Properties

Label 112.2.p.a
Level $112$
Weight $2$
Character orbit 112.p
Analytic conductor $0.894$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 112.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.894324502638\)
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: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -1 + \zeta_{6} ) q^{3} + ( 2 - \zeta_{6} ) q^{5} + ( 1 + 2 \zeta_{6} ) q^{7} + 2 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( -1 + \zeta_{6} ) q^{3} + ( 2 - \zeta_{6} ) q^{5} + ( 1 + 2 \zeta_{6} ) q^{7} + 2 \zeta_{6} q^{9} + ( -1 - \zeta_{6} ) q^{11} + ( -1 + 2 \zeta_{6} ) q^{15} + ( -3 - 3 \zeta_{6} ) q^{17} -7 \zeta_{6} q^{19} + ( -3 + \zeta_{6} ) q^{21} + ( 10 - 5 \zeta_{6} ) q^{23} + ( -2 + 2 \zeta_{6} ) q^{25} -5 q^{27} -6 q^{29} + ( -5 + 5 \zeta_{6} ) q^{31} + ( 2 - \zeta_{6} ) q^{33} + ( 4 + \zeta_{6} ) q^{35} + 5 \zeta_{6} q^{37} + ( 4 - 8 \zeta_{6} ) q^{41} + ( -2 + 4 \zeta_{6} ) q^{43} + ( 2 + 2 \zeta_{6} ) q^{45} -3 \zeta_{6} q^{47} + ( -3 + 8 \zeta_{6} ) q^{49} + ( 6 - 3 \zeta_{6} ) q^{51} + ( 9 - 9 \zeta_{6} ) q^{53} -3 q^{55} + 7 q^{57} + ( -9 + 9 \zeta_{6} ) q^{59} + ( 10 - 5 \zeta_{6} ) q^{61} + ( -4 + 6 \zeta_{6} ) q^{63} + ( 3 + 3 \zeta_{6} ) q^{67} + ( -5 + 10 \zeta_{6} ) q^{69} + ( 2 - 4 \zeta_{6} ) q^{71} + ( 1 + \zeta_{6} ) q^{73} -2 \zeta_{6} q^{75} + ( 1 - 5 \zeta_{6} ) q^{77} + ( -6 + 3 \zeta_{6} ) q^{79} + ( -1 + \zeta_{6} ) q^{81} + 12 q^{83} -9 q^{85} + ( 6 - 6 \zeta_{6} ) q^{87} + ( -14 + 7 \zeta_{6} ) q^{89} -5 \zeta_{6} q^{93} + ( -7 - 7 \zeta_{6} ) q^{95} + ( 4 - 8 \zeta_{6} ) q^{97} + ( 2 - 4 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{3} + 3q^{5} + 4q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - q^{3} + 3q^{5} + 4q^{7} + 2q^{9} - 3q^{11} - 9q^{17} - 7q^{19} - 5q^{21} + 15q^{23} - 2q^{25} - 10q^{27} - 12q^{29} - 5q^{31} + 3q^{33} + 9q^{35} + 5q^{37} + 6q^{45} - 3q^{47} + 2q^{49} + 9q^{51} + 9q^{53} - 6q^{55} + 14q^{57} - 9q^{59} + 15q^{61} - 2q^{63} + 9q^{67} + 3q^{73} - 2q^{75} - 3q^{77} - 9q^{79} - q^{81} + 24q^{83} - 18q^{85} + 6q^{87} - 21q^{89} - 5q^{93} - 21q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(85\)
\(\chi(n)\) \(-1\) \(\zeta_{6}\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
0.500000 + 0.866025i
0.500000 0.866025i
0 −0.500000 + 0.866025i 0 1.50000 0.866025i 0 2.00000 + 1.73205i 0 1.00000 + 1.73205i 0
47.1 0 −0.500000 0.866025i 0 1.50000 + 0.866025i 0 2.00000 1.73205i 0 1.00000 1.73205i 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
28.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 112.2.p.a 2
3.b odd 2 1 1008.2.cs.f 2
4.b odd 2 1 112.2.p.b yes 2
7.b odd 2 1 784.2.p.d 2
7.c even 3 1 784.2.f.b 2
7.c even 3 1 784.2.p.c 2
7.d odd 6 1 112.2.p.b yes 2
7.d odd 6 1 784.2.f.a 2
8.b even 2 1 448.2.p.b 2
8.d odd 2 1 448.2.p.a 2
12.b even 2 1 1008.2.cs.c 2
21.g even 6 1 1008.2.cs.c 2
21.g even 6 1 7056.2.b.b 2
21.h odd 6 1 7056.2.b.m 2
28.d even 2 1 784.2.p.c 2
28.f even 6 1 inner 112.2.p.a 2
28.f even 6 1 784.2.f.b 2
28.g odd 6 1 784.2.f.a 2
28.g odd 6 1 784.2.p.d 2
56.j odd 6 1 448.2.p.a 2
56.j odd 6 1 3136.2.f.b 2
56.k odd 6 1 3136.2.f.b 2
56.m even 6 1 448.2.p.b 2
56.m even 6 1 3136.2.f.a 2
56.p even 6 1 3136.2.f.a 2
84.j odd 6 1 1008.2.cs.f 2
84.j odd 6 1 7056.2.b.m 2
84.n even 6 1 7056.2.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.p.a 2 1.a even 1 1 trivial
112.2.p.a 2 28.f even 6 1 inner
112.2.p.b yes 2 4.b odd 2 1
112.2.p.b yes 2 7.d odd 6 1
448.2.p.a 2 8.d odd 2 1
448.2.p.a 2 56.j odd 6 1
448.2.p.b 2 8.b even 2 1
448.2.p.b 2 56.m even 6 1
784.2.f.a 2 7.d odd 6 1
784.2.f.a 2 28.g odd 6 1
784.2.f.b 2 7.c even 3 1
784.2.f.b 2 28.f even 6 1
784.2.p.c 2 7.c even 3 1
784.2.p.c 2 28.d even 2 1
784.2.p.d 2 7.b odd 2 1
784.2.p.d 2 28.g odd 6 1
1008.2.cs.c 2 12.b even 2 1
1008.2.cs.c 2 21.g even 6 1
1008.2.cs.f 2 3.b odd 2 1
1008.2.cs.f 2 84.j odd 6 1
3136.2.f.a 2 56.m even 6 1
3136.2.f.a 2 56.p even 6 1
3136.2.f.b 2 56.j odd 6 1
3136.2.f.b 2 56.k odd 6 1
7056.2.b.b 2 21.g even 6 1
7056.2.b.b 2 84.n even 6 1
7056.2.b.m 2 21.h odd 6 1
7056.2.b.m 2 84.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(112, [\chi])\).