Properties

 Label 8450.2.a.a Level $8450$ Weight $2$ Character orbit 8450.a Self dual yes Analytic conductor $67.474$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$8450 = 2 \cdot 5^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8450.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$67.4735897080$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 650) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} - 2 q^{3} + q^{4} + 2 q^{6} - 5 q^{7} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 - 2 * q^3 + q^4 + 2 * q^6 - 5 * q^7 - q^8 + q^9 $$q - q^{2} - 2 q^{3} + q^{4} + 2 q^{6} - 5 q^{7} - q^{8} + q^{9} + 3 q^{11} - 2 q^{12} + 5 q^{14} + q^{16} + 3 q^{17} - q^{18} + 4 q^{19} + 10 q^{21} - 3 q^{22} + 6 q^{23} + 2 q^{24} + 4 q^{27} - 5 q^{28} + 9 q^{29} - 5 q^{31} - q^{32} - 6 q^{33} - 3 q^{34} + q^{36} - 2 q^{37} - 4 q^{38} - 10 q^{42} + 2 q^{43} + 3 q^{44} - 6 q^{46} + 9 q^{47} - 2 q^{48} + 18 q^{49} - 6 q^{51} - 9 q^{53} - 4 q^{54} + 5 q^{56} - 8 q^{57} - 9 q^{58} + 9 q^{59} - q^{61} + 5 q^{62} - 5 q^{63} + q^{64} + 6 q^{66} - 5 q^{67} + 3 q^{68} - 12 q^{69} - q^{72} - 14 q^{73} + 2 q^{74} + 4 q^{76} - 15 q^{77} - 16 q^{79} - 11 q^{81} + 15 q^{83} + 10 q^{84} - 2 q^{86} - 18 q^{87} - 3 q^{88} + 6 q^{89} + 6 q^{92} + 10 q^{93} - 9 q^{94} + 2 q^{96} - 8 q^{97} - 18 q^{98} + 3 q^{99}+O(q^{100})$$ q - q^2 - 2 * q^3 + q^4 + 2 * q^6 - 5 * q^7 - q^8 + q^9 + 3 * q^11 - 2 * q^12 + 5 * q^14 + q^16 + 3 * q^17 - q^18 + 4 * q^19 + 10 * q^21 - 3 * q^22 + 6 * q^23 + 2 * q^24 + 4 * q^27 - 5 * q^28 + 9 * q^29 - 5 * q^31 - q^32 - 6 * q^33 - 3 * q^34 + q^36 - 2 * q^37 - 4 * q^38 - 10 * q^42 + 2 * q^43 + 3 * q^44 - 6 * q^46 + 9 * q^47 - 2 * q^48 + 18 * q^49 - 6 * q^51 - 9 * q^53 - 4 * q^54 + 5 * q^56 - 8 * q^57 - 9 * q^58 + 9 * q^59 - q^61 + 5 * q^62 - 5 * q^63 + q^64 + 6 * q^66 - 5 * q^67 + 3 * q^68 - 12 * q^69 - q^72 - 14 * q^73 + 2 * q^74 + 4 * q^76 - 15 * q^77 - 16 * q^79 - 11 * q^81 + 15 * q^83 + 10 * q^84 - 2 * q^86 - 18 * q^87 - 3 * q^88 + 6 * q^89 + 6 * q^92 + 10 * q^93 - 9 * q^94 + 2 * q^96 - 8 * q^97 - 18 * q^98 + 3 * 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 −2.00000 1.00000 0 2.00000 −5.00000 −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
$$2$$ $$1$$
$$5$$ $$-1$$
$$13$$ $$1$$

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 8450.2.a.a 1
5.b even 2 1 8450.2.a.x 1
13.b even 2 1 650.2.a.h yes 1
39.d odd 2 1 5850.2.a.bb 1
52.b odd 2 1 5200.2.a.bc 1
65.d even 2 1 650.2.a.f 1
65.h odd 4 2 650.2.b.b 2
195.e odd 2 1 5850.2.a.bc 1
195.s even 4 2 5850.2.e.ba 2
260.g odd 2 1 5200.2.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
650.2.a.f 1 65.d even 2 1
650.2.a.h yes 1 13.b even 2 1
650.2.b.b 2 65.h odd 4 2
5200.2.a.i 1 260.g odd 2 1
5200.2.a.bc 1 52.b odd 2 1
5850.2.a.bb 1 39.d odd 2 1
5850.2.a.bc 1 195.e odd 2 1
5850.2.e.ba 2 195.s even 4 2
8450.2.a.a 1 1.a even 1 1 trivial
8450.2.a.x 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8450))$$:

 $$T_{3} + 2$$ T3 + 2 $$T_{7} + 5$$ T7 + 5 $$T_{11} - 3$$ T11 - 3 $$T_{17} - 3$$ T17 - 3 $$T_{31} + 5$$ T31 + 5

Hecke characteristic polynomials

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