# Properties

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

# Related objects

## 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: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -2 i q^{2} -3 i q^{3} -4 q^{4} -6 q^{6} -7 i q^{7} + 8 i q^{8} -9 q^{9} +O(q^{10})$$ $$q -2 i q^{2} -3 i q^{3} -4 q^{4} -6 q^{6} -7 i q^{7} + 8 i q^{8} -9 q^{9} -72 q^{11} + 12 i q^{12} -34 i q^{13} -14 q^{14} + 16 q^{16} -6 i q^{17} + 18 i q^{18} -92 q^{19} -21 q^{21} + 144 i q^{22} -180 i q^{23} + 24 q^{24} -68 q^{26} + 27 i q^{27} + 28 i q^{28} + 114 q^{29} + 56 q^{31} -32 i q^{32} + 216 i q^{33} -12 q^{34} + 36 q^{36} + 34 i q^{37} + 184 i q^{38} -102 q^{39} + 6 q^{41} + 42 i q^{42} + 164 i q^{43} + 288 q^{44} -360 q^{46} -168 i q^{47} -48 i q^{48} -49 q^{49} -18 q^{51} + 136 i q^{52} + 654 i q^{53} + 54 q^{54} + 56 q^{56} + 276 i q^{57} -228 i q^{58} + 492 q^{59} -250 q^{61} -112 i q^{62} + 63 i q^{63} -64 q^{64} + 432 q^{66} + 124 i q^{67} + 24 i q^{68} -540 q^{69} + 36 q^{71} -72 i q^{72} + 1010 i q^{73} + 68 q^{74} + 368 q^{76} + 504 i q^{77} + 204 i q^{78} -56 q^{79} + 81 q^{81} -12 i q^{82} + 228 i q^{83} + 84 q^{84} + 328 q^{86} -342 i q^{87} -576 i q^{88} -390 q^{89} -238 q^{91} + 720 i q^{92} -168 i q^{93} -336 q^{94} -96 q^{96} + 70 i q^{97} + 98 i q^{98} + 648 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 8q^{4} - 12q^{6} - 18q^{9} + O(q^{10})$$ $$2q - 8q^{4} - 12q^{6} - 18q^{9} - 144q^{11} - 28q^{14} + 32q^{16} - 184q^{19} - 42q^{21} + 48q^{24} - 136q^{26} + 228q^{29} + 112q^{31} - 24q^{34} + 72q^{36} - 204q^{39} + 12q^{41} + 576q^{44} - 720q^{46} - 98q^{49} - 36q^{51} + 108q^{54} + 112q^{56} + 984q^{59} - 500q^{61} - 128q^{64} + 864q^{66} - 1080q^{69} + 72q^{71} + 136q^{74} + 736q^{76} - 112q^{79} + 162q^{81} + 168q^{84} + 656q^{86} - 780q^{89} - 476q^{91} - 672q^{94} - 192q^{96} + 1296q^{99} + O(q^{100})$$

## 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.a 2
5.b even 2 1 inner 1050.4.g.a 2
5.c odd 4 1 42.4.a.a 1
5.c odd 4 1 1050.4.a.g 1
15.e even 4 1 126.4.a.a 1
20.e even 4 1 336.4.a.l 1
35.f even 4 1 294.4.a.i 1
35.k even 12 2 294.4.e.b 2
35.l odd 12 2 294.4.e.c 2
40.i odd 4 1 1344.4.a.o 1
40.k even 4 1 1344.4.a.a 1
60.l odd 4 1 1008.4.a.b 1
105.k odd 4 1 882.4.a.g 1
105.w odd 12 2 882.4.g.o 2
105.x even 12 2 882.4.g.w 2
140.j odd 4 1 2352.4.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.a.a 1 5.c odd 4 1
126.4.a.a 1 15.e even 4 1
294.4.a.i 1 35.f even 4 1
294.4.e.b 2 35.k even 12 2
294.4.e.c 2 35.l odd 12 2
336.4.a.l 1 20.e even 4 1
882.4.a.g 1 105.k odd 4 1
882.4.g.o 2 105.w odd 12 2
882.4.g.w 2 105.x even 12 2
1008.4.a.b 1 60.l odd 4 1
1050.4.a.g 1 5.c odd 4 1
1050.4.g.a 2 1.a even 1 1 trivial
1050.4.g.a 2 5.b even 2 1 inner
1344.4.a.a 1 40.k even 4 1
1344.4.a.o 1 40.i odd 4 1
2352.4.a.a 1 140.j odd 4 1

## Hecke kernels

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

 $$T_{11} + 72$$ $$T_{13}^{2} + 1156$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + T^{2}$$
$3$ $$9 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$49 + T^{2}$$
$11$ $$( 72 + T )^{2}$$
$13$ $$1156 + T^{2}$$
$17$ $$36 + T^{2}$$
$19$ $$( 92 + T )^{2}$$
$23$ $$32400 + T^{2}$$
$29$ $$( -114 + T )^{2}$$
$31$ $$( -56 + T )^{2}$$
$37$ $$1156 + T^{2}$$
$41$ $$( -6 + T )^{2}$$
$43$ $$26896 + T^{2}$$
$47$ $$28224 + T^{2}$$
$53$ $$427716 + T^{2}$$
$59$ $$( -492 + T )^{2}$$
$61$ $$( 250 + T )^{2}$$
$67$ $$15376 + T^{2}$$
$71$ $$( -36 + T )^{2}$$
$73$ $$1020100 + T^{2}$$
$79$ $$( 56 + T )^{2}$$
$83$ $$51984 + T^{2}$$
$89$ $$( 390 + T )^{2}$$
$97$ $$4900 + T^{2}$$