LMFDB - Embedded newform 315.4.d.c.64.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [315,4,Mod(64,315)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(315, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("315.64");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	315.d (of order $2$, degree $1$, 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:	$18.5856016518$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 55x^{8} + 983x^{6} + 6409x^{4} + 13560x^{2} + 3600 $ x^10 + 55x^8 + 983x^6 + 6409x^4 + 13560x^2 + 3600
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 7^{4} $
Twist minimal:	no (minimal twist has level 35)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			64.10
Root			$4.31366i$ of defining polynomial
Character	$\chi$	$=$	315.64
Dual form			315.4.d.c.64.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+5.31366i q^{2} -20.2350 q^{4} +(2.32771 + 10.9353i) q^{5} -7.00000i q^{7} -65.0123i q^{8} +(-58.1067 + 12.3686i) q^{10} -25.5420 q^{11} -64.1014i q^{13} +37.1956 q^{14} +183.574 q^{16} -27.6952i q^{17} -0.792436 q^{19} +(-47.1011 - 221.276i) q^{20} -135.721i q^{22} -108.606i q^{23} +(-114.164 + 50.9086i) q^{25} +340.613 q^{26} +141.645i q^{28} -234.000 q^{29} +129.204 q^{31} +455.349i q^{32} +147.163 q^{34} +(76.5474 - 16.2940i) q^{35} -38.3108i q^{37} -4.21073i q^{38} +(710.932 - 151.330i) q^{40} +403.216 q^{41} +172.895i q^{43} +516.840 q^{44} +577.097 q^{46} +206.943i q^{47} -49.0000 q^{49} +(-270.511 - 606.626i) q^{50} +1297.09i q^{52} -144.031i q^{53} +(-59.4542 - 279.310i) q^{55} -455.086 q^{56} -1243.40i q^{58} -679.086 q^{59} -574.717 q^{61} +686.544i q^{62} -950.977 q^{64} +(700.971 - 149.209i) q^{65} -515.640i q^{67} +560.411i q^{68} +(86.5805 + 406.747i) q^{70} -556.612 q^{71} -173.243i q^{73} +203.571 q^{74} +16.0349 q^{76} +178.794i q^{77} -79.3290 q^{79} +(427.306 + 2007.44i) q^{80} +2142.55i q^{82} -1043.56i q^{83} +(302.856 - 64.4663i) q^{85} -918.703 q^{86} +1660.54i q^{88} +652.060 q^{89} -448.710 q^{91} +2197.65i q^{92} -1099.62 q^{94} +(-1.84456 - 8.66556i) q^{95} -515.714i q^{97} -260.369i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 36 q^{4} - 6 q^{5} - 16 q^{10} - 84 q^{11} + 56 q^{14} + 148 q^{16} + 72 q^{19} + 68 q^{20} - 362 q^{25} + 620 q^{26} - 88 q^{29} + 120 q^{31} + 964 q^{34} + 28 q^{35} + 1396 q^{40} + 852 q^{41}+ \cdots + 1628 q^{95}+O(q^{100}) $ 10 * q - 36 * q^4 - 6 * q^5 - 16 * q^10 - 84 * q^11 + 56 * q^14 + 148 * q^16 + 72 * q^19 + 68 * q^20 - 362 * q^25 + 620 * q^26 - 88 * q^29 + 120 * q^31 + 964 * q^34 + 28 * q^35 + 1396 * q^40 + 852 * q^41 + 1424 * q^44 + 176 * q^46 - 490 * q^49 - 1644 * q^50 - 1136 * q^55 - 588 * q^56 - 864 * q^59 - 884 * q^61 - 2724 * q^64 - 520 * q^65 + 756 * q^70 - 880 * q^71 - 3024 * q^74 - 2672 * q^76 - 3580 * q^79 + 2316 * q^80 + 1572 * q^85 - 2432 * q^86 + 1492 * q^89 + 476 * q^91 + 1652 * q^94 + 1628 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/315\mathbb{Z}\right)^\times$.

$n$	$127$	$136$	$281$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$			5.31366i			1.87866i	0.343013	+	0.939331i	$0.388553\pi$
$2$			5.31366i			1.87866i	−0.343013	+	0.939331i	$0.611447\pi$
$3$		0			0
$3$		0			0
$4$	−20.2350			−2.52937
$5$	2.32771	+	10.9353i	0.208197	+	0.978087i
$5$	2.32771	+	10.9353i	0.208197	+	0.978087i
$6$		0			0
$7$		−	7.00000i		−	0.377964i
$7$		−	7.00000i		−	0.377964i
$8$		−	65.0123i		−	2.87317i
$9$		0			0
$10$	−58.1067	+	12.3686i	−1.83749	+	0.391131i
$11$	−25.5420			−0.700108			−0.350054	−	0.936730i	$-0.613837\pi$
$11$	−25.5420			−0.700108			−0.350054	+	0.936730i	$0.613837\pi$
$12$		0			0
$13$		−	64.1014i		−	1.36758i	−0.729679	−	0.683790i	$-0.760330\pi$
$13$		−	64.1014i		−	1.36758i	0.729679	−	0.683790i	$-0.239670\pi$
$14$	37.1956			0.710067
$15$		0			0
$16$	183.574			2.86834
$17$		−	27.6952i		−	0.395122i	−0.980291	−	0.197561i	$-0.936698\pi$
$17$		−	27.6952i		−	0.395122i	0.980291	−	0.197561i	$-0.0633020\pi$
$18$		0			0
$19$	−0.792436			−0.00956828			−0.00478414	−	0.999989i	$-0.501523\pi$
$19$	−0.792436			−0.00956828			−0.00478414	+	0.999989i	$0.501523\pi$
$20$	−47.1011	−	221.276i	−0.526606	−	2.47394i
$21$		0			0
$22$		−	135.721i		−	1.31527i
$23$		−	108.606i		−	0.984609i	−0.870423	−	0.492305i	$-0.836155\pi$
$23$		−	108.606i		−	0.984609i	0.870423	−	0.492305i	$-0.163845\pi$
$24$		0			0
$25$	−114.164	+	50.9086i	−0.913308	+	0.407269i
$26$	340.613			2.56922
$27$		0			0
$28$			141.645i			0.956012i
$29$	−234.000			−1.49837			−0.749186	−	0.662360i	$-0.769555\pi$
$29$	−234.000			−1.49837			−0.749186	+	0.662360i	$0.769555\pi$
$30$		0			0
$31$	129.204			0.748570			0.374285	−	0.927314i	$-0.377888\pi$
$31$	129.204			0.748570			0.374285	+	0.927314i	$0.377888\pi$
$32$			455.349i			2.51547i
$33$		0			0
$34$	147.163			0.742300
$35$	76.5474	−	16.2940i	0.369682	−	0.0786909i
$36$		0			0
$37$		−	38.3108i		−	0.170223i	−0.996371	−	0.0851116i	$-0.972875\pi$
$37$		−	38.3108i		−	0.170223i	0.996371	−	0.0851116i	$-0.0271247\pi$
$38$		−	4.21073i		−	0.0179756i
$39$		0			0
$40$	710.932	−	151.330i	2.81021	−	0.598183i
$41$	403.216			1.53590			0.767949	−	0.640511i	$-0.221278\pi$
$41$	403.216			1.53590			0.767949	+	0.640511i	$0.221278\pi$
$42$		0			0
$43$			172.895i			0.613167i	0.951844	+	0.306584i	$0.0991859\pi$
$43$			172.895i			0.613167i	−0.951844	+	0.306584i	$0.900814\pi$
$44$	516.840			1.77083
$45$		0			0
$46$	577.097			1.84975
$47$			206.943i			0.642250i	0.947037	+	0.321125i	$0.104061\pi$
$47$			206.943i			0.642250i	−0.947037	+	0.321125i	$0.895939\pi$
$48$		0			0
$49$	−49.0000			−0.142857
$50$	−270.511	−	606.626i	−0.765120	−	1.71580i
$51$		0			0
$52$			1297.09i			3.45911i
$53$		−	144.031i		−	0.373287i	−0.982428	−	0.186643i	$-0.940239\pi$
$53$		−	144.031i		−	0.373287i	0.982428	−	0.186643i	$-0.0597609\pi$
$54$		0			0
$55$	−59.4542	−	279.310i	−0.145760	−	0.684767i
$56$	−455.086			−1.08595
$57$		0			0
$58$		−	1243.40i		−	2.81493i
$59$	−679.086			−1.49846			−0.749232	−	0.662307i	$-0.769577\pi$
$59$	−679.086			−1.49846			−0.749232	+	0.662307i	$0.769577\pi$
$60$		0			0
$61$	−574.717			−1.20631			−0.603155	−	0.797624i	$-0.706090\pi$
$61$	−574.717			−1.20631			−0.603155	+	0.797624i	$0.706090\pi$
$62$			686.544i			1.40631i
$63$		0			0
$64$	−950.977			−1.85738
$65$	700.971	−	149.209i	1.33761	−	0.284725i
$66$		0			0
$67$		−	515.640i		−	0.940230i	−0.882605	−	0.470115i	$-0.844212\pi$
$67$		−	515.640i		−	0.940230i	0.882605	−	0.470115i	$-0.155788\pi$
$68$			560.411i			0.999409i
$69$		0			0
$70$	86.5805	+	406.747i	0.147834	+	0.694508i
$71$	−556.612			−0.930391			−0.465195	−	0.885208i	$-0.654016\pi$
$71$	−556.612			−0.930391			−0.465195	+	0.885208i	$0.654016\pi$
$72$		0			0
$73$		−	173.243i		−	0.277762i	−0.990309	−	0.138881i	$-0.955650\pi$
$73$		−	173.243i		−	0.277762i	0.990309	−	0.138881i	$-0.0443505\pi$
$74$	203.571			0.319792
$75$		0			0
$76$	16.0349			0.0242017
$77$			178.794i			0.264616i
$78$		0			0
$79$	−79.3290			−0.112977			−0.0564887	−	0.998403i	$-0.517990\pi$
$79$	−79.3290			−0.112977			−0.0564887	+	0.998403i	$0.517990\pi$
$80$	427.306	+	2007.44i	0.597178	+	2.80548i
$81$		0			0
$82$			2142.55i			2.88543i
$83$		−	1043.56i		−	1.38007i	−0.723777	−	0.690034i	$-0.757596\pi$
$83$		−	1043.56i		−	1.38007i	0.723777	−	0.690034i	$-0.242404\pi$
$84$		0			0
$85$	302.856	−	64.4663i	0.386463	−	0.0822630i
$86$	−918.703			−1.15193
$87$		0			0
$88$			1660.54i			2.01153i
$89$	652.060			0.776609			0.388304	−	0.921531i	$-0.373061\pi$
$89$	652.060			0.776609			0.388304	+	0.921531i	$0.373061\pi$
$90$		0			0
$91$	−448.710			−0.516897
$92$			2197.65i			2.49044i
$93$		0			0
$94$	−1099.62			−1.20657
$95$	−1.84456	−	8.66556i	−0.00199208	−	0.00935861i
$96$		0			0
$97$		−	515.714i		−	0.539823i	−0.962885	−	0.269912i	$-0.913005\pi$
$97$		−	515.714i		−	0.539823i	0.962885	−	0.269912i	$-0.0869945\pi$
$98$		−	260.369i		−	0.268380i
$99$		0			0
$100$	2310.09	−	1030.13i	2.31009	−	1.03013i
$101$	536.339			0.528394			0.264197	−	0.964469i	$-0.414893\pi$
$101$	536.339			0.528394			0.264197	+	0.964469i	$0.414893\pi$
$102$		0			0
$103$			381.693i			0.365139i	0.983193	+	0.182570i	$0.0584415\pi$
$103$			381.693i			0.365139i	−0.983193	+	0.182570i	$0.941559\pi$
$104$	−4167.38			−3.92928
$105$		0			0
$106$	765.332			0.701279
$107$		−	1381.12i		−	1.24783i	−0.781492	−	0.623915i	$-0.785541\pi$
$107$		−	1381.12i		−	1.24783i	0.781492	−	0.623915i	$-0.214459\pi$
$108$		0			0
$109$	390.582			0.343220			0.171610	−	0.985165i	$-0.445103\pi$
$109$	390.582			0.343220			0.171610	+	0.985165i	$0.445103\pi$
$110$	1484.16	−	315.919i	1.28644	−	0.273834i
$111$		0			0
$112$		−	1285.02i		−	1.08413i
$113$		−	1643.15i		−	1.36792i	−0.729521	−	0.683958i	$-0.760257\pi$
$113$		−	1643.15i		−	1.36792i	0.729521	−	0.683958i	$-0.239743\pi$
$114$		0			0
$115$	1187.65	−	252.804i	0.963033	−	0.204992i
$116$	4734.99			3.78994
$117$		0			0
$118$		−	3608.43i		−	2.81511i
$119$	−193.866			−0.149342
$120$		0			0
$121$	−678.609			−0.509849
$122$		−	3053.85i		−	2.26625i
$123$		0			0
$124$	−2614.43			−1.89341
$125$	−822.443	−	1129.92i	−0.588492	−	0.808503i
$126$		0			0
$127$			192.032i			0.134174i	0.997747	+	0.0670869i	$0.0213705\pi$
$127$			192.032i			0.134174i	−0.997747	+	0.0670869i	$0.978630\pi$
$128$		−	1410.38i		−	0.973914i
$129$		0			0
$130$	792.848	+	3724.72i	0.534903	+	2.51292i
$131$	−2082.90			−1.38919			−0.694594	−	0.719402i	$-0.744416\pi$
$131$	−2082.90			−1.38919			−0.694594	+	0.719402i	$0.744416\pi$
$132$		0			0
$133$			5.54705i			0.00361647i
$134$	2739.93			1.76637
$135$		0			0
$136$	−1800.53			−1.13525
$137$		−	78.1709i		−	0.0487488i	−0.999703	−	0.0243744i	$-0.992241\pi$
$137$		−	78.1709i		−	0.0487488i	0.999703	−	0.0243744i	$-0.00775939\pi$
$138$		0			0
$139$	−1393.67			−0.850426			−0.425213	−	0.905093i	$-0.639801\pi$
$139$	−1393.67			−0.850426			−0.425213	+	0.905093i	$0.639801\pi$
$140$	−1548.93	+	329.707i	−0.935062	+	0.199038i
$141$		0			0
$142$		−	2957.65i		−	1.74789i
$143$			1637.28i			0.957454i
$144$		0			0
$145$	−544.685	−	2558.88i	−0.311956	−	1.46554i
$146$	920.556			0.521820
$147$		0			0
$148$			775.217i			0.430557i
$149$	32.5002			0.0178693			0.00893463	−	0.999960i	$-0.497156\pi$
$149$	32.5002			0.0178693			0.00893463	+	0.999960i	$0.497156\pi$
$150$		0			0
$151$	466.762			0.251553			0.125777	−	0.992059i	$-0.459858\pi$
$151$	466.762			0.251553			0.125777	+	0.992059i	$0.459858\pi$
$152$			51.5181i			0.0274913i
$153$		0			0
$154$	−950.048			−0.497124
$155$	300.749	+	1412.89i	0.155850	+	0.732167i
$156$		0			0
$157$		−	1673.50i		−	0.850701i	−0.905029	−	0.425351i	$-0.860151\pi$
$157$		−	1673.50i		−	0.850701i	0.905029	−	0.425351i	$-0.139849\pi$
$158$		−	421.527i		−	0.212246i
$159$		0			0
$160$	−4979.39	+	1059.92i	−2.46035	+	0.523712i
$161$	−760.245			−0.372147
$162$		0			0
$163$			1869.36i			0.898279i	0.893462	+	0.449139i	$0.148269\pi$
$163$			1869.36i			0.898279i	−0.893462	+	0.449139i	$0.851731\pi$
$164$	−8159.06			−3.88485
$165$		0			0
$166$	5545.12			2.59268
$167$			46.5250i			0.0215581i	0.999942	+	0.0107791i	$0.00343115\pi$
$167$			46.5250i			0.0215581i	−0.999942	+	0.0107791i	$0.996569\pi$
$168$		0			0
$169$	−1911.99			−0.870275
$170$	342.552	+	1609.28i	0.154544	+	0.726034i
$171$		0			0
$172$		−	3498.52i		−	1.55093i
$173$		−	2496.28i		−	1.09704i	−0.836137	−	0.548521i	$-0.815191\pi$
$173$		−	2496.28i		−	1.09704i	0.836137	−	0.548521i	$-0.184809\pi$
$174$		0			0
$175$	356.360	+	799.145i	0.153933	+	0.345198i
$176$	−4688.83			−2.00815
$177$		0			0
$178$			3464.82i			1.45898i
$179$	2975.70			1.24254			0.621269	−	0.783598i	$-0.286618\pi$
$179$	2975.70			1.24254			0.621269	+	0.783598i	$0.286618\pi$
$180$		0			0
$181$	966.273			0.396809			0.198405	−	0.980120i	$-0.436424\pi$
$181$	966.273			0.396809			0.198405	+	0.980120i	$0.436424\pi$
$182$		−	2384.29i		−	0.971074i
$183$		0			0
$184$	−7060.76			−2.82895
$185$	418.942	−	89.1764i	0.166493	−	0.0354399i
$186$		0			0
$187$			707.389i			0.276628i
$188$		−	4187.48i		−	1.62449i
$189$		0			0
$190$	46.0458	−	9.80136i	0.0175817	−	0.00374245i
$191$	1545.50			0.585488			0.292744	−	0.956191i	$-0.405432\pi$
$191$	1545.50			0.585488			0.292744	+	0.956191i	$0.405432\pi$
$192$		0			0
$193$		−	2304.05i		−	0.859322i	−0.902990	−	0.429661i	$-0.858633\pi$
$193$		−	2304.05i		−	0.859322i	0.902990	−	0.429661i	$-0.141367\pi$
$194$	2740.33			1.01415
$195$		0			0
$196$	991.513			0.361338
$197$		−	222.021i		−	0.0802960i	−0.999194	−	0.0401480i	$-0.987217\pi$
$197$		−	222.021i		−	0.0802960i	0.999194	−	0.0401480i	$-0.0127829\pi$
$198$		0			0
$199$	−3580.56			−1.27547			−0.637736	−	0.770255i	$-0.720129\pi$
$199$	−3580.56			−1.27547			−0.637736	+	0.770255i	$0.720129\pi$
$200$	3309.69	+	7422.04i	1.17015	+	2.62409i
$201$		0			0
$202$			2849.92i			0.992673i
$203$			1638.00i			0.566331i
$204$		0			0
$205$	938.570	+	4409.31i	0.319769	+	1.50224i
$206$	−2028.19			−0.685973
$207$		0			0
$208$		−	11767.3i		−	3.92268i
$209$	20.2404			0.00669883
$210$		0			0
$211$	−4181.04			−1.36415			−0.682073	−	0.731284i	$-0.738921\pi$
$211$	−4181.04			−1.36415			−0.682073	+	0.731284i	$0.738921\pi$
$212$			2914.46i			0.944180i
$213$		0			0
$214$	7338.79			2.34425
$215$	−1890.66	+	402.449i	−0.599731	+	0.127659i
$216$		0			0
$217$		−	904.426i		−	0.282933i
$218$			2075.42i			0.644794i
$219$		0			0
$220$	1203.05	+	5651.83i	0.368681	+	1.73203i
$221$	−1775.30			−0.540361
$222$		0			0
$223$			2361.52i			0.709145i	0.935028	+	0.354573i	$0.115374\pi$
$223$			2361.52i			0.709145i	−0.935028	+	0.354573i	$0.884626\pi$
$224$	3187.44			0.950758
$225$		0			0
$226$	8731.14			2.56985
$227$			586.877i			0.171596i	0.996313	+	0.0857982i	$0.0273440\pi$
$227$			586.877i			0.171596i	−0.996313	+	0.0857982i	$0.972656\pi$
$228$		0			0
$229$	4619.55			1.33305			0.666526	−	0.745482i	$-0.267781\pi$
$229$	4619.55			1.33305			0.666526	+	0.745482i	$0.267781\pi$
$230$	1343.31	+	6310.76i	0.385111	+	1.80921i
$231$		0			0
$232$			15212.9i			4.30507i
$233$			5120.44i			1.43971i	0.694127	+	0.719853i	$0.255791\pi$
$233$			5120.44i			1.43971i	−0.694127	+	0.719853i	$0.744209\pi$
$234$		0			0
$235$	−2262.99	+	481.703i	−0.628176	+	0.133714i
$236$	13741.3			3.79017
$237$		0			0
$238$		−	1030.14i		−	0.280563i
$239$	1127.51			0.305158			0.152579	−	0.988291i	$-0.451242\pi$
$239$	1127.51			0.305158			0.152579	+	0.988291i	$0.451242\pi$
$240$		0			0
$241$	−3549.53			−0.948736			−0.474368	−	0.880327i	$-0.657323\pi$
$241$	−3549.53			−0.948736			−0.474368	+	0.880327i	$0.657323\pi$
$242$		−	3605.89i		−	0.957833i
$243$		0			0
$244$	11629.4			3.05120
$245$	−114.058	−	535.832i	−0.0297424	−	0.139727i
$246$		0			0
$247$			50.7963i			0.0130854i
$248$		−	8399.84i		−	2.15077i
$249$		0			0
$250$	6003.99	−	4370.18i	1.51890	−	1.10558i
$251$	−4717.19			−1.18624			−0.593120	−	0.805114i	$-0.702104\pi$
$251$	−4717.19			−1.18624			−0.593120	+	0.805114i	$0.702104\pi$
$252$		0			0
$253$			2774.02i			0.689333i
$254$	−1020.39			−0.252067
$255$		0			0
$256$	−113.552			−0.0277227
$257$			6260.31i			1.51948i	0.650224	+	0.759742i	$0.274675\pi$
$257$			6260.31i			1.51948i	−0.650224	+	0.759742i	$0.725325\pi$
$258$		0			0
$259$	−268.176			−0.0643383
$260$	−14184.1	+	3019.25i	−3.38331	+	0.720176i
$261$		0			0
$262$		−	11067.8i		−	2.60981i
$263$			5753.04i			1.34885i	0.738344	+	0.674425i	$0.235608\pi$
$263$			5753.04i			1.34885i	−0.738344	+	0.674425i	$0.764392\pi$
$264$		0			0
$265$	1575.03	−	335.262i	0.365107	−	0.0777170i
$266$	−29.4751			−0.00679412
$267$		0			0
$268$			10433.9i			2.37819i
$269$	−7059.21			−1.60003			−0.800014	−	0.599982i	$-0.795174\pi$
$269$	−7059.21			−1.60003			−0.800014	+	0.599982i	$0.795174\pi$
$270$		0			0
$271$	−8534.52			−1.91305			−0.956523	−	0.291658i	$-0.905793\pi$
$271$	−8534.52			−1.91305			−0.956523	+	0.291658i	$0.905793\pi$
$272$		−	5084.11i		−	1.13334i
$273$		0			0
$274$	415.373			0.0915826
$275$	2915.96	−	1300.30i	0.639415	−	0.285132i
$276$		0			0
$277$			1313.94i			0.285008i	0.989794	+	0.142504i	$0.0455154\pi$
$277$			1313.94i			0.285008i	−0.989794	+	0.142504i	$0.954485\pi$
$278$		−	7405.47i		−	1.59766i
$279$		0			0
$280$	−1059.31	−	4976.53i	−0.226092	−	1.06216i
$281$	247.229			0.0524856			0.0262428	−	0.999656i	$-0.491646\pi$
$281$	247.229			0.0524856			0.0262428	+	0.999656i	$0.491646\pi$
$282$		0			0
$283$		−	9074.90i		−	1.90617i	−0.302697	−	0.953087i	$-0.597887\pi$
$283$		−	9074.90i		−	1.90617i	0.302697	−	0.953087i	$-0.402113\pi$
$284$	11263.0			2.35330
$285$		0			0
$286$	−8699.92			−1.79873
$287$		−	2822.51i		−	0.580515i
$288$		0			0
$289$	4145.98			0.843879
$290$	13597.0	−	2894.27i	2.75325	−	0.586060i
$291$		0			0
$292$			3505.57i			0.702562i
$293$			2740.72i			0.546466i	0.961948	+	0.273233i	$0.0880930\pi$
$293$			2740.72i			0.546466i	−0.961948	+	0.273233i	$0.911907\pi$
$294$		0			0
$295$	−1580.71	−	7426.04i	−0.311975	−	1.46563i
$296$	−2490.68			−0.489080
$297$		0			0
$298$			172.695i			0.0335703i
$299$	−6961.83			−1.34653
$300$		0			0
$301$	1210.26			0.231755
$302$			2480.21i			0.472584i
$303$		0			0
$304$	−145.470			−0.0274451
$305$	−1337.77	−	6284.73i	−0.251150	−	1.17988i
$306$		0			0
$307$			6985.46i			1.29864i	0.760517	+	0.649318i	$0.224946\pi$
$307$			6985.46i			1.29864i	−0.760517	+	0.649318i	$0.775054\pi$
$308$		−	3617.88i		−	0.669311i
$309$		0			0
$310$	−7507.60	+	1598.08i	−1.37549	+	0.292789i
$311$	356.841			0.0650630			0.0325315	−	0.999471i	$-0.489643\pi$
$311$	356.841			0.0650630			0.0325315	+	0.999471i	$0.489643\pi$
$312$		0			0
$313$			6630.12i			1.19731i	0.801009	+	0.598653i	$0.204297\pi$
$313$			6630.12i			1.19731i	−0.801009	+	0.598653i	$0.795703\pi$
$314$	8892.42			1.59818
$315$		0			0
$316$	1605.22			0.285762
$317$			2494.05i			0.441891i	0.975286	+	0.220946i	$0.0709144\pi$
$317$			2494.05i			0.441891i	−0.975286	+	0.220946i	$0.929086\pi$
$318$		0			0
$319$	5976.83			1.04902
$320$	−2213.60	−	10399.3i	−0.386700	−	1.81668i
$321$		0			0
$322$		−	4039.68i		−	0.699139i
$323$			21.9467i			0.00378063i
$324$		0			0
$325$	3263.31	+	7318.05i	0.556973	+	1.24902i
$326$	−9933.13			−1.68756
$327$		0			0
$328$		−	26214.0i		−	4.41289i
$329$	1448.60			0.242748
$330$		0			0
$331$	−4682.47			−0.777558			−0.388779	−	0.921331i	$-0.627103\pi$
$331$	−4682.47			−0.777558			−0.388779	+	0.921331i	$0.627103\pi$
$332$			21116.4i			3.49070i
$333$		0			0
$334$	−247.218			−0.0405005
$335$	5638.70	−	1200.26i	0.919627	−	0.195753i
$336$		0			0
$337$			3596.60i			0.581363i	0.956820	+	0.290681i	$0.0938820\pi$
$337$			3596.60i			0.581363i	−0.956820	+	0.290681i	$0.906118\pi$
$338$		−	10159.7i		−	1.63495i
$339$		0			0
$340$	−6128.28	+	1304.47i	−0.977509	+	0.208073i
$341$	−3300.12			−0.524080
$342$		0			0
$343$			343.000i			0.0539949i
$344$	11240.3			1.76173
$345$		0			0
$346$	13264.4			2.06097
$347$		−	1899.51i		−	0.293865i	−0.989147	−	0.146932i	$-0.953060\pi$
$347$		−	1899.51i		−	0.293865i	0.989147	−	0.146932i	$-0.0469400\pi$
$348$		0			0
$349$	1037.55			0.159137			0.0795683	−	0.996829i	$-0.474646\pi$
$349$	1037.55			0.159137			0.0795683	+	0.996829i	$0.474646\pi$
$350$	−4246.38	+	1893.58i	−0.648510	+	0.289188i
$351$		0			0
$352$		−	11630.5i		−	1.76110i
$353$		−	4087.08i		−	0.616242i	−0.951347	−	0.308121i	$-0.900300\pi$
$353$		−	4087.08i		−	0.616242i	0.951347	−	0.308121i	$-0.0997001\pi$
$354$		0			0
$355$	−1295.63	−	6086.75i	−0.193704	−	0.910003i
$356$	−13194.4			−1.96433
$357$		0			0
$358$			15811.8i			2.33431i
$359$	3472.67			0.510530			0.255265	−	0.966871i	$-0.417837\pi$
$359$	3472.67			0.510530			0.255265	+	0.966871i	$0.417837\pi$
$360$		0			0
$361$	−6858.37			−0.999908
$362$			5134.44i			0.745470i
$363$		0			0
$364$	9079.63			1.30742
$365$	1894.48	−	403.260i	0.271675	−	0.0578290i
$366$		0			0
$367$			8769.14i			1.24726i	0.781719	+	0.623631i	$0.214343\pi$
$367$			8769.14i			1.24726i	−0.781719	+	0.623631i	$0.785657\pi$
$368$		−	19937.3i		−	2.82419i
$369$		0			0
$370$	473.853	+	2226.11i	0.0665796	+	0.312784i
$371$	−1008.22			−0.141089
$372$		0			0
$373$		−	11368.9i		−	1.57817i	−0.614284	−	0.789085i	$-0.710555\pi$
$373$		−	11368.9i		−	1.57817i	0.614284	−	0.789085i	$-0.289445\pi$
$374$	−3758.82			−0.519690
$375$		0			0
$376$	13453.8			1.84529
$377$			14999.8i			2.04914i
$378$		0			0
$379$	12137.4			1.64500			0.822501	−	0.568764i	$-0.192578\pi$
$379$	12137.4			1.64500			0.822501	+	0.568764i	$0.192578\pi$
$380$	37.3246	+	175.347i	0.00503871	+	0.0236714i
$381$		0			0
$382$			8212.24i			1.09993i
$383$		−	9869.61i		−	1.31675i	−0.752692	−	0.658373i	$-0.771245\pi$
$383$		−	9869.61i		−	1.31675i	0.752692	−	0.658373i	$-0.228755\pi$
$384$		0			0
$385$	−1955.17	+	416.180i	−0.258817	+	0.0550921i
$386$	12242.9			1.61438
$387$		0			0
$388$			10435.5i			1.36541i
$389$	57.1166			0.00744454			0.00372227	−	0.999993i	$-0.498815\pi$
$389$	57.1166			0.00744454			0.00372227	+	0.999993i	$0.498815\pi$
$390$		0			0
$391$	−3007.88			−0.389041
$392$			3185.60i			0.410452i
$393$		0			0
$394$	1179.74			0.150849
$395$	−184.655	−	867.490i	−0.0235215	−	0.110502i
$396$		0			0
$397$		−	7436.17i		−	0.940078i	−0.882646	−	0.470039i	$-0.844240\pi$
$397$		−	7436.17i		−	0.940078i	0.882646	−	0.470039i	$-0.155760\pi$
$398$		−	19025.8i		−	2.39618i
$399$		0			0
$400$	−20957.4	+	9345.47i	−2.61968	+	1.16818i
$401$	12465.0			1.55230			0.776149	−	0.630550i	$-0.217171\pi$
$401$	12465.0			1.55230			0.776149	+	0.630550i	$0.217171\pi$
$402$		0			0
$403$		−	8282.15i		−	1.02373i
$404$	−10852.8			−1.33650
$405$		0			0
$406$	−8703.79			−1.06394
$407$			978.533i			0.119175i
$408$		0			0
$409$	1708.72			0.206578			0.103289	−	0.994651i	$-0.467063\pi$
$409$	1708.72			0.206578			0.103289	+	0.994651i	$0.467063\pi$
$410$	−23429.6	+	4987.24i	−2.82220	+	0.600737i
$411$		0			0
$412$		−	7723.54i		−	0.923571i
$413$			4753.60i			0.566366i
$414$		0			0
$415$	11411.7	−	2429.10i	1.34983	−	0.287325i
$416$	29188.5			3.44011
$417$		0			0
$418$			107.550i			0.0125848i
$419$	10618.8			1.23810			0.619050	−	0.785352i	$-0.287518\pi$
$419$	10618.8			1.23810			0.619050	+	0.785352i	$0.287518\pi$
$420$		0			0
$421$	13273.5			1.53661			0.768304	−	0.640085i	$-0.221101\pi$
$421$	13273.5			1.53661			0.768304	+	0.640085i	$0.221101\pi$
$422$		−	22216.6i		−	2.56277i
$423$		0			0
$424$	−9363.80			−1.07251
$425$	1409.92	+	3161.78i	0.160921	+	0.360868i
$426$		0			0
$427$			4023.02i			0.455943i
$428$			27946.9i			3.15622i
$429$		0			0
$430$	−2138.47	−	10046.3i	−0.239829	−	1.12669i
$431$	−7918.20			−0.884934			−0.442467	−	0.896785i	$-0.645897\pi$
$431$	−7918.20			−0.884934			−0.442467	+	0.896785i	$0.645897\pi$
$432$		0			0
$433$			4433.34i			0.492038i	0.969265	+	0.246019i	$0.0791226\pi$
$433$			4433.34i			0.492038i	−0.969265	+	0.246019i	$0.920877\pi$
$434$	4805.81			0.531535
$435$		0			0
$436$	−7903.40			−0.868129
$437$			86.0637i			0.00942102i
$438$		0			0
$439$	−12958.4			−1.40882			−0.704408	−	0.709796i	$-0.748787\pi$
$439$	−12958.4			−1.40882			−0.704408	+	0.709796i	$0.748787\pi$
$440$	−18158.6	+	3865.26i	−1.96745	+	0.418793i
$441$		0			0
$442$		−	9433.34i		−	1.01515i
$443$			12040.4i			1.29133i	0.763621	+	0.645664i	$0.223420\pi$
$443$			12040.4i			1.29133i	−0.763621	+	0.645664i	$0.776580\pi$
$444$		0			0
$445$	1517.80	+	7130.50i	0.161687	+	0.759591i
$446$	−12548.3			−1.33224
$447$		0			0
$448$			6656.84i			0.702023i
$449$	11586.3			1.21780			0.608899	−	0.793247i	$-0.291611\pi$
$449$	11586.3			1.21780			0.608899	+	0.793247i	$0.291611\pi$
$450$		0			0
$451$	−10298.9			−1.07529
$452$			33249.1i			3.45997i
$453$		0			0
$454$	−3118.46			−0.322371
$455$	−1044.47	−	4906.80i	−0.107616	−	0.505570i
$456$		0			0
$457$		−	9734.34i		−	0.996396i	−0.867063	−	0.498198i	$-0.833995\pi$
$457$		−	9734.34i		−	0.996396i	0.867063	−	0.498198i	$-0.166005\pi$
$458$			24546.7i			2.50435i
$459$		0			0
$460$	−24032.0	+	5115.48i	−2.43587	+	0.518501i
$461$	1343.41			0.135724			0.0678621	−	0.997695i	$-0.478382\pi$
$461$	1343.41			0.135724			0.0678621	+	0.997695i	$0.478382\pi$
$462$		0			0
$463$		−	6613.72i		−	0.663857i	−0.943305	−	0.331929i	$-0.892301\pi$
$463$		−	6613.72i		−	0.663857i	0.943305	−	0.331929i	$-0.107699\pi$
$464$	−42956.3			−4.29784
$465$		0			0
$466$	−27208.3			−2.70472
$467$		−	14688.9i		−	1.45551i	−0.685838	−	0.727755i	$-0.740564\pi$
$467$		−	14688.9i		−	1.45551i	0.685838	−	0.727755i	$-0.259436\pi$
$468$		0			0
$469$	−3609.48			−0.355374
$470$	−2559.60	−	12024.8i	−0.251204	−	1.18013i
$471$		0			0
$472$			44148.9i			4.30534i
$473$		−	4416.07i		−	0.429283i
$474$		0			0
$475$	90.4673	−	40.3418i	0.00873879	−	0.00389686i
$476$	3922.88			0.377741
$477$		0			0
$478$			5991.22i			0.573288i
$479$	−15298.6			−1.45931			−0.729657	−	0.683813i	$-0.760320\pi$
$479$	−15298.6			−1.45931			−0.729657	+	0.683813i	$0.760320\pi$
$480$		0			0
$481$	−2455.78			−0.232794
$482$		−	18861.0i		−	1.78235i
$483$		0			0
$484$	13731.6			1.28960
$485$	5639.52	−	1200.43i	0.527994	−	0.112389i
$486$		0			0
$487$			9653.80i			0.898266i	0.893465	+	0.449133i	$0.148267\pi$
$487$			9653.80i			0.898266i	−0.893465	+	0.449133i	$0.851733\pi$
$488$			37363.7i			3.46593i
$489$		0			0
$490$	2847.23	−	606.064i	0.262499	−	0.0558758i
$491$	−20142.6			−1.85137			−0.925684	−	0.378297i	$-0.876510\pi$
$491$	−20142.6			−1.85137			−0.925684	+	0.378297i	$0.876510\pi$
$492$		0			0
$493$			6480.69i			0.592039i
$494$	−269.914			−0.0245830
$495$		0			0
$496$	23718.4			2.14715
$497$			3896.29i			0.351655i
$498$		0			0
$499$	1309.29			0.117459			0.0587293	−	0.998274i	$-0.481295\pi$
$499$	1309.29			0.117459			0.0587293	+	0.998274i	$0.481295\pi$
$500$	16642.1	+	22863.8i	1.48851	+	2.04500i
$501$		0			0
$502$		−	25065.5i		−	2.22855i
$503$		−	2186.17i		−	0.193791i	−0.995295	−	0.0968953i	$-0.969109\pi$
$503$		−	2186.17i		−	0.193791i	0.995295	−	0.0968953i	$-0.0308912\pi$
$504$		0			0
$505$	1248.44	+	5865.06i	0.110010	+	0.516815i
$506$	−14740.2			−1.29502
$507$		0			0
$508$		−	3885.76i		−	0.339375i
$509$	3591.11			0.312718			0.156359	−	0.987700i	$-0.450024\pi$
$509$	3591.11			0.312718			0.156359	+	0.987700i	$0.450024\pi$
$510$		0			0
$511$	−1212.70			−0.104984
$512$		−	11886.4i		−	1.02600i
$513$		0			0
$514$	−33265.2			−2.85460
$515$	−4173.94	+	888.470i	−0.357138	+	0.0760207i
$516$		0			0
$517$		−	5285.73i		−	0.449644i
$518$		−	1424.99i		−	0.120870i
$519$		0			0
$520$	−9700.46	−	45571.8i	−0.818064	−	3.84318i
$521$	8605.34			0.723621			0.361811	−	0.932252i	$-0.382159\pi$
$521$	8605.34			0.723621			0.361811	+	0.932252i	$0.382159\pi$
$522$		0			0
$523$			22536.4i			1.88423i	0.335297	+	0.942113i	$0.391163\pi$
$523$			22536.4i			1.88423i	−0.335297	+	0.942113i	$0.608837\pi$
$524$	42147.3			3.51377
$525$		0			0
$526$	−30569.7			−2.53403
$527$		−	3578.32i		−	0.295776i
$528$		0			0
$529$	371.637			0.0305447
$530$	1781.47	+	8369.17i	0.146004	+	0.685912i
$531$		0			0
$532$		−	112.244i		−	0.00914738i
$533$		−	25846.7i		−	2.10046i
$534$		0			0
$535$	15103.0	−	3214.84i	1.22049	−	0.259794i
$536$	−33522.9			−2.70144
$537$		0			0
$538$		−	37510.2i		−	3.00591i
$539$	1251.56			0.100015
$540$		0			0
$541$	8782.44			0.697942			0.348971	−	0.937134i	$-0.386531\pi$
$541$	8782.44			0.697942			0.348971	+	0.937134i	$0.386531\pi$
$542$		−	45349.5i		−	3.59396i
$543$		0			0
$544$	12611.0			0.993917
$545$	909.161	+	4271.15i	0.0714572	+	0.335699i
$546$		0			0
$547$		−	22593.0i		−	1.76601i	−0.469366	−	0.883004i	$-0.655518\pi$
$547$		−	22593.0i		−	1.76601i	0.469366	−	0.883004i	$-0.344482\pi$
$548$			1581.78i			0.123304i
$549$		0			0
$550$	6909.37	+	15494.4i	0.535667	+	1.20124i
$551$	185.430			0.0143368
$552$		0			0
$553$			555.303i			0.0427014i
$554$	−6981.84			−0.535433
$555$		0			0
$556$	28200.8			2.15104
$557$		−	7264.13i		−	0.552587i	−0.961073	−	0.276294i	$-0.910894\pi$
$557$		−	7264.13i		−	0.552587i	0.961073	−	0.276294i	$-0.0891062\pi$
$558$		0			0
$559$	11082.8			0.838555
$560$	14052.1	−	2991.14i	1.06037	−	0.225712i
$561$		0			0
$562$			1313.69i			0.0986027i
$563$		−	3851.42i		−	0.288309i	−0.989555	−	0.144154i	$-0.953954\pi$
$563$		−	3851.42i		−	0.288309i	0.989555	−	0.144154i	$-0.0460462\pi$
$564$		0			0
$565$	17968.4	−	3824.78i	1.33794	−	0.284796i
$566$	48220.9			3.58105
$567$		0			0
$568$			36186.7i			2.67317i
$569$	16580.3			1.22158			0.610792	−	0.791791i	$-0.290851\pi$
$569$	16580.3			1.22158			0.610792	+	0.791791i	$0.290851\pi$
$570$		0			0
$571$	6385.86			0.468021			0.234010	−	0.972234i	$-0.424815\pi$
$571$	6385.86			0.468021			0.234010	+	0.972234i	$0.424815\pi$
$572$		−	33130.2i		−	2.42175i
$573$		0			0
$574$	14997.9			1.09059
$575$	5529.00	+	12398.9i	0.401001	+	0.899252i
$576$		0			0
$577$			11059.1i			0.797912i	0.916970	+	0.398956i	$0.130627\pi$
$577$			11059.1i			0.797912i	−0.916970	+	0.398956i	$0.869373\pi$
$578$			22030.3i			1.58536i
$579$		0			0
$580$	11021.7	+	51778.7i	0.789052	+	3.70689i
$581$	−7304.92			−0.521617
$582$		0			0
$583$			3678.84i			0.261341i
$584$	−11263.0			−0.798055
$585$		0			0
$586$	−14563.2			−1.02662
$587$		−	7871.25i		−	0.553461i	−0.960948	−	0.276730i	$-0.910749\pi$
$587$		−	7871.25i		−	0.553461i	0.960948	−	0.276730i	$-0.0892509\pi$
$588$		0			0
$589$	−102.386			−0.00716253
$590$	39459.4	−	8399.37i	2.75342	−	0.586096i
$591$		0			0
$592$		−	7032.85i		−	0.488258i
$593$			2018.06i			0.139750i	0.997556	+	0.0698750i	$0.0222600\pi$
$593$			2018.06i			0.139750i	−0.997556	+	0.0698750i	$0.977740\pi$
$594$		0			0
$595$	−451.264	−	2120.00i	−0.0310925	−	0.146069i
$596$	−657.640			−0.0451980
$597$		0			0
$598$		−	36992.8i		−	2.52968i
$599$	−1356.67			−0.0925409			−0.0462705	−	0.998929i	$-0.514734\pi$
$599$	−1356.67			−0.0925409			−0.0462705	+	0.998929i	$0.514734\pi$
$600$		0			0
$601$	11178.7			0.758715			0.379358	−	0.925250i	$-0.376145\pi$
$601$	11178.7			0.758715			0.379358	+	0.925250i	$0.376145\pi$
$602$			6430.92i			0.435390i
$603$		0			0
$604$	−9444.91			−0.636272
$605$	−1579.60	−	7420.82i	−0.106149	−	0.498676i
$606$		0			0
$607$		−	9404.67i		−	0.628870i	−0.949279	−	0.314435i	$-0.898185\pi$
$607$		−	9404.67i		−	0.628870i	0.949279	−	0.314435i	$-0.101815\pi$
$608$		−	360.835i		−	0.0240687i
$609$		0			0
$610$	33394.9	−	7108.47i	2.21659	−	0.471825i
$611$	13265.3			0.878328
$612$		0			0
$613$		−	18938.0i		−	1.24779i	−0.781507	−	0.623897i	$-0.785548\pi$
$613$		−	18938.0i		−	1.24779i	0.781507	−	0.623897i	$-0.214452\pi$
$614$	−37118.4			−2.43970
$615$		0			0
$616$	11623.8			0.760286
$617$			17716.9i			1.15600i	0.816036	+	0.578001i	$0.196167\pi$
$617$			17716.9i			1.15600i	−0.816036	+	0.578001i	$0.803833\pi$
$618$		0			0
$619$	6240.33			0.405202			0.202601	−	0.979261i	$-0.435061\pi$
$619$	6240.33			0.405202			0.202601	+	0.979261i	$0.435061\pi$
$620$	−6085.63	−	28589.7i	−0.394202	−	1.85192i
$621$		0			0
$622$			1896.13i			0.122231i
$623$		−	4564.42i		−	0.293531i
$624$		0			0
$625$	10441.6	−	11623.8i	0.668264	−	0.743924i
$626$	−35230.2			−2.24933
$627$		0			0
$628$			33863.2i			2.15174i
$629$	−1061.03			−0.0672589
$630$		0			0
$631$	−25887.7			−1.63323			−0.816617	−	0.577179i	$-0.804153\pi$
$631$	−25887.7			−1.63323			−0.816617	+	0.577179i	$0.804153\pi$
$632$			5157.37i			0.324603i
$633$		0			0
$634$	−13252.5			−0.830165
$635$	−2099.94	+	446.994i	−0.131234	+	0.0279345i
$636$		0			0
$637$			3140.97i			0.195369i
$638$			31758.8i			1.97076i
$639$		0			0
$640$	15423.0	−	3282.95i	0.952573	−	0.202766i
$641$	−18798.4			−1.15834			−0.579168	−	0.815208i	$-0.696622\pi$
$641$	−18798.4			−1.15834			−0.579168	+	0.815208i	$0.696622\pi$
$642$		0			0
$643$		−	2287.70i		−	0.140308i	−0.997536	−	0.0701541i	$-0.977651\pi$
$643$		−	2287.70i		−	0.140308i	0.997536	−	0.0701541i	$-0.0223491\pi$
$644$	15383.5			0.941298
$645$		0			0
$646$	−116.617			−0.00710253
$647$			1769.31i			0.107510i	0.998554	+	0.0537548i	$0.0171189\pi$
$647$			1769.31i			0.107510i	−0.998554	+	0.0537548i	$0.982881\pi$
$648$		0			0
$649$	17345.2			1.04909
$650$	−38885.6	+	17340.1i	−2.34649	+	1.04636i
$651$		0			0
$652$		−	37826.4i		−	2.27208i
$653$		−	3891.48i		−	0.233209i	−0.993178	−	0.116604i	$-0.962799\pi$
$653$		−	3891.48i		−	0.233209i	0.993178	−	0.116604i	$-0.0372010\pi$
$654$		0			0
$655$	−4848.38	−	22777.2i	−0.289224	−	1.35875i
$656$	74019.8			4.40547
$657$		0			0
$658$			7697.37i			0.456040i
$659$	20097.6			1.18800			0.594001	−	0.804465i	$-0.297548\pi$
$659$	20097.6			1.18800			0.594001	+	0.804465i	$0.297548\pi$
$660$		0			0
$661$	−27167.9			−1.59865			−0.799326	−	0.600898i	$-0.794810\pi$
$661$	−27167.9			−1.59865			−0.799326	+	0.600898i	$0.794810\pi$
$662$		−	24881.0i		−	1.46077i
$663$		0			0
$664$	−67844.3			−3.96516
$665$	−60.6589	+	12.9119i	−0.00353722	+	0.000752937i
$666$		0			0
$667$			25414.0i			1.47531i
$668$		−	941.430i		−	0.0545285i
$669$		0			0
$670$	6377.77	+	29962.1i	0.367753	+	1.72767i
$671$	14679.4			0.844548
$672$		0			0
$673$			25909.7i			1.48402i	0.670389	+	0.742010i	$0.266127\pi$
$673$			25909.7i			1.48402i	−0.670389	+	0.742010i	$0.733873\pi$
$674$	−19111.1			−1.09218
$675$		0			0
$676$	38689.1			2.20125
$677$			4359.22i			0.247472i	0.992315	+	0.123736i	$0.0394875\pi$
$677$			4359.22i			0.247472i	−0.992315	+	0.123736i	$0.960512\pi$
$678$		0			0
$679$	−3610.00			−0.204034
$680$	−4191.11	−	19689.4i	−0.236355	−	1.11037i
$681$		0			0
$682$		−	17535.7i		−	0.984569i
$683$		−	29721.9i		−	1.66512i	−0.553937	−	0.832559i	$-0.686875\pi$
$683$		−	29721.9i		−	1.66512i	0.553937	−	0.832559i	$-0.313125\pi$
$684$		0			0
$685$	854.826	−	181.959i	0.0476806	−	0.0101493i
$686$	−1822.58			−0.101438
$687$		0			0
$688$			31738.9i			1.75877i
$689$	−9232.60			−0.510499
$690$		0			0
$691$	−4929.13			−0.271365			−0.135682	−	0.990752i	$-0.543323\pi$
$691$	−4929.13			−0.271365			−0.135682	+	0.990752i	$0.543323\pi$
$692$			50512.0i			2.77482i
$693$		0			0
$694$	10093.4			0.552073
$695$	−3244.05	−	15240.2i	−0.177056	−	0.831791i
$696$		0			0
$697$		−	11167.1i		−	0.606866i
$698$			5513.18i			0.298964i
$699$		0			0
$700$	−7210.93	−	16170.7i	−0.389354	−	0.873133i
$701$	19358.8			1.04304			0.521520	−	0.853239i	$-0.325365\pi$
$701$	19358.8			1.04304			0.521520	+	0.853239i	$0.325365\pi$
$702$		0			0
$703$			30.3589i			0.00162874i
$704$	24289.8			1.30036
$705$		0			0
$706$	21717.3			1.15771
$707$		−	3754.38i		−	0.199714i
$708$		0			0
$709$	17186.3			0.910359			0.455180	−	0.890400i	$-0.349575\pi$
$709$	17186.3			0.910359			0.455180	+	0.890400i	$0.349575\pi$
$710$	32342.9	−	6884.54i	1.70959	−	0.363905i
$711$		0			0
$712$		−	42391.9i		−	2.23133i
$713$		−	14032.4i		−	0.737049i
$714$		0			0
$715$	−17904.2	+	3811.10i	−0.936473	+	0.199339i
$716$	−60213.1			−3.14284
$717$		0			0
$718$			18452.6i			0.959113i
$719$	−15107.3			−0.783598			−0.391799	−	0.920051i	$-0.628147\pi$
$719$	−15107.3			−0.783598			−0.391799	+	0.920051i	$0.628147\pi$
$720$		0			0
$721$	2671.85			0.138010
$722$		−	36443.0i		−	1.87849i
$723$		0			0
$724$	−19552.5			−1.00368
$725$	26714.3	−	11912.6i	1.36848	−	0.610240i
$726$		0			0
$727$			15840.9i			0.808124i	0.914732	+	0.404062i	$0.132402\pi$
$727$			15840.9i			0.808124i	−0.914732	+	0.404062i	$0.867598\pi$
$728$			29171.7i			1.48513i
$729$		0			0
$730$	2142.79	+	10066.6i	0.108641	+	0.510385i
$731$	4788.35			0.242276
$732$		0			0
$733$		−	27639.7i		−	1.39276i	−0.717671	−	0.696382i	$-0.754792\pi$
$733$		−	27639.7i		−	1.39276i	0.717671	−	0.696382i	$-0.245208\pi$
$734$	−46596.2			−2.34318
$735$		0			0
$736$	49453.8			2.47675
$737$			13170.4i			0.658263i
$738$		0			0
$739$	−34874.4			−1.73596			−0.867982	−	0.496597i	$-0.834583\pi$
$739$	−34874.4			−1.73596			−0.867982	+	0.496597i	$0.834583\pi$
$740$	−8477.27	+	1804.48i	−0.421123	+	0.0896406i
$741$		0			0
$742$		−	5357.32i		−	0.265059i
$743$			27686.7i			1.36706i	0.729922	+	0.683530i	$0.239556\pi$
$743$			27686.7i			1.36706i	−0.729922	+	0.683530i	$0.760444\pi$
$744$		0			0
$745$	75.6510	+	355.401i	0.00372032	+	0.0174777i
$746$	60410.2			2.96485
$747$		0			0
$748$		−	14314.0i		−	0.699694i
$749$	−9667.83			−0.471635
$750$		0			0
$751$	4806.85			0.233561			0.116781	−	0.993158i	$-0.462743\pi$
$751$	4806.85			0.233561			0.116781	+	0.993158i	$0.462743\pi$
$752$			37989.3i			1.84219i
$753$		0			0
$754$	−79703.6			−3.84965
$755$	1086.49	+	5104.21i	0.0523726	+	0.246041i
$756$		0			0
$757$			40166.6i			1.92851i	0.264983	+	0.964253i	$0.414634\pi$
$757$			40166.6i			1.92851i	−0.264983	+	0.964253i	$0.585366\pi$
$758$			64493.9i			3.09040i
$759$		0			0
$760$	−563.368	+	119.919i	−0.0268888	+	0.00572359i
$761$	−25912.3			−1.23432			−0.617162	−	0.786836i	$-0.711718\pi$
$761$	−25912.3			−1.23432			−0.617162	+	0.786836i	$0.711718\pi$
$762$		0			0
$763$		−	2734.07i		−	0.129725i
$764$	−31273.1			−1.48092
$765$		0			0
$766$	52443.7			2.47372
$767$			43530.4i			2.04927i
$768$		0			0
$769$	23231.3			1.08939			0.544697	−	0.838633i	$-0.316645\pi$
$769$	23231.3			1.08939			0.544697	+	0.838633i	$0.316645\pi$
$770$	−2211.44	−	10389.1i	−0.103499	−	0.486230i
$771$		0			0
$772$			46622.4i			2.17354i
$773$		−	836.306i		−	0.0389131i	−0.999811	−	0.0194566i	$-0.993806\pi$
$773$		−	836.306i		−	0.0389131i	0.999811	−	0.0194566i	$-0.00619360\pi$
$774$		0			0
$775$	−14750.4	+	6577.58i	−0.683675	+	0.304869i
$776$	−33527.8			−1.55100
$777$		0			0
$778$			303.498i			0.0139858i
$779$	−319.523			−0.0146959
$780$		0			0
$781$	14217.0			0.651374
$782$		−	15982.8i		−	0.730875i
$783$		0			0
$784$	−8995.11			−0.409763
$785$	18300.3	−	3895.43i	0.832060	−	0.177113i
$786$		0			0
$787$		−	31945.6i		−	1.44693i	−0.690359	−	0.723467i	$-0.742547\pi$
$787$		−	31945.6i		−	1.44693i	0.690359	−	0.723467i	$-0.257453\pi$
$788$			4492.58i			0.203098i

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:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	315.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+5.31366i q^{2} -20.2350 q^{4} +(2.32771 + 10.9353i) q^{5} -7.00000i q^{7} -65.0123i q^{8} +(-58.1067 + 12.3686i) q^{10} -25.5420 q^{11} -64.1014i q^{13} +37.1956 q^{14} +183.574 q^{16} -27.6952i q^{17} -0.792436 q^{19} +(-47.1011 - 221.276i) q^{20} -135.721i q^{22} -108.606i q^{23} +(-114.164 + 50.9086i) q^{25} +340.613 q^{26} +141.645i q^{28} -234.000 q^{29} +129.204 q^{31} +455.349i q^{32} +147.163 q^{34} +(76.5474 - 16.2940i) q^{35} -38.3108i q^{37} -4.21073i q^{38} +(710.932 - 151.330i) q^{40} +403.216 q^{41} +172.895i q^{43} +516.840 q^{44} +577.097 q^{46} +206.943i q^{47} -49.0000 q^{49} +(-270.511 - 606.626i) q^{50} +1297.09i q^{52} -144.031i q^{53} +(-59.4542 - 279.310i) q^{55} -455.086 q^{56} -1243.40i q^{58} -679.086 q^{59} -574.717 q^{61} +686.544i q^{62} -950.977 q^{64} +(700.971 - 149.209i) q^{65} -515.640i q^{67} +560.411i q^{68} +(86.5805 + 406.747i) q^{70} -556.612 q^{71} -173.243i q^{73} +203.571 q^{74} +16.0349 q^{76} +178.794i q^{77} -79.3290 q^{79} +(427.306 + 2007.44i) q^{80} +2142.55i q^{82} -1043.56i q^{83} +(302.856 - 64.4663i) q^{85} -918.703 q^{86} +1660.54i q^{88} +652.060 q^{89} -448.710 q^{91} +2197.65i q^{92} -1099.62 q^{94} +(-1.84456 - 8.66556i) q^{95} -515.714i q^{97} -260.369i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 36 q^{4} - 6 q^{5} - 16 q^{10} - 84 q^{11} + 56 q^{14} + 148 q^{16} + 72 q^{19} + 68 q^{20} - 362 q^{25} + 620 q^{26} - 88 q^{29} + 120 q^{31} + 964 q^{34} + 28 q^{35} + 1396 q^{40} + 852 q^{41}+ \cdots + 1628 q^{95}+O(q^{100}) \) 10 * q - 36 * q^4 - 6 * q^5 - 16 * q^10 - 84 * q^11 + 56 * q^14 + 148 * q^16 + 72 * q^19 + 68 * q^20 - 362 * q^25 + 620 * q^26 - 88 * q^29 + 120 * q^31 + 964 * q^34 + 28 * q^35 + 1396 * q^40 + 852 * q^41 + 1424 * q^44 + 176 * q^46 - 490 * q^49 - 1644 * q^50 - 1136 * q^55 - 588 * q^56 - 864 * q^59 - 884 * q^61 - 2724 * q^64 - 520 * q^65 + 756 * q^70 - 880 * q^71 - 3024 * q^74 - 2672 * q^76 - 3580 * q^79 + 2316 * q^80 + 1572 * q^85 - 2432 * q^86 + 1492 * q^89 + 476 * q^91 + 1652 * q^94 + 1628 * q^95

Introduction

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