Properties

 Label 1900.2.a.b Level $1900$ Weight $2$ Character orbit 1900.a Self dual yes Analytic conductor $15.172$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1900 = 2^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1900.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$15.1715763840$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 76) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q - 2 q^{3} + 3 q^{7} + q^{9} + O(q^{10})$$ $$q - 2 q^{3} + 3 q^{7} + q^{9} + 5 q^{11} + 4 q^{13} + 3 q^{17} - q^{19} - 6 q^{21} - 8 q^{23} + 4 q^{27} - 2 q^{29} + 4 q^{31} - 10 q^{33} - 10 q^{37} - 8 q^{39} + 10 q^{41} - q^{43} + q^{47} + 2 q^{49} - 6 q^{51} + 4 q^{53} + 2 q^{57} + 6 q^{59} - 13 q^{61} + 3 q^{63} + 12 q^{67} + 16 q^{69} + 2 q^{71} - 9 q^{73} + 15 q^{77} + 8 q^{79} - 11 q^{81} + 12 q^{83} + 4 q^{87} + 12 q^{89} + 12 q^{91} - 8 q^{93} + 8 q^{97} + 5 q^{99} + O(q^{100})$$

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
0 −2.00000 0 0 0 3.00000 0 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
$$2$$ $$-1$$
$$5$$ $$1$$
$$19$$ $$1$$

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 1900.2.a.b 1
4.b odd 2 1 7600.2.a.p 1
5.b even 2 1 76.2.a.a 1
5.c odd 4 2 1900.2.c.b 2
15.d odd 2 1 684.2.a.b 1
20.d odd 2 1 304.2.a.a 1
35.c odd 2 1 3724.2.a.a 1
40.e odd 2 1 1216.2.a.q 1
40.f even 2 1 1216.2.a.c 1
55.d odd 2 1 9196.2.a.f 1
60.h even 2 1 2736.2.a.q 1
95.d odd 2 1 1444.2.a.a 1
95.h odd 6 2 1444.2.e.c 2
95.i even 6 2 1444.2.e.a 2
380.d even 2 1 5776.2.a.p 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.2.a.a 1 5.b even 2 1
304.2.a.a 1 20.d odd 2 1
684.2.a.b 1 15.d odd 2 1
1216.2.a.c 1 40.f even 2 1
1216.2.a.q 1 40.e odd 2 1
1444.2.a.a 1 95.d odd 2 1
1444.2.e.a 2 95.i even 6 2
1444.2.e.c 2 95.h odd 6 2
1900.2.a.b 1 1.a even 1 1 trivial
1900.2.c.b 2 5.c odd 4 2
2736.2.a.q 1 60.h even 2 1
3724.2.a.a 1 35.c odd 2 1
5776.2.a.p 1 380.d even 2 1
7600.2.a.p 1 4.b odd 2 1
9196.2.a.f 1 55.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1900))$$:

 $$T_{3} + 2$$ $$T_{7} - 3$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$2 + T$$
$5$ $$T$$
$7$ $$-3 + T$$
$11$ $$-5 + T$$
$13$ $$-4 + T$$
$17$ $$-3 + T$$
$19$ $$1 + T$$
$23$ $$8 + T$$
$29$ $$2 + T$$
$31$ $$-4 + T$$
$37$ $$10 + T$$
$41$ $$-10 + T$$
$43$ $$1 + T$$
$47$ $$-1 + T$$
$53$ $$-4 + T$$
$59$ $$-6 + T$$
$61$ $$13 + T$$
$67$ $$-12 + T$$
$71$ $$-2 + T$$
$73$ $$9 + T$$
$79$ $$-8 + T$$
$83$ $$-12 + T$$
$89$ $$-12 + T$$
$97$ $$-8 + T$$