Properties

 Label 740.1.t.a Level $740$ Weight $1$ Character orbit 740.t Analytic conductor $0.369$ Analytic rank $0$ Dimension $2$ Projective image $S_{4}$ CM/RM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$740 = 2^{2} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 740.t (of order $$4$$, degree $$2$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$0.369308109348$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$S_{4}$$ Projective field: Galois closure of 4.0.5065300.1

$q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{5} -i q^{7} +O(q^{10})$$ $$q - q^{3} + q^{5} -i q^{7} -i q^{11} - q^{15} + i q^{21} + q^{25} + q^{27} + ( 1 - i ) q^{31} + i q^{33} -i q^{35} -i q^{37} -i q^{41} + ( -1 + i ) q^{43} + i q^{47} + i q^{53} -i q^{55} + ( -1 + i ) q^{61} - q^{71} + q^{73} - q^{75} - q^{77} + ( 1 + i ) q^{79} - q^{81} + i q^{83} + ( -1 + i ) q^{89} + ( -1 + i ) q^{93} + ( 1 - i ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{5} + O(q^{10})$$ $$2 q - 2 q^{3} + 2 q^{5} - 2 q^{15} + 2 q^{25} + 2 q^{27} + 2 q^{31} - 2 q^{43} - 2 q^{61} - 2 q^{71} + 2 q^{73} - 2 q^{75} - 2 q^{77} + 2 q^{79} - 2 q^{81} - 2 q^{89} - 2 q^{93} + 2 q^{97} + O(q^{100})$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/740\mathbb{Z}\right)^\times$$.

 $$n$$ $$261$$ $$297$$ $$371$$ $$\chi(n)$$ $$-i$$ $$-1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
549.1
 − 1.00000i 1.00000i
0 −1.00000 0 1.00000 0 1.00000i 0 0 0
709.1 0 −1.00000 0 1.00000 0 1.00000i 0 0 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
185.j odd 4 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 740.1.t.a 2
4.b odd 2 1 2960.1.cj.b 2
5.b even 2 1 740.1.t.b yes 2
5.c odd 4 1 3700.1.j.a 2
5.c odd 4 1 3700.1.j.b 2
20.d odd 2 1 2960.1.cj.a 2
37.d odd 4 1 740.1.t.b yes 2
148.g even 4 1 2960.1.cj.a 2
185.f even 4 1 3700.1.j.b 2
185.j odd 4 1 inner 740.1.t.a 2
185.k even 4 1 3700.1.j.a 2
740.k even 4 1 2960.1.cj.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.1.t.a 2 1.a even 1 1 trivial
740.1.t.a 2 185.j odd 4 1 inner
740.1.t.b yes 2 5.b even 2 1
740.1.t.b yes 2 37.d odd 4 1
2960.1.cj.a 2 20.d odd 2 1
2960.1.cj.a 2 148.g even 4 1
2960.1.cj.b 2 4.b odd 2 1
2960.1.cj.b 2 740.k even 4 1
3700.1.j.a 2 5.c odd 4 1
3700.1.j.a 2 185.k even 4 1
3700.1.j.b 2 5.c odd 4 1
3700.1.j.b 2 185.f even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 1$$ acting on $$S_{1}^{\mathrm{new}}(740, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$1 + T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$2 - 2 T + T^{2}$$
$37$ $$1 + T^{2}$$
$41$ $$1 + T^{2}$$
$43$ $$2 + 2 T + T^{2}$$
$47$ $$1 + T^{2}$$
$53$ $$1 + T^{2}$$
$59$ $$T^{2}$$
$61$ $$2 + 2 T + T^{2}$$
$67$ $$T^{2}$$
$71$ $$( 1 + T )^{2}$$
$73$ $$( -1 + T )^{2}$$
$79$ $$2 - 2 T + T^{2}$$
$83$ $$1 + T^{2}$$
$89$ $$2 + 2 T + T^{2}$$
$97$ $$2 - 2 T + T^{2}$$