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Properties

Label 672.3.m.a
Level $672$
Weight $3$
Character orbit 672.m
Analytic conductor $18.311$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [672,3,Mod(127,672)]
 
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([1, 0, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("672.127");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 672.m (of order \(2\), degree \(1\), 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: \(18.3106737650\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.49787136.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{3} + (\beta_{4} + 2) q^{5} - \beta_{6} q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{3} + (\beta_{4} + 2) q^{5} - \beta_{6} q^{7} - 3 q^{9} + (2 \beta_{6} - 2 \beta_{5} + \beta_1) q^{11} + (2 \beta_{7} + \beta_{4} - \beta_{2} - 8) q^{13} + (2 \beta_{5} + \beta_{3} + \beta_1) q^{15} + (2 \beta_{7} + 3 \beta_{4} + 2 \beta_{2} + 8) q^{17} + (2 \beta_{6} - 2 \beta_{5} + 2 \beta_1) q^{19} - \beta_{7} q^{21} + (2 \beta_{6} + 6 \beta_{5} + 3 \beta_1) q^{23} + (2 \beta_{7} + 4 \beta_{4} - 11) q^{25} - 3 \beta_{5} q^{27} + (6 \beta_{7} - 4 \beta_{4} - 2 \beta_{2} + 8) q^{29} + (6 \beta_{6} + 6 \beta_{5} + 2 \beta_1) q^{31} + (2 \beta_{7} - 2 \beta_{4} - \beta_{2} + 6) q^{33} + ( - 2 \beta_{6} + 2 \beta_{3} + \beta_1) q^{35} + ( - 2 \beta_{7} - 2 \beta_{4} + \cdots + 16) q^{37}+ \cdots + ( - 6 \beta_{6} + 6 \beta_{5} - 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{5} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{5} - 24 q^{9} - 64 q^{13} + 64 q^{17} - 88 q^{25} + 64 q^{29} + 48 q^{33} + 128 q^{37} - 48 q^{45} - 56 q^{49} - 160 q^{53} + 48 q^{57} + 32 q^{61} - 32 q^{65} - 144 q^{69} - 112 q^{73} + 112 q^{77} + 72 q^{81} + 336 q^{85} + 352 q^{89} - 144 q^{93} - 240 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 5\nu^{5} + 15\nu^{3} + 42\nu ) / 10 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{5} - \nu^{3} + \nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{7} + 5\nu^{5} - 5\nu^{3} + 16\nu ) / 10 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} - \nu^{5} - 3\nu^{3} - 6\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{6} + 5\nu^{4} + 15\nu^{2} + 26 ) / 10 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -2\nu^{6} - 9 ) / 5 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{6} + 3\nu^{4} + \nu^{2} + 6 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} + \beta_{3} + \beta_{2} + 2\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} + \beta_{6} + 3\beta_{5} - 3 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{4} - 5\beta_{3} - \beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{7} + 3\beta_{6} - \beta_{5} - 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2\beta_{4} + 3\beta_{3} - 3\beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -5\beta_{6} - 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -17\beta_{4} + 3\beta_{3} + 3\beta_{2} - 14\beta_1 ) / 8 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
−0.228425 + 1.39564i
−1.09445 + 0.895644i
1.09445 − 0.895644i
0.228425 − 1.39564i
−0.228425 − 1.39564i
−1.09445 − 0.895644i
1.09445 + 0.895644i
0.228425 + 1.39564i
0 − 1.73205i 0 −2.37780 0 − 2.64575i 0 −3.00000 0
127.2 0 − 1.73205i 0 1.08630 0 2.64575i 0 −3.00000 0
127.3 0 − 1.73205i 0 2.91370 0 2.64575i 0 −3.00000 0
127.4 0 − 1.73205i 0 6.37780 0 − 2.64575i 0 −3.00000 0
127.5 0 1.73205i 0 −2.37780 0 2.64575i 0 −3.00000 0
127.6 0 1.73205i 0 1.08630 0 − 2.64575i 0 −3.00000 0
127.7 0 1.73205i 0 2.91370 0 − 2.64575i 0 −3.00000 0
127.8 0 1.73205i 0 6.37780 0 2.64575i 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 672.3.m.a ✓ 8
3.b odd 2 1 2016.3.m.b 8
4.b odd 2 1 inner 672.3.m.a ✓ 8
8.b even 2 1 1344.3.m.d 8
8.d odd 2 1 1344.3.m.d 8
12.b even 2 1 2016.3.m.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.3.m.a ✓ 8 1.a even 1 1 trivial
672.3.m.a ✓ 8 4.b odd 2 1 inner
1344.3.m.d 8 8.b even 2 1
1344.3.m.d 8 8.d odd 2 1
2016.3.m.b 8 3.b odd 2 1
2016.3.m.b 8 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 8T_{5}^{3} + 4T_{5}^{2} + 48T_{5} - 48 \) acting on \(S_{3}^{\mathrm{new}}(672, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3)^{4} \) Copy content Toggle raw display
$5$ \( (T^{4} - 8 T^{3} + 4 T^{2} + \cdots - 48)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 7)^{4} \) Copy content Toggle raw display
$11$ \( T^{8} + 248 T^{6} + \cdots + 3154176 \) Copy content Toggle raw display
$13$ \( (T^{4} + 32 T^{3} + \cdots - 11504)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 32 T^{3} + \cdots - 77808)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 512 T^{6} + \cdots + 9437184 \) Copy content Toggle raw display
$23$ \( T^{8} + 1336 T^{6} + \cdots + 9535744 \) Copy content Toggle raw display
$29$ \( (T^{4} - 32 T^{3} + \cdots + 6736)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 1792 T^{6} + \cdots + 144769024 \) Copy content Toggle raw display
$37$ \( (T^{4} - 64 T^{3} + \cdots + 751248)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 1340 T^{2} + \cdots - 81392)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 1503036768256 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 3086795997184 \) Copy content Toggle raw display
$53$ \( (T^{4} + 80 T^{3} + \cdots + 5454096)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 247398169378816 \) Copy content Toggle raw display
$61$ \( (T^{4} - 16 T^{3} + \cdots + 12815056)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 18\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 34882285824 \) Copy content Toggle raw display
$73$ \( (T^{4} + 56 T^{3} + \cdots + 1119696)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 26\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 20\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( (T^{4} - 176 T^{3} + \cdots + 7414800)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 120 T^{3} + \cdots + 11486416)^{2} \) Copy content Toggle raw display
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