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Properties

Label 15.18.b.a
Level $15$
Weight $18$
Character orbit 15.b
Analytic conductor $27.483$
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] = [15,18,Mod(4,15)] 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("15.4"); S:= CuspForms(chi, 18); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(15, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 18, names="a")
 
Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 15.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(27.4833131017\)
Analytic rank: \(0\)
Dimension: \(16\)
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: \( x^{16} + 1479925 x^{14} + 856740725236 x^{12} + \cdots + 35\!\cdots\!36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{30}\cdot 3^{60}\cdot 5^{23} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

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

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{3} + \beta_1) q^{3} + (\beta_{2} - 53919) q^{4} + ( - \beta_{5} - 16 \beta_{3} + \cdots + 18296) q^{5} + (\beta_{5} - \beta_{4} + \beta_{2} - 155004) q^{6} + (\beta_{9} + 2 \beta_{5} + \cdots - 12910 \beta_1) q^{7}+ \cdots + (172186884 \beta_{15} + \cdots - 24\!\cdots\!60) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 862698 q^{4} + 292740 q^{5} - 2480058 q^{6} - 688747536 q^{9} - 158256910 q^{10} + 907386144 q^{11} + 38248328748 q^{14} - 11415352680 q^{15} + 64410639650 q^{16} + 44375877728 q^{19} + 124652464020 q^{20}+ \cdots - 39\!\cdots\!24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 1479925 x^{14} + 856740725236 x^{12} + \cdots + 35\!\cdots\!36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 184991 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 63\!\cdots\!73 \nu^{15} + \cdots - 24\!\cdots\!76 \nu ) / 37\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 49\!\cdots\!47 \nu^{15} + \cdots - 49\!\cdots\!72 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 49\!\cdots\!47 \nu^{15} + \cdots - 25\!\cdots\!72 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 16\!\cdots\!41 \nu^{15} + \cdots - 12\!\cdots\!36 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 12\!\cdots\!93 \nu^{15} + \cdots + 40\!\cdots\!92 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 26\!\cdots\!43 \nu^{15} + \cdots + 13\!\cdots\!88 ) / 81\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 33\!\cdots\!87 \nu^{15} + \cdots + 98\!\cdots\!88 ) / 81\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 15\!\cdots\!87 \nu^{15} + \cdots - 89\!\cdots\!88 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 85\!\cdots\!29 \nu^{15} + \cdots + 74\!\cdots\!04 ) / 81\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 99\!\cdots\!81 \nu^{15} + \cdots - 24\!\cdots\!48 ) / 81\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 55\!\cdots\!71 \nu^{15} + \cdots - 19\!\cdots\!04 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 11\!\cdots\!79 \nu^{15} + \cdots + 42\!\cdots\!44 ) / 81\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 59\!\cdots\!03 \nu^{15} + \cdots - 50\!\cdots\!32 ) / 40\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 184991 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + 2\beta_{9} + 19\beta_{5} + \beta_{4} + 95\beta_{3} + 8\beta_{2} - 322123\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 135 \beta_{15} - 155 \beta_{14} + 90 \beta_{13} - 112 \beta_{12} + 75 \beta_{11} + 119 \beta_{10} + \cdots + 59587232771 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 20916 \beta_{14} + 8972 \beta_{13} - 1044 \beta_{12} - 616291 \beta_{11} + 85246 \beta_{10} + \cdots - 1292168 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 88867035 \beta_{15} + 99551895 \beta_{14} - 59392794 \beta_{13} + 74984904 \beta_{12} + \cdots - 22\!\cdots\!47 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 13503625188 \beta_{14} - 5291008476 \beta_{13} + 851290692 \beta_{12} + 301127294955 \beta_{11} + \cdots + 648749126400 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 44514052663275 \beta_{15} - 50084197602759 \beta_{14} + 29548936324122 \beta_{13} + \cdots + 87\!\cdots\!03 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 68\!\cdots\!44 \beta_{14} + \cdots - 30\!\cdots\!04 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 20\!\cdots\!91 \beta_{15} + \cdots - 36\!\cdots\!39 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 31\!\cdots\!40 \beta_{14} + \cdots + 13\!\cdots\!64 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 89\!\cdots\!15 \beta_{15} + \cdots + 15\!\cdots\!51 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 14\!\cdots\!96 \beta_{14} + \cdots - 61\!\cdots\!24 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 38\!\cdots\!03 \beta_{15} + \cdots - 64\!\cdots\!55 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 63\!\cdots\!88 \beta_{14} + \cdots + 27\!\cdots\!44 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/15\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(11\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
− 667.429i
− 632.918i
− 487.903i
− 419.421i
− 413.029i
− 171.991i
− 107.970i
− 89.9038i
89.9038i
107.970i
171.991i
413.029i
419.421i
487.903i
632.918i
667.429i
− 667.429i − 6561.00i −314390. 424492. − 763378.i −4.37900e6 1.80736e7i 1.22352e8i −4.30467e7 −5.09501e8 − 2.83319e8i
4.2 − 632.918i 6561.00i −269514. −873071. + 26201.9i 4.15258e6 1.32183e7i 8.76222e7i −4.30467e7 1.65836e7 + 5.52583e8i
4.3 − 487.903i 6561.00i −106978. 777203. + 398617.i 3.20113e6 5.16658e6i − 1.17557e7i −4.30467e7 1.94487e8 − 3.79200e8i
4.4 − 419.421i − 6561.00i −44841.8 −808072. − 331600.i −2.75182e6 − 1.07280e7i − 3.61668e7i −4.30467e7 −1.39080e8 + 3.38922e8i
4.5 − 413.029i − 6561.00i −39521.0 477316. + 731511.i −2.70988e6 1.47912e7i − 3.78132e7i −4.30467e7 3.02135e8 − 1.97145e8i
4.6 − 171.991i 6561.00i 101491. −301019. + 819956.i 1.12843e6 − 1.93725e7i − 3.99988e7i −4.30467e7 1.41025e8 + 5.17725e7i
4.7 − 107.970i 6561.00i 119414. −423568. − 763891.i 708394. 584538.i − 2.70451e7i −4.30467e7 −8.24776e7 + 4.57328e7i
4.8 − 89.9038i − 6561.00i 122989. 873089. − 25589.6i −589859. − 2.40973e7i − 2.28411e7i −4.30467e7 −2.30061e6 − 7.84941e7i
4.9 89.9038i 6561.00i 122989. 873089. + 25589.6i −589859. 2.40973e7i 2.28411e7i −4.30467e7 −2.30061e6 + 7.84941e7i
4.10 107.970i − 6561.00i 119414. −423568. + 763891.i 708394. − 584538.i 2.70451e7i −4.30467e7 −8.24776e7 − 4.57328e7i
4.11 171.991i − 6561.00i 101491. −301019. − 819956.i 1.12843e6 1.93725e7i 3.99988e7i −4.30467e7 1.41025e8 − 5.17725e7i
4.12 413.029i 6561.00i −39521.0 477316. − 731511.i −2.70988e6 − 1.47912e7i 3.78132e7i −4.30467e7 3.02135e8 + 1.97145e8i
4.13 419.421i 6561.00i −44841.8 −808072. + 331600.i −2.75182e6 1.07280e7i 3.61668e7i −4.30467e7 −1.39080e8 − 3.38922e8i
4.14 487.903i − 6561.00i −106978. 777203. − 398617.i 3.20113e6 − 5.16658e6i 1.17557e7i −4.30467e7 1.94487e8 + 3.79200e8i
4.15 632.918i − 6561.00i −269514. −873071. − 26201.9i 4.15258e6 − 1.32183e7i − 8.76222e7i −4.30467e7 1.65836e7 − 5.52583e8i
4.16 667.429i 6561.00i −314390. 424492. + 763378.i −4.37900e6 − 1.80736e7i − 1.22352e8i −4.30467e7 −5.09501e8 + 2.83319e8i
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 4.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 15.18.b.a ✓ 16
3.b odd 2 1 45.18.b.d 16
5.b even 2 1 inner 15.18.b.a ✓ 16
5.c odd 4 1 75.18.a.k 8
5.c odd 4 1 75.18.a.l 8
15.d odd 2 1 45.18.b.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.18.b.a ✓ 16 1.a even 1 1 trivial
15.18.b.a ✓ 16 5.b even 2 1 inner
45.18.b.d 16 3.b odd 2 1
45.18.b.d 16 15.d odd 2 1
75.18.a.k 8 5.c odd 4 1
75.18.a.l 8 5.c odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{18}^{\mathrm{new}}(15, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 35\!\cdots\!36 \) Copy content Toggle raw display
$3$ \( (T^{2} + 43046721)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 11\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 10\!\cdots\!16)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 53\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots - 36\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 46\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 57\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots - 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 27\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 46\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 67\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 31\!\cdots\!76)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 22\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots - 14\!\cdots\!04)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 83\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 31\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 77\!\cdots\!56 \) Copy content Toggle raw display
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