Properties

Label 56.10.a.a.1.3
Level $56$
Weight $10$
Character 56.1
Self dual yes
Analytic conductor $28.842$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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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] = [56,10,Mod(1,56)] 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("56.1"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(56, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 56 = 2^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 56.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,-92] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(28.8420068252\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 823x - 4578 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-5.79960\) of defining polynomial
Character \(\chi\) \(=\) 56.1

$q$-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)
 
\(f(q)\) \(=\) \(q+189.265 q^{3} -729.944 q^{5} -2401.00 q^{7} +16138.2 q^{9} -33434.1 q^{11} -131907. q^{13} -138153. q^{15} -504408. q^{17} +280047. q^{19} -454425. q^{21} +1.37786e6 q^{23} -1.42031e6 q^{25} -670906. q^{27} +2.14540e6 q^{29} +2.31562e6 q^{31} -6.32790e6 q^{33} +1.75259e6 q^{35} -2.44526e6 q^{37} -2.49653e7 q^{39} -3.40357e7 q^{41} -1.28493e7 q^{43} -1.17800e7 q^{45} -2.22094e7 q^{47} +5.76480e6 q^{49} -9.54666e7 q^{51} +4.76786e7 q^{53} +2.44050e7 q^{55} +5.30030e7 q^{57} -1.04479e7 q^{59} -5.53164e7 q^{61} -3.87478e7 q^{63} +9.62845e7 q^{65} -2.10177e8 q^{67} +2.60780e8 q^{69} +2.00438e8 q^{71} +3.73407e8 q^{73} -2.68814e8 q^{75} +8.02753e7 q^{77} -1.59503e8 q^{79} -4.44627e8 q^{81} +2.02178e8 q^{83} +3.68189e8 q^{85} +4.06049e8 q^{87} -2.91432e8 q^{89} +3.16708e8 q^{91} +4.38265e8 q^{93} -2.04418e8 q^{95} +7.97606e8 q^{97} -5.39566e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 92 q^{3} + 274 q^{5} - 7203 q^{7} + 51871 q^{9} - 45364 q^{11} - 11158 q^{13} - 95584 q^{15} - 55866 q^{17} + 488772 q^{19} + 220892 q^{21} - 253888 q^{23} - 3872619 q^{25} - 10157192 q^{27} - 765318 q^{29}+ \cdots + 168732636 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 189.265 1.34904 0.674519 − 0.738257i \(-0.264351\pi\)
0.674519 + 0.738257i \(0.264351\pi\)
\(4\) 0 0
\(5\) −729.944 −0.522305 −0.261153 − 0.965298i \(-0.584103\pi\)
−0.261153 + 0.965298i \(0.584103\pi\)
\(6\) 0 0
\(7\) −2401.00 −0.377964
\(8\) 0 0
\(9\) 16138.2 0.819905
\(10\) 0 0
\(11\) −33434.1 −0.688530 −0.344265 − 0.938873i \(-0.611872\pi\)
−0.344265 + 0.938873i \(0.611872\pi\)
\(12\) 0 0
\(13\) −131907. −1.28092 −0.640460 − 0.767991i \(-0.721256\pi\)
−0.640460 + 0.767991i \(0.721256\pi\)
\(14\) 0 0
\(15\) −138153. −0.704610
\(16\) 0 0
\(17\) −504408. −1.46474 −0.732371 − 0.680905i \(-0.761587\pi\)
−0.732371 + 0.680905i \(0.761587\pi\)
\(18\) 0 0
\(19\) 280047. 0.492992 0.246496 − 0.969144i \(-0.420721\pi\)
0.246496 + 0.969144i \(0.420721\pi\)
\(20\) 0 0
\(21\) −454425. −0.509889
\(22\) 0 0
\(23\) 1.37786e6 1.02666 0.513332 − 0.858190i \(-0.328411\pi\)
0.513332 + 0.858190i \(0.328411\pi\)
\(24\) 0 0
\(25\) −1.42031e6 −0.727197
\(26\) 0 0
\(27\) −670906. −0.242954
\(28\) 0 0
\(29\) 2.14540e6 0.563271 0.281635 − 0.959521i \(-0.409123\pi\)
0.281635 + 0.959521i \(0.409123\pi\)
\(30\) 0 0
\(31\) 2.31562e6 0.450338 0.225169 − 0.974320i \(-0.427707\pi\)
0.225169 + 0.974320i \(0.427707\pi\)
\(32\) 0 0
\(33\) −6.32790e6 −0.928853
\(34\) 0 0
\(35\) 1.75259e6 0.197413
\(36\) 0 0
\(37\) −2.44526e6 −0.214495 −0.107247 − 0.994232i \(-0.534204\pi\)
−0.107247 + 0.994232i \(0.534204\pi\)
\(38\) 0 0
\(39\) −2.49653e7 −1.72801
\(40\) 0 0
\(41\) −3.40357e7 −1.88108 −0.940541 − 0.339681i \(-0.889681\pi\)
−0.940541 + 0.339681i \(0.889681\pi\)
\(42\) 0 0
\(43\) −1.28493e7 −0.573153 −0.286576 − 0.958057i \(-0.592517\pi\)
−0.286576 + 0.958057i \(0.592517\pi\)
\(44\) 0 0
\(45\) −1.17800e7 −0.428241
\(46\) 0 0
\(47\) −2.22094e7 −0.663891 −0.331946 − 0.943299i \(-0.607705\pi\)
−0.331946 + 0.943299i \(0.607705\pi\)
\(48\) 0 0
\(49\) 5.76480e6 0.142857
\(50\) 0 0
\(51\) −9.54666e7 −1.97599
\(52\) 0 0
\(53\) 4.76786e7 0.830007 0.415003 − 0.909820i \(-0.363780\pi\)
0.415003 + 0.909820i \(0.363780\pi\)
\(54\) 0 0
\(55\) 2.44050e7 0.359623
\(56\) 0 0
\(57\) 5.30030e7 0.665065
\(58\) 0 0
\(59\) −1.04479e7 −0.112252 −0.0561259 − 0.998424i \(-0.517875\pi\)
−0.0561259 + 0.998424i \(0.517875\pi\)
\(60\) 0 0
\(61\) −5.53164e7 −0.511529 −0.255764 − 0.966739i \(-0.582327\pi\)
−0.255764 + 0.966739i \(0.582327\pi\)
\(62\) 0 0
\(63\) −3.87478e7 −0.309895
\(64\) 0 0
\(65\) 9.62845e7 0.669031
\(66\) 0 0
\(67\) −2.10177e8 −1.27423 −0.637115 − 0.770769i \(-0.719872\pi\)
−0.637115 + 0.770769i \(0.719872\pi\)
\(68\) 0 0
\(69\) 2.60780e8 1.38501
\(70\) 0 0
\(71\) 2.00438e8 0.936089 0.468045 − 0.883705i \(-0.344959\pi\)
0.468045 + 0.883705i \(0.344959\pi\)
\(72\) 0 0
\(73\) 3.73407e8 1.53897 0.769485 − 0.638665i \(-0.220513\pi\)
0.769485 + 0.638665i \(0.220513\pi\)
\(74\) 0 0
\(75\) −2.68814e8 −0.981017
\(76\) 0 0
\(77\) 8.02753e7 0.260240
\(78\) 0 0
\(79\) −1.59503e8 −0.460732 −0.230366 − 0.973104i \(-0.573992\pi\)
−0.230366 + 0.973104i \(0.573992\pi\)
\(80\) 0 0
\(81\) −4.44627e8 −1.14766
\(82\) 0 0
\(83\) 2.02178e8 0.467608 0.233804 − 0.972284i \(-0.424883\pi\)
0.233804 + 0.972284i \(0.424883\pi\)
\(84\) 0 0
\(85\) 3.68189e8 0.765043
\(86\) 0 0
\(87\) 4.06049e8 0.759874
\(88\) 0 0
\(89\) −2.91432e8 −0.492360 −0.246180 − 0.969224i \(-0.579175\pi\)
−0.246180 + 0.969224i \(0.579175\pi\)
\(90\) 0 0
\(91\) 3.16708e8 0.484142
\(92\) 0 0
\(93\) 4.38265e8 0.607524
\(94\) 0 0
\(95\) −2.04418e8 −0.257492
\(96\) 0 0
\(97\) 7.97606e8 0.914778 0.457389 − 0.889267i \(-0.348785\pi\)
0.457389 + 0.889267i \(0.348785\pi\)
\(98\) 0 0
\(99\) −5.39566e8 −0.564529
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 56.10.a.a.1.3 ✓ 3
4.3 odd 2 112.10.a.i.1.1 3
7.6 odd 2 392.10.a.d.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.10.a.a.1.3 ✓ 3 1.1 even 1 trivial
112.10.a.i.1.1 3 4.3 odd 2
392.10.a.d.1.1 3 7.6 odd 2