# Properties

 Label 1900.2.c.c Level $1900$ Weight $2$ Character orbit 1900.c Analytic conductor $15.172$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1900 = 2^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1900.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$15.1715763840$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 380) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{7} + 3 q^{9} +O(q^{10})$$ q - b * q^7 + 3 * q^9 $$q - \beta q^{7} + 3 q^{9} - 4 q^{11} + 2 \beta q^{13} + 3 \beta q^{17} - q^{19} + \beta q^{23} + 6 q^{29} - 8 q^{31} + 2 \beta q^{37} + 6 q^{41} + 3 \beta q^{43} + 3 \beta q^{47} + 3 q^{49} - 4 \beta q^{53} + 12 q^{59} + 6 q^{61} - 3 \beta q^{63} + 5 \beta q^{73} + 4 \beta q^{77} + 8 q^{79} + 9 q^{81} - 7 \beta q^{83} - 14 q^{89} + 8 q^{91} + 8 \beta q^{97} - 12 q^{99} +O(q^{100})$$ q - b * q^7 + 3 * q^9 - 4 * q^11 + 2*b * q^13 + 3*b * q^17 - q^19 + b * q^23 + 6 * q^29 - 8 * q^31 + 2*b * q^37 + 6 * q^41 + 3*b * q^43 + 3*b * q^47 + 3 * q^49 - 4*b * q^53 + 12 * q^59 + 6 * q^61 - 3*b * q^63 + 5*b * q^73 + 4*b * q^77 + 8 * q^79 + 9 * q^81 - 7*b * q^83 - 14 * q^89 + 8 * q^91 + 8*b * q^97 - 12 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{9}+O(q^{10})$$ 2 * q + 6 * q^9 $$2 q + 6 q^{9} - 8 q^{11} - 2 q^{19} + 12 q^{29} - 16 q^{31} + 12 q^{41} + 6 q^{49} + 24 q^{59} + 12 q^{61} + 16 q^{79} + 18 q^{81} - 28 q^{89} + 16 q^{91} - 24 q^{99}+O(q^{100})$$ 2 * q + 6 * q^9 - 8 * q^11 - 2 * q^19 + 12 * q^29 - 16 * q^31 + 12 * q^41 + 6 * q^49 + 24 * q^59 + 12 * q^61 + 16 * q^79 + 18 * q^81 - 28 * q^89 + 16 * q^91 - 24 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$951$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1749.1
 1.00000i − 1.00000i
0 0 0 0 0 2.00000i 0 3.00000 0
1749.2 0 0 0 0 0 2.00000i 0 3.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1900.2.c.c 2
5.b even 2 1 inner 1900.2.c.c 2
5.c odd 4 1 380.2.a.a 1
5.c odd 4 1 1900.2.a.c 1
15.e even 4 1 3420.2.a.d 1
20.e even 4 1 1520.2.a.e 1
20.e even 4 1 7600.2.a.j 1
40.i odd 4 1 6080.2.a.m 1
40.k even 4 1 6080.2.a.n 1
95.g even 4 1 7220.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.a.a 1 5.c odd 4 1
1520.2.a.e 1 20.e even 4 1
1900.2.a.c 1 5.c odd 4 1
1900.2.c.c 2 1.a even 1 1 trivial
1900.2.c.c 2 5.b even 2 1 inner
3420.2.a.d 1 15.e even 4 1
6080.2.a.m 1 40.i odd 4 1
6080.2.a.n 1 40.k even 4 1
7220.2.a.d 1 95.g even 4 1
7600.2.a.j 1 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1900, [\chi])$$:

 $$T_{3}$$ T3 $$T_{7}^{2} + 4$$ T7^2 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 4$$
$11$ $$(T + 4)^{2}$$
$13$ $$T^{2} + 16$$
$17$ $$T^{2} + 36$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2} + 4$$
$29$ $$(T - 6)^{2}$$
$31$ $$(T + 8)^{2}$$
$37$ $$T^{2} + 16$$
$41$ $$(T - 6)^{2}$$
$43$ $$T^{2} + 36$$
$47$ $$T^{2} + 36$$
$53$ $$T^{2} + 64$$
$59$ $$(T - 12)^{2}$$
$61$ $$(T - 6)^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 100$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2} + 196$$
$89$ $$(T + 14)^{2}$$
$97$ $$T^{2} + 256$$