LMFDB - Embedded newform 3078.2.a.x.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3078,2,Mod(1,3078)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3078, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3078.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3078 = 2 \cdot 3^{4} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3078.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:	$24.5779537422$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.37354176.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 8x^{4} + 9x^{2} - 2x - 1 $ x^6 - 8x^4 + 9x^2 - 2*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 342)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.244728$ of defining polynomial
Character	$\chi$	$=$	3078.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.14606 q^{5} -1.63489 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.14606 q^{5} -1.63489 q^{7} +1.00000 q^{8} +1.14606 q^{10} +2.20898 q^{11} +4.36908 q^{13} -1.63489 q^{14} +1.00000 q^{16} -1.42591 q^{17} -1.00000 q^{19} +1.14606 q^{20} +2.20898 q^{22} +4.33718 q^{23} -3.68654 q^{25} +4.36908 q^{26} -1.63489 q^{28} +1.40010 q^{29} +7.33149 q^{31} +1.00000 q^{32} -1.42591 q^{34} -1.87369 q^{35} +6.94627 q^{37} -1.00000 q^{38} +1.14606 q^{40} -5.28035 q^{41} -8.75912 q^{43} +2.20898 q^{44} +4.33718 q^{46} +3.07258 q^{47} -4.32712 q^{49} -3.68654 q^{50} +4.36908 q^{52} +5.16929 q^{53} +2.53163 q^{55} -1.63489 q^{56} +1.40010 q^{58} +1.30616 q^{59} -1.07137 q^{61} +7.33149 q^{62} +1.00000 q^{64} +5.00724 q^{65} +7.14158 q^{67} -1.42591 q^{68} -1.87369 q^{70} +12.0806 q^{71} +6.78533 q^{73} +6.94627 q^{74} -1.00000 q^{76} -3.61145 q^{77} +10.8087 q^{79} +1.14606 q^{80} -5.28035 q^{82} -5.19721 q^{83} -1.63418 q^{85} -8.75912 q^{86} +2.20898 q^{88} -7.62362 q^{89} -7.14298 q^{91} +4.33718 q^{92} +3.07258 q^{94} -1.14606 q^{95} +15.5463 q^{97} -4.32712 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{2} + 6 q^{4} + 6 q^{7} + 6 q^{8}+O(q^{10}) $ 6 * q + 6 * q^2 + 6 * q^4 + 6 * q^7 + 6 * q^8	$ 6 q + 6 q^{2} + 6 q^{4} + 6 q^{7} + 6 q^{8} + 6 q^{11} + 6 q^{13} + 6 q^{14} + 6 q^{16} - 6 q^{19} + 6 q^{22} + 12 q^{23} + 30 q^{25} + 6 q^{26} + 6 q^{28} - 6 q^{29} + 6 q^{31} + 6 q^{32} + 12 q^{35} + 6 q^{37} - 6 q^{38} - 6 q^{41} + 12 q^{43} + 6 q^{44} + 12 q^{46} + 6 q^{47} + 18 q^{49} + 30 q^{50} + 6 q^{52} - 18 q^{53} + 24 q^{55} + 6 q^{56} - 6 q^{58} - 12 q^{59} + 12 q^{61} + 6 q^{62} + 6 q^{64} + 6 q^{65} + 18 q^{67} + 12 q^{70} + 18 q^{71} + 24 q^{73} + 6 q^{74} - 6 q^{76} + 6 q^{77} + 18 q^{79} - 6 q^{82} - 12 q^{83} + 36 q^{85} + 12 q^{86} + 6 q^{88} + 6 q^{89} - 12 q^{91} + 12 q^{92} + 6 q^{94} - 18 q^{97} + 18 q^{98}+O(q^{100}) $ 6 * q + 6 * q^2 + 6 * q^4 + 6 * q^7 + 6 * q^8 + 6 * q^11 + 6 * q^13 + 6 * q^14 + 6 * q^16 - 6 * q^19 + 6 * q^22 + 12 * q^23 + 30 * q^25 + 6 * q^26 + 6 * q^28 - 6 * q^29 + 6 * q^31 + 6 * q^32 + 12 * q^35 + 6 * q^37 - 6 * q^38 - 6 * q^41 + 12 * q^43 + 6 * q^44 + 12 * q^46 + 6 * q^47 + 18 * q^49 + 30 * q^50 + 6 * q^52 - 18 * q^53 + 24 * q^55 + 6 * q^56 - 6 * q^58 - 12 * q^59 + 12 * q^61 + 6 * q^62 + 6 * q^64 + 6 * q^65 + 18 * q^67 + 12 * q^70 + 18 * q^71 + 24 * q^73 + 6 * q^74 - 6 * q^76 + 6 * q^77 + 18 * q^79 - 6 * q^82 - 12 * q^83 + 36 * q^85 + 12 * q^86 + 6 * q^88 + 6 * q^89 - 12 * q^91 + 12 * q^92 + 6 * q^94 - 18 * q^97 + 18 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.14606	0.512535	0.256267	−	0.966606i	$-0.417507\pi$
$5$	1.14606	0.512535	0.256267	+	0.966606i	$0.417507\pi$
$6$	0	0
$7$	−1.63489	−0.617932	−0.308966	−	0.951073i	$-0.599983\pi$
$7$	−1.63489	−0.617932	−0.308966	+	0.951073i	$0.599983\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	1.14606	0.362417
$11$	2.20898	0.666033	0.333017	−	0.942921i	$-0.391933\pi$
$11$	2.20898	0.666033	0.333017	+	0.942921i	$0.391933\pi$
$12$	0	0
$13$	4.36908	1.21176	0.605882	−	0.795555i	$-0.292820\pi$
$13$	4.36908	1.21176	0.605882	+	0.795555i	$0.292820\pi$
$14$	−1.63489	−0.436944
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.42591	−0.345834	−0.172917	−	0.984936i	$-0.555319\pi$
$17$	−1.42591	−0.345834	−0.172917	+	0.984936i	$0.555319\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	1.14606	0.256267
$21$	0	0
$22$	2.20898	0.470957
$23$	4.33718	0.904366	0.452183	−	0.891925i	$-0.350645\pi$
$23$	4.33718	0.904366	0.452183	+	0.891925i	$0.350645\pi$
$24$	0	0
$25$	−3.68654	−0.737308
$26$	4.36908	0.856847
$27$	0	0
$28$	−1.63489	−0.308966
$29$	1.40010	0.259993	0.129996	−	0.991514i	$-0.458503\pi$
$29$	1.40010	0.259993	0.129996	+	0.991514i	$0.458503\pi$
$30$	0	0
$31$	7.33149	1.31678	0.658388	−	0.752679i	$-0.271239\pi$
$31$	7.33149	1.31678	0.658388	+	0.752679i	$0.271239\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−1.42591	−0.244542
$35$	−1.87369	−0.316712
$36$	0	0
$37$	6.94627	1.14196	0.570980	−	0.820964i	$-0.306563\pi$
$37$	6.94627	1.14196	0.570980	+	0.820964i	$0.306563\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	1.14606	0.181208
$41$	−5.28035	−0.824652	−0.412326	−	0.911036i	$-0.635284\pi$
$41$	−5.28035	−0.824652	−0.412326	+	0.911036i	$0.635284\pi$
$42$	0	0
$43$	−8.75912	−1.33575	−0.667877	−	0.744272i	$-0.732797\pi$
$43$	−8.75912	−1.33575	−0.667877	+	0.744272i	$0.732797\pi$
$44$	2.20898	0.333017
$45$	0	0
$46$	4.33718	0.639483
$47$	3.07258	0.448182	0.224091	−	0.974568i	$-0.428059\pi$
$47$	3.07258	0.448182	0.224091	+	0.974568i	$0.428059\pi$
$48$	0	0
$49$	−4.32712	−0.618160
$50$	−3.68654	−0.521355
$51$	0	0
$52$	4.36908	0.605882
$53$	5.16929	0.710056	0.355028	−	0.934856i	$-0.384471\pi$
$53$	5.16929	0.710056	0.355028	+	0.934856i	$0.384471\pi$
$54$	0	0
$55$	2.53163	0.341365
$56$	−1.63489	−0.218472
$57$	0	0
$58$	1.40010	0.183843
$59$	1.30616	0.170047	0.0850236	−	0.996379i	$-0.472903\pi$
$59$	1.30616	0.170047	0.0850236	+	0.996379i	$0.472903\pi$
$60$	0	0
$61$	−1.07137	−0.137175	−0.0685873	−	0.997645i	$-0.521849\pi$
$61$	−1.07137	−0.137175	−0.0685873	+	0.997645i	$0.521849\pi$
$62$	7.33149	0.931101
$63$	0	0
$64$	1.00000	0.125000
$65$	5.00724	0.621071
$66$	0	0
$67$	7.14158	0.872483	0.436242	−	0.899830i	$-0.356309\pi$
$67$	7.14158	0.872483	0.436242	+	0.899830i	$0.356309\pi$
$68$	−1.42591	−0.172917
$69$	0	0
$70$	−1.87369	−0.223949
$71$	12.0806	1.43370	0.716849	−	0.697228i	$-0.245584\pi$
$71$	12.0806	1.43370	0.716849	+	0.697228i	$0.245584\pi$
$72$	0	0
$73$	6.78533	0.794163	0.397081	−	0.917783i	$-0.370023\pi$
$73$	6.78533	0.794163	0.397081	+	0.917783i	$0.370023\pi$
$74$	6.94627	0.807488
$75$	0	0
$76$	−1.00000	−0.114708
$77$	−3.61145	−0.411563
$78$	0	0
$79$	10.8087	1.21607	0.608034	−	0.793911i	$-0.291958\pi$
$79$	10.8087	1.21607	0.608034	+	0.793911i	$0.291958\pi$
$80$	1.14606	0.128134
$81$	0	0
$82$	−5.28035	−0.583117
$83$	−5.19721	−0.570468	−0.285234	−	0.958458i	$-0.592071\pi$
$83$	−5.19721	−0.570468	−0.285234	+	0.958458i	$0.592071\pi$
$84$	0	0
$85$	−1.63418	−0.177252
$86$	−8.75912	−0.944520
$87$	0	0
$88$	2.20898	0.235478
$89$	−7.62362	−0.808102	−0.404051	−	0.914736i	$-0.632398\pi$
$89$	−7.62362	−0.808102	−0.404051	+	0.914736i	$0.632398\pi$
$90$	0	0
$91$	−7.14298	−0.748787
$92$	4.33718	0.452183
$93$	0	0
$94$	3.07258	0.316912
$95$	−1.14606	−0.117584
$96$	0	0
$97$	15.5463	1.57849	0.789246	−	0.614077i	$-0.210472\pi$
$97$	15.5463	1.57849	0.789246	+	0.614077i	$0.210472\pi$
$98$	−4.32712	−0.437105
$99$	0	0
$100$	−3.68654	−0.368654
$101$	−7.92531	−0.788597	−0.394299	−	0.918982i	$-0.629012\pi$
$101$	−7.92531	−0.788597	−0.394299	+	0.918982i	$0.629012\pi$
$102$	0	0
$103$	−4.34327	−0.427955	−0.213978	−	0.976839i	$-0.568642\pi$
$103$	−4.34327	−0.427955	−0.213978	+	0.976839i	$0.568642\pi$
$104$	4.36908	0.428423
$105$	0	0
$106$	5.16929	0.502085
$107$	−4.42033	−0.427329	−0.213665	−	0.976907i	$-0.568540\pi$
$107$	−4.42033	−0.427329	−0.213665	+	0.976907i	$0.568540\pi$
$108$	0	0
$109$	18.5327	1.77511	0.887554	−	0.460704i	$-0.152403\pi$
$109$	18.5327	1.77511	0.887554	+	0.460704i	$0.152403\pi$
$110$	2.53163	0.241382
$111$	0	0
$112$	−1.63489	−0.154483
$113$	−10.3412	−0.972814	−0.486407	−	0.873732i	$-0.661693\pi$
$113$	−10.3412	−0.972814	−0.486407	+	0.873732i	$0.661693\pi$
$114$	0	0
$115$	4.97069	0.463519
$116$	1.40010	0.129996
$117$	0	0
$118$	1.30616	0.120242
$119$	2.33121	0.213702
$120$	0	0
$121$	−6.12040	−0.556400
$122$	−1.07137	−0.0969971
$123$	0	0
$124$	7.33149	0.658388
$125$	−9.95532	−0.890431
$126$	0	0
$127$	8.15572	0.723703	0.361852	−	0.932236i	$-0.382145\pi$
$127$	8.15572	0.723703	0.361852	+	0.932236i	$0.382145\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	5.00724	0.439164
$131$	5.73729	0.501269	0.250635	−	0.968082i	$-0.419361\pi$
$131$	5.73729	0.501269	0.250635	+	0.968082i	$0.419361\pi$
$132$	0	0
$133$	1.63489	0.141763
$134$	7.14158	0.616939
$135$	0	0
$136$	−1.42591	−0.122271
$137$	−21.1576	−1.80762	−0.903808	−	0.427937i	$-0.859240\pi$
$137$	−21.1576	−1.80762	−0.903808	+	0.427937i	$0.859240\pi$
$138$	0	0
$139$	18.3618	1.55743	0.778713	−	0.627381i	$-0.215873\pi$
$139$	18.3618	1.55743	0.778713	+	0.627381i	$0.215873\pi$
$140$	−1.87369	−0.158356
$141$	0	0
$142$	12.0806	1.01378
$143$	9.65121	0.807075
$144$	0	0
$145$	1.60461	0.133255
$146$	6.78533	0.561558
$147$	0	0
$148$	6.94627	0.570980
$149$	20.3982	1.67108	0.835541	−	0.549428i	$-0.185154\pi$
$149$	20.3982	1.67108	0.835541	+	0.549428i	$0.185154\pi$
$150$	0	0
$151$	−14.2233	−1.15748	−0.578738	−	0.815514i	$-0.696454\pi$
$151$	−14.2233	−1.15748	−0.578738	+	0.815514i	$0.696454\pi$
$152$	−1.00000	−0.0811107
$153$	0	0
$154$	−3.61145	−0.291019
$155$	8.40235	0.674893
$156$	0	0
$157$	−1.95891	−0.156338	−0.0781691	−	0.996940i	$-0.524907\pi$
$157$	−1.95891	−0.156338	−0.0781691	+	0.996940i	$0.524907\pi$
$158$	10.8087	0.859891
$159$	0	0
$160$	1.14606	0.0906042
$161$	−7.09083	−0.558836
$162$	0	0
$163$	4.31174	0.337722	0.168861	−	0.985640i	$-0.445991\pi$
$163$	4.31174	0.337722	0.168861	+	0.985640i	$0.445991\pi$
$164$	−5.28035	−0.412326
$165$	0	0
$166$	−5.19721	−0.403381
$167$	4.95770	0.383638	0.191819	−	0.981430i	$-0.438561\pi$
$167$	4.95770	0.383638	0.191819	+	0.981430i	$0.438561\pi$
$168$	0	0
$169$	6.08883	0.468372
$170$	−1.63418	−0.125336
$171$	0	0
$172$	−8.75912	−0.667877
$173$	−13.1012	−0.996064	−0.498032	−	0.867159i	$-0.665944\pi$
$173$	−13.1012	−0.996064	−0.498032	+	0.867159i	$0.665944\pi$
$174$	0	0
$175$	6.02710	0.455606
$176$	2.20898	0.166508
$177$	0	0
$178$	−7.62362	−0.571415
$179$	−12.2984	−0.919224	−0.459612	−	0.888120i	$-0.652012\pi$
$179$	−12.2984	−0.919224	−0.459612	+	0.888120i	$0.652012\pi$
$180$	0	0
$181$	−1.39563	−0.103736	−0.0518680	−	0.998654i	$-0.516518\pi$
$181$	−1.39563	−0.103736	−0.0518680	+	0.998654i	$0.516518\pi$
$182$	−7.14298	−0.529473
$183$	0	0
$184$	4.33718	0.319742
$185$	7.96086	0.585294
$186$	0	0
$187$	−3.14981	−0.230337
$188$	3.07258	0.224091
$189$	0	0
$190$	−1.14606	−0.0831441
$191$	−0.104472	−0.00755935	−0.00377968	−	0.999993i	$-0.501203\pi$
$191$	−0.104472	−0.00755935	−0.00377968	+	0.999993i	$0.501203\pi$
$192$	0	0
$193$	−20.9740	−1.50974	−0.754870	−	0.655875i	$-0.772300\pi$
$193$	−20.9740	−1.50974	−0.754870	+	0.655875i	$0.772300\pi$
$194$	15.5463	1.11616
$195$	0	0
$196$	−4.32712	−0.309080
$197$	21.4751	1.53004	0.765018	−	0.644009i	$-0.222730\pi$
$197$	21.4751	1.53004	0.765018	+	0.644009i	$0.222730\pi$
$198$	0	0
$199$	6.38880	0.452890	0.226445	−	0.974024i	$-0.427290\pi$
$199$	6.38880	0.452890	0.226445	+	0.974024i	$0.427290\pi$
$200$	−3.68654	−0.260678
$201$	0	0
$202$	−7.92531	−0.557623
$203$	−2.28902	−0.160658
$204$	0	0
$205$	−6.05161	−0.422663
$206$	−4.34327	−0.302610
$207$	0	0
$208$	4.36908	0.302941
$209$	−2.20898	−0.152798
$210$	0	0
$211$	−0.302582	−0.0208306	−0.0104153	−	0.999946i	$-0.503315\pi$
$211$	−0.302582	−0.0208306	−0.0104153	+	0.999946i	$0.503315\pi$
$212$	5.16929	0.355028
$213$	0	0
$214$	−4.42033	−0.302167
$215$	−10.0385	−0.684620
$216$	0	0
$217$	−11.9862	−0.813677
$218$	18.5327	1.25519
$219$	0	0
$220$	2.53163	0.170683
$221$	−6.22992	−0.419070
$222$	0	0
$223$	21.6610	1.45053	0.725263	−	0.688472i	$-0.241718\pi$
$223$	21.6610	1.45053	0.725263	+	0.688472i	$0.241718\pi$
$224$	−1.63489	−0.109236
$225$	0	0
$226$	−10.3412	−0.687883
$227$	7.50858	0.498362	0.249181	−	0.968457i	$-0.419839\pi$
$227$	7.50858	0.498362	0.249181	+	0.968457i	$0.419839\pi$
$228$	0	0
$229$	−6.11527	−0.404109	−0.202054	−	0.979374i	$-0.564762\pi$
$229$	−6.11527	−0.404109	−0.202054	+	0.979374i	$0.564762\pi$
$230$	4.97069	0.327757
$231$	0	0
$232$	1.40010	0.0919213
$233$	−25.3810	−1.66277	−0.831383	−	0.555700i	$-0.812450\pi$
$233$	−25.3810	−1.66277	−0.831383	+	0.555700i	$0.812450\pi$
$234$	0	0
$235$	3.52137	0.229709
$236$	1.30616	0.0850236
$237$	0	0
$238$	2.33121	0.151110
$239$	6.17409	0.399368	0.199684	−	0.979860i	$-0.436008\pi$
$239$	6.17409	0.399368	0.199684	+	0.979860i	$0.436008\pi$
$240$	0	0
$241$	11.8326	0.762205	0.381102	−	0.924533i	$-0.375545\pi$
$241$	11.8326	0.762205	0.381102	+	0.924533i	$0.375545\pi$
$242$	−6.12040	−0.393434
$243$	0	0
$244$	−1.07137	−0.0685873
$245$	−4.95916	−0.316829
$246$	0	0
$247$	−4.36908	−0.277998
$248$	7.33149	0.465550
$249$	0	0
$250$	−9.95532	−0.629630
$251$	−29.5694	−1.86640	−0.933201	−	0.359354i	$-0.882997\pi$
$251$	−29.5694	−1.86640	−0.933201	+	0.359354i	$0.882997\pi$
$252$	0	0
$253$	9.58076	0.602337
$254$	8.15572	0.511736
$255$	0	0
$256$	1.00000	0.0625000
$257$	−28.2774	−1.76390	−0.881948	−	0.471347i	$-0.843768\pi$
$257$	−28.2774	−1.76390	−0.881948	+	0.471347i	$0.843768\pi$
$258$	0	0
$259$	−11.3564	−0.705653
$260$	5.00724	0.310536
$261$	0	0
$262$	5.73729	0.354451
$263$	−14.8951	−0.918473	−0.459236	−	0.888314i	$-0.651877\pi$
$263$	−14.8951	−0.918473	−0.459236	+	0.888314i	$0.651877\pi$
$264$	0	0
$265$	5.92433	0.363928
$266$	1.63489	0.100242
$267$	0	0
$268$	7.14158	0.436242
$269$	−28.5720	−1.74206	−0.871032	−	0.491227i	$-0.836549\pi$
$269$	−28.5720	−1.74206	−0.871032	+	0.491227i	$0.836549\pi$
$270$	0	0
$271$	−17.7114	−1.07589	−0.537947	−	0.842979i	$-0.680800\pi$
$271$	−17.7114	−1.07589	−0.537947	+	0.842979i	$0.680800\pi$
$272$	−1.42591	−0.0864586
$273$	0	0
$274$	−21.1576	−1.27818
$275$	−8.14350	−0.491072
$276$	0	0
$277$	16.7591	1.00696	0.503479	−	0.864007i	$-0.332053\pi$
$277$	16.7591	1.00696	0.503479	+	0.864007i	$0.332053\pi$
$278$	18.3618	1.10127
$279$	0	0
$280$	−1.87369	−0.111974
$281$	−4.05701	−0.242021	−0.121010	−	0.992651i	$-0.538613\pi$
$281$	−4.05701	−0.242021	−0.121010	+	0.992651i	$0.538613\pi$
$282$	0	0
$283$	19.3861	1.15238	0.576191	−	0.817315i	$-0.304538\pi$
$283$	19.3861	1.15238	0.576191	+	0.817315i	$0.304538\pi$
$284$	12.0806	0.716849
$285$	0	0
$286$	9.65121	0.570688
$287$	8.63281	0.509579
$288$	0	0
$289$	−14.9668	−0.880399
$290$	1.60461	0.0942258
$291$	0	0
$292$	6.78533	0.397081
$293$	−1.03103	−0.0602332	−0.0301166	−	0.999546i	$-0.509588\pi$
$293$	−1.03103	−0.0602332	−0.0301166	+	0.999546i	$0.509588\pi$
$294$	0	0
$295$	1.49694	0.0871552
$296$	6.94627	0.403744
$297$	0	0
$298$	20.3982	1.18163
$299$	18.9495	1.09588
$300$	0	0
$301$	14.3202	0.825404
$302$	−14.2233	−0.818459
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	−1.22786	−0.0703068
$306$	0	0
$307$	−8.79051	−0.501701	−0.250850	−	0.968026i	$-0.580710\pi$
$307$	−8.79051	−0.501701	−0.250850	+	0.968026i	$0.580710\pi$
$308$	−3.61145	−0.205781
$309$	0	0
$310$	8.40235	0.477222
$311$	−30.3994	−1.72379	−0.861895	−	0.507087i	$-0.830722\pi$
$311$	−30.3994	−1.72379	−0.861895	+	0.507087i	$0.830722\pi$
$312$	0	0
$313$	1.06306	0.0600878	0.0300439	−	0.999549i	$-0.490435\pi$
$313$	1.06306	0.0600878	0.0300439	+	0.999549i	$0.490435\pi$
$314$	−1.95891	−0.110548
$315$	0	0
$316$	10.8087	0.608034
$317$	−16.6096	−0.932887	−0.466444	−	0.884551i	$-0.654465\pi$
$317$	−16.6096	−0.932887	−0.466444	+	0.884551i	$0.654465\pi$
$318$	0	0
$319$	3.09280	0.173164
$320$	1.14606	0.0640669
$321$	0	0
$322$	−7.09083	−0.395157
$323$	1.42591	0.0793398
$324$	0	0
$325$	−16.1068	−0.893443
$326$	4.31174	0.238805
$327$	0	0
$328$	−5.28035	−0.291559
$329$	−5.02334	−0.276946
$330$	0	0
$331$	16.7309	0.919611	0.459805	−	0.888020i	$-0.347919\pi$
$331$	16.7309	0.919611	0.459805	+	0.888020i	$0.347919\pi$
$332$	−5.19721	−0.285234
$333$	0	0
$334$	4.95770	0.271273
$335$	8.18471	0.447178
$336$	0	0
$337$	−31.5837	−1.72047	−0.860237	−	0.509895i	$-0.829684\pi$
$337$	−31.5837	−1.72047	−0.860237	+	0.509895i	$0.829684\pi$
$338$	6.08883	0.331189
$339$	0	0
$340$	−1.63418	−0.0886261
$341$	16.1951	0.877016
$342$	0	0
$343$	18.5186	0.999913
$344$	−8.75912	−0.472260
$345$	0	0
$346$	−13.1012	−0.704324
$347$	34.5656	1.85558	0.927789	−	0.373105i	$-0.121707\pi$
$347$	34.5656	1.85558	0.927789	+	0.373105i	$0.121707\pi$
$348$	0	0
$349$	5.03290	0.269405	0.134703	−	0.990886i	$-0.456992\pi$
$349$	5.03290	0.269405	0.134703	+	0.990886i	$0.456992\pi$
$350$	6.02710	0.322162
$351$	0	0
$352$	2.20898	0.117739
$353$	−1.62921	−0.0867140	−0.0433570	−	0.999060i	$-0.513805\pi$
$353$	−1.62921	−0.0867140	−0.0433570	+	0.999060i	$0.513805\pi$
$354$	0	0
$355$	13.8451	0.734821
$356$	−7.62362	−0.404051
$357$	0	0
$358$	−12.2984	−0.649990
$359$	2.96052	0.156250	0.0781252	−	0.996944i	$-0.475107\pi$
$359$	2.96052	0.156250	0.0781252	+	0.996944i	$0.475107\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−1.39563	−0.0733524
$363$	0	0
$364$	−7.14298	−0.374394
$365$	7.77641	0.407036
$366$	0	0
$367$	−28.2325	−1.47372	−0.736862	−	0.676043i	$-0.763693\pi$
$367$	−28.2325	−1.47372	−0.736862	+	0.676043i	$0.763693\pi$
$368$	4.33718	0.226091
$369$	0	0
$370$	7.96086	0.413866
$371$	−8.45123	−0.438766
$372$	0	0
$373$	−9.86453	−0.510766	−0.255383	−	0.966840i	$-0.582202\pi$
$373$	−9.86453	−0.510766	−0.255383	+	0.966840i	$0.582202\pi$
$374$	−3.14981	−0.162873
$375$	0	0
$376$	3.07258	0.158456
$377$	6.11716	0.315050
$378$	0	0
$379$	−4.36079	−0.223999	−0.111999	−	0.993708i	$-0.535725\pi$
$379$	−4.36079	−0.223999	−0.111999	+	0.993708i	$0.535725\pi$
$380$	−1.14606	−0.0587918
$381$	0	0
$382$	−0.104472	−0.00534527
$383$	−33.3710	−1.70518	−0.852588	−	0.522583i	$-0.824968\pi$
$383$	−33.3710	−1.70518	−0.852588	+	0.522583i	$0.824968\pi$
$384$	0	0
$385$	−4.13895	−0.210940
$386$	−20.9740	−1.06755
$387$	0	0
$388$	15.5463	0.789246
$389$	−13.2354	−0.671062	−0.335531	−	0.942029i	$-0.608916\pi$
$389$	−13.2354	−0.671062	−0.335531	+	0.942029i	$0.608916\pi$
$390$	0	0
$391$	−6.18444	−0.312761
$392$	−4.32712	−0.218553
$393$	0	0
$394$	21.4751	1.08190
$395$	12.3874	0.623278
$396$	0	0
$397$	27.6615	1.38829	0.694145	−	0.719835i	$-0.255783\pi$
$397$	27.6615	1.38829	0.694145	+	0.719835i	$0.255783\pi$
$398$	6.38880	0.320242
$399$	0	0
$400$	−3.68654	−0.184327
$401$	7.81861	0.390443	0.195221	−	0.980759i	$-0.437457\pi$
$401$	7.81861	0.390443	0.195221	+	0.980759i	$0.437457\pi$
$402$	0	0
$403$	32.0319	1.59562
$404$	−7.92531	−0.394299
$405$	0	0
$406$	−2.28902	−0.113602
$407$	15.3442	0.760583
$408$	0	0
$409$	19.6064	0.969474	0.484737	−	0.874660i	$-0.338915\pi$
$409$	19.6064	0.969474	0.484737	+	0.874660i	$0.338915\pi$
$410$	−6.05161	−0.298868
$411$	0	0
$412$	−4.34327	−0.213978
$413$	−2.13543	−0.105078
$414$	0	0
$415$	−5.95633	−0.292385
$416$	4.36908	0.214212
$417$	0	0
$418$	−2.20898	−0.108045
$419$	13.1970	0.644718	0.322359	−	0.946618i	$-0.395524\pi$
$419$	13.1970	0.644718	0.322359	+	0.946618i	$0.395524\pi$
$420$	0	0
$421$	−30.3919	−1.48121	−0.740606	−	0.671939i	$-0.765462\pi$
$421$	−30.3919	−1.48121	−0.740606	+	0.671939i	$0.765462\pi$
$422$	−0.302582	−0.0147295
$423$	0	0
$424$	5.16929	0.251043
$425$	5.25668	0.254986
$426$	0	0
$427$	1.75157	0.0847646
$428$	−4.42033	−0.213665
$429$	0	0
$430$	−10.0385	−0.484100
$431$	−12.2675	−0.590903	−0.295452	−	0.955358i	$-0.595470\pi$
$431$	−12.2675	−0.590903	−0.295452	+	0.955358i	$0.595470\pi$
$432$	0	0
$433$	−3.30259	−0.158712	−0.0793561	−	0.996846i	$-0.525286\pi$
$433$	−3.30259	−0.158712	−0.0793561	+	0.996846i	$0.525286\pi$
$434$	−11.9862	−0.575357
$435$	0	0
$436$	18.5327	0.887554
$437$	−4.33718	−0.207476
$438$	0	0
$439$	20.9570	1.00022	0.500112	−	0.865961i	$-0.333292\pi$
$439$	20.9570	1.00022	0.500112	+	0.865961i	$0.333292\pi$
$440$	2.53163	0.120691
$441$	0	0
$442$	−6.22992	−0.296327
$443$	34.9623	1.66111	0.830554	−	0.556938i	$-0.188024\pi$
$443$	34.9623	1.66111	0.830554	+	0.556938i	$0.188024\pi$
$444$	0	0
$445$	−8.73715	−0.414181
$446$	21.6610	1.02568
$447$	0	0
$448$	−1.63489	−0.0772415
$449$	12.8844	0.608055	0.304027	−	0.952663i	$-0.401669\pi$
$449$	12.8844	0.608055	0.304027	+	0.952663i	$0.401669\pi$
$450$	0	0
$451$	−11.6642	−0.549246
$452$	−10.3412	−0.486407
$453$	0	0
$454$	7.50858	0.352395
$455$	−8.18630	−0.383780
$456$	0	0
$457$	−2.71683	−0.127088	−0.0635439	−	0.997979i	$-0.520240\pi$
$457$	−2.71683	−0.127088	−0.0635439	+	0.997979i	$0.520240\pi$
$458$	−6.11527	−0.285748
$459$	0	0
$460$	4.97069	0.231759
$461$	−3.66814	−0.170842	−0.0854211	−	0.996345i	$-0.527224\pi$
$461$	−3.66814	−0.170842	−0.0854211	+	0.996345i	$0.527224\pi$
$462$	0	0
$463$	12.2995	0.571606	0.285803	−	0.958288i	$-0.407740\pi$
$463$	12.2995	0.571606	0.285803	+	0.958288i	$0.407740\pi$
$464$	1.40010	0.0649982
$465$	0	0
$466$	−25.3810	−1.17575
$467$	17.4976	0.809693	0.404847	−	0.914385i	$-0.367325\pi$
$467$	17.4976	0.809693	0.404847	+	0.914385i	$0.367325\pi$
$468$	0	0
$469$	−11.6757	−0.539135
$470$	3.52137	0.162429
$471$	0	0
$472$	1.30616	0.0601208
$473$	−19.3487	−0.889656
$474$	0	0
$475$	3.68654	0.169150
$476$	2.33121	0.106851
$477$	0	0
$478$	6.17409	0.282396
$479$	−11.5196	−0.526345	−0.263172	−	0.964749i	$-0.584769\pi$
$479$	−11.5196	−0.526345	−0.263172	+	0.964749i	$0.584769\pi$
$480$	0	0
$481$	30.3488	1.38379
$482$	11.8326	0.538960
$483$	0	0
$484$	−6.12040	−0.278200
$485$	17.8171	0.809032
$486$	0	0
$487$	−31.3219	−1.41933	−0.709666	−	0.704538i	$-0.751154\pi$
$487$	−31.3219	−1.41933	−0.709666	+	0.704538i	$0.751154\pi$
$488$	−1.07137	−0.0484986
$489$	0	0
$490$	−4.95916	−0.224032
$491$	−36.2018	−1.63377	−0.816883	−	0.576803i	$-0.804300\pi$
$491$	−36.2018	−1.63377	−0.816883	+	0.576803i	$0.804300\pi$
$492$	0	0
$493$	−1.99642	−0.0899144
$494$	−4.36908	−0.196574
$495$	0	0
$496$	7.33149	0.329194
$497$	−19.7504	−0.885928
$498$	0	0
$499$	−13.8719	−0.620993	−0.310496	−	0.950575i	$-0.600495\pi$
$499$	−13.8719	−0.620993	−0.310496	+	0.950575i	$0.600495\pi$
$500$	−9.95532	−0.445216
$501$	0	0
$502$	−29.5694	−1.31975
$503$	−1.02490	−0.0456982	−0.0228491	−	0.999739i	$-0.507274\pi$
$503$	−1.02490	−0.0456982	−0.0228491	+	0.999739i	$0.507274\pi$
$504$	0	0
$505$	−9.08290	−0.404184
$506$	9.58076	0.425917
$507$	0	0
$508$	8.15572	0.361852
$509$	3.89439	0.172616	0.0863078	−	0.996269i	$-0.472493\pi$
$509$	3.89439	0.172616	0.0863078	+	0.996269i	$0.472493\pi$
$510$	0	0
$511$	−11.0933	−0.490738
$512$	1.00000	0.0441942
$513$	0	0
$514$	−28.2774	−1.24726
$515$	−4.97766	−0.219342
$516$	0	0
$517$	6.78727	0.298504
$518$	−11.3564	−0.498972
$519$	0	0
$520$	5.00724	0.219582
$521$	30.3308	1.32882	0.664408	−	0.747370i	$-0.268684\pi$
$521$	30.3308	1.32882	0.664408	+	0.747370i	$0.268684\pi$
$522$	0	0
$523$	13.5519	0.592582	0.296291	−	0.955098i	$-0.404250\pi$
$523$	13.5519	0.592582	0.296291	+	0.955098i	$0.404250\pi$
$524$	5.73729	0.250635
$525$	0	0
$526$	−14.8951	−0.649458
$527$	−10.4541	−0.455386
$528$	0	0
$529$	−4.18883	−0.182123
$530$	5.92433	0.257336
$531$	0	0
$532$	1.63489	0.0708816
$533$	−23.0703	−0.999284
$534$	0	0
$535$	−5.06598	−0.219021
$536$	7.14158	0.308469
$537$	0	0
$538$	−28.5720	−1.23183
$539$	−9.55854	−0.411715
$540$	0	0
$541$	6.78464	0.291694	0.145847	−	0.989307i	$-0.453409\pi$
$541$	6.78464	0.291694	0.145847	+	0.989307i	$0.453409\pi$
$542$	−17.7114	−0.760772
$543$	0	0
$544$	−1.42591	−0.0611354
$545$	21.2396	0.909805
$546$	0	0
$547$	−31.1020	−1.32983	−0.664913	−	0.746921i	$-0.731531\pi$
$547$	−31.1020	−1.32983	−0.664913	+	0.746921i	$0.731531\pi$
$548$	−21.1576	−0.903808
$549$	0	0
$550$	−8.14350	−0.347240
$551$	−1.40010	−0.0596464
$552$	0	0
$553$	−17.6710	−0.751447
$554$	16.7591	0.712027
$555$	0	0
$556$	18.3618	0.778713
$557$	23.6478	1.00199	0.500995	−	0.865450i	$-0.332967\pi$
$557$	23.6478	1.00199	0.500995	+	0.865450i	$0.332967\pi$
$558$	0	0
$559$	−38.2693	−1.61862
$560$	−1.87369	−0.0791779
$561$	0	0
$562$	−4.05701	−0.171135
$563$	−32.2346	−1.35853	−0.679263	−	0.733895i	$-0.737700\pi$
$563$	−32.2346	−1.35853	−0.679263	+	0.733895i	$0.737700\pi$
$564$	0	0
$565$	−11.8516	−0.498601
$566$	19.3861	0.814857
$567$	0	0
$568$	12.0806	0.506889
$569$	29.0233	1.21672	0.608361	−	0.793661i	$-0.291827\pi$
$569$	29.0233	1.21672	0.608361	+	0.793661i	$0.291827\pi$
$570$	0	0
$571$	−18.5735	−0.777277	−0.388638	−	0.921390i	$-0.627054\pi$
$571$	−18.5735	−0.777277	−0.388638	+	0.921390i	$0.627054\pi$
$572$	9.65121	0.403538
$573$	0	0
$574$	8.63281	0.360327
$575$	−15.9892	−0.666796
$576$	0	0
$577$	−9.63391	−0.401065	−0.200533	−	0.979687i	$-0.564267\pi$
$577$	−9.63391	−0.401065	−0.200533	+	0.979687i	$0.564267\pi$
$578$	−14.9668	−0.622536
$579$	0	0
$580$	1.60461	0.0666277
$581$	8.49688	0.352510
$582$	0	0
$583$	11.4189	0.472921
$584$	6.78533	0.280779
$585$	0	0
$586$	−1.03103	−0.0425913
$587$	−25.7526	−1.06292	−0.531461	−	0.847083i	$-0.678357\pi$
$587$	−25.7526	−1.06292	−0.531461	+	0.847083i	$0.678357\pi$
$588$	0	0
$589$	−7.33149	−0.302089
$590$	1.49694	0.0616280
$591$	0	0
$592$	6.94627	0.285490
$593$	−23.1785	−0.951825	−0.475913	−	0.879493i	$-0.657882\pi$
$593$	−23.1785	−0.951825	−0.475913	+	0.879493i	$0.657882\pi$
$594$	0	0
$595$	2.67172	0.109530
$596$	20.3982	0.835541
$597$	0	0
$598$	18.9495	0.774902
$599$	−44.6926	−1.82609	−0.913045	−	0.407858i	$-0.866276\pi$
$599$	−44.6926	−1.82609	−0.913045	+	0.407858i	$0.866276\pi$
$600$	0	0
$601$	−48.3600	−1.97265	−0.986323	−	0.164821i	$-0.947295\pi$
$601$	−48.3600	−1.97265	−0.986323	+	0.164821i	$0.947295\pi$
$602$	14.3202	0.583649
$603$	0	0
$604$	−14.2233	−0.578738
$605$	−7.01436	−0.285174
$606$	0	0
$607$	26.9651	1.09448	0.547240	−	0.836975i	$-0.315678\pi$
$607$	26.9651	1.09448	0.547240	+	0.836975i	$0.315678\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	−1.22786	−0.0497144
$611$	13.4243	0.543091
$612$	0	0
$613$	14.7376	0.595248	0.297624	−	0.954683i	$-0.403806\pi$
$613$	14.7376	0.595248	0.297624	+	0.954683i	$0.403806\pi$
$614$	−8.79051	−0.354756
$615$	0	0
$616$	−3.61145	−0.145509
$617$	15.1460	0.609753	0.304877	−	0.952392i	$-0.401385\pi$
$617$	15.1460	0.609753	0.304877	+	0.952392i	$0.401385\pi$
$618$	0	0
$619$	−1.46174	−0.0587523	−0.0293761	−	0.999568i	$-0.509352\pi$
$619$	−1.46174	−0.0587523	−0.0293761	+	0.999568i	$0.509352\pi$
$620$	8.40235	0.337447
$621$	0	0
$622$	−30.3994	−1.21890
$623$	12.4638	0.499352
$624$	0	0
$625$	7.02327	0.280931
$626$	1.06306	0.0424885
$627$	0	0
$628$	−1.95891	−0.0781691
$629$	−9.90477	−0.394929
$630$	0	0
$631$	−8.53126	−0.339624	−0.169812	−	0.985476i	$-0.554316\pi$
$631$	−8.53126	−0.339624	−0.169812	+	0.985476i	$0.554316\pi$
$632$	10.8087	0.429945
$633$	0	0
$634$	−16.6096	−0.659651
$635$	9.34697	0.370923
$636$	0	0
$637$	−18.9055	−0.749065
$638$	3.09280	0.122445
$639$	0	0
$640$	1.14606	0.0453021
$641$	−9.04072	−0.357087	−0.178543	−	0.983932i	$-0.557139\pi$
$641$	−9.04072	−0.357087	−0.178543	+	0.983932i	$0.557139\pi$
$642$	0	0
$643$	−18.7914	−0.741062	−0.370531	−	0.928820i	$-0.620824\pi$
$643$	−18.7914	−0.741062	−0.370531	+	0.928820i	$0.620824\pi$
$644$	−7.09083	−0.279418
$645$	0	0
$646$	1.42591	0.0561017
$647$	28.8633	1.13473	0.567367	−	0.823465i	$-0.307962\pi$
$647$	28.8633	1.13473	0.567367	+	0.823465i	$0.307962\pi$
$648$	0	0
$649$	2.88528	0.113257
$650$	−16.1068	−0.631760
$651$	0	0
$652$	4.31174	0.168861
$653$	−47.6560	−1.86492	−0.932461	−	0.361270i	$-0.882343\pi$
$653$	−47.6560	−1.86492	−0.932461	+	0.361270i	$0.882343\pi$
$654$	0	0
$655$	6.57529	0.256918
$656$	−5.28035	−0.206163
$657$	0	0
$658$	−5.02334	−0.195830
$659$	49.6770	1.93514	0.967571	−	0.252600i	$-0.0812857\pi$
$659$	49.6770	1.93514	0.967571	+	0.252600i	$0.0812857\pi$
$660$	0	0
$661$	−18.0027	−0.700223	−0.350111	−	0.936708i	$-0.613856\pi$
$661$	−18.0027	−0.700223	−0.350111	+	0.936708i	$0.613856\pi$
$662$	16.7309	0.650263
$663$	0	0
$664$	−5.19721	−0.201691
$665$	1.87369	0.0726586
$666$	0	0
$667$	6.07251	0.235129
$668$	4.95770	0.191819
$669$	0	0
$670$	8.18471	0.316203
$671$	−2.36663	−0.0913629
$672$	0	0
$673$	16.6713	0.642632	0.321316	−	0.946972i	$-0.395875\pi$
$673$	16.6713	0.642632	0.321316	+	0.946972i	$0.395875\pi$
$674$	−31.5837	−1.21656
$675$	0	0
$676$	6.08883	0.234186
$677$	−43.7841	−1.68276	−0.841380	−	0.540444i	$-0.818256\pi$
$677$	−43.7841	−1.68276	−0.841380	+	0.540444i	$0.818256\pi$
$678$	0	0
$679$	−25.4166	−0.975400
$680$	−1.63418	−0.0626681
$681$	0	0
$682$	16.1951	0.620144
$683$	33.3197	1.27494	0.637472	−	0.770474i	$-0.279980\pi$
$683$	33.3197	1.27494	0.637472	+	0.770474i	$0.279980\pi$
$684$	0	0
$685$	−24.2480	−0.926467
$686$	18.5186	0.707045
$687$	0	0
$688$	−8.75912	−0.333938
$689$	22.5850	0.860420
$690$	0	0
$691$	−50.1426	−1.90751	−0.953757	−	0.300578i	$-0.902820\pi$
$691$	−50.1426	−1.90751	−0.953757	+	0.300578i	$0.902820\pi$
$692$	−13.1012	−0.498032
$693$	0	0
$694$	34.5656	1.31209
$695$	21.0438	0.798235
$696$	0	0
$697$	7.52931	0.285193
$698$	5.03290	0.190498
$699$	0	0
$700$	6.02710	0.227803
$701$	−38.8954	−1.46906	−0.734529	−	0.678578i	$-0.762597\pi$
$701$	−38.8954	−1.46906	−0.734529	+	0.678578i	$0.762597\pi$
$702$	0	0
$703$	−6.94627	−0.261984
$704$	2.20898	0.0832541
$705$	0	0
$706$	−1.62921	−0.0613161
$707$	12.9570	0.487299
$708$	0	0
$709$	−40.5790	−1.52398	−0.761989	−	0.647590i	$-0.775777\pi$
$709$	−40.5790	−1.52398	−0.761989	+	0.647590i	$0.775777\pi$
$710$	13.8451	0.519597
$711$	0	0
$712$	−7.62362	−0.285707
$713$	31.7980	1.19085
$714$	0	0
$715$	11.0609	0.413654
$716$	−12.2984	−0.459612
$717$	0	0
$718$	2.96052	0.110486
$719$	31.4899	1.17437	0.587187	−	0.809451i	$-0.300235\pi$
$719$	31.4899	1.17437	0.587187	+	0.809451i	$0.300235\pi$
$720$	0	0
$721$	7.10078	0.264447
$722$	1.00000	0.0372161
$723$	0	0
$724$	−1.39563	−0.0518680
$725$	−5.16154	−0.191695
$726$	0	0
$727$	−2.62612	−0.0973974	−0.0486987	−	0.998814i	$-0.515507\pi$
$727$	−2.62612	−0.0973974	−0.0486987	+	0.998814i	$0.515507\pi$
$728$	−7.14298	−0.264736
$729$	0	0
$730$	7.77641	0.287818
$731$	12.4897	0.461949
$732$	0	0
$733$	−8.08087	−0.298474	−0.149237	−	0.988801i	$-0.547682\pi$
$733$	−8.08087	−0.298474	−0.149237	+	0.988801i	$0.547682\pi$
$734$	−28.2325	−1.04208
$735$	0	0
$736$	4.33718	0.159871
$737$	15.7756	0.581103
$738$	0	0
$739$	46.0684	1.69465	0.847326	−	0.531073i	$-0.178211\pi$
$739$	46.0684	1.69465	0.847326	+	0.531073i	$0.178211\pi$
$740$	7.96086	0.292647
$741$	0	0
$742$	−8.45123	−0.310254
$743$	36.1741	1.32710	0.663550	−	0.748132i	$-0.269049\pi$
$743$	36.1741	1.32710	0.663550	+	0.748132i	$0.269049\pi$
$744$	0	0
$745$	23.3776	0.856488
$746$	−9.86453	−0.361166
$747$	0	0
$748$	−3.14981	−0.115169
$749$	7.22677	0.264060
$750$	0	0
$751$	−22.6617	−0.826937	−0.413468	−	0.910518i	$-0.635683\pi$
$751$	−22.6617	−0.826937	−0.413468	+	0.910518i	$0.635683\pi$
$752$	3.07258	0.112045
$753$	0	0
$754$	6.11716	0.222774
$755$	−16.3008	−0.593247
$756$	0	0
$757$	6.45466	0.234599	0.117299	−	0.993097i	$-0.462576\pi$
$757$	6.45466	0.234599	0.117299	+	0.993097i	$0.462576\pi$
$758$	−4.36079	−0.158391
$759$	0	0
$760$	−1.14606	−0.0415721
$761$	7.46636	0.270655	0.135328	−	0.990801i	$-0.456791\pi$
$761$	7.46636	0.270655	0.135328	+	0.990801i	$0.456791\pi$
$762$	0	0
$763$	−30.2989	−1.09690
$764$	−0.104472	−0.00377968
$765$	0	0
$766$	−33.3710	−1.20574
$767$	5.70670	0.206057
$768$	0	0
$769$	−24.9548	−0.899894	−0.449947	−	0.893055i	$-0.648557\pi$
$769$	−24.9548	−0.899894	−0.449947	+	0.893055i	$0.648557\pi$
$770$	−4.13895	−0.149157
$771$	0	0
$772$	−20.9740	−0.754870
$773$	−52.8951	−1.90250	−0.951252	−	0.308414i	$-0.900202\pi$
$773$	−52.8951	−1.90250	−0.951252	+	0.308414i	$0.900202\pi$
$774$	0	0
$775$	−27.0278	−0.970869
$776$	15.5463	0.558081
$777$	0	0
$778$	−13.2354	−0.474513
$779$	5.28035	0.189188
$780$	0	0
$781$	26.6857	0.954891
$782$	−6.18444	−0.221155
$783$	0	0
$784$	−4.32712	−0.154540
$785$	−2.24504	−0.0801288
$786$	0	0
$787$	36.7031	1.30832	0.654161	−	0.756355i	$-0.273022\pi$
$787$	36.7031	1.30832	0.654161	+	0.756355i	$0.273022\pi$
$788$	21.4751	0.765018
$789$	0	0
$790$	12.3874	0.440724
$791$	16.9067	0.601133
$792$	0	0
$793$	−4.68089	−0.166223
$794$	27.6615	0.981669
$795$	0	0
$796$	6.38880	0.226445
$797$	−13.1782	−0.466797	−0.233398	−	0.972381i	$-0.574985\pi$
$797$	−13.1782	−0.466797	−0.233398	+	0.972381i	$0.574985\pi$
$798$	0	0
$799$	−4.38123	−0.154997
$800$	−3.68654	−0.130339
$801$	0	0
$802$	7.81861	0.276085
$803$	14.9887	0.528939
$804$	0	0
$805$	−8.12654	−0.286423
$806$	32.0319	1.12827
$807$	0	0
$808$	−7.92531	−0.278811
$809$	−53.7680	−1.89038	−0.945191	−	0.326519i	$-0.894124\pi$
$809$	−53.7680	−1.89038	−0.945191	+	0.326519i	$0.894124\pi$
$810$	0	0
$811$	−26.3479	−0.925200	−0.462600	−	0.886567i	$-0.653083\pi$
$811$	−26.3479	−0.925200	−0.462600	+	0.886567i	$0.653083\pi$
$812$	−2.28902	−0.0803289
$813$	0	0
$814$	15.3442	0.537814
$815$	4.94153	0.173094
$816$	0	0
$817$	8.75912	0.306443
$818$	19.6064	0.685522
$819$	0	0
$820$	−6.05161	−0.211332
$821$	32.1171	1.12089	0.560447	−	0.828191i	$-0.310630\pi$
$821$	32.1171	1.12089	0.560447	+	0.828191i	$0.310630\pi$
$822$	0	0
$823$	−51.6597	−1.80074	−0.900372	−	0.435121i	$-0.856706\pi$
$823$	−51.6597	−1.80074	−0.900372	+	0.435121i	$0.856706\pi$
$824$	−4.34327	−0.151305
$825$	0	0
$826$	−2.13543	−0.0743011
$827$	10.0857	0.350713	0.175357	−	0.984505i	$-0.443892\pi$
$827$	10.0857	0.350713	0.175357	+	0.984505i	$0.443892\pi$
$828$	0	0
$829$	15.4641	0.537092	0.268546	−	0.963267i	$-0.413457\pi$
$829$	15.4641	0.537092	0.268546	+	0.963267i	$0.413457\pi$
$830$	−5.95633	−0.206747
$831$	0	0
$832$	4.36908	0.151470
$833$	6.17009	0.213781
$834$	0	0
$835$	5.68184	0.196628
$836$	−2.20898	−0.0763992
$837$	0	0
$838$	13.1970	0.455884
$839$	−24.3670	−0.841242	−0.420621	−	0.907237i	$-0.638188\pi$
$839$	−24.3670	−0.841242	−0.420621	+	0.907237i	$0.638188\pi$
$840$	0	0
$841$	−27.0397	−0.932404
$842$	−30.3919	−1.04738
$843$	0	0
$844$	−0.302582	−0.0104153
$845$	6.97819	0.240057
$846$	0	0
$847$	10.0062	0.343817
$848$	5.16929	0.177514
$849$	0	0
$850$	5.25668	0.180303
$851$	30.1273	1.03275
$852$	0	0
$853$	27.1817	0.930682	0.465341	−	0.885131i	$-0.345932\pi$
$853$	27.1817	0.930682	0.465341	+	0.885131i	$0.345932\pi$
$854$	1.75157	0.0599376
$855$	0	0
$856$	−4.42033	−0.151084
$857$	−43.8568	−1.49812	−0.749060	−	0.662502i	$-0.769495\pi$
$857$	−43.8568	−1.49812	−0.749060	+	0.662502i	$0.769495\pi$
$858$	0	0
$859$	−53.1062	−1.81196	−0.905979	−	0.423322i	$-0.860864\pi$
$859$	−53.1062	−1.81196	−0.905979	+	0.423322i	$0.860864\pi$
$860$	−10.0385	−0.342310
$861$	0	0
$862$	−12.2675	−0.417832
$863$	9.08908	0.309396	0.154698	−	0.987962i	$-0.450560\pi$
$863$	9.08908	0.309396	0.154698	+	0.987962i	$0.450560\pi$
$864$	0	0
$865$	−15.0148	−0.510518
$866$	−3.30259	−0.112227
$867$	0	0
$868$	−11.9862	−0.406839
$869$	23.8761	0.809942
$870$	0	0
$871$	31.2021	1.05724
$872$	18.5327	0.627595
$873$	0	0
$874$	−4.33718	−0.146707
$875$	16.2759	0.550226
$876$	0	0
$877$	9.27182	0.313087	0.156544	−	0.987671i	$-0.449965\pi$
$877$	9.27182	0.313087	0.156544	+	0.987671i	$0.449965\pi$
$878$	20.9570	0.707265
$879$	0	0
$880$	2.53163	0.0853413
$881$	0.681963	0.0229759	0.0114880	−	0.999934i	$-0.496343\pi$
$881$	0.681963	0.0229759	0.0114880	+	0.999934i	$0.496343\pi$
$882$	0	0
$883$	−1.29612	−0.0436179	−0.0218090	−	0.999762i	$-0.506943\pi$
$883$	−1.29612	−0.0436179	−0.0218090	+	0.999762i	$0.506943\pi$
$884$	−6.22992	−0.209535
$885$	0	0
$886$	34.9623	1.17458
$887$	21.4573	0.720467	0.360234	−	0.932862i	$-0.382697\pi$
$887$	21.4573	0.720467	0.360234	+	0.932862i	$0.382697\pi$
$888$	0	0
$889$	−13.3337	−0.447199
$890$	−8.73715	−0.292870
$891$	0	0
$892$	21.6610	0.725263
$893$	−3.07258	−0.102820
$894$	0	0
$895$	−14.0947	−0.471135
$896$	−1.63489	−0.0546180
$897$	0	0
$898$	12.8844	0.429960
$899$	10.2649	0.342352
$900$	0	0
$901$	−7.37094	−0.245562
$902$	−11.6642	−0.388375
$903$	0	0
$904$	−10.3412	−0.343942
$905$	−1.59947	−0.0531683
$906$	0	0
$907$	−3.96221	−0.131563	−0.0657814	−	0.997834i	$-0.520954\pi$
$907$	−3.96221	−0.131563	−0.0657814	+	0.997834i	$0.520954\pi$
$908$	7.50858	0.249181
$909$	0	0
$910$	−8.18630	−0.271373
$911$	−17.1652	−0.568709	−0.284355	−	0.958719i	$-0.591779\pi$
$911$	−17.1652	−0.568709	−0.284355	+	0.958719i	$0.591779\pi$
$912$	0	0
$913$	−11.4805	−0.379950
$914$	−2.71683	−0.0898646
$915$	0	0
$916$	−6.11527	−0.202054
$917$	−9.37986	−0.309750
$918$	0	0
$919$	−1.23791	−0.0408350	−0.0204175	−	0.999792i	$-0.506500\pi$
$919$	−1.23791	−0.0408350	−0.0204175	+	0.999792i	$0.506500\pi$
$920$	4.97069	0.163879
$921$	0	0
$922$	−3.66814	−0.120804
$923$	52.7809	1.73730
$924$	0	0
$925$	−25.6077	−0.841976
$926$	12.2995	0.404186
$927$	0	0
$928$	1.40010	0.0459607
$929$	36.8970	1.21055	0.605276	−	0.796015i	$-0.293063\pi$
$929$	36.8970	1.21055	0.605276	+	0.796015i	$0.293063\pi$
$930$	0	0
$931$	4.32712	0.141816
$932$	−25.3810	−0.831383
$933$	0	0
$934$	17.4976	0.572540
$935$	−3.60988	−0.118056
$936$	0	0
$937$	−29.7583	−0.972162	−0.486081	−	0.873914i	$-0.661574\pi$
$937$	−29.7583	−0.972162	−0.486081	+	0.873914i	$0.661574\pi$
$938$	−11.6757	−0.381226
$939$	0	0
$940$	3.52137	0.114854
$941$	28.8650	0.940972	0.470486	−	0.882408i	$-0.344079\pi$
$941$	28.8650	0.940972	0.470486	+	0.882408i	$0.344079\pi$
$942$	0	0
$943$	−22.9019	−0.745787
$944$	1.30616	0.0425118
$945$	0	0
$946$	−19.3487	−0.629082
$947$	36.0247	1.17065	0.585323	−	0.810800i	$-0.300968\pi$
$947$	36.0247	1.17065	0.585323	+	0.810800i	$0.300968\pi$
$948$	0	0
$949$	29.6456	0.962337
$950$	3.68654	0.119607
$951$	0	0
$952$	2.33121	0.0755550
$953$	−3.29871	−0.106856	−0.0534278	−	0.998572i	$-0.517015\pi$
$953$	−3.29871	−0.106856	−0.0534278	+	0.998572i	$0.517015\pi$
$954$	0	0
$955$	−0.119732	−0.00387443
$956$	6.17409	0.199684
$957$	0	0
$958$	−11.5196	−0.372182
$959$	34.5904	1.11698
$960$	0	0
$961$	22.7508	0.733897
$962$	30.3488	0.978484
$963$	0	0
$964$	11.8326	0.381102
$965$	−24.0375	−0.773794
$966$	0	0
$967$	51.7009	1.66259	0.831294	−	0.555833i	$-0.187601\pi$
$967$	51.7009	1.66259	0.831294	+	0.555833i	$0.187601\pi$
$968$	−6.12040	−0.196717
$969$	0	0
$970$	17.8171	0.572072
$971$	−4.38750	−0.140801	−0.0704007	−	0.997519i	$-0.522428\pi$
$971$	−4.38750	−0.140801	−0.0704007	+	0.997519i	$0.522428\pi$
$972$	0	0
$973$	−30.0195	−0.962382
$974$	−31.3219	−1.00362
$975$	0	0
$976$	−1.07137	−0.0342937
$977$	49.3549	1.57900	0.789502	−	0.613749i	$-0.210339\pi$
$977$	49.3549	1.57900	0.789502	+	0.613749i	$0.210339\pi$
$978$	0	0
$979$	−16.8404	−0.538223
$980$	−4.95916	−0.158414
$981$	0	0
$982$	−36.2018	−1.15525
$983$	−52.4546	−1.67304	−0.836520	−	0.547936i	$-0.815414\pi$
$983$	−52.4546	−1.67304	−0.836520	+	0.547936i	$0.815414\pi$
$984$	0	0
$985$	24.6118	0.784197
$986$	−1.99642	−0.0635791
$987$	0	0
$988$	−4.36908	−0.138999
$989$	−37.9899	−1.20801
$990$	0	0
$991$	−16.9358	−0.537985	−0.268992	−	0.963142i	$-0.586691\pi$
$991$	−16.9358	−0.537985	−0.268992	+	0.963142i	$0.586691\pi$
$992$	7.33149	0.232775
$993$	0	0
$994$	−19.7504	−0.626445
$995$	7.32197	0.232122
$996$	0	0
$997$	59.9848	1.89974	0.949868	−	0.312650i	$-0.101217\pi$
$997$	59.9848	1.89974	0.949868	+	0.312650i	$0.101217\pi$
$998$	−13.8719	−0.439108
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3078.2.a.x.1.4		6
3.2	odd	2		3078.2.a.v.1.3		6
9.2	odd	6		1026.2.e.d.685.4		12
9.4	even	3		342.2.e.c.115.3	✓	12
9.5	odd	6		1026.2.e.d.343.4		12
9.7	even	3		342.2.e.c.229.3	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
342.2.e.c.115.3	✓	12	9.4	even	3
342.2.e.c.229.3	yes	12	9.7	even	3
1026.2.e.d.343.4		12	9.5	odd	6
1026.2.e.d.685.4		12	9.2	odd	6
3078.2.a.v.1.3		6	3.2	odd	2
3078.2.a.x.1.4		6	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:	\( N \)	\(=\)	\( 3078 = 2 \cdot 3^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3078.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.14606 q^{5} -1.63489 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.14606 q^{5} -1.63489 q^{7} +1.00000 q^{8} +1.14606 q^{10} +2.20898 q^{11} +4.36908 q^{13} -1.63489 q^{14} +1.00000 q^{16} -1.42591 q^{17} -1.00000 q^{19} +1.14606 q^{20} +2.20898 q^{22} +4.33718 q^{23} -3.68654 q^{25} +4.36908 q^{26} -1.63489 q^{28} +1.40010 q^{29} +7.33149 q^{31} +1.00000 q^{32} -1.42591 q^{34} -1.87369 q^{35} +6.94627 q^{37} -1.00000 q^{38} +1.14606 q^{40} -5.28035 q^{41} -8.75912 q^{43} +2.20898 q^{44} +4.33718 q^{46} +3.07258 q^{47} -4.32712 q^{49} -3.68654 q^{50} +4.36908 q^{52} +5.16929 q^{53} +2.53163 q^{55} -1.63489 q^{56} +1.40010 q^{58} +1.30616 q^{59} -1.07137 q^{61} +7.33149 q^{62} +1.00000 q^{64} +5.00724 q^{65} +7.14158 q^{67} -1.42591 q^{68} -1.87369 q^{70} +12.0806 q^{71} +6.78533 q^{73} +6.94627 q^{74} -1.00000 q^{76} -3.61145 q^{77} +10.8087 q^{79} +1.14606 q^{80} -5.28035 q^{82} -5.19721 q^{83} -1.63418 q^{85} -8.75912 q^{86} +2.20898 q^{88} -7.62362 q^{89} -7.14298 q^{91} +4.33718 q^{92} +3.07258 q^{94} -1.14606 q^{95} +15.5463 q^{97} -4.32712 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} + 6 q^{4} + 6 q^{7} + 6 q^{8}+O(q^{10}) \) 6 * q + 6 * q^2 + 6 * q^4 + 6 * q^7 + 6 * q^8	\( 6 q + 6 q^{2} + 6 q^{4} + 6 q^{7} + 6 q^{8} + 6 q^{11} + 6 q^{13} + 6 q^{14} + 6 q^{16} - 6 q^{19} + 6 q^{22} + 12 q^{23} + 30 q^{25} + 6 q^{26} + 6 q^{28} - 6 q^{29} + 6 q^{31} + 6 q^{32} + 12 q^{35} + 6 q^{37} - 6 q^{38} - 6 q^{41} + 12 q^{43} + 6 q^{44} + 12 q^{46} + 6 q^{47} + 18 q^{49} + 30 q^{50} + 6 q^{52} - 18 q^{53} + 24 q^{55} + 6 q^{56} - 6 q^{58} - 12 q^{59} + 12 q^{61} + 6 q^{62} + 6 q^{64} + 6 q^{65} + 18 q^{67} + 12 q^{70} + 18 q^{71} + 24 q^{73} + 6 q^{74} - 6 q^{76} + 6 q^{77} + 18 q^{79} - 6 q^{82} - 12 q^{83} + 36 q^{85} + 12 q^{86} + 6 q^{88} + 6 q^{89} - 12 q^{91} + 12 q^{92} + 6 q^{94} - 18 q^{97} + 18 q^{98}+O(q^{100}) \) 6 * q + 6 * q^2 + 6 * q^4 + 6 * q^7 + 6 * q^8 + 6 * q^11 + 6 * q^13 + 6 * q^14 + 6 * q^16 - 6 * q^19 + 6 * q^22 + 12 * q^23 + 30 * q^25 + 6 * q^26 + 6 * q^28 - 6 * q^29 + 6 * q^31 + 6 * q^32 + 12 * q^35 + 6 * q^37 - 6 * q^38 - 6 * q^41 + 12 * q^43 + 6 * q^44 + 12 * q^46 + 6 * q^47 + 18 * q^49 + 30 * q^50 + 6 * q^52 - 18 * q^53 + 24 * q^55 + 6 * q^56 - 6 * q^58 - 12 * q^59 + 12 * q^61 + 6 * q^62 + 6 * q^64 + 6 * q^65 + 18 * q^67 + 12 * q^70 + 18 * q^71 + 24 * q^73 + 6 * q^74 - 6 * q^76 + 6 * q^77 + 18 * q^79 - 6 * q^82 - 12 * q^83 + 36 * q^85 + 12 * q^86 + 6 * q^88 + 6 * q^89 - 12 * q^91 + 12 * q^92 + 6 * q^94 - 18 * q^97 + 18 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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