# Properties

 Label 2450.4.a.c 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 70) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 2 q^{2} - 7 q^{3} + 4 q^{4} + 14 q^{6} - 8 q^{8} + 22 q^{9}+O(q^{10})$$ q - 2 * q^2 - 7 * q^3 + 4 * q^4 + 14 * q^6 - 8 * q^8 + 22 * q^9 $$q - 2 q^{2} - 7 q^{3} + 4 q^{4} + 14 q^{6} - 8 q^{8} + 22 q^{9} - 37 q^{11} - 28 q^{12} + 51 q^{13} + 16 q^{16} + 41 q^{17} - 44 q^{18} + 108 q^{19} + 74 q^{22} + 70 q^{23} + 56 q^{24} - 102 q^{26} + 35 q^{27} - 249 q^{29} + 134 q^{31} - 32 q^{32} + 259 q^{33} - 82 q^{34} + 88 q^{36} + 334 q^{37} - 216 q^{38} - 357 q^{39} - 206 q^{41} + 376 q^{43} - 148 q^{44} - 140 q^{46} - 287 q^{47} - 112 q^{48} - 287 q^{51} + 204 q^{52} + 6 q^{53} - 70 q^{54} - 756 q^{57} + 498 q^{58} + 2 q^{59} + 940 q^{61} - 268 q^{62} + 64 q^{64} - 518 q^{66} - 106 q^{67} + 164 q^{68} - 490 q^{69} + 456 q^{71} - 176 q^{72} + 650 q^{73} - 668 q^{74} + 432 q^{76} + 714 q^{78} - 1239 q^{79} - 839 q^{81} + 412 q^{82} + 428 q^{83} - 752 q^{86} + 1743 q^{87} + 296 q^{88} + 220 q^{89} + 280 q^{92} - 938 q^{93} + 574 q^{94} + 224 q^{96} - 1055 q^{97} - 814 q^{99}+O(q^{100})$$ q - 2 * q^2 - 7 * q^3 + 4 * q^4 + 14 * q^6 - 8 * q^8 + 22 * q^9 - 37 * q^11 - 28 * q^12 + 51 * q^13 + 16 * q^16 + 41 * q^17 - 44 * q^18 + 108 * q^19 + 74 * q^22 + 70 * q^23 + 56 * q^24 - 102 * q^26 + 35 * q^27 - 249 * q^29 + 134 * q^31 - 32 * q^32 + 259 * q^33 - 82 * q^34 + 88 * q^36 + 334 * q^37 - 216 * q^38 - 357 * q^39 - 206 * q^41 + 376 * q^43 - 148 * q^44 - 140 * q^46 - 287 * q^47 - 112 * q^48 - 287 * q^51 + 204 * q^52 + 6 * q^53 - 70 * q^54 - 756 * q^57 + 498 * q^58 + 2 * q^59 + 940 * q^61 - 268 * q^62 + 64 * q^64 - 518 * q^66 - 106 * q^67 + 164 * q^68 - 490 * q^69 + 456 * q^71 - 176 * q^72 + 650 * q^73 - 668 * q^74 + 432 * q^76 + 714 * q^78 - 1239 * q^79 - 839 * q^81 + 412 * q^82 + 428 * q^83 - 752 * q^86 + 1743 * q^87 + 296 * q^88 + 220 * q^89 + 280 * q^92 - 938 * q^93 + 574 * q^94 + 224 * q^96 - 1055 * q^97 - 814 * 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
−2.00000 −7.00000 4.00000 0 14.0000 0 −8.00000 22.0000 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.c 1
5.b even 2 1 2450.4.a.bn 1
5.c odd 4 2 490.4.c.a 2
7.b odd 2 1 350.4.a.i 1
35.c odd 2 1 350.4.a.m 1
35.f even 4 2 70.4.c.a 2
105.k odd 4 2 630.4.g.a 2
140.j odd 4 2 560.4.g.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.4.c.a 2 35.f even 4 2
350.4.a.i 1 7.b odd 2 1
350.4.a.m 1 35.c odd 2 1
490.4.c.a 2 5.c odd 4 2
560.4.g.c 2 140.j odd 4 2
630.4.g.a 2 105.k odd 4 2
2450.4.a.c 1 1.a even 1 1 trivial
2450.4.a.bn 1 5.b even 2 1

## Hecke kernels

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

 $$T_{3} + 7$$ T3 + 7 $$T_{11} + 37$$ T11 + 37 $$T_{19} - 108$$ T19 - 108 $$T_{23} - 70$$ T23 - 70

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T + 7$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T + 37$$
$13$ $$T - 51$$
$17$ $$T - 41$$
$19$ $$T - 108$$
$23$ $$T - 70$$
$29$ $$T + 249$$
$31$ $$T - 134$$
$37$ $$T - 334$$
$41$ $$T + 206$$
$43$ $$T - 376$$
$47$ $$T + 287$$
$53$ $$T - 6$$
$59$ $$T - 2$$
$61$ $$T - 940$$
$67$ $$T + 106$$
$71$ $$T - 456$$
$73$ $$T - 650$$
$79$ $$T + 1239$$
$83$ $$T - 428$$
$89$ $$T - 220$$
$97$ $$T + 1055$$