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Properties

Label 112.6.i.d
Level $112$
Weight $6$
Character orbit 112.i
Analytic conductor $17.963$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [112,6,Mod(65,112)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(112, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("112.65");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 112.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(17.9629878191\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{79})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 79x^{2} + 6241 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (7 \beta_{2} + \beta_1 + 7) q^{3} + ( - 4 \beta_{3} + 35 \beta_{2} - 4 \beta_1) q^{5} + ( - 7 \beta_{3} + 42 \beta_{2} + 21) q^{7} + (14 \beta_{3} + 122 \beta_{2} + 14 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (7 \beta_{2} + \beta_1 + 7) q^{3} + ( - 4 \beta_{3} + 35 \beta_{2} - 4 \beta_1) q^{5} + ( - 7 \beta_{3} + 42 \beta_{2} + 21) q^{7} + (14 \beta_{3} + 122 \beta_{2} + 14 \beta_1) q^{9} + (31 \beta_{2} + 7 \beta_1 + 31) q^{11} + ( - 14 \beta_{3} + 910) q^{13} + (7 \beta_{3} + 1019) q^{15} + ( - 847 \beta_{2} - 22 \beta_1 - 847) q^{17} + (39 \beta_{3} - 413 \beta_{2} + 39 \beta_1) q^{19} + (42 \beta_{3} + 2359 \beta_{2} + \cdots + 2065) q^{21}+ \cdots + (1288 \beta_{3} - 34750) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 14 q^{3} - 70 q^{5} - 244 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 14 q^{3} - 70 q^{5} - 244 q^{9} + 62 q^{11} + 3640 q^{13} + 4076 q^{15} - 1694 q^{17} + 826 q^{19} + 3542 q^{21} - 2734 q^{23} - 6312 q^{25} - 14308 q^{27} - 5704 q^{29} - 2674 q^{31} - 4858 q^{33} + 13286 q^{35} + 9146 q^{37} + 21588 q^{39} + 12264 q^{41} + 32080 q^{43} + 26852 q^{45} - 25326 q^{47} + 56644 q^{49} + 25762 q^{51} - 14958 q^{53} + 31052 q^{55} - 37732 q^{57} - 1106 q^{59} + 28042 q^{61} - 77308 q^{63} - 28308 q^{65} - 102642 q^{67} - 188692 q^{69} + 22112 q^{71} - 35070 q^{73} - 132776 q^{75} + 27062 q^{77} + 101762 q^{79} + 23750 q^{81} + 89264 q^{83} + 7348 q^{85} - 170380 q^{87} + 75474 q^{89} + 123872 q^{91} + 53478 q^{93} + 127502 q^{95} - 16632 q^{97} - 139000 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 79x^{2} + 6241 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 79 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{3} ) / 79 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 79\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 79\beta_{3} ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(15\) \(17\) \(85\)
\(\chi(n)\) \(1\) \(\beta_{2}\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
−4.44410 − 7.69740i
4.44410 + 7.69740i
−4.44410 + 7.69740i
4.44410 − 7.69740i
0 −5.38819 − 9.33263i 0 −53.0528 + 91.8901i 0 −124.435 + 36.3731i 0 63.4347 − 109.872i 0
65.2 0 12.3882 + 21.4570i 0 18.0528 − 31.2683i 0 124.435 + 36.3731i 0 −185.435 + 321.182i 0
81.1 0 −5.38819 + 9.33263i 0 −53.0528 − 91.8901i 0 −124.435 − 36.3731i 0 63.4347 + 109.872i 0
81.2 0 12.3882 − 21.4570i 0 18.0528 + 31.2683i 0 124.435 − 36.3731i 0 −185.435 − 321.182i 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 112.6.i.d 4
4.b odd 2 1 14.6.c.a ✓ 4
7.c even 3 1 inner 112.6.i.d 4
7.c even 3 1 784.6.a.s 2
7.d odd 6 1 784.6.a.bb 2
12.b even 2 1 126.6.g.j 4
28.d even 2 1 98.6.c.e 4
28.f even 6 1 98.6.a.g 2
28.f even 6 1 98.6.c.e 4
28.g odd 6 1 14.6.c.a ✓ 4
28.g odd 6 1 98.6.a.h 2
84.j odd 6 1 882.6.a.bi 2
84.n even 6 1 126.6.g.j 4
84.n even 6 1 882.6.a.ba 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.6.c.a ✓ 4 4.b odd 2 1
14.6.c.a ✓ 4 28.g odd 6 1
98.6.a.g 2 28.f even 6 1
98.6.a.h 2 28.g odd 6 1
98.6.c.e 4 28.d even 2 1
98.6.c.e 4 28.f even 6 1
112.6.i.d 4 1.a even 1 1 trivial
112.6.i.d 4 7.c even 3 1 inner
126.6.g.j 4 12.b even 2 1
126.6.g.j 4 84.n even 6 1
784.6.a.s 2 7.c even 3 1
784.6.a.bb 2 7.d odd 6 1
882.6.a.ba 2 84.n even 6 1
882.6.a.bi 2 84.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 14T_{3}^{3} + 463T_{3}^{2} + 3738T_{3} + 71289 \) acting on \(S_{6}^{\mathrm{new}}(112, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 14 T^{3} + \cdots + 71289 \) Copy content Toggle raw display
$5$ \( T^{4} + 70 T^{3} + \cdots + 14676561 \) Copy content Toggle raw display
$7$ \( T^{4} - 28322 T^{2} + 282475249 \) Copy content Toggle raw display
$11$ \( T^{4} - 62 T^{3} + \cdots + 210917529 \) Copy content Toggle raw display
$13$ \( (T^{2} - 1820 T + 766164)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 318620736225 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 96141544489 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 6792210678969 \) Copy content Toggle raw display
$29$ \( (T^{2} + 2852 T - 15866028)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 691673325561 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 252667333533169 \) Copy content Toggle raw display
$41$ \( (T^{2} - 6132 T + 8842932)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 16040 T - 78380144)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 24\!\cdots\!25 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 85\!\cdots\!81 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 868886159765625 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 38\!\cdots\!25 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 65\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( (T^{2} - 11056 T - 177793920)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 22\!\cdots\!81 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 62\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( (T^{2} - 44632 T - 5608638000)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 36\!\cdots\!25 \) Copy content Toggle raw display
$97$ \( (T^{2} + 8316 T - 349929580)^{2} \) Copy content Toggle raw display
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