# Properties

 Label 418.2.b.b Level $418$ Weight $2$ Character orbit 418.b Analytic conductor $3.338$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$418 = 2 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 418.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.33774680449$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-10})$$ Defining polynomial: $$x^{2} + 10$$ x^2 + 10 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 $$\beta = \sqrt{-10}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/418\mathbb{Z}\right)^\times$$.

 $$n$$ $$287$$ $$343$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
417.1
 − 3.16228i 3.16228i
1.00000 0 1.00000 −2.00000 0 3.16228i 1.00000 3.00000 −2.00000
417.2 1.00000 0 1.00000 −2.00000 0 3.16228i 1.00000 3.00000 −2.00000
 $$n$$: 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
209.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 418.2.b.b yes 2
3.b odd 2 1 3762.2.g.c 2
11.b odd 2 1 418.2.b.a 2
19.b odd 2 1 418.2.b.a 2
33.d even 2 1 3762.2.g.f 2
57.d even 2 1 3762.2.g.f 2
209.d even 2 1 inner 418.2.b.b yes 2
627.b odd 2 1 3762.2.g.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.b.a 2 11.b odd 2 1
418.2.b.a 2 19.b odd 2 1
418.2.b.b yes 2 1.a even 1 1 trivial
418.2.b.b yes 2 209.d even 2 1 inner
3762.2.g.c 2 3.b odd 2 1
3762.2.g.c 2 627.b odd 2 1
3762.2.g.f 2 33.d even 2 1
3762.2.g.f 2 57.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(418, [\chi])$$:

 $$T_{3}$$ T3 $$T_{13} - 6$$ T13 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$T^{2}$$
$5$ $$(T + 2)^{2}$$
$7$ $$T^{2} + 10$$
$11$ $$T^{2} - 2T + 11$$
$13$ $$(T - 6)^{2}$$
$17$ $$T^{2} + 40$$
$19$ $$T^{2} - 6T + 19$$
$23$ $$(T + 4)^{2}$$
$29$ $$(T + 6)^{2}$$
$31$ $$T^{2} + 90$$
$37$ $$T^{2} + 90$$
$41$ $$T^{2}$$
$43$ $$T^{2}$$
$47$ $$(T + 8)^{2}$$
$53$ $$T^{2} + 90$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 10$$
$67$ $$T^{2}$$
$71$ $$T^{2} + 90$$
$73$ $$T^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 40$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 360$$