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Properties

Label 585.3.s.a
Level $585$
Weight $3$
Character orbit 585.s
Analytic conductor $15.940$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [585,3,Mod(226,585)] 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("585.226"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(585, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 3])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 585.s (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,0,0,0,20,36,0,0,0] 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: no
Analytic conductor: \(15.9400954651\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 12 x^{13} + 168 x^{12} - 120 x^{11} + 72 x^{10} - 852 x^{9} + 6994 x^{8} - 8856 x^{7} + \cdots + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{15} - \beta_{12} + \cdots - \beta_1) q^{4} + \beta_{10} q^{5} + ( - \beta_{15} + \beta_{14} - \beta_{9} + \cdots + 1) q^{7} + (\beta_{15} + \beta_{14} + \beta_{9} + \cdots + 2) q^{8}+ \cdots + (4 \beta_{15} - 4 \beta_{12} + \cdots + 25) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 20 q^{7} + 36 q^{8} + 12 q^{13} + 32 q^{16} - 28 q^{19} + 40 q^{20} + 48 q^{22} - 144 q^{26} + 36 q^{28} + 136 q^{29} - 12 q^{31} + 24 q^{32} + 148 q^{34} - 80 q^{35} - 84 q^{37} - 4 q^{41} - 108 q^{44}+ \cdots + 384 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 12 x^{13} + 168 x^{12} - 120 x^{11} + 72 x^{10} - 852 x^{9} + 6994 x^{8} - 8856 x^{7} + \cdots + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 36\!\cdots\!97 \nu^{15} + \cdots - 30\!\cdots\!34 ) / 32\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 78\!\cdots\!52 \nu^{15} + \cdots - 52\!\cdots\!46 ) / 21\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 81\!\cdots\!64 \nu^{15} + \cdots + 51\!\cdots\!86 ) / 21\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 23\!\cdots\!02 \nu^{15} + \cdots - 40\!\cdots\!97 ) / 57\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 14\!\cdots\!75 \nu^{15} + \cdots - 35\!\cdots\!62 ) / 19\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 18\!\cdots\!35 \nu^{15} + \cdots - 13\!\cdots\!44 ) / 19\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 33\!\cdots\!26 \nu^{15} + \cdots + 23\!\cdots\!02 ) / 28\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 10\!\cdots\!10 \nu^{15} + \cdots + 16\!\cdots\!89 ) / 57\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 10\!\cdots\!30 \nu^{15} + \cdots + 17\!\cdots\!11 ) / 57\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 11\!\cdots\!22 \nu^{15} + \cdots + 58\!\cdots\!11 ) / 57\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 13\!\cdots\!45 \nu^{15} + \cdots + 93\!\cdots\!08 ) / 57\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 58\!\cdots\!76 \nu^{15} + \cdots - 93\!\cdots\!97 ) / 19\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 88\!\cdots\!38 \nu^{15} + \cdots + 11\!\cdots\!65 ) / 19\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 47\!\cdots\!51 \nu^{15} + \cdots - 33\!\cdots\!40 ) / 57\!\cdots\!12 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{15} - \beta_{12} - 5\beta_{8} + \beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} + \beta_{14} + \beta_{9} + 2\beta_{8} + \beta_{7} - \beta_{4} + \beta_{3} - 10\beta_{2} - \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{14} + \beta_{13} + \beta_{11} - 10\beta_{7} + 12\beta_{6} + \beta_{5} + 2\beta_{4} + 14\beta_{2} + 14\beta _1 - 41 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 14 \beta_{15} - 14 \beta_{13} - 3 \beta_{12} + 8 \beta_{10} - 34 \beta_{8} + 14 \beta_{7} - 3 \beta_{6} + \cdots + 34 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 103 \beta_{15} + 19 \beta_{14} + 19 \beta_{13} + 129 \beta_{12} - 16 \beta_{11} + 8 \beta_{10} + \cdots + 135 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 173 \beta_{15} - 173 \beta_{14} - 72 \beta_{12} + 12 \beta_{11} - 41 \beta_{9} - 463 \beta_{8} + \cdots - 463 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 288 \beta_{14} - 288 \beta_{13} - 204 \beta_{11} - 172 \beta_{10} - 172 \beta_{9} + 1110 \beta_{7} + \cdots + 4033 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2094 \beta_{15} + 2106 \beta_{13} + 1224 \beta_{12} + 42 \beta_{10} + 5916 \beta_{8} - 2094 \beta_{7} + \cdots - 5916 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 12415 \beta_{15} - 3994 \beta_{14} - 3994 \beta_{13} - 15195 \beta_{12} + 2458 \beta_{11} + \cdots - 17343 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 25331 \beta_{15} + 25643 \beta_{14} + 18126 \beta_{12} - 5532 \beta_{11} - 5413 \beta_{9} + 74038 \beta_{8} + \cdots + 74038 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 52889 \beta_{14} + 52889 \beta_{13} + 29231 \beta_{11} + 36792 \beta_{10} + 36792 \beta_{9} + \cdots - 489661 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 307540 \beta_{15} - 312796 \beta_{13} - 250245 \beta_{12} - 109442 \beta_{10} - 919856 \beta_{8} + \cdots + 919856 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 1672193 \beta_{15} + 681953 \beta_{14} + 681953 \beta_{13} + 1951215 \beta_{12} - 347804 \beta_{11} + \cdots + 2383977 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 3747883 \beta_{15} - 3821263 \beta_{14} - 3318912 \beta_{12} + 1174440 \beta_{11} + 1716281 \beta_{9} + \cdots - 11394137 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/585\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\) \(496\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{8}\)

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
−2.47839 + 2.47839i
−2.03354 + 2.03354i
−0.936959 + 0.936959i
−0.0522614 + 0.0522614i
0.530829 − 0.530829i
1.31421 − 1.31421i
1.55497 − 1.55497i
2.10114 − 2.10114i
−2.47839 − 2.47839i
−2.03354 − 2.03354i
−0.936959 − 0.936959i
−0.0522614 − 0.0522614i
0.530829 + 0.530829i
1.31421 + 1.31421i
1.55497 + 1.55497i
2.10114 + 2.10114i
−2.47839 + 2.47839i 0 − 8.28488i −1.58114 + 1.58114i 0 3.35602 + 3.35602i 10.6196 + 10.6196i 0 − 7.83737i
226.2 −2.03354 + 2.03354i 0 − 4.27056i 1.58114 − 1.58114i 0 −0.803062 − 0.803062i 0.550188 + 0.550188i 0 6.43061i
226.3 −0.936959 + 0.936959i 0 2.24421i −1.58114 + 1.58114i 0 −1.94401 − 1.94401i −5.85057 − 5.85057i 0 − 2.96293i
226.4 −0.0522614 + 0.0522614i 0 3.99454i 1.58114 − 1.58114i 0 9.56240 + 9.56240i −0.417805 − 0.417805i 0 0.165265i
226.5 0.530829 − 0.530829i 0 3.43644i 1.58114 − 1.58114i 0 −4.94074 − 4.94074i 3.94748 + 3.94748i 0 − 1.67863i
226.6 1.31421 − 1.31421i 0 0.545679i −1.58114 + 1.58114i 0 6.15655 + 6.15655i 5.97400 + 5.97400i 0 4.15591i
226.7 1.55497 − 1.55497i 0 − 0.835865i 1.58114 − 1.58114i 0 −5.14316 − 5.14316i 4.92014 + 4.92014i 0 − 4.91725i
226.8 2.10114 − 2.10114i 0 − 4.82957i −1.58114 + 1.58114i 0 3.75600 + 3.75600i −1.74304 − 1.74304i 0 6.64438i
541.1 −2.47839 − 2.47839i 0 8.28488i −1.58114 − 1.58114i 0 3.35602 − 3.35602i 10.6196 − 10.6196i 0 7.83737i
541.2 −2.03354 − 2.03354i 0 4.27056i 1.58114 + 1.58114i 0 −0.803062 + 0.803062i 0.550188 − 0.550188i 0 − 6.43061i
541.3 −0.936959 − 0.936959i 0 − 2.24421i −1.58114 − 1.58114i 0 −1.94401 + 1.94401i −5.85057 + 5.85057i 0 2.96293i
541.4 −0.0522614 − 0.0522614i 0 − 3.99454i 1.58114 + 1.58114i 0 9.56240 − 9.56240i −0.417805 + 0.417805i 0 − 0.165265i
541.5 0.530829 + 0.530829i 0 − 3.43644i 1.58114 + 1.58114i 0 −4.94074 + 4.94074i 3.94748 − 3.94748i 0 1.67863i
541.6 1.31421 + 1.31421i 0 − 0.545679i −1.58114 − 1.58114i 0 6.15655 − 6.15655i 5.97400 − 5.97400i 0 − 4.15591i
541.7 1.55497 + 1.55497i 0 0.835865i 1.58114 + 1.58114i 0 −5.14316 + 5.14316i 4.92014 − 4.92014i 0 4.91725i
541.8 2.10114 + 2.10114i 0 4.82957i −1.58114 − 1.58114i 0 3.75600 − 3.75600i −1.74304 + 1.74304i 0 − 6.64438i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 226.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.d odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.3.s.a 16
3.b odd 2 1 65.3.j.a ✓ 16
13.d odd 4 1 inner 585.3.s.a 16
15.d odd 2 1 325.3.j.b 16
15.e even 4 1 325.3.g.c 16
15.e even 4 1 325.3.g.d 16
39.f even 4 1 65.3.j.a ✓ 16
195.j odd 4 1 325.3.g.c 16
195.n even 4 1 325.3.j.b 16
195.u odd 4 1 325.3.g.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.3.j.a ✓ 16 3.b odd 2 1
65.3.j.a ✓ 16 39.f even 4 1
325.3.g.c 16 15.e even 4 1
325.3.g.c 16 195.j odd 4 1
325.3.g.d 16 15.e even 4 1
325.3.g.d 16 195.u odd 4 1
325.3.j.b 16 15.d odd 2 1
325.3.j.b 16 195.n even 4 1
585.3.s.a 16 1.a even 1 1 trivial
585.3.s.a 16 13.d odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - 12 T_{2}^{13} + 168 T_{2}^{12} - 120 T_{2}^{11} + 72 T_{2}^{10} - 852 T_{2}^{9} + 6994 T_{2}^{8} + \cdots + 81 \) acting on \(S_{3}^{\mathrm{new}}(585, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 12 T^{13} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{4} + 25)^{4} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 221863608576 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 1753124291136 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 66\!\cdots\!41 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 37\!\cdots\!16 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 76\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{8} - 68 T^{7} + \cdots - 261146435856)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 44\!\cdots\!96 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 50\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 75\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 24\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 2684009717184)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 1856471260176)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 18\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 37\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 47\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots - 62456990009856)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 23\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
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