Properties

Label 6003.2.a.i
Level $6003$
Weight $2$
Character orbit 6003.a
Self dual yes
Analytic conductor $47.934$
Analytic rank $1$
Dimension $7$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6003,2,Mod(1,6003)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6003, 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("6003.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6003.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: \(47.9341963334\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 9x^{5} + 10x^{4} + 19x^{3} - 20x^{2} - 5x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2001)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + (\beta_{5} + \beta_{4} + 1) q^{4} + (\beta_{6} - \beta_{3} + 1) q^{5} + ( - \beta_{5} - \beta_{4} - \beta_{2} - 1) q^{7} + ( - \beta_{6} - \beta_{2} + \beta_1 - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + (\beta_{5} + \beta_{4} + 1) q^{4} + (\beta_{6} - \beta_{3} + 1) q^{5} + ( - \beta_{5} - \beta_{4} - \beta_{2} - 1) q^{7} + ( - \beta_{6} - \beta_{2} + \beta_1 - 1) q^{8} + (\beta_{6} - \beta_{5} - 2 \beta_{4} - 1) q^{10} + ( - \beta_{6} + \beta_{5} + 2) q^{11} + (\beta_{5} - \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 1) q^{13} + (\beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - 2 \beta_1 + 1) q^{14} + ( - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_1 - 2) q^{16} + ( - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + 2) q^{17} + (\beta_{4} + 2 \beta_{3} - 3 \beta_1 - 1) q^{19} + (\beta_{6} - 2 \beta_{3} + \beta_{2} - \beta_1 + 2) q^{20} + ( - \beta_{6} + 2 \beta_{4} + 3 \beta_{3} - \beta_{2} + \beta_1 - 2) q^{22} + q^{23} + (\beta_{6} + \beta_{4} - \beta_{2} - \beta_1 - 1) q^{25} + (\beta_{5} - 3 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 1) q^{26} + ( - 3 \beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 4) q^{28} - q^{29} + ( - \beta_{5} + \beta_{4} + \beta_{2} + 2 \beta_1 - 2) q^{31} + ( - \beta_{5} + \beta_{4} - \beta_1) q^{32} + ( - 2 \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_{2} - 4) q^{34} + ( - 2 \beta_{5} - \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 2) q^{35} + ( - 2 \beta_{4} - 2 \beta_{2} - \beta_1 - 3) q^{37} + ( - \beta_{6} - 4 \beta_{5} - 2 \beta_{4} - 3 \beta_{2} - 1) q^{38} + ( - \beta_{6} - 3 \beta_{5} - \beta_{4} + \beta_1 - 4) q^{40} + ( - \beta_{6} - 4 \beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1 - 4) q^{41} + (\beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + 2 \beta_1 + 2) q^{43} + ( - \beta_{6} + 4 \beta_{5} + 4 \beta_{4} + 2 \beta_{3} - \beta_1 + 3) q^{44} + \beta_{3} q^{46} + (\beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1 - 5) q^{47} + (\beta_{6} + 3 \beta_{5} + \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - \beta_1) q^{49} + ( - \beta_{5} - 2 \beta_{4} - 2 \beta_{2} - \beta_1 - 1) q^{50} + ( - 2 \beta_{5} - 2 \beta_{4} + \beta_{2} - 1) q^{52} + ( - \beta_{6} + 2 \beta_{5} + 3 \beta_{4} + 2 \beta_{3} - \beta_{2} - 1) q^{53} + (\beta_{6} + \beta_{5} - 2 \beta_{3} + 2 \beta_{2} - \beta_1) q^{55} + (2 \beta_{5} - 3 \beta_{3} + 3 \beta_{2} + 5) q^{56} - \beta_{3} q^{58} + (2 \beta_{4} + 2 \beta_{3} - 2 \beta_1 - 2) q^{59} + (2 \beta_{6} - 3 \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_{2} - \beta_1 - 1) q^{61} + ( - \beta_{6} + 3 \beta_{5} - \beta_{4} - 2 \beta_{3} + 4 \beta_{2} + 3) q^{62} + (\beta_{6} - 4 \beta_{5} - 5 \beta_{4} - \beta_{3} + \beta_1 + 1) q^{64} + ( - 4 \beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 2) q^{65} + (\beta_{6} - \beta_{5} + 3 \beta_{4} - \beta_{3} - \beta_{2} - 4) q^{67} + (3 \beta_{6} + \beta_{5} - 5 \beta_{3} + 2 \beta_{2}) q^{68} + (\beta_{6} + 4 \beta_{5} + 3 \beta_{4} - \beta_{3} - 4 \beta_1 + 7) q^{70} + (\beta_{5} - \beta_{4} - \beta_{3} - \beta_1 + 3) q^{71} + ( - 4 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} - \beta_{2} - 2 \beta_1 - 6) q^{73} + (2 \beta_{6} + 3 \beta_{4} - 3 \beta_{3} - 3 \beta_{2} - 2 \beta_1) q^{74} + (\beta_{6} + 3 \beta_{5} - 3 \beta_{3} + \beta_{2} - \beta_1 + 2) q^{76} + ( - \beta_{6} - 3 \beta_{5} - 4 \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{77} + ( - 2 \beta_{6} + \beta_{5} + \beta_{4} + \beta_{2} - 2 \beta_1 - 6) q^{79} + ( - 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{2} - \beta_1 - 3) q^{80} + ( - 2 \beta_{6} + \beta_{5} - 2 \beta_{4} - \beta_{3} + 2 \beta_{2} - 5 \beta_1 + 1) q^{82} + ( - \beta_{6} + 3 \beta_{5} + \beta_{4} - \beta_{3} + 5 \beta_{2} + 2 \beta_1 - 1) q^{83} + ( - \beta_{6} + 4 \beta_{5} + 4 \beta_{4} - 2 \beta_{3} + \beta_{2} + 3 \beta_1) q^{85} + (2 \beta_{6} - \beta_{5} - 3 \beta_{3} + 4 \beta_{2} + 4 \beta_1) q^{86} + ( - 3 \beta_{6} - 2 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 2) q^{88} + (\beta_{6} - 3 \beta_{5} - 3 \beta_{4} + 2 \beta_{3} - 3 \beta_1 + 1) q^{89} + ( - 3 \beta_{6} + \beta_{5} + 3 \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1 - 4) q^{91} + (\beta_{5} + \beta_{4} + 1) q^{92} + ( - \beta_{6} + 6 \beta_{5} + 3 \beta_{4} - 3 \beta_{3} + \beta_{2} - \beta_1 + 6) q^{94} + (\beta_{6} + 4 \beta_{5} - \beta_{3} - 2 \beta_1 + 1) q^{95} + (\beta_{6} + \beta_{5} - 2 \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 7) q^{97} + ( - 6 \beta_{5} - 4 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 5 \beta_1 - 7) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + q^{2} + 7 q^{4} + 5 q^{5} - 5 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + q^{2} + 7 q^{4} + 5 q^{5} - 5 q^{7} - 3 q^{8} - 11 q^{10} + 12 q^{11} - 13 q^{13} + 3 q^{14} - 13 q^{16} + 12 q^{17} - 5 q^{19} + 8 q^{20} - q^{22} + 7 q^{23} - 4 q^{25} - 2 q^{26} - 21 q^{28} - 7 q^{29} - 8 q^{31} + 5 q^{32} - 28 q^{34} - 5 q^{35} - 24 q^{37} + 6 q^{38} - 20 q^{40} - 9 q^{41} - q^{43} + 23 q^{44} + q^{46} - 27 q^{47} - 14 q^{49} - 7 q^{50} - 9 q^{52} + q^{53} - 11 q^{55} + 20 q^{56} - q^{58} - 8 q^{59} + q^{61} + 3 q^{64} - 12 q^{65} - 16 q^{67} - 15 q^{68} + 40 q^{70} + 13 q^{71} - 23 q^{73} + 8 q^{74} - 2 q^{76} - 13 q^{77} - 44 q^{79} - 30 q^{80} - 10 q^{82} - 21 q^{83} - 6 q^{86} + 21 q^{88} + 5 q^{89} - 18 q^{91} + 7 q^{92} + 28 q^{94} - 9 q^{95} - 55 q^{97} - 36 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - x^{6} - 9x^{5} + 10x^{4} + 19x^{3} - 20x^{2} - 5x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 2\nu^{5} - 7\nu^{4} - 11\nu^{3} + 14\nu^{2} + 10\nu - 7 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} - 2\nu^{5} + 7\nu^{4} + 15\nu^{3} - 10\nu^{2} - 26\nu - 1 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} - 9\nu^{4} + \nu^{3} + 20\nu^{2} - 2\nu - 5 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{6} - 2\nu^{5} + 25\nu^{4} + 9\nu^{3} - 50\nu^{2} - 6\nu + 5 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{6} - 2\nu^{5} - 25\nu^{4} + 23\nu^{3} + 46\nu^{2} - 46\nu - 5 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} + \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{5} - \beta_{4} + \beta_{3} + 4\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} + 7\beta_{5} + 5\beta_{4} - 2\beta_{3} + 6\beta_{2} - \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{6} - 10\beta_{5} - 9\beta_{4} + 8\beta_{3} - \beta_{2} + 19\beta _1 - 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 9\beta_{6} + 44\beta_{5} + 28\beta_{4} - 19\beta_{3} + 34\beta_{2} - 11\beta _1 + 81 \) Copy content Toggle raw display

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
−2.51747
1.40506
0.136094
−1.64802
−0.325238
2.13025
1.81932
−2.35219 0 3.53279 2.63525 0 −4.33765 −3.60540 0 −6.19860
1.2 −1.75752 0 1.08886 2.67591 0 0.0258051 1.60134 0 −4.70295
1.3 −1.17088 0 −0.629030 −0.418864 0 1.98148 3.07829 0 0.490441
1.4 0.865893 0 −1.25023 −1.66575 0 −0.715958 −2.81435 0 −1.44236
1.5 1.49169 0 0.225141 2.94997 0 1.89422 −2.64754 0 4.40045
1.6 1.55072 0 0.404722 0.920116 0 −2.53796 −2.47382 0 1.42684
1.7 2.37229 0 3.62775 −2.09663 0 −1.30993 3.86149 0 −4.97382
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(23\) \(-1\)
\(29\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6003.2.a.i 7
3.b odd 2 1 2001.2.a.j 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2001.2.a.j 7 3.b odd 2 1
6003.2.a.i 7 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6003))\):

\( T_{2}^{7} - T_{2}^{6} - 10T_{2}^{5} + 10T_{2}^{4} + 29T_{2}^{3} - 29T_{2}^{2} - 24T_{2} + 23 \) Copy content Toggle raw display
\( T_{5}^{7} - 5T_{5}^{6} - 3T_{5}^{5} + 40T_{5}^{4} - 15T_{5}^{3} - 87T_{5}^{2} + 36T_{5} + 28 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} - T^{6} - 10 T^{5} + 10 T^{4} + \cdots + 23 \) Copy content Toggle raw display
$3$ \( T^{7} \) Copy content Toggle raw display
$5$ \( T^{7} - 5 T^{6} - 3 T^{5} + 40 T^{4} + \cdots + 28 \) Copy content Toggle raw display
$7$ \( T^{7} + 5 T^{6} - 5 T^{5} - 38 T^{4} + \cdots - 1 \) Copy content Toggle raw display
$11$ \( T^{7} - 12 T^{6} + 39 T^{5} + 9 T^{4} + \cdots - 23 \) Copy content Toggle raw display
$13$ \( T^{7} + 13 T^{6} + 27 T^{5} + \cdots - 1867 \) Copy content Toggle raw display
$17$ \( T^{7} - 12 T^{6} - 2 T^{5} + \cdots - 12751 \) Copy content Toggle raw display
$19$ \( T^{7} + 5 T^{6} - 62 T^{5} + \cdots - 7244 \) Copy content Toggle raw display
$23$ \( (T - 1)^{7} \) Copy content Toggle raw display
$29$ \( (T + 1)^{7} \) Copy content Toggle raw display
$31$ \( T^{7} + 8 T^{6} - 54 T^{5} + \cdots - 8156 \) Copy content Toggle raw display
$37$ \( T^{7} + 24 T^{6} + 146 T^{5} + \cdots + 85172 \) Copy content Toggle raw display
$41$ \( T^{7} + 9 T^{6} - 187 T^{5} + \cdots + 1065508 \) Copy content Toggle raw display
$43$ \( T^{7} + T^{6} - 115 T^{5} + \cdots - 15268 \) Copy content Toggle raw display
$47$ \( T^{7} + 27 T^{6} + 226 T^{5} + \cdots + 10031 \) Copy content Toggle raw display
$53$ \( T^{7} - T^{6} - 105 T^{5} - 506 T^{4} + \cdots - 596 \) Copy content Toggle raw display
$59$ \( T^{7} + 8 T^{6} - 36 T^{5} - 192 T^{4} + \cdots - 128 \) Copy content Toggle raw display
$61$ \( T^{7} - T^{6} - 134 T^{5} + \cdots - 36740 \) Copy content Toggle raw display
$67$ \( T^{7} + 16 T^{6} - 80 T^{5} + \cdots + 124927 \) Copy content Toggle raw display
$71$ \( T^{7} - 13 T^{6} + 24 T^{5} + 201 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$73$ \( T^{7} + 23 T^{6} - 59 T^{5} + \cdots + 4228 \) Copy content Toggle raw display
$79$ \( T^{7} + 44 T^{6} + 692 T^{5} + \cdots + 127580 \) Copy content Toggle raw display
$83$ \( T^{7} + 21 T^{6} - 41 T^{5} + \cdots + 212788 \) Copy content Toggle raw display
$89$ \( T^{7} - 5 T^{6} - 195 T^{5} + \cdots + 18865 \) Copy content Toggle raw display
$97$ \( T^{7} + 55 T^{6} + 1170 T^{5} + \cdots - 1979204 \) Copy content Toggle raw display
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