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Properties

Label 672.2.c.b
Level $672$
Weight $2$
Character orbit 672.c
Analytic conductor $5.366$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [672,2,Mod(337,672)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(672, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("672.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 672.c (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: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.386672896.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} - 2x^{5} + 2x^{4} - 4x^{3} - 4x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 168)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + \beta_1 q^{5} - q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + \beta_1 q^{5} - q^{7} - q^{9} + ( - \beta_{7} - 2 \beta_{2}) q^{11} + ( - \beta_{6} + \beta_{2}) q^{13} + \beta_{5} q^{15} + ( - \beta_{5} + \beta_{3} + 1) q^{17} + ( - \beta_{7} + 2 \beta_{2} + \beta_1) q^{19} - \beta_{2} q^{21} + ( - \beta_{5} + \beta_{3} - 1) q^{23} + ( - \beta_{5} + \beta_{4} + 2 \beta_{3} - 3) q^{25} - \beta_{2} q^{27} + ( - \beta_{7} + 2 \beta_{6} + \beta_1) q^{29} + (\beta_{5} + \beta_{4} - 2) q^{31} + ( - \beta_{4} + 2) q^{33} - \beta_1 q^{35} + ( - \beta_{7} - 2 \beta_{2} - \beta_1) q^{37} + (\beta_{3} - 1) q^{39} + (\beta_{5} - \beta_{3} - 1) q^{41} + (\beta_{7} - \beta_{6} - \beta_{2} + \beta_1) q^{43} - \beta_1 q^{45} + (2 \beta_{5} - 2 \beta_{4}) q^{47} + q^{49} + (\beta_{6} + \beta_{2} + \beta_1) q^{51} + ( - \beta_{7} + 2 \beta_{2} + \beta_1) q^{53} + ( - 3 \beta_{5} + \beta_{4} + 2) q^{55} + (\beta_{5} - \beta_{4} - 2) q^{57} - 4 \beta_{2} q^{59} + ( - \beta_{6} + \beta_{2} + 2 \beta_1) q^{61} + q^{63} + (4 \beta_{5} - 4) q^{65} + ( - \beta_{7} - \beta_{6} + \cdots + \beta_1) q^{67}+ \cdots + (\beta_{7} + 2 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{7} - 8 q^{9} + 4 q^{15} + 4 q^{17} - 12 q^{23} - 24 q^{25} - 8 q^{31} + 12 q^{33} - 8 q^{39} - 4 q^{41} + 8 q^{49} + 8 q^{55} - 16 q^{57} + 8 q^{63} - 16 q^{65} + 28 q^{71} - 8 q^{73} + 40 q^{79} + 8 q^{81} + 20 q^{89} - 40 q^{95} + 40 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( ( \nu^{5} + \nu^{3} - 2\nu^{2} - 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} - 2\nu^{6} - 3\nu^{5} + 4\nu^{4} - 2\nu^{3} + 4\nu + 24 ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 2\nu^{6} - 3\nu^{5} + 6\nu^{3} + 8\nu^{2} - 12\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} - 2\nu^{6} + \nu^{5} + 4\nu^{4} + 2\nu^{3} + 8\nu^{2} + 20\nu + 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - 2\nu^{6} - \nu^{5} + 6\nu^{3} - 4\nu + 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{7} - 2\nu^{6} - 7\nu^{5} - 4\nu^{4} + 6\nu^{3} - 16\nu^{2} + 20\nu + 24 ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5\nu^{7} + 6\nu^{6} + 3\nu^{5} + 6\nu^{3} - 16\nu^{2} - 36\nu - 40 ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} - \beta_{5} + \beta_{4} - \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{6} + \beta_{5} + \beta_{4} - 3\beta_{2} - 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{6} + \beta_{5} + \beta_{4} + 2\beta_{3} - \beta_{2} + 2\beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{7} - \beta_{6} - \beta_{5} + 3\beta_{4} + 9\beta_{2} - 4 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -3\beta_{6} + \beta_{5} + \beta_{4} - 2\beta_{3} - 5\beta_{2} + 2\beta _1 + 14 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2\beta_{7} + 3\beta_{6} - 9\beta_{5} + 3\beta_{4} + 4\beta_{3} - 3\beta_{2} + 16 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 4\beta_{7} + \beta_{6} + 5\beta_{5} + 5\beta_{4} - 6\beta_{3} - 9\beta_{2} - 10\beta _1 + 2 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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
0.621372 + 1.27039i
−0.835949 − 1.14070i
1.40961 + 0.114062i
−1.19503 + 0.756243i
−1.19503 − 0.756243i
1.40961 − 0.114062i
−0.835949 + 1.14070i
0.621372 − 1.27039i
0 − 1.00000i 0 − 3.69833i 0 −1.00000 0 −1.00000 0
337.2 0 − 1.00000i 0 0.467138i 0 −1.00000 0 −1.00000 0
337.3 0 − 1.00000i 0 1.12875i 0 −1.00000 0 −1.00000 0
337.4 0 − 1.00000i 0 4.10245i 0 −1.00000 0 −1.00000 0
337.5 0 1.00000i 0 − 4.10245i 0 −1.00000 0 −1.00000 0
337.6 0 1.00000i 0 − 1.12875i 0 −1.00000 0 −1.00000 0
337.7 0 1.00000i 0 − 0.467138i 0 −1.00000 0 −1.00000 0
337.8 0 1.00000i 0 3.69833i 0 −1.00000 0 −1.00000 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 672.2.c.b 8
3.b odd 2 1 2016.2.c.e 8
4.b odd 2 1 168.2.c.b ✓ 8
7.b odd 2 1 4704.2.c.c 8
8.b even 2 1 inner 672.2.c.b 8
8.d odd 2 1 168.2.c.b ✓ 8
12.b even 2 1 504.2.c.f 8
16.e even 4 1 5376.2.a.bl 4
16.e even 4 1 5376.2.a.bq 4
16.f odd 4 1 5376.2.a.bm 4
16.f odd 4 1 5376.2.a.bp 4
24.f even 2 1 504.2.c.f 8
24.h odd 2 1 2016.2.c.e 8
28.d even 2 1 1176.2.c.c 8
56.e even 2 1 1176.2.c.c 8
56.h odd 2 1 4704.2.c.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.c.b ✓ 8 4.b odd 2 1
168.2.c.b ✓ 8 8.d odd 2 1
504.2.c.f 8 12.b even 2 1
504.2.c.f 8 24.f even 2 1
672.2.c.b 8 1.a even 1 1 trivial
672.2.c.b 8 8.b even 2 1 inner
1176.2.c.c 8 28.d even 2 1
1176.2.c.c 8 56.e even 2 1
2016.2.c.e 8 3.b odd 2 1
2016.2.c.e 8 24.h odd 2 1
4704.2.c.c 8 7.b odd 2 1
4704.2.c.c 8 56.h odd 2 1
5376.2.a.bl 4 16.e even 4 1
5376.2.a.bm 4 16.f odd 4 1
5376.2.a.bp 4 16.f odd 4 1
5376.2.a.bq 4 16.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 32T_{5}^{6} + 276T_{5}^{4} + 352T_{5}^{2} + 64 \) acting on \(S_{2}^{\mathrm{new}}(672, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + 32 T^{6} + \cdots + 64 \) Copy content Toggle raw display
$7$ \( (T + 1)^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + 64 T^{6} + \cdots + 40000 \) Copy content Toggle raw display
$13$ \( T^{8} + 56 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$17$ \( (T^{4} - 2 T^{3} - 30 T^{2} + \cdots - 8)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 88 T^{6} + \cdots + 16384 \) Copy content Toggle raw display
$23$ \( (T^{4} + 6 T^{3} - 18 T^{2} + \cdots - 64)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 216 T^{6} + \cdots + 6885376 \) Copy content Toggle raw display
$31$ \( (T^{4} + 4 T^{3} + \cdots + 256)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 104 T^{6} + \cdots + 65536 \) Copy content Toggle raw display
$41$ \( (T^{4} + 2 T^{3} - 30 T^{2} + \cdots - 8)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 176 T^{6} + \cdots + 565504 \) Copy content Toggle raw display
$47$ \( (T^{4} - 144 T^{2} + \cdots + 3584)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 88 T^{6} + \cdots + 16384 \) Copy content Toggle raw display
$59$ \( (T^{2} + 16)^{4} \) Copy content Toggle raw display
$61$ \( T^{8} + 216 T^{6} + \cdots + 409600 \) Copy content Toggle raw display
$67$ \( T^{8} + 192 T^{6} + \cdots + 12544 \) Copy content Toggle raw display
$71$ \( (T^{4} - 14 T^{3} + \cdots - 3136)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 4 T^{3} + \cdots - 2608)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 20 T^{3} + \cdots - 2560)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 416 T^{6} + \cdots + 55115776 \) Copy content Toggle raw display
$89$ \( (T^{4} - 10 T^{3} + \cdots - 6280)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 20 T^{3} + \cdots - 13808)^{2} \) Copy content Toggle raw display
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