Properties

Label 112.2.i.b
Level $112$
Weight $2$
Character orbit 112.i
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.i (of order \(3\), degree \(2\), not 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: no (minimal twist has level 28)
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.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{3} - 3 \zeta_{6} q^{5} + ( - 2 \zeta_{6} + 3) q^{7} + 2 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{6} + 1) q^{3} - 3 \zeta_{6} q^{5} + ( - 2 \zeta_{6} + 3) q^{7} + 2 \zeta_{6} q^{9} + (3 \zeta_{6} - 3) q^{11} + 2 q^{13} - 3 q^{15} + (3 \zeta_{6} - 3) q^{17} - \zeta_{6} q^{19} + ( - 3 \zeta_{6} + 1) q^{21} + 3 \zeta_{6} q^{23} + (4 \zeta_{6} - 4) q^{25} + 5 q^{27} - 6 q^{29} + (7 \zeta_{6} - 7) q^{31} + 3 \zeta_{6} q^{33} + ( - 3 \zeta_{6} - 6) q^{35} + \zeta_{6} q^{37} + ( - 2 \zeta_{6} + 2) q^{39} + 6 q^{41} + 4 q^{43} + ( - 6 \zeta_{6} + 6) q^{45} - 9 \zeta_{6} q^{47} + ( - 8 \zeta_{6} + 5) q^{49} + 3 \zeta_{6} q^{51} + (3 \zeta_{6} - 3) q^{53} + 9 q^{55} - q^{57} + ( - 9 \zeta_{6} + 9) q^{59} + \zeta_{6} q^{61} + (2 \zeta_{6} + 4) q^{63} - 6 \zeta_{6} q^{65} + (7 \zeta_{6} - 7) q^{67} + 3 q^{69} + ( - \zeta_{6} + 1) q^{73} + 4 \zeta_{6} q^{75} + (9 \zeta_{6} - 3) q^{77} - 13 \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{81} - 12 q^{83} + 9 q^{85} + (6 \zeta_{6} - 6) q^{87} - 15 \zeta_{6} q^{89} + ( - 4 \zeta_{6} + 6) q^{91} + 7 \zeta_{6} q^{93} + (3 \zeta_{6} - 3) q^{95} - 10 q^{97} - 6 q^{99} +O(q^{100}) \) 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 \( 2 q + q^{3} - 3 q^{5} + 4 q^{7} + 2 q^{9} - 3 q^{11} + 4 q^{13} - 6 q^{15} - 3 q^{17} - q^{19} - q^{21} + 3 q^{23} - 4 q^{25} + 10 q^{27} - 12 q^{29} - 7 q^{31} + 3 q^{33} - 15 q^{35} + q^{37} + 2 q^{39} + 12 q^{41} + 8 q^{43} + 6 q^{45} - 9 q^{47} + 2 q^{49} + 3 q^{51} - 3 q^{53} + 18 q^{55} - 2 q^{57} + 9 q^{59} + q^{61} + 10 q^{63} - 6 q^{65} - 7 q^{67} + 6 q^{69} + q^{73} + 4 q^{75} + 3 q^{77} - 13 q^{79} - q^{81} - 24 q^{83} + 18 q^{85} - 6 q^{87} - 15 q^{89} + 8 q^{91} + 7 q^{93} - 3 q^{95} - 20 q^{97} - 12 q^{99}+O(q^{100}) \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0.500000 + 0.866025i 0 −1.50000 + 2.59808i 0 2.00000 + 1.73205i 0 1.00000 1.73205i 0
81.1 0 0.500000 0.866025i 0 −1.50000 2.59808i 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
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 112.2.i.b 2
3.b odd 2 1 1008.2.s.p 2
4.b odd 2 1 28.2.e.a 2
7.b odd 2 1 784.2.i.d 2
7.c even 3 1 inner 112.2.i.b 2
7.c even 3 1 784.2.a.d 1
7.d odd 6 1 784.2.a.g 1
7.d odd 6 1 784.2.i.d 2
8.b even 2 1 448.2.i.c 2
8.d odd 2 1 448.2.i.e 2
12.b even 2 1 252.2.k.c 2
20.d odd 2 1 700.2.i.c 2
20.e even 4 2 700.2.r.b 4
21.g even 6 1 7056.2.a.bw 1
21.h odd 6 1 1008.2.s.p 2
21.h odd 6 1 7056.2.a.f 1
28.d even 2 1 196.2.e.a 2
28.f even 6 1 196.2.a.a 1
28.f even 6 1 196.2.e.a 2
28.g odd 6 1 28.2.e.a 2
28.g odd 6 1 196.2.a.b 1
36.f odd 6 1 2268.2.i.a 2
36.f odd 6 1 2268.2.l.h 2
36.h even 6 1 2268.2.i.h 2
36.h even 6 1 2268.2.l.a 2
56.j odd 6 1 3136.2.a.k 1
56.k odd 6 1 448.2.i.e 2
56.k odd 6 1 3136.2.a.h 1
56.m even 6 1 3136.2.a.v 1
56.p even 6 1 448.2.i.c 2
56.p even 6 1 3136.2.a.s 1
84.h odd 2 1 1764.2.k.b 2
84.j odd 6 1 1764.2.a.j 1
84.j odd 6 1 1764.2.k.b 2
84.n even 6 1 252.2.k.c 2
84.n even 6 1 1764.2.a.a 1
140.p odd 6 1 700.2.i.c 2
140.p odd 6 1 4900.2.a.g 1
140.s even 6 1 4900.2.a.n 1
140.w even 12 2 700.2.r.b 4
140.w even 12 2 4900.2.e.i 2
140.x odd 12 2 4900.2.e.h 2
252.o even 6 1 2268.2.i.h 2
252.u odd 6 1 2268.2.l.h 2
252.bb even 6 1 2268.2.l.a 2
252.bl odd 6 1 2268.2.i.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.2.e.a 2 4.b odd 2 1
28.2.e.a 2 28.g odd 6 1
112.2.i.b 2 1.a even 1 1 trivial
112.2.i.b 2 7.c even 3 1 inner
196.2.a.a 1 28.f even 6 1
196.2.a.b 1 28.g odd 6 1
196.2.e.a 2 28.d even 2 1
196.2.e.a 2 28.f even 6 1
252.2.k.c 2 12.b even 2 1
252.2.k.c 2 84.n even 6 1
448.2.i.c 2 8.b even 2 1
448.2.i.c 2 56.p even 6 1
448.2.i.e 2 8.d odd 2 1
448.2.i.e 2 56.k odd 6 1
700.2.i.c 2 20.d odd 2 1
700.2.i.c 2 140.p odd 6 1
700.2.r.b 4 20.e even 4 2
700.2.r.b 4 140.w even 12 2
784.2.a.d 1 7.c even 3 1
784.2.a.g 1 7.d odd 6 1
784.2.i.d 2 7.b odd 2 1
784.2.i.d 2 7.d odd 6 1
1008.2.s.p 2 3.b odd 2 1
1008.2.s.p 2 21.h odd 6 1
1764.2.a.a 1 84.n even 6 1
1764.2.a.j 1 84.j odd 6 1
1764.2.k.b 2 84.h odd 2 1
1764.2.k.b 2 84.j odd 6 1
2268.2.i.a 2 36.f odd 6 1
2268.2.i.a 2 252.bl odd 6 1
2268.2.i.h 2 36.h even 6 1
2268.2.i.h 2 252.o even 6 1
2268.2.l.a 2 36.h even 6 1
2268.2.l.a 2 252.bb even 6 1
2268.2.l.h 2 36.f odd 6 1
2268.2.l.h 2 252.u odd 6 1
3136.2.a.h 1 56.k odd 6 1
3136.2.a.k 1 56.j odd 6 1
3136.2.a.s 1 56.p even 6 1
3136.2.a.v 1 56.m even 6 1
4900.2.a.g 1 140.p odd 6 1
4900.2.a.n 1 140.s even 6 1
4900.2.e.h 2 140.x odd 12 2
4900.2.e.i 2 140.w even 12 2
7056.2.a.f 1 21.h odd 6 1
7056.2.a.bw 1 21.g 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
$5$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$7$ \( T^{2} - 4T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$13$ \( (T - 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$19$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$23$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$29$ \( (T + 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$37$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$41$ \( (T - 6)^{2} \) Copy content Toggle raw display
$43$ \( (T - 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$53$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$59$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$61$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$67$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$79$ \( T^{2} + 13T + 169 \) Copy content Toggle raw display
$83$ \( (T + 12)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 15T + 225 \) Copy content Toggle raw display
$97$ \( (T + 10)^{2} \) Copy content Toggle raw display
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