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Properties

Label 616.2.bf.a.593.6
Level $616$
Weight $2$
Character 616.593
Analytic conductor $4.919$
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] = [616,2,Mod(241,616)] 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("616.241"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(616, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 1, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 616 = 2^{3} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 616.bf (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.91878476451\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 593.6
Character \(\chi\) \(=\) 616.593
Dual form 616.2.bf.a.241.6

$q$-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 \(\left(\mathbb{Z}/616\mathbb{Z}\right)^\times\).

\(n\) \(57\) \(309\) \(353\) \(463\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{5}{6}\right)\) \(1\)

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 \(-0.843977\pi\)
−0.0334318 + 0.999441i \(0.510644\pi\)
\(4\) 0 0
\(5\) −0.631951 − 0.364857i −0.282617 − 0.163169i 0.351990 − 0.936004i \(-0.385505\pi\)
−0.634608 + 0.772834i \(0.718838\pi\)
\(6\) 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 0.981317 − 0.192396i \(-0.0616257\pi\)
−0.657278 + 0.753648i \(0.728292\pi\)
\(18\) 0 0
\(19\) 0.0917501 − 0.158916i 0.0210489 − 0.0364578i −0.855309 − 0.518118i \(-0.826633\pi\)
0.876358 + 0.481660i \(0.159966\pi\)
\(20\) 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 \(-0.699572\pi\)
0.994661 + 0.103192i \(0.0329058\pi\)
\(24\) 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 \(-0.922843\pi\)
0.970765 − 0.240030i \(-0.0771574\pi\)
\(30\) 0 0
\(31\) 6.23797 − 3.60149i 1.12037 − 0.646847i 0.178876 − 0.983872i \(-0.442754\pi\)
0.941496 + 0.337024i \(0.109421\pi\)
\(32\) 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 −0.633116 − 0.774057i \(-0.718225\pi\)
0.986911 + 0.161267i \(0.0515579\pi\)
\(38\) 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 \(-0.638954\pi\)
−0.996213 + 0.0869519i \(0.972287\pi\)
\(48\) 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 −0.999983 − 0.00579180i \(-0.998156\pi\)
0.494976 − 0.868907i \(-0.335177\pi\)
\(54\) 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 \(-0.512326\pi\)
0.846019 + 0.533152i \(0.178993\pi\)
\(60\) 0 0
\(61\) −2.59317 + 4.49150i −0.332022 + 0.575078i −0.982908 − 0.184097i \(-0.941064\pi\)
0.650887 + 0.759175i \(0.274397\pi\)
\(62\) 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 0.988549 + 0.150899i \(0.0482168\pi\)
−0.363592 + 0.931558i \(0.618450\pi\)
\(68\) 0 0
\(69\) 7.16627i 0.862717i
\(70\) 0 0
\(71\) −9.55981 −1.13454 −0.567270 − 0.823532i \(-0.692000\pi\)
−0.567270 + 0.823532i \(0.692000\pi\)
\(72\) 0 0
\(73\) 7.41069 + 12.8357i 0.867356 + 1.50230i 0.864689 + 0.502308i \(0.167515\pi\)
0.00266666 + 0.999996i \(0.499151\pi\)
\(74\) 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 \(-0.184328\pi\)
−0.0554577 + 0.998461i \(0.517662\pi\)
\(80\) 0 0
\(81\) 4.96828 + 8.60531i 0.552031 + 0.956146i
\(82\) 0 0
\(83\) −10.0234 −1.10022 −0.550108 − 0.835094i \(-0.685413\pi\)
−0.550108 + 0.835094i \(0.685413\pi\)
\(84\) 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 0.567439 − 0.823416i \(-0.307934\pi\)
−0.429380 + 0.903124i \(0.641268\pi\)
\(90\) 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 \(-0.999048\pi\)
0.999996 − 0.00299215i \(-0.000952433\pi\)
\(98\) 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 616.2.bf.a.593.6 yes 48
4.3 odd 2 1232.2.bn.d.593.19 48
7.3 odd 6 inner 616.2.bf.a.241.5 ✓ 48
11.10 odd 2 inner 616.2.bf.a.593.5 yes 48
28.3 even 6 1232.2.bn.d.241.20 48
44.43 even 2 1232.2.bn.d.593.20 48
77.10 even 6 inner 616.2.bf.a.241.6 yes 48
308.87 odd 6 1232.2.bn.d.241.19 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
616.2.bf.a.241.5 ✓ 48 7.3 odd 6 inner
616.2.bf.a.241.6 yes 48 77.10 even 6 inner
616.2.bf.a.593.5 yes 48 11.10 odd 2 inner
616.2.bf.a.593.6 yes 48 1.1 even 1 trivial
1232.2.bn.d.241.19 48 308.87 odd 6
1232.2.bn.d.241.20 48 28.3 even 6
1232.2.bn.d.593.19 48 4.3 odd 2
1232.2.bn.d.593.20 48 44.43 even 2