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Properties

Label 784.2.x.c.557.1
Level $784$
Weight $2$
Character 784.557
Analytic conductor $6.260$
Analytic rank $0$
Dimension $4$
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] = [784,2,Mod(165,784)] 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("784.165"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(784, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 3, 8])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 784.x (of order \(12\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,2,-2,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.26027151847\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 16)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 557.1
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 784.557
Dual form 784.2.x.c.373.1

$q$-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+(1.36603 + 0.366025i) q^{2} +(-1.36603 + 0.366025i) q^{3} +(1.73205 + 1.00000i) q^{4} +(-1.36603 - 0.366025i) q^{5} -2.00000 q^{6} +(2.00000 + 2.00000i) q^{8} +(-0.866025 + 0.500000i) q^{9} +(-1.73205 - 1.00000i) q^{10} +(0.366025 + 1.36603i) q^{11} +(-2.73205 - 0.732051i) q^{12} +(1.00000 + 1.00000i) q^{13} +2.00000 q^{15} +(2.00000 + 3.46410i) q^{16} +(-1.00000 + 1.73205i) q^{17} +(-1.36603 + 0.366025i) q^{18} +(-1.09808 + 4.09808i) q^{19} +(-2.00000 - 2.00000i) q^{20} +2.00000i q^{22} +(-5.19615 + 3.00000i) q^{23} +(-3.46410 - 2.00000i) q^{24} +(-2.59808 - 1.50000i) q^{25} +(1.00000 + 1.73205i) q^{26} +(4.00000 - 4.00000i) q^{27} +(3.00000 + 3.00000i) q^{29} +(2.73205 + 0.732051i) q^{30} +(-4.00000 + 6.92820i) q^{31} +(1.46410 + 5.46410i) q^{32} +(-1.00000 - 1.73205i) q^{33} +(-2.00000 + 2.00000i) q^{34} -2.00000 q^{36} +(-4.09808 - 1.09808i) q^{37} +(-3.00000 + 5.19615i) q^{38} +(-1.73205 - 1.00000i) q^{39} +(-2.00000 - 3.46410i) q^{40} +(5.00000 - 5.00000i) q^{43} +(-0.732051 + 2.73205i) q^{44} +(1.36603 - 0.366025i) q^{45} +(-8.19615 + 2.19615i) q^{46} +(4.00000 + 6.92820i) q^{47} +(-4.00000 - 4.00000i) q^{48} +(-3.00000 - 3.00000i) q^{50} +(0.732051 - 2.73205i) q^{51} +(0.732051 + 2.73205i) q^{52} +(-1.83013 - 6.83013i) q^{53} +(6.92820 - 4.00000i) q^{54} -2.00000i q^{55} -6.00000i q^{57} +(3.00000 + 5.19615i) q^{58} +(1.09808 + 4.09808i) q^{59} +(3.46410 + 2.00000i) q^{60} +(3.29423 - 12.2942i) q^{61} +(-8.00000 + 8.00000i) q^{62} +8.00000i q^{64} +(-1.00000 - 1.73205i) q^{65} +(-0.732051 - 2.73205i) q^{66} +(6.83013 - 1.83013i) q^{67} +(-3.46410 + 2.00000i) q^{68} +(6.00000 - 6.00000i) q^{69} +10.0000i q^{71} +(-2.73205 - 0.732051i) q^{72} +(3.46410 + 2.00000i) q^{73} +(-5.19615 - 3.00000i) q^{74} +(4.09808 + 1.09808i) q^{75} +(-6.00000 + 6.00000i) q^{76} +(-2.00000 - 2.00000i) q^{78} +(-1.46410 - 5.46410i) q^{80} +(-2.50000 + 4.33013i) q^{81} +(1.00000 + 1.00000i) q^{83} +(2.00000 - 2.00000i) q^{85} +(8.66025 - 5.00000i) q^{86} +(-5.19615 - 3.00000i) q^{87} +(-2.00000 + 3.46410i) q^{88} +(3.46410 - 2.00000i) q^{89} +2.00000 q^{90} -12.0000 q^{92} +(2.92820 - 10.9282i) q^{93} +(2.92820 + 10.9282i) q^{94} +(3.00000 - 5.19615i) q^{95} +(-4.00000 - 6.92820i) q^{96} +2.00000 q^{97} +(-1.00000 - 1.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{3} - 2 q^{5} - 8 q^{6} + 8 q^{8} - 2 q^{11} - 4 q^{12} + 4 q^{13} + 8 q^{15} + 8 q^{16} - 4 q^{17} - 2 q^{18} + 6 q^{19} - 8 q^{20} + 4 q^{26} + 16 q^{27} + 12 q^{29} + 4 q^{30} - 16 q^{31}+ \cdots - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/784\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.36603 + 0.366025i 0.965926 + 0.258819i
\(3\) −1.36603 + 0.366025i −0.788675 + 0.211325i −0.630606 − 0.776103i \(-0.717194\pi\)
−0.158069 + 0.987428i \(0.550527\pi\)
\(4\) 1.73205 + 1.00000i 0.866025 + 0.500000i
\(5\) −1.36603 − 0.366025i −0.610905 − 0.163692i −0.0599153 − 0.998203i \(-0.519083\pi\)
−0.550990 + 0.834512i \(0.685750\pi\)
\(6\) −2.00000 −0.816497
\(7\) 0 0
\(8\) 2.00000 + 2.00000i 0.707107 + 0.707107i
\(9\) −0.866025 + 0.500000i −0.288675 + 0.166667i
\(10\) −1.73205 − 1.00000i −0.547723 − 0.316228i
\(11\) 0.366025 + 1.36603i 0.110361 + 0.411872i 0.998898 − 0.0469323i \(-0.0149445\pi\)
−0.888537 + 0.458804i \(0.848278\pi\)
\(12\) −2.73205 − 0.732051i −0.788675 − 0.211325i
\(13\) 1.00000 + 1.00000i 0.277350 + 0.277350i 0.832050 − 0.554700i \(-0.187167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 2.00000 0.516398
\(16\) 2.00000 + 3.46410i 0.500000 + 0.866025i
\(17\) −1.00000 + 1.73205i −0.242536 + 0.420084i −0.961436 − 0.275029i \(-0.911312\pi\)
0.718900 + 0.695113i \(0.244646\pi\)
\(18\) −1.36603 + 0.366025i −0.321975 + 0.0862730i
\(19\) −1.09808 + 4.09808i −0.251916 + 0.940163i 0.717864 + 0.696183i \(0.245120\pi\)
−0.969780 + 0.243980i \(0.921547\pi\)
\(20\) −2.00000 − 2.00000i −0.447214 − 0.447214i
\(21\) 0 0
\(22\) 2.00000i 0.426401i
\(23\) −5.19615 + 3.00000i −1.08347 + 0.625543i −0.931831 − 0.362892i \(-0.881789\pi\)
−0.151642 + 0.988436i \(0.548456\pi\)
\(24\) −3.46410 − 2.00000i −0.707107 − 0.408248i
\(25\) −2.59808 − 1.50000i −0.519615 − 0.300000i
\(26\) 1.00000 + 1.73205i 0.196116 + 0.339683i
\(27\) 4.00000 − 4.00000i 0.769800 − 0.769800i
\(28\) 0 0
\(29\) 3.00000 + 3.00000i 0.557086 + 0.557086i 0.928477 − 0.371391i \(-0.121119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 2.73205 + 0.732051i 0.498802 + 0.133654i
\(31\) −4.00000 + 6.92820i −0.718421 + 1.24434i 0.243204 + 0.969975i \(0.421802\pi\)
−0.961625 + 0.274367i \(0.911532\pi\)
\(32\) 1.46410 + 5.46410i 0.258819 + 0.965926i
\(33\) −1.00000 − 1.73205i −0.174078 − 0.301511i
\(34\) −2.00000 + 2.00000i −0.342997 + 0.342997i
\(35\) 0 0
\(36\) −2.00000 −0.333333
\(37\) −4.09808 − 1.09808i −0.673720 − 0.180523i −0.0942898 − 0.995545i \(-0.530058\pi\)
−0.579430 + 0.815022i \(0.696725\pi\)
\(38\) −3.00000 + 5.19615i −0.486664 + 0.842927i
\(39\) −1.73205 − 1.00000i −0.277350 − 0.160128i
\(40\) −2.00000 − 3.46410i −0.316228 − 0.547723i
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 5.00000 − 5.00000i 0.762493 − 0.762493i −0.214280 − 0.976772i \(-0.568740\pi\)
0.976772 + 0.214280i \(0.0687403\pi\)
\(44\) −0.732051 + 2.73205i −0.110361 + 0.411872i
\(45\) 1.36603 − 0.366025i 0.203635 − 0.0545638i
\(46\) −8.19615 + 2.19615i −1.20846 + 0.323805i
\(47\) 4.00000 + 6.92820i 0.583460 + 1.01058i 0.995066 + 0.0992202i \(0.0316348\pi\)
−0.411606 + 0.911362i \(0.635032\pi\)
\(48\) −4.00000 − 4.00000i −0.577350 − 0.577350i
\(49\) 0 0
\(50\) −3.00000 − 3.00000i −0.424264 − 0.424264i
\(51\) 0.732051 − 2.73205i 0.102508 − 0.382564i
\(52\) 0.732051 + 2.73205i 0.101517 + 0.378867i
\(53\) −1.83013 − 6.83013i −0.251387 − 0.938190i −0.970065 − 0.242846i \(-0.921919\pi\)
0.718677 − 0.695344i \(-0.244748\pi\)
\(54\) 6.92820 − 4.00000i 0.942809 − 0.544331i
\(55\) − 2.00000i − 0.269680i
\(56\) 0 0
\(57\) − 6.00000i − 0.794719i
\(58\) 3.00000 + 5.19615i 0.393919 + 0.682288i
\(59\) 1.09808 + 4.09808i 0.142957 + 0.533524i 0.999838 + 0.0180090i \(0.00573274\pi\)
−0.856880 + 0.515515i \(0.827601\pi\)
\(60\) 3.46410 + 2.00000i 0.447214 + 0.258199i
\(61\) 3.29423 − 12.2942i 0.421783 − 1.57411i −0.349067 − 0.937098i \(-0.613501\pi\)
0.770850 − 0.637017i \(-0.219832\pi\)
\(62\) −8.00000 + 8.00000i −1.01600 + 1.01600i
\(63\) 0 0
\(64\) 8.00000i 1.00000i
\(65\) −1.00000 − 1.73205i −0.124035 − 0.214834i
\(66\) −0.732051 − 2.73205i −0.0901092 − 0.336292i
\(67\) 6.83013 − 1.83013i 0.834433 − 0.223586i 0.183786 − 0.982966i \(-0.441165\pi\)
0.650647 + 0.759381i \(0.274498\pi\)
\(68\) −3.46410 + 2.00000i −0.420084 + 0.242536i
\(69\) 6.00000 − 6.00000i 0.722315 − 0.722315i
\(70\) 0 0
\(71\) 10.0000i 1.18678i 0.804914 + 0.593391i \(0.202211\pi\)
−0.804914 + 0.593391i \(0.797789\pi\)
\(72\) −2.73205 − 0.732051i −0.321975 − 0.0862730i
\(73\) 3.46410 + 2.00000i 0.405442 + 0.234082i 0.688830 − 0.724923i \(-0.258125\pi\)
−0.283387 + 0.959006i \(0.591458\pi\)
\(74\) −5.19615 − 3.00000i −0.604040 − 0.348743i
\(75\) 4.09808 + 1.09808i 0.473205 + 0.126795i
\(76\) −6.00000 + 6.00000i −0.688247 + 0.688247i
\(77\) 0 0
\(78\) −2.00000 − 2.00000i −0.226455 − 0.226455i
\(79\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(80\) −1.46410 − 5.46410i −0.163692 − 0.610905i
\(81\) −2.50000 + 4.33013i −0.277778 + 0.481125i
\(82\) 0 0
\(83\) 1.00000 + 1.00000i 0.109764 + 0.109764i 0.759856 − 0.650092i \(-0.225269\pi\)
−0.650092 + 0.759856i \(0.725269\pi\)
\(84\) 0 0
\(85\) 2.00000 − 2.00000i 0.216930 − 0.216930i
\(86\) 8.66025 − 5.00000i 0.933859 − 0.539164i
\(87\) −5.19615 − 3.00000i −0.557086 − 0.321634i
\(88\) −2.00000 + 3.46410i −0.213201 + 0.369274i
\(89\) 3.46410 − 2.00000i 0.367194 − 0.212000i −0.305038 − 0.952340i \(-0.598669\pi\)
0.672232 + 0.740341i \(0.265336\pi\)
\(90\) 2.00000 0.210819
\(91\) 0 0
\(92\) −12.0000 −1.25109
\(93\) 2.92820 − 10.9282i 0.303641 − 1.13320i
\(94\) 2.92820 + 10.9282i 0.302021 + 1.12716i
\(95\) 3.00000 − 5.19615i 0.307794 − 0.533114i
\(96\) −4.00000 − 6.92820i −0.408248 − 0.707107i
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) −1.00000 − 1.00000i −0.100504 − 0.100504i
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 784.2.x.c.557.1 4
7.2 even 3 inner 784.2.x.c.765.1 4
7.3 odd 6 16.2.e.a.13.1 yes 2
7.4 even 3 784.2.m.b.589.1 2
7.5 odd 6 784.2.x.f.765.1 4
7.6 odd 2 784.2.x.f.557.1 4
16.5 even 4 inner 784.2.x.c.165.1 4
21.17 even 6 144.2.k.a.109.1 2
28.3 even 6 64.2.e.a.17.1 2
35.3 even 12 400.2.q.b.349.1 2
35.17 even 12 400.2.q.a.349.1 2
35.24 odd 6 400.2.l.c.301.1 2
56.3 even 6 128.2.e.a.33.1 2
56.45 odd 6 128.2.e.b.33.1 2
84.59 odd 6 576.2.k.a.145.1 2
112.3 even 12 128.2.e.a.97.1 2
112.5 odd 12 784.2.x.f.373.1 4
112.37 even 12 inner 784.2.x.c.373.1 4
112.45 odd 12 128.2.e.b.97.1 2
112.53 even 12 784.2.m.b.197.1 2
112.59 even 12 64.2.e.a.49.1 2
112.69 odd 4 784.2.x.f.165.1 4
112.101 odd 12 16.2.e.a.5.1 ✓ 2
140.3 odd 12 1600.2.q.a.849.1 2
140.59 even 6 1600.2.l.a.401.1 2
140.87 odd 12 1600.2.q.b.849.1 2
168.59 odd 6 1152.2.k.a.289.1 2
168.101 even 6 1152.2.k.b.289.1 2
224.3 even 24 1024.2.b.b.513.2 2
224.45 odd 24 1024.2.b.e.513.2 2
224.59 even 24 1024.2.a.e.1.1 2
224.101 odd 24 1024.2.a.b.1.2 2
224.115 even 24 1024.2.b.b.513.1 2
224.157 odd 24 1024.2.b.e.513.1 2
224.171 even 24 1024.2.a.e.1.2 2
224.213 odd 24 1024.2.a.b.1.1 2
336.59 odd 12 576.2.k.a.433.1 2
336.101 even 12 144.2.k.a.37.1 2
336.227 odd 12 1152.2.k.a.865.1 2
336.269 even 12 1152.2.k.b.865.1 2
560.59 even 12 1600.2.l.a.1201.1 2
560.213 even 12 400.2.q.a.149.1 2
560.283 odd 12 1600.2.q.b.49.1 2
560.437 even 12 400.2.q.b.149.1 2
560.507 odd 12 1600.2.q.a.49.1 2
560.549 odd 12 400.2.l.c.101.1 2
672.59 odd 24 9216.2.a.s.1.2 2
672.101 even 24 9216.2.a.d.1.2 2
672.395 odd 24 9216.2.a.s.1.1 2
672.437 even 24 9216.2.a.d.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
16.2.e.a.5.1 ✓ 2 112.101 odd 12
16.2.e.a.13.1 yes 2 7.3 odd 6
64.2.e.a.17.1 2 28.3 even 6
64.2.e.a.49.1 2 112.59 even 12
128.2.e.a.33.1 2 56.3 even 6
128.2.e.a.97.1 2 112.3 even 12
128.2.e.b.33.1 2 56.45 odd 6
128.2.e.b.97.1 2 112.45 odd 12
144.2.k.a.37.1 2 336.101 even 12
144.2.k.a.109.1 2 21.17 even 6
400.2.l.c.101.1 2 560.549 odd 12
400.2.l.c.301.1 2 35.24 odd 6
400.2.q.a.149.1 2 560.213 even 12
400.2.q.a.349.1 2 35.17 even 12
400.2.q.b.149.1 2 560.437 even 12
400.2.q.b.349.1 2 35.3 even 12
576.2.k.a.145.1 2 84.59 odd 6
576.2.k.a.433.1 2 336.59 odd 12
784.2.m.b.197.1 2 112.53 even 12
784.2.m.b.589.1 2 7.4 even 3
784.2.x.c.165.1 4 16.5 even 4 inner
784.2.x.c.373.1 4 112.37 even 12 inner
784.2.x.c.557.1 4 1.1 even 1 trivial
784.2.x.c.765.1 4 7.2 even 3 inner
784.2.x.f.165.1 4 112.69 odd 4
784.2.x.f.373.1 4 112.5 odd 12
784.2.x.f.557.1 4 7.6 odd 2
784.2.x.f.765.1 4 7.5 odd 6
1024.2.a.b.1.1 2 224.213 odd 24
1024.2.a.b.1.2 2 224.101 odd 24
1024.2.a.e.1.1 2 224.59 even 24
1024.2.a.e.1.2 2 224.171 even 24
1024.2.b.b.513.1 2 224.115 even 24
1024.2.b.b.513.2 2 224.3 even 24
1024.2.b.e.513.1 2 224.157 odd 24
1024.2.b.e.513.2 2 224.45 odd 24
1152.2.k.a.289.1 2 168.59 odd 6
1152.2.k.a.865.1 2 336.227 odd 12
1152.2.k.b.289.1 2 168.101 even 6
1152.2.k.b.865.1 2 336.269 even 12
1600.2.l.a.401.1 2 140.59 even 6
1600.2.l.a.1201.1 2 560.59 even 12
1600.2.q.a.49.1 2 560.507 odd 12
1600.2.q.a.849.1 2 140.3 odd 12
1600.2.q.b.49.1 2 560.283 odd 12
1600.2.q.b.849.1 2 140.87 odd 12
9216.2.a.d.1.1 2 672.437 even 24
9216.2.a.d.1.2 2 672.101 even 24
9216.2.a.s.1.1 2 672.395 odd 24
9216.2.a.s.1.2 2 672.59 odd 24