# Properties

 Label 414.2.i.f 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 46) 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.

 $$f(q)$$ $$=$$ $$q - \zeta_{22}^{4} q^{2} + \zeta_{22}^{8} q^{4} + (\zeta_{22}^{9} - \zeta_{22}^{6} + \zeta_{22}^{3} - 2 \zeta_{22}^{2} + \zeta_{22}) q^{5} + ( - \zeta_{22}^{8} - \zeta_{22}^{6} - \zeta_{22}^{5} - \zeta_{22}^{4} - \zeta_{22}^{2}) q^{7} + \zeta_{22} q^{8} +O(q^{10})$$ q - z^4 * q^2 + z^8 * q^4 + (z^9 - z^6 + z^3 - 2*z^2 + z) * q^5 + (-z^8 - z^6 - z^5 - z^4 - z^2) * q^7 + z * q^8 $$q - \zeta_{22}^{4} q^{2} + \zeta_{22}^{8} q^{4} + (\zeta_{22}^{9} - \zeta_{22}^{6} + \zeta_{22}^{3} - 2 \zeta_{22}^{2} + \zeta_{22}) q^{5} + ( - \zeta_{22}^{8} - \zeta_{22}^{6} - \zeta_{22}^{5} - \zeta_{22}^{4} - \zeta_{22}^{2}) q^{7} + \zeta_{22} q^{8} + (\zeta_{22}^{9} - \zeta_{22}^{8} + \zeta_{22}^{6} - \zeta_{22}^{4} + \zeta_{22}^{3} + \zeta_{22} - 1) q^{10} + (\zeta_{22}^{8} + \zeta_{22}^{7} + \zeta_{22}^{6} - \zeta_{22}^{3} - 2 \zeta_{22}^{2} + 2 \zeta_{22} + 1) q^{11} + (\zeta_{22}^{7} - 2 \zeta_{22}^{6} + \zeta_{22}^{5} + 2 \zeta_{22} - 2) q^{13} + (2 \zeta_{22}^{9} + \zeta_{22}^{7} + \zeta_{22}^{5} - \zeta_{22}^{4} + \zeta_{22}^{3} - \zeta_{22}^{2} - 1) q^{14} - \zeta_{22}^{5} q^{16} + ( - 2 \zeta_{22}^{6} - \zeta_{22}^{5} - \zeta_{22}^{4} + 3 \zeta_{22}^{3} - \zeta_{22}^{2} - \zeta_{22} - 2) q^{17} + (\zeta_{22}^{9} + 2 \zeta_{22}^{8} + \zeta_{22}^{7} + \zeta_{22}^{4} + 2 \zeta_{22}^{3} - 2 \zeta_{22}^{2} - \zeta_{22}) q^{19} + ( - \zeta_{22}^{9} + 2 \zeta_{22}^{8} - 2 \zeta_{22}^{7} + \zeta_{22}^{6} - 2 \zeta_{22}^{5} + 2 \zeta_{22}^{4} - \zeta_{22}^{3} + \cdots + 1) q^{20} + \cdots + ( - \zeta_{22}^{9} + \zeta_{22}^{8} + 4 \zeta_{22}^{6} + \zeta_{22}^{5} + 4 \zeta_{22}^{4} - 4 \zeta_{22}^{3} + 4 \zeta_{22}^{2} + \cdots + 4) q^{98} +O(q^{100})$$ q - z^4 * q^2 + z^8 * q^4 + (z^9 - z^6 + z^3 - 2*z^2 + z) * q^5 + (-z^8 - z^6 - z^5 - z^4 - z^2) * q^7 + z * q^8 + (z^9 - z^8 + z^6 - z^4 + z^3 + z - 1) * q^10 + (z^8 + z^7 + z^6 - z^3 - 2*z^2 + 2*z + 1) * q^11 + (z^7 - 2*z^6 + z^5 + 2*z - 2) * q^13 + (2*z^9 + z^7 + z^5 - z^4 + z^3 - z^2 - 1) * q^14 - z^5 * q^16 + (-2*z^6 - z^5 - z^4 + 3*z^3 - z^2 - z - 2) * q^17 + (z^9 + 2*z^8 + z^7 + z^4 + 2*z^3 - 2*z^2 - z) * q^19 + (-z^9 + 2*z^8 - 2*z^7 + z^6 - 2*z^5 + 2*z^4 - z^3 + 2*z^2 - 2*z + 1) * q^20 + (-z^9 + z^8 + 3*z^6 - 3*z^5 - z^3 + z^2 + 2) * q^22 + (3*z^9 - z^8 - z^7 - z^5 - 2*z^4 - 2*z^3 + z^2) * q^23 + (2*z^8 - z^7 - z^6 - 2*z^5 + z^4 - 2*z^3 - z^2 - z + 2) * q^25 + (z^9 - 2*z^8 + 2*z^7 - 2*z^6 + 2*z^3 - 2*z^2 + 2*z - 1) * q^26 + (-z^9 + z^8 - z^7 + z^6 + z^4 + 2*z^2 + 1) * q^28 + (2*z^9 - 2*z^8 - z^5 - 3*z^4 - 3*z^2 - z) * q^29 + (3*z^9 - 5*z^8 + 3*z^7 - 3*z^6 + 5*z^5 - 3*z^4 - 5*z^2 + 2*z - 5) * q^31 + z^9 * q^32 + (3*z^9 - z^8 - z^7 - z^6 + 3*z^5 + 2*z^3 - 2*z^2 + 2*z - 2) * q^34 + (z^9 - z^8 + 3*z^7 - z^6 + z^5 + 2*z^3 - 2*z^2 + 2*z - 2) * q^35 + (-5*z^9 - 5*z^7 + 5*z^6 - z^5 + z^2 - 5*z + 5) * q^37 + (-z^8 - 2*z^7 + 2*z^6 + z^5 + z^2 + 2*z + 1) * q^38 + (z^9 - z^8 - z^6 + z^5 - z^3 + z - 1) * q^40 + (-5*z^9 + 4*z^8 - 4*z^7 + 5*z^6 + 7*z^4 - 5*z^3 + 4*z^2 - 5*z + 7) * q^41 + (z^9 - 3*z^8 - 3*z^6 + 4*z^5 - 4*z^4 + 3*z^3 + 3*z - 1) * q^43 + (3*z^8 - 2*z^7 + 2*z^6 - 3*z^5 + z^4 - 3*z^3 + 2*z^2 - 2*z + 3) * q^44 + (z^9 + 2*z^8 + 2*z^7 - z^6 + 3*z^2 - z - 1) * q^46 + (2*z^8 - 4*z^7 - 2*z^6 + 2*z^5 + 4*z^4 - 2*z^3 - 1) * q^47 + (5*z^8 + 4*z^6 - 3*z^5 + 3*z^4 - 4*z^3 - 5*z) * q^49 + (3*z^9 - 2*z^8 + 3*z^7 + 2*z^5 - 3*z^4 + z^3 - z^2 + 3*z - 2) * q^50 + (2*z^9 - 2*z^8 - z^4 + 2*z^3 - z^2) * q^52 + (6*z^7 - 3*z^6 + 3*z^5 - 3*z^4 + 6*z^3) * q^53 + (3*z^9 - 4*z^6 + 3*z^4 - 3*z^3 + 4*z) * q^55 + (-z^9 - z^7 - z^6 - z^5 - z^3) * q^56 + (z^9 + 3*z^8 + 3*z^6 + z^5 + 2*z^2 - 2*z) * q^58 + (2*z^9 - 4*z^8 + 4*z^7 - 3*z^6 + 4*z^5 - 4*z^4 + 2*z^3 + 2*z - 2) * q^59 + (-5*z^9 - z^8 - 2*z^7 + 2*z^6 + z^5 + 5*z^4 + 6*z^2 - 5*z + 6) * q^61 + (-2*z^9 + 3*z^7 + 2*z^6 + z^5 + 2*z^4 + 3*z^3 - 2*z) * q^62 + z^2 * q^64 + (-3*z^9 + 3*z^8 - 3*z^7 + 2*z^5 + z^4 - 4*z^3 + 4*z^2 - z - 2) * q^65 + (-2*z^8 + 5*z^7 - 2*z^5 + z^4 - 2*z^3 + 5*z - 2) * q^67 + (-2*z^9 - z^8 - z^7 + z^6 - z^5 + z^4 + z^3 + 2*z^2 - 2) * q^68 + (-z^8 - z^7 + z^6 - z^5 + z^4 + z^3 + 2) * q^70 + (-8*z^8 + 3*z^7 - 4*z^6 + 2*z^5 - 5*z^4 + 2*z^3 - 4*z^2 + 3*z - 8) * q^71 + (-2*z^8 + 2*z^5 - z^4 + 3*z^3 - 3*z^2 + z - 2) * q^73 + (-4*z^9 + 5*z^8 - 5*z^7 + 4*z^6 - 5*z^3 - 5*z) * q^74 + (-3*z^9 + 2*z^8 - 2*z^7 + z^6 - 4*z^5 + z^4 - 2*z^3 + 2*z^2 - 3*z) * q^76 + (-2*z^9 + z^8 + z^7 - z^6 - z^5 + 2*z^4 + z^2 + 3*z + 1) * q^77 + (2*z^9 - 2*z^8 + 2*z^7 - 5*z^6 + 2*z^5 - 2*z^4 + 2*z^3 + z - 1) * q^79 + (-z^8 + 2*z^7 - z^6 + z^3 - 1) * q^80 + (-5*z^9 - 2*z^8 + z^6 - 2*z^4 - 5*z^3 - z + 1) * q^82 + (-5*z^9 - 6*z^7 + 7*z^6 - 9*z^5 + 5*z^4 - 5*z^3 + 9*z^2 - 7*z + 6) * q^83 + (-7*z^9 + 4*z^8 + 6*z^6 - 8*z^5 + 6*z^4 + 4*z^2 - 7*z) * q^85 + (-z^9 + z^8 - 3*z^6 - 2*z^4 + 3*z^3 - 2*z^2 - 3) * q^86 + (z^9 + z^8 + z^7 - z^4 - 2*z^3 + 2*z^2 + z) * q^88 + (-3*z^9 - z^8 - 4*z^7 + 4*z^6 - 4*z^3 + 4*z^2 + z + 3) * q^89 + (-3*z^9 + 3*z^8 - 2*z^7 - z^6 + z^5 + 2*z^4 - 3*z^3 + 3*z^2 - 1) * q^91 + (z^9 - z^8 + z^7 - 4*z^6 + 2*z^5 + z^3 + 3*z + 1) * q^92 + (-6*z^8 + 4*z^7 - 2*z^6 + 2*z^5 - z^4 + 2*z^3 - 2*z^2 + 4*z - 6) * q^94 + (-6*z^9 + 7*z^8 - 3*z^7 + 2*z^6 - 10*z^5 + 10*z^4 - 2*z^3 + 3*z^2 - 7*z + 6) * q^95 + (2*z^9 + 3*z^8 - 3*z^7 - 2*z^6 - 3*z^4 + 3*z^2 - 3) * q^97 + (-z^9 + z^8 + 4*z^6 + z^5 + 4*z^4 - 4*z^3 + 4*z^2 + z + 4) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q + q^{2} - q^{4} + 6 q^{5} + 3 q^{7} + q^{8}+O(q^{10})$$ 10 * q + q^2 - q^4 + 6 * q^5 + 3 * q^7 + q^8 $$10 q + q^{2} - q^{4} + 6 q^{5} + 3 q^{7} + q^{8} - 6 q^{10} + 12 q^{11} - 14 q^{13} - 3 q^{14} - q^{16} - 15 q^{17} + 2 q^{19} - 5 q^{20} + 10 q^{22} + q^{23} + 13 q^{25} + 3 q^{26} + 3 q^{28} + 8 q^{29} - 21 q^{31} + q^{32} - 7 q^{34} - 7 q^{35} + 28 q^{37} + 9 q^{38} - 6 q^{40} + 31 q^{41} + 11 q^{43} + 12 q^{44} - 12 q^{46} - 18 q^{47} - 24 q^{49} - 2 q^{50} + 8 q^{52} + 21 q^{53} + 5 q^{55} - 3 q^{56} - 8 q^{58} + 5 q^{59} + 37 q^{61} - q^{62} - q^{64} - 37 q^{65} - 13 q^{67} - 26 q^{68} + 18 q^{70} - 49 q^{71} - 8 q^{73} - 28 q^{74} - 20 q^{76} + 8 q^{77} + 8 q^{79} - 5 q^{80} + 2 q^{82} + 7 q^{83} - 42 q^{85} - 22 q^{86} - q^{88} + 13 q^{89} - 24 q^{91} + 23 q^{92} - 37 q^{94} + 10 q^{95} - 32 q^{97} + 24 q^{98}+O(q^{100})$$ 10 * q + q^2 - q^4 + 6 * q^5 + 3 * q^7 + q^8 - 6 * q^10 + 12 * q^11 - 14 * q^13 - 3 * q^14 - q^16 - 15 * q^17 + 2 * q^19 - 5 * q^20 + 10 * q^22 + q^23 + 13 * q^25 + 3 * q^26 + 3 * q^28 + 8 * q^29 - 21 * q^31 + q^32 - 7 * q^34 - 7 * q^35 + 28 * q^37 + 9 * q^38 - 6 * q^40 + 31 * q^41 + 11 * q^43 + 12 * q^44 - 12 * q^46 - 18 * q^47 - 24 * q^49 - 2 * q^50 + 8 * q^52 + 21 * q^53 + 5 * q^55 - 3 * q^56 - 8 * q^58 + 5 * q^59 + 37 * q^61 - q^62 - q^64 - 37 * q^65 - 13 * q^67 - 26 * q^68 + 18 * q^70 - 49 * q^71 - 8 * q^73 - 28 * q^74 - 20 * q^76 + 8 * q^77 + 8 * q^79 - 5 * q^80 + 2 * q^82 + 7 * q^83 - 42 * q^85 - 22 * q^86 - q^88 + 13 * q^89 - 24 * q^91 + 23 * q^92 - 37 * q^94 + 10 * q^95 - 32 * q^97 + 24 * 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 −0.174863 1.21620i 0 0.260554 0.570534i 0.654861 0.755750i 0 −0.510424 1.11767i
73.1 −0.415415 0.909632i 0 −0.654861 + 0.755750i −0.767092 0.492980i 0 −0.601808 4.18567i 0.959493 + 0.281733i 0 −0.129769 + 0.902563i
127.1 −0.841254 + 0.540641i 0 0.415415 0.909632i 3.26024 0.957293i 0 −0.297176 0.342959i 0.142315 + 0.989821i 0 −2.22514 + 2.56794i
163.1 −0.841254 0.540641i 0 0.415415 + 0.909632i 3.26024 + 0.957293i 0 −0.297176 + 0.342959i 0.142315 0.989821i 0 −2.22514 2.56794i
271.1 0.959493 + 0.281733i 0 0.841254 + 0.540641i −0.174863 + 1.21620i 0 0.260554 + 0.570534i 0.654861 + 0.755750i 0 −0.510424 + 1.11767i
289.1 0.654861 + 0.755750i 0 −0.142315 + 0.989821i −0.985691 + 2.15836i 0 0.381761 + 0.112095i −0.841254 + 0.540641i 0 −2.27667 + 0.668491i
307.1 0.142315 0.989821i 0 −0.959493 0.281733i 1.66741 + 1.92429i 0 1.75667 + 1.12894i −0.415415 + 0.909632i 0 2.14200 1.37658i
325.1 0.142315 + 0.989821i 0 −0.959493 + 0.281733i 1.66741 1.92429i 0 1.75667 1.12894i −0.415415 0.909632i 0 2.14200 + 1.37658i
361.1 0.654861 0.755750i 0 −0.142315 0.989821i −0.985691 2.15836i 0 0.381761 0.112095i −0.841254 0.540641i 0 −2.27667 0.668491i
397.1 −0.415415 + 0.909632i 0 −0.654861 0.755750i −0.767092 + 0.492980i 0 −0.601808 + 4.18567i 0.959493 0.281733i 0 −0.129769 0.902563i
 $$n$$: 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.f 10
3.b odd 2 1 46.2.c.a 10
12.b even 2 1 368.2.m.b 10
23.c even 11 1 inner 414.2.i.f 10
23.c even 11 1 9522.2.a.bp 5
23.d odd 22 1 9522.2.a.bu 5
69.g even 22 1 1058.2.a.l 5
69.h odd 22 1 46.2.c.a 10
69.h odd 22 1 1058.2.a.m 5
276.j odd 22 1 8464.2.a.bw 5
276.o even 22 1 368.2.m.b 10
276.o even 22 1 8464.2.a.bx 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
46.2.c.a 10 3.b odd 2 1
46.2.c.a 10 69.h odd 22 1
368.2.m.b 10 12.b even 2 1
368.2.m.b 10 276.o even 22 1
414.2.i.f 10 1.a even 1 1 trivial
414.2.i.f 10 23.c even 11 1 inner
1058.2.a.l 5 69.g even 22 1
1058.2.a.m 5 69.h odd 22 1
8464.2.a.bw 5 276.j odd 22 1
8464.2.a.bx 5 276.o even 22 1
9522.2.a.bp 5 23.c even 11 1
9522.2.a.bu 5 23.d odd 22 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{10} - 6 T_{5}^{9} + 14 T_{5}^{8} - 29 T_{5}^{7} + 86 T_{5}^{6} - 153 T_{5}^{5} + 126 T_{5}^{4} + 201 T_{5}^{3} + 587 T_{5}^{2} + 713 T_{5} + 529$$ acting on $$S_{2}^{\mathrm{new}}(414, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10} - T^{9} + T^{8} - T^{7} + T^{6} - T^{5} + \cdots + 1$$
$3$ $$T^{10}$$
$5$ $$T^{10} - 6 T^{9} + 14 T^{8} - 29 T^{7} + \cdots + 529$$
$7$ $$T^{10} - 3 T^{9} + 20 T^{8} - 71 T^{7} + \cdots + 1$$
$11$ $$T^{10} - 12 T^{9} + 56 T^{8} + \cdots + 109561$$
$13$ $$T^{10} + 14 T^{9} + 86 T^{8} + 258 T^{7} + \cdots + 1$$
$17$ $$T^{10} + 15 T^{9} + 137 T^{8} + \cdots + 214369$$
$19$ $$T^{10} - 2 T^{9} + 37 T^{8} + \cdots + 4489$$
$23$ $$T^{10} - T^{9} + 78 T^{8} + \cdots + 6436343$$
$29$ $$T^{10} - 8 T^{9} - 2 T^{8} - 259 T^{7} + \cdots + 4489$$
$31$ $$T^{10} + 21 T^{9} + 232 T^{8} + \cdots + 20529961$$
$37$ $$T^{10} - 28 T^{9} + 509 T^{8} + \cdots + 49857721$$
$41$ $$T^{10} - 31 T^{9} + \cdots + 172475689$$
$43$ $$T^{10} - 11 T^{9} + 66 T^{8} + \cdots + 7027801$$
$47$ $$(T^{5} + 9 T^{4} - 82 T^{3} - 922 T^{2} + \cdots + 529)^{2}$$
$53$ $$T^{10} - 21 T^{9} + 144 T^{8} + \cdots + 31236921$$
$59$ $$T^{10} - 5 T^{9} - 41 T^{8} + \cdots + 4489$$
$61$ $$T^{10} - 37 T^{9} + \cdots + 349727401$$
$67$ $$T^{10} + 13 T^{9} + 202 T^{8} + \cdots + 94109401$$
$71$ $$T^{10} + 49 T^{9} + 1180 T^{8} + \cdots + 8300161$$
$73$ $$T^{10} + 8 T^{9} + 42 T^{8} + \cdots + 7921$$
$79$ $$T^{10} - 8 T^{9} + 9 T^{8} + \cdots + 17161$$
$83$ $$T^{10} - 7 T^{9} + 49 T^{8} + \cdots + 667137241$$
$89$ $$T^{10} - 13 T^{9} + 147 T^{8} + \cdots + 26739241$$
$97$ $$T^{10} + 32 T^{9} + 628 T^{8} + \cdots + 5031049$$