Properties

Label 154.2.f.a
Level $154$
Weight $2$
Character orbit 154.f
Analytic conductor $1.230$
Analytic rank $0$
Dimension $4$
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(15,154)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(154, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("154.15");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 154 = 2 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 154.f (of order \(5\), degree \(4\), 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: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(45\) \(57\)
\(\chi(n)\) \(1\) \(-\zeta_{10}^{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1
0.809017 0.587785i
−0.309017 + 0.951057i
0.809017 + 0.587785i
−0.309017 0.951057i
−0.809017 0.587785i −0.309017 + 0.951057i 0.309017 + 0.951057i 0.309017 0.224514i 0.809017 0.587785i 0.309017 + 0.951057i 0.309017 0.951057i 1.61803 + 1.17557i −0.381966
71.1 0.309017 + 0.951057i 0.809017 + 0.587785i −0.809017 + 0.587785i −0.809017 + 2.48990i −0.309017 + 0.951057i −0.809017 + 0.587785i −0.809017 0.587785i −0.618034 1.90211i −2.61803
113.1 −0.809017 + 0.587785i −0.309017 0.951057i 0.309017 0.951057i 0.309017 + 0.224514i 0.809017 + 0.587785i 0.309017 0.951057i 0.309017 + 0.951057i 1.61803 1.17557i −0.381966
141.1 0.309017 0.951057i 0.809017 0.587785i −0.809017 0.587785i −0.809017 2.48990i −0.309017 0.951057i −0.809017 0.587785i −0.809017 + 0.587785i −0.618034 + 1.90211i −2.61803
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 154.2.f.a 4
11.c even 5 1 inner 154.2.f.a 4
11.c even 5 1 1694.2.a.q 2
11.d odd 10 1 1694.2.a.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.f.a 4 1.a even 1 1 trivial
154.2.f.a 4 11.c even 5 1 inner
1694.2.a.k 2 11.d odd 10 1
1694.2.a.q 2 11.c even 5 1

Hecke kernels

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

\( T_{3}^{4} - T_{3}^{3} + T_{3}^{2} - T_{3} + 1 \) Copy content Toggle raw display
\( T_{5}^{4} + T_{5}^{3} + 6T_{5}^{2} - 4T_{5} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + T^{3} + T^{2} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{4} - T^{3} + T^{2} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{4} + T^{3} + 6 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{4} + T^{3} + T^{2} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{4} - 4 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$13$ \( T^{4} - 11 T^{3} + \cdots + 361 \) Copy content Toggle raw display
$17$ \( T^{4} - 7 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$19$ \( T^{4} - 15 T^{3} + \cdots + 2025 \) Copy content Toggle raw display
$23$ \( (T^{2} + 12 T + 31)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 5 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$31$ \( T^{4} - 3 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$37$ \( T^{4} - 12 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$41$ \( T^{4} - 8 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$43$ \( (T^{2} + 17 T + 71)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$53$ \( T^{4} + 9 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$59$ \( T^{4} + 20 T^{3} + \cdots + 3025 \) Copy content Toggle raw display
$61$ \( T^{4} - 18 T^{3} + \cdots + 9801 \) Copy content Toggle raw display
$67$ \( (T^{2} + 4 T - 41)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} - 3 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$73$ \( T^{4} - 11 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{4} - 25 T^{3} + \cdots + 3025 \) Copy content Toggle raw display
$83$ \( T^{4} + 19 T^{3} + \cdots + 3481 \) Copy content Toggle raw display
$89$ \( (T^{2} - 5 T - 25)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 18 T^{3} + \cdots + 1936 \) Copy content Toggle raw display
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