Properties

Label 1014.4.a.g
Level $1014$
Weight $4$
Character orbit 1014.a
Self dual yes
Analytic conductor $59.828$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1014,4,Mod(1,1014)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1014, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1014.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1014 = 2 \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1014.a (trivial)

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: yes
Analytic conductor: \(59.8279367458\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 6)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 2 q^{2} - 3 q^{3} + 4 q^{4} - 6 q^{5} - 6 q^{6} + 16 q^{7} + 8 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} - 3 q^{3} + 4 q^{4} - 6 q^{5} - 6 q^{6} + 16 q^{7} + 8 q^{8} + 9 q^{9} - 12 q^{10} - 12 q^{11} - 12 q^{12} + 32 q^{14} + 18 q^{15} + 16 q^{16} - 126 q^{17} + 18 q^{18} - 20 q^{19} - 24 q^{20} - 48 q^{21} - 24 q^{22} + 168 q^{23} - 24 q^{24} - 89 q^{25} - 27 q^{27} + 64 q^{28} + 30 q^{29} + 36 q^{30} + 88 q^{31} + 32 q^{32} + 36 q^{33} - 252 q^{34} - 96 q^{35} + 36 q^{36} - 254 q^{37} - 40 q^{38} - 48 q^{40} - 42 q^{41} - 96 q^{42} - 52 q^{43} - 48 q^{44} - 54 q^{45} + 336 q^{46} + 96 q^{47} - 48 q^{48} - 87 q^{49} - 178 q^{50} + 378 q^{51} + 198 q^{53} - 54 q^{54} + 72 q^{55} + 128 q^{56} + 60 q^{57} + 60 q^{58} + 660 q^{59} + 72 q^{60} - 538 q^{61} + 176 q^{62} + 144 q^{63} + 64 q^{64} + 72 q^{66} - 884 q^{67} - 504 q^{68} - 504 q^{69} - 192 q^{70} - 792 q^{71} + 72 q^{72} - 218 q^{73} - 508 q^{74} + 267 q^{75} - 80 q^{76} - 192 q^{77} - 520 q^{79} - 96 q^{80} + 81 q^{81} - 84 q^{82} + 492 q^{83} - 192 q^{84} + 756 q^{85} - 104 q^{86} - 90 q^{87} - 96 q^{88} - 810 q^{89} - 108 q^{90} + 672 q^{92} - 264 q^{93} + 192 q^{94} + 120 q^{95} - 96 q^{96} - 1154 q^{97} - 174 q^{98} - 108 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
2.00000 −3.00000 4.00000 −6.00000 −6.00000 16.0000 8.00000 9.00000 −12.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(13\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1014.4.a.g 1
13.b even 2 1 6.4.a.a 1
13.d odd 4 2 1014.4.b.d 2
39.d odd 2 1 18.4.a.a 1
52.b odd 2 1 48.4.a.c 1
65.d even 2 1 150.4.a.i 1
65.h odd 4 2 150.4.c.d 2
91.b odd 2 1 294.4.a.e 1
91.r even 6 2 294.4.e.h 2
91.s odd 6 2 294.4.e.g 2
104.e even 2 1 192.4.a.i 1
104.h odd 2 1 192.4.a.c 1
117.n odd 6 2 162.4.c.c 2
117.t even 6 2 162.4.c.f 2
143.d odd 2 1 726.4.a.f 1
156.h even 2 1 144.4.a.c 1
195.e odd 2 1 450.4.a.h 1
195.s even 4 2 450.4.c.e 2
208.o odd 4 2 768.4.d.c 2
208.p even 4 2 768.4.d.n 2
221.b even 2 1 1734.4.a.d 1
247.d odd 2 1 2166.4.a.i 1
260.g odd 2 1 1200.4.a.b 1
260.p even 4 2 1200.4.f.j 2
273.g even 2 1 882.4.a.n 1
273.w odd 6 2 882.4.g.i 2
273.ba even 6 2 882.4.g.f 2
312.b odd 2 1 576.4.a.q 1
312.h even 2 1 576.4.a.r 1
364.h even 2 1 2352.4.a.e 1
429.e even 2 1 2178.4.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.4.a.a 1 13.b even 2 1
18.4.a.a 1 39.d odd 2 1
48.4.a.c 1 52.b odd 2 1
144.4.a.c 1 156.h even 2 1
150.4.a.i 1 65.d even 2 1
150.4.c.d 2 65.h odd 4 2
162.4.c.c 2 117.n odd 6 2
162.4.c.f 2 117.t even 6 2
192.4.a.c 1 104.h odd 2 1
192.4.a.i 1 104.e even 2 1
294.4.a.e 1 91.b odd 2 1
294.4.e.g 2 91.s odd 6 2
294.4.e.h 2 91.r even 6 2
450.4.a.h 1 195.e odd 2 1
450.4.c.e 2 195.s even 4 2
576.4.a.q 1 312.b odd 2 1
576.4.a.r 1 312.h even 2 1
726.4.a.f 1 143.d odd 2 1
768.4.d.c 2 208.o odd 4 2
768.4.d.n 2 208.p even 4 2
882.4.a.n 1 273.g even 2 1
882.4.g.f 2 273.ba even 6 2
882.4.g.i 2 273.w odd 6 2
1014.4.a.g 1 1.a even 1 1 trivial
1014.4.b.d 2 13.d odd 4 2
1200.4.a.b 1 260.g odd 2 1
1200.4.f.j 2 260.p even 4 2
1734.4.a.d 1 221.b even 2 1
2166.4.a.i 1 247.d odd 2 1
2178.4.a.e 1 429.e even 2 1
2352.4.a.e 1 364.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1014))\):

\( T_{5} + 6 \) Copy content Toggle raw display
\( T_{7} - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 2 \) Copy content Toggle raw display
$3$ \( T + 3 \) Copy content Toggle raw display
$5$ \( T + 6 \) Copy content Toggle raw display
$7$ \( T - 16 \) Copy content Toggle raw display
$11$ \( T + 12 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T + 126 \) Copy content Toggle raw display
$19$ \( T + 20 \) Copy content Toggle raw display
$23$ \( T - 168 \) Copy content Toggle raw display
$29$ \( T - 30 \) Copy content Toggle raw display
$31$ \( T - 88 \) Copy content Toggle raw display
$37$ \( T + 254 \) Copy content Toggle raw display
$41$ \( T + 42 \) Copy content Toggle raw display
$43$ \( T + 52 \) Copy content Toggle raw display
$47$ \( T - 96 \) Copy content Toggle raw display
$53$ \( T - 198 \) Copy content Toggle raw display
$59$ \( T - 660 \) Copy content Toggle raw display
$61$ \( T + 538 \) Copy content Toggle raw display
$67$ \( T + 884 \) Copy content Toggle raw display
$71$ \( T + 792 \) Copy content Toggle raw display
$73$ \( T + 218 \) Copy content Toggle raw display
$79$ \( T + 520 \) Copy content Toggle raw display
$83$ \( T - 492 \) Copy content Toggle raw display
$89$ \( T + 810 \) Copy content Toggle raw display
$97$ \( T + 1154 \) Copy content Toggle raw display
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