LMFDB - Embedded newform 8046.2.a.q.1.8

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:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 2 x^{13} - 42 x^{12} + 70 x^{11} + 680 x^{10} - 882 x^{9} - 5559 x^{8} + 5066 x^{7} + \cdots - 7083 $ x^14 - 2x^13 - 42x^12 + 70x^11 + 680x^10 - 882x^9 - 5559x^8 + 5066x^7 + 24455x^6 - 12990x^5 - 55580x^4 + 9808x^3 + 53551x^2 + 6282*x - 7083
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-0.541630$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +0.541630 q^{5} +3.83971 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +0.541630 q^{5} +3.83971 q^{7} -1.00000 q^{8} -0.541630 q^{10} +4.61398 q^{11} -0.851387 q^{13} -3.83971 q^{14} +1.00000 q^{16} +6.23362 q^{17} +6.81016 q^{19} +0.541630 q^{20} -4.61398 q^{22} -7.84623 q^{23} -4.70664 q^{25} +0.851387 q^{26} +3.83971 q^{28} -3.10642 q^{29} -6.26522 q^{31} -1.00000 q^{32} -6.23362 q^{34} +2.07970 q^{35} +10.0388 q^{37} -6.81016 q^{38} -0.541630 q^{40} -2.40500 q^{41} +1.56645 q^{43} +4.61398 q^{44} +7.84623 q^{46} -1.46207 q^{47} +7.74335 q^{49} +4.70664 q^{50} -0.851387 q^{52} +9.31424 q^{53} +2.49907 q^{55} -3.83971 q^{56} +3.10642 q^{58} +7.20189 q^{59} +10.2457 q^{61} +6.26522 q^{62} +1.00000 q^{64} -0.461136 q^{65} +10.2356 q^{67} +6.23362 q^{68} -2.07970 q^{70} -7.51292 q^{71} +9.36139 q^{73} -10.0388 q^{74} +6.81016 q^{76} +17.7163 q^{77} +8.95017 q^{79} +0.541630 q^{80} +2.40500 q^{82} -4.80371 q^{83} +3.37631 q^{85} -1.56645 q^{86} -4.61398 q^{88} -6.21905 q^{89} -3.26908 q^{91} -7.84623 q^{92} +1.46207 q^{94} +3.68858 q^{95} -7.03070 q^{97} -7.74335 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 14 q^{2} + 14 q^{4} - 2 q^{5} + 4 q^{7} - 14 q^{8}+O(q^{10}) $ 14 * q - 14 * q^2 + 14 * q^4 - 2 * q^5 + 4 * q^7 - 14 * q^8	$ 14 q - 14 q^{2} + 14 q^{4} - 2 q^{5} + 4 q^{7} - 14 q^{8} + 2 q^{10} - 2 q^{11} + 4 q^{13} - 4 q^{14} + 14 q^{16} - 9 q^{17} + 14 q^{19} - 2 q^{20} + 2 q^{22} - 30 q^{23} + 18 q^{25} - 4 q^{26} + 4 q^{28} - 6 q^{29} + 11 q^{31} - 14 q^{32} + 9 q^{34} + 18 q^{35} + 13 q^{37} - 14 q^{38} + 2 q^{40} + 2 q^{41} + 12 q^{43} - 2 q^{44} + 30 q^{46} - 21 q^{47} + 32 q^{49} - 18 q^{50} + 4 q^{52} - 22 q^{53} - 7 q^{55} - 4 q^{56} + 6 q^{58} - 14 q^{59} + 31 q^{61} - 11 q^{62} + 14 q^{64} + 24 q^{67} - 9 q^{68} - 18 q^{70} - 28 q^{71} + 24 q^{73} - 13 q^{74} + 14 q^{76} - 16 q^{77} + 65 q^{79} - 2 q^{80} - 2 q^{82} + 15 q^{83} - 19 q^{85} - 12 q^{86} + 2 q^{88} + 11 q^{89} + 68 q^{91} - 30 q^{92} + 21 q^{94} - 8 q^{95} + 23 q^{97} - 32 q^{98}+O(q^{100}) $ 14 * q - 14 * q^2 + 14 * q^4 - 2 * q^5 + 4 * q^7 - 14 * q^8 + 2 * q^10 - 2 * q^11 + 4 * q^13 - 4 * q^14 + 14 * q^16 - 9 * q^17 + 14 * q^19 - 2 * q^20 + 2 * q^22 - 30 * q^23 + 18 * q^25 - 4 * q^26 + 4 * q^28 - 6 * q^29 + 11 * q^31 - 14 * q^32 + 9 * q^34 + 18 * q^35 + 13 * q^37 - 14 * q^38 + 2 * q^40 + 2 * q^41 + 12 * q^43 - 2 * q^44 + 30 * q^46 - 21 * q^47 + 32 * q^49 - 18 * q^50 + 4 * q^52 - 22 * q^53 - 7 * q^55 - 4 * q^56 + 6 * q^58 - 14 * q^59 + 31 * q^61 - 11 * q^62 + 14 * q^64 + 24 * q^67 - 9 * q^68 - 18 * q^70 - 28 * q^71 + 24 * q^73 - 13 * q^74 + 14 * q^76 - 16 * q^77 + 65 * q^79 - 2 * q^80 - 2 * q^82 + 15 * q^83 - 19 * q^85 - 12 * q^86 + 2 * q^88 + 11 * q^89 + 68 * q^91 - 30 * q^92 + 21 * q^94 - 8 * q^95 + 23 * q^97 - 32 * 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
$5$	0.541630	0.242224	0.121112	−	0.992639i	$-0.461354\pi$
$5$	0.541630	0.242224	0.121112	+	0.992639i	$0.461354\pi$
$6$	0	0
$7$	3.83971	1.45127	0.725636	−	0.688078i	$-0.241546\pi$
$7$	3.83971	1.45127	0.725636	+	0.688078i	$0.241546\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−0.541630	−0.171278
$11$	4.61398	1.39117	0.695584	−	0.718445i	$-0.255146\pi$
$11$	4.61398	1.39117	0.695584	+	0.718445i	$0.255146\pi$
$12$	0	0
$13$	−0.851387	−0.236132	−0.118066	−	0.993006i	$-0.537669\pi$
$13$	−0.851387	−0.236132	−0.118066	+	0.993006i	$0.537669\pi$
$14$	−3.83971	−1.02620
$15$	0	0
$16$	1.00000	0.250000
$17$	6.23362	1.51187	0.755937	−	0.654644i	$-0.227181\pi$
$17$	6.23362	1.51187	0.755937	+	0.654644i	$0.227181\pi$
$18$	0	0
$19$	6.81016	1.56236	0.781179	−	0.624307i	$-0.214619\pi$
$19$	6.81016	1.56236	0.781179	+	0.624307i	$0.214619\pi$
$20$	0.541630	0.121112
$21$	0	0
$22$	−4.61398	−0.983704
$23$	−7.84623	−1.63605	−0.818026	−	0.575181i	$-0.804932\pi$
$23$	−7.84623	−1.63605	−0.818026	+	0.575181i	$0.804932\pi$
$24$	0	0
$25$	−4.70664	−0.941327
$26$	0.851387	0.166971
$27$	0	0
$28$	3.83971	0.725636
$29$	−3.10642	−0.576848	−0.288424	−	0.957503i	$-0.593131\pi$
$29$	−3.10642	−0.576848	−0.288424	+	0.957503i	$0.593131\pi$
$30$	0	0
$31$	−6.26522	−1.12527	−0.562633	−	0.826707i	$-0.690212\pi$
$31$	−6.26522	−1.12527	−0.562633	+	0.826707i	$0.690212\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−6.23362	−1.06906
$35$	2.07970	0.351533
$36$	0	0
$37$	10.0388	1.65036	0.825181	−	0.564869i	$-0.191073\pi$
$37$	10.0388	1.65036	0.825181	+	0.564869i	$0.191073\pi$
$38$	−6.81016	−1.10475
$39$	0	0
$40$	−0.541630	−0.0856392
$41$	−2.40500	−0.375598	−0.187799	−	0.982207i	$-0.560135\pi$
$41$	−2.40500	−0.375598	−0.187799	+	0.982207i	$0.560135\pi$
$42$	0	0
$43$	1.56645	0.238881	0.119440	−	0.992841i	$-0.461890\pi$
$43$	1.56645	0.238881	0.119440	+	0.992841i	$0.461890\pi$
$44$	4.61398	0.695584
$45$	0	0
$46$	7.84623	1.15686
$47$	−1.46207	−0.213264	−0.106632	−	0.994299i	$-0.534007\pi$
$47$	−1.46207	−0.213264	−0.106632	+	0.994299i	$0.534007\pi$
$48$	0	0
$49$	7.74335	1.10619
$50$	4.70664	0.665619
$51$	0	0
$52$	−0.851387	−0.118066
$53$	9.31424	1.27941	0.639704	−	0.768621i	$-0.279057\pi$
$53$	9.31424	1.27941	0.639704	+	0.768621i	$0.279057\pi$
$54$	0	0
$55$	2.49907	0.336974
$56$	−3.83971	−0.513102
$57$	0	0
$58$	3.10642	0.407893
$59$	7.20189	0.937607	0.468803	−	0.883303i	$-0.344685\pi$
$59$	7.20189	0.937607	0.468803	+	0.883303i	$0.344685\pi$
$60$	0	0
$61$	10.2457	1.31183	0.655915	−	0.754835i	$-0.272283\pi$
$61$	10.2457	1.31183	0.655915	+	0.754835i	$0.272283\pi$
$62$	6.26522	0.795683
$63$	0	0
$64$	1.00000	0.125000
$65$	−0.461136	−0.0571969
$66$	0	0
$67$	10.2356	1.25048	0.625239	−	0.780433i	$-0.285001\pi$
$67$	10.2356	1.25048	0.625239	+	0.780433i	$0.285001\pi$
$68$	6.23362	0.755937
$69$	0	0
$70$	−2.07970	−0.248572
$71$	−7.51292	−0.891620	−0.445810	−	0.895128i	$-0.647084\pi$
$71$	−7.51292	−0.891620	−0.445810	+	0.895128i	$0.647084\pi$
$72$	0	0
$73$	9.36139	1.09567	0.547834	−	0.836587i	$-0.315453\pi$
$73$	9.36139	1.09567	0.547834	+	0.836587i	$0.315453\pi$
$74$	−10.0388	−1.16698
$75$	0	0
$76$	6.81016	0.781179
$77$	17.7163	2.01896
$78$	0	0
$79$	8.95017	1.00697	0.503487	−	0.864003i	$-0.332050\pi$
$79$	8.95017	1.00697	0.503487	+	0.864003i	$0.332050\pi$
$80$	0.541630	0.0605560
$81$	0	0
$82$	2.40500	0.265588
$83$	−4.80371	−0.527276	−0.263638	−	0.964622i	$-0.584922\pi$
$83$	−4.80371	−0.527276	−0.263638	+	0.964622i	$0.584922\pi$
$84$	0	0
$85$	3.37631	0.366213
$86$	−1.56645	−0.168914
$87$	0	0
$88$	−4.61398	−0.491852
$89$	−6.21905	−0.659218	−0.329609	−	0.944118i	$-0.606917\pi$
$89$	−6.21905	−0.659218	−0.329609	+	0.944118i	$0.606917\pi$
$90$	0	0
$91$	−3.26908	−0.342692
$92$	−7.84623	−0.818026
$93$	0	0
$94$	1.46207	0.150801
$95$	3.68858	0.378441
$96$	0	0
$97$	−7.03070	−0.713860	−0.356930	−	0.934131i	$-0.616176\pi$
$97$	−7.03070	−0.713860	−0.356930	+	0.934131i	$0.616176\pi$
$98$	−7.74335	−0.782196
$99$	0	0
$100$	−4.70664	−0.470664
$101$	−16.0390	−1.59594	−0.797969	−	0.602698i	$-0.794092\pi$
$101$	−16.0390	−1.59594	−0.797969	+	0.602698i	$0.794092\pi$
$102$	0	0
$103$	3.77932	0.372388	0.186194	−	0.982513i	$-0.440385\pi$
$103$	3.77932	0.372388	0.186194	+	0.982513i	$0.440385\pi$
$104$	0.851387	0.0834853
$105$	0	0
$106$	−9.31424	−0.904679
$107$	16.5149	1.59656	0.798279	−	0.602287i	$-0.205744\pi$
$107$	16.5149	1.59656	0.798279	+	0.602287i	$0.205744\pi$
$108$	0	0
$109$	7.76752	0.743993	0.371997	−	0.928234i	$-0.378673\pi$
$109$	7.76752	0.743993	0.371997	+	0.928234i	$0.378673\pi$
$110$	−2.49907	−0.238277
$111$	0	0
$112$	3.83971	0.362818
$113$	−17.9157	−1.68537	−0.842686	−	0.538406i	$-0.819027\pi$
$113$	−17.9157	−1.68537	−0.842686	+	0.538406i	$0.819027\pi$
$114$	0	0
$115$	−4.24975	−0.396291
$116$	−3.10642	−0.288424
$117$	0	0
$118$	−7.20189	−0.662988
$119$	23.9353	2.19414
$120$	0	0
$121$	10.2888	0.935348
$122$	−10.2457	−0.927604
$123$	0	0
$124$	−6.26522	−0.562633
$125$	−5.25740	−0.470236
$126$	0	0
$127$	−5.71054	−0.506728	−0.253364	−	0.967371i	$-0.581537\pi$
$127$	−5.71054	−0.506728	−0.253364	+	0.967371i	$0.581537\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0.461136	0.0404443
$131$	−4.15378	−0.362918	−0.181459	−	0.983399i	$-0.558082\pi$
$131$	−4.15378	−0.362918	−0.181459	+	0.983399i	$0.558082\pi$
$132$	0	0
$133$	26.1490	2.26741
$134$	−10.2356	−0.884222
$135$	0	0
$136$	−6.23362	−0.534528
$137$	−10.6321	−0.908365	−0.454183	−	0.890909i	$-0.650069\pi$
$137$	−10.6321	−0.908365	−0.454183	+	0.890909i	$0.650069\pi$
$138$	0	0
$139$	−16.1763	−1.37206	−0.686029	−	0.727574i	$-0.740648\pi$
$139$	−16.1763	−1.37206	−0.686029	+	0.727574i	$0.740648\pi$
$140$	2.07970	0.175767
$141$	0	0
$142$	7.51292	0.630470
$143$	−3.92828	−0.328500
$144$	0	0
$145$	−1.68253	−0.139726
$146$	−9.36139	−0.774754
$147$	0	0
$148$	10.0388	0.825181
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	−7.78058	−0.633175	−0.316587	−	0.948563i	$-0.602537\pi$
$151$	−7.78058	−0.633175	−0.316587	+	0.948563i	$0.602537\pi$
$152$	−6.81016	−0.552377
$153$	0	0
$154$	−17.7163	−1.42762
$155$	−3.39343	−0.272567
$156$	0	0
$157$	7.01357	0.559744	0.279872	−	0.960037i	$-0.409708\pi$
$157$	7.01357	0.559744	0.279872	+	0.960037i	$0.409708\pi$
$158$	−8.95017	−0.712038
$159$	0	0
$160$	−0.541630	−0.0428196
$161$	−30.1272	−2.37436
$162$	0	0
$163$	0.522099	0.0408939	0.0204470	−	0.999791i	$-0.493491\pi$
$163$	0.522099	0.0408939	0.0204470	+	0.999791i	$0.493491\pi$
$164$	−2.40500	−0.187799
$165$	0	0
$166$	4.80371	0.372840
$167$	0.694263	0.0537237	0.0268618	−	0.999639i	$-0.491449\pi$
$167$	0.694263	0.0537237	0.0268618	+	0.999639i	$0.491449\pi$
$168$	0	0
$169$	−12.2751	−0.944242
$170$	−3.37631	−0.258951
$171$	0	0
$172$	1.56645	0.119440
$173$	20.2344	1.53840	0.769198	−	0.639011i	$-0.220656\pi$
$173$	20.2344	1.53840	0.769198	+	0.639011i	$0.220656\pi$
$174$	0	0
$175$	−18.0721	−1.36612
$176$	4.61398	0.347792
$177$	0	0
$178$	6.21905	0.466138
$179$	2.97140	0.222093	0.111046	−	0.993815i	$-0.464580\pi$
$179$	2.97140	0.222093	0.111046	+	0.993815i	$0.464580\pi$
$180$	0	0
$181$	−12.7209	−0.945540	−0.472770	−	0.881186i	$-0.656746\pi$
$181$	−12.7209	−0.945540	−0.472770	+	0.881186i	$0.656746\pi$
$182$	3.26908	0.242320
$183$	0	0
$184$	7.84623	0.578432
$185$	5.43729	0.399757
$186$	0	0
$187$	28.7618	2.10327
$188$	−1.46207	−0.106632
$189$	0	0
$190$	−3.68858	−0.267598
$191$	22.4661	1.62559	0.812794	−	0.582552i	$-0.197946\pi$
$191$	22.4661	1.62559	0.812794	+	0.582552i	$0.197946\pi$
$192$	0	0
$193$	23.5808	1.69738	0.848692	−	0.528888i	$-0.177391\pi$
$193$	23.5808	1.69738	0.848692	+	0.528888i	$0.177391\pi$
$194$	7.03070	0.504775
$195$	0	0
$196$	7.74335	0.553096
$197$	−19.3337	−1.37747	−0.688735	−	0.725013i	$-0.741834\pi$
$197$	−19.3337	−1.37747	−0.688735	+	0.725013i	$0.741834\pi$
$198$	0	0
$199$	23.5740	1.67111	0.835557	−	0.549404i	$-0.185145\pi$
$199$	23.5740	1.67111	0.835557	+	0.549404i	$0.185145\pi$
$200$	4.70664	0.332810
$201$	0	0
$202$	16.0390	1.12850
$203$	−11.9277	−0.837163
$204$	0	0
$205$	−1.30262	−0.0909789
$206$	−3.77932	−0.263318
$207$	0	0
$208$	−0.851387	−0.0590331
$209$	31.4219	2.17350
$210$	0	0
$211$	15.8528	1.09135	0.545676	−	0.837996i	$-0.316273\pi$
$211$	15.8528	1.09135	0.545676	+	0.837996i	$0.316273\pi$
$212$	9.31424	0.639704
$213$	0	0
$214$	−16.5149	−1.12894
$215$	0.848434	0.0578627
$216$	0	0
$217$	−24.0566	−1.63307
$218$	−7.76752	−0.526083
$219$	0	0
$220$	2.49907	0.168487
$221$	−5.30722	−0.357002
$222$	0	0
$223$	−21.7748	−1.45815	−0.729075	−	0.684434i	$-0.760049\pi$
$223$	−21.7748	−1.45815	−0.729075	+	0.684434i	$0.760049\pi$
$224$	−3.83971	−0.256551
$225$	0	0
$226$	17.9157	1.19174
$227$	−7.12157	−0.472675	−0.236338	−	0.971671i	$-0.575947\pi$
$227$	−7.12157	−0.472675	−0.236338	+	0.971671i	$0.575947\pi$
$228$	0	0
$229$	7.15412	0.472757	0.236379	−	0.971661i	$-0.424039\pi$
$229$	7.15412	0.472757	0.236379	+	0.971661i	$0.424039\pi$
$230$	4.24975	0.280220
$231$	0	0
$232$	3.10642	0.203946
$233$	3.66468	0.240081	0.120041	−	0.992769i	$-0.461697\pi$
$233$	3.66468	0.240081	0.120041	+	0.992769i	$0.461697\pi$
$234$	0	0
$235$	−0.791899	−0.0516578
$236$	7.20189	0.468803
$237$	0	0
$238$	−23.9353	−1.55149
$239$	−1.59313	−0.103051	−0.0515254	−	0.998672i	$-0.516408\pi$
$239$	−1.59313	−0.103051	−0.0515254	+	0.998672i	$0.516408\pi$
$240$	0	0
$241$	−8.11827	−0.522944	−0.261472	−	0.965211i	$-0.584208\pi$
$241$	−8.11827	−0.522944	−0.261472	+	0.965211i	$0.584208\pi$
$242$	−10.2888	−0.661391
$243$	0	0
$244$	10.2457	0.655915
$245$	4.19403	0.267946
$246$	0	0
$247$	−5.79808	−0.368923
$248$	6.26522	0.397842
$249$	0	0
$250$	5.25740	0.332507
$251$	3.90033	0.246187	0.123093	−	0.992395i	$-0.460719\pi$
$251$	3.90033	0.246187	0.123093	+	0.992395i	$0.460719\pi$
$252$	0	0
$253$	−36.2024	−2.27602
$254$	5.71054	0.358311
$255$	0	0
$256$	1.00000	0.0625000
$257$	−4.94048	−0.308179	−0.154089	−	0.988057i	$-0.549244\pi$
$257$	−4.94048	−0.308179	−0.154089	+	0.988057i	$0.549244\pi$
$258$	0	0
$259$	38.5459	2.39512
$260$	−0.461136	−0.0285985
$261$	0	0
$262$	4.15378	0.256622
$263$	−3.80892	−0.234868	−0.117434	−	0.993081i	$-0.537467\pi$
$263$	−3.80892	−0.234868	−0.117434	+	0.993081i	$0.537467\pi$
$264$	0	0
$265$	5.04487	0.309904
$266$	−26.1490	−1.60330
$267$	0	0
$268$	10.2356	0.625239
$269$	15.1636	0.924543	0.462272	−	0.886738i	$-0.347034\pi$
$269$	15.1636	0.924543	0.462272	+	0.886738i	$0.347034\pi$
$270$	0	0
$271$	0.778862	0.0473125	0.0236563	−	0.999720i	$-0.492469\pi$
$271$	0.778862	0.0473125	0.0236563	+	0.999720i	$0.492469\pi$
$272$	6.23362	0.377969
$273$	0	0
$274$	10.6321	0.642311
$275$	−21.7163	−1.30954
$276$	0	0
$277$	−26.1963	−1.57398	−0.786991	−	0.616965i	$-0.788362\pi$
$277$	−26.1963	−1.57398	−0.786991	+	0.616965i	$0.788362\pi$
$278$	16.1763	0.970192
$279$	0	0
$280$	−2.07970	−0.124286
$281$	−21.8702	−1.30467	−0.652333	−	0.757933i	$-0.726209\pi$
$281$	−21.8702	−1.30467	−0.652333	+	0.757933i	$0.726209\pi$
$282$	0	0
$283$	−1.62401	−0.0965372	−0.0482686	−	0.998834i	$-0.515370\pi$
$283$	−1.62401	−0.0965372	−0.0482686	+	0.998834i	$0.515370\pi$
$284$	−7.51292	−0.445810
$285$	0	0
$286$	3.92828	0.232284
$287$	−9.23449	−0.545095
$288$	0	0
$289$	21.8580	1.28577
$290$	1.68253	0.0988015
$291$	0	0
$292$	9.36139	0.547834
$293$	2.34720	0.137125	0.0685623	−	0.997647i	$-0.478159\pi$
$293$	2.34720	0.137125	0.0685623	+	0.997647i	$0.478159\pi$
$294$	0	0
$295$	3.90076	0.227111
$296$	−10.0388	−0.583491
$297$	0	0
$298$	1.00000	0.0579284
$299$	6.68018	0.386325
$300$	0	0
$301$	6.01469	0.346681
$302$	7.78058	0.447722
$303$	0	0
$304$	6.81016	0.390589
$305$	5.54939	0.317757
$306$	0	0
$307$	24.5965	1.40380	0.701899	−	0.712277i	$-0.252336\pi$
$307$	24.5965	1.40380	0.701899	+	0.712277i	$0.252336\pi$
$308$	17.7163	1.00948
$309$	0	0
$310$	3.39343	0.192734
$311$	3.58440	0.203253	0.101626	−	0.994823i	$-0.467595\pi$
$311$	3.58440	0.203253	0.101626	+	0.994823i	$0.467595\pi$
$312$	0	0
$313$	−19.4563	−1.09974	−0.549868	−	0.835252i	$-0.685322\pi$
$313$	−19.4563	−1.09974	−0.549868	+	0.835252i	$0.685322\pi$
$314$	−7.01357	−0.395799
$315$	0	0
$316$	8.95017	0.503487
$317$	−9.82199	−0.551658	−0.275829	−	0.961207i	$-0.588952\pi$
$317$	−9.82199	−0.551658	−0.275829	+	0.961207i	$0.588952\pi$
$318$	0	0
$319$	−14.3330	−0.802492
$320$	0.541630	0.0302780
$321$	0	0
$322$	30.1272	1.67892
$323$	42.4519	2.36209
$324$	0	0
$325$	4.00717	0.222278
$326$	−0.522099	−0.0289164
$327$	0	0
$328$	2.40500	0.132794
$329$	−5.61391	−0.309505
$330$	0	0
$331$	18.7639	1.03136	0.515678	−	0.856782i	$-0.327540\pi$
$331$	18.7639	1.03136	0.515678	+	0.856782i	$0.327540\pi$
$332$	−4.80371	−0.263638
$333$	0	0
$334$	−0.694263	−0.0379884
$335$	5.54391	0.302896
$336$	0	0
$337$	−30.7997	−1.67777	−0.838883	−	0.544311i	$-0.816791\pi$
$337$	−30.7997	−1.67777	−0.838883	+	0.544311i	$0.816791\pi$
$338$	12.2751	0.667680
$339$	0	0
$340$	3.37631	0.183106
$341$	−28.9076	−1.56543
$342$	0	0
$343$	2.85423	0.154114
$344$	−1.56645	−0.0844571
$345$	0	0
$346$	−20.2344	−1.08781
$347$	11.5057	0.617660	0.308830	−	0.951117i	$-0.400063\pi$
$347$	11.5057	0.617660	0.308830	+	0.951117i	$0.400063\pi$
$348$	0	0
$349$	−3.43190	−0.183705	−0.0918526	−	0.995773i	$-0.529279\pi$
$349$	−3.43190	−0.183705	−0.0918526	+	0.995773i	$0.529279\pi$
$350$	18.0721	0.965995
$351$	0	0
$352$	−4.61398	−0.245926
$353$	−10.7842	−0.573985	−0.286992	−	0.957933i	$-0.592655\pi$
$353$	−10.7842	−0.573985	−0.286992	+	0.957933i	$0.592655\pi$
$354$	0	0
$355$	−4.06922	−0.215972
$356$	−6.21905	−0.329609
$357$	0	0
$358$	−2.97140	−0.157043
$359$	−4.35125	−0.229650	−0.114825	−	0.993386i	$-0.536631\pi$
$359$	−4.35125	−0.229650	−0.114825	+	0.993386i	$0.536631\pi$
$360$	0	0
$361$	27.3783	1.44096
$362$	12.7209	0.668597
$363$	0	0
$364$	−3.26908	−0.171346
$365$	5.07041	0.265397
$366$	0	0
$367$	−35.1762	−1.83618	−0.918092	−	0.396367i	$-0.870271\pi$
$367$	−35.1762	−1.83618	−0.918092	+	0.396367i	$0.870271\pi$
$368$	−7.84623	−0.409013
$369$	0	0
$370$	−5.43729	−0.282671
$371$	35.7639	1.85677
$372$	0	0
$373$	−15.7885	−0.817498	−0.408749	−	0.912647i	$-0.634035\pi$
$373$	−15.7885	−0.817498	−0.408749	+	0.912647i	$0.634035\pi$
$374$	−28.7618	−1.48724
$375$	0	0
$376$	1.46207	0.0754003
$377$	2.64476	0.136212
$378$	0	0
$379$	1.54608	0.0794168	0.0397084	−	0.999211i	$-0.487357\pi$
$379$	1.54608	0.0794168	0.0397084	+	0.999211i	$0.487357\pi$
$380$	3.68858	0.189220
$381$	0	0
$382$	−22.4661	−1.14946
$383$	−23.0028	−1.17539	−0.587695	−	0.809083i	$-0.699964\pi$
$383$	−23.0028	−1.17539	−0.587695	+	0.809083i	$0.699964\pi$
$384$	0	0
$385$	9.59569	0.489042
$386$	−23.5808	−1.20023
$387$	0	0
$388$	−7.03070	−0.356930
$389$	−5.10201	−0.258682	−0.129341	−	0.991600i	$-0.541286\pi$
$389$	−5.10201	−0.258682	−0.129341	+	0.991600i	$0.541286\pi$
$390$	0	0
$391$	−48.9104	−2.47351
$392$	−7.74335	−0.391098
$393$	0	0
$394$	19.3337	0.974019
$395$	4.84768	0.243913
$396$	0	0
$397$	4.63306	0.232526	0.116263	−	0.993218i	$-0.462908\pi$
$397$	4.63306	0.232526	0.116263	+	0.993218i	$0.462908\pi$
$398$	−23.5740	−1.18166
$399$	0	0
$400$	−4.70664	−0.235332
$401$	−8.09165	−0.404078	−0.202039	−	0.979378i	$-0.564757\pi$
$401$	−8.09165	−0.404078	−0.202039	+	0.979378i	$0.564757\pi$
$402$	0	0
$403$	5.33412	0.265712
$404$	−16.0390	−0.797969
$405$	0	0
$406$	11.9277	0.591964
$407$	46.3186	2.29593
$408$	0	0
$409$	17.2833	0.854606	0.427303	−	0.904109i	$-0.359464\pi$
$409$	17.2833	0.854606	0.427303	+	0.904109i	$0.359464\pi$
$410$	1.30262	0.0643318
$411$	0	0
$412$	3.77932	0.186194
$413$	27.6532	1.36072
$414$	0	0
$415$	−2.60183	−0.127719
$416$	0.851387	0.0417427
$417$	0	0
$418$	−31.4219	−1.53690
$419$	7.15760	0.349671	0.174836	−	0.984598i	$-0.444061\pi$
$419$	7.15760	0.349671	0.174836	+	0.984598i	$0.444061\pi$
$420$	0	0
$421$	17.2363	0.840045	0.420023	−	0.907514i	$-0.362022\pi$
$421$	17.2363	0.840045	0.420023	+	0.907514i	$0.362022\pi$
$422$	−15.8528	−0.771703
$423$	0	0
$424$	−9.31424	−0.452339
$425$	−29.3394	−1.42317
$426$	0	0
$427$	39.3405	1.90382
$428$	16.5149	0.798279
$429$	0	0
$430$	−0.848434	−0.0409151
$431$	−23.4748	−1.13074	−0.565371	−	0.824837i	$-0.691267\pi$
$431$	−23.4748	−1.13074	−0.565371	+	0.824837i	$0.691267\pi$
$432$	0	0
$433$	17.1686	0.825068	0.412534	−	0.910942i	$-0.364644\pi$
$433$	17.1686	0.825068	0.412534	+	0.910942i	$0.364644\pi$
$434$	24.0566	1.15475
$435$	0	0
$436$	7.76752	0.371997
$437$	−53.4341	−2.55610
$438$	0	0
$439$	−39.6526	−1.89252	−0.946259	−	0.323411i	$-0.895170\pi$
$439$	−39.6526	−1.89252	−0.946259	+	0.323411i	$0.895170\pi$
$440$	−2.49907	−0.119138
$441$	0	0
$442$	5.30722	0.252439
$443$	29.8126	1.41644	0.708220	−	0.705992i	$-0.249498\pi$
$443$	29.8126	1.41644	0.708220	+	0.705992i	$0.249498\pi$
$444$	0	0
$445$	−3.36842	−0.159679
$446$	21.7748	1.03107
$447$	0	0
$448$	3.83971	0.181409
$449$	33.1854	1.56612	0.783059	−	0.621947i	$-0.213658\pi$
$449$	33.1854	1.56612	0.783059	+	0.621947i	$0.213658\pi$
$450$	0	0
$451$	−11.0966	−0.522520
$452$	−17.9157	−0.842686
$453$	0	0
$454$	7.12157	0.334232
$455$	−1.77063	−0.0830083
$456$	0	0
$457$	−1.83444	−0.0858117	−0.0429058	−	0.999079i	$-0.513662\pi$
$457$	−1.83444	−0.0858117	−0.0429058	+	0.999079i	$0.513662\pi$
$458$	−7.15412	−0.334290
$459$	0	0
$460$	−4.24975	−0.198146
$461$	24.9472	1.16191	0.580954	−	0.813936i	$-0.302680\pi$
$461$	24.9472	1.16191	0.580954	+	0.813936i	$0.302680\pi$
$462$	0	0
$463$	13.8121	0.641904	0.320952	−	0.947095i	$-0.395997\pi$
$463$	13.8121	0.641904	0.320952	+	0.947095i	$0.395997\pi$
$464$	−3.10642	−0.144212
$465$	0	0
$466$	−3.66468	−0.169763
$467$	−18.4587	−0.854166	−0.427083	−	0.904212i	$-0.640459\pi$
$467$	−18.4587	−0.854166	−0.427083	+	0.904212i	$0.640459\pi$
$468$	0	0
$469$	39.3017	1.81479
$470$	0.791899	0.0365276
$471$	0	0
$472$	−7.20189	−0.331494
$473$	7.22756	0.332323
$474$	0	0
$475$	−32.0529	−1.47069
$476$	23.9353	1.09707
$477$	0	0
$478$	1.59313	0.0728679
$479$	29.4723	1.34663	0.673313	−	0.739358i	$-0.264871\pi$
$479$	29.4723	1.34663	0.673313	+	0.739358i	$0.264871\pi$
$480$	0	0
$481$	−8.54686	−0.389703
$482$	8.11827	0.369777
$483$	0	0
$484$	10.2888	0.467674
$485$	−3.80804	−0.172914
$486$	0	0
$487$	41.5276	1.88179	0.940897	−	0.338694i	$-0.109985\pi$
$487$	41.5276	1.88179	0.940897	+	0.338694i	$0.109985\pi$
$488$	−10.2457	−0.463802
$489$	0	0
$490$	−4.19403	−0.189467
$491$	−17.4020	−0.785342	−0.392671	−	0.919679i	$-0.628449\pi$
$491$	−17.4020	−0.785342	−0.392671	+	0.919679i	$0.628449\pi$
$492$	0	0
$493$	−19.3642	−0.872121
$494$	5.79808	0.260868
$495$	0	0
$496$	−6.26522	−0.281317
$497$	−28.8474	−1.29398
$498$	0	0
$499$	−25.9772	−1.16290	−0.581449	−	0.813583i	$-0.697514\pi$
$499$	−25.9772	−1.16290	−0.581449	+	0.813583i	$0.697514\pi$
$500$	−5.25740	−0.235118
$501$	0	0
$502$	−3.90033	−0.174080
$503$	−44.3888	−1.97920	−0.989599	−	0.143852i	$-0.954051\pi$
$503$	−44.3888	−1.97920	−0.989599	+	0.143852i	$0.954051\pi$
$504$	0	0
$505$	−8.68719	−0.386575
$506$	36.2024	1.60939
$507$	0	0
$508$	−5.71054	−0.253364
$509$	24.2597	1.07529	0.537646	−	0.843171i	$-0.319314\pi$
$509$	24.2597	1.07529	0.537646	+	0.843171i	$0.319314\pi$
$510$	0	0
$511$	35.9450	1.59011
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	4.94048	0.217915
$515$	2.04699	0.0902013
$516$	0	0
$517$	−6.74595	−0.296687
$518$	−38.5459	−1.69361
$519$	0	0
$520$	0.461136	0.0202222
$521$	31.8479	1.39528	0.697641	−	0.716447i	$-0.254233\pi$
$521$	31.8479	1.39528	0.697641	+	0.716447i	$0.254233\pi$
$522$	0	0
$523$	12.0751	0.528009	0.264004	−	0.964521i	$-0.414957\pi$
$523$	12.0751	0.528009	0.264004	+	0.964521i	$0.414957\pi$
$524$	−4.15378	−0.181459
$525$	0	0
$526$	3.80892	0.166077
$527$	−39.0550	−1.70126
$528$	0	0
$529$	38.5633	1.67667
$530$	−5.04487	−0.219135
$531$	0	0
$532$	26.1490	1.13370
$533$	2.04759	0.0886908
$534$	0	0
$535$	8.94498	0.386725
$536$	−10.2356	−0.442111
$537$	0	0
$538$	−15.1636	−0.653751
$539$	35.7277	1.53890
$540$	0	0
$541$	−8.82224	−0.379298	−0.189649	−	0.981852i	$-0.560735\pi$
$541$	−8.82224	−0.379298	−0.189649	+	0.981852i	$0.560735\pi$
$542$	−0.778862	−0.0334550
$543$	0	0
$544$	−6.23362	−0.267264
$545$	4.20712	0.180213
$546$	0	0
$547$	24.0941	1.03019	0.515094	−	0.857134i	$-0.327757\pi$
$547$	24.0941	1.03019	0.515094	+	0.857134i	$0.327757\pi$
$548$	−10.6321	−0.454183
$549$	0	0
$550$	21.7163	0.925988
$551$	−21.1552	−0.901242
$552$	0	0
$553$	34.3660	1.46139
$554$	26.1963	1.11297
$555$	0	0
$556$	−16.1763	−0.686029
$557$	−44.0332	−1.86575	−0.932873	−	0.360204i	$-0.882707\pi$
$557$	−44.0332	−1.86575	−0.932873	+	0.360204i	$0.882707\pi$
$558$	0	0
$559$	−1.33365	−0.0564075
$560$	2.07970	0.0878833
$561$	0	0
$562$	21.8702	0.922538
$563$	21.5238	0.907118	0.453559	−	0.891226i	$-0.350154\pi$
$563$	21.5238	0.907118	0.453559	+	0.891226i	$0.350154\pi$
$564$	0	0
$565$	−9.70370	−0.408238
$566$	1.62401	0.0682621
$567$	0	0
$568$	7.51292	0.315235
$569$	24.0393	1.00778	0.503889	−	0.863768i	$-0.331902\pi$
$569$	24.0393	1.00778	0.503889	+	0.863768i	$0.331902\pi$
$570$	0	0
$571$	18.7608	0.785116	0.392558	−	0.919727i	$-0.371590\pi$
$571$	18.7608	0.785116	0.392558	+	0.919727i	$0.371590\pi$
$572$	−3.92828	−0.164250
$573$	0	0
$574$	9.23449	0.385440
$575$	36.9294	1.54006
$576$	0	0
$577$	37.4087	1.55734	0.778672	−	0.627431i	$-0.215894\pi$
$577$	37.4087	1.55734	0.778672	+	0.627431i	$0.215894\pi$
$578$	−21.8580	−0.909174
$579$	0	0
$580$	−1.68253	−0.0698632
$581$	−18.4448	−0.765221
$582$	0	0
$583$	42.9757	1.77987
$584$	−9.36139	−0.387377
$585$	0	0
$586$	−2.34720	−0.0969618
$587$	−7.57658	−0.312719	−0.156360	−	0.987700i	$-0.549976\pi$
$587$	−7.57658	−0.312719	−0.156360	+	0.987700i	$0.549976\pi$
$588$	0	0
$589$	−42.6671	−1.75807
$590$	−3.90076	−0.160592
$591$	0	0
$592$	10.0388	0.412590
$593$	22.9668	0.943135	0.471567	−	0.881830i	$-0.343688\pi$
$593$	22.9668	0.943135	0.471567	+	0.881830i	$0.343688\pi$
$594$	0	0
$595$	12.9641	0.531474
$596$	−1.00000	−0.0409616
$597$	0	0
$598$	−6.68018	−0.273173
$599$	−5.40537	−0.220857	−0.110429	−	0.993884i	$-0.535222\pi$
$599$	−5.40537	−0.220857	−0.110429	+	0.993884i	$0.535222\pi$
$600$	0	0
$601$	22.4064	0.913975	0.456987	−	0.889473i	$-0.348928\pi$
$601$	22.4064	0.913975	0.456987	+	0.889473i	$0.348928\pi$
$602$	−6.01469	−0.245141
$603$	0	0
$604$	−7.78058	−0.316587
$605$	5.57274	0.226564
$606$	0	0
$607$	−24.9916	−1.01438	−0.507189	−	0.861835i	$-0.669316\pi$
$607$	−24.9916	−1.01438	−0.507189	+	0.861835i	$0.669316\pi$
$608$	−6.81016	−0.276188
$609$	0	0
$610$	−5.54939	−0.224688
$611$	1.24478	0.0503586
$612$	0	0
$613$	16.4400	0.664005	0.332003	−	0.943278i	$-0.392276\pi$
$613$	16.4400	0.664005	0.332003	+	0.943278i	$0.392276\pi$
$614$	−24.5965	−0.992635
$615$	0	0
$616$	−17.7163	−0.713812
$617$	21.2953	0.857317	0.428658	−	0.903467i	$-0.358986\pi$
$617$	21.2953	0.857317	0.428658	+	0.903467i	$0.358986\pi$
$618$	0	0
$619$	43.8122	1.76096	0.880480	−	0.474084i	$-0.157221\pi$
$619$	43.8122	1.76096	0.880480	+	0.474084i	$0.157221\pi$
$620$	−3.39343	−0.136283
$621$	0	0
$622$	−3.58440	−0.143721
$623$	−23.8793	−0.956705
$624$	0	0
$625$	20.6856	0.827425
$626$	19.4563	0.777631
$627$	0	0
$628$	7.01357	0.279872
$629$	62.5778	2.49514
$630$	0	0
$631$	−38.4267	−1.52974	−0.764871	−	0.644183i	$-0.777198\pi$
$631$	−38.4267	−1.52974	−0.764871	+	0.644183i	$0.777198\pi$
$632$	−8.95017	−0.356019
$633$	0	0
$634$	9.82199	0.390081
$635$	−3.09300	−0.122742
$636$	0	0
$637$	−6.59258	−0.261208
$638$	14.3330	0.567447
$639$	0	0
$640$	−0.541630	−0.0214098
$641$	−27.2355	−1.07574	−0.537869	−	0.843028i	$-0.680771\pi$
$641$	−27.2355	−1.07574	−0.537869	+	0.843028i	$0.680771\pi$
$642$	0	0
$643$	−15.0663	−0.594158	−0.297079	−	0.954853i	$-0.596012\pi$
$643$	−15.0663	−0.594158	−0.297079	+	0.954853i	$0.596012\pi$
$644$	−30.1272	−1.18718
$645$	0	0
$646$	−42.4519	−1.67025
$647$	16.4431	0.646446	0.323223	−	0.946323i	$-0.395234\pi$
$647$	16.4431	0.646446	0.323223	+	0.946323i	$0.395234\pi$
$648$	0	0
$649$	33.2294	1.30437
$650$	−4.00717	−0.157174
$651$	0	0
$652$	0.522099	0.0204470
$653$	−10.8821	−0.425851	−0.212925	−	0.977068i	$-0.568299\pi$
$653$	−10.8821	−0.425851	−0.212925	+	0.977068i	$0.568299\pi$
$654$	0	0
$655$	−2.24981	−0.0879075
$656$	−2.40500	−0.0938995
$657$	0	0
$658$	5.61391	0.218853
$659$	−28.1001	−1.09462	−0.547312	−	0.836928i	$-0.684349\pi$
$659$	−28.1001	−1.09462	−0.547312	+	0.836928i	$0.684349\pi$
$660$	0	0
$661$	35.2578	1.37137	0.685685	−	0.727898i	$-0.259503\pi$
$661$	35.2578	1.37137	0.685685	+	0.727898i	$0.259503\pi$
$662$	−18.7639	−0.729279
$663$	0	0
$664$	4.80371	0.186420
$665$	14.1631	0.549221
$666$	0	0
$667$	24.3737	0.943753
$668$	0.694263	0.0268618
$669$	0	0
$670$	−5.54391	−0.214180
$671$	47.2736	1.82498
$672$	0	0
$673$	−30.3046	−1.16816	−0.584078	−	0.811697i	$-0.698544\pi$
$673$	−30.3046	−1.16816	−0.584078	+	0.811697i	$0.698544\pi$
$674$	30.7997	1.18636
$675$	0	0
$676$	−12.2751	−0.472121
$677$	23.3210	0.896300	0.448150	−	0.893958i	$-0.352083\pi$
$677$	23.3210	0.896300	0.448150	+	0.893958i	$0.352083\pi$
$678$	0	0
$679$	−26.9958	−1.03600
$680$	−3.37631	−0.129476
$681$	0	0
$682$	28.9076	1.10693
$683$	3.97935	0.152266	0.0761329	−	0.997098i	$-0.475743\pi$
$683$	3.97935	0.152266	0.0761329	+	0.997098i	$0.475743\pi$
$684$	0	0
$685$	−5.75869	−0.220028
$686$	−2.85423	−0.108975
$687$	0	0
$688$	1.56645	0.0597202
$689$	−7.93002	−0.302110
$690$	0	0
$691$	19.2003	0.730414	0.365207	−	0.930926i	$-0.380998\pi$
$691$	19.2003	0.730414	0.365207	+	0.930926i	$0.380998\pi$
$692$	20.2344	0.769198
$693$	0	0
$694$	−11.5057	−0.436752
$695$	−8.76158	−0.332346
$696$	0	0
$697$	−14.9919	−0.567857
$698$	3.43190	0.129899
$699$	0	0
$700$	−18.0721	−0.683061
$701$	12.5080	0.472420	0.236210	−	0.971702i	$-0.424095\pi$
$701$	12.5080	0.472420	0.236210	+	0.971702i	$0.424095\pi$
$702$	0	0
$703$	68.3655	2.57845
$704$	4.61398	0.173896
$705$	0	0
$706$	10.7842	0.405869
$707$	−61.5850	−2.31614
$708$	0	0
$709$	0.732678	0.0275163	0.0137582	−	0.999905i	$-0.495621\pi$
$709$	0.732678	0.0275163	0.0137582	+	0.999905i	$0.495621\pi$
$710$	4.06922	0.152715
$711$	0	0
$712$	6.21905	0.233069
$713$	49.1583	1.84099
$714$	0	0
$715$	−2.12767	−0.0795705
$716$	2.97140	0.111046
$717$	0	0
$718$	4.35125	0.162387
$719$	−38.2311	−1.42578	−0.712889	−	0.701276i	$-0.752614\pi$
$719$	−38.2311	−1.42578	−0.712889	+	0.701276i	$0.752614\pi$
$720$	0	0
$721$	14.5115	0.540436
$722$	−27.3783	−1.01891
$723$	0	0
$724$	−12.7209	−0.472770
$725$	14.6208	0.543002
$726$	0	0
$727$	26.0760	0.967104	0.483552	−	0.875316i	$-0.339346\pi$
$727$	26.0760	0.967104	0.483552	+	0.875316i	$0.339346\pi$
$728$	3.26908	0.121160
$729$	0	0
$730$	−5.07041	−0.187664
$731$	9.76463	0.361158
$732$	0	0
$733$	25.1808	0.930075	0.465037	−	0.885291i	$-0.346041\pi$
$733$	25.1808	0.930075	0.465037	+	0.885291i	$0.346041\pi$
$734$	35.1762	1.29838
$735$	0	0
$736$	7.84623	0.289216
$737$	47.2269	1.73963
$738$	0	0
$739$	6.15880	0.226555	0.113277	−	0.993563i	$-0.463865\pi$
$739$	6.15880	0.226555	0.113277	+	0.993563i	$0.463865\pi$
$740$	5.43729	0.199879
$741$	0	0
$742$	−35.7639	−1.31294
$743$	−4.68882	−0.172016	−0.0860080	−	0.996294i	$-0.527411\pi$
$743$	−4.68882	−0.172016	−0.0860080	+	0.996294i	$0.527411\pi$
$744$	0	0
$745$	−0.541630	−0.0198438
$746$	15.7885	0.578058
$747$	0	0
$748$	28.7618	1.05164
$749$	63.4125	2.31704
$750$	0	0
$751$	−0.0118276	−0.000431594 0	−0.000215797	−	1.00000i	$-0.500069\pi$
$751$	−0.0118276	−0.000431594 0	−0.000215797		1.00000i	$0.500069\pi$
$752$	−1.46207	−0.0533161
$753$	0	0
$754$	−2.64476	−0.0963166
$755$	−4.21419	−0.153370
$756$	0	0
$757$	−30.9076	−1.12336	−0.561678	−	0.827356i	$-0.689844\pi$
$757$	−30.9076	−1.12336	−0.561678	+	0.827356i	$0.689844\pi$
$758$	−1.54608	−0.0561561
$759$	0	0
$760$	−3.68858	−0.133799
$761$	0.909169	0.0329573	0.0164787	−	0.999864i	$-0.494754\pi$
$761$	0.909169	0.0329573	0.0164787	+	0.999864i	$0.494754\pi$
$762$	0	0
$763$	29.8250	1.07974
$764$	22.4661	0.812794
$765$	0	0
$766$	23.0028	0.831126
$767$	−6.13160	−0.221399
$768$	0	0
$769$	−18.3228	−0.660736	−0.330368	−	0.943852i	$-0.607173\pi$
$769$	−18.3228	−0.660736	−0.330368	+	0.943852i	$0.607173\pi$
$770$	−9.59569	−0.345805
$771$	0	0
$772$	23.5808	0.848692
$773$	−23.4991	−0.845203	−0.422601	−	0.906316i	$-0.638883\pi$
$773$	−23.4991	−0.845203	−0.422601	+	0.906316i	$0.638883\pi$
$774$	0	0
$775$	29.4881	1.05924
$776$	7.03070	0.252387
$777$	0	0
$778$	5.10201	0.182916
$779$	−16.3784	−0.586818
$780$	0	0
$781$	−34.6645	−1.24039
$782$	48.9104	1.74903
$783$	0	0
$784$	7.74335	0.276548
$785$	3.79876	0.135584
$786$	0	0
$787$	−23.8069	−0.848625	−0.424313	−	0.905516i	$-0.639484\pi$
$787$	−23.8069	−0.848625	−0.424313	+	0.905516i	$0.639484\pi$
$788$	−19.3337	−0.688735
$789$	0	0
$790$	−4.84768	−0.172473
$791$	−68.7912	−2.44593
$792$	0	0
$793$	−8.72307	−0.309765
$794$	−4.63306	−0.164421
$795$	0	0
$796$	23.5740	0.835557
$797$	6.86554	0.243190	0.121595	−	0.992580i	$-0.461199\pi$
$797$	6.86554	0.243190	0.121595	+	0.992580i	$0.461199\pi$
$798$	0	0
$799$	−9.11397	−0.322429
$800$	4.70664	0.166405
$801$	0	0
$802$	8.09165	0.285726
$803$	43.1933	1.52426
$804$	0	0
$805$	−16.3178	−0.575127
$806$	−5.33412	−0.187886
$807$	0	0
$808$	16.0390	0.564249
$809$	25.5716	0.899049	0.449524	−	0.893268i	$-0.351594\pi$
$809$	25.5716	0.899049	0.449524	+	0.893268i	$0.351594\pi$
$810$	0	0
$811$	−9.94545	−0.349232	−0.174616	−	0.984637i	$-0.555868\pi$
$811$	−9.94545	−0.349232	−0.174616	+	0.984637i	$0.555868\pi$
$812$	−11.9277	−0.418582
$813$	0	0
$814$	−46.3186	−1.62347
$815$	0.282784	0.00990549
$816$	0	0
$817$	10.6677	0.373217
$818$	−17.2833	−0.604298
$819$	0	0
$820$	−1.30262	−0.0454894
$821$	20.0838	0.700930	0.350465	−	0.936576i	$-0.386024\pi$
$821$	20.0838	0.700930	0.350465	+	0.936576i	$0.386024\pi$
$822$	0	0
$823$	10.9500	0.381693	0.190846	−	0.981620i	$-0.438877\pi$
$823$	10.9500	0.381693	0.190846	+	0.981620i	$0.438877\pi$
$824$	−3.77932	−0.131659
$825$	0	0
$826$	−27.6532	−0.962176
$827$	11.0928	0.385735	0.192867	−	0.981225i	$-0.438221\pi$
$827$	11.0928	0.385735	0.192867	+	0.981225i	$0.438221\pi$
$828$	0	0
$829$	−48.4675	−1.68334	−0.841672	−	0.539989i	$-0.818428\pi$
$829$	−48.4675	−1.68334	−0.841672	+	0.539989i	$0.818428\pi$
$830$	2.60183	0.0903109
$831$	0	0
$832$	−0.851387	−0.0295165
$833$	48.2691	1.67242
$834$	0	0
$835$	0.376033	0.0130132
$836$	31.4219	1.08675
$837$	0	0
$838$	−7.15760	−0.247255
$839$	18.1006	0.624901	0.312451	−	0.949934i	$-0.398850\pi$
$839$	18.1006	0.624901	0.312451	+	0.949934i	$0.398850\pi$
$840$	0	0
$841$	−19.3502	−0.667247
$842$	−17.2363	−0.594002
$843$	0	0
$844$	15.8528	0.545676
$845$	−6.64858	−0.228718
$846$	0	0
$847$	39.5061	1.35745
$848$	9.31424	0.319852
$849$	0	0
$850$	29.3394	1.00633
$851$	−78.7664	−2.70008
$852$	0	0
$853$	−26.8058	−0.917813	−0.458907	−	0.888485i	$-0.651759\pi$
$853$	−26.8058	−0.917813	−0.458907	+	0.888485i	$0.651759\pi$
$854$	−39.3405	−1.34621
$855$	0	0
$856$	−16.5149	−0.564469
$857$	−35.9763	−1.22893	−0.614464	−	0.788945i	$-0.710628\pi$
$857$	−35.9763	−1.22893	−0.614464	+	0.788945i	$0.710628\pi$
$858$	0	0
$859$	−23.9498	−0.817157	−0.408579	−	0.912723i	$-0.633975\pi$
$859$	−23.9498	−0.817157	−0.408579	+	0.912723i	$0.633975\pi$
$860$	0.848434	0.0289314
$861$	0	0
$862$	23.4748	0.799555
$863$	41.8310	1.42394	0.711972	−	0.702208i	$-0.247802\pi$
$863$	41.8310	1.42394	0.711972	+	0.702208i	$0.247802\pi$
$864$	0	0
$865$	10.9596	0.372637
$866$	−17.1686	−0.583411
$867$	0	0
$868$	−24.0566	−0.816534
$869$	41.2959	1.40087
$870$	0	0
$871$	−8.71446	−0.295278
$872$	−7.76752	−0.263041
$873$	0	0
$874$	53.4341	1.80743
$875$	−20.1869	−0.682441
$876$	0	0
$877$	13.3371	0.450362	0.225181	−	0.974317i	$-0.427703\pi$
$877$	13.3371	0.450362	0.225181	+	0.974317i	$0.427703\pi$
$878$	39.6526	1.33821
$879$	0	0
$880$	2.49907	0.0842436
$881$	−42.3819	−1.42788	−0.713942	−	0.700205i	$-0.753092\pi$
$881$	−42.3819	−1.42788	−0.713942	+	0.700205i	$0.753092\pi$
$882$	0	0
$883$	−11.9589	−0.402450	−0.201225	−	0.979545i	$-0.564492\pi$
$883$	−11.9589	−0.402450	−0.201225	+	0.979545i	$0.564492\pi$
$884$	−5.30722	−0.178501
$885$	0	0
$886$	−29.8126	−1.00157
$887$	6.52603	0.219123	0.109561	−	0.993980i	$-0.465055\pi$
$887$	6.52603	0.219123	0.109561	+	0.993980i	$0.465055\pi$
$888$	0	0
$889$	−21.9268	−0.735401
$890$	3.36842	0.112910
$891$	0	0
$892$	−21.7748	−0.729075
$893$	−9.95691	−0.333195
$894$	0	0
$895$	1.60940	0.0537962
$896$	−3.83971	−0.128276
$897$	0	0
$898$	−33.1854	−1.10741
$899$	19.4624	0.649107
$900$	0	0
$901$	58.0614	1.93431
$902$	11.0966	0.369477
$903$	0	0
$904$	17.9157	0.595869
$905$	−6.89004	−0.229033
$906$	0	0
$907$	−6.92030	−0.229785	−0.114892	−	0.993378i	$-0.536652\pi$
$907$	−6.92030	−0.229785	−0.114892	+	0.993378i	$0.536652\pi$
$908$	−7.12157	−0.236338
$909$	0	0
$910$	1.77063	0.0586958
$911$	−12.5060	−0.414342	−0.207171	−	0.978305i	$-0.566426\pi$
$911$	−12.5060	−0.414342	−0.207171	+	0.978305i	$0.566426\pi$
$912$	0	0
$913$	−22.1642	−0.733529
$914$	1.83444	0.0606780
$915$	0	0
$916$	7.15412	0.236379
$917$	−15.9493	−0.526693
$918$	0	0
$919$	−11.6095	−0.382962	−0.191481	−	0.981496i	$-0.561329\pi$
$919$	−11.6095	−0.382962	−0.191481	+	0.981496i	$0.561329\pi$
$920$	4.24975	0.140110
$921$	0	0
$922$	−24.9472	−0.821593
$923$	6.39640	0.210540
$924$	0	0
$925$	−47.2488	−1.55353
$926$	−13.8121	−0.453895
$927$	0	0
$928$	3.10642	0.101973
$929$	12.3787	0.406132	0.203066	−	0.979165i	$-0.434909\pi$
$929$	12.3787	0.406132	0.203066	+	0.979165i	$0.434909\pi$
$930$	0	0
$931$	52.7334	1.72827
$932$	3.66468	0.120041
$933$	0	0
$934$	18.4587	0.603987
$935$	15.5783	0.509463
$936$	0	0
$937$	−10.6111	−0.346650	−0.173325	−	0.984865i	$-0.555451\pi$
$937$	−10.6111	−0.346650	−0.173325	+	0.984865i	$0.555451\pi$
$938$	−39.3017	−1.28325
$939$	0	0
$940$	−0.791899	−0.0258289
$941$	−7.21646	−0.235250	−0.117625	−	0.993058i	$-0.537528\pi$
$941$	−7.21646	−0.235250	−0.117625	+	0.993058i	$0.537528\pi$
$942$	0	0
$943$	18.8702	0.614498
$944$	7.20189	0.234402
$945$	0	0
$946$	−7.22756	−0.234988
$947$	−26.0747	−0.847313	−0.423657	−	0.905823i	$-0.639254\pi$
$947$	−26.0747	−0.847313	−0.423657	+	0.905823i	$0.639254\pi$
$948$	0	0
$949$	−7.97017	−0.258723
$950$	32.0529	1.03993
$951$	0	0
$952$	−23.9353	−0.775747
$953$	−34.3472	−1.11262	−0.556308	−	0.830976i	$-0.687782\pi$
$953$	−34.3472	−1.11262	−0.556308	+	0.830976i	$0.687782\pi$
$954$	0	0
$955$	12.1683	0.393756
$956$	−1.59313	−0.0515254
$957$	0	0
$958$	−29.4723	−0.952208
$959$	−40.8243	−1.31829
$960$	0	0
$961$	8.25293	0.266224
$962$	8.54686	0.275562
$963$	0	0
$964$	−8.11827	−0.261472
$965$	12.7721	0.411147
$966$	0	0
$967$	−28.8792	−0.928692	−0.464346	−	0.885654i	$-0.653711\pi$
$967$	−28.8792	−0.928692	−0.464346	+	0.885654i	$0.653711\pi$
$968$	−10.2888	−0.330696
$969$	0	0
$970$	3.80804	0.122269
$971$	−33.9389	−1.08915	−0.544576	−	0.838712i	$-0.683309\pi$
$971$	−33.9389	−1.08915	−0.544576	+	0.838712i	$0.683309\pi$
$972$	0	0
$973$	−62.1123	−1.99123
$974$	−41.5276	−1.33063
$975$	0	0
$976$	10.2457	0.327957
$977$	39.6364	1.26808	0.634041	−	0.773299i	$-0.281395\pi$
$977$	39.6364	1.26808	0.634041	+	0.773299i	$0.281395\pi$
$978$	0	0
$979$	−28.6946	−0.917083
$980$	4.19403	0.133973
$981$	0	0
$982$	17.4020	0.555321
$983$	47.8453	1.52603	0.763014	−	0.646381i	$-0.223719\pi$
$983$	47.8453	1.52603	0.763014	+	0.646381i	$0.223719\pi$
$984$	0	0
$985$	−10.4717	−0.333657
$986$	19.3642	0.616683
$987$	0	0
$988$	−5.79808	−0.184461
$989$	−12.2907	−0.390822
$990$	0	0
$991$	−54.7222	−1.73831	−0.869154	−	0.494542i	$-0.835336\pi$
$991$	−54.7222	−1.73831	−0.869154	+	0.494542i	$0.835336\pi$
$992$	6.26522	0.198921
$993$	0	0
$994$	28.8474	0.914984
$995$	12.7684	0.404784
$996$	0	0
$997$	−6.63055	−0.209992	−0.104996	−	0.994473i	$-0.533483\pi$
$997$	−6.63055	−0.209992	−0.104996	+	0.994473i	$0.533483\pi$
$998$	25.9772	0.822293
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.q.1.8	✓	14
3.2	odd	2		8046.2.a.r.1.7	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.q.1.8	✓	14	1.1	even	1	trivial
8046.2.a.r.1.7	yes	14	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +0.541630 q^{5} +3.83971 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +0.541630 q^{5} +3.83971 q^{7} -1.00000 q^{8} -0.541630 q^{10} +4.61398 q^{11} -0.851387 q^{13} -3.83971 q^{14} +1.00000 q^{16} +6.23362 q^{17} +6.81016 q^{19} +0.541630 q^{20} -4.61398 q^{22} -7.84623 q^{23} -4.70664 q^{25} +0.851387 q^{26} +3.83971 q^{28} -3.10642 q^{29} -6.26522 q^{31} -1.00000 q^{32} -6.23362 q^{34} +2.07970 q^{35} +10.0388 q^{37} -6.81016 q^{38} -0.541630 q^{40} -2.40500 q^{41} +1.56645 q^{43} +4.61398 q^{44} +7.84623 q^{46} -1.46207 q^{47} +7.74335 q^{49} +4.70664 q^{50} -0.851387 q^{52} +9.31424 q^{53} +2.49907 q^{55} -3.83971 q^{56} +3.10642 q^{58} +7.20189 q^{59} +10.2457 q^{61} +6.26522 q^{62} +1.00000 q^{64} -0.461136 q^{65} +10.2356 q^{67} +6.23362 q^{68} -2.07970 q^{70} -7.51292 q^{71} +9.36139 q^{73} -10.0388 q^{74} +6.81016 q^{76} +17.7163 q^{77} +8.95017 q^{79} +0.541630 q^{80} +2.40500 q^{82} -4.80371 q^{83} +3.37631 q^{85} -1.56645 q^{86} -4.61398 q^{88} -6.21905 q^{89} -3.26908 q^{91} -7.84623 q^{92} +1.46207 q^{94} +3.68858 q^{95} -7.03070 q^{97} -7.74335 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 14 q^{2} + 14 q^{4} - 2 q^{5} + 4 q^{7} - 14 q^{8}+O(q^{10}) \) 14 * q - 14 * q^2 + 14 * q^4 - 2 * q^5 + 4 * q^7 - 14 * q^8	\( 14 q - 14 q^{2} + 14 q^{4} - 2 q^{5} + 4 q^{7} - 14 q^{8} + 2 q^{10} - 2 q^{11} + 4 q^{13} - 4 q^{14} + 14 q^{16} - 9 q^{17} + 14 q^{19} - 2 q^{20} + 2 q^{22} - 30 q^{23} + 18 q^{25} - 4 q^{26} + 4 q^{28} - 6 q^{29} + 11 q^{31} - 14 q^{32} + 9 q^{34} + 18 q^{35} + 13 q^{37} - 14 q^{38} + 2 q^{40} + 2 q^{41} + 12 q^{43} - 2 q^{44} + 30 q^{46} - 21 q^{47} + 32 q^{49} - 18 q^{50} + 4 q^{52} - 22 q^{53} - 7 q^{55} - 4 q^{56} + 6 q^{58} - 14 q^{59} + 31 q^{61} - 11 q^{62} + 14 q^{64} + 24 q^{67} - 9 q^{68} - 18 q^{70} - 28 q^{71} + 24 q^{73} - 13 q^{74} + 14 q^{76} - 16 q^{77} + 65 q^{79} - 2 q^{80} - 2 q^{82} + 15 q^{83} - 19 q^{85} - 12 q^{86} + 2 q^{88} + 11 q^{89} + 68 q^{91} - 30 q^{92} + 21 q^{94} - 8 q^{95} + 23 q^{97} - 32 q^{98}+O(q^{100}) \) 14 * q - 14 * q^2 + 14 * q^4 - 2 * q^5 + 4 * q^7 - 14 * q^8 + 2 * q^10 - 2 * q^11 + 4 * q^13 - 4 * q^14 + 14 * q^16 - 9 * q^17 + 14 * q^19 - 2 * q^20 + 2 * q^22 - 30 * q^23 + 18 * q^25 - 4 * q^26 + 4 * q^28 - 6 * q^29 + 11 * q^31 - 14 * q^32 + 9 * q^34 + 18 * q^35 + 13 * q^37 - 14 * q^38 + 2 * q^40 + 2 * q^41 + 12 * q^43 - 2 * q^44 + 30 * q^46 - 21 * q^47 + 32 * q^49 - 18 * q^50 + 4 * q^52 - 22 * q^53 - 7 * q^55 - 4 * q^56 + 6 * q^58 - 14 * q^59 + 31 * q^61 - 11 * q^62 + 14 * q^64 + 24 * q^67 - 9 * q^68 - 18 * q^70 - 28 * q^71 + 24 * q^73 - 13 * q^74 + 14 * q^76 - 16 * q^77 + 65 * q^79 - 2 * q^80 - 2 * q^82 + 15 * q^83 - 19 * q^85 - 12 * q^86 + 2 * q^88 + 11 * q^89 + 68 * q^91 - 30 * q^92 + 21 * q^94 - 8 * q^95 + 23 * q^97 - 32 * q^98

Introduction

L-functions

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