Properties

Label 416.2.ba.c
Level $416$
Weight $2$
Character orbit 416.ba
Analytic conductor $3.322$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [416,2,Mod(17,416)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(416, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("416.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 416 = 2^{5} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 416.ba (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.32177672409\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: 16.0.8607891481591137382656.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 3 x^{15} + 5 x^{14} - 6 x^{13} + 6 x^{12} - 20 x^{10} + 48 x^{9} - 76 x^{8} + 96 x^{7} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 104)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
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.

\(f(q)\) \(=\) \( q + (\beta_{12} + \beta_{8} - \beta_1) q^{3} + (\beta_{12} + \beta_{8} - \beta_{3} - \beta_1) q^{5} + (\beta_{11} + \beta_{2} - 1) q^{7} + (\beta_{10} - \beta_{6} - \beta_{5}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{12} + \beta_{8} - \beta_1) q^{3} + (\beta_{12} + \beta_{8} - \beta_{3} - \beta_1) q^{5} + (\beta_{11} + \beta_{2} - 1) q^{7} + (\beta_{10} - \beta_{6} - \beta_{5}) q^{9} + (\beta_{12} - \beta_{8} - 2 \beta_{4} + \cdots + \beta_1) q^{11}+ \cdots + (\beta_{14} + \beta_{13} + \cdots + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 18 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 18 q^{7} + 2 q^{9} + 36 q^{15} + 8 q^{17} + 2 q^{23} - 12 q^{25} - 30 q^{33} + 14 q^{39} + 24 q^{41} - 14 q^{49} - 4 q^{55} + 6 q^{65} - 6 q^{71} - 32 q^{79} + 12 q^{81} - 34 q^{87} - 30 q^{89} - 28 q^{95} - 30 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 3 x^{15} + 5 x^{14} - 6 x^{13} + 6 x^{12} - 20 x^{10} + 48 x^{9} - 76 x^{8} + 96 x^{7} + \cdots + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - \nu^{15} - 3 \nu^{14} + 9 \nu^{13} - 14 \nu^{12} + 12 \nu^{11} - 10 \nu^{10} - 4 \nu^{9} + \cdots + 32 \nu ) / 320 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{15} + \nu^{14} - 3 \nu^{13} + 2 \nu^{12} - 6 \nu^{11} + 8 \nu^{10} + 12 \nu^{9} - 32 \nu^{8} + \cdots + 512 ) / 128 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 9 \nu^{15} + 5 \nu^{14} - 7 \nu^{13} - 6 \nu^{12} - 26 \nu^{11} + 44 \nu^{10} - 4 \nu^{9} + \cdots + 1664 ) / 640 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{15} + 10 \nu^{14} - 12 \nu^{13} + 19 \nu^{12} - 26 \nu^{11} + 14 \nu^{10} + 36 \nu^{9} + \cdots + 384 ) / 320 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3 \nu^{15} + \nu^{14} - 3 \nu^{13} - 8 \nu^{12} + 6 \nu^{11} + 8 \nu^{10} - 4 \nu^{9} - 16 \nu^{8} + \cdots + 512 ) / 128 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - \nu^{15} + \nu^{14} - \nu^{13} + 4 \nu^{12} - 6 \nu^{11} - 2 \nu^{10} + 8 \nu^{9} - 12 \nu^{8} + \cdots - 64 ) / 64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9 \nu^{15} - 15 \nu^{14} + 13 \nu^{13} - 26 \nu^{12} + 54 \nu^{11} + 4 \nu^{10} - 124 \nu^{9} + \cdots - 1536 ) / 640 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3 \nu^{15} - 10 \nu^{14} + 16 \nu^{13} - 27 \nu^{12} + 28 \nu^{11} - 12 \nu^{10} - 68 \nu^{9} + \cdots - 832 ) / 320 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - \nu^{15} + 2 \nu^{14} - 2 \nu^{13} + 3 \nu^{12} - 2 \nu^{11} - 8 \nu^{10} + 20 \nu^{9} - 32 \nu^{8} + \cdots + 64 ) / 64 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{15} - 2 \nu^{14} + 2 \nu^{13} - 5 \nu^{12} + 4 \nu^{11} + 2 \nu^{10} - 12 \nu^{9} + 12 \nu^{8} + \cdots + 128 \nu ) / 64 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - \nu^{15} + 2 \nu^{14} - 2 \nu^{13} + 3 \nu^{12} - 6 \nu^{11} + 4 \nu^{10} - 8 \nu^{8} + 20 \nu^{7} + \cdots + 64 ) / 64 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 7 \nu^{15} - 29 \nu^{14} + 27 \nu^{13} - 22 \nu^{12} + 46 \nu^{11} + 20 \nu^{10} - 172 \nu^{9} + \cdots - 1920 ) / 640 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 5 \nu^{15} - 16 \nu^{14} + 2 \nu^{13} - \nu^{12} + 6 \nu^{11} + 42 \nu^{10} - 120 \nu^{9} + \cdots - 1088 ) / 320 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 12 \nu^{15} + 7 \nu^{14} + 7 \nu^{13} - 5 \nu^{12} + 26 \nu^{11} - 86 \nu^{10} + 112 \nu^{9} + \cdots - 576 ) / 320 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 7 \nu^{15} + 17 \nu^{14} - 19 \nu^{13} + 22 \nu^{12} - 26 \nu^{11} - 28 \nu^{10} + 132 \nu^{9} + \cdots + 1024 ) / 128 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{8} + \beta_{4} - \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{12} - \beta_{10} + \beta_{6} + \beta_{5} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - \beta_{15} + \beta_{14} + \beta_{13} - \beta_{12} - \beta_{10} + \beta_{9} - \beta_{8} + \beta_{6} + \cdots + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{14} + 3\beta_{12} + 2\beta_{8} - 4\beta_{7} - \beta_{6} + \beta_{5} + 2\beta_{4} - 3\beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{15} + 2 \beta_{14} - \beta_{13} + 6 \beta_{12} + 2 \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} + \cdots - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( \beta_{15} + \beta_{14} - \beta_{13} + \beta_{12} - 2 \beta_{11} + 3 \beta_{10} - 5 \beta_{9} - 3 \beta_{8} + \cdots + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 2 \beta_{15} + \beta_{14} - 2 \beta_{13} - 5 \beta_{12} + 2 \beta_{10} - 2 \beta_{9} + 2 \beta_{8} + \cdots - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 3 \beta_{15} - 3 \beta_{13} + 6 \beta_{12} - 2 \beta_{11} - 7 \beta_{10} - 11 \beta_{9} + 7 \beta_{8} + \cdots - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 3 \beta_{15} + 5 \beta_{14} + 5 \beta_{13} - 11 \beta_{12} - 14 \beta_{11} - 3 \beta_{10} - 19 \beta_{9} + \cdots - 15 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 3 \beta_{14} - 11 \beta_{12} - 12 \beta_{11} - 32 \beta_{9} - 24 \beta_{8} - 8 \beta_{7} - 3 \beta_{6} + \cdots - 35 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 9 \beta_{15} + 14 \beta_{14} - 7 \beta_{13} + 4 \beta_{12} + 14 \beta_{11} - 15 \beta_{10} + \cdots - 24 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 7 \beta_{15} + 9 \beta_{14} - 9 \beta_{13} + 21 \beta_{12} - 10 \beta_{11} - 9 \beta_{10} - \beta_{9} + \cdots + 81 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 22 \beta_{15} + 21 \beta_{14} - 42 \beta_{13} + 15 \beta_{12} + 22 \beta_{10} - 10 \beta_{9} + \cdots + 15 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 13 \beta_{15} - 67 \beta_{13} + 34 \beta_{12} + 14 \beta_{11} - 51 \beta_{10} - 43 \beta_{9} + \cdots + 14 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 73 \beta_{15} + 9 \beta_{14} + 9 \beta_{13} - 87 \beta_{12} - 62 \beta_{11} - 103 \beta_{10} + \cdots + 61 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/416\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(287\) \(353\)
\(\chi(n)\) \(-1\) \(1\) \(-\beta_{9}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
−0.608487 + 1.27661i
1.21425 0.724984i
−1.41366 0.0395346i
0.867327 + 1.11702i
1.40104 + 0.192615i
−0.741068 1.20450i
−0.0207305 + 1.41406i
0.801337 1.16527i
−0.608487 1.27661i
1.21425 + 0.724984i
−1.41366 + 0.0395346i
0.867327 1.11702i
1.40104 0.192615i
−0.741068 + 1.20450i
−0.0207305 1.41406i
0.801337 + 1.16527i
0 −2.28316 1.31818i 0 −3.40672 0 −1.30715 + 0.754684i 0 1.97521 + 3.42116i 0
17.2 0 −1.95589 1.12924i 0 −0.642566 0 −0.306483 + 0.176948i 0 1.05034 + 1.81925i 0
17.3 0 −0.779193 0.449867i 0 0.893415 0 −3.65473 + 2.11006i 0 −1.09524 1.89701i 0
17.4 0 −0.323315 0.186666i 0 −2.04528 0 0.768362 0.443614i 0 −1.43031 2.47737i 0
17.5 0 0.323315 + 0.186666i 0 2.04528 0 0.768362 0.443614i 0 −1.43031 2.47737i 0
17.6 0 0.779193 + 0.449867i 0 −0.893415 0 −3.65473 + 2.11006i 0 −1.09524 1.89701i 0
17.7 0 1.95589 + 1.12924i 0 0.642566 0 −0.306483 + 0.176948i 0 1.05034 + 1.81925i 0
17.8 0 2.28316 + 1.31818i 0 3.40672 0 −1.30715 + 0.754684i 0 1.97521 + 3.42116i 0
49.1 0 −2.28316 + 1.31818i 0 −3.40672 0 −1.30715 0.754684i 0 1.97521 3.42116i 0
49.2 0 −1.95589 + 1.12924i 0 −0.642566 0 −0.306483 0.176948i 0 1.05034 1.81925i 0
49.3 0 −0.779193 + 0.449867i 0 0.893415 0 −3.65473 2.11006i 0 −1.09524 + 1.89701i 0
49.4 0 −0.323315 + 0.186666i 0 −2.04528 0 0.768362 + 0.443614i 0 −1.43031 + 2.47737i 0
49.5 0 0.323315 0.186666i 0 2.04528 0 0.768362 + 0.443614i 0 −1.43031 + 2.47737i 0
49.6 0 0.779193 0.449867i 0 −0.893415 0 −3.65473 2.11006i 0 −1.09524 + 1.89701i 0
49.7 0 1.95589 1.12924i 0 0.642566 0 −0.306483 0.176948i 0 1.05034 1.81925i 0
49.8 0 2.28316 1.31818i 0 3.40672 0 −1.30715 0.754684i 0 1.97521 3.42116i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
13.e even 6 1 inner
104.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 416.2.ba.c 16
4.b odd 2 1 104.2.s.c 16
8.b even 2 1 inner 416.2.ba.c 16
8.d odd 2 1 104.2.s.c 16
12.b even 2 1 936.2.dg.d 16
13.e even 6 1 inner 416.2.ba.c 16
24.f even 2 1 936.2.dg.d 16
52.i odd 6 1 104.2.s.c 16
104.p odd 6 1 104.2.s.c 16
104.s even 6 1 inner 416.2.ba.c 16
156.r even 6 1 936.2.dg.d 16
312.ba even 6 1 936.2.dg.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.2.s.c 16 4.b odd 2 1
104.2.s.c 16 8.d odd 2 1
104.2.s.c 16 52.i odd 6 1
104.2.s.c 16 104.p odd 6 1
416.2.ba.c 16 1.a even 1 1 trivial
416.2.ba.c 16 8.b even 2 1 inner
416.2.ba.c 16 13.e even 6 1 inner
416.2.ba.c 16 104.s even 6 1 inner
936.2.dg.d 16 12.b even 2 1
936.2.dg.d 16 24.f even 2 1
936.2.dg.d 16 156.r even 6 1
936.2.dg.d 16 312.ba even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - 13T_{3}^{14} + 122T_{3}^{12} - 541T_{3}^{10} + 1750T_{3}^{8} - 1541T_{3}^{6} + 1037T_{3}^{4} - 140T_{3}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(416, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} - 13 T^{14} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( (T^{8} - 17 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + 9 T^{7} + 30 T^{6} + \cdots + 4)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + 55 T^{14} + \cdots + 14776336 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 815730721 \) Copy content Toggle raw display
$17$ \( (T^{8} - 4 T^{7} + \cdots + 5329)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 31713911056 \) Copy content Toggle raw display
$23$ \( (T^{8} - T^{7} + 22 T^{6} + \cdots + 256)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 12594450625 \) Copy content Toggle raw display
$31$ \( (T^{8} + 132 T^{6} + \cdots + 16384)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 2947295521 \) Copy content Toggle raw display
$41$ \( (T^{8} - 12 T^{7} + \cdots + 81)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} - 145 T^{14} + \cdots + 4096 \) Copy content Toggle raw display
$47$ \( (T^{8} + 204 T^{6} + \cdots + 64)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 31 T^{6} + \cdots + 400)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 121173610000 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 18\!\cdots\!81 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 2897022976 \) Copy content Toggle raw display
$71$ \( (T^{8} + 3 T^{7} + \cdots + 1295044)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 307 T^{6} + \cdots + 9610000)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 8 T^{3} - 4 T^{2} + \cdots + 40)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} - 560 T^{6} + \cdots + 189337600)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 15 T^{7} + \cdots + 67600)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 15 T^{7} + \cdots + 473344)^{2} \) Copy content Toggle raw display
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