Properties

Label 6003.2.a.o
Level $6003$
Weight $2$
Character orbit 6003.a
Self dual yes
Analytic conductor $47.934$
Analytic rank $1$
Dimension $13$
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: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 4 x^{12} - 11 x^{11} + 58 x^{10} + 24 x^{9} - 298 x^{8} + 97 x^{7} + 641 x^{6} - 402 x^{5} - 547 x^{4} + 352 x^{3} + 219 x^{2} - 88 x - 40 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 667)
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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + (\beta_{8} - 1) q^{5} + \beta_{9} q^{7} + ( - \beta_{7} + \beta_{6} - \beta_{2} - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + (\beta_{8} - 1) q^{5} + \beta_{9} q^{7} + ( - \beta_{7} + \beta_{6} - \beta_{2} - \beta_1) q^{8} + ( - \beta_{8} - \beta_{4} + \beta_1) q^{10} + ( - \beta_{11} - 1) q^{11} + (\beta_{8} + \beta_{4} + 1) q^{13} + ( - \beta_{12} + 2 \beta_{10} - \beta_{9} - \beta_{4} - \beta_{3} - \beta_1 + 1) q^{14} + (\beta_{7} - \beta_{6} + \beta_{3} + \beta_{2} + \beta_1) q^{16} + (\beta_{11} - \beta_{10} - \beta_{6} + \beta_{4} + \beta_1 - 2) q^{17} + ( - \beta_{11} - \beta_{10} - \beta_{7} + 2 \beta_{6} - \beta_1) q^{19} + (\beta_{11} + \beta_{9} - \beta_{6} - \beta_{5} + \beta_{4} - 2 \beta_{2} + \beta_1 - 2) q^{20} + (\beta_{11} + \beta_{7} + \beta_{4} - \beta_{2} + 3 \beta_1 - 2) q^{22} + q^{23} + (\beta_{12} - \beta_{9} - 2 \beta_{8} + \beta_{6} + \beta_{5} - \beta_{4} - \beta_{2} + 1) q^{25} + ( - \beta_{11} - \beta_{9} - 2 \beta_{8} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} - 2 \beta_1 + 1) q^{26} + (2 \beta_{12} + \beta_{11} - 2 \beta_{10} + 2 \beta_{9} + \beta_{8} - 2 \beta_{6} - \beta_{5} + \beta_{4} + \cdots + 1) q^{28}+ \cdots + (\beta_{12} + 2 \beta_{11} - 2 \beta_{10} + 4 \beta_{9} + 2 \beta_{8} - \beta_{6} - 2 \beta_{5} + \cdots + 3 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q - 4 q^{2} + 12 q^{4} - 16 q^{5} + q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 13 q - 4 q^{2} + 12 q^{4} - 16 q^{5} + q^{7} - 6 q^{8} + 10 q^{10} - 10 q^{11} + 7 q^{13} + 12 q^{14} + 2 q^{16} - 26 q^{17} - 25 q^{20} - 15 q^{22} + 13 q^{23} + 19 q^{25} + 15 q^{26} + 5 q^{28} - 13 q^{29} - 6 q^{31} - 16 q^{32} + 11 q^{34} - q^{35} + 15 q^{37} - 8 q^{38} + 14 q^{40} - 9 q^{41} + q^{43} - 29 q^{44} - 4 q^{46} - 15 q^{47} + 4 q^{49} - 31 q^{50} - 8 q^{52} - 43 q^{53} - 3 q^{55} + 5 q^{56} + 4 q^{58} + 9 q^{59} + 20 q^{61} - 11 q^{62} - 16 q^{64} + 25 q^{65} + q^{67} - 21 q^{68} - 2 q^{70} - 17 q^{71} + 26 q^{73} - 11 q^{74} + 8 q^{76} - 17 q^{77} + 5 q^{79} - 10 q^{80} - 25 q^{82} - 4 q^{83} + 20 q^{85} + 13 q^{86} - 32 q^{88} - 48 q^{89} - 9 q^{91} + 12 q^{92} - 65 q^{94} - 8 q^{95} + 30 q^{97} - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - 4 x^{12} - 11 x^{11} + 58 x^{10} + 24 x^{9} - 298 x^{8} + 97 x^{7} + 641 x^{6} - 402 x^{5} - 547 x^{4} + 352 x^{3} + 219 x^{2} - 88 x - 40 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - \nu^{3} - 6\nu^{2} + 4\nu + 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 6 \nu^{12} - 22 \nu^{11} - 67 \nu^{10} + 313 \nu^{9} + 147 \nu^{8} - 1530 \nu^{7} + 604 \nu^{6} + 2882 \nu^{5} - 2433 \nu^{4} - 1566 \nu^{3} + 1894 \nu^{2} + 153 \nu - 363 ) / 19 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 11 \nu^{12} + 34 \nu^{11} + 145 \nu^{10} - 482 \nu^{9} - 621 \nu^{8} + 2387 \nu^{7} + 818 \nu^{6} - 4758 \nu^{5} + 233 \nu^{4} + 3232 \nu^{3} - 236 \nu^{2} - 613 \nu - 28 ) / 19 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11 \nu^{12} + 34 \nu^{11} + 145 \nu^{10} - 482 \nu^{9} - 621 \nu^{8} + 2387 \nu^{7} + 818 \nu^{6} - 4777 \nu^{5} + 233 \nu^{4} + 3365 \nu^{3} - 198 \nu^{2} - 765 \nu - 104 ) / 19 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 11 \nu^{12} + 34 \nu^{11} + 145 \nu^{10} - 482 \nu^{9} - 621 \nu^{8} + 2387 \nu^{7} + 818 \nu^{6} - 4777 \nu^{5} + 233 \nu^{4} + 3384 \nu^{3} - 217 \nu^{2} - 860 \nu - 47 ) / 19 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 9 \nu^{12} + 33 \nu^{11} + 110 \nu^{10} - 479 \nu^{9} - 382 \nu^{8} + 2447 \nu^{7} + 44 \nu^{6} - 5121 \nu^{5} + 1379 \nu^{4} + 3869 \nu^{3} - 865 \nu^{2} - 942 \nu + 3 ) / 19 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 10 \nu^{12} - 24 \nu^{11} - 156 \nu^{10} + 357 \nu^{9} + 910 \nu^{8} - 1923 \nu^{7} - 2464 \nu^{6} + 4512 \nu^{5} + 3203 \nu^{4} - 4377 \nu^{3} - 2024 \nu^{2} + 1338 \nu + 516 ) / 19 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 31 \nu^{12} + 82 \nu^{11} + 457 \nu^{10} - 1196 \nu^{9} - 2422 \nu^{8} + 6176 \nu^{7} + 5613 \nu^{6} - 13307 \nu^{5} - 6078 \nu^{4} + 10979 \nu^{3} + 4344 \nu^{2} - 2985 \nu - 1326 ) / 38 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 21 \nu^{12} + 58 \nu^{11} + 301 \nu^{10} - 839 \nu^{9} - 1512 \nu^{8} + 4272 \nu^{7} + 3092 \nu^{6} - 8947 \nu^{5} - 2400 \nu^{4} + 6887 \nu^{3} + 1313 \nu^{2} - 1628 \nu - 449 ) / 19 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 31 \nu^{12} + 82 \nu^{11} + 457 \nu^{10} - 1196 \nu^{9} - 2422 \nu^{8} + 6195 \nu^{7} + 5575 \nu^{6} - 13478 \nu^{5} - 5774 \nu^{4} + 11397 \nu^{3} + 3736 \nu^{2} - 3175 \nu - 1136 ) / 19 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - \beta_{6} + \beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} - \beta_{6} + \beta_{3} + 7\beta_{2} + \beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 7\beta_{7} - 8\beta_{6} + \beta_{5} + 9\beta_{2} + 27\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{12} - \beta_{11} + \beta_{9} + 9\beta_{7} - 10\beta_{6} + \beta_{5} + 9\beta_{3} + 44\beta_{2} + 12\beta _1 + 74 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3 \beta_{12} - 2 \beta_{11} - 2 \beta_{10} + 2 \beta_{9} + 43 \beta_{7} - 54 \beta_{6} + 11 \beta_{5} + 2 \beta_{3} + 67 \beta_{2} + 151 \beta _1 + 28 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 16 \beta_{12} - 13 \beta_{11} - 4 \beta_{10} + 15 \beta_{9} + 65 \beta_{7} - 80 \beta_{6} + 14 \beta_{5} + 64 \beta_{3} + 274 \beta_{2} + 106 \beta _1 + 412 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 45 \beta_{12} - 28 \beta_{11} - 30 \beta_{10} + 33 \beta_{9} + 261 \beta_{7} - 352 \beta_{6} + 91 \beta_{5} + 2 \beta_{4} + 33 \beta_{3} + 473 \beta_{2} + 869 \beta _1 + 264 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 169 \beta_{12} - 117 \beta_{11} - 66 \beta_{10} + 156 \beta_{9} + 2 \beta_{8} + 445 \beta_{7} - 599 \beta_{6} + 137 \beta_{5} + 5 \beta_{4} + 428 \beta_{3} + 1720 \beta_{2} + 837 \beta _1 + 2367 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 462 \beta_{12} - 266 \beta_{11} - 312 \beta_{10} + 364 \beta_{9} + 7 \beta_{8} + 1603 \beta_{7} - 2290 \beta_{6} + 680 \beta_{5} + 41 \beta_{4} + 361 \beta_{3} + 3268 \beta_{2} + 5135 \beta _1 + 2139 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1506 \beta_{12} - 912 \beta_{11} - 728 \beta_{10} + 1397 \beta_{9} + 48 \beta_{8} + 3002 \beta_{7} - 4350 \beta_{6} + 1157 \beta_{5} + 105 \beta_{4} + 2823 \beta_{3} + 10918 \beta_{2} + 6258 \beta _1 + 13929 \) 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
2.60496
2.43510
2.24788
1.46878
1.30511
1.18857
0.775068
−0.397523
−0.552582
−0.806558
−1.78056
−2.22202
−2.26622
−2.60496 0 4.78583 −1.68070 0 2.80821 −7.25699 0 4.37815
1.2 −2.43510 0 3.92972 −3.49556 0 −4.87125 −4.69905 0 8.51205
1.3 −2.24788 0 3.05297 0.269352 0 −0.523800 −2.36696 0 −0.605470
1.4 −1.46878 0 0.157302 −4.31834 0 1.97692 2.70651 0 6.34267
1.5 −1.30511 0 −0.296685 −0.572849 0 1.21746 2.99743 0 0.747631
1.6 −1.18857 0 −0.587311 2.45282 0 −2.24002 3.07519 0 −2.91534
1.7 −0.775068 0 −1.39927 −3.48646 0 0.0624539 2.63467 0 2.70224
1.8 0.397523 0 −1.84198 3.16986 0 −1.67469 −1.52727 0 1.26009
1.9 0.552582 0 −1.69465 −3.05371 0 −3.72348 −2.04160 0 −1.68743
1.10 0.806558 0 −1.34946 −2.04515 0 4.05430 −2.70154 0 −1.64953
1.11 1.78056 0 1.17038 −0.267809 0 2.76268 −1.47718 0 −0.476849
1.12 2.22202 0 2.93739 −2.84468 0 −1.80107 2.08291 0 −6.32095
1.13 2.26622 0 3.13576 −0.126764 0 2.95229 2.57389 0 −0.287275
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
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.o 13
3.b odd 2 1 667.2.a.c 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
667.2.a.c 13 3.b odd 2 1
6003.2.a.o 13 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6003))\):

\( T_{2}^{13} + 4 T_{2}^{12} - 11 T_{2}^{11} - 58 T_{2}^{10} + 24 T_{2}^{9} + 298 T_{2}^{8} + 97 T_{2}^{7} - 641 T_{2}^{6} - 402 T_{2}^{5} + 547 T_{2}^{4} + 352 T_{2}^{3} - 219 T_{2}^{2} - 88 T_{2} + 40 \) Copy content Toggle raw display
\( T_{5}^{13} + 16 T_{5}^{12} + 86 T_{5}^{11} + 72 T_{5}^{10} - 1022 T_{5}^{9} - 3861 T_{5}^{8} - 2653 T_{5}^{7} + 12522 T_{5}^{6} + 31627 T_{5}^{5} + 27980 T_{5}^{4} + 7871 T_{5}^{3} - 1214 T_{5}^{2} - 736 T_{5} - 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{13} + 4 T^{12} - 11 T^{11} - 58 T^{10} + \cdots + 40 \) Copy content Toggle raw display
$3$ \( T^{13} \) Copy content Toggle raw display
$5$ \( T^{13} + 16 T^{12} + 86 T^{11} + 72 T^{10} + \cdots - 64 \) Copy content Toggle raw display
$7$ \( T^{13} - T^{12} - 47 T^{11} + 62 T^{10} + \cdots - 896 \) Copy content Toggle raw display
$11$ \( T^{13} + 10 T^{12} - 14 T^{11} + \cdots + 28240 \) Copy content Toggle raw display
$13$ \( T^{13} - 7 T^{12} - 36 T^{11} + \cdots + 1694 \) Copy content Toggle raw display
$17$ \( T^{13} + 26 T^{12} + 222 T^{11} + \cdots + 551222 \) Copy content Toggle raw display
$19$ \( T^{13} - 116 T^{11} - 239 T^{10} + \cdots + 11924 \) Copy content Toggle raw display
$23$ \( (T - 1)^{13} \) Copy content Toggle raw display
$29$ \( (T + 1)^{13} \) Copy content Toggle raw display
$31$ \( T^{13} + 6 T^{12} - 184 T^{11} + \cdots + 70389088 \) Copy content Toggle raw display
$37$ \( T^{13} - 15 T^{12} - 77 T^{11} + \cdots + 99546086 \) Copy content Toggle raw display
$41$ \( T^{13} + 9 T^{12} - 221 T^{11} + \cdots - 171294376 \) Copy content Toggle raw display
$43$ \( T^{13} - T^{12} - 250 T^{11} + \cdots - 47372716 \) Copy content Toggle raw display
$47$ \( T^{13} + 15 T^{12} - 73 T^{11} + \cdots + 739168 \) Copy content Toggle raw display
$53$ \( T^{13} + 43 T^{12} + \cdots - 530954104 \) Copy content Toggle raw display
$59$ \( T^{13} - 9 T^{12} + \cdots + 91791625948 \) Copy content Toggle raw display
$61$ \( T^{13} - 20 T^{12} + \cdots - 7874543600 \) Copy content Toggle raw display
$67$ \( T^{13} - T^{12} - 526 T^{11} + \cdots - 6179333792 \) Copy content Toggle raw display
$71$ \( T^{13} + 17 T^{12} + \cdots + 59462318104 \) Copy content Toggle raw display
$73$ \( T^{13} - 26 T^{12} + \cdots + 7929287008 \) Copy content Toggle raw display
$79$ \( T^{13} - 5 T^{12} + \cdots + 20724897088 \) Copy content Toggle raw display
$83$ \( T^{13} + 4 T^{12} + \cdots - 27841625168 \) Copy content Toggle raw display
$89$ \( T^{13} + 48 T^{12} + \cdots - 376563331102 \) Copy content Toggle raw display
$97$ \( T^{13} - 30 T^{12} + \cdots - 32043114320 \) Copy content Toggle raw display
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