Properties

Label 1300.1.bc.a
Level $1300$
Weight $1$
Character orbit 1300.bc
Analytic conductor $0.649$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -4
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1300.bc (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.648784516423\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 52)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.676.1
Artin image: $C_6\times S_3$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{2} + \zeta_{6}^{2} q^{4} - q^{8} + \zeta_{6}^{2} q^{9} +O(q^{10})\) \( q + \zeta_{6} q^{2} + \zeta_{6}^{2} q^{4} - q^{8} + \zeta_{6}^{2} q^{9} + \zeta_{6} q^{13} -\zeta_{6} q^{16} + \zeta_{6}^{2} q^{17} - q^{18} + \zeta_{6}^{2} q^{26} + \zeta_{6} q^{29} -\zeta_{6}^{2} q^{32} - q^{34} -\zeta_{6} q^{36} -\zeta_{6} q^{37} + \zeta_{6} q^{41} -\zeta_{6} q^{49} - q^{52} + q^{53} + \zeta_{6}^{2} q^{58} -\zeta_{6}^{2} q^{61} + q^{64} -\zeta_{6} q^{68} -\zeta_{6}^{2} q^{72} + q^{73} -\zeta_{6}^{2} q^{74} -\zeta_{6} q^{81} + \zeta_{6}^{2} q^{82} -2 \zeta_{6} q^{89} -2 \zeta_{6}^{2} q^{97} -\zeta_{6}^{2} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{4} - 2 q^{8} - q^{9} + O(q^{10}) \) \( 2 q + q^{2} - q^{4} - 2 q^{8} - q^{9} + q^{13} - q^{16} - q^{17} - 2 q^{18} - q^{26} + q^{29} + q^{32} - 2 q^{34} - q^{36} - q^{37} + q^{41} - q^{49} - 2 q^{52} + 2 q^{53} - q^{58} + q^{61} + 2 q^{64} - q^{68} + q^{72} + 2 q^{73} + q^{74} - q^{81} - q^{82} - 2 q^{89} + 2 q^{97} + q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1300\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(651\) \(677\)
\(\chi(n)\) \(\zeta_{6}^{2}\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
451.1
0.500000 0.866025i
0.500000 + 0.866025i
0.500000 0.866025i 0 −0.500000 0.866025i 0 0 0 −1.00000 −0.500000 0.866025i 0
1251.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 0 0 −1.00000 −0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
13.c even 3 1 inner
52.j odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1300.1.bc.a 2
4.b odd 2 1 CM 1300.1.bc.a 2
5.b even 2 1 52.1.j.a 2
5.c odd 4 2 1300.1.w.a 4
13.c even 3 1 inner 1300.1.bc.a 2
15.d odd 2 1 468.1.br.a 2
20.d odd 2 1 52.1.j.a 2
20.e even 4 2 1300.1.w.a 4
35.c odd 2 1 2548.1.bn.a 2
35.i odd 6 1 2548.1.q.a 2
35.i odd 6 1 2548.1.bi.a 2
35.j even 6 1 2548.1.q.b 2
35.j even 6 1 2548.1.bi.b 2
40.e odd 2 1 832.1.bb.a 2
40.f even 2 1 832.1.bb.a 2
52.j odd 6 1 inner 1300.1.bc.a 2
60.h even 2 1 468.1.br.a 2
65.d even 2 1 676.1.j.a 2
65.g odd 4 2 676.1.i.a 4
65.l even 6 1 676.1.c.a 1
65.l even 6 1 676.1.j.a 2
65.n even 6 1 52.1.j.a 2
65.n even 6 1 676.1.c.b 1
65.q odd 12 2 1300.1.w.a 4
65.s odd 12 2 676.1.b.a 2
65.s odd 12 2 676.1.i.a 4
80.k odd 4 2 3328.1.v.b 4
80.q even 4 2 3328.1.v.b 4
140.c even 2 1 2548.1.bn.a 2
140.p odd 6 1 2548.1.q.b 2
140.p odd 6 1 2548.1.bi.b 2
140.s even 6 1 2548.1.q.a 2
140.s even 6 1 2548.1.bi.a 2
195.x odd 6 1 468.1.br.a 2
260.g odd 2 1 676.1.j.a 2
260.u even 4 2 676.1.i.a 4
260.v odd 6 1 52.1.j.a 2
260.v odd 6 1 676.1.c.b 1
260.w odd 6 1 676.1.c.a 1
260.w odd 6 1 676.1.j.a 2
260.bc even 12 2 676.1.b.a 2
260.bc even 12 2 676.1.i.a 4
260.bj even 12 2 1300.1.w.a 4
455.y odd 6 1 2548.1.q.a 2
455.ba even 6 1 2548.1.q.b 2
455.bm even 6 1 2548.1.bi.b 2
455.bp odd 6 1 2548.1.bn.a 2
455.bw odd 6 1 2548.1.bi.a 2
520.bv even 6 1 832.1.bb.a 2
520.bx odd 6 1 832.1.bb.a 2
780.br even 6 1 468.1.br.a 2
1040.ec even 12 2 3328.1.v.b 4
1040.fb odd 12 2 3328.1.v.b 4
1820.bt even 6 1 2548.1.bi.a 2
1820.ck odd 6 1 2548.1.q.b 2
1820.dj even 6 1 2548.1.bn.a 2
1820.dr even 6 1 2548.1.q.a 2
1820.dv odd 6 1 2548.1.bi.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.1.j.a 2 5.b even 2 1
52.1.j.a 2 20.d odd 2 1
52.1.j.a 2 65.n even 6 1
52.1.j.a 2 260.v odd 6 1
468.1.br.a 2 15.d odd 2 1
468.1.br.a 2 60.h even 2 1
468.1.br.a 2 195.x odd 6 1
468.1.br.a 2 780.br even 6 1
676.1.b.a 2 65.s odd 12 2
676.1.b.a 2 260.bc even 12 2
676.1.c.a 1 65.l even 6 1
676.1.c.a 1 260.w odd 6 1
676.1.c.b 1 65.n even 6 1
676.1.c.b 1 260.v odd 6 1
676.1.i.a 4 65.g odd 4 2
676.1.i.a 4 65.s odd 12 2
676.1.i.a 4 260.u even 4 2
676.1.i.a 4 260.bc even 12 2
676.1.j.a 2 65.d even 2 1
676.1.j.a 2 65.l even 6 1
676.1.j.a 2 260.g odd 2 1
676.1.j.a 2 260.w odd 6 1
832.1.bb.a 2 40.e odd 2 1
832.1.bb.a 2 40.f even 2 1
832.1.bb.a 2 520.bv even 6 1
832.1.bb.a 2 520.bx odd 6 1
1300.1.w.a 4 5.c odd 4 2
1300.1.w.a 4 20.e even 4 2
1300.1.w.a 4 65.q odd 12 2
1300.1.w.a 4 260.bj even 12 2
1300.1.bc.a 2 1.a even 1 1 trivial
1300.1.bc.a 2 4.b odd 2 1 CM
1300.1.bc.a 2 13.c even 3 1 inner
1300.1.bc.a 2 52.j odd 6 1 inner
2548.1.q.a 2 35.i odd 6 1
2548.1.q.a 2 140.s even 6 1
2548.1.q.a 2 455.y odd 6 1
2548.1.q.a 2 1820.dr even 6 1
2548.1.q.b 2 35.j even 6 1
2548.1.q.b 2 140.p odd 6 1
2548.1.q.b 2 455.ba even 6 1
2548.1.q.b 2 1820.ck odd 6 1
2548.1.bi.a 2 35.i odd 6 1
2548.1.bi.a 2 140.s even 6 1
2548.1.bi.a 2 455.bw odd 6 1
2548.1.bi.a 2 1820.bt even 6 1
2548.1.bi.b 2 35.j even 6 1
2548.1.bi.b 2 140.p odd 6 1
2548.1.bi.b 2 455.bm even 6 1
2548.1.bi.b 2 1820.dv odd 6 1
2548.1.bn.a 2 35.c odd 2 1
2548.1.bn.a 2 140.c even 2 1
2548.1.bn.a 2 455.bp odd 6 1
2548.1.bn.a 2 1820.dj even 6 1
3328.1.v.b 4 80.k odd 4 2
3328.1.v.b 4 80.q even 4 2
3328.1.v.b 4 1040.ec even 12 2
3328.1.v.b 4 1040.fb odd 12 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1300, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( 1 - T + T^{2} \)
$17$ \( 1 + T + T^{2} \)
$19$ \( T^{2} \)
$23$ \( T^{2} \)
$29$ \( 1 - T + T^{2} \)
$31$ \( T^{2} \)
$37$ \( 1 + T + T^{2} \)
$41$ \( 1 - T + T^{2} \)
$43$ \( T^{2} \)
$47$ \( T^{2} \)
$53$ \( ( -1 + T )^{2} \)
$59$ \( T^{2} \)
$61$ \( 1 - T + T^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( -1 + T )^{2} \)
$79$ \( T^{2} \)
$83$ \( T^{2} \)
$89$ \( 4 + 2 T + T^{2} \)
$97$ \( 4 - 2 T + T^{2} \)
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