# Properties

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

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## 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: yes 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.

 $$f(q)$$ $$=$$ $$q + 2 i q^{3} - q^{9} +O(q^{10})$$ $$q + 2 i q^{3} - q^{9} - q^{11} + 2 i q^{13} + 4 i q^{17} -5 i q^{23} + 4 i q^{27} + 3 q^{29} -10 q^{31} -2 i q^{33} + 5 i q^{37} -4 q^{39} -10 q^{41} + 5 i q^{43} -4 i q^{47} -8 q^{51} -10 i q^{53} + 10 q^{59} + 10 q^{61} + 5 i q^{67} + 10 q^{69} + 3 q^{71} + 10 i q^{73} -13 q^{79} -11 q^{81} + 10 i q^{83} + 6 i q^{87} -10 q^{89} -20 i q^{93} + 6 i q^{97} + q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{9} + O(q^{10})$$ $$2 q - 2 q^{9} - 2 q^{11} + 6 q^{29} - 20 q^{31} - 8 q^{39} - 20 q^{41} - 16 q^{51} + 20 q^{59} + 20 q^{61} + 20 q^{69} + 6 q^{71} - 26 q^{79} - 22 q^{81} - 20 q^{89} + 2 q^{99} + O(q^{100})$$

## 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.d 2
5.b even 2 1 inner 4900.2.e.d 2
5.c odd 4 1 4900.2.a.d yes 1
5.c odd 4 1 4900.2.a.r yes 1
7.b odd 2 1 4900.2.e.e 2
35.c odd 2 1 4900.2.e.e 2
35.f even 4 1 4900.2.a.c 1
35.f even 4 1 4900.2.a.s yes 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4900.2.a.c 1 35.f even 4 1
4900.2.a.d yes 1 5.c odd 4 1
4900.2.a.r yes 1 5.c odd 4 1
4900.2.a.s yes 1 35.f even 4 1
4900.2.e.d 2 1.a even 1 1 trivial
4900.2.e.d 2 5.b even 2 1 inner
4900.2.e.e 2 7.b odd 2 1
4900.2.e.e 2 35.c odd 2 1

## Hecke kernels

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

 $$T_{3}^{2} + 4$$ $$T_{11} + 1$$ $$T_{19}$$ $$T_{31} + 10$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$4 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$( 1 + T )^{2}$$
$13$ $$4 + T^{2}$$
$17$ $$16 + T^{2}$$
$19$ $$T^{2}$$
$23$ $$25 + T^{2}$$
$29$ $$( -3 + T )^{2}$$
$31$ $$( 10 + T )^{2}$$
$37$ $$25 + T^{2}$$
$41$ $$( 10 + T )^{2}$$
$43$ $$25 + T^{2}$$
$47$ $$16 + T^{2}$$
$53$ $$100 + T^{2}$$
$59$ $$( -10 + T )^{2}$$
$61$ $$( -10 + T )^{2}$$
$67$ $$25 + T^{2}$$
$71$ $$( -3 + T )^{2}$$
$73$ $$100 + T^{2}$$
$79$ $$( 13 + T )^{2}$$
$83$ $$100 + T^{2}$$
$89$ $$( 10 + T )^{2}$$
$97$ $$36 + T^{2}$$
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