Properties

Label 1014.2.e.a
Level $1014$
Weight $2$
Character orbit 1014.e
Analytic conductor $8.097$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 78)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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.

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

Character values

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

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
529.1
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i −3.00000 −0.500000 0.866025i 1.00000 + 1.73205i 1.00000 −0.500000 0.866025i 1.50000 2.59808i
991.1 −0.500000 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i −3.00000 −0.500000 + 0.866025i 1.00000 1.73205i 1.00000 −0.500000 + 0.866025i 1.50000 + 2.59808i
\(n\): e.g. 2-40 or 990-1000
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.a 2
13.b even 2 1 78.2.e.a 2
13.c even 3 1 1014.2.a.f 1
13.c even 3 1 inner 1014.2.e.a 2
13.d odd 4 2 1014.2.i.b 4
13.e even 6 1 78.2.e.a 2
13.e even 6 1 1014.2.a.c 1
13.f odd 12 2 1014.2.b.c 2
13.f odd 12 2 1014.2.i.b 4
39.d odd 2 1 234.2.h.a 2
39.h odd 6 1 234.2.h.a 2
39.h odd 6 1 3042.2.a.i 1
39.i odd 6 1 3042.2.a.h 1
39.k even 12 2 3042.2.b.h 2
52.b odd 2 1 624.2.q.g 2
52.i odd 6 1 624.2.q.g 2
52.i odd 6 1 8112.2.a.m 1
52.j odd 6 1 8112.2.a.c 1
65.d even 2 1 1950.2.i.m 2
65.h odd 4 2 1950.2.z.g 4
65.l even 6 1 1950.2.i.m 2
65.r odd 12 2 1950.2.z.g 4
156.h even 2 1 1872.2.t.c 2
156.r even 6 1 1872.2.t.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.e.a 2 13.b even 2 1
78.2.e.a 2 13.e even 6 1
234.2.h.a 2 39.d odd 2 1
234.2.h.a 2 39.h odd 6 1
624.2.q.g 2 52.b odd 2 1
624.2.q.g 2 52.i odd 6 1
1014.2.a.c 1 13.e even 6 1
1014.2.a.f 1 13.c even 3 1
1014.2.b.c 2 13.f odd 12 2
1014.2.e.a 2 1.a even 1 1 trivial
1014.2.e.a 2 13.c even 3 1 inner
1014.2.i.b 4 13.d odd 4 2
1014.2.i.b 4 13.f odd 12 2
1872.2.t.c 2 156.h even 2 1
1872.2.t.c 2 156.r even 6 1
1950.2.i.m 2 65.d even 2 1
1950.2.i.m 2 65.l even 6 1
1950.2.z.g 4 65.h odd 4 2
1950.2.z.g 4 65.r odd 12 2
3042.2.a.h 1 39.i odd 6 1
3042.2.a.i 1 39.h odd 6 1
3042.2.b.h 2 39.k even 12 2
8112.2.a.c 1 52.j odd 6 1
8112.2.a.m 1 52.i odd 6 1

Hecke kernels

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

\( T_{5} + 3 \)
\( T_{7}^{2} - 2 T_{7} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( 1 + T + T^{2} \)
$5$ \( ( 3 + T )^{2} \)
$7$ \( 4 - 2 T + T^{2} \)
$11$ \( 36 - 6 T + T^{2} \)
$13$ \( T^{2} \)
$17$ \( 9 - 3 T + T^{2} \)
$19$ \( 4 - 2 T + T^{2} \)
$23$ \( 36 - 6 T + T^{2} \)
$29$ \( 9 + 3 T + T^{2} \)
$31$ \( ( -4 + T )^{2} \)
$37$ \( 49 + 7 T + T^{2} \)
$41$ \( 9 + 3 T + T^{2} \)
$43$ \( 100 - 10 T + T^{2} \)
$47$ \( ( 6 + T )^{2} \)
$53$ \( ( -3 + T )^{2} \)
$59$ \( T^{2} \)
$61$ \( 49 - 7 T + T^{2} \)
$67$ \( 100 + 10 T + T^{2} \)
$71$ \( 36 - 6 T + T^{2} \)
$73$ \( ( -13 + T )^{2} \)
$79$ \( ( 4 + T )^{2} \)
$83$ \( ( -6 + T )^{2} \)
$89$ \( 324 - 18 T + T^{2} \)
$97$ \( 196 - 14 T + T^{2} \)
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