# Properties

 Label 4200.2.t.j Level $4200$ Weight $2$ Character orbit 4200.t Analytic conductor $33.537$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4200 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4200.t (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$33.5371688489$$ 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 168) 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.

 $$f(q)$$ $$=$$ $$q -i q^{3} -i q^{7} - q^{9} +O(q^{10})$$ $$q -i q^{3} -i q^{7} - q^{9} + 6 i q^{13} + 2 i q^{17} -4 q^{19} - q^{21} -4 i q^{23} + i q^{27} + 10 q^{29} -8 q^{31} -6 i q^{37} + 6 q^{39} -2 q^{41} -4 i q^{43} -8 i q^{47} - q^{49} + 2 q^{51} -10 i q^{53} + 4 i q^{57} -12 q^{59} -2 q^{61} + i q^{63} -12 i q^{67} -4 q^{69} -12 q^{71} -14 i q^{73} + 8 q^{79} + q^{81} + 12 i q^{83} -10 i q^{87} + 2 q^{89} + 6 q^{91} + 8 i q^{93} -10 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{9} + O(q^{10})$$ $$2q - 2q^{9} - 8q^{19} - 2q^{21} + 20q^{29} - 16q^{31} + 12q^{39} - 4q^{41} - 2q^{49} + 4q^{51} - 24q^{59} - 4q^{61} - 8q^{69} - 24q^{71} + 16q^{79} + 2q^{81} + 4q^{89} + 12q^{91} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4200\mathbb{Z}\right)^\times$$.

 $$n$$ $$1177$$ $$2101$$ $$2801$$ $$3151$$ $$3601$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1849.1
 1.00000i − 1.00000i
0 1.00000i 0 0 0 1.00000i 0 −1.00000 0
1849.2 0 1.00000i 0 0 0 1.00000i 0 −1.00000 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 4200.2.t.j 2
5.b even 2 1 inner 4200.2.t.j 2
5.c odd 4 1 168.2.a.a 1
5.c odd 4 1 4200.2.a.t 1
15.e even 4 1 504.2.a.e 1
20.e even 4 1 336.2.a.e 1
20.e even 4 1 8400.2.a.y 1
35.f even 4 1 1176.2.a.f 1
35.k even 12 2 1176.2.q.d 2
35.l odd 12 2 1176.2.q.f 2
40.i odd 4 1 1344.2.a.m 1
40.k even 4 1 1344.2.a.b 1
60.l odd 4 1 1008.2.a.b 1
80.i odd 4 1 5376.2.c.d 2
80.j even 4 1 5376.2.c.bb 2
80.s even 4 1 5376.2.c.bb 2
80.t odd 4 1 5376.2.c.d 2
105.k odd 4 1 3528.2.a.v 1
105.w odd 12 2 3528.2.s.g 2
105.x even 12 2 3528.2.s.w 2
120.q odd 4 1 4032.2.a.bc 1
120.w even 4 1 4032.2.a.bh 1
140.j odd 4 1 2352.2.a.c 1
140.w even 12 2 2352.2.q.d 2
140.x odd 12 2 2352.2.q.w 2
280.s even 4 1 9408.2.a.be 1
280.y odd 4 1 9408.2.a.da 1
420.w even 4 1 7056.2.a.bq 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.a.a 1 5.c odd 4 1
336.2.a.e 1 20.e even 4 1
504.2.a.e 1 15.e even 4 1
1008.2.a.b 1 60.l odd 4 1
1176.2.a.f 1 35.f even 4 1
1176.2.q.d 2 35.k even 12 2
1176.2.q.f 2 35.l odd 12 2
1344.2.a.b 1 40.k even 4 1
1344.2.a.m 1 40.i odd 4 1
2352.2.a.c 1 140.j odd 4 1
2352.2.q.d 2 140.w even 12 2
2352.2.q.w 2 140.x odd 12 2
3528.2.a.v 1 105.k odd 4 1
3528.2.s.g 2 105.w odd 12 2
3528.2.s.w 2 105.x even 12 2
4032.2.a.bc 1 120.q odd 4 1
4032.2.a.bh 1 120.w even 4 1
4200.2.a.t 1 5.c odd 4 1
4200.2.t.j 2 1.a even 1 1 trivial
4200.2.t.j 2 5.b even 2 1 inner
5376.2.c.d 2 80.i odd 4 1
5376.2.c.d 2 80.t odd 4 1
5376.2.c.bb 2 80.j even 4 1
5376.2.c.bb 2 80.s even 4 1
7056.2.a.bq 1 420.w even 4 1
8400.2.a.y 1 20.e even 4 1
9408.2.a.be 1 280.s even 4 1
9408.2.a.da 1 280.y odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(4200, [\chi])$$:

 $$T_{11}$$ $$T_{13}^{2} + 36$$ $$T_{17}^{2} + 4$$ $$T_{19} + 4$$ $$T_{29} - 10$$ $$T_{31} + 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + T^{2}$$
$5$ 1
$7$ $$1 + T^{2}$$
$11$ $$( 1 + 11 T^{2} )^{2}$$
$13$ $$( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )$$
$17$ $$( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} )$$
$19$ $$( 1 + 4 T + 19 T^{2} )^{2}$$
$23$ $$1 - 30 T^{2} + 529 T^{4}$$
$29$ $$( 1 - 10 T + 29 T^{2} )^{2}$$
$31$ $$( 1 + 8 T + 31 T^{2} )^{2}$$
$37$ $$1 - 38 T^{2} + 1369 T^{4}$$
$41$ $$( 1 + 2 T + 41 T^{2} )^{2}$$
$43$ $$1 - 70 T^{2} + 1849 T^{4}$$
$47$ $$1 - 30 T^{2} + 2209 T^{4}$$
$53$ $$1 - 6 T^{2} + 2809 T^{4}$$
$59$ $$( 1 + 12 T + 59 T^{2} )^{2}$$
$61$ $$( 1 + 2 T + 61 T^{2} )^{2}$$
$67$ $$1 + 10 T^{2} + 4489 T^{4}$$
$71$ $$( 1 + 12 T + 71 T^{2} )^{2}$$
$73$ $$1 + 50 T^{2} + 5329 T^{4}$$
$79$ $$( 1 - 8 T + 79 T^{2} )^{2}$$
$83$ $$1 - 22 T^{2} + 6889 T^{4}$$
$89$ $$( 1 - 2 T + 89 T^{2} )^{2}$$
$97$ $$1 - 94 T^{2} + 9409 T^{4}$$