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Properties

Label 1232.2.bn.d.241.19
Level $1232$
Weight $2$
Character 1232.241
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 241.19
Character \(\chi\) \(=\) 1232.241
Dual form 1232.2.bn.d.593.19

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.58602 + 0.915689i) q^{3} +(-0.631951 + 0.364857i) q^{5} +(-2.46576 - 0.959177i) q^{7} +(0.176973 + 0.306526i) q^{9} +(0.819296 - 3.21384i) q^{11} -6.43416 q^{13} -1.33638 q^{15} +(1.33605 - 2.31410i) q^{17} +(-0.0917501 - 0.158916i) q^{19} +(-3.03244 - 3.77914i) q^{21} +(-1.95652 - 3.38880i) q^{23} +(-2.23376 + 3.86898i) q^{25} -4.84593i q^{27} +2.58521i q^{29} +(-6.23797 - 3.60149i) q^{31} +(4.24230 - 4.34699i) q^{33} +(1.90820 - 0.293498i) q^{35} +(2.15205 + 3.72746i) q^{37} +(-10.2047 - 5.89169i) q^{39} -6.21853 q^{41} -6.37145i q^{43} +(-0.223676 - 0.129140i) q^{45} +(9.72828 - 5.61663i) q^{47} +(5.15996 + 4.73020i) q^{49} +(4.23800 - 2.44681i) q^{51} +(-3.67651 + 6.36790i) q^{53} +(0.654837 + 2.32991i) q^{55} -0.336058i q^{57} +(-6.20103 - 3.58017i) q^{59} +(-2.59317 - 4.49150i) q^{61} +(-0.142360 - 0.925567i) q^{63} +(4.06608 - 2.34755i) q^{65} +(-5.11549 + 8.86029i) q^{67} -7.16627i q^{69} +9.55981 q^{71} +(7.41069 - 12.8357i) q^{73} +(-7.08557 + 4.09086i) q^{75} +(-5.10283 + 7.13871i) q^{77} +(-6.94618 + 4.01038i) q^{79} +(4.96828 - 8.60531i) q^{81} +10.0234 q^{83} +1.94987i q^{85} +(-2.36724 + 4.10019i) q^{87} +(1.30245 - 0.751968i) q^{89} +(15.8651 + 6.17150i) q^{91} +(-6.59569 - 11.4241i) q^{93} +(0.115963 + 0.0669514i) q^{95} +0.0589386i q^{97} +(1.13012 - 0.317626i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q + 24 q^{9} + 2 q^{11} + 8 q^{15} - 12 q^{23} + 20 q^{25} - 12 q^{31} - 30 q^{33} + 4 q^{37} + 48 q^{45} - 48 q^{49} + 20 q^{53} - 8 q^{67} + 16 q^{71} + 24 q^{75} + 34 q^{77} - 24 q^{81} - 24 q^{89}+ \cdots + 8 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) 1.58602 + 0.915689i 0.915689 + 0.528673i 0.882257 − 0.470768i \(-0.156023\pi\)
0.0334318 + 0.999441i \(0.489356\pi\)
\(4\) 0 0
\(5\) −0.631951 + 0.364857i −0.282617 + 0.163169i −0.634608 − 0.772834i \(-0.718838\pi\)
0.351990 + 0.936004i \(0.385505\pi\)
\(6\) 0 0
\(7\) −2.46576 − 0.959177i −0.931970 − 0.362535i
\(8\) 0 0
\(9\) 0.176973 + 0.306526i 0.0589909 + 0.102175i
\(10\) 0 0
\(11\) 0.819296 − 3.21384i 0.247027 − 0.969009i
\(12\) 0 0
\(13\) −6.43416 −1.78452 −0.892258 − 0.451527i \(-0.850880\pi\)
−0.892258 + 0.451527i \(0.850880\pi\)
\(14\) 0 0
\(15\) −1.33638 −0.345053
\(16\) 0 0
\(17\) 1.33605 − 2.31410i 0.324039 − 0.561252i −0.657278 − 0.753648i \(-0.728292\pi\)
0.981317 + 0.192396i \(0.0616257\pi\)
\(18\) 0 0
\(19\) −0.0917501 − 0.158916i −0.0210489 − 0.0364578i 0.855309 − 0.518118i \(-0.173367\pi\)
−0.876358 + 0.481660i \(0.840034\pi\)
\(20\) 0 0
\(21\) −3.03244 − 3.77914i −0.661733 − 0.824677i
\(22\) 0 0
\(23\) −1.95652 − 3.38880i −0.407963 − 0.706613i 0.586698 − 0.809806i \(-0.300428\pi\)
−0.994661 + 0.103192i \(0.967094\pi\)
\(24\) 0 0
\(25\) −2.23376 + 3.86898i −0.446752 + 0.773797i
\(26\) 0 0
\(27\) − 4.84593i − 0.932599i
\(28\) 0 0
\(29\) 2.58521i 0.480061i 0.970765 + 0.240030i \(0.0771574\pi\)
−0.970765 + 0.240030i \(0.922843\pi\)
\(30\) 0 0
\(31\) −6.23797 − 3.60149i −1.12037 − 0.646847i −0.178876 − 0.983872i \(-0.557246\pi\)
−0.941496 + 0.337024i \(0.890579\pi\)
\(32\) 0 0
\(33\) 4.24230 − 4.34699i 0.738489 − 0.756714i
\(34\) 0 0
\(35\) 1.90820 − 0.293498i 0.322545 − 0.0496102i
\(36\) 0 0
\(37\) 2.15205 + 3.72746i 0.353794 + 0.612790i 0.986911 − 0.161267i \(-0.0515579\pi\)
−0.633116 + 0.774057i \(0.718225\pi\)
\(38\) 0 0
\(39\) −10.2047 − 5.89169i −1.63406 − 0.943426i
\(40\) 0 0
\(41\) −6.21853 −0.971171 −0.485586 − 0.874189i \(-0.661394\pi\)
−0.485586 + 0.874189i \(0.661394\pi\)
\(42\) 0 0
\(43\) − 6.37145i − 0.971637i −0.874060 − 0.485819i \(-0.838522\pi\)
0.874060 − 0.485819i \(-0.161478\pi\)
\(44\) 0 0
\(45\) −0.223676 − 0.129140i −0.0333437 − 0.0192510i
\(46\) 0 0
\(47\) 9.72828 − 5.61663i 1.41902 − 0.819269i 0.422804 − 0.906221i \(-0.361046\pi\)
0.996213 + 0.0869519i \(0.0277127\pi\)
\(48\) 0 0
\(49\) 5.15996 + 4.73020i 0.737137 + 0.675743i
\(50\) 0 0
\(51\) 4.23800 − 2.44681i 0.593438 − 0.342622i
\(52\) 0 0
\(53\) −3.67651 + 6.36790i −0.505007 + 0.874699i 0.494976 + 0.868907i \(0.335177\pi\)
−0.999983 + 0.00579180i \(0.998156\pi\)
\(54\) 0 0
\(55\) 0.654837 + 2.32991i 0.0882982 + 0.314166i
\(56\) 0 0
\(57\) − 0.336058i − 0.0445120i
\(58\) 0 0
\(59\) −6.20103 − 3.58017i −0.807306 − 0.466098i 0.0387135 − 0.999250i \(-0.487674\pi\)
−0.846019 + 0.533152i \(0.821007\pi\)
\(60\) 0 0
\(61\) −2.59317 − 4.49150i −0.332022 − 0.575078i 0.650887 − 0.759175i \(-0.274397\pi\)
−0.982908 + 0.184097i \(0.941064\pi\)
\(62\) 0 0
\(63\) −0.142360 − 0.925567i −0.0179357 − 0.116611i
\(64\) 0 0
\(65\) 4.06608 − 2.34755i 0.504335 − 0.291178i
\(66\) 0 0
\(67\) −5.11549 + 8.86029i −0.624957 + 1.08246i 0.363592 + 0.931558i \(0.381550\pi\)
−0.988549 + 0.150899i \(0.951783\pi\)
\(68\) 0 0
\(69\) − 7.16627i − 0.862717i
\(70\) 0 0
\(71\) 9.55981 1.13454 0.567270 − 0.823532i \(-0.308000\pi\)
0.567270 + 0.823532i \(0.308000\pi\)
\(72\) 0 0
\(73\) 7.41069 − 12.8357i 0.867356 − 1.50230i 0.00266666 − 0.999996i \(-0.499151\pi\)
0.864689 − 0.502308i \(-0.167515\pi\)
\(74\) 0 0
\(75\) −7.08557 + 4.09086i −0.818171 + 0.472371i
\(76\) 0 0
\(77\) −5.10283 + 7.13871i −0.581521 + 0.813531i
\(78\) 0 0
\(79\) −6.94618 + 4.01038i −0.781506 + 0.451203i −0.836964 − 0.547258i \(-0.815672\pi\)
0.0554577 + 0.998461i \(0.482338\pi\)
\(80\) 0 0
\(81\) 4.96828 − 8.60531i 0.552031 − 0.956146i
\(82\) 0 0
\(83\) 10.0234 1.10022 0.550108 − 0.835094i \(-0.314587\pi\)
0.550108 + 0.835094i \(0.314587\pi\)
\(84\) 0 0
\(85\) 1.94987i 0.211493i
\(86\) 0 0
\(87\) −2.36724 + 4.10019i −0.253795 + 0.439586i
\(88\) 0 0
\(89\) 1.30245 − 0.751968i 0.138059 − 0.0797085i −0.429380 − 0.903124i \(-0.641268\pi\)
0.567439 + 0.823416i \(0.307934\pi\)
\(90\) 0 0
\(91\) 15.8651 + 6.17150i 1.66312 + 0.646949i
\(92\) 0 0
\(93\) −6.59569 − 11.4241i −0.683942 − 1.18462i
\(94\) 0 0
\(95\) 0.115963 + 0.0669514i 0.0118976 + 0.00686906i
\(96\) 0 0
\(97\) 0.0589386i 0.00598430i 0.999996 + 0.00299215i \(0.000952433\pi\)
−0.999996 + 0.00299215i \(0.999048\pi\)
\(98\) 0 0
\(99\) 1.13012 − 0.317626i 0.113581 − 0.0319226i
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.241.19 48
4.3 odd 2 616.2.bf.a.241.6 yes 48
7.5 odd 6 inner 1232.2.bn.d.593.20 48
11.10 odd 2 inner 1232.2.bn.d.241.20 48
28.19 even 6 616.2.bf.a.593.5 yes 48
44.43 even 2 616.2.bf.a.241.5 ✓ 48
77.54 even 6 inner 1232.2.bn.d.593.19 48
308.131 odd 6 616.2.bf.a.593.6 yes 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
616.2.bf.a.241.5 ✓ 48 44.43 even 2
616.2.bf.a.241.6 yes 48 4.3 odd 2
616.2.bf.a.593.5 yes 48 28.19 even 6
616.2.bf.a.593.6 yes 48 308.131 odd 6
1232.2.bn.d.241.19 48 1.1 even 1 trivial
1232.2.bn.d.241.20 48 11.10 odd 2 inner
1232.2.bn.d.593.19 48 77.54 even 6 inner
1232.2.bn.d.593.20 48 7.5 odd 6 inner