LMFDB - Embedded newform 420.3.p.a.349.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [420,3,Mod(349,420)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(420, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("420.349"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	420.p (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$11.4441711031$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 88 x^{14} + 3876 x^{12} + 102922 x^{10} + 1866070 x^{8} + 23190492 x^{6} + 203608845 x^{4} + \cdots + 3839661225 $ x^16 + 88x^14 + 3876x^12 + 102922x^10 + 1866070x^8 + 23190492x^6 + 203608845x^4 + 1131190758*x^2 + 3839661225
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{13}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			349.8
Root			$-1.73205 - 2.62461i$ of defining polynomial
Character	$\chi$	$=$	420.349
Dual form			420.3.p.a.349.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.73205 q^{3} +(4.25575 + 2.62461i) q^{5} +(2.56785 + 6.51200i) q^{7} +3.00000 q^{9} +8.37580 q^{11} -0.0883033 q^{13} +(-7.37118 - 4.54596i) q^{15} -29.7704 q^{17} -17.8840i q^{19} +(-4.44765 - 11.2791i) q^{21} +39.2955i q^{23} +(11.2229 + 22.3394i) q^{25} -5.19615 q^{27} +43.7168 q^{29} +53.8656i q^{31} -14.5073 q^{33} +(-6.16331 + 34.4531i) q^{35} -27.0440i q^{37} +0.152946 q^{39} -46.0785i q^{41} +74.8906i q^{43} +(12.7673 + 7.87383i) q^{45} +27.7687 q^{47} +(-35.8123 + 33.4437i) q^{49} +51.5639 q^{51} -5.38329i q^{53} +(35.6453 + 21.9832i) q^{55} +30.9761i q^{57} +1.93853i q^{59} +74.8723i q^{61} +(7.70356 + 19.5360i) q^{63} +(-0.375797 - 0.231762i) q^{65} +30.6588i q^{67} -68.0617i q^{69} -72.3104 q^{71} -34.0300 q^{73} +(-19.4385 - 38.6929i) q^{75} +(21.5078 + 54.5432i) q^{77} +125.748 q^{79} +9.00000 q^{81} -92.4285 q^{83} +(-126.695 - 78.1357i) q^{85} -75.7197 q^{87} -141.387i q^{89} +(-0.226750 - 0.575031i) q^{91} -93.2979i q^{93} +(46.9386 - 76.1100i) q^{95} +107.878 q^{97} +25.1274 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 48 q^{9} - 24 q^{11} - 24 q^{15} - 12 q^{21} - 48 q^{25} - 32 q^{29} + 76 q^{35} + 72 q^{39} - 88 q^{49} + 24 q^{51} + 152 q^{65} + 168 q^{71} + 16 q^{79} + 144 q^{81} - 416 q^{85} - 568 q^{91} + 136 q^{95}+ \cdots - 72 q^{99}+O(q^{100}) $ 16 * q + 48 * q^9 - 24 * q^11 - 24 * q^15 - 12 * q^21 - 48 * q^25 - 32 * q^29 + 76 * q^35 + 72 * q^39 - 88 * q^49 + 24 * q^51 + 152 * q^65 + 168 * q^71 + 16 * q^79 + 144 * q^81 - 416 * q^85 - 568 * q^91 + 136 * q^95 - 72 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/420\mathbb{Z}\right)^\times$.

$n$	$211$	$241$	$281$	$337$
$\chi(n)$	$1$	$-1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−1.73205			−0.577350
$3$	−1.73205			−0.577350
$4$		0			0
$5$	4.25575	+	2.62461i	0.851150	+	0.524922i
$5$	4.25575	+	2.62461i	0.851150	+	0.524922i
$6$		0			0
$7$	2.56785	+	6.51200i	0.366836	+	0.930286i
$7$	2.56785	+	6.51200i	0.366836	+	0.930286i
$8$		0			0
$9$	3.00000			0.333333
$10$		0			0
$11$	8.37580			0.761436			0.380718	−	0.924691i	$-0.375677\pi$
$11$	8.37580			0.761436			0.380718	+	0.924691i	$0.375677\pi$
$12$		0			0
$13$	−0.0883033			−0.00679256			−0.00339628	−	0.999994i	$-0.501081\pi$
$13$	−0.0883033			−0.00679256			−0.00339628	+	0.999994i	$0.501081\pi$
$14$		0			0
$15$	−7.37118	−	4.54596i	−0.491412	−	0.303064i
$16$		0			0
$17$	−29.7704			−1.75120			−0.875600	−	0.483037i	$-0.839534\pi$
$17$	−29.7704			−1.75120			−0.875600	+	0.483037i	$0.839534\pi$
$18$		0			0
$19$		−	17.8840i		−	0.941265i	−0.882329	−	0.470633i	$-0.844026\pi$
$19$		−	17.8840i		−	0.941265i	0.882329	−	0.470633i	$-0.155974\pi$
$20$		0			0
$21$	−4.44765	−	11.2791i	−0.211793	−	0.537101i
$22$		0			0
$23$			39.2955i			1.70850i	0.519864	+	0.854249i	$0.325983\pi$
$23$			39.2955i			1.70850i	−0.519864	+	0.854249i	$0.674017\pi$
$24$		0			0
$25$	11.2229	+	22.3394i	0.448914	+	0.893575i
$26$		0			0
$27$	−5.19615			−0.192450
$28$		0			0
$29$	43.7168			1.50748			0.753738	−	0.657175i	$-0.228249\pi$
$29$	43.7168			1.50748			0.753738	+	0.657175i	$0.228249\pi$
$30$		0			0
$31$			53.8656i			1.73760i	0.495164	+	0.868799i	$0.335108\pi$
$31$			53.8656i			1.73760i	−0.495164	+	0.868799i	$0.664892\pi$
$32$		0			0
$33$	−14.5073			−0.439615
$34$		0			0
$35$	−6.16331	+	34.4531i	−0.176095	+	0.984373i
$36$		0			0
$37$		−	27.0440i		−	0.730918i	−0.930827	−	0.365459i	$-0.880912\pi$
$37$		−	27.0440i		−	0.730918i	0.930827	−	0.365459i	$-0.119088\pi$
$38$		0			0
$39$	0.152946			0.00392169
$40$		0			0
$41$		−	46.0785i		−	1.12387i	−0.827183	−	0.561933i	$-0.810058\pi$
$41$		−	46.0785i		−	1.12387i	0.827183	−	0.561933i	$-0.189942\pi$
$42$		0			0
$43$			74.8906i			1.74164i	0.491601	+	0.870821i	$0.336412\pi$
$43$			74.8906i			1.74164i	−0.491601	+	0.870821i	$0.663588\pi$
$44$		0			0
$45$	12.7673	+	7.87383i	0.283717	+	0.174974i
$46$		0			0
$47$	27.7687			0.590824			0.295412	−	0.955370i	$-0.404543\pi$
$47$	27.7687			0.590824			0.295412	+	0.955370i	$0.404543\pi$
$48$		0			0
$49$	−35.8123	+	33.4437i	−0.730863	+	0.682525i
$50$		0			0
$51$	51.5639			1.01106
$52$		0			0
$53$		−	5.38329i		−	0.101572i	−0.998710	−	0.0507858i	$-0.983827\pi$
$53$		−	5.38329i		−	0.101572i	0.998710	−	0.0507858i	$-0.0161726\pi$
$54$		0			0
$55$	35.6453	+	21.9832i	0.648097	+	0.399694i
$56$		0			0
$57$			30.9761i			0.543440i
$58$		0			0
$59$			1.93853i			0.0328565i	0.999865	+	0.0164282i	$0.00522951\pi$
$59$			1.93853i			0.0328565i	−0.999865	+	0.0164282i	$0.994770\pi$
$60$		0			0
$61$			74.8723i			1.22741i	0.789534	+	0.613707i	$0.210323\pi$
$61$			74.8723i			1.22741i	−0.789534	+	0.613707i	$0.789677\pi$
$62$		0			0
$63$	7.70356	+	19.5360i	0.122279	+	0.310095i
$64$		0			0
$65$	−0.375797	−	0.231762i	−0.00578149	−	0.00356556i
$66$		0			0
$67$			30.6588i			0.457594i	0.973474	+	0.228797i	$0.0734791\pi$
$67$			30.6588i			0.457594i	−0.973474	+	0.228797i	$0.926521\pi$
$68$		0			0
$69$		−	68.0617i		−	0.986402i
$70$		0			0
$71$	−72.3104			−1.01846			−0.509228	−	0.860631i	$-0.670069\pi$
$71$	−72.3104			−1.01846			−0.509228	+	0.860631i	$0.670069\pi$
$72$		0			0
$73$	−34.0300			−0.466165			−0.233082	−	0.972457i	$-0.574881\pi$
$73$	−34.0300			−0.466165			−0.233082	+	0.972457i	$0.574881\pi$
$74$		0			0
$75$	−19.4385	−	38.6929i	−0.259181	−	0.515906i
$76$		0			0
$77$	21.5078	+	54.5432i	0.279322	+	0.708353i
$78$		0			0
$79$	125.748			1.59175			0.795873	−	0.605464i	$-0.207012\pi$
$79$	125.748			1.59175			0.795873	+	0.605464i	$0.207012\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$		0			0
$83$	−92.4285			−1.11360			−0.556798	−	0.830648i	$-0.687970\pi$
$83$	−92.4285			−1.11360			−0.556798	+	0.830648i	$0.687970\pi$
$84$		0			0
$85$	−126.695	−	78.1357i	−1.49053	−	0.919243i
$86$		0			0
$87$	−75.7197			−0.870342
$88$		0			0
$89$		−	141.387i		−	1.58862i	−0.607514	−	0.794309i	$-0.707833\pi$
$89$		−	141.387i		−	1.58862i	0.607514	−	0.794309i	$-0.292167\pi$
$90$		0			0
$91$	−0.226750	−	0.575031i	−0.00249176	−	0.00631902i
$92$		0			0
$93$		−	93.2979i		−	1.00320i
$94$		0			0
$95$	46.9386	−	76.1100i	0.494091	−	0.801158i
$96$		0			0
$97$	107.878			1.11214			0.556070	−	0.831135i	$-0.312309\pi$
$97$	107.878			1.11214			0.556070	+	0.831135i	$0.312309\pi$
$98$		0			0
$99$	25.1274			0.253812
$100$		0			0
$101$		−	0.600487i		−	0.00594541i	−0.999996	−	0.00297271i	$-0.999054\pi$
$101$		−	0.600487i		−	0.00594541i	0.999996	−	0.00297271i	$-0.000946243\pi$
$102$		0			0
$103$	105.893			1.02808			0.514041	−	0.857765i	$-0.328148\pi$
$103$	105.893			1.02808			0.514041	+	0.857765i	$0.328148\pi$
$104$		0			0
$105$	10.6752	−	59.6745i	0.101668	−	0.568328i
$106$		0			0
$107$		−	119.448i		−	1.11634i	−0.829728	−	0.558168i	$-0.811504\pi$
$107$		−	119.448i		−	1.11634i	0.829728	−	0.558168i	$-0.188496\pi$
$108$		0			0
$109$	−68.8319			−0.631485			−0.315742	−	0.948845i	$-0.602254\pi$
$109$	−68.8319			−0.631485			−0.315742	+	0.948845i	$0.602254\pi$
$110$		0			0
$111$			46.8415i			0.421996i
$112$		0			0
$113$		−	82.4292i		−	0.729462i	−0.931113	−	0.364731i	$-0.881161\pi$
$113$		−	82.4292i		−	0.729462i	0.931113	−	0.364731i	$-0.118839\pi$
$114$		0			0
$115$	−103.135	+	167.232i	−0.896828	+	1.45419i
$116$		0			0
$117$	−0.264910			−0.00226419
$118$		0			0
$119$	−76.4460	−	193.865i	−0.642404	−	1.62912i
$120$		0			0
$121$	−50.8460			−0.420215
$122$		0			0
$123$			79.8104i			0.648865i
$124$		0			0
$125$	−10.8705	+	124.526i	−0.0869637	+	0.996211i
$126$		0			0
$127$			77.8951i			0.613348i	0.951815	+	0.306674i	$0.0992161\pi$
$127$			77.8951i			0.613348i	−0.951815	+	0.306674i	$0.900784\pi$
$128$		0			0
$129$		−	129.714i		−	1.00554i
$130$		0			0
$131$		−	40.8293i		−	0.311674i	−0.987783	−	0.155837i	$-0.950193\pi$
$131$		−	40.8293i		−	0.311674i	0.987783	−	0.155837i	$-0.0498075\pi$
$132$		0			0
$133$	116.461	−	45.9236i	0.875645	−	0.345290i
$134$		0			0
$135$	−22.1135	−	13.6379i	−0.163804	−	0.101021i
$136$		0			0
$137$		−	233.755i		−	1.70624i	−0.521712	−	0.853122i	$-0.674706\pi$
$137$		−	233.755i		−	1.70624i	0.521712	−	0.853122i	$-0.325294\pi$
$138$		0			0
$139$		−	93.7217i		−	0.674257i	−0.941459	−	0.337128i	$-0.890544\pi$
$139$		−	93.7217i		−	0.674257i	0.941459	−	0.337128i	$-0.109456\pi$
$140$		0			0
$141$	−48.0968			−0.341112
$142$		0			0
$143$	−0.739611			−0.00517210
$144$		0			0
$145$	186.048	+	114.740i	1.28309	+	0.791307i
$146$		0			0
$147$	62.0287	−	57.9262i	0.421964	−	0.394056i
$148$		0			0
$149$	141.672			0.950818			0.475409	−	0.879765i	$-0.342300\pi$
$149$	141.672			0.950818			0.475409	+	0.879765i	$0.342300\pi$
$150$		0			0
$151$	−81.1008			−0.537091			−0.268546	−	0.963267i	$-0.586543\pi$
$151$	−81.1008			−0.537091			−0.268546	+	0.963267i	$0.586543\pi$
$152$		0			0
$153$	−89.3112			−0.583733
$154$		0			0
$155$	−141.376	+	229.238i	−0.912103	+	1.47896i
$156$		0			0
$157$	217.806			1.38730			0.693649	−	0.720313i	$-0.256002\pi$
$157$	217.806			1.38730			0.693649	+	0.720313i	$0.256002\pi$
$158$		0			0
$159$			9.32413i			0.0586423i
$160$		0			0
$161$	−255.892	+	100.905i	−1.58939	+	0.626739i
$162$		0			0
$163$		−	149.171i		−	0.915159i	−0.889169	−	0.457580i	$-0.848716\pi$
$163$		−	149.171i		−	0.915159i	0.889169	−	0.457580i	$-0.151284\pi$
$164$		0			0
$165$	−61.7395	−	38.0760i	−0.374179	−	0.230764i
$166$		0			0
$167$	0.343560			0.00205725			0.00102862	−	0.999999i	$-0.499673\pi$
$167$	0.343560			0.00205725			0.00102862	+	0.999999i	$0.499673\pi$
$168$		0			0
$169$	−168.992			−0.999954
$170$		0			0
$171$		−	53.6521i		−	0.313755i
$172$		0			0
$173$	−2.95065			−0.0170558			−0.00852790	−	0.999964i	$-0.502715\pi$
$173$	−2.95065			−0.0170558			−0.00852790	+	0.999964i	$0.502715\pi$
$174$		0			0
$175$	−116.655	+	130.447i	−0.666602	+	0.745414i
$176$		0			0
$177$		−	3.35764i		−	0.0189697i
$178$		0			0
$179$	306.545			1.71254			0.856272	−	0.516526i	$-0.172775\pi$
$179$	306.545			1.71254			0.856272	+	0.516526i	$0.172775\pi$
$180$		0			0
$181$			270.883i			1.49659i	0.663364	+	0.748297i	$0.269128\pi$
$181$			270.883i			1.49659i	−0.663364	+	0.748297i	$0.730872\pi$
$182$		0			0
$183$		−	129.683i		−	0.708648i
$184$		0			0
$185$	70.9799	−	115.092i	0.383675	−	0.622121i
$186$		0			0
$187$	−249.351			−1.33343
$188$		0			0
$189$	−13.3430	−	33.8373i	−0.0705976	−	0.179034i
$190$		0			0
$191$	340.931			1.78498			0.892490	−	0.451067i	$-0.148956\pi$
$191$	340.931			1.78498			0.892490	+	0.451067i	$0.148956\pi$
$192$		0			0
$193$		−	181.872i		−	0.942344i	−0.882041	−	0.471172i	$-0.843831\pi$
$193$		−	181.872i		−	0.942344i	0.882041	−	0.471172i	$-0.156169\pi$
$194$		0			0
$195$	0.650899	+	0.401423i	0.00333795	+	0.00205858i
$196$		0			0
$197$		−	98.7921i		−	0.501483i	−0.968054	−	0.250741i	$-0.919326\pi$
$197$		−	98.7921i		−	0.501483i	0.968054	−	0.250741i	$-0.0806744\pi$
$198$		0			0
$199$		−	181.839i		−	0.913765i	−0.889527	−	0.456882i	$-0.848966\pi$
$199$		−	181.839i		−	0.913765i	0.889527	−	0.456882i	$-0.151034\pi$
$200$		0			0
$201$		−	53.1025i		−	0.264192i
$202$		0			0
$203$	112.258	+	284.684i	0.552997	+	1.40238i
$204$		0			0
$205$	120.938	−	196.099i	0.589942	−	0.956580i
$206$		0			0
$207$			117.886i			0.569499i
$208$		0			0
$209$		−	149.793i		−	0.716713i
$210$		0			0
$211$	10.0016			0.0474009			0.0237004	−	0.999719i	$-0.492455\pi$
$211$	10.0016			0.0474009			0.0237004	+	0.999719i	$0.492455\pi$
$212$		0			0
$213$	125.245			0.588006
$214$		0			0
$215$	−196.559	+	318.716i	−0.914226	+	1.48240i
$216$		0			0
$217$	−350.772	+	138.319i	−1.61646	+	0.637414i
$218$		0			0
$219$	58.9417			0.269140
$220$		0			0
$221$	2.62883			0.0118951
$222$		0			0
$223$	68.5318			0.307318			0.153659	−	0.988124i	$-0.450894\pi$
$223$	68.5318			0.307318			0.153659	+	0.988124i	$0.450894\pi$
$224$		0			0
$225$	33.6686	+	67.0181i	0.149638	+	0.297858i
$226$		0			0
$227$	17.8883			0.0788031			0.0394015	−	0.999223i	$-0.487455\pi$
$227$	17.8883			0.0788031			0.0394015	+	0.999223i	$0.487455\pi$
$228$		0			0
$229$		−	119.591i		−	0.522230i	−0.965308	−	0.261115i	$-0.915910\pi$
$229$		−	119.591i		−	0.522230i	0.965308	−	0.261115i	$-0.0840901\pi$
$230$		0			0
$231$	−37.2526	−	94.4716i	−0.161267	−	0.408968i
$232$		0			0
$233$		−	157.824i		−	0.677357i	−0.940902	−	0.338679i	$-0.890020\pi$
$233$		−	157.824i		−	0.677357i	0.940902	−	0.338679i	$-0.109980\pi$
$234$		0			0
$235$	118.177	+	72.8820i	0.502880	+	0.310136i
$236$		0			0
$237$	−217.802			−0.918995
$238$		0			0
$239$	−299.375			−1.25261			−0.626307	−	0.779576i	$-0.715435\pi$
$239$	−299.375			−1.25261			−0.626307	+	0.779576i	$0.715435\pi$
$240$		0			0
$241$			73.5375i			0.305135i	0.988293	+	0.152568i	$0.0487542\pi$
$241$			73.5375i			0.305135i	−0.988293	+	0.152568i	$0.951246\pi$
$242$		0			0
$243$	−15.5885			−0.0641500
$244$		0			0
$245$	−240.185	+	48.3349i	−0.980346	+	0.197285i
$246$		0			0
$247$			1.57922i			0.00639360i
$248$		0			0
$249$	160.091			0.642935
$250$		0			0
$251$			113.219i			0.451073i	0.974235	+	0.225536i	$0.0724134\pi$
$251$			113.219i			0.451073i	−0.974235	+	0.225536i	$0.927587\pi$
$252$		0			0
$253$			329.131i			1.30091i
$254$		0			0
$255$	219.443	+	135.335i	0.860561	+	0.530725i
$256$		0			0
$257$	−24.6272			−0.0958257			−0.0479128	−	0.998852i	$-0.515257\pi$
$257$	−24.6272			−0.0958257			−0.0479128	+	0.998852i	$0.515257\pi$
$258$		0			0
$259$	176.110	−	69.4450i	0.679963	−	0.268127i
$260$		0			0
$261$	131.150			0.502492
$262$		0			0
$263$		−	81.4679i		−	0.309764i	−0.987933	−	0.154882i	$-0.950500\pi$
$263$		−	81.4679i		−	0.309764i	0.987933	−	0.154882i	$-0.0494997\pi$
$264$		0			0
$265$	14.1290	−	22.9100i	0.0533171	−	0.0864526i
$266$		0			0
$267$			244.890i			0.917189i
$268$		0			0
$269$		−	328.768i		−	1.22218i	−0.791559	−	0.611092i	$-0.790730\pi$
$269$		−	328.768i		−	1.22218i	0.791559	−	0.611092i	$-0.209270\pi$
$270$		0			0
$271$		−	161.616i		−	0.596370i	−0.954508	−	0.298185i	$-0.903619\pi$
$271$		−	161.616i		−	0.596370i	0.954508	−	0.298185i	$-0.0963813\pi$
$272$		0			0
$273$	0.392742	+	0.995983i	0.00143862	+	0.00364829i
$274$		0			0
$275$	94.0003	+	187.110i	0.341819	+	0.680400i
$276$		0			0
$277$			393.057i			1.41898i	0.704717	+	0.709489i	$0.251074\pi$
$277$			393.057i			1.41898i	−0.704717	+	0.709489i	$0.748926\pi$
$278$		0			0
$279$			161.597i			0.579200i
$280$		0			0
$281$	165.651			0.589505			0.294753	−	0.955574i	$-0.404763\pi$
$281$	165.651			0.589505			0.294753	+	0.955574i	$0.404763\pi$
$282$		0			0
$283$	−423.072			−1.49495			−0.747476	−	0.664289i	$-0.768735\pi$
$283$	−423.072			−1.49495			−0.747476	+	0.664289i	$0.768735\pi$
$284$		0			0
$285$	−81.3001	+	131.826i	−0.285263	+	0.462549i
$286$		0			0
$287$	300.063	−	118.323i	1.04552	−	0.412275i
$288$		0			0
$289$	597.277			2.06670
$290$		0			0
$291$	−186.850			−0.642095
$292$		0			0
$293$	−99.9270			−0.341048			−0.170524	−	0.985354i	$-0.554546\pi$
$293$	−99.9270			−0.341048			−0.170524	+	0.985354i	$0.554546\pi$
$294$		0			0
$295$	−5.08789	+	8.24991i	−0.0172471	+	0.0279658i
$296$		0			0
$297$	−43.5219			−0.146538
$298$		0			0
$299$		−	3.46992i		−	0.0116051i
$300$		0			0
$301$	−487.687	+	192.308i	−1.62022	+	0.638897i
$302$		0			0
$303$			1.04007i			0.00343259i
$304$		0			0
$305$	−196.511	+	318.638i	−0.644297	+	1.04471i
$306$		0			0
$307$	287.175			0.935422			0.467711	−	0.883881i	$-0.345079\pi$
$307$	287.175			0.935422			0.467711	+	0.883881i	$0.345079\pi$
$308$		0			0
$309$	−183.411			−0.593564
$310$		0			0
$311$			187.061i			0.601482i	0.953706	+	0.300741i	$0.0972339\pi$
$311$			187.061i			0.601482i	−0.953706	+	0.300741i	$0.902766\pi$
$312$		0			0
$313$	274.169			0.875940			0.437970	−	0.898990i	$-0.355698\pi$
$313$	274.169			0.875940			0.437970	+	0.898990i	$0.355698\pi$
$314$		0			0
$315$	−18.4899	+	103.359i	−0.0586982	+	0.328124i
$316$		0			0
$317$		−	84.2117i		−	0.265652i	−0.991139	−	0.132826i	$-0.957595\pi$
$317$		−	84.2117i		−	0.265652i	0.991139	−	0.132826i	$-0.0424051\pi$
$318$		0			0
$319$	366.163			1.14785
$320$		0			0
$321$			206.890i			0.644517i
$322$		0			0
$323$			532.415i			1.64834i
$324$		0			0
$325$	−0.991015	−	1.97264i	−0.00304928	−	0.00606966i
$326$		0			0
$327$	119.220			0.364588
$328$		0			0
$329$	71.3060	+	180.830i	0.216735	+	0.549635i
$330$		0			0
$331$	568.223			1.71669			0.858343	−	0.513076i	$-0.171494\pi$
$331$	568.223			1.71669			0.858343	+	0.513076i	$0.171494\pi$
$332$		0			0
$333$		−	81.1319i		−	0.243639i
$334$		0			0
$335$	−80.4673	+	130.476i	−0.240201	+	0.389481i
$336$		0			0
$337$			127.801i			0.379232i	0.981858	+	0.189616i	$0.0607243\pi$
$337$			127.801i			0.379232i	−0.981858	+	0.189616i	$0.939276\pi$
$338$		0			0
$339$			142.771i			0.421155i
$340$		0			0
$341$			451.167i			1.32307i
$342$		0			0
$343$	−309.746	−	147.331i	−0.903050	−	0.429536i
$344$		0			0
$345$	178.635	−	289.654i	0.517784	−	0.839576i
$346$		0			0
$347$		−	301.061i		−	0.867612i	−0.901006	−	0.433806i	$-0.857170\pi$
$347$		−	301.061i		−	0.867612i	0.901006	−	0.433806i	$-0.142830\pi$
$348$		0			0
$349$		−	612.654i		−	1.75546i	−0.479159	−	0.877728i	$-0.659058\pi$
$349$		−	612.654i		−	1.75546i	0.479159	−	0.877728i	$-0.340942\pi$
$350$		0			0
$351$	0.458837			0.00130723
$352$		0			0
$353$	−63.4467			−0.179736			−0.0898678	−	0.995954i	$-0.528644\pi$
$353$	−63.4467			−0.179736			−0.0898678	+	0.995954i	$0.528644\pi$
$354$		0			0
$355$	−307.735	−	189.787i	−0.866860	−	0.534610i
$356$		0			0
$357$	132.408	+	335.784i	0.370892	+	0.940571i
$358$		0			0
$359$	−172.375			−0.480154			−0.240077	−	0.970754i	$-0.577173\pi$
$359$	−172.375			−0.480154			−0.240077	+	0.970754i	$0.577173\pi$
$360$		0			0
$361$	41.1612			0.114020
$362$		0			0
$363$	88.0679			0.242611
$364$		0			0
$365$	−144.823	−	89.3155i	−0.396776	−	0.244700i
$366$		0			0
$367$	−162.503			−0.442789			−0.221394	−	0.975184i	$-0.571061\pi$
$367$	−162.503			−0.442789			−0.221394	+	0.975184i	$0.571061\pi$
$368$		0			0
$369$		−	138.236i		−	0.374622i
$370$		0			0
$371$	35.0560	−	13.8235i	0.0944905	−	0.0372601i
$372$		0			0
$373$		−	84.5145i		−	0.226580i	−0.993562	−	0.113290i	$-0.963861\pi$
$373$		−	84.5145i		−	0.226580i	0.993562	−	0.113290i	$-0.0361390\pi$
$374$		0			0
$375$	18.8282	−	215.686i	0.0502085	−	0.575163i
$376$		0			0
$377$	−3.86034			−0.0102396
$378$		0			0
$379$	−570.247			−1.50461			−0.752304	−	0.658816i	$-0.771058\pi$
$379$	−570.247			−1.50461			−0.752304	+	0.658816i	$0.771058\pi$
$380$		0			0
$381$		−	134.918i		−	0.354116i
$382$		0			0
$383$	−91.0688			−0.237777			−0.118889	−	0.992908i	$-0.537933\pi$
$383$	−91.0688			−0.237777			−0.118889	+	0.992908i	$0.537933\pi$
$384$		0			0
$385$	−51.6226	+	288.572i	−0.134085	+	0.749537i
$386$		0			0
$387$			224.672i			0.580547i
$388$		0			0
$389$	62.7679			0.161357			0.0806786	−	0.996740i	$-0.474291\pi$
$389$	62.7679			0.161357			0.0806786	+	0.996740i	$0.474291\pi$
$390$		0			0
$391$		−	1169.84i		−	2.99192i
$392$		0			0
$393$			70.7184i			0.179945i
$394$		0			0
$395$	535.152	+	330.039i	1.35482	+	0.835542i
$396$		0			0
$397$	−175.124			−0.441119			−0.220559	−	0.975374i	$-0.570788\pi$
$397$	−175.124			−0.441119			−0.220559	+	0.975374i	$0.570788\pi$
$398$		0			0
$399$	−201.716	+	79.5420i	−0.505554	+	0.199353i
$400$		0			0
$401$	70.8623			0.176714			0.0883569	−	0.996089i	$-0.471838\pi$
$401$	70.8623			0.176714			0.0883569	+	0.996089i	$0.471838\pi$
$402$		0			0
$403$		−	4.75651i		−	0.0118027i
$404$		0			0
$405$	38.3018	+	23.6215i	0.0945723	+	0.0583247i
$406$		0			0
$407$		−	226.515i		−	0.556548i
$408$		0			0
$409$			425.778i			1.04102i	0.853855	+	0.520511i	$0.174259\pi$
$409$			425.778i			1.04102i	−0.853855	+	0.520511i	$0.825741\pi$
$410$		0			0
$411$			404.876i			0.985100i
$412$		0			0
$413$	−12.6237	+	4.97787i	−0.0305659	+	0.0120529i
$414$		0			0
$415$	−393.353	−	242.589i	−0.947838	−	0.584551i
$416$		0			0
$417$			162.331i			0.389282i
$418$		0			0
$419$			393.611i			0.939406i	0.882825	+	0.469703i	$0.155639\pi$
$419$			393.611i			0.939406i	−0.882825	+	0.469703i	$0.844361\pi$
$420$		0			0
$421$	157.500			0.374110			0.187055	−	0.982349i	$-0.440106\pi$
$421$	157.500			0.374110			0.187055	+	0.982349i	$0.440106\pi$
$422$		0			0
$423$	83.3061			0.196941
$424$		0			0
$425$	−334.109	−	665.052i	−0.786138	−	1.56483i
$426$		0			0
$427$	−487.568	+	192.261i	−1.14185	+	0.450260i
$428$		0			0
$429$	1.28104			0.00298611
$430$		0			0
$431$	−316.064			−0.733327			−0.366663	−	0.930354i	$-0.619500\pi$
$431$	−316.064			−0.733327			−0.366663	+	0.930354i	$0.619500\pi$
$432$		0			0
$433$	530.639			1.22549			0.612747	−	0.790279i	$-0.290064\pi$
$433$	530.639			1.22549			0.612747	+	0.790279i	$0.290064\pi$
$434$		0			0
$435$	−322.244	−	198.735i	−0.740792	−	0.456861i
$436$		0			0
$437$	702.761			1.60815
$438$		0			0
$439$		−	154.699i		−	0.352389i	−0.984355	−	0.176194i	$-0.943621\pi$
$439$		−	154.699i		−	0.352389i	0.984355	−	0.176194i	$-0.0563787\pi$
$440$		0			0
$441$	−107.437	+	100.331i	−0.243621	+	0.227508i
$442$		0			0
$443$		−	94.9327i		−	0.214295i	−0.994243	−	0.107147i	$-0.965828\pi$
$443$		−	94.9327i		−	0.214295i	0.994243	−	0.107147i	$-0.0341717\pi$
$444$		0			0
$445$	371.086	−	601.708i	0.833901	−	1.35215i
$446$		0			0
$447$	−245.383			−0.548955
$448$		0			0
$449$	64.1731			0.142924			0.0714622	−	0.997443i	$-0.477233\pi$
$449$	64.1731			0.142924			0.0714622	+	0.997443i	$0.477233\pi$
$450$		0			0
$451$		−	385.944i		−	0.855753i
$452$		0			0
$453$	140.471			0.310090
$454$		0			0
$455$	0.544241	−	3.04232i	0.00119613	−	0.00668642i
$456$		0			0
$457$		−	270.716i		−	0.592376i	−0.955130	−	0.296188i	$-0.904285\pi$
$457$		−	270.716i		−	0.592376i	0.955130	−	0.296188i	$-0.0957154\pi$
$458$		0			0
$459$	154.692			0.337019
$460$		0			0
$461$		−	73.7555i		−	0.159990i	−0.996795	−	0.0799951i	$-0.974510\pi$
$461$		−	73.7555i		−	0.159990i	0.996795	−	0.0799951i	$-0.0254905\pi$
$462$		0			0
$463$		−	729.376i		−	1.57533i	−0.616106	−	0.787663i	$-0.711291\pi$
$463$		−	729.376i		−	1.57533i	0.616106	−	0.787663i	$-0.288709\pi$
$464$		0			0
$465$	244.870	−	397.053i	0.526603	−	0.853877i
$466$		0			0
$467$	491.236			1.05190			0.525949	−	0.850516i	$-0.323710\pi$
$467$	491.236			1.05190			0.525949	+	0.850516i	$0.323710\pi$
$468$		0			0
$469$	−199.650	+	78.7272i	−0.425693	+	0.167862i
$470$		0			0
$471$	−377.251			−0.800957
$472$		0			0
$473$			627.268i			1.32615i
$474$		0			0
$475$	399.518	−	200.710i	0.841091	−	0.422547i
$476$		0			0
$477$		−	16.1499i		−	0.0338572i
$478$		0			0
$479$		−	32.7275i		−	0.0683246i	−0.999416	−	0.0341623i	$-0.989124\pi$
$479$		−	32.7275i		−	0.0683246i	0.999416	−	0.0341623i	$-0.0108763\pi$
$480$		0			0
$481$			2.38807i			0.00496481i
$482$		0			0
$483$	443.218	−	174.773i	0.917635	−	0.361848i
$484$		0			0
$485$	459.100	+	283.137i	0.946599	+	0.583787i
$486$		0			0
$487$		−	892.345i		−	1.83233i	−0.400800	−	0.916165i	$-0.631268\pi$
$487$		−	892.345i		−	1.83233i	0.400800	−	0.916165i	$-0.368732\pi$
$488$		0			0
$489$			258.372i			0.528367i
$490$		0			0
$491$	−186.106			−0.379035			−0.189517	−	0.981877i	$-0.560692\pi$
$491$	−186.106			−0.379035			−0.189517	+	0.981877i	$0.560692\pi$
$492$		0			0
$493$	−1301.47			−2.63989
$494$		0			0
$495$	106.936	+	65.9496i	0.216032	+	0.133231i
$496$		0			0
$497$	−185.683	−	470.885i	−0.373607	−	0.947456i
$498$		0			0
$499$	−189.705			−0.380171			−0.190085	−	0.981768i	$-0.560876\pi$
$499$	−189.705			−0.380171			−0.190085	+	0.981768i	$0.560876\pi$
$500$		0			0
$501$	−0.595064			−0.00118775
$502$		0			0
$503$	82.3835			0.163784			0.0818921	−	0.996641i	$-0.473904\pi$
$503$	82.3835			0.163784			0.0818921	+	0.996641i	$0.473904\pi$
$504$		0			0
$505$	1.57604	−	2.55552i	0.00312088	−	0.00506044i
$506$		0			0
$507$	292.703			0.577324
$508$		0			0
$509$			612.287i			1.20292i	0.798902	+	0.601461i	$0.205414\pi$
$509$			612.287i			1.20292i	−0.798902	+	0.601461i	$0.794586\pi$
$510$		0			0
$511$	−87.3841	−	221.603i	−0.171006	−	0.433666i
$512$		0			0
$513$			92.9282i			0.181147i
$514$		0			0
$515$	450.652	+	277.927i	0.875053	+	0.539663i
$516$		0			0
$517$	232.585			0.449874
$518$		0			0
$519$	5.11068			0.00984717
$520$		0			0
$521$			190.073i			0.364823i	0.983222	+	0.182412i	$0.0583903\pi$
$521$			190.073i			0.364823i	−0.983222	+	0.182412i	$0.941610\pi$
$522$		0			0
$523$	324.679			0.620800			0.310400	−	0.950606i	$-0.399537\pi$
$523$	324.679			0.620800			0.310400	+	0.950606i	$0.399537\pi$
$524$		0			0
$525$	202.053	−	225.942i	0.384863	−	0.430365i
$526$		0			0
$527$		−	1603.60i		−	3.04288i
$528$		0			0
$529$	−1015.13			−1.91897
$530$		0			0
$531$			5.81560i			0.0109522i
$532$		0			0
$533$			4.06889i			0.00763393i
$534$		0			0
$535$	313.504	−	508.341i	0.585989	−	0.950170i
$536$		0			0
$537$	−530.952			−0.988737
$538$		0			0
$539$	−299.956	+	280.118i	−0.556505	+	0.519699i
$540$		0			0
$541$	−2.39602			−0.00442888			−0.00221444	−	0.999998i	$-0.500705\pi$
$541$	−2.39602			−0.00442888			−0.00221444	+	0.999998i	$0.500705\pi$
$542$		0			0
$543$		−	469.184i		−	0.864059i
$544$		0			0
$545$	−292.931	−	180.657i	−0.537489	−	0.331480i
$546$		0			0
$547$			495.383i			0.905637i	0.891603	+	0.452818i	$0.149581\pi$
$547$			495.383i			0.905637i	−0.891603	+	0.452818i	$0.850419\pi$
$548$		0			0
$549$			224.617i			0.409138i
$550$		0			0
$551$		−	781.833i		−	1.41893i
$552$		0			0
$553$	322.902	+	818.870i	0.583910	+	1.48078i
$554$		0			0
$555$	−122.941	+	199.346i	−0.221515	+	0.359182i
$556$		0			0
$557$		−	180.812i		−	0.324618i	−0.986740	−	0.162309i	$-0.948106\pi$
$557$		−	180.812i		−	0.324618i	0.986740	−	0.162309i	$-0.0518941\pi$
$558$		0			0
$559$		−	6.61309i		−	0.0118302i
$560$		0			0
$561$	431.888			0.769855
$562$		0			0
$563$	−937.810			−1.66574			−0.832868	−	0.553471i	$-0.813303\pi$
$563$	−937.810			−1.66574			−0.832868	+	0.553471i	$0.813303\pi$
$564$		0			0
$565$	216.344	−	350.798i	0.382910	−	0.620882i
$566$		0			0
$567$	23.1107	+	58.6080i	0.0407596	+	0.103365i
$568$		0			0
$569$	−27.5357			−0.0483931			−0.0241965	−	0.999707i	$-0.507703\pi$
$569$	−27.5357			−0.0483931			−0.0241965	+	0.999707i	$0.507703\pi$
$570$		0			0
$571$	404.489			0.708386			0.354193	−	0.935172i	$-0.384756\pi$
$571$	404.489			0.708386			0.354193	+	0.935172i	$0.384756\pi$
$572$		0			0
$573$	−590.510			−1.03056
$574$		0			0
$575$	−877.836	+	441.007i	−1.52667	+	0.766969i
$576$		0			0
$577$	−480.904			−0.833456			−0.416728	−	0.909031i	$-0.636823\pi$
$577$	−480.904			−0.833456			−0.416728	+	0.909031i	$0.636823\pi$
$578$		0			0
$579$			315.012i			0.544063i
$580$		0			0
$581$	−237.343	−	601.894i	−0.408507	−	1.03596i
$582$		0			0
$583$		−	45.0894i		−	0.0773402i
$584$		0			0
$585$	−1.12739	−	0.695285i	−0.00192716	−	0.00118852i
$586$		0			0
$587$	−335.196			−0.571033			−0.285516	−	0.958374i	$-0.592165\pi$
$587$	−335.196			−0.571033			−0.285516	+	0.958374i	$0.592165\pi$
$588$		0			0
$589$	963.334			1.63554
$590$		0			0
$591$			171.113i			0.289531i
$592$		0			0
$593$	1125.32			1.89768			0.948840	−	0.315757i	$-0.102258\pi$
$593$	1125.32			1.89768			0.948840	+	0.315757i	$0.102258\pi$
$594$		0			0
$595$	183.484	−	1025.68i	0.308377	−	1.72383i
$596$		0			0
$597$			314.955i			0.527562i
$598$		0			0
$599$	−643.405			−1.07413			−0.537066	−	0.843540i	$-0.680467\pi$
$599$	−643.405			−1.07413			−0.537066	+	0.843540i	$0.680467\pi$
$600$		0			0
$601$			521.270i			0.867338i	0.901072	+	0.433669i	$0.142781\pi$
$601$			521.270i			0.867338i	−0.901072	+	0.433669i	$0.857219\pi$
$602$		0			0
$603$			91.9763i			0.152531i
$604$		0			0
$605$	−216.388	−	133.451i	−0.357666	−	0.220580i
$606$		0			0
$607$	82.3135			0.135607			0.0678035	−	0.997699i	$-0.478401\pi$
$607$	82.3135			0.135607			0.0678035	+	0.997699i	$0.478401\pi$
$608$		0			0
$609$	−194.437	−	493.087i	−0.319273	−	0.809666i
$610$		0			0
$611$	−2.45207			−0.00401321
$612$		0			0
$613$			228.128i			0.372150i	0.982536	+	0.186075i	$0.0595767\pi$
$613$			228.128i			0.372150i	−0.982536	+	0.186075i	$0.940423\pi$
$614$		0			0
$615$	−209.471	+	339.653i	−0.340603	+	0.552281i
$616$		0			0
$617$		−	516.126i		−	0.836509i	−0.908330	−	0.418255i	$-0.862642\pi$
$617$		−	516.126i		−	0.836509i	0.908330	−	0.418255i	$-0.137358\pi$
$618$		0			0
$619$			683.396i			1.10403i	0.833833	+	0.552016i	$0.186141\pi$
$619$			683.396i			1.10403i	−0.833833	+	0.552016i	$0.813859\pi$
$620$		0			0
$621$		−	204.185i		−	0.328801i
$622$		0			0
$623$	920.712	−	363.061i	1.47787	−	0.582763i
$624$		0			0
$625$	−373.095	+	501.423i	−0.596952	+	0.802277i
$626$		0			0
$627$			259.449i			0.413795i
$628$		0			0
$629$			805.110i			1.27998i
$630$		0			0
$631$	13.6990			0.0217100			0.0108550	−	0.999941i	$-0.496545\pi$
$631$	13.6990			0.0217100			0.0108550	+	0.999941i	$0.496545\pi$
$632$		0			0
$633$	−17.3233			−0.0273669
$634$		0			0
$635$	−204.444	+	331.502i	−0.321960	+	0.522051i
$636$		0			0
$637$	3.16234	−	2.95319i	0.00496443	−	0.00463609i
$638$		0			0
$639$	−216.931			−0.339486
$640$		0			0
$641$	−655.326			−1.02235			−0.511175	−	0.859477i	$-0.670789\pi$
$641$	−655.326			−1.02235			−0.511175	+	0.859477i	$0.670789\pi$
$642$		0			0
$643$	588.998			0.916016			0.458008	−	0.888948i	$-0.348563\pi$
$643$	588.998			0.916016			0.458008	+	0.888948i	$0.348563\pi$
$644$		0			0
$645$	340.449	−	552.032i	0.527829	−	0.855863i
$646$		0			0
$647$	1014.59			1.56815			0.784075	−	0.620666i	$-0.213138\pi$
$647$	1014.59			1.56815			0.784075	+	0.620666i	$0.213138\pi$
$648$		0			0
$649$			16.2368i			0.0250181i
$650$		0			0
$651$	607.556	−	239.575i	0.933265	−	0.368011i
$652$		0			0
$653$			741.204i			1.13508i	0.823348	+	0.567538i	$0.192104\pi$
$653$			741.204i			1.13508i	−0.823348	+	0.567538i	$0.807896\pi$
$654$		0			0
$655$	107.161	−	173.759i	0.163605	−	0.265282i
$656$		0			0
$657$	−102.090			−0.155388
$658$		0			0
$659$	257.591			0.390882			0.195441	−	0.980715i	$-0.437386\pi$
$659$	257.591			0.390882			0.195441	+	0.980715i	$0.437386\pi$
$660$		0			0
$661$			1183.96i			1.79117i	0.444893	+	0.895584i	$0.353242\pi$
$661$			1183.96i			1.79117i	−0.444893	+	0.895584i	$0.646758\pi$
$662$		0			0
$663$	−4.55326			−0.00686766
$664$		0			0
$665$	616.160	+	110.225i	0.926556	+	0.165752i
$666$		0			0
$667$			1717.87i			2.57552i
$668$		0			0
$669$	−118.701			−0.177430
$670$		0			0
$671$			627.115i			0.934598i
$672$		0			0
$673$		−	987.972i		−	1.46801i	−0.679143	−	0.734006i	$-0.737648\pi$
$673$		−	987.972i		−	1.46801i	0.679143	−	0.734006i	$-0.262352\pi$
$674$		0			0
$675$	−58.3156	−	116.079i	−0.0863935	−	0.171969i
$676$		0			0
$677$	−657.422			−0.971082			−0.485541	−	0.874214i	$-0.661377\pi$
$677$	−657.422			−0.971082			−0.485541	+	0.874214i	$0.661377\pi$
$678$		0			0
$679$	277.014	+	702.499i	0.407973	+	1.03461i
$680$		0			0
$681$	−30.9834			−0.0454970
$682$		0			0
$683$			255.185i			0.373624i	0.982396	+	0.186812i	$0.0598154\pi$
$683$			255.185i			0.373624i	−0.982396	+	0.186812i	$0.940185\pi$
$684$		0			0
$685$	613.517	−	994.805i	0.895645	−	1.45227i
$686$		0			0
$687$			207.137i			0.301510i
$688$		0			0
$689$			0.475362i			0.000689931i
$690$		0			0
$691$		−	442.576i		−	0.640486i	−0.947336	−	0.320243i	$-0.896236\pi$
$691$		−	442.576i		−	0.640486i	0.947336	−	0.320243i	$-0.103764\pi$
$692$		0			0
$693$	64.5234	+	163.630i	0.0931074	+	0.236118i
$694$		0			0
$695$	245.983	−	398.856i	0.353932	−	0.573894i
$696$		0			0
$697$			1371.78i			1.96812i
$698$		0			0
$699$			273.360i			0.391072i
$700$		0			0
$701$	771.107			1.10001			0.550005	−	0.835161i	$-0.314626\pi$
$701$	771.107			1.10001			0.550005	+	0.835161i	$0.314626\pi$
$702$		0			0
$703$	−483.656			−0.687988
$704$		0			0
$705$	−204.688	−	126.235i	−0.290338	−	0.179057i
$706$		0			0
$707$	3.91037	−	1.54196i	0.00553093	−	0.00218099i
$708$		0			0
$709$	−431.982			−0.609284			−0.304642	−	0.952467i	$-0.598537\pi$
$709$	−431.982			−0.609284			−0.304642	+	0.952467i	$0.598537\pi$
$710$		0			0
$711$	377.244			0.530582
$712$		0			0
$713$	−2116.67			−2.96868
$714$		0			0
$715$	−3.14760	−	1.94119i	−0.00440224	−	0.00271495i
$716$		0			0
$717$	518.533			0.723197
$718$		0			0
$719$			767.053i			1.06683i	0.845853	+	0.533416i	$0.179092\pi$
$719$			767.053i			1.06683i	−0.845853	+	0.533416i	$0.820908\pi$
$720$		0			0
$721$	271.916	+	689.572i	0.377138	+	0.956411i
$722$		0			0
$723$		−	127.371i		−	0.176170i
$724$		0			0
$725$	490.627	+	976.606i	0.676727	+	1.34704i
$726$		0			0
$727$	230.714			0.317350			0.158675	−	0.987331i	$-0.449278\pi$
$727$	230.714			0.317350			0.158675	+	0.987331i	$0.449278\pi$
$728$		0			0
$729$	27.0000			0.0370370
$730$		0			0
$731$		−	2229.52i		−	3.04996i
$732$		0			0
$733$	632.429			0.862795			0.431398	−	0.902162i	$-0.358021\pi$
$733$	632.429			0.862795			0.431398	+	0.902162i	$0.358021\pi$
$734$		0			0
$735$	416.012	−	83.7186i	0.566003	−	0.113903i
$736$		0			0
$737$			256.792i			0.348428i
$738$		0			0
$739$	466.036			0.630631			0.315316	−	0.948987i	$-0.397890\pi$
$739$	466.036			0.630631			0.315316	+	0.948987i	$0.397890\pi$
$740$		0			0
$741$		−	2.73529i		−	0.00369135i
$742$		0			0
$743$			632.427i			0.851180i	0.904916	+	0.425590i	$0.139933\pi$
$743$			632.427i			0.851180i	−0.904916	+	0.425590i	$0.860067\pi$
$744$		0			0
$745$	602.920	+	371.833i	0.809289	+	0.499105i
$746$		0			0
$747$	−277.285			−0.371199
$748$		0			0
$749$	777.845	−	306.725i	1.03851	−	0.409513i
$750$		0			0
$751$	−387.564			−0.516064			−0.258032	−	0.966136i	$-0.583074\pi$
$751$	−387.564			−0.516064			−0.258032	+	0.966136i	$0.583074\pi$
$752$		0			0
$753$		−	196.101i		−	0.260427i
$754$		0			0
$755$	−345.145	−	212.858i	−0.457146	−	0.281931i
$756$		0			0
$757$			343.268i			0.453458i	0.973958	+	0.226729i	$0.0728032\pi$
$757$			343.268i			0.453458i	−0.973958	+	0.226729i	$0.927197\pi$
$758$		0			0
$759$		−	570.071i		−	0.751082i
$760$		0			0
$761$		−	67.4358i		−	0.0886147i	−0.999018	−	0.0443073i	$-0.985892\pi$
$761$		−	67.4358i		−	0.0886147i	0.999018	−	0.0443073i	$-0.0141081\pi$
$762$		0			0
$763$	−176.750	−	448.233i	−0.231651	−	0.587461i
$764$		0			0
$765$	−380.086	−	234.407i	−0.496845	−	0.306414i
$766$		0			0
$767$		−	0.171179i		−	0.000223180i
$768$		0			0
$769$			1474.43i			1.91733i	0.284529	+	0.958667i	$0.408163\pi$
$769$			1474.43i			1.91733i	−0.284529	+	0.958667i	$0.591837\pi$
$770$		0			0
$771$	42.6556			0.0553250
$772$		0			0
$773$	−670.136			−0.866929			−0.433465	−	0.901171i	$-0.642709\pi$
$773$	−670.136			−0.866929			−0.433465	+	0.901171i	$0.642709\pi$
$774$		0			0
$775$	−1203.32	+	604.525i	−1.55267	+	0.780032i
$776$		0			0
$777$	−305.032	+	120.282i	−0.392577	+	0.154803i
$778$		0			0
$779$	−824.070			−1.05786
$780$		0			0
$781$	−605.658			−0.775490
$782$		0			0
$783$	−227.159			−0.290114
$784$		0			0
$785$	926.928	+	571.655i	1.18080	+	0.728223i
$786$		0			0
$787$	−1177.11			−1.49569			−0.747846	−	0.663873i	$-0.768912\pi$
$787$	−1177.11			−1.49569			−0.747846	+	0.663873i	$0.768912\pi$
$788$		0			0
$789$			141.107i			0.178842i
$790$		0			0
$791$	536.779	−	211.666i	0.678608	−	0.267593i
$792$		0			0
$793$		−	6.61147i		−	0.00833729i
$794$		0			0
$795$	−24.4722	+	39.6812i	−0.0307827	+	0.0499135i
$796$		0			0
$797$	370.289			0.464604			0.232302	−	0.972644i	$-0.425374\pi$
$797$	370.289			0.464604			0.232302	+	0.972644i	$0.425374\pi$
$798$		0			0
$799$	−826.686			−1.03465
$800$		0			0
$801$		−	424.161i		−	0.529540i
$802$		0			0
$803$	−285.029			−0.354955
$804$		0			0
$805$	−1353.85	−	242.190i	−1.68180	−	0.300857i
$806$		0			0
$807$			569.442i			0.705628i
$808$		0			0
$809$	−260.603			−0.322130			−0.161065	−	0.986944i	$-0.551493\pi$
$809$	−260.603			−0.322130			−0.161065	+	0.986944i	$0.551493\pi$
$810$		0			0
$811$		−	867.045i		−	1.06911i	−0.845135	−	0.534553i	$-0.820480\pi$
$811$		−	867.045i		−	1.06911i	0.845135	−	0.534553i	$-0.179520\pi$
$812$		0			0
$813$			279.928i			0.344315i
$814$		0			0
$815$	391.515	−	634.835i	0.480387	−	0.778938i
$816$		0			0
$817$	1339.35			1.63935
$818$		0			0
$819$	−0.680250	−	1.72509i	−0.000830586	−	0.00210634i
$820$		0			0
$821$	−70.3996			−0.0857486			−0.0428743	−	0.999080i	$-0.513652\pi$
$821$	−70.3996			−0.0857486			−0.0428743	+	0.999080i	$0.513652\pi$
$822$		0			0
$823$		−	123.305i		−	0.149824i	−0.997190	−	0.0749118i	$-0.976132\pi$
$823$		−	123.305i		−	0.149824i	0.997190	−	0.0749118i	$-0.0238675\pi$
$824$		0			0
$825$	−162.813	−	324.084i	−0.197349	−	0.392829i
$826$		0			0
$827$		−	760.741i		−	0.919880i	−0.887950	−	0.459940i	$-0.847871\pi$
$827$		−	760.741i		−	0.919880i	0.887950	−	0.459940i	$-0.152129\pi$
$828$		0			0
$829$			802.031i			0.967468i	0.875215	+	0.483734i	$0.160720\pi$
$829$			802.031i			0.967468i	−0.875215	+	0.483734i	$0.839280\pi$
$830$		0			0
$831$		−	680.794i		−	0.819247i
$832$		0			0
$833$	1066.15	−	995.633i	1.27989	−	1.19524i
$834$		0			0
$835$	1.46211	+	0.901712i	0.00175103	+	0.00107989i
$836$		0			0
$837$		−	279.894i		−	0.334401i
$838$		0			0
$839$		−	442.366i		−	0.527254i	−0.964625	−	0.263627i	$-0.915081\pi$
$839$		−	442.366i		−	0.527254i	0.964625	−	0.263627i	$-0.0849188\pi$
$840$		0			0
$841$	1070.16			1.27248
$842$		0			0
$843$	−286.916			−0.340351
$844$		0			0
$845$	−719.189	−	443.539i	−0.851111	−	0.524898i
$846$		0			0
$847$	−130.565	−	331.109i	−0.154150	−	0.390920i
$848$		0			0
$849$	732.781			0.863111
$850$		0			0
$851$	1062.71			1.24877
$852$		0			0
$853$	648.240			0.759953			0.379977	−	0.924996i	$-0.375932\pi$
$853$	648.240			0.759953			0.379977	+	0.924996i	$0.375932\pi$
$854$		0			0
$855$	140.816	−	228.330i	0.164697	−	0.267053i
$856$		0			0
$857$	336.227			0.392331			0.196165	−	0.980571i	$-0.437151\pi$
$857$	336.227			0.392331			0.196165	+	0.980571i	$0.437151\pi$
$858$		0			0
$859$			1188.57i			1.38366i	0.722058	+	0.691832i	$0.243196\pi$
$859$			1188.57i			1.38366i	−0.722058	+	0.691832i	$0.756804\pi$
$860$		0			0
$861$	−519.725	+	204.941i	−0.603629	+	0.238027i
$862$		0			0
$863$			1071.22i			1.24128i	0.784096	+	0.620639i	$0.213127\pi$
$863$			1071.22i			1.24128i	−0.784096	+	0.620639i	$0.786873\pi$
$864$		0			0
$865$	−12.5572	−	7.74431i	−0.0145170	−	0.00895296i
$866$		0			0
$867$	−1034.51			−1.19321
$868$		0			0
$869$	1053.24			1.21201
$870$		0			0
$871$		−	2.70727i		−	0.00310823i
$872$		0			0
$873$	323.633			0.370713
$874$		0			0
$875$	−838.830	+	248.977i	−0.958663	+	0.284545i
$876$		0			0
$877$		−	163.935i		−	0.186927i	−0.995623	−	0.0934635i	$-0.970206\pi$
$877$		−	163.935i		−	0.186927i	0.995623	−	0.0934635i	$-0.0297938\pi$
$878$		0			0
$879$	173.079			0.196904
$880$		0			0
$881$			1088.98i			1.23608i	0.786148	+	0.618038i	$0.212072\pi$
$881$			1088.98i			1.23608i	−0.786148	+	0.618038i	$0.787928\pi$
$882$		0			0
$883$			1148.47i			1.30064i	0.759660	+	0.650321i	$0.225365\pi$
$883$			1148.47i			1.30064i	−0.759660	+	0.650321i	$0.774635\pi$
$884$		0			0
$885$	8.81248	−	14.2893i	0.00995761	−	0.0161461i
$886$		0			0
$887$	−468.656			−0.528361			−0.264180	−	0.964473i	$-0.585101\pi$
$887$	−468.656			−0.528361			−0.264180	+	0.964473i	$0.585101\pi$
$888$		0			0
$889$	−507.253	+	200.023i	−0.570588	+	0.224998i
$890$		0			0
$891$	75.3822			0.0846040
$892$		0			0
$893$		−	496.617i		−	0.556122i
$894$		0			0
$895$	1304.58	+	804.562i	1.45763	+	0.898951i
$896$		0			0
$897$			6.01008i			0.00670020i
$898$		0			0
$899$			2354.83i			2.61939i
$900$		0			0
$901$			160.263i			0.177872i
$902$		0			0
$903$	844.699	−	333.087i	0.935437	−	0.368867i
$904$		0			0
$905$	−710.963	+	1152.81i	−0.785595	+	1.27383i
$906$		0			0
$907$			61.3264i			0.0676145i	0.999428	+	0.0338073i	$0.0107632\pi$
$907$			61.3264i			0.0676145i	−0.999428	+	0.0338073i	$0.989237\pi$
$908$		0			0
$909$		−	1.80146i		−	0.00198180i
$910$		0			0
$911$	86.0805			0.0944901			0.0472451	−	0.998883i	$-0.484956\pi$
$911$	86.0805			0.0944901			0.0472451	+	0.998883i	$0.484956\pi$
$912$		0			0
$913$	−774.162			−0.847932
$914$		0			0
$915$	340.366	−	551.897i	0.371985	−	0.603166i
$916$		0			0
$917$	265.880	−	104.844i	0.289946	−	0.114333i
$918$		0			0
$919$	−1330.80			−1.44810			−0.724048	−	0.689750i	$-0.757721\pi$
$919$	−1330.80			−1.44810			−0.724048	+	0.689750i	$0.757721\pi$
$920$		0			0
$921$	−497.401			−0.540066
$922$		0			0
$923$	6.38525			0.00691793
$924$		0			0
$925$	604.146	−	303.511i	0.653130	−	0.328120i
$926$		0			0
$927$	317.678			0.342694
$928$		0			0
$929$		−	1382.16i		−	1.48779i	−0.668297	−	0.743895i	$-0.732976\pi$
$929$		−	1382.16i		−	1.48779i	0.668297	−	0.743895i	$-0.267024\pi$
$930$		0			0
$931$	598.109	+	640.468i	0.642437	+	0.687935i
$932$		0			0
$933$		−	323.999i		−	0.347266i
$934$		0			0
$935$	−1061.18	−	654.449i	−1.13495	−	0.699945i
$936$		0			0
$937$	1470.54			1.56941			0.784704	−	0.619871i	$-0.212815\pi$
$937$	1470.54			1.56941			0.784704	+	0.619871i	$0.212815\pi$
$938$		0			0
$939$	−474.875			−0.505724
$940$		0			0
$941$			709.900i			0.754410i	0.926130	+	0.377205i	$0.123115\pi$
$941$			709.900i			0.754410i	−0.926130	+	0.377205i	$0.876885\pi$
$942$		0			0
$943$	1810.68			1.92012
$944$		0			0
$945$	32.0255	−	179.023i	0.0338894	−	0.189443i
$946$		0			0
$947$			1170.44i			1.23594i	0.786201	+	0.617971i	$0.212045\pi$
$947$			1170.44i			1.23594i	−0.786201	+	0.617971i	$0.787955\pi$
$948$		0			0
$949$	3.00496			0.00316645
$950$		0			0
$951$			145.859i			0.153374i
$952$		0			0
$953$		−	1354.92i		−	1.42174i	−0.703323	−	0.710871i	$-0.748301\pi$
$953$		−	1354.92i		−	1.42174i	0.703323	−	0.710871i	$-0.251699\pi$
$954$		0			0
$955$	1450.92	+	894.812i	1.51929	+	0.936975i
$956$		0			0
$957$	−634.213			−0.662710
$958$		0			0
$959$	1522.21	−	600.249i	1.58729	−	0.625912i
$960$		0			0
$961$	−1940.50			−2.01925
$962$		0			0
$963$		−	358.344i		−	0.372112i
$964$		0			0
$965$	477.344	−	774.004i	0.494657	−	0.802077i
$966$		0			0
$967$		−	564.951i		−	0.584231i	−0.956383	−	0.292115i	$-0.905641\pi$
$967$		−	564.951i		−	0.584231i	0.956383	−	0.292115i	$-0.0943591\pi$
$968$		0			0
$969$		−	922.170i		−	0.951672i
$970$		0			0
$971$			579.120i			0.596416i	0.954501	+	0.298208i	$0.0963889\pi$
$971$			579.120i			0.596416i	−0.954501	+	0.298208i	$0.903611\pi$
$972$		0			0
$973$	610.316	−	240.664i	0.627251	−	0.247342i
$974$		0			0
$975$	1.71649	+	3.41671i	0.00176050	+	0.00350432i
$976$		0			0
$977$		−	304.418i		−	0.311584i	−0.987790	−	0.155792i	$-0.950207\pi$
$977$		−	304.418i		−	0.311584i	0.987790	−	0.155792i	$-0.0497930\pi$
$978$		0			0
$979$		−	1184.23i		−	1.20963i
$980$		0			0
$981$	−206.496			−0.210495
$982$		0			0
$983$	−764.954			−0.778183			−0.389092	−	0.921199i	$-0.627211\pi$
$983$	−764.954			−0.778183			−0.389092	+	0.921199i	$0.627211\pi$
$984$		0			0
$985$	259.291	−	420.435i	0.263239	−	0.426837i
$986$		0			0
$987$	−123.506	−	313.206i	−0.125132	−	0.317332i
$988$		0			0
$989$	−2942.86			−2.97559
$990$		0			0
$991$	−395.387			−0.398978			−0.199489	−	0.979900i	$-0.563928\pi$
$991$	−395.387			−0.398978			−0.199489	+	0.979900i	$0.563928\pi$
$992$		0			0
$993$	−984.192			−0.991129
$994$		0			0
$995$	477.257	−	773.862i	0.479655	−	0.777751i
$996$		0			0
$997$	−1308.65			−1.31258			−0.656292	−	0.754507i	$-0.727876\pi$
$997$	−1308.65			−1.31258			−0.656292	+	0.754507i	$0.727876\pi$
$998$		0			0
$999$			140.525i			0.140665i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	420.3.p.a.349.8	yes	16
3.2	odd	2		1260.3.p.e.1189.1		16
4.3	odd	2		1680.3.bd.b.769.16		16
5.2	odd	4		2100.3.j.g.601.9		16
5.3	odd	4		2100.3.j.g.601.8		16
5.4	even	2	inner	420.3.p.a.349.10	yes	16
7.6	odd	2	inner	420.3.p.a.349.9	yes	16
15.14	odd	2		1260.3.p.e.1189.15		16
20.19	odd	2		1680.3.bd.b.769.2		16
21.20	even	2		1260.3.p.e.1189.16		16
28.27	even	2		1680.3.bd.b.769.1		16
35.13	even	4		2100.3.j.g.601.16		16
35.27	even	4		2100.3.j.g.601.1		16
35.34	odd	2	inner	420.3.p.a.349.7	✓	16
105.104	even	2		1260.3.p.e.1189.2		16
140.139	even	2		1680.3.bd.b.769.15		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.3.p.a.349.7	✓	16	35.34	odd	2	inner
420.3.p.a.349.8	yes	16	1.1	even	1	trivial
420.3.p.a.349.9	yes	16	7.6	odd	2	inner
420.3.p.a.349.10	yes	16	5.4	even	2	inner
1260.3.p.e.1189.1		16	3.2	odd	2
1260.3.p.e.1189.2		16	105.104	even	2
1260.3.p.e.1189.15		16	15.14	odd	2
1260.3.p.e.1189.16		16	21.20	even	2
1680.3.bd.b.769.1		16	28.27	even	2
1680.3.bd.b.769.2		16	20.19	odd	2
1680.3.bd.b.769.15		16	140.139	even	2
1680.3.bd.b.769.16		16	4.3	odd	2
2100.3.j.g.601.1		16	35.27	even	4
2100.3.j.g.601.8		16	5.3	odd	4
2100.3.j.g.601.9		16	5.2	odd	4
2100.3.j.g.601.16		16	35.13	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	420.p (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.73205 q^{3} +(4.25575 + 2.62461i) q^{5} +(2.56785 + 6.51200i) q^{7} +3.00000 q^{9} +8.37580 q^{11} -0.0883033 q^{13} +(-7.37118 - 4.54596i) q^{15} -29.7704 q^{17} -17.8840i q^{19} +(-4.44765 - 11.2791i) q^{21} +39.2955i q^{23} +(11.2229 + 22.3394i) q^{25} -5.19615 q^{27} +43.7168 q^{29} +53.8656i q^{31} -14.5073 q^{33} +(-6.16331 + 34.4531i) q^{35} -27.0440i q^{37} +0.152946 q^{39} -46.0785i q^{41} +74.8906i q^{43} +(12.7673 + 7.87383i) q^{45} +27.7687 q^{47} +(-35.8123 + 33.4437i) q^{49} +51.5639 q^{51} -5.38329i q^{53} +(35.6453 + 21.9832i) q^{55} +30.9761i q^{57} +1.93853i q^{59} +74.8723i q^{61} +(7.70356 + 19.5360i) q^{63} +(-0.375797 - 0.231762i) q^{65} +30.6588i q^{67} -68.0617i q^{69} -72.3104 q^{71} -34.0300 q^{73} +(-19.4385 - 38.6929i) q^{75} +(21.5078 + 54.5432i) q^{77} +125.748 q^{79} +9.00000 q^{81} -92.4285 q^{83} +(-126.695 - 78.1357i) q^{85} -75.7197 q^{87} -141.387i q^{89} +(-0.226750 - 0.575031i) q^{91} -93.2979i q^{93} +(46.9386 - 76.1100i) q^{95} +107.878 q^{97} +25.1274 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 48 q^{9} - 24 q^{11} - 24 q^{15} - 12 q^{21} - 48 q^{25} - 32 q^{29} + 76 q^{35} + 72 q^{39} - 88 q^{49} + 24 q^{51} + 152 q^{65} + 168 q^{71} + 16 q^{79} + 144 q^{81} - 416 q^{85} - 568 q^{91} + 136 q^{95}+ \cdots - 72 q^{99}+O(q^{100}) \) 16 * q + 48 * q^9 - 24 * q^11 - 24 * q^15 - 12 * q^21 - 48 * q^25 - 32 * q^29 + 76 * q^35 + 72 * q^39 - 88 * q^49 + 24 * q^51 + 152 * q^65 + 168 * q^71 + 16 * q^79 + 144 * q^81 - 416 * q^85 - 568 * q^91 + 136 * q^95 - 72 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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