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Properties

Label 1156.4.a.j.1.8
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 1.8
Root \(-4.10622\) 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)
 
\(f(q)\) \(=\) \(q+4.10622 q^{3} -11.3149 q^{5} +32.1558 q^{7} -10.1390 q^{9} +45.0701 q^{11} -84.8358 q^{13} -46.4614 q^{15} -126.562 q^{19} +132.039 q^{21} +177.112 q^{23} +3.02668 q^{25} -152.501 q^{27} -190.458 q^{29} -16.2456 q^{31} +185.067 q^{33} -363.839 q^{35} +33.0143 q^{37} -348.354 q^{39} -276.654 q^{41} +102.518 q^{43} +114.721 q^{45} -102.641 q^{47} +690.997 q^{49} -382.196 q^{53} -509.963 q^{55} -519.693 q^{57} +339.613 q^{59} -54.9196 q^{61} -326.027 q^{63} +959.907 q^{65} -306.975 q^{67} +727.260 q^{69} +113.927 q^{71} +830.644 q^{73} +12.4282 q^{75} +1449.26 q^{77} -211.048 q^{79} -352.449 q^{81} -1035.81 q^{83} -782.064 q^{87} +251.975 q^{89} -2727.96 q^{91} -66.7080 q^{93} +1432.04 q^{95} -389.861 q^{97} -456.964 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 30 q^{5} + 18 q^{7} + 108 q^{9} - 66 q^{11} - 72 q^{13} - 138 q^{15} + 138 q^{19} - 42 q^{21} - 132 q^{23} + 444 q^{25} - 222 q^{27} - 564 q^{29} - 54 q^{31} - 390 q^{33} + 678 q^{35} - 474 q^{37}+ \cdots - 6978 q^{99}+O(q^{100}) \) 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
\(3\) 4.10622 0.790242 0.395121 − 0.918629i \(-0.370703\pi\)
0.395121 + 0.918629i \(0.370703\pi\)
\(4\) 0 0
\(5\) −11.3149 −1.01203 −0.506017 − 0.862523i \(-0.668883\pi\)
−0.506017 + 0.862523i \(0.668883\pi\)
\(6\) 0 0
\(7\) 32.1558 1.73625 0.868125 − 0.496345i \(-0.165325\pi\)
0.868125 + 0.496345i \(0.165325\pi\)
\(8\) 0 0
\(9\) −10.1390 −0.375518
\(10\) 0 0
\(11\) 45.0701 1.23538 0.617688 − 0.786423i \(-0.288070\pi\)
0.617688 + 0.786423i \(0.288070\pi\)
\(12\) 0 0
\(13\) −84.8358 −1.80994 −0.904970 − 0.425476i \(-0.860107\pi\)
−0.904970 + 0.425476i \(0.860107\pi\)
\(14\) 0 0
\(15\) −46.4614 −0.799752
\(16\) 0 0
\(17\) 0 0
\(18\) 0 0
\(19\) −126.562 −1.52818 −0.764089 − 0.645110i \(-0.776811\pi\)
−0.764089 + 0.645110i \(0.776811\pi\)
\(20\) 0 0
\(21\) 132.039 1.37206
\(22\) 0 0
\(23\) 177.112 1.60567 0.802835 − 0.596202i \(-0.203324\pi\)
0.802835 + 0.596202i \(0.203324\pi\)
\(24\) 0 0
\(25\) 3.02668 0.0242135
\(26\) 0 0
\(27\) −152.501 −1.08699
\(28\) 0 0
\(29\) −190.458 −1.21956 −0.609780 − 0.792571i \(-0.708742\pi\)
−0.609780 + 0.792571i \(0.708742\pi\)
\(30\) 0 0
\(31\) −16.2456 −0.0941225 −0.0470613 − 0.998892i \(-0.514986\pi\)
−0.0470613 + 0.998892i \(0.514986\pi\)
\(32\) 0 0
\(33\) 185.067 0.976246
\(34\) 0 0
\(35\) −363.839 −1.75715
\(36\) 0 0
\(37\) 33.0143 0.146689 0.0733447 − 0.997307i \(-0.476633\pi\)
0.0733447 + 0.997307i \(0.476633\pi\)
\(38\) 0 0
\(39\) −348.354 −1.43029
\(40\) 0 0
\(41\) −276.654 −1.05381 −0.526905 − 0.849924i \(-0.676647\pi\)
−0.526905 + 0.849924i \(0.676647\pi\)
\(42\) 0 0
\(43\) 102.518 0.363576 0.181788 − 0.983338i \(-0.441812\pi\)
0.181788 + 0.983338i \(0.441812\pi\)
\(44\) 0 0
\(45\) 114.721 0.380037
\(46\) 0 0
\(47\) −102.641 −0.318547 −0.159274 − 0.987234i \(-0.550915\pi\)
−0.159274 + 0.987234i \(0.550915\pi\)
\(48\) 0 0
\(49\) 690.997 2.01457
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −382.196 −0.990541 −0.495270 − 0.868739i \(-0.664931\pi\)
−0.495270 + 0.868739i \(0.664931\pi\)
\(54\) 0 0
\(55\) −509.963 −1.25024
\(56\) 0 0
\(57\) −519.693 −1.20763
\(58\) 0 0
\(59\) 339.613 0.749387 0.374694 − 0.927149i \(-0.377748\pi\)
0.374694 + 0.927149i \(0.377748\pi\)
\(60\) 0 0
\(61\) −54.9196 −0.115274 −0.0576372 − 0.998338i \(-0.518357\pi\)
−0.0576372 + 0.998338i \(0.518357\pi\)
\(62\) 0 0
\(63\) −326.027 −0.651993
\(64\) 0 0
\(65\) 959.907 1.83172
\(66\) 0 0
\(67\) −306.975 −0.559746 −0.279873 − 0.960037i \(-0.590292\pi\)
−0.279873 + 0.960037i \(0.590292\pi\)
\(68\) 0 0
\(69\) 727.260 1.26887
\(70\) 0 0
\(71\) 113.927 0.190432 0.0952162 − 0.995457i \(-0.469646\pi\)
0.0952162 + 0.995457i \(0.469646\pi\)
\(72\) 0 0
\(73\) 830.644 1.33177 0.665887 − 0.746052i \(-0.268053\pi\)
0.665887 + 0.746052i \(0.268053\pi\)
\(74\) 0 0
\(75\) 12.4282 0.0191345
\(76\) 0 0
\(77\) 1449.26 2.14492
\(78\) 0 0
\(79\) −211.048 −0.300566 −0.150283 − 0.988643i \(-0.548018\pi\)
−0.150283 + 0.988643i \(0.548018\pi\)
\(80\) 0 0
\(81\) −352.449 −0.483469
\(82\) 0 0
\(83\) −1035.81 −1.36981 −0.684906 − 0.728632i \(-0.740157\pi\)
−0.684906 + 0.728632i \(0.740157\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −782.064 −0.963748
\(88\) 0 0
\(89\) 251.975 0.300104 0.150052 − 0.988678i \(-0.452056\pi\)
0.150052 + 0.988678i \(0.452056\pi\)
\(90\) 0 0
\(91\) −2727.96 −3.14251
\(92\) 0 0
\(93\) −66.7080 −0.0743796
\(94\) 0 0
\(95\) 1432.04 1.54657
\(96\) 0 0
\(97\) −389.861 −0.408087 −0.204043 − 0.978962i \(-0.565408\pi\)
−0.204043 + 0.978962i \(0.565408\pi\)
\(98\) 0 0
\(99\) −456.964 −0.463905
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.1.8 ✓ 12
17.4 even 4 1156.4.b.h.577.10 24
17.13 even 4 1156.4.b.h.577.15 24
17.16 even 2 1156.4.a.k.1.5 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1156.4.a.j.1.8 ✓ 12 1.1 even 1 trivial
1156.4.a.k.1.5 yes 12 17.16 even 2
1156.4.b.h.577.10 24 17.4 even 4
1156.4.b.h.577.15 24 17.13 even 4