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Properties

Label 912.2.d.b.191.5
Level $912$
Weight $2$
Character 912.191
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(191,912)] 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("912.191"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(912, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.d (of order \(2\), degree \(1\), minimal)

Newform invariants

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

Embedding invariants

Embedding label 191.5
Character \(\chi\) \(=\) 912.191
Dual form 912.2.d.b.191.6

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.50956 - 0.849252i) q^{3} +0.771899i q^{5} -1.48496i q^{7} +(1.55754 + 2.56399i) q^{9} +4.34856 q^{11} -5.42004 q^{13} +(0.655537 - 1.16523i) q^{15} +7.36768i q^{17} -1.00000i q^{19} +(-1.26110 + 2.24163i) q^{21} +1.65261 q^{23} +4.40417 q^{25} +(-0.173726 - 5.19325i) q^{27} -5.57672i q^{29} -1.97544i q^{31} +(-6.56441 - 3.69302i) q^{33} +1.14624 q^{35} +10.2727 q^{37} +(8.18188 + 4.60298i) q^{39} +3.92937i q^{41} +8.99665i q^{43} +(-1.97914 + 1.20227i) q^{45} +3.62677 q^{47} +4.79490 q^{49} +(6.25702 - 11.1220i) q^{51} +2.72185i q^{53} +3.35665i q^{55} +(-0.849252 + 1.50956i) q^{57} +1.94462 q^{59} +12.2553 q^{61} +(3.80742 - 2.31288i) q^{63} -4.18372i q^{65} -10.8690i q^{67} +(-2.49471 - 1.40348i) q^{69} +16.0982 q^{71} +4.95000 q^{73} +(-6.64836 - 3.74025i) q^{75} -6.45743i q^{77} -3.32493i q^{79} +(-4.14813 + 7.98706i) q^{81} -17.0770 q^{83} -5.68711 q^{85} +(-4.73604 + 8.41839i) q^{87} +4.94478i q^{89} +8.04853i q^{91} +(-1.67765 + 2.98205i) q^{93} +0.771899 q^{95} +5.51162 q^{97} +(6.77307 + 11.1497i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 4 q^{9} - 12 q^{21} - 64 q^{25} + 12 q^{33} + 64 q^{37} - 16 q^{45} - 8 q^{49} - 24 q^{61} - 8 q^{69} - 16 q^{73} - 4 q^{81} - 8 q^{85} + 32 q^{93} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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
\(3\) −1.50956 − 0.849252i −0.871545 − 0.490316i
\(4\) 0 0
\(5\) 0.771899i 0.345204i 0.984992 + 0.172602i \(0.0552174\pi\)
−0.984992 + 0.172602i \(0.944783\pi\)
\(6\) 0 0
\(7\) − 1.48496i − 0.561261i −0.959816 − 0.280631i \(-0.909456\pi\)
0.959816 − 0.280631i \(-0.0905436\pi\)
\(8\) 0 0
\(9\) 1.55754 + 2.56399i 0.519181 + 0.854665i
\(10\) 0 0
\(11\) 4.34856 1.31114 0.655570 − 0.755134i \(-0.272428\pi\)
0.655570 + 0.755134i \(0.272428\pi\)
\(12\) 0 0
\(13\) −5.42004 −1.50325 −0.751625 − 0.659591i \(-0.770729\pi\)
−0.751625 + 0.659591i \(0.770729\pi\)
\(14\) 0 0
\(15\) 0.655537 − 1.16523i 0.169259 − 0.300861i
\(16\) 0 0
\(17\) 7.36768i 1.78693i 0.449138 + 0.893463i \(0.351731\pi\)
−0.449138 + 0.893463i \(0.648269\pi\)
\(18\) 0 0
\(19\) − 1.00000i − 0.229416i
\(20\) 0 0
\(21\) −1.26110 + 2.24163i −0.275195 + 0.489164i
\(22\) 0 0
\(23\) 1.65261 0.344592 0.172296 − 0.985045i \(-0.444881\pi\)
0.172296 + 0.985045i \(0.444881\pi\)
\(24\) 0 0
\(25\) 4.40417 0.880834
\(26\) 0 0
\(27\) −0.173726 − 5.19325i −0.0334336 − 0.999441i
\(28\) 0 0
\(29\) − 5.57672i − 1.03557i −0.855510 − 0.517786i \(-0.826757\pi\)
0.855510 − 0.517786i \(-0.173243\pi\)
\(30\) 0 0
\(31\) − 1.97544i − 0.354800i −0.984139 − 0.177400i \(-0.943231\pi\)
0.984139 − 0.177400i \(-0.0567686\pi\)
\(32\) 0 0
\(33\) −6.56441 − 3.69302i −1.14272 − 0.642873i
\(34\) 0 0
\(35\) 1.14624 0.193749
\(36\) 0 0
\(37\) 10.2727 1.68882 0.844411 − 0.535696i \(-0.179951\pi\)
0.844411 + 0.535696i \(0.179951\pi\)
\(38\) 0 0
\(39\) 8.18188 + 4.60298i 1.31015 + 0.737067i
\(40\) 0 0
\(41\) 3.92937i 0.613665i 0.951764 + 0.306832i \(0.0992691\pi\)
−0.951764 + 0.306832i \(0.900731\pi\)
\(42\) 0 0
\(43\) 8.99665i 1.37198i 0.727613 + 0.685988i \(0.240630\pi\)
−0.727613 + 0.685988i \(0.759370\pi\)
\(44\) 0 0
\(45\) −1.97914 + 1.20227i −0.295033 + 0.179223i
\(46\) 0 0
\(47\) 3.62677 0.529019 0.264509 − 0.964383i \(-0.414790\pi\)
0.264509 + 0.964383i \(0.414790\pi\)
\(48\) 0 0
\(49\) 4.79490 0.684986
\(50\) 0 0
\(51\) 6.25702 − 11.1220i 0.876158 − 1.55739i
\(52\) 0 0
\(53\) 2.72185i 0.373875i 0.982372 + 0.186937i \(0.0598562\pi\)
−0.982372 + 0.186937i \(0.940144\pi\)
\(54\) 0 0
\(55\) 3.35665i 0.452611i
\(56\) 0 0
\(57\) −0.849252 + 1.50956i −0.112486 + 0.199946i
\(58\) 0 0
\(59\) 1.94462 0.253168 0.126584 − 0.991956i \(-0.459599\pi\)
0.126584 + 0.991956i \(0.459599\pi\)
\(60\) 0 0
\(61\) 12.2553 1.56913 0.784566 − 0.620046i \(-0.212886\pi\)
0.784566 + 0.620046i \(0.212886\pi\)
\(62\) 0 0
\(63\) 3.80742 − 2.31288i 0.479690 − 0.291396i
\(64\) 0 0
\(65\) − 4.18372i − 0.518927i
\(66\) 0 0
\(67\) − 10.8690i − 1.32786i −0.747793 − 0.663932i \(-0.768886\pi\)
0.747793 − 0.663932i \(-0.231114\pi\)
\(68\) 0 0
\(69\) −2.49471 − 1.40348i −0.300328 − 0.168959i
\(70\) 0 0
\(71\) 16.0982 1.91051 0.955254 − 0.295787i \(-0.0955819\pi\)
0.955254 + 0.295787i \(0.0955819\pi\)
\(72\) 0 0
\(73\) 4.95000 0.579354 0.289677 − 0.957125i \(-0.406452\pi\)
0.289677 + 0.957125i \(0.406452\pi\)
\(74\) 0 0
\(75\) −6.64836 − 3.74025i −0.767687 − 0.431887i
\(76\) 0 0
\(77\) − 6.45743i − 0.735892i
\(78\) 0 0
\(79\) − 3.32493i − 0.374084i −0.982352 − 0.187042i \(-0.940110\pi\)
0.982352 − 0.187042i \(-0.0598900\pi\)
\(80\) 0 0
\(81\) −4.14813 + 7.98706i −0.460903 + 0.887451i
\(82\) 0 0
\(83\) −17.0770 −1.87444 −0.937220 − 0.348740i \(-0.886610\pi\)
−0.937220 + 0.348740i \(0.886610\pi\)
\(84\) 0 0
\(85\) −5.68711 −0.616853
\(86\) 0 0
\(87\) −4.73604 + 8.41839i −0.507757 + 0.902547i
\(88\) 0 0
\(89\) 4.94478i 0.524146i 0.965048 + 0.262073i \(0.0844061\pi\)
−0.965048 + 0.262073i \(0.915594\pi\)
\(90\) 0 0
\(91\) 8.04853i 0.843715i
\(92\) 0 0
\(93\) −1.67765 + 2.98205i −0.173964 + 0.309224i
\(94\) 0 0
\(95\) 0.771899 0.0791952
\(96\) 0 0
\(97\) 5.51162 0.559621 0.279810 − 0.960055i \(-0.409728\pi\)
0.279810 + 0.960055i \(0.409728\pi\)
\(98\) 0 0
\(99\) 6.77307 + 11.1497i 0.680719 + 1.12059i
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.d.b.191.5 ✓ 24
3.2 odd 2 inner 912.2.d.b.191.19 yes 24
4.3 odd 2 inner 912.2.d.b.191.20 yes 24
12.11 even 2 inner 912.2.d.b.191.6 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
912.2.d.b.191.5 ✓ 24 1.1 even 1 trivial
912.2.d.b.191.6 yes 24 12.11 even 2 inner
912.2.d.b.191.19 yes 24 3.2 odd 2 inner
912.2.d.b.191.20 yes 24 4.3 odd 2 inner