# Properties

 Label 338.6.b.a Level $338$ Weight $6$ Character orbit 338.b Analytic conductor $54.210$ 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] = [338,6,Mod(337,338)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(338, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

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

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

chi := DirichletCharacter("338.337");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 338.b (of order $$2$$, degree $$1$$, 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: $$54.2097310968$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 26) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$\beta = 2i$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 2 \beta q^{2} - 16 q^{4} - 7 \beta q^{5} + 85 \beta q^{7} + 32 \beta q^{8} - 243 q^{9} +O(q^{10})$$ q - 2*b * q^2 - 16 * q^4 - 7*b * q^5 + 85*b * q^7 + 32*b * q^8 - 243 * q^9 $$q - 2 \beta q^{2} - 16 q^{4} - 7 \beta q^{5} + 85 \beta q^{7} + 32 \beta q^{8} - 243 q^{9} - 56 q^{10} + 125 \beta q^{11} + 680 q^{14} + 256 q^{16} - 1062 q^{17} + 486 \beta q^{18} - 39 \beta q^{19} + 112 \beta q^{20} + 1000 q^{22} - 1576 q^{23} + 2929 q^{25} - 1360 \beta q^{28} + 2578 q^{29} - 4327 \beta q^{31} - 512 \beta q^{32} + 2124 \beta q^{34} + 2380 q^{35} + 3888 q^{36} - 5493 \beta q^{37} - 312 q^{38} + 896 q^{40} + 525 \beta q^{41} + 5900 q^{43} - 2000 \beta q^{44} + 1701 \beta q^{45} + 3152 \beta q^{46} + 2981 \beta q^{47} - 12093 q^{49} - 5858 \beta q^{50} + 29046 q^{53} + 3500 q^{55} - 10880 q^{56} - 5156 \beta q^{58} + 6961 \beta q^{59} - 32882 q^{61} - 34616 q^{62} - 20655 \beta q^{63} - 4096 q^{64} - 34783 \beta q^{67} + 16992 q^{68} - 4760 \beta q^{70} - 25271 \beta q^{71} - 7776 \beta q^{72} + 23375 \beta q^{73} - 43944 q^{74} + 624 \beta q^{76} - 42500 q^{77} - 19348 q^{79} - 1792 \beta q^{80} + 59049 q^{81} + 4200 q^{82} - 43719 \beta q^{83} + 7434 \beta q^{85} - 11800 \beta q^{86} - 16000 q^{88} - 47085 \beta q^{89} + 13608 q^{90} + 25216 q^{92} + 23848 q^{94} - 1092 q^{95} + 91393 \beta q^{97} + 24186 \beta q^{98} - 30375 \beta q^{99} +O(q^{100})$$ q - 2*b * q^2 - 16 * q^4 - 7*b * q^5 + 85*b * q^7 + 32*b * q^8 - 243 * q^9 - 56 * q^10 + 125*b * q^11 + 680 * q^14 + 256 * q^16 - 1062 * q^17 + 486*b * q^18 - 39*b * q^19 + 112*b * q^20 + 1000 * q^22 - 1576 * q^23 + 2929 * q^25 - 1360*b * q^28 + 2578 * q^29 - 4327*b * q^31 - 512*b * q^32 + 2124*b * q^34 + 2380 * q^35 + 3888 * q^36 - 5493*b * q^37 - 312 * q^38 + 896 * q^40 + 525*b * q^41 + 5900 * q^43 - 2000*b * q^44 + 1701*b * q^45 + 3152*b * q^46 + 2981*b * q^47 - 12093 * q^49 - 5858*b * q^50 + 29046 * q^53 + 3500 * q^55 - 10880 * q^56 - 5156*b * q^58 + 6961*b * q^59 - 32882 * q^61 - 34616 * q^62 - 20655*b * q^63 - 4096 * q^64 - 34783*b * q^67 + 16992 * q^68 - 4760*b * q^70 - 25271*b * q^71 - 7776*b * q^72 + 23375*b * q^73 - 43944 * q^74 + 624*b * q^76 - 42500 * q^77 - 19348 * q^79 - 1792*b * q^80 + 59049 * q^81 + 4200 * q^82 - 43719*b * q^83 + 7434*b * q^85 - 11800*b * q^86 - 16000 * q^88 - 47085*b * q^89 + 13608 * q^90 + 25216 * q^92 + 23848 * q^94 - 1092 * q^95 + 91393*b * q^97 + 24186*b * q^98 - 30375*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 32 q^{4} - 486 q^{9}+O(q^{10})$$ 2 * q - 32 * q^4 - 486 * q^9 $$2 q - 32 q^{4} - 486 q^{9} - 112 q^{10} + 1360 q^{14} + 512 q^{16} - 2124 q^{17} + 2000 q^{22} - 3152 q^{23} + 5858 q^{25} + 5156 q^{29} + 4760 q^{35} + 7776 q^{36} - 624 q^{38} + 1792 q^{40} + 11800 q^{43} - 24186 q^{49} + 58092 q^{53} + 7000 q^{55} - 21760 q^{56} - 65764 q^{61} - 69232 q^{62} - 8192 q^{64} + 33984 q^{68} - 87888 q^{74} - 85000 q^{77} - 38696 q^{79} + 118098 q^{81} + 8400 q^{82} - 32000 q^{88} + 27216 q^{90} + 50432 q^{92} + 47696 q^{94} - 2184 q^{95}+O(q^{100})$$ 2 * q - 32 * q^4 - 486 * q^9 - 112 * q^10 + 1360 * q^14 + 512 * q^16 - 2124 * q^17 + 2000 * q^22 - 3152 * q^23 + 5858 * q^25 + 5156 * q^29 + 4760 * q^35 + 7776 * q^36 - 624 * q^38 + 1792 * q^40 + 11800 * q^43 - 24186 * q^49 + 58092 * q^53 + 7000 * q^55 - 21760 * q^56 - 65764 * q^61 - 69232 * q^62 - 8192 * q^64 + 33984 * q^68 - 87888 * q^74 - 85000 * q^77 - 38696 * q^79 + 118098 * q^81 + 8400 * q^82 - 32000 * q^88 + 27216 * q^90 + 50432 * q^92 + 47696 * q^94 - 2184 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/338\mathbb{Z}\right)^\times$$.

 $$n$$ $$171$$ $$\chi(n)$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
337.1
 1.00000i − 1.00000i
4.00000i 0 −16.0000 14.0000i 0 170.000i 64.0000i −243.000 −56.0000
337.2 4.00000i 0 −16.0000 14.0000i 0 170.000i 64.0000i −243.000 −56.0000
 $$n$$: 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
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 338.6.b.a 2
13.b even 2 1 inner 338.6.b.a 2
13.d odd 4 1 26.6.a.a 1
13.d odd 4 1 338.6.a.d 1
39.f even 4 1 234.6.a.g 1
52.f even 4 1 208.6.a.b 1
65.f even 4 1 650.6.b.a 2
65.g odd 4 1 650.6.a.a 1
65.k even 4 1 650.6.b.a 2
104.j odd 4 1 832.6.a.d 1
104.m even 4 1 832.6.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.6.a.a 1 13.d odd 4 1
208.6.a.b 1 52.f even 4 1
234.6.a.g 1 39.f even 4 1
338.6.a.d 1 13.d odd 4 1
338.6.b.a 2 1.a even 1 1 trivial
338.6.b.a 2 13.b even 2 1 inner
650.6.a.a 1 65.g odd 4 1
650.6.b.a 2 65.f even 4 1
650.6.b.a 2 65.k even 4 1
832.6.a.d 1 104.j odd 4 1
832.6.a.e 1 104.m even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}$$ acting on $$S_{6}^{\mathrm{new}}(338, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 16$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 196$$
$7$ $$T^{2} + 28900$$
$11$ $$T^{2} + 62500$$
$13$ $$T^{2}$$
$17$ $$(T + 1062)^{2}$$
$19$ $$T^{2} + 6084$$
$23$ $$(T + 1576)^{2}$$
$29$ $$(T - 2578)^{2}$$
$31$ $$T^{2} + 74891716$$
$37$ $$T^{2} + 120692196$$
$41$ $$T^{2} + 1102500$$
$43$ $$(T - 5900)^{2}$$
$47$ $$T^{2} + 35545444$$
$53$ $$(T - 29046)^{2}$$
$59$ $$T^{2} + 193822084$$
$61$ $$(T + 32882)^{2}$$
$67$ $$T^{2} + 4839428356$$
$71$ $$T^{2} + 2554493764$$
$73$ $$T^{2} + 2185562500$$
$79$ $$(T + 19348)^{2}$$
$83$ $$T^{2} + 7645403844$$
$89$ $$T^{2} + 8867988900$$
$97$ $$T^{2} + 33410721796$$