Properties

Label 1274.2.h.c
Level $1274$
Weight $2$
Character orbit 1274.h
Analytic conductor $10.173$
Analytic rank $0$
Dimension $2$
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] = [1274,2,Mod(263,1274)]
 
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([4, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1274.263");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1274 = 2 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1274.h (of order \(3\), degree \(2\), not 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.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{2} - 3 q^{3} + (\zeta_{6} - 1) q^{4} + (3 \zeta_{6} - 3) q^{5} - 3 \zeta_{6} q^{6} - q^{8} + 6 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{6} q^{2} - 3 q^{3} + (\zeta_{6} - 1) q^{4} + (3 \zeta_{6} - 3) q^{5} - 3 \zeta_{6} q^{6} - q^{8} + 6 q^{9} - 3 q^{10} + 4 q^{11} + ( - 3 \zeta_{6} + 3) q^{12} + ( - 3 \zeta_{6} + 4) q^{13} + ( - 9 \zeta_{6} + 9) q^{15} - \zeta_{6} q^{16} + ( - 2 \zeta_{6} + 2) q^{17} + 6 \zeta_{6} q^{18} + 4 q^{19} - 3 \zeta_{6} q^{20} + 4 \zeta_{6} q^{22} + \zeta_{6} q^{23} + 3 q^{24} - 4 \zeta_{6} q^{25} + (\zeta_{6} + 3) q^{26} - 9 q^{27} + ( - 2 \zeta_{6} + 2) q^{29} + 9 q^{30} + 10 \zeta_{6} q^{31} + ( - \zeta_{6} + 1) q^{32} - 12 q^{33} + 2 q^{34} + (6 \zeta_{6} - 6) q^{36} - 4 \zeta_{6} q^{37} + 4 \zeta_{6} q^{38} + (9 \zeta_{6} - 12) q^{39} + ( - 3 \zeta_{6} + 3) q^{40} + ( - 12 \zeta_{6} + 12) q^{41} + (4 \zeta_{6} - 4) q^{44} + (18 \zeta_{6} - 18) q^{45} + (\zeta_{6} - 1) q^{46} + (10 \zeta_{6} - 10) q^{47} + 3 \zeta_{6} q^{48} + ( - 4 \zeta_{6} + 4) q^{50} + (6 \zeta_{6} - 6) q^{51} + (4 \zeta_{6} - 1) q^{52} + 4 \zeta_{6} q^{53} - 9 \zeta_{6} q^{54} + (12 \zeta_{6} - 12) q^{55} - 12 q^{57} + 2 q^{58} + ( - 9 \zeta_{6} + 9) q^{59} + 9 \zeta_{6} q^{60} - 13 q^{61} + (10 \zeta_{6} - 10) q^{62} + q^{64} + (12 \zeta_{6} - 3) q^{65} - 12 \zeta_{6} q^{66} + 2 q^{67} + 2 \zeta_{6} q^{68} - 3 \zeta_{6} q^{69} + 15 \zeta_{6} q^{71} - 6 q^{72} - 2 \zeta_{6} q^{73} + ( - 4 \zeta_{6} + 4) q^{74} + 12 \zeta_{6} q^{75} + (4 \zeta_{6} - 4) q^{76} + ( - 3 \zeta_{6} - 9) q^{78} + (4 \zeta_{6} - 4) q^{79} + 3 q^{80} + 9 q^{81} + 12 q^{82} + 8 q^{83} + 6 \zeta_{6} q^{85} + (6 \zeta_{6} - 6) q^{87} - 4 q^{88} + 14 \zeta_{6} q^{89} - 18 q^{90} - q^{92} - 30 \zeta_{6} q^{93} - 10 q^{94} + (12 \zeta_{6} - 12) q^{95} + (3 \zeta_{6} - 3) q^{96} + 2 \zeta_{6} q^{97} + 24 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 6 q^{3} - q^{4} - 3 q^{5} - 3 q^{6} - 2 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - 6 q^{3} - q^{4} - 3 q^{5} - 3 q^{6} - 2 q^{8} + 12 q^{9} - 6 q^{10} + 8 q^{11} + 3 q^{12} + 5 q^{13} + 9 q^{15} - q^{16} + 2 q^{17} + 6 q^{18} + 8 q^{19} - 3 q^{20} + 4 q^{22} + q^{23} + 6 q^{24} - 4 q^{25} + 7 q^{26} - 18 q^{27} + 2 q^{29} + 18 q^{30} + 10 q^{31} + q^{32} - 24 q^{33} + 4 q^{34} - 6 q^{36} - 4 q^{37} + 4 q^{38} - 15 q^{39} + 3 q^{40} + 12 q^{41} - 4 q^{44} - 18 q^{45} - q^{46} - 10 q^{47} + 3 q^{48} + 4 q^{50} - 6 q^{51} + 2 q^{52} + 4 q^{53} - 9 q^{54} - 12 q^{55} - 24 q^{57} + 4 q^{58} + 9 q^{59} + 9 q^{60} - 26 q^{61} - 10 q^{62} + 2 q^{64} + 6 q^{65} - 12 q^{66} + 4 q^{67} + 2 q^{68} - 3 q^{69} + 15 q^{71} - 12 q^{72} - 2 q^{73} + 4 q^{74} + 12 q^{75} - 4 q^{76} - 21 q^{78} - 4 q^{79} + 6 q^{80} + 18 q^{81} + 24 q^{82} + 16 q^{83} + 6 q^{85} - 6 q^{87} - 8 q^{88} + 14 q^{89} - 36 q^{90} - 2 q^{92} - 30 q^{93} - 20 q^{94} - 12 q^{95} - 3 q^{96} + 2 q^{97} + 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
263.1
0.500000 0.866025i
0.500000 + 0.866025i
0.500000 0.866025i −3.00000 −0.500000 0.866025i −1.50000 2.59808i −1.50000 + 2.59808i 0 −1.00000 6.00000 −3.00000
373.1 0.500000 + 0.866025i −3.00000 −0.500000 + 0.866025i −1.50000 + 2.59808i −1.50000 2.59808i 0 −1.00000 6.00000 −3.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.g 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.h.c 2
7.b odd 2 1 1274.2.h.n 2
7.c even 3 1 1274.2.e.l 2
7.c even 3 1 1274.2.g.k 2
7.d odd 6 1 182.2.g.a 2
7.d odd 6 1 1274.2.e.a 2
13.c even 3 1 1274.2.e.l 2
21.g even 6 1 1638.2.r.n 2
28.f even 6 1 1456.2.s.g 2
91.g even 3 1 inner 1274.2.h.c 2
91.h even 3 1 1274.2.g.k 2
91.m odd 6 1 1274.2.h.n 2
91.m odd 6 1 2366.2.a.g 1
91.n odd 6 1 1274.2.e.a 2
91.p odd 6 1 2366.2.a.p 1
91.v odd 6 1 182.2.g.a 2
91.w even 12 2 2366.2.d.h 2
273.r even 6 1 1638.2.r.n 2
364.ba even 6 1 1456.2.s.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.g.a 2 7.d odd 6 1
182.2.g.a 2 91.v odd 6 1
1274.2.e.a 2 7.d odd 6 1
1274.2.e.a 2 91.n odd 6 1
1274.2.e.l 2 7.c even 3 1
1274.2.e.l 2 13.c even 3 1
1274.2.g.k 2 7.c even 3 1
1274.2.g.k 2 91.h even 3 1
1274.2.h.c 2 1.a even 1 1 trivial
1274.2.h.c 2 91.g even 3 1 inner
1274.2.h.n 2 7.b odd 2 1
1274.2.h.n 2 91.m odd 6 1
1456.2.s.g 2 28.f even 6 1
1456.2.s.g 2 364.ba even 6 1
1638.2.r.n 2 21.g even 6 1
1638.2.r.n 2 273.r even 6 1
2366.2.a.g 1 91.m odd 6 1
2366.2.a.p 1 91.p odd 6 1
2366.2.d.h 2 91.w even 12 2

Hecke kernels

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

\( T_{3} + 3 \) Copy content Toggle raw display
\( T_{5}^{2} + 3T_{5} + 9 \) Copy content Toggle raw display
\( T_{11} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( (T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T - 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 5T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$19$ \( (T - 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$29$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$31$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$37$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$41$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$53$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$59$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$61$ \( (T + 13)^{2} \) Copy content Toggle raw display
$67$ \( (T - 2)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 15T + 225 \) Copy content Toggle raw display
$73$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$79$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$83$ \( (T - 8)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 14T + 196 \) Copy content Toggle raw display
$97$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
show more
show less