Properties

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

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 672.q (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{6} ) q^{3} + ( -2 - \zeta_{6} ) q^{7} -\zeta_{6} q^{9} +O(q^{10})\) \( q + ( 1 - \zeta_{6} ) q^{3} + ( -2 - \zeta_{6} ) q^{7} -\zeta_{6} q^{9} + ( 2 - 2 \zeta_{6} ) q^{11} + q^{13} + ( 2 - 2 \zeta_{6} ) q^{17} -5 \zeta_{6} q^{19} + ( -3 + 2 \zeta_{6} ) q^{21} -6 \zeta_{6} q^{23} + ( 5 - 5 \zeta_{6} ) q^{25} - q^{27} -8 q^{29} + ( 3 - 3 \zeta_{6} ) q^{31} -2 \zeta_{6} q^{33} + 9 \zeta_{6} q^{37} + ( 1 - \zeta_{6} ) q^{39} + 2 q^{41} + q^{43} -8 \zeta_{6} q^{47} + ( 3 + 5 \zeta_{6} ) q^{49} -2 \zeta_{6} q^{51} + ( -6 + 6 \zeta_{6} ) q^{53} -5 q^{57} + ( -6 + 6 \zeta_{6} ) q^{59} + 2 \zeta_{6} q^{61} + ( -1 + 3 \zeta_{6} ) q^{63} + ( 5 - 5 \zeta_{6} ) q^{67} -6 q^{69} + 4 q^{71} + ( 11 - 11 \zeta_{6} ) q^{73} -5 \zeta_{6} q^{75} + ( -6 + 4 \zeta_{6} ) q^{77} + 5 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} + ( -8 + 8 \zeta_{6} ) q^{87} -12 \zeta_{6} q^{89} + ( -2 - \zeta_{6} ) q^{91} -3 \zeta_{6} q^{93} + 18 q^{97} -2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} - 5q^{7} - q^{9} + O(q^{10}) \) \( 2q + q^{3} - 5q^{7} - q^{9} + 2q^{11} + 2q^{13} + 2q^{17} - 5q^{19} - 4q^{21} - 6q^{23} + 5q^{25} - 2q^{27} - 16q^{29} + 3q^{31} - 2q^{33} + 9q^{37} + q^{39} + 4q^{41} + 2q^{43} - 8q^{47} + 11q^{49} - 2q^{51} - 6q^{53} - 10q^{57} - 6q^{59} + 2q^{61} + q^{63} + 5q^{67} - 12q^{69} + 8q^{71} + 11q^{73} - 5q^{75} - 8q^{77} + 5q^{79} - q^{81} - 8q^{87} - 12q^{89} - 5q^{91} - 3q^{93} + 36q^{97} - 4q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-\zeta_{6}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0.500000 0.866025i 0 0 0 −2.50000 0.866025i 0 −0.500000 0.866025i 0
289.1 0 0.500000 + 0.866025i 0 0 0 −2.50000 + 0.866025i 0 −0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 672.2.q.h yes 2
3.b odd 2 1 2016.2.s.e 2
4.b odd 2 1 672.2.q.d 2
7.c even 3 1 inner 672.2.q.h yes 2
7.c even 3 1 4704.2.a.g 1
7.d odd 6 1 4704.2.a.y 1
8.b even 2 1 1344.2.q.e 2
8.d odd 2 1 1344.2.q.q 2
12.b even 2 1 2016.2.s.h 2
21.h odd 6 1 2016.2.s.e 2
28.f even 6 1 4704.2.a.j 1
28.g odd 6 1 672.2.q.d 2
28.g odd 6 1 4704.2.a.ba 1
56.j odd 6 1 9408.2.a.x 1
56.k odd 6 1 1344.2.q.q 2
56.k odd 6 1 9408.2.a.t 1
56.m even 6 1 9408.2.a.ci 1
56.p even 6 1 1344.2.q.e 2
56.p even 6 1 9408.2.a.cn 1
84.n even 6 1 2016.2.s.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.q.d 2 4.b odd 2 1
672.2.q.d 2 28.g odd 6 1
672.2.q.h yes 2 1.a even 1 1 trivial
672.2.q.h yes 2 7.c even 3 1 inner
1344.2.q.e 2 8.b even 2 1
1344.2.q.e 2 56.p even 6 1
1344.2.q.q 2 8.d odd 2 1
1344.2.q.q 2 56.k odd 6 1
2016.2.s.e 2 3.b odd 2 1
2016.2.s.e 2 21.h odd 6 1
2016.2.s.h 2 12.b even 2 1
2016.2.s.h 2 84.n even 6 1
4704.2.a.g 1 7.c even 3 1
4704.2.a.j 1 28.f even 6 1
4704.2.a.y 1 7.d odd 6 1
4704.2.a.ba 1 28.g odd 6 1
9408.2.a.t 1 56.k odd 6 1
9408.2.a.x 1 56.j odd 6 1
9408.2.a.ci 1 56.m even 6 1
9408.2.a.cn 1 56.p even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(672, [\chi])\):

\( T_{5} \)
\( T_{11}^{2} - 2 T_{11} + 4 \)
\( T_{13} - 1 \)