Properties

Label 28.2.f.a
Level $28$
Weight $2$
Character orbit 28.f
Analytic conductor $0.224$
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] = [28,2,Mod(3,28)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(28, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("28.3");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 28 = 2^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 28.f (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: \(0.223581125660\)
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: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12}) q^{2} + ( - 2 \zeta_{12}^{3} + \zeta_{12}) q^{3} + 2 \zeta_{12} q^{4} + (\zeta_{12}^{2} - 2) q^{5} + (\zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12} - 1) q^{6} + (3 \zeta_{12}^{3} - 2 \zeta_{12}) q^{7} + ( - 2 \zeta_{12}^{3} - 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12}) q^{2} + ( - 2 \zeta_{12}^{3} + \zeta_{12}) q^{3} + 2 \zeta_{12} q^{4} + (\zeta_{12}^{2} - 2) q^{5} + (\zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12} - 1) q^{6} + (3 \zeta_{12}^{3} - 2 \zeta_{12}) q^{7} + ( - 2 \zeta_{12}^{3} - 2) q^{8} + ( - 2 \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12} + 1) q^{10} + \zeta_{12} q^{11} + ( - 2 \zeta_{12}^{2} + 4) q^{12} + ( - 4 \zeta_{12}^{2} + 2) q^{13} + ( - \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 3 \zeta_{12} + 2) q^{14} + 3 \zeta_{12}^{3} q^{15} + 4 \zeta_{12}^{2} q^{16} + ( - \zeta_{12}^{2} - 1) q^{17} + ( - 3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{19} + (2 \zeta_{12}^{3} - 4 \zeta_{12}) q^{20} + (5 \zeta_{12}^{2} - 1) q^{21} + ( - \zeta_{12}^{3} - 1) q^{22} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{23} + (4 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} - 2) q^{24} + (2 \zeta_{12}^{2} - 2) q^{25} + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 2 \zeta_{12} - 4) q^{26} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12}) q^{27} + (2 \zeta_{12}^{2} - 6) q^{28} + 4 q^{29} + ( - 3 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 3 \zeta_{12}) q^{30} + (2 \zeta_{12}^{3} - \zeta_{12}) q^{31} + ( - 4 \zeta_{12}^{2} - 4 \zeta_{12} + 4) q^{32} + ( - \zeta_{12}^{2} + 2) q^{33} + ( - \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 2 \zeta_{12} - 1) q^{34} + ( - 5 \zeta_{12}^{3} + \zeta_{12}) q^{35} - 3 \zeta_{12}^{2} q^{37} + (6 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 3 \zeta_{12} + 3) q^{38} - 6 \zeta_{12} q^{39} + (2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} + 4) q^{40} + (4 \zeta_{12}^{2} - 2) q^{41} + ( - \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 4 \zeta_{12} + 5) q^{42} + 2 \zeta_{12}^{3} q^{43} + 2 \zeta_{12}^{2} q^{44} + (\zeta_{12}^{2} - \zeta_{12} - 1) q^{46} + (5 \zeta_{12}^{3} + 5 \zeta_{12}) q^{47} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12}) q^{48} + ( - 8 \zeta_{12}^{2} + 3) q^{49} + ( - 2 \zeta_{12}^{3} + 2) q^{50} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{51} + ( - 8 \zeta_{12}^{3} + 4 \zeta_{12}) q^{52} + ( - \zeta_{12}^{2} + 1) q^{53} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 3 \zeta_{12} - 6) q^{54} + (\zeta_{12}^{3} - 2 \zeta_{12}) q^{55} + ( - 6 \zeta_{12}^{3} + 4 \zeta_{12}^{2} + 4 \zeta_{12} + 2) q^{56} - 9 q^{57} + (4 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 4 \zeta_{12}) q^{58} + (6 \zeta_{12}^{3} - 3 \zeta_{12}) q^{59} + (6 \zeta_{12}^{2} - 6) q^{60} + (3 \zeta_{12}^{2} - 6) q^{61} + ( - \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} + 1) q^{62} + 8 \zeta_{12}^{3} q^{64} + 6 \zeta_{12}^{2} q^{65} + (2 \zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12} - 1) q^{66} + 3 \zeta_{12} q^{67} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{68} + ( - 2 \zeta_{12}^{2} + 1) q^{69} + (4 \zeta_{12}^{3} + 5 \zeta_{12}^{2} - 5 \zeta_{12} - 1) q^{70} - 14 \zeta_{12}^{3} q^{71} + (5 \zeta_{12}^{2} + 5) q^{73} + (3 \zeta_{12}^{2} + 3 \zeta_{12} - 3) q^{74} + (2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{75} + ( - 12 \zeta_{12}^{2} + 6) q^{76} + (\zeta_{12}^{2} - 3) q^{77} + (6 \zeta_{12}^{3} + 6) q^{78} + ( - 9 \zeta_{12}^{3} + 9 \zeta_{12}) q^{79} + ( - 4 \zeta_{12}^{2} - 4) q^{80} + ( - 9 \zeta_{12}^{2} + 9) q^{81} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 4) q^{82} + (8 \zeta_{12}^{3} - 16 \zeta_{12}) q^{83} + (10 \zeta_{12}^{3} - 2 \zeta_{12}) q^{84} + 3 q^{85} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12}) q^{86} + ( - 8 \zeta_{12}^{3} + 4 \zeta_{12}) q^{87} + ( - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 2) q^{88} + ( - 9 \zeta_{12}^{2} + 18) q^{89} + (2 \zeta_{12}^{3} + 8 \zeta_{12}) q^{91} + 2 q^{92} + 3 \zeta_{12}^{2} q^{93} + ( - 10 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + 5 \zeta_{12} - 5) q^{94} + 9 \zeta_{12} q^{95} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 4 \zeta_{12} - 8) q^{96} + (20 \zeta_{12}^{2} - 10) q^{97} + (3 \zeta_{12}^{3} + 5 \zeta_{12}^{2} + 5 \zeta_{12} - 8) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 6 q^{5} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} - 6 q^{5} - 8 q^{8} + 6 q^{10} + 12 q^{12} + 2 q^{14} + 8 q^{16} - 6 q^{17} + 6 q^{21} - 4 q^{22} - 12 q^{24} - 4 q^{25} - 12 q^{26} - 20 q^{28} + 16 q^{29} - 6 q^{30} + 8 q^{32} + 6 q^{33} - 6 q^{37} + 18 q^{38} + 12 q^{40} + 12 q^{42} + 4 q^{44} - 2 q^{46} - 4 q^{49} + 8 q^{50} + 2 q^{53} - 18 q^{54} + 16 q^{56} - 36 q^{57} - 8 q^{58} - 12 q^{60} - 18 q^{61} + 12 q^{65} - 6 q^{66} + 6 q^{70} + 30 q^{73} - 6 q^{74} - 10 q^{77} + 24 q^{78} - 24 q^{80} + 18 q^{81} + 12 q^{82} + 12 q^{85} - 4 q^{86} + 4 q^{88} + 54 q^{89} + 8 q^{92} + 6 q^{93} - 30 q^{94} - 24 q^{96} - 22 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/28\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\)
\(\chi(n)\) \(-1\) \(\zeta_{12}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
−1.36603 0.366025i 0.866025 1.50000i 1.73205 + 1.00000i −1.50000 + 0.866025i −1.73205 + 1.73205i −1.73205 + 2.00000i −2.00000 2.00000i 0 2.36603 0.633975i
3.2 0.366025 1.36603i −0.866025 + 1.50000i −1.73205 1.00000i −1.50000 + 0.866025i 1.73205 + 1.73205i 1.73205 2.00000i −2.00000 + 2.00000i 0 0.633975 + 2.36603i
19.1 −1.36603 + 0.366025i 0.866025 + 1.50000i 1.73205 1.00000i −1.50000 0.866025i −1.73205 1.73205i −1.73205 2.00000i −2.00000 + 2.00000i 0 2.36603 + 0.633975i
19.2 0.366025 + 1.36603i −0.866025 1.50000i −1.73205 + 1.00000i −1.50000 0.866025i 1.73205 1.73205i 1.73205 + 2.00000i −2.00000 2.00000i 0 0.633975 2.36603i
\(n\): 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
7.d odd 6 1 inner
28.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 28.2.f.a 4
3.b odd 2 1 252.2.bf.e 4
4.b odd 2 1 inner 28.2.f.a 4
5.b even 2 1 700.2.p.a 4
5.c odd 4 1 700.2.t.a 4
5.c odd 4 1 700.2.t.b 4
7.b odd 2 1 196.2.f.a 4
7.c even 3 1 196.2.d.b 4
7.c even 3 1 196.2.f.a 4
7.d odd 6 1 inner 28.2.f.a 4
7.d odd 6 1 196.2.d.b 4
8.b even 2 1 448.2.p.d 4
8.d odd 2 1 448.2.p.d 4
12.b even 2 1 252.2.bf.e 4
20.d odd 2 1 700.2.p.a 4
20.e even 4 1 700.2.t.a 4
20.e even 4 1 700.2.t.b 4
21.g even 6 1 252.2.bf.e 4
21.g even 6 1 1764.2.b.a 4
21.h odd 6 1 1764.2.b.a 4
28.d even 2 1 196.2.f.a 4
28.f even 6 1 inner 28.2.f.a 4
28.f even 6 1 196.2.d.b 4
28.g odd 6 1 196.2.d.b 4
28.g odd 6 1 196.2.f.a 4
35.i odd 6 1 700.2.p.a 4
35.k even 12 1 700.2.t.a 4
35.k even 12 1 700.2.t.b 4
56.j odd 6 1 448.2.p.d 4
56.j odd 6 1 3136.2.f.e 4
56.k odd 6 1 3136.2.f.e 4
56.m even 6 1 448.2.p.d 4
56.m even 6 1 3136.2.f.e 4
56.p even 6 1 3136.2.f.e 4
84.j odd 6 1 252.2.bf.e 4
84.j odd 6 1 1764.2.b.a 4
84.n even 6 1 1764.2.b.a 4
140.s even 6 1 700.2.p.a 4
140.x odd 12 1 700.2.t.a 4
140.x odd 12 1 700.2.t.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.2.f.a 4 1.a even 1 1 trivial
28.2.f.a 4 4.b odd 2 1 inner
28.2.f.a 4 7.d odd 6 1 inner
28.2.f.a 4 28.f even 6 1 inner
196.2.d.b 4 7.c even 3 1
196.2.d.b 4 7.d odd 6 1
196.2.d.b 4 28.f even 6 1
196.2.d.b 4 28.g odd 6 1
196.2.f.a 4 7.b odd 2 1
196.2.f.a 4 7.c even 3 1
196.2.f.a 4 28.d even 2 1
196.2.f.a 4 28.g odd 6 1
252.2.bf.e 4 3.b odd 2 1
252.2.bf.e 4 12.b even 2 1
252.2.bf.e 4 21.g even 6 1
252.2.bf.e 4 84.j odd 6 1
448.2.p.d 4 8.b even 2 1
448.2.p.d 4 8.d odd 2 1
448.2.p.d 4 56.j odd 6 1
448.2.p.d 4 56.m even 6 1
700.2.p.a 4 5.b even 2 1
700.2.p.a 4 20.d odd 2 1
700.2.p.a 4 35.i odd 6 1
700.2.p.a 4 140.s even 6 1
700.2.t.a 4 5.c odd 4 1
700.2.t.a 4 20.e even 4 1
700.2.t.a 4 35.k even 12 1
700.2.t.a 4 140.x odd 12 1
700.2.t.b 4 5.c odd 4 1
700.2.t.b 4 20.e even 4 1
700.2.t.b 4 35.k even 12 1
700.2.t.b 4 140.x odd 12 1
1764.2.b.a 4 21.g even 6 1
1764.2.b.a 4 21.h odd 6 1
1764.2.b.a 4 84.j odd 6 1
1764.2.b.a 4 84.n even 6 1
3136.2.f.e 4 56.j odd 6 1
3136.2.f.e 4 56.k odd 6 1
3136.2.f.e 4 56.m even 6 1
3136.2.f.e 4 56.p even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(28, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4 \) Copy content Toggle raw display
$3$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$5$ \( (T^{2} + 3 T + 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 2T^{2} + 49 \) Copy content Toggle raw display
$11$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$13$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 3 T + 3)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 27T^{2} + 729 \) Copy content Toggle raw display
$23$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$29$ \( (T - 4)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$37$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 75T^{2} + 5625 \) Copy content Toggle raw display
$53$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 27T^{2} + 729 \) Copy content Toggle raw display
$61$ \( (T^{2} + 9 T + 27)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$71$ \( (T^{2} + 196)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 15 T + 75)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 81T^{2} + 6561 \) Copy content Toggle raw display
$83$ \( (T^{2} - 192)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 27 T + 243)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 300)^{2} \) Copy content Toggle raw display
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