Properties

Label 1274.2.h.e
Level $1274$
Weight $2$
Character orbit 1274.h
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(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.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{2} - 2 q^{3} + (\zeta_{6} - 1) q^{4} + ( - 3 \zeta_{6} + 3) q^{5} - 2 \zeta_{6} q^{6} - q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{6} q^{2} - 2 q^{3} + (\zeta_{6} - 1) q^{4} + ( - 3 \zeta_{6} + 3) q^{5} - 2 \zeta_{6} q^{6} - q^{8} + q^{9} + 3 q^{10} - 6 q^{11} + ( - 2 \zeta_{6} + 2) q^{12} + (3 \zeta_{6} + 1) q^{13} + (6 \zeta_{6} - 6) q^{15} - \zeta_{6} q^{16} + ( - 3 \zeta_{6} + 3) q^{17} + \zeta_{6} q^{18} - 4 q^{19} + 3 \zeta_{6} q^{20} - 6 \zeta_{6} q^{22} + 6 \zeta_{6} q^{23} + 2 q^{24} - 4 \zeta_{6} q^{25} + (4 \zeta_{6} - 3) q^{26} + 4 q^{27} + (3 \zeta_{6} - 3) q^{29} - 6 q^{30} + 10 \zeta_{6} q^{31} + ( - \zeta_{6} + 1) q^{32} + 12 q^{33} + 3 q^{34} + (\zeta_{6} - 1) q^{36} + \zeta_{6} q^{37} - 4 \zeta_{6} q^{38} + ( - 6 \zeta_{6} - 2) q^{39} + (3 \zeta_{6} - 3) q^{40} + ( - 3 \zeta_{6} + 3) q^{41} + 10 \zeta_{6} q^{43} + ( - 6 \zeta_{6} + 6) q^{44} + ( - 3 \zeta_{6} + 3) q^{45} + (6 \zeta_{6} - 6) q^{46} + 2 \zeta_{6} q^{48} + ( - 4 \zeta_{6} + 4) q^{50} + (6 \zeta_{6} - 6) q^{51} + (\zeta_{6} - 4) q^{52} + 9 \zeta_{6} q^{53} + 4 \zeta_{6} q^{54} + (18 \zeta_{6} - 18) q^{55} + 8 q^{57} - 3 q^{58} + ( - 6 \zeta_{6} + 6) q^{59} - 6 \zeta_{6} q^{60} - 7 q^{61} + (10 \zeta_{6} - 10) q^{62} + q^{64} + ( - 3 \zeta_{6} + 12) q^{65} + 12 \zeta_{6} q^{66} + 2 q^{67} + 3 \zeta_{6} q^{68} - 12 \zeta_{6} q^{69} - q^{72} + 7 \zeta_{6} q^{73} + (\zeta_{6} - 1) q^{74} + 8 \zeta_{6} q^{75} + ( - 4 \zeta_{6} + 4) q^{76} + ( - 8 \zeta_{6} + 6) q^{78} + (14 \zeta_{6} - 14) q^{79} - 3 q^{80} - 11 q^{81} + 3 q^{82} - 18 q^{83} - 9 \zeta_{6} q^{85} + (10 \zeta_{6} - 10) q^{86} + ( - 6 \zeta_{6} + 6) q^{87} + 6 q^{88} + 6 \zeta_{6} q^{89} + 3 q^{90} - 6 q^{92} - 20 \zeta_{6} q^{93} + (12 \zeta_{6} - 12) q^{95} + (2 \zeta_{6} - 2) q^{96} - 2 \zeta_{6} q^{97} - 6 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 4 q^{3} - q^{4} + 3 q^{5} - 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - 4 q^{3} - q^{4} + 3 q^{5} - 2 q^{6} - 2 q^{8} + 2 q^{9} + 6 q^{10} - 12 q^{11} + 2 q^{12} + 5 q^{13} - 6 q^{15} - q^{16} + 3 q^{17} + q^{18} - 8 q^{19} + 3 q^{20} - 6 q^{22} + 6 q^{23} + 4 q^{24} - 4 q^{25} - 2 q^{26} + 8 q^{27} - 3 q^{29} - 12 q^{30} + 10 q^{31} + q^{32} + 24 q^{33} + 6 q^{34} - q^{36} + q^{37} - 4 q^{38} - 10 q^{39} - 3 q^{40} + 3 q^{41} + 10 q^{43} + 6 q^{44} + 3 q^{45} - 6 q^{46} + 2 q^{48} + 4 q^{50} - 6 q^{51} - 7 q^{52} + 9 q^{53} + 4 q^{54} - 18 q^{55} + 16 q^{57} - 6 q^{58} + 6 q^{59} - 6 q^{60} - 14 q^{61} - 10 q^{62} + 2 q^{64} + 21 q^{65} + 12 q^{66} + 4 q^{67} + 3 q^{68} - 12 q^{69} - 2 q^{72} + 7 q^{73} - q^{74} + 8 q^{75} + 4 q^{76} + 4 q^{78} - 14 q^{79} - 6 q^{80} - 22 q^{81} + 6 q^{82} - 36 q^{83} - 9 q^{85} - 10 q^{86} + 6 q^{87} + 12 q^{88} + 6 q^{89} + 6 q^{90} - 12 q^{92} - 20 q^{93} - 12 q^{95} - 2 q^{96} - 2 q^{97} - 12 q^{99}+O(q^{100}) \) 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 −2.00000 −0.500000 0.866025i 1.50000 + 2.59808i −1.00000 + 1.73205i 0 −1.00000 1.00000 3.00000
373.1 0.500000 + 0.866025i −2.00000 −0.500000 + 0.866025i 1.50000 2.59808i −1.00000 1.73205i 0 −1.00000 1.00000 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.e 2
7.b odd 2 1 1274.2.h.l 2
7.c even 3 1 182.2.g.c 2
7.c even 3 1 1274.2.e.k 2
7.d odd 6 1 1274.2.e.b 2
7.d odd 6 1 1274.2.g.c 2
13.c even 3 1 1274.2.e.k 2
21.h odd 6 1 1638.2.r.m 2
28.g odd 6 1 1456.2.s.a 2
91.g even 3 1 inner 1274.2.h.e 2
91.g even 3 1 2366.2.a.a 1
91.h even 3 1 182.2.g.c 2
91.m odd 6 1 1274.2.h.l 2
91.n odd 6 1 1274.2.e.b 2
91.u even 6 1 2366.2.a.k 1
91.v odd 6 1 1274.2.g.c 2
91.bd odd 12 2 2366.2.d.a 2
273.s odd 6 1 1638.2.r.m 2
364.bi odd 6 1 1456.2.s.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.g.c 2 7.c even 3 1
182.2.g.c 2 91.h even 3 1
1274.2.e.b 2 7.d odd 6 1
1274.2.e.b 2 91.n odd 6 1
1274.2.e.k 2 7.c even 3 1
1274.2.e.k 2 13.c even 3 1
1274.2.g.c 2 7.d odd 6 1
1274.2.g.c 2 91.v odd 6 1
1274.2.h.e 2 1.a even 1 1 trivial
1274.2.h.e 2 91.g even 3 1 inner
1274.2.h.l 2 7.b odd 2 1
1274.2.h.l 2 91.m odd 6 1
1456.2.s.a 2 28.g odd 6 1
1456.2.s.a 2 364.bi odd 6 1
1638.2.r.m 2 21.h odd 6 1
1638.2.r.m 2 273.s odd 6 1
2366.2.a.a 1 91.g even 3 1
2366.2.a.k 1 91.u even 6 1
2366.2.d.a 2 91.bd odd 12 2

Hecke kernels

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

\( T_{3} + 2 \) Copy content Toggle raw display
\( T_{5}^{2} - 3T_{5} + 9 \) Copy content Toggle raw display
\( T_{11} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( (T + 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T + 6)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 5T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$19$ \( (T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$29$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$31$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$37$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$41$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$43$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$59$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$61$ \( (T + 7)^{2} \) Copy content Toggle raw display
$67$ \( (T - 2)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$79$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$83$ \( (T + 18)^{2} \) 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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