# Properties

 Label 240.3.c.a Level $240$ Weight $3$ Character orbit 240.c Self dual yes Analytic conductor $6.540$ Analytic rank $0$ Dimension $1$ CM discriminant -15 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$240 = 2^{4} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 240.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$6.53952634465$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 3 q^{3} - 5 q^{5} + 9 q^{9} + O(q^{10})$$ $$q - 3 q^{3} - 5 q^{5} + 9 q^{9} + 15 q^{15} + 14 q^{17} + 22 q^{19} + 34 q^{23} + 25 q^{25} - 27 q^{27} - 2 q^{31} - 45 q^{45} - 14 q^{47} + 49 q^{49} - 42 q^{51} + 86 q^{53} - 66 q^{57} - 118 q^{61} - 102 q^{69} - 75 q^{75} - 98 q^{79} + 81 q^{81} + 154 q^{83} - 70 q^{85} + 6 q^{93} - 110 q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/240\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$97$$ $$161$$ $$181$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$1$$

## 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
 0
0 −3.00000 0 −5.00000 0 0 0 9.00000 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
15.d odd 2 1 CM by $$\Q(\sqrt{-15})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.3.c.a 1
3.b odd 2 1 240.3.c.b 1
4.b odd 2 1 15.3.d.a 1
5.b even 2 1 240.3.c.b 1
5.c odd 4 2 1200.3.l.l 2
8.b even 2 1 960.3.c.d 1
8.d odd 2 1 960.3.c.b 1
12.b even 2 1 15.3.d.b yes 1
15.d odd 2 1 CM 240.3.c.a 1
15.e even 4 2 1200.3.l.l 2
20.d odd 2 1 15.3.d.b yes 1
20.e even 4 2 75.3.c.d 2
24.f even 2 1 960.3.c.c 1
24.h odd 2 1 960.3.c.a 1
36.f odd 6 2 405.3.h.b 2
36.h even 6 2 405.3.h.a 2
40.e odd 2 1 960.3.c.c 1
40.f even 2 1 960.3.c.a 1
60.h even 2 1 15.3.d.a 1
60.l odd 4 2 75.3.c.d 2
120.i odd 2 1 960.3.c.d 1
120.m even 2 1 960.3.c.b 1
180.n even 6 2 405.3.h.b 2
180.p odd 6 2 405.3.h.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.3.d.a 1 4.b odd 2 1
15.3.d.a 1 60.h even 2 1
15.3.d.b yes 1 12.b even 2 1
15.3.d.b yes 1 20.d odd 2 1
75.3.c.d 2 20.e even 4 2
75.3.c.d 2 60.l odd 4 2
240.3.c.a 1 1.a even 1 1 trivial
240.3.c.a 1 15.d odd 2 1 CM
240.3.c.b 1 3.b odd 2 1
240.3.c.b 1 5.b even 2 1
405.3.h.a 2 36.h even 6 2
405.3.h.a 2 180.p odd 6 2
405.3.h.b 2 36.f odd 6 2
405.3.h.b 2 180.n even 6 2
960.3.c.a 1 24.h odd 2 1
960.3.c.a 1 40.f even 2 1
960.3.c.b 1 8.d odd 2 1
960.3.c.b 1 120.m even 2 1
960.3.c.c 1 24.f even 2 1
960.3.c.c 1 40.e odd 2 1
960.3.c.d 1 8.b even 2 1
960.3.c.d 1 120.i odd 2 1
1200.3.l.l 2 5.c odd 4 2
1200.3.l.l 2 15.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(240, [\chi])$$:

 $$T_{7}$$ $$T_{17} - 14$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$3 + T$$
$5$ $$5 + T$$
$7$ $$T$$
$11$ $$T$$
$13$ $$T$$
$17$ $$-14 + T$$
$19$ $$-22 + T$$
$23$ $$-34 + T$$
$29$ $$T$$
$31$ $$2 + T$$
$37$ $$T$$
$41$ $$T$$
$43$ $$T$$
$47$ $$14 + T$$
$53$ $$-86 + T$$
$59$ $$T$$
$61$ $$118 + T$$
$67$ $$T$$
$71$ $$T$$
$73$ $$T$$
$79$ $$98 + T$$
$83$ $$-154 + T$$
$89$ $$T$$
$97$ $$T$$