LMFDB - Embedded newform 2664.2.r.n.433.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2664,2,Mod(433,2664)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2664, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))

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

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

chi := DirichletCharacter("2664.433");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2664 = 2^{3} \cdot 3^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2664.r (of order $3$, degree $2$, 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:	$21.2721470985$
Analytic rank:	$0$
Dimension:	$10$
Relative dimension:	$5$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} + 9x^{8} + 4x^{7} + 54x^{6} + 6x^{5} + 98x^{4} - 8x^{3} + 148x^{2} - 24x + 4 $ x^10 - x^9 + 9x^8 + 4x^7 + 54x^6 + 6x^5 + 98x^4 - 8x^3 + 148x^2 - 24x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 296)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			433.3
Root			$0.657199 - 1.13830i$ of defining polynomial
Character	$\chi$	$=$	2664.433
Dual form			2664.2.r.n.1009.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.883889 + 1.53094i) q^{5} +(0.597948 - 1.03568i) q^{7} +0.628794 q^{11} +(-2.91932 + 5.05641i) q^{13} +(-0.500000 - 0.866025i) q^{17} +(1.60492 - 2.77980i) q^{19} -2.01395 q^{23} +(0.937481 + 1.62376i) q^{25} +4.68823 q^{29} +0.628794 q^{31} +(1.05704 + 1.83085i) q^{35} +(-5.81237 - 1.79342i) q^{37} +(-4.20526 + 7.28372i) q^{41} -0.767778 q^{43} -11.9047 q^{47} +(2.78492 + 4.82362i) q^{49} +(-5.19168 - 8.99225i) q^{53} +(-0.555784 + 0.962647i) q^{55} +(1.29052 + 2.23525i) q^{59} +(-4.45933 + 7.72378i) q^{61} +(-5.16070 - 8.93860i) q^{65} +(0.837143 - 1.44997i) q^{67} +(5.80779 - 10.0594i) q^{71} -9.89329 q^{73} +(0.375986 - 0.651227i) q^{77} +(-5.68494 + 9.84661i) q^{79} +(8.10516 + 14.0385i) q^{83} +1.76778 q^{85} +(-6.20526 - 10.7478i) q^{89} +(3.49120 + 6.04694i) q^{91} +(2.83714 + 4.91408i) q^{95} +9.87569 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 3 q^{5} + 3 q^{7} - 16 q^{11} + q^{13} - 5 q^{17} - 3 q^{19} + 12 q^{23} - 12 q^{25} - 16 q^{31} - 5 q^{35} - 12 q^{37} - 9 q^{41} + 4 q^{43} - 32 q^{47} - 18 q^{49} - 5 q^{53} + 4 q^{55} + 5 q^{59}+ \cdots - 36 q^{97}+O(q^{100}) $ 10 * q - 3 * q^5 + 3 * q^7 - 16 * q^11 + q^13 - 5 * q^17 - 3 * q^19 + 12 * q^23 - 12 * q^25 - 16 * q^31 - 5 * q^35 - 12 * q^37 - 9 * q^41 + 4 * q^43 - 32 * q^47 - 18 * q^49 - 5 * q^53 + 4 * q^55 + 5 * q^59 + 15 * q^61 - 23 * q^65 + q^67 + 17 * q^71 - 36 * q^73 + 4 * q^77 + 21 * q^79 + 15 * q^83 + 6 * q^85 - 29 * q^89 - q^91 + 21 * q^95 - 36 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2664\mathbb{Z}\right)^\times$.

$n$	$1297$	$1333$	$1999$	$2369$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−0.883889	+	1.53094i	−0.395287	+	0.684657i	−0.993138	−	0.116950i	$-0.962688\pi$
$5$	−0.883889	+	1.53094i	−0.395287	+	0.684657i	0.597851	+	0.801607i	$0.296022\pi$
$6$		0			0
$7$	0.597948	−	1.03568i	0.226003	−	0.391449i	−0.730617	−	0.682788i	$-0.760767\pi$
$7$	0.597948	−	1.03568i	0.226003	−	0.391449i	0.956620	+	0.291339i	$0.0941007\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.628794			0.189589			0.0947943	−	0.995497i	$-0.469781\pi$
$11$	0.628794			0.189589			0.0947943	+	0.995497i	$0.469781\pi$
$12$		0			0
$13$	−2.91932	+	5.05641i	−0.809673	+	1.40239i	0.103418	+	0.994638i	$0.467022\pi$
$13$	−2.91932	+	5.05641i	−0.809673	+	1.40239i	−0.913091	+	0.407757i	$0.866311\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.500000	−	0.866025i	−0.121268	−	0.210042i	0.799000	−	0.601331i	$-0.205363\pi$
$17$	−0.500000	−	0.866025i	−0.121268	−	0.210042i	−0.920268	+	0.391289i	$0.872029\pi$
$18$		0			0
$19$	1.60492	−	2.77980i	0.368194	−	0.637731i	−0.621089	−	0.783740i	$-0.713310\pi$
$19$	1.60492	−	2.77980i	0.368194	−	0.637731i	0.989283	+	0.146009i	$0.0466429\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.01395			−0.419937			−0.209968	−	0.977708i	$-0.567336\pi$
$23$	−2.01395			−0.419937			−0.209968	+	0.977708i	$0.567336\pi$
$24$		0			0
$25$	0.937481	+	1.62376i	0.187496	+	0.324753i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4.68823			0.870583			0.435291	−	0.900290i	$-0.356645\pi$
$29$	4.68823			0.870583			0.435291	+	0.900290i	$0.356645\pi$
$30$		0			0
$31$	0.628794			0.112935			0.0564674	−	0.998404i	$-0.482016\pi$
$31$	0.628794			0.112935			0.0564674	+	0.998404i	$0.482016\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.05704	+	1.83085i	0.178672	+	0.309469i
$36$		0			0
$37$	−5.81237	−	1.79342i	−0.955548	−	0.294836i
$37$	−5.81237	−	1.79342i	−0.955548	−	0.294836i
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−4.20526	+	7.28372i	−0.656751	+	1.13753i	0.324701	+	0.945817i	$0.394736\pi$
$41$	−4.20526	+	7.28372i	−0.656751	+	1.13753i	−0.981452	+	0.191709i	$0.938597\pi$
$42$		0			0
$43$	−0.767778			−0.117085			−0.0585425	−	0.998285i	$-0.518645\pi$
$43$	−0.767778			−0.117085			−0.0585425	+	0.998285i	$0.518645\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−11.9047			−1.73648			−0.868240	−	0.496144i	$-0.834749\pi$
$47$	−11.9047			−1.73648			−0.868240	+	0.496144i	$0.834749\pi$
$48$		0			0
$49$	2.78492	+	4.82362i	0.397845	+	0.689088i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−5.19168	−	8.99225i	−0.713132	−	1.23518i	−0.963676	−	0.267075i	$-0.913943\pi$
$53$	−5.19168	−	8.99225i	−0.713132	−	1.23518i	0.250544	−	0.968105i	$-0.419391\pi$
$54$		0			0
$55$	−0.555784	+	0.962647i	−0.0749419	+	0.129803i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.29052	+	2.23525i	0.168012	+	0.291005i	0.937721	−	0.347390i	$-0.112932\pi$
$59$	1.29052	+	2.23525i	0.168012	+	0.291005i	−0.769709	+	0.638395i	$0.779599\pi$
$60$		0			0
$61$	−4.45933	+	7.72378i	−0.570959	+	0.988929i	0.425509	+	0.904954i	$0.360095\pi$
$61$	−4.45933	+	7.72378i	−0.570959	+	0.988929i	−0.996468	+	0.0839752i	$0.973238\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−5.16070	−	8.93860i	−0.640107	−	1.10870i
$66$		0			0
$67$	0.837143	−	1.44997i	0.102273	−	0.177143i	−0.810348	−	0.585949i	$-0.800722\pi$
$67$	0.837143	−	1.44997i	0.102273	−	0.177143i	0.912621	+	0.408807i	$0.134055\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	5.80779	−	10.0594i	0.689258	−	1.19383i	−0.282821	−	0.959173i	$-0.591270\pi$
$71$	5.80779	−	10.0594i	0.689258	−	1.19383i	0.972078	−	0.234657i	$-0.0753966\pi$
$72$		0			0
$73$	−9.89329			−1.15792			−0.578961	−	0.815355i	$-0.696542\pi$
$73$	−9.89329			−1.15792			−0.578961	+	0.815355i	$0.696542\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.375986	−	0.651227i	0.0428476	−	0.0742142i
$78$		0			0
$79$	−5.68494	+	9.84661i	−0.639606	+	1.10783i	0.345913	+	0.938267i	$0.387569\pi$
$79$	−5.68494	+	9.84661i	−0.639606	+	1.10783i	−0.985519	+	0.169564i	$0.945764\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	8.10516	+	14.0385i	0.889657	+	1.54093i	0.840282	+	0.542150i	$0.182390\pi$
$83$	8.10516	+	14.0385i	0.889657	+	1.54093i	0.0493752	+	0.998780i	$0.484277\pi$
$84$		0			0
$85$	1.76778			0.191742
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−6.20526	−	10.7478i	−0.657756	−	1.13927i	−0.981195	−	0.193018i	$-0.938172\pi$
$89$	−6.20526	−	10.7478i	−0.657756	−	1.13927i	0.323439	−	0.946249i	$-0.395161\pi$
$90$		0			0
$91$	3.49120	+	6.04694i	0.365977	+	0.633891i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.83714	+	4.91408i	0.291085	+	0.504174i
$96$		0			0
$97$	9.87569			1.00272			0.501362	−	0.865237i	$-0.332832\pi$
$97$	9.87569			1.00272			0.501362	+	0.865237i	$0.332832\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1.99820			−0.198829			−0.0994144	−	0.995046i	$-0.531697\pi$
$101$	−1.99820			−0.198829			−0.0994144	+	0.995046i	$0.531697\pi$
$102$		0			0
$103$	−14.1966			−1.39884			−0.699418	−	0.714713i	$-0.746557\pi$
$103$	−14.1966			−1.39884			−0.699418	+	0.714713i	$0.746557\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.67441	+	2.90017i	−0.161872	+	0.280370i	−0.935540	−	0.353221i	$-0.885086\pi$
$107$	−1.67441	+	2.90017i	−0.161872	+	0.280370i	0.773668	+	0.633591i	$0.218420\pi$
$108$		0			0
$109$	1.03965	+	1.80072i	0.0995801	+	0.172478i	0.911511	−	0.411276i	$-0.134917\pi$
$109$	1.03965	+	1.80072i	0.0995801	+	0.172478i	−0.811931	+	0.583754i	$0.801583\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	3.08902	+	5.35034i	0.290591	+	0.503318i	0.973950	−	0.226765i	$-0.0728149\pi$
$113$	3.08902	+	5.35034i	0.290591	+	0.503318i	−0.683359	+	0.730083i	$0.739482\pi$
$114$		0			0
$115$	1.78010	−	3.08323i	0.165996	−	0.287513i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1.19590			−0.109628
$120$		0			0
$121$	−10.6046			−0.964056
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−12.1534			−1.08703
$126$		0			0
$127$	−8.92629	−	15.4608i	−0.792080	−	1.37192i	−0.924677	−	0.380753i	$-0.875665\pi$
$127$	−8.92629	−	15.4608i	−0.792080	−	1.37192i	0.132597	−	0.991170i	$-0.457669\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	7.64071	+	13.2341i	0.667572	+	1.15627i	0.978581	+	0.205862i	$0.0659998\pi$
$131$	7.64071	+	13.2341i	0.667572	+	1.15627i	−0.311009	+	0.950407i	$0.600667\pi$
$132$		0			0
$133$	−1.91932	−	3.32436i	−0.166426	−	0.288258i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−17.7690			−1.51811			−0.759053	−	0.651028i	$-0.774338\pi$
$137$	−17.7690			−1.51811			−0.759053	+	0.651028i	$0.774338\pi$
$138$		0			0
$139$	9.53806	+	16.5204i	0.809007	+	1.40124i	0.913552	+	0.406722i	$0.133328\pi$
$139$	9.53806	+	16.5204i	0.809007	+	1.40124i	−0.104545	+	0.994520i	$0.533339\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.83565	+	3.17944i	−0.153505	+	0.265878i
$144$		0			0
$145$	−4.14388	+	7.17740i	−0.344130	+	0.596051i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−2.76088			−0.226180			−0.113090	−	0.993585i	$-0.536075\pi$
$149$	−2.76088			−0.226180			−0.113090	+	0.993585i	$0.536075\pi$
$150$		0			0
$151$	−3.35441	+	5.81001i	−0.272978	+	0.472812i	−0.969623	−	0.244604i	$-0.921342\pi$
$151$	−3.35441	+	5.81001i	−0.272978	+	0.472812i	0.696645	+	0.717416i	$0.254675\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−0.555784	+	0.962647i	−0.0446417	+	0.0773216i
$156$		0			0
$157$	−7.22711	−	12.5177i	−0.576786	−	0.999022i	−0.995845	−	0.0910636i	$-0.970973\pi$
$157$	−7.22711	−	12.5177i	−0.576786	−	0.999022i	0.419059	−	0.907959i	$-0.362360\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−1.20423	+	2.08580i	−0.0949070	+	0.164384i
$162$		0			0
$163$	−3.79076	−	6.56579i	−0.296915	−	0.514272i	0.678513	−	0.734588i	$-0.262625\pi$
$163$	−3.79076	−	6.56579i	−0.296915	−	0.514272i	−0.975429	+	0.220316i	$0.929291\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1.69626	+	2.93801i	−0.131261	+	0.227350i	−0.924163	−	0.381999i	$-0.875236\pi$
$167$	−1.69626	+	2.93801i	−0.131261	+	0.227350i	0.792902	+	0.609349i	$0.208569\pi$
$168$		0			0
$169$	−10.5448	−	18.2642i	−0.811141	−	1.40494i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	4.06754	+	7.04518i	0.309249	+	0.535635i	0.978198	−	0.207673i	$-0.0665890\pi$
$173$	4.06754	+	7.04518i	0.309249	+	0.535635i	−0.668949	+	0.743308i	$0.733256\pi$
$174$		0			0
$175$	2.24226			0.169499
$176$		0			0
$177$		0			0
$178$		0			0
$179$	6.31914			0.472315			0.236157	−	0.971715i	$-0.424112\pi$
$179$	6.31914			0.472315			0.236157	+	0.971715i	$0.424112\pi$
$180$		0			0
$181$	0.730960	−	1.26606i	0.0543318	−	0.0941055i	−0.837580	−	0.546314i	$-0.816030\pi$
$181$	0.730960	−	1.26606i	0.0543318	−	0.0941055i	0.891912	+	0.452209i	$0.149364\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	7.88310	−	7.31321i	0.579577	−	0.537678i
$186$		0			0
$187$	−0.314397	−	0.544552i	−0.0229910	−	0.0398216i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−9.05717			−0.655354			−0.327677	−	0.944790i	$-0.606266\pi$
$191$	−9.05717			−0.655354			−0.327677	+	0.944790i	$0.606266\pi$
$192$		0			0
$193$	−10.5195			−0.757210			−0.378605	−	0.925558i	$-0.623596\pi$
$193$	−10.5195			−0.757210			−0.378605	+	0.925558i	$0.623596\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	6.55735	+	11.3577i	0.467192	+	0.809200i	0.999297	−	0.0374781i	$-0.0119325\pi$
$197$	6.55735	+	11.3577i	0.467192	+	0.809200i	−0.532106	+	0.846678i	$0.678599\pi$
$198$		0			0
$199$	−3.87749			−0.274868			−0.137434	−	0.990511i	$-0.543885\pi$
$199$	−3.87749			−0.274868			−0.137434	+	0.990511i	$0.543885\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	2.80332	−	4.85549i	0.196754	−	0.340789i
$204$		0			0
$205$	−7.43396	−	12.8760i	−0.519210	−	0.899299i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1.00916	−	1.74792i	0.0698054	−	0.120906i
$210$		0			0
$211$	−16.2850			−1.12111			−0.560553	−	0.828119i	$-0.689411\pi$
$211$	−16.2850			−1.12111			−0.560553	+	0.828119i	$0.689411\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	0.678630	−	1.17542i	0.0462822	−	0.0801631i
$216$		0			0
$217$	0.375986	−	0.651227i	0.0255236	−	0.0442082i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	5.83864			0.392749
$222$		0			0
$223$	−12.3331			−0.825884			−0.412942	−	0.910757i	$-0.635499\pi$
$223$	−12.3331			−0.825884			−0.412942	+	0.910757i	$0.635499\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	7.69889	−	13.3349i	0.510993	−	0.885066i	−0.488926	−	0.872325i	$-0.662611\pi$
$227$	7.69889	−	13.3349i	0.510993	−	0.885066i	0.999919	−	0.0127407i	$-0.00405559\pi$
$228$		0			0
$229$	−6.60973	+	11.4484i	−0.436783	+	0.756531i	−0.997439	−	0.0715178i	$-0.977216\pi$
$229$	−6.60973	+	11.4484i	−0.436783	+	0.756531i	0.560656	+	0.828049i	$0.310549\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	6.07338			0.397880			0.198940	−	0.980012i	$-0.436250\pi$
$233$	6.07338			0.397880			0.198940	+	0.980012i	$0.436250\pi$
$234$		0			0
$235$	10.5224	−	18.2254i	0.686408	−	1.18889i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	5.24069	+	9.07714i	0.338992	+	0.587151i	0.984243	−	0.176819i	$-0.0565807\pi$
$239$	5.24069	+	9.07714i	0.338992	+	0.587151i	−0.645251	+	0.763970i	$0.723247\pi$
$240$		0			0
$241$	−12.5179	+	21.6816i	−0.806347	+	1.39663i	0.109031	+	0.994038i	$0.465225\pi$
$241$	−12.5179	+	21.6816i	−0.806347	+	1.39663i	−0.915378	+	0.402595i	$0.868108\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−9.84623			−0.629052
$246$		0			0
$247$	9.37055	+	16.2303i	0.596234	+	1.03271i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	10.2581			0.647483			0.323742	−	0.946146i	$-0.395059\pi$
$251$	10.2581			0.647483			0.323742	+	0.946146i	$0.395059\pi$
$252$		0			0
$253$	−1.26636			−0.0796152
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−12.3216	−	21.3417i	−0.768601	−	1.33126i	−0.938322	−	0.345763i	$-0.887620\pi$
$257$	−12.3216	−	21.3417i	−0.768601	−	1.33126i	0.169721	−	0.985492i	$-0.445713\pi$
$258$		0			0
$259$	−5.33289	+	4.94737i	−0.331370	+	0.307414i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−8.02172	+	13.8940i	−0.494641	+	0.856743i	−0.999981	−	0.00617757i	$-0.998034\pi$
$263$	−8.02172	+	13.8940i	−0.494641	+	0.856743i	0.505340	+	0.862920i	$0.331367\pi$
$264$		0			0
$265$	18.3555			1.12757
$266$		0			0
$267$		0			0
$268$		0			0
$269$	2.03666			0.124177			0.0620887	−	0.998071i	$-0.480224\pi$
$269$	2.03666			0.124177			0.0620887	+	0.998071i	$0.480224\pi$
$270$		0			0
$271$	14.2253	+	24.6389i	0.864124	+	1.49671i	0.867914	+	0.496714i	$0.165460\pi$
$271$	14.2253	+	24.6389i	0.864124	+	1.49671i	−0.00379005	+	0.999993i	$0.501206\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.589483	+	1.02101i	0.0355471	+	0.0615695i
$276$		0			0
$277$	8.49921	−	14.7211i	0.510668	−	0.884503i	−0.489255	−	0.872141i	$-0.662731\pi$
$277$	8.49921	−	14.7211i	0.510668	−	0.884503i	0.999924	−	0.0123628i	$-0.00393529\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−2.61635	−	4.53165i	−0.156078	−	0.270336i	0.777373	−	0.629040i	$-0.216552\pi$
$281$	−2.61635	−	4.53165i	−0.156078	−	0.270336i	−0.933451	+	0.358705i	$0.883219\pi$
$282$		0			0
$283$	−1.03466	+	1.79208i	−0.0615040	+	0.106528i	−0.895138	−	0.445789i	$-0.852923\pi$
$283$	−1.03466	+	1.79208i	−0.0615040	+	0.106528i	0.833634	+	0.552317i	$0.186256\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	5.02905	+	8.71057i	0.296855	+	0.514169i
$288$		0			0
$289$	8.00000	−	13.8564i	0.470588	−	0.815083i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	2.43396	−	4.21575i	0.142194	−	0.246287i	−0.786129	−	0.618063i	$-0.787918\pi$
$293$	2.43396	−	4.21575i	0.142194	−	0.246287i	0.928322	+	0.371776i	$0.121251\pi$
$294$		0			0
$295$	−4.56272			−0.265652
$296$		0			0
$297$		0			0
$298$		0			0
$299$	5.87935	−	10.1833i	0.340011	−	0.588917i
$300$		0			0
$301$	−0.459091	+	0.795169i	−0.0264616	+	0.0458328i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−7.88310	−	13.6539i	−0.451385	−	0.781822i
$306$		0			0
$307$	23.3475			1.33251			0.666256	−	0.745723i	$-0.267896\pi$
$307$	23.3475			1.33251			0.666256	+	0.745723i	$0.267896\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	12.7946	+	22.1608i	0.725514	+	1.25663i	0.958762	+	0.284209i	$0.0917311\pi$
$311$	12.7946	+	22.1608i	0.725514	+	1.25663i	−0.233249	+	0.972417i	$0.574936\pi$
$312$		0			0
$313$	12.8114	+	22.1900i	0.724144	+	1.25425i	0.959325	+	0.282303i	$0.0910983\pi$
$313$	12.8114	+	22.1900i	0.724144	+	1.25425i	−0.235181	+	0.971952i	$0.575568\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	13.4108	+	23.2281i	0.753223	+	1.30462i	0.946253	+	0.323428i	$0.104835\pi$
$317$	13.4108	+	23.2281i	0.753223	+	1.30462i	−0.193030	+	0.981193i	$0.561831\pi$
$318$		0			0
$319$	2.94793			0.165053
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−3.20984			−0.178600
$324$		0			0
$325$	−10.9472			−0.607242
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−7.11840	+	12.3294i	−0.392450	+	0.679743i
$330$		0			0
$331$	16.7532	+	29.0173i	0.920838	+	1.59494i	0.798122	+	0.602496i	$0.205827\pi$
$331$	16.7532	+	29.0173i	0.920838	+	1.59494i	0.122716	+	0.992442i	$0.460840\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	1.47988	+	2.56323i	0.0808546	+	0.140044i
$336$		0			0
$337$	7.31244	−	12.6655i	0.398334	−	0.689935i	−0.595186	−	0.803588i	$-0.702922\pi$
$337$	7.31244	−	12.6655i	0.398334	−	0.689935i	0.993521	+	0.113653i	$0.0362551\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	0.395382			0.0214111
$342$		0			0
$343$	15.0322			0.811663
$344$		0			0
$345$		0			0
$346$		0			0
$347$	30.7694			1.65179			0.825895	−	0.563824i	$-0.190670\pi$
$347$	30.7694			1.65179			0.825895	+	0.563824i	$0.190670\pi$
$348$		0			0
$349$	−6.80207	−	11.7815i	−0.364107	−	0.630651i	0.624526	−	0.781004i	$-0.285292\pi$
$349$	−6.80207	−	11.7815i	−0.364107	−	0.630651i	−0.988632	+	0.150353i	$0.951959\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−7.63411	−	13.2227i	−0.406322	−	0.703771i	0.588152	−	0.808750i	$-0.299856\pi$
$353$	−7.63411	−	13.2227i	−0.406322	−	0.703771i	−0.994474	+	0.104979i	$0.966522\pi$
$354$		0			0
$355$	10.2669	+	17.7828i	0.544909	+	0.943811i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−2.92580			−0.154418			−0.0772090	−	0.997015i	$-0.524601\pi$
$359$	−2.92580			−0.154418			−0.0772090	+	0.997015i	$0.524601\pi$
$360$		0			0
$361$	4.34846	+	7.53175i	0.228866	+	0.396408i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	8.74457	−	15.1460i	0.457712	−	0.792780i
$366$		0			0
$367$	11.4986	−	19.9161i	0.600220	−	1.03961i	−0.392567	−	0.919723i	$-0.628413\pi$
$367$	11.4986	−	19.9161i	0.600220	−	1.03961i	0.992787	−	0.119889i	$-0.0382538\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−12.4174			−0.644680
$372$		0			0
$373$	2.14400	−	3.71352i	0.111012	−	0.192279i	−0.805166	−	0.593049i	$-0.797924\pi$
$373$	2.14400	−	3.71352i	0.111012	−	0.192279i	0.916179	+	0.400770i	$0.131257\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−13.6864	+	23.7056i	−0.704887	+	1.22090i
$378$		0			0
$379$	−18.1555	−	31.4463i	−0.932587	−	1.61529i	−0.778881	−	0.627172i	$-0.784212\pi$
$379$	−18.1555	−	31.4463i	−0.932587	−	1.61529i	−0.153706	−	0.988117i	$-0.549121\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	16.9403	−	29.3415i	0.865611	−	1.49928i	−0.000828397	−	1.00000i	$-0.500264\pi$
$383$	16.9403	−	29.3415i	0.865611	−	1.49928i	0.866439	−	0.499282i	$-0.166403\pi$
$384$		0			0
$385$	0.664660	+	1.15122i	0.0338742	+	0.0586719i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	5.41944	−	9.38675i	0.274777	−	0.475927i	−0.695302	−	0.718718i	$-0.744729\pi$
$389$	5.41944	−	9.38675i	0.274777	−	0.475927i	0.970079	+	0.242790i	$0.0780627\pi$
$390$		0			0
$391$	1.00697	+	1.74413i	0.0509248	+	0.0882043i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−10.0497	−	17.4066i	−0.505656	−	0.875822i
$396$		0			0
$397$	−28.1216			−1.41138			−0.705691	−	0.708520i	$-0.749363\pi$
$397$	−28.1216			−1.41138			−0.705691	+	0.708520i	$0.749363\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	6.19915			0.309571			0.154785	−	0.987948i	$-0.450531\pi$
$401$	6.19915			0.309571			0.154785	+	0.987948i	$0.450531\pi$
$402$		0			0
$403$	−1.83565	+	3.17944i	−0.0914403	+	0.158379i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−3.65479	−	1.12769i	−0.181161	−	0.0558975i
$408$		0			0
$409$	−11.7159	−	20.2926i	−0.579315	−	1.00340i	−0.995558	−	0.0941498i	$-0.969987\pi$
$409$	−11.7159	−	20.2926i	−0.579315	−	1.00340i	0.416243	−	0.909253i	$-0.363347\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	3.08666			0.151885
$414$		0			0
$415$	−28.6562			−1.40668
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−8.85723	−	15.3412i	−0.432704	−	0.749466i	0.564401	−	0.825501i	$-0.309107\pi$
$419$	−8.85723	−	15.3412i	−0.432704	−	0.749466i	−0.997105	+	0.0760351i	$0.975774\pi$
$420$		0			0
$421$	27.3203			1.33151			0.665756	−	0.746169i	$-0.268109\pi$
$421$	27.3203			1.33151			0.665756	+	0.746169i	$0.268109\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	0.937481	−	1.62376i	0.0454745	−	0.0787642i
$426$		0			0
$427$	5.33289	+	9.23684i	0.258077	+	0.447002i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−13.4098	+	23.2264i	−0.645925	+	1.11878i	0.338162	+	0.941088i	$0.390195\pi$
$431$	−13.4098	+	23.2264i	−0.645925	+	1.11878i	−0.984087	+	0.177687i	$0.943138\pi$
$432$		0			0
$433$	11.3381			0.544874			0.272437	−	0.962174i	$-0.412170\pi$
$433$	11.3381			0.544874			0.272437	+	0.962174i	$0.412170\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−3.23222	+	5.59837i	−0.154618	+	0.267807i
$438$		0			0
$439$	9.62993	−	16.6795i	0.459611	−	0.796070i	−0.539329	−	0.842095i	$-0.681322\pi$
$439$	9.62993	−	16.6795i	0.459611	−	0.796070i	0.998940	+	0.0460248i	$0.0146553\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−20.7574			−0.986216			−0.493108	−	0.869968i	$-0.664139\pi$
$443$	−20.7574			−0.986216			−0.493108	+	0.869968i	$0.664139\pi$
$444$		0			0
$445$	21.9390			1.04001
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−10.3286	+	17.8897i	−0.487437	+	0.844266i	−0.999896	−	0.0144458i	$-0.995402\pi$
$449$	−10.3286	+	17.8897i	−0.487437	+	0.844266i	0.512458	+	0.858712i	$0.328735\pi$
$450$		0			0
$451$	−2.64424	+	4.57996i	−0.124513	+	0.215662i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−12.3433			−0.578664
$456$		0			0
$457$	5.31147	−	9.19974i	0.248460	−	0.430346i	−0.714639	−	0.699494i	$-0.753409\pi$
$457$	5.31147	−	9.19974i	0.248460	−	0.430346i	0.963099	+	0.269148i	$0.0867421\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−7.57302	−	13.1169i	−0.352711	−	0.610913i	0.634013	−	0.773323i	$-0.281407\pi$
$461$	−7.57302	−	13.1169i	−0.352711	−	0.610913i	−0.986723	+	0.162410i	$0.948073\pi$
$462$		0			0
$463$	5.44600	−	9.43275i	0.253097	−	0.438377i	−0.711280	−	0.702909i	$-0.751884\pi$
$463$	5.44600	−	9.43275i	0.253097	−	0.438377i	0.964377	+	0.264532i	$0.0852174\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	24.1884			1.11931			0.559654	−	0.828727i	$-0.310934\pi$
$467$	24.1884			1.11931			0.559654	+	0.828727i	$0.310934\pi$
$468$		0			0
$469$	−1.00114	−	1.73402i	−0.0462281	−	0.0800695i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−0.482774			−0.0221980
$474$		0			0
$475$	6.01833			0.276140
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−9.02676	−	15.6348i	−0.412443	−	0.714372i	0.582713	−	0.812678i	$-0.301991\pi$
$479$	−9.02676	−	15.6348i	−0.412443	−	0.714372i	−0.995156	+	0.0983056i	$0.968658\pi$
$480$		0			0
$481$	26.0364	−	24.1542i	1.18716	−	1.10134i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−8.72901	+	15.1191i	−0.396364	+	0.686523i
$486$		0			0
$487$	15.2208			0.689720			0.344860	−	0.938654i	$-0.387926\pi$
$487$	15.2208			0.689720			0.344860	+	0.938654i	$0.387926\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−18.6105			−0.839879			−0.419939	−	0.907552i	$-0.637949\pi$
$491$	−18.6105			−0.839879			−0.419939	+	0.907552i	$0.637949\pi$
$492$		0			0
$493$	−2.34412	−	4.06013i	−0.105574	−	0.182859i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−6.94551	−	12.0300i	−0.311549	−	0.539618i
$498$		0			0
$499$	−6.67830	+	11.5672i	−0.298962	+	0.517817i	−0.975899	−	0.218224i	$-0.929974\pi$
$499$	−6.67830	+	11.5672i	−0.298962	+	0.517817i	0.676937	+	0.736041i	$0.263307\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	4.28610	+	7.42374i	0.191108	+	0.331008i	0.945618	−	0.325280i	$-0.105459\pi$
$503$	4.28610	+	7.42374i	0.191108	+	0.331008i	−0.754510	+	0.656289i	$0.772125\pi$
$504$		0			0
$505$	1.76619	−	3.05913i	0.0785945	−	0.136130i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	12.3473	+	21.3861i	0.547283	+	0.947922i	0.998459	+	0.0554872i	$0.0176712\pi$
$509$	12.3473	+	21.3861i	0.547283	+	0.947922i	−0.451176	+	0.892435i	$0.648995\pi$
$510$		0			0
$511$	−5.91567	+	10.2462i	−0.261694	+	0.453267i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	12.5482	−	21.7342i	0.552941	−	0.957723i
$516$		0			0
$517$	−7.48562			−0.329217
$518$		0			0
$519$		0			0
$520$		0			0
$521$	1.55347	−	2.69068i	0.0680586	−	0.117881i	−0.829988	−	0.557781i	$-0.811653\pi$
$521$	1.55347	−	2.69068i	0.0680586	−	0.117881i	0.898047	+	0.439900i	$0.144986\pi$
$522$		0			0
$523$	17.6268	−	30.5305i	0.770765	−	1.33500i	−0.166380	−	0.986062i	$-0.553208\pi$
$523$	17.6268	−	30.5305i	0.770765	−	1.33500i	0.937144	−	0.348942i	$-0.113459\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−0.314397	−	0.544552i	−0.0136954	−	0.0237210i
$528$		0			0
$529$	−18.9440			−0.823653
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−24.5530	−	42.5270i	−1.06351	−	1.84205i
$534$		0			0
$535$	−2.95999	−	5.12685i	−0.127971	−	0.221653i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1.75114	+	3.03306i	0.0754269	+	0.130643i
$540$		0			0
$541$	10.8483			0.466404			0.233202	−	0.972428i	$-0.425080\pi$
$541$	10.8483			0.466404			0.233202	+	0.972428i	$0.425080\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−3.67573			−0.157451
$546$		0			0
$547$	41.8843			1.79084			0.895421	−	0.445221i	$-0.146875\pi$
$547$	41.8843			1.79084			0.895421	+	0.445221i	$0.146875\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	7.52424	−	13.0324i	0.320543	−	0.555197i
$552$		0			0
$553$	6.79860	+	11.7755i	0.289106	+	0.500746i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−5.57901	−	9.66314i	−0.236390	−	0.409440i	0.723285	−	0.690549i	$-0.242631\pi$
$557$	−5.57901	−	9.66314i	−0.236390	−	0.409440i	−0.959676	+	0.281109i	$0.909298\pi$
$558$		0			0
$559$	2.24139	−	3.88220i	0.0948006	−	0.164199i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	41.5032			1.74915			0.874577	−	0.484887i	$-0.161139\pi$
$563$	41.5032			1.74915			0.874577	+	0.484887i	$0.161139\pi$
$564$		0			0
$565$	−10.9214			−0.459467
$566$		0			0
$567$		0			0
$568$		0			0
$569$	32.9432			1.38105			0.690526	−	0.723308i	$-0.257379\pi$
$569$	32.9432			1.38105			0.690526	+	0.723308i	$0.257379\pi$
$570$		0			0
$571$	−2.51196	−	4.35085i	−0.105122	−	0.182077i	0.808666	−	0.588268i	$-0.200190\pi$
$571$	−2.51196	−	4.35085i	−0.105122	−	0.182077i	−0.913788	+	0.406191i	$0.866857\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1.88804	−	3.27017i	−0.0787365	−	0.136376i
$576$		0			0
$577$	16.4003	+	28.4061i	0.682753	+	1.18256i	0.974137	+	0.225957i	$0.0725509\pi$
$577$	16.4003	+	28.4061i	0.682753	+	1.18256i	−0.291384	+	0.956606i	$0.594116\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	19.3858			0.804261
$582$		0			0
$583$	−3.26450	−	5.65428i	−0.135202	−	0.234176i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−11.5744	+	20.0474i	−0.477726	+	0.827446i	−0.999674	−	0.0255317i	$-0.991872\pi$
$587$	−11.5744	+	20.0474i	−0.477726	+	0.827446i	0.521948	+	0.852977i	$0.325205\pi$
$588$		0			0
$589$	1.00916	−	1.74792i	0.0415819	−	0.0720220i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−28.6917			−1.17823			−0.589113	−	0.808051i	$-0.700523\pi$
$593$	−28.6917			−1.17823			−0.589113	+	0.808051i	$0.700523\pi$
$594$		0			0
$595$	1.05704	−	1.83085i	0.0433344	−	0.0750573i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	0.422970	−	0.732606i	0.0172821	−	0.0299335i	−0.857255	−	0.514892i	$-0.827832\pi$
$599$	0.422970	−	0.732606i	0.0172821	−	0.0299335i	0.874537	+	0.484959i	$0.161165\pi$
$600$		0			0
$601$	20.4989	+	35.5052i	0.836169	+	1.44829i	0.893075	+	0.449908i	$0.148543\pi$
$601$	20.4989	+	35.5052i	0.836169	+	1.44829i	−0.0569062	+	0.998380i	$0.518124\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	9.37330	−	16.2350i	0.381079	−	0.660048i
$606$		0			0
$607$	7.72152	+	13.3741i	0.313407	+	0.542837i	0.979098	−	0.203391i	$-0.0651963\pi$
$607$	7.72152	+	13.3741i	0.313407	+	0.542837i	−0.665691	+	0.746228i	$0.731863\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	34.7536	−	60.1951i	1.40598	−	2.43523i
$612$		0			0
$613$	19.0465	+	32.9894i	0.769279	+	1.33243i	0.937954	+	0.346759i	$0.112718\pi$
$613$	19.0465	+	32.9894i	0.769279	+	1.33243i	−0.168675	+	0.985672i	$0.553949\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	15.1851	+	26.3014i	0.611329	+	1.05885i	0.991017	+	0.133738i	$0.0426982\pi$
$617$	15.1851	+	26.3014i	0.611329	+	1.05885i	−0.379687	+	0.925115i	$0.623968\pi$
$618$		0			0
$619$	12.4445			0.500188			0.250094	−	0.968222i	$-0.419538\pi$
$619$	12.4445			0.500188			0.250094	+	0.968222i	$0.419538\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−14.8417			−0.594620
$624$		0			0
$625$	6.05485	−	10.4873i	0.242194	−	0.419493i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	1.35304	+	5.93037i	0.0539493	+	0.236459i
$630$		0			0
$631$	4.07849	+	7.06415i	0.162362	+	0.281219i	0.935715	−	0.352756i	$-0.114755\pi$
$631$	4.07849	+	7.06415i	0.162362	+	0.281219i	−0.773353	+	0.633975i	$0.781422\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	31.5594			1.25240
$636$		0			0
$637$	−32.5202			−1.28850
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−3.48370	−	6.03395i	−0.137598	−	0.238327i	0.788989	−	0.614407i	$-0.210605\pi$
$641$	−3.48370	−	6.03395i	−0.137598	−	0.238327i	−0.926587	+	0.376081i	$0.877272\pi$
$642$		0			0
$643$	−20.1146			−0.793241			−0.396621	−	0.917983i	$-0.629817\pi$
$643$	−20.1146			−0.793241			−0.396621	+	0.917983i	$0.629817\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−9.99452	+	17.3110i	−0.392925	+	0.680566i	−0.992834	−	0.119502i	$-0.961870\pi$
$647$	−9.99452	+	17.3110i	−0.392925	+	0.680566i	0.599909	+	0.800068i	$0.295204\pi$
$648$		0			0
$649$	0.811474	+	1.40551i	0.0318531	+	0.0551712i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−15.3946	+	26.6641i	−0.602435	+	1.04345i	0.390016	+	0.920808i	$0.372470\pi$
$653$	−15.3946	+	26.6641i	−0.602435	+	1.04345i	−0.992451	+	0.122641i	$0.960864\pi$
$654$		0			0
$655$	−27.0142			−1.05553
$656$		0			0
$657$		0			0
$658$		0			0
$659$	6.40235	−	11.0892i	0.249400	−	0.431973i	−0.713960	−	0.700187i	$-0.753100\pi$
$659$	6.40235	−	11.0892i	0.249400	−	0.431973i	0.963359	+	0.268214i	$0.0864333\pi$
$660$		0			0
$661$	11.8458	−	20.5175i	0.460747	−	0.798037i	−0.538251	−	0.842784i	$-0.680915\pi$
$661$	11.8458	−	20.5175i	0.460747	−	0.798037i	0.998998	+	0.0447472i	$0.0142482\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	6.78585			0.263144
$666$		0			0
$667$	−9.44184			−0.365590
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−2.80400	+	4.85667i	−0.108247	+	0.187490i
$672$		0			0
$673$	10.9460	−	18.9590i	0.421937	−	0.730816i	−0.574192	−	0.818721i	$-0.694684\pi$
$673$	10.9460	−	18.9590i	0.421937	−	0.730816i	0.996129	+	0.0879047i	$0.0280171\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−30.9548			−1.18969			−0.594845	−	0.803841i	$-0.702786\pi$
$677$	−30.9548			−1.18969			−0.594845	+	0.803841i	$0.702786\pi$
$678$		0			0
$679$	5.90515	−	10.2280i	0.226619	−	0.392515i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−7.31212	−	12.6650i	−0.279790	−	0.484611i	0.691542	−	0.722336i	$-0.256932\pi$
$683$	−7.31212	−	12.6650i	−0.279790	−	0.484611i	−0.971333	+	0.237725i	$0.923598\pi$
$684$		0			0
$685$	15.7058	−	27.2033i	0.600088	−	1.03938i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	60.6246			2.30961
$690$		0			0
$691$	−14.2504	−	24.6824i	−0.542110	−	0.938963i	−0.998783	−	0.0493278i	$-0.984292\pi$
$691$	−14.2504	−	24.6824i	−0.542110	−	0.938963i	0.456672	−	0.889635i	$-0.349041\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−33.7223			−1.27916
$696$		0			0
$697$	8.41052			0.318571
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−4.71426	−	8.16533i	−0.178055	−	0.308400i	0.763159	−	0.646210i	$-0.223647\pi$
$701$	−4.71426	−	8.16533i	−0.178055	−	0.308400i	−0.941214	+	0.337810i	$0.890314\pi$
$702$		0			0
$703$	−14.3137	+	13.2790i	−0.539853	+	0.500826i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1.19482	+	2.06949i	−0.0449359	+	0.0778313i
$708$		0			0
$709$	22.2276			0.834777			0.417388	−	0.908728i	$-0.362945\pi$
$709$	22.2276			0.834777			0.417388	+	0.908728i	$0.362945\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−1.26636			−0.0474255
$714$		0			0
$715$	−3.24502	−	5.62054i	−0.121357	−	0.210196i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	10.5233	+	18.2269i	0.392452	+	0.679747i	0.992772	−	0.120012i	$-0.0382935\pi$
$719$	10.5233	+	18.2269i	0.392452	+	0.679747i	−0.600320	+	0.799760i	$0.704960\pi$
$720$		0			0
$721$	−8.48884	+	14.7031i	−0.316141	+	0.547572i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	4.39513	+	7.61258i	0.163231	+	0.282724i
$726$		0			0
$727$	20.4073	−	35.3466i	0.756867	−	1.31093i	−0.187574	−	0.982250i	$-0.560063\pi$
$727$	20.4073	−	35.3466i	0.756867	−	1.31093i	0.944441	−	0.328681i	$-0.106604\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	0.383889	+	0.664915i	0.0141986	+	0.0245928i
$732$		0			0
$733$	−8.09809	+	14.0263i	−0.299110	+	0.518073i	−0.975933	−	0.218073i	$-0.930023\pi$
$733$	−8.09809	+	14.0263i	−0.299110	+	0.518073i	0.676823	+	0.736146i	$0.263356\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	0.526391	−	0.911735i	0.0193898	−	0.0335842i
$738$		0			0
$739$	31.3025			1.15148			0.575741	−	0.817632i	$-0.304714\pi$
$739$	31.3025			1.15148			0.575741	+	0.817632i	$0.304714\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−14.6168	+	25.3171i	−0.536239	+	0.928793i	0.462863	+	0.886430i	$0.346822\pi$
$743$	−14.6168	+	25.3171i	−0.536239	+	0.928793i	−0.999102	+	0.0423635i	$0.986511\pi$
$744$		0			0
$745$	2.44031	−	4.22675i	0.0894062	−	0.154856i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	2.00242	+	3.46830i	0.0731669	+	0.126729i
$750$		0			0
$751$	4.06848			0.148461			0.0742304	−	0.997241i	$-0.476350\pi$
$751$	4.06848			0.148461			0.0742304	+	0.997241i	$0.476350\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−5.92985	−	10.2708i	−0.215809	−	0.373793i
$756$		0			0
$757$	5.56856	+	9.64503i	0.202393	+	0.350555i	0.949299	−	0.314375i	$-0.101795\pi$
$757$	5.56856	+	9.64503i	0.202393	+	0.350555i	−0.746906	+	0.664930i	$0.768462\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	4.42071	+	7.65689i	0.160250	+	0.277562i	0.934958	−	0.354757i	$-0.115436\pi$
$761$	4.42071	+	7.65689i	0.160250	+	0.277562i	−0.774708	+	0.632319i	$0.782103\pi$
$762$		0			0
$763$	2.48662			0.0900216
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−15.0698			−0.544139
$768$		0			0
$769$	−6.07005			−0.218892			−0.109446	−	0.993993i	$-0.534908\pi$
$769$	−6.07005			−0.218892			−0.109446	+	0.993993i	$0.534908\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	0.352621	−	0.610757i	0.0126829	−	0.0219674i	−0.859614	−	0.510943i	$-0.829296\pi$
$773$	0.352621	−	0.610757i	0.0126829	−	0.0219674i	0.872297	+	0.488976i	$0.162629\pi$
$774$		0			0
$775$	0.589483	+	1.02101i	0.0211748	+	0.0366759i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	13.4982	+	23.3796i	0.483624	+	0.837661i
$780$		0			0
$781$	3.65190	−	6.32528i	0.130675	−	0.226336i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	25.5518			0.911984
$786$		0			0
$787$	−17.2276			−0.614096			−0.307048	−	0.951694i	$-0.599341\pi$
$787$	−17.2276			−0.614096			−0.307048	+	0.951694i	$0.599341\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	7.38829			0.262697
$792$		0			0
$793$	−26.0364	−	45.0964i	−0.924580	−	1.60142i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	22.9290	+	39.7141i	0.812185	+	1.40675i	0.911331	+	0.411673i	$0.135056\pi$
$797$	22.9290	+	39.7141i	0.812185	+	1.40675i	−0.0991461	+	0.995073i	$0.531611\pi$
$798$		0			0
$799$	5.95236	+	10.3098i	0.210579	+	0.364734i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−6.22085			−0.219529
$804$		0			0
$805$	−2.12882	−	3.68722i	−0.0750310	−	0.129958i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	8.50823	−	14.7367i	0.299133	−	0.518114i	−0.676805	−	0.736163i	$-0.736636\pi$
$809$	8.50823	−	14.7367i	0.299133	−	0.518114i	0.975938	+	0.218049i	$0.0699691\pi$
$810$		0			0
$811$	9.73127	−	16.8550i	0.341711	−	0.591861i	−0.643040	−	0.765833i	$-0.722327\pi$
$811$	9.73127	−	16.8550i	0.341711	−	0.591861i	0.984751	+	0.173972i	$0.0556603\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	13.4024			0.469467
$816$		0			0
$817$	−1.23222	+	2.13427i	−0.0431100	+	0.0746687i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	23.9274	−	41.4435i	0.835073	−	1.44639i	−0.0588987	−	0.998264i	$-0.518759\pi$
$821$	23.9274	−	41.4435i	0.835073	−	1.44639i	0.893971	−	0.448124i	$-0.147908\pi$
$822$		0			0
$823$	−7.21568	−	12.4979i	−0.251523	−	0.435650i	0.712423	−	0.701751i	$-0.247598\pi$
$823$	−7.21568	−	12.4979i	−0.251523	−	0.435650i	−0.963945	+	0.266101i	$0.914265\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−9.05625	+	15.6859i	−0.314917	+	0.545452i	−0.979420	−	0.201834i	$-0.935310\pi$
$827$	−9.05625	+	15.6859i	−0.314917	+	0.545452i	0.664503	+	0.747286i	$0.268643\pi$
$828$		0			0
$829$	20.7127	+	35.8754i	0.719381	+	1.24600i	0.961245	+	0.275695i	$0.0889079\pi$
$829$	20.7127	+	35.8754i	0.719381	+	1.24600i	−0.241864	+	0.970310i	$0.577759\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	2.78492	−	4.82362i	0.0964916	−	0.167128i
$834$		0			0
$835$	−2.99861	−	5.19375i	−0.103771	−	0.179737i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−1.06229	−	1.83994i	−0.0366743	−	0.0635217i	0.847106	−	0.531424i	$-0.178343\pi$
$839$	−1.06229	−	1.83994i	−0.0366743	−	0.0635217i	−0.883780	+	0.467903i	$0.845010\pi$
$840$		0			0
$841$	−7.02049			−0.242086
$842$		0			0
$843$		0			0
$844$		0			0
$845$	37.2818			1.28253
$846$		0			0
$847$	−6.34101	+	10.9829i	−0.217880	+	0.377379i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	11.7058	+	3.61184i	0.401270	+	0.123812i
$852$		0			0
$853$	−14.6351	−	25.3487i	−0.501096	−	0.867924i	−0.999999	−	0.00126632i	$-0.999597\pi$
$853$	−14.6351	−	25.3487i	−0.501096	−	0.867924i	0.498903	−	0.866658i	$-0.333736\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	0.908844			0.0310455			0.0155228	−	0.999880i	$-0.495059\pi$
$857$	0.908844			0.0310455			0.0155228	+	0.999880i	$0.495059\pi$
$858$		0			0
$859$	11.8389			0.403938			0.201969	−	0.979392i	$-0.435266\pi$
$859$	11.8389			0.403938			0.201969	+	0.979392i	$0.435266\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−5.86015	−	10.1501i	−0.199482	−	0.345513i	0.748879	−	0.662707i	$-0.230593\pi$
$863$	−5.86015	−	10.1501i	−0.199482	−	0.345513i	−0.948361	+	0.317194i	$0.897259\pi$
$864$		0			0
$865$	−14.3810			−0.488969
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−3.57466	+	6.19149i	−0.121262	+	0.210032i
$870$		0			0
$871$	4.88777	+	8.46587i	0.165616	+	0.286855i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−7.26710	+	12.5870i	−0.245673	+	0.425518i
$876$		0			0
$877$	−20.0129			−0.675789			−0.337895	−	0.941184i	$-0.609715\pi$
$877$	−20.0129			−0.675789			−0.337895	+	0.941184i	$0.609715\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	23.4902	−	40.6863i	0.791406	−	1.37076i	−0.133690	−	0.991023i	$-0.542683\pi$
$881$	23.4902	−	40.6863i	0.791406	−	1.37076i	0.925096	−	0.379733i	$-0.123984\pi$
$882$		0			0
$883$	−8.81576	+	15.2693i	−0.296674	+	0.513854i	−0.975373	−	0.220562i	$-0.929211\pi$
$883$	−8.81576	+	15.2693i	−0.296674	+	0.513854i	0.678699	+	0.734416i	$0.262544\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−16.4698			−0.553001			−0.276501	−	0.961014i	$-0.589175\pi$
$887$	−16.4698			−0.553001			−0.276501	+	0.961014i	$0.589175\pi$
$888$		0			0
$889$	−21.3498			−0.716050
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−19.1061	+	33.0928i	−0.639362	+	1.10741i
$894$		0			0
$895$	−5.58542	+	9.67422i	−0.186700	+	0.323374i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	2.94793			0.0983191
$900$		0			0
$901$	−5.19168	+	8.99225i	−0.172960	+	0.299575i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1.29217	+	2.23811i	0.0429533	+	0.0743974i
$906$		0			0
$907$	−3.52059	+	6.09785i	−0.116899	+	0.202476i	−0.918537	−	0.395334i	$-0.870629\pi$
$907$	−3.52059	+	6.09785i	−0.116899	+	0.202476i	0.801638	+	0.597810i	$0.203962\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−3.13128			−0.103744			−0.0518719	−	0.998654i	$-0.516519\pi$
$911$	−3.13128			−0.103744			−0.0518719	+	0.998654i	$0.516519\pi$
$912$		0			0
$913$	5.09648	+	8.82736i	0.168669	+	0.292143i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	18.2750			0.603493
$918$		0			0
$919$	−35.2994			−1.16442			−0.582210	−	0.813038i	$-0.697812\pi$
$919$	−35.2994			−1.16442			−0.582210	+	0.813038i	$0.697812\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	33.9096	+	58.7331i	1.11615	+	1.93322i
$924$		0			0
$925$	−2.53690	−	11.1192i	−0.0834129	−	0.365598i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−12.8384	+	22.2368i	−0.421215	+	0.729567i	−0.996059	−	0.0886970i	$-0.971730\pi$
$929$	−12.8384	+	22.2368i	−0.421215	+	0.729567i	0.574843	+	0.818264i	$0.305063\pi$
$930$		0			0
$931$	17.8783			0.585937
$932$		0			0
$933$		0			0
$934$		0			0
$935$	1.11157			0.0363522
$936$		0			0
$937$	29.5162	+	51.1235i	0.964252	+	1.67013i	0.711612	+	0.702572i	$0.247965\pi$
$937$	29.5162	+	51.1235i	0.964252	+	1.67013i	0.252639	+	0.967560i	$0.418701\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−26.0269	−	45.0799i	−0.848452	−	1.46956i	−0.882589	−	0.470145i	$-0.844201\pi$
$941$	−26.0269	−	45.0799i	−0.848452	−	1.46956i	0.0341365	−	0.999417i	$-0.489132\pi$
$942$		0			0
$943$	8.46916	−	14.6690i	0.275794	−	0.477689i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	6.96068	+	12.0562i	0.226192	+	0.391775i	0.956676	−	0.291154i	$-0.0940391\pi$
$947$	6.96068	+	12.0562i	0.226192	+	0.391775i	−0.730485	+	0.682929i	$0.760706\pi$
$948$		0			0
$949$	28.8817	−	50.0245i	0.937538	−	1.62386i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	14.3770	+	24.9017i	0.465718	+	0.806646i	0.999234	−	0.0391435i	$-0.0124630\pi$
$953$	14.3770	+	24.9017i	0.465718	+	0.806646i	−0.533516	+	0.845790i	$0.679130\pi$
$954$		0			0
$955$	8.00553	−	13.8660i	0.259053	−	0.448693i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−10.6249	+	18.4029i	−0.343097	+	0.594261i
$960$		0			0
$961$	−30.6046			−0.987246
$962$		0			0
$963$		0			0
$964$		0			0
$965$	9.29807	−	16.1047i	0.299315	−	0.518429i
$966$		0			0
$967$	19.0404	−	32.9790i	0.612300	−	1.06053i	−0.378552	−	0.925580i	$-0.623578\pi$
$967$	19.0404	−	32.9790i	0.612300	−	1.06053i	0.990852	−	0.134954i	$-0.0430887\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−0.0548866	−	0.0950663i	−0.00176139	−	0.00305082i	0.865143	−	0.501525i	$-0.167227\pi$
$971$	−0.0548866	−	0.0950663i	−0.00176139	−	0.00305082i	−0.866905	+	0.498474i	$0.833894\pi$
$972$		0			0
$973$	22.8130			0.731353
$974$		0			0
$975$		0			0
$976$		0			0
$977$	0.767944	+	1.33012i	0.0245687	+	0.0425543i	0.878048	−	0.478572i	$-0.158845\pi$
$977$	0.767944	+	1.33012i	0.0245687	+	0.0425543i	−0.853480	+	0.521126i	$0.825512\pi$
$978$		0			0
$979$	−3.90183	−	6.75817i	−0.124703	−	0.215992i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−9.04473	−	15.6659i	−0.288482	−	0.499666i	0.684966	−	0.728575i	$-0.259817\pi$
$983$	−9.04473	−	15.6659i	−0.288482	−	0.499666i	−0.973448	+	0.228910i	$0.926484\pi$
$984$		0			0
$985$	−23.1839			−0.738699
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1.54626			0.0491683
$990$		0			0
$991$	−19.2773			−0.612364			−0.306182	−	0.951973i	$-0.599052\pi$
$991$	−19.2773			−0.612364			−0.306182	+	0.951973i	$0.599052\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	3.42727	−	5.93620i	0.108652	−	0.188190i
$996$		0			0
$997$	20.3347	+	35.2207i	0.644006	+	1.11545i	0.984530	+	0.175215i	$0.0560621\pi$
$997$	20.3347	+	35.2207i	0.644006	+	1.11545i	−0.340524	+	0.940236i	$0.610605\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2664.2.r.n.433.3		10
3.2	odd	2		296.2.i.c.137.5	yes	10
12.11	even	2		592.2.i.h.433.1		10
37.10	even	3	inner	2664.2.r.n.1009.3		10
111.47	odd	6		296.2.i.c.121.5	✓	10
444.47	even	6		592.2.i.h.417.1		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
296.2.i.c.121.5	✓	10	111.47	odd	6
296.2.i.c.137.5	yes	10	3.2	odd	2
592.2.i.h.417.1		10	444.47	even	6
592.2.i.h.433.1		10	12.11	even	2
2664.2.r.n.433.3		10	1.1	even	1	trivial
2664.2.r.n.1009.3		10	37.10	even	3	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:	\( N \)	\(=\)	\( 2664 = 2^{3} \cdot 3^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2664.r (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.883889 + 1.53094i) q^{5} +(0.597948 - 1.03568i) q^{7} +0.628794 q^{11} +(-2.91932 + 5.05641i) q^{13} +(-0.500000 - 0.866025i) q^{17} +(1.60492 - 2.77980i) q^{19} -2.01395 q^{23} +(0.937481 + 1.62376i) q^{25} +4.68823 q^{29} +0.628794 q^{31} +(1.05704 + 1.83085i) q^{35} +(-5.81237 - 1.79342i) q^{37} +(-4.20526 + 7.28372i) q^{41} -0.767778 q^{43} -11.9047 q^{47} +(2.78492 + 4.82362i) q^{49} +(-5.19168 - 8.99225i) q^{53} +(-0.555784 + 0.962647i) q^{55} +(1.29052 + 2.23525i) q^{59} +(-4.45933 + 7.72378i) q^{61} +(-5.16070 - 8.93860i) q^{65} +(0.837143 - 1.44997i) q^{67} +(5.80779 - 10.0594i) q^{71} -9.89329 q^{73} +(0.375986 - 0.651227i) q^{77} +(-5.68494 + 9.84661i) q^{79} +(8.10516 + 14.0385i) q^{83} +1.76778 q^{85} +(-6.20526 - 10.7478i) q^{89} +(3.49120 + 6.04694i) q^{91} +(2.83714 + 4.91408i) q^{95} +9.87569 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 3 q^{5} + 3 q^{7} - 16 q^{11} + q^{13} - 5 q^{17} - 3 q^{19} + 12 q^{23} - 12 q^{25} - 16 q^{31} - 5 q^{35} - 12 q^{37} - 9 q^{41} + 4 q^{43} - 32 q^{47} - 18 q^{49} - 5 q^{53} + 4 q^{55} + 5 q^{59}+ \cdots - 36 q^{97}+O(q^{100}) \) 10 * q - 3 * q^5 + 3 * q^7 - 16 * q^11 + q^13 - 5 * q^17 - 3 * q^19 + 12 * q^23 - 12 * q^25 - 16 * q^31 - 5 * q^35 - 12 * q^37 - 9 * q^41 + 4 * q^43 - 32 * q^47 - 18 * q^49 - 5 * q^53 + 4 * q^55 + 5 * q^59 + 15 * q^61 - 23 * q^65 + q^67 + 17 * q^71 - 36 * q^73 + 4 * q^77 + 21 * q^79 + 15 * q^83 + 6 * q^85 - 29 * q^89 - q^91 + 21 * q^95 - 36 * q^97

Introduction

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