LMFDB - Embedded newform 6012.2.h.a.3005.15

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			3005.15
Character	$\chi$	$=$	6012.3005
Dual form			6012.2.h.a.3005.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.08738 q^{5} +3.97701 q^{7} +O(q^{10})$	$q-2.08738 q^{5} +3.97701 q^{7} -5.31004i q^{11} +4.83536i q^{13} +6.52168 q^{17} +7.92418 q^{19} -1.37306 q^{23} -0.642863 q^{25} +3.16157i q^{29} +5.86836 q^{31} -8.30151 q^{35} -8.85961i q^{37} -7.58734 q^{41} +1.05520i q^{43} +0.619938i q^{47} +8.81660 q^{49} -0.405697 q^{53} +11.0840i q^{55} +10.8869 q^{59} -1.50582 q^{61} -10.0932i q^{65} +14.1578i q^{67} -6.00226 q^{71} -3.66335i q^{73} -21.1181i q^{77} +6.01121i q^{79} +6.48181 q^{83} -13.6132 q^{85} +3.50411i q^{89} +19.2303i q^{91} -16.5407 q^{95} -16.1278 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−2.08738			−0.933503			−0.466751	−	0.884389i	$-0.654576\pi$
$5$	−2.08738			−0.933503			−0.466751	+	0.884389i	$0.654576\pi$
$6$		0			0
$7$	3.97701			1.50317			0.751584	−	0.659637i	$-0.229290\pi$
$7$	3.97701			1.50317			0.751584	+	0.659637i	$0.229290\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	5.31004i		−	1.60104i	−0.599308	−	0.800518i	$-0.704558\pi$
$11$		−	5.31004i		−	1.60104i	0.599308	−	0.800518i	$-0.295442\pi$
$12$		0			0
$13$			4.83536i			1.34109i	0.741870	+	0.670544i	$0.233939\pi$
$13$			4.83536i			1.34109i	−0.741870	+	0.670544i	$0.766061\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	6.52168			1.58174			0.790870	−	0.611985i	$-0.209629\pi$
$17$	6.52168			1.58174			0.790870	+	0.611985i	$0.209629\pi$
$18$		0			0
$19$	7.92418			1.81793			0.908966	−	0.416870i	$-0.136873\pi$
$19$	7.92418			1.81793			0.908966	+	0.416870i	$0.136873\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.37306			−0.286303			−0.143152	−	0.989701i	$-0.545724\pi$
$23$	−1.37306			−0.286303			−0.143152	+	0.989701i	$0.545724\pi$
$24$		0			0
$25$	−0.642863			−0.128573
$26$		0			0
$27$		0			0
$28$		0			0
$29$			3.16157i			0.587088i	0.955945	+	0.293544i	$0.0948348\pi$
$29$			3.16157i			0.587088i	−0.955945	+	0.293544i	$0.905165\pi$
$30$		0			0
$31$	5.86836			1.05399			0.526994	−	0.849869i	$-0.323319\pi$
$31$	5.86836			1.05399			0.526994	+	0.849869i	$0.323319\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−8.30151			−1.40321
$36$		0			0
$37$		−	8.85961i		−	1.45651i	−0.685305	−	0.728256i	$-0.740331\pi$
$37$		−	8.85961i		−	1.45651i	0.685305	−	0.728256i	$-0.259669\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−7.58734			−1.18494			−0.592471	−	0.805591i	$-0.701848\pi$
$41$	−7.58734			−1.18494			−0.592471	+	0.805591i	$0.701848\pi$
$42$		0			0
$43$			1.05520i			0.160917i	0.996758	+	0.0804583i	$0.0256384\pi$
$43$			1.05520i			0.160917i	−0.996758	+	0.0804583i	$0.974362\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			0.619938i			0.0904272i	0.998977	+	0.0452136i	$0.0143969\pi$
$47$			0.619938i			0.0904272i	−0.998977	+	0.0452136i	$0.985603\pi$
$48$		0			0
$49$	8.81660			1.25951
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−0.405697			−0.0557267			−0.0278634	−	0.999612i	$-0.508870\pi$
$53$	−0.405697			−0.0557267			−0.0278634	+	0.999612i	$0.508870\pi$
$54$		0			0
$55$			11.0840i			1.49457i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	10.8869			1.41736			0.708679	−	0.705531i	$-0.249292\pi$
$59$	10.8869			1.41736			0.708679	+	0.705531i	$0.249292\pi$
$60$		0			0
$61$	−1.50582			−0.192801			−0.0964005	−	0.995343i	$-0.530733\pi$
$61$	−1.50582			−0.192801			−0.0964005	+	0.995343i	$0.530733\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	10.0932i		−	1.25191i
$66$		0			0
$67$			14.1578i			1.72965i	0.502075	+	0.864824i	$0.332570\pi$
$67$			14.1578i			1.72965i	−0.502075	+	0.864824i	$0.667430\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−6.00226			−0.712337			−0.356169	−	0.934422i	$-0.615917\pi$
$71$	−6.00226			−0.712337			−0.356169	+	0.934422i	$0.615917\pi$
$72$		0			0
$73$		−	3.66335i		−	0.428763i	−0.976750	−	0.214381i	$-0.931226\pi$
$73$		−	3.66335i		−	0.428763i	0.976750	−	0.214381i	$-0.0687735\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	21.1181i		−	2.40663i
$78$		0			0
$79$			6.01121i			0.676314i	0.941090	+	0.338157i	$0.109804\pi$
$79$			6.01121i			0.676314i	−0.941090	+	0.338157i	$0.890196\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	6.48181			0.711471			0.355735	−	0.934587i	$-0.384230\pi$
$83$	6.48181			0.711471			0.355735	+	0.934587i	$0.384230\pi$
$84$		0			0
$85$	−13.6132			−1.47656
$86$		0			0
$87$		0			0
$88$		0			0
$89$			3.50411i			0.371435i	0.982603	+	0.185718i	$0.0594610\pi$
$89$			3.50411i			0.371435i	−0.982603	+	0.185718i	$0.940539\pi$
$90$		0			0
$91$			19.2303i			2.01588i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−16.5407			−1.69704
$96$		0			0
$97$	−16.1278			−1.63753			−0.818765	−	0.574129i	$-0.805341\pi$
$97$	−16.1278			−1.63753			−0.818765	+	0.574129i	$0.805341\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1.20758			−0.120159			−0.0600794	−	0.998194i	$-0.519135\pi$
$101$	−1.20758			−0.120159			−0.0600794	+	0.998194i	$0.519135\pi$
$102$		0			0
$103$			2.79454i			0.275354i	0.990477	+	0.137677i	$0.0439637\pi$
$103$			2.79454i			0.275354i	−0.990477	+	0.137677i	$0.956036\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	10.9620i		−	1.05974i	−0.848079	−	0.529870i	$-0.822241\pi$
$107$		−	10.9620i		−	1.05974i	0.848079	−	0.529870i	$-0.177759\pi$
$108$		0			0
$109$			0.681898i			0.0653140i	0.999467	+	0.0326570i	$0.0103969\pi$
$109$			0.681898i			0.0653140i	−0.999467	+	0.0326570i	$0.989603\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−15.5097			−1.45903			−0.729516	−	0.683964i	$-0.760255\pi$
$113$	−15.5097			−1.45903			−0.729516	+	0.683964i	$0.760255\pi$
$114$		0			0
$115$	2.86610			0.267265
$116$		0			0
$117$		0			0
$118$		0			0
$119$	25.9368			2.37762
$120$		0			0
$121$	−17.1965			−1.56332
$122$		0			0
$123$		0			0
$124$		0			0
$125$	11.7788			1.05353
$126$		0			0
$127$	20.8656			1.85153			0.925763	−	0.378105i	$-0.123424\pi$
$127$	20.8656			1.85153			0.925763	+	0.378105i	$0.123424\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	4.87818			0.426208			0.213104	−	0.977029i	$-0.431643\pi$
$131$	4.87818			0.426208			0.213104	+	0.977029i	$0.431643\pi$
$132$		0			0
$133$	31.5145			2.73266
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	1.07973i		−	0.0922476i	−0.998936	−	0.0461238i	$-0.985313\pi$
$137$		−	1.07973i		−	0.0922476i	0.998936	−	0.0461238i	$-0.0146869\pi$
$138$		0			0
$139$		−	11.8376i		−	1.00405i	−0.864852	−	0.502027i	$-0.832588\pi$
$139$		−	11.8376i		−	1.00405i	0.864852	−	0.502027i	$-0.167412\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	25.6759			2.14713
$144$		0			0
$145$		−	6.59938i		−	0.548049i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	13.8597			1.13543			0.567714	−	0.823226i	$-0.307828\pi$
$149$	13.8597			1.13543			0.567714	+	0.823226i	$0.307828\pi$
$150$		0			0
$151$		−	18.1640i		−	1.47817i	−0.673613	−	0.739084i	$-0.735258\pi$
$151$		−	18.1640i		−	1.47817i	0.673613	−	0.739084i	$-0.264742\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−12.2495			−0.983901
$156$		0			0
$157$	6.90136			0.550789			0.275394	−	0.961331i	$-0.411192\pi$
$157$	6.90136			0.550789			0.275394	+	0.961331i	$0.411192\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−5.46068			−0.430362
$162$		0			0
$163$			3.64394i			0.285415i	0.989765	+	0.142708i	$0.0455808\pi$
$163$			3.64394i			0.285415i	−0.989765	+	0.142708i	$0.954419\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−5.22495	−	11.8195i	−0.404319	−	0.914618i
$167$	−5.22495	−	11.8195i	−0.404319	−	0.914618i
$168$		0			0
$169$	−10.3807			−0.798515
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	5.34258i		−	0.406189i	−0.979159	−	0.203094i	$-0.934900\pi$
$173$		−	5.34258i		−	0.406189i	0.979159	−	0.203094i	$-0.0650998\pi$
$174$		0			0
$175$	−2.55667			−0.193266
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	15.1556i		−	1.13279i	−0.824135	−	0.566393i	$-0.808338\pi$
$179$		−	15.1556i		−	1.13279i	0.824135	−	0.566393i	$-0.191662\pi$
$180$		0			0
$181$	13.0023			0.966455			0.483227	−	0.875495i	$-0.339464\pi$
$181$	13.0023			0.966455			0.483227	+	0.875495i	$0.339464\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			18.4933i			1.35966i
$186$		0			0
$187$		−	34.6304i		−	2.53242i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			22.0965i			1.59885i	0.600768	+	0.799423i	$0.294861\pi$
$191$			22.0965i			1.59885i	−0.600768	+	0.799423i	$0.705139\pi$
$192$		0			0
$193$			18.9583i			1.36465i	0.731048	+	0.682326i	$0.239031\pi$
$193$			18.9583i			1.36465i	−0.731048	+	0.682326i	$0.760969\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	9.22603			0.657328			0.328664	−	0.944447i	$-0.393402\pi$
$197$	9.22603			0.657328			0.328664	+	0.944447i	$0.393402\pi$
$198$		0			0
$199$	−12.7024			−0.900448			−0.450224	−	0.892916i	$-0.648656\pi$
$199$	−12.7024			−0.900448			−0.450224	+	0.892916i	$0.648656\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			12.5736i			0.882493i
$204$		0			0
$205$	15.8376			1.10615
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	42.0777i		−	2.91058i
$210$		0			0
$211$	−2.64849			−0.182330			−0.0911649	−	0.995836i	$-0.529059\pi$
$211$	−2.64849			−0.182330			−0.0911649	+	0.995836i	$0.529059\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	2.20260i		−	0.150216i
$216$		0			0
$217$	23.3385			1.58432
$218$		0			0
$219$		0			0
$220$		0			0
$221$			31.5346i			2.12125i
$222$		0			0
$223$	14.0637			0.941772			0.470886	−	0.882194i	$-0.343934\pi$
$223$	14.0637			0.941772			0.470886	+	0.882194i	$0.343934\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	27.1134			1.79958			0.899790	−	0.436323i	$-0.143720\pi$
$227$	27.1134			1.79958			0.899790	+	0.436323i	$0.143720\pi$
$228$		0			0
$229$	−5.40537			−0.357197			−0.178599	−	0.983922i	$-0.557156\pi$
$229$	−5.40537			−0.357197			−0.178599	+	0.983922i	$0.557156\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	20.2157i		−	1.32437i	−0.749339	−	0.662187i	$-0.769629\pi$
$233$		−	20.2157i		−	1.32437i	0.749339	−	0.662187i	$-0.230371\pi$
$234$		0			0
$235$		−	1.29404i		−	0.0844141i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			20.0580i			1.29744i	0.761026	+	0.648721i	$0.224696\pi$
$239$			20.0580i			1.29744i	−0.761026	+	0.648721i	$0.775304\pi$
$240$		0			0
$241$			6.58002i			0.423857i	0.977285	+	0.211928i	$0.0679743\pi$
$241$			6.58002i			0.423857i	−0.977285	+	0.211928i	$0.932026\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−18.4036			−1.17576
$246$		0			0
$247$			38.3163i			2.43801i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			9.32718i			0.588726i	0.955694	+	0.294363i	$0.0951075\pi$
$251$			9.32718i			0.588726i	−0.955694	+	0.294363i	$0.904893\pi$
$252$		0			0
$253$			7.29102i			0.458382i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−24.6036			−1.53473			−0.767367	−	0.641209i	$-0.778433\pi$
$257$	−24.6036			−1.53473			−0.767367	+	0.641209i	$0.778433\pi$
$258$		0			0
$259$		−	35.2348i		−	2.18938i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			20.2834i			1.25073i	0.780333	+	0.625364i	$0.215050\pi$
$263$			20.2834i			1.25073i	−0.780333	+	0.625364i	$0.784950\pi$
$264$		0			0
$265$	0.846841			0.0520210
$266$		0			0
$267$		0			0
$268$		0			0
$269$	10.1372			0.618077			0.309039	−	0.951049i	$-0.399993\pi$
$269$	10.1372			0.618077			0.309039	+	0.951049i	$0.399993\pi$
$270$		0			0
$271$		−	31.6624i		−	1.92335i	−0.274183	−	0.961677i	$-0.588408\pi$
$271$		−	31.6624i		−	1.92335i	0.274183	−	0.961677i	$-0.411592\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			3.41363i			0.205850i
$276$		0			0
$277$			26.9874i			1.62151i	0.585382	+	0.810757i	$0.300944\pi$
$277$			26.9874i			1.62151i	−0.585382	+	0.810757i	$0.699056\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	4.31730i		−	0.257549i	−0.991674	−	0.128774i	$-0.958896\pi$
$281$		−	4.31730i		−	0.257549i	0.991674	−	0.128774i	$-0.0411043\pi$
$282$		0			0
$283$	20.0419			1.19137			0.595683	−	0.803220i	$-0.296881\pi$
$283$	20.0419			1.19137			0.595683	+	0.803220i	$0.296881\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−30.1749			−1.78117
$288$		0			0
$289$	25.5323			1.50190
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	18.6388i		−	1.08889i	−0.838796	−	0.544446i	$-0.816740\pi$
$293$		−	18.6388i		−	1.08889i	0.838796	−	0.544446i	$-0.183260\pi$
$294$		0			0
$295$	−22.7251			−1.32311
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	6.63925i		−	0.383958i
$300$		0			0
$301$			4.19654i			0.241885i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	3.14322			0.179980
$306$		0			0
$307$		−	0.835081i		−	0.0476606i	−0.999716	−	0.0238303i	$-0.992414\pi$
$307$		−	0.835081i		−	0.0476606i	0.999716	−	0.0238303i	$-0.00758614\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	4.28717i		−	0.243103i	−0.992585	−	0.121552i	$-0.961213\pi$
$311$		−	4.28717i		−	0.243103i	0.992585	−	0.121552i	$-0.0387870\pi$
$312$		0			0
$313$			13.9566i			0.788872i	0.918923	+	0.394436i	$0.129060\pi$
$313$			13.9566i			0.788872i	−0.918923	+	0.394436i	$0.870940\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			28.9136i			1.62395i	0.583692	+	0.811975i	$0.301608\pi$
$317$			28.9136i			1.62395i	−0.583692	+	0.811975i	$0.698392\pi$
$318$		0			0
$319$	16.7880			0.939950
$320$		0			0
$321$		0			0
$322$		0			0
$323$	51.6790			2.87549
$324$		0			0
$325$		−	3.10848i		−	0.172427i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			2.46550i			0.135927i
$330$		0			0
$331$		−	30.6221i		−	1.68314i	−0.540145	−	0.841572i	$-0.681631\pi$
$331$		−	30.6221i		−	1.68314i	0.540145	−	0.841572i	$-0.318369\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	29.5526i		−	1.61463i
$336$		0			0
$337$	−22.2130			−1.21002			−0.605011	−	0.796217i	$-0.706831\pi$
$337$	−22.2130			−1.21002			−0.605011	+	0.796217i	$0.706831\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	31.1612i		−	1.68747i
$342$		0			0
$343$	7.22464			0.390094
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−8.23540			−0.442099			−0.221050	−	0.975263i	$-0.570948\pi$
$347$	−8.23540			−0.442099			−0.221050	+	0.975263i	$0.570948\pi$
$348$		0			0
$349$		−	20.8450i		−	1.11581i	−0.829905	−	0.557904i	$-0.811606\pi$
$349$		−	20.8450i		−	1.11581i	0.829905	−	0.557904i	$-0.188394\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	19.4771i		−	1.03666i	−0.855179	−	0.518332i	$-0.826553\pi$
$353$		−	19.4771i		−	1.03666i	0.855179	−	0.518332i	$-0.173447\pi$
$354$		0			0
$355$	12.5290			0.664969
$356$		0			0
$357$		0			0
$358$		0			0
$359$			8.74825i			0.461715i	0.972988	+	0.230857i	$0.0741531\pi$
$359$			8.74825i			0.461715i	−0.972988	+	0.230857i	$0.925847\pi$
$360$		0			0
$361$	43.7926			2.30488
$362$		0			0
$363$		0			0
$364$		0			0
$365$			7.64679i			0.400251i
$366$		0			0
$367$	26.2691			1.37123			0.685617	−	0.727962i	$-0.259532\pi$
$367$	26.2691			1.37123			0.685617	+	0.727962i	$0.259532\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−1.61346			−0.0837666
$372$		0			0
$373$			8.02553i			0.415546i	0.978177	+	0.207773i	$0.0666216\pi$
$373$			8.02553i			0.415546i	−0.978177	+	0.207773i	$0.933378\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−15.2873			−0.787337
$378$		0			0
$379$			32.6197i			1.67556i	0.546007	+	0.837781i	$0.316147\pi$
$379$			32.6197i			1.67556i	−0.546007	+	0.837781i	$0.683853\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			26.8165i			1.37026i	0.728422	+	0.685129i	$0.240254\pi$
$383$			26.8165i			1.37026i	−0.728422	+	0.685129i	$0.759746\pi$
$384$		0			0
$385$			44.0813i			2.24659i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	25.3033			1.28293			0.641463	−	0.767154i	$-0.278328\pi$
$389$	25.3033			1.28293			0.641463	+	0.767154i	$0.278328\pi$
$390$		0			0
$391$	−8.95467			−0.452857
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	12.5477i		−	0.631341i
$396$		0			0
$397$	2.37174			0.119034			0.0595172	−	0.998227i	$-0.481044\pi$
$397$	2.37174			0.119034			0.0595172	+	0.998227i	$0.481044\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	17.9123			0.894495			0.447248	−	0.894410i	$-0.352404\pi$
$401$	17.9123			0.894495			0.447248	+	0.894410i	$0.352404\pi$
$402$		0			0
$403$			28.3756i			1.41349i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−47.0449			−2.33193
$408$		0			0
$409$	12.2284			0.604653			0.302327	−	0.953204i	$-0.402237\pi$
$409$	12.2284			0.604653			0.302327	+	0.953204i	$0.402237\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	43.2974			2.13053
$414$		0			0
$415$	−13.5300			−0.664160
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	19.0883i		−	0.932523i	−0.884647	−	0.466261i	$-0.845601\pi$
$419$		−	19.0883i		−	0.932523i	0.884647	−	0.466261i	$-0.154399\pi$
$420$		0			0
$421$	−14.5942			−0.711277			−0.355639	−	0.934624i	$-0.615737\pi$
$421$	−14.5942			−0.711277			−0.355639	+	0.934624i	$0.615737\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−4.19255			−0.203368
$426$		0			0
$427$	−5.98867			−0.289812
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	27.1700i		−	1.30873i	−0.756178	−	0.654366i	$-0.772936\pi$
$431$		−	27.1700i		−	1.30873i	0.756178	−	0.654366i	$-0.227064\pi$
$432$		0			0
$433$	27.3677			1.31521			0.657603	−	0.753365i	$-0.271570\pi$
$433$	27.3677			1.31521			0.657603	+	0.753365i	$0.271570\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−10.8804			−0.520480
$438$		0			0
$439$			11.3349i			0.540984i	0.962722	+	0.270492i	$0.0871864\pi$
$439$			11.3349i			0.540984i	−0.962722	+	0.270492i	$0.912814\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	7.10877			0.337748			0.168874	−	0.985638i	$-0.445987\pi$
$443$	7.10877			0.337748			0.168874	+	0.985638i	$0.445987\pi$
$444$		0			0
$445$		−	7.31440i		−	0.346736i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	7.51446i		−	0.354629i	−0.984154	−	0.177315i	$-0.943259\pi$
$449$		−	7.51446i		−	0.354629i	0.984154	−	0.177315i	$-0.0567410\pi$
$450$		0			0
$451$			40.2890i			1.89714i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	40.1408i		−	1.88183i
$456$		0			0
$457$		−	12.3693i		−	0.578610i	−0.957237	−	0.289305i	$-0.906576\pi$
$457$		−	12.3693i		−	0.578610i	0.957237	−	0.289305i	$-0.0934241\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			13.7961i			0.642549i	0.946986	+	0.321275i	$0.104111\pi$
$461$			13.7961i			0.642549i	−0.946986	+	0.321275i	$0.895889\pi$
$462$		0			0
$463$		−	0.371515i		−	0.0172657i	−0.999963	−	0.00863287i	$-0.997252\pi$
$463$		−	0.371515i		−	0.0172657i	0.999963	−	0.00863287i	$-0.00274796\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	11.8305i		−	0.547450i	−0.961808	−	0.273725i	$-0.911744\pi$
$467$		−	11.8305i		−	0.547450i	0.961808	−	0.273725i	$-0.0882559\pi$
$468$		0			0
$469$			56.3056i			2.59995i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	5.60315			0.257633
$474$		0			0
$475$	−5.09417			−0.233736
$476$		0			0
$477$		0			0
$478$		0			0
$479$	31.1013			1.42105			0.710527	−	0.703670i	$-0.248457\pi$
$479$	31.1013			1.42105			0.710527	+	0.703670i	$0.248457\pi$
$480$		0			0
$481$	42.8394			1.95331
$482$		0			0
$483$		0			0
$484$		0			0
$485$	33.6648			1.52864
$486$		0			0
$487$		−	33.2095i		−	1.50486i	−0.658670	−	0.752432i	$-0.728881\pi$
$487$		−	33.2095i		−	1.50486i	0.658670	−	0.752432i	$-0.271119\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	32.8757i		−	1.48366i	−0.670587	−	0.741831i	$-0.733958\pi$
$491$		−	32.8757i		−	1.48366i	0.670587	−	0.741831i	$-0.266042\pi$
$492$		0			0
$493$			20.6187i			0.928621i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−23.8710			−1.07076
$498$		0			0
$499$		−	42.1052i		−	1.88489i	−0.334364	−	0.942444i	$-0.608521\pi$
$499$		−	42.1052i		−	1.88489i	0.334364	−	0.942444i	$-0.391479\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			31.5903i			1.40854i	0.709932	+	0.704270i	$0.248726\pi$
$503$			31.5903i			1.40854i	−0.709932	+	0.704270i	$0.751274\pi$
$504$		0			0
$505$	2.52067			0.112169
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	3.32671i		−	0.147454i	−0.997278	−	0.0737270i	$-0.976511\pi$
$509$		−	3.32671i		−	0.147454i	0.997278	−	0.0737270i	$-0.0234894\pi$
$510$		0			0
$511$		−	14.5692i		−	0.644502i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	5.83326i		−	0.257044i
$516$		0			0
$517$	3.29189			0.144777
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−8.84464			−0.387491			−0.193745	−	0.981052i	$-0.562064\pi$
$521$	−8.84464			−0.387491			−0.193745	+	0.981052i	$0.562064\pi$
$522$		0			0
$523$	−9.09660			−0.397767			−0.198883	−	0.980023i	$-0.563731\pi$
$523$	−9.09660			−0.397767			−0.198883	+	0.980023i	$0.563731\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	38.2715			1.66713
$528$		0			0
$529$	−21.1147			−0.918030
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	36.6875i		−	1.58911i
$534$		0			0
$535$			22.8819i			0.989269i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	46.8165i		−	2.01653i
$540$		0			0
$541$		−	27.1309i		−	1.16645i	−0.812312	−	0.583224i	$-0.801791\pi$
$541$		−	27.1309i		−	1.16645i	0.812312	−	0.583224i	$-0.198209\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	1.42338i		−	0.0609708i
$546$		0			0
$547$		−	23.5346i		−	1.00627i	−0.864209	−	0.503133i	$-0.832181\pi$
$547$		−	23.5346i		−	1.00627i	0.864209	−	0.503133i	$-0.167819\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			25.0528i			1.06729i
$552$		0			0
$553$			23.9066i			1.01661i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			11.1446i			0.472211i	0.971727	+	0.236106i	$0.0758711\pi$
$557$			11.1446i			0.472211i	−0.971727	+	0.236106i	$0.924129\pi$
$558$		0			0
$559$	−5.10227			−0.215803
$560$		0			0
$561$		0			0
$562$		0			0
$563$			13.5797i			0.572318i	0.958182	+	0.286159i	$0.0923785\pi$
$563$			13.5797i			0.572318i	−0.958182	+	0.286159i	$0.907621\pi$
$564$		0			0
$565$	32.3746			1.36201
$566$		0			0
$567$		0			0
$568$		0			0
$569$	12.2712			0.514437			0.257218	−	0.966353i	$-0.417194\pi$
$569$	12.2712			0.514437			0.257218	+	0.966353i	$0.417194\pi$
$570$		0			0
$571$			17.3601i			0.726498i	0.931692	+	0.363249i	$0.118333\pi$
$571$			17.3601i			0.726498i	−0.931692	+	0.363249i	$0.881667\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	0.882692			0.0368108
$576$		0			0
$577$	−11.6366			−0.484440			−0.242220	−	0.970221i	$-0.577876\pi$
$577$	−11.6366			−0.484440			−0.242220	+	0.970221i	$0.577876\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	25.7782			1.06946
$582$		0			0
$583$			2.15426i			0.0892205i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−10.1689			−0.419714			−0.209857	−	0.977732i	$-0.567300\pi$
$587$	−10.1689			−0.419714			−0.209857	+	0.977732i	$0.567300\pi$
$588$		0			0
$589$	46.5019			1.91608
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−30.7427			−1.26245			−0.631226	−	0.775599i	$-0.717448\pi$
$593$	−30.7427			−1.26245			−0.631226	+	0.775599i	$0.717448\pi$
$594$		0			0
$595$	−54.1398			−2.21951
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	0.884078i		−	0.0361225i	−0.999837	−	0.0180612i	$-0.994251\pi$
$599$		−	0.884078i		−	0.0361225i	0.999837	−	0.0180612i	$-0.00574938\pi$
$600$		0			0
$601$	8.35635			0.340863			0.170431	−	0.985370i	$-0.445484\pi$
$601$	8.35635			0.340863			0.170431	+	0.985370i	$0.445484\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	35.8956			1.45936
$606$		0			0
$607$		−	35.0284i		−	1.42176i	−0.703313	−	0.710880i	$-0.748297\pi$
$607$		−	35.0284i		−	1.42176i	0.703313	−	0.710880i	$-0.251703\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−2.99762			−0.121271
$612$		0			0
$613$	27.7213			1.11965			0.559827	−	0.828610i	$-0.310868\pi$
$613$	27.7213			1.11965			0.559827	+	0.828610i	$0.310868\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	19.4355i		−	0.782445i	−0.920296	−	0.391223i	$-0.872052\pi$
$617$		−	19.4355i		−	0.782445i	0.920296	−	0.391223i	$-0.127948\pi$
$618$		0			0
$619$			19.6060i			0.788030i	0.919104	+	0.394015i	$0.128914\pi$
$619$			19.6060i			0.788030i	−0.919104	+	0.394015i	$0.871086\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			13.9359i			0.558330i
$624$		0			0
$625$	−21.3724			−0.854896
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	57.7795i		−	2.30382i
$630$		0			0
$631$	14.5937			0.580967			0.290483	−	0.956880i	$-0.406184\pi$
$631$	14.5937			0.580967			0.290483	+	0.956880i	$0.406184\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−43.5544			−1.72840
$636$		0			0
$637$			42.6314i			1.68912i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−42.9143			−1.69501			−0.847506	−	0.530786i	$-0.821897\pi$
$641$	−42.9143			−1.69501			−0.847506	+	0.530786i	$0.821897\pi$
$642$		0			0
$643$			24.5082i			0.966509i	0.875480	+	0.483255i	$0.160545\pi$
$643$			24.5082i			0.966509i	−0.875480	+	0.483255i	$0.839455\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−18.7375			−0.736648			−0.368324	−	0.929698i	$-0.620068\pi$
$647$	−18.7375			−0.736648			−0.368324	+	0.929698i	$0.620068\pi$
$648$		0			0
$649$		−	57.8100i		−	2.26924i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			27.0973i			1.06040i	0.847873	+	0.530200i	$0.177883\pi$
$653$			27.0973i			1.06040i	−0.847873	+	0.530200i	$0.822117\pi$
$654$		0			0
$655$	−10.1826			−0.397867
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−20.6136			−0.802994			−0.401497	−	0.915860i	$-0.631510\pi$
$659$	−20.6136			−0.802994			−0.401497	+	0.915860i	$0.631510\pi$
$660$		0			0
$661$		−	33.9954i		−	1.32227i	−0.750268	−	0.661134i	$-0.770076\pi$
$661$		−	33.9954i		−	1.32227i	0.750268	−	0.661134i	$-0.229924\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−65.7827			−2.55094
$666$		0			0
$667$		−	4.34103i		−	0.168085i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			7.99598i			0.308681i
$672$		0			0
$673$		−	35.7944i		−	1.37977i	−0.723917	−	0.689887i	$-0.757660\pi$
$673$		−	35.7944i		−	1.37977i	0.723917	−	0.689887i	$-0.242340\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			25.8702i			0.994274i	0.867672	+	0.497137i	$0.165615\pi$
$677$			25.8702i			0.994274i	−0.867672	+	0.497137i	$0.834385\pi$
$678$		0			0
$679$	−64.1404			−2.46148
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−8.26099			−0.316098			−0.158049	−	0.987431i	$-0.550520\pi$
$683$	−8.26099			−0.316098			−0.158049	+	0.987431i	$0.550520\pi$
$684$		0			0
$685$			2.25380i			0.0861134i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	1.96169i		−	0.0747344i
$690$		0			0
$691$			16.2248i			0.617220i	0.951189	+	0.308610i	$0.0998638\pi$
$691$			16.2248i			0.617220i	−0.951189	+	0.308610i	$0.900136\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			24.7096i			0.937287i
$696$		0			0
$697$	−49.4822			−1.87427
$698$		0			0
$699$		0			0
$700$		0			0
$701$			9.38880i			0.354610i	0.984156	+	0.177305i	$0.0567379\pi$
$701$			9.38880i			0.354610i	−0.984156	+	0.177305i	$0.943262\pi$
$702$		0			0
$703$		−	70.2052i		−	2.64784i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−4.80256			−0.180619
$708$		0			0
$709$		−	23.9016i		−	0.897645i	−0.893621	−	0.448822i	$-0.851844\pi$
$709$		−	23.9016i		−	0.897645i	0.893621	−	0.448822i	$-0.148156\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−8.05762			−0.301760
$714$		0			0
$715$	−53.5953			−2.00435
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−13.8522			−0.516600			−0.258300	−	0.966065i	$-0.583162\pi$
$719$	−13.8522			−0.516600			−0.258300	+	0.966065i	$0.583162\pi$
$720$		0			0
$721$			11.1139i			0.413904i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	2.03246i		−	0.0754835i
$726$		0			0
$727$			29.0448i			1.07721i	0.842558	+	0.538606i	$0.181049\pi$
$727$			29.0448i			1.07721i	−0.842558	+	0.538606i	$0.818951\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			6.88168i			0.254528i
$732$		0			0
$733$	−17.0241			−0.628798			−0.314399	−	0.949291i	$-0.601803\pi$
$733$	−17.0241			−0.628798			−0.314399	+	0.949291i	$0.601803\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	75.1784			2.76923
$738$		0			0
$739$			8.03430i			0.295547i	0.989021	+	0.147773i	$0.0472106\pi$
$739$			8.03430i			0.295547i	−0.989021	+	0.147773i	$0.952789\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			48.2156i			1.76886i	0.466674	+	0.884430i	$0.345452\pi$
$743$			48.2156i			1.76886i	−0.466674	+	0.884430i	$0.654548\pi$
$744$		0			0
$745$	−28.9303			−1.05992
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	43.5961i		−	1.59297i
$750$		0			0
$751$		−	34.2525i		−	1.24989i	−0.780668	−	0.624945i	$-0.785121\pi$
$751$		−	34.2525i		−	1.24989i	0.780668	−	0.624945i	$-0.214879\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			37.9152i			1.37987i
$756$		0			0
$757$	19.1088			0.694523			0.347261	−	0.937768i	$-0.387112\pi$
$757$	19.1088			0.694523			0.347261	+	0.937768i	$0.387112\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			54.0783i			1.96034i	0.198163	+	0.980169i	$0.436503\pi$
$761$			54.0783i			1.96034i	−0.198163	+	0.980169i	$0.563497\pi$
$762$		0			0
$763$			2.71192i			0.0981780i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			52.6422i			1.90080i
$768$		0			0
$769$			49.4481i			1.78314i	0.452880	+	0.891571i	$0.350396\pi$
$769$			49.4481i			1.78314i	−0.452880	+	0.891571i	$0.649604\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−35.6309			−1.28156			−0.640778	−	0.767726i	$-0.721388\pi$
$773$	−35.6309			−1.28156			−0.640778	+	0.767726i	$0.721388\pi$
$774$		0			0
$775$	−3.77255			−0.135514
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−60.1234			−2.15415
$780$		0			0
$781$			31.8722i			1.14048i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−14.4057			−0.514163
$786$		0			0
$787$			15.8163i			0.563790i	0.959445	+	0.281895i	$0.0909630\pi$
$787$			15.8163i			0.563790i	−0.959445	+	0.281895i	$0.909037\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−61.6823			−2.19317
$792$		0			0
$793$		−	7.28120i		−	0.258563i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−15.5032			−0.549151			−0.274576	−	0.961566i	$-0.588537\pi$
$797$	−15.5032			−0.549151			−0.274576	+	0.961566i	$0.588537\pi$
$798$		0			0
$799$			4.04304i			0.143032i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−19.4525			−0.686465
$804$		0			0
$805$	11.3985			0.401744
$806$		0			0
$807$		0			0
$808$		0			0
$809$			54.5804i			1.91895i	0.281801	+	0.959473i	$0.409068\pi$
$809$			54.5804i			1.91895i	−0.281801	+	0.959473i	$0.590932\pi$
$810$		0			0
$811$		−	30.0062i		−	1.05366i	−0.849971	−	0.526830i	$-0.823381\pi$
$811$		−	30.0062i		−	1.05366i	0.849971	−	0.526830i	$-0.176619\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	7.60626i		−	0.266436i
$816$		0			0
$817$			8.36160i			0.292535i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−26.6206			−0.929066			−0.464533	−	0.885556i	$-0.653778\pi$
$821$	−26.6206			−0.929066			−0.464533	+	0.885556i	$0.653778\pi$
$822$		0			0
$823$			48.5037i			1.69073i	0.534186	+	0.845367i	$0.320618\pi$
$823$			48.5037i			1.69073i	−0.534186	+	0.845367i	$0.679382\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−26.2081			−0.911345			−0.455673	−	0.890147i	$-0.650601\pi$
$827$	−26.2081			−0.911345			−0.455673	+	0.890147i	$0.650601\pi$
$828$		0			0
$829$		−	48.1328i		−	1.67172i	−0.548942	−	0.835861i	$-0.684969\pi$
$829$		−	48.1328i		−	1.67172i	0.548942	−	0.835861i	$-0.315031\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	57.4990			1.99222
$834$		0			0
$835$	10.9064	+	24.6717i	0.377433	+	0.853798i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	18.6091i		−	0.642457i	−0.947002	−	0.321228i	$-0.895904\pi$
$839$		−	18.6091i		−	0.642457i	0.947002	−	0.321228i	$-0.104096\pi$
$840$		0			0
$841$	19.0045			0.655327
$842$		0			0
$843$		0			0
$844$		0			0
$845$	21.6684			0.745416
$846$		0			0
$847$	−68.3907			−2.34993
$848$		0			0
$849$		0			0
$850$		0			0
$851$			12.1648i			0.417004i
$852$		0			0
$853$	40.0355			1.37079			0.685394	−	0.728172i	$-0.259630\pi$
$853$	40.0355			1.37079			0.685394	+	0.728172i	$0.259630\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	54.4291i		−	1.85926i	−0.368490	−	0.929632i	$-0.620125\pi$
$857$		−	54.4291i		−	1.85926i	0.368490	−	0.929632i	$-0.379875\pi$
$858$		0			0
$859$	−2.82159			−0.0962715			−0.0481357	−	0.998841i	$-0.515328\pi$
$859$	−2.82159			−0.0962715			−0.0481357	+	0.998841i	$0.515328\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			16.3174i			0.555452i	0.960660	+	0.277726i	$0.0895807\pi$
$863$			16.3174i			0.555452i	−0.960660	+	0.277726i	$0.910419\pi$
$864$		0			0
$865$			11.1520i			0.379178i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	31.9198			1.08280
$870$		0			0
$871$	−68.4579			−2.31961
$872$		0			0
$873$		0			0
$874$		0			0
$875$	46.8443			1.58363
$876$		0			0
$877$	3.53406			0.119337			0.0596684	−	0.998218i	$-0.480996\pi$
$877$	3.53406			0.119337			0.0596684	+	0.998218i	$0.480996\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−39.1515			−1.31905			−0.659524	−	0.751683i	$-0.729242\pi$
$881$	−39.1515			−1.31905			−0.659524	+	0.751683i	$0.729242\pi$
$882$		0			0
$883$	−32.2513			−1.08534			−0.542671	−	0.839945i	$-0.682587\pi$
$883$	−32.2513			−1.08534			−0.542671	+	0.839945i	$0.682587\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	7.29902			0.245077			0.122539	−	0.992464i	$-0.460896\pi$
$887$	7.29902			0.245077			0.122539	+	0.992464i	$0.460896\pi$
$888$		0			0
$889$	82.9828			2.78315
$890$		0			0
$891$		0			0
$892$		0			0
$893$			4.91250i			0.164391i
$894$		0			0
$895$			31.6355i			1.05746i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			18.5532i			0.618784i
$900$		0			0
$901$	−2.64582			−0.0881451
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−27.1407			−0.902188
$906$		0			0
$907$	−52.2728			−1.73569			−0.867844	−	0.496836i	$-0.834495\pi$
$907$	−52.2728			−1.73569			−0.867844	+	0.496836i	$0.834495\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	34.5021i		−	1.14311i	−0.820565	−	0.571553i	$-0.806341\pi$
$911$		−	34.5021i		−	1.14311i	0.820565	−	0.571553i	$-0.193659\pi$
$912$		0			0
$913$		−	34.4186i		−	1.13909i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	19.4005			0.640663
$918$		0			0
$919$	−43.0449			−1.41992			−0.709960	−	0.704242i	$-0.751287\pi$
$919$	−43.0449			−1.41992			−0.709960	+	0.704242i	$0.751287\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	29.0231i		−	0.955306i
$924$		0			0
$925$			5.69552i			0.187268i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			40.6479i			1.33361i	0.745230	+	0.666807i	$0.232340\pi$
$929$			40.6479i			1.33361i	−0.745230	+	0.666807i	$0.767660\pi$
$930$		0			0
$931$	69.8643			2.28971
$932$		0			0
$933$		0			0
$934$		0			0
$935$			72.2866i			2.36402i
$936$		0			0
$937$			16.4182i			0.536358i	0.963369	+	0.268179i	$0.0864219\pi$
$937$			16.4182i			0.536358i	−0.963369	+	0.268179i	$0.913578\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	51.3496			1.67395			0.836974	−	0.547243i	$-0.184323\pi$
$941$	51.3496			1.67395			0.836974	+	0.547243i	$0.184323\pi$
$942$		0			0
$943$	10.4179			0.339253
$944$		0			0
$945$		0			0
$946$		0			0
$947$			26.9408i			0.875458i	0.899107	+	0.437729i	$0.144217\pi$
$947$			26.9408i			0.875458i	−0.899107	+	0.437729i	$0.855783\pi$
$948$		0			0
$949$	17.7136			0.575008
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−48.4198			−1.56847			−0.784236	−	0.620463i	$-0.786945\pi$
$953$	−48.4198			−1.56847			−0.784236	+	0.620463i	$0.786945\pi$
$954$		0			0
$955$		−	46.1237i		−	1.49253i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	4.29410i		−	0.138664i
$960$		0			0
$961$	3.43760			0.110890
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	39.5732i		−	1.27391i
$966$		0			0
$967$	16.8327			0.541303			0.270652	−	0.962677i	$-0.412761\pi$
$967$	16.8327			0.541303			0.270652	+	0.962677i	$0.412761\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	0.572490			0.0183721			0.00918604	−	0.999958i	$-0.497076\pi$
$971$	0.572490			0.0183721			0.00918604	+	0.999958i	$0.497076\pi$
$972$		0			0
$973$		−	47.0783i		−	1.50926i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.66292			0.0532016			0.0266008	−	0.999646i	$-0.491532\pi$
$977$	1.66292			0.0532016			0.0266008	+	0.999646i	$0.491532\pi$
$978$		0			0
$979$	18.6070			0.594682
$980$		0			0
$981$		0			0
$982$		0			0
$983$	3.19139			0.101790			0.0508948	−	0.998704i	$-0.483793\pi$
$983$	3.19139			0.101790			0.0508948	+	0.998704i	$0.483793\pi$
$984$		0			0
$985$	−19.2582			−0.613617
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	1.44886i		−	0.0460709i
$990$		0			0
$991$		−	18.6097i		−	0.591156i	−0.955319	−	0.295578i	$-0.904488\pi$
$991$		−	18.6097i		−	0.591156i	0.955319	−	0.295578i	$-0.0955122\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	26.5146			0.840571
$996$		0			0
$997$	−8.93965			−0.283122			−0.141561	−	0.989930i	$-0.545212\pi$
$997$	−8.93965			−0.283122			−0.141561	+	0.989930i	$0.545212\pi$
$998$		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.3005.15	✓	56
3.2	odd	2	inner	6012.2.h.a.3005.41	yes	56
167.166	odd	2	inner	6012.2.h.a.3005.42	yes	56
501.500	even	2	inner	6012.2.h.a.3005.16	yes	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6012.2.h.a.3005.15	✓	56	1.1	even	1	trivial
6012.2.h.a.3005.16	yes	56	501.500	even	2	inner
6012.2.h.a.3005.41	yes	56	3.2	odd	2	inner
6012.2.h.a.3005.42	yes	56	167.166	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)

\(f(q)\)	\(=\)	\(q-2.08738 q^{5} +3.97701 q^{7} +O(q^{10})\)	\(q-2.08738 q^{5} +3.97701 q^{7} -5.31004i q^{11} +4.83536i q^{13} +6.52168 q^{17} +7.92418 q^{19} -1.37306 q^{23} -0.642863 q^{25} +3.16157i q^{29} +5.86836 q^{31} -8.30151 q^{35} -8.85961i q^{37} -7.58734 q^{41} +1.05520i q^{43} +0.619938i q^{47} +8.81660 q^{49} -0.405697 q^{53} +11.0840i q^{55} +10.8869 q^{59} -1.50582 q^{61} -10.0932i q^{65} +14.1578i q^{67} -6.00226 q^{71} -3.66335i q^{73} -21.1181i q^{77} +6.01121i q^{79} +6.48181 q^{83} -13.6132 q^{85} +3.50411i q^{89} +19.2303i q^{91} -16.5407 q^{95} -16.1278 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q+O(q^{10}) \) 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

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