# Properties

 Label 4900.2.e.g Level $4900$ Weight $2$ Character orbit 4900.e Analytic conductor $39.127$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4900 = 2^{2} \cdot 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4900.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$39.1266969904$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 700) 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.

 $$f(q)$$ $$=$$ $$q + 2 i q^{3} - q^{9} +O(q^{10})$$ $$q + 2 i q^{3} - q^{9} + 3 q^{11} + 4 i q^{13} + 2 q^{19} -3 i q^{23} + 4 i q^{27} -9 q^{29} -8 q^{31} + 6 i q^{33} -5 i q^{37} -8 q^{39} + 6 q^{41} + 11 i q^{43} + 6 i q^{47} + 6 i q^{53} + 4 i q^{57} + 10 q^{61} -5 i q^{67} + 6 q^{69} + 15 q^{71} + 10 i q^{73} + 7 q^{79} -11 q^{81} -12 i q^{83} -18 i q^{87} -12 q^{89} -16 i q^{93} + 8 i q^{97} -3 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{9} + O(q^{10})$$ $$2q - 2q^{9} + 6q^{11} + 4q^{19} - 18q^{29} - 16q^{31} - 16q^{39} + 12q^{41} + 20q^{61} + 12q^{69} + 30q^{71} + 14q^{79} - 22q^{81} - 24q^{89} - 6q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4900\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$1177$$ $$2451$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2549.1
 − 1.00000i 1.00000i
0 2.00000i 0 0 0 0 0 −1.00000 0
2549.2 0 2.00000i 0 0 0 0 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 4900.2.e.g 2
5.b even 2 1 inner 4900.2.e.g 2
5.c odd 4 1 4900.2.a.f 1
5.c odd 4 1 4900.2.a.t 1
7.b odd 2 1 700.2.e.b 2
21.c even 2 1 6300.2.k.d 2
28.d even 2 1 2800.2.g.e 2
35.c odd 2 1 700.2.e.b 2
35.f even 4 1 700.2.a.c 1
35.f even 4 1 700.2.a.i yes 1
105.g even 2 1 6300.2.k.d 2
105.k odd 4 1 6300.2.a.e 1
105.k odd 4 1 6300.2.a.s 1
140.c even 2 1 2800.2.g.e 2
140.j odd 4 1 2800.2.a.e 1
140.j odd 4 1 2800.2.a.ba 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
700.2.a.c 1 35.f even 4 1
700.2.a.i yes 1 35.f even 4 1
700.2.e.b 2 7.b odd 2 1
700.2.e.b 2 35.c odd 2 1
2800.2.a.e 1 140.j odd 4 1
2800.2.a.ba 1 140.j odd 4 1
2800.2.g.e 2 28.d even 2 1
2800.2.g.e 2 140.c even 2 1
4900.2.a.f 1 5.c odd 4 1
4900.2.a.t 1 5.c odd 4 1
4900.2.e.g 2 1.a even 1 1 trivial
4900.2.e.g 2 5.b even 2 1 inner
6300.2.a.e 1 105.k odd 4 1
6300.2.a.s 1 105.k odd 4 1
6300.2.k.d 2 21.c even 2 1
6300.2.k.d 2 105.g even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(4900, [\chi])$$:

 $$T_{3}^{2} + 4$$ $$T_{11} - 3$$ $$T_{19} - 2$$ $$T_{31} + 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$4 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$( -3 + T )^{2}$$
$13$ $$16 + T^{2}$$
$17$ $$T^{2}$$
$19$ $$( -2 + T )^{2}$$
$23$ $$9 + T^{2}$$
$29$ $$( 9 + T )^{2}$$
$31$ $$( 8 + T )^{2}$$
$37$ $$25 + T^{2}$$
$41$ $$( -6 + T )^{2}$$
$43$ $$121 + T^{2}$$
$47$ $$36 + T^{2}$$
$53$ $$36 + T^{2}$$
$59$ $$T^{2}$$
$61$ $$( -10 + T )^{2}$$
$67$ $$25 + T^{2}$$
$71$ $$( -15 + T )^{2}$$
$73$ $$100 + T^{2}$$
$79$ $$( -7 + T )^{2}$$
$83$ $$144 + T^{2}$$
$89$ $$( 12 + T )^{2}$$
$97$ $$64 + T^{2}$$