Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9,26,Mod(1,9)] 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("9.1"); S:= CuspForms(chi, 26); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 26, names="a")
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(35.6397101957\)
Analytic rank: \(0\)
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: \( x^{3} - x^{2} - 783420x + 148321440 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{6}\cdot 3^{5}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 3)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-967.357\) of defining polynomial
Character \(\chi\) \(=\) 9.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1864.10 q^{2} -3.00796e7 q^{4} -6.53169e8 q^{5} -4.71716e10 q^{7} -1.18620e11 q^{8} -1.21757e12 q^{10} -1.60651e13 q^{11} +1.23331e14 q^{13} -8.79325e13 q^{14} +7.88183e14 q^{16} -1.37928e15 q^{17} +2.03673e14 q^{19} +1.96471e16 q^{20} -2.99469e16 q^{22} -1.96123e14 q^{23} +1.28607e17 q^{25} +2.29901e17 q^{26} +1.41890e18 q^{28} -1.83596e18 q^{29} -4.66849e18 q^{31} +5.44948e18 q^{32} -2.57111e18 q^{34} +3.08110e19 q^{35} +4.09928e19 q^{37} +3.79666e17 q^{38} +7.74790e19 q^{40} +5.64898e19 q^{41} +1.41068e20 q^{43} +4.83231e20 q^{44} -3.65593e17 q^{46} +1.37300e21 q^{47} +8.84088e20 q^{49} +2.39736e20 q^{50} -3.70974e21 q^{52} -4.94248e21 q^{53} +1.04932e22 q^{55} +5.59550e21 q^{56} -3.42242e21 q^{58} -3.33610e21 q^{59} -2.98992e22 q^{61} -8.70253e21 q^{62} -1.62887e22 q^{64} -8.05561e22 q^{65} +3.37278e22 q^{67} +4.14881e22 q^{68} +5.74348e22 q^{70} -1.69417e23 q^{71} -2.16683e23 q^{73} +7.64147e22 q^{74} -6.12638e21 q^{76} +7.57815e23 q^{77} +4.45944e23 q^{79} -5.14817e23 q^{80} +1.05303e23 q^{82} +1.39666e24 q^{83} +9.00902e23 q^{85} +2.62965e23 q^{86} +1.90564e24 q^{88} -1.32461e24 q^{89} -5.81772e24 q^{91} +5.89930e21 q^{92} +2.55940e24 q^{94} -1.33033e23 q^{95} -5.44255e24 q^{97} +1.64803e24 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3678 q^{2} + 90747588 q^{4} + 163152750 q^{5} - 9622572744 q^{7} + 268012830648 q^{8} + 8363188874700 q^{10} + 5946998130780 q^{11} + 248137774407690 q^{13} - 754361641264848 q^{14} + 23\!\cdots\!16 q^{16}+ \cdots - 24\!\cdots\!50 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1864.10 0.321806 0.160903 − 0.986970i \(-0.448559\pi\)
0.160903 + 0.986970i \(0.448559\pi\)
\(3\) 0 0
\(4\) −3.00796e7 −0.896441
\(5\) −6.53169e8 −1.19647 −0.598234 − 0.801322i \(-0.704130\pi\)
−0.598234 + 0.801322i \(0.704130\pi\)
\(6\) 0 0
\(7\) −4.71716e10 −1.28812 −0.644058 − 0.764977i \(-0.722750\pi\)
−0.644058 + 0.764977i \(0.722750\pi\)
\(8\) −1.18620e11 −0.610286
\(9\) 0 0
\(10\) −1.21757e12 −0.385030
\(11\) −1.60651e13 −1.54339 −0.771693 − 0.635995i \(-0.780590\pi\)
−0.771693 + 0.635995i \(0.780590\pi\)
\(12\) 0 0
\(13\) 1.23331e14 1.46818 0.734092 − 0.679050i \(-0.237608\pi\)
0.734092 + 0.679050i \(0.237608\pi\)
\(14\) −8.79325e13 −0.414523
\(15\) 0 0
\(16\) 7.88183e14 0.700047
\(17\) −1.37928e15 −0.574170 −0.287085 − 0.957905i \(-0.592686\pi\)
−0.287085 + 0.957905i \(0.592686\pi\)
\(18\) 0 0
\(19\) 2.03673e14 0.0211112 0.0105556 − 0.999944i \(-0.496640\pi\)
0.0105556 + 0.999944i \(0.496640\pi\)
\(20\) 1.96471e16 1.07256
\(21\) 0 0
\(22\) −2.99469e16 −0.496671
\(23\) −1.96123e14 −0.00186608 −0.000933042 − 1.00000i \(-0.500297\pi\)
−0.000933042 1.00000i \(0.500297\pi\)
\(24\) 0 0
\(25\) 1.28607e17 0.431534
\(26\) 2.29901e17 0.472471
\(27\) 0 0
\(28\) 1.41890e18 1.15472
\(29\) −1.83596e18 −0.963583 −0.481792 − 0.876286i \(-0.660014\pi\)
−0.481792 + 0.876286i \(0.660014\pi\)
\(30\) 0 0
\(31\) −4.66849e18 −1.06452 −0.532262 − 0.846580i \(-0.678658\pi\)
−0.532262 + 0.846580i \(0.678658\pi\)
\(32\) 5.44948e18 0.835565
\(33\) 0 0
\(34\) −2.57111e18 −0.184771
\(35\) 3.08110e19 1.54119
\(36\) 0 0
\(37\) 4.09928e19 1.02373 0.511866 − 0.859066i \(-0.328955\pi\)
0.511866 + 0.859066i \(0.328955\pi\)
\(38\) 3.79666e17 0.00679370
\(39\) 0 0
\(40\) 7.74790e19 0.730187
\(41\) 5.64898e19 0.390996 0.195498 − 0.980704i \(-0.437368\pi\)
0.195498 + 0.980704i \(0.437368\pi\)
\(42\) 0 0
\(43\) 1.41068e20 0.538361 0.269180 − 0.963090i \(-0.413247\pi\)
0.269180 + 0.963090i \(0.413247\pi\)
\(44\) 4.83231e20 1.38355
\(45\) 0 0
\(46\) −3.65593e17 −0.000600517 0
\(47\) 1.37300e21 1.72364 0.861820 − 0.507215i \(-0.169325\pi\)
0.861820 + 0.507215i \(0.169325\pi\)
\(48\) 0 0
\(49\) 8.84088e20 0.659241
\(50\) 2.39736e20 0.138870
\(51\) 0 0
\(52\) −3.70974e21 −1.31614
\(53\) −4.94248e21 −1.38196 −0.690982 − 0.722872i \(-0.742822\pi\)
−0.690982 + 0.722872i \(0.742822\pi\)
\(54\) 0 0
\(55\) 1.04932e22 1.84661
\(56\) 5.59550e21 0.786119
\(57\) 0 0
\(58\) −3.42242e21 −0.310087
\(59\) −3.33610e21 −0.244112 −0.122056 − 0.992523i \(-0.538949\pi\)
−0.122056 + 0.992523i \(0.538949\pi\)
\(60\) 0 0
\(61\) −2.98992e22 −1.44224 −0.721119 − 0.692811i \(-0.756372\pi\)
−0.721119 + 0.692811i \(0.756372\pi\)
\(62\) −8.70253e21 −0.342570
\(63\) 0 0
\(64\) −1.62887e22 −0.431157
\(65\) −8.05561e22 −1.75663
\(66\) 0 0
\(67\) 3.37278e22 0.503563 0.251781 − 0.967784i \(-0.418984\pi\)
0.251781 + 0.967784i \(0.418984\pi\)
\(68\) 4.14881e22 0.514709
\(69\) 0 0
\(70\) 5.74348e22 0.495963
\(71\) −1.69417e23 −1.22526 −0.612629 − 0.790370i \(-0.709888\pi\)
−0.612629 + 0.790370i \(0.709888\pi\)
\(72\) 0 0
\(73\) −2.16683e23 −1.10736 −0.553680 − 0.832730i \(-0.686777\pi\)
−0.553680 + 0.832730i \(0.686777\pi\)
\(74\) 7.64147e22 0.329443
\(75\) 0 0
\(76\) −6.12638e21 −0.0189249
\(77\) 7.57815e23 1.98806
\(78\) 0 0
\(79\) 4.45944e23 0.849066 0.424533 − 0.905413i \(-0.360438\pi\)
0.424533 + 0.905413i \(0.360438\pi\)
\(80\) −5.14817e23 −0.837583
\(81\) 0 0
\(82\) 1.05303e23 0.125825
\(83\) 1.39666e24 1.43422 0.717109 − 0.696961i \(-0.245465\pi\)
0.717109 + 0.696961i \(0.245465\pi\)
\(84\) 0 0
\(85\) 9.00902e23 0.686975
\(86\) 2.62965e23 0.173248
\(87\) 0 0
\(88\) 1.90564e24 0.941907
\(89\) −1.32461e24 −0.568476 −0.284238 − 0.958754i \(-0.591741\pi\)
−0.284238 + 0.958754i \(0.591741\pi\)
\(90\) 0 0
\(91\) −5.81772e24 −1.89119
\(92\) 5.89930e21 0.00167283
\(93\) 0 0
\(94\) 2.55940e24 0.554678
\(95\) −1.33033e23 −0.0252588
\(96\) 0 0
\(97\) −5.44255e24 −0.796445 −0.398222 − 0.917289i \(-0.630373\pi\)
−0.398222 + 0.917289i \(0.630373\pi\)
\(98\) 1.64803e24 0.212148
\(99\) 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 9.26.a.c.1.2 3
3.2 odd 2 3.26.a.b.1.2 ✓ 3
12.11 even 2 48.26.a.i.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.26.a.b.1.2 ✓ 3 3.2 odd 2
9.26.a.c.1.2 3 1.1 even 1 trivial
48.26.a.i.1.3 3 12.11 even 2