LMFDB - Embedded newform 480.4.b.a.431.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [480,4,Mod(431,480)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(480, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("480.431");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 480 = 2^{5} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	480.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$28.3209168028$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			431.15
Character	$\chi$	$=$	480.431
Dual form			480.4.b.a.431.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.49310 - 4.97701i) q^{3} -5.00000 q^{5} -14.5956i q^{7} +(-22.5413 - 14.8623i) q^{9} +O(q^{10})$	\(q+(1.49310 - 4.97701i) q^{3} -5.00000 q^{5} -14.5956i q^{7} +(-22.5413 - 14.8623i) q^{9} +58.0556i q^{11} +43.9441i q^{13} +(-7.46548 + 24.8851i) q^{15} +78.8045i q^{17} -103.306 q^{19} +(-72.6425 - 21.7926i) q^{21} +163.945 q^{23} +25.0000 q^{25} +(-107.626 + 89.9977i) q^{27} +247.735 q^{29} -153.271i q^{31} +(288.943 + 86.6825i) q^{33} +72.9780i q^{35} +153.638i q^{37} +(218.711 + 65.6128i) q^{39} +27.0586i q^{41} -246.934 q^{43} +(112.707 + 74.3116i) q^{45} -411.901 q^{47} +129.968 q^{49} +(392.211 + 117.663i) q^{51} +80.1737 q^{53} -290.278i q^{55} +(-154.246 + 514.157i) q^{57} +513.612i q^{59} +27.0460i q^{61} +(-216.924 + 329.004i) q^{63} -219.721i q^{65} +453.089 q^{67} +(244.786 - 815.958i) q^{69} +446.258 q^{71} +524.821 q^{73} +(37.3274 - 124.425i) q^{75} +847.356 q^{77} +93.4259i q^{79} +(287.223 + 670.033i) q^{81} +1249.95i q^{83} -394.023i q^{85} +(369.892 - 1232.98i) q^{87} -267.674i q^{89} +641.391 q^{91} +(-762.832 - 228.848i) q^{93} +516.532 q^{95} -897.223 q^{97} +(862.840 - 1308.65i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 120 q^{5}+O(q^{10}) $ 24 * q - 120 * q^5	$ 24 q - 120 q^{5} - 12 q^{19} + 4 q^{21} + 228 q^{23} + 600 q^{25} - 132 q^{27} + 116 q^{33} - 656 q^{39} - 924 q^{47} - 816 q^{49} + 700 q^{51} - 528 q^{53} - 172 q^{57} + 476 q^{63} - 1632 q^{67} - 980 q^{69} + 216 q^{71} - 216 q^{73} + 152 q^{81} + 252 q^{87} + 1800 q^{91} + 60 q^{95} + 792 q^{97} + 1328 q^{99}+O(q^{100}) $ 24 * q - 120 * q^5 - 12 * q^19 + 4 * q^21 + 228 * q^23 + 600 * q^25 - 132 * q^27 + 116 * q^33 - 656 * q^39 - 924 * q^47 - 816 * q^49 + 700 * q^51 - 528 * q^53 - 172 * q^57 + 476 * q^63 - 1632 * q^67 - 980 * q^69 + 216 * q^71 - 216 * q^73 + 152 * q^81 + 252 * q^87 + 1800 * q^91 + 60 * q^95 + 792 * q^97 + 1328 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/480\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$161$	$421$
$\chi(n)$	$-1$	$1$	$-1$	$-1$

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
$3$	1.49310	−	4.97701i	0.287346	−	0.957827i
$3$	1.49310	−	4.97701i	0.287346	−	0.957827i
$4$		0			0
$5$	−5.00000			−0.447214
$5$	−5.00000			−0.447214
$6$		0			0
$7$		−	14.5956i		−	0.788089i	−0.919091	−	0.394044i	$-0.871076\pi$
$7$		−	14.5956i		−	0.788089i	0.919091	−	0.394044i	$-0.128924\pi$
$8$		0			0
$9$	−22.5413	−	14.8623i	−0.834864	−	0.550456i
$10$		0			0
$11$			58.0556i			1.59131i	0.605750	+	0.795655i	$0.292873\pi$
$11$			58.0556i			1.59131i	−0.605750	+	0.795655i	$0.707127\pi$
$12$		0			0
$13$			43.9441i			0.937532i	0.883322	+	0.468766i	$0.155301\pi$
$13$			43.9441i			0.937532i	−0.883322	+	0.468766i	$0.844699\pi$
$14$		0			0
$15$	−7.46548	+	24.8851i	−0.128505	+	0.428353i
$16$		0			0
$17$			78.8045i			1.12429i	0.827039	+	0.562144i	$0.190023\pi$
$17$			78.8045i			1.12429i	−0.827039	+	0.562144i	$0.809977\pi$
$18$		0			0
$19$	−103.306			−1.24737			−0.623687	−	0.781674i	$-0.714366\pi$
$19$	−103.306			−1.24737			−0.623687	+	0.781674i	$0.714366\pi$
$20$		0			0
$21$	−72.6425	−	21.7926i	−0.754852	−	0.226454i
$22$		0			0
$23$	163.945			1.48630			0.743151	−	0.669123i	$-0.233330\pi$
$23$	163.945			1.48630			0.743151	+	0.669123i	$0.233330\pi$
$24$		0			0
$25$	25.0000			0.200000
$26$		0			0
$27$	−107.626	+	89.9977i	−0.767137	+	0.641484i
$28$		0			0
$29$	247.735			1.58632			0.793160	−	0.609013i	$-0.208434\pi$
$29$	247.735			1.58632			0.793160	+	0.609013i	$0.208434\pi$
$30$		0			0
$31$		−	153.271i		−	0.888009i	−0.896025	−	0.444004i	$-0.853557\pi$
$31$		−	153.271i		−	0.888009i	0.896025	−	0.444004i	$-0.146443\pi$
$32$		0			0
$33$	288.943	+	86.6825i	1.52420	+	0.457257i
$34$		0			0
$35$			72.9780i			0.352444i
$36$		0			0
$37$			153.638i			0.682646i	0.939946	+	0.341323i	$0.110875\pi$
$37$			153.638i			0.682646i	−0.939946	+	0.341323i	$0.889125\pi$
$38$		0			0
$39$	218.711	+	65.6128i	0.897993	+	0.269396i
$40$		0			0
$41$			27.0586i			0.103070i	0.998671	+	0.0515348i	$0.0164113\pi$
$41$			27.0586i			0.103070i	−0.998671	+	0.0515348i	$0.983589\pi$
$42$		0			0
$43$	−246.934			−0.875746			−0.437873	−	0.899037i	$-0.644268\pi$
$43$	−246.934			−0.875746			−0.437873	+	0.899037i	$0.644268\pi$
$44$		0			0
$45$	112.707	+	74.3116i	0.373363	+	0.246171i
$46$		0			0
$47$	−411.901			−1.27834			−0.639170	−	0.769065i	$-0.720722\pi$
$47$	−411.901			−1.27834			−0.639170	+	0.769065i	$0.720722\pi$
$48$		0			0
$49$	129.968			0.378916
$50$		0			0
$51$	392.211	+	117.663i	1.07687	+	0.323060i
$52$		0			0
$53$	80.1737			0.207787			0.103893	−	0.994588i	$-0.466870\pi$
$53$	80.1737			0.207787			0.103893	+	0.994588i	$0.466870\pi$
$54$		0			0
$55$		−	290.278i		−	0.711656i
$56$		0			0
$57$	−154.246	+	514.157i	−0.358428	+	1.19477i
$58$		0			0
$59$			513.612i			1.13333i	0.823947	+	0.566666i	$0.191767\pi$
$59$			513.612i			1.13333i	−0.823947	+	0.566666i	$0.808233\pi$
$60$		0			0
$61$			27.0460i			0.0567686i	0.999597	+	0.0283843i	$0.00903622\pi$
$61$			27.0460i			0.0567686i	−0.999597	+	0.0283843i	$0.990964\pi$
$62$		0			0
$63$	−216.924	+	329.004i	−0.433808	+	0.657947i
$64$		0			0
$65$		−	219.721i		−	0.419277i
$66$		0			0
$67$	453.089			0.826173			0.413087	−	0.910692i	$-0.364451\pi$
$67$	453.089			0.826173			0.413087	+	0.910692i	$0.364451\pi$
$68$		0			0
$69$	244.786	−	815.958i	0.427084	−	1.42362i
$70$		0			0
$71$	446.258			0.745930			0.372965	−	0.927845i	$-0.378341\pi$
$71$	446.258			0.745930			0.372965	+	0.927845i	$0.378341\pi$
$72$		0			0
$73$	524.821			0.841447			0.420724	−	0.907189i	$-0.361776\pi$
$73$	524.821			0.841447			0.420724	+	0.907189i	$0.361776\pi$
$74$		0			0
$75$	37.3274	−	124.425i	0.0574693	−	0.191565i
$76$		0			0
$77$	847.356			1.25409
$78$		0			0
$79$			93.4259i			0.133054i	0.997785	+	0.0665268i	$0.0211918\pi$
$79$			93.4259i			0.133054i	−0.997785	+	0.0665268i	$0.978808\pi$
$80$		0			0
$81$	287.223	+	670.033i	0.393996	+	0.919112i
$82$		0			0
$83$			1249.95i			1.65301i	0.562933	+	0.826503i	$0.309673\pi$
$83$			1249.95i			1.65301i	−0.562933	+	0.826503i	$0.690327\pi$
$84$		0			0
$85$		−	394.023i		−	0.502797i
$86$		0			0
$87$	369.892	−	1232.98i	0.455823	−	1.51942i
$88$		0			0
$89$		−	267.674i		−	0.318803i	−0.987214	−	0.159401i	$-0.949044\pi$
$89$		−	267.674i		−	0.318803i	0.987214	−	0.159401i	$-0.0509564\pi$
$90$		0			0
$91$	641.391			0.738858
$92$		0			0
$93$	−762.832	−	228.848i	−0.850559	−	0.255166i
$94$		0			0
$95$	516.532			0.557843
$96$		0			0
$97$	−897.223			−0.939167			−0.469584	−	0.882888i	$-0.655596\pi$
$97$	−897.223			−0.939167			−0.469584	+	0.882888i	$0.655596\pi$
$98$		0			0
$99$	862.840	−	1308.65i	0.875946	−	1.32853i
$100$		0			0
$101$	−1086.03			−1.06994			−0.534972	−	0.844870i	$-0.679678\pi$
$101$	−1086.03			−1.06994			−0.534972	+	0.844870i	$0.679678\pi$
$102$		0			0
$103$			602.598i			0.576464i	0.957561	+	0.288232i	$0.0930674\pi$
$103$			602.598i			0.576464i	−0.957561	+	0.288232i	$0.906933\pi$
$104$		0			0
$105$	363.213	+	108.963i	0.337580	+	0.101273i
$106$		0			0
$107$		−	209.243i		−	0.189050i	−0.995522	−	0.0945249i	$-0.969867\pi$
$107$		−	209.243i		−	0.189050i	0.995522	−	0.0945249i	$-0.0301332\pi$
$108$		0			0
$109$			866.438i			0.761374i	0.924704	+	0.380687i	$0.124312\pi$
$109$			866.438i			0.761374i	−0.924704	+	0.380687i	$0.875688\pi$
$110$		0			0
$111$	764.657	+	229.396i	0.653856	+	0.196156i
$112$		0			0
$113$			1455.92i			1.21205i	0.795445	+	0.606026i	$0.207237\pi$
$113$			1455.92i			1.21205i	−0.795445	+	0.606026i	$0.792763\pi$
$114$		0			0
$115$	−819.727			−0.664695
$116$		0			0
$117$	653.112	−	990.560i	0.516070	−	0.782712i
$118$		0			0
$119$	1150.20			0.886039
$120$		0			0
$121$	−2039.45			−1.53227
$122$		0			0
$123$	134.671	+	40.4011i	0.0987228	+	0.0296167i
$124$		0			0
$125$	−125.000			−0.0894427
$126$		0			0
$127$			1710.12i			1.19487i	0.801917	+	0.597435i	$0.203814\pi$
$127$			1710.12i			1.19487i	−0.801917	+	0.597435i	$0.796186\pi$
$128$		0			0
$129$	−368.696	+	1228.99i	−0.251642	+	0.838813i
$130$		0			0
$131$			994.011i			0.662955i	0.943463	+	0.331478i	$0.107547\pi$
$131$			994.011i			0.662955i	−0.943463	+	0.331478i	$0.892453\pi$
$132$		0			0
$133$			1507.82i			0.983041i
$134$		0			0
$135$	538.131	−	449.988i	0.343074	−	0.286880i
$136$		0			0
$137$			218.135i			0.136033i	0.997684	+	0.0680166i	$0.0216671\pi$
$137$			218.135i			0.136033i	−0.997684	+	0.0680166i	$0.978333\pi$
$138$		0			0
$139$	−871.961			−0.532078			−0.266039	−	0.963962i	$-0.585715\pi$
$139$	−871.961			−0.532078			−0.266039	+	0.963962i	$0.585715\pi$
$140$		0			0
$141$	−615.008	+	2050.04i	−0.367326	+	1.22443i
$142$		0			0
$143$	−2551.20			−1.49190
$144$		0			0
$145$	−1238.68			−0.709424
$146$		0			0
$147$	194.055	−	646.854i	0.108880	−	0.362936i
$148$		0			0
$149$	2649.54			1.45677			0.728385	−	0.685168i	$-0.240271\pi$
$149$	2649.54			1.45677			0.728385	+	0.685168i	$0.240271\pi$
$150$		0			0
$151$			717.633i			0.386756i	0.981124	+	0.193378i	$0.0619443\pi$
$151$			717.633i			0.386756i	−0.981124	+	0.193378i	$0.938056\pi$
$152$		0			0
$153$	1171.22	−	1776.36i	0.618871	−	0.938628i
$154$		0			0
$155$			766.355i			0.397130i
$156$		0			0
$157$		−	3251.75i		−	1.65298i	−0.562951	−	0.826490i	$-0.690334\pi$
$157$		−	3251.75i		−	1.65298i	0.562951	−	0.826490i	$-0.309666\pi$
$158$		0			0
$159$	119.707	−	399.026i	0.0597068	−	0.199024i
$160$		0			0
$161$		−	2392.88i		−	1.17134i
$162$		0			0
$163$	180.616			0.0867912			0.0433956	−	0.999058i	$-0.486182\pi$
$163$	180.616			0.0867912			0.0433956	+	0.999058i	$0.486182\pi$
$164$		0			0
$165$	−1444.72	−	433.413i	−0.681643	−	0.204492i
$166$		0			0
$167$	−3583.90			−1.66066			−0.830332	−	0.557269i	$-0.811849\pi$
$167$	−3583.90			−1.66066			−0.830332	+	0.557269i	$0.811849\pi$
$168$		0			0
$169$	265.912			0.121034
$170$		0			0
$171$	2328.66	+	1535.37i	1.04139	+	0.686625i
$172$		0			0
$173$	529.974			0.232909			0.116454	−	0.993196i	$-0.462847\pi$
$173$	529.974			0.232909			0.116454	+	0.993196i	$0.462847\pi$
$174$		0			0
$175$		−	364.890i		−	0.157618i
$176$		0			0
$177$	2556.26	+	766.872i	1.08554	+	0.325659i
$178$		0			0
$179$		−	2272.25i		−	0.948805i	−0.880308	−	0.474402i	$-0.842664\pi$
$179$		−	2272.25i		−	0.948805i	0.880308	−	0.474402i	$-0.157336\pi$
$180$		0			0
$181$			768.299i			0.315509i	0.987478	+	0.157755i	$0.0504255\pi$
$181$			768.299i			0.315509i	−0.987478	+	0.157755i	$0.949574\pi$
$182$		0			0
$183$	134.608	+	40.3823i	0.0543745	+	0.0163123i
$184$		0			0
$185$		−	768.189i		−	0.305288i
$186$		0			0
$187$	−4575.04			−1.78909
$188$		0			0
$189$	1313.57	+	1570.87i	0.505546	+	0.604572i
$190$		0			0
$191$	1338.21			0.506962			0.253481	−	0.967340i	$-0.418424\pi$
$191$	1338.21			0.506962			0.253481	+	0.967340i	$0.418424\pi$
$192$		0			0
$193$	−858.809			−0.320303			−0.160151	−	0.987092i	$-0.551198\pi$
$193$	−858.809			−0.320303			−0.160151	+	0.987092i	$0.551198\pi$
$194$		0			0
$195$	−1093.55	−	328.064i	−0.401595	−	0.120478i
$196$		0			0
$197$	−3040.41			−1.09960			−0.549798	−	0.835298i	$-0.685295\pi$
$197$	−3040.41			−1.09960			−0.549798	+	0.835298i	$0.685295\pi$
$198$		0			0
$199$			1636.95i			0.583118i	0.956553	+	0.291559i	$0.0941740\pi$
$199$			1636.95i			0.583118i	−0.956553	+	0.291559i	$0.905826\pi$
$200$		0			0
$201$	676.505	−	2255.03i	0.237398	−	0.791331i
$202$		0			0
$203$		−	3615.85i		−	1.25016i
$204$		0			0
$205$		−	135.293i		−	0.0460941i
$206$		0			0
$207$	−3695.55	−	2436.61i	−1.24086	−	0.818144i
$208$		0			0
$209$		−	5997.51i		−	1.98496i
$210$		0			0
$211$	−6036.05			−1.96938			−0.984689	−	0.174322i	$-0.944227\pi$
$211$	−6036.05			−1.96938			−0.984689	+	0.174322i	$0.944227\pi$
$212$		0			0
$213$	666.305	−	2221.03i	0.214340	−	0.714472i
$214$		0			0
$215$	1234.67			0.391646
$216$		0			0
$217$	−2237.08			−0.699830
$218$		0			0
$219$	783.608	−	2612.04i	0.241787	−	0.805961i
$220$		0			0
$221$	−3463.00			−1.05406
$222$		0			0
$223$			1524.61i			0.457828i	0.973447	+	0.228914i	$0.0735175\pi$
$223$			1524.61i			0.457828i	−0.973447	+	0.228914i	$0.926483\pi$
$224$		0			0
$225$	−563.533	−	371.558i	−0.166973	−	0.110091i
$226$		0			0
$227$		−	472.571i		−	0.138174i	−0.997611	−	0.0690872i	$-0.977991\pi$
$227$		−	472.571i		−	0.138174i	0.997611	−	0.0690872i	$-0.0220087\pi$
$228$		0			0
$229$			3068.70i			0.885526i	0.896639	+	0.442763i	$0.146002\pi$
$229$			3068.70i			0.885526i	−0.896639	+	0.442763i	$0.853998\pi$
$230$		0			0
$231$	1265.18	−	4217.30i	0.360359	−	1.20120i
$232$		0			0
$233$			141.515i			0.0397895i	0.999802	+	0.0198947i	$0.00633311\pi$
$233$			141.515i			0.0397895i	−0.999802	+	0.0198947i	$0.993667\pi$
$234$		0			0
$235$	2059.51			0.571691
$236$		0			0
$237$	464.982	+	139.494i	0.127442	+	0.0382325i
$238$		0			0
$239$	−3911.18			−1.05855			−0.529274	−	0.848451i	$-0.677536\pi$
$239$	−3911.18			−1.05855			−0.529274	+	0.848451i	$0.677536\pi$
$240$		0			0
$241$	4114.37			1.09971			0.549855	−	0.835260i	$-0.314683\pi$
$241$	4114.37			1.09971			0.549855	+	0.835260i	$0.314683\pi$
$242$		0			0
$243$	3763.61	−	429.092i	0.993563	−	0.113277i
$244$		0			0
$245$	−649.841			−0.169456
$246$		0			0
$247$		−	4539.71i		−	1.16945i
$248$		0			0
$249$	6221.00	+	1866.29i	1.58329	+	0.474985i
$250$		0			0
$251$		−	5279.91i		−	1.32775i	−0.747844	−	0.663874i	$-0.768911\pi$
$251$		−	5279.91i		−	1.32775i	0.747844	−	0.663874i	$-0.231089\pi$
$252$		0			0
$253$			9517.94i			2.36517i
$254$		0			0
$255$	−1961.06	−	588.313i	−0.481592	−	0.144477i
$256$		0			0
$257$			2278.07i			0.552928i	0.961024	+	0.276464i	$0.0891626\pi$
$257$			2278.07i			0.552928i	−0.961024	+	0.276464i	$0.910837\pi$
$258$		0			0
$259$	2242.44			0.537985
$260$		0			0
$261$	−5584.28	−	3681.92i	−1.32436	−	0.873199i
$262$		0			0
$263$	6017.98			1.41097			0.705484	−	0.708726i	$-0.250730\pi$
$263$	6017.98			1.41097			0.705484	+	0.708726i	$0.250730\pi$
$264$		0			0
$265$	−400.869			−0.0929252
$266$		0			0
$267$	−1332.22	−	399.663i	−0.305358	−	0.0916067i
$268$		0			0
$269$	−3008.84			−0.681979			−0.340989	−	0.940067i	$-0.610762\pi$
$269$	−3008.84			−0.681979			−0.340989	+	0.940067i	$0.610762\pi$
$270$		0			0
$271$			2907.26i			0.651673i	0.945426	+	0.325836i	$0.105646\pi$
$271$			2907.26i			0.651673i	−0.945426	+	0.325836i	$0.894354\pi$
$272$		0			0
$273$	957.659	−	3192.21i	0.212308	−	0.707698i
$274$		0			0
$275$			1451.39i			0.318262i
$276$		0			0
$277$		−	7971.57i		−	1.72912i	−0.502532	−	0.864558i	$-0.667598\pi$
$277$		−	7971.57i		−	1.72912i	0.502532	−	0.864558i	$-0.332402\pi$
$278$		0			0
$279$	−2277.96	+	3454.93i	−0.488810	+	0.741367i
$280$		0			0
$281$			913.436i			0.193918i	0.995288	+	0.0969591i	$0.0309116\pi$
$281$			913.436i			0.193918i	−0.995288	+	0.0969591i	$0.969088\pi$
$282$		0			0
$283$	1838.40			0.386154			0.193077	−	0.981184i	$-0.438153\pi$
$283$	1838.40			0.386154			0.193077	+	0.981184i	$0.438153\pi$
$284$		0			0
$285$	771.231	−	2570.79i	0.160294	−	0.534317i
$286$		0			0
$287$	394.937			0.0812279
$288$		0			0
$289$	−1297.15			−0.264024
$290$		0			0
$291$	−1339.64	+	4465.49i	−0.269866	+	0.899559i
$292$		0			0
$293$	−742.598			−0.148065			−0.0740324	−	0.997256i	$-0.523587\pi$
$293$	−742.598			−0.148065			−0.0740324	+	0.997256i	$0.523587\pi$
$294$		0			0
$295$		−	2568.06i		−	0.506842i
$296$		0			0
$297$	−5224.87	−	6248.31i	−1.02080	−	1.22075i
$298$		0			0
$299$			7204.44i			1.39346i
$300$		0			0
$301$			3604.15i			0.690166i
$302$		0			0
$303$	−1621.55	+	5405.20i	−0.307445	+	1.02482i
$304$		0			0
$305$		−	135.230i		−	0.0253877i
$306$		0			0
$307$	−8317.65			−1.54630			−0.773149	−	0.634225i	$-0.781319\pi$
$307$	−8317.65			−1.54630			−0.773149	+	0.634225i	$0.781319\pi$
$308$		0			0
$309$	2999.14	+	899.736i	0.552152	+	0.165645i
$310$		0			0
$311$	3016.33			0.549970			0.274985	−	0.961449i	$-0.411327\pi$
$311$	3016.33			0.549970			0.274985	+	0.961449i	$0.411327\pi$
$312$		0			0
$313$	−9400.48			−1.69759			−0.848797	−	0.528720i	$-0.822672\pi$
$313$	−9400.48			−1.69759			−0.848797	+	0.528720i	$0.822672\pi$
$314$		0			0
$315$	1084.62	−	1645.02i	0.194005	−	0.294243i
$316$		0			0
$317$	5801.88			1.02797			0.513985	−	0.857799i	$-0.328169\pi$
$317$	5801.88			1.02797			0.513985	+	0.857799i	$0.328169\pi$
$318$		0			0
$319$			14382.4i			2.52433i
$320$		0			0
$321$	−1041.41	−	312.420i	−0.181077	−	0.0543228i
$322$		0			0
$323$		−	8141.01i		−	1.40241i
$324$		0			0
$325$			1098.60i			0.187506i
$326$		0			0
$327$	4312.28	+	1293.68i	0.729264	+	0.218778i
$328$		0			0
$329$			6011.95i			1.00745i
$330$		0			0
$331$	5997.26			0.995888			0.497944	−	0.867209i	$-0.334088\pi$
$331$	5997.26			0.995888			0.497944	+	0.867209i	$0.334088\pi$
$332$		0			0
$333$	2283.41	−	3463.20i	0.375766	−	0.569916i
$334$		0			0
$335$	−2265.44			−0.369476
$336$		0			0
$337$	−8013.40			−1.29531			−0.647653	−	0.761936i	$-0.724249\pi$
$337$	−8013.40			−1.29531			−0.647653	+	0.761936i	$0.724249\pi$
$338$		0			0
$339$	7246.16	+	2173.83i	1.16094	+	0.348279i
$340$		0			0
$341$	8898.23			1.41310
$342$		0			0
$343$		−	6903.26i		−	1.08671i
$344$		0			0
$345$	−1223.93	+	4079.79i	−0.190998	+	0.636663i
$346$		0			0
$347$			249.070i			0.0385325i	0.999814	+	0.0192662i	$0.00613301\pi$
$347$			249.070i			0.0385325i	−0.999814	+	0.0192662i	$0.993867\pi$
$348$		0			0
$349$			8681.38i			1.33153i	0.746162	+	0.665764i	$0.231894\pi$
$349$			8681.38i			1.33153i	−0.746162	+	0.665764i	$0.768106\pi$
$350$		0			0
$351$	−3954.87	−	4729.55i	−0.601411	−	0.719215i
$352$		0			0
$353$			6747.03i			1.01730i	0.860972	+	0.508652i	$0.169856\pi$
$353$			6747.03i			1.01730i	−0.860972	+	0.508652i	$0.830144\pi$
$354$		0			0
$355$	−2231.29			−0.333590
$356$		0			0
$357$	1717.36	−	5724.56i	0.254600	−	0.848672i
$358$		0			0
$359$	1559.98			0.229339			0.114669	−	0.993404i	$-0.463419\pi$
$359$	1559.98			0.229339			0.114669	+	0.993404i	$0.463419\pi$
$360$		0			0
$361$	3813.21			0.555942
$362$		0			0
$363$	−3045.09	+	10150.4i	−0.440292	+	1.46765i
$364$		0			0
$365$	−2624.10			−0.376307
$366$		0			0
$367$		−	13670.5i		−	1.94440i	−0.234144	−	0.972202i	$-0.575229\pi$
$367$		−	13670.5i		−	1.94440i	0.234144	−	0.972202i	$-0.424771\pi$
$368$		0			0
$369$	402.154	−	609.938i	0.0567352	−	0.0860491i
$370$		0			0
$371$		−	1170.18i		−	0.163755i
$372$		0			0
$373$		−	10962.3i		−	1.52173i	−0.648909	−	0.760866i	$-0.724774\pi$
$373$		−	10962.3i		−	1.52173i	0.648909	−	0.760866i	$-0.275226\pi$
$374$		0			0
$375$	−186.637	+	622.127i	−0.0257010	+	0.0856706i
$376$		0			0
$377$			10886.5i			1.48723i
$378$		0			0
$379$	221.741			0.0300529			0.0150265	−	0.999887i	$-0.495217\pi$
$379$	221.741			0.0300529			0.0150265	+	0.999887i	$0.495217\pi$
$380$		0			0
$381$	8511.29	+	2553.37i	1.14448	+	0.343342i
$382$		0			0
$383$	567.176			0.0756694			0.0378347	−	0.999284i	$-0.487954\pi$
$383$	567.176			0.0756694			0.0378347	+	0.999284i	$0.487954\pi$
$384$		0			0
$385$	−4236.78			−0.560848
$386$		0			0
$387$	5566.22	+	3670.01i	0.731129	+	0.482060i
$388$		0			0
$389$	−10143.0			−1.32204			−0.661019	−	0.750369i	$-0.729876\pi$
$389$	−10143.0			−1.32204			−0.661019	+	0.750369i	$0.729876\pi$
$390$		0			0
$391$			12919.6i			1.67103i
$392$		0			0
$393$	4947.21	+	1484.15i	0.634996	+	0.190498i
$394$		0			0
$395$		−	467.129i		−	0.0595034i
$396$		0			0
$397$			4548.29i			0.574992i	0.957782	+	0.287496i	$0.0928228\pi$
$397$			4548.29i			0.574992i	−0.957782	+	0.287496i	$0.907177\pi$
$398$		0			0
$399$	7504.44	+	2251.32i	0.941583	+	0.282473i
$400$		0			0
$401$		−	141.751i		−	0.0176526i	−0.999961	−	0.00882632i	$-0.997190\pi$
$401$		−	141.751i		−	0.0176526i	0.999961	−	0.00882632i	$-0.00280954\pi$
$402$		0			0
$403$	6735.36			0.832537
$404$		0			0
$405$	−1436.12	−	3350.16i	−0.176201	−	0.411039i
$406$		0			0
$407$	−8919.53			−1.08630
$408$		0			0
$409$	10767.4			1.30175			0.650873	−	0.759186i	$-0.274403\pi$
$409$	10767.4			1.30175			0.650873	+	0.759186i	$0.274403\pi$
$410$		0			0
$411$	1085.66	+	325.697i	0.130296	+	0.0390886i
$412$		0			0
$413$	7496.48			0.893167
$414$		0			0
$415$		−	6249.73i		−	0.739246i
$416$		0			0
$417$	−1301.92	+	4339.76i	−0.152891	+	0.509638i
$418$		0			0
$419$			1276.55i			0.148840i	0.997227	+	0.0744198i	$0.0237105\pi$
$419$			1276.55i			0.148840i	−0.997227	+	0.0744198i	$0.976290\pi$
$420$		0			0
$421$		−	14268.0i		−	1.65173i	−0.563869	−	0.825864i	$-0.690688\pi$
$421$		−	14268.0i		−	1.65173i	0.563869	−	0.825864i	$-0.309312\pi$
$422$		0			0
$423$	9284.81	+	6121.81i	1.06724	+	0.703670i
$424$		0			0
$425$			1970.11i			0.224858i
$426$		0			0
$427$	394.753			0.0447387
$428$		0			0
$429$	−3809.19	+	12697.4i	−0.428693	+	1.42899i
$430$		0			0
$431$	15402.6			1.72139			0.860695	−	0.509120i	$-0.170029\pi$
$431$	15402.6			1.72139			0.860695	+	0.509120i	$0.170029\pi$
$432$		0			0
$433$	9528.41			1.05752			0.528760	−	0.848771i	$-0.322657\pi$
$433$	9528.41			1.05752			0.528760	+	0.848771i	$0.322657\pi$
$434$		0			0
$435$	−1849.46	+	6164.91i	−0.203850	+	0.679505i
$436$		0			0
$437$	−16936.6			−1.85398
$438$		0			0
$439$			2811.08i			0.305616i	0.988256	+	0.152808i	$0.0488317\pi$
$439$			2811.08i			0.305616i	−0.988256	+	0.152808i	$0.951168\pi$
$440$		0			0
$441$	−2929.66	−	1931.63i	−0.316344	−	0.208577i
$442$		0			0
$443$			4207.40i			0.451241i	0.974215	+	0.225620i	$0.0724409\pi$
$443$			4207.40i			0.451241i	−0.974215	+	0.225620i	$0.927559\pi$
$444$		0			0
$445$			1338.37i			0.142573i
$446$		0			0
$447$	3956.02	−	13186.8i	0.418598	−	1.39533i
$448$		0			0
$449$			5099.95i			0.536040i	0.963413	+	0.268020i	$0.0863693\pi$
$449$			5099.95i			0.536040i	−0.963413	+	0.268020i	$0.913631\pi$
$450$		0			0
$451$	−1570.91			−0.164016
$452$		0			0
$453$	3571.67	+	1071.49i	0.370445	+	0.111133i
$454$		0			0
$455$	−3206.96			−0.330427
$456$		0			0
$457$	8660.31			0.886460			0.443230	−	0.896408i	$-0.353833\pi$
$457$	8660.31			0.886460			0.443230	+	0.896408i	$0.353833\pi$
$458$		0			0
$459$	−7092.22	−	8481.44i	−0.721213	−	0.862483i
$460$		0			0
$461$	11202.5			1.13179			0.565893	−	0.824479i	$-0.308532\pi$
$461$	11202.5			1.13179			0.565893	+	0.824479i	$0.308532\pi$
$462$		0			0
$463$		−	2975.15i		−	0.298633i	−0.988789	−	0.149316i	$-0.952293\pi$
$463$		−	2975.15i		−	0.298633i	0.988789	−	0.149316i	$-0.0477073\pi$
$464$		0			0
$465$	3814.16	+	1144.24i	0.380381	+	0.114114i
$466$		0			0
$467$			13454.8i			1.33322i	0.745406	+	0.666611i	$0.232256\pi$
$467$			13454.8i			1.33322i	−0.745406	+	0.666611i	$0.767744\pi$
$468$		0			0
$469$		−	6613.11i		−	0.651098i
$470$		0			0
$471$	−16184.0	−	4855.17i	−1.58327	−	0.474978i
$472$		0			0
$473$		−	14335.9i		−	1.39358i
$474$		0			0
$475$	−2582.66			−0.249475
$476$		0			0
$477$	−1807.22	−	1191.57i	−0.173474	−	0.114378i
$478$		0			0
$479$	11128.5			1.06153			0.530765	−	0.847519i	$-0.321905\pi$
$479$	11128.5			1.06153			0.530765	+	0.847519i	$0.321905\pi$
$480$		0			0
$481$	−6751.48			−0.640002
$482$		0			0
$483$	−11909.4	−	3572.80i	−1.12194	−	0.336580i
$484$		0			0
$485$	4486.12			0.420008
$486$		0			0
$487$		−	2167.65i		−	0.201695i	−0.994902	−	0.100848i	$-0.967845\pi$
$487$		−	2167.65i		−	0.201695i	0.994902	−	0.100848i	$-0.0321555\pi$
$488$		0			0
$489$	269.677	−	898.930i	0.0249391	−	0.0831309i
$490$		0			0
$491$			4706.59i			0.432597i	0.976327	+	0.216299i	$0.0693985\pi$
$491$			4706.59i			0.432597i	−0.976327	+	0.216299i	$0.930601\pi$
$492$		0			0
$493$			19522.7i			1.78348i
$494$		0			0
$495$	−4314.20	+	6543.25i	−0.391735	+	0.594136i
$496$		0			0
$497$		−	6513.40i		−	0.587859i
$498$		0			0
$499$	21594.1			1.93724			0.968620	−	0.248546i	$-0.0799528\pi$
$499$	21594.1			1.93724			0.968620	+	0.248546i	$0.0799528\pi$
$500$		0			0
$501$	−5351.11	+	17837.1i	−0.477186	+	1.59063i
$502$		0			0
$503$	−3910.09			−0.346605			−0.173303	−	0.984869i	$-0.555444\pi$
$503$	−3910.09			−0.346605			−0.173303	+	0.984869i	$0.555444\pi$
$504$		0			0
$505$	5430.17			0.478494
$506$		0			0
$507$	397.032	−	1323.45i	0.0347787	−	0.115930i
$508$		0			0
$509$	18113.4			1.57733			0.788667	−	0.614821i	$-0.210772\pi$
$509$	18113.4			1.57733			0.788667	+	0.614821i	$0.210772\pi$
$510$		0			0
$511$		−	7660.08i		−	0.663135i
$512$		0			0
$513$	11118.5	−	9297.33i	0.956906	−	0.800170i
$514$		0			0
$515$		−	3012.99i		−	0.257802i
$516$		0			0
$517$		−	23913.2i		−	2.03424i
$518$		0			0
$519$	791.302	−	2637.69i	0.0669254	−	0.223086i
$520$		0			0
$521$			16469.0i			1.38487i	0.721480	+	0.692436i	$0.243462\pi$
$521$			16469.0i			1.38487i	−0.721480	+	0.692436i	$0.756538\pi$
$522$		0			0
$523$	−380.198			−0.0317876			−0.0158938	−	0.999874i	$-0.505059\pi$
$523$	−380.198			−0.0317876			−0.0158938	+	0.999874i	$0.505059\pi$
$524$		0			0
$525$	−1816.06	−	544.816i	−0.150970	−	0.0452909i
$526$		0			0
$527$	12078.4			0.998378
$528$		0			0
$529$	14711.1			1.20910
$530$		0			0
$531$	7633.47	−	11577.5i	0.623850	−	0.946179i
$532$		0			0
$533$	−1189.07			−0.0966310
$534$		0			0
$535$			1046.22i			0.0845456i
$536$		0			0
$537$	−11309.0	−	3392.69i	−0.908791	−	0.272636i
$538$		0			0
$539$			7545.38i			0.602973i
$540$		0			0
$541$		−	5985.34i		−	0.475656i	−0.971307	−	0.237828i	$-0.923565\pi$
$541$		−	5985.34i		−	0.475656i	0.971307	−	0.237828i	$-0.0764354\pi$
$542$		0			0
$543$	3823.83	+	1147.14i	0.302203	+	0.0906605i
$544$		0			0
$545$		−	4332.19i		−	0.340497i
$546$		0			0
$547$	−6496.76			−0.507827			−0.253914	−	0.967227i	$-0.581718\pi$
$547$	−6496.76			−0.507827			−0.253914	+	0.967227i	$0.581718\pi$
$548$		0			0
$549$	401.966	−	609.653i	0.0312486	−	0.0473941i
$550$		0			0
$551$	−25592.6			−1.97873
$552$		0			0
$553$	1363.61			0.104858
$554$		0			0
$555$	−3823.29	−	1146.98i	−0.292413	−	0.0877235i
$556$		0			0
$557$	−22280.1			−1.69486			−0.847431	−	0.530905i	$-0.821852\pi$
$557$	−22280.1			−1.69486			−0.847431	+	0.530905i	$0.821852\pi$
$558$		0			0
$559$		−	10851.3i		−	0.821040i
$560$		0			0
$561$	−6830.97	+	22770.0i	−0.514089	+	1.71364i
$562$		0			0
$563$			10370.5i			0.776317i	0.921593	+	0.388158i	$0.126889\pi$
$563$			10370.5i			0.776317i	−0.921593	+	0.388158i	$0.873111\pi$
$564$		0			0
$565$		−	7279.62i		−	0.542046i
$566$		0			0
$567$	9779.53	−	4192.20i	0.724342	−	0.310504i
$568$		0			0
$569$		−	15791.9i		−	1.16350i	−0.813368	−	0.581750i	$-0.802368\pi$
$569$		−	15791.9i		−	1.16350i	0.813368	−	0.581750i	$-0.197632\pi$
$570$		0			0
$571$	−14459.2			−1.05972			−0.529858	−	0.848087i	$-0.677755\pi$
$571$	−14459.2			−1.05972			−0.529858	+	0.848087i	$0.677755\pi$
$572$		0			0
$573$	1998.08	−	6660.31i	0.145674	−	0.485582i
$574$		0			0
$575$	4098.63			0.297261
$576$		0			0
$577$	−1179.27			−0.0850845			−0.0425423	−	0.999095i	$-0.513546\pi$
$577$	−1179.27			−0.0850845			−0.0425423	+	0.999095i	$0.513546\pi$
$578$		0			0
$579$	−1282.28	+	4274.30i	−0.0920378	+	0.306795i
$580$		0			0
$581$	18243.7			1.30271
$582$		0			0
$583$			4654.53i			0.330654i
$584$		0			0
$585$	−3265.56	+	4952.80i	−0.230794	+	0.350039i
$586$		0			0
$587$			3761.00i			0.264451i	0.991220	+	0.132226i	$0.0422124\pi$
$587$			3761.00i			0.264451i	−0.991220	+	0.132226i	$0.957788\pi$
$588$		0			0
$589$			15833.9i			1.10768i
$590$		0			0
$591$	−4539.63	+	15132.2i	−0.315965	+	1.05322i
$592$		0			0
$593$			7912.85i			0.547962i	0.961735	+	0.273981i	$0.0883406\pi$
$593$			7912.85i			0.547962i	−0.961735	+	0.273981i	$0.911659\pi$
$594$		0			0
$595$	−5751.00			−0.396249
$596$		0			0
$597$	8147.14	+	2444.13i	0.558526	+	0.167557i
$598$		0			0
$599$	−13018.4			−0.888012			−0.444006	−	0.896024i	$-0.646443\pi$
$599$	−13018.4			−0.888012			−0.444006	+	0.896024i	$0.646443\pi$
$600$		0			0
$601$	6784.23			0.460457			0.230228	−	0.973137i	$-0.426053\pi$
$601$	6784.23			0.460457			0.230228	+	0.973137i	$0.426053\pi$
$602$		0			0
$603$	−10213.2	−	6733.95i	−0.689742	−	0.454772i
$604$		0			0
$605$	10197.2			0.685251
$606$		0			0
$607$			16589.8i			1.10932i	0.832077	+	0.554661i	$0.187152\pi$
$607$			16589.8i			1.10932i	−0.832077	+	0.554661i	$0.812848\pi$
$608$		0			0
$609$	−17996.1	−	5398.80i	−1.19744	−	0.359229i
$610$		0			0
$611$		−	18100.7i		−	1.19848i
$612$		0			0
$613$			16207.7i			1.06790i	0.845516	+	0.533950i	$0.179293\pi$
$613$			16207.7i			1.06790i	−0.845516	+	0.533950i	$0.820707\pi$
$614$		0			0
$615$	−673.356	−	202.006i	−0.0441502	−	0.0132450i
$616$		0			0
$617$		−	17461.9i		−	1.13936i	−0.821865	−	0.569682i	$-0.807066\pi$
$617$		−	17461.9i		−	1.13936i	0.821865	−	0.569682i	$-0.192934\pi$
$618$		0			0
$619$	−5700.52			−0.370151			−0.185075	−	0.982724i	$-0.559253\pi$
$619$	−5700.52			−0.370151			−0.185075	+	0.982724i	$0.559253\pi$
$620$		0			0
$621$	−17644.8	+	14754.7i	−1.14020	+	0.953439i
$622$		0			0
$623$	−3906.87			−0.251245
$624$		0			0
$625$	625.000			0.0400000
$626$		0			0
$627$	−29849.7	−	8954.86i	−1.90125	−	0.570371i
$628$		0			0
$629$	−12107.3			−0.767491
$630$		0			0
$631$			8186.74i			0.516496i	0.966079	+	0.258248i	$0.0831452\pi$
$631$			8186.74i			0.516496i	−0.966079	+	0.258248i	$0.916855\pi$
$632$		0			0
$633$	−9012.39	+	30041.5i	−0.565893	+	1.88632i
$634$		0			0
$635$		−	8550.60i		−	0.534363i
$636$		0			0
$637$			5711.34i			0.355246i
$638$		0			0
$639$	−10059.2	−	6632.42i	−0.622750	−	0.410602i
$640$		0			0
$641$		−	27307.8i		−	1.68267i	−0.540510	−	0.841337i	$-0.681769\pi$
$641$		−	27307.8i		−	1.68267i	0.540510	−	0.841337i	$-0.318231\pi$
$642$		0			0
$643$	11306.2			0.693427			0.346714	−	0.937971i	$-0.387298\pi$
$643$	11306.2			0.693427			0.346714	+	0.937971i	$0.387298\pi$
$644$		0			0
$645$	1843.48	−	6144.97i	0.112538	−	0.375129i
$646$		0			0
$647$	−4998.32			−0.303716			−0.151858	−	0.988402i	$-0.548526\pi$
$647$	−4998.32			−0.303716			−0.151858	+	0.988402i	$0.548526\pi$
$648$		0			0
$649$	−29818.1			−1.80348
$650$		0			0
$651$	−3340.18	+	11134.0i	−0.201094	+	0.670316i
$652$		0			0
$653$	−23047.6			−1.38120			−0.690599	−	0.723238i	$-0.742653\pi$
$653$	−23047.6			−1.38120			−0.690599	+	0.723238i	$0.742653\pi$
$654$		0			0
$655$		−	4970.06i		−	0.296483i
$656$		0			0
$657$	−11830.2	−	7800.05i	−0.702494	−	0.463180i
$658$		0			0
$659$		−	18811.4i		−	1.11197i	−0.831191	−	0.555986i	$-0.812341\pi$
$659$		−	18811.4i		−	1.11197i	0.831191	−	0.555986i	$-0.187659\pi$
$660$		0			0
$661$		−	30855.3i		−	1.81563i	−0.419370	−	0.907815i	$-0.637749\pi$
$661$		−	30855.3i		−	1.81563i	0.419370	−	0.907815i	$-0.362251\pi$
$662$		0			0
$663$	−5170.58	+	17235.4i	−0.302879	+	1.00960i
$664$		0			0
$665$		−	7539.10i		−	0.439629i
$666$		0			0
$667$	40615.0			2.35775
$668$		0			0
$669$	7588.02	+	2276.39i	0.438520	+	0.131555i
$670$		0			0
$671$	−1570.17			−0.0903365
$672$		0			0
$673$	12293.3			0.704117			0.352058	−	0.935978i	$-0.385482\pi$
$673$	12293.3			0.704117			0.352058	+	0.935978i	$0.385482\pi$
$674$		0			0
$675$	−2690.66	+	2249.94i	−0.153427	+	0.128297i
$676$		0			0
$677$	296.947			0.0168576			0.00842881	−	0.999964i	$-0.497317\pi$
$677$	296.947			0.0168576			0.00842881	+	0.999964i	$0.497317\pi$
$678$		0			0
$679$			13095.5i			0.740147i
$680$		0			0
$681$	−2351.99	−	705.593i	−0.132347	−	0.0397039i
$682$		0			0
$683$		−	6305.85i		−	0.353275i	−0.984276	−	0.176638i	$-0.943478\pi$
$683$		−	6305.85i		−	0.353275i	0.984276	−	0.176638i	$-0.0565220\pi$
$684$		0			0
$685$		−	1090.68i		−	0.0608359i
$686$		0			0
$687$	15273.0	+	4581.86i	0.848181	+	0.254453i
$688$		0			0
$689$			3523.17i			0.194807i
$690$		0			0
$691$	27248.7			1.50013			0.750064	−	0.661365i	$-0.230022\pi$
$691$	27248.7			1.50013			0.750064	+	0.661365i	$0.230022\pi$
$692$		0			0
$693$	−19100.5	−	12593.7i	−1.04700	−	0.690323i
$694$		0			0
$695$	4359.81			0.237952
$696$		0			0
$697$	−2132.34			−0.115880
$698$		0			0
$699$	704.321	+	211.295i	0.0381114	+	0.0114334i
$700$		0			0
$701$	17919.3			0.965481			0.482741	−	0.875763i	$-0.339641\pi$
$701$	17919.3			0.965481			0.482741	+	0.875763i	$0.339641\pi$
$702$		0			0
$703$		−	15871.8i		−	0.851515i
$704$		0			0
$705$	3075.04	−	10250.2i	0.164273	−	0.547581i
$706$		0			0
$707$			15851.3i			0.843211i
$708$		0			0
$709$		−	12485.0i		−	0.661329i	−0.943748	−	0.330665i	$-0.892727\pi$
$709$		−	12485.0i		−	0.661329i	0.943748	−	0.330665i	$-0.107273\pi$
$710$		0			0
$711$	1388.52	−	2105.94i	0.0732402	−	0.111082i
$712$		0			0
$713$		−	25128.1i		−	1.31985i
$714$		0			0
$715$	12756.0			0.667200
$716$		0			0
$717$	−5839.76	+	19466.0i	−0.304170	+	1.01391i
$718$		0			0
$719$	7727.83			0.400834			0.200417	−	0.979711i	$-0.435770\pi$
$719$	7727.83			0.400834			0.200417	+	0.979711i	$0.435770\pi$
$720$		0			0
$721$	8795.28			0.454304
$722$		0			0
$723$	6143.15	−	20477.3i	0.315998	−	1.05333i
$724$		0			0
$725$	6193.38			0.317264
$726$		0			0
$727$		−	12917.1i		−	0.658964i	−0.944162	−	0.329482i	$-0.893126\pi$
$727$		−	12917.1i		−	0.658964i	0.944162	−	0.329482i	$-0.106874\pi$
$728$		0			0
$729$	3483.84	−	19372.2i	0.176997	−	0.984211i
$730$		0			0
$731$		−	19459.5i		−	0.984591i
$732$		0			0
$733$			5574.53i			0.280900i	0.990088	+	0.140450i	$0.0448550\pi$
$733$			5574.53i			0.280900i	−0.990088	+	0.140450i	$0.955145\pi$
$734$		0			0
$735$	−970.275	+	3234.27i	−0.0486927	+	0.162310i
$736$		0			0
$737$			26304.3i			1.31470i
$738$		0			0
$739$	−17289.1			−0.860607			−0.430304	−	0.902684i	$-0.641594\pi$
$739$	−17289.1			−0.860607			−0.430304	+	0.902684i	$0.641594\pi$
$740$		0			0
$741$	−22594.2	−	6778.22i	−1.12013	−	0.336038i
$742$		0			0
$743$	−13359.7			−0.659649			−0.329824	−	0.944042i	$-0.606990\pi$
$743$	−13359.7			−0.659649			−0.329824	+	0.944042i	$0.606990\pi$
$744$		0			0
$745$	−13247.7			−0.651487
$746$		0			0
$747$	18577.1	−	28175.5i	0.909907	−	1.38003i
$748$		0			0
$749$	−3054.03			−0.148988
$750$		0			0
$751$		−	8468.05i		−	0.411456i	−0.978609	−	0.205728i	$-0.934044\pi$
$751$		−	8468.05i		−	0.411456i	0.978609	−	0.205728i	$-0.0659563\pi$
$752$		0			0
$753$	−26278.2	−	7883.41i	−1.27175	−	0.381524i
$754$		0			0
$755$		−	3588.16i		−	0.172962i
$756$		0			0
$757$			6277.34i			0.301392i	0.988580	+	0.150696i	$0.0481515\pi$
$757$			6277.34i			0.301392i	−0.988580	+	0.150696i	$0.951849\pi$
$758$		0			0
$759$	47370.9	+	14211.2i	2.26542	+	0.679623i
$760$		0			0
$761$		−	11037.7i		−	0.525778i	−0.964826	−	0.262889i	$-0.915325\pi$
$761$		−	11037.7i		−	0.525778i	0.964826	−	0.262889i	$-0.0846752\pi$
$762$		0			0
$763$	12646.2			0.600030
$764$		0			0
$765$	−5856.09	+	8881.79i	−0.276768	+	0.419767i
$766$		0			0
$767$	−22570.3			−1.06254
$768$		0			0
$769$	5492.77			0.257574			0.128787	−	0.991672i	$-0.458892\pi$
$769$	5492.77			0.257574			0.128787	+	0.991672i	$0.458892\pi$
$770$		0			0
$771$	11338.0	+	3401.38i	0.529609	+	0.158882i
$772$		0			0
$773$	−21115.4			−0.982497			−0.491248	−	0.871020i	$-0.663459\pi$
$773$	−21115.4			−0.982497			−0.491248	+	0.871020i	$0.663459\pi$
$774$		0			0
$775$		−	3831.77i		−	0.177602i
$776$		0			0
$777$	3348.17	−	11160.6i	0.154588	−	0.515297i
$778$		0			0
$779$		−	2795.33i		−	0.128566i
$780$		0			0
$781$			25907.7i			1.18701i
$782$		0			0
$783$	−26662.8	+	22295.6i	−1.21692	+	1.01760i
$784$		0			0
$785$			16258.7i			0.739235i
$786$		0			0
$787$	−38018.5			−1.72200			−0.861000	−	0.508605i	$-0.830161\pi$
$787$	−38018.5			−1.72200			−0.861000	+	0.508605i	$0.830161\pi$
$788$		0			0
$789$	8985.42	−	29951.6i	0.405436	−	1.35146i
$790$		0			0
$791$	21250.1			0.955205
$792$		0			0
$793$	−1188.51			−0.0532224
$794$		0			0
$795$	−598.535	+	1995.13i	−0.0267017	+	0.0890062i
$796$		0			0
$797$	−5023.82			−0.223278			−0.111639	−	0.993749i	$-0.535610\pi$
$797$	−5023.82			−0.223278			−0.111639	+	0.993749i	$0.535610\pi$
$798$		0			0
$799$		−	32459.7i		−	1.43722i
$800$		0			0
$801$	−3978.26	+	6033.74i	−0.175487	+	0.266157i
$802$		0			0
$803$			30468.8i			1.33900i
$804$		0			0
$805$			11964.4i			0.523839i
$806$		0			0
$807$	−4492.49	+	14975.0i	−0.195964	+	0.653218i
$808$		0			0
$809$			9928.57i			0.431483i	0.976450	+	0.215742i	$0.0692169\pi$
$809$			9928.57i			0.431483i	−0.976450	+	0.215742i	$0.930783\pi$
$810$		0			0
$811$	−16918.7			−0.732549			−0.366274	−	0.930507i	$-0.619367\pi$
$811$	−16918.7			−0.732549			−0.366274	+	0.930507i	$0.619367\pi$
$812$		0			0
$813$	14469.5	+	4340.81i	0.624190	+	0.187256i
$814$		0			0
$815$	−903.082			−0.0388142
$816$		0			0
$817$	25509.9			1.09238
$818$		0			0
$819$	−14457.8	−	9532.56i	−0.616846	−	0.406709i
$820$		0			0
$821$	−18362.5			−0.780579			−0.390289	−	0.920692i	$-0.627625\pi$
$821$	−18362.5			−0.780579			−0.390289	+	0.920692i	$0.627625\pi$
$822$		0			0
$823$		−	35007.5i		−	1.48273i	−0.671103	−	0.741364i	$-0.734179\pi$
$823$		−	35007.5i		−	1.48273i	0.671103	−	0.741364i	$-0.265821\pi$
$824$		0			0
$825$	7223.59	+	2167.06i	0.304840	+	0.0914514i
$826$		0			0
$827$		−	21104.7i		−	0.887404i	−0.896174	−	0.443702i	$-0.853665\pi$
$827$		−	21104.7i		−	0.887404i	0.896174	−	0.443702i	$-0.146335\pi$
$828$		0			0
$829$		−	44840.3i		−	1.87861i	−0.343085	−	0.939304i	$-0.611472\pi$
$829$		−	44840.3i		−	1.87861i	0.343085	−	0.939304i	$-0.388528\pi$
$830$		0			0
$831$	−39674.6	−	11902.3i	−1.65619	−	0.496855i
$832$		0			0
$833$			10242.1i			0.426011i
$834$		0			0
$835$	17919.5			0.742671
$836$		0			0
$837$	13794.0	+	16496.0i	0.569643	+	0.681224i
$838$		0			0
$839$	−19998.2			−0.822900			−0.411450	−	0.911432i	$-0.634978\pi$
$839$	−19998.2			−0.822900			−0.411450	+	0.911432i	$0.634978\pi$
$840$		0			0
$841$	36983.8			1.51641
$842$		0			0
$843$	4546.18	+	1363.85i	0.185740	+	0.0557217i
$844$		0			0
$845$	−1329.56			−0.0541281
$846$		0			0
$847$			29767.0i			1.20756i
$848$		0			0
$849$	2744.91	−	9149.75i	0.110960	−	0.369869i
$850$		0			0
$851$			25188.2i			1.01462i
$852$		0			0
$853$		−	27500.9i		−	1.10388i	−0.833882	−	0.551942i	$-0.813887\pi$
$853$		−	27500.9i		−	1.10388i	0.833882	−	0.551942i	$-0.186113\pi$
$854$		0			0
$855$	−11643.3	−	7676.86i	−0.465723	−	0.307068i
$856$		0			0
$857$		−	1250.31i		−	0.0498365i	−0.999689	−	0.0249183i	$-0.992067\pi$
$857$		−	1250.31i		−	0.0498365i	0.999689	−	0.0249183i	$-0.00793255\pi$
$858$		0			0
$859$	−15260.3			−0.606142			−0.303071	−	0.952968i	$-0.598012\pi$
$859$	−15260.3			−0.606142			−0.303071	+	0.952968i	$0.598012\pi$
$860$		0			0
$861$	589.679	−	1965.61i	0.0233406	−	0.0778023i
$862$		0			0
$863$	15513.7			0.611926			0.305963	−	0.952043i	$-0.401022\pi$
$863$	15513.7			0.611926			0.305963	+	0.952043i	$0.401022\pi$
$864$		0			0
$865$	−2649.87			−0.104160
$866$		0			0
$867$	−1936.77	+	6455.93i	−0.0758664	+	0.252889i
$868$		0			0
$869$	−5423.89			−0.211730
$870$		0			0
$871$			19910.6i			0.774564i
$872$		0			0
$873$	20224.6	+	13334.8i	0.784077	+	0.516970i
$874$		0			0
$875$			1824.45i			0.0704888i
$876$		0			0
$877$			17948.8i			0.691091i	0.938402	+	0.345545i	$0.112306\pi$
$877$			17948.8i			0.691091i	−0.938402	+	0.345545i	$0.887694\pi$
$878$		0			0
$879$	−1108.77	+	3695.92i	−0.0425459	+	0.141821i
$880$		0			0
$881$		−	50006.7i		−	1.91234i	−0.292817	−	0.956169i	$-0.594593\pi$
$881$		−	50006.7i		−	1.91234i	0.292817	−	0.956169i	$-0.405407\pi$
$882$		0			0
$883$	2670.53			0.101779			0.0508894	−	0.998704i	$-0.483794\pi$
$883$	2670.53			0.101779			0.0508894	+	0.998704i	$0.483794\pi$
$884$		0			0
$885$	−12781.3	−	3834.36i	−0.485467	−	0.145639i
$886$		0			0
$887$	−27123.1			−1.02672			−0.513362	−	0.858172i	$-0.671600\pi$
$887$	−27123.1			−1.02672			−0.513362	+	0.858172i	$0.671600\pi$
$888$		0			0
$889$	24960.2			0.941664
$890$		0			0
$891$	−38899.1	+	16674.9i	−1.46259	+	0.626970i
$892$		0			0
$893$	42552.0			1.59457
$894$		0			0
$895$			11361.3i			0.424318i
$896$		0			0
$897$	35856.6	+	10756.9i	1.33469	+	0.400405i
$898$		0			0
$899$		−	37970.6i		−	1.40867i
$900$		0			0
$901$			6318.05i			0.233612i
$902$		0			0
$903$	17937.9	+	5381.34i	0.661059	+	0.198317i
$904$		0			0
$905$		−	3841.49i		−	0.141100i
$906$		0			0
$907$	33013.5			1.20860			0.604298	−	0.796759i	$-0.293454\pi$
$907$	33013.5			1.20860			0.604298	+	0.796759i	$0.293454\pi$
$908$		0			0
$909$	24480.6	+	16141.0i	0.893258	+	0.588957i
$910$		0			0
$911$	47241.1			1.71808			0.859038	−	0.511912i	$-0.171062\pi$
$911$	47241.1			1.71808			0.859038	+	0.511912i	$0.171062\pi$
$912$		0			0
$913$	−72566.3			−2.63044
$914$		0			0
$915$	−673.041	−	201.911i	−0.0243170	−	0.00729506i
$916$		0			0
$917$	14508.2			0.522468
$918$		0			0
$919$		−	50644.8i		−	1.81787i	−0.416942	−	0.908933i	$-0.636898\pi$
$919$		−	50644.8i		−	1.81787i	0.416942	−	0.908933i	$-0.363102\pi$
$920$		0			0
$921$	−12419.0	+	41397.1i	−0.444323	+	1.48108i
$922$		0			0
$923$			19610.4i			0.699333i
$924$		0			0
$925$			3840.94i			0.136529i
$926$		0			0
$927$	8956.00	−	13583.4i	0.317318	−	0.481269i
$928$		0			0
$929$			36463.1i			1.28774i	0.765133	+	0.643872i	$0.222673\pi$
$929$			36463.1i			1.28774i	−0.765133	+	0.643872i	$0.777327\pi$
$930$		0			0
$931$	−13426.5			−0.472650
$932$		0			0
$933$	4503.67	−	15012.3i	0.158032	−	0.526776i
$934$		0			0
$935$	22875.2			0.800106
$936$		0			0
$937$	−1429.44			−0.0498376			−0.0249188	−	0.999689i	$-0.507933\pi$
$937$	−1429.44			−0.0498376			−0.0249188	+	0.999689i	$0.507933\pi$
$938$		0			0
$939$	−14035.8	+	46786.3i	−0.487797	+	1.62600i
$940$		0			0
$941$	8397.60			0.290918			0.145459	−	0.989364i	$-0.453534\pi$
$941$	8397.60			0.290918			0.145459	+	0.989364i	$0.453534\pi$
$942$		0			0
$943$			4436.14i			0.153193i
$944$		0			0
$945$	−6567.85	−	7854.36i	−0.226087	−	0.270373i
$946$		0			0
$947$			43156.1i			1.48087i	0.672127	+	0.740436i	$0.265381\pi$
$947$			43156.1i			1.48087i	−0.672127	+	0.740436i	$0.734619\pi$
$948$		0			0
$949$			23062.8i			0.788883i
$950$		0			0
$951$	8662.77	−	28876.1i	0.295383	−	0.984616i
$952$		0			0
$953$			20652.4i			0.701991i	0.936377	+	0.350995i	$0.114157\pi$
$953$			20652.4i			0.701991i	−0.936377	+	0.350995i	$0.885843\pi$
$954$		0			0
$955$	−6691.07			−0.226720
$956$		0			0
$957$	71581.5	+	21474.3i	2.41787	+	0.725356i
$958$		0			0
$959$	3183.81			0.107206
$960$		0			0
$961$	6299.02			0.211440
$962$		0			0
$963$	−3109.84	+	4716.63i	−0.104064	+	0.157831i
$964$		0			0
$965$	4294.05			0.143244
$966$		0			0
$967$			27587.9i			0.917443i	0.888580	+	0.458721i	$0.151692\pi$
$967$			27587.9i			0.917443i	−0.888580	+	0.458721i	$0.848308\pi$
$968$		0			0
$969$	−40517.9	−	12155.3i	−1.34326	−	0.402977i
$970$		0			0
$971$			13285.7i			0.439093i	0.975602	+	0.219547i	$0.0704578\pi$
$971$			13285.7i			0.439093i	−0.975602	+	0.219547i	$0.929542\pi$
$972$		0			0
$973$			12726.8i			0.419324i
$974$		0			0
$975$	5467.77	+	1640.32i	0.179599	+	0.0538793i
$976$		0			0
$977$			20541.2i			0.672642i	0.941747	+	0.336321i	$0.109183\pi$
$977$			20541.2i			0.672642i	−0.941747	+	0.336321i	$0.890817\pi$
$978$		0			0
$979$	15540.0			0.507314
$980$		0			0
$981$	12877.3	−	19530.7i	0.419103	−	0.635644i
$982$		0			0
$983$	−28203.9			−0.915123			−0.457562	−	0.889178i	$-0.651277\pi$
$983$	−28203.9			−0.915123			−0.457562	+	0.889178i	$0.651277\pi$
$984$		0			0
$985$	15202.1			0.491754
$986$		0			0
$987$	29921.6	+	8976.42i	0.964958	+	0.289486i
$988$		0			0
$989$	−40483.7			−1.30162
$990$		0			0
$991$			30236.1i			0.969206i	0.874734	+	0.484603i	$0.161036\pi$
$991$			30236.1i			0.969206i	−0.874734	+	0.484603i	$0.838964\pi$
$992$		0			0
$993$	8954.48	−	29848.4i	0.286165	−	0.953888i
$994$		0			0
$995$		−	8184.76i		−	0.260778i
$996$		0			0
$997$		−	848.710i		−	0.0269598i	−0.999909	−	0.0134799i	$-0.995709\pi$
$997$		−	848.710i		−	0.0269598i	0.999909	−	0.0134799i	$-0.00429091\pi$
$998$		0			0
$999$	−13827.0	−	16535.5i	−0.437906	−	0.523683i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	480.4.b.a.431.15		24
3.2	odd	2		480.4.b.b.431.16		24
4.3	odd	2		120.4.b.b.11.20	yes	24
8.3	odd	2		480.4.b.b.431.15		24
8.5	even	2		120.4.b.a.11.6	yes	24
12.11	even	2		120.4.b.a.11.5	✓	24
24.5	odd	2		120.4.b.b.11.19	yes	24
24.11	even	2	inner	480.4.b.a.431.16		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.4.b.a.11.5	✓	24	12.11	even	2
120.4.b.a.11.6	yes	24	8.5	even	2
120.4.b.b.11.19	yes	24	24.5	odd	2
120.4.b.b.11.20	yes	24	4.3	odd	2
480.4.b.a.431.15		24	1.1	even	1	trivial
480.4.b.a.431.16		24	24.11	even	2	inner
480.4.b.b.431.15		24	8.3	odd	2
480.4.b.b.431.16		24	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	480.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.49310 - 4.97701i) q^{3} -5.00000 q^{5} -14.5956i q^{7} +(-22.5413 - 14.8623i) q^{9} +O(q^{10})\)	\(q+(1.49310 - 4.97701i) q^{3} -5.00000 q^{5} -14.5956i q^{7} +(-22.5413 - 14.8623i) q^{9} +58.0556i q^{11} +43.9441i q^{13} +(-7.46548 + 24.8851i) q^{15} +78.8045i q^{17} -103.306 q^{19} +(-72.6425 - 21.7926i) q^{21} +163.945 q^{23} +25.0000 q^{25} +(-107.626 + 89.9977i) q^{27} +247.735 q^{29} -153.271i q^{31} +(288.943 + 86.6825i) q^{33} +72.9780i q^{35} +153.638i q^{37} +(218.711 + 65.6128i) q^{39} +27.0586i q^{41} -246.934 q^{43} +(112.707 + 74.3116i) q^{45} -411.901 q^{47} +129.968 q^{49} +(392.211 + 117.663i) q^{51} +80.1737 q^{53} -290.278i q^{55} +(-154.246 + 514.157i) q^{57} +513.612i q^{59} +27.0460i q^{61} +(-216.924 + 329.004i) q^{63} -219.721i q^{65} +453.089 q^{67} +(244.786 - 815.958i) q^{69} +446.258 q^{71} +524.821 q^{73} +(37.3274 - 124.425i) q^{75} +847.356 q^{77} +93.4259i q^{79} +(287.223 + 670.033i) q^{81} +1249.95i q^{83} -394.023i q^{85} +(369.892 - 1232.98i) q^{87} -267.674i q^{89} +641.391 q^{91} +(-762.832 - 228.848i) q^{93} +516.532 q^{95} -897.223 q^{97} +(862.840 - 1308.65i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 120 q^{5}+O(q^{10}) \) 24 * q - 120 * q^5	\( 24 q - 120 q^{5} - 12 q^{19} + 4 q^{21} + 228 q^{23} + 600 q^{25} - 132 q^{27} + 116 q^{33} - 656 q^{39} - 924 q^{47} - 816 q^{49} + 700 q^{51} - 528 q^{53} - 172 q^{57} + 476 q^{63} - 1632 q^{67} - 980 q^{69} + 216 q^{71} - 216 q^{73} + 152 q^{81} + 252 q^{87} + 1800 q^{91} + 60 q^{95} + 792 q^{97} + 1328 q^{99}+O(q^{100}) \) 24 * q - 120 * q^5 - 12 * q^19 + 4 * q^21 + 228 * q^23 + 600 * q^25 - 132 * q^27 + 116 * q^33 - 656 * q^39 - 924 * q^47 - 816 * q^49 + 700 * q^51 - 528 * q^53 - 172 * q^57 + 476 * q^63 - 1632 * q^67 - 980 * q^69 + 216 * q^71 - 216 * q^73 + 152 * q^81 + 252 * q^87 + 1800 * q^91 + 60 * q^95 + 792 * q^97 + 1328 * q^99

Introduction

L-functions

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Database

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