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Properties

Label 252.8.a.g.1.1
Level $252$
Weight $8$
Character 252.1
Self dual yes
Analytic conductor $78.721$
Analytic rank $1$
Dimension $4$
Inner twists $2$

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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] = [252,8,Mod(1,252)] 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("252.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(252, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 252.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,-1372] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(78.7210264220\)
Analytic rank: \(1\)
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: \( x^{4} - x^{3} - 14292x^{2} + 540043x + 5027477 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-133.838\) of defining polynomial
Character \(\chi\) \(=\) 252.1

$q$-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)
 
\(f(q)\) \(=\) \(q-537.098 q^{5} -343.000 q^{7} +2419.02 q^{11} +5190.96 q^{13} +25751.6 q^{17} -34136.3 q^{19} +93525.7 q^{23} +210349. q^{25} +170556. q^{29} -285569. q^{31} +184225. q^{35} -540970. q^{37} -219356. q^{41} +876643. q^{43} -532161. q^{47} +117649. q^{49} -1.70818e6 q^{53} -1.29925e6 q^{55} -200256. q^{59} -941914. q^{61} -2.78805e6 q^{65} -2.43568e6 q^{67} +4.15568e6 q^{71} +5.31689e6 q^{73} -829724. q^{77} -3.41181e6 q^{79} +5.76711e6 q^{83} -1.38311e7 q^{85} -5.37254e6 q^{89} -1.78050e6 q^{91} +1.83346e7 q^{95} -7.26128e6 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 1372 q^{7} + 13272 q^{13} - 9184 q^{19} + 264524 q^{25} - 445536 q^{31} - 852808 q^{37} + 150224 q^{43} + 470596 q^{49} - 2627296 q^{55} - 1302840 q^{61} - 6071696 q^{67} + 3631768 q^{73} - 16898688 q^{79}+ \cdots - 36492008 q^{97}+O(q^{100}) \) 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
\(3\) 0 0
\(4\) 0 0
\(5\) −537.098 −1.92158 −0.960790 − 0.277278i \(-0.910568\pi\)
−0.960790 + 0.277278i \(0.910568\pi\)
\(6\) 0 0
\(7\) −343.000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2419.02 0.547980 0.273990 − 0.961733i \(-0.411656\pi\)
0.273990 + 0.961733i \(0.411656\pi\)
\(12\) 0 0
\(13\) 5190.96 0.655309 0.327654 − 0.944798i \(-0.393742\pi\)
0.327654 + 0.944798i \(0.393742\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 25751.6 1.27126 0.635628 − 0.771996i \(-0.280741\pi\)
0.635628 + 0.771996i \(0.280741\pi\)
\(18\) 0 0
\(19\) −34136.3 −1.14177 −0.570886 − 0.821029i \(-0.693400\pi\)
−0.570886 + 0.821029i \(0.693400\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 93525.7 1.60282 0.801408 − 0.598118i \(-0.204085\pi\)
0.801408 + 0.598118i \(0.204085\pi\)
\(24\) 0 0
\(25\) 210349. 2.69247
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 170556. 1.29859 0.649296 − 0.760536i \(-0.275064\pi\)
0.649296 + 0.760536i \(0.275064\pi\)
\(30\) 0 0
\(31\) −285569. −1.72165 −0.860827 − 0.508898i \(-0.830053\pi\)
−0.860827 + 0.508898i \(0.830053\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 184225. 0.726289
\(36\) 0 0
\(37\) −540970. −1.75577 −0.877884 − 0.478873i \(-0.841046\pi\)
−0.877884 + 0.478873i \(0.841046\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −219356. −0.497056 −0.248528 − 0.968625i \(-0.579947\pi\)
−0.248528 + 0.968625i \(0.579947\pi\)
\(42\) 0 0
\(43\) 876643. 1.68145 0.840723 − 0.541465i \(-0.182130\pi\)
0.840723 + 0.541465i \(0.182130\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −532161. −0.747653 −0.373827 − 0.927499i \(-0.621955\pi\)
−0.373827 + 0.927499i \(0.621955\pi\)
\(48\) 0 0
\(49\) 117649. 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.70818e6 −1.57604 −0.788022 − 0.615647i \(-0.788895\pi\)
−0.788022 + 0.615647i \(0.788895\pi\)
\(54\) 0 0
\(55\) −1.29925e6 −1.05299
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −200256. −0.126941 −0.0634706 − 0.997984i \(-0.520217\pi\)
−0.0634706 + 0.997984i \(0.520217\pi\)
\(60\) 0 0
\(61\) −941914. −0.531321 −0.265660 − 0.964067i \(-0.585590\pi\)
−0.265660 + 0.964067i \(0.585590\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.78805e6 −1.25923
\(66\) 0 0
\(67\) −2.43568e6 −0.989367 −0.494684 − 0.869073i \(-0.664716\pi\)
−0.494684 + 0.869073i \(0.664716\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 4.15568e6 1.37796 0.688981 − 0.724779i \(-0.258058\pi\)
0.688981 + 0.724779i \(0.258058\pi\)
\(72\) 0 0
\(73\) 5.31689e6 1.59966 0.799830 − 0.600226i \(-0.204923\pi\)
0.799830 + 0.600226i \(0.204923\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −829724. −0.207117
\(78\) 0 0
\(79\) −3.41181e6 −0.778556 −0.389278 − 0.921120i \(-0.627275\pi\)
−0.389278 + 0.921120i \(0.627275\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 5.76711e6 1.10709 0.553547 − 0.832818i \(-0.313274\pi\)
0.553547 + 0.832818i \(0.313274\pi\)
\(84\) 0 0
\(85\) −1.38311e7 −2.44282
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −5.37254e6 −0.807820 −0.403910 − 0.914799i \(-0.632349\pi\)
−0.403910 + 0.914799i \(0.632349\pi\)
\(90\) 0 0
\(91\) −1.78050e6 −0.247683
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.83346e7 2.19401
\(96\) 0 0
\(97\) −7.26128e6 −0.807815 −0.403908 − 0.914800i \(-0.632348\pi\)
−0.403908 + 0.914800i \(0.632348\pi\)
\(98\) 0 0
\(99\) 0 0
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 252.8.a.g.1.1 ✓ 4
3.2 odd 2 inner 252.8.a.g.1.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.8.a.g.1.1 ✓ 4 1.1 even 1 trivial
252.8.a.g.1.4 yes 4 3.2 odd 2 inner