Properties

Label 154.2.f.e
Level $154$
Weight $2$
Character orbit 154.f
Analytic conductor $1.230$
Analytic rank $0$
Dimension $8$
Inner twists $2$

Related objects

Downloads

Learn more

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: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.58140625.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 4x^{6} - 7x^{5} + 11x^{4} + 5x^{3} - 10x^{2} - 25x + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 4x^{6} - 7x^{5} + 11x^{4} + 5x^{3} - 10x^{2} - 25x + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 406\nu^{7} - 714\nu^{6} + 747\nu^{5} - 1896\nu^{4} + 2103\nu^{3} + 4949\nu^{2} + 1065\nu - 7800 ) / 1355 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 420\nu^{7} - 776\nu^{6} + 698\nu^{5} - 1924\nu^{4} + 2297\nu^{3} + 5129\nu^{2} + 1055\nu - 10265 ) / 1355 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 728\nu^{7} - 1327\nu^{6} + 1246\nu^{5} - 3353\nu^{4} + 3584\nu^{3} + 8547\nu^{2} + 2190\nu - 15715 ) / 1355 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -857\nu^{7} + 1666\nu^{6} - 1743\nu^{5} + 4424\nu^{4} - 4907\nu^{3} - 9470\nu^{2} - 2485\nu + 18200 ) / 1355 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 891\nu^{7} - 1623\nu^{6} + 1624\nu^{5} - 4492\nu^{4} + 4991\nu^{3} + 9520\nu^{2} + 3235\nu - 18960 ) / 1355 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 955\nu^{7} - 1829\nu^{6} + 1942\nu^{5} - 4891\nu^{4} + 5723\nu^{3} + 9646\nu^{2} + 2415\nu - 20550 ) / 1355 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{5} - 3\beta_{4} + 4\beta_{3} + \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{7} - 5\beta_{6} + 4\beta_{5} + 2\beta_{4} + 2\beta_{3} + 2\beta_{2} + 4\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{7} - 6\beta_{6} - 7\beta_{3} + 11\beta_{2} + 4\beta _1 - 13 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 7\beta_{6} + 7\beta_{5} - 8\beta_{4} - 7\beta_{3} + 21\beta_{2} - 2\beta _1 - 21 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 23\beta_{7} + 38\beta_{5} + 23\beta_{4} - 23\beta_{3} + 12\beta_{2} \) Copy content Toggle raw display

Character values

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

\(n\) \(45\) \(57\)
\(\chi(n)\) \(1\) \(\beta_{3}\)

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
−0.357358 1.86824i
1.66637 + 0.917186i
1.17421 + 0.0566033i
−0.983224 0.644389i
−0.357358 + 1.86824i
1.66637 0.917186i
1.17421 0.0566033i
−0.983224 + 0.644389i
0.809017 + 0.587785i −1.02988 + 3.16963i 0.309017 + 0.951057i 2.47539 1.79848i −2.69625 + 1.95894i 0.309017 + 0.951057i −0.309017 + 0.951057i −6.55888 4.76530i 3.05975
15.2 0.809017 + 0.587785i 0.220859 0.679734i 0.309017 + 0.951057i 0.451659 0.328150i 0.578217 0.420099i 0.309017 + 0.951057i −0.309017 + 0.951057i 2.01379 + 1.46310i 0.558282
71.1 −0.309017 0.951057i −1.59089 1.15585i −0.809017 + 0.587785i −1.29224 + 3.97711i −0.607666 + 1.87020i −0.809017 + 0.587785i 0.809017 + 0.587785i 0.267892 + 0.824488i 4.18178
71.2 −0.309017 0.951057i 1.89991 + 1.38036i −0.809017 + 0.587785i 0.865190 2.66278i 0.725700 2.23347i −0.809017 + 0.587785i 0.809017 + 0.587785i 0.777193 + 2.39195i −2.79981
113.1 0.809017 0.587785i −1.02988 3.16963i 0.309017 0.951057i 2.47539 + 1.79848i −2.69625 1.95894i 0.309017 0.951057i −0.309017 0.951057i −6.55888 + 4.76530i 3.05975
113.2 0.809017 0.587785i 0.220859 + 0.679734i 0.309017 0.951057i 0.451659 + 0.328150i 0.578217 + 0.420099i 0.309017 0.951057i −0.309017 0.951057i 2.01379 1.46310i 0.558282
141.1 −0.309017 + 0.951057i −1.59089 + 1.15585i −0.809017 0.587785i −1.29224 3.97711i −0.607666 1.87020i −0.809017 0.587785i 0.809017 0.587785i 0.267892 0.824488i 4.18178
141.2 −0.309017 + 0.951057i 1.89991 1.38036i −0.809017 0.587785i 0.865190 + 2.66278i 0.725700 + 2.23347i −0.809017 0.587785i 0.809017 0.587785i 0.777193 2.39195i −2.79981
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 15.2
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.e 8
11.c even 5 1 inner 154.2.f.e 8
11.c even 5 1 1694.2.a.x 4
11.d odd 10 1 1694.2.a.z 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.f.e 8 1.a even 1 1 trivial
154.2.f.e 8 11.c even 5 1 inner
1694.2.a.x 4 11.c even 5 1
1694.2.a.z 4 11.d odd 10 1

Hecke kernels

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

\( T_{3}^{8} + T_{3}^{7} + 7T_{3}^{6} - 12T_{3}^{5} + 5T_{3}^{4} + 72T_{3}^{3} + 202T_{3}^{2} - 66T_{3} + 121 \) Copy content Toggle raw display
\( T_{5}^{8} - 5T_{5}^{7} + 30T_{5}^{6} - 130T_{5}^{5} + 485T_{5}^{4} - 1150T_{5}^{3} + 2100T_{5}^{2} - 1400T_{5} + 400 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} + T^{7} + \cdots + 121 \) Copy content Toggle raw display
$5$ \( T^{8} - 5 T^{7} + \cdots + 400 \) Copy content Toggle raw display
$7$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 5 T^{7} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( T^{8} - 3 T^{7} + \cdots + 80656 \) Copy content Toggle raw display
$17$ \( T^{8} + 7 T^{7} + \cdots + 6241 \) Copy content Toggle raw display
$19$ \( T^{8} + 14 T^{7} + \cdots + 121 \) Copy content Toggle raw display
$23$ \( (T^{4} + 10 T^{3} + \cdots - 20)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 23 T^{7} + \cdots + 4096 \) Copy content Toggle raw display
$31$ \( T^{8} - 3 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$37$ \( T^{8} - 6 T^{7} + \cdots + 4145296 \) Copy content Toggle raw display
$41$ \( T^{8} + 2 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$43$ \( (T^{4} + 8 T^{3} - 21 T^{2} + \cdots - 29)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 34 T^{7} + \cdots + 839056 \) Copy content Toggle raw display
$53$ \( T^{8} - 13 T^{7} + \cdots + 4096 \) Copy content Toggle raw display
$59$ \( T^{8} + 3 T^{7} + \cdots + 44102881 \) Copy content Toggle raw display
$61$ \( T^{8} + 16 T^{7} + \cdots + 246016 \) Copy content Toggle raw display
$67$ \( (T^{4} - 9 T^{3} + \cdots + 341)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 45 T^{7} + \cdots + 15366400 \) Copy content Toggle raw display
$73$ \( T^{8} - T^{7} + \cdots + 52113961 \) Copy content Toggle raw display
$79$ \( T^{8} + 21 T^{7} + \cdots + 38416 \) Copy content Toggle raw display
$83$ \( T^{8} - 12 T^{7} + \cdots + 11881 \) Copy content Toggle raw display
$89$ \( (T^{4} - 6 T^{3} + \cdots - 649)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 5 T^{7} + \cdots + 2016400 \) Copy content Toggle raw display
show more
show less