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Properties

Label 912.2.bh.g.239.12
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 239.12
Character \(\chi\) \(=\) 912.239
Dual form 912.2.bh.g.767.12

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.69963 - 0.333549i) q^{3} +(1.84530 + 1.06538i) q^{5} -2.59062i q^{7} +(2.77749 - 1.13382i) q^{9} +0.785637 q^{11} +(0.884423 + 1.53187i) q^{13} +(3.49168 + 1.19526i) q^{15} +(1.24775 + 0.720391i) q^{17} +(-3.25921 - 2.89440i) q^{19} +(-0.864099 - 4.40310i) q^{21} +(2.49723 + 4.32532i) q^{23} +(-0.229918 - 0.398230i) q^{25} +(4.34252 - 2.85351i) q^{27} +(-2.97987 + 1.72043i) q^{29} -4.93620i q^{31} +(1.33529 - 0.262049i) q^{33} +(2.76000 - 4.78047i) q^{35} -2.10614 q^{37} +(2.01415 + 2.30861i) q^{39} +(5.14815 + 2.97229i) q^{41} +(-5.29328 - 3.05607i) q^{43} +(6.33325 + 0.866855i) q^{45} +(2.78097 + 4.81678i) q^{47} +0.288682 q^{49} +(2.36101 + 0.808212i) q^{51} +(-11.4360 + 6.60255i) q^{53} +(1.44973 + 0.837005i) q^{55} +(-6.50488 - 3.83230i) q^{57} +(5.76494 - 9.98517i) q^{59} +(5.26557 + 9.12023i) q^{61} +(-2.93730 - 7.19542i) q^{63} +3.76900i q^{65} +(3.64415 - 2.10395i) q^{67} +(5.68707 + 6.51850i) q^{69} +(-3.09477 + 5.36030i) q^{71} +(-5.73490 + 9.93314i) q^{73} +(-0.523605 - 0.600154i) q^{75} -2.03529i q^{77} +(-3.99019 - 2.30374i) q^{79} +(6.42890 - 6.29835i) q^{81} -5.64041 q^{83} +(1.53499 + 2.65867i) q^{85} +(-4.49084 + 3.91803i) q^{87} +(-2.23758 + 1.29187i) q^{89} +(3.96848 - 2.29121i) q^{91} +(-1.64647 - 8.38972i) q^{93} +(-2.93057 - 8.81334i) q^{95} +(3.67173 - 6.35962i) q^{97} +(2.18210 - 0.890772i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 6 q^{3} - 8 q^{9} - 6 q^{13} - 9 q^{15} + 6 q^{19} - 12 q^{21} + 26 q^{25} + 9 q^{33} - 32 q^{37} + 42 q^{43} - 10 q^{45} - 32 q^{49} + 51 q^{51} - 24 q^{55} - 9 q^{57} - 6 q^{61} + 12 q^{67} - 14 q^{69}+ \cdots - 33 q^{99}+O(q^{100}) \) 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
\(3\) 1.69963 − 0.333549i 0.981282 − 0.192575i
\(4\) 0 0
\(5\) 1.84530 + 1.06538i 0.825242 + 0.476454i 0.852221 − 0.523182i \(-0.175255\pi\)
−0.0269787 + 0.999636i \(0.508589\pi\)
\(6\) 0 0
\(7\) − 2.59062i − 0.979163i −0.871958 − 0.489581i \(-0.837150\pi\)
0.871958 − 0.489581i \(-0.162850\pi\)
\(8\) 0 0
\(9\) 2.77749 − 1.13382i 0.925830 − 0.377940i
\(10\) 0 0
\(11\) 0.785637 0.236879 0.118439 − 0.992961i \(-0.462211\pi\)
0.118439 + 0.992961i \(0.462211\pi\)
\(12\) 0 0
\(13\) 0.884423 + 1.53187i 0.245295 + 0.424863i 0.962214 − 0.272293i \(-0.0877819\pi\)
−0.716920 + 0.697156i \(0.754449\pi\)
\(14\) 0 0
\(15\) 3.49168 + 1.19526i 0.901548 + 0.308615i
\(16\) 0 0
\(17\) 1.24775 + 0.720391i 0.302625 + 0.174721i 0.643621 − 0.765344i \(-0.277431\pi\)
−0.340997 + 0.940065i \(0.610764\pi\)
\(18\) 0 0
\(19\) −3.25921 − 2.89440i −0.747714 − 0.664021i
\(20\) 0 0
\(21\) −0.864099 − 4.40310i −0.188562 − 0.960835i
\(22\) 0 0
\(23\) 2.49723 + 4.32532i 0.520707 + 0.901892i 0.999710 + 0.0240783i \(0.00766512\pi\)
−0.479003 + 0.877813i \(0.659002\pi\)
\(24\) 0 0
\(25\) −0.229918 − 0.398230i −0.0459836 − 0.0796459i
\(26\) 0 0
\(27\) 4.34252 − 2.85351i 0.835719 − 0.549157i
\(28\) 0 0
\(29\) −2.97987 + 1.72043i −0.553349 + 0.319476i −0.750472 − 0.660903i \(-0.770174\pi\)
0.197123 + 0.980379i \(0.436840\pi\)
\(30\) 0 0
\(31\) − 4.93620i − 0.886568i −0.896381 − 0.443284i \(-0.853813\pi\)
0.896381 − 0.443284i \(-0.146187\pi\)
\(32\) 0 0
\(33\) 1.33529 − 0.262049i 0.232445 − 0.0456168i
\(34\) 0 0
\(35\) 2.76000 − 4.78047i 0.466526 − 0.808046i
\(36\) 0 0
\(37\) −2.10614 −0.346248 −0.173124 − 0.984900i \(-0.555386\pi\)
−0.173124 + 0.984900i \(0.555386\pi\)
\(38\) 0 0
\(39\) 2.01415 + 2.30861i 0.322521 + 0.369673i
\(40\) 0 0
\(41\) 5.14815 + 2.97229i 0.804006 + 0.464193i 0.844870 − 0.534971i \(-0.179678\pi\)
−0.0408638 + 0.999165i \(0.513011\pi\)
\(42\) 0 0
\(43\) −5.29328 − 3.05607i −0.807217 − 0.466047i 0.0387715 − 0.999248i \(-0.487656\pi\)
−0.845988 + 0.533201i \(0.820989\pi\)
\(44\) 0 0
\(45\) 6.33325 + 0.866855i 0.944105 + 0.129223i
\(46\) 0 0
\(47\) 2.78097 + 4.81678i 0.405646 + 0.702599i 0.994396 − 0.105716i \(-0.0337133\pi\)
−0.588751 + 0.808315i \(0.700380\pi\)
\(48\) 0 0
\(49\) 0.288682 0.0412403
\(50\) 0 0
\(51\) 2.36101 + 0.808212i 0.330607 + 0.113172i
\(52\) 0 0
\(53\) −11.4360 + 6.60255i −1.57085 + 0.906930i −0.574784 + 0.818305i \(0.694914\pi\)
−0.996065 + 0.0886249i \(0.971753\pi\)
\(54\) 0 0
\(55\) 1.44973 + 0.837005i 0.195482 + 0.112862i
\(56\) 0 0
\(57\) −6.50488 − 3.83230i −0.861592 − 0.507601i
\(58\) 0 0
\(59\) 5.76494 − 9.98517i 0.750532 − 1.29996i −0.197034 − 0.980397i \(-0.563131\pi\)
0.947565 − 0.319562i \(-0.103536\pi\)
\(60\) 0 0
\(61\) 5.26557 + 9.12023i 0.674187 + 1.16773i 0.976706 + 0.214583i \(0.0688392\pi\)
−0.302519 + 0.953143i \(0.597827\pi\)
\(62\) 0 0
\(63\) −2.93730 − 7.19542i −0.370065 − 0.906538i
\(64\) 0 0
\(65\) 3.76900i 0.467487i
\(66\) 0 0
\(67\) 3.64415 − 2.10395i 0.445204 − 0.257039i −0.260599 − 0.965447i \(-0.583920\pi\)
0.705802 + 0.708409i \(0.250587\pi\)
\(68\) 0 0
\(69\) 5.68707 + 6.51850i 0.684643 + 0.784735i
\(70\) 0 0
\(71\) −3.09477 + 5.36030i −0.367281 + 0.636150i −0.989139 − 0.146980i \(-0.953045\pi\)
0.621858 + 0.783130i \(0.286378\pi\)
\(72\) 0 0
\(73\) −5.73490 + 9.93314i −0.671220 + 1.16259i 0.306339 + 0.951923i \(0.400896\pi\)
−0.977559 + 0.210664i \(0.932437\pi\)
\(74\) 0 0
\(75\) −0.523605 − 0.600154i −0.0604607 − 0.0692998i
\(76\) 0 0
\(77\) − 2.03529i − 0.231943i
\(78\) 0 0
\(79\) −3.99019 − 2.30374i −0.448931 − 0.259190i 0.258448 − 0.966025i \(-0.416789\pi\)
−0.707379 + 0.706835i \(0.750122\pi\)
\(80\) 0 0
\(81\) 6.42890 − 6.29835i 0.714322 − 0.699817i
\(82\) 0 0
\(83\) −5.64041 −0.619116 −0.309558 − 0.950881i \(-0.600181\pi\)
−0.309558 + 0.950881i \(0.600181\pi\)
\(84\) 0 0
\(85\) 1.53499 + 2.65867i 0.166493 + 0.288374i
\(86\) 0 0
\(87\) −4.49084 + 3.91803i −0.481468 + 0.420057i
\(88\) 0 0
\(89\) −2.23758 + 1.29187i −0.237183 + 0.136937i −0.613881 − 0.789398i \(-0.710393\pi\)
0.376699 + 0.926336i \(0.377059\pi\)
\(90\) 0 0
\(91\) 3.96848 − 2.29121i 0.416010 − 0.240184i
\(92\) 0 0
\(93\) −1.64647 − 8.38972i −0.170731 − 0.869974i
\(94\) 0 0
\(95\) −2.93057 − 8.81334i −0.300670 − 0.904229i
\(96\) 0 0
\(97\) 3.67173 − 6.35962i 0.372808 − 0.645722i −0.617189 − 0.786815i \(-0.711728\pi\)
0.989996 + 0.141093i \(0.0450618\pi\)
\(98\) 0 0
\(99\) 2.18210 − 0.890772i 0.219309 − 0.0895259i
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.239.12 yes 24
3.2 odd 2 inner 912.2.bh.g.239.7 yes 24
4.3 odd 2 912.2.bh.e.239.1 ✓ 24
12.11 even 2 912.2.bh.e.239.6 yes 24
19.7 even 3 912.2.bh.e.767.6 yes 24
57.26 odd 6 912.2.bh.e.767.1 yes 24
76.7 odd 6 inner 912.2.bh.g.767.7 yes 24
228.83 even 6 inner 912.2.bh.g.767.12 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
912.2.bh.e.239.1 ✓ 24 4.3 odd 2
912.2.bh.e.239.6 yes 24 12.11 even 2
912.2.bh.e.767.1 yes 24 57.26 odd 6
912.2.bh.e.767.6 yes 24 19.7 even 3
912.2.bh.g.239.7 yes 24 3.2 odd 2 inner
912.2.bh.g.239.12 yes 24 1.1 even 1 trivial
912.2.bh.g.767.7 yes 24 76.7 odd 6 inner
912.2.bh.g.767.12 yes 24 228.83 even 6 inner