LMFDB - Embedded newform 2100.3.j.g.601.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2100,3,Mod(601,2100)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2100, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 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("2100.601"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2100.j (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,-48,0,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$57.2208555157$
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} - 56 x^{14} + 2052 x^{12} - 43310 x^{10} + 663499 x^{8} - 6680748 x^{6} + 49052709 x^{4} + \cdots + 601475625 $ x^16 - 56x^14 + 2052x^12 - 43310x^10 + 663499x^8 - 6680748x^6 + 49052709x^4 - 213293925*x^2 + 601475625
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{20}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 420)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			601.8
Root			$2.27298 + 1.31230i$ of defining polynomial
Character	$\chi$	$=$	2100.601
Dual form			2100.3.j.g.601.16

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} +(6.51200 - 2.56785i) q^{7} -3.00000 q^{9} +8.37580 q^{11} -0.0883033i q^{13} +29.7704i q^{17} +17.8840i q^{19} +(-4.44765 - 11.2791i) q^{21} -39.2955 q^{23} +5.19615i q^{27} -43.7168 q^{29} +53.8656i q^{31} -14.5073i q^{33} -27.0440 q^{37} -0.152946 q^{39} -46.0785i q^{41} -74.8906 q^{43} -27.7687i q^{47} +(35.8123 - 33.4437i) q^{49} +51.5639 q^{51} +5.38329 q^{53} +30.9761 q^{57} -1.93853i q^{59} +74.8723i q^{61} +(-19.5360 + 7.70356i) q^{63} +30.6588 q^{67} +68.0617i q^{69} -72.3104 q^{71} -34.0300i q^{73} +(54.5432 - 21.5078i) q^{77} -125.748 q^{79} +9.00000 q^{81} -92.4285i q^{83} +75.7197i q^{87} +141.387i q^{89} +(-0.226750 - 0.575031i) q^{91} +93.2979 q^{93} -107.878i q^{97} -25.1274 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 48 q^{9} - 24 q^{11} - 12 q^{21} + 32 q^{29} - 72 q^{39} + 88 q^{49} + 24 q^{51} + 168 q^{71} - 16 q^{79} + 144 q^{81} - 568 q^{91} + 72 q^{99}+O(q^{100}) $ 16 * q - 48 * q^9 - 24 * q^11 - 12 * q^21 + 32 * q^29 - 72 * q^39 + 88 * q^49 + 24 * q^51 + 168 * q^71 - 16 * q^79 + 144 * q^81 - 568 * q^91 + 72 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times$.

$n$	$701$	$1051$	$1177$	$1501$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$		−	1.73205i		−	0.577350i
$3$		−	1.73205i		−	0.577350i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	6.51200	−	2.56785i	0.930286	−	0.366836i
$7$	6.51200	−	2.56785i	0.930286	−	0.366836i
$8$		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
$13$		−	0.0883033i		−	0.00679256i	−0.999994	−	0.00339628i	$-0.998919\pi$
$13$		−	0.0883033i		−	0.00679256i	0.999994	−	0.00339628i	$-0.00108107\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			29.7704i			1.75120i	0.483037	+	0.875600i	$0.339534\pi$
$17$			29.7704i			1.75120i	−0.483037	+	0.875600i	$0.660466\pi$
$18$		0			0
$19$			17.8840i			0.941265i	0.882329	+	0.470633i	$0.155974\pi$
$19$			17.8840i			0.941265i	−0.882329	+	0.470633i	$0.844026\pi$
$20$		0			0
$21$	−4.44765	−	11.2791i	−0.211793	−	0.537101i
$22$		0			0
$23$	−39.2955			−1.70850			−0.854249	−	0.519864i	$-0.825983\pi$
$23$	−39.2955			−1.70850			−0.854249	+	0.519864i	$0.825983\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$			5.19615i			0.192450i
$28$		0			0
$29$	−43.7168			−1.50748			−0.753738	−	0.657175i	$-0.771751\pi$
$29$	−43.7168			−1.50748			−0.753738	+	0.657175i	$0.771751\pi$
$30$		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
$33$		−	14.5073i		−	0.439615i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−27.0440			−0.730918			−0.365459	−	0.930827i	$-0.619088\pi$
$37$	−27.0440			−0.730918			−0.365459	+	0.930827i	$0.619088\pi$
$38$		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
$43$	−74.8906			−1.74164			−0.870821	−	0.491601i	$-0.836412\pi$
$43$	−74.8906			−1.74164			−0.870821	+	0.491601i	$0.836412\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	27.7687i		−	0.590824i	−0.955370	−	0.295412i	$-0.904543\pi$
$47$		−	27.7687i		−	0.590824i	0.955370	−	0.295412i	$-0.0954569\pi$
$48$		0			0
$49$	35.8123	−	33.4437i	0.730863	−	0.682525i
$50$		0			0
$51$	51.5639			1.01106
$52$		0			0
$53$	5.38329			0.101572			0.0507858	−	0.998710i	$-0.483827\pi$
$53$	5.38329			0.101572			0.0507858	+	0.998710i	$0.483827\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	30.9761			0.543440
$58$		0			0
$59$		−	1.93853i		−	0.0328565i	−0.999865	−	0.0164282i	$-0.994770\pi$
$59$		−	1.93853i		−	0.0328565i	0.999865	−	0.0164282i	$-0.00522951\pi$
$60$		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
$63$	−19.5360	+	7.70356i	−0.310095	+	0.122279i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	30.6588			0.457594			0.228797	−	0.973474i	$-0.426521\pi$
$67$	30.6588			0.457594			0.228797	+	0.973474i	$0.426521\pi$
$68$		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
$73$		−	34.0300i		−	0.466165i	−0.972457	−	0.233082i	$-0.925119\pi$
$73$		−	34.0300i		−	0.466165i	0.972457	−	0.233082i	$-0.0748812\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	54.5432	−	21.5078i	0.708353	−	0.279322i
$78$		0			0
$79$	−125.748			−1.59175			−0.795873	−	0.605464i	$-0.792988\pi$
$79$	−125.748			−1.59175			−0.795873	+	0.605464i	$0.792988\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$		0			0
$83$		−	92.4285i		−	1.11360i	−0.830648	−	0.556798i	$-0.812030\pi$
$83$		−	92.4285i		−	1.11360i	0.830648	−	0.556798i	$-0.187970\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$			75.7197i			0.870342i
$88$		0			0
$89$			141.387i			1.58862i	0.607514	+	0.794309i	$0.292167\pi$
$89$			141.387i			1.58862i	−0.607514	+	0.794309i	$0.707833\pi$
$90$		0			0
$91$	−0.226750	−	0.575031i	−0.00249176	−	0.00631902i
$92$		0			0
$93$	93.2979			1.00320
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	107.878i		−	1.11214i	−0.831135	−	0.556070i	$-0.812309\pi$
$97$		−	107.878i		−	1.11214i	0.831135	−	0.556070i	$-0.187691\pi$
$98$		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
$103$			105.893i			1.02808i	0.857765	+	0.514041i	$0.171852\pi$
$103$			105.893i			1.02808i	−0.857765	+	0.514041i	$0.828148\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−119.448			−1.11634			−0.558168	−	0.829728i	$-0.688496\pi$
$107$	−119.448			−1.11634			−0.558168	+	0.829728i	$0.688496\pi$
$108$		0			0
$109$	68.8319			0.631485			0.315742	−	0.948845i	$-0.397746\pi$
$109$	68.8319			0.631485			0.315742	+	0.948845i	$0.397746\pi$
$110$		0			0
$111$			46.8415i			0.421996i
$112$		0			0
$113$	82.4292			0.729462			0.364731	−	0.931113i	$-0.381161\pi$
$113$	82.4292			0.729462			0.364731	+	0.931113i	$0.381161\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$			0.264910i			0.00226419i
$118$		0			0
$119$	76.4460	+	193.865i	0.642404	+	1.62912i
$120$		0			0
$121$	−50.8460			−0.420215
$122$		0			0
$123$	−79.8104			−0.648865
$124$		0			0
$125$		0			0
$126$		0			0
$127$	77.8951			0.613348			0.306674	−	0.951815i	$-0.400784\pi$
$127$	77.8951			0.613348			0.306674	+	0.951815i	$0.400784\pi$
$128$		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
$133$	45.9236	+	116.461i	0.345290	+	0.875645i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−233.755			−1.70624			−0.853122	−	0.521712i	$-0.825294\pi$
$137$	−233.755			−1.70624			−0.853122	+	0.521712i	$0.825294\pi$
$138$		0			0
$139$			93.7217i			0.674257i	0.941459	+	0.337128i	$0.109456\pi$
$139$			93.7217i			0.674257i	−0.941459	+	0.337128i	$0.890544\pi$
$140$		0			0
$141$	−48.0968			−0.341112
$142$		0			0
$143$		−	0.739611i		−	0.00517210i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−57.9262	−	62.0287i	−0.394056	−	0.421964i
$148$		0			0
$149$	−141.672			−0.950818			−0.475409	−	0.879765i	$-0.657700\pi$
$149$	−141.672			−0.950818			−0.475409	+	0.879765i	$0.657700\pi$
$150$		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
$153$		−	89.3112i		−	0.583733i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	217.806i		−	1.38730i	−0.720313	−	0.693649i	$-0.756002\pi$
$157$		−	217.806i		−	1.38730i	0.720313	−	0.693649i	$-0.243998\pi$
$158$		0			0
$159$		−	9.32413i		−	0.0586423i
$160$		0			0
$161$	−255.892	+	100.905i	−1.58939	+	0.626739i
$162$		0			0
$163$	149.171			0.915159			0.457580	−	0.889169i	$-0.348716\pi$
$163$	149.171			0.915159			0.457580	+	0.889169i	$0.348716\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	0.343560i		−	0.00205725i	−0.999999	−	0.00102862i	$-0.999673\pi$
$167$		−	0.343560i		−	0.00205725i	0.999999	−	0.00102862i	$-0.000327421\pi$
$168$		0			0
$169$	168.992			0.999954
$170$		0			0
$171$		−	53.6521i		−	0.313755i
$172$		0			0
$173$		−	2.95065i		−	0.0170558i	−0.999964	−	0.00852790i	$-0.997285\pi$
$173$		−	2.95065i		−	0.0170558i	0.999964	−	0.00852790i	$-0.00271455\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−3.35764			−0.0189697
$178$		0			0
$179$	−306.545			−1.71254			−0.856272	−	0.516526i	$-0.827225\pi$
$179$	−306.545			−1.71254			−0.856272	+	0.516526i	$0.827225\pi$
$180$		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
$183$	129.683			0.708648
$184$		0			0
$185$		0			0
$186$		0			0
$187$			249.351i			1.33343i
$188$		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
$193$	181.872			0.942344			0.471172	−	0.882041i	$-0.343831\pi$
$193$	181.872			0.942344			0.471172	+	0.882041i	$0.343831\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−98.7921			−0.501483			−0.250741	−	0.968054i	$-0.580674\pi$
$197$	−98.7921			−0.501483			−0.250741	+	0.968054i	$0.580674\pi$
$198$		0			0
$199$			181.839i			0.913765i	0.889527	+	0.456882i	$0.151034\pi$
$199$			181.839i			0.913765i	−0.889527	+	0.456882i	$0.848966\pi$
$200$		0			0
$201$		−	53.1025i		−	0.264192i
$202$		0			0
$203$	−284.684	+	112.258i	−1.40238	+	0.552997i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	117.886			0.569499
$208$		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
$213$			125.245i			0.588006i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	138.319	+	350.772i	0.637414	+	1.61646i
$218$		0			0
$219$	−58.9417			−0.269140
$220$		0			0
$221$	2.62883			0.0118951
$222$		0			0
$223$			68.5318i			0.307318i	0.988124	+	0.153659i	$0.0491056\pi$
$223$			68.5318i			0.307318i	−0.988124	+	0.153659i	$0.950894\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	17.8883i		−	0.0788031i	−0.999223	−	0.0394015i	$-0.987455\pi$
$227$		−	17.8883i		−	0.0788031i	0.999223	−	0.0394015i	$-0.0125451\pi$
$228$		0			0
$229$			119.591i			0.522230i	0.965308	+	0.261115i	$0.0840901\pi$
$229$			119.591i			0.522230i	−0.965308	+	0.261115i	$0.915910\pi$
$230$		0			0
$231$	−37.2526	−	94.4716i	−0.161267	−	0.408968i
$232$		0			0
$233$	157.824			0.677357			0.338679	−	0.940902i	$-0.390020\pi$
$233$	157.824			0.677357			0.338679	+	0.940902i	$0.390020\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$			217.802i			0.918995i
$238$		0			0
$239$	299.375			1.25261			0.626307	−	0.779576i	$-0.284565\pi$
$239$	299.375			1.25261			0.626307	+	0.779576i	$0.284565\pi$
$240$		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
$243$		−	15.5885i		−	0.0641500i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	1.57922			0.00639360
$248$		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
$253$	−329.131			−1.30091
$254$		0			0
$255$		0			0
$256$		0			0
$257$			24.6272i			0.0958257i	0.998852	+	0.0479128i	$0.0152570\pi$
$257$			24.6272i			0.0958257i	−0.998852	+	0.0479128i	$0.984743\pi$
$258$		0			0
$259$	−176.110	+	69.4450i	−0.679963	+	0.268127i
$260$		0			0
$261$	131.150			0.502492
$262$		0			0
$263$	81.4679			0.309764			0.154882	−	0.987933i	$-0.450500\pi$
$263$	81.4679			0.309764			0.154882	+	0.987933i	$0.450500\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	244.890			0.917189
$268$		0			0
$269$			328.768i			1.22218i	0.791559	+	0.611092i	$0.209270\pi$
$269$			328.768i			1.22218i	−0.791559	+	0.611092i	$0.790730\pi$
$270$		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
$273$	−0.995983	+	0.392742i	−0.00364829	+	0.00143862i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	393.057			1.41898			0.709489	−	0.704717i	$-0.248926\pi$
$277$	393.057			1.41898			0.709489	+	0.704717i	$0.248926\pi$
$278$		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
$283$		−	423.072i		−	1.49495i	−0.664289	−	0.747476i	$-0.731265\pi$
$283$		−	423.072i		−	1.49495i	0.664289	−	0.747476i	$-0.268735\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−118.323	−	300.063i	−0.412275	−	1.04552i
$288$		0			0
$289$	−597.277			−2.06670
$290$		0			0
$291$	−186.850			−0.642095
$292$		0			0
$293$		−	99.9270i		−	0.341048i	−0.985354	−	0.170524i	$-0.945454\pi$
$293$		−	99.9270i		−	0.341048i	0.985354	−	0.170524i	$-0.0545460\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$			43.5219i			0.146538i
$298$		0			0
$299$			3.46992i			0.0116051i
$300$		0			0
$301$	−487.687	+	192.308i	−1.62022	+	0.638897i
$302$		0			0
$303$	−1.04007			−0.00343259
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	287.175i		−	0.935422i	−0.883881	−	0.467711i	$-0.845079\pi$
$307$		−	287.175i		−	0.935422i	0.883881	−	0.467711i	$-0.154921\pi$
$308$		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
$313$			274.169i			0.875940i	0.898990	+	0.437970i	$0.144302\pi$
$313$			274.169i			0.875940i	−0.898990	+	0.437970i	$0.855698\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−84.2117			−0.265652			−0.132826	−	0.991139i	$-0.542405\pi$
$317$	−84.2117			−0.265652			−0.132826	+	0.991139i	$0.542405\pi$
$318$		0			0
$319$	−366.163			−1.14785
$320$		0			0
$321$			206.890i			0.644517i
$322$		0			0
$323$	−532.415			−1.64834
$324$		0			0
$325$		0			0
$326$		0			0
$327$		−	119.220i		−	0.364588i
$328$		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
$333$	81.1319			0.243639
$334$		0			0
$335$		0			0
$336$		0			0
$337$	127.801			0.379232			0.189616	−	0.981858i	$-0.439276\pi$
$337$	127.801			0.379232			0.189616	+	0.981858i	$0.439276\pi$
$338$		0			0
$339$		−	142.771i		−	0.421155i
$340$		0			0
$341$			451.167i			1.32307i
$342$		0			0
$343$	147.331	−	309.746i	0.429536	−	0.903050i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−301.061			−0.867612			−0.433806	−	0.901006i	$-0.642830\pi$
$347$	−301.061			−0.867612			−0.433806	+	0.901006i	$0.642830\pi$
$348$		0			0
$349$			612.654i			1.75546i	0.479159	+	0.877728i	$0.340942\pi$
$349$			612.654i			1.75546i	−0.479159	+	0.877728i	$0.659058\pi$
$350$		0			0
$351$	0.458837			0.00130723
$352$		0			0
$353$		−	63.4467i		−	0.179736i	−0.995954	−	0.0898678i	$-0.971356\pi$
$353$		−	63.4467i		−	0.179736i	0.995954	−	0.0898678i	$-0.0286444\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	335.784	−	132.408i	0.940571	−	0.370892i
$358$		0			0
$359$	172.375			0.480154			0.240077	−	0.970754i	$-0.422827\pi$
$359$	172.375			0.480154			0.240077	+	0.970754i	$0.422827\pi$
$360$		0			0
$361$	41.1612			0.114020
$362$		0			0
$363$			88.0679i			0.242611i
$364$		0			0
$365$		0			0
$366$		0			0
$367$			162.503i			0.442789i	0.975184	+	0.221394i	$0.0710608\pi$
$367$			162.503i			0.442789i	−0.975184	+	0.221394i	$0.928939\pi$
$368$		0			0
$369$			138.236i			0.374622i
$370$		0			0
$371$	35.0560	−	13.8235i	0.0944905	−	0.0372601i
$372$		0			0
$373$	84.5145			0.226580			0.113290	−	0.993562i	$-0.463861\pi$
$373$	84.5145			0.226580			0.113290	+	0.993562i	$0.463861\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			3.86034i			0.0102396i
$378$		0			0
$379$	570.247			1.50461			0.752304	−	0.658816i	$-0.228942\pi$
$379$	570.247			1.50461			0.752304	+	0.658816i	$0.228942\pi$
$380$		0			0
$381$		−	134.918i		−	0.354116i
$382$		0			0
$383$		−	91.0688i		−	0.237777i	−0.992908	−	0.118889i	$-0.962067\pi$
$383$		−	91.0688i		−	0.237777i	0.992908	−	0.118889i	$-0.0379332\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	224.672			0.580547
$388$		0			0
$389$	−62.7679			−0.161357			−0.0806786	−	0.996740i	$-0.525709\pi$
$389$	−62.7679			−0.161357			−0.0806786	+	0.996740i	$0.525709\pi$
$390$		0			0
$391$		−	1169.84i		−	2.99192i
$392$		0			0
$393$	−70.7184			−0.179945
$394$		0			0
$395$		0			0
$396$		0			0
$397$			175.124i			0.441119i	0.975374	+	0.220559i	$0.0707883\pi$
$397$			175.124i			0.441119i	−0.975374	+	0.220559i	$0.929212\pi$
$398$		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
$403$	4.75651			0.0118027
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−226.515			−0.556548
$408$		0			0
$409$		−	425.778i		−	1.04102i	−0.853855	−	0.520511i	$-0.825741\pi$
$409$		−	425.778i		−	1.04102i	0.853855	−	0.520511i	$-0.174259\pi$
$410$		0			0
$411$			404.876i			0.985100i
$412$		0			0
$413$	−4.97787	−	12.6237i	−0.0120529	−	0.0305659i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	162.331			0.389282
$418$		0			0
$419$		−	393.611i		−	0.939406i	−0.882825	−	0.469703i	$-0.844361\pi$
$419$		−	393.611i		−	0.939406i	0.882825	−	0.469703i	$-0.155639\pi$
$420$		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
$423$			83.3061i			0.196941i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	192.261	+	487.568i	0.450260	+	1.14185i
$428$		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
$433$			530.639i			1.22549i	0.790279	+	0.612747i	$0.209936\pi$
$433$			530.639i			1.22549i	−0.790279	+	0.612747i	$0.790064\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	702.761i		−	1.60815i
$438$		0			0
$439$			154.699i			0.352389i	0.984355	+	0.176194i	$0.0563787\pi$
$439$			154.699i			0.352389i	−0.984355	+	0.176194i	$0.943621\pi$
$440$		0			0
$441$	−107.437	+	100.331i	−0.243621	+	0.227508i
$442$		0			0
$443$	94.9327			0.214295			0.107147	−	0.994243i	$-0.465828\pi$
$443$	94.9327			0.214295			0.107147	+	0.994243i	$0.465828\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$			245.383i			0.548955i
$448$		0			0
$449$	−64.1731			−0.142924			−0.0714622	−	0.997443i	$-0.522767\pi$
$449$	−64.1731			−0.142924			−0.0714622	+	0.997443i	$0.522767\pi$
$450$		0			0
$451$		−	385.944i		−	0.855753i
$452$		0			0
$453$			140.471i			0.310090i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−270.716			−0.592376			−0.296188	−	0.955130i	$-0.595715\pi$
$457$	−270.716			−0.592376			−0.296188	+	0.955130i	$0.595715\pi$
$458$		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
$463$	729.376			1.57533			0.787663	−	0.616106i	$-0.211291\pi$
$463$	729.376			1.57533			0.787663	+	0.616106i	$0.211291\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	491.236i		−	1.05190i	−0.850516	−	0.525949i	$-0.823710\pi$
$467$		−	491.236i		−	1.05190i	0.850516	−	0.525949i	$-0.176290\pi$
$468$		0			0
$469$	199.650	−	78.7272i	0.425693	−	0.167862i
$470$		0			0
$471$	−377.251			−0.800957
$472$		0			0
$473$	−627.268			−1.32615
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−16.1499			−0.0338572
$478$		0			0
$479$			32.7275i			0.0683246i	0.999416	+	0.0341623i	$0.0108763\pi$
$479$			32.7275i			0.0683246i	−0.999416	+	0.0341623i	$0.989124\pi$
$480$		0			0
$481$			2.38807i			0.00496481i
$482$		0			0
$483$	174.773	+	443.218i	0.361848	+	0.917635i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−892.345			−1.83233			−0.916165	−	0.400800i	$-0.868732\pi$
$487$	−892.345			−1.83233			−0.916165	+	0.400800i	$0.868732\pi$
$488$		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
$493$		−	1301.47i		−	2.63989i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−470.885	+	185.683i	−0.947456	+	0.373607i
$498$		0			0
$499$	189.705			0.380171			0.190085	−	0.981768i	$-0.439124\pi$
$499$	189.705			0.380171			0.190085	+	0.981768i	$0.439124\pi$
$500$		0			0
$501$	−0.595064			−0.00118775
$502$		0			0
$503$			82.3835i			0.163784i	0.996641	+	0.0818921i	$0.0260963\pi$
$503$			82.3835i			0.163784i	−0.996641	+	0.0818921i	$0.973904\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		−	292.703i		−	0.577324i
$508$		0			0
$509$		−	612.287i		−	1.20292i	−0.798902	−	0.601461i	$-0.794586\pi$
$509$		−	612.287i		−	1.20292i	0.798902	−	0.601461i	$-0.205414\pi$
$510$		0			0
$511$	−87.3841	−	221.603i	−0.171006	−	0.433666i
$512$		0			0
$513$	−92.9282			−0.181147
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	232.585i		−	0.449874i
$518$		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
$523$			324.679i			0.620800i	0.950606	+	0.310400i	$0.100463\pi$
$523$			324.679i			0.620800i	−0.950606	+	0.310400i	$0.899537\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1603.60			−3.04288
$528$		0			0
$529$	1015.13			1.91897
$530$		0			0
$531$			5.81560i			0.0109522i
$532$		0			0
$533$	−4.06889			−0.00763393
$534$		0			0
$535$		0			0
$536$		0			0
$537$			530.952i			0.988737i
$538$		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
$543$	469.184			0.864059
$544$		0			0
$545$		0			0
$546$		0			0
$547$	495.383			0.905637			0.452818	−	0.891603i	$-0.350419\pi$
$547$	495.383			0.905637			0.452818	+	0.891603i	$0.350419\pi$
$548$		0			0
$549$		−	224.617i		−	0.409138i
$550$		0			0
$551$		−	781.833i		−	1.41893i
$552$		0			0
$553$	−818.870	+	322.902i	−1.48078	+	0.583910i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−180.812			−0.324618			−0.162309	−	0.986740i	$-0.551894\pi$
$557$	−180.812			−0.324618			−0.162309	+	0.986740i	$0.551894\pi$
$558$		0			0
$559$			6.61309i			0.0118302i
$560$		0			0
$561$	431.888			0.769855
$562$		0			0
$563$		−	937.810i		−	1.66574i	−0.553471	−	0.832868i	$-0.686697\pi$
$563$		−	937.810i		−	1.66574i	0.553471	−	0.832868i	$-0.313303\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	58.6080	−	23.1107i	0.103365	−	0.0407596i
$568$		0			0
$569$	27.5357			0.0483931			0.0241965	−	0.999707i	$-0.492297\pi$
$569$	27.5357			0.0483931			0.0241965	+	0.999707i	$0.492297\pi$
$570$		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
$573$		−	590.510i		−	1.03056i
$574$		0			0
$575$		0			0
$576$		0			0
$577$			480.904i			0.833456i	0.909031	+	0.416728i	$0.136823\pi$
$577$			480.904i			0.833456i	−0.909031	+	0.416728i	$0.863177\pi$
$578$		0			0
$579$		−	315.012i		−	0.544063i
$580$		0			0
$581$	−237.343	−	601.894i	−0.408507	−	1.03596i
$582$		0			0
$583$	45.0894			0.0773402
$584$		0			0
$585$		0			0
$586$		0			0
$587$			335.196i			0.571033i	0.958374	+	0.285516i	$0.0921651\pi$
$587$			335.196i			0.571033i	−0.958374	+	0.285516i	$0.907835\pi$
$588$		0			0
$589$	−963.334			−1.63554
$590$		0			0
$591$			171.113i			0.289531i
$592$		0			0
$593$			1125.32i			1.89768i	0.315757	+	0.948840i	$0.397742\pi$
$593$			1125.32i			1.89768i	−0.315757	+	0.948840i	$0.602258\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	314.955			0.527562
$598$		0			0
$599$	643.405			1.07413			0.537066	−	0.843540i	$-0.319533\pi$
$599$	643.405			1.07413			0.537066	+	0.843540i	$0.319533\pi$
$600$		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
$603$	−91.9763			−0.152531
$604$		0			0
$605$		0			0
$606$		0			0
$607$		−	82.3135i		−	0.135607i	−0.997699	−	0.0678035i	$-0.978401\pi$
$607$		−	82.3135i		−	0.135607i	0.997699	−	0.0678035i	$-0.0215991\pi$
$608$		0			0
$609$	194.437	+	493.087i	0.319273	+	0.809666i
$610$		0			0
$611$	−2.45207			−0.00401321
$612$		0			0
$613$	−228.128			−0.372150			−0.186075	−	0.982536i	$-0.559577\pi$
$613$	−228.128			−0.372150			−0.186075	+	0.982536i	$0.559577\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−516.126			−0.836509			−0.418255	−	0.908330i	$-0.637358\pi$
$617$	−516.126			−0.836509			−0.418255	+	0.908330i	$0.637358\pi$
$618$		0			0
$619$		−	683.396i		−	1.10403i	−0.833833	−	0.552016i	$-0.813859\pi$
$619$		−	683.396i		−	1.10403i	0.833833	−	0.552016i	$-0.186141\pi$
$620$		0			0
$621$		−	204.185i		−	0.328801i
$622$		0			0
$623$	363.061	+	920.712i	0.582763	+	1.47787i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	259.449			0.413795
$628$		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
$633$		−	17.3233i		−	0.0273669i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−2.95319	−	3.16234i	−0.00463609	−	0.00496443i
$638$		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
$643$			588.998i			0.916016i	0.888948	+	0.458008i	$0.151437\pi$
$643$			588.998i			0.916016i	−0.888948	+	0.458008i	$0.848563\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	1014.59i		−	1.56815i	−0.620666	−	0.784075i	$-0.713138\pi$
$647$		−	1014.59i		−	1.56815i	0.620666	−	0.784075i	$-0.286862\pi$
$648$		0			0
$649$		−	16.2368i		−	0.0250181i
$650$		0			0
$651$	607.556	−	239.575i	0.933265	−	0.368011i
$652$		0			0
$653$	−741.204			−1.13508			−0.567538	−	0.823348i	$-0.692104\pi$
$653$	−741.204			−1.13508			−0.567538	+	0.823348i	$0.692104\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$			102.090i			0.155388i
$658$		0			0
$659$	−257.591			−0.390882			−0.195441	−	0.980715i	$-0.562614\pi$
$659$	−257.591			−0.390882			−0.195441	+	0.980715i	$0.562614\pi$
$660$		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
$663$		−	4.55326i		−	0.00686766i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	1717.87			2.57552
$668$		0			0
$669$	118.701			0.177430
$670$		0			0
$671$			627.115i			0.934598i
$672$		0			0
$673$	987.972			1.46801			0.734006	−	0.679143i	$-0.237648\pi$
$673$	987.972			1.46801			0.734006	+	0.679143i	$0.237648\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			657.422i			0.971082i	0.874214	+	0.485541i	$0.161377\pi$
$677$			657.422i			0.971082i	−0.874214	+	0.485541i	$0.838623\pi$
$678$		0			0
$679$	−277.014	−	702.499i	−0.407973	−	1.03461i
$680$		0			0
$681$	−30.9834			−0.0454970
$682$		0			0
$683$	−255.185			−0.373624			−0.186812	−	0.982396i	$-0.559815\pi$
$683$	−255.185			−0.373624			−0.186812	+	0.982396i	$0.559815\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	207.137			0.301510
$688$		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
$693$	−163.630	+	64.5234i	−0.236118	+	0.0931074i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	1371.78			1.96812
$698$		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
$703$		−	483.656i		−	0.687988i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1.54196	−	3.91037i	−0.00218099	−	0.00553093i
$708$		0			0
$709$	431.982			0.609284			0.304642	−	0.952467i	$-0.401463\pi$
$709$	431.982			0.609284			0.304642	+	0.952467i	$0.401463\pi$
$710$		0			0
$711$	377.244			0.530582
$712$		0			0
$713$		−	2116.67i		−	2.96868i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		−	518.533i		−	0.723197i
$718$		0			0
$719$		−	767.053i		−	1.06683i	−0.845853	−	0.533416i	$-0.820908\pi$
$719$		−	767.053i		−	1.06683i	0.845853	−	0.533416i	$-0.179092\pi$
$720$		0			0
$721$	271.916	+	689.572i	0.377138	+	0.956411i
$722$		0			0
$723$	127.371			0.176170
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	230.714i		−	0.317350i	−0.987331	−	0.158675i	$-0.949278\pi$
$727$		−	230.714i		−	0.317350i	0.987331	−	0.158675i	$-0.0507222\pi$
$728$		0			0
$729$	−27.0000			−0.0370370
$730$		0			0
$731$		−	2229.52i		−	3.04996i
$732$		0			0
$733$			632.429i			0.862795i	0.902162	+	0.431398i	$0.141979\pi$
$733$			632.429i			0.862795i	−0.902162	+	0.431398i	$0.858021\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	256.792			0.348428
$738$		0			0
$739$	−466.036			−0.630631			−0.315316	−	0.948987i	$-0.602110\pi$
$739$	−466.036			−0.630631			−0.315316	+	0.948987i	$0.602110\pi$
$740$		0			0
$741$		−	2.73529i		−	0.00369135i
$742$		0			0
$743$	−632.427			−0.851180			−0.425590	−	0.904916i	$-0.639933\pi$
$743$	−632.427			−0.851180			−0.425590	+	0.904916i	$0.639933\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$			277.285i			0.371199i
$748$		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
$753$	196.101			0.260427
$754$		0			0
$755$		0			0
$756$		0			0
$757$	343.268			0.453458			0.226729	−	0.973958i	$-0.427197\pi$
$757$	343.268			0.453458			0.226729	+	0.973958i	$0.427197\pi$
$758$		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
$763$	448.233	−	176.750i	0.587461	−	0.231651i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−0.171179			−0.000223180
$768$		0			0
$769$		−	1474.43i		−	1.91733i	−0.284529	−	0.958667i	$-0.591837\pi$
$769$		−	1474.43i		−	1.91733i	0.284529	−	0.958667i	$-0.408163\pi$
$770$		0			0
$771$	42.6556			0.0553250
$772$		0			0
$773$		−	670.136i		−	0.866929i	−0.901171	−	0.433465i	$-0.857291\pi$
$773$		−	670.136i		−	0.866929i	0.901171	−	0.433465i	$-0.142709\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	120.282	+	305.032i	0.154803	+	0.392577i
$778$		0			0
$779$	824.070			1.05786
$780$		0			0
$781$	−605.658			−0.775490
$782$		0			0
$783$		−	227.159i		−	0.290114i
$784$		0			0
$785$		0			0
$786$		0			0
$787$			1177.11i			1.49569i	0.663873	+	0.747846i	$0.268912\pi$
$787$			1177.11i			1.49569i	−0.663873	+	0.747846i	$0.731088\pi$
$788$		0			0
$789$		−	141.107i		−	0.178842i
$790$		0			0
$791$	536.779	−	211.666i	0.678608	−	0.267593i
$792$		0			0
$793$	6.61147			0.00833729
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	370.289i		−	0.464604i	−0.972644	−	0.232302i	$-0.925374\pi$
$797$		−	370.289i		−	0.464604i	0.972644	−	0.232302i	$-0.0746257\pi$
$798$		0			0
$799$	826.686			1.03465
$800$		0			0
$801$		−	424.161i		−	0.529540i
$802$		0			0
$803$		−	285.029i		−	0.354955i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	569.442			0.705628
$808$		0			0
$809$	260.603			0.322130			0.161065	−	0.986944i	$-0.448507\pi$
$809$	260.603			0.322130			0.161065	+	0.986944i	$0.448507\pi$
$810$		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
$813$	−279.928			−0.344315
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	1339.35i		−	1.63935i
$818$		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
$823$	123.305			0.149824			0.0749118	−	0.997190i	$-0.476132\pi$
$823$	123.305			0.149824			0.0749118	+	0.997190i	$0.476132\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−760.741			−0.919880			−0.459940	−	0.887950i	$-0.652129\pi$
$827$	−760.741			−0.919880			−0.459940	+	0.887950i	$0.652129\pi$
$828$		0			0
$829$		−	802.031i		−	0.967468i	−0.875215	−	0.483734i	$-0.839280\pi$
$829$		−	802.031i		−	0.967468i	0.875215	−	0.483734i	$-0.160720\pi$
$830$		0			0
$831$		−	680.794i		−	0.819247i
$832$		0			0
$833$	995.633	+	1066.15i	1.19524	+	1.27989i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−279.894			−0.334401
$838$		0			0
$839$			442.366i			0.527254i	0.964625	+	0.263627i	$0.0849188\pi$
$839$			442.366i			0.527254i	−0.964625	+	0.263627i	$0.915081\pi$
$840$		0			0
$841$	1070.16			1.27248
$842$		0			0
$843$		−	286.916i		−	0.340351i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−331.109	+	130.565i	−0.390920	+	0.154150i
$848$		0			0
$849$	−732.781			−0.863111
$850$		0			0
$851$	1062.71			1.24877
$852$		0			0
$853$			648.240i			0.759953i	0.924996	+	0.379977i	$0.124068\pi$
$853$			648.240i			0.759953i	−0.924996	+	0.379977i	$0.875932\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	336.227i		−	0.392331i	−0.980571	−	0.196165i	$-0.937151\pi$
$857$		−	336.227i		−	0.392331i	0.980571	−	0.196165i	$-0.0628489\pi$
$858$		0			0
$859$		−	1188.57i		−	1.38366i	−0.722058	−	0.691832i	$-0.756804\pi$
$859$		−	1188.57i		−	1.38366i	0.722058	−	0.691832i	$-0.243196\pi$
$860$		0			0
$861$	−519.725	+	204.941i	−0.603629	+	0.238027i
$862$		0			0
$863$	−1071.22			−1.24128			−0.620639	−	0.784096i	$-0.713127\pi$
$863$	−1071.22			−1.24128			−0.620639	+	0.784096i	$0.713127\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			1034.51i			1.19321i
$868$		0			0
$869$	−1053.24			−1.21201
$870$		0			0
$871$		−	2.70727i		−	0.00310823i
$872$		0			0
$873$			323.633i			0.370713i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−163.935			−0.186927			−0.0934635	−	0.995623i	$-0.529794\pi$
$877$	−163.935			−0.186927			−0.0934635	+	0.995623i	$0.529794\pi$
$878$		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
$883$	−1148.47			−1.30064			−0.650321	−	0.759660i	$-0.725365\pi$
$883$	−1148.47			−1.30064			−0.650321	+	0.759660i	$0.725365\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			468.656i			0.528361i	0.964473	+	0.264180i	$0.0851014\pi$
$887$			468.656i			0.528361i	−0.964473	+	0.264180i	$0.914899\pi$
$888$		0			0
$889$	507.253	−	200.023i	0.570588	−	0.224998i
$890$		0			0
$891$	75.3822			0.0846040
$892$		0			0
$893$	496.617			0.556122
$894$		0			0
$895$		0			0
$896$		0			0
$897$	6.01008			0.00670020
$898$		0			0
$899$		−	2354.83i		−	2.61939i
$900$		0			0
$901$			160.263i			0.177872i
$902$		0			0
$903$	333.087	+	844.699i	0.368867	+	0.935437i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	61.3264			0.0676145			0.0338073	−	0.999428i	$-0.489237\pi$
$907$	61.3264			0.0676145			0.0338073	+	0.999428i	$0.489237\pi$
$908$		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
$913$		−	774.162i		−	0.847932i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−104.844	−	265.880i	−0.114333	−	0.289946i
$918$		0			0
$919$	1330.80			1.44810			0.724048	−	0.689750i	$-0.242279\pi$
$919$	1330.80			1.44810			0.724048	+	0.689750i	$0.242279\pi$
$920$		0			0
$921$	−497.401			−0.540066
$922$		0			0
$923$			6.38525i			0.00691793i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		−	317.678i		−	0.342694i
$928$		0			0
$929$			1382.16i			1.48779i	0.668297	+	0.743895i	$0.267024\pi$
$929$			1382.16i			1.48779i	−0.668297	+	0.743895i	$0.732976\pi$
$930$		0			0
$931$	598.109	+	640.468i	0.642437	+	0.687935i
$932$		0			0
$933$	323.999			0.347266
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	1470.54i		−	1.56941i	−0.619871	−	0.784704i	$-0.712815\pi$
$937$		−	1470.54i		−	1.56941i	0.619871	−	0.784704i	$-0.287185\pi$
$938$		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
$943$			1810.68i			1.92012i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1170.44			1.23594			0.617971	−	0.786201i	$-0.287955\pi$
$947$	1170.44			1.23594			0.617971	+	0.786201i	$0.287955\pi$
$948$		0			0
$949$	−3.00496			−0.00316645
$950$		0			0
$951$			145.859i			0.153374i
$952$		0			0
$953$	1354.92			1.42174			0.710871	−	0.703323i	$-0.248301\pi$
$953$	1354.92			1.42174			0.710871	+	0.703323i	$0.248301\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$			634.213i			0.662710i
$958$		0			0
$959$	−1522.21	+	600.249i	−1.58729	+	0.625912i
$960$		0			0
$961$	−1940.50			−2.01925
$962$		0			0
$963$	358.344			0.372112
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−564.951			−0.584231			−0.292115	−	0.956383i	$-0.594359\pi$
$967$	−564.951			−0.584231			−0.292115	+	0.956383i	$0.594359\pi$
$968$		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
$973$	240.664	+	610.316i	0.247342	+	0.627251i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−304.418			−0.311584			−0.155792	−	0.987790i	$-0.549793\pi$
$977$	−304.418			−0.311584			−0.155792	+	0.987790i	$0.549793\pi$
$978$		0			0
$979$			1184.23i			1.20963i
$980$		0			0
$981$	−206.496			−0.210495
$982$		0			0
$983$		−	764.954i		−	0.778183i	−0.921199	−	0.389092i	$-0.872789\pi$
$983$		−	764.954i		−	0.778183i	0.921199	−	0.389092i	$-0.127211\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−313.206	+	123.506i	−0.317332	+	0.125132i
$988$		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
$993$		−	984.192i		−	0.991129i
$994$		0			0
$995$		0			0
$996$		0			0
$997$			1308.65i			1.31258i	0.754507	+	0.656292i	$0.227876\pi$
$997$			1308.65i			1.31258i	−0.754507	+	0.656292i	$0.772124\pi$
$998$		0			0
$999$		−	140.525i		−	0.140665i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2100.3.j.g.601.8		16
5.2	odd	4		420.3.p.a.349.8	yes	16
5.3	odd	4		420.3.p.a.349.10	yes	16
5.4	even	2	inner	2100.3.j.g.601.9		16
7.6	odd	2	inner	2100.3.j.g.601.16		16
15.2	even	4		1260.3.p.e.1189.1		16
15.8	even	4		1260.3.p.e.1189.15		16
20.3	even	4		1680.3.bd.b.769.2		16
20.7	even	4		1680.3.bd.b.769.16		16
35.13	even	4		420.3.p.a.349.7	✓	16
35.27	even	4		420.3.p.a.349.9	yes	16
35.34	odd	2	inner	2100.3.j.g.601.1		16
105.62	odd	4		1260.3.p.e.1189.16		16
105.83	odd	4		1260.3.p.e.1189.2		16
140.27	odd	4		1680.3.bd.b.769.1		16
140.83	odd	4		1680.3.bd.b.769.15		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.3.p.a.349.7	✓	16	35.13	even	4
420.3.p.a.349.8	yes	16	5.2	odd	4
420.3.p.a.349.9	yes	16	35.27	even	4
420.3.p.a.349.10	yes	16	5.3	odd	4
1260.3.p.e.1189.1		16	15.2	even	4
1260.3.p.e.1189.2		16	105.83	odd	4
1260.3.p.e.1189.15		16	15.8	even	4
1260.3.p.e.1189.16		16	105.62	odd	4
1680.3.bd.b.769.1		16	140.27	odd	4
1680.3.bd.b.769.2		16	20.3	even	4
1680.3.bd.b.769.15		16	140.83	odd	4
1680.3.bd.b.769.16		16	20.7	even	4
2100.3.j.g.601.1		16	35.34	odd	2	inner
2100.3.j.g.601.8		16	1.1	even	1	trivial
2100.3.j.g.601.9		16	5.4	even	2	inner
2100.3.j.g.601.16		16	7.6	odd	2	inner

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:	\( N \)	\(=\)	\( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2100.j (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} +(6.51200 - 2.56785i) q^{7} -3.00000 q^{9} +8.37580 q^{11} -0.0883033i q^{13} +29.7704i q^{17} +17.8840i q^{19} +(-4.44765 - 11.2791i) q^{21} -39.2955 q^{23} +5.19615i q^{27} -43.7168 q^{29} +53.8656i q^{31} -14.5073i q^{33} -27.0440 q^{37} -0.152946 q^{39} -46.0785i q^{41} -74.8906 q^{43} -27.7687i q^{47} +(35.8123 - 33.4437i) q^{49} +51.5639 q^{51} +5.38329 q^{53} +30.9761 q^{57} -1.93853i q^{59} +74.8723i q^{61} +(-19.5360 + 7.70356i) q^{63} +30.6588 q^{67} +68.0617i q^{69} -72.3104 q^{71} -34.0300i q^{73} +(54.5432 - 21.5078i) q^{77} -125.748 q^{79} +9.00000 q^{81} -92.4285i q^{83} +75.7197i q^{87} +141.387i q^{89} +(-0.226750 - 0.575031i) q^{91} +93.2979 q^{93} -107.878i q^{97} -25.1274 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 48 q^{9} - 24 q^{11} - 12 q^{21} + 32 q^{29} - 72 q^{39} + 88 q^{49} + 24 q^{51} + 168 q^{71} - 16 q^{79} + 144 q^{81} - 568 q^{91} + 72 q^{99}+O(q^{100}) \) 16 * q - 48 * q^9 - 24 * q^11 - 12 * q^21 + 32 * q^29 - 72 * q^39 + 88 * q^49 + 24 * q^51 + 168 * q^71 - 16 * q^79 + 144 * q^81 - 568 * q^91 + 72 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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