Properties

Label 304.2.i.e
Level $304$
Weight $2$
Character orbit 304.i
Analytic conductor $2.427$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [304,2,Mod(49,304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(304, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("304.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 304.i (of order \(3\), degree \(2\), not 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: \(2.42745222145\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 7x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 38)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{2} + \beta_1 - 1) q^{5} + (\beta_{3} - 1) q^{7} + 4 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - \beta_{2} + \beta_1 - 1) q^{5} + (\beta_{3} - 1) q^{7} + 4 \beta_{2} q^{9} + ( - \beta_{3} + 2) q^{11} + 2 \beta_{2} q^{13} + ( - \beta_{3} + 7 \beta_{2} - \beta_1) q^{15} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 - 4) q^{19} + ( - 7 \beta_{2} - \beta_1 - 7) q^{21} + (\beta_{3} - \beta_{2} + \beta_1) q^{23} + ( - 2 \beta_{3} + 3 \beta_{2} - 2 \beta_1) q^{25} + \beta_{3} q^{27} + (\beta_{3} - \beta_{2} + \beta_1) q^{29} + ( - \beta_{3} + 3) q^{31} + (7 \beta_{2} + 2 \beta_1 + 7) q^{33} + ( - 6 \beta_{2} - 6) q^{35} + (\beta_{3} + 3) q^{37} + 2 \beta_{3} q^{39} + ( - 5 \beta_{2} + 2 \beta_1 - 5) q^{41} + (6 \beta_{2} + 2 \beta_1 + 6) q^{43} + (4 \beta_{3} + 4) q^{45} + (\beta_{3} - 7 \beta_{2} + \beta_1) q^{47} + ( - 2 \beta_{3} + 1) q^{49} + ( - 4 \beta_{3} - 2 \beta_{2} - 4 \beta_1) q^{53} + (5 \beta_{2} + \beta_1 + 5) q^{55} + ( - 2 \beta_{3} - 4 \beta_1 + 7) q^{57} - 3 \beta_1 q^{59} + (3 \beta_{3} - 7 \beta_{2} + 3 \beta_1) q^{61} + ( - 4 \beta_{3} - 4 \beta_{2} - 4 \beta_1) q^{63} + (2 \beta_{3} + 2) q^{65} + ( - \beta_{3} + 2 \beta_{2} - \beta_1) q^{67} + ( - \beta_{3} - 7) q^{69} + ( - 8 \beta_{2} + 2 \beta_1 - 8) q^{71} + ( - 7 \beta_{2} + 2 \beta_1 - 7) q^{73} + (3 \beta_{3} + 14) q^{75} + (3 \beta_{3} - 9) q^{77} + ( - 4 \beta_{2} - 4) q^{79} + (5 \beta_{2} + 5) q^{81} + 3 \beta_{3} q^{83} + ( - \beta_{3} - 7) q^{87} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{91} + (7 \beta_{2} + 3 \beta_1 + 7) q^{93} + ( - \beta_{3} + 4 \beta_{2} - 4 \beta_1 + 9) q^{95} + (9 \beta_{2} - 2 \beta_1 + 9) q^{97} + (4 \beta_{3} + 8 \beta_{2} + 4 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{5} - 4 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{5} - 4 q^{7} - 8 q^{9} + 8 q^{11} - 4 q^{13} - 14 q^{15} - 12 q^{19} - 14 q^{21} + 2 q^{23} - 6 q^{25} + 2 q^{29} + 12 q^{31} + 14 q^{33} - 12 q^{35} + 12 q^{37} - 10 q^{41} + 12 q^{43} + 16 q^{45} + 14 q^{47} + 4 q^{49} + 4 q^{53} + 10 q^{55} + 28 q^{57} + 14 q^{61} + 8 q^{63} + 8 q^{65} - 4 q^{67} - 28 q^{69} - 16 q^{71} - 14 q^{73} + 56 q^{75} - 36 q^{77} - 8 q^{79} + 10 q^{81} - 28 q^{87} + 4 q^{91} + 14 q^{93} + 28 q^{95} + 18 q^{97} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 7x^{2} + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 7\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 7\beta_{3} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/304\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(191\) \(229\)
\(\chi(n)\) \(\beta_{2}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−1.32288 + 2.29129i
1.32288 2.29129i
−1.32288 2.29129i
1.32288 + 2.29129i
0 −1.32288 + 2.29129i 0 −1.82288 + 3.15731i 0 1.64575 0 −2.00000 3.46410i 0
49.2 0 1.32288 2.29129i 0 0.822876 1.42526i 0 −3.64575 0 −2.00000 3.46410i 0
273.1 0 −1.32288 2.29129i 0 −1.82288 3.15731i 0 1.64575 0 −2.00000 + 3.46410i 0
273.2 0 1.32288 + 2.29129i 0 0.822876 + 1.42526i 0 −3.64575 0 −2.00000 + 3.46410i 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
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 304.2.i.e 4
3.b odd 2 1 2736.2.s.v 4
4.b odd 2 1 38.2.c.b 4
8.b even 2 1 1216.2.i.k 4
8.d odd 2 1 1216.2.i.l 4
12.b even 2 1 342.2.g.f 4
19.c even 3 1 inner 304.2.i.e 4
19.c even 3 1 5776.2.a.ba 2
19.d odd 6 1 5776.2.a.z 2
20.d odd 2 1 950.2.e.k 4
20.e even 4 2 950.2.j.g 8
57.h odd 6 1 2736.2.s.v 4
76.d even 2 1 722.2.c.j 4
76.f even 6 1 722.2.a.g 2
76.f even 6 1 722.2.c.j 4
76.g odd 6 1 38.2.c.b 4
76.g odd 6 1 722.2.a.j 2
76.k even 18 6 722.2.e.o 12
76.l odd 18 6 722.2.e.n 12
152.k odd 6 1 1216.2.i.l 4
152.p even 6 1 1216.2.i.k 4
228.m even 6 1 342.2.g.f 4
228.m even 6 1 6498.2.a.ba 2
228.n odd 6 1 6498.2.a.bg 2
380.p odd 6 1 950.2.e.k 4
380.v even 12 2 950.2.j.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.c.b 4 4.b odd 2 1
38.2.c.b 4 76.g odd 6 1
304.2.i.e 4 1.a even 1 1 trivial
304.2.i.e 4 19.c even 3 1 inner
342.2.g.f 4 12.b even 2 1
342.2.g.f 4 228.m even 6 1
722.2.a.g 2 76.f even 6 1
722.2.a.j 2 76.g odd 6 1
722.2.c.j 4 76.d even 2 1
722.2.c.j 4 76.f even 6 1
722.2.e.n 12 76.l odd 18 6
722.2.e.o 12 76.k even 18 6
950.2.e.k 4 20.d odd 2 1
950.2.e.k 4 380.p odd 6 1
950.2.j.g 8 20.e even 4 2
950.2.j.g 8 380.v even 12 2
1216.2.i.k 4 8.b even 2 1
1216.2.i.k 4 152.p even 6 1
1216.2.i.l 4 8.d odd 2 1
1216.2.i.l 4 152.k odd 6 1
2736.2.s.v 4 3.b odd 2 1
2736.2.s.v 4 57.h odd 6 1
5776.2.a.z 2 19.d odd 6 1
5776.2.a.ba 2 19.c even 3 1
6498.2.a.ba 2 228.m even 6 1
6498.2.a.bg 2 228.n odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(304, [\chi])\):

\( T_{3}^{4} + 7T_{3}^{2} + 49 \) Copy content Toggle raw display
\( T_{5}^{4} + 2T_{5}^{3} + 10T_{5}^{2} - 12T_{5} + 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 7T^{2} + 49 \) Copy content Toggle raw display
$5$ \( T^{4} + 2 T^{3} + 10 T^{2} - 12 T + 36 \) Copy content Toggle raw display
$7$ \( (T^{2} + 2 T - 6)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 4 T - 3)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} + 12 T^{3} + 67 T^{2} + \cdots + 361 \) Copy content Toggle raw display
$23$ \( T^{4} - 2 T^{3} + 10 T^{2} + 12 T + 36 \) Copy content Toggle raw display
$29$ \( T^{4} - 2 T^{3} + 10 T^{2} + 12 T + 36 \) Copy content Toggle raw display
$31$ \( (T^{2} - 6 T + 2)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 6 T + 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 10 T^{3} + 103 T^{2} - 30 T + 9 \) Copy content Toggle raw display
$43$ \( T^{4} - 12 T^{3} + 136 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$47$ \( T^{4} - 14 T^{3} + 154 T^{2} + \cdots + 1764 \) Copy content Toggle raw display
$53$ \( T^{4} - 4 T^{3} + 124 T^{2} + \cdots + 11664 \) Copy content Toggle raw display
$59$ \( T^{4} + 63T^{2} + 3969 \) Copy content Toggle raw display
$61$ \( T^{4} - 14 T^{3} + 210 T^{2} + \cdots + 196 \) Copy content Toggle raw display
$67$ \( T^{4} + 4 T^{3} + 19 T^{2} - 12 T + 9 \) Copy content Toggle raw display
$71$ \( T^{4} + 16 T^{3} + 220 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$73$ \( T^{4} + 14 T^{3} + 175 T^{2} + \cdots + 441 \) Copy content Toggle raw display
$79$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 63)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} - 18 T^{3} + 271 T^{2} + \cdots + 2809 \) Copy content Toggle raw display
show more
show less