# Properties

 Label 2450.4.a.bs Level $2450$ Weight $4$ Character orbit 2450.a Self dual yes Analytic conductor $144.555$ 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: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2450.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$144.554679514$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{22})$$ Defining polynomial: $$x^{2} - 22$$ x^2 - 22 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 98) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{22}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + \beta q^{3} + 4 q^{4} - 2 \beta q^{6} - 8 q^{8} + 61 q^{9} +O(q^{10})$$ q - 2 * q^2 + b * q^3 + 4 * q^4 - 2*b * q^6 - 8 * q^8 + 61 * q^9 $$q - 2 q^{2} + \beta q^{3} + 4 q^{4} - 2 \beta q^{6} - 8 q^{8} + 61 q^{9} + 20 q^{11} + 4 \beta q^{12} - 7 \beta q^{13} + 16 q^{16} - 6 \beta q^{17} - 122 q^{18} + \beta q^{19} - 40 q^{22} - 48 q^{23} - 8 \beta q^{24} + 14 \beta q^{26} + 34 \beta q^{27} - 166 q^{29} - 22 \beta q^{31} - 32 q^{32} + 20 \beta q^{33} + 12 \beta q^{34} + 244 q^{36} + 78 q^{37} - 2 \beta q^{38} - 616 q^{39} + 42 \beta q^{41} - 436 q^{43} + 80 q^{44} + 96 q^{46} - 22 \beta q^{47} + 16 \beta q^{48} - 528 q^{51} - 28 \beta q^{52} - 62 q^{53} - 68 \beta q^{54} + 88 q^{57} + 332 q^{58} - 71 \beta q^{59} + 29 \beta q^{61} + 44 \beta q^{62} + 64 q^{64} - 40 \beta q^{66} - 580 q^{67} - 24 \beta q^{68} - 48 \beta q^{69} - 544 q^{71} - 488 q^{72} + 64 \beta q^{73} - 156 q^{74} + 4 \beta q^{76} + 1232 q^{78} - 680 q^{79} + 1345 q^{81} - 84 \beta q^{82} - 21 \beta q^{83} + 872 q^{86} - 166 \beta q^{87} - 160 q^{88} - 160 \beta q^{89} - 192 q^{92} - 1936 q^{93} + 44 \beta q^{94} - 32 \beta q^{96} + 70 \beta q^{97} + 1220 q^{99} +O(q^{100})$$ q - 2 * q^2 + b * q^3 + 4 * q^4 - 2*b * q^6 - 8 * q^8 + 61 * q^9 + 20 * q^11 + 4*b * q^12 - 7*b * q^13 + 16 * q^16 - 6*b * q^17 - 122 * q^18 + b * q^19 - 40 * q^22 - 48 * q^23 - 8*b * q^24 + 14*b * q^26 + 34*b * q^27 - 166 * q^29 - 22*b * q^31 - 32 * q^32 + 20*b * q^33 + 12*b * q^34 + 244 * q^36 + 78 * q^37 - 2*b * q^38 - 616 * q^39 + 42*b * q^41 - 436 * q^43 + 80 * q^44 + 96 * q^46 - 22*b * q^47 + 16*b * q^48 - 528 * q^51 - 28*b * q^52 - 62 * q^53 - 68*b * q^54 + 88 * q^57 + 332 * q^58 - 71*b * q^59 + 29*b * q^61 + 44*b * q^62 + 64 * q^64 - 40*b * q^66 - 580 * q^67 - 24*b * q^68 - 48*b * q^69 - 544 * q^71 - 488 * q^72 + 64*b * q^73 - 156 * q^74 + 4*b * q^76 + 1232 * q^78 - 680 * q^79 + 1345 * q^81 - 84*b * q^82 - 21*b * q^83 + 872 * q^86 - 166*b * q^87 - 160 * q^88 - 160*b * q^89 - 192 * q^92 - 1936 * q^93 + 44*b * q^94 - 32*b * q^96 + 70*b * q^97 + 1220 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{2} + 8 q^{4} - 16 q^{8} + 122 q^{9}+O(q^{10})$$ 2 * q - 4 * q^2 + 8 * q^4 - 16 * q^8 + 122 * q^9 $$2 q - 4 q^{2} + 8 q^{4} - 16 q^{8} + 122 q^{9} + 40 q^{11} + 32 q^{16} - 244 q^{18} - 80 q^{22} - 96 q^{23} - 332 q^{29} - 64 q^{32} + 488 q^{36} + 156 q^{37} - 1232 q^{39} - 872 q^{43} + 160 q^{44} + 192 q^{46} - 1056 q^{51} - 124 q^{53} + 176 q^{57} + 664 q^{58} + 128 q^{64} - 1160 q^{67} - 1088 q^{71} - 976 q^{72} - 312 q^{74} + 2464 q^{78} - 1360 q^{79} + 2690 q^{81} + 1744 q^{86} - 320 q^{88} - 384 q^{92} - 3872 q^{93} + 2440 q^{99}+O(q^{100})$$ 2 * q - 4 * q^2 + 8 * q^4 - 16 * q^8 + 122 * q^9 + 40 * q^11 + 32 * q^16 - 244 * q^18 - 80 * q^22 - 96 * q^23 - 332 * q^29 - 64 * q^32 + 488 * q^36 + 156 * q^37 - 1232 * q^39 - 872 * q^43 + 160 * q^44 + 192 * q^46 - 1056 * q^51 - 124 * q^53 + 176 * q^57 + 664 * q^58 + 128 * q^64 - 1160 * q^67 - 1088 * q^71 - 976 * q^72 - 312 * q^74 + 2464 * q^78 - 1360 * q^79 + 2690 * q^81 + 1744 * q^86 - 320 * q^88 - 384 * q^92 - 3872 * q^93 + 2440 * 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
 −4.69042 4.69042
−2.00000 −9.38083 4.00000 0 18.7617 0 −8.00000 61.0000 0
1.2 −2.00000 9.38083 4.00000 0 −18.7617 0 −8.00000 61.0000 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2450.4.a.bs 2
5.b even 2 1 98.4.a.h 2
7.b odd 2 1 inner 2450.4.a.bs 2
15.d odd 2 1 882.4.a.w 2
20.d odd 2 1 784.4.a.z 2
35.c odd 2 1 98.4.a.h 2
35.i odd 6 2 98.4.c.g 4
35.j even 6 2 98.4.c.g 4
105.g even 2 1 882.4.a.w 2
105.o odd 6 2 882.4.g.bi 4
105.p even 6 2 882.4.g.bi 4
140.c even 2 1 784.4.a.z 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.4.a.h 2 5.b even 2 1
98.4.a.h 2 35.c odd 2 1
98.4.c.g 4 35.i odd 6 2
98.4.c.g 4 35.j even 6 2
784.4.a.z 2 20.d odd 2 1
784.4.a.z 2 140.c even 2 1
882.4.a.w 2 15.d odd 2 1
882.4.a.w 2 105.g even 2 1
882.4.g.bi 4 105.o odd 6 2
882.4.g.bi 4 105.p even 6 2
2450.4.a.bs 2 1.a even 1 1 trivial
2450.4.a.bs 2 7.b odd 2 1 inner

## Hecke kernels

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

 $$T_{3}^{2} - 88$$ T3^2 - 88 $$T_{11} - 20$$ T11 - 20 $$T_{19}^{2} - 88$$ T19^2 - 88 $$T_{23} + 48$$ T23 + 48

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 2)^{2}$$
$3$ $$T^{2} - 88$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$(T - 20)^{2}$$
$13$ $$T^{2} - 4312$$
$17$ $$T^{2} - 3168$$
$19$ $$T^{2} - 88$$
$23$ $$(T + 48)^{2}$$
$29$ $$(T + 166)^{2}$$
$31$ $$T^{2} - 42592$$
$37$ $$(T - 78)^{2}$$
$41$ $$T^{2} - 155232$$
$43$ $$(T + 436)^{2}$$
$47$ $$T^{2} - 42592$$
$53$ $$(T + 62)^{2}$$
$59$ $$T^{2} - 443608$$
$61$ $$T^{2} - 74008$$
$67$ $$(T + 580)^{2}$$
$71$ $$(T + 544)^{2}$$
$73$ $$T^{2} - 360448$$
$79$ $$(T + 680)^{2}$$
$83$ $$T^{2} - 38808$$
$89$ $$T^{2} - 2252800$$
$97$ $$T^{2} - 431200$$