Properties

Label 1014.2.e.l
Level $1014$
Weight $2$
Character orbit 1014.e
Analytic conductor $8.097$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1014 = 2 \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1014.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.09683076496\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.64827.1
Defining polynomial: \(x^{6} - x^{5} + 3 x^{4} + 5 x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} + 3 x^{4} + 5 x^{2} - 2 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{5} + 3 \nu^{4} - 9 \nu^{3} + 5 \nu^{2} - 2 \nu + 6 \)\()/13\)
\(\beta_{3}\)\(=\)\((\)\( -3 \nu^{5} + 9 \nu^{4} - 14 \nu^{3} + 15 \nu^{2} - 6 \nu + 18 \)\()/13\)
\(\beta_{4}\)\(=\)\((\)\( -4 \nu^{5} - \nu^{4} - 10 \nu^{3} - 6 \nu^{2} - 34 \nu - 2 \)\()/13\)
\(\beta_{5}\)\(=\)\((\)\( -6 \nu^{5} + 5 \nu^{4} - 15 \nu^{3} - 9 \nu^{2} - 25 \nu + 10 \)\()/13\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_{1}\)
\(\nu^{3}\)\(=\)\(\beta_{3} - 3 \beta_{2}\)
\(\nu^{4}\)\(=\)\(2 \beta_{5} - 3 \beta_{4} - 4 \beta_{1} - 2\)
\(\nu^{5}\)\(=\)\(\beta_{5} - 4 \beta_{4} - 4 \beta_{3} + 9 \beta_{2} - 9 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(-\beta_{5}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
529.1
−0.623490 + 1.07992i
0.222521 0.385418i
0.900969 1.56052i
−0.623490 1.07992i
0.222521 + 0.385418i
0.900969 + 1.56052i
−0.500000 + 0.866025i 0.500000 0.866025i −0.500000 0.866025i −4.29590 0.500000 + 0.866025i −2.17845 3.77318i 1.00000 −0.500000 0.866025i 2.14795 3.72036i
529.2 −0.500000 + 0.866025i 0.500000 0.866025i −0.500000 0.866025i 2.13706 0.500000 + 0.866025i 0.0244587 + 0.0423637i 1.00000 −0.500000 0.866025i −1.06853 + 1.85075i
529.3 −0.500000 + 0.866025i 0.500000 0.866025i −0.500000 0.866025i 3.15883 0.500000 + 0.866025i −2.34601 4.06341i 1.00000 −0.500000 0.866025i −1.57942 + 2.73563i
991.1 −0.500000 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i −4.29590 0.500000 0.866025i −2.17845 + 3.77318i 1.00000 −0.500000 + 0.866025i 2.14795 + 3.72036i
991.2 −0.500000 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i 2.13706 0.500000 0.866025i 0.0244587 0.0423637i 1.00000 −0.500000 + 0.866025i −1.06853 1.85075i
991.3 −0.500000 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i 3.15883 0.500000 0.866025i −2.34601 + 4.06341i 1.00000 −0.500000 + 0.866025i −1.57942 2.73563i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 991.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1014.2.e.l 6
13.b even 2 1 1014.2.e.n 6
13.c even 3 1 1014.2.a.n yes 3
13.c even 3 1 inner 1014.2.e.l 6
13.d odd 4 2 1014.2.i.h 12
13.e even 6 1 1014.2.a.l 3
13.e even 6 1 1014.2.e.n 6
13.f odd 12 2 1014.2.b.f 6
13.f odd 12 2 1014.2.i.h 12
39.h odd 6 1 3042.2.a.bh 3
39.i odd 6 1 3042.2.a.ba 3
39.k even 12 2 3042.2.b.o 6
52.i odd 6 1 8112.2.a.cj 3
52.j odd 6 1 8112.2.a.cm 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1014.2.a.l 3 13.e even 6 1
1014.2.a.n yes 3 13.c even 3 1
1014.2.b.f 6 13.f odd 12 2
1014.2.e.l 6 1.a even 1 1 trivial
1014.2.e.l 6 13.c even 3 1 inner
1014.2.e.n 6 13.b even 2 1
1014.2.e.n 6 13.e even 6 1
1014.2.i.h 12 13.d odd 4 2
1014.2.i.h 12 13.f odd 12 2
3042.2.a.ba 3 39.i odd 6 1
3042.2.a.bh 3 39.h odd 6 1
3042.2.b.o 6 39.k even 12 2
8112.2.a.cj 3 52.i odd 6 1
8112.2.a.cm 3 52.j odd 6 1

Hecke kernels

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

\( T_{5}^{3} - T_{5}^{2} - 16 T_{5} + 29 \)
\( T_{7}^{6} + 9 T_{7}^{5} + 61 T_{7}^{4} + 182 T_{7}^{3} + 409 T_{7}^{2} - 20 T_{7} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{3} \)
$3$ \( ( 1 - T + T^{2} )^{3} \)
$5$ \( ( 29 - 16 T - T^{2} + T^{3} )^{2} \)
$7$ \( 1 - 20 T + 409 T^{2} + 182 T^{3} + 61 T^{4} + 9 T^{5} + T^{6} \)
$11$ \( 1 - 8 T + 59 T^{2} - 42 T^{3} + 33 T^{4} + 5 T^{5} + T^{6} \)
$13$ \( T^{6} \)
$17$ \( 64 + 96 T + 208 T^{2} - 112 T^{3} + 52 T^{4} - 8 T^{5} + T^{6} \)
$19$ \( 4096 - 2048 T + 1280 T^{2} + 48 T^{4} - 4 T^{5} + T^{6} \)
$23$ \( 3136 + 1568 T + 784 T^{2} + 112 T^{3} + 28 T^{4} + T^{6} \)
$29$ \( 841 - 696 T + 895 T^{2} + 322 T^{3} + 97 T^{4} + 11 T^{5} + T^{6} \)
$31$ \( ( 125 - 50 T - 5 T^{2} + T^{3} )^{2} \)
$37$ \( 64 - 352 T + 2000 T^{2} + 336 T^{3} + 108 T^{4} - 8 T^{5} + T^{6} \)
$41$ \( 53824 + 14848 T + 4560 T^{2} + 336 T^{3} + 68 T^{4} + 2 T^{5} + T^{6} \)
$43$ \( 10816 - 2080 T + 1648 T^{2} + 448 T^{3} + 124 T^{4} + 12 T^{5} + T^{6} \)
$47$ \( ( 64 - 32 T - 4 T^{2} + T^{3} )^{2} \)
$53$ \( ( -43 - 36 T - 5 T^{2} + T^{3} )^{2} \)
$59$ \( 27889 - 6012 T + 2131 T^{2} - 154 T^{3} + 61 T^{4} - 5 T^{5} + T^{6} \)
$61$ \( 107584 + 49856 T + 15888 T^{2} + 2688 T^{3} + 332 T^{4} + 22 T^{5} + T^{6} \)
$67$ \( 10816 + 1664 T + 880 T^{2} + 112 T^{3} + 52 T^{4} + 6 T^{5} + T^{6} \)
$71$ \( 64 + 640 T + 6544 T^{2} - 1456 T^{3} + 244 T^{4} - 18 T^{5} + T^{6} \)
$73$ \( ( -13 - 30 T + 13 T^{2} + T^{3} )^{2} \)
$79$ \( ( -533 + 276 T - 31 T^{2} + T^{3} )^{2} \)
$83$ \( ( 1567 - 128 T - 13 T^{2} + T^{3} )^{2} \)
$89$ \( 3136 - 3136 T + 3920 T^{2} + 672 T^{3} + 252 T^{4} - 14 T^{5} + T^{6} \)
$97$ \( 9409 - 8730 T + 5869 T^{2} - 1876 T^{3} + 439 T^{4} - 23 T^{5} + T^{6} \)
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