LMFDB - Embedded newform 7800.2.a.bw.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7800,2,Mod(1,7800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 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("7800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7800.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:	$62.2833135766$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.10304.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 7x^{2} + 8x + 8 $ x^4 - 2x^3 - 7x^2 + 8*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1560)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.692297$ of defining polynomial
Character	$\chi$	$=$	7800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -2.82843 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.82843 q^{7} +1.00000 q^{9} -2.97906 q^{11} -1.00000 q^{13} +1.44383 q^{17} +4.17113 q^{19} -2.82843 q^{21} +6.21302 q^{23} +1.00000 q^{27} -0.828427 q^{29} +1.65685 q^{31} -2.97906 q^{33} -0.0418875 q^{37} -1.00000 q^{39} -4.36365 q^{41} -8.99956 q^{43} -4.40554 q^{47} +1.00000 q^{49} +1.44383 q^{51} -6.72730 q^{53} +4.17113 q^{57} -2.97906 q^{59} +6.27226 q^{61} -2.82843 q^{63} +0.727302 q^{67} +6.21302 q^{69} -8.32176 q^{71} -6.87031 q^{73} +8.42604 q^{77} +6.11190 q^{79} +1.00000 q^{81} +14.7059 q^{83} -0.828427 q^{87} +14.7478 q^{89} +2.82843 q^{91} +1.65685 q^{93} -6.78654 q^{97} -2.97906 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 4 * q^9	$ 4 q + 4 q^{3} + 4 q^{9} - 8 q^{11} - 4 q^{13} + 4 q^{17} - 12 q^{19} + 4 q^{23} + 4 q^{27} + 8 q^{29} - 16 q^{31} - 8 q^{33} - 8 q^{37} - 4 q^{39} - 4 q^{41} + 4 q^{43} - 12 q^{47} + 4 q^{49} + 4 q^{51} - 12 q^{57} - 8 q^{59} + 12 q^{61} - 24 q^{67} + 4 q^{69} - 12 q^{71} - 24 q^{73} - 8 q^{77} - 12 q^{79} + 4 q^{81} - 12 q^{83} + 8 q^{87} - 4 q^{89} - 16 q^{93} - 8 q^{97} - 8 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 4 * q^9 - 8 * q^11 - 4 * q^13 + 4 * q^17 - 12 * q^19 + 4 * q^23 + 4 * q^27 + 8 * q^29 - 16 * q^31 - 8 * q^33 - 8 * q^37 - 4 * q^39 - 4 * q^41 + 4 * q^43 - 12 * q^47 + 4 * q^49 + 4 * q^51 - 12 * q^57 - 8 * q^59 + 12 * q^61 - 24 * q^67 + 4 * q^69 - 12 * q^71 - 24 * q^73 - 8 * q^77 - 12 * q^79 + 4 * q^81 - 12 * q^83 + 8 * q^87 - 4 * q^89 - 16 * q^93 - 8 * q^97 - 8 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.82843	−1.06904	−0.534522	−	0.845154i	$-0.679509\pi$
$7$	−2.82843	−1.06904	−0.534522	+	0.845154i	$0.679509\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.97906	−0.898219	−0.449110	−	0.893477i	$-0.648259\pi$
$11$	−2.97906	−0.898219	−0.449110	+	0.893477i	$0.648259\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.44383	0.350181	0.175090	−	0.984552i	$-0.443978\pi$
$17$	1.44383	0.350181	0.175090	+	0.984552i	$0.443978\pi$
$18$	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
$23$	6.21302	1.29550	0.647752	−	0.761851i	$-0.275709\pi$
$23$	6.21302	1.29550	0.647752	+	0.761851i	$0.275709\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−0.828427	−0.153835	−0.0769175	−	0.997037i	$-0.524508\pi$
$29$	−0.828427	−0.153835	−0.0769175	+	0.997037i	$0.524508\pi$
$30$	0	0
$31$	1.65685	0.297580	0.148790	−	0.988869i	$-0.452462\pi$
$31$	1.65685	0.297580	0.148790	+	0.988869i	$0.452462\pi$
$32$	0	0
$33$	−2.97906	−0.518587
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−0.0418875	−0.00688626	−0.00344313	−	0.999994i	$-0.501096\pi$
$37$	−0.0418875	−0.00688626	−0.00344313	+	0.999994i	$0.501096\pi$
$38$	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
$43$	−8.99956	−1.37242	−0.686210	−	0.727403i	$-0.740727\pi$
$43$	−8.99956	−1.37242	−0.686210	+	0.727403i	$0.740727\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.40554	−0.642614	−0.321307	−	0.946975i	$-0.604122\pi$
$47$	−4.40554	−0.642614	−0.321307	+	0.946975i	$0.604122\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.44383	0.202177
$52$	0	0
$53$	−6.72730	−0.924066	−0.462033	−	0.886863i	$-0.652880\pi$
$53$	−6.72730	−0.924066	−0.462033	+	0.886863i	$0.652880\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.17113	0.552480
$58$	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
$63$	−2.82843	−0.356348
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0.727302	0.0888540	0.0444270	−	0.999013i	$-0.485854\pi$
$67$	0.727302	0.0888540	0.0444270	+	0.999013i	$0.485854\pi$
$68$	0	0
$69$	6.21302	0.747960
$70$	0	0
$71$	−8.32176	−0.987612	−0.493806	−	0.869572i	$-0.664395\pi$
$71$	−8.32176	−0.987612	−0.493806	+	0.869572i	$0.664395\pi$
$72$	0	0
$73$	−6.87031	−0.804110	−0.402055	−	0.915616i	$-0.631704\pi$
$73$	−6.87031	−0.804110	−0.402055	+	0.915616i	$0.631704\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	8.42604	0.960237
$78$	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
$83$	14.7059	1.61418	0.807092	−	0.590425i	$-0.201040\pi$
$83$	14.7059	1.61418	0.807092	+	0.590425i	$0.201040\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−0.828427	−0.0888167
$88$	0	0
$89$	14.7478	1.56326	0.781632	−	0.623740i	$-0.214387\pi$
$89$	14.7478	1.56326	0.781632	+	0.623740i	$0.214387\pi$
$90$	0	0
$91$	2.82843	0.296500
$92$	0	0
$93$	1.65685	0.171808
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−6.78654	−0.689069	−0.344534	−	0.938774i	$-0.611963\pi$
$97$	−6.78654	−0.689069	−0.344534	+	0.938774i	$0.611963\pi$
$98$	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
$103$	2.81108	0.276984	0.138492	−	0.990364i	$-0.455775\pi$
$103$	2.81108	0.276984	0.138492	+	0.990364i	$0.455775\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−18.3842	−1.77726	−0.888632	−	0.458621i	$-0.848343\pi$
$107$	−18.3842	−1.77726	−0.888632	+	0.458621i	$0.848343\pi$
$108$	0	0
$109$	1.44383	0.138294	0.0691470	−	0.997606i	$-0.477972\pi$
$109$	1.44383	0.138294	0.0691470	+	0.997606i	$0.477972\pi$
$110$	0	0
$111$	−0.0418875	−0.00397579
$112$	0	0
$113$	−3.40194	−0.320028	−0.160014	−	0.987115i	$-0.551154\pi$
$113$	−3.40194	−0.320028	−0.160014	+	0.987115i	$0.551154\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−4.08378	−0.374359
$120$	0	0
$121$	−2.12522	−0.193202
$122$	0	0
$123$	−4.36365	−0.393457
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−15.8106	−1.40297	−0.701484	−	0.712686i	$-0.747479\pi$
$127$	−15.8106	−1.40297	−0.701484	+	0.712686i	$0.747479\pi$
$128$	0	0
$129$	−8.99956	−0.792367
$130$	0	0
$131$	−19.1707	−1.67495	−0.837476	−	0.546475i	$-0.815970\pi$
$131$	−19.1707	−1.67495	−0.837476	+	0.546475i	$0.815970\pi$
$132$	0	0
$133$	−11.7977	−1.02299
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1.67824	−0.143381	−0.0716907	−	0.997427i	$-0.522839\pi$
$137$	−1.67824	−0.143381	−0.0716907	+	0.997427i	$0.522839\pi$
$138$	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
$143$	2.97906	0.249121
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	0.677798	0.0555274	0.0277637	−	0.999615i	$-0.491161\pi$
$149$	0.677798	0.0555274	0.0277637	+	0.999615i	$0.491161\pi$
$150$	0	0
$151$	−17.9991	−1.46475	−0.732374	−	0.680903i	$-0.761588\pi$
$151$	−17.9991	−1.46475	−0.732374	+	0.680903i	$0.761588\pi$
$152$	0	0
$153$	1.44383	0.116727
$154$	0	0
$155$	0	0
$156$	0	0
$157$	12.8392	1.02468	0.512340	−	0.858783i	$-0.328779\pi$
$157$	12.8392	1.02468	0.512340	+	0.858783i	$0.328779\pi$
$158$	0	0
$159$	−6.72730	−0.533510
$160$	0	0
$161$	−17.5731	−1.38495
$162$	0	0
$163$	−15.3137	−1.19946	−0.599731	−	0.800202i	$-0.704726\pi$
$163$	−15.3137	−1.19946	−0.599731	+	0.800202i	$0.704726\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−15.3922	−1.19109	−0.595543	−	0.803324i	$-0.703063\pi$
$167$	−15.3922	−1.19109	−0.595543	+	0.803324i	$0.703063\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	4.17113	0.318975
$172$	0	0
$173$	14.9706	1.13819	0.569095	−	0.822272i	$-0.307294\pi$
$173$	14.9706	1.13819	0.569095	+	0.822272i	$0.307294\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−2.97906	−0.223920
$178$	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
$183$	6.27226	0.463659
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−4.30126	−0.314539
$188$	0	0
$189$	−2.82843	−0.205738
$190$	0	0
$191$	−10.8877	−0.787804	−0.393902	−	0.919152i	$-0.628875\pi$
$191$	−10.8877	−0.787804	−0.393902	+	0.919152i	$0.628875\pi$
$192$	0	0
$193$	−4.94690	−0.356086	−0.178043	−	0.984023i	$-0.556977\pi$
$193$	−4.94690	−0.356086	−0.178043	+	0.984023i	$0.556977\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−26.3628	−1.87827	−0.939135	−	0.343549i	$-0.888371\pi$
$197$	−26.3628	−1.87827	−0.939135	+	0.343549i	$0.888371\pi$
$198$	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
$203$	2.34315	0.164457
$204$	0	0
$205$	0	0
$206$	0	0
$207$	6.21302	0.431835
$208$	0	0
$209$	−12.4260	−0.859527
$210$	0	0
$211$	9.68585	0.666802	0.333401	−	0.942785i	$-0.391804\pi$
$211$	9.68585	0.666802	0.333401	+	0.942785i	$0.391804\pi$
$212$	0	0
$213$	−8.32176	−0.570198
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−4.68629	−0.318126
$218$	0	0
$219$	−6.87031	−0.464253
$220$	0	0
$221$	−1.44383	−0.0971227
$222$	0	0
$223$	−9.21258	−0.616920	−0.308460	−	0.951237i	$-0.599814\pi$
$223$	−9.21258	−0.616920	−0.308460	+	0.951237i	$0.599814\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−20.3218	−1.34880	−0.674401	−	0.738365i	$-0.735598\pi$
$227$	−20.3218	−1.34880	−0.674401	+	0.738365i	$0.735598\pi$
$228$	0	0
$229$	−16.4715	−1.08847	−0.544234	−	0.838933i	$-0.683180\pi$
$229$	−16.4715	−1.08847	−0.544234	+	0.838933i	$0.683180\pi$
$230$	0	0
$231$	8.42604	0.554393
$232$	0	0
$233$	−2.37339	−0.155486	−0.0777428	−	0.996973i	$-0.524771\pi$
$233$	−2.37339	−0.155486	−0.0777428	+	0.996973i	$0.524771\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	6.11190	0.397010
$238$	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
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−4.17113	−0.265403
$248$	0	0
$249$	14.7059	0.931950
$250$	0	0
$251$	14.6262	0.923196	0.461598	−	0.887089i	$-0.347276\pi$
$251$	14.6262	0.923196	0.461598	+	0.887089i	$0.347276\pi$
$252$	0	0
$253$	−18.5089	−1.16365
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−24.9403	−1.55573	−0.777867	−	0.628429i	$-0.783698\pi$
$257$	−24.9403	−1.55573	−0.777867	+	0.628429i	$0.783698\pi$
$258$	0	0
$259$	0.118476	0.00736173
$260$	0	0
$261$	−0.828427	−0.0512784
$262$	0	0
$263$	9.01691	0.556007	0.278003	−	0.960580i	$-0.410327\pi$
$263$	9.01691	0.556007	0.278003	+	0.960580i	$0.410327\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	14.7478	0.902551
$268$	0	0
$269$	8.28391	0.505079	0.252539	−	0.967587i	$-0.418734\pi$
$269$	8.28391	0.505079	0.252539	+	0.967587i	$0.418734\pi$
$270$	0	0
$271$	−19.9572	−1.21232	−0.606158	−	0.795344i	$-0.707290\pi$
$271$	−19.9572	−1.21232	−0.606158	+	0.795344i	$0.707290\pi$
$272$	0	0
$273$	2.82843	0.171184
$274$	0	0
$275$	0	0
$276$	0	0
$277$	22.6145	1.35878	0.679388	−	0.733780i	$-0.262245\pi$
$277$	22.6145	1.35878	0.679388	+	0.733780i	$0.262245\pi$
$278$	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
$283$	11.2299	0.667550	0.333775	−	0.942653i	$-0.391677\pi$
$283$	11.2299	0.667550	0.333775	+	0.942653i	$0.391677\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	12.3423	0.728541
$288$	0	0
$289$	−14.9153	−0.877373
$290$	0	0
$291$	−6.78654	−0.397834
$292$	0	0
$293$	−27.6774	−1.61693	−0.808464	−	0.588545i	$-0.799701\pi$
$293$	−27.6774	−1.61693	−0.808464	+	0.588545i	$0.799701\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−2.97906	−0.172862
$298$	0	0
$299$	−6.21302	−0.359308
$300$	0	0
$301$	25.4546	1.46718
$302$	0	0
$303$	12.1421	0.697547
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−24.9858	−1.42601	−0.713007	−	0.701157i	$-0.752667\pi$
$307$	−24.9858	−1.42601	−0.713007	+	0.701157i	$0.752667\pi$
$308$	0	0
$309$	2.81108	0.159917
$310$	0	0
$311$	18.3842	1.04247	0.521235	−	0.853413i	$-0.325472\pi$
$311$	18.3842	1.04247	0.521235	+	0.853413i	$0.325472\pi$
$312$	0	0
$313$	8.61453	0.486922	0.243461	−	0.969911i	$-0.421717\pi$
$313$	8.61453	0.486922	0.243461	+	0.969911i	$0.421717\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−15.0491	−0.845240	−0.422620	−	0.906307i	$-0.638889\pi$
$317$	−15.0491	−0.845240	−0.422620	+	0.906307i	$0.638889\pi$
$318$	0	0
$319$	2.46793	0.138178
$320$	0	0
$321$	−18.3842	−1.02610
$322$	0	0
$323$	6.02242	0.335096
$324$	0	0
$325$	0	0
$326$	0	0
$327$	1.44383	0.0798441
$328$	0	0
$329$	12.4607	0.686983
$330$	0	0
$331$	22.6738	1.24626	0.623131	−	0.782117i	$-0.285860\pi$
$331$	22.6738	1.24626	0.623131	+	0.782117i	$0.285860\pi$
$332$	0	0
$333$	−0.0418875	−0.00229542
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−27.9282	−1.52135	−0.760674	−	0.649134i	$-0.775131\pi$
$337$	−27.9282	−1.52135	−0.760674	+	0.649134i	$0.775131\pi$
$338$	0	0
$339$	−3.40194	−0.184768
$340$	0	0
$341$	−4.93586	−0.267292
$342$	0	0
$343$	16.9706	0.916324
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−19.7398	−1.05969	−0.529843	−	0.848096i	$-0.677749\pi$
$347$	−19.7398	−1.05969	−0.529843	+	0.848096i	$0.677749\pi$
$348$	0	0
$349$	30.7157	1.64417	0.822086	−	0.569364i	$-0.192810\pi$
$349$	30.7157	1.64417	0.822086	+	0.569364i	$0.192810\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	17.0072	0.905201	0.452600	−	0.891713i	$-0.350496\pi$
$353$	17.0072	0.905201	0.452600	+	0.891713i	$0.350496\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−4.08378	−0.216136
$358$	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
$363$	−2.12522	−0.111545
$364$	0	0
$365$	0	0
$366$	0	0
$367$	34.1810	1.78424	0.892118	−	0.451803i	$-0.149219\pi$
$367$	34.1810	1.78424	0.892118	+	0.451803i	$0.149219\pi$
$368$	0	0
$369$	−4.36365	−0.227163
$370$	0	0
$371$	19.0277	0.987868
$372$	0	0
$373$	−18.5098	−0.958402	−0.479201	−	0.877705i	$-0.659074\pi$
$373$	−18.5098	−0.958402	−0.479201	+	0.877705i	$0.659074\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.828427	0.0426662
$378$	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
$383$	0.664032	0.0339304	0.0169652	−	0.999856i	$-0.494600\pi$
$383$	0.664032	0.0339304	0.0169652	+	0.999856i	$0.494600\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−8.99956	−0.457473
$388$	0	0
$389$	2.32580	0.117923	0.0589613	−	0.998260i	$-0.481221\pi$
$389$	2.32580	0.117923	0.0589613	+	0.998260i	$0.481221\pi$
$390$	0	0
$391$	8.97056	0.453661
$392$	0	0
$393$	−19.1707	−0.967034
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−18.2433	−0.915603	−0.457802	−	0.889054i	$-0.651363\pi$
$397$	−18.2433	−0.915603	−0.457802	+	0.889054i	$0.651363\pi$
$398$	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
$403$	−1.65685	−0.0825338
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.124785	0.00618538
$408$	0	0
$409$	−10.8111	−0.534573	−0.267287	−	0.963617i	$-0.586127\pi$
$409$	−10.8111	−0.534573	−0.267287	+	0.963617i	$0.586127\pi$
$410$	0	0
$411$	−1.67824	−0.0827813
$412$	0	0
$413$	8.42604	0.414618
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−12.3423	−0.604403
$418$	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
$423$	−4.40554	−0.214205
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−17.7406	−0.858529
$428$	0	0
$429$	2.97906	0.143830
$430$	0	0
$431$	−8.48931	−0.408916	−0.204458	−	0.978875i	$-0.565543\pi$
$431$	−8.48931	−0.408916	−0.204458	+	0.978875i	$0.565543\pi$
$432$	0	0
$433$	13.8453	0.665365	0.332682	−	0.943039i	$-0.392046\pi$
$433$	13.8453	0.665365	0.332682	+	0.943039i	$0.392046\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	25.9153	1.23970
$438$	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
$443$	−17.6721	−0.839626	−0.419813	−	0.907611i	$-0.637904\pi$
$443$	−17.6721	−0.839626	−0.419813	+	0.907611i	$0.637904\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0.677798	0.0320587
$448$	0	0
$449$	3.33509	0.157393	0.0786963	−	0.996899i	$-0.474924\pi$
$449$	3.33509	0.157393	0.0786963	+	0.996899i	$0.474924\pi$
$450$	0	0
$451$	12.9996	0.612125
$452$	0	0
$453$	−17.9991	−0.845673
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−38.9266	−1.82091	−0.910454	−	0.413611i	$-0.864267\pi$
$457$	−38.9266	−1.82091	−0.910454	+	0.413611i	$0.864267\pi$
$458$	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
$463$	10.2259	0.475238	0.237619	−	0.971358i	$-0.423633\pi$
$463$	10.2259	0.475238	0.237619	+	0.971358i	$0.423633\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−5.61584	−0.259870	−0.129935	−	0.991522i	$-0.541477\pi$
$467$	−5.61584	−0.259870	−0.129935	+	0.991522i	$0.541477\pi$
$468$	0	0
$469$	−2.05712	−0.0949890
$470$	0	0
$471$	12.8392	0.591599
$472$	0	0
$473$	26.8102	1.23273
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−6.72730	−0.308022
$478$	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
$483$	−17.5731	−0.799603
$484$	0	0
$485$	0	0
$486$	0	0
$487$	18.3596	0.831954	0.415977	−	0.909375i	$-0.363440\pi$
$487$	18.3596	0.831954	0.415977	+	0.909375i	$0.363440\pi$
$488$	0	0
$489$	−15.3137	−0.692510
$490$	0	0
$491$	−33.7981	−1.52529	−0.762644	−	0.646819i	$-0.776099\pi$
$491$	−33.7981	−1.52529	−0.762644	+	0.646819i	$0.776099\pi$
$492$	0	0
$493$	−1.19611	−0.0538701
$494$	0	0
$495$	0	0
$496$	0	0
$497$	23.5375	1.05580
$498$	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
$503$	−20.2121	−0.901215	−0.450607	−	0.892722i	$-0.648793\pi$
$503$	−20.2121	−0.901215	−0.450607	+	0.892722i	$0.648793\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	6.02708	0.267146	0.133573	−	0.991039i	$-0.457355\pi$
$509$	6.02708	0.267146	0.133573	+	0.991039i	$0.457355\pi$
$510$	0	0
$511$	19.4322	0.859629
$512$	0	0
$513$	4.17113	0.184160
$514$	0	0
$515$	0	0
$516$	0	0
$517$	13.1243	0.577208
$518$	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
$523$	−21.6859	−0.948256	−0.474128	−	0.880456i	$-0.657237\pi$
$523$	−21.6859	−0.948256	−0.474128	+	0.880456i	$0.657237\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.39222	0.104207
$528$	0	0
$529$	15.6016	0.678332
$530$	0	0
$531$	−2.97906	−0.129280
$532$	0	0
$533$	4.36365	0.189011
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−7.63950	−0.329669
$538$	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
$543$	15.0286	0.644937
$544$	0	0
$545$	0	0
$546$	0	0
$547$	14.1466	0.604865	0.302432	−	0.953171i	$-0.402201\pi$
$547$	14.1466	0.604865	0.302432	+	0.953171i	$0.402201\pi$
$548$	0	0
$549$	6.27226	0.267694
$550$	0	0
$551$	−3.45548	−0.147208
$552$	0	0
$553$	−17.2871	−0.735120
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−29.7603	−1.26098	−0.630491	−	0.776196i	$-0.717147\pi$
$557$	−29.7603	−1.26098	−0.630491	+	0.776196i	$0.717147\pi$
$558$	0	0
$559$	8.99956	0.380641
$560$	0	0
$561$	−4.30126	−0.181599
$562$	0	0
$563$	−28.6498	−1.20745	−0.603723	−	0.797194i	$-0.706317\pi$
$563$	−28.6498	−1.20745	−0.603723	+	0.797194i	$0.706317\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−2.82843	−0.118783
$568$	0	0
$569$	39.7326	1.66568	0.832838	−	0.553517i	$-0.186715\pi$
$569$	39.7326	1.66568	0.832838	+	0.553517i	$0.186715\pi$
$570$	0	0
$571$	30.5726	1.27943	0.639713	−	0.768614i	$-0.279053\pi$
$571$	30.5726	1.27943	0.639713	+	0.768614i	$0.279053\pi$
$572$	0	0
$573$	−10.8877	−0.454839
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−35.3116	−1.47004	−0.735020	−	0.678045i	$-0.762827\pi$
$577$	−35.3116	−1.47004	−0.735020	+	0.678045i	$0.762827\pi$
$578$	0	0
$579$	−4.94690	−0.205586
$580$	0	0
$581$	−41.5946	−1.72564
$582$	0	0
$583$	20.0410	0.830014
$584$	0	0
$585$	0	0
$586$	0	0
$587$	19.5517	0.806985	0.403492	−	0.914983i	$-0.367796\pi$
$587$	19.5517	0.806985	0.403492	+	0.914983i	$0.367796\pi$
$588$	0	0
$589$	6.91096	0.284761
$590$	0	0
$591$	−26.3628	−1.08442
$592$	0	0
$593$	4.66491	0.191565	0.0957824	−	0.995402i	$-0.469465\pi$
$593$	4.66491	0.191565	0.0957824	+	0.995402i	$0.469465\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	6.11190	0.250143
$598$	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
$603$	0.727302	0.0296180
$604$	0	0
$605$	0	0
$606$	0	0
$607$	34.2648	1.39077	0.695383	−	0.718640i	$-0.255235\pi$
$607$	34.2648	1.39077	0.695383	+	0.718640i	$0.255235\pi$
$608$	0	0
$609$	2.34315	0.0949491
$610$	0	0
$611$	4.40554	0.178229
$612$	0	0
$613$	33.7203	1.36195	0.680975	−	0.732307i	$-0.261556\pi$
$613$	33.7203	1.36195	0.680975	+	0.732307i	$0.261556\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−35.0901	−1.41267	−0.706337	−	0.707876i	$-0.749653\pi$
$617$	−35.0901	−1.41267	−0.706337	+	0.707876i	$0.749653\pi$
$618$	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
$623$	−41.7131	−1.67120
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−12.4260	−0.496248
$628$	0	0
$629$	−0.0604785	−0.00241144
$630$	0	0
$631$	−28.7683	−1.14525	−0.572624	−	0.819818i	$-0.694075\pi$
$631$	−28.7683	−1.14525	−0.572624	+	0.819818i	$0.694075\pi$
$632$	0	0
$633$	9.68585	0.384978
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	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
$643$	−27.2709	−1.07546	−0.537731	−	0.843117i	$-0.680718\pi$
$643$	−27.2709	−1.07546	−0.537731	+	0.843117i	$0.680718\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−4.22913	−0.166264	−0.0831322	−	0.996539i	$-0.526492\pi$
$647$	−4.22913	−0.166264	−0.0831322	+	0.996539i	$0.526492\pi$
$648$	0	0
$649$	8.87478	0.348365
$650$	0	0
$651$	−4.68629	−0.183670
$652$	0	0
$653$	−22.1400	−0.866406	−0.433203	−	0.901296i	$-0.642617\pi$
$653$	−22.1400	−0.866406	−0.433203	+	0.901296i	$0.642617\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−6.87031	−0.268037
$658$	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
$663$	−1.44383	−0.0560738
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−5.14704	−0.199294
$668$	0	0
$669$	−9.21258	−0.356179
$670$	0	0
$671$	−18.6854	−0.721342
$672$	0	0
$673$	23.6213	0.910533	0.455267	−	0.890355i	$-0.349544\pi$
$673$	23.6213	0.910533	0.455267	+	0.890355i	$0.349544\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	18.8111	0.722968	0.361484	−	0.932378i	$-0.382270\pi$
$677$	18.8111	0.722968	0.361484	+	0.932378i	$0.382270\pi$
$678$	0	0
$679$	19.1952	0.736645
$680$	0	0
$681$	−20.3218	−0.778732
$682$	0	0
$683$	−33.7827	−1.29266	−0.646329	−	0.763059i	$-0.723697\pi$
$683$	−33.7827	−1.29266	−0.646329	+	0.763059i	$0.723697\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−16.4715	−0.628428
$688$	0	0
$689$	6.72730	0.256290
$690$	0	0
$691$	−47.3236	−1.80027	−0.900137	−	0.435606i	$-0.856534\pi$
$691$	−47.3236	−1.80027	−0.900137	+	0.435606i	$0.856534\pi$
$692$	0	0
$693$	8.42604	0.320079
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−6.30038	−0.238644
$698$	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
$703$	−0.174718	−0.00658963
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−34.3431	−1.29161
$708$	0	0
$709$	46.8404	1.75913	0.879565	−	0.475779i	$-0.157834\pi$
$709$	46.8404	1.75913	0.879565	+	0.475779i	$0.157834\pi$
$710$	0	0
$711$	6.11190	0.229214
$712$	0	0
$713$	10.2941	0.385516
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−13.6363	−0.509259
$718$	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
$723$	1.07045	0.0398104
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−10.8383	−0.401971	−0.200986	−	0.979594i	$-0.564414\pi$
$727$	−10.8383	−0.401971	−0.200986	+	0.979594i	$0.564414\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−12.9939	−0.480595
$732$	0	0
$733$	21.1952	0.782864	0.391432	−	0.920207i	$-0.371980\pi$
$733$	21.1952	0.782864	0.391432	+	0.920207i	$0.371980\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.16667	−0.0798104
$738$	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
$743$	21.3913	0.784772	0.392386	−	0.919801i	$-0.371650\pi$
$743$	21.3913	0.784772	0.392386	+	0.919801i	$0.371650\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	14.7059	0.538061
$748$	0	0
$749$	51.9982	1.89997
$750$	0	0
$751$	18.2247	0.665028	0.332514	−	0.943098i	$-0.392103\pi$
$751$	18.2247	0.665028	0.332514	+	0.943098i	$0.392103\pi$
$752$	0	0
$753$	14.6262	0.533007
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−39.3708	−1.43096	−0.715479	−	0.698635i	$-0.753791\pi$
$757$	−39.3708	−1.43096	−0.715479	+	0.698635i	$0.753791\pi$
$758$	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
$763$	−4.08378	−0.147843
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.97906	0.107567
$768$	0	0
$769$	−12.6274	−0.455356	−0.227678	−	0.973736i	$-0.573113\pi$
$769$	−12.6274	−0.455356	−0.227678	+	0.973736i	$0.573113\pi$
$770$	0	0
$771$	−24.9403	−0.898204
$772$	0	0
$773$	−34.7825	−1.25104	−0.625520	−	0.780208i	$-0.715113\pi$
$773$	−34.7825	−1.25104	−0.625520	+	0.780208i	$0.715113\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0.118476	0.00425029
$778$	0	0
$779$	−18.2014	−0.652132
$780$	0	0
$781$	24.7910	0.887092
$782$	0	0
$783$	−0.828427	−0.0296056
$784$	0	0
$785$	0	0
$786$	0	0
$787$	36.7835	1.31119	0.655596	−	0.755112i	$-0.272418\pi$
$787$	36.7835	1.31119	0.655596	+	0.755112i	$0.272418\pi$
$788$	0	0
$789$	9.01691	0.321011
$790$	0	0
$791$	9.62215	0.342124
$792$	0	0
$793$	−6.27226	−0.222734
$794$	0	0
$795$	0	0
$796$	0	0
$797$	42.4413	1.50335	0.751674	−	0.659535i	$-0.229247\pi$
$797$	42.4413	1.50335	0.751674	+	0.659535i	$0.229247\pi$
$798$	0	0
$799$	−6.36086	−0.225031
$800$	0	0
$801$	14.7478	0.521088
$802$	0	0
$803$	20.4671	0.722267
$804$	0	0
$805$	0	0
$806$	0	0
$807$	8.28391	0.291607
$808$	0	0
$809$	55.3771	1.94696	0.973478	−	0.228780i	$-0.0734735\pi$
$809$	55.3771	1.94696	0.973478	+	0.228780i	$0.0734735\pi$
$810$	0	0
$811$	−17.1159	−0.601021	−0.300511	−	0.953778i	$-0.597157\pi$
$811$	−17.1159	−0.601021	−0.300511	+	0.953778i	$0.597157\pi$
$812$	0	0
$813$	−19.9572	−0.699931
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−37.5384	−1.31330
$818$	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
$823$	−0.0998847	−0.00348176	−0.00174088	−	0.999998i	$-0.500554\pi$
$823$	−0.0998847	−0.00348176	−0.00174088	+	0.999998i	$0.500554\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	9.53311	0.331499	0.165749	−	0.986168i	$-0.446996\pi$
$827$	9.53311	0.331499	0.165749	+	0.986168i	$0.446996\pi$
$828$	0	0
$829$	50.6361	1.75866	0.879332	−	0.476210i	$-0.157990\pi$
$829$	50.6361	1.75866	0.879332	+	0.476210i	$0.157990\pi$
$830$	0	0
$831$	22.6145	0.784489
$832$	0	0
$833$	1.44383	0.0500258
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.65685	0.0572693
$838$	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
$843$	−1.89572	−0.0652921
$844$	0	0
$845$	0	0
$846$	0	0
$847$	6.01104	0.206542
$848$	0	0
$849$	11.2299	0.385410
$850$	0	0
$851$	−0.260248	−0.00892119
$852$	0	0
$853$	31.6979	1.08531	0.542657	−	0.839954i	$-0.317418\pi$
$853$	31.6979	1.08531	0.542657	+	0.839954i	$0.317418\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	28.8912	0.986906	0.493453	−	0.869772i	$-0.335734\pi$
$857$	28.8912	0.986906	0.493453	+	0.869772i	$0.335734\pi$
$858$	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
$863$	4.56502	0.155395	0.0776976	−	0.996977i	$-0.475243\pi$
$863$	4.56502	0.155395	0.0776976	+	0.996977i	$0.475243\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−14.9153	−0.506552
$868$	0	0
$869$	−18.2077	−0.617653
$870$	0	0
$871$	−0.727302	−0.0246437
$872$	0	0
$873$	−6.78654	−0.229690
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−13.5312	−0.456916	−0.228458	−	0.973554i	$-0.573368\pi$
$877$	−13.5312	−0.456916	−0.228458	+	0.973554i	$0.573368\pi$
$878$	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
$883$	47.3700	1.59413	0.797063	−	0.603896i	$-0.206386\pi$
$883$	47.3700	1.59413	0.797063	+	0.603896i	$0.206386\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0.606101	0.0203509	0.0101754	−	0.999948i	$-0.496761\pi$
$887$	0.606101	0.0203509	0.0101754	+	0.999948i	$0.496761\pi$
$888$	0	0
$889$	44.7192	1.49984
$890$	0	0
$891$	−2.97906	−0.0998021
$892$	0	0
$893$	−18.3761	−0.614932
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−6.21302	−0.207447
$898$	0	0
$899$	−1.37258	−0.0457782
$900$	0	0
$901$	−9.71310	−0.323590
$902$	0	0
$903$	25.4546	0.847076
$904$	0	0
$905$	0	0
$906$	0	0
$907$	11.8582	0.393746	0.196873	−	0.980429i	$-0.436921\pi$
$907$	11.8582	0.393746	0.196873	+	0.980429i	$0.436921\pi$
$908$	0	0
$909$	12.1421	0.402729
$910$	0	0
$911$	−33.5642	−1.11203	−0.556015	−	0.831172i	$-0.687670\pi$
$911$	−33.5642	−1.11203	−0.556015	+	0.831172i	$0.687670\pi$
$912$	0	0
$913$	−43.8098	−1.44989
$914$	0	0
$915$	0	0
$916$	0	0
$917$	54.2229	1.79060
$918$	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
$923$	8.32176	0.273914
$924$	0	0
$925$	0	0
$926$	0	0
$927$	2.81108	0.0923279
$928$	0	0
$929$	−36.0349	−1.18227	−0.591133	−	0.806574i	$-0.701319\pi$
$929$	−36.0349	−1.18227	−0.591133	+	0.806574i	$0.701319\pi$
$930$	0	0
$931$	4.17113	0.136703
$932$	0	0
$933$	18.3842	0.601870
$934$	0	0
$935$	0	0
$936$	0	0
$937$	41.5851	1.35853	0.679263	−	0.733895i	$-0.262300\pi$
$937$	41.5851	1.35853	0.679263	+	0.733895i	$0.262300\pi$
$938$	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
$943$	−27.1115	−0.882871
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−11.3922	−0.370197	−0.185099	−	0.982720i	$-0.559260\pi$
$947$	−11.3922	−0.370197	−0.185099	+	0.982720i	$0.559260\pi$
$948$	0	0
$949$	6.87031	0.223020
$950$	0	0
$951$	−15.0491	−0.487999
$952$	0	0
$953$	56.7986	1.83989	0.919943	−	0.392053i	$-0.128235\pi$
$953$	56.7986	1.83989	0.919943	+	0.392053i	$0.128235\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	2.46793	0.0797769
$958$	0	0
$959$	4.74677	0.153281
$960$	0	0
$961$	−28.2548	−0.911446
$962$	0	0
$963$	−18.3842	−0.592421
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−25.7224	−0.827177	−0.413588	−	0.910464i	$-0.635725\pi$
$967$	−25.7224	−0.827177	−0.413588	+	0.910464i	$0.635725\pi$
$968$	0	0
$969$	6.02242	0.193468
$970$	0	0
$971$	−32.8703	−1.05486	−0.527429	−	0.849599i	$-0.676844\pi$
$971$	−32.8703	−1.05486	−0.527429	+	0.849599i	$0.676844\pi$
$972$	0	0
$973$	34.9092	1.11914
$974$	0	0
$975$	0	0
$976$	0	0
$977$	30.1453	0.964433	0.482217	−	0.876052i	$-0.339832\pi$
$977$	30.1453	0.964433	0.482217	+	0.876052i	$0.339832\pi$
$978$	0	0
$979$	−43.9345	−1.40415
$980$	0	0
$981$	1.44383	0.0460980
$982$	0	0
$983$	−32.4965	−1.03648	−0.518238	−	0.855236i	$-0.673412\pi$
$983$	−32.4965	−1.03648	−0.518238	+	0.855236i	$0.673412\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	12.4607	0.396630
$988$	0	0
$989$	−55.9145	−1.77798
$990$	0	0
$991$	37.4893	1.19089	0.595443	−	0.803397i	$-0.296976\pi$
$991$	37.4893	1.19089	0.595443	+	0.803397i	$0.296976\pi$
$992$	0	0
$993$	22.6738	0.719530
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−18.8659	−0.597488	−0.298744	−	0.954333i	$-0.596568\pi$
$997$	−18.8659	−0.597488	−0.298744	+	0.954333i	$0.596568\pi$
$998$	0	0
$999$	−0.0418875	−0.00132526

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.l.e.1249.3	✓	8	5.2	odd	4
1560.2.l.e.1249.7	yes	8	5.3	odd	4
3120.2.l.o.1249.3		8	20.3	even	4
3120.2.l.o.1249.7		8	20.7	even	4
4680.2.l.f.2809.3		8	15.8	even	4
4680.2.l.f.2809.4		8	15.2	even	4
7800.2.a.bv.1.3		4	5.4	even	2
7800.2.a.bw.1.1		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7800.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.82843 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.82843 q^{7} +1.00000 q^{9} -2.97906 q^{11} -1.00000 q^{13} +1.44383 q^{17} +4.17113 q^{19} -2.82843 q^{21} +6.21302 q^{23} +1.00000 q^{27} -0.828427 q^{29} +1.65685 q^{31} -2.97906 q^{33} -0.0418875 q^{37} -1.00000 q^{39} -4.36365 q^{41} -8.99956 q^{43} -4.40554 q^{47} +1.00000 q^{49} +1.44383 q^{51} -6.72730 q^{53} +4.17113 q^{57} -2.97906 q^{59} +6.27226 q^{61} -2.82843 q^{63} +0.727302 q^{67} +6.21302 q^{69} -8.32176 q^{71} -6.87031 q^{73} +8.42604 q^{77} +6.11190 q^{79} +1.00000 q^{81} +14.7059 q^{83} -0.828427 q^{87} +14.7478 q^{89} +2.82843 q^{91} +1.65685 q^{93} -6.78654 q^{97} -2.97906 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 4 * q^9	\( 4 q + 4 q^{3} + 4 q^{9} - 8 q^{11} - 4 q^{13} + 4 q^{17} - 12 q^{19} + 4 q^{23} + 4 q^{27} + 8 q^{29} - 16 q^{31} - 8 q^{33} - 8 q^{37} - 4 q^{39} - 4 q^{41} + 4 q^{43} - 12 q^{47} + 4 q^{49} + 4 q^{51} - 12 q^{57} - 8 q^{59} + 12 q^{61} - 24 q^{67} + 4 q^{69} - 12 q^{71} - 24 q^{73} - 8 q^{77} - 12 q^{79} + 4 q^{81} - 12 q^{83} + 8 q^{87} - 4 q^{89} - 16 q^{93} - 8 q^{97} - 8 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 4 * q^9 - 8 * q^11 - 4 * q^13 + 4 * q^17 - 12 * q^19 + 4 * q^23 + 4 * q^27 + 8 * q^29 - 16 * q^31 - 8 * q^33 - 8 * q^37 - 4 * q^39 - 4 * q^41 + 4 * q^43 - 12 * q^47 + 4 * q^49 + 4 * q^51 - 12 * q^57 - 8 * q^59 + 12 * q^61 - 24 * q^67 + 4 * q^69 - 12 * q^71 - 24 * q^73 - 8 * q^77 - 12 * q^79 + 4 * q^81 - 12 * q^83 + 8 * q^87 - 4 * q^89 - 16 * q^93 - 8 * q^97 - 8 * q^99

Introduction

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