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Properties

Label 6003.2.a.p
Level $6003$
Weight $2$
Character orbit 6003.a
Self dual yes
Analytic conductor $47.934$
Analytic rank $0$
Dimension $14$
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: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 2 x^{13} - 18 x^{12} + 34 x^{11} + 124 x^{10} - 216 x^{9} - 420 x^{8} + 647 x^{7} + 750 x^{6} + \cdots - 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} - \beta_{11} q^{5} + ( - \beta_{13} - \beta_{11} + \beta_{3}) q^{7} + (\beta_{4} + \beta_{3} + \beta_1 + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} - \beta_{11} q^{5} + ( - \beta_{13} - \beta_{11} + \beta_{3}) q^{7} + (\beta_{4} + \beta_{3} + \beta_1 + 1) q^{8} + ( - \beta_{11} + \beta_{10} - 1) q^{10} + (\beta_{6} + 1) q^{11} + ( - \beta_{8} + 1) q^{13} + ( - \beta_{13} - \beta_{11} + \cdots - \beta_1) q^{14}+ \cdots + (\beta_{13} - \beta_{12} - \beta_{11} + \cdots - 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 2 q^{2} + 12 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 2 q^{2} + 12 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8} - 5 q^{10} + 12 q^{11} + 13 q^{13} + 9 q^{14} + 14 q^{17} - 9 q^{19} + 2 q^{20} - 9 q^{22} + 14 q^{23} + 13 q^{25} + 16 q^{26} + 3 q^{28} + 14 q^{29} - 28 q^{31} + 4 q^{32} + 14 q^{34} + 9 q^{35} - 12 q^{37} - 2 q^{38} - 20 q^{40} + 25 q^{41} + 5 q^{43} + 37 q^{44} + 2 q^{46} + 17 q^{47} + 17 q^{49} + 44 q^{50} + 25 q^{52} + 17 q^{53} + q^{55} + 54 q^{56} + 2 q^{58} + 18 q^{59} - 13 q^{61} + 8 q^{62} + 20 q^{64} + 16 q^{65} + 2 q^{67} + 19 q^{68} + 14 q^{70} + 55 q^{71} + 19 q^{73} - 4 q^{74} - 32 q^{76} + 19 q^{77} - 68 q^{79} + 2 q^{80} - 12 q^{82} + 21 q^{83} + 16 q^{85} + 22 q^{86} - 25 q^{88} + 17 q^{89} - 30 q^{91} + 12 q^{92} + 16 q^{94} + 55 q^{95} + 25 q^{97} + 31 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 2 x^{13} - 18 x^{12} + 34 x^{11} + 124 x^{10} - 216 x^{9} - 420 x^{8} + 647 x^{7} + 750 x^{6} + \cdots - 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{13} - 2 \nu^{12} - 18 \nu^{11} + 34 \nu^{10} + 124 \nu^{9} - 216 \nu^{8} - 420 \nu^{7} + \cdots - 128 ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{13} + 2 \nu^{12} + 18 \nu^{11} - 34 \nu^{10} - 124 \nu^{9} + 216 \nu^{8} + 420 \nu^{7} + \cdots + 112 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 21 \nu^{13} + 142 \nu^{12} + 210 \nu^{11} - 2450 \nu^{10} + 108 \nu^{9} + 15640 \nu^{8} + \cdots + 8496 ) / 368 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 11 \nu^{13} - 47 \nu^{12} - 156 \nu^{11} + 808 \nu^{10} + 686 \nu^{9} - 5152 \nu^{8} - 664 \nu^{7} + \cdots - 2124 ) / 92 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 14 \nu^{13} - 5 \nu^{12} + 278 \nu^{11} + 130 \nu^{10} - 2090 \nu^{9} - 1196 \nu^{8} + 7444 \nu^{7} + \cdots - 1604 ) / 92 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 49 \nu^{13} - 86 \nu^{12} - 858 \nu^{11} + 1362 \nu^{10} + 5636 \nu^{9} - 7728 \nu^{8} - 17452 \nu^{7} + \cdots + 48 ) / 184 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 97 \nu^{13} - 266 \nu^{12} - 1522 \nu^{11} + 4386 \nu^{10} + 8412 \nu^{9} - 26680 \nu^{8} + \cdots - 6544 ) / 368 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 10 \nu^{13} - 26 \nu^{12} - 169 \nu^{11} + 446 \nu^{10} + 1046 \nu^{9} - 2852 \nu^{8} - 2912 \nu^{7} + \cdots - 1266 ) / 46 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 12 \nu^{13} - 22 \nu^{12} - 212 \nu^{11} + 365 \nu^{10} + 1407 \nu^{9} - 2231 \nu^{8} - 4419 \nu^{7} + \cdots - 544 ) / 46 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 157 \nu^{13} - 146 \nu^{12} - 2858 \nu^{11} + 2002 \nu^{10} + 19748 \nu^{9} - 8832 \nu^{8} + \cdots + 5088 ) / 368 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 177 \nu^{13} + 198 \nu^{12} + 3242 \nu^{11} - 2986 \nu^{10} - 22484 \nu^{9} + 15640 \nu^{8} + \cdots - 6144 ) / 368 \) 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
\(\nu^{3}\)\(=\) \( \beta_{4} + \beta_{3} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} + \beta_{5} + \beta_{4} + 7\beta_{2} + \beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{13} + \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} + 9\beta_{4} + 9\beta_{3} + 30\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{13} + \beta_{12} + \beta_{11} - \beta_{9} + 11\beta_{6} + 10\beta_{5} + 11\beta_{4} + 45\beta_{2} + 12\beta _1 + 78 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 14 \beta_{13} + \beta_{12} + 3 \beta_{11} - \beta_{9} + 13 \beta_{8} - 12 \beta_{7} + 13 \beta_{6} + \cdots + 58 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 16 \beta_{13} + 13 \beta_{12} + 17 \beta_{11} - 2 \beta_{10} - 14 \beta_{9} + 3 \beta_{8} - 2 \beta_{7} + \cdots + 475 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 137 \beta_{13} + 16 \beta_{12} + 50 \beta_{11} - 4 \beta_{10} - 16 \beta_{9} + 120 \beta_{8} + \cdots + 418 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 177 \beta_{13} + 121 \beta_{12} + 195 \beta_{11} - 34 \beta_{10} - 136 \beta_{9} + 51 \beta_{8} + \cdots + 3043 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1168 \beta_{13} + 174 \beta_{12} + 558 \beta_{11} - 74 \beta_{10} - 172 \beta_{9} + 970 \beta_{8} + \cdots + 3035 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1673 \beta_{13} + 998 \beta_{12} + 1878 \beta_{11} - 384 \beta_{10} - 1144 \beta_{9} + 575 \beta_{8} + \cdots + 20110 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 9303 \beta_{13} + 1611 \beta_{12} + 5255 \beta_{11} - 880 \beta_{10} - 1573 \beta_{9} + 7354 \beta_{8} + \cdots + 22213 \) 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
−2.55124
−1.98240
−1.86019
−1.37642
−0.894954
−0.656871
−0.429717
0.630703
1.14192
1.35292
1.57408
2.05175
2.31598
2.68441
−2.55124 0 4.50880 1.64740 0 −2.91756 −6.40055 0 −4.20290
1.2 −1.98240 0 1.92989 0.252916 0 −1.21499 0.138982 0 −0.501379
1.3 −1.86019 0 1.46029 −0.608499 0 0.188796 1.00396 0 1.13192
1.4 −1.37642 0 −0.105481 1.48313 0 4.92788 2.89802 0 −2.04140
1.5 −0.894954 0 −1.19906 −0.474904 0 −1.99579 2.86301 0 0.425017
1.6 −0.656871 0 −1.56852 −3.35329 0 −2.47996 2.34406 0 2.20268
1.7 −0.429717 0 −1.81534 3.93237 0 0.264416 1.63952 0 −1.68981
1.8 0.630703 0 −1.60221 −2.56371 0 2.03354 −2.27193 0 −1.61694
1.9 1.14192 0 −0.696014 4.45694 0 −4.52974 −3.07864 0 5.08948
1.10 1.35292 0 −0.169600 0.738177 0 3.44152 −2.93530 0 0.998696
1.11 1.57408 0 0.477741 −0.564555 0 −3.58089 −2.39616 0 −0.888657
1.12 2.05175 0 2.20969 −3.75134 0 −2.35999 0.430233 0 −7.69683
1.13 2.31598 0 3.36376 2.86683 0 3.60131 3.15843 0 6.63952
1.14 2.68441 0 5.20606 −1.06146 0 1.62145 8.60637 0 −2.84940
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
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.p 14
3.b odd 2 1 2001.2.a.m ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2001.2.a.m ✓ 14 3.b odd 2 1
6003.2.a.p 14 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6003))\):

\( T_{2}^{14} - 2 T_{2}^{13} - 18 T_{2}^{12} + 34 T_{2}^{11} + 124 T_{2}^{10} - 216 T_{2}^{9} - 420 T_{2}^{8} + \cdots - 64 \) Copy content Toggle raw display
\( T_{5}^{14} - 3 T_{5}^{13} - 37 T_{5}^{12} + 96 T_{5}^{11} + 479 T_{5}^{10} - 1025 T_{5}^{9} - 2540 T_{5}^{8} + \cdots - 128 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} - 2 T^{13} + \cdots - 64 \) Copy content Toggle raw display
$3$ \( T^{14} \) Copy content Toggle raw display
$5$ \( T^{14} - 3 T^{13} + \cdots - 128 \) Copy content Toggle raw display
$7$ \( T^{14} + 3 T^{13} + \cdots - 6752 \) Copy content Toggle raw display
$11$ \( T^{14} - 12 T^{13} + \cdots + 120704 \) Copy content Toggle raw display
$13$ \( T^{14} - 13 T^{13} + \cdots + 18424 \) Copy content Toggle raw display
$17$ \( T^{14} - 14 T^{13} + \cdots + 740744 \) Copy content Toggle raw display
$19$ \( T^{14} + 9 T^{13} + \cdots + 1130464 \) Copy content Toggle raw display
$23$ \( (T - 1)^{14} \) Copy content Toggle raw display
$29$ \( (T - 1)^{14} \) Copy content Toggle raw display
$31$ \( T^{14} + 28 T^{13} + \cdots + 36610048 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 5876204288 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 245620864 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 11926919072 \) Copy content Toggle raw display
$47$ \( T^{14} - 17 T^{13} + \cdots - 24729856 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots - 371147830784 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots - 513236992 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 268644352 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 1310603264 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 17005748224 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots - 202777216 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots - 78594928544 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 2370202624 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots - 2726914213064 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots - 3664836644864 \) Copy content Toggle raw display
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