Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [225,6,Mod(1,225)] 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("225.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 225.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,24,0,0,0] 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: \(36.0863594579\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{11}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 5)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.31662\) of defining polynomial
Character \(\chi\) \(=\) 225.1

$q$-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)
 
\(f(q)\) \(=\) \(q+6.63325 q^{2} +12.0000 q^{4} +59.6992 q^{7} -132.665 q^{8} -252.000 q^{11} -119.398 q^{13} +396.000 q^{14} -1264.00 q^{16} -689.858 q^{17} +220.000 q^{19} -1671.58 q^{22} -2434.40 q^{23} -792.000 q^{26} +716.391 q^{28} -6930.00 q^{29} +6752.00 q^{31} -4139.15 q^{32} -4576.00 q^{34} -13969.6 q^{37} +1459.31 q^{38} +198.000 q^{41} -417.895 q^{43} -3024.00 q^{44} -16148.0 q^{46} -10540.2 q^{47} -13243.0 q^{49} -1432.78 q^{52} +5823.99 q^{53} -7920.00 q^{56} -45968.4 q^{58} -24660.0 q^{59} -5698.00 q^{61} +44787.7 q^{62} +12992.0 q^{64} +43640.1 q^{67} -8278.30 q^{68} -53352.0 q^{71} +70922.7 q^{73} -92664.0 q^{74} +2640.00 q^{76} -15044.2 q^{77} -51920.0 q^{79} +1313.38 q^{82} +61841.8 q^{83} -2772.00 q^{86} +33431.6 q^{88} -9990.00 q^{89} -7128.00 q^{91} -29212.8 q^{92} -69916.0 q^{94} +101250. q^{97} -87844.1 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 24 q^{4} - 504 q^{11} + 792 q^{14} - 2528 q^{16} + 440 q^{19} - 1584 q^{26} - 13860 q^{29} + 13504 q^{31} - 9152 q^{34} + 396 q^{41} - 6048 q^{44} - 32296 q^{46} - 26486 q^{49} - 15840 q^{56} - 49320 q^{59}+ \cdots - 139832 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 6.63325 1.17260 0.586302 − 0.810093i \(-0.300583\pi\)
0.586302 + 0.810093i \(0.300583\pi\)
\(3\) 0 0
\(4\) 12.0000 0.375000
\(5\) 0 0
\(6\) 0 0
\(7\) 59.6992 0.460494 0.230247 − 0.973132i \(-0.426047\pi\)
0.230247 + 0.973132i \(0.426047\pi\)
\(8\) −132.665 −0.732877
\(9\) 0 0
\(10\) 0 0
\(11\) −252.000 −0.627941 −0.313970 − 0.949433i \(-0.601659\pi\)
−0.313970 + 0.949433i \(0.601659\pi\)
\(12\) 0 0
\(13\) −119.398 −0.195948 −0.0979739 − 0.995189i \(-0.531236\pi\)
−0.0979739 + 0.995189i \(0.531236\pi\)
\(14\) 396.000 0.539977
\(15\) 0 0
\(16\) −1264.00 −1.23438
\(17\) −689.858 −0.578945 −0.289473 − 0.957186i \(-0.593480\pi\)
−0.289473 + 0.957186i \(0.593480\pi\)
\(18\) 0 0
\(19\) 220.000 0.139810 0.0699051 − 0.997554i \(-0.477730\pi\)
0.0699051 + 0.997554i \(0.477730\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −1671.58 −0.736326
\(23\) −2434.40 −0.959561 −0.479781 − 0.877388i \(-0.659284\pi\)
−0.479781 + 0.877388i \(0.659284\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −792.000 −0.229769
\(27\) 0 0
\(28\) 716.391 0.172685
\(29\) −6930.00 −1.53016 −0.765082 − 0.643932i \(-0.777302\pi\)
−0.765082 + 0.643932i \(0.777302\pi\)
\(30\) 0 0
\(31\) 6752.00 1.26191 0.630955 − 0.775820i \(-0.282663\pi\)
0.630955 + 0.775820i \(0.282663\pi\)
\(32\) −4139.15 −0.714556
\(33\) 0 0
\(34\) −4576.00 −0.678873
\(35\) 0 0
\(36\) 0 0
\(37\) −13969.6 −1.67757 −0.838785 − 0.544464i \(-0.816733\pi\)
−0.838785 + 0.544464i \(0.816733\pi\)
\(38\) 1459.31 0.163942
\(39\) 0 0
\(40\) 0 0
\(41\) 198.000 0.0183952 0.00919762 − 0.999958i \(-0.497072\pi\)
0.00919762 + 0.999958i \(0.497072\pi\)
\(42\) 0 0
\(43\) −417.895 −0.0344664 −0.0172332 − 0.999851i \(-0.505486\pi\)
−0.0172332 + 0.999851i \(0.505486\pi\)
\(44\) −3024.00 −0.235478
\(45\) 0 0
\(46\) −16148.0 −1.12519
\(47\) −10540.2 −0.695994 −0.347997 − 0.937496i \(-0.613138\pi\)
−0.347997 + 0.937496i \(0.613138\pi\)
\(48\) 0 0
\(49\) −13243.0 −0.787945
\(50\) 0 0
\(51\) 0 0
\(52\) −1432.78 −0.0734804
\(53\) 5823.99 0.284794 0.142397 − 0.989810i \(-0.454519\pi\)
0.142397 + 0.989810i \(0.454519\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −7920.00 −0.337485
\(57\) 0 0
\(58\) −45968.4 −1.79428
\(59\) −24660.0 −0.922281 −0.461140 − 0.887327i \(-0.652560\pi\)
−0.461140 + 0.887327i \(0.652560\pi\)
\(60\) 0 0
\(61\) −5698.00 −0.196064 −0.0980320 − 0.995183i \(-0.531255\pi\)
−0.0980320 + 0.995183i \(0.531255\pi\)
\(62\) 44787.7 1.47972
\(63\) 0 0
\(64\) 12992.0 0.396484
\(65\) 0 0
\(66\) 0 0
\(67\) 43640.1 1.18768 0.593840 − 0.804583i \(-0.297611\pi\)
0.593840 + 0.804583i \(0.297611\pi\)
\(68\) −8278.30 −0.217104
\(69\) 0 0
\(70\) 0 0
\(71\) −53352.0 −1.25604 −0.628022 − 0.778196i \(-0.716135\pi\)
−0.628022 + 0.778196i \(0.716135\pi\)
\(72\) 0 0
\(73\) 70922.7 1.55768 0.778840 − 0.627223i \(-0.215808\pi\)
0.778840 + 0.627223i \(0.215808\pi\)
\(74\) −92664.0 −1.96712
\(75\) 0 0
\(76\) 2640.00 0.0524288
\(77\) −15044.2 −0.289163
\(78\) 0 0
\(79\) −51920.0 −0.935981 −0.467990 − 0.883734i \(-0.655022\pi\)
−0.467990 + 0.883734i \(0.655022\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 1313.38 0.0215703
\(83\) 61841.8 0.985342 0.492671 − 0.870216i \(-0.336021\pi\)
0.492671 + 0.870216i \(0.336021\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −2772.00 −0.0404154
\(87\) 0 0
\(88\) 33431.6 0.460204
\(89\) −9990.00 −0.133687 −0.0668437 − 0.997763i \(-0.521293\pi\)
−0.0668437 + 0.997763i \(0.521293\pi\)
\(90\) 0 0
\(91\) −7128.00 −0.0902328
\(92\) −29212.8 −0.359836
\(93\) 0 0
\(94\) −69916.0 −0.816125
\(95\) 0 0
\(96\) 0 0
\(97\) 101250. 1.09261 0.546305 − 0.837586i \(-0.316034\pi\)
0.546305 + 0.837586i \(0.316034\pi\)
\(98\) −87844.1 −0.923948
\(99\) 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 225.6.a.n.1.2 2
3.2 odd 2 25.6.a.c.1.1 2
5.2 odd 4 45.6.b.b.19.2 2
5.3 odd 4 45.6.b.b.19.1 2
5.4 even 2 inner 225.6.a.n.1.1 2
12.11 even 2 400.6.a.t.1.2 2
15.2 even 4 5.6.b.a.4.1 ✓ 2
15.8 even 4 5.6.b.a.4.2 yes 2
15.14 odd 2 25.6.a.c.1.2 2
20.3 even 4 720.6.f.f.289.1 2
20.7 even 4 720.6.f.f.289.2 2
60.23 odd 4 80.6.c.a.49.2 2
60.47 odd 4 80.6.c.a.49.1 2
60.59 even 2 400.6.a.t.1.1 2
105.62 odd 4 245.6.b.a.99.1 2
105.83 odd 4 245.6.b.a.99.2 2
120.53 even 4 320.6.c.f.129.2 2
120.77 even 4 320.6.c.f.129.1 2
120.83 odd 4 320.6.c.g.129.1 2
120.107 odd 4 320.6.c.g.129.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5.6.b.a.4.1 ✓ 2 15.2 even 4
5.6.b.a.4.2 yes 2 15.8 even 4
25.6.a.c.1.1 2 3.2 odd 2
25.6.a.c.1.2 2 15.14 odd 2
45.6.b.b.19.1 2 5.3 odd 4
45.6.b.b.19.2 2 5.2 odd 4
80.6.c.a.49.1 2 60.47 odd 4
80.6.c.a.49.2 2 60.23 odd 4
225.6.a.n.1.1 2 5.4 even 2 inner
225.6.a.n.1.2 2 1.1 even 1 trivial
245.6.b.a.99.1 2 105.62 odd 4
245.6.b.a.99.2 2 105.83 odd 4
320.6.c.f.129.1 2 120.77 even 4
320.6.c.f.129.2 2 120.53 even 4
320.6.c.g.129.1 2 120.83 odd 4
320.6.c.g.129.2 2 120.107 odd 4
400.6.a.t.1.1 2 60.59 even 2
400.6.a.t.1.2 2 12.11 even 2
720.6.f.f.289.1 2 20.3 even 4
720.6.f.f.289.2 2 20.7 even 4