# Properties

 Label 98.6.c.a Level $98$ Weight $6$ Character orbit 98.c Analytic conductor $15.718$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [98,6,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, 6, names="a")

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

chi := DirichletCharacter("98.67");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$98 = 2 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ 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: $$15.7176143417$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 14) 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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 4 \zeta_{6} q^{2} + (8 \zeta_{6} - 8) q^{3} + (16 \zeta_{6} - 16) q^{4} - 10 \zeta_{6} q^{5} + 32 q^{6} + 64 q^{8} + 179 \zeta_{6} q^{9} +O(q^{10})$$ q - 4*z * q^2 + (8*z - 8) * q^3 + (16*z - 16) * q^4 - 10*z * q^5 + 32 * q^6 + 64 * q^8 + 179*z * q^9 $$q - 4 \zeta_{6} q^{2} + (8 \zeta_{6} - 8) q^{3} + (16 \zeta_{6} - 16) q^{4} - 10 \zeta_{6} q^{5} + 32 q^{6} + 64 q^{8} + 179 \zeta_{6} q^{9} + (40 \zeta_{6} - 40) q^{10} + ( - 340 \zeta_{6} + 340) q^{11} - 128 \zeta_{6} q^{12} - 294 q^{13} + 80 q^{15} - 256 \zeta_{6} q^{16} + (1226 \zeta_{6} - 1226) q^{17} + ( - 716 \zeta_{6} + 716) q^{18} - 2432 \zeta_{6} q^{19} + 160 q^{20} - 1360 q^{22} - 2000 \zeta_{6} q^{23} + (512 \zeta_{6} - 512) q^{24} + ( - 3025 \zeta_{6} + 3025) q^{25} + 1176 \zeta_{6} q^{26} - 3376 q^{27} - 6746 q^{29} - 320 \zeta_{6} q^{30} + (8856 \zeta_{6} - 8856) q^{31} + (1024 \zeta_{6} - 1024) q^{32} + 2720 \zeta_{6} q^{33} + 4904 q^{34} - 2864 q^{36} - 9182 \zeta_{6} q^{37} + (9728 \zeta_{6} - 9728) q^{38} + ( - 2352 \zeta_{6} + 2352) q^{39} - 640 \zeta_{6} q^{40} - 14574 q^{41} + 8108 q^{43} + 5440 \zeta_{6} q^{44} + ( - 1790 \zeta_{6} + 1790) q^{45} + (8000 \zeta_{6} - 8000) q^{46} + 312 \zeta_{6} q^{47} + 2048 q^{48} - 12100 q^{50} - 9808 \zeta_{6} q^{51} + ( - 4704 \zeta_{6} + 4704) q^{52} + ( - 14634 \zeta_{6} + 14634) q^{53} + 13504 \zeta_{6} q^{54} - 3400 q^{55} + 19456 q^{57} + 26984 \zeta_{6} q^{58} + ( - 27656 \zeta_{6} + 27656) q^{59} + (1280 \zeta_{6} - 1280) q^{60} - 34338 \zeta_{6} q^{61} + 35424 q^{62} + 4096 q^{64} + 2940 \zeta_{6} q^{65} + ( - 10880 \zeta_{6} + 10880) q^{66} + (12316 \zeta_{6} - 12316) q^{67} - 19616 \zeta_{6} q^{68} + 16000 q^{69} + 36920 q^{71} + 11456 \zeta_{6} q^{72} + ( - 61718 \zeta_{6} + 61718) q^{73} + (36728 \zeta_{6} - 36728) q^{74} + 24200 \zeta_{6} q^{75} + 38912 q^{76} - 9408 q^{78} + 64752 \zeta_{6} q^{79} + (2560 \zeta_{6} - 2560) q^{80} + (16489 \zeta_{6} - 16489) q^{81} + 58296 \zeta_{6} q^{82} - 77056 q^{83} + 12260 q^{85} - 32432 \zeta_{6} q^{86} + ( - 53968 \zeta_{6} + 53968) q^{87} + ( - 21760 \zeta_{6} + 21760) q^{88} + 8166 \zeta_{6} q^{89} - 7160 q^{90} + 32000 q^{92} - 70848 \zeta_{6} q^{93} + ( - 1248 \zeta_{6} + 1248) q^{94} + (24320 \zeta_{6} - 24320) q^{95} - 8192 \zeta_{6} q^{96} + 20650 q^{97} + 60860 q^{99} +O(q^{100})$$ q - 4*z * q^2 + (8*z - 8) * q^3 + (16*z - 16) * q^4 - 10*z * q^5 + 32 * q^6 + 64 * q^8 + 179*z * q^9 + (40*z - 40) * q^10 + (-340*z + 340) * q^11 - 128*z * q^12 - 294 * q^13 + 80 * q^15 - 256*z * q^16 + (1226*z - 1226) * q^17 + (-716*z + 716) * q^18 - 2432*z * q^19 + 160 * q^20 - 1360 * q^22 - 2000*z * q^23 + (512*z - 512) * q^24 + (-3025*z + 3025) * q^25 + 1176*z * q^26 - 3376 * q^27 - 6746 * q^29 - 320*z * q^30 + (8856*z - 8856) * q^31 + (1024*z - 1024) * q^32 + 2720*z * q^33 + 4904 * q^34 - 2864 * q^36 - 9182*z * q^37 + (9728*z - 9728) * q^38 + (-2352*z + 2352) * q^39 - 640*z * q^40 - 14574 * q^41 + 8108 * q^43 + 5440*z * q^44 + (-1790*z + 1790) * q^45 + (8000*z - 8000) * q^46 + 312*z * q^47 + 2048 * q^48 - 12100 * q^50 - 9808*z * q^51 + (-4704*z + 4704) * q^52 + (-14634*z + 14634) * q^53 + 13504*z * q^54 - 3400 * q^55 + 19456 * q^57 + 26984*z * q^58 + (-27656*z + 27656) * q^59 + (1280*z - 1280) * q^60 - 34338*z * q^61 + 35424 * q^62 + 4096 * q^64 + 2940*z * q^65 + (-10880*z + 10880) * q^66 + (12316*z - 12316) * q^67 - 19616*z * q^68 + 16000 * q^69 + 36920 * q^71 + 11456*z * q^72 + (-61718*z + 61718) * q^73 + (36728*z - 36728) * q^74 + 24200*z * q^75 + 38912 * q^76 - 9408 * q^78 + 64752*z * q^79 + (2560*z - 2560) * q^80 + (16489*z - 16489) * q^81 + 58296*z * q^82 - 77056 * q^83 + 12260 * q^85 - 32432*z * q^86 + (-53968*z + 53968) * q^87 + (-21760*z + 21760) * q^88 + 8166*z * q^89 - 7160 * q^90 + 32000 * q^92 - 70848*z * q^93 + (-1248*z + 1248) * q^94 + (24320*z - 24320) * q^95 - 8192*z * q^96 + 20650 * q^97 + 60860 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{2} - 8 q^{3} - 16 q^{4} - 10 q^{5} + 64 q^{6} + 128 q^{8} + 179 q^{9}+O(q^{10})$$ 2 * q - 4 * q^2 - 8 * q^3 - 16 * q^4 - 10 * q^5 + 64 * q^6 + 128 * q^8 + 179 * q^9 $$2 q - 4 q^{2} - 8 q^{3} - 16 q^{4} - 10 q^{5} + 64 q^{6} + 128 q^{8} + 179 q^{9} - 40 q^{10} + 340 q^{11} - 128 q^{12} - 588 q^{13} + 160 q^{15} - 256 q^{16} - 1226 q^{17} + 716 q^{18} - 2432 q^{19} + 320 q^{20} - 2720 q^{22} - 2000 q^{23} - 512 q^{24} + 3025 q^{25} + 1176 q^{26} - 6752 q^{27} - 13492 q^{29} - 320 q^{30} - 8856 q^{31} - 1024 q^{32} + 2720 q^{33} + 9808 q^{34} - 5728 q^{36} - 9182 q^{37} - 9728 q^{38} + 2352 q^{39} - 640 q^{40} - 29148 q^{41} + 16216 q^{43} + 5440 q^{44} + 1790 q^{45} - 8000 q^{46} + 312 q^{47} + 4096 q^{48} - 24200 q^{50} - 9808 q^{51} + 4704 q^{52} + 14634 q^{53} + 13504 q^{54} - 6800 q^{55} + 38912 q^{57} + 26984 q^{58} + 27656 q^{59} - 1280 q^{60} - 34338 q^{61} + 70848 q^{62} + 8192 q^{64} + 2940 q^{65} + 10880 q^{66} - 12316 q^{67} - 19616 q^{68} + 32000 q^{69} + 73840 q^{71} + 11456 q^{72} + 61718 q^{73} - 36728 q^{74} + 24200 q^{75} + 77824 q^{76} - 18816 q^{78} + 64752 q^{79} - 2560 q^{80} - 16489 q^{81} + 58296 q^{82} - 154112 q^{83} + 24520 q^{85} - 32432 q^{86} + 53968 q^{87} + 21760 q^{88} + 8166 q^{89} - 14320 q^{90} + 64000 q^{92} - 70848 q^{93} + 1248 q^{94} - 24320 q^{95} - 8192 q^{96} + 41300 q^{97} + 121720 q^{99}+O(q^{100})$$ 2 * q - 4 * q^2 - 8 * q^3 - 16 * q^4 - 10 * q^5 + 64 * q^6 + 128 * q^8 + 179 * q^9 - 40 * q^10 + 340 * q^11 - 128 * q^12 - 588 * q^13 + 160 * q^15 - 256 * q^16 - 1226 * q^17 + 716 * q^18 - 2432 * q^19 + 320 * q^20 - 2720 * q^22 - 2000 * q^23 - 512 * q^24 + 3025 * q^25 + 1176 * q^26 - 6752 * q^27 - 13492 * q^29 - 320 * q^30 - 8856 * q^31 - 1024 * q^32 + 2720 * q^33 + 9808 * q^34 - 5728 * q^36 - 9182 * q^37 - 9728 * q^38 + 2352 * q^39 - 640 * q^40 - 29148 * q^41 + 16216 * q^43 + 5440 * q^44 + 1790 * q^45 - 8000 * q^46 + 312 * q^47 + 4096 * q^48 - 24200 * q^50 - 9808 * q^51 + 4704 * q^52 + 14634 * q^53 + 13504 * q^54 - 6800 * q^55 + 38912 * q^57 + 26984 * q^58 + 27656 * q^59 - 1280 * q^60 - 34338 * q^61 + 70848 * q^62 + 8192 * q^64 + 2940 * q^65 + 10880 * q^66 - 12316 * q^67 - 19616 * q^68 + 32000 * q^69 + 73840 * q^71 + 11456 * q^72 + 61718 * q^73 - 36728 * q^74 + 24200 * q^75 + 77824 * q^76 - 18816 * q^78 + 64752 * q^79 - 2560 * q^80 - 16489 * q^81 + 58296 * q^82 - 154112 * q^83 + 24520 * q^85 - 32432 * q^86 + 53968 * q^87 + 21760 * q^88 + 8166 * q^89 - 14320 * q^90 + 64000 * q^92 - 70848 * q^93 + 1248 * q^94 - 24320 * q^95 - 8192 * q^96 + 41300 * q^97 + 121720 * q^99

## Character values

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

 $$n$$ $$3$$ $$\chi(n)$$ $$-\zeta_{6}$$

## 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
 0.5 + 0.866025i 0.5 − 0.866025i
−2.00000 3.46410i −4.00000 + 6.92820i −8.00000 + 13.8564i −5.00000 8.66025i 32.0000 0 64.0000 89.5000 + 155.019i −20.0000 + 34.6410i
79.1 −2.00000 + 3.46410i −4.00000 6.92820i −8.00000 13.8564i −5.00000 + 8.66025i 32.0000 0 64.0000 89.5000 155.019i −20.0000 34.6410i
 $$n$$: 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 98.6.c.a 2
7.b odd 2 1 98.6.c.b 2
7.c even 3 1 14.6.a.b 1
7.c even 3 1 inner 98.6.c.a 2
7.d odd 6 1 98.6.a.b 1
7.d odd 6 1 98.6.c.b 2
21.g even 6 1 882.6.a.g 1
21.h odd 6 1 126.6.a.c 1
28.f even 6 1 784.6.a.h 1
28.g odd 6 1 112.6.a.d 1
35.j even 6 1 350.6.a.b 1
35.l odd 12 2 350.6.c.f 2
56.k odd 6 1 448.6.a.k 1
56.p even 6 1 448.6.a.f 1
84.n even 6 1 1008.6.a.n 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.6.a.b 1 7.c even 3 1
98.6.a.b 1 7.d odd 6 1
98.6.c.a 2 1.a even 1 1 trivial
98.6.c.a 2 7.c even 3 1 inner
98.6.c.b 2 7.b odd 2 1
98.6.c.b 2 7.d odd 6 1
112.6.a.d 1 28.g odd 6 1
126.6.a.c 1 21.h odd 6 1
350.6.a.b 1 35.j even 6 1
350.6.c.f 2 35.l odd 12 2
448.6.a.f 1 56.p even 6 1
448.6.a.k 1 56.k odd 6 1
784.6.a.h 1 28.f even 6 1
882.6.a.g 1 21.g even 6 1
1008.6.a.n 1 84.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 8T_{3} + 64$$ acting on $$S_{6}^{\mathrm{new}}(98, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 4T + 16$$
$3$ $$T^{2} + 8T + 64$$
$5$ $$T^{2} + 10T + 100$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 340T + 115600$$
$13$ $$(T + 294)^{2}$$
$17$ $$T^{2} + 1226 T + 1503076$$
$19$ $$T^{2} + 2432 T + 5914624$$
$23$ $$T^{2} + 2000 T + 4000000$$
$29$ $$(T + 6746)^{2}$$
$31$ $$T^{2} + 8856 T + 78428736$$
$37$ $$T^{2} + 9182 T + 84309124$$
$41$ $$(T + 14574)^{2}$$
$43$ $$(T - 8108)^{2}$$
$47$ $$T^{2} - 312T + 97344$$
$53$ $$T^{2} - 14634 T + 214153956$$
$59$ $$T^{2} - 27656 T + 764854336$$
$61$ $$T^{2} + \cdots + 1179098244$$
$67$ $$T^{2} + 12316 T + 151683856$$
$71$ $$(T - 36920)^{2}$$
$73$ $$T^{2} + \cdots + 3809111524$$
$79$ $$T^{2} + \cdots + 4192821504$$
$83$ $$(T + 77056)^{2}$$
$89$ $$T^{2} - 8166 T + 66683556$$
$97$ $$(T - 20650)^{2}$$