Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [304,2,Mod(31,304)]
 
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([3, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("304.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 304.n (of order \(6\), 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: \(2.42745222145\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + \zeta_{12}^{2} q^{5} - 2 \zeta_{12}^{3} q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + \zeta_{12}^{2} q^{5} - 2 \zeta_{12}^{3} q^{7} - 2 \zeta_{12}^{3} q^{11} + ( - \zeta_{12}^{2} - 1) q^{13} + ( - 2 \zeta_{12}^{3} + \zeta_{12}) q^{15} - \zeta_{12}^{2} q^{17} + ( - 3 \zeta_{12}^{3} - 2 \zeta_{12}) q^{19} + (2 \zeta_{12}^{2} - 4) q^{21} + \zeta_{12} q^{23} + ( - 4 \zeta_{12}^{2} + 4) q^{25} + (3 \zeta_{12}^{3} - 6 \zeta_{12}) q^{27} + ( - \zeta_{12}^{2} - 1) q^{29} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12}) q^{31} + (2 \zeta_{12}^{2} - 4) q^{33} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{35} + (4 \zeta_{12}^{2} - 2) q^{37} + 3 \zeta_{12}^{3} q^{39} + ( - 3 \zeta_{12}^{2} + 6) q^{41} + (9 \zeta_{12}^{3} - 9 \zeta_{12}) q^{43} + 11 \zeta_{12} q^{47} + 3 q^{49} + (2 \zeta_{12}^{3} - \zeta_{12}) q^{51} + (5 \zeta_{12}^{2} + 5) q^{53} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{55} + (7 \zeta_{12}^{2} - 8) q^{57} + (5 \zeta_{12}^{3} + 5 \zeta_{12}) q^{59} + ( - 9 \zeta_{12}^{2} + 9) q^{61} + ( - 2 \zeta_{12}^{2} + 1) q^{65} + (6 \zeta_{12}^{3} - 3 \zeta_{12}) q^{67} + ( - 2 \zeta_{12}^{2} + 1) q^{69} + (9 \zeta_{12}^{3} + 9 \zeta_{12}) q^{71} + \zeta_{12}^{2} q^{73} + (4 \zeta_{12}^{3} - 8 \zeta_{12}) q^{75} - 4 q^{77} + (\zeta_{12}^{3} + \zeta_{12}) q^{79} + 9 \zeta_{12}^{2} q^{81} - 8 \zeta_{12}^{3} q^{83} + ( - \zeta_{12}^{2} + 1) q^{85} + 3 \zeta_{12}^{3} q^{87} + ( - 9 \zeta_{12}^{2} - 9) q^{89} + (4 \zeta_{12}^{3} - 2 \zeta_{12}) q^{91} - 6 \zeta_{12}^{2} q^{93} + ( - 5 \zeta_{12}^{3} + 3 \zeta_{12}) q^{95} + (9 \zeta_{12}^{2} - 18) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{5} - 6 q^{13} - 2 q^{17} - 12 q^{21} + 8 q^{25} - 6 q^{29} - 12 q^{33} + 18 q^{41} + 12 q^{49} + 30 q^{53} - 18 q^{57} + 18 q^{61} + 2 q^{73} - 16 q^{77} + 18 q^{81} + 2 q^{85} - 54 q^{89} - 12 q^{93} - 54 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(97\) \(191\) \(229\)
\(\chi(n)\) \(1 - \zeta_{12}^{2}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0 −0.866025 1.50000i 0 0.500000 + 0.866025i 0 2.00000i 0 0 0
31.2 0 0.866025 + 1.50000i 0 0.500000 + 0.866025i 0 2.00000i 0 0 0
255.1 0 −0.866025 + 1.50000i 0 0.500000 0.866025i 0 2.00000i 0 0 0
255.2 0 0.866025 1.50000i 0 0.500000 0.866025i 0 2.00000i 0 0 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
4.b odd 2 1 inner
19.d odd 6 1 inner
76.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 304.2.n.c 4
3.b odd 2 1 2736.2.bm.k 4
4.b odd 2 1 inner 304.2.n.c 4
8.b even 2 1 1216.2.n.c 4
8.d odd 2 1 1216.2.n.c 4
12.b even 2 1 2736.2.bm.k 4
19.d odd 6 1 inner 304.2.n.c 4
57.f even 6 1 2736.2.bm.k 4
76.f even 6 1 inner 304.2.n.c 4
152.l odd 6 1 1216.2.n.c 4
152.o even 6 1 1216.2.n.c 4
228.n odd 6 1 2736.2.bm.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
304.2.n.c 4 1.a even 1 1 trivial
304.2.n.c 4 4.b odd 2 1 inner
304.2.n.c 4 19.d odd 6 1 inner
304.2.n.c 4 76.f even 6 1 inner
1216.2.n.c 4 8.b even 2 1
1216.2.n.c 4 8.d odd 2 1
1216.2.n.c 4 152.l odd 6 1
1216.2.n.c 4 152.o even 6 1
2736.2.bm.k 4 3.b odd 2 1
2736.2.bm.k 4 12.b even 2 1
2736.2.bm.k 4 57.f even 6 1
2736.2.bm.k 4 228.n odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 3T_{3}^{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(304, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 3 T + 3)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 26T^{2} + 361 \) Copy content Toggle raw display
$23$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$29$ \( (T^{2} + 3 T + 3)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 9 T + 27)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 81T^{2} + 6561 \) Copy content Toggle raw display
$47$ \( T^{4} - 121 T^{2} + 14641 \) Copy content Toggle raw display
$53$ \( (T^{2} - 15 T + 75)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 75T^{2} + 5625 \) Copy content Toggle raw display
$61$ \( (T^{2} - 9 T + 81)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 27T^{2} + 729 \) Copy content Toggle raw display
$71$ \( T^{4} + 243 T^{2} + 59049 \) Copy content Toggle raw display
$73$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$83$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 27 T + 243)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 27 T + 243)^{2} \) Copy content Toggle raw display
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