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Properties

Label 9522.2.a.t.1.1
Level $9522$
Weight $2$
Character 9522.1
Self dual yes
Analytic conductor $76.034$
Analytic rank $1$
Dimension $2$
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 = [2,-2,0,2,0,0,0,-2,0,0,0,0,-10,0,0,2,0,0,0,0,0,0,0,0,-4,10,0, 0,6] 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: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1058)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.73205 q^{5} -3.46410 q^{7} -1.00000 q^{8} +1.73205 q^{10} -3.46410 q^{11} -5.00000 q^{13} +3.46410 q^{14} +1.00000 q^{16} +6.92820 q^{17} +3.46410 q^{19} -1.73205 q^{20} +3.46410 q^{22} -2.00000 q^{25} +5.00000 q^{26} -3.46410 q^{28} +3.00000 q^{29} -8.00000 q^{31} -1.00000 q^{32} -6.92820 q^{34} +6.00000 q^{35} -3.46410 q^{38} +1.73205 q^{40} +9.00000 q^{41} +6.92820 q^{43} -3.46410 q^{44} +6.00000 q^{47} +5.00000 q^{49} +2.00000 q^{50} -5.00000 q^{52} +1.73205 q^{53} +6.00000 q^{55} +3.46410 q^{56} -3.00000 q^{58} +6.00000 q^{59} +5.19615 q^{61} +8.00000 q^{62} +1.00000 q^{64} +8.66025 q^{65} -6.92820 q^{67} +6.92820 q^{68} -6.00000 q^{70} -6.00000 q^{71} -11.0000 q^{73} +3.46410 q^{76} +12.0000 q^{77} -6.92820 q^{79} -1.73205 q^{80} -9.00000 q^{82} -12.0000 q^{85} -6.92820 q^{86} +3.46410 q^{88} +1.73205 q^{89} +17.3205 q^{91} -6.00000 q^{94} -6.00000 q^{95} +12.1244 q^{97} -5.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 10 q^{13} + 2 q^{16} - 4 q^{25} + 10 q^{26} + 6 q^{29} - 16 q^{31} - 2 q^{32} + 12 q^{35} + 18 q^{41} + 12 q^{47} + 10 q^{49} + 4 q^{50} - 10 q^{52} + 12 q^{55} - 6 q^{58}+ \cdots - 10 q^{98}+O(q^{100}) \) 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
\(5\) −1.73205 −0.774597 −0.387298 − 0.921954i \(-0.626592\pi\)
−0.387298 + 0.921954i \(0.626592\pi\)
\(6\) 0 0
\(7\) −3.46410 −1.30931 −0.654654 − 0.755929i \(-0.727186\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 1.73205 0.547723
\(11\) −3.46410 −1.04447 −0.522233 − 0.852803i \(-0.674901\pi\)
−0.522233 + 0.852803i \(0.674901\pi\)
\(12\) 0 0
\(13\) −5.00000 −1.38675 −0.693375 − 0.720577i \(-0.743877\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 3.46410 0.925820
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.92820 1.68034 0.840168 − 0.542326i \(-0.182456\pi\)
0.840168 + 0.542326i \(0.182456\pi\)
\(18\) 0 0
\(19\) 3.46410 0.794719 0.397360 − 0.917663i \(-0.369927\pi\)
0.397360 + 0.917663i \(0.369927\pi\)
\(20\) −1.73205 −0.387298
\(21\) 0 0
\(22\) 3.46410 0.738549
\(23\) 0 0
\(24\) 0 0
\(25\) −2.00000 −0.400000
\(26\) 5.00000 0.980581
\(27\) 0 0
\(28\) −3.46410 −0.654654
\(29\) 3.00000 0.557086 0.278543 − 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 − 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −6.92820 −1.18818
\(35\) 6.00000 1.01419
\(36\) 0 0
\(37\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) −3.46410 −0.561951
\(39\) 0 0
\(40\) 1.73205 0.273861
\(41\) 9.00000 1.40556 0.702782 − 0.711405i \(-0.251941\pi\)
0.702782 + 0.711405i \(0.251941\pi\)
\(42\) 0 0
\(43\) 6.92820 1.05654 0.528271 − 0.849076i \(-0.322841\pi\)
0.528271 + 0.849076i \(0.322841\pi\)
\(44\) −3.46410 −0.522233
\(45\) 0 0
\(46\) 0 0
\(47\) 6.00000 0.875190 0.437595 − 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) 5.00000 0.714286
\(50\) 2.00000 0.282843
\(51\) 0 0
\(52\) −5.00000 −0.693375
\(53\) 1.73205 0.237915 0.118958 − 0.992899i \(-0.462045\pi\)
0.118958 + 0.992899i \(0.462045\pi\)
\(54\) 0 0
\(55\) 6.00000 0.809040
\(56\) 3.46410 0.462910
\(57\) 0 0
\(58\) −3.00000 −0.393919
\(59\) 6.00000 0.781133 0.390567 − 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) 5.19615 0.665299 0.332650 − 0.943051i \(-0.392057\pi\)
0.332650 + 0.943051i \(0.392057\pi\)
\(62\) 8.00000 1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 8.66025 1.07417
\(66\) 0 0
\(67\) −6.92820 −0.846415 −0.423207 − 0.906033i \(-0.639096\pi\)
−0.423207 + 0.906033i \(0.639096\pi\)
\(68\) 6.92820 0.840168
\(69\) 0 0
\(70\) −6.00000 −0.717137
\(71\) −6.00000 −0.712069 −0.356034 − 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) −11.0000 −1.28745 −0.643726 − 0.765256i \(-0.722612\pi\)
−0.643726 + 0.765256i \(0.722612\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 3.46410 0.397360
\(77\) 12.0000 1.36753
\(78\) 0 0
\(79\) −6.92820 −0.779484 −0.389742 − 0.920924i \(-0.627436\pi\)
−0.389742 + 0.920924i \(0.627436\pi\)
\(80\) −1.73205 −0.193649
\(81\) 0 0
\(82\) −9.00000 −0.993884
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −12.0000 −1.30158
\(86\) −6.92820 −0.747087
\(87\) 0 0
\(88\) 3.46410 0.369274
\(89\) 1.73205 0.183597 0.0917985 − 0.995778i \(-0.470738\pi\)
0.0917985 + 0.995778i \(0.470738\pi\)
\(90\) 0 0
\(91\) 17.3205 1.81568
\(92\) 0 0
\(93\) 0 0
\(94\) −6.00000 −0.618853
\(95\) −6.00000 −0.615587
\(96\) 0 0
\(97\) 12.1244 1.23104 0.615521 − 0.788121i \(-0.288946\pi\)
0.615521 + 0.788121i \(0.288946\pi\)
\(98\) −5.00000 −0.505076
\(99\) 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.t.1.1 2
3.2 odd 2 1058.2.a.g.1.2 yes 2
12.11 even 2 8464.2.a.bh.1.2 2
23.22 odd 2 inner 9522.2.a.t.1.2 2
69.68 even 2 1058.2.a.g.1.1 ✓ 2
276.275 odd 2 8464.2.a.bh.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1058.2.a.g.1.1 ✓ 2 69.68 even 2
1058.2.a.g.1.2 yes 2 3.2 odd 2
8464.2.a.bh.1.1 2 276.275 odd 2
8464.2.a.bh.1.2 2 12.11 even 2
9522.2.a.t.1.1 2 1.1 even 1 trivial
9522.2.a.t.1.2 2 23.22 odd 2 inner