Properties

Label 56.6.a.c
Level $56$
Weight $6$
Character orbit 56.a
Self dual yes
Analytic conductor $8.981$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [56,6,Mod(1,56)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(56, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("56.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 56 = 2^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 56.a (trivial)

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: yes
Analytic conductor: \(8.98149390953\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{177}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 44 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \sqrt{177}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 13) q^{3} + ( - 5 \beta - 31) q^{5} + 49 q^{7} + (26 \beta + 103) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 13) q^{3} + ( - 5 \beta - 31) q^{5} + 49 q^{7} + (26 \beta + 103) q^{9} + ( - 6 \beta + 486) q^{11} + ( - 71 \beta + 39) q^{13} + (96 \beta + 1288) q^{15} + (6 \beta + 280) q^{17} + (37 \beta + 1321) q^{19} + ( - 49 \beta - 637) q^{21} + ( - 248 \beta + 1136) q^{23} + (310 \beta + 2261) q^{25} + ( - 198 \beta - 2782) q^{27} + ( - 14 \beta - 3904) q^{29} + (130 \beta + 2722) q^{31} + ( - 408 \beta - 5256) q^{33} + ( - 245 \beta - 1519) q^{35} + (650 \beta + 288) q^{37} + (884 \beta + 12060) q^{39} + (378 \beta - 8444) q^{41} + ( - 338 \beta - 4198) q^{43} + ( - 1321 \beta - 26203) q^{45} + (1238 \beta - 2266) q^{47} + 2401 q^{49} + ( - 358 \beta - 4702) q^{51} + ( - 1152 \beta + 710) q^{53} + ( - 2244 \beta - 9756) q^{55} + ( - 1802 \beta - 23722) q^{57} + (325 \beta + 17073) q^{59} + (3347 \beta + 9553) q^{61} + (1274 \beta + 5047) q^{63} + (2006 \beta + 61626) q^{65} + (1240 \beta + 28476) q^{67} + (2088 \beta + 29128) q^{69} + (700 \beta - 3612) q^{71} + ( - 1260 \beta + 64414) q^{73} + ( - 6291 \beta - 84263) q^{75} + ( - 294 \beta + 23814) q^{77} + ( - 1548 \beta + 26404) q^{79} + ( - 962 \beta + 46183) q^{81} + (5209 \beta - 42243) q^{83} + ( - 1586 \beta - 13990) q^{85} + (4086 \beta + 53230) q^{87} + ( - 2312 \beta - 65486) q^{89} + ( - 3479 \beta + 1911) q^{91} + ( - 4412 \beta - 58396) q^{93} + ( - 7752 \beta - 73696) q^{95} + (350 \beta + 97312) q^{97} + (12018 \beta + 22446) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 26 q^{3} - 62 q^{5} + 98 q^{7} + 206 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 26 q^{3} - 62 q^{5} + 98 q^{7} + 206 q^{9} + 972 q^{11} + 78 q^{13} + 2576 q^{15} + 560 q^{17} + 2642 q^{19} - 1274 q^{21} + 2272 q^{23} + 4522 q^{25} - 5564 q^{27} - 7808 q^{29} + 5444 q^{31} - 10512 q^{33} - 3038 q^{35} + 576 q^{37} + 24120 q^{39} - 16888 q^{41} - 8396 q^{43} - 52406 q^{45} - 4532 q^{47} + 4802 q^{49} - 9404 q^{51} + 1420 q^{53} - 19512 q^{55} - 47444 q^{57} + 34146 q^{59} + 19106 q^{61} + 10094 q^{63} + 123252 q^{65} + 56952 q^{67} + 58256 q^{69} - 7224 q^{71} + 128828 q^{73} - 168526 q^{75} + 47628 q^{77} + 52808 q^{79} + 92366 q^{81} - 84486 q^{83} - 27980 q^{85} + 106460 q^{87} - 130972 q^{89} + 3822 q^{91} - 116792 q^{93} - 147392 q^{95} + 194624 q^{97} + 44892 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
7.15207
−6.15207
0 −26.3041 0 −97.5207 0 49.0000 0 448.908 0
1.2 0 0.304135 0 35.5207 0 49.0000 0 −242.908 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(7\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 56.6.a.c 2
3.b odd 2 1 504.6.a.s 2
4.b odd 2 1 112.6.a.k 2
7.b odd 2 1 392.6.a.f 2
7.c even 3 2 392.6.i.l 4
7.d odd 6 2 392.6.i.g 4
8.b even 2 1 448.6.a.z 2
8.d odd 2 1 448.6.a.q 2
12.b even 2 1 1008.6.a.bt 2
28.d even 2 1 784.6.a.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.6.a.c 2 1.a even 1 1 trivial
112.6.a.k 2 4.b odd 2 1
392.6.a.f 2 7.b odd 2 1
392.6.i.g 4 7.d odd 6 2
392.6.i.l 4 7.c even 3 2
448.6.a.q 2 8.d odd 2 1
448.6.a.z 2 8.b even 2 1
504.6.a.s 2 3.b odd 2 1
784.6.a.p 2 28.d even 2 1
1008.6.a.bt 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 26T_{3} - 8 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(56))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 26T - 8 \) Copy content Toggle raw display
$5$ \( T^{2} + 62T - 3464 \) Copy content Toggle raw display
$7$ \( (T - 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 972T + 229824 \) Copy content Toggle raw display
$13$ \( T^{2} - 78T - 890736 \) Copy content Toggle raw display
$17$ \( T^{2} - 560T + 72028 \) Copy content Toggle raw display
$19$ \( T^{2} - 2642 T + 1502728 \) Copy content Toggle raw display
$23$ \( T^{2} - 2272 T - 9595712 \) Copy content Toggle raw display
$29$ \( T^{2} + 7808 T + 15206524 \) Copy content Toggle raw display
$31$ \( T^{2} - 5444 T + 4417984 \) Copy content Toggle raw display
$37$ \( T^{2} - 576 T - 74699556 \) Copy content Toggle raw display
$41$ \( T^{2} + 16888 T + 46010668 \) Copy content Toggle raw display
$43$ \( T^{2} + 8396 T - 2597984 \) Copy content Toggle raw display
$47$ \( T^{2} + 4532 T - 266143232 \) Copy content Toggle raw display
$53$ \( T^{2} - 1420 T - 234393308 \) Copy content Toggle raw display
$59$ \( T^{2} - 34146 T + 272791704 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots - 1891566584 \) Copy content Toggle raw display
$67$ \( T^{2} - 56952 T + 538727376 \) Copy content Toggle raw display
$71$ \( T^{2} + 7224 T - 73683456 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 3868158196 \) Copy content Toggle raw display
$79$ \( T^{2} - 52808 T + 273025408 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 3018190488 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 3342290308 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 9447942844 \) Copy content Toggle raw display
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