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Properties

Label 1156.4.a.j.1.7
Level $1156$
Weight $4$
Character 1156.1
Self dual yes
Analytic conductor $68.206$
Analytic rank $1$
Dimension $12$
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] = [1156,4,Mod(1,1156)] 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("1156.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1156, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1156 = 2^{2} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1156.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,-30] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(68.2062079666\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 216 x^{10} - 74 x^{9} + 17391 x^{8} + 9408 x^{7} - 659646 x^{6} - 424698 x^{5} + \cdots + 168035561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3\cdot 17^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(-2.20468\) of defining polynomial
Character \(\chi\) \(=\) 1156.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.20468 q^{3} +8.02973 q^{5} -0.312143 q^{7} -22.1394 q^{9} +3.79618 q^{11} +6.12306 q^{13} +17.7029 q^{15} -91.3719 q^{19} -0.688174 q^{21} +91.3037 q^{23} -60.5235 q^{25} -108.336 q^{27} +4.36196 q^{29} -64.4116 q^{31} +8.36935 q^{33} -2.50642 q^{35} -154.878 q^{37} +13.4994 q^{39} +263.550 q^{41} -263.352 q^{43} -177.773 q^{45} +6.44784 q^{47} -342.903 q^{49} -224.572 q^{53} +30.4823 q^{55} -201.445 q^{57} +350.775 q^{59} -204.738 q^{61} +6.91066 q^{63} +49.1665 q^{65} -404.747 q^{67} +201.295 q^{69} -835.282 q^{71} -339.506 q^{73} -133.435 q^{75} -1.18495 q^{77} -725.932 q^{79} +358.917 q^{81} -206.632 q^{83} +9.61670 q^{87} +1274.61 q^{89} -1.91127 q^{91} -142.007 q^{93} -733.691 q^{95} +1199.16 q^{97} -84.0452 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 30 q^{5} + 18 q^{7} + 108 q^{9} - 66 q^{11} - 72 q^{13} - 138 q^{15} + 138 q^{19} - 42 q^{21} - 132 q^{23} + 444 q^{25} - 222 q^{27} - 564 q^{29} - 54 q^{31} - 390 q^{33} + 678 q^{35} - 474 q^{37}+ \cdots - 6978 q^{99}+O(q^{100}) \) 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 0
\(3\) 2.20468 0.424290 0.212145 − 0.977238i \(-0.431955\pi\)
0.212145 + 0.977238i \(0.431955\pi\)
\(4\) 0 0
\(5\) 8.02973 0.718201 0.359100 − 0.933299i \(-0.383084\pi\)
0.359100 + 0.933299i \(0.383084\pi\)
\(6\) 0 0
\(7\) −0.312143 −0.0168541 −0.00842707 − 0.999964i \(-0.502682\pi\)
−0.00842707 + 0.999964i \(0.502682\pi\)
\(8\) 0 0
\(9\) −22.1394 −0.819978
\(10\) 0 0
\(11\) 3.79618 0.104054 0.0520269 − 0.998646i \(-0.483432\pi\)
0.0520269 + 0.998646i \(0.483432\pi\)
\(12\) 0 0
\(13\) 6.12306 0.130633 0.0653166 − 0.997865i \(-0.479194\pi\)
0.0653166 + 0.997865i \(0.479194\pi\)
\(14\) 0 0
\(15\) 17.7029 0.304725
\(16\) 0 0
\(17\) 0 0
\(18\) 0 0
\(19\) −91.3719 −1.10327 −0.551635 − 0.834085i \(-0.685996\pi\)
−0.551635 + 0.834085i \(0.685996\pi\)
\(20\) 0 0
\(21\) −0.688174 −0.00715104
\(22\) 0 0
\(23\) 91.3037 0.827745 0.413872 − 0.910335i \(-0.364176\pi\)
0.413872 + 0.910335i \(0.364176\pi\)
\(24\) 0 0
\(25\) −60.5235 −0.484188
\(26\) 0 0
\(27\) −108.336 −0.772199
\(28\) 0 0
\(29\) 4.36196 0.0279309 0.0139654 − 0.999902i \(-0.495555\pi\)
0.0139654 + 0.999902i \(0.495555\pi\)
\(30\) 0 0
\(31\) −64.4116 −0.373183 −0.186591 − 0.982438i \(-0.559744\pi\)
−0.186591 + 0.982438i \(0.559744\pi\)
\(32\) 0 0
\(33\) 8.36935 0.0441490
\(34\) 0 0
\(35\) −2.50642 −0.0121046
\(36\) 0 0
\(37\) −154.878 −0.688155 −0.344077 − 0.938941i \(-0.611808\pi\)
−0.344077 + 0.938941i \(0.611808\pi\)
\(38\) 0 0
\(39\) 13.4994 0.0554264
\(40\) 0 0
\(41\) 263.550 1.00389 0.501947 − 0.864899i \(-0.332617\pi\)
0.501947 + 0.864899i \(0.332617\pi\)
\(42\) 0 0
\(43\) −263.352 −0.933972 −0.466986 − 0.884265i \(-0.654660\pi\)
−0.466986 + 0.884265i \(0.654660\pi\)
\(44\) 0 0
\(45\) −177.773 −0.588909
\(46\) 0 0
\(47\) 6.44784 0.0200109 0.0100055 − 0.999950i \(-0.496815\pi\)
0.0100055 + 0.999950i \(0.496815\pi\)
\(48\) 0 0
\(49\) −342.903 −0.999716
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −224.572 −0.582026 −0.291013 − 0.956719i \(-0.593992\pi\)
−0.291013 + 0.956719i \(0.593992\pi\)
\(54\) 0 0
\(55\) 30.4823 0.0747315
\(56\) 0 0
\(57\) −201.445 −0.468107
\(58\) 0 0
\(59\) 350.775 0.774017 0.387009 − 0.922076i \(-0.373508\pi\)
0.387009 + 0.922076i \(0.373508\pi\)
\(60\) 0 0
\(61\) −204.738 −0.429738 −0.214869 − 0.976643i \(-0.568932\pi\)
−0.214869 + 0.976643i \(0.568932\pi\)
\(62\) 0 0
\(63\) 6.91066 0.0138200
\(64\) 0 0
\(65\) 49.1665 0.0938209
\(66\) 0 0
\(67\) −404.747 −0.738026 −0.369013 − 0.929424i \(-0.620304\pi\)
−0.369013 + 0.929424i \(0.620304\pi\)
\(68\) 0 0
\(69\) 201.295 0.351204
\(70\) 0 0
\(71\) −835.282 −1.39619 −0.698097 − 0.716004i \(-0.745969\pi\)
−0.698097 + 0.716004i \(0.745969\pi\)
\(72\) 0 0
\(73\) −339.506 −0.544331 −0.272166 − 0.962250i \(-0.587740\pi\)
−0.272166 + 0.962250i \(0.587740\pi\)
\(74\) 0 0
\(75\) −133.435 −0.205436
\(76\) 0 0
\(77\) −1.18495 −0.00175374
\(78\) 0 0
\(79\) −725.932 −1.03385 −0.516923 − 0.856032i \(-0.672923\pi\)
−0.516923 + 0.856032i \(0.672923\pi\)
\(80\) 0 0
\(81\) 358.917 0.492342
\(82\) 0 0
\(83\) −206.632 −0.273262 −0.136631 − 0.990622i \(-0.543627\pi\)
−0.136631 + 0.990622i \(0.543627\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 9.61670 0.0118508
\(88\) 0 0
\(89\) 1274.61 1.51807 0.759037 − 0.651048i \(-0.225670\pi\)
0.759037 + 0.651048i \(0.225670\pi\)
\(90\) 0 0
\(91\) −1.91127 −0.00220171
\(92\) 0 0
\(93\) −142.007 −0.158338
\(94\) 0 0
\(95\) −733.691 −0.792370
\(96\) 0 0
\(97\) 1199.16 1.25522 0.627610 − 0.778528i \(-0.284033\pi\)
0.627610 + 0.778528i \(0.284033\pi\)
\(98\) 0 0
\(99\) −84.0452 −0.0853219
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 1156.4.a.j.1.7 ✓ 12
17.4 even 4 1156.4.b.h.577.11 24
17.13 even 4 1156.4.b.h.577.14 24
17.16 even 2 1156.4.a.k.1.6 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1156.4.a.j.1.7 ✓ 12 1.1 even 1 trivial
1156.4.a.k.1.6 yes 12 17.16 even 2
1156.4.b.h.577.11 24 17.4 even 4
1156.4.b.h.577.14 24 17.13 even 4