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Properties

Label 9522.2.a.cd.1.8
Level $9522$
Weight $2$
Character 9522.1
Self dual yes
Analytic conductor $76.034$
Analytic rank $1$
Dimension $8$
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] = [9522,2,Mod(1,9522)] 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("9522.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9522, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9522 = 2 \cdot 3^{2} \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9522.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-8,0,8,0,0,0,-8,0,0,0,0,12,0,0,8,0,0,0,0,0,0,0,0,8,-12,0,0, 12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(76.0335528047\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: 8.8.546984493056.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 20x^{6} + 129x^{4} - 320x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(2.27550\) of defining polynomial
Character \(\chi\) \(=\) 9522.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +3.68971 q^{5} -1.75786 q^{7} -1.00000 q^{8} -3.68971 q^{10} -3.04470 q^{11} -2.39592 q^{13} +1.75786 q^{14} +1.00000 q^{16} +3.38834 q^{17} -3.51572 q^{19} +3.68971 q^{20} +3.04470 q^{22} +8.61396 q^{25} +2.39592 q^{26} -1.75786 q^{28} +3.48599 q^{29} -1.57605 q^{31} -1.00000 q^{32} -3.38834 q^{34} -6.48599 q^{35} -10.4285 q^{37} +3.51572 q^{38} -3.68971 q^{40} -3.17788 q^{41} +11.7463 q^{43} -3.04470 q^{44} -9.10833 q^{47} -3.90993 q^{49} -8.61396 q^{50} -2.39592 q^{52} +13.4575 q^{53} -11.2341 q^{55} +1.75786 q^{56} -3.48599 q^{58} -0.604079 q^{59} +10.5628 q^{61} +1.57605 q^{62} +1.00000 q^{64} -8.84025 q^{65} -11.2431 q^{67} +3.38834 q^{68} +6.48599 q^{70} +7.90018 q^{71} +13.3656 q^{73} +10.4285 q^{74} -3.51572 q^{76} +5.35215 q^{77} -11.5558 q^{79} +3.68971 q^{80} +3.17788 q^{82} -17.8035 q^{83} +12.5020 q^{85} -11.7463 q^{86} +3.04470 q^{88} +8.55774 q^{89} +4.21169 q^{91} +9.10833 q^{94} -12.9720 q^{95} -10.1656 q^{97} +3.90993 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} + 8 q^{4} - 8 q^{8} + 12 q^{13} + 8 q^{16} + 8 q^{25} - 12 q^{26} + 12 q^{29} - 12 q^{31} - 8 q^{32} - 36 q^{35} - 24 q^{41} - 48 q^{47} - 16 q^{49} - 8 q^{50} + 12 q^{52} + 12 q^{55} - 12 q^{58}+ \cdots + 16 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 3.68971 1.65009 0.825044 − 0.565068i \(-0.191150\pi\)
0.825044 + 0.565068i \(0.191150\pi\)
\(6\) 0 0
\(7\) −1.75786 −0.664408 −0.332204 − 0.943208i \(-0.607792\pi\)
−0.332204 + 0.943208i \(0.607792\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −3.68971 −1.16679
\(11\) −3.04470 −0.918012 −0.459006 − 0.888433i \(-0.651794\pi\)
−0.459006 + 0.888433i \(0.651794\pi\)
\(12\) 0 0
\(13\) −2.39592 −0.664509 −0.332255 − 0.943190i \(-0.607809\pi\)
−0.332255 + 0.943190i \(0.607809\pi\)
\(14\) 1.75786 0.469807
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.38834 0.821794 0.410897 − 0.911682i \(-0.365216\pi\)
0.410897 + 0.911682i \(0.365216\pi\)
\(18\) 0 0
\(19\) −3.51572 −0.806561 −0.403280 − 0.915076i \(-0.632130\pi\)
−0.403280 + 0.915076i \(0.632130\pi\)
\(20\) 3.68971 0.825044
\(21\) 0 0
\(22\) 3.04470 0.649132
\(23\) 0 0
\(24\) 0 0
\(25\) 8.61396 1.72279
\(26\) 2.39592 0.469879
\(27\) 0 0
\(28\) −1.75786 −0.332204
\(29\) 3.48599 0.647332 0.323666 − 0.946171i \(-0.395085\pi\)
0.323666 + 0.946171i \(0.395085\pi\)
\(30\) 0 0
\(31\) −1.57605 −0.283067 −0.141534 − 0.989933i \(-0.545203\pi\)
−0.141534 + 0.989933i \(0.545203\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −3.38834 −0.581096
\(35\) −6.48599 −1.09633
\(36\) 0 0
\(37\) −10.4285 −1.71443 −0.857214 − 0.514960i \(-0.827807\pi\)
−0.857214 + 0.514960i \(0.827807\pi\)
\(38\) 3.51572 0.570325
\(39\) 0 0
\(40\) −3.68971 −0.583394
\(41\) −3.17788 −0.496302 −0.248151 − 0.968721i \(-0.579823\pi\)
−0.248151 + 0.968721i \(0.579823\pi\)
\(42\) 0 0
\(43\) 11.7463 1.79129 0.895643 − 0.444773i \(-0.146716\pi\)
0.895643 + 0.444773i \(0.146716\pi\)
\(44\) −3.04470 −0.459006
\(45\) 0 0
\(46\) 0 0
\(47\) −9.10833 −1.32859 −0.664294 − 0.747472i \(-0.731268\pi\)
−0.664294 + 0.747472i \(0.731268\pi\)
\(48\) 0 0
\(49\) −3.90993 −0.558562
\(50\) −8.61396 −1.21820
\(51\) 0 0
\(52\) −2.39592 −0.332255
\(53\) 13.4575 1.84853 0.924264 − 0.381753i \(-0.124680\pi\)
0.924264 + 0.381753i \(0.124680\pi\)
\(54\) 0 0
\(55\) −11.2341 −1.51480
\(56\) 1.75786 0.234904
\(57\) 0 0
\(58\) −3.48599 −0.457733
\(59\) −0.604079 −0.0786443 −0.0393222 − 0.999227i \(-0.512520\pi\)
−0.0393222 + 0.999227i \(0.512520\pi\)
\(60\) 0 0
\(61\) 10.5628 1.35243 0.676215 − 0.736704i \(-0.263619\pi\)
0.676215 + 0.736704i \(0.263619\pi\)
\(62\) 1.57605 0.200159
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −8.84025 −1.09650
\(66\) 0 0
\(67\) −11.2431 −1.37357 −0.686783 − 0.726862i \(-0.740978\pi\)
−0.686783 + 0.726862i \(0.740978\pi\)
\(68\) 3.38834 0.410897
\(69\) 0 0
\(70\) 6.48599 0.775224
\(71\) 7.90018 0.937579 0.468789 − 0.883310i \(-0.344690\pi\)
0.468789 + 0.883310i \(0.344690\pi\)
\(72\) 0 0
\(73\) 13.3656 1.56433 0.782165 − 0.623071i \(-0.214115\pi\)
0.782165 + 0.623071i \(0.214115\pi\)
\(74\) 10.4285 1.21228
\(75\) 0 0
\(76\) −3.51572 −0.403280
\(77\) 5.35215 0.609934
\(78\) 0 0
\(79\) −11.5558 −1.30013 −0.650066 − 0.759878i \(-0.725259\pi\)
−0.650066 + 0.759878i \(0.725259\pi\)
\(80\) 3.68971 0.412522
\(81\) 0 0
\(82\) 3.17788 0.350939
\(83\) −17.8035 −1.95419 −0.977096 − 0.212798i \(-0.931742\pi\)
−0.977096 + 0.212798i \(0.931742\pi\)
\(84\) 0 0
\(85\) 12.5020 1.35603
\(86\) −11.7463 −1.26663
\(87\) 0 0
\(88\) 3.04470 0.324566
\(89\) 8.55774 0.907119 0.453559 − 0.891226i \(-0.350154\pi\)
0.453559 + 0.891226i \(0.350154\pi\)
\(90\) 0 0
\(91\) 4.21169 0.441505
\(92\) 0 0
\(93\) 0 0
\(94\) 9.10833 0.939453
\(95\) −12.9720 −1.33090
\(96\) 0 0
\(97\) −10.1656 −1.03216 −0.516079 − 0.856541i \(-0.672609\pi\)
−0.516079 + 0.856541i \(0.672609\pi\)
\(98\) 3.90993 0.394963
\(99\) 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 9522.2.a.cd.1.8 yes 8
3.2 odd 2 9522.2.a.cf.1.1 yes 8
23.22 odd 2 inner 9522.2.a.cd.1.1 ✓ 8
69.68 even 2 9522.2.a.cf.1.8 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9522.2.a.cd.1.1 ✓ 8 23.22 odd 2 inner
9522.2.a.cd.1.8 yes 8 1.1 even 1 trivial
9522.2.a.cf.1.1 yes 8 3.2 odd 2
9522.2.a.cf.1.8 yes 8 69.68 even 2