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Properties

Label 56.6.a.a.1.1
Level $56$
Weight $6$
Character 56.1
Self dual yes
Analytic conductor $8.981$
Analytic rank $1$
Dimension $1$
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] = [56,6,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, 6); 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, 6, names="a")
 
Level: \( N \) \(=\) \( 56 = 2^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 56.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-6] 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: \(8.98149390953\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
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)
 
\(f(q)\) \(=\) \(q-6.00000 q^{3} +4.00000 q^{5} +49.0000 q^{7} -207.000 q^{9} -240.000 q^{11} -744.000 q^{13} -24.0000 q^{15} -1042.00 q^{17} -986.000 q^{19} -294.000 q^{21} +184.000 q^{23} -3109.00 q^{25} +2700.00 q^{27} -734.000 q^{29} +5140.00 q^{31} +1440.00 q^{33} +196.000 q^{35} -6054.00 q^{37} +4464.00 q^{39} +7598.00 q^{41} +13016.0 q^{43} -828.000 q^{45} +14668.0 q^{47} +2401.00 q^{49} +6252.00 q^{51} -14522.0 q^{53} -960.000 q^{55} +5916.00 q^{57} -13362.0 q^{59} +9676.00 q^{61} -10143.0 q^{63} -2976.00 q^{65} -62124.0 q^{67} -1104.00 q^{69} -2112.00 q^{71} -28910.0 q^{73} +18654.0 q^{75} -11760.0 q^{77} -101768. q^{79} +34101.0 q^{81} -23922.0 q^{83} -4168.00 q^{85} +4404.00 q^{87} +141674. q^{89} -36456.0 q^{91} -30840.0 q^{93} -3944.00 q^{95} +99982.0 q^{97} +49680.0 q^{99} +O(q^{100})\)

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
\(3\) −6.00000 −0.384900 −0.192450 − 0.981307i \(-0.561643\pi\)
−0.192450 + 0.981307i \(0.561643\pi\)
\(4\) 0 0
\(5\) 4.00000 0.0715542 0.0357771 − 0.999360i \(-0.488609\pi\)
0.0357771 + 0.999360i \(0.488609\pi\)
\(6\) 0 0
\(7\) 49.0000 0.377964
\(8\) 0 0
\(9\) −207.000 −0.851852
\(10\) 0 0
\(11\) −240.000 −0.598039 −0.299020 − 0.954247i \(-0.596660\pi\)
−0.299020 + 0.954247i \(0.596660\pi\)
\(12\) 0 0
\(13\) −744.000 −1.22100 −0.610498 − 0.792017i \(-0.709031\pi\)
−0.610498 + 0.792017i \(0.709031\pi\)
\(14\) 0 0
\(15\) −24.0000 −0.0275412
\(16\) 0 0
\(17\) −1042.00 −0.874471 −0.437236 − 0.899347i \(-0.644042\pi\)
−0.437236 + 0.899347i \(0.644042\pi\)
\(18\) 0 0
\(19\) −986.000 −0.626604 −0.313302 − 0.949654i \(-0.601435\pi\)
−0.313302 + 0.949654i \(0.601435\pi\)
\(20\) 0 0
\(21\) −294.000 −0.145479
\(22\) 0 0
\(23\) 184.000 0.0725268 0.0362634 − 0.999342i \(-0.488454\pi\)
0.0362634 + 0.999342i \(0.488454\pi\)
\(24\) 0 0
\(25\) −3109.00 −0.994880
\(26\) 0 0
\(27\) 2700.00 0.712778
\(28\) 0 0
\(29\) −734.000 −0.162069 −0.0810347 − 0.996711i \(-0.525822\pi\)
−0.0810347 + 0.996711i \(0.525822\pi\)
\(30\) 0 0
\(31\) 5140.00 0.960636 0.480318 − 0.877094i \(-0.340521\pi\)
0.480318 + 0.877094i \(0.340521\pi\)
\(32\) 0 0
\(33\) 1440.00 0.230185
\(34\) 0 0
\(35\) 196.000 0.0270449
\(36\) 0 0
\(37\) −6054.00 −0.727006 −0.363503 − 0.931593i \(-0.618419\pi\)
−0.363503 + 0.931593i \(0.618419\pi\)
\(38\) 0 0
\(39\) 4464.00 0.469962
\(40\) 0 0
\(41\) 7598.00 0.705894 0.352947 − 0.935643i \(-0.385180\pi\)
0.352947 + 0.935643i \(0.385180\pi\)
\(42\) 0 0
\(43\) 13016.0 1.07351 0.536755 − 0.843738i \(-0.319650\pi\)
0.536755 + 0.843738i \(0.319650\pi\)
\(44\) 0 0
\(45\) −828.000 −0.0609536
\(46\) 0 0
\(47\) 14668.0 0.968559 0.484280 − 0.874913i \(-0.339082\pi\)
0.484280 + 0.874913i \(0.339082\pi\)
\(48\) 0 0
\(49\) 2401.00 0.142857
\(50\) 0 0
\(51\) 6252.00 0.336584
\(52\) 0 0
\(53\) −14522.0 −0.710128 −0.355064 − 0.934842i \(-0.615541\pi\)
−0.355064 + 0.934842i \(0.615541\pi\)
\(54\) 0 0
\(55\) −960.000 −0.0427922
\(56\) 0 0
\(57\) 5916.00 0.241180
\(58\) 0 0
\(59\) −13362.0 −0.499737 −0.249868 − 0.968280i \(-0.580387\pi\)
−0.249868 + 0.968280i \(0.580387\pi\)
\(60\) 0 0
\(61\) 9676.00 0.332944 0.166472 − 0.986046i \(-0.446762\pi\)
0.166472 + 0.986046i \(0.446762\pi\)
\(62\) 0 0
\(63\) −10143.0 −0.321970
\(64\) 0 0
\(65\) −2976.00 −0.0873674
\(66\) 0 0
\(67\) −62124.0 −1.69072 −0.845361 − 0.534195i \(-0.820615\pi\)
−0.845361 + 0.534195i \(0.820615\pi\)
\(68\) 0 0
\(69\) −1104.00 −0.0279156
\(70\) 0 0
\(71\) −2112.00 −0.0497219 −0.0248610 − 0.999691i \(-0.507914\pi\)
−0.0248610 + 0.999691i \(0.507914\pi\)
\(72\) 0 0
\(73\) −28910.0 −0.634952 −0.317476 − 0.948266i \(-0.602835\pi\)
−0.317476 + 0.948266i \(0.602835\pi\)
\(74\) 0 0
\(75\) 18654.0 0.382929
\(76\) 0 0
\(77\) −11760.0 −0.226038
\(78\) 0 0
\(79\) −101768. −1.83461 −0.917304 − 0.398186i \(-0.869640\pi\)
−0.917304 + 0.398186i \(0.869640\pi\)
\(80\) 0 0
\(81\) 34101.0 0.577503
\(82\) 0 0
\(83\) −23922.0 −0.381156 −0.190578 − 0.981672i \(-0.561036\pi\)
−0.190578 + 0.981672i \(0.561036\pi\)
\(84\) 0 0
\(85\) −4168.00 −0.0625721
\(86\) 0 0
\(87\) 4404.00 0.0623805
\(88\) 0 0
\(89\) 141674. 1.89590 0.947949 − 0.318421i \(-0.103153\pi\)
0.947949 + 0.318421i \(0.103153\pi\)
\(90\) 0 0
\(91\) −36456.0 −0.461493
\(92\) 0 0
\(93\) −30840.0 −0.369749
\(94\) 0 0
\(95\) −3944.00 −0.0448361
\(96\) 0 0
\(97\) 99982.0 1.07893 0.539464 − 0.842009i \(-0.318627\pi\)
0.539464 + 0.842009i \(0.318627\pi\)
\(98\) 0 0
\(99\) 49680.0 0.509441
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 56.6.a.a.1.1 ✓ 1
3.2 odd 2 504.6.a.e.1.1 1
4.3 odd 2 112.6.a.f.1.1 1
7.2 even 3 392.6.i.d.361.1 2
7.3 odd 6 392.6.i.c.177.1 2
7.4 even 3 392.6.i.d.177.1 2
7.5 odd 6 392.6.i.c.361.1 2
7.6 odd 2 392.6.a.c.1.1 1
8.3 odd 2 448.6.a.g.1.1 1
8.5 even 2 448.6.a.j.1.1 1
12.11 even 2 1008.6.a.p.1.1 1
28.27 even 2 784.6.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.6.a.a.1.1 ✓ 1 1.1 even 1 trivial
112.6.a.f.1.1 1 4.3 odd 2
392.6.a.c.1.1 1 7.6 odd 2
392.6.i.c.177.1 2 7.3 odd 6
392.6.i.c.361.1 2 7.5 odd 6
392.6.i.d.177.1 2 7.4 even 3
392.6.i.d.361.1 2 7.2 even 3
448.6.a.g.1.1 1 8.3 odd 2
448.6.a.j.1.1 1 8.5 even 2
504.6.a.e.1.1 1 3.2 odd 2
784.6.a.e.1.1 1 28.27 even 2
1008.6.a.p.1.1 1 12.11 even 2