Properties

Label 1014.2.e.c
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.

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

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 −2.00000 0.500000 + 0.866025i 2.00000 + 3.46410i 1.00000 −0.500000 0.866025i 1.00000 1.73205i
991.1 −0.500000 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i −2.00000 0.500000 0.866025i 2.00000 3.46410i 1.00000 −0.500000 + 0.866025i 1.00000 + 1.73205i
\(n\): 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.c 2
13.b even 2 1 1014.2.e.f 2
13.c even 3 1 1014.2.a.d 1
13.c even 3 1 inner 1014.2.e.c 2
13.d odd 4 2 1014.2.i.d 4
13.e even 6 1 78.2.a.a 1
13.e even 6 1 1014.2.e.f 2
13.f odd 12 2 1014.2.b.b 2
13.f odd 12 2 1014.2.i.d 4
39.h odd 6 1 234.2.a.c 1
39.i odd 6 1 3042.2.a.f 1
39.k even 12 2 3042.2.b.g 2
52.i odd 6 1 624.2.a.h 1
52.j odd 6 1 8112.2.a.v 1
65.l even 6 1 1950.2.a.w 1
65.r odd 12 2 1950.2.e.i 2
91.t odd 6 1 3822.2.a.j 1
104.p odd 6 1 2496.2.a.b 1
104.s even 6 1 2496.2.a.t 1
117.l even 6 1 2106.2.e.q 2
117.m odd 6 1 2106.2.e.j 2
117.r even 6 1 2106.2.e.q 2
117.v odd 6 1 2106.2.e.j 2
143.i odd 6 1 9438.2.a.t 1
156.r even 6 1 1872.2.a.c 1
195.y odd 6 1 5850.2.a.d 1
195.bf even 12 2 5850.2.e.bb 2
312.ba even 6 1 7488.2.a.bk 1
312.bg odd 6 1 7488.2.a.bz 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.a.a 1 13.e even 6 1
234.2.a.c 1 39.h odd 6 1
624.2.a.h 1 52.i odd 6 1
1014.2.a.d 1 13.c even 3 1
1014.2.b.b 2 13.f odd 12 2
1014.2.e.c 2 1.a even 1 1 trivial
1014.2.e.c 2 13.c even 3 1 inner
1014.2.e.f 2 13.b even 2 1
1014.2.e.f 2 13.e even 6 1
1014.2.i.d 4 13.d odd 4 2
1014.2.i.d 4 13.f odd 12 2
1872.2.a.c 1 156.r even 6 1
1950.2.a.w 1 65.l even 6 1
1950.2.e.i 2 65.r odd 12 2
2106.2.e.j 2 117.m odd 6 1
2106.2.e.j 2 117.v odd 6 1
2106.2.e.q 2 117.l even 6 1
2106.2.e.q 2 117.r even 6 1
2496.2.a.b 1 104.p odd 6 1
2496.2.a.t 1 104.s even 6 1
3042.2.a.f 1 39.i odd 6 1
3042.2.b.g 2 39.k even 12 2
3822.2.a.j 1 91.t odd 6 1
5850.2.a.d 1 195.y odd 6 1
5850.2.e.bb 2 195.bf even 12 2
7488.2.a.bk 1 312.ba even 6 1
7488.2.a.bz 1 312.bg odd 6 1
8112.2.a.v 1 52.j odd 6 1
9438.2.a.t 1 143.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 \(S_{2}^{\mathrm{new}}(1014, [\chi])\):

\( T_{5} + 2 \)
\( T_{7}^{2} - 4 T_{7} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( 1 - T + T^{2} \)
$5$ \( ( 2 + T )^{2} \)
$7$ \( 16 - 4 T + T^{2} \)
$11$ \( 16 + 4 T + T^{2} \)
$13$ \( T^{2} \)
$17$ \( 4 + 2 T + T^{2} \)
$19$ \( 64 + 8 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( 36 + 6 T + T^{2} \)
$31$ \( ( -4 + T )^{2} \)
$37$ \( 4 + 2 T + T^{2} \)
$41$ \( 100 + 10 T + T^{2} \)
$43$ \( 16 + 4 T + T^{2} \)
$47$ \( ( 8 + T )^{2} \)
$53$ \( ( 10 + T )^{2} \)
$59$ \( 16 - 4 T + T^{2} \)
$61$ \( 4 - 2 T + T^{2} \)
$67$ \( 256 + 16 T + T^{2} \)
$71$ \( 64 + 8 T + T^{2} \)
$73$ \( ( 2 + T )^{2} \)
$79$ \( ( -8 + T )^{2} \)
$83$ \( ( 12 + T )^{2} \)
$89$ \( 196 - 14 T + T^{2} \)
$97$ \( 100 - 10 T + T^{2} \)
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