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Properties

Label 9522.2.a.cd.1.4
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 1.4
Root \(-1.75786\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -0.343645 q^{5} +2.27550 q^{7} -1.00000 q^{8} +0.343645 q^{10} +3.94128 q^{11} +5.39592 q^{13} -2.27550 q^{14} +1.00000 q^{16} -7.63099 q^{17} +4.55099 q^{19} -0.343645 q^{20} -3.94128 q^{22} -4.88191 q^{25} -5.39592 q^{26} +2.27550 q^{28} -2.21804 q^{29} +2.04015 q^{31} -1.00000 q^{32} +7.63099 q^{34} -0.781962 q^{35} -7.47584 q^{37} -4.55099 q^{38} +0.343645 q^{40} -1.09007 q^{41} -2.22570 q^{43} +3.94128 q^{44} -13.2840 q^{47} -1.82212 q^{49} +4.88191 q^{50} +5.39592 q^{52} -5.62855 q^{53} -1.35440 q^{55} -2.27550 q^{56} +2.21804 q^{58} -8.39592 q^{59} -4.48986 q^{61} -2.04015 q^{62} +1.00000 q^{64} -1.85428 q^{65} -3.17641 q^{67} -7.63099 q^{68} +0.781962 q^{70} -3.50787 q^{71} -12.0977 q^{73} +7.47584 q^{74} +4.55099 q^{76} +8.96836 q^{77} -7.52246 q^{79} -0.343645 q^{80} +1.09007 q^{82} +5.31585 q^{83} +2.62235 q^{85} +2.22570 q^{86} -3.94128 q^{88} +12.5911 q^{89} +12.2784 q^{91} +13.2840 q^{94} -1.56392 q^{95} -13.1182 q^{97} +1.82212 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} + 8 q^{4} - 8 q^{8} + 12 q^{13} + 8 q^{16} + 8 q^{25} - 12 q^{26} + 12 q^{29} - 12 q^{31} - 8 q^{32} - 36 q^{35} - 24 q^{41} - 48 q^{47} - 16 q^{49} - 8 q^{50} + 12 q^{52} + 12 q^{55} - 12 q^{58}+ \cdots + 16 q^{98}+O(q^{100}) \) 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
\(5\) −0.343645 −0.153683 −0.0768413 − 0.997043i \(-0.524483\pi\)
−0.0768413 + 0.997043i \(0.524483\pi\)
\(6\) 0 0
\(7\) 2.27550 0.860057 0.430028 − 0.902815i \(-0.358504\pi\)
0.430028 + 0.902815i \(0.358504\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 0.343645 0.108670
\(11\) 3.94128 1.18834 0.594170 − 0.804340i \(-0.297481\pi\)
0.594170 + 0.804340i \(0.297481\pi\)
\(12\) 0 0
\(13\) 5.39592 1.49656 0.748280 − 0.663383i \(-0.230880\pi\)
0.748280 + 0.663383i \(0.230880\pi\)
\(14\) −2.27550 −0.608152
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −7.63099 −1.85079 −0.925393 − 0.379009i \(-0.876265\pi\)
−0.925393 + 0.379009i \(0.876265\pi\)
\(18\) 0 0
\(19\) 4.55099 1.04407 0.522035 − 0.852924i \(-0.325173\pi\)
0.522035 + 0.852924i \(0.325173\pi\)
\(20\) −0.343645 −0.0768413
\(21\) 0 0
\(22\) −3.94128 −0.840283
\(23\) 0 0
\(24\) 0 0
\(25\) −4.88191 −0.976382
\(26\) −5.39592 −1.05823
\(27\) 0 0
\(28\) 2.27550 0.430028
\(29\) −2.21804 −0.411879 −0.205940 − 0.978565i \(-0.566025\pi\)
−0.205940 + 0.978565i \(0.566025\pi\)
\(30\) 0 0
\(31\) 2.04015 0.366423 0.183211 − 0.983074i \(-0.441351\pi\)
0.183211 + 0.983074i \(0.441351\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 7.63099 1.30870
\(35\) −0.781962 −0.132176
\(36\) 0 0
\(37\) −7.47584 −1.22902 −0.614510 − 0.788909i \(-0.710646\pi\)
−0.614510 + 0.788909i \(0.710646\pi\)
\(38\) −4.55099 −0.738268
\(39\) 0 0
\(40\) 0.343645 0.0543350
\(41\) −1.09007 −0.170240 −0.0851198 − 0.996371i \(-0.527127\pi\)
−0.0851198 + 0.996371i \(0.527127\pi\)
\(42\) 0 0
\(43\) −2.22570 −0.339416 −0.169708 − 0.985494i \(-0.554282\pi\)
−0.169708 + 0.985494i \(0.554282\pi\)
\(44\) 3.94128 0.594170
\(45\) 0 0
\(46\) 0 0
\(47\) −13.2840 −1.93767 −0.968833 − 0.247714i \(-0.920321\pi\)
−0.968833 + 0.247714i \(0.920321\pi\)
\(48\) 0 0
\(49\) −1.82212 −0.260302
\(50\) 4.88191 0.690406
\(51\) 0 0
\(52\) 5.39592 0.748280
\(53\) −5.62855 −0.773141 −0.386570 − 0.922260i \(-0.626340\pi\)
−0.386570 + 0.922260i \(0.626340\pi\)
\(54\) 0 0
\(55\) −1.35440 −0.182627
\(56\) −2.27550 −0.304076
\(57\) 0 0
\(58\) 2.21804 0.291243
\(59\) −8.39592 −1.09306 −0.546528 − 0.837441i \(-0.684051\pi\)
−0.546528 + 0.837441i \(0.684051\pi\)
\(60\) 0 0
\(61\) −4.48986 −0.574868 −0.287434 − 0.957800i \(-0.592802\pi\)
−0.287434 + 0.957800i \(0.592802\pi\)
\(62\) −2.04015 −0.259100
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −1.85428 −0.229995
\(66\) 0 0
\(67\) −3.17641 −0.388061 −0.194030 − 0.980996i \(-0.562156\pi\)
−0.194030 + 0.980996i \(0.562156\pi\)
\(68\) −7.63099 −0.925393
\(69\) 0 0
\(70\) 0.781962 0.0934624
\(71\) −3.50787 −0.416308 −0.208154 − 0.978096i \(-0.566745\pi\)
−0.208154 + 0.978096i \(0.566745\pi\)
\(72\) 0 0
\(73\) −12.0977 −1.41593 −0.707964 − 0.706248i \(-0.750386\pi\)
−0.707964 + 0.706248i \(0.750386\pi\)
\(74\) 7.47584 0.869049
\(75\) 0 0
\(76\) 4.55099 0.522035
\(77\) 8.96836 1.02204
\(78\) 0 0
\(79\) −7.52246 −0.846343 −0.423172 − 0.906050i \(-0.639083\pi\)
−0.423172 + 0.906050i \(0.639083\pi\)
\(80\) −0.343645 −0.0384206
\(81\) 0 0
\(82\) 1.09007 0.120378
\(83\) 5.31585 0.583491 0.291745 − 0.956496i \(-0.405764\pi\)
0.291745 + 0.956496i \(0.405764\pi\)
\(84\) 0 0
\(85\) 2.62235 0.284434
\(86\) 2.22570 0.240003
\(87\) 0 0
\(88\) −3.94128 −0.420141
\(89\) 12.5911 1.33465 0.667327 − 0.744765i \(-0.267439\pi\)
0.667327 + 0.744765i \(0.267439\pi\)
\(90\) 0 0
\(91\) 12.2784 1.28713
\(92\) 0 0
\(93\) 0 0
\(94\) 13.2840 1.37014
\(95\) −1.56392 −0.160455
\(96\) 0 0
\(97\) −13.1182 −1.33195 −0.665975 − 0.745974i \(-0.731984\pi\)
−0.665975 + 0.745974i \(0.731984\pi\)
\(98\) 1.82212 0.184062
\(99\) 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.1.4 ✓ 8
3.2 odd 2 9522.2.a.cf.1.5 yes 8
23.22 odd 2 inner 9522.2.a.cd.1.5 yes 8
69.68 even 2 9522.2.a.cf.1.4 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9522.2.a.cd.1.4 ✓ 8 1.1 even 1 trivial
9522.2.a.cd.1.5 yes 8 23.22 odd 2 inner
9522.2.a.cf.1.4 yes 8 69.68 even 2
9522.2.a.cf.1.5 yes 8 3.2 odd 2