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Properties

Label 1232.2.bn.d.593.8
Level $1232$
Weight $2$
Character 1232.593
Analytic conductor $9.838$
Analytic rank $0$
Dimension $48$
Inner twists $4$

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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] = [1232,2,Mod(241,1232)] 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("1232.241"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1232, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 1, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1232.bn (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48,0,0,0,0] 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: no
Analytic conductor: \(9.83756952902\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 616)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 593.8
Character \(\chi\) \(=\) 1232.593
Dual form 1232.2.bn.d.241.8

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.19884 + 0.692153i) q^{3} +(-2.04863 - 1.18278i) q^{5} +(-2.46903 - 0.950736i) q^{7} +(-0.541847 + 0.938507i) q^{9} +(-3.14036 + 1.06683i) q^{11} +3.00583 q^{13} +3.27465 q^{15} +(0.342735 + 0.593634i) q^{17} +(-2.06807 + 3.58200i) q^{19} +(3.61804 - 0.569161i) q^{21} +(-3.48628 + 6.03841i) q^{23} +(0.297914 + 0.516002i) q^{25} -5.65309i q^{27} -5.94917i q^{29} +(6.48060 - 3.74158i) q^{31} +(3.02639 - 3.45258i) q^{33} +(3.93361 + 4.86801i) q^{35} +(5.83377 - 10.1044i) q^{37} +(-3.60352 + 2.08049i) q^{39} +2.13747 q^{41} -11.1370i q^{43} +(2.22009 - 1.28177i) q^{45} +(11.2046 + 6.46898i) q^{47} +(5.19220 + 4.69479i) q^{49} +(-0.821772 - 0.474450i) q^{51} +(4.98299 + 8.63078i) q^{53} +(7.69525 + 1.52880i) q^{55} -5.72568i q^{57} +(3.50849 - 2.02563i) q^{59} +(-6.76582 + 11.7187i) q^{61} +(2.23011 - 1.80205i) q^{63} +(-6.15782 - 3.55522i) q^{65} +(-2.22520 - 3.85416i) q^{67} -9.65215i q^{69} +9.67463 q^{71} +(2.61522 + 4.52970i) q^{73} +(-0.714305 - 0.412404i) q^{75} +(8.76792 + 0.351614i) q^{77} +(-9.32635 - 5.38457i) q^{79} +(2.28726 + 3.96165i) q^{81} -11.9979 q^{83} -1.62151i q^{85} +(4.11774 + 7.13214i) q^{87} +(0.643974 + 0.371799i) q^{89} +(-7.42147 - 2.85775i) q^{91} +(-5.17949 + 8.97114i) q^{93} +(8.47339 - 4.89211i) q^{95} +4.34845i q^{97} +(0.700366 - 3.52531i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q + 24 q^{9} + 2 q^{11} + 8 q^{15} - 12 q^{23} + 20 q^{25} - 12 q^{31} - 30 q^{33} + 4 q^{37} + 48 q^{45} - 48 q^{49} + 20 q^{53} - 8 q^{67} + 16 q^{71} + 24 q^{75} + 34 q^{77} - 24 q^{81} - 24 q^{89}+ \cdots + 8 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1232\mathbb{Z}\right)^\times\).

\(n\) \(309\) \(353\) \(463\) \(673\)
\(\chi(n)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(1\) \(-1\)

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
\(3\) −1.19884 + 0.692153i −0.692153 + 0.399615i −0.804418 − 0.594063i \(-0.797523\pi\)
0.112265 + 0.993678i \(0.464189\pi\)
\(4\) 0 0
\(5\) −2.04863 − 1.18278i −0.916174 − 0.528953i −0.0337613 − 0.999430i \(-0.510749\pi\)
−0.882412 + 0.470477i \(0.844082\pi\)
\(6\) 0 0
\(7\) −2.46903 − 0.950736i −0.933205 − 0.359344i
\(8\) 0 0
\(9\) −0.541847 + 0.938507i −0.180616 + 0.312836i
\(10\) 0 0
\(11\) −3.14036 + 1.06683i −0.946854 + 0.321662i
\(12\) 0 0
\(13\) 3.00583 0.833666 0.416833 − 0.908983i \(-0.363140\pi\)
0.416833 + 0.908983i \(0.363140\pi\)
\(14\) 0 0
\(15\) 3.27465 0.845510
\(16\) 0 0
\(17\) 0.342735 + 0.593634i 0.0831254 + 0.143977i 0.904591 − 0.426281i \(-0.140176\pi\)
−0.821465 + 0.570258i \(0.806843\pi\)
\(18\) 0 0
\(19\) −2.06807 + 3.58200i −0.474447 + 0.821766i −0.999572 − 0.0292589i \(-0.990685\pi\)
0.525125 + 0.851025i \(0.324019\pi\)
\(20\) 0 0
\(21\) 3.61804 − 0.569161i 0.789520 − 0.124201i
\(22\) 0 0
\(23\) −3.48628 + 6.03841i −0.726939 + 1.25910i 0.231232 + 0.972899i \(0.425724\pi\)
−0.958171 + 0.286197i \(0.907609\pi\)
\(24\) 0 0
\(25\) 0.297914 + 0.516002i 0.0595828 + 0.103200i
\(26\) 0 0
\(27\) − 5.65309i − 1.08794i
\(28\) 0 0
\(29\) − 5.94917i − 1.10473i −0.833601 − 0.552367i \(-0.813725\pi\)
0.833601 − 0.552367i \(-0.186275\pi\)
\(30\) 0 0
\(31\) 6.48060 − 3.74158i 1.16395 − 0.672007i 0.211703 − 0.977334i \(-0.432099\pi\)
0.952247 + 0.305327i \(0.0987659\pi\)
\(32\) 0 0
\(33\) 3.02639 − 3.45258i 0.526828 − 0.601017i
\(34\) 0 0
\(35\) 3.93361 + 4.86801i 0.664902 + 0.822844i
\(36\) 0 0
\(37\) 5.83377 − 10.1044i 0.959067 − 1.66115i 0.234292 − 0.972166i \(-0.424723\pi\)
0.724775 − 0.688986i \(-0.241944\pi\)
\(38\) 0 0
\(39\) −3.60352 + 2.08049i −0.577025 + 0.333145i
\(40\) 0 0
\(41\) 2.13747 0.333816 0.166908 − 0.985972i \(-0.446622\pi\)
0.166908 + 0.985972i \(0.446622\pi\)
\(42\) 0 0
\(43\) − 11.1370i − 1.69837i −0.528092 − 0.849187i \(-0.677092\pi\)
0.528092 − 0.849187i \(-0.322908\pi\)
\(44\) 0 0
\(45\) 2.22009 − 1.28177i 0.330951 − 0.191075i
\(46\) 0 0
\(47\) 11.2046 + 6.46898i 1.63436 + 0.943598i 0.982726 + 0.185065i \(0.0592494\pi\)
0.651634 + 0.758534i \(0.274084\pi\)
\(48\) 0 0
\(49\) 5.19220 + 4.69479i 0.741743 + 0.670684i
\(50\) 0 0
\(51\) −0.821772 − 0.474450i −0.115071 − 0.0664363i
\(52\) 0 0
\(53\) 4.98299 + 8.63078i 0.684466 + 1.18553i 0.973604 + 0.228242i \(0.0732978\pi\)
−0.289139 + 0.957287i \(0.593369\pi\)
\(54\) 0 0
\(55\) 7.69525 + 1.52880i 1.03763 + 0.206143i
\(56\) 0 0
\(57\) − 5.72568i − 0.758384i
\(58\) 0 0
\(59\) 3.50849 − 2.02563i 0.456767 − 0.263714i −0.253917 − 0.967226i \(-0.581719\pi\)
0.710684 + 0.703512i \(0.248386\pi\)
\(60\) 0 0
\(61\) −6.76582 + 11.7187i −0.866274 + 1.50043i −0.000497294 1.00000i \(0.500158\pi\)
−0.865777 + 0.500431i \(0.833175\pi\)
\(62\) 0 0
\(63\) 2.23011 − 1.80205i 0.280967 − 0.227037i
\(64\) 0 0
\(65\) −6.15782 − 3.55522i −0.763783 − 0.440970i
\(66\) 0 0
\(67\) −2.22520 − 3.85416i −0.271851 − 0.470860i 0.697485 − 0.716600i \(-0.254303\pi\)
−0.969336 + 0.245740i \(0.920969\pi\)
\(68\) 0 0
\(69\) − 9.65215i − 1.16198i
\(70\) 0 0
\(71\) 9.67463 1.14817 0.574084 − 0.818797i \(-0.305358\pi\)
0.574084 + 0.818797i \(0.305358\pi\)
\(72\) 0 0
\(73\) 2.61522 + 4.52970i 0.306089 + 0.530161i 0.977503 − 0.210921i \(-0.0676463\pi\)
−0.671414 + 0.741082i \(0.734313\pi\)
\(74\) 0 0
\(75\) −0.714305 − 0.412404i −0.0824809 − 0.0476203i
\(76\) 0 0
\(77\) 8.76792 + 0.351614i 0.999197 + 0.0400701i
\(78\) 0 0
\(79\) −9.32635 − 5.38457i −1.04930 − 0.605812i −0.126844 − 0.991923i \(-0.540485\pi\)
−0.922452 + 0.386111i \(0.873818\pi\)
\(80\) 0 0
\(81\) 2.28726 + 3.96165i 0.254140 + 0.440183i
\(82\) 0 0
\(83\) −11.9979 −1.31694 −0.658472 − 0.752605i \(-0.728797\pi\)
−0.658472 + 0.752605i \(0.728797\pi\)
\(84\) 0 0
\(85\) − 1.62151i − 0.175878i
\(86\) 0 0
\(87\) 4.11774 + 7.13214i 0.441468 + 0.764645i
\(88\) 0 0
\(89\) 0.643974 + 0.371799i 0.0682611 + 0.0394106i 0.533742 − 0.845647i \(-0.320785\pi\)
−0.465481 + 0.885058i \(0.654119\pi\)
\(90\) 0 0
\(91\) −7.42147 − 2.85775i −0.777981 − 0.299573i
\(92\) 0 0
\(93\) −5.17949 + 8.97114i −0.537088 + 0.930264i
\(94\) 0 0
\(95\) 8.47339 − 4.89211i 0.869352 − 0.501920i
\(96\) 0 0
\(97\) 4.34845i 0.441518i 0.975328 + 0.220759i \(0.0708534\pi\)
−0.975328 + 0.220759i \(0.929147\pi\)
\(98\) 0 0
\(99\) 0.700366 − 3.52531i 0.0703894 − 0.354307i
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 1232.2.bn.d.593.8 48
4.3 odd 2 616.2.bf.a.593.18 yes 48
7.3 odd 6 inner 1232.2.bn.d.241.7 48
11.10 odd 2 inner 1232.2.bn.d.593.7 48
28.3 even 6 616.2.bf.a.241.17 ✓ 48
44.43 even 2 616.2.bf.a.593.17 yes 48
77.10 even 6 inner 1232.2.bn.d.241.8 48
308.87 odd 6 616.2.bf.a.241.18 yes 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
616.2.bf.a.241.17 ✓ 48 28.3 even 6
616.2.bf.a.241.18 yes 48 308.87 odd 6
616.2.bf.a.593.17 yes 48 44.43 even 2
616.2.bf.a.593.18 yes 48 4.3 odd 2
1232.2.bn.d.241.7 48 7.3 odd 6 inner
1232.2.bn.d.241.8 48 77.10 even 6 inner
1232.2.bn.d.593.7 48 11.10 odd 2 inner
1232.2.bn.d.593.8 48 1.1 even 1 trivial