Properties

Label 1014.2.a.n
Level $1014$
Weight $2$
Character orbit 1014.a
Self dual yes
Analytic conductor $8.097$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1014 = 2 \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1014.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.09683076496\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} + ( -\beta_{1} - 2 \beta_{2} ) q^{5} - q^{6} + ( 2 + 2 \beta_{1} - \beta_{2} ) q^{7} + q^{8} + q^{9} +O(q^{10})\) \( q + q^{2} - q^{3} + q^{4} + ( -\beta_{1} - 2 \beta_{2} ) q^{5} - q^{6} + ( 2 + 2 \beta_{1} - \beta_{2} ) q^{7} + q^{8} + q^{9} + ( -\beta_{1} - 2 \beta_{2} ) q^{10} + ( 3 - 3 \beta_{1} + \beta_{2} ) q^{11} - q^{12} + ( 2 + 2 \beta_{1} - \beta_{2} ) q^{14} + ( \beta_{1} + 2 \beta_{2} ) q^{15} + q^{16} + ( -2 + 2 \beta_{2} ) q^{17} + q^{18} + ( -4 + 4 \beta_{1} - 4 \beta_{2} ) q^{19} + ( -\beta_{1} - 2 \beta_{2} ) q^{20} + ( -2 - 2 \beta_{1} + \beta_{2} ) q^{21} + ( 3 - 3 \beta_{1} + \beta_{2} ) q^{22} + ( 2 - 2 \beta_{1} + 4 \beta_{2} ) q^{23} - q^{24} + ( 5 + 4 \beta_{1} + \beta_{2} ) q^{25} - q^{27} + ( 2 + 2 \beta_{1} - \beta_{2} ) q^{28} + ( 5 - \beta_{1} + 3 \beta_{2} ) q^{29} + ( \beta_{1} + 2 \beta_{2} ) q^{30} + ( 5 - 5 \beta_{1} + 5 \beta_{2} ) q^{31} + q^{32} + ( -3 + 3 \beta_{1} - \beta_{2} ) q^{33} + ( -2 + 2 \beta_{2} ) q^{34} + ( -5 - 11 \beta_{2} ) q^{35} + q^{36} + ( -2 \beta_{1} + 6 \beta_{2} ) q^{37} + ( -4 + 4 \beta_{1} - 4 \beta_{2} ) q^{38} + ( -\beta_{1} - 2 \beta_{2} ) q^{40} + ( -2 + 6 \beta_{1} - 2 \beta_{2} ) q^{41} + ( -2 - 2 \beta_{1} + \beta_{2} ) q^{42} + ( 6 - 2 \beta_{1} + 4 \beta_{2} ) q^{43} + ( 3 - 3 \beta_{1} + \beta_{2} ) q^{44} + ( -\beta_{1} - 2 \beta_{2} ) q^{45} + ( 2 - 2 \beta_{1} + 4 \beta_{2} ) q^{46} + ( 4 - 4 \beta_{1} + 4 \beta_{2} ) q^{47} - q^{48} + ( 2 + 9 \beta_{1} - 5 \beta_{2} ) q^{49} + ( 5 + 4 \beta_{1} + \beta_{2} ) q^{50} + ( 2 - 2 \beta_{2} ) q^{51} + ( 4 - 5 \beta_{1} + 2 \beta_{2} ) q^{53} - q^{54} + ( 9 - 5 \beta_{1} + 4 \beta_{2} ) q^{55} + ( 2 + 2 \beta_{1} - \beta_{2} ) q^{56} + ( 4 - 4 \beta_{1} + 4 \beta_{2} ) q^{57} + ( 5 - \beta_{1} + 3 \beta_{2} ) q^{58} + ( -4 + 2 \beta_{1} - 5 \beta_{2} ) q^{59} + ( \beta_{1} + 2 \beta_{2} ) q^{60} + ( 8 + 2 \beta_{2} ) q^{61} + ( 5 - 5 \beta_{1} + 5 \beta_{2} ) q^{62} + ( 2 + 2 \beta_{1} - \beta_{2} ) q^{63} + q^{64} + ( -3 + 3 \beta_{1} - \beta_{2} ) q^{66} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{67} + ( -2 + 2 \beta_{2} ) q^{68} + ( -2 + 2 \beta_{1} - 4 \beta_{2} ) q^{69} + ( -5 - 11 \beta_{2} ) q^{70} + ( -8 + 2 \beta_{1} - 4 \beta_{2} ) q^{71} + q^{72} + ( -8 + 7 \beta_{1} - 4 \beta_{2} ) q^{73} + ( -2 \beta_{1} + 6 \beta_{2} ) q^{74} + ( -5 - 4 \beta_{1} - \beta_{2} ) q^{75} + ( -4 + 4 \beta_{1} - 4 \beta_{2} ) q^{76} + ( -2 - \beta_{1} - \beta_{2} ) q^{77} + ( 10 - 2 \beta_{1} - 3 \beta_{2} ) q^{79} + ( -\beta_{1} - 2 \beta_{2} ) q^{80} + q^{81} + ( -2 + 6 \beta_{1} - 2 \beta_{2} ) q^{82} + ( 10 \beta_{1} - 3 \beta_{2} ) q^{83} + ( -2 - 2 \beta_{1} + \beta_{2} ) q^{84} + ( -6 - 2 \beta_{1} + 6 \beta_{2} ) q^{85} + ( 6 - 2 \beta_{1} + 4 \beta_{2} ) q^{86} + ( -5 + \beta_{1} - 3 \beta_{2} ) q^{87} + ( 3 - 3 \beta_{1} + \beta_{2} ) q^{88} + ( -6 \beta_{1} + 8 \beta_{2} ) q^{89} + ( -\beta_{1} - 2 \beta_{2} ) q^{90} + ( 2 - 2 \beta_{1} + 4 \beta_{2} ) q^{92} + ( -5 + 5 \beta_{1} - 5 \beta_{2} ) q^{93} + ( 4 - 4 \beta_{1} + 4 \beta_{2} ) q^{94} + ( -4 + 12 \beta_{1} - 8 \beta_{2} ) q^{95} - q^{96} + ( -8 - 3 \beta_{1} - 4 \beta_{2} ) q^{97} + ( 2 + 9 \beta_{1} - 5 \beta_{2} ) q^{98} + ( 3 - 3 \beta_{1} + \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} + q^{5} - 3 q^{6} + 9 q^{7} + 3 q^{8} + 3 q^{9} + O(q^{10}) \) \( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} + q^{5} - 3 q^{6} + 9 q^{7} + 3 q^{8} + 3 q^{9} + q^{10} + 5 q^{11} - 3 q^{12} + 9 q^{14} - q^{15} + 3 q^{16} - 8 q^{17} + 3 q^{18} - 4 q^{19} + q^{20} - 9 q^{21} + 5 q^{22} - 3 q^{24} + 18 q^{25} - 3 q^{27} + 9 q^{28} + 11 q^{29} - q^{30} + 5 q^{31} + 3 q^{32} - 5 q^{33} - 8 q^{34} - 4 q^{35} + 3 q^{36} - 8 q^{37} - 4 q^{38} + q^{40} + 2 q^{41} - 9 q^{42} + 12 q^{43} + 5 q^{44} + q^{45} + 4 q^{47} - 3 q^{48} + 20 q^{49} + 18 q^{50} + 8 q^{51} + 5 q^{53} - 3 q^{54} + 18 q^{55} + 9 q^{56} + 4 q^{57} + 11 q^{58} - 5 q^{59} - q^{60} + 22 q^{61} + 5 q^{62} + 9 q^{63} + 3 q^{64} - 5 q^{66} + 6 q^{67} - 8 q^{68} - 4 q^{70} - 18 q^{71} + 3 q^{72} - 13 q^{73} - 8 q^{74} - 18 q^{75} - 4 q^{76} - 6 q^{77} + 31 q^{79} + q^{80} + 3 q^{81} + 2 q^{82} + 13 q^{83} - 9 q^{84} - 26 q^{85} + 12 q^{86} - 11 q^{87} + 5 q^{88} - 14 q^{89} + q^{90} - 5 q^{93} + 4 q^{94} + 8 q^{95} - 3 q^{96} - 23 q^{97} + 20 q^{98} + 5 q^{99} + O(q^{100}) \)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
1.80194
−1.24698
0.445042
1.00000 −1.00000 1.00000 −4.29590 −1.00000 4.35690 1.00000 1.00000 −4.29590
1.2 1.00000 −1.00000 1.00000 2.13706 −1.00000 −0.0489173 1.00000 1.00000 2.13706
1.3 1.00000 −1.00000 1.00000 3.15883 −1.00000 4.69202 1.00000 1.00000 3.15883
\(n\): 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 1014.2.a.n yes 3
3.b odd 2 1 3042.2.a.ba 3
4.b odd 2 1 8112.2.a.cm 3
13.b even 2 1 1014.2.a.l 3
13.c even 3 2 1014.2.e.l 6
13.d odd 4 2 1014.2.b.f 6
13.e even 6 2 1014.2.e.n 6
13.f odd 12 4 1014.2.i.h 12
39.d odd 2 1 3042.2.a.bh 3
39.f even 4 2 3042.2.b.o 6
52.b odd 2 1 8112.2.a.cj 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1014.2.a.l 3 13.b even 2 1
1014.2.a.n yes 3 1.a even 1 1 trivial
1014.2.b.f 6 13.d odd 4 2
1014.2.e.l 6 13.c even 3 2
1014.2.e.n 6 13.e even 6 2
1014.2.i.h 12 13.f odd 12 4
3042.2.a.ba 3 3.b odd 2 1
3042.2.a.bh 3 39.d odd 2 1
3042.2.b.o 6 39.f even 4 2
8112.2.a.cj 3 52.b odd 2 1
8112.2.a.cm 3 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1014))\):

\( T_{5}^{3} - T_{5}^{2} - 16 T_{5} + 29 \)
\( T_{7}^{3} - 9 T_{7}^{2} + 20 T_{7} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( 29 - 16 T - T^{2} + T^{3} \)
$7$ \( 1 + 20 T - 9 T^{2} + T^{3} \)
$11$ \( -1 - 8 T - 5 T^{2} + T^{3} \)
$13$ \( T^{3} \)
$17$ \( -8 + 12 T + 8 T^{2} + T^{3} \)
$19$ \( -64 - 32 T + 4 T^{2} + T^{3} \)
$23$ \( 56 - 28 T + T^{3} \)
$29$ \( 29 + 24 T - 11 T^{2} + T^{3} \)
$31$ \( 125 - 50 T - 5 T^{2} + T^{3} \)
$37$ \( -8 - 44 T + 8 T^{2} + T^{3} \)
$41$ \( 232 - 64 T - 2 T^{2} + T^{3} \)
$43$ \( 104 + 20 T - 12 T^{2} + T^{3} \)
$47$ \( 64 - 32 T - 4 T^{2} + T^{3} \)
$53$ \( -43 - 36 T - 5 T^{2} + T^{3} \)
$59$ \( -167 - 36 T + 5 T^{2} + T^{3} \)
$61$ \( -328 + 152 T - 22 T^{2} + T^{3} \)
$67$ \( 104 - 16 T - 6 T^{2} + T^{3} \)
$71$ \( -8 + 80 T + 18 T^{2} + T^{3} \)
$73$ \( -13 - 30 T + 13 T^{2} + T^{3} \)
$79$ \( -533 + 276 T - 31 T^{2} + T^{3} \)
$83$ \( 1567 - 128 T - 13 T^{2} + T^{3} \)
$89$ \( -56 - 56 T + 14 T^{2} + T^{3} \)
$97$ \( 97 + 90 T + 23 T^{2} + T^{3} \)
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