# Properties

 Label 1050.2.s.e Level 1050 Weight 2 Character orbit 1050.s Analytic conductor 8.384 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: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 1050.s (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.38429221223$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.303595776.1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 210) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{5} q^{2} + ( -\beta_{1} + \beta_{7} ) q^{3} + ( 1 + \beta_{4} ) q^{4} + ( \beta_{2} - \beta_{4} ) q^{6} + ( 3 \beta_{3} + 2 \beta_{5} ) q^{7} + ( \beta_{3} + \beta_{5} ) q^{8} + ( 2 - \beta_{6} ) q^{9} +O(q^{10})$$ $$q + \beta_{5} q^{2} + ( -\beta_{1} + \beta_{7} ) q^{3} + ( 1 + \beta_{4} ) q^{4} + ( \beta_{2} - \beta_{4} ) q^{6} + ( 3 \beta_{3} + 2 \beta_{5} ) q^{7} + ( \beta_{3} + \beta_{5} ) q^{8} + ( 2 - \beta_{6} ) q^{9} + ( -2 + \beta_{2} + \beta_{4} - \beta_{6} ) q^{11} + \beta_{7} q^{12} + ( -2 \beta_{3} - 2 \beta_{5} ) q^{13} + ( -1 + 2 \beta_{4} ) q^{14} + \beta_{4} q^{16} + ( 2 \beta_{3} + 4 \beta_{7} ) q^{17} + ( -\beta_{1} + 3 \beta_{5} ) q^{18} + ( 4 + 2 \beta_{4} ) q^{19} + ( 3 - \beta_{2} + \beta_{4} + 3 \beta_{6} ) q^{21} + ( -\beta_{1} + 2 \beta_{3} - \beta_{5} + \beta_{7} ) q^{22} + ( -\beta_{1} + \beta_{3} - \beta_{5} + 2 \beta_{7} ) q^{23} + ( 1 + \beta_{6} ) q^{24} -2 \beta_{4} q^{26} + ( -2 \beta_{1} - 3 \beta_{3} - 3 \beta_{5} + 2 \beta_{7} ) q^{27} + ( 2 \beta_{3} - \beta_{5} ) q^{28} + ( 1 + 2 \beta_{6} ) q^{29} + ( 3 \beta_{2} - 3 \beta_{4} - 3 \beta_{6} ) q^{31} + \beta_{3} q^{32} + ( 3 \beta_{1} - 3 \beta_{3} - 2 \beta_{7} ) q^{33} + ( 2 + 4 \beta_{6} ) q^{34} + ( 2 + \beta_{2} + 2 \beta_{4} - \beta_{6} ) q^{36} + ( -6 \beta_{1} + 2 \beta_{3} + 4 \beta_{5} ) q^{37} + ( 2 \beta_{3} + 4 \beta_{5} ) q^{38} + ( -2 - 2 \beta_{6} ) q^{39} + ( -4 - 2 \beta_{2} + \beta_{4} + \beta_{6} ) q^{41} + ( 3 \beta_{1} - \beta_{7} ) q^{42} + ( -3 \beta_{1} + 5 \beta_{3} - 2 \beta_{5} + 3 \beta_{7} ) q^{43} + ( -3 + \beta_{2} - 2 \beta_{4} ) q^{44} + ( -1 + \beta_{2} - 2 \beta_{4} + \beta_{6} ) q^{46} + ( 2 \beta_{1} + 6 \beta_{3} + 2 \beta_{5} ) q^{47} + \beta_{1} q^{48} + ( -8 - 5 \beta_{4} ) q^{49} + ( 10 + 2 \beta_{2} + 10 \beta_{4} - 2 \beta_{6} ) q^{51} -2 \beta_{3} q^{52} + ( 2 \beta_{1} + \beta_{3} - \beta_{5} - \beta_{7} ) q^{53} + ( 2 \beta_{2} - 5 \beta_{4} ) q^{54} + ( -3 - \beta_{4} ) q^{56} + ( -2 \beta_{1} + 4 \beta_{7} ) q^{57} + ( 2 \beta_{1} - \beta_{5} ) q^{58} + ( -6 - 3 \beta_{2} - 3 \beta_{4} - 3 \beta_{6} ) q^{59} + ( -9 - 3 \beta_{2} - 3 \beta_{4} ) q^{61} + ( -3 \beta_{1} + 3 \beta_{5} + 3 \beta_{7} ) q^{62} + ( -2 \beta_{1} + 9 \beta_{3} + 6 \beta_{5} + 3 \beta_{7} ) q^{63} - q^{64} + ( 4 - 3 \beta_{2} + 3 \beta_{4} + \beta_{6} ) q^{66} + ( -3 \beta_{5} + 3 \beta_{7} ) q^{67} + ( 4 \beta_{1} - 2 \beta_{5} ) q^{68} + ( 5 - \beta_{2} + 4 \beta_{4} - \beta_{6} ) q^{69} + ( -4 - 6 \beta_{4} - 2 \beta_{6} ) q^{71} + ( -\beta_{1} + 3 \beta_{3} + 3 \beta_{5} + \beta_{7} ) q^{72} + 2 \beta_{3} q^{73} + ( -4 + 6 \beta_{2} - 2 \beta_{4} - 6 \beta_{6} ) q^{74} + ( 2 + 4 \beta_{4} ) q^{76} + ( \beta_{1} - 5 \beta_{3} - 8 \beta_{5} + 2 \beta_{7} ) q^{77} -2 \beta_{1} q^{78} + ( -3 + 3 \beta_{2} - 2 \beta_{4} - 6 \beta_{6} ) q^{79} + ( 1 - 5 \beta_{6} ) q^{81} + ( \beta_{1} - \beta_{3} - 5 \beta_{5} - 2 \beta_{7} ) q^{82} + ( -4 \beta_{1} - \beta_{3} + 5 \beta_{5} + 4 \beta_{7} ) q^{83} + ( 2 - 3 \beta_{2} + 3 \beta_{4} + 2 \beta_{6} ) q^{84} + ( -7 + 3 \beta_{2} - 5 \beta_{4} ) q^{86} + ( -\beta_{1} + 6 \beta_{3} + 6 \beta_{5} + \beta_{7} ) q^{87} + ( -\beta_{3} - 3 \beta_{5} + \beta_{7} ) q^{88} + ( 1 - \beta_{2} - \beta_{4} + 2 \beta_{6} ) q^{89} + ( 6 + 2 \beta_{4} ) q^{91} + ( \beta_{1} - \beta_{3} - 2 \beta_{5} + \beta_{7} ) q^{92} + ( -3 \beta_{1} - 9 \beta_{3} ) q^{93} + ( -2 - 2 \beta_{2} + 4 \beta_{4} + 2 \beta_{6} ) q^{94} + ( 1 - \beta_{2} + \beta_{4} + \beta_{6} ) q^{96} + ( 3 \beta_{1} + 8 \beta_{3} + 5 \beta_{5} + 3 \beta_{7} ) q^{97} + ( -5 \beta_{3} - 8 \beta_{5} ) q^{98} + ( -7 + 4 \beta_{2} - \beta_{4} - \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{4} + 2q^{6} + 20q^{9} + O(q^{10})$$ $$8q + 4q^{4} + 2q^{6} + 20q^{9} - 18q^{11} - 16q^{14} - 4q^{16} + 24q^{19} + 10q^{21} + 4q^{24} + 8q^{26} + 18q^{31} + 10q^{36} - 8q^{39} - 36q^{41} - 18q^{44} - 6q^{46} - 44q^{49} + 44q^{51} + 16q^{54} - 20q^{56} - 18q^{59} - 54q^{61} - 8q^{64} + 22q^{66} + 30q^{69} - 12q^{74} + 2q^{79} + 28q^{81} + 2q^{84} - 42q^{86} + 6q^{89} + 40q^{91} - 36q^{94} + 2q^{96} - 56q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 5 x^{6} + 16 x^{4} + 45 x^{2} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} + 32 \nu^{4} + 16 \nu^{2} + 45$$$$)/144$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} + 32 \nu^{5} + 16 \nu^{3} + 45 \nu$$$$)/432$$ $$\beta_{4}$$ $$=$$ $$($$$$-5 \nu^{6} - 16 \nu^{4} - 80 \nu^{2} - 225$$$$)/144$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} + 13 \nu$$$$)/48$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{6} - 13$$$$)/16$$ $$\beta_{7}$$ $$=$$ $$($$$$5 \nu^{7} + 16 \nu^{5} + 80 \nu^{3} + 225 \nu$$$$)/144$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} - 2 \beta_{4} - \beta_{2} - 2$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{7} - 3 \beta_{5} - 3 \beta_{3} - 2 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} + 5 \beta_{2}$$ $$\nu^{5}$$ $$=$$ $$-\beta_{7} + 15 \beta_{3}$$ $$\nu^{6}$$ $$=$$ $$-16 \beta_{6} - 13$$ $$\nu^{7}$$ $$=$$ $$48 \beta_{5} - 13 \beta_{1}$$

## Character values

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

 $$n$$ $$127$$ $$451$$ $$701$$ $$\chi(n)$$ $$1$$ $$-\beta_{4}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
101.1
 0.396143 + 1.68614i −1.26217 − 1.18614i 1.26217 + 1.18614i −0.396143 − 1.68614i 0.396143 − 1.68614i −1.26217 + 1.18614i 1.26217 − 1.18614i −0.396143 + 1.68614i
−0.866025 + 0.500000i −1.65831 0.500000i 0.500000 0.866025i 0 1.68614 0.396143i 0.866025 + 2.50000i 1.00000i 2.50000 + 1.65831i 0
101.2 −0.866025 + 0.500000i 1.65831 0.500000i 0.500000 0.866025i 0 −1.18614 + 1.26217i 0.866025 + 2.50000i 1.00000i 2.50000 1.65831i 0
101.3 0.866025 0.500000i −1.65831 + 0.500000i 0.500000 0.866025i 0 −1.18614 + 1.26217i −0.866025 2.50000i 1.00000i 2.50000 1.65831i 0
101.4 0.866025 0.500000i 1.65831 + 0.500000i 0.500000 0.866025i 0 1.68614 0.396143i −0.866025 2.50000i 1.00000i 2.50000 + 1.65831i 0
551.1 −0.866025 0.500000i −1.65831 + 0.500000i 0.500000 + 0.866025i 0 1.68614 + 0.396143i 0.866025 2.50000i 1.00000i 2.50000 1.65831i 0
551.2 −0.866025 0.500000i 1.65831 + 0.500000i 0.500000 + 0.866025i 0 −1.18614 1.26217i 0.866025 2.50000i 1.00000i 2.50000 + 1.65831i 0
551.3 0.866025 + 0.500000i −1.65831 0.500000i 0.500000 + 0.866025i 0 −1.18614 1.26217i −0.866025 + 2.50000i 1.00000i 2.50000 + 1.65831i 0
551.4 0.866025 + 0.500000i 1.65831 0.500000i 0.500000 + 0.866025i 0 1.68614 + 0.396143i −0.866025 + 2.50000i 1.00000i 2.50000 1.65831i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 551.4 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
21.g even 6 1 inner
105.p even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.2.s.e 8
3.b odd 2 1 1050.2.s.d 8
5.b even 2 1 inner 1050.2.s.e 8
5.c odd 4 1 210.2.t.a 4
5.c odd 4 1 210.2.t.d yes 4
7.d odd 6 1 1050.2.s.d 8
15.d odd 2 1 1050.2.s.d 8
15.e even 4 1 210.2.t.b yes 4
15.e even 4 1 210.2.t.c yes 4
21.g even 6 1 inner 1050.2.s.e 8
35.i odd 6 1 1050.2.s.d 8
35.k even 12 1 210.2.t.b yes 4
35.k even 12 1 210.2.t.c yes 4
35.k even 12 1 1470.2.d.b 4
35.k even 12 1 1470.2.d.c 4
35.l odd 12 1 1470.2.d.a 4
35.l odd 12 1 1470.2.d.d 4
105.p even 6 1 inner 1050.2.s.e 8
105.w odd 12 1 210.2.t.a 4
105.w odd 12 1 210.2.t.d yes 4
105.w odd 12 1 1470.2.d.a 4
105.w odd 12 1 1470.2.d.d 4
105.x even 12 1 1470.2.d.b 4
105.x even 12 1 1470.2.d.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.t.a 4 5.c odd 4 1
210.2.t.a 4 105.w odd 12 1
210.2.t.b yes 4 15.e even 4 1
210.2.t.b yes 4 35.k even 12 1
210.2.t.c yes 4 15.e even 4 1
210.2.t.c yes 4 35.k even 12 1
210.2.t.d yes 4 5.c odd 4 1
210.2.t.d yes 4 105.w odd 12 1
1050.2.s.d 8 3.b odd 2 1
1050.2.s.d 8 7.d odd 6 1
1050.2.s.d 8 15.d odd 2 1
1050.2.s.d 8 35.i odd 6 1
1050.2.s.e 8 1.a even 1 1 trivial
1050.2.s.e 8 5.b even 2 1 inner
1050.2.s.e 8 21.g even 6 1 inner
1050.2.s.e 8 105.p even 6 1 inner
1470.2.d.a 4 35.l odd 12 1
1470.2.d.a 4 105.w odd 12 1
1470.2.d.b 4 35.k even 12 1
1470.2.d.b 4 105.x even 12 1
1470.2.d.c 4 35.k even 12 1
1470.2.d.c 4 105.x even 12 1
1470.2.d.d 4 35.l odd 12 1
1470.2.d.d 4 105.w odd 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1050, [\chi])$$:

 $$T_{11}^{4} + 9 T_{11}^{3} + 31 T_{11}^{2} + 36 T_{11} + 16$$ $$T_{37}^{8} + 204 T_{37}^{6} + 32400 T_{37}^{4} + 1880064 T_{37}^{2} + 84934656$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} )^{2}$$
$3$ $$( 1 - 5 T^{2} + 9 T^{4} )^{2}$$
$5$ 1
$7$ $$( 1 + 11 T^{2} + 49 T^{4} )^{2}$$
$11$ $$( 1 + 9 T + 53 T^{2} + 234 T^{3} + 852 T^{4} + 2574 T^{5} + 6413 T^{6} + 11979 T^{7} + 14641 T^{8} )^{2}$$
$13$ $$( 1 - 22 T^{2} + 169 T^{4} )^{4}$$
$17$ $$( 1 + 10 T^{2} - 189 T^{4} + 2890 T^{6} + 83521 T^{8} )^{2}$$
$19$ $$( 1 - 7 T + 19 T^{2} )^{4}( 1 + T + 19 T^{2} )^{4}$$
$23$ $$1 + 71 T^{2} + 2797 T^{4} + 84206 T^{6} + 2089006 T^{8} + 44544974 T^{10} + 782715277 T^{12} + 10510548119 T^{14} + 78310985281 T^{16}$$
$29$ $$( 1 - 47 T^{2} + 841 T^{4} )^{4}$$
$31$ $$( 1 - 9 T + 71 T^{2} - 396 T^{3} + 1812 T^{4} - 12276 T^{5} + 68231 T^{6} - 268119 T^{7} + 923521 T^{8} )^{2}$$
$37$ $$1 + 56 T^{2} + 802 T^{4} - 22624 T^{6} - 719789 T^{8} - 30972256 T^{10} + 1503077122 T^{12} + 143680678904 T^{14} + 3512479453921 T^{16}$$
$41$ $$( 1 + 9 T + 94 T^{2} + 369 T^{3} + 1681 T^{4} )^{4}$$
$43$ $$( 1 + 49 T^{2} + 660 T^{4} + 90601 T^{6} + 3418801 T^{8} )^{2}$$
$47$ $$1 - 112 T^{2} + 6178 T^{4} - 218176 T^{6} + 7936579 T^{8} - 481950784 T^{10} + 30146669218 T^{12} - 1207272116848 T^{14} + 23811286661761 T^{16}$$
$53$ $$1 + 191 T^{2} + 21817 T^{4} + 1727786 T^{6} + 104667286 T^{8} + 4853350874 T^{10} + 172146623977 T^{12} + 4233392975639 T^{14} + 62259690411361 T^{16}$$
$59$ $$( 1 + 9 T + 17 T^{2} - 486 T^{3} - 4164 T^{4} - 28674 T^{5} + 59177 T^{6} + 1848411 T^{7} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 + 27 T + 401 T^{2} + 4266 T^{3} + 36066 T^{4} + 260226 T^{5} + 1492121 T^{6} + 6128487 T^{7} + 13845841 T^{8} )^{2}$$
$67$ $$1 - 205 T^{2} + 23209 T^{4} - 2016790 T^{6} + 145240510 T^{8} - 9053370310 T^{10} + 467687367289 T^{12} - 18543968344645 T^{14} + 406067677556641 T^{16}$$
$71$ $$( 1 - 208 T^{2} + 19710 T^{4} - 1048528 T^{6} + 25411681 T^{8} )^{2}$$
$73$ $$( 1 + 142 T^{2} + 14835 T^{4} + 756718 T^{6} + 28398241 T^{8} )^{2}$$
$79$ $$( 1 - T - 83 T^{2} + 74 T^{3} + 736 T^{4} + 5846 T^{5} - 518003 T^{6} - 493039 T^{7} + 38950081 T^{8} )^{2}$$
$83$ $$( 1 + 190 T^{2} + 18051 T^{4} + 1308910 T^{6} + 47458321 T^{8} )^{2}$$
$89$ $$( 1 - 3 T - 163 T^{2} + 18 T^{3} + 20862 T^{4} + 1602 T^{5} - 1291123 T^{6} - 2114907 T^{7} + 62742241 T^{8} )^{2}$$
$97$ $$( 1 - 155 T^{2} + 12276 T^{4} - 1458395 T^{6} + 88529281 T^{8} )^{2}$$