LMFDB - Embedded newform 6120.2.h.l.1801.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6120,2,Mod(1801,6120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6120, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("6120.1801");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6120 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6120.h (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:	$48.8684460370$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.399424.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 $ x^6 - 2x^5 + 3x^4 - 6x^3 + 6x^2 - 8*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 2040)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1801.4
Root			$0.264658 - 1.38923i$ of defining polynomial
Character	$\chi$	$=$	6120.1801
Dual form			6120.2.h.l.1801.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{5} -4.77846i q^{7} -4.77846i q^{11} -6.49828 q^{13} +(-3.71982 + 1.77846i) q^{17} +1.83709 q^{19} +4.49828i q^{23} -1.00000 q^{25} -6.77846i q^{29} +0.941367i q^{31} +4.77846 q^{35} -6.77846i q^{37} +4.89572i q^{41} -11.1138 q^{43} +7.60256 q^{47} -15.8337 q^{49} -0.280176 q^{53} +4.77846 q^{55} +6.38101 q^{59} +12.4983i q^{61} -6.49828i q^{65} -10.0552 q^{67} +12.9966i q^{71} -6.45264i q^{73} -22.8337 q^{77} -5.11383i q^{79} +13.4396 q^{83} +(-1.77846 - 3.71982i) q^{85} +0.941367 q^{89} +31.0518i q^{91} +1.83709i q^{95} +17.1138i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{13} - 4 q^{17} - 4 q^{19} - 6 q^{25} + 12 q^{35} + 24 q^{47} - 10 q^{49} - 20 q^{53} + 12 q^{55} + 8 q^{67} - 52 q^{77} + 44 q^{83} + 6 q^{85} + 4 q^{89}+O(q^{100}) $ 6 * q - 4 * q^13 - 4 * q^17 - 4 * q^19 - 6 * q^25 + 12 * q^35 + 24 * q^47 - 10 * q^49 - 20 * q^53 + 12 * q^55 + 8 * q^67 - 52 * q^77 + 44 * q^83 + 6 * q^85 + 4 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6120\mathbb{Z}\right)^\times$.

$n$	$1361$	$3061$	$4321$	$4591$	$4897$
$\chi(n)$	$1$	$1$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$		−	4.77846i		−	1.80609i	−0.429549	−	0.903044i	$-0.641327\pi$
$7$		−	4.77846i		−	1.80609i	0.429549	−	0.903044i	$-0.358673\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	4.77846i		−	1.44076i	−0.693580	−	0.720380i	$-0.743968\pi$
$11$		−	4.77846i		−	1.44076i	0.693580	−	0.720380i	$-0.256032\pi$
$12$		0			0
$13$	−6.49828			−1.80230			−0.901149	−	0.433509i	$-0.857275\pi$
$13$	−6.49828			−1.80230			−0.901149	+	0.433509i	$0.857275\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.71982	+	1.77846i	−0.902190	+	0.431339i
$17$	−3.71982	+	1.77846i	−0.902190	+	0.431339i
$18$		0			0
$19$	1.83709			0.421457			0.210729	−	0.977545i	$-0.432416\pi$
$19$	1.83709			0.421457			0.210729	+	0.977545i	$0.432416\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			4.49828i			0.937956i	0.883210	+	0.468978i	$0.155378\pi$
$23$			4.49828i			0.937956i	−0.883210	+	0.468978i	$0.844622\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	6.77846i		−	1.25873i	−0.777111	−	0.629364i	$-0.783315\pi$
$29$		−	6.77846i		−	1.25873i	0.777111	−	0.629364i	$-0.216685\pi$
$30$		0			0
$31$			0.941367i			0.169074i	0.996420	+	0.0845372i	$0.0269412\pi$
$31$			0.941367i			0.169074i	−0.996420	+	0.0845372i	$0.973059\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	4.77846			0.807707
$36$		0			0
$37$		−	6.77846i		−	1.11437i	−0.830388	−	0.557186i	$-0.811881\pi$
$37$		−	6.77846i		−	1.11437i	0.830388	−	0.557186i	$-0.188119\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			4.89572i			0.764584i	0.924042	+	0.382292i	$0.124865\pi$
$41$			4.89572i			0.764584i	−0.924042	+	0.382292i	$0.875135\pi$
$42$		0			0
$43$	−11.1138			−1.69484			−0.847421	−	0.530921i	$-0.821846\pi$
$43$	−11.1138			−1.69484			−0.847421	+	0.530921i	$0.821846\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	7.60256			1.10895			0.554474	−	0.832201i	$-0.312920\pi$
$47$	7.60256			1.10895			0.554474	+	0.832201i	$0.312920\pi$
$48$		0			0
$49$	−15.8337			−2.26195
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−0.280176			−0.0384851			−0.0192426	−	0.999815i	$-0.506125\pi$
$53$	−0.280176			−0.0384851			−0.0192426	+	0.999815i	$0.506125\pi$
$54$		0			0
$55$	4.77846			0.644327
$56$		0			0
$57$		0			0
$58$		0			0
$59$	6.38101			0.830737			0.415369	−	0.909653i	$-0.363653\pi$
$59$	6.38101			0.830737			0.415369	+	0.909653i	$0.363653\pi$
$60$		0			0
$61$			12.4983i			1.60024i	0.599839	+	0.800120i	$0.295231\pi$
$61$			12.4983i			1.60024i	−0.599839	+	0.800120i	$0.704769\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	6.49828i		−	0.806013i
$66$		0			0
$67$	−10.0552			−1.22844			−0.614219	−	0.789136i	$-0.710529\pi$
$67$	−10.0552			−1.22844			−0.614219	+	0.789136i	$0.710529\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			12.9966i			1.54241i	0.636588	+	0.771204i	$0.280345\pi$
$71$			12.9966i			1.54241i	−0.636588	+	0.771204i	$0.719655\pi$
$72$		0			0
$73$		−	6.45264i		−	0.755224i	−0.925964	−	0.377612i	$-0.876745\pi$
$73$		−	6.45264i		−	0.755224i	0.925964	−	0.377612i	$-0.123255\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−22.8337			−2.60214
$78$		0			0
$79$		−	5.11383i		−	0.575351i	−0.957728	−	0.287675i	$-0.907118\pi$
$79$		−	5.11383i		−	0.575351i	0.957728	−	0.287675i	$-0.0928824\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	13.4396			1.47519			0.737597	−	0.675242i	$-0.235961\pi$
$83$	13.4396			1.47519			0.737597	+	0.675242i	$0.235961\pi$
$84$		0			0
$85$	−1.77846	−	3.71982i	−0.192901	−	0.403472i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	0.941367			0.0997847			0.0498923	−	0.998755i	$-0.484112\pi$
$89$	0.941367			0.0997847			0.0498923	+	0.998755i	$0.484112\pi$
$90$		0			0
$91$			31.0518i			3.25511i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			1.83709i			0.188481i
$96$		0			0
$97$			17.1138i			1.73765i	0.495123	+	0.868823i	$0.335123\pi$
$97$			17.1138i			1.73765i	−0.495123	+	0.868823i	$0.664877\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	7.55691			0.751941			0.375971	−	0.926632i	$-0.377309\pi$
$101$	7.55691			0.751941			0.375971	+	0.926632i	$0.377309\pi$
$102$		0			0
$103$	−0.325819			−0.0321039			−0.0160520	−	0.999871i	$-0.505110\pi$
$103$	−0.325819			−0.0321039			−0.0160520	+	0.999871i	$0.505110\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	2.44309i		−	0.236182i	−0.993003	−	0.118091i	$-0.962323\pi$
$107$		−	2.44309i		−	0.236182i	0.993003	−	0.118091i	$-0.0376775\pi$
$108$		0			0
$109$			11.1138i			1.06451i	0.846584	+	0.532256i	$0.178656\pi$
$109$			11.1138i			1.06451i	−0.846584	+	0.532256i	$0.821344\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	6.17246i		−	0.580656i	−0.956927	−	0.290328i	$-0.906236\pi$
$113$		−	6.17246i		−	0.580656i	0.956927	−	0.290328i	$-0.0937645\pi$
$114$		0			0
$115$	−4.49828			−0.419467
$116$		0			0
$117$		0			0
$118$		0			0
$119$	8.49828	+	17.7750i	0.779036	+	1.62943i
$120$		0			0
$121$	−11.8337			−1.07579
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$	8.49828			0.754101			0.377050	−	0.926193i	$-0.376938\pi$
$127$	8.49828			0.754101			0.377050	+	0.926193i	$0.376938\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			11.1138i			0.971020i	0.874231	+	0.485510i	$0.161366\pi$
$131$			11.1138i			0.971020i	−0.874231	+	0.485510i	$0.838634\pi$
$132$		0			0
$133$		−	8.77846i		−	0.761189i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1.27674			−0.109079			−0.0545396	−	0.998512i	$-0.517369\pi$
$137$	−1.27674			−0.109079			−0.0545396	+	0.998512i	$0.517369\pi$
$138$		0			0
$139$		−	17.1138i		−	1.45158i	−0.687918	−	0.725788i	$-0.741475\pi$
$139$		−	17.1138i		−	1.45158i	0.687918	−	0.725788i	$-0.258525\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			31.0518i			2.59668i
$144$		0			0
$145$	6.77846			0.562920
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1.17590			0.0963334			0.0481667	−	0.998839i	$-0.484662\pi$
$149$	1.17590			0.0963334			0.0481667	+	0.998839i	$0.484662\pi$
$150$		0			0
$151$	−5.27674			−0.429415			−0.214707	−	0.976678i	$-0.568880\pi$
$151$	−5.27674			−0.429415			−0.214707	+	0.976678i	$0.568880\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−0.941367			−0.0756124
$156$		0			0
$157$	−1.00344			−0.0800831			−0.0400415	−	0.999198i	$-0.512749\pi$
$157$	−1.00344			−0.0800831			−0.0400415	+	0.999198i	$0.512749\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	21.4948			1.69403
$162$		0			0
$163$		−	2.33537i		−	0.182920i	−0.995809	−	0.0914602i	$-0.970847\pi$
$163$		−	2.33537i		−	0.182920i	0.995809	−	0.0914602i	$-0.0291534\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			20.7620i			1.60661i	0.595565	+	0.803307i	$0.296928\pi$
$167$			20.7620i			1.60661i	−0.595565	+	0.803307i	$0.703072\pi$
$168$		0			0
$169$	29.2277			2.24828
$170$		0			0
$171$		0			0
$172$		0			0
$173$			8.61555i			0.655028i	0.944846	+	0.327514i	$0.106211\pi$
$173$			8.61555i			0.655028i	−0.944846	+	0.327514i	$0.893789\pi$
$174$		0			0
$175$			4.77846i			0.361217i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	16.9966			1.27038			0.635191	−	0.772355i	$-0.280921\pi$
$179$	16.9966			1.27038			0.635191	+	0.772355i	$0.280921\pi$
$180$		0			0
$181$		−	7.17590i		−	0.533380i	−0.963782	−	0.266690i	$-0.914070\pi$
$181$		−	7.17590i		−	0.533380i	0.963782	−	0.266690i	$-0.0859301\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	6.77846			0.498362
$186$		0			0
$187$	8.49828	+	17.7750i	0.621456	+	1.29984i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−10.6155			−0.768114			−0.384057	−	0.923309i	$-0.625473\pi$
$191$	−10.6155			−0.768114			−0.384057	+	0.923309i	$0.625473\pi$
$192$		0			0
$193$		−	5.11383i		−	0.368101i	−0.982917	−	0.184051i	$-0.941079\pi$
$193$		−	5.11383i		−	0.368101i	0.982917	−	0.184051i	$-0.0589211\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			14.7620i			1.05175i	0.850562	+	0.525876i	$0.176262\pi$
$197$			14.7620i			1.05175i	−0.850562	+	0.525876i	$0.823738\pi$
$198$		0			0
$199$			7.55691i			0.535695i	0.963461	+	0.267848i	$0.0863124\pi$
$199$			7.55691i			0.535695i	−0.963461	+	0.267848i	$0.913688\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−32.3906			−2.27337
$204$		0			0
$205$	−4.89572			−0.341932
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	8.77846i		−	0.607219i
$210$		0			0
$211$		−	13.3484i		−	0.918939i	−0.888194	−	0.459470i	$-0.848040\pi$
$211$		−	13.3484i		−	0.918939i	0.888194	−	0.459470i	$-0.151960\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	11.1138i		−	0.757957i
$216$		0			0
$217$	4.49828			0.305363
$218$		0			0
$219$		0			0
$220$		0			0
$221$	24.1725	−	11.5569i	1.62602	−	0.777402i
$222$		0			0
$223$	−10.5535			−0.706713			−0.353357	−	0.935489i	$-0.614960\pi$
$223$	−10.5535			−0.706713			−0.353357	+	0.935489i	$0.614960\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	18.0552i		−	1.19837i	−0.800612	−	0.599183i	$-0.795492\pi$
$227$		−	18.0552i		−	1.19837i	0.800612	−	0.599183i	$-0.204508\pi$
$228$		0			0
$229$	27.2767			1.80250			0.901249	−	0.433302i	$-0.142652\pi$
$229$	27.2767			1.80250			0.901249	+	0.433302i	$0.142652\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	20.6155i		−	1.35057i	−0.737557	−	0.675285i	$-0.764021\pi$
$233$		−	20.6155i		−	1.35057i	0.737557	−	0.675285i	$-0.235979\pi$
$234$		0			0
$235$			7.60256i			0.495936i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−25.4036			−1.64322			−0.821610	−	0.570050i	$-0.806924\pi$
$239$	−25.4036			−1.64322			−0.821610	+	0.570050i	$0.806924\pi$
$240$		0			0
$241$			15.0518i			0.969569i	0.874634	+	0.484784i	$0.161102\pi$
$241$			15.0518i			0.969569i	−0.874634	+	0.484784i	$0.838898\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		−	15.8337i		−	1.01157i
$246$		0			0
$247$	−11.9379			−0.759592
$248$		0			0
$249$		0			0
$250$		0			0
$251$	19.9379			1.25847			0.629235	−	0.777215i	$-0.283368\pi$
$251$	19.9379			1.25847			0.629235	+	0.777215i	$0.283368\pi$
$252$		0			0
$253$	21.4948			1.35137
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−24.6707			−1.53892			−0.769459	−	0.638696i	$-0.779474\pi$
$257$	−24.6707			−1.53892			−0.769459	+	0.638696i	$0.779474\pi$
$258$		0			0
$259$	−32.3906			−2.01265
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−24.0388			−1.48229			−0.741147	−	0.671343i	$-0.765718\pi$
$263$	−24.0388			−1.48229			−0.741147	+	0.671343i	$0.765718\pi$
$264$		0			0
$265$		−	0.280176i		−	0.0172111i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			20.8957i			1.27403i	0.770849	+	0.637017i	$0.219832\pi$
$269$			20.8957i			1.27403i	−0.770849	+	0.637017i	$0.780168\pi$
$270$		0			0
$271$	−31.2242			−1.89674			−0.948368	−	0.317172i	$-0.897267\pi$
$271$	−31.2242			−1.89674			−0.948368	+	0.317172i	$0.897267\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			4.77846i			0.288152i
$276$		0			0
$277$		−	7.88273i		−	0.473628i	−0.971555	−	0.236814i	$-0.923897\pi$
$277$		−	7.88273i		−	0.473628i	0.971555	−	0.236814i	$-0.0761031\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−22.0000			−1.31241			−0.656205	−	0.754583i	$-0.727839\pi$
$281$	−22.0000			−1.31241			−0.656205	+	0.754583i	$0.727839\pi$
$282$		0			0
$283$			26.4458i			1.57204i	0.618203	+	0.786019i	$0.287861\pi$
$283$			26.4458i			1.57204i	−0.618203	+	0.786019i	$0.712139\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	23.3940			1.38090
$288$		0			0
$289$	10.6742	−	13.2311i	0.627893	−	0.778300i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−16.4819			−0.962880			−0.481440	−	0.876479i	$-0.659886\pi$
$293$	−16.4819			−0.962880			−0.481440	+	0.876479i	$0.659886\pi$
$294$		0			0
$295$			6.38101i			0.371517i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	29.2311i		−	1.69048i
$300$		0			0
$301$			53.1070i			3.06103i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−12.4983			−0.715649
$306$		0			0
$307$	3.76547			0.214907			0.107453	−	0.994210i	$-0.465730\pi$
$307$	3.76547			0.214907			0.107453	+	0.994210i	$0.465730\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	11.9836i		−	0.679526i	−0.940511	−	0.339763i	$-0.889653\pi$
$311$		−	11.9836i		−	0.679526i	0.940511	−	0.339763i	$-0.110347\pi$
$312$		0			0
$313$			12.5699i			0.710493i	0.934773	+	0.355246i	$0.115603\pi$
$313$			12.5699i			0.710493i	−0.934773	+	0.355246i	$0.884397\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			9.50172i			0.533670i	0.963742	+	0.266835i	$0.0859778\pi$
$317$			9.50172i			0.533670i	−0.963742	+	0.266835i	$0.914022\pi$
$318$		0			0
$319$	−32.3906			−1.81352
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−6.83365	+	3.26719i	−0.380235	+	0.181791i
$324$		0			0
$325$	6.49828			0.360460
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	36.3285i		−	2.00285i
$330$		0			0
$331$	21.6026			1.18738			0.593692	−	0.804692i	$-0.297670\pi$
$331$	21.6026			1.18738			0.593692	+	0.804692i	$0.297670\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	10.0552i		−	0.549374i
$336$		0			0
$337$		−	4.33537i		−	0.236163i	−0.993004	−	0.118081i	$-0.962326\pi$
$337$		−	4.33537i		−	0.236163i	0.993004	−	0.118081i	$-0.0376744\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	4.49828			0.243596
$342$		0			0
$343$			42.2112i			2.27919i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			28.1725i			1.51238i	0.654354	+	0.756188i	$0.272941\pi$
$347$			28.1725i			1.51238i	−0.654354	+	0.756188i	$0.727059\pi$
$348$		0			0
$349$	−19.1855			−1.02697			−0.513487	−	0.858097i	$-0.671646\pi$
$349$	−19.1855			−1.02697			−0.513487	+	0.858097i	$0.671646\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−11.1595			−0.593959			−0.296979	−	0.954884i	$-0.595979\pi$
$353$	−11.1595			−0.593959			−0.296979	+	0.954884i	$0.595979\pi$
$354$		0			0
$355$	−12.9966			−0.689786
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−15.0518			−0.794401			−0.397201	−	0.917732i	$-0.630018\pi$
$359$	−15.0518			−0.794401			−0.397201	+	0.917732i	$0.630018\pi$
$360$		0			0
$361$	−15.6251			−0.822374
$362$		0			0
$363$		0			0
$364$		0			0
$365$	6.45264			0.337746
$366$		0			0
$367$			11.7655i			0.614152i	0.951685	+	0.307076i	$0.0993506\pi$
$367$			11.7655i			0.614152i	−0.951685	+	0.307076i	$0.900649\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			1.33881i			0.0695075i
$372$		0			0
$373$	−1.11383			−0.0576719			−0.0288359	−	0.999584i	$-0.509180\pi$
$373$	−1.11383			−0.0576719			−0.0288359	+	0.999584i	$0.509180\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			44.0483i			2.26860i
$378$		0			0
$379$		−	31.9931i		−	1.64338i	−0.569937	−	0.821688i	$-0.693033\pi$
$379$		−	31.9931i		−	1.64338i	0.569937	−	0.821688i	$-0.306967\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−10.3906			−0.530933			−0.265467	−	0.964120i	$-0.585526\pi$
$383$	−10.3906			−0.530933			−0.265467	+	0.964120i	$0.585526\pi$
$384$		0			0
$385$		−	22.8337i		−	1.16371i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−2.99656			−0.151932			−0.0759659	−	0.997110i	$-0.524204\pi$
$389$	−2.99656			−0.151932			−0.0759659	+	0.997110i	$0.524204\pi$
$390$		0			0
$391$	−8.00000	−	16.7328i	−0.404577	−	0.846215i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	5.11383			0.257305
$396$		0			0
$397$		−	19.8663i		−	0.997061i	−0.866872	−	0.498531i	$-0.833873\pi$
$397$		−	19.8663i		−	0.997061i	0.866872	−	0.498531i	$-0.166127\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			19.6837i			0.982959i	0.870889	+	0.491479i	$0.163544\pi$
$401$			19.6837i			0.982959i	−0.870889	+	0.491479i	$0.836456\pi$
$402$		0			0
$403$		−	6.11727i		−	0.304723i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−32.3906			−1.60554
$408$		0			0
$409$	−17.9544			−0.887786			−0.443893	−	0.896080i	$-0.646403\pi$
$409$	−17.9544			−0.887786			−0.443893	+	0.896080i	$0.646403\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	30.4914i		−	1.50038i
$414$		0			0
$415$			13.4396i			0.659726i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	3.45608i		−	0.168840i	−0.996430	−	0.0844202i	$-0.973096\pi$
$419$		−	3.45608i		−	0.168840i	0.996430	−	0.0844202i	$-0.0269038\pi$
$420$		0			0
$421$	28.0388			1.36653			0.683263	−	0.730172i	$-0.260560\pi$
$421$	28.0388			1.36653			0.683263	+	0.730172i	$0.260560\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	3.71982	−	1.77846i	0.180438	−	0.0862678i
$426$		0			0
$427$	59.7225			2.89017
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	8.68717i		−	0.418446i	−0.977868	−	0.209223i	$-0.932907\pi$
$431$		−	8.68717i		−	0.418446i	0.977868	−	0.209223i	$-0.0670935\pi$
$432$		0			0
$433$	2.23453			0.107385			0.0536924	−	0.998558i	$-0.482901\pi$
$433$	2.23453			0.107385			0.0536924	+	0.998558i	$0.482901\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			8.26375i			0.395309i
$438$		0			0
$439$			8.05520i			0.384453i	0.981351	+	0.192227i	$0.0615709\pi$
$439$			8.05520i			0.384453i	−0.981351	+	0.192227i	$0.938429\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−6.09129			−0.289406			−0.144703	−	0.989475i	$-0.546223\pi$
$443$	−6.09129			−0.289406			−0.144703	+	0.989475i	$0.546223\pi$
$444$		0			0
$445$			0.941367i			0.0446251i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	12.8793i		−	0.607812i	−0.952702	−	0.303906i	$-0.901709\pi$
$449$		−	12.8793i		−	0.607812i	0.952702	−	0.303906i	$-0.0982908\pi$
$450$		0			0
$451$	23.3940			1.10158
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−31.0518			−1.45573
$456$		0			0
$457$	−29.7034			−1.38947			−0.694733	−	0.719268i	$-0.744478\pi$
$457$	−29.7034			−1.38947			−0.694733	+	0.719268i	$0.744478\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	9.05176			0.421582			0.210791	−	0.977531i	$-0.432396\pi$
$461$	9.05176			0.421582			0.210791	+	0.977531i	$0.432396\pi$
$462$		0			0
$463$	−0.996562			−0.0463142			−0.0231571	−	0.999732i	$-0.507372\pi$
$463$	−0.996562			−0.0463142			−0.0231571	+	0.999732i	$0.507372\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	1.71982			0.0795840			0.0397920	−	0.999208i	$-0.487330\pi$
$467$	1.71982			0.0795840			0.0397920	+	0.999208i	$0.487330\pi$
$468$		0			0
$469$			48.0483i			2.21867i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			53.1070i			2.44186i
$474$		0			0
$475$	−1.83709			−0.0842915
$476$		0			0
$477$		0			0
$478$		0			0
$479$			12.7620i			0.583112i	0.956554	+	0.291556i	$0.0941730\pi$
$479$			12.7620i			0.583112i	−0.956554	+	0.291556i	$0.905827\pi$
$480$		0			0
$481$			44.0483i			2.00843i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−17.1138			−0.777099
$486$		0			0
$487$			13.2311i			0.599558i	0.954009	+	0.299779i	$0.0969130\pi$
$487$			13.2311i			0.599558i	−0.954009	+	0.299779i	$0.903087\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−14.8793			−0.671493			−0.335747	−	0.941952i	$-0.608989\pi$
$491$	−14.8793			−0.671493			−0.335747	+	0.941952i	$0.608989\pi$
$492$		0			0
$493$	12.0552	+	25.2147i	0.542939	+	1.13561i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	62.1035			2.78572
$498$		0			0
$499$			1.73625i			0.0777253i	0.999245	+	0.0388626i	$0.0123735\pi$
$499$			1.73625i			0.0777253i	−0.999245	+	0.0388626i	$0.987627\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			14.9706i			0.667505i	0.942661	+	0.333753i	$0.108315\pi$
$503$			14.9706i			0.667505i	−0.942661	+	0.333753i	$0.891685\pi$
$504$		0			0
$505$			7.55691i			0.336278i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−11.3845			−0.504607			−0.252303	−	0.967648i	$-0.581188\pi$
$509$	−11.3845			−0.504607			−0.252303	+	0.967648i	$0.581188\pi$
$510$		0			0
$511$	−30.8337			−1.36400
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	0.325819i		−	0.0143573i
$516$		0			0
$517$		−	36.3285i		−	1.59773i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	35.5405i		−	1.55706i	−0.627609	−	0.778528i	$-0.715966\pi$
$521$		−	35.5405i		−	1.55706i	0.627609	−	0.778528i	$-0.284034\pi$
$522$		0			0
$523$	−30.1173			−1.31694			−0.658468	−	0.752609i	$-0.728795\pi$
$523$	−30.1173			−1.31694			−0.658468	+	0.752609i	$0.728795\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1.67418	−	3.50172i	−0.0729285	−	0.152537i
$528$		0			0
$529$	2.76547			0.120238
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	31.8138i		−	1.37801i
$534$		0			0
$535$	2.44309			0.105624
$536$		0			0
$537$		0			0
$538$		0			0
$539$			75.6604i			3.25893i
$540$		0			0
$541$		−	6.27062i		−	0.269595i	−0.990873	−	0.134798i	$-0.956962\pi$
$541$		−	6.27062i		−	0.269595i	0.990873	−	0.134798i	$-0.0430384\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−11.1138			−0.476064
$546$		0			0
$547$		−	1.99045i		−	0.0851054i	−0.999094	−	0.0425527i	$-0.986451\pi$
$547$		−	1.99045i		−	0.0851054i	0.999094	−	0.0425527i	$-0.0135490\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	12.4526i		−	0.530500i
$552$		0			0
$553$	−24.4362			−1.03913
$554$		0			0
$555$		0			0
$556$		0			0
$557$	7.32926			0.310551			0.155275	−	0.987871i	$-0.450374\pi$
$557$	7.32926			0.310551			0.155275	+	0.987871i	$0.450374\pi$
$558$		0			0
$559$	72.2208			3.05461
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−29.8302			−1.25719			−0.628597	−	0.777731i	$-0.716370\pi$
$563$	−29.8302			−1.25719			−0.628597	+	0.777731i	$0.716370\pi$
$564$		0			0
$565$	6.17246			0.259677
$566$		0			0
$567$		0			0
$568$		0			0
$569$	23.9018			1.00202			0.501009	−	0.865442i	$-0.332963\pi$
$569$	23.9018			1.00202			0.501009	+	0.865442i	$0.332963\pi$
$570$		0			0
$571$			35.7294i			1.49523i	0.664134	+	0.747614i	$0.268801\pi$
$571$			35.7294i			1.49523i	−0.664134	+	0.747614i	$0.731199\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	4.49828i		−	0.187591i
$576$		0			0
$577$	−45.4588			−1.89247			−0.946236	−	0.323476i	$-0.895148\pi$
$577$	−45.4588			−1.89247			−0.946236	+	0.323476i	$0.895148\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	64.2208i		−	2.66433i
$582$		0			0
$583$			1.33881i			0.0554478i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	29.0682			1.19977			0.599886	−	0.800085i	$-0.295213\pi$
$587$	29.0682			1.19977			0.599886	+	0.800085i	$0.295213\pi$
$588$		0			0
$589$			1.72938i			0.0712577i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	0.280176			0.0115055			0.00575273	−	0.999983i	$-0.498169\pi$
$593$	0.280176			0.0115055			0.00575273	+	0.999983i	$0.498169\pi$
$594$		0			0
$595$	−17.7750	+	8.49828i	−0.728705	+	0.348396i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	20.5604			0.840073			0.420037	−	0.907507i	$-0.362017\pi$
$599$	20.5604			0.840073			0.420037	+	0.907507i	$0.362017\pi$
$600$		0			0
$601$		−	32.8432i		−	1.33970i	−0.742495	−	0.669851i	$-0.766358\pi$
$601$		−	32.8432i		−	1.33970i	0.742495	−	0.669851i	$-0.233642\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	11.8337i		−	0.481106i
$606$		0			0
$607$		−	43.5665i		−	1.76831i	−0.467195	−	0.884154i	$-0.654735\pi$
$607$		−	43.5665i		−	1.76831i	0.467195	−	0.884154i	$-0.345265\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−49.4036			−1.99865
$612$		0			0
$613$	−40.3189			−1.62847			−0.814233	−	0.580538i	$-0.802842\pi$
$613$	−40.3189			−1.62847			−0.814233	+	0.580538i	$0.802842\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	19.2932i		−	0.776714i	−0.921509	−	0.388357i	$-0.873043\pi$
$617$		−	19.2932i		−	0.776714i	0.921509	−	0.388357i	$-0.126957\pi$
$618$		0			0
$619$			7.99312i			0.321271i	0.987014	+	0.160635i	$0.0513543\pi$
$619$			7.99312i			0.321271i	−0.987014	+	0.160635i	$0.948646\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	4.49828i		−	0.180220i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	12.0552	+	25.2147i	0.480672	+	1.00537i
$630$		0			0
$631$	−21.7458			−0.865687			−0.432843	−	0.901469i	$-0.642490\pi$
$631$	−21.7458			−0.865687			−0.432843	+	0.901469i	$0.642490\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			8.49828i			0.337244i
$636$		0			0
$637$	102.892			4.07671
$638$		0			0
$639$		0			0
$640$		0			0
$641$			14.3614i			0.567239i	0.958937	+	0.283620i	$0.0915353\pi$
$641$			14.3614i			0.567239i	−0.958937	+	0.283620i	$0.908465\pi$
$642$		0			0
$643$		−	34.4458i		−	1.35841i	−0.733949	−	0.679204i	$-0.762325\pi$
$643$		−	34.4458i		−	1.35841i	0.733949	−	0.679204i	$-0.237675\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	15.3224			0.602385			0.301192	−	0.953563i	$-0.402615\pi$
$647$	15.3224			0.602385			0.301192	+	0.953563i	$0.402615\pi$
$648$		0			0
$649$		−	30.4914i		−	1.19689i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	17.7846i		−	0.695964i	−0.937501	−	0.347982i	$-0.886867\pi$
$653$		−	17.7846i		−	0.695964i	0.937501	−	0.347982i	$-0.113133\pi$
$654$		0			0
$655$	−11.1138			−0.434253
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−39.6604			−1.54495			−0.772475	−	0.635045i	$-0.780982\pi$
$659$	−39.6604			−1.54495			−0.772475	+	0.635045i	$0.780982\pi$
$660$		0			0
$661$	26.1560			1.01735			0.508676	−	0.860958i	$-0.330135\pi$
$661$	26.1560			1.01735			0.508676	+	0.860958i	$0.330135\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	8.77846			0.340414
$666$		0			0
$667$	30.4914			1.18063
$668$		0			0
$669$		0			0
$670$		0			0
$671$	59.7225			2.30556
$672$		0			0
$673$			6.00000i			0.231283i	0.993291	+	0.115642i	$0.0368924\pi$
$673$			6.00000i			0.231283i	−0.993291	+	0.115642i	$0.963108\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			2.67074i			0.102645i	0.998682	+	0.0513225i	$0.0163436\pi$
$677$			2.67074i			0.102645i	−0.998682	+	0.0513225i	$0.983656\pi$
$678$		0			0
$679$	81.7777			3.13834
$680$		0			0
$681$		0			0
$682$		0			0
$683$			3.00344i			0.114923i	0.998348	+	0.0574617i	$0.0183007\pi$
$683$			3.00344i			0.114923i	−0.998348	+	0.0574617i	$0.981699\pi$
$684$		0			0
$685$		−	1.27674i		−	0.0487817i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1.82066			0.0693617
$690$		0			0
$691$		−	7.23109i		−	0.275084i	−0.990496	−	0.137542i	$-0.956080\pi$
$691$		−	7.23109i		−	0.275084i	0.990496	−	0.137542i	$-0.0439202\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	17.1138			0.649165
$696$		0			0
$697$	−8.70683	−	18.2112i	−0.329795	−	0.689799i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−1.41043			−0.0532713			−0.0266356	−	0.999645i	$-0.508479\pi$
$701$	−1.41043			−0.0532713			−0.0266356	+	0.999645i	$0.508479\pi$
$702$		0			0
$703$		−	12.4526i		−	0.469660i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	36.1104i		−	1.35807i
$708$		0			0
$709$			6.20855i			0.233167i	0.993181	+	0.116584i	$0.0371943\pi$
$709$			6.20855i			0.233167i	−0.993181	+	0.116584i	$0.962806\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−4.23453			−0.158584
$714$		0			0
$715$	−31.0518			−1.16127
$716$		0			0
$717$		0			0
$718$		0			0
$719$			33.9767i			1.26712i	0.773695	+	0.633558i	$0.218406\pi$
$719$			33.9767i			1.26712i	−0.773695	+	0.633558i	$0.781594\pi$
$720$		0			0
$721$			1.55691i			0.0579825i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			6.77846i			0.251746i
$726$		0			0
$727$	−33.1398			−1.22909			−0.614544	−	0.788883i	$-0.710660\pi$
$727$	−33.1398			−1.22909			−0.614544	+	0.788883i	$0.710660\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	41.3415	−	19.7655i	1.52907	−	0.731052i
$732$		0			0
$733$	15.7103			0.580272			0.290136	−	0.956985i	$-0.406299\pi$
$733$	15.7103			0.580272			0.290136	+	0.956985i	$0.406299\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			48.0483i			1.76988i
$738$		0			0
$739$	25.2311			0.928141			0.464070	−	0.885798i	$-0.346388\pi$
$739$	25.2311			0.928141			0.464070	+	0.885798i	$0.346388\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			0.0912868i			0.00334899i	0.999999	+	0.00167449i	$0.000533008\pi$
$743$			0.0912868i			0.00334899i	−0.999999	+	0.00167449i	$0.999467\pi$
$744$		0			0
$745$			1.17590i			0.0430816i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−11.6742			−0.426565
$750$		0			0
$751$			17.6121i			0.642675i	0.946965	+	0.321338i	$0.104132\pi$
$751$			17.6121i			0.642675i	−0.946965	+	0.321338i	$0.895868\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	5.27674i		−	0.192040i
$756$		0			0
$757$	0.117266			0.00426212			0.00213106	−	0.999998i	$-0.499322\pi$
$757$	0.117266			0.00426212			0.00213106	+	0.999998i	$0.499322\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−13.0846			−0.474317			−0.237158	−	0.971471i	$-0.576216\pi$
$761$	−13.0846			−0.474317			−0.237158	+	0.971471i	$0.576216\pi$
$762$		0			0
$763$	53.1070			1.92260
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−41.4656			−1.49724
$768$		0			0
$769$	−16.6904			−0.601871			−0.300936	−	0.953644i	$-0.597299\pi$
$769$	−16.6904			−0.601871			−0.300936	+	0.953644i	$0.597299\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−0.371463			−0.0133606			−0.00668029	−	0.999978i	$-0.502126\pi$
$773$	−0.371463			−0.0133606			−0.00668029	+	0.999978i	$0.502126\pi$
$774$		0			0
$775$		−	0.941367i		−	0.0338149i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			8.99389i			0.322239i
$780$		0			0
$781$	62.1035			2.22224
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	1.00344i		−	0.0358142i
$786$		0			0
$787$			7.13026i			0.254166i	0.991892	+	0.127083i	$0.0405615\pi$
$787$			7.13026i			0.254166i	−0.991892	+	0.127083i	$0.959439\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−29.4948			−1.04872
$792$		0			0
$793$		−	81.2173i		−	2.88411i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	3.28361			0.116312			0.0581558	−	0.998308i	$-0.481478\pi$
$797$	3.28361			0.116312			0.0581558	+	0.998308i	$0.481478\pi$
$798$		0			0
$799$	−28.2802	+	13.5208i	−1.00048	+	0.478332i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−30.8337			−1.08810
$804$		0			0
$805$			21.4948i			0.757594i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	14.9966i		−	0.527251i	−0.964625	−	0.263626i	$-0.915082\pi$
$809$		−	14.9966i		−	0.527251i	0.964625	−	0.263626i	$-0.0849183\pi$
$810$		0			0
$811$		−	18.8241i		−	0.661004i	−0.943805	−	0.330502i	$-0.892782\pi$
$811$		−	18.8241i		−	0.661004i	0.943805	−	0.330502i	$-0.107218\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	2.33537			0.0818045
$816$		0			0
$817$	−20.4171			−0.714304
$818$		0			0
$819$		0			0
$820$		0			0
$821$			5.34836i			0.186659i	0.995635	+	0.0933295i	$0.0297510\pi$
$821$			5.34836i			0.186659i	−0.995635	+	0.0933295i	$0.970249\pi$
$822$		0			0
$823$			34.8957i			1.21639i	0.793788	+	0.608194i	$0.208106\pi$
$823$			34.8957i			1.21639i	−0.793788	+	0.608194i	$0.791894\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			17.9639i			0.624666i	0.949973	+	0.312333i	$0.101111\pi$
$827$			17.9639i			0.624666i	−0.949973	+	0.312333i	$0.898889\pi$
$828$		0			0
$829$	−54.5922			−1.89607			−0.948034	−	0.318171i	$-0.896932\pi$
$829$	−54.5922			−1.89607			−0.948034	+	0.318171i	$0.896932\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	58.8984	−	28.1595i	2.04071	−	0.975668i
$834$		0			0
$835$	−20.7620			−0.718500
$836$		0			0
$837$		0			0
$838$		0			0
$839$			31.2406i			1.07855i	0.842131	+	0.539273i	$0.181301\pi$
$839$			31.2406i			1.07855i	−0.842131	+	0.539273i	$0.818699\pi$
$840$		0			0
$841$	−16.9475			−0.584396
$842$		0			0
$843$		0			0
$844$		0			0
$845$			29.2277i			1.00546i
$846$		0			0
$847$			56.5466i			1.94296i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	30.4914			1.04523
$852$		0			0
$853$		−	16.2277i		−	0.555624i	−0.960635	−	0.277812i	$-0.910391\pi$
$853$		−	16.2277i		−	0.555624i	0.960635	−	0.277812i	$-0.0896093\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	32.3189i		−	1.10399i	−0.833846	−	0.551997i	$-0.813866\pi$
$857$		−	32.3189i		−	1.10399i	0.833846	−	0.551997i	$-0.186134\pi$
$858$		0			0
$859$	25.1526			0.858196			0.429098	−	0.903258i	$-0.358832\pi$
$859$	25.1526			0.858196			0.429098	+	0.903258i	$0.358832\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	28.3974			0.966660			0.483330	−	0.875438i	$-0.339427\pi$
$863$	28.3974			0.966660			0.483330	+	0.875438i	$0.339427\pi$
$864$		0			0
$865$	−8.61555			−0.292937
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−24.4362			−0.828942
$870$		0			0
$871$	65.3415			2.21401
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−4.77846			−0.161541
$876$		0			0
$877$		−	33.4423i		−	1.12927i	−0.825342	−	0.564634i	$-0.809018\pi$
$877$		−	33.4423i		−	1.12927i	0.825342	−	0.564634i	$-0.190982\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	38.8888i		−	1.31020i	−0.755543	−	0.655099i	$-0.772627\pi$
$881$		−	38.8888i		−	1.31020i	0.755543	−	0.655099i	$-0.227373\pi$
$882$		0			0
$883$	−31.2863			−1.05287			−0.526434	−	0.850216i	$-0.676471\pi$
$883$	−31.2863			−1.05287			−0.526434	+	0.850216i	$0.676471\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			22.6639i			0.760978i	0.924785	+	0.380489i	$0.124244\pi$
$887$			22.6639i			0.760978i	−0.924785	+	0.380489i	$0.875756\pi$
$888$		0			0
$889$		−	40.6087i		−	1.36197i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	13.9666			0.467374
$894$		0			0
$895$			16.9966i			0.568132i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	6.38101			0.212819
$900$		0			0
$901$	1.04221	−	0.498281i	0.0347209	−	0.0166001i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	7.17590			0.238535
$906$		0			0
$907$			18.8629i			0.626331i	0.949699	+	0.313166i	$0.101389\pi$
$907$			18.8629i			0.626331i	−0.949699	+	0.313166i	$0.898611\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			50.9148i			1.68688i	0.537221	+	0.843442i	$0.319474\pi$
$911$			50.9148i			1.68688i	−0.537221	+	0.843442i	$0.680526\pi$
$912$		0			0
$913$		−	64.2208i		−	2.12540i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	53.1070			1.75375
$918$		0			0
$919$	−5.18545			−0.171052			−0.0855261	−	0.996336i	$-0.527257\pi$
$919$	−5.18545			−0.171052			−0.0855261	+	0.996336i	$0.527257\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	84.4553i		−	2.77988i
$924$		0			0
$925$			6.77846i			0.222874i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			50.2372i			1.64823i	0.566423	+	0.824115i	$0.308327\pi$
$929$			50.2372i			1.64823i	−0.566423	+	0.824115i	$0.691673\pi$
$930$		0			0
$931$	−29.0878			−0.953316
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−17.7750	+	8.49828i	−0.581305	+	0.277924i
$936$		0			0
$937$	53.1950			1.73781			0.868903	−	0.494983i	$-0.164826\pi$
$937$	53.1950			1.73781			0.868903	+	0.494983i	$0.164826\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			31.3415i			1.02170i	0.859669	+	0.510852i	$0.170670\pi$
$941$			31.3415i			1.02170i	−0.859669	+	0.510852i	$0.829330\pi$
$942$		0			0
$943$	−22.0223			−0.717146
$944$		0			0
$945$		0			0
$946$		0			0
$947$			16.7949i			0.545760i	0.962048	+	0.272880i	$0.0879763\pi$
$947$			16.7949i			0.545760i	−0.962048	+	0.272880i	$0.912024\pi$
$948$		0			0
$949$			41.9311i			1.36114i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−39.3803			−1.27565			−0.637826	−	0.770181i	$-0.720166\pi$
$953$	−39.3803			−1.27565			−0.637826	+	0.770181i	$0.720166\pi$
$954$		0			0
$955$		−	10.6155i		−	0.343511i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			6.10084i			0.197006i
$960$		0			0
$961$	30.1138			0.971414
$962$		0			0
$963$		0			0
$964$		0			0
$965$	5.11383			0.164620
$966$		0			0
$967$	−2.28973			−0.0736327			−0.0368163	−	0.999322i	$-0.511722\pi$
$967$	−2.28973			−0.0736327			−0.0368163	+	0.999322i	$0.511722\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	50.8172			1.63080			0.815401	−	0.578896i	$-0.196516\pi$
$971$	50.8172			1.63080			0.815401	+	0.578896i	$0.196516\pi$
$972$		0			0
$973$	−81.7777			−2.62167
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−31.2571			−1.00000			−0.500001	−	0.866025i	$-0.666667\pi$
$977$	−31.2571			−1.00000			−0.500001	+	0.866025i	$0.666667\pi$
$978$		0			0
$979$		−	4.49828i		−	0.143766i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	3.72250i		−	0.118729i	−0.998236	−	0.0593647i	$-0.981093\pi$
$983$		−	3.72250i		−	0.118729i	0.998236	−	0.0593647i	$-0.0189075\pi$
$984$		0			0
$985$	−14.7620			−0.470357
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	49.9931i		−	1.58969i
$990$		0			0
$991$			16.1465i			0.512910i	0.966556	+	0.256455i	$0.0825545\pi$
$991$			16.1465i			0.512910i	−0.966556	+	0.256455i	$0.917445\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−7.55691			−0.239570
$996$		0			0
$997$		−	61.5336i		−	1.94879i	−0.224845	−	0.974395i	$-0.572188\pi$
$997$		−	61.5336i		−	1.94879i	0.224845	−	0.974395i	$-0.427812\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6120.2.h.l.1801.4		6
3.2	odd	2		2040.2.h.i.1801.4	yes	6
12.11	even	2		4080.2.h.r.3841.3		6
17.16	even	2	inner	6120.2.h.l.1801.3		6
51.50	odd	2		2040.2.h.i.1801.3	✓	6
204.203	even	2		4080.2.h.r.3841.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2040.2.h.i.1801.3	✓	6	51.50	odd	2
2040.2.h.i.1801.4	yes	6	3.2	odd	2
4080.2.h.r.3841.3		6	12.11	even	2
4080.2.h.r.3841.4		6	204.203	even	2
6120.2.h.l.1801.3		6	17.16	even	2	inner
6120.2.h.l.1801.4		6	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6120 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6120.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{5} -4.77846i q^{7} -4.77846i q^{11} -6.49828 q^{13} +(-3.71982 + 1.77846i) q^{17} +1.83709 q^{19} +4.49828i q^{23} -1.00000 q^{25} -6.77846i q^{29} +0.941367i q^{31} +4.77846 q^{35} -6.77846i q^{37} +4.89572i q^{41} -11.1138 q^{43} +7.60256 q^{47} -15.8337 q^{49} -0.280176 q^{53} +4.77846 q^{55} +6.38101 q^{59} +12.4983i q^{61} -6.49828i q^{65} -10.0552 q^{67} +12.9966i q^{71} -6.45264i q^{73} -22.8337 q^{77} -5.11383i q^{79} +13.4396 q^{83} +(-1.77846 - 3.71982i) q^{85} +0.941367 q^{89} +31.0518i q^{91} +1.83709i q^{95} +17.1138i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{13} - 4 q^{17} - 4 q^{19} - 6 q^{25} + 12 q^{35} + 24 q^{47} - 10 q^{49} - 20 q^{53} + 12 q^{55} + 8 q^{67} - 52 q^{77} + 44 q^{83} + 6 q^{85} + 4 q^{89}+O(q^{100}) \) 6 * q - 4 * q^13 - 4 * q^17 - 4 * q^19 - 6 * q^25 + 12 * q^35 + 24 * q^47 - 10 * q^49 - 20 * q^53 + 12 * q^55 + 8 * q^67 - 52 * q^77 + 44 * q^83 + 6 * q^85 + 4 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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