# Properties

 Label 2450.4.a.bo Level $2450$ Weight $4$ Character orbit 2450.a Self dual yes Analytic conductor $144.555$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2450 = 2 \cdot 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2450.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$144.554679514$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 14) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 2q^{2} + 8q^{3} + 4q^{4} + 16q^{6} + 8q^{8} + 37q^{9} + O(q^{10})$$ $$q + 2q^{2} + 8q^{3} + 4q^{4} + 16q^{6} + 8q^{8} + 37q^{9} - 28q^{11} + 32q^{12} + 18q^{13} + 16q^{16} + 74q^{17} + 74q^{18} - 80q^{19} - 56q^{22} + 112q^{23} + 64q^{24} + 36q^{26} + 80q^{27} + 190q^{29} - 72q^{31} + 32q^{32} - 224q^{33} + 148q^{34} + 148q^{36} + 346q^{37} - 160q^{38} + 144q^{39} - 162q^{41} + 412q^{43} - 112q^{44} + 224q^{46} + 24q^{47} + 128q^{48} + 592q^{51} + 72q^{52} - 318q^{53} + 160q^{54} - 640q^{57} + 380q^{58} + 200q^{59} + 198q^{61} - 144q^{62} + 64q^{64} - 448q^{66} + 716q^{67} + 296q^{68} + 896q^{69} + 392q^{71} + 296q^{72} + 538q^{73} + 692q^{74} - 320q^{76} + 288q^{78} + 240q^{79} - 359q^{81} - 324q^{82} - 1072q^{83} + 824q^{86} + 1520q^{87} - 224q^{88} - 810q^{89} + 448q^{92} - 576q^{93} + 48q^{94} + 256q^{96} + 1354q^{97} - 1036q^{99} + O(q^{100})$$

## 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
2.00000 8.00000 4.00000 0 16.0000 0 8.00000 37.0000 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 2450.4.a.bo 1
5.b even 2 1 98.4.a.a 1
7.b odd 2 1 350.4.a.l 1
15.d odd 2 1 882.4.a.i 1
20.d odd 2 1 784.4.a.s 1
35.c odd 2 1 14.4.a.a 1
35.f even 4 2 350.4.c.b 2
35.i odd 6 2 98.4.c.d 2
35.j even 6 2 98.4.c.f 2
105.g even 2 1 126.4.a.h 1
105.o odd 6 2 882.4.g.k 2
105.p even 6 2 882.4.g.b 2
140.c even 2 1 112.4.a.a 1
280.c odd 2 1 448.4.a.b 1
280.n even 2 1 448.4.a.o 1
385.h even 2 1 1694.4.a.g 1
420.o odd 2 1 1008.4.a.s 1
455.h odd 2 1 2366.4.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.a.a 1 35.c odd 2 1
98.4.a.a 1 5.b even 2 1
98.4.c.d 2 35.i odd 6 2
98.4.c.f 2 35.j even 6 2
112.4.a.a 1 140.c even 2 1
126.4.a.h 1 105.g even 2 1
350.4.a.l 1 7.b odd 2 1
350.4.c.b 2 35.f even 4 2
448.4.a.b 1 280.c odd 2 1
448.4.a.o 1 280.n even 2 1
784.4.a.s 1 20.d odd 2 1
882.4.a.i 1 15.d odd 2 1
882.4.g.b 2 105.p even 6 2
882.4.g.k 2 105.o odd 6 2
1008.4.a.s 1 420.o odd 2 1
1694.4.a.g 1 385.h even 2 1
2366.4.a.h 1 455.h odd 2 1
2450.4.a.bo 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(2450))$$:

 $$T_{3} - 8$$ $$T_{11} + 28$$ $$T_{19} + 80$$ $$T_{23} - 112$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-2 + T$$
$3$ $$-8 + T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$28 + T$$
$13$ $$-18 + T$$
$17$ $$-74 + T$$
$19$ $$80 + T$$
$23$ $$-112 + T$$
$29$ $$-190 + T$$
$31$ $$72 + T$$
$37$ $$-346 + T$$
$41$ $$162 + T$$
$43$ $$-412 + T$$
$47$ $$-24 + T$$
$53$ $$318 + T$$
$59$ $$-200 + T$$
$61$ $$-198 + T$$
$67$ $$-716 + T$$
$71$ $$-392 + T$$
$73$ $$-538 + T$$
$79$ $$-240 + T$$
$83$ $$1072 + T$$
$89$ $$810 + T$$
$97$ $$-1354 + T$$