LMFDB - Embedded newform 252.6.k.f.109.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [252,6,Mod(37,252)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("252.37"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(252, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	252.k (of order $3$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$40.4167225929$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 703x^{6} + 2770x^{5} + 427565x^{4} + 718170x^{3} + 42175732x^{2} - 40929504x + 3559792896 $ x^8 - 2x^7 + 703x^6 + 2770x^5 + 427565x^4 + 718170x^3 + 42175732x^2 - 40929504*x + 3559792896
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{3}\cdot 7 $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			109.3
Root			$-5.49618 + 9.51967i$ of defining polynomial
Character	$\chi$	$=$	252.109
Dual form			252.6.k.f.37.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(23.0577 + 39.9371i) q^{5} +(112.271 + 64.8240i) q^{7} +(-315.582 + 546.605i) q^{11} -1079.22 q^{13} +(80.5778 - 139.565i) q^{17} +(-588.428 - 1019.19i) q^{19} +(1081.73 + 1873.61i) q^{23} +(499.186 - 864.615i) q^{25} +4492.01 q^{29} +(-159.130 + 275.621i) q^{31} +(-0.166415 + 5978.48i) q^{35} +(-7593.41 - 13152.2i) q^{37} -20587.2 q^{41} -455.118 q^{43} +(-10381.4 - 17981.1i) q^{47} +(8402.69 + 14555.8i) q^{49} +(-9650.03 + 16714.3i) q^{53} -29106.4 q^{55} +(3184.15 - 5515.11i) q^{59} +(24572.6 + 42560.9i) q^{61} +(-24884.4 - 43101.1i) q^{65} +(-17027.0 + 29491.7i) q^{67} -62962.4 q^{71} +(4433.88 - 7679.70i) q^{73} +(-70864.0 + 40910.7i) q^{77} +(-17206.6 - 29802.7i) q^{79} +7041.42 q^{83} +7431.75 q^{85} +(-10121.4 - 17530.7i) q^{89} +(-121166. - 69959.7i) q^{91} +(27135.6 - 47000.2i) q^{95} +54066.6 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 42 q^{7} + 462 q^{11} - 1204 q^{13} - 228 q^{17} + 358 q^{19} + 2148 q^{23} - 5454 q^{25} + 11064 q^{29} + 830 q^{31} - 7692 q^{35} - 3914 q^{37} + 16632 q^{41} - 29036 q^{43} - 41700 q^{47} + 41876 q^{49}+ \cdots - 433356 q^{97}+O(q^{100}) $ 8 * q - 42 * q^7 + 462 * q^11 - 1204 * q^13 - 228 * q^17 + 358 * q^19 + 2148 * q^23 - 5454 * q^25 + 11064 * q^29 + 830 * q^31 - 7692 * q^35 - 3914 * q^37 + 16632 * q^41 - 29036 * q^43 - 41700 * q^47 + 41876 * q^49 - 22164 * q^53 + 7784 * q^55 - 32886 * q^59 + 83732 * q^61 + 93192 * q^65 - 80034 * q^67 - 89544 * q^71 - 22470 * q^73 - 138732 * q^77 - 75286 * q^79 + 34836 * q^83 + 278504 * q^85 - 28944 * q^89 - 12058 * q^91 + 144120 * q^95 - 433356 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$.

$n$	$29$	$73$	$127$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	23.0577	+	39.9371i	0.412469	+	0.714416i	0.995159	−	0.0982777i	$-0.0313333\pi$
$5$	23.0577	+	39.9371i	0.412469	+	0.714416i	−0.582691	+	0.812694i	$0.698000\pi$
$6$		0			0
$7$	112.271	+	64.8240i	0.866011	+	0.500024i
$7$	112.271	+	64.8240i	0.866011	+	0.500024i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−315.582	+	546.605i	−0.786378	+	1.36205i	0.141795	+	0.989896i	$0.454713\pi$
$11$	−315.582	+	546.605i	−0.786378	+	1.36205i	−0.928173	+	0.372150i	$0.878621\pi$
$12$		0			0
$13$	−1079.22			−1.77114			−0.885571	−	0.464503i	$-0.846233\pi$
$13$	−1079.22			−1.77114			−0.885571	+	0.464503i	$0.846233\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	80.5778	−	139.565i	0.0676228	−	0.117126i	−0.830232	−	0.557419i	$-0.811792\pi$
$17$	80.5778	−	139.565i	0.0676228	−	0.117126i	0.897854	+	0.440292i	$0.145125\pi$
$18$		0			0
$19$	−588.428	−	1019.19i	−0.373946	−	0.647694i	0.616222	−	0.787572i	$-0.288662\pi$
$19$	−588.428	−	1019.19i	−0.373946	−	0.647694i	−0.990169	+	0.139878i	$0.955329\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1081.73	+	1873.61i	0.426382	+	0.738516i	0.996548	−	0.0830136i	$-0.0264545\pi$
$23$	1081.73	+	1873.61i	0.426382	+	0.738516i	−0.570166	+	0.821529i	$0.693121\pi$
$24$		0			0
$25$	499.186	−	864.615i	0.159739	−	0.276677i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4492.01			0.991850			0.495925	−	0.868365i	$-0.334829\pi$
$29$	4492.01			0.991850			0.495925	+	0.868365i	$0.334829\pi$
$30$		0			0
$31$	−159.130	+	275.621i	−0.0297405	+	0.0515120i	−0.880513	−	0.474023i	$-0.842801\pi$
$31$	−159.130	+	275.621i	−0.0297405	+	0.0515120i	0.850772	+	0.525535i	$0.176135\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.166415	+	5978.48i	−2.29627e−5	+	0.824937i
$36$		0			0
$37$	−7593.41	−	13152.2i	−0.911869	−	1.57940i	−0.811422	−	0.584461i	$-0.801306\pi$
$37$	−7593.41	−	13152.2i	−0.911869	−	1.57940i	−0.100447	−	0.994942i	$-0.532027\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−20587.2			−1.91266			−0.956330	−	0.292289i	$-0.905583\pi$
$41$	−20587.2			−1.91266			−0.956330	+	0.292289i	$0.905583\pi$
$42$		0			0
$43$	−455.118			−0.0375364			−0.0187682	−	0.999824i	$-0.505974\pi$
$43$	−455.118			−0.0375364			−0.0187682	+	0.999824i	$0.505974\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−10381.4	−	17981.1i	−0.685504	−	1.18733i	−0.973278	−	0.229630i	$-0.926248\pi$
$47$	−10381.4	−	17981.1i	−0.685504	−	1.18733i	0.287774	−	0.957698i	$-0.407085\pi$
$48$		0			0
$49$	8402.69	+	14555.8i	0.499952	+	0.866053i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−9650.03	+	16714.3i	−0.471888	+	0.817334i	−0.999483	−	0.0321622i	$-0.989761\pi$
$53$	−9650.03	+	16714.3i	−0.471888	+	0.817334i	0.527595	+	0.849496i	$0.323094\pi$
$54$		0			0
$55$	−29106.4			−1.29742
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3184.15	−	5515.11i	0.119087	−	0.206264i	−0.800319	−	0.599574i	$-0.795337\pi$
$59$	3184.15	−	5515.11i	0.119087	−	0.206264i	0.919406	+	0.393310i	$0.128670\pi$
$60$		0			0
$61$	24572.6	+	42560.9i	0.845524	+	1.46449i	0.885165	+	0.465276i	$0.154045\pi$
$61$	24572.6	+	42560.9i	0.845524	+	1.46449i	−0.0396416	+	0.999214i	$0.512622\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−24884.4	−	43101.1i	−0.730541	−	1.26533i
$66$		0			0
$67$	−17027.0	+	29491.7i	−0.463395	+	0.802624i	−0.999128	−	0.0417639i	$-0.986702\pi$
$67$	−17027.0	+	29491.7i	−0.463395	+	0.802624i	0.535732	+	0.844388i	$0.320036\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−62962.4			−1.48230			−0.741149	−	0.671341i	$-0.765719\pi$
$71$	−62962.4			−1.48230			−0.741149	+	0.671341i	$0.765719\pi$
$72$		0			0
$73$	4433.88	−	7679.70i	0.0973815	−	0.168670i	−0.813219	−	0.581958i	$-0.802287\pi$
$73$	4433.88	−	7679.70i	0.0973815	−	0.168670i	0.910600	+	0.413289i	$0.135620\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−70864.0	+	40910.7i	−1.36207	+	0.786340i
$78$		0			0
$79$	−17206.6	−	29802.7i	−0.310190	−	0.537265i	0.668213	−	0.743970i	$-0.267059\pi$
$79$	−17206.6	−	29802.7i	−0.310190	−	0.537265i	−0.978403	+	0.206705i	$0.933726\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	7041.42			0.112193			0.0560964	−	0.998425i	$-0.482135\pi$
$83$	7041.42			0.112193			0.0560964	+	0.998425i	$0.482135\pi$
$84$		0			0
$85$	7431.75			0.111569
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−10121.4	−	17530.7i	−0.135445	−	0.234598i	0.790322	−	0.612692i	$-0.209913\pi$
$89$	−10121.4	−	17530.7i	−0.135445	−	0.234598i	−0.925768	+	0.378093i	$0.876580\pi$
$90$		0			0
$91$	−121166.	−	69959.7i	−1.53383	−	0.885614i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	27135.6	−	47000.2i	0.308482	−	0.534307i
$96$		0			0
$97$	54066.6			0.583444			0.291722	−	0.956503i	$-0.405772\pi$
$97$	54066.6			0.583444			0.291722	+	0.956503i	$0.405772\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−45743.2	+	79229.5i	−0.446193	+	0.772829i	−0.998134	−	0.0610540i	$-0.980554\pi$
$101$	−45743.2	+	79229.5i	−0.446193	+	0.772829i	0.551942	+	0.833883i	$0.313887\pi$
$102$		0			0
$103$	−37690.7	−	65282.2i	−0.350059	−	0.606320i	0.636201	−	0.771524i	$-0.280505\pi$
$103$	−37690.7	−	65282.2i	−0.350059	−	0.606320i	−0.986259	+	0.165204i	$0.947172\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	76066.3	+	131751.i	0.642293	+	1.11248i	0.984920	+	0.173012i	$0.0553500\pi$
$107$	76066.3	+	131751.i	0.642293	+	1.11248i	−0.342627	+	0.939472i	$0.611317\pi$
$108$		0			0
$109$	−38336.0	+	66399.8i	−0.309058	+	0.535304i	−0.978157	−	0.207869i	$-0.933347\pi$
$109$	−38336.0	+	66399.8i	−0.309058	+	0.535304i	0.669099	+	0.743174i	$0.266680\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−228515.			−1.68352			−0.841760	−	0.539852i	$-0.818480\pi$
$113$	−228515.			−1.68352			−0.841760	+	0.539852i	$0.818480\pi$
$114$		0			0
$115$	−49884.4	+	86402.3i	−0.351739	+	0.609229i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	18093.7	−	10445.7i	0.117128	−	0.0676195i
$120$		0			0
$121$	−118659.	−	205524.i	−0.736780	−	1.27614i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	190151.			1.08849
$126$		0			0
$127$	122111.			0.671809			0.335905	−	0.941896i	$-0.390958\pi$
$127$	122111.			0.671809			0.335905	+	0.941896i	$0.390958\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	37897.7	+	65640.7i	0.192945	+	0.334191i	0.946225	−	0.323509i	$-0.104863\pi$
$131$	37897.7	+	65640.7i	0.192945	+	0.334191i	−0.753280	+	0.657700i	$0.771529\pi$
$132$		0			0
$133$	4.24689	−	152570.i	2.08181e−5	−	0.747892i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−120806.	+	209242.i	−0.549903	+	0.952460i	0.448378	+	0.893844i	$0.352002\pi$
$137$	−120806.	+	209242.i	−0.549903	+	0.952460i	−0.998281	+	0.0586154i	$0.981331\pi$
$138$		0			0
$139$	125657.			0.551634			0.275817	−	0.961210i	$-0.411052\pi$
$139$	125657.			0.551634			0.275817	+	0.961210i	$0.411052\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	340584.	−	589910.i	1.39279	−	2.41238i
$144$		0			0
$145$	103575.	+	179398.i	0.409107	+	0.708594i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	18023.3	+	31217.2i	0.0665071	+	0.115194i	0.897362	−	0.441296i	$-0.145481\pi$
$149$	18023.3	+	31217.2i	0.0665071	+	0.115194i	−0.830854	+	0.556490i	$0.812148\pi$
$150$		0			0
$151$	−75334.0	+	130482.i	−0.268874	+	0.465703i	−0.968571	−	0.248736i	$-0.919985\pi$
$151$	−75334.0	+	130482.i	−0.268874	+	0.465703i	0.699697	+	0.714439i	$0.253318\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−14676.7			−0.0490680
$156$		0			0
$157$	−213207.	+	369285.i	−0.690323	+	1.19567i	0.281409	+	0.959588i	$0.409198\pi$
$157$	−213207.	+	369285.i	−0.690323	+	1.19567i	−0.971732	+	0.236086i	$0.924135\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−7.80722	+	280475.i	−2.37373e−5	+	0.852765i
$162$		0			0
$163$	96255.0	+	166718.i	0.283762	+	0.491490i	0.972308	−	0.233702i	$-0.0750841\pi$
$163$	96255.0	+	166718.i	0.283762	+	0.491490i	−0.688546	+	0.725192i	$0.741751\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−164987.			−0.457782			−0.228891	−	0.973452i	$-0.573510\pi$
$167$	−164987.			−0.457782			−0.228891	+	0.973452i	$0.573510\pi$
$168$		0			0
$169$	793433.			2.13695
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−164749.	−	285353.i	−0.418511	−	0.724883i	0.577279	−	0.816547i	$-0.304115\pi$
$173$	−164749.	−	285353.i	−0.418511	−	0.724883i	−0.995790	+	0.0916644i	$0.970781\pi$
$174$		0			0
$175$	112092.	−	64712.2i	0.276681	−	0.159732i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	184174.	−	318999.i	0.429631	−	0.744143i	−0.567209	−	0.823574i	$-0.691977\pi$
$179$	184174.	−	318999.i	0.429631	−	0.744143i	0.996840	+	0.0794308i	$0.0253103\pi$
$180$		0			0
$181$	−79607.3			−0.180616			−0.0903080	−	0.995914i	$-0.528785\pi$
$181$	−79607.3			−0.180616			−0.0903080	+	0.995914i	$0.528785\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	350173.	−	606517.i	0.752234	−	1.30291i
$186$		0			0
$187$	50857.9	+	88088.4i	0.106354	+	0.184211i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	282957.	+	490096.i	0.561225	+	0.972070i	0.997390	+	0.0722036i	$0.0230031\pi$
$191$	282957.	+	490096.i	0.561225	+	0.972070i	−0.436165	+	0.899867i	$0.643664\pi$
$192$		0			0
$193$	19332.9	−	33485.6i	0.0373597	−	0.0647089i	−0.846741	−	0.532005i	$-0.821439\pi$
$193$	19332.9	−	33485.6i	0.0373597	−	0.0647089i	0.884101	+	0.467296i	$0.154772\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	334957.			0.614927			0.307463	−	0.951560i	$-0.400520\pi$
$197$	334957.			0.614927			0.307463	+	0.951560i	$0.400520\pi$
$198$		0			0
$199$	300123.	−	519828.i	0.537237	−	0.930522i	−0.461814	−	0.886977i	$-0.652801\pi$
$199$	300123.	−	519828.i	0.537237	−	0.930522i	0.999051	−	0.0435454i	$-0.0138653\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	504324.	+	291191.i	0.858954	+	0.495949i
$204$		0			0
$205$	−474693.	−	822193.i	−0.788912	−	1.36644i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	742790.			1.17625
$210$		0			0
$211$	1.06504e6			1.64687			0.823433	−	0.567414i	$-0.192056\pi$
$211$	1.06504e6			1.64687			0.823433	+	0.567414i	$0.192056\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−10494.0	−	18176.1i	−0.0154826	−	0.0268166i
$216$		0			0
$217$	−35732.6	+	20628.9i	−0.0515128	+	0.0297390i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−86961.6	+	150622.i	−0.119770	+	0.207447i
$222$		0			0
$223$	−1.37380e6			−1.84995			−0.924976	−	0.380027i	$-0.875915\pi$
$223$	−1.37380e6			−1.84995			−0.924976	+	0.380027i	$0.875915\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	162440.	−	281354.i	0.209232	−	0.362400i	−0.742241	−	0.670133i	$-0.766237\pi$
$227$	162440.	−	281354.i	0.209232	−	0.362400i	0.951473	+	0.307733i	$0.0995703\pi$
$228$		0			0
$229$	411196.	+	712212.i	0.518155	+	0.897472i	0.999778	+	0.0210926i	$0.00671448\pi$
$229$	411196.	+	712212.i	0.518155	+	0.897472i	−0.481622	+	0.876379i	$0.659952\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	569070.	+	985657.i	0.686713	+	1.18942i	0.972895	+	0.231247i	$0.0742806\pi$
$233$	569070.	+	985657.i	0.686713	+	1.18942i	−0.286182	+	0.958175i	$0.592386\pi$
$234$		0			0
$235$	478741.	−	829204.i	0.565498	−	0.979471i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−483125.			−0.547097			−0.273549	−	0.961858i	$-0.588197\pi$
$239$	−483125.			−0.547097			−0.273549	+	0.961858i	$0.588197\pi$
$240$		0			0
$241$	−507406.	+	878853.i	−0.562747	+	0.974706i	0.434509	+	0.900668i	$0.356922\pi$
$241$	−507406.	+	878853.i	−0.562747	+	0.974706i	−0.997255	+	0.0740384i	$0.976411\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−387568.	+	671201.i	−0.412508	+	0.714393i
$246$		0			0
$247$	635046.	+	1.09993e6i	0.662312	+	1.14716i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−415812.			−0.416593			−0.208297	−	0.978066i	$-0.566792\pi$
$251$	−415812.			−0.416593			−0.208297	+	0.978066i	$0.566792\pi$
$252$		0			0
$253$	−1.36550e6			−1.34119
$254$		0			0
$255$		0			0
$256$		0			0
$257$	505998.	+	876414.i	0.477876	+	0.827706i	0.999678	−	0.0253604i	$-0.00807332\pi$
$257$	505998.	+	876414.i	0.477876	+	0.827706i	−0.521802	+	0.853067i	$0.674740\pi$
$258$		0			0
$259$	54.8043	−	1.96885e6i	5.07651e−5	−	1.82374i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	1.02885e6	−	1.78202e6i	0.917195	−	1.58863i	0.113539	−	0.993534i	$-0.463781\pi$
$263$	1.02885e6	−	1.78202e6i	0.917195	−	1.58863i	0.803656	−	0.595094i	$-0.202885\pi$
$264$		0			0
$265$	−890030.			−0.778556
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−403998.	+	699745.i	−0.340407	+	0.589602i	−0.984508	−	0.175338i	$-0.943898\pi$
$269$	−403998.	+	699745.i	−0.340407	+	0.589602i	0.644101	+	0.764940i	$0.277232\pi$
$270$		0			0
$271$	−98011.4	−	169761.i	−0.0810687	−	0.140415i	0.822641	−	0.568562i	$-0.192500\pi$
$271$	−98011.4	−	169761.i	−0.0810687	−	0.140415i	−0.903709	+	0.428147i	$0.859167\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	315069.	+	545715.i	0.251231	+	0.435145i
$276$		0			0
$277$	−151292.	+	262046.i	−0.118472	+	0.205200i	−0.919162	−	0.393879i	$-0.871133\pi$
$277$	−151292.	+	262046.i	−0.118472	+	0.205200i	0.800690	+	0.599079i	$0.204466\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−646014.			−0.488063			−0.244032	−	0.969767i	$-0.578470\pi$
$281$	−646014.			−0.488063			−0.244032	+	0.969767i	$0.578470\pi$
$282$		0			0
$283$	553748.	−	959119.i	0.411004	−	0.711879i	−0.583996	−	0.811757i	$-0.698512\pi$
$283$	553748.	−	959119.i	0.411004	−	0.711879i	0.995000	+	0.0998771i	$0.0318450\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−2.31135e6	−	1.33455e6i	−1.65639	−	0.956376i
$288$		0			0
$289$	696943.	+	1.20714e6i	0.490854	+	0.850185i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1.89396e6			−1.28885			−0.644425	−	0.764668i	$-0.722903\pi$
$293$	−1.89396e6			−1.28885			−0.644425	+	0.764668i	$0.722903\pi$
$294$		0			0
$295$	293677.			0.196478
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1.16743e6	−	2.02205e6i	−0.755184	−	1.30802i
$300$		0			0
$301$	−51096.7	−	29502.6i	−0.0325070	−	0.0187691i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.13317e6	+	1.96271e6i	−0.697504	+	1.20811i
$306$		0			0
$307$	1.97803e6			1.19781			0.598905	−	0.800820i	$-0.295603\pi$
$307$	1.97803e6			1.19781			0.598905	+	0.800820i	$0.295603\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	1.29393e6	−	2.24116e6i	0.758596	−	1.31393i	−0.184970	−	0.982744i	$-0.559219\pi$
$311$	1.29393e6	−	2.24116e6i	0.758596	−	1.31393i	0.943566	−	0.331183i	$-0.107448\pi$
$312$		0			0
$313$	70678.5	+	122419.i	0.0407781	+	0.0706297i	0.885694	−	0.464269i	$-0.153683\pi$
$313$	70678.5	+	122419.i	0.0407781	+	0.0706297i	−0.844916	+	0.534899i	$0.820350\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	586423.	+	1.01571e6i	0.327765	+	0.567706i	0.982068	−	0.188527i	$-0.0603712\pi$
$317$	586423.	+	1.01571e6i	0.327765	+	0.567706i	−0.654303	+	0.756233i	$0.727038\pi$
$318$		0			0
$319$	−1.41760e6	+	2.45536e6i	−0.779969	+	1.35095i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−189657.			−0.101149
$324$		0			0
$325$	−538734.	+	933114.i	−0.282921	+	0.490034i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	74.9260	−	2.69172e6i	3.81630e−5	−	1.37101i
$330$		0			0
$331$	−653305.	−	1.13156e6i	−0.327752	−	0.567684i	0.654313	−	0.756224i	$-0.272958\pi$
$331$	−653305.	−	1.13156e6i	−0.327752	−	0.567684i	−0.982066	+	0.188540i	$0.939624\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1.57041e6			−0.764544
$336$		0			0
$337$	−265059.			−0.127136			−0.0635679	−	0.997978i	$-0.520248\pi$
$337$	−265059.			−0.127136			−0.0635679	+	0.997978i	$0.520248\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−100437.	−	173962.i	−0.0467745	−	0.0810157i
$342$		0			0
$343$	−181.953	+	2.17889e6i	−8.35072e−5	+	1.00000i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1.52791e6	+	2.64642e6i	−0.681200	+	1.17987i	0.293414	+	0.955985i	$0.405208\pi$
$347$	−1.52791e6	+	2.64642e6i	−0.681200	+	1.17987i	−0.974615	+	0.223888i	$0.928125\pi$
$348$		0			0
$349$	−1.47164e6			−0.646753			−0.323377	−	0.946270i	$-0.604818\pi$
$349$	−1.47164e6			−0.646753			−0.323377	+	0.946270i	$0.604818\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−192592.	+	333578.i	−0.0822623	+	0.142482i	−0.904221	−	0.427064i	$-0.859548\pi$
$353$	−192592.	+	333578.i	−0.0822623	+	0.142482i	0.821959	+	0.569547i	$0.192881\pi$
$354$		0			0
$355$	−1.45177e6	−	2.51454e6i	−0.611401	−	1.05898i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−820939.	−	1.42191e6i	−0.336182	−	0.582285i	0.647529	−	0.762041i	$-0.275802\pi$
$359$	−820939.	−	1.42191e6i	−0.336182	−	0.582285i	−0.983711	+	0.179756i	$0.942469\pi$
$360$		0			0
$361$	545555.	−	944929.i	0.220328	−	0.381620i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	408940.			0.160667
$366$		0			0
$367$	561616.	−	972747.i	0.217658	−	0.376994i	−0.736434	−	0.676510i	$-0.763492\pi$
$367$	561616.	−	972747.i	0.217658	−	0.376994i	0.954091	+	0.299516i	$0.0968250\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−2.16691e6	+	1.25099e6i	−0.817347	+	0.471865i
$372$		0			0
$373$	2.11052e6	+	3.65554e6i	0.785450	+	1.36044i	0.928730	+	0.370757i	$0.120902\pi$
$373$	2.11052e6	+	3.65554e6i	0.785450	+	1.36044i	−0.143280	+	0.989682i	$0.545765\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−4.84789e6			−1.75671
$378$		0			0
$379$	−4.49923e6			−1.60894			−0.804471	−	0.593993i	$-0.797551\pi$
$379$	−4.49923e6			−1.60894			−0.804471	+	0.593993i	$0.797551\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	2.21557e6	+	3.83748e6i	0.771771	+	1.33675i	0.936591	+	0.350423i	$0.113962\pi$
$383$	2.21557e6	+	3.83748e6i	0.771771	+	1.33675i	−0.164820	+	0.986324i	$0.552704\pi$
$384$		0			0
$385$	−3.26781e6	−	1.88680e6i	−1.12358	−	0.648743i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−2.72111e6	+	4.71311e6i	−0.911744	+	1.57919i	−0.100144	+	0.994973i	$0.531930\pi$
$389$	−2.72111e6	+	4.71311e6i	−0.911744	+	1.57919i	−0.811600	+	0.584214i	$0.801403\pi$
$390$		0			0
$391$	348654.			0.115333
$392$		0			0
$393$		0			0
$394$		0			0
$395$	793489.	−	1.37436e6i	0.255887	−	0.443209i
$396$		0			0
$397$	−1.34397e6	−	2.32782e6i	−0.427969	−	0.741264i	0.568723	−	0.822529i	$-0.307438\pi$
$397$	−1.34397e6	−	2.32782e6i	−0.427969	−	0.741264i	−0.996693	+	0.0812644i	$0.974104\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	221647.	+	383904.i	0.0688337	+	0.119223i	0.898388	−	0.439202i	$-0.144739\pi$
$401$	221647.	+	383904.i	0.0688337	+	0.119223i	−0.829554	+	0.558426i	$0.811406\pi$
$402$		0			0
$403$	171737.	−	297457.i	0.0526746	−	0.0912351i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	9.58539e6			2.86829
$408$		0			0
$409$	2.44789e6	−	4.23987e6i	0.723575	−	1.25327i	−0.235983	−	0.971757i	$-0.575831\pi$
$409$	2.44789e6	−	4.23987e6i	0.723575	−	1.25327i	0.959558	−	0.281511i	$-0.0908355\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	715001.	−	412779.i	0.206268	−	0.119081i
$414$		0			0
$415$	162359.	+	281214.i	0.0462760	+	0.0801523i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	2.23186e6			0.621058			0.310529	−	0.950564i	$-0.399494\pi$
$419$	2.23186e6			0.621058			0.310529	+	0.950564i	$0.399494\pi$
$420$		0			0
$421$	5.48208e6			1.50744			0.753721	−	0.657195i	$-0.228257\pi$
$421$	5.48208e6			1.50744			0.753721	+	0.657195i	$0.228257\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−80446.6	−	139338.i	−0.0216041	−	0.0374193i
$426$		0			0
$427$	−177.349	+	6.37126e6i	−4.70715e−5	+	1.69105i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	2.31183e6	−	4.00421e6i	0.599463	−	1.03830i	−0.393437	−	0.919352i	$-0.628714\pi$
$431$	2.31183e6	−	4.00421e6i	0.599463	−	1.03830i	0.992900	−	0.118949i	$-0.0379526\pi$
$432$		0			0
$433$	−3.08314e6			−0.790267			−0.395134	−	0.918624i	$-0.629302\pi$
$433$	−3.08314e6			−0.790267			−0.395134	+	0.918624i	$0.629302\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1.27304e6	−	2.20497e6i	0.318888	−	0.552330i
$438$		0			0
$439$	−217235.	−	376262.i	−0.0537983	−	0.0931815i	0.837872	−	0.545867i	$-0.183799\pi$
$439$	−217235.	−	376262.i	−0.0537983	−	0.0931815i	−0.891670	+	0.452685i	$0.850466\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	860317.	+	1.49011e6i	0.208281	+	0.360753i	0.951173	−	0.308658i	$-0.0998799\pi$
$443$	860317.	+	1.49011e6i	0.208281	+	0.360753i	−0.742892	+	0.669411i	$0.766547\pi$
$444$		0			0
$445$	466751.	−	808437.i	0.111734	−	0.193529i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	4.06508e6			0.951598			0.475799	−	0.879554i	$-0.342159\pi$
$449$	4.06508e6			0.951598			0.475799	+	0.879554i	$0.342159\pi$
$450$		0			0
$451$	6.49696e6	−	1.12531e7i	1.50407	−	2.60513i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	179.600	−	6.45213e6i	4.06703e−5	−	1.46108i
$456$		0			0
$457$	−1.13092e6	−	1.95881e6i	−0.253303	−	0.438734i	0.711130	−	0.703061i	$-0.248184\pi$
$457$	−1.13092e6	−	1.95881e6i	−0.253303	−	0.438734i	−0.964433	+	0.264326i	$0.914850\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−7.80980e6			−1.71154			−0.855771	−	0.517355i	$-0.826917\pi$
$461$	−7.80980e6			−1.71154			−0.855771	+	0.517355i	$0.826917\pi$
$462$		0			0
$463$	−525518.			−0.113929			−0.0569647	−	0.998376i	$-0.518142\pi$
$463$	−525518.			−0.113929			−0.0569647	+	0.998376i	$0.518142\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	2.79688e6	+	4.84434e6i	0.593447	+	1.02788i	0.993764	+	0.111503i	$0.0355665\pi$
$467$	2.79688e6	+	4.84434e6i	0.593447	+	1.02788i	−0.400318	+	0.916376i	$0.631100\pi$
$468$		0			0
$469$	−3.82341e6	+	2.20731e6i	−0.802637	+	0.463373i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	143627.	−	248770.i	0.0295178	−	0.0511264i
$474$		0			0
$475$	−1.17494e6			−0.238936
$476$		0			0
$477$		0			0
$478$		0			0
$479$	906232.	−	1.56964e6i	0.180468	−	0.312580i	−0.761572	−	0.648080i	$-0.775572\pi$
$479$	906232.	−	1.56964e6i	0.180468	−	0.312580i	0.942040	+	0.335500i	$0.108905\pi$
$480$		0			0
$481$	8.19499e6	+	1.41941e7i	1.61505	+	2.79735i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1.24665e6	+	2.15926e6i	0.240652	+	0.416822i
$486$		0			0
$487$	−720340.	+	1.24767e6i	−0.137631	+	0.238383i	−0.926599	−	0.376050i	$-0.877282\pi$
$487$	−720340.	+	1.24767e6i	−0.137631	+	0.238383i	0.788969	+	0.614433i	$0.210615\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−5.25026e6			−0.982827			−0.491413	−	0.870926i	$-0.663520\pi$
$491$	−5.25026e6			−0.982827			−0.491413	+	0.870926i	$0.663520\pi$
$492$		0			0
$493$	361957.	−	626927.i	0.0670717	−	0.116172i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−7.06887e6	−	4.08148e6i	−1.28369	−	0.741185i
$498$		0			0
$499$	4.07080e6	+	7.05083e6i	0.731861	+	1.26762i	0.956087	+	0.293083i	$0.0946811\pi$
$499$	4.07080e6	+	7.05083e6i	0.731861	+	1.26762i	−0.224227	+	0.974537i	$0.571986\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	7.98029e6			1.40637			0.703183	−	0.711009i	$-0.251761\pi$
$503$	7.98029e6			1.40637			0.703183	+	0.711009i	$0.251761\pi$
$504$		0			0
$505$	−4.21893e6			−0.736162
$506$		0			0
$507$		0			0
$508$		0			0
$509$	4.07141e6	+	7.05189e6i	0.696547	+	1.20646i	0.969656	+	0.244472i	$0.0786148\pi$
$509$	4.07141e6	+	7.05189e6i	0.696547	+	1.20646i	−0.273109	+	0.961983i	$0.588052\pi$
$510$		0			0
$511$	995626.	−	574788.i	0.168672	−	0.0973768i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1.73812e6	−	3.01051e6i	0.288776	−	0.500176i
$516$		0			0
$517$	1.31047e7			2.15626
$518$		0			0
$519$		0			0
$520$		0			0
$521$	2.55984e6	−	4.43377e6i	0.413160	−	0.715613i	−0.582074	−	0.813136i	$-0.697759\pi$
$521$	2.55984e6	−	4.43377e6i	0.413160	−	0.715613i	0.995233	+	0.0975227i	$0.0310919\pi$
$522$		0			0
$523$	−2.36736e6	−	4.10040e6i	−0.378452	−	0.655498i	0.612385	−	0.790560i	$-0.290210\pi$
$523$	−2.36736e6	−	4.10040e6i	−0.378452	−	0.655498i	−0.990837	+	0.135061i	$0.956877\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	25644.7	+	44417.9i	0.00402226	+	0.00696677i
$528$		0			0
$529$	877893.	−	1.52055e6i	0.136396	−	0.236245i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	2.22182e7			3.38759
$534$		0			0
$535$	−3.50783e6	+	6.07574e6i	−0.529851	+	0.917729i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−1.06080e7	−	590.562i	−1.57276	−	8.75576e-5i
$540$		0			0
$541$	4.10226e6	+	7.10533e6i	0.602602	+	1.04374i	0.992426	+	0.122848i	$0.0392026\pi$
$541$	4.10226e6	+	7.10533e6i	0.602602	+	1.04374i	−0.389824	+	0.920890i	$0.627464\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−3.53575e6			−0.509907
$546$		0			0
$547$	−1.57733e6			−0.225400			−0.112700	−	0.993629i	$-0.535950\pi$
$547$	−1.57733e6			−0.225400			−0.112700	+	0.993629i	$0.535950\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−2.64323e6	−	4.57820e6i	−0.370899	−	0.642415i
$552$		0			0
$553$	124.186	−	4.46139e6i	1.72687e−5	−	0.620380i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−1.05205e6	+	1.82221e6i	−0.143681	+	0.248863i	−0.928880	−	0.370380i	$-0.879227\pi$
$557$	−1.05205e6	+	1.82221e6i	−0.143681	+	0.248863i	0.785199	+	0.619244i	$0.212561\pi$
$558$		0			0
$559$	491175.			0.0664824
$560$		0			0
$561$		0			0
$562$		0			0
$563$	3.84583e6	−	6.66118e6i	0.511351	−	0.885687i	−0.488562	−	0.872529i	$-0.662478\pi$
$563$	3.84583e6	−	6.66118e6i	0.511351	−	0.885687i	0.999913	−	0.0131574i	$-0.00418826\pi$
$564$		0			0
$565$	−5.26903e6	−	9.12622e6i	−0.694399	−	1.20273i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	3.65070e6	+	6.32319e6i	0.472710	+	0.818758i	0.999512	−	0.0312298i	$-0.00994238\pi$
$569$	3.65070e6	+	6.32319e6i	0.472710	+	0.818758i	−0.526802	+	0.849988i	$0.676609\pi$
$570$		0			0
$571$	−1.70967e6	+	2.96124e6i	−0.219443	+	0.380087i	−0.954638	−	0.297769i	$-0.903757\pi$
$571$	−1.70967e6	+	2.96124e6i	−0.219443	+	0.380087i	0.735195	+	0.677856i	$0.237091\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	2.15994e6			0.272440
$576$		0			0
$577$	5.46447e6	−	9.46473e6i	0.683295	−	1.18350i	−0.290674	−	0.956822i	$-0.593880\pi$
$577$	5.46447e6	−	9.46473e6i	0.683295	−	1.18350i	0.973969	−	0.226680i	$-0.0727870\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	790549.	+	456453.i	0.0971602	+	0.0560991i
$582$		0			0
$583$	−6.09076e6	−	1.05495e7i	−0.742164	−	1.28547i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−2.25268e6			−0.269839			−0.134919	−	0.990857i	$-0.543078\pi$
$587$	−2.25268e6			−0.269839			−0.134919	+	0.990857i	$0.543078\pi$
$588$		0			0
$589$	374546.			0.0444853
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−2.84120e6	−	4.92111e6i	−0.331792	−	0.574680i	0.651072	−	0.759016i	$-0.274320\pi$
$593$	−2.84120e6	−	4.92111e6i	−0.331792	−	0.574680i	−0.982863	+	0.184336i	$0.940986\pi$
$594$		0			0
$595$	834372.	+	481756.i	0.0966201	+	0.0557872i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−6.40132e6	+	1.10874e7i	−0.728958	+	1.26259i	0.228366	+	0.973575i	$0.426662\pi$
$599$	−6.40132e6	+	1.10874e7i	−0.728958	+	1.26259i	−0.957324	+	0.289017i	$0.906672\pi$
$600$		0			0
$601$	−9.25870e6			−1.04560			−0.522798	−	0.852457i	$-0.675112\pi$
$601$	−9.25870e6			−1.04560			−0.522798	+	0.852457i	$0.675112\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	5.47201e6	−	9.47780e6i	0.607797	−	1.05274i
$606$		0			0
$607$	200660.	+	347553.i	0.0221049	+	0.0382868i	0.876866	−	0.480734i	$-0.159630\pi$
$607$	200660.	+	347553.i	0.0221049	+	0.0382868i	−0.854761	+	0.519021i	$0.826297\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	1.12038e7	+	1.94056e7i	1.21413	+	2.10293i
$612$		0			0
$613$	−3.41198e6	+	5.90973e6i	−0.366738	+	0.635208i	−0.989053	−	0.147558i	$-0.952859\pi$
$613$	−3.41198e6	+	5.90973e6i	−0.366738	+	0.635208i	0.622316	+	0.782766i	$0.286192\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	336274.			0.0355615			0.0177808	−	0.999842i	$-0.494340\pi$
$617$	336274.			0.0355615			0.0177808	+	0.999842i	$0.494340\pi$
$618$		0			0
$619$	4.80575e6	−	8.32381e6i	0.504121	−	0.873163i	−0.495868	−	0.868398i	$-0.665150\pi$
$619$	4.80575e6	−	8.32381e6i	0.504121	−	0.873163i	0.999989	−	0.00476520i	$-0.00151681\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	73.0495	−	2.62431e6i	7.54045e−6	−	0.270891i
$624$		0			0
$625$	2.82448e6	+	4.89215e6i	0.289227	+	0.500956i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−2.44744e6			−0.246652
$630$		0			0
$631$	1.28813e7			1.28791			0.643957	−	0.765062i	$-0.277292\pi$
$631$	1.28813e7			1.28791			0.643957	+	0.765062i	$0.277292\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	2.81560e6	+	4.87676e6i	0.277100	+	0.479951i
$636$		0			0
$637$	−9.06839e6	−	1.57089e7i	−0.885486	−	1.53390i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−7.47479e6	+	1.29467e7i	−0.718545	+	1.24456i	0.243031	+	0.970018i	$0.421858\pi$
$641$	−7.47479e6	+	1.29467e7i	−0.718545	+	1.24456i	−0.961576	+	0.274538i	$0.911475\pi$
$642$		0			0
$643$	−8.03892e6			−0.766780			−0.383390	−	0.923587i	$-0.625243\pi$
$643$	−8.03892e6			−0.766780			−0.383390	+	0.923587i	$0.625243\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−3.34388e6	+	5.79176e6i	−0.314043	+	0.543939i	−0.979234	−	0.202735i	$-0.935017\pi$
$647$	−3.34388e6	+	5.79176e6i	−0.314043	+	0.543939i	0.665190	+	0.746674i	$0.268350\pi$
$648$		0			0
$649$	2.00973e6	+	3.48095e6i	0.187294	+	0.324404i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−522213.	−	904499.i	−0.0479253	−	0.0830090i	0.841068	−	0.540930i	$-0.181928\pi$
$653$	−522213.	−	904499.i	−0.0479253	−	0.0830090i	−0.888993	+	0.457921i	$0.848594\pi$
$654$		0			0
$655$	−1.74766e6	+	3.02704e6i	−0.159168	+	0.275687i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−2.10237e7			−1.88580			−0.942902	−	0.333071i	$-0.891915\pi$
$659$	−2.10237e7			−1.88580			−0.942902	+	0.333071i	$0.891915\pi$
$660$		0			0
$661$	−4.73340e6	+	8.19849e6i	−0.421376	+	0.729845i	−0.996074	−	0.0885207i	$-0.971786\pi$
$661$	−4.73340e6	+	8.19849e6i	−0.421376	+	0.729845i	0.574698	+	0.818365i	$0.305119\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	6.09329e6	−	3.51773e6i	0.534315	−	0.308467i
$666$		0			0
$667$	4.85915e6	+	8.41629e6i	0.422908	+	0.732497i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−3.10187e7			−2.65960
$672$		0			0
$673$	−1.95188e7			−1.66117			−0.830587	−	0.556889i	$-0.811995\pi$
$673$	−1.95188e7			−1.66117			−0.830587	+	0.556889i	$0.811995\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−8.52722e6	−	1.47696e7i	−0.715049	−	1.23850i	−0.962941	−	0.269713i	$-0.913071\pi$
$677$	−8.52722e6	−	1.47696e7i	−0.715049	−	1.23850i	0.247892	−	0.968788i	$-0.420262\pi$
$678$		0			0
$679$	6.07012e6	+	3.50481e6i	0.505270	+	0.291736i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	5.47375e6	−	9.48080e6i	0.448986	−	0.777667i	−0.549334	−	0.835603i	$-0.685119\pi$
$683$	5.47375e6	−	9.48080e6i	0.448986	−	0.777667i	0.998320	+	0.0579361i	$0.0184520\pi$
$684$		0			0
$685$	−1.11420e7			−0.907270
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1.04146e7	−	1.80385e7i	0.835781	−	1.44762i
$690$		0			0
$691$	−5.22690e6	−	9.05326e6i	−0.416437	−	0.721290i	0.579141	−	0.815227i	$-0.303388\pi$
$691$	−5.22690e6	−	9.05326e6i	−0.416437	−	0.721290i	−0.995578	+	0.0939372i	$0.970055\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	2.89737e6	+	5.01839e6i	0.227532	+	0.394096i
$696$		0			0
$697$	−1.65887e6	+	2.87325e6i	−0.129339	+	0.224022i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−8.71564e6			−0.669891			−0.334946	−	0.942237i	$-0.608718\pi$
$701$	−8.71564e6			−0.669891			−0.334946	+	0.942237i	$0.608718\pi$
$702$		0			0
$703$	−8.93634e6	+	1.54782e7i	−0.681980	+	1.18122i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1.02716e7	+	5.92994e6i	−0.772841	+	0.446171i
$708$		0			0
$709$	469220.	+	812712.i	0.0350559	+	0.0607185i	0.883021	−	0.469333i	$-0.155506\pi$
$709$	469220.	+	812712.i	0.0350559	+	0.0607185i	−0.847965	+	0.530052i	$0.822172\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−688542.			−0.0507232
$714$		0			0
$715$	3.14124e7			2.29792
$716$		0			0
$717$		0			0
$718$		0			0
$719$	3.29537e6	+	5.70774e6i	0.237729	+	0.411758i	0.960062	−	0.279787i	$-0.0902638\pi$
$719$	3.29537e6	+	5.70774e6i	0.237729	+	0.411758i	−0.722334	+	0.691545i	$0.756930\pi$
$720$		0			0
$721$	272.027	−	9.77258e6i	1.94883e−5	−	0.700118i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	2.24235e6	−	3.88386e6i	0.158438	−	0.274422i
$726$		0			0
$727$	−2.32586e7			−1.63210			−0.816052	−	0.577979i	$-0.803842\pi$
$727$	−2.32586e7			−1.63210			−0.816052	+	0.577979i	$0.803842\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−36672.4	+	63518.5i	−0.00253832	+	0.00439650i
$732$		0			0
$733$	1.01427e7	+	1.75677e7i	0.697259	+	1.20769i	0.969413	+	0.245434i	$0.0789305\pi$
$733$	1.01427e7	+	1.75677e7i	0.697259	+	1.20769i	−0.272154	+	0.962254i	$0.587736\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1.07469e7	−	1.86141e7i	−0.728807	−	1.26233i
$738$		0			0
$739$	−296714.	+	513923.i	−0.0199860	+	0.0346168i	−0.875845	−	0.482592i	$-0.839696\pi$
$739$	−296714.	+	513923.i	−0.0199860	+	0.0346168i	0.855859	+	0.517209i	$0.173029\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−7.70228e6			−0.511855			−0.255928	−	0.966696i	$-0.582381\pi$
$743$	−7.70228e6			−0.511855			−0.255928	+	0.966696i	$0.582381\pi$
$744$		0			0
$745$	−831151.	+	1.43959e6i	−0.0548642	+	0.0950276i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−548.997	+	1.97228e7i	−3.57574e−5	+	1.28459i
$750$		0			0
$751$	1.34166e7	+	2.32383e7i	0.868048	+	1.50350i	0.863989	+	0.503511i	$0.167959\pi$
$751$	1.34166e7	+	2.32383e7i	0.868048	+	1.50350i	0.00405860	+	0.999992i	$0.498708\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−6.94811e6			−0.443608
$756$		0			0
$757$	2.34943e7			1.49013			0.745063	−	0.666994i	$-0.232419\pi$
$757$	2.34943e7			1.49013			0.745063	+	0.666994i	$0.232419\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−6.30959e6	−	1.09285e7i	−0.394947	−	0.684069i	0.598147	−	0.801386i	$-0.295904\pi$
$761$	−6.30959e6	−	1.09285e7i	−0.394947	−	0.684069i	−0.993094	+	0.117317i	$0.962571\pi$
$762$		0			0
$763$	−8.60833e6	+	4.96970e6i	−0.535313	+	0.309043i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−3.43642e6	+	5.95205e6i	−0.210920	+	0.365324i
$768$		0			0
$769$	2.73674e6			0.166885			0.0834425	−	0.996513i	$-0.473408\pi$
$769$	2.73674e6			0.166885			0.0834425	+	0.996513i	$0.473408\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	1.07752e7	−	1.86633e7i	0.648602	−	1.12341i	−0.334856	−	0.942269i	$-0.608688\pi$
$773$	1.07752e7	−	1.86633e7i	0.648602	−	1.12341i	0.983457	−	0.181141i	$-0.0579792\pi$
$774$		0			0
$775$	158871.	+	275172.i	0.00950145	+	0.0164570i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1.21141e7	+	2.09822e7i	0.715232	+	1.23882i
$780$		0			0
$781$	1.98698e7	−	3.44156e7i	1.16565	−	2.01896i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1.96642e7			−1.13895
$786$		0			0
$787$	−1.44945e6	+	2.51053e6i	−0.0834195	+	0.144487i	−0.904717	−	0.426014i	$-0.859917\pi$
$787$	−1.44945e6	+	2.51053e6i	−0.0834195	+	0.144487i	0.821297	+	0.570501i	$0.193251\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−2.56557e7	−	1.48133e7i	−1.45795	−	0.841801i
$792$		0			0
$793$	−2.65193e7	−	4.59328e7i	−1.49754	−	2.59382i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1.57797e7			0.879937			0.439969	−	0.898013i	$-0.354990\pi$
$797$	1.57797e7			0.879937			0.439969	+	0.898013i	$0.354990\pi$
$798$		0			0
$799$	−3.34603e6			−0.185423
$800$		0			0
$801$		0			0
$802$		0			0
$803$	2.79851e6	+	4.84716e6i	0.153157	+	0.265276i
$804$		0			0
$805$	−1.12015e7	+	6.46679e6i	−0.609239	+	0.351722i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	3.97780e6	−	6.88975e6i	0.213684	−	0.370111i	−0.739181	−	0.673507i	$-0.764787\pi$
$809$	3.97780e6	−	6.88975e6i	0.213684	−	0.370111i	0.952865	+	0.303396i	$0.0981205\pi$
$810$		0			0
$811$	−2.48725e7			−1.32791			−0.663953	−	0.747774i	$-0.731123\pi$
$811$	−2.48725e7			−1.32791			−0.663953	+	0.747774i	$0.731123\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−4.43883e6	+	7.68829e6i	−0.234086	+	0.405448i
$816$		0			0
$817$	267804.	+	463850.i	0.0140366	+	0.0243121i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1.09584e7	+	1.89805e7i	0.567401	+	0.982768i	0.996822	+	0.0796631i	$0.0253844\pi$
$821$	1.09584e7	+	1.89805e7i	0.567401	+	0.982768i	−0.429421	+	0.903105i	$0.641282\pi$
$822$		0			0
$823$	−9.75359e6	+	1.68937e7i	−0.501955	+	0.869412i	0.498042	+	0.867153i	$0.334053\pi$
$823$	−9.75359e6	+	1.68937e7i	−0.501955	+	0.869412i	−0.999997	+	0.00225934i	$0.999281\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−9.90134e6			−0.503420			−0.251710	−	0.967803i	$-0.580993\pi$
$827$	−9.90134e6			−0.503420			−0.251710	+	0.967803i	$0.580993\pi$
$828$		0			0
$829$	−1.19894e7	+	2.07663e7i	−0.605916	+	1.04948i	0.385990	+	0.922503i	$0.373860\pi$
$829$	−1.19894e7	+	2.07663e7i	−0.605916	+	1.04948i	−0.991906	+	0.126974i	$0.959474\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	2.70854e6	+	150.788i	0.135246	+	7.52932e-6i
$834$		0			0
$835$	−3.80422e6	−	6.58911e6i	−0.188821	−	0.327047i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	1.26483e6			0.0620334			0.0310167	−	0.999519i	$-0.490125\pi$
$839$	1.26483e6			0.0620334			0.0310167	+	0.999519i	$0.490125\pi$
$840$		0			0
$841$	−332952.			−0.0162327
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1.82947e7	+	3.16874e7i	0.881423	+	1.52667i
$846$		0			0
$847$	856.404	−	3.07664e7i	4.10176e−5	−	1.47356i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	1.64280e7	−	2.84542e7i	0.777610	−	1.34686i
$852$		0			0
$853$	2.75502e7			1.29644			0.648219	−	0.761454i	$-0.275514\pi$
$853$	2.75502e7			1.29644			0.648219	+	0.761454i	$0.275514\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1.79537e7	−	3.10967e7i	0.835030	−	1.44631i	−0.0589756	−	0.998259i	$-0.518783\pi$
$857$	1.79537e7	−	3.10967e7i	0.835030	−	1.44631i	0.894006	−	0.448055i	$-0.147883\pi$
$858$		0			0
$859$	1.17170e7	+	2.02945e7i	0.541795	+	0.938416i	0.998801	+	0.0489529i	$0.0155884\pi$
$859$	1.17170e7	+	2.02945e7i	0.541795	+	0.938416i	−0.457006	+	0.889464i	$0.651078\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1.67705e7	+	2.90473e7i	0.766511	+	1.32764i	0.939444	+	0.342702i	$0.111342\pi$
$863$	1.67705e7	+	2.90473e7i	0.766511	+	1.32764i	−0.172933	+	0.984934i	$0.555325\pi$
$864$		0			0
$865$	7.59746e6	−	1.31592e7i	0.345245	−	0.597983i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	2.17204e7			0.975705
$870$		0			0
$871$	1.83760e7	−	3.18281e7i	0.820739	−	1.42156i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	2.13485e7	+	1.23263e7i	0.942642	+	0.544270i
$876$		0			0
$877$	7.72784e6	+	1.33850e7i	0.339281	+	0.587651i	0.984298	−	0.176517i	$-0.0564831\pi$
$877$	7.72784e6	+	1.33850e7i	0.339281	+	0.587651i	−0.645017	+	0.764168i	$0.723150\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1.02425e7			−0.444599			−0.222299	−	0.974978i	$-0.571356\pi$
$881$	−1.02425e7			−0.444599			−0.222299	+	0.974978i	$0.571356\pi$
$882$		0			0
$883$	−6.85044e6			−0.295676			−0.147838	−	0.989012i	$-0.547231\pi$
$883$	−6.85044e6			−0.295676			−0.147838	+	0.989012i	$0.547231\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	3.04380e6	+	5.27201e6i	0.129899	+	0.224992i	0.923637	−	0.383268i	$-0.125201\pi$
$887$	3.04380e6	+	5.27201e6i	0.129899	+	0.224992i	−0.793738	+	0.608260i	$0.791868\pi$
$888$		0			0
$889$	1.37096e7	+	7.91573e6i	0.581794	+	0.335921i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1.22174e7	+	2.11611e7i	−0.512684	+	0.887994i
$894$		0			0
$895$	1.69865e7			0.708837
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−714814.	+	1.23809e6i	−0.0294981	+	0.0510922i
$900$		0			0
$901$	1.55516e6	+	2.69361e6i	0.0638208	+	0.110541i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−1.83556e6	−	3.17928e6i	−0.0744984	−	0.129035i
$906$		0			0
$907$	−2.31252e7	+	4.00540e7i	−0.933398	+	1.61669i	−0.155932	+	0.987768i	$0.549838\pi$
$907$	−2.31252e7	+	4.00540e7i	−0.933398	+	1.61669i	−0.777466	+	0.628925i	$0.783495\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	4.09674e7			1.63547			0.817734	−	0.575596i	$-0.195230\pi$
$911$	4.09674e7			1.63547			0.817734	+	0.575596i	$0.195230\pi$
$912$		0			0
$913$	−2.22215e6	+	3.84887e6i	−0.0882259	+	0.152812i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−273.521	+	9.82624e6i	−1.07415e−5	+	0.385890i
$918$		0			0
$919$	−1.52928e7	−	2.64880e7i	−0.597309	−	1.03457i	−0.993217	−	0.116279i	$-0.962903\pi$
$919$	−1.52928e7	−	2.64880e7i	−0.597309	−	1.03457i	0.395907	−	0.918290i	$-0.370430\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	6.79506e7			2.62536
$924$		0			0
$925$	−1.51621e7			−0.582646
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1.58426e7	−	2.74401e7i	−0.602263	−	1.04315i	−0.992478	−	0.122427i	$-0.960932\pi$
$929$	−1.58426e7	−	2.74401e7i	−0.602263	−	1.04315i	0.390214	−	0.920724i	$-0.372401\pi$
$930$		0			0
$931$	9.89066e6	−	1.71289e7i	0.373982	−	0.647673i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−2.34533e6	+	4.06223e6i	−0.0877354	+	0.151962i
$936$		0			0
$937$	2.43042e7			0.904342			0.452171	−	0.891931i	$-0.350650\pi$
$937$	2.43042e7			0.904342			0.452171	+	0.891931i	$0.350650\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1.24230e7	+	2.15172e7i	−0.457352	+	0.792157i	−0.998820	−	0.0485643i	$-0.984535\pi$
$941$	−1.24230e7	+	2.15172e7i	−0.457352	+	0.792157i	0.541468	+	0.840721i	$0.317869\pi$
$942$		0			0
$943$	−2.22698e7	−	3.85724e7i	−0.815525	−	1.41253i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−2.45809e7	−	4.25753e7i	−0.890681	−	1.54270i	−0.839061	−	0.544038i	$-0.816895\pi$
$947$	−2.45809e7	−	4.25753e7i	−0.890681	−	1.54270i	−0.0516202	−	0.998667i	$-0.516439\pi$
$948$		0			0
$949$	−4.78515e6	+	8.28812e6i	−0.172477	+	0.298738i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1.84741e7			−0.658918			−0.329459	−	0.944170i	$-0.606866\pi$
$953$	−1.84741e7			−0.658918			−0.329459	+	0.944170i	$0.606866\pi$
$954$		0			0
$955$	−1.30487e7	+	2.26010e7i	−0.462975	+	0.801897i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−2.71269e7	+	1.56607e7i	−0.952475	+	0.549876i
$960$		0			0
$961$	1.42639e7	+	2.47059e7i	0.498231	+	0.862961i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1.78309e6			0.0616388
$966$		0			0
$967$	3.78820e7			1.30277			0.651384	−	0.758748i	$-0.274189\pi$
$967$	3.78820e7			1.30277			0.651384	+	0.758748i	$0.274189\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1.86311e7	−	3.22699e7i	−0.634146	−	1.09837i	−0.986695	−	0.162580i	$-0.948018\pi$
$971$	−1.86311e7	−	3.22699e7i	−0.634146	−	1.09837i	0.352549	−	0.935793i	$-0.385315\pi$
$972$		0			0
$973$	1.41077e7	+	8.14562e6i	0.477721	+	0.275830i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1.85295e6	+	3.20941e6i	−0.0621052	+	0.107569i	−0.895406	−	0.445250i	$-0.853115\pi$
$977$	−1.85295e6	+	3.20941e6i	−0.0621052	+	0.107569i	0.833301	+	0.552819i	$0.186448\pi$
$978$		0			0
$979$	1.27765e7			0.426045
$980$		0			0
$981$		0			0
$982$		0			0
$983$	4.14382e6	−	7.17731e6i	0.136778	−	0.236907i	−0.789497	−	0.613754i	$-0.789659\pi$
$983$	4.14382e6	−	7.17731e6i	0.136778	−	0.236907i	0.926275	+	0.376847i	$0.122992\pi$
$984$		0			0
$985$	7.72333e6	+	1.33772e7i	0.253638	+	0.439314i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−492315.	−	852714.i	−0.0160049	−	0.0277213i
$990$		0			0
$991$	−8.12492e6	+	1.40728e7i	−0.262806	+	0.455193i	−0.966986	−	0.254828i	$-0.917981\pi$
$991$	−8.12492e6	+	1.40728e7i	−0.262806	+	0.455193i	0.704181	+	0.710021i	$0.251315\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	2.76805e7			0.886374
$996$		0			0
$997$	2.17929e7	−	3.77464e7i	0.694348	−	1.20265i	−0.276053	−	0.961143i	$-0.589026\pi$
$997$	2.17929e7	−	3.77464e7i	0.694348	−	1.20265i	0.970400	−	0.241503i	$-0.0776403\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.6.k.f.109.3		8
3.2	odd	2		84.6.i.c.25.2	✓	8
7.2	even	3	inner	252.6.k.f.37.3		8
12.11	even	2		336.6.q.i.193.2		8
21.2	odd	6		84.6.i.c.37.2	yes	8
21.5	even	6		588.6.i.o.373.3		8
21.11	odd	6		588.6.a.n.1.3		4
21.17	even	6		588.6.a.p.1.2		4
21.20	even	2		588.6.i.o.361.3		8
84.23	even	6		336.6.q.i.289.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.6.i.c.25.2	✓	8	3.2	odd	2
84.6.i.c.37.2	yes	8	21.2	odd	6
252.6.k.f.37.3		8	7.2	even	3	inner
252.6.k.f.109.3		8	1.1	even	1	trivial
336.6.q.i.193.2		8	12.11	even	2
336.6.q.i.289.2		8	84.23	even	6
588.6.a.n.1.3		4	21.11	odd	6
588.6.a.p.1.2		4	21.17	even	6
588.6.i.o.361.3		8	21.20	even	2
588.6.i.o.373.3		8	21.5	even	6

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	252.k (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(23.0577 + 39.9371i) q^{5} +(112.271 + 64.8240i) q^{7} +(-315.582 + 546.605i) q^{11} -1079.22 q^{13} +(80.5778 - 139.565i) q^{17} +(-588.428 - 1019.19i) q^{19} +(1081.73 + 1873.61i) q^{23} +(499.186 - 864.615i) q^{25} +4492.01 q^{29} +(-159.130 + 275.621i) q^{31} +(-0.166415 + 5978.48i) q^{35} +(-7593.41 - 13152.2i) q^{37} -20587.2 q^{41} -455.118 q^{43} +(-10381.4 - 17981.1i) q^{47} +(8402.69 + 14555.8i) q^{49} +(-9650.03 + 16714.3i) q^{53} -29106.4 q^{55} +(3184.15 - 5515.11i) q^{59} +(24572.6 + 42560.9i) q^{61} +(-24884.4 - 43101.1i) q^{65} +(-17027.0 + 29491.7i) q^{67} -62962.4 q^{71} +(4433.88 - 7679.70i) q^{73} +(-70864.0 + 40910.7i) q^{77} +(-17206.6 - 29802.7i) q^{79} +7041.42 q^{83} +7431.75 q^{85} +(-10121.4 - 17530.7i) q^{89} +(-121166. - 69959.7i) q^{91} +(27135.6 - 47000.2i) q^{95} +54066.6 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 42 q^{7} + 462 q^{11} - 1204 q^{13} - 228 q^{17} + 358 q^{19} + 2148 q^{23} - 5454 q^{25} + 11064 q^{29} + 830 q^{31} - 7692 q^{35} - 3914 q^{37} + 16632 q^{41} - 29036 q^{43} - 41700 q^{47} + 41876 q^{49}+ \cdots - 433356 q^{97}+O(q^{100}) \) 8 * q - 42 * q^7 + 462 * q^11 - 1204 * q^13 - 228 * q^17 + 358 * q^19 + 2148 * q^23 - 5454 * q^25 + 11064 * q^29 + 830 * q^31 - 7692 * q^35 - 3914 * q^37 + 16632 * q^41 - 29036 * q^43 - 41700 * q^47 + 41876 * q^49 - 22164 * q^53 + 7784 * q^55 - 32886 * q^59 + 83732 * q^61 + 93192 * q^65 - 80034 * q^67 - 89544 * q^71 - 22470 * q^73 - 138732 * q^77 - 75286 * q^79 + 34836 * q^83 + 278504 * q^85 - 28944 * q^89 - 12058 * q^91 + 144120 * q^95 - 433356 * q^97

Introduction

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