LMFDB - Embedded newform 3120.2.l.n.1249.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3120,2,Mod(1249,3120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3120, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0]))

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

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

chi := DirichletCharacter("3120.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3120.l (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$24.9133254306$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.57815240704.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 2x^{6} + 89x^{4} - 170x^{3} + 162x^{2} - 72x + 16 $ x^8 - 2x^7 + 2x^6 + 89x^4 - 170x^3 + 162x^2 - 72x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1560)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.3
Root			$0.353624 - 0.353624i$ of defining polynomial
Character	$\chi$	$=$	3120.1249
Dual form			3120.2.l.n.1249.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +(-0.353624 + 2.20793i) q^{5} +3.09417i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +(-0.353624 + 2.20793i) q^{5} +3.09417i q^{7} -1.00000 q^{9} -5.51003 q^{11} -1.00000i q^{13} +(2.20793 + 0.353624i) q^{15} -6.21728i q^{17} +3.09417 q^{21} +4.21728i q^{23} +(-4.74990 - 1.56155i) q^{25} +1.00000i q^{27} +1.70861 q^{29} +5.51003i q^{33} +(-6.83172 - 1.09417i) q^{35} -4.02893i q^{37} -1.00000 q^{39} +0.198578 q^{41} -5.70861i q^{43} +(0.353624 - 2.20793i) q^{45} +6.41586i q^{47} -2.57391 q^{49} -6.21728 q^{51} -4.02893i q^{53} +(1.94848 - 12.1658i) q^{55} -7.53896 q^{59} +11.9259 q^{61} -3.09417i q^{63} +(2.20793 + 0.353624i) q^{65} -4.83172i q^{67} +4.21728 q^{69} -7.98977 q^{71} -9.31145i q^{73} +(-1.56155 + 4.74990i) q^{75} -17.0490i q^{77} +8.51139 q^{79} +1.00000 q^{81} -14.7926i q^{83} +(13.7273 + 2.19858i) q^{85} -1.70861i q^{87} +9.40699 q^{89} +3.09417 q^{91} +14.3026i q^{97} +5.51003 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{5} - 8 q^{9}+O(q^{10}) $ 8 * q - 2 * q^5 - 8 * q^9	$ 8 q - 2 q^{5} - 8 q^{9} - 2 q^{11} + 2 q^{15} + 14 q^{21} - 16 q^{29} + 8 q^{35} - 8 q^{39} + 14 q^{41} + 2 q^{45} - 18 q^{49} - 6 q^{51} - 10 q^{55} + 4 q^{59} + 22 q^{61} + 2 q^{65} - 10 q^{69} - 30 q^{71} + 4 q^{75} - 2 q^{79} + 8 q^{81} + 24 q^{85} - 18 q^{89} + 14 q^{91} + 2 q^{99}+O(q^{100}) $ 8 * q - 2 * q^5 - 8 * q^9 - 2 * q^11 + 2 * q^15 + 14 * q^21 - 16 * q^29 + 8 * q^35 - 8 * q^39 + 14 * q^41 + 2 * q^45 - 18 * q^49 - 6 * q^51 - 10 * q^55 + 4 * q^59 + 22 * q^61 + 2 * q^65 - 10 * q^69 - 30 * q^71 + 4 * q^75 - 2 * q^79 + 8 * q^81 + 24 * q^85 - 18 * q^89 + 14 * q^91 + 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3120\mathbb{Z}\right)^\times$.

$n$	$1951$	$2081$	$2341$	$2497$	$2641$
$\chi(n)$	$1$	$1$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$	−0.353624	+	2.20793i	−0.158145	+	0.987416i
$5$	−0.353624	+	2.20793i	−0.158145	+	0.987416i
$6$		0			0
$7$			3.09417i			1.16949i	0.811218	+	0.584744i	$0.198805\pi$
$7$			3.09417i			1.16949i	−0.811218	+	0.584744i	$0.801195\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−5.51003			−1.66134			−0.830669	−	0.556767i	$-0.812042\pi$
$11$	−5.51003			−1.66134			−0.830669	+	0.556767i	$0.812042\pi$
$12$		0			0
$13$		−	1.00000i		−	0.277350i
$13$		−	1.00000i		−	0.277350i
$14$		0			0
$15$	2.20793	+	0.353624i	0.570085	+	0.0913053i
$16$		0			0
$17$		−	6.21728i		−	1.50791i	−0.656925	−	0.753956i	$-0.728143\pi$
$17$		−	6.21728i		−	1.50791i	0.656925	−	0.753956i	$-0.271857\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$	3.09417			0.675204
$22$		0			0
$23$			4.21728i			0.879364i	0.898154	+	0.439682i	$0.144909\pi$
$23$			4.21728i			0.879364i	−0.898154	+	0.439682i	$0.855091\pi$
$24$		0			0
$25$	−4.74990	−	1.56155i	−0.949980	−	0.312311i
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	1.70861			0.317281			0.158640	−	0.987336i	$-0.449289\pi$
$29$	1.70861			0.317281			0.158640	+	0.987336i	$0.449289\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$		0			0
$33$			5.51003i			0.959173i
$34$		0			0
$35$	−6.83172	−	1.09417i	−1.15477	−	0.184949i
$36$		0			0
$37$		−	4.02893i		−	0.662352i	−0.943569	−	0.331176i	$-0.892555\pi$
$37$		−	4.02893i		−	0.662352i	0.943569	−	0.331176i	$-0.107445\pi$
$38$		0			0
$39$	−1.00000			−0.160128
$40$		0			0
$41$	0.198578			0.0310127			0.0155064	−	0.999880i	$-0.495064\pi$
$41$	0.198578			0.0310127			0.0155064	+	0.999880i	$0.495064\pi$
$42$		0			0
$43$		−	5.70861i		−	0.870555i	−0.900296	−	0.435277i	$-0.856650\pi$
$43$		−	5.70861i		−	0.870555i	0.900296	−	0.435277i	$-0.143350\pi$
$44$		0			0
$45$	0.353624	−	2.20793i	0.0527151	−	0.329139i
$46$		0			0
$47$			6.41586i			0.935849i	0.883768	+	0.467925i	$0.154998\pi$
$47$			6.41586i			0.935849i	−0.883768	+	0.467925i	$0.845002\pi$
$48$		0			0
$49$	−2.57391			−0.367702
$50$		0			0
$51$	−6.21728			−0.870593
$52$		0			0
$53$		−	4.02893i		−	0.553416i	−0.960954	−	0.276708i	$-0.910756\pi$
$53$		−	4.02893i		−	0.553416i	0.960954	−	0.276708i	$-0.0892435\pi$
$54$		0			0
$55$	1.94848	−	12.1658i	0.262733	−	1.64043i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−7.53896			−0.981489			−0.490745	−	0.871303i	$-0.663275\pi$
$59$	−7.53896			−0.981489			−0.490745	+	0.871303i	$0.663275\pi$
$60$		0			0
$61$	11.9259			1.52695			0.763477	−	0.645835i	$-0.223491\pi$
$61$	11.9259			1.52695			0.763477	+	0.645835i	$0.223491\pi$
$62$		0			0
$63$		−	3.09417i		−	0.389829i
$64$		0			0
$65$	2.20793	+	0.353624i	0.273860	+	0.0438616i
$66$		0			0
$67$		−	4.83172i		−	0.590288i	−0.955453	−	0.295144i	$-0.904632\pi$
$67$		−	4.83172i		−	0.590288i	0.955453	−	0.295144i	$-0.0953676\pi$
$68$		0			0
$69$	4.21728			0.507701
$70$		0			0
$71$	−7.98977			−0.948211			−0.474106	−	0.880468i	$-0.657228\pi$
$71$	−7.98977			−0.948211			−0.474106	+	0.880468i	$0.657228\pi$
$72$		0			0
$73$		−	9.31145i		−	1.08982i	−0.838494	−	0.544912i	$-0.816563\pi$
$73$		−	9.31145i		−	1.08982i	0.838494	−	0.544912i	$-0.183437\pi$
$74$		0			0
$75$	−1.56155	+	4.74990i	−0.180313	+	0.548471i
$76$		0			0
$77$		−	17.0490i		−	1.94291i
$78$		0			0
$79$	8.51139			0.957607			0.478803	−	0.877922i	$-0.341071\pi$
$79$	8.51139			0.957607			0.478803	+	0.877922i	$0.341071\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	14.7926i		−	1.62369i	−0.583871	−	0.811847i	$-0.698462\pi$
$83$		−	14.7926i		−	1.62369i	0.583871	−	0.811847i	$-0.301538\pi$
$84$		0			0
$85$	13.7273	+	2.19858i	1.48894	+	0.238469i
$86$		0			0
$87$		−	1.70861i		−	0.183182i
$88$		0			0
$89$	9.40699			0.997139			0.498569	−	0.866850i	$-0.333859\pi$
$89$	9.40699			0.997139			0.498569	+	0.866850i	$0.333859\pi$
$90$		0			0
$91$	3.09417			0.324358
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$			14.3026i			1.45221i	0.687585	+	0.726104i	$0.258671\pi$
$97$			14.3026i			1.45221i	−0.687585	+	0.726104i	$0.741329\pi$
$98$		0			0
$99$	5.51003			0.553779
$100$		0			0
$101$	−6.72595			−0.669257			−0.334628	−	0.942350i	$-0.608611\pi$
$101$	−6.72595			−0.669257			−0.334628	+	0.942350i	$0.608611\pi$
$102$		0			0
$103$		−	17.0779i		−	1.68274i	−0.540461	−	0.841369i	$-0.681750\pi$
$103$		−	17.0779i		−	1.68274i	0.540461	−	0.841369i	$-0.318250\pi$
$104$		0			0
$105$	−1.09417	+	6.83172i	−0.106780	+	0.666707i
$106$		0			0
$107$		−	7.63450i		−	0.738055i	−0.929418	−	0.369027i	$-0.879691\pi$
$107$		−	7.63450i		−	0.738055i	0.929418	−	0.369027i	$-0.120309\pi$
$108$		0			0
$109$	5.02006			0.480835			0.240417	−	0.970670i	$-0.422716\pi$
$109$	5.02006			0.480835			0.240417	+	0.970670i	$0.422716\pi$
$110$		0			0
$111$	−4.02893			−0.382409
$112$		0			0
$113$		−	16.6229i		−	1.56375i	−0.623434	−	0.781876i	$-0.714263\pi$
$113$		−	16.6229i		−	1.56375i	0.623434	−	0.781876i	$-0.285737\pi$
$114$		0			0
$115$	−9.31145	−	1.49133i	−0.868297	−	0.139067i
$116$		0			0
$117$			1.00000i			0.0924500i
$118$		0			0
$119$	19.2373			1.76348
$120$		0			0
$121$	19.3604			1.76004
$122$		0			0
$123$		−	0.198578i		−	0.0179052i
$124$		0			0
$125$	5.12748	−	9.93524i	0.458615	−	0.888635i
$126$		0			0
$127$			2.87689i			0.255283i	0.991820	+	0.127642i	$0.0407407\pi$
$127$			2.87689i			0.255283i	−0.991820	+	0.127642i	$0.959259\pi$
$128$		0			0
$129$	−5.70861			−0.502615
$130$		0			0
$131$	16.5199			1.44335			0.721674	−	0.692233i	$-0.243373\pi$
$131$	16.5199			1.44335			0.721674	+	0.692233i	$0.243373\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−2.20793	−	0.353624i	−0.190028	−	0.0304351i
$136$		0			0
$137$			12.8504i			1.09789i	0.835860	+	0.548943i	$0.184969\pi$
$137$			12.8504i			1.09789i	−0.835860	+	0.548943i	$0.815031\pi$
$138$		0			0
$139$	−17.0490			−1.44608			−0.723038	−	0.690808i	$-0.757255\pi$
$139$	−17.0490			−1.44608			−0.723038	+	0.690808i	$0.757255\pi$
$140$		0			0
$141$	6.41586			0.540313
$142$		0			0
$143$			5.51003i			0.460772i
$144$		0			0
$145$	−0.604205	+	3.77249i	−0.0501765	+	0.313288i
$146$		0			0
$147$			2.57391i			0.212293i
$148$		0			0
$149$	−19.5679			−1.60306			−0.801532	−	0.597952i	$-0.795981\pi$
$149$	−19.5679			−1.60306			−0.801532	+	0.597952i	$0.795981\pi$
$150$		0			0
$151$	9.22887			0.751035			0.375518	−	0.926815i	$-0.377465\pi$
$151$	9.22887			0.751035			0.375518	+	0.926815i	$0.377465\pi$
$152$		0			0
$153$			6.21728i			0.502637i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	3.89423i		−	0.310794i	−0.987852	−	0.155397i	$-0.950334\pi$
$157$		−	3.89423i		−	0.310794i	0.987852	−	0.155397i	$-0.0496656\pi$
$158$		0			0
$159$	−4.02893			−0.319515
$160$		0			0
$161$	−13.0490			−1.02840
$162$		0			0
$163$		−	0.237741i		−	0.0186213i	−0.999957	−	0.00931067i	$-0.997036\pi$
$163$		−	0.237741i		−	0.0186213i	0.999957	−	0.00931067i	$-0.00296372\pi$
$164$		0			0
$165$	−12.1658	−	1.94848i	−0.947103	−	0.151689i
$166$		0			0
$167$			11.8304i			0.915460i	0.889091	+	0.457730i	$0.151337\pi$
$167$			11.8304i			0.915460i	−0.889091	+	0.457730i	$0.848663\pi$
$168$		0			0
$169$	−1.00000			−0.0769231
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	11.6634i		−	0.886754i	−0.896335	−	0.443377i	$-0.853780\pi$
$173$		−	11.6634i		−	0.886754i	0.896335	−	0.443377i	$-0.146220\pi$
$174$		0			0
$175$	4.83172	−	14.6970i	0.365243	−	1.11099i
$176$		0			0
$177$			7.53896i			0.566663i
$178$		0			0
$179$	−12.5403			−0.937308			−0.468654	−	0.883382i	$-0.655261\pi$
$179$	−12.5403			−0.937308			−0.468654	+	0.883382i	$0.655261\pi$
$180$		0			0
$181$	−5.44343			−0.404607			−0.202303	−	0.979323i	$-0.564843\pi$
$181$	−5.44343			−0.404607			−0.202303	+	0.979323i	$0.564843\pi$
$182$		0			0
$183$		−	11.9259i		−	0.881587i
$184$		0			0
$185$	8.89560	+	1.42473i	0.654017	+	0.104748i
$186$		0			0
$187$			34.2574i			2.50515i
$188$		0			0
$189$	−3.09417			−0.225068
$190$		0			0
$191$	−5.22887			−0.378348			−0.189174	−	0.981944i	$-0.560581\pi$
$191$	−5.22887			−0.378348			−0.189174	+	0.981944i	$0.560581\pi$
$192$		0			0
$193$		−	17.5287i		−	1.26175i	−0.775886	−	0.630873i	$-0.782697\pi$
$193$		−	17.5287i		−	1.26175i	0.775886	−	0.630873i	$-0.217303\pi$
$194$		0			0
$195$	0.353624	−	2.20793i	0.0253235	−	0.158113i
$196$		0			0
$197$		−	22.4764i		−	1.60138i	−0.599079	−	0.800690i	$-0.704466\pi$
$197$		−	22.4764i		−	1.60138i	0.599079	−	0.800690i	$-0.295534\pi$
$198$		0			0
$199$	−16.9143			−1.19902			−0.599511	−	0.800366i	$-0.704638\pi$
$199$	−16.9143			−1.19902			−0.599511	+	0.800366i	$0.704638\pi$
$200$		0			0
$201$	−4.83172			−0.340803
$202$		0			0
$203$			5.28674i			0.371056i
$204$		0			0
$205$	−0.0702221	+	0.438447i	−0.00490452	+	0.0306225i
$206$		0			0
$207$		−	4.21728i		−	0.293121i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	11.8391			0.815037			0.407518	−	0.913197i	$-0.366394\pi$
$211$	11.8391			0.815037			0.407518	+	0.913197i	$0.366394\pi$
$212$		0			0
$213$			7.98977i			0.547450i
$214$		0			0
$215$	12.6042	+	2.01870i	0.859600	+	0.137674i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−9.31145			−0.629210
$220$		0			0
$221$	−6.21728			−0.418219
$222$		0			0
$223$			4.48246i			0.300168i	0.988673	+	0.150084i	$0.0479544\pi$
$223$			4.48246i			0.300168i	−0.988673	+	0.150084i	$0.952046\pi$
$224$		0			0
$225$	4.74990	+	1.56155i	0.316660	+	0.104104i
$226$		0			0
$227$			6.35799i			0.421995i	0.977487	+	0.210997i	$0.0676712\pi$
$227$			6.35799i			0.421995i	−0.977487	+	0.210997i	$0.932329\pi$
$228$		0			0
$229$	−6.45502			−0.426560			−0.213280	−	0.976991i	$-0.568415\pi$
$229$	−6.45502			−0.426560			−0.213280	+	0.976991i	$0.568415\pi$
$230$		0			0
$231$	−17.0490			−1.12174
$232$		0			0
$233$			6.40290i			0.419468i	0.977758	+	0.209734i	$0.0672598\pi$
$233$			6.40290i			0.419468i	−0.977758	+	0.209734i	$0.932740\pi$
$234$		0			0
$235$	−14.1658	−	2.26880i	−0.924072	−	0.148000i
$236$		0			0
$237$		−	8.51139i		−	0.552874i
$238$		0			0
$239$	−1.55793			−0.100774			−0.0503872	−	0.998730i	$-0.516046\pi$
$239$	−1.55793			−0.100774			−0.0503872	+	0.998730i	$0.516046\pi$
$240$		0			0
$241$	−14.1911			−0.914127			−0.457064	−	0.889434i	$-0.651099\pi$
$241$	−14.1911			−0.914127			−0.457064	+	0.889434i	$0.651099\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	0.910196	−	5.68301i	0.0581503	−	0.363074i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−14.7926			−0.937440
$250$		0			0
$251$	−22.7082			−1.43333			−0.716665	−	0.697418i	$-0.754332\pi$
$251$	−22.7082			−1.43333			−0.716665	+	0.697418i	$0.754332\pi$
$252$		0			0
$253$		−	23.2373i		−	1.46092i
$254$		0			0
$255$	2.19858	−	13.7273i	0.137680	−	0.859638i
$256$		0			0
$257$		−	21.4172i		−	1.33597i	−0.744175	−	0.667985i	$-0.767157\pi$
$257$		−	21.4172i		−	1.33597i	0.744175	−	0.667985i	$-0.232843\pi$
$258$		0			0
$259$	12.4662			0.774613
$260$		0			0
$261$	−1.70861			−0.105760
$262$		0			0
$263$		−	26.4951i		−	1.63376i	−0.576807	−	0.816880i	$-0.695702\pi$
$263$		−	26.4951i		−	1.63376i	0.576807	−	0.816880i	$-0.304298\pi$
$264$		0			0
$265$	8.89560	+	1.42473i	0.546452	+	0.0875203i
$266$		0			0
$267$		−	9.40699i		−	0.575698i
$268$		0			0
$269$	−14.7259			−0.897857			−0.448928	−	0.893568i	$-0.648194\pi$
$269$	−14.7259			−0.897857			−0.448928	+	0.893568i	$0.648194\pi$
$270$		0			0
$271$	25.8491			1.57022			0.785109	−	0.619357i	$-0.212607\pi$
$271$	25.8491			1.57022			0.785109	+	0.619357i	$0.212607\pi$
$272$		0			0
$273$		−	3.09417i		−	0.187268i
$274$		0			0
$275$	26.1721	+	8.60421i	1.57824	+	0.518853i
$276$		0			0
$277$			20.1227i			1.20906i	0.796584	+	0.604528i	$0.206638\pi$
$277$			20.1227i			1.20906i	−0.796584	+	0.604528i	$0.793362\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−16.3954			−0.978067			−0.489034	−	0.872265i	$-0.662650\pi$
$281$	−16.3954			−0.978067			−0.489034	+	0.872265i	$0.662650\pi$
$282$		0			0
$283$		−	14.5650i		−	0.865802i	−0.901441	−	0.432901i	$-0.857490\pi$
$283$		−	14.5650i		−	0.865802i	0.901441	−	0.432901i	$-0.142510\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			0.614436i			0.0362690i
$288$		0			0
$289$	−21.6546			−1.27380
$290$		0			0
$291$	14.3026			0.838432
$292$		0			0
$293$			26.6995i			1.55980i	0.625904	+	0.779900i	$0.284730\pi$
$293$			26.6995i			1.55980i	−0.625904	+	0.779900i	$0.715270\pi$
$294$		0			0
$295$	2.66596	−	16.6455i	0.155218	−	0.969138i
$296$		0			0
$297$		−	5.51003i		−	0.319724i
$298$		0			0
$299$	4.21728			0.243892
$300$		0			0
$301$	17.6634			1.01810
$302$		0			0
$303$			6.72595i			0.386396i
$304$		0			0
$305$	−4.21728	+	26.3315i	−0.241481	+	1.50774i
$306$		0			0
$307$		−	11.4230i		−	0.651943i	−0.945380	−	0.325972i	$-0.894309\pi$
$307$		−	11.4230i		−	0.651943i	0.945380	−	0.325972i	$-0.105691\pi$
$308$		0			0
$309$	−17.0779			−0.971529
$310$		0			0
$311$	−19.4546			−1.10317			−0.551585	−	0.834119i	$-0.685977\pi$
$311$	−19.4546			−1.10317			−0.551585	+	0.834119i	$0.685977\pi$
$312$		0			0
$313$			32.5805i			1.84156i	0.390087	+	0.920778i	$0.372445\pi$
$313$			32.5805i			1.84156i	−0.390087	+	0.920778i	$0.627555\pi$
$314$		0			0
$315$	6.83172	+	1.09417i	0.384924	+	0.0616497i
$316$		0			0
$317$		−	31.3781i		−	1.76237i	−0.472774	−	0.881184i	$-0.656747\pi$
$317$		−	31.3781i		−	1.76237i	0.472774	−	0.881184i	$-0.343253\pi$
$318$		0			0
$319$	−9.41450			−0.527111
$320$		0			0
$321$	−7.63450			−0.426116
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−1.56155	+	4.74990i	−0.0866194	+	0.263477i
$326$		0			0
$327$		−	5.02006i		−	0.277610i
$328$		0			0
$329$	−19.8518			−1.09446
$330$		0			0
$331$	30.4951			1.67616			0.838082	−	0.545544i	$-0.183677\pi$
$331$	30.4951			1.67616			0.838082	+	0.545544i	$0.183677\pi$
$332$		0			0
$333$			4.02893i			0.220784i
$334$		0			0
$335$	10.6681	+	1.70861i	0.582860	+	0.0933513i
$336$		0			0
$337$		−	15.8765i		−	0.864848i	−0.901670	−	0.432424i	$-0.857658\pi$
$337$		−	15.8765i		−	0.864848i	0.901670	−	0.432424i	$-0.142342\pi$
$338$		0			0
$339$	−16.6229			−0.902832
$340$		0			0
$341$		0			0
$342$		0			0
$343$			13.6951i			0.739465i
$344$		0			0
$345$	−1.49133	+	9.31145i	−0.0802905	+	0.501312i
$346$		0			0
$347$			21.1069i			1.13308i	0.824036	+	0.566538i	$0.191717\pi$
$347$			21.1069i			1.13308i	−0.824036	+	0.566538i	$0.808283\pi$
$348$		0			0
$349$	−4.82899			−0.258490			−0.129245	−	0.991613i	$-0.541255\pi$
$349$	−4.82899			−0.258490			−0.129245	+	0.991613i	$0.541255\pi$
$350$		0			0
$351$	1.00000			0.0533761
$352$		0			0
$353$			19.4359i			1.03447i	0.855844	+	0.517235i	$0.173039\pi$
$353$			19.4359i			1.03447i	−0.855844	+	0.517235i	$0.826961\pi$
$354$		0			0
$355$	2.82537	−	17.6408i	0.149955	−	0.936279i
$356$		0			0
$357$		−	19.2373i		−	1.01815i
$358$		0			0
$359$	−27.2271			−1.43699			−0.718496	−	0.695531i	$-0.755169\pi$
$359$	−27.2271			−1.43699			−0.718496	+	0.695531i	$0.755169\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		0			0
$363$		−	19.3604i		−	1.01616i
$364$		0			0
$365$	20.5590	+	3.29275i	1.07611	+	0.172351i
$366$		0			0
$367$		−	15.7008i		−	0.819577i	−0.912181	−	0.409788i	$-0.865603\pi$
$367$		−	15.7008i		−	0.819577i	0.912181	−	0.409788i	$-0.134397\pi$
$368$		0			0
$369$	−0.198578			−0.0103376
$370$		0			0
$371$	12.4662			0.647214
$372$		0			0
$373$			3.86952i			0.200356i	0.994970	+	0.100178i	$0.0319412\pi$
$373$			3.86952i			0.200356i	−0.994970	+	0.100178i	$0.968059\pi$
$374$		0			0
$375$	−9.93524	−	5.12748i	−0.513054	−	0.264782i
$376$		0			0
$377$		−	1.70861i		−	0.0879979i
$378$		0			0
$379$	−7.30149			−0.375052			−0.187526	−	0.982260i	$-0.560047\pi$
$379$	−7.30149			−0.375052			−0.187526	+	0.982260i	$0.560047\pi$
$380$		0			0
$381$	2.87689			0.147388
$382$		0			0
$383$			37.2094i			1.90131i	0.310250	+	0.950655i	$0.399587\pi$
$383$			37.2094i			1.90131i	−0.310250	+	0.950655i	$0.600413\pi$
$384$		0			0
$385$	37.6430	+	6.02893i	1.91846	+	0.307263i
$386$		0			0
$387$			5.70861i			0.290185i
$388$		0			0
$389$	−30.1227			−1.52728			−0.763641	−	0.645641i	$-0.776590\pi$
$389$	−30.1227			−1.52728			−0.763641	+	0.645641i	$0.776590\pi$
$390$		0			0
$391$	26.2200			1.32600
$392$		0			0
$393$		−	16.5199i		−	0.833317i
$394$		0			0
$395$	−3.00983	+	18.7926i	−0.151441	+	0.945556i
$396$		0			0
$397$			1.25780i			0.0631274i	0.999502	+	0.0315637i	$0.0100487\pi$
$397$			1.25780i			0.0631274i	−0.999502	+	0.0315637i	$0.989951\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	20.6443			1.03093			0.515464	−	0.856911i	$-0.327619\pi$
$401$	20.6443			1.03093			0.515464	+	0.856911i	$0.327619\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−0.353624	+	2.20793i	−0.0175717	+	0.109713i
$406$		0			0
$407$			22.1995i			1.10039i
$408$		0			0
$409$	31.3820			1.55174			0.775870	−	0.630893i	$-0.217311\pi$
$409$	31.3820			1.55174			0.775870	+	0.630893i	$0.217311\pi$
$410$		0			0
$411$	12.8504			0.633864
$412$		0			0
$413$		−	23.3269i		−	1.14784i
$414$		0			0
$415$	32.6609	+	5.23100i	1.60326	+	0.256780i
$416$		0			0
$417$			17.0490i			0.834893i
$418$		0			0
$419$	−19.5549			−0.955321			−0.477661	−	0.878544i	$-0.658515\pi$
$419$	−19.5549			−0.955321			−0.477661	+	0.878544i	$0.658515\pi$
$420$		0			0
$421$	−5.07793			−0.247483			−0.123742	−	0.992314i	$-0.539489\pi$
$421$	−5.07793			−0.247483			−0.123742	+	0.992314i	$0.539489\pi$
$422$		0			0
$423$		−	6.41586i		−	0.311950i
$424$		0			0
$425$	−9.70861	+	29.5315i	−0.470937	+	1.43249i
$426$		0			0
$427$			36.9008i			1.78575i
$428$		0			0
$429$	5.51003			0.266027
$430$		0			0
$431$	−7.52900			−0.362659			−0.181330	−	0.983422i	$-0.558040\pi$
$431$	−7.52900			−0.362659			−0.181330	+	0.983422i	$0.558040\pi$
$432$		0			0
$433$			37.4821i			1.80127i	0.434573	+	0.900637i	$0.356899\pi$
$433$			37.4821i			1.80127i	−0.434573	+	0.900637i	$0.643101\pi$
$434$		0			0
$435$	3.77249	+	0.604205i	0.180877	+	0.0289694i
$436$		0			0
$437$		0			0
$438$		0			0
$439$	13.4836			0.643535			0.321767	−	0.946819i	$-0.395723\pi$
$439$	13.4836			0.643535			0.321767	+	0.946819i	$0.395723\pi$
$440$		0			0
$441$	2.57391			0.122567
$442$		0			0
$443$		−	6.61171i		−	0.314132i	−0.987588	−	0.157066i	$-0.949796\pi$
$443$		−	6.61171i		−	0.314132i	0.987588	−	0.157066i	$-0.0502035\pi$
$444$		0			0
$445$	−3.32654	+	20.7700i	−0.157693	+	0.984591i
$446$		0			0
$447$			19.5679i			0.925530i
$448$		0			0
$449$	−5.24183			−0.247377			−0.123689	−	0.992321i	$-0.539472\pi$
$449$	−5.24183			−0.247377			−0.123689	+	0.992321i	$0.539472\pi$
$450$		0			0
$451$	−1.09417			−0.0515226
$452$		0			0
$453$		−	9.22887i		−	0.433610i
$454$		0			0
$455$	−1.09417	+	6.83172i	−0.0512957	+	0.320276i
$456$		0			0
$457$		−	8.69702i		−	0.406829i	−0.979093	−	0.203415i	$-0.934796\pi$
$457$		−	8.69702i		−	0.406829i	0.979093	−	0.203415i	$-0.0652039\pi$
$458$		0			0
$459$	6.21728			0.290198
$460$		0			0
$461$	−3.11287			−0.144981			−0.0724905	−	0.997369i	$-0.523095\pi$
$461$	−3.11287			−0.144981			−0.0724905	+	0.997369i	$0.523095\pi$
$462$		0			0
$463$		−	33.8328i		−	1.57234i	−0.618008	−	0.786172i	$-0.712060\pi$
$463$		−	33.8328i		−	1.57234i	0.618008	−	0.786172i	$-0.287940\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	28.0112i		−	1.29620i	−0.761554	−	0.648102i	$-0.775563\pi$
$467$		−	28.0112i		−	1.29620i	0.761554	−	0.648102i	$-0.224437\pi$
$468$		0			0
$469$	14.9502			0.690335
$470$		0			0
$471$	−3.89423			−0.179437
$472$		0			0
$473$			31.4546i			1.44629i
$474$		0			0
$475$		0			0
$476$		0			0
$477$			4.02893i			0.184472i
$478$		0			0
$479$	0.424328			0.0193880			0.00969401	−	0.999953i	$-0.496914\pi$
$479$	0.424328			0.0193880			0.00969401	+	0.999953i	$0.496914\pi$
$480$		0			0
$481$	−4.02893			−0.183703
$482$		0			0
$483$			13.0490i			0.593750i
$484$		0			0
$485$	−31.5791	−	5.05774i	−1.43393	−	0.229660i
$486$		0			0
$487$			32.5720i			1.47598i	0.674813	+	0.737989i	$0.264224\pi$
$487$			32.5720i			1.47598i	−0.674813	+	0.737989i	$0.735776\pi$
$488$		0			0
$489$	−0.237741			−0.0107510
$490$		0			0
$491$	1.36659			0.0616734			0.0308367	−	0.999524i	$-0.490183\pi$
$491$	1.36659			0.0616734			0.0308367	+	0.999524i	$0.490183\pi$
$492$		0			0
$493$		−	10.6229i		−	0.478432i
$494$		0			0
$495$	−1.94848	+	12.1658i	−0.0875776	+	0.546810i
$496$		0			0
$497$		−	24.7217i		−	1.10892i
$498$		0			0
$499$	17.6430			0.789808			0.394904	−	0.918722i	$-0.370778\pi$
$499$	17.6430			0.789808			0.394904	+	0.918722i	$0.370778\pi$
$500$		0			0
$501$	11.8304			0.528541
$502$		0			0
$503$			24.6202i			1.09776i	0.835901	+	0.548880i	$0.184946\pi$
$503$			24.6202i			1.09776i	−0.835901	+	0.548880i	$0.815054\pi$
$504$		0			0
$505$	2.37846	−	14.8504i	0.105840	−	0.660835i
$506$		0			0
$507$			1.00000i			0.0444116i
$508$		0			0
$509$	−1.41204			−0.0625876			−0.0312938	−	0.999510i	$-0.509963\pi$
$509$	−1.41204			−0.0625876			−0.0312938	+	0.999510i	$0.509963\pi$
$510$		0			0
$511$	28.8113			1.27453
$512$		0			0
$513$		0			0
$514$		0			0
$515$	37.7068	+	6.03916i	1.66156	+	0.266117i
$516$		0			0
$517$		−	35.3516i		−	1.55476i
$518$		0			0
$519$	−11.6634			−0.511968
$520$		0			0
$521$	38.5298			1.68802			0.844011	−	0.536326i	$-0.180188\pi$
$521$	38.5298			1.68802			0.844011	+	0.536326i	$0.180188\pi$
$522$		0			0
$523$		−	2.23353i		−	0.0976653i	−0.998807	−	0.0488326i	$-0.984450\pi$
$523$		−	2.23353i		−	0.0976653i	0.998807	−	0.0488326i	$-0.0155501\pi$
$524$		0			0
$525$	−14.6970	−	4.83172i	−0.641430	−	0.210873i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	5.21455			0.226720
$530$		0			0
$531$	7.53896			0.327163
$532$		0			0
$533$		−	0.198578i		−	0.00860139i
$534$		0			0
$535$	16.8564	+	2.69974i	0.728767	+	0.116720i
$536$		0			0
$537$			12.5403i			0.541155i
$538$		0			0
$539$	14.1823			0.610876
$540$		0			0
$541$	−37.8892			−1.62898			−0.814492	−	0.580175i	$-0.802984\pi$
$541$	−37.8892			−1.62898			−0.814492	+	0.580175i	$0.802984\pi$
$542$		0			0
$543$			5.44343i			0.233600i
$544$		0			0
$545$	−1.77521	+	11.0839i	−0.0760418	+	0.474784i
$546$		0			0
$547$		−	43.8645i		−	1.87551i	−0.347299	−	0.937755i	$-0.612901\pi$
$547$		−	43.8645i		−	1.87551i	0.347299	−	0.937755i	$-0.387099\pi$
$548$		0			0
$549$	−11.9259			−0.508985
$550$		0			0
$551$		0			0
$552$		0			0
$553$			26.3357i			1.11991i
$554$		0			0
$555$	1.42473	−	8.89560i	0.0604763	−	0.377597i
$556$		0			0
$557$		−	13.4537i		−	0.570050i	−0.958520	−	0.285025i	$-0.907998\pi$
$557$		−	13.4537i		−	0.570050i	0.958520	−	0.285025i	$-0.0920019\pi$
$558$		0			0
$559$	−5.70861			−0.241448
$560$		0			0
$561$	34.2574			1.44635
$562$		0			0
$563$		−	31.6140i		−	1.33237i	−0.745785	−	0.666186i	$-0.767926\pi$
$563$		−	31.6140i		−	1.33237i	0.745785	−	0.666186i	$-0.232074\pi$
$564$		0			0
$565$	36.7022	+	5.87826i	1.54407	+	0.247300i
$566$		0			0
$567$			3.09417i			0.129943i
$568$		0			0
$569$	22.3562			0.937222			0.468611	−	0.883405i	$-0.344755\pi$
$569$	22.3562			0.937222			0.468611	+	0.883405i	$0.344755\pi$
$570$		0			0
$571$	12.6596			0.529788			0.264894	−	0.964277i	$-0.414663\pi$
$571$	12.6596			0.529788			0.264894	+	0.964277i	$0.414663\pi$
$572$		0			0
$573$			5.22887i			0.218439i
$574$		0			0
$575$	6.58550	−	20.0317i	0.274635	−	0.835378i
$576$		0			0
$577$		−	7.48861i		−	0.311755i	−0.987776	−	0.155877i	$-0.950180\pi$
$577$		−	7.48861i		−	0.311755i	0.987776	−	0.155877i	$-0.0498205\pi$
$578$		0			0
$579$	−17.5287			−0.728469
$580$		0			0
$581$	45.7707			1.89889
$582$		0			0
$583$			22.1995i			0.919411i
$584$		0			0
$585$	−2.20793	−	0.353624i	−0.0912866	−	0.0146205i
$586$		0			0
$587$			4.41858i			0.182374i	0.995834	+	0.0911872i	$0.0290662\pi$
$587$			4.41858i			0.182374i	−0.995834	+	0.0911872i	$0.970934\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$	−22.4764			−0.924557
$592$		0			0
$593$		−	12.5810i		−	0.516641i	−0.966059	−	0.258320i	$-0.916831\pi$
$593$		−	12.5810i		−	0.516641i	0.966059	−	0.258320i	$-0.0831690\pi$
$594$		0			0
$595$	−6.80278	+	42.4747i	−0.278887	+	1.74129i
$596$		0			0
$597$			16.9143i			0.692256i
$598$		0			0
$599$	−2.64337			−0.108005			−0.0540025	−	0.998541i	$-0.517198\pi$
$599$	−2.64337			−0.108005			−0.0540025	+	0.998541i	$0.517198\pi$
$600$		0			0
$601$	−10.4740			−0.427243			−0.213622	−	0.976916i	$-0.568526\pi$
$601$	−10.4740			−0.427243			−0.213622	+	0.976916i	$0.568526\pi$
$602$		0			0
$603$			4.83172i			0.196763i
$604$		0			0
$605$	−6.84632	+	42.7465i	−0.278342	+	1.73789i
$606$		0			0
$607$		−	12.0197i		−	0.487863i	−0.969792	−	0.243932i	$-0.921563\pi$
$607$		−	12.0197i		−	0.487863i	0.969792	−	0.243932i	$-0.0784372\pi$
$608$		0			0
$609$	5.28674			0.214229
$610$		0			0
$611$	6.41586			0.259558
$612$		0			0
$613$		−	34.7676i		−	1.40425i	−0.712054	−	0.702124i	$-0.752235\pi$
$613$		−	34.7676i		−	1.40425i	0.712054	−	0.702124i	$-0.247765\pi$
$614$		0			0
$615$	0.438447	+	0.0702221i	0.0176799	+	0.00283163i
$616$		0			0
$617$			31.6644i			1.27476i	0.770549	+	0.637380i	$0.219982\pi$
$617$			31.6644i			1.27476i	−0.770549	+	0.637380i	$0.780018\pi$
$618$		0			0
$619$	22.6024			0.908469			0.454234	−	0.890882i	$-0.349913\pi$
$619$	22.6024			0.908469			0.454234	+	0.890882i	$0.349913\pi$
$620$		0			0
$621$	−4.21728			−0.169234
$622$		0			0
$623$			29.1069i			1.16614i
$624$		0			0
$625$	20.1231	+	14.8344i	0.804924	+	0.593378i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−25.0490			−0.998769
$630$		0			0
$631$	−18.6229			−0.741366			−0.370683	−	0.928759i	$-0.620876\pi$
$631$	−18.6229			−0.741366			−0.370683	+	0.928759i	$0.620876\pi$
$632$		0			0
$633$		−	11.8391i		−	0.470562i
$634$		0			0
$635$	−6.35198	−	1.01734i	−0.252071	−	0.0403718i
$636$		0			0
$637$			2.57391i			0.101982i
$638$		0			0
$639$	7.98977			0.316070
$640$		0			0
$641$	−8.77658			−0.346654			−0.173327	−	0.984864i	$-0.555452\pi$
$641$	−8.77658			−0.346654			−0.173327	+	0.984864i	$0.555452\pi$
$642$		0			0
$643$		−	30.6920i		−	1.21037i	−0.796084	−	0.605186i	$-0.793099\pi$
$643$		−	30.6920i		−	1.21037i	0.796084	−	0.605186i	$-0.206901\pi$
$644$		0			0
$645$	2.01870	−	12.6042i	0.0794863	−	0.496290i
$646$		0			0
$647$		−	36.2023i		−	1.42326i	−0.702555	−	0.711629i	$-0.747958\pi$
$647$		−	36.2023i		−	1.42326i	0.702555	−	0.711629i	$-0.252042\pi$
$648$		0			0
$649$	41.5399			1.63058
$650$		0			0
$651$		0			0
$652$		0			0
$653$			42.8746i			1.67781i	0.544277	+	0.838906i	$0.316804\pi$
$653$			42.8746i			1.67781i	−0.544277	+	0.838906i	$0.683196\pi$
$654$		0			0
$655$	−5.84182	+	36.4747i	−0.228259	+	1.42518i
$656$		0			0
$657$			9.31145i			0.363274i
$658$		0			0
$659$	−3.72907			−0.145264			−0.0726320	−	0.997359i	$-0.523140\pi$
$659$	−3.72907			−0.145264			−0.0726320	+	0.997359i	$0.523140\pi$
$660$		0			0
$661$	−13.2057			−0.513642			−0.256821	−	0.966459i	$-0.582675\pi$
$661$	−13.2057			−0.513642			−0.256821	+	0.966459i	$0.582675\pi$
$662$		0			0
$663$			6.21728i			0.241459i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			7.20569i			0.279005i
$668$		0			0
$669$	4.48246			0.173302
$670$		0			0
$671$	−65.7120			−2.53678
$672$		0			0
$673$			0.901611i			0.0347546i	0.999849	+	0.0173773i	$0.00553164\pi$
$673$			0.901611i			0.0347546i	−0.999849	+	0.0173773i	$0.994468\pi$
$674$		0			0
$675$	1.56155	−	4.74990i	0.0601042	−	0.182824i
$676$		0			0
$677$		−	10.0057i		−	0.384552i	−0.981341	−	0.192276i	$-0.938413\pi$
$677$		−	10.0057i		−	0.384552i	0.981341	−	0.192276i	$-0.0615869\pi$
$678$		0			0
$679$	−44.2547			−1.69834
$680$		0			0
$681$	6.35799			0.243639
$682$		0			0
$683$		−	16.9110i		−	0.647082i	−0.946214	−	0.323541i	$-0.895127\pi$
$683$		−	16.9110i		−	0.647082i	0.946214	−	0.323541i	$-0.104873\pi$
$684$		0			0
$685$	−28.3728	−	4.54421i	−1.08407	−	0.173626i
$686$		0			0
$687$			6.45502i			0.246274i
$688$		0			0
$689$	−4.02893			−0.153490
$690$		0			0
$691$	−43.1003			−1.63961			−0.819807	−	0.572640i	$-0.805919\pi$
$691$	−43.1003			−1.63961			−0.819807	+	0.572640i	$0.805919\pi$
$692$		0			0
$693$			17.0490i			0.647638i
$694$		0			0
$695$	6.02893	−	37.6430i	0.228690	−	1.42788i
$696$		0			0
$697$		−	1.23462i		−	0.0467645i
$698$		0			0
$699$	6.40290			0.242180
$700$		0			0
$701$	34.9170			1.31880			0.659399	−	0.751793i	$-0.270811\pi$
$701$	34.9170			1.31880			0.659399	+	0.751793i	$0.270811\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	−2.26880	+	14.1658i	−0.0854480	+	0.533513i
$706$		0			0
$707$		−	20.8113i		−	0.782688i
$708$		0			0
$709$	−20.8140			−0.781685			−0.390843	−	0.920457i	$-0.627816\pi$
$709$	−20.8140			−0.781685			−0.390843	+	0.920457i	$0.627816\pi$
$710$		0			0
$711$	−8.51139			−0.319202
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−12.1658	−	1.94848i	−0.454974	−	0.0728690i
$716$		0			0
$717$			1.55793i			0.0581821i
$718$		0			0
$719$	10.6024			0.395404			0.197702	−	0.980262i	$-0.436652\pi$
$719$	10.6024			0.395404			0.197702	+	0.980262i	$0.436652\pi$
$720$		0			0
$721$	52.8421			1.96794
$722$		0			0
$723$			14.1911i			0.527772i
$724$		0			0
$725$	−8.11573	−	2.66808i	−0.301411	−	0.0990902i
$726$		0			0
$727$			1.06524i			0.0395076i	0.999805	+	0.0197538i	$0.00628824\pi$
$727$			1.06524i			0.0395076i	−0.999805	+	0.0197538i	$0.993712\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−35.4920			−1.31272
$732$		0			0
$733$			29.1216i			1.07563i	0.843062	+	0.537816i	$0.180750\pi$
$733$			29.1216i			1.07563i	−0.843062	+	0.537816i	$0.819250\pi$
$734$		0			0
$735$	−5.68301	−	0.910196i	−0.209621	−	0.0335731i
$736$		0			0
$737$			26.6229i			0.980667i
$738$		0			0
$739$	−41.7966			−1.53751			−0.768757	−	0.639541i	$-0.779125\pi$
$739$	−41.7966			−1.53751			−0.768757	+	0.639541i	$0.779125\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			32.8531i			1.20526i	0.798019	+	0.602632i	$0.205881\pi$
$743$			32.8531i			1.20526i	−0.798019	+	0.602632i	$0.794119\pi$
$744$		0			0
$745$	6.91968	−	43.2045i	0.253517	−	1.58289i
$746$		0			0
$747$			14.7926i			0.541231i
$748$		0			0
$749$	23.6225			0.863146
$750$		0			0
$751$	20.2956			0.740597			0.370299	−	0.928913i	$-0.379255\pi$
$751$	20.2956			0.740597			0.370299	+	0.928913i	$0.379255\pi$
$752$		0			0
$753$			22.7082i			0.827533i
$754$		0			0
$755$	−3.26355	+	20.3767i	−0.118773	+	0.741584i
$756$		0			0
$757$		−	1.62331i		−	0.0590000i	−0.999565	−	0.0295000i	$-0.990608\pi$
$757$		−	1.62331i		−	0.0590000i	0.999565	−	0.0295000i	$-0.00939151\pi$
$758$		0			0
$759$	−23.2373			−0.843462
$760$		0			0
$761$	−50.4410			−1.82848			−0.914242	−	0.405169i	$-0.867213\pi$
$761$	−50.4410			−1.82848			−0.914242	+	0.405169i	$0.867213\pi$
$762$		0			0
$763$			15.5329i			0.562330i
$764$		0			0
$765$	−13.7273	−	2.19858i	−0.496312	−	0.0794898i
$766$		0			0
$767$			7.53896i			0.272216i
$768$		0			0
$769$	6.75339			0.243533			0.121767	−	0.992559i	$-0.461144\pi$
$769$	6.75339			0.243533			0.121767	+	0.992559i	$0.461144\pi$
$770$		0			0
$771$	−21.4172			−0.771322
$772$		0			0
$773$		−	36.5660i		−	1.31519i	−0.753373	−	0.657594i	$-0.771574\pi$
$773$		−	36.5660i		−	1.31519i	0.753373	−	0.657594i	$-0.228426\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		−	12.4662i		−	0.447223i
$778$		0			0
$779$		0			0
$780$		0			0
$781$	44.0239			1.57530
$782$		0			0
$783$			1.70861i			0.0610607i
$784$		0			0
$785$	8.59819	+	1.37709i	0.306883	+	0.0491506i
$786$		0			0
$787$		−	6.21426i		−	0.221514i	−0.993847	−	0.110757i	$-0.964672\pi$
$787$		−	6.21426i		−	0.221514i	0.993847	−	0.110757i	$-0.0353276\pi$
$788$		0			0
$789$	−26.4951			−0.943252
$790$		0			0
$791$	51.4342			1.82879
$792$		0			0
$793$		−	11.9259i		−	0.423501i
$794$		0			0
$795$	1.42473	−	8.89560i	0.0505298	−	0.315494i
$796$		0			0
$797$			39.5788i			1.40195i	0.713184	+	0.700977i	$0.247252\pi$
$797$			39.5788i			1.40195i	−0.713184	+	0.700977i	$0.752748\pi$
$798$		0			0
$799$	39.8892			1.41118
$800$		0			0
$801$	−9.40699			−0.332380
$802$		0			0
$803$			51.3064i			1.81056i
$804$		0			0
$805$	4.61444	−	28.8113i	0.162638	−	1.01546i
$806$		0			0
$807$			14.7259i			0.518378i
$808$		0			0
$809$	3.47236			0.122082			0.0610408	−	0.998135i	$-0.480558\pi$
$809$	3.47236			0.122082			0.0610408	+	0.998135i	$0.480558\pi$
$810$		0			0
$811$	29.2404			1.02677			0.513384	−	0.858159i	$-0.328392\pi$
$811$	29.2404			1.02677			0.513384	+	0.858159i	$0.328392\pi$
$812$		0			0
$813$		−	25.8491i		−	0.906566i
$814$		0			0
$815$	0.524916	+	0.0840710i	0.0183870	+	0.00294488i
$816$		0			0
$817$		0			0
$818$		0			0
$819$	−3.09417			−0.108119
$820$		0			0
$821$	48.3417			1.68714			0.843569	−	0.537020i	$-0.180450\pi$
$821$	48.3417			1.68714			0.843569	+	0.537020i	$0.180450\pi$
$822$		0			0
$823$		−	9.33889i		−	0.325533i	−0.986665	−	0.162767i	$-0.947958\pi$
$823$		−	9.33889i		−	0.325533i	0.986665	−	0.162767i	$-0.0520418\pi$
$824$		0			0
$825$	8.60421	−	26.1721i	0.299560	−	0.911196i
$826$		0			0
$827$			45.8175i			1.59323i	0.604486	+	0.796616i	$0.293379\pi$
$827$			45.8175i			1.59323i	−0.604486	+	0.796616i	$0.706621\pi$
$828$		0			0
$829$	37.8819			1.31569			0.657847	−	0.753151i	$-0.271467\pi$
$829$	37.8819			1.31569			0.657847	+	0.753151i	$0.271467\pi$
$830$		0			0
$831$	20.1227			0.698049
$832$		0			0
$833$			16.0027i			0.554462i
$834$		0			0
$835$	−26.1206	−	4.18350i	−0.903940	−	0.144776i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−32.6678			−1.12782			−0.563909	−	0.825837i	$-0.690703\pi$
$839$	−32.6678			−1.12782			−0.563909	+	0.825837i	$0.690703\pi$
$840$		0			0
$841$	−26.0807			−0.899333
$842$		0			0
$843$			16.3954i			0.564687i
$844$		0			0
$845$	0.353624	−	2.20793i	0.0121650	−	0.0759551i
$846$		0			0
$847$			59.9046i			2.05835i
$848$		0			0
$849$	−14.5650			−0.499871
$850$		0			0
$851$	16.9911			0.582448
$852$		0			0
$853$		−	26.4867i		−	0.906887i	−0.891285	−	0.453443i	$-0.850195\pi$
$853$		−	26.4867i		−	0.906887i	0.891285	−	0.453443i	$-0.149805\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			43.1292i			1.47327i	0.676292	+	0.736634i	$0.263586\pi$
$857$			43.1292i			1.47327i	−0.676292	+	0.736634i	$0.736414\pi$
$858$		0			0
$859$	−41.1690			−1.40467			−0.702334	−	0.711848i	$-0.747859\pi$
$859$	−41.1690			−1.40467			−0.702334	+	0.711848i	$0.747859\pi$
$860$		0			0
$861$	0.614436			0.0209399
$862$		0			0
$863$			7.88822i			0.268518i	0.990946	+	0.134259i	$0.0428654\pi$
$863$			7.88822i			0.268518i	−0.990946	+	0.134259i	$0.957135\pi$
$864$		0			0
$865$	25.7520	+	4.12447i	0.875595	+	0.140236i
$866$		0			0
$867$			21.6546i			0.735428i
$868$		0			0
$869$	−46.8980			−1.59091
$870$		0			0
$871$	−4.83172			−0.163716
$872$		0			0
$873$		−	14.3026i		−	0.484069i
$874$		0			0
$875$	30.7414	+	15.8653i	1.03925	+	0.536345i
$876$		0			0
$877$		−	6.03740i		−	0.203869i	−0.994791	−	0.101934i	$-0.967497\pi$
$877$		−	6.03740i		−	0.203869i	0.994791	−	0.101934i	$-0.0325031\pi$
$878$		0			0
$879$	26.6995			0.900551
$880$		0			0
$881$	−41.4546			−1.39664			−0.698321	−	0.715785i	$-0.746069\pi$
$881$	−41.4546			−1.39664			−0.698321	+	0.715785i	$0.746069\pi$
$882$		0			0
$883$			35.0575i			1.17978i	0.807484	+	0.589889i	$0.200828\pi$
$883$			35.0575i			1.17978i	−0.807484	+	0.589889i	$0.799172\pi$
$884$		0			0
$885$	−16.6455	−	2.66596i	−0.559532	−	0.0896152i
$886$		0			0
$887$		−	26.8661i		−	0.902075i	−0.892505	−	0.451038i	$-0.851054\pi$
$887$		−	26.8661i		−	0.902075i	0.892505	−	0.451038i	$-0.148946\pi$
$888$		0			0
$889$	−8.90161			−0.298550
$890$		0			0
$891$	−5.51003			−0.184593
$892$		0			0
$893$		0			0
$894$		0			0
$895$	4.43456	−	27.6881i	0.148231	−	0.925513i
$896$		0			0
$897$		−	4.21728i		−	0.140811i
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−25.0490			−0.834503
$902$		0			0
$903$		−	17.6634i		−	0.587802i
$904$		0			0
$905$	1.92493	−	12.0187i	0.0639867	−	0.399515i
$906$		0			0
$907$			38.2137i			1.26887i	0.772978	+	0.634433i	$0.218766\pi$
$907$			38.2137i			1.26887i	−0.772978	+	0.634433i	$0.781234\pi$
$908$		0			0
$909$	6.72595			0.223086
$910$		0			0
$911$	49.8919			1.65299			0.826496	−	0.562942i	$-0.190331\pi$
$911$	49.8919			1.65299			0.826496	+	0.562942i	$0.190331\pi$
$912$		0			0
$913$			81.5074i			2.69750i
$914$		0			0
$915$	26.3315	+	4.21728i	0.870493	+	0.139419i
$916$		0			0
$917$			51.1153i			1.68798i
$918$		0			0
$919$	5.41599			0.178657			0.0893284	−	0.996002i	$-0.471528\pi$
$919$	5.41599			0.178657			0.0893284	+	0.996002i	$0.471528\pi$
$920$		0			0
$921$	−11.4230			−0.376400
$922$		0			0
$923$			7.98977i			0.262986i
$924$		0			0
$925$	−6.29139	+	19.1370i	−0.206860	+	0.629221i
$926$		0			0
$927$			17.0779i			0.560913i
$928$		0			0
$929$	58.7311			1.92691			0.963453	−	0.267878i	$-0.0863224\pi$
$929$	58.7311			1.92691			0.963453	+	0.267878i	$0.0863224\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$			19.4546i			0.636916i
$934$		0			0
$935$	−75.6379	−	12.1142i	−2.47362	−	0.396178i
$936$		0			0
$937$		−	6.01541i		−	0.196515i	−0.995161	−	0.0982574i	$-0.968673\pi$
$937$		−	6.01541i		−	0.196515i	0.995161	−	0.0982574i	$-0.0313269\pi$
$938$		0			0
$939$	32.5805			1.06322
$940$		0			0
$941$	40.7389			1.32805			0.664025	−	0.747710i	$-0.268847\pi$
$941$	40.7389			1.32805			0.664025	+	0.747710i	$0.268847\pi$
$942$		0			0
$943$			0.837461i			0.0272715i
$944$		0			0
$945$	1.09417	−	6.83172i	0.0355935	−	0.222236i
$946$		0			0
$947$		−	3.69715i		−	0.120141i	−0.998194	−	0.0600705i	$-0.980867\pi$
$947$		−	3.69715i		−	0.120141i	0.998194	−	0.0600705i	$-0.0191326\pi$
$948$		0			0
$949$	−9.31145			−0.302263
$950$		0			0
$951$	−31.3781			−1.01750
$952$		0			0
$953$			1.12460i			0.0364292i	0.999834	+	0.0182146i	$0.00579822\pi$
$953$			1.12460i			0.0364292i	−0.999834	+	0.0182146i	$0.994202\pi$
$954$		0			0
$955$	1.84905	−	11.5450i	0.0598340	−	0.373587i
$956$		0			0
$957$			9.41450i			0.304327i
$958$		0			0
$959$	−39.7614			−1.28396
$960$		0			0
$961$	−31.0000			−1.00000
$962$		0			0
$963$			7.63450i			0.246018i
$964$		0			0
$965$	38.7022	+	6.19858i	1.24587	+	0.199539i
$966$		0			0
$967$			5.85956i			0.188431i	0.995552	+	0.0942153i	$0.0300342\pi$
$967$			5.85956i			0.188431i	−0.995552	+	0.0942153i	$0.969966\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−33.0737			−1.06138			−0.530692	−	0.847565i	$-0.678068\pi$
$971$	−33.0737			−1.06138			−0.530692	+	0.847565i	$0.678068\pi$
$972$		0			0
$973$		−	52.7526i		−	1.69117i
$974$		0			0
$975$	4.74990	+	1.56155i	0.152119	+	0.0500097i
$976$		0			0
$977$		−	21.1488i		−	0.676610i	−0.941037	−	0.338305i	$-0.890147\pi$
$977$		−	21.1488i		−	0.676610i	0.941037	−	0.338305i	$-0.109853\pi$
$978$		0			0
$979$	−51.8328			−1.65658
$980$		0			0
$981$	−5.02006			−0.160278
$982$		0			0
$983$		−	32.8960i		−	1.04922i	−0.851343	−	0.524610i	$-0.824211\pi$
$983$		−	32.8960i		−	1.04922i	0.851343	−	0.524610i	$-0.175789\pi$
$984$		0			0
$985$	49.6264	+	7.94821i	1.58123	+	0.253251i
$986$		0			0
$987$			19.8518i			0.631889i
$988$		0			0
$989$	24.0748			0.765534
$990$		0			0
$991$	18.6322			0.591871			0.295935	−	0.955208i	$-0.404369\pi$
$991$	18.6322			0.591871			0.295935	+	0.955208i	$0.404369\pi$
$992$		0			0
$993$		−	30.4951i		−	0.967734i
$994$		0			0
$995$	5.98130	−	37.3456i	0.189620	−	1.18393i
$996$		0			0
$997$			3.08298i			0.0976389i	0.998808	+	0.0488195i	$0.0155459\pi$
$997$			3.08298i			0.0976389i	−0.998808	+	0.0488195i	$0.984454\pi$
$998$		0			0
$999$	4.02893			0.127470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3120.2.l.n.1249.3		8
4.3	odd	2		1560.2.l.d.1249.7	yes	8
5.4	even	2	inner	3120.2.l.n.1249.7		8
12.11	even	2		4680.2.l.g.2809.3		8
20.3	even	4		7800.2.a.bt.1.2		4
20.7	even	4		7800.2.a.by.1.3		4
20.19	odd	2		1560.2.l.d.1249.3	✓	8
60.59	even	2		4680.2.l.g.2809.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.l.d.1249.3	✓	8	20.19	odd	2
1560.2.l.d.1249.7	yes	8	4.3	odd	2
3120.2.l.n.1249.3		8	1.1	even	1	trivial
3120.2.l.n.1249.7		8	5.4	even	2	inner
4680.2.l.g.2809.3		8	12.11	even	2
4680.2.l.g.2809.4		8	60.59	even	2
7800.2.a.bt.1.2		4	20.3	even	4
7800.2.a.by.1.3		4	20.7	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3120.l (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +(-0.353624 + 2.20793i) q^{5} +3.09417i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +(-0.353624 + 2.20793i) q^{5} +3.09417i q^{7} -1.00000 q^{9} -5.51003 q^{11} -1.00000i q^{13} +(2.20793 + 0.353624i) q^{15} -6.21728i q^{17} +3.09417 q^{21} +4.21728i q^{23} +(-4.74990 - 1.56155i) q^{25} +1.00000i q^{27} +1.70861 q^{29} +5.51003i q^{33} +(-6.83172 - 1.09417i) q^{35} -4.02893i q^{37} -1.00000 q^{39} +0.198578 q^{41} -5.70861i q^{43} +(0.353624 - 2.20793i) q^{45} +6.41586i q^{47} -2.57391 q^{49} -6.21728 q^{51} -4.02893i q^{53} +(1.94848 - 12.1658i) q^{55} -7.53896 q^{59} +11.9259 q^{61} -3.09417i q^{63} +(2.20793 + 0.353624i) q^{65} -4.83172i q^{67} +4.21728 q^{69} -7.98977 q^{71} -9.31145i q^{73} +(-1.56155 + 4.74990i) q^{75} -17.0490i q^{77} +8.51139 q^{79} +1.00000 q^{81} -14.7926i q^{83} +(13.7273 + 2.19858i) q^{85} -1.70861i q^{87} +9.40699 q^{89} +3.09417 q^{91} +14.3026i q^{97} +5.51003 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{5} - 8 q^{9}+O(q^{10}) \) 8 * q - 2 * q^5 - 8 * q^9	\( 8 q - 2 q^{5} - 8 q^{9} - 2 q^{11} + 2 q^{15} + 14 q^{21} - 16 q^{29} + 8 q^{35} - 8 q^{39} + 14 q^{41} + 2 q^{45} - 18 q^{49} - 6 q^{51} - 10 q^{55} + 4 q^{59} + 22 q^{61} + 2 q^{65} - 10 q^{69} - 30 q^{71} + 4 q^{75} - 2 q^{79} + 8 q^{81} + 24 q^{85} - 18 q^{89} + 14 q^{91} + 2 q^{99}+O(q^{100}) \) 8 * q - 2 * q^5 - 8 * q^9 - 2 * q^11 + 2 * q^15 + 14 * q^21 - 16 * q^29 + 8 * q^35 - 8 * q^39 + 14 * q^41 + 2 * q^45 - 18 * q^49 - 6 * q^51 - 10 * q^55 + 4 * q^59 + 22 * q^61 + 2 * q^65 - 10 * q^69 - 30 * q^71 + 4 * q^75 - 2 * q^79 + 8 * q^81 + 24 * q^85 - 18 * q^89 + 14 * q^91 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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