LMFDB - Embedded newform 3100.3.d.h.1301.8

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:	$24$
Twist minimal:	no (minimal twist has level 620)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1301.8
Character	$\chi$	$=$	3100.1301
Dual form			3100.3.d.h.1301.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.662053i q^{3} +8.05015 q^{7} +8.56169 q^{9} +O(q^{10})$	$q+0.662053i q^{3} +8.05015 q^{7} +8.56169 q^{9} +14.8318i q^{11} -7.12511i q^{13} +26.3657i q^{17} +10.7364 q^{19} +5.32962i q^{21} -6.66349i q^{23} +11.6268i q^{27} -5.50735i q^{29} +(0.544361 + 30.9952i) q^{31} -9.81941 q^{33} +30.4173i q^{37} +4.71720 q^{39} +27.2630 q^{41} +34.3776i q^{43} -22.6712 q^{47} +15.8049 q^{49} -17.4555 q^{51} -42.2112i q^{53} +7.10807i q^{57} -55.6539 q^{59} +100.878i q^{61} +68.9228 q^{63} -21.3271 q^{67} +4.41158 q^{69} -78.2254 q^{71} +11.2443i q^{73} +119.398i q^{77} -28.9400i q^{79} +69.3577 q^{81} -123.515i q^{83} +3.64615 q^{87} -78.1423i q^{89} -57.3582i q^{91} +(-20.5205 + 0.360396i) q^{93} -24.3968 q^{97} +126.985i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 44 q^{9}+O(q^{10}) $ 24 * q - 44 * q^9	$ 24 q - 44 q^{9} - 8 q^{19} - 24 q^{31} + 280 q^{39} - 248 q^{41} + 644 q^{49} - 100 q^{51} - 152 q^{59} + 288 q^{69} - 352 q^{71} - 368 q^{81}+O(q^{100}) $ 24 * q - 44 * q^9 - 8 * q^19 - 24 * q^31 + 280 * q^39 - 248 * q^41 + 644 * q^49 - 100 * q^51 - 152 * q^59 + 288 * q^69 - 352 * q^71 - 368 * q^81

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
$3$			0.662053i			0.220684i	0.993894	+	0.110342i	$0.0351946\pi$
$3$			0.662053i			0.220684i	−0.993894	+	0.110342i	$0.964805\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	8.05015			1.15002			0.575010	−	0.818146i	$-0.304998\pi$
$7$	8.05015			1.15002			0.575010	+	0.818146i	$0.304998\pi$
$8$		0			0
$9$	8.56169			0.951298
$10$		0			0
$11$			14.8318i			1.34834i	0.738575	+	0.674171i	$0.235499\pi$
$11$			14.8318i			1.34834i	−0.738575	+	0.674171i	$0.764501\pi$
$12$		0			0
$13$		−	7.12511i		−	0.548085i	−0.961718	−	0.274043i	$-0.911639\pi$
$13$		−	7.12511i		−	0.548085i	0.961718	−	0.274043i	$-0.0883610\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			26.3657i			1.55092i	0.631394	+	0.775462i	$0.282483\pi$
$17$			26.3657i			1.55092i	−0.631394	+	0.775462i	$0.717517\pi$
$18$		0			0
$19$	10.7364			0.565074			0.282537	−	0.959256i	$-0.408824\pi$
$19$	10.7364			0.565074			0.282537	+	0.959256i	$0.408824\pi$
$20$		0			0
$21$			5.32962i			0.253791i
$22$		0			0
$23$		−	6.66349i		−	0.289717i	−0.989452	−	0.144858i	$-0.953727\pi$
$23$		−	6.66349i		−	0.289717i	0.989452	−	0.144858i	$-0.0462727\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$			11.6268i			0.430621i
$28$		0			0
$29$		−	5.50735i		−	0.189909i	−0.995482	−	0.0949543i	$-0.969730\pi$
$29$		−	5.50735i		−	0.189909i	0.995482	−	0.0949543i	$-0.0302705\pi$
$30$		0			0
$31$	0.544361	+	30.9952i	0.0175600	+	0.999846i
$31$	0.544361	+	30.9952i	0.0175600	+	0.999846i
$32$		0			0
$33$	−9.81941			−0.297558
$34$		0			0
$35$		0			0
$36$		0			0
$37$			30.4173i			0.822089i	0.911615	+	0.411044i	$0.134836\pi$
$37$			30.4173i			0.822089i	−0.911615	+	0.411044i	$0.865164\pi$
$38$		0			0
$39$	4.71720			0.120954
$40$		0			0
$41$	27.2630			0.664952			0.332476	−	0.943112i	$-0.392116\pi$
$41$	27.2630			0.664952			0.332476	+	0.943112i	$0.392116\pi$
$42$		0			0
$43$			34.3776i			0.799479i	0.916629	+	0.399739i	$0.130899\pi$
$43$			34.3776i			0.799479i	−0.916629	+	0.399739i	$0.869101\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−22.6712			−0.482367			−0.241183	−	0.970480i	$-0.577536\pi$
$47$	−22.6712			−0.482367			−0.241183	+	0.970480i	$0.577536\pi$
$48$		0			0
$49$	15.8049			0.322548
$50$		0			0
$51$	−17.4555			−0.342265
$52$		0			0
$53$		−	42.2112i		−	0.796437i	−0.917290	−	0.398219i	$-0.869629\pi$
$53$		−	42.2112i		−	0.796437i	0.917290	−	0.398219i	$-0.130371\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$			7.10807i			0.124703i
$58$		0			0
$59$	−55.6539			−0.943286			−0.471643	−	0.881790i	$-0.656339\pi$
$59$	−55.6539			−0.943286			−0.471643	+	0.881790i	$0.656339\pi$
$60$		0			0
$61$			100.878i			1.65373i	0.562401	+	0.826865i	$0.309878\pi$
$61$			100.878i			1.65373i	−0.562401	+	0.826865i	$0.690122\pi$
$62$		0			0
$63$	68.9228			1.09401
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−21.3271			−0.318315			−0.159158	−	0.987253i	$-0.550878\pi$
$67$	−21.3271			−0.318315			−0.159158	+	0.987253i	$0.550878\pi$
$68$		0			0
$69$	4.41158			0.0639359
$70$		0			0
$71$	−78.2254			−1.10177			−0.550883	−	0.834582i	$-0.685709\pi$
$71$	−78.2254			−1.10177			−0.550883	+	0.834582i	$0.685709\pi$
$72$		0			0
$73$			11.2443i			0.154032i	0.997030	+	0.0770160i	$0.0245393\pi$
$73$			11.2443i			0.154032i	−0.997030	+	0.0770160i	$0.975461\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			119.398i			1.55062i
$78$		0			0
$79$		−	28.9400i		−	0.366329i	−0.983082	−	0.183164i	$-0.941366\pi$
$79$		−	28.9400i		−	0.366329i	0.983082	−	0.183164i	$-0.0586341\pi$
$80$		0			0
$81$	69.3577			0.856267
$82$		0			0
$83$		−	123.515i		−	1.48814i	−0.668104	−	0.744068i	$-0.732894\pi$
$83$		−	123.515i		−	1.48814i	0.668104	−	0.744068i	$-0.267106\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	3.64615			0.0419098
$88$		0			0
$89$		−	78.1423i		−	0.878003i	−0.898486	−	0.439001i	$-0.855332\pi$
$89$		−	78.1423i		−	0.878003i	0.898486	−	0.439001i	$-0.144668\pi$
$90$		0			0
$91$		−	57.3582i		−	0.630310i
$92$		0			0
$93$	−20.5205	+	0.360396i	−0.220650	+	0.00387522i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−24.3968			−0.251514			−0.125757	−	0.992061i	$-0.540136\pi$
$97$	−24.3968			−0.251514			−0.125757	+	0.992061i	$0.540136\pi$
$98$		0			0
$99$			126.985i			1.28268i
$100$		0			0
$101$	27.3184			0.270479			0.135240	−	0.990813i	$-0.456820\pi$
$101$	27.3184			0.270479			0.135240	+	0.990813i	$0.456820\pi$
$102$		0			0
$103$	−7.89886			−0.0766879			−0.0383440	−	0.999265i	$-0.512208\pi$
$103$	−7.89886			−0.0766879			−0.0383440	+	0.999265i	$0.512208\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	132.000			1.23364			0.616821	−	0.787103i	$-0.288420\pi$
$107$	132.000			1.23364			0.616821	+	0.787103i	$0.288420\pi$
$108$		0			0
$109$	−157.192			−1.44213			−0.721066	−	0.692867i	$-0.756347\pi$
$109$	−157.192			−1.44213			−0.721066	+	0.692867i	$0.756347\pi$
$110$		0			0
$111$	−20.1378			−0.181422
$112$		0			0
$113$	−4.30617			−0.0381077			−0.0190538	−	0.999818i	$-0.506065\pi$
$113$	−4.30617			−0.0381077			−0.0190538	+	0.999818i	$0.506065\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		−	61.0030i		−	0.521393i
$118$		0			0
$119$			212.248i			1.78360i
$120$		0			0
$121$	−98.9813			−0.818027
$122$		0			0
$123$			18.0496i			0.146744i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	140.526i		−	1.10650i	−0.833014	−	0.553251i	$-0.813387\pi$
$127$		−	140.526i		−	1.10650i	0.833014	−	0.553251i	$-0.186613\pi$
$128$		0			0
$129$	−22.7598			−0.176432
$130$		0			0
$131$	9.50981			0.0725940			0.0362970	−	0.999341i	$-0.488444\pi$
$131$	9.50981			0.0725940			0.0362970	+	0.999341i	$0.488444\pi$
$132$		0			0
$133$	86.4297			0.649847
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	185.339i		−	1.35284i	−0.736517	−	0.676419i	$-0.763531\pi$
$137$		−	185.339i		−	1.35284i	0.736517	−	0.676419i	$-0.236469\pi$
$138$		0			0
$139$			123.803i			0.890671i	0.895364	+	0.445335i	$0.146916\pi$
$139$			123.803i			0.890671i	−0.895364	+	0.445335i	$0.853084\pi$
$140$		0			0
$141$		−	15.0095i		−	0.106451i
$142$		0			0
$143$	105.678			0.739007
$144$		0			0
$145$		0			0
$146$		0			0
$147$			10.4636i			0.0711813i
$148$		0			0
$149$	190.006			1.27521			0.637605	−	0.770364i	$-0.279925\pi$
$149$	190.006			1.27521			0.637605	+	0.770364i	$0.279925\pi$
$150$		0			0
$151$			38.6263i			0.255803i	0.991787	+	0.127902i	$0.0408242\pi$
$151$			38.6263i			0.255803i	−0.991787	+	0.127902i	$0.959176\pi$
$152$		0			0
$153$			225.735i			1.47539i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	184.991			1.17829			0.589143	−	0.808029i	$-0.299465\pi$
$157$	184.991			1.17829			0.589143	+	0.808029i	$0.299465\pi$
$158$		0			0
$159$	27.9460			0.175761
$160$		0			0
$161$		−	53.6420i		−	0.333180i
$162$		0			0
$163$	−234.011			−1.43565			−0.717827	−	0.696222i	$-0.754863\pi$
$163$	−234.011			−1.43565			−0.717827	+	0.696222i	$0.754863\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	192.032i		−	1.14989i	−0.818192	−	0.574945i	$-0.805023\pi$
$167$		−	192.032i		−	1.14989i	0.818192	−	0.574945i	$-0.194977\pi$
$168$		0			0
$169$	118.233			0.699602
$170$		0			0
$171$	91.9218			0.537554
$172$		0			0
$173$	190.245			1.09968			0.549842	−	0.835269i	$-0.314688\pi$
$173$	190.245			1.09968			0.549842	+	0.835269i	$0.314688\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		−	36.8458i		−	0.208168i
$178$		0			0
$179$			271.846i			1.51869i	0.650688	+	0.759345i	$0.274481\pi$
$179$			271.846i			1.51869i	−0.650688	+	0.759345i	$0.725519\pi$
$180$		0			0
$181$			5.78601i			0.0319669i	0.999872	+	0.0159834i	$0.00508790\pi$
$181$			5.78601i			0.0319669i	−0.999872	+	0.0159834i	$0.994912\pi$
$182$		0			0
$183$	−66.7862			−0.364952
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−391.050			−2.09118
$188$		0			0
$189$			93.5971i			0.495223i
$190$		0			0
$191$	203.298			1.06439			0.532194	−	0.846622i	$-0.321368\pi$
$191$	203.298			1.06439			0.532194	+	0.846622i	$0.321368\pi$
$192$		0			0
$193$	−315.180			−1.63306			−0.816529	−	0.577304i	$-0.804105\pi$
$193$	−315.180			−1.63306			−0.816529	+	0.577304i	$0.804105\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	23.1674i		−	0.117601i	−0.998270	−	0.0588006i	$-0.981272\pi$
$197$		−	23.1674i		−	0.117601i	0.998270	−	0.0588006i	$-0.0187276\pi$
$198$		0			0
$199$			141.839i			0.712758i	0.934341	+	0.356379i	$0.115989\pi$
$199$			141.839i			0.712758i	−0.934341	+	0.356379i	$0.884011\pi$
$200$		0			0
$201$		−	14.1197i		−	0.0702472i
$202$		0			0
$203$		−	44.3350i		−	0.218399i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		−	57.0507i		−	0.275607i
$208$		0			0
$209$			159.240i			0.761914i
$210$		0			0
$211$	−286.662			−1.35859			−0.679294	−	0.733866i	$-0.737714\pi$
$211$	−286.662			−1.35859			−0.679294	+	0.733866i	$0.737714\pi$
$212$		0			0
$213$		−	51.7893i		−	0.243142i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	4.38219	+	249.516i	0.0201944	+	1.14984i
$218$		0			0
$219$	−7.44435			−0.0339924
$220$		0			0
$221$	187.859			0.850039
$222$		0			0
$223$		−	63.4396i		−	0.284482i	−0.989832	−	0.142241i	$-0.954569\pi$
$223$		−	63.4396i		−	0.284482i	0.989832	−	0.142241i	$-0.0454309\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	11.9019			0.0524312			0.0262156	−	0.999656i	$-0.491654\pi$
$227$	11.9019			0.0524312			0.0262156	+	0.999656i	$0.491654\pi$
$228$		0			0
$229$			356.022i			1.55468i	0.629081	+	0.777340i	$0.283432\pi$
$229$			356.022i			1.55468i	−0.629081	+	0.777340i	$0.716568\pi$
$230$		0			0
$231$	−79.0477			−0.342198
$232$		0			0
$233$	201.013			0.862718			0.431359	−	0.902180i	$-0.358034\pi$
$233$	201.013			0.862718			0.431359	+	0.902180i	$0.358034\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	19.1598			0.0808430
$238$		0			0
$239$			272.587i			1.14053i	0.821461	+	0.570265i	$0.193160\pi$
$239$			272.587i			1.14053i	−0.821461	+	0.570265i	$0.806840\pi$
$240$		0			0
$241$			120.703i			0.500843i	0.968137	+	0.250422i	$0.0805693\pi$
$241$			120.703i			0.500843i	−0.968137	+	0.250422i	$0.919431\pi$
$242$		0			0
$243$			150.559i			0.619585i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		−	76.4981i		−	0.309709i
$248$		0			0
$249$	81.7736			0.328408
$250$		0			0
$251$			0.519867i			0.00207118i	0.999999	+	0.00103559i	$0.000329639\pi$
$251$			0.519867i			0.00207118i	−0.999999	+	0.00103559i	$0.999670\pi$
$252$		0			0
$253$	98.8313			0.390638
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−255.224			−0.993091			−0.496546	−	0.868011i	$-0.665398\pi$
$257$	−255.224			−0.993091			−0.496546	+	0.868011i	$0.665398\pi$
$258$		0			0
$259$			244.864i			0.945419i
$260$		0			0
$261$		−	47.1522i		−	0.180660i
$262$		0			0
$263$			324.000i			1.23194i	0.787770	+	0.615970i	$0.211236\pi$
$263$			324.000i			1.23194i	−0.787770	+	0.615970i	$0.788764\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	51.7343			0.193761
$268$		0			0
$269$		−	75.4742i		−	0.280573i	−0.990111	−	0.140287i	$-0.955198\pi$
$269$		−	75.4742i		−	0.280573i	0.990111	−	0.140287i	$-0.0448024\pi$
$270$		0			0
$271$			379.458i			1.40021i	0.714038	+	0.700107i	$0.246864\pi$
$271$			379.458i			1.40021i	−0.714038	+	0.700107i	$0.753136\pi$
$272$		0			0
$273$	37.9741			0.139099
$274$		0			0
$275$		0			0
$276$		0			0
$277$			240.276i			0.867423i	0.901052	+	0.433711i	$0.142796\pi$
$277$			240.276i			0.867423i	−0.901052	+	0.433711i	$0.857204\pi$
$278$		0			0
$279$	4.66065	+	265.371i	0.0167048	+	0.951152i
$280$		0			0
$281$	−6.47121			−0.0230292			−0.0115146	−	0.999934i	$-0.503665\pi$
$281$	−6.47121			−0.0230292			−0.0115146	+	0.999934i	$0.503665\pi$
$282$		0			0
$283$	140.562			0.496686			0.248343	−	0.968672i	$-0.420114\pi$
$283$	140.562			0.496686			0.248343	+	0.968672i	$0.420114\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	219.471			0.764708
$288$		0			0
$289$	−406.151			−1.40537
$290$		0			0
$291$		−	16.1520i		−	0.0555051i
$292$		0			0
$293$	−270.986			−0.924867			−0.462434	−	0.886654i	$-0.653024\pi$
$293$	−270.986			−0.924867			−0.462434	+	0.886654i	$0.653024\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−172.445			−0.580624
$298$		0			0
$299$	−47.4781			−0.158790
$300$		0			0
$301$			276.745i			0.919417i
$302$		0			0
$303$			18.0862i			0.0596905i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	96.0564			0.312887			0.156444	−	0.987687i	$-0.449997\pi$
$307$	96.0564			0.312887			0.156444	+	0.987687i	$0.449997\pi$
$308$		0			0
$309$		−	5.22946i		−	0.0169238i
$310$		0			0
$311$	160.058			0.514656			0.257328	−	0.966324i	$-0.417158\pi$
$311$	160.058			0.514656			0.257328	+	0.966324i	$0.417158\pi$
$312$		0			0
$313$		−	128.846i		−	0.411649i	−0.978589	−	0.205824i	$-0.934012\pi$
$313$		−	128.846i		−	0.411649i	0.978589	−	0.205824i	$-0.0659876\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	596.199			1.88075			0.940377	−	0.340133i	$-0.110472\pi$
$317$	596.199			1.88075			0.940377	+	0.340133i	$0.110472\pi$
$318$		0			0
$319$	81.6837			0.256062
$320$		0			0
$321$			87.3908i			0.272245i
$322$		0			0
$323$			283.073i			0.876388i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		−	104.070i		−	0.318256i
$328$		0			0
$329$	−182.507			−0.554732
$330$		0			0
$331$			129.482i			0.391185i	0.980685	+	0.195593i	$0.0626630\pi$
$331$			129.482i			0.391185i	−0.980685	+	0.195593i	$0.937337\pi$
$332$		0			0
$333$			260.423i			0.782052i
$334$		0			0
$335$		0			0
$336$		0			0
$337$			655.496i			1.94509i	0.232714	+	0.972545i	$0.425240\pi$
$337$			655.496i			1.94509i	−0.232714	+	0.972545i	$0.574760\pi$
$338$		0			0
$339$		−	2.85091i		−	0.00840976i
$340$		0			0
$341$	−459.714	+	8.07384i	−1.34813	+	0.0236769i
$342$		0			0
$343$	−267.226			−0.779084
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	458.708i		−	1.32192i	−0.750419	−	0.660962i	$-0.770148\pi$
$347$		−	458.708i		−	1.32192i	0.750419	−	0.660962i	$-0.229852\pi$
$348$		0			0
$349$	15.7444			0.0451129			0.0225564	−	0.999746i	$-0.492819\pi$
$349$	15.7444			0.0451129			0.0225564	+	0.999746i	$0.492819\pi$
$350$		0			0
$351$	82.8420			0.236017
$352$		0			0
$353$		−	620.487i		−	1.75775i	−0.477049	−	0.878877i	$-0.658294\pi$
$353$		−	620.487i		−	1.75775i	0.477049	−	0.878877i	$-0.341706\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−140.519			−0.393611
$358$		0			0
$359$	−70.8303			−0.197299			−0.0986494	−	0.995122i	$-0.531452\pi$
$359$	−70.8303			−0.197299			−0.0986494	+	0.995122i	$0.531452\pi$
$360$		0			0
$361$	−245.730			−0.680691
$362$		0			0
$363$		−	65.5308i		−	0.180526i
$364$		0			0
$365$		0			0
$366$		0			0
$367$			312.134i			0.850502i	0.905075	+	0.425251i	$0.139814\pi$
$367$			312.134i			0.850502i	−0.905075	+	0.425251i	$0.860186\pi$
$368$		0			0
$369$	233.417			0.632568
$370$		0			0
$371$		−	339.806i		−	0.915920i
$372$		0			0
$373$	471.910			1.26517			0.632587	−	0.774489i	$-0.281993\pi$
$373$	471.910			1.26517			0.632587	+	0.774489i	$0.281993\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−39.2405			−0.104086
$378$		0			0
$379$	−247.630			−0.653377			−0.326688	−	0.945132i	$-0.605933\pi$
$379$	−247.630			−0.653377			−0.326688	+	0.945132i	$0.605933\pi$
$380$		0			0
$381$	93.0355			0.244188
$382$		0			0
$383$		−	235.094i		−	0.613823i	−0.951738	−	0.306911i	$-0.900705\pi$
$383$		−	235.094i		−	0.613823i	0.951738	−	0.306911i	$-0.0992955\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$			294.330i			0.760543i
$388$		0			0
$389$			672.061i			1.72766i	0.503780	+	0.863832i	$0.331942\pi$
$389$			672.061i			1.72766i	−0.503780	+	0.863832i	$0.668058\pi$
$390$		0			0
$391$	175.688			0.449329
$392$		0			0
$393$			6.29599i			0.0160203i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−21.8998			−0.0551632			−0.0275816	−	0.999620i	$-0.508781\pi$
$397$	−21.8998			−0.0551632			−0.0275816	+	0.999620i	$0.508781\pi$
$398$		0			0
$399$			57.2210i			0.143411i
$400$		0			0
$401$			112.124i			0.279611i	0.990179	+	0.139805i	$0.0446477\pi$
$401$			112.124i			0.279611i	−0.990179	+	0.139805i	$0.955352\pi$
$402$		0			0
$403$	220.844	−	3.87863i	0.548001	−	0.00962440i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−451.142			−1.10846
$408$		0			0
$409$		−	646.615i		−	1.58097i	−0.612484	−	0.790483i	$-0.709830\pi$
$409$		−	646.615i		−	1.58097i	0.612484	−	0.790483i	$-0.290170\pi$
$410$		0			0
$411$	122.704			0.298550
$412$		0			0
$413$	−448.022			−1.08480
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−81.9642			−0.196557
$418$		0			0
$419$	−76.6718			−0.182987			−0.0914937	−	0.995806i	$-0.529164\pi$
$419$	−76.6718			−0.182987			−0.0914937	+	0.995806i	$0.529164\pi$
$420$		0			0
$421$	−598.861			−1.42247			−0.711236	−	0.702953i	$-0.751864\pi$
$421$	−598.861			−1.42247			−0.711236	+	0.702953i	$0.751864\pi$
$422$		0			0
$423$	−194.104			−0.458875
$424$		0			0
$425$		0			0
$426$		0			0
$427$			812.079i			1.90182i
$428$		0			0
$429$			69.9644i			0.163087i
$430$		0			0
$431$	433.382			1.00553			0.502763	−	0.864424i	$-0.332317\pi$
$431$	433.382			1.00553			0.502763	+	0.864424i	$0.332317\pi$
$432$		0			0
$433$		−	472.936i		−	1.09223i	−0.837710	−	0.546115i	$-0.816106\pi$
$433$		−	472.936i		−	1.09223i	0.837710	−	0.546115i	$-0.183894\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	71.5419i		−	0.163712i
$438$		0			0
$439$	403.672			0.919526			0.459763	−	0.888042i	$-0.347934\pi$
$439$	403.672			0.919526			0.459763	+	0.888042i	$0.347934\pi$
$440$		0			0
$441$	135.316			0.306840
$442$		0			0
$443$	−395.134			−0.891950			−0.445975	−	0.895045i	$-0.647143\pi$
$443$	−395.134			−0.891950			−0.445975	+	0.895045i	$0.647143\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$			125.794i			0.281419i
$448$		0			0
$449$		−	168.750i		−	0.375836i	−0.982185	−	0.187918i	$-0.939826\pi$
$449$		−	168.750i		−	0.375836i	0.982185	−	0.187918i	$-0.0601740\pi$
$450$		0			0
$451$			404.359i			0.896583i
$452$		0			0
$453$	−25.5726			−0.0564517
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	183.891i		−	0.402386i	−0.979552	−	0.201193i	$-0.935518\pi$
$457$		−	183.891i		−	0.402386i	0.979552	−	0.201193i	$-0.0644819\pi$
$458$		0			0
$459$	−306.548			−0.667860
$460$		0			0
$461$		−	169.485i		−	0.367646i	−0.982959	−	0.183823i	$-0.941153\pi$
$461$		−	169.485i		−	0.367646i	0.982959	−	0.183823i	$-0.0588473\pi$
$462$		0			0
$463$		−	665.325i		−	1.43699i	−0.695534	−	0.718493i	$-0.744832\pi$
$463$		−	665.325i		−	1.43699i	0.695534	−	0.718493i	$-0.255168\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	454.044			0.972257			0.486128	−	0.873887i	$-0.338409\pi$
$467$	454.044			0.972257			0.486128	+	0.873887i	$0.338409\pi$
$468$		0			0
$469$	−171.687			−0.366069
$470$		0			0
$471$			122.474i			0.260029i
$472$		0			0
$473$	−509.880			−1.07797
$474$		0			0
$475$		0			0
$476$		0			0
$477$		−	361.399i		−	0.757650i
$478$		0			0
$479$	708.201			1.47850			0.739250	−	0.673432i	$-0.235180\pi$
$479$	708.201			1.47850			0.739250	+	0.673432i	$0.235180\pi$
$480$		0			0
$481$	216.727			0.450575
$482$		0			0
$483$	35.5139			0.0735277
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	757.573i		−	1.55559i	−0.628517	−	0.777796i	$-0.716338\pi$
$487$		−	757.573i		−	1.55559i	0.628517	−	0.777796i	$-0.283662\pi$
$488$		0			0
$489$		−	154.928i		−	0.316826i
$490$		0			0
$491$			725.518i			1.47763i	0.673907	+	0.738816i	$0.264615\pi$
$491$			725.518i			1.47763i	−0.673907	+	0.738816i	$0.735385\pi$
$492$		0			0
$493$	145.205			0.294534
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−629.726			−1.26705
$498$		0			0
$499$		−	884.066i		−	1.77168i	−0.463995	−	0.885838i	$-0.653584\pi$
$499$		−	884.066i		−	1.77168i	0.463995	−	0.885838i	$-0.346416\pi$
$500$		0			0
$501$	127.135			0.253763
$502$		0			0
$503$	308.976			0.614266			0.307133	−	0.951667i	$-0.400630\pi$
$503$	308.976			0.614266			0.307133	+	0.951667i	$0.400630\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$			78.2763i			0.154391i
$508$		0			0
$509$			11.6472i			0.0228825i	0.999935	+	0.0114412i	$0.00364193\pi$
$509$			11.6472i			0.0228825i	−0.999935	+	0.0114412i	$0.996358\pi$
$510$		0			0
$511$			90.5186i			0.177140i
$512$		0			0
$513$			124.830i			0.243333i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	336.254i		−	0.650395i
$518$		0			0
$519$			125.952i			0.242683i
$520$		0			0
$521$	−99.3399			−0.190672			−0.0953358	−	0.995445i	$-0.530392\pi$
$521$	−99.3399			−0.190672			−0.0953358	+	0.995445i	$0.530392\pi$
$522$		0			0
$523$			793.172i			1.51658i	0.651916	+	0.758291i	$0.273965\pi$
$523$			793.172i			1.51658i	−0.651916	+	0.758291i	$0.726035\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−817.211	+	14.3525i	−1.55069	+	0.0272343i
$528$		0			0
$529$	484.598			0.916064
$530$		0			0
$531$	−476.491			−0.897347
$532$		0			0
$533$		−	194.252i		−	0.364450i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−179.976			−0.335151
$538$		0			0
$539$			234.414i			0.434905i
$540$		0			0
$541$	704.672			1.30254			0.651268	−	0.758848i	$-0.274237\pi$
$541$	704.672			1.30254			0.651268	+	0.758848i	$0.274237\pi$
$542$		0			0
$543$	−3.83064			−0.00705459
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−270.733			−0.494942			−0.247471	−	0.968895i	$-0.579600\pi$
$547$	−270.733			−0.494942			−0.247471	+	0.968895i	$0.579600\pi$
$548$		0			0
$549$			863.682i			1.57319i
$550$		0			0
$551$		−	59.1291i		−	0.107312i
$552$		0			0
$553$		−	232.971i		−	0.421286i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	1047.43i		−	1.88048i	−0.340515	−	0.940239i	$-0.610601\pi$
$557$		−	1047.43i		−	1.88048i	0.340515	−	0.940239i	$-0.389399\pi$
$558$		0			0
$559$	244.944			0.438183
$560$		0			0
$561$		−	258.896i		−	0.461490i
$562$		0			0
$563$	1003.55			1.78250			0.891251	−	0.453510i	$-0.149828\pi$
$563$	1003.55			1.78250			0.891251	+	0.453510i	$0.149828\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	558.339			0.984725
$568$		0			0
$569$			455.807i			0.801066i	0.916282	+	0.400533i	$0.131175\pi$
$569$			455.807i			0.801066i	−0.916282	+	0.400533i	$0.868825\pi$
$570$		0			0
$571$			706.187i			1.23675i	0.785882	+	0.618377i	$0.212210\pi$
$571$			706.187i			1.23675i	−0.785882	+	0.618377i	$0.787790\pi$
$572$		0			0
$573$			134.594i			0.234894i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	1137.81			1.97194			0.985969	−	0.166927i	$-0.0533843\pi$
$577$	1137.81			1.97194			0.985969	+	0.166927i	$0.0533843\pi$
$578$		0			0
$579$		−	208.666i		−	0.360390i
$580$		0			0
$581$		−	994.316i		−	1.71139i
$582$		0			0
$583$	626.066			1.07387
$584$		0			0
$585$		0			0
$586$		0			0
$587$			630.730i			1.07450i	0.843424	+	0.537249i	$0.180536\pi$
$587$			630.730i			1.07450i	−0.843424	+	0.537249i	$0.819464\pi$
$588$		0			0
$589$	5.84448	+	332.777i	0.00992272	+	0.564987i
$590$		0			0
$591$	15.3381			0.0259527
$592$		0			0
$593$	−304.441			−0.513392			−0.256696	−	0.966492i	$-0.582634\pi$
$593$	−304.441			−0.513392			−0.256696	+	0.966492i	$0.582634\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−93.9048			−0.157295
$598$		0			0
$599$	701.717			1.17148			0.585740	−	0.810499i	$-0.300804\pi$
$599$	701.717			1.17148			0.585740	+	0.810499i	$0.300804\pi$
$600$		0			0
$601$			314.899i			0.523959i	0.965073	+	0.261980i	$0.0843753\pi$
$601$			314.899i			0.523959i	−0.965073	+	0.261980i	$0.915625\pi$
$602$		0			0
$603$	−182.596			−0.302813
$604$		0			0
$605$		0			0
$606$		0			0
$607$	973.580			1.60392			0.801961	−	0.597377i	$-0.203790\pi$
$607$	973.580			1.60392			0.801961	+	0.597377i	$0.203790\pi$
$608$		0			0
$609$	29.3521			0.0481972
$610$		0			0
$611$			161.535i			0.264378i
$612$		0			0
$613$			585.418i			0.955005i	0.878630	+	0.477503i	$0.158458\pi$
$613$			585.418i			0.955005i	−0.878630	+	0.477503i	$0.841542\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	529.977			0.858957			0.429479	−	0.903077i	$-0.358697\pi$
$617$	529.977			0.858957			0.429479	+	0.903077i	$0.358697\pi$
$618$		0			0
$619$			117.342i			0.189568i	0.995498	+	0.0947839i	$0.0302160\pi$
$619$			117.342i			0.189568i	−0.995498	+	0.0947839i	$0.969784\pi$
$620$		0			0
$621$	77.4748			0.124758
$622$		0			0
$623$		−	629.057i		−	1.00972i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−105.425			−0.168142
$628$		0			0
$629$	−801.974			−1.27500
$630$		0			0
$631$			558.447i			0.885020i	0.896764	+	0.442510i	$0.145912\pi$
$631$			558.447i			0.885020i	−0.896764	+	0.442510i	$0.854088\pi$
$632$		0			0
$633$		−	189.785i		−	0.299819i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		−	112.611i		−	0.176784i
$638$		0			0
$639$	−669.742			−1.04811
$640$		0			0
$641$		−	1124.12i		−	1.75369i	−0.480770	−	0.876847i	$-0.659643\pi$
$641$		−	1124.12i		−	1.75369i	0.480770	−	0.876847i	$-0.340357\pi$
$642$		0			0
$643$			138.217i			0.214957i	0.994207	+	0.107478i	$0.0342777\pi$
$643$			138.217i			0.214957i	−0.994207	+	0.107478i	$0.965722\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	309.147i		−	0.477817i	−0.971042	−	0.238908i	$-0.923210\pi$
$647$		−	309.147i		−	0.477817i	0.971042	−	0.238908i	$-0.0767895\pi$
$648$		0			0
$649$		−	825.446i		−	1.27187i
$650$		0			0
$651$	−165.193	+	2.90124i	−0.253752	+	0.00445659i
$652$		0			0
$653$	568.866			0.871158			0.435579	−	0.900150i	$-0.356544\pi$
$653$	568.866			0.871158			0.435579	+	0.900150i	$0.356544\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$			96.2705i			0.146530i
$658$		0			0
$659$	−1245.80			−1.89044			−0.945219	−	0.326436i	$-0.894152\pi$
$659$	−1245.80			−1.89044			−0.945219	+	0.326436i	$0.894152\pi$
$660$		0			0
$661$	−379.992			−0.574874			−0.287437	−	0.957800i	$-0.592803\pi$
$661$	−379.992			−0.574874			−0.287437	+	0.957800i	$0.592803\pi$
$662$		0			0
$663$			124.372i			0.187590i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−36.6981			−0.0550197
$668$		0			0
$669$	42.0003			0.0627808
$670$		0			0
$671$	−1496.19			−2.22979
$672$		0			0
$673$		−	602.763i		−	0.895635i	−0.894125	−	0.447818i	$-0.852201\pi$
$673$		−	602.763i		−	0.895635i	0.894125	−	0.447818i	$-0.147799\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			439.109i			0.648611i	0.945952	+	0.324305i	$0.105131\pi$
$677$			439.109i			0.648611i	−0.945952	+	0.324305i	$0.894869\pi$
$678$		0			0
$679$	−196.398			−0.289246
$680$		0			0
$681$			7.87967i			0.0115707i
$682$		0			0
$683$	−795.860			−1.16524			−0.582621	−	0.812744i	$-0.697973\pi$
$683$	−795.860			−1.16524			−0.582621	+	0.812744i	$0.697973\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−235.705			−0.343093
$688$		0			0
$689$	−300.759			−0.436516
$690$		0			0
$691$	1209.16			1.74987			0.874933	−	0.484243i	$-0.160905\pi$
$691$	1209.16			1.74987			0.874933	+	0.484243i	$0.160905\pi$
$692$		0			0
$693$			1022.25i			1.47510i
$694$		0			0
$695$		0			0
$696$		0			0
$697$			718.809i			1.03129i
$698$		0			0
$699$			133.081i			0.190388i
$700$		0			0
$701$	586.751			0.837020			0.418510	−	0.908212i	$-0.362552\pi$
$701$	586.751			0.837020			0.418510	+	0.908212i	$0.362552\pi$
$702$		0			0
$703$			326.572i			0.464541i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	219.917			0.311057
$708$		0			0
$709$		−	631.408i		−	0.890561i	−0.895391	−	0.445280i	$-0.853104\pi$
$709$		−	631.408i		−	0.890561i	0.895391	−	0.445280i	$-0.146896\pi$
$710$		0			0
$711$		−	247.775i		−	0.348488i
$712$		0			0
$713$	206.536	−	3.62734i	0.289672	−	0.00508744i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−180.467			−0.251697
$718$		0			0
$719$		−	546.760i		−	0.760446i	−0.924895	−	0.380223i	$-0.875847\pi$
$719$		−	546.760i		−	0.760446i	0.924895	−	0.380223i	$-0.124153\pi$
$720$		0			0
$721$	−63.5870			−0.0881927
$722$		0			0
$723$	−79.9119			−0.110528
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−993.901			−1.36713			−0.683563	−	0.729891i	$-0.739571\pi$
$727$	−993.901			−1.36713			−0.683563	+	0.729891i	$0.739571\pi$
$728$		0			0
$729$	524.541			0.719535
$730$		0			0
$731$	−906.390			−1.23993
$732$		0			0
$733$	569.055			0.776336			0.388168	−	0.921589i	$-0.373108\pi$
$733$	569.055			0.776336			0.388168	+	0.921589i	$0.373108\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	316.319i		−	0.429198i
$738$		0			0
$739$			879.306i			1.18986i	0.803778	+	0.594930i	$0.202820\pi$
$739$			879.306i			1.18986i	−0.803778	+	0.594930i	$0.797180\pi$
$740$		0			0
$741$	50.6458			0.0683479
$742$		0			0
$743$			943.673i			1.27008i	0.772477	+	0.635042i	$0.219017\pi$
$743$			943.673i			1.27008i	−0.772477	+	0.635042i	$0.780983\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		−	1057.50i		−	1.41566i
$748$		0			0
$749$	1062.62			1.41871
$750$		0			0
$751$	245.490			0.326884			0.163442	−	0.986553i	$-0.447740\pi$
$751$	245.490			0.326884			0.163442	+	0.986553i	$0.447740\pi$
$752$		0			0
$753$	−0.344179			−0.000457077
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	712.011i		−	0.940570i	−0.882515	−	0.470285i	$-0.844151\pi$
$757$		−	712.011i		−	0.940570i	0.882515	−	0.470285i	$-0.155849\pi$
$758$		0			0
$759$			65.4315i			0.0862075i
$760$		0			0
$761$		−	1377.65i		−	1.81032i	−0.425071	−	0.905160i	$-0.639751\pi$
$761$		−	1377.65i		−	1.81032i	0.425071	−	0.905160i	$-0.360249\pi$
$762$		0			0
$763$	−1265.42			−1.65848
$764$		0			0
$765$		0			0
$766$		0			0
$767$			396.540i			0.517002i
$768$		0			0
$769$	−421.796			−0.548499			−0.274250	−	0.961659i	$-0.588429\pi$
$769$	−421.796			−0.548499			−0.274250	+	0.961659i	$0.588429\pi$
$770$		0			0
$771$		−	168.972i		−	0.219160i
$772$		0			0
$773$			567.102i			0.733637i	0.930292	+	0.366819i	$0.119553\pi$
$773$			567.102i			0.733637i	−0.930292	+	0.366819i	$0.880447\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−162.113			−0.208639
$778$		0			0
$779$	292.707			0.375747
$780$		0			0
$781$		−	1160.22i		−	1.48556i
$782$		0			0
$783$	64.0326			0.0817785
$784$		0			0
$785$		0			0
$786$		0			0
$787$			715.550i			0.909212i	0.890693	+	0.454606i	$0.150220\pi$
$787$			715.550i			0.909212i	−0.890693	+	0.454606i	$0.849780\pi$
$788$		0			0
$789$	−214.505			−0.271870
$790$		0			0
$791$	−34.6653			−0.0438246
$792$		0			0
$793$	718.764			0.906385
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	781.911i		−	0.981068i	−0.871422	−	0.490534i	$-0.836802\pi$
$797$		−	781.911i		−	0.981068i	0.871422	−	0.490534i	$-0.163198\pi$
$798$		0			0
$799$		−	597.743i		−	0.748114i
$800$		0			0
$801$		−	669.030i		−	0.835243i
$802$		0			0
$803$	−166.773			−0.207688
$804$		0			0
$805$		0			0
$806$		0			0
$807$	49.9679			0.0619181
$808$		0			0
$809$		−	1232.69i		−	1.52372i	−0.647739	−	0.761862i	$-0.724285\pi$
$809$		−	1232.69i		−	1.52372i	0.647739	−	0.761862i	$-0.275715\pi$
$810$		0			0
$811$	−366.014			−0.451312			−0.225656	−	0.974207i	$-0.572453\pi$
$811$	−366.014			−0.451312			−0.225656	+	0.974207i	$0.572453\pi$
$812$		0			0
$813$	−251.221			−0.309005
$814$		0			0
$815$		0			0
$816$		0			0
$817$			369.092i			0.451765i
$818$		0			0
$819$		−	491.083i		−	0.599613i
$820$		0			0
$821$		−	579.420i		−	0.705750i	−0.935671	−	0.352875i	$-0.885204\pi$
$821$		−	579.420i		−	0.705750i	0.935671	−	0.352875i	$-0.114796\pi$
$822$		0			0
$823$			213.733i			0.259699i	0.991534	+	0.129850i	$0.0414495\pi$
$823$			213.733i			0.259699i	−0.991534	+	0.129850i	$0.958551\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			1360.90i			1.64559i	0.568340	+	0.822794i	$0.307586\pi$
$827$			1360.90i			1.64559i	−0.568340	+	0.822794i	$0.692414\pi$
$828$		0			0
$829$			509.748i			0.614896i	0.951565	+	0.307448i	$0.0994749\pi$
$829$			509.748i			0.614896i	−0.951565	+	0.307448i	$0.900525\pi$
$830$		0			0
$831$	−159.075			−0.191427
$832$		0			0
$833$			416.707i			0.500248i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−360.374	+	6.32916i	−0.430554	+	0.00756172i
$838$		0			0
$839$	−904.240			−1.07776			−0.538880	−	0.842383i	$-0.681152\pi$
$839$	−904.240			−1.07776			−0.538880	+	0.842383i	$0.681152\pi$
$840$		0			0
$841$	810.669			0.963935
$842$		0			0
$843$		−	4.28428i		−	0.00508218i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−796.814			−0.940749
$848$		0			0
$849$			93.0596i			0.109611i
$850$		0			0
$851$	202.685			0.238173
$852$		0			0
$853$	−905.373			−1.06140			−0.530699	−	0.847560i	$-0.678071\pi$
$853$	−905.373			−1.06140			−0.530699	+	0.847560i	$0.678071\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−690.248			−0.805424			−0.402712	−	0.915327i	$-0.631932\pi$
$857$	−690.248			−0.805424			−0.402712	+	0.915327i	$0.631932\pi$
$858$		0			0
$859$			451.723i			0.525871i	0.964813	+	0.262936i	$0.0846907\pi$
$859$			451.723i			0.525871i	−0.964813	+	0.262936i	$0.915309\pi$
$860$		0			0
$861$			145.302i			0.168759i
$862$		0			0
$863$			132.226i			0.153217i	0.997061	+	0.0766083i	$0.0244091\pi$
$863$			132.226i			0.153217i	−0.997061	+	0.0766083i	$0.975591\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		−	268.894i		−	0.310142i
$868$		0			0
$869$	429.231			0.493937
$870$		0			0
$871$			151.958i			0.174464i
$872$		0			0
$873$	−208.878			−0.239265
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−247.457			−0.282163			−0.141081	−	0.989998i	$-0.545058\pi$
$877$	−247.457			−0.282163			−0.141081	+	0.989998i	$0.545058\pi$
$878$		0			0
$879$		−	179.407i		−	0.204104i
$880$		0			0
$881$		−	989.137i		−	1.12274i	−0.827564	−	0.561372i	$-0.810274\pi$
$881$		−	989.137i		−	1.12274i	0.827564	−	0.561372i	$-0.189726\pi$
$882$		0			0
$883$		−	1193.22i		−	1.35132i	−0.737213	−	0.675661i	$-0.763858\pi$
$883$		−	1193.22i		−	1.35132i	0.737213	−	0.675661i	$-0.236142\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1399.51			1.57780			0.788898	−	0.614524i	$-0.210652\pi$
$887$	1399.51			1.57780			0.788898	+	0.614524i	$0.210652\pi$
$888$		0			0
$889$		−	1131.25i		−	1.27250i
$890$		0			0
$891$			1028.70i			1.15454i
$892$		0			0
$893$	−243.408			−0.272573
$894$		0			0
$895$		0			0
$896$		0			0
$897$		−	31.4330i		−	0.0350424i
$898$		0			0
$899$	170.701	−	2.99799i	0.189879	−	0.00333480i
$900$		0			0
$901$	1112.93			1.23521
$902$		0			0
$903$	−183.220			−0.202901
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−701.236			−0.773138			−0.386569	−	0.922260i	$-0.626340\pi$
$907$	−701.236			−0.773138			−0.386569	+	0.922260i	$0.626340\pi$
$908$		0			0
$909$	233.891			0.257306
$910$		0			0
$911$			306.403i			0.336337i	0.985758	+	0.168169i	$0.0537853\pi$
$911$			306.403i			0.336337i	−0.985758	+	0.168169i	$0.946215\pi$
$912$		0			0
$913$	1831.95			2.00652
$914$		0			0
$915$		0			0
$916$		0			0
$917$	76.5554			0.0834846
$918$		0			0
$919$	1008.81			1.09773			0.548865	−	0.835911i	$-0.315060\pi$
$919$	1008.81			1.09773			0.548865	+	0.835911i	$0.315060\pi$
$920$		0			0
$921$			63.5944i			0.0690493i
$922$		0			0
$923$			557.365i			0.603862i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−67.6275			−0.0729531
$928$		0			0
$929$		−	1732.51i		−	1.86492i	−0.361274	−	0.932460i	$-0.617658\pi$
$929$		−	1732.51i		−	1.86492i	0.361274	−	0.932460i	$-0.382342\pi$
$930$		0			0
$931$	169.687			0.182264
$932$		0			0
$933$			105.967i			0.113577i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1275.33			1.36108			0.680538	−	0.732713i	$-0.261746\pi$
$937$	1275.33			1.36108			0.680538	+	0.732713i	$0.261746\pi$
$938$		0			0
$939$	85.3029			0.0908444
$940$		0			0
$941$		−	1110.88i		−	1.18053i	−0.807210	−	0.590265i	$-0.799023\pi$
$941$		−	1110.88i		−	1.18053i	0.807210	−	0.590265i	$-0.200977\pi$
$942$		0			0
$943$		−	181.667i		−	0.192648i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			1029.36i			1.08697i	0.839419	+	0.543485i	$0.182895\pi$
$947$			1029.36i			1.08697i	−0.839419	+	0.543485i	$0.817105\pi$
$948$		0			0
$949$	80.1172			0.0844228
$950$		0			0
$951$			394.715i			0.415053i
$952$		0			0
$953$			202.156i			0.212126i	0.994359	+	0.106063i	$0.0338245\pi$
$953$			202.156i			0.212126i	−0.994359	+	0.106063i	$0.966175\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$			54.0789i			0.0565088i
$958$		0			0
$959$		−	1492.00i		−	1.55579i
$960$		0			0
$961$	−960.407	+	33.7452i	−0.999383	+	0.0351147i
$962$		0			0
$963$	1130.14			1.17356
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	274.969i		−	0.284353i	−0.989841	−	0.142176i	$-0.954590\pi$
$967$		−	274.969i		−	0.284353i	0.989841	−	0.142176i	$-0.0454101\pi$
$968$		0			0
$969$	−187.409			−0.193405
$970$		0			0
$971$	−684.427			−0.704868			−0.352434	−	0.935837i	$-0.614646\pi$
$971$	−684.427			−0.704868			−0.352434	+	0.935837i	$0.614646\pi$
$972$		0			0
$973$			996.634i			1.02429i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1777.11			1.81894			0.909472	−	0.415765i	$-0.136486\pi$
$977$	1777.11			1.81894			0.909472	+	0.415765i	$0.136486\pi$
$978$		0			0
$979$	1158.99			1.18385
$980$		0			0
$981$	−1345.83			−1.37190
$982$		0			0
$983$		−	899.542i		−	0.915099i	−0.889184	−	0.457549i	$-0.848727\pi$
$983$		−	899.542i		−	0.915099i	0.889184	−	0.457549i	$-0.151273\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		−	120.829i		−	0.122421i
$988$		0			0
$989$	229.075			0.231622
$990$		0			0
$991$			519.695i			0.524415i	0.965012	+	0.262207i	$0.0844505\pi$
$991$			519.695i			0.524415i	−0.965012	+	0.262207i	$0.915550\pi$
$992$		0			0
$993$	−85.7241			−0.0863284
$994$		0			0
$995$		0			0
$996$		0			0
$997$	1390.36			1.39454			0.697269	−	0.716809i	$-0.254398\pi$
$997$	1390.36			1.39454			0.697269	+	0.716809i	$0.254398\pi$
$998$		0			0
$999$	−353.654			−0.354008

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3100.3.d.h.1301.8		24
5.2	odd	4		620.3.f.c.309.14	yes	24
5.3	odd	4		620.3.f.c.309.11	✓	24
5.4	even	2	inner	3100.3.d.h.1301.17		24
31.30	odd	2	inner	3100.3.d.h.1301.7		24
155.92	even	4		620.3.f.c.309.12	yes	24
155.123	even	4		620.3.f.c.309.13	yes	24
155.154	odd	2	inner	3100.3.d.h.1301.18		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
620.3.f.c.309.11	✓	24	5.3	odd	4
620.3.f.c.309.12	yes	24	155.92	even	4
620.3.f.c.309.13	yes	24	155.123	even	4
620.3.f.c.309.14	yes	24	5.2	odd	4
3100.3.d.h.1301.7		24	31.30	odd	2	inner
3100.3.d.h.1301.8		24	1.1	even	1	trivial
3100.3.d.h.1301.17		24	5.4	even	2	inner
3100.3.d.h.1301.18		24	155.154	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3100 = 2^{2} \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	3100.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.662053i q^{3} +8.05015 q^{7} +8.56169 q^{9} +O(q^{10})\)	\(q+0.662053i q^{3} +8.05015 q^{7} +8.56169 q^{9} +14.8318i q^{11} -7.12511i q^{13} +26.3657i q^{17} +10.7364 q^{19} +5.32962i q^{21} -6.66349i q^{23} +11.6268i q^{27} -5.50735i q^{29} +(0.544361 + 30.9952i) q^{31} -9.81941 q^{33} +30.4173i q^{37} +4.71720 q^{39} +27.2630 q^{41} +34.3776i q^{43} -22.6712 q^{47} +15.8049 q^{49} -17.4555 q^{51} -42.2112i q^{53} +7.10807i q^{57} -55.6539 q^{59} +100.878i q^{61} +68.9228 q^{63} -21.3271 q^{67} +4.41158 q^{69} -78.2254 q^{71} +11.2443i q^{73} +119.398i q^{77} -28.9400i q^{79} +69.3577 q^{81} -123.515i q^{83} +3.64615 q^{87} -78.1423i q^{89} -57.3582i q^{91} +(-20.5205 + 0.360396i) q^{93} -24.3968 q^{97} +126.985i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 44 q^{9}+O(q^{10}) \) 24 * q - 44 * q^9	\( 24 q - 44 q^{9} - 8 q^{19} - 24 q^{31} + 280 q^{39} - 248 q^{41} + 644 q^{49} - 100 q^{51} - 152 q^{59} + 288 q^{69} - 352 q^{71} - 368 q^{81}+O(q^{100}) \) 24 * q - 44 * q^9 - 8 * q^19 - 24 * q^31 + 280 * q^39 - 248 * q^41 + 644 * q^49 - 100 * q^51 - 152 * q^59 + 288 * q^69 - 352 * q^71 - 368 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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