# Properties

 Label 414.2.i.e 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.

 $$f(q)$$ $$=$$ $$q - \zeta_{22}^{4} q^{2} + \zeta_{22}^{8} q^{4} + ( - \zeta_{22}^{9} + \zeta_{22}^{6} + \zeta_{22}^{4} - 2 \zeta_{22}^{3} + \zeta_{22}^{2} - 2 \zeta_{22} + 1) q^{5} + (\zeta_{22}^{9} + \zeta_{22}^{7} + \zeta_{22}^{6} - 2 \zeta_{22}^{5} + \zeta_{22}^{4} + \zeta_{22}^{3} + \zeta_{22}) q^{7} + \zeta_{22} q^{8} +O(q^{10})$$ q - z^4 * q^2 + z^8 * q^4 + (-z^9 + z^6 + z^4 - 2*z^3 + z^2 - 2*z + 1) * q^5 + (z^9 + z^7 + z^6 - 2*z^5 + z^4 + z^3 + z) * q^7 + z * q^8 $$q - \zeta_{22}^{4} q^{2} + \zeta_{22}^{8} q^{4} + ( - \zeta_{22}^{9} + \zeta_{22}^{6} + \zeta_{22}^{4} - 2 \zeta_{22}^{3} + \zeta_{22}^{2} - 2 \zeta_{22} + 1) q^{5} + (\zeta_{22}^{9} + \zeta_{22}^{7} + \zeta_{22}^{6} - 2 \zeta_{22}^{5} + \zeta_{22}^{4} + \zeta_{22}^{3} + \zeta_{22}) q^{7} + \zeta_{22} q^{8} + ( - \zeta_{22}^{9} + \zeta_{22}^{7} + \zeta_{22}^{5} - \zeta_{22}^{3} - \zeta_{22} + 1) q^{10} + (2 \zeta_{22}^{7} + \zeta_{22}^{3} + 2 \zeta_{22}^{2} - 2 \zeta_{22} - 1) q^{11} + ( - 2 \zeta_{22}^{8} + \zeta_{22}^{7} + 2 \zeta_{22}^{6} + \zeta_{22}^{5} - 2 \zeta_{22}^{4} - \zeta_{22} + 1) q^{13} + (\zeta_{22}^{9} - 2 \zeta_{22}^{7} + \zeta_{22}^{6} - 2 \zeta_{22}^{5} + \zeta_{22}^{4} - \zeta_{22}^{3} + 2 \zeta_{22}^{2} - \zeta_{22} + 2) q^{14} - \zeta_{22}^{5} q^{16} + (3 \zeta_{22}^{6} + \zeta_{22}^{4} - \zeta_{22}^{3} + \zeta_{22}^{2} + 3) q^{17} + (3 \zeta_{22}^{8} + \zeta_{22}^{5} - \zeta_{22}^{3} + \zeta_{22}^{2} - 1) q^{19} + ( - \zeta_{22}^{9} + \zeta_{22}^{7} + \zeta_{22}^{5} - \zeta_{22}^{4} - \zeta_{22}^{2} + 1) q^{20} + ( - \zeta_{22}^{7} - 2 \zeta_{22}^{6} + 2 \zeta_{22}^{5} + \zeta_{22}^{4} + 2) q^{22} + ( - \zeta_{22}^{8} + \zeta_{22}^{7} - \zeta_{22}^{6} - \zeta_{22}^{4} + 5 \zeta_{22}^{3} + \zeta_{22}) q^{23} + ( - 3 \zeta_{22}^{8} - 3 \zeta_{22}^{7} + 2 \zeta_{22}^{6} - 2 \zeta_{22}^{5} + 2 \zeta_{22}^{4} - 2 \zeta_{22}^{3} + \cdots - 3) q^{25} + \cdots + (3 \zeta_{22}^{9} - 3 \zeta_{22}^{8} + 2 \zeta_{22}^{5} - 5 \zeta_{22}^{4} + 6 \zeta_{22}^{3} - 5 \zeta_{22}^{2} + 2 \zeta_{22}) q^{98} +O(q^{100})$$ q - z^4 * q^2 + z^8 * q^4 + (-z^9 + z^6 + z^4 - 2*z^3 + z^2 - 2*z + 1) * q^5 + (z^9 + z^7 + z^6 - 2*z^5 + z^4 + z^3 + z) * q^7 + z * q^8 + (-z^9 + z^7 + z^5 - z^3 - z + 1) * q^10 + (2*z^7 + z^3 + 2*z^2 - 2*z - 1) * q^11 + (-2*z^8 + z^7 + 2*z^6 + z^5 - 2*z^4 - z + 1) * q^13 + (z^9 - 2*z^7 + z^6 - 2*z^5 + z^4 - z^3 + 2*z^2 - z + 2) * q^14 - z^5 * q^16 + (3*z^6 + z^4 - z^3 + z^2 + 3) * q^17 + (3*z^8 + z^5 - z^3 + z^2 - 1) * q^19 + (-z^9 + z^7 + z^5 - z^4 - z^2 + 1) * q^20 + (-z^7 - 2*z^6 + 2*z^5 + z^4 + 2) * q^22 + (-z^8 + z^7 - z^6 - z^4 + 5*z^3 + z) * q^23 + (-3*z^8 - 3*z^7 + 2*z^6 - 2*z^5 + 2*z^4 - 2*z^3 + 2*z^2 - 3*z - 3) * q^25 + (-3*z^9 + 4*z^8 - 2*z^7 + 2*z^6 - z^5 + z^4 - 2*z^3 + 2*z^2 - 4*z + 3) * q^26 + (z^9 - z^6 - z^4 - z^3 + 2*z^2 - z - 1) * q^28 + (-3*z^9 + 3*z^8 - z^5 + 2*z^4 - z^3 + 2*z^2 - z) * q^29 + (-3*z^9 + z^8 - 2*z^7 + 2*z^6 - z^5 + 3*z^4 + 2*z^2 + 2*z + 2) * q^31 + z^9 * q^32 + (-3*z^9 + 2*z^8 - 2*z^7 + 2*z^6 - 3*z^5 - 3*z^3 + 3*z^2 - 3*z + 3) * q^34 + (-4*z^9 + 7*z^8 - 5*z^7 + 7*z^6 - 4*z^5 - 3*z^3 + 3) * q^35 + (-2*z^9 + z^7 + 2*z^6 - z^4 + z^3 - 2*z - 1) * q^37 + (-z^9 + z^7 - z^6 + z^4 + 3*z) * q^38 + (-z^9 + z^8 + z^6 - z^4 - z^2 + 1) * q^40 + (4*z^9 - 3*z^8 + 3*z^7 - 4*z^6 - z^4 + 3*z^3 + z^2 + 3*z - 1) * q^41 + (4*z^9 - z^8 + 2*z^7 - 7*z^6 + 4*z^5 - 4*z^4 + 7*z^3 - 2*z^2 + z - 4) * q^43 + (-3*z^8 + 2*z^7 - 2*z^6 + 2*z^5 - 4*z^4 + 2*z^3 - 2*z^2 + 2*z - 3) * q^44 + (z^9 - 4*z^7 - z^6 - z^4 + z^3 - z^2) * q^46 + (5*z^9 - 3*z^8 - z^6 + z^5 + 3*z^3 - 5*z^2 - 1) * q^47 + (z^9 - 4*z^8 + 6*z^7 - 6*z^6 + 3*z^5 - 3*z^4 + 6*z^3 - 6*z^2 + 4*z - 1) * q^49 + (z^5 + 5*z^4 - 2*z^3 + 2*z^2 - 5*z - 1) * q^50 + (-z^9 + z^8 + 2*z^5 - z^4 - 2*z^3 - z^2 + 2*z) * q^52 + (4*z^9 - 2*z^8 + 4*z^7 - 6*z^6 + 3*z^5 - 6*z^4 + 4*z^3 - 2*z^2 + 4*z) * q^53 + (2*z^9 - z^7 + z^6 - 5*z^5 + 5*z^4 - 5*z^3 + 5*z^2 - z + 1) * q^55 + (z^9 + 2*z^7 - 3*z^6 + 2*z^5 + z^3 + z - 1) * q^56 + (z^9 - 2*z^8 + z^7 - 2*z^6 + z^5 - 3*z^2 + 3*z) * q^58 + (-4*z^9 - z^8 - 2*z^7 - 2*z^5 - z^4 - 4*z^3 + 3*z - 3) * q^59 + (6*z^8 + z^7 - z^6 - 6*z^5 + z^2 - z + 1) * q^61 + (-z^9 - z^8 - 2*z^7 - 4*z^5 - 2*z^3 - z^2 - z) * q^62 + z^2 * q^64 + (-2*z^9 - 3*z^8 - 2*z^7 + 2*z^5 + 6*z^4 - 3*z^3 + 3*z^2 - 6*z - 2) * q^65 + (-z^8 + 4*z^7 - 4*z^6 + 5*z^5 + 3*z^4 + 5*z^3 - 4*z^2 + 4*z - 1) * q^67 + (z^9 + 2*z^8 + z^7 - z^6 + z^5 - z^4 - 2*z^3 - z^2) * q^68 + (-3*z^9 + 7*z^8 - 4*z^7 + 7*z^6 - 7*z^5 + 4*z^4 - 7*z^3 + 3*z^2 + 2) * q^70 + (-5*z^8 - z^7 - 3*z^6 - z^5 + z^4 - z^3 - 3*z^2 - z - 5) * q^71 + (-2*z^9 - 4*z^8 - 2*z^7 + 7*z^5 - 4*z^4 + 3*z^3 - 3*z^2 + 4*z - 7) * q^73 + (-2*z^9 + 3*z^8 - 3*z^7 + 2*z^6 + 3*z^4 - 2*z^3 - 2*z + 3) * q^74 + (z^9 - 2*z^8 + z^7 - z^6 - 2*z^5 - z^4 + z^3 - 2*z^2 + z) * q^76 + (3*z^9 - z^8 - 5*z^7 + 5*z^6 + z^5 - 3*z^4 - 5*z^2 + 3*z - 5) * q^77 + (-z^9 + 4*z^8 + z^7 + 9*z^6 + z^5 + 4*z^4 - z^3 - z + 1) * q^79 + (-z^9 + 2*z^8 - z^7 + 2*z^6 - z^5 - z^3 + 1) * q^80 + (4*z^9 - 3*z^8 + z^7 - 5*z^6 + z^5 - 3*z^4 + 4*z^3 + z - 1) * q^82 + (z^9 - 2*z^7 + z^6 - z + 2) * q^83 + (-6*z^9 + z^8 - 5*z^7 + 6*z^6 - z^5 + 6*z^4 - 5*z^3 + z^2 - 6*z) * q^85 + (3*z^9 - 3*z^8 - 5*z^6 + 6*z^5 - 3*z^4 + 7*z^3 - 3*z^2 + 6*z - 5) * q^86 + (2*z^8 + z^4 + 2*z^3 - 2*z^2 - z) * q^88 + (5*z^9 - z^8 - 5*z^7 + 2*z^6 - z^5 + z^4 - 2*z^3 + 5*z^2 + z - 5) * q^89 + (4*z^9 + z^8 - 2*z^7 + 8*z^6 - 8*z^5 + 2*z^4 - z^3 - 4*z^2 + 8) * q^91 + (z^9 + z^5 - z^4 + z^3 + z - 5) * q^92 + (-z^8 - 2*z^7 + 4*z^6 + z^5 + z^3 + 4*z^2 - 2*z - 1) * q^94 + (-2*z^9 - z^8 + 3*z^7 + z^5 - z^4 - 3*z^2 + z + 2) * q^95 + (3*z^9 - 3*z^6 - z^4 - 7*z^3 + 4*z^2 - 7*z - 1) * q^97 + (3*z^9 - 3*z^8 + 2*z^5 - 5*z^4 + 6*z^3 - 5*z^2 + 2*z) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q + q^{2} - q^{4} + 2 q^{5} + q^{8}+O(q^{10})$$ 10 * q + q^2 - q^4 + 2 * q^5 + q^8 $$10 q + q^{2} - q^{4} + 2 q^{5} + q^{8} + 9 q^{10} - 11 q^{11} + 13 q^{13} + 11 q^{14} - q^{16} + 24 q^{17} - 14 q^{19} + 13 q^{20} + 22 q^{22} + 10 q^{23} - 43 q^{25} + 9 q^{26} - 11 q^{28} - 13 q^{29} + 8 q^{31} + q^{32} + 9 q^{34} - 13 q^{37} + 3 q^{38} + 9 q^{40} + 10 q^{41} - 8 q^{43} - 11 q^{44} + q^{46} + 8 q^{47} + 29 q^{49} - 23 q^{50} + 2 q^{52} + 35 q^{53} - 11 q^{55} + 13 q^{58} - 37 q^{59} - 2 q^{61} - 8 q^{62} - q^{64} - 37 q^{65} + 14 q^{67} + 2 q^{68} - 22 q^{70} - 44 q^{71} - 49 q^{73} + 13 q^{74} + 8 q^{76} - 44 q^{77} - 8 q^{79} + 2 q^{80} + 12 q^{82} + 17 q^{83} - 37 q^{85} - 14 q^{86} - 59 q^{89} + 66 q^{91} - 45 q^{92} - 19 q^{94} + 28 q^{95} - 21 q^{97} + 26 q^{98}+O(q^{100})$$ 10 * q + q^2 - q^4 + 2 * q^5 + q^8 + 9 * q^10 - 11 * q^11 + 13 * q^13 + 11 * q^14 - q^16 + 24 * q^17 - 14 * q^19 + 13 * q^20 + 22 * q^22 + 10 * q^23 - 43 * q^25 + 9 * q^26 - 11 * q^28 - 13 * q^29 + 8 * q^31 + q^32 + 9 * q^34 - 13 * q^37 + 3 * q^38 + 9 * q^40 + 10 * q^41 - 8 * q^43 - 11 * q^44 + q^46 + 8 * q^47 + 29 * q^49 - 23 * q^50 + 2 * q^52 + 35 * q^53 - 11 * q^55 + 13 * q^58 - 37 * q^59 - 2 * q^61 - 8 * q^62 - q^64 - 37 * q^65 + 14 * q^67 + 2 * q^68 - 22 * q^70 - 44 * q^71 - 49 * q^73 + 13 * q^74 + 8 * q^76 - 44 * q^77 - 8 * q^79 + 2 * q^80 + 12 * q^82 + 17 * q^83 - 37 * q^85 - 14 * q^86 - 59 * q^89 + 66 * q^91 - 45 * q^92 - 19 * q^94 + 28 * q^95 - 21 * q^97 + 26 * 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 0.544078 + 3.78415i 0 1.20217 2.63238i 0.654861 0.755750i 0 1.58816 + 3.47758i
73.1 −0.415415 0.909632i 0 −0.654861 + 0.755750i −0.273100 0.175511i 0 0.346156 + 2.40757i 0.959493 + 0.281733i 0 −0.0462003 + 0.321330i
127.1 −0.841254 + 0.540641i 0 0.415415 0.909632i −0.186393 + 0.0547299i 0 −1.27819 1.47511i 0.142315 + 0.989821i 0 0.127214 0.146813i
163.1 −0.841254 0.540641i 0 0.415415 + 0.909632i −0.186393 0.0547299i 0 −1.27819 + 1.47511i 0.142315 0.989821i 0 0.127214 + 0.146813i
271.1 0.959493 + 0.281733i 0 0.841254 + 0.540641i 0.544078 3.78415i 0 1.20217 + 2.63238i 0.654861 + 0.755750i 0 1.58816 3.47758i
289.1 0.654861 + 0.755750i 0 −0.142315 + 0.989821i 1.61435 3.53494i 0 −3.99283 1.17240i −0.841254 + 0.540641i 0 3.72871 1.09485i
307.1 0.142315 0.989821i 0 −0.959493 0.281733i −0.698939 0.806618i 0 3.72270 + 2.39243i −0.415415 + 0.909632i 0 −0.897877 + 0.577031i
325.1 0.142315 + 0.989821i 0 −0.959493 + 0.281733i −0.698939 + 0.806618i 0 3.72270 2.39243i −0.415415 0.909632i 0 −0.897877 0.577031i
361.1 0.654861 0.755750i 0 −0.142315 0.989821i 1.61435 + 3.53494i 0 −3.99283 + 1.17240i −0.841254 0.540641i 0 3.72871 + 1.09485i
397.1 −0.415415 + 0.909632i 0 −0.654861 0.755750i −0.273100 + 0.175511i 0 0.346156 2.40757i 0.959493 0.281733i 0 −0.0462003 0.321330i
 $$n$$: 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.e 10
3.b odd 2 1 138.2.e.b 10
23.c even 11 1 inner 414.2.i.e 10
23.c even 11 1 9522.2.a.bs 5
23.d odd 22 1 9522.2.a.br 5
69.g even 22 1 3174.2.a.bb 5
69.h odd 22 1 138.2.e.b 10
69.h odd 22 1 3174.2.a.ba 5

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{10} - 2T_{5}^{9} + 26T_{5}^{8} + 3T_{5}^{7} + 159T_{5}^{6} + 386T_{5}^{5} + 526T_{5}^{4} + 323T_{5}^{3} + 102T_{5}^{2} + 16T_{5} + 1$$ acting on $$S_{2}^{\mathrm{new}}(414, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10} - T^{9} + T^{8} - T^{7} + T^{6} - T^{5} + \cdots + 1$$
$3$ $$T^{10}$$
$5$ $$T^{10} - 2 T^{9} + 26 T^{8} + 3 T^{7} + \cdots + 1$$
$7$ $$T^{10} - 11 T^{8} + 55 T^{7} + \cdots + 64009$$
$11$ $$T^{10} + 11 T^{9} + 55 T^{8} + \cdots + 64009$$
$13$ $$T^{10} - 13 T^{9} + 81 T^{8} + \cdots + 11881$$
$17$ $$T^{10} - 24 T^{9} + 279 T^{8} + \cdots + 279841$$
$19$ $$T^{10} + 14 T^{9} + 64 T^{8} + \cdots + 4489$$
$23$ $$T^{10} - 10 T^{9} + 34 T^{8} + \cdots + 6436343$$
$29$ $$T^{10} + 13 T^{9} + 70 T^{8} + \cdots + 529$$
$31$ $$T^{10} - 8 T^{9} - 46 T^{8} + 137 T^{7} + \cdots + 529$$
$37$ $$T^{10} + 13 T^{9} + 70 T^{8} + \cdots + 139129$$
$41$ $$T^{10} - 10 T^{9} + 177 T^{8} + \cdots + 434281$$
$43$ $$T^{10} + 8 T^{9} + 31 T^{8} + \cdots + 50794129$$
$47$ $$(T^{5} - 4 T^{4} - 97 T^{3} + 82 T^{2} + \cdots + 4817)^{2}$$
$53$ $$T^{10} - 35 T^{9} + 587 T^{8} + \cdots + 279841$$
$59$ $$T^{10} + 37 T^{9} + \cdots + 127893481$$
$61$ $$T^{10} + 2 T^{9} + 147 T^{8} + \cdots + 11485321$$
$67$ $$T^{10} - 14 T^{9} + \cdots + 1051640041$$
$71$ $$T^{10} + 44 T^{9} + 968 T^{8} + \cdots + 25938649$$
$73$ $$T^{10} + 49 T^{9} + \cdots + 310499641$$
$79$ $$T^{10} + 8 T^{9} + 306 T^{8} + \cdots + 23203489$$
$83$ $$T^{10} - 17 T^{9} + 135 T^{8} - 590 T^{7} + \cdots + 1$$
$89$ $$T^{10} + 59 T^{9} + \cdots + 3044501329$$
$97$ $$T^{10} + 21 T^{9} + 397 T^{8} + \cdots + 9078169$$