LMFDB - Embedded newform 6012.2.h.a.3005.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6012,2,Mod(3005,6012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6012, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6012.3005");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6012 = 2^{2} \cdot 3^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6012.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:	$48.0060616952$
Analytic rank:	$0$
Dimension:	$56$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3005.20
Character	$\chi$	$=$	6012.3005
Dual form			6012.2.h.a.3005.19

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.15018 q^{5} -0.123775 q^{7} +O(q^{10})$	$q-1.15018 q^{5} -0.123775 q^{7} -0.794122i q^{11} -0.814543i q^{13} -4.10883 q^{17} +2.35843 q^{19} -4.07130 q^{23} -3.67709 q^{25} -4.97503i q^{29} +8.23579 q^{31} +0.142364 q^{35} -4.60513i q^{37} -1.20278 q^{41} +10.0697i q^{43} +8.74772i q^{47} -6.98468 q^{49} +6.14215 q^{53} +0.913381i q^{55} +5.67351 q^{59} +5.76977 q^{61} +0.936870i q^{65} -1.63628i q^{67} +6.18282 q^{71} +10.9470i q^{73} +0.0982927i q^{77} -7.10481i q^{79} -11.9780 q^{83} +4.72589 q^{85} +2.78947i q^{89} +0.100820i q^{91} -2.71262 q^{95} +13.6099 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q+O(q^{10}) $ 56 * q	$ 56 q + 8 q^{19} + 64 q^{25} - 8 q^{31} + 56 q^{49} - 8 q^{61} + 32 q^{85} - 48 q^{97}+O(q^{100}) $ 56 * q + 8 * q^19 + 64 * q^25 - 8 * q^31 + 56 * q^49 - 8 * q^61 + 32 * q^85 - 48 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6012\mathbb{Z}\right)^\times$.

$n$	$3007$	$3341$	$4681$
$\chi(n)$	$1$	$-1$	$-1$

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			0
$3$		0			0
$4$		0			0
$5$	−1.15018			−0.514375			−0.257188	−	0.966361i	$-0.582796\pi$
$5$	−1.15018			−0.514375			−0.257188	+	0.966361i	$0.582796\pi$
$6$		0			0
$7$	−0.123775			−0.0467827			−0.0233914	−	0.999726i	$-0.507446\pi$
$7$	−0.123775			−0.0467827			−0.0233914	+	0.999726i	$0.507446\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	0.794122i		−	0.239437i	−0.992808	−	0.119718i	$-0.961801\pi$
$11$		−	0.794122i		−	0.239437i	0.992808	−	0.119718i	$-0.0381991\pi$
$12$		0			0
$13$		−	0.814543i		−	0.225914i	−0.993600	−	0.112957i	$-0.963968\pi$
$13$		−	0.814543i		−	0.225914i	0.993600	−	0.112957i	$-0.0360322\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.10883			−0.996538			−0.498269	−	0.867022i	$-0.666031\pi$
$17$	−4.10883			−0.996538			−0.498269	+	0.867022i	$0.666031\pi$
$18$		0			0
$19$	2.35843			0.541062			0.270531	−	0.962711i	$-0.412801\pi$
$19$	2.35843			0.541062			0.270531	+	0.962711i	$0.412801\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.07130			−0.848925			−0.424463	−	0.905445i	$-0.639537\pi$
$23$	−4.07130			−0.848925			−0.424463	+	0.905445i	$0.639537\pi$
$24$		0			0
$25$	−3.67709			−0.735418
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	4.97503i		−	0.923840i	−0.886922	−	0.461920i	$-0.847161\pi$
$29$		−	4.97503i		−	0.923840i	0.886922	−	0.461920i	$-0.152839\pi$
$30$		0			0
$31$	8.23579			1.47919			0.739596	−	0.673051i	$-0.235017\pi$
$31$	8.23579			1.47919			0.739596	+	0.673051i	$0.235017\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.142364			0.0240639
$36$		0			0
$37$		−	4.60513i		−	0.757079i	−0.925585	−	0.378539i	$-0.876426\pi$
$37$		−	4.60513i		−	0.757079i	0.925585	−	0.378539i	$-0.123574\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−1.20278			−0.187842			−0.0939210	−	0.995580i	$-0.529940\pi$
$41$	−1.20278			−0.187842			−0.0939210	+	0.995580i	$0.529940\pi$
$42$		0			0
$43$			10.0697i			1.53561i	0.640681	+	0.767807i	$0.278652\pi$
$43$			10.0697i			1.53561i	−0.640681	+	0.767807i	$0.721348\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			8.74772i			1.27599i	0.770042	+	0.637993i	$0.220235\pi$
$47$			8.74772i			1.27599i	−0.770042	+	0.637993i	$0.779765\pi$
$48$		0			0
$49$	−6.98468			−0.997811
$50$		0			0
$51$		0			0
$52$		0			0
$53$	6.14215			0.843690			0.421845	−	0.906668i	$-0.361383\pi$
$53$	6.14215			0.843690			0.421845	+	0.906668i	$0.361383\pi$
$54$		0			0
$55$			0.913381i			0.123160i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	5.67351			0.738628			0.369314	−	0.929305i	$-0.379593\pi$
$59$	5.67351			0.738628			0.369314	+	0.929305i	$0.379593\pi$
$60$		0			0
$61$	5.76977			0.738743			0.369371	−	0.929282i	$-0.379573\pi$
$61$	5.76977			0.738743			0.369371	+	0.929282i	$0.379573\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			0.936870i			0.116204i
$66$		0			0
$67$		−	1.63628i		−	0.199903i	−0.994992	−	0.0999515i	$-0.968131\pi$
$67$		−	1.63628i		−	0.199903i	0.994992	−	0.0999515i	$-0.0318688\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	6.18282			0.733766			0.366883	−	0.930267i	$-0.380425\pi$
$71$	6.18282			0.733766			0.366883	+	0.930267i	$0.380425\pi$
$72$		0			0
$73$			10.9470i			1.28124i	0.767856	+	0.640622i	$0.221324\pi$
$73$			10.9470i			1.28124i	−0.767856	+	0.640622i	$0.778676\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			0.0982927i			0.0112015i
$78$		0			0
$79$		−	7.10481i		−	0.799354i	−0.916656	−	0.399677i	$-0.869122\pi$
$79$		−	7.10481i		−	0.799354i	0.916656	−	0.399677i	$-0.130878\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−11.9780			−1.31476			−0.657378	−	0.753561i	$-0.728335\pi$
$83$	−11.9780			−1.31476			−0.657378	+	0.753561i	$0.728335\pi$
$84$		0			0
$85$	4.72589			0.512595
$86$		0			0
$87$		0			0
$88$		0			0
$89$			2.78947i			0.295683i	0.989011	+	0.147841i	$0.0472325\pi$
$89$			2.78947i			0.295683i	−0.989011	+	0.147841i	$0.952767\pi$
$90$		0			0
$91$			0.100820i			0.0105689i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.71262			−0.278309
$96$		0			0
$97$	13.6099			1.38188			0.690938	−	0.722914i	$-0.257198\pi$
$97$	13.6099			1.38188			0.690938	+	0.722914i	$0.257198\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−16.8665			−1.67828			−0.839141	−	0.543913i	$-0.816942\pi$
$101$	−16.8665			−1.67828			−0.839141	+	0.543913i	$0.816942\pi$
$102$		0			0
$103$			7.67764i			0.756500i	0.925703	+	0.378250i	$0.123474\pi$
$103$			7.67764i			0.756500i	−0.925703	+	0.378250i	$0.876526\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			5.59639i			0.541023i	0.962717	+	0.270512i	$0.0871928\pi$
$107$			5.59639i			0.541023i	−0.962717	+	0.270512i	$0.912807\pi$
$108$		0			0
$109$			7.09482i			0.679561i	0.940505	+	0.339780i	$0.110353\pi$
$109$			7.09482i			0.679561i	−0.940505	+	0.339780i	$0.889647\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−11.5035			−1.08216			−0.541080	−	0.840971i	$-0.681984\pi$
$113$	−11.5035			−1.08216			−0.541080	+	0.840971i	$0.681984\pi$
$114$		0			0
$115$	4.68272			0.436666
$116$		0			0
$117$		0			0
$118$		0			0
$119$	0.508573			0.0466208
$120$		0			0
$121$	10.3694			0.942670
$122$		0			0
$123$		0			0
$124$		0			0
$125$	9.98020			0.892656
$126$		0			0
$127$	−20.7013			−1.83694			−0.918472	−	0.395485i	$-0.870577\pi$
$127$	−20.7013			−1.83694			−0.918472	+	0.395485i	$0.870577\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−0.790714			−0.0690850			−0.0345425	−	0.999403i	$-0.510997\pi$
$131$	−0.790714			−0.0690850			−0.0345425	+	0.999403i	$0.510997\pi$
$132$		0			0
$133$	−0.291916			−0.0253123
$134$		0			0
$135$		0			0
$136$		0			0
$137$			8.47688i			0.724229i	0.932134	+	0.362115i	$0.117945\pi$
$137$			8.47688i			0.724229i	−0.932134	+	0.362115i	$0.882055\pi$
$138$		0			0
$139$			4.17420i			0.354051i	0.984206	+	0.177026i	$0.0566476\pi$
$139$			4.17420i			0.354051i	−0.984206	+	0.177026i	$0.943352\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−0.646846			−0.0540920
$144$		0			0
$145$			5.72217i			0.475200i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1.51667			−0.124250			−0.0621252	−	0.998068i	$-0.519788\pi$
$149$	−1.51667			−0.124250			−0.0621252	+	0.998068i	$0.519788\pi$
$150$		0			0
$151$		−	4.54670i		−	0.370005i	−0.982738	−	0.185002i	$-0.940771\pi$
$151$		−	4.54670i		−	0.370005i	0.982738	−	0.185002i	$-0.0592293\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−9.47263			−0.760860
$156$		0			0
$157$	14.9852			1.19595			0.597974	−	0.801515i	$-0.295972\pi$
$157$	14.9852			1.19595			0.597974	+	0.801515i	$0.295972\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	0.503927			0.0397150
$162$		0			0
$163$			5.16688i			0.404701i	0.979313	+	0.202351i	$0.0648580\pi$
$163$			5.16688i			0.404701i	−0.979313	+	0.202351i	$0.935142\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	9.57244	+	8.68150i	0.740738	+	0.671794i
$167$	9.57244	+	8.68150i	0.740738	+	0.671794i
$168$		0			0
$169$	12.3365			0.948963
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	2.57331i		−	0.195645i	−0.995204	−	0.0978226i	$-0.968812\pi$
$173$		−	2.57331i		−	0.195645i	0.995204	−	0.0978226i	$-0.0311878\pi$
$174$		0			0
$175$	0.455133			0.0344049
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	20.5854i		−	1.53863i	−0.638872	−	0.769313i	$-0.720599\pi$
$179$		−	20.5854i		−	1.53863i	0.638872	−	0.769313i	$-0.279401\pi$
$180$		0			0
$181$	14.6593			1.08962			0.544810	−	0.838559i	$-0.316602\pi$
$181$	14.6593			1.08962			0.544810	+	0.838559i	$0.316602\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			5.29672i			0.389423i
$186$		0			0
$187$			3.26291i			0.238608i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			8.56029i			0.619401i	0.950834	+	0.309700i	$0.100229\pi$
$191$			8.56029i			0.619401i	−0.950834	+	0.309700i	$0.899771\pi$
$192$		0			0
$193$			15.4273i			1.11048i	0.831690	+	0.555241i	$0.187374\pi$
$193$			15.4273i			1.11048i	−0.831690	+	0.555241i	$0.812626\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−11.5232			−0.820992			−0.410496	−	0.911862i	$-0.634644\pi$
$197$	−11.5232			−0.820992			−0.410496	+	0.911862i	$0.634644\pi$
$198$		0			0
$199$	−2.53256			−0.179529			−0.0897643	−	0.995963i	$-0.528611\pi$
$199$	−2.53256			−0.179529			−0.0897643	+	0.995963i	$0.528611\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			0.615786i			0.0432197i
$204$		0			0
$205$	1.38341			0.0966213
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	1.87288i		−	0.129550i
$210$		0			0
$211$	−0.916720			−0.0631096			−0.0315548	−	0.999502i	$-0.510046\pi$
$211$	−0.916720			−0.0631096			−0.0315548	+	0.999502i	$0.510046\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	11.5819i		−	0.789882i
$216$		0			0
$217$	−1.01939			−0.0692006
$218$		0			0
$219$		0			0
$220$		0			0
$221$			3.34682i			0.225132i
$222$		0			0
$223$	8.01407			0.536662			0.268331	−	0.963327i	$-0.413528\pi$
$223$	8.01407			0.536662			0.268331	+	0.963327i	$0.413528\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	5.99466			0.397879			0.198940	−	0.980012i	$-0.436250\pi$
$227$	5.99466			0.397879			0.198940	+	0.980012i	$0.436250\pi$
$228$		0			0
$229$	−0.823194			−0.0543982			−0.0271991	−	0.999630i	$-0.508659\pi$
$229$	−0.823194			−0.0543982			−0.0271991	+	0.999630i	$0.508659\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			12.4559i			0.816015i	0.912979	+	0.408008i	$0.133776\pi$
$233$			12.4559i			0.816015i	−0.912979	+	0.408008i	$0.866224\pi$
$234$		0			0
$235$		−	10.0614i		−	0.656336i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			4.60828i			0.298085i	0.988831	+	0.149042i	$0.0476190\pi$
$239$			4.60828i			0.298085i	−0.988831	+	0.149042i	$0.952381\pi$
$240$		0			0
$241$			30.8197i			1.98527i	0.121137	+	0.992636i	$0.461346\pi$
$241$			30.8197i			1.98527i	−0.121137	+	0.992636i	$0.538654\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	8.03363			0.513250
$246$		0			0
$247$		−	1.92105i		−	0.122233i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			17.6086i			1.11145i	0.831367	+	0.555724i	$0.187559\pi$
$251$			17.6086i			1.11145i	−0.831367	+	0.555724i	$0.812441\pi$
$252$		0			0
$253$			3.23311i			0.203264i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−25.2238			−1.57342			−0.786710	−	0.617323i	$-0.788217\pi$
$257$	−25.2238			−1.57342			−0.786710	+	0.617323i	$0.788217\pi$
$258$		0			0
$259$			0.570002i			0.0354182i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			22.8911i			1.41153i	0.708448	+	0.705763i	$0.249396\pi$
$263$			22.8911i			1.41153i	−0.708448	+	0.705763i	$0.750604\pi$
$264$		0			0
$265$	−7.06457			−0.433973
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−17.1408			−1.04509			−0.522545	−	0.852612i	$-0.675017\pi$
$269$	−17.1408			−1.04509			−0.522545	+	0.852612i	$0.675017\pi$
$270$		0			0
$271$			15.2752i			0.927900i	0.885861	+	0.463950i	$0.153568\pi$
$271$			15.2752i			0.927900i	−0.885861	+	0.463950i	$0.846432\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			2.92006i			0.176086i
$276$		0			0
$277$			27.2829i			1.63927i	0.572888	+	0.819634i	$0.305823\pi$
$277$			27.2829i			1.63927i	−0.572888	+	0.819634i	$0.694177\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			9.76299i			0.582411i	0.956660	+	0.291206i	$0.0940564\pi$
$281$			9.76299i			0.582411i	−0.956660	+	0.291206i	$0.905944\pi$
$282$		0			0
$283$	18.7180			1.11267			0.556334	−	0.830959i	$-0.312207\pi$
$283$	18.7180			1.11267			0.556334	+	0.830959i	$0.312207\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	0.148874			0.00878776
$288$		0			0
$289$	−0.117498			−0.00691166
$290$		0			0
$291$		0			0
$292$		0			0
$293$			11.3088i			0.660667i	0.943864	+	0.330334i	$0.107161\pi$
$293$			11.3088i			0.660667i	−0.943864	+	0.330334i	$0.892839\pi$
$294$		0			0
$295$	−6.52555			−0.379932
$296$		0			0
$297$		0			0
$298$		0			0
$299$			3.31625i			0.191784i
$300$		0			0
$301$		−	1.24638i		−	0.0718402i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−6.63626			−0.379991
$306$		0			0
$307$			22.2817i			1.27169i	0.771818	+	0.635843i	$0.219348\pi$
$307$			22.2817i			1.27169i	−0.771818	+	0.635843i	$0.780652\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	22.5078i		−	1.27630i	−0.769912	−	0.638150i	$-0.779700\pi$
$311$		−	22.5078i		−	1.27630i	0.769912	−	0.638150i	$-0.220300\pi$
$312$		0			0
$313$		−	28.3219i		−	1.60085i	−0.599434	−	0.800424i	$-0.704608\pi$
$313$		−	28.3219i		−	1.60085i	0.599434	−	0.800424i	$-0.295392\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			31.3547i			1.76106i	0.473994	+	0.880528i	$0.342812\pi$
$317$			31.3547i			1.76106i	−0.473994	+	0.880528i	$0.657188\pi$
$318$		0			0
$319$	−3.95078			−0.221201
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−9.69041			−0.539189
$324$		0			0
$325$			2.99515i			0.166141i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	1.08275i		−	0.0596941i
$330$		0			0
$331$			0.436333i			0.0239830i	0.999928	+	0.0119915i	$0.00381711\pi$
$331$			0.436333i			0.0239830i	−0.999928	+	0.0119915i	$0.996183\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			1.88201i			0.102825i
$336$		0			0
$337$	−6.04105			−0.329077			−0.164539	−	0.986371i	$-0.552614\pi$
$337$	−6.04105			−0.329077			−0.164539	+	0.986371i	$0.552614\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	6.54022i		−	0.354173i
$342$		0			0
$343$	1.73096			0.0934631
$344$		0			0
$345$		0			0
$346$		0			0
$347$	36.3162			1.94956			0.974778	−	0.223175i	$-0.0716421\pi$
$347$	36.3162			1.94956			0.974778	+	0.223175i	$0.0716421\pi$
$348$		0			0
$349$			24.4596i			1.30929i	0.755936	+	0.654646i	$0.227182\pi$
$349$			24.4596i			1.30929i	−0.755936	+	0.654646i	$0.772818\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	15.6082i		−	0.830742i	−0.909652	−	0.415371i	$-0.863652\pi$
$353$		−	15.6082i		−	0.830742i	0.909652	−	0.415371i	$-0.136348\pi$
$354$		0			0
$355$	−7.11135			−0.377431
$356$		0			0
$357$		0			0
$358$		0			0
$359$			21.7241i			1.14655i	0.819362	+	0.573277i	$0.194328\pi$
$359$			21.7241i			1.14655i	−0.819362	+	0.573277i	$0.805672\pi$
$360$		0			0
$361$	−13.4378			−0.707252
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	12.5910i		−	0.659041i
$366$		0			0
$367$	3.18168			0.166082			0.0830411	−	0.996546i	$-0.473537\pi$
$367$	3.18168			0.166082			0.0830411	+	0.996546i	$0.473537\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−0.760248			−0.0394701
$372$		0			0
$373$			4.76949i			0.246955i	0.992347	+	0.123477i	$0.0394047\pi$
$373$			4.76949i			0.246955i	−0.992347	+	0.123477i	$0.960595\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−4.05238			−0.208708
$378$		0			0
$379$		−	37.7989i		−	1.94160i	−0.239889	−	0.970800i	$-0.577111\pi$
$379$		−	37.7989i		−	1.94160i	0.239889	−	0.970800i	$-0.422889\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			10.2238i			0.522410i	0.965283	+	0.261205i	$0.0841198\pi$
$383$			10.2238i			0.522410i	−0.965283	+	0.261205i	$0.915880\pi$
$384$		0			0
$385$		−	0.113054i		−	0.00576177i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−16.2159			−0.822181			−0.411090	−	0.911595i	$-0.634852\pi$
$389$	−16.2159			−0.822181			−0.411090	+	0.911595i	$0.634852\pi$
$390$		0			0
$391$	16.7283			0.845987
$392$		0			0
$393$		0			0
$394$		0			0
$395$			8.17180i			0.411168i
$396$		0			0
$397$	16.4032			0.823253			0.411626	−	0.911353i	$-0.364961\pi$
$397$	16.4032			0.823253			0.411626	+	0.911353i	$0.364961\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	26.7573			1.33620			0.668098	−	0.744073i	$-0.267109\pi$
$401$	26.7573			1.33620			0.668098	+	0.744073i	$0.267109\pi$
$402$		0			0
$403$		−	6.70841i		−	0.334170i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−3.65703			−0.181272
$408$		0			0
$409$	8.02809			0.396964			0.198482	−	0.980105i	$-0.436399\pi$
$409$	8.02809			0.396964			0.198482	+	0.980105i	$0.436399\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−0.702241			−0.0345550
$414$		0			0
$415$	13.7768			0.676278
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	37.5741i		−	1.83561i	−0.397028	−	0.917806i	$-0.629959\pi$
$419$		−	37.5741i		−	1.83561i	0.397028	−	0.917806i	$-0.370041\pi$
$420$		0			0
$421$	0.251731			0.0122686			0.00613432	−	0.999981i	$-0.498047\pi$
$421$	0.251731			0.0122686			0.00613432	+	0.999981i	$0.498047\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	15.1085			0.732872
$426$		0			0
$427$	−0.714155			−0.0345604
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	36.2347i		−	1.74537i	−0.488287	−	0.872683i	$-0.662378\pi$
$431$		−	36.2347i		−	1.74537i	0.488287	−	0.872683i	$-0.337622\pi$
$432$		0			0
$433$	34.5240			1.65912			0.829560	−	0.558418i	$-0.188591\pi$
$433$	34.5240			1.65912			0.829560	+	0.558418i	$0.188591\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−9.60190			−0.459321
$438$		0			0
$439$		−	4.48008i		−	0.213823i	−0.994269	−	0.106911i	$-0.965904\pi$
$439$		−	4.48008i		−	0.213823i	0.994269	−	0.106911i	$-0.0340961\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−19.2513			−0.914659			−0.457330	−	0.889297i	$-0.651194\pi$
$443$	−19.2513			−0.914659			−0.457330	+	0.889297i	$0.651194\pi$
$444$		0			0
$445$		−	3.20838i		−	0.152092i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			14.5088i			0.684712i	0.939570	+	0.342356i	$0.111225\pi$
$449$			14.5088i			0.684712i	−0.939570	+	0.342356i	$0.888775\pi$
$450$		0			0
$451$			0.955150i			0.0449762i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	0.115962i		−	0.00543636i
$456$		0			0
$457$		−	23.3890i		−	1.09409i	−0.837103	−	0.547045i	$-0.815753\pi$
$457$		−	23.3890i		−	1.09409i	0.837103	−	0.547045i	$-0.184247\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			36.1915i			1.68560i	0.538224	+	0.842802i	$0.319096\pi$
$461$			36.1915i			1.68560i	−0.538224	+	0.842802i	$0.680904\pi$
$462$		0			0
$463$			35.9351i			1.67005i	0.550214	+	0.835023i	$0.314546\pi$
$463$			35.9351i			1.67005i	−0.550214	+	0.835023i	$0.685454\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			24.9986i			1.15680i	0.815755	+	0.578398i	$0.196322\pi$
$467$			24.9986i			1.15680i	−0.815755	+	0.578398i	$0.803678\pi$
$468$		0			0
$469$			0.202531i			0.00935200i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	7.99656			0.367682
$474$		0			0
$475$	−8.67217			−0.397906
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−14.4680			−0.661059			−0.330529	−	0.943796i	$-0.607227\pi$
$479$	−14.4680			−0.661059			−0.330529	+	0.943796i	$0.607227\pi$
$480$		0			0
$481$	−3.75108			−0.171034
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−15.6538			−0.710803
$486$		0			0
$487$			32.4111i			1.46869i	0.678777	+	0.734344i	$0.262510\pi$
$487$			32.4111i			1.46869i	−0.678777	+	0.734344i	$0.737490\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	11.1482i		−	0.503112i	−0.967843	−	0.251556i	$-0.919058\pi$
$491$		−	11.1482i		−	0.503112i	0.967843	−	0.251556i	$-0.0809423\pi$
$492$		0			0
$493$			20.4416i			0.920641i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−0.765282			−0.0343276
$498$		0			0
$499$			4.33292i			0.193968i	0.995286	+	0.0969840i	$0.0309196\pi$
$499$			4.33292i			0.193968i	−0.995286	+	0.0969840i	$0.969080\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	19.1302i		−	0.852974i	−0.904494	−	0.426487i	$-0.859751\pi$
$503$		−	19.1302i		−	0.852974i	0.904494	−	0.426487i	$-0.140249\pi$
$504$		0			0
$505$	19.3995			0.863267
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	4.13258i		−	0.183173i	−0.995797	−	0.0915866i	$-0.970806\pi$
$509$		−	4.13258i		−	0.183173i	0.995797	−	0.0915866i	$-0.0291938\pi$
$510$		0			0
$511$		−	1.35496i		−	0.0599401i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	8.83065i		−	0.389125i
$516$		0			0
$517$	6.94675			0.305518
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−32.0773			−1.40533			−0.702666	−	0.711520i	$-0.748007\pi$
$521$	−32.0773			−1.40533			−0.702666	+	0.711520i	$0.748007\pi$
$522$		0			0
$523$	−20.6678			−0.903741			−0.451871	−	0.892083i	$-0.649243\pi$
$523$	−20.6678			−0.903741			−0.451871	+	0.892083i	$0.649243\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−33.8395			−1.47407
$528$		0			0
$529$	−6.42449			−0.279326
$530$		0			0
$531$		0			0
$532$		0			0
$533$			0.979713i			0.0424361i
$534$		0			0
$535$		−	6.43684i		−	0.278289i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			5.54668i			0.238913i
$540$		0			0
$541$			29.8851i			1.28486i	0.766345	+	0.642430i	$0.222074\pi$
$541$			29.8851i			1.28486i	−0.766345	+	0.642430i	$0.777926\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	8.16031i		−	0.349549i
$546$		0			0
$547$		−	2.10073i		−	0.0898207i	−0.998991	−	0.0449103i	$-0.985700\pi$
$547$		−	2.10073i		−	0.0898207i	0.998991	−	0.0449103i	$-0.0143002\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	11.7333i		−	0.499854i
$552$		0			0
$553$			0.879401i			0.0373959i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			11.2694i			0.477500i	0.971081	+	0.238750i	$0.0767376\pi$
$557$			11.2694i			0.477500i	−0.971081	+	0.238750i	$0.923262\pi$
$558$		0			0
$559$	8.20220			0.346916
$560$		0			0
$561$		0			0
$562$		0			0
$563$			25.6691i			1.08183i	0.841079	+	0.540913i	$0.181921\pi$
$563$			25.6691i			1.08183i	−0.841079	+	0.540913i	$0.818079\pi$
$564$		0			0
$565$	13.2311			0.556636
$566$		0			0
$567$		0			0
$568$		0			0
$569$	28.5577			1.19720			0.598601	−	0.801047i	$-0.295724\pi$
$569$	28.5577			1.19720			0.598601	+	0.801047i	$0.295724\pi$
$570$		0			0
$571$		−	24.9317i		−	1.04336i	−0.853142	−	0.521679i	$-0.825306\pi$
$571$		−	24.9317i		−	1.04336i	0.853142	−	0.521679i	$-0.174694\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	14.9705			0.624315
$576$		0			0
$577$	14.3062			0.595575			0.297787	−	0.954632i	$-0.403751\pi$
$577$	14.3062			0.595575			0.297787	+	0.954632i	$0.403751\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	1.48258			0.0615079
$582$		0			0
$583$		−	4.87762i		−	0.202010i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−37.9790			−1.56756			−0.783780	−	0.621039i	$-0.786711\pi$
$587$	−37.9790			−1.56756			−0.783780	+	0.621039i	$0.786711\pi$
$588$		0			0
$589$	19.4236			0.800334
$590$		0			0
$591$		0			0
$592$		0			0
$593$	21.9951			0.903231			0.451615	−	0.892213i	$-0.350848\pi$
$593$	21.9951			0.903231			0.451615	+	0.892213i	$0.350848\pi$
$594$		0			0
$595$	−0.584949			−0.0239806
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	10.0813i		−	0.411912i	−0.978561	−	0.205956i	$-0.933970\pi$
$599$		−	10.0813i		−	0.411912i	0.978561	−	0.205956i	$-0.0660304\pi$
$600$		0			0
$601$	−37.3683			−1.52428			−0.762142	−	0.647410i	$-0.775852\pi$
$601$	−37.3683			−1.52428			−0.762142	+	0.647410i	$0.775852\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−11.9266			−0.484886
$606$		0			0
$607$		−	10.1385i		−	0.411511i	−0.978603	−	0.205755i	$-0.934035\pi$
$607$		−	10.1385i		−	0.411511i	0.978603	−	0.205755i	$-0.0659651\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	7.12539			0.288263
$612$		0			0
$613$	18.2559			0.737347			0.368674	−	0.929559i	$-0.379812\pi$
$613$	18.2559			0.737347			0.368674	+	0.929559i	$0.379812\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	41.1896i		−	1.65823i	−0.559078	−	0.829115i	$-0.688845\pi$
$617$		−	41.1896i		−	1.65823i	0.559078	−	0.829115i	$-0.311155\pi$
$618$		0			0
$619$		−	0.482020i		−	0.0193740i	−0.999953	−	0.00968701i	$-0.996916\pi$
$619$		−	0.482020i		−	0.0193740i	0.999953	−	0.00968701i	$-0.00308352\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	0.345268i		−	0.0138329i
$624$		0			0
$625$	6.90644			0.276258
$626$		0			0
$627$		0			0
$628$		0			0
$629$			18.9217i			0.754458i
$630$		0			0
$631$	−32.9041			−1.30989			−0.654946	−	0.755676i	$-0.727309\pi$
$631$	−32.9041			−1.30989			−0.654946	+	0.755676i	$0.727309\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	23.8102			0.944879
$636$		0			0
$637$			5.68932i			0.225419i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−49.4849			−1.95454			−0.977268	−	0.212008i	$-0.932000\pi$
$641$	−49.4849			−1.95454			−0.977268	+	0.212008i	$0.932000\pi$
$642$		0			0
$643$		−	34.0548i		−	1.34299i	−0.741009	−	0.671495i	$-0.765652\pi$
$643$		−	34.0548i		−	1.34299i	0.741009	−	0.671495i	$-0.234348\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	31.4190			1.23521			0.617603	−	0.786490i	$-0.288104\pi$
$647$	31.4190			1.23521			0.617603	+	0.786490i	$0.288104\pi$
$648$		0			0
$649$		−	4.50546i		−	0.176855i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			43.7737i			1.71300i	0.516148	+	0.856499i	$0.327365\pi$
$653$			43.7737i			1.71300i	−0.516148	+	0.856499i	$0.672635\pi$
$654$		0			0
$655$	0.909462			0.0355356
$656$		0			0
$657$		0			0
$658$		0			0
$659$	41.8148			1.62887			0.814437	−	0.580253i	$-0.197046\pi$
$659$	41.8148			1.62887			0.814437	+	0.580253i	$0.197046\pi$
$660$		0			0
$661$		−	30.3568i		−	1.18074i	−0.807131	−	0.590372i	$-0.798981\pi$
$661$		−	30.3568i		−	1.18074i	0.807131	−	0.590372i	$-0.201019\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	0.335756			0.0130200
$666$		0			0
$667$			20.2549i			0.784271i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	4.58190i		−	0.176882i
$672$		0			0
$673$		−	19.8641i		−	0.765707i	−0.923809	−	0.382853i	$-0.874941\pi$
$673$		−	19.8641i		−	0.765707i	0.923809	−	0.382853i	$-0.125059\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	42.0025i		−	1.61429i	−0.590356	−	0.807143i	$-0.701013\pi$
$677$		−	42.0025i		−	1.61429i	0.590356	−	0.807143i	$-0.298987\pi$
$678$		0			0
$679$	−1.68457			−0.0646479
$680$		0			0
$681$		0			0
$682$		0			0
$683$	19.1373			0.732269			0.366135	−	0.930562i	$-0.380681\pi$
$683$	19.1373			0.732269			0.366135	+	0.930562i	$0.380681\pi$
$684$		0			0
$685$		−	9.74993i		−	0.372526i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	5.00305i		−	0.190601i
$690$		0			0
$691$			33.3900i			1.27022i	0.772423	+	0.635109i	$0.219045\pi$
$691$			33.3900i			1.27022i	−0.772423	+	0.635109i	$0.780955\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	4.80108i		−	0.182115i
$696$		0			0
$697$	4.94200			0.187192
$698$		0			0
$699$		0			0
$700$		0			0
$701$			5.30339i			0.200306i	0.994972	+	0.100153i	$0.0319333\pi$
$701$			5.30339i			0.200306i	−0.994972	+	0.100153i	$0.968067\pi$
$702$		0			0
$703$		−	10.8609i		−	0.409626i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	2.08766			0.0785146
$708$		0			0
$709$			25.2444i			0.948074i	0.880505	+	0.474037i	$0.157204\pi$
$709$			25.2444i			0.948074i	−0.880505	+	0.474037i	$0.842796\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−33.5304			−1.25572
$714$		0			0
$715$	0.743989			0.0278236
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−19.1715			−0.714975			−0.357487	−	0.933918i	$-0.616366\pi$
$719$	−19.1715			−0.714975			−0.357487	+	0.933918i	$0.616366\pi$
$720$		0			0
$721$		−	0.950303i		−	0.0353911i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			18.2936i			0.679408i
$726$		0			0
$727$		−	9.95814i		−	0.369327i	−0.982802	−	0.184663i	$-0.940881\pi$
$727$		−	9.95814i		−	0.369327i	0.982802	−	0.184663i	$-0.0591195\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	41.3747i		−	1.53030i
$732$		0			0
$733$	−7.86615			−0.290543			−0.145271	−	0.989392i	$-0.546406\pi$
$733$	−7.86615			−0.290543			−0.145271	+	0.989392i	$0.546406\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1.29940			−0.0478641
$738$		0			0
$739$		−	47.6692i		−	1.75354i	−0.480910	−	0.876770i	$-0.659694\pi$
$739$		−	47.6692i		−	1.75354i	0.480910	−	0.876770i	$-0.340306\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			7.00382i			0.256945i	0.991713	+	0.128473i	$0.0410074\pi$
$743$			7.00382i			0.256945i	−0.991713	+	0.128473i	$0.958993\pi$
$744$		0			0
$745$	1.74444			0.0639113
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	0.692695i		−	0.0253105i
$750$		0			0
$751$		−	42.7064i		−	1.55838i	−0.626789	−	0.779189i	$-0.715631\pi$
$751$		−	42.7064i		−	1.55838i	0.626789	−	0.779189i	$-0.284369\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			5.22951i			0.190321i
$756$		0			0
$757$	−36.1458			−1.31374			−0.656870	−	0.754004i	$-0.728120\pi$
$757$	−36.1458			−1.31374			−0.656870	+	0.754004i	$0.728120\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	21.1228i		−	0.765702i	−0.923810	−	0.382851i	$-0.874942\pi$
$761$		−	21.1228i		−	0.765702i	0.923810	−	0.382851i	$-0.125058\pi$
$762$		0			0
$763$		−	0.878165i		−	0.0317917i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	4.62132i		−	0.166866i
$768$		0			0
$769$			44.7684i			1.61439i	0.590284	+	0.807195i	$0.299016\pi$
$769$			44.7684i			1.61439i	−0.590284	+	0.807195i	$0.700984\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	16.3112			0.586673			0.293337	−	0.956009i	$-0.405234\pi$
$773$	16.3112			0.586673			0.293337	+	0.956009i	$0.405234\pi$
$774$		0			0
$775$	−30.2837			−1.08782
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−2.83666			−0.101634
$780$		0			0
$781$		−	4.90991i		−	0.175691i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−17.2356			−0.615166
$786$		0			0
$787$			26.6793i			0.951016i	0.879711	+	0.475508i	$0.157736\pi$
$787$			26.6793i			0.951016i	−0.879711	+	0.475508i	$0.842264\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	1.42385			0.0506264
$792$		0			0
$793$		−	4.69973i		−	0.166892i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	36.7400			1.30140			0.650698	−	0.759337i	$-0.274476\pi$
$797$	36.7400			1.30140			0.650698	+	0.759337i	$0.274476\pi$
$798$		0			0
$799$		−	35.9429i		−	1.27157i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	8.69321			0.306777
$804$		0			0
$805$	−0.579606			−0.0204284
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	15.9591i		−	0.561094i	−0.959840	−	0.280547i	$-0.909484\pi$
$809$		−	15.9591i		−	0.561094i	0.959840	−	0.280547i	$-0.0905158\pi$
$810$		0			0
$811$			3.14320i			0.110373i	0.998476	+	0.0551863i	$0.0175753\pi$
$811$			3.14320i			0.110373i	−0.998476	+	0.0551863i	$0.982425\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	5.94283i		−	0.208168i
$816$		0			0
$817$			23.7487i			0.830862i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−9.31500			−0.325096			−0.162548	−	0.986701i	$-0.551971\pi$
$821$	−9.31500			−0.325096			−0.162548	+	0.986701i	$0.551971\pi$
$822$		0			0
$823$		−	35.9340i		−	1.25258i	−0.779590	−	0.626290i	$-0.784573\pi$
$823$		−	35.9340i		−	1.25258i	0.779590	−	0.626290i	$-0.215427\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	7.92795			0.275682			0.137841	−	0.990454i	$-0.455984\pi$
$827$	7.92795			0.275682			0.137841	+	0.990454i	$0.455984\pi$
$828$		0			0
$829$			17.3199i			0.601546i	0.953696	+	0.300773i	$0.0972447\pi$
$829$			17.3199i			0.601546i	−0.953696	+	0.300773i	$0.902755\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	28.6989			0.994357
$834$		0			0
$835$	−11.0100	−	9.98527i	−0.381017	−	0.345555i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			29.1645i			1.00687i	0.864033	+	0.503435i	$0.167931\pi$
$839$			29.1645i			1.00687i	−0.864033	+	0.503435i	$0.832069\pi$
$840$		0			0
$841$	4.24909			0.146520
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−14.1892			−0.488123
$846$		0			0
$847$	−1.28347			−0.0441007
$848$		0			0
$849$		0			0
$850$		0			0
$851$			18.7489i			0.642703i
$852$		0			0
$853$	−27.7054			−0.948615			−0.474308	−	0.880359i	$-0.657302\pi$
$853$	−27.7054			−0.948615			−0.474308	+	0.880359i	$0.657302\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			2.92893i			0.100050i	0.998748	+	0.0500251i	$0.0159301\pi$
$857$			2.92893i			0.100050i	−0.998748	+	0.0500251i	$0.984070\pi$
$858$		0			0
$859$	−39.8078			−1.35822			−0.679112	−	0.734035i	$-0.737635\pi$
$859$	−39.8078			−1.35822			−0.679112	+	0.734035i	$0.737635\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	52.6436i		−	1.79201i	−0.444045	−	0.896004i	$-0.646457\pi$
$863$		−	52.6436i		−	1.79201i	0.444045	−	0.896004i	$-0.353543\pi$
$864$		0			0
$865$			2.95977i			0.100635i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−5.64208			−0.191395
$870$		0			0
$871$	−1.33282			−0.0451608
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1.23530			−0.0417609
$876$		0			0
$877$	24.1243			0.814620			0.407310	−	0.913290i	$-0.366467\pi$
$877$	24.1243			0.814620			0.407310	+	0.913290i	$0.366467\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	28.6431			0.965011			0.482505	−	0.875893i	$-0.339727\pi$
$881$	28.6431			0.965011			0.482505	+	0.875893i	$0.339727\pi$
$882$		0			0
$883$	44.7933			1.50741			0.753707	−	0.657211i	$-0.228264\pi$
$883$	44.7933			1.50741			0.753707	+	0.657211i	$0.228264\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	10.7187			0.359897			0.179949	−	0.983676i	$-0.442407\pi$
$887$	10.7187			0.359897			0.179949	+	0.983676i	$0.442407\pi$
$888$		0			0
$889$	2.56231			0.0859373
$890$		0			0
$891$		0			0
$892$		0			0
$893$			20.6309i			0.690387i
$894$		0			0
$895$			23.6769i			0.791431i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	40.9733i		−	1.36654i
$900$		0			0
$901$	−25.2371			−0.840769
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−16.8609			−0.560474
$906$		0			0
$907$	−47.1394			−1.56524			−0.782620	−	0.622500i	$-0.786117\pi$
$907$	−47.1394			−1.56524			−0.782620	+	0.622500i	$0.786117\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			57.6610i			1.91039i	0.295972	+	0.955197i	$0.404357\pi$
$911$			57.6610i			1.91039i	−0.295972	+	0.955197i	$0.595643\pi$
$912$		0			0
$913$			9.51199i			0.314801i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	0.0978710			0.00323199
$918$		0			0
$919$	25.0905			0.827661			0.413830	−	0.910354i	$-0.364191\pi$
$919$	25.0905			0.827661			0.413830	+	0.910354i	$0.364191\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	5.03618i		−	0.165768i
$924$		0			0
$925$			16.9335i			0.556769i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			45.5642i			1.49491i	0.664310	+	0.747457i	$0.268725\pi$
$929$			45.5642i			1.49491i	−0.664310	+	0.747457i	$0.731275\pi$
$930$		0			0
$931$	−16.4729			−0.539877
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	3.75293i		−	0.122734i
$936$		0			0
$937$		−	24.6125i		−	0.804055i	−0.915628	−	0.402028i	$-0.868306\pi$
$937$		−	24.6125i		−	0.804055i	0.915628	−	0.402028i	$-0.131694\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−38.4101			−1.25213			−0.626067	−	0.779770i	$-0.715336\pi$
$941$	−38.4101			−1.25213			−0.626067	+	0.779770i	$0.715336\pi$
$942$		0			0
$943$	4.89686			0.159464
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	22.1038i		−	0.718278i	−0.933284	−	0.359139i	$-0.883070\pi$
$947$		−	22.1038i		−	0.718278i	0.933284	−	0.359139i	$-0.116930\pi$
$948$		0			0
$949$	8.91677			0.289451
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−11.7540			−0.380749			−0.190374	−	0.981712i	$-0.560970\pi$
$953$	−11.7540			−0.380749			−0.190374	+	0.981712i	$0.560970\pi$
$954$		0			0
$955$		−	9.84586i		−	0.318605i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	1.04923i		−	0.0338814i
$960$		0			0
$961$	36.8282			1.18801
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	17.7442i		−	0.571205i
$966$		0			0
$967$	4.29230			0.138031			0.0690155	−	0.997616i	$-0.478014\pi$
$967$	4.29230			0.138031			0.0690155	+	0.997616i	$0.478014\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	43.6096			1.39950			0.699750	−	0.714388i	$-0.253295\pi$
$971$	43.6096			1.39950			0.699750	+	0.714388i	$0.253295\pi$
$972$		0			0
$973$		−	0.516664i		−	0.0165635i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	34.1019			1.09102			0.545508	−	0.838105i	$-0.316337\pi$
$977$	34.1019			1.09102			0.545508	+	0.838105i	$0.316337\pi$
$978$		0			0
$979$	2.21518			0.0707973
$980$		0			0
$981$		0			0
$982$		0			0
$983$	11.1052			0.354202			0.177101	−	0.984193i	$-0.443328\pi$
$983$	11.1052			0.354202			0.177101	+	0.984193i	$0.443328\pi$
$984$		0			0
$985$	13.2537			0.422298
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	40.9968i		−	1.30362i
$990$		0			0
$991$		−	26.8198i		−	0.851958i	−0.904733	−	0.425979i	$-0.859930\pi$
$991$		−	26.8198i		−	0.851958i	0.904733	−	0.425979i	$-0.140070\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	2.91290			0.0923451
$996$		0			0
$997$	19.1424			0.606245			0.303122	−	0.952952i	$-0.401971\pi$
$997$	19.1424			0.606245			0.303122	+	0.952952i	$0.401971\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6012.2.h.a.3005.20	yes	56
3.2	odd	2	inner	6012.2.h.a.3005.38	yes	56
167.166	odd	2	inner	6012.2.h.a.3005.37	yes	56
501.500	even	2	inner	6012.2.h.a.3005.19	✓	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6012.2.h.a.3005.19	✓	56	501.500	even	2	inner
6012.2.h.a.3005.20	yes	56	1.1	even	1	trivial
6012.2.h.a.3005.37	yes	56	167.166	odd	2	inner
6012.2.h.a.3005.38	yes	56	3.2	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:	\( N \)	\(=\)	\( 6012 = 2^{2} \cdot 3^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6012.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.15018 q^{5} -0.123775 q^{7} +O(q^{10})\)	\(q-1.15018 q^{5} -0.123775 q^{7} -0.794122i q^{11} -0.814543i q^{13} -4.10883 q^{17} +2.35843 q^{19} -4.07130 q^{23} -3.67709 q^{25} -4.97503i q^{29} +8.23579 q^{31} +0.142364 q^{35} -4.60513i q^{37} -1.20278 q^{41} +10.0697i q^{43} +8.74772i q^{47} -6.98468 q^{49} +6.14215 q^{53} +0.913381i q^{55} +5.67351 q^{59} +5.76977 q^{61} +0.936870i q^{65} -1.63628i q^{67} +6.18282 q^{71} +10.9470i q^{73} +0.0982927i q^{77} -7.10481i q^{79} -11.9780 q^{83} +4.72589 q^{85} +2.78947i q^{89} +0.100820i q^{91} -2.71262 q^{95} +13.6099 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q+O(q^{10}) \) 56 * q	\( 56 q + 8 q^{19} + 64 q^{25} - 8 q^{31} + 56 q^{49} - 8 q^{61} + 32 q^{85} - 48 q^{97}+O(q^{100}) \) 56 * q + 8 * q^19 + 64 * q^25 - 8 * q^31 + 56 * q^49 - 8 * q^61 + 32 * q^85 - 48 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more