# 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 $2$

# Related objects

## 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: $$x^{2} + 1$$ x^2 + 1 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.

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

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{4}^{\mathrm{new}}(560, [\chi])$$:

 $$T_{3}^{2} + 49$$ T3^2 + 49 $$T_{11} - 37$$ T11 - 37

## Hecke characteristic polynomials

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