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Properties

Label 616.2.bf.a.241.17
Level $616$
Weight $2$
Character 616.241
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 241.17
Character \(\chi\) \(=\) 616.241
Dual form 616.2.bf.a.593.17

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.19884 + 0.692153i) q^{3} +(-2.04863 + 1.18278i) q^{5} +(-2.46903 + 0.950736i) q^{7} +(-0.541847 - 0.938507i) q^{9} +(-2.49408 - 2.18622i) q^{11} -3.00583 q^{13} -3.27465 q^{15} +(-0.342735 + 0.593634i) q^{17} +(-2.06807 - 3.58200i) q^{19} +(-3.61804 - 0.569161i) q^{21} +(3.48628 + 6.03841i) q^{23} +(0.297914 - 0.516002i) q^{25} -5.65309i q^{27} -5.94917i q^{29} +(-6.48060 - 3.74158i) q^{31} +(-1.47682 - 4.34722i) q^{33} +(3.93361 - 4.86801i) q^{35} +(5.83377 + 10.1044i) q^{37} +(-3.60352 - 2.08049i) q^{39} -2.13747 q^{41} +11.1370i q^{43} +(2.22009 + 1.28177i) q^{45} +(-11.2046 + 6.46898i) q^{47} +(5.19220 - 4.69479i) q^{49} +(-0.821772 + 0.474450i) q^{51} +(4.98299 - 8.63078i) q^{53} +(7.69525 + 1.52880i) q^{55} -5.72568i q^{57} +(-3.50849 - 2.02563i) q^{59} +(6.76582 + 11.7187i) q^{61} +(2.23011 + 1.80205i) q^{63} +(6.15782 - 3.55522i) q^{65} +(2.22520 - 3.85416i) q^{67} +9.65215i q^{69} -9.67463 q^{71} +(-2.61522 + 4.52970i) q^{73} +(0.714305 - 0.412404i) q^{75} +(8.23648 + 3.02661i) q^{77} +(-9.32635 + 5.38457i) q^{79} +(2.28726 - 3.96165i) q^{81} -11.9979 q^{83} -1.62151i q^{85} +(4.11774 - 7.13214i) q^{87} +(0.643974 - 0.371799i) q^{89} +(7.42147 - 2.85775i) q^{91} +(-5.17949 - 8.97114i) q^{93} +(8.47339 + 4.89211i) q^{95} -4.34845i q^{97} +(-0.700366 + 3.52531i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q + 24 q^{9} - 2 q^{11} - 8 q^{15} + 12 q^{23} + 20 q^{25} + 12 q^{31} - 30 q^{33} + 4 q^{37} + 48 q^{45} - 48 q^{49} + 20 q^{53} + 8 q^{67} - 16 q^{71} - 24 q^{75} + 34 q^{77} - 24 q^{81} - 24 q^{89}+ \cdots + 8 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/616\mathbb{Z}\right)^\times\).

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

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
\(3\) 1.19884 + 0.692153i 0.692153 + 0.399615i 0.804418 − 0.594063i \(-0.202477\pi\)
−0.112265 + 0.993678i \(0.535811\pi\)
\(4\) 0 0
\(5\) −2.04863 + 1.18278i −0.916174 + 0.528953i −0.882412 − 0.470477i \(-0.844082\pi\)
−0.0337613 + 0.999430i \(0.510749\pi\)
\(6\) 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
\(11\) −2.49408 − 2.18622i −0.751995 − 0.659169i
\(12\) 0 0
\(13\) −3.00583 −0.833666 −0.416833 − 0.908983i \(-0.636860\pi\)
−0.416833 + 0.908983i \(0.636860\pi\)
\(14\) 0 0
\(15\) −3.27465 −0.845510
\(16\) 0 0
\(17\) −0.342735 + 0.593634i −0.0831254 + 0.143977i −0.904591 − 0.426281i \(-0.859824\pi\)
0.821465 + 0.570258i \(0.193157\pi\)
\(18\) 0 0
\(19\) −2.06807 − 3.58200i −0.474447 − 0.821766i 0.525125 − 0.851025i \(-0.324019\pi\)
−0.999572 + 0.0292589i \(0.990685\pi\)
\(20\) 0 0
\(21\) −3.61804 − 0.569161i −0.789520 − 0.124201i
\(22\) 0 0
\(23\) 3.48628 + 6.03841i 0.726939 + 1.25910i 0.958171 + 0.286197i \(0.0923910\pi\)
−0.231232 + 0.972899i \(0.574276\pi\)
\(24\) 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 \(-0.567901\pi\)
−0.952247 + 0.305327i \(0.901234\pi\)
\(32\) 0 0
\(33\) −1.47682 − 4.34722i −0.257082 − 0.756754i
\(34\) 0 0
\(35\) 3.93361 − 4.86801i 0.664902 − 0.822844i
\(36\) 0 0
\(37\) 5.83377 + 10.1044i 0.959067 + 1.66115i 0.724775 + 0.688986i \(0.241944\pi\)
0.234292 + 0.972166i \(0.424723\pi\)
\(38\) 0 0
\(39\) −3.60352 − 2.08049i −0.577025 − 0.333145i
\(40\) 0 0
\(41\) −2.13747 −0.333816 −0.166908 − 0.985972i \(-0.553378\pi\)
−0.166908 + 0.985972i \(0.553378\pi\)
\(42\) 0 0
\(43\) 11.1370i 1.69837i 0.528092 + 0.849187i \(0.322908\pi\)
−0.528092 + 0.849187i \(0.677092\pi\)
\(44\) 0 0
\(45\) 2.22009 + 1.28177i 0.330951 + 0.191075i
\(46\) 0 0
\(47\) −11.2046 + 6.46898i −1.63436 + 0.943598i −0.651634 + 0.758534i \(0.725916\pi\)
−0.982726 + 0.185065i \(0.940751\pi\)
\(48\) 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 −0.289139 − 0.957287i \(-0.593369\pi\)
0.973604 − 0.228242i \(-0.0732978\pi\)
\(54\) 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 \(-0.418281\pi\)
−0.710684 + 0.703512i \(0.751614\pi\)
\(60\) 0 0
\(61\) 6.76582 + 11.7187i 0.866274 + 1.50043i 0.865777 + 0.500431i \(0.166825\pi\)
0.000497294 1.00000i \(0.499842\pi\)
\(62\) 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 \(-0.745697\pi\)
0.969336 + 0.245740i \(0.0790308\pi\)
\(68\) 0 0
\(69\) 9.65215i 1.16198i
\(70\) 0 0
\(71\) −9.67463 −1.14817 −0.574084 − 0.818797i \(-0.694642\pi\)
−0.574084 + 0.818797i \(0.694642\pi\)
\(72\) 0 0
\(73\) −2.61522 + 4.52970i −0.306089 + 0.530161i −0.977503 − 0.210921i \(-0.932354\pi\)
0.671414 + 0.741082i \(0.265687\pi\)
\(74\) 0 0
\(75\) 0.714305 − 0.412404i 0.0824809 − 0.0476203i
\(76\) 0 0
\(77\) 8.23648 + 3.02661i 0.938634 + 0.344915i
\(78\) 0 0
\(79\) −9.32635 + 5.38457i −1.04930 + 0.605812i −0.922452 − 0.386111i \(-0.873818\pi\)
−0.126844 + 0.991923i \(0.540485\pi\)
\(80\) 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 −0.465481 − 0.885058i \(-0.654119\pi\)
0.533742 + 0.845647i \(0.320785\pi\)
\(90\) 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 \(-0.929147\pi\)
0.975328 − 0.220759i \(-0.0708534\pi\)
\(98\) 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 616.2.bf.a.241.17 ✓ 48
4.3 odd 2 1232.2.bn.d.241.7 48
7.5 odd 6 inner 616.2.bf.a.593.18 yes 48
11.10 odd 2 inner 616.2.bf.a.241.18 yes 48
28.19 even 6 1232.2.bn.d.593.8 48
44.43 even 2 1232.2.bn.d.241.8 48
77.54 even 6 inner 616.2.bf.a.593.17 yes 48
308.131 odd 6 1232.2.bn.d.593.7 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
616.2.bf.a.241.17 ✓ 48 1.1 even 1 trivial
616.2.bf.a.241.18 yes 48 11.10 odd 2 inner
616.2.bf.a.593.17 yes 48 77.54 even 6 inner
616.2.bf.a.593.18 yes 48 7.5 odd 6 inner
1232.2.bn.d.241.7 48 4.3 odd 2
1232.2.bn.d.241.8 48 44.43 even 2
1232.2.bn.d.593.7 48 308.131 odd 6
1232.2.bn.d.593.8 48 28.19 even 6