Properties

 Label 2075.2.a.d Level $2075$ Weight $2$ Character orbit 2075.a Self dual yes Analytic conductor $16.569$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$2075 = 5^{2} \cdot 83$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2075.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$16.5689584194$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 83) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} - q^{4} + q^{6} + 3 q^{7} - 3 q^{8} - 2 q^{9}+O(q^{10})$$ q + q^2 + q^3 - q^4 + q^6 + 3 * q^7 - 3 * q^8 - 2 * q^9 $$q + q^{2} + q^{3} - q^{4} + q^{6} + 3 q^{7} - 3 q^{8} - 2 q^{9} + 3 q^{11} - q^{12} + 6 q^{13} + 3 q^{14} - q^{16} - 5 q^{17} - 2 q^{18} + 2 q^{19} + 3 q^{21} + 3 q^{22} + 4 q^{23} - 3 q^{24} + 6 q^{26} - 5 q^{27} - 3 q^{28} - 7 q^{29} + 5 q^{31} + 5 q^{32} + 3 q^{33} - 5 q^{34} + 2 q^{36} + 11 q^{37} + 2 q^{38} + 6 q^{39} - 2 q^{41} + 3 q^{42} + 8 q^{43} - 3 q^{44} + 4 q^{46} - q^{48} + 2 q^{49} - 5 q^{51} - 6 q^{52} - 6 q^{53} - 5 q^{54} - 9 q^{56} + 2 q^{57} - 7 q^{58} + 5 q^{59} + 5 q^{61} + 5 q^{62} - 6 q^{63} + 7 q^{64} + 3 q^{66} + 2 q^{67} + 5 q^{68} + 4 q^{69} + 2 q^{71} + 6 q^{72} + 11 q^{74} - 2 q^{76} + 9 q^{77} + 6 q^{78} + 14 q^{79} + q^{81} - 2 q^{82} + q^{83} - 3 q^{84} + 8 q^{86} - 7 q^{87} - 9 q^{88} + 18 q^{91} - 4 q^{92} + 5 q^{93} + 5 q^{96} + 8 q^{97} + 2 q^{98} - 6 q^{99}+O(q^{100})$$ q + q^2 + q^3 - q^4 + q^6 + 3 * q^7 - 3 * q^8 - 2 * q^9 + 3 * q^11 - q^12 + 6 * q^13 + 3 * q^14 - q^16 - 5 * q^17 - 2 * q^18 + 2 * q^19 + 3 * q^21 + 3 * q^22 + 4 * q^23 - 3 * q^24 + 6 * q^26 - 5 * q^27 - 3 * q^28 - 7 * q^29 + 5 * q^31 + 5 * q^32 + 3 * q^33 - 5 * q^34 + 2 * q^36 + 11 * q^37 + 2 * q^38 + 6 * q^39 - 2 * q^41 + 3 * q^42 + 8 * q^43 - 3 * q^44 + 4 * q^46 - q^48 + 2 * q^49 - 5 * q^51 - 6 * q^52 - 6 * q^53 - 5 * q^54 - 9 * q^56 + 2 * q^57 - 7 * q^58 + 5 * q^59 + 5 * q^61 + 5 * q^62 - 6 * q^63 + 7 * q^64 + 3 * q^66 + 2 * q^67 + 5 * q^68 + 4 * q^69 + 2 * q^71 + 6 * q^72 + 11 * q^74 - 2 * q^76 + 9 * q^77 + 6 * q^78 + 14 * q^79 + q^81 - 2 * q^82 + q^83 - 3 * q^84 + 8 * q^86 - 7 * q^87 - 9 * q^88 + 18 * q^91 - 4 * q^92 + 5 * q^93 + 5 * q^96 + 8 * q^97 + 2 * q^98 - 6 * 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 1.00000 −1.00000 0 1.00000 3.00000 −3.00000 −2.00000 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
$$5$$ $$1$$
$$83$$ $$-1$$

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 2075.2.a.d 1
5.b even 2 1 83.2.a.a 1
15.d odd 2 1 747.2.a.d 1
20.d odd 2 1 1328.2.a.c 1
35.c odd 2 1 4067.2.a.a 1
40.e odd 2 1 5312.2.a.h 1
40.f even 2 1 5312.2.a.l 1
415.d odd 2 1 6889.2.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
83.2.a.a 1 5.b even 2 1
747.2.a.d 1 15.d odd 2 1
1328.2.a.c 1 20.d odd 2 1
2075.2.a.d 1 1.a even 1 1 trivial
4067.2.a.a 1 35.c odd 2 1
5312.2.a.h 1 40.e odd 2 1
5312.2.a.l 1 40.f even 2 1
6889.2.a.a 1 415.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2075))$$:

 $$T_{2} - 1$$ T2 - 1 $$T_{3} - 1$$ T3 - 1

Hecke characteristic polynomials

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