# Properties

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

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## 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: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} - q^{4} -i q^{8} + 3 q^{9} +O(q^{10})$$ $$q + i q^{2} - q^{4} -i q^{8} + 3 q^{9} + 4 q^{11} -6 i q^{13} + q^{16} -2 i q^{17} + 3 i q^{18} + 4 i q^{22} + 6 q^{26} -6 q^{29} -8 q^{31} + i q^{32} + 2 q^{34} -3 q^{36} -10 i q^{37} -2 q^{41} -4 i q^{43} -4 q^{44} -8 i q^{47} + 6 i q^{52} + 2 i q^{53} -6 i q^{58} -8 q^{59} + 14 q^{61} -8 i q^{62} - q^{64} -12 i q^{67} + 2 i q^{68} -16 q^{71} -3 i q^{72} + 2 i q^{73} + 10 q^{74} + 8 q^{79} + 9 q^{81} -2 i q^{82} + 8 i q^{83} + 4 q^{86} -4 i q^{88} + 10 q^{89} + 8 q^{94} -2 i q^{97} + 12 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + 6q^{9} + O(q^{10})$$ $$2q - 2q^{4} + 6q^{9} + 8q^{11} + 2q^{16} + 12q^{26} - 12q^{29} - 16q^{31} + 4q^{34} - 6q^{36} - 4q^{41} - 8q^{44} - 16q^{59} + 28q^{61} - 2q^{64} - 32q^{71} + 20q^{74} + 16q^{79} + 18q^{81} + 8q^{86} + 20q^{89} + 16q^{94} + 24q^{99} + O(q^{100})$$

## 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 0 −1.00000 0 0 0 1.00000i 3.00000 0
99.2 1.00000i 0 −1.00000 0 0 0 1.00000i 3.00000 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.k 2
5.b even 2 1 inner 2450.2.c.k 2
5.c odd 4 1 490.2.a.h 1
5.c odd 4 1 2450.2.a.l 1
7.b odd 2 1 350.2.c.b 2
15.e even 4 1 4410.2.a.b 1
20.e even 4 1 3920.2.a.t 1
21.c even 2 1 3150.2.g.c 2
28.d even 2 1 2800.2.g.n 2
35.c odd 2 1 350.2.c.b 2
35.f even 4 1 70.2.a.a 1
35.f even 4 1 350.2.a.b 1
35.k even 12 2 490.2.e.d 2
35.l odd 12 2 490.2.e.c 2
105.g even 2 1 3150.2.g.c 2
105.k odd 4 1 630.2.a.d 1
105.k odd 4 1 3150.2.a.bj 1
140.c even 2 1 2800.2.g.n 2
140.j odd 4 1 560.2.a.d 1
140.j odd 4 1 2800.2.a.m 1
280.s even 4 1 2240.2.a.n 1
280.y odd 4 1 2240.2.a.q 1
385.l odd 4 1 8470.2.a.j 1
420.w even 4 1 5040.2.a.bm 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.a.a 1 35.f even 4 1
350.2.a.b 1 35.f even 4 1
350.2.c.b 2 7.b odd 2 1
350.2.c.b 2 35.c odd 2 1
490.2.a.h 1 5.c odd 4 1
490.2.e.c 2 35.l odd 12 2
490.2.e.d 2 35.k even 12 2
560.2.a.d 1 140.j odd 4 1
630.2.a.d 1 105.k odd 4 1
2240.2.a.n 1 280.s even 4 1
2240.2.a.q 1 280.y odd 4 1
2450.2.a.l 1 5.c odd 4 1
2450.2.c.k 2 1.a even 1 1 trivial
2450.2.c.k 2 5.b even 2 1 inner
2800.2.a.m 1 140.j odd 4 1
2800.2.g.n 2 28.d even 2 1
2800.2.g.n 2 140.c even 2 1
3150.2.a.bj 1 105.k odd 4 1
3150.2.g.c 2 21.c even 2 1
3150.2.g.c 2 105.g even 2 1
3920.2.a.t 1 20.e even 4 1
4410.2.a.b 1 15.e even 4 1
5040.2.a.bm 1 420.w even 4 1
8470.2.a.j 1 385.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2450, [\chi])$$:

 $$T_{3}$$ $$T_{11} - 4$$ $$T_{13}^{2} + 36$$ $$T_{19}$$ $$T_{31} + 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$( -4 + T )^{2}$$
$13$ $$36 + T^{2}$$
$17$ $$4 + T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$( 6 + T )^{2}$$
$31$ $$( 8 + T )^{2}$$
$37$ $$100 + T^{2}$$
$41$ $$( 2 + T )^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$64 + T^{2}$$
$53$ $$4 + T^{2}$$
$59$ $$( 8 + T )^{2}$$
$61$ $$( -14 + T )^{2}$$
$67$ $$144 + T^{2}$$
$71$ $$( 16 + T )^{2}$$
$73$ $$4 + T^{2}$$
$79$ $$( -8 + T )^{2}$$
$83$ $$64 + T^{2}$$
$89$ $$( -10 + T )^{2}$$
$97$ $$4 + T^{2}$$
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