LMFDB - Embedded newform 6336.2.f.j.3169.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6336,2,Mod(3169,6336)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6336, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6336.3169"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 6336 = 2^{6} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6336.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-8,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(41)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$50.5932147207$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.3057647616.6
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 30x^{4} + 9 $ x^8 + 30*x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3169.4
Root			$0.524648 - 0.524648i$ of defining polynomial
Character	$\chi$	$=$	6336.3169
Dual form			6336.2.f.j.3169.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.04930i q^{5} +1.04930 q^{7} -1.00000i q^{11} -3.61953i q^{13} +4.89898 q^{17} -2.89898i q^{19} +5.71812 q^{23} +3.89898 q^{25} +7.23907i q^{29} +2.09859 q^{31} -1.10102i q^{35} +11.4362i q^{37} +1.10102 q^{41} +2.89898i q^{43} +5.71812 q^{47} -5.89898 q^{49} -8.28836i q^{53} -1.04930 q^{55} -9.79796i q^{59} +5.71812i q^{61} -3.79796 q^{65} +7.79796i q^{67} -15.0558 q^{71} +2.00000 q^{73} -1.04930i q^{77} +12.4855 q^{79} -1.10102i q^{83} -5.14048i q^{85} +12.0000 q^{89} -3.79796i q^{91} -3.04189 q^{95} -10.6969 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{25} + 48 q^{41} - 8 q^{49} + 48 q^{65} + 16 q^{73} + 96 q^{89} + 32 q^{97}+O(q^{100}) $ 8 * q - 8 * q^25 + 48 * q^41 - 8 * q^49 + 48 * q^65 + 16 * q^73 + 96 * q^89 + 32 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6336\mathbb{Z}\right)^\times$.

$n$	$1729$	$3521$	$4159$	$4357$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$5$	− 1.04930i	− 0.469259i	−0.972085	−	0.234630i	$-0.924612\pi$
$5$	− 1.04930i	− 0.469259i	0.972085	−	0.234630i	$-0.0753877\pi$
$6$	0	0
$7$	1.04930	0.396596	0.198298	−	0.980142i	$-0.436459\pi$
$7$	1.04930	0.396596	0.198298	+	0.980142i	$0.436459\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 1.00000i	− 0.301511i
$11$	− 1.00000i	− 0.301511i
$12$	0	0
$13$	− 3.61953i	− 1.00388i	−0.864903	−	0.501939i	$-0.832620\pi$
$13$	− 3.61953i	− 1.00388i	0.864903	−	0.501939i	$-0.167380\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.89898	1.18818	0.594089	−	0.804400i	$-0.297513\pi$
$17$	4.89898	1.18818	0.594089	+	0.804400i	$0.297513\pi$
$18$	0	0
$19$	− 2.89898i	− 0.665072i	−0.943091	−	0.332536i	$-0.892096\pi$
$19$	− 2.89898i	− 0.665072i	0.943091	−	0.332536i	$-0.107904\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.71812	1.19231	0.596156	−	0.802869i	$-0.296694\pi$
$23$	5.71812	1.19231	0.596156	+	0.802869i	$0.296694\pi$
$24$	0	0
$25$	3.89898	0.779796
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7.23907i	1.34426i	0.740433	+	0.672130i	$0.234621\pi$
$29$	7.23907i	1.34426i	−0.740433	+	0.672130i	$0.765379\pi$
$30$	0	0
$31$	2.09859	0.376918	0.188459	−	0.982081i	$-0.439651\pi$
$31$	2.09859	0.376918	0.188459	+	0.982081i	$0.439651\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 1.10102i	− 0.186106i
$36$	0	0
$37$	11.4362i	1.88011i	0.341026	+	0.940054i	$0.389225\pi$
$37$	11.4362i	1.88011i	−0.341026	+	0.940054i	$0.610775\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.10102	0.171951	0.0859753	−	0.996297i	$-0.472599\pi$
$41$	1.10102	0.171951	0.0859753	+	0.996297i	$0.472599\pi$
$42$	0	0
$43$	2.89898i	0.442090i	0.975264	+	0.221045i	$0.0709468\pi$
$43$	2.89898i	0.442090i	−0.975264	+	0.221045i	$0.929053\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.71812	0.834074	0.417037	−	0.908889i	$-0.363069\pi$
$47$	5.71812	0.834074	0.417037	+	0.908889i	$0.363069\pi$
$48$	0	0
$49$	−5.89898	−0.842711
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 8.28836i	− 1.13849i	−0.822167	−	0.569247i	$-0.807235\pi$
$53$	− 8.28836i	− 1.13849i	0.822167	−	0.569247i	$-0.192765\pi$
$54$	0	0
$55$	−1.04930	−0.141487
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 9.79796i	− 1.27559i	−0.770208	−	0.637793i	$-0.779848\pi$
$59$	− 9.79796i	− 1.27559i	0.770208	−	0.637793i	$-0.220152\pi$
$60$	0	0
$61$	5.71812i	0.732131i	0.930589	+	0.366065i	$0.119295\pi$
$61$	5.71812i	0.732131i	−0.930589	+	0.366065i	$0.880705\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.79796	−0.471079
$66$	0	0
$67$	7.79796i	0.952672i	0.879263	+	0.476336i	$0.158035\pi$
$67$	7.79796i	0.952672i	−0.879263	+	0.476336i	$0.841965\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−15.0558	−1.78679	−0.893396	−	0.449270i	$-0.851684\pi$
$71$	−15.0558	−1.78679	−0.893396	+	0.449270i	$0.851684\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 1.04930i	− 0.119578i
$78$	0	0
$79$	12.4855	1.40473	0.702367	−	0.711815i	$-0.252127\pi$
$79$	12.4855	1.40473	0.702367	+	0.711815i	$0.252127\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 1.10102i	− 0.120853i	−0.998173	−	0.0604264i	$-0.980754\pi$
$83$	− 1.10102i	− 0.120853i	0.998173	−	0.0604264i	$-0.0192460\pi$
$84$	0	0
$85$	− 5.14048i	− 0.557563i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	0	0
$91$	− 3.79796i	− 0.398134i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−3.04189	−0.312091
$96$	0	0
$97$	−10.6969	−1.08611	−0.543055	−	0.839697i	$-0.682732\pi$
$97$	−10.6969	−1.08611	−0.543055	+	0.839697i	$0.682732\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 9.33766i	− 0.929132i	−0.885539	−	0.464566i	$-0.846210\pi$
$101$	− 9.33766i	− 0.929132i	0.885539	−	0.464566i	$-0.153790\pi$
$102$	0	0
$103$	−14.4781	−1.42657	−0.713286	−	0.700873i	$-0.752794\pi$
$103$	−14.4781	−1.42657	−0.713286	+	0.700873i	$0.752794\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 12.0000i	− 1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$	− 12.0000i	− 1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$	0	0
$109$	− 7.81671i	− 0.748705i	−0.927286	−	0.374353i	$-0.877865\pi$
$109$	− 7.81671i	− 0.748705i	0.927286	−	0.374353i	$-0.122135\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	9.79796	0.921714	0.460857	−	0.887474i	$-0.347542\pi$
$113$	9.79796	0.921714	0.460857	+	0.887474i	$0.347542\pi$
$114$	0	0
$115$	− 6.00000i	− 0.559503i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	5.14048	0.471227
$120$	0	0
$121$	−1.00000	−0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 9.33766i	− 0.835185i
$126$	0	0
$127$	−10.3870	−0.921693	−0.460846	−	0.887480i	$-0.652454\pi$
$127$	−10.3870	−0.921693	−0.460846	+	0.887480i	$0.652454\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 20.6969i	− 1.80830i	−0.427215	−	0.904150i	$-0.640505\pi$
$131$	− 20.6969i	− 1.80830i	0.427215	−	0.904150i	$-0.359495\pi$
$132$	0	0
$133$	− 3.04189i	− 0.263765i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	21.7980	1.86233	0.931163	−	0.364604i	$-0.118796\pi$
$137$	21.7980	1.86233	0.931163	+	0.364604i	$0.118796\pi$
$138$	0	0
$139$	6.89898i	0.585164i	0.956240	+	0.292582i	$0.0945144\pi$
$139$	6.89898i	0.585164i	−0.956240	+	0.292582i	$0.905486\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−3.61953	−0.302681
$144$	0	0
$145$	7.59592	0.630807
$146$	0	0
$147$	0	0
$148$	0	0
$149$	15.6334i	1.28074i	0.768066	+	0.640370i	$0.221219\pi$
$149$	15.6334i	1.28074i	−0.768066	+	0.640370i	$0.778781\pi$
$150$	0	0
$151$	17.6260	1.43439	0.717193	−	0.696875i	$-0.245427\pi$
$151$	17.6260	1.43439	0.717193	+	0.696875i	$0.245427\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 2.20204i	− 0.176872i
$156$	0	0
$157$	3.04189i	0.242769i	0.992606	+	0.121384i	$0.0387334\pi$
$157$	3.04189i	0.242769i	−0.992606	+	0.121384i	$0.961267\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	6.00000	0.472866
$162$	0	0
$163$	− 4.00000i	− 0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$	− 4.00000i	− 0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−4.19718	−0.324788	−0.162394	−	0.986726i	$-0.551921\pi$
$167$	−4.19718	−0.324788	−0.162394	+	0.986726i	$0.551921\pi$
$168$	0	0
$169$	−0.101021	−0.00777081
$170$	0	0
$171$	0	0
$172$	0	0
$173$	15.6334i	1.18859i	0.804248	+	0.594294i	$0.202568\pi$
$173$	15.6334i	1.18859i	−0.804248	+	0.594294i	$0.797432\pi$
$174$	0	0
$175$	4.09118	0.309264
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 2.20204i	− 0.164588i	−0.996608	−	0.0822941i	$-0.973775\pi$
$179$	− 2.20204i	− 0.164588i	0.996608	−	0.0822941i	$-0.0262247\pi$
$180$	0	0
$181$	− 4.19718i	− 0.311974i	−0.987759	−	0.155987i	$-0.950144\pi$
$181$	− 4.19718i	− 0.311974i	0.987759	−	0.155987i	$-0.0498558\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	12.0000	0.882258
$186$	0	0
$187$	− 4.89898i	− 0.358249i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.52094	0.110052	0.0550258	−	0.998485i	$-0.482476\pi$
$191$	1.52094	0.110052	0.0550258	+	0.998485i	$0.482476\pi$
$192$	0	0
$193$	11.7980	0.849236	0.424618	−	0.905373i	$-0.360408\pi$
$193$	11.7980	0.849236	0.424618	+	0.905373i	$0.360408\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	13.5348i	0.964317i	0.876084	+	0.482159i	$0.160147\pi$
$197$	13.5348i	0.964317i	−0.876084	+	0.482159i	$0.839853\pi$
$198$	0	0
$199$	16.5767	1.17509	0.587546	−	0.809190i	$-0.300094\pi$
$199$	16.5767	1.17509	0.587546	+	0.809190i	$0.300094\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	7.59592i	0.533129i
$204$	0	0
$205$	− 1.15530i	− 0.0806893i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−2.89898	−0.200527
$210$	0	0
$211$	− 16.6969i	− 1.14946i	−0.818341	−	0.574732i	$-0.805106\pi$
$211$	− 16.6969i	− 1.14946i	0.818341	−	0.574732i	$-0.194894\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	3.04189	0.207455
$216$	0	0
$217$	2.20204	0.149484
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 17.7320i	− 1.19278i
$222$	0	0
$223$	−18.6753	−1.25059	−0.625296	−	0.780388i	$-0.715022\pi$
$223$	−18.6753	−1.25059	−0.625296	+	0.780388i	$0.715022\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 21.7980i	− 1.44678i	−0.690439	−	0.723391i	$-0.742583\pi$
$227$	− 21.7980i	− 1.44678i	0.690439	−	0.723391i	$-0.257417\pi$
$228$	0	0
$229$	11.4362i	0.755728i	0.925861	+	0.377864i	$0.123341\pi$
$229$	11.4362i	0.755728i	−0.925861	+	0.377864i	$0.876659\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	22.8990	1.50016	0.750081	−	0.661346i	$-0.230014\pi$
$233$	22.8990	1.50016	0.750081	+	0.661346i	$0.230014\pi$
$234$	0	0
$235$	− 6.00000i	− 0.391397i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	14.4781	0.936513	0.468256	−	0.883593i	$-0.344882\pi$
$239$	14.4781	0.936513	0.468256	+	0.883593i	$0.344882\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	6.18977i	0.395450i
$246$	0	0
$247$	−10.4930	−0.667651
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 21.7980i	− 1.37587i	−0.725770	−	0.687937i	$-0.758516\pi$
$251$	− 21.7980i	− 1.37587i	0.725770	−	0.687937i	$-0.241484\pi$
$252$	0	0
$253$	− 5.71812i	− 0.359495i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−7.59592	−0.473820	−0.236910	−	0.971532i	$-0.576135\pi$
$257$	−7.59592	−0.473820	−0.236910	+	0.971532i	$0.576135\pi$
$258$	0	0
$259$	12.0000i	0.745644i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−15.6334	−0.963998	−0.481999	−	0.876172i	$-0.660089\pi$
$263$	−15.6334	−0.963998	−0.481999	+	0.876172i	$0.660089\pi$
$264$	0	0
$265$	−8.69694	−0.534249
$266$	0	0
$267$	0	0
$268$	0	0
$269$	5.24648i	0.319883i	0.987126	+	0.159942i	$0.0511306\pi$
$269$	5.24648i	0.319883i	−0.987126	+	0.159942i	$0.948869\pi$
$270$	0	0
$271$	−28.1190	−1.70811	−0.854053	−	0.520186i	$-0.825863\pi$
$271$	−28.1190	−1.70811	−0.854053	+	0.520186i	$0.825863\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 3.89898i	− 0.235117i
$276$	0	0
$277$	2.67624i	0.160800i	0.996763	+	0.0803998i	$0.0256197\pi$
$277$	2.67624i	0.160800i	−0.996763	+	0.0803998i	$0.974380\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	19.1010	1.13947	0.569736	−	0.821828i	$-0.307046\pi$
$281$	19.1010	1.13947	0.569736	+	0.821828i	$0.307046\pi$
$282$	0	0
$283$	18.8990i	1.12343i	0.827332	+	0.561714i	$0.189858\pi$
$283$	18.8990i	1.12343i	−0.827332	+	0.561714i	$0.810142\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.15530	0.0681949
$288$	0	0
$289$	7.00000	0.411765
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 17.7320i	− 1.03592i	−0.855406	−	0.517958i	$-0.826692\pi$
$293$	− 17.7320i	− 1.03592i	0.855406	−	0.517958i	$-0.173308\pi$
$294$	0	0
$295$	−10.2810	−0.598580
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 20.6969i	− 1.19693i
$300$	0	0
$301$	3.04189i	0.175331i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	6.00000	0.343559
$306$	0	0
$307$	12.6969i	0.724653i	0.932051	+	0.362326i	$0.118017\pi$
$307$	12.6969i	0.724653i	−0.932051	+	0.362326i	$0.881983\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−6.66142	−0.377734	−0.188867	−	0.982003i	$-0.560482\pi$
$311$	−6.66142	−0.377734	−0.188867	+	0.982003i	$0.560482\pi$
$312$	0	0
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	21.8232i	1.22571i	0.790194	+	0.612856i	$0.209980\pi$
$317$	21.8232i	1.22571i	−0.790194	+	0.612856i	$0.790020\pi$
$318$	0	0
$319$	7.23907	0.405310
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 14.2020i	− 0.790223i
$324$	0	0
$325$	− 14.1125i	− 0.782820i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	6.00000	0.330791
$330$	0	0
$331$	− 4.00000i	− 0.219860i	−0.993939	−	0.109930i	$-0.964937\pi$
$331$	− 4.00000i	− 0.219860i	0.993939	−	0.109930i	$-0.0350627\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	8.18236	0.447050
$336$	0	0
$337$	0.202041	0.0110059	0.00550294	−	0.999985i	$-0.498248\pi$
$337$	0.202041	0.0110059	0.00550294	+	0.999985i	$0.498248\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 2.09859i	− 0.113645i
$342$	0	0
$343$	−13.5348	−0.730813
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 1.10102i	− 0.0591059i	−0.999563	−	0.0295529i	$-0.990592\pi$
$347$	− 1.10102i	− 0.0591059i	0.999563	−	0.0295529i	$-0.00940836\pi$
$348$	0	0
$349$	− 22.2948i	− 1.19342i	−0.802459	−	0.596708i	$-0.796475\pi$
$349$	− 22.2948i	− 1.19342i	0.802459	−	0.596708i	$-0.203525\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	7.59592	0.404290	0.202145	−	0.979356i	$-0.435209\pi$
$353$	7.59592	0.404290	0.202145	+	0.979356i	$0.435209\pi$
$354$	0	0
$355$	15.7980i	0.838469i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	30.1116	1.58923	0.794614	−	0.607115i	$-0.207673\pi$
$359$	30.1116	1.58923	0.794614	+	0.607115i	$0.207673\pi$
$360$	0	0
$361$	10.5959	0.557680
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 2.09859i	− 0.109845i
$366$	0	0
$367$	27.0697	1.41303	0.706513	−	0.707700i	$-0.250267\pi$
$367$	27.0697	1.41303	0.706513	+	0.707700i	$0.250267\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 8.69694i	− 0.451523i
$372$	0	0
$373$	3.61953i	0.187412i	0.995600	+	0.0937062i	$0.0298714\pi$
$373$	3.61953i	0.187412i	−0.995600	+	0.0937062i	$0.970129\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	26.2020	1.34947
$378$	0	0
$379$	− 10.2020i	− 0.524044i	−0.965062	−	0.262022i	$-0.915611\pi$
$379$	− 10.2020i	− 0.524044i	0.965062	−	0.262022i	$-0.0843892\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−13.9005	−0.710282	−0.355141	−	0.934813i	$-0.615567\pi$
$383$	−13.9005	−0.710282	−0.355141	+	0.934813i	$0.615567\pi$
$384$	0	0
$385$	−1.10102	−0.0561132
$386$	0	0
$387$	0	0
$388$	0	0
$389$	6.18977i	0.313834i	0.987612	+	0.156917i	$0.0501555\pi$
$389$	6.18977i	0.313834i	−0.987612	+	0.156917i	$0.949845\pi$
$390$	0	0
$391$	28.0130	1.41668
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 13.1010i	− 0.659184i
$396$	0	0
$397$	− 18.6753i	− 0.937287i	−0.883387	−	0.468644i	$-0.844743\pi$
$397$	− 18.6753i	− 0.937287i	0.883387	−	0.468644i	$-0.155257\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2.20204	−0.109965	−0.0549823	−	0.998487i	$-0.517510\pi$
$401$	−2.20204	−0.109965	−0.0549823	+	0.998487i	$0.517510\pi$
$402$	0	0
$403$	− 7.59592i	− 0.378380i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	11.4362	0.566874
$408$	0	0
$409$	−29.5959	−1.46342	−0.731712	−	0.681614i	$-0.761278\pi$
$409$	−29.5959	−1.46342	−0.731712	+	0.681614i	$0.761278\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 10.2810i	− 0.505893i
$414$	0	0
$415$	−1.15530	−0.0567112
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	19.8306i	0.966485i	0.875487	+	0.483242i	$0.160541\pi$
$421$	19.8306i	0.966485i	−0.875487	+	0.483242i	$0.839459\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	19.1010	0.926536
$426$	0	0
$427$	6.00000i	0.290360i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	1.15530	0.0556486	0.0278243	−	0.999613i	$-0.491142\pi$
$431$	1.15530	0.0556486	0.0278243	+	0.999613i	$0.491142\pi$
$432$	0	0
$433$	−21.5959	−1.03783	−0.518917	−	0.854825i	$-0.673665\pi$
$433$	−21.5959	−1.03783	−0.518917	+	0.854825i	$0.673665\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 16.5767i	− 0.792972i
$438$	0	0
$439$	−26.9637	−1.28691	−0.643453	−	0.765486i	$-0.722499\pi$
$439$	−26.9637	−1.28691	−0.643453	+	0.765486i	$0.722499\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 9.79796i	− 0.465515i	−0.972535	−	0.232758i	$-0.925225\pi$
$443$	− 9.79796i	− 0.465515i	0.972535	−	0.232758i	$-0.0747749\pi$
$444$	0	0
$445$	− 12.5915i	− 0.596896i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	− 1.10102i	− 0.0518450i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−3.98518	−0.186828
$456$	0	0
$457$	14.0000	0.654892	0.327446	−	0.944870i	$-0.393812\pi$
$457$	14.0000	0.654892	0.327446	+	0.944870i	$0.393812\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 23.8158i	− 1.10921i	−0.832113	−	0.554606i	$-0.812869\pi$
$461$	− 23.8158i	− 1.10921i	0.832113	−	0.554606i	$-0.187131\pi$
$462$	0	0
$463$	8.39436	0.390119	0.195060	−	0.980791i	$-0.437510\pi$
$463$	8.39436	0.390119	0.195060	+	0.980791i	$0.437510\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 31.5959i	− 1.46208i	−0.682332	−	0.731042i	$-0.739034\pi$
$467$	− 31.5959i	− 1.46208i	0.682332	−	0.731042i	$-0.260966\pi$
$468$	0	0
$469$	8.18236i	0.377826i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	2.89898	0.133295
$474$	0	0
$475$	− 11.3031i	− 0.518620i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−42.7031	−1.95115	−0.975577	−	0.219656i	$-0.929506\pi$
$479$	−42.7031	−1.95115	−0.975577	+	0.219656i	$0.929506\pi$
$480$	0	0
$481$	41.3939	1.88740
$482$	0	0
$483$	0	0
$484$	0	0
$485$	11.2242i	0.509667i
$486$	0	0
$487$	24.9711	1.13155	0.565774	−	0.824560i	$-0.308577\pi$
$487$	24.9711	1.13155	0.565774	+	0.824560i	$0.308577\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	17.3939i	0.784975i	0.919757	+	0.392487i	$0.128385\pi$
$491$	17.3939i	0.784975i	−0.919757	+	0.392487i	$0.871615\pi$
$492$	0	0
$493$	35.4640i	1.59722i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−15.7980	−0.708635
$498$	0	0
$499$	− 8.00000i	− 0.358129i	−0.983837	−	0.179065i	$-0.942693\pi$
$499$	− 8.00000i	− 0.358129i	0.983837	−	0.179065i	$-0.0573071\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−41.5478	−1.85253	−0.926263	−	0.376879i	$-0.876997\pi$
$503$	−41.5478	−1.85253	−0.926263	+	0.376879i	$0.876997\pi$
$504$	0	0
$505$	−9.79796	−0.436003
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 18.7813i	− 0.832467i	−0.909258	−	0.416234i	$-0.863350\pi$
$509$	− 18.7813i	− 0.832467i	0.909258	−	0.416234i	$-0.136650\pi$
$510$	0	0
$511$	2.09859	0.0928362
$512$	0	0
$513$	0	0
$514$	0	0
$515$	15.1918i	0.669432i
$516$	0	0
$517$	− 5.71812i	− 0.251483i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	33.7980	1.48072	0.740358	−	0.672213i	$-0.234656\pi$
$521$	33.7980	1.48072	0.740358	+	0.672213i	$0.234656\pi$
$522$	0	0
$523$	− 4.69694i	− 0.205383i	−0.994713	−	0.102691i	$-0.967255\pi$
$523$	− 4.69694i	− 0.205383i	0.994713	−	0.102691i	$-0.0327454\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	10.2810	0.447845
$528$	0	0
$529$	9.69694	0.421606
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 3.98518i	− 0.172617i
$534$	0	0
$535$	−12.5915	−0.544380
$536$	0	0
$537$	0	0
$538$	0	0
$539$	5.89898i	0.254087i
$540$	0	0
$541$	19.2530i	0.827749i	0.910334	+	0.413875i	$0.135825\pi$
$541$	19.2530i	0.827749i	−0.910334	+	0.413875i	$0.864175\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−8.20204	−0.351337
$546$	0	0
$547$	18.8990i	0.808062i	0.914745	+	0.404031i	$0.132391\pi$
$547$	18.8990i	0.808062i	−0.914745	+	0.404031i	$0.867609\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	20.9859	0.894030
$552$	0	0
$553$	13.1010	0.557112
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 32.2102i	− 1.36479i	−0.730985	−	0.682394i	$-0.760939\pi$
$557$	− 32.2102i	− 1.36479i	0.730985	−	0.682394i	$-0.239061\pi$
$558$	0	0
$559$	10.4930	0.443805
$560$	0	0
$561$	0	0
$562$	0	0
$563$	21.7980i	0.918674i	0.888262	+	0.459337i	$0.151913\pi$
$563$	21.7980i	0.918674i	−0.888262	+	0.459337i	$0.848087\pi$
$564$	0	0
$565$	− 10.2810i	− 0.432523i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	14.6969	0.616128	0.308064	−	0.951366i	$-0.400319\pi$
$569$	14.6969	0.616128	0.308064	+	0.951366i	$0.400319\pi$
$570$	0	0
$571$	− 42.0908i	− 1.76145i	−0.473632	−	0.880723i	$-0.657057\pi$
$571$	− 42.0908i	− 1.76145i	0.473632	−	0.880723i	$-0.342943\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	22.2948	0.929759
$576$	0	0
$577$	−27.1010	−1.12823	−0.564115	−	0.825696i	$-0.690783\pi$
$577$	−27.1010	−1.12823	−0.564115	+	0.825696i	$0.690783\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 1.15530i	− 0.0479297i
$582$	0	0
$583$	−8.28836	−0.343269
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 9.79796i	− 0.404405i	−0.979344	−	0.202203i	$-0.935190\pi$
$587$	− 9.79796i	− 0.404405i	0.979344	−	0.202203i	$-0.0648099\pi$
$588$	0	0
$589$	− 6.08377i	− 0.250677i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	4.89898	0.201177	0.100588	−	0.994928i	$-0.467927\pi$
$593$	4.89898	0.201177	0.100588	+	0.994928i	$0.467927\pi$
$594$	0	0
$595$	− 5.39388i	− 0.221127i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	22.2948	0.910943	0.455471	−	0.890250i	$-0.349471\pi$
$599$	22.2948	0.910943	0.455471	+	0.890250i	$0.349471\pi$
$600$	0	0
$601$	31.3939	1.28058	0.640291	−	0.768132i	$-0.278814\pi$
$601$	31.3939	1.28058	0.640291	+	0.768132i	$0.278814\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.04930i	0.0426599i
$606$	0	0
$607$	5.24648	0.212948	0.106474	−	0.994315i	$-0.466044\pi$
$607$	5.24648	0.212948	0.106474	+	0.994315i	$0.466044\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 20.6969i	− 0.837309i
$612$	0	0
$613$	12.9572i	0.523336i	0.965158	+	0.261668i	$0.0842725\pi$
$613$	12.9572i	0.523336i	−0.965158	+	0.261668i	$0.915727\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	2.20204	0.0886508	0.0443254	−	0.999017i	$-0.485886\pi$
$617$	2.20204	0.0886508	0.0443254	+	0.999017i	$0.485886\pi$
$618$	0	0
$619$	14.0000i	0.562708i	0.959604	+	0.281354i	$0.0907834\pi$
$619$	14.0000i	0.562708i	−0.959604	+	0.281354i	$0.909217\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	12.5915	0.504470
$624$	0	0
$625$	9.69694	0.387878
$626$	0	0
$627$	0	0
$628$	0	0
$629$	56.0259i	2.23390i
$630$	0	0
$631$	−12.3795	−0.492822	−0.246411	−	0.969165i	$-0.579251\pi$
$631$	−12.3795	−0.492822	−0.246411	+	0.969165i	$0.579251\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	10.8990i	0.432513i
$636$	0	0
$637$	21.3516i	0.845979i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−43.5959	−1.72194	−0.860968	−	0.508660i	$-0.830141\pi$
$641$	−43.5959	−1.72194	−0.860968	+	0.508660i	$0.830141\pi$
$642$	0	0
$643$	− 34.0000i	− 1.34083i	−0.741987	−	0.670415i	$-0.766116\pi$
$643$	− 34.0000i	− 1.34083i	0.741987	−	0.670415i	$-0.233884\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−4.77483	−0.187718	−0.0938589	−	0.995586i	$-0.529920\pi$
$647$	−4.77483	−0.187718	−0.0938589	+	0.995586i	$0.529920\pi$
$648$	0	0
$649$	−9.79796	−0.384604
$650$	0	0
$651$	0	0
$652$	0	0
$653$	32.1042i	1.25633i	0.778079	+	0.628166i	$0.216194\pi$
$653$	32.1042i	1.25633i	−0.778079	+	0.628166i	$0.783806\pi$
$654$	0	0
$655$	−21.7172	−0.848561
$656$	0	0
$657$	0	0
$658$	0	0
$659$	31.5959i	1.23080i	0.788214	+	0.615401i	$0.211006\pi$
$659$	31.5959i	1.23080i	−0.788214	+	0.615401i	$0.788994\pi$
$660$	0	0
$661$	19.8306i	0.771321i	0.922641	+	0.385661i	$0.126026\pi$
$661$	19.8306i	0.771321i	−0.922641	+	0.385661i	$0.873974\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−3.19184	−0.123774
$666$	0	0
$667$	41.3939i	1.60278i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	5.71812	0.220746
$672$	0	0
$673$	−43.3939	−1.67271	−0.836356	−	0.548187i	$-0.815318\pi$
$673$	−43.3939	−1.67271	−0.836356	+	0.548187i	$0.815318\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 0.943295i	− 0.0362538i	−0.999836	−	0.0181269i	$-0.994230\pi$
$677$	− 0.943295i	− 0.0362538i	0.999836	−	0.0181269i	$-0.00577028\pi$
$678$	0	0
$679$	−11.2242	−0.430747
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 39.1918i	− 1.49963i	−0.661645	−	0.749817i	$-0.730142\pi$
$683$	− 39.1918i	− 1.49963i	0.661645	−	0.749817i	$-0.269858\pi$
$684$	0	0
$685$	− 22.8725i	− 0.873913i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−30.0000	−1.14291
$690$	0	0
$691$	− 2.40408i	− 0.0914556i	−0.998954	−	0.0457278i	$-0.985439\pi$
$691$	− 2.40408i	− 0.0914556i	0.998954	−	0.0457278i	$-0.0145607\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	7.23907	0.274593
$696$	0	0
$697$	5.39388	0.204308
$698$	0	0
$699$	0	0
$700$	0	0
$701$	9.33766i	0.352678i	0.984329	+	0.176339i	$0.0564256\pi$
$701$	9.33766i	0.352678i	−0.984329	+	0.176339i	$0.943574\pi$
$702$	0	0
$703$	33.1534	1.25041
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 9.79796i	− 0.368490i
$708$	0	0
$709$	− 3.04189i	− 0.114240i	−0.998367	−	0.0571202i	$-0.981808\pi$
$709$	− 3.04189i	− 0.114240i	0.998367	−	0.0571202i	$-0.0181918\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	12.0000	0.449404
$714$	0	0
$715$	3.79796i	0.142036i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	2.46424	0.0919006	0.0459503	−	0.998944i	$-0.485368\pi$
$719$	2.46424	0.0919006	0.0459503	+	0.998944i	$0.485368\pi$
$720$	0	0
$721$	−15.1918	−0.565774
$722$	0	0
$723$	0	0
$724$	0	0
$725$	28.2250i	1.04825i
$726$	0	0
$727$	6.29577	0.233497	0.116749	−	0.993161i	$-0.462753\pi$
$727$	6.29577	0.233497	0.116749	+	0.993161i	$0.462753\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	14.2020i	0.525281i
$732$	0	0
$733$	− 22.2948i	− 0.823479i	−0.911302	−	0.411739i	$-0.864921\pi$
$733$	− 22.2948i	− 0.823479i	0.911302	−	0.411739i	$-0.135079\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	7.79796	0.287242
$738$	0	0
$739$	− 11.3031i	− 0.415790i	−0.978151	−	0.207895i	$-0.933339\pi$
$739$	− 11.3031i	− 0.415790i	0.978151	−	0.207895i	$-0.0666612\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	21.7172	0.796727	0.398363	−	0.917228i	$-0.369578\pi$
$743$	21.7172	0.796727	0.398363	+	0.917228i	$0.369578\pi$
$744$	0	0
$745$	16.4041	0.600999
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 12.5915i	− 0.460085i
$750$	0	0
$751$	−12.3795	−0.451736	−0.225868	−	0.974158i	$-0.572522\pi$
$751$	−12.3795	−0.451736	−0.225868	+	0.974158i	$0.572522\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 18.4949i	− 0.673098i
$756$	0	0
$757$	− 30.1116i	− 1.09442i	−0.836994	−	0.547212i	$-0.815689\pi$
$757$	− 30.1116i	− 1.09442i	0.836994	−	0.547212i	$-0.184311\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−4.89898	−0.177588	−0.0887939	−	0.996050i	$-0.528301\pi$
$761$	−4.89898	−0.177588	−0.0887939	+	0.996050i	$0.528301\pi$
$762$	0	0
$763$	− 8.20204i	− 0.296934i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−35.4640	−1.28053
$768$	0	0
$769$	−10.0000	−0.360609	−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000	−0.360609	−0.180305	+	0.983611i	$0.557708\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 38.3999i	− 1.38115i	−0.723261	−	0.690575i	$-0.757358\pi$
$773$	− 38.3999i	− 1.38115i	0.723261	−	0.690575i	$-0.242642\pi$
$774$	0	0
$775$	8.18236	0.293919
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 3.19184i	− 0.114359i
$780$	0	0
$781$	15.0558i	0.538738i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	3.19184	0.113922
$786$	0	0
$787$	28.6969i	1.02294i	0.859303	+	0.511468i	$0.170898\pi$
$787$	28.6969i	1.02294i	−0.859303	+	0.511468i	$0.829102\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	10.2810	0.365549
$792$	0	0
$793$	20.6969	0.734970
$794$	0	0
$795$	0	0
$796$	0	0
$797$	39.5552i	1.40112i	0.713595	+	0.700559i	$0.247066\pi$
$797$	39.5552i	1.40112i	−0.713595	+	0.700559i	$0.752934\pi$
$798$	0	0
$799$	28.0130	0.991028
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 2.00000i	− 0.0705785i
$804$	0	0
$805$	− 6.29577i	− 0.221897i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1.10102	0.0387098	0.0193549	−	0.999813i	$-0.493839\pi$
$809$	1.10102	0.0387098	0.0193549	+	0.999813i	$0.493839\pi$
$810$	0	0
$811$	48.6969i	1.70998i	0.518644	+	0.854990i	$0.326437\pi$
$811$	48.6969i	1.70998i	−0.518644	+	0.854990i	$0.673563\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−4.19718	−0.147021
$816$	0	0
$817$	8.40408	0.294022
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0.943295i	0.0329212i	0.999865	+	0.0164606i	$0.00523981\pi$
$821$	0.943295i	0.0329212i	−0.999865	+	0.0164606i	$0.994760\pi$
$822$	0	0
$823$	33.1534	1.15566	0.577828	−	0.816158i	$-0.303900\pi$
$823$	33.1534	1.15566	0.577828	+	0.816158i	$0.303900\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 3.30306i	− 0.114859i	−0.998350	−	0.0574294i	$-0.981710\pi$
$827$	− 3.30306i	− 0.114859i	0.998350	−	0.0574294i	$-0.0182904\pi$
$828$	0	0
$829$	12.5915i	0.437322i	0.975801	+	0.218661i	$0.0701689\pi$
$829$	12.5915i	0.437322i	−0.975801	+	0.218661i	$0.929831\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−28.8990	−1.00129
$834$	0	0
$835$	4.40408i	0.152410i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−20.1963	−0.697252	−0.348626	−	0.937262i	$-0.613352\pi$
$839$	−20.1963	−0.697252	−0.348626	+	0.937262i	$0.613352\pi$
$840$	0	0
$841$	−23.4041	−0.807037
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0.106000i	0.00364652i
$846$	0	0
$847$	−1.04930	−0.0360542
$848$	0	0
$849$	0	0
$850$	0	0
$851$	65.3939i	2.24167i
$852$	0	0
$853$	19.0410i	0.651950i	0.945378	+	0.325975i	$0.105693\pi$
$853$	19.0410i	0.651950i	−0.945378	+	0.325975i	$0.894307\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	40.2929	1.37638	0.688189	−	0.725532i	$-0.258406\pi$
$857$	40.2929	1.37638	0.688189	+	0.725532i	$0.258406\pi$
$858$	0	0
$859$	− 5.79796i	− 0.197824i	−0.995096	−	0.0989119i	$-0.968464\pi$
$859$	− 5.79796i	− 0.197824i	0.995096	−	0.0989119i	$-0.0315362\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−2.67624	−0.0911002	−0.0455501	−	0.998962i	$-0.514504\pi$
$863$	−2.67624	−0.0911002	−0.0455501	+	0.998962i	$0.514504\pi$
$864$	0	0
$865$	16.4041	0.557756
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 12.4855i	− 0.423543i
$870$	0	0
$871$	28.2250	0.956367
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 9.79796i	− 0.331231i
$876$	0	0
$877$	50.3078i	1.69877i	0.527770	+	0.849387i	$0.323028\pi$
$877$	50.3078i	1.69877i	−0.527770	+	0.849387i	$0.676972\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−48.9898	−1.65051	−0.825254	−	0.564762i	$-0.808968\pi$
$881$	−48.9898	−1.65051	−0.825254	+	0.564762i	$0.808968\pi$
$882$	0	0
$883$	43.3939i	1.46032i	0.683276	+	0.730160i	$0.260555\pi$
$883$	43.3939i	1.46032i	−0.683276	+	0.730160i	$0.739445\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−48.7869	−1.63810	−0.819051	−	0.573720i	$-0.805500\pi$
$887$	−48.7869	−1.63810	−0.819051	+	0.573720i	$0.805500\pi$
$888$	0	0
$889$	−10.8990	−0.365540
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 16.5767i	− 0.554719i
$894$	0	0
$895$	−2.31059	−0.0772345
$896$	0	0
$897$	0	0
$898$	0	0
$899$	15.1918i	0.506676i
$900$	0	0
$901$	− 40.6045i	− 1.35273i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−4.40408	−0.146397
$906$	0	0
$907$	− 53.1918i	− 1.76621i	−0.469179	−	0.883103i	$-0.655450\pi$
$907$	− 53.1918i	− 1.76621i	0.469179	−	0.883103i	$-0.344550\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−5.71812	−0.189450	−0.0947249	−	0.995503i	$-0.530197\pi$
$911$	−5.71812	−0.189450	−0.0947249	+	0.995503i	$0.530197\pi$
$912$	0	0
$913$	−1.10102	−0.0364385
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 21.7172i	− 0.717165i
$918$	0	0
$919$	−11.5422	−0.380744	−0.190372	−	0.981712i	$-0.560969\pi$
$919$	−11.5422	−0.380744	−0.190372	+	0.981712i	$0.560969\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	54.4949i	1.79372i
$924$	0	0
$925$	44.5897i	1.46610i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−45.7980	−1.50258	−0.751291	−	0.659971i	$-0.770569\pi$
$929$	−45.7980	−1.50258	−0.751291	+	0.659971i	$0.770569\pi$
$930$	0	0
$931$	17.1010i	0.560463i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−5.14048	−0.168112
$936$	0	0
$937$	−26.0000	−0.849383	−0.424691	−	0.905338i	$-0.639617\pi$
$937$	−26.0000	−0.849383	−0.424691	+	0.905338i	$0.639617\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	40.6045i	1.32367i	0.749650	+	0.661835i	$0.230222\pi$
$941$	40.6045i	1.32367i	−0.749650	+	0.661835i	$0.769778\pi$
$942$	0	0
$943$	6.29577	0.205019
$944$	0	0
$945$	0	0
$946$	0	0
$947$	41.3939i	1.34512i	0.740042	+	0.672560i	$0.234805\pi$
$947$	41.3939i	1.34512i	−0.740042	+	0.672560i	$0.765195\pi$
$948$	0	0
$949$	− 7.23907i	− 0.234990i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	20.6969	0.670440	0.335220	−	0.942140i	$-0.391189\pi$
$953$	20.6969	0.670440	0.335220	+	0.942140i	$0.391189\pi$
$954$	0	0
$955$	− 1.59592i	− 0.0516427i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	22.8725	0.738591
$960$	0	0
$961$	−26.5959	−0.857933
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 12.3795i	− 0.398512i
$966$	0	0
$967$	−18.5693	−0.597149	−0.298574	−	0.954386i	$-0.596511\pi$
$967$	−18.5693	−0.597149	−0.298574	+	0.954386i	$0.596511\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	7.59592i	0.243765i	0.992545	+	0.121882i	$0.0388930\pi$
$971$	7.59592i	0.243765i	−0.992545	+	0.121882i	$0.961107\pi$
$972$	0	0
$973$	7.23907i	0.232074i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−45.7980	−1.46521	−0.732603	−	0.680656i	$-0.761695\pi$
$977$	−45.7980	−1.46521	−0.732603	+	0.680656i	$0.761695\pi$
$978$	0	0
$979$	− 12.0000i	− 0.383522i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	20.4083	0.650922	0.325461	−	0.945555i	$-0.394480\pi$
$983$	20.4083	0.650922	0.325461	+	0.945555i	$0.394480\pi$
$984$	0	0
$985$	14.2020	0.452515
$986$	0	0
$987$	0	0
$988$	0	0
$989$	16.5767i	0.527109i
$990$	0	0
$991$	−58.1245	−1.84639	−0.923193	−	0.384336i	$-0.874430\pi$
$991$	−58.1245	−1.84639	−0.923193	+	0.384336i	$0.874430\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 17.3939i	− 0.551423i
$996$	0	0
$997$	− 54.5050i	− 1.72619i	−0.505041	−	0.863095i	$-0.668523\pi$
$997$	− 54.5050i	− 1.72619i	0.505041	−	0.863095i	$-0.331477\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6336.2.f.j.3169.4	yes	8
3.2	odd	2		6336.2.f.i.3169.6	yes	8
4.3	odd	2	inner	6336.2.f.j.3169.3	yes	8
8.3	odd	2	inner	6336.2.f.j.3169.5	yes	8
8.5	even	2	inner	6336.2.f.j.3169.6	yes	8
12.11	even	2		6336.2.f.i.3169.5	yes	8
24.5	odd	2		6336.2.f.i.3169.4	yes	8
24.11	even	2		6336.2.f.i.3169.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6336.2.f.i.3169.3	✓	8	24.11	even	2
6336.2.f.i.3169.4	yes	8	24.5	odd	2
6336.2.f.i.3169.5	yes	8	12.11	even	2
6336.2.f.i.3169.6	yes	8	3.2	odd	2
6336.2.f.j.3169.3	yes	8	4.3	odd	2	inner
6336.2.f.j.3169.4	yes	8	1.1	even	1	trivial
6336.2.f.j.3169.5	yes	8	8.3	odd	2	inner
6336.2.f.j.3169.6	yes	8	8.5	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6336 = 2^{6} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6336.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.04930i q^{5} +1.04930 q^{7} -1.00000i q^{11} -3.61953i q^{13} +4.89898 q^{17} -2.89898i q^{19} +5.71812 q^{23} +3.89898 q^{25} +7.23907i q^{29} +2.09859 q^{31} -1.10102i q^{35} +11.4362i q^{37} +1.10102 q^{41} +2.89898i q^{43} +5.71812 q^{47} -5.89898 q^{49} -8.28836i q^{53} -1.04930 q^{55} -9.79796i q^{59} +5.71812i q^{61} -3.79796 q^{65} +7.79796i q^{67} -15.0558 q^{71} +2.00000 q^{73} -1.04930i q^{77} +12.4855 q^{79} -1.10102i q^{83} -5.14048i q^{85} +12.0000 q^{89} -3.79796i q^{91} -3.04189 q^{95} -10.6969 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{25} + 48 q^{41} - 8 q^{49} + 48 q^{65} + 16 q^{73} + 96 q^{89} + 32 q^{97}+O(q^{100}) \) 8 * q - 8 * q^25 + 48 * q^41 - 8 * q^49 + 48 * q^65 + 16 * q^73 + 96 * q^89 + 32 * q^97

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