# Properties

 Label 98.2.c.c Level $98$ Weight $2$ Character orbit 98.c Analytic conductor $0.783$ Analytic rank $0$ Dimension $4$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [98,2,Mod(67,98)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(98, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("98.67");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$0.782533939809$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 2x^{2} + 4$$ 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

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + \beta_1 q^{3} + ( - \beta_{2} - 1) q^{4} + (2 \beta_{3} + 2 \beta_1) q^{5} + \beta_{3} q^{6} + q^{8} - \beta_{2} q^{9}+O(q^{10})$$ q + b2 * q^2 + b1 * q^3 + (-b2 - 1) * q^4 + (2*b3 + 2*b1) * q^5 + b3 * q^6 + q^8 - b2 * q^9 $$q + \beta_{2} q^{2} + \beta_1 q^{3} + ( - \beta_{2} - 1) q^{4} + (2 \beta_{3} + 2 \beta_1) q^{5} + \beta_{3} q^{6} + q^{8} - \beta_{2} q^{9} - 2 \beta_1 q^{10} + (2 \beta_{2} + 2) q^{11} + ( - \beta_{3} - \beta_1) q^{12} - 4 q^{15} + \beta_{2} q^{16} + \beta_1 q^{17} + (\beta_{2} + 1) q^{18} + ( - 5 \beta_{3} - 5 \beta_1) q^{19} - 2 \beta_{3} q^{20} - 2 q^{22} - 4 \beta_{2} q^{23} + \beta_1 q^{24} + ( - 3 \beta_{2} - 3) q^{25} - 4 \beta_{3} q^{27} + 2 q^{29} - 4 \beta_{2} q^{30} - 6 \beta_1 q^{31} + ( - \beta_{2} - 1) q^{32} + (2 \beta_{3} + 2 \beta_1) q^{33} + \beta_{3} q^{34} - q^{36} + 10 \beta_{2} q^{37} + 5 \beta_1 q^{38} + (2 \beta_{3} + 2 \beta_1) q^{40} + 7 \beta_{3} q^{41} + 2 q^{43} - 2 \beta_{2} q^{44} + 2 \beta_1 q^{45} + (4 \beta_{2} + 4) q^{46} + (2 \beta_{3} + 2 \beta_1) q^{47} + \beta_{3} q^{48} + 3 q^{50} + 2 \beta_{2} q^{51} + (2 \beta_{2} + 2) q^{53} + (4 \beta_{3} + 4 \beta_1) q^{54} + 4 \beta_{3} q^{55} + 10 q^{57} + 2 \beta_{2} q^{58} + \beta_1 q^{59} + (4 \beta_{2} + 4) q^{60} + (2 \beta_{3} + 2 \beta_1) q^{61} - 6 \beta_{3} q^{62} + q^{64} - 2 \beta_1 q^{66} + ( - 12 \beta_{2} - 12) q^{67} + ( - \beta_{3} - \beta_1) q^{68} - 4 \beta_{3} q^{69} - 12 q^{71} - \beta_{2} q^{72} + \beta_1 q^{73} + ( - 10 \beta_{2} - 10) q^{74} + ( - 3 \beta_{3} - 3 \beta_1) q^{75} + 5 \beta_{3} q^{76} - 4 \beta_{2} q^{79} - 2 \beta_1 q^{80} + (5 \beta_{2} + 5) q^{81} + ( - 7 \beta_{3} - 7 \beta_1) q^{82} - 7 \beta_{3} q^{83} - 4 q^{85} + 2 \beta_{2} q^{86} + 2 \beta_1 q^{87} + (2 \beta_{2} + 2) q^{88} + ( - 5 \beta_{3} - 5 \beta_1) q^{89} + 2 \beta_{3} q^{90} - 4 q^{92} - 12 \beta_{2} q^{93} - 2 \beta_1 q^{94} + (20 \beta_{2} + 20) q^{95} + ( - \beta_{3} - \beta_1) q^{96} - 7 \beta_{3} q^{97} + 2 q^{99}+O(q^{100})$$ q + b2 * q^2 + b1 * q^3 + (-b2 - 1) * q^4 + (2*b3 + 2*b1) * q^5 + b3 * q^6 + q^8 - b2 * q^9 - 2*b1 * q^10 + (2*b2 + 2) * q^11 + (-b3 - b1) * q^12 - 4 * q^15 + b2 * q^16 + b1 * q^17 + (b2 + 1) * q^18 + (-5*b3 - 5*b1) * q^19 - 2*b3 * q^20 - 2 * q^22 - 4*b2 * q^23 + b1 * q^24 + (-3*b2 - 3) * q^25 - 4*b3 * q^27 + 2 * q^29 - 4*b2 * q^30 - 6*b1 * q^31 + (-b2 - 1) * q^32 + (2*b3 + 2*b1) * q^33 + b3 * q^34 - q^36 + 10*b2 * q^37 + 5*b1 * q^38 + (2*b3 + 2*b1) * q^40 + 7*b3 * q^41 + 2 * q^43 - 2*b2 * q^44 + 2*b1 * q^45 + (4*b2 + 4) * q^46 + (2*b3 + 2*b1) * q^47 + b3 * q^48 + 3 * q^50 + 2*b2 * q^51 + (2*b2 + 2) * q^53 + (4*b3 + 4*b1) * q^54 + 4*b3 * q^55 + 10 * q^57 + 2*b2 * q^58 + b1 * q^59 + (4*b2 + 4) * q^60 + (2*b3 + 2*b1) * q^61 - 6*b3 * q^62 + q^64 - 2*b1 * q^66 + (-12*b2 - 12) * q^67 + (-b3 - b1) * q^68 - 4*b3 * q^69 - 12 * q^71 - b2 * q^72 + b1 * q^73 + (-10*b2 - 10) * q^74 + (-3*b3 - 3*b1) * q^75 + 5*b3 * q^76 - 4*b2 * q^79 - 2*b1 * q^80 + (5*b2 + 5) * q^81 + (-7*b3 - 7*b1) * q^82 - 7*b3 * q^83 - 4 * q^85 + 2*b2 * q^86 + 2*b1 * q^87 + (2*b2 + 2) * q^88 + (-5*b3 - 5*b1) * q^89 + 2*b3 * q^90 - 4 * q^92 - 12*b2 * q^93 - 2*b1 * q^94 + (20*b2 + 20) * q^95 + (-b3 - b1) * q^96 - 7*b3 * q^97 + 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} - 2 q^{4} + 4 q^{8} + 2 q^{9}+O(q^{10})$$ 4 * q - 2 * q^2 - 2 * q^4 + 4 * q^8 + 2 * q^9 $$4 q - 2 q^{2} - 2 q^{4} + 4 q^{8} + 2 q^{9} + 4 q^{11} - 16 q^{15} - 2 q^{16} + 2 q^{18} - 8 q^{22} + 8 q^{23} - 6 q^{25} + 8 q^{29} + 8 q^{30} - 2 q^{32} - 4 q^{36} - 20 q^{37} + 8 q^{43} + 4 q^{44} + 8 q^{46} + 12 q^{50} - 4 q^{51} + 4 q^{53} + 40 q^{57} - 4 q^{58} + 8 q^{60} + 4 q^{64} - 24 q^{67} - 48 q^{71} + 2 q^{72} - 20 q^{74} + 8 q^{79} + 10 q^{81} - 16 q^{85} - 4 q^{86} + 4 q^{88} - 16 q^{92} + 24 q^{93} + 40 q^{95} + 8 q^{99}+O(q^{100})$$ 4 * q - 2 * q^2 - 2 * q^4 + 4 * q^8 + 2 * q^9 + 4 * q^11 - 16 * q^15 - 2 * q^16 + 2 * q^18 - 8 * q^22 + 8 * q^23 - 6 * q^25 + 8 * q^29 + 8 * q^30 - 2 * q^32 - 4 * q^36 - 20 * q^37 + 8 * q^43 + 4 * q^44 + 8 * q^46 + 12 * q^50 - 4 * q^51 + 4 * q^53 + 40 * q^57 - 4 * q^58 + 8 * q^60 + 4 * q^64 - 24 * q^67 - 48 * q^71 + 2 * q^72 - 20 * q^74 + 8 * q^79 + 10 * q^81 - 16 * q^85 - 4 * q^86 + 4 * q^88 - 16 * q^92 + 24 * q^93 + 40 * q^95 + 8 * q^99

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*b3

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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 $$T_{3}^{4} + 2T_{3}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(98, [\chi])$$.

## Hecke characteristic polynomials

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