LMFDB - Embedded newform 3120.2.l.o.1249.7

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:	8.0.1698758656.6
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 18x^{6} + 97x^{4} + 176x^{2} + 64 $ x^8 + 18x^6 + 97x^4 + 176*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 1560)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.7
Root			$1.69230i$ of defining polynomial
Character	$\chi$	$=$	3120.1249
Dual form			3120.2.l.o.1249.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(0.489528 - 2.18183i) q^{5} +2.82843i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +(0.489528 - 2.18183i) q^{5} +2.82843i q^{7} -1.00000 q^{9} +2.97906 q^{11} +1.00000i q^{13} +(2.18183 + 0.489528i) q^{15} +1.44383i q^{17} +4.17113 q^{19} -2.82843 q^{21} +6.21302i q^{23} +(-4.52072 - 2.13613i) q^{25} -1.00000i q^{27} +0.828427 q^{29} -1.65685 q^{31} +2.97906i q^{33} +(6.17113 + 1.38459i) q^{35} -0.0418875i q^{37} -1.00000 q^{39} -4.36365 q^{41} -8.99956i q^{43} +(-0.489528 + 2.18183i) q^{45} +4.40554i q^{47} -1.00000 q^{49} -1.44383 q^{51} +6.72730i q^{53} +(1.45833 - 6.49978i) q^{55} +4.17113i q^{57} -2.97906 q^{59} +6.27226 q^{61} -2.82843i q^{63} +(2.18183 + 0.489528i) q^{65} -0.727302i q^{67} -6.21302 q^{69} +8.32176 q^{71} +6.87031i q^{73} +(2.13613 - 4.52072i) q^{75} +8.42604i q^{77} +6.11190 q^{79} +1.00000 q^{81} +14.7059i q^{83} +(3.15019 + 0.706797i) q^{85} +0.828427i q^{87} -14.7478 q^{89} -2.82843 q^{91} -1.65685i q^{93} +(2.04189 - 9.10069i) q^{95} -6.78654i q^{97} -2.97906 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^9	$ 8 q - 8 q^{9} + 16 q^{11} + 4 q^{15} - 24 q^{19} - 4 q^{25} - 16 q^{29} + 32 q^{31} - 8 q^{35} - 8 q^{39} - 8 q^{41} - 8 q^{49} - 8 q^{51} + 36 q^{55} - 16 q^{59} + 24 q^{61} + 4 q^{65} - 8 q^{69} + 24 q^{71} + 4 q^{75} - 24 q^{79} + 8 q^{81} - 40 q^{85} + 8 q^{89} + 32 q^{95} - 16 q^{99}+O(q^{100}) $ 8 * q - 8 * q^9 + 16 * q^11 + 4 * q^15 - 24 * q^19 - 4 * q^25 - 16 * q^29 + 32 * q^31 - 8 * q^35 - 8 * q^39 - 8 * q^41 - 8 * q^49 - 8 * q^51 + 36 * q^55 - 16 * q^59 + 24 * q^61 + 4 * q^65 - 8 * q^69 + 24 * q^71 + 4 * q^75 - 24 * q^79 + 8 * q^81 - 40 * q^85 + 8 * q^89 + 32 * q^95 - 16 * 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
$5$	0.489528	−	2.18183i	0.218924	−	0.975742i
$5$	0.489528	−	2.18183i	0.218924	−	0.975742i
$6$		0			0
$7$			2.82843i			1.06904i	0.845154	+	0.534522i	$0.179509\pi$
$7$			2.82843i			1.06904i	−0.845154	+	0.534522i	$0.820491\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	2.97906			0.898219			0.449110	−	0.893477i	$-0.351741\pi$
$11$	2.97906			0.898219			0.449110	+	0.893477i	$0.351741\pi$
$12$		0			0
$13$			1.00000i			0.277350i
$13$			1.00000i			0.277350i
$14$		0			0
$15$	2.18183	+	0.489528i	0.563345	+	0.126396i
$16$		0			0
$17$			1.44383i			0.350181i	0.984552	+	0.175090i	$0.0560218\pi$
$17$			1.44383i			0.350181i	−0.984552	+	0.175090i	$0.943978\pi$
$18$		0			0
$19$	4.17113			0.956924			0.478462	−	0.878108i	$-0.341194\pi$
$19$	4.17113			0.956924			0.478462	+	0.878108i	$0.341194\pi$
$20$		0			0
$21$	−2.82843			−0.617213
$22$		0			0
$23$			6.21302i			1.29550i	0.761851	+	0.647752i	$0.224291\pi$
$23$			6.21302i			1.29550i	−0.761851	+	0.647752i	$0.775709\pi$
$24$		0			0
$25$	−4.52072	−	2.13613i	−0.904145	−	0.427226i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	0.828427			0.153835			0.0769175	−	0.997037i	$-0.475492\pi$
$29$	0.828427			0.153835			0.0769175	+	0.997037i	$0.475492\pi$
$30$		0			0
$31$	−1.65685			−0.297580			−0.148790	−	0.988869i	$-0.547538\pi$
$31$	−1.65685			−0.297580			−0.148790	+	0.988869i	$0.547538\pi$
$32$		0			0
$33$			2.97906i			0.518587i
$34$		0			0
$35$	6.17113	+	1.38459i	1.04311	+	0.234039i
$36$		0			0
$37$		−	0.0418875i		−	0.00688626i	−0.999994	−	0.00344313i	$-0.998904\pi$
$37$		−	0.0418875i		−	0.00688626i	0.999994	−	0.00344313i	$-0.00109599\pi$
$38$		0			0
$39$	−1.00000			−0.160128
$40$		0			0
$41$	−4.36365			−0.681488			−0.340744	−	0.940156i	$-0.610679\pi$
$41$	−4.36365			−0.681488			−0.340744	+	0.940156i	$0.610679\pi$
$42$		0			0
$43$		−	8.99956i		−	1.37242i	−0.727403	−	0.686210i	$-0.759273\pi$
$43$		−	8.99956i		−	1.37242i	0.727403	−	0.686210i	$-0.240727\pi$
$44$		0			0
$45$	−0.489528	+	2.18183i	−0.0729745	+	0.325247i
$46$		0			0
$47$			4.40554i			0.642614i	0.946975	+	0.321307i	$0.104122\pi$
$47$			4.40554i			0.642614i	−0.946975	+	0.321307i	$0.895878\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−1.44383			−0.202177
$52$		0			0
$53$			6.72730i			0.924066i	0.886863	+	0.462033i	$0.152880\pi$
$53$			6.72730i			0.924066i	−0.886863	+	0.462033i	$0.847120\pi$
$54$		0			0
$55$	1.45833	−	6.49978i	0.196641	−	0.876430i
$56$		0			0
$57$			4.17113i			0.552480i
$58$		0			0
$59$	−2.97906			−0.387840			−0.193920	−	0.981017i	$-0.562120\pi$
$59$	−2.97906			−0.387840			−0.193920	+	0.981017i	$0.562120\pi$
$60$		0			0
$61$	6.27226			0.803081			0.401540	−	0.915841i	$-0.368475\pi$
$61$	6.27226			0.803081			0.401540	+	0.915841i	$0.368475\pi$
$62$		0			0
$63$		−	2.82843i		−	0.356348i
$64$		0			0
$65$	2.18183	+	0.489528i	0.270622	+	0.0607185i
$66$		0			0
$67$		−	0.727302i		−	0.0888540i	−0.999013	−	0.0444270i	$-0.985854\pi$
$67$		−	0.727302i		−	0.0888540i	0.999013	−	0.0444270i	$-0.0141462\pi$
$68$		0			0
$69$	−6.21302			−0.747960
$70$		0			0
$71$	8.32176			0.987612			0.493806	−	0.869572i	$-0.335605\pi$
$71$	8.32176			0.987612			0.493806	+	0.869572i	$0.335605\pi$
$72$		0			0
$73$			6.87031i			0.804110i	0.915616	+	0.402055i	$0.131704\pi$
$73$			6.87031i			0.804110i	−0.915616	+	0.402055i	$0.868296\pi$
$74$		0			0
$75$	2.13613	−	4.52072i	0.246659	−	0.522008i
$76$		0			0
$77$			8.42604i			0.960237i
$78$		0			0
$79$	6.11190			0.687642			0.343821	−	0.939035i	$-0.388279\pi$
$79$	6.11190			0.687642			0.343821	+	0.939035i	$0.388279\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			14.7059i			1.61418i	0.590425	+	0.807092i	$0.298960\pi$
$83$			14.7059i			1.61418i	−0.590425	+	0.807092i	$0.701040\pi$
$84$		0			0
$85$	3.15019	+	0.706797i	0.341686	+	0.0766629i
$86$		0			0
$87$			0.828427i			0.0888167i
$88$		0			0
$89$	−14.7478			−1.56326			−0.781632	−	0.623740i	$-0.785613\pi$
$89$	−14.7478			−1.56326			−0.781632	+	0.623740i	$0.785613\pi$
$90$		0			0
$91$	−2.82843			−0.296500
$92$		0			0
$93$		−	1.65685i		−	0.171808i
$94$		0			0
$95$	2.04189	−	9.10069i	0.209493	−	0.933711i
$96$		0			0
$97$		−	6.78654i		−	0.689069i	−0.938774	−	0.344534i	$-0.888037\pi$
$97$		−	6.78654i		−	0.689069i	0.938774	−	0.344534i	$-0.111963\pi$
$98$		0			0
$99$	−2.97906			−0.299406
$100$		0			0
$101$	12.1421			1.20819			0.604094	−	0.796913i	$-0.293535\pi$
$101$	12.1421			1.20819			0.604094	+	0.796913i	$0.293535\pi$
$102$		0			0
$103$			2.81108i			0.276984i	0.990364	+	0.138492i	$0.0442255\pi$
$103$			2.81108i			0.276984i	−0.990364	+	0.138492i	$0.955775\pi$
$104$		0			0
$105$	−1.38459	+	6.17113i	−0.135123	+	0.602241i
$106$		0			0
$107$			18.3842i			1.77726i	0.458621	+	0.888632i	$0.348343\pi$
$107$			18.3842i			1.77726i	−0.458621	+	0.888632i	$0.651657\pi$
$108$		0			0
$109$	−1.44383			−0.138294			−0.0691470	−	0.997606i	$-0.522028\pi$
$109$	−1.44383			−0.138294			−0.0691470	+	0.997606i	$0.522028\pi$
$110$		0			0
$111$	0.0418875			0.00397579
$112$		0			0
$113$			3.40194i			0.320028i	0.987115	+	0.160014i	$0.0511539\pi$
$113$			3.40194i			0.320028i	−0.987115	+	0.160014i	$0.948846\pi$
$114$		0			0
$115$	13.5557	+	3.04145i	1.26408	+	0.283617i
$116$		0			0
$117$		−	1.00000i		−	0.0924500i
$118$		0			0
$119$	−4.08378			−0.374359
$120$		0			0
$121$	−2.12522			−0.193202
$122$		0			0
$123$		−	4.36365i		−	0.393457i
$124$		0			0
$125$	−6.87368	+	8.81774i	−0.614801	+	0.788682i
$126$		0			0
$127$			15.8106i			1.40297i	0.712686	+	0.701484i	$0.247479\pi$
$127$			15.8106i			1.40297i	−0.712686	+	0.701484i	$0.752521\pi$
$128$		0			0
$129$	8.99956			0.792367
$130$		0			0
$131$	19.1707			1.67495			0.837476	−	0.546475i	$-0.184030\pi$
$131$	19.1707			1.67495			0.837476	+	0.546475i	$0.184030\pi$
$132$		0			0
$133$			11.7977i			1.02299i
$134$		0			0
$135$	−2.18183	−	0.489528i	−0.187782	−	0.0421319i
$136$		0			0
$137$		−	1.67824i		−	0.143381i	−0.997427	−	0.0716907i	$-0.977161\pi$
$137$		−	1.67824i		−	0.143381i	0.997427	−	0.0716907i	$-0.0228395\pi$
$138$		0			0
$139$	−12.3423			−1.04686			−0.523429	−	0.852069i	$-0.675347\pi$
$139$	−12.3423			−1.04686			−0.523429	+	0.852069i	$0.675347\pi$
$140$		0			0
$141$	−4.40554			−0.371013
$142$		0			0
$143$			2.97906i			0.249121i
$144$		0			0
$145$	0.405538	−	1.80748i	0.0336781	−	0.150103i
$146$		0			0
$147$		−	1.00000i		−	0.0824786i
$148$		0			0
$149$	−0.677798			−0.0555274			−0.0277637	−	0.999615i	$-0.508839\pi$
$149$	−0.677798			−0.0555274			−0.0277637	+	0.999615i	$0.508839\pi$
$150$		0			0
$151$	17.9991			1.46475			0.732374	−	0.680903i	$-0.238412\pi$
$151$	17.9991			1.46475			0.732374	+	0.680903i	$0.238412\pi$
$152$		0			0
$153$		−	1.44383i		−	0.116727i
$154$		0			0
$155$	−0.811077	+	3.61497i	−0.0651473	+	0.290361i
$156$		0			0
$157$			12.8392i			1.02468i	0.858783	+	0.512340i	$0.171221\pi$
$157$			12.8392i			1.02468i	−0.858783	+	0.512340i	$0.828779\pi$
$158$		0			0
$159$	−6.72730			−0.533510
$160$		0			0
$161$	−17.5731			−1.38495
$162$		0			0
$163$		−	15.3137i		−	1.19946i	−0.800202	−	0.599731i	$-0.795274\pi$
$163$		−	15.3137i		−	1.19946i	0.800202	−	0.599731i	$-0.204726\pi$
$164$		0			0
$165$	6.49978	+	1.45833i	0.506007	+	0.113531i
$166$		0			0
$167$			15.3922i			1.19109i	0.803324	+	0.595543i	$0.203063\pi$
$167$			15.3922i			1.19109i	−0.803324	+	0.595543i	$0.796937\pi$
$168$		0			0
$169$	−1.00000			−0.0769231
$170$		0			0
$171$	−4.17113			−0.318975
$172$		0			0
$173$		−	14.9706i		−	1.13819i	−0.822272	−	0.569095i	$-0.807294\pi$
$173$		−	14.9706i		−	1.13819i	0.822272	−	0.569095i	$-0.192706\pi$
$174$		0			0
$175$	6.04189	−	12.7865i	0.456724	−	0.966572i
$176$		0			0
$177$		−	2.97906i		−	0.223920i
$178$		0			0
$179$	−7.63950			−0.571003			−0.285502	−	0.958378i	$-0.592160\pi$
$179$	−7.63950			−0.571003			−0.285502	+	0.958378i	$0.592160\pi$
$180$		0			0
$181$	15.0286			1.11706			0.558532	−	0.829483i	$-0.311365\pi$
$181$	15.0286			1.11706			0.558532	+	0.829483i	$0.311365\pi$
$182$		0			0
$183$			6.27226i			0.463659i
$184$		0			0
$185$	−0.0913912	−	0.0205051i	−0.00671922	−	0.00150757i
$186$		0			0
$187$			4.30126i			0.314539i
$188$		0			0
$189$	2.82843			0.205738
$190$		0			0
$191$	10.8877			0.787804			0.393902	−	0.919152i	$-0.371125\pi$
$191$	10.8877			0.787804			0.393902	+	0.919152i	$0.371125\pi$
$192$		0			0
$193$			4.94690i			0.356086i	0.984023	+	0.178043i	$0.0569766\pi$
$193$			4.94690i			0.356086i	−0.984023	+	0.178043i	$0.943023\pi$
$194$		0			0
$195$	−0.489528	+	2.18183i	−0.0350558	+	0.156244i
$196$		0			0
$197$		−	26.3628i		−	1.87827i	−0.343549	−	0.939135i	$-0.611629\pi$
$197$		−	26.3628i		−	1.87827i	0.343549	−	0.939135i	$-0.388371\pi$
$198$		0			0
$199$	6.11190			0.433261			0.216630	−	0.976254i	$-0.430493\pi$
$199$	6.11190			0.433261			0.216630	+	0.976254i	$0.430493\pi$
$200$		0			0
$201$	0.727302			0.0512999
$202$		0			0
$203$			2.34315i			0.164457i
$204$		0			0
$205$	−2.13613	+	9.52072i	−0.149194	+	0.664956i
$206$		0			0
$207$		−	6.21302i		−	0.431835i
$208$		0			0
$209$	12.4260			0.859527
$210$		0			0
$211$	−9.68585			−0.666802			−0.333401	−	0.942785i	$-0.608196\pi$
$211$	−9.68585			−0.666802			−0.333401	+	0.942785i	$0.608196\pi$
$212$		0			0
$213$			8.32176i			0.570198i
$214$		0			0
$215$	−19.6355	−	4.40554i	−1.33913	−	0.300455i
$216$		0			0
$217$		−	4.68629i		−	0.318126i
$218$		0			0
$219$	−6.87031			−0.464253
$220$		0			0
$221$	−1.44383			−0.0971227
$222$		0			0
$223$		−	9.21258i		−	0.616920i	−0.951237	−	0.308460i	$-0.900186\pi$
$223$		−	9.21258i		−	0.616920i	0.951237	−	0.308460i	$-0.0998136\pi$
$224$		0			0
$225$	4.52072	+	2.13613i	0.301382	+	0.142409i
$226$		0			0
$227$			20.3218i			1.34880i	0.738365	+	0.674401i	$0.235598\pi$
$227$			20.3218i			1.34880i	−0.738365	+	0.674401i	$0.764402\pi$
$228$		0			0
$229$	16.4715			1.08847			0.544234	−	0.838933i	$-0.316820\pi$
$229$	16.4715			1.08847			0.544234	+	0.838933i	$0.316820\pi$
$230$		0			0
$231$	−8.42604			−0.554393
$232$		0			0
$233$			2.37339i			0.155486i	0.996973	+	0.0777428i	$0.0247713\pi$
$233$			2.37339i			0.155486i	−0.996973	+	0.0777428i	$0.975229\pi$
$234$		0			0
$235$	9.61212	+	2.15663i	0.627025	+	0.140683i
$236$		0			0
$237$			6.11190i			0.397010i
$238$		0			0
$239$	−13.6363			−0.882062			−0.441031	−	0.897492i	$-0.645387\pi$
$239$	−13.6363			−0.882062			−0.441031	+	0.897492i	$0.645387\pi$
$240$		0			0
$241$	1.07045			0.0689536			0.0344768	−	0.999405i	$-0.489024\pi$
$241$	1.07045			0.0689536			0.0344768	+	0.999405i	$0.489024\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	−0.489528	+	2.18183i	−0.0312748	+	0.139392i
$246$		0			0
$247$			4.17113i			0.265403i
$248$		0			0
$249$	−14.7059			−0.931950
$250$		0			0
$251$	−14.6262			−0.923196			−0.461598	−	0.887089i	$-0.652724\pi$
$251$	−14.6262			−0.923196			−0.461598	+	0.887089i	$0.652724\pi$
$252$		0			0
$253$			18.5089i			1.16365i
$254$		0			0
$255$	−0.706797	+	3.15019i	−0.0442613	+	0.197273i
$256$		0			0
$257$		−	24.9403i		−	1.55573i	−0.628429	−	0.777867i	$-0.716302\pi$
$257$		−	24.9403i		−	1.55573i	0.628429	−	0.777867i	$-0.283698\pi$
$258$		0			0
$259$	0.118476			0.00736173
$260$		0			0
$261$	−0.828427			−0.0512784
$262$		0			0
$263$			9.01691i			0.556007i	0.960580	+	0.278003i	$0.0896726\pi$
$263$			9.01691i			0.556007i	−0.960580	+	0.278003i	$0.910327\pi$
$264$		0			0
$265$	14.6778	+	3.29320i	0.901650	+	0.202300i
$266$		0			0
$267$		−	14.7478i		−	0.902551i
$268$		0			0
$269$	−8.28391			−0.505079			−0.252539	−	0.967587i	$-0.581266\pi$
$269$	−8.28391			−0.505079			−0.252539	+	0.967587i	$0.581266\pi$
$270$		0			0
$271$	19.9572			1.21232			0.606158	−	0.795344i	$-0.292710\pi$
$271$	19.9572			1.21232			0.606158	+	0.795344i	$0.292710\pi$
$272$		0			0
$273$		−	2.82843i		−	0.171184i
$274$		0			0
$275$	−13.4675	−	6.36365i	−0.812120	−	0.383743i
$276$		0			0
$277$			22.6145i			1.35878i	0.733780	+	0.679388i	$0.237755\pi$
$277$			22.6145i			1.35878i	−0.733780	+	0.679388i	$0.762245\pi$
$278$		0			0
$279$	1.65685			0.0991933
$280$		0			0
$281$	−1.89572			−0.113089			−0.0565446	−	0.998400i	$-0.518008\pi$
$281$	−1.89572			−0.113089			−0.0565446	+	0.998400i	$0.518008\pi$
$282$		0			0
$283$			11.2299i			0.667550i	0.942653	+	0.333775i	$0.108323\pi$
$283$			11.2299i			0.667550i	−0.942653	+	0.333775i	$0.891677\pi$
$284$		0			0
$285$	9.10069	+	2.04189i	0.539078	+	0.120951i
$286$		0			0
$287$		−	12.3423i		−	0.728541i
$288$		0			0
$289$	14.9153			0.877373
$290$		0			0
$291$	6.78654			0.397834
$292$		0			0
$293$			27.6774i			1.61693i	0.588545	+	0.808464i	$0.299701\pi$
$293$			27.6774i			1.61693i	−0.588545	+	0.808464i	$0.700299\pi$
$294$		0			0
$295$	−1.45833	+	6.49978i	−0.0849074	+	0.378432i
$296$		0			0
$297$		−	2.97906i		−	0.172862i
$298$		0			0
$299$	−6.21302			−0.359308
$300$		0			0
$301$	25.4546			1.46718
$302$		0			0
$303$			12.1421i			0.697547i
$304$		0			0
$305$	3.07045	−	13.6850i	0.175813	−	0.783599i
$306$		0			0
$307$			24.9858i			1.42601i	0.701157	+	0.713007i	$0.252667\pi$
$307$			24.9858i			1.42601i	−0.701157	+	0.713007i	$0.747333\pi$
$308$		0			0
$309$	−2.81108			−0.159917
$310$		0			0
$311$	−18.3842			−1.04247			−0.521235	−	0.853413i	$-0.674528\pi$
$311$	−18.3842			−1.04247			−0.521235	+	0.853413i	$0.674528\pi$
$312$		0			0
$313$		−	8.61453i		−	0.486922i	−0.969911	−	0.243461i	$-0.921717\pi$
$313$		−	8.61453i		−	0.486922i	0.969911	−	0.243461i	$-0.0782828\pi$
$314$		0			0
$315$	−6.17113	−	1.38459i	−0.347704	−	0.0780131i
$316$		0			0
$317$		−	15.0491i		−	0.845240i	−0.906307	−	0.422620i	$-0.861111\pi$
$317$		−	15.0491i		−	0.845240i	0.906307	−	0.422620i	$-0.138889\pi$
$318$		0			0
$319$	2.46793			0.138178
$320$		0			0
$321$	−18.3842			−1.02610
$322$		0			0
$323$			6.02242i			0.335096i
$324$		0			0
$325$	2.13613	−	4.52072i	0.118491	−	0.250765i
$326$		0			0
$327$		−	1.44383i		−	0.0798441i
$328$		0			0
$329$	−12.4607			−0.686983
$330$		0			0
$331$	−22.6738			−1.24626			−0.623131	−	0.782117i	$-0.714140\pi$
$331$	−22.6738			−1.24626			−0.623131	+	0.782117i	$0.714140\pi$
$332$		0			0
$333$			0.0418875i			0.00229542i
$334$		0			0
$335$	−1.58685	−	0.356035i	−0.0866986	−	0.0194523i
$336$		0			0
$337$		−	27.9282i		−	1.52135i	−0.649134	−	0.760674i	$-0.724869\pi$
$337$		−	27.9282i		−	1.52135i	0.649134	−	0.760674i	$-0.275131\pi$
$338$		0			0
$339$	−3.40194			−0.184768
$340$		0			0
$341$	−4.93586			−0.267292
$342$		0			0
$343$			16.9706i			0.916324i
$344$		0			0
$345$	−3.04145	+	13.5557i	−0.163746	+	0.729816i
$346$		0			0
$347$			19.7398i			1.05969i	0.848096	+	0.529843i	$0.177749\pi$
$347$			19.7398i			1.05969i	−0.848096	+	0.529843i	$0.822251\pi$
$348$		0			0
$349$	−30.7157			−1.64417			−0.822086	−	0.569364i	$-0.807190\pi$
$349$	−30.7157			−1.64417			−0.822086	+	0.569364i	$0.807190\pi$
$350$		0			0
$351$	1.00000			0.0533761
$352$		0			0
$353$		−	17.0072i		−	0.905201i	−0.891713	−	0.452600i	$-0.850496\pi$
$353$		−	17.0072i		−	0.905201i	0.891713	−	0.452600i	$-0.149504\pi$
$354$		0			0
$355$	4.07374	−	18.1566i	0.216212	−	0.963654i
$356$		0			0
$357$		−	4.08378i		−	0.216136i
$358$		0			0
$359$	−7.90203			−0.417053			−0.208527	−	0.978017i	$-0.566867\pi$
$359$	−7.90203			−0.417053			−0.208527	+	0.978017i	$0.566867\pi$
$360$		0			0
$361$	−1.60164			−0.0842968
$362$		0			0
$363$		−	2.12522i		−	0.111545i
$364$		0			0
$365$	14.9898	+	3.36321i	0.784603	+	0.176039i
$366$		0			0
$367$		−	34.1810i		−	1.78424i	−0.451803	−	0.892118i	$-0.649219\pi$
$367$		−	34.1810i		−	1.78424i	0.451803	−	0.892118i	$-0.350781\pi$
$368$		0			0
$369$	4.36365			0.227163
$370$		0			0
$371$	−19.0277			−0.987868
$372$		0			0
$373$			18.5098i			0.958402i	0.877705	+	0.479201i	$0.159074\pi$
$373$			18.5098i			0.958402i	−0.877705	+	0.479201i	$0.840926\pi$
$374$		0			0
$375$	−8.81774	−	6.87368i	−0.455346	−	0.354956i
$376$		0			0
$377$			0.828427i			0.0426662i
$378$		0			0
$379$	−12.7994			−0.657462			−0.328731	−	0.944424i	$-0.606621\pi$
$379$	−12.7994			−0.657462			−0.328731	+	0.944424i	$0.606621\pi$
$380$		0			0
$381$	−15.8106			−0.810004
$382$		0			0
$383$			0.664032i			0.0339304i	0.999856	+	0.0169652i	$0.00540046\pi$
$383$			0.664032i			0.0339304i	−0.999856	+	0.0169652i	$0.994600\pi$
$384$		0			0
$385$	18.3842	+	4.12479i	0.936943	+	0.210219i
$386$		0			0
$387$			8.99956i			0.457473i
$388$		0			0
$389$	−2.32580			−0.117923			−0.0589613	−	0.998260i	$-0.518779\pi$
$389$	−2.32580			−0.117923			−0.0589613	+	0.998260i	$0.518779\pi$
$390$		0			0
$391$	−8.97056			−0.453661
$392$		0			0
$393$			19.1707i			0.967034i
$394$		0			0
$395$	2.99195	−	13.3351i	0.150541	−	0.670961i
$396$		0			0
$397$		−	18.2433i		−	0.915603i	−0.889054	−	0.457802i	$-0.848637\pi$
$397$		−	18.2433i		−	0.915603i	0.889054	−	0.457802i	$-0.151363\pi$
$398$		0			0
$399$	−11.7977			−0.590626
$400$		0			0
$401$	16.9919			0.848537			0.424269	−	0.905536i	$-0.360531\pi$
$401$	16.9919			0.848537			0.424269	+	0.905536i	$0.360531\pi$
$402$		0			0
$403$		−	1.65685i		−	0.0825338i
$404$		0			0
$405$	0.489528	−	2.18183i	0.0243248	−	0.108416i
$406$		0			0
$407$		−	0.124785i		−	0.00618538i
$408$		0			0
$409$	10.8111			0.534573			0.267287	−	0.963617i	$-0.413873\pi$
$409$	10.8111			0.534573			0.267287	+	0.963617i	$0.413873\pi$
$410$		0			0
$411$	1.67824			0.0827813
$412$		0			0
$413$		−	8.42604i		−	0.414618i
$414$		0			0
$415$	32.0857	+	7.19896i	1.57503	+	0.353383i
$416$		0			0
$417$		−	12.3423i		−	0.604403i
$418$		0			0
$419$	−5.08780			−0.248555			−0.124278	−	0.992247i	$-0.539661\pi$
$419$	−5.08780			−0.248555			−0.124278	+	0.992247i	$0.539661\pi$
$420$		0			0
$421$	12.0945			0.589452			0.294726	−	0.955582i	$-0.404772\pi$
$421$	12.0945			0.589452			0.294726	+	0.955582i	$0.404772\pi$
$422$		0			0
$423$		−	4.40554i		−	0.214205i
$424$		0			0
$425$	3.08421	−	6.52717i	0.149606	−	0.316614i
$426$		0			0
$427$			17.7406i			0.858529i
$428$		0			0
$429$	−2.97906			−0.143830
$430$		0			0
$431$	8.48931			0.408916			0.204458	−	0.978875i	$-0.434457\pi$
$431$	8.48931			0.408916			0.204458	+	0.978875i	$0.434457\pi$
$432$		0			0
$433$		−	13.8453i		−	0.665365i	−0.943039	−	0.332682i	$-0.892046\pi$
$433$		−	13.8453i		−	0.665365i	0.943039	−	0.332682i	$-0.107954\pi$
$434$		0			0
$435$	1.80748	+	0.405538i	0.0866622	+	0.0194441i
$436$		0			0
$437$			25.9153i			1.23970i
$438$		0			0
$439$	0.602516			0.0287565			0.0143783	−	0.999897i	$-0.495423\pi$
$439$	0.602516			0.0287565			0.0143783	+	0.999897i	$0.495423\pi$
$440$		0			0
$441$	1.00000			0.0476190
$442$		0			0
$443$		−	17.6721i		−	0.839626i	−0.907611	−	0.419813i	$-0.862096\pi$
$443$		−	17.6721i		−	0.839626i	0.907611	−	0.419813i	$-0.137904\pi$
$444$		0			0
$445$	−7.21947	+	32.1771i	−0.342236	+	1.52534i
$446$		0			0
$447$		−	0.677798i		−	0.0320587i
$448$		0			0
$449$	−3.33509			−0.157393			−0.0786963	−	0.996899i	$-0.525076\pi$
$449$	−3.33509			−0.157393			−0.0786963	+	0.996899i	$0.525076\pi$
$450$		0			0
$451$	−12.9996			−0.612125
$452$		0			0
$453$			17.9991i			0.845673i
$454$		0			0
$455$	−1.38459	+	6.17113i	−0.0649108	+	0.289307i
$456$		0			0
$457$		−	38.9266i		−	1.82091i	−0.413611	−	0.910454i	$-0.635733\pi$
$457$		−	38.9266i		−	1.82091i	0.413611	−	0.910454i	$-0.364267\pi$
$458$		0			0
$459$	1.44383			0.0673923
$460$		0			0
$461$	28.3918			1.32234			0.661168	−	0.750238i	$-0.270061\pi$
$461$	28.3918			1.32234			0.661168	+	0.750238i	$0.270061\pi$
$462$		0			0
$463$			10.2259i			0.475238i	0.971358	+	0.237619i	$0.0763670\pi$
$463$			10.2259i			0.475238i	−0.971358	+	0.237619i	$0.923633\pi$
$464$		0			0
$465$	−3.61497	−	0.811077i	−0.167640	−	0.0376128i
$466$		0			0
$467$			5.61584i			0.259870i	0.991522	+	0.129935i	$0.0414769\pi$
$467$			5.61584i			0.259870i	−0.991522	+	0.129935i	$0.958523\pi$
$468$		0			0
$469$	2.05712			0.0949890
$470$		0			0
$471$	−12.8392			−0.591599
$472$		0			0
$473$		−	26.8102i		−	1.23273i
$474$		0			0
$475$	−18.8565	−	8.91008i	−0.865198	−	0.408823i
$476$		0			0
$477$		−	6.72730i		−	0.308022i
$478$		0			0
$479$	−8.95006			−0.408939			−0.204469	−	0.978873i	$-0.565547\pi$
$479$	−8.95006			−0.408939			−0.204469	+	0.978873i	$0.565547\pi$
$480$		0			0
$481$	0.0418875			0.00190991
$482$		0			0
$483$		−	17.5731i		−	0.799603i
$484$		0			0
$485$	−14.8070	−	3.32220i	−0.672353	−	0.150853i
$486$		0			0
$487$		−	18.3596i		−	0.831954i	−0.909375	−	0.415977i	$-0.863440\pi$
$487$		−	18.3596i		−	0.831954i	0.909375	−	0.415977i	$-0.136560\pi$
$488$		0			0
$489$	15.3137			0.692510
$490$		0			0
$491$	33.7981			1.52529			0.762644	−	0.646819i	$-0.223901\pi$
$491$	33.7981			1.52529			0.762644	+	0.646819i	$0.223901\pi$
$492$		0			0
$493$			1.19611i			0.0538701i
$494$		0			0
$495$	−1.45833	+	6.49978i	−0.0655471	+	0.292143i
$496$		0			0
$497$			23.5375i			1.05580i
$498$		0			0
$499$	5.72898			0.256464			0.128232	−	0.991744i	$-0.459070\pi$
$499$	5.72898			0.256464			0.128232	+	0.991744i	$0.459070\pi$
$500$		0			0
$501$	−15.3922			−0.687673
$502$		0			0
$503$		−	20.2121i		−	0.901215i	−0.892722	−	0.450607i	$-0.851207\pi$
$503$		−	20.2121i		−	0.901215i	0.892722	−	0.450607i	$-0.148793\pi$
$504$		0			0
$505$	5.94392	−	26.4920i	0.264501	−	1.17888i
$506$		0			0
$507$		−	1.00000i		−	0.0444116i
$508$		0			0
$509$	−6.02708			−0.267146			−0.133573	−	0.991039i	$-0.542645\pi$
$509$	−6.02708			−0.267146			−0.133573	+	0.991039i	$0.542645\pi$
$510$		0			0
$511$	−19.4322			−0.859629
$512$		0			0
$513$		−	4.17113i		−	0.184160i
$514$		0			0
$515$	6.13328	+	1.37610i	0.270265	+	0.0606383i
$516$		0			0
$517$			13.1243i			0.577208i
$518$		0			0
$519$	14.9706			0.657135
$520$		0			0
$521$	−1.95899			−0.0858249			−0.0429125	−	0.999079i	$-0.513664\pi$
$521$	−1.95899			−0.0858249			−0.0429125	+	0.999079i	$0.513664\pi$
$522$		0			0
$523$		−	21.6859i		−	0.948256i	−0.880456	−	0.474128i	$-0.842763\pi$
$523$		−	21.6859i		−	0.948256i	0.880456	−	0.474128i	$-0.157237\pi$
$524$		0			0
$525$	12.7865	+	6.04189i	0.558050	+	0.263690i
$526$		0			0
$527$		−	2.39222i		−	0.104207i
$528$		0			0
$529$	−15.6016			−0.678332
$530$		0			0
$531$	2.97906			0.129280
$532$		0			0
$533$		−	4.36365i		−	0.189011i
$534$		0			0
$535$	40.1110	+	8.99956i	1.73415	+	0.389085i
$536$		0			0
$537$		−	7.63950i		−	0.329669i
$538$		0			0
$539$	−2.97906			−0.128317
$540$		0			0
$541$	39.3245			1.69069			0.845346	−	0.534220i	$-0.179394\pi$
$541$	39.3245			1.69069			0.845346	+	0.534220i	$0.179394\pi$
$542$		0			0
$543$			15.0286i			0.644937i
$544$		0			0
$545$	−0.706797	+	3.15019i	−0.0302758	+	0.134939i
$546$		0			0
$547$		−	14.1466i		−	0.604865i	−0.953171	−	0.302432i	$-0.902201\pi$
$547$		−	14.1466i		−	0.604865i	0.953171	−	0.302432i	$-0.0977987\pi$
$548$		0			0
$549$	−6.27226			−0.267694
$550$		0			0
$551$	3.45548			0.147208
$552$		0			0
$553$			17.2871i			0.735120i
$554$		0			0
$555$	0.0205051	−	0.0913912i	0.000870394	−	0.00387934i
$556$		0			0
$557$		−	29.7603i		−	1.26098i	−0.776196	−	0.630491i	$-0.782853\pi$
$557$		−	29.7603i		−	1.26098i	0.776196	−	0.630491i	$-0.217147\pi$
$558$		0			0
$559$	8.99956			0.380641
$560$		0			0
$561$	−4.30126			−0.181599
$562$		0			0
$563$		−	28.6498i		−	1.20745i	−0.797194	−	0.603723i	$-0.793683\pi$
$563$		−	28.6498i		−	1.20745i	0.797194	−	0.603723i	$-0.206317\pi$
$564$		0			0
$565$	7.42245	+	1.66535i	0.312265	+	0.0700617i
$566$		0			0
$567$			2.82843i			0.118783i
$568$		0			0
$569$	−39.7326			−1.66568			−0.832838	−	0.553517i	$-0.813285\pi$
$569$	−39.7326			−1.66568			−0.832838	+	0.553517i	$0.813285\pi$
$570$		0			0
$571$	−30.5726			−1.27943			−0.639713	−	0.768614i	$-0.720947\pi$
$571$	−30.5726			−1.27943			−0.639713	+	0.768614i	$0.720947\pi$
$572$		0			0
$573$			10.8877i			0.454839i
$574$		0			0
$575$	13.2718	−	28.0874i	0.553473	−	1.17132i
$576$		0			0
$577$		−	35.3116i		−	1.47004i	−0.678045	−	0.735020i	$-0.737173\pi$
$577$		−	35.3116i		−	1.47004i	0.678045	−	0.735020i	$-0.262827\pi$
$578$		0			0
$579$	−4.94690			−0.205586
$580$		0			0
$581$	−41.5946			−1.72564
$582$		0			0
$583$			20.0410i			0.830014i
$584$		0			0
$585$	−2.18183	−	0.489528i	−0.0902074	−	0.0202395i
$586$		0			0
$587$		−	19.5517i		−	0.806985i	−0.914983	−	0.403492i	$-0.867796\pi$
$587$		−	19.5517i		−	0.806985i	0.914983	−	0.403492i	$-0.132204\pi$
$588$		0			0
$589$	−6.91096			−0.284761
$590$		0			0
$591$	26.3628			1.08442
$592$		0			0
$593$		−	4.66491i		−	0.191565i	−0.995402	−	0.0957824i	$-0.969465\pi$
$593$		−	4.66491i		−	0.191565i	0.995402	−	0.0957824i	$-0.0305353\pi$
$594$		0			0
$595$	−1.99912	+	8.91008i	−0.0819560	+	0.365278i
$596$		0			0
$597$			6.11190i			0.250143i
$598$		0			0
$599$	−19.6497			−0.802864			−0.401432	−	0.915889i	$-0.631487\pi$
$599$	−19.6497			−0.802864			−0.401432	+	0.915889i	$0.631487\pi$
$600$		0			0
$601$	44.2714			1.80587			0.902934	−	0.429780i	$-0.141409\pi$
$601$	44.2714			1.80587			0.902934	+	0.429780i	$0.141409\pi$
$602$		0			0
$603$			0.727302i			0.0296180i
$604$		0			0
$605$	−1.04036	+	4.63687i	−0.0422965	+	0.188515i
$606$		0			0
$607$		−	34.2648i		−	1.39077i	−0.718640	−	0.695383i	$-0.755235\pi$
$607$		−	34.2648i		−	1.39077i	0.718640	−	0.695383i	$-0.244765\pi$
$608$		0			0
$609$	−2.34315			−0.0949491
$610$		0			0
$611$	−4.40554			−0.178229
$612$		0			0
$613$		−	33.7203i		−	1.36195i	−0.732307	−	0.680975i	$-0.761556\pi$
$613$		−	33.7203i		−	1.36195i	0.732307	−	0.680975i	$-0.238444\pi$
$614$		0			0
$615$	−9.52072	−	2.13613i	−0.383913	−	0.0861371i
$616$		0			0
$617$		−	35.0901i		−	1.41267i	−0.707876	−	0.706337i	$-0.750347\pi$
$617$		−	35.0901i		−	1.41267i	0.707876	−	0.706337i	$-0.249653\pi$
$618$		0			0
$619$	2.63907			0.106073			0.0530365	−	0.998593i	$-0.483110\pi$
$619$	2.63907			0.106073			0.0530365	+	0.998593i	$0.483110\pi$
$620$		0			0
$621$	6.21302			0.249320
$622$		0			0
$623$		−	41.7131i		−	1.67120i
$624$		0			0
$625$	15.8739	+	19.3137i	0.634956	+	0.772548i
$626$		0			0
$627$			12.4260i			0.496248i
$628$		0			0
$629$	0.0604785			0.00241144
$630$		0			0
$631$	28.7683			1.14525			0.572624	−	0.819818i	$-0.305925\pi$
$631$	28.7683			1.14525			0.572624	+	0.819818i	$0.305925\pi$
$632$		0			0
$633$		−	9.68585i		−	0.384978i
$634$		0			0
$635$	34.4961	+	7.73975i	1.36893	+	0.307143i
$636$		0			0
$637$		−	1.00000i		−	0.0396214i
$638$		0			0
$639$	−8.32176			−0.329204
$640$		0			0
$641$	35.1944			1.39009			0.695047	−	0.718965i	$-0.255384\pi$
$641$	35.1944			1.39009			0.695047	+	0.718965i	$0.255384\pi$
$642$		0			0
$643$		−	27.2709i		−	1.07546i	−0.843117	−	0.537731i	$-0.819282\pi$
$643$		−	27.2709i		−	1.07546i	0.843117	−	0.537731i	$-0.180718\pi$
$644$		0			0
$645$	4.40554	−	19.6355i	0.173468	−	0.773146i
$646$		0			0
$647$			4.22913i			0.166264i	0.996539	+	0.0831322i	$0.0264924\pi$
$647$			4.22913i			0.166264i	−0.996539	+	0.0831322i	$0.973508\pi$
$648$		0			0
$649$	−8.87478			−0.348365
$650$		0			0
$651$	4.68629			0.183670
$652$		0			0
$653$			22.1400i			0.866406i	0.901296	+	0.433203i	$0.142617\pi$
$653$			22.1400i			0.866406i	−0.901296	+	0.433203i	$0.857383\pi$
$654$		0			0
$655$	9.38459	−	41.8271i	0.366686	−	1.63432i
$656$		0			0
$657$		−	6.87031i		−	0.268037i
$658$		0			0
$659$	27.0932			1.05540			0.527701	−	0.849430i	$-0.323054\pi$
$659$	27.0932			1.05540			0.527701	+	0.849430i	$0.323054\pi$
$660$		0			0
$661$	19.0597			0.741335			0.370668	−	0.928766i	$-0.379129\pi$
$661$	19.0597			0.741335			0.370668	+	0.928766i	$0.379129\pi$
$662$		0			0
$663$		−	1.44383i		−	0.0560738i
$664$		0			0
$665$	25.7406	+	5.77533i	0.998179	+	0.223958i
$666$		0			0
$667$			5.14704i			0.199294i
$668$		0			0
$669$	9.21258			0.356179
$670$		0			0
$671$	18.6854			0.721342
$672$		0			0
$673$		−	23.6213i		−	0.910533i	−0.890355	−	0.455267i	$-0.849544\pi$
$673$		−	23.6213i		−	0.910533i	0.890355	−	0.455267i	$-0.150456\pi$
$674$		0			0
$675$	−2.13613	+	4.52072i	−0.0822197	+	0.174003i
$676$		0			0
$677$			18.8111i			0.722968i	0.932378	+	0.361484i	$0.117730\pi$
$677$			18.8111i			0.722968i	−0.932378	+	0.361484i	$0.882270\pi$
$678$		0			0
$679$	19.1952			0.736645
$680$		0			0
$681$	−20.3218			−0.778732
$682$		0			0
$683$		−	33.7827i		−	1.29266i	−0.763059	−	0.646329i	$-0.776303\pi$
$683$		−	33.7827i		−	1.29266i	0.763059	−	0.646329i	$-0.223697\pi$
$684$		0			0
$685$	−3.66162	−	0.821544i	−0.139903	−	0.0313896i
$686$		0			0
$687$			16.4715i			0.628428i
$688$		0			0
$689$	−6.72730			−0.256290
$690$		0			0
$691$	47.3236			1.80027			0.900137	−	0.435606i	$-0.143466\pi$
$691$	47.3236			1.80027			0.900137	+	0.435606i	$0.143466\pi$
$692$		0			0
$693$		−	8.42604i		−	0.320079i
$694$		0			0
$695$	−6.04189	+	26.9287i	−0.229182	+	1.02146i
$696$		0			0
$697$		−	6.30038i		−	0.238644i
$698$		0			0
$699$	−2.37339			−0.0897697
$700$		0			0
$701$	−3.29636			−0.124502			−0.0622509	−	0.998061i	$-0.519828\pi$
$701$	−3.29636			−0.124502			−0.0622509	+	0.998061i	$0.519828\pi$
$702$		0			0
$703$		−	0.174718i		−	0.00658963i
$704$		0			0
$705$	−2.15663	+	9.61212i	−0.0812236	+	0.362013i
$706$		0			0
$707$			34.3431i			1.29161i
$708$		0			0
$709$	−46.8404			−1.75913			−0.879565	−	0.475779i	$-0.842166\pi$
$709$	−46.8404			−1.75913			−0.879565	+	0.475779i	$0.842166\pi$
$710$		0			0
$711$	−6.11190			−0.229214
$712$		0			0
$713$		−	10.2941i		−	0.385516i
$714$		0			0
$715$	6.49978	+	1.45833i	0.243078	+	0.0545385i
$716$		0			0
$717$		−	13.6363i		−	0.509259i
$718$		0			0
$719$	−20.0152			−0.746442			−0.373221	−	0.927742i	$-0.621747\pi$
$719$	−20.0152			−0.746442			−0.373221	+	0.927742i	$0.621747\pi$
$720$		0			0
$721$	−7.95093			−0.296108
$722$		0			0
$723$			1.07045i			0.0398104i
$724$		0			0
$725$	−3.74509	−	1.76963i	−0.139089	−	0.0657223i
$726$		0			0
$727$			10.8383i			0.401971i	0.979594	+	0.200986i	$0.0644144\pi$
$727$			10.8383i			0.401971i	−0.979594	+	0.200986i	$0.935586\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	12.9939			0.480595
$732$		0			0
$733$		−	21.1952i		−	0.782864i	−0.920207	−	0.391432i	$-0.871980\pi$
$733$		−	21.1952i		−	0.782864i	0.920207	−	0.391432i	$-0.128020\pi$
$734$		0			0
$735$	−2.18183	−	0.489528i	−0.0804778	−	0.0180565i
$736$		0			0
$737$		−	2.16667i		−	0.0798104i
$738$		0			0
$739$	7.07403			0.260222			0.130111	−	0.991499i	$-0.458467\pi$
$739$	7.07403			0.260222			0.130111	+	0.991499i	$0.458467\pi$
$740$		0			0
$741$	−4.17113			−0.153230
$742$		0			0
$743$			21.3913i			0.784772i	0.919801	+	0.392386i	$0.128350\pi$
$743$			21.3913i			0.784772i	−0.919801	+	0.392386i	$0.871650\pi$
$744$		0			0
$745$	−0.331801	+	1.47884i	−0.0121563	+	0.0541804i
$746$		0			0
$747$		−	14.7059i		−	0.538061i
$748$		0			0
$749$	−51.9982			−1.89997
$750$		0			0
$751$	−18.2247			−0.665028			−0.332514	−	0.943098i	$-0.607897\pi$
$751$	−18.2247			−0.665028			−0.332514	+	0.943098i	$0.607897\pi$
$752$		0			0
$753$		−	14.6262i		−	0.533007i
$754$		0			0
$755$	8.81108	−	39.2709i	0.320668	−	1.42922i
$756$		0			0
$757$		−	39.3708i		−	1.43096i	−0.698635	−	0.715479i	$-0.746209\pi$
$757$		−	39.3708i		−	1.43096i	0.698635	−	0.715479i	$-0.253791\pi$
$758$		0			0
$759$	−18.5089			−0.671832
$760$		0			0
$761$	−13.8610			−0.502462			−0.251231	−	0.967927i	$-0.580835\pi$
$761$	−13.8610			−0.502462			−0.251231	+	0.967927i	$0.580835\pi$
$762$		0			0
$763$		−	4.08378i		−	0.147843i
$764$		0			0
$765$	−3.15019	−	0.706797i	−0.113895	−	0.0255543i
$766$		0			0
$767$		−	2.97906i		−	0.107567i
$768$		0			0
$769$	12.6274			0.455356			0.227678	−	0.973736i	$-0.426887\pi$
$769$	12.6274			0.455356			0.227678	+	0.973736i	$0.426887\pi$
$770$		0			0
$771$	24.9403			0.898204
$772$		0			0
$773$			34.7825i			1.25104i	0.780208	+	0.625520i	$0.215113\pi$
$773$			34.7825i			1.25104i	−0.780208	+	0.625520i	$0.784887\pi$
$774$		0			0
$775$	7.49018	+	3.53926i	0.269055	+	0.127134i
$776$		0			0
$777$			0.118476i			0.00425029i
$778$		0			0
$779$	−18.2014			−0.652132
$780$		0			0
$781$	24.7910			0.887092
$782$		0			0
$783$		−	0.828427i		−	0.0296056i
$784$		0			0
$785$	28.0129	+	6.28515i	0.999823	+	0.224327i
$786$		0			0
$787$		−	36.7835i		−	1.31119i	−0.755112	−	0.655596i	$-0.772418\pi$
$787$		−	36.7835i		−	1.31119i	0.755112	−	0.655596i	$-0.227582\pi$
$788$		0			0
$789$	−9.01691			−0.321011
$790$		0			0
$791$	−9.62215			−0.342124
$792$		0			0
$793$			6.27226i			0.222734i
$794$		0			0
$795$	−3.29320	+	14.6778i	−0.116798	+	0.520568i
$796$		0			0
$797$			42.4413i			1.50335i	0.659535	+	0.751674i	$0.270753\pi$
$797$			42.4413i			1.50335i	−0.659535	+	0.751674i	$0.729247\pi$
$798$		0			0
$799$	−6.36086			−0.225031
$800$		0			0
$801$	14.7478			0.521088
$802$		0			0
$803$			20.4671i			0.722267i
$804$		0			0
$805$	−8.60252	+	38.3414i	−0.303199	+	1.35136i
$806$		0			0
$807$		−	8.28391i		−	0.291607i
$808$		0			0
$809$	−55.3771			−1.94696			−0.973478	−	0.228780i	$-0.926526\pi$
$809$	−55.3771			−1.94696			−0.973478	+	0.228780i	$0.926526\pi$
$810$		0			0
$811$	17.1159			0.601021			0.300511	−	0.953778i	$-0.402843\pi$
$811$	17.1159			0.601021			0.300511	+	0.953778i	$0.402843\pi$
$812$		0			0
$813$			19.9572i			0.699931i
$814$		0			0
$815$	−33.4118	−	7.49649i	−1.17037	−	0.262591i
$816$		0			0
$817$		−	37.5384i		−	1.31330i
$818$		0			0
$819$	2.82843			0.0988332
$820$		0			0
$821$	15.5460			0.542559			0.271279	−	0.962501i	$-0.412553\pi$
$821$	15.5460			0.542559			0.271279	+	0.962501i	$0.412553\pi$
$822$		0			0
$823$		−	0.0998847i		−	0.00348176i	−0.999998	−	0.00174088i	$-0.999446\pi$
$823$		−	0.0998847i		−	0.00348176i	0.999998	−	0.00174088i	$-0.000554140\pi$
$824$		0			0
$825$	6.36365	−	13.4675i	0.221554	−	0.468878i
$826$		0			0
$827$		−	9.53311i		−	0.331499i	−0.986168	−	0.165749i	$-0.946996\pi$
$827$		−	9.53311i		−	0.331499i	0.986168	−	0.165749i	$-0.0530042\pi$
$828$		0			0
$829$	−50.6361			−1.75866			−0.879332	−	0.476210i	$-0.842010\pi$
$829$	−50.6361			−1.75866			−0.879332	+	0.476210i	$0.842010\pi$
$830$		0			0
$831$	−22.6145			−0.784489
$832$		0			0
$833$		−	1.44383i		−	0.0500258i
$834$		0			0
$835$	33.5831	+	7.53492i	1.16219	+	0.260757i
$836$		0			0
$837$			1.65685i			0.0572693i
$838$		0			0
$839$	−43.2310			−1.49250			−0.746249	−	0.665666i	$-0.768147\pi$
$839$	−43.2310			−1.49250			−0.746249	+	0.665666i	$0.768147\pi$
$840$		0			0
$841$	−28.3137			−0.976335
$842$		0			0
$843$		−	1.89572i		−	0.0652921i
$844$		0			0
$845$	−0.489528	+	2.18183i	−0.0168403	+	0.0750571i
$846$		0			0
$847$		−	6.01104i		−	0.206542i
$848$		0			0
$849$	−11.2299			−0.385410
$850$		0			0
$851$	0.260248			0.00892119
$852$		0			0
$853$		−	31.6979i		−	1.08531i	−0.839954	−	0.542657i	$-0.817418\pi$
$853$		−	31.6979i		−	1.08531i	0.839954	−	0.542657i	$-0.182582\pi$
$854$		0			0
$855$	−2.04189	+	9.10069i	−0.0698311	+	0.311237i
$856$		0			0
$857$			28.8912i			0.986906i	0.869772	+	0.493453i	$0.164266\pi$
$857$			28.8912i			0.986906i	−0.869772	+	0.493453i	$0.835734\pi$
$858$		0			0
$859$	49.7099			1.69608			0.848040	−	0.529933i	$-0.177783\pi$
$859$	49.7099			1.69608			0.848040	+	0.529933i	$0.177783\pi$
$860$		0			0
$861$	12.3423			0.420623
$862$		0			0
$863$			4.56502i			0.155395i	0.996977	+	0.0776976i	$0.0247569\pi$
$863$			4.56502i			0.155395i	−0.996977	+	0.0776976i	$0.975243\pi$
$864$		0			0
$865$	−32.6632	−	7.32851i	−1.11058	−	0.249177i
$866$		0			0
$867$			14.9153i			0.506552i
$868$		0			0
$869$	18.2077			0.617653
$870$		0			0
$871$	0.727302			0.0246437
$872$		0			0
$873$			6.78654i			0.229690i
$874$		0			0
$875$	−24.9403	−	19.4417i	−0.843137	−	0.657250i
$876$		0			0
$877$		−	13.5312i		−	0.456916i	−0.973554	−	0.228458i	$-0.926632\pi$
$877$		−	13.5312i		−	0.456916i	0.973554	−	0.228458i	$-0.0733683\pi$
$878$		0			0
$879$	−27.6774			−0.933534
$880$		0			0
$881$	−28.0634			−0.945481			−0.472740	−	0.881202i	$-0.656735\pi$
$881$	−28.0634			−0.945481			−0.472740	+	0.881202i	$0.656735\pi$
$882$		0			0
$883$			47.3700i			1.59413i	0.603896	+	0.797063i	$0.293614\pi$
$883$			47.3700i			1.59413i	−0.603896	+	0.797063i	$0.706386\pi$
$884$		0			0
$885$	−6.49978	−	1.45833i	−0.218488	−	0.0490213i
$886$		0			0
$887$		−	0.606101i		−	0.0203509i	−0.999948	−	0.0101754i	$-0.996761\pi$
$887$		−	0.606101i		−	0.0203509i	0.999948	−	0.0101754i	$-0.00323900\pi$
$888$		0			0
$889$	−44.7192			−1.49984
$890$		0			0
$891$	2.97906			0.0998021
$892$		0			0
$893$			18.3761i			0.614932i
$894$		0			0
$895$	−3.73975	+	16.6681i	−0.125006	+	0.557152i
$896$		0			0
$897$		−	6.21302i		−	0.207447i
$898$		0			0
$899$	−1.37258			−0.0457782
$900$		0			0
$901$	−9.71310			−0.323590
$902$		0			0
$903$			25.4546i			0.847076i
$904$		0			0
$905$	7.35690	−	32.7897i	0.244552	−	1.08997i
$906$		0			0
$907$		−	11.8582i		−	0.393746i	−0.980429	−	0.196873i	$-0.936921\pi$
$907$		−	11.8582i		−	0.393746i	0.980429	−	0.196873i	$-0.0630787\pi$
$908$		0			0
$909$	−12.1421			−0.402729
$910$		0			0
$911$	33.5642			1.11203			0.556015	−	0.831172i	$-0.312330\pi$
$911$	33.5642			1.11203			0.556015	+	0.831172i	$0.312330\pi$
$912$		0			0
$913$			43.8098i			1.44989i
$914$		0			0
$915$	13.6850	+	3.07045i	0.452411	+	0.101506i
$916$		0			0
$917$			54.2229i			1.79060i
$918$		0			0
$919$	−19.2242			−0.634149			−0.317074	−	0.948401i	$-0.602701\pi$
$919$	−19.2242			−0.634149			−0.317074	+	0.948401i	$0.602701\pi$
$920$		0			0
$921$	−24.9858			−0.823310
$922$		0			0
$923$			8.32176i			0.273914i
$924$		0			0
$925$	−0.0894772	+	0.189362i	−0.00294199	+	0.00622618i
$926$		0			0
$927$		−	2.81108i		−	0.0923279i
$928$		0			0
$929$	36.0349			1.18227			0.591133	−	0.806574i	$-0.298681\pi$
$929$	36.0349			1.18227			0.591133	+	0.806574i	$0.298681\pi$
$930$		0			0
$931$	−4.17113			−0.136703
$932$		0			0
$933$		−	18.3842i		−	0.601870i
$934$		0			0
$935$	9.38459	+	2.10559i	0.306909	+	0.0688601i
$936$		0			0
$937$			41.5851i			1.35853i	0.733895	+	0.679263i	$0.237700\pi$
$937$			41.5851i			1.35853i	−0.733895	+	0.679263i	$0.762300\pi$
$938$		0			0
$939$	8.61453			0.281124
$940$		0			0
$941$	−56.4933			−1.84163			−0.920814	−	0.390002i	$-0.872474\pi$
$941$	−56.4933			−1.84163			−0.920814	+	0.390002i	$0.872474\pi$
$942$		0			0
$943$		−	27.1115i		−	0.882871i
$944$		0			0
$945$	1.38459	−	6.17113i	0.0450409	−	0.200747i
$946$		0			0
$947$			11.3922i			0.370197i	0.982720	+	0.185099i	$0.0592604\pi$
$947$			11.3922i			0.370197i	−0.982720	+	0.185099i	$0.940740\pi$
$948$		0			0
$949$	−6.87031			−0.223020
$950$		0			0
$951$	15.0491			0.487999
$952$		0			0
$953$		−	56.7986i		−	1.83989i	−0.392053	−	0.919943i	$-0.628235\pi$
$953$		−	56.7986i		−	1.83989i	0.392053	−	0.919943i	$-0.371765\pi$
$954$		0			0
$955$	5.32982	−	23.7550i	0.172469	−	0.768693i
$956$		0			0
$957$			2.46793i			0.0797769i
$958$		0			0
$959$	4.74677			0.153281
$960$		0			0
$961$	−28.2548			−0.911446
$962$		0			0
$963$		−	18.3842i		−	0.592421i
$964$		0			0
$965$	10.7933	+	2.42165i	0.347448	+	0.0779556i
$966$		0			0
$967$			25.7224i			0.827177i	0.910464	+	0.413588i	$0.135725\pi$
$967$			25.7224i			0.827177i	−0.910464	+	0.413588i	$0.864275\pi$
$968$		0			0
$969$	−6.02242			−0.193468
$970$		0			0
$971$	32.8703			1.05486			0.527429	−	0.849599i	$-0.323156\pi$
$971$	32.8703			1.05486			0.527429	+	0.849599i	$0.323156\pi$
$972$		0			0
$973$		−	34.9092i		−	1.11914i
$974$		0			0
$975$	4.52072	+	2.13613i	0.144779	+	0.0684109i
$976$		0			0
$977$			30.1453i			0.964433i	0.876052	+	0.482217i	$0.160168\pi$
$977$			30.1453i			0.964433i	−0.876052	+	0.482217i	$0.839832\pi$
$978$		0			0
$979$	−43.9345			−1.40415
$980$		0			0
$981$	1.44383			0.0460980
$982$		0			0
$983$		−	32.4965i		−	1.03648i	−0.855236	−	0.518238i	$-0.826588\pi$
$983$		−	32.4965i		−	1.03648i	0.855236	−	0.518238i	$-0.173412\pi$
$984$		0			0
$985$	−57.5190	−	12.9053i	−1.83271	−	0.411198i
$986$		0			0
$987$		−	12.4607i		−	0.396630i
$988$		0			0
$989$	55.9145			1.77798
$990$		0			0
$991$	−37.4893			−1.19089			−0.595443	−	0.803397i	$-0.703024\pi$
$991$	−37.4893			−1.19089			−0.595443	+	0.803397i	$0.703024\pi$
$992$		0			0
$993$		−	22.6738i		−	0.719530i
$994$		0			0
$995$	2.99195	−	13.3351i	0.0948510	−	0.422751i
$996$		0			0
$997$		−	18.8659i		−	0.597488i	−0.954333	−	0.298744i	$-0.903432\pi$
$997$		−	18.8659i		−	0.597488i	0.954333	−	0.298744i	$-0.0965676\pi$
$998$		0			0
$999$	−0.0418875			−0.00132526

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3120.2.l.o.1249.7		8
4.3	odd	2		1560.2.l.e.1249.3	✓	8
5.4	even	2	inner	3120.2.l.o.1249.3		8
12.11	even	2		4680.2.l.f.2809.4		8
20.3	even	4		7800.2.a.bw.1.1		4
20.7	even	4		7800.2.a.bv.1.3		4
20.19	odd	2		1560.2.l.e.1249.7	yes	8
60.59	even	2		4680.2.l.f.2809.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.l.e.1249.3	✓	8	4.3	odd	2
1560.2.l.e.1249.7	yes	8	20.19	odd	2
3120.2.l.o.1249.3		8	5.4	even	2	inner
3120.2.l.o.1249.7		8	1.1	even	1	trivial
4680.2.l.f.2809.3		8	60.59	even	2
4680.2.l.f.2809.4		8	12.11	even	2
7800.2.a.bv.1.3		4	20.7	even	4
7800.2.a.bw.1.1		4	20.3	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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(0.489528 - 2.18183i) q^{5} +2.82843i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +(0.489528 - 2.18183i) q^{5} +2.82843i q^{7} -1.00000 q^{9} +2.97906 q^{11} +1.00000i q^{13} +(2.18183 + 0.489528i) q^{15} +1.44383i q^{17} +4.17113 q^{19} -2.82843 q^{21} +6.21302i q^{23} +(-4.52072 - 2.13613i) q^{25} -1.00000i q^{27} +0.828427 q^{29} -1.65685 q^{31} +2.97906i q^{33} +(6.17113 + 1.38459i) q^{35} -0.0418875i q^{37} -1.00000 q^{39} -4.36365 q^{41} -8.99956i q^{43} +(-0.489528 + 2.18183i) q^{45} +4.40554i q^{47} -1.00000 q^{49} -1.44383 q^{51} +6.72730i q^{53} +(1.45833 - 6.49978i) q^{55} +4.17113i q^{57} -2.97906 q^{59} +6.27226 q^{61} -2.82843i q^{63} +(2.18183 + 0.489528i) q^{65} -0.727302i q^{67} -6.21302 q^{69} +8.32176 q^{71} +6.87031i q^{73} +(2.13613 - 4.52072i) q^{75} +8.42604i q^{77} +6.11190 q^{79} +1.00000 q^{81} +14.7059i q^{83} +(3.15019 + 0.706797i) q^{85} +0.828427i q^{87} -14.7478 q^{89} -2.82843 q^{91} -1.65685i q^{93} +(2.04189 - 9.10069i) q^{95} -6.78654i q^{97} -2.97906 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^9	\( 8 q - 8 q^{9} + 16 q^{11} + 4 q^{15} - 24 q^{19} - 4 q^{25} - 16 q^{29} + 32 q^{31} - 8 q^{35} - 8 q^{39} - 8 q^{41} - 8 q^{49} - 8 q^{51} + 36 q^{55} - 16 q^{59} + 24 q^{61} + 4 q^{65} - 8 q^{69} + 24 q^{71} + 4 q^{75} - 24 q^{79} + 8 q^{81} - 40 q^{85} + 8 q^{89} + 32 q^{95} - 16 q^{99}+O(q^{100}) \) 8 * q - 8 * q^9 + 16 * q^11 + 4 * q^15 - 24 * q^19 - 4 * q^25 - 16 * q^29 + 32 * q^31 - 8 * q^35 - 8 * q^39 - 8 * q^41 - 8 * q^49 - 8 * q^51 + 36 * q^55 - 16 * q^59 + 24 * q^61 + 4 * q^65 - 8 * q^69 + 24 * q^71 + 4 * q^75 - 24 * q^79 + 8 * q^81 - 40 * q^85 + 8 * q^89 + 32 * q^95 - 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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