i	− 0.341743i	0.985293	−	0.170872i	$-0.0546583\pi$
$138$	0	0
$139$	−18.0000	−1.52674	−0.763370	−	0.645961i	$-0.776457\pi$
$139$	−18.0000	−1.52674	−0.763370	+	0.645961i	$0.776457\pi$
$140$	0	0
$141$	21.0000	1.76852
$142$	0	0
$143$	3.00000i	0.250873i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	27.0000i	2.22692i
$148$	0	0
$149$	−9.00000	−0.737309	−0.368654	−	0.929567i	$-0.620181\pi$
$149$	−9.00000	−0.737309	−0.368654	+	0.929567i	$0.620181\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	0	0
$153$	12.0000i	0.970143i
$154$	0	0
$155$	27.0000i	2.16869i
$156$	0	0
$157$	4.00000i	0.319235i	0.987179	+	0.159617i	$0.0510260\pi$
$157$	4.00000i	0.319235i	−0.987179	+	0.159617i	$0.948974\pi$
$158$	0	0
$159$	− 15.0000i	− 1.18958i
$160$	0	0
$161$	−24.0000	−1.89146
$162$	0	0
$163$	7.00000i	0.548282i	0.961689	+	0.274141i	$0.0883936\pi$
$163$	7.00000i	0.548282i	−0.961689	+	0.274141i	$0.911606\pi$
$164$	0	0
$165$	9.00000	0.700649
$166$	0	0
$167$	−2.00000	−0.154765	−0.0773823	−	0.997001i	$-0.524656\pi$
$167$	−2.00000	−0.154765	−0.0773823	+	0.997001i	$0.524656\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	0	0
$171$	24.0000i	1.83533i
$172$	0	0
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	0	0
$175$	16.0000	1.20949
$176$	0	0
$177$	− 30.0000i	− 2.25494i
$178$	0	0
$179$	−10.0000	−0.747435	−0.373718	−	0.927543i	$-0.621917\pi$
$179$	−10.0000	−0.747435	−0.373718	+	0.927543i	$0.621917\pi$
$180$	0	0
$181$	17.0000	1.26360	0.631800	−	0.775131i	$-0.282316\pi$
$181$	17.0000	1.26360	0.631800	+	0.775131i	$0.282316\pi$
$182$	0	0
$183$	30.0000	2.21766
$184$	0	0
$185$	24.0000i	1.76452i
$186$	0	0
$187$	2.00000	0.146254
$188$	0	0
$189$	− 36.0000i	− 2.61861i
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	8.00000i	0.575853i	0.957653	+	0.287926i	$0.0929658\pi$
$193$	8.00000i	0.575853i	−0.957653	+	0.287926i	$0.907034\pi$
$194$	0	0
$195$	− 27.0000i	− 1.93351i
$196$	0	0
$197$	26.0000	1.85242	0.926212	−	0.377004i	$-0.123046\pi$
$197$	26.0000	1.85242	0.926212	+	0.377004i	$0.123046\pi$
$198$	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
$201$	− 24.0000i	− 1.69283i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	24.0000i	1.67623i
$206$	0	0
$207$	36.0000	2.50217
$208$	0	0
$209$	4.00000	0.276686
$210$	0	0
$211$	− 13.0000i	− 0.894957i	−0.894295	−	0.447478i	$-0.852322\pi$
$211$	− 13.0000i	− 0.894957i	0.894295	−	0.447478i	$-0.147678\pi$
$212$	0	0
$213$	6.00000i	0.411113i
$214$	0	0
$215$	− 15.0000i	− 1.02299i
$216$	0	0
$217$	− 36.0000i	− 2.44384i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 6.00000i	− 0.403604i
$222$	0	0
$223$	14.0000	0.937509	0.468755	−	0.883328i	$-0.344703\pi$
$223$	14.0000	0.937509	0.468755	+	0.883328i	$0.344703\pi$
$224$	0	0
$225$	−24.0000	−1.60000
$226$	0	0
$227$	30.0000	1.99117	0.995585	−	0.0938647i	$-0.0299221\pi$
$227$	30.0000	1.99117	0.995585	+	0.0938647i	$0.0299221\pi$
$228$	0	0
$229$	20.0000i	1.32164i	0.750546	+	0.660819i	$0.229791\pi$
$229$	20.0000i	1.32164i	−0.750546	+	0.660819i	$0.770209\pi$
$230$	0	0
$231$	−12.0000	−0.789542
$232$	0	0
$233$	23.0000	1.50678	0.753390	−	0.657574i	$-0.228417\pi$
$233$	23.0000	1.50678	0.753390	+	0.657574i	$0.228417\pi$
$234$	0	0
$235$	21.0000i	1.36989i
$236$	0	0
$237$	−3.00000	−0.194871
$238$	0	0
$239$	−22.0000	−1.42306	−0.711531	−	0.702655i	$-0.751998\pi$
$239$	−22.0000	−1.42306	−0.711531	+	0.702655i	$0.751998\pi$
$240$	0	0
$241$	−17.0000	−1.09507	−0.547533	−	0.836784i	$-0.684433\pi$
$241$	−17.0000	−1.09507	−0.547533	+	0.836784i	$0.684433\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−27.0000	−1.72497
$246$	0	0
$247$	− 12.0000i	− 0.763542i
$248$	0	0
$249$	18.0000i	1.14070i
$250$	0	0
$251$	31.0000i	1.95670i	0.206951	+	0.978351i	$0.433646\pi$
$251$	31.0000i	1.95670i	−0.206951	+	0.978351i	$0.566354\pi$
$252$	0	0
$253$	− 6.00000i	− 0.377217i
$254$	0	0
$255$	−18.0000	−1.12720
$256$	0	0
$257$	−27.0000	−1.68421	−0.842107	−	0.539311i	$-0.818685\pi$
$257$	−27.0000	−1.68421	−0.842107	+	0.539311i	$0.818685\pi$
$258$	0	0
$259$	− 32.0000i	− 1.98838i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 13.0000i	− 0.801614i	−0.916162	−	0.400807i	$-0.868730\pi$
$263$	− 13.0000i	− 0.801614i	0.916162	−	0.400807i	$-0.131270\pi$
$264$	0	0
$265$	15.0000	0.921443
$266$	0	0
$267$	36.0000	2.20316
$268$	0	0
$269$	14.0000i	0.853595i	0.904347	+	0.426798i	$0.140358\pi$
$269$	14.0000i	0.853595i	−0.904347	+	0.426798i	$0.859642\pi$
$270$	0	0
$271$	− 5.00000i	− 0.303728i	−0.988401	−	0.151864i	$-0.951472\pi$
$271$	− 5.00000i	− 0.303728i	0.988401	−	0.151864i	$-0.0485276\pi$
$272$	0	0
$273$	36.0000i	2.17882i
$274$	0	0
$275$	4.00000i	0.241209i
$276$	0	0
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	0	0
$279$	54.0000i	3.23290i
$280$	0	0
$281$	−1.00000	−0.0596550	−0.0298275	−	0.999555i	$-0.509496\pi$
$281$	−1.00000	−0.0596550	−0.0298275	+	0.999555i	$0.509496\pi$
$282$	0	0
$283$	6.00000	0.356663	0.178331	−	0.983970i	$-0.442930\pi$
$283$	6.00000	0.356663	0.178331	+	0.983970i	$0.442930\pi$
$284$	0	0
$285$	−36.0000	−2.13246
$286$	0	0
$287$	− 32.0000i	− 1.88890i
$288$	0	0
$289$	13.0000	0.764706
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 2.00000i	− 0.116841i	−0.998292	−	0.0584206i	$-0.981394\pi$
$293$	− 2.00000i	− 0.116841i	0.998292	−	0.0584206i	$-0.0186065\pi$
$294$	0	0
$295$	30.0000	1.74667
$296$	0	0
$297$	9.00000	0.522233
$298$	0	0
$299$	−18.0000	−1.04097
$300$	0	0
$301$	20.0000i	1.15278i
$302$	0	0
$303$	−12.0000	−0.689382
$304$	0	0
$305$	30.0000i	1.71780i
$306$	0	0
$307$	− 7.00000i	− 0.399511i	−0.979846	−	0.199756i	$-0.935985\pi$
$307$	− 7.00000i	− 0.399511i	0.979846	−	0.199756i	$-0.0640148\pi$
$308$	0	0
$309$	18.0000i	1.02398i
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	1.00000	0.0565233	0.0282617	−	0.999601i	$-0.491003\pi$
$313$	1.00000	0.0565233	0.0282617	+	0.999601i	$0.491003\pi$
$314$	0	0
$315$	72.0000	4.05674
$316$	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
$319$	0	0
$320$	0	0
$321$	− 24.0000i	− 1.33955i
$322$	0	0
$323$	−8.00000	−0.445132
$324$	0	0
$325$	12.0000	0.665640
$326$	0	0
$327$	15.0000i	0.829502i
$328$	0	0
$329$	− 28.0000i	− 1.54369i
$330$	0	0
$331$	3.00000i	0.164895i	0.996595	+	0.0824475i	$0.0262737\pi$
$331$	3.00000i	0.164895i	−0.996595	+	0.0824475i	$0.973726\pi$
$332$	0	0
$333$	48.0000i	2.63038i
$334$	0	0
$335$	24.0000	1.31126
$336$	0	0
$337$	− 14.0000i	− 0.762629i	−0.924445	−	0.381314i	$-0.875472\pi$
$337$	− 14.0000i	− 0.762629i	0.924445	−	0.381314i	$-0.124528\pi$
$338$	0	0
$339$	18.0000	0.977626
$340$	0	0
$341$	9.00000	0.487377
$342$	0	0
$343$	8.00000	0.431959
$344$	0	0
$345$	54.0000i	2.90726i
$346$	0	0
$347$	6.00000	0.322097	0.161048	−	0.986947i	$-0.448512\pi$
$347$	6.00000	0.322097	0.161048	+	0.986947i	$0.448512\pi$
$348$	0	0
$349$	35.0000	1.87351	0.936754	−	0.349990i	$-0.113815\pi$
$349$	35.0000	1.87351	0.936754	+	0.349990i	$0.113815\pi$
$350$	0	0
$351$	− 27.0000i	− 1.44115i
$352$	0	0
$353$	−22.0000	−1.17094	−0.585471	−	0.810693i	$-0.699090\pi$
$353$	−22.0000	−1.17094	−0.585471	+	0.810693i	$0.699090\pi$
$354$	0	0
$355$	−6.00000	−0.318447
$356$	0	0
$357$	24.0000	1.27021
$358$	0	0
$359$	− 3.00000i	− 0.158334i	−0.996861	−	0.0791670i	$-0.974774\pi$
$359$	− 3.00000i	− 0.158334i	0.996861	−	0.0791670i	$-0.0252260\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	30.0000i	1.57459i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 20.0000i	− 1.04399i	−0.852948	−	0.521996i	$-0.825188\pi$
$367$	− 20.0000i	− 1.04399i	0.852948	−	0.521996i	$-0.174812\pi$
$368$	0	0
$369$	48.0000i	2.49878i
$370$	0	0
$371$	−20.0000	−1.03835
$372$	0	0
$373$	−19.0000	−0.983783	−0.491891	−	0.870657i	$-0.663694\pi$
$373$	−19.0000	−0.983783	−0.491891	+	0.870657i	$0.663694\pi$
$374$	0	0
$375$	9.00000i	0.464758i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 32.0000i	− 1.64373i	−0.569683	−	0.821865i	$-0.692934\pi$
$379$	− 32.0000i	− 1.64373i	0.569683	−	0.821865i	$-0.307066\pi$
$380$	0	0
$381$	−24.0000	−1.22956
$382$	0	0
$383$	16.0000	0.817562	0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000	0.817562	0.408781	+	0.912633i	$0.365954\pi$
$384$	0	0
$385$	− 12.0000i	− 0.611577i
$386$	0	0
$387$	− 30.0000i	− 1.52499i
$388$	0	0
$389$	4.00000i	0.202808i	0.994845	+	0.101404i	$0.0323335\pi$
$389$	4.00000i	0.202808i	−0.994845	+	0.101404i	$0.967667\pi$
$390$	0	0
$391$	12.0000i	0.606866i
$392$	0	0
$393$	−36.0000	−1.81596
$394$	0	0
$395$	− 3.00000i	− 0.150946i
$396$	0	0
$397$	13.0000	0.652451	0.326226	−	0.945292i	$-0.394223\pi$
$397$	13.0000	0.652451	0.326226	+	0.945292i	$0.394223\pi$
$398$	0	0
$399$	48.0000	2.40301
$400$	0	0
$401$	−5.00000	−0.249688	−0.124844	−	0.992176i	$-0.539843\pi$
$401$	−5.00000	−0.249688	−0.124844	+	0.992176i	$0.539843\pi$
$402$	0	0
$403$	− 27.0000i	− 1.34497i
$404$	0	0
$405$	−27.0000	−1.34164
$406$	0	0
$407$	8.00000	0.396545
$408$	0	0
$409$	26.0000i	1.28562i	0.766027	+	0.642809i	$0.222231\pi$
$409$	26.0000i	1.28562i	−0.766027	+	0.642809i	$0.777769\pi$
$410$	0	0
$411$	12.0000	0.591916
$412$	0	0
$413$	−40.0000	−1.96827
$414$	0	0
$415$	−18.0000	−0.883585
$416$	0	0
$417$	− 54.0000i	− 2.64439i
$418$	0	0
$419$	−10.0000	−0.488532	−0.244266	−	0.969708i	$-0.578547\pi$
$419$	−10.0000	−0.488532	−0.244266	+	0.969708i	$0.578547\pi$
$420$	0	0
$421$	2.00000i	0.0974740i	0.998812	+	0.0487370i	$0.0155196\pi$
$421$	2.00000i	0.0974740i	−0.998812	+	0.0487370i	$0.984480\pi$
$422$	0	0
$423$	42.0000i	2.04211i
$424$	0	0
$425$	− 8.00000i	− 0.388057i
$426$	0	0
$427$	− 40.0000i	− 1.93574i
$428$	0	0
$429$	−9.00000	−0.434524
$430$	0	0
$431$	−14.0000	−0.674356	−0.337178	−	0.941441i	$-0.609472\pi$
$431$	−14.0000	−0.674356	−0.337178	+	0.941441i	$0.609472\pi$
$432$	0	0
$433$	− 28.0000i	− 1.34559i	−0.739827	−	0.672797i	$-0.765093\pi$
$433$	− 28.0000i	− 1.34559i	0.739827	−	0.672797i	$-0.234907\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	24.0000i	1.14808i
$438$	0	0
$439$	−22.0000	−1.05000	−0.525001	−	0.851101i	$-0.675935\pi$
$439$	−22.0000	−1.05000	−0.525001	+	0.851101i	$0.675935\pi$
$440$	0	0
$441$	−54.0000	−2.57143
$442$	0	0
$443$	− 12.0000i	− 0.570137i	−0.958507	−	0.285069i	$-0.907984\pi$
$443$	− 12.0000i	− 0.570137i	0.958507	−	0.285069i	$-0.0920164\pi$
$444$	0	0
$445$	36.0000i	1.70656i
$446$	0	0
$447$	− 27.0000i	− 1.27706i
$448$	0	0
$449$	− 22.0000i	− 1.03824i	−0.854700	−	0.519122i	$-0.826259\pi$
$449$	− 22.0000i	− 1.03824i	0.854700	−	0.519122i	$-0.173741\pi$
$450$	0	0
$451$	8.00000	0.376705
$452$	0	0
$453$	− 12.0000i	− 0.563809i
$454$	0	0
$455$	−36.0000	−1.68771
$456$	0	0
$457$	−38.0000	−1.77757	−0.888783	−	0.458329i	$-0.848448\pi$
$457$	−38.0000	−1.77757	−0.888783	+	0.458329i	$0.848448\pi$
$458$	0	0
$459$	−18.0000	−0.840168
$460$	0	0
$461$	− 42.0000i	− 1.95614i	−0.208288	−	0.978068i	$-0.566789\pi$
$461$	− 42.0000i	− 1.95614i	0.208288	−	0.978068i	$-0.433211\pi$
$462$	0	0
$463$	10.0000	0.464739	0.232370	−	0.972628i	$-0.425352\pi$
$463$	10.0000	0.464739	0.232370	+	0.972628i	$0.425352\pi$
$464$	0	0
$465$	−81.0000	−3.75629
$466$	0	0
$467$	13.0000i	0.601568i	0.953692	+	0.300784i	$0.0972484\pi$
$467$	13.0000i	0.601568i	−0.953692	+	0.300784i	$0.902752\pi$
$468$	0	0
$469$	−32.0000	−1.47762
$470$	0	0
$471$	−12.0000	−0.552931
$472$	0	0
$473$	−5.00000	−0.229900
$474$	0	0
$475$	− 16.0000i	− 0.734130i
$476$	0	0
$477$	30.0000	1.37361
$478$	0	0
$479$	− 17.0000i	− 0.776750i	−0.921501	−	0.388375i	$-0.873037\pi$
$479$	− 17.0000i	− 0.776750i	0.921501	−	0.388375i	$-0.126963\pi$
$480$	0	0
$481$	− 24.0000i	− 1.09431i
$482$	0	0
$483$	− 72.0000i	− 3.27611i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	32.0000	1.45006	0.725029	−	0.688718i	$-0.241826\pi$
$487$	32.0000	1.45006	0.725029	+	0.688718i	$0.241826\pi$
$488$	0	0
$489$	−21.0000	−0.949653
$490$	0	0
$491$	17.0000i	0.767199i	0.923499	+	0.383600i	$0.125316\pi$
$491$	17.0000i	0.767199i	−0.923499	+	0.383600i	$0.874684\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	18.0000i	0.809040i
$496$	0	0
$497$	8.00000	0.358849
$498$	0	0
$499$	−10.0000	−0.447661	−0.223831	−	0.974628i	$-0.571856\pi$
$499$	−10.0000	−0.447661	−0.223831	+	0.974628i	$0.571856\pi$
$500$	0	0
$501$	− 6.00000i	− 0.268060i
$502$	0	0
$503$	23.0000i	1.02552i	0.858532	+	0.512760i	$0.171377\pi$
$503$	23.0000i	1.02552i	−0.858532	+	0.512760i	$0.828623\pi$
$504$	0	0
$505$	− 12.0000i	− 0.533993i
$506$	0	0
$507$	− 12.0000i	− 0.532939i
$508$	0	0
$509$	15.0000	0.664863	0.332432	−	0.943127i	$-0.392131\pi$
$509$	15.0000	0.664863	0.332432	+	0.943127i	$0.392131\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−36.0000	−1.58944
$514$	0	0
$515$	−18.0000	−0.793175
$516$	0	0
$517$	7.00000	0.307860
$518$	0	0
$519$	6.00000i	0.263371i
$520$	0	0
$521$	−39.0000	−1.70862	−0.854311	−	0.519763i	$-0.826020\pi$
$521$	−39.0000	−1.70862	−0.854311	+	0.519763i	$0.826020\pi$
$522$	0	0
$523$	−20.0000	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	0	0
$525$	48.0000i	2.09489i
$526$	0	0
$527$	−18.0000	−0.784092
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	60.0000	2.60378
$532$	0	0
$533$	− 24.0000i	− 1.03956i
$534$	0	0
$535$	24.0000	1.03761
$536$	0	0
$537$	− 30.0000i	− 1.29460i
$538$	0	0
$539$	9.00000i	0.387657i
$540$	0	0
$541$	− 40.0000i	− 1.71973i	−0.510518	−	0.859867i	$-0.670546\pi$
$541$	− 40.0000i	− 1.71973i	0.510518	−	0.859867i	$-0.329454\pi$
$542$	0	0
$543$	51.0000i	2.18862i
$544$	0	0
$545$	−15.0000	−0.642529
$546$	0	0
$547$	−8.00000	−0.342055	−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000	−0.342055	−0.171028	+	0.985266i	$0.554709\pi$
$548$	0	0
$549$	60.0000i	2.56074i
$550$	0	0
$551$	0	0
$552$	0	0
$553$	4.00000i	0.170097i
$554$	0	0
$555$	−72.0000	−3.05623
$556$	0	0
$557$	−42.0000	−1.77960	−0.889799	−	0.456354i	$-0.849155\pi$
$557$	−42.0000	−1.77960	−0.889799	+	0.456354i	$0.849155\pi$
$558$	0	0
$559$	15.0000i	0.634432i
$560$	0	0
$561$	6.00000i	0.253320i
$562$	0	0
$563$	− 33.0000i	− 1.39078i	−0.718631	−	0.695392i	$-0.755231\pi$
$563$	− 33.0000i	− 1.39078i	0.718631	−	0.695392i	$-0.244769\pi$
$564$	0	0
$565$	18.0000i	0.757266i
$566$	0	0
$567$	36.0000	1.51186
$568$	0	0
$569$	− 18.0000i	− 0.754599i	−0.926091	−	0.377300i	$-0.876853\pi$
$569$	− 18.0000i	− 0.754599i	0.926091	−	0.377300i	$-0.123147\pi$
$570$	0	0
$571$	0	0		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−24.0000	−1.00087
$576$	0	0
$577$	− 14.0000i	− 0.582828i	−0.956597	−	0.291414i	$-0.905874\pi$
$577$	− 14.0000i	− 0.582828i	0.956597	−	0.291414i	$-0.0941257\pi$
$578$	0	0
$579$	−24.0000	−0.997406
$580$	0	0
$581$	24.0000	0.995688
$582$	0	0
$583$	− 5.00000i	− 0.207079i
$584$	0	0
$585$	54.0000	2.23263
$586$	0	0
$587$	42.0000	1.73353	0.866763	−	0.498721i	$-0.166197\pi$
$587$	42.0000	1.73353	0.866763	+	0.498721i	$0.166197\pi$
$588$	0	0
$589$	−36.0000	−1.48335
$590$	0	0
$591$	78.0000i	3.20849i
$592$	0	0
$593$	29.0000	1.19089	0.595444	−	0.803397i	$-0.296976\pi$
$593$	29.0000	1.19089	0.595444	+	0.803397i	$0.296976\pi$
$594$	0	0
$595$	24.0000i	0.983904i
$596$	0	0
$597$	− 60.0000i	− 2.45564i
$598$	0	0
$599$	− 7.00000i	− 0.286012i	−0.989722	−	0.143006i	$-0.954323\pi$
$599$	− 7.00000i	− 0.286012i	0.989722	−	0.143006i	$-0.0456769\pi$
$600$	0	0
$601$	28.0000i	1.14214i	0.820900	+	0.571072i	$0.193472\pi$
$601$	28.0000i	1.14214i	−0.820900	+	0.571072i	$0.806528\pi$
$602$	0	0
$603$	48.0000	1.95471
$604$	0	0
$605$	−30.0000	−1.21967
$606$	0	0
$607$	47.0000i	1.90767i	0.300329	+	0.953836i	$0.402903\pi$
$607$	47.0000i	1.90767i	−0.300329	+	0.953836i	$0.597097\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 21.0000i	− 0.849569i
$612$	0	0
$613$	21.0000	0.848182	0.424091	−	0.905620i	$-0.360594\pi$
$613$	21.0000	0.848182	0.424091	+	0.905620i	$0.360594\pi$
$614$	0	0
$615$	−72.0000	−2.90332
$616$	0	0
$617$	− 36.0000i	− 1.44931i	−0.689114	−	0.724653i	$-0.742000\pi$
$617$	− 36.0000i	− 1.44931i	0.689114	−	0.724653i	$-0.258000\pi$
$618$	0	0
$619$	− 41.0000i	− 1.64793i	−0.566641	−	0.823965i	$-0.691757\pi$
$619$	− 41.0000i	− 1.64793i	0.566641	−	0.823965i	$-0.308243\pi$
$620$	0	0
$621$	54.0000i	2.16695i
$622$	0	0
$623$	− 48.0000i	− 1.92308i
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	12.0000i	0.479234i
$628$	0	0
$629$	−16.0000	−0.637962
$630$	0	0
$631$	14.0000	0.557331	0.278666	−	0.960388i	$-0.410108\pi$
$631$	14.0000	0.557331	0.278666	+	0.960388i	$0.410108\pi$
$632$	0	0
$633$	39.0000	1.55011
$634$	0	0
$635$	− 24.0000i	− 0.952411i
$636$	0	0
$637$	27.0000	1.06978
$638$	0	0
$639$	−12.0000	−0.474713
$640$	0	0
$641$	− 42.0000i	− 1.65890i	−0.558581	−	0.829450i	$-0.688654\pi$
$641$	− 42.0000i	− 1.65890i	0.558581	−	0.829450i	$-0.311346\pi$
$642$	0	0
$643$	16.0000	0.630978	0.315489	−	0.948929i	$-0.397831\pi$
$643$	16.0000	0.630978	0.315489	+	0.948929i	$0.397831\pi$
$644$	0	0
$645$	45.0000	1.77187
$646$	0	0
$647$	−22.0000	−0.864909	−0.432455	−	0.901656i	$-0.642352\pi$
$647$	−22.0000	−0.864909	−0.432455	+	0.901656i	$0.642352\pi$
$648$	0	0
$649$	− 10.0000i	− 0.392534i
$650$	0	0
$651$	108.000	4.23285
$652$	0	0
$653$	4.00000i	0.156532i	0.996933	+	0.0782660i	$0.0249384\pi$
$653$	4.00000i	0.156532i	−0.996933	+	0.0782660i	$0.975062\pi$
$654$	0	0
$655$	− 36.0000i	− 1.40664i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	15.0000i	0.584317i	0.956370	+	0.292159i	$0.0943735\pi$
$659$	15.0000i	0.584317i	−0.956370	+	0.292159i	$0.905627\pi$
$660$	0	0
$661$	−38.0000	−1.47803	−0.739014	−	0.673690i	$-0.764708\pi$
$661$	−38.0000	−1.47803	−0.739014	+	0.673690i	$0.764708\pi$
$662$	0	0
$663$	18.0000	0.699062
$664$	0	0
$665$	48.0000i	1.86136i
$666$	0	0
$667$	0	0
$668$	0	0
$669$	42.0000i	1.62381i
$670$	0	0
$671$	10.0000	0.386046
$672$	0	0
$673$	−25.0000	−0.963679	−0.481840	−	0.876259i	$-0.660031\pi$
$673$	−25.0000	−0.963679	−0.481840	+	0.876259i	$0.660031\pi$
$674$	0	0
$675$	− 36.0000i	− 1.38564i
$676$	0	0
$677$	− 6.00000i	− 0.230599i	−0.993331	−	0.115299i	$-0.963217\pi$
$677$	− 6.00000i	− 0.230599i	0.993331	−	0.115299i	$-0.0367827\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	90.0000i	3.44881i
$682$	0	0
$683$	8.00000	0.306111	0.153056	−	0.988218i	$-0.451089\pi$
$683$	8.00000	0.306111	0.153056	+	0.988218i	$0.451089\pi$
$684$	0	0
$685$	12.0000i	0.458496i
$686$	0	0
$687$	−60.0000	−2.28914
$688$	0	0
$689$	−15.0000	−0.571454
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	0	0
$693$	− 24.0000i	− 0.911685i
$694$	0	0
$695$	54.0000	2.04834
$696$	0	0
$697$	−16.0000	−0.606043
$698$	0	0
$699$	69.0000i	2.60982i
$700$	0	0
$701$	35.0000	1.32193	0.660966	−	0.750416i	$-0.270147\pi$
$701$	35.0000	1.32193	0.660966	+	0.750416i	$0.270147\pi$
$702$	0	0
$703$	−32.0000	−1.20690
$704$	0	0
$705$	−63.0000	−2.37272
$706$	0	0
$707$	16.0000i	0.601742i
$708$	0	0
$709$	−29.0000	−1.08912	−0.544559	−	0.838723i	$-0.683303\pi$
$709$	−29.0000	−1.08912	−0.544559	+	0.838723i	$0.683303\pi$
$710$	0	0
$711$	− 6.00000i	− 0.225018i
$712$	0	0
$713$	54.0000i	2.02232i
$714$	0	0
$715$	− 9.00000i	− 0.336581i
$716$	0	0
$717$	− 66.0000i	− 2.46482i
$718$	0	0
$719$	6.00000	0.223762	0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000	0.223762	0.111881	+	0.993722i	$0.464312\pi$
$720$	0	0
$721$	24.0000	0.893807
$722$	0	0
$723$	− 51.0000i	− 1.89671i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 32.0000i	− 1.18681i	−0.804902	−	0.593407i	$-0.797782\pi$
$727$	− 32.0000i	− 1.18681i	0.804902	−	0.593407i	$-0.202218\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	10.0000	0.369863
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	− 81.0000i	− 2.98773i
$736$	0	0
$737$	− 8.00000i	− 0.294684i
$738$	0	0
$739$	− 1.00000i	− 0.0367856i	−0.999831	−	0.0183928i	$-0.994145\pi$
$739$	− 1.00000i	− 0.0367856i	0.999831	−	0.0183928i	$-0.00585494\pi$
$740$	0	0
$741$	36.0000	1.32249
$742$	0	0
$743$	8.00000i	0.293492i	0.989174	+	0.146746i	$0.0468799\pi$
$743$	8.00000i	0.293492i	−0.989174	+	0.146746i	$0.953120\pi$
$744$	0	0
$745$	27.0000	0.989203
$746$	0	0
$747$	−36.0000	−1.31717
$748$	0	0
$749$	−32.0000	−1.16925
$750$	0	0
$751$	44.0000i	1.60558i	0.596260	+	0.802791i	$0.296653\pi$
$751$	44.0000i	1.60558i	−0.596260	+	0.802791i	$0.703347\pi$
$752$	0	0
$753$	−93.0000	−3.38911
$754$	0	0
$755$	12.0000	0.436725
$756$	0	0
$757$	34.0000i	1.23575i	0.786276	+	0.617876i	$0.212006\pi$
$757$	34.0000i	1.23575i	−0.786276	+	0.617876i	$0.787994\pi$
$758$	0	0
$759$	18.0000	0.653359
$760$	0	0
$761$	−14.0000	−0.507500	−0.253750	−	0.967270i	$-0.581664\pi$
$761$	−14.0000	−0.507500	−0.253750	+	0.967270i	$0.581664\pi$
$762$	0	0
$763$	20.0000	0.724049
$764$	0	0
$765$	− 36.0000i	− 1.30158i
$766$	0	0
$767$	−30.0000	−1.08324
$768$	0	0
$769$	6.00000i	0.216366i	0.994131	+	0.108183i	$0.0345032\pi$
$769$	6.00000i	0.216366i	−0.994131	+	0.108183i	$0.965497\pi$
$770$	0	0
$771$	− 81.0000i	− 2.91714i
$772$	0	0
$773$	− 36.0000i	− 1.29483i	−0.762138	−	0.647415i	$-0.775850\pi$
$773$	− 36.0000i	− 1.29483i	0.762138	−	0.647415i	$-0.224150\pi$
$774$	0	0
$775$	− 36.0000i	− 1.29316i
$776$	0	0
$777$	96.0000	3.44398
$778$	0	0
$779$	−32.0000	−1.14652
$780$	0	0
$781$	2.00000i	0.0715656i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 12.0000i	− 0.428298i
$786$	0	0
$787$	14.0000	0.499046	0.249523	−	0.968369i	$-0.419726\pi$
$787$	14.0000	0.499046	0.249523	+	0.968369i	$0.419726\pi$
$788$	0	0
$789$	39.0000	1.38844
$790$	0	0
$791$	− 24.0000i	− 0.853342i
$792$	0	0
$793$	− 30.0000i	− 1.06533i
$794$	0	0
$795$	45.0000i	1.59599i
$796$	0	0
$797$	30.0000i	1.06265i	0.847167	+	0.531327i	$0.178307\pi$
$797$	30.0000i	1.06265i	−0.847167	+	0.531327i	$0.821693\pi$
$798$	0	0
$799$	−14.0000	−0.495284
$800$	0	0
$801$	72.0000i	2.54399i
$802$	0	0
$803$	0	0
$804$	0	0
$805$	72.0000	2.53767
$806$	0	0
$807$	−42.0000	−1.47847
$808$	0	0
$809$	− 36.0000i	− 1.26569i	−0.774277	−	0.632846i	$-0.781886\pi$
$809$	− 36.0000i	− 1.26569i	0.774277	−	0.632846i	$-0.218114\pi$
$810$	0	0
$811$	24.0000	0.842754	0.421377	−	0.906886i	$-0.361547\pi$
$811$	24.0000	0.842754	0.421377	+	0.906886i	$0.361547\pi$
$812$	0	0
$813$	15.0000	0.526073
$814$	0	0
$815$	− 21.0000i	− 0.735598i
$816$	0	0
$817$	20.0000	0.699711
$818$	0	0
$819$	−72.0000	−2.51588
$820$	0	0
$821$	43.0000	1.50071	0.750355	−	0.661035i	$-0.229882\pi$
$821$	43.0000	1.50071	0.750355	+	0.661035i	$0.229882\pi$
$822$	0	0
$823$	− 32.0000i	− 1.11545i	−0.830026	−	0.557725i	$-0.811674\pi$
$823$	− 32.0000i	− 1.11545i	0.830026	−	0.557725i	$-0.188326\pi$
$824$	0	0
$825$	−12.0000	−0.417786
$826$	0	0
$827$	− 13.0000i	− 0.452054i	−0.974121	−	0.226027i	$-0.927426\pi$
$827$	− 13.0000i	− 0.452054i	0.974121	−	0.226027i	$-0.0725738\pi$
$828$	0	0
$829$	− 14.0000i	− 0.486240i	−0.969996	−	0.243120i	$-0.921829\pi$
$829$	− 14.0000i	− 0.486240i	0.969996	−	0.243120i	$-0.0781709\pi$
$830$	0	0
$831$	− 30.0000i	− 1.04069i
$832$	0	0
$833$	− 18.0000i	− 0.623663i
$834$	0	0
$835$	6.00000	0.207639
$836$	0	0
$837$	−81.0000	−2.79977
$838$	0	0
$839$	9.00000i	0.310715i	0.987858	+	0.155357i	$0.0496529\pi$
$839$	9.00000i	0.310715i	−0.987858	+	0.155357i	$0.950347\pi$
$840$	0	0
$841$	0	0
$842$	0	0
$843$	− 3.00000i	− 0.103325i
$844$	0	0
$845$	12.0000	0.412813
$846$	0	0
$847$	40.0000	1.37442
$848$	0	0
$849$	18.0000i	0.617758i
$850$	0	0
$851$	48.0000i	1.64542i
$852$	0	0
$853$	− 42.0000i	− 1.43805i	−0.694983	−	0.719026i	$-0.744588\pi$
$853$	− 42.0000i	− 1.43805i	0.694983	−	0.719026i	$-0.255412\pi$
$854$	0	0
$855$	− 72.0000i	− 2.46235i
$856$	0	0
$857$	49.0000	1.67381	0.836904	−	0.547350i	$-0.184363\pi$
$857$	49.0000	1.67381	0.836904	+	0.547350i	$0.184363\pi$
$858$	0	0
$859$	9.00000i	0.307076i	0.988143	+	0.153538i	$0.0490668\pi$
$859$	9.00000i	0.307076i	−0.988143	+	0.153538i	$0.950933\pi$
$860$	0	0
$861$	96.0000	3.27167
$862$	0	0
$863$	−54.0000	−1.83818	−0.919091	−	0.394046i	$-0.871075\pi$
$863$	−54.0000	−1.83818	−0.919091	+	0.394046i	$0.871075\pi$
$864$	0	0
$865$	−6.00000	−0.204006
$866$	0	0
$867$	39.0000i	1.32451i
$868$	0	0
$869$	−1.00000	−0.0339227
$870$	0	0
$871$	−24.0000	−0.813209
$872$	0	0
$873$	0	0
$874$	0	0
$875$	12.0000	0.405674
$876$	0	0
$877$	−29.0000	−0.979260	−0.489630	−	0.871930i	$-0.662868\pi$
$877$	−29.0000	−0.979260	−0.489630	+	0.871930i	$0.662868\pi$
$878$	0	0
$879$	6.00000	0.202375
$880$	0	0
$881$	42.0000i	1.41502i	0.706705	+	0.707508i	$0.250181\pi$
$881$	42.0000i	1.41502i	−0.706705	+	0.707508i	$0.749819\pi$
$882$	0	0
$883$	24.0000	0.807664	0.403832	−	0.914833i	$-0.367678\pi$
$883$	24.0000	0.807664	0.403832	+	0.914833i	$0.367678\pi$
$884$	0	0
$885$	90.0000i	3.02532i
$886$	0	0
$887$	55.0000i	1.84672i	0.383936	+	0.923360i	$0.374568\pi$
$887$	55.0000i	1.84672i	−0.383936	+	0.923360i	$0.625432\pi$
$888$	0	0
$889$	32.0000i	1.07325i
$890$	0	0
$891$	9.00000i	0.301511i
$892$	0	0
$893$	−28.0000	−0.936984
$894$	0	0
$895$	30.0000	1.00279
$896$	0	0
$897$	− 54.0000i	− 1.80301i
$898$	0	0
$899$	0	0
$900$	0	0
$901$	10.0000i	0.333148i
$902$	0	0
$903$	−60.0000	−1.99667
$904$	0	0
$905$	−51.0000	−1.69530
$906$	0	0
$907$	44.0000i	1.46100i	0.682915	+	0.730498i	$0.260712\pi$
$907$	44.0000i	1.46100i	−0.682915	+	0.730498i	$0.739288\pi$
$908$	0	0
$909$	− 24.0000i	− 0.796030i
$910$	0	0
$911$	51.0000i	1.68971i	0.534999	+	0.844853i	$0.320312\pi$
$911$	51.0000i	1.68971i	−0.534999	+	0.844853i	$0.679688\pi$
$912$	0	0
$913$	6.00000i	0.198571i
$914$	0	0
$915$	−90.0000	−2.97531
$916$	0	0
$917$	48.0000i	1.58510i
$918$	0	0
$919$	−18.0000	−0.593765	−0.296883	−	0.954914i	$-0.595947\pi$
$919$	−18.0000	−0.593765	−0.296883	+	0.954914i	$0.595947\pi$
$920$	0	0
$921$	21.0000	0.691974
$922$	0	0
$923$	6.00000	0.197492
$924$	0	0
$925$	− 32.0000i	− 1.05215i
$926$	0	0
$927$	−36.0000	−1.18240
$928$	0	0
$929$	42.0000	1.37798	0.688988	−	0.724773i	$-0.258055\pi$
$929$	42.0000	1.37798	0.688988	+	0.724773i	$0.258055\pi$
$930$	0	0
$931$	− 36.0000i	− 1.17985i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−6.00000	−0.196221
$936$	0	0
$937$	22.0000	0.718709	0.359354	−	0.933201i	$-0.382997\pi$
$937$	22.0000	0.718709	0.359354	+	0.933201i	$0.382997\pi$
$938$	0	0
$939$	3.00000i	0.0979013i
$940$	0	0
$941$	−27.0000	−0.880175	−0.440087	−	0.897955i	$-0.645053\pi$
$941$	−27.0000	−0.880175	−0.440087	+	0.897955i	$0.645053\pi$
$942$	0	0
$943$	48.0000i	1.56310i
$944$	0	0
$945$	108.000i	3.51324i
$946$	0	0
$947$	− 27.0000i	− 0.877382i	−0.898638	−	0.438691i	$-0.855442\pi$
$947$	− 27.0000i	− 0.877382i	0.898638	−	0.438691i	$-0.144558\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−54.0000	−1.75107
$952$	0	0
$953$	3.00000	0.0971795	0.0485898	−	0.998819i	$-0.484527\pi$
$953$	3.00000	0.0971795	0.0485898	+	0.998819i	$0.484527\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 16.0000i	− 0.516667i
$960$	0	0
$961$	−50.0000	−1.61290
$962$	0	0
$963$	48.0000	1.54678
$964$	0	0
$965$	− 24.0000i	− 0.772587i
$966$	0	0
$967$	17.0000i	0.546683i	0.961917	+	0.273342i	$0.0881289\pi$
$967$	17.0000i	0.546683i	−0.961917	+	0.273342i	$0.911871\pi$
$968$	0	0
$969$	− 24.0000i	− 0.770991i
$970$	0	0
$971$	4.00000i	0.128366i	0.997938	+	0.0641831i	$0.0204442\pi$
$971$	4.00000i	0.128366i	−0.997938	+	0.0641831i	$0.979556\pi$
$972$	0	0
$973$	−72.0000	−2.30821
$974$	0	0
$975$	36.0000i	1.15292i
$976$	0	0
$977$	25.0000	0.799821	0.399910	−	0.916554i	$-0.369041\pi$
$977$	25.0000	0.799821	0.399910	+	0.916554i	$0.369041\pi$
$978$	0	0
$979$	12.0000	0.383522
$980$	0	0
$981$	−30.0000	−0.957826
$982$	0	0
$983$	45.0000i	1.43528i	0.696416	+	0.717639i	$0.254777\pi$
$983$	45.0000i	1.43528i	−0.696416	+	0.717639i	$0.745223\pi$
$984$	0	0
$985$	−78.0000	−2.48529
$986$	0	0
$987$	84.0000	2.67375
$988$	0	0
$989$	− 30.0000i	− 0.953945i
$990$	0	0
$991$	34.0000	1.08005	0.540023	−	0.841650i	$-0.318416\pi$
$991$	34.0000	1.08005	0.540023	+	0.841650i	$0.318416\pi$
$992$	0	0
$993$	−9.00000	−0.285606
$994$	0	0
$995$	60.0000	1.90213
$996$	0	0
$997$	8.00000i	0.253363i	0.991943	+	0.126681i	$0.0404325\pi$
$997$	8.00000i	0.253363i	−0.991943	+	0.126681i	$0.959567\pi$
$998$	0	0
$999$	−72.0000	−2.27798

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3364.2.c.b.1681.2		2
29.12	odd	4		116.2.a.a.1.1	✓	1
29.17	odd	4		3364.2.a.c.1.1		1
29.28	even	2	inner	3364.2.c.b.1681.1		2
87.41	even	4		1044.2.a.c.1.1		1
116.99	even	4		464.2.a.g.1.1		1
145.12	even	4		2900.2.c.a.349.2		2
145.99	odd	4		2900.2.a.e.1.1		1
145.128	even	4		2900.2.c.a.349.1		2
203.41	even	4		5684.2.a.k.1.1		1
232.99	even	4		1856.2.a.a.1.1		1
232.157	odd	4		1856.2.a.o.1.1		1
348.215	odd	4		4176.2.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
116.2.a.a.1.1	✓	1	29.12	odd	4
464.2.a.g.1.1		1	116.99	even	4
1044.2.a.c.1.1		1	87.41	even	4
1856.2.a.a.1.1		1	232.99	even	4
1856.2.a.o.1.1		1	232.157	odd	4
2900.2.a.e.1.1		1	145.99	odd	4
2900.2.c.a.349.1		2	145.128	even	4
2900.2.c.a.349.2		2	145.12	even	4
3364.2.a.c.1.1		1	29.17	odd	4
3364.2.c.b.1681.1		2	29.28	even	2	inner
3364.2.c.b.1681.2		2	1.1	even	1	trivial
4176.2.a.c.1.1		1	348.215	odd	4
5684.2.a.k.1.1		1	203.41	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:	\( N \)	\(=\)	\( 3364 = 2^{2} \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3364.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} -3.00000 q^{5} +4.00000 q^{7} -6.00000 q^{9} +O(q^{10})\)	\(q+3.00000i q^{3} -3.00000 q^{5} +4.00000 q^{7} -6.00000 q^{9} +1.00000i q^{11} +3.00000 q^{13} -9.00000i q^{15} -2.00000i q^{17} -4.00000i q^{19} +12.0000i q^{21} -6.00000 q^{23} +4.00000 q^{25} -9.00000i q^{27} -9.00000i q^{31} -3.00000 q^{33} -12.0000 q^{35} -8.00000i q^{37} +9.00000i q^{39} -8.00000i q^{41} +5.00000i q^{43} +18.0000 q^{45} -7.00000i q^{47} +9.00000 q^{49} +6.00000 q^{51} -5.00000 q^{53} -3.00000i q^{55} +12.0000 q^{57} -10.0000 q^{59} -10.0000i q^{61} -24.0000 q^{63} -9.00000 q^{65} -8.00000 q^{67} -18.0000i q^{69} +2.00000 q^{71} +12.0000i q^{75} +4.00000i q^{77} +1.00000i q^{79} +9.00000 q^{81} +6.00000 q^{83} +6.00000i q^{85} -12.0000i q^{89} +12.0000 q^{91} +27.0000 q^{93} +12.0000i q^{95} -6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{5} + 8 q^{7} - 12 q^{9}+O(q^{10}) \) 2 * q - 6 * q^5 + 8 * q^7 - 12 * q^9	\( 2 q - 6 q^{5} + 8 q^{7} - 12 q^{9} + 6 q^{13} - 12 q^{23} + 8 q^{25} - 6 q^{33} - 24 q^{35} + 36 q^{45} + 18 q^{49} + 12 q^{51} - 10 q^{53} + 24 q^{57} - 20 q^{59} - 48 q^{63} - 18 q^{65} - 16 q^{67} + 4 q^{71} + 18 q^{81} + 12 q^{83} + 24 q^{91} + 54 q^{93}+O(q^{100}) \) 2 * q - 6 * q^5 + 8 * q^7 - 12 * q^9 + 6 * q^13 - 12 * q^23 + 8 * q^25 - 6 * q^33 - 24 * q^35 + 36 * q^45 + 18 * q^49 + 12 * q^51 - 10 * q^53 + 24 * q^57 - 20 * q^59 - 48 * q^63 - 18 * q^65 - 16 * q^67 + 4 * q^71 + 18 * q^81 + 12 * q^83 + 24 * q^91 + 54 * q^93

Introduction

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