# Properties

 Label 418.2.e.g Level $418$ Weight $2$ Character orbit 418.e Analytic conductor $3.338$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$418 = 2 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 418.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.33774680449$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2x^{2} + 4$$ x^4 - 2*x^2 + 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} - \beta_{2} q^{3} + (\beta_1 - 1) q^{4} + (\beta_{3} - \beta_{2}) q^{6} + \beta_{3} q^{7} - q^{8} + (3 \beta_1 - 3) q^{9}+O(q^{10})$$ q + b1 * q^2 - b2 * q^3 + (b1 - 1) * q^4 + (b3 - b2) * q^6 + b3 * q^7 - q^8 + (3*b1 - 3) * q^9 $$q + \beta_1 q^{2} - \beta_{2} q^{3} + (\beta_1 - 1) q^{4} + (\beta_{3} - \beta_{2}) q^{6} + \beta_{3} q^{7} - q^{8} + (3 \beta_1 - 3) q^{9} - q^{11} + \beta_{3} q^{12} + (2 \beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{13} + \beta_{2} q^{14} - \beta_1 q^{16} + ( - \beta_{2} + 2 \beta_1) q^{17} - 3 q^{18} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{19} - 6 \beta_1 q^{21} - \beta_1 q^{22} + (2 \beta_{3} - 2 \beta_{2} - 4 \beta_1 + 4) q^{23} + \beta_{2} q^{24} + ( - 5 \beta_1 + 5) q^{25} + (2 \beta_{3} + 1) q^{26} + ( - \beta_{3} + \beta_{2}) q^{28} + ( - 2 \beta_{3} + 2 \beta_{2} + 3 \beta_1 - 3) q^{29} + 2 q^{31} + ( - \beta_1 + 1) q^{32} + \beta_{2} q^{33} + (\beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{34} - 3 \beta_1 q^{36} + ( - \beta_{3} + 6) q^{37} + (\beta_{3} - 2 \beta_{2} + 1) q^{38} + ( - \beta_{3} - 12) q^{39} + (\beta_{2} + 10 \beta_1) q^{41} + ( - 6 \beta_1 + 6) q^{42} + (\beta_{2} - 7 \beta_1) q^{43} + ( - \beta_1 + 1) q^{44} + (2 \beta_{3} + 4) q^{46} + (\beta_{3} - \beta_{2} - 9 \beta_1 + 9) q^{47} + ( - \beta_{3} + \beta_{2}) q^{48} - q^{49} + 5 q^{50} + (2 \beta_{3} - 2 \beta_{2} + 6 \beta_1 - 6) q^{51} + (2 \beta_{2} + \beta_1) q^{52} + ( - 4 \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 2) q^{53} - \beta_{3} q^{56} + ( - \beta_{3} + 12 \beta_1 - 6) q^{57} + ( - 2 \beta_{3} - 3) q^{58} + (2 \beta_{2} + 2 \beta_1) q^{59} + (2 \beta_{3} - 2 \beta_{2} + 3 \beta_1 - 3) q^{61} + 2 \beta_1 q^{62} + ( - 3 \beta_{3} + 3 \beta_{2}) q^{63} + q^{64} + ( - \beta_{3} + \beta_{2}) q^{66} + ( - \beta_{3} + \beta_{2} + 12 \beta_1 - 12) q^{67} + (\beta_{3} - 2) q^{68} + ( - 4 \beta_{3} - 12) q^{69} + (\beta_{2} - 5 \beta_1) q^{71} + ( - 3 \beta_1 + 3) q^{72} + ( - \beta_{2} + 12 \beta_1) q^{73} + ( - \beta_{2} + 6 \beta_1) q^{74} - 5 \beta_{3} q^{75} + (2 \beta_{3} - \beta_{2} + \beta_1) q^{76} - \beta_{3} q^{77} + ( - \beta_{2} - 12 \beta_1) q^{78} + (\beta_{2} - 2 \beta_1) q^{79} + 9 \beta_1 q^{81} + ( - \beta_{3} + \beta_{2} + 10 \beta_1 - 10) q^{82} + (\beta_{3} - 3) q^{83} + 6 q^{84} + ( - \beta_{3} + \beta_{2} - 7 \beta_1 + 7) q^{86} + (3 \beta_{3} + 12) q^{87} + q^{88} + ( - 9 \beta_1 + 9) q^{89} + (\beta_{3} - \beta_{2} - 12 \beta_1 + 12) q^{91} + (2 \beta_{2} + 4 \beta_1) q^{92} - 2 \beta_{2} q^{93} + (\beta_{3} + 9) q^{94} - \beta_{3} q^{96} + (2 \beta_{2} + \beta_1) q^{97} - \beta_1 q^{98} + ( - 3 \beta_1 + 3) q^{99}+O(q^{100})$$ q + b1 * q^2 - b2 * q^3 + (b1 - 1) * q^4 + (b3 - b2) * q^6 + b3 * q^7 - q^8 + (3*b1 - 3) * q^9 - q^11 + b3 * q^12 + (2*b3 - 2*b2 - b1 + 1) * q^13 + b2 * q^14 - b1 * q^16 + (-b2 + 2*b1) * q^17 - 3 * q^18 + (-b3 - b2 - b1 + 1) * q^19 - 6*b1 * q^21 - b1 * q^22 + (2*b3 - 2*b2 - 4*b1 + 4) * q^23 + b2 * q^24 + (-5*b1 + 5) * q^25 + (2*b3 + 1) * q^26 + (-b3 + b2) * q^28 + (-2*b3 + 2*b2 + 3*b1 - 3) * q^29 + 2 * q^31 + (-b1 + 1) * q^32 + b2 * q^33 + (b3 - b2 + 2*b1 - 2) * q^34 - 3*b1 * q^36 + (-b3 + 6) * q^37 + (b3 - 2*b2 + 1) * q^38 + (-b3 - 12) * q^39 + (b2 + 10*b1) * q^41 + (-6*b1 + 6) * q^42 + (b2 - 7*b1) * q^43 + (-b1 + 1) * q^44 + (2*b3 + 4) * q^46 + (b3 - b2 - 9*b1 + 9) * q^47 + (-b3 + b2) * q^48 - q^49 + 5 * q^50 + (2*b3 - 2*b2 + 6*b1 - 6) * q^51 + (2*b2 + b1) * q^52 + (-4*b3 + 4*b2 + 2*b1 - 2) * q^53 - b3 * q^56 + (-b3 + 12*b1 - 6) * q^57 + (-2*b3 - 3) * q^58 + (2*b2 + 2*b1) * q^59 + (2*b3 - 2*b2 + 3*b1 - 3) * q^61 + 2*b1 * q^62 + (-3*b3 + 3*b2) * q^63 + q^64 + (-b3 + b2) * q^66 + (-b3 + b2 + 12*b1 - 12) * q^67 + (b3 - 2) * q^68 + (-4*b3 - 12) * q^69 + (b2 - 5*b1) * q^71 + (-3*b1 + 3) * q^72 + (-b2 + 12*b1) * q^73 + (-b2 + 6*b1) * q^74 - 5*b3 * q^75 + (2*b3 - b2 + b1) * q^76 - b3 * q^77 + (-b2 - 12*b1) * q^78 + (b2 - 2*b1) * q^79 + 9*b1 * q^81 + (-b3 + b2 + 10*b1 - 10) * q^82 + (b3 - 3) * q^83 + 6 * q^84 + (-b3 + b2 - 7*b1 + 7) * q^86 + (3*b3 + 12) * q^87 + q^88 + (-9*b1 + 9) * q^89 + (b3 - b2 - 12*b1 + 12) * q^91 + (2*b2 + 4*b1) * q^92 - 2*b2 * q^93 + (b3 + 9) * q^94 - b3 * q^96 + (2*b2 + b1) * q^97 - b1 * q^98 + (-3*b1 + 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} - 2 q^{4} - 4 q^{8} - 6 q^{9}+O(q^{10})$$ 4 * q + 2 * q^2 - 2 * q^4 - 4 * q^8 - 6 * q^9 $$4 q + 2 q^{2} - 2 q^{4} - 4 q^{8} - 6 q^{9} - 4 q^{11} + 2 q^{13} - 2 q^{16} + 4 q^{17} - 12 q^{18} + 2 q^{19} - 12 q^{21} - 2 q^{22} + 8 q^{23} + 10 q^{25} + 4 q^{26} - 6 q^{29} + 8 q^{31} + 2 q^{32} - 4 q^{34} - 6 q^{36} + 24 q^{37} + 4 q^{38} - 48 q^{39} + 20 q^{41} + 12 q^{42} - 14 q^{43} + 2 q^{44} + 16 q^{46} + 18 q^{47} - 4 q^{49} + 20 q^{50} - 12 q^{51} + 2 q^{52} - 4 q^{53} - 12 q^{58} + 4 q^{59} - 6 q^{61} + 4 q^{62} + 4 q^{64} - 24 q^{67} - 8 q^{68} - 48 q^{69} - 10 q^{71} + 6 q^{72} + 24 q^{73} + 12 q^{74} + 2 q^{76} - 24 q^{78} - 4 q^{79} + 18 q^{81} - 20 q^{82} - 12 q^{83} + 24 q^{84} + 14 q^{86} + 48 q^{87} + 4 q^{88} + 18 q^{89} + 24 q^{91} + 8 q^{92} + 36 q^{94} + 2 q^{97} - 2 q^{98} + 6 q^{99}+O(q^{100})$$ 4 * q + 2 * q^2 - 2 * q^4 - 4 * q^8 - 6 * q^9 - 4 * q^11 + 2 * q^13 - 2 * q^16 + 4 * q^17 - 12 * q^18 + 2 * q^19 - 12 * q^21 - 2 * q^22 + 8 * q^23 + 10 * q^25 + 4 * q^26 - 6 * q^29 + 8 * q^31 + 2 * q^32 - 4 * q^34 - 6 * q^36 + 24 * q^37 + 4 * q^38 - 48 * q^39 + 20 * q^41 + 12 * q^42 - 14 * q^43 + 2 * q^44 + 16 * q^46 + 18 * q^47 - 4 * q^49 + 20 * q^50 - 12 * q^51 + 2 * q^52 - 4 * q^53 - 12 * q^58 + 4 * q^59 - 6 * q^61 + 4 * q^62 + 4 * q^64 - 24 * q^67 - 8 * q^68 - 48 * q^69 - 10 * q^71 + 6 * q^72 + 24 * q^73 + 12 * q^74 + 2 * q^76 - 24 * q^78 - 4 * q^79 + 18 * q^81 - 20 * q^82 - 12 * q^83 + 24 * q^84 + 14 * q^86 + 48 * q^87 + 4 * q^88 + 18 * q^89 + 24 * q^91 + 8 * q^92 + 36 * q^94 + 2 * q^97 - 2 * q^98 + 6 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 2\nu ) / 2$$ (v^3 + 2*v) / 2 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 4\nu ) / 2$$ (-v^3 + 4*v) / 2
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 3$$ (b3 + b2) / 3 $$\nu^{2}$$ $$=$$ $$2\beta_1$$ 2*b1 $$\nu^{3}$$ $$=$$ $$( -2\beta_{3} + 4\beta_{2} ) / 3$$ (-2*b3 + 4*b2) / 3

## Character values

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

 $$n$$ $$287$$ $$343$$ $$\chi(n)$$ $$-1 + \beta_{1}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
45.1
 1.22474 + 0.707107i −1.22474 − 0.707107i 1.22474 − 0.707107i −1.22474 + 0.707107i
0.500000 + 0.866025i −1.22474 2.12132i −0.500000 + 0.866025i 0 1.22474 2.12132i 2.44949 −1.00000 −1.50000 + 2.59808i 0
45.2 0.500000 + 0.866025i 1.22474 + 2.12132i −0.500000 + 0.866025i 0 −1.22474 + 2.12132i −2.44949 −1.00000 −1.50000 + 2.59808i 0
353.1 0.500000 0.866025i −1.22474 + 2.12132i −0.500000 0.866025i 0 1.22474 + 2.12132i 2.44949 −1.00000 −1.50000 2.59808i 0
353.2 0.500000 0.866025i 1.22474 2.12132i −0.500000 0.866025i 0 −1.22474 2.12132i −2.44949 −1.00000 −1.50000 2.59808i 0
 $$n$$: 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
19.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 418.2.e.g 4
19.c even 3 1 inner 418.2.e.g 4
19.c even 3 1 7942.2.a.v 2
19.d odd 6 1 7942.2.a.y 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.e.g 4 1.a even 1 1 trivial
418.2.e.g 4 19.c even 3 1 inner
7942.2.a.v 2 19.c even 3 1
7942.2.a.y 2 19.d odd 6 1

## Hecke kernels

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

 $$T_{3}^{4} + 6T_{3}^{2} + 36$$ T3^4 + 6*T3^2 + 36 $$T_{5}$$ T5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - T + 1)^{2}$$
$3$ $$T^{4} + 6T^{2} + 36$$
$5$ $$T^{4}$$
$7$ $$(T^{2} - 6)^{2}$$
$11$ $$(T + 1)^{4}$$
$13$ $$T^{4} - 2 T^{3} + 27 T^{2} + 46 T + 529$$
$17$ $$T^{4} - 4 T^{3} + 18 T^{2} + 8 T + 4$$
$19$ $$T^{4} - 2 T^{3} - 15 T^{2} - 38 T + 361$$
$23$ $$T^{4} - 8 T^{3} + 72 T^{2} + 64 T + 64$$
$29$ $$T^{4} + 6 T^{3} + 51 T^{2} - 90 T + 225$$
$31$ $$(T - 2)^{4}$$
$37$ $$(T^{2} - 12 T + 30)^{2}$$
$41$ $$T^{4} - 20 T^{3} + 306 T^{2} + \cdots + 8836$$
$43$ $$T^{4} + 14 T^{3} + 153 T^{2} + \cdots + 1849$$
$47$ $$T^{4} - 18 T^{3} + 249 T^{2} + \cdots + 5625$$
$53$ $$T^{4} + 4 T^{3} + 108 T^{2} + \cdots + 8464$$
$59$ $$T^{4} - 4 T^{3} + 36 T^{2} + 80 T + 400$$
$61$ $$T^{4} + 6 T^{3} + 51 T^{2} - 90 T + 225$$
$67$ $$T^{4} + 24 T^{3} + 438 T^{2} + \cdots + 19044$$
$71$ $$T^{4} + 10 T^{3} + 81 T^{2} + \cdots + 361$$
$73$ $$T^{4} - 24 T^{3} + 438 T^{2} + \cdots + 19044$$
$79$ $$T^{4} + 4 T^{3} + 18 T^{2} - 8 T + 4$$
$83$ $$(T^{2} + 6 T + 3)^{2}$$
$89$ $$(T^{2} - 9 T + 81)^{2}$$
$97$ $$T^{4} - 2 T^{3} + 27 T^{2} + 46 T + 529$$