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Properties

Label 475.2.g.c
Level $475$
Weight $2$
Character orbit 475.g
Analytic conductor $3.793$
Analytic rank $0$
Dimension $16$
CM discriminant -95
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [475,2,Mod(18,475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(475, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("475.18");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.g (of order \(4\), degree \(2\), 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: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 96x^{12} + 2394x^{8} + 9184x^{4} + 2025 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{9} q^{2} - \beta_{3} q^{3} + ( - \beta_{10} - 2 \beta_{7}) q^{4} + ( - \beta_{6} + \beta_{5}) q^{6} + (\beta_{4} - 2 \beta_1) q^{8} + (\beta_{11} + \beta_{10} + 3 \beta_{7}) q^{9} + (\beta_{6} - \beta_{2}) q^{11}+ \cdots + (2 \beta_{12} - 3 \beta_{11} + \cdots - 4 \beta_{7}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 64 q^{16} - 48 q^{26} + 208 q^{36} - 208 q^{66} - 144 q^{81} + 304 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 96x^{12} + 2394x^{8} + 9184x^{4} + 2025 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{12} + 104\nu^{8} + 3757\nu^{4} + 30744 ) / 8496 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{13} + 104\nu^{9} + 2813\nu^{5} + 9976\nu ) / 2832 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{13} + 104\nu^{9} + 3757\nu^{5} + 47736\nu ) / 8496 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{12} - 104\nu^{8} - 2695\nu^{4} - 5256 ) / 1062 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -10\nu^{12} - 863\nu^{8} - 17746\nu^{4} - 27603 ) / 8496 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 8\nu^{14} + 773\nu^{10} + 19672\nu^{6} + 92257\nu^{2} ) / 42480 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -16\nu^{13} - 1487\nu^{9} - 34624\nu^{5} - 87459\nu ) / 8496 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 8\nu^{15} + 773\nu^{11} + 19672\nu^{7} + 92257\nu^{3} ) / 42480 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -8\nu^{14} - 773\nu^{10} - 19672\nu^{6} - 81637\nu^{2} ) / 10620 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 11\nu^{14} + 1026\nu^{10} + 24099\nu^{6} + 66194\nu^{2} ) / 10620 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( \nu^{14} + 96\nu^{10} + 2349\nu^{6} + 7024\nu^{2} ) / 720 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -107\nu^{15} - 10302\nu^{11} - 256623\nu^{7} - 925478\nu^{3} ) / 127440 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 8\nu^{15} + 773\nu^{11} + 19672\nu^{7} + 85177\nu^{3} ) / 7080 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 173\nu^{15} + 16458\nu^{11} + 401217\nu^{7} + 1322642\nu^{3} ) / 63720 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + 4\beta_{7} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{14} + 6\beta_{9} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} + 8\beta_{2} - 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{4} - 3\beta_{3} - 40\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -12\beta_{12} + 3\beta_{11} - 58\beta_{10} - 160\beta_{7} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3\beta_{15} + 70\beta_{14} + 42\beta_{13} - 276\beta_{9} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 48\beta_{6} - 112\beta_{5} - 416\beta_{2} + 1107 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 48\beta_{8} - 528\beta_{4} + 432\beta_{3} + 1939\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 960\beta_{12} - 528\beta_{11} + 2995\beta_{10} + 7804\beta_{7} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -528\beta_{15} - 3955\beta_{14} - 3936\beta_{13} + 13794\beta_{9} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -4992\beta_{6} + 7891\beta_{5} + 21704\beta_{2} - 55704 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( -4992\beta_{8} + 29595\beta_{4} - 33657\beta_{3} - 99112\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( -63252\beta_{12} + 43641\beta_{11} - 158302\beta_{10} - 401440\beta_{7} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 43641\beta_{15} + 221554\beta_{14} + 277038\beta_{13} - 718044\beta_{9} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(77\) \(401\)
\(\chi(n)\) \(\beta_{7}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
18.1
−1.93849 + 1.93849i
−1.71874 + 1.71874i
−1.02270 + 1.02270i
−0.492178 + 0.492178i
0.492178 − 0.492178i
1.02270 − 1.02270i
1.71874 − 1.71874i
1.93849 − 1.93849i
−1.93849 − 1.93849i
−1.71874 − 1.71874i
−1.02270 − 1.02270i
−0.492178 − 0.492178i
0.492178 + 0.492178i
1.02270 + 1.02270i
1.71874 + 1.71874i
1.93849 + 1.93849i
−1.93849 − 1.93849i 1.83729 − 1.83729i 5.51552i 0 −7.12315 0 6.81482 − 6.81482i − 3.75127i 0
18.2 −1.71874 − 1.71874i −2.44467 + 2.44467i 3.90817i 0 8.40351 0 3.27965 − 3.27965i − 8.95278i 0
18.3 −1.02270 − 1.02270i −0.153655 + 0.153655i 0.0918331i 0 0.314285 0 −1.95148 + 1.95148i 2.95278i 0
18.4 −0.492178 − 0.492178i 1.61999 − 1.61999i − 1.51552i 0 −1.59464 0 −1.73026 + 1.73026i − 2.24873i 0
18.5 0.492178 + 0.492178i −1.61999 + 1.61999i − 1.51552i 0 −1.59464 0 1.73026 − 1.73026i − 2.24873i 0
18.6 1.02270 + 1.02270i 0.153655 − 0.153655i 0.0918331i 0 0.314285 0 1.95148 − 1.95148i 2.95278i 0
18.7 1.71874 + 1.71874i 2.44467 − 2.44467i 3.90817i 0 8.40351 0 −3.27965 + 3.27965i − 8.95278i 0
18.8 1.93849 + 1.93849i −1.83729 + 1.83729i 5.51552i 0 −7.12315 0 −6.81482 + 6.81482i − 3.75127i 0
132.1 −1.93849 + 1.93849i 1.83729 + 1.83729i − 5.51552i 0 −7.12315 0 6.81482 + 6.81482i 3.75127i 0
132.2 −1.71874 + 1.71874i −2.44467 − 2.44467i − 3.90817i 0 8.40351 0 3.27965 + 3.27965i 8.95278i 0
132.3 −1.02270 + 1.02270i −0.153655 − 0.153655i − 0.0918331i 0 0.314285 0 −1.95148 − 1.95148i − 2.95278i 0
132.4 −0.492178 + 0.492178i 1.61999 + 1.61999i 1.51552i 0 −1.59464 0 −1.73026 − 1.73026i 2.24873i 0
132.5 0.492178 − 0.492178i −1.61999 − 1.61999i 1.51552i 0 −1.59464 0 1.73026 + 1.73026i 2.24873i 0
132.6 1.02270 − 1.02270i 0.153655 + 0.153655i − 0.0918331i 0 0.314285 0 1.95148 + 1.95148i − 2.95278i 0
132.7 1.71874 − 1.71874i 2.44467 + 2.44467i − 3.90817i 0 8.40351 0 −3.27965 − 3.27965i 8.95278i 0
132.8 1.93849 − 1.93849i −1.83729 − 1.83729i − 5.51552i 0 −7.12315 0 −6.81482 − 6.81482i 3.75127i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 18.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
95.d odd 2 1 CM by \(\Q(\sqrt{-95}) \)
5.b even 2 1 inner
5.c odd 4 2 inner
19.b odd 2 1 inner
95.g even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.g.c ✓ 16
5.b even 2 1 inner 475.2.g.c ✓ 16
5.c odd 4 2 inner 475.2.g.c ✓ 16
19.b odd 2 1 inner 475.2.g.c ✓ 16
95.d odd 2 1 CM 475.2.g.c ✓ 16
95.g even 4 2 inner 475.2.g.c ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
475.2.g.c ✓ 16 1.a even 1 1 trivial
475.2.g.c ✓ 16 5.b even 2 1 inner
475.2.g.c ✓ 16 5.c odd 4 2 inner
475.2.g.c ✓ 16 19.b odd 2 1 inner
475.2.g.c ✓ 16 95.d odd 2 1 CM
475.2.g.c ✓ 16 95.g even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 96T_{2}^{12} + 2394T_{2}^{8} + 9184T_{2}^{4} + 2025 \) acting on \(S_{2}^{\mathrm{new}}(475, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 96 T^{12} + \cdots + 2025 \) Copy content Toggle raw display
$3$ \( T^{16} + 216 T^{12} + \cdots + 400 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{4} - 44 T^{2} + 180)^{4} \) Copy content Toggle raw display
$13$ \( T^{16} + 4056 T^{12} + \cdots + 20250000 \) Copy content Toggle raw display
$17$ \( T^{16} \) Copy content Toggle raw display
$19$ \( (T^{2} + 19)^{8} \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( T^{16} \) Copy content Toggle raw display
$31$ \( T^{16} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 48798886784400 \) Copy content Toggle raw display
$41$ \( T^{16} \) Copy content Toggle raw display
$43$ \( T^{16} \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 392602496400 \) Copy content Toggle raw display
$59$ \( T^{16} \) Copy content Toggle raw display
$61$ \( (T^{4} - 244 T^{2} + 14580)^{4} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{16} \) Copy content Toggle raw display
$73$ \( T^{16} \) Copy content Toggle raw display
$79$ \( T^{16} \) Copy content Toggle raw display
$83$ \( T^{16} \) Copy content Toggle raw display
$89$ \( T^{16} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
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