LMFDB - Embedded newform 3380.2.f.j.3041.16

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:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} + 39x^{16} + 605x^{14} + 4764x^{12} + 20080x^{10} + 43783x^{8} + 43791x^{6} + 14071x^{4} + 378x^{2} + 1 $ x^18 + 39x^16 + 605x^14 + 4764x^12 + 20080x^10 + 43783x^8 + 43791x^6 + 14071x^4 + 378x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3041.16
Root			$-2.79462i$ of defining polynomial
Character	$\chi$	$=$	3380.3041
Dual form			3380.2.f.j.3041.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.79462 q^{3} +1.00000i q^{5} -1.22908i q^{7} +4.80991 q^{9} +O(q^{10})$	$q+2.79462 q^{3} +1.00000i q^{5} -1.22908i q^{7} +4.80991 q^{9} -3.73479i q^{11} +2.79462i q^{15} -4.95611 q^{17} -3.06501i q^{19} -3.43481i q^{21} +1.04385 q^{23} -1.00000 q^{25} +5.05802 q^{27} +9.02219 q^{29} -6.57334i q^{31} -10.4373i q^{33} +1.22908 q^{35} +0.619519i q^{37} -8.41141i q^{41} +8.50043 q^{43} +4.80991i q^{45} +5.65693i q^{47} +5.48936 q^{49} -13.8504 q^{51} +9.83977 q^{53} +3.73479 q^{55} -8.56555i q^{57} -12.1840i q^{59} +11.5333 q^{61} -5.91176i q^{63} -7.73375i q^{67} +2.91718 q^{69} -5.36427i q^{71} +14.3772i q^{73} -2.79462 q^{75} -4.59035 q^{77} -11.8261 q^{79} -0.294490 q^{81} +12.6989i q^{83} -4.95611i q^{85} +25.2136 q^{87} +13.5729i q^{89} -18.3700i q^{93} +3.06501 q^{95} +13.9803i q^{97} -17.9640i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 2 q^{3} + 24 q^{9}+O(q^{10}) $ 18 * q - 2 * q^3 + 24 * q^9	$ 18 q - 2 q^{3} + 24 q^{9} - 26 q^{17} - 24 q^{23} - 18 q^{25} - 8 q^{27} + 32 q^{29} + 2 q^{35} - 2 q^{43} - 40 q^{49} - 22 q^{51} + 60 q^{53} - 14 q^{55} + 42 q^{61} - 30 q^{69} + 2 q^{75} - 92 q^{77} + 62 q^{79} + 82 q^{81} + 56 q^{87} + 8 q^{95}+O(q^{100}) $ 18 * q - 2 * q^3 + 24 * q^9 - 26 * q^17 - 24 * q^23 - 18 * q^25 - 8 * q^27 + 32 * q^29 + 2 * q^35 - 2 * q^43 - 40 * q^49 - 22 * q^51 + 60 * q^53 - 14 * q^55 + 42 * q^61 - 30 * q^69 + 2 * q^75 - 92 * q^77 + 62 * q^79 + 82 * q^81 + 56 * 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
$3$	2.79462	1.61348	0.806738	−	0.590910i	$-0.201231\pi$
$3$	2.79462	1.61348	0.806738	+	0.590910i	$0.201231\pi$
$4$	0	0
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	− 1.22908i	− 0.464548i	−0.972650	−	0.232274i	$-0.925383\pi$
$7$	− 1.22908i	− 0.464548i	0.972650	−	0.232274i	$-0.0746167\pi$
$8$	0	0
$9$	4.80991	1.60330
$10$	0	0
$11$	− 3.73479i	− 1.12608i	−0.826429	−	0.563040i	$-0.809632\pi$
$11$	− 3.73479i	− 1.12608i	0.826429	−	0.563040i	$-0.190368\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	2.79462i	0.721568i
$16$	0	0
$17$	−4.95611	−1.20203	−0.601016	−	0.799237i	$-0.705237\pi$
$17$	−4.95611	−1.20203	−0.601016	+	0.799237i	$0.705237\pi$
$18$	0	0
$19$	− 3.06501i	− 0.703162i	−0.936158	−	0.351581i	$-0.885644\pi$
$19$	− 3.06501i	− 0.703162i	0.936158	−	0.351581i	$-0.114356\pi$
$20$	0	0
$21$	− 3.43481i	− 0.749537i
$22$	0	0
$23$	1.04385	0.217659	0.108829	−	0.994060i	$-0.465290\pi$
$23$	1.04385	0.217659	0.108829	+	0.994060i	$0.465290\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	5.05802	0.973416
$28$	0	0
$29$	9.02219	1.67538	0.837689	−	0.546147i	$-0.183906\pi$
$29$	9.02219	1.67538	0.837689	+	0.546147i	$0.183906\pi$
$30$	0	0
$31$	− 6.57334i	− 1.18061i	−0.807182	−	0.590303i	$-0.799008\pi$
$31$	− 6.57334i	− 1.18061i	0.807182	−	0.590303i	$-0.200992\pi$
$32$	0	0
$33$	− 10.4373i	− 1.81690i
$34$	0	0
$35$	1.22908	0.207752
$36$	0	0
$37$	0.619519i	0.101848i	0.998703	+	0.0509241i	$0.0162167\pi$
$37$	0.619519i	0.101848i	−0.998703	+	0.0509241i	$0.983783\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 8.41141i	− 1.31364i	−0.754047	−	0.656821i	$-0.771901\pi$
$41$	− 8.41141i	− 1.31364i	0.754047	−	0.656821i	$-0.228099\pi$
$42$	0	0
$43$	8.50043	1.29630	0.648152	−	0.761511i	$-0.275542\pi$
$43$	8.50043	1.29630	0.648152	+	0.761511i	$0.275542\pi$
$44$	0	0
$45$	4.80991i	0.717019i
$46$	0	0
$47$	5.65693i	0.825148i	0.910924	+	0.412574i	$0.135370\pi$
$47$	5.65693i	0.825148i	−0.910924	+	0.412574i	$0.864630\pi$
$48$	0	0
$49$	5.48936	0.784195
$50$	0	0
$51$	−13.8504	−1.93945
$52$	0	0
$53$	9.83977	1.35160	0.675798	−	0.737087i	$-0.263799\pi$
$53$	9.83977	1.35160	0.675798	+	0.737087i	$0.263799\pi$
$54$	0	0
$55$	3.73479	0.503599
$56$	0	0
$57$	− 8.56555i	− 1.13453i
$58$	0	0
$59$	− 12.1840i	− 1.58622i	−0.609078	−	0.793110i	$-0.708461\pi$
$59$	− 12.1840i	− 1.58622i	0.609078	−	0.793110i	$-0.291539\pi$
$60$	0	0
$61$	11.5333	1.47668	0.738342	−	0.674426i	$-0.235609\pi$
$61$	11.5333	1.47668	0.738342	+	0.674426i	$0.235609\pi$
$62$	0	0
$63$	− 5.91176i	− 0.744812i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 7.73375i	− 0.944827i	−0.881377	−	0.472414i	$-0.843383\pi$
$67$	− 7.73375i	− 0.944827i	0.881377	−	0.472414i	$-0.156617\pi$
$68$	0	0
$69$	2.91718	0.351187
$70$	0	0
$71$	− 5.36427i	− 0.636622i	−0.947986	−	0.318311i	$-0.896884\pi$
$71$	− 5.36427i	− 0.636622i	0.947986	−	0.318311i	$-0.103116\pi$
$72$	0	0
$73$	14.3772i	1.68273i	0.540471	+	0.841363i	$0.318246\pi$
$73$	14.3772i	1.68273i	−0.540471	+	0.841363i	$0.681754\pi$
$74$	0	0
$75$	−2.79462	−0.322695
$76$	0	0
$77$	−4.59035	−0.523119
$78$	0	0
$79$	−11.8261	−1.33054	−0.665271	−	0.746602i	$-0.731684\pi$
$79$	−11.8261	−1.33054	−0.665271	+	0.746602i	$0.731684\pi$
$80$	0	0
$81$	−0.294490	−0.0327211
$82$	0	0
$83$	12.6989i	1.39389i	0.717127	+	0.696943i	$0.245457\pi$
$83$	12.6989i	1.39389i	−0.717127	+	0.696943i	$0.754543\pi$
$84$	0	0
$85$	− 4.95611i	− 0.537565i
$86$	0	0
$87$	25.2136	2.70318
$88$	0	0
$89$	13.5729i	1.43872i	0.694635	+	0.719362i	$0.255566\pi$
$89$	13.5729i	1.43872i	−0.694635	+	0.719362i	$0.744434\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 18.3700i	− 1.90488i
$94$	0	0
$95$	3.06501	0.314463
$96$	0	0
$97$	13.9803i	1.41948i	0.704462	+	0.709742i	$0.251189\pi$
$97$	13.9803i	1.41948i	−0.704462	+	0.709742i	$0.748811\pi$
$98$	0	0
$99$	− 17.9640i	− 1.80545i
$100$	0	0
$101$	−7.43227	−0.739538	−0.369769	−	0.929124i	$-0.620563\pi$
$101$	−7.43227	−0.739538	−0.369769	+	0.929124i	$0.620563\pi$
$102$	0	0
$103$	−16.4706	−1.62289	−0.811447	−	0.584427i	$-0.801319\pi$
$103$	−16.4706	−1.62289	−0.811447	+	0.584427i	$0.801319\pi$
$104$	0	0
$105$	3.43481	0.335203
$106$	0	0
$107$	−8.76920	−0.847750	−0.423875	−	0.905721i	$-0.639330\pi$
$107$	−8.76920	−0.847750	−0.423875	+	0.905721i	$0.639330\pi$
$108$	0	0
$109$	− 7.17891i	− 0.687615i	−0.939040	−	0.343808i	$-0.888283\pi$
$109$	− 7.17891i	− 0.687615i	0.939040	−	0.343808i	$-0.111717\pi$
$110$	0	0
$111$	1.73132i	0.164330i
$112$	0	0
$113$	−10.0341	−0.943931	−0.471966	−	0.881617i	$-0.656455\pi$
$113$	−10.0341	−0.943931	−0.471966	+	0.881617i	$0.656455\pi$
$114$	0	0
$115$	1.04385i	0.0973399i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	6.09145i	0.558402i
$120$	0	0
$121$	−2.94864	−0.268058
$122$	0	0
$123$	− 23.5067i	− 2.11953i
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	−1.60663	−0.142565	−0.0712827	−	0.997456i	$-0.522709\pi$
$127$	−1.60663	−0.142565	−0.0712827	+	0.997456i	$0.522709\pi$
$128$	0	0
$129$	23.7555	2.09155
$130$	0	0
$131$	−10.8166	−0.945051	−0.472525	−	0.881317i	$-0.656657\pi$
$131$	−10.8166	−0.945051	−0.472525	+	0.881317i	$0.656657\pi$
$132$	0	0
$133$	−3.76714	−0.326653
$134$	0	0
$135$	5.05802i	0.435325i
$136$	0	0
$137$	− 7.73123i	− 0.660523i	−0.943889	−	0.330262i	$-0.892863\pi$
$137$	− 7.73123i	− 0.660523i	0.943889	−	0.330262i	$-0.107137\pi$
$138$	0	0
$139$	12.9186	1.09574	0.547869	−	0.836564i	$-0.315439\pi$
$139$	12.9186	1.09574	0.547869	+	0.836564i	$0.315439\pi$
$140$	0	0
$141$	15.8090i	1.33136i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	9.02219i	0.749252i
$146$	0	0
$147$	15.3407	1.26528
$148$	0	0
$149$	17.0466i	1.39651i	0.715849	+	0.698255i	$0.246040\pi$
$149$	17.0466i	1.39651i	−0.715849	+	0.698255i	$0.753960\pi$
$150$	0	0
$151$	− 2.41977i	− 0.196918i	−0.995141	−	0.0984588i	$-0.968609\pi$
$151$	− 2.41977i	− 0.196918i	0.995141	−	0.0984588i	$-0.0313913\pi$
$152$	0	0
$153$	−23.8384	−1.92722
$154$	0	0
$155$	6.57334	0.527983
$156$	0	0
$157$	19.3659	1.54557	0.772785	−	0.634668i	$-0.218863\pi$
$157$	19.3659	1.54557	0.772785	+	0.634668i	$0.218863\pi$
$158$	0	0
$159$	27.4984	2.18077
$160$	0	0
$161$	− 1.28298i	− 0.101113i
$162$	0	0
$163$	8.19218i	0.641661i	0.947137	+	0.320831i	$0.103962\pi$
$163$	8.19218i	0.641661i	−0.947137	+	0.320831i	$0.896038\pi$
$164$	0	0
$165$	10.4373	0.812544
$166$	0	0
$167$	0.119894i	0.00927770i	0.999989	+	0.00463885i	$0.00147660\pi$
$167$	0.119894i	0.00927770i	−0.999989	+	0.00463885i	$0.998523\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	− 14.7424i	− 1.12738i
$172$	0	0
$173$	13.3720	1.01665	0.508326	−	0.861165i	$-0.330265\pi$
$173$	13.3720	1.01665	0.508326	+	0.861165i	$0.330265\pi$
$174$	0	0
$175$	1.22908i	0.0929097i
$176$	0	0
$177$	− 34.0496i	− 2.55933i
$178$	0	0
$179$	−14.3046	−1.06918	−0.534588	−	0.845113i	$-0.679533\pi$
$179$	−14.3046	−1.06918	−0.534588	+	0.845113i	$0.679533\pi$
$180$	0	0
$181$	9.65346	0.717536	0.358768	−	0.933427i	$-0.383197\pi$
$181$	9.65346	0.717536	0.358768	+	0.933427i	$0.383197\pi$
$182$	0	0
$183$	32.2311	2.38259
$184$	0	0
$185$	−0.619519	−0.0455479
$186$	0	0
$187$	18.5100i	1.35359i
$188$	0	0
$189$	− 6.21670i	− 0.452199i
$190$	0	0
$191$	2.60265	0.188321	0.0941607	−	0.995557i	$-0.469983\pi$
$191$	2.60265	0.188321	0.0941607	+	0.995557i	$0.469983\pi$
$192$	0	0
$193$	2.44821i	0.176226i	0.996110	+	0.0881132i	$0.0280837\pi$
$193$	2.44821i	0.176226i	−0.996110	+	0.0881132i	$0.971916\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0.851216i	0.0606466i	0.999540	+	0.0303233i	$0.00965369\pi$
$197$	0.851216i	0.0606466i	−0.999540	+	0.0303233i	$0.990346\pi$
$198$	0	0
$199$	−9.06160	−0.642360	−0.321180	−	0.947018i	$-0.604079\pi$
$199$	−9.06160	−0.642360	−0.321180	+	0.947018i	$0.604079\pi$
$200$	0	0
$201$	− 21.6129i	− 1.52446i
$202$	0	0
$203$	− 11.0890i	− 0.778294i
$204$	0	0
$205$	8.41141	0.587478
$206$	0	0
$207$	5.02085	0.348973
$208$	0	0
$209$	−11.4472	−0.791817
$210$	0	0
$211$	−16.4317	−1.13121	−0.565603	−	0.824678i	$-0.691356\pi$
$211$	−16.4317	−1.13121	−0.565603	+	0.824678i	$0.691356\pi$
$212$	0	0
$213$	− 14.9911i	− 1.02717i
$214$	0	0
$215$	8.50043i	0.579725i
$216$	0	0
$217$	−8.07915	−0.548449
$218$	0	0
$219$	40.1789i	2.71504i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 23.2006i	− 1.55363i	−0.629729	−	0.776815i	$-0.716834\pi$
$223$	− 23.2006i	− 1.55363i	0.629729	−	0.776815i	$-0.283166\pi$
$224$	0	0
$225$	−4.80991	−0.320661
$226$	0	0
$227$	5.33774i	0.354278i	0.984186	+	0.177139i	$0.0566842\pi$
$227$	5.33774i	0.354278i	−0.984186	+	0.177139i	$0.943316\pi$
$228$	0	0
$229$	− 16.1535i	− 1.06745i	−0.845657	−	0.533726i	$-0.820791\pi$
$229$	− 16.1535i	− 1.06745i	0.845657	−	0.533726i	$-0.179209\pi$
$230$	0	0
$231$	−12.8283	−0.844040
$232$	0	0
$233$	5.09722	0.333930	0.166965	−	0.985963i	$-0.446603\pi$
$233$	5.09722	0.333930	0.166965	+	0.985963i	$0.446603\pi$
$234$	0	0
$235$	−5.65693	−0.369017
$236$	0	0
$237$	−33.0495	−2.14680
$238$	0	0
$239$	17.7527i	1.14833i	0.818741	+	0.574163i	$0.194672\pi$
$239$	17.7527i	1.14833i	−0.818741	+	0.574163i	$0.805328\pi$
$240$	0	0
$241$	− 3.17008i	− 0.204203i	−0.994774	−	0.102101i	$-0.967443\pi$
$241$	− 3.17008i	− 0.204203i	0.994774	−	0.102101i	$-0.0325566\pi$
$242$	0	0
$243$	−15.9970	−1.02621
$244$	0	0
$245$	5.48936i	0.350703i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	35.4886i	2.24900i
$250$	0	0
$251$	18.4648	1.16549	0.582744	−	0.812656i	$-0.301979\pi$
$251$	18.4648	1.16549	0.582744	+	0.812656i	$0.301979\pi$
$252$	0	0
$253$	− 3.89858i	− 0.245101i
$254$	0	0
$255$	− 13.8504i	− 0.867348i
$256$	0	0
$257$	6.02478	0.375816	0.187908	−	0.982187i	$-0.439829\pi$
$257$	6.02478	0.375816	0.187908	+	0.982187i	$0.439829\pi$
$258$	0	0
$259$	0.761438	0.0473134
$260$	0	0
$261$	43.3959	2.68614
$262$	0	0
$263$	−28.3677	−1.74923	−0.874614	−	0.484820i	$-0.838885\pi$
$263$	−28.3677	−1.74923	−0.874614	+	0.484820i	$0.838885\pi$
$264$	0	0
$265$	9.83977i	0.604452i
$266$	0	0
$267$	37.9311i	2.32135i
$268$	0	0
$269$	0.738744	0.0450420	0.0225210	−	0.999746i	$-0.492831\pi$
$269$	0.738744	0.0450420	0.0225210	+	0.999746i	$0.492831\pi$
$270$	0	0
$271$	− 3.61053i	− 0.219324i	−0.993969	−	0.109662i	$-0.965023\pi$
$271$	− 3.61053i	− 0.219324i	0.993969	−	0.109662i	$-0.0349769\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.73479i	0.225216i
$276$	0	0
$277$	19.9027	1.19583	0.597917	−	0.801558i	$-0.295995\pi$
$277$	19.9027	1.19583	0.597917	+	0.801558i	$0.295995\pi$
$278$	0	0
$279$	− 31.6172i	− 1.89287i
$280$	0	0
$281$	27.1022i	1.61678i	0.588645	+	0.808391i	$0.299661\pi$
$281$	27.1022i	1.61678i	−0.588645	+	0.808391i	$0.700339\pi$
$282$	0	0
$283$	−3.45523	−0.205392	−0.102696	−	0.994713i	$-0.532747\pi$
$283$	−3.45523	−0.205392	−0.102696	+	0.994713i	$0.532747\pi$
$284$	0	0
$285$	8.56555	0.507379
$286$	0	0
$287$	−10.3383	−0.610250
$288$	0	0
$289$	7.56298	0.444881
$290$	0	0
$291$	39.0696i	2.29030i
$292$	0	0
$293$	− 13.8410i	− 0.808600i	−0.914626	−	0.404300i	$-0.867515\pi$
$293$	− 13.8410i	− 0.808600i	0.914626	−	0.404300i	$-0.132485\pi$
$294$	0	0
$295$	12.1840	0.709379
$296$	0	0
$297$	− 18.8906i	− 1.09614i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 10.4477i	− 0.602196i
$302$	0	0
$303$	−20.7704	−1.19323
$304$	0	0
$305$	11.5333i	0.660393i
$306$	0	0
$307$	13.4284i	0.766401i	0.923665	+	0.383201i	$0.125178\pi$
$307$	13.4284i	0.766401i	−0.923665	+	0.383201i	$0.874822\pi$
$308$	0	0
$309$	−46.0290	−2.61850
$310$	0	0
$311$	−18.7934	−1.06567	−0.532837	−	0.846218i	$-0.678874\pi$
$311$	−18.7934	−1.06567	−0.532837	+	0.846218i	$0.678874\pi$
$312$	0	0
$313$	−22.1388	−1.25136	−0.625679	−	0.780081i	$-0.715178\pi$
$313$	−22.1388	−1.25136	−0.625679	+	0.780081i	$0.715178\pi$
$314$	0	0
$315$	5.91176	0.333090
$316$	0	0
$317$	21.7038i	1.21901i	0.792783	+	0.609504i	$0.208631\pi$
$317$	21.7038i	1.21901i	−0.792783	+	0.609504i	$0.791369\pi$
$318$	0	0
$319$	− 33.6960i	− 1.88661i
$320$	0	0
$321$	−24.5066	−1.36782
$322$	0	0
$323$	15.1905i	0.845223i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 20.0623i	− 1.10945i
$328$	0	0
$329$	6.95281	0.383321
$330$	0	0
$331$	13.1827i	0.724588i	0.932064	+	0.362294i	$0.118006\pi$
$331$	13.1827i	0.724588i	−0.932064	+	0.362294i	$0.881994\pi$
$332$	0	0
$333$	2.97983i	0.163294i
$334$	0	0
$335$	7.73375	0.422540
$336$	0	0
$337$	−18.9196	−1.03062	−0.515309	−	0.857004i	$-0.672323\pi$
$337$	−18.9196	−1.03062	−0.515309	+	0.857004i	$0.672323\pi$
$338$	0	0
$339$	−28.0416	−1.52301
$340$	0	0
$341$	−24.5500	−1.32946
$342$	0	0
$343$	− 15.3504i	− 0.828845i
$344$	0	0
$345$	2.91718i	0.157056i
$346$	0	0
$347$	−1.48879	−0.0799226	−0.0399613	−	0.999201i	$-0.512723\pi$
$347$	−1.48879	−0.0799226	−0.0399613	+	0.999201i	$0.512723\pi$
$348$	0	0
$349$	15.5379i	0.831725i	0.909428	+	0.415862i	$0.136520\pi$
$349$	15.5379i	0.831725i	−0.909428	+	0.415862i	$0.863480\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 18.4163i	− 0.980199i	−0.871667	−	0.490099i	$-0.836960\pi$
$353$	− 18.4163i	− 0.980199i	0.871667	−	0.490099i	$-0.163040\pi$
$354$	0	0
$355$	5.36427	0.284706
$356$	0	0
$357$	17.0233i	0.900968i
$358$	0	0
$359$	− 2.26366i	− 0.119472i	−0.998214	−	0.0597358i	$-0.980974\pi$
$359$	− 2.26366i	− 0.119472i	0.998214	−	0.0597358i	$-0.0190258\pi$
$360$	0	0
$361$	9.60571	0.505564
$362$	0	0
$363$	−8.24034	−0.432505
$364$	0	0
$365$	−14.3772	−0.752538
$366$	0	0
$367$	22.4716	1.17301	0.586504	−	0.809947i	$-0.300504\pi$
$367$	22.4716	1.17301	0.586504	+	0.809947i	$0.300504\pi$
$368$	0	0
$369$	− 40.4581i	− 2.10617i
$370$	0	0
$371$	− 12.0939i	− 0.627882i
$372$	0	0
$373$	14.6760	0.759896	0.379948	−	0.925008i	$-0.375942\pi$
$373$	14.6760	0.759896	0.379948	+	0.925008i	$0.375942\pi$
$374$	0	0
$375$	− 2.79462i	− 0.144314i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	31.2728i	1.60638i	0.595725	+	0.803188i	$0.296865\pi$
$379$	31.2728i	1.60638i	−0.595725	+	0.803188i	$0.703135\pi$
$380$	0	0
$381$	−4.48992	−0.230026
$382$	0	0
$383$	− 26.2659i	− 1.34212i	−0.741401	−	0.671062i	$-0.765838\pi$
$383$	− 26.2659i	− 1.34212i	0.741401	−	0.671062i	$-0.234162\pi$
$384$	0	0
$385$	− 4.59035i	− 0.233946i
$386$	0	0
$387$	40.8863	2.07837
$388$	0	0
$389$	18.8290	0.954667	0.477333	−	0.878722i	$-0.341603\pi$
$389$	18.8290	0.954667	0.477333	+	0.878722i	$0.341603\pi$
$390$	0	0
$391$	−5.17345	−0.261633
$392$	0	0
$393$	−30.2283	−1.52482
$394$	0	0
$395$	− 11.8261i	− 0.595037i
$396$	0	0
$397$	− 23.8701i	− 1.19800i	−0.800747	−	0.599002i	$-0.795564\pi$
$397$	− 23.8701i	− 1.19800i	0.800747	−	0.599002i	$-0.204436\pi$
$398$	0	0
$399$	−10.5277	−0.527046
$400$	0	0
$401$	− 2.56312i	− 0.127996i	−0.997950	−	0.0639980i	$-0.979615\pi$
$401$	− 2.56312i	− 0.127996i	0.997950	−	0.0639980i	$-0.0203851\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 0.294490i	− 0.0146333i
$406$	0	0
$407$	2.31377	0.114689
$408$	0	0
$409$	18.9442i	0.936732i	0.883535	+	0.468366i	$0.155157\pi$
$409$	18.9442i	0.936732i	−0.883535	+	0.468366i	$0.844843\pi$
$410$	0	0
$411$	− 21.6059i	− 1.06574i
$412$	0	0
$413$	−14.9751	−0.736876
$414$	0	0
$415$	−12.6989	−0.623365
$416$	0	0
$417$	36.1025	1.76795
$418$	0	0
$419$	11.9453	0.583567	0.291783	−	0.956484i	$-0.405751\pi$
$419$	11.9453	0.583567	0.291783	+	0.956484i	$0.405751\pi$
$420$	0	0
$421$	− 25.0406i	− 1.22040i	−0.792247	−	0.610201i	$-0.791089\pi$
$421$	− 25.0406i	− 1.22040i	0.792247	−	0.610201i	$-0.208911\pi$
$422$	0	0
$423$	27.2093i	1.32296i
$424$	0	0
$425$	4.95611	0.240406
$426$	0	0
$427$	− 14.1753i	− 0.685991i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	8.58228i	0.413394i	0.978405	+	0.206697i	$0.0662714\pi$
$431$	8.58228i	0.413394i	−0.978405	+	0.206697i	$0.933729\pi$
$432$	0	0
$433$	−12.5767	−0.604399	−0.302200	−	0.953245i	$-0.597721\pi$
$433$	−12.5767	−0.604399	−0.302200	+	0.953245i	$0.597721\pi$
$434$	0	0
$435$	25.2136i	1.20890i
$436$	0	0
$437$	− 3.19943i	− 0.153049i
$438$	0	0
$439$	33.2154	1.58528	0.792642	−	0.609687i	$-0.208705\pi$
$439$	33.2154	1.58528	0.792642	+	0.609687i	$0.208705\pi$
$440$	0	0
$441$	26.4034	1.25730
$442$	0	0
$443$	−36.7404	−1.74559	−0.872795	−	0.488087i	$-0.837695\pi$
$443$	−36.7404	−1.74559	−0.872795	+	0.488087i	$0.837695\pi$
$444$	0	0
$445$	−13.5729	−0.643417
$446$	0	0
$447$	47.6387i	2.25323i
$448$	0	0
$449$	5.12272i	0.241756i	0.992667	+	0.120878i	$0.0385710\pi$
$449$	5.12272i	0.241756i	−0.992667	+	0.120878i	$0.961429\pi$
$450$	0	0
$451$	−31.4148	−1.47927
$452$	0	0
$453$	− 6.76233i	− 0.317722i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	24.0427i	1.12467i	0.826910	+	0.562334i	$0.190097\pi$
$457$	24.0427i	1.12467i	−0.826910	+	0.562334i	$0.809903\pi$
$458$	0	0
$459$	−25.0681	−1.17008
$460$	0	0
$461$	− 4.22878i	− 0.196954i	−0.995139	−	0.0984770i	$-0.968603\pi$
$461$	− 4.22878i	− 0.196954i	0.995139	−	0.0984770i	$-0.0313971\pi$
$462$	0	0
$463$	29.4028i	1.36647i	0.730200	+	0.683233i	$0.239427\pi$
$463$	29.4028i	1.36647i	−0.730200	+	0.683233i	$0.760573\pi$
$464$	0	0
$465$	18.3700	0.851888
$466$	0	0
$467$	5.06469	0.234366	0.117183	−	0.993110i	$-0.462614\pi$
$467$	5.06469	0.234366	0.117183	+	0.993110i	$0.462614\pi$
$468$	0	0
$469$	−9.50539	−0.438918
$470$	0	0
$471$	54.1204	2.49374
$472$	0	0
$473$	− 31.7473i	− 1.45974i
$474$	0	0
$475$	3.06501i	0.140632i
$476$	0	0
$477$	47.3284	2.16702
$478$	0	0
$479$	7.18242i	0.328173i	0.986446	+	0.164087i	$0.0524677\pi$
$479$	7.18242i	0.328173i	−0.986446	+	0.164087i	$0.947532\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	− 3.58544i	− 0.163143i
$484$	0	0
$485$	−13.9803	−0.634813
$486$	0	0
$487$	19.0308i	0.862366i	0.902264	+	0.431183i	$0.141904\pi$
$487$	19.0308i	0.862366i	−0.902264	+	0.431183i	$0.858096\pi$
$488$	0	0
$489$	22.8941i	1.03530i
$490$	0	0
$491$	−9.89677	−0.446635	−0.223318	−	0.974746i	$-0.571689\pi$
$491$	−9.89677	−0.446635	−0.223318	+	0.974746i	$0.571689\pi$
$492$	0	0
$493$	−44.7149	−2.01386
$494$	0	0
$495$	17.9640	0.807422
$496$	0	0
$497$	−6.59311	−0.295742
$498$	0	0
$499$	− 11.7475i	− 0.525890i	−0.964811	−	0.262945i	$-0.915306\pi$
$499$	− 11.7475i	− 0.525890i	0.964811	−	0.262945i	$-0.0846938\pi$
$500$	0	0
$501$	0.335059i	0.0149693i
$502$	0	0
$503$	−7.50291	−0.334538	−0.167269	−	0.985911i	$-0.553495\pi$
$503$	−7.50291	−0.334538	−0.167269	+	0.985911i	$0.553495\pi$
$504$	0	0
$505$	− 7.43227i	− 0.330732i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 3.05609i	− 0.135459i	−0.997704	−	0.0677294i	$-0.978425\pi$
$509$	− 3.05609i	− 0.135459i	0.997704	−	0.0677294i	$-0.0215755\pi$
$510$	0	0
$511$	17.6707	0.781707
$512$	0	0
$513$	− 15.5029i	− 0.684469i
$514$	0	0
$515$	− 16.4706i	− 0.725780i
$516$	0	0
$517$	21.1274	0.929183
$518$	0	0
$519$	37.3696	1.64034
$520$	0	0
$521$	4.66278	0.204280	0.102140	−	0.994770i	$-0.467431\pi$
$521$	4.66278	0.204280	0.102140	+	0.994770i	$0.467431\pi$
$522$	0	0
$523$	37.7910	1.65249	0.826243	−	0.563314i	$-0.190474\pi$
$523$	37.7910	1.65249	0.826243	+	0.563314i	$0.190474\pi$
$524$	0	0
$525$	3.43481i	0.149907i
$526$	0	0
$527$	32.5782i	1.41913i
$528$	0	0
$529$	−21.9104	−0.952625
$530$	0	0
$531$	− 58.6039i	− 2.54319i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 8.76920i	− 0.379125i
$536$	0	0
$537$	−39.9760	−1.72509
$538$	0	0
$539$	− 20.5016i	− 0.883067i
$540$	0	0
$541$	38.5091i	1.65564i	0.560997	+	0.827818i	$0.310418\pi$
$541$	38.5091i	1.65564i	−0.560997	+	0.827818i	$0.689582\pi$
$542$	0	0
$543$	26.9778	1.15773
$544$	0	0
$545$	7.17891	0.307511
$546$	0	0
$547$	6.31857	0.270163	0.135081	−	0.990835i	$-0.456870\pi$
$547$	6.31857	0.270163	0.135081	+	0.990835i	$0.456870\pi$
$548$	0	0
$549$	55.4740	2.36757
$550$	0	0
$551$	− 27.6531i	− 1.17806i
$552$	0	0
$553$	14.5352i	0.618102i
$554$	0	0
$555$	−1.73132	−0.0734905
$556$	0	0
$557$	− 25.4127i	− 1.07677i	−0.842699	−	0.538385i	$-0.819034\pi$
$557$	− 25.4127i	− 1.07677i	0.842699	−	0.538385i	$-0.180966\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	51.7285i	2.18398i
$562$	0	0
$563$	−3.72041	−0.156797	−0.0783983	−	0.996922i	$-0.524981\pi$
$563$	−3.72041	−0.156797	−0.0783983	+	0.996922i	$0.524981\pi$
$564$	0	0
$565$	− 10.0341i	− 0.422139i
$566$	0	0
$567$	0.361952i	0.0152005i
$568$	0	0
$569$	−6.96568	−0.292016	−0.146008	−	0.989283i	$-0.546643\pi$
$569$	−6.96568	−0.292016	−0.146008	+	0.989283i	$0.546643\pi$
$570$	0	0
$571$	−9.94609	−0.416231	−0.208116	−	0.978104i	$-0.566733\pi$
$571$	−9.94609	−0.416231	−0.208116	+	0.978104i	$0.566733\pi$
$572$	0	0
$573$	7.27343	0.303852
$574$	0	0
$575$	−1.04385	−0.0435317
$576$	0	0
$577$	− 13.3817i	− 0.557087i	−0.960424	−	0.278544i	$-0.910148\pi$
$577$	− 13.3817i	− 0.557087i	0.960424	−	0.278544i	$-0.0898517\pi$
$578$	0	0
$579$	6.84183i	0.284337i
$580$	0	0
$581$	15.6080	0.647527
$582$	0	0
$583$	− 36.7495i	− 1.52201i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0.400413i	0.0165268i	0.999966	+	0.00826340i	$0.00263035\pi$
$587$	0.400413i	0.0165268i	−0.999966	+	0.00826340i	$0.997370\pi$
$588$	0	0
$589$	−20.1473	−0.830157
$590$	0	0
$591$	2.37883i	0.0978518i
$592$	0	0
$593$	16.8256i	0.690945i	0.938429	+	0.345472i	$0.112281\pi$
$593$	16.8256i	0.690945i	−0.938429	+	0.345472i	$0.887719\pi$
$594$	0	0
$595$	−6.09145	−0.249725
$596$	0	0
$597$	−25.3237	−1.03643
$598$	0	0
$599$	−5.94610	−0.242951	−0.121475	−	0.992594i	$-0.538763\pi$
$599$	−5.94610	−0.242951	−0.121475	+	0.992594i	$0.538763\pi$
$600$	0	0
$601$	41.0898	1.67609	0.838045	−	0.545601i	$-0.183699\pi$
$601$	41.0898	1.67609	0.838045	+	0.545601i	$0.183699\pi$
$602$	0	0
$603$	− 37.1986i	− 1.51485i
$604$	0	0
$605$	− 2.94864i	− 0.119879i
$606$	0	0
$607$	34.3906	1.39587	0.697935	−	0.716161i	$-0.254102\pi$
$607$	34.3906	1.39587	0.697935	+	0.716161i	$0.254102\pi$
$608$	0	0
$609$	− 30.9895i	− 1.25576i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 13.1645i	− 0.531709i	−0.964013	−	0.265854i	$-0.914346\pi$
$613$	− 13.1645i	− 0.531709i	0.964013	−	0.265854i	$-0.0856540\pi$
$614$	0	0
$615$	23.5067	0.947882
$616$	0	0
$617$	36.4559i	1.46766i	0.679333	+	0.733830i	$0.262269\pi$
$617$	36.4559i	1.46766i	−0.679333	+	0.733830i	$0.737731\pi$
$618$	0	0
$619$	44.4340i	1.78595i	0.450105	+	0.892976i	$0.351387\pi$
$619$	44.4340i	1.78595i	−0.450105	+	0.892976i	$0.648613\pi$
$620$	0	0
$621$	5.27983	0.211872
$622$	0	0
$623$	16.6822	0.668357
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−31.9905	−1.27758
$628$	0	0
$629$	− 3.07040i	− 0.122425i
$630$	0	0
$631$	− 22.7501i	− 0.905666i	−0.891595	−	0.452833i	$-0.850413\pi$
$631$	− 22.7501i	− 0.905666i	0.891595	−	0.452833i	$-0.149587\pi$
$632$	0	0
$633$	−45.9204	−1.82517
$634$	0	0
$635$	− 1.60663i	− 0.0637572i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 25.8017i	− 1.02070i
$640$	0	0
$641$	−36.2272	−1.43089	−0.715444	−	0.698670i	$-0.753775\pi$
$641$	−36.2272	−1.43089	−0.715444	+	0.698670i	$0.753775\pi$
$642$	0	0
$643$	− 3.55026i	− 0.140008i	−0.997547	−	0.0700042i	$-0.977699\pi$
$643$	− 3.55026i	− 0.140008i	0.997547	−	0.0700042i	$-0.0223013\pi$
$644$	0	0
$645$	23.7555i	0.935372i
$646$	0	0
$647$	13.2766	0.521958	0.260979	−	0.965344i	$-0.415955\pi$
$647$	13.2766	0.521958	0.260979	+	0.965344i	$0.415955\pi$
$648$	0	0
$649$	−45.5046	−1.78621
$650$	0	0
$651$	−22.5782	−0.884909
$652$	0	0
$653$	12.3080	0.481651	0.240825	−	0.970568i	$-0.422582\pi$
$653$	12.3080	0.481651	0.240825	+	0.970568i	$0.422582\pi$
$654$	0	0
$655$	− 10.8166i	− 0.422640i
$656$	0	0
$657$	69.1531i	2.69792i
$658$	0	0
$659$	−13.0293	−0.507551	−0.253776	−	0.967263i	$-0.581673\pi$
$659$	−13.0293	−0.507551	−0.253776	+	0.967263i	$0.581673\pi$
$660$	0	0
$661$	27.5909i	1.07316i	0.843849	+	0.536581i	$0.180284\pi$
$661$	27.5909i	1.07316i	−0.843849	+	0.536581i	$0.819716\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 3.76714i	− 0.146083i
$666$	0	0
$667$	9.41785	0.364661
$668$	0	0
$669$	− 64.8370i	− 2.50674i
$670$	0	0
$671$	− 43.0743i	− 1.66287i
$672$	0	0
$673$	−11.5736	−0.446131	−0.223065	−	0.974803i	$-0.571606\pi$
$673$	−11.5736	−0.446131	−0.223065	+	0.974803i	$0.571606\pi$
$674$	0	0
$675$	−5.05802	−0.194683
$676$	0	0
$677$	9.62321	0.369850	0.184925	−	0.982753i	$-0.440796\pi$
$677$	9.62321	0.369850	0.184925	+	0.982753i	$0.440796\pi$
$678$	0	0
$679$	17.1829	0.659419
$680$	0	0
$681$	14.9170i	0.571619i
$682$	0	0
$683$	− 33.3432i	− 1.27584i	−0.770102	−	0.637921i	$-0.779795\pi$
$683$	− 33.3432i	− 1.27584i	0.770102	−	0.637921i	$-0.220205\pi$
$684$	0	0
$685$	7.73123	0.295395
$686$	0	0
$687$	− 45.1429i	− 1.72231i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 43.1681i	− 1.64219i	−0.570791	−	0.821095i	$-0.693363\pi$
$691$	− 43.1681i	− 1.64219i	0.570791	−	0.821095i	$-0.306637\pi$
$692$	0	0
$693$	−22.0792	−0.838719
$694$	0	0
$695$	12.9186i	0.490029i
$696$	0	0
$697$	41.6878i	1.57904i
$698$	0	0
$699$	14.2448	0.538788
$700$	0	0
$701$	−16.4423	−0.621017	−0.310508	−	0.950571i	$-0.600499\pi$
$701$	−16.4423	−0.621017	−0.310508	+	0.950571i	$0.600499\pi$
$702$	0	0
$703$	1.89883	0.0716158
$704$	0	0
$705$	−15.8090	−0.595400
$706$	0	0
$707$	9.13485i	0.343551i
$708$	0	0
$709$	− 25.8445i	− 0.970611i	−0.874345	−	0.485306i	$-0.838708\pi$
$709$	− 25.8445i	− 0.970611i	0.874345	−	0.485306i	$-0.161292\pi$
$710$	0	0
$711$	−56.8826	−2.13326
$712$	0	0
$713$	− 6.86161i	− 0.256969i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	49.6120i	1.85279i
$718$	0	0
$719$	7.66800	0.285968	0.142984	−	0.989725i	$-0.454330\pi$
$719$	7.66800	0.285968	0.142984	+	0.989725i	$0.454330\pi$
$720$	0	0
$721$	20.2436i	0.753912i
$722$	0	0
$723$	− 8.85917i	− 0.329476i
$724$	0	0
$725$	−9.02219	−0.335076
$726$	0	0
$727$	35.4370	1.31428	0.657142	−	0.753767i	$-0.271765\pi$
$727$	35.4370	1.31428	0.657142	+	0.753767i	$0.271765\pi$
$728$	0	0
$729$	−43.8222	−1.62304
$730$	0	0
$731$	−42.1290	−1.55820
$732$	0	0
$733$	48.0345i	1.77419i	0.461582	+	0.887097i	$0.347282\pi$
$733$	48.0345i	1.77419i	−0.461582	+	0.887097i	$0.652718\pi$
$734$	0	0
$735$	15.3407i	0.565850i
$736$	0	0
$737$	−28.8839	−1.06395
$738$	0	0
$739$	− 20.3284i	− 0.747794i	−0.927470	−	0.373897i	$-0.878021\pi$
$739$	− 20.3284i	− 0.747794i	0.927470	−	0.373897i	$-0.121979\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	39.5385i	1.45053i	0.688472	+	0.725263i	$0.258282\pi$
$743$	39.5385i	1.45053i	−0.688472	+	0.725263i	$0.741718\pi$
$744$	0	0
$745$	−17.0466	−0.624538
$746$	0	0
$747$	61.0806i	2.23482i
$748$	0	0
$749$	10.7780i	0.393821i
$750$	0	0
$751$	36.1294	1.31838	0.659191	−	0.751976i	$-0.270899\pi$
$751$	36.1294	1.31838	0.659191	+	0.751976i	$0.270899\pi$
$752$	0	0
$753$	51.6021	1.88049
$754$	0	0
$755$	2.41977	0.0880643
$756$	0	0
$757$	−11.2361	−0.408382	−0.204191	−	0.978931i	$-0.565456\pi$
$757$	−11.2361	−0.408382	−0.204191	+	0.978931i	$0.565456\pi$
$758$	0	0
$759$	− 10.8950i	− 0.395465i
$760$	0	0
$761$	43.4661i	1.57564i	0.615902	+	0.787822i	$0.288792\pi$
$761$	43.4661i	1.57564i	−0.615902	+	0.787822i	$0.711208\pi$
$762$	0	0
$763$	−8.82345	−0.319430
$764$	0	0
$765$	− 23.8384i	− 0.861880i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	− 1.98755i	− 0.0716729i	−0.999358	−	0.0358364i	$-0.988590\pi$
$769$	− 1.98755i	− 0.0716729i	0.999358	−	0.0358364i	$-0.0114095\pi$
$770$	0	0
$771$	16.8370	0.606369
$772$	0	0
$773$	46.0457i	1.65615i	0.560619	+	0.828074i	$0.310563\pi$
$773$	46.0457i	1.65615i	−0.560619	+	0.828074i	$0.689437\pi$
$774$	0	0
$775$	6.57334i	0.236121i
$776$	0	0
$777$	2.12793	0.0763391
$778$	0	0
$779$	−25.7811	−0.923702
$780$	0	0
$781$	−20.0344	−0.716887
$782$	0	0
$783$	45.6344	1.63084
$784$	0	0
$785$	19.3659i	0.691200i
$786$	0	0
$787$	− 29.7703i	− 1.06120i	−0.847623	−	0.530598i	$-0.821967\pi$
$787$	− 29.7703i	− 1.06120i	0.847623	−	0.530598i	$-0.178033\pi$
$788$	0	0
$789$	−79.2770	−2.82234
$790$	0	0
$791$	12.3327i	0.438502i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	27.4984i	0.975269i
$796$	0	0
$797$	51.7725	1.83388	0.916938	−	0.399030i	$-0.130653\pi$
$797$	51.7725	1.83388	0.916938	+	0.399030i	$0.130653\pi$
$798$	0	0
$799$	− 28.0363i	− 0.991854i
$800$	0	0
$801$	65.2844i	2.30671i
$802$	0	0
$803$	53.6958	1.89489
$804$	0	0
$805$	1.28298	0.0452191
$806$	0	0
$807$	2.06451	0.0726742
$808$	0	0
$809$	−19.4923	−0.685312	−0.342656	−	0.939461i	$-0.611327\pi$
$809$	−19.4923	−0.685312	−0.342656	+	0.939461i	$0.611327\pi$
$810$	0	0
$811$	− 54.9207i	− 1.92853i	−0.264945	−	0.964264i	$-0.585354\pi$
$811$	− 54.9207i	− 1.92853i	0.264945	−	0.964264i	$-0.414646\pi$
$812$	0	0
$813$	− 10.0901i	− 0.353874i
$814$	0	0
$815$	−8.19218	−0.286960
$816$	0	0
$817$	− 26.0539i	− 0.911511i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 21.6079i	− 0.754120i	−0.926189	−	0.377060i	$-0.876935\pi$
$821$	− 21.6079i	− 0.754120i	0.926189	−	0.377060i	$-0.123065\pi$
$822$	0	0
$823$	26.0715	0.908797	0.454399	−	0.890799i	$-0.349854\pi$
$823$	26.0715	0.908797	0.454399	+	0.890799i	$0.349854\pi$
$824$	0	0
$825$	10.4373i	0.363381i
$826$	0	0
$827$	35.6861i	1.24093i	0.784235	+	0.620463i	$0.213055\pi$
$827$	35.6861i	1.24093i	−0.784235	+	0.620463i	$0.786945\pi$
$828$	0	0
$829$	−45.9700	−1.59661	−0.798303	−	0.602256i	$-0.794268\pi$
$829$	−45.9700	−1.59661	−0.798303	+	0.602256i	$0.794268\pi$
$830$	0	0
$831$	55.6204	1.92945
$832$	0	0
$833$	−27.2059	−0.942627
$834$	0	0
$835$	−0.119894	−0.00414912
$836$	0	0
$837$	− 33.2480i	− 1.14922i
$838$	0	0
$839$	21.5917i	0.745427i	0.927946	+	0.372714i	$0.121573\pi$
$839$	21.5917i	0.745427i	−0.927946	+	0.372714i	$0.878427\pi$
$840$	0	0
$841$	52.3999	1.80689
$842$	0	0
$843$	75.7404i	2.60864i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	3.62411i	0.124526i
$848$	0	0
$849$	−9.65605	−0.331395
$850$	0	0
$851$	0.646687i	0.0221682i
$852$	0	0
$853$	− 9.78391i	− 0.334995i	−0.985873	−	0.167497i	$-0.946431\pi$
$853$	− 9.78391i	− 0.334995i	0.985873	−	0.167497i	$-0.0535686\pi$
$854$	0	0
$855$	14.7424	0.504180
$856$	0	0
$857$	−43.4611	−1.48460	−0.742301	−	0.670067i	$-0.766265\pi$
$857$	−43.4611	−1.48460	−0.742301	+	0.670067i	$0.766265\pi$
$858$	0	0
$859$	−4.30641	−0.146933	−0.0734664	−	0.997298i	$-0.523406\pi$
$859$	−4.30641	−0.146933	−0.0734664	+	0.997298i	$0.523406\pi$
$860$	0	0
$861$	−28.8916	−0.984624
$862$	0	0
$863$	20.2021i	0.687688i	0.939027	+	0.343844i	$0.111729\pi$
$863$	20.2021i	0.687688i	−0.939027	+	0.343844i	$0.888271\pi$
$864$	0	0
$865$	13.3720i	0.454660i
$866$	0	0
$867$	21.1357	0.717805
$868$	0	0
$869$	44.1681i	1.49830i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	67.2440i	2.27586i
$874$	0	0
$875$	−1.22908	−0.0415505
$876$	0	0
$877$	28.9831i	0.978689i	0.872091	+	0.489344i	$0.162764\pi$
$877$	28.9831i	0.978689i	−0.872091	+	0.489344i	$0.837236\pi$
$878$	0	0
$879$	− 38.6804i	− 1.30466i
$880$	0	0
$881$	40.9072	1.37820	0.689100	−	0.724667i	$-0.258006\pi$
$881$	40.9072	1.37820	0.689100	+	0.724667i	$0.258006\pi$
$882$	0	0
$883$	25.8593	0.870234	0.435117	−	0.900374i	$-0.356707\pi$
$883$	25.8593	0.870234	0.435117	+	0.900374i	$0.356707\pi$
$884$	0	0
$885$	34.0496	1.14457
$886$	0	0
$887$	−48.3313	−1.62281	−0.811404	−	0.584486i	$-0.801296\pi$
$887$	−48.3313	−1.62281	−0.811404	+	0.584486i	$0.801296\pi$
$888$	0	0
$889$	1.97468i	0.0662285i
$890$	0	0
$891$	1.09986i	0.0368466i
$892$	0	0
$893$	17.3385	0.580212
$894$	0	0
$895$	− 14.3046i	− 0.478150i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 59.3059i	− 1.97796i
$900$	0	0
$901$	−48.7669	−1.62466
$902$	0	0
$903$	− 29.1974i	− 0.971628i
$904$	0	0
$905$	9.65346i	0.320892i
$906$	0	0
$907$	44.4104	1.47462	0.737311	−	0.675553i	$-0.236095\pi$
$907$	44.4104	1.47462	0.737311	+	0.675553i	$0.236095\pi$
$908$	0	0
$909$	−35.7485	−1.18570
$910$	0	0
$911$	−45.6542	−1.51259	−0.756296	−	0.654230i	$-0.772993\pi$
$911$	−45.6542	−1.51259	−0.756296	+	0.654230i	$0.772993\pi$
$912$	0	0
$913$	47.4277	1.56963
$914$	0	0
$915$	32.2311i	1.06553i
$916$	0	0
$917$	13.2945i	0.439022i
$918$	0	0
$919$	−26.9213	−0.888053	−0.444026	−	0.896014i	$-0.646450\pi$
$919$	−26.9213	−0.888053	−0.444026	+	0.896014i	$0.646450\pi$
$920$	0	0
$921$	37.5274i	1.23657i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 0.619519i	− 0.0203696i
$926$	0	0
$927$	−79.2220	−2.60199
$928$	0	0
$929$	− 7.00807i	− 0.229927i	−0.993370	−	0.114964i	$-0.963325\pi$
$929$	− 7.00807i	− 0.229927i	0.993370	−	0.114964i	$-0.0366751\pi$
$930$	0	0
$931$	− 16.8250i	− 0.551416i
$932$	0	0
$933$	−52.5204	−1.71944
$934$	0	0
$935$	−18.5100	−0.605342
$936$	0	0
$937$	−34.8660	−1.13902	−0.569512	−	0.821983i	$-0.692868\pi$
$937$	−34.8660	−1.13902	−0.569512	+	0.821983i	$0.692868\pi$
$938$	0	0
$939$	−61.8695	−2.01903
$940$	0	0
$941$	38.5412i	1.25641i	0.778048	+	0.628204i	$0.216210\pi$
$941$	38.5412i	1.25641i	−0.778048	+	0.628204i	$0.783790\pi$
$942$	0	0
$943$	− 8.78029i	− 0.285926i
$944$	0	0
$945$	6.21670	0.202229
$946$	0	0
$947$	21.8022i	0.708475i	0.935156	+	0.354237i	$0.115260\pi$
$947$	21.8022i	0.708475i	−0.935156	+	0.354237i	$0.884740\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	60.6540i	1.96684i
$952$	0	0
$953$	−7.43281	−0.240772	−0.120386	−	0.992727i	$-0.538413\pi$
$953$	−7.43281	−0.240772	−0.120386	+	0.992727i	$0.538413\pi$
$954$	0	0
$955$	2.60265i	0.0842199i
$956$	0	0
$957$	− 94.1675i	− 3.04400i
$958$	0	0
$959$	−9.50229	−0.306845
$960$	0	0
$961$	−12.2088	−0.393831
$962$	0	0
$963$	−42.1791	−1.35920
$964$	0	0
$965$	−2.44821	−0.0788108
$966$	0	0
$967$	− 10.0827i	− 0.324237i	−0.986771	−	0.162118i	$-0.948167\pi$
$967$	− 10.0827i	− 0.324237i	0.986771	−	0.162118i	$-0.0518327\pi$
$968$	0	0
$969$	42.4517i	1.36375i
$970$	0	0
$971$	−18.7318	−0.601131	−0.300565	−	0.953761i	$-0.597175\pi$
$971$	−18.7318	−0.601131	−0.300565	+	0.953761i	$0.597175\pi$
$972$	0	0
$973$	− 15.8779i	− 0.509023i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	34.3147i	1.09782i	0.835880	+	0.548912i	$0.184958\pi$
$977$	34.3147i	1.09782i	−0.835880	+	0.548912i	$0.815042\pi$
$978$	0	0
$979$	50.6919	1.62012
$980$	0	0
$981$	− 34.5299i	− 1.10246i
$982$	0	0
$983$	− 21.9179i	− 0.699074i	−0.936923	−	0.349537i	$-0.886339\pi$
$983$	− 21.9179i	− 0.699074i	0.936923	−	0.349537i	$-0.113661\pi$
$984$	0	0
$985$	−0.851216	−0.0271220
$986$	0	0
$987$	19.4305	0.618479
$988$	0	0
$989$	8.87321	0.282152
$990$	0	0
$991$	−4.18345	−0.132892	−0.0664459	−	0.997790i	$-0.521166\pi$
$991$	−4.18345	−0.132892	−0.0664459	+	0.997790i	$0.521166\pi$
$992$	0	0
$993$	36.8407i	1.16911i
$994$	0	0
$995$	− 9.06160i	− 0.287272i
$996$	0	0
$997$	−8.42055	−0.266682	−0.133341	−	0.991070i	$-0.542570\pi$
$997$	−8.42055	−0.266682	−0.133341	+	0.991070i	$0.542570\pi$
$998$	0	0
$999$	3.13354i	0.0991407i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3380.2.f.j.3041.16		18
13.5	odd	4		3380.2.a.s.1.8	yes	9
13.8	odd	4		3380.2.a.r.1.8	✓	9
13.12	even	2	inner	3380.2.f.j.3041.15		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3380.2.a.r.1.8	✓	9	13.8	odd	4
3380.2.a.s.1.8	yes	9	13.5	odd	4
3380.2.f.j.3041.15		18	13.12	even	2	inner
3380.2.f.j.3041.16		18	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+2.79462 q^{3} +1.00000i q^{5} -1.22908i q^{7} +4.80991 q^{9} +O(q^{10})\)	\(q+2.79462 q^{3} +1.00000i q^{5} -1.22908i q^{7} +4.80991 q^{9} -3.73479i q^{11} +2.79462i q^{15} -4.95611 q^{17} -3.06501i q^{19} -3.43481i q^{21} +1.04385 q^{23} -1.00000 q^{25} +5.05802 q^{27} +9.02219 q^{29} -6.57334i q^{31} -10.4373i q^{33} +1.22908 q^{35} +0.619519i q^{37} -8.41141i q^{41} +8.50043 q^{43} +4.80991i q^{45} +5.65693i q^{47} +5.48936 q^{49} -13.8504 q^{51} +9.83977 q^{53} +3.73479 q^{55} -8.56555i q^{57} -12.1840i q^{59} +11.5333 q^{61} -5.91176i q^{63} -7.73375i q^{67} +2.91718 q^{69} -5.36427i q^{71} +14.3772i q^{73} -2.79462 q^{75} -4.59035 q^{77} -11.8261 q^{79} -0.294490 q^{81} +12.6989i q^{83} -4.95611i q^{85} +25.2136 q^{87} +13.5729i q^{89} -18.3700i q^{93} +3.06501 q^{95} +13.9803i q^{97} -17.9640i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 2 q^{3} + 24 q^{9}+O(q^{10}) \) 18 * q - 2 * q^3 + 24 * q^9	\( 18 q - 2 q^{3} + 24 q^{9} - 26 q^{17} - 24 q^{23} - 18 q^{25} - 8 q^{27} + 32 q^{29} + 2 q^{35} - 2 q^{43} - 40 q^{49} - 22 q^{51} + 60 q^{53} - 14 q^{55} + 42 q^{61} - 30 q^{69} + 2 q^{75} - 92 q^{77} + 62 q^{79} + 82 q^{81} + 56 q^{87} + 8 q^{95}+O(q^{100}) \) 18 * q - 2 * q^3 + 24 * q^9 - 26 * q^17 - 24 * q^23 - 18 * q^25 - 8 * q^27 + 32 * q^29 + 2 * q^35 - 2 * q^43 - 40 * q^49 - 22 * q^51 + 60 * q^53 - 14 * q^55 + 42 * q^61 - 30 * q^69 + 2 * q^75 - 92 * q^77 + 62 * q^79 + 82 * q^81 + 56 * q^87 + 8 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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