Properties

Label 112.2.p.b
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: \( x^{2} - x + 1 \) Copy content Toggle raw display
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.

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

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.b yes 2
3.b odd 2 1 1008.2.cs.c 2
4.b odd 2 1 112.2.p.a 2
7.b odd 2 1 784.2.p.c 2
7.c even 3 1 784.2.f.a 2
7.c even 3 1 784.2.p.d 2
7.d odd 6 1 112.2.p.a 2
7.d odd 6 1 784.2.f.b 2
8.b even 2 1 448.2.p.a 2
8.d odd 2 1 448.2.p.b 2
12.b even 2 1 1008.2.cs.f 2
21.g even 6 1 1008.2.cs.f 2
21.g even 6 1 7056.2.b.m 2
21.h odd 6 1 7056.2.b.b 2
28.d even 2 1 784.2.p.d 2
28.f even 6 1 inner 112.2.p.b yes 2
28.f even 6 1 784.2.f.a 2
28.g odd 6 1 784.2.f.b 2
28.g odd 6 1 784.2.p.c 2
56.j odd 6 1 448.2.p.b 2
56.j odd 6 1 3136.2.f.a 2
56.k odd 6 1 3136.2.f.a 2
56.m even 6 1 448.2.p.a 2
56.m even 6 1 3136.2.f.b 2
56.p even 6 1 3136.2.f.b 2
84.j odd 6 1 1008.2.cs.c 2
84.j odd 6 1 7056.2.b.b 2
84.n even 6 1 7056.2.b.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.p.a 2 4.b odd 2 1
112.2.p.a 2 7.d odd 6 1
112.2.p.b yes 2 1.a even 1 1 trivial
112.2.p.b yes 2 28.f even 6 1 inner
448.2.p.a 2 8.b even 2 1
448.2.p.a 2 56.m even 6 1
448.2.p.b 2 8.d odd 2 1
448.2.p.b 2 56.j odd 6 1
784.2.f.a 2 7.c even 3 1
784.2.f.a 2 28.f even 6 1
784.2.f.b 2 7.d odd 6 1
784.2.f.b 2 28.g odd 6 1
784.2.p.c 2 7.b odd 2 1
784.2.p.c 2 28.g odd 6 1
784.2.p.d 2 7.c even 3 1
784.2.p.d 2 28.d even 2 1
1008.2.cs.c 2 3.b odd 2 1
1008.2.cs.c 2 84.j odd 6 1
1008.2.cs.f 2 12.b even 2 1
1008.2.cs.f 2 21.g even 6 1
3136.2.f.a 2 56.j odd 6 1
3136.2.f.a 2 56.k odd 6 1
3136.2.f.b 2 56.m even 6 1
3136.2.f.b 2 56.p even 6 1
7056.2.b.b 2 21.h odd 6 1
7056.2.b.b 2 84.j odd 6 1
7056.2.b.m 2 21.g even 6 1
7056.2.b.m 2 84.n even 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 \(S_{2}^{\mathrm{new}}(112, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$7$ \( T^{2} + 4T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 9T + 27 \) Copy content Toggle raw display
$19$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$23$ \( T^{2} + 15T + 75 \) Copy content Toggle raw display
$29$ \( (T + 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$37$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$41$ \( T^{2} + 48 \) Copy content Toggle raw display
$43$ \( T^{2} + 12 \) Copy content Toggle raw display
$47$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$53$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$59$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$61$ \( T^{2} - 15T + 75 \) Copy content Toggle raw display
$67$ \( T^{2} + 9T + 27 \) Copy content Toggle raw display
$71$ \( T^{2} + 12 \) Copy content Toggle raw display
$73$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$79$ \( T^{2} - 9T + 27 \) Copy content Toggle raw display
$83$ \( (T + 12)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 21T + 147 \) Copy content Toggle raw display
$97$ \( T^{2} + 48 \) Copy content Toggle raw display
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