# Properties

 Label 3700.1.t.a Level 3700 Weight 1 Character orbit 3700.t 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.t (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.84654054674$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 148) Projective image $$S_{4}$$ Projective field Galois closure of 4.0.202612.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q - q^{3} + i q^{7} +O(q^{10})$$ $$q - q^{3} + i q^{7} -i q^{11} + ( -1 + i ) q^{17} + ( -1 - i ) q^{19} -i q^{21} + ( 1 - i ) q^{23} + q^{27} + ( 1 - i ) q^{29} + i q^{33} + q^{37} + i q^{41} -i q^{47} + ( 1 - i ) q^{51} + i q^{53} + ( 1 + i ) q^{57} + ( -1 + i ) q^{69} + q^{71} - q^{73} + q^{77} + ( 1 + i ) q^{79} - q^{81} -i q^{83} + ( -1 + i ) q^{87} + ( 1 - i ) q^{89} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} + O(q^{10})$$ $$2q - 2q^{3} - 2q^{17} - 2q^{19} + 2q^{23} + 2q^{27} + 2q^{29} + 2q^{37} + 2q^{51} + 2q^{57} - 2q^{69} + 2q^{71} - 2q^{73} + 2q^{77} + 2q^{79} - 2q^{81} - 2q^{87} + 2q^{89} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
549.1
 − 1.00000i 1.00000i
0 −1.00000 0 0 0 1.00000i 0 0 0
1449.1 0 −1.00000 0 0 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 3700.1.t.a 2
5.b even 2 1 3700.1.t.b 2
5.c odd 4 1 148.1.f.a 2
5.c odd 4 1 3700.1.j.c 2
15.e even 4 1 1332.1.o.a 2
20.e even 4 1 592.1.k.b 2
37.d odd 4 1 3700.1.t.b 2
40.i odd 4 1 2368.1.k.a 2
40.k even 4 1 2368.1.k.b 2
185.f even 4 1 148.1.f.a 2
185.j odd 4 1 inner 3700.1.t.a 2
185.k even 4 1 3700.1.j.c 2
555.u odd 4 1 1332.1.o.a 2
740.p odd 4 1 592.1.k.b 2
1480.bc odd 4 1 2368.1.k.b 2
1480.bk even 4 1 2368.1.k.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
148.1.f.a 2 5.c odd 4 1
148.1.f.a 2 185.f even 4 1
592.1.k.b 2 20.e even 4 1
592.1.k.b 2 740.p odd 4 1
1332.1.o.a 2 15.e even 4 1
1332.1.o.a 2 555.u odd 4 1
2368.1.k.a 2 40.i odd 4 1
2368.1.k.a 2 1480.bk even 4 1
2368.1.k.b 2 40.k even 4 1
2368.1.k.b 2 1480.bc odd 4 1
3700.1.j.c 2 5.c odd 4 1
3700.1.j.c 2 185.k even 4 1
3700.1.t.a 2 1.a even 1 1 trivial
3700.1.t.a 2 185.j odd 4 1 inner
3700.1.t.b 2 5.b even 2 1
3700.1.t.b 2 37.d odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 1$$ acting on $$S_{1}^{\mathrm{new}}(3700, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 + T + T^{2} )^{2}$$
$5$ 1
$7$ $$1 - T^{2} + T^{4}$$
$11$ $$1 - T^{2} + T^{4}$$
$13$ $$1 + T^{4}$$
$17$ $$( 1 + T )^{2}( 1 + T^{2} )$$
$19$ $$( 1 + T )^{2}( 1 + T^{2} )$$
$23$ $$( 1 - T )^{2}( 1 + T^{2} )$$
$29$ $$( 1 - T )^{2}( 1 + T^{2} )$$
$31$ $$1 + T^{4}$$
$37$ $$( 1 - T )^{2}$$
$41$ $$1 - T^{2} + T^{4}$$
$43$ $$1 + T^{4}$$
$47$ $$1 - T^{2} + T^{4}$$
$53$ $$1 - T^{2} + T^{4}$$
$59$ $$1 + T^{4}$$
$61$ $$1 + T^{4}$$
$67$ $$( 1 + T^{2} )^{2}$$
$71$ $$( 1 - T + T^{2} )^{2}$$
$73$ $$( 1 + T + T^{2} )^{2}$$
$79$ $$( 1 - T )^{2}( 1 + T^{2} )$$
$83$ $$1 - T^{2} + T^{4}$$
$89$ $$( 1 - T )^{2}( 1 + T^{2} )$$
$97$ $$1 + T^{4}$$