Properties

Label 1014.2.b.c
Level $1014$
Weight $2$
Character orbit 1014.b
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.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

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

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
1.00000i
1.00000i
1.00000i 1.00000 −1.00000 3.00000i 1.00000i 2.00000i 1.00000i 1.00000 3.00000
337.2 1.00000i 1.00000 −1.00000 3.00000i 1.00000i 2.00000i 1.00000i 1.00000 3.00000
\(n\): 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

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(1014, [\chi])\).

Hecke characteristic polynomials

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