LMFDB - Embedded newform 3380.2.a.r.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3380,2,Mod(1,3380)]

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, 0]))

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

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

chi := DirichletCharacter("3380.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3380 = 2^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3380.a (trivial)

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:	yes
Analytic conductor:	$26.9894358832$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 19x^{7} + 16x^{6} + 106x^{5} - 87x^{4} - 153x^{3} + 149x^{2} - 26x + 1 $ x^9 - x^8 - 19x^7 + 16x^6 + 106x^5 - 87x^4 - 153x^3 + 149x^2 - 26*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-2.79462$ of defining polynomial
Character	$\chi$	$=$	3380.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.79462 q^{3} -1.00000 q^{5} -1.22908 q^{7} +4.80991 q^{9} +O(q^{10})$	$q+2.79462 q^{3} -1.00000 q^{5} -1.22908 q^{7} +4.80991 q^{9} -3.73479 q^{11} -2.79462 q^{15} +4.95611 q^{17} +3.06501 q^{19} -3.43481 q^{21} -1.04385 q^{23} +1.00000 q^{25} +5.05802 q^{27} +9.02219 q^{29} +6.57334 q^{31} -10.4373 q^{33} +1.22908 q^{35} +0.619519 q^{37} +8.41141 q^{41} -8.50043 q^{43} -4.80991 q^{45} +5.65693 q^{47} -5.48936 q^{49} +13.8504 q^{51} +9.83977 q^{53} +3.73479 q^{55} +8.56555 q^{57} -12.1840 q^{59} +11.5333 q^{61} -5.91176 q^{63} +7.73375 q^{67} -2.91718 q^{69} +5.36427 q^{71} +14.3772 q^{73} +2.79462 q^{75} +4.59035 q^{77} -11.8261 q^{79} -0.294490 q^{81} -12.6989 q^{83} -4.95611 q^{85} +25.2136 q^{87} +13.5729 q^{89} +18.3700 q^{93} -3.06501 q^{95} -13.9803 q^{97} -17.9640 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - q^{3} - 9 q^{5} - q^{7} + 12 q^{9}+O(q^{10}) $ 9 * q - q^3 - 9 * q^5 - q^7 + 12 * q^9	$ 9 q - q^{3} - 9 q^{5} - q^{7} + 12 q^{9} + 7 q^{11} + q^{15} + 13 q^{17} + 4 q^{19} + 3 q^{21} + 12 q^{23} + 9 q^{25} - 4 q^{27} + 16 q^{29} - 13 q^{31} - 34 q^{33} + q^{35} - q^{37} + 6 q^{41} + q^{43} - 12 q^{45} + 2 q^{47} + 20 q^{49} + 11 q^{51} + 30 q^{53} - 7 q^{55} - 38 q^{57} - 15 q^{59} + 21 q^{61} + 17 q^{63} + 7 q^{67} + 15 q^{69} + 7 q^{71} + 28 q^{73} - q^{75} + 46 q^{77} + 31 q^{79} + 41 q^{81} - 45 q^{83} - 13 q^{85} + 28 q^{87} + 41 q^{89} + 11 q^{93} - 4 q^{95} - 8 q^{97} + 81 q^{99}+O(q^{100}) $ 9 * q - q^3 - 9 * q^5 - q^7 + 12 * q^9 + 7 * q^11 + q^15 + 13 * q^17 + 4 * q^19 + 3 * q^21 + 12 * q^23 + 9 * q^25 - 4 * q^27 + 16 * q^29 - 13 * q^31 - 34 * q^33 + q^35 - q^37 + 6 * q^41 + q^43 - 12 * q^45 + 2 * q^47 + 20 * q^49 + 11 * q^51 + 30 * q^53 - 7 * q^55 - 38 * q^57 - 15 * q^59 + 21 * q^61 + 17 * q^63 + 7 * q^67 + 15 * q^69 + 7 * q^71 + 28 * q^73 - q^75 + 46 * q^77 + 31 * q^79 + 41 * q^81 - 45 * q^83 - 13 * q^85 + 28 * q^87 + 41 * q^89 + 11 * q^93 - 4 * q^95 - 8 * q^97 + 81 * q^99

Display 100 coefficients 10 coefficients

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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.22908	−0.464548	−0.232274	−	0.972650i	$-0.574617\pi$
$7$	−1.22908	−0.464548	−0.232274	+	0.972650i	$0.574617\pi$
$8$	0	0
$9$	4.80991	1.60330
$10$	0	0
$11$	−3.73479	−1.12608	−0.563040	−	0.826429i	$-0.690368\pi$
$11$	−3.73479	−1.12608	−0.563040	+	0.826429i	$0.690368\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−2.79462	−0.721568
$16$	0	0
$17$	4.95611	1.20203	0.601016	−	0.799237i	$-0.294763\pi$
$17$	4.95611	1.20203	0.601016	+	0.799237i	$0.294763\pi$
$18$	0	0
$19$	3.06501	0.703162	0.351581	−	0.936158i	$-0.385644\pi$
$19$	3.06501	0.703162	0.351581	+	0.936158i	$0.385644\pi$
$20$	0	0
$21$	−3.43481	−0.749537
$22$	0	0
$23$	−1.04385	−0.217659	−0.108829	−	0.994060i	$-0.534710\pi$
$23$	−1.04385	−0.217659	−0.108829	+	0.994060i	$0.534710\pi$
$24$	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
$31$	6.57334	1.18061	0.590303	−	0.807182i	$-0.299008\pi$
$31$	6.57334	1.18061	0.590303	+	0.807182i	$0.299008\pi$
$32$	0	0
$33$	−10.4373	−1.81690
$34$	0	0
$35$	1.22908	0.207752
$36$	0	0
$37$	0.619519	0.101848	0.0509241	−	0.998703i	$-0.483783\pi$
$37$	0.619519	0.101848	0.0509241	+	0.998703i	$0.483783\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	8.41141	1.31364	0.656821	−	0.754047i	$-0.271901\pi$
$41$	8.41141	1.31364	0.656821	+	0.754047i	$0.271901\pi$
$42$	0	0
$43$	−8.50043	−1.29630	−0.648152	−	0.761511i	$-0.724458\pi$
$43$	−8.50043	−1.29630	−0.648152	+	0.761511i	$0.724458\pi$
$44$	0	0
$45$	−4.80991	−0.717019
$46$	0	0
$47$	5.65693	0.825148	0.412574	−	0.910924i	$-0.364630\pi$
$47$	5.65693	0.825148	0.412574	+	0.910924i	$0.364630\pi$
$48$	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
$57$	8.56555	1.13453
$58$	0	0
$59$	−12.1840	−1.58622	−0.793110	−	0.609078i	$-0.791539\pi$
$59$	−12.1840	−1.58622	−0.793110	+	0.609078i	$0.791539\pi$
$60$	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
$63$	−5.91176	−0.744812
$64$	0	0
$65$	0	0
$66$	0	0
$67$	7.73375	0.944827	0.472414	−	0.881377i	$-0.343383\pi$
$67$	7.73375	0.944827	0.472414	+	0.881377i	$0.343383\pi$
$68$	0	0
$69$	−2.91718	−0.351187
$70$	0	0
$71$	5.36427	0.636622	0.318311	−	0.947986i	$-0.396884\pi$
$71$	5.36427	0.636622	0.318311	+	0.947986i	$0.396884\pi$
$72$	0	0
$73$	14.3772	1.68273	0.841363	−	0.540471i	$-0.181754\pi$
$73$	14.3772	1.68273	0.841363	+	0.540471i	$0.181754\pi$
$74$	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
$83$	−12.6989	−1.39389	−0.696943	−	0.717127i	$-0.745457\pi$
$83$	−12.6989	−1.39389	−0.696943	+	0.717127i	$0.745457\pi$
$84$	0	0
$85$	−4.95611	−0.537565
$86$	0	0
$87$	25.2136	2.70318
$88$	0	0
$89$	13.5729	1.43872	0.719362	−	0.694635i	$-0.244434\pi$
$89$	13.5729	1.43872	0.719362	+	0.694635i	$0.244434\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	18.3700	1.90488
$94$	0	0
$95$	−3.06501	−0.314463
$96$	0	0
$97$	−13.9803	−1.41948	−0.709742	−	0.704462i	$-0.751189\pi$
$97$	−13.9803	−1.41948	−0.709742	+	0.704462i	$0.751189\pi$
$98$	0	0
$99$	−17.9640	−1.80545
$100$	0	0
$101$	7.43227	0.739538	0.369769	−	0.929124i	$-0.379437\pi$
$101$	7.43227	0.739538	0.369769	+	0.929124i	$0.379437\pi$
$102$	0	0
$103$	16.4706	1.62289	0.811447	−	0.584427i	$-0.198681\pi$
$103$	16.4706	1.62289	0.811447	+	0.584427i	$0.198681\pi$
$104$	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
$109$	7.17891	0.687615	0.343808	−	0.939040i	$-0.388283\pi$
$109$	7.17891	0.687615	0.343808	+	0.939040i	$0.388283\pi$
$110$	0	0
$111$	1.73132	0.164330
$112$	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
$115$	1.04385	0.0973399
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−6.09145	−0.558402
$120$	0	0
$121$	2.94864	0.268058
$122$	0	0
$123$	23.5067	2.11953
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	1.60663	0.142565	0.0712827	−	0.997456i	$-0.477291\pi$
$127$	1.60663	0.142565	0.0712827	+	0.997456i	$0.477291\pi$
$128$	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
$135$	−5.05802	−0.435325
$136$	0	0
$137$	−7.73123	−0.660523	−0.330262	−	0.943889i	$-0.607137\pi$
$137$	−7.73123	−0.660523	−0.330262	+	0.943889i	$0.607137\pi$
$138$	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
$141$	15.8090	1.33136
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−9.02219	−0.749252
$146$	0	0
$147$	−15.3407	−1.26528
$148$	0	0
$149$	−17.0466	−1.39651	−0.698255	−	0.715849i	$-0.746040\pi$
$149$	−17.0466	−1.39651	−0.698255	+	0.715849i	$0.746040\pi$
$150$	0	0
$151$	−2.41977	−0.196918	−0.0984588	−	0.995141i	$-0.531391\pi$
$151$	−2.41977	−0.196918	−0.0984588	+	0.995141i	$0.531391\pi$
$152$	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
$161$	1.28298	0.101113
$162$	0	0
$163$	8.19218	0.641661	0.320831	−	0.947137i	$-0.396038\pi$
$163$	8.19218	0.641661	0.320831	+	0.947137i	$0.396038\pi$
$164$	0	0
$165$	10.4373	0.812544
$166$	0	0
$167$	0.119894	0.00927770	0.00463885	−	0.999989i	$-0.498523\pi$
$167$	0.119894	0.00927770	0.00463885	+	0.999989i	$0.498523\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	14.7424	1.12738
$172$	0	0
$173$	−13.3720	−1.01665	−0.508326	−	0.861165i	$-0.669735\pi$
$173$	−13.3720	−1.01665	−0.508326	+	0.861165i	$0.669735\pi$
$174$	0	0
$175$	−1.22908	−0.0929097
$176$	0	0
$177$	−34.0496	−2.55933
$178$	0	0
$179$	14.3046	1.06918	0.534588	−	0.845113i	$-0.320467\pi$
$179$	14.3046	1.06918	0.534588	+	0.845113i	$0.320467\pi$
$180$	0	0
$181$	−9.65346	−0.717536	−0.358768	−	0.933427i	$-0.616803\pi$
$181$	−9.65346	−0.717536	−0.358768	+	0.933427i	$0.616803\pi$
$182$	0	0
$183$	32.2311	2.38259
$184$	0	0
$185$	−0.619519	−0.0455479
$186$	0	0
$187$	−18.5100	−1.35359
$188$	0	0
$189$	−6.21670	−0.452199
$190$	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
$193$	2.44821	0.176226	0.0881132	−	0.996110i	$-0.471916\pi$
$193$	2.44821	0.176226	0.0881132	+	0.996110i	$0.471916\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−0.851216	−0.0606466	−0.0303233	−	0.999540i	$-0.509654\pi$
$197$	−0.851216	−0.0606466	−0.0303233	+	0.999540i	$0.509654\pi$
$198$	0	0
$199$	9.06160	0.642360	0.321180	−	0.947018i	$-0.395921\pi$
$199$	9.06160	0.642360	0.321180	+	0.947018i	$0.395921\pi$
$200$	0	0
$201$	21.6129	1.52446
$202$	0	0
$203$	−11.0890	−0.778294
$204$	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
$213$	14.9911	1.02717
$214$	0	0
$215$	8.50043	0.579725
$216$	0	0
$217$	−8.07915	−0.548449
$218$	0	0
$219$	40.1789	2.71504
$220$	0	0
$221$	0	0
$222$	0	0
$223$	23.2006	1.55363	0.776815	−	0.629729i	$-0.216834\pi$
$223$	23.2006	1.55363	0.776815	+	0.629729i	$0.216834\pi$
$224$	0	0
$225$	4.80991	0.320661
$226$	0	0
$227$	−5.33774	−0.354278	−0.177139	−	0.984186i	$-0.556684\pi$
$227$	−5.33774	−0.354278	−0.177139	+	0.984186i	$0.556684\pi$
$228$	0	0
$229$	−16.1535	−1.06745	−0.533726	−	0.845657i	$-0.679209\pi$
$229$	−16.1535	−1.06745	−0.533726	+	0.845657i	$0.679209\pi$
$230$	0	0
$231$	12.8283	0.844040
$232$	0	0
$233$	−5.09722	−0.333930	−0.166965	−	0.985963i	$-0.553397\pi$
$233$	−5.09722	−0.333930	−0.166965	+	0.985963i	$0.553397\pi$
$234$	0	0
$235$	−5.65693	−0.369017
$236$	0	0
$237$	−33.0495	−2.14680
$238$	0	0
$239$	−17.7527	−1.14833	−0.574163	−	0.818741i	$-0.694672\pi$
$239$	−17.7527	−1.14833	−0.574163	+	0.818741i	$0.694672\pi$
$240$	0	0
$241$	−3.17008	−0.204203	−0.102101	−	0.994774i	$-0.532557\pi$
$241$	−3.17008	−0.204203	−0.102101	+	0.994774i	$0.532557\pi$
$242$	0	0
$243$	−15.9970	−1.02621
$244$	0	0
$245$	5.48936	0.350703
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−35.4886	−2.24900
$250$	0	0
$251$	−18.4648	−1.16549	−0.582744	−	0.812656i	$-0.698021\pi$
$251$	−18.4648	−1.16549	−0.582744	+	0.812656i	$0.698021\pi$
$252$	0	0
$253$	3.89858	0.245101
$254$	0	0
$255$	−13.8504	−0.867348
$256$	0	0
$257$	−6.02478	−0.375816	−0.187908	−	0.982187i	$-0.560171\pi$
$257$	−6.02478	−0.375816	−0.187908	+	0.982187i	$0.560171\pi$
$258$	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
$265$	−9.83977	−0.604452
$266$	0	0
$267$	37.9311	2.32135
$268$	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
$271$	−3.61053	−0.219324	−0.109662	−	0.993969i	$-0.534977\pi$
$271$	−3.61053	−0.219324	−0.109662	+	0.993969i	$0.534977\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3.73479	−0.225216
$276$	0	0
$277$	−19.9027	−1.19583	−0.597917	−	0.801558i	$-0.704005\pi$
$277$	−19.9027	−1.19583	−0.597917	+	0.801558i	$0.704005\pi$
$278$	0	0
$279$	31.6172	1.89287
$280$	0	0
$281$	27.1022	1.61678	0.808391	−	0.588645i	$-0.200339\pi$
$281$	27.1022	1.61678	0.808391	+	0.588645i	$0.200339\pi$
$282$	0	0
$283$	3.45523	0.205392	0.102696	−	0.994713i	$-0.467253\pi$
$283$	3.45523	0.205392	0.102696	+	0.994713i	$0.467253\pi$
$284$	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
$291$	−39.0696	−2.29030
$292$	0	0
$293$	−13.8410	−0.808600	−0.404300	−	0.914626i	$-0.632485\pi$
$293$	−13.8410	−0.808600	−0.404300	+	0.914626i	$0.632485\pi$
$294$	0	0
$295$	12.1840	0.709379
$296$	0	0
$297$	−18.8906	−1.09614
$298$	0	0
$299$	0	0
$300$	0	0
$301$	10.4477	0.602196
$302$	0	0
$303$	20.7704	1.19323
$304$	0	0
$305$	−11.5333	−0.660393
$306$	0	0
$307$	13.4284	0.766401	0.383201	−	0.923665i	$-0.374822\pi$
$307$	13.4284	0.766401	0.383201	+	0.923665i	$0.374822\pi$
$308$	0	0
$309$	46.0290	2.61850
$310$	0	0
$311$	18.7934	1.06567	0.532837	−	0.846218i	$-0.321126\pi$
$311$	18.7934	1.06567	0.532837	+	0.846218i	$0.321126\pi$
$312$	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
$317$	−21.7038	−1.21901	−0.609504	−	0.792783i	$-0.708631\pi$
$317$	−21.7038	−1.21901	−0.609504	+	0.792783i	$0.708631\pi$
$318$	0	0
$319$	−33.6960	−1.88661
$320$	0	0
$321$	−24.5066	−1.36782
$322$	0	0
$323$	15.1905	0.845223
$324$	0	0
$325$	0	0
$326$	0	0
$327$	20.0623	1.10945
$328$	0	0
$329$	−6.95281	−0.383321
$330$	0	0
$331$	−13.1827	−0.724588	−0.362294	−	0.932064i	$-0.618006\pi$
$331$	−13.1827	−0.724588	−0.362294	+	0.932064i	$0.618006\pi$
$332$	0	0
$333$	2.97983	0.163294
$334$	0	0
$335$	−7.73375	−0.422540
$336$	0	0
$337$	18.9196	1.03062	0.515309	−	0.857004i	$-0.327677\pi$
$337$	18.9196	1.03062	0.515309	+	0.857004i	$0.327677\pi$
$338$	0	0
$339$	−28.0416	−1.52301
$340$	0	0
$341$	−24.5500	−1.32946
$342$	0	0
$343$	15.3504	0.828845
$344$	0	0
$345$	2.91718	0.157056
$346$	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
$349$	15.5379	0.831725	0.415862	−	0.909428i	$-0.363480\pi$
$349$	15.5379	0.831725	0.415862	+	0.909428i	$0.363480\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	18.4163	0.980199	0.490099	−	0.871667i	$-0.336960\pi$
$353$	18.4163	0.980199	0.490099	+	0.871667i	$0.336960\pi$
$354$	0	0
$355$	−5.36427	−0.284706
$356$	0	0
$357$	−17.0233	−0.900968
$358$	0	0
$359$	−2.26366	−0.119472	−0.0597358	−	0.998214i	$-0.519026\pi$
$359$	−2.26366	−0.119472	−0.0597358	+	0.998214i	$0.519026\pi$
$360$	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
$369$	40.4581	2.10617
$370$	0	0
$371$	−12.0939	−0.627882
$372$	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
$375$	−2.79462	−0.144314
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−31.2728	−1.60638	−0.803188	−	0.595725i	$-0.796865\pi$
$379$	−31.2728	−1.60638	−0.803188	+	0.595725i	$0.796865\pi$
$380$	0	0
$381$	4.48992	0.230026
$382$	0	0
$383$	26.2659	1.34212	0.671062	−	0.741401i	$-0.265838\pi$
$383$	26.2659	1.34212	0.671062	+	0.741401i	$0.265838\pi$
$384$	0	0
$385$	−4.59035	−0.233946
$386$	0	0
$387$	−40.8863	−2.07837
$388$	0	0
$389$	−18.8290	−0.954667	−0.477333	−	0.878722i	$-0.658397\pi$
$389$	−18.8290	−0.954667	−0.477333	+	0.878722i	$0.658397\pi$
$390$	0	0
$391$	−5.17345	−0.261633
$392$	0	0
$393$	−30.2283	−1.52482
$394$	0	0
$395$	11.8261	0.595037
$396$	0	0
$397$	−23.8701	−1.19800	−0.599002	−	0.800747i	$-0.704436\pi$
$397$	−23.8701	−1.19800	−0.599002	+	0.800747i	$0.704436\pi$
$398$	0	0
$399$	−10.5277	−0.527046
$400$	0	0
$401$	−2.56312	−0.127996	−0.0639980	−	0.997950i	$-0.520385\pi$
$401$	−2.56312	−0.127996	−0.0639980	+	0.997950i	$0.520385\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0.294490	0.0146333
$406$	0	0
$407$	−2.31377	−0.114689
$408$	0	0
$409$	−18.9442	−0.936732	−0.468366	−	0.883535i	$-0.655157\pi$
$409$	−18.9442	−0.936732	−0.468366	+	0.883535i	$0.655157\pi$
$410$	0	0
$411$	−21.6059	−1.06574
$412$	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
$421$	25.0406	1.22040	0.610201	−	0.792247i	$-0.291089\pi$
$421$	25.0406	1.22040	0.610201	+	0.792247i	$0.291089\pi$
$422$	0	0
$423$	27.2093	1.32296
$424$	0	0
$425$	4.95611	0.240406
$426$	0	0
$427$	−14.1753	−0.685991
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−8.58228	−0.413394	−0.206697	−	0.978405i	$-0.566271\pi$
$431$	−8.58228	−0.413394	−0.206697	+	0.978405i	$0.566271\pi$
$432$	0	0
$433$	12.5767	0.604399	0.302200	−	0.953245i	$-0.402279\pi$
$433$	12.5767	0.604399	0.302200	+	0.953245i	$0.402279\pi$
$434$	0	0
$435$	−25.2136	−1.20890
$436$	0	0
$437$	−3.19943	−0.153049
$438$	0	0
$439$	−33.2154	−1.58528	−0.792642	−	0.609687i	$-0.791295\pi$
$439$	−33.2154	−1.58528	−0.792642	+	0.609687i	$0.791295\pi$
$440$	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
$447$	−47.6387	−2.25323
$448$	0	0
$449$	5.12272	0.241756	0.120878	−	0.992667i	$-0.461429\pi$
$449$	5.12272	0.241756	0.120878	+	0.992667i	$0.461429\pi$
$450$	0	0
$451$	−31.4148	−1.47927
$452$	0	0
$453$	−6.76233	−0.317722
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−24.0427	−1.12467	−0.562334	−	0.826910i	$-0.690097\pi$
$457$	−24.0427	−1.12467	−0.562334	+	0.826910i	$0.690097\pi$
$458$	0	0
$459$	25.0681	1.17008
$460$	0	0
$461$	4.22878	0.196954	0.0984770	−	0.995139i	$-0.468603\pi$
$461$	4.22878	0.196954	0.0984770	+	0.995139i	$0.468603\pi$
$462$	0	0
$463$	29.4028	1.36647	0.683233	−	0.730200i	$-0.260573\pi$
$463$	29.4028	1.36647	0.683233	+	0.730200i	$0.260573\pi$
$464$	0	0
$465$	−18.3700	−0.851888
$466$	0	0
$467$	−5.06469	−0.234366	−0.117183	−	0.993110i	$-0.537386\pi$
$467$	−5.06469	−0.234366	−0.117183	+	0.993110i	$0.537386\pi$
$468$	0	0
$469$	−9.50539	−0.438918
$470$	0	0
$471$	54.1204	2.49374
$472$	0	0
$473$	31.7473	1.45974
$474$	0	0
$475$	3.06501	0.140632
$476$	0	0
$477$	47.3284	2.16702
$478$	0	0
$479$	7.18242	0.328173	0.164087	−	0.986446i	$-0.447532\pi$
$479$	7.18242	0.328173	0.164087	+	0.986446i	$0.447532\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	3.58544	0.163143
$484$	0	0
$485$	13.9803	0.634813
$486$	0	0
$487$	−19.0308	−0.862366	−0.431183	−	0.902264i	$-0.641904\pi$
$487$	−19.0308	−0.862366	−0.431183	+	0.902264i	$0.641904\pi$
$488$	0	0
$489$	22.8941	1.03530
$490$	0	0
$491$	9.89677	0.446635	0.223318	−	0.974746i	$-0.428311\pi$
$491$	9.89677	0.446635	0.223318	+	0.974746i	$0.428311\pi$
$492$	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
$499$	11.7475	0.525890	0.262945	−	0.964811i	$-0.415306\pi$
$499$	11.7475	0.525890	0.262945	+	0.964811i	$0.415306\pi$
$500$	0	0
$501$	0.335059	0.0149693
$502$	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
$505$	−7.43227	−0.330732
$506$	0	0
$507$	0	0
$508$	0	0
$509$	3.05609	0.135459	0.0677294	−	0.997704i	$-0.478425\pi$
$509$	3.05609	0.135459	0.0677294	+	0.997704i	$0.478425\pi$
$510$	0	0
$511$	−17.6707	−0.781707
$512$	0	0
$513$	15.5029	0.684469
$514$	0	0
$515$	−16.4706	−0.725780
$516$	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
$525$	−3.43481	−0.149907
$526$	0	0
$527$	32.5782	1.41913
$528$	0	0
$529$	−21.9104	−0.952625
$530$	0	0
$531$	−58.6039	−2.54319
$532$	0	0
$533$	0	0
$534$	0	0
$535$	8.76920	0.379125
$536$	0	0
$537$	39.9760	1.72509
$538$	0	0
$539$	20.5016	0.883067
$540$	0	0
$541$	38.5091	1.65564	0.827818	−	0.560997i	$-0.189582\pi$
$541$	38.5091	1.65564	0.827818	+	0.560997i	$0.189582\pi$
$542$	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
$551$	27.6531	1.17806
$552$	0	0
$553$	14.5352	0.618102
$554$	0	0
$555$	−1.73132	−0.0734905
$556$	0	0
$557$	−25.4127	−1.07677	−0.538385	−	0.842699i	$-0.680966\pi$
$557$	−25.4127	−1.07677	−0.538385	+	0.842699i	$0.680966\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−51.7285	−2.18398
$562$	0	0
$563$	3.72041	0.156797	0.0783983	−	0.996922i	$-0.475019\pi$
$563$	3.72041	0.156797	0.0783983	+	0.996922i	$0.475019\pi$
$564$	0	0
$565$	10.0341	0.422139
$566$	0	0
$567$	0.361952	0.0152005
$568$	0	0
$569$	6.96568	0.292016	0.146008	−	0.989283i	$-0.453357\pi$
$569$	6.96568	0.292016	0.146008	+	0.989283i	$0.453357\pi$
$570$	0	0
$571$	9.94609	0.416231	0.208116	−	0.978104i	$-0.433267\pi$
$571$	9.94609	0.416231	0.208116	+	0.978104i	$0.433267\pi$
$572$	0	0
$573$	7.27343	0.303852
$574$	0	0
$575$	−1.04385	−0.0435317
$576$	0	0
$577$	13.3817	0.557087	0.278544	−	0.960424i	$-0.410148\pi$
$577$	13.3817	0.557087	0.278544	+	0.960424i	$0.410148\pi$
$578$	0	0
$579$	6.84183	0.284337
$580$	0	0
$581$	15.6080	0.647527
$582$	0	0
$583$	−36.7495	−1.52201
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−0.400413	−0.0165268	−0.00826340	−	0.999966i	$-0.502630\pi$
$587$	−0.400413	−0.0165268	−0.00826340	+	0.999966i	$0.502630\pi$
$588$	0	0
$589$	20.1473	0.830157
$590$	0	0
$591$	−2.37883	−0.0978518
$592$	0	0
$593$	16.8256	0.690945	0.345472	−	0.938429i	$-0.387719\pi$
$593$	16.8256	0.690945	0.345472	+	0.938429i	$0.387719\pi$
$594$	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
$603$	37.1986	1.51485
$604$	0	0
$605$	−2.94864	−0.119879
$606$	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
$609$	−30.9895	−1.25576
$610$	0	0
$611$	0	0
$612$	0	0
$613$	13.1645	0.531709	0.265854	−	0.964013i	$-0.414346\pi$
$613$	13.1645	0.531709	0.265854	+	0.964013i	$0.414346\pi$
$614$	0	0
$615$	−23.5067	−0.947882
$616$	0	0
$617$	−36.4559	−1.46766	−0.733830	−	0.679333i	$-0.762269\pi$
$617$	−36.4559	−1.46766	−0.733830	+	0.679333i	$0.762269\pi$
$618$	0	0
$619$	44.4340	1.78595	0.892976	−	0.450105i	$-0.148613\pi$
$619$	44.4340	1.78595	0.892976	+	0.450105i	$0.148613\pi$
$620$	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
$629$	3.07040	0.122425
$630$	0	0
$631$	−22.7501	−0.905666	−0.452833	−	0.891595i	$-0.649587\pi$
$631$	−22.7501	−0.905666	−0.452833	+	0.891595i	$0.649587\pi$
$632$	0	0
$633$	−45.9204	−1.82517
$634$	0	0
$635$	−1.60663	−0.0637572
$636$	0	0
$637$	0	0
$638$	0	0
$639$	25.8017	1.02070
$640$	0	0
$641$	36.2272	1.43089	0.715444	−	0.698670i	$-0.246225\pi$
$641$	36.2272	1.43089	0.715444	+	0.698670i	$0.246225\pi$
$642$	0	0
$643$	3.55026	0.140008	0.0700042	−	0.997547i	$-0.477699\pi$
$643$	3.55026	0.140008	0.0700042	+	0.997547i	$0.477699\pi$
$644$	0	0
$645$	23.7555	0.935372
$646$	0	0
$647$	−13.2766	−0.521958	−0.260979	−	0.965344i	$-0.584045\pi$
$647$	−13.2766	−0.521958	−0.260979	+	0.965344i	$0.584045\pi$
$648$	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
$655$	10.8166	0.422640
$656$	0	0
$657$	69.1531	2.69792
$658$	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
$661$	27.5909	1.07316	0.536581	−	0.843849i	$-0.319716\pi$
$661$	27.5909	1.07316	0.536581	+	0.843849i	$0.319716\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3.76714	0.146083
$666$	0	0
$667$	−9.41785	−0.364661
$668$	0	0
$669$	64.8370	2.50674
$670$	0	0
$671$	−43.0743	−1.66287
$672$	0	0
$673$	11.5736	0.446131	0.223065	−	0.974803i	$-0.428394\pi$
$673$	11.5736	0.446131	0.223065	+	0.974803i	$0.428394\pi$
$674$	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
$681$	−14.9170	−0.571619
$682$	0	0
$683$	−33.3432	−1.27584	−0.637921	−	0.770102i	$-0.720205\pi$
$683$	−33.3432	−1.27584	−0.637921	+	0.770102i	$0.720205\pi$
$684$	0	0
$685$	7.73123	0.295395
$686$	0	0
$687$	−45.1429	−1.72231
$688$	0	0
$689$	0	0
$690$	0	0
$691$	43.1681	1.64219	0.821095	−	0.570791i	$-0.193363\pi$
$691$	43.1681	1.64219	0.821095	+	0.570791i	$0.193363\pi$
$692$	0	0
$693$	22.0792	0.838719
$694$	0	0
$695$	−12.9186	−0.490029
$696$	0	0
$697$	41.6878	1.57904
$698$	0	0
$699$	−14.2448	−0.538788
$700$	0	0
$701$	16.4423	0.621017	0.310508	−	0.950571i	$-0.399501\pi$
$701$	16.4423	0.621017	0.310508	+	0.950571i	$0.399501\pi$
$702$	0	0
$703$	1.89883	0.0716158
$704$	0	0
$705$	−15.8090	−0.595400
$706$	0	0
$707$	−9.13485	−0.343551
$708$	0	0
$709$	−25.8445	−0.970611	−0.485306	−	0.874345i	$-0.661292\pi$
$709$	−25.8445	−0.970611	−0.485306	+	0.874345i	$0.661292\pi$
$710$	0	0
$711$	−56.8826	−2.13326
$712$	0	0
$713$	−6.86161	−0.256969
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−49.6120	−1.85279
$718$	0	0
$719$	−7.66800	−0.285968	−0.142984	−	0.989725i	$-0.545670\pi$
$719$	−7.66800	−0.285968	−0.142984	+	0.989725i	$0.545670\pi$
$720$	0	0
$721$	−20.2436	−0.753912
$722$	0	0
$723$	−8.85917	−0.329476
$724$	0	0
$725$	9.02219	0.335076
$726$	0	0
$727$	−35.4370	−1.31428	−0.657142	−	0.753767i	$-0.728235\pi$
$727$	−35.4370	−1.31428	−0.657142	+	0.753767i	$0.728235\pi$
$728$	0	0
$729$	−43.8222	−1.62304
$730$	0	0
$731$	−42.1290	−1.55820
$732$	0	0
$733$	−48.0345	−1.77419	−0.887097	−	0.461582i	$-0.847282\pi$
$733$	−48.0345	−1.77419	−0.887097	+	0.461582i	$0.847282\pi$
$734$	0	0
$735$	15.3407	0.565850
$736$	0	0
$737$	−28.8839	−1.06395
$738$	0	0
$739$	−20.3284	−0.747794	−0.373897	−	0.927470i	$-0.621979\pi$
$739$	−20.3284	−0.747794	−0.373897	+	0.927470i	$0.621979\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−39.5385	−1.45053	−0.725263	−	0.688472i	$-0.758282\pi$
$743$	−39.5385	−1.45053	−0.725263	+	0.688472i	$0.758282\pi$
$744$	0	0
$745$	17.0466	0.624538
$746$	0	0
$747$	−61.0806	−2.23482
$748$	0	0
$749$	10.7780	0.393821
$750$	0	0
$751$	−36.1294	−1.31838	−0.659191	−	0.751976i	$-0.729101\pi$
$751$	−36.1294	−1.31838	−0.659191	+	0.751976i	$0.729101\pi$
$752$	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
$759$	10.8950	0.395465
$760$	0	0
$761$	43.4661	1.57564	0.787822	−	0.615902i	$-0.211208\pi$
$761$	43.4661	1.57564	0.787822	+	0.615902i	$0.211208\pi$
$762$	0	0
$763$	−8.82345	−0.319430
$764$	0	0
$765$	−23.8384	−0.861880
$766$	0	0
$767$	0	0
$768$	0	0
$769$	1.98755	0.0716729	0.0358364	−	0.999358i	$-0.488590\pi$
$769$	1.98755	0.0716729	0.0358364	+	0.999358i	$0.488590\pi$
$770$	0	0
$771$	−16.8370	−0.606369
$772$	0	0
$773$	−46.0457	−1.65615	−0.828074	−	0.560619i	$-0.810563\pi$
$773$	−46.0457	−1.65615	−0.828074	+	0.560619i	$0.810563\pi$
$774$	0	0
$775$	6.57334	0.236121
$776$	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
$785$	−19.3659	−0.691200
$786$	0	0
$787$	−29.7703	−1.06120	−0.530598	−	0.847623i	$-0.678033\pi$
$787$	−29.7703	−1.06120	−0.530598	+	0.847623i	$0.678033\pi$
$788$	0	0
$789$	−79.2770	−2.82234
$790$	0	0
$791$	12.3327	0.438502
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−27.4984	−0.975269
$796$	0	0
$797$	−51.7725	−1.83388	−0.916938	−	0.399030i	$-0.869347\pi$
$797$	−51.7725	−1.83388	−0.916938	+	0.399030i	$0.869347\pi$
$798$	0	0
$799$	28.0363	0.991854
$800$	0	0
$801$	65.2844	2.30671
$802$	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
$811$	54.9207	1.92853	0.964264	−	0.264945i	$-0.0853538\pi$
$811$	54.9207	1.92853	0.964264	+	0.264945i	$0.0853538\pi$
$812$	0	0
$813$	−10.0901	−0.353874
$814$	0	0
$815$	−8.19218	−0.286960
$816$	0	0
$817$	−26.0539	−0.911511
$818$	0	0
$819$	0	0
$820$	0	0
$821$	21.6079	0.754120	0.377060	−	0.926189i	$-0.376935\pi$
$821$	21.6079	0.754120	0.377060	+	0.926189i	$0.376935\pi$
$822$	0	0
$823$	−26.0715	−0.908797	−0.454399	−	0.890799i	$-0.650146\pi$
$823$	−26.0715	−0.908797	−0.454399	+	0.890799i	$0.650146\pi$
$824$	0	0
$825$	−10.4373	−0.363381
$826$	0	0
$827$	35.6861	1.24093	0.620463	−	0.784235i	$-0.286945\pi$
$827$	35.6861	1.24093	0.620463	+	0.784235i	$0.286945\pi$
$828$	0	0
$829$	45.9700	1.59661	0.798303	−	0.602256i	$-0.205732\pi$
$829$	45.9700	1.59661	0.798303	+	0.602256i	$0.205732\pi$
$830$	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
$837$	33.2480	1.14922
$838$	0	0
$839$	21.5917	0.745427	0.372714	−	0.927946i	$-0.378427\pi$
$839$	21.5917	0.745427	0.372714	+	0.927946i	$0.378427\pi$
$840$	0	0
$841$	52.3999	1.80689
$842$	0	0
$843$	75.7404	2.60864
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−3.62411	−0.124526
$848$	0	0
$849$	9.65605	0.331395
$850$	0	0
$851$	−0.646687	−0.0221682
$852$	0	0
$853$	−9.78391	−0.334995	−0.167497	−	0.985873i	$-0.553569\pi$
$853$	−9.78391	−0.334995	−0.167497	+	0.985873i	$0.553569\pi$
$854$	0	0
$855$	−14.7424	−0.504180
$856$	0	0
$857$	43.4611	1.48460	0.742301	−	0.670067i	$-0.233735\pi$
$857$	43.4611	1.48460	0.742301	+	0.670067i	$0.233735\pi$
$858$	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
$863$	−20.2021	−0.687688	−0.343844	−	0.939027i	$-0.611729\pi$
$863$	−20.2021	−0.687688	−0.343844	+	0.939027i	$0.611729\pi$
$864$	0	0
$865$	13.3720	0.454660
$866$	0	0
$867$	21.1357	0.717805
$868$	0	0
$869$	44.1681	1.49830
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−67.2440	−2.27586
$874$	0	0
$875$	1.22908	0.0415505
$876$	0	0
$877$	−28.9831	−0.978689	−0.489344	−	0.872091i	$-0.662764\pi$
$877$	−28.9831	−0.978689	−0.489344	+	0.872091i	$0.662764\pi$
$878$	0	0
$879$	−38.6804	−1.30466
$880$	0	0
$881$	−40.9072	−1.37820	−0.689100	−	0.724667i	$-0.741994\pi$
$881$	−40.9072	−1.37820	−0.689100	+	0.724667i	$0.741994\pi$
$882$	0	0
$883$	−25.8593	−0.870234	−0.435117	−	0.900374i	$-0.643293\pi$
$883$	−25.8593	−0.870234	−0.435117	+	0.900374i	$0.643293\pi$
$884$	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
$889$	−1.97468	−0.0662285
$890$	0	0
$891$	1.09986	0.0368466
$892$	0	0
$893$	17.3385	0.580212
$894$	0	0
$895$	−14.3046	−0.478150
$896$	0	0
$897$	0	0
$898$	0	0
$899$	59.3059	1.97796
$900$	0	0
$901$	48.7669	1.62466
$902$	0	0
$903$	29.1974	0.971628
$904$	0	0
$905$	9.65346	0.320892
$906$	0	0
$907$	−44.4104	−1.47462	−0.737311	−	0.675553i	$-0.763905\pi$
$907$	−44.4104	−1.47462	−0.737311	+	0.675553i	$0.763905\pi$
$908$	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
$915$	−32.2311	−1.06553
$916$	0	0
$917$	13.2945	0.439022
$918$	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
$921$	37.5274	1.23657
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0.619519	0.0203696
$926$	0	0
$927$	79.2220	2.60199
$928$	0	0
$929$	7.00807	0.229927	0.114964	−	0.993370i	$-0.463325\pi$
$929$	7.00807	0.229927	0.114964	+	0.993370i	$0.463325\pi$
$930$	0	0
$931$	−16.8250	−0.551416
$932$	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
$941$	−38.5412	−1.25641	−0.628204	−	0.778048i	$-0.716210\pi$
$941$	−38.5412	−1.25641	−0.628204	+	0.778048i	$0.716210\pi$
$942$	0	0
$943$	−8.78029	−0.285926
$944$	0	0
$945$	6.21670	0.202229
$946$	0	0
$947$	21.8022	0.708475	0.354237	−	0.935156i	$-0.384740\pi$
$947$	21.8022	0.708475	0.354237	+	0.935156i	$0.384740\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−60.6540	−1.96684
$952$	0	0
$953$	7.43281	0.240772	0.120386	−	0.992727i	$-0.461587\pi$
$953$	7.43281	0.240772	0.120386	+	0.992727i	$0.461587\pi$
$954$	0	0
$955$	−2.60265	−0.0842199
$956$	0	0
$957$	−94.1675	−3.04400
$958$	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
$967$	10.0827	0.324237	0.162118	−	0.986771i	$-0.448167\pi$
$967$	10.0827	0.324237	0.162118	+	0.986771i	$0.448167\pi$
$968$	0	0
$969$	42.4517	1.36375
$970$	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
$973$	−15.8779	−0.509023
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−34.3147	−1.09782	−0.548912	−	0.835880i	$-0.684958\pi$
$977$	−34.3147	−1.09782	−0.548912	+	0.835880i	$0.684958\pi$
$978$	0	0
$979$	−50.6919	−1.62012
$980$	0	0
$981$	34.5299	1.10246
$982$	0	0
$983$	−21.9179	−0.699074	−0.349537	−	0.936923i	$-0.613661\pi$
$983$	−21.9179	−0.699074	−0.349537	+	0.936923i	$0.613661\pi$
$984$	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
$993$	−36.8407	−1.16911
$994$	0	0
$995$	−9.06160	−0.287272
$996$	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
$999$	3.13354	0.0991407

Twists

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

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

\(f(q)\)	\(=\)	\(q+2.79462 q^{3} -1.00000 q^{5} -1.22908 q^{7} +4.80991 q^{9} +O(q^{10})\)	\(q+2.79462 q^{3} -1.00000 q^{5} -1.22908 q^{7} +4.80991 q^{9} -3.73479 q^{11} -2.79462 q^{15} +4.95611 q^{17} +3.06501 q^{19} -3.43481 q^{21} -1.04385 q^{23} +1.00000 q^{25} +5.05802 q^{27} +9.02219 q^{29} +6.57334 q^{31} -10.4373 q^{33} +1.22908 q^{35} +0.619519 q^{37} +8.41141 q^{41} -8.50043 q^{43} -4.80991 q^{45} +5.65693 q^{47} -5.48936 q^{49} +13.8504 q^{51} +9.83977 q^{53} +3.73479 q^{55} +8.56555 q^{57} -12.1840 q^{59} +11.5333 q^{61} -5.91176 q^{63} +7.73375 q^{67} -2.91718 q^{69} +5.36427 q^{71} +14.3772 q^{73} +2.79462 q^{75} +4.59035 q^{77} -11.8261 q^{79} -0.294490 q^{81} -12.6989 q^{83} -4.95611 q^{85} +25.2136 q^{87} +13.5729 q^{89} +18.3700 q^{93} -3.06501 q^{95} -13.9803 q^{97} -17.9640 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - q^{3} - 9 q^{5} - q^{7} + 12 q^{9}+O(q^{10}) \) 9 * q - q^3 - 9 * q^5 - q^7 + 12 * q^9	\( 9 q - q^{3} - 9 q^{5} - q^{7} + 12 q^{9} + 7 q^{11} + q^{15} + 13 q^{17} + 4 q^{19} + 3 q^{21} + 12 q^{23} + 9 q^{25} - 4 q^{27} + 16 q^{29} - 13 q^{31} - 34 q^{33} + q^{35} - q^{37} + 6 q^{41} + q^{43} - 12 q^{45} + 2 q^{47} + 20 q^{49} + 11 q^{51} + 30 q^{53} - 7 q^{55} - 38 q^{57} - 15 q^{59} + 21 q^{61} + 17 q^{63} + 7 q^{67} + 15 q^{69} + 7 q^{71} + 28 q^{73} - q^{75} + 46 q^{77} + 31 q^{79} + 41 q^{81} - 45 q^{83} - 13 q^{85} + 28 q^{87} + 41 q^{89} + 11 q^{93} - 4 q^{95} - 8 q^{97} + 81 q^{99}+O(q^{100}) \) 9 * q - q^3 - 9 * q^5 - q^7 + 12 * q^9 + 7 * q^11 + q^15 + 13 * q^17 + 4 * q^19 + 3 * q^21 + 12 * q^23 + 9 * q^25 - 4 * q^27 + 16 * q^29 - 13 * q^31 - 34 * q^33 + q^35 - q^37 + 6 * q^41 + q^43 - 12 * q^45 + 2 * q^47 + 20 * q^49 + 11 * q^51 + 30 * q^53 - 7 * q^55 - 38 * q^57 - 15 * q^59 + 21 * q^61 + 17 * q^63 + 7 * q^67 + 15 * q^69 + 7 * q^71 + 28 * q^73 - q^75 + 46 * q^77 + 31 * q^79 + 41 * q^81 - 45 * q^83 - 13 * q^85 + 28 * q^87 + 41 * q^89 + 11 * q^93 - 4 * q^95 - 8 * q^97 + 81 * q^99

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