↑

Properties

Label 5733.2.a.ca
Level $5733$
Weight $2$
Character orbit 5733.a
Self dual yes
Analytic conductor $45.778$
Analytic rank $1$
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] = [5733,2,Mod(1,5733)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5733, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5733.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5733 = 3^{2} \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5733.a (trivial)

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: yes
Analytic conductor: \(45.7782354788\)
Analytic rank: \(1\)
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: \( x^{12} - 16x^{10} + 92x^{8} - 228x^{6} + 225x^{4} - 60x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} - \beta_{10} q^{5} + (\beta_{3} + \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} - \beta_{10} q^{5} + (\beta_{3} + \beta_1) q^{8} + ( - \beta_{4} - \beta_{2} - 1) q^{10} + (\beta_{11} + \beta_{10} - \beta_{6}) q^{11} - q^{13} + ( - \beta_{5} + \beta_{4}) q^{16} + ( - \beta_{11} - \beta_{10} + \beta_{3}) q^{17} + (\beta_{7} - \beta_{5} - \beta_{2} - 3) q^{19} + (\beta_{10} + \beta_{8} + \cdots - 3 \beta_1) q^{20}+ \cdots + ( - \beta_{7} + \beta_{5} - 2 \beta_{4} + \cdots - 5) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 8 q^{4} - 8 q^{10} - 12 q^{13} - 28 q^{19} + 16 q^{25} - 12 q^{31} - 24 q^{34} - 32 q^{40} - 20 q^{43} - 8 q^{46} - 8 q^{52} - 56 q^{55} + 8 q^{58} - 24 q^{61} - 40 q^{64} + 8 q^{67} - 12 q^{73} - 64 q^{76} - 20 q^{79} - 40 q^{82} + 32 q^{85} + 8 q^{88} - 8 q^{94} - 68 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 16x^{10} + 92x^{8} - 228x^{6} + 225x^{4} - 60x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{10} - 18\nu^{8} + 116\nu^{6} - 304\nu^{4} + 245\nu^{2} + 2 ) / 12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{10} - 18\nu^{8} + 116\nu^{6} - 316\nu^{4} + 317\nu^{2} - 46 ) / 12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{11} + 18\nu^{9} - 116\nu^{7} + 316\nu^{5} - 317\nu^{3} + 46\nu ) / 12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{10} + 15\nu^{8} - 77\nu^{6} + 154\nu^{4} - 95\nu^{2} + 7 ) / 3 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3\nu^{11} - 46\nu^{9} + 248\nu^{7} - 556\nu^{5} + 467\nu^{3} - 90\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 5\nu^{10} - 78\nu^{8} + 436\nu^{6} - 1040\nu^{4} + 937\nu^{2} - 146 ) / 12 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 11\nu^{11} - 174\nu^{9} + 976\nu^{7} - 2276\nu^{5} + 1867\nu^{3} - 146\nu ) / 24 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -23\nu^{11} + 366\nu^{9} - 2080\nu^{7} + 5012\nu^{5} - 4543\nu^{3} + 746\nu ) / 24 \) 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
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{5} + \beta_{4} + 6\beta_{2} + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{10} - \beta_{8} + \beta_{6} + 7\beta_{3} + 26\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{9} + \beta_{7} - 10\beta_{5} + 9\beta_{4} + 34\beta_{2} + 72 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -2\beta_{11} + 8\beta_{10} - 12\beta_{8} + 13\beta_{6} + 43\beta_{3} + 141\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 13\beta_{9} + 12\beta_{7} - 80\beta_{5} + 63\beta_{4} + 192\beta_{2} + 389 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -25\beta_{11} + 51\beta_{10} - 100\beta_{8} + 118\beta_{6} + 255\beta_{3} + 785\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 118\beta_{9} + 100\beta_{7} - 584\beta_{5} + 406\beta_{4} + 1091\beta_{2} + 2169 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -218\beta_{11} + 306\beta_{10} - 724\beta_{8} + 920\beta_{6} + 1497\beta_{3} + 4451\beta_1 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.46169
−2.16360
−1.80357
−1.28254
−0.509646
−0.318530
0.318530
0.509646
1.28254
1.80357
2.16360
2.46169
−2.46169 0 4.05990 2.84709 0 0 −5.07082 0 −7.00865
1.2 −2.16360 0 2.68115 0.888156 0 0 −1.47372 0 −1.92161
1.3 −1.80357 0 1.25286 −2.82630 0 0 1.34752 0 5.09742
1.4 −1.28254 0 −0.355099 −1.64096 0 0 3.02050 0 2.10459
1.5 −0.509646 0 −1.74026 4.28332 0 0 1.90621 0 −2.18298
1.6 −0.318530 0 −1.89854 0.278702 0 0 1.24180 0 −0.0887749
1.7 0.318530 0 −1.89854 −0.278702 0 0 −1.24180 0 −0.0887749
1.8 0.509646 0 −1.74026 −4.28332 0 0 −1.90621 0 −2.18298
1.9 1.28254 0 −0.355099 1.64096 0 0 −3.02050 0 2.10459
1.10 1.80357 0 1.25286 2.82630 0 0 −1.34752 0 5.09742
1.11 2.16360 0 2.68115 −0.888156 0 0 1.47372 0 −1.92161
1.12 2.46169 0 4.05990 −2.84709 0 0 5.07082 0 −7.00865
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(1\)
\(13\) \(1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5733.2.a.ca ✓ 12
3.b odd 2 1 inner 5733.2.a.ca ✓ 12
7.b odd 2 1 5733.2.a.cb yes 12
21.c even 2 1 5733.2.a.cb yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5733.2.a.ca ✓ 12 1.a even 1 1 trivial
5733.2.a.ca ✓ 12 3.b odd 2 1 inner
5733.2.a.cb yes 12 7.b odd 2 1
5733.2.a.cb yes 12 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5733))\):

\( T_{2}^{12} - 16T_{2}^{10} + 92T_{2}^{8} - 228T_{2}^{6} + 225T_{2}^{4} - 60T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{5}^{12} - 38T_{5}^{10} + 485T_{5}^{8} - 2552T_{5}^{6} + 5096T_{5}^{4} - 2904T_{5}^{2} + 196 \) Copy content Toggle raw display
\( T_{11}^{12} - 56T_{11}^{10} + 1108T_{11}^{8} - 9120T_{11}^{6} + 27844T_{11}^{4} - 18016T_{11}^{2} + 64 \) Copy content Toggle raw display
\( T_{17}^{12} - 60T_{17}^{10} + 900T_{17}^{8} - 3648T_{17}^{6} + 5888T_{17}^{4} - 4096T_{17}^{2} + 1024 \) Copy content Toggle raw display
\( T_{19}^{6} + 14T_{19}^{5} + 31T_{19}^{4} - 256T_{19}^{3} - 1072T_{19}^{2} - 384T_{19} + 1472 \) Copy content Toggle raw display
\( T_{31}^{6} + 6T_{31}^{5} - 49T_{31}^{4} - 208T_{31}^{3} + 440T_{31}^{2} - 160T_{31} - 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 16 T^{10} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - 38 T^{10} + \cdots + 196 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} - 56 T^{10} + \cdots + 64 \) Copy content Toggle raw display
$13$ \( (T + 1)^{12} \) Copy content Toggle raw display
$17$ \( T^{12} - 60 T^{10} + \cdots + 1024 \) Copy content Toggle raw display
$19$ \( (T^{6} + 14 T^{5} + \cdots + 1472)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} - 146 T^{10} + \cdots + 1597696 \) Copy content Toggle raw display
$29$ \( T^{12} - 114 T^{10} + \cdots + 1024 \) Copy content Toggle raw display
$31$ \( (T^{6} + 6 T^{5} - 49 T^{4} + \cdots - 32)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 136 T^{4} + \cdots - 2176)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} - 184 T^{10} + \cdots + 17909824 \) Copy content Toggle raw display
$43$ \( (T^{6} + 10 T^{5} + \cdots - 2032)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} - 230 T^{10} + \cdots + 722500 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 66357760000 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 339149056 \) Copy content Toggle raw display
$61$ \( (T^{6} + 12 T^{5} + \cdots - 12800)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} - 4 T^{5} + \cdots - 7616)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 6091490304 \) Copy content Toggle raw display
$73$ \( (T^{6} + 6 T^{5} + \cdots + 9200)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 10 T^{5} + \cdots + 6404)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 5379048964 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 19941393796 \) Copy content Toggle raw display
$97$ \( (T^{6} + 34 T^{5} + \cdots + 255600)^{2} \) Copy content Toggle raw display
show more
show less