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Properties

Label 3009.2.a.h
Level $3009$
Weight $2$
Character orbit 3009.a
Self dual yes
Analytic conductor $24.027$
Analytic rank $0$
Dimension $18$
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: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 7 x^{17} - 6 x^{16} + 140 x^{15} - 149 x^{14} - 1055 x^{13} + 2010 x^{12} + 3596 x^{11} + \cdots + 101 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + \beta_{15} q^{5} - \beta_1 q^{6} - \beta_{12} q^{7} + (\beta_{3} + \beta_1 + 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + \beta_{15} q^{5} - \beta_1 q^{6} - \beta_{12} q^{7} + (\beta_{3} + \beta_1 + 1) q^{8} + q^{9} + (2 \beta_{15} - \beta_{14} + \cdots + \beta_1) q^{10}+ \cdots + ( - \beta_{14} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 7 q^{2} - 18 q^{3} + 25 q^{4} + 2 q^{5} - 7 q^{6} + 3 q^{7} + 21 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 7 q^{2} - 18 q^{3} + 25 q^{4} + 2 q^{5} - 7 q^{6} + 3 q^{7} + 21 q^{8} + 18 q^{9} - 4 q^{10} + 20 q^{11} - 25 q^{12} + 6 q^{13} + 2 q^{14} - 2 q^{15} + 31 q^{16} + 18 q^{17} + 7 q^{18} + 11 q^{19} - 2 q^{20} - 3 q^{21} - q^{22} + 23 q^{23} - 21 q^{24} + 14 q^{25} + 9 q^{26} - 18 q^{27} - 9 q^{28} + 13 q^{29} + 4 q^{30} + 17 q^{31} + 39 q^{32} - 20 q^{33} + 7 q^{34} + 24 q^{35} + 25 q^{36} - 2 q^{37} + 10 q^{38} - 6 q^{39} + 26 q^{40} + 16 q^{41} - 2 q^{42} + 9 q^{43} + 14 q^{44} + 2 q^{45} + 8 q^{46} + 37 q^{47} - 31 q^{48} + 15 q^{49} + 37 q^{50} - 18 q^{51} + 3 q^{52} + 26 q^{53} - 7 q^{54} + 22 q^{55} + 23 q^{56} - 11 q^{57} - 32 q^{58} - 18 q^{59} + 2 q^{60} + 2 q^{61} + 20 q^{62} + 3 q^{63} + 3 q^{64} - 6 q^{65} + q^{66} + 6 q^{67} + 25 q^{68} - 23 q^{69} + 28 q^{70} + 59 q^{71} + 21 q^{72} - 41 q^{73} + 22 q^{74} - 14 q^{75} + 17 q^{76} - 12 q^{77} - 9 q^{78} + 36 q^{79} - 11 q^{80} + 18 q^{81} - 21 q^{82} + 49 q^{83} + 9 q^{84} + 2 q^{85} + 25 q^{86} - 13 q^{87} - 2 q^{88} + 52 q^{89} - 4 q^{90} + 21 q^{91} + 17 q^{92} - 17 q^{93} - 10 q^{94} + 61 q^{95} - 39 q^{96} + q^{97} + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 7 x^{17} - 6 x^{16} + 140 x^{15} - 149 x^{14} - 1055 x^{13} + 2010 x^{12} + 3596 x^{11} + \cdots + 101 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu - 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 457 \nu^{17} + 280318 \nu^{16} - 1399684 \nu^{15} - 4613334 \nu^{14} + 30135073 \nu^{13} + \cdots + 26520773 ) / 2103540 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 15221 \nu^{17} + 20549 \nu^{16} + 528509 \nu^{15} - 836833 \nu^{14} - 6951280 \nu^{13} + \cdots + 2276086 ) / 420708 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 46319 \nu^{17} - 382539 \nu^{16} - 151478 \nu^{15} + 8177357 \nu^{14} - 10334259 \nu^{13} + \cdots - 29526579 ) / 1051770 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 229021 \nu^{17} + 857526 \nu^{16} + 4963432 \nu^{15} - 19712668 \nu^{14} - 43251159 \nu^{13} + \cdots + 15759741 ) / 2103540 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 392543 \nu^{17} + 2228608 \nu^{16} + 4890446 \nu^{15} - 46386834 \nu^{14} + 3461233 \nu^{13} + \cdots + 23718413 ) / 2103540 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 401402 \nu^{17} + 1923427 \nu^{16} + 6749939 \nu^{15} - 41603571 \nu^{14} - 34307063 \nu^{13} + \cdots - 986653 ) / 2103540 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 476524 \nu^{17} + 2242869 \nu^{16} + 8309263 \nu^{15} - 49458247 \nu^{14} - 45736131 \nu^{13} + \cdots + 47925039 ) / 2103540 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 257418 \nu^{17} - 1054588 \nu^{16} - 5097876 \nu^{15} + 23468539 \nu^{14} + 38486747 \nu^{13} + \cdots + 3283627 ) / 1051770 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 302656 \nu^{17} - 1366031 \nu^{16} - 5490817 \nu^{15} + 30201828 \nu^{14} + 33825454 \nu^{13} + \cdots - 22117156 ) / 1051770 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 139193 \nu^{17} + 685869 \nu^{16} + 2323607 \nu^{15} - 15130835 \nu^{14} - 11010144 \nu^{13} + \cdots + 15026274 ) / 420708 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 775531 \nu^{17} - 3655771 \nu^{16} - 13251067 \nu^{15} + 79406423 \nu^{14} + 70052744 \nu^{13} + \cdots - 15029246 ) / 2103540 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 856394 \nu^{17} + 4115389 \nu^{16} + 14094773 \nu^{15} - 88277187 \nu^{14} - 66602411 \nu^{13} + \cdots + 14260439 ) / 2103540 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 335419 \nu^{17} + 1645269 \nu^{16} + 5443963 \nu^{15} - 35540187 \nu^{14} - 23940966 \nu^{13} + \cdots + 20451424 ) / 701180 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 257431 \nu^{17} + 1294175 \nu^{16} + 4018489 \nu^{15} - 27709785 \nu^{14} - 15125500 \nu^{13} + \cdots + 13830382 ) / 420708 \) 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
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{16} - \beta_{12} + \beta_{9} + \beta_{4} + \beta_{3} + 6\beta_{2} + \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{16} + \beta_{14} - 3 \beta_{12} + \beta_{11} + 2 \beta_{9} - \beta_{7} + \beta_{6} + \beta_{4} + \cdots + 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{17} - 11 \beta_{16} - \beta_{13} - 11 \beta_{12} + \beta_{11} + 11 \beta_{9} - \beta_{7} + \cdots + 85 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3 \beta_{17} - 16 \beta_{16} + \beta_{15} + 11 \beta_{14} - \beta_{13} - 35 \beta_{12} + 12 \beta_{11} + \cdots + 64 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 16 \beta_{17} - 98 \beta_{16} + 4 \beta_{15} + 4 \beta_{14} - 13 \beta_{13} - 96 \beta_{12} + 15 \beta_{11} + \cdots + 515 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 47 \beta_{17} - 175 \beta_{16} + 24 \beta_{15} + 95 \beta_{14} - 16 \beta_{13} - 302 \beta_{12} + \cdots + 517 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 177 \beta_{17} - 815 \beta_{16} + 85 \beta_{15} + 75 \beta_{14} - 119 \beta_{13} - 761 \beta_{12} + \cdots + 3280 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 515 \beta_{17} - 1656 \beta_{16} + 357 \beta_{15} + 770 \beta_{14} - 168 \beta_{13} - 2326 \beta_{12} + \cdots + 4159 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1697 \beta_{17} - 6574 \beta_{16} + 1170 \beta_{15} + 939 \beta_{14} - 947 \beta_{13} - 5711 \beta_{12} + \cdots + 21748 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 4909 \beta_{17} - 14588 \beta_{16} + 4253 \beta_{15} + 6146 \beta_{14} - 1474 \beta_{13} - 16935 \beta_{12} + \cdots + 33174 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 15168 \beta_{17} - 52234 \beta_{16} + 13261 \beta_{15} + 9895 \beta_{14} - 7013 \beta_{13} + \cdots + 148967 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 43688 \beta_{17} - 123407 \beta_{16} + 44632 \beta_{15} + 49064 \beta_{14} - 11735 \beta_{13} + \cdots + 262337 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 130413 \beta_{17} - 411866 \beta_{16} + 134699 \beta_{15} + 95023 \beta_{14} - 49753 \beta_{13} + \cdots + 1047120 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 374294 \beta_{17} - 1018350 \beta_{16} + 432285 \beta_{15} + 393345 \beta_{14} - 87922 \beta_{13} + \cdots + 2059125 \) 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
−2.41443
−2.38255
−2.08517
−1.80034
−1.13260
−0.655555
−0.606060
−0.133638
0.436795
0.476012
1.26951
1.55205
2.08761
2.14674
2.21369
2.59282
2.64952
2.78560
−2.41443 −1.00000 3.82947 −0.459274 2.41443 −2.93249 −4.41712 1.00000 1.10889
1.2 −2.38255 −1.00000 3.67652 −2.54868 2.38255 −1.58471 −3.99440 1.00000 6.07235
1.3 −2.08517 −1.00000 2.34792 1.84948 2.08517 4.40170 −0.725477 1.00000 −3.85647
1.4 −1.80034 −1.00000 1.24123 3.40722 1.80034 1.35642 1.36604 1.00000 −6.13416
1.5 −1.13260 −1.00000 −0.717213 −0.514136 1.13260 −2.23606 3.07752 1.00000 0.582312
1.6 −0.655555 −1.00000 −1.57025 −1.95856 0.655555 2.69194 2.34049 1.00000 1.28395
1.7 −0.606060 −1.00000 −1.63269 3.34012 0.606060 −3.51111 2.20163 1.00000 −2.02431
1.8 −0.133638 −1.00000 −1.98214 0.300059 0.133638 3.41091 0.532164 1.00000 −0.0400992
1.9 0.436795 −1.00000 −1.80921 −2.47163 −0.436795 1.30965 −1.66384 1.00000 −1.07959
1.10 0.476012 −1.00000 −1.77341 1.99783 −0.476012 1.80349 −1.79619 1.00000 0.950988
1.11 1.26951 −1.00000 −0.388352 0.877447 −1.26951 −3.80776 −3.03203 1.00000 1.11393
1.12 1.55205 −1.00000 0.408854 −3.27535 −1.55205 2.82178 −2.46954 1.00000 −5.08350
1.13 2.08761 −1.00000 2.35812 3.03245 −2.08761 2.51872 0.747614 1.00000 6.33058
1.14 2.14674 −1.00000 2.60849 −3.79239 −2.14674 −4.31978 1.30627 1.00000 −8.14128
1.15 2.21369 −1.00000 2.90040 3.00500 −2.21369 −0.797773 1.99321 1.00000 6.65212
1.16 2.59282 −1.00000 4.72269 −3.14647 −2.59282 −2.20065 7.05944 1.00000 −8.15821
1.17 2.64952 −1.00000 5.01997 1.04992 −2.64952 3.80520 8.00147 1.00000 2.78178
1.18 2.78560 −1.00000 5.75959 1.30698 −2.78560 0.270526 10.4727 1.00000 3.64072
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
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.h ✓ 18
3.b odd 2 1 9027.2.a.p 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3009.2.a.h ✓ 18 1.a even 1 1 trivial
9027.2.a.p 18 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3009))\):

\( T_{2}^{18} - 7 T_{2}^{17} - 6 T_{2}^{16} + 140 T_{2}^{15} - 149 T_{2}^{14} - 1055 T_{2}^{13} + \cdots + 101 \) Copy content Toggle raw display
\( T_{5}^{18} - 2 T_{5}^{17} - 50 T_{5}^{16} + 106 T_{5}^{15} + 999 T_{5}^{14} - 2263 T_{5}^{13} + \cdots + 15763 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} - 7 T^{17} + \cdots + 101 \) Copy content Toggle raw display
$3$ \( (T + 1)^{18} \) Copy content Toggle raw display
$5$ \( T^{18} - 2 T^{17} + \cdots + 15763 \) Copy content Toggle raw display
$7$ \( T^{18} - 3 T^{17} + \cdots + 998127 \) Copy content Toggle raw display
$11$ \( T^{18} - 20 T^{17} + \cdots - 58457 \) Copy content Toggle raw display
$13$ \( T^{18} - 6 T^{17} + \cdots + 299520 \) Copy content Toggle raw display
$17$ \( (T - 1)^{18} \) Copy content Toggle raw display
$19$ \( T^{18} - 11 T^{17} + \cdots + 67216576 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 16882361407 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 52427160719 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 2021739328 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots - 95474439835 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 457634663015517 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots - 250573936343 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 16040548932160 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 5158422720 \) Copy content Toggle raw display
$59$ \( (T + 1)^{18} \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 68686237080095 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots - 1552402017 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 100123195809728 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots - 3607413219365 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots - 190663615 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots - 24\!\cdots\!72 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 7627593959685 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 36783231093440 \) Copy content Toggle raw display
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