# Properties

 Label 370.2.b.b Level $370$ Weight $2$ Character orbit 370.b 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.b (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: $$x^{2} + 1$$ 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} - q^{4} + (2 i + 1) q^{5} + i q^{7} - i q^{8} + 3 q^{9} +O(q^{10})$$ q + i * q^2 - q^4 + (2*i + 1) * q^5 + i * q^7 - i * q^8 + 3 * q^9 $$q + i q^{2} - q^{4} + (2 i + 1) q^{5} + i q^{7} - i q^{8} + 3 q^{9} + (i - 2) q^{10} - 3 q^{11} + 4 i q^{13} - q^{14} + q^{16} - 3 i q^{17} + 3 i q^{18} + ( - 2 i - 1) q^{20} - 3 i q^{22} + 8 i q^{23} + (4 i - 3) q^{25} - 4 q^{26} - i q^{28} + 3 q^{29} - 7 q^{31} + i q^{32} + 3 q^{34} + (i - 2) q^{35} - 3 q^{36} - i q^{37} + ( - i + 2) q^{40} + 11 q^{41} - 11 i q^{43} + 3 q^{44} + (6 i + 3) q^{45} - 8 q^{46} + 4 i q^{47} + 6 q^{49} + ( - 3 i - 4) q^{50} - 4 i q^{52} - 11 i q^{53} + ( - 6 i - 3) q^{55} + q^{56} + 3 i q^{58} + 12 q^{59} - 15 q^{61} - 7 i q^{62} + 3 i q^{63} - q^{64} + (4 i - 8) q^{65} - 4 i q^{67} + 3 i q^{68} + ( - 2 i - 1) q^{70} + 6 q^{71} - 3 i q^{72} - 2 i q^{73} + q^{74} - 3 i q^{77} + 8 q^{79} + (2 i + 1) q^{80} + 9 q^{81} + 11 i q^{82} - 12 i q^{83} + ( - 3 i + 6) q^{85} + 11 q^{86} + 3 i q^{88} + (3 i - 6) q^{90} - 4 q^{91} - 8 i q^{92} - 4 q^{94} - i q^{97} + 6 i q^{98} - 9 q^{99} +O(q^{100})$$ q + i * q^2 - q^4 + (2*i + 1) * q^5 + i * q^7 - i * q^8 + 3 * q^9 + (i - 2) * q^10 - 3 * q^11 + 4*i * q^13 - q^14 + q^16 - 3*i * q^17 + 3*i * q^18 + (-2*i - 1) * q^20 - 3*i * q^22 + 8*i * q^23 + (4*i - 3) * q^25 - 4 * q^26 - i * q^28 + 3 * q^29 - 7 * q^31 + i * q^32 + 3 * q^34 + (i - 2) * q^35 - 3 * q^36 - i * q^37 + (-i + 2) * q^40 + 11 * q^41 - 11*i * q^43 + 3 * q^44 + (6*i + 3) * q^45 - 8 * q^46 + 4*i * q^47 + 6 * q^49 + (-3*i - 4) * q^50 - 4*i * q^52 - 11*i * q^53 + (-6*i - 3) * q^55 + q^56 + 3*i * q^58 + 12 * q^59 - 15 * q^61 - 7*i * q^62 + 3*i * q^63 - q^64 + (4*i - 8) * q^65 - 4*i * q^67 + 3*i * q^68 + (-2*i - 1) * q^70 + 6 * q^71 - 3*i * q^72 - 2*i * q^73 + q^74 - 3*i * q^77 + 8 * q^79 + (2*i + 1) * q^80 + 9 * q^81 + 11*i * q^82 - 12*i * q^83 + (-3*i + 6) * q^85 + 11 * q^86 + 3*i * q^88 + (3*i - 6) * q^90 - 4 * q^91 - 8*i * q^92 - 4 * q^94 - i * q^97 + 6*i * q^98 - 9 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 2 q^{5} + 6 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 + 2 * q^5 + 6 * q^9 $$2 q - 2 q^{4} + 2 q^{5} + 6 q^{9} - 4 q^{10} - 6 q^{11} - 2 q^{14} + 2 q^{16} - 2 q^{20} - 6 q^{25} - 8 q^{26} + 6 q^{29} - 14 q^{31} + 6 q^{34} - 4 q^{35} - 6 q^{36} + 4 q^{40} + 22 q^{41} + 6 q^{44} + 6 q^{45} - 16 q^{46} + 12 q^{49} - 8 q^{50} - 6 q^{55} + 2 q^{56} + 24 q^{59} - 30 q^{61} - 2 q^{64} - 16 q^{65} - 2 q^{70} + 12 q^{71} + 2 q^{74} + 16 q^{79} + 2 q^{80} + 18 q^{81} + 12 q^{85} + 22 q^{86} - 12 q^{90} - 8 q^{91} - 8 q^{94} - 18 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 + 2 * q^5 + 6 * q^9 - 4 * q^10 - 6 * q^11 - 2 * q^14 + 2 * q^16 - 2 * q^20 - 6 * q^25 - 8 * q^26 + 6 * q^29 - 14 * q^31 + 6 * q^34 - 4 * q^35 - 6 * q^36 + 4 * q^40 + 22 * q^41 + 6 * q^44 + 6 * q^45 - 16 * q^46 + 12 * q^49 - 8 * q^50 - 6 * q^55 + 2 * q^56 + 24 * q^59 - 30 * q^61 - 2 * q^64 - 16 * q^65 - 2 * q^70 + 12 * q^71 + 2 * q^74 + 16 * q^79 + 2 * q^80 + 18 * q^81 + 12 * q^85 + 22 * q^86 - 12 * q^90 - 8 * q^91 - 8 * q^94 - 18 * q^99

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
149.1
 − 1.00000i 1.00000i
1.00000i 0 −1.00000 1.00000 2.00000i 0 1.00000i 1.00000i 3.00000 −2.00000 1.00000i
149.2 1.00000i 0 −1.00000 1.00000 + 2.00000i 0 1.00000i 1.00000i 3.00000 −2.00000 + 1.00000i
 $$n$$: 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
5.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.b.b 2
3.b odd 2 1 3330.2.d.c 2
5.b even 2 1 inner 370.2.b.b 2
5.c odd 4 1 1850.2.a.d 1
5.c odd 4 1 1850.2.a.l 1
15.d odd 2 1 3330.2.d.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.b.b 2 1.a even 1 1 trivial
370.2.b.b 2 5.b even 2 1 inner
1850.2.a.d 1 5.c odd 4 1
1850.2.a.l 1 5.c odd 4 1
3330.2.d.c 2 3.b odd 2 1
3330.2.d.c 2 15.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 $$S_{2}^{\mathrm{new}}(370, [\chi])$$:

 $$T_{3}$$ T3 $$T_{7}^{2} + 1$$ T7^2 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 2T + 5$$
$7$ $$T^{2} + 1$$
$11$ $$(T + 3)^{2}$$
$13$ $$T^{2} + 16$$
$17$ $$T^{2} + 9$$
$19$ $$T^{2}$$
$23$ $$T^{2} + 64$$
$29$ $$(T - 3)^{2}$$
$31$ $$(T + 7)^{2}$$
$37$ $$T^{2} + 1$$
$41$ $$(T - 11)^{2}$$
$43$ $$T^{2} + 121$$
$47$ $$T^{2} + 16$$
$53$ $$T^{2} + 121$$
$59$ $$(T - 12)^{2}$$
$61$ $$(T + 15)^{2}$$
$67$ $$T^{2} + 16$$
$71$ $$(T - 6)^{2}$$
$73$ $$T^{2} + 4$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2} + 144$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 1$$