Properties

Label 1014.2.i.b
Level $1014$
Weight $2$
Character orbit 1014.i
Analytic conductor $8.097$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1014 = 2 \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1014.i (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.09683076496\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 78)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(1 - \zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
−0.866025 + 0.500000i −0.500000 0.866025i 0.500000 0.866025i 3.00000i 0.866025 + 0.500000i 1.73205 + 1.00000i 1.00000i −0.500000 + 0.866025i −1.50000 2.59808i
361.2 0.866025 0.500000i −0.500000 0.866025i 0.500000 0.866025i 3.00000i −0.866025 0.500000i −1.73205 1.00000i 1.00000i −0.500000 + 0.866025i −1.50000 2.59808i
823.1 −0.866025 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 3.00000i 0.866025 0.500000i 1.73205 1.00000i 1.00000i −0.500000 0.866025i −1.50000 + 2.59808i
823.2 0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 3.00000i −0.866025 + 0.500000i −1.73205 + 1.00000i 1.00000i −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.b even 2 1 inner
13.c even 3 1 inner
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1014.2.i.b 4
13.b even 2 1 inner 1014.2.i.b 4
13.c even 3 1 1014.2.b.c 2
13.c even 3 1 inner 1014.2.i.b 4
13.d odd 4 1 78.2.e.a 2
13.d odd 4 1 1014.2.e.a 2
13.e even 6 1 1014.2.b.c 2
13.e even 6 1 inner 1014.2.i.b 4
13.f odd 12 1 78.2.e.a 2
13.f odd 12 1 1014.2.a.c 1
13.f odd 12 1 1014.2.a.f 1
13.f odd 12 1 1014.2.e.a 2
39.f even 4 1 234.2.h.a 2
39.h odd 6 1 3042.2.b.h 2
39.i odd 6 1 3042.2.b.h 2
39.k even 12 1 234.2.h.a 2
39.k even 12 1 3042.2.a.h 1
39.k even 12 1 3042.2.a.i 1
52.f even 4 1 624.2.q.g 2
52.l even 12 1 624.2.q.g 2
52.l even 12 1 8112.2.a.c 1
52.l even 12 1 8112.2.a.m 1
65.f even 4 1 1950.2.z.g 4
65.g odd 4 1 1950.2.i.m 2
65.k even 4 1 1950.2.z.g 4
65.o even 12 1 1950.2.z.g 4
65.s odd 12 1 1950.2.i.m 2
65.t even 12 1 1950.2.z.g 4
156.l odd 4 1 1872.2.t.c 2
156.v odd 12 1 1872.2.t.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.e.a 2 13.d odd 4 1
78.2.e.a 2 13.f odd 12 1
234.2.h.a 2 39.f even 4 1
234.2.h.a 2 39.k even 12 1
624.2.q.g 2 52.f even 4 1
624.2.q.g 2 52.l even 12 1
1014.2.a.c 1 13.f odd 12 1
1014.2.a.f 1 13.f odd 12 1
1014.2.b.c 2 13.c even 3 1
1014.2.b.c 2 13.e even 6 1
1014.2.e.a 2 13.d odd 4 1
1014.2.e.a 2 13.f odd 12 1
1014.2.i.b 4 1.a even 1 1 trivial
1014.2.i.b 4 13.b even 2 1 inner
1014.2.i.b 4 13.c even 3 1 inner
1014.2.i.b 4 13.e even 6 1 inner
1872.2.t.c 2 156.l odd 4 1
1872.2.t.c 2 156.v odd 12 1
1950.2.i.m 2 65.g odd 4 1
1950.2.i.m 2 65.s odd 12 1
1950.2.z.g 4 65.f even 4 1
1950.2.z.g 4 65.k even 4 1
1950.2.z.g 4 65.o even 12 1
1950.2.z.g 4 65.t even 12 1
3042.2.a.h 1 39.k even 12 1
3042.2.a.i 1 39.k even 12 1
3042.2.b.h 2 39.h odd 6 1
3042.2.b.h 2 39.i odd 6 1
8112.2.a.c 1 52.l even 12 1
8112.2.a.m 1 52.l even 12 1

Hecke kernels

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

\( T_{5}^{2} + 9 \)
\( T_{7}^{4} - 4 T_{7}^{2} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( ( 1 + T + T^{2} )^{2} \)
$5$ \( ( 9 + T^{2} )^{2} \)
$7$ \( 16 - 4 T^{2} + T^{4} \)
$11$ \( 1296 - 36 T^{2} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( ( 9 + 3 T + T^{2} )^{2} \)
$19$ \( 16 - 4 T^{2} + T^{4} \)
$23$ \( ( 36 + 6 T + T^{2} )^{2} \)
$29$ \( ( 9 + 3 T + T^{2} )^{2} \)
$31$ \( ( 16 + T^{2} )^{2} \)
$37$ \( 2401 - 49 T^{2} + T^{4} \)
$41$ \( 81 - 9 T^{2} + T^{4} \)
$43$ \( ( 100 + 10 T + T^{2} )^{2} \)
$47$ \( ( 36 + T^{2} )^{2} \)
$53$ \( ( -3 + T )^{4} \)
$59$ \( T^{4} \)
$61$ \( ( 49 - 7 T + T^{2} )^{2} \)
$67$ \( 10000 - 100 T^{2} + T^{4} \)
$71$ \( 1296 - 36 T^{2} + T^{4} \)
$73$ \( ( 169 + T^{2} )^{2} \)
$79$ \( ( 4 + T )^{4} \)
$83$ \( ( 36 + T^{2} )^{2} \)
$89$ \( 104976 - 324 T^{2} + T^{4} \)
$97$ \( 38416 - 196 T^{2} + T^{4} \)
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