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: \( x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + (\beta_{5} - 1) q^{2} + ( - \beta_{5} + 1) q^{3} - \beta_{5} q^{4} + (\beta_{3} + 2 \beta_{2}) q^{5} + \beta_{5} q^{6} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - \beta_{2} + \beta_1) q^{7} + q^{8} - \beta_{5} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} - 1) q^{2} + ( - \beta_{5} + 1) q^{3} - \beta_{5} q^{4} + (\beta_{3} + 2 \beta_{2}) q^{5} + \beta_{5} q^{6} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - \beta_{2} + \beta_1) q^{7} + q^{8} - \beta_{5} q^{9} + (\beta_{4} - 2 \beta_1) q^{10} + (3 \beta_{5} + 3 \beta_{4} + \beta_1 - 3) q^{11} - q^{12} + ( - 2 \beta_{3} + \beta_{2} + 2) q^{14} + ( - \beta_{4} + 2 \beta_1) q^{15} + (\beta_{5} - 1) q^{16} + (2 \beta_{5} + 2 \beta_{2} - 2 \beta_1) q^{17} + q^{18} + (4 \beta_{5} + 4 \beta_{4} + 4 \beta_{3} - 4 \beta_{2} + 4 \beta_1) q^{19} + ( - \beta_{4} - \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{20} + (2 \beta_{3} - \beta_{2} - 2) q^{21} + ( - 3 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} + \beta_{2} - \beta_1) q^{22} + (2 \beta_{5} + 2 \beta_{4} + 4 \beta_1 - 2) q^{23} + ( - \beta_{5} + 1) q^{24} + ( - 4 \beta_{3} - \beta_{2} + 5) q^{25} - q^{27} + (2 \beta_{5} - 2 \beta_{4} - \beta_1 - 2) q^{28} + (5 \beta_{5} + \beta_{4} + 3 \beta_1 - 5) q^{29} + (\beta_{4} + \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{30} + (5 \beta_{3} - 5 \beta_{2} + 5) q^{31} - \beta_{5} q^{32} + (3 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} - \beta_{2} + \beta_1) q^{33} + ( - 2 \beta_{2} - 2) q^{34} + (5 \beta_{5} - 11 \beta_{2} + 11 \beta_1) q^{35} + (\beta_{5} - 1) q^{36} + (2 \beta_{4} + 6 \beta_1) q^{37} + ( - 4 \beta_{3} + 4 \beta_{2} - 4) q^{38} + (\beta_{3} + 2 \beta_{2}) q^{40} + ( - 2 \beta_{5} - 6 \beta_{4} - 2 \beta_1 + 2) q^{41} + ( - 2 \beta_{5} + 2 \beta_{4} + \beta_1 + 2) q^{42} + ( - 6 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 4 \beta_{2} - 4 \beta_1) q^{43} + (3 \beta_{3} - \beta_{2} + 3) q^{44} + ( - \beta_{4} - \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{45} + ( - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 4 \beta_{2} - 4 \beta_1) q^{46} + (4 \beta_{3} - 4 \beta_{2} + 4) q^{47} + \beta_{5} q^{48} + (2 \beta_{5} - 9 \beta_{4} - 5 \beta_1 - 2) q^{49} + (5 \beta_{5} - 4 \beta_{4} + \beta_1 - 5) q^{50} + (2 \beta_{2} + 2) q^{51} + (5 \beta_{3} - 2 \beta_{2} + 4) q^{53} + ( - \beta_{5} + 1) q^{54} + (9 \beta_{5} + 5 \beta_{4} + 4 \beta_1 - 9) q^{55} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - \beta_{2} + \beta_1) q^{56} + (4 \beta_{3} - 4 \beta_{2} + 4) q^{57} + ( - 5 \beta_{5} - \beta_{4} - \beta_{3} + 3 \beta_{2} - 3 \beta_1) q^{58} + (4 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 5 \beta_{2} + 5 \beta_1) q^{59} + ( - \beta_{3} - 2 \beta_{2}) q^{60} + ( - 8 \beta_{5} + 2 \beta_{2} - 2 \beta_1) q^{61} + (5 \beta_{5} + 5 \beta_{4} + 5 \beta_1 - 5) q^{62} + (2 \beta_{5} - 2 \beta_{4} - \beta_1 - 2) q^{63} + q^{64} + ( - 3 \beta_{3} + \beta_{2} - 3) q^{66} + (2 \beta_{5} + 2 \beta_{4} - 2 \beta_1 - 2) q^{67} + ( - 2 \beta_{5} + 2 \beta_1 + 2) q^{68} + (2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} + 4 \beta_1) q^{69} + (11 \beta_{2} - 5) q^{70} + (8 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} + 4 \beta_1) q^{71} - \beta_{5} q^{72} + ( - 7 \beta_{3} + 4 \beta_{2} - 8) q^{73} + ( - 2 \beta_{4} - 2 \beta_{3} + 6 \beta_{2} - 6 \beta_1) q^{74} + ( - 5 \beta_{5} + 4 \beta_{4} - \beta_1 + 5) q^{75} + ( - 4 \beta_{5} - 4 \beta_{4} - 4 \beta_1 + 4) q^{76} + (\beta_{3} + \beta_{2} - 2) q^{77} + (2 \beta_{3} + 3 \beta_{2} + 10) q^{79} + (\beta_{4} - 2 \beta_1) q^{80} + (\beta_{5} - 1) q^{81} + (2 \beta_{5} + 6 \beta_{4} + 6 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{82} + ( - 10 \beta_{3} + 3 \beta_{2}) q^{83} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1) q^{84} + (6 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 6 \beta_{2} - 6 \beta_1) q^{85} + (2 \beta_{3} - 4 \beta_{2} + 6) q^{86} + (5 \beta_{5} + \beta_{4} + \beta_{3} - 3 \beta_{2} + 3 \beta_1) q^{87} + (3 \beta_{5} + 3 \beta_{4} + \beta_1 - 3) q^{88} + (6 \beta_{4} + 8 \beta_1) q^{89} + (\beta_{3} + 2 \beta_{2}) q^{90} + (2 \beta_{3} - 4 \beta_{2} + 2) q^{92} + ( - 5 \beta_{5} - 5 \beta_{4} - 5 \beta_1 + 5) q^{93} + (4 \beta_{5} + 4 \beta_{4} + 4 \beta_1 - 4) q^{94} + (4 \beta_{5} + 12 \beta_{4} + 12 \beta_{3} - 8 \beta_{2} + 8 \beta_1) q^{95} - q^{96} + (8 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} - 4 \beta_{2} + 4 \beta_1) q^{97} + ( - 2 \beta_{5} + 9 \beta_{4} + 9 \beta_{3} - 5 \beta_{2} + 5 \beta_1) q^{98} + (3 \beta_{3} - \beta_{2} + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{2} + 3 q^{3} - 3 q^{4} + 2 q^{5} + 3 q^{6} - 9 q^{7} + 6 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{2} + 3 q^{3} - 3 q^{4} + 2 q^{5} + 3 q^{6} - 9 q^{7} + 6 q^{8} - 3 q^{9} - q^{10} - 5 q^{11} - 6 q^{12} + 18 q^{14} + q^{15} - 3 q^{16} + 8 q^{17} + 6 q^{18} + 4 q^{19} - q^{20} - 18 q^{21} - 5 q^{22} + 3 q^{24} + 36 q^{25} - 6 q^{27} - 9 q^{28} - 11 q^{29} + q^{30} + 10 q^{31} - 3 q^{32} + 5 q^{33} - 16 q^{34} + 4 q^{35} - 3 q^{36} + 8 q^{37} - 8 q^{38} + 2 q^{40} - 2 q^{41} + 9 q^{42} - 12 q^{43} + 10 q^{44} - q^{45} + 8 q^{47} + 3 q^{48} - 20 q^{49} - 18 q^{50} + 16 q^{51} + 10 q^{53} + 3 q^{54} - 18 q^{55} - 9 q^{56} + 8 q^{57} - 11 q^{58} + 5 q^{59} - 2 q^{60} - 22 q^{61} - 5 q^{62} - 9 q^{63} + 6 q^{64} - 10 q^{66} - 6 q^{67} + 8 q^{68} - 8 q^{70} + 18 q^{71} - 3 q^{72} - 26 q^{73} + 8 q^{74} + 18 q^{75} + 4 q^{76} - 12 q^{77} + 62 q^{79} - q^{80} - 3 q^{81} - 2 q^{82} + 26 q^{83} + 9 q^{84} + 26 q^{85} + 24 q^{86} + 11 q^{87} - 5 q^{88} + 14 q^{89} + 2 q^{90} + 5 q^{93} - 4 q^{94} - 8 q^{95} - 6 q^{96} + 23 q^{97} - 20 q^{98} + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 3\nu^{4} - 9\nu^{3} + 5\nu^{2} - 2\nu + 6 ) / 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{5} + 9\nu^{4} - 14\nu^{3} + 15\nu^{2} - 6\nu + 18 ) / 13 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -4\nu^{5} - \nu^{4} - 10\nu^{3} - 6\nu^{2} - 34\nu - 2 ) / 13 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -6\nu^{5} + 5\nu^{4} - 15\nu^{3} - 9\nu^{2} - 25\nu + 10 ) / 13 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} - 3\beta_{4} - 4\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} - 4\beta_{4} - 4\beta_{3} + 9\beta_{2} - 9\beta_1 \) Copy content Toggle raw display

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 \(S_{2}^{\mathrm{new}}(1014, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$5$ \( (T^{3} - T^{2} - 16 T + 29)^{2} \) Copy content Toggle raw display
$7$ \( T^{6} + 9 T^{5} + 61 T^{4} + 182 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{6} + 5 T^{5} + 33 T^{4} - 42 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( T^{6} - 8 T^{5} + 52 T^{4} - 112 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$19$ \( T^{6} - 4 T^{5} + 48 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$23$ \( T^{6} + 28 T^{4} + 112 T^{3} + \cdots + 3136 \) Copy content Toggle raw display
$29$ \( T^{6} + 11 T^{5} + 97 T^{4} + \cdots + 841 \) Copy content Toggle raw display
$31$ \( (T^{3} - 5 T^{2} - 50 T + 125)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} - 8 T^{5} + 108 T^{4} + 336 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$41$ \( T^{6} + 2 T^{5} + 68 T^{4} + \cdots + 53824 \) Copy content Toggle raw display
$43$ \( T^{6} + 12 T^{5} + 124 T^{4} + \cdots + 10816 \) Copy content Toggle raw display
$47$ \( (T^{3} - 4 T^{2} - 32 T + 64)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} - 5 T^{2} - 36 T - 43)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} - 5 T^{5} + 61 T^{4} + \cdots + 27889 \) Copy content Toggle raw display
$61$ \( T^{6} + 22 T^{5} + 332 T^{4} + \cdots + 107584 \) Copy content Toggle raw display
$67$ \( T^{6} + 6 T^{5} + 52 T^{4} + \cdots + 10816 \) Copy content Toggle raw display
$71$ \( T^{6} - 18 T^{5} + 244 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$73$ \( (T^{3} + 13 T^{2} - 30 T - 13)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 31 T^{2} + 276 T - 533)^{2} \) Copy content Toggle raw display
$83$ \( (T^{3} - 13 T^{2} - 128 T + 1567)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} - 14 T^{5} + 252 T^{4} + \cdots + 3136 \) Copy content Toggle raw display
$97$ \( T^{6} - 23 T^{5} + 439 T^{4} + \cdots + 9409 \) Copy content Toggle raw display
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