# Properties

 Label 338.4.c.a 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: $$\Z[a_1, \ldots, a_{9}]$$ 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.

 $$f(q)$$ $$=$$ $$q + (2 \zeta_{6} - 2) q^{2} + (4 \zeta_{6} - 4) q^{3} - 4 \zeta_{6} q^{4} - 18 q^{5} - 8 \zeta_{6} q^{6} - 20 \zeta_{6} q^{7} + 8 q^{8} + 11 \zeta_{6} q^{9} +O(q^{10})$$ q + (2*z - 2) * q^2 + (4*z - 4) * q^3 - 4*z * q^4 - 18 * q^5 - 8*z * q^6 - 20*z * q^7 + 8 * q^8 + 11*z * q^9 $$q + (2 \zeta_{6} - 2) q^{2} + (4 \zeta_{6} - 4) q^{3} - 4 \zeta_{6} q^{4} - 18 q^{5} - 8 \zeta_{6} q^{6} - 20 \zeta_{6} q^{7} + 8 q^{8} + 11 \zeta_{6} q^{9} + ( - 36 \zeta_{6} + 36) q^{10} + ( - 48 \zeta_{6} + 48) q^{11} + 16 q^{12} + 40 q^{14} + ( - 72 \zeta_{6} + 72) q^{15} + (16 \zeta_{6} - 16) q^{16} - 66 \zeta_{6} q^{17} - 22 q^{18} + 16 \zeta_{6} q^{19} + 72 \zeta_{6} q^{20} + 80 q^{21} + 96 \zeta_{6} q^{22} + (168 \zeta_{6} - 168) q^{23} + (32 \zeta_{6} - 32) q^{24} + 199 q^{25} - 152 q^{27} + (80 \zeta_{6} - 80) q^{28} + (6 \zeta_{6} - 6) q^{29} + 144 \zeta_{6} q^{30} + 20 q^{31} - 32 \zeta_{6} q^{32} + 192 \zeta_{6} q^{33} + 132 q^{34} + 360 \zeta_{6} q^{35} + ( - 44 \zeta_{6} + 44) q^{36} + (254 \zeta_{6} - 254) q^{37} - 32 q^{38} - 144 q^{40} + ( - 390 \zeta_{6} + 390) q^{41} + (160 \zeta_{6} - 160) q^{42} + 124 \zeta_{6} q^{43} - 192 q^{44} - 198 \zeta_{6} q^{45} - 336 \zeta_{6} q^{46} - 468 q^{47} - 64 \zeta_{6} q^{48} + (57 \zeta_{6} - 57) q^{49} + (398 \zeta_{6} - 398) q^{50} + 264 q^{51} + 558 q^{53} + ( - 304 \zeta_{6} + 304) q^{54} + (864 \zeta_{6} - 864) q^{55} - 160 \zeta_{6} q^{56} - 64 q^{57} - 12 \zeta_{6} q^{58} + 96 \zeta_{6} q^{59} - 288 q^{60} + 826 \zeta_{6} q^{61} + (40 \zeta_{6} - 40) q^{62} + ( - 220 \zeta_{6} + 220) q^{63} + 64 q^{64} - 384 q^{66} + ( - 160 \zeta_{6} + 160) q^{67} + (264 \zeta_{6} - 264) q^{68} - 672 \zeta_{6} q^{69} - 720 q^{70} + 420 \zeta_{6} q^{71} + 88 \zeta_{6} q^{72} + 362 q^{73} - 508 \zeta_{6} q^{74} + (796 \zeta_{6} - 796) q^{75} + ( - 64 \zeta_{6} + 64) q^{76} - 960 q^{77} + 776 q^{79} + ( - 288 \zeta_{6} + 288) q^{80} + ( - 311 \zeta_{6} + 311) q^{81} + 780 \zeta_{6} q^{82} - 320 \zeta_{6} q^{84} + 1188 \zeta_{6} q^{85} - 248 q^{86} - 24 \zeta_{6} q^{87} + ( - 384 \zeta_{6} + 384) q^{88} + (1626 \zeta_{6} - 1626) q^{89} + 396 q^{90} + 672 q^{92} + (80 \zeta_{6} - 80) q^{93} + ( - 936 \zeta_{6} + 936) q^{94} - 288 \zeta_{6} q^{95} + 128 q^{96} + 1294 \zeta_{6} q^{97} - 114 \zeta_{6} q^{98} + 528 q^{99} +O(q^{100})$$ q + (2*z - 2) * q^2 + (4*z - 4) * q^3 - 4*z * q^4 - 18 * q^5 - 8*z * q^6 - 20*z * q^7 + 8 * q^8 + 11*z * q^9 + (-36*z + 36) * q^10 + (-48*z + 48) * q^11 + 16 * q^12 + 40 * q^14 + (-72*z + 72) * q^15 + (16*z - 16) * q^16 - 66*z * q^17 - 22 * q^18 + 16*z * q^19 + 72*z * q^20 + 80 * q^21 + 96*z * q^22 + (168*z - 168) * q^23 + (32*z - 32) * q^24 + 199 * q^25 - 152 * q^27 + (80*z - 80) * q^28 + (6*z - 6) * q^29 + 144*z * q^30 + 20 * q^31 - 32*z * q^32 + 192*z * q^33 + 132 * q^34 + 360*z * q^35 + (-44*z + 44) * q^36 + (254*z - 254) * q^37 - 32 * q^38 - 144 * q^40 + (-390*z + 390) * q^41 + (160*z - 160) * q^42 + 124*z * q^43 - 192 * q^44 - 198*z * q^45 - 336*z * q^46 - 468 * q^47 - 64*z * q^48 + (57*z - 57) * q^49 + (398*z - 398) * q^50 + 264 * q^51 + 558 * q^53 + (-304*z + 304) * q^54 + (864*z - 864) * q^55 - 160*z * q^56 - 64 * q^57 - 12*z * q^58 + 96*z * q^59 - 288 * q^60 + 826*z * q^61 + (40*z - 40) * q^62 + (-220*z + 220) * q^63 + 64 * q^64 - 384 * q^66 + (-160*z + 160) * q^67 + (264*z - 264) * q^68 - 672*z * q^69 - 720 * q^70 + 420*z * q^71 + 88*z * q^72 + 362 * q^73 - 508*z * q^74 + (796*z - 796) * q^75 + (-64*z + 64) * q^76 - 960 * q^77 + 776 * q^79 + (-288*z + 288) * q^80 + (-311*z + 311) * q^81 + 780*z * q^82 - 320*z * q^84 + 1188*z * q^85 - 248 * q^86 - 24*z * q^87 + (-384*z + 384) * q^88 + (1626*z - 1626) * q^89 + 396 * q^90 + 672 * q^92 + (80*z - 80) * q^93 + (-936*z + 936) * q^94 - 288*z * q^95 + 128 * q^96 + 1294*z * q^97 - 114*z * q^98 + 528 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 4 q^{3} - 4 q^{4} - 36 q^{5} - 8 q^{6} - 20 q^{7} + 16 q^{8} + 11 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - 4 * q^3 - 4 * q^4 - 36 * q^5 - 8 * q^6 - 20 * q^7 + 16 * q^8 + 11 * q^9 $$2 q - 2 q^{2} - 4 q^{3} - 4 q^{4} - 36 q^{5} - 8 q^{6} - 20 q^{7} + 16 q^{8} + 11 q^{9} + 36 q^{10} + 48 q^{11} + 32 q^{12} + 80 q^{14} + 72 q^{15} - 16 q^{16} - 66 q^{17} - 44 q^{18} + 16 q^{19} + 72 q^{20} + 160 q^{21} + 96 q^{22} - 168 q^{23} - 32 q^{24} + 398 q^{25} - 304 q^{27} - 80 q^{28} - 6 q^{29} + 144 q^{30} + 40 q^{31} - 32 q^{32} + 192 q^{33} + 264 q^{34} + 360 q^{35} + 44 q^{36} - 254 q^{37} - 64 q^{38} - 288 q^{40} + 390 q^{41} - 160 q^{42} + 124 q^{43} - 384 q^{44} - 198 q^{45} - 336 q^{46} - 936 q^{47} - 64 q^{48} - 57 q^{49} - 398 q^{50} + 528 q^{51} + 1116 q^{53} + 304 q^{54} - 864 q^{55} - 160 q^{56} - 128 q^{57} - 12 q^{58} + 96 q^{59} - 576 q^{60} + 826 q^{61} - 40 q^{62} + 220 q^{63} + 128 q^{64} - 768 q^{66} + 160 q^{67} - 264 q^{68} - 672 q^{69} - 1440 q^{70} + 420 q^{71} + 88 q^{72} + 724 q^{73} - 508 q^{74} - 796 q^{75} + 64 q^{76} - 1920 q^{77} + 1552 q^{79} + 288 q^{80} + 311 q^{81} + 780 q^{82} - 320 q^{84} + 1188 q^{85} - 496 q^{86} - 24 q^{87} + 384 q^{88} - 1626 q^{89} + 792 q^{90} + 1344 q^{92} - 80 q^{93} + 936 q^{94} - 288 q^{95} + 256 q^{96} + 1294 q^{97} - 114 q^{98} + 1056 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 - 4 * q^3 - 4 * q^4 - 36 * q^5 - 8 * q^6 - 20 * q^7 + 16 * q^8 + 11 * q^9 + 36 * q^10 + 48 * q^11 + 32 * q^12 + 80 * q^14 + 72 * q^15 - 16 * q^16 - 66 * q^17 - 44 * q^18 + 16 * q^19 + 72 * q^20 + 160 * q^21 + 96 * q^22 - 168 * q^23 - 32 * q^24 + 398 * q^25 - 304 * q^27 - 80 * q^28 - 6 * q^29 + 144 * q^30 + 40 * q^31 - 32 * q^32 + 192 * q^33 + 264 * q^34 + 360 * q^35 + 44 * q^36 - 254 * q^37 - 64 * q^38 - 288 * q^40 + 390 * q^41 - 160 * q^42 + 124 * q^43 - 384 * q^44 - 198 * q^45 - 336 * q^46 - 936 * q^47 - 64 * q^48 - 57 * q^49 - 398 * q^50 + 528 * q^51 + 1116 * q^53 + 304 * q^54 - 864 * q^55 - 160 * q^56 - 128 * q^57 - 12 * q^58 + 96 * q^59 - 576 * q^60 + 826 * q^61 - 40 * q^62 + 220 * q^63 + 128 * q^64 - 768 * q^66 + 160 * q^67 - 264 * q^68 - 672 * q^69 - 1440 * q^70 + 420 * q^71 + 88 * q^72 + 724 * q^73 - 508 * q^74 - 796 * q^75 + 64 * q^76 - 1920 * q^77 + 1552 * q^79 + 288 * q^80 + 311 * q^81 + 780 * q^82 - 320 * q^84 + 1188 * q^85 - 496 * q^86 - 24 * q^87 + 384 * q^88 - 1626 * q^89 + 792 * q^90 + 1344 * q^92 - 80 * q^93 + 936 * q^94 - 288 * q^95 + 256 * q^96 + 1294 * q^97 - 114 * q^98 + 1056 * 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 −2.00000 + 3.46410i −2.00000 3.46410i −18.0000 −4.00000 6.92820i −10.0000 17.3205i 8.00000 5.50000 + 9.52628i 18.0000 31.1769i
315.1 −1.00000 1.73205i −2.00000 3.46410i −2.00000 + 3.46410i −18.0000 −4.00000 + 6.92820i −10.0000 + 17.3205i 8.00000 5.50000 9.52628i 18.0000 + 31.1769i
 $$n$$: 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.a 2
13.b even 2 1 338.4.c.e 2
13.c even 3 1 26.4.a.c 1
13.c even 3 1 inner 338.4.c.a 2
13.d odd 4 2 338.4.e.a 4
13.e even 6 1 338.4.a.c 1
13.e even 6 1 338.4.c.e 2
13.f odd 12 2 338.4.b.d 2
13.f odd 12 2 338.4.e.a 4
39.i odd 6 1 234.4.a.e 1
52.j odd 6 1 208.4.a.b 1
65.n even 6 1 650.4.a.b 1
65.q odd 12 2 650.4.b.f 2
91.n odd 6 1 1274.4.a.d 1
104.n odd 6 1 832.4.a.o 1
104.r even 6 1 832.4.a.d 1
156.p even 6 1 1872.4.a.q 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.4.a.c 1 13.c even 3 1
208.4.a.b 1 52.j odd 6 1
234.4.a.e 1 39.i odd 6 1
338.4.a.c 1 13.e even 6 1
338.4.b.d 2 13.f odd 12 2
338.4.c.a 2 1.a even 1 1 trivial
338.4.c.a 2 13.c even 3 1 inner
338.4.c.e 2 13.b even 2 1
338.4.c.e 2 13.e even 6 1
338.4.e.a 4 13.d odd 4 2
338.4.e.a 4 13.f odd 12 2
650.4.a.b 1 65.n even 6 1
650.4.b.f 2 65.q odd 12 2
832.4.a.d 1 104.r even 6 1
832.4.a.o 1 104.n odd 6 1
1274.4.a.d 1 91.n odd 6 1
1872.4.a.q 1 156.p even 6 1

## Hecke kernels

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

 $$T_{3}^{2} + 4T_{3} + 16$$ T3^2 + 4*T3 + 16 $$T_{5} + 18$$ T5 + 18

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T + 4$$
$3$ $$T^{2} + 4T + 16$$
$5$ $$(T + 18)^{2}$$
$7$ $$T^{2} + 20T + 400$$
$11$ $$T^{2} - 48T + 2304$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 66T + 4356$$
$19$ $$T^{2} - 16T + 256$$
$23$ $$T^{2} + 168T + 28224$$
$29$ $$T^{2} + 6T + 36$$
$31$ $$(T - 20)^{2}$$
$37$ $$T^{2} + 254T + 64516$$
$41$ $$T^{2} - 390T + 152100$$
$43$ $$T^{2} - 124T + 15376$$
$47$ $$(T + 468)^{2}$$
$53$ $$(T - 558)^{2}$$
$59$ $$T^{2} - 96T + 9216$$
$61$ $$T^{2} - 826T + 682276$$
$67$ $$T^{2} - 160T + 25600$$
$71$ $$T^{2} - 420T + 176400$$
$73$ $$(T - 362)^{2}$$
$79$ $$(T - 776)^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2} + 1626 T + 2643876$$
$97$ $$T^{2} - 1294 T + 1674436$$