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Properties

Label 3009.2.a.e
Level $3009$
Weight $2$
Character orbit 3009.a
Self dual yes
Analytic conductor $24.027$
Analytic rank $1$
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] = [3009,2,Mod(1,3009)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3009, 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("3009.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3009 = 3 \cdot 17 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3009.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: \(24.0269859682\)
Analytic rank: \(1\)
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: \( x^{14} - 17 x^{12} - x^{11} + 109 x^{10} + 13 x^{9} - 333 x^{8} - 61 x^{7} + 510 x^{6} + 125 x^{5} + \cdots - 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + q^{3} + \beta_{2} q^{4} - \beta_{11} q^{5} - \beta_1 q^{6} + ( - \beta_{13} - \beta_{2}) q^{7} + (\beta_{12} - \beta_{10} + \beta_{9} + \cdots - 1) q^{8}+ \cdots + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + q^{3} + \beta_{2} q^{4} - \beta_{11} q^{5} - \beta_1 q^{6} + ( - \beta_{13} - \beta_{2}) q^{7} + (\beta_{12} - \beta_{10} + \beta_{9} + \cdots - 1) q^{8}+ \cdots + (\beta_{13} + \beta_{11} - \beta_{3} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 14 q^{3} + 6 q^{4} - 5 q^{5} - 11 q^{7} - 3 q^{8} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 14 q^{3} + 6 q^{4} - 5 q^{5} - 11 q^{7} - 3 q^{8} + 14 q^{9} - 10 q^{10} - 7 q^{11} + 6 q^{12} - 15 q^{13} - 3 q^{14} - 5 q^{15} - 6 q^{16} - 14 q^{17} - 9 q^{19} - 13 q^{20} - 11 q^{21} - 5 q^{22} - 20 q^{23} - 3 q^{24} - 9 q^{25} - 19 q^{26} + 14 q^{27} - 34 q^{28} - 4 q^{29} - 10 q^{30} - 35 q^{31} + 4 q^{32} - 7 q^{33} - 11 q^{35} + 6 q^{36} - 28 q^{37} + q^{38} - 15 q^{39} - 31 q^{40} - 10 q^{41} - 3 q^{42} - 6 q^{43} - 33 q^{44} - 5 q^{45} - 30 q^{46} + 8 q^{47} - 6 q^{48} - 13 q^{49} + 39 q^{50} - 14 q^{51} - 4 q^{52} - 4 q^{53} - 27 q^{55} + 31 q^{56} - 9 q^{57} - 25 q^{58} + 14 q^{59} - 13 q^{60} - 22 q^{61} - 15 q^{62} - 11 q^{63} - 7 q^{64} + 13 q^{65} - 5 q^{66} - 20 q^{67} - 6 q^{68} - 20 q^{69} + 18 q^{70} - 62 q^{71} - 3 q^{72} - 31 q^{73} - 23 q^{74} - 9 q^{75} + 8 q^{76} - 15 q^{77} - 19 q^{78} - 47 q^{79} - 7 q^{80} + 14 q^{81} + 19 q^{83} - 34 q^{84} + 5 q^{85} - 30 q^{86} - 4 q^{87} - 11 q^{88} - q^{89} - 10 q^{90} - 28 q^{91} + 32 q^{92} - 35 q^{93} + q^{94} - 28 q^{95} + 4 q^{96} - 38 q^{97} - 30 q^{98} - 7 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 17 x^{12} - x^{11} + 109 x^{10} + 13 x^{9} - 333 x^{8} - 61 x^{7} + 510 x^{6} + 125 x^{5} + \cdots - 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 229 \nu^{13} + 177 \nu^{12} + 3695 \nu^{11} - 2667 \nu^{10} - 21916 \nu^{9} + 14628 \nu^{8} + \cdots + 1693 ) / 173 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 545 \nu^{13} - 422 \nu^{12} - 8963 \nu^{11} + 6385 \nu^{10} + 54834 \nu^{9} - 35213 \nu^{8} + \cdots - 7082 ) / 173 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 623 \nu^{13} - 477 \nu^{12} - 10248 \nu^{11} + 7202 \nu^{10} + 62771 \nu^{9} - 39647 \nu^{8} + \cdots - 8087 ) / 173 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 715 \nu^{13} + 533 \nu^{12} + 11808 \nu^{11} - 8037 \nu^{10} - 72727 \nu^{9} + 44105 \nu^{8} + \cdots + 8953 ) / 173 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 883 \nu^{13} - 718 \nu^{12} - 14416 \nu^{11} + 10848 \nu^{10} + 87267 \nu^{9} - 59617 \nu^{8} + \cdots - 10399 ) / 173 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 895 \nu^{13} - 793 \nu^{12} - 14627 \nu^{11} + 12025 \nu^{10} + 88701 \nu^{9} - 66261 \nu^{8} + \cdots - 10181 ) / 173 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 932 \nu^{13} + 635 \nu^{12} + 15292 \nu^{11} - 9527 \nu^{10} - 93209 \nu^{9} + 52089 \nu^{8} + \cdots + 10691 ) / 173 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1086 \nu^{13} + 819 \nu^{12} + 17798 \nu^{11} - 12320 \nu^{10} - 108325 \nu^{9} + 67404 \nu^{8} + \cdots + 12103 ) / 173 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1191 \nu^{13} - 913 \nu^{12} - 19601 \nu^{11} + 13839 \nu^{10} + 120094 \nu^{9} - 76580 \nu^{8} + \cdots - 15126 ) / 173 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1364 \nu^{13} - 1086 \nu^{12} - 22369 \nu^{11} + 16434 \nu^{10} + 136356 \nu^{9} - 90593 \nu^{8} + \cdots - 16337 ) / 173 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 1958 \nu^{13} + 1425 \nu^{12} + 32208 \nu^{11} - 21498 \nu^{10} - 197132 \nu^{9} + 118278 \nu^{8} + \cdots + 23538 ) / 173 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{12} + \beta_{10} - \beta_{9} + \beta_{8} + \beta_{5} - \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{13} - \beta_{12} + \beta_{8} + \beta_{6} - \beta_{4} + \beta_{3} + 5\beta_{2} + 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 7 \beta_{12} + 7 \beta_{10} - 8 \beta_{9} + 8 \beta_{8} - \beta_{7} + \beta_{6} + 7 \beta_{5} + \cdots + 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 8 \beta_{13} - 10 \beta_{12} + \beta_{11} - \beta_{9} + 9 \beta_{8} + \beta_{7} + 9 \beta_{6} + \cdots + 40 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2 \beta_{13} - 44 \beta_{12} + 3 \beta_{11} + 41 \beta_{10} - 54 \beta_{9} + 53 \beta_{8} - 10 \beta_{7} + \cdots + 33 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 53 \beta_{13} - 73 \beta_{12} + 10 \beta_{11} + \beta_{10} - 13 \beta_{9} + 65 \beta_{8} + 10 \beta_{7} + \cdots + 214 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 25 \beta_{13} - 270 \beta_{12} + 39 \beta_{11} + 232 \beta_{10} - 347 \beta_{9} + 334 \beta_{8} + \cdots + 183 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 334 \beta_{13} - 483 \beta_{12} + 76 \beta_{11} + 18 \beta_{10} - 120 \beta_{9} + 439 \beta_{8} + \cdots + 1176 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 214 \beta_{13} - 1644 \beta_{12} + 348 \beta_{11} + 1310 \beta_{10} - 2183 \beta_{9} + 2068 \beta_{8} + \cdots + 1035 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 2063 \beta_{13} - 3079 \beta_{12} + 530 \beta_{11} + 200 \beta_{10} - 966 \beta_{9} + 2881 \beta_{8} + \cdots + 6566 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 1566 \beta_{13} - 9991 \beta_{12} + 2666 \beta_{11} + 7444 \beta_{10} - 13576 \beta_{9} + 12714 \beta_{8} + \cdots + 5975 \) 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
2.48639
2.19816
1.69658
1.32779
1.14252
0.689071
0.201460
−0.698700
−0.751517
−0.824916
−1.23810
−1.62437
−2.20542
−2.39896
−2.48639 1.00000 4.18214 0.801546 −2.48639 −4.85131 −5.42565 1.00000 −1.99296
1.2 −2.19816 1.00000 2.83190 0.561564 −2.19816 0.959620 −1.82865 1.00000 −1.23441
1.3 −1.69658 1.00000 0.878394 −2.55880 −1.69658 −0.248570 1.90290 1.00000 4.34122
1.4 −1.32779 1.00000 −0.236974 0.238659 −1.32779 −2.58799 2.97023 1.00000 −0.316889
1.5 −1.14252 1.00000 −0.694639 2.10786 −1.14252 0.0804509 3.07869 1.00000 −2.40828
1.6 −0.689071 1.00000 −1.52518 −2.80437 −0.689071 1.78283 2.42910 1.00000 1.93241
1.7 −0.201460 1.00000 −1.95941 1.13905 −0.201460 3.48995 0.797662 1.00000 −0.229472
1.8 0.698700 1.00000 −1.51182 0.633377 0.698700 −3.80301 −2.45371 1.00000 0.442541
1.9 0.751517 1.00000 −1.43522 −3.24488 0.751517 2.72814 −2.58163 1.00000 −2.43858
1.10 0.824916 1.00000 −1.31951 −0.130051 0.824916 −1.62236 −2.73832 1.00000 −0.107281
1.11 1.23810 1.00000 −0.467110 3.08954 1.23810 −1.49132 −3.05453 1.00000 3.82515
1.12 1.62437 1.00000 0.638571 1.12563 1.62437 −1.97394 −2.21146 1.00000 1.82844
1.13 2.20542 1.00000 2.86386 −3.37806 2.20542 −2.67123 1.90517 1.00000 −7.45002
1.14 2.39896 1.00000 3.75501 −2.58106 2.39896 −0.791248 4.21019 1.00000 −6.19187
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(17\) \(1\)
\(59\) \(-1\)

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 3009.2.a.e ✓ 14
3.b odd 2 1 9027.2.a.k 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3009.2.a.e ✓ 14 1.a even 1 1 trivial
9027.2.a.k 14 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3009))\):

\( T_{2}^{14} - 17 T_{2}^{12} + T_{2}^{11} + 109 T_{2}^{10} - 13 T_{2}^{9} - 333 T_{2}^{8} + 61 T_{2}^{7} + \cdots - 9 \) Copy content Toggle raw display
\( T_{5}^{14} + 5 T_{5}^{13} - 18 T_{5}^{12} - 103 T_{5}^{11} + 127 T_{5}^{10} + 743 T_{5}^{9} - 688 T_{5}^{8} + \cdots + 15 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} - 17 T^{12} + \cdots - 9 \) Copy content Toggle raw display
$3$ \( (T - 1)^{14} \) Copy content Toggle raw display
$5$ \( T^{14} + 5 T^{13} + \cdots + 15 \) Copy content Toggle raw display
$7$ \( T^{14} + 11 T^{13} + \cdots - 157 \) Copy content Toggle raw display
$11$ \( T^{14} + 7 T^{13} + \cdots - 33 \) Copy content Toggle raw display
$13$ \( T^{14} + 15 T^{13} + \cdots - 2999 \) Copy content Toggle raw display
$17$ \( (T + 1)^{14} \) Copy content Toggle raw display
$19$ \( T^{14} + 9 T^{13} + \cdots - 859225 \) Copy content Toggle raw display
$23$ \( T^{14} + 20 T^{13} + \cdots - 52209 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 207810777 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots - 2499296459 \) Copy content Toggle raw display
$37$ \( T^{14} + 28 T^{13} + \cdots + 22224787 \) Copy content Toggle raw display
$41$ \( T^{14} + 10 T^{13} + \cdots - 17564271 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 363357143 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots - 237895525317 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 1492790421 \) Copy content Toggle raw display
$59$ \( (T - 1)^{14} \) Copy content Toggle raw display
$61$ \( T^{14} + 22 T^{13} + \cdots - 94911293 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 296153246987 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 3777552735 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 410820746527 \) Copy content Toggle raw display
$79$ \( T^{14} + 47 T^{13} + \cdots - 85746263 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 7189459791 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots - 2619638114805 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots - 4541851231 \) Copy content Toggle raw display
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