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: \(x^{2} - x + 1\)
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.

\(f(q)\) \(=\) \( q + ( -1 + \zeta_{6} ) q^{3} -7 \zeta_{6} q^{5} + ( 1 + 18 \zeta_{6} ) q^{7} + 26 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( -1 + \zeta_{6} ) q^{3} -7 \zeta_{6} q^{5} + ( 1 + 18 \zeta_{6} ) q^{7} + 26 \zeta_{6} q^{9} + ( 35 - 35 \zeta_{6} ) q^{11} + 66 q^{13} + 7 q^{15} + ( -59 + 59 \zeta_{6} ) q^{17} + 137 \zeta_{6} q^{19} + ( -19 + \zeta_{6} ) q^{21} -7 \zeta_{6} q^{23} + ( 76 - 76 \zeta_{6} ) q^{25} -53 q^{27} + 106 q^{29} + ( 75 - 75 \zeta_{6} ) q^{31} + 35 \zeta_{6} q^{33} + ( 126 - 133 \zeta_{6} ) q^{35} -11 \zeta_{6} q^{37} + ( -66 + 66 \zeta_{6} ) q^{39} -498 q^{41} -260 q^{43} + ( 182 - 182 \zeta_{6} ) q^{45} -171 \zeta_{6} q^{47} + ( -323 + 360 \zeta_{6} ) q^{49} -59 \zeta_{6} q^{51} + ( 417 - 417 \zeta_{6} ) q^{53} -245 q^{55} -137 q^{57} + ( -17 + 17 \zeta_{6} ) q^{59} -51 \zeta_{6} q^{61} + ( -468 + 494 \zeta_{6} ) q^{63} -462 \zeta_{6} q^{65} + ( 439 - 439 \zeta_{6} ) q^{67} + 7 q^{69} + 784 q^{71} + ( -295 + 295 \zeta_{6} ) q^{73} + 76 \zeta_{6} q^{75} + ( 665 - 35 \zeta_{6} ) q^{77} -495 \zeta_{6} q^{79} + ( -649 + 649 \zeta_{6} ) q^{81} -932 q^{83} + 413 q^{85} + ( -106 + 106 \zeta_{6} ) q^{87} + 873 \zeta_{6} q^{89} + ( 66 + 1188 \zeta_{6} ) q^{91} + 75 \zeta_{6} q^{93} + ( 959 - 959 \zeta_{6} ) q^{95} -290 q^{97} + 910 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{3} - 7q^{5} + 20q^{7} + 26q^{9} + O(q^{10}) \) \( 2q - q^{3} - 7q^{5} + 20q^{7} + 26q^{9} + 35q^{11} + 132q^{13} + 14q^{15} - 59q^{17} + 137q^{19} - 37q^{21} - 7q^{23} + 76q^{25} - 106q^{27} + 212q^{29} + 75q^{31} + 35q^{33} + 119q^{35} - 11q^{37} - 66q^{39} - 996q^{41} - 520q^{43} + 182q^{45} - 171q^{47} - 286q^{49} - 59q^{51} + 417q^{53} - 490q^{55} - 274q^{57} - 17q^{59} - 51q^{61} - 442q^{63} - 462q^{65} + 439q^{67} + 14q^{69} + 1568q^{71} - 295q^{73} + 76q^{75} + 1295q^{77} - 495q^{79} - 649q^{81} - 1864q^{83} + 826q^{85} - 106q^{87} + 873q^{89} + 1320q^{91} + 75q^{93} + 959q^{95} - 580q^{97} + 1820q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(15\) \(17\) \(85\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(1\)

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 \(S_{4}^{\mathrm{new}}(112, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + T - 26 T^{2} + 27 T^{3} + 729 T^{4} \)
$5$ \( 1 + 7 T - 76 T^{2} + 875 T^{3} + 15625 T^{4} \)
$7$ \( 1 - 20 T + 343 T^{2} \)
$11$ \( 1 - 35 T - 106 T^{2} - 46585 T^{3} + 1771561 T^{4} \)
$13$ \( ( 1 - 66 T + 2197 T^{2} )^{2} \)
$17$ \( 1 + 59 T - 1432 T^{2} + 289867 T^{3} + 24137569 T^{4} \)
$19$ \( 1 - 137 T + 11910 T^{2} - 939683 T^{3} + 47045881 T^{4} \)
$23$ \( 1 + 7 T - 12118 T^{2} + 85169 T^{3} + 148035889 T^{4} \)
$29$ \( ( 1 - 106 T + 24389 T^{2} )^{2} \)
$31$ \( 1 - 75 T - 24166 T^{2} - 2234325 T^{3} + 887503681 T^{4} \)
$37$ \( 1 + 11 T - 50532 T^{2} + 557183 T^{3} + 2565726409 T^{4} \)
$41$ \( ( 1 + 498 T + 68921 T^{2} )^{2} \)
$43$ \( ( 1 + 260 T + 79507 T^{2} )^{2} \)
$47$ \( 1 + 171 T - 74582 T^{2} + 17753733 T^{3} + 10779215329 T^{4} \)
$53$ \( 1 - 417 T + 25012 T^{2} - 62081709 T^{3} + 22164361129 T^{4} \)
$59$ \( 1 + 17 T - 205090 T^{2} + 3491443 T^{3} + 42180533641 T^{4} \)
$61$ \( 1 + 51 T - 224380 T^{2} + 11576031 T^{3} + 51520374361 T^{4} \)
$67$ \( 1 - 439 T - 108042 T^{2} - 132034957 T^{3} + 90458382169 T^{4} \)
$71$ \( ( 1 - 784 T + 357911 T^{2} )^{2} \)
$73$ \( 1 + 295 T - 301992 T^{2} + 114760015 T^{3} + 151334226289 T^{4} \)
$79$ \( 1 + 495 T - 248014 T^{2} + 244054305 T^{3} + 243087455521 T^{4} \)
$83$ \( ( 1 + 932 T + 571787 T^{2} )^{2} \)
$89$ \( 1 - 873 T + 57160 T^{2} - 615437937 T^{3} + 496981290961 T^{4} \)
$97$ \( ( 1 + 290 T + 912673 T^{2} )^{2} \)
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