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Properties

Label 7803.2.a.bv
Level $7803$
Weight $2$
Character orbit 7803.a
Self dual yes
Analytic conductor $62.307$
Analytic rank $1$
Dimension $12$
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] = [7803,2,Mod(1,7803)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7803, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("7803.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7803 = 3^{3} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7803.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: \(62.3072686972\)
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: \( x^{12} - 18x^{10} + 115x^{8} - 318x^{6} + 395x^{4} - 208x^{2} + 34 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 459)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + (\beta_{11} - \beta_1) q^{5} + ( - \beta_{9} - 1) q^{7} + (\beta_{3} + \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + (\beta_{11} - \beta_1) q^{5} + ( - \beta_{9} - 1) q^{7} + (\beta_{3} + \beta_1) q^{8} + (\beta_{10} + \beta_{4} - \beta_{2} - 2) q^{10} + (\beta_{11} - \beta_{8} - \beta_{5}) q^{11} + (\beta_{9} - \beta_{6}) q^{13} + ( - \beta_{8} - \beta_{3} - 2 \beta_1) q^{14} + ( - \beta_{10} + \beta_{9} - \beta_{4} + \cdots + 2) q^{16}+ \cdots + ( - \beta_{11} - \beta_{8} + \cdots + 3 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{4} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{4} - 8 q^{7} - 24 q^{10} - 4 q^{13} + 20 q^{16} - 4 q^{19} + 28 q^{22} - 48 q^{28} - 8 q^{31} - 12 q^{37} - 44 q^{40} - 20 q^{43} - 36 q^{46} + 28 q^{49} + 8 q^{52} - 4 q^{55} - 28 q^{58} - 48 q^{61} + 40 q^{64} - 8 q^{67} + 44 q^{70} - 4 q^{73} - 28 q^{76} - 40 q^{79} - 76 q^{82} + 40 q^{88} - 52 q^{91} + 28 q^{94} - 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 18x^{10} + 115x^{8} - 318x^{6} + 395x^{4} - 208x^{2} + 34 \) : 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
\(\beta_{4}\)\(=\) \( ( -\nu^{10} + 19\nu^{8} - 134\nu^{6} + 419\nu^{4} - 517\nu^{2} + 164 ) / 33 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{11} + 19\nu^{9} - 134\nu^{7} + 419\nu^{5} - 517\nu^{3} + 164\nu ) / 33 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 13\nu^{10} - 214\nu^{8} + 1181\nu^{6} - 2477\nu^{4} + 1771\nu^{2} - 317 ) / 33 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2\nu^{11} + 38\nu^{9} - 257\nu^{7} + 739\nu^{5} - 847\nu^{3} + 262\nu ) / 11 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 16\nu^{11} - 271\nu^{9} + 1550\nu^{7} - 3404\nu^{5} + 2497\nu^{3} - 314\nu ) / 33 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 16\nu^{10} - 271\nu^{8} + 1550\nu^{6} - 3404\nu^{4} + 2530\nu^{2} - 446 ) / 33 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 17\nu^{10} - 290\nu^{8} + 1684\nu^{6} - 3856\nu^{4} + 3278\nu^{2} - 775 ) / 33 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 17\nu^{11} - 290\nu^{9} + 1684\nu^{7} - 3856\nu^{5} + 3278\nu^{3} - 775\nu ) / 33 \) 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
\(\nu^{4}\)\(=\) \( -\beta_{10} + \beta_{9} - \beta_{4} + 7\beta_{2} + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{11} + \beta_{8} - \beta_{5} + 8\beta_{3} + 31\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -10\beta_{10} + 9\beta_{9} + \beta_{6} - 13\beta_{4} + 46\beta_{2} + 99 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -9\beta_{11} + 9\beta_{8} + \beta_{7} - 15\beta_{5} + 55\beta_{3} + 200\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -80\beta_{10} + 63\beta_{9} + 18\beta_{6} - 118\beta_{4} + 302\beta_{2} + 638 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -62\beta_{11} + 63\beta_{8} + 18\beta_{7} - 154\beta_{5} + 365\beta_{3} + 1305\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -599\beta_{10} + 410\beta_{9} + 208\beta_{6} - 952\beta_{4} + 1990\beta_{2} + 4173 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -391\beta_{11} + 410\beta_{8} + 208\beta_{7} - 1368\beta_{5} + 2400\beta_{3} + 8563\beta_1 \) 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
−2.62597
−2.47149
−1.61349
−1.10929
−0.934753
−0.537009
0.537009
0.934753
1.10929
1.61349
2.47149
2.62597
−2.62597 0 4.89571 2.81046 0 −3.79635 −7.60403 0 −7.38018
1.2 −2.47149 0 4.10827 1.30344 0 −2.43990 −5.21058 0 −3.22145
1.3 −1.61349 0 0.603350 −0.815992 0 2.52607 2.25348 0 1.31659
1.4 −1.10929 0 −0.769469 3.10245 0 4.31051 3.07215 0 −3.44153
1.5 −0.934753 0 −1.12624 −2.13241 0 −2.51416 2.92226 0 1.99328
1.6 −0.537009 0 −1.71162 2.35884 0 −2.08617 1.99317 0 −1.26672
1.7 0.537009 0 −1.71162 −2.35884 0 −2.08617 −1.99317 0 −1.26672
1.8 0.934753 0 −1.12624 2.13241 0 −2.51416 −2.92226 0 1.99328
1.9 1.10929 0 −0.769469 −3.10245 0 4.31051 −3.07215 0 −3.44153
1.10 1.61349 0 0.603350 0.815992 0 2.52607 −2.25348 0 1.31659
1.11 2.47149 0 4.10827 −1.30344 0 −2.43990 5.21058 0 −3.22145
1.12 2.62597 0 4.89571 −2.81046 0 −3.79635 7.60403 0 −7.38018
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(17\) \(1\)

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 7803.2.a.bv 12
3.b odd 2 1 inner 7803.2.a.bv 12
17.b even 2 1 7803.2.a.bw 12
17.d even 8 2 459.2.f.c ✓ 24
51.c odd 2 1 7803.2.a.bw 12
51.g odd 8 2 459.2.f.c ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
459.2.f.c ✓ 24 17.d even 8 2
459.2.f.c ✓ 24 51.g odd 8 2
7803.2.a.bv 12 1.a even 1 1 trivial
7803.2.a.bv 12 3.b odd 2 1 inner
7803.2.a.bw 12 17.b even 2 1
7803.2.a.bw 12 51.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7803))\):

\( T_{2}^{12} - 18T_{2}^{10} + 115T_{2}^{8} - 318T_{2}^{6} + 395T_{2}^{4} - 208T_{2}^{2} + 34 \) Copy content Toggle raw display
\( T_{5}^{12} - 30T_{5}^{10} + 345T_{5}^{8} - 1902T_{5}^{6} + 5105T_{5}^{4} - 5920T_{5}^{2} + 2176 \) Copy content Toggle raw display
\( T_{7}^{6} + 4T_{7}^{5} - 20T_{7}^{4} - 102T_{7}^{3} + 4T_{7}^{2} + 488T_{7} + 529 \) Copy content Toggle raw display
\( T_{11}^{12} - 60T_{11}^{10} + 1244T_{11}^{8} - 10672T_{11}^{6} + 32773T_{11}^{4} - 12916T_{11}^{2} + 34 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 18 T^{10} + \cdots + 34 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - 30 T^{10} + \cdots + 2176 \) Copy content Toggle raw display
$7$ \( (T^{6} + 4 T^{5} + \cdots + 529)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} - 60 T^{10} + \cdots + 34 \) Copy content Toggle raw display
$13$ \( (T^{6} + 2 T^{5} - 33 T^{4} + \cdots - 64)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} \) Copy content Toggle raw display
$19$ \( (T^{6} + 2 T^{5} + \cdots - 2007)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} - 146 T^{10} + \cdots + 44808736 \) Copy content Toggle raw display
$29$ \( T^{12} - 64 T^{10} + \cdots + 34 \) Copy content Toggle raw display
$31$ \( (T^{6} + 4 T^{5} + \cdots - 2338)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 6 T^{5} + \cdots - 2524)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} - 202 T^{10} + \cdots + 39304 \) Copy content Toggle raw display
$43$ \( (T^{6} + 10 T^{5} + \cdots + 34146)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 27343917954 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 451972744 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 4331476546 \) Copy content Toggle raw display
$61$ \( (T^{6} + 24 T^{5} + \cdots - 49409)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 4 T^{5} + \cdots - 8257)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 391072539136 \) Copy content Toggle raw display
$73$ \( (T^{6} + 2 T^{5} + \cdots + 2231)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 20 T^{5} + \cdots - 4063)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 2840480479714 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 162395230914 \) Copy content Toggle raw display
$97$ \( (T^{6} + 12 T^{5} + \cdots - 32743)^{2} \) Copy content Toggle raw display
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