# Properties

 Label 2450.2.c.r Level $2450$ Weight $2$ Character orbit 2450.c Analytic conductor $19.563$ Analytic rank $1$ 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: $$1$$ 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 350) 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.

 $$f(q)$$ $$=$$ $$q -i q^{2} + 3 i q^{3} - q^{4} + 3 q^{6} + i q^{8} -6 q^{9} +O(q^{10})$$ $$q -i q^{2} + 3 i q^{3} - q^{4} + 3 q^{6} + i q^{8} -6 q^{9} -5 q^{11} -3 i q^{12} + 6 i q^{13} + q^{16} -i q^{17} + 6 i q^{18} -3 q^{19} + 5 i q^{22} -3 q^{24} + 6 q^{26} -9 i q^{27} + 6 q^{29} + 4 q^{31} -i q^{32} -15 i q^{33} - q^{34} + 6 q^{36} -8 i q^{37} + 3 i q^{38} -18 q^{39} -11 q^{41} -8 i q^{43} + 5 q^{44} + 2 i q^{47} + 3 i q^{48} + 3 q^{51} -6 i q^{52} + 4 i q^{53} -9 q^{54} -9 i q^{57} -6 i q^{58} + 4 q^{59} + 2 q^{61} -4 i q^{62} - q^{64} -15 q^{66} -9 i q^{67} + i q^{68} -10 q^{71} -6 i q^{72} + 7 i q^{73} -8 q^{74} + 3 q^{76} + 18 i q^{78} + 2 q^{79} + 9 q^{81} + 11 i q^{82} -11 i q^{83} -8 q^{86} + 18 i q^{87} -5 i q^{88} -11 q^{89} + 12 i q^{93} + 2 q^{94} + 3 q^{96} -10 i q^{97} + 30 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + 6q^{6} - 12q^{9} + O(q^{10})$$ $$2q - 2q^{4} + 6q^{6} - 12q^{9} - 10q^{11} + 2q^{16} - 6q^{19} - 6q^{24} + 12q^{26} + 12q^{29} + 8q^{31} - 2q^{34} + 12q^{36} - 36q^{39} - 22q^{41} + 10q^{44} + 6q^{51} - 18q^{54} + 8q^{59} + 4q^{61} - 2q^{64} - 30q^{66} - 20q^{71} - 16q^{74} + 6q^{76} + 4q^{79} + 18q^{81} - 16q^{86} - 22q^{89} + 4q^{94} + 6q^{96} + 60q^{99} + O(q^{100})$$

## 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 3.00000i −1.00000 0 3.00000 0 1.00000i −6.00000 0
99.2 1.00000i 3.00000i −1.00000 0 3.00000 0 1.00000i −6.00000 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.r 2
5.b even 2 1 inner 2450.2.c.r 2
5.c odd 4 1 2450.2.a.a 1
5.c odd 4 1 2450.2.a.bg 1
7.b odd 2 1 350.2.c.a 2
21.c even 2 1 3150.2.g.v 2
28.d even 2 1 2800.2.g.a 2
35.c odd 2 1 350.2.c.a 2
35.f even 4 1 350.2.a.c 1
35.f even 4 1 350.2.a.d yes 1
105.g even 2 1 3150.2.g.v 2
105.k odd 4 1 3150.2.a.j 1
105.k odd 4 1 3150.2.a.bq 1
140.c even 2 1 2800.2.g.a 2
140.j odd 4 1 2800.2.a.b 1
140.j odd 4 1 2800.2.a.bg 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.2.a.c 1 35.f even 4 1
350.2.a.d yes 1 35.f even 4 1
350.2.c.a 2 7.b odd 2 1
350.2.c.a 2 35.c odd 2 1
2450.2.a.a 1 5.c odd 4 1
2450.2.a.bg 1 5.c odd 4 1
2450.2.c.r 2 1.a even 1 1 trivial
2450.2.c.r 2 5.b even 2 1 inner
2800.2.a.b 1 140.j odd 4 1
2800.2.a.bg 1 140.j odd 4 1
2800.2.g.a 2 28.d even 2 1
2800.2.g.a 2 140.c even 2 1
3150.2.a.j 1 105.k odd 4 1
3150.2.a.bq 1 105.k odd 4 1
3150.2.g.v 2 21.c even 2 1
3150.2.g.v 2 105.g even 2 1

## Hecke kernels

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

 $$T_{3}^{2} + 9$$ $$T_{11} + 5$$ $$T_{13}^{2} + 36$$ $$T_{19} + 3$$ $$T_{31} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$9 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$( 5 + T )^{2}$$
$13$ $$36 + T^{2}$$
$17$ $$1 + T^{2}$$
$19$ $$( 3 + T )^{2}$$
$23$ $$T^{2}$$
$29$ $$( -6 + T )^{2}$$
$31$ $$( -4 + T )^{2}$$
$37$ $$64 + T^{2}$$
$41$ $$( 11 + T )^{2}$$
$43$ $$64 + T^{2}$$
$47$ $$4 + T^{2}$$
$53$ $$16 + T^{2}$$
$59$ $$( -4 + T )^{2}$$
$61$ $$( -2 + T )^{2}$$
$67$ $$81 + T^{2}$$
$71$ $$( 10 + T )^{2}$$
$73$ $$49 + T^{2}$$
$79$ $$( -2 + T )^{2}$$
$83$ $$121 + T^{2}$$
$89$ $$( 11 + T )^{2}$$
$97$ $$100 + T^{2}$$