LMFDB - Embedded newform 3380.2.f.b.3041.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3380,2,Mod(3041,3380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3380, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3380.3041");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3380 = 2^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3380.f (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:	$26.9894358832$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 20)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3041.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	3380.3041
Dual form			3380.2.f.b.3041.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{3} -1.00000i q^{5} -2.00000i q^{7} +1.00000 q^{9} +O(q^{10})$	$q-2.00000 q^{3} -1.00000i q^{5} -2.00000i q^{7} +1.00000 q^{9} +2.00000i q^{15} +6.00000 q^{17} -4.00000i q^{19} +4.00000i q^{21} -6.00000 q^{23} -1.00000 q^{25} +4.00000 q^{27} +6.00000 q^{29} -4.00000i q^{31} -2.00000 q^{35} -2.00000i q^{37} +6.00000i q^{41} +10.0000 q^{43} -1.00000i q^{45} +6.00000i q^{47} +3.00000 q^{49} -12.0000 q^{51} -6.00000 q^{53} +8.00000i q^{57} -12.0000i q^{59} +2.00000 q^{61} -2.00000i q^{63} +2.00000i q^{67} +12.0000 q^{69} -12.0000i q^{71} -2.00000i q^{73} +2.00000 q^{75} +8.00000 q^{79} -11.0000 q^{81} +6.00000i q^{83} -6.00000i q^{85} -12.0000 q^{87} +6.00000i q^{89} +8.00000i q^{93} -4.00000 q^{95} +2.00000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{3} + 2 q^{9}+O(q^{10}) $ 2 * q - 4 * q^3 + 2 * q^9	$ 2 q - 4 q^{3} + 2 q^{9} + 12 q^{17} - 12 q^{23} - 2 q^{25} + 8 q^{27} + 12 q^{29} - 4 q^{35} + 20 q^{43} + 6 q^{49} - 24 q^{51} - 12 q^{53} + 4 q^{61} + 24 q^{69} + 4 q^{75} + 16 q^{79} - 22 q^{81} - 24 q^{87} - 8 q^{95}+O(q^{100}) $ 2 * q - 4 * q^3 + 2 * q^9 + 12 * q^17 - 12 * q^23 - 2 * q^25 + 8 * q^27 + 12 * q^29 - 4 * q^35 + 20 * q^43 + 6 * q^49 - 24 * q^51 - 12 * q^53 + 4 * q^61 + 24 * q^69 + 4 * q^75 + 16 * q^79 - 22 * q^81 - 24 * q^87 - 8 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3380\mathbb{Z}\right)^\times$.

$n$	$677$	$1691$	$1861$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	0	0
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	0	0
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	2.00000i	0.516398i
$16$	0	0
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	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
$21$	4.00000i	0.872872i
$22$	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
$25$	−1.00000	−0.200000
$26$	0	0
$27$	4.00000	0.769800
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	− 4.00000i	− 0.718421i	−0.933257	−	0.359211i	$-0.883046\pi$
$31$	− 4.00000i	− 0.718421i	0.933257	−	0.359211i	$-0.116954\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.00000	−0.338062
$36$	0	0
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.00000i	0.937043i	0.883452	+	0.468521i	$0.155213\pi$
$41$	6.00000i	0.937043i	−0.883452	+	0.468521i	$0.844787\pi$
$42$	0	0
$43$	10.0000	1.52499	0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000	1.52499	0.762493	+	0.646997i	$0.223975\pi$
$44$	0	0
$45$	− 1.00000i	− 0.149071i
$46$	0	0
$47$	6.00000i	0.875190i	0.899172	+	0.437595i	$0.144170\pi$
$47$	6.00000i	0.875190i	−0.899172	+	0.437595i	$0.855830\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	−12.0000	−1.68034
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	8.00000i	1.05963i
$58$	0	0
$59$	− 12.0000i	− 1.56227i	−0.624364	−	0.781133i	$-0.714642\pi$
$59$	− 12.0000i	− 1.56227i	0.624364	−	0.781133i	$-0.285358\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	− 2.00000i	− 0.251976i
$64$	0	0
$65$	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	$0.961015\pi$
$68$	0	0
$69$	12.0000	1.44463
$70$	0	0
$71$	− 12.0000i	− 1.42414i	−0.702109	−	0.712069i	$-0.747758\pi$
$71$	− 12.0000i	− 1.42414i	0.702109	−	0.712069i	$-0.252242\pi$
$72$	0	0
$73$	− 2.00000i	− 0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$	− 2.00000i	− 0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$	0	0
$75$	2.00000	0.230940
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	6.00000i	0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$	6.00000i	0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$	0	0
$85$	− 6.00000i	− 0.650791i
$86$	0	0
$87$	−12.0000	−1.28654
$88$	0	0
$89$	6.00000i	0.635999i	0.948091	+	0.317999i	$0.103011\pi$
$89$	6.00000i	0.635999i	−0.948091	+	0.317999i	$0.896989\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	8.00000i	0.829561i
$94$	0	0
$95$	−4.00000	−0.410391
$96$	0	0
$97$	2.00000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	2.00000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	−14.0000	−1.37946	−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000	−1.37946	−0.689730	+	0.724066i	$0.742271\pi$
$104$	0	0
$105$	4.00000	0.390360
$106$	0	0
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	0	0
$109$	2.00000i	0.191565i	0.995402	+	0.0957826i	$0.0305354\pi$
$109$	2.00000i	0.191565i	−0.995402	+	0.0957826i	$0.969465\pi$
$110$	0	0
$111$	4.00000i	0.379663i
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	6.00000i	0.559503i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 12.0000i	− 1.10004i
$120$	0	0
$121$	11.0000	1.00000
$122$	0	0
$123$	− 12.0000i	− 1.08200i
$124$	0	0
$125$	1.00000i	0.0894427i
$126$	0	0
$127$	−2.00000	−0.177471	−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000	−0.177471	−0.0887357	+	0.996055i	$0.528283\pi$
$128$	0	0
$129$	−20.0000	−1.76090
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	−8.00000	−0.693688
$134$	0	0
$135$	− 4.00000i	− 0.344265i
$136$	0	0
$137$	− 18.0000i	− 1.53784i	−0.639343	−	0.768922i	$-0.720793\pi$
$137$	− 18.0000i	− 1.53784i	0.639343	−	0.768922i	$-0.279207\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	− 12.0000i	− 1.01058i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	− 6.00000i	− 0.498273i
$146$	0	0
$147$	−6.00000	−0.494872
$148$	0	0
$149$	− 6.00000i	− 0.491539i	−0.969328	−	0.245770i	$-0.920959\pi$
$149$	− 6.00000i	− 0.491539i	0.969328	−	0.245770i	$-0.0790407\pi$
$150$	0	0
$151$	− 20.0000i	− 1.62758i	−0.581161	−	0.813788i	$-0.697401\pi$
$151$	− 20.0000i	− 1.62758i	0.581161	−	0.813788i	$-0.302599\pi$
$152$	0	0
$153$	6.00000	0.485071
$154$	0	0
$155$	−4.00000	−0.321288
$156$	0	0
$157$	−22.0000	−1.75579	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−22.0000	−1.75579	−0.877896	+	0.478852i	$0.841053\pi$
$158$	0	0
$159$	12.0000	0.951662
$160$	0	0
$161$	12.0000i	0.945732i
$162$	0	0
$163$	10.0000i	0.783260i	0.920123	+	0.391630i	$0.128089\pi$
$163$	10.0000i	0.783260i	−0.920123	+	0.391630i	$0.871911\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 18.0000i	− 1.39288i	−0.717614	−	0.696441i	$-0.754766\pi$
$167$	− 18.0000i	− 1.39288i	0.717614	−	0.696441i	$-0.245234\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	− 4.00000i	− 0.305888i
$172$	0	0
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	2.00000i	0.151186i
$176$	0	0
$177$	24.0000i	1.80395i
$178$	0	0
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	0	0
$183$	−4.00000	−0.295689
$184$	0	0
$185$	−2.00000	−0.147043
$186$	0	0
$187$	0	0
$188$	0	0
$189$	− 8.00000i	− 0.581914i
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	0	0
$193$	− 26.0000i	− 1.87152i	−0.352636	−	0.935760i	$-0.614715\pi$
$193$	− 26.0000i	− 1.87152i	0.352636	−	0.935760i	$-0.385285\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	18.0000i	1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$	18.0000i	1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	− 4.00000i	− 0.282138i
$202$	0	0
$203$	− 12.0000i	− 0.842235i
$204$	0	0
$205$	6.00000	0.419058
$206$	0	0
$207$	−6.00000	−0.417029
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−16.0000	−1.10149	−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000	−1.10149	−0.550743	+	0.834675i	$0.685655\pi$
$212$	0	0
$213$	24.0000i	1.64445i
$214$	0	0
$215$	− 10.0000i	− 0.681994i
$216$	0	0
$217$	−8.00000	−0.543075
$218$	0	0
$219$	4.00000i	0.270295i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 10.0000i	− 0.669650i	−0.942280	−	0.334825i	$-0.891323\pi$
$223$	− 10.0000i	− 0.669650i	0.942280	−	0.334825i	$-0.108677\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	0	0
$227$	− 6.00000i	− 0.398234i	−0.979976	−	0.199117i	$-0.936193\pi$
$227$	− 6.00000i	− 0.398234i	0.979976	−	0.199117i	$-0.0638074\pi$
$228$	0	0
$229$	− 14.0000i	− 0.925146i	−0.886581	−	0.462573i	$-0.846926\pi$
$229$	− 14.0000i	− 0.925146i	0.886581	−	0.462573i	$-0.153074\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	6.00000	0.391397
$236$	0	0
$237$	−16.0000	−1.03931
$238$	0	0
$239$	− 24.0000i	− 1.55243i	−0.630468	−	0.776215i	$-0.717137\pi$
$239$	− 24.0000i	− 1.55243i	0.630468	−	0.776215i	$-0.282863\pi$
$240$	0	0
$241$	− 14.0000i	− 0.901819i	−0.892570	−	0.450910i	$-0.851100\pi$
$241$	− 14.0000i	− 0.901819i	0.892570	−	0.450910i	$-0.148900\pi$
$242$	0	0
$243$	10.0000	0.641500
$244$	0	0
$245$	− 3.00000i	− 0.191663i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	− 12.0000i	− 0.760469i
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	12.0000i	0.751469i
$256$	0	0
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	−4.00000	−0.248548
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	−18.0000	−1.10993	−0.554964	−	0.831875i	$-0.687268\pi$
$263$	−18.0000	−1.10993	−0.554964	+	0.831875i	$0.687268\pi$
$264$	0	0
$265$	6.00000i	0.368577i
$266$	0	0
$267$	− 12.0000i	− 0.734388i
$268$	0	0
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	− 20.0000i	− 1.21491i	−0.794353	−	0.607457i	$-0.792190\pi$
$271$	− 20.0000i	− 1.21491i	0.794353	−	0.607457i	$-0.207810\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−26.0000	−1.56219	−0.781094	−	0.624413i	$-0.785338\pi$
$277$	−26.0000	−1.56219	−0.781094	+	0.624413i	$0.785338\pi$
$278$	0	0
$279$	− 4.00000i	− 0.239474i
$280$	0	0
$281$	− 6.00000i	− 0.357930i	−0.983855	−	0.178965i	$-0.942725\pi$
$281$	− 6.00000i	− 0.357930i	0.983855	−	0.178965i	$-0.0572749\pi$
$282$	0	0
$283$	−14.0000	−0.832214	−0.416107	−	0.909316i	$-0.636606\pi$
$283$	−14.0000	−0.832214	−0.416107	+	0.909316i	$0.636606\pi$
$284$	0	0
$285$	8.00000	0.473879
$286$	0	0
$287$	12.0000	0.708338
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	− 4.00000i	− 0.234484i
$292$	0	0
$293$	30.0000i	1.75262i	0.481749	+	0.876309i	$0.340002\pi$
$293$	30.0000i	1.75262i	−0.481749	+	0.876309i	$0.659998\pi$
$294$	0	0
$295$	−12.0000	−0.698667
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 20.0000i	− 1.15278i
$302$	0	0
$303$	12.0000	0.689382
$304$	0	0
$305$	− 2.00000i	− 0.114520i
$306$	0	0
$307$	− 2.00000i	− 0.114146i	−0.998370	−	0.0570730i	$-0.981823\pi$
$307$	− 2.00000i	− 0.114146i	0.998370	−	0.0570730i	$-0.0181768\pi$
$308$	0	0
$309$	28.0000	1.59286
$310$	0	0
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	0	0
$315$	−2.00000	−0.112687
$316$	0	0
$317$	− 6.00000i	− 0.336994i	−0.985702	−	0.168497i	$-0.946109\pi$
$317$	− 6.00000i	− 0.336994i	0.985702	−	0.168497i	$-0.0538913\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	− 24.0000i	− 1.33540i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 4.00000i	− 0.221201i
$328$	0	0
$329$	12.0000	0.661581
$330$	0	0
$331$	8.00000i	0.439720i	0.975531	+	0.219860i	$0.0705600\pi$
$331$	8.00000i	0.439720i	−0.975531	+	0.219860i	$0.929440\pi$
$332$	0	0
$333$	− 2.00000i	− 0.109599i
$334$	0	0
$335$	2.00000	0.109272
$336$	0	0
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	0	0
$339$	12.0000	0.651751
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 20.0000i	− 1.07990i
$344$	0	0
$345$	− 12.0000i	− 0.646058i
$346$	0	0
$347$	−30.0000	−1.61048	−0.805242	−	0.592946i	$-0.797965\pi$
$347$	−30.0000	−1.61048	−0.805242	+	0.592946i	$0.797965\pi$
$348$	0	0
$349$	10.0000i	0.535288i	0.963518	+	0.267644i	$0.0862451\pi$
$349$	10.0000i	0.535288i	−0.963518	+	0.267644i	$0.913755\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	18.0000i	0.958043i	0.877803	+	0.479022i	$0.159008\pi$
$353$	18.0000i	0.958043i	−0.877803	+	0.479022i	$0.840992\pi$
$354$	0	0
$355$	−12.0000	−0.636894
$356$	0	0
$357$	24.0000i	1.27021i
$358$	0	0
$359$	− 24.0000i	− 1.26667i	−0.773877	−	0.633336i	$-0.781685\pi$
$359$	− 24.0000i	− 1.26667i	0.773877	−	0.633336i	$-0.218315\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	−22.0000	−1.15470
$364$	0	0
$365$	−2.00000	−0.104685
$366$	0	0
$367$	−22.0000	−1.14839	−0.574195	−	0.818718i	$-0.694685\pi$
$367$	−22.0000	−1.14839	−0.574195	+	0.818718i	$0.694685\pi$
$368$	0	0
$369$	6.00000i	0.312348i
$370$	0	0
$371$	12.0000i	0.623009i
$372$	0	0
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	0	0
$375$	− 2.00000i	− 0.103280i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 28.0000i	− 1.43826i	−0.694874	−	0.719132i	$-0.744540\pi$
$379$	− 28.0000i	− 1.43826i	0.694874	−	0.719132i	$-0.255460\pi$
$380$	0	0
$381$	4.00000	0.204926
$382$	0	0
$383$	6.00000i	0.306586i	0.988181	+	0.153293i	$0.0489878\pi$
$383$	6.00000i	0.306586i	−0.988181	+	0.153293i	$0.951012\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	10.0000	0.508329
$388$	0	0
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	−36.0000	−1.82060
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 8.00000i	− 0.402524i
$396$	0	0
$397$	− 2.00000i	− 0.100377i	−0.998740	−	0.0501886i	$-0.984018\pi$
$397$	− 2.00000i	− 0.100377i	0.998740	−	0.0501886i	$-0.0159822\pi$
$398$	0	0
$399$	16.0000	0.801002
$400$	0	0
$401$	30.0000i	1.49813i	0.662497	+	0.749064i	$0.269497\pi$
$401$	30.0000i	1.49813i	−0.662497	+	0.749064i	$0.730503\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	11.0000i	0.546594i
$406$	0	0
$407$	0	0
$408$	0	0
$409$	− 34.0000i	− 1.68119i	−0.541663	−	0.840596i	$-0.682205\pi$
$409$	− 34.0000i	− 1.68119i	0.541663	−	0.840596i	$-0.317795\pi$
$410$	0	0
$411$	36.0000i	1.77575i
$412$	0	0
$413$	−24.0000	−1.18096
$414$	0	0
$415$	6.00000	0.294528
$416$	0	0
$417$	8.00000	0.391762
$418$	0	0
$419$	36.0000	1.75872	0.879358	−	0.476162i	$-0.157972\pi$
$419$	36.0000	1.75872	0.879358	+	0.476162i	$0.157972\pi$
$420$	0	0
$421$	26.0000i	1.26716i	0.773676	+	0.633581i	$0.218416\pi$
$421$	26.0000i	1.26716i	−0.773676	+	0.633581i	$0.781584\pi$
$422$	0	0
$423$	6.00000i	0.291730i
$424$	0	0
$425$	−6.00000	−0.291043
$426$	0	0
$427$	− 4.00000i	− 0.193574i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	36.0000i	1.73406i	0.498257	+	0.867029i	$0.333974\pi$
$431$	36.0000i	1.73406i	−0.498257	+	0.867029i	$0.666026\pi$
$432$	0	0
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	0	0
$435$	12.0000i	0.575356i
$436$	0	0
$437$	24.0000i	1.14808i
$438$	0	0
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	3.00000	0.142857
$442$	0	0
$443$	6.00000	0.285069	0.142534	−	0.989790i	$-0.454475\pi$
$443$	6.00000	0.285069	0.142534	+	0.989790i	$0.454475\pi$
$444$	0	0
$445$	6.00000	0.284427
$446$	0	0
$447$	12.0000i	0.567581i
$448$	0	0
$449$	− 6.00000i	− 0.283158i	−0.989927	−	0.141579i	$-0.954782\pi$
$449$	− 6.00000i	− 0.283158i	0.989927	−	0.141579i	$-0.0452178\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	40.0000i	1.87936i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	26.0000i	1.21623i	0.793849	+	0.608114i	$0.208074\pi$
$457$	26.0000i	1.21623i	−0.793849	+	0.608114i	$0.791926\pi$
$458$	0	0
$459$	24.0000	1.12022
$460$	0	0
$461$	30.0000i	1.39724i	0.715493	+	0.698620i	$0.246202\pi$
$461$	30.0000i	1.39724i	−0.715493	+	0.698620i	$0.753798\pi$
$462$	0	0
$463$	− 14.0000i	− 0.650635i	−0.945605	−	0.325318i	$-0.894529\pi$
$463$	− 14.0000i	− 0.650635i	0.945605	−	0.325318i	$-0.105471\pi$
$464$	0	0
$465$	8.00000	0.370991
$466$	0	0
$467$	30.0000	1.38823	0.694117	−	0.719862i	$-0.255795\pi$
$467$	30.0000	1.38823	0.694117	+	0.719862i	$0.255795\pi$
$468$	0	0
$469$	4.00000	0.184703
$470$	0	0
$471$	44.0000	2.02741
$472$	0	0
$473$	0	0
$474$	0	0
$475$	4.00000i	0.183533i
$476$	0	0
$477$	−6.00000	−0.274721
$478$	0	0
$479$	24.0000i	1.09659i	0.836286	+	0.548294i	$0.184723\pi$
$479$	24.0000i	1.09659i	−0.836286	+	0.548294i	$0.815277\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	− 24.0000i	− 1.09204i
$484$	0	0
$485$	2.00000	0.0908153
$486$	0	0
$487$	26.0000i	1.17817i	0.808070	+	0.589086i	$0.200512\pi$
$487$	26.0000i	1.17817i	−0.808070	+	0.589086i	$0.799488\pi$
$488$	0	0
$489$	− 20.0000i	− 0.904431i
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	36.0000	1.62136
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−24.0000	−1.07655
$498$	0	0
$499$	− 4.00000i	− 0.179065i	−0.995984	−	0.0895323i	$-0.971463\pi$
$499$	− 4.00000i	− 0.179065i	0.995984	−	0.0895323i	$-0.0285372\pi$
$500$	0	0
$501$	36.0000i	1.60836i
$502$	0	0
$503$	−18.0000	−0.802580	−0.401290	−	0.915951i	$-0.631438\pi$
$503$	−18.0000	−0.802580	−0.401290	+	0.915951i	$0.631438\pi$
$504$	0	0
$505$	6.00000i	0.266996i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	6.00000i	0.265945i	0.991120	+	0.132973i	$0.0424523\pi$
$509$	6.00000i	0.265945i	−0.991120	+	0.132973i	$0.957548\pi$
$510$	0	0
$511$	−4.00000	−0.176950
$512$	0	0
$513$	− 16.0000i	− 0.706417i
$514$	0	0
$515$	14.0000i	0.616914i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−12.0000	−0.526742
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	0	0
$523$	14.0000	0.612177	0.306089	−	0.952003i	$-0.400980\pi$
$523$	14.0000	0.612177	0.306089	+	0.952003i	$0.400980\pi$
$524$	0	0
$525$	− 4.00000i	− 0.174574i
$526$	0	0
$527$	− 24.0000i	− 1.04546i
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	− 12.0000i	− 0.520756i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	6.00000i	0.259403i
$536$	0	0
$537$	−24.0000	−1.03568
$538$	0	0
$539$	0	0
$540$	0	0
$541$	− 14.0000i	− 0.601907i	−0.953639	−	0.300954i	$-0.902695\pi$
$541$	− 14.0000i	− 0.601907i	0.953639	−	0.300954i	$-0.0973049\pi$
$542$	0	0
$543$	−20.0000	−0.858282
$544$	0	0
$545$	2.00000	0.0856706
$546$	0	0
$547$	26.0000	1.11168	0.555840	−	0.831289i	$-0.312397\pi$
$547$	26.0000	1.11168	0.555840	+	0.831289i	$0.312397\pi$
$548$	0	0
$549$	2.00000	0.0853579
$550$	0	0
$551$	− 24.0000i	− 1.02243i
$552$	0	0
$553$	− 16.0000i	− 0.680389i
$554$	0	0
$555$	4.00000	0.169791
$556$	0	0
$557$	30.0000i	1.27114i	0.772043	+	0.635570i	$0.219235\pi$
$557$	30.0000i	1.27114i	−0.772043	+	0.635570i	$0.780765\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	18.0000	0.758610	0.379305	−	0.925272i	$-0.376163\pi$
$563$	18.0000	0.758610	0.379305	+	0.925272i	$0.376163\pi$
$564$	0	0
$565$	6.00000i	0.252422i
$566$	0	0
$567$	22.0000i	0.923913i
$568$	0	0
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	0	0
$571$	−8.00000	−0.334790	−0.167395	−	0.985890i	$-0.553535\pi$
$571$	−8.00000	−0.334790	−0.167395	+	0.985890i	$0.553535\pi$
$572$	0	0
$573$	24.0000	1.00261
$574$	0	0
$575$	6.00000	0.250217
$576$	0	0
$577$	− 22.0000i	− 0.915872i	−0.888985	−	0.457936i	$-0.848589\pi$
$577$	− 22.0000i	− 0.915872i	0.888985	−	0.457936i	$-0.151411\pi$
$578$	0	0
$579$	52.0000i	2.16105i
$580$	0	0
$581$	12.0000	0.497844
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 6.00000i	− 0.247647i	−0.992304	−	0.123823i	$-0.960484\pi$
$587$	− 6.00000i	− 0.247647i	0.992304	−	0.123823i	$-0.0395156\pi$
$588$	0	0
$589$	−16.0000	−0.659269
$590$	0	0
$591$	− 36.0000i	− 1.48084i
$592$	0	0
$593$	− 18.0000i	− 0.739171i	−0.929197	−	0.369586i	$-0.879500\pi$
$593$	− 18.0000i	− 0.739171i	0.929197	−	0.369586i	$-0.120500\pi$
$594$	0	0
$595$	−12.0000	−0.491952
$596$	0	0
$597$	16.0000	0.654836
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	0	0
$603$	2.00000i	0.0814463i
$604$	0	0
$605$	− 11.0000i	− 0.447214i
$606$	0	0
$607$	−22.0000	−0.892952	−0.446476	−	0.894795i	$-0.647321\pi$
$607$	−22.0000	−0.892952	−0.446476	+	0.894795i	$0.647321\pi$
$608$	0	0
$609$	24.0000i	0.972529i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	2.00000i	0.0807792i	0.999184	+	0.0403896i	$0.0128599\pi$
$613$	2.00000i	0.0807792i	−0.999184	+	0.0403896i	$0.987140\pi$
$614$	0	0
$615$	−12.0000	−0.483887
$616$	0	0
$617$	− 6.00000i	− 0.241551i	−0.992680	−	0.120775i	$-0.961462\pi$
$617$	− 6.00000i	− 0.241551i	0.992680	−	0.120775i	$-0.0385381\pi$
$618$	0	0
$619$	− 20.0000i	− 0.803868i	−0.915669	−	0.401934i	$-0.868338\pi$
$619$	− 20.0000i	− 0.803868i	0.915669	−	0.401934i	$-0.131662\pi$
$620$	0	0
$621$	−24.0000	−0.963087
$622$	0	0
$623$	12.0000	0.480770
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 12.0000i	− 0.478471i
$630$	0	0
$631$	28.0000i	1.11466i	0.830290	+	0.557331i	$0.188175\pi$
$631$	28.0000i	1.11466i	−0.830290	+	0.557331i	$0.811825\pi$
$632$	0	0
$633$	32.0000	1.27189
$634$	0	0
$635$	2.00000i	0.0793676i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 12.0000i	− 0.474713i
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	0	0
$643$	14.0000i	0.552106i	0.961142	+	0.276053i	$0.0890266\pi$
$643$	14.0000i	0.552106i	−0.961142	+	0.276053i	$0.910973\pi$
$644$	0	0
$645$	20.0000i	0.787499i
$646$	0	0
$647$	−42.0000	−1.65119	−0.825595	−	0.564263i	$-0.809160\pi$
$647$	−42.0000	−1.65119	−0.825595	+	0.564263i	$0.809160\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	16.0000	0.627089
$652$	0	0
$653$	42.0000	1.64359	0.821794	−	0.569785i	$-0.192974\pi$
$653$	42.0000	1.64359	0.821794	+	0.569785i	$0.192974\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 2.00000i	− 0.0780274i
$658$	0	0
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	0	0
$661$	22.0000i	0.855701i	0.903850	+	0.427850i	$0.140729\pi$
$661$	22.0000i	0.855701i	−0.903850	+	0.427850i	$0.859271\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	8.00000i	0.310227i
$666$	0	0
$667$	−36.0000	−1.39393
$668$	0	0
$669$	20.0000i	0.773245i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	46.0000	1.77317	0.886585	−	0.462566i	$-0.153071\pi$
$673$	46.0000	1.77317	0.886585	+	0.462566i	$0.153071\pi$
$674$	0	0
$675$	−4.00000	−0.153960
$676$	0	0
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	0	0
$679$	4.00000	0.153506
$680$	0	0
$681$	12.0000i	0.459841i
$682$	0	0
$683$	42.0000i	1.60709i	0.595247	+	0.803543i	$0.297054\pi$
$683$	42.0000i	1.60709i	−0.595247	+	0.803543i	$0.702946\pi$
$684$	0	0
$685$	−18.0000	−0.687745
$686$	0	0
$687$	28.0000i	1.06827i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	8.00000i	0.304334i	0.988355	+	0.152167i	$0.0486252\pi$
$691$	8.00000i	0.304334i	−0.988355	+	0.152167i	$0.951375\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4.00000i	0.151729i
$696$	0	0
$697$	36.0000i	1.36360i
$698$	0	0
$699$	−12.0000	−0.453882
$700$	0	0
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	0	0
$703$	−8.00000	−0.301726
$704$	0	0
$705$	−12.0000	−0.451946
$706$	0	0
$707$	12.0000i	0.451306i
$708$	0	0
$709$	34.0000i	1.27690i	0.769665	+	0.638448i	$0.220423\pi$
$709$	34.0000i	1.27690i	−0.769665	+	0.638448i	$0.779577\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	0	0
$713$	24.0000i	0.898807i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	48.0000i	1.79259i
$718$	0	0
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	0	0
$721$	28.0000i	1.04277i
$722$	0	0
$723$	28.0000i	1.04133i
$724$	0	0
$725$	−6.00000	−0.222834
$726$	0	0
$727$	46.0000	1.70605	0.853023	−	0.521874i	$-0.174767\pi$
$727$	46.0000	1.70605	0.853023	+	0.521874i	$0.174767\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	60.0000	2.21918
$732$	0	0
$733$	− 22.0000i	− 0.812589i	−0.913742	−	0.406294i	$-0.866821\pi$
$733$	− 22.0000i	− 0.812589i	0.913742	−	0.406294i	$-0.133179\pi$
$734$	0	0
$735$	6.00000i	0.221313i
$736$	0	0
$737$	0	0
$738$	0	0
$739$	− 20.0000i	− 0.735712i	−0.929883	−	0.367856i	$-0.880092\pi$
$739$	− 20.0000i	− 0.735712i	0.929883	−	0.367856i	$-0.119908\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	6.00000i	0.220119i	0.993925	+	0.110059i	$0.0351041\pi$
$743$	6.00000i	0.220119i	−0.993925	+	0.110059i	$0.964896\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	0	0
$747$	6.00000i	0.219529i
$748$	0	0
$749$	12.0000i	0.438470i
$750$	0	0
$751$	4.00000	0.145962	0.0729810	−	0.997333i	$-0.476749\pi$
$751$	4.00000	0.145962	0.0729810	+	0.997333i	$0.476749\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−20.0000	−0.727875
$756$	0	0
$757$	−22.0000	−0.799604	−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000	−0.799604	−0.399802	+	0.916602i	$0.630921\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 42.0000i	− 1.52250i	−0.648459	−	0.761249i	$-0.724586\pi$
$761$	− 42.0000i	− 1.52250i	0.648459	−	0.761249i	$-0.275414\pi$
$762$	0	0
$763$	4.00000	0.144810
$764$	0	0
$765$	− 6.00000i	− 0.216930i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	2.00000i	0.0721218i	0.999350	+	0.0360609i	$0.0114810\pi$
$769$	2.00000i	0.0721218i	−0.999350	+	0.0360609i	$0.988519\pi$
$770$	0	0
$771$	−12.0000	−0.432169
$772$	0	0
$773$	− 30.0000i	− 1.07903i	−0.841978	−	0.539513i	$-0.818609\pi$
$773$	− 30.0000i	− 1.07903i	0.841978	−	0.539513i	$-0.181391\pi$
$774$	0	0
$775$	4.00000i	0.143684i
$776$	0	0
$777$	8.00000	0.286998
$778$	0	0
$779$	24.0000	0.859889
$780$	0	0
$781$	0	0
$782$	0	0
$783$	24.0000	0.857690
$784$	0	0
$785$	22.0000i	0.785214i
$786$	0	0
$787$	− 26.0000i	− 0.926800i	−0.886149	−	0.463400i	$-0.846629\pi$
$787$	− 26.0000i	− 0.926800i	0.886149	−	0.463400i	$-0.153371\pi$
$788$	0	0
$789$	36.0000	1.28163
$790$	0	0
$791$	12.0000i	0.426671i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 12.0000i	− 0.425596i
$796$	0	0
$797$	−42.0000	−1.48772	−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000	−1.48772	−0.743858	+	0.668338i	$0.767006\pi$
$798$	0	0
$799$	36.0000i	1.27359i
$800$	0	0
$801$	6.00000i	0.212000i
$802$	0	0
$803$	0	0
$804$	0	0
$805$	12.0000	0.422944
$806$	0	0
$807$	−36.0000	−1.26726
$808$	0	0
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	0	0
$811$	− 16.0000i	− 0.561836i	−0.959732	−	0.280918i	$-0.909361\pi$
$811$	− 16.0000i	− 0.561836i	0.959732	−	0.280918i	$-0.0906389\pi$
$812$	0	0
$813$	40.0000i	1.40286i
$814$	0	0
$815$	10.0000	0.350285
$816$	0	0
$817$	− 40.0000i	− 1.39942i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 54.0000i	− 1.88461i	−0.334751	−	0.942306i	$-0.608652\pi$
$821$	− 54.0000i	− 1.88461i	0.334751	−	0.942306i	$-0.391348\pi$
$822$	0	0
$823$	−38.0000	−1.32460	−0.662298	−	0.749240i	$-0.730419\pi$
$823$	−38.0000	−1.32460	−0.662298	+	0.749240i	$0.730419\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	30.0000i	1.04320i	0.853189	+	0.521601i	$0.174665\pi$
$827$	30.0000i	1.04320i	−0.853189	+	0.521601i	$0.825335\pi$
$828$	0	0
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	0	0
$831$	52.0000	1.80386
$832$	0	0
$833$	18.0000	0.623663
$834$	0	0
$835$	−18.0000	−0.622916
$836$	0	0
$837$	− 16.0000i	− 0.553041i
$838$	0	0
$839$	48.0000i	1.65714i	0.559883	+	0.828572i	$0.310846\pi$
$839$	48.0000i	1.65714i	−0.559883	+	0.828572i	$0.689154\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	12.0000i	0.413302i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 22.0000i	− 0.755929i
$848$	0	0
$849$	28.0000	0.960958
$850$	0	0
$851$	12.0000i	0.411355i
$852$	0	0
$853$	− 50.0000i	− 1.71197i	−0.517003	−	0.855984i	$-0.672952\pi$
$853$	− 50.0000i	− 1.71197i	0.517003	−	0.855984i	$-0.327048\pi$
$854$	0	0
$855$	−4.00000	−0.136797
$856$	0	0
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	0	0
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	0	0
$861$	−24.0000	−0.817918
$862$	0	0
$863$	6.00000i	0.204242i	0.994772	+	0.102121i	$0.0325630\pi$
$863$	6.00000i	0.204242i	−0.994772	+	0.102121i	$0.967437\pi$
$864$	0	0
$865$	− 6.00000i	− 0.204006i
$866$	0	0
$867$	−38.0000	−1.29055
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	2.00000i	0.0676897i
$874$	0	0
$875$	2.00000	0.0676123
$876$	0	0
$877$	26.0000i	0.877958i	0.898497	+	0.438979i	$0.144660\pi$
$877$	26.0000i	0.877958i	−0.898497	+	0.438979i	$0.855340\pi$
$878$	0	0
$879$	− 60.0000i	− 2.02375i
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	0	0
$883$	−14.0000	−0.471138	−0.235569	−	0.971858i	$-0.575695\pi$
$883$	−14.0000	−0.471138	−0.235569	+	0.971858i	$0.575695\pi$
$884$	0	0
$885$	24.0000	0.806751
$886$	0	0
$887$	18.0000	0.604381	0.302190	−	0.953248i	$-0.402282\pi$
$887$	18.0000	0.604381	0.302190	+	0.953248i	$0.402282\pi$
$888$	0	0
$889$	4.00000i	0.134156i
$890$	0	0
$891$	0	0
$892$	0	0
$893$	24.0000	0.803129
$894$	0	0
$895$	− 12.0000i	− 0.401116i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 24.0000i	− 0.800445i
$900$	0	0
$901$	−36.0000	−1.19933
$902$	0	0
$903$	40.0000i	1.33112i
$904$	0	0
$905$	− 10.0000i	− 0.332411i
$906$	0	0
$907$	46.0000	1.52740	0.763702	−	0.645568i	$-0.223379\pi$
$907$	46.0000	1.52740	0.763702	+	0.645568i	$0.223379\pi$
$908$	0	0
$909$	−6.00000	−0.199007
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	4.00000i	0.132236i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	4.00000i	0.131804i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	2.00000i	0.0657596i
$926$	0	0
$927$	−14.0000	−0.459820
$928$	0	0
$929$	− 42.0000i	− 1.37798i	−0.724773	−	0.688988i	$-0.758055\pi$
$929$	− 42.0000i	− 1.37798i	0.724773	−	0.688988i	$-0.241945\pi$
$930$	0	0
$931$	− 12.0000i	− 0.393284i
$932$	0	0
$933$	24.0000	0.785725
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	0	0
$939$	44.0000	1.43589
$940$	0	0
$941$	− 18.0000i	− 0.586783i	−0.955992	−	0.293392i	$-0.905216\pi$
$941$	− 18.0000i	− 0.586783i	0.955992	−	0.293392i	$-0.0947840\pi$
$942$	0	0
$943$	− 36.0000i	− 1.17232i
$944$	0	0
$945$	−8.00000	−0.260240
$946$	0	0
$947$	− 18.0000i	− 0.584921i	−0.956278	−	0.292461i	$-0.905526\pi$
$947$	− 18.0000i	− 0.584921i	0.956278	−	0.292461i	$-0.0944741\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	12.0000i	0.389127i
$952$	0	0
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	0	0
$955$	12.0000i	0.388311i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−36.0000	−1.16250
$960$	0	0
$961$	15.0000	0.483871
$962$	0	0
$963$	−6.00000	−0.193347
$964$	0	0
$965$	−26.0000	−0.836970
$966$	0	0
$967$	− 22.0000i	− 0.707472i	−0.935345	−	0.353736i	$-0.884911\pi$
$967$	− 22.0000i	− 0.707472i	0.935345	−	0.353736i	$-0.115089\pi$
$968$	0	0
$969$	48.0000i	1.54198i
$970$	0	0
$971$	−24.0000	−0.770197	−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000	−0.770197	−0.385098	+	0.922876i	$0.625832\pi$
$972$	0	0
$973$	8.00000i	0.256468i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	18.0000i	0.575871i	0.957650	+	0.287936i	$0.0929689\pi$
$977$	18.0000i	0.575871i	−0.957650	+	0.287936i	$0.907031\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	2.00000i	0.0638551i
$982$	0	0
$983$	18.0000i	0.574111i	0.957914	+	0.287055i	$0.0926764\pi$
$983$	18.0000i	0.574111i	−0.957914	+	0.287055i	$0.907324\pi$
$984$	0	0
$985$	18.0000	0.573528
$986$	0	0
$987$	−24.0000	−0.763928
$988$	0	0
$989$	−60.0000	−1.90789
$990$	0	0
$991$	−4.00000	−0.127064	−0.0635321	−	0.997980i	$-0.520237\pi$
$991$	−4.00000	−0.127064	−0.0635321	+	0.997980i	$0.520237\pi$
$992$	0	0
$993$	− 16.0000i	− 0.507745i
$994$	0	0
$995$	8.00000i	0.253617i
$996$	0	0
$997$	26.0000	0.823428	0.411714	−	0.911313i	$-0.364930\pi$
$997$	26.0000	0.823428	0.411714	+	0.911313i	$0.364930\pi$
$998$	0	0
$999$	− 8.00000i	− 0.253109i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3380.2.f.b.3041.1		2
13.5	odd	4		20.2.a.a.1.1	✓	1
13.8	odd	4		3380.2.a.c.1.1		1
13.12	even	2	inner	3380.2.f.b.3041.2		2
39.5	even	4		180.2.a.a.1.1		1
52.31	even	4		80.2.a.b.1.1		1
65.18	even	4		100.2.c.a.49.1		2
65.44	odd	4		100.2.a.a.1.1		1
65.57	even	4		100.2.c.a.49.2		2
91.5	even	12		980.2.i.c.361.1		2
91.18	odd	12		980.2.i.i.961.1		2
91.31	even	12		980.2.i.c.961.1		2
91.44	odd	12		980.2.i.i.361.1		2
91.83	even	4		980.2.a.h.1.1		1
104.5	odd	4		320.2.a.f.1.1		1
104.83	even	4		320.2.a.a.1.1		1
117.5	even	12		1620.2.i.b.541.1		2
117.31	odd	12		1620.2.i.h.541.1		2
117.70	odd	12		1620.2.i.h.1081.1		2
117.83	even	12		1620.2.i.b.1081.1		2
143.109	even	4		2420.2.a.a.1.1		1
156.83	odd	4		720.2.a.h.1.1		1
195.44	even	4		900.2.a.b.1.1		1
195.83	odd	4		900.2.d.c.649.1		2
195.122	odd	4		900.2.d.c.649.2		2
208.5	odd	4		1280.2.d.c.641.2		2
208.83	even	4		1280.2.d.g.641.2		2
208.109	odd	4		1280.2.d.c.641.1		2
208.187	even	4		1280.2.d.g.641.1		2
221.135	odd	4		5780.2.a.f.1.1		1
221.174	odd	4		5780.2.c.a.5201.2		2
221.200	odd	4		5780.2.c.a.5201.1		2
247.18	even	4		7220.2.a.f.1.1		1
260.83	odd	4		400.2.c.b.49.2		2
260.187	odd	4		400.2.c.b.49.1		2
260.239	even	4		400.2.a.c.1.1		1
273.83	odd	4		8820.2.a.g.1.1		1
312.5	even	4		2880.2.a.m.1.1		1
312.83	odd	4		2880.2.a.f.1.1		1
364.83	odd	4		3920.2.a.h.1.1		1
455.83	odd	4		4900.2.e.f.2549.2		2
455.174	even	4		4900.2.a.e.1.1		1
455.447	odd	4		4900.2.e.f.2549.1		2
520.83	odd	4		1600.2.c.e.449.1		2
520.109	odd	4		1600.2.a.c.1.1		1
520.187	odd	4		1600.2.c.e.449.2		2
520.213	even	4		1600.2.c.d.449.2		2
520.317	even	4		1600.2.c.d.449.1		2
520.499	even	4		1600.2.a.w.1.1		1
572.395	odd	4		9680.2.a.ba.1.1		1
780.83	even	4		3600.2.f.j.2449.2		2
780.239	odd	4		3600.2.a.be.1.1		1
780.707	even	4		3600.2.f.j.2449.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.2.a.a.1.1	✓	1	13.5	odd	4
80.2.a.b.1.1		1	52.31	even	4
100.2.a.a.1.1		1	65.44	odd	4
100.2.c.a.49.1		2	65.18	even	4
100.2.c.a.49.2		2	65.57	even	4
180.2.a.a.1.1		1	39.5	even	4
320.2.a.a.1.1		1	104.83	even	4
320.2.a.f.1.1		1	104.5	odd	4
400.2.a.c.1.1		1	260.239	even	4
400.2.c.b.49.1		2	260.187	odd	4
400.2.c.b.49.2		2	260.83	odd	4
720.2.a.h.1.1		1	156.83	odd	4
900.2.a.b.1.1		1	195.44	even	4
900.2.d.c.649.1		2	195.83	odd	4
900.2.d.c.649.2		2	195.122	odd	4
980.2.a.h.1.1		1	91.83	even	4
980.2.i.c.361.1		2	91.5	even	12
980.2.i.c.961.1		2	91.31	even	12
980.2.i.i.361.1		2	91.44	odd	12
980.2.i.i.961.1		2	91.18	odd	12
1280.2.d.c.641.1		2	208.109	odd	4
1280.2.d.c.641.2		2	208.5	odd	4
1280.2.d.g.641.1		2	208.187	even	4
1280.2.d.g.641.2		2	208.83	even	4
1600.2.a.c.1.1		1	520.109	odd	4
1600.2.a.w.1.1		1	520.499	even	4
1600.2.c.d.449.1		2	520.317	even	4
1600.2.c.d.449.2		2	520.213	even	4
1600.2.c.e.449.1		2	520.83	odd	4
1600.2.c.e.449.2		2	520.187	odd	4
1620.2.i.b.541.1		2	117.5	even	12
1620.2.i.b.1081.1		2	117.83	even	12
1620.2.i.h.541.1		2	117.31	odd	12
1620.2.i.h.1081.1		2	117.70	odd	12
2420.2.a.a.1.1		1	143.109	even	4
2880.2.a.f.1.1		1	312.83	odd	4
2880.2.a.m.1.1		1	312.5	even	4
3380.2.a.c.1.1		1	13.8	odd	4
3380.2.f.b.3041.1		2	1.1	even	1	trivial
3380.2.f.b.3041.2		2	13.12	even	2	inner
3600.2.a.be.1.1		1	780.239	odd	4
3600.2.f.j.2449.1		2	780.707	even	4
3600.2.f.j.2449.2		2	780.83	even	4
3920.2.a.h.1.1		1	364.83	odd	4
4900.2.a.e.1.1		1	455.174	even	4
4900.2.e.f.2549.1		2	455.447	odd	4
4900.2.e.f.2549.2		2	455.83	odd	4
5780.2.a.f.1.1		1	221.135	odd	4
5780.2.c.a.5201.1		2	221.200	odd	4
5780.2.c.a.5201.2		2	221.174	odd	4
7220.2.a.f.1.1		1	247.18	even	4
8820.2.a.g.1.1		1	273.83	odd	4
9680.2.a.ba.1.1		1	572.395	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:	\( N \)	\(=\)	\( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3380.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000 q^{3} -1.00000i q^{5} -2.00000i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-2.00000 q^{3} -1.00000i q^{5} -2.00000i q^{7} +1.00000 q^{9} +2.00000i q^{15} +6.00000 q^{17} -4.00000i q^{19} +4.00000i q^{21} -6.00000 q^{23} -1.00000 q^{25} +4.00000 q^{27} +6.00000 q^{29} -4.00000i q^{31} -2.00000 q^{35} -2.00000i q^{37} +6.00000i q^{41} +10.0000 q^{43} -1.00000i q^{45} +6.00000i q^{47} +3.00000 q^{49} -12.0000 q^{51} -6.00000 q^{53} +8.00000i q^{57} -12.0000i q^{59} +2.00000 q^{61} -2.00000i q^{63} +2.00000i q^{67} +12.0000 q^{69} -12.0000i q^{71} -2.00000i q^{73} +2.00000 q^{75} +8.00000 q^{79} -11.0000 q^{81} +6.00000i q^{83} -6.00000i q^{85} -12.0000 q^{87} +6.00000i q^{89} +8.00000i q^{93} -4.00000 q^{95} +2.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{3} + 2 q^{9}+O(q^{10}) \) 2 * q - 4 * q^3 + 2 * q^9	\( 2 q - 4 q^{3} + 2 q^{9} + 12 q^{17} - 12 q^{23} - 2 q^{25} + 8 q^{27} + 12 q^{29} - 4 q^{35} + 20 q^{43} + 6 q^{49} - 24 q^{51} - 12 q^{53} + 4 q^{61} + 24 q^{69} + 4 q^{75} + 16 q^{79} - 22 q^{81} - 24 q^{87} - 8 q^{95}+O(q^{100}) \) 2 * q - 4 * q^3 + 2 * q^9 + 12 * q^17 - 12 * q^23 - 2 * q^25 + 8 * q^27 + 12 * q^29 - 4 * q^35 + 20 * q^43 + 6 * q^49 - 24 * q^51 - 12 * q^53 + 4 * q^61 + 24 * q^69 + 4 * q^75 + 16 * q^79 - 22 * q^81 - 24 * q^87 - 8 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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