# Properties

 Label 5054.2.a.c Level $5054$ Weight $2$ Character orbit 5054.a Self dual yes Analytic conductor $40.356$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5054 = 2 \cdot 7 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5054.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$40.3563931816$$ 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

 $$f(q)$$ $$=$$ $$q + q^{2} + 2 q^{3} + q^{4} + 2 q^{6} + q^{7} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 + 2 * q^3 + q^4 + 2 * q^6 + q^7 + q^8 + q^9 $$q + q^{2} + 2 q^{3} + q^{4} + 2 q^{6} + q^{7} + q^{8} + q^{9} + 2 q^{12} + 4 q^{13} + q^{14} + q^{16} + 6 q^{17} + q^{18} + 2 q^{21} + 2 q^{24} - 5 q^{25} + 4 q^{26} - 4 q^{27} + q^{28} + 6 q^{29} + 4 q^{31} + q^{32} + 6 q^{34} + q^{36} - 2 q^{37} + 8 q^{39} - 6 q^{41} + 2 q^{42} + 8 q^{43} - 12 q^{47} + 2 q^{48} + q^{49} - 5 q^{50} + 12 q^{51} + 4 q^{52} - 6 q^{53} - 4 q^{54} + q^{56} + 6 q^{58} + 6 q^{59} + 8 q^{61} + 4 q^{62} + q^{63} + q^{64} + 4 q^{67} + 6 q^{68} + q^{72} + 2 q^{73} - 2 q^{74} - 10 q^{75} + 8 q^{78} - 8 q^{79} - 11 q^{81} - 6 q^{82} - 6 q^{83} + 2 q^{84} + 8 q^{86} + 12 q^{87} + 6 q^{89} + 4 q^{91} + 8 q^{93} - 12 q^{94} + 2 q^{96} + 10 q^{97} + q^{98}+O(q^{100})$$ q + q^2 + 2 * q^3 + q^4 + 2 * q^6 + q^7 + q^8 + q^9 + 2 * q^12 + 4 * q^13 + q^14 + q^16 + 6 * q^17 + q^18 + 2 * q^21 + 2 * q^24 - 5 * q^25 + 4 * q^26 - 4 * q^27 + q^28 + 6 * q^29 + 4 * q^31 + q^32 + 6 * q^34 + q^36 - 2 * q^37 + 8 * q^39 - 6 * q^41 + 2 * q^42 + 8 * q^43 - 12 * q^47 + 2 * q^48 + q^49 - 5 * q^50 + 12 * q^51 + 4 * q^52 - 6 * q^53 - 4 * q^54 + q^56 + 6 * q^58 + 6 * q^59 + 8 * q^61 + 4 * q^62 + q^63 + q^64 + 4 * q^67 + 6 * q^68 + q^72 + 2 * q^73 - 2 * q^74 - 10 * q^75 + 8 * q^78 - 8 * q^79 - 11 * q^81 - 6 * q^82 - 6 * q^83 + 2 * q^84 + 8 * q^86 + 12 * q^87 + 6 * q^89 + 4 * q^91 + 8 * q^93 - 12 * q^94 + 2 * q^96 + 10 * q^97 + q^98

## 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 2.00000 1.00000 0 2.00000 1.00000 1.00000 1.00000 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
$$2$$ $$-1$$
$$7$$ $$-1$$
$$19$$ $$-1$$

## 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 5054.2.a.c 1
19.b odd 2 1 14.2.a.a 1
57.d even 2 1 126.2.a.b 1
76.d even 2 1 112.2.a.c 1
95.d odd 2 1 350.2.a.f 1
95.g even 4 2 350.2.c.d 2
133.c even 2 1 98.2.a.a 1
133.o even 6 2 98.2.c.a 2
133.r odd 6 2 98.2.c.b 2
152.b even 2 1 448.2.a.a 1
152.g odd 2 1 448.2.a.g 1
171.l even 6 2 1134.2.f.f 2
171.o odd 6 2 1134.2.f.l 2
209.d even 2 1 1694.2.a.e 1
228.b odd 2 1 1008.2.a.h 1
247.d odd 2 1 2366.2.a.j 1
247.i even 4 2 2366.2.d.b 2
285.b even 2 1 3150.2.a.i 1
285.j odd 4 2 3150.2.g.j 2
304.j odd 4 2 1792.2.b.c 2
304.m even 4 2 1792.2.b.g 2
323.c odd 2 1 4046.2.a.f 1
380.d even 2 1 2800.2.a.g 1
380.j odd 4 2 2800.2.g.h 2
399.h odd 2 1 882.2.a.i 1
399.s odd 6 2 882.2.g.d 2
399.w even 6 2 882.2.g.c 2
437.b even 2 1 7406.2.a.a 1
456.l odd 2 1 4032.2.a.r 1
456.p even 2 1 4032.2.a.w 1
532.b odd 2 1 784.2.a.b 1
532.t even 6 2 784.2.i.c 2
532.bh odd 6 2 784.2.i.i 2
665.g even 2 1 2450.2.a.t 1
665.n odd 4 2 2450.2.c.c 2
1064.f even 2 1 3136.2.a.e 1
1064.p odd 2 1 3136.2.a.z 1
1596.p even 2 1 7056.2.a.bd 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.2.a.a 1 19.b odd 2 1
98.2.a.a 1 133.c even 2 1
98.2.c.a 2 133.o even 6 2
98.2.c.b 2 133.r odd 6 2
112.2.a.c 1 76.d even 2 1
126.2.a.b 1 57.d even 2 1
350.2.a.f 1 95.d odd 2 1
350.2.c.d 2 95.g even 4 2
448.2.a.a 1 152.b even 2 1
448.2.a.g 1 152.g odd 2 1
784.2.a.b 1 532.b odd 2 1
784.2.i.c 2 532.t even 6 2
784.2.i.i 2 532.bh odd 6 2
882.2.a.i 1 399.h odd 2 1
882.2.g.c 2 399.w even 6 2
882.2.g.d 2 399.s odd 6 2
1008.2.a.h 1 228.b odd 2 1
1134.2.f.f 2 171.l even 6 2
1134.2.f.l 2 171.o odd 6 2
1694.2.a.e 1 209.d even 2 1
1792.2.b.c 2 304.j odd 4 2
1792.2.b.g 2 304.m even 4 2
2366.2.a.j 1 247.d odd 2 1
2366.2.d.b 2 247.i even 4 2
2450.2.a.t 1 665.g even 2 1
2450.2.c.c 2 665.n odd 4 2
2800.2.a.g 1 380.d even 2 1
2800.2.g.h 2 380.j odd 4 2
3136.2.a.e 1 1064.f even 2 1
3136.2.a.z 1 1064.p odd 2 1
3150.2.a.i 1 285.b even 2 1
3150.2.g.j 2 285.j odd 4 2
4032.2.a.r 1 456.l odd 2 1
4032.2.a.w 1 456.p even 2 1
4046.2.a.f 1 323.c odd 2 1
5054.2.a.c 1 1.a even 1 1 trivial
7056.2.a.bd 1 1596.p even 2 1
7406.2.a.a 1 437.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5054))$$:

 $$T_{3} - 2$$ T3 - 2 $$T_{5}$$ T5 $$T_{13} - 4$$ T13 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T - 2$$
$5$ $$T$$
$7$ $$T - 1$$
$11$ $$T$$
$13$ $$T - 4$$
$17$ $$T - 6$$
$19$ $$T$$
$23$ $$T$$
$29$ $$T - 6$$
$31$ $$T - 4$$
$37$ $$T + 2$$
$41$ $$T + 6$$
$43$ $$T - 8$$
$47$ $$T + 12$$
$53$ $$T + 6$$
$59$ $$T - 6$$
$61$ $$T - 8$$
$67$ $$T - 4$$
$71$ $$T$$
$73$ $$T - 2$$
$79$ $$T + 8$$
$83$ $$T + 6$$
$89$ $$T - 6$$
$97$ $$T - 10$$