Properties

Label 560.5.p.d
Level $560$
Weight $5$
Character orbit 560.p
Analytic conductor $57.887$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(57.8871793270\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-6}) \)
Defining polynomial: \( x^{2} + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{-6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 5 q^{3} + (5 \beta + 5) q^{5} + (7 \beta - 35) q^{7} - 56 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 5 q^{3} + (5 \beta + 5) q^{5} + (7 \beta - 35) q^{7} - 56 q^{9} - 89 q^{11} - 5 q^{13} + ( - 25 \beta - 25) q^{15} - 485 q^{17} - 45 \beta q^{19} + ( - 35 \beta + 175) q^{21} - 143 \beta q^{23} + (50 \beta - 575) q^{25} + 685 q^{27} + 191 q^{29} + 215 \beta q^{31} + 445 q^{33} + ( - 140 \beta - 1015) q^{35} - 333 \beta q^{37} + 25 q^{39} + 595 \beta q^{41} + 77 \beta q^{43} + ( - 280 \beta - 280) q^{45} + 2195 q^{47} + ( - 490 \beta + 49) q^{49} + 2425 q^{51} + 324 \beta q^{53} + ( - 445 \beta - 445) q^{55} + 225 \beta q^{57} + 740 \beta q^{59} + 395 \beta q^{61} + ( - 392 \beta + 1960) q^{63} + ( - 25 \beta - 25) q^{65} + 418 \beta q^{67} + 715 \beta q^{69} - 4454 q^{71} + 8650 q^{73} + ( - 250 \beta + 2875) q^{75} + ( - 623 \beta + 3115) q^{77} - 5561 q^{79} + 1111 q^{81} - 1990 q^{83} + ( - 2425 \beta - 2425) q^{85} - 955 q^{87} - 165 \beta q^{89} + ( - 35 \beta + 175) q^{91} - 1075 \beta q^{93} + ( - 225 \beta + 5400) q^{95} + 9235 q^{97} + 4984 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 10 q^{3} + 10 q^{5} - 70 q^{7} - 112 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 10 q^{3} + 10 q^{5} - 70 q^{7} - 112 q^{9} - 178 q^{11} - 10 q^{13} - 50 q^{15} - 970 q^{17} + 350 q^{21} - 1150 q^{25} + 1370 q^{27} + 382 q^{29} + 890 q^{33} - 2030 q^{35} + 50 q^{39} - 560 q^{45} + 4390 q^{47} + 98 q^{49} + 4850 q^{51} - 890 q^{55} + 3920 q^{63} - 50 q^{65} - 8908 q^{71} + 17300 q^{73} + 5750 q^{75} + 6230 q^{77} - 11122 q^{79} + 2222 q^{81} - 3980 q^{83} - 4850 q^{85} - 1910 q^{87} + 350 q^{91} + 10800 q^{95} + 18470 q^{97} + 9968 q^{99}+O(q^{100}) \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1
2.44949i
2.44949i
0 −5.00000 0 5.00000 24.4949i 0 −35.0000 34.2929i 0 −56.0000 0
209.2 0 −5.00000 0 5.00000 + 24.4949i 0 −35.0000 + 34.2929i 0 −56.0000 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
35.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.5.p.d 2
4.b odd 2 1 35.5.c.d yes 2
5.b even 2 1 560.5.p.e 2
7.b odd 2 1 560.5.p.e 2
12.b even 2 1 315.5.e.c 2
20.d odd 2 1 35.5.c.c 2
20.e even 4 2 175.5.d.e 4
28.d even 2 1 35.5.c.c 2
35.c odd 2 1 inner 560.5.p.d 2
60.h even 2 1 315.5.e.d 2
84.h odd 2 1 315.5.e.d 2
140.c even 2 1 35.5.c.d yes 2
140.j odd 4 2 175.5.d.e 4
420.o odd 2 1 315.5.e.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.5.c.c 2 20.d odd 2 1
35.5.c.c 2 28.d even 2 1
35.5.c.d yes 2 4.b odd 2 1
35.5.c.d yes 2 140.c even 2 1
175.5.d.e 4 20.e even 4 2
175.5.d.e 4 140.j odd 4 2
315.5.e.c 2 12.b even 2 1
315.5.e.c 2 420.o odd 2 1
315.5.e.d 2 60.h even 2 1
315.5.e.d 2 84.h odd 2 1
560.5.p.d 2 1.a even 1 1 trivial
560.5.p.d 2 35.c odd 2 1 inner
560.5.p.e 2 5.b even 2 1
560.5.p.e 2 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 5 \) acting on \(S_{5}^{\mathrm{new}}(560, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 5)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 10T + 625 \) Copy content Toggle raw display
$7$ \( T^{2} + 70T + 2401 \) Copy content Toggle raw display
$11$ \( (T + 89)^{2} \) Copy content Toggle raw display
$13$ \( (T + 5)^{2} \) Copy content Toggle raw display
$17$ \( (T + 485)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 48600 \) Copy content Toggle raw display
$23$ \( T^{2} + 490776 \) Copy content Toggle raw display
$29$ \( (T - 191)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 1109400 \) Copy content Toggle raw display
$37$ \( T^{2} + 2661336 \) Copy content Toggle raw display
$41$ \( T^{2} + 8496600 \) Copy content Toggle raw display
$43$ \( T^{2} + 142296 \) Copy content Toggle raw display
$47$ \( (T - 2195)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 2519424 \) Copy content Toggle raw display
$59$ \( T^{2} + 13142400 \) Copy content Toggle raw display
$61$ \( T^{2} + 3744600 \) Copy content Toggle raw display
$67$ \( T^{2} + 4193376 \) Copy content Toggle raw display
$71$ \( (T + 4454)^{2} \) Copy content Toggle raw display
$73$ \( (T - 8650)^{2} \) Copy content Toggle raw display
$79$ \( (T + 5561)^{2} \) Copy content Toggle raw display
$83$ \( (T + 1990)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 653400 \) Copy content Toggle raw display
$97$ \( (T - 9235)^{2} \) Copy content Toggle raw display
show more
show less