# Properties

 Label 3700.1.j.a Level $3700$ Weight $1$ Character orbit 3700.j Analytic conductor $1.847$ Analytic rank $0$ Dimension $2$ Projective image $S_{4}$ CM/RM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3700 = 2^{2} \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3700.j (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.84654054674$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 740) Projective image: $$S_{4}$$ Projective field: Galois closure of 4.0.5065300.1

## $q$-expansion

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

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3700\mathbb{Z}\right)^\times$$.

 $$n$$ $$1001$$ $$1777$$ $$1851$$ $$\chi(n)$$ $$-i$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
401.1
 − 1.00000i 1.00000i
0 1.00000i 0 0 0 −1.00000 0 0 0
1301.1 0 1.00000i 0 0 0 −1.00000 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
37.d odd 4 1 inner

## Twists

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.1.t.a 2 5.c odd 4 1
740.1.t.a 2 185.k even 4 1
740.1.t.b yes 2 5.c odd 4 1
740.1.t.b yes 2 185.f even 4 1
2960.1.cj.a 2 20.e even 4 1
2960.1.cj.a 2 740.p odd 4 1
2960.1.cj.b 2 20.e even 4 1
2960.1.cj.b 2 740.s odd 4 1
3700.1.j.a 2 1.a even 1 1 trivial
3700.1.j.a 2 37.d odd 4 1 inner
3700.1.j.b 2 5.b even 2 1
3700.1.j.b 2 185.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 $$S_{1}^{\mathrm{new}}(3700, [\chi])$$:

 $$T_{7} + 1$$ $$T_{17}$$

## Hecke characteristic polynomials

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