# Properties

 Label 1050.3.e.a Level $1050$ Weight $3$ Character orbit 1050.e Analytic conductor $28.610$ Analytic rank $0$ Dimension $4$ 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.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$28.6104277578$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{7})$$ Defining polynomial: $$x^{4} + 8x^{2} + 9$$ x^4 + 8*x^2 + 9 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 42) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{3} + \beta_1) q^{3} - 2 q^{4} + (\beta_{2} - 2) q^{6} - \beta_{3} q^{7} - 2 \beta_1 q^{8} + (2 \beta_{2} + 5) q^{9}+O(q^{10})$$ q + b1 * q^2 + (b3 + b1) * q^3 - 2 * q^4 + (b2 - 2) * q^6 - b3 * q^7 - 2*b1 * q^8 + (2*b2 + 5) * q^9 $$q + \beta_1 q^{2} + (\beta_{3} + \beta_1) q^{3} - 2 q^{4} + (\beta_{2} - 2) q^{6} - \beta_{3} q^{7} - 2 \beta_1 q^{8} + (2 \beta_{2} + 5) q^{9} + ( - 2 \beta_{2} + 5 \beta_1) q^{11} + ( - 2 \beta_{3} - 2 \beta_1) q^{12} + ( - 4 \beta_{3} - 10) q^{13} - \beta_{2} q^{14} + 4 q^{16} + (5 \beta_{2} - 2 \beta_1) q^{17} + ( - 4 \beta_{3} + 5 \beta_1) q^{18} - 16 q^{19} + ( - \beta_{2} - 7) q^{21} + (4 \beta_{3} - 10) q^{22} + ( - 10 \beta_{2} + \beta_1) q^{23} + ( - 2 \beta_{2} + 4) q^{24} + ( - 4 \beta_{2} - 10 \beta_1) q^{26} + (\beta_{3} + 19 \beta_1) q^{27} + 2 \beta_{3} q^{28} + (2 \beta_{2} - 20 \beta_1) q^{29} + (10 \beta_{3} - 32) q^{31} + 4 \beta_1 q^{32} + (4 \beta_{3} + 5 \beta_{2} - 14 \beta_1 - 10) q^{33} + ( - 10 \beta_{3} + 4) q^{34} + ( - 4 \beta_{2} - 10) q^{36} - 20 q^{37} - 16 \beta_1 q^{38} + ( - 10 \beta_{3} - 4 \beta_{2} - 10 \beta_1 - 28) q^{39} + (9 \beta_{2} + 30 \beta_1) q^{41} + (2 \beta_{3} - 7 \beta_1) q^{42} + ( - 12 \beta_{3} - 20) q^{43} + (4 \beta_{2} - 10 \beta_1) q^{44} + (20 \beta_{3} - 2) q^{46} - 6 \beta_1 q^{47} + (4 \beta_{3} + 4 \beta_1) q^{48} + 7 q^{49} + ( - 10 \beta_{3} - 2 \beta_{2} + 35 \beta_1 + 4) q^{51} + (8 \beta_{3} + 20) q^{52} - 36 \beta_1 q^{53} + (\beta_{2} - 38) q^{54} + 2 \beta_{2} q^{56} + ( - 16 \beta_{3} - 16 \beta_1) q^{57} + ( - 4 \beta_{3} + 40) q^{58} + (8 \beta_{2} - 20 \beta_1) q^{59} + ( - 20 \beta_{3} - 14) q^{61} + (10 \beta_{2} - 32 \beta_1) q^{62} + ( - 5 \beta_{3} - 14 \beta_1) q^{63} - 8 q^{64} + ( - 10 \beta_{3} + 4 \beta_{2} - 10 \beta_1 + 28) q^{66} + (4 \beta_{3} - 60) q^{67} + ( - 10 \beta_{2} + 4 \beta_1) q^{68} + (20 \beta_{3} + \beta_{2} - 70 \beta_1 - 2) q^{69} + ( - 14 \beta_{2} - 25 \beta_1) q^{71} + (8 \beta_{3} - 10 \beta_1) q^{72} + (16 \beta_{3} - 30) q^{73} - 20 \beta_1 q^{74} + 32 q^{76} + ( - 5 \beta_{2} + 14 \beta_1) q^{77} + (8 \beta_{3} - 10 \beta_{2} - 28 \beta_1 + 20) q^{78} + ( - 20 \beta_{3} - 32) q^{79} + (20 \beta_{2} - 31) q^{81} + ( - 18 \beta_{3} - 60) q^{82} + (20 \beta_{2} - 50 \beta_1) q^{83} + (2 \beta_{2} + 14) q^{84} + ( - 12 \beta_{2} - 20 \beta_1) q^{86} + ( - 4 \beta_{3} - 20 \beta_{2} + 14 \beta_1 + 40) q^{87} + ( - 8 \beta_{3} + 20) q^{88} + (3 \beta_{2} - 30 \beta_1) q^{89} + (10 \beta_{3} + 28) q^{91} + (20 \beta_{2} - 2 \beta_1) q^{92} + ( - 32 \beta_{3} + 10 \beta_{2} - 32 \beta_1 + 70) q^{93} + 12 q^{94} + (4 \beta_{2} - 8) q^{96} + ( - 8 \beta_{3} + 90) q^{97} + 7 \beta_1 q^{98} + ( - 20 \beta_{3} - 10 \beta_{2} + 25 \beta_1 + 56) q^{99}+O(q^{100})$$ q + b1 * q^2 + (b3 + b1) * q^3 - 2 * q^4 + (b2 - 2) * q^6 - b3 * q^7 - 2*b1 * q^8 + (2*b2 + 5) * q^9 + (-2*b2 + 5*b1) * q^11 + (-2*b3 - 2*b1) * q^12 + (-4*b3 - 10) * q^13 - b2 * q^14 + 4 * q^16 + (5*b2 - 2*b1) * q^17 + (-4*b3 + 5*b1) * q^18 - 16 * q^19 + (-b2 - 7) * q^21 + (4*b3 - 10) * q^22 + (-10*b2 + b1) * q^23 + (-2*b2 + 4) * q^24 + (-4*b2 - 10*b1) * q^26 + (b3 + 19*b1) * q^27 + 2*b3 * q^28 + (2*b2 - 20*b1) * q^29 + (10*b3 - 32) * q^31 + 4*b1 * q^32 + (4*b3 + 5*b2 - 14*b1 - 10) * q^33 + (-10*b3 + 4) * q^34 + (-4*b2 - 10) * q^36 - 20 * q^37 - 16*b1 * q^38 + (-10*b3 - 4*b2 - 10*b1 - 28) * q^39 + (9*b2 + 30*b1) * q^41 + (2*b3 - 7*b1) * q^42 + (-12*b3 - 20) * q^43 + (4*b2 - 10*b1) * q^44 + (20*b3 - 2) * q^46 - 6*b1 * q^47 + (4*b3 + 4*b1) * q^48 + 7 * q^49 + (-10*b3 - 2*b2 + 35*b1 + 4) * q^51 + (8*b3 + 20) * q^52 - 36*b1 * q^53 + (b2 - 38) * q^54 + 2*b2 * q^56 + (-16*b3 - 16*b1) * q^57 + (-4*b3 + 40) * q^58 + (8*b2 - 20*b1) * q^59 + (-20*b3 - 14) * q^61 + (10*b2 - 32*b1) * q^62 + (-5*b3 - 14*b1) * q^63 - 8 * q^64 + (-10*b3 + 4*b2 - 10*b1 + 28) * q^66 + (4*b3 - 60) * q^67 + (-10*b2 + 4*b1) * q^68 + (20*b3 + b2 - 70*b1 - 2) * q^69 + (-14*b2 - 25*b1) * q^71 + (8*b3 - 10*b1) * q^72 + (16*b3 - 30) * q^73 - 20*b1 * q^74 + 32 * q^76 + (-5*b2 + 14*b1) * q^77 + (8*b3 - 10*b2 - 28*b1 + 20) * q^78 + (-20*b3 - 32) * q^79 + (20*b2 - 31) * q^81 + (-18*b3 - 60) * q^82 + (20*b2 - 50*b1) * q^83 + (2*b2 + 14) * q^84 + (-12*b2 - 20*b1) * q^86 + (-4*b3 - 20*b2 + 14*b1 + 40) * q^87 + (-8*b3 + 20) * q^88 + (3*b2 - 30*b1) * q^89 + (10*b3 + 28) * q^91 + (20*b2 - 2*b1) * q^92 + (-32*b3 + 10*b2 - 32*b1 + 70) * q^93 + 12 * q^94 + (4*b2 - 8) * q^96 + (-8*b3 + 90) * q^97 + 7*b1 * q^98 + (-20*b3 - 10*b2 + 25*b1 + 56) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{4} - 8 q^{6} + 20 q^{9}+O(q^{10})$$ 4 * q - 8 * q^4 - 8 * q^6 + 20 * q^9 $$4 q - 8 q^{4} - 8 q^{6} + 20 q^{9} - 40 q^{13} + 16 q^{16} - 64 q^{19} - 28 q^{21} - 40 q^{22} + 16 q^{24} - 128 q^{31} - 40 q^{33} + 16 q^{34} - 40 q^{36} - 80 q^{37} - 112 q^{39} - 80 q^{43} - 8 q^{46} + 28 q^{49} + 16 q^{51} + 80 q^{52} - 152 q^{54} + 160 q^{58} - 56 q^{61} - 32 q^{64} + 112 q^{66} - 240 q^{67} - 8 q^{69} - 120 q^{73} + 128 q^{76} + 80 q^{78} - 128 q^{79} - 124 q^{81} - 240 q^{82} + 56 q^{84} + 160 q^{87} + 80 q^{88} + 112 q^{91} + 280 q^{93} + 48 q^{94} - 32 q^{96} + 360 q^{97} + 224 q^{99}+O(q^{100})$$ 4 * q - 8 * q^4 - 8 * q^6 + 20 * q^9 - 40 * q^13 + 16 * q^16 - 64 * q^19 - 28 * q^21 - 40 * q^22 + 16 * q^24 - 128 * q^31 - 40 * q^33 + 16 * q^34 - 40 * q^36 - 80 * q^37 - 112 * q^39 - 80 * q^43 - 8 * q^46 + 28 * q^49 + 16 * q^51 + 80 * q^52 - 152 * q^54 + 160 * q^58 - 56 * q^61 - 32 * q^64 + 112 * q^66 - 240 * q^67 - 8 * q^69 - 120 * q^73 + 128 * q^76 + 80 * q^78 - 128 * q^79 - 124 * q^81 - 240 * q^82 + 56 * q^84 + 160 * q^87 + 80 * q^88 + 112 * q^91 + 280 * q^93 + 48 * q^94 - 32 * q^96 + 360 * q^97 + 224 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 8x^{2} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 5\nu ) / 3$$ (v^3 + 5*v) / 3 $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 11\nu ) / 3$$ (v^3 + 11*v) / 3 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 4$$ v^2 + 4
 $$\nu$$ $$=$$ $$( \beta_{2} - \beta_1 ) / 2$$ (b2 - b1) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 4$$ b3 - 4 $$\nu^{3}$$ $$=$$ $$( -5\beta_{2} + 11\beta_1 ) / 2$$ (-5*b2 + 11*b1) / 2

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
701.1
 2.57794i − 1.16372i − 2.57794i 1.16372i
1.41421i −2.64575 1.41421i −2.00000 0 −2.00000 + 3.74166i 2.64575 2.82843i 5.00000 + 7.48331i 0
701.2 1.41421i 2.64575 1.41421i −2.00000 0 −2.00000 3.74166i −2.64575 2.82843i 5.00000 7.48331i 0
701.3 1.41421i −2.64575 + 1.41421i −2.00000 0 −2.00000 3.74166i 2.64575 2.82843i 5.00000 7.48331i 0
701.4 1.41421i 2.64575 + 1.41421i −2.00000 0 −2.00000 + 3.74166i −2.64575 2.82843i 5.00000 + 7.48331i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.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.e.a 4
3.b odd 2 1 inner 1050.3.e.a 4
5.b even 2 1 42.3.b.a 4
5.c odd 4 2 1050.3.c.a 8
15.d odd 2 1 42.3.b.a 4
15.e even 4 2 1050.3.c.a 8
20.d odd 2 1 336.3.d.b 4
35.c odd 2 1 294.3.b.h 4
35.i odd 6 2 294.3.h.g 8
35.j even 6 2 294.3.h.d 8
40.e odd 2 1 1344.3.d.e 4
40.f even 2 1 1344.3.d.c 4
45.h odd 6 2 1134.3.q.a 8
45.j even 6 2 1134.3.q.a 8
60.h even 2 1 336.3.d.b 4
105.g even 2 1 294.3.b.h 4
105.o odd 6 2 294.3.h.d 8
105.p even 6 2 294.3.h.g 8
120.i odd 2 1 1344.3.d.c 4
120.m even 2 1 1344.3.d.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.3.b.a 4 5.b even 2 1
42.3.b.a 4 15.d odd 2 1
294.3.b.h 4 35.c odd 2 1
294.3.b.h 4 105.g even 2 1
294.3.h.d 8 35.j even 6 2
294.3.h.d 8 105.o odd 6 2
294.3.h.g 8 35.i odd 6 2
294.3.h.g 8 105.p even 6 2
336.3.d.b 4 20.d odd 2 1
336.3.d.b 4 60.h even 2 1
1050.3.c.a 8 5.c odd 4 2
1050.3.c.a 8 15.e even 4 2
1050.3.e.a 4 1.a even 1 1 trivial
1050.3.e.a 4 3.b odd 2 1 inner
1134.3.q.a 8 45.h odd 6 2
1134.3.q.a 8 45.j even 6 2
1344.3.d.c 4 40.f even 2 1
1344.3.d.c 4 120.i odd 2 1
1344.3.d.e 4 40.e odd 2 1
1344.3.d.e 4 120.m even 2 1

## Hecke kernels

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

 $$T_{11}^{4} + 212T_{11}^{2} + 36$$ T11^4 + 212*T11^2 + 36 $$T_{13}^{2} + 20T_{13} - 12$$ T13^2 + 20*T13 - 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 2)^{2}$$
$3$ $$T^{4} - 10T^{2} + 81$$
$5$ $$T^{4}$$
$7$ $$(T^{2} - 7)^{2}$$
$11$ $$T^{4} + 212T^{2} + 36$$
$13$ $$(T^{2} + 20 T - 12)^{2}$$
$17$ $$T^{4} + 716 T^{2} + 116964$$
$19$ $$(T + 16)^{4}$$
$23$ $$T^{4} + 2804 T^{2} + \cdots + 1954404$$
$29$ $$T^{4} + 1712 T^{2} + 553536$$
$31$ $$(T^{2} + 64 T + 324)^{2}$$
$37$ $$(T + 20)^{4}$$
$41$ $$T^{4} + 5868 T^{2} + 443556$$
$43$ $$(T^{2} + 40 T - 608)^{2}$$
$47$ $$(T^{2} + 72)^{2}$$
$53$ $$(T^{2} + 2592)^{2}$$
$59$ $$T^{4} + 3392 T^{2} + 9216$$
$61$ $$(T^{2} + 28 T - 2604)^{2}$$
$67$ $$(T^{2} + 120 T + 3488)^{2}$$
$71$ $$T^{4} + 7988 T^{2} + \cdots + 2232036$$
$73$ $$(T^{2} + 60 T - 892)^{2}$$
$79$ $$(T^{2} + 64 T - 1776)^{2}$$
$83$ $$T^{4} + 21200 T^{2} + \cdots + 360000$$
$89$ $$T^{4} + 3852 T^{2} + \cdots + 2802276$$
$97$ $$(T^{2} - 180 T + 7652)^{2}$$