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Properties

Label 7569.2.a.bm.1.2
Level $7569$
Weight $2$
Character 7569.1
Self dual yes
Analytic conductor $60.439$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

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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] = [7569,2,Mod(1,7569)] 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("7569.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7569, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7569 = 3^{2} \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7569.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [9,5,0,11,4,0,5,24,0,0,-1,0,1,9,0,35,2,0,-9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(60.4387692899\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 4x^{8} - 6x^{7} + 33x^{6} + 6x^{5} - 90x^{4} + 21x^{3} + 84x^{2} - 36x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 87)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.21072\) of defining polynomial
Character \(\chi\) \(=\) 7569.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.21072 q^{2} -0.534161 q^{4} -1.79844 q^{5} +4.27693 q^{7} +3.06816 q^{8} +2.17740 q^{10} +1.05626 q^{11} -5.28193 q^{13} -5.17816 q^{14} -2.64635 q^{16} +5.61769 q^{17} -6.41175 q^{19} +0.960655 q^{20} -1.27883 q^{22} -2.04633 q^{23} -1.76563 q^{25} +6.39492 q^{26} -2.28457 q^{28} -3.82471 q^{31} -2.93233 q^{32} -6.80144 q^{34} -7.69178 q^{35} +0.350297 q^{37} +7.76282 q^{38} -5.51788 q^{40} -3.62240 q^{41} -1.74900 q^{43} -0.564212 q^{44} +2.47753 q^{46} +6.98005 q^{47} +11.2921 q^{49} +2.13768 q^{50} +2.82140 q^{52} +7.81869 q^{53} -1.89961 q^{55} +13.1223 q^{56} -0.382668 q^{59} +5.46644 q^{61} +4.63064 q^{62} +8.84292 q^{64} +9.49920 q^{65} +7.81166 q^{67} -3.00075 q^{68} +9.31258 q^{70} +15.6115 q^{71} +2.54289 q^{73} -0.424110 q^{74} +3.42491 q^{76} +4.51754 q^{77} -10.2680 q^{79} +4.75929 q^{80} +4.38571 q^{82} -1.52139 q^{83} -10.1030 q^{85} +2.11755 q^{86} +3.24076 q^{88} -17.9444 q^{89} -22.5904 q^{91} +1.09307 q^{92} -8.45087 q^{94} +11.5311 q^{95} +6.80968 q^{97} -13.6716 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 5 q^{2} + 11 q^{4} + 4 q^{5} + 5 q^{7} + 24 q^{8} - q^{11} + q^{13} + 9 q^{14} + 35 q^{16} + 2 q^{17} - 9 q^{19} + 18 q^{20} - 4 q^{22} + 4 q^{23} + q^{25} - 8 q^{26} + 40 q^{28} - 8 q^{31} + 43 q^{32}+ \cdots - 30 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.21072 −0.856107 −0.428054 − 0.903753i \(-0.640801\pi\)
−0.428054 + 0.903753i \(0.640801\pi\)
\(3\) 0 0
\(4\) −0.534161 −0.267081
\(5\) −1.79844 −0.804285 −0.402142 − 0.915577i \(-0.631734\pi\)
−0.402142 + 0.915577i \(0.631734\pi\)
\(6\) 0 0
\(7\) 4.27693 1.61653 0.808264 − 0.588821i \(-0.200408\pi\)
0.808264 + 0.588821i \(0.200408\pi\)
\(8\) 3.06816 1.08476
\(9\) 0 0
\(10\) 2.17740 0.688554
\(11\) 1.05626 0.318474 0.159237 − 0.987240i \(-0.449097\pi\)
0.159237 + 0.987240i \(0.449097\pi\)
\(12\) 0 0
\(13\) −5.28193 −1.46494 −0.732471 − 0.680798i \(-0.761633\pi\)
−0.732471 + 0.680798i \(0.761633\pi\)
\(14\) −5.17816 −1.38392
\(15\) 0 0
\(16\) −2.64635 −0.661587
\(17\) 5.61769 1.36249 0.681245 − 0.732056i \(-0.261439\pi\)
0.681245 + 0.732056i \(0.261439\pi\)
\(18\) 0 0
\(19\) −6.41175 −1.47096 −0.735478 − 0.677548i \(-0.763042\pi\)
−0.735478 + 0.677548i \(0.763042\pi\)
\(20\) 0.960655 0.214809
\(21\) 0 0
\(22\) −1.27883 −0.272648
\(23\) −2.04633 −0.426689 −0.213345 − 0.976977i \(-0.568436\pi\)
−0.213345 + 0.976977i \(0.568436\pi\)
\(24\) 0 0
\(25\) −1.76563 −0.353126
\(26\) 6.39492 1.25415
\(27\) 0 0
\(28\) −2.28457 −0.431743
\(29\) 0 0
\(30\) 0 0
\(31\) −3.82471 −0.686937 −0.343469 − 0.939164i \(-0.611602\pi\)
−0.343469 + 0.939164i \(0.611602\pi\)
\(32\) −2.93233 −0.518367
\(33\) 0 0
\(34\) −6.80144 −1.16644
\(35\) −7.69178 −1.30015
\(36\) 0 0
\(37\) 0.350297 0.0575884 0.0287942 − 0.999585i \(-0.490833\pi\)
0.0287942 + 0.999585i \(0.490833\pi\)
\(38\) 7.76282 1.25930
\(39\) 0 0
\(40\) −5.51788 −0.872453
\(41\) −3.62240 −0.565725 −0.282862 − 0.959161i \(-0.591284\pi\)
−0.282862 + 0.959161i \(0.591284\pi\)
\(42\) 0 0
\(43\) −1.74900 −0.266721 −0.133360 − 0.991068i \(-0.542577\pi\)
−0.133360 + 0.991068i \(0.542577\pi\)
\(44\) −0.564212 −0.0850582
\(45\) 0 0
\(46\) 2.47753 0.365292
\(47\) 6.98005 1.01814 0.509072 − 0.860724i \(-0.329989\pi\)
0.509072 + 0.860724i \(0.329989\pi\)
\(48\) 0 0
\(49\) 11.2921 1.61316
\(50\) 2.13768 0.302314
\(51\) 0 0
\(52\) 2.82140 0.391258
\(53\) 7.81869 1.07398 0.536990 − 0.843589i \(-0.319561\pi\)
0.536990 + 0.843589i \(0.319561\pi\)
\(54\) 0 0
\(55\) −1.89961 −0.256144
\(56\) 13.1223 1.75354
\(57\) 0 0
\(58\) 0 0
\(59\) −0.382668 −0.0498191 −0.0249096 − 0.999690i \(-0.507930\pi\)
−0.0249096 + 0.999690i \(0.507930\pi\)
\(60\) 0 0
\(61\) 5.46644 0.699906 0.349953 − 0.936767i \(-0.386197\pi\)
0.349953 + 0.936767i \(0.386197\pi\)
\(62\) 4.63064 0.588092
\(63\) 0 0
\(64\) 8.84292 1.10537
\(65\) 9.49920 1.17823
\(66\) 0 0
\(67\) 7.81166 0.954346 0.477173 − 0.878809i \(-0.341662\pi\)
0.477173 + 0.878809i \(0.341662\pi\)
\(68\) −3.00075 −0.363895
\(69\) 0 0
\(70\) 9.31258 1.11307
\(71\) 15.6115 1.85274 0.926370 − 0.376615i \(-0.122912\pi\)
0.926370 + 0.376615i \(0.122912\pi\)
\(72\) 0 0
\(73\) 2.54289 0.297623 0.148811 − 0.988866i \(-0.452455\pi\)
0.148811 + 0.988866i \(0.452455\pi\)
\(74\) −0.424110 −0.0493018
\(75\) 0 0
\(76\) 3.42491 0.392864
\(77\) 4.51754 0.514822
\(78\) 0 0
\(79\) −10.2680 −1.15524 −0.577619 − 0.816306i \(-0.696018\pi\)
−0.577619 + 0.816306i \(0.696018\pi\)
\(80\) 4.75929 0.532104
\(81\) 0 0
\(82\) 4.38571 0.484321
\(83\) −1.52139 −0.166994 −0.0834971 − 0.996508i \(-0.526609\pi\)
−0.0834971 + 0.996508i \(0.526609\pi\)
\(84\) 0 0
\(85\) −10.1030 −1.09583
\(86\) 2.11755 0.228341
\(87\) 0 0
\(88\) 3.24076 0.345467
\(89\) −17.9444 −1.90210 −0.951049 − 0.309041i \(-0.899992\pi\)
−0.951049 + 0.309041i \(0.899992\pi\)
\(90\) 0 0
\(91\) −22.5904 −2.36812
\(92\) 1.09307 0.113960
\(93\) 0 0
\(94\) −8.45087 −0.871641
\(95\) 11.5311 1.18307
\(96\) 0 0
\(97\) 6.80968 0.691418 0.345709 − 0.938342i \(-0.387638\pi\)
0.345709 + 0.938342i \(0.387638\pi\)
\(98\) −13.6716 −1.38104
\(99\) 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 7569.2.a.bm.1.2 9
3.2 odd 2 2523.2.a.o.1.8 9
29.9 even 14 261.2.k.c.226.3 18
29.13 even 14 261.2.k.c.82.3 18
29.28 even 2 7569.2.a.bj.1.8 9
87.38 odd 14 87.2.g.a.52.1 ✓ 18
87.71 odd 14 87.2.g.a.82.1 yes 18
87.86 odd 2 2523.2.a.r.1.2 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
87.2.g.a.52.1 ✓ 18 87.38 odd 14
87.2.g.a.82.1 yes 18 87.71 odd 14
261.2.k.c.82.3 18 29.13 even 14
261.2.k.c.226.3 18 29.9 even 14
2523.2.a.o.1.8 9 3.2 odd 2
2523.2.a.r.1.2 9 87.86 odd 2
7569.2.a.bj.1.8 9 29.28 even 2
7569.2.a.bm.1.2 9 1.1 even 1 trivial