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Properties

Label 912.2.bh.g.239.10
Level $912$
Weight $2$
Character 912.239
Analytic conductor $7.282$
Analytic rank $0$
Dimension $24$
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] = [912,2,Mod(239,912)] 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("912.239"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(912, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 3, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.bh (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,0,6,0,0,0,0,0,-8,0,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.28235666434\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 239.10
Character \(\chi\) \(=\) 912.239
Dual form 912.2.bh.g.767.10

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.34700 + 1.08885i) q^{3} +(1.83758 + 1.06093i) q^{5} +2.54579i q^{7} +(0.628815 + 2.93336i) q^{9} +5.05331 q^{11} +(-0.176690 - 0.306037i) q^{13} +(1.32003 + 3.42992i) q^{15} +(-5.55778 - 3.20879i) q^{17} +(1.08376 + 4.22202i) q^{19} +(-2.77198 + 3.42918i) q^{21} +(-4.18455 - 7.24786i) q^{23} +(-0.248860 - 0.431039i) q^{25} +(-2.34697 + 4.63592i) q^{27} +(6.34651 - 3.66416i) q^{29} -0.0429449i q^{31} +(6.80680 + 5.50229i) q^{33} +(-2.70090 + 4.67810i) q^{35} -10.4921 q^{37} +(0.0952260 - 0.604620i) q^{39} +(-1.59714 - 0.922110i) q^{41} +(8.24082 + 4.75784i) q^{43} +(-1.95659 + 6.05742i) q^{45} +(3.63041 + 6.28806i) q^{47} +0.518962 q^{49} +(-3.99245 - 10.3738i) q^{51} +(-7.17104 + 4.14020i) q^{53} +(9.28587 + 5.36120i) q^{55} +(-3.13732 + 6.86711i) q^{57} +(5.19853 - 9.00411i) q^{59} +(-3.53243 - 6.11835i) q^{61} +(-7.46771 + 1.60083i) q^{63} -0.749824i q^{65} +(9.04144 - 5.22008i) q^{67} +(2.25523 - 14.3192i) q^{69} +(-3.21081 + 5.56129i) q^{71} +(3.74704 - 6.49007i) q^{73} +(0.134121 - 0.851580i) q^{75} +12.8647i q^{77} +(-3.45627 - 1.99548i) q^{79} +(-8.20918 + 3.68908i) q^{81} +2.54553 q^{83} +(-6.80859 - 11.7928i) q^{85} +(12.5385 + 1.97477i) q^{87} +(8.38838 - 4.84303i) q^{89} +(0.779105 - 0.449816i) q^{91} +(0.0467605 - 0.0578467i) q^{93} +(-2.48776 + 8.90811i) q^{95} +(-2.20496 + 3.81911i) q^{97} +(3.17760 + 14.8232i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 6 q^{3} - 8 q^{9} - 6 q^{13} - 9 q^{15} + 6 q^{19} - 12 q^{21} + 26 q^{25} + 9 q^{33} - 32 q^{37} + 42 q^{43} - 10 q^{45} - 32 q^{49} + 51 q^{51} - 24 q^{55} - 9 q^{57} - 6 q^{61} + 12 q^{67} - 14 q^{69}+ \cdots - 33 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(-1\) \(-1\)

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
\(3\) 1.34700 + 1.08885i 0.777690 + 0.628647i
\(4\) 0 0
\(5\) 1.83758 + 1.06093i 0.821792 + 0.474462i 0.851034 − 0.525111i \(-0.175976\pi\)
−0.0292421 + 0.999572i \(0.509309\pi\)
\(6\) 0 0
\(7\) 2.54579i 0.962218i 0.876661 + 0.481109i \(0.159766\pi\)
−0.876661 + 0.481109i \(0.840234\pi\)
\(8\) 0 0
\(9\) 0.628815 + 2.93336i 0.209605 + 0.977786i
\(10\) 0 0
\(11\) 5.05331 1.52363 0.761815 − 0.647795i \(-0.224309\pi\)
0.761815 + 0.647795i \(0.224309\pi\)
\(12\) 0 0
\(13\) −0.176690 − 0.306037i −0.0490051 − 0.0848793i 0.840482 − 0.541839i \(-0.182272\pi\)
−0.889487 + 0.456960i \(0.848938\pi\)
\(14\) 0 0
\(15\) 1.32003 + 3.42992i 0.340831 + 0.885602i
\(16\) 0 0
\(17\) −5.55778 − 3.20879i −1.34796 − 0.778246i −0.360000 − 0.932952i \(-0.617223\pi\)
−0.987960 + 0.154707i \(0.950557\pi\)
\(18\) 0 0
\(19\) 1.08376 + 4.22202i 0.248632 + 0.968598i
\(20\) 0 0
\(21\) −2.77198 + 3.42918i −0.604896 + 0.748307i
\(22\) 0 0
\(23\) −4.18455 − 7.24786i −0.872540 − 1.51128i −0.859361 − 0.511370i \(-0.829138\pi\)
−0.0131794 − 0.999913i \(-0.504195\pi\)
\(24\) 0 0
\(25\) −0.248860 − 0.431039i −0.0497721 − 0.0862078i
\(26\) 0 0
\(27\) −2.34697 + 4.63592i −0.451675 + 0.892183i
\(28\) 0 0
\(29\) 6.34651 − 3.66416i 1.17852 − 0.680417i 0.222846 − 0.974854i \(-0.428465\pi\)
0.955671 + 0.294436i \(0.0951318\pi\)
\(30\) 0 0
\(31\) − 0.0429449i − 0.00771313i −0.999993 − 0.00385656i \(-0.998772\pi\)
0.999993 − 0.00385656i \(-0.00122759\pi\)
\(32\) 0 0
\(33\) 6.80680 + 5.50229i 1.18491 + 0.957826i
\(34\) 0 0
\(35\) −2.70090 + 4.67810i −0.456535 + 0.790743i
\(36\) 0 0
\(37\) −10.4921 −1.72490 −0.862449 − 0.506145i \(-0.831070\pi\)
−0.862449 + 0.506145i \(0.831070\pi\)
\(38\) 0 0
\(39\) 0.0952260 − 0.604620i 0.0152484 − 0.0968167i
\(40\) 0 0
\(41\) −1.59714 − 0.922110i −0.249431 − 0.144009i 0.370072 − 0.929003i \(-0.379333\pi\)
−0.619504 + 0.784994i \(0.712666\pi\)
\(42\) 0 0
\(43\) 8.24082 + 4.75784i 1.25671 + 0.725564i 0.972434 − 0.233178i \(-0.0749124\pi\)
0.284279 + 0.958742i \(0.408246\pi\)
\(44\) 0 0
\(45\) −1.95659 + 6.05742i −0.291670 + 0.902986i
\(46\) 0 0
\(47\) 3.63041 + 6.28806i 0.529550 + 0.917208i 0.999406 + 0.0344647i \(0.0109726\pi\)
−0.469856 + 0.882743i \(0.655694\pi\)
\(48\) 0 0
\(49\) 0.518962 0.0741374
\(50\) 0 0
\(51\) −3.99245 − 10.3738i −0.559054 − 1.45263i
\(52\) 0 0
\(53\) −7.17104 + 4.14020i −0.985018 + 0.568700i −0.903781 − 0.427994i \(-0.859220\pi\)
−0.0812367 + 0.996695i \(0.525887\pi\)
\(54\) 0 0
\(55\) 9.28587 + 5.36120i 1.25211 + 0.722904i
\(56\) 0 0
\(57\) −3.13732 + 6.86711i −0.415548 + 0.909571i
\(58\) 0 0
\(59\) 5.19853 − 9.00411i 0.676790 − 1.17224i −0.299152 − 0.954206i \(-0.596704\pi\)
0.975942 − 0.218030i \(-0.0699630\pi\)
\(60\) 0 0
\(61\) −3.53243 − 6.11835i −0.452282 − 0.783375i 0.546246 − 0.837625i \(-0.316057\pi\)
−0.998527 + 0.0542501i \(0.982723\pi\)
\(62\) 0 0
\(63\) −7.46771 + 1.60083i −0.940843 + 0.201686i
\(64\) 0 0
\(65\) − 0.749824i − 0.0930042i
\(66\) 0 0
\(67\) 9.04144 − 5.22008i 1.10459 − 0.637734i 0.167165 − 0.985929i \(-0.446539\pi\)
0.937422 + 0.348195i \(0.113205\pi\)
\(68\) 0 0
\(69\) 2.25523 − 14.3192i 0.271498 − 1.72383i
\(70\) 0 0
\(71\) −3.21081 + 5.56129i −0.381053 + 0.660004i −0.991213 − 0.132275i \(-0.957772\pi\)
0.610160 + 0.792278i \(0.291105\pi\)
\(72\) 0 0
\(73\) 3.74704 − 6.49007i 0.438558 − 0.759605i −0.559020 − 0.829154i \(-0.688823\pi\)
0.997579 + 0.0695489i \(0.0221560\pi\)
\(74\) 0 0
\(75\) 0.134121 − 0.851580i 0.0154870 − 0.0983320i
\(76\) 0 0
\(77\) 12.8647i 1.46606i
\(78\) 0 0
\(79\) −3.45627 − 1.99548i −0.388861 − 0.224509i 0.292805 − 0.956172i \(-0.405411\pi\)
−0.681667 + 0.731663i \(0.738745\pi\)
\(80\) 0 0
\(81\) −8.20918 + 3.68908i −0.912132 + 0.409898i
\(82\) 0 0
\(83\) 2.54553 0.279408 0.139704 − 0.990193i \(-0.455385\pi\)
0.139704 + 0.990193i \(0.455385\pi\)
\(84\) 0 0
\(85\) −6.80859 − 11.7928i −0.738496 − 1.27911i
\(86\) 0 0
\(87\) 12.5385 + 1.97477i 1.34426 + 0.211718i
\(88\) 0 0
\(89\) 8.38838 − 4.84303i 0.889166 − 0.513360i 0.0154966 − 0.999880i \(-0.495067\pi\)
0.873670 + 0.486520i \(0.161734\pi\)
\(90\) 0 0
\(91\) 0.779105 − 0.449816i 0.0816723 − 0.0471536i
\(92\) 0 0
\(93\) 0.0467605 − 0.0578467i 0.00484884 − 0.00599842i
\(94\) 0 0
\(95\) −2.48776 + 8.90811i −0.255239 + 0.913952i
\(96\) 0 0
\(97\) −2.20496 + 3.81911i −0.223880 + 0.387771i −0.955983 − 0.293423i \(-0.905206\pi\)
0.732103 + 0.681194i \(0.238539\pi\)
\(98\) 0 0
\(99\) 3.17760 + 14.8232i 0.319360 + 1.48978i
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 912.2.bh.g.239.10 yes 24
3.2 odd 2 inner 912.2.bh.g.239.11 yes 24
4.3 odd 2 912.2.bh.e.239.3 yes 24
12.11 even 2 912.2.bh.e.239.2 ✓ 24
19.7 even 3 912.2.bh.e.767.2 yes 24
57.26 odd 6 912.2.bh.e.767.3 yes 24
76.7 odd 6 inner 912.2.bh.g.767.11 yes 24
228.83 even 6 inner 912.2.bh.g.767.10 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
912.2.bh.e.239.2 ✓ 24 12.11 even 2
912.2.bh.e.239.3 yes 24 4.3 odd 2
912.2.bh.e.767.2 yes 24 19.7 even 3
912.2.bh.e.767.3 yes 24 57.26 odd 6
912.2.bh.g.239.10 yes 24 1.1 even 1 trivial
912.2.bh.g.239.11 yes 24 3.2 odd 2 inner
912.2.bh.g.767.10 yes 24 228.83 even 6 inner
912.2.bh.g.767.11 yes 24 76.7 odd 6 inner