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Properties

Label 416.2.w.c
Level $416$
Weight $2$
Character orbit 416.w
Analytic conductor $3.322$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [416,2,Mod(225,416)]
 
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, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("416.225");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 416 = 2^{5} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 416.w (of order \(6\), degree \(2\), 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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.56070144.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{3} + (\beta_{5} - \beta_{4}) q^{5} + \beta_{3} q^{7} + (2 \beta_{5} - \beta_{4} - 2 \beta_{2} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{3} + (\beta_{5} - \beta_{4}) q^{5} + \beta_{3} q^{7} + (2 \beta_{5} - \beta_{4} - 2 \beta_{2} - 2) q^{9} + ( - \beta_{6} + 2 \beta_1) q^{11} + (4 \beta_{2} + 1) q^{13} + (\beta_{7} + \beta_{6} - 2 \beta_1) q^{15} + (2 \beta_{5} - \beta_{4} - 3 \beta_{2} - 3) q^{17} + \beta_{3} q^{19} + (2 \beta_{5} - 2 \beta_{4} + 2 \beta_{2} + 1) q^{21} + ( - \beta_{7} - 2 \beta_{6} - 2 \beta_{3}) q^{23} + q^{25} + (\beta_{7} - \beta_{3} - 2 \beta_1) q^{27} + 3 \beta_{2} q^{29} + ( - 2 \beta_{7} - 2 \beta_{3}) q^{31} + ( - 3 \beta_{4} - 5 \beta_{2} + 5) q^{33} + ( - 2 \beta_{7} + \beta_{6} + \cdots - \beta_1) q^{35}+ \cdots + (3 \beta_{7} + 10 \beta_{6} + \cdots - 5 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{9} - 8 q^{13} - 12 q^{17} + 8 q^{25} - 12 q^{29} + 60 q^{33} + 12 q^{37} - 36 q^{41} - 48 q^{45} + 16 q^{49} + 48 q^{53} - 12 q^{61} + 28 q^{69} - 24 q^{77} - 4 q^{81} - 48 q^{85} - 36 q^{89} - 24 q^{93} + 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{4} - 2\nu^{3} + 7\nu^{2} - 6\nu + 6 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 8\nu^{7} - 28\nu^{6} + 114\nu^{5} - 215\nu^{4} + 378\nu^{3} - 366\nu^{2} + 266\nu - 97 ) / 37 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -24\nu^{7} + 47\nu^{6} - 194\nu^{5} + 127\nu^{4} - 172\nu^{3} - 234\nu^{2} + 238\nu - 153 ) / 37 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -6\nu^{7} - 16\nu^{6} + 44\nu^{5} - 292\nu^{4} + 512\nu^{3} - 854\nu^{2} + 670\nu - 362 ) / 37 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 6\nu^{7} - 58\nu^{6} + 178\nu^{5} - 522\nu^{4} + 746\nu^{3} - 996\nu^{2} + 588\nu - 304 ) / 37 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -18\nu^{7} + 63\nu^{6} - 238\nu^{5} + 456\nu^{4} - 758\nu^{3} + 805\nu^{2} - 654\nu + 283 ) / 37 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -24\nu^{7} + 121\nu^{6} - 416\nu^{5} + 978\nu^{4} - 1504\nu^{3} + 1727\nu^{2} - 1094\nu + 365 ) / 37 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - 2\beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{6} - \beta_{4} + \beta_{3} + \beta _1 - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5\beta_{7} + 3\beta_{6} - 7\beta_{5} + 4\beta_{4} - 2\beta_{3} - 6\beta_{2} - 16 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{7} + 2\beta_{6} - 2\beta_{5} + 4\beta_{4} - 3\beta_{3} - 3\beta_{2} - \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 22\beta_{7} - 3\beta_{6} + 26\beta_{5} - \beta_{4} - 3\beta_{3} + 30\beta_{2} - \beta _1 + 82 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 21\beta_{7} - 15\beta_{6} + 31\beta_{5} - 40\beta_{4} + 28\beta_{3} + 60\beta_{2} + 4\beta _1 + 20 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -71\beta_{7} - 14\beta_{6} - 71\beta_{5} - 83\beta_{4} + 69\beta_{3} - 28\beta_{2} - 7\beta _1 - 334 ) / 4 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(261\) \(287\) \(353\)
\(\chi(n)\) \(1\) \(1\) \(1 + \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
225.1
0.500000 − 1.56488i
0.500000 − 1.19293i
0.500000 + 2.19293i
0.500000 + 0.564882i
0.500000 + 1.56488i
0.500000 + 1.19293i
0.500000 − 2.19293i
0.500000 − 0.564882i
0 −1.45466 + 2.51954i 0 − 2.00000i 0 −0.675108 + 0.389774i 0 −2.73205 − 4.73205i 0
225.2 0 −0.619657 + 1.07328i 0 2.00000i 0 4.00552 − 2.31259i 0 0.732051 + 1.26795i 0
225.3 0 0.619657 − 1.07328i 0 2.00000i 0 −4.00552 + 2.31259i 0 0.732051 + 1.26795i 0
225.4 0 1.45466 − 2.51954i 0 − 2.00000i 0 0.675108 − 0.389774i 0 −2.73205 − 4.73205i 0
257.1 0 −1.45466 − 2.51954i 0 2.00000i 0 −0.675108 − 0.389774i 0 −2.73205 + 4.73205i 0
257.2 0 −0.619657 − 1.07328i 0 − 2.00000i 0 4.00552 + 2.31259i 0 0.732051 − 1.26795i 0
257.3 0 0.619657 + 1.07328i 0 − 2.00000i 0 −4.00552 − 2.31259i 0 0.732051 − 1.26795i 0
257.4 0 1.45466 + 2.51954i 0 2.00000i 0 0.675108 + 0.389774i 0 −2.73205 + 4.73205i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 225.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
13.e even 6 1 inner
52.i odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 416.2.w.c ✓ 8
4.b odd 2 1 inner 416.2.w.c ✓ 8
8.b even 2 1 832.2.w.h 8
8.d odd 2 1 832.2.w.h 8
13.e even 6 1 inner 416.2.w.c ✓ 8
13.f odd 12 1 5408.2.a.bg 4
13.f odd 12 1 5408.2.a.bk 4
52.i odd 6 1 inner 416.2.w.c ✓ 8
52.l even 12 1 5408.2.a.bg 4
52.l even 12 1 5408.2.a.bk 4
104.p odd 6 1 832.2.w.h 8
104.s even 6 1 832.2.w.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.2.w.c ✓ 8 1.a even 1 1 trivial
416.2.w.c ✓ 8 4.b odd 2 1 inner
416.2.w.c ✓ 8 13.e even 6 1 inner
416.2.w.c ✓ 8 52.i odd 6 1 inner
832.2.w.h 8 8.b even 2 1
832.2.w.h 8 8.d odd 2 1
832.2.w.h 8 104.p odd 6 1
832.2.w.h 8 104.s even 6 1
5408.2.a.bg 4 13.f odd 12 1
5408.2.a.bg 4 52.l even 12 1
5408.2.a.bk 4 13.f odd 12 1
5408.2.a.bk 4 52.l even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 10T_{3}^{6} + 87T_{3}^{4} + 130T_{3}^{2} + 169 \) acting on \(S_{2}^{\mathrm{new}}(416, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 10 T^{6} + \cdots + 169 \) Copy content Toggle raw display
$5$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} - 22 T^{6} + \cdots + 169 \) Copy content Toggle raw display
$11$ \( T^{8} - 30 T^{6} + \cdots + 13689 \) Copy content Toggle raw display
$13$ \( (T^{2} + 2 T + 13)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + 6 T^{3} + 39 T^{2} + \cdots + 9)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} - 22 T^{6} + \cdots + 169 \) Copy content Toggle raw display
$23$ \( T^{8} + 82 T^{6} + \cdots + 2474329 \) Copy content Toggle raw display
$29$ \( (T^{2} + 3 T + 9)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 88 T^{2} + 208)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 6 T^{3} + \cdots + 1089)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 18 T^{3} + \cdots + 529)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 90 T^{6} + \cdots + 1108809 \) Copy content Toggle raw display
$47$ \( (T^{4} + 192 T^{2} + 7488)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 12 T + 24)^{4} \) Copy content Toggle raw display
$59$ \( T^{8} - 246 T^{6} + \cdots + 200420649 \) Copy content Toggle raw display
$61$ \( (T^{2} + 3 T + 9)^{4} \) Copy content Toggle raw display
$67$ \( T^{8} - 22 T^{6} + \cdots + 169 \) Copy content Toggle raw display
$71$ \( T^{8} - 270 T^{6} + \cdots + 89813529 \) Copy content Toggle raw display
$73$ \( (T^{2} + 36)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 120 T^{2} + 1872)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 120 T^{2} + 1872)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 18 T^{3} + \cdots + 121)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 6 T^{3} + \cdots + 19881)^{2} \) Copy content Toggle raw display
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