# Properties

 Label 370.2.h.b Level $370$ Weight $2$ Character orbit 370.h 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.h (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: $$\Z[a_1, a_2, a_3]$$ 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.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
117.1
 − 1.00000i 1.00000i
1.00000 1.00000 + 1.00000i 1.00000 −1.00000 + 2.00000i 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 1.00000i −1.00000 + 2.00000i
253.1 1.00000 1.00000 1.00000i 1.00000 −1.00000 2.00000i 1.00000 1.00000i 1.00000 1.00000i 1.00000 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.k 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.h.b yes 2
5.c odd 4 1 370.2.g.a 2
37.d odd 4 1 370.2.g.a 2
185.k even 4 1 inner 370.2.h.b yes 2

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

## 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

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$2 - 2 T + T^{2}$$
$5$ $$5 + 2 T + T^{2}$$
$7$ $$2 - 2 T + T^{2}$$
$11$ $$4 + T^{2}$$
$13$ $$( -2 + T )^{2}$$
$17$ $$16 + T^{2}$$
$19$ $$50 + 10 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$18 + 6 T + T^{2}$$
$31$ $$98 - 14 T + T^{2}$$
$37$ $$37 + 2 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$( -4 + T )^{2}$$
$47$ $$98 + 14 T + T^{2}$$
$53$ $$2 - 2 T + T^{2}$$
$59$ $$2 + 2 T + T^{2}$$
$61$ $$18 + 6 T + T^{2}$$
$67$ $$18 - 6 T + T^{2}$$
$71$ $$( 8 + T )^{2}$$
$73$ $$162 - 18 T + T^{2}$$
$79$ $$2 + 2 T + T^{2}$$
$83$ $$50 + 10 T + T^{2}$$
$89$ $$50 - 10 T + T^{2}$$
$97$ $$64 + T^{2}$$