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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [15,13,Mod(11,15)] 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("15.11"); S:= CuspForms(chi, 13); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(15, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 13, names="a")
 
Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 15.c (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: \(13.7099072591\)
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: \( x^{16} + 48241 x^{14} + 888112936 x^{12} + 8047740261076 x^{10} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{30}\cdot 5^{28} \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} - \beta_1 + 57) q^{3} + ( - \beta_{3} - 2 \beta_{2} + \cdots - 2018) q^{4} + ( - \beta_{5} - \beta_{2} + 11 \beta_1) q^{5} + ( - 2 \beta_{5} - \beta_{4} + \cdots - 3342) q^{6}+ \cdots + (196330 \beta_{15} + \cdots - 100503781501) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 920 q^{3} - 32306 q^{4} - 53474 q^{6} - 45760 q^{7} - 1527800 q^{9} + 1031250 q^{10} - 16090820 q^{12} - 708400 q^{13} + 6500000 q^{15} + 136466818 q^{16} + 220351880 q^{18} - 220253680 q^{19} + 134279736 q^{21}+ \cdots - 1607313356800 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 48241 x^{14} + 888112936 x^{12} + 8047740261076 x^{10} + \cdots + 17\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 39\!\cdots\!23 \nu^{15} + \cdots - 68\!\cdots\!00 \nu ) / 52\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 59\!\cdots\!45 \nu^{15} + \cdots - 24\!\cdots\!00 ) / 87\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 50\!\cdots\!57 \nu^{15} + \cdots + 28\!\cdots\!00 ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 15\!\cdots\!44 \nu^{15} + \cdots + 48\!\cdots\!00 ) / 46\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 38\!\cdots\!71 \nu^{15} + \cdots + 20\!\cdots\!00 ) / 74\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 12\!\cdots\!83 \nu^{15} + \cdots - 31\!\cdots\!00 ) / 74\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 14\!\cdots\!23 \nu^{15} + \cdots + 10\!\cdots\!00 ) / 74\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 18\!\cdots\!93 \nu^{15} + \cdots - 60\!\cdots\!00 ) / 93\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 43\!\cdots\!67 \nu^{15} + \cdots + 30\!\cdots\!00 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 14\!\cdots\!31 \nu^{15} + \cdots + 44\!\cdots\!00 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 85\!\cdots\!87 \nu^{15} + \cdots - 56\!\cdots\!00 ) / 74\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 90\!\cdots\!87 \nu^{15} + \cdots + 94\!\cdots\!00 ) / 74\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 96\!\cdots\!93 \nu^{15} + \cdots - 54\!\cdots\!00 ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 48\!\cdots\!17 \nu^{15} + \cdots - 26\!\cdots\!00 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 91\!\cdots\!77 \nu^{15} + \cdots + 36\!\cdots\!00 ) / 24\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -4\beta_{5} - 4\beta_{2} - 3081\beta_1 ) / 3125 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 16 \beta_{11} - 8 \beta_{10} + 8 \beta_{9} + 16 \beta_{8} + 8 \beta_{7} + 8 \beta_{5} + \cdots - 18839994 ) / 3125 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 768 \beta_{15} - 132 \beta_{14} - 864 \beta_{13} + 252 \beta_{12} - 300 \beta_{11} + 324 \beta_{10} + \cdots - 97595 ) / 3125 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 23625 \beta_{14} - 77500 \beta_{13} + 163875 \beta_{12} + 225083 \beta_{11} + 233979 \beta_{10} + \cdots + 215157068797 ) / 3125 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 6764144 \beta_{15} - 151569 \beta_{14} + 4971912 \beta_{13} - 2822016 \beta_{12} + 3872650 \beta_{11} + \cdots + 737312885 ) / 625 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 553932375 \beta_{14} + 2229341250 \beta_{13} - 4140047625 \beta_{12} - 4251976833 \beta_{11} + \cdots - 30\!\cdots\!47 ) / 3125 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 797565348432 \beta_{15} + 35172160707 \beta_{14} - 571407098736 \beta_{13} + 347352645048 \beta_{12} + \cdots - 90685661854905 ) / 3125 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 10528322595375 \beta_{14} - 48163179003750 \beta_{13} + 82119336749625 \beta_{12} + \cdots + 48\!\cdots\!79 ) / 3125 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 16\!\cdots\!96 \beta_{15} - 744850748805771 \beta_{14} + \cdots + 19\!\cdots\!65 ) / 3125 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 38\!\cdots\!75 \beta_{14} + \cdots - 16\!\cdots\!59 ) / 625 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 30\!\cdots\!56 \beta_{15} + \cdots - 37\!\cdots\!65 ) / 3125 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 34\!\cdots\!75 \beta_{14} + \cdots + 13\!\cdots\!11 ) / 3125 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 57\!\cdots\!52 \beta_{15} + \cdots + 69\!\cdots\!05 ) / 3125 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 60\!\cdots\!75 \beta_{14} + \cdots - 23\!\cdots\!23 ) / 3125 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 20\!\cdots\!08 \beta_{15} + \cdots - 25\!\cdots\!45 ) / 625 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
− 133.518i
− 109.074i
− 68.4259i
− 83.9699i
− 64.8245i
− 32.4240i
− 30.3754i
− 24.6488i
24.6488i
30.3754i
32.4240i
64.8245i
83.9699i
68.4259i
109.074i
133.518i
− 124.574i 549.819 + 478.686i −11422.6 − 6987.71i 59631.7 − 68493.1i −124156. 912706.i 73161.3 + 526381.i −870486.
11.2 − 118.018i −274.499 − 675.345i −9832.29 6987.71i −79703.0 + 32395.8i 80183.4 676986.i −380742. + 370763.i 824677.
11.3 − 77.3702i 728.907 − 11.6559i −1890.15 6987.71i −901.823 − 56395.7i 140164. − 170667.i 531169. − 16992.2i 540641.
11.4 − 75.0256i 345.574 − 641.888i −1532.85 − 6987.71i −48158.0 − 25926.9i −47419.9 − 192302.i −292599. − 443639.i −524258.
11.5 − 73.7688i −16.6838 + 728.809i −1345.83 6987.71i 53763.3 + 1230.74i −82132.6 − 202877.i −530884. − 24318.6i 515475.
11.6 − 41.3682i −572.114 − 451.804i 2384.67 6987.71i −18690.3 + 23667.3i −204687. − 268094.i 123187. + 516967.i 289069.
11.7 − 21.4312i 263.348 + 679.771i 3636.71 − 6987.71i 14568.3 − 5643.86i 90117.1 − 165721.i −392737. + 358033.i −149755.
11.8 − 15.7045i −564.352 − 461.463i 3849.37 − 6987.71i −7247.06 + 8862.87i 125051. − 124778.i 105544. + 520855.i −109739.
11.9 15.7045i −564.352 + 461.463i 3849.37 6987.71i −7247.06 − 8862.87i 125051. 124778.i 105544. − 520855.i −109739.
11.10 21.4312i 263.348 − 679.771i 3636.71 6987.71i 14568.3 + 5643.86i 90117.1 165721.i −392737. − 358033.i −149755.
11.11 41.3682i −572.114 + 451.804i 2384.67 − 6987.71i −18690.3 − 23667.3i −204687. 268094.i 123187. − 516967.i 289069.
11.12 73.7688i −16.6838 − 728.809i −1345.83 − 6987.71i 53763.3 − 1230.74i −82132.6 202877.i −530884. + 24318.6i 515475.
11.13 75.0256i 345.574 + 641.888i −1532.85 6987.71i −48158.0 + 25926.9i −47419.9 192302.i −292599. + 443639.i −524258.
11.14 77.3702i 728.907 + 11.6559i −1890.15 − 6987.71i −901.823 + 56395.7i 140164. 170667.i 531169. + 16992.2i 540641.
11.15 118.018i −274.499 + 675.345i −9832.29 − 6987.71i −79703.0 − 32395.8i 80183.4 − 676986.i −380742. − 370763.i 824677.
11.16 124.574i 549.819 − 478.686i −11422.6 6987.71i 59631.7 + 68493.1i −124156. − 912706.i 73161.3 − 526381.i −870486.
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 11.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 15.13.c.a ✓ 16
3.b odd 2 1 inner 15.13.c.a ✓ 16
5.b even 2 1 75.13.c.d 16
5.c odd 4 2 75.13.d.c 32
15.d odd 2 1 75.13.c.d 16
15.e even 4 2 75.13.d.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.13.c.a ✓ 16 1.a even 1 1 trivial
15.13.c.a ✓ 16 3.b odd 2 1 inner
75.13.c.d 16 5.b even 2 1
75.13.c.d 16 15.d odd 2 1
75.13.d.c 32 5.c odd 4 2
75.13.d.c 32 15.e even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{13}^{\mathrm{new}}(15, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 76\!\cdots\!00 \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 63\!\cdots\!21 \) Copy content Toggle raw display
$5$ \( (T^{2} + 48828125)^{8} \) Copy content Toggle raw display
$7$ \( (T^{8} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 83\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots - 70\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 84\!\cdots\!16)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 45\!\cdots\!76)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots - 49\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots - 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 85\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 68\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 99\!\cdots\!56)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 63\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots - 82\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots - 11\!\cdots\!64)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots - 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
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