LMFDB - Embedded newform 124.3.c.b.61.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [124,3,Mod(61,124)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(124, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("124.61");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 124 = 2^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	124.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$3.37875527807$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.63368.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 11x^{2} + 8 $ x^4 + 11*x^2 + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			61.4
Root			$0.884878i$ of defining polynomial
Character	$\chi$	$=$	124.61
Dual form			124.3.c.b.61.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.52040i q^{3} +9.21699 q^{5} -3.21699 q^{7} -11.4340 q^{9} +O(q^{10})$	$q+4.52040i q^{3} +9.21699 q^{5} -3.21699 q^{7} -11.4340 q^{9} +8.05991i q^{11} -15.5230i q^{13} +41.6644i q^{15} +25.5446i q^{17} -15.6510 q^{19} -14.5421i q^{21} -23.5829i q^{23} +59.9529 q^{25} -11.0026i q^{27} -38.7218i q^{29} +(4.56602 - 30.6619i) q^{31} -36.4340 q^{33} -29.6510 q^{35} +2.94265i q^{37} +70.1699 q^{39} +6.51893 q^{41} +4.52040i q^{43} -105.387 q^{45} +72.1699 q^{47} -38.6510 q^{49} -115.472 q^{51} -71.3455i q^{53} +74.2881i q^{55} -70.7486i q^{57} +42.9529 q^{59} +83.9258i q^{61} +36.7830 q^{63} -143.075i q^{65} +82.8680 q^{67} +106.604 q^{69} -74.2549 q^{71} -5.11726i q^{73} +271.011i q^{75} -25.9287i q^{77} -8.23109i q^{79} -53.1699 q^{81} -103.585i q^{83} +235.445i q^{85} +175.038 q^{87} +114.759i q^{89} +49.9372i q^{91} +(138.604 + 20.6402i) q^{93} -144.255 q^{95} -21.8209 q^{97} -92.1568i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 18 q^{5} + 6 q^{7} - 8 q^{9}+O(q^{10}) $ 4 * q + 18 * q^5 + 6 * q^7 - 8 * q^9	$ 4 q + 18 q^{5} + 6 q^{7} - 8 q^{9} - 6 q^{19} + 70 q^{25} + 56 q^{31} - 108 q^{33} - 62 q^{35} + 92 q^{39} - 106 q^{41} - 214 q^{45} + 100 q^{47} - 98 q^{49} - 160 q^{51} + 2 q^{59} + 166 q^{63} + 256 q^{67} + 200 q^{69} - 14 q^{71} - 24 q^{81} + 436 q^{87} + 328 q^{93} - 294 q^{95} + 158 q^{97}+O(q^{100}) $ 4 * q + 18 * q^5 + 6 * q^7 - 8 * q^9 - 6 * q^19 + 70 * q^25 + 56 * q^31 - 108 * q^33 - 62 * q^35 + 92 * q^39 - 106 * q^41 - 214 * q^45 + 100 * q^47 - 98 * q^49 - 160 * q^51 + 2 * q^59 + 166 * q^63 + 256 * q^67 + 200 * q^69 - 14 * q^71 - 24 * q^81 + 436 * q^87 + 328 * q^93 - 294 * q^95 + 158 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/124\mathbb{Z}\right)^\times$.

$n$	$63$	$65$
$\chi(n)$	$1$	$-1$

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
$3$			4.52040i			1.50680i	0.657563	+	0.753399i	$0.271587\pi$
$3$			4.52040i			1.50680i	−0.657563	+	0.753399i	$0.728413\pi$
$4$		0			0
$5$	9.21699			1.84340			0.921699	−	0.387906i	$-0.126801\pi$
$5$	9.21699			1.84340			0.921699	+	0.387906i	$0.126801\pi$
$6$		0			0
$7$	−3.21699			−0.459570			−0.229785	−	0.973241i	$-0.573802\pi$
$7$	−3.21699			−0.459570			−0.229785	+	0.973241i	$0.573802\pi$
$8$		0			0
$9$	−11.4340			−1.27044
$10$		0			0
$11$			8.05991i			0.732719i	0.930473	+	0.366359i	$0.119396\pi$
$11$			8.05991i			0.732719i	−0.930473	+	0.366359i	$0.880604\pi$
$12$		0			0
$13$		−	15.5230i		−	1.19407i	−0.802214	−	0.597037i	$-0.796345\pi$
$13$		−	15.5230i		−	1.19407i	0.802214	−	0.597037i	$-0.203655\pi$
$14$		0			0
$15$			41.6644i			2.77763i
$16$		0			0
$17$			25.5446i			1.50263i	0.659946	+	0.751313i	$0.270579\pi$
$17$			25.5446i			1.50263i	−0.659946	+	0.751313i	$0.729421\pi$
$18$		0			0
$19$	−15.6510			−0.823735			−0.411868	−	0.911244i	$-0.635123\pi$
$19$	−15.6510			−0.823735			−0.411868	+	0.911244i	$0.635123\pi$
$20$		0			0
$21$		−	14.5421i		−	0.692480i
$22$		0			0
$23$		−	23.5829i		−	1.02534i	−0.858585	−	0.512671i	$-0.828656\pi$
$23$		−	23.5829i		−	1.02534i	0.858585	−	0.512671i	$-0.171344\pi$
$24$		0			0
$25$	59.9529			2.39812
$26$		0			0
$27$		−	11.0026i		−	0.407502i
$28$		0			0
$29$		−	38.7218i		−	1.33523i	−0.744505	−	0.667617i	$-0.767314\pi$
$29$		−	38.7218i		−	1.33523i	0.744505	−	0.667617i	$-0.232686\pi$
$30$		0			0
$31$	4.56602	−	30.6619i	0.147291	−	0.989093i
$31$	4.56602	−	30.6619i	0.147291	−	0.989093i
$32$		0			0
$33$	−36.4340			−1.10406
$34$		0			0
$35$	−29.6510			−0.847171
$36$		0			0
$37$			2.94265i			0.0795311i	0.999209	+	0.0397655i	$0.0126611\pi$
$37$			2.94265i			0.0795311i	−0.999209	+	0.0397655i	$0.987339\pi$
$38$		0			0
$39$	70.1699			1.79923
$40$		0			0
$41$	6.51893			0.158998			0.0794992	−	0.996835i	$-0.474668\pi$
$41$	6.51893			0.158998			0.0794992	+	0.996835i	$0.474668\pi$
$42$		0			0
$43$			4.52040i			0.105125i	0.998618	+	0.0525627i	$0.0167390\pi$
$43$			4.52040i			0.105125i	−0.998618	+	0.0525627i	$0.983261\pi$
$44$		0			0
$45$	−105.387			−2.34193
$46$		0			0
$47$	72.1699			1.53553			0.767765	−	0.640732i	$-0.221369\pi$
$47$	72.1699			1.53553			0.767765	+	0.640732i	$0.221369\pi$
$48$		0			0
$49$	−38.6510			−0.788795
$50$		0			0
$51$	−115.472			−2.26415
$52$		0			0
$53$		−	71.3455i		−	1.34614i	−0.739578	−	0.673070i	$-0.764975\pi$
$53$		−	71.3455i		−	1.34614i	0.739578	−	0.673070i	$-0.235025\pi$
$54$		0			0
$55$			74.2881i			1.35069i
$56$		0			0
$57$		−	70.7486i		−	1.24120i
$58$		0			0
$59$	42.9529			0.728016			0.364008	−	0.931396i	$-0.381408\pi$
$59$	42.9529			0.728016			0.364008	+	0.931396i	$0.381408\pi$
$60$		0			0
$61$			83.9258i			1.37583i	0.725790	+	0.687916i	$0.241474\pi$
$61$			83.9258i			1.37583i	−0.725790	+	0.687916i	$0.758526\pi$
$62$		0			0
$63$	36.7830			0.583857
$64$		0			0
$65$		−	143.075i		−	2.20115i
$66$		0			0
$67$	82.8680			1.23684			0.618418	−	0.785850i	$-0.287774\pi$
$67$	82.8680			1.23684			0.618418	+	0.785850i	$0.287774\pi$
$68$		0			0
$69$	106.604			1.54498
$70$		0			0
$71$	−74.2549			−1.04584			−0.522922	−	0.852381i	$-0.675158\pi$
$71$	−74.2549			−1.04584			−0.522922	+	0.852381i	$0.675158\pi$
$72$		0			0
$73$		−	5.11726i		−	0.0700994i	−0.999386	−	0.0350497i	$-0.988841\pi$
$73$		−	5.11726i		−	0.0700994i	0.999386	−	0.0350497i	$-0.0111590\pi$
$74$		0			0
$75$			271.011i			3.61348i
$76$		0			0
$77$		−	25.9287i		−	0.336736i
$78$		0			0
$79$		−	8.23109i		−	0.104191i	−0.998642	−	0.0520955i	$-0.983410\pi$
$79$		−	8.23109i		−	0.104191i	0.998642	−	0.0520955i	$-0.0165900\pi$
$80$		0			0
$81$	−53.1699			−0.656419
$82$		0			0
$83$		−	103.585i		−	1.24801i	−0.781419	−	0.624007i	$-0.785504\pi$
$83$		−	103.585i		−	1.24801i	0.781419	−	0.624007i	$-0.214496\pi$
$84$		0			0
$85$			235.445i			2.76994i
$86$		0			0
$87$	175.038			2.01193
$88$		0			0
$89$			114.759i			1.28943i	0.764425	+	0.644713i	$0.223023\pi$
$89$			114.759i			1.28943i	−0.764425	+	0.644713i	$0.776977\pi$
$90$		0			0
$91$			49.9372i			0.548760i
$92$		0			0
$93$	138.604	+	20.6402i	1.49036	+	0.221938i
$94$		0			0
$95$	−144.255			−1.51847
$96$		0			0
$97$	−21.8209			−0.224957			−0.112479	−	0.993654i	$-0.535879\pi$
$97$	−21.8209			−0.224957			−0.112479	+	0.993654i	$0.535879\pi$
$98$		0			0
$99$		−	92.1568i		−	0.930877i
$100$		0			0
$101$	−121.387			−1.20185			−0.600925	−	0.799305i	$-0.705201\pi$
$101$	−121.387			−1.20185			−0.600925	+	0.799305i	$0.705201\pi$
$102$		0			0
$103$	−145.387			−1.41152			−0.705762	−	0.708449i	$-0.749395\pi$
$103$	−145.387			−1.41152			−0.705762	+	0.708449i	$0.749395\pi$
$104$		0			0
$105$		−	134.034i		−	1.27652i
$106$		0			0
$107$	−119.651			−1.11823			−0.559117	−	0.829089i	$-0.688860\pi$
$107$	−119.651			−1.11823			−0.559117	+	0.829089i	$0.688860\pi$
$108$		0			0
$109$	−59.9908			−0.550374			−0.275187	−	0.961391i	$-0.588740\pi$
$109$	−59.9908			−0.550374			−0.275187	+	0.961391i	$0.588740\pi$
$110$		0			0
$111$	−13.3019			−0.119837
$112$		0			0
$113$	66.8587			0.591670			0.295835	−	0.955239i	$-0.404402\pi$
$113$	66.8587			0.591670			0.295835	+	0.955239i	$0.404402\pi$
$114$		0			0
$115$		−	217.363i		−	1.89011i
$116$		0			0
$117$			177.489i			1.51700i
$118$		0			0
$119$		−	82.1768i		−	0.690562i
$120$		0			0
$121$	56.0379			0.463123
$122$		0			0
$123$			29.4682i			0.239579i
$124$		0			0
$125$	322.161			2.57729
$126$		0			0
$127$		−	31.4716i		−	0.247808i	−0.992294	−	0.123904i	$-0.960459\pi$
$127$		−	31.4716i		−	0.247808i	0.992294	−	0.123904i	$-0.0395415\pi$
$128$		0			0
$129$	−20.4340			−0.158403
$130$		0			0
$131$	−93.1320			−0.710932			−0.355466	−	0.934689i	$-0.615678\pi$
$131$	−93.1320			−0.710932			−0.355466	+	0.934689i	$0.615678\pi$
$132$		0			0
$133$	50.3490			0.378564
$134$		0			0
$135$		−	101.410i		−	0.751189i
$136$		0			0
$137$		−	45.5880i		−	0.332759i	−0.986062	−	0.166379i	$-0.946792\pi$
$137$		−	45.5880i		−	0.332759i	0.986062	−	0.166379i	$-0.0532077\pi$
$138$		0			0
$139$			224.613i			1.61592i	0.589236	+	0.807961i	$0.299429\pi$
$139$			224.613i			1.61592i	−0.589236	+	0.807961i	$0.700571\pi$
$140$		0			0
$141$			326.237i			2.31373i
$142$		0			0
$143$	125.114			0.874920
$144$		0			0
$145$		−	356.898i		−	2.46137i
$146$		0			0
$147$		−	174.718i		−	1.18856i
$148$		0			0
$149$	−247.548			−1.66139			−0.830697	−	0.556725i	$-0.812058\pi$
$149$	−247.548			−1.66139			−0.830697	+	0.556725i	$0.812058\pi$
$150$		0			0
$151$			7.88873i			0.0522433i	0.999659	+	0.0261216i	$0.00831572\pi$
$151$			7.88873i			0.0522433i	−0.999659	+	0.0261216i	$0.991684\pi$
$152$		0			0
$153$		−	292.077i		−	1.90900i
$154$		0			0
$155$	42.0850	−	282.610i	0.271516	−	1.82329i
$156$		0			0
$157$	204.689			1.30375			0.651875	−	0.758326i	$-0.273983\pi$
$157$	204.689			1.30375			0.651875	+	0.758326i	$0.273983\pi$
$158$		0			0
$159$	322.510			2.02836
$160$		0			0
$161$			75.8659i			0.471216i
$162$		0			0
$163$	−148.179			−0.909074			−0.454537	−	0.890728i	$-0.650195\pi$
$163$	−148.179			−0.909074			−0.454537	+	0.890728i	$0.650195\pi$
$164$		0			0
$165$	−335.812			−2.03522
$166$		0			0
$167$		−	69.5965i		−	0.416746i	−0.978049	−	0.208373i	$-0.933183\pi$
$167$		−	69.5965i		−	0.416746i	0.978049	−	0.208373i	$-0.0668167\pi$
$168$		0			0
$169$	−71.9621			−0.425811
$170$		0			0
$171$	178.953			1.04651
$172$		0			0
$173$	−114.340			−0.660924			−0.330462	−	0.943819i	$-0.607205\pi$
$173$	−114.340			−0.660924			−0.330462	+	0.943819i	$0.607205\pi$
$174$		0			0
$175$	−192.868			−1.10210
$176$		0			0
$177$			194.164i			1.09697i
$178$		0			0
$179$			80.7703i			0.451231i	0.974216	+	0.225615i	$0.0724392\pi$
$179$			80.7703i			0.451231i	−0.974216	+	0.225615i	$0.927561\pi$
$180$		0			0
$181$		−	3.32667i		−	0.0183794i	−0.999958	−	0.00918970i	$-0.997075\pi$
$181$		−	3.32667i		−	0.0183794i	0.999958	−	0.00918970i	$-0.00292521\pi$
$182$		0			0
$183$	−379.378			−2.07310
$184$		0			0
$185$			27.1224i			0.146607i
$186$		0			0
$187$	−205.887			−1.10100
$188$		0			0
$189$			35.3951i			0.187276i
$190$		0			0
$191$	−95.5568			−0.500297			−0.250149	−	0.968207i	$-0.580480\pi$
$191$	−95.5568			−0.500297			−0.250149	+	0.968207i	$0.580480\pi$
$192$		0			0
$193$	77.5568			0.401849			0.200924	−	0.979607i	$-0.435605\pi$
$193$	77.5568			0.401849			0.200924	+	0.979607i	$0.435605\pi$
$194$		0			0
$195$	646.755			3.31669
$196$		0			0
$197$			305.638i			1.55146i	0.631064	+	0.775731i	$0.282619\pi$
$197$			305.638i			1.55146i	−0.631064	+	0.775731i	$0.717381\pi$
$198$		0			0
$199$		−	181.158i		−	0.910343i	−0.890404	−	0.455171i	$-0.849578\pi$
$199$		−	181.158i		−	0.910343i	0.890404	−	0.455171i	$-0.150422\pi$
$200$		0			0
$201$			374.596i			1.86366i
$202$		0			0
$203$			124.568i			0.613634i
$204$		0			0
$205$	60.0850			0.293097
$206$		0			0
$207$			269.646i			1.30264i
$208$		0			0
$209$		−	126.145i		−	0.603567i
$210$		0			0
$211$	68.0092			0.322319			0.161159	−	0.986928i	$-0.448477\pi$
$211$	68.0092			0.322319			0.161159	+	0.986928i	$0.448477\pi$
$212$		0			0
$213$		−	335.661i		−	1.57587i
$214$		0			0
$215$			41.6644i			0.193788i
$216$		0			0
$217$	−14.6888	+	98.6390i	−0.0676905	+	0.454558i
$218$		0			0
$219$	23.1320			0.105626
$220$		0			0
$221$	396.528			1.79425
$222$		0			0
$223$			263.293i			1.18069i	0.807152	+	0.590344i	$0.201008\pi$
$223$			263.293i			1.18069i	−0.807152	+	0.590344i	$0.798992\pi$
$224$		0			0
$225$	−685.500			−3.04667
$226$		0			0
$227$	164.416			0.724298			0.362149	−	0.932120i	$-0.382043\pi$
$227$	164.416			0.724298			0.362149	+	0.932120i	$0.382043\pi$
$228$		0			0
$229$			298.559i			1.30375i	0.758326	+	0.651876i	$0.226018\pi$
$229$			298.559i			1.30375i	−0.758326	+	0.651876i	$0.773982\pi$
$230$		0			0
$231$	117.208			0.507393
$232$		0			0
$233$	65.3112			0.280305			0.140153	−	0.990130i	$-0.455241\pi$
$233$	65.3112			0.280305			0.140153	+	0.990130i	$0.455241\pi$
$234$		0			0
$235$	665.189			2.83059
$236$		0			0
$237$	37.2078			0.156995
$238$		0			0
$239$		−	192.202i		−	0.804194i	−0.915597	−	0.402097i	$-0.868281\pi$
$239$		−	192.202i		−	0.804194i	0.915597	−	0.402097i	$-0.131719\pi$
$240$		0			0
$241$		−	38.5506i		−	0.159961i	−0.996796	−	0.0799805i	$-0.974514\pi$
$241$		−	38.5506i		−	0.159961i	0.996796	−	0.0799805i	$-0.0254858\pi$
$242$		0			0
$243$		−	339.372i		−	1.39659i
$244$		0			0
$245$	−356.246			−1.45406
$246$		0			0
$247$			242.949i			0.983601i
$248$		0			0
$249$	468.246			1.88050
$250$		0			0
$251$		−	139.281i		−	0.554904i	−0.960739	−	0.277452i	$-0.910510\pi$
$251$		−	139.281i		−	0.554904i	0.960739	−	0.277452i	$-0.0894900\pi$
$252$		0			0
$253$	190.076			0.751287
$254$		0			0
$255$	−1064.30			−4.17374
$256$		0			0
$257$	273.217			1.06310			0.531551	−	0.847027i	$-0.321610\pi$
$257$	273.217			1.06310			0.531551	+	0.847027i	$0.321610\pi$
$258$		0			0
$259$		−	9.46648i		−	0.0365501i
$260$		0			0
$261$			442.744i			1.69634i
$262$		0			0
$263$		−	388.287i		−	1.47638i	−0.674595	−	0.738188i	$-0.735682\pi$
$263$		−	388.287i		−	1.47638i	0.674595	−	0.738188i	$-0.264318\pi$
$264$		0			0
$265$		−	657.590i		−	2.48147i
$266$		0			0
$267$	−518.755			−1.94290
$268$		0			0
$269$			192.374i			0.715144i	0.933886	+	0.357572i	$0.116395\pi$
$269$			192.374i			0.715144i	−0.933886	+	0.357572i	$0.883605\pi$
$270$		0			0
$271$			276.299i			1.01955i	0.860306	+	0.509777i	$0.170272\pi$
$271$			276.299i			1.01955i	−0.860306	+	0.509777i	$0.829728\pi$
$272$		0			0
$273$	−225.736			−0.826872
$274$		0			0
$275$			483.215i			1.75715i
$276$		0			0
$277$			220.731i			0.796864i	0.917198	+	0.398432i	$0.130446\pi$
$277$			220.731i			0.796864i	−0.917198	+	0.398432i	$0.869554\pi$
$278$		0			0
$279$	−52.2078	+	350.587i	−0.187125	+	1.25659i
$280$		0			0
$281$	287.293			1.02239			0.511197	−	0.859464i	$-0.329202\pi$
$281$	287.293			1.02239			0.511197	+	0.859464i	$0.329202\pi$
$282$		0			0
$283$	24.2641			0.0857388			0.0428694	−	0.999081i	$-0.486350\pi$
$283$	24.2641			0.0857388			0.0428694	+	0.999081i	$0.486350\pi$
$284$		0			0
$285$		−	652.089i		−	2.28803i
$286$		0			0
$287$	−20.9713			−0.0730709
$288$		0			0
$289$	−363.528			−1.25788
$290$		0			0
$291$		−	98.6390i		−	0.338966i
$292$		0			0
$293$	187.359			0.639451			0.319726	−	0.947510i	$-0.396409\pi$
$293$	187.359			0.639451			0.319726	+	0.947510i	$0.396409\pi$
$294$		0			0
$295$	395.897			1.34202
$296$		0			0
$297$	88.6796			0.298585
$298$		0			0
$299$	−366.076			−1.22433
$300$		0			0
$301$		−	14.5421i		−	0.0483125i
$302$		0			0
$303$		−	548.717i		−	1.81095i
$304$		0			0
$305$			773.543i			2.53621i
$306$		0			0
$307$	−491.123			−1.59975			−0.799874	−	0.600168i	$-0.795101\pi$
$307$	−491.123			−1.59975			−0.799874	+	0.600168i	$0.795101\pi$
$308$		0			0
$309$		−	657.206i		−	2.12688i
$310$		0			0
$311$	4.19756			0.0134970			0.00674848	−	0.999977i	$-0.497852\pi$
$311$	4.19756			0.0134970			0.00674848	+	0.999977i	$0.497852\pi$
$312$		0			0
$313$		−	309.816i		−	0.989828i	−0.868942	−	0.494914i	$-0.835200\pi$
$313$		−	309.816i		−	0.989828i	0.868942	−	0.494914i	$-0.164800\pi$
$314$		0			0
$315$	339.029			1.07628
$316$		0			0
$317$	−196.859			−0.621006			−0.310503	−	0.950572i	$-0.600497\pi$
$317$	−196.859			−0.621006			−0.310503	+	0.950572i	$0.600497\pi$
$318$		0			0
$319$	312.094			0.978352
$320$		0			0
$321$		−	540.870i		−	1.68495i
$322$		0			0
$323$		−	399.798i		−	1.23777i
$324$		0			0
$325$		−	930.646i		−	2.86353i
$326$		0			0
$327$		−	271.182i		−	0.829303i
$328$		0			0
$329$	−232.170			−0.705684
$330$		0			0
$331$			495.152i			1.49593i	0.663739	+	0.747964i	$0.268969\pi$
$331$			495.152i			1.49593i	−0.663739	+	0.747964i	$0.731031\pi$
$332$		0			0
$333$		−	33.6462i		−	0.101040i
$334$		0			0
$335$	763.793			2.27998
$336$		0			0
$337$		−	271.608i		−	0.805958i	−0.915209	−	0.402979i	$-0.867975\pi$
$337$		−	271.608i		−	0.805958i	0.915209	−	0.402979i	$-0.132025\pi$
$338$		0			0
$339$			302.228i			0.891528i
$340$		0			0
$341$	247.132	+	36.8017i	0.724727	+	0.107923i
$342$		0			0
$343$	281.972			0.822077
$344$		0			0
$345$	982.567			2.84802
$346$		0			0
$347$			300.562i			0.866174i	0.901352	+	0.433087i	$0.142576\pi$
$347$			300.562i			0.866174i	−0.901352	+	0.433087i	$0.857424\pi$
$348$		0			0
$349$	−617.472			−1.76926			−0.884630	−	0.466293i	$-0.845589\pi$
$349$	−617.472			−1.76926			−0.884630	+	0.466293i	$0.845589\pi$
$350$		0			0
$351$	−170.792			−0.486588
$352$		0			0
$353$			449.652i			1.27380i	0.770946	+	0.636901i	$0.219784\pi$
$353$			449.652i			1.27380i	−0.770946	+	0.636901i	$0.780216\pi$
$354$		0			0
$355$	−684.406			−1.92791
$356$		0			0
$357$	371.472			1.04054
$358$		0			0
$359$	−294.312			−0.819811			−0.409906	−	0.912128i	$-0.634438\pi$
$359$	−294.312			−0.819811			−0.409906	+	0.912128i	$0.634438\pi$
$360$		0			0
$361$	−116.047			−0.321460
$362$		0			0
$363$			253.313i			0.697833i
$364$		0			0
$365$		−	47.1657i		−	0.129221i
$366$		0			0
$367$			59.7044i			0.162682i	0.996686	+	0.0813411i	$0.0259203\pi$
$367$			59.7044i			0.162682i	−0.996686	+	0.0813411i	$0.974080\pi$
$368$		0			0
$369$	−74.5374			−0.201998
$370$		0			0
$371$			229.518i			0.618646i
$372$		0			0
$373$	−149.727			−0.401412			−0.200706	−	0.979652i	$-0.564324\pi$
$373$	−149.727			−0.401412			−0.200706	+	0.979652i	$0.564324\pi$
$374$		0			0
$375$			1456.29i			3.88345i
$376$		0			0
$377$	−601.077			−1.59437
$378$		0			0
$379$	221.284			0.583862			0.291931	−	0.956439i	$-0.405702\pi$
$379$	221.284			0.583862			0.291931	+	0.956439i	$0.405702\pi$
$380$		0			0
$381$	142.264			0.373397
$382$		0			0
$383$		−	186.275i		−	0.486359i	−0.969981	−	0.243179i	$-0.921810\pi$
$383$		−	186.275i		−	0.486359i	0.969981	−	0.243179i	$-0.0781904\pi$
$384$		0			0
$385$		−	238.984i		−	0.620738i
$386$		0			0
$387$		−	51.6861i		−	0.133556i
$388$		0			0
$389$			140.174i			0.360344i	0.983635	+	0.180172i	$0.0576655\pi$
$389$			140.174i			0.360344i	−0.983635	+	0.180172i	$0.942335\pi$
$390$		0			0
$391$	602.416			1.54070
$392$		0			0
$393$		−	420.994i		−	1.07123i
$394$		0			0
$395$		−	75.8659i		−	0.192065i
$396$		0			0
$397$	106.085			0.267217			0.133608	−	0.991034i	$-0.457344\pi$
$397$	106.085			0.267217			0.133608	+	0.991034i	$0.457344\pi$
$398$		0			0
$399$			227.598i			0.570420i
$400$		0			0
$401$			97.9543i			0.244275i	0.992513	+	0.122138i	$0.0389749\pi$
$401$			97.9543i			0.244275i	−0.992513	+	0.122138i	$0.961025\pi$
$402$		0			0
$403$	−475.963	−	70.8781i	−1.18105	−	0.175876i
$404$		0			0
$405$	−490.067			−1.21004
$406$		0			0
$407$	−23.7175			−0.0582739
$408$		0			0
$409$		−	11.7289i		−	0.0286771i	−0.999897	−	0.0143386i	$-0.995436\pi$
$409$		−	11.7289i		−	0.0286771i	0.999897	−	0.0143386i	$-0.00456426\pi$
$410$		0			0
$411$	206.076			0.501401
$412$		0			0
$413$	−138.179			−0.334574
$414$		0			0
$415$		−	954.743i		−	2.30059i
$416$		0			0
$417$	−1015.34			−2.43487
$418$		0			0
$419$	5.74514			0.0137116			0.00685578	−	0.999976i	$-0.497818\pi$
$419$	5.74514			0.0137116			0.00685578	+	0.999976i	$0.497818\pi$
$420$		0			0
$421$	35.8209			0.0850852			0.0425426	−	0.999095i	$-0.486454\pi$
$421$	35.8209			0.0850852			0.0425426	+	0.999095i	$0.486454\pi$
$422$		0			0
$423$	−825.189			−1.95080
$424$		0			0
$425$			1531.48i			3.60347i
$426$		0			0
$427$		−	269.988i		−	0.632291i
$428$		0			0
$429$			565.563i			1.31833i
$430$		0			0
$431$	216.170			0.501554			0.250777	−	0.968045i	$-0.419314\pi$
$431$	216.170			0.501554			0.250777	+	0.968045i	$0.419314\pi$
$432$		0			0
$433$		−	274.847i		−	0.634750i	−0.948300	−	0.317375i	$-0.897199\pi$
$433$		−	274.847i		−	0.634750i	0.948300	−	0.317375i	$-0.102801\pi$
$434$		0			0
$435$	1613.32			3.70879
$436$		0			0
$437$			369.095i			0.844610i
$438$		0			0
$439$	549.463			1.25162			0.625812	−	0.779974i	$-0.284768\pi$
$439$	549.463			1.25162			0.625812	+	0.779974i	$0.284768\pi$
$440$		0			0
$441$	441.934			1.00212
$442$		0			0
$443$	151.669			0.342369			0.171184	−	0.985239i	$-0.445241\pi$
$443$	151.669			0.342369			0.171184	+	0.985239i	$0.445241\pi$
$444$		0			0
$445$			1057.73i			2.37692i
$446$		0			0
$447$		−	1119.01i		−	2.50339i
$448$		0			0
$449$			345.854i			0.770277i	0.922859	+	0.385138i	$0.125846\pi$
$449$			345.854i			0.770277i	−0.922859	+	0.385138i	$0.874154\pi$
$450$		0			0
$451$			52.5420i			0.116501i
$452$		0			0
$453$	−35.6602			−0.0787201
$454$		0			0
$455$			460.271i			1.01158i
$456$		0			0
$457$		−	707.060i		−	1.54718i	−0.633688	−	0.773589i	$-0.718460\pi$
$457$		−	707.060i		−	1.54718i	0.633688	−	0.773589i	$-0.281540\pi$
$458$		0			0
$459$	281.056			0.612323
$460$		0			0
$461$		−	860.967i		−	1.86761i	−0.357787	−	0.933803i	$-0.616469\pi$
$461$		−	860.967i		−	1.86761i	0.357787	−	0.933803i	$-0.383531\pi$
$462$		0			0
$463$			336.471i			0.726719i	0.931649	+	0.363360i	$0.118370\pi$
$463$			336.471i			0.726719i	−0.931649	+	0.363360i	$0.881630\pi$
$464$		0			0
$465$	1277.51	+	190.241i	2.74733	+	0.409120i
$466$		0			0
$467$	−429.727			−0.920186			−0.460093	−	0.887871i	$-0.652184\pi$
$467$	−429.727			−0.920186			−0.460093	+	0.887871i	$0.652184\pi$
$468$		0			0
$469$	−266.585			−0.568412
$470$		0			0
$471$			925.275i			1.96449i
$472$		0			0
$473$	−36.4340			−0.0770274
$474$		0			0
$475$	−938.321			−1.97541
$476$		0			0
$477$			815.763i			1.71019i
$478$		0			0
$479$	232.652			0.485703			0.242852	−	0.970063i	$-0.421917\pi$
$479$	232.652			0.485703			0.242852	+	0.970063i	$0.421917\pi$
$480$		0			0
$481$	45.6786			0.0949660
$482$		0			0
$483$	−342.944			−0.710028
$484$		0			0
$485$	−201.123			−0.414686
$486$		0			0
$487$			293.955i			0.603604i	0.953371	+	0.301802i	$0.0975882\pi$
$487$			293.955i			0.603604i	−0.953371	+	0.301802i	$0.902412\pi$
$488$		0			0
$489$		−	669.828i		−	1.36979i
$490$		0			0
$491$		−	554.348i		−	1.12902i	−0.825427	−	0.564509i	$-0.809066\pi$
$491$		−	554.348i		−	1.12902i	0.825427	−	0.564509i	$-0.190934\pi$
$492$		0			0
$493$	989.134			2.00636
$494$		0			0
$495$		−	849.409i		−	1.71598i
$496$		0			0
$497$	238.877			0.480638
$498$		0			0
$499$			49.4237i			0.0990454i	0.998773	+	0.0495227i	$0.0157700\pi$
$499$			49.4237i			0.0990454i	−0.998773	+	0.0495227i	$0.984230\pi$
$500$		0			0
$501$	314.604			0.627952
$502$		0			0
$503$	302.048			0.600493			0.300247	−	0.953862i	$-0.402931\pi$
$503$	302.048			0.600493			0.300247	+	0.953862i	$0.402931\pi$
$504$		0			0
$505$	−1118.82			−2.21549
$506$		0			0
$507$		−	325.297i		−	0.641612i
$508$		0			0
$509$		−	904.593i		−	1.77720i	−0.458687	−	0.888598i	$-0.651680\pi$
$509$		−	904.593i		−	1.77720i	0.458687	−	0.888598i	$-0.348320\pi$
$510$		0			0
$511$			16.4622i			0.0322156i
$512$		0			0
$513$			172.201i			0.335674i
$514$		0			0
$515$	−1340.03			−2.60200
$516$		0			0
$517$			581.683i			1.12511i
$518$		0			0
$519$		−	516.861i		−	0.995879i
$520$		0			0
$521$	238.340			0.457466			0.228733	−	0.973489i	$-0.426542\pi$
$521$	238.340			0.457466			0.228733	+	0.973489i	$0.426542\pi$
$522$		0			0
$523$		−	328.027i		−	0.627203i	−0.949555	−	0.313601i	$-0.898464\pi$
$523$		−	328.027i		−	0.627203i	0.949555	−	0.313601i	$-0.101536\pi$
$524$		0			0
$525$		−	871.840i		−	1.66065i
$526$		0			0
$527$	783.247	+	116.637i	1.48624	+	0.221323i
$528$		0			0
$529$	−27.1515			−0.0513260
$530$		0			0
$531$	−491.123			−0.924902
$532$		0			0
$533$		−	101.193i		−	0.189856i
$534$		0			0
$535$	−1102.82			−2.06135
$536$		0			0
$537$	−365.114			−0.679914
$538$		0			0
$539$		−	311.523i		−	0.577965i
$540$		0			0
$541$	−842.067			−1.55650			−0.778250	−	0.627955i	$-0.783892\pi$
$541$	−842.067			−1.55650			−0.778250	+	0.627955i	$0.783892\pi$
$542$		0			0
$543$	15.0379			0.0276940
$544$		0			0
$545$	−552.934			−1.01456
$546$		0			0
$547$	700.803			1.28118			0.640588	−	0.767885i	$-0.278691\pi$
$547$	700.803			1.28118			0.640588	+	0.767885i	$0.278691\pi$
$548$		0			0
$549$		−	959.606i		−	1.74792i
$550$		0			0
$551$			606.034i			1.09988i
$552$		0			0
$553$			26.4793i			0.0478831i
$554$		0			0
$555$	−122.604			−0.220908
$556$		0			0
$557$			37.2274i			0.0668355i	0.999441	+	0.0334178i	$0.0106392\pi$
$557$			37.2274i			0.0668355i	−0.999441	+	0.0334178i	$0.989361\pi$
$558$		0			0
$559$	70.1699			0.125528
$560$		0			0
$561$		−	930.693i		−	1.65899i
$562$		0			0
$563$	−240.896			−0.427879			−0.213939	−	0.976847i	$-0.568629\pi$
$563$	−240.896			−0.427879			−0.213939	+	0.976847i	$0.568629\pi$
$564$		0			0
$565$	616.236			1.09068
$566$		0			0
$567$	171.047			0.301670
$568$		0			0
$569$			129.301i			0.227242i	0.993524	+	0.113621i	$0.0362450\pi$
$569$			129.301i			0.227242i	−0.993524	+	0.113621i	$0.963755\pi$
$570$		0			0
$571$			1092.40i			1.91314i	0.291499	+	0.956571i	$0.405846\pi$
$571$			1092.40i			1.91314i	−0.291499	+	0.956571i	$0.594154\pi$
$572$		0			0
$573$		−	431.955i		−	0.753847i
$574$		0			0
$575$		−	1413.86i		−	2.45889i
$576$		0			0
$577$	1052.62			1.82431			0.912153	−	0.409850i	$-0.134419\pi$
$577$	1052.62			1.82431			0.912153	+	0.409850i	$0.134419\pi$
$578$		0			0
$579$			350.587i			0.605505i
$580$		0			0
$581$			333.232i			0.573549i
$582$		0			0
$583$	575.038			0.986343
$584$		0			0
$585$			1635.92i			2.79644i
$586$		0			0
$587$			625.906i			1.06628i	0.846027	+	0.533140i	$0.178988\pi$
$587$			625.906i			1.06628i	−0.846027	+	0.533140i	$0.821012\pi$
$588$		0			0
$589$	−71.4626	+	479.888i	−0.121329	+	0.814751i
$590$		0			0
$591$	−1381.60			−2.33774
$592$		0			0
$593$	500.971			0.844808			0.422404	−	0.906408i	$-0.361186\pi$
$593$	500.971			0.844808			0.422404	+	0.906408i	$0.361186\pi$
$594$		0			0
$595$		−	757.423i		−	1.27298i
$596$		0			0
$597$	818.907			1.37170
$598$		0			0
$599$	305.991			0.510836			0.255418	−	0.966831i	$-0.417787\pi$
$599$	305.991			0.510836			0.255418	+	0.966831i	$0.417787\pi$
$600$		0			0
$601$			592.727i			0.986235i	0.869963	+	0.493117i	$0.164143\pi$
$601$			592.727i			0.986235i	−0.869963	+	0.493117i	$0.835857\pi$
$602$		0			0
$603$	−947.511			−1.57133
$604$		0			0
$605$	516.500			0.853720
$606$		0			0
$607$	853.095			1.40543			0.702714	−	0.711472i	$-0.251971\pi$
$607$	853.095			1.40543			0.702714	+	0.711472i	$0.251971\pi$
$608$		0			0
$609$	−563.095			−0.924623
$610$		0			0
$611$		−	1120.29i		−	1.83354i
$612$		0			0
$613$		−	129.856i		−	0.211837i	−0.994375	−	0.105919i	$-0.966222\pi$
$613$		−	129.856i		−	0.211837i	0.994375	−	0.105919i	$-0.0337783\pi$
$614$		0			0
$615$			271.608i			0.441639i
$616$		0			0
$617$	−368.907			−0.597904			−0.298952	−	0.954268i	$-0.596637\pi$
$617$	−368.907			−0.597904			−0.298952	+	0.954268i	$0.596637\pi$
$618$		0			0
$619$			454.339i			0.733989i	0.930223	+	0.366995i	$0.119613\pi$
$619$			454.339i			0.733989i	−0.930223	+	0.366995i	$0.880387\pi$
$620$		0			0
$621$	−259.472			−0.417829
$622$		0			0
$623$		−	369.178i		−	0.592581i
$624$		0			0
$625$	1470.53			2.35285
$626$		0			0
$627$	570.227			0.909453
$628$		0			0
$629$	−75.1689			−0.119505
$630$		0			0
$631$		−	962.590i		−	1.52550i	−0.646694	−	0.762749i	$-0.723849\pi$
$631$		−	962.590i		−	1.52550i	0.646694	−	0.762749i	$-0.276151\pi$
$632$		0			0
$633$			307.429i			0.485669i
$634$		0			0
$635$		−	290.073i		−	0.456808i
$636$		0			0
$637$			599.977i			0.941880i
$638$		0			0
$639$	849.029			1.32868
$640$		0			0
$641$			53.6434i			0.0836870i	0.999124	+	0.0418435i	$0.0133231\pi$
$641$			53.6434i			0.0836870i	−0.999124	+	0.0418435i	$0.986677\pi$
$642$		0			0
$643$		−	285.086i		−	0.443368i	−0.975119	−	0.221684i	$-0.928845\pi$
$643$		−	285.086i		−	0.443368i	0.975119	−	0.221684i	$-0.0711553\pi$
$644$		0			0
$645$	−188.340			−0.292000
$646$		0			0
$647$		−	997.134i		−	1.54116i	−0.637340	−	0.770582i	$-0.719965\pi$
$647$		−	997.134i		−	1.54116i	0.637340	−	0.770582i	$-0.280035\pi$
$648$		0			0
$649$			346.197i			0.533431i
$650$		0			0
$651$	−445.887	−	66.3994i	−0.684927	−	0.101996i
$652$		0			0
$653$	205.660			0.314947			0.157473	−	0.987523i	$-0.449665\pi$
$653$	205.660			0.314947			0.157473	+	0.987523i	$0.449665\pi$
$654$		0			0
$655$	−858.397			−1.31053
$656$		0			0
$657$			58.5106i			0.0890573i
$658$		0			0
$659$	1070.54			1.62449			0.812245	−	0.583316i	$-0.198245\pi$
$659$	1070.54			1.62449			0.812245	+	0.583316i	$0.198245\pi$
$660$		0			0
$661$	−773.162			−1.16968			−0.584842	−	0.811147i	$-0.698844\pi$
$661$	−773.162			−1.16968			−0.584842	+	0.811147i	$0.698844\pi$
$662$		0			0
$663$			1792.46i			2.70357i
$664$		0			0
$665$	464.067			0.697844
$666$		0			0
$667$	−913.171			−1.36907
$668$		0			0
$669$	−1190.19			−1.77906
$670$		0			0
$671$	−676.434			−1.00810
$672$		0			0
$673$			1126.90i			1.67445i	0.546862	+	0.837223i	$0.315822\pi$
$673$			1126.90i			1.67445i	−0.546862	+	0.837223i	$0.684178\pi$
$674$		0			0
$675$		−	659.635i		−	0.977238i
$676$		0			0
$677$		−	116.207i		−	0.171650i	−0.996310	−	0.0858250i	$-0.972647\pi$
$677$		−	116.207i		−	0.171650i	0.996310	−	0.0858250i	$-0.0273526\pi$
$678$		0			0
$679$	70.1976			0.103384
$680$		0			0
$681$			743.223i			1.09137i
$682$		0			0
$683$	1016.16			1.48779			0.743895	−	0.668296i	$-0.232976\pi$
$683$	1016.16			1.48779			0.743895	+	0.668296i	$0.232976\pi$
$684$		0			0
$685$		−	420.184i		−	0.613407i
$686$		0			0
$687$	−1349.60			−1.96449
$688$		0			0
$689$	−1107.49			−1.60739
$690$		0			0
$691$	1008.84			1.45997			0.729986	−	0.683462i	$-0.239527\pi$
$691$	1008.84			1.45997			0.729986	+	0.683462i	$0.239527\pi$
$692$		0			0
$693$			296.468i			0.427803i
$694$		0			0
$695$			2070.26i			2.97879i
$696$		0			0
$697$			166.524i			0.238915i
$698$		0			0
$699$			295.232i			0.422364i
$700$		0			0
$701$	−594.632			−0.848262			−0.424131	−	0.905601i	$-0.639420\pi$
$701$	−594.632			−0.848262			−0.424131	+	0.905601i	$0.639420\pi$
$702$		0			0
$703$		−	46.0553i		−	0.0655126i
$704$		0			0
$705$			3006.92i			4.26513i
$706$		0			0
$707$	390.500			0.552335
$708$		0			0
$709$		−	249.172i		−	0.351442i	−0.984440	−	0.175721i	$-0.943774\pi$
$709$		−	249.172i		−	0.351442i	0.984440	−	0.175721i	$-0.0562257\pi$
$710$		0			0
$711$			94.1141i			0.132369i
$712$		0			0
$713$	−723.095	−	107.680i	−1.01416	−	0.151024i
$714$		0			0
$715$	1153.17			1.61283
$716$		0			0
$717$	868.831			1.21176
$718$		0			0
$719$		−	667.700i		−	0.928651i	−0.885665	−	0.464325i	$-0.846297\pi$
$719$		−	667.700i		−	0.928651i	0.885665	−	0.464325i	$-0.153703\pi$
$720$		0			0
$721$	467.708			0.648694
$722$		0			0
$723$	174.264			0.241029
$724$		0			0
$725$		−	2321.48i		−	3.20205i
$726$		0			0
$727$	393.217			0.540876			0.270438	−	0.962737i	$-0.412831\pi$
$727$	393.217			0.540876			0.270438	+	0.962737i	$0.412831\pi$
$728$		0			0
$729$	1055.57			1.44797
$730$		0			0
$731$	−115.472			−0.157964
$732$		0			0
$733$	−584.445			−0.797333			−0.398667	−	0.917096i	$-0.630527\pi$
$733$	−584.445			−0.797333			−0.398667	+	0.917096i	$0.630527\pi$
$734$		0			0
$735$		−	1610.37i		−	2.19098i
$736$		0			0
$737$			667.908i			0.906253i
$738$		0			0
$739$			301.506i			0.407992i	0.978972	+	0.203996i	$0.0653930\pi$
$739$			301.506i			0.407992i	−0.978972	+	0.203996i	$0.934607\pi$
$740$		0			0
$741$	−1098.23			−1.48209
$742$		0			0
$743$			988.819i			1.33085i	0.746466	+	0.665423i	$0.231749\pi$
$743$			988.819i			1.33085i	−0.746466	+	0.665423i	$0.768251\pi$
$744$		0			0
$745$	−2281.64			−3.06261
$746$		0			0
$747$			1184.39i			1.58553i
$748$		0			0
$749$	384.916			0.513907
$750$		0			0
$751$	−315.708			−0.420384			−0.210192	−	0.977660i	$-0.567409\pi$
$751$	−315.708			−0.420384			−0.210192	+	0.977660i	$0.567409\pi$
$752$		0			0
$753$	629.605			0.836129
$754$		0			0
$755$			72.7104i			0.0963051i
$756$		0			0
$757$		−	425.088i		−	0.561543i	−0.959775	−	0.280772i	$-0.909410\pi$
$757$		−	425.088i		−	0.561543i	0.959775	−	0.280772i	$-0.0905904\pi$
$758$		0			0
$759$			859.218i			1.13204i
$760$		0			0
$761$		−	148.234i		−	0.194788i	−0.995246	−	0.0973941i	$-0.968949\pi$
$761$		−	148.234i		−	0.194788i	0.995246	−	0.0973941i	$-0.0310507\pi$
$762$		0			0
$763$	192.990			0.252935
$764$		0			0
$765$		−	2692.07i		−	3.51904i
$766$		0			0
$767$		−	666.756i		−	0.869304i
$768$		0			0
$769$	−311.933			−0.405635			−0.202818	−	0.979217i	$-0.565010\pi$
$769$	−311.933			−0.405635			−0.202818	+	0.979217i	$0.565010\pi$
$770$		0			0
$771$			1235.05i			1.60188i
$772$		0			0
$773$		−	864.247i		−	1.11804i	−0.829153	−	0.559021i	$-0.811177\pi$
$773$		−	864.247i		−	1.11804i	0.829153	−	0.559021i	$-0.188823\pi$
$774$		0			0
$775$	273.746	−	1838.27i	0.353221	−	2.37196i
$776$		0			0
$777$	42.7922			0.0550737
$778$		0			0
$779$	−102.028			−0.130973
$780$		0			0
$781$		−	598.487i		−	0.766309i
$782$		0			0
$783$	−426.039			−0.544111
$784$		0			0
$785$	1886.62			2.40333
$786$		0			0
$787$		−	1348.53i		−	1.71350i	−0.515730	−	0.856751i	$-0.672479\pi$
$787$		−	1348.53i		−	1.71350i	0.515730	−	0.856751i	$-0.327521\pi$
$788$		0			0
$789$	1755.21			2.22460
$790$		0			0
$791$	−215.084			−0.271914
$792$		0			0
$793$	1302.78			1.64284
$794$		0			0
$795$	2972.57			3.73908
$796$		0			0
$797$			349.056i			0.437962i	0.975729	+	0.218981i	$0.0702733\pi$
$797$			349.056i			0.437962i	−0.975729	+	0.218981i	$0.929727\pi$
$798$		0			0
$799$			1843.55i			2.30733i
$800$		0			0
$801$		−	1312.15i		−	1.63814i
$802$		0			0
$803$	41.2446			0.0513632
$804$		0			0
$805$			699.255i			0.868640i
$806$		0			0
$807$	−869.605			−1.07758
$808$		0			0
$809$		−	285.974i		−	0.353491i	−0.984257	−	0.176745i	$-0.943443\pi$
$809$		−	285.974i		−	0.353491i	0.984257	−	0.176745i	$-0.0565569\pi$
$810$		0			0
$811$	165.775			0.204408			0.102204	−	0.994763i	$-0.467411\pi$
$811$	165.775			0.204408			0.102204	+	0.994763i	$0.467411\pi$
$812$		0			0
$813$	−1248.98			−1.53626
$814$		0			0
$815$	−1365.77			−1.67579
$816$		0			0
$817$		−	70.7486i		−	0.0865956i
$818$		0			0
$819$		−	570.981i		−	0.697168i
$820$		0			0
$821$			424.496i			0.517048i	0.966005	+	0.258524i	$0.0832360\pi$
$821$			424.496i			0.517048i	−0.966005	+	0.258524i	$0.916764\pi$
$822$		0			0
$823$			1338.93i			1.62689i	0.581642	+	0.813445i	$0.302410\pi$
$823$			1338.93i			1.62689i	−0.581642	+	0.813445i	$0.697590\pi$
$824$		0			0
$825$	−2184.32			−2.64766
$826$		0			0
$827$		−	699.125i		−	0.845375i	−0.906275	−	0.422688i	$-0.861087\pi$
$827$		−	699.125i		−	0.845375i	0.906275	−	0.422688i	$-0.138913\pi$
$828$		0			0
$829$		−	1419.19i		−	1.71193i	−0.517032	−	0.855966i	$-0.672963\pi$
$829$		−	1419.19i		−	1.71193i	0.517032	−	0.855966i	$-0.327037\pi$
$830$		0			0
$831$	−997.793			−1.20071
$832$		0			0
$833$		−	987.325i		−	1.18526i
$834$		0			0
$835$		−	641.471i		−	0.768228i
$836$		0			0
$837$	−337.359	−	50.2379i	−0.403058	−	0.0600214i
$838$		0			0
$839$	−411.114			−0.490004			−0.245002	−	0.969523i	$-0.578789\pi$
$839$	−411.114			−0.490004			−0.245002	+	0.969523i	$0.578789\pi$
$840$		0			0
$841$	−658.378			−0.782851
$842$		0			0
$843$			1298.68i			1.54054i
$844$		0			0
$845$	−663.274			−0.784940
$846$		0			0
$847$	−180.273			−0.212837
$848$		0			0
$849$			109.683i			0.129191i
$850$		0			0
$851$	69.3961			0.0815465
$852$		0			0
$853$	683.171			0.800904			0.400452	−	0.916318i	$-0.368853\pi$
$853$	683.171			0.800904			0.400452	+	0.916318i	$0.368853\pi$
$854$		0			0
$855$	1649.41			1.92913
$856$		0			0
$857$	540.227			0.630370			0.315185	−	0.949030i	$-0.397933\pi$
$857$	540.227			0.630370			0.315185	+	0.949030i	$0.397933\pi$
$858$		0			0
$859$			1137.82i			1.32458i	0.749246	+	0.662291i	$0.230416\pi$
$859$			1137.82i			1.32458i	−0.749246	+	0.662291i	$0.769584\pi$
$860$		0			0
$861$		−	94.7988i		−	0.110103i
$862$		0			0
$863$		−	436.646i		−	0.505963i	−0.967471	−	0.252982i	$-0.918589\pi$
$863$		−	436.646i		−	0.505963i	0.967471	−	0.252982i	$-0.0814112\pi$
$864$		0			0
$865$	−1053.87			−1.21835
$866$		0			0
$867$		−	1643.29i		−	1.89538i
$868$		0			0
$869$	66.3418			0.0763427
$870$		0			0
$871$		−	1286.36i		−	1.47687i
$872$		0			0
$873$	249.500			0.285796
$874$		0			0
$875$	−1036.39			−1.18444
$876$		0			0
$877$	−73.7636			−0.0841090			−0.0420545	−	0.999115i	$-0.513390\pi$
$877$	−73.7636			−0.0841090			−0.0420545	+	0.999115i	$0.513390\pi$
$878$		0			0
$879$			846.938i			0.963524i
$880$		0			0
$881$		−	109.341i		−	0.124110i	−0.998073	−	0.0620550i	$-0.980235\pi$
$881$		−	109.341i		−	0.124110i	0.998073	−	0.0620550i	$-0.0197654\pi$
$882$		0			0
$883$			1598.39i			1.81018i	0.425221	+	0.905089i	$0.360196\pi$
$883$			1598.39i			1.81018i	−0.425221	+	0.905089i	$0.639804\pi$
$884$		0			0
$885$			1789.61i			2.02216i
$886$		0			0
$887$	1514.29			1.70721			0.853604	−	0.520922i	$-0.174412\pi$
$887$	1514.29			1.70721			0.853604	+	0.520922i	$0.174412\pi$
$888$		0			0
$889$			101.244i			0.113885i
$890$		0			0
$891$		−	428.545i		−	0.480970i
$892$		0			0
$893$	−1129.53			−1.26487
$894$		0			0
$895$			744.459i			0.831798i
$896$		0			0
$897$		−	1654.81i		−	1.84482i
$898$		0			0
$899$	−1187.28	−	176.804i	−1.32067	−	0.196668i
$900$		0			0
$901$	1822.49			2.02275
$902$		0			0
$903$	65.7359			0.0727973
$904$		0			0
$905$		−	30.6619i		−	0.0338805i
$906$		0			0
$907$	−1501.96			−1.65596			−0.827980	−	0.560757i	$-0.810510\pi$
$907$	−1501.96			−1.65596			−0.827980	+	0.560757i	$0.810510\pi$
$908$		0			0
$909$	1387.94			1.52688
$910$		0			0
$911$		−	426.662i		−	0.468344i	−0.972195	−	0.234172i	$-0.924762\pi$
$911$		−	426.662i		−	0.468344i	0.972195	−	0.234172i	$-0.0752379\pi$
$912$		0			0
$913$	834.886			0.914443
$914$		0			0
$915$	−3496.72			−3.82155
$916$		0			0
$917$	299.605			0.326723
$918$		0			0
$919$	−1692.49			−1.84167			−0.920834	−	0.389954i	$-0.872491\pi$
$919$	−1692.49			−1.84167			−0.920834	+	0.389954i	$0.872491\pi$
$920$		0			0
$921$		−	2220.07i		−	2.41050i
$922$		0			0
$923$			1152.65i			1.24881i
$924$		0			0
$925$			176.420i			0.190725i
$926$		0			0
$927$	1662.35			1.79326
$928$		0			0
$929$		−	1196.29i		−	1.28772i	−0.765144	−	0.643859i	$-0.777332\pi$
$929$		−	1196.29i		−	1.28772i	0.765144	−	0.643859i	$-0.222668\pi$
$930$		0			0
$931$	604.925			0.649759
$932$		0			0
$933$			18.9746i			0.0203372i
$934$		0			0
$935$	−1897.66			−2.02959
$936$		0			0
$937$	1226.11			1.30855			0.654277	−	0.756255i	$-0.272973\pi$
$937$	1226.11			1.30855			0.654277	+	0.756255i	$0.272973\pi$
$938$		0			0
$939$	1400.49			1.49147
$940$		0			0
$941$			387.181i			0.411457i	0.978609	+	0.205728i	$0.0659563\pi$
$941$			387.181i			0.411457i	−0.978609	+	0.205728i	$0.934044\pi$
$942$		0			0
$943$		−	153.735i		−	0.163028i
$944$		0			0
$945$			326.237i			0.345224i
$946$		0			0
$947$			385.636i			0.407218i	0.979052	+	0.203609i	$0.0652672\pi$
$947$			385.636i			0.407218i	−0.979052	+	0.203609i	$0.934733\pi$
$948$		0			0
$949$	−79.4350			−0.0837039
$950$		0			0
$951$		−	889.880i		−	0.935730i
$952$		0			0
$953$			708.546i			0.743490i	0.928335	+	0.371745i	$0.121240\pi$
$953$			708.546i			0.743490i	−0.928335	+	0.371745i	$0.878760\pi$
$954$		0			0
$955$	−880.746			−0.922247
$956$		0			0
$957$			1410.79i			1.47418i
$958$		0			0
$959$			146.656i			0.152926i
$960$		0			0
$961$	−919.303	−	280.006i	−0.956611	−	0.291369i
$962$		0			0
$963$	1368.09			1.42065
$964$		0			0
$965$	714.840			0.740767
$966$		0			0
$967$			699.130i			0.722988i	0.932374	+	0.361494i	$0.117733\pi$
$967$			699.130i			0.722988i	−0.932374	+	0.361494i	$0.882267\pi$
$968$		0			0
$969$	1807.25			1.86506
$970$		0			0
$971$	872.452			0.898509			0.449255	−	0.893404i	$-0.351690\pi$
$971$	872.452			0.898509			0.449255	+	0.893404i	$0.351690\pi$
$972$		0			0
$973$		−	722.579i		−	0.742630i
$974$		0			0
$975$	4206.89			4.31476
$976$		0			0
$977$	827.633			0.847116			0.423558	−	0.905869i	$-0.360781\pi$
$977$	827.633			0.847116			0.423558	+	0.905869i	$0.360781\pi$
$978$		0			0
$979$	−924.946			−0.944786
$980$		0			0
$981$	685.933			0.699219
$982$		0			0
$983$		−	265.857i		−	0.270454i	−0.990815	−	0.135227i	$-0.956824\pi$
$983$		−	265.857i		−	0.270454i	0.990815	−	0.135227i	$-0.0431764\pi$
$984$		0			0
$985$			2817.06i			2.85996i
$986$		0			0
$987$		−	1049.50i		−	1.06332i
$988$		0			0
$989$	106.604			0.107790
$990$		0			0
$991$			1209.46i			1.22045i	0.792229	+	0.610223i	$0.208920\pi$
$991$			1209.46i			1.22045i	−0.792229	+	0.610223i	$0.791080\pi$
$992$		0			0
$993$	−2238.28			−2.25406
$994$		0			0
$995$		−	1669.73i		−	1.67812i
$996$		0			0
$997$	−870.333			−0.872951			−0.436476	−	0.899716i	$-0.643774\pi$
$997$	−870.333			−0.872951			−0.436476	+	0.899716i	$0.643774\pi$
$998$		0			0
$999$	32.3767			0.0324091

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.3.c.b.61.4	yes	4
3.2	odd	2		1116.3.h.d.433.1		4
4.3	odd	2		496.3.e.e.433.1		4
5.2	odd	4		3100.3.f.b.1549.7		8
5.3	odd	4		3100.3.f.b.1549.2		8
5.4	even	2		3100.3.d.b.1301.1		4
31.30	odd	2	inner	124.3.c.b.61.1	✓	4
93.92	even	2		1116.3.h.d.433.2		4
124.123	even	2		496.3.e.e.433.4		4
155.92	even	4		3100.3.f.b.1549.1		8
155.123	even	4		3100.3.f.b.1549.8		8
155.154	odd	2		3100.3.d.b.1301.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.3.c.b.61.1	✓	4	31.30	odd	2	inner
124.3.c.b.61.4	yes	4	1.1	even	1	trivial
496.3.e.e.433.1		4	4.3	odd	2
496.3.e.e.433.4		4	124.123	even	2
1116.3.h.d.433.1		4	3.2	odd	2
1116.3.h.d.433.2		4	93.92	even	2
3100.3.d.b.1301.1		4	5.4	even	2
3100.3.d.b.1301.4		4	155.154	odd	2
3100.3.f.b.1549.1		8	155.92	even	4
3100.3.f.b.1549.2		8	5.3	odd	4
3100.3.f.b.1549.7		8	5.2	odd	4
3100.3.f.b.1549.8		8	155.123	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 124 = 2^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	124.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.52040i q^{3} +9.21699 q^{5} -3.21699 q^{7} -11.4340 q^{9} +O(q^{10})\)	\(q+4.52040i q^{3} +9.21699 q^{5} -3.21699 q^{7} -11.4340 q^{9} +8.05991i q^{11} -15.5230i q^{13} +41.6644i q^{15} +25.5446i q^{17} -15.6510 q^{19} -14.5421i q^{21} -23.5829i q^{23} +59.9529 q^{25} -11.0026i q^{27} -38.7218i q^{29} +(4.56602 - 30.6619i) q^{31} -36.4340 q^{33} -29.6510 q^{35} +2.94265i q^{37} +70.1699 q^{39} +6.51893 q^{41} +4.52040i q^{43} -105.387 q^{45} +72.1699 q^{47} -38.6510 q^{49} -115.472 q^{51} -71.3455i q^{53} +74.2881i q^{55} -70.7486i q^{57} +42.9529 q^{59} +83.9258i q^{61} +36.7830 q^{63} -143.075i q^{65} +82.8680 q^{67} +106.604 q^{69} -74.2549 q^{71} -5.11726i q^{73} +271.011i q^{75} -25.9287i q^{77} -8.23109i q^{79} -53.1699 q^{81} -103.585i q^{83} +235.445i q^{85} +175.038 q^{87} +114.759i q^{89} +49.9372i q^{91} +(138.604 + 20.6402i) q^{93} -144.255 q^{95} -21.8209 q^{97} -92.1568i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 18 q^{5} + 6 q^{7} - 8 q^{9}+O(q^{10}) \) 4 * q + 18 * q^5 + 6 * q^7 - 8 * q^9	\( 4 q + 18 q^{5} + 6 q^{7} - 8 q^{9} - 6 q^{19} + 70 q^{25} + 56 q^{31} - 108 q^{33} - 62 q^{35} + 92 q^{39} - 106 q^{41} - 214 q^{45} + 100 q^{47} - 98 q^{49} - 160 q^{51} + 2 q^{59} + 166 q^{63} + 256 q^{67} + 200 q^{69} - 14 q^{71} - 24 q^{81} + 436 q^{87} + 328 q^{93} - 294 q^{95} + 158 q^{97}+O(q^{100}) \) 4 * q + 18 * q^5 + 6 * q^7 - 8 * q^9 - 6 * q^19 + 70 * q^25 + 56 * q^31 - 108 * q^33 - 62 * q^35 + 92 * q^39 - 106 * q^41 - 214 * q^45 + 100 * q^47 - 98 * q^49 - 160 * q^51 + 2 * q^59 + 166 * q^63 + 256 * q^67 + 200 * q^69 - 14 * q^71 - 24 * q^81 + 436 * q^87 + 328 * q^93 - 294 * q^95 + 158 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more