# Properties

 Label 338.8.b.a Level $338$ Weight $8$ Character orbit 338.b Analytic conductor $105.586$ Analytic rank $1$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [338,8,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, 8, names="a")

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

chi := DirichletCharacter("338.337");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$8$$ 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: $$105.586138614$$ Analytic rank: $$1$$ 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: $$1$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 8 i q^{2} - 87 q^{3} - 64 q^{4} - 321 i q^{5} + 696 i q^{6} - 181 i q^{7} + 512 i q^{8} + 5382 q^{9} +O(q^{10})$$ q - 8*i * q^2 - 87 * q^3 - 64 * q^4 - 321*i * q^5 + 696*i * q^6 - 181*i * q^7 + 512*i * q^8 + 5382 * q^9 $$q - 8 i q^{2} - 87 q^{3} - 64 q^{4} - 321 i q^{5} + 696 i q^{6} - 181 i q^{7} + 512 i q^{8} + 5382 q^{9} - 2568 q^{10} + 7782 i q^{11} + 5568 q^{12} - 1448 q^{14} + 27927 i q^{15} + 4096 q^{16} - 9069 q^{17} - 43056 i q^{18} + 37150 i q^{19} + 20544 i q^{20} + 15747 i q^{21} + 62256 q^{22} - 19008 q^{23} - 44544 i q^{24} - 24916 q^{25} - 277965 q^{27} + 11584 i q^{28} + 174750 q^{29} + 223416 q^{30} - 29012 i q^{31} - 32768 i q^{32} - 677034 i q^{33} + 72552 i q^{34} - 58101 q^{35} - 344448 q^{36} + 323669 i q^{37} + 297200 q^{38} + 164352 q^{40} - 795312 i q^{41} + 125976 q^{42} + 314137 q^{43} - 498048 i q^{44} - 1727622 i q^{45} + 152064 i q^{46} - 447441 i q^{47} - 356352 q^{48} + 790782 q^{49} + 199328 i q^{50} + 789003 q^{51} - 1469232 q^{53} + 2223720 i q^{54} + 2498022 q^{55} + 92672 q^{56} - 3232050 i q^{57} - 1398000 i q^{58} + 1627770 i q^{59} - 1787328 i q^{60} - 2399608 q^{61} - 232096 q^{62} - 974142 i q^{63} - 262144 q^{64} - 5416272 q^{66} + 64066 i q^{67} + 580416 q^{68} + 1653696 q^{69} + 464808 i q^{70} + 322383 i q^{71} + 2755584 i q^{72} - 4454782 i q^{73} + 2589352 q^{74} + 2167692 q^{75} - 2377600 i q^{76} + 1408542 q^{77} + 753560 q^{79} - 1314816 i q^{80} + 12412521 q^{81} - 6362496 q^{82} + 1219092 i q^{83} - 1007808 i q^{84} + 2911149 i q^{85} - 2513096 i q^{86} - 15203250 q^{87} - 3984384 q^{88} + 3390330 i q^{89} - 13820976 q^{90} + 1216512 q^{92} + 2524044 i q^{93} - 3579528 q^{94} + 11925150 q^{95} + 2850816 i q^{96} - 1628774 i q^{97} - 6326256 i q^{98} + 41882724 i q^{99} +O(q^{100})$$ q - 8*i * q^2 - 87 * q^3 - 64 * q^4 - 321*i * q^5 + 696*i * q^6 - 181*i * q^7 + 512*i * q^8 + 5382 * q^9 - 2568 * q^10 + 7782*i * q^11 + 5568 * q^12 - 1448 * q^14 + 27927*i * q^15 + 4096 * q^16 - 9069 * q^17 - 43056*i * q^18 + 37150*i * q^19 + 20544*i * q^20 + 15747*i * q^21 + 62256 * q^22 - 19008 * q^23 - 44544*i * q^24 - 24916 * q^25 - 277965 * q^27 + 11584*i * q^28 + 174750 * q^29 + 223416 * q^30 - 29012*i * q^31 - 32768*i * q^32 - 677034*i * q^33 + 72552*i * q^34 - 58101 * q^35 - 344448 * q^36 + 323669*i * q^37 + 297200 * q^38 + 164352 * q^40 - 795312*i * q^41 + 125976 * q^42 + 314137 * q^43 - 498048*i * q^44 - 1727622*i * q^45 + 152064*i * q^46 - 447441*i * q^47 - 356352 * q^48 + 790782 * q^49 + 199328*i * q^50 + 789003 * q^51 - 1469232 * q^53 + 2223720*i * q^54 + 2498022 * q^55 + 92672 * q^56 - 3232050*i * q^57 - 1398000*i * q^58 + 1627770*i * q^59 - 1787328*i * q^60 - 2399608 * q^61 - 232096 * q^62 - 974142*i * q^63 - 262144 * q^64 - 5416272 * q^66 + 64066*i * q^67 + 580416 * q^68 + 1653696 * q^69 + 464808*i * q^70 + 322383*i * q^71 + 2755584*i * q^72 - 4454782*i * q^73 + 2589352 * q^74 + 2167692 * q^75 - 2377600*i * q^76 + 1408542 * q^77 + 753560 * q^79 - 1314816*i * q^80 + 12412521 * q^81 - 6362496 * q^82 + 1219092*i * q^83 - 1007808*i * q^84 + 2911149*i * q^85 - 2513096*i * q^86 - 15203250 * q^87 - 3984384 * q^88 + 3390330*i * q^89 - 13820976 * q^90 + 1216512 * q^92 + 2524044*i * q^93 - 3579528 * q^94 + 11925150 * q^95 + 2850816*i * q^96 - 1628774*i * q^97 - 6326256*i * q^98 + 41882724*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 174 q^{3} - 128 q^{4} + 10764 q^{9}+O(q^{10})$$ 2 * q - 174 * q^3 - 128 * q^4 + 10764 * q^9 $$2 q - 174 q^{3} - 128 q^{4} + 10764 q^{9} - 5136 q^{10} + 11136 q^{12} - 2896 q^{14} + 8192 q^{16} - 18138 q^{17} + 124512 q^{22} - 38016 q^{23} - 49832 q^{25} - 555930 q^{27} + 349500 q^{29} + 446832 q^{30} - 116202 q^{35} - 688896 q^{36} + 594400 q^{38} + 328704 q^{40} + 251952 q^{42} + 628274 q^{43} - 712704 q^{48} + 1581564 q^{49} + 1578006 q^{51} - 2938464 q^{53} + 4996044 q^{55} + 185344 q^{56} - 4799216 q^{61} - 464192 q^{62} - 524288 q^{64} - 10832544 q^{66} + 1160832 q^{68} + 3307392 q^{69} + 5178704 q^{74} + 4335384 q^{75} + 2817084 q^{77} + 1507120 q^{79} + 24825042 q^{81} - 12724992 q^{82} - 30406500 q^{87} - 7968768 q^{88} - 27641952 q^{90} + 2433024 q^{92} - 7159056 q^{94} + 23850300 q^{95}+O(q^{100})$$ 2 * q - 174 * q^3 - 128 * q^4 + 10764 * q^9 - 5136 * q^10 + 11136 * q^12 - 2896 * q^14 + 8192 * q^16 - 18138 * q^17 + 124512 * q^22 - 38016 * q^23 - 49832 * q^25 - 555930 * q^27 + 349500 * q^29 + 446832 * q^30 - 116202 * q^35 - 688896 * q^36 + 594400 * q^38 + 328704 * q^40 + 251952 * q^42 + 628274 * q^43 - 712704 * q^48 + 1581564 * q^49 + 1578006 * q^51 - 2938464 * q^53 + 4996044 * q^55 + 185344 * q^56 - 4799216 * q^61 - 464192 * q^62 - 524288 * q^64 - 10832544 * q^66 + 1160832 * q^68 + 3307392 * q^69 + 5178704 * q^74 + 4335384 * q^75 + 2817084 * q^77 + 1507120 * q^79 + 24825042 * q^81 - 12724992 * q^82 - 30406500 * q^87 - 7968768 * q^88 - 27641952 * q^90 + 2433024 * q^92 - 7159056 * q^94 + 23850300 * 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
8.00000i −87.0000 −64.0000 321.000i 696.000i 181.000i 512.000i 5382.00 −2568.00
337.2 8.00000i −87.0000 −64.0000 321.000i 696.000i 181.000i 512.000i 5382.00 −2568.00
 $$n$$: 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 338.8.b.a 2
13.b even 2 1 inner 338.8.b.a 2
13.d odd 4 1 26.8.a.b 1
13.d odd 4 1 338.8.a.a 1
39.f even 4 1 234.8.a.a 1
52.f even 4 1 208.8.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.8.a.b 1 13.d odd 4 1
208.8.a.e 1 52.f even 4 1
234.8.a.a 1 39.f even 4 1
338.8.a.a 1 13.d odd 4 1
338.8.b.a 2 1.a even 1 1 trivial
338.8.b.a 2 13.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 87$$ acting on $$S_{8}^{\mathrm{new}}(338, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 64$$
$3$ $$(T + 87)^{2}$$
$5$ $$T^{2} + 103041$$
$7$ $$T^{2} + 32761$$
$11$ $$T^{2} + 60559524$$
$13$ $$T^{2}$$
$17$ $$(T + 9069)^{2}$$
$19$ $$T^{2} + 1380122500$$
$23$ $$(T + 19008)^{2}$$
$29$ $$(T - 174750)^{2}$$
$31$ $$T^{2} + 841696144$$
$37$ $$T^{2} + 104761621561$$
$41$ $$T^{2} + 632521177344$$
$43$ $$(T - 314137)^{2}$$
$47$ $$T^{2} + 200203448481$$
$53$ $$(T + 1469232)^{2}$$
$59$ $$T^{2} + 2649635172900$$
$61$ $$(T + 2399608)^{2}$$
$67$ $$T^{2} + 4104452356$$
$71$ $$T^{2} + 103930798689$$
$73$ $$T^{2} + 19845082667524$$
$79$ $$(T - 753560)^{2}$$
$83$ $$T^{2} + 1486185304464$$
$89$ $$T^{2} + 11494337508900$$
$97$ $$T^{2} + 2652904743076$$