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Properties

Label 576.2.r.c.97.3
Level $576$
Weight $2$
Character 576.97
Analytic conductor $4.599$
Analytic rank $0$
Dimension $8$
CM discriminant -8
Inner twists $8$

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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] = [576,2,Mod(97,576)] 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("576.97"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(576, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 576.r (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.59938315643\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label 97.3
Root \(0.965926 - 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 576.97
Dual form 576.2.r.c.481.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.158919 + 1.72474i) q^{3} +(-2.94949 + 0.548188i) q^{9} +(4.71940 + 2.72474i) q^{11} -1.89898 q^{17} +8.34847i q^{19} +(-2.50000 + 4.33013i) q^{25} +(-1.41421 - 5.00000i) q^{27} +(-3.94949 + 8.57277i) q^{33} +(-6.39898 - 11.0834i) q^{41} +(2.03383 + 1.17423i) q^{43} +(3.50000 + 6.06218i) q^{49} +(-0.301783 - 3.27526i) q^{51} +(-14.3990 + 1.32673i) q^{57} +(8.00853 - 4.62372i) q^{59} +(-12.4261 + 7.17423i) q^{67} +13.6969 q^{73} +(-7.86566 - 3.62372i) q^{75} +(8.39898 - 3.23375i) q^{81} +(15.5885 + 9.00000i) q^{83} +18.0000 q^{89} +(9.84847 - 17.0580i) q^{97} +(-15.4135 - 5.44949i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{9} + 24 q^{17} - 20 q^{25} - 12 q^{33} - 12 q^{41} + 28 q^{49} - 76 q^{57} - 8 q^{73} + 28 q^{81} + 144 q^{89} + 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). 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
\(3\) 0.158919 + 1.72474i 0.0917517 + 0.995782i
\(4\) 0 0
\(5\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(6\) 0 0
\(7\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(8\) 0 0
\(9\) −2.94949 + 0.548188i −0.983163 + 0.182729i
\(10\) 0 0
\(11\) 4.71940 + 2.72474i 1.42295 + 0.821541i 0.996550 − 0.0829925i \(-0.0264478\pi\)
0.426401 + 0.904534i \(0.359781\pi\)
\(12\) 0 0
\(13\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.89898 −0.460570 −0.230285 − 0.973123i \(-0.573966\pi\)
−0.230285 + 0.973123i \(0.573966\pi\)
\(18\) 0 0
\(19\) 8.34847i 1.91527i 0.287984 + 0.957635i \(0.407015\pi\)
−0.287984 + 0.957635i \(0.592985\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(24\) 0 0
\(25\) −2.50000 + 4.33013i −0.500000 + 0.866025i
\(26\) 0 0
\(27\) −1.41421 − 5.00000i −0.272166 − 0.962250i
\(28\) 0 0
\(29\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(30\) 0 0
\(31\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(32\) 0 0
\(33\) −3.94949 + 8.57277i −0.687518 + 1.49233i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.39898 − 11.0834i −0.999353 − 1.73093i −0.530831 − 0.847477i \(-0.678120\pi\)
−0.468521 − 0.883452i \(-0.655213\pi\)
\(42\) 0 0
\(43\) 2.03383 + 1.17423i 0.310157 + 0.179069i 0.646997 − 0.762493i \(-0.276025\pi\)
−0.336840 + 0.941562i \(0.609358\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) 0 0
\(49\) 3.50000 + 6.06218i 0.500000 + 0.866025i
\(50\) 0 0
\(51\) −0.301783 − 3.27526i −0.0422581 − 0.458627i
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −14.3990 + 1.32673i −1.90719 + 0.175729i
\(58\) 0 0
\(59\) 8.00853 − 4.62372i 1.04262 − 0.601958i 0.122047 − 0.992524i \(-0.461054\pi\)
0.920575 + 0.390567i \(0.127721\pi\)
\(60\) 0 0
\(61\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −12.4261 + 7.17423i −1.51809 + 0.876472i −0.518321 + 0.855186i \(0.673443\pi\)
−0.999773 + 0.0212861i \(0.993224\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 13.6969 1.60311 0.801553 − 0.597924i \(-0.204008\pi\)
0.801553 + 0.597924i \(0.204008\pi\)
\(74\) 0 0
\(75\) −7.86566 − 3.62372i −0.908248 − 0.418432i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(80\) 0 0
\(81\) 8.39898 − 3.23375i 0.933220 − 0.359306i
\(82\) 0 0
\(83\) 15.5885 + 9.00000i 1.71106 + 0.987878i 0.933143 + 0.359506i \(0.117055\pi\)
0.777913 + 0.628372i \(0.216279\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 18.0000 1.90800 0.953998 − 0.299813i \(-0.0969242\pi\)
0.953998 + 0.299813i \(0.0969242\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 9.84847 − 17.0580i 0.999961 − 1.73198i 0.492287 − 0.870433i \(-0.336161\pi\)
0.507673 − 0.861550i \(-0.330506\pi\)
\(98\) 0 0
\(99\) −15.4135 − 5.44949i −1.54911 − 0.547694i
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 576.2.r.c.97.3 yes 8
3.2 odd 2 1728.2.r.c.289.1 8
4.3 odd 2 inner 576.2.r.c.97.2 ✓ 8
8.3 odd 2 CM 576.2.r.c.97.3 yes 8
8.5 even 2 inner 576.2.r.c.97.2 ✓ 8
9.2 odd 6 5184.2.d.e.2593.4 4
9.4 even 3 inner 576.2.r.c.481.2 yes 8
9.5 odd 6 1728.2.r.c.1441.4 8
9.7 even 3 5184.2.d.l.2593.1 4
12.11 even 2 1728.2.r.c.289.4 8
24.5 odd 2 1728.2.r.c.289.4 8
24.11 even 2 1728.2.r.c.289.1 8
36.7 odd 6 5184.2.d.l.2593.4 4
36.11 even 6 5184.2.d.e.2593.1 4
36.23 even 6 1728.2.r.c.1441.1 8
36.31 odd 6 inner 576.2.r.c.481.3 yes 8
72.5 odd 6 1728.2.r.c.1441.1 8
72.11 even 6 5184.2.d.e.2593.4 4
72.13 even 6 inner 576.2.r.c.481.3 yes 8
72.29 odd 6 5184.2.d.e.2593.1 4
72.43 odd 6 5184.2.d.l.2593.1 4
72.59 even 6 1728.2.r.c.1441.4 8
72.61 even 6 5184.2.d.l.2593.4 4
72.67 odd 6 inner 576.2.r.c.481.2 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
576.2.r.c.97.2 ✓ 8 4.3 odd 2 inner
576.2.r.c.97.2 ✓ 8 8.5 even 2 inner
576.2.r.c.97.3 yes 8 1.1 even 1 trivial
576.2.r.c.97.3 yes 8 8.3 odd 2 CM
576.2.r.c.481.2 yes 8 9.4 even 3 inner
576.2.r.c.481.2 yes 8 72.67 odd 6 inner
576.2.r.c.481.3 yes 8 36.31 odd 6 inner
576.2.r.c.481.3 yes 8 72.13 even 6 inner
1728.2.r.c.289.1 8 3.2 odd 2
1728.2.r.c.289.1 8 24.11 even 2
1728.2.r.c.289.4 8 12.11 even 2
1728.2.r.c.289.4 8 24.5 odd 2
1728.2.r.c.1441.1 8 36.23 even 6
1728.2.r.c.1441.1 8 72.5 odd 6
1728.2.r.c.1441.4 8 9.5 odd 6
1728.2.r.c.1441.4 8 72.59 even 6
5184.2.d.e.2593.1 4 36.11 even 6
5184.2.d.e.2593.1 4 72.29 odd 6
5184.2.d.e.2593.4 4 9.2 odd 6
5184.2.d.e.2593.4 4 72.11 even 6
5184.2.d.l.2593.1 4 9.7 even 3
5184.2.d.l.2593.1 4 72.43 odd 6
5184.2.d.l.2593.4 4 36.7 odd 6
5184.2.d.l.2593.4 4 72.61 even 6