Properties

Label 110.2.f.c
Level $110$
Weight $2$
Character orbit 110.f
Analytic conductor $0.878$
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] = [110,2,Mod(43,110)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(110, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("110.43");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 110 = 2 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 110.f (of order \(4\), degree \(2\), 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: \(0.878354422234\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{8} q^{2} + (\zeta_{8}^{2} + \zeta_{8} + 1) q^{3} + \zeta_{8}^{2} q^{4} + ( - \zeta_{8}^{3} - 2 \zeta_{8}) q^{5} + (\zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8}) q^{6} - 3 \zeta_{8} q^{7} + \zeta_{8}^{3} q^{8} + (2 \zeta_{8}^{3} + 2 \zeta_{8}) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{8} q^{2} + (\zeta_{8}^{2} + \zeta_{8} + 1) q^{3} + \zeta_{8}^{2} q^{4} + ( - \zeta_{8}^{3} - 2 \zeta_{8}) q^{5} + (\zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8}) q^{6} - 3 \zeta_{8} q^{7} + \zeta_{8}^{3} q^{8} + (2 \zeta_{8}^{3} + 2 \zeta_{8}) q^{9} + ( - 2 \zeta_{8}^{2} + 1) q^{10} + (\zeta_{8}^{3} - 3 \zeta_{8}^{2} - \zeta_{8}) q^{11} + (\zeta_{8}^{3} + \zeta_{8}^{2} - 1) q^{12} + (3 \zeta_{8}^{2} - 3) q^{13} - 3 \zeta_{8}^{2} q^{14} + ( - 3 \zeta_{8}^{3} - 2 \zeta_{8}^{2} - \zeta_{8} + 1) q^{15} - q^{16} + (3 \zeta_{8}^{2} + 3 \zeta_{8} + 3) q^{17} + (2 \zeta_{8}^{2} - 2) q^{18} + 3 q^{19} + ( - 2 \zeta_{8}^{3} + \zeta_{8}) q^{20} + ( - 3 \zeta_{8}^{3} - 3 \zeta_{8}^{2} - 3 \zeta_{8}) q^{21} + ( - 3 \zeta_{8}^{3} - \zeta_{8}^{2} - 1) q^{22} + ( - 3 \zeta_{8}^{2} + 4 \zeta_{8} - 3) q^{23} + (\zeta_{8}^{3} - \zeta_{8} - 1) q^{24} + (3 \zeta_{8}^{2} - 4) q^{25} + (3 \zeta_{8}^{3} - 3 \zeta_{8}) q^{26} + (\zeta_{8}^{3} - \zeta_{8}^{2} + 1) q^{27} - 3 \zeta_{8}^{3} q^{28} + ( - 3 \zeta_{8}^{3} + 3 \zeta_{8} - 3) q^{29} + ( - 2 \zeta_{8}^{3} - \zeta_{8}^{2} + \zeta_{8} + 3) q^{30} + (3 \zeta_{8}^{3} - 3 \zeta_{8} - 3) q^{31} - \zeta_{8} q^{32} + ( - 3 \zeta_{8}^{3} - 4 \zeta_{8}^{2} - 2 \zeta_{8} + 2) q^{33} + (3 \zeta_{8}^{3} + 3 \zeta_{8}^{2} + 3 \zeta_{8}) q^{34} + (6 \zeta_{8}^{2} - 3) q^{35} + (2 \zeta_{8}^{3} - 2 \zeta_{8}) q^{36} + ( - 3 \zeta_{8}^{3} + 2 \zeta_{8}^{2} - 2) q^{37} + 3 \zeta_{8} q^{38} + (3 \zeta_{8}^{3} - 3 \zeta_{8} - 6) q^{39} + (\zeta_{8}^{2} + 2) q^{40} + (3 \zeta_{8}^{3} - 6 \zeta_{8}^{2} + 3 \zeta_{8}) q^{41} + ( - 3 \zeta_{8}^{3} - 3 \zeta_{8}^{2} + 3) q^{42} + (6 \zeta_{8}^{3} - 3 \zeta_{8}^{2} + 3) q^{43} + ( - \zeta_{8}^{3} - \zeta_{8} + 3) q^{44} + ( - 2 \zeta_{8}^{2} + 6) q^{45} + ( - 3 \zeta_{8}^{3} + 4 \zeta_{8}^{2} - 3 \zeta_{8}) q^{46} + ( - 2 \zeta_{8}^{3} - 3 \zeta_{8}^{2} + 3) q^{47} + ( - \zeta_{8}^{2} - \zeta_{8} - 1) q^{48} + 2 \zeta_{8}^{2} q^{49} + (3 \zeta_{8}^{3} - 4 \zeta_{8}) q^{50} + (6 \zeta_{8}^{3} + 9 \zeta_{8}^{2} + 6 \zeta_{8}) q^{51} + ( - 3 \zeta_{8}^{2} - 3) q^{52} + (6 \zeta_{8}^{2} + 5 \zeta_{8} + 6) q^{53} + ( - \zeta_{8}^{3} + \zeta_{8} - 1) q^{54} + (6 \zeta_{8}^{3} + 3 \zeta_{8}^{2} - 3 \zeta_{8} + 1) q^{55} + 3 q^{56} + (3 \zeta_{8}^{2} + 3 \zeta_{8} + 3) q^{57} + (3 \zeta_{8}^{2} - 3 \zeta_{8} + 3) q^{58} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{59} + ( - \zeta_{8}^{3} + \zeta_{8}^{2} + 3 \zeta_{8} + 2) q^{60} + ( - 3 \zeta_{8}^{3} - 3 \zeta_{8}^{2} - 3 \zeta_{8}) q^{61} + ( - 3 \zeta_{8}^{2} - 3 \zeta_{8} - 3) q^{62} + ( - 6 \zeta_{8}^{2} + 6) q^{63} - \zeta_{8}^{2} q^{64} + ( - 3 \zeta_{8}^{3} + 9 \zeta_{8}) q^{65} + ( - 4 \zeta_{8}^{3} - 2 \zeta_{8}^{2} + 2 \zeta_{8} + 3) q^{66} + (4 \zeta_{8}^{2} - 4) q^{67} + (3 \zeta_{8}^{3} + 3 \zeta_{8}^{2} - 3) q^{68} + (\zeta_{8}^{3} - 2 \zeta_{8}^{2} + \zeta_{8}) q^{69} + (6 \zeta_{8}^{3} - 3 \zeta_{8}) q^{70} + (3 \zeta_{8}^{3} - 3 \zeta_{8} + 3) q^{71} + ( - 2 \zeta_{8}^{2} - 2) q^{72} + (6 \zeta_{8}^{2} - 6) q^{73} + (2 \zeta_{8}^{3} - 2 \zeta_{8} + 3) q^{74} + (3 \zeta_{8}^{3} - \zeta_{8}^{2} - 4 \zeta_{8} - 7) q^{75} + 3 \zeta_{8}^{2} q^{76} + (9 \zeta_{8}^{3} + 3 \zeta_{8}^{2} + 3) q^{77} + ( - 3 \zeta_{8}^{2} - 6 \zeta_{8} - 3) q^{78} + (3 \zeta_{8}^{3} - 3 \zeta_{8} - 6) q^{79} + (\zeta_{8}^{3} + 2 \zeta_{8}) q^{80} + ( - 6 \zeta_{8}^{3} + 6 \zeta_{8} + 1) q^{81} + ( - 6 \zeta_{8}^{3} + 3 \zeta_{8}^{2} - 3) q^{82} + ( - 6 \zeta_{8}^{3} - 3 \zeta_{8}^{2} + 3) q^{83} + ( - 3 \zeta_{8}^{3} + 3 \zeta_{8} + 3) q^{84} + ( - 9 \zeta_{8}^{3} - 6 \zeta_{8}^{2} - 3 \zeta_{8} + 3) q^{85} + ( - 3 \zeta_{8}^{3} + 3 \zeta_{8} - 6) q^{86} + 3 \zeta_{8} q^{87} + ( - \zeta_{8}^{2} + 3 \zeta_{8} + 1) q^{88} + ( - 6 \zeta_{8}^{3} + 3 \zeta_{8}^{2} - 6 \zeta_{8}) q^{89} + ( - 2 \zeta_{8}^{3} + 6 \zeta_{8}) q^{90} + ( - 9 \zeta_{8}^{3} + 9 \zeta_{8}) q^{91} + (4 \zeta_{8}^{3} - 3 \zeta_{8}^{2} + 3) q^{92} + ( - 6 \zeta_{8}^{2} - 9 \zeta_{8} - 6) q^{93} + ( - 3 \zeta_{8}^{3} + 3 \zeta_{8} + 2) q^{94} + ( - 3 \zeta_{8}^{3} - 6 \zeta_{8}) q^{95} + ( - \zeta_{8}^{3} - \zeta_{8}^{2} - \zeta_{8}) q^{96} + (6 \zeta_{8}^{3} - 2 \zeta_{8}^{2} + 2) q^{97} + 2 \zeta_{8}^{3} q^{98} + ( - 6 \zeta_{8}^{3} - 4 \zeta_{8}^{2} + 6 \zeta_{8}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} + 4 q^{10} - 4 q^{12} - 12 q^{13} + 4 q^{15} - 4 q^{16} + 12 q^{17} - 8 q^{18} + 12 q^{19} - 4 q^{22} - 12 q^{23} - 4 q^{24} - 16 q^{25} + 4 q^{27} - 12 q^{29} + 12 q^{30} - 12 q^{31} + 8 q^{33} - 12 q^{35} - 8 q^{37} - 24 q^{39} + 8 q^{40} + 12 q^{42} + 12 q^{43} + 12 q^{44} + 24 q^{45} + 12 q^{47} - 4 q^{48} - 12 q^{52} + 24 q^{53} - 4 q^{54} + 4 q^{55} + 12 q^{56} + 12 q^{57} + 12 q^{58} + 8 q^{60} - 12 q^{62} + 24 q^{63} + 12 q^{66} - 16 q^{67} - 12 q^{68} + 12 q^{71} - 8 q^{72} - 24 q^{73} + 12 q^{74} - 28 q^{75} + 12 q^{77} - 12 q^{78} - 24 q^{79} + 4 q^{81} - 12 q^{82} + 12 q^{83} + 12 q^{84} + 12 q^{85} - 24 q^{86} + 4 q^{88} + 12 q^{92} - 24 q^{93} + 8 q^{94} + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/110\mathbb{Z}\right)^\times\).

\(n\) \(67\) \(101\)
\(\chi(n)\) \(-\zeta_{8}^{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
−0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i 0.292893 + 0.292893i 1.00000i 0.707107 + 2.12132i 0.414214i 2.12132 + 2.12132i 0.707107 0.707107i 2.82843i 1.00000 2.00000i
43.2 0.707107 + 0.707107i 1.70711 + 1.70711i 1.00000i −0.707107 2.12132i 2.41421i −2.12132 2.12132i −0.707107 + 0.707107i 2.82843i 1.00000 2.00000i
87.1 −0.707107 + 0.707107i 0.292893 0.292893i 1.00000i 0.707107 2.12132i 0.414214i 2.12132 2.12132i 0.707107 + 0.707107i 2.82843i 1.00000 + 2.00000i
87.2 0.707107 0.707107i 1.70711 1.70711i 1.00000i −0.707107 + 2.12132i 2.41421i −2.12132 + 2.12132i −0.707107 0.707107i 2.82843i 1.00000 + 2.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
55.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 110.2.f.c yes 4
3.b odd 2 1 990.2.m.d 4
4.b odd 2 1 880.2.bd.b 4
5.b even 2 1 550.2.f.b 4
5.c odd 4 1 110.2.f.b 4
5.c odd 4 1 550.2.f.a 4
11.b odd 2 1 110.2.f.b 4
15.e even 4 1 990.2.m.c 4
20.e even 4 1 880.2.bd.c 4
33.d even 2 1 990.2.m.c 4
44.c even 2 1 880.2.bd.c 4
55.d odd 2 1 550.2.f.a 4
55.e even 4 1 inner 110.2.f.c yes 4
55.e even 4 1 550.2.f.b 4
165.l odd 4 1 990.2.m.d 4
220.i odd 4 1 880.2.bd.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.2.f.b 4 5.c odd 4 1
110.2.f.b 4 11.b odd 2 1
110.2.f.c yes 4 1.a even 1 1 trivial
110.2.f.c yes 4 55.e even 4 1 inner
550.2.f.a 4 5.c odd 4 1
550.2.f.a 4 55.d odd 2 1
550.2.f.b 4 5.b even 2 1
550.2.f.b 4 55.e even 4 1
880.2.bd.b 4 4.b odd 2 1
880.2.bd.b 4 220.i odd 4 1
880.2.bd.c 4 20.e even 4 1
880.2.bd.c 4 44.c even 2 1
990.2.m.c 4 15.e even 4 1
990.2.m.c 4 33.d even 2 1
990.2.m.d 4 3.b odd 2 1
990.2.m.d 4 165.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(110, [\chi])\):

\( T_{3}^{4} - 4T_{3}^{3} + 8T_{3}^{2} - 4T_{3} + 1 \) Copy content Toggle raw display
\( T_{13}^{2} + 6T_{13} + 18 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} - 4 T^{3} + 8 T^{2} - 4 T + 1 \) Copy content Toggle raw display
$5$ \( T^{4} + 8T^{2} + 25 \) Copy content Toggle raw display
$7$ \( T^{4} + 81 \) Copy content Toggle raw display
$11$ \( T^{4} + 14T^{2} + 121 \) Copy content Toggle raw display
$13$ \( (T^{2} + 6 T + 18)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 12 T^{3} + 72 T^{2} - 108 T + 81 \) Copy content Toggle raw display
$19$ \( (T - 3)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 12 T^{3} + 72 T^{2} + 24 T + 4 \) Copy content Toggle raw display
$29$ \( (T^{2} + 6 T - 9)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 6 T - 9)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 8 T^{3} + 32 T^{2} - 8 T + 1 \) Copy content Toggle raw display
$41$ \( T^{4} + 108T^{2} + 324 \) Copy content Toggle raw display
$43$ \( T^{4} - 12 T^{3} + 72 T^{2} + \cdots + 324 \) Copy content Toggle raw display
$47$ \( T^{4} - 12 T^{3} + 72 T^{2} + \cdots + 196 \) Copy content Toggle raw display
$53$ \( T^{4} - 24 T^{3} + 288 T^{2} + \cdots + 2209 \) Copy content Toggle raw display
$59$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 54T^{2} + 81 \) Copy content Toggle raw display
$67$ \( (T^{2} + 8 T + 32)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 6 T - 9)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 12 T + 72)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 12 T + 18)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 12 T^{3} + 72 T^{2} + \cdots + 324 \) Copy content Toggle raw display
$89$ \( T^{4} + 162T^{2} + 3969 \) Copy content Toggle raw display
$97$ \( T^{4} - 8 T^{3} + 32 T^{2} + 224 T + 784 \) Copy content Toggle raw display
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