# Properties

 Label 98.6.c.c 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.

 $$f(q)$$ $$=$$ $$q + 4 \zeta_{6} q^{2} + (10 \zeta_{6} - 10) q^{3} + (16 \zeta_{6} - 16) q^{4} - 84 \zeta_{6} q^{5} - 40 q^{6} - 64 q^{8} + 143 \zeta_{6} q^{9} +O(q^{10})$$ q + 4*z * q^2 + (10*z - 10) * q^3 + (16*z - 16) * q^4 - 84*z * q^5 - 40 * q^6 - 64 * q^8 + 143*z * q^9 $$q + 4 \zeta_{6} q^{2} + (10 \zeta_{6} - 10) q^{3} + (16 \zeta_{6} - 16) q^{4} - 84 \zeta_{6} q^{5} - 40 q^{6} - 64 q^{8} + 143 \zeta_{6} q^{9} + ( - 336 \zeta_{6} + 336) q^{10} + ( - 336 \zeta_{6} + 336) q^{11} - 160 \zeta_{6} q^{12} + 584 q^{13} + 840 q^{15} - 256 \zeta_{6} q^{16} + ( - 1458 \zeta_{6} + 1458) q^{17} + (572 \zeta_{6} - 572) q^{18} - 470 \zeta_{6} q^{19} + 1344 q^{20} + 1344 q^{22} + 4200 \zeta_{6} q^{23} + ( - 640 \zeta_{6} + 640) q^{24} + (3931 \zeta_{6} - 3931) q^{25} + 2336 \zeta_{6} q^{26} - 3860 q^{27} + 4866 q^{29} + 3360 \zeta_{6} q^{30} + ( - 7372 \zeta_{6} + 7372) q^{31} + ( - 1024 \zeta_{6} + 1024) q^{32} + 3360 \zeta_{6} q^{33} + 5832 q^{34} - 2288 q^{36} - 14330 \zeta_{6} q^{37} + ( - 1880 \zeta_{6} + 1880) q^{38} + (5840 \zeta_{6} - 5840) q^{39} + 5376 \zeta_{6} q^{40} + 6222 q^{41} + 3704 q^{43} + 5376 \zeta_{6} q^{44} + ( - 12012 \zeta_{6} + 12012) q^{45} + (16800 \zeta_{6} - 16800) q^{46} + 1812 \zeta_{6} q^{47} + 2560 q^{48} - 15724 q^{50} + 14580 \zeta_{6} q^{51} + (9344 \zeta_{6} - 9344) q^{52} + ( - 37242 \zeta_{6} + 37242) q^{53} - 15440 \zeta_{6} q^{54} - 28224 q^{55} + 4700 q^{57} + 19464 \zeta_{6} q^{58} + (34302 \zeta_{6} - 34302) q^{59} + (13440 \zeta_{6} - 13440) q^{60} - 24476 \zeta_{6} q^{61} + 29488 q^{62} + 4096 q^{64} - 49056 \zeta_{6} q^{65} + (13440 \zeta_{6} - 13440) q^{66} + ( - 17452 \zeta_{6} + 17452) q^{67} + 23328 \zeta_{6} q^{68} - 42000 q^{69} + 28224 q^{71} - 9152 \zeta_{6} q^{72} + (3602 \zeta_{6} - 3602) q^{73} + ( - 57320 \zeta_{6} + 57320) q^{74} - 39310 \zeta_{6} q^{75} + 7520 q^{76} - 23360 q^{78} - 42872 \zeta_{6} q^{79} + (21504 \zeta_{6} - 21504) q^{80} + ( - 3851 \zeta_{6} + 3851) q^{81} + 24888 \zeta_{6} q^{82} - 35202 q^{83} - 122472 q^{85} + 14816 \zeta_{6} q^{86} + (48660 \zeta_{6} - 48660) q^{87} + (21504 \zeta_{6} - 21504) q^{88} - 26730 \zeta_{6} q^{89} + 48048 q^{90} - 67200 q^{92} + 73720 \zeta_{6} q^{93} + (7248 \zeta_{6} - 7248) q^{94} + (39480 \zeta_{6} - 39480) q^{95} + 10240 \zeta_{6} q^{96} - 16978 q^{97} + 48048 q^{99} +O(q^{100})$$ q + 4*z * q^2 + (10*z - 10) * q^3 + (16*z - 16) * q^4 - 84*z * q^5 - 40 * q^6 - 64 * q^8 + 143*z * q^9 + (-336*z + 336) * q^10 + (-336*z + 336) * q^11 - 160*z * q^12 + 584 * q^13 + 840 * q^15 - 256*z * q^16 + (-1458*z + 1458) * q^17 + (572*z - 572) * q^18 - 470*z * q^19 + 1344 * q^20 + 1344 * q^22 + 4200*z * q^23 + (-640*z + 640) * q^24 + (3931*z - 3931) * q^25 + 2336*z * q^26 - 3860 * q^27 + 4866 * q^29 + 3360*z * q^30 + (-7372*z + 7372) * q^31 + (-1024*z + 1024) * q^32 + 3360*z * q^33 + 5832 * q^34 - 2288 * q^36 - 14330*z * q^37 + (-1880*z + 1880) * q^38 + (5840*z - 5840) * q^39 + 5376*z * q^40 + 6222 * q^41 + 3704 * q^43 + 5376*z * q^44 + (-12012*z + 12012) * q^45 + (16800*z - 16800) * q^46 + 1812*z * q^47 + 2560 * q^48 - 15724 * q^50 + 14580*z * q^51 + (9344*z - 9344) * q^52 + (-37242*z + 37242) * q^53 - 15440*z * q^54 - 28224 * q^55 + 4700 * q^57 + 19464*z * q^58 + (34302*z - 34302) * q^59 + (13440*z - 13440) * q^60 - 24476*z * q^61 + 29488 * q^62 + 4096 * q^64 - 49056*z * q^65 + (13440*z - 13440) * q^66 + (-17452*z + 17452) * q^67 + 23328*z * q^68 - 42000 * q^69 + 28224 * q^71 - 9152*z * q^72 + (3602*z - 3602) * q^73 + (-57320*z + 57320) * q^74 - 39310*z * q^75 + 7520 * q^76 - 23360 * q^78 - 42872*z * q^79 + (21504*z - 21504) * q^80 + (-3851*z + 3851) * q^81 + 24888*z * q^82 - 35202 * q^83 - 122472 * q^85 + 14816*z * q^86 + (48660*z - 48660) * q^87 + (21504*z - 21504) * q^88 - 26730*z * q^89 + 48048 * q^90 - 67200 * q^92 + 73720*z * q^93 + (7248*z - 7248) * q^94 + (39480*z - 39480) * q^95 + 10240*z * q^96 - 16978 * q^97 + 48048 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{2} - 10 q^{3} - 16 q^{4} - 84 q^{5} - 80 q^{6} - 128 q^{8} + 143 q^{9}+O(q^{10})$$ 2 * q + 4 * q^2 - 10 * q^3 - 16 * q^4 - 84 * q^5 - 80 * q^6 - 128 * q^8 + 143 * q^9 $$2 q + 4 q^{2} - 10 q^{3} - 16 q^{4} - 84 q^{5} - 80 q^{6} - 128 q^{8} + 143 q^{9} + 336 q^{10} + 336 q^{11} - 160 q^{12} + 1168 q^{13} + 1680 q^{15} - 256 q^{16} + 1458 q^{17} - 572 q^{18} - 470 q^{19} + 2688 q^{20} + 2688 q^{22} + 4200 q^{23} + 640 q^{24} - 3931 q^{25} + 2336 q^{26} - 7720 q^{27} + 9732 q^{29} + 3360 q^{30} + 7372 q^{31} + 1024 q^{32} + 3360 q^{33} + 11664 q^{34} - 4576 q^{36} - 14330 q^{37} + 1880 q^{38} - 5840 q^{39} + 5376 q^{40} + 12444 q^{41} + 7408 q^{43} + 5376 q^{44} + 12012 q^{45} - 16800 q^{46} + 1812 q^{47} + 5120 q^{48} - 31448 q^{50} + 14580 q^{51} - 9344 q^{52} + 37242 q^{53} - 15440 q^{54} - 56448 q^{55} + 9400 q^{57} + 19464 q^{58} - 34302 q^{59} - 13440 q^{60} - 24476 q^{61} + 58976 q^{62} + 8192 q^{64} - 49056 q^{65} - 13440 q^{66} + 17452 q^{67} + 23328 q^{68} - 84000 q^{69} + 56448 q^{71} - 9152 q^{72} - 3602 q^{73} + 57320 q^{74} - 39310 q^{75} + 15040 q^{76} - 46720 q^{78} - 42872 q^{79} - 21504 q^{80} + 3851 q^{81} + 24888 q^{82} - 70404 q^{83} - 244944 q^{85} + 14816 q^{86} - 48660 q^{87} - 21504 q^{88} - 26730 q^{89} + 96096 q^{90} - 134400 q^{92} + 73720 q^{93} - 7248 q^{94} - 39480 q^{95} + 10240 q^{96} - 33956 q^{97} + 96096 q^{99}+O(q^{100})$$ 2 * q + 4 * q^2 - 10 * q^3 - 16 * q^4 - 84 * q^5 - 80 * q^6 - 128 * q^8 + 143 * q^9 + 336 * q^10 + 336 * q^11 - 160 * q^12 + 1168 * q^13 + 1680 * q^15 - 256 * q^16 + 1458 * q^17 - 572 * q^18 - 470 * q^19 + 2688 * q^20 + 2688 * q^22 + 4200 * q^23 + 640 * q^24 - 3931 * q^25 + 2336 * q^26 - 7720 * q^27 + 9732 * q^29 + 3360 * q^30 + 7372 * q^31 + 1024 * q^32 + 3360 * q^33 + 11664 * q^34 - 4576 * q^36 - 14330 * q^37 + 1880 * q^38 - 5840 * q^39 + 5376 * q^40 + 12444 * q^41 + 7408 * q^43 + 5376 * q^44 + 12012 * q^45 - 16800 * q^46 + 1812 * q^47 + 5120 * q^48 - 31448 * q^50 + 14580 * q^51 - 9344 * q^52 + 37242 * q^53 - 15440 * q^54 - 56448 * q^55 + 9400 * q^57 + 19464 * q^58 - 34302 * q^59 - 13440 * q^60 - 24476 * q^61 + 58976 * q^62 + 8192 * q^64 - 49056 * q^65 - 13440 * q^66 + 17452 * q^67 + 23328 * q^68 - 84000 * q^69 + 56448 * q^71 - 9152 * q^72 - 3602 * q^73 + 57320 * q^74 - 39310 * q^75 + 15040 * q^76 - 46720 * q^78 - 42872 * q^79 - 21504 * q^80 + 3851 * q^81 + 24888 * q^82 - 70404 * q^83 - 244944 * q^85 + 14816 * q^86 - 48660 * q^87 - 21504 * q^88 - 26730 * q^89 + 96096 * q^90 - 134400 * q^92 + 73720 * q^93 - 7248 * q^94 - 39480 * q^95 + 10240 * q^96 - 33956 * q^97 + 96096 * 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 −5.00000 + 8.66025i −8.00000 + 13.8564i −42.0000 72.7461i −40.0000 0 −64.0000 71.5000 + 123.842i 168.000 290.985i
79.1 2.00000 3.46410i −5.00000 8.66025i −8.00000 13.8564i −42.0000 + 72.7461i −40.0000 0 −64.0000 71.5000 123.842i 168.000 + 290.985i
 $$n$$: 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.c 2
7.b odd 2 1 98.6.c.d 2
7.c even 3 1 14.6.a.a 1
7.c even 3 1 inner 98.6.c.c 2
7.d odd 6 1 98.6.a.a 1
7.d odd 6 1 98.6.c.d 2
21.g even 6 1 882.6.a.x 1
21.h odd 6 1 126.6.a.f 1
28.f even 6 1 784.6.a.i 1
28.g odd 6 1 112.6.a.c 1
35.j even 6 1 350.6.a.i 1
35.l odd 12 2 350.6.c.d 2
56.k odd 6 1 448.6.a.l 1
56.p even 6 1 448.6.a.e 1
84.n even 6 1 1008.6.a.b 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 10T_{3} + 100$$ acting on $$S_{6}^{\mathrm{new}}(98, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 4T + 16$$
$3$ $$T^{2} + 10T + 100$$
$5$ $$T^{2} + 84T + 7056$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 336T + 112896$$
$13$ $$(T - 584)^{2}$$
$17$ $$T^{2} - 1458 T + 2125764$$
$19$ $$T^{2} + 470T + 220900$$
$23$ $$T^{2} - 4200 T + 17640000$$
$29$ $$(T - 4866)^{2}$$
$31$ $$T^{2} - 7372 T + 54346384$$
$37$ $$T^{2} + 14330 T + 205348900$$
$41$ $$(T - 6222)^{2}$$
$43$ $$(T - 3704)^{2}$$
$47$ $$T^{2} - 1812 T + 3283344$$
$53$ $$T^{2} + \cdots + 1386966564$$
$59$ $$T^{2} + \cdots + 1176627204$$
$61$ $$T^{2} + 24476 T + 599074576$$
$67$ $$T^{2} - 17452 T + 304572304$$
$71$ $$(T - 28224)^{2}$$
$73$ $$T^{2} + 3602 T + 12974404$$
$79$ $$T^{2} + \cdots + 1838008384$$
$83$ $$(T + 35202)^{2}$$
$89$ $$T^{2} + 26730 T + 714492900$$
$97$ $$(T + 16978)^{2}$$