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Properties

Label 56.10.a.c.1.2
Level $56$
Weight $10$
Character 56.1
Self dual yes
Analytic conductor $28.842$
Analytic rank $0$
Dimension $4$
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 = [4,0,70] 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: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 5576x^{2} - 170673x - 607824 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{11}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-43.0532\) 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)
 
\(f(q)\) \(=\) \(q-125.863 q^{3} +1742.38 q^{5} +2401.00 q^{7} -3841.54 q^{9} +20811.9 q^{11} -32117.7 q^{13} -219301. q^{15} -650381. q^{17} +941018. q^{19} -302197. q^{21} +522726. q^{23} +1.08277e6 q^{25} +2.96087e6 q^{27} +4.31490e6 q^{29} +4.82414e6 q^{31} -2.61945e6 q^{33} +4.18346e6 q^{35} +3.13068e6 q^{37} +4.04243e6 q^{39} +1.56871e7 q^{41} -2.35598e7 q^{43} -6.69342e6 q^{45} +5.41852e7 q^{47} +5.76480e6 q^{49} +8.18589e7 q^{51} +4.83442e7 q^{53} +3.62623e7 q^{55} -1.18439e8 q^{57} -3.54395e7 q^{59} +1.75556e8 q^{61} -9.22354e6 q^{63} -5.59613e7 q^{65} +1.35352e7 q^{67} -6.57917e7 q^{69} +7.20247e7 q^{71} -2.35970e8 q^{73} -1.36280e8 q^{75} +4.99694e7 q^{77} +2.65731e8 q^{79} -2.97050e8 q^{81} +9.28236e6 q^{83} -1.13321e9 q^{85} -5.43085e8 q^{87} -2.37979e8 q^{89} -7.71147e7 q^{91} -6.07180e8 q^{93} +1.63961e9 q^{95} -1.51412e9 q^{97} -7.99498e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 70 q^{3} + 1022 q^{5} + 9604 q^{7} + 12176 q^{9} - 31476 q^{11} + 11466 q^{13} + 98504 q^{15} + 397012 q^{17} + 246610 q^{19} + 168070 q^{21} + 1976552 q^{23} + 1086512 q^{25} + 3139108 q^{27} + 3718940 q^{29}+ \cdots + 1279453212 q^{99}+O(q^{100}) \) 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
\(3\) −125.863 −0.897123 −0.448561 − 0.893752i \(-0.648063\pi\)
−0.448561 + 0.893752i \(0.648063\pi\)
\(4\) 0 0
\(5\) 1742.38 1.24675 0.623373 − 0.781925i \(-0.285762\pi\)
0.623373 + 0.781925i \(0.285762\pi\)
\(6\) 0 0
\(7\) 2401.00 0.377964
\(8\) 0 0
\(9\) −3841.54 −0.195170
\(10\) 0 0
\(11\) 20811.9 0.428593 0.214297 − 0.976769i \(-0.431254\pi\)
0.214297 + 0.976769i \(0.431254\pi\)
\(12\) 0 0
\(13\) −32117.7 −0.311889 −0.155944 − 0.987766i \(-0.549842\pi\)
−0.155944 + 0.987766i \(0.549842\pi\)
\(14\) 0 0
\(15\) −219301. −1.11848
\(16\) 0 0
\(17\) −650381. −1.88863 −0.944317 − 0.329036i \(-0.893276\pi\)
−0.944317 + 0.329036i \(0.893276\pi\)
\(18\) 0 0
\(19\) 941018. 1.65656 0.828279 − 0.560316i \(-0.189320\pi\)
0.828279 + 0.560316i \(0.189320\pi\)
\(20\) 0 0
\(21\) −302197. −0.339081
\(22\) 0 0
\(23\) 522726. 0.389492 0.194746 − 0.980854i \(-0.437612\pi\)
0.194746 + 0.980854i \(0.437612\pi\)
\(24\) 0 0
\(25\) 1.08277e6 0.554376
\(26\) 0 0
\(27\) 2.96087e6 1.07221
\(28\) 0 0
\(29\) 4.31490e6 1.13287 0.566434 − 0.824107i \(-0.308323\pi\)
0.566434 + 0.824107i \(0.308323\pi\)
\(30\) 0 0
\(31\) 4.82414e6 0.938192 0.469096 − 0.883147i \(-0.344580\pi\)
0.469096 + 0.883147i \(0.344580\pi\)
\(32\) 0 0
\(33\) −2.61945e6 −0.384501
\(34\) 0 0
\(35\) 4.18346e6 0.471226
\(36\) 0 0
\(37\) 3.13068e6 0.274619 0.137309 − 0.990528i \(-0.456155\pi\)
0.137309 + 0.990528i \(0.456155\pi\)
\(38\) 0 0
\(39\) 4.04243e6 0.279803
\(40\) 0 0
\(41\) 1.56871e7 0.866995 0.433497 − 0.901155i \(-0.357279\pi\)
0.433497 + 0.901155i \(0.357279\pi\)
\(42\) 0 0
\(43\) −2.35598e7 −1.05091 −0.525453 − 0.850823i \(-0.676104\pi\)
−0.525453 + 0.850823i \(0.676104\pi\)
\(44\) 0 0
\(45\) −6.69342e6 −0.243328
\(46\) 0 0
\(47\) 5.41852e7 1.61972 0.809861 − 0.586621i \(-0.199542\pi\)
0.809861 + 0.586621i \(0.199542\pi\)
\(48\) 0 0
\(49\) 5.76480e6 0.142857
\(50\) 0 0
\(51\) 8.18589e7 1.69434
\(52\) 0 0
\(53\) 4.83442e7 0.841594 0.420797 − 0.907155i \(-0.361750\pi\)
0.420797 + 0.907155i \(0.361750\pi\)
\(54\) 0 0
\(55\) 3.62623e7 0.534347
\(56\) 0 0
\(57\) −1.18439e8 −1.48614
\(58\) 0 0
\(59\) −3.54395e7 −0.380762 −0.190381 − 0.981710i \(-0.560972\pi\)
−0.190381 + 0.981710i \(0.560972\pi\)
\(60\) 0 0
\(61\) 1.75556e8 1.62342 0.811711 − 0.584060i \(-0.198537\pi\)
0.811711 + 0.584060i \(0.198537\pi\)
\(62\) 0 0
\(63\) −9.22354e6 −0.0737675
\(64\) 0 0
\(65\) −5.59613e7 −0.388846
\(66\) 0 0
\(67\) 1.35352e7 0.0820591 0.0410295 − 0.999158i \(-0.486936\pi\)
0.0410295 + 0.999158i \(0.486936\pi\)
\(68\) 0 0
\(69\) −6.57917e7 −0.349422
\(70\) 0 0
\(71\) 7.20247e7 0.336371 0.168186 − 0.985755i \(-0.446209\pi\)
0.168186 + 0.985755i \(0.446209\pi\)
\(72\) 0 0
\(73\) −2.35970e8 −0.972534 −0.486267 − 0.873810i \(-0.661642\pi\)
−0.486267 + 0.873810i \(0.661642\pi\)
\(74\) 0 0
\(75\) −1.36280e8 −0.497343
\(76\) 0 0
\(77\) 4.99694e7 0.161993
\(78\) 0 0
\(79\) 2.65731e8 0.767573 0.383786 − 0.923422i \(-0.374620\pi\)
0.383786 + 0.923422i \(0.374620\pi\)
\(80\) 0 0
\(81\) −2.97050e8 −0.766738
\(82\) 0 0
\(83\) 9.28236e6 0.0214688 0.0107344 − 0.999942i \(-0.496583\pi\)
0.0107344 + 0.999942i \(0.496583\pi\)
\(84\) 0 0
\(85\) −1.13321e9 −2.35465
\(86\) 0 0
\(87\) −5.43085e8 −1.01632
\(88\) 0 0
\(89\) −2.37979e8 −0.402053 −0.201027 − 0.979586i \(-0.564428\pi\)
−0.201027 + 0.979586i \(0.564428\pi\)
\(90\) 0 0
\(91\) −7.71147e7 −0.117883
\(92\) 0 0
\(93\) −6.07180e8 −0.841674
\(94\) 0 0
\(95\) 1.63961e9 2.06531
\(96\) 0 0
\(97\) −1.51412e9 −1.73656 −0.868278 − 0.496078i \(-0.834773\pi\)
−0.868278 + 0.496078i \(0.834773\pi\)
\(98\) 0 0
\(99\) −7.99498e7 −0.0836487
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.c.1.2 ✓ 4
4.3 odd 2 112.10.a.k.1.3 4
7.6 odd 2 392.10.a.f.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.10.a.c.1.2 ✓ 4 1.1 even 1 trivial
112.10.a.k.1.3 4 4.3 odd 2
392.10.a.f.1.3 4 7.6 odd 2