Properties

Label 112.2.p.c
Level $112$
Weight $2$
Character orbit 112.p
Analytic conductor $0.894$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{7})\)
Defining polynomial: \( x^{4} + 7x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{2} - 2) q^{5} - \beta_{3} q^{7} + 4 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - \beta_{2} - 2) q^{5} - \beta_{3} q^{7} + 4 \beta_{2} q^{9} + (2 \beta_{3} + \beta_1) q^{11} + ( - 4 \beta_{2} - 2) q^{13} + ( - \beta_{3} - 2 \beta_1) q^{15} + ( - 3 \beta_{2} + 3) q^{17} + ( - \beta_{3} - \beta_1) q^{19} + (7 \beta_{2} + 7) q^{21} + (\beta_{3} - \beta_1) q^{23} + ( - 2 \beta_{2} - 2) q^{25} + \beta_{3} q^{27} - \beta_1 q^{31} + ( - 7 \beta_{2} - 14) q^{33} + (\beta_{3} - \beta_1) q^{35} + 7 \beta_{2} q^{37} + ( - 4 \beta_{3} - 2 \beta_1) q^{39} + (4 \beta_{2} + 2) q^{41} + (2 \beta_{3} + 4 \beta_1) q^{43} + ( - 4 \beta_{2} + 4) q^{45} + (3 \beta_{3} + 3 \beta_1) q^{47} + 7 q^{49} + ( - 3 \beta_{3} + 3 \beta_1) q^{51} + ( - 3 \beta_{2} - 3) q^{53} - 3 \beta_{3} q^{55} + 7 q^{57} - 3 \beta_1 q^{59} + (\beta_{2} + 2) q^{61} + (4 \beta_{3} + 4 \beta_1) q^{63} + 6 \beta_{2} q^{65} + ( - 2 \beta_{3} - \beta_1) q^{67} + ( - 14 \beta_{2} - 7) q^{69} + (2 \beta_{3} + 4 \beta_1) q^{71} + (3 \beta_{2} - 3) q^{73} + ( - 2 \beta_{3} - 2 \beta_1) q^{75} + (7 \beta_{2} - 7) q^{77} + (\beta_{3} - \beta_1) q^{79} + (5 \beta_{2} + 5) q^{81} - 9 q^{85} + (\beta_{2} + 2) q^{89} + ( - 2 \beta_{3} - 4 \beta_1) q^{91} - 7 \beta_{2} q^{93} + (2 \beta_{3} + \beta_1) q^{95} + (4 \beta_{2} + 2) q^{97} + ( - 4 \beta_{3} - 8 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{5} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{5} - 8 q^{9} + 18 q^{17} + 14 q^{21} - 4 q^{25} - 42 q^{33} - 14 q^{37} + 24 q^{45} + 28 q^{49} - 6 q^{53} + 28 q^{57} + 6 q^{61} - 12 q^{65} - 18 q^{73} - 42 q^{77} + 10 q^{81} - 36 q^{85} + 6 q^{89} + 14 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 7x^{2} + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 7\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 7\beta_{3} \) Copy content Toggle raw display

Character values

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

\(n\) \(15\) \(17\) \(85\)
\(\chi(n)\) \(-1\) \(-\beta_{2}\) \(1\)

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
−1.32288 + 2.29129i
1.32288 2.29129i
−1.32288 2.29129i
1.32288 + 2.29129i
0 −1.32288 + 2.29129i 0 −1.50000 + 0.866025i 0 −2.64575 0 −2.00000 3.46410i 0
31.2 0 1.32288 2.29129i 0 −1.50000 + 0.866025i 0 2.64575 0 −2.00000 3.46410i 0
47.1 0 −1.32288 2.29129i 0 −1.50000 0.866025i 0 −2.64575 0 −2.00000 + 3.46410i 0
47.2 0 1.32288 + 2.29129i 0 −1.50000 0.866025i 0 2.64575 0 −2.00000 + 3.46410i 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
4.b odd 2 1 inner
7.d odd 6 1 inner
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.c 4
3.b odd 2 1 1008.2.cs.q 4
4.b odd 2 1 inner 112.2.p.c 4
7.b odd 2 1 784.2.p.g 4
7.c even 3 1 784.2.f.d 4
7.c even 3 1 784.2.p.g 4
7.d odd 6 1 inner 112.2.p.c 4
7.d odd 6 1 784.2.f.d 4
8.b even 2 1 448.2.p.c 4
8.d odd 2 1 448.2.p.c 4
12.b even 2 1 1008.2.cs.q 4
21.g even 6 1 1008.2.cs.q 4
21.g even 6 1 7056.2.b.s 4
21.h odd 6 1 7056.2.b.s 4
28.d even 2 1 784.2.p.g 4
28.f even 6 1 inner 112.2.p.c 4
28.f even 6 1 784.2.f.d 4
28.g odd 6 1 784.2.f.d 4
28.g odd 6 1 784.2.p.g 4
56.j odd 6 1 448.2.p.c 4
56.j odd 6 1 3136.2.f.f 4
56.k odd 6 1 3136.2.f.f 4
56.m even 6 1 448.2.p.c 4
56.m even 6 1 3136.2.f.f 4
56.p even 6 1 3136.2.f.f 4
84.j odd 6 1 1008.2.cs.q 4
84.j odd 6 1 7056.2.b.s 4
84.n even 6 1 7056.2.b.s 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.p.c 4 1.a even 1 1 trivial
112.2.p.c 4 4.b odd 2 1 inner
112.2.p.c 4 7.d odd 6 1 inner
112.2.p.c 4 28.f even 6 1 inner
448.2.p.c 4 8.b even 2 1
448.2.p.c 4 8.d odd 2 1
448.2.p.c 4 56.j odd 6 1
448.2.p.c 4 56.m even 6 1
784.2.f.d 4 7.c even 3 1
784.2.f.d 4 7.d odd 6 1
784.2.f.d 4 28.f even 6 1
784.2.f.d 4 28.g odd 6 1
784.2.p.g 4 7.b odd 2 1
784.2.p.g 4 7.c even 3 1
784.2.p.g 4 28.d even 2 1
784.2.p.g 4 28.g odd 6 1
1008.2.cs.q 4 3.b odd 2 1
1008.2.cs.q 4 12.b even 2 1
1008.2.cs.q 4 21.g even 6 1
1008.2.cs.q 4 84.j odd 6 1
3136.2.f.f 4 56.j odd 6 1
3136.2.f.f 4 56.k odd 6 1
3136.2.f.f 4 56.m even 6 1
3136.2.f.f 4 56.p even 6 1
7056.2.b.s 4 21.g even 6 1
7056.2.b.s 4 21.h odd 6 1
7056.2.b.s 4 84.j odd 6 1
7056.2.b.s 4 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 7T_{3}^{2} + 49 \) acting on \(S_{2}^{\mathrm{new}}(112, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 7T^{2} + 49 \) Copy content Toggle raw display
$5$ \( (T^{2} + 3 T + 3)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 21T^{2} + 441 \) Copy content Toggle raw display
$13$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 9 T + 27)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 7T^{2} + 49 \) Copy content Toggle raw display
$23$ \( T^{4} - 21T^{2} + 441 \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 7T^{2} + 49 \) Copy content Toggle raw display
$37$ \( (T^{2} + 7 T + 49)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 84)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 63T^{2} + 3969 \) Copy content Toggle raw display
$53$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 63T^{2} + 3969 \) Copy content Toggle raw display
$61$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 21T^{2} + 441 \) Copy content Toggle raw display
$71$ \( (T^{2} + 84)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 9 T + 27)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 21T^{2} + 441 \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
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