# Properties

 Label 414.2.i.a Level $414$ Weight $2$ Character orbit 414.i Analytic conductor $3.306$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$414 = 2 \cdot 3^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 414.i (of order $$11$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.30580664368$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\Q(\zeta_{22})$$ Defining polynomial: $$x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$$ x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 138) Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{22}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{22}^{4} q^{2} + \zeta_{22}^{8} q^{4} + (\zeta_{22}^{9} - 2 \zeta_{22}^{8} + 2 \zeta_{22}^{7} - \zeta_{22}^{6} - \zeta_{22}^{4} - \zeta_{22}^{2} - 1) q^{5} + ( - \zeta_{22}^{9} + \zeta_{22}^{7} - \zeta_{22}^{6} - \zeta_{22}^{4} + \zeta_{22}^{3} - \zeta_{22}) q^{7} - \zeta_{22} q^{8} +O(q^{10})$$ q + z^4 * q^2 + z^8 * q^4 + (z^9 - 2*z^8 + 2*z^7 - z^6 - z^4 - z^2 - 1) * q^5 + (-z^9 + z^7 - z^6 - z^4 + z^3 - z) * q^7 - z * q^8 $$q + \zeta_{22}^{4} q^{2} + \zeta_{22}^{8} q^{4} + (\zeta_{22}^{9} - 2 \zeta_{22}^{8} + 2 \zeta_{22}^{7} - \zeta_{22}^{6} - \zeta_{22}^{4} - \zeta_{22}^{2} - 1) q^{5} + ( - \zeta_{22}^{9} + \zeta_{22}^{7} - \zeta_{22}^{6} - \zeta_{22}^{4} + \zeta_{22}^{3} - \zeta_{22}) q^{7} - \zeta_{22} q^{8} + ( - \zeta_{22}^{9} - \zeta_{22}^{7} - \zeta_{22}^{5} - \zeta_{22}^{3} + \zeta_{22} - 1) q^{10} + ( - 2 \zeta_{22}^{9} - 2 \zeta_{22}^{5} - \zeta_{22}^{3} + 1) q^{11} + ( - 2 \zeta_{22}^{9} + 2 \zeta_{22}^{8} - 3 \zeta_{22}^{7} - 3 \zeta_{22}^{5} + 2 \zeta_{22}^{4} - 2 \zeta_{22}^{3} + \cdots + 3) q^{13} + \cdots + (5 \zeta_{22}^{9} - 5 \zeta_{22}^{8} + 2 \zeta_{22}^{5} + \zeta_{22}^{4} + 2 \zeta_{22}^{3} + \zeta_{22}^{2} + 2 \zeta_{22}) q^{98} +O(q^{100})$$ q + z^4 * q^2 + z^8 * q^4 + (z^9 - 2*z^8 + 2*z^7 - z^6 - z^4 - z^2 - 1) * q^5 + (-z^9 + z^7 - z^6 - z^4 + z^3 - z) * q^7 - z * q^8 + (-z^9 - z^7 - z^5 - z^3 + z - 1) * q^10 + (-2*z^9 - 2*z^5 - z^3 + 1) * q^11 + (-2*z^9 + 2*z^8 - 3*z^7 - 3*z^5 + 2*z^4 - 2*z^3 - 3*z + 3) * q^13 + (-z^9 + z^6 - 2*z^5 + z^4 - z^3 + 2*z^2 - z) * q^14 - z^5 * q^16 + (-4*z^9 + 4*z^8 + z^6 + z^4 + z^3 + z^2 + 1) * q^17 + (2*z^9 - 3*z^8 + 2*z^7 + z^5 + z^3 - z^2 - 1) * q^19 + (-z^9 - z^7 + z^5 - z^4 + z^2 + 1) * q^20 + (-2*z^9 - z^7 + z^4 + 2*z^2) * q^22 + (-z^8 - z^7 + z^6 - 2*z^5 + z^4 - 3*z^3 - z + 4) * q^23 + (z^8 + z^7 + 2*z^6 - 2*z^5 + 2*z^4 - 2*z^3 + 2*z^2 + z + 1) * q^25 + (-3*z^9 + 2*z^8 - 2*z^7 - 3*z^5 + 3*z^4 + 2*z^2 - 2*z + 3) * q^26 + (-z^9 + z^6 - z^4 + z^3 + z - 1) * q^28 + (-z^9 + z^8 - 2*z^6 - z^5 - 5*z^3 - z - 2) * q^29 + (3*z^9 - 5*z^8 + 4*z^7 - 4*z^6 + 5*z^5 - 3*z^4 - 4*z^2 + 4*z - 4) * q^31 - z^9 * q^32 + (z^9 + 2*z^7 + z^5 + z^3 + 3*z^2 - 3*z - 1) * q^34 + (4*z^9 - 3*z^8 + 3*z^7 - 3*z^6 + 4*z^5 - z^3 + 1) * q^35 + (4*z^9 + z^7 + 2*z^6 + z^4 - z^3 - 2*z - 1) * q^37 + (z^9 + z^7 - z^6 - z^4 - 2*z^2 + 3*z - 2) * q^38 + (z^9 - z^8 + z^6 + z^4 + z^2 + 1) * q^40 + (-2*z^9 + 5*z^8 - 5*z^7 + 2*z^6 + 3*z^4 - z^3 + z^2 - z + 3) * q^41 + (-2*z^9 - z^8 + 4*z^7 - z^6 + 2*z^5 - 2*z^4 + z^3 - 4*z^2 + z + 2) * q^43 + (z^8 + 2*z^6 + 2*z^2 + 1) * q^44 + (-z^9 - 2*z^7 - z^6 + 3*z^4 + z^3 - z^2 + 2*z) * q^46 + (-z^9 + z^8 - 2*z^7 + 5*z^6 - 5*z^5 + 2*z^4 - z^3 + z^2 + 1) * q^47 + (-3*z^9 - 2*z^7 + 2*z^6 + 3*z^5 - 3*z^4 - 2*z^3 + 2*z^2 + 3) * q^49 + (3*z^5 - z^4 + 2*z^3 - 2*z^2 + z - 3) * q^50 + (-3*z^9 + 3*z^8 + 2*z^6 - 2*z^5 + 3*z^4 + 3*z^2 - 2*z + 2) * q^52 + (-4*z^9 + 2*z^7 + 2*z^6 - z^5 + 2*z^4 + 2*z^3 - 4*z) * q^53 + (4*z^9 + 5*z^7 - 3*z^6 + 5*z^5 - z^4 + z^3 - 5*z^2 + 3*z - 5) * q^55 + (z^9 - 2*z^8 + 2*z^7 - z^6 + 2*z^5 - 2*z^4 + z^3 + z - 1) * q^56 + (-3*z^9 + 2*z^8 - 7*z^7 + 2*z^6 - 3*z^5 - 2*z^3 + 3*z^2 - 3*z + 2) * q^58 + (6*z^9 - 5*z^8 + 2*z^7 - 4*z^6 + 2*z^5 - 5*z^4 + 6*z^3 + z - 1) * q^59 + (2*z^9 + 2*z^8 - z^7 + z^6 - 2*z^5 - 2*z^4 - z^2 + 7*z - 1) * q^61 + (z^9 + z^8 - 4*z^7 - 4*z^3 + z^2 + z) * q^62 + z^2 * q^64 + (8*z^9 - 5*z^8 + 8*z^7 + 8*z^5 - 6*z^4 + 5*z^3 - 5*z^2 + 6*z - 8) * q^65 + (-z^8 - 6*z^7 + 4*z^6 - z^5 + z^4 - z^3 + 4*z^2 - 6*z - 1) * q^67 + (z^9 + z^7 + 3*z^6 - 3*z^5 - z^4 - z^2 - 2) * q^68 + (z^9 + 3*z^8 - 4*z^7 + 3*z^6 - 3*z^5 + 4*z^4 - 3*z^3 - z^2) * q^70 + (-z^8 - 5*z^7 + 3*z^6 + 3*z^5 + 3*z^4 + 3*z^3 + 3*z^2 - 5*z - 1) * q^71 + (-6*z^9 + 8*z^8 - 6*z^7 - 9*z^5 + 4*z^4 - 5*z^3 + 5*z^2 - 4*z + 9) * q^73 + (2*z^9 - z^8 + z^7 - 2*z^6 - 3*z^4 + 2*z^3 - 6*z^2 + 2*z - 3) * q^74 + (-z^9 - z^7 - z^6 + 2*z^5 - z^4 - z^3 - z) * q^76 + (3*z^9 - 3*z^8 + z^7 - z^6 + 3*z^5 - 3*z^4 - 3*z^2 + 3*z - 3) * q^77 + (z^9 + 6*z^8 - 7*z^7 + z^6 - 7*z^5 + 6*z^4 + z^3 - z + 1) * q^79 + (z^9 + z^7 + z^5 + z^3 - 2*z^2 + 2*z - 1) * q^80 + (2*z^9 + z^8 + z^7 - z^6 + z^5 + z^4 + 2*z^3 - 3*z + 3) * q^82 + (9*z^9 + 4*z^7 - 5*z^6 + 6*z^5 - 4*z^4 + 4*z^3 - 6*z^2 + 5*z - 4) * q^83 + (-3*z^8 - z^7 - 10*z^6 + 9*z^5 - 10*z^4 - z^3 - 3*z^2) * q^85 + (z^9 - z^8 - 3*z^6 + 3*z^4 - z^3 + 3*z^2 - 3) * q^86 + (2*z^9 - 2*z^8 + 2*z^7 + 2*z^5 - z^4 + 2*z^3 - 2*z^2 + z - 2) * q^88 + (-z^9 + z^8 - 3*z^7 - 4*z^6 + z^5 - z^4 + 4*z^3 + 3*z^2 - z + 1) * q^89 + (-2*z^9 - 3*z^8 + 6*z^7 + 2*z^6 - 2*z^5 - 6*z^4 + 3*z^3 + 2*z^2 - 8) * q^91 + (-z^9 + 4*z^8 + z^5 + z^4 - z^3 + 2*z^2 - z + 3) * q^92 + (-3*z^8 + 4*z^7 - 4*z^6 + 5*z^5 - 4*z^4 + 5*z^3 - 4*z^2 + 4*z - 3) * q^94 + (-2*z^9 - z^8 - z^7 + 2*z^6 - 7*z^5 + 7*z^4 - 2*z^3 + z^2 + z + 2) * q^95 + (-z^9 + z^6 + 3*z^4 + 5*z^3 - 4*z^2 + 5*z + 3) * q^97 + (5*z^9 - 5*z^8 + 2*z^5 + z^4 + 2*z^3 + z^2 + 2*z) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q - q^{2} - q^{4} - 2 q^{5} + 2 q^{7} - q^{8}+O(q^{10})$$ 10 * q - q^2 - q^4 - 2 * q^5 + 2 * q^7 - q^8 $$10 q - q^{2} - q^{4} - 2 q^{5} + 2 q^{7} - q^{8} - 13 q^{10} + 5 q^{11} + 13 q^{13} - 9 q^{14} - q^{16} + 9 q^{20} - 6 q^{22} + 32 q^{23} + q^{25} + 13 q^{26} - 9 q^{28} - 27 q^{29} - 8 q^{31} - q^{32} - 11 q^{34} + 26 q^{35} - 11 q^{37} - 11 q^{38} + 9 q^{40} + 10 q^{41} + 34 q^{43} + 5 q^{44} - q^{46} - 8 q^{47} + 25 q^{49} - 21 q^{50} + 2 q^{52} - 9 q^{53} - 23 q^{55} + 2 q^{56} - 5 q^{58} + 21 q^{59} - 4 q^{61} - 8 q^{62} - q^{64} - 29 q^{65} - 32 q^{67} - 22 q^{68} - 18 q^{70} - 22 q^{71} + 43 q^{73} - 11 q^{74} - 10 q^{77} - 16 q^{79} - 2 q^{80} + 32 q^{82} + 3 q^{83} + 33 q^{85} - 32 q^{86} - 6 q^{88} + 11 q^{89} - 70 q^{91} + 21 q^{92} + 3 q^{94} + 39 q^{97} + 14 q^{98}+O(q^{100})$$ 10 * q - q^2 - q^4 - 2 * q^5 + 2 * q^7 - q^8 - 13 * q^10 + 5 * q^11 + 13 * q^13 - 9 * q^14 - q^16 + 9 * q^20 - 6 * q^22 + 32 * q^23 + q^25 + 13 * q^26 - 9 * q^28 - 27 * q^29 - 8 * q^31 - q^32 - 11 * q^34 + 26 * q^35 - 11 * q^37 - 11 * q^38 + 9 * q^40 + 10 * q^41 + 34 * q^43 + 5 * q^44 - q^46 - 8 * q^47 + 25 * q^49 - 21 * q^50 + 2 * q^52 - 9 * q^53 - 23 * q^55 + 2 * q^56 - 5 * q^58 + 21 * q^59 - 4 * q^61 - 8 * q^62 - q^64 - 29 * q^65 - 32 * q^67 - 22 * q^68 - 18 * q^70 - 22 * q^71 + 43 * q^73 - 11 * q^74 - 10 * q^77 - 16 * q^79 - 2 * q^80 + 32 * q^82 + 3 * q^83 + 33 * q^85 - 32 * q^86 - 6 * q^88 + 11 * q^89 - 70 * q^91 + 21 * q^92 + 3 * q^94 + 39 * q^97 + 14 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/414\mathbb{Z}\right)^\times$$.

 $$n$$ $$47$$ $$235$$ $$\chi(n)$$ $$1$$ $$\zeta_{22}^{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
55.1
 0.654861 − 0.755750i 0.959493 + 0.281733i 0.142315 + 0.989821i 0.142315 − 0.989821i 0.654861 + 0.755750i −0.841254 + 0.540641i −0.415415 + 0.909632i −0.415415 − 0.909632i −0.841254 − 0.540641i 0.959493 − 0.281733i
−0.959493 + 0.281733i 0 0.841254 0.540641i 0.0651865 + 0.453382i 0 −0.134858 + 0.295298i −0.654861 + 0.755750i 0 −0.190279 0.416652i
73.1 0.415415 + 0.909632i 0 −0.654861 + 0.755750i −2.47672 1.59169i 0 −0.151894 1.05645i −0.959493 0.281733i 0 0.418986 2.91411i
127.1 0.841254 0.540641i 0 0.415415 0.909632i −1.78074 + 0.522874i 0 −2.54487 2.93694i −0.142315 0.989821i 0 −1.21537 + 1.40261i
163.1 0.841254 + 0.540641i 0 0.415415 + 0.909632i −1.78074 0.522874i 0 −2.54487 + 2.93694i −0.142315 + 0.989821i 0 −1.21537 1.40261i
271.1 −0.959493 0.281733i 0 0.841254 + 0.540641i 0.0651865 0.453382i 0 −0.134858 0.295298i −0.654861 0.755750i 0 −0.190279 + 0.416652i
289.1 −0.654861 0.755750i 0 −0.142315 + 0.989821i 1.37787 3.01713i 0 3.66820 + 1.07708i 0.841254 0.540641i 0 −3.18251 + 0.934468i
307.1 −0.142315 + 0.989821i 0 −0.959493 0.281733i 1.81440 + 2.09393i 0 0.163423 + 0.105026i 0.415415 0.909632i 0 −2.33083 + 1.49793i
325.1 −0.142315 0.989821i 0 −0.959493 + 0.281733i 1.81440 2.09393i 0 0.163423 0.105026i 0.415415 + 0.909632i 0 −2.33083 1.49793i
361.1 −0.654861 + 0.755750i 0 −0.142315 0.989821i 1.37787 + 3.01713i 0 3.66820 1.07708i 0.841254 + 0.540641i 0 −3.18251 0.934468i
397.1 0.415415 0.909632i 0 −0.654861 0.755750i −2.47672 + 1.59169i 0 −0.151894 + 1.05645i −0.959493 + 0.281733i 0 0.418986 + 2.91411i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 397.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 414.2.i.a 10
3.b odd 2 1 138.2.e.d 10
23.c even 11 1 inner 414.2.i.a 10
23.c even 11 1 9522.2.a.bx 5
23.d odd 22 1 9522.2.a.by 5
69.g even 22 1 3174.2.a.w 5
69.h odd 22 1 138.2.e.d 10
69.h odd 22 1 3174.2.a.x 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.2.e.d 10 3.b odd 2 1
138.2.e.d 10 69.h odd 22 1
414.2.i.a 10 1.a even 1 1 trivial
414.2.i.a 10 23.c even 11 1 inner
3174.2.a.w 5 69.g even 22 1
3174.2.a.x 5 69.h odd 22 1
9522.2.a.bx 5 23.c even 11 1
9522.2.a.by 5 23.d odd 22 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{10} + 2 T_{5}^{9} + 4 T_{5}^{8} + 41 T_{5}^{7} + 137 T_{5}^{6} + 76 T_{5}^{5} + 460 T_{5}^{4} + 2185 T_{5}^{3} + 2324 T_{5}^{2} + 138 T_{5} + 529$$ acting on $$S_{2}^{\mathrm{new}}(414, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10} + T^{9} + T^{8} + T^{7} + T^{6} + T^{5} + \cdots + 1$$
$3$ $$T^{10}$$
$5$ $$T^{10} + 2 T^{9} + 4 T^{8} + 41 T^{7} + \cdots + 529$$
$7$ $$T^{10} - 2 T^{9} - 7 T^{8} - 41 T^{7} + \cdots + 1$$
$11$ $$T^{10} - 5 T^{9} + 25 T^{8} - 37 T^{7} + \cdots + 529$$
$13$ $$T^{10} - 13 T^{9} + 125 T^{8} + \cdots + 529$$
$17$ $$T^{10} + 55 T^{8} + 121 T^{7} + \cdots + 7745089$$
$19$ $$T^{10} - 33 T^{7} - 165 T^{6} + \cdots + 64009$$
$23$ $$T^{10} - 32 T^{9} + 496 T^{8} + \cdots + 6436343$$
$29$ $$T^{10} + 27 T^{9} + 322 T^{8} + \cdots + 7921$$
$31$ $$T^{10} + 8 T^{9} + 108 T^{8} + \cdots + 8300161$$
$37$ $$T^{10} + 11 T^{9} + 66 T^{8} + \cdots + 64009$$
$41$ $$T^{10} - 10 T^{9} + 23 T^{8} + \cdots + 1515361$$
$43$ $$T^{10} - 34 T^{9} + 595 T^{8} + \cdots + 109561$$
$47$ $$(T^{5} + 4 T^{4} - 75 T^{3} + 28 T^{2} + \cdots - 857)^{2}$$
$53$ $$T^{10} + 9 T^{9} + 103 T^{8} + \cdots + 8427409$$
$59$ $$T^{10} - 21 T^{9} + 199 T^{8} + \cdots + 978121$$
$61$ $$T^{10} + 4 T^{9} + 5 T^{8} + \cdots + 9078169$$
$67$ $$T^{10} + 32 T^{9} + 628 T^{8} + \cdots + 83375161$$
$71$ $$T^{10} + 22 T^{9} + 198 T^{8} + \cdots + 64009$$
$73$ $$T^{10} - 43 T^{9} + 1013 T^{8} + \cdots + 21132409$$
$79$ $$T^{10} + 16 T^{9} + 14 T^{8} + \cdots + 72471169$$
$83$ $$T^{10} - 3 T^{9} - 79 T^{8} + \cdots + 16752649$$
$89$ $$T^{10} - 11 T^{9} + \cdots + 517972081$$
$97$ $$T^{10} - 39 T^{9} + 773 T^{8} + \cdots + 5650129$$