Properties

Label 112.4.i.b
Level $112$
Weight $4$
Character orbit 112.i
Analytic conductor $6.608$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 112.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.60821392064\)
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 14)
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.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{3} - 7 \zeta_{6} q^{5} + (18 \zeta_{6} + 1) q^{7} + 26 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} - 1) q^{3} - 7 \zeta_{6} q^{5} + (18 \zeta_{6} + 1) q^{7} + 26 \zeta_{6} q^{9} + ( - 35 \zeta_{6} + 35) q^{11} + 66 q^{13} + 7 q^{15} + (59 \zeta_{6} - 59) q^{17} + 137 \zeta_{6} q^{19} + (\zeta_{6} - 19) q^{21} - 7 \zeta_{6} q^{23} + ( - 76 \zeta_{6} + 76) q^{25} - 53 q^{27} + 106 q^{29} + ( - 75 \zeta_{6} + 75) q^{31} + 35 \zeta_{6} q^{33} + ( - 133 \zeta_{6} + 126) q^{35} - 11 \zeta_{6} q^{37} + (66 \zeta_{6} - 66) q^{39} - 498 q^{41} - 260 q^{43} + ( - 182 \zeta_{6} + 182) q^{45} - 171 \zeta_{6} q^{47} + (360 \zeta_{6} - 323) q^{49} - 59 \zeta_{6} q^{51} + ( - 417 \zeta_{6} + 417) q^{53} - 245 q^{55} - 137 q^{57} + (17 \zeta_{6} - 17) q^{59} - 51 \zeta_{6} q^{61} + (494 \zeta_{6} - 468) q^{63} - 462 \zeta_{6} q^{65} + ( - 439 \zeta_{6} + 439) q^{67} + 7 q^{69} + 784 q^{71} + (295 \zeta_{6} - 295) q^{73} + 76 \zeta_{6} q^{75} + ( - 35 \zeta_{6} + 665) q^{77} - 495 \zeta_{6} q^{79} + (649 \zeta_{6} - 649) q^{81} - 932 q^{83} + 413 q^{85} + (106 \zeta_{6} - 106) q^{87} + 873 \zeta_{6} q^{89} + (1188 \zeta_{6} + 66) q^{91} + 75 \zeta_{6} q^{93} + ( - 959 \zeta_{6} + 959) q^{95} - 290 q^{97} + 910 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} - 7 q^{5} + 20 q^{7} + 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} - 7 q^{5} + 20 q^{7} + 26 q^{9} + 35 q^{11} + 132 q^{13} + 14 q^{15} - 59 q^{17} + 137 q^{19} - 37 q^{21} - 7 q^{23} + 76 q^{25} - 106 q^{27} + 212 q^{29} + 75 q^{31} + 35 q^{33} + 119 q^{35} - 11 q^{37} - 66 q^{39} - 996 q^{41} - 520 q^{43} + 182 q^{45} - 171 q^{47} - 286 q^{49} - 59 q^{51} + 417 q^{53} - 490 q^{55} - 274 q^{57} - 17 q^{59} - 51 q^{61} - 442 q^{63} - 462 q^{65} + 439 q^{67} + 14 q^{69} + 1568 q^{71} - 295 q^{73} + 76 q^{75} + 1295 q^{77} - 495 q^{79} - 649 q^{81} - 1864 q^{83} + 826 q^{85} - 106 q^{87} + 873 q^{89} + 1320 q^{91} + 75 q^{93} + 959 q^{95} - 580 q^{97} + 1820 q^{99}+O(q^{100}) \) 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 −3.50000 + 6.06218i 0 10.0000 15.5885i 0 13.0000 22.5167i 0
81.1 0 −0.500000 + 0.866025i 0 −3.50000 6.06218i 0 10.0000 + 15.5885i 0 13.0000 + 22.5167i 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 112.4.i.b 2
4.b odd 2 1 14.4.c.b 2
7.c even 3 1 inner 112.4.i.b 2
7.c even 3 1 784.4.a.l 1
7.d odd 6 1 784.4.a.j 1
8.b even 2 1 448.4.i.d 2
8.d odd 2 1 448.4.i.c 2
12.b even 2 1 126.4.g.c 2
20.d odd 2 1 350.4.e.b 2
20.e even 4 2 350.4.j.d 4
28.d even 2 1 98.4.c.e 2
28.f even 6 1 98.4.a.c 1
28.f even 6 1 98.4.c.e 2
28.g odd 6 1 14.4.c.b 2
28.g odd 6 1 98.4.a.b 1
56.k odd 6 1 448.4.i.c 2
56.p even 6 1 448.4.i.d 2
84.h odd 2 1 882.4.g.d 2
84.j odd 6 1 882.4.a.p 1
84.j odd 6 1 882.4.g.d 2
84.n even 6 1 126.4.g.c 2
84.n even 6 1 882.4.a.k 1
140.p odd 6 1 350.4.e.b 2
140.p odd 6 1 2450.4.a.bh 1
140.s even 6 1 2450.4.a.bf 1
140.w even 12 2 350.4.j.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.c.b 2 4.b odd 2 1
14.4.c.b 2 28.g odd 6 1
98.4.a.b 1 28.g odd 6 1
98.4.a.c 1 28.f even 6 1
98.4.c.e 2 28.d even 2 1
98.4.c.e 2 28.f even 6 1
112.4.i.b 2 1.a even 1 1 trivial
112.4.i.b 2 7.c even 3 1 inner
126.4.g.c 2 12.b even 2 1
126.4.g.c 2 84.n even 6 1
350.4.e.b 2 20.d odd 2 1
350.4.e.b 2 140.p odd 6 1
350.4.j.d 4 20.e even 4 2
350.4.j.d 4 140.w even 12 2
448.4.i.c 2 8.d odd 2 1
448.4.i.c 2 56.k odd 6 1
448.4.i.d 2 8.b even 2 1
448.4.i.d 2 56.p even 6 1
784.4.a.j 1 7.d odd 6 1
784.4.a.l 1 7.c even 3 1
882.4.a.k 1 84.n even 6 1
882.4.a.p 1 84.j odd 6 1
882.4.g.d 2 84.h odd 2 1
882.4.g.d 2 84.j odd 6 1
2450.4.a.bf 1 140.s even 6 1
2450.4.a.bh 1 140.p odd 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 \(S_{4}^{\mathrm{new}}(112, [\chi])\). 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
$5$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$7$ \( T^{2} - 20T + 343 \) Copy content Toggle raw display
$11$ \( T^{2} - 35T + 1225 \) Copy content Toggle raw display
$13$ \( (T - 66)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 59T + 3481 \) Copy content Toggle raw display
$19$ \( T^{2} - 137T + 18769 \) Copy content Toggle raw display
$23$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$29$ \( (T - 106)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 75T + 5625 \) Copy content Toggle raw display
$37$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$41$ \( (T + 498)^{2} \) Copy content Toggle raw display
$43$ \( (T + 260)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 171T + 29241 \) Copy content Toggle raw display
$53$ \( T^{2} - 417T + 173889 \) Copy content Toggle raw display
$59$ \( T^{2} + 17T + 289 \) Copy content Toggle raw display
$61$ \( T^{2} + 51T + 2601 \) Copy content Toggle raw display
$67$ \( T^{2} - 439T + 192721 \) Copy content Toggle raw display
$71$ \( (T - 784)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 295T + 87025 \) Copy content Toggle raw display
$79$ \( T^{2} + 495T + 245025 \) Copy content Toggle raw display
$83$ \( (T + 932)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 873T + 762129 \) Copy content Toggle raw display
$97$ \( (T + 290)^{2} \) Copy content Toggle raw display
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