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Properties

Label 1170.2.bs.e
Level $1170$
Weight $2$
Character orbit 1170.bs
Analytic conductor $9.342$
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] = [1170,2,Mod(361,1170)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1170, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 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("1170.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1170.bs (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: \(9.34249703649\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 390)
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.

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

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1170\mathbb{Z}\right)^\times\).

\(n\) \(911\) \(937\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(1 - \zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
0.866025 + 0.500000i
−0.866025 − 0.500000i
0.866025 − 0.500000i
−0.866025 + 0.500000i
−0.866025 + 0.500000i 0 0.500000 − 0.866025i − 1.00000i 0 1.73205 + 1.00000i 1.00000i 0 0.500000 + 0.866025i
361.2 0.866025 − 0.500000i 0 0.500000 − 0.866025i 1.00000i 0 −1.73205 − 1.00000i − 1.00000i 0 0.500000 + 0.866025i
901.1 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 1.00000i 0 1.73205 − 1.00000i − 1.00000i 0 0.500000 − 0.866025i
901.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i − 1.00000i 0 −1.73205 + 1.00000i 1.00000i 0 0.500000 − 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1170.2.bs.e 4
3.b odd 2 1 390.2.bb.b ✓ 4
13.e even 6 1 inner 1170.2.bs.e 4
15.d odd 2 1 1950.2.bc.b 4
15.e even 4 1 1950.2.y.c 4
15.e even 4 1 1950.2.y.f 4
39.h odd 6 1 390.2.bb.b ✓ 4
39.h odd 6 1 5070.2.b.o 4
39.i odd 6 1 5070.2.b.o 4
39.k even 12 1 5070.2.a.y 2
39.k even 12 1 5070.2.a.bg 2
195.y odd 6 1 1950.2.bc.b 4
195.bf even 12 1 1950.2.y.c 4
195.bf even 12 1 1950.2.y.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.bb.b ✓ 4 3.b odd 2 1
390.2.bb.b ✓ 4 39.h odd 6 1
1170.2.bs.e 4 1.a even 1 1 trivial
1170.2.bs.e 4 13.e even 6 1 inner
1950.2.y.c 4 15.e even 4 1
1950.2.y.c 4 195.bf even 12 1
1950.2.y.f 4 15.e even 4 1
1950.2.y.f 4 195.bf even 12 1
1950.2.bc.b 4 15.d odd 2 1
1950.2.bc.b 4 195.y odd 6 1
5070.2.a.y 2 39.k even 12 1
5070.2.a.bg 2 39.k even 12 1
5070.2.b.o 4 39.h odd 6 1
5070.2.b.o 4 39.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1170, [\chi])\):

\( T_{7}^{4} - 4T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{11}^{4} + 12T_{11}^{3} + 51T_{11}^{2} + 36T_{11} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$11$ \( T^{4} + 12 T^{3} + 51 T^{2} + 36 T + 9 \) Copy content Toggle raw display
$13$ \( (T^{2} - 2 T + 13)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - 12 T^{3} + 44 T^{2} + 48 T + 16 \) Copy content Toggle raw display
$23$ \( T^{4} - 4 T^{3} + 15 T^{2} - 4 T + 1 \) Copy content Toggle raw display
$29$ \( T^{4} - 4 T^{3} + 15 T^{2} - 4 T + 1 \) Copy content Toggle raw display
$31$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 18 T^{3} + 119 T^{2} + \cdots + 121 \) Copy content Toggle raw display
$41$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$43$ \( T^{4} + 10 T^{3} + 123 T^{2} + \cdots + 529 \) Copy content Toggle raw display
$47$ \( T^{4} + 122T^{2} + 1369 \) Copy content Toggle raw display
$53$ \( (T^{2} - 12 T - 12)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 12 T^{3} + 35 T^{2} + \cdots + 169 \) Copy content Toggle raw display
$61$ \( T^{4} + 108 T^{2} + 11664 \) Copy content Toggle raw display
$67$ \( T^{4} + 12 T^{3} - 4 T^{2} + \cdots + 2704 \) Copy content Toggle raw display
$71$ \( T^{4} - 36 T^{3} + 536 T^{2} + \cdots + 10816 \) Copy content Toggle raw display
$73$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 14 T + 1)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 104T^{2} + 1936 \) Copy content Toggle raw display
$89$ \( T^{4} + 12 T^{3} + 44 T^{2} - 48 T + 16 \) Copy content Toggle raw display
$97$ \( T^{4} + 12 T^{3} + 44 T^{2} - 48 T + 16 \) Copy content Toggle raw display
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