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Properties

Label 8.24.a.a.1.3
Level $8$
Weight $24$
Character 8.1
Self dual yes
Analytic conductor $26.816$
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] = [8,24,Mod(1,8)] 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("8.1"); S:= CuspForms(chi, 24); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 24, names="a")
 
Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 24 \)
Character orbit: \([\chi]\) \(=\) 8.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,-213948] 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: \(26.8163229876\)
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: \( x^{3} - 166408x - 10560732 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{21}\cdot 3^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-371.453\) of defining polynomial
Character \(\chi\) \(=\) 8.1

$q$-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)
 
\(f(q)\) \(=\) \(q+356598. q^{3} +8.06163e7 q^{5} -9.58726e9 q^{7} +3.30186e10 q^{9} -1.65112e12 q^{11} -3.34485e12 q^{13} +2.87476e13 q^{15} +5.54986e13 q^{17} +1.98436e14 q^{19} -3.41879e15 q^{21} +2.67336e15 q^{23} -5.42194e15 q^{25} -2.17969e16 q^{27} -1.00677e16 q^{29} -2.05413e17 q^{31} -5.88785e17 q^{33} -7.72889e17 q^{35} +1.65927e18 q^{37} -1.19276e18 q^{39} -6.38124e18 q^{41} +6.62433e18 q^{43} +2.66184e18 q^{45} -3.90892e17 q^{47} +6.45468e19 q^{49} +1.97907e19 q^{51} +5.42200e19 q^{53} -1.33107e20 q^{55} +7.07619e19 q^{57} -1.72156e20 q^{59} +1.88983e19 q^{61} -3.16558e20 q^{63} -2.69649e20 q^{65} -9.12457e20 q^{67} +9.53314e20 q^{69} -3.13315e21 q^{71} +4.37526e20 q^{73} -1.93345e21 q^{75} +1.58297e22 q^{77} +1.12495e22 q^{79} -1.08812e22 q^{81} -3.90258e21 q^{83} +4.47409e21 q^{85} -3.59011e21 q^{87} -7.96034e21 q^{89} +3.20679e22 q^{91} -7.32497e22 q^{93} +1.59972e22 q^{95} -6.63497e22 q^{97} -5.45176e22 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 213948 q^{3} + 95628618 q^{5} - 8647912920 q^{7} + 174509823951 q^{9} + 35420906796 q^{11} + 3164858452338 q^{13} - 19825526344392 q^{15} - 30233487828906 q^{17} - 382754784400236 q^{19} + 27788918984928 q^{21}+ \cdots + 19\!\cdots\!00 q^{99}+O(q^{100}) \) 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
\(3\) 356598. 1.16221 0.581104 − 0.813829i \(-0.302621\pi\)
0.581104 + 0.813829i \(0.302621\pi\)
\(4\) 0 0
\(5\) 8.06163e7 0.738359 0.369180 − 0.929358i \(-0.379639\pi\)
0.369180 + 0.929358i \(0.379639\pi\)
\(6\) 0 0
\(7\) −9.58726e9 −1.83260 −0.916299 − 0.400496i \(-0.868838\pi\)
−0.916299 + 0.400496i \(0.868838\pi\)
\(8\) 0 0
\(9\) 3.30186e10 0.350728
\(10\) 0 0
\(11\) −1.65112e12 −1.74487 −0.872433 − 0.488733i \(-0.837459\pi\)
−0.872433 + 0.488733i \(0.837459\pi\)
\(12\) 0 0
\(13\) −3.34485e12 −0.517640 −0.258820 − 0.965926i \(-0.583334\pi\)
−0.258820 + 0.965926i \(0.583334\pi\)
\(14\) 0 0
\(15\) 2.87476e13 0.858127
\(16\) 0 0
\(17\) 5.54986e13 0.392753 0.196377 − 0.980529i \(-0.437082\pi\)
0.196377 + 0.980529i \(0.437082\pi\)
\(18\) 0 0
\(19\) 1.98436e14 0.390800 0.195400 − 0.980724i \(-0.437399\pi\)
0.195400 + 0.980724i \(0.437399\pi\)
\(20\) 0 0
\(21\) −3.41879e15 −2.12986
\(22\) 0 0
\(23\) 2.67336e15 0.585043 0.292521 − 0.956259i \(-0.405506\pi\)
0.292521 + 0.956259i \(0.405506\pi\)
\(24\) 0 0
\(25\) −5.42194e15 −0.454826
\(26\) 0 0
\(27\) −2.17969e16 −0.754590
\(28\) 0 0
\(29\) −1.00677e16 −0.153233 −0.0766164 − 0.997061i \(-0.524412\pi\)
−0.0766164 + 0.997061i \(0.524412\pi\)
\(30\) 0 0
\(31\) −2.05413e17 −1.45201 −0.726003 − 0.687692i \(-0.758624\pi\)
−0.726003 + 0.687692i \(0.758624\pi\)
\(32\) 0 0
\(33\) −5.88785e17 −2.02790
\(34\) 0 0
\(35\) −7.72889e17 −1.35312
\(36\) 0 0
\(37\) 1.65927e18 1.53319 0.766595 − 0.642131i \(-0.221949\pi\)
0.766595 + 0.642131i \(0.221949\pi\)
\(38\) 0 0
\(39\) −1.19276e18 −0.601606
\(40\) 0 0
\(41\) −6.38124e18 −1.81088 −0.905442 − 0.424470i \(-0.860461\pi\)
−0.905442 + 0.424470i \(0.860461\pi\)
\(42\) 0 0
\(43\) 6.62433e18 1.08706 0.543531 − 0.839389i \(-0.317087\pi\)
0.543531 + 0.839389i \(0.317087\pi\)
\(44\) 0 0
\(45\) 2.66184e18 0.258963
\(46\) 0 0
\(47\) −3.90892e17 −0.0230638 −0.0115319 − 0.999934i \(-0.503671\pi\)
−0.0115319 + 0.999934i \(0.503671\pi\)
\(48\) 0 0
\(49\) 6.45468e19 2.35841
\(50\) 0 0
\(51\) 1.97907e19 0.456461
\(52\) 0 0
\(53\) 5.42200e19 0.803503 0.401751 − 0.915749i \(-0.368402\pi\)
0.401751 + 0.915749i \(0.368402\pi\)
\(54\) 0 0
\(55\) −1.33107e20 −1.28834
\(56\) 0 0
\(57\) 7.07619e19 0.454191
\(58\) 0 0
\(59\) −1.72156e20 −0.743230 −0.371615 − 0.928387i \(-0.621196\pi\)
−0.371615 + 0.928387i \(0.621196\pi\)
\(60\) 0 0
\(61\) 1.88983e19 0.0556069 0.0278035 − 0.999613i \(-0.491149\pi\)
0.0278035 + 0.999613i \(0.491149\pi\)
\(62\) 0 0
\(63\) −3.16558e20 −0.642742
\(64\) 0 0
\(65\) −2.69649e20 −0.382204
\(66\) 0 0
\(67\) −9.12457e20 −0.912751 −0.456375 − 0.889787i \(-0.650853\pi\)
−0.456375 + 0.889787i \(0.650853\pi\)
\(68\) 0 0
\(69\) 9.53314e20 0.679941
\(70\) 0 0
\(71\) −3.13315e21 −1.60883 −0.804414 − 0.594069i \(-0.797521\pi\)
−0.804414 + 0.594069i \(0.797521\pi\)
\(72\) 0 0
\(73\) 4.37526e20 0.163227 0.0816134 − 0.996664i \(-0.473993\pi\)
0.0816134 + 0.996664i \(0.473993\pi\)
\(74\) 0 0
\(75\) −1.93345e21 −0.528602
\(76\) 0 0
\(77\) 1.58297e22 3.19764
\(78\) 0 0
\(79\) 1.12495e22 1.69209 0.846045 − 0.533112i \(-0.178978\pi\)
0.846045 + 0.533112i \(0.178978\pi\)
\(80\) 0 0
\(81\) −1.08812e22 −1.22772
\(82\) 0 0
\(83\) −3.90258e21 −0.332624 −0.166312 − 0.986073i \(-0.553186\pi\)
−0.166312 + 0.986073i \(0.553186\pi\)
\(84\) 0 0
\(85\) 4.47409e21 0.289993
\(86\) 0 0
\(87\) −3.59011e21 −0.178088
\(88\) 0 0
\(89\) −7.96034e21 −0.304051 −0.152025 − 0.988377i \(-0.548580\pi\)
−0.152025 + 0.988377i \(0.548580\pi\)
\(90\) 0 0
\(91\) 3.20679e22 0.948626
\(92\) 0 0
\(93\) −7.32497e22 −1.68753
\(94\) 0 0
\(95\) 1.59972e22 0.288551
\(96\) 0 0
\(97\) −6.63497e22 −0.941812 −0.470906 − 0.882183i \(-0.656073\pi\)
−0.470906 + 0.882183i \(0.656073\pi\)
\(98\) 0 0
\(99\) −5.45176e22 −0.611973
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 8.24.a.a.1.3 ✓ 3
4.3 odd 2 16.24.a.e.1.1 3
8.3 odd 2 64.24.a.h.1.3 3
8.5 even 2 64.24.a.k.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.24.a.a.1.3 ✓ 3 1.1 even 1 trivial
16.24.a.e.1.1 3 4.3 odd 2
64.24.a.h.1.3 3 8.3 odd 2
64.24.a.k.1.1 3 8.5 even 2