# Properties

 Label 414.2.i.c 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{22}^{4} q^{2} + \zeta_{22}^{8} q^{4} + (\zeta_{22}^{9} - \zeta_{22}^{6} + \zeta_{22}^{3} + \zeta_{22}) q^{5} + (\zeta_{22}^{8} - 2 \zeta_{22}^{7} + \zeta_{22}^{6} + \zeta_{22}^{5} + \zeta_{22}^{4} - 2 \zeta_{22}^{3} + \zeta_{22}^{2}) q^{7} - \zeta_{22} q^{8} +O(q^{10})$$ q + z^4 * q^2 + z^8 * q^4 + (z^9 - z^6 + z^3 + z) * q^5 + (z^8 - 2*z^7 + z^6 + z^5 + z^4 - 2*z^3 + z^2) * q^7 - z * q^8 $$q + \zeta_{22}^{4} q^{2} + \zeta_{22}^{8} q^{4} + (\zeta_{22}^{9} - \zeta_{22}^{6} + \zeta_{22}^{3} + \zeta_{22}) q^{5} + (\zeta_{22}^{8} - 2 \zeta_{22}^{7} + \zeta_{22}^{6} + \zeta_{22}^{5} + \zeta_{22}^{4} - 2 \zeta_{22}^{3} + \zeta_{22}^{2}) q^{7} - \zeta_{22} q^{8} + ( - \zeta_{22}^{9} + \zeta_{22}^{8} + \zeta_{22}^{6} + \zeta_{22}^{4} - \zeta_{22}^{3} - \zeta_{22} + 1) q^{10} + (\zeta_{22}^{8} - \zeta_{22}^{7} + \zeta_{22}^{6} - \zeta_{22}^{3} + 2 \zeta_{22}^{2} - 2 \zeta_{22} + 1) q^{11} + (4 \zeta_{22}^{9} - 2 \zeta_{22}^{8} + 3 \zeta_{22}^{7} - 2 \zeta_{22}^{6} + 3 \zeta_{22}^{5} - 2 \zeta_{22}^{4} + \cdots - 2) q^{13} + \cdots + (\zeta_{22}^{9} - \zeta_{22}^{8} - 4 \zeta_{22}^{6} + 7 \zeta_{22}^{5} + 7 \zeta_{22} - 4) q^{98} +O(q^{100})$$ q + z^4 * q^2 + z^8 * q^4 + (z^9 - z^6 + z^3 + z) * q^5 + (z^8 - 2*z^7 + z^6 + z^5 + z^4 - 2*z^3 + 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 + (4*z^9 - 2*z^8 + 3*z^7 - 2*z^6 + 3*z^5 - 2*z^4 + 4*z^3 + 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 + 3*z^4 - z^3 + 3*z^2 - z + 2) * q^17 + (-3*z^9 - 3*z^7 - 2*z^5 + 3*z^4 - 2*z^3 + 2*z^2 - 3*z + 2) * q^19 + (z^9 - z^6 + z^3 - 1) * q^20 + (z^9 - z^8 + z^6 - z^5 + z^3 - z^2) * q^22 + (z^9 + z^8 + z^7 - 2*z^6 + z^5 + 2*z^4 + z^2 - 2*z - 2) * q^23 + (-2*z^8 - z^7 - z^6 + 2*z^5 - 3*z^4 + 2*z^3 - z^2 - z - 2) * q^25 + (z^9 + 2*z^7 + 2*z^6 - 2*z^3 - 2*z^2 - 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 - 2*z^6 + 3*z^5 - 3*z^4 + 4*z^3 - 3*z^2 + 3*z - 2) * q^29 + (-z^9 + z^8 + 3*z^7 - 3*z^6 - z^5 + z^4 + z^2 + 1) * q^31 - z^9 * q^32 + (z^9 + z^8 + z^7 + z^6 + z^5 + 2*z^3 - 2*z^2 + 2*z - 2) * q^34 + (z^9 + z^8 - z^7 + z^6 + z^5 - 2*z^3 + 2) * q^35 + (5*z^9 + 3*z^7 - z^6 - z^5 - 2*z^4 + 2*z^3 + z^2 + z - 3) * q^37 + (-2*z^9 + 3*z^8 - 2*z^7 + 2*z^6 - 3*z^5 + 2*z^4 + 3*z^2 + 3) * q^38 + (-z^9 + z^8 + z^6 - z^5 - z^3 - z + 1) * q^40 + (3*z^9 - 3*z^6 - z^4 - z^3 - z - 1) * q^41 + (-3*z^9 + 5*z^8 - 8*z^7 + 5*z^6 - 6*z^5 + 6*z^4 - 5*z^3 + 8*z^2 - 5*z + 3) * q^43 + (-z^8 + 2*z^7 - 2*z^6 + z^5 - z^4 + z^3 - 2*z^2 + 2*z - 1) * q^44 + (-z^9 + 4*z^8 - 2*z^7 + 3*z^6 - 4*z^5 - 2*z^3 + z^2 - 3*z + 1) * q^46 + (-2*z^9 - 2*z^8 + 4*z^6 - 4*z^5 + 2*z^3 + 2*z^2 + 3) * q^47 + (-7*z^8 + 4*z^7 + z^5 - z^4 - 4*z^2 + 7*z) * q^49 + (z^9 - 2*z^8 + z^7 - 2*z^5 - z^4 - z^3 + z^2 + z + 2) * q^50 + (2*z^9 - 2*z^8 - 4*z^6 + 2*z^5 - 3*z^4 + 2*z^3 - 3*z^2 + 2*z - 4) * q^52 + (-3*z^6 + z^5 - 3*z^4) * q^53 + (z^9 - z^4 + z^3) * q^55 + (-z^9 + 2*z^8 - z^7 - z^6 - z^5 + 2*z^4 - z^3) * q^56 + (z^9 - z^8 + 2*z^7 - z^6 + z^5 - 2*z^3 + 2) * q^58 + (-2*z^9 - 2*z^8 - 2*z^7 + 3*z^6 - 2*z^5 - 2*z^4 - 2*z^3 + 4*z - 4) * q^59 + (z^9 - 3*z^8 + 2*z^7 - 2*z^6 + 3*z^5 - z^4 - 2*z^2 + 3*z - 2) * q^61 + (-4*z^9 + 4*z^8 - 3*z^7 + 4*z^6 - 3*z^5 + 4*z^4 - 3*z^3 + 4*z^2 - 4*z) * q^62 + z^2 * q^64 + (z^9 - 5*z^8 + z^7 + 6*z^5 + z^4 + 4*z^3 - 4*z^2 - z - 6) * q^65 + (-4*z^8 + 5*z^7 - 2*z^6 + 6*z^5 - 9*z^4 + 6*z^3 - 2*z^2 + 5*z - 4) * q^67 + (2*z^9 - z^8 + 3*z^7 - 3*z^6 + 3*z^5 - 3*z^4 + z^3 - 2*z^2 - 2) * q^68 + (2*z^9 - z^8 - z^7 - z^6 + z^5 + z^4 + z^3 - 2*z^2) * q^70 + (z^7 - 4*z^6 - z^4 - 4*z^2 + z) * q^71 + (-4*z^9 + 2*z^8 - 4*z^7 + 2*z^5 - z^4 - z^3 + z^2 + z - 2) * q^73 + (-2*z^9 - z^8 + z^7 + 2*z^6 - 2*z^4 - z^3 - 4*z^2 - z - 2) * q^74 + (-z^9 + 2*z^7 + z^6 + 2*z^5 + z^4 + 2*z^3 - z) * q^76 + (-2*z^9 - z^8 + 5*z^7 - 5*z^6 + z^5 + 2*z^4 - 5*z^2 + 7*z - 5) * q^77 + (-6*z^9 + 2*z^8 - 6*z^7 + z^6 - 6*z^5 + 2*z^4 - 6*z^3 - 7*z + 7) * q^79 + (-z^8 - z^6 + z^3 - 1) * q^80 + (-3*z^9 + 2*z^8 - 4*z^7 + 3*z^6 - 4*z^5 + 2*z^4 - 3*z^3 - 3*z + 3) * q^82 + (-7*z^9 - 4*z^7 + 3*z^6 - 9*z^5 + z^4 - z^3 + 9*z^2 - 3*z + 4) * q^83 + (z^9 + 2*z^7 + 2*z^5 + 2*z^3 + z) * q^85 + (-z^9 + z^8 + 3*z^6 - 2*z^4 + 5*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 + (-7*z^9 + 3*z^8 + 4*z^6 - 4*z^3 - 3*z + 7) * q^89 + (-3*z^9 + 5*z^8 + 3*z^6 - 3*z^5 - 5*z^3 + 3*z^2 - 3) * q^91 + (-z^9 - 3*z^8 + z^7 - 2*z^6 - 2*z^4 + 3*z^3 - 2*z^2 - z - 1) * q^92 + (-4*z^8 + 6*z^7 - 2*z^6 + 4*z^5 - z^4 + 4*z^3 - 2*z^2 + 6*z - 4) * q^94 + (-8*z^9 + 5*z^8 - 3*z^7 + 6*z^6 - 4*z^5 + 4*z^4 - 6*z^3 + 3*z^2 - 5*z + 8) * q^95 + (2*z^9 + 3*z^8 - 3*z^7 - 2*z^6 + z^4 - 9*z^2 + 1) * q^97 + (z^9 - z^8 - 4*z^6 + 7*z^5 + 7*z - 4) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q - q^{2} - q^{4} + 4 q^{5} - 7 q^{7} - q^{8}+O(q^{10})$$ 10 * q - q^2 - q^4 + 4 * q^5 - 7 * q^7 - q^8 $$10 q - q^{2} - q^{4} + 4 q^{5} - 7 q^{7} - q^{8} + 4 q^{10} + 2 q^{11} + 2 q^{13} + 15 q^{14} - q^{16} + 9 q^{17} + 2 q^{19} - 7 q^{20} + 2 q^{22} - 21 q^{23} - 11 q^{25} - 9 q^{26} + 15 q^{28} + 2 q^{29} + 11 q^{31} - q^{32} - 13 q^{34} + 17 q^{35} - 18 q^{37} + 13 q^{38} + 4 q^{40} - 5 q^{41} - 21 q^{43} + 2 q^{44} - 10 q^{46} + 22 q^{47} + 24 q^{49} + 22 q^{50} - 20 q^{52} + 7 q^{53} + 3 q^{55} - 7 q^{56} + 24 q^{58} - 43 q^{59} - 3 q^{61} - 33 q^{62} - q^{64} - 41 q^{65} - q^{67} - 2 q^{68} + 6 q^{70} + 11 q^{71} - 28 q^{73} - 18 q^{74} + 2 q^{76} - 30 q^{77} + 34 q^{79} - 7 q^{80} + 6 q^{82} + 3 q^{83} + 8 q^{85} + 34 q^{86} - 9 q^{88} + 49 q^{89} - 52 q^{91} + q^{92} - 11 q^{94} + 36 q^{95} + 16 q^{97} - 20 q^{98}+O(q^{100})$$ 10 * q - q^2 - q^4 + 4 * q^5 - 7 * q^7 - q^8 + 4 * q^10 + 2 * q^11 + 2 * q^13 + 15 * q^14 - q^16 + 9 * q^17 + 2 * q^19 - 7 * q^20 + 2 * q^22 - 21 * q^23 - 11 * q^25 - 9 * q^26 + 15 * q^28 + 2 * q^29 + 11 * q^31 - q^32 - 13 * q^34 + 17 * q^35 - 18 * q^37 + 13 * q^38 + 4 * q^40 - 5 * q^41 - 21 * q^43 + 2 * q^44 - 10 * q^46 + 22 * q^47 + 24 * q^49 + 22 * q^50 - 20 * q^52 + 7 * q^53 + 3 * q^55 - 7 * q^56 + 24 * q^58 - 43 * q^59 - 3 * q^61 - 33 * q^62 - q^64 - 41 * q^65 - q^67 - 2 * q^68 + 6 * q^70 + 11 * q^71 - 28 * q^73 - 18 * q^74 + 2 * q^76 - 30 * q^77 + 34 * q^79 - 7 * q^80 + 6 * q^82 + 3 * q^83 + 8 * q^85 + 34 * q^86 - 9 * q^88 + 49 * q^89 - 52 * q^91 + q^92 - 11 * q^94 + 36 * q^95 + 16 * q^97 - 20 * 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 −0.459493 3.19584i 0 −0.497033 + 1.08835i −0.654861 + 0.755750i 0 1.34125 + 2.93694i
73.1 0.415415 + 0.909632i 0 −0.654861 + 0.755750i 0.915415 + 0.588302i 0 0.122916 + 0.854902i −0.959493 0.281733i 0 −0.154861 + 1.07708i
127.1 0.841254 0.540641i 0 0.415415 0.909632i 1.34125 0.393828i 0 2.81051 + 3.24350i −0.142315 0.989821i 0 0.915415 1.05645i
163.1 0.841254 + 0.540641i 0 0.415415 + 0.909632i 1.34125 + 0.393828i 0 2.81051 3.24350i −0.142315 + 0.989821i 0 0.915415 + 1.05645i
271.1 −0.959493 0.281733i 0 0.841254 + 0.540641i −0.459493 + 3.19584i 0 −0.497033 1.08835i −0.654861 0.755750i 0 1.34125 2.93694i
289.1 −0.654861 0.755750i 0 −0.142315 + 0.989821i −0.154861 + 0.339098i 0 −1.97611 0.580239i 0.841254 0.540641i 0 0.357685 0.105026i
307.1 −0.142315 + 0.989821i 0 −0.959493 0.281733i 0.357685 + 0.412791i 0 −3.96028 2.54512i 0.415415 0.909632i 0 −0.459493 + 0.295298i
325.1 −0.142315 0.989821i 0 −0.959493 + 0.281733i 0.357685 0.412791i 0 −3.96028 + 2.54512i 0.415415 + 0.909632i 0 −0.459493 0.295298i
361.1 −0.654861 + 0.755750i 0 −0.142315 0.989821i −0.154861 0.339098i 0 −1.97611 + 0.580239i 0.841254 + 0.540641i 0 0.357685 + 0.105026i
397.1 0.415415 0.909632i 0 −0.654861 0.755750i 0.915415 0.588302i 0 0.122916 0.854902i −0.959493 + 0.281733i 0 −0.154861 1.07708i
 $$n$$: 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.c 10
3.b odd 2 1 46.2.c.b 10
12.b even 2 1 368.2.m.a 10
23.c even 11 1 inner 414.2.i.c 10
23.c even 11 1 9522.2.a.bz 5
23.d odd 22 1 9522.2.a.bw 5
69.g even 22 1 1058.2.a.k 5
69.h odd 22 1 46.2.c.b 10
69.h odd 22 1 1058.2.a.j 5
276.j odd 22 1 8464.2.a.bv 5
276.o even 22 1 368.2.m.a 10
276.o even 22 1 8464.2.a.bu 5

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{10} - 4T_{5}^{9} + 16T_{5}^{8} - 53T_{5}^{7} + 102T_{5}^{6} - 111T_{5}^{5} + 70T_{5}^{4} - 27T_{5}^{3} + 9T_{5}^{2} - 3T_{5} + 1$$ acting on $$S_{2}^{\mathrm{new}}(414, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10} + T^{9} + T^{8} + T^{7} + T^{6} + T^{5} + \cdots + 1$$
$3$ $$T^{10}$$
$5$ $$T^{10} - 4 T^{9} + 16 T^{8} - 53 T^{7} + \cdots + 1$$
$7$ $$T^{10} + 7 T^{9} + 16 T^{8} + \cdots + 1849$$
$11$ $$T^{10} - 2 T^{9} + 26 T^{8} - 74 T^{7} + \cdots + 1$$
$13$ $$T^{10} - 2 T^{9} + 48 T^{8} + \cdots + 436921$$
$17$ $$T^{10} - 9 T^{9} + 81 T^{8} - 454 T^{7} + \cdots + 1$$
$19$ $$T^{10} - 2 T^{9} + 15 T^{8} + \cdots + 139129$$
$23$ $$T^{10} + 21 T^{9} + 210 T^{8} + \cdots + 6436343$$
$29$ $$T^{10} - 2 T^{9} + 48 T^{8} + 25 T^{7} + \cdots + 529$$
$31$ $$T^{10} - 11 T^{9} + 110 T^{8} + \cdots + 2076481$$
$37$ $$T^{10} + 18 T^{9} + 225 T^{8} + \cdots + 14645929$$
$41$ $$T^{10} + 5 T^{9} + 25 T^{8} + \cdots + 1985281$$
$43$ $$T^{10} + 21 T^{9} + 232 T^{8} + \cdots + 53333809$$
$47$ $$(T^{5} - 11 T^{4} - 66 T^{3} + 726 T^{2} + \cdots - 3883)^{2}$$
$53$ $$T^{10} - 7 T^{9} + 16 T^{8} + \cdots + 1849$$
$59$ $$T^{10} + 43 T^{9} + 859 T^{8} + \cdots + 8300161$$
$61$ $$T^{10} + 3 T^{9} + 53 T^{8} + 104 T^{7} + \cdots + 529$$
$67$ $$T^{10} + T^{9} - 54 T^{8} + \cdots + 157609$$
$71$ $$T^{10} - 11 T^{9} + 110 T^{8} + \cdots + 2076481$$
$73$ $$T^{10} + 28 T^{9} + 366 T^{8} + \cdots + 34774609$$
$79$ $$T^{10} - 34 T^{9} + \cdots + 118613881$$
$83$ $$T^{10} - 3 T^{9} - 277 T^{8} + \cdots + 923126689$$
$89$ $$T^{10} - 49 T^{9} + 1191 T^{8} + \cdots + 380689$$
$97$ $$T^{10} - 16 T^{9} + \cdots + 788093329$$