↑

Properties

Label 1375.2.b.c
Level $1375$
Weight $2$
Character orbit 1375.b
Analytic conductor $10.979$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1375,2,Mod(749,1375)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1375, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1375.749");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1375 = 5^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1375.b (of order \(2\), degree \(1\), 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: \(10.9794302779\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 16x^{10} + 91x^{8} + 219x^{6} + 195x^{4} + 28x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{9} + \beta_{4}) q^{3} + ( - \beta_{5} + \beta_{3}) q^{4} + ( - \beta_{3} + \beta_{2}) q^{6} + (\beta_{11} - \beta_{6} - \beta_{4}) q^{7} + ( - \beta_{6} + \beta_{4} - \beta_1) q^{8} + ( - \beta_{8} - \beta_{7} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{9} + \beta_{4}) q^{3} + ( - \beta_{5} + \beta_{3}) q^{4} + ( - \beta_{3} + \beta_{2}) q^{6} + (\beta_{11} - \beta_{6} - \beta_{4}) q^{7} + ( - \beta_{6} + \beta_{4} - \beta_1) q^{8} + ( - \beta_{8} - \beta_{7} - 3) q^{9} - q^{11} + (\beta_{11} - \beta_{10} + \cdots + 2 \beta_1) q^{12}+ \cdots + (\beta_{8} + \beta_{7} + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 8 q^{4} + 6 q^{6} - 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 8 q^{4} + 6 q^{6} - 32 q^{9} - 12 q^{11} - 6 q^{14} + 4 q^{16} - 6 q^{19} + 34 q^{21} - 32 q^{24} + 18 q^{26} - 38 q^{29} - 28 q^{31} + 8 q^{34} + 22 q^{36} + 52 q^{41} + 8 q^{44} + 10 q^{46} - 22 q^{49} + 44 q^{51} + 50 q^{54} + 20 q^{56} - 10 q^{59} + 10 q^{61} + 38 q^{64} - 6 q^{66} - 52 q^{69} + 30 q^{71} - 78 q^{74} - 56 q^{76} + 2 q^{79} + 60 q^{81} - 124 q^{84} + 42 q^{86} - 38 q^{89} - 50 q^{91} - 30 q^{94} + 62 q^{96} + 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 16x^{10} + 91x^{8} + 219x^{6} + 195x^{4} + 28x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{8} - 12\nu^{6} - 42\nu^{4} - 40\nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{10} - 30\nu^{8} - 157\nu^{6} - 341\nu^{4} - 259\nu^{2} - 7 ) / 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{11} - 30\nu^{9} - 157\nu^{7} - 341\nu^{5} - 259\nu^{3} + 3\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{10} - 30\nu^{8} - 157\nu^{6} - 341\nu^{4} - 264\nu^{2} - 17 ) / 5 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -2\nu^{11} - 30\nu^{9} - 157\nu^{7} - 341\nu^{5} - 264\nu^{3} - 22\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2\nu^{10} + 35\nu^{8} + 217\nu^{6} + 556\nu^{4} + 494\nu^{2} + 37 ) / 5 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3\nu^{10} + 50\nu^{8} + 298\nu^{6} + 749\nu^{4} + 671\nu^{2} + 53 ) / 5 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( \nu^{11} + 16\nu^{9} + 91\nu^{7} + 219\nu^{5} + 195\nu^{3} + 28\nu \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -6\nu^{11} - 95\nu^{9} - 531\nu^{7} - 1238\nu^{5} - 1022\nu^{3} - 81\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -17\nu^{11} - 270\nu^{9} - 1517\nu^{7} - 3566\nu^{5} - 2974\nu^{3} - 212\nu ) / 5 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{3} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{6} + \beta_{4} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} + 7\beta_{5} - 6\beta_{3} + \beta_{2} + 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{11} - 2\beta_{10} + \beta_{9} + 9\beta_{6} - 9\beta_{4} + 27\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{8} - 11\beta_{7} - 44\beta_{5} + 36\beta_{3} - 9\beta_{2} - 39 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -11\beta_{11} + 24\beta_{10} - 9\beta_{9} - 66\beta_{6} + 65\beta_{4} - 155\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -24\beta_{8} + 90\beta_{7} + 274\beta_{5} - 220\beta_{3} + 65\beta_{2} + 212 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 89\beta_{11} - 203\beta_{10} + 65\beta_{9} + 454\beta_{6} - 439\beta_{4} + 926\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 203\beta_{8} - 657\beta_{7} - 1720\beta_{5} + 1365\beta_{3} - 439\beta_{2} - 1227 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -642\beta_{11} + 1502\beta_{10} - 439\beta_{9} - 3034\beta_{6} + 2885\beta_{4} - 5677\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1375\mathbb{Z}\right)^\times\).

\(n\) \(376\) \(1002\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
749.1
− 2.53099i
− 2.01035i
− 1.92880i
− 1.29062i
− 0.331443i
− 0.238201i
0.238201i
0.331443i
1.29062i
1.92880i
2.01035i
2.53099i
− 2.53099i 2.38927i −4.40593 0 6.04724 − 1.06922i 6.08941i −2.70863 0
749.2 − 2.01035i 1.82331i −2.04150 0 3.66549 − 4.27127i 0.0834224i −0.324466 0
749.3 − 1.92880i − 3.14194i −1.72028 0 −6.06018 4.01358i − 0.539527i −6.87178 0
749.4 − 1.29062i − 1.44023i 0.334304 0 −1.85878 0.605652i − 3.01270i 0.925745 0
749.5 − 0.331443i 1.19146i 1.89015 0 0.394902 − 2.85130i − 1.28936i 1.58041 0
749.6 − 0.238201i 3.40606i 1.94326 0 0.811328 3.00125i − 0.939288i −8.60128 0
749.7 0.238201i − 3.40606i 1.94326 0 0.811328 − 3.00125i 0.939288i −8.60128 0
749.8 0.331443i − 1.19146i 1.89015 0 0.394902 2.85130i 1.28936i 1.58041 0
749.9 1.29062i 1.44023i 0.334304 0 −1.85878 − 0.605652i 3.01270i 0.925745 0
749.10 1.92880i 3.14194i −1.72028 0 −6.06018 − 4.01358i 0.539527i −6.87178 0
749.11 2.01035i − 1.82331i −2.04150 0 3.66549 4.27127i − 0.0834224i −0.324466 0
749.12 2.53099i − 2.38927i −4.40593 0 6.04724 1.06922i − 6.08941i −2.70863 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 749.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1375.2.b.c 12
5.b even 2 1 inner 1375.2.b.c 12
5.c odd 4 2 1375.2.a.h ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1375.2.a.h ✓ 12 5.c odd 4 2
1375.2.b.c 12 1.a even 1 1 trivial
1375.2.b.c 12 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 16T_{2}^{10} + 91T_{2}^{8} + 219T_{2}^{6} + 195T_{2}^{4} + 28T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(1375, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 16 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{12} + 34 T^{10} + \cdots + 6400 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + 53 T^{10} + \cdots + 9025 \) Copy content Toggle raw display
$11$ \( (T + 1)^{12} \) Copy content Toggle raw display
$13$ \( T^{12} + 78 T^{10} + \cdots + 160000 \) Copy content Toggle raw display
$17$ \( T^{12} + 108 T^{10} + \cdots + 15452761 \) Copy content Toggle raw display
$19$ \( (T^{6} + 3 T^{5} + \cdots - 304)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 126 T^{10} + \cdots + 256 \) Copy content Toggle raw display
$29$ \( (T^{6} + 19 T^{5} + \cdots - 304)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 14 T^{5} + \cdots + 17461)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + 219 T^{10} + \cdots + 3748096 \) Copy content Toggle raw display
$41$ \( (T^{6} - 26 T^{5} + \cdots + 14384)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 295461721 \) Copy content Toggle raw display
$47$ \( T^{12} + 354 T^{10} + \cdots + 3748096 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 1198821376 \) Copy content Toggle raw display
$59$ \( (T^{6} + 5 T^{5} + \cdots - 330641)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 5 T^{5} + \cdots + 211280)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 2302464256 \) Copy content Toggle raw display
$71$ \( (T^{6} - 15 T^{5} + \cdots + 4451)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + 389 T^{10} + \cdots + 4456321 \) Copy content Toggle raw display
$79$ \( (T^{6} - T^{5} + \cdots - 15856)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 1739243990416 \) Copy content Toggle raw display
$89$ \( (T^{6} + 19 T^{5} + \cdots - 42655)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 52495974400 \) Copy content Toggle raw display
show more
show less