# Properties

 Label 98.2.c.c Level $98$ Weight $2$ Character orbit 98.c Analytic conductor $0.783$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$98 = 2 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 98.c (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.782533939809$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + \beta_{1} q^{3} + ( -1 - \beta_{2} ) q^{4} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{5} + \beta_{3} q^{6} + q^{8} -\beta_{2} q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{2} + \beta_{1} q^{3} + ( -1 - \beta_{2} ) q^{4} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{5} + \beta_{3} q^{6} + q^{8} -\beta_{2} q^{9} -2 \beta_{1} q^{10} + ( 2 + 2 \beta_{2} ) q^{11} + ( -\beta_{1} - \beta_{3} ) q^{12} -4 q^{15} + \beta_{2} q^{16} + \beta_{1} q^{17} + ( 1 + \beta_{2} ) q^{18} + ( -5 \beta_{1} - 5 \beta_{3} ) q^{19} -2 \beta_{3} q^{20} -2 q^{22} -4 \beta_{2} q^{23} + \beta_{1} q^{24} + ( -3 - 3 \beta_{2} ) q^{25} -4 \beta_{3} q^{27} + 2 q^{29} -4 \beta_{2} q^{30} -6 \beta_{1} q^{31} + ( -1 - \beta_{2} ) q^{32} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{33} + \beta_{3} q^{34} - q^{36} + 10 \beta_{2} q^{37} + 5 \beta_{1} q^{38} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{40} + 7 \beta_{3} q^{41} + 2 q^{43} -2 \beta_{2} q^{44} + 2 \beta_{1} q^{45} + ( 4 + 4 \beta_{2} ) q^{46} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{47} + \beta_{3} q^{48} + 3 q^{50} + 2 \beta_{2} q^{51} + ( 2 + 2 \beta_{2} ) q^{53} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{54} + 4 \beta_{3} q^{55} + 10 q^{57} + 2 \beta_{2} q^{58} + \beta_{1} q^{59} + ( 4 + 4 \beta_{2} ) q^{60} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{61} -6 \beta_{3} q^{62} + q^{64} -2 \beta_{1} q^{66} + ( -12 - 12 \beta_{2} ) q^{67} + ( -\beta_{1} - \beta_{3} ) q^{68} -4 \beta_{3} q^{69} -12 q^{71} -\beta_{2} q^{72} + \beta_{1} q^{73} + ( -10 - 10 \beta_{2} ) q^{74} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{75} + 5 \beta_{3} q^{76} -4 \beta_{2} q^{79} -2 \beta_{1} q^{80} + ( 5 + 5 \beta_{2} ) q^{81} + ( -7 \beta_{1} - 7 \beta_{3} ) q^{82} -7 \beta_{3} q^{83} -4 q^{85} + 2 \beta_{2} q^{86} + 2 \beta_{1} q^{87} + ( 2 + 2 \beta_{2} ) q^{88} + ( -5 \beta_{1} - 5 \beta_{3} ) q^{89} + 2 \beta_{3} q^{90} -4 q^{92} -12 \beta_{2} q^{93} -2 \beta_{1} q^{94} + ( 20 + 20 \beta_{2} ) q^{95} + ( -\beta_{1} - \beta_{3} ) q^{96} -7 \beta_{3} q^{97} + 2 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{2} - 2q^{4} + 4q^{8} + 2q^{9} + O(q^{10})$$ $$4q - 2q^{2} - 2q^{4} + 4q^{8} + 2q^{9} + 4q^{11} - 16q^{15} - 2q^{16} + 2q^{18} - 8q^{22} + 8q^{23} - 6q^{25} + 8q^{29} + 8q^{30} - 2q^{32} - 4q^{36} - 20q^{37} + 8q^{43} + 4q^{44} + 8q^{46} + 12q^{50} - 4q^{51} + 4q^{53} + 40q^{57} - 4q^{58} + 8q^{60} + 4q^{64} - 24q^{67} - 48q^{71} + 2q^{72} - 20q^{74} + 8q^{79} + 10q^{81} - 16q^{85} - 4q^{86} + 4q^{88} - 16q^{92} + 24q^{93} + 40q^{95} + 8q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/98\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 −0.707107 + 1.22474i 0.707107 − 1.22474i −0.707107 − 1.22474i 0.707107 + 1.22474i
−0.500000 0.866025i −0.707107 + 1.22474i −0.500000 + 0.866025i 1.41421 + 2.44949i 1.41421 0 1.00000 0.500000 + 0.866025i 1.41421 2.44949i
67.2 −0.500000 0.866025i 0.707107 1.22474i −0.500000 + 0.866025i −1.41421 2.44949i −1.41421 0 1.00000 0.500000 + 0.866025i −1.41421 + 2.44949i
79.1 −0.500000 + 0.866025i −0.707107 1.22474i −0.500000 0.866025i 1.41421 2.44949i 1.41421 0 1.00000 0.500000 0.866025i 1.41421 + 2.44949i
79.2 −0.500000 + 0.866025i 0.707107 + 1.22474i −0.500000 0.866025i −1.41421 + 2.44949i −1.41421 0 1.00000 0.500000 0.866025i −1.41421 2.44949i
 $$n$$: 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
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 98.2.c.c 4
3.b odd 2 1 882.2.g.l 4
4.b odd 2 1 784.2.i.m 4
7.b odd 2 1 inner 98.2.c.c 4
7.c even 3 1 98.2.a.b 2
7.c even 3 1 inner 98.2.c.c 4
7.d odd 6 1 98.2.a.b 2
7.d odd 6 1 inner 98.2.c.c 4
21.c even 2 1 882.2.g.l 4
21.g even 6 1 882.2.a.n 2
21.g even 6 1 882.2.g.l 4
21.h odd 6 1 882.2.a.n 2
21.h odd 6 1 882.2.g.l 4
28.d even 2 1 784.2.i.m 4
28.f even 6 1 784.2.a.l 2
28.f even 6 1 784.2.i.m 4
28.g odd 6 1 784.2.a.l 2
28.g odd 6 1 784.2.i.m 4
35.i odd 6 1 2450.2.a.bj 2
35.j even 6 1 2450.2.a.bj 2
35.k even 12 2 2450.2.c.v 4
35.l odd 12 2 2450.2.c.v 4
56.j odd 6 1 3136.2.a.bn 2
56.k odd 6 1 3136.2.a.bm 2
56.m even 6 1 3136.2.a.bm 2
56.p even 6 1 3136.2.a.bn 2
84.j odd 6 1 7056.2.a.cl 2
84.n even 6 1 7056.2.a.cl 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.2.a.b 2 7.c even 3 1
98.2.a.b 2 7.d odd 6 1
98.2.c.c 4 1.a even 1 1 trivial
98.2.c.c 4 7.b odd 2 1 inner
98.2.c.c 4 7.c even 3 1 inner
98.2.c.c 4 7.d odd 6 1 inner
784.2.a.l 2 28.f even 6 1
784.2.a.l 2 28.g odd 6 1
784.2.i.m 4 4.b odd 2 1
784.2.i.m 4 28.d even 2 1
784.2.i.m 4 28.f even 6 1
784.2.i.m 4 28.g odd 6 1
882.2.a.n 2 21.g even 6 1
882.2.a.n 2 21.h odd 6 1
882.2.g.l 4 3.b odd 2 1
882.2.g.l 4 21.c even 2 1
882.2.g.l 4 21.g even 6 1
882.2.g.l 4 21.h odd 6 1
2450.2.a.bj 2 35.i odd 6 1
2450.2.a.bj 2 35.j even 6 1
2450.2.c.v 4 35.k even 12 2
2450.2.c.v 4 35.l odd 12 2
3136.2.a.bm 2 56.k odd 6 1
3136.2.a.bm 2 56.m even 6 1
3136.2.a.bn 2 56.j odd 6 1
3136.2.a.bn 2 56.p even 6 1
7056.2.a.cl 2 84.j odd 6 1
7056.2.a.cl 2 84.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 2 T_{3}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(98, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} )^{2}$$
$3$ $$4 + 2 T^{2} + T^{4}$$
$5$ $$64 + 8 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 4 - 2 T + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$4 + 2 T^{2} + T^{4}$$
$19$ $$2500 + 50 T^{2} + T^{4}$$
$23$ $$( 16 - 4 T + T^{2} )^{2}$$
$29$ $$( -2 + T )^{4}$$
$31$ $$5184 + 72 T^{2} + T^{4}$$
$37$ $$( 100 + 10 T + T^{2} )^{2}$$
$41$ $$( -98 + T^{2} )^{2}$$
$43$ $$( -2 + T )^{4}$$
$47$ $$64 + 8 T^{2} + T^{4}$$
$53$ $$( 4 - 2 T + T^{2} )^{2}$$
$59$ $$4 + 2 T^{2} + T^{4}$$
$61$ $$64 + 8 T^{2} + T^{4}$$
$67$ $$( 144 + 12 T + T^{2} )^{2}$$
$71$ $$( 12 + T )^{4}$$
$73$ $$4 + 2 T^{2} + T^{4}$$
$79$ $$( 16 - 4 T + T^{2} )^{2}$$
$83$ $$( -98 + T^{2} )^{2}$$
$89$ $$2500 + 50 T^{2} + T^{4}$$
$97$ $$( -98 + T^{2} )^{2}$$