Properties

Label 56.10.a.a.1.1
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 1.1
Root \(31.1447\) 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)
 
\(f(q)\) \(=\) \(q-273.944 q^{3} -187.226 q^{5} -2401.00 q^{7} +55362.2 q^{9} +6322.34 q^{11} +70415.3 q^{13} +51289.4 q^{15} +164999. q^{17} +482060. q^{19} +657739. q^{21} -571132. q^{23} -1.91807e6 q^{25} -9.77410e6 q^{27} +2.49049e6 q^{29} -8.74543e6 q^{31} -1.73196e6 q^{33} +449529. q^{35} +1.74181e7 q^{37} -1.92898e7 q^{39} -3.27790e7 q^{41} +1.01909e7 q^{43} -1.03652e7 q^{45} +2.96277e7 q^{47} +5.76480e6 q^{49} -4.52005e7 q^{51} +5.85308e7 q^{53} -1.18370e6 q^{55} -1.32057e8 q^{57} -1.04666e8 q^{59} -2.06399e8 q^{61} -1.32925e8 q^{63} -1.31836e7 q^{65} +1.79752e8 q^{67} +1.56458e8 q^{69} -1.16985e8 q^{71} +2.81318e8 q^{73} +5.25444e8 q^{75} -1.51799e7 q^{77} -3.44616e8 q^{79} +1.58786e9 q^{81} -2.85676e8 q^{83} -3.08921e7 q^{85} -6.82254e8 q^{87} +9.14231e8 q^{89} -1.69067e8 q^{91} +2.39576e9 q^{93} -9.02541e7 q^{95} -1.47825e9 q^{97} +3.50018e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 92 q^{3} + 274 q^{5} - 7203 q^{7} + 51871 q^{9} - 45364 q^{11} - 11158 q^{13} - 95584 q^{15} - 55866 q^{17} + 488772 q^{19} + 220892 q^{21} - 253888 q^{23} - 3872619 q^{25} - 10157192 q^{27} - 765318 q^{29}+ \cdots + 168732636 q^{99}+O(q^{100}) \) 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
\(3\) −273.944 −1.95261 −0.976306 − 0.216396i \(-0.930570\pi\)
−0.976306 + 0.216396i \(0.930570\pi\)
\(4\) 0 0
\(5\) −187.226 −0.133968 −0.0669840 − 0.997754i \(-0.521338\pi\)
−0.0669840 + 0.997754i \(0.521338\pi\)
\(6\) 0 0
\(7\) −2401.00 −0.377964
\(8\) 0 0
\(9\) 55362.2 2.81269
\(10\) 0 0
\(11\) 6322.34 0.130200 0.0650999 − 0.997879i \(-0.479263\pi\)
0.0650999 + 0.997879i \(0.479263\pi\)
\(12\) 0 0
\(13\) 70415.3 0.683789 0.341895 − 0.939738i \(-0.388931\pi\)
0.341895 + 0.939738i \(0.388931\pi\)
\(14\) 0 0
\(15\) 51289.4 0.261587
\(16\) 0 0
\(17\) 164999. 0.479139 0.239569 − 0.970879i \(-0.422994\pi\)
0.239569 + 0.970879i \(0.422994\pi\)
\(18\) 0 0
\(19\) 482060. 0.848613 0.424307 − 0.905519i \(-0.360518\pi\)
0.424307 + 0.905519i \(0.360518\pi\)
\(20\) 0 0
\(21\) 657739. 0.738018
\(22\) 0 0
\(23\) −571132. −0.425560 −0.212780 − 0.977100i \(-0.568252\pi\)
−0.212780 + 0.977100i \(0.568252\pi\)
\(24\) 0 0
\(25\) −1.91807e6 −0.982053
\(26\) 0 0
\(27\) −9.77410e6 −3.53948
\(28\) 0 0
\(29\) 2.49049e6 0.653874 0.326937 − 0.945046i \(-0.393984\pi\)
0.326937 + 0.945046i \(0.393984\pi\)
\(30\) 0 0
\(31\) −8.74543e6 −1.70080 −0.850400 − 0.526136i \(-0.823640\pi\)
−0.850400 + 0.526136i \(0.823640\pi\)
\(32\) 0 0
\(33\) −1.73196e6 −0.254230
\(34\) 0 0
\(35\) 449529. 0.0506351
\(36\) 0 0
\(37\) 1.74181e7 1.52790 0.763948 − 0.645278i \(-0.223258\pi\)
0.763948 + 0.645278i \(0.223258\pi\)
\(38\) 0 0
\(39\) −1.92898e7 −1.33517
\(40\) 0 0
\(41\) −3.27790e7 −1.81163 −0.905813 − 0.423677i \(-0.860739\pi\)
−0.905813 + 0.423677i \(0.860739\pi\)
\(42\) 0 0
\(43\) 1.01909e7 0.454576 0.227288 − 0.973828i \(-0.427014\pi\)
0.227288 + 0.973828i \(0.427014\pi\)
\(44\) 0 0
\(45\) −1.03652e7 −0.376810
\(46\) 0 0
\(47\) 2.96277e7 0.885639 0.442820 − 0.896611i \(-0.353978\pi\)
0.442820 + 0.896611i \(0.353978\pi\)
\(48\) 0 0
\(49\) 5.76480e6 0.142857
\(50\) 0 0
\(51\) −4.52005e7 −0.935572
\(52\) 0 0
\(53\) 5.85308e7 1.01893 0.509463 − 0.860492i \(-0.329844\pi\)
0.509463 + 0.860492i \(0.329844\pi\)
\(54\) 0 0
\(55\) −1.18370e6 −0.0174426
\(56\) 0 0
\(57\) −1.32057e8 −1.65701
\(58\) 0 0
\(59\) −1.04666e8 −1.12453 −0.562263 − 0.826958i \(-0.690069\pi\)
−0.562263 + 0.826958i \(0.690069\pi\)
\(60\) 0 0
\(61\) −2.06399e8 −1.90864 −0.954318 − 0.298792i \(-0.903416\pi\)
−0.954318 + 0.298792i \(0.903416\pi\)
\(62\) 0 0
\(63\) −1.32925e8 −1.06310
\(64\) 0 0
\(65\) −1.31836e7 −0.0916058
\(66\) 0 0
\(67\) 1.79752e8 1.08977 0.544887 − 0.838510i \(-0.316573\pi\)
0.544887 + 0.838510i \(0.316573\pi\)
\(68\) 0 0
\(69\) 1.56458e8 0.830954
\(70\) 0 0
\(71\) −1.16985e8 −0.546345 −0.273172 − 0.961965i \(-0.588073\pi\)
−0.273172 + 0.961965i \(0.588073\pi\)
\(72\) 0 0
\(73\) 2.81318e8 1.15943 0.579715 − 0.814819i \(-0.303164\pi\)
0.579715 + 0.814819i \(0.303164\pi\)
\(74\) 0 0
\(75\) 5.25444e8 1.91757
\(76\) 0 0
\(77\) −1.51799e7 −0.0492109
\(78\) 0 0
\(79\) −3.44616e8 −0.995435 −0.497718 − 0.867339i \(-0.665828\pi\)
−0.497718 + 0.867339i \(0.665828\pi\)
\(80\) 0 0
\(81\) 1.58786e9 4.09854
\(82\) 0 0
\(83\) −2.85676e8 −0.660728 −0.330364 − 0.943854i \(-0.607172\pi\)
−0.330364 + 0.943854i \(0.607172\pi\)
\(84\) 0 0
\(85\) −3.08921e7 −0.0641892
\(86\) 0 0
\(87\) −6.82254e8 −1.27676
\(88\) 0 0
\(89\) 9.14231e8 1.54455 0.772273 − 0.635290i \(-0.219120\pi\)
0.772273 + 0.635290i \(0.219120\pi\)
\(90\) 0 0
\(91\) −1.69067e8 −0.258448
\(92\) 0 0
\(93\) 2.39576e9 3.32100
\(94\) 0 0
\(95\) −9.02541e7 −0.113687
\(96\) 0 0
\(97\) −1.47825e9 −1.69541 −0.847706 − 0.530466i \(-0.822017\pi\)
−0.847706 + 0.530466i \(0.822017\pi\)
\(98\) 0 0
\(99\) 3.50018e8 0.366212
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.1.1 ✓ 3
4.3 odd 2 112.10.a.i.1.3 3
7.6 odd 2 392.10.a.d.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.10.a.a.1.1 ✓ 3 1.1 even 1 trivial
112.10.a.i.1.3 3 4.3 odd 2
392.10.a.d.1.3 3 7.6 odd 2