# Properties

 Label 338.4.c.g Level $338$ Weight $4$ Character orbit 338.c Analytic conductor $19.943$ 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] = [338,4,Mod(191,338)]

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([4]))

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

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

chi := DirichletCharacter("338.191");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 338.c (of order $$3$$, 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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 26) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - 2 \zeta_{6} + 2) q^{2} + ( - \zeta_{6} + 1) q^{3} - 4 \zeta_{6} q^{4} - 17 q^{5} - 2 \zeta_{6} q^{6} - 35 \zeta_{6} q^{7} - 8 q^{8} + 26 \zeta_{6} q^{9} +O(q^{10})$$ q + (-2*z + 2) * q^2 + (-z + 1) * q^3 - 4*z * q^4 - 17 * q^5 - 2*z * q^6 - 35*z * q^7 - 8 * q^8 + 26*z * q^9 $$q + ( - 2 \zeta_{6} + 2) q^{2} + ( - \zeta_{6} + 1) q^{3} - 4 \zeta_{6} q^{4} - 17 q^{5} - 2 \zeta_{6} q^{6} - 35 \zeta_{6} q^{7} - 8 q^{8} + 26 \zeta_{6} q^{9} + (34 \zeta_{6} - 34) q^{10} + ( - 2 \zeta_{6} + 2) q^{11} - 4 q^{12} - 70 q^{14} + (17 \zeta_{6} - 17) q^{15} + (16 \zeta_{6} - 16) q^{16} + 19 \zeta_{6} q^{17} + 52 q^{18} + 94 \zeta_{6} q^{19} + 68 \zeta_{6} q^{20} - 35 q^{21} - 4 \zeta_{6} q^{22} + ( - 72 \zeta_{6} + 72) q^{23} + (8 \zeta_{6} - 8) q^{24} + 164 q^{25} + 53 q^{27} + (140 \zeta_{6} - 140) q^{28} + (246 \zeta_{6} - 246) q^{29} + 34 \zeta_{6} q^{30} + 100 q^{31} + 32 \zeta_{6} q^{32} - 2 \zeta_{6} q^{33} + 38 q^{34} + 595 \zeta_{6} q^{35} + ( - 104 \zeta_{6} + 104) q^{36} + (11 \zeta_{6} - 11) q^{37} + 188 q^{38} + 136 q^{40} + (280 \zeta_{6} - 280) q^{41} + (70 \zeta_{6} - 70) q^{42} - 241 \zeta_{6} q^{43} - 8 q^{44} - 442 \zeta_{6} q^{45} - 144 \zeta_{6} q^{46} - 137 q^{47} + 16 \zeta_{6} q^{48} + (882 \zeta_{6} - 882) q^{49} + ( - 328 \zeta_{6} + 328) q^{50} + 19 q^{51} - 232 q^{53} + ( - 106 \zeta_{6} + 106) q^{54} + (34 \zeta_{6} - 34) q^{55} + 280 \zeta_{6} q^{56} + 94 q^{57} + 492 \zeta_{6} q^{58} - 386 \zeta_{6} q^{59} + 68 q^{60} - 64 \zeta_{6} q^{61} + ( - 200 \zeta_{6} + 200) q^{62} + ( - 910 \zeta_{6} + 910) q^{63} + 64 q^{64} - 4 q^{66} + (670 \zeta_{6} - 670) q^{67} + ( - 76 \zeta_{6} + 76) q^{68} - 72 \zeta_{6} q^{69} + 1190 q^{70} + 55 \zeta_{6} q^{71} - 208 \zeta_{6} q^{72} + 838 q^{73} + 22 \zeta_{6} q^{74} + ( - 164 \zeta_{6} + 164) q^{75} + ( - 376 \zeta_{6} + 376) q^{76} - 70 q^{77} + 1016 q^{79} + ( - 272 \zeta_{6} + 272) q^{80} + (649 \zeta_{6} - 649) q^{81} + 560 \zeta_{6} q^{82} - 420 q^{83} + 140 \zeta_{6} q^{84} - 323 \zeta_{6} q^{85} - 482 q^{86} + 246 \zeta_{6} q^{87} + (16 \zeta_{6} - 16) q^{88} + (934 \zeta_{6} - 934) q^{89} - 884 q^{90} - 288 q^{92} + ( - 100 \zeta_{6} + 100) q^{93} + (274 \zeta_{6} - 274) q^{94} - 1598 \zeta_{6} q^{95} + 32 q^{96} - 1154 \zeta_{6} q^{97} + 1764 \zeta_{6} q^{98} + 52 q^{99} +O(q^{100})$$ q + (-2*z + 2) * q^2 + (-z + 1) * q^3 - 4*z * q^4 - 17 * q^5 - 2*z * q^6 - 35*z * q^7 - 8 * q^8 + 26*z * q^9 + (34*z - 34) * q^10 + (-2*z + 2) * q^11 - 4 * q^12 - 70 * q^14 + (17*z - 17) * q^15 + (16*z - 16) * q^16 + 19*z * q^17 + 52 * q^18 + 94*z * q^19 + 68*z * q^20 - 35 * q^21 - 4*z * q^22 + (-72*z + 72) * q^23 + (8*z - 8) * q^24 + 164 * q^25 + 53 * q^27 + (140*z - 140) * q^28 + (246*z - 246) * q^29 + 34*z * q^30 + 100 * q^31 + 32*z * q^32 - 2*z * q^33 + 38 * q^34 + 595*z * q^35 + (-104*z + 104) * q^36 + (11*z - 11) * q^37 + 188 * q^38 + 136 * q^40 + (280*z - 280) * q^41 + (70*z - 70) * q^42 - 241*z * q^43 - 8 * q^44 - 442*z * q^45 - 144*z * q^46 - 137 * q^47 + 16*z * q^48 + (882*z - 882) * q^49 + (-328*z + 328) * q^50 + 19 * q^51 - 232 * q^53 + (-106*z + 106) * q^54 + (34*z - 34) * q^55 + 280*z * q^56 + 94 * q^57 + 492*z * q^58 - 386*z * q^59 + 68 * q^60 - 64*z * q^61 + (-200*z + 200) * q^62 + (-910*z + 910) * q^63 + 64 * q^64 - 4 * q^66 + (670*z - 670) * q^67 + (-76*z + 76) * q^68 - 72*z * q^69 + 1190 * q^70 + 55*z * q^71 - 208*z * q^72 + 838 * q^73 + 22*z * q^74 + (-164*z + 164) * q^75 + (-376*z + 376) * q^76 - 70 * q^77 + 1016 * q^79 + (-272*z + 272) * q^80 + (649*z - 649) * q^81 + 560*z * q^82 - 420 * q^83 + 140*z * q^84 - 323*z * q^85 - 482 * q^86 + 246*z * q^87 + (16*z - 16) * q^88 + (934*z - 934) * q^89 - 884 * q^90 - 288 * q^92 + (-100*z + 100) * q^93 + (274*z - 274) * q^94 - 1598*z * q^95 + 32 * q^96 - 1154*z * q^97 + 1764*z * q^98 + 52 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + q^{3} - 4 q^{4} - 34 q^{5} - 2 q^{6} - 35 q^{7} - 16 q^{8} + 26 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + q^3 - 4 * q^4 - 34 * q^5 - 2 * q^6 - 35 * q^7 - 16 * q^8 + 26 * q^9 $$2 q + 2 q^{2} + q^{3} - 4 q^{4} - 34 q^{5} - 2 q^{6} - 35 q^{7} - 16 q^{8} + 26 q^{9} - 34 q^{10} + 2 q^{11} - 8 q^{12} - 140 q^{14} - 17 q^{15} - 16 q^{16} + 19 q^{17} + 104 q^{18} + 94 q^{19} + 68 q^{20} - 70 q^{21} - 4 q^{22} + 72 q^{23} - 8 q^{24} + 328 q^{25} + 106 q^{27} - 140 q^{28} - 246 q^{29} + 34 q^{30} + 200 q^{31} + 32 q^{32} - 2 q^{33} + 76 q^{34} + 595 q^{35} + 104 q^{36} - 11 q^{37} + 376 q^{38} + 272 q^{40} - 280 q^{41} - 70 q^{42} - 241 q^{43} - 16 q^{44} - 442 q^{45} - 144 q^{46} - 274 q^{47} + 16 q^{48} - 882 q^{49} + 328 q^{50} + 38 q^{51} - 464 q^{53} + 106 q^{54} - 34 q^{55} + 280 q^{56} + 188 q^{57} + 492 q^{58} - 386 q^{59} + 136 q^{60} - 64 q^{61} + 200 q^{62} + 910 q^{63} + 128 q^{64} - 8 q^{66} - 670 q^{67} + 76 q^{68} - 72 q^{69} + 2380 q^{70} + 55 q^{71} - 208 q^{72} + 1676 q^{73} + 22 q^{74} + 164 q^{75} + 376 q^{76} - 140 q^{77} + 2032 q^{79} + 272 q^{80} - 649 q^{81} + 560 q^{82} - 840 q^{83} + 140 q^{84} - 323 q^{85} - 964 q^{86} + 246 q^{87} - 16 q^{88} - 934 q^{89} - 1768 q^{90} - 576 q^{92} + 100 q^{93} - 274 q^{94} - 1598 q^{95} + 64 q^{96} - 1154 q^{97} + 1764 q^{98} + 104 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + q^3 - 4 * q^4 - 34 * q^5 - 2 * q^6 - 35 * q^7 - 16 * q^8 + 26 * q^9 - 34 * q^10 + 2 * q^11 - 8 * q^12 - 140 * q^14 - 17 * q^15 - 16 * q^16 + 19 * q^17 + 104 * q^18 + 94 * q^19 + 68 * q^20 - 70 * q^21 - 4 * q^22 + 72 * q^23 - 8 * q^24 + 328 * q^25 + 106 * q^27 - 140 * q^28 - 246 * q^29 + 34 * q^30 + 200 * q^31 + 32 * q^32 - 2 * q^33 + 76 * q^34 + 595 * q^35 + 104 * q^36 - 11 * q^37 + 376 * q^38 + 272 * q^40 - 280 * q^41 - 70 * q^42 - 241 * q^43 - 16 * q^44 - 442 * q^45 - 144 * q^46 - 274 * q^47 + 16 * q^48 - 882 * q^49 + 328 * q^50 + 38 * q^51 - 464 * q^53 + 106 * q^54 - 34 * q^55 + 280 * q^56 + 188 * q^57 + 492 * q^58 - 386 * q^59 + 136 * q^60 - 64 * q^61 + 200 * q^62 + 910 * q^63 + 128 * q^64 - 8 * q^66 - 670 * q^67 + 76 * q^68 - 72 * q^69 + 2380 * q^70 + 55 * q^71 - 208 * q^72 + 1676 * q^73 + 22 * q^74 + 164 * q^75 + 376 * q^76 - 140 * q^77 + 2032 * q^79 + 272 * q^80 - 649 * q^81 + 560 * q^82 - 840 * q^83 + 140 * q^84 - 323 * q^85 - 964 * q^86 + 246 * q^87 - 16 * q^88 - 934 * q^89 - 1768 * q^90 - 576 * q^92 + 100 * q^93 - 274 * q^94 - 1598 * q^95 + 64 * q^96 - 1154 * q^97 + 1764 * q^98 + 104 * q^99

## Character values

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

 $$n$$ $$171$$ $$\chi(n)$$ $$-\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
191.1
 0.5 + 0.866025i 0.5 − 0.866025i
1.00000 1.73205i 0.500000 0.866025i −2.00000 3.46410i −17.0000 −1.00000 1.73205i −17.5000 30.3109i −8.00000 13.0000 + 22.5167i −17.0000 + 29.4449i
315.1 1.00000 + 1.73205i 0.500000 + 0.866025i −2.00000 + 3.46410i −17.0000 −1.00000 + 1.73205i −17.5000 + 30.3109i −8.00000 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.c even 3 1 inner

## Twists

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.4.a.b 1 13.e even 6 1
208.4.a.e 1 52.i odd 6 1
234.4.a.a 1 39.h odd 6 1
338.4.a.b 1 13.c even 3 1
338.4.b.b 2 13.f odd 12 2
338.4.c.c 2 13.b even 2 1
338.4.c.c 2 13.e even 6 1
338.4.c.g 2 1.a even 1 1 trivial
338.4.c.g 2 13.c even 3 1 inner
338.4.e.c 4 13.d odd 4 2
338.4.e.c 4 13.f odd 12 2
650.4.a.c 1 65.l even 6 1
650.4.b.d 2 65.r odd 12 2
832.4.a.g 1 104.p odd 6 1
832.4.a.j 1 104.s even 6 1
1274.4.a.f 1 91.t odd 6 1
1872.4.a.b 1 156.r even 6 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 $$T_{5} + 17$$ T5 + 17

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T + 4$$
$3$ $$T^{2} - T + 1$$
$5$ $$(T + 17)^{2}$$
$7$ $$T^{2} + 35T + 1225$$
$11$ $$T^{2} - 2T + 4$$
$13$ $$T^{2}$$
$17$ $$T^{2} - 19T + 361$$
$19$ $$T^{2} - 94T + 8836$$
$23$ $$T^{2} - 72T + 5184$$
$29$ $$T^{2} + 246T + 60516$$
$31$ $$(T - 100)^{2}$$
$37$ $$T^{2} + 11T + 121$$
$41$ $$T^{2} + 280T + 78400$$
$43$ $$T^{2} + 241T + 58081$$
$47$ $$(T + 137)^{2}$$
$53$ $$(T + 232)^{2}$$
$59$ $$T^{2} + 386T + 148996$$
$61$ $$T^{2} + 64T + 4096$$
$67$ $$T^{2} + 670T + 448900$$
$71$ $$T^{2} - 55T + 3025$$
$73$ $$(T - 838)^{2}$$
$79$ $$(T - 1016)^{2}$$
$83$ $$(T + 420)^{2}$$
$89$ $$T^{2} + 934T + 872356$$
$97$ $$T^{2} + 1154 T + 1331716$$