LMFDB - Embedded newform 3100.3.d.b.1301.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3100,3,Mod(1301,3100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3100, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3100.1301");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 3100 = 2^{2} \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	3100.d (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:	$84.4688819517$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 124)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1301.4
Root			$0.884878i$ of defining polynomial
Character	$\chi$	$=$	3100.1301
Dual form			3100.3.d.b.1301.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.52040i q^{3} +3.21699 q^{7} -11.4340 q^{9} +O(q^{10})$	$q+4.52040i q^{3} +3.21699 q^{7} -11.4340 q^{9} -8.05991i q^{11} -15.5230i q^{13} +25.5446i q^{17} -15.6510 q^{19} +14.5421i q^{21} -23.5829i q^{23} -11.0026i q^{27} +38.7218i q^{29} +(4.56602 + 30.6619i) q^{31} +36.4340 q^{33} +2.94265i q^{37} +70.1699 q^{39} +6.51893 q^{41} +4.52040i q^{43} -72.1699 q^{47} -38.6510 q^{49} -115.472 q^{51} -71.3455i q^{53} -70.7486i q^{57} +42.9529 q^{59} -83.9258i q^{61} -36.7830 q^{63} -82.8680 q^{67} +106.604 q^{69} -74.2549 q^{71} -5.11726i q^{73} -25.9287i q^{77} +8.23109i q^{79} -53.1699 q^{81} -103.585i q^{83} -175.038 q^{87} -114.759i q^{89} -49.9372i q^{91} +(-138.604 + 20.6402i) q^{93} +21.8209 q^{97} +92.1568i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{7} - 8 q^{9}+O(q^{10}) $ 4 * q - 6 * q^7 - 8 * q^9	$ 4 q - 6 q^{7} - 8 q^{9} - 6 q^{19} + 56 q^{31} + 108 q^{33} + 92 q^{39} - 106 q^{41} - 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} - 158 q^{97}+O(q^{100}) $ 4 * q - 6 * q^7 - 8 * q^9 - 6 * q^19 + 56 * q^31 + 108 * q^33 + 92 * q^39 - 106 * q^41 - 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 - 158 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3100\mathbb{Z}\right)^\times$.

$n$	$1551$	$1801$	$2977$
$\chi(n)$	$1$	$-1$	$1$

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
$3$			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
$5$		0			0
$5$		0			0
$6$		0			0
$7$	3.21699			0.459570			0.229785	−	0.973241i	$-0.426198\pi$
$7$	3.21699			0.459570			0.229785	+	0.973241i	$0.426198\pi$
$8$		0			0
$9$	−11.4340			−1.27044
$10$		0			0
$11$		−	8.05991i		−	0.732719i	−0.930473	−	0.366359i	$-0.880604\pi$
$11$		−	8.05991i		−	0.732719i	0.930473	−	0.366359i	$-0.119396\pi$
$12$		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
$15$		0			0
$16$		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
$25$		0			0
$26$		0			0
$27$		−	11.0026i		−	0.407502i
$28$		0			0
$29$			38.7218i			1.33523i	0.744505	+	0.667617i	$0.232686\pi$
$29$			38.7218i			1.33523i	−0.744505	+	0.667617i	$0.767314\pi$
$30$		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
$35$		0			0
$36$		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
$45$		0			0
$46$		0			0
$47$	−72.1699			−1.53553			−0.767765	−	0.640732i	$-0.778631\pi$
$47$	−72.1699			−1.53553			−0.767765	+	0.640732i	$0.778631\pi$
$48$		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
$55$		0			0
$56$		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	$-0.758526\pi$
$61$		−	83.9258i		−	1.37583i	0.725790	−	0.687916i	$-0.241474\pi$
$62$		0			0
$63$	−36.7830			−0.583857
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−82.8680			−1.23684			−0.618418	−	0.785850i	$-0.712226\pi$
$67$	−82.8680			−1.23684			−0.618418	+	0.785850i	$0.712226\pi$
$68$		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
$75$		0			0
$76$		0			0
$77$		−	25.9287i		−	0.336736i
$78$		0			0
$79$			8.23109i			0.104191i	0.998642	+	0.0520955i	$0.0165900\pi$
$79$			8.23109i			0.104191i	−0.998642	+	0.0520955i	$0.983410\pi$
$80$		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
$85$		0			0
$86$		0			0
$87$	−175.038			−2.01193
$88$		0			0
$89$		−	114.759i		−	1.28943i	−0.764425	−	0.644713i	$-0.776977\pi$
$89$		−	114.759i		−	1.28943i	0.764425	−	0.644713i	$-0.223023\pi$
$90$		0			0
$91$		−	49.9372i		−	0.548760i
$92$		0			0
$93$	−138.604	+	20.6402i	−1.49036	+	0.221938i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	21.8209			0.224957			0.112479	−	0.993654i	$-0.464121\pi$
$97$	21.8209			0.224957			0.112479	+	0.993654i	$0.464121\pi$
$98$		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	$-0.250605\pi$
$103$	145.387			1.41152			0.705762	+	0.708449i	$0.250605\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	119.651			1.11823			0.559117	−	0.829089i	$-0.311140\pi$
$107$	119.651			1.11823			0.559117	+	0.829089i	$0.311140\pi$
$108$		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	$-0.595598\pi$
$113$	−66.8587			−0.591670			−0.295835	+	0.955239i	$0.595598\pi$
$114$		0			0
$115$		0			0
$116$		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
$125$		0			0
$126$		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
$135$		0			0
$136$		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	$-0.700571\pi$
$139$		−	224.613i		−	1.61592i	0.589236	−	0.807961i	$-0.299429\pi$
$140$		0			0
$141$		−	326.237i		−	2.31373i
$142$		0			0
$143$	−125.114			−0.874920
$144$		0			0
$145$		0			0
$146$		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	$-0.991684\pi$
$151$		−	7.88873i		−	0.0522433i	0.999659	−	0.0261216i	$-0.00831572\pi$
$152$		0			0
$153$		−	292.077i		−	1.90900i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−204.689			−1.30375			−0.651875	−	0.758326i	$-0.726017\pi$
$157$	−204.689			−1.30375			−0.651875	+	0.758326i	$0.726017\pi$
$158$		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	$-0.349805\pi$
$163$	148.179			0.909074			0.454537	+	0.890728i	$0.349805\pi$
$164$		0			0
$165$		0			0
$166$		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	$-0.392795\pi$
$173$	114.340			0.660924			0.330462	+	0.943819i	$0.392795\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			194.164i			1.09697i
$178$		0			0
$179$		−	80.7703i		−	0.451231i	−0.974216	−	0.225615i	$-0.927561\pi$
$179$		−	80.7703i		−	0.451231i	0.974216	−	0.225615i	$-0.0724392\pi$
$180$		0			0
$181$			3.32667i			0.0183794i	0.999958	+	0.00918970i	$0.00292521\pi$
$181$			3.32667i			0.0183794i	−0.999958	+	0.00918970i	$0.997075\pi$
$182$		0			0
$183$	379.378			2.07310
$184$		0			0
$185$		0			0
$186$		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	$-0.564395\pi$
$193$	−77.5568			−0.401849			−0.200924	+	0.979607i	$0.564395\pi$
$194$		0			0
$195$		0			0
$196$		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	$0.150422\pi$
$199$			181.158i			0.910343i	−0.890404	+	0.455171i	$0.849578\pi$
$200$		0			0
$201$		−	374.596i		−	1.86366i
$202$		0			0
$203$			124.568i			0.613634i
$204$		0			0
$205$		0			0
$206$		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
$215$		0			0
$216$		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
$225$		0			0
$226$		0			0
$227$	−164.416			−0.724298			−0.362149	−	0.932120i	$-0.617957\pi$
$227$	−164.416			−0.724298			−0.362149	+	0.932120i	$0.617957\pi$
$228$		0			0
$229$		−	298.559i		−	1.30375i	−0.758326	−	0.651876i	$-0.773982\pi$
$229$		−	298.559i		−	1.30375i	0.758326	−	0.651876i	$-0.226018\pi$
$230$		0			0
$231$	117.208			0.507393
$232$		0			0
$233$	−65.3112			−0.280305			−0.140153	−	0.990130i	$-0.544759\pi$
$233$	−65.3112			−0.280305			−0.140153	+	0.990130i	$0.544759\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−37.2078			−0.156995
$238$		0			0
$239$			192.202i			0.804194i	0.915597	+	0.402097i	$0.131719\pi$
$239$			192.202i			0.804194i	−0.915597	+	0.402097i	$0.868281\pi$
$240$		0			0
$241$			38.5506i			0.159961i	0.996796	+	0.0799805i	$0.0254858\pi$
$241$			38.5506i			0.159961i	−0.996796	+	0.0799805i	$0.974514\pi$
$242$		0			0
$243$		−	339.372i		−	1.39659i
$244$		0			0
$245$		0			0
$246$		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	$0.0894900\pi$
$251$			139.281i			0.554904i	−0.960739	+	0.277452i	$0.910510\pi$
$252$		0			0
$253$	−190.076			−0.751287
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−273.217			−1.06310			−0.531551	−	0.847027i	$-0.678390\pi$
$257$	−273.217			−1.06310			−0.531551	+	0.847027i	$0.678390\pi$
$258$		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
$265$		0			0
$266$		0			0
$267$	518.755			1.94290
$268$		0			0
$269$		−	192.374i		−	0.715144i	−0.933886	−	0.357572i	$-0.883605\pi$
$269$		−	192.374i		−	0.715144i	0.933886	−	0.357572i	$-0.116395\pi$
$270$		0			0
$271$		−	276.299i		−	1.01955i	−0.860306	−	0.509777i	$-0.829728\pi$
$271$		−	276.299i		−	1.01955i	0.860306	−	0.509777i	$-0.170272\pi$
$272$		0			0
$273$	225.736			0.826872
$274$		0			0
$275$		0			0
$276$		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	$-0.513650\pi$
$283$	−24.2641			−0.0857388			−0.0428694	+	0.999081i	$0.513650\pi$
$284$		0			0
$285$		0			0
$286$		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	$-0.603591\pi$
$293$	−187.359			−0.639451			−0.319726	+	0.947510i	$0.603591\pi$
$294$		0			0
$295$		0			0
$296$		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
$305$		0			0
$306$		0			0
$307$	491.123			1.59975			0.799874	−	0.600168i	$-0.204899\pi$
$307$	491.123			1.59975			0.799874	+	0.600168i	$0.204899\pi$
$308$		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
$315$		0			0
$316$		0			0
$317$	196.859			0.621006			0.310503	−	0.950572i	$-0.399503\pi$
$317$	196.859			0.621006			0.310503	+	0.950572i	$0.399503\pi$
$318$		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
$325$		0			0
$326$		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	$-0.731031\pi$
$331$		−	495.152i		−	1.49593i	0.663739	−	0.747964i	$-0.268969\pi$
$332$		0			0
$333$		−	33.6462i		−	0.101040i
$334$		0			0
$335$		0			0
$336$		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
$345$		0			0
$346$		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
$355$		0			0
$356$		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
$365$		0			0
$366$		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	$-0.435676\pi$
$373$	149.727			0.401412			0.200706	+	0.979652i	$0.435676\pi$
$374$		0			0
$375$		0			0
$376$		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
$385$		0			0
$386$		0			0
$387$		−	51.6861i		−	0.133556i
$388$		0			0
$389$		−	140.174i		−	0.360344i	−0.983635	−	0.180172i	$-0.942335\pi$
$389$		−	140.174i		−	0.360344i	0.983635	−	0.180172i	$-0.0576655\pi$
$390$		0			0
$391$	602.416			1.54070
$392$		0			0
$393$		−	420.994i		−	1.07123i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−106.085			−0.267217			−0.133608	−	0.991034i	$-0.542656\pi$
$397$	−106.085			−0.267217			−0.133608	+	0.991034i	$0.542656\pi$
$398$		0			0
$399$		−	227.598i		−	0.570420i
$400$		0			0
$401$		−	97.9543i		−	0.244275i	−0.992513	−	0.122138i	$-0.961025\pi$
$401$		−	97.9543i		−	0.244275i	0.992513	−	0.122138i	$-0.0389749\pi$
$402$		0			0
$403$	475.963	−	70.8781i	1.18105	−	0.175876i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	23.7175			0.0582739
$408$		0			0
$409$			11.7289i			0.0286771i	0.999897	+	0.0143386i	$0.00456426\pi$
$409$			11.7289i			0.0286771i	−0.999897	+	0.0143386i	$0.995436\pi$
$410$		0			0
$411$	206.076			0.501401
$412$		0			0
$413$	138.179			0.334574
$414$		0			0
$415$		0			0
$416$		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
$425$		0			0
$426$		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
$435$		0			0
$436$		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	$-0.554759\pi$
$443$	−151.669			−0.342369			−0.171184	+	0.985239i	$0.554759\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		−	1119.01i		−	2.50339i
$448$		0			0
$449$		−	345.854i		−	0.770277i	−0.922859	−	0.385138i	$-0.874154\pi$
$449$		−	345.854i		−	0.770277i	0.922859	−	0.385138i	$-0.125846\pi$
$450$		0			0
$451$		−	52.5420i		−	0.116501i
$452$		0			0
$453$	35.6602			0.0787201
$454$		0			0
$455$		0			0
$456$		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	$0.383531\pi$
$461$			860.967i			1.86761i	−0.357787	+	0.933803i	$0.616469\pi$
$462$		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
$465$		0			0
$466$		0			0
$467$	429.727			0.920186			0.460093	−	0.887871i	$-0.347816\pi$
$467$	429.727			0.920186			0.460093	+	0.887871i	$0.347816\pi$
$468$		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
$475$		0			0
$476$		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
$485$		0			0
$486$		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	$0.190934\pi$
$491$			554.348i			1.12902i	−0.825427	+	0.564509i	$0.809066\pi$
$492$		0			0
$493$	−989.134			−2.00636
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−238.877			−0.480638
$498$		0			0
$499$		−	49.4237i		−	0.0990454i	−0.998773	−	0.0495227i	$-0.984230\pi$
$499$		−	49.4237i		−	0.0990454i	0.998773	−	0.0495227i	$-0.0157700\pi$
$500$		0			0
$501$	314.604			0.627952
$502$		0			0
$503$	−302.048			−0.600493			−0.300247	−	0.953862i	$-0.597069\pi$
$503$	−302.048			−0.600493			−0.300247	+	0.953862i	$0.597069\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		−	325.297i		−	0.641612i
$508$		0			0
$509$			904.593i			1.77720i	0.458687	+	0.888598i	$0.348320\pi$
$509$			904.593i			1.77720i	−0.458687	+	0.888598i	$0.651680\pi$
$510$		0			0
$511$		−	16.4622i		−	0.0322156i
$512$		0			0
$513$			172.201i			0.335674i
$514$		0			0
$515$		0			0
$516$		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
$525$		0			0
$526$		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
$535$		0			0
$536$		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
$545$		0			0
$546$		0			0
$547$	−700.803			−1.28118			−0.640588	−	0.767885i	$-0.721309\pi$
$547$	−700.803			−1.28118			−0.640588	+	0.767885i	$0.721309\pi$
$548$		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
$555$		0			0
$556$		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	$-0.431371\pi$
$563$	240.896			0.427879			0.213939	+	0.976847i	$0.431371\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−171.047			−0.301670
$568$		0			0
$569$		−	129.301i		−	0.227242i	−0.993524	−	0.113621i	$-0.963755\pi$
$569$		−	129.301i		−	0.227242i	0.993524	−	0.113621i	$-0.0362450\pi$
$570$		0			0
$571$		−	1092.40i		−	1.91314i	−0.291499	−	0.956571i	$-0.594154\pi$
$571$		−	1092.40i		−	1.91314i	0.291499	−	0.956571i	$-0.405846\pi$
$572$		0			0
$573$		−	431.955i		−	0.753847i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−1052.62			−1.82431			−0.912153	−	0.409850i	$-0.865581\pi$
$577$	−1052.62			−1.82431			−0.912153	+	0.409850i	$0.865581\pi$
$578$		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
$585$		0			0
$586$		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	$-0.638814\pi$
$593$	−500.971			−0.844808			−0.422404	+	0.906408i	$0.638814\pi$
$594$		0			0
$595$		0			0
$596$		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	$-0.835857\pi$
$601$		−	592.727i		−	0.986235i	0.869963	−	0.493117i	$-0.164143\pi$
$602$		0			0
$603$	947.511			1.57133
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−853.095			−1.40543			−0.702714	−	0.711472i	$-0.748029\pi$
$607$	−853.095			−1.40543			−0.702714	+	0.711472i	$0.748029\pi$
$608$		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
$615$		0			0
$616$		0			0
$617$	368.907			0.597904			0.298952	−	0.954268i	$-0.403363\pi$
$617$	368.907			0.597904			0.298952	+	0.954268i	$0.403363\pi$
$618$		0			0
$619$		−	454.339i		−	0.733989i	−0.930223	−	0.366995i	$-0.880387\pi$
$619$		−	454.339i		−	0.733989i	0.930223	−	0.366995i	$-0.119613\pi$
$620$		0			0
$621$	−259.472			−0.417829
$622$		0			0
$623$		−	369.178i		−	0.592581i
$624$		0			0
$625$		0			0
$626$		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	$0.276151\pi$
$631$			962.590i			1.52550i	−0.646694	+	0.762749i	$0.723849\pi$
$632$		0			0
$633$			307.429i			0.485669i
$634$		0			0
$635$		0			0
$636$		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	$-0.986677\pi$
$641$		−	53.6434i		−	0.0836870i	0.999124	−	0.0418435i	$-0.0133231\pi$
$642$		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
$645$		0			0
$646$		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	$-0.550335\pi$
$653$	−205.660			−0.314947			−0.157473	+	0.987523i	$0.550335\pi$
$654$		0			0
$655$		0			0
$656$		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
$665$		0			0
$666$		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
$675$		0			0
$676$		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	$-0.767024\pi$
$683$	−1016.16			−1.48779			−0.743895	+	0.668296i	$0.767024\pi$
$684$		0			0
$685$		0			0
$686$		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
$695$		0			0
$696$		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
$705$		0			0
$706$		0			0
$707$	−390.500			−0.552335
$708$		0			0
$709$			249.172i			0.351442i	0.984440	+	0.175721i	$0.0562257\pi$
$709$			249.172i			0.351442i	−0.984440	+	0.175721i	$0.943774\pi$
$710$		0			0
$711$		−	94.1141i		−	0.132369i
$712$		0			0
$713$	723.095	−	107.680i	1.01416	−	0.151024i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−868.831			−1.21176
$718$		0			0
$719$			667.700i			0.928651i	0.885665	+	0.464325i	$0.153703\pi$
$719$			667.700i			0.928651i	−0.885665	+	0.464325i	$0.846297\pi$
$720$		0			0
$721$	467.708			0.648694
$722$		0			0
$723$	−174.264			−0.241029
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−393.217			−0.540876			−0.270438	−	0.962737i	$-0.587169\pi$
$727$	−393.217			−0.540876			−0.270438	+	0.962737i	$0.587169\pi$
$728$		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	$-0.369473\pi$
$733$	584.445			0.797333			0.398667	+	0.917096i	$0.369473\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			667.908i			0.906253i
$738$		0			0
$739$		−	301.506i		−	0.407992i	−0.978972	−	0.203996i	$-0.934607\pi$
$739$		−	301.506i		−	0.407992i	0.978972	−	0.203996i	$-0.0653930\pi$
$740$		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
$745$		0			0
$746$		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
$755$		0			0
$756$		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	$0.0310507\pi$
$761$			148.234i			0.194788i	−0.995246	+	0.0973941i	$0.968949\pi$
$762$		0			0
$763$	−192.990			−0.252935
$764$		0			0
$765$		0			0
$766$		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
$775$		0			0
$776$		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
$785$		0			0
$786$		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
$795$		0			0
$796$		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
$805$		0			0
$806$		0			0
$807$	869.605			1.07758
$808$		0			0
$809$			285.974i			0.353491i	0.984257	+	0.176745i	$0.0565569\pi$
$809$			285.974i			0.353491i	−0.984257	+	0.176745i	$0.943443\pi$
$810$		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
$815$		0			0
$816$		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	$-0.916764\pi$
$821$		−	424.496i		−	0.517048i	0.966005	−	0.258524i	$-0.0832360\pi$
$822$		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
$825$		0			0
$826$		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	$0.327037\pi$
$829$			1419.19i			1.71193i	−0.517032	+	0.855966i	$0.672963\pi$
$830$		0			0
$831$	−997.793			−1.20071
$832$		0			0
$833$		−	987.325i		−	1.18526i
$834$		0			0
$835$		0			0
$836$		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
$845$		0			0
$846$		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	$-0.631147\pi$
$853$	−683.171			−0.800904			−0.400452	+	0.916318i	$0.631147\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−540.227			−0.630370			−0.315185	−	0.949030i	$-0.602067\pi$
$857$	−540.227			−0.630370			−0.315185	+	0.949030i	$0.602067\pi$
$858$		0			0
$859$		−	1137.82i		−	1.32458i	−0.749246	−	0.662291i	$-0.769584\pi$
$859$		−	1137.82i		−	1.32458i	0.749246	−	0.662291i	$-0.230416\pi$
$860$		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
$865$		0			0
$866$		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
$875$		0			0
$876$		0			0
$877$	73.7636			0.0841090			0.0420545	−	0.999115i	$-0.486610\pi$
$877$	73.7636			0.0841090			0.0420545	+	0.999115i	$0.486610\pi$
$878$		0			0
$879$		−	846.938i		−	0.963524i
$880$		0			0
$881$			109.341i			0.124110i	0.998073	+	0.0620550i	$0.0197654\pi$
$881$			109.341i			0.124110i	−0.998073	+	0.0620550i	$0.980235\pi$
$882$		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
$885$		0			0
$886$		0			0
$887$	−1514.29			−1.70721			−0.853604	−	0.520922i	$-0.825588\pi$
$887$	−1514.29			−1.70721			−0.853604	+	0.520922i	$0.825588\pi$
$888$		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
$895$		0			0
$896$		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
$905$		0			0
$906$		0			0
$907$	1501.96			1.65596			0.827980	−	0.560757i	$-0.189490\pi$
$907$	1501.96			1.65596			0.827980	+	0.560757i	$0.189490\pi$
$908$		0			0
$909$	1387.94			1.52688
$910$		0			0
$911$			426.662i			0.468344i	0.972195	+	0.234172i	$0.0752379\pi$
$911$			426.662i			0.468344i	−0.972195	+	0.234172i	$0.924762\pi$
$912$		0			0
$913$	−834.886			−0.914443
$914$		0			0
$915$		0			0
$916$		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
$925$		0			0
$926$		0			0
$927$	−1662.35			−1.79326
$928$		0			0
$929$			1196.29i			1.28772i	0.765144	+	0.643859i	$0.222668\pi$
$929$			1196.29i			1.28772i	−0.765144	+	0.643859i	$0.777332\pi$
$930$		0			0
$931$	604.925			0.649759
$932$		0			0
$933$			18.9746i			0.0203372i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−1226.11			−1.30855			−0.654277	−	0.756255i	$-0.727027\pi$
$937$	−1226.11			−1.30855			−0.654277	+	0.756255i	$0.727027\pi$
$938$		0			0
$939$	1400.49			1.49147
$940$		0			0
$941$		−	387.181i		−	0.411457i	−0.978609	−	0.205728i	$-0.934044\pi$
$941$		−	387.181i		−	0.411457i	0.978609	−	0.205728i	$-0.0659563\pi$
$942$		0			0
$943$		−	153.735i		−	0.163028i
$944$		0			0
$945$		0			0
$946$		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
$955$		0			0
$956$		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
$965$		0			0
$966$		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
$975$		0			0
$976$		0			0
$977$	−827.633			−0.847116			−0.423558	−	0.905869i	$-0.639219\pi$
$977$	−827.633			−0.847116			−0.423558	+	0.905869i	$0.639219\pi$
$978$		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
$985$		0			0
$986$		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	$-0.791080\pi$
$991$		−	1209.46i		−	1.22045i	0.792229	−	0.610223i	$-0.208920\pi$
$992$		0			0
$993$	2238.28			2.25406
$994$		0			0
$995$		0			0
$996$		0			0
$997$	870.333			0.872951			0.436476	−	0.899716i	$-0.356226\pi$
$997$	870.333			0.872951			0.436476	+	0.899716i	$0.356226\pi$
$998$		0			0
$999$	32.3767			0.0324091

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.3.c.b.61.1	✓	4	5.4	even	2
124.3.c.b.61.4	yes	4	155.154	odd	2
496.3.e.e.433.1		4	620.619	even	2
496.3.e.e.433.4		4	20.19	odd	2
1116.3.h.d.433.1		4	465.464	even	2
1116.3.h.d.433.2		4	15.14	odd	2
3100.3.d.b.1301.1		4	31.30	odd	2	inner
3100.3.d.b.1301.4		4	1.1	even	1	trivial
3100.3.f.b.1549.1		8	5.3	odd	4
3100.3.f.b.1549.2		8	155.92	even	4
3100.3.f.b.1549.7		8	155.123	even	4
3100.3.f.b.1549.8		8	5.2	odd	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:	\( N \)	\(=\)	\( 3100 = 2^{2} \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	3100.d (of order \(2\), degree \(1\), minimal)

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more