Properties

Label 1014.2.i.c
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: \( x^{4} - x^{2} + 1 \) 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: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

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

Character values

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

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

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 2.00000i 0.866025 + 0.500000i −1.73205 1.00000i 1.00000i −0.500000 + 0.866025i 1.00000 + 1.73205i
361.2 0.866025 0.500000i −0.500000 0.866025i 0.500000 0.866025i 2.00000i −0.866025 0.500000i 1.73205 + 1.00000i 1.00000i −0.500000 + 0.866025i 1.00000 + 1.73205i
823.1 −0.866025 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 2.00000i 0.866025 0.500000i −1.73205 + 1.00000i 1.00000i −0.500000 0.866025i 1.00000 1.73205i
823.2 0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 2.00000i −0.866025 + 0.500000i 1.73205 1.00000i 1.00000i −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.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.c 4
13.b even 2 1 inner 1014.2.i.c 4
13.c even 3 1 78.2.b.a 2
13.c even 3 1 inner 1014.2.i.c 4
13.d odd 4 1 1014.2.e.b 2
13.d odd 4 1 1014.2.e.e 2
13.e even 6 1 78.2.b.a 2
13.e even 6 1 inner 1014.2.i.c 4
13.f odd 12 1 1014.2.a.b 1
13.f odd 12 1 1014.2.a.g 1
13.f odd 12 1 1014.2.e.b 2
13.f odd 12 1 1014.2.e.e 2
39.h odd 6 1 234.2.b.a 2
39.i odd 6 1 234.2.b.a 2
39.k even 12 1 3042.2.a.c 1
39.k even 12 1 3042.2.a.n 1
52.i odd 6 1 624.2.c.a 2
52.j odd 6 1 624.2.c.a 2
52.l even 12 1 8112.2.a.g 1
52.l even 12 1 8112.2.a.j 1
65.l even 6 1 1950.2.b.c 2
65.n even 6 1 1950.2.b.c 2
65.q odd 12 1 1950.2.f.d 2
65.q odd 12 1 1950.2.f.g 2
65.r odd 12 1 1950.2.f.d 2
65.r odd 12 1 1950.2.f.g 2
91.n odd 6 1 3822.2.c.d 2
91.t odd 6 1 3822.2.c.d 2
104.n odd 6 1 2496.2.c.m 2
104.p odd 6 1 2496.2.c.m 2
104.r even 6 1 2496.2.c.f 2
104.s even 6 1 2496.2.c.f 2
156.p even 6 1 1872.2.c.b 2
156.r even 6 1 1872.2.c.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.b.a 2 13.c even 3 1
78.2.b.a 2 13.e even 6 1
234.2.b.a 2 39.h odd 6 1
234.2.b.a 2 39.i odd 6 1
624.2.c.a 2 52.i odd 6 1
624.2.c.a 2 52.j odd 6 1
1014.2.a.b 1 13.f odd 12 1
1014.2.a.g 1 13.f odd 12 1
1014.2.e.b 2 13.d odd 4 1
1014.2.e.b 2 13.f odd 12 1
1014.2.e.e 2 13.d odd 4 1
1014.2.e.e 2 13.f odd 12 1
1014.2.i.c 4 1.a even 1 1 trivial
1014.2.i.c 4 13.b even 2 1 inner
1014.2.i.c 4 13.c even 3 1 inner
1014.2.i.c 4 13.e even 6 1 inner
1872.2.c.b 2 156.p even 6 1
1872.2.c.b 2 156.r even 6 1
1950.2.b.c 2 65.l even 6 1
1950.2.b.c 2 65.n even 6 1
1950.2.f.d 2 65.q odd 12 1
1950.2.f.d 2 65.r odd 12 1
1950.2.f.g 2 65.q odd 12 1
1950.2.f.g 2 65.r odd 12 1
2496.2.c.f 2 104.r even 6 1
2496.2.c.f 2 104.s even 6 1
2496.2.c.m 2 104.n odd 6 1
2496.2.c.m 2 104.p odd 6 1
3042.2.a.c 1 39.k even 12 1
3042.2.a.n 1 39.k even 12 1
3822.2.c.d 2 91.n odd 6 1
3822.2.c.d 2 91.t odd 6 1
8112.2.a.g 1 52.l even 12 1
8112.2.a.j 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 \(S_{2}^{\mathrm{new}}(1014, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$23$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 10 T + 100)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 64T^{2} + 4096 \) Copy content Toggle raw display
$41$ \( T^{4} - 100 T^{2} + 10000 \) Copy content Toggle raw display
$43$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$53$ \( (T + 6)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$61$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$97$ \( T^{4} - 144 T^{2} + 20736 \) Copy content Toggle raw display
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