# Properties

 Label 1050.4.g.o Level 1050 Weight 4 Character orbit 1050.g Analytic conductor 61.952 Analytic rank 0 Dimension 2 CM no Inner twists 2

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1050.g (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$61.9520055060$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 210) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -2 i q^{2} + 3 i q^{3} -4 q^{4} + 6 q^{6} + 7 i q^{7} + 8 i q^{8} -9 q^{9} +O(q^{10})$$ $$q -2 i q^{2} + 3 i q^{3} -4 q^{4} + 6 q^{6} + 7 i q^{7} + 8 i q^{8} -9 q^{9} + 12 q^{11} -12 i q^{12} -2 i q^{13} + 14 q^{14} + 16 q^{16} -18 i q^{17} + 18 i q^{18} -56 q^{19} -21 q^{21} -24 i q^{22} + 156 i q^{23} -24 q^{24} -4 q^{26} -27 i q^{27} -28 i q^{28} + 186 q^{29} -52 q^{31} -32 i q^{32} + 36 i q^{33} -36 q^{34} + 36 q^{36} -178 i q^{37} + 112 i q^{38} + 6 q^{39} -138 q^{41} + 42 i q^{42} + 412 i q^{43} -48 q^{44} + 312 q^{46} -456 i q^{47} + 48 i q^{48} -49 q^{49} + 54 q^{51} + 8 i q^{52} + 198 i q^{53} -54 q^{54} -56 q^{56} -168 i q^{57} -372 i q^{58} -348 q^{59} + 110 q^{61} + 104 i q^{62} -63 i q^{63} -64 q^{64} + 72 q^{66} -196 i q^{67} + 72 i q^{68} -468 q^{69} -936 q^{71} -72 i q^{72} -542 i q^{73} -356 q^{74} + 224 q^{76} + 84 i q^{77} -12 i q^{78} -992 q^{79} + 81 q^{81} + 276 i q^{82} + 276 i q^{83} + 84 q^{84} + 824 q^{86} + 558 i q^{87} + 96 i q^{88} -630 q^{89} + 14 q^{91} -624 i q^{92} -156 i q^{93} -912 q^{94} + 96 q^{96} + 110 i q^{97} + 98 i q^{98} -108 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 8q^{4} + 12q^{6} - 18q^{9} + O(q^{10})$$ $$2q - 8q^{4} + 12q^{6} - 18q^{9} + 24q^{11} + 28q^{14} + 32q^{16} - 112q^{19} - 42q^{21} - 48q^{24} - 8q^{26} + 372q^{29} - 104q^{31} - 72q^{34} + 72q^{36} + 12q^{39} - 276q^{41} - 96q^{44} + 624q^{46} - 98q^{49} + 108q^{51} - 108q^{54} - 112q^{56} - 696q^{59} + 220q^{61} - 128q^{64} + 144q^{66} - 936q^{69} - 1872q^{71} - 712q^{74} + 448q^{76} - 1984q^{79} + 162q^{81} + 168q^{84} + 1648q^{86} - 1260q^{89} + 28q^{91} - 1824q^{94} + 192q^{96} - 216q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
799.1
 1.00000i − 1.00000i
2.00000i 3.00000i −4.00000 0 6.00000 7.00000i 8.00000i −9.00000 0
799.2 2.00000i 3.00000i −4.00000 0 6.00000 7.00000i 8.00000i −9.00000 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.4.g.o 2
5.b even 2 1 inner 1050.4.g.o 2
5.c odd 4 1 210.4.a.a 1
5.c odd 4 1 1050.4.a.t 1
15.e even 4 1 630.4.a.v 1
20.e even 4 1 1680.4.a.n 1
35.f even 4 1 1470.4.a.n 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.4.a.a 1 5.c odd 4 1
630.4.a.v 1 15.e even 4 1
1050.4.a.t 1 5.c odd 4 1
1050.4.g.o 2 1.a even 1 1 trivial
1050.4.g.o 2 5.b even 2 1 inner
1470.4.a.n 1 35.f even 4 1
1680.4.a.n 1 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(1050, [\chi])$$:

 $$T_{11} - 12$$ $$T_{13}^{2} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 4 T^{2}$$
$3$ $$1 + 9 T^{2}$$
$5$ 1
$7$ $$1 + 49 T^{2}$$
$11$ $$( 1 - 12 T + 1331 T^{2} )^{2}$$
$13$ $$1 - 4390 T^{2} + 4826809 T^{4}$$
$17$ $$1 - 9502 T^{2} + 24137569 T^{4}$$
$19$ $$( 1 + 56 T + 6859 T^{2} )^{2}$$
$23$ $$1 + 2 T^{2} + 148035889 T^{4}$$
$29$ $$( 1 - 186 T + 24389 T^{2} )^{2}$$
$31$ $$( 1 + 52 T + 29791 T^{2} )^{2}$$
$37$ $$1 - 69622 T^{2} + 2565726409 T^{4}$$
$41$ $$( 1 + 138 T + 68921 T^{2} )^{2}$$
$43$ $$1 + 10730 T^{2} + 6321363049 T^{4}$$
$47$ $$1 + 290 T^{2} + 10779215329 T^{4}$$
$53$ $$1 - 258550 T^{2} + 22164361129 T^{4}$$
$59$ $$( 1 + 348 T + 205379 T^{2} )^{2}$$
$61$ $$( 1 - 110 T + 226981 T^{2} )^{2}$$
$67$ $$1 - 563110 T^{2} + 90458382169 T^{4}$$
$71$ $$( 1 + 936 T + 357911 T^{2} )^{2}$$
$73$ $$1 - 484270 T^{2} + 151334226289 T^{4}$$
$79$ $$( 1 + 992 T + 493039 T^{2} )^{2}$$
$83$ $$1 - 1067398 T^{2} + 326940373369 T^{4}$$
$89$ $$( 1 + 630 T + 704969 T^{2} )^{2}$$
$97$ $$1 - 1813246 T^{2} + 832972004929 T^{4}$$
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