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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [18,16,Mod(7,18)] 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("18.7"); S:= CuspForms(chi, 16); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(18, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 16, names="a")
 
Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 18.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(25.6848309180\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 7 x^{13} + 9044747118 x^{12} - 54268482617 x^{11} + \cdots + 75\!\cdots\!07 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{20}\cdot 3^{43}\cdot 5^{2}\cdot 47^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 128 \beta_1 - 128) q^{2} + (\beta_{3} + \beta_{2} - 480 \beta_1 - 240) q^{3} + 16384 \beta_1 q^{4} + ( - \beta_{4} - \beta_{3} - 12 \beta_{2} + \cdots + 7) q^{5} + ( - 128 \beta_{2} + 30720 \beta_1 - 30592) q^{6}+ \cdots + ( - 313562385 \beta_{13} + \cdots - 199840032842808) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 896 q^{2} - 114688 q^{4} + 74529 q^{5} - 643968 q^{6} + 592525 q^{7} + 29360128 q^{8} - 1381752 q^{9} - 19079424 q^{10} - 35640111 q^{11} + 82427904 q^{12} + 384947593 q^{13} + 75843200 q^{14} + 2199530781 q^{15}+ \cdots - 22\!\cdots\!11 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 7 x^{13} + 9044747118 x^{12} - 54268482617 x^{11} + \cdots + 75\!\cdots\!07 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 68\!\cdots\!62 \nu^{13} + \cdots - 35\!\cdots\!29 ) / 70\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 22\!\cdots\!73 \nu^{13} + \cdots + 89\!\cdots\!70 ) / 87\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 51\!\cdots\!49 \nu^{13} + \cdots + 13\!\cdots\!03 ) / 87\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 32\!\cdots\!25 \nu^{13} + \cdots + 72\!\cdots\!57 ) / 87\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 12\!\cdots\!64 \nu^{13} + \cdots + 36\!\cdots\!09 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 54\!\cdots\!66 \nu^{13} + \cdots - 53\!\cdots\!63 ) / 87\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 66\!\cdots\!40 \nu^{13} + \cdots + 25\!\cdots\!92 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 29\!\cdots\!15 \nu^{13} + \cdots + 35\!\cdots\!99 ) / 87\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 54\!\cdots\!41 \nu^{13} + \cdots + 19\!\cdots\!94 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 30\!\cdots\!25 \nu^{13} + \cdots - 11\!\cdots\!04 ) / 29\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 21\!\cdots\!65 \nu^{13} + \cdots + 12\!\cdots\!37 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 76\!\cdots\!13 \nu^{13} + \cdots - 28\!\cdots\!86 ) / 43\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 15\!\cdots\!67 \nu^{13} + \cdots + 37\!\cdots\!28 ) / 87\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{6} - 2\beta_{4} - 13\beta_{3} - 23\beta_{2} + 25\beta _1 + 21 ) / 9 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 15125 \beta_{10} + 1499 \beta_{9} + 15122 \beta_{8} + 863 \beta_{7} - 18113 \beta_{6} + \cdots - 34884970973 ) / 27 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 179593890 \beta_{13} - 781275389 \beta_{12} + 552946313 \beta_{11} + 128944229 \beta_{10} + \cdots + 448714184969772 ) / 81 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 1077563340 \beta_{13} - 4687652334 \beta_{12} + 3317677878 \beta_{11} - 465138247379733 \beta_{10} + \cdots + 73\!\cdots\!66 ) / 243 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 21\!\cdots\!46 \beta_{13} + \cdots - 37\!\cdots\!57 ) / 243 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 63\!\cdots\!88 \beta_{13} + \cdots - 19\!\cdots\!23 ) / 243 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 63\!\cdots\!84 \beta_{13} + \cdots + 91\!\cdots\!93 ) / 243 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 25\!\cdots\!40 \beta_{13} + \cdots + 53\!\cdots\!31 ) / 243 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 62\!\cdots\!94 \beta_{13} + \cdots - 77\!\cdots\!98 ) / 81 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 94\!\cdots\!76 \beta_{13} + \cdots - 14\!\cdots\!40 ) / 243 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 57\!\cdots\!66 \beta_{13} + \cdots + 63\!\cdots\!51 ) / 243 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 34\!\cdots\!88 \beta_{13} + \cdots + 42\!\cdots\!27 ) / 243 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 17\!\cdots\!80 \beta_{13} + \cdots - 18\!\cdots\!17 ) / 243 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/18\mathbb{Z}\right)^\times\).

\(n\) \(11\)
\(\chi(n)\) \(\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
0.500000 + 11956.9i
0.500000 − 26764.6i
0.500000 − 18478.7i
0.500000 + 43930.8i
0.500000 − 53313.3i
0.500000 + 54207.6i
0.500000 − 11544.8i
0.500000 − 11956.9i
0.500000 + 26764.6i
0.500000 + 18478.7i
0.500000 − 43930.8i
0.500000 + 53313.3i
0.500000 − 54207.6i
0.500000 + 11544.8i
−64.0000 + 110.851i −3787.56 + 57.5919i −8192.00 − 14189.0i 36390.6 + 63030.3i 236020. − 423541.i −1.31720e6 + 2.28146e6i 2.09715e6 1.43423e7 − 436265.i −9.31599e6
7.2 −64.0000 + 110.851i −3215.20 − 2002.85i −8192.00 − 14189.0i −64210.7 − 111216.i 427791. − 228227.i 1.17716e6 − 2.03890e6i 2.09715e6 6.32613e6 + 1.28791e7i 1.64380e7
7.3 −64.0000 + 110.851i −1338.89 + 3543.49i −8192.00 − 14189.0i −42683.2 − 73929.5i −307111. − 375200.i −1.14680e6 + 1.98632e6i 2.09715e6 −1.07637e7 − 9.48864e6i 1.09269e7
7.4 −64.0000 + 110.851i 325.579 − 3773.98i −8192.00 − 14189.0i 119461. + 206913.i 397513. + 277625.i 338758. − 586746.i 2.09715e6 −1.41369e7 − 2.45746e6i −3.05821e7
7.5 −64.0000 + 110.851i 1716.13 + 3376.95i −8192.00 − 14189.0i −133186. − 230685.i −484172. − 25889.9i 1.73900e6 − 3.01203e6i 2.09715e6 −8.45871e6 + 1.15906e7i 3.40957e7
7.6 −64.0000 + 110.851i 2647.81 + 2708.88i −8192.00 − 14189.0i 146161. + 253159.i −469742. + 120145.i −382634. + 662741.i 2.09715e6 −327103. + 1.43452e7i −3.74173e7
7.7 −64.0000 + 110.851i 3652.12 − 1005.43i −8192.00 − 14189.0i −24668.4 − 42727.0i −122282. + 469190.i −112016. + 194017.i 2.09715e6 1.23271e7 − 7.34394e6i 6.31512e6
13.1 −64.0000 − 110.851i −3787.56 − 57.5919i −8192.00 + 14189.0i 36390.6 − 63030.3i 236020. + 423541.i −1.31720e6 − 2.28146e6i 2.09715e6 1.43423e7 + 436265.i −9.31599e6
13.2 −64.0000 − 110.851i −3215.20 + 2002.85i −8192.00 + 14189.0i −64210.7 + 111216.i 427791. + 228227.i 1.17716e6 + 2.03890e6i 2.09715e6 6.32613e6 − 1.28791e7i 1.64380e7
13.3 −64.0000 − 110.851i −1338.89 − 3543.49i −8192.00 + 14189.0i −42683.2 + 73929.5i −307111. + 375200.i −1.14680e6 − 1.98632e6i 2.09715e6 −1.07637e7 + 9.48864e6i 1.09269e7
13.4 −64.0000 − 110.851i 325.579 + 3773.98i −8192.00 + 14189.0i 119461. − 206913.i 397513. − 277625.i 338758. + 586746.i 2.09715e6 −1.41369e7 + 2.45746e6i −3.05821e7
13.5 −64.0000 − 110.851i 1716.13 − 3376.95i −8192.00 + 14189.0i −133186. + 230685.i −484172. + 25889.9i 1.73900e6 + 3.01203e6i 2.09715e6 −8.45871e6 − 1.15906e7i 3.40957e7
13.6 −64.0000 − 110.851i 2647.81 − 2708.88i −8192.00 + 14189.0i 146161. − 253159.i −469742. − 120145.i −382634. − 662741.i 2.09715e6 −327103. − 1.43452e7i −3.74173e7
13.7 −64.0000 − 110.851i 3652.12 + 1005.43i −8192.00 + 14189.0i −24668.4 + 42727.0i −122282. − 469190.i −112016. − 194017.i 2.09715e6 1.23271e7 + 7.34394e6i 6.31512e6
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 7.7
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 18.16.c.a ✓ 14
3.b odd 2 1 54.16.c.a 14
9.c even 3 1 inner 18.16.c.a ✓ 14
9.d odd 6 1 54.16.c.a 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.16.c.a ✓ 14 1.a even 1 1 trivial
18.16.c.a ✓ 14 9.c even 3 1 inner
54.16.c.a 14 3.b odd 2 1
54.16.c.a 14 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{14} - 74529 T_{5}^{13} + 125278126767 T_{5}^{12} + \cdots + 53\!\cdots\!00 \) acting on \(S_{16}^{\mathrm{new}}(18, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 128 T + 16384)^{7} \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots + 12\!\cdots\!43 \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 53\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 33\!\cdots\!16 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 66\!\cdots\!25 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 94\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( (T^{7} + \cdots + 40\!\cdots\!28)^{2} \) Copy content Toggle raw display
$19$ \( (T^{7} + \cdots - 44\!\cdots\!60)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 28\!\cdots\!96 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{7} + \cdots + 10\!\cdots\!32)^{2} \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 18\!\cdots\!89 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 24\!\cdots\!89 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{7} + \cdots + 75\!\cdots\!60)^{2} \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 43\!\cdots\!49 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 72\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 90\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( (T^{7} + \cdots + 98\!\cdots\!84)^{2} \) Copy content Toggle raw display
$73$ \( (T^{7} + \cdots - 42\!\cdots\!96)^{2} \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 19\!\cdots\!76 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 82\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( (T^{7} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 75\!\cdots\!21 \) Copy content Toggle raw display
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