# Properties

 Label 370.2.g.a Level $370$ Weight $2$ Character orbit 370.g 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.g (of order $$4$$, degree $$2$$, 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: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + i q^{2} + ( -1 - i ) q^{3} - q^{4} + ( 2 + i ) q^{5} + ( 1 - i ) q^{6} + ( 1 + i ) q^{7} -i q^{8} -i q^{9} +O(q^{10})$$ $$q + i q^{2} + ( -1 - i ) q^{3} - q^{4} + ( 2 + i ) q^{5} + ( 1 - i ) q^{6} + ( 1 + i ) q^{7} -i q^{8} -i q^{9} + ( -1 + 2 i ) q^{10} + 2 i q^{11} + ( 1 + i ) q^{12} -2 i q^{13} + ( -1 + i ) q^{14} + ( -1 - 3 i ) q^{15} + q^{16} + 4 q^{17} + q^{18} + ( 5 + 5 i ) q^{19} + ( -2 - i ) q^{20} -2 i q^{21} -2 q^{22} + ( -1 + i ) q^{24} + ( 3 + 4 i ) q^{25} + 2 q^{26} + ( -4 + 4 i ) q^{27} + ( -1 - i ) q^{28} + ( 3 - 3 i ) q^{29} + ( 3 - i ) q^{30} + ( 7 + 7 i ) q^{31} + i q^{32} + ( 2 - 2 i ) q^{33} + 4 i q^{34} + ( 1 + 3 i ) q^{35} + i q^{36} + ( -6 - i ) q^{37} + ( -5 + 5 i ) q^{38} + ( -2 + 2 i ) q^{39} + ( 1 - 2 i ) q^{40} + 2 q^{42} -4 i q^{43} -2 i q^{44} + ( 1 - 2 i ) q^{45} + ( -7 - 7 i ) q^{47} + ( -1 - i ) q^{48} -5 i q^{49} + ( -4 + 3 i ) q^{50} + ( -4 - 4 i ) q^{51} + 2 i q^{52} + ( 1 - i ) q^{53} + ( -4 - 4 i ) q^{54} + ( -2 + 4 i ) q^{55} + ( 1 - i ) q^{56} -10 i q^{57} + ( 3 + 3 i ) q^{58} + ( 1 + i ) q^{59} + ( 1 + 3 i ) q^{60} + ( -3 - 3 i ) q^{61} + ( -7 + 7 i ) q^{62} + ( 1 - i ) q^{63} - q^{64} + ( 2 - 4 i ) q^{65} + ( 2 + 2 i ) q^{66} + ( -3 + 3 i ) q^{67} -4 q^{68} + ( -3 + i ) q^{70} -8 q^{71} - q^{72} + ( -9 - 9 i ) q^{73} + ( 1 - 6 i ) q^{74} + ( 1 - 7 i ) q^{75} + ( -5 - 5 i ) q^{76} + ( -2 + 2 i ) q^{77} + ( -2 - 2 i ) q^{78} + ( 1 + i ) q^{79} + ( 2 + i ) q^{80} + 5 q^{81} + ( -5 + 5 i ) q^{83} + 2 i q^{84} + ( 8 + 4 i ) q^{85} + 4 q^{86} -6 q^{87} + 2 q^{88} + ( -5 + 5 i ) q^{89} + ( 2 + i ) q^{90} + ( 2 - 2 i ) q^{91} -14 i q^{93} + ( 7 - 7 i ) q^{94} + ( 5 + 15 i ) q^{95} + ( 1 - i ) q^{96} + 8 q^{97} + 5 q^{98} + 2 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} - 2q^{4} + 4q^{5} + 2q^{6} + 2q^{7} + O(q^{10})$$ $$2q - 2q^{3} - 2q^{4} + 4q^{5} + 2q^{6} + 2q^{7} - 2q^{10} + 2q^{12} - 2q^{14} - 2q^{15} + 2q^{16} + 8q^{17} + 2q^{18} + 10q^{19} - 4q^{20} - 4q^{22} - 2q^{24} + 6q^{25} + 4q^{26} - 8q^{27} - 2q^{28} + 6q^{29} + 6q^{30} + 14q^{31} + 4q^{33} + 2q^{35} - 12q^{37} - 10q^{38} - 4q^{39} + 2q^{40} + 4q^{42} + 2q^{45} - 14q^{47} - 2q^{48} - 8q^{50} - 8q^{51} + 2q^{53} - 8q^{54} - 4q^{55} + 2q^{56} + 6q^{58} + 2q^{59} + 2q^{60} - 6q^{61} - 14q^{62} + 2q^{63} - 2q^{64} + 4q^{65} + 4q^{66} - 6q^{67} - 8q^{68} - 6q^{70} - 16q^{71} - 2q^{72} - 18q^{73} + 2q^{74} + 2q^{75} - 10q^{76} - 4q^{77} - 4q^{78} + 2q^{79} + 4q^{80} + 10q^{81} - 10q^{83} + 16q^{85} + 8q^{86} - 12q^{87} + 4q^{88} - 10q^{89} + 4q^{90} + 4q^{91} + 14q^{94} + 10q^{95} + 2q^{96} + 16q^{97} + 10q^{98} + 4q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$261$$ $$297$$ $$\chi(n)$$ $$i$$ $$i$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
43.1
 − 1.00000i 1.00000i
1.00000i −1.00000 + 1.00000i −1.00000 2.00000 1.00000i 1.00000 + 1.00000i 1.00000 1.00000i 1.00000i 1.00000i −1.00000 2.00000i
327.1 1.00000i −1.00000 1.00000i −1.00000 2.00000 + 1.00000i 1.00000 1.00000i 1.00000 + 1.00000i 1.00000i 1.00000i −1.00000 + 2.00000i
 $$n$$: 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.f even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 370.2.g.a 2
5.c odd 4 1 370.2.h.b yes 2
37.d odd 4 1 370.2.h.b yes 2
185.f even 4 1 inner 370.2.g.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.g.a 2 1.a even 1 1 trivial
370.2.g.a 2 185.f even 4 1 inner
370.2.h.b yes 2 5.c odd 4 1
370.2.h.b yes 2 37.d odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 2 T_{3} + 2$$ acting on $$S_{2}^{\mathrm{new}}(370, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$2 + 2 T + T^{2}$$
$5$ $$5 - 4 T + T^{2}$$
$7$ $$2 - 2 T + T^{2}$$
$11$ $$4 + T^{2}$$
$13$ $$4 + T^{2}$$
$17$ $$( -4 + T )^{2}$$
$19$ $$50 - 10 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$18 - 6 T + T^{2}$$
$31$ $$98 - 14 T + T^{2}$$
$37$ $$37 + 12 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$98 + 14 T + T^{2}$$
$53$ $$2 - 2 T + T^{2}$$
$59$ $$2 - 2 T + T^{2}$$
$61$ $$18 + 6 T + T^{2}$$
$67$ $$18 + 6 T + T^{2}$$
$71$ $$( 8 + T )^{2}$$
$73$ $$162 + 18 T + T^{2}$$
$79$ $$2 - 2 T + T^{2}$$
$83$ $$50 + 10 T + T^{2}$$
$89$ $$50 + 10 T + T^{2}$$
$97$ $$( -8 + T )^{2}$$