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Properties

Label 720.2.w.e.593.4
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 593.4
Root \(1.14412 - 1.14412i\) of defining polynomial
Character \(\chi\) \(=\) 720.593
Dual form 720.2.w.e.17.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.58114 + 1.58114i) q^{5} +(3.23607 - 3.23607i) q^{7} -4.57649i q^{11} +(-4.23607 - 4.23607i) q^{13} +(1.74806 + 1.74806i) q^{17} -2.47214i q^{19} +(-2.82843 + 2.82843i) q^{23} +5.00000i q^{25} +5.99070 q^{29} -1.52786 q^{31} +10.2333 q^{35} +(2.23607 - 2.23607i) q^{37} +7.07107i q^{41} +(-2.47214 - 2.47214i) q^{43} +(1.74806 + 1.74806i) q^{47} -13.9443i q^{49} +(7.23607 - 7.23607i) q^{55} -1.08036 q^{59} +10.4721 q^{61} -13.3956i q^{65} +(1.52786 - 1.52786i) q^{67} +12.6491i q^{71} +(-0.527864 - 0.527864i) q^{73} +(-14.8098 - 14.8098i) q^{77} +14.4721i q^{79} +(-1.08036 + 1.08036i) q^{83} +5.52786i q^{85} +0.746512 q^{89} -27.4164 q^{91} +(3.90879 - 3.90879i) q^{95} +(1.00000 - 1.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{7} - 16 q^{13} - 48 q^{31} + 16 q^{43} + 40 q^{55} + 48 q^{61} + 48 q^{67} - 40 q^{73} - 112 q^{91} + 8 q^{97}+O(q^{100}) \) 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
\(7\) 3.23607 − 3.23607i 1.22312 − 1.22312i 0.256601 − 0.966517i \(-0.417397\pi\)
0.966517 − 0.256601i \(-0.0826028\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 4.57649i − 1.37986i −0.723874 − 0.689932i \(-0.757640\pi\)
0.723874 − 0.689932i \(-0.242360\pi\)
\(12\) 0 0
\(13\) −4.23607 − 4.23607i −1.17487 − 1.17487i −0.981033 − 0.193841i \(-0.937905\pi\)
−0.193841 − 0.981033i \(-0.562095\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.74806 + 1.74806i 0.423968 + 0.423968i 0.886567 − 0.462600i \(-0.153083\pi\)
−0.462600 + 0.886567i \(0.653083\pi\)
\(18\) 0 0
\(19\) − 2.47214i − 0.567147i −0.958951 − 0.283573i \(-0.908480\pi\)
0.958951 − 0.283573i \(-0.0915200\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.82843 + 2.82843i −0.589768 + 0.589768i −0.937568 − 0.347801i \(-0.886929\pi\)
0.347801 + 0.937568i \(0.386929\pi\)
\(24\) 0 0
\(25\) 5.00000i 1.00000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 5.99070 1.11245 0.556223 − 0.831033i \(-0.312250\pi\)
0.556223 + 0.831033i \(0.312250\pi\)
\(30\) 0 0
\(31\) −1.52786 −0.274412 −0.137206 − 0.990543i \(-0.543812\pi\)
−0.137206 + 0.990543i \(0.543812\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 10.2333 1.72975
\(36\) 0 0
\(37\) 2.23607 − 2.23607i 0.367607 − 0.367607i −0.498997 − 0.866604i \(-0.666298\pi\)
0.866604 + 0.498997i \(0.166298\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 7.07107i 1.10432i 0.833740 + 0.552158i \(0.186195\pi\)
−0.833740 + 0.552158i \(0.813805\pi\)
\(42\) 0 0
\(43\) −2.47214 − 2.47214i −0.376997 − 0.376997i 0.493020 − 0.870018i \(-0.335893\pi\)
−0.870018 + 0.493020i \(0.835893\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.74806 + 1.74806i 0.254981 + 0.254981i 0.823009 − 0.568028i \(-0.192293\pi\)
−0.568028 + 0.823009i \(0.692293\pi\)
\(48\) 0 0
\(49\) − 13.9443i − 1.99204i
\(50\) 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
\(55\) 7.23607 − 7.23607i 0.975711 − 0.975711i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −1.08036 −0.140651 −0.0703256 − 0.997524i \(-0.522404\pi\)
−0.0703256 + 0.997524i \(0.522404\pi\)
\(60\) 0 0
\(61\) 10.4721 1.34082 0.670410 − 0.741991i \(-0.266118\pi\)
0.670410 + 0.741991i \(0.266118\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 13.3956i − 1.66152i
\(66\) 0 0
\(67\) 1.52786 − 1.52786i 0.186658 − 0.186658i −0.607591 − 0.794250i \(-0.707864\pi\)
0.794250 + 0.607591i \(0.207864\pi\)
\(68\) 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
\(73\) −0.527864 − 0.527864i −0.0617818 − 0.0617818i 0.675541 − 0.737323i \(-0.263910\pi\)
−0.737323 + 0.675541i \(0.763910\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −14.8098 − 14.8098i −1.68774 − 1.68774i
\(78\) 0 0
\(79\) 14.4721i 1.62824i 0.580695 + 0.814121i \(0.302781\pi\)
−0.580695 + 0.814121i \(0.697219\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −1.08036 + 1.08036i −0.118585 + 0.118585i −0.763909 − 0.645324i \(-0.776722\pi\)
0.645324 + 0.763909i \(0.276722\pi\)
\(84\) 0 0
\(85\) 5.52786i 0.599581i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0.746512 0.0791302 0.0395651 − 0.999217i \(-0.487403\pi\)
0.0395651 + 0.999217i \(0.487403\pi\)
\(90\) 0 0
\(91\) −27.4164 −2.87402
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.90879 − 3.90879i 0.401033 − 0.401033i
\(96\) 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.593.4 8
3.2 odd 2 inner 720.2.w.e.593.2 8
4.3 odd 2 360.2.s.b.233.3 yes 8
5.2 odd 4 inner 720.2.w.e.17.2 8
5.3 odd 4 3600.2.w.j.1457.1 8
5.4 even 2 3600.2.w.j.593.1 8
8.3 odd 2 2880.2.w.m.2753.1 8
8.5 even 2 2880.2.w.o.2753.2 8
12.11 even 2 360.2.s.b.233.1 yes 8
15.2 even 4 inner 720.2.w.e.17.4 8
15.8 even 4 3600.2.w.j.1457.2 8
15.14 odd 2 3600.2.w.j.593.2 8
20.3 even 4 1800.2.s.e.1457.4 8
20.7 even 4 360.2.s.b.17.1 ✓ 8
20.19 odd 2 1800.2.s.e.593.4 8
24.5 odd 2 2880.2.w.o.2753.4 8
24.11 even 2 2880.2.w.m.2753.3 8
40.27 even 4 2880.2.w.m.2177.3 8
40.37 odd 4 2880.2.w.o.2177.4 8
60.23 odd 4 1800.2.s.e.1457.3 8
60.47 odd 4 360.2.s.b.17.3 yes 8
60.59 even 2 1800.2.s.e.593.3 8
120.77 even 4 2880.2.w.o.2177.2 8
120.107 odd 4 2880.2.w.m.2177.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
360.2.s.b.17.1 ✓ 8 20.7 even 4
360.2.s.b.17.3 yes 8 60.47 odd 4
360.2.s.b.233.1 yes 8 12.11 even 2
360.2.s.b.233.3 yes 8 4.3 odd 2
720.2.w.e.17.2 8 5.2 odd 4 inner
720.2.w.e.17.4 8 15.2 even 4 inner
720.2.w.e.593.2 8 3.2 odd 2 inner
720.2.w.e.593.4 8 1.1 even 1 trivial
1800.2.s.e.593.3 8 60.59 even 2
1800.2.s.e.593.4 8 20.19 odd 2
1800.2.s.e.1457.3 8 60.23 odd 4
1800.2.s.e.1457.4 8 20.3 even 4
2880.2.w.m.2177.1 8 120.107 odd 4
2880.2.w.m.2177.3 8 40.27 even 4
2880.2.w.m.2753.1 8 8.3 odd 2
2880.2.w.m.2753.3 8 24.11 even 2
2880.2.w.o.2177.2 8 120.77 even 4
2880.2.w.o.2177.4 8 40.37 odd 4
2880.2.w.o.2753.2 8 8.5 even 2
2880.2.w.o.2753.4 8 24.5 odd 2
3600.2.w.j.593.1 8 5.4 even 2
3600.2.w.j.593.2 8 15.14 odd 2
3600.2.w.j.1457.1 8 5.3 odd 4
3600.2.w.j.1457.2 8 15.8 even 4