# Properties

 Label 1050.3.c.a Level $1050$ Weight $3$ Character orbit 1050.c Analytic conductor $28.610$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1050.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$28.6104277578$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.157351936.1 Defining polynomial: $$x^{8} + x^{4} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{8}\cdot 5^{4}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{2} + ( -\beta_{1} + \beta_{3} ) q^{3} + 2 q^{4} + ( -2 + \beta_{5} ) q^{6} -\beta_{1} q^{7} -2 \beta_{3} q^{8} + ( -5 - 2 \beta_{5} ) q^{9} +O(q^{10})$$ $$q -\beta_{3} q^{2} + ( -\beta_{1} + \beta_{3} ) q^{3} + 2 q^{4} + ( -2 + \beta_{5} ) q^{6} -\beta_{1} q^{7} -2 \beta_{3} q^{8} + ( -5 - 2 \beta_{5} ) q^{9} + ( -2 \beta_{5} - \beta_{6} ) q^{11} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{12} + ( 4 \beta_{1} - \beta_{2} ) q^{13} + \beta_{5} q^{14} + 4 q^{16} + ( 2 \beta_{3} + \beta_{7} ) q^{17} + ( 4 \beta_{1} + 5 \beta_{3} ) q^{18} + 16 q^{19} + ( -7 - \beta_{5} ) q^{21} + ( 4 \beta_{1} + \beta_{2} ) q^{22} + ( \beta_{3} + 2 \beta_{7} ) q^{23} + ( -4 + 2 \beta_{5} ) q^{24} + ( -4 \beta_{5} + 2 \beta_{6} ) q^{26} + ( \beta_{1} - 19 \beta_{3} ) q^{27} -2 \beta_{1} q^{28} + ( -2 \beta_{5} - 4 \beta_{6} ) q^{29} + ( -32 - \beta_{4} ) q^{31} -4 \beta_{3} q^{32} + ( -4 \beta_{1} - \beta_{2} - 14 \beta_{3} - \beta_{7} ) q^{33} + ( -4 - \beta_{4} ) q^{34} + ( -10 - 4 \beta_{5} ) q^{36} + 2 \beta_{2} q^{37} -16 \beta_{3} q^{38} + ( 28 - \beta_{4} + 4 \beta_{5} - 2 \beta_{6} ) q^{39} + ( 9 \beta_{5} - 6 \beta_{6} ) q^{41} + ( 2 \beta_{1} + 7 \beta_{3} ) q^{42} + ( 12 \beta_{1} - 2 \beta_{2} ) q^{43} + ( -4 \beta_{5} - 2 \beta_{6} ) q^{44} + ( -2 - 2 \beta_{4} ) q^{46} + 6 \beta_{3} q^{47} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{48} -7 q^{49} + ( 4 + \beta_{4} - 2 \beta_{5} - 7 \beta_{6} ) q^{51} + ( 8 \beta_{1} - 2 \beta_{2} ) q^{52} -36 \beta_{3} q^{53} + ( 38 - \beta_{5} ) q^{54} + 2 \beta_{5} q^{56} + ( -16 \beta_{1} + 16 \beta_{3} ) q^{57} + ( 4 \beta_{1} + 4 \beta_{2} ) q^{58} + ( -8 \beta_{5} - 4 \beta_{6} ) q^{59} + ( -14 + 2 \beta_{4} ) q^{61} + ( 32 \beta_{3} + 2 \beta_{7} ) q^{62} + ( 5 \beta_{1} - 14 \beta_{3} ) q^{63} + 8 q^{64} + ( 28 + \beta_{4} + 4 \beta_{5} + 2 \beta_{6} ) q^{66} + ( 4 \beta_{1} + 6 \beta_{2} ) q^{67} + ( 4 \beta_{3} + 2 \beta_{7} ) q^{68} + ( 2 + 2 \beta_{4} - \beta_{5} - 14 \beta_{6} ) q^{69} + ( -14 \beta_{5} + 5 \beta_{6} ) q^{71} + ( 8 \beta_{1} + 10 \beta_{3} ) q^{72} + ( -16 \beta_{1} - 3 \beta_{2} ) q^{73} -4 \beta_{6} q^{74} + 32 q^{76} + ( -14 \beta_{3} - \beta_{7} ) q^{77} + ( -8 \beta_{1} + 2 \beta_{2} - 28 \beta_{3} + 2 \beta_{7} ) q^{78} + ( 32 - 2 \beta_{4} ) q^{79} + ( -31 + 20 \beta_{5} ) q^{81} + ( -18 \beta_{1} + 6 \beta_{2} ) q^{82} + ( -50 \beta_{3} - 4 \beta_{7} ) q^{83} + ( -14 - 2 \beta_{5} ) q^{84} + ( -12 \beta_{5} + 4 \beta_{6} ) q^{86} + ( -4 \beta_{1} - 4 \beta_{2} - 14 \beta_{3} - 4 \beta_{7} ) q^{87} + ( 8 \beta_{1} + 2 \beta_{2} ) q^{88} + ( -3 \beta_{5} - 6 \beta_{6} ) q^{89} + ( 28 - \beta_{4} ) q^{91} + ( 2 \beta_{3} + 4 \beta_{7} ) q^{92} + ( 32 \beta_{1} + 7 \beta_{2} - 32 \beta_{3} - 2 \beta_{7} ) q^{93} -12 q^{94} + ( -8 + 4 \beta_{5} ) q^{96} + ( -8 \beta_{1} - 9 \beta_{2} ) q^{97} + 7 \beta_{3} q^{98} + ( -56 - 2 \beta_{4} + 10 \beta_{5} + 5 \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 16 q^{4} - 16 q^{6} - 40 q^{9} + O(q^{10})$$ $$8 q + 16 q^{4} - 16 q^{6} - 40 q^{9} + 32 q^{16} + 128 q^{19} - 56 q^{21} - 32 q^{24} - 256 q^{31} - 32 q^{34} - 80 q^{36} + 224 q^{39} - 16 q^{46} - 56 q^{49} + 32 q^{51} + 304 q^{54} - 112 q^{61} + 64 q^{64} + 224 q^{66} + 16 q^{69} + 256 q^{76} + 256 q^{79} - 248 q^{81} - 112 q^{84} + 224 q^{91} - 96 q^{94} - 64 q^{96} - 448 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + x^{4} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$2 \nu^{4} + 1$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$5 \nu^{6} + 25 \nu^{2}$$$$)/6$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{7} - 4 \nu^{5} + 7 \nu^{3} + 4 \nu$$$$)/24$$ $$\beta_{4}$$ $$=$$ $$($$$$-5 \nu^{6} + 15 \nu^{2}$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$($$$$5 \nu^{7} + 4 \nu^{5} + 13 \nu^{3} + 44 \nu$$$$)/24$$ $$\beta_{6}$$ $$=$$ $$($$$$-5 \nu^{7} + 20 \nu^{5} + 35 \nu^{3} - 20 \nu$$$$)/24$$ $$\beta_{7}$$ $$=$$ $$($$$$-25 \nu^{7} + 20 \nu^{5} - 65 \nu^{3} + 220 \nu$$$$)/24$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} - \beta_{6} + 5 \beta_{5} + 5 \beta_{3}$$$$)/20$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{4} + 3 \beta_{2}$$$$)/20$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{7} + 5 \beta_{6} + 5 \beta_{5} + 25 \beta_{3}$$$$)/20$$ $$\nu^{4}$$ $$=$$ $$($$$$3 \beta_{1} - 1$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$\beta_{7} + 11 \beta_{6} + 5 \beta_{5} - 55 \beta_{3}$$$$)/20$$ $$\nu^{6}$$ $$=$$ $$($$$$-5 \beta_{4} + 9 \beta_{2}$$$$)/20$$ $$\nu^{7}$$ $$=$$ $$($$$$-7 \beta_{7} - 13 \beta_{6} + 35 \beta_{5} - 65 \beta_{3}$$$$)/20$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 1.28897 + 0.581861i −0.581861 + 1.28897i −0.581861 − 1.28897i 1.28897 − 0.581861i 0.581861 − 1.28897i −1.28897 − 0.581861i −1.28897 + 0.581861i 0.581861 + 1.28897i
−1.41421 1.41421 2.64575i 2.00000 0 −2.00000 + 3.74166i 2.64575i −2.82843 −5.00000 7.48331i 0
449.2 −1.41421 1.41421 2.64575i 2.00000 0 −2.00000 + 3.74166i 2.64575i −2.82843 −5.00000 7.48331i 0
449.3 −1.41421 1.41421 + 2.64575i 2.00000 0 −2.00000 3.74166i 2.64575i −2.82843 −5.00000 + 7.48331i 0
449.4 −1.41421 1.41421 + 2.64575i 2.00000 0 −2.00000 3.74166i 2.64575i −2.82843 −5.00000 + 7.48331i 0
449.5 1.41421 −1.41421 2.64575i 2.00000 0 −2.00000 3.74166i 2.64575i 2.82843 −5.00000 + 7.48331i 0
449.6 1.41421 −1.41421 2.64575i 2.00000 0 −2.00000 3.74166i 2.64575i 2.82843 −5.00000 + 7.48331i 0
449.7 1.41421 −1.41421 + 2.64575i 2.00000 0 −2.00000 + 3.74166i 2.64575i 2.82843 −5.00000 7.48331i 0
449.8 1.41421 −1.41421 + 2.64575i 2.00000 0 −2.00000 + 3.74166i 2.64575i 2.82843 −5.00000 7.48331i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 449.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d 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.c.a 8
3.b odd 2 1 inner 1050.3.c.a 8
5.b even 2 1 inner 1050.3.c.a 8
5.c odd 4 1 42.3.b.a 4
5.c odd 4 1 1050.3.e.a 4
15.d odd 2 1 inner 1050.3.c.a 8
15.e even 4 1 42.3.b.a 4
15.e even 4 1 1050.3.e.a 4
20.e even 4 1 336.3.d.b 4
35.f even 4 1 294.3.b.h 4
35.k even 12 2 294.3.h.g 8
35.l odd 12 2 294.3.h.d 8
40.i odd 4 1 1344.3.d.c 4
40.k even 4 1 1344.3.d.e 4
45.k odd 12 2 1134.3.q.a 8
45.l even 12 2 1134.3.q.a 8
60.l odd 4 1 336.3.d.b 4
105.k odd 4 1 294.3.b.h 4
105.w odd 12 2 294.3.h.g 8
105.x even 12 2 294.3.h.d 8
120.q odd 4 1 1344.3.d.e 4
120.w even 4 1 1344.3.d.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.3.b.a 4 5.c odd 4 1
42.3.b.a 4 15.e even 4 1
294.3.b.h 4 35.f even 4 1
294.3.b.h 4 105.k odd 4 1
294.3.h.d 8 35.l odd 12 2
294.3.h.d 8 105.x even 12 2
294.3.h.g 8 35.k even 12 2
294.3.h.g 8 105.w odd 12 2
336.3.d.b 4 20.e even 4 1
336.3.d.b 4 60.l odd 4 1
1050.3.c.a 8 1.a even 1 1 trivial
1050.3.c.a 8 3.b odd 2 1 inner
1050.3.c.a 8 5.b even 2 1 inner
1050.3.c.a 8 15.d odd 2 1 inner
1050.3.e.a 4 5.c odd 4 1
1050.3.e.a 4 15.e even 4 1
1134.3.q.a 8 45.k odd 12 2
1134.3.q.a 8 45.l even 12 2
1344.3.d.c 4 40.i odd 4 1
1344.3.d.c 4 120.w even 4 1
1344.3.d.e 4 40.k even 4 1
1344.3.d.e 4 120.q odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{4} + 212 T_{11}^{2} + 36$$ acting on $$S_{3}^{\mathrm{new}}(1050, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -2 + T^{2} )^{4}$$
$3$ $$( 81 + 10 T^{2} + T^{4} )^{2}$$
$5$ $$T^{8}$$
$7$ $$( 7 + T^{2} )^{4}$$
$11$ $$( 36 + 212 T^{2} + T^{4} )^{2}$$
$13$ $$( 144 + 424 T^{2} + T^{4} )^{2}$$
$17$ $$( 116964 - 716 T^{2} + T^{4} )^{2}$$
$19$ $$( -16 + T )^{8}$$
$23$ $$( 1954404 - 2804 T^{2} + T^{4} )^{2}$$
$29$ $$( 553536 + 1712 T^{2} + T^{4} )^{2}$$
$31$ $$( 324 + 64 T + T^{2} )^{4}$$
$37$ $$( 400 + T^{2} )^{4}$$
$41$ $$( 443556 + 5868 T^{2} + T^{4} )^{2}$$
$43$ $$( 369664 + 2816 T^{2} + T^{4} )^{2}$$
$47$ $$( -72 + T^{2} )^{4}$$
$53$ $$( -2592 + T^{2} )^{4}$$
$59$ $$( 9216 + 3392 T^{2} + T^{4} )^{2}$$
$61$ $$( -2604 + 28 T + T^{2} )^{4}$$
$67$ $$( 12166144 + 7424 T^{2} + T^{4} )^{2}$$
$71$ $$( 2232036 + 7988 T^{2} + T^{4} )^{2}$$
$73$ $$( 795664 + 5384 T^{2} + T^{4} )^{2}$$
$79$ $$( -1776 - 64 T + T^{2} )^{4}$$
$83$ $$( 360000 - 21200 T^{2} + T^{4} )^{2}$$
$89$ $$( 2802276 + 3852 T^{2} + T^{4} )^{2}$$
$97$ $$( 58553104 + 17096 T^{2} + T^{4} )^{2}$$