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Properties

Label 945.2.b.a
Level $945$
Weight $2$
Character orbit 945.b
Analytic conductor $7.546$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [945,2,Mod(566,945)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(945, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.566");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.b (of order \(2\), degree \(1\), 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: \(7.54586299101\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 14x^{8} + 63x^{6} + 110x^{4} + 73x^{2} + 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} - 1) q^{4} - q^{5} - \beta_{8} q^{7} + (\beta_{9} + \beta_{8} + \beta_{5} + \cdots - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} - 1) q^{4} - q^{5} - \beta_{8} q^{7} + (\beta_{9} + \beta_{8} + \beta_{5} + \cdots - \beta_1) q^{8}+ \cdots + (2 \beta_{9} - 2 \beta_{5} + 2 \beta_{4} + \cdots - 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 8 q^{4} - 10 q^{5} + 3 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 8 q^{4} - 10 q^{5} + 3 q^{7} - 6 q^{14} + 12 q^{16} + 8 q^{20} + 32 q^{22} + 10 q^{25} + 24 q^{26} - 22 q^{28} - 3 q^{35} + 30 q^{37} + 48 q^{38} + 24 q^{43} - 16 q^{46} - 12 q^{47} - 5 q^{49} + 24 q^{56} - 16 q^{58} - 36 q^{59} + 48 q^{62} - 48 q^{64} + 14 q^{67} + 60 q^{68} + 6 q^{70} + 42 q^{77} - 34 q^{79} - 12 q^{80} - 80 q^{88} - 12 q^{89} - 21 q^{91} - 30 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 14x^{8} + 63x^{6} + 110x^{4} + 73x^{2} + 12 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 12\nu^{5} + 38\nu^{3} + 25\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{8} + 13\nu^{6} + 49\nu^{4} + 52\nu^{2} + 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{9} + 13\nu^{7} + 50\nu^{5} + 61\nu^{3} + 19\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2\nu^{8} + 25\nu^{6} + 89\nu^{4} + 91\nu^{2} + 16 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{9} + 2\nu^{8} + 12\nu^{7} + 26\nu^{6} + 39\nu^{5} + 100\nu^{4} + 30\nu^{3} + 122\nu^{2} - \nu + 34 ) / 4 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{9} + 2\nu^{8} - 12\nu^{7} + 26\nu^{6} - 39\nu^{5} + 100\nu^{4} - 30\nu^{3} + 122\nu^{2} + \nu + 34 ) / 4 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -\nu^{9} - 2\nu^{8} - 12\nu^{7} - 26\nu^{6} - 37\nu^{5} - 100\nu^{4} - 12\nu^{3} - 122\nu^{2} + 31\nu - 34 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + \beta_{8} + \beta_{5} - \beta_{3} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} + \beta_{7} - 2\beta_{4} - 9\beta_{2} + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -7\beta_{9} - 9\beta_{8} + 2\beta_{7} - 9\beta_{5} + 9\beta_{3} + 30\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -9\beta_{8} - 9\beta_{7} - 2\beta_{6} + 22\beta_{4} + 68\beta_{2} - 101 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 46\beta_{9} + 70\beta_{8} - 24\beta_{7} + 70\beta_{5} - 68\beta_{3} - 195\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 68\beta_{8} + 68\beta_{7} + 26\beta_{6} - 186\beta_{4} - 495\beta_{2} + 679 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -309\beta_{9} - 521\beta_{8} + 212\beta_{7} - 519\beta_{5} + 495\beta_{3} + 1321\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/945\mathbb{Z}\right)^\times\).

\(n\) \(136\) \(596\) \(757\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
566.1
− 2.67094i
− 1.97525i
− 1.26571i
− 1.05960i
− 0.489586i
0.489586i
1.05960i
1.26571i
1.97525i
2.67094i
− 2.67094i 0 −5.13393 −1.00000 0 1.64848 − 2.06942i 8.37055i 0 2.67094i
566.2 − 1.97525i 0 −1.90161 −1.00000 0 0.123123 + 2.64288i − 0.194346i 0 1.97525i
566.3 − 1.26571i 0 0.397988 −1.00000 0 −0.368446 − 2.61997i − 3.03515i 0 1.26571i
566.4 − 1.05960i 0 0.877248 −1.00000 0 2.63463 + 0.242363i − 3.04873i 0 1.05960i
566.5 − 0.489586i 0 1.76031 −1.00000 0 −2.53778 + 0.748102i − 1.84099i 0 0.489586i
566.6 0.489586i 0 1.76031 −1.00000 0 −2.53778 − 0.748102i 1.84099i 0 − 0.489586i
566.7 1.05960i 0 0.877248 −1.00000 0 2.63463 − 0.242363i 3.04873i 0 − 1.05960i
566.8 1.26571i 0 0.397988 −1.00000 0 −0.368446 + 2.61997i 3.03515i 0 − 1.26571i
566.9 1.97525i 0 −1.90161 −1.00000 0 0.123123 − 2.64288i 0.194346i 0 − 1.97525i
566.10 2.67094i 0 −5.13393 −1.00000 0 1.64848 + 2.06942i − 8.37055i 0 − 2.67094i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 566.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 945.2.b.a ✓ 10
3.b odd 2 1 945.2.b.b yes 10
7.b odd 2 1 945.2.b.b yes 10
21.c even 2 1 inner 945.2.b.a ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
945.2.b.a ✓ 10 1.a even 1 1 trivial
945.2.b.a ✓ 10 21.c even 2 1 inner
945.2.b.b yes 10 3.b odd 2 1
945.2.b.b yes 10 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\):

\( T_{2}^{10} + 14T_{2}^{8} + 63T_{2}^{6} + 110T_{2}^{4} + 73T_{2}^{2} + 12 \) Copy content Toggle raw display
\( T_{17}^{5} - 45T_{17}^{3} - 102T_{17}^{2} + 93T_{17} + 252 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 14 T^{8} + \cdots + 12 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( (T + 1)^{10} \) Copy content Toggle raw display
$7$ \( T^{10} - 3 T^{9} + \cdots + 16807 \) Copy content Toggle raw display
$11$ \( T^{10} + 68 T^{8} + \cdots + 122412 \) Copy content Toggle raw display
$13$ \( T^{10} + 69 T^{8} + \cdots + 41067 \) Copy content Toggle raw display
$17$ \( (T^{5} - 45 T^{3} + \cdots + 252)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} + 117 T^{8} + \cdots + 1783323 \) Copy content Toggle raw display
$23$ \( T^{10} + 182 T^{8} + \cdots + 1929612 \) Copy content Toggle raw display
$29$ \( T^{10} + 110 T^{8} + \cdots + 2028 \) Copy content Toggle raw display
$31$ \( T^{10} + 180 T^{8} + \cdots + 5370732 \) Copy content Toggle raw display
$37$ \( (T^{5} - 15 T^{4} + \cdots + 9551)^{2} \) Copy content Toggle raw display
$41$ \( (T^{5} - 21 T^{3} + \cdots - 54)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} - 12 T^{4} + \cdots + 524)^{2} \) Copy content Toggle raw display
$47$ \( (T^{5} + 6 T^{4} + \cdots + 13968)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 102667500 \) Copy content Toggle raw display
$59$ \( (T^{5} + 18 T^{4} + \cdots - 2016)^{2} \) Copy content Toggle raw display
$61$ \( T^{10} + 261 T^{8} + \cdots + 7555707 \) Copy content Toggle raw display
$67$ \( (T^{5} - 7 T^{4} + \cdots + 151)^{2} \) Copy content Toggle raw display
$71$ \( T^{10} + 182 T^{8} + \cdots + 475212 \) Copy content Toggle raw display
$73$ \( T^{10} + 255 T^{8} + \cdots + 25456707 \) Copy content Toggle raw display
$79$ \( (T^{5} + 17 T^{4} + \cdots + 2383)^{2} \) Copy content Toggle raw display
$83$ \( (T^{5} - 294 T^{3} + \cdots - 50814)^{2} \) Copy content Toggle raw display
$89$ \( (T^{5} + 6 T^{4} + \cdots + 28008)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + 573 T^{8} + \cdots + 10750347 \) Copy content Toggle raw display
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