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Properties

Label 225.6.b.f.199.2
Level $225$
Weight $6$
Character 225.199
Analytic conductor $36.086$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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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] = [225,6,Mod(199,225)] 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("225.199"); 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, 1])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 225.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,64,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(36.0863594579\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 199.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 225.199
Dual form 225.6.b.f.199.1

$q$-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)
 
\(f(q)\) \(=\) \(q+32.0000 q^{4} +25.0000i q^{7} -775.000i q^{13} +1024.00 q^{16} +1711.00 q^{19} +800.000i q^{28} +2723.00 q^{31} -16550.0i q^{37} +22475.0i q^{43} +16182.0 q^{49} -24800.0i q^{52} +56927.0 q^{61} +32768.0 q^{64} -73475.0i q^{67} -1450.00i q^{73} +54752.0 q^{76} +100564. q^{79} +19375.0 q^{91} -177725. i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 64 q^{4} + 2048 q^{16} + 3422 q^{19} + 5446 q^{31} + 32364 q^{49} + 113854 q^{61} + 65536 q^{64} + 109504 q^{76} + 201128 q^{79} + 38750 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) 0 0
\(4\) 32.0000 1.00000
\(5\) 0 0
\(6\) 0 0
\(7\) 25.0000i 0.192839i 0.995341 + 0.0964195i \(0.0307390\pi\)
−0.995341 + 0.0964195i \(0.969261\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) − 775.000i − 1.27187i −0.771742 − 0.635936i \(-0.780614\pi\)
0.771742 − 0.635936i \(-0.219386\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1024.00 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 1711.00 1.08734 0.543671 − 0.839299i \(-0.317034\pi\)
0.543671 + 0.839299i \(0.317034\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 800.000i 0.192839i
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 2723.00 0.508913 0.254456 − 0.967084i \(-0.418103\pi\)
0.254456 + 0.967084i \(0.418103\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 16550.0i − 1.98744i −0.111902 − 0.993719i \(-0.535694\pi\)
0.111902 − 0.993719i \(-0.464306\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 22475.0i 1.85365i 0.375489 + 0.926827i \(0.377475\pi\)
−0.375489 + 0.926827i \(0.622525\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 16182.0 0.962813
\(50\) 0 0
\(51\) 0 0
\(52\) − 24800.0i − 1.27187i
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 56927.0 1.95882 0.979408 − 0.201890i \(-0.0647084\pi\)
0.979408 + 0.201890i \(0.0647084\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 32768.0 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) − 73475.0i − 1.99964i −0.0188789 − 0.999822i \(-0.506010\pi\)
0.0188789 − 0.999822i \(-0.493990\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) − 1450.00i − 0.0318464i −0.999873 − 0.0159232i \(-0.994931\pi\)
0.999873 − 0.0159232i \(-0.00506873\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 54752.0 1.08734
\(77\) 0 0
\(78\) 0 0
\(79\) 100564. 1.81290 0.906452 − 0.422309i \(-0.138780\pi\)
0.906452 + 0.422309i \(0.138780\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 19375.0 0.245266
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 177725.i − 1.91787i −0.283626 − 0.958935i \(-0.591537\pi\)
0.283626 − 0.958935i \(-0.408463\pi\)
\(98\) 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 225.6.b.f.199.2 2
3.2 odd 2 CM 225.6.b.f.199.2 2
5.2 odd 4 225.6.a.d.1.1 ✓ 1
5.3 odd 4 225.6.a.e.1.1 yes 1
5.4 even 2 inner 225.6.b.f.199.1 2
15.2 even 4 225.6.a.d.1.1 ✓ 1
15.8 even 4 225.6.a.e.1.1 yes 1
15.14 odd 2 inner 225.6.b.f.199.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
225.6.a.d.1.1 ✓ 1 5.2 odd 4
225.6.a.d.1.1 ✓ 1 15.2 even 4
225.6.a.e.1.1 yes 1 5.3 odd 4
225.6.a.e.1.1 yes 1 15.8 even 4
225.6.b.f.199.1 2 5.4 even 2 inner
225.6.b.f.199.1 2 15.14 odd 2 inner
225.6.b.f.199.2 2 1.1 even 1 trivial
225.6.b.f.199.2 2 3.2 odd 2 CM