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 1.3
Root \(-7.73394\) 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)
 
\(f(q)\) \(=\) \(q+6.16284 q^{5} -343.000 q^{7} -2336.31 q^{11} +1445.04 q^{13} +32024.2 q^{17} +29544.3 q^{19} -79563.9 q^{23} -78087.0 q^{25} -31968.2 q^{29} +62801.4 q^{31} -2113.85 q^{35} +114566. q^{37} +842829. q^{41} -801531. q^{43} -704927. q^{47} +117649. q^{49} -1.83417e6 q^{53} -14398.3 q^{55} +2.29700e6 q^{59} +290494. q^{61} +8905.54 q^{65} -600173. q^{67} +381707. q^{71} -3.50101e6 q^{73} +801353. q^{77} -5.03754e6 q^{79} -5.75444e6 q^{83} +197360. q^{85} +4.78032e6 q^{89} -495648. q^{91} +182077. q^{95} -1.09847e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 1372 q^{7} + 13272 q^{13} - 9184 q^{19} + 264524 q^{25} - 445536 q^{31} - 852808 q^{37} + 150224 q^{43} + 470596 q^{49} - 2627296 q^{55} - 1302840 q^{61} - 6071696 q^{67} + 3631768 q^{73} - 16898688 q^{79}+ \cdots - 36492008 q^{97}+O(q^{100}) \) 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
\(5\) 6.16284 0.0220488 0.0110244 − 0.999939i \(-0.496491\pi\)
0.0110244 + 0.999939i \(0.496491\pi\)
\(6\) 0 0
\(7\) −343.000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2336.31 −0.529244 −0.264622 − 0.964352i \(-0.585247\pi\)
−0.264622 + 0.964352i \(0.585247\pi\)
\(12\) 0 0
\(13\) 1445.04 0.182422 0.0912111 − 0.995832i \(-0.470926\pi\)
0.0912111 + 0.995832i \(0.470926\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 32024.2 1.58091 0.790455 − 0.612520i \(-0.209844\pi\)
0.790455 + 0.612520i \(0.209844\pi\)
\(18\) 0 0
\(19\) 29544.3 0.988181 0.494091 − 0.869410i \(-0.335501\pi\)
0.494091 + 0.869410i \(0.335501\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −79563.9 −1.36354 −0.681771 − 0.731566i \(-0.738790\pi\)
−0.681771 + 0.731566i \(0.738790\pi\)
\(24\) 0 0
\(25\) −78087.0 −0.999514
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −31968.2 −0.243402 −0.121701 − 0.992567i \(-0.538835\pi\)
−0.121701 + 0.992567i \(0.538835\pi\)
\(30\) 0 0
\(31\) 62801.4 0.378620 0.189310 − 0.981917i \(-0.439375\pi\)
0.189310 + 0.981917i \(0.439375\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2113.85 −0.00833367
\(36\) 0 0
\(37\) 114566. 0.371835 0.185918 − 0.982565i \(-0.440474\pi\)
0.185918 + 0.982565i \(0.440474\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 842829. 1.90984 0.954918 − 0.296869i \(-0.0959425\pi\)
0.954918 + 0.296869i \(0.0959425\pi\)
\(42\) 0 0
\(43\) −801531. −1.53738 −0.768689 − 0.639623i \(-0.779090\pi\)
−0.768689 + 0.639623i \(0.779090\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −704927. −0.990380 −0.495190 − 0.868785i \(-0.664902\pi\)
−0.495190 + 0.868785i \(0.664902\pi\)
\(48\) 0 0
\(49\) 117649. 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.83417e6 −1.69228 −0.846142 − 0.532958i \(-0.821080\pi\)
−0.846142 + 0.532958i \(0.821080\pi\)
\(54\) 0 0
\(55\) −14398.3 −0.0116692
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.29700e6 1.45606 0.728029 − 0.685547i \(-0.240437\pi\)
0.728029 + 0.685547i \(0.240437\pi\)
\(60\) 0 0
\(61\) 290494. 0.163864 0.0819319 − 0.996638i \(-0.473891\pi\)
0.0819319 + 0.996638i \(0.473891\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 8905.54 0.00402220
\(66\) 0 0
\(67\) −600173. −0.243789 −0.121895 − 0.992543i \(-0.538897\pi\)
−0.121895 + 0.992543i \(0.538897\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 381707. 0.126569 0.0632843 − 0.997996i \(-0.479843\pi\)
0.0632843 + 0.997996i \(0.479843\pi\)
\(72\) 0 0
\(73\) −3.50101e6 −1.05333 −0.526663 − 0.850074i \(-0.676557\pi\)
−0.526663 + 0.850074i \(0.676557\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 801353. 0.200035
\(78\) 0 0
\(79\) −5.03754e6 −1.14954 −0.574769 − 0.818316i \(-0.694908\pi\)
−0.574769 + 0.818316i \(0.694908\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −5.75444e6 −1.10466 −0.552332 − 0.833624i \(-0.686262\pi\)
−0.552332 + 0.833624i \(0.686262\pi\)
\(84\) 0 0
\(85\) 197360. 0.0348572
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 4.78032e6 0.718774 0.359387 − 0.933189i \(-0.382986\pi\)
0.359387 + 0.933189i \(0.382986\pi\)
\(90\) 0 0
\(91\) −495648. −0.0689491
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 182077. 0.0217882
\(96\) 0 0
\(97\) −1.09847e7 −1.22205 −0.611024 − 0.791612i \(-0.709242\pi\)
−0.611024 + 0.791612i \(0.709242\pi\)
\(98\) 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.1.3 yes 4
3.2 odd 2 inner 252.8.a.g.1.2 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.8.a.g.1.2 ✓ 4 3.2 odd 2 inner
252.8.a.g.1.3 yes 4 1.1 even 1 trivial