# Properties

 Label 1050.3.f.b Level $1050$ Weight $3$ Character orbit 1050.f Analytic conductor $28.610$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1050.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$28.6104277578$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.3317760000.3 Defining polynomial: $$x^{8} - 4x^{6} + 7x^{4} - 36x^{2} + 81$$ x^8 - 4*x^6 + 7*x^4 - 36*x^2 + 81 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 210) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + \beta_{5} q^{3} + 2 q^{4} - \beta_{3} q^{6} + ( - 2 \beta_{6} - \beta_{4} + \beta_{3} + 2 \beta_1) q^{7} + 2 \beta_1 q^{8} - 3 q^{9}+O(q^{10})$$ q + b1 * q^2 + b5 * q^3 + 2 * q^4 - b3 * q^6 + (-2*b6 - b4 + b3 + 2*b1) * q^7 + 2*b1 * q^8 - 3 * q^9 $$q + \beta_1 q^{2} + \beta_{5} q^{3} + 2 q^{4} - \beta_{3} q^{6} + ( - 2 \beta_{6} - \beta_{4} + \beta_{3} + 2 \beta_1) q^{7} + 2 \beta_1 q^{8} - 3 q^{9} + ( - \beta_{2} - 6 \beta_1 - 2) q^{11} + 2 \beta_{5} q^{12} + ( - \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{3}) q^{13} + (2 \beta_{7} - 2 \beta_{5} - \beta_{2} + 4) q^{14} + 4 q^{16} + (2 \beta_{7} - 6 \beta_{6}) q^{17} - 3 \beta_1 q^{18} + ( - 2 \beta_{7} - 2 \beta_{6} + 6 \beta_{5} - 7 \beta_{3}) q^{19} + ( - 3 \beta_{6} + 2 \beta_{4} - 2 \beta_{3} + 3 \beta_1) q^{21} + ( - 2 \beta_{4} - 2 \beta_1 - 12) q^{22} + ( - 2 \beta_{4} - 4 \beta_{2} + 8 \beta_1) q^{23} - 2 \beta_{3} q^{24} + ( - 2 \beta_{7} + 2 \beta_{6} - 4 \beta_{5} - 2 \beta_{3}) q^{26} - 3 \beta_{5} q^{27} + ( - 4 \beta_{6} - 2 \beta_{4} + 2 \beta_{3} + 4 \beta_1) q^{28} + ( - 4 \beta_{4} - 14 \beta_1 - 18) q^{29} + ( - 2 \beta_{7} + 2 \beta_{6} + 14 \beta_{5} - \beta_{3}) q^{31} + 4 \beta_1 q^{32} + (3 \beta_{7} - 2 \beta_{5} + 6 \beta_{3}) q^{33} + (6 \beta_{7} - 4 \beta_{6}) q^{34} - 6 q^{36} + ( - 8 \beta_{4} - 4 \beta_{2} - 10 \beta_1 + 6) q^{37} + (2 \beta_{7} + 4 \beta_{6} + 14 \beta_{5} - 6 \beta_{3}) q^{38} + ( - 2 \beta_{4} - \beta_{2} + 6 \beta_1 - 6) q^{39} + ( - 2 \beta_{7} + 8 \beta_{6} + 4 \beta_{5} + 4 \beta_{3}) q^{41} + (3 \beta_{7} + 4 \beta_{5} + 2 \beta_{2} + 6) q^{42} + (4 \beta_{4} - 8 \beta_{2} - 4 \beta_1 + 8) q^{43} + ( - 2 \beta_{2} - 12 \beta_1 - 4) q^{44} + ( - 8 \beta_{4} - 2 \beta_{2} + 16) q^{46} + (8 \beta_{7} - 4 \beta_{6} - 6 \beta_{5} - 14 \beta_{3}) q^{47} + 4 \beta_{5} q^{48} + (2 \beta_{7} + 12 \beta_{5} - 8 \beta_{2} - 3) q^{49} + (6 \beta_{4} + 2 \beta_{2}) q^{51} + ( - 2 \beta_{7} + 4 \beta_{6} + 4 \beta_{5} + 4 \beta_{3}) q^{52} + (14 \beta_{4} + 2 \beta_{2} + 11 \beta_1 - 16) q^{53} + 3 \beta_{3} q^{54} + (4 \beta_{7} - 4 \beta_{5} - 2 \beta_{2} + 8) q^{56} + (2 \beta_{4} - 2 \beta_{2} - 21 \beta_1 - 18) q^{57} + ( - 4 \beta_{2} - 18 \beta_1 - 28) q^{58} + (8 \beta_{7} + 24 \beta_{5} + 14 \beta_{3}) q^{59} + ( - 10 \beta_{7} - 16 \beta_{6} - 28 \beta_{5} + 16 \beta_{3}) q^{61} + ( - 2 \beta_{7} + 4 \beta_{6} + 2 \beta_{5} - 14 \beta_{3}) q^{62} + (6 \beta_{6} + 3 \beta_{4} - 3 \beta_{3} - 6 \beta_1) q^{63} + 8 q^{64} + ( - 6 \beta_{6} - 12 \beta_{5} + 2 \beta_{3}) q^{66} + (6 \beta_{4} + 4 \beta_{2} - 36 \beta_1 + 24) q^{67} + (4 \beta_{7} - 12 \beta_{6}) q^{68} + (12 \beta_{7} - 6 \beta_{6} - 8 \beta_{3}) q^{69} + ( - 4 \beta_{4} + 15 \beta_{2} + 14 \beta_1 + 22) q^{71} - 6 \beta_1 q^{72} + (9 \beta_{7} + 14 \beta_{6} + 38 \beta_{5} - 2 \beta_{3}) q^{73} + ( - 8 \beta_{4} - 8 \beta_{2} + 6 \beta_1 - 20) q^{74} + ( - 4 \beta_{7} - 4 \beta_{6} + 12 \beta_{5} - 14 \beta_{3}) q^{76} + ( - 12 \beta_{7} + 10 \beta_{6} + 12 \beta_{5} - 2 \beta_{4} - 12 \beta_{3} + \cdots - 24) q^{77}+ \cdots + (3 \beta_{2} + 18 \beta_1 + 6) q^{99}+O(q^{100})$$ q + b1 * q^2 + b5 * q^3 + 2 * q^4 - b3 * q^6 + (-2*b6 - b4 + b3 + 2*b1) * q^7 + 2*b1 * q^8 - 3 * q^9 + (-b2 - 6*b1 - 2) * q^11 + 2*b5 * q^12 + (-b7 + 2*b6 + 2*b5 + 2*b3) * q^13 + (2*b7 - 2*b5 - b2 + 4) * q^14 + 4 * q^16 + (2*b7 - 6*b6) * q^17 - 3*b1 * q^18 + (-2*b7 - 2*b6 + 6*b5 - 7*b3) * q^19 + (-3*b6 + 2*b4 - 2*b3 + 3*b1) * q^21 + (-2*b4 - 2*b1 - 12) * q^22 + (-2*b4 - 4*b2 + 8*b1) * q^23 - 2*b3 * q^24 + (-2*b7 + 2*b6 - 4*b5 - 2*b3) * q^26 - 3*b5 * q^27 + (-4*b6 - 2*b4 + 2*b3 + 4*b1) * q^28 + (-4*b4 - 14*b1 - 18) * q^29 + (-2*b7 + 2*b6 + 14*b5 - b3) * q^31 + 4*b1 * q^32 + (3*b7 - 2*b5 + 6*b3) * q^33 + (6*b7 - 4*b6) * q^34 - 6 * q^36 + (-8*b4 - 4*b2 - 10*b1 + 6) * q^37 + (2*b7 + 4*b6 + 14*b5 - 6*b3) * q^38 + (-2*b4 - b2 + 6*b1 - 6) * q^39 + (-2*b7 + 8*b6 + 4*b5 + 4*b3) * q^41 + (3*b7 + 4*b5 + 2*b2 + 6) * q^42 + (4*b4 - 8*b2 - 4*b1 + 8) * q^43 + (-2*b2 - 12*b1 - 4) * q^44 + (-8*b4 - 2*b2 + 16) * q^46 + (8*b7 - 4*b6 - 6*b5 - 14*b3) * q^47 + 4*b5 * q^48 + (2*b7 + 12*b5 - 8*b2 - 3) * q^49 + (6*b4 + 2*b2) * q^51 + (-2*b7 + 4*b6 + 4*b5 + 4*b3) * q^52 + (14*b4 + 2*b2 + 11*b1 - 16) * q^53 + 3*b3 * q^54 + (4*b7 - 4*b5 - 2*b2 + 8) * q^56 + (2*b4 - 2*b2 - 21*b1 - 18) * q^57 + (-4*b2 - 18*b1 - 28) * q^58 + (8*b7 + 24*b5 + 14*b3) * q^59 + (-10*b7 - 16*b6 - 28*b5 + 16*b3) * q^61 + (-2*b7 + 4*b6 + 2*b5 - 14*b3) * q^62 + (6*b6 + 3*b4 - 3*b3 - 6*b1) * q^63 + 8 * q^64 + (-6*b6 - 12*b5 + 2*b3) * q^66 + (6*b4 + 4*b2 - 36*b1 + 24) * q^67 + (4*b7 - 12*b6) * q^68 + (12*b7 - 6*b6 - 8*b3) * q^69 + (-4*b4 + 15*b2 + 14*b1 + 22) * q^71 - 6*b1 * q^72 + (9*b7 + 14*b6 + 38*b5 - 2*b3) * q^73 + (-8*b4 - 8*b2 + 6*b1 - 20) * q^74 + (-4*b7 - 4*b6 + 12*b5 - 14*b3) * q^76 + (-12*b7 + 10*b6 + 12*b5 - 2*b4 - 12*b3 + 6*b2 + 11*b1 - 24) * q^77 + (-2*b4 - 2*b2 - 6*b1 + 12) * q^78 + (-8*b2 + 16*b1 - 36) * q^79 + 9 * q^81 + (-8*b7 + 4*b6 - 8*b5 - 4*b3) * q^82 + (20*b7 + 4*b6 + 2*b5 - 6*b3) * q^83 + (-6*b6 + 4*b4 - 4*b3 + 6*b1) * q^84 + (-16*b4 + 4*b2 + 8*b1 - 8) * q^86 + (-12*b6 - 18*b5 + 14*b3) * q^87 + (-4*b4 - 4*b1 - 24) * q^88 + (32*b7 - 10*b6 + 4*b3) * q^89 + (-10*b7 - 2*b6 - 18*b5 + 6*b4 + b3 - 2*b2 + 16*b1 + 8) * q^91 + (-4*b4 - 8*b2 + 16*b1) * q^92 + (-2*b4 - 2*b2 - 3*b1 - 42) * q^93 + (4*b7 - 16*b6 + 28*b5 + 6*b3) * q^94 - 4*b3 * q^96 + (-3*b7 - 34*b6 + 30*b5 + 6*b3) * q^97 + (-4*b6 - 16*b4 - 12*b3 - 3*b1) * q^98 + (3*b2 + 18*b1 + 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 16 q^{4} - 24 q^{9}+O(q^{10})$$ 8 * q + 16 * q^4 - 24 * q^9 $$8 q + 16 q^{4} - 24 q^{9} - 16 q^{11} + 32 q^{14} + 32 q^{16} - 96 q^{22} - 144 q^{29} - 48 q^{36} + 48 q^{37} - 48 q^{39} + 48 q^{42} + 64 q^{43} - 32 q^{44} + 128 q^{46} - 24 q^{49} - 128 q^{53} + 64 q^{56} - 144 q^{57} - 224 q^{58} + 64 q^{64} + 192 q^{67} + 176 q^{71} - 160 q^{74} - 192 q^{77} + 96 q^{78} - 288 q^{79} + 72 q^{81} - 64 q^{86} - 192 q^{88} + 64 q^{91} - 336 q^{93} + 48 q^{99}+O(q^{100})$$ 8 * q + 16 * q^4 - 24 * q^9 - 16 * q^11 + 32 * q^14 + 32 * q^16 - 96 * q^22 - 144 * q^29 - 48 * q^36 + 48 * q^37 - 48 * q^39 + 48 * q^42 + 64 * q^43 - 32 * q^44 + 128 * q^46 - 24 * q^49 - 128 * q^53 + 64 * q^56 - 144 * q^57 - 224 * q^58 + 64 * q^64 + 192 * q^67 + 176 * q^71 - 160 * q^74 - 192 * q^77 + 96 * q^78 - 288 * q^79 + 72 * q^81 - 64 * q^86 - 192 * q^88 + 64 * q^91 - 336 * q^93 + 48 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4x^{6} + 7x^{4} - 36x^{2} + 81$$ :

 $$\beta_{1}$$ $$=$$ $$( -4\nu^{7} + 7\nu^{5} + 35\nu^{3} + 81\nu ) / 189$$ (-4*v^7 + 7*v^5 + 35*v^3 + 81*v) / 189 $$\beta_{2}$$ $$=$$ $$( -10\nu^{7} + 49\nu^{5} - 133\nu^{3} + 801\nu ) / 189$$ (-10*v^7 + 49*v^5 - 133*v^3 + 801*v) / 189 $$\beta_{3}$$ $$=$$ $$( 2\nu^{7} + \nu^{5} + 5\nu^{3} - 63\nu ) / 27$$ (2*v^7 + v^5 + 5*v^3 - 63*v) / 27 $$\beta_{4}$$ $$=$$ $$( -\nu^{6} + 4\nu^{4} + 2\nu^{2} + 18 ) / 9$$ (-v^6 + 4*v^4 + 2*v^2 + 18) / 9 $$\beta_{5}$$ $$=$$ $$( -8\nu^{6} + 14\nu^{4} - 56\nu^{2} + 225 ) / 63$$ (-8*v^6 + 14*v^4 - 56*v^2 + 225) / 63 $$\beta_{6}$$ $$=$$ $$( -\nu^{6} + 22 ) / 7$$ (-v^6 + 22) / 7 $$\beta_{7}$$ $$=$$ $$( 4\nu^{7} - 7\nu^{5} + 19\nu^{3} - 81\nu ) / 27$$ (4*v^7 - 7*v^5 + 19*v^3 - 81*v) / 27
 $$\nu$$ $$=$$ $$( \beta_{7} - \beta_{3} + \beta_{2} + \beta_1 ) / 4$$ (b7 - b3 + b2 + b1) / 4 $$\nu^{2}$$ $$=$$ $$( \beta_{6} - 2\beta_{5} + \beta_{4} + 2 ) / 2$$ (b6 - 2*b5 + b4 + 2) / 2 $$\nu^{3}$$ $$=$$ $$( \beta_{7} + 7\beta_1 ) / 2$$ (b7 + 7*b1) / 2 $$\nu^{4}$$ $$=$$ $$( -4\beta_{6} + \beta_{5} + 4\beta_{4} + 1 ) / 2$$ (-4*b6 + b5 + 4*b4 + 1) / 2 $$\nu^{5}$$ $$=$$ $$( -5\beta_{7} + 19\beta_{3} + 5\beta_{2} + 19\beta_1 ) / 4$$ (-5*b7 + 19*b3 + 5*b2 + 19*b1) / 4 $$\nu^{6}$$ $$=$$ $$-7\beta_{6} + 22$$ -7*b6 + 22 $$\nu^{7}$$ $$=$$ $$( 29\beta_{7} + 13\beta_{3} + 29\beta_{2} - 13\beta_1 ) / 4$$ (29*b7 + 13*b3 + 29*b2 - 13*b1) / 4

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$451$$ $$701$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
601.1
 −1.72286 − 0.178197i 1.01575 + 1.40294i −1.72286 + 0.178197i 1.01575 − 1.40294i 1.72286 + 0.178197i −1.01575 − 1.40294i 1.72286 − 0.178197i −1.01575 + 1.40294i
−1.41421 1.73205i 2.00000 0 2.44949i −6.70141 + 2.02265i −2.82843 −3.00000 0
601.2 −1.41421 1.73205i 2.00000 0 2.44949i 1.04456 6.92163i −2.82843 −3.00000 0
601.3 −1.41421 1.73205i 2.00000 0 2.44949i −6.70141 2.02265i −2.82843 −3.00000 0
601.4 −1.41421 1.73205i 2.00000 0 2.44949i 1.04456 + 6.92163i −2.82843 −3.00000 0
601.5 1.41421 1.73205i 2.00000 0 2.44949i −1.04456 + 6.92163i 2.82843 −3.00000 0
601.6 1.41421 1.73205i 2.00000 0 2.44949i 6.70141 2.02265i 2.82843 −3.00000 0
601.7 1.41421 1.73205i 2.00000 0 2.44949i −1.04456 6.92163i 2.82843 −3.00000 0
601.8 1.41421 1.73205i 2.00000 0 2.44949i 6.70141 + 2.02265i 2.82843 −3.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 601.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.3.f.b 8
5.b even 2 1 210.3.f.a 8
5.c odd 4 2 1050.3.h.b 16
7.b odd 2 1 inner 1050.3.f.b 8
15.d odd 2 1 630.3.f.c 8
20.d odd 2 1 1680.3.s.a 8
35.c odd 2 1 210.3.f.a 8
35.f even 4 2 1050.3.h.b 16
105.g even 2 1 630.3.f.c 8
140.c even 2 1 1680.3.s.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.f.a 8 5.b even 2 1
210.3.f.a 8 35.c odd 2 1
630.3.f.c 8 15.d odd 2 1
630.3.f.c 8 105.g even 2 1
1050.3.f.b 8 1.a even 1 1 trivial
1050.3.f.b 8 7.b odd 2 1 inner
1050.3.h.b 16 5.c odd 4 2
1050.3.h.b 16 35.f even 4 2
1680.3.s.a 8 20.d odd 2 1
1680.3.s.a 8 140.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1050, [\chi])$$:

 $$T_{11}^{4} + 8T_{11}^{3} - 180T_{11}^{2} - 784T_{11} + 964$$ T11^4 + 8*T11^3 - 180*T11^2 - 784*T11 + 964 $$T_{23}^{4} - 1336T_{23}^{2} - 15360T_{23} + 54544$$ T23^4 - 1336*T23^2 - 15360*T23 + 54544

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 2)^{4}$$
$3$ $$(T^{2} + 3)^{4}$$
$5$ $$T^{8}$$
$7$ $$T^{8} + 12 T^{6} - 2842 T^{4} + \cdots + 5764801$$
$11$ $$(T^{4} + 8 T^{3} - 180 T^{2} - 784 T + 964)^{2}$$
$13$ $$T^{8} + 264 T^{6} + \cdots + 14714896$$
$17$ $$(T^{4} + 440 T^{2} + 19600)^{2}$$
$19$ $$T^{8} + 1848 T^{6} + \cdots + 2869102096$$
$23$ $$(T^{4} - 1336 T^{2} - 15360 T + 54544)^{2}$$
$29$ $$(T^{4} + 72 T^{3} + 680 T^{2} + \cdots - 281456)^{2}$$
$31$ $$T^{8} + 2616 T^{6} + \cdots + 79120688656$$
$37$ $$(T^{4} - 24 T^{3} - 3064 T^{2} + \cdots - 883184)^{2}$$
$41$ $$T^{8} + 2016 T^{6} + \cdots + 7126061056$$
$43$ $$(T^{4} - 32 T^{3} - 4000 T^{2} + \cdots + 2654464)^{2}$$
$47$ $$T^{8} + 8016 T^{6} + \cdots + 1081300500736$$
$53$ $$(T^{4} + 64 T^{3} - 5068 T^{2} + \cdots + 3722116)^{2}$$
$59$ $$T^{8} + 14176 T^{6} + \cdots + 9016903929856$$
$61$ $$T^{8} + 24672 T^{6} + \cdots + 11\!\cdots\!76$$
$67$ $$(T^{4} - 96 T^{3} - 3768 T^{2} + \cdots - 7371504)^{2}$$
$71$ $$(T^{4} - 88 T^{3} - 11860 T^{2} + \cdots + 25706884)^{2}$$
$73$ $$T^{8} + 24584 T^{6} + \cdots + 483086161936$$
$79$ $$(T^{4} + 144 T^{3} + 2912 T^{2} + \cdots - 2641664)^{2}$$
$83$ $$T^{8} + \cdots + 177346111242496$$
$89$ $$T^{8} + 43344 T^{6} + \cdots + 86\!\cdots\!96$$
$97$ $$T^{8} + 35144 T^{6} + \cdots + 76019496215056$$