# Properties

 Label 1050.4.g.n 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.

 $$f(q)$$ $$=$$ $$q + 2 i q^{2} -3 i q^{3} -4 q^{4} + 6 q^{6} -7 i q^{7} -8 i q^{8} -9 q^{9} +O(q^{10})$$ $$q + 2 i q^{2} -3 i q^{3} -4 q^{4} + 6 q^{6} -7 i q^{7} -8 i q^{8} -9 q^{9} -8 q^{11} + 12 i q^{12} + 42 i q^{13} + 14 q^{14} + 16 q^{16} -2 i q^{17} -18 i q^{18} + 124 q^{19} -21 q^{21} -16 i q^{22} -76 i q^{23} -24 q^{24} -84 q^{26} + 27 i q^{27} + 28 i q^{28} -254 q^{29} -72 q^{31} + 32 i q^{32} + 24 i q^{33} + 4 q^{34} + 36 q^{36} + 398 i q^{37} + 248 i q^{38} + 126 q^{39} + 462 q^{41} -42 i q^{42} -212 i q^{43} + 32 q^{44} + 152 q^{46} -264 i q^{47} -48 i q^{48} -49 q^{49} -6 q^{51} -168 i q^{52} + 162 i q^{53} -54 q^{54} -56 q^{56} -372 i q^{57} -508 i q^{58} + 772 q^{59} + 30 q^{61} -144 i q^{62} + 63 i q^{63} -64 q^{64} -48 q^{66} -764 i q^{67} + 8 i q^{68} -228 q^{69} -236 q^{71} + 72 i q^{72} -418 i q^{73} -796 q^{74} -496 q^{76} + 56 i q^{77} + 252 i q^{78} -552 q^{79} + 81 q^{81} + 924 i q^{82} -1036 i q^{83} + 84 q^{84} + 424 q^{86} + 762 i q^{87} + 64 i q^{88} -30 q^{89} + 294 q^{91} + 304 i q^{92} + 216 i q^{93} + 528 q^{94} + 96 q^{96} -1190 i q^{97} -98 i q^{98} + 72 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 8q^{4} + 12q^{6} - 18q^{9} + O(q^{10})$$ $$2q - 8q^{4} + 12q^{6} - 18q^{9} - 16q^{11} + 28q^{14} + 32q^{16} + 248q^{19} - 42q^{21} - 48q^{24} - 168q^{26} - 508q^{29} - 144q^{31} + 8q^{34} + 72q^{36} + 252q^{39} + 924q^{41} + 64q^{44} + 304q^{46} - 98q^{49} - 12q^{51} - 108q^{54} - 112q^{56} + 1544q^{59} + 60q^{61} - 128q^{64} - 96q^{66} - 456q^{69} - 472q^{71} - 1592q^{74} - 992q^{76} - 1104q^{79} + 162q^{81} + 168q^{84} + 848q^{86} - 60q^{89} + 588q^{91} + 1056q^{94} + 192q^{96} + 144q^{99} + O(q^{100})$$

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.a.b 1 5.c odd 4 1
126.4.a.c 1 15.e even 4 1
294.4.a.h 1 35.f even 4 1
294.4.e.a 2 35.l odd 12 2
294.4.e.d 2 35.k even 12 2
336.4.a.d 1 20.e even 4 1
882.4.a.d 1 105.k odd 4 1
882.4.g.r 2 105.w odd 12 2
882.4.g.s 2 105.x even 12 2
1008.4.a.j 1 60.l odd 4 1
1050.4.a.d 1 5.c odd 4 1
1050.4.g.n 2 1.a even 1 1 trivial
1050.4.g.n 2 5.b even 2 1 inner
1344.4.a.f 1 40.i odd 4 1
1344.4.a.t 1 40.k even 4 1
2352.4.a.ba 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 $$S_{4}^{\mathrm{new}}(1050, [\chi])$$:

 $$T_{11} + 8$$ $$T_{13}^{2} + 1764$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + T^{2}$$
$3$ $$9 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$49 + T^{2}$$
$11$ $$( 8 + T )^{2}$$
$13$ $$1764 + T^{2}$$
$17$ $$4 + T^{2}$$
$19$ $$( -124 + T )^{2}$$
$23$ $$5776 + T^{2}$$
$29$ $$( 254 + T )^{2}$$
$31$ $$( 72 + T )^{2}$$
$37$ $$158404 + T^{2}$$
$41$ $$( -462 + T )^{2}$$
$43$ $$44944 + T^{2}$$
$47$ $$69696 + T^{2}$$
$53$ $$26244 + T^{2}$$
$59$ $$( -772 + T )^{2}$$
$61$ $$( -30 + T )^{2}$$
$67$ $$583696 + T^{2}$$
$71$ $$( 236 + T )^{2}$$
$73$ $$174724 + T^{2}$$
$79$ $$( 552 + T )^{2}$$
$83$ $$1073296 + T^{2}$$
$89$ $$( 30 + T )^{2}$$
$97$ $$1416100 + T^{2}$$