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Properties

Label 9675.2.a.cb.1.3
Level $9675$
Weight $2$
Character 9675.1
Self dual yes
Analytic conductor $77.255$
Analytic rank $0$
Dimension $5$
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] = [9675,2,Mod(1,9675)] 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("9675.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9675, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9675 = 3^{2} \cdot 5^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9675.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,0,0,0,0,0,0,-3,0,0,9,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(77.2552639556\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.24217.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 5x^{3} - x^{2} + 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1075)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.369680\) of defining polynomial
Character \(\chi\) \(=\) 9675.1

$q$-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)
 
\(f(q)\) \(=\) \(q+0.369680 q^{2} -1.86334 q^{4} -1.49366 q^{7} -1.42820 q^{8} -3.68692 q^{11} -3.61627 q^{13} -0.552175 q^{14} +3.19870 q^{16} -5.00064 q^{17} -6.14424 q^{19} -1.36298 q^{22} +1.06002 q^{23} -1.33686 q^{26} +2.78318 q^{28} -0.426517 q^{29} -10.0955 q^{31} +4.03889 q^{32} -1.84864 q^{34} -8.49430 q^{37} -2.27141 q^{38} +10.0225 q^{41} +1.00000 q^{43} +6.86997 q^{44} +0.391870 q^{46} -12.7077 q^{47} -4.76899 q^{49} +6.73832 q^{52} -1.96199 q^{53} +2.13324 q^{56} -0.157675 q^{58} +8.00909 q^{59} -14.9538 q^{61} -3.73211 q^{62} -4.90429 q^{64} -3.94582 q^{67} +9.31788 q^{68} -1.62426 q^{71} +8.77264 q^{73} -3.14018 q^{74} +11.4488 q^{76} +5.50699 q^{77} +5.75850 q^{79} +3.70513 q^{82} -1.38813 q^{83} +0.369680 q^{86} +5.26566 q^{88} -10.8191 q^{89} +5.40146 q^{91} -1.97518 q^{92} -4.69778 q^{94} -4.27085 q^{97} -1.76300 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 3 q^{8} + 9 q^{11} + q^{13} + 7 q^{14} - 2 q^{16} + 3 q^{17} - 11 q^{19} - 13 q^{22} + 5 q^{26} + 15 q^{28} + 22 q^{29} - 5 q^{31} + 4 q^{32} - 7 q^{34} - 7 q^{37} - 3 q^{38} + 21 q^{41} + 5 q^{43}+ \cdots + 25 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.369680 0.261404 0.130702 − 0.991422i \(-0.458277\pi\)
0.130702 + 0.991422i \(0.458277\pi\)
\(3\) 0 0
\(4\) −1.86334 −0.931668
\(5\) 0 0
\(6\) 0 0
\(7\) −1.49366 −0.564549 −0.282274 − 0.959334i \(-0.591089\pi\)
−0.282274 + 0.959334i \(0.591089\pi\)
\(8\) −1.42820 −0.504945
\(9\) 0 0
\(10\) 0 0
\(11\) −3.68692 −1.11165 −0.555824 − 0.831300i \(-0.687597\pi\)
−0.555824 + 0.831300i \(0.687597\pi\)
\(12\) 0 0
\(13\) −3.61627 −1.00297 −0.501486 − 0.865166i \(-0.667213\pi\)
−0.501486 + 0.865166i \(0.667213\pi\)
\(14\) −0.552175 −0.147575
\(15\) 0 0
\(16\) 3.19870 0.799674
\(17\) −5.00064 −1.21283 −0.606417 − 0.795147i \(-0.707394\pi\)
−0.606417 + 0.795147i \(0.707394\pi\)
\(18\) 0 0
\(19\) −6.14424 −1.40959 −0.704793 − 0.709413i \(-0.748960\pi\)
−0.704793 + 0.709413i \(0.748960\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −1.36298 −0.290589
\(23\) 1.06002 0.221030 0.110515 − 0.993874i \(-0.464750\pi\)
0.110515 + 0.993874i \(0.464750\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −1.33686 −0.262180
\(27\) 0 0
\(28\) 2.78318 0.525972
\(29\) −0.426517 −0.0792023 −0.0396011 − 0.999216i \(-0.512609\pi\)
−0.0396011 + 0.999216i \(0.512609\pi\)
\(30\) 0 0
\(31\) −10.0955 −1.81320 −0.906602 − 0.421987i \(-0.861333\pi\)
−0.906602 + 0.421987i \(0.861333\pi\)
\(32\) 4.03889 0.713982
\(33\) 0 0
\(34\) −1.84864 −0.317039
\(35\) 0 0
\(36\) 0 0
\(37\) −8.49430 −1.39645 −0.698227 − 0.715876i \(-0.746027\pi\)
−0.698227 + 0.715876i \(0.746027\pi\)
\(38\) −2.27141 −0.368471
\(39\) 0 0
\(40\) 0 0
\(41\) 10.0225 1.56525 0.782627 − 0.622491i \(-0.213879\pi\)
0.782627 + 0.622491i \(0.213879\pi\)
\(42\) 0 0
\(43\) 1.00000 0.152499
\(44\) 6.86997 1.03569
\(45\) 0 0
\(46\) 0.391870 0.0577781
\(47\) −12.7077 −1.85361 −0.926804 − 0.375546i \(-0.877455\pi\)
−0.926804 + 0.375546i \(0.877455\pi\)
\(48\) 0 0
\(49\) −4.76899 −0.681285
\(50\) 0 0
\(51\) 0 0
\(52\) 6.73832 0.934437
\(53\) −1.96199 −0.269500 −0.134750 − 0.990880i \(-0.543023\pi\)
−0.134750 + 0.990880i \(0.543023\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 2.13324 0.285066
\(57\) 0 0
\(58\) −0.157675 −0.0207038
\(59\) 8.00909 1.04269 0.521347 − 0.853345i \(-0.325430\pi\)
0.521347 + 0.853345i \(0.325430\pi\)
\(60\) 0 0
\(61\) −14.9538 −1.91463 −0.957317 − 0.289039i \(-0.906664\pi\)
−0.957317 + 0.289039i \(0.906664\pi\)
\(62\) −3.73211 −0.473978
\(63\) 0 0
\(64\) −4.90429 −0.613036
\(65\) 0 0
\(66\) 0 0
\(67\) −3.94582 −0.482058 −0.241029 − 0.970518i \(-0.577485\pi\)
−0.241029 + 0.970518i \(0.577485\pi\)
\(68\) 9.31788 1.12996
\(69\) 0 0
\(70\) 0 0
\(71\) −1.62426 −0.192765 −0.0963824 − 0.995344i \(-0.530727\pi\)
−0.0963824 + 0.995344i \(0.530727\pi\)
\(72\) 0 0
\(73\) 8.77264 1.02676 0.513380 − 0.858161i \(-0.328393\pi\)
0.513380 + 0.858161i \(0.328393\pi\)
\(74\) −3.14018 −0.365038
\(75\) 0 0
\(76\) 11.4488 1.31327
\(77\) 5.50699 0.627580
\(78\) 0 0
\(79\) 5.75850 0.647882 0.323941 − 0.946077i \(-0.394992\pi\)
0.323941 + 0.946077i \(0.394992\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 3.70513 0.409163
\(83\) −1.38813 −0.152367 −0.0761834 − 0.997094i \(-0.524273\pi\)
−0.0761834 + 0.997094i \(0.524273\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0.369680 0.0398637
\(87\) 0 0
\(88\) 5.26566 0.561321
\(89\) −10.8191 −1.14682 −0.573412 − 0.819267i \(-0.694381\pi\)
−0.573412 + 0.819267i \(0.694381\pi\)
\(90\) 0 0
\(91\) 5.40146 0.566227
\(92\) −1.97518 −0.205927
\(93\) 0 0
\(94\) −4.69778 −0.484539
\(95\) 0 0
\(96\) 0 0
\(97\) −4.27085 −0.433639 −0.216820 − 0.976212i \(-0.569568\pi\)
−0.216820 + 0.976212i \(0.569568\pi\)
\(98\) −1.76300 −0.178090
\(99\) 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 9675.2.a.cb.1.3 5
3.2 odd 2 1075.2.a.o.1.3 yes 5
5.4 even 2 9675.2.a.cc.1.3 5
15.2 even 4 1075.2.b.i.474.5 10
15.8 even 4 1075.2.b.i.474.6 10
15.14 odd 2 1075.2.a.n.1.3 ✓ 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1075.2.a.n.1.3 ✓ 5 15.14 odd 2
1075.2.a.o.1.3 yes 5 3.2 odd 2
1075.2.b.i.474.5 10 15.2 even 4
1075.2.b.i.474.6 10 15.8 even 4
9675.2.a.cb.1.3 5 1.1 even 1 trivial
9675.2.a.cc.1.3 5 5.4 even 2