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Properties

Label 475.2.p.e
Level $475$
Weight $2$
Character orbit 475.p
Analytic conductor $3.793$
Analytic rank $0$
Dimension $16$
CM no
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(107,475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(475, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([3, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("475.107");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.p (of order \(12\), degree \(4\), 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: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: 16.0.14096583954457373039394816.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 127x^{12} + 13728x^{8} - 304927x^{4} + 5764801 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{14} q^{2} + (\beta_{3} + \beta_1) q^{3} - \beta_{5} q^{4} - \beta_{6} q^{6} + ( - 2 \beta_{14} - \beta_{9}) q^{7} + (3 \beta_{7} - 3 \beta_{3}) q^{8} + ( - \beta_{10} - 4 \beta_{5}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{14} q^{2} + (\beta_{3} + \beta_1) q^{3} - \beta_{5} q^{4} - \beta_{6} q^{6} + ( - 2 \beta_{14} - \beta_{9}) q^{7} + (3 \beta_{7} - 3 \beta_{3}) q^{8} + ( - \beta_{10} - 4 \beta_{5}) q^{9} + (\beta_{4} - 1) q^{11} + ( - \beta_{9} + \beta_{8}) q^{12} + ( - \beta_{11} - \beta_{7}) q^{13} + ( - \beta_{12} - \beta_{5}) q^{14} - \beta_{2} q^{16} + ( - \beta_{15} + \beta_{9} - \beta_{8}) q^{17} + (\beta_{11} + 4 \beta_{7} + \cdots - \beta_1) q^{18}+ \cdots + (6 \beta_{10} + 11 \beta_{5}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{6} - 24 q^{11} - 8 q^{16} + 12 q^{21} + 24 q^{26} + 36 q^{36} - 12 q^{41} - 180 q^{51} - 36 q^{61} + 64 q^{66} + 96 q^{71} + 56 q^{76} + 40 q^{81} + 60 q^{86} - 36 q^{91} - 40 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 127x^{12} + 13728x^{8} - 304927x^{4} + 5764801 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 127\nu^{12} - 13728\nu^{8} + 1743456\nu^{4} - 5764801 ) / 32960928 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{13} + 669761\nu ) / 1441440 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{12} - 669761 ) / 205920 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{14} - 2111201\nu^{2} ) / 10090080 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -827\nu^{12} + 121836\nu^{8} - 11353056\nu^{4} + 252174629 ) / 61801740 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9017\nu^{13} - 974688\nu^{9} + 90824448\nu^{5} - 409300871\nu ) / 3460897440 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{15} + 2111201\nu^{3} ) / 10090080 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -\nu^{15} - 849941\nu^{3} ) / 8828820 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -\nu^{14} - 849941\nu^{2} ) / 1261260 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 127\nu^{13} - 13728\nu^{9} + 1556178\nu^{5} - 5764801\nu ) / 19664190 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -127\nu^{14} + 13728\nu^{10} - 1556178\nu^{6} + 5764801\nu^{2} ) / 137649330 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 66263\nu^{14} - 9238944\nu^{10} + 909658464\nu^{6} - 20205377801\nu^{2} ) / 24226282080 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 85471\nu^{15} - 9238944\nu^{11} + 909658464\nu^{7} - 3879711073\nu^{3} ) / 169583974560 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( -\nu^{15} + 127\nu^{11} - 13728\nu^{7} + 304927\nu^{3} ) / 823543 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} - 8\beta_{5} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 7\beta_{9} + 8\beta_{8} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 15\beta_{6} + 15\beta_{4} + 71\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 71\beta_{11} - 176\beta_{7} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -176\beta_{13} - 673\beta_{12} + 176\beta_{10} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -673\beta_{15} - 1905\beta_{14} - 673\beta_{9} + 673\beta_{8} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1905\beta_{6} + 6616\beta_{2} - 6616 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 6616\beta_{11} - 19951\beta_{7} + 13335\beta_{3} - 6616\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -19951\beta_{13} - 66263\beta_{12} + 66263\beta_{5} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -66263\beta_{15} - 205920\beta_{14} - 205920\beta_{9} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -205920\beta_{4} - 669761 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 1441440\beta_{3} - 669761\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( -2111201\beta_{10} + 6799528\beta_{5} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( -14778407\beta_{9} - 6799528\beta_{8} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(77\) \(401\)
\(\chi(n)\) \(\beta_{12}\) \(1 - \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
−0.826301 + 3.08380i
0.567482 − 2.11787i
−0.567482 + 2.11787i
0.826301 − 3.08380i
−0.826301 − 3.08380i
0.567482 + 2.11787i
−0.567482 − 2.11787i
0.826301 + 3.08380i
−3.08380 + 0.826301i
2.11787 − 0.567482i
−2.11787 + 0.567482i
3.08380 − 0.826301i
−3.08380 − 0.826301i
2.11787 + 0.567482i
−2.11787 − 0.567482i
3.08380 + 0.826301i
−0.965926 − 0.258819i −0.567482 + 2.11787i −0.866025 − 0.500000i 0 1.09629 − 1.89883i 1.22474 + 1.22474i 2.12132 + 2.12132i −1.56527 − 0.903709i 0
107.2 −0.965926 − 0.258819i 0.826301 − 3.08380i −0.866025 − 0.500000i 0 −1.59629 + 2.76486i 1.22474 + 1.22474i 2.12132 + 2.12132i −6.22896 − 3.59629i 0
107.3 0.965926 + 0.258819i −0.826301 + 3.08380i −0.866025 − 0.500000i 0 −1.59629 + 2.76486i −1.22474 − 1.22474i −2.12132 − 2.12132i −6.22896 − 3.59629i 0
107.4 0.965926 + 0.258819i 0.567482 − 2.11787i −0.866025 − 0.500000i 0 1.09629 − 1.89883i −1.22474 − 1.22474i −2.12132 − 2.12132i −1.56527 − 0.903709i 0
293.1 −0.965926 + 0.258819i −0.567482 − 2.11787i −0.866025 + 0.500000i 0 1.09629 + 1.89883i 1.22474 − 1.22474i 2.12132 − 2.12132i −1.56527 + 0.903709i 0
293.2 −0.965926 + 0.258819i 0.826301 + 3.08380i −0.866025 + 0.500000i 0 −1.59629 − 2.76486i 1.22474 − 1.22474i 2.12132 − 2.12132i −6.22896 + 3.59629i 0
293.3 0.965926 − 0.258819i −0.826301 − 3.08380i −0.866025 + 0.500000i 0 −1.59629 − 2.76486i −1.22474 + 1.22474i −2.12132 + 2.12132i −6.22896 + 3.59629i 0
293.4 0.965926 − 0.258819i 0.567482 + 2.11787i −0.866025 + 0.500000i 0 1.09629 + 1.89883i −1.22474 + 1.22474i −2.12132 + 2.12132i −1.56527 + 0.903709i 0
407.1 −0.258819 − 0.965926i −2.11787 + 0.567482i 0.866025 − 0.500000i 0 1.09629 + 1.89883i 1.22474 + 1.22474i −2.12132 − 2.12132i 1.56527 − 0.903709i 0
407.2 −0.258819 − 0.965926i 3.08380 − 0.826301i 0.866025 − 0.500000i 0 −1.59629 − 2.76486i 1.22474 + 1.22474i −2.12132 − 2.12132i 6.22896 − 3.59629i 0
407.3 0.258819 + 0.965926i −3.08380 + 0.826301i 0.866025 − 0.500000i 0 −1.59629 − 2.76486i −1.22474 − 1.22474i 2.12132 + 2.12132i 6.22896 − 3.59629i 0
407.4 0.258819 + 0.965926i 2.11787 − 0.567482i 0.866025 − 0.500000i 0 1.09629 + 1.89883i −1.22474 − 1.22474i 2.12132 + 2.12132i 1.56527 − 0.903709i 0
468.1 −0.258819 + 0.965926i −2.11787 − 0.567482i 0.866025 + 0.500000i 0 1.09629 − 1.89883i 1.22474 − 1.22474i −2.12132 + 2.12132i 1.56527 + 0.903709i 0
468.2 −0.258819 + 0.965926i 3.08380 + 0.826301i 0.866025 + 0.500000i 0 −1.59629 + 2.76486i 1.22474 − 1.22474i −2.12132 + 2.12132i 6.22896 + 3.59629i 0
468.3 0.258819 − 0.965926i −3.08380 − 0.826301i 0.866025 + 0.500000i 0 −1.59629 + 2.76486i −1.22474 + 1.22474i 2.12132 − 2.12132i 6.22896 + 3.59629i 0
468.4 0.258819 − 0.965926i 2.11787 + 0.567482i 0.866025 + 0.500000i 0 1.09629 − 1.89883i −1.22474 + 1.22474i 2.12132 − 2.12132i 1.56527 + 0.903709i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner
19.d odd 6 1 inner
95.h odd 6 1 inner
95.l even 12 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.p.e ✓ 16
5.b even 2 1 inner 475.2.p.e ✓ 16
5.c odd 4 2 inner 475.2.p.e ✓ 16
19.d odd 6 1 inner 475.2.p.e ✓ 16
95.h odd 6 1 inner 475.2.p.e ✓ 16
95.l even 12 2 inner 475.2.p.e ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
475.2.p.e ✓ 16 1.a even 1 1 trivial
475.2.p.e ✓ 16 5.b even 2 1 inner
475.2.p.e ✓ 16 5.c odd 4 2 inner
475.2.p.e ✓ 16 19.d odd 6 1 inner
475.2.p.e ✓ 16 95.h odd 6 1 inner
475.2.p.e ✓ 16 95.l even 12 2 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(475, [\chi])\):

\( T_{2}^{8} - T_{2}^{4} + 1 \) Copy content Toggle raw display
\( T_{3}^{16} - 127T_{3}^{12} + 13728T_{3}^{8} - 304927T_{3}^{4} + 5764801 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - T^{4} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} - 127 T^{12} + \cdots + 5764801 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{4} + 9)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 3 T - 5)^{8} \) Copy content Toggle raw display
$13$ \( T^{16} - 311 T^{12} + \cdots + 390625 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 37822859361 \) Copy content Toggle raw display
$19$ \( (T^{4} - 37 T^{2} + 361)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} - 9 T^{4} + 81)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 45 T^{6} + \cdots + 194481)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 57 T^{2} + 225)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + 4976 T^{4} + 160000)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 3 T^{3} + \cdots + 441)^{4} \) Copy content Toggle raw display
$43$ \( T^{16} - 6543 T^{12} + \cdots + 6561 \) Copy content Toggle raw display
$47$ \( (T^{8} - 7569 T^{4} + 57289761)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 2401 T^{4} + 5764801)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 87 T^{2} + 7569)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + 9 T^{3} + \cdots + 2025)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} - 13456 T^{4} + 181063936)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 24 T^{3} + \cdots + 1521)^{4} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 5352009260481 \) Copy content Toggle raw display
$79$ \( (T^{8} + 405 T^{6} + \cdots + 1275989841)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 2799 T^{4} + 50625)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 27 T^{2} + 729)^{4} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 815730721 \) Copy content Toggle raw display
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