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Properties

Label 416.2.i.g
Level $416$
Weight $2$
Character orbit 416.i
Analytic conductor $3.322$
Analytic rank $0$
Dimension $8$
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(289,416)]
 
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, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("416.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 416 = 2^{5} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 416.i (of order \(3\), 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 \(\Q(\zeta_{3})\)
Coefficient field: 8.0.6927565824.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 5x^{6} + 23x^{4} + 10x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{7} - \beta_{2}) q^{3} + (\beta_{3} - 1) q^{5} + ( - \beta_{2} + \beta_1) q^{7} + (\beta_{5} + \beta_{4}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{7} - \beta_{2}) q^{3} + (\beta_{3} - 1) q^{5} + ( - \beta_{2} + \beta_1) q^{7} + (\beta_{5} + \beta_{4}) q^{9} + (\beta_{7} + \beta_{6} + \beta_{2}) q^{11} + ( - \beta_{5} - \beta_{4} + 2 \beta_{3} - 1) q^{13} + (2 \beta_{7} - \beta_{6} + 2 \beta_{2}) q^{15} + (2 \beta_{5} + 3 \beta_{4}) q^{17} + (\beta_{2} - 2 \beta_1) q^{19} + (3 \beta_{3} - 4) q^{21} + ( - 3 \beta_{7} - 3 \beta_{2}) q^{23} - \beta_{3} q^{25} + ( - \beta_{7} - \beta_{6} + \beta_1) q^{27} + (5 \beta_{4} + 5) q^{29} + (2 \beta_{7} + \beta_{6} - \beta_1) q^{31} + (\beta_{5} - 4 \beta_{4}) q^{33} + 4 \beta_{2} q^{35} + ( - 2 \beta_{5} + \beta_{4} + 2 \beta_{3} + 1) q^{37} + (\beta_{7} - \beta_{6} + 3 \beta_{2} - \beta_1) q^{39} + (2 \beta_{5} + \beta_{4} - 2 \beta_{3} + 1) q^{41} + ( - 3 \beta_{2} + \beta_1) q^{43} + ( - \beta_{5} - 5 \beta_{4}) q^{45} + ( - 4 \beta_{7} + \beta_{6} - \beta_1) q^{47} + ( - \beta_{5} - 5 \beta_{4} + \beta_{3} - 5) q^{49} + (5 \beta_{7} - 2 \beta_{6} + 2 \beta_1) q^{51} + (\beta_{3} - 1) q^{53} + 2 \beta_{6} q^{55} + ( - 5 \beta_{3} + 4) q^{57} + ( - 5 \beta_{2} - \beta_1) q^{59} + ( - 4 \beta_{5} - 5 \beta_{4}) q^{61} + (4 \beta_{7} + 4 \beta_{2}) q^{63} + (\beta_{5} + 5 \beta_{4} - \beta_{3} + 9) q^{65} + ( - 3 \beta_{7} + 2 \beta_{6} - 3 \beta_{2}) q^{67} + (3 \beta_{5} + 12 \beta_{4}) q^{69} + ( - \beta_{2} + 2 \beta_1) q^{71} + ( - 3 \beta_{3} - 9) q^{73} + ( - \beta_{7} + \beta_{6} - \beta_{2}) q^{75} + ( - 5 \beta_{3} - 4) q^{77} + ( - 6 \beta_{7} - 2 \beta_{6} + 2 \beta_1) q^{79} + (2 \beta_{5} + 7 \beta_{4} - 2 \beta_{3} + 7) q^{81} + ( - 2 \beta_{6} + 2 \beta_1) q^{83} + ( - 3 \beta_{5} - 11 \beta_{4}) q^{85} - 5 \beta_{2} q^{87} + ( - 3 \beta_{5} - 6 \beta_{4} + \cdots - 6) q^{89}+ \cdots + (2 \beta_{6} - 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{5} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{5} - 2 q^{9} + 2 q^{13} - 8 q^{17} - 20 q^{21} - 4 q^{25} + 20 q^{29} + 18 q^{33} + 8 q^{37} + 18 q^{45} - 18 q^{49} - 4 q^{53} + 12 q^{57} + 12 q^{61} + 50 q^{65} - 42 q^{69} - 84 q^{73} - 52 q^{77} + 24 q^{81} + 38 q^{85} - 18 q^{89} - 32 q^{93} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 5x^{6} + 23x^{4} + 10x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - 82\nu ) / 23 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} - 36 ) / 23 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{6} - 23\nu^{4} - 115\nu^{2} - 50 ) / 46 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -4\nu^{6} - 23\nu^{4} - 92\nu^{2} - 40 ) / 23 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5\nu^{7} + 23\nu^{5} + 115\nu^{3} + 50\nu ) / 23 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -10\nu^{7} - 46\nu^{5} - 207\nu^{3} - 8\nu ) / 23 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - 2\beta_{4} - \beta_{3} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 2\beta_{6} - 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -5\beta_{5} + 8\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -5\beta_{7} - 9\beta_{6} - 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 23\beta_{3} + 36 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 23\beta_{2} + 41\beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(261\) \(287\) \(353\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
0.331077 − 0.573442i
−1.06789 + 1.84964i
1.06789 − 1.84964i
−0.331077 + 0.573442i
0.331077 + 0.573442i
−1.06789 − 1.84964i
1.06789 + 1.84964i
−0.331077 − 0.573442i
0 −1.17915 − 2.04234i 0 −2.56155 0 1.84130 − 3.18923i 0 −1.28078 + 2.21837i 0
289.2 0 −0.599676 − 1.03867i 0 1.56155 0 −1.53610 + 2.66061i 0 0.780776 − 1.35234i 0
289.3 0 0.599676 + 1.03867i 0 1.56155 0 1.53610 − 2.66061i 0 0.780776 − 1.35234i 0
289.4 0 1.17915 + 2.04234i 0 −2.56155 0 −1.84130 + 3.18923i 0 −1.28078 + 2.21837i 0
321.1 0 −1.17915 + 2.04234i 0 −2.56155 0 1.84130 + 3.18923i 0 −1.28078 − 2.21837i 0
321.2 0 −0.599676 + 1.03867i 0 1.56155 0 −1.53610 − 2.66061i 0 0.780776 + 1.35234i 0
321.3 0 0.599676 − 1.03867i 0 1.56155 0 1.53610 + 2.66061i 0 0.780776 + 1.35234i 0
321.4 0 1.17915 − 2.04234i 0 −2.56155 0 −1.84130 − 3.18923i 0 −1.28078 − 2.21837i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
13.c even 3 1 inner
52.j 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.i.g ✓ 8
4.b odd 2 1 inner 416.2.i.g ✓ 8
8.b even 2 1 832.2.i.q 8
8.d odd 2 1 832.2.i.q 8
13.c even 3 1 inner 416.2.i.g ✓ 8
13.c even 3 1 5408.2.a.bh 4
13.e even 6 1 5408.2.a.bi 4
52.i odd 6 1 5408.2.a.bi 4
52.j odd 6 1 inner 416.2.i.g ✓ 8
52.j odd 6 1 5408.2.a.bh 4
104.n odd 6 1 832.2.i.q 8
104.r even 6 1 832.2.i.q 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.2.i.g ✓ 8 1.a even 1 1 trivial
416.2.i.g ✓ 8 4.b odd 2 1 inner
416.2.i.g ✓ 8 13.c even 3 1 inner
416.2.i.g ✓ 8 52.j odd 6 1 inner
832.2.i.q 8 8.b even 2 1
832.2.i.q 8 8.d odd 2 1
832.2.i.q 8 104.n odd 6 1
832.2.i.q 8 104.r even 6 1
5408.2.a.bh 4 13.c even 3 1
5408.2.a.bh 4 52.j odd 6 1
5408.2.a.bi 4 13.e even 6 1
5408.2.a.bi 4 52.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 7T_{3}^{6} + 41T_{3}^{4} + 56T_{3}^{2} + 64 \) 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
$3$ \( T^{8} + 7 T^{6} + \cdots + 64 \) Copy content Toggle raw display
$5$ \( (T^{2} + T - 4)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} + 23 T^{6} + \cdots + 16384 \) Copy content Toggle raw display
$11$ \( T^{8} + 31 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$13$ \( (T^{4} - T^{3} - 12 T^{2} + \cdots + 169)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 4 T^{3} + \cdots + 169)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 79 T^{6} + \cdots + 1827904 \) Copy content Toggle raw display
$23$ \( T^{8} + 63 T^{6} + \cdots + 419904 \) Copy content Toggle raw display
$29$ \( (T^{2} - 5 T + 25)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 56 T^{2} + 512)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 4 T^{3} + \cdots + 169)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 17 T^{2} + 289)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 71 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$47$ \( (T^{4} - 116 T^{2} + 32)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + T - 4)^{4} \) Copy content Toggle raw display
$59$ \( T^{8} + 215 T^{6} + \cdots + 133448704 \) Copy content Toggle raw display
$61$ \( (T^{4} - 6 T^{3} + \cdots + 3481)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 119 T^{6} + \cdots + 5345344 \) Copy content Toggle raw display
$71$ \( T^{8} + 79 T^{6} + \cdots + 1827904 \) Copy content Toggle raw display
$73$ \( (T^{2} + 21 T + 72)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 380 T^{2} + 32768)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 80 T^{2} + 512)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 9 T^{3} + \cdots + 324)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + T^{3} + \cdots + 11236)^{2} \) Copy content Toggle raw display
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