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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Newspace parameters

Level: \( N \) \(=\) \( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6003.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(47.9341963334\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - x^{6} - 9 x^{5} + 10 x^{4} + 19 x^{3} - 20 x^{2} - 5 x + 1\)
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

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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} + 2 \nu^{5} - 7 \nu^{4} - 11 \nu^{3} + 14 \nu^{2} + 10 \nu - 7 \)\()/4\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{6} - 2 \nu^{5} + 7 \nu^{4} + 15 \nu^{3} - 10 \nu^{2} - 26 \nu - 1 \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{6} - 9 \nu^{4} + \nu^{3} + 20 \nu^{2} - 2 \nu - 5 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{6} - 2 \nu^{5} + 25 \nu^{4} + 9 \nu^{3} - 50 \nu^{2} - 6 \nu + 5 \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( 3 \nu^{6} - 2 \nu^{5} - 25 \nu^{4} + 23 \nu^{3} + 46 \nu^{2} - 46 \nu - 5 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} + \beta_{4} + \beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(-\beta_{5} - \beta_{4} + \beta_{3} + 4 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(\beta_{6} + 7 \beta_{5} + 5 \beta_{4} - 2 \beta_{3} + 6 \beta_{2} - \beta_{1} + 15\)
\(\nu^{5}\)\(=\)\(-\beta_{6} - 10 \beta_{5} - 9 \beta_{4} + 8 \beta_{3} - \beta_{2} + 19 \beta_{1} - 11\)
\(\nu^{6}\)\(=\)\(9 \beta_{6} + 44 \beta_{5} + 28 \beta_{4} - 19 \beta_{3} + 34 \beta_{2} - 11 \beta_{1} + 81\)

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.

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6003))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 23 - 24 T - 29 T^{2} + 29 T^{3} + 10 T^{4} - 10 T^{5} - T^{6} + T^{7} \)
$3$ \( T^{7} \)
$5$ \( 28 + 36 T - 87 T^{2} - 15 T^{3} + 40 T^{4} - 3 T^{5} - 5 T^{6} + T^{7} \)
$7$ \( -1 + 37 T + 68 T^{2} - 3 T^{3} - 38 T^{4} - 5 T^{5} + 5 T^{6} + T^{7} \)
$11$ \( -23 + 90 T + 2 T^{2} - 150 T^{3} + 9 T^{4} + 39 T^{5} - 12 T^{6} + T^{7} \)
$13$ \( -1867 - 4597 T - 4314 T^{2} - 1849 T^{3} - 294 T^{4} + 27 T^{5} + 13 T^{6} + T^{7} \)
$17$ \( -12751 + 14263 T - 3350 T^{2} - 1068 T^{3} + 443 T^{4} - 2 T^{5} - 12 T^{6} + T^{7} \)
$19$ \( -7244 - 3664 T + 3993 T^{2} + 909 T^{3} - 306 T^{4} - 62 T^{5} + 5 T^{6} + T^{7} \)
$23$ \( ( -1 + T )^{7} \)
$29$ \( ( 1 + T )^{7} \)
$31$ \( -8156 + 6494 T + 7247 T^{2} + 278 T^{3} - 515 T^{4} - 54 T^{5} + 8 T^{6} + T^{7} \)
$37$ \( 85172 + 64270 T - 4649 T^{2} - 6432 T^{3} - 483 T^{4} + 146 T^{5} + 24 T^{6} + T^{7} \)
$41$ \( 1065508 + 685908 T + 136507 T^{2} + 1335 T^{3} - 2446 T^{4} - 187 T^{5} + 9 T^{6} + T^{7} \)
$43$ \( -15268 + 7782 T + 13761 T^{2} + 2793 T^{3} - 380 T^{4} - 115 T^{5} + T^{6} + T^{7} \)
$47$ \( 10031 - 4296 T - 12110 T^{2} - 3545 T^{3} + 326 T^{4} + 226 T^{5} + 27 T^{6} + T^{7} \)
$53$ \( -596 + 1262 T + 955 T^{2} - 525 T^{3} - 506 T^{4} - 105 T^{5} - T^{6} + T^{7} \)
$59$ \( -128 + 128 T + 640 T^{2} + 192 T^{3} - 192 T^{4} - 36 T^{5} + 8 T^{6} + T^{7} \)
$61$ \( -36740 - 16402 T + 7949 T^{2} + 3219 T^{3} - 263 T^{4} - 134 T^{5} - T^{6} + T^{7} \)
$67$ \( 124927 + 60419 T - 16308 T^{2} - 14086 T^{3} - 2557 T^{4} - 80 T^{5} + 16 T^{6} + T^{7} \)
$71$ \( 16 + 4 T - 367 T^{2} - 479 T^{3} + 201 T^{4} + 24 T^{5} - 13 T^{6} + T^{7} \)
$73$ \( 4228 + 55498 T - 11447 T^{2} - 20226 T^{3} - 3953 T^{4} - 59 T^{5} + 23 T^{6} + T^{7} \)
$79$ \( 127580 - 162018 T - 78309 T^{2} + 772 T^{3} + 4205 T^{4} + 692 T^{5} + 44 T^{6} + T^{7} \)
$83$ \( 212788 + 184452 T + 32991 T^{2} - 7657 T^{3} - 2362 T^{4} - 41 T^{5} + 21 T^{6} + T^{7} \)
$89$ \( 18865 - 48069 T + 18732 T^{2} + 11085 T^{3} + 274 T^{4} - 195 T^{5} - 5 T^{6} + T^{7} \)
$97$ \( -1979204 - 998308 T - 62983 T^{2} + 46413 T^{3} + 11671 T^{4} + 1170 T^{5} + 55 T^{6} + T^{7} \)
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