# Properties

 Label 3800.2.d.f Level $3800$ Weight $2$ Character orbit 3800.d Analytic conductor $30.343$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Learn more

## 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: $$0$$ 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: $$1$$ Twist minimal: no (minimal twist has level 152) 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.

 $$f(q)$$ $$=$$ $$q + i q^{3} - 3 i q^{7} + 2 q^{9} +O(q^{10})$$ q + i * q^3 - 3*i * q^7 + 2 * q^9 $$q + i q^{3} - 3 i q^{7} + 2 q^{9} + 2 q^{11} + i q^{13} + 5 i q^{17} - q^{19} + 3 q^{21} - i q^{23} + 5 i q^{27} + 3 q^{29} + 4 q^{31} + 2 i q^{33} - 2 i q^{37} - q^{39} - 8 q^{41} - 8 i q^{43} + 8 i q^{47} - 2 q^{49} - 5 q^{51} + 9 i q^{53} - i q^{57} - q^{59} + 14 q^{61} - 6 i q^{63} - 13 i q^{67} + q^{69} + 10 q^{71} + 9 i q^{73} - 6 i q^{77} + 10 q^{79} + q^{81} + 10 i q^{83} + 3 i q^{87} + 12 q^{89} + 3 q^{91} + 4 i q^{93} - 14 i q^{97} + 4 q^{99} +O(q^{100})$$ q + i * q^3 - 3*i * q^7 + 2 * q^9 + 2 * q^11 + i * q^13 + 5*i * q^17 - q^19 + 3 * q^21 - i * q^23 + 5*i * q^27 + 3 * q^29 + 4 * q^31 + 2*i * q^33 - 2*i * q^37 - q^39 - 8 * q^41 - 8*i * q^43 + 8*i * q^47 - 2 * q^49 - 5 * q^51 + 9*i * q^53 - i * q^57 - q^59 + 14 * q^61 - 6*i * q^63 - 13*i * q^67 + q^69 + 10 * q^71 + 9*i * q^73 - 6*i * q^77 + 10 * q^79 + q^81 + 10*i * q^83 + 3*i * q^87 + 12 * q^89 + 3 * q^91 + 4*i * q^93 - 14*i * q^97 + 4 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{9}+O(q^{10})$$ 2 * q + 4 * q^9 $$2 q + 4 q^{9} + 4 q^{11} - 2 q^{19} + 6 q^{21} + 6 q^{29} + 8 q^{31} - 2 q^{39} - 16 q^{41} - 4 q^{49} - 10 q^{51} - 2 q^{59} + 28 q^{61} + 2 q^{69} + 20 q^{71} + 20 q^{79} + 2 q^{81} + 24 q^{89} + 6 q^{91} + 8 q^{99}+O(q^{100})$$ 2 * q + 4 * q^9 + 4 * q^11 - 2 * q^19 + 6 * q^21 + 6 * q^29 + 8 * q^31 - 2 * q^39 - 16 * q^41 - 4 * q^49 - 10 * q^51 - 2 * q^59 + 28 * q^61 + 2 * q^69 + 20 * q^71 + 20 * q^79 + 2 * q^81 + 24 * q^89 + 6 * q^91 + 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 1.00000i 0 0 0 3.00000i 0 2.00000 0
3649.2 0 1.00000i 0 0 0 3.00000i 0 2.00000 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.f 2
5.b even 2 1 inner 3800.2.d.f 2
5.c odd 4 1 152.2.a.b 1
5.c odd 4 1 3800.2.a.d 1
15.e even 4 1 1368.2.a.g 1
20.e even 4 1 304.2.a.b 1
20.e even 4 1 7600.2.a.o 1
35.f even 4 1 7448.2.a.g 1
40.i odd 4 1 1216.2.a.f 1
40.k even 4 1 1216.2.a.l 1
60.l odd 4 1 2736.2.a.k 1
95.g even 4 1 2888.2.a.b 1
380.j odd 4 1 5776.2.a.l 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.2.a.b 1 5.c odd 4 1
304.2.a.b 1 20.e even 4 1
1216.2.a.f 1 40.i odd 4 1
1216.2.a.l 1 40.k even 4 1
1368.2.a.g 1 15.e even 4 1
2736.2.a.k 1 60.l odd 4 1
2888.2.a.b 1 95.g even 4 1
3800.2.a.d 1 5.c odd 4 1
3800.2.d.f 2 1.a even 1 1 trivial
3800.2.d.f 2 5.b even 2 1 inner
5776.2.a.l 1 380.j odd 4 1
7448.2.a.g 1 35.f even 4 1
7600.2.a.o 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 $$S_{2}^{\mathrm{new}}(3800, [\chi])$$:

 $$T_{3}^{2} + 1$$ T3^2 + 1 $$T_{7}^{2} + 9$$ T7^2 + 9 $$T_{11} - 2$$ T11 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 9$$
$11$ $$(T - 2)^{2}$$
$13$ $$T^{2} + 1$$
$17$ $$T^{2} + 25$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2} + 1$$
$29$ $$(T - 3)^{2}$$
$31$ $$(T - 4)^{2}$$
$37$ $$T^{2} + 4$$
$41$ $$(T + 8)^{2}$$
$43$ $$T^{2} + 64$$
$47$ $$T^{2} + 64$$
$53$ $$T^{2} + 81$$
$59$ $$(T + 1)^{2}$$
$61$ $$(T - 14)^{2}$$
$67$ $$T^{2} + 169$$
$71$ $$(T - 10)^{2}$$
$73$ $$T^{2} + 81$$
$79$ $$(T - 10)^{2}$$
$83$ $$T^{2} + 100$$
$89$ $$(T - 12)^{2}$$
$97$ $$T^{2} + 196$$
show more
show less