# Properties

 Label 3900.2.h.b Level $3900$ Weight $2$ Character orbit 3900.h Analytic conductor $31.142$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3900.h (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$31.1416567883$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 156) 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.

 $$f(q)$$ $$=$$ $$q -i q^{3} + 2 i q^{7} - q^{9} +O(q^{10})$$ $$q -i q^{3} + 2 i q^{7} - q^{9} -4 q^{11} + i q^{13} -2 i q^{17} + 2 q^{19} + 2 q^{21} + i q^{27} + 6 q^{29} -10 q^{31} + 4 i q^{33} -10 i q^{37} + q^{39} + 8 q^{41} + 4 i q^{43} + 4 i q^{47} + 3 q^{49} -2 q^{51} -10 i q^{53} -2 i q^{57} + 8 q^{59} -14 q^{61} -2 i q^{63} -2 i q^{67} + 16 q^{71} -10 i q^{73} -8 i q^{77} + 16 q^{79} + q^{81} -6 i q^{87} + 4 q^{89} -2 q^{91} + 10 i q^{93} + 2 i q^{97} + 4 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{9} + O(q^{10})$$ $$2q - 2q^{9} - 8q^{11} + 4q^{19} + 4q^{21} + 12q^{29} - 20q^{31} + 2q^{39} + 16q^{41} + 6q^{49} - 4q^{51} + 16q^{59} - 28q^{61} + 32q^{71} + 32q^{79} + 2q^{81} + 8q^{89} - 4q^{91} + 8q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3900\mathbb{Z}\right)^\times$$.

 $$n$$ $$301$$ $$1301$$ $$1951$$ $$3277$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1249.1
 1.00000i − 1.00000i
0 1.00000i 0 0 0 2.00000i 0 −1.00000 0
1249.2 0 1.00000i 0 0 0 2.00000i 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 3900.2.h.b 2
5.b even 2 1 inner 3900.2.h.b 2
5.c odd 4 1 156.2.a.a 1
5.c odd 4 1 3900.2.a.m 1
15.e even 4 1 468.2.a.d 1
20.e even 4 1 624.2.a.e 1
35.f even 4 1 7644.2.a.k 1
40.i odd 4 1 2496.2.a.bc 1
40.k even 4 1 2496.2.a.o 1
45.k odd 12 2 4212.2.i.l 2
45.l even 12 2 4212.2.i.b 2
60.l odd 4 1 1872.2.a.s 1
65.f even 4 1 2028.2.b.a 2
65.h odd 4 1 2028.2.a.c 1
65.k even 4 1 2028.2.b.a 2
65.o even 12 2 2028.2.q.h 4
65.q odd 12 2 2028.2.i.e 2
65.r odd 12 2 2028.2.i.g 2
65.t even 12 2 2028.2.q.h 4
120.q odd 4 1 7488.2.a.d 1
120.w even 4 1 7488.2.a.c 1
195.j odd 4 1 6084.2.b.j 2
195.s even 4 1 6084.2.a.b 1
195.u odd 4 1 6084.2.b.j 2
260.p even 4 1 8112.2.a.bi 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.2.a.a 1 5.c odd 4 1
468.2.a.d 1 15.e even 4 1
624.2.a.e 1 20.e even 4 1
1872.2.a.s 1 60.l odd 4 1
2028.2.a.c 1 65.h odd 4 1
2028.2.b.a 2 65.f even 4 1
2028.2.b.a 2 65.k even 4 1
2028.2.i.e 2 65.q odd 12 2
2028.2.i.g 2 65.r odd 12 2
2028.2.q.h 4 65.o even 12 2
2028.2.q.h 4 65.t even 12 2
2496.2.a.o 1 40.k even 4 1
2496.2.a.bc 1 40.i odd 4 1
3900.2.a.m 1 5.c odd 4 1
3900.2.h.b 2 1.a even 1 1 trivial
3900.2.h.b 2 5.b even 2 1 inner
4212.2.i.b 2 45.l even 12 2
4212.2.i.l 2 45.k odd 12 2
6084.2.a.b 1 195.s even 4 1
6084.2.b.j 2 195.j odd 4 1
6084.2.b.j 2 195.u odd 4 1
7488.2.a.c 1 120.w even 4 1
7488.2.a.d 1 120.q odd 4 1
7644.2.a.k 1 35.f even 4 1
8112.2.a.bi 1 260.p even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3900, [\chi])$$:

 $$T_{7}^{2} + 4$$ $$T_{11} + 4$$ $$T_{19} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$4 + T^{2}$$
$11$ $$( 4 + T )^{2}$$
$13$ $$1 + T^{2}$$
$17$ $$4 + T^{2}$$
$19$ $$( -2 + T )^{2}$$
$23$ $$T^{2}$$
$29$ $$( -6 + T )^{2}$$
$31$ $$( 10 + T )^{2}$$
$37$ $$100 + T^{2}$$
$41$ $$( -8 + T )^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$16 + T^{2}$$
$53$ $$100 + T^{2}$$
$59$ $$( -8 + T )^{2}$$
$61$ $$( 14 + T )^{2}$$
$67$ $$4 + T^{2}$$
$71$ $$( -16 + T )^{2}$$
$73$ $$100 + T^{2}$$
$79$ $$( -16 + T )^{2}$$
$83$ $$T^{2}$$
$89$ $$( -4 + T )^{2}$$
$97$ $$4 + T^{2}$$