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

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

Character values

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

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

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

\( T_{5} + 2 \) Copy content Toggle raw display
\( T_{7}^{2} - 4T_{7} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

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