LMFDB - Embedded newform 3024.2.r.d.2017.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3024,2,Mod(1009,3024)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3024, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2, 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("3024.1009"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3024 = 2^{4} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3024.r (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,2,0,1,0,0,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$24.1467615712$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 504)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			2017.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	3024.2017
Dual form			3024.2.r.d.1009.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 + 1.73205i) q^{5} +(0.500000 - 0.866025i) q^{7} +(1.50000 - 2.59808i) q^{11} +(3.00000 + 5.19615i) q^{13} -7.00000 q^{17} -1.00000 q^{19} +(2.00000 + 3.46410i) q^{23} +(0.500000 - 0.866025i) q^{25} +(-2.00000 + 3.46410i) q^{29} +(5.00000 + 8.66025i) q^{31} +2.00000 q^{35} -6.00000 q^{37} +(3.50000 + 6.06218i) q^{41} +(5.50000 - 9.52628i) q^{43} +(-4.00000 + 6.92820i) q^{47} +(-0.500000 - 0.866025i) q^{49} +4.00000 q^{53} +6.00000 q^{55} +(-4.50000 - 7.79423i) q^{59} +(-2.00000 + 3.46410i) q^{61} +(-6.00000 + 10.3923i) q^{65} +(4.50000 + 7.79423i) q^{67} +13.0000 q^{73} +(-1.50000 - 2.59808i) q^{77} +(5.00000 - 8.66025i) q^{79} +(-7.00000 - 12.1244i) q^{85} -10.0000 q^{89} +6.00000 q^{91} +(-1.00000 - 1.73205i) q^{95} +(-3.50000 + 6.06218i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} + q^{7} + 3 q^{11} + 6 q^{13} - 14 q^{17} - 2 q^{19} + 4 q^{23} + q^{25} - 4 q^{29} + 10 q^{31} + 4 q^{35} - 12 q^{37} + 7 q^{41} + 11 q^{43} - 8 q^{47} - q^{49} + 8 q^{53} + 12 q^{55}+ \cdots - 7 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 + q^7 + 3 * q^11 + 6 * q^13 - 14 * q^17 - 2 * q^19 + 4 * q^23 + q^25 - 4 * q^29 + 10 * q^31 + 4 * q^35 - 12 * q^37 + 7 * q^41 + 11 * q^43 - 8 * q^47 - q^49 + 8 * q^53 + 12 * q^55 - 9 * q^59 - 4 * q^61 - 12 * q^65 + 9 * q^67 + 26 * q^73 - 3 * q^77 + 10 * q^79 - 14 * q^85 - 20 * q^89 + 12 * q^91 - 2 * q^95 - 7 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times$.

$n$	$757$	$785$	$1135$	$2593$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$	$1$

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
$5$	1.00000	+	1.73205i	0.447214	+	0.774597i	0.998203	−	0.0599153i	$-0.0190830\pi$
$5$	1.00000	+	1.73205i	0.447214	+	0.774597i	−0.550990	+	0.834512i	$0.685750\pi$
$6$		0			0
$7$	0.500000	−	0.866025i	0.188982	−	0.327327i
$7$	0.500000	−	0.866025i	0.188982	−	0.327327i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.50000	−	2.59808i	0.452267	−	0.783349i	−0.546259	−	0.837616i	$-0.683949\pi$
$11$	1.50000	−	2.59808i	0.452267	−	0.783349i	0.998526	+	0.0542666i	$0.0172821\pi$
$12$		0			0
$13$	3.00000	+	5.19615i	0.832050	+	1.44115i	0.896410	+	0.443227i	$0.146166\pi$
$13$	3.00000	+	5.19615i	0.832050	+	1.44115i	−0.0643593	+	0.997927i	$0.520500\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−7.00000			−1.69775			−0.848875	−	0.528594i	$-0.822719\pi$
$17$	−7.00000			−1.69775			−0.848875	+	0.528594i	$0.822719\pi$
$18$		0			0
$19$	−1.00000			−0.229416			−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000			−0.229416			−0.114708	+	0.993399i	$0.536593\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.00000	+	3.46410i	0.417029	+	0.722315i	0.995639	−	0.0932891i	$-0.0297381\pi$
$23$	2.00000	+	3.46410i	0.417029	+	0.722315i	−0.578610	+	0.815604i	$0.696405\pi$
$24$		0			0
$25$	0.500000	−	0.866025i	0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.00000	+	3.46410i	−0.371391	+	0.643268i	−0.989780	−	0.142605i	$-0.954452\pi$
$29$	−2.00000	+	3.46410i	−0.371391	+	0.643268i	0.618389	+	0.785872i	$0.287786\pi$
$30$		0			0
$31$	5.00000	+	8.66025i	0.898027	+	1.55543i	0.830014	+	0.557743i	$0.188333\pi$
$31$	5.00000	+	8.66025i	0.898027	+	1.55543i	0.0680129	+	0.997684i	$0.478334\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.00000			0.338062
$36$		0			0
$37$	−6.00000			−0.986394			−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000			−0.986394			−0.493197	+	0.869918i	$0.664172\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.50000	+	6.06218i	0.546608	+	0.946753i	0.998504	+	0.0546823i	$0.0174146\pi$
$41$	3.50000	+	6.06218i	0.546608	+	0.946753i	−0.451896	+	0.892071i	$0.649252\pi$
$42$		0			0
$43$	5.50000	−	9.52628i	0.838742	−	1.45274i	−0.0522047	−	0.998636i	$-0.516625\pi$
$43$	5.50000	−	9.52628i	0.838742	−	1.45274i	0.890947	−	0.454108i	$-0.150042\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.00000	+	6.92820i	−0.583460	+	1.01058i	0.411606	+	0.911362i	$0.364968\pi$
$47$	−4.00000	+	6.92820i	−0.583460	+	1.01058i	−0.995066	+	0.0992202i	$0.968365\pi$
$48$		0			0
$49$	−0.500000	−	0.866025i	−0.0714286	−	0.123718i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	4.00000			0.549442			0.274721	−	0.961524i	$-0.411414\pi$
$53$	4.00000			0.549442			0.274721	+	0.961524i	$0.411414\pi$
$54$		0			0
$55$	6.00000			0.809040
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.50000	−	7.79423i	−0.585850	−	1.01472i	−0.994769	−	0.102151i	$-0.967427\pi$
$59$	−4.50000	−	7.79423i	−0.585850	−	1.01472i	0.408919	−	0.912571i	$-0.365906\pi$
$60$		0			0
$61$	−2.00000	+	3.46410i	−0.256074	+	0.443533i	−0.965187	−	0.261562i	$-0.915762\pi$
$61$	−2.00000	+	3.46410i	−0.256074	+	0.443533i	0.709113	+	0.705095i	$0.249096\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−6.00000	+	10.3923i	−0.744208	+	1.28901i
$66$		0			0
$67$	4.50000	+	7.79423i	0.549762	+	0.952217i	0.998290	+	0.0584478i	$0.0186151\pi$
$67$	4.50000	+	7.79423i	0.549762	+	0.952217i	−0.448528	+	0.893769i	$0.648052\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	13.0000			1.52153			0.760767	−	0.649025i	$-0.224823\pi$
$73$	13.0000			1.52153			0.760767	+	0.649025i	$0.224823\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.50000	−	2.59808i	−0.170941	−	0.296078i
$78$		0			0
$79$	5.00000	−	8.66025i	0.562544	−	0.974355i	−0.434730	−	0.900561i	$-0.643156\pi$
$79$	5.00000	−	8.66025i	0.562544	−	0.974355i	0.997274	−	0.0737937i	$-0.0235106\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$83$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$84$		0			0
$85$	−7.00000	−	12.1244i	−0.759257	−	1.31507i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−10.0000			−1.06000			−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000			−1.06000			−0.529999	+	0.847998i	$0.677808\pi$
$90$		0			0
$91$	6.00000			0.628971
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.00000	−	1.73205i	−0.102598	−	0.177705i
$96$		0			0
$97$	−3.50000	+	6.06218i	−0.355371	+	0.615521i	−0.987181	−	0.159602i	$-0.948979\pi$
$97$	−3.50000	+	6.06218i	−0.355371	+	0.615521i	0.631810	+	0.775123i	$0.282312\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−6.00000	+	10.3923i	−0.597022	+	1.03407i	0.396236	+	0.918149i	$0.370316\pi$
$101$	−6.00000	+	10.3923i	−0.597022	+	1.03407i	−0.993258	+	0.115924i	$0.963017\pi$
$102$		0			0
$103$	3.00000	+	5.19615i	0.295599	+	0.511992i	0.975124	−	0.221660i	$-0.0711475\pi$
$103$	3.00000	+	5.19615i	0.295599	+	0.511992i	−0.679525	+	0.733652i	$0.737814\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−9.00000			−0.870063			−0.435031	−	0.900415i	$-0.643263\pi$
$107$	−9.00000			−0.870063			−0.435031	+	0.900415i	$0.643263\pi$
$108$		0			0
$109$	−2.00000			−0.191565			−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000			−0.191565			−0.0957826	+	0.995402i	$0.530535\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	3.00000	+	5.19615i	0.282216	+	0.488813i	0.971930	−	0.235269i	$-0.0755971\pi$
$113$	3.00000	+	5.19615i	0.282216	+	0.488813i	−0.689714	+	0.724082i	$0.742264\pi$
$114$		0			0
$115$	−4.00000	+	6.92820i	−0.373002	+	0.646058i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−3.50000	+	6.06218i	−0.320844	+	0.555719i
$120$		0			0
$121$	1.00000	+	1.73205i	0.0909091	+	0.157459i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	12.0000			1.07331
$126$		0			0
$127$	4.00000			0.354943			0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000			0.354943			0.177471	+	0.984126i	$0.443208\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	2.00000	+	3.46410i	0.174741	+	0.302660i	0.940072	−	0.340977i	$-0.110758\pi$
$131$	2.00000	+	3.46410i	0.174741	+	0.302660i	−0.765331	+	0.643637i	$0.777425\pi$
$132$		0			0
$133$	−0.500000	+	0.866025i	−0.0433555	+	0.0750939i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	2.50000	−	4.33013i	0.213589	−	0.369948i	−0.739246	−	0.673436i	$-0.764818\pi$
$137$	2.50000	−	4.33013i	0.213589	−	0.369948i	0.952835	+	0.303488i	$0.0981512\pi$
$138$		0			0
$139$	1.50000	+	2.59808i	0.127228	+	0.220366i	0.922602	−	0.385754i	$-0.126059\pi$
$139$	1.50000	+	2.59808i	0.127228	+	0.220366i	−0.795373	+	0.606120i	$0.792725\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	18.0000			1.50524
$144$		0			0
$145$	−8.00000			−0.664364
$146$		0			0
$147$		0			0
$148$		0			0
$149$	8.00000	+	13.8564i	0.655386	+	1.13516i	0.981797	+	0.189933i	$0.0608272\pi$
$149$	8.00000	+	13.8564i	0.655386	+	1.13516i	−0.326411	+	0.945228i	$0.605840\pi$
$150$		0			0
$151$	5.00000	−	8.66025i	0.406894	−	0.704761i	−0.587646	−	0.809118i	$-0.699945\pi$
$151$	5.00000	−	8.66025i	0.406894	−	0.704761i	0.994540	+	0.104357i	$0.0332784\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−10.0000	+	17.3205i	−0.803219	+	1.39122i
$156$		0			0
$157$	3.00000	+	5.19615i	0.239426	+	0.414698i	0.960550	−	0.278108i	$-0.0897074\pi$
$157$	3.00000	+	5.19615i	0.239426	+	0.414698i	−0.721124	+	0.692806i	$0.756374\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	4.00000			0.315244
$162$		0			0
$163$	−4.00000			−0.313304			−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000			−0.313304			−0.156652	+	0.987654i	$0.550070\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−6.00000	−	10.3923i	−0.464294	−	0.804181i	0.534875	−	0.844931i	$-0.320359\pi$
$167$	−6.00000	−	10.3923i	−0.464294	−	0.804181i	−0.999169	+	0.0407502i	$0.987025\pi$
$168$		0			0
$169$	−11.5000	+	19.9186i	−0.884615	+	1.53220i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−11.0000	+	19.0526i	−0.836315	+	1.44854i	0.0566411	+	0.998395i	$0.481961\pi$
$173$	−11.0000	+	19.0526i	−0.836315	+	1.44854i	−0.892956	+	0.450145i	$0.851372\pi$
$174$		0			0
$175$	−0.500000	−	0.866025i	−0.0377964	−	0.0654654i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−16.0000			−1.19590			−0.597948	−	0.801535i	$-0.704017\pi$
$179$	−16.0000			−1.19590			−0.597948	+	0.801535i	$0.704017\pi$
$180$		0			0
$181$	8.00000			0.594635			0.297318	−	0.954779i	$-0.403908\pi$
$181$	8.00000			0.594635			0.297318	+	0.954779i	$0.403908\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−6.00000	−	10.3923i	−0.441129	−	0.764057i
$186$		0			0
$187$	−10.5000	+	18.1865i	−0.767836	+	1.32993i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	10.0000	−	17.3205i	0.723575	−	1.25327i	−0.235983	−	0.971757i	$-0.575831\pi$
$191$	10.0000	−	17.3205i	0.723575	−	1.25327i	0.959558	−	0.281511i	$-0.0908356\pi$
$192$		0			0
$193$	7.50000	+	12.9904i	0.539862	+	0.935068i	0.998911	+	0.0466572i	$0.0148568\pi$
$193$	7.50000	+	12.9904i	0.539862	+	0.935068i	−0.459049	+	0.888411i	$0.651810\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	6.00000			0.427482			0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000			0.427482			0.213741	+	0.976890i	$0.431435\pi$
$198$		0			0
$199$	−14.0000			−0.992434			−0.496217	−	0.868199i	$-0.665278\pi$
$199$	−14.0000			−0.992434			−0.496217	+	0.868199i	$0.665278\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	2.00000	+	3.46410i	0.140372	+	0.243132i
$204$		0			0
$205$	−7.00000	+	12.1244i	−0.488901	+	0.846802i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1.50000	+	2.59808i	−0.103757	+	0.179713i
$210$		0			0
$211$	4.00000	+	6.92820i	0.275371	+	0.476957i	0.970229	−	0.242190i	$-0.0778659\pi$
$211$	4.00000	+	6.92820i	0.275371	+	0.476957i	−0.694857	+	0.719148i	$0.744533\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	22.0000			1.50039
$216$		0			0
$217$	10.0000			0.678844
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−21.0000	−	36.3731i	−1.41261	−	2.44672i
$222$		0			0
$223$	−6.00000	+	10.3923i	−0.401790	+	0.695920i	−0.993942	−	0.109906i	$-0.964945\pi$
$223$	−6.00000	+	10.3923i	−0.401790	+	0.695920i	0.592152	+	0.805826i	$0.298278\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−5.50000	+	9.52628i	−0.365048	+	0.632281i	−0.988784	−	0.149354i	$-0.952281\pi$
$227$	−5.50000	+	9.52628i	−0.365048	+	0.632281i	0.623736	+	0.781635i	$0.285614\pi$
$228$		0			0
$229$	−7.00000	−	12.1244i	−0.462573	−	0.801200i	0.536515	−	0.843891i	$-0.319740\pi$
$229$	−7.00000	−	12.1244i	−0.462573	−	0.801200i	−0.999088	+	0.0426906i	$0.986407\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	13.0000			0.851658			0.425829	−	0.904804i	$-0.359982\pi$
$233$	13.0000			0.851658			0.425829	+	0.904804i	$0.359982\pi$
$234$		0			0
$235$	−16.0000			−1.04372
$236$		0			0
$237$		0			0
$238$		0			0
$239$	5.00000	+	8.66025i	0.323423	+	0.560185i	0.981192	−	0.193035i	$-0.0618330\pi$
$239$	5.00000	+	8.66025i	0.323423	+	0.560185i	−0.657769	+	0.753220i	$0.728500\pi$
$240$		0			0
$241$	14.5000	−	25.1147i	0.934027	−	1.61778i	0.157667	−	0.987492i	$-0.449603\pi$
$241$	14.5000	−	25.1147i	0.934027	−	1.61778i	0.776360	−	0.630290i	$-0.217064\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.00000	−	1.73205i	0.0638877	−	0.110657i
$246$		0			0
$247$	−3.00000	−	5.19615i	−0.190885	−	0.330623i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−5.00000			−0.315597			−0.157799	−	0.987471i	$-0.550440\pi$
$251$	−5.00000			−0.315597			−0.157799	+	0.987471i	$0.550440\pi$
$252$		0			0
$253$	12.0000			0.754434
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−13.5000	−	23.3827i	−0.842107	−	1.45857i	−0.888110	−	0.459631i	$-0.847982\pi$
$257$	−13.5000	−	23.3827i	−0.842107	−	1.45857i	0.0460033	−	0.998941i	$-0.485352\pi$
$258$		0			0
$259$	−3.00000	+	5.19615i	−0.186411	+	0.322873i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	3.00000	−	5.19615i	0.184988	−	0.320408i	−0.758585	−	0.651575i	$-0.774109\pi$
$263$	3.00000	−	5.19615i	0.184988	−	0.320408i	0.943572	+	0.331166i	$0.107442\pi$
$264$		0			0
$265$	4.00000	+	6.92820i	0.245718	+	0.425596i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	12.0000			0.731653			0.365826	−	0.930683i	$-0.380786\pi$
$269$	12.0000			0.731653			0.365826	+	0.930683i	$0.380786\pi$
$270$		0			0
$271$	−26.0000			−1.57939			−0.789694	−	0.613501i	$-0.789761\pi$
$271$	−26.0000			−1.57939			−0.789694	+	0.613501i	$0.789761\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.50000	−	2.59808i	−0.0904534	−	0.156670i
$276$		0			0
$277$	−5.00000	+	8.66025i	−0.300421	+	0.520344i	−0.976231	−	0.216731i	$-0.930460\pi$
$277$	−5.00000	+	8.66025i	−0.300421	+	0.520344i	0.675810	+	0.737075i	$0.263794\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	3.00000	−	5.19615i	0.178965	−	0.309976i	−0.762561	−	0.646916i	$-0.776058\pi$
$281$	3.00000	−	5.19615i	0.178965	−	0.309976i	0.941526	+	0.336939i	$0.109392\pi$
$282$		0			0
$283$	10.0000	+	17.3205i	0.594438	+	1.02960i	0.993626	+	0.112728i	$0.0359589\pi$
$283$	10.0000	+	17.3205i	0.594438	+	1.02960i	−0.399188	+	0.916869i	$0.630708\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	7.00000			0.413197
$288$		0			0
$289$	32.0000			1.88235
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$293$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$294$		0			0
$295$	9.00000	−	15.5885i	0.524000	−	0.907595i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−12.0000	+	20.7846i	−0.693978	+	1.20201i
$300$		0			0
$301$	−5.50000	−	9.52628i	−0.317015	−	0.549086i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−8.00000			−0.458079
$306$		0			0
$307$	17.0000			0.970241			0.485121	−	0.874447i	$-0.338776\pi$
$307$	17.0000			0.970241			0.485121	+	0.874447i	$0.338776\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	15.0000	+	25.9808i	0.850572	+	1.47323i	0.880693	+	0.473688i	$0.157077\pi$
$311$	15.0000	+	25.9808i	0.850572	+	1.47323i	−0.0301210	+	0.999546i	$0.509589\pi$
$312$		0			0
$313$	6.50000	−	11.2583i	0.367402	−	0.636358i	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	6.50000	−	11.2583i	0.367402	−	0.636358i	0.989158	+	0.146852i	$0.0469141\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	9.00000	−	15.5885i	0.505490	−	0.875535i	−0.494489	−	0.869184i	$-0.664645\pi$
$317$	9.00000	−	15.5885i	0.505490	−	0.875535i	0.999980	−	0.00635137i	$-0.00202172\pi$
$318$		0			0
$319$	6.00000	+	10.3923i	0.335936	+	0.581857i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	7.00000			0.389490
$324$		0			0
$325$	6.00000			0.332820
$326$		0			0
$327$		0			0
$328$		0			0
$329$	4.00000	+	6.92820i	0.220527	+	0.381964i
$330$		0			0
$331$	16.0000	−	27.7128i	0.879440	−	1.52323i	0.0274825	−	0.999622i	$-0.491251\pi$
$331$	16.0000	−	27.7128i	0.879440	−	1.52323i	0.851957	−	0.523612i	$-0.175416\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−9.00000	+	15.5885i	−0.491723	+	0.851688i
$336$		0			0
$337$	12.5000	+	21.6506i	0.680918	+	1.17939i	0.974701	+	0.223513i	$0.0717525\pi$
$337$	12.5000	+	21.6506i	0.680918	+	1.17939i	−0.293783	+	0.955872i	$0.594914\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	30.0000			1.62459
$342$		0			0
$343$	−1.00000			−0.0539949
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−4.50000	−	7.79423i	−0.241573	−	0.418416i	0.719590	−	0.694399i	$-0.244330\pi$
$347$	−4.50000	−	7.79423i	−0.241573	−	0.418416i	−0.961162	+	0.275983i	$0.910997\pi$
$348$		0			0
$349$	−13.0000	+	22.5167i	−0.695874	+	1.20529i	0.274011	+	0.961727i	$0.411649\pi$
$349$	−13.0000	+	22.5167i	−0.695874	+	1.20529i	−0.969885	+	0.243563i	$0.921684\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	10.5000	−	18.1865i	0.558859	−	0.967972i	−0.438733	−	0.898617i	$-0.644573\pi$
$353$	10.5000	−	18.1865i	0.558859	−	0.967972i	0.997592	−	0.0693543i	$-0.0220939\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−6.00000			−0.316668			−0.158334	−	0.987386i	$-0.550612\pi$
$359$	−6.00000			−0.316668			−0.158334	+	0.987386i	$0.550612\pi$
$360$		0			0
$361$	−18.0000			−0.947368
$362$		0			0
$363$		0			0
$364$		0			0
$365$	13.0000	+	22.5167i	0.680451	+	1.17858i
$366$		0			0
$367$	1.00000	−	1.73205i	0.0521996	−	0.0904123i	−0.838745	−	0.544524i	$-0.816710\pi$
$367$	1.00000	−	1.73205i	0.0521996	−	0.0904123i	0.890945	+	0.454112i	$0.150043\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	2.00000	−	3.46410i	0.103835	−	0.179847i
$372$		0			0
$373$	−7.00000	−	12.1244i	−0.362446	−	0.627775i	0.625917	−	0.779890i	$-0.284725\pi$
$373$	−7.00000	−	12.1244i	−0.362446	−	0.627775i	−0.988363	+	0.152115i	$0.951392\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−24.0000			−1.23606
$378$		0			0
$379$	−27.0000			−1.38690			−0.693448	−	0.720506i	$-0.743909\pi$
$379$	−27.0000			−1.38690			−0.693448	+	0.720506i	$0.743909\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−6.00000	−	10.3923i	−0.306586	−	0.531022i	0.671027	−	0.741433i	$-0.265853\pi$
$383$	−6.00000	−	10.3923i	−0.306586	−	0.531022i	−0.977613	+	0.210411i	$0.932520\pi$
$384$		0			0
$385$	3.00000	−	5.19615i	0.152894	−	0.264820i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−4.00000	+	6.92820i	−0.202808	+	0.351274i	−0.949432	−	0.313972i	$-0.898340\pi$
$389$	−4.00000	+	6.92820i	−0.202808	+	0.351274i	0.746624	+	0.665246i	$0.231673\pi$
$390$		0			0
$391$	−14.0000	−	24.2487i	−0.708010	−	1.22631i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	20.0000			1.00631
$396$		0			0
$397$	34.0000			1.70641			0.853206	−	0.521575i	$-0.174655\pi$
$397$	34.0000			1.70641			0.853206	+	0.521575i	$0.174655\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	16.5000	+	28.5788i	0.823971	+	1.42716i	0.902703	+	0.430263i	$0.141579\pi$
$401$	16.5000	+	28.5788i	0.823971	+	1.42716i	−0.0787327	+	0.996896i	$0.525087\pi$
$402$		0			0
$403$	−30.0000	+	51.9615i	−1.49441	+	2.58839i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−9.00000	+	15.5885i	−0.446113	+	0.772691i
$408$		0			0
$409$	−7.50000	−	12.9904i	−0.370851	−	0.642333i	0.618846	−	0.785513i	$-0.287601\pi$
$409$	−7.50000	−	12.9904i	−0.370851	−	0.642333i	−0.989697	+	0.143180i	$0.954267\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−9.00000			−0.442861
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−10.0000	−	17.3205i	−0.488532	−	0.846162i	0.511381	−	0.859354i	$-0.329134\pi$
$419$	−10.0000	−	17.3205i	−0.488532	−	0.846162i	−0.999913	+	0.0131919i	$0.995801\pi$
$420$		0			0
$421$	10.0000	−	17.3205i	0.487370	−	0.844150i	−0.512524	−	0.858673i	$-0.671290\pi$
$421$	10.0000	−	17.3205i	0.487370	−	0.844150i	0.999895	+	0.0145228i	$0.00462290\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−3.50000	+	6.06218i	−0.169775	+	0.294059i
$426$		0			0
$427$	2.00000	+	3.46410i	0.0967868	+	0.167640i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$	−5.00000			−0.240285			−0.120142	−	0.992757i	$-0.538335\pi$
$433$	−5.00000			−0.240285			−0.120142	+	0.992757i	$0.538335\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−2.00000	−	3.46410i	−0.0956730	−	0.165710i
$438$		0			0
$439$	12.0000	−	20.7846i	0.572729	−	0.991995i	−0.423556	−	0.905870i	$-0.639218\pi$
$439$	12.0000	−	20.7846i	0.572729	−	0.991995i	0.996284	−	0.0861252i	$-0.0274485\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−5.50000	+	9.52628i	−0.261313	+	0.452607i	−0.966591	−	0.256323i	$-0.917489\pi$
$443$	−5.50000	+	9.52628i	−0.261313	+	0.452607i	0.705278	+	0.708931i	$0.250822\pi$
$444$		0			0
$445$	−10.0000	−	17.3205i	−0.474045	−	0.821071i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	23.0000			1.08544			0.542719	−	0.839915i	$-0.317395\pi$
$449$	23.0000			1.08544			0.542719	+	0.839915i	$0.317395\pi$
$450$		0			0
$451$	21.0000			0.988851
$452$		0			0
$453$		0			0
$454$		0			0
$455$	6.00000	+	10.3923i	0.281284	+	0.487199i
$456$		0			0
$457$	−12.5000	+	21.6506i	−0.584725	+	1.01277i	0.410184	+	0.912003i	$0.365464\pi$
$457$	−12.5000	+	21.6506i	−0.584725	+	1.01277i	−0.994910	+	0.100771i	$0.967869\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	15.0000	−	25.9808i	0.698620	−	1.21004i	−0.270326	−	0.962769i	$-0.587131\pi$
$461$	15.0000	−	25.9808i	0.698620	−	1.21004i	0.968945	−	0.247276i	$-0.0795353\pi$
$462$		0			0
$463$	−8.00000	−	13.8564i	−0.371792	−	0.643962i	0.618050	−	0.786139i	$-0.287923\pi$
$463$	−8.00000	−	13.8564i	−0.371792	−	0.643962i	−0.989841	+	0.142177i	$0.954590\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−21.0000			−0.971764			−0.485882	−	0.874024i	$-0.661502\pi$
$467$	−21.0000			−0.971764			−0.485882	+	0.874024i	$0.661502\pi$
$468$		0			0
$469$	9.00000			0.415581
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−16.5000	−	28.5788i	−0.758671	−	1.31406i
$474$		0			0
$475$	−0.500000	+	0.866025i	−0.0229416	+	0.0397360i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	10.0000	−	17.3205i	0.456912	−	0.791394i	−0.541884	−	0.840453i	$-0.682289\pi$
$479$	10.0000	−	17.3205i	0.456912	−	0.791394i	0.998796	+	0.0490589i	$0.0156222\pi$
$480$		0			0
$481$	−18.0000	−	31.1769i	−0.820729	−	1.42154i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−14.0000			−0.635707
$486$		0			0
$487$	2.00000			0.0906287			0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000			0.0906287			0.0453143	+	0.998973i	$0.485571\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−10.5000	−	18.1865i	−0.473858	−	0.820747i	0.525694	−	0.850674i	$-0.323806\pi$
$491$	−10.5000	−	18.1865i	−0.473858	−	0.820747i	−0.999552	+	0.0299272i	$0.990472\pi$
$492$		0			0
$493$	14.0000	−	24.2487i	0.630528	−	1.09211i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−8.50000	−	14.7224i	−0.380512	−	0.659067i	0.610623	−	0.791921i	$-0.290919\pi$
$499$	−8.50000	−	14.7224i	−0.380512	−	0.659067i	−0.991136	+	0.132855i	$0.957586\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	22.0000			0.980932			0.490466	−	0.871460i	$-0.336827\pi$
$503$	22.0000			0.980932			0.490466	+	0.871460i	$0.336827\pi$
$504$		0			0
$505$	−24.0000			−1.06799
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−21.0000	−	36.3731i	−0.930809	−	1.61221i	−0.781943	−	0.623350i	$-0.785771\pi$
$509$	−21.0000	−	36.3731i	−0.930809	−	1.61221i	−0.148866	−	0.988857i	$-0.547562\pi$
$510$		0			0
$511$	6.50000	−	11.2583i	0.287543	−	0.498039i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−6.00000	+	10.3923i	−0.264392	+	0.457940i
$516$		0			0
$517$	12.0000	+	20.7846i	0.527759	+	0.914106i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−3.00000			−0.131432			−0.0657162	−	0.997838i	$-0.520933\pi$
$521$	−3.00000			−0.131432			−0.0657162	+	0.997838i	$0.520933\pi$
$522$		0			0
$523$	−20.0000			−0.874539			−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000			−0.874539			−0.437269	+	0.899331i	$0.644054\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−35.0000	−	60.6218i	−1.52462	−	2.64073i
$528$		0			0
$529$	3.50000	−	6.06218i	0.152174	−	0.263573i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−21.0000	+	36.3731i	−0.909611	+	1.57549i
$534$		0			0
$535$	−9.00000	−	15.5885i	−0.389104	−	0.673948i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−3.00000			−0.129219
$540$		0			0
$541$	−32.0000			−1.37579			−0.687894	−	0.725811i	$-0.741464\pi$
$541$	−32.0000			−1.37579			−0.687894	+	0.725811i	$0.741464\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−2.00000	−	3.46410i	−0.0856706	−	0.148386i
$546$		0			0
$547$	−8.50000	+	14.7224i	−0.363434	+	0.629486i	−0.988524	−	0.151067i	$-0.951729\pi$
$547$	−8.50000	+	14.7224i	−0.363434	+	0.629486i	0.625090	+	0.780553i	$0.285062\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	2.00000	−	3.46410i	0.0852029	−	0.147576i
$552$		0			0
$553$	−5.00000	−	8.66025i	−0.212622	−	0.368271i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−4.00000			−0.169485			−0.0847427	−	0.996403i	$-0.527007\pi$
$557$	−4.00000			−0.169485			−0.0847427	+	0.996403i	$0.527007\pi$
$558$		0			0
$559$	66.0000			2.79150
$560$		0			0
$561$		0			0
$562$		0			0
$563$	8.50000	+	14.7224i	0.358232	+	0.620477i	0.987666	−	0.156578i	$-0.0500463\pi$
$563$	8.50000	+	14.7224i	0.358232	+	0.620477i	−0.629433	+	0.777055i	$0.716713\pi$
$564$		0			0
$565$	−6.00000	+	10.3923i	−0.252422	+	0.437208i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−7.50000	+	12.9904i	−0.314416	+	0.544585i	−0.979313	−	0.202350i	$-0.935142\pi$
$569$	−7.50000	+	12.9904i	−0.314416	+	0.544585i	0.664897	+	0.746935i	$0.268475\pi$
$570$		0			0
$571$	−18.5000	−	32.0429i	−0.774201	−	1.34096i	−0.935243	−	0.354008i	$-0.884819\pi$
$571$	−18.5000	−	32.0429i	−0.774201	−	1.34096i	0.161042	−	0.986948i	$-0.448515\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	4.00000			0.166812
$576$		0			0
$577$	−15.0000			−0.624458			−0.312229	−	0.950007i	$-0.601076\pi$
$577$	−15.0000			−0.624458			−0.312229	+	0.950007i	$0.601076\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	6.00000	−	10.3923i	0.248495	−	0.430405i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	11.5000	−	19.9186i	0.474656	−	0.822128i	−0.524923	−	0.851150i	$-0.675906\pi$
$587$	11.5000	−	19.9186i	0.474656	−	0.822128i	0.999579	+	0.0290218i	$0.00923921\pi$
$588$		0			0
$589$	−5.00000	−	8.66025i	−0.206021	−	0.356840i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	6.00000			0.246390			0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000			0.246390			0.123195	+	0.992382i	$0.460686\pi$
$594$		0			0
$595$	−14.0000			−0.573944
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−8.00000	−	13.8564i	−0.326871	−	0.566157i	0.655018	−	0.755613i	$-0.272661\pi$
$599$	−8.00000	−	13.8564i	−0.326871	−	0.566157i	−0.981889	+	0.189456i	$0.939328\pi$
$600$		0			0
$601$	11.5000	−	19.9186i	0.469095	−	0.812496i	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	11.5000	−	19.9186i	0.469095	−	0.812496i	0.999376	+	0.0353259i	$0.0112469\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−2.00000	+	3.46410i	−0.0813116	+	0.140836i
$606$		0			0
$607$	−4.00000	−	6.92820i	−0.162355	−	0.281207i	0.773358	−	0.633970i	$-0.218576\pi$
$607$	−4.00000	−	6.92820i	−0.162355	−	0.281207i	−0.935713	+	0.352763i	$0.885242\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−48.0000			−1.94187
$612$		0			0
$613$	−14.0000			−0.565455			−0.282727	−	0.959200i	$-0.591239\pi$
$613$	−14.0000			−0.565455			−0.282727	+	0.959200i	$0.591239\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−4.50000	−	7.79423i	−0.181163	−	0.313784i	0.761114	−	0.648618i	$-0.224653\pi$
$617$	−4.50000	−	7.79423i	−0.181163	−	0.313784i	−0.942277	+	0.334835i	$0.891320\pi$
$618$		0			0
$619$	10.5000	−	18.1865i	0.422031	−	0.730978i	−0.574107	−	0.818780i	$-0.694651\pi$
$619$	10.5000	−	18.1865i	0.422031	−	0.730978i	0.996138	+	0.0878015i	$0.0279841\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−5.00000	+	8.66025i	−0.200321	+	0.346966i
$624$		0			0
$625$	9.50000	+	16.4545i	0.380000	+	0.658179i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	42.0000			1.67465
$630$		0			0
$631$	16.0000			0.636950			0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000			0.636950			0.318475	+	0.947931i	$0.396829\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	4.00000	+	6.92820i	0.158735	+	0.274937i
$636$		0			0
$637$	3.00000	−	5.19615i	0.118864	−	0.205879i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	12.5000	−	21.6506i	0.493720	−	0.855149i	−0.506254	−	0.862385i	$-0.668970\pi$
$641$	12.5000	−	21.6506i	0.493720	−	0.855149i	0.999974	+	0.00723604i	$0.00230332\pi$
$642$		0			0
$643$	12.5000	+	21.6506i	0.492952	+	0.853818i	0.999967	−	0.00811944i	$-0.00258453\pi$
$643$	12.5000	+	21.6506i	0.492952	+	0.853818i	−0.507015	+	0.861937i	$0.669251\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	12.0000			0.471769			0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000			0.471769			0.235884	+	0.971781i	$0.424201\pi$
$648$		0			0
$649$	−27.0000			−1.05984
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−11.0000	−	19.0526i	−0.430463	−	0.745584i	0.566450	−	0.824096i	$-0.308316\pi$
$653$	−11.0000	−	19.0526i	−0.430463	−	0.745584i	−0.996913	+	0.0785119i	$0.974983\pi$
$654$		0			0
$655$	−4.00000	+	6.92820i	−0.156293	+	0.270707i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	20.0000	−	34.6410i	0.779089	−	1.34942i	−0.153378	−	0.988168i	$-0.549015\pi$
$659$	20.0000	−	34.6410i	0.779089	−	1.34942i	0.932467	−	0.361255i	$-0.117652\pi$
$660$		0			0
$661$	14.0000	+	24.2487i	0.544537	+	0.943166i	0.998636	+	0.0522143i	$0.0166279\pi$
$661$	14.0000	+	24.2487i	0.544537	+	0.943166i	−0.454099	+	0.890951i	$0.650039\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−2.00000			−0.0775567
$666$		0			0
$667$	−16.0000			−0.619522
$668$		0			0
$669$		0			0
$670$		0			0
$671$	6.00000	+	10.3923i	0.231627	+	0.401190i
$672$		0			0
$673$	25.0000	−	43.3013i	0.963679	−	1.66914i	0.250557	−	0.968102i	$-0.419386\pi$
$673$	25.0000	−	43.3013i	0.963679	−	1.66914i	0.713123	−	0.701039i	$-0.247280\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	3.00000	−	5.19615i	0.115299	−	0.199704i	−0.802600	−	0.596518i	$-0.796551\pi$
$677$	3.00000	−	5.19615i	0.115299	−	0.199704i	0.917899	+	0.396813i	$0.129884\pi$
$678$		0			0
$679$	3.50000	+	6.06218i	0.134318	+	0.232645i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	27.0000			1.03313			0.516563	−	0.856249i	$-0.327211\pi$
$683$	27.0000			1.03313			0.516563	+	0.856249i	$0.327211\pi$
$684$		0			0
$685$	10.0000			0.382080
$686$		0			0
$687$		0			0
$688$		0			0
$689$	12.0000	+	20.7846i	0.457164	+	0.791831i
$690$		0			0
$691$	−8.00000	+	13.8564i	−0.304334	+	0.527123i	−0.977113	−	0.212721i	$-0.931767\pi$
$691$	−8.00000	+	13.8564i	−0.304334	+	0.527123i	0.672779	+	0.739844i	$0.265101\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−3.00000	+	5.19615i	−0.113796	+	0.197101i
$696$		0			0
$697$	−24.5000	−	42.4352i	−0.928004	−	1.60735i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−16.0000			−0.604312			−0.302156	−	0.953259i	$-0.597706\pi$
$701$	−16.0000			−0.604312			−0.302156	+	0.953259i	$0.597706\pi$
$702$		0			0
$703$	6.00000			0.226294
$704$		0			0
$705$		0			0
$706$		0			0
$707$	6.00000	+	10.3923i	0.225653	+	0.390843i
$708$		0			0
$709$	−14.0000	+	24.2487i	−0.525781	+	0.910679i	0.473768	+	0.880650i	$0.342894\pi$
$709$	−14.0000	+	24.2487i	−0.525781	+	0.910679i	−0.999549	+	0.0300298i	$0.990440\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−20.0000	+	34.6410i	−0.749006	+	1.29732i
$714$		0			0
$715$	18.0000	+	31.1769i	0.673162	+	1.16595i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	30.0000			1.11881			0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000			1.11881			0.559406	+	0.828894i	$0.311029\pi$
$720$		0			0
$721$	6.00000			0.223452
$722$		0			0
$723$		0			0
$724$		0			0
$725$	2.00000	+	3.46410i	0.0742781	+	0.128654i
$726$		0			0
$727$	13.0000	−	22.5167i	0.482143	−	0.835097i	−0.517647	−	0.855595i	$-0.673192\pi$
$727$	13.0000	−	22.5167i	0.482143	−	0.835097i	0.999790	+	0.0204978i	$0.00652512\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−38.5000	+	66.6840i	−1.42397	+	2.46640i
$732$		0			0
$733$	−11.0000	−	19.0526i	−0.406294	−	0.703722i	0.588177	−	0.808732i	$-0.299846\pi$
$733$	−11.0000	−	19.0526i	−0.406294	−	0.703722i	−0.994471	+	0.105010i	$0.966513\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	27.0000			0.994558
$738$		0			0
$739$	−19.0000			−0.698926			−0.349463	−	0.936950i	$-0.613636\pi$
$739$	−19.0000			−0.698926			−0.349463	+	0.936950i	$0.613636\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1.00000	−	1.73205i	−0.0366864	−	0.0635428i	0.847099	−	0.531435i	$-0.178347\pi$
$743$	−1.00000	−	1.73205i	−0.0366864	−	0.0635428i	−0.883786	+	0.467892i	$0.845014\pi$
$744$		0			0
$745$	−16.0000	+	27.7128i	−0.586195	+	1.01532i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−4.50000	+	7.79423i	−0.164426	+	0.284795i
$750$		0			0
$751$	−17.0000	−	29.4449i	−0.620339	−	1.07446i	−0.989423	−	0.145062i	$-0.953662\pi$
$751$	−17.0000	−	29.4449i	−0.620339	−	1.07446i	0.369084	−	0.929396i	$-0.379672\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	20.0000			0.727875
$756$		0			0
$757$	16.0000			0.581530			0.290765	−	0.956795i	$-0.406090\pi$
$757$	16.0000			0.581530			0.290765	+	0.956795i	$0.406090\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−3.00000	−	5.19615i	−0.108750	−	0.188360i	0.806514	−	0.591215i	$-0.201351\pi$
$761$	−3.00000	−	5.19615i	−0.108750	−	0.188360i	−0.915264	+	0.402854i	$0.868018\pi$
$762$		0			0
$763$	−1.00000	+	1.73205i	−0.0362024	+	0.0627044i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	27.0000	−	46.7654i	0.974913	−	1.68860i
$768$		0			0
$769$	13.0000	+	22.5167i	0.468792	+	0.811972i	0.999364	−	0.0356685i	$-0.0113561\pi$
$769$	13.0000	+	22.5167i	0.468792	+	0.811972i	−0.530572	+	0.847640i	$0.678023\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−4.00000			−0.143870			−0.0719350	−	0.997409i	$-0.522917\pi$
$773$	−4.00000			−0.143870			−0.0719350	+	0.997409i	$0.522917\pi$
$774$		0			0
$775$	10.0000			0.359211
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−3.50000	−	6.06218i	−0.125401	−	0.217200i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−6.00000	+	10.3923i	−0.214149	+	0.370917i
$786$		0			0
$787$	2.00000	+	3.46410i	0.0712923	+	0.123482i	0.899468	−	0.436987i	$-0.143954\pi$
$787$	2.00000	+	3.46410i	0.0712923	+	0.123482i	−0.828176	+	0.560469i	$0.810621\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	6.00000			0.213335
$792$		0			0
$793$	−24.0000			−0.852265
$794$		0			0
$795$		0			0
$796$		0			0
$797$	6.00000	+	10.3923i	0.212531	+	0.368114i	0.952506	−	0.304520i	$-0.0984960\pi$
$797$	6.00000	+	10.3923i	0.212531	+	0.368114i	−0.739975	+	0.672634i	$0.765163\pi$
$798$		0			0
$799$	28.0000	−	48.4974i	0.990569	−	1.71572i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	19.5000	−	33.7750i	0.688140	−	1.19189i
$804$		0			0
$805$	4.00000	+	6.92820i	0.140981	+	0.244187i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−45.0000			−1.58212			−0.791058	−	0.611741i	$-0.790469\pi$
$809$	−45.0000			−1.58212			−0.791058	+	0.611741i	$0.790469\pi$
$810$		0			0
$811$	−15.0000			−0.526721			−0.263361	−	0.964697i	$-0.584831\pi$
$811$	−15.0000			−0.526721			−0.263361	+	0.964697i	$0.584831\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−4.00000	−	6.92820i	−0.140114	−	0.242684i
$816$		0			0
$817$	−5.50000	+	9.52628i	−0.192421	+	0.333282i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−9.00000	+	15.5885i	−0.314102	+	0.544041i	−0.979246	−	0.202674i	$-0.935037\pi$
$821$	−9.00000	+	15.5885i	−0.314102	+	0.544041i	0.665144	+	0.746715i	$0.268370\pi$
$822$		0			0
$823$	−23.0000	−	39.8372i	−0.801730	−	1.38864i	−0.918477	−	0.395475i	$-0.870580\pi$
$823$	−23.0000	−	39.8372i	−0.801730	−	1.38864i	0.116747	−	0.993162i	$-0.462753\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	36.0000			1.25184			0.625921	−	0.779886i	$-0.284723\pi$
$827$	36.0000			1.25184			0.625921	+	0.779886i	$0.284723\pi$
$828$		0			0
$829$	−28.0000			−0.972480			−0.486240	−	0.873825i	$-0.661632\pi$
$829$	−28.0000			−0.972480			−0.486240	+	0.873825i	$0.661632\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	3.50000	+	6.06218i	0.121268	+	0.210042i
$834$		0			0
$835$	12.0000	−	20.7846i	0.415277	−	0.719281i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	18.0000	−	31.1769i	0.621429	−	1.07635i	−0.367791	−	0.929909i	$-0.619886\pi$
$839$	18.0000	−	31.1769i	0.621429	−	1.07635i	0.989220	−	0.146438i	$-0.0467809\pi$
$840$		0			0
$841$	6.50000	+	11.2583i	0.224138	+	0.388218i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−46.0000			−1.58245
$846$		0			0
$847$	2.00000			0.0687208
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−12.0000	−	20.7846i	−0.411355	−	0.712487i
$852$		0			0
$853$	22.0000	−	38.1051i	0.753266	−	1.30469i	−0.192966	−	0.981205i	$-0.561811\pi$
$853$	22.0000	−	38.1051i	0.753266	−	1.30469i	0.946232	−	0.323489i	$-0.104856\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	15.0000	−	25.9808i	0.512390	−	0.887486i	−0.487507	−	0.873119i	$-0.662093\pi$
$857$	15.0000	−	25.9808i	0.512390	−	0.887486i	0.999897	−	0.0143666i	$-0.00457319\pi$
$858$		0			0
$859$	−5.50000	−	9.52628i	−0.187658	−	0.325032i	0.756811	−	0.653633i	$-0.226756\pi$
$859$	−5.50000	−	9.52628i	−0.187658	−	0.325032i	−0.944469	+	0.328601i	$0.893423\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	38.0000			1.29354			0.646768	−	0.762687i	$-0.276120\pi$
$863$	38.0000			1.29354			0.646768	+	0.762687i	$0.276120\pi$
$864$		0			0
$865$	−44.0000			−1.49604
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−15.0000	−	25.9808i	−0.508840	−	0.881337i
$870$		0			0
$871$	−27.0000	+	46.7654i	−0.914860	+	1.58458i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	6.00000	−	10.3923i	0.202837	−	0.351324i
$876$		0			0
$877$	−16.0000	−	27.7128i	−0.540282	−	0.935795i	−0.998888	−	0.0471555i	$-0.984984\pi$
$877$	−16.0000	−	27.7128i	−0.540282	−	0.935795i	0.458606	−	0.888640i	$-0.348349\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	38.0000			1.28025			0.640126	−	0.768270i	$-0.278882\pi$
$881$	38.0000			1.28025			0.640126	+	0.768270i	$0.278882\pi$
$882$		0			0
$883$	−1.00000			−0.0336527			−0.0168263	−	0.999858i	$-0.505356\pi$
$883$	−1.00000			−0.0336527			−0.0168263	+	0.999858i	$0.505356\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−9.00000	−	15.5885i	−0.302190	−	0.523409i	0.674441	−	0.738328i	$-0.264385\pi$
$887$	−9.00000	−	15.5885i	−0.302190	−	0.523409i	−0.976632	+	0.214919i	$0.931051\pi$
$888$		0			0
$889$	2.00000	−	3.46410i	0.0670778	−	0.116182i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	4.00000	−	6.92820i	0.133855	−	0.231843i
$894$		0			0
$895$	−16.0000	−	27.7128i	−0.534821	−	0.926337i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−40.0000			−1.33407
$900$		0			0
$901$	−28.0000			−0.932815
$902$		0			0
$903$		0			0
$904$		0			0
$905$	8.00000	+	13.8564i	0.265929	+	0.460603i
$906$		0			0
$907$	−27.5000	+	47.6314i	−0.913123	+	1.58157i	−0.103495	+	0.994630i	$0.533003\pi$
$907$	−27.5000	+	47.6314i	−0.913123	+	1.58157i	−0.809627	+	0.586945i	$0.800331\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	16.0000	−	27.7128i	0.530104	−	0.918166i	−0.469280	−	0.883050i	$-0.655486\pi$
$911$	16.0000	−	27.7128i	0.530104	−	0.918166i	0.999383	−	0.0351168i	$-0.0111803\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	4.00000			0.132092
$918$		0			0
$919$	8.00000			0.263896			0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000			0.263896			0.131948	+	0.991257i	$0.457877\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−3.00000	+	5.19615i	−0.0986394	+	0.170848i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	23.0000	−	39.8372i	0.754606	−	1.30702i	−0.190965	−	0.981597i	$-0.561162\pi$
$929$	23.0000	−	39.8372i	0.754606	−	1.30702i	0.945570	−	0.325418i	$-0.105505\pi$
$930$		0			0
$931$	0.500000	+	0.866025i	0.0163868	+	0.0283828i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−42.0000			−1.37355
$936$		0			0
$937$	−30.0000			−0.980057			−0.490029	−	0.871706i	$-0.663014\pi$
$937$	−30.0000			−0.980057			−0.490029	+	0.871706i	$0.663014\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−6.00000	−	10.3923i	−0.195594	−	0.338779i	0.751501	−	0.659732i	$-0.229330\pi$
$941$	−6.00000	−	10.3923i	−0.195594	−	0.338779i	−0.947095	+	0.320953i	$0.895997\pi$
$942$		0			0
$943$	−14.0000	+	24.2487i	−0.455903	+	0.789647i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	8.50000	−	14.7224i	0.276213	−	0.478415i	−0.694228	−	0.719756i	$-0.744254\pi$
$947$	8.50000	−	14.7224i	0.276213	−	0.478415i	0.970440	+	0.241341i	$0.0775872\pi$
$948$		0			0
$949$	39.0000	+	67.5500i	1.26599	+	2.19277i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	35.0000			1.13376			0.566881	−	0.823800i	$-0.308150\pi$
$953$	35.0000			1.13376			0.566881	+	0.823800i	$0.308150\pi$
$954$		0			0
$955$	40.0000			1.29437
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−2.50000	−	4.33013i	−0.0807292	−	0.139827i
$960$		0			0
$961$	−34.5000	+	59.7558i	−1.11290	+	1.92760i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−15.0000	+	25.9808i	−0.482867	+	0.836350i
$966$		0			0
$967$	3.00000	+	5.19615i	0.0964735	+	0.167097i	0.910223	−	0.414119i	$-0.135910\pi$
$967$	3.00000	+	5.19615i	0.0964735	+	0.167097i	−0.813749	+	0.581216i	$0.802577\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	4.00000			0.128366			0.0641831	−	0.997938i	$-0.479556\pi$
$971$	4.00000			0.128366			0.0641831	+	0.997938i	$0.479556\pi$
$972$		0			0
$973$	3.00000			0.0961756
$974$		0			0
$975$		0			0
$976$		0			0
$977$	16.5000	+	28.5788i	0.527882	+	0.914318i	0.999472	+	0.0325001i	$0.0103469\pi$
$977$	16.5000	+	28.5788i	0.527882	+	0.914318i	−0.471590	+	0.881818i	$0.656320\pi$
$978$		0			0
$979$	−15.0000	+	25.9808i	−0.479402	+	0.830349i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	18.0000	−	31.1769i	0.574111	−	0.994389i	−0.422027	−	0.906583i	$-0.638681\pi$
$983$	18.0000	−	31.1769i	0.574111	−	0.994389i	0.996138	−	0.0878058i	$-0.0279855\pi$
$984$		0			0
$985$	6.00000	+	10.3923i	0.191176	+	0.331126i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	44.0000			1.39912
$990$		0			0
$991$	−36.0000			−1.14358			−0.571789	−	0.820401i	$-0.693750\pi$
$991$	−36.0000			−1.14358			−0.571789	+	0.820401i	$0.693750\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−14.0000	−	24.2487i	−0.443830	−	0.768736i
$996$		0			0
$997$	19.0000	−	32.9090i	0.601736	−	1.04224i	−0.390822	−	0.920466i	$-0.627809\pi$
$997$	19.0000	−	32.9090i	0.601736	−	1.04224i	0.992558	−	0.121771i	$-0.0388574\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3024.2.r.d.2017.1		2
3.2	odd	2		1008.2.r.b.673.1		2
4.3	odd	2		1512.2.r.b.505.1		2
9.2	odd	6		9072.2.a.s.1.1		1
9.4	even	3	inner	3024.2.r.d.1009.1		2
9.5	odd	6		1008.2.r.b.337.1		2
9.7	even	3		9072.2.a.d.1.1		1
12.11	even	2		504.2.r.b.169.1	✓	2
36.7	odd	6		4536.2.a.c.1.1		1
36.11	even	6		4536.2.a.h.1.1		1
36.23	even	6		504.2.r.b.337.1	yes	2
36.31	odd	6		1512.2.r.b.1009.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.r.b.169.1	✓	2	12.11	even	2
504.2.r.b.337.1	yes	2	36.23	even	6
1008.2.r.b.337.1		2	9.5	odd	6
1008.2.r.b.673.1		2	3.2	odd	2
1512.2.r.b.505.1		2	4.3	odd	2
1512.2.r.b.1009.1		2	36.31	odd	6
3024.2.r.d.1009.1		2	9.4	even	3	inner
3024.2.r.d.2017.1		2	1.1	even	1	trivial
4536.2.a.c.1.1		1	36.7	odd	6
4536.2.a.h.1.1		1	36.11	even	6
9072.2.a.d.1.1		1	9.7	even	3
9072.2.a.s.1.1		1	9.2	odd	6

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:	\( N \)	\(=\)	\( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3024.r (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.73205i) q^{5} +(0.500000 - 0.866025i) q^{7} +(1.50000 - 2.59808i) q^{11} +(3.00000 + 5.19615i) q^{13} -7.00000 q^{17} -1.00000 q^{19} +(2.00000 + 3.46410i) q^{23} +(0.500000 - 0.866025i) q^{25} +(-2.00000 + 3.46410i) q^{29} +(5.00000 + 8.66025i) q^{31} +2.00000 q^{35} -6.00000 q^{37} +(3.50000 + 6.06218i) q^{41} +(5.50000 - 9.52628i) q^{43} +(-4.00000 + 6.92820i) q^{47} +(-0.500000 - 0.866025i) q^{49} +4.00000 q^{53} +6.00000 q^{55} +(-4.50000 - 7.79423i) q^{59} +(-2.00000 + 3.46410i) q^{61} +(-6.00000 + 10.3923i) q^{65} +(4.50000 + 7.79423i) q^{67} +13.0000 q^{73} +(-1.50000 - 2.59808i) q^{77} +(5.00000 - 8.66025i) q^{79} +(-7.00000 - 12.1244i) q^{85} -10.0000 q^{89} +6.00000 q^{91} +(-1.00000 - 1.73205i) q^{95} +(-3.50000 + 6.06218i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} + q^{7} + 3 q^{11} + 6 q^{13} - 14 q^{17} - 2 q^{19} + 4 q^{23} + q^{25} - 4 q^{29} + 10 q^{31} + 4 q^{35} - 12 q^{37} + 7 q^{41} + 11 q^{43} - 8 q^{47} - q^{49} + 8 q^{53} + 12 q^{55}+ \cdots - 7 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 + q^7 + 3 * q^11 + 6 * q^13 - 14 * q^17 - 2 * q^19 + 4 * q^23 + q^25 - 4 * q^29 + 10 * q^31 + 4 * q^35 - 12 * q^37 + 7 * q^41 + 11 * q^43 - 8 * q^47 - q^49 + 8 * q^53 + 12 * q^55 - 9 * q^59 - 4 * q^61 - 12 * q^65 + 9 * q^67 + 26 * q^73 - 3 * q^77 + 10 * q^79 - 14 * q^85 - 20 * q^89 + 12 * q^91 - 2 * q^95 - 7 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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