Properties

Label 1274.2.g.b
Level $1274$
Weight $2$
Character orbit 1274.g
Analytic conductor $10.173$
Analytic rank $0$
Dimension $2$
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] = [1274,2,Mod(295,1274)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1274, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([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("1274.295");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1274 = 2 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1274.g (of order \(3\), 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: \(10.1729412175\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 182)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{2} + (2 \zeta_{6} - 2) q^{3} - \zeta_{6} q^{4} - q^{5} + 2 \zeta_{6} q^{6} - q^{8} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{6} + 1) q^{2} + (2 \zeta_{6} - 2) q^{3} - \zeta_{6} q^{4} - q^{5} + 2 \zeta_{6} q^{6} - q^{8} - \zeta_{6} q^{9} + (\zeta_{6} - 1) q^{10} + (2 \zeta_{6} - 2) q^{11} + 2 q^{12} + (\zeta_{6} + 3) q^{13} + ( - 2 \zeta_{6} + 2) q^{15} + (\zeta_{6} - 1) q^{16} + \zeta_{6} q^{17} - q^{18} - 4 \zeta_{6} q^{19} + \zeta_{6} q^{20} + 2 \zeta_{6} q^{22} + (2 \zeta_{6} - 2) q^{23} + ( - 2 \zeta_{6} + 2) q^{24} - 4 q^{25} + ( - 3 \zeta_{6} + 4) q^{26} - 4 q^{27} + ( - 5 \zeta_{6} + 5) q^{29} - 2 \zeta_{6} q^{30} - 6 q^{31} + \zeta_{6} q^{32} - 4 \zeta_{6} q^{33} + q^{34} + (\zeta_{6} - 1) q^{36} + (7 \zeta_{6} - 7) q^{37} - 4 q^{38} + (6 \zeta_{6} - 8) q^{39} + q^{40} + (7 \zeta_{6} - 7) q^{41} + 2 \zeta_{6} q^{43} + 2 q^{44} + \zeta_{6} q^{45} + 2 \zeta_{6} q^{46} - 2 \zeta_{6} q^{48} + (4 \zeta_{6} - 4) q^{50} - 2 q^{51} + ( - 4 \zeta_{6} + 1) q^{52} - 9 q^{53} + (4 \zeta_{6} - 4) q^{54} + ( - 2 \zeta_{6} + 2) q^{55} + 8 q^{57} - 5 \zeta_{6} q^{58} - 6 \zeta_{6} q^{59} - 2 q^{60} + 5 \zeta_{6} q^{61} + (6 \zeta_{6} - 6) q^{62} + q^{64} + ( - \zeta_{6} - 3) q^{65} - 4 q^{66} + (10 \zeta_{6} - 10) q^{67} + ( - \zeta_{6} + 1) q^{68} - 4 \zeta_{6} q^{69} - 16 \zeta_{6} q^{71} + \zeta_{6} q^{72} + 3 q^{73} + 7 \zeta_{6} q^{74} + ( - 8 \zeta_{6} + 8) q^{75} + (4 \zeta_{6} - 4) q^{76} + (8 \zeta_{6} - 2) q^{78} - 10 q^{79} + ( - \zeta_{6} + 1) q^{80} + ( - 11 \zeta_{6} + 11) q^{81} + 7 \zeta_{6} q^{82} - 14 q^{83} - \zeta_{6} q^{85} + 2 q^{86} + 10 \zeta_{6} q^{87} + ( - 2 \zeta_{6} + 2) q^{88} + (6 \zeta_{6} - 6) q^{89} + q^{90} + 2 q^{92} + ( - 12 \zeta_{6} + 12) q^{93} + 4 \zeta_{6} q^{95} - 2 q^{96} + 2 \zeta_{6} q^{97} + 2 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 2 q^{3} - q^{4} - 2 q^{5} + 2 q^{6} - 2 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - 2 q^{3} - q^{4} - 2 q^{5} + 2 q^{6} - 2 q^{8} - q^{9} - q^{10} - 2 q^{11} + 4 q^{12} + 7 q^{13} + 2 q^{15} - q^{16} + q^{17} - 2 q^{18} - 4 q^{19} + q^{20} + 2 q^{22} - 2 q^{23} + 2 q^{24} - 8 q^{25} + 5 q^{26} - 8 q^{27} + 5 q^{29} - 2 q^{30} - 12 q^{31} + q^{32} - 4 q^{33} + 2 q^{34} - q^{36} - 7 q^{37} - 8 q^{38} - 10 q^{39} + 2 q^{40} - 7 q^{41} + 2 q^{43} + 4 q^{44} + q^{45} + 2 q^{46} - 2 q^{48} - 4 q^{50} - 4 q^{51} - 2 q^{52} - 18 q^{53} - 4 q^{54} + 2 q^{55} + 16 q^{57} - 5 q^{58} - 6 q^{59} - 4 q^{60} + 5 q^{61} - 6 q^{62} + 2 q^{64} - 7 q^{65} - 8 q^{66} - 10 q^{67} + q^{68} - 4 q^{69} - 16 q^{71} + q^{72} + 6 q^{73} + 7 q^{74} + 8 q^{75} - 4 q^{76} + 4 q^{78} - 20 q^{79} + q^{80} + 11 q^{81} + 7 q^{82} - 28 q^{83} - q^{85} + 4 q^{86} + 10 q^{87} + 2 q^{88} - 6 q^{89} + 2 q^{90} + 4 q^{92} + 12 q^{93} + 4 q^{95} - 4 q^{96} + 2 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1274\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(885\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
295.1
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 0.866025i −1.00000 + 1.73205i −0.500000 0.866025i −1.00000 1.00000 + 1.73205i 0 −1.00000 −0.500000 0.866025i −0.500000 + 0.866025i
393.1 0.500000 + 0.866025i −1.00000 1.73205i −0.500000 + 0.866025i −1.00000 1.00000 1.73205i 0 −1.00000 −0.500000 + 0.866025i −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.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1274.2.g.b 2
7.b odd 2 1 182.2.g.d 2
7.c even 3 1 1274.2.e.c 2
7.c even 3 1 1274.2.h.m 2
7.d odd 6 1 1274.2.e.j 2
7.d odd 6 1 1274.2.h.d 2
13.c even 3 1 inner 1274.2.g.b 2
21.c even 2 1 1638.2.r.e 2
28.d even 2 1 1456.2.s.b 2
91.g even 3 1 1274.2.e.c 2
91.h even 3 1 1274.2.h.m 2
91.m odd 6 1 1274.2.e.j 2
91.n odd 6 1 182.2.g.d 2
91.n odd 6 1 2366.2.a.b 1
91.t odd 6 1 2366.2.a.i 1
91.v odd 6 1 1274.2.h.d 2
91.bc even 12 2 2366.2.d.c 2
273.bn even 6 1 1638.2.r.e 2
364.v even 6 1 1456.2.s.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.g.d 2 7.b odd 2 1
182.2.g.d 2 91.n odd 6 1
1274.2.e.c 2 7.c even 3 1
1274.2.e.c 2 91.g even 3 1
1274.2.e.j 2 7.d odd 6 1
1274.2.e.j 2 91.m odd 6 1
1274.2.g.b 2 1.a even 1 1 trivial
1274.2.g.b 2 13.c even 3 1 inner
1274.2.h.d 2 7.d odd 6 1
1274.2.h.d 2 91.v odd 6 1
1274.2.h.m 2 7.c even 3 1
1274.2.h.m 2 91.h even 3 1
1456.2.s.b 2 28.d even 2 1
1456.2.s.b 2 364.v even 6 1
1638.2.r.e 2 21.c even 2 1
1638.2.r.e 2 273.bn even 6 1
2366.2.a.b 1 91.n odd 6 1
2366.2.a.i 1 91.t odd 6 1
2366.2.d.c 2 91.bc even 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1274, [\chi])\):

\( T_{3}^{2} + 2T_{3} + 4 \) Copy content Toggle raw display
\( T_{5} + 1 \) Copy content Toggle raw display
\( T_{11}^{2} + 2T_{11} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$13$ \( T^{2} - 7T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$19$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$23$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$29$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$31$ \( (T + 6)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$41$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$43$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( (T + 9)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$61$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$67$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$71$ \( T^{2} + 16T + 256 \) Copy content Toggle raw display
$73$ \( (T - 3)^{2} \) Copy content Toggle raw display
$79$ \( (T + 10)^{2} \) Copy content Toggle raw display
$83$ \( (T + 14)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$97$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
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