# Properties

 Label 2450.2.c.e Level 2450 Weight 2 Character orbit 2450.c Analytic conductor 19.563 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$2450 = 2 \cdot 5^{2} \cdot 7^{2}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 2450.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$19.5633484952$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 70) 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.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2450\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$1177$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
99.1
 − 1.00000i 1.00000i
1.00000i 2.00000i −1.00000 0 −2.00000 0 1.00000i −1.00000 0
99.2 1.00000i 2.00000i −1.00000 0 −2.00000 0 1.00000i −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 2450.2.c.e 2
5.b even 2 1 inner 2450.2.c.e 2
5.c odd 4 1 490.2.a.d 1
5.c odd 4 1 2450.2.a.v 1
7.b odd 2 1 2450.2.c.q 2
7.d odd 6 2 350.2.j.d 4
15.e even 4 1 4410.2.a.bg 1
20.e even 4 1 3920.2.a.e 1
35.c odd 2 1 2450.2.c.q 2
35.f even 4 1 490.2.a.a 1
35.f even 4 1 2450.2.a.bf 1
35.i odd 6 2 350.2.j.d 4
35.k even 12 2 70.2.e.d 2
35.k even 12 2 350.2.e.b 2
35.l odd 12 2 490.2.e.g 2
105.k odd 4 1 4410.2.a.x 1
105.w odd 12 2 630.2.k.d 2
140.j odd 4 1 3920.2.a.bh 1
140.x odd 12 2 560.2.q.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.e.d 2 35.k even 12 2
350.2.e.b 2 35.k even 12 2
350.2.j.d 4 7.d odd 6 2
350.2.j.d 4 35.i odd 6 2
490.2.a.a 1 35.f even 4 1
490.2.a.d 1 5.c odd 4 1
490.2.e.g 2 35.l odd 12 2
560.2.q.b 2 140.x odd 12 2
630.2.k.d 2 105.w odd 12 2
2450.2.a.v 1 5.c odd 4 1
2450.2.a.bf 1 35.f even 4 1
2450.2.c.e 2 1.a even 1 1 trivial
2450.2.c.e 2 5.b even 2 1 inner
2450.2.c.q 2 7.b odd 2 1
2450.2.c.q 2 35.c odd 2 1
3920.2.a.e 1 20.e even 4 1
3920.2.a.bh 1 140.j odd 4 1
4410.2.a.x 1 105.k odd 4 1
4410.2.a.bg 1 15.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 $$S_{2}^{\mathrm{new}}(2450, [\chi])$$:

 $$T_{3}^{2} + 4$$ $$T_{11} - 3$$ $$T_{13}^{2} + 1$$ $$T_{19} + 1$$ $$T_{31} + 8$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$1 - 2 T^{2} + 9 T^{4}$$
$5$ 1
$7$ 1
$11$ $$( 1 - 3 T + 11 T^{2} )^{2}$$
$13$ $$1 - 25 T^{2} + 169 T^{4}$$
$17$ $$1 + 2 T^{2} + 289 T^{4}$$
$19$ $$( 1 + T + 19 T^{2} )^{2}$$
$23$ $$1 + 35 T^{2} + 529 T^{4}$$
$29$ $$( 1 + 6 T + 29 T^{2} )^{2}$$
$31$ $$( 1 + 8 T + 31 T^{2} )^{2}$$
$37$ $$1 - 25 T^{2} + 1369 T^{4}$$
$41$ $$( 1 + 3 T + 41 T^{2} )^{2}$$
$43$ $$1 - 82 T^{2} + 1849 T^{4}$$
$47$ $$1 - 13 T^{2} + 2209 T^{4}$$
$53$ $$1 - 25 T^{2} + 2809 T^{4}$$
$59$ $$( 1 + 59 T^{2} )^{2}$$
$61$ $$( 1 + 8 T + 61 T^{2} )^{2}$$
$67$ $$1 - 70 T^{2} + 4489 T^{4}$$
$71$ $$( 1 + 71 T^{2} )^{2}$$
$73$ $$1 - 130 T^{2} + 5329 T^{4}$$
$79$ $$( 1 - 10 T + 79 T^{2} )^{2}$$
$83$ $$( 1 - 83 T^{2} )^{2}$$
$89$ $$( 1 - 6 T + 89 T^{2} )^{2}$$
$97$ $$1 - 94 T^{2} + 9409 T^{4}$$