Properties

Label 308.2.j.a
Level $308$
Weight $2$
Character orbit 308.j
Analytic conductor $2.459$
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] = [308,2,Mod(113,308)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(308, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("308.113");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 308 = 2^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 308.j (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: \(2.45939238226\)
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: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + (2 \zeta_{10}^{3} + 2 \zeta_{10}) q^{3} - 2 \zeta_{10} q^{5} + \zeta_{10}^{3} q^{7} + (5 \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 5) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (2 \zeta_{10}^{3} + 2 \zeta_{10}) q^{3} - 2 \zeta_{10} q^{5} + \zeta_{10}^{3} q^{7} + (5 \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 5) q^{9} + (2 \zeta_{10}^{3} + \zeta_{10} + 2) q^{11} + (2 \zeta_{10}^{3} - 2) q^{13} + ( - 4 \zeta_{10}^{3} - 4 \zeta_{10} + 4) q^{15} + 4 \zeta_{10} q^{17} - 2 \zeta_{10}^{2} q^{19} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 2) q^{21} + ( - \zeta_{10}^{3} + \zeta_{10}^{2}) q^{23} - \zeta_{10}^{2} q^{25} + ( - 4 \zeta_{10}^{2} - 8 \zeta_{10} - 4) q^{27} + ( - \zeta_{10}^{3} + 3 \zeta_{10} - 3) q^{29} + ( - 2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} + 2) q^{31} + (10 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 6 \zeta_{10} - 6) q^{33} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{35} + ( - 8 \zeta_{10}^{3} - 3 \zeta_{10} + 3) q^{37} + ( - 4 \zeta_{10}^{2} - 4 \zeta_{10} - 4) q^{39} + (6 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 6 \zeta_{10}) q^{41} + ( - 3 \zeta_{10}^{3} + 3 \zeta_{10}^{2} + 9) q^{43} + ( - 8 \zeta_{10}^{3} + 8 \zeta_{10}^{2} + 10) q^{45} + 10 \zeta_{10}^{2} q^{47} - \zeta_{10} q^{49} + (8 \zeta_{10}^{3} + 8 \zeta_{10} - 8) q^{51} + ( - 9 \zeta_{10}^{3} + 8 \zeta_{10}^{2} - 8 \zeta_{10} + 9) q^{53} + ( - 4 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 8 \zeta_{10} + 4) q^{55} + ( - 4 \zeta_{10}^{3} + 4) q^{57} + (6 \zeta_{10} - 6) q^{59} + ( - 4 \zeta_{10}^{2} + 6 \zeta_{10} - 4) q^{61} + ( - 4 \zeta_{10}^{3} - \zeta_{10}^{2} - 4 \zeta_{10}) q^{63} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} + 4) q^{65} + (\zeta_{10}^{3} - \zeta_{10}^{2} + 1) q^{67} + 2 \zeta_{10}^{2} q^{69} + (5 \zeta_{10}^{2} - 10 \zeta_{10} + 5) q^{71} + (6 \zeta_{10}^{3} + 14 \zeta_{10} - 14) q^{73} + ( - 2 \zeta_{10}^{3} + 2) q^{75} + (3 \zeta_{10}^{3} - \zeta_{10}^{2} - \zeta_{10} - 1) q^{77} + ( - 3 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} + 3) q^{79} + ( - 17 \zeta_{10}^{3} - 12 \zeta_{10} + 12) q^{81} + ( - 8 \zeta_{10}^{2} - 2 \zeta_{10} - 8) q^{83} - 8 \zeta_{10}^{2} q^{85} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 4) q^{87} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 8) q^{89} + ( - 2 \zeta_{10}^{3} - 2 \zeta_{10}) q^{91} + (4 \zeta_{10}^{2} + 8 \zeta_{10} + 4) q^{93} + 4 \zeta_{10}^{3} q^{95} + ( - 8 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10} + 8) q^{97} + (6 \zeta_{10}^{3} - 8 \zeta_{10}^{2} - 6 \zeta_{10} - 15) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 2 q^{5} + q^{7} - 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} - 2 q^{5} + q^{7} - 13 q^{9} + 11 q^{11} - 6 q^{13} + 8 q^{15} + 4 q^{17} + 2 q^{19} - 4 q^{21} - 2 q^{23} + q^{25} - 20 q^{27} - 10 q^{29} + 10 q^{31} - 4 q^{33} + 2 q^{35} + q^{37} - 16 q^{39} + 10 q^{41} + 30 q^{43} + 24 q^{45} - 10 q^{47} - q^{49} - 16 q^{51} + 11 q^{53} + 2 q^{55} + 12 q^{57} - 18 q^{59} - 6 q^{61} - 7 q^{63} + 8 q^{65} + 6 q^{67} - 2 q^{69} + 5 q^{71} - 36 q^{73} + 6 q^{75} - q^{77} + 13 q^{79} + 19 q^{81} - 26 q^{83} + 8 q^{85} - 20 q^{87} - 36 q^{89} - 4 q^{91} + 20 q^{93} + 4 q^{95} + 32 q^{97} - 52 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/308\mathbb{Z}\right)^\times\).

\(n\) \(45\) \(57\) \(155\)
\(\chi(n)\) \(1\) \(-\zeta_{10}^{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1
0.809017 + 0.587785i
−0.309017 0.951057i
0.809017 0.587785i
−0.309017 + 0.951057i
0 1.00000 + 3.07768i 0 −1.61803 1.17557i 0 −0.309017 + 0.951057i 0 −6.04508 + 4.39201i 0
141.1 0 1.00000 0.726543i 0 0.618034 + 1.90211i 0 0.809017 + 0.587785i 0 −0.454915 + 1.40008i 0
169.1 0 1.00000 3.07768i 0 −1.61803 + 1.17557i 0 −0.309017 0.951057i 0 −6.04508 4.39201i 0
225.1 0 1.00000 + 0.726543i 0 0.618034 1.90211i 0 0.809017 0.587785i 0 −0.454915 1.40008i 0
\(n\): 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 308.2.j.a 4
11.c even 5 1 inner 308.2.j.a 4
11.c even 5 1 3388.2.a.k 2
11.d odd 10 1 3388.2.a.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
308.2.j.a 4 1.a even 1 1 trivial
308.2.j.a 4 11.c even 5 1 inner
3388.2.a.k 2 11.c even 5 1
3388.2.a.l 2 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 4T_{3}^{3} + 16T_{3}^{2} - 24T_{3} + 16 \) acting on \(S_{2}^{\mathrm{new}}(308, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 4 T^{3} + 16 T^{2} - 24 T + 16 \) Copy content Toggle raw display
$5$ \( T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16 \) Copy content Toggle raw display
$7$ \( T^{4} - T^{3} + T^{2} - T + 1 \) Copy content Toggle raw display
$11$ \( T^{4} - 11 T^{3} + 51 T^{2} + \cdots + 121 \) Copy content Toggle raw display
$13$ \( T^{4} + 6 T^{3} + 16 T^{2} + 16 T + 16 \) Copy content Toggle raw display
$17$ \( T^{4} - 4 T^{3} + 16 T^{2} - 64 T + 256 \) Copy content Toggle raw display
$19$ \( T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16 \) Copy content Toggle raw display
$23$ \( (T^{2} + T - 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 10 T^{3} + 40 T^{2} + 25 T + 25 \) Copy content Toggle raw display
$31$ \( T^{4} - 10 T^{3} + 60 T^{2} + \cdots + 400 \) Copy content Toggle raw display
$37$ \( T^{4} - T^{3} + 76 T^{2} + 434 T + 961 \) Copy content Toggle raw display
$41$ \( T^{4} - 10 T^{3} + 160 T^{2} + \cdots + 400 \) Copy content Toggle raw display
$43$ \( (T^{2} - 15 T + 45)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 10 T^{3} + 100 T^{2} + \cdots + 10000 \) Copy content Toggle raw display
$53$ \( T^{4} - 11 T^{3} + 76 T^{2} + \cdots + 5041 \) Copy content Toggle raw display
$59$ \( T^{4} + 18 T^{3} + 144 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$61$ \( T^{4} + 6 T^{3} + 76 T^{2} + 56 T + 16 \) Copy content Toggle raw display
$67$ \( (T^{2} - 3 T + 1)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} - 5 T^{3} + 150 T^{2} + \cdots + 625 \) Copy content Toggle raw display
$73$ \( T^{4} + 36 T^{3} + 736 T^{2} + \cdots + 59536 \) Copy content Toggle raw display
$79$ \( T^{4} - 13 T^{3} + 94 T^{2} + \cdots + 961 \) Copy content Toggle raw display
$83$ \( T^{4} + 26 T^{3} + 276 T^{2} + \cdots + 1936 \) Copy content Toggle raw display
$89$ \( (T^{2} + 18 T + 76)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 32 T^{3} + 544 T^{2} + \cdots + 30976 \) Copy content Toggle raw display
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