Properties

Label 560.4.g.c
Level $560$
Weight $4$
Character orbit 560.g
Analytic conductor $33.041$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 560.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(33.0410696032\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
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.

\(f(q)\) \(=\) \( q + 7 i q^{3} + ( - 5 i + 10) q^{5} + 7 i q^{7} - 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 7 i q^{3} + ( - 5 i + 10) q^{5} + 7 i q^{7} - 22 q^{9} + 37 q^{11} + 51 i q^{13} + (70 i + 35) q^{15} - 41 i q^{17} - 108 q^{19} - 49 q^{21} + 70 i q^{23} + ( - 100 i + 75) q^{25} + 35 i q^{27} + 249 q^{29} + 134 q^{31} + 259 i q^{33} + (70 i + 35) q^{35} + 334 i q^{37} - 357 q^{39} + 206 q^{41} + 376 i q^{43} + (110 i - 220) q^{45} - 287 i q^{47} - 49 q^{49} + 287 q^{51} - 6 i q^{53} + ( - 185 i + 370) q^{55} - 756 i q^{57} - 2 q^{59} - 940 q^{61} - 154 i q^{63} + (510 i + 255) q^{65} + 106 i q^{67} - 490 q^{69} - 456 q^{71} + 650 i q^{73} + (525 i + 700) q^{75} + 259 i q^{77} - 1239 q^{79} - 839 q^{81} - 428 i q^{83} + ( - 410 i - 205) q^{85} + 1743 i q^{87} + 220 q^{89} - 357 q^{91} + 938 i q^{93} + (540 i - 1080) q^{95} + 1055 i q^{97} - 814 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 20 q^{5} - 44 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 20 q^{5} - 44 q^{9} + 74 q^{11} + 70 q^{15} - 216 q^{19} - 98 q^{21} + 150 q^{25} + 498 q^{29} + 268 q^{31} + 70 q^{35} - 714 q^{39} + 412 q^{41} - 440 q^{45} - 98 q^{49} + 574 q^{51} + 740 q^{55} - 4 q^{59} - 1880 q^{61} + 510 q^{65} - 980 q^{69} - 912 q^{71} + 1400 q^{75} - 2478 q^{79} - 1678 q^{81} - 410 q^{85} + 440 q^{89} - 714 q^{91} - 2160 q^{95} - 1628 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/560\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(351\) \(421\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
1.00000i
1.00000i
0 7.00000i 0 10.0000 + 5.00000i 0 7.00000i 0 −22.0000 0
449.2 0 7.00000i 0 10.0000 5.00000i 0 7.00000i 0 −22.0000 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.4.g.c 2
4.b odd 2 1 70.4.c.a 2
5.b even 2 1 inner 560.4.g.c 2
12.b even 2 1 630.4.g.a 2
20.d odd 2 1 70.4.c.a 2
20.e even 4 1 350.4.a.i 1
20.e even 4 1 350.4.a.m 1
28.d even 2 1 490.4.c.a 2
60.h even 2 1 630.4.g.a 2
140.c even 2 1 490.4.c.a 2
140.j odd 4 1 2450.4.a.c 1
140.j odd 4 1 2450.4.a.bn 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.4.c.a 2 4.b odd 2 1
70.4.c.a 2 20.d odd 2 1
350.4.a.i 1 20.e even 4 1
350.4.a.m 1 20.e even 4 1
490.4.c.a 2 28.d even 2 1
490.4.c.a 2 140.c even 2 1
560.4.g.c 2 1.a even 1 1 trivial
560.4.g.c 2 5.b even 2 1 inner
630.4.g.a 2 12.b even 2 1
630.4.g.a 2 60.h even 2 1
2450.4.a.c 1 140.j odd 4 1
2450.4.a.bn 1 140.j odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(560, [\chi])\):

\( T_{3}^{2} + 49 \) Copy content Toggle raw display
\( T_{11} - 37 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 49 \) Copy content Toggle raw display
$5$ \( T^{2} - 20T + 125 \) Copy content Toggle raw display
$7$ \( T^{2} + 49 \) Copy content Toggle raw display
$11$ \( (T - 37)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 2601 \) Copy content Toggle raw display
$17$ \( T^{2} + 1681 \) Copy content Toggle raw display
$19$ \( (T + 108)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 4900 \) Copy content Toggle raw display
$29$ \( (T - 249)^{2} \) Copy content Toggle raw display
$31$ \( (T - 134)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 111556 \) Copy content Toggle raw display
$41$ \( (T - 206)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 141376 \) Copy content Toggle raw display
$47$ \( T^{2} + 82369 \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( (T + 2)^{2} \) Copy content Toggle raw display
$61$ \( (T + 940)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 11236 \) Copy content Toggle raw display
$71$ \( (T + 456)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 422500 \) Copy content Toggle raw display
$79$ \( (T + 1239)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 183184 \) Copy content Toggle raw display
$89$ \( (T - 220)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 1113025 \) Copy content Toggle raw display
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