# Properties

 Label 338.8.b.b Level $338$ Weight $8$ Character orbit 338.b Analytic conductor $105.586$ 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] = [338,8,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: $$0$$ 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: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q - 8 i q^{2} - 39 q^{3} - 64 q^{4} + 385 i q^{5} + 312 i q^{6} + 293 i q^{7} + 512 i q^{8} - 666 q^{9} +O(q^{10})$$ q - 8*i * q^2 - 39 * q^3 - 64 * q^4 + 385*i * q^5 + 312*i * q^6 + 293*i * q^7 + 512*i * q^8 - 666 * q^9 $$q - 8 i q^{2} - 39 q^{3} - 64 q^{4} + 385 i q^{5} + 312 i q^{6} + 293 i q^{7} + 512 i q^{8} - 666 q^{9} + 3080 q^{10} + 5402 i q^{11} + 2496 q^{12} + 2344 q^{14} - 15015 i q^{15} + 4096 q^{16} + 21011 q^{17} + 5328 i q^{18} - 27326 i q^{19} - 24640 i q^{20} - 11427 i q^{21} + 43216 q^{22} + 63072 q^{23} - 19968 i q^{24} - 70100 q^{25} + 111267 q^{27} - 18752 i q^{28} + 122238 q^{29} - 120120 q^{30} - 208396 i q^{31} - 32768 i q^{32} - 210678 i q^{33} - 168088 i q^{34} - 112805 q^{35} + 42624 q^{36} + 442379 i q^{37} - 218608 q^{38} - 197120 q^{40} + 58000 i q^{41} - 91416 q^{42} + 202025 q^{43} - 345728 i q^{44} - 256410 i q^{45} - 504576 i q^{46} - 588511 i q^{47} - 159744 q^{48} + 737694 q^{49} + 560800 i q^{50} - 819429 q^{51} + 1684336 q^{53} - 890136 i q^{54} - 2079770 q^{55} - 150016 q^{56} + 1065714 i q^{57} - 977904 i q^{58} + 442630 i q^{59} + 960960 i q^{60} - 1083608 q^{61} - 1667168 q^{62} - 195138 i q^{63} - 262144 q^{64} - 1685424 q^{66} + 3443486 i q^{67} - 1344704 q^{68} - 2459808 q^{69} + 902440 i q^{70} + 2084705 i q^{71} - 340992 i q^{72} - 5937890 i q^{73} + 3539032 q^{74} + 2733900 q^{75} + 1748864 i q^{76} - 1582786 q^{77} - 6609256 q^{79} + 1576960 i q^{80} - 2882871 q^{81} + 464000 q^{82} - 142740 i q^{83} + 731328 i q^{84} + 8089235 i q^{85} - 1616200 i q^{86} - 4767282 q^{87} - 2765824 q^{88} + 6985286 i q^{89} - 2051280 q^{90} - 4036608 q^{92} + 8127444 i q^{93} - 4708088 q^{94} + 10520510 q^{95} + 1277952 i q^{96} - 200762 i q^{97} - 5901552 i q^{98} - 3597732 i q^{99} +O(q^{100})$$ q - 8*i * q^2 - 39 * q^3 - 64 * q^4 + 385*i * q^5 + 312*i * q^6 + 293*i * q^7 + 512*i * q^8 - 666 * q^9 + 3080 * q^10 + 5402*i * q^11 + 2496 * q^12 + 2344 * q^14 - 15015*i * q^15 + 4096 * q^16 + 21011 * q^17 + 5328*i * q^18 - 27326*i * q^19 - 24640*i * q^20 - 11427*i * q^21 + 43216 * q^22 + 63072 * q^23 - 19968*i * q^24 - 70100 * q^25 + 111267 * q^27 - 18752*i * q^28 + 122238 * q^29 - 120120 * q^30 - 208396*i * q^31 - 32768*i * q^32 - 210678*i * q^33 - 168088*i * q^34 - 112805 * q^35 + 42624 * q^36 + 442379*i * q^37 - 218608 * q^38 - 197120 * q^40 + 58000*i * q^41 - 91416 * q^42 + 202025 * q^43 - 345728*i * q^44 - 256410*i * q^45 - 504576*i * q^46 - 588511*i * q^47 - 159744 * q^48 + 737694 * q^49 + 560800*i * q^50 - 819429 * q^51 + 1684336 * q^53 - 890136*i * q^54 - 2079770 * q^55 - 150016 * q^56 + 1065714*i * q^57 - 977904*i * q^58 + 442630*i * q^59 + 960960*i * q^60 - 1083608 * q^61 - 1667168 * q^62 - 195138*i * q^63 - 262144 * q^64 - 1685424 * q^66 + 3443486*i * q^67 - 1344704 * q^68 - 2459808 * q^69 + 902440*i * q^70 + 2084705*i * q^71 - 340992*i * q^72 - 5937890*i * q^73 + 3539032 * q^74 + 2733900 * q^75 + 1748864*i * q^76 - 1582786 * q^77 - 6609256 * q^79 + 1576960*i * q^80 - 2882871 * q^81 + 464000 * q^82 - 142740*i * q^83 + 731328*i * q^84 + 8089235*i * q^85 - 1616200*i * q^86 - 4767282 * q^87 - 2765824 * q^88 + 6985286*i * q^89 - 2051280 * q^90 - 4036608 * q^92 + 8127444*i * q^93 - 4708088 * q^94 + 10520510 * q^95 + 1277952*i * q^96 - 200762*i * q^97 - 5901552*i * q^98 - 3597732*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 78 q^{3} - 128 q^{4} - 1332 q^{9}+O(q^{10})$$ 2 * q - 78 * q^3 - 128 * q^4 - 1332 * q^9 $$2 q - 78 q^{3} - 128 q^{4} - 1332 q^{9} + 6160 q^{10} + 4992 q^{12} + 4688 q^{14} + 8192 q^{16} + 42022 q^{17} + 86432 q^{22} + 126144 q^{23} - 140200 q^{25} + 222534 q^{27} + 244476 q^{29} - 240240 q^{30} - 225610 q^{35} + 85248 q^{36} - 437216 q^{38} - 394240 q^{40} - 182832 q^{42} + 404050 q^{43} - 319488 q^{48} + 1475388 q^{49} - 1638858 q^{51} + 3368672 q^{53} - 4159540 q^{55} - 300032 q^{56} - 2167216 q^{61} - 3334336 q^{62} - 524288 q^{64} - 3370848 q^{66} - 2689408 q^{68} - 4919616 q^{69} + 7078064 q^{74} + 5467800 q^{75} - 3165572 q^{77} - 13218512 q^{79} - 5765742 q^{81} + 928000 q^{82} - 9534564 q^{87} - 5531648 q^{88} - 4102560 q^{90} - 8073216 q^{92} - 9416176 q^{94} + 21041020 q^{95}+O(q^{100})$$ 2 * q - 78 * q^3 - 128 * q^4 - 1332 * q^9 + 6160 * q^10 + 4992 * q^12 + 4688 * q^14 + 8192 * q^16 + 42022 * q^17 + 86432 * q^22 + 126144 * q^23 - 140200 * q^25 + 222534 * q^27 + 244476 * q^29 - 240240 * q^30 - 225610 * q^35 + 85248 * q^36 - 437216 * q^38 - 394240 * q^40 - 182832 * q^42 + 404050 * q^43 - 319488 * q^48 + 1475388 * q^49 - 1638858 * q^51 + 3368672 * q^53 - 4159540 * q^55 - 300032 * q^56 - 2167216 * q^61 - 3334336 * q^62 - 524288 * q^64 - 3370848 * q^66 - 2689408 * q^68 - 4919616 * q^69 + 7078064 * q^74 + 5467800 * q^75 - 3165572 * q^77 - 13218512 * q^79 - 5765742 * q^81 + 928000 * q^82 - 9534564 * q^87 - 5531648 * q^88 - 4102560 * q^90 - 8073216 * q^92 - 9416176 * q^94 + 21041020 * 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 −39.0000 −64.0000 385.000i 312.000i 293.000i 512.000i −666.000 3080.00
337.2 8.00000i −39.0000 −64.0000 385.000i 312.000i 293.000i 512.000i −666.000 3080.00
 $$n$$: 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.b 2
13.b even 2 1 inner 338.8.b.b 2
13.d odd 4 1 26.8.a.a 1
13.d odd 4 1 338.8.a.c 1
39.f even 4 1 234.8.a.d 1
52.f even 4 1 208.8.a.c 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 39$$ acting on $$S_{8}^{\mathrm{new}}(338, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 64$$
$3$ $$(T + 39)^{2}$$
$5$ $$T^{2} + 148225$$
$7$ $$T^{2} + 85849$$
$11$ $$T^{2} + 29181604$$
$13$ $$T^{2}$$
$17$ $$(T - 21011)^{2}$$
$19$ $$T^{2} + 746710276$$
$23$ $$(T - 63072)^{2}$$
$29$ $$(T - 122238)^{2}$$
$31$ $$T^{2} + 43428892816$$
$37$ $$T^{2} + 195699179641$$
$41$ $$T^{2} + 3364000000$$
$43$ $$(T - 202025)^{2}$$
$47$ $$T^{2} + 346345197121$$
$53$ $$(T - 1684336)^{2}$$
$59$ $$T^{2} + 195921316900$$
$61$ $$(T + 1083608)^{2}$$
$67$ $$T^{2} + 11857595832196$$
$71$ $$T^{2} + 4345994937025$$
$73$ $$T^{2} + 35258537652100$$
$79$ $$(T + 6609256)^{2}$$
$83$ $$T^{2} + 20374707600$$
$89$ $$T^{2} + 48794220501796$$
$97$ $$T^{2} + 40305380644$$