Properties

Label 5780.2.c.a
Level 5780
Weight 2
Character orbit 5780.c
Analytic conductor 46.154
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 5780 = 2^{2} \cdot 5 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5780.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(46.1535323683\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 20)
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.

\(f(q)\) \(=\) \( q + 2 i q^{3} + i q^{5} + 2 i q^{7} - q^{9} +O(q^{10})\) \( q + 2 i q^{3} + i q^{5} + 2 i q^{7} - q^{9} + 2 q^{13} -2 q^{15} + 4 q^{19} -4 q^{21} + 6 i q^{23} - q^{25} + 4 i q^{27} -6 i q^{29} + 4 i q^{31} -2 q^{35} -2 i q^{37} + 4 i q^{39} + 6 i q^{41} + 10 q^{43} -i q^{45} -6 q^{47} + 3 q^{49} + 6 q^{53} + 8 i q^{57} -12 q^{59} + 2 i q^{61} -2 i q^{63} + 2 i q^{65} + 2 q^{67} -12 q^{69} + 12 i q^{71} -2 i q^{73} -2 i q^{75} + 8 i q^{79} -11 q^{81} -6 q^{83} + 12 q^{87} -6 q^{89} + 4 i q^{91} -8 q^{93} + 4 i q^{95} -2 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{9} + 4q^{13} - 4q^{15} + 8q^{19} - 8q^{21} - 2q^{25} - 4q^{35} + 20q^{43} - 12q^{47} + 6q^{49} + 12q^{53} - 24q^{59} + 4q^{67} - 24q^{69} - 22q^{81} - 12q^{83} + 24q^{87} - 12q^{89} - 16q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5780\mathbb{Z}\right)^\times\).

\(n\) \(581\) \(1157\) \(2891\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5201.1
1.00000i
1.00000i
0 2.00000i 0 1.00000i 0 2.00000i 0 −1.00000 0
5201.2 0 2.00000i 0 1.00000i 0 2.00000i 0 −1.00000 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
17.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5780.2.c.a 2
17.b even 2 1 inner 5780.2.c.a 2
17.c even 4 1 20.2.a.a 1
17.c even 4 1 5780.2.a.f 1
51.f odd 4 1 180.2.a.a 1
68.f odd 4 1 80.2.a.b 1
85.f odd 4 1 100.2.c.a 2
85.i odd 4 1 100.2.c.a 2
85.j even 4 1 100.2.a.a 1
119.f odd 4 1 980.2.a.h 1
119.m odd 12 2 980.2.i.c 2
119.n even 12 2 980.2.i.i 2
136.i even 4 1 320.2.a.f 1
136.j odd 4 1 320.2.a.a 1
153.m odd 12 2 1620.2.i.b 2
153.n even 12 2 1620.2.i.h 2
187.f odd 4 1 2420.2.a.a 1
204.l even 4 1 720.2.a.h 1
221.h odd 4 1 3380.2.f.b 2
221.i odd 4 1 3380.2.f.b 2
221.k even 4 1 3380.2.a.c 1
255.i odd 4 1 900.2.a.b 1
255.k even 4 1 900.2.d.c 2
255.r even 4 1 900.2.d.c 2
272.i odd 4 1 1280.2.d.g 2
272.j even 4 1 1280.2.d.c 2
272.s even 4 1 1280.2.d.c 2
272.t odd 4 1 1280.2.d.g 2
323.g odd 4 1 7220.2.a.f 1
340.i even 4 1 400.2.c.b 2
340.n odd 4 1 400.2.a.c 1
340.s even 4 1 400.2.c.b 2
357.l even 4 1 8820.2.a.g 1
408.q even 4 1 2880.2.a.f 1
408.t odd 4 1 2880.2.a.m 1
476.k even 4 1 3920.2.a.h 1
595.l even 4 1 4900.2.e.f 2
595.r even 4 1 4900.2.e.f 2
595.u odd 4 1 4900.2.a.e 1
680.s odd 4 1 1600.2.c.d 2
680.t even 4 1 1600.2.c.e 2
680.bc odd 4 1 1600.2.a.w 1
680.be even 4 1 1600.2.a.c 1
680.bk odd 4 1 1600.2.c.d 2
680.bl even 4 1 1600.2.c.e 2
748.j even 4 1 9680.2.a.ba 1
1020.q odd 4 1 3600.2.f.j 2
1020.ba even 4 1 3600.2.a.be 1
1020.bl odd 4 1 3600.2.f.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.2.a.a 1 17.c even 4 1
80.2.a.b 1 68.f odd 4 1
100.2.a.a 1 85.j even 4 1
100.2.c.a 2 85.f odd 4 1
100.2.c.a 2 85.i odd 4 1
180.2.a.a 1 51.f odd 4 1
320.2.a.a 1 136.j odd 4 1
320.2.a.f 1 136.i even 4 1
400.2.a.c 1 340.n odd 4 1
400.2.c.b 2 340.i even 4 1
400.2.c.b 2 340.s even 4 1
720.2.a.h 1 204.l even 4 1
900.2.a.b 1 255.i odd 4 1
900.2.d.c 2 255.k even 4 1
900.2.d.c 2 255.r even 4 1
980.2.a.h 1 119.f odd 4 1
980.2.i.c 2 119.m odd 12 2
980.2.i.i 2 119.n even 12 2
1280.2.d.c 2 272.j even 4 1
1280.2.d.c 2 272.s even 4 1
1280.2.d.g 2 272.i odd 4 1
1280.2.d.g 2 272.t odd 4 1
1600.2.a.c 1 680.be even 4 1
1600.2.a.w 1 680.bc odd 4 1
1600.2.c.d 2 680.s odd 4 1
1600.2.c.d 2 680.bk odd 4 1
1600.2.c.e 2 680.t even 4 1
1600.2.c.e 2 680.bl even 4 1
1620.2.i.b 2 153.m odd 12 2
1620.2.i.h 2 153.n even 12 2
2420.2.a.a 1 187.f odd 4 1
2880.2.a.f 1 408.q even 4 1
2880.2.a.m 1 408.t odd 4 1
3380.2.a.c 1 221.k even 4 1
3380.2.f.b 2 221.h odd 4 1
3380.2.f.b 2 221.i odd 4 1
3600.2.a.be 1 1020.ba even 4 1
3600.2.f.j 2 1020.q odd 4 1
3600.2.f.j 2 1020.bl odd 4 1
3920.2.a.h 1 476.k even 4 1
4900.2.a.e 1 595.u odd 4 1
4900.2.e.f 2 595.l even 4 1
4900.2.e.f 2 595.r even 4 1
5780.2.a.f 1 17.c even 4 1
5780.2.c.a 2 1.a even 1 1 trivial
5780.2.c.a 2 17.b even 2 1 inner
7220.2.a.f 1 323.g odd 4 1
8820.2.a.g 1 357.l even 4 1
9680.2.a.ba 1 748.j even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(5780, [\chi])\):

\( T_{3}^{2} + 4 \)
\( T_{7}^{2} + 4 \)
\( T_{11} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 2 T^{2} + 9 T^{4} \)
$5$ \( 1 + T^{2} \)
$7$ \( 1 - 10 T^{2} + 49 T^{4} \)
$11$ \( ( 1 - 11 T^{2} )^{2} \)
$13$ \( ( 1 - 2 T + 13 T^{2} )^{2} \)
$17$ 1
$19$ \( ( 1 - 4 T + 19 T^{2} )^{2} \)
$23$ \( 1 - 10 T^{2} + 529 T^{4} \)
$29$ \( 1 - 22 T^{2} + 841 T^{4} \)
$31$ \( 1 - 46 T^{2} + 961 T^{4} \)
$37$ \( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \)
$41$ \( 1 - 46 T^{2} + 1681 T^{4} \)
$43$ \( ( 1 - 10 T + 43 T^{2} )^{2} \)
$47$ \( ( 1 + 6 T + 47 T^{2} )^{2} \)
$53$ \( ( 1 - 6 T + 53 T^{2} )^{2} \)
$59$ \( ( 1 + 12 T + 59 T^{2} )^{2} \)
$61$ \( 1 - 118 T^{2} + 3721 T^{4} \)
$67$ \( ( 1 - 2 T + 67 T^{2} )^{2} \)
$71$ \( 1 + 2 T^{2} + 5041 T^{4} \)
$73$ \( 1 - 142 T^{2} + 5329 T^{4} \)
$79$ \( 1 - 94 T^{2} + 6241 T^{4} \)
$83$ \( ( 1 + 6 T + 83 T^{2} )^{2} \)
$89$ \( ( 1 + 6 T + 89 T^{2} )^{2} \)
$97$ \( 1 - 190 T^{2} + 9409 T^{4} \)
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