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Properties

Label 1156.4.a.j.1.6
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 1.6
Root \(1.14445\) 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)
 
\(f(q)\) \(=\) \(q-1.14445 q^{3} +4.44584 q^{5} +23.8731 q^{7} -25.6902 q^{9} +35.8559 q^{11} -4.00455 q^{13} -5.08803 q^{15} -144.209 q^{19} -27.3215 q^{21} -146.305 q^{23} -105.235 q^{25} +60.3013 q^{27} +44.3405 q^{29} -90.5083 q^{31} -41.0353 q^{33} +106.136 q^{35} -142.148 q^{37} +4.58301 q^{39} +167.813 q^{41} +109.737 q^{43} -114.215 q^{45} -237.183 q^{47} +226.923 q^{49} +612.843 q^{53} +159.410 q^{55} +165.040 q^{57} -420.210 q^{59} -346.867 q^{61} -613.305 q^{63} -17.8036 q^{65} +279.957 q^{67} +167.439 q^{69} +501.325 q^{71} -1086.85 q^{73} +120.436 q^{75} +855.991 q^{77} -598.857 q^{79} +624.625 q^{81} -842.938 q^{83} -50.7454 q^{87} -1343.43 q^{89} -95.6010 q^{91} +103.582 q^{93} -641.131 q^{95} -562.857 q^{97} -921.148 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 30 q^{5} + 18 q^{7} + 108 q^{9} - 66 q^{11} - 72 q^{13} - 138 q^{15} + 138 q^{19} - 42 q^{21} - 132 q^{23} + 444 q^{25} - 222 q^{27} - 564 q^{29} - 54 q^{31} - 390 q^{33} + 678 q^{35} - 474 q^{37}+ \cdots - 6978 q^{99}+O(q^{100}) \) 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
\(3\) −1.14445 −0.220249 −0.110125 − 0.993918i \(-0.535125\pi\)
−0.110125 + 0.993918i \(0.535125\pi\)
\(4\) 0 0
\(5\) 4.44584 0.397648 0.198824 − 0.980035i \(-0.436288\pi\)
0.198824 + 0.980035i \(0.436288\pi\)
\(6\) 0 0
\(7\) 23.8731 1.28902 0.644512 − 0.764594i \(-0.277061\pi\)
0.644512 + 0.764594i \(0.277061\pi\)
\(8\) 0 0
\(9\) −25.6902 −0.951490
\(10\) 0 0
\(11\) 35.8559 0.982816 0.491408 − 0.870930i \(-0.336482\pi\)
0.491408 + 0.870930i \(0.336482\pi\)
\(12\) 0 0
\(13\) −4.00455 −0.0854357 −0.0427178 − 0.999087i \(-0.513602\pi\)
−0.0427178 + 0.999087i \(0.513602\pi\)
\(14\) 0 0
\(15\) −5.08803 −0.0875816
\(16\) 0 0
\(17\) 0 0
\(18\) 0 0
\(19\) −144.209 −1.74126 −0.870629 − 0.491941i \(-0.836288\pi\)
−0.870629 + 0.491941i \(0.836288\pi\)
\(20\) 0 0
\(21\) −27.3215 −0.283907
\(22\) 0 0
\(23\) −146.305 −1.32638 −0.663189 − 0.748452i \(-0.730798\pi\)
−0.663189 + 0.748452i \(0.730798\pi\)
\(24\) 0 0
\(25\) −105.235 −0.841876
\(26\) 0 0
\(27\) 60.3013 0.429814
\(28\) 0 0
\(29\) 44.3405 0.283925 0.141962 − 0.989872i \(-0.454659\pi\)
0.141962 + 0.989872i \(0.454659\pi\)
\(30\) 0 0
\(31\) −90.5083 −0.524380 −0.262190 − 0.965016i \(-0.584445\pi\)
−0.262190 + 0.965016i \(0.584445\pi\)
\(32\) 0 0
\(33\) −41.0353 −0.216464
\(34\) 0 0
\(35\) 106.136 0.512577
\(36\) 0 0
\(37\) −142.148 −0.631596 −0.315798 − 0.948826i \(-0.602272\pi\)
−0.315798 + 0.948826i \(0.602272\pi\)
\(38\) 0 0
\(39\) 4.58301 0.0188171
\(40\) 0 0
\(41\) 167.813 0.639218 0.319609 − 0.947550i \(-0.396448\pi\)
0.319609 + 0.947550i \(0.396448\pi\)
\(42\) 0 0
\(43\) 109.737 0.389179 0.194590 − 0.980885i \(-0.437662\pi\)
0.194590 + 0.980885i \(0.437662\pi\)
\(44\) 0 0
\(45\) −114.215 −0.378358
\(46\) 0 0
\(47\) −237.183 −0.736100 −0.368050 − 0.929806i \(-0.619975\pi\)
−0.368050 + 0.929806i \(0.619975\pi\)
\(48\) 0 0
\(49\) 226.923 0.661583
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 612.843 1.58831 0.794156 − 0.607715i \(-0.207914\pi\)
0.794156 + 0.607715i \(0.207914\pi\)
\(54\) 0 0
\(55\) 159.410 0.390814
\(56\) 0 0
\(57\) 165.040 0.383511
\(58\) 0 0
\(59\) −420.210 −0.927233 −0.463616 − 0.886036i \(-0.653448\pi\)
−0.463616 + 0.886036i \(0.653448\pi\)
\(60\) 0 0
\(61\) −346.867 −0.728062 −0.364031 − 0.931387i \(-0.618600\pi\)
−0.364031 + 0.931387i \(0.618600\pi\)
\(62\) 0 0
\(63\) −613.305 −1.22649
\(64\) 0 0
\(65\) −17.8036 −0.0339733
\(66\) 0 0
\(67\) 279.957 0.510480 0.255240 − 0.966878i \(-0.417846\pi\)
0.255240 + 0.966878i \(0.417846\pi\)
\(68\) 0 0
\(69\) 167.439 0.292134
\(70\) 0 0
\(71\) 501.325 0.837976 0.418988 − 0.907992i \(-0.362385\pi\)
0.418988 + 0.907992i \(0.362385\pi\)
\(72\) 0 0
\(73\) −1086.85 −1.74255 −0.871273 − 0.490798i \(-0.836705\pi\)
−0.871273 + 0.490798i \(0.836705\pi\)
\(74\) 0 0
\(75\) 120.436 0.185423
\(76\) 0 0
\(77\) 855.991 1.26687
\(78\) 0 0
\(79\) −598.857 −0.852870 −0.426435 − 0.904518i \(-0.640231\pi\)
−0.426435 + 0.904518i \(0.640231\pi\)
\(80\) 0 0
\(81\) 624.625 0.856824
\(82\) 0 0
\(83\) −842.938 −1.11475 −0.557376 − 0.830260i \(-0.688192\pi\)
−0.557376 + 0.830260i \(0.688192\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −50.7454 −0.0625342
\(88\) 0 0
\(89\) −1343.43 −1.60003 −0.800017 − 0.599978i \(-0.795176\pi\)
−0.800017 + 0.599978i \(0.795176\pi\)
\(90\) 0 0
\(91\) −95.6010 −0.110129
\(92\) 0 0
\(93\) 103.582 0.115494
\(94\) 0 0
\(95\) −641.131 −0.692407
\(96\) 0 0
\(97\) −562.857 −0.589170 −0.294585 − 0.955625i \(-0.595181\pi\)
−0.294585 + 0.955625i \(0.595181\pi\)
\(98\) 0 0
\(99\) −921.148 −0.935140
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.1.6 ✓ 12
17.4 even 4 1156.4.b.h.577.13 24
17.13 even 4 1156.4.b.h.577.12 24
17.16 even 2 1156.4.a.k.1.7 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1156.4.a.j.1.6 ✓ 12 1.1 even 1 trivial
1156.4.a.k.1.7 yes 12 17.16 even 2
1156.4.b.h.577.12 24 17.13 even 4
1156.4.b.h.577.13 24 17.4 even 4