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Properties

Label 513.2.e.b
Level $513$
Weight $2$
Character orbit 513.e
Analytic conductor $4.096$
Analytic rank $0$
Dimension $18$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [513,2,Mod(172,513)] 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("513.172"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(513, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 513 = 3^{3} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 513.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [18,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.09632562369\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(\zeta_{3})\)
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: \( x^{18} - 3 x^{17} + 18 x^{16} - 33 x^{15} + 150 x^{14} - 237 x^{13} + 816 x^{12} - 966 x^{11} + \cdots + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{6} \)
Twist minimal: no (minimal twist has level 171)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{4} - \beta_1) q^{2} + ( - \beta_{16} + \beta_{7} + \beta_{3}) q^{4} + (\beta_{12} - \beta_{7} + \beta_{2}) q^{5} + (\beta_{14} - \beta_{9}) q^{7} + ( - \beta_{10} + \beta_{3} - 1) q^{8} + (\beta_{15} + \beta_{14} + \beta_{12} + \cdots - 1) q^{10}+ \cdots + ( - 2 \beta_{15} + \beta_{14} + \cdots + 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 3 q^{2} - 9 q^{4} + 9 q^{5} - 12 q^{8} + 12 q^{11} + 3 q^{14} + 3 q^{16} - 24 q^{17} + 18 q^{19} + 18 q^{20} + 15 q^{23} - 18 q^{25} - 36 q^{26} + 24 q^{29} + 3 q^{31} + 15 q^{32} - 18 q^{35} - 24 q^{37}+ \cdots - 30 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 3 x^{17} + 18 x^{16} - 33 x^{15} + 150 x^{14} - 237 x^{13} + 816 x^{12} - 966 x^{11} + \cdots + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 793657634671741 \nu^{17} + \cdots + 81\!\cdots\!98 ) / 15\!\cdots\!83 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 10\!\cdots\!47 \nu^{17} + \cdots + 10\!\cdots\!75 ) / 34\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 22\!\cdots\!83 \nu^{17} + \cdots - 58\!\cdots\!89 ) / 31\!\cdots\!66 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 13\!\cdots\!04 \nu^{17} + \cdots + 90\!\cdots\!56 ) / 15\!\cdots\!83 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 73\!\cdots\!65 \nu^{17} + \cdots - 11\!\cdots\!79 ) / 52\!\cdots\!61 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 72\!\cdots\!69 \nu^{17} + \cdots - 16\!\cdots\!33 ) / 31\!\cdots\!66 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 79\!\cdots\!09 \nu^{17} + \cdots - 30\!\cdots\!75 ) / 31\!\cdots\!66 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 82\!\cdots\!53 \nu^{17} + \cdots - 29\!\cdots\!97 ) / 31\!\cdots\!66 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 10\!\cdots\!55 \nu^{17} + \cdots + 84\!\cdots\!65 ) / 31\!\cdots\!66 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 64\!\cdots\!29 \nu^{17} + \cdots + 15\!\cdots\!53 ) / 15\!\cdots\!83 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 73\!\cdots\!94 \nu^{17} + \cdots - 56\!\cdots\!91 ) / 15\!\cdots\!83 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 15\!\cdots\!57 \nu^{17} + \cdots - 56\!\cdots\!17 ) / 31\!\cdots\!66 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 27\!\cdots\!81 \nu^{17} + \cdots - 15\!\cdots\!56 ) / 52\!\cdots\!61 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 20\!\cdots\!81 \nu^{17} + \cdots + 64\!\cdots\!03 ) / 31\!\cdots\!66 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 72\!\cdots\!69 \nu^{17} + \cdots + 15\!\cdots\!33 ) / 10\!\cdots\!22 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 23\!\cdots\!87 \nu^{17} + \cdots - 65\!\cdots\!07 ) / 31\!\cdots\!66 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{16} - 3\beta_{7} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{10} - 4\beta_{4} + \beta_{3} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{16} - \beta_{12} - \beta_{11} + 14\beta_{7} + \beta_{6} + 6\beta_{3} - \beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 8 \beta_{16} + \beta_{14} - 7 \beta_{11} + 7 \beta_{10} - \beta_{9} - \beta_{8} + 10 \beta_{7} + \cdots + 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{17} - \beta_{15} + \beta_{14} + 8 \beta_{12} + 9 \beta_{10} - 8 \beta_{8} - 8 \beta_{6} + \cdots + 72 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - \beta_{17} + 55 \beta_{16} + \beta_{13} + 12 \beta_{12} + 43 \beta_{11} + 10 \beta_{9} + \cdots + 98 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 195 \beta_{16} + 9 \beta_{15} - 12 \beta_{14} + 12 \beta_{13} + 66 \beta_{11} - 66 \beta_{10} + \cdots - 387 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 15 \beta_{17} - 12 \beta_{15} - 75 \beta_{14} - 102 \beta_{12} - 258 \beta_{10} + 84 \beta_{8} + \cdots - 522 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 102 \beta_{17} - 1137 \beta_{16} - 102 \beta_{13} - 354 \beta_{12} - 450 \beta_{11} - 105 \beta_{9} + \cdots - 657 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2298 \beta_{16} + 102 \beta_{15} + 510 \beta_{14} - 153 \beta_{13} - 1545 \beta_{11} + 1545 \beta_{10} + \cdots + 3411 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 765 \beta_{17} - 255 \beta_{15} + 813 \beta_{14} + 2310 \beta_{12} + 2958 \beta_{10} - 1749 \beta_{8} + \cdots + 12066 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 1323 \beta_{17} + 14454 \beta_{16} + 1323 \beta_{13} + 5352 \beta_{12} + 9285 \beta_{11} + \cdots + 17154 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 40125 \beta_{16} + 888 \beta_{15} - 5907 \beta_{14} + 5421 \beta_{13} + 19038 \beta_{11} - 19038 \beta_{10} + \cdots - 69132 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 10440 \beta_{17} - 5499 \beta_{15} - 21366 \beta_{14} - 36351 \beta_{12} - 56070 \beta_{10} + \cdots - 135117 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 37305 \beta_{17} - 241398 \beta_{16} - 37305 \beta_{13} - 97992 \beta_{12} - 120942 \beta_{11} + \cdots - 195192 \beta_1 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 558342 \beta_{16} + 38493 \beta_{15} + 136107 \beta_{14} - 77787 \beta_{13} - 340200 \beta_{11} + \cdots + 833256 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/513\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(325\)
\(\chi(n)\) \(\beta_{7}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
172.1
−1.05841 + 1.83322i
−0.995207 + 1.72375i
−0.608496 + 1.05395i
−0.102535 + 0.177596i
0.361114 − 0.625468i
0.541887 − 0.938575i
0.909577 − 1.57543i
1.20534 − 2.08772i
1.24673 − 2.15939i
−1.05841 − 1.83322i
−0.995207 − 1.72375i
−0.608496 − 1.05395i
−0.102535 − 0.177596i
0.361114 + 0.625468i
0.541887 + 0.938575i
0.909577 + 1.57543i
1.20534 + 2.08772i
1.24673 + 2.15939i
−1.05841 − 1.83322i 0 −1.24046 + 2.14854i 2.05812 − 3.56477i 0 −1.22345 − 2.11909i 1.01801 0 −8.71334
172.2 −0.995207 − 1.72375i 0 −0.980875 + 1.69892i 0.327903 − 0.567945i 0 1.14086 + 1.97603i −0.0761342 0 −1.30533
172.3 −0.608496 − 1.05395i 0 0.259465 − 0.449407i −0.854539 + 1.48010i 0 1.28695 + 2.22907i −3.06552 0 2.07993
172.4 −0.102535 − 0.177596i 0 0.978973 − 1.69563i 0.487677 − 0.844681i 0 −1.50630 − 2.60898i −0.811659 0 −0.200016
172.5 0.361114 + 0.625468i 0 0.739193 − 1.28032i 1.83658 − 3.18105i 0 1.04195 + 1.80471i 2.51219 0 2.65286
172.6 0.541887 + 0.938575i 0 0.412718 − 0.714848i −1.14985 + 1.99159i 0 0.839351 + 1.45380i 3.06213 0 −2.49234
172.7 0.909577 + 1.57543i 0 −0.654662 + 1.13391i 0.845876 − 1.46510i 0 −2.49887 − 4.32816i 1.25645 0 3.07756
172.8 1.20534 + 2.08772i 0 −1.90570 + 3.30078i −1.03888 + 1.79940i 0 −0.352395 − 0.610366i −4.36674 0 −5.00884
172.9 1.24673 + 2.15939i 0 −2.10865 + 3.65229i 1.98711 − 3.44177i 0 1.27190 + 2.20299i −5.52873 0 9.90951
343.1 −1.05841 + 1.83322i 0 −1.24046 − 2.14854i 2.05812 + 3.56477i 0 −1.22345 + 2.11909i 1.01801 0 −8.71334
343.2 −0.995207 + 1.72375i 0 −0.980875 − 1.69892i 0.327903 + 0.567945i 0 1.14086 − 1.97603i −0.0761342 0 −1.30533
343.3 −0.608496 + 1.05395i 0 0.259465 + 0.449407i −0.854539 − 1.48010i 0 1.28695 − 2.22907i −3.06552 0 2.07993
343.4 −0.102535 + 0.177596i 0 0.978973 + 1.69563i 0.487677 + 0.844681i 0 −1.50630 + 2.60898i −0.811659 0 −0.200016
343.5 0.361114 − 0.625468i 0 0.739193 + 1.28032i 1.83658 + 3.18105i 0 1.04195 − 1.80471i 2.51219 0 2.65286
343.6 0.541887 − 0.938575i 0 0.412718 + 0.714848i −1.14985 − 1.99159i 0 0.839351 − 1.45380i 3.06213 0 −2.49234
343.7 0.909577 − 1.57543i 0 −0.654662 − 1.13391i 0.845876 + 1.46510i 0 −2.49887 + 4.32816i 1.25645 0 3.07756
343.8 1.20534 − 2.08772i 0 −1.90570 − 3.30078i −1.03888 − 1.79940i 0 −0.352395 + 0.610366i −4.36674 0 −5.00884
343.9 1.24673 − 2.15939i 0 −2.10865 − 3.65229i 1.98711 + 3.44177i 0 1.27190 − 2.20299i −5.52873 0 9.90951
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 172.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 513.2.e.b 18
3.b odd 2 1 171.2.e.a ✓ 18
9.c even 3 1 inner 513.2.e.b 18
9.c even 3 1 1539.2.a.n 9
9.d odd 6 1 171.2.e.a ✓ 18
9.d odd 6 1 1539.2.a.q 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.2.e.a ✓ 18 3.b odd 2 1
171.2.e.a ✓ 18 9.d odd 6 1
513.2.e.b 18 1.a even 1 1 trivial
513.2.e.b 18 9.c even 3 1 inner
1539.2.a.n 9 9.c even 3 1
1539.2.a.q 9 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{18} - 3 T_{2}^{17} + 18 T_{2}^{16} - 33 T_{2}^{15} + 150 T_{2}^{14} - 237 T_{2}^{13} + 816 T_{2}^{12} + \cdots + 81 \) acting on \(S_{2}^{\mathrm{new}}(513, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} - 3 T^{17} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( T^{18} \) Copy content Toggle raw display
$5$ \( T^{18} - 9 T^{17} + \cdots + 281961 \) Copy content Toggle raw display
$7$ \( T^{18} + 33 T^{16} + \cdots + 1841449 \) Copy content Toggle raw display
$11$ \( T^{18} - 12 T^{17} + \cdots + 164025 \) Copy content Toggle raw display
$13$ \( T^{18} + 54 T^{16} + \cdots + 600625 \) Copy content Toggle raw display
$17$ \( (T^{9} + 12 T^{8} + \cdots - 75492)^{2} \) Copy content Toggle raw display
$19$ \( (T - 1)^{18} \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 28333978929 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 16119603369 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 84129582601 \) Copy content Toggle raw display
$37$ \( (T^{9} + 12 T^{8} + \cdots + 189172)^{2} \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 55471825139481 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 403559079169 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 2830133601 \) Copy content Toggle raw display
$53$ \( (T^{9} + 21 T^{8} + \cdots + 19116)^{2} \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 7410627225 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 46262728392241 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 19\!\cdots\!49 \) Copy content Toggle raw display
$71$ \( (T^{9} - 267 T^{7} + \cdots - 5854032)^{2} \) Copy content Toggle raw display
$73$ \( (T^{9} - 12 T^{8} + \cdots + 2073724)^{2} \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 13\!\cdots\!41 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 998270748225 \) Copy content Toggle raw display
$89$ \( (T^{9} + 12 T^{8} + \cdots - 14801940)^{2} \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 10\!\cdots\!09 \) Copy content Toggle raw display
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