Properties

Label 6003.2.a.n
Level $6003$
Weight $2$
Character orbit 6003.a
Self dual yes
Analytic conductor $47.934$
Analytic rank $0$
Dimension $12$
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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} - 13 x^{10} + 41 x^{9} + 54 x^{8} - 188 x^{7} - 77 x^{6} + 342 x^{5} + 13 x^{4} - 215 x^{3} + 9 x^{2} + 37 x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{8} - \beta_{6} - \beta_{4} + 1) q^{4} + (\beta_{8} + 1) q^{5} + ( - \beta_{10} + \beta_{6} + \beta_1 - 1) q^{7} + (\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{8} - \beta_{6} - \beta_{4} + 1) q^{4} + (\beta_{8} + 1) q^{5} + ( - \beta_{10} + \beta_{6} + \beta_1 - 1) q^{7} + (\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1) q^{8} + (\beta_{11} + \beta_{10} + \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + \beta_{3} + 2 \beta_1 - 1) q^{10} + ( - \beta_{10} - \beta_{5} + \beta_{2} + 1) q^{11} + ( - \beta_{11} - \beta_{10} + 2 \beta_{9} + \beta_{6} + \beta_{3} + \beta_1 - 2) q^{13} + ( - \beta_{7} - \beta_{6} - \beta_{4} + \beta_{2} - 2 \beta_1 + 2) q^{14} + (\beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{2} + 1) q^{16} + (\beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} + 2) q^{17} + (\beta_{11} + \beta_{10} + \beta_{7} + \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 1) q^{19} + (\beta_{10} + 3 \beta_{8} - 2 \beta_{6} + 2 \beta_{5} - \beta_{4} - \beta_{2} + 2) q^{20} + ( - \beta_{10} - \beta_{9} - \beta_{7} - 2 \beta_{6} - \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + \cdots + 1) q^{22}+ \cdots + (\beta_{11} + 2 \beta_{10} - \beta_{8} + 3 \beta_{7} + 4 \beta_{6} + 4 \beta_{4} - \beta_{2} + \beta_1 - 8) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{2} + 11 q^{4} + 16 q^{5} - 7 q^{7} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 3 q^{2} + 11 q^{4} + 16 q^{5} - 7 q^{7} + 9 q^{8} + 6 q^{11} - 15 q^{13} + 8 q^{14} + 17 q^{16} + 18 q^{17} - 6 q^{19} + 39 q^{20} - 5 q^{22} + 12 q^{23} + 14 q^{25} + 3 q^{26} - 19 q^{28} + 12 q^{29} + 16 q^{31} + 21 q^{32} - 7 q^{34} + 11 q^{35} - q^{37} + 24 q^{38} + 30 q^{40} - 3 q^{41} - 23 q^{43} - 23 q^{44} + 3 q^{46} + 35 q^{47} + 3 q^{49} + 2 q^{50} + 45 q^{53} + 17 q^{55} + 17 q^{56} + 3 q^{58} + 11 q^{59} + 4 q^{61} + 7 q^{62} + 15 q^{64} - 5 q^{65} - 19 q^{67} - q^{68} + 14 q^{70} - 19 q^{71} + 10 q^{73} + 15 q^{74} - 4 q^{76} + 39 q^{77} + 17 q^{79} + 90 q^{80} - 3 q^{82} + 12 q^{83} + 14 q^{85} - 17 q^{86} - 2 q^{88} + 20 q^{89} + 11 q^{91} + 11 q^{92} + 13 q^{94} - 12 q^{95} - 12 q^{97} - 75 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} - 13 x^{10} + 41 x^{9} + 54 x^{8} - 188 x^{7} - 77 x^{6} + 342 x^{5} + 13 x^{4} - 215 x^{3} + 9 x^{2} + 37 x - 5 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 69 \nu^{11} - 2268 \nu^{10} + 4126 \nu^{9} + 30343 \nu^{8} - 62480 \nu^{7} - 131017 \nu^{6} + 277030 \nu^{5} + 194815 \nu^{4} - 433403 \nu^{3} - 31183 \nu^{2} + 158295 \nu - 20718 ) / 6469 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 127 \nu^{11} - 607 \nu^{10} + 4875 \nu^{9} + 7810 \nu^{8} - 51132 \nu^{7} - 30082 \nu^{6} + 209382 \nu^{5} + 24036 \nu^{4} - 328362 \nu^{3} + 49707 \nu^{2} + 117880 \nu - 29932 ) / 6469 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 353 \nu^{11} - 210 \nu^{10} + 8049 \nu^{9} + 773 \nu^{8} - 63527 \nu^{7} + 13625 \nu^{6} + 210856 \nu^{5} - 84627 \nu^{4} - 276488 \nu^{3} + 135717 \nu^{2} + 84019 \nu - 25638 ) / 6469 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 549 \nu^{11} - 1451 \nu^{10} - 8798 \nu^{9} + 21760 \nu^{8} + 52179 \nu^{7} - 114560 \nu^{6} - 143208 \nu^{5} + 255406 \nu^{4} + 177916 \nu^{3} - 216607 \nu^{2} + \cdots + 34852 ) / 6469 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 874 \nu^{11} + 2852 \nu^{10} + 10271 \nu^{9} - 37175 \nu^{8} - 32306 \nu^{7} + 154428 \nu^{6} + 1464 \nu^{5} - 218601 \nu^{4} + 86000 \nu^{3} + 34877 \nu^{2} + \cdots + 23075 ) / 6469 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1153 \nu^{11} - 2741 \nu^{10} - 16651 \nu^{9} + 36674 \nu^{8} + 84275 \nu^{7} - 163016 \nu^{6} - 182412 \nu^{5} + 282647 \nu^{4} + 156809 \nu^{3} - 153652 \nu^{2} + \cdots + 10438 ) / 6469 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1227 \nu^{11} + 2642 \nu^{10} + 18320 \nu^{9} - 36402 \nu^{8} - 95833 \nu^{7} + 168053 \nu^{6} + 212320 \nu^{5} - 303228 \nu^{4} - 190488 \nu^{3} + 177063 \nu^{2} + \cdots - 21970 ) / 6469 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1231 \nu^{11} - 3336 \nu^{10} - 16487 \nu^{9} + 44255 \nu^{8} + 73554 \nu^{7} - 190180 \nu^{6} - 130539 \nu^{5} + 291927 \nu^{4} + 89329 \nu^{3} - 89336 \nu^{2} + \cdots - 10451 ) / 6469 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2257 \nu^{11} - 6684 \nu^{10} - 28263 \nu^{9} + 88739 \nu^{8} + 106697 \nu^{7} - 383278 \nu^{6} - 103569 \nu^{5} + 605079 \nu^{4} - 88693 \nu^{3} - 238564 \nu^{2} + \cdots + 8869 ) / 6469 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 2687 \nu^{11} + 10130 \nu^{10} + 28520 \nu^{9} - 134015 \nu^{8} - 63186 \nu^{7} + 578643 \nu^{6} - 126167 \nu^{5} - 926609 \nu^{4} + 441457 \nu^{3} + 391735 \nu^{2} + \cdots - 20106 ) / 6469 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} - \beta_{6} - \beta_{4} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{10} - \beta_{9} + 7\beta_{8} - \beta_{7} - 7\beta_{6} + \beta_{5} - 6\beta_{4} - \beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{11} + 2 \beta_{10} + 2 \beta_{8} + 2 \beta_{7} + 2 \beta_{6} + 10 \beta_{5} + 10 \beta_{4} + 10 \beta_{3} - 9 \beta_{2} + 30 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{11} + 11 \beta_{10} - 8 \beta_{9} + 47 \beta_{8} - 10 \beta_{7} - 45 \beta_{6} + 11 \beta_{5} - 35 \beta_{4} + \beta_{3} - 10 \beta_{2} + 2 \beta _1 + 86 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 10 \beta_{11} + 21 \beta_{10} + 3 \beta_{9} + 24 \beta_{8} + 20 \beta_{7} + 22 \beta_{6} + 78 \beta_{5} + 76 \beta_{4} + 78 \beta_{3} - 68 \beta_{2} + 192 \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 12 \beta_{11} + 90 \beta_{10} - 50 \beta_{9} + 314 \beta_{8} - 77 \beta_{7} - 283 \beta_{6} + 94 \beta_{5} - 209 \beta_{4} + 19 \beta_{3} - 81 \beta_{2} + 32 \beta _1 + 527 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 74 \beta_{11} + 167 \beta_{10} + 43 \beta_{9} + 214 \beta_{8} + 148 \beta_{7} + 178 \beta_{6} + 559 \beta_{5} + 524 \beta_{4} + 565 \beta_{3} - 488 \beta_{2} + 1261 \beta _1 + 40 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 99 \beta_{11} + 661 \beta_{10} - 284 \beta_{9} + 2098 \beta_{8} - 546 \beta_{7} - 1773 \beta_{6} + 724 \beta_{5} - 1281 \beta_{4} + 227 \beta_{3} - 622 \beta_{2} + 351 \beta _1 + 3354 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 491 \beta_{11} + 1212 \beta_{10} + 431 \beta_{9} + 1715 \beta_{8} + 980 \beta_{7} + 1286 \beta_{6} + 3859 \beta_{5} + 3450 \beta_{4} + 3981 \beta_{3} - 3436 \beta_{2} + 8380 \beta _1 + 515 \) 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
−2.52122
−1.89304
−1.65670
−0.724122
−0.613795
0.147789
0.431373
1.08419
1.49364
1.99588
2.58646
2.66955
−2.52122 0 4.35654 3.14716 0 −3.79117 −5.94135 0 −7.93467
1.2 −1.89304 0 1.58359 4.13664 0 2.17357 0.788273 0 −7.83081
1.3 −1.65670 0 0.744648 −1.18716 0 −3.31827 2.07974 0 1.96676
1.4 −0.724122 0 −1.47565 1.17886 0 −4.09140 2.51679 0 −0.853636
1.5 −0.613795 0 −1.62326 0.782514 0 3.32687 2.22394 0 −0.480303
1.6 0.147789 0 −1.97816 −0.429801 0 0.404424 −0.587929 0 −0.0635200
1.7 0.431373 0 −1.81392 3.65141 0 2.43060 −1.64522 0 1.57512
1.8 1.08419 0 −0.824542 −0.786014 0 −4.63453 −3.06233 0 −0.852185
1.9 1.49364 0 0.230961 1.29771 0 1.16738 −2.64231 0 1.93831
1.10 1.99588 0 1.98352 −2.38535 0 −0.101071 −0.0328940 0 −4.76086
1.11 2.58646 0 4.68978 3.69848 0 −0.298143 6.95702 0 9.56597
1.12 2.66955 0 5.12647 2.89555 0 −0.268248 8.34627 0 7.72982
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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.n 12
3.b odd 2 1 667.2.a.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
667.2.a.b 12 3.b odd 2 1
6003.2.a.n 12 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6003))\):

\( T_{2}^{12} - 3 T_{2}^{11} - 13 T_{2}^{10} + 41 T_{2}^{9} + 54 T_{2}^{8} - 188 T_{2}^{7} - 77 T_{2}^{6} + 342 T_{2}^{5} + 13 T_{2}^{4} - 215 T_{2}^{3} + 9 T_{2}^{2} + 37 T_{2} - 5 \) Copy content Toggle raw display
\( T_{5}^{12} - 16 T_{5}^{11} + 91 T_{5}^{10} - 162 T_{5}^{9} - 396 T_{5}^{8} + 1943 T_{5}^{7} - 1627 T_{5}^{6} - 2991 T_{5}^{5} + 4808 T_{5}^{4} + 660 T_{5}^{3} - 3214 T_{5}^{2} + 300 T_{5} + 583 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 3 T^{11} - 13 T^{10} + 41 T^{9} + \cdots - 5 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - 16 T^{11} + 91 T^{10} + \cdots + 583 \) Copy content Toggle raw display
$7$ \( T^{12} + 7 T^{11} - 19 T^{10} - 182 T^{9} + \cdots - 16 \) Copy content Toggle raw display
$11$ \( T^{12} - 6 T^{11} - 63 T^{10} + \cdots + 266075 \) Copy content Toggle raw display
$13$ \( T^{12} + 15 T^{11} + 10 T^{10} + \cdots + 3090655 \) Copy content Toggle raw display
$17$ \( T^{12} - 18 T^{11} + 71 T^{10} + \cdots - 9596 \) Copy content Toggle raw display
$19$ \( T^{12} + 6 T^{11} - 119 T^{10} + \cdots - 149440 \) Copy content Toggle raw display
$23$ \( (T - 1)^{12} \) Copy content Toggle raw display
$29$ \( (T - 1)^{12} \) Copy content Toggle raw display
$31$ \( T^{12} - 16 T^{11} + 25 T^{10} + \cdots + 169 \) Copy content Toggle raw display
$37$ \( T^{12} + T^{11} - 300 T^{10} + \cdots - 362875340 \) Copy content Toggle raw display
$41$ \( T^{12} + 3 T^{11} - 225 T^{10} + \cdots - 27754352 \) Copy content Toggle raw display
$43$ \( T^{12} + 23 T^{11} + \cdots - 229352509 \) Copy content Toggle raw display
$47$ \( T^{12} - 35 T^{11} + 222 T^{10} + \cdots - 40804117 \) Copy content Toggle raw display
$53$ \( T^{12} - 45 T^{11} + \cdots - 142956229 \) Copy content Toggle raw display
$59$ \( T^{12} - 11 T^{11} + \cdots + 8891929600 \) Copy content Toggle raw display
$61$ \( T^{12} - 4 T^{11} - 288 T^{10} + \cdots - 18119916 \) Copy content Toggle raw display
$67$ \( T^{12} + 19 T^{11} + \cdots - 409376228 \) Copy content Toggle raw display
$71$ \( T^{12} + 19 T^{11} + \cdots - 226642180 \) Copy content Toggle raw display
$73$ \( T^{12} - 10 T^{11} + \cdots + 2131153580 \) Copy content Toggle raw display
$79$ \( T^{12} - 17 T^{11} - 222 T^{10} + \cdots - 36136523 \) Copy content Toggle raw display
$83$ \( T^{12} - 12 T^{11} + \cdots + 14844146512 \) Copy content Toggle raw display
$89$ \( T^{12} - 20 T^{11} + \cdots - 12272416784 \) Copy content Toggle raw display
$97$ \( T^{12} + 12 T^{11} + \cdots - 6155578420 \) Copy content Toggle raw display
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