LMFDB - Embedded newform 8046.2.a.i.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8046,2,Mod(1,8046)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8046, 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("8046.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8046 = 2 \cdot 3^{3} \cdot 149 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8046.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:	$64.2476334663$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5 x^{11} - 23 x^{10} + 142 x^{9} + 104 x^{8} - 1302 x^{7} + 607 x^{6} + 4323 x^{5} - 4461 x^{4} + \cdots - 553 $ x^12 - 5x^11 - 23x^10 + 142x^9 + 104x^8 - 1302x^7 + 607x^6 + 4323x^5 - 4461x^4 - 3333x^3 + 4805x^2 - 224*x - 553
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			1.6
Root			$1.20508$ of defining polynomial
Character	$\chi$	$=$	8046.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.20508 q^{5} +4.22185 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -1.20508 q^{5} +4.22185 q^{7} -1.00000 q^{8} +1.20508 q^{10} -1.67831 q^{11} -3.39389 q^{13} -4.22185 q^{14} +1.00000 q^{16} -6.30450 q^{17} +0.549921 q^{19} -1.20508 q^{20} +1.67831 q^{22} +1.62182 q^{23} -3.54779 q^{25} +3.39389 q^{26} +4.22185 q^{28} +5.50042 q^{29} +5.74587 q^{31} -1.00000 q^{32} +6.30450 q^{34} -5.08766 q^{35} +6.81324 q^{37} -0.549921 q^{38} +1.20508 q^{40} +1.19094 q^{41} -3.93505 q^{43} -1.67831 q^{44} -1.62182 q^{46} -1.55685 q^{47} +10.8240 q^{49} +3.54779 q^{50} -3.39389 q^{52} -5.73700 q^{53} +2.02250 q^{55} -4.22185 q^{56} -5.50042 q^{58} -1.95825 q^{59} +0.935725 q^{61} -5.74587 q^{62} +1.00000 q^{64} +4.08991 q^{65} +9.79140 q^{67} -6.30450 q^{68} +5.08766 q^{70} +2.12729 q^{71} +7.02740 q^{73} -6.81324 q^{74} +0.549921 q^{76} -7.08559 q^{77} -12.4006 q^{79} -1.20508 q^{80} -1.19094 q^{82} -1.69109 q^{83} +7.59742 q^{85} +3.93505 q^{86} +1.67831 q^{88} +1.92164 q^{89} -14.3285 q^{91} +1.62182 q^{92} +1.55685 q^{94} -0.662698 q^{95} -0.826873 q^{97} -10.8240 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{2} + 12 q^{4} - 5 q^{5} + 6 q^{7} - 12 q^{8}+O(q^{10}) $ 12 * q - 12 * q^2 + 12 * q^4 - 5 * q^5 + 6 * q^7 - 12 * q^8	$ 12 q - 12 q^{2} + 12 q^{4} - 5 q^{5} + 6 q^{7} - 12 q^{8} + 5 q^{10} - 6 q^{11} + 3 q^{13} - 6 q^{14} + 12 q^{16} - 6 q^{17} + 8 q^{19} - 5 q^{20} + 6 q^{22} - 11 q^{23} + 11 q^{25} - 3 q^{26} + 6 q^{28} - 29 q^{29} + 2 q^{31} - 12 q^{32} + 6 q^{34} - 4 q^{35} + 5 q^{37} - 8 q^{38} + 5 q^{40} - 22 q^{41} + 9 q^{43} - 6 q^{44} + 11 q^{46} - 15 q^{47} + 14 q^{49} - 11 q^{50} + 3 q^{52} - 12 q^{53} + 13 q^{55} - 6 q^{56} + 29 q^{58} - 34 q^{59} - 4 q^{61} - 2 q^{62} + 12 q^{64} - 12 q^{65} + q^{67} - 6 q^{68} + 4 q^{70} - 21 q^{71} - 2 q^{73} - 5 q^{74} + 8 q^{76} - 34 q^{77} + 9 q^{79} - 5 q^{80} + 22 q^{82} - 10 q^{83} + 5 q^{85} - 9 q^{86} + 6 q^{88} + 2 q^{89} + 17 q^{91} - 11 q^{92} + 15 q^{94} - 69 q^{95} - 13 q^{97} - 14 q^{98}+O(q^{100}) $ 12 * q - 12 * q^2 + 12 * q^4 - 5 * q^5 + 6 * q^7 - 12 * q^8 + 5 * q^10 - 6 * q^11 + 3 * q^13 - 6 * q^14 + 12 * q^16 - 6 * q^17 + 8 * q^19 - 5 * q^20 + 6 * q^22 - 11 * q^23 + 11 * q^25 - 3 * q^26 + 6 * q^28 - 29 * q^29 + 2 * q^31 - 12 * q^32 + 6 * q^34 - 4 * q^35 + 5 * q^37 - 8 * q^38 + 5 * q^40 - 22 * q^41 + 9 * q^43 - 6 * q^44 + 11 * q^46 - 15 * q^47 + 14 * q^49 - 11 * q^50 + 3 * q^52 - 12 * q^53 + 13 * q^55 - 6 * q^56 + 29 * q^58 - 34 * q^59 - 4 * q^61 - 2 * q^62 + 12 * q^64 - 12 * q^65 + q^67 - 6 * q^68 + 4 * q^70 - 21 * q^71 - 2 * q^73 - 5 * q^74 + 8 * q^76 - 34 * q^77 + 9 * q^79 - 5 * q^80 + 22 * q^82 - 10 * q^83 + 5 * q^85 - 9 * q^86 + 6 * q^88 + 2 * q^89 + 17 * q^91 - 11 * q^92 + 15 * q^94 - 69 * q^95 - 13 * q^97 - 14 * 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
$5$	−1.20508	−0.538927	−0.269464	−	0.963011i	$-0.586846\pi$
$5$	−1.20508	−0.538927	−0.269464	+	0.963011i	$0.586846\pi$
$6$	0	0
$7$	4.22185	1.59571	0.797855	−	0.602849i	$-0.205968\pi$
$7$	4.22185	1.59571	0.797855	+	0.602849i	$0.205968\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.20508	0.381079
$11$	−1.67831	−0.506030	−0.253015	−	0.967462i	$-0.581422\pi$
$11$	−1.67831	−0.506030	−0.253015	+	0.967462i	$0.581422\pi$
$12$	0	0
$13$	−3.39389	−0.941297	−0.470648	−	0.882321i	$-0.655980\pi$
$13$	−3.39389	−0.941297	−0.470648	+	0.882321i	$0.655980\pi$
$14$	−4.22185	−1.12834
$15$	0	0
$16$	1.00000	0.250000
$17$	−6.30450	−1.52907	−0.764533	−	0.644584i	$-0.777031\pi$
$17$	−6.30450	−1.52907	−0.764533	+	0.644584i	$0.777031\pi$
$18$	0	0
$19$	0.549921	0.126161	0.0630803	−	0.998008i	$-0.479908\pi$
$19$	0.549921	0.126161	0.0630803	+	0.998008i	$0.479908\pi$
$20$	−1.20508	−0.269464
$21$	0	0
$22$	1.67831	0.357817
$23$	1.62182	0.338173	0.169087	−	0.985601i	$-0.445918\pi$
$23$	1.62182	0.338173	0.169087	+	0.985601i	$0.445918\pi$
$24$	0	0
$25$	−3.54779	−0.709557
$26$	3.39389	0.665597
$27$	0	0
$28$	4.22185	0.797855
$29$	5.50042	1.02140	0.510701	−	0.859758i	$-0.329386\pi$
$29$	5.50042	1.02140	0.510701	+	0.859758i	$0.329386\pi$
$30$	0	0
$31$	5.74587	1.03199	0.515995	−	0.856592i	$-0.327422\pi$
$31$	5.74587	1.03199	0.515995	+	0.856592i	$0.327422\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	6.30450	1.08121
$35$	−5.08766	−0.859972
$36$	0	0
$37$	6.81324	1.12009	0.560045	−	0.828462i	$-0.310784\pi$
$37$	6.81324	1.12009	0.560045	+	0.828462i	$0.310784\pi$
$38$	−0.549921	−0.0892089
$39$	0	0
$40$	1.20508	0.190540
$41$	1.19094	0.185993	0.0929967	−	0.995666i	$-0.470355\pi$
$41$	1.19094	0.185993	0.0929967	+	0.995666i	$0.470355\pi$
$42$	0	0
$43$	−3.93505	−0.600090	−0.300045	−	0.953925i	$-0.597002\pi$
$43$	−3.93505	−0.600090	−0.300045	+	0.953925i	$0.597002\pi$
$44$	−1.67831	−0.253015
$45$	0	0
$46$	−1.62182	−0.239125
$47$	−1.55685	−0.227090	−0.113545	−	0.993533i	$-0.536221\pi$
$47$	−1.55685	−0.227090	−0.113545	+	0.993533i	$0.536221\pi$
$48$	0	0
$49$	10.8240	1.54629
$50$	3.54779	0.501733
$51$	0	0
$52$	−3.39389	−0.470648
$53$	−5.73700	−0.788038	−0.394019	−	0.919102i	$-0.628916\pi$
$53$	−5.73700	−0.788038	−0.394019	+	0.919102i	$0.628916\pi$
$54$	0	0
$55$	2.02250	0.272714
$56$	−4.22185	−0.564169
$57$	0	0
$58$	−5.50042	−0.722241
$59$	−1.95825	−0.254942	−0.127471	−	0.991842i	$-0.540686\pi$
$59$	−1.95825	−0.254942	−0.127471	+	0.991842i	$0.540686\pi$
$60$	0	0
$61$	0.935725	0.119807	0.0599036	−	0.998204i	$-0.480921\pi$
$61$	0.935725	0.119807	0.0599036	+	0.998204i	$0.480921\pi$
$62$	−5.74587	−0.729727
$63$	0	0
$64$	1.00000	0.125000
$65$	4.08991	0.507291
$66$	0	0
$67$	9.79140	1.19621	0.598105	−	0.801418i	$-0.295921\pi$
$67$	9.79140	1.19621	0.598105	+	0.801418i	$0.295921\pi$
$68$	−6.30450	−0.764533
$69$	0	0
$70$	5.08766	0.608092
$71$	2.12729	0.252463	0.126231	−	0.992001i	$-0.459712\pi$
$71$	2.12729	0.252463	0.126231	+	0.992001i	$0.459712\pi$
$72$	0	0
$73$	7.02740	0.822495	0.411247	−	0.911524i	$-0.365093\pi$
$73$	7.02740	0.822495	0.411247	+	0.911524i	$0.365093\pi$
$74$	−6.81324	−0.792023
$75$	0	0
$76$	0.549921	0.0630803
$77$	−7.08559	−0.807477
$78$	0	0
$79$	−12.4006	−1.39517	−0.697586	−	0.716501i	$-0.745743\pi$
$79$	−12.4006	−1.39517	−0.697586	+	0.716501i	$0.745743\pi$
$80$	−1.20508	−0.134732
$81$	0	0
$82$	−1.19094	−0.131517
$83$	−1.69109	−0.185622	−0.0928108	−	0.995684i	$-0.529585\pi$
$83$	−1.69109	−0.185622	−0.0928108	+	0.995684i	$0.529585\pi$
$84$	0	0
$85$	7.59742	0.824056
$86$	3.93505	0.424328
$87$	0	0
$88$	1.67831	0.178909
$89$	1.92164	0.203693	0.101846	−	0.994800i	$-0.467525\pi$
$89$	1.92164	0.203693	0.101846	+	0.994800i	$0.467525\pi$
$90$	0	0
$91$	−14.3285	−1.50204
$92$	1.62182	0.169087
$93$	0	0
$94$	1.55685	0.160577
$95$	−0.662698	−0.0679914
$96$	0	0
$97$	−0.826873	−0.0839562	−0.0419781	−	0.999119i	$-0.513366\pi$
$97$	−0.826873	−0.0839562	−0.0419781	+	0.999119i	$0.513366\pi$
$98$	−10.8240	−1.09339
$99$	0	0
$100$	−3.54779	−0.354779
$101$	−12.5987	−1.25362	−0.626810	−	0.779172i	$-0.715640\pi$
$101$	−12.5987	−1.25362	−0.626810	+	0.779172i	$0.715640\pi$
$102$	0	0
$103$	12.9030	1.27137	0.635685	−	0.771949i	$-0.280718\pi$
$103$	12.9030	1.27137	0.635685	+	0.771949i	$0.280718\pi$
$104$	3.39389	0.332799
$105$	0	0
$106$	5.73700	0.557227
$107$	3.39084	0.327805	0.163903	−	0.986477i	$-0.447592\pi$
$107$	3.39084	0.327805	0.163903	+	0.986477i	$0.447592\pi$
$108$	0	0
$109$	−16.8130	−1.61040	−0.805199	−	0.593005i	$-0.797942\pi$
$109$	−16.8130	−1.61040	−0.805199	+	0.593005i	$0.797942\pi$
$110$	−2.02250	−0.192838
$111$	0	0
$112$	4.22185	0.398927
$113$	−13.1890	−1.24072	−0.620360	−	0.784318i	$-0.713013\pi$
$113$	−13.1890	−1.24072	−0.620360	+	0.784318i	$0.713013\pi$
$114$	0	0
$115$	−1.95442	−0.182251
$116$	5.50042	0.510701
$117$	0	0
$118$	1.95825	0.180271
$119$	−26.6167	−2.43995
$120$	0	0
$121$	−8.18327	−0.743933
$122$	−0.935725	−0.0847165
$123$	0	0
$124$	5.74587	0.515995
$125$	10.3008	0.921327
$126$	0	0
$127$	−14.3011	−1.26901	−0.634507	−	0.772917i	$-0.718797\pi$
$127$	−14.3011	−1.26901	−0.634507	+	0.772917i	$0.718797\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−4.08991	−0.358709
$131$	−6.41294	−0.560302	−0.280151	−	0.959956i	$-0.590384\pi$
$131$	−6.41294	−0.560302	−0.280151	+	0.959956i	$0.590384\pi$
$132$	0	0
$133$	2.32168	0.201316
$134$	−9.79140	−0.845848
$135$	0	0
$136$	6.30450	0.540607
$137$	5.47243	0.467541	0.233771	−	0.972292i	$-0.424893\pi$
$137$	5.47243	0.467541	0.233771	+	0.972292i	$0.424893\pi$
$138$	0	0
$139$	−13.1838	−1.11823	−0.559116	−	0.829089i	$-0.688860\pi$
$139$	−13.1838	−1.11823	−0.559116	+	0.829089i	$0.688860\pi$
$140$	−5.08766	−0.429986
$141$	0	0
$142$	−2.12729	−0.178518
$143$	5.69601	0.476325
$144$	0	0
$145$	−6.62844	−0.550462
$146$	−7.02740	−0.581591
$147$	0	0
$148$	6.81324	0.560045
$149$	1.00000	0.0819232
$149$	1.00000	0.0819232
$150$	0	0
$151$	−14.7932	−1.20385	−0.601926	−	0.798552i	$-0.705600\pi$
$151$	−14.7932	−1.20385	−0.601926	+	0.798552i	$0.705600\pi$
$152$	−0.549921	−0.0446045
$153$	0	0
$154$	7.08559	0.570973
$155$	−6.92423	−0.556167
$156$	0	0
$157$	7.94423	0.634018	0.317009	−	0.948422i	$-0.397321\pi$
$157$	7.94423	0.634018	0.317009	+	0.948422i	$0.397321\pi$
$158$	12.4006	0.986536
$159$	0	0
$160$	1.20508	0.0952698
$161$	6.84709	0.539626
$162$	0	0
$163$	−6.47556	−0.507205	−0.253602	−	0.967309i	$-0.581615\pi$
$163$	−6.47556	−0.507205	−0.253602	+	0.967309i	$0.581615\pi$
$164$	1.19094	0.0929967
$165$	0	0
$166$	1.69109	0.131254
$167$	−15.1075	−1.16905	−0.584526	−	0.811375i	$-0.698720\pi$
$167$	−15.1075	−1.16905	−0.584526	+	0.811375i	$0.698720\pi$
$168$	0	0
$169$	−1.48148	−0.113960
$170$	−7.59742	−0.582696
$171$	0	0
$172$	−3.93505	−0.300045
$173$	−3.20709	−0.243830	−0.121915	−	0.992541i	$-0.538904\pi$
$173$	−3.20709	−0.243830	−0.121915	+	0.992541i	$0.538904\pi$
$174$	0	0
$175$	−14.9782	−1.13225
$176$	−1.67831	−0.126508
$177$	0	0
$178$	−1.92164	−0.144033
$179$	−3.34006	−0.249648	−0.124824	−	0.992179i	$-0.539837\pi$
$179$	−3.34006	−0.249648	−0.124824	+	0.992179i	$0.539837\pi$
$180$	0	0
$181$	21.9744	1.63335	0.816673	−	0.577100i	$-0.195816\pi$
$181$	21.9744	1.63335	0.816673	+	0.577100i	$0.195816\pi$
$182$	14.3285	1.06210
$183$	0	0
$184$	−1.62182	−0.119562
$185$	−8.21049	−0.603647
$186$	0	0
$187$	10.5809	0.773754
$188$	−1.55685	−0.113545
$189$	0	0
$190$	0.662698	0.0480771
$191$	9.42084	0.681668	0.340834	−	0.940124i	$-0.389291\pi$
$191$	9.42084	0.681668	0.340834	+	0.940124i	$0.389291\pi$
$192$	0	0
$193$	−7.22241	−0.519881	−0.259940	−	0.965625i	$-0.583703\pi$
$193$	−7.22241	−0.519881	−0.259940	+	0.965625i	$0.583703\pi$
$194$	0.826873	0.0593660
$195$	0	0
$196$	10.8240	0.773145
$197$	−8.67259	−0.617896	−0.308948	−	0.951079i	$-0.599977\pi$
$197$	−8.67259	−0.617896	−0.308948	+	0.951079i	$0.599977\pi$
$198$	0	0
$199$	−17.2431	−1.22233	−0.611166	−	0.791502i	$-0.709299\pi$
$199$	−17.2431	−1.22233	−0.611166	+	0.791502i	$0.709299\pi$
$200$	3.54779	0.250866
$201$	0	0
$202$	12.5987	0.886443
$203$	23.2220	1.62986
$204$	0	0
$205$	−1.43517	−0.100237
$206$	−12.9030	−0.898994
$207$	0	0
$208$	−3.39389	−0.235324
$209$	−0.922939	−0.0638410
$210$	0	0
$211$	19.1995	1.32175	0.660873	−	0.750498i	$-0.270186\pi$
$211$	19.1995	1.32175	0.660873	+	0.750498i	$0.270186\pi$
$212$	−5.73700	−0.394019
$213$	0	0
$214$	−3.39084	−0.231793
$215$	4.74205	0.323405
$216$	0	0
$217$	24.2582	1.64676
$218$	16.8130	1.13872
$219$	0	0
$220$	2.02250	0.136357
$221$	21.3968	1.43931
$222$	0	0
$223$	−11.9857	−0.802620	−0.401310	−	0.915942i	$-0.631445\pi$
$223$	−11.9857	−0.802620	−0.401310	+	0.915942i	$0.631445\pi$
$224$	−4.22185	−0.282084
$225$	0	0
$226$	13.1890	0.877321
$227$	−14.0722	−0.934007	−0.467003	−	0.884256i	$-0.654666\pi$
$227$	−14.0722	−0.934007	−0.467003	+	0.884256i	$0.654666\pi$
$228$	0	0
$229$	3.19113	0.210876	0.105438	−	0.994426i	$-0.466376\pi$
$229$	3.19113	0.210876	0.105438	+	0.994426i	$0.466376\pi$
$230$	1.95442	0.128871
$231$	0	0
$232$	−5.50042	−0.361120
$233$	12.7452	0.834968	0.417484	−	0.908684i	$-0.362912\pi$
$233$	12.7452	0.834968	0.417484	+	0.908684i	$0.362912\pi$
$234$	0	0
$235$	1.87613	0.122385
$236$	−1.95825	−0.127471
$237$	0	0
$238$	26.6167	1.72530
$239$	−10.0062	−0.647248	−0.323624	−	0.946186i	$-0.604901\pi$
$239$	−10.0062	−0.647248	−0.323624	+	0.946186i	$0.604901\pi$
$240$	0	0
$241$	−21.4573	−1.38219	−0.691093	−	0.722766i	$-0.742871\pi$
$241$	−21.4573	−1.38219	−0.691093	+	0.722766i	$0.742871\pi$
$242$	8.18327	0.526040
$243$	0	0
$244$	0.935725	0.0599036
$245$	−13.0438	−0.833338
$246$	0	0
$247$	−1.86637	−0.118754
$248$	−5.74587	−0.364863
$249$	0	0
$250$	−10.3008	−0.651477
$251$	25.5353	1.61177	0.805886	−	0.592071i	$-0.201690\pi$
$251$	25.5353	1.61177	0.805886	+	0.592071i	$0.201690\pi$
$252$	0	0
$253$	−2.72192	−0.171126
$254$	14.3011	0.897328
$255$	0	0
$256$	1.00000	0.0625000
$257$	−14.4673	−0.902443	−0.451221	−	0.892412i	$-0.649012\pi$
$257$	−14.4673	−0.902443	−0.451221	+	0.892412i	$0.649012\pi$
$258$	0	0
$259$	28.7645	1.78734
$260$	4.08991	0.253645
$261$	0	0
$262$	6.41294	0.396193
$263$	3.88603	0.239623	0.119812	−	0.992797i	$-0.461771\pi$
$263$	3.88603	0.239623	0.119812	+	0.992797i	$0.461771\pi$
$264$	0	0
$265$	6.91354	0.424695
$266$	−2.32168	−0.142352
$267$	0	0
$268$	9.79140	0.598105
$269$	21.4971	1.31070	0.655351	−	0.755324i	$-0.272521\pi$
$269$	21.4971	1.31070	0.655351	+	0.755324i	$0.272521\pi$
$270$	0	0
$271$	−25.5835	−1.55409	−0.777044	−	0.629446i	$-0.783282\pi$
$271$	−25.5835	−1.55409	−0.777044	+	0.629446i	$0.783282\pi$
$272$	−6.30450	−0.382267
$273$	0	0
$274$	−5.47243	−0.330602
$275$	5.95429	0.359057
$276$	0	0
$277$	20.6290	1.23948	0.619739	−	0.784808i	$-0.287239\pi$
$277$	20.6290	1.23948	0.619739	+	0.784808i	$0.287239\pi$
$278$	13.1838	0.790710
$279$	0	0
$280$	5.08766	0.304046
$281$	−21.2735	−1.26907	−0.634536	−	0.772894i	$-0.718809\pi$
$281$	−21.2735	−1.26907	−0.634536	+	0.772894i	$0.718809\pi$
$282$	0	0
$283$	−17.0236	−1.01195	−0.505973	−	0.862549i	$-0.668867\pi$
$283$	−17.0236	−1.01195	−0.505973	+	0.862549i	$0.668867\pi$
$284$	2.12729	0.126231
$285$	0	0
$286$	−5.69601	−0.336812
$287$	5.02797	0.296791
$288$	0	0
$289$	22.7468	1.33805
$290$	6.62844	0.389235
$291$	0	0
$292$	7.02740	0.411247
$293$	−1.85605	−0.108431	−0.0542157	−	0.998529i	$-0.517266\pi$
$293$	−1.85605	−0.108431	−0.0542157	+	0.998529i	$0.517266\pi$
$294$	0	0
$295$	2.35984	0.137395
$296$	−6.81324	−0.396012
$297$	0	0
$298$	−1.00000	−0.0579284
$299$	−5.50429	−0.318321
$300$	0	0
$301$	−16.6132	−0.957569
$302$	14.7932	0.851252
$303$	0	0
$304$	0.549921	0.0315401
$305$	−1.12762	−0.0645674
$306$	0	0
$307$	4.35716	0.248676	0.124338	−	0.992240i	$-0.460319\pi$
$307$	4.35716	0.248676	0.124338	+	0.992240i	$0.460319\pi$
$308$	−7.08559	−0.403739
$309$	0	0
$310$	6.92423	0.393270
$311$	−27.7324	−1.57256	−0.786281	−	0.617870i	$-0.787996\pi$
$311$	−27.7324	−1.57256	−0.786281	+	0.617870i	$0.787996\pi$
$312$	0	0
$313$	21.7725	1.23065	0.615326	−	0.788273i	$-0.289024\pi$
$313$	21.7725	1.23065	0.615326	+	0.788273i	$0.289024\pi$
$314$	−7.94423	−0.448319
$315$	0	0
$316$	−12.4006	−0.697586
$317$	8.32791	0.467742	0.233871	−	0.972268i	$-0.424861\pi$
$317$	8.32791	0.467742	0.233871	+	0.972268i	$0.424861\pi$
$318$	0	0
$319$	−9.23143	−0.516861
$320$	−1.20508	−0.0673659
$321$	0	0
$322$	−6.84709	−0.381573
$323$	−3.46698	−0.192908
$324$	0	0
$325$	12.0408	0.667904
$326$	6.47556	0.358648
$327$	0	0
$328$	−1.19094	−0.0657586
$329$	−6.57281	−0.362370
$330$	0	0
$331$	2.10160	0.115515	0.0577573	−	0.998331i	$-0.481605\pi$
$331$	2.10160	0.115515	0.0577573	+	0.998331i	$0.481605\pi$
$332$	−1.69109	−0.0928108
$333$	0	0
$334$	15.1075	0.826645
$335$	−11.7994	−0.644670
$336$	0	0
$337$	6.36786	0.346880	0.173440	−	0.984844i	$-0.444512\pi$
$337$	6.36786	0.346880	0.173440	+	0.984844i	$0.444512\pi$
$338$	1.48148	0.0805821
$339$	0	0
$340$	7.59742	0.412028
$341$	−9.64337	−0.522218
$342$	0	0
$343$	16.1445	0.871720
$344$	3.93505	0.212164
$345$	0	0
$346$	3.20709	0.172414
$347$	−10.7654	−0.577916	−0.288958	−	0.957342i	$-0.593309\pi$
$347$	−10.7654	−0.577916	−0.288958	+	0.957342i	$0.593309\pi$
$348$	0	0
$349$	6.24987	0.334548	0.167274	−	0.985910i	$-0.446504\pi$
$349$	6.24987	0.334548	0.167274	+	0.985910i	$0.446504\pi$
$350$	14.9782	0.800620
$351$	0	0
$352$	1.67831	0.0894544
$353$	33.6545	1.79125	0.895625	−	0.444810i	$-0.146729\pi$
$353$	33.6545	1.79125	0.895625	+	0.444810i	$0.146729\pi$
$354$	0	0
$355$	−2.56355	−0.136059
$356$	1.92164	0.101846
$357$	0	0
$358$	3.34006	0.176528
$359$	−26.9210	−1.42084	−0.710419	−	0.703779i	$-0.751494\pi$
$359$	−26.9210	−1.42084	−0.710419	+	0.703779i	$0.751494\pi$
$360$	0	0
$361$	−18.6976	−0.984084
$362$	−21.9744	−1.15495
$363$	0	0
$364$	−14.3285	−0.751018
$365$	−8.46856	−0.443265
$366$	0	0
$367$	−26.5963	−1.38832	−0.694158	−	0.719823i	$-0.744223\pi$
$367$	−26.5963	−1.38832	−0.694158	+	0.719823i	$0.744223\pi$
$368$	1.62182	0.0845433
$369$	0	0
$370$	8.21049	0.426843
$371$	−24.2208	−1.25748
$372$	0	0
$373$	19.0090	0.984251	0.492125	−	0.870524i	$-0.336220\pi$
$373$	19.0090	0.984251	0.492125	+	0.870524i	$0.336220\pi$
$374$	−10.5809	−0.547127
$375$	0	0
$376$	1.55685	0.0802886
$377$	−18.6679	−0.961443
$378$	0	0
$379$	16.8543	0.865745	0.432873	−	0.901455i	$-0.357500\pi$
$379$	16.8543	0.865745	0.432873	+	0.901455i	$0.357500\pi$
$380$	−0.662698	−0.0339957
$381$	0	0
$382$	−9.42084	−0.482012
$383$	−16.1653	−0.826008	−0.413004	−	0.910729i	$-0.635521\pi$
$383$	−16.1653	−0.826008	−0.413004	+	0.910729i	$0.635521\pi$
$384$	0	0
$385$	8.53869	0.435172
$386$	7.22241	0.367611
$387$	0	0
$388$	−0.826873	−0.0419781
$389$	−2.90557	−0.147318	−0.0736590	−	0.997283i	$-0.523468\pi$
$389$	−2.90557	−0.147318	−0.0736590	+	0.997283i	$0.523468\pi$
$390$	0	0
$391$	−10.2248	−0.517090
$392$	−10.8240	−0.546696
$393$	0	0
$394$	8.67259	0.436919
$395$	14.9436	0.751897
$396$	0	0
$397$	−6.01866	−0.302068	−0.151034	−	0.988529i	$-0.548260\pi$
$397$	−6.01866	−0.302068	−0.151034	+	0.988529i	$0.548260\pi$
$398$	17.2431	0.864319
$399$	0	0
$400$	−3.54779	−0.177389
$401$	25.7566	1.28622	0.643112	−	0.765772i	$-0.277643\pi$
$401$	25.7566	1.28622	0.643112	+	0.765772i	$0.277643\pi$
$402$	0	0
$403$	−19.5009	−0.971408
$404$	−12.5987	−0.626810
$405$	0	0
$406$	−23.2220	−1.15249
$407$	−11.4347	−0.566799
$408$	0	0
$409$	−29.2259	−1.44513	−0.722564	−	0.691304i	$-0.757037\pi$
$409$	−29.2259	−1.44513	−0.722564	+	0.691304i	$0.757037\pi$
$410$	1.43517	0.0708782
$411$	0	0
$412$	12.9030	0.635685
$413$	−8.26743	−0.406814
$414$	0	0
$415$	2.03790	0.100037
$416$	3.39389	0.166399
$417$	0	0
$418$	0.922939	0.0451424
$419$	−31.8164	−1.55433	−0.777167	−	0.629295i	$-0.783344\pi$
$419$	−31.8164	−1.55433	−0.777167	+	0.629295i	$0.783344\pi$
$420$	0	0
$421$	−26.5671	−1.29480	−0.647399	−	0.762151i	$-0.724143\pi$
$421$	−26.5671	−1.29480	−0.647399	+	0.762151i	$0.724143\pi$
$422$	−19.1995	−0.934616
$423$	0	0
$424$	5.73700	0.278613
$425$	22.3670	1.08496
$426$	0	0
$427$	3.95049	0.191178
$428$	3.39084	0.163903
$429$	0	0
$430$	−4.74205	−0.228682
$431$	28.3811	1.36707	0.683535	−	0.729918i	$-0.260442\pi$
$431$	28.3811	1.36707	0.683535	+	0.729918i	$0.260442\pi$
$432$	0	0
$433$	−1.75201	−0.0841962	−0.0420981	−	0.999113i	$-0.513404\pi$
$433$	−1.75201	−0.0841962	−0.0420981	+	0.999113i	$0.513404\pi$
$434$	−24.2582	−1.16443
$435$	0	0
$436$	−16.8130	−0.805199
$437$	0.891874	0.0426641
$438$	0	0
$439$	−4.57507	−0.218356	−0.109178	−	0.994022i	$-0.534822\pi$
$439$	−4.57507	−0.218356	−0.109178	+	0.994022i	$0.534822\pi$
$440$	−2.02250	−0.0964188
$441$	0	0
$442$	−21.3968	−1.01774
$443$	−5.48904	−0.260792	−0.130396	−	0.991462i	$-0.541625\pi$
$443$	−5.48904	−0.260792	−0.130396	+	0.991462i	$0.541625\pi$
$444$	0	0
$445$	−2.31572	−0.109776
$446$	11.9857	0.567538
$447$	0	0
$448$	4.22185	0.199464
$449$	5.86084	0.276590	0.138295	−	0.990391i	$-0.455838\pi$
$449$	5.86084	0.276590	0.138295	+	0.990391i	$0.455838\pi$
$450$	0	0
$451$	−1.99877	−0.0941183
$452$	−13.1890	−0.620360
$453$	0	0
$454$	14.0722	0.660443
$455$	17.2670	0.809489
$456$	0	0
$457$	−12.9582	−0.606158	−0.303079	−	0.952965i	$-0.598015\pi$
$457$	−12.9582	−0.606158	−0.303079	+	0.952965i	$0.598015\pi$
$458$	−3.19113	−0.149112
$459$	0	0
$460$	−1.95442	−0.0911254
$461$	−5.48261	−0.255351	−0.127675	−	0.991816i	$-0.540752\pi$
$461$	−5.48261	−0.255351	−0.127675	+	0.991816i	$0.540752\pi$
$462$	0	0
$463$	−26.5140	−1.23221	−0.616105	−	0.787664i	$-0.711290\pi$
$463$	−26.5140	−1.23221	−0.616105	+	0.787664i	$0.711290\pi$
$464$	5.50042	0.255351
$465$	0	0
$466$	−12.7452	−0.590412
$467$	32.5252	1.50509	0.752544	−	0.658542i	$-0.228827\pi$
$467$	32.5252	1.50509	0.752544	+	0.658542i	$0.228827\pi$
$468$	0	0
$469$	41.3378	1.90880
$470$	−1.87613	−0.0865395
$471$	0	0
$472$	1.95825	0.0901357
$473$	6.60425	0.303664
$474$	0	0
$475$	−1.95100	−0.0895181
$476$	−26.6167	−1.21997
$477$	0	0
$478$	10.0062	0.457673
$479$	−38.1095	−1.74127	−0.870635	−	0.491930i	$-0.836292\pi$
$479$	−38.1095	−1.74127	−0.870635	+	0.491930i	$0.836292\pi$
$480$	0	0
$481$	−23.1234	−1.05434
$482$	21.4573	0.977353
$483$	0	0
$484$	−8.18327	−0.371967
$485$	0.996447	0.0452463
$486$	0	0
$487$	18.6992	0.847342	0.423671	−	0.905816i	$-0.360741\pi$
$487$	18.6992	0.847342	0.423671	+	0.905816i	$0.360741\pi$
$488$	−0.935725	−0.0423583
$489$	0	0
$490$	13.0438	0.589259
$491$	14.9990	0.676896	0.338448	−	0.940985i	$-0.390098\pi$
$491$	14.9990	0.676896	0.338448	+	0.940985i	$0.390098\pi$
$492$	0	0
$493$	−34.6774	−1.56179
$494$	1.86637	0.0839721
$495$	0	0
$496$	5.74587	0.257997
$497$	8.98110	0.402858
$498$	0	0
$499$	−32.2730	−1.44474	−0.722369	−	0.691508i	$-0.756947\pi$
$499$	−32.2730	−1.44474	−0.722369	+	0.691508i	$0.756947\pi$
$500$	10.3008	0.460664
$501$	0	0
$502$	−25.5353	−1.13969
$503$	8.87585	0.395755	0.197877	−	0.980227i	$-0.436595\pi$
$503$	8.87585	0.395755	0.197877	+	0.980227i	$0.436595\pi$
$504$	0	0
$505$	15.1825	0.675610
$506$	2.72192	0.121004
$507$	0	0
$508$	−14.3011	−0.634507
$509$	−33.2900	−1.47555	−0.737777	−	0.675045i	$-0.764124\pi$
$509$	−33.2900	−1.47555	−0.737777	+	0.675045i	$0.764124\pi$
$510$	0	0
$511$	29.6686	1.31246
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	14.4673	0.638124
$515$	−15.5491	−0.685176
$516$	0	0
$517$	2.61289	0.114915
$518$	−28.7645	−1.26384
$519$	0	0
$520$	−4.08991	−0.179354
$521$	23.0651	1.01050	0.505249	−	0.862974i	$-0.331401\pi$
$521$	23.0651	1.01050	0.505249	+	0.862974i	$0.331401\pi$
$522$	0	0
$523$	−7.97526	−0.348734	−0.174367	−	0.984681i	$-0.555788\pi$
$523$	−7.97526	−0.348734	−0.174367	+	0.984681i	$0.555788\pi$
$524$	−6.41294	−0.280151
$525$	0	0
$526$	−3.88603	−0.169439
$527$	−36.2249	−1.57798
$528$	0	0
$529$	−20.3697	−0.885639
$530$	−6.91354	−0.300305
$531$	0	0
$532$	2.32168	0.100658
$533$	−4.04192	−0.175075
$534$	0	0
$535$	−4.08623	−0.176663
$536$	−9.79140	−0.422924
$537$	0	0
$538$	−21.4971	−0.926807
$539$	−18.1661	−0.782470
$540$	0	0
$541$	14.7417	0.633794	0.316897	−	0.948460i	$-0.397359\pi$
$541$	14.7417	0.633794	0.316897	+	0.948460i	$0.397359\pi$
$542$	25.5835	1.09891
$543$	0	0
$544$	6.30450	0.270303
$545$	20.2610	0.867888
$546$	0	0
$547$	−1.49522	−0.0639311	−0.0319656	−	0.999489i	$-0.510177\pi$
$547$	−1.49522	−0.0639311	−0.0319656	+	0.999489i	$0.510177\pi$
$548$	5.47243	0.233771
$549$	0	0
$550$	−5.95429	−0.253892
$551$	3.02480	0.128861
$552$	0	0
$553$	−52.3533	−2.22629
$554$	−20.6290	−0.876443
$555$	0	0
$556$	−13.1838	−0.559116
$557$	−36.9339	−1.56494	−0.782470	−	0.622689i	$-0.786040\pi$
$557$	−36.9339	−1.56494	−0.782470	+	0.622689i	$0.786040\pi$
$558$	0	0
$559$	13.3551	0.564863
$560$	−5.08766	−0.214993
$561$	0	0
$562$	21.2735	0.897369
$563$	−3.08471	−0.130005	−0.0650025	−	0.997885i	$-0.520706\pi$
$563$	−3.08471	−0.130005	−0.0650025	+	0.997885i	$0.520706\pi$
$564$	0	0
$565$	15.8938	0.668658
$566$	17.0236	0.715555
$567$	0	0
$568$	−2.12729	−0.0892591
$569$	−15.4745	−0.648724	−0.324362	−	0.945933i	$-0.605150\pi$
$569$	−15.4745	−0.648724	−0.324362	+	0.945933i	$0.605150\pi$
$570$	0	0
$571$	14.6179	0.611739	0.305870	−	0.952073i	$-0.401053\pi$
$571$	14.6179	0.611739	0.305870	+	0.952073i	$0.401053\pi$
$572$	5.69601	0.238162
$573$	0	0
$574$	−5.02797	−0.209863
$575$	−5.75388	−0.239953
$576$	0	0
$577$	−6.51193	−0.271095	−0.135548	−	0.990771i	$-0.543279\pi$
$577$	−6.51193	−0.271095	−0.135548	+	0.990771i	$0.543279\pi$
$578$	−22.7468	−0.946141
$579$	0	0
$580$	−6.62844	−0.275231
$581$	−7.13954	−0.296198
$582$	0	0
$583$	9.62848	0.398771
$584$	−7.02740	−0.290796
$585$	0	0
$586$	1.85605	0.0766726
$587$	12.1750	0.502516	0.251258	−	0.967920i	$-0.419156\pi$
$587$	12.1750	0.502516	0.251258	+	0.967920i	$0.419156\pi$
$588$	0	0
$589$	3.15978	0.130196
$590$	−2.35984	−0.0971532
$591$	0	0
$592$	6.81324	0.280022
$593$	−27.9994	−1.14980	−0.574898	−	0.818225i	$-0.694958\pi$
$593$	−27.9994	−1.14980	−0.574898	+	0.818225i	$0.694958\pi$
$594$	0	0
$595$	32.0752	1.31495
$596$	1.00000	0.0409616
$597$	0	0
$598$	5.50429	0.225087
$599$	−17.2016	−0.702838	−0.351419	−	0.936218i	$-0.614301\pi$
$599$	−17.2016	−0.702838	−0.351419	+	0.936218i	$0.614301\pi$
$600$	0	0
$601$	2.52455	0.102979	0.0514893	−	0.998674i	$-0.483603\pi$
$601$	2.52455	0.102979	0.0514893	+	0.998674i	$0.483603\pi$
$602$	16.6132	0.677104
$603$	0	0
$604$	−14.7932	−0.601926
$605$	9.86148	0.400926
$606$	0	0
$607$	44.8101	1.81879	0.909393	−	0.415938i	$-0.136547\pi$
$607$	44.8101	1.81879	0.909393	+	0.415938i	$0.136547\pi$
$608$	−0.549921	−0.0223022
$609$	0	0
$610$	1.12762	0.0456561
$611$	5.28380	0.213760
$612$	0	0
$613$	−26.8267	−1.08352	−0.541760	−	0.840533i	$-0.682242\pi$
$613$	−26.8267	−1.08352	−0.541760	+	0.840533i	$0.682242\pi$
$614$	−4.35716	−0.175841
$615$	0	0
$616$	7.08559	0.285486
$617$	1.61687	0.0650929	0.0325465	−	0.999470i	$-0.489638\pi$
$617$	1.61687	0.0650929	0.0325465	+	0.999470i	$0.489638\pi$
$618$	0	0
$619$	−20.8905	−0.839659	−0.419830	−	0.907603i	$-0.637910\pi$
$619$	−20.8905	−0.839659	−0.419830	+	0.907603i	$0.637910\pi$
$620$	−6.92423	−0.278084
$621$	0	0
$622$	27.7324	1.11197
$623$	8.11286	0.325035
$624$	0	0
$625$	5.32572	0.213029
$626$	−21.7725	−0.870203
$627$	0	0
$628$	7.94423	0.317009
$629$	−42.9541	−1.71269
$630$	0	0
$631$	−31.6368	−1.25944	−0.629721	−	0.776821i	$-0.716831\pi$
$631$	−31.6368	−1.25944	−0.629721	+	0.776821i	$0.716831\pi$
$632$	12.4006	0.493268
$633$	0	0
$634$	−8.32791	−0.330743
$635$	17.2339	0.683906
$636$	0	0
$637$	−36.7356	−1.45552
$638$	9.23143	0.365476
$639$	0	0
$640$	1.20508	0.0476349
$641$	−23.6784	−0.935241	−0.467620	−	0.883929i	$-0.654889\pi$
$641$	−23.6784	−0.935241	−0.467620	+	0.883929i	$0.654889\pi$
$642$	0	0
$643$	35.3177	1.39279	0.696396	−	0.717657i	$-0.254786\pi$
$643$	35.3177	1.39279	0.696396	+	0.717657i	$0.254786\pi$
$644$	6.84709	0.269813
$645$	0	0
$646$	3.46698	0.136406
$647$	28.5654	1.12302	0.561512	−	0.827469i	$-0.310220\pi$
$647$	28.5654	1.12302	0.561512	+	0.827469i	$0.310220\pi$
$648$	0	0
$649$	3.28655	0.129008
$650$	−12.0408	−0.472279
$651$	0	0
$652$	−6.47556	−0.253602
$653$	−36.6555	−1.43444	−0.717221	−	0.696846i	$-0.754586\pi$
$653$	−36.6555	−1.43444	−0.717221	+	0.696846i	$0.754586\pi$
$654$	0	0
$655$	7.72810	0.301962
$656$	1.19094	0.0464983
$657$	0	0
$658$	6.57281	0.256235
$659$	−19.8343	−0.772633	−0.386317	−	0.922366i	$-0.626253\pi$
$659$	−19.8343	−0.772633	−0.386317	+	0.922366i	$0.626253\pi$
$660$	0	0
$661$	−13.2857	−0.516752	−0.258376	−	0.966044i	$-0.583187\pi$
$661$	−13.2857	−0.516752	−0.258376	+	0.966044i	$0.583187\pi$
$662$	−2.10160	−0.0816811
$663$	0	0
$664$	1.69109	0.0656271
$665$	−2.79781	−0.108494
$666$	0	0
$667$	8.92071	0.345411
$668$	−15.1075	−0.584526
$669$	0	0
$670$	11.7994	0.455851
$671$	−1.57044	−0.0606261
$672$	0	0
$673$	16.2624	0.626868	0.313434	−	0.949610i	$-0.398521\pi$
$673$	16.2624	0.626868	0.313434	+	0.949610i	$0.398521\pi$
$674$	−6.36786	−0.245281
$675$	0	0
$676$	−1.48148	−0.0569801
$677$	3.23669	0.124396	0.0621980	−	0.998064i	$-0.480189\pi$
$677$	3.23669	0.124396	0.0621980	+	0.998064i	$0.480189\pi$
$678$	0	0
$679$	−3.49094	−0.133970
$680$	−7.59742	−0.291348
$681$	0	0
$682$	9.64337	0.369264
$683$	28.2461	1.08081	0.540404	−	0.841405i	$-0.318271\pi$
$683$	28.2461	1.08081	0.540404	+	0.841405i	$0.318271\pi$
$684$	0	0
$685$	−6.59471	−0.251971
$686$	−16.1445	−0.616399
$687$	0	0
$688$	−3.93505	−0.150022
$689$	19.4708	0.741778
$690$	0	0
$691$	23.2577	0.884765	0.442383	−	0.896826i	$-0.354133\pi$
$691$	23.2577	0.884765	0.442383	+	0.896826i	$0.354133\pi$
$692$	−3.20709	−0.121915
$693$	0	0
$694$	10.7654	0.408648
$695$	15.8875	0.602646
$696$	0	0
$697$	−7.50828	−0.284396
$698$	−6.24987	−0.236561
$699$	0	0
$700$	−14.9782	−0.566124
$701$	17.7509	0.670444	0.335222	−	0.942139i	$-0.391189\pi$
$701$	17.7509	0.670444	0.335222	+	0.942139i	$0.391189\pi$
$702$	0	0
$703$	3.74674	0.141311
$704$	−1.67831	−0.0632538
$705$	0	0
$706$	−33.6545	−1.26660
$707$	−53.1900	−2.00041
$708$	0	0
$709$	−9.16195	−0.344084	−0.172042	−	0.985090i	$-0.555037\pi$
$709$	−9.16195	−0.344084	−0.172042	+	0.985090i	$0.555037\pi$
$710$	2.56355	0.0962084
$711$	0	0
$712$	−1.92164	−0.0720163
$713$	9.31878	0.348991
$714$	0	0
$715$	−6.86414	−0.256704
$716$	−3.34006	−0.124824
$717$	0	0
$718$	26.9210	1.00468
$719$	−29.0077	−1.08180	−0.540902	−	0.841086i	$-0.681917\pi$
$719$	−29.0077	−1.08180	−0.540902	+	0.841086i	$0.681917\pi$
$720$	0	0
$721$	54.4745	2.02874
$722$	18.6976	0.695852
$723$	0	0
$724$	21.9744	0.816673
$725$	−19.5143	−0.724744
$726$	0	0
$727$	5.42724	0.201285	0.100643	−	0.994923i	$-0.467910\pi$
$727$	5.42724	0.201285	0.100643	+	0.994923i	$0.467910\pi$
$728$	14.3285	0.531050
$729$	0	0
$730$	8.46856	0.313436
$731$	24.8086	0.917577
$732$	0	0
$733$	−37.0677	−1.36913	−0.684563	−	0.728954i	$-0.740007\pi$
$733$	−37.0677	−1.36913	−0.684563	+	0.728954i	$0.740007\pi$
$734$	26.5963	0.981688
$735$	0	0
$736$	−1.62182	−0.0597811
$737$	−16.4330	−0.605318
$738$	0	0
$739$	33.3686	1.22748	0.613741	−	0.789507i	$-0.289664\pi$
$739$	33.3686	1.22748	0.613741	+	0.789507i	$0.289664\pi$
$740$	−8.21049	−0.301824
$741$	0	0
$742$	24.2208	0.889172
$743$	−16.6388	−0.610418	−0.305209	−	0.952285i	$-0.598726\pi$
$743$	−16.6388	−0.610418	−0.305209	+	0.952285i	$0.598726\pi$
$744$	0	0
$745$	−1.20508	−0.0441507
$746$	−19.0090	−0.695970
$747$	0	0
$748$	10.5809	0.386877
$749$	14.3156	0.523082
$750$	0	0
$751$	−36.5532	−1.33384	−0.666922	−	0.745127i	$-0.732389\pi$
$751$	−36.5532	−1.33384	−0.666922	+	0.745127i	$0.732389\pi$
$752$	−1.55685	−0.0567726
$753$	0	0
$754$	18.6679	0.679843
$755$	17.8269	0.648789
$756$	0	0
$757$	16.6750	0.606064	0.303032	−	0.952980i	$-0.402001\pi$
$757$	16.6750	0.606064	0.303032	+	0.952980i	$0.402001\pi$
$758$	−16.8543	−0.612174
$759$	0	0
$760$	0.662698	0.0240386
$761$	4.80780	0.174283	0.0871413	−	0.996196i	$-0.472227\pi$
$761$	4.80780	0.174283	0.0871413	+	0.996196i	$0.472227\pi$
$762$	0	0
$763$	−70.9822	−2.56973
$764$	9.42084	0.340834
$765$	0	0
$766$	16.1653	0.584076
$767$	6.64608	0.239976
$768$	0	0
$769$	0.0529014	0.00190767	0.000953836	−	1.00000i	$-0.499696\pi$
$769$	0.0529014	0.00190767	0.000953836		1.00000i	$0.499696\pi$
$770$	−8.53869	−0.307713
$771$	0	0
$772$	−7.22241	−0.259940
$773$	14.4129	0.518396	0.259198	−	0.965824i	$-0.416542\pi$
$773$	14.4129	0.518396	0.259198	+	0.965824i	$0.416542\pi$
$774$	0	0
$775$	−20.3851	−0.732256
$776$	0.826873	0.0296830
$777$	0	0
$778$	2.90557	0.104170
$779$	0.654922	0.0234650
$780$	0	0
$781$	−3.57026	−0.127754
$782$	10.2248	0.365637
$783$	0	0
$784$	10.8240	0.386572
$785$	−9.57342	−0.341690
$786$	0	0
$787$	15.4753	0.551634	0.275817	−	0.961210i	$-0.411052\pi$
$787$	15.4753	0.551634	0.275817	+	0.961210i	$0.411052\pi$
$788$	−8.67259	−0.308948
$789$	0	0
$790$	−14.9436	−0.531671
$791$	−55.6821	−1.97983
$792$	0	0
$793$	−3.17575	−0.112774
$794$	6.01866	0.213594
$795$	0	0
$796$	−17.2431	−0.611166
$797$	−35.2849	−1.24986	−0.624928	−	0.780682i	$-0.714872\pi$
$797$	−35.2849	−1.24986	−0.624928	+	0.780682i	$0.714872\pi$
$798$	0	0
$799$	9.81519	0.347237
$800$	3.54779	0.125433
$801$	0	0
$802$	−25.7566	−0.909498
$803$	−11.7942	−0.416207
$804$	0	0
$805$	−8.25128	−0.290819
$806$	19.5009	0.686889
$807$	0	0
$808$	12.5987	0.443222
$809$	8.50051	0.298862	0.149431	−	0.988772i	$-0.452256\pi$
$809$	8.50051	0.298862	0.149431	+	0.988772i	$0.452256\pi$
$810$	0	0
$811$	36.4644	1.28044	0.640219	−	0.768193i	$-0.278844\pi$
$811$	36.4644	1.28044	0.640219	+	0.768193i	$0.278844\pi$
$812$	23.2220	0.814931
$813$	0	0
$814$	11.4347	0.400788
$815$	7.80355	0.273347
$816$	0	0
$817$	−2.16397	−0.0757076
$818$	29.2259	1.02186
$819$	0	0
$820$	−1.43517	−0.0501185
$821$	−2.74277	−0.0957232	−0.0478616	−	0.998854i	$-0.515241\pi$
$821$	−2.74277	−0.0957232	−0.0478616	+	0.998854i	$0.515241\pi$
$822$	0	0
$823$	−24.6796	−0.860277	−0.430138	−	0.902763i	$-0.641535\pi$
$823$	−24.6796	−0.860277	−0.430138	+	0.902763i	$0.641535\pi$
$824$	−12.9030	−0.449497
$825$	0	0
$826$	8.26743	0.287661
$827$	−46.4246	−1.61434	−0.807171	−	0.590318i	$-0.799002\pi$
$827$	−46.4246	−1.61434	−0.807171	+	0.590318i	$0.799002\pi$
$828$	0	0
$829$	21.5246	0.747579	0.373790	−	0.927513i	$-0.378058\pi$
$829$	21.5246	0.747579	0.373790	+	0.927513i	$0.378058\pi$
$830$	−2.03790	−0.0707365
$831$	0	0
$832$	−3.39389	−0.117662
$833$	−68.2401	−2.36438
$834$	0	0
$835$	18.2057	0.630035
$836$	−0.922939	−0.0319205
$837$	0	0
$838$	31.8164	1.09908
$839$	55.2405	1.90712	0.953558	−	0.301211i	$-0.0973908\pi$
$839$	55.2405	1.90712	0.953558	+	0.301211i	$0.0973908\pi$
$840$	0	0
$841$	1.25466	0.0432641
$842$	26.5671	0.915561
$843$	0	0
$844$	19.1995	0.660873
$845$	1.78530	0.0614163
$846$	0	0
$847$	−34.5485	−1.18710
$848$	−5.73700	−0.197009
$849$	0	0
$850$	−22.3670	−0.767183
$851$	11.0499	0.378784
$852$	0	0
$853$	−47.8251	−1.63750	−0.818750	−	0.574151i	$-0.805332\pi$
$853$	−47.8251	−1.63750	−0.818750	+	0.574151i	$0.805332\pi$
$854$	−3.95049	−0.135183
$855$	0	0
$856$	−3.39084	−0.115897
$857$	−27.4067	−0.936196	−0.468098	−	0.883676i	$-0.655061\pi$
$857$	−27.4067	−0.936196	−0.468098	+	0.883676i	$0.655061\pi$
$858$	0	0
$859$	5.60596	0.191273	0.0956364	−	0.995416i	$-0.469511\pi$
$859$	5.60596	0.191273	0.0956364	+	0.995416i	$0.469511\pi$
$860$	4.74205	0.161702
$861$	0	0
$862$	−28.3811	−0.966665
$863$	25.8722	0.880700	0.440350	−	0.897826i	$-0.354854\pi$
$863$	25.8722	0.880700	0.440350	+	0.897826i	$0.354854\pi$
$864$	0	0
$865$	3.86479	0.131407
$866$	1.75201	0.0595357
$867$	0	0
$868$	24.2582	0.823378
$869$	20.8120	0.705999
$870$	0	0
$871$	−33.2310	−1.12599
$872$	16.8130	0.569362
$873$	0	0
$874$	−0.891874	−0.0301681
$875$	43.4882	1.47017
$876$	0	0
$877$	8.26910	0.279228	0.139614	−	0.990206i	$-0.455414\pi$
$877$	8.26910	0.279228	0.139614	+	0.990206i	$0.455414\pi$
$878$	4.57507	0.154401
$879$	0	0
$880$	2.02250	0.0681784
$881$	7.19072	0.242262	0.121131	−	0.992637i	$-0.461348\pi$
$881$	7.19072	0.242262	0.121131	+	0.992637i	$0.461348\pi$
$882$	0	0
$883$	16.3882	0.551506	0.275753	−	0.961229i	$-0.411073\pi$
$883$	16.3882	0.551506	0.275753	+	0.961229i	$0.411073\pi$
$884$	21.3968	0.719653
$885$	0	0
$886$	5.48904	0.184408
$887$	26.1452	0.877870	0.438935	−	0.898519i	$-0.355356\pi$
$887$	26.1452	0.877870	0.438935	+	0.898519i	$0.355356\pi$
$888$	0	0
$889$	−60.3769	−2.02498
$890$	2.31572	0.0776231
$891$	0	0
$892$	−11.9857	−0.401310
$893$	−0.856146	−0.0286498
$894$	0	0
$895$	4.02503	0.134542
$896$	−4.22185	−0.141042
$897$	0	0
$898$	−5.86084	−0.195579
$899$	31.6047	1.05408
$900$	0	0
$901$	36.1690	1.20496
$902$	1.99877	0.0665517
$903$	0	0
$904$	13.1890	0.438660
$905$	−26.4809	−0.880255
$906$	0	0
$907$	−46.5624	−1.54608	−0.773040	−	0.634357i	$-0.781265\pi$
$907$	−46.5624	−1.54608	−0.773040	+	0.634357i	$0.781265\pi$
$908$	−14.0722	−0.467003
$909$	0	0
$910$	−17.2670	−0.572395
$911$	−9.44555	−0.312945	−0.156472	−	0.987682i	$-0.550012\pi$
$911$	−9.44555	−0.312945	−0.156472	+	0.987682i	$0.550012\pi$
$912$	0	0
$913$	2.83818	0.0939301
$914$	12.9582	0.428619
$915$	0	0
$916$	3.19113	0.105438
$917$	−27.0745	−0.894079
$918$	0	0
$919$	−28.7329	−0.947811	−0.473906	−	0.880576i	$-0.657156\pi$
$919$	−28.7329	−0.947811	−0.473906	+	0.880576i	$0.657156\pi$
$920$	1.95442	0.0644354
$921$	0	0
$922$	5.48261	0.180560
$923$	−7.21980	−0.237643
$924$	0	0
$925$	−24.1719	−0.794768
$926$	26.5140	0.871305
$927$	0	0
$928$	−5.50042	−0.180560
$929$	52.1305	1.71035	0.855173	−	0.518343i	$-0.173451\pi$
$929$	52.1305	1.71035	0.855173	+	0.518343i	$0.173451\pi$
$930$	0	0
$931$	5.95236	0.195081
$932$	12.7452	0.417484
$933$	0	0
$934$	−32.5252	−1.06426
$935$	−12.7508	−0.416997
$936$	0	0
$937$	−2.75008	−0.0898410	−0.0449205	−	0.998991i	$-0.514303\pi$
$937$	−2.75008	−0.0898410	−0.0449205	+	0.998991i	$0.514303\pi$
$938$	−41.3378	−1.34973
$939$	0	0
$940$	1.87613	0.0611926
$941$	18.1078	0.590296	0.295148	−	0.955452i	$-0.404631\pi$
$941$	18.1078	0.590296	0.295148	+	0.955452i	$0.404631\pi$
$942$	0	0
$943$	1.93149	0.0628980
$944$	−1.95825	−0.0637355
$945$	0	0
$946$	−6.60425	−0.214723
$947$	−0.785550	−0.0255269	−0.0127635	−	0.999919i	$-0.504063\pi$
$947$	−0.785550	−0.0255269	−0.0127635	+	0.999919i	$0.504063\pi$
$948$	0	0
$949$	−23.8502	−0.774211
$950$	1.95100	0.0632989
$951$	0	0
$952$	26.6167	0.862652
$953$	36.5226	1.18308	0.591541	−	0.806275i	$-0.298520\pi$
$953$	36.5226	1.18308	0.591541	+	0.806275i	$0.298520\pi$
$954$	0	0
$955$	−11.3528	−0.367370
$956$	−10.0062	−0.323624
$957$	0	0
$958$	38.1095	1.23126
$959$	23.1038	0.746061
$960$	0	0
$961$	2.01507	0.0650021
$962$	23.1234	0.745529
$963$	0	0
$964$	−21.4573	−0.691093
$965$	8.70357	0.280178
$966$	0	0
$967$	39.2382	1.26181	0.630907	−	0.775858i	$-0.282683\pi$
$967$	39.2382	1.26181	0.630907	+	0.775858i	$0.282683\pi$
$968$	8.18327	0.263020
$969$	0	0
$970$	−0.996447	−0.0319940
$971$	39.5725	1.26994	0.634971	−	0.772536i	$-0.281012\pi$
$971$	39.5725	1.26994	0.634971	+	0.772536i	$0.281012\pi$
$972$	0	0
$973$	−55.6599	−1.78438
$974$	−18.6992	−0.599161
$975$	0	0
$976$	0.935725	0.0299518
$977$	37.3943	1.19635	0.598174	−	0.801366i	$-0.295893\pi$
$977$	37.3943	1.19635	0.598174	+	0.801366i	$0.295893\pi$
$978$	0	0
$979$	−3.22510	−0.103075
$980$	−13.0438	−0.416669
$981$	0	0
$982$	−14.9990	−0.478638
$983$	−4.76717	−0.152049	−0.0760245	−	0.997106i	$-0.524223\pi$
$983$	−4.76717	−0.152049	−0.0760245	+	0.997106i	$0.524223\pi$
$984$	0	0
$985$	10.4511	0.333001
$986$	34.6774	1.10435
$987$	0	0
$988$	−1.86637	−0.0593772
$989$	−6.38195	−0.202934
$990$	0	0
$991$	33.8296	1.07463	0.537316	−	0.843381i	$-0.319438\pi$
$991$	33.8296	1.07463	0.537316	+	0.843381i	$0.319438\pi$
$992$	−5.74587	−0.182432
$993$	0	0
$994$	−8.98110	−0.284863
$995$	20.7793	0.658748
$996$	0	0
$997$	−33.6855	−1.06683	−0.533416	−	0.845853i	$-0.679092\pi$
$997$	−33.6855	−1.06683	−0.533416	+	0.845853i	$0.679092\pi$
$998$	32.2730	1.02158
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.i.1.6	✓	12
3.2	odd	2		8046.2.a.p.1.7	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.i.1.6	✓	12	1.1	even	1	trivial
8046.2.a.p.1.7	yes	12	3.2	odd	2

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:	\( N \)	\(=\)	\( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8046.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.20508 q^{5} +4.22185 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.20508 q^{5} +4.22185 q^{7} -1.00000 q^{8} +1.20508 q^{10} -1.67831 q^{11} -3.39389 q^{13} -4.22185 q^{14} +1.00000 q^{16} -6.30450 q^{17} +0.549921 q^{19} -1.20508 q^{20} +1.67831 q^{22} +1.62182 q^{23} -3.54779 q^{25} +3.39389 q^{26} +4.22185 q^{28} +5.50042 q^{29} +5.74587 q^{31} -1.00000 q^{32} +6.30450 q^{34} -5.08766 q^{35} +6.81324 q^{37} -0.549921 q^{38} +1.20508 q^{40} +1.19094 q^{41} -3.93505 q^{43} -1.67831 q^{44} -1.62182 q^{46} -1.55685 q^{47} +10.8240 q^{49} +3.54779 q^{50} -3.39389 q^{52} -5.73700 q^{53} +2.02250 q^{55} -4.22185 q^{56} -5.50042 q^{58} -1.95825 q^{59} +0.935725 q^{61} -5.74587 q^{62} +1.00000 q^{64} +4.08991 q^{65} +9.79140 q^{67} -6.30450 q^{68} +5.08766 q^{70} +2.12729 q^{71} +7.02740 q^{73} -6.81324 q^{74} +0.549921 q^{76} -7.08559 q^{77} -12.4006 q^{79} -1.20508 q^{80} -1.19094 q^{82} -1.69109 q^{83} +7.59742 q^{85} +3.93505 q^{86} +1.67831 q^{88} +1.92164 q^{89} -14.3285 q^{91} +1.62182 q^{92} +1.55685 q^{94} -0.662698 q^{95} -0.826873 q^{97} -10.8240 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{2} + 12 q^{4} - 5 q^{5} + 6 q^{7} - 12 q^{8}+O(q^{10}) \) 12 * q - 12 * q^2 + 12 * q^4 - 5 * q^5 + 6 * q^7 - 12 * q^8	\( 12 q - 12 q^{2} + 12 q^{4} - 5 q^{5} + 6 q^{7} - 12 q^{8} + 5 q^{10} - 6 q^{11} + 3 q^{13} - 6 q^{14} + 12 q^{16} - 6 q^{17} + 8 q^{19} - 5 q^{20} + 6 q^{22} - 11 q^{23} + 11 q^{25} - 3 q^{26} + 6 q^{28} - 29 q^{29} + 2 q^{31} - 12 q^{32} + 6 q^{34} - 4 q^{35} + 5 q^{37} - 8 q^{38} + 5 q^{40} - 22 q^{41} + 9 q^{43} - 6 q^{44} + 11 q^{46} - 15 q^{47} + 14 q^{49} - 11 q^{50} + 3 q^{52} - 12 q^{53} + 13 q^{55} - 6 q^{56} + 29 q^{58} - 34 q^{59} - 4 q^{61} - 2 q^{62} + 12 q^{64} - 12 q^{65} + q^{67} - 6 q^{68} + 4 q^{70} - 21 q^{71} - 2 q^{73} - 5 q^{74} + 8 q^{76} - 34 q^{77} + 9 q^{79} - 5 q^{80} + 22 q^{82} - 10 q^{83} + 5 q^{85} - 9 q^{86} + 6 q^{88} + 2 q^{89} + 17 q^{91} - 11 q^{92} + 15 q^{94} - 69 q^{95} - 13 q^{97} - 14 q^{98}+O(q^{100}) \) 12 * q - 12 * q^2 + 12 * q^4 - 5 * q^5 + 6 * q^7 - 12 * q^8 + 5 * q^10 - 6 * q^11 + 3 * q^13 - 6 * q^14 + 12 * q^16 - 6 * q^17 + 8 * q^19 - 5 * q^20 + 6 * q^22 - 11 * q^23 + 11 * q^25 - 3 * q^26 + 6 * q^28 - 29 * q^29 + 2 * q^31 - 12 * q^32 + 6 * q^34 - 4 * q^35 + 5 * q^37 - 8 * q^38 + 5 * q^40 - 22 * q^41 + 9 * q^43 - 6 * q^44 + 11 * q^46 - 15 * q^47 + 14 * q^49 - 11 * q^50 + 3 * q^52 - 12 * q^53 + 13 * q^55 - 6 * q^56 + 29 * q^58 - 34 * q^59 - 4 * q^61 - 2 * q^62 + 12 * q^64 - 12 * q^65 + q^67 - 6 * q^68 + 4 * q^70 - 21 * q^71 - 2 * q^73 - 5 * q^74 + 8 * q^76 - 34 * q^77 + 9 * q^79 - 5 * q^80 + 22 * q^82 - 10 * q^83 + 5 * q^85 - 9 * q^86 + 6 * q^88 + 2 * q^89 + 17 * q^91 - 11 * q^92 + 15 * q^94 - 69 * q^95 - 13 * q^97 - 14 * q^98

Introduction

L-functions

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