Properties

Label 1300.1.w.a
Level $1300$
Weight $1$
Character orbit 1300.w
Analytic conductor $0.649$
Analytic rank $0$
Dimension $4$
Projective image $D_{3}$
CM discriminant -4
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1300.w (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.648784516423\)
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 52)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.676.1
Artin image: $C_{12}\times S_3$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{12}^{5} q^{2} -\zeta_{12}^{4} q^{4} + \zeta_{12}^{3} q^{8} -\zeta_{12}^{4} q^{9} +O(q^{10})\) \( q + \zeta_{12}^{5} q^{2} -\zeta_{12}^{4} q^{4} + \zeta_{12}^{3} q^{8} -\zeta_{12}^{4} q^{9} -\zeta_{12}^{5} q^{13} -\zeta_{12}^{2} q^{16} -\zeta_{12} q^{17} + \zeta_{12}^{3} q^{18} + \zeta_{12}^{4} q^{26} -\zeta_{12}^{2} q^{29} + \zeta_{12} q^{32} + q^{34} -\zeta_{12}^{2} q^{36} -\zeta_{12}^{5} q^{37} + \zeta_{12}^{2} q^{41} + \zeta_{12}^{2} q^{49} -\zeta_{12}^{3} q^{52} -\zeta_{12}^{3} q^{53} + \zeta_{12} q^{58} -\zeta_{12}^{4} q^{61} - q^{64} + \zeta_{12}^{5} q^{68} + \zeta_{12} q^{72} -\zeta_{12}^{3} q^{73} + \zeta_{12}^{4} q^{74} -\zeta_{12}^{2} q^{81} -\zeta_{12} q^{82} + 2 \zeta_{12}^{2} q^{89} + 2 \zeta_{12} q^{97} -\zeta_{12} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{4} + 2q^{9} + O(q^{10}) \) \( 4q + 2q^{4} + 2q^{9} - 2q^{16} - 2q^{26} - 2q^{29} + 4q^{34} - 2q^{36} + 2q^{41} + 2q^{49} + 2q^{61} - 4q^{64} - 2q^{74} - 2q^{81} + 4q^{89} + O(q^{100}) \)

Character values

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

\(n\) \(301\) \(651\) \(677\)
\(\chi(n)\) \(\zeta_{12}^{4}\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
399.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 0.500000i 0 0.500000 + 0.866025i 0 0 0 1.00000i 0.500000 + 0.866025i 0
399.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0 0 0 1.00000i 0.500000 + 0.866025i 0
1199.1 −0.866025 + 0.500000i 0 0.500000 0.866025i 0 0 0 1.00000i 0.500000 0.866025i 0
1199.2 0.866025 0.500000i 0 0.500000 0.866025i 0 0 0 1.00000i 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 \(\Q(\sqrt{-1}) \)
5.b even 2 1 inner
13.c even 3 1 inner
20.d odd 2 1 inner
52.j odd 6 1 inner
65.n even 6 1 inner
260.v 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.w.a 4
4.b odd 2 1 CM 1300.1.w.a 4
5.b even 2 1 inner 1300.1.w.a 4
5.c odd 4 1 52.1.j.a 2
5.c odd 4 1 1300.1.bc.a 2
13.c even 3 1 inner 1300.1.w.a 4
15.e even 4 1 468.1.br.a 2
20.d odd 2 1 inner 1300.1.w.a 4
20.e even 4 1 52.1.j.a 2
20.e even 4 1 1300.1.bc.a 2
35.f even 4 1 2548.1.bn.a 2
35.k even 12 1 2548.1.q.a 2
35.k even 12 1 2548.1.bi.a 2
35.l odd 12 1 2548.1.q.b 2
35.l odd 12 1 2548.1.bi.b 2
40.i odd 4 1 832.1.bb.a 2
40.k even 4 1 832.1.bb.a 2
52.j odd 6 1 inner 1300.1.w.a 4
60.l odd 4 1 468.1.br.a 2
65.f even 4 1 676.1.i.a 4
65.h odd 4 1 676.1.j.a 2
65.k even 4 1 676.1.i.a 4
65.n even 6 1 inner 1300.1.w.a 4
65.o even 12 1 676.1.b.a 2
65.o even 12 1 676.1.i.a 4
65.q odd 12 1 52.1.j.a 2
65.q odd 12 1 676.1.c.b 1
65.q odd 12 1 1300.1.bc.a 2
65.r odd 12 1 676.1.c.a 1
65.r odd 12 1 676.1.j.a 2
65.t even 12 1 676.1.b.a 2
65.t even 12 1 676.1.i.a 4
80.i odd 4 1 3328.1.v.b 4
80.j even 4 1 3328.1.v.b 4
80.s even 4 1 3328.1.v.b 4
80.t odd 4 1 3328.1.v.b 4
140.j odd 4 1 2548.1.bn.a 2
140.w even 12 1 2548.1.q.b 2
140.w even 12 1 2548.1.bi.b 2
140.x odd 12 1 2548.1.q.a 2
140.x odd 12 1 2548.1.bi.a 2
195.bl even 12 1 468.1.br.a 2
260.l odd 4 1 676.1.i.a 4
260.p even 4 1 676.1.j.a 2
260.s odd 4 1 676.1.i.a 4
260.v odd 6 1 inner 1300.1.w.a 4
260.be odd 12 1 676.1.b.a 2
260.be odd 12 1 676.1.i.a 4
260.bg even 12 1 676.1.c.a 1
260.bg even 12 1 676.1.j.a 2
260.bj even 12 1 52.1.j.a 2
260.bj even 12 1 676.1.c.b 1
260.bj even 12 1 1300.1.bc.a 2
260.bl odd 12 1 676.1.b.a 2
260.bl odd 12 1 676.1.i.a 4
455.cq odd 12 1 2548.1.q.b 2
455.cs even 12 1 2548.1.q.a 2
455.cx odd 12 1 2548.1.bi.b 2
455.db even 12 1 2548.1.bi.a 2
455.dc even 12 1 2548.1.bn.a 2
520.cm even 12 1 832.1.bb.a 2
520.cq odd 12 1 832.1.bb.a 2
780.cj odd 12 1 468.1.br.a 2
1040.dp even 12 1 3328.1.v.b 4
1040.dr odd 12 1 3328.1.v.b 4
1040.fh even 12 1 3328.1.v.b 4
1040.fj odd 12 1 3328.1.v.b 4
1820.er odd 12 1 2548.1.bi.a 2
1820.es even 12 1 2548.1.bi.b 2
1820.gu even 12 1 2548.1.q.b 2
1820.gz odd 12 1 2548.1.q.a 2
1820.hb odd 12 1 2548.1.bn.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.1.j.a 2 5.c odd 4 1
52.1.j.a 2 20.e even 4 1
52.1.j.a 2 65.q odd 12 1
52.1.j.a 2 260.bj even 12 1
468.1.br.a 2 15.e even 4 1
468.1.br.a 2 60.l odd 4 1
468.1.br.a 2 195.bl even 12 1
468.1.br.a 2 780.cj odd 12 1
676.1.b.a 2 65.o even 12 1
676.1.b.a 2 65.t even 12 1
676.1.b.a 2 260.be odd 12 1
676.1.b.a 2 260.bl odd 12 1
676.1.c.a 1 65.r odd 12 1
676.1.c.a 1 260.bg even 12 1
676.1.c.b 1 65.q odd 12 1
676.1.c.b 1 260.bj even 12 1
676.1.i.a 4 65.f even 4 1
676.1.i.a 4 65.k even 4 1
676.1.i.a 4 65.o even 12 1
676.1.i.a 4 65.t even 12 1
676.1.i.a 4 260.l odd 4 1
676.1.i.a 4 260.s odd 4 1
676.1.i.a 4 260.be odd 12 1
676.1.i.a 4 260.bl odd 12 1
676.1.j.a 2 65.h odd 4 1
676.1.j.a 2 65.r odd 12 1
676.1.j.a 2 260.p even 4 1
676.1.j.a 2 260.bg even 12 1
832.1.bb.a 2 40.i odd 4 1
832.1.bb.a 2 40.k even 4 1
832.1.bb.a 2 520.cm even 12 1
832.1.bb.a 2 520.cq odd 12 1
1300.1.w.a 4 1.a even 1 1 trivial
1300.1.w.a 4 4.b odd 2 1 CM
1300.1.w.a 4 5.b even 2 1 inner
1300.1.w.a 4 13.c even 3 1 inner
1300.1.w.a 4 20.d odd 2 1 inner
1300.1.w.a 4 52.j odd 6 1 inner
1300.1.w.a 4 65.n even 6 1 inner
1300.1.w.a 4 260.v odd 6 1 inner
1300.1.bc.a 2 5.c odd 4 1
1300.1.bc.a 2 20.e even 4 1
1300.1.bc.a 2 65.q odd 12 1
1300.1.bc.a 2 260.bj even 12 1
2548.1.q.a 2 35.k even 12 1
2548.1.q.a 2 140.x odd 12 1
2548.1.q.a 2 455.cs even 12 1
2548.1.q.a 2 1820.gz odd 12 1
2548.1.q.b 2 35.l odd 12 1
2548.1.q.b 2 140.w even 12 1
2548.1.q.b 2 455.cq odd 12 1
2548.1.q.b 2 1820.gu even 12 1
2548.1.bi.a 2 35.k even 12 1
2548.1.bi.a 2 140.x odd 12 1
2548.1.bi.a 2 455.db even 12 1
2548.1.bi.a 2 1820.er odd 12 1
2548.1.bi.b 2 35.l odd 12 1
2548.1.bi.b 2 140.w even 12 1
2548.1.bi.b 2 455.cx odd 12 1
2548.1.bi.b 2 1820.es even 12 1
2548.1.bn.a 2 35.f even 4 1
2548.1.bn.a 2 140.j odd 4 1
2548.1.bn.a 2 455.dc even 12 1
2548.1.bn.a 2 1820.hb odd 12 1
3328.1.v.b 4 80.i odd 4 1
3328.1.v.b 4 80.j even 4 1
3328.1.v.b 4 80.s even 4 1
3328.1.v.b 4 80.t odd 4 1
3328.1.v.b 4 1040.dp even 12 1
3328.1.v.b 4 1040.dr odd 12 1
3328.1.v.b 4 1040.fh even 12 1
3328.1.v.b 4 1040.fj odd 12 1

Hecke kernels

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

Hecke characteristic polynomials

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