LMFDB - Embedded newform 4140.2.f.c.829.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,2,Mod(829,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4140.829");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4140.f (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:	$33.0580664368$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 27x^{12} + 283x^{10} + 1441x^{8} + 3596x^{6} + 3740x^{4} + 772x^{2} + 4 $ x^14 + 27x^12 + 283x^10 + 1441x^8 + 3596x^6 + 3740x^4 + 772x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 1380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			829.10
Root			$2.47928i$ of defining polynomial
Character	$\chi$	$=$	4140.829
Dual form			4140.2.f.c.829.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.258163 + 2.22111i) q^{5} +2.14682i q^{7} +O(q^{10})$	$q+(0.258163 + 2.22111i) q^{5} +2.14682i q^{7} -4.93726 q^{11} +0.809778i q^{13} +6.88250i q^{17} +4.52399 q^{19} -1.00000i q^{23} +(-4.86670 + 1.14682i) q^{25} +4.44027 q^{29} +4.11654 q^{31} +(-4.76833 + 0.554230i) q^{35} +4.35588i q^{37} -10.6103 q^{41} -9.56695i q^{43} +8.41962i q^{47} +2.39116 q^{49} +3.13546i q^{53} +(-1.27462 - 10.9662i) q^{55} +6.26048 q^{59} -9.98788 q^{61} +(-1.79861 + 0.209055i) q^{65} +0.0193386i q^{67} -0.834347 q^{71} -1.48386i q^{73} -10.5994i q^{77} -11.9246 q^{79} +15.3247i q^{83} +(-15.2868 + 1.77681i) q^{85} -1.06455 q^{89} -1.73845 q^{91} +(1.16793 + 10.0483i) q^{95} -10.8113i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q+O(q^{10}) $ 14 * q	$ 14 q + 4 q^{19} - 6 q^{25} + 30 q^{29} + 6 q^{31} + 14 q^{35} - 46 q^{41} - 20 q^{49} - 16 q^{55} + 10 q^{59} + 64 q^{61} + 36 q^{65} - 42 q^{71} - 32 q^{79} - 42 q^{85} + 52 q^{89} + 28 q^{91} + 44 q^{95}+O(q^{100}) $ 14 * q + 4 * q^19 - 6 * q^25 + 30 * q^29 + 6 * q^31 + 14 * q^35 - 46 * q^41 - 20 * q^49 - 16 * q^55 + 10 * q^59 + 64 * q^61 + 36 * q^65 - 42 * q^71 - 32 * q^79 - 42 * q^85 + 52 * q^89 + 28 * q^91 + 44 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$2071$	$3961$
$\chi(n)$	$1$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.258163	+	2.22111i	0.115454	+	0.993313i
$5$	0.258163	+	2.22111i	0.115454	+	0.993313i
$6$		0			0
$7$			2.14682i			0.811422i	0.914001	+	0.405711i	$0.132976\pi$
$7$			2.14682i			0.811422i	−0.914001	+	0.405711i	$0.867024\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−4.93726			−1.48864			−0.744320	−	0.667823i	$-0.767226\pi$
$11$	−4.93726			−1.48864			−0.744320	+	0.667823i	$0.767226\pi$
$12$		0			0
$13$			0.809778i			0.224592i	0.993675	+	0.112296i	$0.0358205\pi$
$13$			0.809778i			0.224592i	−0.993675	+	0.112296i	$0.964180\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			6.88250i			1.66925i	0.550817	+	0.834626i	$0.314316\pi$
$17$			6.88250i			1.66925i	−0.550817	+	0.834626i	$0.685684\pi$
$18$		0			0
$19$	4.52399			1.03787			0.518937	−	0.854812i	$-0.326328\pi$
$19$	4.52399			1.03787			0.518937	+	0.854812i	$0.326328\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	1.00000i		−	0.208514i
$23$		−	1.00000i		−	0.208514i
$24$		0			0
$25$	−4.86670	+	1.14682i	−0.973341	+	0.229364i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4.44027			0.824538			0.412269	−	0.911062i	$-0.364736\pi$
$29$	4.44027			0.824538			0.412269	+	0.911062i	$0.364736\pi$
$30$		0			0
$31$	4.11654			0.739353			0.369677	−	0.929160i	$-0.379468\pi$
$31$	4.11654			0.739353			0.369677	+	0.929160i	$0.379468\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−4.76833	+	0.554230i	−0.805996	+	0.0936820i
$36$		0			0
$37$			4.35588i			0.716101i	0.933702	+	0.358051i	$0.116559\pi$
$37$			4.35588i			0.716101i	−0.933702	+	0.358051i	$0.883441\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−10.6103			−1.65706			−0.828529	−	0.559946i	$-0.810822\pi$
$41$	−10.6103			−1.65706			−0.828529	+	0.559946i	$0.810822\pi$
$42$		0			0
$43$		−	9.56695i		−	1.45895i	−0.684010	−	0.729473i	$-0.739766\pi$
$43$		−	9.56695i		−	1.45895i	0.684010	−	0.729473i	$-0.260234\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			8.41962i			1.22813i	0.789256	+	0.614064i	$0.210466\pi$
$47$			8.41962i			1.22813i	−0.789256	+	0.614064i	$0.789534\pi$
$48$		0			0
$49$	2.39116			0.341595
$50$		0			0
$51$		0			0
$52$		0			0
$53$			3.13546i			0.430689i	0.976538	+	0.215345i	$0.0690875\pi$
$53$			3.13546i			0.430689i	−0.976538	+	0.215345i	$0.930913\pi$
$54$		0			0
$55$	−1.27462	−	10.9662i	−0.171870	−	1.47869i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	6.26048			0.815045			0.407523	−	0.913195i	$-0.366393\pi$
$59$	6.26048			0.815045			0.407523	+	0.913195i	$0.366393\pi$
$60$		0			0
$61$	−9.98788			−1.27882			−0.639408	−	0.768867i	$-0.720821\pi$
$61$	−9.98788			−1.27882			−0.639408	+	0.768867i	$0.720821\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.79861	+	0.209055i	−0.223090	+	0.0259301i
$66$		0			0
$67$			0.0193386i			0.00236259i	0.999999	+	0.00118129i	$0.000376018\pi$
$67$			0.0193386i			0.00236259i	−0.999999	+	0.00118129i	$0.999624\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−0.834347			−0.0990187			−0.0495094	−	0.998774i	$-0.515766\pi$
$71$	−0.834347			−0.0990187			−0.0495094	+	0.998774i	$0.515766\pi$
$72$		0			0
$73$		−	1.48386i		−	0.173673i	−0.996223	−	0.0868365i	$-0.972324\pi$
$73$		−	1.48386i		−	0.173673i	0.996223	−	0.0868365i	$-0.0276758\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	10.5994i		−	1.20791i
$78$		0			0
$79$	−11.9246			−1.34162			−0.670811	−	0.741629i	$-0.734054\pi$
$79$	−11.9246			−1.34162			−0.670811	+	0.741629i	$0.734054\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			15.3247i			1.68211i	0.540951	+	0.841054i	$0.318064\pi$
$83$			15.3247i			1.68211i	−0.540951	+	0.841054i	$0.681936\pi$
$84$		0			0
$85$	−15.2868	+	1.77681i	−1.65809	+	0.192722i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1.06455			−0.112842			−0.0564212	−	0.998407i	$-0.517969\pi$
$89$	−1.06455			−0.112842			−0.0564212	+	0.998407i	$0.517969\pi$
$90$		0			0
$91$	−1.73845			−0.182239
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.16793	+	10.0483i	0.119827	+	1.03093i
$96$		0			0
$97$		−	10.8113i		−	1.09772i	−0.835914	−	0.548860i	$-0.815062\pi$
$97$		−	10.8113i		−	1.09772i	0.835914	−	0.548860i	$-0.184938\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−15.3221			−1.52461			−0.762303	−	0.647221i	$-0.775931\pi$
$101$	−15.3221			−1.52461			−0.762303	+	0.647221i	$0.775931\pi$
$102$		0			0
$103$		−	10.8116i		−	1.06530i	−0.846336	−	0.532649i	$-0.821197\pi$
$103$		−	10.8116i		−	1.06530i	0.846336	−	0.532649i	$-0.178803\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	14.8991i		−	1.44035i	−0.693794	−	0.720173i	$-0.744062\pi$
$107$		−	14.8991i		−	1.44035i	0.693794	−	0.720173i	$-0.255938\pi$
$108$		0			0
$109$	−13.6942			−1.31167			−0.655835	−	0.754904i	$-0.727683\pi$
$109$	−13.6942			−1.31167			−0.655835	+	0.754904i	$0.727683\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	6.78165i		−	0.637964i	−0.947761	−	0.318982i	$-0.896659\pi$
$113$		−	6.78165i		−	0.637964i	0.947761	−	0.318982i	$-0.103341\pi$
$114$		0			0
$115$	2.22111	−	0.258163i	0.207120	−	0.0240738i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−14.7755			−1.35447
$120$		0			0
$121$	13.3765			1.21605
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−3.80362	−	10.5134i	−0.340206	−	0.940351i
$126$		0			0
$127$			10.3039i			0.914319i	0.889385	+	0.457160i	$0.151133\pi$
$127$			10.3039i			0.914319i	−0.889385	+	0.457160i	$0.848867\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	6.15644			0.537891			0.268945	−	0.963155i	$-0.413325\pi$
$131$	6.15644			0.537891			0.268945	+	0.963155i	$0.413325\pi$
$132$		0			0
$133$			9.71220i			0.842154i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			1.36435i			0.116564i	0.998300	+	0.0582821i	$0.0185623\pi$
$137$			1.36435i			0.116564i	−0.998300	+	0.0582821i	$0.981438\pi$
$138$		0			0
$139$	7.33723			0.622336			0.311168	−	0.950355i	$-0.399280\pi$
$139$	7.33723			0.622336			0.311168	+	0.950355i	$0.399280\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	3.99809i		−	0.334337i
$144$		0			0
$145$	1.14631	+	9.86235i	0.0951963	+	0.819024i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−19.9729			−1.63625			−0.818123	−	0.575043i	$-0.804985\pi$
$149$	−19.9729			−1.63625			−0.818123	+	0.575043i	$0.804985\pi$
$150$		0			0
$151$	1.24302			0.101156			0.0505778	−	0.998720i	$-0.483894\pi$
$151$	1.24302			0.101156			0.0505778	+	0.998720i	$0.483894\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.06274	+	9.14332i	0.0853613	+	0.734409i
$156$		0			0
$157$			0.781547i			0.0623742i	0.999514	+	0.0311871i	$0.00992878\pi$
$157$			0.781547i			0.0623742i	−0.999514	+	0.0311871i	$0.990071\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	2.14682			0.169193
$162$		0			0
$163$		−	6.46142i		−	0.506098i	−0.967453	−	0.253049i	$-0.918567\pi$
$163$		−	6.46142i		−	0.506098i	0.967453	−	0.253049i	$-0.0814333\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			2.91455i			0.225534i	0.993621	+	0.112767i	$0.0359714\pi$
$167$			2.91455i			0.225534i	−0.993621	+	0.112767i	$0.964029\pi$
$168$		0			0
$169$	12.3443			0.949558
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	25.9868i		−	1.97574i	−0.155295	−	0.987868i	$-0.549633\pi$
$173$		−	25.9868i		−	1.97574i	0.155295	−	0.987868i	$-0.450367\pi$
$174$		0			0
$175$	−2.46202	−	10.4479i	−0.186111	−	0.789790i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	21.0777			1.57542			0.787709	−	0.616047i	$-0.211267\pi$
$179$	21.0777			1.57542			0.787709	+	0.616047i	$0.211267\pi$
$180$		0			0
$181$	0.631114			0.0469103			0.0234552	−	0.999725i	$-0.492533\pi$
$181$	0.631114			0.0469103			0.0234552	+	0.999725i	$0.492533\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−9.67490	+	1.12453i	−0.711313	+	0.0826768i
$186$		0			0
$187$		−	33.9807i		−	2.48491i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−19.9625			−1.44444			−0.722219	−	0.691664i	$-0.756878\pi$
$191$	−19.9625			−1.44444			−0.722219	+	0.691664i	$0.756878\pi$
$192$		0			0
$193$			14.1301i			1.01711i	0.861030	+	0.508554i	$0.169820\pi$
$193$			14.1301i			1.01711i	−0.861030	+	0.508554i	$0.830180\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			7.31530i			0.521193i	0.965448	+	0.260597i	$0.0839193\pi$
$197$			7.31530i			0.521193i	−0.965448	+	0.260597i	$0.916081\pi$
$198$		0			0
$199$	4.95333			0.351132			0.175566	−	0.984468i	$-0.443824\pi$
$199$	4.95333			0.351132			0.175566	+	0.984468i	$0.443824\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			9.53247i			0.669048i
$204$		0			0
$205$	−2.73920	−	23.5668i	−0.191314	−	1.64598i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−22.3361			−1.54502
$210$		0			0
$211$	6.34717			0.436957			0.218479	−	0.975842i	$-0.429891\pi$
$211$	6.34717			0.436957			0.218479	+	0.975842i	$0.429891\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	21.2493	−	2.46983i	1.44919	−	0.168441i
$216$		0			0
$217$			8.83748i			0.599927i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−5.57330			−0.374901
$222$		0			0
$223$			21.4867i			1.43886i	0.694567	+	0.719428i	$0.255596\pi$
$223$			21.4867i			1.43886i	−0.694567	+	0.719428i	$0.744404\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	15.4284i		−	1.02402i	−0.858979	−	0.512010i	$-0.828901\pi$
$227$		−	15.4284i		−	1.02402i	0.858979	−	0.512010i	$-0.171099\pi$
$228$		0			0
$229$	16.7704			1.10822			0.554109	−	0.832444i	$-0.313059\pi$
$229$	16.7704			1.10822			0.554109	+	0.832444i	$0.313059\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	17.6505i		−	1.15632i	−0.815922	−	0.578162i	$-0.803770\pi$
$233$		−	17.6505i		−	1.15632i	0.815922	−	0.578162i	$-0.196230\pi$
$234$		0			0
$235$	−18.7009	+	2.17364i	−1.21992	+	0.141792i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−4.97588			−0.321863			−0.160931	−	0.986966i	$-0.551450\pi$
$239$	−4.97588			−0.321863			−0.160931	+	0.986966i	$0.551450\pi$
$240$		0			0
$241$	−8.30385			−0.534898			−0.267449	−	0.963572i	$-0.586181\pi$
$241$	−8.30385			−0.534898			−0.267449	+	0.963572i	$0.586181\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	0.617310	+	5.31105i	0.0394385	+	0.339310i
$246$		0			0
$247$			3.66343i			0.233098i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−24.4211			−1.54145			−0.770724	−	0.637169i	$-0.780105\pi$
$251$	−24.4211			−1.54145			−0.770724	+	0.637169i	$0.780105\pi$
$252$		0			0
$253$			4.93726i			0.310403i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	17.0108i		−	1.06111i	−0.847652	−	0.530553i	$-0.821984\pi$
$257$		−	17.0108i		−	1.06111i	0.847652	−	0.530553i	$-0.178016\pi$
$258$		0			0
$259$	−9.35128			−0.581060
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	6.66989i		−	0.411283i	−0.978627	−	0.205641i	$-0.934072\pi$
$263$		−	6.66989i		−	0.411283i	0.978627	−	0.205641i	$-0.0659281\pi$
$264$		0			0
$265$	−6.96423	+	0.809461i	−0.427809	+	0.0497248i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	31.2203			1.90354			0.951769	−	0.306815i	$-0.0992634\pi$
$269$	31.2203			1.90354			0.951769	+	0.306815i	$0.0992634\pi$
$270$		0			0
$271$	6.55525			0.398203			0.199101	−	0.979979i	$-0.436198\pi$
$271$	6.55525			0.398203			0.199101	+	0.979979i	$0.436198\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	24.0282	−	5.66215i	1.44895	−	0.341440i
$276$		0			0
$277$			5.42877i			0.326183i	0.986611	+	0.163092i	$0.0521466\pi$
$277$			5.42877i			0.326183i	−0.986611	+	0.163092i	$0.947853\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−12.9964			−0.775302			−0.387651	−	0.921806i	$-0.626713\pi$
$281$	−12.9964			−0.775302			−0.387651	+	0.921806i	$0.626713\pi$
$282$		0			0
$283$			30.2102i			1.79581i	0.440188	+	0.897906i	$0.354912\pi$
$283$			30.2102i			1.79581i	−0.440188	+	0.897906i	$0.645088\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	22.7785i		−	1.34457i
$288$		0			0
$289$	−30.3688			−1.78640
$290$		0			0
$291$		0			0
$292$		0			0
$293$			9.57791i			0.559548i	0.960066	+	0.279774i	$0.0902595\pi$
$293$			9.57791i			0.559548i	−0.960066	+	0.279774i	$0.909741\pi$
$294$		0			0
$295$	1.61623	+	13.9053i	0.0941003	+	0.809595i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	0.809778			0.0468307
$300$		0			0
$301$	20.5385			1.18382
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−2.57850	−	22.1842i	−0.147645	−	1.27027i
$306$		0			0
$307$		−	16.3237i		−	0.931645i	−0.884878	−	0.465822i	$-0.845759\pi$
$307$		−	16.3237i		−	0.931645i	0.884878	−	0.465822i	$-0.154241\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−0.265582			−0.0150598			−0.00752989	−	0.999972i	$-0.502397\pi$
$311$	−0.265582			−0.0150598			−0.00752989	+	0.999972i	$0.502397\pi$
$312$		0			0
$313$			1.73093i			0.0978378i	0.998803	+	0.0489189i	$0.0155776\pi$
$313$			1.73093i			0.0978378i	−0.998803	+	0.0489189i	$0.984422\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			13.1977i			0.741256i	0.928781	+	0.370628i	$0.120857\pi$
$317$			13.1977i			0.741256i	−0.928781	+	0.370628i	$0.879143\pi$
$318$		0			0
$319$	−21.9228			−1.22744
$320$		0			0
$321$		0			0
$322$		0			0
$323$			31.1364i			1.73247i
$324$		0			0
$325$	−0.928670	−	3.94095i	−0.0515133	−	0.218605i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−18.0754			−0.996530
$330$		0			0
$331$	−9.37015			−0.515030			−0.257515	−	0.966274i	$-0.582904\pi$
$331$	−9.37015			−0.515030			−0.257515	+	0.966274i	$0.582904\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−0.0429533	+	0.00499252i	−0.00234679	+	0.000272770i
$336$		0			0
$337$			14.0469i			0.765185i	0.923917	+	0.382592i	$0.124969\pi$
$337$			14.0469i			0.765185i	−0.923917	+	0.382592i	$0.875031\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−20.3244			−1.10063
$342$		0			0
$343$			20.1611i			1.08860i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			28.9655i			1.55495i	0.628914	+	0.777475i	$0.283500\pi$
$347$			28.9655i			1.55495i	−0.628914	+	0.777475i	$0.716500\pi$
$348$		0			0
$349$	−32.3176			−1.72992			−0.864960	−	0.501840i	$-0.832656\pi$
$349$	−32.3176			−1.72992			−0.864960	+	0.501840i	$0.832656\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			2.29639i			0.122225i	0.998131	+	0.0611123i	$0.0194648\pi$
$353$			2.29639i			0.122225i	−0.998131	+	0.0611123i	$0.980535\pi$
$354$		0			0
$355$	−0.215398	−	1.85318i	−0.0114321	−	0.0983566i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−20.0321			−1.05725			−0.528627	−	0.848854i	$-0.677293\pi$
$359$	−20.0321			−1.05725			−0.528627	+	0.848854i	$0.677293\pi$
$360$		0			0
$361$	1.46650			0.0771843
$362$		0			0
$363$		0			0
$364$		0			0
$365$	3.29583	−	0.383079i	0.172512	−	0.0200512i
$366$		0			0
$367$		−	15.8562i		−	0.827686i	−0.910348	−	0.413843i	$-0.864186\pi$
$367$		−	15.8562i		−	0.827686i	0.910348	−	0.413843i	$-0.135814\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−6.73128			−0.349471
$372$		0			0
$373$			28.1922i			1.45974i	0.683588	+	0.729868i	$0.260419\pi$
$373$			28.1922i			1.45974i	−0.683588	+	0.729868i	$0.739581\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			3.59564i			0.185185i
$378$		0			0
$379$	4.83894			0.248560			0.124280	−	0.992247i	$-0.460338\pi$
$379$	4.83894			0.248560			0.124280	+	0.992247i	$0.460338\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			21.8319i			1.11556i	0.829990	+	0.557778i	$0.188346\pi$
$383$			21.8319i			1.11556i	−0.829990	+	0.557778i	$0.811654\pi$
$384$		0			0
$385$	23.5425	−	2.73638i	1.19984	−	0.139459i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	14.9774			0.759383			0.379691	−	0.925113i	$-0.376030\pi$
$389$	14.9774			0.759383			0.379691	+	0.925113i	$0.376030\pi$
$390$		0			0
$391$	6.88250			0.348063
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−3.07849	−	26.4859i	−0.154896	−	1.33265i
$396$		0			0
$397$			32.5240i			1.63233i	0.577818	+	0.816166i	$0.303905\pi$
$397$			32.5240i			1.63233i	−0.577818	+	0.816166i	$0.696095\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−10.5912			−0.528899			−0.264450	−	0.964400i	$-0.585190\pi$
$401$	−10.5912			−0.528899			−0.264450	+	0.964400i	$0.585190\pi$
$402$		0			0
$403$			3.33349i			0.166053i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	21.5061i		−	1.06602i
$408$		0			0
$409$	−34.4601			−1.70394			−0.851972	−	0.523588i	$-0.824593\pi$
$409$	−34.4601			−1.70394			−0.851972	+	0.523588i	$0.824593\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			13.4401i			0.661346i
$414$		0			0
$415$	−34.0380	+	3.95628i	−1.67086	+	0.194206i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−30.8017			−1.50476			−0.752381	−	0.658729i	$-0.771095\pi$
$419$	−30.8017			−1.50476			−0.752381	+	0.658729i	$0.771095\pi$
$420$		0			0
$421$	−5.11875			−0.249473			−0.124736	−	0.992190i	$-0.539808\pi$
$421$	−5.11875			−0.249473			−0.124736	+	0.992190i	$0.539808\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−7.89299	−	33.4951i	−0.382866	−	1.62475i
$426$		0			0
$427$		−	21.4422i		−	1.03766i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−23.0874			−1.11208			−0.556040	−	0.831156i	$-0.687680\pi$
$431$	−23.0874			−1.11208			−0.556040	+	0.831156i	$0.687680\pi$
$432$		0			0
$433$			17.0941i			0.821488i	0.911751	+	0.410744i	$0.134731\pi$
$433$			17.0941i			0.821488i	−0.911751	+	0.410744i	$0.865269\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	4.52399i		−	0.216412i
$438$		0			0
$439$	35.5924			1.69873			0.849366	−	0.527804i	$-0.176985\pi$
$439$	35.5924			1.69873			0.849366	+	0.527804i	$0.176985\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			14.4578i			0.686909i	0.939169	+	0.343454i	$0.111597\pi$
$443$			14.4578i			0.686909i	−0.939169	+	0.343454i	$0.888403\pi$
$444$		0			0
$445$	−0.274828	−	2.36449i	−0.0130281	−	0.112088i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	4.38273			0.206834			0.103417	−	0.994638i	$-0.467022\pi$
$449$	4.38273			0.206834			0.103417	+	0.994638i	$0.467022\pi$
$450$		0			0
$451$	52.3860			2.46676
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−0.448803	−	3.86129i	−0.0210402	−	0.181020i
$456$		0			0
$457$			27.3530i			1.27952i	0.768575	+	0.639759i	$0.220966\pi$
$457$			27.3530i			1.27952i	−0.768575	+	0.639759i	$0.779034\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	16.7743			0.781258			0.390629	−	0.920548i	$-0.372258\pi$
$461$	16.7743			0.781258			0.390629	+	0.920548i	$0.372258\pi$
$462$		0			0
$463$			1.09362i			0.0508249i	0.999677	+	0.0254124i	$0.00808990\pi$
$463$			1.09362i			0.0508249i	−0.999677	+	0.0254124i	$0.991910\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	27.9347i		−	1.29267i	−0.763056	−	0.646333i	$-0.776302\pi$
$467$		−	27.9347i		−	1.29267i	0.763056	−	0.646333i	$-0.223698\pi$
$468$		0			0
$469$	−0.0415165			−0.00191705
$470$		0			0
$471$		0			0
$472$		0			0
$473$			47.2345i			2.17184i
$474$		0			0
$475$	−22.0169	+	5.18821i	−1.01021	+	0.238051i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	33.2489			1.51918			0.759590	−	0.650402i	$-0.225400\pi$
$479$	33.2489			1.51918			0.759590	+	0.650402i	$0.225400\pi$
$480$		0			0
$481$	−3.52729			−0.160831
$482$		0			0
$483$		0			0
$484$		0			0
$485$	24.0131	−	2.79108i	1.09038	−	0.126736i
$486$		0			0
$487$		−	4.41792i		−	0.200195i	−0.994978	−	0.100098i	$-0.968085\pi$
$487$		−	4.41792i		−	0.200195i	0.994978	−	0.100098i	$-0.0319155\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	30.7675			1.38852			0.694259	−	0.719725i	$-0.255732\pi$
$491$	30.7675			1.38852			0.694259	+	0.719725i	$0.255732\pi$
$492$		0			0
$493$			30.5602i			1.37636i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	1.79119i		−	0.0803460i
$498$		0			0
$499$	4.11654			0.184282			0.0921409	−	0.995746i	$-0.470629\pi$
$499$	4.11654			0.184282			0.0921409	+	0.995746i	$0.470629\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	23.2852i		−	1.03823i	−0.854703	−	0.519117i	$-0.826261\pi$
$503$		−	23.2852i		−	1.03823i	0.854703	−	0.519117i	$-0.173739\pi$
$504$		0			0
$505$	−3.95560	−	34.0321i	−0.176022	−	1.51441i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	41.3086			1.83097			0.915486	−	0.402350i	$-0.131807\pi$
$509$	41.3086			1.83097			0.915486	+	0.402350i	$0.131807\pi$
$510$		0			0
$511$	3.18559			0.140922
$512$		0			0
$513$		0			0
$514$		0			0
$515$	24.0138	−	2.79115i	1.05817	−	0.122993i
$516$		0			0
$517$		−	41.5698i		−	1.82824i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	28.0113			1.22720			0.613599	−	0.789618i	$-0.289721\pi$
$521$	28.0113			1.22720			0.613599	+	0.789618i	$0.289721\pi$
$522$		0			0
$523$		−	5.14519i		−	0.224983i	−0.993653	−	0.112492i	$-0.964117\pi$
$523$		−	5.14519i		−	0.224983i	0.993653	−	0.112492i	$-0.0358832\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			28.3321i			1.23417i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	8.59203i		−	0.372162i
$534$		0			0
$535$	33.0925	−	3.84639i	1.43071	−	0.166294i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−11.8058			−0.508511
$540$		0			0
$541$	15.8177			0.680056			0.340028	−	0.940415i	$-0.389564\pi$
$541$	15.8177			0.680056			0.340028	+	0.940415i	$0.389564\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−3.53535	−	30.4165i	−0.151438	−	1.30290i
$546$		0			0
$547$		−	18.8429i		−	0.805663i	−0.915274	−	0.402832i	$-0.868026\pi$
$547$		−	18.8429i		−	0.805663i	0.915274	−	0.402832i	$-0.131974\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	20.0878			0.855767
$552$		0			0
$553$		−	25.6000i		−	1.08862i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			18.5073i			0.784178i	0.919927	+	0.392089i	$0.128247\pi$
$557$			18.5073i			0.784178i	−0.919927	+	0.392089i	$0.871753\pi$
$558$		0			0
$559$	7.74710			0.327668
$560$		0			0
$561$		0			0
$562$		0			0
$563$			30.8657i			1.30083i	0.759578	+	0.650416i	$0.225405\pi$
$563$			30.8657i			1.30083i	−0.759578	+	0.650416i	$0.774595\pi$
$564$		0			0
$565$	15.0628	−	1.75077i	0.633698	−	0.0736555i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−2.60351			−0.109145			−0.0545723	−	0.998510i	$-0.517380\pi$
$569$	−2.60351			−0.109145			−0.0545723	+	0.998510i	$0.517380\pi$
$570$		0			0
$571$	6.18614			0.258882			0.129441	−	0.991587i	$-0.458682\pi$
$571$	6.18614			0.258882			0.129441	+	0.991587i	$0.458682\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1.14682	+	4.86670i	0.0478257	+	0.202956i
$576$		0			0
$577$			11.1413i			0.463817i	0.972738	+	0.231909i	$0.0744970\pi$
$577$			11.1413i			0.463817i	−0.972738	+	0.231909i	$0.925503\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−32.8994			−1.36490
$582$		0			0
$583$		−	15.4806i		−	0.641141i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	2.10459i		−	0.0868657i	−0.999056	−	0.0434328i	$-0.986171\pi$
$587$		−	2.10459i		−	0.0868657i	0.999056	−	0.0434328i	$-0.0138295\pi$
$588$		0			0
$589$	18.6232			0.767356
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	19.2952i		−	0.792360i	−0.918173	−	0.396180i	$-0.870336\pi$
$593$		−	19.2952i		−	0.792360i	0.918173	−	0.396180i	$-0.129664\pi$
$594$		0			0
$595$	−3.81449	−	32.8181i	−0.156379	−	1.34541i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	22.8923			0.935355			0.467678	−	0.883899i	$-0.345091\pi$
$599$	22.8923			0.935355			0.467678	+	0.883899i	$0.345091\pi$
$600$		0			0
$601$	−43.8245			−1.78764			−0.893819	−	0.448428i	$-0.851984\pi$
$601$	−43.8245			−1.78764			−0.893819	+	0.448428i	$0.851984\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	3.45333	+	29.7108i	0.140398	+	1.20792i
$606$		0			0
$607$			4.78090i			0.194051i	0.995282	+	0.0970254i	$0.0309328\pi$
$607$			4.78090i			0.194051i	−0.995282	+	0.0970254i	$0.969067\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−6.81802			−0.275828
$612$		0			0
$613$		−	27.3166i		−	1.10331i	−0.834074	−	0.551653i	$-0.813997\pi$
$613$		−	27.3166i		−	1.10331i	0.834074	−	0.551653i	$-0.186003\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			36.1036i			1.45348i	0.686915	+	0.726738i	$0.258964\pi$
$617$			36.1036i			1.45348i	−0.686915	+	0.726738i	$0.741036\pi$
$618$		0			0
$619$	24.6819			0.992051			0.496025	−	0.868308i	$-0.334792\pi$
$619$	24.6819			0.992051			0.496025	+	0.868308i	$0.334792\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	2.28540i		−	0.0915628i
$624$		0			0
$625$	22.3696	−	11.1625i	0.894784	−	0.446499i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−29.9793			−1.19535
$630$		0			0
$631$	−28.7083			−1.14286			−0.571430	−	0.820651i	$-0.693611\pi$
$631$	−28.7083			−1.14286			−0.571430	+	0.820651i	$0.693611\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−22.8860	+	2.66008i	−0.908205	+	0.105562i
$636$		0			0
$637$			1.93631i			0.0767195i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−17.3375			−0.684791			−0.342396	−	0.939556i	$-0.611238\pi$
$641$	−17.3375			−0.684791			−0.342396	+	0.939556i	$0.611238\pi$
$642$		0			0
$643$			31.8678i			1.25674i	0.777913	+	0.628372i	$0.216278\pi$
$643$			31.8678i			1.25674i	−0.777913	+	0.628372i	$0.783722\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	6.12123i		−	0.240650i	−0.992735	−	0.120325i	$-0.961606\pi$
$647$		−	6.12123i		−	0.240650i	0.992735	−	0.120325i	$-0.0383937\pi$
$648$		0			0
$649$	−30.9096			−1.21331
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	11.7049i		−	0.458050i	−0.973421	−	0.229025i	$-0.926446\pi$
$653$		−	11.7049i		−	0.458050i	0.973421	−	0.229025i	$-0.0735537\pi$
$654$		0			0
$655$	1.58937	+	13.6742i	0.0621017	+	0.534294i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	45.1416			1.75847			0.879234	−	0.476390i	$-0.158055\pi$
$659$	45.1416			1.75847			0.879234	+	0.476390i	$0.158055\pi$
$660$		0			0
$661$	15.3302			0.596276			0.298138	−	0.954523i	$-0.403634\pi$
$661$	15.3302			0.596276			0.298138	+	0.954523i	$0.403634\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−21.5719	+	2.50733i	−0.836523	+	0.0972302i
$666$		0			0
$667$		−	4.44027i		−	0.171928i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	49.3128			1.90370
$672$		0			0
$673$		−	9.66306i		−	0.372484i	−0.982504	−	0.186242i	$-0.940369\pi$
$673$		−	9.66306i		−	0.372484i	0.982504	−	0.186242i	$-0.0596308\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			6.46075i			0.248307i	0.992263	+	0.124153i	$0.0396215\pi$
$677$			6.46075i			0.248307i	−0.992263	+	0.124153i	$0.960379\pi$
$678$		0			0
$679$	23.2099			0.890714
$680$		0			0
$681$		0			0
$682$		0			0
$683$			37.1528i			1.42161i	0.703389	+	0.710806i	$0.251669\pi$
$683$			37.1528i			1.42161i	−0.703389	+	0.710806i	$0.748331\pi$
$684$		0			0
$685$	−3.03038	+	0.352225i	−0.115785	+	0.0134578i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−2.53903			−0.0967294
$690$		0			0
$691$	−13.0169			−0.495185			−0.247593	−	0.968864i	$-0.579639\pi$
$691$	−13.0169			−0.495185			−0.247593	+	0.968864i	$0.579639\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1.89420	+	16.2968i	0.0718512	+	0.618174i
$696$		0			0
$697$		−	73.0257i		−	2.76605i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−34.9650			−1.32061			−0.660305	−	0.750998i	$-0.729573\pi$
$701$	−34.9650			−1.32061			−0.660305	+	0.750998i	$0.729573\pi$
$702$		0			0
$703$			19.7059i			0.743224i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	32.8938i		−	1.23710i
$708$		0			0
$709$	−45.6647			−1.71497			−0.857487	−	0.514505i	$-0.827976\pi$
$709$	−45.6647			−1.71497			−0.857487	+	0.514505i	$0.827976\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	4.11654i		−	0.154166i
$714$		0			0
$715$	8.88021	−	1.03216i	0.332101	−	0.0386005i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	6.08688			0.227002			0.113501	−	0.993538i	$-0.463793\pi$
$719$	6.08688			0.227002			0.113501	+	0.993538i	$0.463793\pi$
$720$		0			0
$721$	23.2105			0.864406
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−21.6095	+	5.09219i	−0.802556	+	0.189119i
$726$		0			0
$727$			10.4501i			0.387574i	0.981044	+	0.193787i	$0.0620771\pi$
$727$			10.4501i			0.387574i	−0.981044	+	0.193787i	$0.937923\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	65.8445			2.43535
$732$		0			0
$733$			42.4009i			1.56611i	0.621952	+	0.783056i	$0.286340\pi$
$733$			42.4009i			1.56611i	−0.621952	+	0.783056i	$0.713660\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	0.0954797i		−	0.00351704i
$738$		0			0
$739$	44.8917			1.65137			0.825684	−	0.564133i	$-0.190790\pi$
$739$	44.8917			1.65137			0.825684	+	0.564133i	$0.190790\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	24.6557i		−	0.904529i	−0.891884	−	0.452264i	$-0.850616\pi$
$743$		−	24.6557i		−	0.904529i	0.891884	−	0.452264i	$-0.149384\pi$
$744$		0			0
$745$	−5.15628	−	44.3622i	−0.188911	−	1.62530i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	31.9856			1.16873
$750$		0			0
$751$	28.5628			1.04227			0.521135	−	0.853474i	$-0.325509\pi$
$751$	28.5628			1.04227			0.521135	+	0.853474i	$0.325509\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	0.320902	+	2.76089i	0.0116788	+	0.100479i
$756$		0			0
$757$			20.4821i			0.744435i	0.928145	+	0.372218i	$0.121402\pi$
$757$			20.4821i			0.744435i	−0.928145	+	0.372218i	$0.878598\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−27.3485			−0.991383			−0.495692	−	0.868499i	$-0.665085\pi$
$761$	−27.3485			−0.991383			−0.495692	+	0.868499i	$0.665085\pi$
$762$		0			0
$763$		−	29.3991i		−	1.06432i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			5.06960i			0.183053i
$768$		0			0
$769$	51.3059			1.85014			0.925068	−	0.379800i	$-0.124007\pi$
$769$	51.3059			1.85014			0.925068	+	0.379800i	$0.124007\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	4.98839i		−	0.179420i	−0.995968	−	0.0897099i	$-0.971406\pi$
$773$		−	4.98839i		−	0.179420i	0.995968	−	0.0897099i	$-0.0285940\pi$
$774$		0			0
$775$	−20.0340	+	4.72094i	−0.719643	+	0.169581i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−48.0011			−1.71982
$780$		0			0
$781$	4.11939			0.147403
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1.73591	+	0.201767i	−0.0619571	+	0.00720136i
$786$		0			0
$787$		−	26.7729i		−	0.954352i	−0.878808	−	0.477176i	$-0.841660\pi$
$787$		−	26.7729i		−	0.954352i	0.878808	−	0.477176i	$-0.158340\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	14.5590			0.517658
$792$		0			0
$793$		−	8.08797i		−	0.287212i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			4.59170i			0.162646i	0.996688	+	0.0813232i	$0.0259146\pi$
$797$			4.59170i			0.162646i	−0.996688	+	0.0813232i	$0.974085\pi$
$798$		0			0
$799$	−57.9480			−2.05005
$800$		0			0
$801$		0			0
$802$		0			0
$803$			7.32621i			0.258536i
$804$		0			0
$805$	0.554230	+	4.76833i	0.0195340	+	0.168062i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	15.0503			0.529141			0.264571	−	0.964366i	$-0.414770\pi$
$809$	15.0503			0.529141			0.264571	+	0.964366i	$0.414770\pi$
$810$		0			0
$811$	45.7341			1.60594			0.802970	−	0.596019i	$-0.203252\pi$
$811$	45.7341			1.60594			0.802970	+	0.596019i	$0.203252\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	14.3516	−	1.66810i	0.502713	−	0.0584311i
$816$		0			0
$817$		−	43.2808i		−	1.51420i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	4.70052			0.164049			0.0820246	−	0.996630i	$-0.473861\pi$
$821$	4.70052			0.164049			0.0820246	+	0.996630i	$0.473861\pi$
$822$		0			0
$823$			52.8036i			1.84062i	0.391193	+	0.920309i	$0.372062\pi$
$823$			52.8036i			1.84062i	−0.391193	+	0.920309i	$0.627938\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			13.9187i			0.484000i	0.970276	+	0.242000i	$0.0778034\pi$
$827$			13.9187i			0.484000i	−0.970276	+	0.242000i	$0.922197\pi$
$828$		0			0
$829$	11.0940			0.385310			0.192655	−	0.981267i	$-0.438290\pi$
$829$	11.0940			0.385310			0.192655	+	0.981267i	$0.438290\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			16.4572i			0.570208i
$834$		0			0
$835$	−6.47354	+	0.752429i	−0.224026	+	0.0260389i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−13.1144			−0.452760			−0.226380	−	0.974039i	$-0.572689\pi$
$839$	−13.1144			−0.452760			−0.226380	+	0.974039i	$0.572689\pi$
$840$		0			0
$841$	−9.28398			−0.320137
$842$		0			0
$843$		0			0
$844$		0			0
$845$	3.18683	+	27.4180i	0.109630	+	0.943209i
$846$		0			0
$847$			28.7170i			0.986728i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	4.35588			0.149317
$852$		0			0
$853$			7.46994i			0.255766i	0.991789	+	0.127883i	$0.0408182\pi$
$853$			7.46994i			0.255766i	−0.991789	+	0.127883i	$0.959182\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			22.3864i			0.764705i	0.924017	+	0.382353i	$0.124886\pi$
$857$			22.3864i			0.764705i	−0.924017	+	0.382353i	$0.875114\pi$
$858$		0			0
$859$	44.3843			1.51437			0.757187	−	0.653199i	$-0.226573\pi$
$859$	44.3843			1.51437			0.757187	+	0.653199i	$0.226573\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			18.9699i			0.645743i	0.946443	+	0.322871i	$0.104648\pi$
$863$			18.9699i			0.645743i	−0.946443	+	0.322871i	$0.895352\pi$
$864$		0			0
$865$	57.7196	−	6.70882i	1.96252	−	0.228107i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	58.8748			1.99719
$870$		0			0
$871$	−0.0156600			−0.000530618
$872$		0			0
$873$		0			0
$874$		0			0
$875$	22.5705	−	8.16570i	0.763021	−	0.276051i
$876$		0			0
$877$		−	11.3079i		−	0.381840i	−0.981606	−	0.190920i	$-0.938853\pi$
$877$		−	11.3079i		−	0.381840i	0.981606	−	0.190920i	$-0.0611471\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−32.4624			−1.09369			−0.546844	−	0.837235i	$-0.684171\pi$
$881$	−32.4624			−1.09369			−0.546844	+	0.837235i	$0.684171\pi$
$882$		0			0
$883$		−	13.9682i		−	0.470067i	−0.971987	−	0.235033i	$-0.924480\pi$
$883$		−	13.9682i		−	0.470067i	0.971987	−	0.235033i	$-0.0755200\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	13.5685i		−	0.455585i	−0.973710	−	0.227793i	$-0.926849\pi$
$887$		−	13.5685i		−	0.455585i	0.973710	−	0.227793i	$-0.0731509\pi$
$888$		0			0
$889$	−22.1205			−0.741899
$890$		0			0
$891$		0			0
$892$		0			0
$893$			38.0903i			1.27464i
$894$		0			0
$895$	5.44148	+	46.8159i	0.181889	+	1.56488i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	18.2786			0.609625
$900$		0			0
$901$	−21.5798			−0.718929
$902$		0			0
$903$		0			0
$904$		0			0
$905$	0.162930	+	1.40178i	0.00541599	+	0.0465966i
$906$		0			0
$907$			4.61972i			0.153395i	0.997054	+	0.0766977i	$0.0244376\pi$
$907$			4.61972i			0.153395i	−0.997054	+	0.0766977i	$0.975562\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	9.27716			0.307366			0.153683	−	0.988120i	$-0.450887\pi$
$911$	9.27716			0.307366			0.153683	+	0.988120i	$0.450887\pi$
$912$		0			0
$913$		−	75.6622i		−	2.50405i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			13.2168i			0.436456i
$918$		0			0
$919$	−18.6035			−0.613674			−0.306837	−	0.951762i	$-0.599271\pi$
$919$	−18.6035			−0.613674			−0.306837	+	0.951762i	$0.599271\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	0.675636i		−	0.0222388i
$924$		0			0
$925$	−4.99541	−	21.1988i	−0.164248	−	0.697011i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−31.7560			−1.04188			−0.520940	−	0.853594i	$-0.674418\pi$
$929$	−31.7560			−1.04188			−0.520940	+	0.853594i	$0.674418\pi$
$930$		0			0
$931$	10.8176			0.354533
$932$		0			0
$933$		0			0
$934$		0			0
$935$	75.4750	−	8.77257i	2.46830	−	0.286894i
$936$		0			0
$937$		−	28.3410i		−	0.925859i	−0.886395	−	0.462929i	$-0.846798\pi$
$937$		−	28.3410i		−	0.925859i	0.886395	−	0.462929i	$-0.153202\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	6.22197			0.202831			0.101415	−	0.994844i	$-0.467663\pi$
$941$	6.22197			0.202831			0.101415	+	0.994844i	$0.467663\pi$
$942$		0			0
$943$			10.6103i			0.345520i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			23.5290i			0.764588i	0.924041	+	0.382294i	$0.124866\pi$
$947$			23.5290i			0.764588i	−0.924041	+	0.382294i	$0.875134\pi$
$948$		0			0
$949$	1.20160			0.0390056
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	4.57509i		−	0.148202i	−0.997251	−	0.0741009i	$-0.976391\pi$
$953$		−	4.57509i		−	0.148202i	0.997251	−	0.0741009i	$-0.0236087\pi$
$954$		0			0
$955$	−5.15359	−	44.3391i	−0.166766	−	1.43478i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−2.92901			−0.0945827
$960$		0			0
$961$	−14.0541			−0.453357
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−31.3846	+	3.64788i	−1.01031	+	0.117429i
$966$		0			0
$967$		−	7.67627i		−	0.246852i	−0.992354	−	0.123426i	$-0.960612\pi$
$967$		−	7.67627i		−	0.246852i	0.992354	−	0.123426i	$-0.0393882\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	54.2752			1.74178			0.870888	−	0.491482i	$-0.163545\pi$
$971$	54.2752			1.74178			0.870888	+	0.491482i	$0.163545\pi$
$972$		0			0
$973$			15.7517i			0.504977i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	41.5052i		−	1.32787i	−0.747791	−	0.663934i	$-0.768885\pi$
$977$		−	41.5052i		−	1.32787i	0.747791	−	0.663934i	$-0.231115\pi$
$978$		0			0
$979$	5.25597			0.167982
$980$		0			0
$981$		0			0
$982$		0			0
$983$			17.9133i			0.571346i	0.958327	+	0.285673i	$0.0922171\pi$
$983$			17.9133i			0.571346i	−0.958327	+	0.285673i	$0.907783\pi$
$984$		0			0
$985$	−16.2481	+	1.88854i	−0.517708	+	0.0601739i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−9.56695			−0.304211
$990$		0			0
$991$	4.24270			0.134774			0.0673869	−	0.997727i	$-0.478534\pi$
$991$	4.24270			0.134774			0.0673869	+	0.997727i	$0.478534\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1.27877	+	11.0019i	0.0405396	+	0.348784i
$996$		0			0
$997$		−	14.2633i		−	0.451722i	−0.974160	−	0.225861i	$-0.927481\pi$
$997$		−	14.2633i		−	0.451722i	0.974160	−	0.225861i	$-0.0725195\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.2.f.c.829.10		14
3.2	odd	2		1380.2.f.b.829.3	✓	14
5.4	even	2	inner	4140.2.f.c.829.9		14
15.2	even	4		6900.2.a.bc.1.2		7
15.8	even	4		6900.2.a.bd.1.6		7
15.14	odd	2		1380.2.f.b.829.10	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.2.f.b.829.3	✓	14	3.2	odd	2
1380.2.f.b.829.10	yes	14	15.14	odd	2
4140.2.f.c.829.9		14	5.4	even	2	inner
4140.2.f.c.829.10		14	1.1	even	1	trivial
6900.2.a.bc.1.2		7	15.2	even	4
6900.2.a.bd.1.6		7	15.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4140.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.258163 + 2.22111i) q^{5} +2.14682i q^{7} +O(q^{10})\)	\(q+(0.258163 + 2.22111i) q^{5} +2.14682i q^{7} -4.93726 q^{11} +0.809778i q^{13} +6.88250i q^{17} +4.52399 q^{19} -1.00000i q^{23} +(-4.86670 + 1.14682i) q^{25} +4.44027 q^{29} +4.11654 q^{31} +(-4.76833 + 0.554230i) q^{35} +4.35588i q^{37} -10.6103 q^{41} -9.56695i q^{43} +8.41962i q^{47} +2.39116 q^{49} +3.13546i q^{53} +(-1.27462 - 10.9662i) q^{55} +6.26048 q^{59} -9.98788 q^{61} +(-1.79861 + 0.209055i) q^{65} +0.0193386i q^{67} -0.834347 q^{71} -1.48386i q^{73} -10.5994i q^{77} -11.9246 q^{79} +15.3247i q^{83} +(-15.2868 + 1.77681i) q^{85} -1.06455 q^{89} -1.73845 q^{91} +(1.16793 + 10.0483i) q^{95} -10.8113i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q+O(q^{10}) \) 14 * q	\( 14 q + 4 q^{19} - 6 q^{25} + 30 q^{29} + 6 q^{31} + 14 q^{35} - 46 q^{41} - 20 q^{49} - 16 q^{55} + 10 q^{59} + 64 q^{61} + 36 q^{65} - 42 q^{71} - 32 q^{79} - 42 q^{85} + 52 q^{89} + 28 q^{91} + 44 q^{95}+O(q^{100}) \) 14 * q + 4 * q^19 - 6 * q^25 + 30 * q^29 + 6 * q^31 + 14 * q^35 - 46 * q^41 - 20 * q^49 - 16 * q^55 + 10 * q^59 + 64 * q^61 + 36 * q^65 - 42 * q^71 - 32 * q^79 - 42 * q^85 + 52 * q^89 + 28 * q^91 + 44 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more