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Properties

Label 720.2.w.e.593.3
Level $720$
Weight $2$
Character 720.593
Analytic conductor $5.749$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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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] = [720,2,Mod(17,720)] 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("720.17"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(720, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 2, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.w (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,8,0,0,0,0,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.74922894553\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{2}, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 7x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 360)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 593.3
Root \(0.437016 - 0.437016i\) of defining polynomial
Character \(\chi\) \(=\) 720.593
Dual form 720.2.w.e.17.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.58114 + 1.58114i) q^{5} +(-1.23607 + 1.23607i) q^{7} -1.74806i q^{11} +(0.236068 + 0.236068i) q^{13} +(4.57649 + 4.57649i) q^{17} +6.47214i q^{19} +(2.82843 - 2.82843i) q^{23} +5.00000i q^{25} +0.333851 q^{29} -10.4721 q^{31} -3.90879 q^{35} +(-2.23607 + 2.23607i) q^{37} -7.07107i q^{41} +(6.47214 + 6.47214i) q^{43} +(4.57649 + 4.57649i) q^{47} +3.94427i q^{49} +(2.76393 - 2.76393i) q^{55} +7.40492 q^{59} +1.52786 q^{61} +0.746512i q^{65} +(10.4721 - 10.4721i) q^{67} +12.6491i q^{71} +(-9.47214 - 9.47214i) q^{73} +(2.16073 + 2.16073i) q^{77} +5.52786i q^{79} +(7.40492 - 7.40492i) q^{83} +14.4721i q^{85} -13.3956 q^{89} -0.583592 q^{91} +(-10.2333 + 10.2333i) q^{95} +(1.00000 - 1.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{7} - 16 q^{13} - 48 q^{31} + 16 q^{43} + 40 q^{55} + 48 q^{61} + 48 q^{67} - 40 q^{73} - 112 q^{91} + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/720\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(271\) \(577\) \(641\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{3}{4}\right)\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 1.58114 + 1.58114i 0.707107 + 0.707107i
\(6\) 0 0
\(7\) −1.23607 + 1.23607i −0.467190 + 0.467190i −0.901003 − 0.433813i \(-0.857168\pi\)
0.433813 + 0.901003i \(0.357168\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 1.74806i − 0.527061i −0.964651 − 0.263531i \(-0.915113\pi\)
0.964651 − 0.263531i \(-0.0848870\pi\)
\(12\) 0 0
\(13\) 0.236068 + 0.236068i 0.0654735 + 0.0654735i 0.739085 − 0.673612i \(-0.235258\pi\)
−0.673612 + 0.739085i \(0.735258\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.57649 + 4.57649i 1.10996 + 1.10996i 0.993155 + 0.116808i \(0.0372661\pi\)
0.116808 + 0.993155i \(0.462734\pi\)
\(18\) 0 0
\(19\) 6.47214i 1.48481i 0.669951 + 0.742405i \(0.266315\pi\)
−0.669951 + 0.742405i \(0.733685\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.82843 − 2.82843i 0.589768 − 0.589768i −0.347801 − 0.937568i \(-0.613071\pi\)
0.937568 + 0.347801i \(0.113071\pi\)
\(24\) 0 0
\(25\) 5.00000i 1.00000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.333851 0.0619945 0.0309972 − 0.999519i \(-0.490132\pi\)
0.0309972 + 0.999519i \(0.490132\pi\)
\(30\) 0 0
\(31\) −10.4721 −1.88085 −0.940426 − 0.340000i \(-0.889573\pi\)
−0.940426 + 0.340000i \(0.889573\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.90879 −0.660706
\(36\) 0 0
\(37\) −2.23607 + 2.23607i −0.367607 + 0.367607i −0.866604 − 0.498997i \(-0.833702\pi\)
0.498997 + 0.866604i \(0.333702\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 7.07107i − 1.10432i −0.833740 − 0.552158i \(-0.813805\pi\)
0.833740 − 0.552158i \(-0.186195\pi\)
\(42\) 0 0
\(43\) 6.47214 + 6.47214i 0.986991 + 0.986991i 0.999916 − 0.0129250i \(-0.00411427\pi\)
−0.0129250 + 0.999916i \(0.504114\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.57649 + 4.57649i 0.667550 + 0.667550i 0.957148 − 0.289598i \(-0.0935217\pi\)
−0.289598 + 0.957148i \(0.593522\pi\)
\(48\) 0 0
\(49\) 3.94427i 0.563467i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(54\) 0 0
\(55\) 2.76393 − 2.76393i 0.372689 − 0.372689i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 7.40492 0.964038 0.482019 − 0.876161i \(-0.339904\pi\)
0.482019 + 0.876161i \(0.339904\pi\)
\(60\) 0 0
\(61\) 1.52786 0.195623 0.0978115 − 0.995205i \(-0.468816\pi\)
0.0978115 + 0.995205i \(0.468816\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.746512i 0.0925935i
\(66\) 0 0
\(67\) 10.4721 − 10.4721i 1.27938 − 1.27938i 0.338357 − 0.941018i \(-0.390129\pi\)
0.941018 − 0.338357i \(-0.109871\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 12.6491i 1.50117i 0.660772 + 0.750587i \(0.270229\pi\)
−0.660772 + 0.750587i \(0.729771\pi\)
\(72\) 0 0
\(73\) −9.47214 − 9.47214i −1.10863 − 1.10863i −0.993331 − 0.115299i \(-0.963217\pi\)
−0.115299 − 0.993331i \(-0.536783\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.16073 + 2.16073i 0.246238 + 0.246238i
\(78\) 0 0
\(79\) 5.52786i 0.621933i 0.950421 + 0.310967i \(0.100653\pi\)
−0.950421 + 0.310967i \(0.899347\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 7.40492 − 7.40492i 0.812795 − 0.812795i −0.172257 − 0.985052i \(-0.555106\pi\)
0.985052 + 0.172257i \(0.0551059\pi\)
\(84\) 0 0
\(85\) 14.4721i 1.56972i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −13.3956 −1.41993 −0.709967 − 0.704235i \(-0.751290\pi\)
−0.709967 + 0.704235i \(0.751290\pi\)
\(90\) 0 0
\(91\) −0.583592 −0.0611771
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −10.2333 + 10.2333i −1.04992 + 1.04992i
\(96\) 0 0
\(97\) 1.00000 − 1.00000i 0.101535 − 0.101535i −0.654515 − 0.756049i \(-0.727127\pi\)
0.756049 + 0.654515i \(0.227127\pi\)
\(98\) 0 0
\(99\) 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 720.2.w.e.593.3 8
3.2 odd 2 inner 720.2.w.e.593.1 8
4.3 odd 2 360.2.s.b.233.4 yes 8
5.2 odd 4 inner 720.2.w.e.17.1 8
5.3 odd 4 3600.2.w.j.1457.3 8
5.4 even 2 3600.2.w.j.593.3 8
8.3 odd 2 2880.2.w.m.2753.2 8
8.5 even 2 2880.2.w.o.2753.1 8
12.11 even 2 360.2.s.b.233.2 yes 8
15.2 even 4 inner 720.2.w.e.17.3 8
15.8 even 4 3600.2.w.j.1457.4 8
15.14 odd 2 3600.2.w.j.593.4 8
20.3 even 4 1800.2.s.e.1457.2 8
20.7 even 4 360.2.s.b.17.2 ✓ 8
20.19 odd 2 1800.2.s.e.593.2 8
24.5 odd 2 2880.2.w.o.2753.3 8
24.11 even 2 2880.2.w.m.2753.4 8
40.27 even 4 2880.2.w.m.2177.4 8
40.37 odd 4 2880.2.w.o.2177.3 8
60.23 odd 4 1800.2.s.e.1457.1 8
60.47 odd 4 360.2.s.b.17.4 yes 8
60.59 even 2 1800.2.s.e.593.1 8
120.77 even 4 2880.2.w.o.2177.1 8
120.107 odd 4 2880.2.w.m.2177.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
360.2.s.b.17.2 ✓ 8 20.7 even 4
360.2.s.b.17.4 yes 8 60.47 odd 4
360.2.s.b.233.2 yes 8 12.11 even 2
360.2.s.b.233.4 yes 8 4.3 odd 2
720.2.w.e.17.1 8 5.2 odd 4 inner
720.2.w.e.17.3 8 15.2 even 4 inner
720.2.w.e.593.1 8 3.2 odd 2 inner
720.2.w.e.593.3 8 1.1 even 1 trivial
1800.2.s.e.593.1 8 60.59 even 2
1800.2.s.e.593.2 8 20.19 odd 2
1800.2.s.e.1457.1 8 60.23 odd 4
1800.2.s.e.1457.2 8 20.3 even 4
2880.2.w.m.2177.2 8 120.107 odd 4
2880.2.w.m.2177.4 8 40.27 even 4
2880.2.w.m.2753.2 8 8.3 odd 2
2880.2.w.m.2753.4 8 24.11 even 2
2880.2.w.o.2177.1 8 120.77 even 4
2880.2.w.o.2177.3 8 40.37 odd 4
2880.2.w.o.2753.1 8 8.5 even 2
2880.2.w.o.2753.3 8 24.5 odd 2
3600.2.w.j.593.3 8 5.4 even 2
3600.2.w.j.593.4 8 15.14 odd 2
3600.2.w.j.1457.3 8 5.3 odd 4
3600.2.w.j.1457.4 8 15.8 even 4