Properties

Label 1210.2.b.i
Level $1210$
Weight $2$
Character orbit 1210.b
Analytic conductor $9.662$
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] = [1210,2,Mod(969,1210)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1210, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1210.969");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1210 = 2 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1210.b (of order \(2\), degree \(1\), 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: \(9.66189864457\)
Analytic rank: \(0\)
Dimension: \(4\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{12}^{3} q^{2} + (\zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{3} - q^{4} + ( - 2 \zeta_{12}^{2} - \zeta_{12} + 2) q^{5} + ( - \zeta_{12}^{3} + 2 \zeta_{12} - 1) q^{6} + (3 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{7} - \zeta_{12}^{3} q^{8} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12} - 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12}^{3} q^{2} + (\zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{3} - q^{4} + ( - 2 \zeta_{12}^{2} - \zeta_{12} + 2) q^{5} + ( - \zeta_{12}^{3} + 2 \zeta_{12} - 1) q^{6} + (3 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{7} - \zeta_{12}^{3} q^{8} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12} - 1) q^{9} + ( - \zeta_{12}^{2} + 2 \zeta_{12} + 1) q^{10} + ( - \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 1) q^{12} + ( - \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 2) q^{13} + ( - \zeta_{12}^{3} + 2 \zeta_{12} - 3) q^{14} + (2 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + \zeta_{12} - 1) q^{15} + q^{16} + (2 \zeta_{12}^{2} - 1) q^{17} + ( - \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 2) q^{18} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12} - 1) q^{19} + (2 \zeta_{12}^{2} + \zeta_{12} - 2) q^{20} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12} - 6) q^{21} + ( - \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{23} + (\zeta_{12}^{3} - 2 \zeta_{12} + 1) q^{24} + (4 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 4 \zeta_{12}) q^{25} + (2 \zeta_{12}^{3} - 4 \zeta_{12} + 1) q^{26} - 4 \zeta_{12}^{3} q^{27} + ( - 3 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 1) q^{28} + (\zeta_{12}^{3} - 2 \zeta_{12} - 2) q^{29} + ( - 4 \zeta_{12}^{3} + \zeta_{12}^{2} + 3 \zeta_{12} - 3) q^{30} + ( - \zeta_{12}^{3} + 2 \zeta_{12} + 7) q^{31} + \zeta_{12}^{3} q^{32} + (\zeta_{12}^{3} - 2 \zeta_{12}) q^{34} + (2 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + 5 \zeta_{12} + 1) q^{35} + (2 \zeta_{12}^{3} - 4 \zeta_{12} + 1) q^{36} + ( - \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 4) q^{37} + ( - \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 3) q^{38} + (3 \zeta_{12}^{3} - 6 \zeta_{12} + 7) q^{39} + (\zeta_{12}^{2} - 2 \zeta_{12} - 1) q^{40} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12} + 7) q^{41} + ( - 6 \zeta_{12}^{3} + 8 \zeta_{12}^{2} - 4) q^{42} + ( - 6 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 2) q^{43} + ( - 8 \zeta_{12}^{3} + 5 \zeta_{12} - 4) q^{45} + ( - \zeta_{12}^{3} + 2 \zeta_{12} + 1) q^{46} + (5 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 1) q^{47} + (\zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{48} + ( - 6 \zeta_{12}^{3} + 12 \zeta_{12} - 5) q^{49} + ( - 3 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 3 \zeta_{12}) q^{50} + (\zeta_{12}^{3} - 2 \zeta_{12} + 3) q^{51} + (\zeta_{12}^{3} - 4 \zeta_{12}^{2} + 2) q^{52} - 3 \zeta_{12}^{3} q^{53} + 4 q^{54} + (\zeta_{12}^{3} - 2 \zeta_{12} + 3) q^{56} + ( - 10 \zeta_{12}^{3} + 8 \zeta_{12}^{2} - 4) q^{57} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{58} + (8 \zeta_{12}^{3} - 16 \zeta_{12}) q^{59} + ( - 2 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - \zeta_{12} + 1) q^{60} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12} + 2) q^{61} + (7 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 1) q^{62} + ( - 9 \zeta_{12}^{3} + 14 \zeta_{12}^{2} - 7) q^{63} - q^{64} + ( - 4 \zeta_{12}^{3} + 5 \zeta_{12}^{2} + 3) q^{65} + (9 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{67} + ( - 2 \zeta_{12}^{2} + 1) q^{68} - 2 q^{69} + ( - 4 \zeta_{12}^{3} + 5 \zeta_{12}^{2} + 5 \zeta_{12} - 7) q^{70} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12}) q^{71} + (\zeta_{12}^{3} - 4 \zeta_{12}^{2} + 2) q^{72} + ( - 6 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 4) q^{73} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12} + 1) q^{74} + (\zeta_{12}^{3} - \zeta_{12}^{2} + 7 \zeta_{12} - 6) q^{75} + (3 \zeta_{12}^{3} - 6 \zeta_{12} + 1) q^{76} + (7 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 3) q^{78} + (\zeta_{12}^{3} - 2 \zeta_{12} + 9) q^{79} + ( - 2 \zeta_{12}^{2} - \zeta_{12} + 2) q^{80} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12} + 1) q^{81} + (7 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 2) q^{82} + (5 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{83} + (4 \zeta_{12}^{3} - 8 \zeta_{12} + 6) q^{84} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + \zeta_{12} + 2) q^{85} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12} + 6) q^{86} + (\zeta_{12}^{3} + 2 \zeta_{12}^{2} - 1) q^{87} + (2 \zeta_{12}^{3} - 4 \zeta_{12} + 3) q^{89} + ( - 4 \zeta_{12}^{3} + 5 \zeta_{12}^{2} + 3) q^{90} + (7 \zeta_{12}^{3} - 14 \zeta_{12} + 9) q^{91} + (\zeta_{12}^{3} + 2 \zeta_{12}^{2} - 1) q^{92} + (4 \zeta_{12}^{3} - 12 \zeta_{12}^{2} + 6) q^{93} + (\zeta_{12}^{3} - 2 \zeta_{12} - 5) q^{94} + ( - 12 \zeta_{12}^{3} - \zeta_{12}^{2} + 7 \zeta_{12} - 5) q^{95} + ( - \zeta_{12}^{3} + 2 \zeta_{12} - 1) q^{96} + ( - 12 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 3) q^{97} + ( - 5 \zeta_{12}^{3} + 12 \zeta_{12}^{2} - 6) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 4 q^{5} - 4 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 4 q^{5} - 4 q^{6} - 4 q^{9} + 2 q^{10} - 12 q^{14} - 10 q^{15} + 4 q^{16} - 4 q^{19} - 4 q^{20} - 24 q^{21} + 4 q^{24} - 6 q^{25} + 4 q^{26} - 8 q^{29} - 10 q^{30} + 28 q^{31} - 6 q^{35} + 4 q^{36} + 28 q^{39} - 2 q^{40} + 28 q^{41} - 16 q^{45} + 4 q^{46} - 20 q^{49} - 8 q^{50} + 12 q^{51} + 16 q^{54} + 12 q^{56} + 10 q^{60} + 8 q^{61} - 4 q^{64} + 22 q^{65} - 8 q^{69} - 18 q^{70} + 4 q^{74} - 26 q^{75} + 4 q^{76} + 36 q^{79} + 4 q^{80} + 4 q^{81} + 24 q^{84} + 12 q^{85} + 24 q^{86} + 12 q^{89} + 22 q^{90} + 36 q^{91} - 20 q^{94} - 22 q^{95} - 4 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1210\mathbb{Z}\right)^\times\).

\(n\) \(727\) \(1091\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
969.1
−0.866025 0.500000i
0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
1.00000i 2.73205i −1.00000 1.86603 1.23205i −2.73205 4.73205i 1.00000i −4.46410 −1.23205 1.86603i
969.2 1.00000i 0.732051i −1.00000 0.133975 + 2.23205i 0.732051 1.26795i 1.00000i 2.46410 2.23205 0.133975i
969.3 1.00000i 0.732051i −1.00000 0.133975 2.23205i 0.732051 1.26795i 1.00000i 2.46410 2.23205 + 0.133975i
969.4 1.00000i 2.73205i −1.00000 1.86603 + 1.23205i −2.73205 4.73205i 1.00000i −4.46410 −1.23205 + 1.86603i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1210.2.b.i 4
5.b even 2 1 inner 1210.2.b.i 4
5.c odd 4 1 6050.2.a.cd 2
5.c odd 4 1 6050.2.a.cn 2
11.b odd 2 1 1210.2.b.j yes 4
55.d odd 2 1 1210.2.b.j yes 4
55.e even 4 1 6050.2.a.bt 2
55.e even 4 1 6050.2.a.cu 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1210.2.b.i 4 1.a even 1 1 trivial
1210.2.b.i 4 5.b even 2 1 inner
1210.2.b.j yes 4 11.b odd 2 1
1210.2.b.j yes 4 55.d odd 2 1
6050.2.a.bt 2 55.e even 4 1
6050.2.a.cd 2 5.c odd 4 1
6050.2.a.cn 2 5.c odd 4 1
6050.2.a.cu 2 55.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1210, [\chi])\):

\( T_{3}^{4} + 8T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{13}^{4} + 26T_{13}^{2} + 121 \) Copy content Toggle raw display
\( T_{19}^{2} + 2T_{19} - 26 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 8T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{4} - 4 T^{3} + 11 T^{2} - 20 T + 25 \) Copy content Toggle raw display
$7$ \( T^{4} + 24T^{2} + 36 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 26T^{2} + 121 \) Copy content Toggle raw display
$17$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 2 T - 26)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 8T^{2} + 4 \) Copy content Toggle raw display
$29$ \( (T^{2} + 4 T + 1)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 14 T + 46)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 98T^{2} + 2209 \) Copy content Toggle raw display
$41$ \( (T^{2} - 14 T + 37)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 96T^{2} + 576 \) Copy content Toggle raw display
$47$ \( T^{4} + 56T^{2} + 484 \) Copy content Toggle raw display
$53$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 192)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 4 T - 44)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 168T^{2} + 6084 \) Copy content Toggle raw display
$71$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 168T^{2} + 144 \) Copy content Toggle raw display
$79$ \( (T^{2} - 18 T + 78)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 56T^{2} + 484 \) Copy content Toggle raw display
$89$ \( (T^{2} - 6 T - 3)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 342 T^{2} + 13689 \) Copy content Toggle raw display
show more
show less