# Properties

 Label 98.6.a.a Level $98$ Weight $6$ Character orbit 98.a Self dual yes Analytic conductor $15.718$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [98,6,Mod(1,98)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(98, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("98.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$98 = 2 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 98.a (trivial)

## 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: yes Analytic conductor: $$15.7176143417$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 14) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - 4 q^{2} - 10 q^{3} + 16 q^{4} - 84 q^{5} + 40 q^{6} - 64 q^{8} - 143 q^{9}+O(q^{10})$$ q - 4 * q^2 - 10 * q^3 + 16 * q^4 - 84 * q^5 + 40 * q^6 - 64 * q^8 - 143 * q^9 $$q - 4 q^{2} - 10 q^{3} + 16 q^{4} - 84 q^{5} + 40 q^{6} - 64 q^{8} - 143 q^{9} + 336 q^{10} - 336 q^{11} - 160 q^{12} - 584 q^{13} + 840 q^{15} + 256 q^{16} + 1458 q^{17} + 572 q^{18} - 470 q^{19} - 1344 q^{20} + 1344 q^{22} - 4200 q^{23} + 640 q^{24} + 3931 q^{25} + 2336 q^{26} + 3860 q^{27} + 4866 q^{29} - 3360 q^{30} + 7372 q^{31} - 1024 q^{32} + 3360 q^{33} - 5832 q^{34} - 2288 q^{36} + 14330 q^{37} + 1880 q^{38} + 5840 q^{39} + 5376 q^{40} - 6222 q^{41} + 3704 q^{43} - 5376 q^{44} + 12012 q^{45} + 16800 q^{46} + 1812 q^{47} - 2560 q^{48} - 15724 q^{50} - 14580 q^{51} - 9344 q^{52} - 37242 q^{53} - 15440 q^{54} + 28224 q^{55} + 4700 q^{57} - 19464 q^{58} - 34302 q^{59} + 13440 q^{60} - 24476 q^{61} - 29488 q^{62} + 4096 q^{64} + 49056 q^{65} - 13440 q^{66} - 17452 q^{67} + 23328 q^{68} + 42000 q^{69} + 28224 q^{71} + 9152 q^{72} - 3602 q^{73} - 57320 q^{74} - 39310 q^{75} - 7520 q^{76} - 23360 q^{78} + 42872 q^{79} - 21504 q^{80} - 3851 q^{81} + 24888 q^{82} + 35202 q^{83} - 122472 q^{85} - 14816 q^{86} - 48660 q^{87} + 21504 q^{88} - 26730 q^{89} - 48048 q^{90} - 67200 q^{92} - 73720 q^{93} - 7248 q^{94} + 39480 q^{95} + 10240 q^{96} + 16978 q^{97} + 48048 q^{99}+O(q^{100})$$ q - 4 * q^2 - 10 * q^3 + 16 * q^4 - 84 * q^5 + 40 * q^6 - 64 * q^8 - 143 * q^9 + 336 * q^10 - 336 * q^11 - 160 * q^12 - 584 * q^13 + 840 * q^15 + 256 * q^16 + 1458 * q^17 + 572 * q^18 - 470 * q^19 - 1344 * q^20 + 1344 * q^22 - 4200 * q^23 + 640 * q^24 + 3931 * q^25 + 2336 * q^26 + 3860 * q^27 + 4866 * q^29 - 3360 * q^30 + 7372 * q^31 - 1024 * q^32 + 3360 * q^33 - 5832 * q^34 - 2288 * q^36 + 14330 * q^37 + 1880 * q^38 + 5840 * q^39 + 5376 * q^40 - 6222 * q^41 + 3704 * q^43 - 5376 * q^44 + 12012 * q^45 + 16800 * q^46 + 1812 * q^47 - 2560 * q^48 - 15724 * q^50 - 14580 * q^51 - 9344 * q^52 - 37242 * q^53 - 15440 * q^54 + 28224 * q^55 + 4700 * q^57 - 19464 * q^58 - 34302 * q^59 + 13440 * q^60 - 24476 * q^61 - 29488 * q^62 + 4096 * q^64 + 49056 * q^65 - 13440 * q^66 - 17452 * q^67 + 23328 * q^68 + 42000 * q^69 + 28224 * q^71 + 9152 * q^72 - 3602 * q^73 - 57320 * q^74 - 39310 * q^75 - 7520 * q^76 - 23360 * q^78 + 42872 * q^79 - 21504 * q^80 - 3851 * q^81 + 24888 * q^82 + 35202 * q^83 - 122472 * q^85 - 14816 * q^86 - 48660 * q^87 + 21504 * q^88 - 26730 * q^89 - 48048 * q^90 - 67200 * q^92 - 73720 * q^93 - 7248 * q^94 + 39480 * q^95 + 10240 * q^96 + 16978 * q^97 + 48048 * q^99

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
−4.00000 −10.0000 16.0000 −84.0000 40.0000 0 −64.0000 −143.000 336.000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$+1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 98.6.a.a 1
3.b odd 2 1 882.6.a.x 1
4.b odd 2 1 784.6.a.i 1
7.b odd 2 1 14.6.a.a 1
7.c even 3 2 98.6.c.d 2
7.d odd 6 2 98.6.c.c 2
21.c even 2 1 126.6.a.f 1
28.d even 2 1 112.6.a.c 1
35.c odd 2 1 350.6.a.i 1
35.f even 4 2 350.6.c.d 2
56.e even 2 1 448.6.a.l 1
56.h odd 2 1 448.6.a.e 1
84.h odd 2 1 1008.6.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.6.a.a 1 7.b odd 2 1
98.6.a.a 1 1.a even 1 1 trivial
98.6.c.c 2 7.d odd 6 2
98.6.c.d 2 7.c even 3 2
112.6.a.c 1 28.d even 2 1
126.6.a.f 1 21.c even 2 1
350.6.a.i 1 35.c odd 2 1
350.6.c.d 2 35.f even 4 2
448.6.a.e 1 56.h odd 2 1
448.6.a.l 1 56.e even 2 1
784.6.a.i 1 4.b odd 2 1
882.6.a.x 1 3.b odd 2 1
1008.6.a.b 1 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 10$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(98))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 4$$
$3$ $$T + 10$$
$5$ $$T + 84$$
$7$ $$T$$
$11$ $$T + 336$$
$13$ $$T + 584$$
$17$ $$T - 1458$$
$19$ $$T + 470$$
$23$ $$T + 4200$$
$29$ $$T - 4866$$
$31$ $$T - 7372$$
$37$ $$T - 14330$$
$41$ $$T + 6222$$
$43$ $$T - 3704$$
$47$ $$T - 1812$$
$53$ $$T + 37242$$
$59$ $$T + 34302$$
$61$ $$T + 24476$$
$67$ $$T + 17452$$
$71$ $$T - 28224$$
$73$ $$T + 3602$$
$79$ $$T - 42872$$
$83$ $$T - 35202$$
$89$ $$T + 26730$$
$97$ $$T - 16978$$