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Properties

Label 375.2.e.a
Level $375$
Weight $2$
Character orbit 375.e
Analytic conductor $2.994$
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] = [375,2,Mod(68,375)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(375, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("375.68");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 375 = 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 375.e (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: \(2.99439007580\)
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: \( x^{16} - 9x^{12} + 121x^{8} - 729x^{4} + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{7} q^{2} + (\beta_{11} - \beta_1) q^{3} - \beta_{2} q^{4} + (\beta_{6} + \beta_{5} - 2) q^{6} + (\beta_{13} + 2 \beta_{9} + \beta_{7}) q^{7} + ( - \beta_{11} - \beta_{8}) q^{8} + ( - \beta_{12} + 2 \beta_{10} + \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{2} + (\beta_{11} - \beta_1) q^{3} - \beta_{2} q^{4} + (\beta_{6} + \beta_{5} - 2) q^{6} + (\beta_{13} + 2 \beta_{9} + \beta_{7}) q^{7} + ( - \beta_{11} - \beta_{8}) q^{8} + ( - \beta_{12} + 2 \beta_{10} + \beta_{2}) q^{9} + ( - 2 \beta_{4} + 1) q^{11} + (\beta_{13} + \beta_{9}) q^{12} + (\beta_{11} - 2 \beta_1) q^{13} + ( - 2 \beta_{15} - 2 \beta_{12} + \cdots + \beta_{2}) q^{14}+ \cdots + (4 \beta_{15} - \beta_{12} + \cdots - 9 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 20 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 20 q^{6} + 56 q^{16} - 28 q^{21} - 8 q^{31} - 4 q^{36} + 80 q^{46} - 20 q^{51} - 88 q^{61} - 20 q^{66} - 48 q^{76} + 36 q^{81} - 56 q^{91} - 40 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 9x^{12} + 121x^{8} - 729x^{4} + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{14} - 99\nu^{10} - 121\nu^{6} - 4320\nu^{2} ) / 18711 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{13} - 99\nu^{9} - 121\nu^{5} - 4320\nu ) / 6237 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5\nu^{12} + 198\nu^{8} - 1177\nu^{4} + 14175 ) / 6237 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{12} + 220\nu^{4} - 234 ) / 693 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{12} - 9\nu^{8} + 40\nu^{4} - 81 ) / 567 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 17\nu^{15} - 396\nu^{11} + 2057\nu^{7} - 65853\nu^{3} ) / 168399 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 16\nu^{13} + 99\nu^{9} - 737\nu^{5} + 7047\nu ) / 18711 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -26\nu^{15} - 495\nu^{11} - 3146\nu^{7} + 26973\nu^{3} ) / 168399 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 20\nu^{14} - 99\nu^{10} + 2420\nu^{6} + 3240\nu^{2} ) / 56133 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -20\nu^{13} + 99\nu^{9} - 2420\nu^{5} + 15471\nu ) / 18711 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -40\nu^{14} + 198\nu^{10} - 4840\nu^{6} + 49653\nu^{2} ) / 56133 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -62\nu^{15} + 396\nu^{11} + 517\nu^{7} + 1539\nu^{3} ) / 168399 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( -\nu^{15} + 9\nu^{11} - 121\nu^{7} + 729\nu^{3} ) / 2187 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( -65\nu^{14} + 99\nu^{10} + 154\nu^{6} - 11421\nu^{2} ) / 56133 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{12} + 2\beta_{10} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{14} + \beta_{9} - 3\beta_{7} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{6} + 3\beta_{5} - \beta_{4} + 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -5\beta_{11} - 7\beta_{8} - 4\beta_{3} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 7\beta_{15} - \beta_{12} + 20\beta_{10} - 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -21\beta_{14} + 21\beta_{13} + \beta_{9} - 17\beta_{7} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -23\beta_{6} + 17\beta_{5} + 20\beta_{4} - 43 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 9\beta_{11} - 60\beta_{3} - 49\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -40\beta_{12} - 107\beta_{10} - 180\beta_{2} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 67\beta_{14} - 220\beta_{9} - 33\beta_{7} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 440\beta_{6} + 33\beta_{5} + 220\beta_{4} - 426 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( -286\beta_{11} + 847\beta_{8} + 187\beta_{3} + 47\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( -847\beta_{15} - 239\beta_{12} - 467\beta_{10} - 286\beta_{2} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 228\beta_{14} - 2541\beta_{13} - 1372\beta_{9} - 427\beta_{7} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/375\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(251\)
\(\chi(n)\) \(-\beta_{10}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
68.1
1.69609 + 0.351096i
−0.351096 − 1.69609i
1.56769 + 0.736438i
−0.736438 − 1.56769i
0.736438 + 1.56769i
−1.56769 − 0.736438i
0.351096 + 1.69609i
−1.69609 − 0.351096i
1.69609 − 0.351096i
−0.351096 + 1.69609i
1.56769 − 0.736438i
−0.736438 + 1.56769i
0.736438 − 1.56769i
−1.56769 + 0.736438i
0.351096 − 1.69609i
−1.69609 + 0.351096i
−1.34500 − 1.34500i −0.351096 − 1.69609i 1.61803i 0 −1.80902 + 2.75346i 3.31242 − 3.31242i −0.513743 + 0.513743i −2.75346 + 1.19098i 0
68.2 −1.34500 − 1.34500i 1.69609 + 0.351096i 1.61803i 0 −1.80902 − 2.75346i −3.31242 + 3.31242i −0.513743 + 0.513743i 2.75346 + 1.19098i 0
68.3 −0.831254 − 0.831254i −0.736438 − 1.56769i − 0.618034i 0 −0.690983 + 1.91532i −1.42403 + 1.42403i −2.17625 + 2.17625i −1.91532 + 2.30902i 0
68.4 −0.831254 − 0.831254i 1.56769 + 0.736438i − 0.618034i 0 −0.690983 − 1.91532i 1.42403 − 1.42403i −2.17625 + 2.17625i 1.91532 + 2.30902i 0
68.5 0.831254 + 0.831254i −1.56769 − 0.736438i − 0.618034i 0 −0.690983 − 1.91532i −1.42403 + 1.42403i 2.17625 − 2.17625i 1.91532 + 2.30902i 0
68.6 0.831254 + 0.831254i 0.736438 + 1.56769i − 0.618034i 0 −0.690983 + 1.91532i 1.42403 − 1.42403i 2.17625 − 2.17625i −1.91532 + 2.30902i 0
68.7 1.34500 + 1.34500i −1.69609 − 0.351096i 1.61803i 0 −1.80902 − 2.75346i 3.31242 − 3.31242i 0.513743 − 0.513743i 2.75346 + 1.19098i 0
68.8 1.34500 + 1.34500i 0.351096 + 1.69609i 1.61803i 0 −1.80902 + 2.75346i −3.31242 + 3.31242i 0.513743 − 0.513743i −2.75346 + 1.19098i 0
182.1 −1.34500 + 1.34500i −0.351096 + 1.69609i − 1.61803i 0 −1.80902 − 2.75346i 3.31242 + 3.31242i −0.513743 − 0.513743i −2.75346 − 1.19098i 0
182.2 −1.34500 + 1.34500i 1.69609 − 0.351096i − 1.61803i 0 −1.80902 + 2.75346i −3.31242 − 3.31242i −0.513743 − 0.513743i 2.75346 − 1.19098i 0
182.3 −0.831254 + 0.831254i −0.736438 + 1.56769i 0.618034i 0 −0.690983 − 1.91532i −1.42403 − 1.42403i −2.17625 − 2.17625i −1.91532 − 2.30902i 0
182.4 −0.831254 + 0.831254i 1.56769 − 0.736438i 0.618034i 0 −0.690983 + 1.91532i 1.42403 + 1.42403i −2.17625 − 2.17625i 1.91532 − 2.30902i 0
182.5 0.831254 − 0.831254i −1.56769 + 0.736438i 0.618034i 0 −0.690983 + 1.91532i −1.42403 − 1.42403i 2.17625 + 2.17625i 1.91532 − 2.30902i 0
182.6 0.831254 − 0.831254i 0.736438 − 1.56769i 0.618034i 0 −0.690983 − 1.91532i 1.42403 + 1.42403i 2.17625 + 2.17625i −1.91532 − 2.30902i 0
182.7 1.34500 − 1.34500i −1.69609 + 0.351096i − 1.61803i 0 −1.80902 + 2.75346i 3.31242 + 3.31242i 0.513743 + 0.513743i 2.75346 − 1.19098i 0
182.8 1.34500 − 1.34500i 0.351096 − 1.69609i − 1.61803i 0 −1.80902 − 2.75346i −3.31242 − 3.31242i 0.513743 + 0.513743i −2.75346 − 1.19098i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 68.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
15.d odd 2 1 inner
15.e even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 375.2.e.a ✓ 16
3.b odd 2 1 inner 375.2.e.a ✓ 16
5.b even 2 1 inner 375.2.e.a ✓ 16
5.c odd 4 2 inner 375.2.e.a ✓ 16
15.d odd 2 1 inner 375.2.e.a ✓ 16
15.e even 4 2 inner 375.2.e.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
375.2.e.a ✓ 16 1.a even 1 1 trivial
375.2.e.a ✓ 16 3.b odd 2 1 inner
375.2.e.a ✓ 16 5.b even 2 1 inner
375.2.e.a ✓ 16 5.c odd 4 2 inner
375.2.e.a ✓ 16 15.d odd 2 1 inner
375.2.e.a ✓ 16 15.e even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 15T_{2}^{4} + 25 \) acting on \(S_{2}^{\mathrm{new}}(375, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 15 T^{4} + 25)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} - 9 T^{12} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} + 498 T^{4} + 7921)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 50 T^{2} + 445)^{4} \) Copy content Toggle raw display
$13$ \( (T^{8} + 183 T^{4} + 7921)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 90 T^{4} + 25)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 27 T^{2} + 81)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} + 240 T^{4} + 6400)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 50 T^{2} + 445)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + T - 11)^{8} \) Copy content Toggle raw display
$37$ \( (T^{8} + 2928 T^{4} + 2027776)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 50 T^{2} + 445)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} + 5298 T^{4} + 7921)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 3015 T^{4} + 366025)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 1215 T^{4} + 164025)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 45 T^{2} + 445)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 11 T + 29)^{8} \) Copy content Toggle raw display
$67$ \( (T^{8} + 783 T^{4} + 7921)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 210 T^{2} + 445)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} + 35583 T^{4} + 115971361)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 162 T^{2} + 81)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + 20640 T^{4} + 93702400)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 340 T^{2} + 7120)^{4} \) Copy content Toggle raw display
$97$ \( (T^{8} + 783 T^{4} + 7921)^{2} \) Copy content Toggle raw display
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