# Properties

 Label 2450.2.a.bc Level $2450$ Weight $2$ Character orbit 2450.a Self dual yes Analytic conductor $19.563$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2450 = 2 \cdot 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2450.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$19.5633484952$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 70) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} + q^{6} + q^{8} - 2 q^{9}+O(q^{10})$$ q + q^2 + q^3 + q^4 + q^6 + q^8 - 2 * q^9 $$q + q^{2} + q^{3} + q^{4} + q^{6} + q^{8} - 2 q^{9} - 6 q^{11} + q^{12} - 4 q^{13} + q^{16} - 2 q^{18} - 2 q^{19} - 6 q^{22} + 3 q^{23} + q^{24} - 4 q^{26} - 5 q^{27} - 3 q^{29} - 8 q^{31} + q^{32} - 6 q^{33} - 2 q^{36} + 4 q^{37} - 2 q^{38} - 4 q^{39} - 9 q^{41} + 7 q^{43} - 6 q^{44} + 3 q^{46} + q^{48} - 4 q^{52} + 6 q^{53} - 5 q^{54} - 2 q^{57} - 3 q^{58} + 6 q^{59} - 5 q^{61} - 8 q^{62} + q^{64} - 6 q^{66} - 5 q^{67} + 3 q^{69} - 6 q^{71} - 2 q^{72} - 16 q^{73} + 4 q^{74} - 2 q^{76} - 4 q^{78} + 2 q^{79} + q^{81} - 9 q^{82} + 3 q^{83} + 7 q^{86} - 3 q^{87} - 6 q^{88} + 15 q^{89} + 3 q^{92} - 8 q^{93} + q^{96} + 14 q^{97} + 12 q^{99}+O(q^{100})$$ q + q^2 + q^3 + q^4 + q^6 + q^8 - 2 * q^9 - 6 * q^11 + q^12 - 4 * q^13 + q^16 - 2 * q^18 - 2 * q^19 - 6 * q^22 + 3 * q^23 + q^24 - 4 * q^26 - 5 * q^27 - 3 * q^29 - 8 * q^31 + q^32 - 6 * q^33 - 2 * q^36 + 4 * q^37 - 2 * q^38 - 4 * q^39 - 9 * q^41 + 7 * q^43 - 6 * q^44 + 3 * q^46 + q^48 - 4 * q^52 + 6 * q^53 - 5 * q^54 - 2 * q^57 - 3 * q^58 + 6 * q^59 - 5 * q^61 - 8 * q^62 + q^64 - 6 * q^66 - 5 * q^67 + 3 * q^69 - 6 * q^71 - 2 * q^72 - 16 * q^73 + 4 * q^74 - 2 * q^76 - 4 * q^78 + 2 * q^79 + q^81 - 9 * q^82 + 3 * q^83 + 7 * q^86 - 3 * q^87 - 6 * q^88 + 15 * q^89 + 3 * q^92 - 8 * q^93 + q^96 + 14 * q^97 + 12 * q^99

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
1.00000 1.00000 1.00000 0 1.00000 0 1.00000 −2.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2450.2.a.bc 1
5.b even 2 1 490.2.a.b 1
5.c odd 4 2 2450.2.c.l 2
7.b odd 2 1 2450.2.a.w 1
7.d odd 6 2 350.2.e.e 2
15.d odd 2 1 4410.2.a.bd 1
20.d odd 2 1 3920.2.a.bc 1
35.c odd 2 1 490.2.a.c 1
35.f even 4 2 2450.2.c.g 2
35.i odd 6 2 70.2.e.c 2
35.j even 6 2 490.2.e.h 2
35.k even 12 4 350.2.j.b 4
105.g even 2 1 4410.2.a.bm 1
105.p even 6 2 630.2.k.b 2
140.c even 2 1 3920.2.a.p 1
140.s even 6 2 560.2.q.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.e.c 2 35.i odd 6 2
350.2.e.e 2 7.d odd 6 2
350.2.j.b 4 35.k even 12 4
490.2.a.b 1 5.b even 2 1
490.2.a.c 1 35.c odd 2 1
490.2.e.h 2 35.j even 6 2
560.2.q.g 2 140.s even 6 2
630.2.k.b 2 105.p even 6 2
2450.2.a.w 1 7.b odd 2 1
2450.2.a.bc 1 1.a even 1 1 trivial
2450.2.c.g 2 35.f even 4 2
2450.2.c.l 2 5.c odd 4 2
3920.2.a.p 1 140.c even 2 1
3920.2.a.bc 1 20.d odd 2 1
4410.2.a.bd 1 15.d odd 2 1
4410.2.a.bm 1 105.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2450))$$:

 $$T_{3} - 1$$ T3 - 1 $$T_{11} + 6$$ T11 + 6 $$T_{13} + 4$$ T13 + 4 $$T_{17}$$ T17 $$T_{19} + 2$$ T19 + 2 $$T_{23} - 3$$ T23 - 3 $$T_{37} - 4$$ T37 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T - 1$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T + 6$$
$13$ $$T + 4$$
$17$ $$T$$
$19$ $$T + 2$$
$23$ $$T - 3$$
$29$ $$T + 3$$
$31$ $$T + 8$$
$37$ $$T - 4$$
$41$ $$T + 9$$
$43$ $$T - 7$$
$47$ $$T$$
$53$ $$T - 6$$
$59$ $$T - 6$$
$61$ $$T + 5$$
$67$ $$T + 5$$
$71$ $$T + 6$$
$73$ $$T + 16$$
$79$ $$T - 2$$
$83$ $$T - 3$$
$89$ $$T - 15$$
$97$ $$T - 14$$