# Properties

 Label 370.2.d.a Level $370$ Weight $2$ Character orbit 370.d Analytic conductor $2.954$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$370 = 2 \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 370.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.95446487479$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} - q^{4} + i q^{5} -2 q^{7} -i q^{8} -3 q^{9} +O(q^{10})$$ $$q + i q^{2} - q^{4} + i q^{5} -2 q^{7} -i q^{8} -3 q^{9} - q^{10} -4 q^{11} + 2 i q^{13} -2 i q^{14} + q^{16} + 6 i q^{17} -3 i q^{18} + 2 i q^{19} -i q^{20} -4 i q^{22} -8 i q^{23} - q^{25} -2 q^{26} + 2 q^{28} -2 i q^{29} + 8 i q^{31} + i q^{32} -6 q^{34} -2 i q^{35} + 3 q^{36} + ( -6 + i ) q^{37} -2 q^{38} + q^{40} -2 q^{41} -4 i q^{43} + 4 q^{44} -3 i q^{45} + 8 q^{46} + 6 q^{47} -3 q^{49} -i q^{50} -2 i q^{52} + 12 q^{53} -4 i q^{55} + 2 i q^{56} + 2 q^{58} + 2 i q^{59} + 14 i q^{61} -8 q^{62} + 6 q^{63} - q^{64} -2 q^{65} + 16 q^{67} -6 i q^{68} + 2 q^{70} -12 q^{71} + 3 i q^{72} + 6 q^{73} + ( -1 - 6 i ) q^{74} -2 i q^{76} + 8 q^{77} + i q^{80} + 9 q^{81} -2 i q^{82} -12 q^{83} -6 q^{85} + 4 q^{86} + 4 i q^{88} + 4 i q^{89} + 3 q^{90} -4 i q^{91} + 8 i q^{92} + 6 i q^{94} -2 q^{95} + 10 i q^{97} -3 i q^{98} + 12 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} - 4q^{7} - 6q^{9} + O(q^{10})$$ $$2q - 2q^{4} - 4q^{7} - 6q^{9} - 2q^{10} - 8q^{11} + 2q^{16} - 2q^{25} - 4q^{26} + 4q^{28} - 12q^{34} + 6q^{36} - 12q^{37} - 4q^{38} + 2q^{40} - 4q^{41} + 8q^{44} + 16q^{46} + 12q^{47} - 6q^{49} + 24q^{53} + 4q^{58} - 16q^{62} + 12q^{63} - 2q^{64} - 4q^{65} + 32q^{67} + 4q^{70} - 24q^{71} + 12q^{73} - 2q^{74} + 16q^{77} + 18q^{81} - 24q^{83} - 12q^{85} + 8q^{86} + 6q^{90} - 4q^{95} + 24q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/370\mathbb{Z}\right)^\times$$.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
221.1
 − 1.00000i 1.00000i
1.00000i 0 −1.00000 1.00000i 0 −2.00000 1.00000i −3.00000 −1.00000
221.2 1.00000i 0 −1.00000 1.00000i 0 −2.00000 1.00000i −3.00000 −1.00000
 $$n$$: 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.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 370.2.d.a 2
3.b odd 2 1 3330.2.h.c 2
4.b odd 2 1 2960.2.p.e 2
5.b even 2 1 1850.2.d.c 2
5.c odd 4 1 1850.2.c.a 2
5.c odd 4 1 1850.2.c.d 2
37.b even 2 1 inner 370.2.d.a 2
111.d odd 2 1 3330.2.h.c 2
148.b odd 2 1 2960.2.p.e 2
185.d even 2 1 1850.2.d.c 2
185.h odd 4 1 1850.2.c.a 2
185.h odd 4 1 1850.2.c.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.d.a 2 1.a even 1 1 trivial
370.2.d.a 2 37.b even 2 1 inner
1850.2.c.a 2 5.c odd 4 1
1850.2.c.a 2 185.h odd 4 1
1850.2.c.d 2 5.c odd 4 1
1850.2.c.d 2 185.h odd 4 1
1850.2.d.c 2 5.b even 2 1
1850.2.d.c 2 185.d even 2 1
2960.2.p.e 2 4.b odd 2 1
2960.2.p.e 2 148.b odd 2 1
3330.2.h.c 2 3.b odd 2 1
3330.2.h.c 2 111.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(370, [\chi])$$:

 $$T_{3}$$ $$T_{7} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$1 + T^{2}$$
$7$ $$( 2 + T )^{2}$$
$11$ $$( 4 + T )^{2}$$
$13$ $$4 + T^{2}$$
$17$ $$36 + T^{2}$$
$19$ $$4 + T^{2}$$
$23$ $$64 + T^{2}$$
$29$ $$4 + T^{2}$$
$31$ $$64 + T^{2}$$
$37$ $$37 + 12 T + T^{2}$$
$41$ $$( 2 + T )^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$( -6 + T )^{2}$$
$53$ $$( -12 + T )^{2}$$
$59$ $$4 + T^{2}$$
$61$ $$196 + T^{2}$$
$67$ $$( -16 + T )^{2}$$
$71$ $$( 12 + T )^{2}$$
$73$ $$( -6 + T )^{2}$$
$79$ $$T^{2}$$
$83$ $$( 12 + T )^{2}$$
$89$ $$16 + T^{2}$$
$97$ $$100 + T^{2}$$