Properties

Label 1014.2.b.b
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: \( x^{2} + 1 \) 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: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

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

Character values

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

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

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 2.00000i 1.00000i 4.00000i 1.00000i 1.00000 2.00000
337.2 1.00000i −1.00000 −1.00000 2.00000i 1.00000i 4.00000i 1.00000i 1.00000 2.00000
\(n\): 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.b 2
3.b odd 2 1 3042.2.b.g 2
13.b even 2 1 inner 1014.2.b.b 2
13.c even 3 2 1014.2.i.d 4
13.d odd 4 1 78.2.a.a 1
13.d odd 4 1 1014.2.a.d 1
13.e even 6 2 1014.2.i.d 4
13.f odd 12 2 1014.2.e.c 2
13.f odd 12 2 1014.2.e.f 2
39.d odd 2 1 3042.2.b.g 2
39.f even 4 1 234.2.a.c 1
39.f even 4 1 3042.2.a.f 1
52.f even 4 1 624.2.a.h 1
52.f even 4 1 8112.2.a.v 1
65.f even 4 1 1950.2.e.i 2
65.g odd 4 1 1950.2.a.w 1
65.k even 4 1 1950.2.e.i 2
91.i even 4 1 3822.2.a.j 1
104.j odd 4 1 2496.2.a.t 1
104.m even 4 1 2496.2.a.b 1
117.y odd 12 2 2106.2.e.q 2
117.z even 12 2 2106.2.e.j 2
143.g even 4 1 9438.2.a.t 1
156.l odd 4 1 1872.2.a.c 1
195.j odd 4 1 5850.2.e.bb 2
195.n even 4 1 5850.2.a.d 1
195.u odd 4 1 5850.2.e.bb 2
312.w odd 4 1 7488.2.a.bk 1
312.y even 4 1 7488.2.a.bz 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.a.a 1 13.d odd 4 1
234.2.a.c 1 39.f even 4 1
624.2.a.h 1 52.f even 4 1
1014.2.a.d 1 13.d odd 4 1
1014.2.b.b 2 1.a even 1 1 trivial
1014.2.b.b 2 13.b even 2 1 inner
1014.2.e.c 2 13.f odd 12 2
1014.2.e.f 2 13.f odd 12 2
1014.2.i.d 4 13.c even 3 2
1014.2.i.d 4 13.e even 6 2
1872.2.a.c 1 156.l odd 4 1
1950.2.a.w 1 65.g odd 4 1
1950.2.e.i 2 65.f even 4 1
1950.2.e.i 2 65.k even 4 1
2106.2.e.j 2 117.z even 12 2
2106.2.e.q 2 117.y odd 12 2
2496.2.a.b 1 104.m even 4 1
2496.2.a.t 1 104.j odd 4 1
3042.2.a.f 1 39.f even 4 1
3042.2.b.g 2 3.b odd 2 1
3042.2.b.g 2 39.d odd 2 1
3822.2.a.j 1 91.i even 4 1
5850.2.a.d 1 195.n even 4 1
5850.2.e.bb 2 195.j odd 4 1
5850.2.e.bb 2 195.u odd 4 1
7488.2.a.bk 1 312.w odd 4 1
7488.2.a.bz 1 312.y even 4 1
8112.2.a.v 1 52.f even 4 1
9438.2.a.t 1 143.g even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(1014, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 4 \) Copy content Toggle raw display
$7$ \( T^{2} + 16 \) Copy content Toggle raw display
$11$ \( T^{2} + 16 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T + 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 64 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 16 \) Copy content Toggle raw display
$37$ \( T^{2} + 4 \) Copy content Toggle raw display
$41$ \( T^{2} + 100 \) Copy content Toggle raw display
$43$ \( (T + 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 64 \) Copy content Toggle raw display
$53$ \( (T + 10)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 16 \) Copy content Toggle raw display
$61$ \( (T + 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 256 \) Copy content Toggle raw display
$71$ \( T^{2} + 64 \) Copy content Toggle raw display
$73$ \( T^{2} + 4 \) Copy content Toggle raw display
$79$ \( (T - 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 144 \) Copy content Toggle raw display
$89$ \( T^{2} + 196 \) Copy content Toggle raw display
$97$ \( T^{2} + 100 \) Copy content Toggle raw display
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