Properties

Label 6003.2.a.m
Level $6003$
Weight $2$
Character orbit 6003.a
Self dual yes
Analytic conductor $47.934$
Analytic rank $1$
Dimension $11$
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: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - 2 x^{10} - 18 x^{9} + 30 x^{8} + 124 x^{7} - 152 x^{6} - 408 x^{5} + 285 x^{4} + 634 x^{3} - 93 x^{2} - 369 x - 108 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
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 \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + \beta_{4} q^{5} - \beta_{6} q^{7} + ( - \beta_{3} - \beta_1 - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + \beta_{4} q^{5} - \beta_{6} q^{7} + ( - \beta_{3} - \beta_1 - 1) q^{8} + ( - \beta_{9} + \beta_{8} - \beta_{7} - \beta_{4} + 1) q^{10} + (\beta_{9} + \beta_{6} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 2) q^{11} + \beta_{7} q^{13} + ( - \beta_{10} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{14} + (\beta_{9} + \beta_{7} + \beta_{5} + \beta_{3} + \beta_{2} + \beta_1 + 2) q^{16} + (\beta_{10} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_1 - 2) q^{17} + ( - \beta_{10} + \beta_{9} + \beta_{6} - \beta_{4} - \beta_{2} + \beta_1 - 1) q^{19} + (\beta_{10} - \beta_{6} - \beta_{5} + 2 \beta_{4} - 2) q^{20} + (\beta_{10} - \beta_{9} - 2 \beta_{8} - \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - \beta_{2} - 3) q^{22} - q^{23} + ( - \beta_{9} + \beta_{8} - \beta_{7} - \beta_{3} + \beta_{2} + 1) q^{25} + ( - \beta_{10} + \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_1 + 1) q^{26} + (\beta_{10} + \beta_{7} - \beta_{6} + \beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1 + 1) q^{28} + q^{29} + (\beta_{6} + \beta_{5} - \beta_{3} + 4) q^{31} + ( - \beta_{10} - \beta_{9} - \beta_{8} + \beta_{5} + \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - \beta_1 - 2) q^{32} + ( - \beta_{7} + 2 \beta_{6} - \beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1 + 1) q^{34} + ( - \beta_{8} - \beta_{6} + \beta_{3} - 2 \beta_1 - 2) q^{35} + ( - \beta_{9} - \beta_{8} - \beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} - 2) q^{37} + (\beta_{10} - \beta_{9} - 3 \beta_{8} - 4 \beta_{6} - 2 \beta_{5} + \beta_{4} + \beta_{2} - 2 \beta_1 - 1) q^{38} + ( - \beta_{10} + \beta_{8} - \beta_{7} + 3 \beta_{6} + \beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} + \cdots - 1) q^{40}+ \cdots + ( - \beta_{10} - \beta_{9} - 3 \beta_{7} + \beta_{6} - 7 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} + \cdots - 9) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q - 2 q^{2} + 18 q^{4} - 2 q^{5} + 3 q^{7} - 18 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 11 q - 2 q^{2} + 18 q^{4} - 2 q^{5} + 3 q^{7} - 18 q^{8} + 14 q^{10} - 11 q^{11} - 5 q^{13} - 17 q^{14} + 20 q^{16} - 15 q^{17} - 6 q^{19} - 21 q^{20} - 10 q^{22} - 11 q^{23} + 3 q^{25} + 5 q^{26} + 7 q^{28} + 11 q^{29} + 35 q^{31} - 28 q^{32} + 28 q^{34} - 15 q^{35} - 28 q^{37} + 2 q^{38} - q^{40} - 10 q^{41} - 6 q^{43} - 18 q^{44} + 2 q^{46} - 15 q^{47} + 22 q^{49} - 15 q^{50} - 36 q^{52} + 7 q^{53} - 12 q^{55} - 56 q^{56} - 2 q^{58} + 20 q^{59} - 20 q^{61} + 11 q^{62} + 36 q^{64} - 11 q^{65} - 39 q^{67} - 35 q^{68} + 38 q^{70} - 49 q^{71} - 3 q^{73} - 37 q^{74} - 18 q^{76} - 25 q^{77} + 41 q^{79} - 51 q^{80} - 19 q^{82} - 13 q^{83} - 62 q^{86} - 40 q^{88} - 34 q^{89} + 2 q^{91} - 18 q^{92} - 14 q^{94} - 25 q^{95} - 11 q^{97} - 53 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{11} - 2 x^{10} - 18 x^{9} + 30 x^{8} + 124 x^{7} - 152 x^{6} - 408 x^{5} + 285 x^{4} + 634 x^{3} - 93 x^{2} - 369 x - 108 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu - 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{10} + 5 \nu^{9} + 9 \nu^{8} - 63 \nu^{7} - 13 \nu^{6} + 251 \nu^{5} - 27 \nu^{4} - 360 \nu^{3} - 4 \nu^{2} + 165 \nu + 48 ) / 6 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 2 \nu^{10} + 7 \nu^{9} + 21 \nu^{8} - 87 \nu^{7} - 59 \nu^{6} + 343 \nu^{5} + 63 \nu^{4} - 498 \nu^{3} - 170 \nu^{2} + 261 \nu + 153 ) / 9 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - \nu^{10} - \nu^{9} + 24 \nu^{8} + 6 \nu^{7} - 187 \nu^{6} + 23 \nu^{5} + 567 \nu^{4} - 150 \nu^{3} - 598 \nu^{2} + 117 \nu + 162 ) / 9 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{8} - 2\nu^{7} - 12\nu^{6} + 22\nu^{5} + 44\nu^{4} - 68\nu^{3} - 57\nu^{2} + 52\nu + 31 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 4 \nu^{10} + 14 \nu^{9} + 42 \nu^{8} - 174 \nu^{7} - 109 \nu^{6} + 686 \nu^{5} + 18 \nu^{4} - 996 \nu^{3} - 7 \nu^{2} + 504 \nu + 135 ) / 9 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2 \nu^{10} - 7 \nu^{9} - 30 \nu^{8} + 105 \nu^{7} + 167 \nu^{6} - 541 \nu^{5} - 450 \nu^{4} + 1101 \nu^{3} + 620 \nu^{2} - 693 \nu - 369 ) / 9 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - \nu^{10} + 5 \nu^{9} + 15 \nu^{8} - 75 \nu^{7} - 91 \nu^{6} + 389 \nu^{5} + 303 \nu^{4} - 822 \nu^{3} - 538 \nu^{2} + 591 \nu + 354 ) / 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} + \beta_{7} + \beta_{5} + \beta_{3} + 7\beta_{2} + \beta _1 + 22 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{10} + \beta_{9} + \beta_{8} - \beta_{5} - \beta_{4} + 10\beta_{3} + 2\beta_{2} + 29\beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 12\beta_{9} + \beta_{8} + 12\beta_{7} + 10\beta_{5} + 12\beta_{3} + 47\beta_{2} + 14\beta _1 + 135 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 12 \beta_{10} + 16 \beta_{9} + 12 \beta_{8} + 3 \beta_{7} + \beta_{6} - 10 \beta_{5} - 10 \beta_{4} + 82 \beta_{3} + 25 \beta_{2} + 184 \beta _1 + 89 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2 \beta_{10} + 110 \beta_{9} + 14 \beta_{8} + 107 \beta_{7} + 2 \beta_{6} + 78 \beta_{5} + 2 \beta_{4} + 112 \beta_{3} + 319 \beta_{2} + 142 \beta _1 + 875 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 105 \beta_{10} + 172 \beta_{9} + 106 \beta_{8} + 53 \beta_{7} + 15 \beta_{6} - 73 \beta_{5} - 71 \beta_{4} + 629 \beta_{3} + 240 \beta_{2} + 1229 \beta _1 + 752 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 38 \beta_{10} + 910 \beta_{9} + 138 \beta_{8} + 856 \beta_{7} + 30 \beta_{6} + 559 \beta_{5} + 36 \beta_{4} + 954 \beta_{3} + 2194 \beta_{2} + 1266 \beta _1 + 5861 \) 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
2.72479
2.70316
2.24285
1.76484
1.17662
−0.467085
−0.661934
−1.05971
−1.94502
−1.96728
−2.51124
−2.72479 0 5.42450 1.02492 0 4.32726 −9.33107 0 −2.79269
1.2 −2.70316 0 5.30708 −2.79988 0 0.880738 −8.93955 0 7.56852
1.3 −2.24285 0 3.03039 −3.33222 0 0.336371 −2.31101 0 7.47368
1.4 −1.76484 0 1.11465 2.54815 0 −4.97592 1.56250 0 −4.49707
1.5 −1.17662 0 −0.615555 −2.85352 0 3.62966 3.07753 0 3.35752
1.6 0.467085 0 −1.78183 0.0105419 0 −1.85912 −1.76644 0 0.00492395
1.7 0.661934 0 −1.56184 1.16115 0 4.80000 −2.35770 0 0.768607
1.8 1.05971 0 −0.877023 1.30384 0 0.720797 −3.04880 0 1.38169
1.9 1.94502 0 1.78310 −0.890641 0 −3.69089 −0.421884 0 −1.73231
1.10 1.96728 0 1.87020 3.90206 0 −0.839519 −0.255353 0 7.67646
1.11 2.51124 0 4.30634 −2.07440 0 −0.329384 5.79178 0 −5.20933
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
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.m 11
3.b odd 2 1 2001.2.a.l 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2001.2.a.l 11 3.b odd 2 1
6003.2.a.m 11 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6003))\):

\( T_{2}^{11} + 2 T_{2}^{10} - 18 T_{2}^{9} - 30 T_{2}^{8} + 124 T_{2}^{7} + 152 T_{2}^{6} - 408 T_{2}^{5} - 285 T_{2}^{4} + 634 T_{2}^{3} + 93 T_{2}^{2} - 369 T_{2} + 108 \) Copy content Toggle raw display
\( T_{5}^{11} + 2 T_{5}^{10} - 27 T_{5}^{9} - 53 T_{5}^{8} + 233 T_{5}^{7} + 387 T_{5}^{6} - 862 T_{5}^{5} - 896 T_{5}^{4} + 1499 T_{5}^{3} + 470 T_{5}^{2} - 764 T_{5} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{11} + 2 T^{10} - 18 T^{9} - 30 T^{8} + \cdots + 108 \) Copy content Toggle raw display
$3$ \( T^{11} \) Copy content Toggle raw display
$5$ \( T^{11} + 2 T^{10} - 27 T^{9} - 53 T^{8} + \cdots + 8 \) Copy content Toggle raw display
$7$ \( T^{11} - 3 T^{10} - 45 T^{9} + 124 T^{8} + \cdots + 152 \) Copy content Toggle raw display
$11$ \( T^{11} + 11 T^{10} - 14 T^{9} + \cdots + 5372 \) Copy content Toggle raw display
$13$ \( T^{11} + 5 T^{10} - 59 T^{9} + \cdots - 5683 \) Copy content Toggle raw display
$17$ \( T^{11} + 15 T^{10} - 1033 T^{8} + \cdots + 14592 \) Copy content Toggle raw display
$19$ \( T^{11} + 6 T^{10} - 129 T^{9} + \cdots - 49408 \) Copy content Toggle raw display
$23$ \( (T + 1)^{11} \) Copy content Toggle raw display
$29$ \( (T - 1)^{11} \) Copy content Toggle raw display
$31$ \( T^{11} - 35 T^{10} + 399 T^{9} + \cdots + 756944 \) Copy content Toggle raw display
$37$ \( T^{11} + 28 T^{10} + 210 T^{9} + \cdots + 759412 \) Copy content Toggle raw display
$41$ \( T^{11} + 10 T^{10} - 223 T^{9} + \cdots + 5506272 \) Copy content Toggle raw display
$43$ \( T^{11} + 6 T^{10} - 224 T^{9} + \cdots + 7091488 \) Copy content Toggle raw display
$47$ \( T^{11} + 15 T^{10} + \cdots + 958100256 \) Copy content Toggle raw display
$53$ \( T^{11} - 7 T^{10} - 331 T^{9} + \cdots + 80314464 \) Copy content Toggle raw display
$59$ \( T^{11} - 20 T^{10} - 190 T^{9} + \cdots - 88926336 \) Copy content Toggle raw display
$61$ \( T^{11} + 20 T^{10} - 132 T^{9} + \cdots - 40052104 \) Copy content Toggle raw display
$67$ \( T^{11} + 39 T^{10} + 335 T^{9} + \cdots - 26096464 \) Copy content Toggle raw display
$71$ \( T^{11} + 49 T^{10} + 780 T^{9} + \cdots - 66188592 \) Copy content Toggle raw display
$73$ \( T^{11} + 3 T^{10} + \cdots - 42266597504 \) Copy content Toggle raw display
$79$ \( T^{11} - 41 T^{10} + \cdots - 360776992 \) Copy content Toggle raw display
$83$ \( T^{11} + 13 T^{10} - 411 T^{9} + \cdots + 15116544 \) Copy content Toggle raw display
$89$ \( T^{11} + 34 T^{10} + 222 T^{9} + \cdots - 18922248 \) Copy content Toggle raw display
$97$ \( T^{11} + 11 T^{10} - 340 T^{9} + \cdots - 580896 \) Copy content Toggle raw display
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