# Properties

 Label 338.4.e.c Level $338$ Weight $4$ Character orbit 338.e Analytic conductor $19.943$ Analytic rank $0$ Dimension $4$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [338,4,Mod(23,338)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(338, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([5]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("338.23");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 338.e (of order $$6$$, 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: $$19.9426455819$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - 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: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{12} q^{2} + ( - \zeta_{12}^{2} + 1) q^{3} + 4 \zeta_{12}^{2} q^{4} - 17 \zeta_{12}^{3} q^{5} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{6} + (35 \zeta_{12}^{3} - 35 \zeta_{12}) q^{7} + 8 \zeta_{12}^{3} q^{8} + 26 \zeta_{12}^{2} q^{9} +O(q^{10})$$ q + 2*z * q^2 + (-z^2 + 1) * q^3 + 4*z^2 * q^4 - 17*z^3 * q^5 + (-2*z^3 + 2*z) * q^6 + (35*z^3 - 35*z) * q^7 + 8*z^3 * q^8 + 26*z^2 * q^9 $$q + 2 \zeta_{12} q^{2} + ( - \zeta_{12}^{2} + 1) q^{3} + 4 \zeta_{12}^{2} q^{4} - 17 \zeta_{12}^{3} q^{5} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{6} + (35 \zeta_{12}^{3} - 35 \zeta_{12}) q^{7} + 8 \zeta_{12}^{3} q^{8} + 26 \zeta_{12}^{2} q^{9} + ( - 34 \zeta_{12}^{2} + 34) q^{10} - 2 \zeta_{12} q^{11} + 4 q^{12} - 70 q^{14} - 17 \zeta_{12} q^{15} + (16 \zeta_{12}^{2} - 16) q^{16} - 19 \zeta_{12}^{2} q^{17} + 52 \zeta_{12}^{3} q^{18} + (94 \zeta_{12}^{3} - 94 \zeta_{12}) q^{19} + ( - 68 \zeta_{12}^{3} + 68 \zeta_{12}) q^{20} + 35 \zeta_{12}^{3} q^{21} - 4 \zeta_{12}^{2} q^{22} + (72 \zeta_{12}^{2} - 72) q^{23} + 8 \zeta_{12} q^{24} - 164 q^{25} + 53 q^{27} - 140 \zeta_{12} q^{28} + (246 \zeta_{12}^{2} - 246) q^{29} - 34 \zeta_{12}^{2} q^{30} + 100 \zeta_{12}^{3} q^{31} + (32 \zeta_{12}^{3} - 32 \zeta_{12}) q^{32} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{33} - 38 \zeta_{12}^{3} q^{34} + 595 \zeta_{12}^{2} q^{35} + (104 \zeta_{12}^{2} - 104) q^{36} + 11 \zeta_{12} q^{37} - 188 q^{38} + 136 q^{40} - 280 \zeta_{12} q^{41} + (70 \zeta_{12}^{2} - 70) q^{42} + 241 \zeta_{12}^{2} q^{43} - 8 \zeta_{12}^{3} q^{44} + ( - 442 \zeta_{12}^{3} + 442 \zeta_{12}) q^{45} + (144 \zeta_{12}^{3} - 144 \zeta_{12}) q^{46} + 137 \zeta_{12}^{3} q^{47} + 16 \zeta_{12}^{2} q^{48} + ( - 882 \zeta_{12}^{2} + 882) q^{49} - 328 \zeta_{12} q^{50} - 19 q^{51} - 232 q^{53} + 106 \zeta_{12} q^{54} + (34 \zeta_{12}^{2} - 34) q^{55} - 280 \zeta_{12}^{2} q^{56} + 94 \zeta_{12}^{3} q^{57} + (492 \zeta_{12}^{3} - 492 \zeta_{12}) q^{58} + (386 \zeta_{12}^{3} - 386 \zeta_{12}) q^{59} - 68 \zeta_{12}^{3} q^{60} - 64 \zeta_{12}^{2} q^{61} + (200 \zeta_{12}^{2} - 200) q^{62} - 910 \zeta_{12} q^{63} - 64 q^{64} - 4 q^{66} - 670 \zeta_{12} q^{67} + ( - 76 \zeta_{12}^{2} + 76) q^{68} + 72 \zeta_{12}^{2} q^{69} + 1190 \zeta_{12}^{3} q^{70} + (55 \zeta_{12}^{3} - 55 \zeta_{12}) q^{71} + (208 \zeta_{12}^{3} - 208 \zeta_{12}) q^{72} - 838 \zeta_{12}^{3} q^{73} + 22 \zeta_{12}^{2} q^{74} + (164 \zeta_{12}^{2} - 164) q^{75} - 376 \zeta_{12} q^{76} + 70 q^{77} + 1016 q^{79} + 272 \zeta_{12} q^{80} + (649 \zeta_{12}^{2} - 649) q^{81} - 560 \zeta_{12}^{2} q^{82} - 420 \zeta_{12}^{3} q^{83} + (140 \zeta_{12}^{3} - 140 \zeta_{12}) q^{84} + (323 \zeta_{12}^{3} - 323 \zeta_{12}) q^{85} + 482 \zeta_{12}^{3} q^{86} + 246 \zeta_{12}^{2} q^{87} + ( - 16 \zeta_{12}^{2} + 16) q^{88} + 934 \zeta_{12} q^{89} + 884 q^{90} - 288 q^{92} + 100 \zeta_{12} q^{93} + (274 \zeta_{12}^{2} - 274) q^{94} + 1598 \zeta_{12}^{2} q^{95} + 32 \zeta_{12}^{3} q^{96} + ( - 1154 \zeta_{12}^{3} + 1154 \zeta_{12}) q^{97} + ( - 1764 \zeta_{12}^{3} + 1764 \zeta_{12}) q^{98} - 52 \zeta_{12}^{3} q^{99} +O(q^{100})$$ q + 2*z * q^2 + (-z^2 + 1) * q^3 + 4*z^2 * q^4 - 17*z^3 * q^5 + (-2*z^3 + 2*z) * q^6 + (35*z^3 - 35*z) * q^7 + 8*z^3 * q^8 + 26*z^2 * q^9 + (-34*z^2 + 34) * q^10 - 2*z * q^11 + 4 * q^12 - 70 * q^14 - 17*z * q^15 + (16*z^2 - 16) * q^16 - 19*z^2 * q^17 + 52*z^3 * q^18 + (94*z^3 - 94*z) * q^19 + (-68*z^3 + 68*z) * q^20 + 35*z^3 * q^21 - 4*z^2 * q^22 + (72*z^2 - 72) * q^23 + 8*z * q^24 - 164 * q^25 + 53 * q^27 - 140*z * q^28 + (246*z^2 - 246) * q^29 - 34*z^2 * q^30 + 100*z^3 * q^31 + (32*z^3 - 32*z) * q^32 + (2*z^3 - 2*z) * q^33 - 38*z^3 * q^34 + 595*z^2 * q^35 + (104*z^2 - 104) * q^36 + 11*z * q^37 - 188 * q^38 + 136 * q^40 - 280*z * q^41 + (70*z^2 - 70) * q^42 + 241*z^2 * q^43 - 8*z^3 * q^44 + (-442*z^3 + 442*z) * q^45 + (144*z^3 - 144*z) * q^46 + 137*z^3 * q^47 + 16*z^2 * q^48 + (-882*z^2 + 882) * q^49 - 328*z * q^50 - 19 * q^51 - 232 * q^53 + 106*z * q^54 + (34*z^2 - 34) * q^55 - 280*z^2 * q^56 + 94*z^3 * q^57 + (492*z^3 - 492*z) * q^58 + (386*z^3 - 386*z) * q^59 - 68*z^3 * q^60 - 64*z^2 * q^61 + (200*z^2 - 200) * q^62 - 910*z * q^63 - 64 * q^64 - 4 * q^66 - 670*z * q^67 + (-76*z^2 + 76) * q^68 + 72*z^2 * q^69 + 1190*z^3 * q^70 + (55*z^3 - 55*z) * q^71 + (208*z^3 - 208*z) * q^72 - 838*z^3 * q^73 + 22*z^2 * q^74 + (164*z^2 - 164) * q^75 - 376*z * q^76 + 70 * q^77 + 1016 * q^79 + 272*z * q^80 + (649*z^2 - 649) * q^81 - 560*z^2 * q^82 - 420*z^3 * q^83 + (140*z^3 - 140*z) * q^84 + (323*z^3 - 323*z) * q^85 + 482*z^3 * q^86 + 246*z^2 * q^87 + (-16*z^2 + 16) * q^88 + 934*z * q^89 + 884 * q^90 - 288 * q^92 + 100*z * q^93 + (274*z^2 - 274) * q^94 + 1598*z^2 * q^95 + 32*z^3 * q^96 + (-1154*z^3 + 1154*z) * q^97 + (-1764*z^3 + 1764*z) * q^98 - 52*z^3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{3} + 8 q^{4} + 52 q^{9}+O(q^{10})$$ 4 * q + 2 * q^3 + 8 * q^4 + 52 * q^9 $$4 q + 2 q^{3} + 8 q^{4} + 52 q^{9} + 68 q^{10} + 16 q^{12} - 280 q^{14} - 32 q^{16} - 38 q^{17} - 8 q^{22} - 144 q^{23} - 656 q^{25} + 212 q^{27} - 492 q^{29} - 68 q^{30} + 1190 q^{35} - 208 q^{36} - 752 q^{38} + 544 q^{40} - 140 q^{42} + 482 q^{43} + 32 q^{48} + 1764 q^{49} - 76 q^{51} - 928 q^{53} - 68 q^{55} - 560 q^{56} - 128 q^{61} - 400 q^{62} - 256 q^{64} - 16 q^{66} + 152 q^{68} + 144 q^{69} + 44 q^{74} - 328 q^{75} + 280 q^{77} + 4064 q^{79} - 1298 q^{81} - 1120 q^{82} + 492 q^{87} + 32 q^{88} + 3536 q^{90} - 1152 q^{92} - 548 q^{94} + 3196 q^{95}+O(q^{100})$$ 4 * q + 2 * q^3 + 8 * q^4 + 52 * q^9 + 68 * q^10 + 16 * q^12 - 280 * q^14 - 32 * q^16 - 38 * q^17 - 8 * q^22 - 144 * q^23 - 656 * q^25 + 212 * q^27 - 492 * q^29 - 68 * q^30 + 1190 * q^35 - 208 * q^36 - 752 * q^38 + 544 * q^40 - 140 * q^42 + 482 * q^43 + 32 * q^48 + 1764 * q^49 - 76 * q^51 - 928 * q^53 - 68 * q^55 - 560 * q^56 - 128 * q^61 - 400 * q^62 - 256 * q^64 - 16 * q^66 + 152 * q^68 + 144 * q^69 + 44 * q^74 - 328 * q^75 + 280 * q^77 + 4064 * q^79 - 1298 * q^81 - 1120 * q^82 + 492 * q^87 + 32 * q^88 + 3536 * q^90 - 1152 * q^92 - 548 * q^94 + 3196 * q^95

## Character values

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

 $$n$$ $$171$$ $$\chi(n)$$ $$\zeta_{12}^{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
23.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
−1.73205 + 1.00000i 0.500000 + 0.866025i 2.00000 3.46410i 17.0000i −1.73205 1.00000i 30.3109 + 17.5000i 8.00000i 13.0000 22.5167i 17.0000 + 29.4449i
23.2 1.73205 1.00000i 0.500000 + 0.866025i 2.00000 3.46410i 17.0000i 1.73205 + 1.00000i −30.3109 17.5000i 8.00000i 13.0000 22.5167i 17.0000 + 29.4449i
147.1 −1.73205 1.00000i 0.500000 0.866025i 2.00000 + 3.46410i 17.0000i −1.73205 + 1.00000i 30.3109 17.5000i 8.00000i 13.0000 + 22.5167i 17.0000 29.4449i
147.2 1.73205 + 1.00000i 0.500000 0.866025i 2.00000 + 3.46410i 17.0000i 1.73205 1.00000i −30.3109 + 17.5000i 8.00000i 13.0000 + 22.5167i 17.0000 29.4449i
 $$n$$: 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
13.c even 3 1 inner
13.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 338.4.e.c 4
13.b even 2 1 inner 338.4.e.c 4
13.c even 3 1 338.4.b.b 2
13.c even 3 1 inner 338.4.e.c 4
13.d odd 4 1 338.4.c.c 2
13.d odd 4 1 338.4.c.g 2
13.e even 6 1 338.4.b.b 2
13.e even 6 1 inner 338.4.e.c 4
13.f odd 12 1 26.4.a.b 1
13.f odd 12 1 338.4.a.b 1
13.f odd 12 1 338.4.c.c 2
13.f odd 12 1 338.4.c.g 2
39.k even 12 1 234.4.a.a 1
52.l even 12 1 208.4.a.e 1
65.o even 12 1 650.4.b.d 2
65.s odd 12 1 650.4.a.c 1
65.t even 12 1 650.4.b.d 2
91.bc even 12 1 1274.4.a.f 1
104.u even 12 1 832.4.a.g 1
104.x odd 12 1 832.4.a.j 1
156.v odd 12 1 1872.4.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.4.a.b 1 13.f odd 12 1
208.4.a.e 1 52.l even 12 1
234.4.a.a 1 39.k even 12 1
338.4.a.b 1 13.f odd 12 1
338.4.b.b 2 13.c even 3 1
338.4.b.b 2 13.e even 6 1
338.4.c.c 2 13.d odd 4 1
338.4.c.c 2 13.f odd 12 1
338.4.c.g 2 13.d odd 4 1
338.4.c.g 2 13.f odd 12 1
338.4.e.c 4 1.a even 1 1 trivial
338.4.e.c 4 13.b even 2 1 inner
338.4.e.c 4 13.c even 3 1 inner
338.4.e.c 4 13.e even 6 1 inner
650.4.a.c 1 65.s odd 12 1
650.4.b.d 2 65.o even 12 1
650.4.b.d 2 65.t even 12 1
832.4.a.g 1 104.u even 12 1
832.4.a.j 1 104.x odd 12 1
1274.4.a.f 1 91.bc even 12 1
1872.4.a.b 1 156.v odd 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(338, [\chi])$$:

 $$T_{3}^{2} - T_{3} + 1$$ T3^2 - T3 + 1 $$T_{5}^{2} + 289$$ T5^2 + 289

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 4T^{2} + 16$$
$3$ $$(T^{2} - T + 1)^{2}$$
$5$ $$(T^{2} + 289)^{2}$$
$7$ $$T^{4} - 1225 T^{2} + 1500625$$
$11$ $$T^{4} - 4T^{2} + 16$$
$13$ $$T^{4}$$
$17$ $$(T^{2} + 19 T + 361)^{2}$$
$19$ $$T^{4} - 8836 T^{2} + 78074896$$
$23$ $$(T^{2} + 72 T + 5184)^{2}$$
$29$ $$(T^{2} + 246 T + 60516)^{2}$$
$31$ $$(T^{2} + 10000)^{2}$$
$37$ $$T^{4} - 121 T^{2} + 14641$$
$41$ $$T^{4} + \cdots + 6146560000$$
$43$ $$(T^{2} - 241 T + 58081)^{2}$$
$47$ $$(T^{2} + 18769)^{2}$$
$53$ $$(T + 232)^{4}$$
$59$ $$T^{4} + \cdots + 22199808016$$
$61$ $$(T^{2} + 64 T + 4096)^{2}$$
$67$ $$T^{4} + \cdots + 201511210000$$
$71$ $$T^{4} - 3025 T^{2} + 9150625$$
$73$ $$(T^{2} + 702244)^{2}$$
$79$ $$(T - 1016)^{4}$$
$83$ $$(T^{2} + 176400)^{2}$$
$89$ $$T^{4} + \cdots + 761004990736$$
$97$ $$T^{4} + \cdots + 1773467504656$$