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Properties

Label 416.2.e.c
Level $416$
Weight $2$
Character orbit 416.e
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(337,416)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(416, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("416.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 416 = 2^{5} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 416.e (of order \(2\), degree \(1\), 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: \(8\)
Coefficient field: 8.0.4521217600.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + x^{6} - 2x^{4} + 4x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 104)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + \beta_{4} q^{5} + \beta_{3} q^{7} + ( - \beta_{5} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + \beta_{4} q^{5} + \beta_{3} q^{7} + ( - \beta_{5} - 3) q^{9} + (\beta_{6} + \beta_{4}) q^{11} + (\beta_{7} + \beta_{6}) q^{13} + ( - \beta_{3} + \beta_1) q^{15} + (\beta_{5} - 2) q^{17} + ( - 3 \beta_{6} + \beta_{4}) q^{19} + ( - 2 \beta_{6} + \beta_{4}) q^{21} - 4 q^{23} + ( - \beta_{5} + 1) q^{25} + ( - 2 \beta_{7} - \beta_{2}) q^{27} - 2 \beta_{7} q^{29} + ( - 2 \beta_{3} - \beta_1) q^{31} + 2 \beta_1 q^{33} + (2 \beta_{7} - \beta_{2}) q^{35} + (2 \beta_{6} - \beta_{4}) q^{37} + ( - 2 \beta_{5} + \beta_{3} + \beta_1 - 2) q^{39} + \beta_{2} q^{43} - 4 \beta_{6} q^{45} + \beta_{3} q^{47} + (\beta_{5} + 3) q^{49} + (2 \beta_{7} - \beta_{2}) q^{51} + 4 \beta_{2} q^{53} + 8 q^{55} + ( - 4 \beta_{3} - 2 \beta_1) q^{57} + (3 \beta_{6} - \beta_{4}) q^{59} - \beta_1 q^{63} + (\beta_{5} + 4 \beta_{3} + 2) q^{65} + (5 \beta_{6} - 3 \beta_{4}) q^{67} - 4 \beta_{2} q^{69} + (\beta_{3} - 2 \beta_1) q^{71} + (4 \beta_{3} - 2 \beta_1) q^{73} - 2 \beta_{7} q^{75} + 2 \beta_{7} q^{77} + 4 \beta_{5} q^{79} + (2 \beta_{5} + 1) q^{81} + ( - \beta_{6} + 3 \beta_{4}) q^{83} + (4 \beta_{6} - 5 \beta_{4}) q^{85} + (4 \beta_{5} + 4) q^{87} + ( - 4 \beta_{3} + 2 \beta_1) q^{89} + (\beta_{6} - 3 \beta_{4} + \beta_{2}) q^{91} + 10 \beta_{6} q^{93} - 4 \beta_{5} q^{95} - 2 \beta_1 q^{97} + ( - 9 \beta_{6} - \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 20 q^{9} - 20 q^{17} - 32 q^{23} + 12 q^{25} - 8 q^{39} + 20 q^{49} + 64 q^{55} + 12 q^{65} - 16 q^{79} + 16 q^{87} + 16 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + x^{6} - 2x^{4} + 4x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 3\nu^{3} ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + \nu^{4} + 2\nu^{2} + 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + \nu^{5} - 2\nu^{3} + 12\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} - 3\nu^{5} + 2\nu^{3} + 12\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} - \nu^{4} + 6\nu^{2} - 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} - \nu^{5} + 2\nu^{3} + 4\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{6} + 3\nu^{4} + 2\nu^{2} - 8 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{4} - \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{7} - \beta_{5} + \beta_{2} + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -3\beta_{4} + 3\beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -2\beta_{7} - \beta_{5} + 5\beta_{2} - 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -12\beta_{6} + 5\beta_{4} - \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(261\) \(287\) \(353\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
−1.29437 + 0.569745i
1.29437 − 0.569745i
0.273147 − 1.38758i
−0.273147 + 1.38758i
0.273147 + 1.38758i
−0.273147 − 1.38758i
−1.29437 − 0.569745i
1.29437 + 0.569745i
0 − 2.94984i 0 −1.81616 0 1.13949i 0 −5.70156 0
337.2 0 − 2.94984i 0 1.81616 0 − 1.13949i 0 −5.70156 0
337.3 0 − 1.51606i 0 −3.11473 0 − 2.77517i 0 0.701562 0
337.4 0 − 1.51606i 0 3.11473 0 2.77517i 0 0.701562 0
337.5 0 1.51606i 0 −3.11473 0 2.77517i 0 0.701562 0
337.6 0 1.51606i 0 3.11473 0 − 2.77517i 0 0.701562 0
337.7 0 2.94984i 0 −1.81616 0 − 1.13949i 0 −5.70156 0
337.8 0 2.94984i 0 1.81616 0 1.13949i 0 −5.70156 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 416.2.e.c 8
3.b odd 2 1 3744.2.m.g 8
4.b odd 2 1 104.2.e.c ✓ 8
8.b even 2 1 inner 416.2.e.c 8
8.d odd 2 1 104.2.e.c ✓ 8
12.b even 2 1 936.2.m.f 8
13.b even 2 1 inner 416.2.e.c 8
24.f even 2 1 936.2.m.f 8
24.h odd 2 1 3744.2.m.g 8
39.d odd 2 1 3744.2.m.g 8
52.b odd 2 1 104.2.e.c ✓ 8
104.e even 2 1 inner 416.2.e.c 8
104.h odd 2 1 104.2.e.c ✓ 8
156.h even 2 1 936.2.m.f 8
312.b odd 2 1 3744.2.m.g 8
312.h even 2 1 936.2.m.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.2.e.c ✓ 8 4.b odd 2 1
104.2.e.c ✓ 8 8.d odd 2 1
104.2.e.c ✓ 8 52.b odd 2 1
104.2.e.c ✓ 8 104.h odd 2 1
416.2.e.c 8 1.a even 1 1 trivial
416.2.e.c 8 8.b even 2 1 inner
416.2.e.c 8 13.b even 2 1 inner
416.2.e.c 8 104.e even 2 1 inner
936.2.m.f 8 12.b even 2 1
936.2.m.f 8 24.f even 2 1
936.2.m.f 8 156.h even 2 1
936.2.m.f 8 312.h even 2 1
3744.2.m.g 8 3.b odd 2 1
3744.2.m.g 8 24.h odd 2 1
3744.2.m.g 8 39.d odd 2 1
3744.2.m.g 8 312.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(416, [\chi])\):

\( T_{3}^{4} + 11T_{3}^{2} + 20 \) Copy content Toggle raw display
\( T_{5}^{4} - 13T_{5}^{2} + 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 11 T^{2} + 20)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} - 13 T^{2} + 32)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 9 T^{2} + 10)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 26 T^{2} + 128)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 24 T^{6} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( (T^{2} + 5 T - 4)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} - 58 T^{2} + 800)^{2} \) Copy content Toggle raw display
$23$ \( (T + 4)^{8} \) Copy content Toggle raw display
$29$ \( (T^{4} + 76 T^{2} + 1280)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 90 T^{2} + 1000)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 29 T^{2} + 200)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( (T^{4} + 11 T^{2} + 20)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 9 T^{2} + 10)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 176 T^{2} + 5120)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 58 T^{2} + 800)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} - 202 T^{2} + 8192)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 185 T^{2} + 6250)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 296 T^{2} + 16000)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 4 T - 160)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 106 T^{2} + 800)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 296 T^{2} + 16000)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 184 T^{2} + 2560)^{2} \) Copy content Toggle raw display
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