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Properties

Label 585.2.j.b.406.1
Level $585$
Weight $2$
Character 585.406
Analytic conductor $4.671$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [585,2,Mod(406,585)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("585.406"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(585, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.j (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,2,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 195)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 406.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 585.406
Dual form 585.2.j.b.451.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.00000 + 1.73205i) q^{2} +(-1.00000 + 1.73205i) q^{4} +1.00000 q^{5} +(-2.50000 + 4.33013i) q^{7} +(1.00000 + 1.73205i) q^{10} +(1.00000 + 1.73205i) q^{11} +(-2.50000 - 2.59808i) q^{13} -10.0000 q^{14} +(2.00000 + 3.46410i) q^{16} +(1.00000 - 1.73205i) q^{17} +(-1.00000 + 1.73205i) q^{20} +(-2.00000 + 3.46410i) q^{22} +(3.00000 + 5.19615i) q^{23} +1.00000 q^{25} +(2.00000 - 6.92820i) q^{26} +(-5.00000 - 8.66025i) q^{28} +(-2.00000 - 3.46410i) q^{29} -7.00000 q^{31} +(-4.00000 + 6.92820i) q^{32} +4.00000 q^{34} +(-2.50000 + 4.33013i) q^{35} +(1.00000 + 1.73205i) q^{37} +(3.00000 + 5.19615i) q^{41} +(-0.500000 + 0.866025i) q^{43} -4.00000 q^{44} +(-6.00000 + 10.3923i) q^{46} +8.00000 q^{47} +(-9.00000 - 15.5885i) q^{49} +(1.00000 + 1.73205i) q^{50} +(7.00000 - 1.73205i) q^{52} +4.00000 q^{53} +(1.00000 + 1.73205i) q^{55} +(4.00000 - 6.92820i) q^{58} +(6.00000 - 10.3923i) q^{59} +(6.50000 - 11.2583i) q^{61} +(-7.00000 - 12.1244i) q^{62} -8.00000 q^{64} +(-2.50000 - 2.59808i) q^{65} +(3.50000 + 6.06218i) q^{67} +(2.00000 + 3.46410i) q^{68} -10.0000 q^{70} +(6.00000 - 10.3923i) q^{71} +15.0000 q^{73} +(-2.00000 + 3.46410i) q^{74} -10.0000 q^{77} +3.00000 q^{79} +(2.00000 + 3.46410i) q^{80} +(-6.00000 + 10.3923i) q^{82} -8.00000 q^{83} +(1.00000 - 1.73205i) q^{85} -2.00000 q^{86} +(7.00000 + 12.1244i) q^{89} +(17.5000 - 4.33013i) q^{91} -12.0000 q^{92} +(8.00000 + 13.8564i) q^{94} +(2.50000 - 4.33013i) q^{97} +(18.0000 - 31.1769i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{4} + 2 q^{5} - 5 q^{7} + 2 q^{10} + 2 q^{11} - 5 q^{13} - 20 q^{14} + 4 q^{16} + 2 q^{17} - 2 q^{20} - 4 q^{22} + 6 q^{23} + 2 q^{25} + 4 q^{26} - 10 q^{28} - 4 q^{29} - 14 q^{31} - 8 q^{32}+ \cdots + 36 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/585\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\) \(496\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

Coefficient data

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



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 + 1.73205i 0.707107 + 1.22474i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(3\) 0 0
\(4\) −1.00000 + 1.73205i −0.500000 + 0.866025i
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −2.50000 + 4.33013i −0.944911 + 1.63663i −0.188982 + 0.981981i \(0.560519\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 1.00000 + 1.73205i 0.316228 + 0.547723i
\(11\) 1.00000 + 1.73205i 0.301511 + 0.522233i 0.976478 − 0.215615i \(-0.0691756\pi\)
−0.674967 + 0.737848i \(0.735842\pi\)
\(12\) 0 0
\(13\) −2.50000 − 2.59808i −0.693375 − 0.720577i
\(14\) −10.0000 −2.67261
\(15\) 0 0
\(16\) 2.00000 + 3.46410i 0.500000 + 0.866025i
\(17\) 1.00000 − 1.73205i 0.242536 − 0.420084i −0.718900 − 0.695113i \(-0.755354\pi\)
0.961436 + 0.275029i \(0.0886875\pi\)
\(18\) 0 0
\(19\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(20\) −1.00000 + 1.73205i −0.223607 + 0.387298i
\(21\) 0 0
\(22\) −2.00000 + 3.46410i −0.426401 + 0.738549i
\(23\) 3.00000 + 5.19615i 0.625543 + 1.08347i 0.988436 + 0.151642i \(0.0484560\pi\)
−0.362892 + 0.931831i \(0.618211\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 2.00000 − 6.92820i 0.392232 − 1.35873i
\(27\) 0 0
\(28\) −5.00000 − 8.66025i −0.944911 − 1.63663i
\(29\) −2.00000 − 3.46410i −0.371391 − 0.643268i 0.618389 − 0.785872i \(-0.287786\pi\)
−0.989780 + 0.142605i \(0.954452\pi\)
\(30\) 0 0
\(31\) −7.00000 −1.25724 −0.628619 − 0.777714i \(-0.716379\pi\)
−0.628619 + 0.777714i \(0.716379\pi\)
\(32\) −4.00000 + 6.92820i −0.707107 + 1.22474i
\(33\) 0 0
\(34\) 4.00000 0.685994
\(35\) −2.50000 + 4.33013i −0.422577 + 0.731925i
\(36\) 0 0
\(37\) 1.00000 + 1.73205i 0.164399 + 0.284747i 0.936442 − 0.350823i \(-0.114098\pi\)
−0.772043 + 0.635571i \(0.780765\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.00000 + 5.19615i 0.468521 + 0.811503i 0.999353 − 0.0359748i \(-0.0114536\pi\)
−0.530831 + 0.847477i \(0.678120\pi\)
\(42\) 0 0
\(43\) −0.500000 + 0.866025i −0.0762493 + 0.132068i −0.901629 − 0.432511i \(-0.857628\pi\)
0.825380 + 0.564578i \(0.190961\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) −6.00000 + 10.3923i −0.884652 + 1.53226i
\(47\) 8.00000 1.16692 0.583460 − 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) −9.00000 − 15.5885i −1.28571 − 2.22692i
\(50\) 1.00000 + 1.73205i 0.141421 + 0.244949i
\(51\) 0 0
\(52\) 7.00000 − 1.73205i 0.970725 − 0.240192i
\(53\) 4.00000 0.549442 0.274721 − 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) 0 0
\(55\) 1.00000 + 1.73205i 0.134840 + 0.233550i
\(56\) 0 0
\(57\) 0 0
\(58\) 4.00000 − 6.92820i 0.525226 − 0.909718i
\(59\) 6.00000 − 10.3923i 0.781133 − 1.35296i −0.150148 − 0.988663i \(-0.547975\pi\)
0.931282 − 0.364299i \(-0.118692\pi\)
\(60\) 0 0
\(61\) 6.50000 − 11.2583i 0.832240 − 1.44148i −0.0640184 − 0.997949i \(-0.520392\pi\)
0.896258 − 0.443533i \(-0.146275\pi\)
\(62\) −7.00000 − 12.1244i −0.889001 − 1.53979i
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) −2.50000 − 2.59808i −0.310087 − 0.322252i
\(66\) 0 0
\(67\) 3.50000 + 6.06218i 0.427593 + 0.740613i 0.996659 − 0.0816792i \(-0.0260283\pi\)
−0.569066 + 0.822292i \(0.692695\pi\)
\(68\) 2.00000 + 3.46410i 0.242536 + 0.420084i
\(69\) 0 0
\(70\) −10.0000 −1.19523
\(71\) 6.00000 − 10.3923i 0.712069 − 1.23334i −0.252010 − 0.967725i \(-0.581092\pi\)
0.964079 − 0.265615i \(-0.0855750\pi\)
\(72\) 0 0
\(73\) 15.0000 1.75562 0.877809 − 0.479012i \(-0.159005\pi\)
0.877809 + 0.479012i \(0.159005\pi\)
\(74\) −2.00000 + 3.46410i −0.232495 + 0.402694i
\(75\) 0 0
\(76\) 0 0
\(77\) −10.0000 −1.13961
\(78\) 0 0
\(79\) 3.00000 0.337526 0.168763 − 0.985657i \(-0.446023\pi\)
0.168763 + 0.985657i \(0.446023\pi\)
\(80\) 2.00000 + 3.46410i 0.223607 + 0.387298i
\(81\) 0 0
\(82\) −6.00000 + 10.3923i −0.662589 + 1.14764i
\(83\) −8.00000 −0.878114 −0.439057 − 0.898459i \(-0.644687\pi\)
−0.439057 + 0.898459i \(0.644687\pi\)
\(84\) 0 0
\(85\) 1.00000 − 1.73205i 0.108465 − 0.187867i
\(86\) −2.00000 −0.215666
\(87\) 0 0
\(88\) 0 0
\(89\) 7.00000 + 12.1244i 0.741999 + 1.28518i 0.951584 + 0.307389i \(0.0994552\pi\)
−0.209585 + 0.977790i \(0.567211\pi\)
\(90\) 0 0
\(91\) 17.5000 − 4.33013i 1.83450 − 0.453921i
\(92\) −12.0000 −1.25109
\(93\) 0 0
\(94\) 8.00000 + 13.8564i 0.825137 + 1.42918i
\(95\) 0 0
\(96\) 0 0
\(97\) 2.50000 − 4.33013i 0.253837 − 0.439658i −0.710742 − 0.703452i \(-0.751641\pi\)
0.964579 + 0.263795i \(0.0849741\pi\)
\(98\) 18.0000 − 31.1769i 1.81827 − 3.14934i
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 585.2.j.b.406.1 2
3.2 odd 2 195.2.i.a.16.1 ✓ 2
13.3 even 3 7605.2.a.a.1.1 1
13.9 even 3 inner 585.2.j.b.451.1 2
13.10 even 6 7605.2.a.s.1.1 1
15.2 even 4 975.2.bb.f.874.2 4
15.8 even 4 975.2.bb.f.874.1 4
15.14 odd 2 975.2.i.i.601.1 2
39.23 odd 6 2535.2.a.c.1.1 1
39.29 odd 6 2535.2.a.m.1.1 1
39.35 odd 6 195.2.i.a.61.1 yes 2
195.74 odd 6 975.2.i.i.451.1 2
195.113 even 12 975.2.bb.f.724.2 4
195.152 even 12 975.2.bb.f.724.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.i.a.16.1 ✓ 2 3.2 odd 2
195.2.i.a.61.1 yes 2 39.35 odd 6
585.2.j.b.406.1 2 1.1 even 1 trivial
585.2.j.b.451.1 2 13.9 even 3 inner
975.2.i.i.451.1 2 195.74 odd 6
975.2.i.i.601.1 2 15.14 odd 2
975.2.bb.f.724.1 4 195.152 even 12
975.2.bb.f.724.2 4 195.113 even 12
975.2.bb.f.874.1 4 15.8 even 4
975.2.bb.f.874.2 4 15.2 even 4
2535.2.a.c.1.1 1 39.23 odd 6
2535.2.a.m.1.1 1 39.29 odd 6
7605.2.a.a.1.1 1 13.3 even 3
7605.2.a.s.1.1 1 13.10 even 6