# Properties

 Label 3800.2.d.c Level $3800$ Weight $2$ Character orbit 3800.d Analytic conductor $30.343$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3800 = 2^{3} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3800.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$30.3431527681$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 760) 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} - 2 \beta q^{7} - q^{9} +O(q^{10})$$ q + b * q^3 - 2*b * q^7 - q^9 $$q + \beta q^{3} - 2 \beta q^{7} - q^{9} - 4 q^{11} - 3 \beta q^{17} + q^{19} + 8 q^{21} + 4 \beta q^{23} + 2 \beta q^{27} + 6 q^{29} - 8 q^{31} - 4 \beta q^{33} + 4 \beta q^{37} - 2 q^{41} - 6 \beta q^{47} - 9 q^{49} + 12 q^{51} + 2 \beta q^{53} + \beta q^{57} - 8 q^{59} - 14 q^{61} + 2 \beta q^{63} + \beta q^{67} - 16 q^{69} - 8 q^{71} - \beta q^{73} + 8 \beta q^{77} - 4 q^{79} - 11 q^{81} + 6 \beta q^{83} + 6 \beta q^{87} - 6 q^{89} - 8 \beta q^{93} + 4 q^{99} +O(q^{100})$$ q + b * q^3 - 2*b * q^7 - q^9 - 4 * q^11 - 3*b * q^17 + q^19 + 8 * q^21 + 4*b * q^23 + 2*b * q^27 + 6 * q^29 - 8 * q^31 - 4*b * q^33 + 4*b * q^37 - 2 * q^41 - 6*b * q^47 - 9 * q^49 + 12 * q^51 + 2*b * q^53 + b * q^57 - 8 * q^59 - 14 * q^61 + 2*b * q^63 + b * q^67 - 16 * q^69 - 8 * q^71 - b * q^73 + 8*b * q^77 - 4 * q^79 - 11 * q^81 + 6*b * q^83 + 6*b * q^87 - 6 * q^89 - 8*b * q^93 + 4 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^9 $$2 q - 2 q^{9} - 8 q^{11} + 2 q^{19} + 16 q^{21} + 12 q^{29} - 16 q^{31} - 4 q^{41} - 18 q^{49} + 24 q^{51} - 16 q^{59} - 28 q^{61} - 32 q^{69} - 16 q^{71} - 8 q^{79} - 22 q^{81} - 12 q^{89} + 8 q^{99}+O(q^{100})$$ 2 * q - 2 * q^9 - 8 * q^11 + 2 * q^19 + 16 * q^21 + 12 * q^29 - 16 * q^31 - 4 * q^41 - 18 * q^49 + 24 * q^51 - 16 * q^59 - 28 * q^61 - 32 * q^69 - 16 * q^71 - 8 * q^79 - 22 * q^81 - 12 * q^89 + 8 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times$$.

 $$n$$ $$401$$ $$951$$ $$1901$$ $$1977$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3649.1
 − 1.00000i 1.00000i
0 2.00000i 0 0 0 4.00000i 0 −1.00000 0
3649.2 0 2.00000i 0 0 0 4.00000i 0 −1.00000 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 3800.2.d.c 2
5.b even 2 1 inner 3800.2.d.c 2
5.c odd 4 1 760.2.a.d 1
5.c odd 4 1 3800.2.a.b 1
15.e even 4 1 6840.2.a.o 1
20.e even 4 1 1520.2.a.c 1
20.e even 4 1 7600.2.a.s 1
40.i odd 4 1 6080.2.a.e 1
40.k even 4 1 6080.2.a.s 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.a.d 1 5.c odd 4 1
1520.2.a.c 1 20.e even 4 1
3800.2.a.b 1 5.c odd 4 1
3800.2.d.c 2 1.a even 1 1 trivial
3800.2.d.c 2 5.b even 2 1 inner
6080.2.a.e 1 40.i odd 4 1
6080.2.a.s 1 40.k even 4 1
6840.2.a.o 1 15.e even 4 1
7600.2.a.s 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 $$S_{2}^{\mathrm{new}}(3800, [\chi])$$:

 $$T_{3}^{2} + 4$$ T3^2 + 4 $$T_{7}^{2} + 16$$ T7^2 + 16 $$T_{11} + 4$$ T11 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 4$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 16$$
$11$ $$(T + 4)^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 36$$
$19$ $$(T - 1)^{2}$$
$23$ $$T^{2} + 64$$
$29$ $$(T - 6)^{2}$$
$31$ $$(T + 8)^{2}$$
$37$ $$T^{2} + 64$$
$41$ $$(T + 2)^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2} + 144$$
$53$ $$T^{2} + 16$$
$59$ $$(T + 8)^{2}$$
$61$ $$(T + 14)^{2}$$
$67$ $$T^{2} + 4$$
$71$ $$(T + 8)^{2}$$
$73$ $$T^{2} + 4$$
$79$ $$(T + 4)^{2}$$
$83$ $$T^{2} + 144$$
$89$ $$(T + 6)^{2}$$
$97$ $$T^{2}$$