Properties

Label 1110.6.a.a
Level $1110$
Weight $6$
Character orbit 1110.a
Self dual yes
Analytic conductor $178.026$
Analytic rank $1$
Dimension $1$
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] = [1110,6,Mod(1,1110)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1110, 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("1110.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1110.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: \(178.026039992\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
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)
 
\(f(q)\) \(=\) \( q + 4 q^{2} - 9 q^{3} + 16 q^{4} + 25 q^{5} - 36 q^{6} + 33 q^{7} + 64 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{2} - 9 q^{3} + 16 q^{4} + 25 q^{5} - 36 q^{6} + 33 q^{7} + 64 q^{8} + 81 q^{9} + 100 q^{10} + 39 q^{11} - 144 q^{12} - 314 q^{13} + 132 q^{14} - 225 q^{15} + 256 q^{16} - 1949 q^{17} + 324 q^{18} + 922 q^{19} + 400 q^{20} - 297 q^{21} + 156 q^{22} + 3844 q^{23} - 576 q^{24} + 625 q^{25} - 1256 q^{26} - 729 q^{27} + 528 q^{28} - 8313 q^{29} - 900 q^{30} + 5985 q^{31} + 1024 q^{32} - 351 q^{33} - 7796 q^{34} + 825 q^{35} + 1296 q^{36} - 1369 q^{37} + 3688 q^{38} + 2826 q^{39} + 1600 q^{40} - 13607 q^{41} - 1188 q^{42} + 847 q^{43} + 624 q^{44} + 2025 q^{45} + 15376 q^{46} + 2904 q^{47} - 2304 q^{48} - 15718 q^{49} + 2500 q^{50} + 17541 q^{51} - 5024 q^{52} - 33851 q^{53} - 2916 q^{54} + 975 q^{55} + 2112 q^{56} - 8298 q^{57} - 33252 q^{58} - 2186 q^{59} - 3600 q^{60} + 19893 q^{61} + 23940 q^{62} + 2673 q^{63} + 4096 q^{64} - 7850 q^{65} - 1404 q^{66} - 18596 q^{67} - 31184 q^{68} - 34596 q^{69} + 3300 q^{70} - 17740 q^{71} + 5184 q^{72} - 44536 q^{73} - 5476 q^{74} - 5625 q^{75} + 14752 q^{76} + 1287 q^{77} + 11304 q^{78} + 79732 q^{79} + 6400 q^{80} + 6561 q^{81} - 54428 q^{82} + 36254 q^{83} - 4752 q^{84} - 48725 q^{85} + 3388 q^{86} + 74817 q^{87} + 2496 q^{88} - 57970 q^{89} + 8100 q^{90} - 10362 q^{91} + 61504 q^{92} - 53865 q^{93} + 11616 q^{94} + 23050 q^{95} - 9216 q^{96} + 85531 q^{97} - 62872 q^{98} + 3159 q^{99}+O(q^{100}) \) 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
0
4.00000 −9.00000 16.0000 25.0000 −36.0000 33.0000 64.0000 81.0000 100.000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(-1\)
\(37\) \(1\)

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 1110.6.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.6.a.a 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} - 33 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(1110))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 4 \) Copy content Toggle raw display
$3$ \( T + 9 \) Copy content Toggle raw display
$5$ \( T - 25 \) Copy content Toggle raw display
$7$ \( T - 33 \) Copy content Toggle raw display
$11$ \( T - 39 \) Copy content Toggle raw display
$13$ \( T + 314 \) Copy content Toggle raw display
$17$ \( T + 1949 \) Copy content Toggle raw display
$19$ \( T - 922 \) Copy content Toggle raw display
$23$ \( T - 3844 \) Copy content Toggle raw display
$29$ \( T + 8313 \) Copy content Toggle raw display
$31$ \( T - 5985 \) Copy content Toggle raw display
$37$ \( T + 1369 \) Copy content Toggle raw display
$41$ \( T + 13607 \) Copy content Toggle raw display
$43$ \( T - 847 \) Copy content Toggle raw display
$47$ \( T - 2904 \) Copy content Toggle raw display
$53$ \( T + 33851 \) Copy content Toggle raw display
$59$ \( T + 2186 \) Copy content Toggle raw display
$61$ \( T - 19893 \) Copy content Toggle raw display
$67$ \( T + 18596 \) Copy content Toggle raw display
$71$ \( T + 17740 \) Copy content Toggle raw display
$73$ \( T + 44536 \) Copy content Toggle raw display
$79$ \( T - 79732 \) Copy content Toggle raw display
$83$ \( T - 36254 \) Copy content Toggle raw display
$89$ \( T + 57970 \) Copy content Toggle raw display
$97$ \( T - 85531 \) Copy content Toggle raw display
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