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Properties

Label 7865.2.a.bg
Level $7865$
Weight $2$
Character orbit 7865.a
Self dual yes
Analytic conductor $62.802$
Analytic rank $1$
Dimension $18$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [7865,2,Mod(1,7865)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7865, 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("7865.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 7865 = 5 \cdot 11^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7865.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [18,-2,-4,16,18,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(62.8023411897\)
Analytic rank: \(1\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 2 x^{17} - 24 x^{16} + 48 x^{15} + 230 x^{14} - 466 x^{13} - 1116 x^{12} + 2346 x^{11} + \cdots + 13 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{10} q^{3} + (\beta_{2} + 1) q^{4} + q^{5} + ( - \beta_{11} + \beta_1) q^{6} + (\beta_{4} - 1) q^{7} + ( - \beta_{3} - \beta_1) q^{8} + ( - \beta_{17} + \beta_{7}) q^{9} - \beta_1 q^{10}+ \cdots + (2 \beta_{16} + \beta_{15} - \beta_{14} + \cdots - 3) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 2 q^{2} - 4 q^{3} + 16 q^{4} + 18 q^{5} - 2 q^{6} - 20 q^{7} + 10 q^{9} - 2 q^{10} - 20 q^{12} + 18 q^{13} + 10 q^{14} - 4 q^{15} + 8 q^{16} - 6 q^{17} - 38 q^{19} + 16 q^{20} - 22 q^{21} - 2 q^{23}+ \cdots - 36 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 2 x^{17} - 24 x^{16} + 48 x^{15} + 230 x^{14} - 466 x^{13} - 1116 x^{12} + 2346 x^{11} + \cdots + 13 \) : 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
\(\beta_{4}\)\(=\) \( ( - 75937 \nu^{17} - 3120436 \nu^{16} + 4173952 \nu^{15} + 78986990 \nu^{14} - 75066214 \nu^{13} + \cdots - 260518903 ) / 79208321 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 5759455 \nu^{17} + 10048517 \nu^{16} + 150185219 \nu^{15} - 242415395 \nu^{14} + \cdots + 377659019 ) / 79208321 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 7059337 \nu^{17} + 15435763 \nu^{16} + 173262749 \nu^{15} - 368550030 \nu^{14} + \cdots + 107187709 ) / 79208321 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 561419 \nu^{17} - 844855 \nu^{16} - 13880438 \nu^{15} + 19321024 \nu^{14} + 138678278 \nu^{13} + \cdots + 14229697 ) / 4659313 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 12166646 \nu^{17} + 5426375 \nu^{16} + 307847249 \nu^{15} - 114503139 \nu^{14} + \cdots + 24431179 ) / 79208321 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 12816433 \nu^{17} - 16713999 \nu^{16} - 318599412 \nu^{15} + 386058112 \nu^{14} + \cdots + 161709347 ) / 79208321 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 20365912 \nu^{17} + 14776968 \nu^{16} + 507036320 \nu^{15} - 331303152 \nu^{14} + \cdots + 390858704 ) / 79208321 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 25954856 \nu^{17} + 18254432 \nu^{16} + 646260624 \nu^{15} - 411647521 \nu^{14} + \cdots + 264756856 ) / 79208321 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 27371792 \nu^{17} + 15304993 \nu^{16} + 676765154 \nu^{15} - 330036289 \nu^{14} + \cdots + 356988614 ) / 79208321 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 28411617 \nu^{17} - 20100839 \nu^{16} - 710862889 \nu^{15} + 457098561 \nu^{14} + \cdots - 169826443 ) / 79208321 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 131997 \nu^{17} - 101010 \nu^{16} - 3294623 \nu^{15} + 2280684 \nu^{14} + 33198936 \nu^{13} + \cdots - 1980170 ) / 315571 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 2227866 \nu^{17} + 2113411 \nu^{16} + 54651963 \nu^{15} - 48089038 \nu^{14} - 539472495 \nu^{13} + \cdots + 23033125 ) / 4659313 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 162984 \nu^{17} - 126695 \nu^{16} - 4055172 \nu^{15} + 2839626 \nu^{14} + 40682766 \nu^{13} + \cdots - 1715961 ) / 315571 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 45899239 \nu^{17} - 30127742 \nu^{16} - 1145005672 \nu^{15} + 666445102 \nu^{14} + \cdots - 143663255 ) / 79208321 \) 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
\(\nu^{4}\)\(=\) \( \beta_{14} + \beta_{12} + \beta_{5} + \beta_{3} + 6\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{17} + \beta_{16} - \beta_{14} - \beta_{11} - \beta_{8} + 8\beta_{3} + 29\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 11 \beta_{14} - 2 \beta_{13} + 11 \beta_{12} - 2 \beta_{10} - \beta_{9} - 2 \beta_{8} - \beta_{7} + \cdots + 86 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 11 \beta_{17} + 13 \beta_{16} + \beta_{15} - 13 \beta_{14} - 11 \beta_{11} - \beta_{10} - 2 \beta_{9} + \cdots + 13 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - \beta_{17} - 3 \beta_{16} + 92 \beta_{14} - 26 \beta_{13} + 91 \beta_{12} - 2 \beta_{11} - 27 \beta_{10} + \cdots + 523 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 91 \beta_{17} + 120 \beta_{16} + 15 \beta_{15} - 120 \beta_{14} - \beta_{13} + 3 \beta_{12} + \cdots + 118 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 18 \beta_{17} - 44 \beta_{16} + 3 \beta_{15} + 694 \beta_{14} - 241 \beta_{13} + 683 \beta_{12} + \cdots + 3283 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 686 \beta_{17} + 968 \beta_{16} + 154 \beta_{15} - 963 \beta_{14} - 24 \beta_{13} + 58 \beta_{12} + \cdots + 944 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 212 \beta_{17} - 427 \beta_{16} + 60 \beta_{15} + 4973 \beta_{14} - 1961 \beta_{13} + 4912 \beta_{12} + \cdots + 21023 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 4972 \beta_{17} + 7294 \beta_{16} + 1350 \beta_{15} - 7184 \beta_{14} - 345 \beta_{13} + 728 \beta_{12} + \cdots + 7164 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 2078 \beta_{17} - 3463 \beta_{16} + 761 \beta_{15} + 34625 \beta_{14} - 14959 \beta_{13} + \cdots + 136505 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 35346 \beta_{17} + 52871 \beta_{16} + 10886 \beta_{15} - 51348 \beta_{14} - 3907 \beta_{13} + \cdots + 53141 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 18429 \beta_{17} - 25407 \beta_{16} + 7855 \beta_{15} + 237002 \beta_{14} - 110054 \beta_{13} + \cdots + 895552 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 248625 \beta_{17} + 374316 \beta_{16} + 83554 \beta_{15} - 357250 \beta_{14} - 38591 \beta_{13} + \cdots + 390491 \) 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
2.64354
2.39484
2.38240
1.86579
1.36661
1.33594
1.08513
0.874482
0.539172
0.0520508
−0.379916
−0.707274
−0.884711
−1.63930
−1.99540
−2.10939
−2.25322
−2.57074
−2.64354 −2.04244 4.98831 1.00000 5.39928 0.928128 −7.89971 1.17157 −2.64354
1.2 −2.39484 2.09411 3.73524 1.00000 −5.01505 −2.72154 −4.15560 1.38531 −2.39484
1.3 −2.38240 −0.126169 3.67583 1.00000 0.300585 −4.61981 −3.99251 −2.98408 −2.38240
1.4 −1.86579 −2.27975 1.48116 1.00000 4.25352 1.97913 0.968053 2.19725 −1.86579
1.5 −1.36661 0.732615 −0.132375 1.00000 −1.00120 0.588593 2.91413 −2.46327 −1.36661
1.6 −1.33594 2.64061 −0.215253 1.00000 −3.52771 −0.113669 2.95945 3.97285 −1.33594
1.7 −1.08513 −2.13577 −0.822490 1.00000 2.31759 −5.04455 3.06277 1.56152 −1.08513
1.8 −0.874482 −1.40090 −1.23528 1.00000 1.22506 −3.11484 2.82920 −1.03749 −0.874482
1.9 −0.539172 −2.44736 −1.70929 1.00000 1.31955 3.00429 1.99995 2.98959 −0.539172
1.10 −0.0520508 1.79183 −1.99729 1.00000 −0.0932661 −4.01083 0.208062 0.210655 −0.0520508
1.11 0.379916 2.63503 −1.85566 1.00000 1.00109 −2.79558 −1.46483 3.94337 0.379916
1.12 0.707274 0.0372111 −1.49976 1.00000 0.0263184 −3.49630 −2.47529 −2.99862 0.707274
1.13 0.884711 −0.0254781 −1.21729 1.00000 −0.0225408 0.775097 −2.84637 −2.99935 0.884711
1.14 1.63930 1.14904 0.687319 1.00000 1.88363 1.97679 −2.15188 −1.67970 1.63930
1.15 1.99540 −3.28811 1.98161 1.00000 −6.56108 −0.568028 −0.0366971 7.81166 1.99540
1.16 2.10939 −0.996146 2.44952 1.00000 −2.10126 2.16100 0.948222 −2.00769 2.10939
1.17 2.25322 1.68410 3.07699 1.00000 3.79465 −4.22010 2.42669 −0.163801 2.25322
1.18 2.57074 −2.02243 4.60873 1.00000 −5.19916 −0.707776 6.70637 1.09023 2.57074
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( -1 \)
\(11\) \( +1 \)
\(13\) \( -1 \)

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 7865.2.a.bg ✓ 18
11.b odd 2 1 7865.2.a.bh yes 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7865.2.a.bg ✓ 18 1.a even 1 1 trivial
7865.2.a.bh yes 18 11.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7865))\):

\( T_{2}^{18} + 2 T_{2}^{17} - 24 T_{2}^{16} - 48 T_{2}^{15} + 230 T_{2}^{14} + 466 T_{2}^{13} - 1116 T_{2}^{12} + \cdots + 13 \) Copy content Toggle raw display
\( T_{3}^{18} + 4 T_{3}^{17} - 24 T_{3}^{16} - 106 T_{3}^{15} + 218 T_{3}^{14} + 1132 T_{3}^{13} - 898 T_{3}^{12} + \cdots + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + 2 T^{17} + \cdots + 13 \) Copy content Toggle raw display
$3$ \( T^{18} + 4 T^{17} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T - 1)^{18} \) Copy content Toggle raw display
$7$ \( T^{18} + 20 T^{17} + \cdots - 16064 \) Copy content Toggle raw display
$11$ \( T^{18} \) Copy content Toggle raw display
$13$ \( (T - 1)^{18} \) Copy content Toggle raw display
$17$ \( T^{18} + 6 T^{17} + \cdots - 7303703 \) Copy content Toggle raw display
$19$ \( T^{18} + 38 T^{17} + \cdots - 25622384 \) Copy content Toggle raw display
$23$ \( T^{18} + 2 T^{17} + \cdots + 91652989 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots - 25533773507 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 8687269350916 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 21975639988 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots - 7719454796 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 56400948577 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 894514270852 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots - 129214631711 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots - 8295399887708 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots - 70284859091 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 244894018580356 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 16057176136564 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots - 238670787056 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots - 86605742580071 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots - 5700216594428 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots - 20247682874288 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 32707282993552 \) Copy content Toggle raw display
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