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Properties

Label 9065.2.a.t
Level $9065$
Weight $2$
Character orbit 9065.a
Self dual yes
Analytic conductor $72.384$
Analytic rank $1$
Dimension $19$
CM no
Inner twists $1$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9065,2,Mod(1,9065)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9065, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9065.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 9065 = 5 \cdot 7^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9065.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [19,1,-7,19,-19,-4,0,-3,18,-1,13] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(72.3843894323\)
Analytic rank: \(1\)
Dimension: \(19\)
Coefficient field: \(\mathbb{Q}[x]/(x^{19} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{19} - x^{18} - 28 x^{17} + 28 x^{16} + 320 x^{15} - 320 x^{14} - 1920 x^{13} + 1924 x^{12} + \cdots + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{18}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{9} q^{3} + (\beta_{2} + 1) q^{4} - q^{5} + ( - \beta_{11} - \beta_1) q^{6} + (\beta_{3} + \beta_1) q^{8} + ( - \beta_{15} + \beta_{9} + 1) q^{9} - \beta_1 q^{10} + (\beta_{16} + \beta_{10} + \cdots - \beta_1) q^{11}+ \cdots + ( - \beta_{17} + 2 \beta_{16} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 19 q + q^{2} - 7 q^{3} + 19 q^{4} - 19 q^{5} - 4 q^{6} - 3 q^{8} + 18 q^{9} - q^{10} + 13 q^{11} - 16 q^{12} - 9 q^{13} + 7 q^{15} + 23 q^{16} - 3 q^{17} + 16 q^{18} - 20 q^{19} - 19 q^{20} - 24 q^{22}+ \cdots + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{19} - x^{18} - 28 x^{17} + 28 x^{16} + 320 x^{15} - 320 x^{14} - 1920 x^{13} + 1924 x^{12} + \cdots + 18 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 7\nu^{2} + 6 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 2624248 \nu^{18} + 18948490 \nu^{17} + 95008225 \nu^{16} - 419944228 \nu^{15} + \cdots - 5962306659 ) / 2229563793 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11004863 \nu^{18} + 44078579 \nu^{17} + 314704415 \nu^{16} - 1158331601 \nu^{15} + \cdots + 4755243851 ) / 743187931 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 96730114 \nu^{18} - 26793352 \nu^{17} - 2590969348 \nu^{16} + 815634322 \nu^{15} + \cdots + 11088856365 ) / 2229563793 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 57506159 \nu^{18} + 113680079 \nu^{17} + 1536273262 \nu^{16} - 3081136661 \nu^{15} + \cdots - 2505501323 ) / 743187931 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 195223802 \nu^{18} - 33378319 \nu^{17} + 5585357333 \nu^{16} + 841303285 \nu^{15} + \cdots + 3557095491 ) / 2229563793 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 220285568 \nu^{18} + 347764181 \nu^{17} + 6033515966 \nu^{16} - 9455352923 \nu^{15} + \cdots - 394600374 ) / 2229563793 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 76200707 \nu^{18} + 39696959 \nu^{17} + 2102523247 \nu^{16} - 1076186549 \nu^{15} + \cdots + 1171342812 ) / 743187931 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 230500127 \nu^{18} + 327296369 \nu^{17} + 5878930373 \nu^{16} - 8868930395 \nu^{15} + \cdots - 18377141415 ) / 2229563793 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 252869075 \nu^{18} + 55775188 \nu^{17} - 7227514151 \nu^{16} - 1693640542 \nu^{15} + \cdots - 21557480895 ) / 2229563793 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 259442672 \nu^{18} + 125388053 \nu^{17} + 6991001510 \nu^{16} - 3364760225 \nu^{15} + \cdots + 5062545609 ) / 2229563793 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 151254431 \nu^{18} - 19110451 \nu^{17} + 4253174725 \nu^{16} + 542088930 \nu^{15} + \cdots + 8582927526 ) / 743187931 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 166930282 \nu^{18} + 53825972 \nu^{17} - 4760605277 \nu^{16} - 1441064295 \nu^{15} + \cdots - 3430367382 ) / 743187931 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 571727897 \nu^{18} - 70937051 \nu^{17} - 15846903200 \nu^{16} + 1726565285 \nu^{15} + \cdots - 14815274583 ) / 2229563793 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 731738372 \nu^{18} + 396315491 \nu^{17} + 20212437611 \nu^{16} - 10940834627 \nu^{15} + \cdots - 3101619378 ) / 2229563793 \) 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
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 7\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{17} - \beta_{16} - \beta_{15} - \beta_{13} + \beta_{12} + \beta_{11} - \beta_{10} + \beta_{6} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{18} - \beta_{14} + \beta_{12} + 2 \beta_{11} + 2 \beta_{9} + \beta_{8} + 2 \beta_{7} + \cdots + 84 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 11 \beta_{17} - 11 \beta_{16} - 13 \beta_{15} + 2 \beta_{14} - 11 \beta_{13} + 11 \beta_{12} + 11 \beta_{11} + \cdots + 10 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 12 \beta_{18} + 3 \beta_{17} + \beta_{16} + 3 \beta_{15} - 14 \beta_{14} - 3 \beta_{13} + 12 \beta_{12} + \cdots + 495 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2 \beta_{18} + 91 \beta_{17} - 93 \beta_{16} - 127 \beta_{15} + 32 \beta_{14} - 95 \beta_{13} + \cdots + 76 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 106 \beta_{18} + 50 \beta_{17} + 14 \beta_{16} + 52 \beta_{15} - 139 \beta_{14} - 48 \beta_{13} + \cdots + 3012 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 44 \beta_{18} + 685 \beta_{17} - 719 \beta_{16} - 1103 \beta_{15} + 347 \beta_{14} - 760 \beta_{13} + \cdots + 514 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 844 \beta_{18} + 550 \beta_{17} + 137 \beta_{16} + 607 \beta_{15} - 1203 \beta_{14} - 505 \beta_{13} + \cdots + 18773 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 596 \beta_{18} + 4957 \beta_{17} - 5338 \beta_{16} - 8986 \beta_{15} + 3204 \beta_{14} - 5871 \beta_{13} + \cdots + 3191 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 6430 \beta_{18} + 5056 \beta_{17} + 1187 \beta_{16} + 5995 \beta_{15} - 9716 \beta_{14} + \cdots + 119301 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 6458 \beta_{18} + 35199 \beta_{17} - 38802 \beta_{16} - 70397 \beta_{15} + 27196 \beta_{14} + \cdots + 18039 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 47963 \beta_{18} + 42132 \beta_{17} + 9841 \beta_{16} + 54117 \beta_{15} - 75510 \beta_{14} + \cdots + 770546 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 61656 \beta_{18} + 247404 \beta_{17} - 278736 \beta_{16} - 537555 \beta_{15} + 219659 \beta_{14} + \cdots + 88418 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 353883 \beta_{18} + 330419 \beta_{17} + 80444 \beta_{16} + 462522 \beta_{15} - 573749 \beta_{14} + \cdots + 5045804 \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
−2.69796
−2.37617
−2.32502
−2.10885
−1.63170
−1.11438
−1.08766
−0.323507
−0.219909
0.144911
0.529297
0.865885
0.949915
1.37918
1.44566
2.08786
2.43044
2.46034
2.59167
−2.69796 −0.948658 5.27899 −1.00000 2.55944 0 −8.84659 −2.10005 2.69796
1.2 −2.37617 −2.79525 3.64620 −1.00000 6.64199 0 −3.91166 4.81340 2.37617
1.3 −2.32502 2.00053 3.40570 −1.00000 −4.65126 0 −3.26829 1.00211 2.32502
1.4 −2.10885 0.417448 2.44725 −1.00000 −0.880335 0 −0.943190 −2.82574 2.10885
1.5 −1.63170 1.00839 0.662438 −1.00000 −1.64538 0 2.18250 −1.98316 1.63170
1.6 −1.11438 1.63380 −0.758158 −1.00000 −1.82067 0 3.07364 −0.330709 1.11438
1.7 −1.08766 −2.74821 −0.816989 −1.00000 2.98913 0 3.06393 4.55267 1.08766
1.8 −0.323507 −3.04658 −1.89534 −1.00000 0.985589 0 1.26017 6.28164 0.323507
1.9 −0.219909 −0.728279 −1.95164 −1.00000 0.160155 0 0.869000 −2.46961 0.219909
1.10 0.144911 0.479413 −1.97900 −1.00000 0.0694721 0 −0.576600 −2.77016 −0.144911
1.11 0.529297 2.26862 −1.71984 −1.00000 1.20078 0 −1.96890 2.14665 −0.529297
1.12 0.865885 −1.25442 −1.25024 −1.00000 −1.08618 0 −2.81434 −1.42644 −0.865885
1.13 0.949915 2.61269 −1.09766 −1.00000 2.48183 0 −2.94251 3.82615 −0.949915
1.14 1.37918 −1.89146 −0.0978605 −1.00000 −2.60867 0 −2.89333 0.577628 −1.37918
1.15 1.44566 −2.69503 0.0899401 −1.00000 −3.89610 0 −2.76130 4.26319 −1.44566
1.16 2.08786 2.31029 2.35916 −1.00000 4.82355 0 0.749868 2.33742 −2.08786
1.17 2.43044 0.704571 3.90703 −1.00000 1.71242 0 4.63491 −2.50358 −2.43044
1.18 2.46034 −1.37545 4.05326 −1.00000 −3.38406 0 5.05171 −1.10815 −2.46034
1.19 2.59167 −2.95241 4.71677 −1.00000 −7.65169 0 7.04099 5.71674 −2.59167
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.19
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( +1 \)
\(7\) \( -1 \)
\(37\) \( -1 \)

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 9065.2.a.t ✓ 19
7.b odd 2 1 9065.2.a.u yes 19
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9065.2.a.t ✓ 19 1.a even 1 1 trivial
9065.2.a.u yes 19 7.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9065))\):

\( T_{2}^{19} - T_{2}^{18} - 28 T_{2}^{17} + 28 T_{2}^{16} + 320 T_{2}^{15} - 320 T_{2}^{14} - 1920 T_{2}^{13} + \cdots + 18 \) Copy content Toggle raw display
\( T_{3}^{19} + 7 T_{3}^{18} - 13 T_{3}^{17} - 181 T_{3}^{16} - 65 T_{3}^{15} + 1903 T_{3}^{14} + \cdots - 2672 \) Copy content Toggle raw display
\( T_{11}^{19} - 13 T_{11}^{18} - 55 T_{11}^{17} + 1295 T_{11}^{16} - 671 T_{11}^{15} - 50125 T_{11}^{14} + \cdots + 209498112 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{19} - T^{18} + \cdots + 18 \) Copy content Toggle raw display
$3$ \( T^{19} + 7 T^{18} + \cdots - 2672 \) Copy content Toggle raw display
$5$ \( (T + 1)^{19} \) Copy content Toggle raw display
$7$ \( T^{19} \) Copy content Toggle raw display
$11$ \( T^{19} + \cdots + 209498112 \) Copy content Toggle raw display
$13$ \( T^{19} + \cdots + 544195584 \) Copy content Toggle raw display
$17$ \( T^{19} + 3 T^{18} + \cdots + 30872576 \) Copy content Toggle raw display
$19$ \( T^{19} + 20 T^{18} + \cdots - 306712 \) Copy content Toggle raw display
$23$ \( T^{19} + \cdots + 19426531392 \) Copy content Toggle raw display
$29$ \( T^{19} + \cdots - 4120617848832 \) Copy content Toggle raw display
$31$ \( T^{19} + \cdots + 1478422181376 \) Copy content Toggle raw display
$37$ \( (T - 1)^{19} \) Copy content Toggle raw display
$41$ \( T^{19} + \cdots + 1169300127744 \) Copy content Toggle raw display
$43$ \( T^{19} + \cdots + 12621737375744 \) Copy content Toggle raw display
$47$ \( T^{19} + \cdots + 23493710396304 \) Copy content Toggle raw display
$53$ \( T^{19} + \cdots + 108869810176 \) Copy content Toggle raw display
$59$ \( T^{19} + \cdots - 17600524909128 \) Copy content Toggle raw display
$61$ \( T^{19} + \cdots + 97670092944096 \) Copy content Toggle raw display
$67$ \( T^{19} + \cdots + 2850553856 \) Copy content Toggle raw display
$71$ \( T^{19} + \cdots - 883956500224 \) Copy content Toggle raw display
$73$ \( T^{19} + \cdots - 20307926532096 \) Copy content Toggle raw display
$79$ \( T^{19} + \cdots - 135407001378816 \) Copy content Toggle raw display
$83$ \( T^{19} + \cdots - 60\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{19} + \cdots - 30094926336 \) Copy content Toggle raw display
$97$ \( T^{19} + \cdots - 23\!\cdots\!32 \) Copy content Toggle raw display
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