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Properties

Label 475.2.j.b
Level $475$
Weight $2$
Character orbit 475.j
Analytic conductor $3.793$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [475,2,Mod(49,475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(475, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("475.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.j (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.50712647503417344.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 13x^{10} + 119x^{8} - 552x^{6} + 1863x^{4} - 2450x^{2} + 2401 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{11} q^{3} + ( - \beta_{6} - \beta_{4} + 3 \beta_{3} + \cdots + 2) q^{4}+ \cdots + (\beta_{5} + \beta_{3} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{11} q^{3} + ( - \beta_{6} - \beta_{4} + 3 \beta_{3} + \cdots + 2) q^{4}+ \cdots + (3 \beta_{6} - 4 \beta_{5} + 3 \beta_{4} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 14 q^{4} + 12 q^{6} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 14 q^{4} + 12 q^{6} + 8 q^{9} - 20 q^{11} + 14 q^{14} - 6 q^{16} + 24 q^{21} - 42 q^{24} - 20 q^{26} - 4 q^{29} - 4 q^{31} - 50 q^{34} - 6 q^{36} + 20 q^{39} + 4 q^{41} - 36 q^{44} - 96 q^{46} + 28 q^{49} + 12 q^{51} + 20 q^{54} + 60 q^{56} + 12 q^{59} + 18 q^{61} + 32 q^{64} - 58 q^{66} + 20 q^{69} + 58 q^{71} - 14 q^{74} - 38 q^{76} - 48 q^{79} + 42 q^{81} + 112 q^{84} + 64 q^{86} - 28 q^{89} + 20 q^{91} - 84 q^{94} + 68 q^{96} - 26 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 13x^{10} + 119x^{8} - 552x^{6} + 1863x^{4} - 2450x^{2} + 2401 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 736\nu^{10} - 31772\nu^{8} + 290836\nu^{6} - 1755683\nu^{4} + 4553172\nu^{2} - 5987800 ) / 1328929 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 850\nu^{10} - 9867\nu^{8} + 90321\nu^{6} - 370073\nu^{4} + 1414017\nu^{2} - 1859550 ) / 1328929 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 169\nu^{10} - 1547\nu^{8} + 14161\nu^{6} - 24219\nu^{4} + 31850\nu^{2} + 467838 ) / 189847 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -200\nu^{10} + 3917\nu^{8} - 21252\nu^{6} + 87076\nu^{4} + 37412\nu^{2} + 124852 ) / 189847 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 9\nu^{10} - 26\nu^{8} + 238\nu^{6} + 289\nu^{4} + 3726\nu^{2} - 4900 ) / 5131 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 52\nu^{11} - 476\nu^{9} + 2271\nu^{7} - 7452\nu^{5} + 9800\nu^{3} - 164812\nu ) / 189847 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -100\nu^{11} + 859\nu^{9} - 10626\nu^{7} + 43538\nu^{5} - 200461\nu^{3} + 62426\nu ) / 251419 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -850\nu^{11} + 9867\nu^{9} - 90321\nu^{7} + 370073\nu^{5} - 1414017\nu^{3} + 1859550\nu ) / 1328929 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 8973\nu^{11} - 159328\nu^{9} + 1458464\nu^{7} - 7813716\nu^{5} + 22832928\nu^{3} - 30027200\nu ) / 9302503 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 273\nu^{11} - 2499\nu^{9} + 18703\nu^{7} - 39123\nu^{5} + 51450\nu^{3} + 328061\nu ) / 189847 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{6} - \beta_{4} + 5\beta_{3} - \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} - \beta_{10} - 5\beta_{9} + 2\beta_{8} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{6} + 28\beta_{3} - 7\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -6\beta_{10} - 29\beta_{9} + 19\beta_{8} - 19\beta_{7} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -13\beta_{6} + 13\beta_{5} + 41\beta_{4} - 122 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 28\beta_{11} - 147\beta_{7} - 176\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 232\beta_{6} + 119\beta_{5} + 232\beta_{4} - 964\beta_{3} + 232\beta_{2} - 732 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 113\beta_{11} + 113\beta_{10} + 1083\beta_{9} - 1059\beta_{8} - 1083\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2255\beta_{6} - 5754\beta_{3} + 1309\beta_{2} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 363\beta_{10} + 6700\beta_{9} - 7348\beta_{8} + 7348\beta_{7} \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/475\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(-1\) \(-1 - \beta_{3}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−2.17114 − 1.25351i
−1.97899 − 1.14257i
−1.05818 − 0.610938i
1.05818 + 0.610938i
1.97899 + 1.14257i
2.17114 + 1.25351i
−2.17114 + 1.25351i
−1.97899 + 1.14257i
−1.05818 + 0.610938i
1.05818 − 0.610938i
1.97899 − 1.14257i
2.17114 − 1.25351i
−2.17114 − 1.25351i −1.05818 − 0.610938i 2.14257 + 3.71104i 0 1.53163 + 2.65287i 0.221876i − 5.72889i −0.753509 − 1.30512i 0
49.2 −1.97899 − 1.14257i −2.17114 − 1.25351i 1.61094 + 2.79023i 0 2.86445 + 4.96137i 3.50702i − 2.79216i 1.64257 + 2.84502i 0
49.3 −1.05818 − 0.610938i 1.97899 + 1.14257i −0.253509 − 0.439091i 0 −1.39608 − 2.41808i − 1.28514i 3.06327i 1.11094 + 1.92420i 0
49.4 1.05818 + 0.610938i −1.97899 − 1.14257i −0.253509 − 0.439091i 0 −1.39608 − 2.41808i 1.28514i − 3.06327i 1.11094 + 1.92420i 0
49.5 1.97899 + 1.14257i 2.17114 + 1.25351i 1.61094 + 2.79023i 0 2.86445 + 4.96137i − 3.50702i 2.79216i 1.64257 + 2.84502i 0
49.6 2.17114 + 1.25351i 1.05818 + 0.610938i 2.14257 + 3.71104i 0 1.53163 + 2.65287i − 0.221876i 5.72889i −0.753509 − 1.30512i 0
349.1 −2.17114 + 1.25351i −1.05818 + 0.610938i 2.14257 − 3.71104i 0 1.53163 − 2.65287i − 0.221876i 5.72889i −0.753509 + 1.30512i 0
349.2 −1.97899 + 1.14257i −2.17114 + 1.25351i 1.61094 − 2.79023i 0 2.86445 − 4.96137i − 3.50702i 2.79216i 1.64257 − 2.84502i 0
349.3 −1.05818 + 0.610938i 1.97899 − 1.14257i −0.253509 + 0.439091i 0 −1.39608 + 2.41808i 1.28514i − 3.06327i 1.11094 − 1.92420i 0
349.4 1.05818 − 0.610938i −1.97899 + 1.14257i −0.253509 + 0.439091i 0 −1.39608 + 2.41808i − 1.28514i 3.06327i 1.11094 − 1.92420i 0
349.5 1.97899 − 1.14257i 2.17114 − 1.25351i 1.61094 − 2.79023i 0 2.86445 − 4.96137i 3.50702i − 2.79216i 1.64257 − 2.84502i 0
349.6 2.17114 − 1.25351i 1.05818 − 0.610938i 2.14257 − 3.71104i 0 1.53163 − 2.65287i 0.221876i − 5.72889i −0.753509 + 1.30512i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
19.c even 3 1 inner
95.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.j.b 12
5.b even 2 1 inner 475.2.j.b 12
5.c odd 4 1 95.2.e.b ✓ 6
5.c odd 4 1 475.2.e.d 6
15.e even 4 1 855.2.k.g 6
19.c even 3 1 inner 475.2.j.b 12
20.e even 4 1 1520.2.q.j 6
95.i even 6 1 inner 475.2.j.b 12
95.l even 12 1 1805.2.a.g 3
95.l even 12 1 9025.2.a.ba 3
95.m odd 12 1 95.2.e.b ✓ 6
95.m odd 12 1 475.2.e.d 6
95.m odd 12 1 1805.2.a.h 3
95.m odd 12 1 9025.2.a.z 3
285.v even 12 1 855.2.k.g 6
380.v even 12 1 1520.2.q.j 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.e.b ✓ 6 5.c odd 4 1
95.2.e.b ✓ 6 95.m odd 12 1
475.2.e.d 6 5.c odd 4 1
475.2.e.d 6 95.m odd 12 1
475.2.j.b 12 1.a even 1 1 trivial
475.2.j.b 12 5.b even 2 1 inner
475.2.j.b 12 19.c even 3 1 inner
475.2.j.b 12 95.i even 6 1 inner
855.2.k.g 6 15.e even 4 1
855.2.k.g 6 285.v even 12 1
1520.2.q.j 6 20.e even 4 1
1520.2.q.j 6 380.v even 12 1
1805.2.a.g 3 95.l even 12 1
1805.2.a.h 3 95.m odd 12 1
9025.2.a.z 3 95.m odd 12 1
9025.2.a.ba 3 95.l even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 13T_{2}^{10} + 119T_{2}^{8} - 552T_{2}^{6} + 1863T_{2}^{4} - 2450T_{2}^{2} + 2401 \) acting on \(S_{2}^{\mathrm{new}}(475, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 13 T^{10} + \cdots + 2401 \) Copy content Toggle raw display
$3$ \( T^{12} - 13 T^{10} + \cdots + 2401 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{6} + 14 T^{4} + 21 T^{2} + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T^{3} + 5 T^{2} + 2 T - 1)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} - 25 T^{2} + 625)^{3} \) Copy content Toggle raw display
$17$ \( T^{12} - 89 T^{10} + \cdots + 2401 \) Copy content Toggle raw display
$19$ \( (T^{6} - 133 T^{3} + 6859)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} - 94 T^{10} + \cdots + 5764801 \) Copy content Toggle raw display
$29$ \( (T^{6} + 2 T^{5} + 9 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + T^{2} - 6 T - 7)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} + 242 T^{4} + \cdots + 51529)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 2 T^{5} + \cdots + 1369)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 214358881 \) Copy content Toggle raw display
$47$ \( T^{12} - 50 T^{10} + \cdots + 5764801 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 9354951841 \) Copy content Toggle raw display
$59$ \( (T^{6} - 6 T^{5} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 9 T^{5} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} - 184 T^{10} + \cdots + 59969536 \) Copy content Toggle raw display
$71$ \( (T^{6} - 29 T^{5} + \cdots + 218089)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} - 250 T^{10} + \cdots + 35153041 \) Copy content Toggle raw display
$79$ \( (T^{6} + 24 T^{5} + \cdots + 61504)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 117 T^{4} + \cdots + 5929)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 14 T^{5} + \cdots + 3136)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 214358881 \) Copy content Toggle raw display
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