# Properties

 Label 414.2.i.d 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.

 $$f(q)$$ $$=$$ $$q - \zeta_{22}^{4} q^{2} + \zeta_{22}^{8} q^{4} + (\zeta_{22}^{9} - \zeta_{22}^{6} - \zeta_{22}^{4} + \zeta_{22}^{2} - 1) q^{5} + (\zeta_{22}^{9} + \zeta_{22}^{7} - \zeta_{22}^{6} + 2 \zeta_{22}^{5} - \zeta_{22}^{4} + \zeta_{22}^{3} + \zeta_{22}) q^{7} + \zeta_{22} q^{8} +O(q^{10})$$ q - z^4 * q^2 + z^8 * q^4 + (z^9 - z^6 - z^4 + z^2 - 1) * q^5 + (z^9 + z^7 - z^6 + 2*z^5 - z^4 + z^3 + z) * q^7 + z * q^8 $$q - \zeta_{22}^{4} q^{2} + \zeta_{22}^{8} q^{4} + (\zeta_{22}^{9} - \zeta_{22}^{6} - \zeta_{22}^{4} + \zeta_{22}^{2} - 1) q^{5} + (\zeta_{22}^{9} + \zeta_{22}^{7} - \zeta_{22}^{6} + 2 \zeta_{22}^{5} - \zeta_{22}^{4} + \zeta_{22}^{3} + \zeta_{22}) q^{7} + \zeta_{22} q^{8} + (\zeta_{22}^{9} + \zeta_{22}^{7} - 2 \zeta_{22}^{6} + \zeta_{22}^{5} + \zeta_{22}^{3} + \zeta_{22} - 1) q^{10} + ( - 2 \zeta_{22}^{8} - 2 \zeta_{22}^{7} - 2 \zeta_{22}^{6} + \zeta_{22}^{3} - 1) q^{11} + ( - 2 \zeta_{22}^{9} - \zeta_{22}^{7} - \zeta_{22}^{5} - 2 \zeta_{22}^{3} - \zeta_{22} + 1) q^{13} + ( - \zeta_{22}^{9} - \zeta_{22}^{6} - \zeta_{22}^{4} + \zeta_{22}^{3} + \zeta_{22}) q^{14} - \zeta_{22}^{5} q^{16} + ( - \zeta_{22}^{6} + 2 \zeta_{22}^{5} - \zeta_{22}^{4} - \zeta_{22}^{3} - \zeta_{22}^{2} + 2 \zeta_{22} - 1) q^{17} + ( - 2 \zeta_{22}^{9} - \zeta_{22}^{8} - 2 \zeta_{22}^{7} - \zeta_{22}^{5} - 3 \zeta_{22}^{3} + 3 \zeta_{22}^{2} + 1) q^{19} + (\zeta_{22}^{9} - 2 \zeta_{22}^{8} + \zeta_{22}^{7} - 2 \zeta_{22}^{6} + \zeta_{22}^{5} - \zeta_{22}^{4} + 2 \zeta_{22}^{3} - \zeta_{22}^{2} + \cdots - 1) q^{20} + \cdots + ( - \zeta_{22}^{9} + \zeta_{22}^{8} + 2 \zeta_{22}^{5} - \zeta_{22}^{4} - 2 \zeta_{22}^{3} - \zeta_{22}^{2} + 2 \zeta_{22}) q^{98} +O(q^{100})$$ q - z^4 * q^2 + z^8 * q^4 + (z^9 - z^6 - z^4 + z^2 - 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 - 2*z^6 + z^5 + z^3 + z - 1) * q^10 + (-2*z^8 - 2*z^7 - 2*z^6 + z^3 - 1) * q^11 + (-2*z^9 - z^7 - z^5 - 2*z^3 - z + 1) * q^13 + (-z^9 - z^6 - z^4 + z^3 + z) * q^14 - z^5 * q^16 + (-z^6 + 2*z^5 - z^4 - z^3 - z^2 + 2*z - 1) * q^17 + (-2*z^9 - z^8 - 2*z^7 - z^5 - 3*z^3 + 3*z^2 + 1) * q^19 + (z^9 - 2*z^8 + z^7 - 2*z^6 + z^5 - z^4 + 2*z^3 - z^2 + 2*z - 1) * q^20 + (2*z^9 - 2*z^8 + z^7 - 2*z^6 + 2*z^5 - z^4 + 2*z^3 - 2*z^2 - 4) * q^22 + (2*z^9 + z^8 + 3*z^7 - 3*z^6 + 2*z^5 - z^4 - z^3 - 2*z^2 + z) * q^23 + (-3*z^8 + z^7 - 2*z^6 + 2*z^5 - 2*z^4 + 2*z^3 - 2*z^2 + z - 3) * q^25 + (z^9 + 2*z^7 + z^5 - z^4 - 2*z^2 - 1) * q^26 + (z^9 - z^6 - z^4 + z^3 - 2*z^2 + z - 1) * q^28 + (-z^9 + z^8 + 2*z^6 + 3*z^5 - 2*z^4 - z^3 - 2*z^2 + 3*z + 2) * q^29 + (z^9 + 3*z^8 - 4*z^7 + 4*z^6 - 3*z^5 - z^4 + 2*z^2 + 2) * q^31 + z^9 * q^32 + (-z^9 + 2*z^7 - z^5 + z^3 - z^2 + z - 1) * q^34 + (-3*z^8 + z^7 - 3*z^6 + z^3 - 1) * q^35 + (2*z^9 - z^7 + z^4 - z^3 + 1) * q^37 + (z^9 + 3*z^7 - 3*z^6 - z^4 - 2*z^2 - z - 2) * q^38 + (z^9 - z^8 - z^6 - z^4 + 2*z^3 - z^2 - 1) * q^40 + (3*z^8 - 3*z^7 - z^4 - 3*z^3 + 3*z^2 - 3*z - 1) * q^41 + (2*z^9 - z^8 - 2*z^7 + 3*z^6 - 2*z^5 + 2*z^4 - 3*z^3 + 2*z^2 + z - 2) * q^43 + (-z^8 + 2*z^5 + 2*z^4 + 2*z^3 - 1) * q^44 + (z^9 - 2*z^8 + 4*z^7 - z^6 + 2*z^5 - 3*z^4 + 3*z^3 - z^2 + 4*z) * q^46 + (-5*z^9 + z^8 + 5*z^6 - 5*z^5 - z^3 + 5*z^2 + 7) * q^47 + (-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^49 + (z^5 + z^4 + 2*z^3 - 2*z^2 - z - 1) * q^50 + (-z^9 + z^8 + 2*z^6 + z^4 + z^2 + 2) * q^52 + (-2*z^9 - 2*z^8 + 6*z^6 - 3*z^5 + 6*z^4 - 2*z^2 - 2*z) * q^53 + (-4*z^9 + z^7 + 5*z^6 + 3*z^5 + 3*z^4 - 3*z^3 - 3*z^2 - 5*z - 1) * q^55 + (z^9 + z^6 + z^3 + z - 1) * q^56 + (-5*z^9 + 4*z^8 - z^7 + 4*z^6 - 5*z^5 - 2*z^3 + z^2 - z + 2) * q^58 + (2*z^9 - 3*z^8 - 2*z^7 + 2*z^6 - 2*z^5 - 3*z^4 + 2*z^3 + z - 1) * q^59 + (-2*z^9 - 6*z^8 + 5*z^7 - 5*z^6 + 6*z^5 + 2*z^4 - z^2 + 3*z - 1) * q^61 + (-z^9 + 5*z^8 - 4*z^7 + 2*z^6 - 4*z^5 + 2*z^4 - 4*z^3 + 5*z^2 - z) * q^62 + z^2 * q^64 + (4*z^9 + z^8 + 4*z^7 - 2*z^4 + z^3 - z^2 + 2*z) * q^65 + (5*z^8 - 6*z^7 - 7*z^5 + z^4 - 7*z^3 - 6*z + 5) * q^67 + (z^9 - z^7 + z^6 - z^5 + z^4 - z^2 + 2) * q^68 + (3*z^9 - 3*z^8 + 2*z^7 - 3*z^6 + 3*z^5 - 2*z^4 + 3*z^3 - 3*z^2 - 2) * q^70 + (z^8 - 7*z^7 + 3*z^6 - 5*z^5 + 7*z^4 - 5*z^3 + 3*z^2 - 7*z + 1) * q^71 + (6*z^9 + 6*z^7 - z^5 + 4*z^4 + 3*z^3 - 3*z^2 - 4*z + 1) * q^73 + (-z^8 + z^7 - z^4 + 2*z^2 - 1) * q^74 + (3*z^9 - 2*z^8 + 3*z^7 - z^6 + 4*z^5 - z^4 + 3*z^3 - 2*z^2 + 3*z) * q^76 + (-5*z^9 - z^8 - 3*z^7 + 3*z^6 + z^5 + 5*z^4 + 3*z^2 - z + 3) * q^77 + (5*z^9 - 2*z^8 + 5*z^7 - 5*z^6 + 5*z^5 - 2*z^4 + 5*z^3 + 9*z - 9) * q^79 + (z^9 - z^7 + z^5 + z^3 - 1) * q^80 + (z^8 + 3*z^7 - 3*z^6 + 3*z^5 + z^4 + 3*z - 3) * q^82 + (-5*z^9 + 5*z^6 + 2*z^5 - 2*z^4 + 2*z^3 - 2*z^2 - 5*z) * q^83 + (2*z^9 - 3*z^8 + 5*z^7 - 2*z^6 - z^5 - 2*z^4 + 5*z^3 - 3*z^2 + 2*z) * q^85 + (-z^9 + z^8 + z^6 - 4*z^5 + 5*z^4 - 3*z^3 + 5*z^2 - 4*z + 1) * q^86 + (-2*z^9 - 2*z^8 - 2*z^7 + z^4 - z) * q^88 + (z^9 + 3*z^8 + z^7 - 4*z^6 + z^5 - z^4 + 4*z^3 - z^2 - 3*z - 1) * q^89 + (-4*z^9 + z^8 - 2*z^7 + 2*z^6 - 2*z^5 + 2*z^4 - z^3 + 4*z^2 + 6) * q^91 + (-z^9 + 2*z^8 - 2*z^7 - 3*z^5 - z^4 + z^3 - z + 3) * q^92 + (5*z^8 - 4*z^7 - 5*z^5 - 2*z^4 - 5*z^3 - 4*z + 5) * q^94 + (4*z^9 - z^8 + 5*z^7 - 2*z^6 - z^5 + z^4 + 2*z^3 - 5*z^2 + z - 4) * q^95 + (-5*z^9 + 8*z^8 - 8*z^7 + 5*z^6 + 11*z^4 - 7*z^3 + 8*z^2 - 7*z + 11) * q^97 + (-z^9 + z^8 + 2*z^5 - z^4 - 2*z^3 - z^2 + 2*z) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q + q^{2} - q^{4} - 8 q^{5} + 8 q^{7} + q^{8}+O(q^{10})$$ 10 * q + q^2 - q^4 - 8 * q^5 + 8 * q^7 + q^8 $$10 q + q^{2} - q^{4} - 8 q^{5} + 8 q^{7} + q^{8} - 3 q^{10} - 7 q^{11} + 3 q^{13} + 3 q^{14} - q^{16} - 4 q^{17} + 3 q^{20} - 26 q^{22} + 12 q^{23} - 15 q^{25} - 3 q^{26} - 3 q^{28} + 25 q^{29} + 6 q^{31} + q^{32} - 7 q^{34} - 2 q^{35} + 9 q^{37} - 11 q^{38} - 3 q^{40} - 24 q^{41} - 30 q^{43} - 7 q^{44} + 21 q^{46} + 48 q^{47} + 9 q^{49} - 7 q^{50} + 14 q^{52} - 15 q^{53} - 23 q^{55} - 8 q^{56} - 3 q^{58} - 5 q^{59} + 12 q^{61} - 28 q^{62} - q^{64} + 13 q^{65} + 18 q^{67} + 18 q^{68} + 2 q^{70} - 28 q^{71} + 19 q^{73} - 9 q^{74} + 22 q^{76} + 12 q^{77} - 52 q^{79} - 8 q^{80} - 20 q^{82} - 7 q^{83} + 23 q^{85} - 14 q^{86} - 4 q^{88} - 3 q^{89} + 42 q^{91} + 23 q^{92} + 29 q^{94} - 22 q^{95} + 51 q^{97} + 2 q^{98}+O(q^{100})$$ 10 * q + q^2 - q^4 - 8 * q^5 + 8 * q^7 + q^8 - 3 * q^10 - 7 * q^11 + 3 * q^13 + 3 * q^14 - q^16 - 4 * q^17 + 3 * q^20 - 26 * q^22 + 12 * q^23 - 15 * q^25 - 3 * q^26 - 3 * q^28 + 25 * q^29 + 6 * q^31 + q^32 - 7 * q^34 - 2 * q^35 + 9 * q^37 - 11 * q^38 - 3 * q^40 - 24 * q^41 - 30 * q^43 - 7 * q^44 + 21 * q^46 + 48 * q^47 + 9 * q^49 - 7 * q^50 + 14 * q^52 - 15 * q^53 - 23 * q^55 - 8 * q^56 - 3 * q^58 - 5 * q^59 + 12 * q^61 - 28 * q^62 - q^64 + 13 * q^65 + 18 * q^67 + 18 * q^68 + 2 * q^70 - 28 * q^71 + 19 * q^73 - 9 * q^74 + 22 * q^76 + 12 * q^77 - 52 * q^79 - 8 * q^80 - 20 * q^82 - 7 * q^83 + 23 * q^85 - 14 * q^86 - 4 * q^88 - 3 * q^89 + 42 * q^91 + 23 * q^92 + 29 * q^94 - 22 * q^95 + 51 * q^97 + 2 * 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 −0.455922 3.17101i 0 0.628663 1.37658i 0.654861 0.755750i 0 −1.33083 2.91411i
73.1 −0.415415 0.909632i 0 −0.654861 + 0.755750i −1.27310 0.818172i 0 0.369215 + 2.56794i 0.959493 + 0.281733i 0 −0.215370 + 1.49793i
127.1 −0.841254 + 0.540641i 0 0.415415 0.909632i −1.18639 + 0.348356i 0 0.968468 + 1.11767i 0.142315 + 0.989821i 0 0.809721 0.934468i
163.1 −0.841254 0.540641i 0 0.415415 + 0.909632i −1.18639 0.348356i 0 0.968468 1.11767i 0.142315 0.989821i 0 0.809721 + 0.934468i
271.1 0.959493 + 0.281733i 0 0.841254 + 0.540641i −0.455922 + 3.17101i 0 0.628663 + 1.37658i 0.654861 + 0.755750i 0 −1.33083 + 2.91411i
289.1 0.654861 + 0.755750i 0 −0.142315 + 0.989821i 0.614354 1.34525i 0 3.07385 + 0.902563i −0.841254 + 0.540641i 0 1.41899 0.416652i
307.1 0.142315 0.989821i 0 −0.959493 0.281733i −1.69894 1.96068i 0 −1.04019 0.668491i −0.415415 + 0.909632i 0 −2.18251 + 1.40261i
325.1 0.142315 + 0.989821i 0 −0.959493 + 0.281733i −1.69894 + 1.96068i 0 −1.04019 + 0.668491i −0.415415 0.909632i 0 −2.18251 1.40261i
361.1 0.654861 0.755750i 0 −0.142315 0.989821i 0.614354 + 1.34525i 0 3.07385 0.902563i −0.841254 0.540641i 0 1.41899 + 0.416652i
397.1 −0.415415 + 0.909632i 0 −0.654861 0.755750i −1.27310 + 0.818172i 0 0.369215 2.56794i 0.959493 0.281733i 0 −0.215370 1.49793i
 $$n$$: 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.d 10
3.b odd 2 1 138.2.e.a 10
23.c even 11 1 inner 414.2.i.d 10
23.c even 11 1 9522.2.a.bt 5
23.d odd 22 1 9522.2.a.bq 5
69.g even 22 1 3174.2.a.bd 5
69.h odd 22 1 138.2.e.a 10
69.h odd 22 1 3174.2.a.bc 5

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{10} + 8 T_{5}^{9} + 42 T_{5}^{8} + 149 T_{5}^{7} + 389 T_{5}^{6} + 736 T_{5}^{5} + 1092 T_{5}^{4} + 1465 T_{5}^{3} + 1754 T_{5}^{2} + 1426 T_{5} + 529$$ acting on $$S_{2}^{\mathrm{new}}(414, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10} - T^{9} + T^{8} - T^{7} + T^{6} - T^{5} + \cdots + 1$$
$3$ $$T^{10}$$
$5$ $$T^{10} + 8 T^{9} + 42 T^{8} + 149 T^{7} + \cdots + 529$$
$7$ $$T^{10} - 8 T^{9} + 31 T^{8} - 83 T^{7} + \cdots + 529$$
$11$ $$T^{10} + 7 T^{9} + 5 T^{8} + 123 T^{7} + \cdots + 1$$
$13$ $$T^{10} - 3 T^{9} + 31 T^{8} - 38 T^{7} + \cdots + 1$$
$17$ $$T^{10} + 4 T^{9} + 5 T^{8} + 97 T^{7} + \cdots + 1849$$
$19$ $$T^{10} + 22 T^{8} + 165 T^{7} + \cdots + 64009$$
$23$ $$T^{10} - 12 T^{9} - 10 T^{8} + \cdots + 6436343$$
$29$ $$T^{10} - 25 T^{9} + 372 T^{8} + \cdots + 2866249$$
$31$ $$T^{10} - 6 T^{9} + 58 T^{8} + \cdots + 896809$$
$37$ $$T^{10} - 9 T^{9} + 26 T^{8} - 69 T^{7} + \cdots + 529$$
$41$ $$T^{10} + 24 T^{9} + 301 T^{8} + \cdots + 529$$
$43$ $$T^{10} + 30 T^{9} + 427 T^{8} + \cdots + 192721$$
$47$ $$(T^{5} - 24 T^{4} + 83 T^{3} + 1520 T^{2} + \cdots + 10649)^{2}$$
$53$ $$T^{10} + 15 T^{9} + \cdots + 361342081$$
$59$ $$T^{10} + 5 T^{9} - 19 T^{8} + \cdots + 31730689$$
$61$ $$T^{10} - 12 T^{9} + \cdots + 2801902489$$
$67$ $$T^{10} - 18 T^{9} + \cdots + 1804635361$$
$71$ $$T^{10} + 28 T^{9} + \cdots + 1490654881$$
$73$ $$T^{10} - 19 T^{9} + \cdots + 236452129$$
$79$ $$T^{10} + 52 T^{9} + \cdots + 2417590561$$
$83$ $$T^{10} + 7 T^{9} + 269 T^{8} + \cdots + 659102929$$
$89$ $$T^{10} + 3 T^{9} + 97 T^{8} + \cdots + 92871769$$
$97$ $$T^{10} - 51 T^{9} + \cdots + 22314683161$$