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Properties

Label 855.2.bs.d
Level $855$
Weight $2$
Character orbit 855.bs
Analytic conductor $6.827$
Analytic rank $0$
Dimension $18$
Inner twists $2$

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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] = [855,2,Mod(226,855)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("855.226"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(855, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 0, 10])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 855.bs (of order \(9\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [18,3,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.82720937282\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{9})\)
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: \( x^{18} - 3 x^{17} + 18 x^{16} - 33 x^{15} + 153 x^{14} - 249 x^{13} + 827 x^{12} - 1005 x^{11} + \cdots + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 285)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + (\beta_{16} + \beta_{14} + \cdots - \beta_{2}) q^{4} + ( - \beta_{16} + \beta_{5}) q^{5} + ( - \beta_{17} + 2 \beta_{14} + \cdots + 2) q^{7} + ( - \beta_{17} - 2 \beta_{16} + \cdots - \beta_1) q^{8}+ \cdots + (\beta_{16} - \beta_{14} + \cdots - 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 3 q^{2} + 3 q^{4} + 6 q^{7} + 6 q^{8} + 3 q^{10} + 21 q^{14} + 15 q^{16} - 18 q^{17} + 18 q^{20} - 51 q^{22} - 9 q^{23} - 15 q^{26} + 27 q^{28} - 3 q^{29} - 12 q^{31} - 3 q^{32} - 51 q^{34} + 3 q^{35}+ \cdots + 39 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 3 x^{17} + 18 x^{16} - 33 x^{15} + 153 x^{14} - 249 x^{13} + 827 x^{12} - 1005 x^{11} + \cdots + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2342072914873 \nu^{17} + 147632814895751 \nu^{16} - 373304123398556 \nu^{15} + \cdots + 34\!\cdots\!76 ) / 73\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 13283395398204 \nu^{17} - 121074428301483 \nu^{16} + 251389872810379 \nu^{15} + \cdots - 15\!\cdots\!04 ) / 73\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 18425857346745 \nu^{17} - 96521761344807 \nu^{16} + 154380607690744 \nu^{15} + \cdots - 17\!\cdots\!48 ) / 73\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 27618008993732 \nu^{17} + 101279884327941 \nu^{16} - 400602400542369 \nu^{15} + \cdots - 42\!\cdots\!76 ) / 73\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 15815116869905 \nu^{17} - 71321968456356 \nu^{16} + 330388936703219 \nu^{15} + \cdots + 61\!\cdots\!16 ) / 36\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 50448667766885 \nu^{17} - 193464442001871 \nu^{16} + \cdots + 19\!\cdots\!76 ) / 73\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 10414163535874 \nu^{17} - 26944248526843 \nu^{16} + 172733855614584 \nu^{15} + \cdots + 14\!\cdots\!32 ) / 60\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 279052692123213 \nu^{17} + \cdots - 30\!\cdots\!88 ) / 14\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 160924614496095 \nu^{17} - 490490989978051 \nu^{16} + \cdots - 850137305485056 ) / 73\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 183898506568409 \nu^{17} - 658771761316034 \nu^{16} + \cdots + 11\!\cdots\!88 ) / 73\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 204807903420767 \nu^{17} + 485129329506016 \nu^{16} + \cdots - 78\!\cdots\!40 ) / 73\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 534949949510327 \nu^{17} + 899554040817963 \nu^{16} + \cdots - 33\!\cdots\!08 ) / 14\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 179979630050129 \nu^{17} + 623252198437379 \nu^{16} + \cdots - 75\!\cdots\!28 ) / 48\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 306761868767665 \nu^{17} + \cdots - 13\!\cdots\!92 ) / 73\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 504087157023427 \nu^{17} + \cdots + 24\!\cdots\!72 ) / 73\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 629708829049013 \nu^{17} + \cdots - 11\!\cdots\!68 ) / 36\!\cdots\!16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{16} + \beta_{15} + 2 \beta_{14} + \beta_{12} - \beta_{10} + \beta_{7} - \beta_{6} - 2 \beta_{5} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{17} + 2 \beta_{16} - \beta_{14} + \beta_{13} - \beta_{11} - \beta_{10} + 4 \beta_{8} + \beta_{7} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 5 \beta_{16} - 6 \beta_{15} - 9 \beta_{14} - 5 \beta_{12} - \beta_{11} + 7 \beta_{10} + \beta_{9} + \cdots - 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 7 \beta_{17} - 16 \beta_{16} - 8 \beta_{15} - \beta_{14} - 8 \beta_{13} - \beta_{11} + \cdots - 29 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 13 \beta_{17} - 24 \beta_{16} + 13 \beta_{14} - 9 \beta_{13} + 9 \beta_{11} - 4 \beta_{9} - 18 \beta_{8} + \cdots + 76 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 17 \beta_{17} - 4 \beta_{16} + 59 \beta_{15} + 80 \beta_{14} + 17 \beta_{13} + 13 \beta_{12} + \cdots + 80 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 68 \beta_{17} + 290 \beta_{16} + 227 \beta_{15} + 225 \beta_{14} + 125 \beta_{13} + 136 \beta_{12} + \cdots - 102 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 464 \beta_{17} + 814 \beta_{16} - 464 \beta_{14} + 278 \beta_{13} - 30 \beta_{12} - 278 \beta_{11} + \cdots - 649 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 576 \beta_{17} - 148 \beta_{16} - 1506 \beta_{15} - 2454 \beta_{14} - 576 \beta_{13} - 724 \beta_{12} + \cdots - 2454 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1814 \beta_{17} - 5128 \beta_{16} - 3267 \beta_{15} - 1953 \beta_{14} - 3535 \beta_{13} - 884 \beta_{12} + \cdots - 268 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 8797 \beta_{17} - 14152 \beta_{16} + 8797 \beta_{14} - 3689 \beta_{13} - 822 \beta_{12} + \cdots + 19444 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 14695 \beta_{17} - 7621 \beta_{16} + 24595 \beta_{15} + 43465 \beta_{14} + 14695 \beta_{13} + \cdots + 43465 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 27462 \beta_{17} + 94557 \beta_{16} + 74507 \beta_{15} + 62108 \beta_{14} + 69998 \beta_{13} + \cdots - 4509 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 206190 \beta_{17} + 327308 \beta_{16} - 206190 \beta_{14} + 86035 \beta_{13} + 12027 \beta_{12} + \cdots - 319180 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 342394 \beta_{17} + 167670 \beta_{16} - 544909 \beta_{15} - 992041 \beta_{14} - 342394 \beta_{13} + \cdots - 992041 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 618917 \beta_{17} - 1906110 \beta_{16} - 1412527 \beta_{15} - 1049652 \beta_{14} - 1577055 \beta_{13} + \cdots - 164528 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/855\mathbb{Z}\right)^\times\).

\(n\) \(172\) \(191\) \(496\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{12}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
226.1
0.805041 + 1.39437i
−0.151088 − 0.261692i
−1.09365 − 1.89425i
−0.407048 − 0.705028i
0.289798 + 0.501945i
1.38329 + 2.39594i
0.805041 − 1.39437i
−0.151088 + 0.261692i
−1.09365 + 1.89425i
1.11017 + 1.92287i
0.616036 + 1.06701i
−1.05256 − 1.82308i
−0.407048 + 0.705028i
0.289798 − 0.501945i
1.38329 − 2.39594i
1.11017 − 1.92287i
0.616036 − 1.06701i
−1.05256 + 1.82308i
−0.279588 + 1.58562i 0 −0.556638 − 0.202600i −0.939693 + 0.342020i 0 −1.19940 − 2.07742i −1.13321 + 1.96277i 0 −0.279588 − 1.58562i
226.2 0.0524723 − 0.297585i 0 1.79358 + 0.652810i −0.939693 + 0.342020i 0 0.815100 + 1.41179i 0.590556 − 1.02287i 0 0.0524723 + 0.297585i
226.3 0.379819 − 2.15406i 0 −2.61633 − 0.952266i −0.939693 + 0.342020i 0 1.55795 + 2.69845i −0.857680 + 1.48554i 0 0.379819 + 2.15406i
271.1 −0.765001 + 0.278437i 0 −1.02439 + 0.859566i 0.766044 + 0.642788i 0 1.14885 + 1.98986i 1.35842 − 2.35285i 0 −0.765001 − 0.278437i
271.2 0.544642 − 0.198233i 0 −1.27475 + 1.06964i 0.766044 + 0.642788i 0 −1.86160 − 3.22438i −1.06184 + 1.83916i 0 0.544642 + 0.198233i
271.3 2.59974 − 0.946229i 0 4.33123 − 3.63433i 0.766044 + 0.642788i 0 0.773058 + 1.33898i 5.05459 − 8.75480i 0 2.59974 + 0.946229i
541.1 −0.279588 − 1.58562i 0 −0.556638 + 0.202600i −0.939693 − 0.342020i 0 −1.19940 + 2.07742i −1.13321 − 1.96277i 0 −0.279588 + 1.58562i
541.2 0.0524723 + 0.297585i 0 1.79358 − 0.652810i −0.939693 − 0.342020i 0 0.815100 − 1.41179i 0.590556 + 1.02287i 0 0.0524723 − 0.297585i
541.3 0.379819 + 2.15406i 0 −2.61633 + 0.952266i −0.939693 − 0.342020i 0 1.55795 − 2.69845i −0.857680 − 1.48554i 0 0.379819 − 2.15406i
586.1 −1.70088 − 1.42720i 0 0.508770 + 2.88538i 0.173648 − 0.984808i 0 −0.824194 − 1.42755i 1.03233 − 1.78806i 0 −1.70088 + 1.42720i
586.2 −0.943822 − 0.791961i 0 −0.0836982 − 0.474676i 0.173648 − 0.984808i 0 2.02067 + 3.49991i −1.52900 + 2.64831i 0 −0.943822 + 0.791961i
586.3 1.61261 + 1.35314i 0 0.422224 + 2.39455i 0.173648 − 0.984808i 0 0.569563 + 0.986512i −0.454171 + 0.786647i 0 1.61261 − 1.35314i
631.1 −0.765001 − 0.278437i 0 −1.02439 − 0.859566i 0.766044 − 0.642788i 0 1.14885 − 1.98986i 1.35842 + 2.35285i 0 −0.765001 + 0.278437i
631.2 0.544642 + 0.198233i 0 −1.27475 − 1.06964i 0.766044 − 0.642788i 0 −1.86160 + 3.22438i −1.06184 − 1.83916i 0 0.544642 − 0.198233i
631.3 2.59974 + 0.946229i 0 4.33123 + 3.63433i 0.766044 − 0.642788i 0 0.773058 − 1.33898i 5.05459 + 8.75480i 0 2.59974 − 0.946229i
766.1 −1.70088 + 1.42720i 0 0.508770 − 2.88538i 0.173648 + 0.984808i 0 −0.824194 + 1.42755i 1.03233 + 1.78806i 0 −1.70088 − 1.42720i
766.2 −0.943822 + 0.791961i 0 −0.0836982 + 0.474676i 0.173648 + 0.984808i 0 2.02067 − 3.49991i −1.52900 − 2.64831i 0 −0.943822 − 0.791961i
766.3 1.61261 − 1.35314i 0 0.422224 − 2.39455i 0.173648 + 0.984808i 0 0.569563 − 0.986512i −0.454171 − 0.786647i 0 1.61261 + 1.35314i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 226.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 855.2.bs.d 18
3.b odd 2 1 285.2.u.b ✓ 18
19.e even 9 1 inner 855.2.bs.d 18
57.j even 18 1 5415.2.a.bk 9
57.l odd 18 1 285.2.u.b ✓ 18
57.l odd 18 1 5415.2.a.bl 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.u.b ✓ 18 3.b odd 2 1
285.2.u.b ✓ 18 57.l odd 18 1
855.2.bs.d 18 1.a even 1 1 trivial
855.2.bs.d 18 19.e even 9 1 inner
5415.2.a.bk 9 57.j even 18 1
5415.2.a.bl 9 57.l odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{18} - 3 T_{2}^{17} + 3 T_{2}^{16} - 6 T_{2}^{15} + 21 T_{2}^{14} - 39 T_{2}^{13} + 185 T_{2}^{12} + \cdots + 64 \) acting on \(S_{2}^{\mathrm{new}}(855, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} - 3 T^{17} + \cdots + 64 \) Copy content Toggle raw display
$3$ \( T^{18} \) Copy content Toggle raw display
$5$ \( (T^{6} + T^{3} + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{18} - 6 T^{17} + \cdots + 1495729 \) Copy content Toggle raw display
$11$ \( T^{18} + 57 T^{16} + \cdots + 2050624 \) Copy content Toggle raw display
$13$ \( T^{18} - 12 T^{16} + \cdots + 201601 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 32345303104 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 322687697779 \) Copy content Toggle raw display
$23$ \( T^{18} + 9 T^{17} + \cdots + 11505664 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 28334315584 \) Copy content Toggle raw display
$31$ \( T^{18} + 12 T^{17} + \cdots + 81 \) Copy content Toggle raw display
$37$ \( (T^{9} + 6 T^{8} + \cdots + 1232499)^{2} \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 1261524326976 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 1096868161 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 4544917056 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 19\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 47939978304 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 7111294223401 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 1038008455929 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 6322158144 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 94\!\cdots\!89 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 8554893765625 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 133852497247296 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 16370828103744 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 32\!\cdots\!21 \) Copy content Toggle raw display
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