Properties

Label 154.2.e.a
Level $154$
Weight $2$
Character orbit 154.e
Analytic conductor $1.230$
Analytic rank $1$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [154,2,Mod(23,154)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(154, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("154.23");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 154 = 2 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 154.e (of order \(3\), 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: \(1.22969619113\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/154\mathbb{Z}\right)^\times\).

\(n\) \(45\) \(57\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1
0.500000 0.866025i
0.500000 + 0.866025i
−0.500000 + 0.866025i −1.50000 2.59808i −0.500000 0.866025i −1.00000 + 1.73205i 3.00000 −2.50000 + 0.866025i 1.00000 −3.00000 + 5.19615i −1.00000 1.73205i
67.1 −0.500000 0.866025i −1.50000 + 2.59808i −0.500000 + 0.866025i −1.00000 1.73205i 3.00000 −2.50000 0.866025i 1.00000 −3.00000 5.19615i −1.00000 + 1.73205i
\(n\): 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 154.2.e.a 2
3.b odd 2 1 1386.2.k.o 2
4.b odd 2 1 1232.2.q.e 2
7.b odd 2 1 1078.2.e.f 2
7.c even 3 1 inner 154.2.e.a 2
7.c even 3 1 1078.2.a.m 1
7.d odd 6 1 1078.2.a.g 1
7.d odd 6 1 1078.2.e.f 2
21.g even 6 1 9702.2.a.y 1
21.h odd 6 1 1386.2.k.o 2
21.h odd 6 1 9702.2.a.i 1
28.f even 6 1 8624.2.a.be 1
28.g odd 6 1 1232.2.q.e 2
28.g odd 6 1 8624.2.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.e.a 2 1.a even 1 1 trivial
154.2.e.a 2 7.c even 3 1 inner
1078.2.a.g 1 7.d odd 6 1
1078.2.a.m 1 7.c even 3 1
1078.2.e.f 2 7.b odd 2 1
1078.2.e.f 2 7.d odd 6 1
1232.2.q.e 2 4.b odd 2 1
1232.2.q.e 2 28.g odd 6 1
1386.2.k.o 2 3.b odd 2 1
1386.2.k.o 2 21.h odd 6 1
8624.2.a.b 1 28.g odd 6 1
8624.2.a.be 1 28.f even 6 1
9702.2.a.i 1 21.h odd 6 1
9702.2.a.y 1 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(154, [\chi])\):

\( T_{3}^{2} + 3T_{3} + 9 \) Copy content Toggle raw display
\( T_{13} + 7 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$5$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$7$ \( T^{2} + 5T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$13$ \( (T + 7)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$29$ \( (T + 5)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$37$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$41$ \( (T - 4)^{2} \) Copy content Toggle raw display
$43$ \( (T + 8)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$53$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$59$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$61$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$67$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$71$ \( (T + 2)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$79$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$83$ \( (T - 6)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$97$ \( (T - 7)^{2} \) Copy content Toggle raw display
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