Properties

Label 112.2.m.b
Level $112$
Weight $2$
Character orbit 112.m
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.m (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.894324502638\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (i - 1) q^{2} - 2 i q^{4} + (2 i + 2) q^{5} - i q^{7} + (2 i + 2) q^{8} + 3 i q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (i - 1) q^{2} - 2 i q^{4} + (2 i + 2) q^{5} - i q^{7} + (2 i + 2) q^{8} + 3 i q^{9} - 4 q^{10} + (i + 1) q^{11} + (i + 1) q^{14} - 4 q^{16} - 2 q^{17} + ( - 3 i - 3) q^{18} + ( - 2 i + 2) q^{19} + ( - 4 i + 4) q^{20} - 2 q^{22} - 6 i q^{23} + 3 i q^{25} - 2 q^{28} + ( - 7 i + 7) q^{29} - 8 q^{31} + ( - 4 i + 4) q^{32} + ( - 2 i + 2) q^{34} + ( - 2 i + 2) q^{35} + 6 q^{36} + ( - 5 i - 5) q^{37} + 4 i q^{38} + 8 i q^{40} + 10 i q^{41} + ( - i - 1) q^{43} + ( - 2 i + 2) q^{44} + (6 i - 6) q^{45} + (6 i + 6) q^{46} - 12 q^{47} - q^{49} + ( - 3 i - 3) q^{50} + ( - i - 1) q^{53} + 4 i q^{55} + ( - 2 i + 2) q^{56} + 14 i q^{58} + (8 i + 8) q^{59} + ( - 6 i + 6) q^{61} + ( - 8 i + 8) q^{62} + 3 q^{63} + 8 i q^{64} + ( - 3 i + 3) q^{67} + 4 i q^{68} + 4 i q^{70} + (6 i - 6) q^{72} - 6 i q^{73} + 10 q^{74} + ( - 4 i - 4) q^{76} + ( - i + 1) q^{77} + 10 q^{79} + ( - 8 i - 8) q^{80} - 9 q^{81} + ( - 10 i - 10) q^{82} + (10 i - 10) q^{83} + ( - 4 i - 4) q^{85} + 2 q^{86} + 4 i q^{88} - 14 i q^{89} - 12 i q^{90} - 12 q^{92} + ( - 12 i + 12) q^{94} + 8 q^{95} - 2 q^{97} + ( - i + 1) q^{98} + (3 i - 3) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 4 q^{5} + 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 4 q^{5} + 4 q^{8} - 8 q^{10} + 2 q^{11} + 2 q^{14} - 8 q^{16} - 4 q^{17} - 6 q^{18} + 4 q^{19} + 8 q^{20} - 4 q^{22} - 4 q^{28} + 14 q^{29} - 16 q^{31} + 8 q^{32} + 4 q^{34} + 4 q^{35} + 12 q^{36} - 10 q^{37} - 2 q^{43} + 4 q^{44} - 12 q^{45} + 12 q^{46} - 24 q^{47} - 2 q^{49} - 6 q^{50} - 2 q^{53} + 4 q^{56} + 16 q^{59} + 12 q^{61} + 16 q^{62} + 6 q^{63} + 6 q^{67} - 12 q^{72} + 20 q^{74} - 8 q^{76} + 2 q^{77} + 20 q^{79} - 16 q^{80} - 18 q^{81} - 20 q^{82} - 20 q^{83} - 8 q^{85} + 4 q^{86} - 24 q^{92} + 24 q^{94} + 16 q^{95} - 4 q^{97} + 2 q^{98} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(15\) \(17\) \(85\)
\(\chi(n)\) \(1\) \(1\) \(i\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1
1.00000i
1.00000i
−1.00000 1.00000i 0 2.00000i 2.00000 2.00000i 0 1.00000i 2.00000 2.00000i 3.00000i −4.00000
85.1 −1.00000 + 1.00000i 0 2.00000i 2.00000 + 2.00000i 0 1.00000i 2.00000 + 2.00000i 3.00000i −4.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 112.2.m.b 2
4.b odd 2 1 448.2.m.a 2
7.b odd 2 1 784.2.m.a 2
7.c even 3 2 784.2.x.d 4
7.d odd 6 2 784.2.x.e 4
8.b even 2 1 896.2.m.b 2
8.d odd 2 1 896.2.m.c 2
16.e even 4 1 inner 112.2.m.b 2
16.e even 4 1 896.2.m.b 2
16.f odd 4 1 448.2.m.a 2
16.f odd 4 1 896.2.m.c 2
32.g even 8 2 7168.2.a.b 2
32.h odd 8 2 7168.2.a.k 2
112.l odd 4 1 784.2.m.a 2
112.w even 12 2 784.2.x.d 4
112.x odd 12 2 784.2.x.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.m.b 2 1.a even 1 1 trivial
112.2.m.b 2 16.e even 4 1 inner
448.2.m.a 2 4.b odd 2 1
448.2.m.a 2 16.f odd 4 1
784.2.m.a 2 7.b odd 2 1
784.2.m.a 2 112.l odd 4 1
784.2.x.d 4 7.c even 3 2
784.2.x.d 4 112.w even 12 2
784.2.x.e 4 7.d odd 6 2
784.2.x.e 4 112.x odd 12 2
896.2.m.b 2 8.b even 2 1
896.2.m.b 2 16.e even 4 1
896.2.m.c 2 8.d odd 2 1
896.2.m.c 2 16.f odd 4 1
7168.2.a.b 2 32.g even 8 2
7168.2.a.k 2 32.h odd 8 2

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 4T + 8 \) Copy content Toggle raw display
$7$ \( T^{2} + 1 \) Copy content Toggle raw display
$11$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T + 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 4T + 8 \) Copy content Toggle raw display
$23$ \( T^{2} + 36 \) Copy content Toggle raw display
$29$ \( T^{2} - 14T + 98 \) Copy content Toggle raw display
$31$ \( (T + 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 10T + 50 \) Copy content Toggle raw display
$41$ \( T^{2} + 100 \) Copy content Toggle raw display
$43$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$47$ \( (T + 12)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$59$ \( T^{2} - 16T + 128 \) Copy content Toggle raw display
$61$ \( T^{2} - 12T + 72 \) Copy content Toggle raw display
$67$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 36 \) Copy content Toggle raw display
$79$ \( (T - 10)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 20T + 200 \) Copy content Toggle raw display
$89$ \( T^{2} + 196 \) Copy content Toggle raw display
$97$ \( (T + 2)^{2} \) Copy content Toggle raw display
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