# Properties

 Label 980.2.e.e Level $980$ Weight $2$ Character orbit 980.e Analytic conductor $7.825$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$980 = 2^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 980.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.82533939809$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{3} + ( \beta_{1} + \beta_{2} ) q^{5} +O(q^{10})$$ $$q -\beta_{2} q^{3} + ( \beta_{1} + \beta_{2} ) q^{5} - q^{11} -\beta_{2} q^{13} + ( 3 + \beta_{3} ) q^{15} -3 \beta_{2} q^{17} + 2 \beta_{1} q^{19} -\beta_{3} q^{23} + ( -1 - 2 \beta_{3} ) q^{25} -3 \beta_{2} q^{27} + 7 q^{29} + 5 \beta_{1} q^{31} + \beta_{2} q^{33} -3 \beta_{3} q^{37} -3 q^{39} + 5 \beta_{1} q^{41} + 4 \beta_{3} q^{43} -7 \beta_{2} q^{47} -9 q^{51} -5 \beta_{3} q^{53} + ( -\beta_{1} - \beta_{2} ) q^{55} + 2 \beta_{3} q^{57} + 5 \beta_{1} q^{59} -10 \beta_{1} q^{61} + ( 3 + \beta_{3} ) q^{65} + 5 \beta_{3} q^{67} + 3 \beta_{1} q^{69} -10 q^{71} + ( 6 \beta_{1} + \beta_{2} ) q^{75} -3 q^{79} -9 q^{81} + ( 9 + 3 \beta_{3} ) q^{85} -7 \beta_{2} q^{87} + 5 \beta_{3} q^{93} + ( 4 - 2 \beta_{3} ) q^{95} + 3 \beta_{2} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 4q^{11} + 12q^{15} - 4q^{25} + 28q^{29} - 12q^{39} - 36q^{51} + 12q^{65} - 40q^{71} - 12q^{79} - 36q^{81} + 36q^{85} + 16q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3}$$$$/2$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 1$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 4 \nu$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 1$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/980\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$197$$ $$491$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
589.1
 0.707107 + 1.22474i −0.707107 − 1.22474i 0.707107 − 1.22474i −0.707107 + 1.22474i
0 1.73205i 0 −1.41421 + 1.73205i 0 0 0 0 0
589.2 0 1.73205i 0 1.41421 + 1.73205i 0 0 0 0 0
589.3 0 1.73205i 0 −1.41421 1.73205i 0 0 0 0 0
589.4 0 1.73205i 0 1.41421 1.73205i 0 0 0 0 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
7.b odd 2 1 inner
35.c odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.2.e.e 4
5.b even 2 1 inner 980.2.e.e 4
5.c odd 4 2 4900.2.a.bh 4
7.b odd 2 1 inner 980.2.e.e 4
7.c even 3 1 980.2.q.a 4
7.c even 3 1 980.2.q.h 4
7.d odd 6 1 980.2.q.a 4
7.d odd 6 1 980.2.q.h 4
35.c odd 2 1 inner 980.2.e.e 4
35.f even 4 2 4900.2.a.bh 4
35.i odd 6 1 980.2.q.a 4
35.i odd 6 1 980.2.q.h 4
35.j even 6 1 980.2.q.a 4
35.j even 6 1 980.2.q.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.2.e.e 4 1.a even 1 1 trivial
980.2.e.e 4 5.b even 2 1 inner
980.2.e.e 4 7.b odd 2 1 inner
980.2.e.e 4 35.c odd 2 1 inner
980.2.q.a 4 7.c even 3 1
980.2.q.a 4 7.d odd 6 1
980.2.q.a 4 35.i odd 6 1
980.2.q.a 4 35.j even 6 1
980.2.q.h 4 7.c even 3 1
980.2.q.h 4 7.d odd 6 1
980.2.q.h 4 35.i odd 6 1
980.2.q.h 4 35.j even 6 1
4900.2.a.bh 4 5.c odd 4 2
4900.2.a.bh 4 35.f even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(980, [\chi])$$:

 $$T_{3}^{2} + 3$$ $$T_{11} + 1$$ $$T_{19}^{2} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 3 + T^{2} )^{2}$$
$5$ $$25 + 2 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 1 + T )^{4}$$
$13$ $$( 3 + T^{2} )^{2}$$
$17$ $$( 27 + T^{2} )^{2}$$
$19$ $$( -8 + T^{2} )^{2}$$
$23$ $$( 6 + T^{2} )^{2}$$
$29$ $$( -7 + T )^{4}$$
$31$ $$( -50 + T^{2} )^{2}$$
$37$ $$( 54 + T^{2} )^{2}$$
$41$ $$( -50 + T^{2} )^{2}$$
$43$ $$( 96 + T^{2} )^{2}$$
$47$ $$( 147 + T^{2} )^{2}$$
$53$ $$( 150 + T^{2} )^{2}$$
$59$ $$( -50 + T^{2} )^{2}$$
$61$ $$( -200 + T^{2} )^{2}$$
$67$ $$( 150 + T^{2} )^{2}$$
$71$ $$( 10 + T )^{4}$$
$73$ $$T^{4}$$
$79$ $$( 3 + T )^{4}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$( 27 + T^{2} )^{2}$$
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