Properties

Label 1274.2.f.a
Level $1274$
Weight $2$
Character orbit 1274.f
Analytic conductor $10.173$
Analytic rank $0$
Dimension $2$
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(79,1274)]
 
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([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1274.79");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1274 = 2 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1274.f (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 26)
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.

\(f(q)\) \(=\) \( q - \zeta_{6} q^{2} + (3 \zeta_{6} - 3) q^{3} + (\zeta_{6} - 1) q^{4} - \zeta_{6} q^{5} + 3 q^{6} + q^{8} - 6 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{6} q^{2} + (3 \zeta_{6} - 3) q^{3} + (\zeta_{6} - 1) q^{4} - \zeta_{6} q^{5} + 3 q^{6} + q^{8} - 6 \zeta_{6} q^{9} + (\zeta_{6} - 1) q^{10} + ( - 2 \zeta_{6} + 2) q^{11} - 3 \zeta_{6} q^{12} + q^{13} + 3 q^{15} - \zeta_{6} q^{16} + (3 \zeta_{6} - 3) q^{17} + (6 \zeta_{6} - 6) q^{18} + 6 \zeta_{6} q^{19} + q^{20} - 2 q^{22} + 4 \zeta_{6} q^{23} + (3 \zeta_{6} - 3) q^{24} + ( - 4 \zeta_{6} + 4) q^{25} - \zeta_{6} q^{26} + 9 q^{27} + 2 q^{29} - 3 \zeta_{6} q^{30} + ( - 4 \zeta_{6} + 4) q^{31} + (\zeta_{6} - 1) q^{32} + 6 \zeta_{6} q^{33} + 3 q^{34} + 6 q^{36} - 3 \zeta_{6} q^{37} + ( - 6 \zeta_{6} + 6) q^{38} + (3 \zeta_{6} - 3) q^{39} - \zeta_{6} q^{40} - 5 q^{43} + 2 \zeta_{6} q^{44} + (6 \zeta_{6} - 6) q^{45} + ( - 4 \zeta_{6} + 4) q^{46} + 13 \zeta_{6} q^{47} + 3 q^{48} - 4 q^{50} - 9 \zeta_{6} q^{51} + (\zeta_{6} - 1) q^{52} + (12 \zeta_{6} - 12) q^{53} - 9 \zeta_{6} q^{54} - 2 q^{55} - 18 q^{57} - 2 \zeta_{6} q^{58} + (10 \zeta_{6} - 10) q^{59} + (3 \zeta_{6} - 3) q^{60} - 8 \zeta_{6} q^{61} - 4 q^{62} + q^{64} - \zeta_{6} q^{65} + ( - 6 \zeta_{6} + 6) q^{66} + ( - 2 \zeta_{6} + 2) q^{67} - 3 \zeta_{6} q^{68} - 12 q^{69} - 5 q^{71} - 6 \zeta_{6} q^{72} + (10 \zeta_{6} - 10) q^{73} + (3 \zeta_{6} - 3) q^{74} + 12 \zeta_{6} q^{75} - 6 q^{76} + 3 q^{78} + 4 \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{80} + (9 \zeta_{6} - 9) q^{81} + 3 q^{85} + 5 \zeta_{6} q^{86} + (6 \zeta_{6} - 6) q^{87} + ( - 2 \zeta_{6} + 2) q^{88} + 6 \zeta_{6} q^{89} + 6 q^{90} - 4 q^{92} + 12 \zeta_{6} q^{93} + ( - 13 \zeta_{6} + 13) q^{94} + ( - 6 \zeta_{6} + 6) q^{95} - 3 \zeta_{6} q^{96} - 14 q^{97} - 12 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - 3 q^{3} - q^{4} - q^{5} + 6 q^{6} + 2 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - 3 q^{3} - q^{4} - q^{5} + 6 q^{6} + 2 q^{8} - 6 q^{9} - q^{10} + 2 q^{11} - 3 q^{12} + 2 q^{13} + 6 q^{15} - q^{16} - 3 q^{17} - 6 q^{18} + 6 q^{19} + 2 q^{20} - 4 q^{22} + 4 q^{23} - 3 q^{24} + 4 q^{25} - q^{26} + 18 q^{27} + 4 q^{29} - 3 q^{30} + 4 q^{31} - q^{32} + 6 q^{33} + 6 q^{34} + 12 q^{36} - 3 q^{37} + 6 q^{38} - 3 q^{39} - q^{40} - 10 q^{43} + 2 q^{44} - 6 q^{45} + 4 q^{46} + 13 q^{47} + 6 q^{48} - 8 q^{50} - 9 q^{51} - q^{52} - 12 q^{53} - 9 q^{54} - 4 q^{55} - 36 q^{57} - 2 q^{58} - 10 q^{59} - 3 q^{60} - 8 q^{61} - 8 q^{62} + 2 q^{64} - q^{65} + 6 q^{66} + 2 q^{67} - 3 q^{68} - 24 q^{69} - 10 q^{71} - 6 q^{72} - 10 q^{73} - 3 q^{74} + 12 q^{75} - 12 q^{76} + 6 q^{78} + 4 q^{79} - q^{80} - 9 q^{81} + 6 q^{85} + 5 q^{86} - 6 q^{87} + 2 q^{88} + 6 q^{89} + 12 q^{90} - 8 q^{92} + 12 q^{93} + 13 q^{94} + 6 q^{95} - 3 q^{96} - 28 q^{97} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
0.500000 0.866025i
0.500000 + 0.866025i
−0.500000 + 0.866025i −1.50000 2.59808i −0.500000 0.866025i −0.500000 + 0.866025i 3.00000 0 1.00000 −3.00000 + 5.19615i −0.500000 0.866025i
1145.1 −0.500000 0.866025i −1.50000 + 2.59808i −0.500000 + 0.866025i −0.500000 0.866025i 3.00000 0 1.00000 −3.00000 5.19615i −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
7.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.f.a 2
7.b odd 2 1 1274.2.f.l 2
7.c even 3 1 1274.2.a.o 1
7.c even 3 1 inner 1274.2.f.a 2
7.d odd 6 1 26.2.a.b 1
7.d odd 6 1 1274.2.f.l 2
21.g even 6 1 234.2.a.b 1
28.f even 6 1 208.2.a.d 1
35.i odd 6 1 650.2.a.g 1
35.k even 12 2 650.2.b.a 2
56.j odd 6 1 832.2.a.j 1
56.m even 6 1 832.2.a.a 1
63.i even 6 1 2106.2.e.t 2
63.k odd 6 1 2106.2.e.h 2
63.s even 6 1 2106.2.e.t 2
63.t odd 6 1 2106.2.e.h 2
77.i even 6 1 3146.2.a.a 1
84.j odd 6 1 1872.2.a.m 1
91.l odd 6 1 338.2.c.g 2
91.m odd 6 1 338.2.c.c 2
91.p odd 6 1 338.2.c.g 2
91.s odd 6 1 338.2.a.a 1
91.v odd 6 1 338.2.c.c 2
91.w even 12 2 338.2.e.d 4
91.ba even 12 2 338.2.e.d 4
91.bb even 12 2 338.2.b.a 2
105.p even 6 1 5850.2.a.bn 1
105.w odd 12 2 5850.2.e.v 2
112.v even 12 2 3328.2.b.k 2
112.x odd 12 2 3328.2.b.g 2
119.h odd 6 1 7514.2.a.i 1
133.o even 6 1 9386.2.a.f 1
140.s even 6 1 5200.2.a.c 1
168.ba even 6 1 7488.2.a.w 1
168.be odd 6 1 7488.2.a.v 1
273.ba even 6 1 3042.2.a.l 1
273.cb odd 12 2 3042.2.b.f 2
364.x even 6 1 2704.2.a.n 1
364.bw odd 12 2 2704.2.f.j 2
455.bf odd 6 1 8450.2.a.y 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.a.b 1 7.d odd 6 1
208.2.a.d 1 28.f even 6 1
234.2.a.b 1 21.g even 6 1
338.2.a.a 1 91.s odd 6 1
338.2.b.a 2 91.bb even 12 2
338.2.c.c 2 91.m odd 6 1
338.2.c.c 2 91.v odd 6 1
338.2.c.g 2 91.l odd 6 1
338.2.c.g 2 91.p odd 6 1
338.2.e.d 4 91.w even 12 2
338.2.e.d 4 91.ba even 12 2
650.2.a.g 1 35.i odd 6 1
650.2.b.a 2 35.k even 12 2
832.2.a.a 1 56.m even 6 1
832.2.a.j 1 56.j odd 6 1
1274.2.a.o 1 7.c even 3 1
1274.2.f.a 2 1.a even 1 1 trivial
1274.2.f.a 2 7.c even 3 1 inner
1274.2.f.l 2 7.b odd 2 1
1274.2.f.l 2 7.d odd 6 1
1872.2.a.m 1 84.j odd 6 1
2106.2.e.h 2 63.k odd 6 1
2106.2.e.h 2 63.t odd 6 1
2106.2.e.t 2 63.i even 6 1
2106.2.e.t 2 63.s even 6 1
2704.2.a.n 1 364.x even 6 1
2704.2.f.j 2 364.bw odd 12 2
3042.2.a.l 1 273.ba even 6 1
3042.2.b.f 2 273.cb odd 12 2
3146.2.a.a 1 77.i even 6 1
3328.2.b.g 2 112.x odd 12 2
3328.2.b.k 2 112.v even 12 2
5200.2.a.c 1 140.s even 6 1
5850.2.a.bn 1 105.p even 6 1
5850.2.e.v 2 105.w odd 12 2
7488.2.a.v 1 168.be odd 6 1
7488.2.a.w 1 168.ba even 6 1
7514.2.a.i 1 119.h odd 6 1
8450.2.a.y 1 455.bf odd 6 1
9386.2.a.f 1 133.o even 6 1

Hecke kernels

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

\( T_{3}^{2} + 3T_{3} + 9 \) Copy content Toggle raw display
\( T_{5}^{2} + T_{5} + 1 \) 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
$3$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$5$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$13$ \( (T - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$19$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$23$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$29$ \( (T - 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$37$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T + 5)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 13T + 169 \) Copy content Toggle raw display
$53$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$59$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$61$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$67$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$71$ \( (T + 5)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$79$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$97$ \( (T + 14)^{2} \) Copy content Toggle raw display
show more
show less