Properties

Label 112.3.d.a
Level $112$
Weight $3$
Character orbit 112.d
Analytic conductor $3.052$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 112.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.05177896084\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-7}) \)
Defining polynomial: \(x^{2} - x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -2 \beta q^{3} + 8 q^{5} + \beta q^{7} -19 q^{9} +O(q^{10})\) \( q -2 \beta q^{3} + 8 q^{5} + \beta q^{7} -19 q^{9} -4 \beta q^{11} -4 q^{13} -16 \beta q^{15} -2 q^{17} + 10 \beta q^{19} + 14 q^{21} + 8 \beta q^{23} + 39 q^{25} + 20 \beta q^{27} + 14 q^{29} -12 \beta q^{31} -56 q^{33} + 8 \beta q^{35} + 14 q^{37} + 8 \beta q^{39} + 46 q^{41} + 4 \beta q^{43} -152 q^{45} + 12 \beta q^{47} -7 q^{49} + 4 \beta q^{51} -22 q^{53} -32 \beta q^{55} + 140 q^{57} + 34 \beta q^{59} + 48 q^{61} -19 \beta q^{63} -32 q^{65} -24 \beta q^{67} + 112 q^{69} + 32 \beta q^{71} -110 q^{73} -78 \beta q^{75} + 28 q^{77} -48 \beta q^{79} + 109 q^{81} -14 \beta q^{83} -16 q^{85} -28 \beta q^{87} -134 q^{89} -4 \beta q^{91} -168 q^{93} + 80 \beta q^{95} -178 q^{97} + 76 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 16q^{5} - 38q^{9} + O(q^{10}) \) \( 2q + 16q^{5} - 38q^{9} - 8q^{13} - 4q^{17} + 28q^{21} + 78q^{25} + 28q^{29} - 112q^{33} + 28q^{37} + 92q^{41} - 304q^{45} - 14q^{49} - 44q^{53} + 280q^{57} + 96q^{61} - 64q^{65} + 224q^{69} - 220q^{73} + 56q^{77} + 218q^{81} - 32q^{85} - 268q^{89} - 336q^{93} - 356q^{97} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1
0.500000 + 1.32288i
0.500000 1.32288i
0 5.29150i 0 8.00000 0 2.64575i 0 −19.0000 0
15.2 0 5.29150i 0 8.00000 0 2.64575i 0 −19.0000 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 112.3.d.a 2
3.b odd 2 1 1008.3.m.a 2
4.b odd 2 1 inner 112.3.d.a 2
7.b odd 2 1 784.3.d.d 2
7.c even 3 2 784.3.r.k 4
7.d odd 6 2 784.3.r.m 4
8.b even 2 1 448.3.d.a 2
8.d odd 2 1 448.3.d.a 2
12.b even 2 1 1008.3.m.a 2
16.e even 4 2 1792.3.g.b 4
16.f odd 4 2 1792.3.g.b 4
28.d even 2 1 784.3.d.d 2
28.f even 6 2 784.3.r.m 4
28.g odd 6 2 784.3.r.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.3.d.a 2 1.a even 1 1 trivial
112.3.d.a 2 4.b odd 2 1 inner
448.3.d.a 2 8.b even 2 1
448.3.d.a 2 8.d odd 2 1
784.3.d.d 2 7.b odd 2 1
784.3.d.d 2 28.d even 2 1
784.3.r.k 4 7.c even 3 2
784.3.r.k 4 28.g odd 6 2
784.3.r.m 4 7.d odd 6 2
784.3.r.m 4 28.f even 6 2
1008.3.m.a 2 3.b odd 2 1
1008.3.m.a 2 12.b even 2 1
1792.3.g.b 4 16.e even 4 2
1792.3.g.b 4 16.f odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 28 \) acting on \(S_{3}^{\mathrm{new}}(112, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 28 + T^{2} \)
$5$ \( ( -8 + T )^{2} \)
$7$ \( 7 + T^{2} \)
$11$ \( 112 + T^{2} \)
$13$ \( ( 4 + T )^{2} \)
$17$ \( ( 2 + T )^{2} \)
$19$ \( 700 + T^{2} \)
$23$ \( 448 + T^{2} \)
$29$ \( ( -14 + T )^{2} \)
$31$ \( 1008 + T^{2} \)
$37$ \( ( -14 + T )^{2} \)
$41$ \( ( -46 + T )^{2} \)
$43$ \( 112 + T^{2} \)
$47$ \( 1008 + T^{2} \)
$53$ \( ( 22 + T )^{2} \)
$59$ \( 8092 + T^{2} \)
$61$ \( ( -48 + T )^{2} \)
$67$ \( 4032 + T^{2} \)
$71$ \( 7168 + T^{2} \)
$73$ \( ( 110 + T )^{2} \)
$79$ \( 16128 + T^{2} \)
$83$ \( 1372 + T^{2} \)
$89$ \( ( 134 + T )^{2} \)
$97$ \( ( 178 + T )^{2} \)
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