# Properties

 Label 1050.3.h.a Level $1050$ Weight $3$ Character orbit 1050.h 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.h (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$28.6104277578$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{8}$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} - \beta_{2} q^{3} - 2 q^{4} - \beta_{7} q^{6} + (\beta_{5} - 2 \beta_{3} - \beta_{2} + 3 \beta_1) q^{7} + 2 \beta_1 q^{8} + 3 q^{9}+O(q^{10})$$ q - b1 * q^2 - b2 * q^3 - 2 * q^4 - b7 * q^6 + (b5 - 2*b3 - b2 + 3*b1) * q^7 + 2*b1 * q^8 + 3 * q^9 $$q - \beta_1 q^{2} - \beta_{2} q^{3} - 2 q^{4} - \beta_{7} q^{6} + (\beta_{5} - 2 \beta_{3} - \beta_{2} + 3 \beta_1) q^{7} + 2 \beta_1 q^{8} + 3 q^{9} + ( - 3 \beta_{6} - 6) q^{11} + 2 \beta_{2} q^{12} + ( - 2 \beta_{3} + 8 \beta_{2}) q^{13} + ( - \beta_{7} - 2 \beta_{6} - 2 \beta_{4} + 6) q^{14} + 4 q^{16} + (11 \beta_{3} + 2 \beta_{2}) q^{17} - 3 \beta_1 q^{18} + ( - 8 \beta_{7} + \beta_{4}) q^{19} + (3 \beta_{7} + 6 \beta_{6} - \beta_{4} + 3) q^{21} + ( - 3 \beta_{5} + 6 \beta_1) q^{22} + ( - 3 \beta_{5} - 9 \beta_1) q^{23} + 2 \beta_{7} q^{24} + (8 \beta_{7} - 2 \beta_{4}) q^{26} - 3 \beta_{2} q^{27} + ( - 2 \beta_{5} + 4 \beta_{3} + 2 \beta_{2} - 6 \beta_1) q^{28} - 30 q^{29} + ( - 12 \beta_{7} - 6 \beta_{4}) q^{31} - 4 \beta_1 q^{32} + (3 \beta_{3} + 6 \beta_{2}) q^{33} + (2 \beta_{7} + 11 \beta_{4}) q^{34} - 6 q^{36} + ( - 10 \beta_{5} - 36 \beta_1) q^{37} + ( - 2 \beta_{3} + 16 \beta_{2}) q^{38} + (6 \beta_{6} - 24) q^{39} + ( - 7 \beta_{7} + 7 \beta_{4}) q^{41} + (6 \beta_{5} + 2 \beta_{3} - 6 \beta_{2} - 3 \beta_1) q^{42} + (16 \beta_{5} - 30 \beta_1) q^{43} + (6 \beta_{6} + 12) q^{44} + (6 \beta_{6} - 18) q^{46} + ( - 4 \beta_{3} - 28 \beta_{2}) q^{47} - 4 \beta_{2} q^{48} + ( - 2 \beta_{7} + 24 \beta_{6} + 10 \beta_{4} + 5) q^{49} + ( - 33 \beta_{6} - 6) q^{51} + (4 \beta_{3} - 16 \beta_{2}) q^{52} + (27 \beta_{5} - 12 \beta_1) q^{53} - 3 \beta_{7} q^{54} + (2 \beta_{7} + 4 \beta_{6} + 4 \beta_{4} - 12) q^{56} + ( - 3 \beta_{5} - 24 \beta_1) q^{57} + 30 \beta_1 q^{58} + ( - 20 \beta_{7} + 14 \beta_{4}) q^{59} + ( - 4 \beta_{7} + 2 \beta_{4}) q^{61} + (12 \beta_{3} + 24 \beta_{2}) q^{62} + (3 \beta_{5} - 6 \beta_{3} - 3 \beta_{2} + 9 \beta_1) q^{63} - 8 q^{64} + (6 \beta_{7} + 3 \beta_{4}) q^{66} + (22 \beta_{5} + 12 \beta_1) q^{67} + ( - 22 \beta_{3} - 4 \beta_{2}) q^{68} + ( - 9 \beta_{7} + 3 \beta_{4}) q^{69} + (57 \beta_{6} - 30) q^{71} + 6 \beta_1 q^{72} + (26 \beta_{3} + 4 \beta_{2}) q^{73} + (20 \beta_{6} - 72) q^{74} + (16 \beta_{7} - 2 \beta_{4}) q^{76} + (3 \beta_{5} + 15 \beta_{3} + 18 \beta_{2} - 12 \beta_1) q^{77} + (6 \beta_{5} + 24 \beta_1) q^{78} + ( - 72 \beta_{6} - 32) q^{79} + 9 q^{81} + ( - 14 \beta_{3} + 14 \beta_{2}) q^{82} + (20 \beta_{3} + 32 \beta_{2}) q^{83} + ( - 6 \beta_{7} - 12 \beta_{6} + 2 \beta_{4} - 6) q^{84} + ( - 32 \beta_{6} - 60) q^{86} + 30 \beta_{2} q^{87} + (6 \beta_{5} - 12 \beta_1) q^{88} + (21 \beta_{7} + 27 \beta_{4}) q^{89} + ( - 28 \beta_{7} - 42 \beta_{6} + 14 \beta_{4}) q^{91} + (6 \beta_{5} + 18 \beta_1) q^{92} + (18 \beta_{5} - 36 \beta_1) q^{93} + ( - 28 \beta_{7} - 4 \beta_{4}) q^{94} - 4 \beta_{7} q^{96} + ( - 10 \beta_{3} - 44 \beta_{2}) q^{97} + (24 \beta_{5} - 20 \beta_{3} + 4 \beta_{2} - 5 \beta_1) q^{98} + ( - 9 \beta_{6} - 18) q^{99}+O(q^{100})$$ q - b1 * q^2 - b2 * q^3 - 2 * q^4 - b7 * q^6 + (b5 - 2*b3 - b2 + 3*b1) * q^7 + 2*b1 * q^8 + 3 * q^9 + (-3*b6 - 6) * q^11 + 2*b2 * q^12 + (-2*b3 + 8*b2) * q^13 + (-b7 - 2*b6 - 2*b4 + 6) * q^14 + 4 * q^16 + (11*b3 + 2*b2) * q^17 - 3*b1 * q^18 + (-8*b7 + b4) * q^19 + (3*b7 + 6*b6 - b4 + 3) * q^21 + (-3*b5 + 6*b1) * q^22 + (-3*b5 - 9*b1) * q^23 + 2*b7 * q^24 + (8*b7 - 2*b4) * q^26 - 3*b2 * q^27 + (-2*b5 + 4*b3 + 2*b2 - 6*b1) * q^28 - 30 * q^29 + (-12*b7 - 6*b4) * q^31 - 4*b1 * q^32 + (3*b3 + 6*b2) * q^33 + (2*b7 + 11*b4) * q^34 - 6 * q^36 + (-10*b5 - 36*b1) * q^37 + (-2*b3 + 16*b2) * q^38 + (6*b6 - 24) * q^39 + (-7*b7 + 7*b4) * q^41 + (6*b5 + 2*b3 - 6*b2 - 3*b1) * q^42 + (16*b5 - 30*b1) * q^43 + (6*b6 + 12) * q^44 + (6*b6 - 18) * q^46 + (-4*b3 - 28*b2) * q^47 - 4*b2 * q^48 + (-2*b7 + 24*b6 + 10*b4 + 5) * q^49 + (-33*b6 - 6) * q^51 + (4*b3 - 16*b2) * q^52 + (27*b5 - 12*b1) * q^53 - 3*b7 * q^54 + (2*b7 + 4*b6 + 4*b4 - 12) * q^56 + (-3*b5 - 24*b1) * q^57 + 30*b1 * q^58 + (-20*b7 + 14*b4) * q^59 + (-4*b7 + 2*b4) * q^61 + (12*b3 + 24*b2) * q^62 + (3*b5 - 6*b3 - 3*b2 + 9*b1) * q^63 - 8 * q^64 + (6*b7 + 3*b4) * q^66 + (22*b5 + 12*b1) * q^67 + (-22*b3 - 4*b2) * q^68 + (-9*b7 + 3*b4) * q^69 + (57*b6 - 30) * q^71 + 6*b1 * q^72 + (26*b3 + 4*b2) * q^73 + (20*b6 - 72) * q^74 + (16*b7 - 2*b4) * q^76 + (3*b5 + 15*b3 + 18*b2 - 12*b1) * q^77 + (6*b5 + 24*b1) * q^78 + (-72*b6 - 32) * q^79 + 9 * q^81 + (-14*b3 + 14*b2) * q^82 + (20*b3 + 32*b2) * q^83 + (-6*b7 - 12*b6 + 2*b4 - 6) * q^84 + (-32*b6 - 60) * q^86 + 30*b2 * q^87 + (6*b5 - 12*b1) * q^88 + (21*b7 + 27*b4) * q^89 + (-28*b7 - 42*b6 + 14*b4) * q^91 + (6*b5 + 18*b1) * q^92 + (18*b5 - 36*b1) * q^93 + (-28*b7 - 4*b4) * q^94 - 4*b7 * q^96 + (-10*b3 - 44*b2) * q^97 + (24*b5 - 20*b3 + 4*b2 - 5*b1) * q^98 + (-9*b6 - 18) * 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 $$8 q - 16 q^{4} + 24 q^{9} - 48 q^{11} + 48 q^{14} + 32 q^{16} + 24 q^{21} - 240 q^{29} - 48 q^{36} - 192 q^{39} + 96 q^{44} - 144 q^{46} + 40 q^{49} - 48 q^{51} - 96 q^{56} - 64 q^{64} - 240 q^{71} - 576 q^{74} - 256 q^{79} + 72 q^{81} - 48 q^{84} - 480 q^{86} - 144 q^{99}+O(q^{100})$$ 8 * q - 16 * q^4 + 24 * q^9 - 48 * q^11 + 48 * q^14 + 32 * q^16 + 24 * q^21 - 240 * q^29 - 48 * q^36 - 192 * q^39 + 96 * q^44 - 144 * q^46 + 40 * q^49 - 48 * q^51 - 96 * q^56 - 64 * q^64 - 240 * q^71 - 576 * q^74 - 256 * q^79 + 72 * q^81 - 48 * q^84 - 480 * q^86 - 144 * q^99

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$-\zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24}$$ -v^5 - v^3 + v $$\beta_{2}$$ $$=$$ $$-\zeta_{24}^{6} + 2\zeta_{24}^{2}$$ -v^6 + 2*v^2 $$\beta_{3}$$ $$=$$ $$-2\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24}$$ -2*v^7 + v^5 + v^3 + v $$\beta_{4}$$ $$=$$ $$4\zeta_{24}^{4} - 2$$ 4*v^4 - 2 $$\beta_{5}$$ $$=$$ $$2\zeta_{24}^{6}$$ 2*v^6 $$\beta_{6}$$ $$=$$ $$-\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24}$$ -v^5 + v^3 + v $$\beta_{7}$$ $$=$$ $$2\zeta_{24}^{7} + \zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24}$$ 2*v^7 + v^5 - v^3 + v
 $$\zeta_{24}$$ $$=$$ $$( \beta_{7} + \beta_{6} + \beta_{3} + \beta_1 ) / 4$$ (b7 + b6 + b3 + b1) / 4 $$\zeta_{24}^{2}$$ $$=$$ $$( \beta_{5} + 2\beta_{2} ) / 4$$ (b5 + 2*b2) / 4 $$\zeta_{24}^{3}$$ $$=$$ $$( \beta_{6} - \beta_1 ) / 2$$ (b6 - b1) / 2 $$\zeta_{24}^{4}$$ $$=$$ $$( \beta_{4} + 2 ) / 4$$ (b4 + 2) / 4 $$\zeta_{24}^{5}$$ $$=$$ $$( \beta_{7} - \beta_{6} + \beta_{3} - \beta_1 ) / 4$$ (b7 - b6 + b3 - b1) / 4 $$\zeta_{24}^{6}$$ $$=$$ $$( \beta_{5} ) / 2$$ (b5) / 2 $$\zeta_{24}^{7}$$ $$=$$ $$( \beta_{7} + \beta_{6} - \beta_{3} - \beta_1 ) / 4$$ (b7 + b6 - b3 - 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
349.1
 0.965926 − 0.258819i −0.965926 − 0.258819i 0.258819 + 0.965926i −0.258819 + 0.965926i 0.965926 + 0.258819i −0.965926 + 0.258819i 0.258819 − 0.965926i −0.258819 − 0.965926i
1.41421i −1.73205 −2.00000 0 2.44949i −6.63103 + 2.24264i 2.82843i 3.00000 0
349.2 1.41421i −1.73205 −2.00000 0 2.44949i 3.16693 + 6.24264i 2.82843i 3.00000 0
349.3 1.41421i 1.73205 −2.00000 0 2.44949i −3.16693 + 6.24264i 2.82843i 3.00000 0
349.4 1.41421i 1.73205 −2.00000 0 2.44949i 6.63103 + 2.24264i 2.82843i 3.00000 0
349.5 1.41421i −1.73205 −2.00000 0 2.44949i −6.63103 2.24264i 2.82843i 3.00000 0
349.6 1.41421i −1.73205 −2.00000 0 2.44949i 3.16693 6.24264i 2.82843i 3.00000 0
349.7 1.41421i 1.73205 −2.00000 0 2.44949i −3.16693 6.24264i 2.82843i 3.00000 0
349.8 1.41421i 1.73205 −2.00000 0 2.44949i 6.63103 2.24264i 2.82843i 3.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 349.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.b odd 2 1 inner
35.c 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.h.a 8
5.b even 2 1 inner 1050.3.h.a 8
5.c odd 4 1 42.3.c.a 4
5.c odd 4 1 1050.3.f.a 4
7.b odd 2 1 inner 1050.3.h.a 8
15.e even 4 1 126.3.c.b 4
20.e even 4 1 336.3.f.c 4
35.c odd 2 1 inner 1050.3.h.a 8
35.f even 4 1 42.3.c.a 4
35.f even 4 1 1050.3.f.a 4
35.k even 12 1 294.3.g.b 4
35.k even 12 1 294.3.g.c 4
35.l odd 12 1 294.3.g.b 4
35.l odd 12 1 294.3.g.c 4
40.i odd 4 1 1344.3.f.f 4
40.k even 4 1 1344.3.f.e 4
60.l odd 4 1 1008.3.f.g 4
105.k odd 4 1 126.3.c.b 4
105.w odd 12 1 882.3.n.a 4
105.w odd 12 1 882.3.n.d 4
105.x even 12 1 882.3.n.a 4
105.x even 12 1 882.3.n.d 4
140.j odd 4 1 336.3.f.c 4
280.s even 4 1 1344.3.f.f 4
280.y odd 4 1 1344.3.f.e 4
420.w even 4 1 1008.3.f.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.3.c.a 4 5.c odd 4 1
42.3.c.a 4 35.f even 4 1
126.3.c.b 4 15.e even 4 1
126.3.c.b 4 105.k odd 4 1
294.3.g.b 4 35.k even 12 1
294.3.g.b 4 35.l odd 12 1
294.3.g.c 4 35.k even 12 1
294.3.g.c 4 35.l odd 12 1
336.3.f.c 4 20.e even 4 1
336.3.f.c 4 140.j odd 4 1
882.3.n.a 4 105.w odd 12 1
882.3.n.a 4 105.x even 12 1
882.3.n.d 4 105.w odd 12 1
882.3.n.d 4 105.x even 12 1
1008.3.f.g 4 60.l odd 4 1
1008.3.f.g 4 420.w even 4 1
1050.3.f.a 4 5.c odd 4 1
1050.3.f.a 4 35.f even 4 1
1050.3.h.a 8 1.a even 1 1 trivial
1050.3.h.a 8 5.b even 2 1 inner
1050.3.h.a 8 7.b odd 2 1 inner
1050.3.h.a 8 35.c odd 2 1 inner
1344.3.f.e 4 40.k even 4 1
1344.3.f.e 4 280.y odd 4 1
1344.3.f.f 4 40.i odd 4 1
1344.3.f.f 4 280.s even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{2} + 12T_{11} + 18$$ acting on $$S_{3}^{\mathrm{new}}(1050, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 2)^{4}$$
$3$ $$(T^{2} - 3)^{4}$$
$5$ $$T^{8}$$
$7$ $$T^{8} - 20 T^{6} + 294 T^{4} + \cdots + 5764801$$
$11$ $$(T^{2} + 12 T + 18)^{4}$$
$13$ $$(T^{4} - 432 T^{2} + 28224)^{2}$$
$17$ $$(T^{4} - 1476 T^{2} + 509796)^{2}$$
$19$ $$(T^{4} + 792 T^{2} + 138384)^{2}$$
$23$ $$(T^{4} + 396 T^{2} + 15876)^{2}$$
$29$ $$(T + 30)^{8}$$
$31$ $$(T^{4} + 2592 T^{2} + 186624)^{2}$$
$37$ $$(T^{4} + 5984 T^{2} + 4804864)^{2}$$
$41$ $$(T^{4} + 1764 T^{2} + 86436)^{2}$$
$43$ $$(T^{4} + 5648 T^{2} + 602176)^{2}$$
$47$ $$(T^{4} - 4896 T^{2} + 5089536)^{2}$$
$53$ $$(T^{4} + 6408 T^{2} + 6906384)^{2}$$
$59$ $$(T^{4} + 9504 T^{2} + 2304)^{2}$$
$61$ $$(T^{4} + 288 T^{2} + 2304)^{2}$$
$67$ $$(T^{4} + 4448 T^{2} + 2715904)^{2}$$
$71$ $$(T^{2} + 60 T - 5598)^{4}$$
$73$ $$(T^{4} - 8208 T^{2} + 16064064)^{2}$$
$79$ $$(T^{2} + 64 T - 9344)^{4}$$
$83$ $$(T^{4} - 10944 T^{2} + 451584)^{2}$$
$89$ $$(T^{4} + 22788 T^{2} + 37234404)^{2}$$
$97$ $$(T^{4} - 12816 T^{2} + 27123264)^{2}$$