# Properties

 Label 418.2.f.b Level $418$ Weight $2$ Character orbit 418.f 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.f (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.33774680449$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ x^4 - x^3 + x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{2} - \zeta_{10}^{3} q^{4} + (3 \zeta_{10}^{2} - \zeta_{10} + 3) q^{5} + ( - \zeta_{10} + 1) q^{7} + \zeta_{10}^{2} q^{8} + ( - 3 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 3 \zeta_{10} + 3) q^{9} +O(q^{10})$$ q + (z^3 - z^2 + z - 1) * q^2 - z^3 * q^4 + (3*z^2 - z + 3) * q^5 + (-z + 1) * q^7 + z^2 * q^8 + (-3*z^3 + 3*z^2 - 3*z + 3) * q^9 $$q + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{2} - \zeta_{10}^{3} q^{4} + (3 \zeta_{10}^{2} - \zeta_{10} + 3) q^{5} + ( - \zeta_{10} + 1) q^{7} + \zeta_{10}^{2} q^{8} + ( - 3 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 3 \zeta_{10} + 3) q^{9} + (3 \zeta_{10}^{3} - 3 \zeta_{10}^{2} - 2) q^{10} + ( - 4 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{11} + (4 \zeta_{10}^{2} - 4 \zeta_{10}) q^{13} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10}) q^{14} - \zeta_{10} q^{16} + ( - 5 \zeta_{10}^{2} + 4 \zeta_{10} - 5) q^{17} + 3 \zeta_{10}^{3} q^{18} + \zeta_{10}^{2} q^{19} + ( - 2 \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} + 2) q^{20} + (2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - \zeta_{10}) q^{22} + (3 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 3) q^{23} + (3 \zeta_{10}^{3} + 5 \zeta_{10}^{2} + 3 \zeta_{10}) q^{25} + ( - 4 \zeta_{10} + 4) q^{26} + ( - \zeta_{10}^{2} + \zeta_{10} - 1) q^{28} + ( - 2 \zeta_{10} + 2) q^{29} + ( - 2 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10} + 2) q^{31} + q^{32} + ( - 5 \zeta_{10}^{3} + 5 \zeta_{10}^{2} + 1) q^{34} + ( - 3 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10} + 3) q^{35} - 3 \zeta_{10}^{2} q^{36} + (8 \zeta_{10}^{3} + 2 \zeta_{10} - 2) q^{37} - \zeta_{10} q^{38} + (2 \zeta_{10}^{3} + 3 \zeta_{10} - 3) q^{40} + (4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} + 4 \zeta_{10}) q^{41} + (\zeta_{10}^{3} - \zeta_{10}^{2} - 1) q^{43} + ( - 2 \zeta_{10}^{2} - 2 \zeta_{10} + 1) q^{44} + ( - 9 \zeta_{10}^{3} + 9 \zeta_{10}^{2} + 6) q^{45} + (3 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 6 \zeta_{10} - 3) q^{46} + ( - \zeta_{10}^{3} - 10 \zeta_{10}^{2} - \zeta_{10}) q^{47} + (\zeta_{10}^{2} + 5 \zeta_{10} + 1) q^{49} + ( - 3 \zeta_{10}^{2} - 5 \zeta_{10} - 3) q^{50} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10}) q^{52} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{53} + ( - 8 \zeta_{10}^{3} + 4 \zeta_{10}^{2} + 5 \zeta_{10} + 5) q^{55} + ( - \zeta_{10}^{3} + \zeta_{10}^{2}) q^{56} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10}) q^{58} + 10 \zeta_{10}^{3} q^{59} + ( - 5 \zeta_{10}^{2} - 2 \zeta_{10} - 5) q^{61} + (2 \zeta_{10}^{3} + 6 \zeta_{10} - 6) q^{62} + ( - 3 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 3 \zeta_{10}) q^{63} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{64} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 12) q^{65} - 8 q^{67} + (\zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10} - 1) q^{68} + (3 \zeta_{10}^{3} - \zeta_{10} + 1) q^{70} + ( - 4 \zeta_{10}^{2} - 2 \zeta_{10} - 4) q^{71} + 3 \zeta_{10} q^{72} + (6 \zeta_{10}^{3} + 4 \zeta_{10} - 4) q^{73} + ( - 2 \zeta_{10}^{3} - 6 \zeta_{10}^{2} - 2 \zeta_{10}) q^{74} + q^{76} + ( - 3 \zeta_{10}^{3} + \zeta_{10}^{2} - 2) q^{77} + ( - 12 \zeta_{10}^{3} + 12 \zeta_{10}^{2} - 12 \zeta_{10} + 12) q^{79} + ( - 3 \zeta_{10}^{3} + \zeta_{10}^{2} - 3 \zeta_{10}) q^{80} - 9 \zeta_{10}^{3} q^{81} + ( - 4 \zeta_{10}^{2} - 4 \zeta_{10} - 4) q^{82} + ( - 3 \zeta_{10}^{2} + \zeta_{10} - 3) q^{83} + (2 \zeta_{10}^{3} - 19 \zeta_{10}^{2} + 2 \zeta_{10}) q^{85} + ( - \zeta_{10}^{3} + 1) q^{86} + (\zeta_{10}^{3} - \zeta_{10}^{2} + 3 \zeta_{10} + 1) q^{88} + (10 \zeta_{10}^{3} - 10 \zeta_{10}^{2} - 8) q^{89} + (6 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 3 \zeta_{10} - 6) q^{90} + ( - 4 \zeta_{10}^{3} + 8 \zeta_{10}^{2} - 4 \zeta_{10}) q^{91} + ( - 3 \zeta_{10}^{3} + 3 \zeta_{10} - 3) q^{92} + (\zeta_{10}^{2} + 10 \zeta_{10} + 1) q^{94} + (2 \zeta_{10}^{3} + 3 \zeta_{10} - 3) q^{95} + ( - 10 \zeta_{10}^{3} + 14 \zeta_{10}^{2} - 14 \zeta_{10} + 10) q^{97} + (\zeta_{10}^{3} - \zeta_{10}^{2} - 6) q^{98} + ( - 6 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 3 \zeta_{10}) q^{99} +O(q^{100})$$ q + (z^3 - z^2 + z - 1) * q^2 - z^3 * q^4 + (3*z^2 - z + 3) * q^5 + (-z + 1) * q^7 + z^2 * q^8 + (-3*z^3 + 3*z^2 - 3*z + 3) * q^9 + (3*z^3 - 3*z^2 - 2) * q^10 + (-4*z^3 + 3*z^2 - 2*z + 2) * q^11 + (4*z^2 - 4*z) * q^13 + (z^3 - z^2 + z) * q^14 - z * q^16 + (-5*z^2 + 4*z - 5) * q^17 + 3*z^3 * q^18 + z^2 * q^19 + (-2*z^3 - z^2 + z + 2) * q^20 + (2*z^3 + 2*z^2 - z) * q^22 + (3*z^3 - 3*z^2 + 3) * q^23 + (3*z^3 + 5*z^2 + 3*z) * q^25 + (-4*z + 4) * q^26 + (-z^2 + z - 1) * q^28 + (-2*z + 2) * q^29 + (-2*z^3 - 4*z^2 + 4*z + 2) * q^31 + q^32 + (-5*z^3 + 5*z^2 + 1) * q^34 + (-3*z^3 + 4*z^2 - 4*z + 3) * q^35 - 3*z^2 * q^36 + (8*z^3 + 2*z - 2) * q^37 - z * q^38 + (2*z^3 + 3*z - 3) * q^40 + (4*z^3 + 4*z^2 + 4*z) * q^41 + (z^3 - z^2 - 1) * q^43 + (-2*z^2 - 2*z + 1) * q^44 + (-9*z^3 + 9*z^2 + 6) * q^45 + (3*z^3 - 6*z^2 + 6*z - 3) * q^46 + (-z^3 - 10*z^2 - z) * q^47 + (z^2 + 5*z + 1) * q^49 + (-3*z^2 - 5*z - 3) * q^50 + (4*z^3 - 4*z^2 + 4*z) * q^52 + (2*z^3 - 2*z^2 + 2*z - 2) * q^53 + (-8*z^3 + 4*z^2 + 5*z + 5) * q^55 + (-z^3 + z^2) * q^56 + (2*z^3 - 2*z^2 + 2*z) * q^58 + 10*z^3 * q^59 + (-5*z^2 - 2*z - 5) * q^61 + (2*z^3 + 6*z - 6) * q^62 + (-3*z^3 + 3*z^2 - 3*z) * q^63 + (z^3 - z^2 + z - 1) * q^64 + (-4*z^3 + 4*z^2 - 12) * q^65 - 8 * q^67 + (z^3 + 4*z^2 - 4*z - 1) * q^68 + (3*z^3 - z + 1) * q^70 + (-4*z^2 - 2*z - 4) * q^71 + 3*z * q^72 + (6*z^3 + 4*z - 4) * q^73 + (-2*z^3 - 6*z^2 - 2*z) * q^74 + q^76 + (-3*z^3 + z^2 - 2) * q^77 + (-12*z^3 + 12*z^2 - 12*z + 12) * q^79 + (-3*z^3 + z^2 - 3*z) * q^80 - 9*z^3 * q^81 + (-4*z^2 - 4*z - 4) * q^82 + (-3*z^2 + z - 3) * q^83 + (2*z^3 - 19*z^2 + 2*z) * q^85 + (-z^3 + 1) * q^86 + (z^3 - z^2 + 3*z + 1) * q^88 + (10*z^3 - 10*z^2 - 8) * q^89 + (6*z^3 + 3*z^2 - 3*z - 6) * q^90 + (-4*z^3 + 8*z^2 - 4*z) * q^91 + (-3*z^3 + 3*z - 3) * q^92 + (z^2 + 10*z + 1) * q^94 + (2*z^3 + 3*z - 3) * q^95 + (-10*z^3 + 14*z^2 - 14*z + 10) * q^97 + (z^3 - z^2 - 6) * q^98 + (-6*z^3 - 6*z^2 + 3*z) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - q^{2} - q^{4} + 8 q^{5} + 3 q^{7} - q^{8} + 3 q^{9}+O(q^{10})$$ 4 * q - q^2 - q^4 + 8 * q^5 + 3 * q^7 - q^8 + 3 * q^9 $$4 q - q^{2} - q^{4} + 8 q^{5} + 3 q^{7} - q^{8} + 3 q^{9} - 2 q^{10} - q^{11} - 8 q^{13} + 3 q^{14} - q^{16} - 11 q^{17} + 3 q^{18} - q^{19} + 8 q^{20} - q^{22} + 18 q^{23} + q^{25} + 12 q^{26} - 2 q^{28} + 6 q^{29} + 14 q^{31} + 4 q^{32} - 6 q^{34} + q^{35} + 3 q^{36} + 2 q^{37} - q^{38} - 7 q^{40} + 4 q^{41} - 2 q^{43} + 4 q^{44} + 6 q^{45} + 3 q^{46} + 8 q^{47} + 8 q^{49} - 14 q^{50} + 12 q^{52} - 2 q^{53} + 13 q^{55} - 2 q^{56} + 6 q^{58} + 10 q^{59} - 17 q^{61} - 16 q^{62} - 9 q^{63} - q^{64} - 56 q^{65} - 32 q^{67} - 11 q^{68} + 6 q^{70} - 14 q^{71} + 3 q^{72} - 6 q^{73} + 2 q^{74} + 4 q^{76} - 12 q^{77} + 12 q^{79} - 7 q^{80} - 9 q^{81} - 16 q^{82} - 8 q^{83} + 23 q^{85} + 3 q^{86} + 9 q^{88} - 12 q^{89} - 24 q^{90} - 16 q^{91} - 12 q^{92} + 13 q^{94} - 7 q^{95} + 2 q^{97} - 22 q^{98} + 3 q^{99}+O(q^{100})$$ 4 * q - q^2 - q^4 + 8 * q^5 + 3 * q^7 - q^8 + 3 * q^9 - 2 * q^10 - q^11 - 8 * q^13 + 3 * q^14 - q^16 - 11 * q^17 + 3 * q^18 - q^19 + 8 * q^20 - q^22 + 18 * q^23 + q^25 + 12 * q^26 - 2 * q^28 + 6 * q^29 + 14 * q^31 + 4 * q^32 - 6 * q^34 + q^35 + 3 * q^36 + 2 * q^37 - q^38 - 7 * q^40 + 4 * q^41 - 2 * q^43 + 4 * q^44 + 6 * q^45 + 3 * q^46 + 8 * q^47 + 8 * q^49 - 14 * q^50 + 12 * q^52 - 2 * q^53 + 13 * q^55 - 2 * q^56 + 6 * q^58 + 10 * q^59 - 17 * q^61 - 16 * q^62 - 9 * q^63 - q^64 - 56 * q^65 - 32 * q^67 - 11 * q^68 + 6 * q^70 - 14 * q^71 + 3 * q^72 - 6 * q^73 + 2 * q^74 + 4 * q^76 - 12 * q^77 + 12 * q^79 - 7 * q^80 - 9 * q^81 - 16 * q^82 - 8 * q^83 + 23 * q^85 + 3 * q^86 + 9 * q^88 - 12 * q^89 - 24 * q^90 - 16 * q^91 - 12 * q^92 + 13 * q^94 - 7 * q^95 + 2 * q^97 - 22 * q^98 + 3 * q^99

## Character values

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

 $$n$$ $$287$$ $$343$$ $$\chi(n)$$ $$1$$ $$-\zeta_{10}^{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
115.1
 −0.309017 + 0.951057i 0.809017 − 0.587785i −0.309017 − 0.951057i 0.809017 + 0.587785i
0.309017 + 0.951057i 0 −0.809017 + 0.587785i 0.881966 2.71441i 0 1.30902 0.951057i −0.809017 0.587785i −0.927051 2.85317i 2.85410
191.1 −0.809017 0.587785i 0 0.309017 + 0.951057i 3.11803 2.26538i 0 0.190983 + 0.587785i 0.309017 0.951057i 2.42705 + 1.76336i −3.85410
229.1 0.309017 0.951057i 0 −0.809017 0.587785i 0.881966 + 2.71441i 0 1.30902 + 0.951057i −0.809017 + 0.587785i −0.927051 + 2.85317i 2.85410
267.1 −0.809017 + 0.587785i 0 0.309017 0.951057i 3.11803 + 2.26538i 0 0.190983 0.587785i 0.309017 + 0.951057i 2.42705 1.76336i −3.85410
 $$n$$: 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
11.c even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 418.2.f.b 4
11.c even 5 1 inner 418.2.f.b 4
11.c even 5 1 4598.2.a.bh 2
11.d odd 10 1 4598.2.a.z 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.f.b 4 1.a even 1 1 trivial
418.2.f.b 4 11.c even 5 1 inner
4598.2.a.z 2 11.d odd 10 1
4598.2.a.bh 2 11.c even 5 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}$$ acting on $$S_{2}^{\mathrm{new}}(418, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + T^{3} + T^{2} + T + 1$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 8 T^{3} + 34 T^{2} - 77 T + 121$$
$7$ $$T^{4} - 3 T^{3} + 4 T^{2} - 2 T + 1$$
$11$ $$T^{4} + T^{3} - 9 T^{2} + 11 T + 121$$
$13$ $$T^{4} + 8 T^{3} + 64 T^{2} + 192 T + 256$$
$17$ $$T^{4} + 11 T^{3} + 96 T^{2} + \cdots + 841$$
$19$ $$T^{4} + T^{3} + T^{2} + T + 1$$
$23$ $$(T^{2} - 9 T + 9)^{2}$$
$29$ $$T^{4} - 6 T^{3} + 16 T^{2} - 16 T + 16$$
$31$ $$T^{4} - 14 T^{3} + 136 T^{2} + \cdots + 1936$$
$37$ $$T^{4} - 2 T^{3} + 64 T^{2} + \cdots + 1936$$
$41$ $$T^{4} - 4 T^{3} + 96 T^{2} + 256 T + 256$$
$43$ $$(T^{2} + T - 1)^{2}$$
$47$ $$T^{4} - 8 T^{3} + 114 T^{2} + \cdots + 11881$$
$53$ $$T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16$$
$59$ $$T^{4} - 10 T^{3} + 100 T^{2} + \cdots + 10000$$
$61$ $$T^{4} + 17 T^{3} + 114 T^{2} + \cdots + 121$$
$67$ $$(T + 8)^{4}$$
$71$ $$T^{4} + 14 T^{3} + 76 T^{2} + 24 T + 16$$
$73$ $$T^{4} + 6 T^{3} + 76 T^{2} + 56 T + 16$$
$79$ $$T^{4} - 12 T^{3} + 144 T^{2} + \cdots + 20736$$
$83$ $$T^{4} + 8 T^{3} + 34 T^{2} + 77 T + 121$$
$89$ $$(T^{2} + 6 T - 116)^{2}$$
$97$ $$T^{4} - 2 T^{3} + 204 T^{2} + \cdots + 15376$$