# Properties

 Label 338.4.c.b 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.

 $$f(q)$$ $$=$$ $$q + (2 \zeta_{6} - 2) q^{2} + (3 \zeta_{6} - 3) q^{3} - 4 \zeta_{6} q^{4} - 11 q^{5} - 6 \zeta_{6} q^{6} + 19 \zeta_{6} q^{7} + 8 q^{8} + 18 \zeta_{6} q^{9} +O(q^{10})$$ q + (2*z - 2) * q^2 + (3*z - 3) * q^3 - 4*z * q^4 - 11 * q^5 - 6*z * q^6 + 19*z * q^7 + 8 * q^8 + 18*z * q^9 $$q + (2 \zeta_{6} - 2) q^{2} + (3 \zeta_{6} - 3) q^{3} - 4 \zeta_{6} q^{4} - 11 q^{5} - 6 \zeta_{6} q^{6} + 19 \zeta_{6} q^{7} + 8 q^{8} + 18 \zeta_{6} q^{9} + ( - 22 \zeta_{6} + 22) q^{10} + (38 \zeta_{6} - 38) q^{11} + 12 q^{12} - 38 q^{14} + ( - 33 \zeta_{6} + 33) q^{15} + (16 \zeta_{6} - 16) q^{16} + 51 \zeta_{6} q^{17} - 36 q^{18} + 90 \zeta_{6} q^{19} + 44 \zeta_{6} q^{20} - 57 q^{21} - 76 \zeta_{6} q^{22} + ( - 52 \zeta_{6} + 52) q^{23} + (24 \zeta_{6} - 24) q^{24} - 4 q^{25} - 135 q^{27} + ( - 76 \zeta_{6} + 76) q^{28} + ( - 190 \zeta_{6} + 190) q^{29} + 66 \zeta_{6} q^{30} - 292 q^{31} - 32 \zeta_{6} q^{32} - 114 \zeta_{6} q^{33} - 102 q^{34} - 209 \zeta_{6} q^{35} + ( - 72 \zeta_{6} + 72) q^{36} + (441 \zeta_{6} - 441) q^{37} - 180 q^{38} - 88 q^{40} + ( - 312 \zeta_{6} + 312) q^{41} + ( - 114 \zeta_{6} + 114) q^{42} - 373 \zeta_{6} q^{43} + 152 q^{44} - 198 \zeta_{6} q^{45} + 104 \zeta_{6} q^{46} + 41 q^{47} - 48 \zeta_{6} q^{48} + (18 \zeta_{6} - 18) q^{49} + ( - 8 \zeta_{6} + 8) q^{50} - 153 q^{51} + 468 q^{53} + ( - 270 \zeta_{6} + 270) q^{54} + ( - 418 \zeta_{6} + 418) q^{55} + 152 \zeta_{6} q^{56} - 270 q^{57} + 380 \zeta_{6} q^{58} + 530 \zeta_{6} q^{59} - 132 q^{60} - 592 \zeta_{6} q^{61} + ( - 584 \zeta_{6} + 584) q^{62} + (342 \zeta_{6} - 342) q^{63} + 64 q^{64} + 228 q^{66} + (206 \zeta_{6} - 206) q^{67} + ( - 204 \zeta_{6} + 204) q^{68} + 156 \zeta_{6} q^{69} + 418 q^{70} - 863 \zeta_{6} q^{71} + 144 \zeta_{6} q^{72} + 322 q^{73} - 882 \zeta_{6} q^{74} + ( - 12 \zeta_{6} + 12) q^{75} + ( - 360 \zeta_{6} + 360) q^{76} - 722 q^{77} - 460 q^{79} + ( - 176 \zeta_{6} + 176) q^{80} + (81 \zeta_{6} - 81) q^{81} + 624 \zeta_{6} q^{82} - 528 q^{83} + 228 \zeta_{6} q^{84} - 561 \zeta_{6} q^{85} + 746 q^{86} + 570 \zeta_{6} q^{87} + (304 \zeta_{6} - 304) q^{88} + ( - 870 \zeta_{6} + 870) q^{89} + 396 q^{90} - 208 q^{92} + ( - 876 \zeta_{6} + 876) q^{93} + (82 \zeta_{6} - 82) q^{94} - 990 \zeta_{6} q^{95} + 96 q^{96} - 346 \zeta_{6} q^{97} - 36 \zeta_{6} q^{98} - 684 q^{99} +O(q^{100})$$ q + (2*z - 2) * q^2 + (3*z - 3) * q^3 - 4*z * q^4 - 11 * q^5 - 6*z * q^6 + 19*z * q^7 + 8 * q^8 + 18*z * q^9 + (-22*z + 22) * q^10 + (38*z - 38) * q^11 + 12 * q^12 - 38 * q^14 + (-33*z + 33) * q^15 + (16*z - 16) * q^16 + 51*z * q^17 - 36 * q^18 + 90*z * q^19 + 44*z * q^20 - 57 * q^21 - 76*z * q^22 + (-52*z + 52) * q^23 + (24*z - 24) * q^24 - 4 * q^25 - 135 * q^27 + (-76*z + 76) * q^28 + (-190*z + 190) * q^29 + 66*z * q^30 - 292 * q^31 - 32*z * q^32 - 114*z * q^33 - 102 * q^34 - 209*z * q^35 + (-72*z + 72) * q^36 + (441*z - 441) * q^37 - 180 * q^38 - 88 * q^40 + (-312*z + 312) * q^41 + (-114*z + 114) * q^42 - 373*z * q^43 + 152 * q^44 - 198*z * q^45 + 104*z * q^46 + 41 * q^47 - 48*z * q^48 + (18*z - 18) * q^49 + (-8*z + 8) * q^50 - 153 * q^51 + 468 * q^53 + (-270*z + 270) * q^54 + (-418*z + 418) * q^55 + 152*z * q^56 - 270 * q^57 + 380*z * q^58 + 530*z * q^59 - 132 * q^60 - 592*z * q^61 + (-584*z + 584) * q^62 + (342*z - 342) * q^63 + 64 * q^64 + 228 * q^66 + (206*z - 206) * q^67 + (-204*z + 204) * q^68 + 156*z * q^69 + 418 * q^70 - 863*z * q^71 + 144*z * q^72 + 322 * q^73 - 882*z * q^74 + (-12*z + 12) * q^75 + (-360*z + 360) * q^76 - 722 * q^77 - 460 * q^79 + (-176*z + 176) * q^80 + (81*z - 81) * q^81 + 624*z * q^82 - 528 * q^83 + 228*z * q^84 - 561*z * q^85 + 746 * q^86 + 570*z * q^87 + (304*z - 304) * q^88 + (-870*z + 870) * q^89 + 396 * q^90 - 208 * q^92 + (-876*z + 876) * q^93 + (82*z - 82) * q^94 - 990*z * q^95 + 96 * q^96 - 346*z * q^97 - 36*z * q^98 - 684 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 3 q^{3} - 4 q^{4} - 22 q^{5} - 6 q^{6} + 19 q^{7} + 16 q^{8} + 18 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - 3 * q^3 - 4 * q^4 - 22 * q^5 - 6 * q^6 + 19 * q^7 + 16 * q^8 + 18 * q^9 $$2 q - 2 q^{2} - 3 q^{3} - 4 q^{4} - 22 q^{5} - 6 q^{6} + 19 q^{7} + 16 q^{8} + 18 q^{9} + 22 q^{10} - 38 q^{11} + 24 q^{12} - 76 q^{14} + 33 q^{15} - 16 q^{16} + 51 q^{17} - 72 q^{18} + 90 q^{19} + 44 q^{20} - 114 q^{21} - 76 q^{22} + 52 q^{23} - 24 q^{24} - 8 q^{25} - 270 q^{27} + 76 q^{28} + 190 q^{29} + 66 q^{30} - 584 q^{31} - 32 q^{32} - 114 q^{33} - 204 q^{34} - 209 q^{35} + 72 q^{36} - 441 q^{37} - 360 q^{38} - 176 q^{40} + 312 q^{41} + 114 q^{42} - 373 q^{43} + 304 q^{44} - 198 q^{45} + 104 q^{46} + 82 q^{47} - 48 q^{48} - 18 q^{49} + 8 q^{50} - 306 q^{51} + 936 q^{53} + 270 q^{54} + 418 q^{55} + 152 q^{56} - 540 q^{57} + 380 q^{58} + 530 q^{59} - 264 q^{60} - 592 q^{61} + 584 q^{62} - 342 q^{63} + 128 q^{64} + 456 q^{66} - 206 q^{67} + 204 q^{68} + 156 q^{69} + 836 q^{70} - 863 q^{71} + 144 q^{72} + 644 q^{73} - 882 q^{74} + 12 q^{75} + 360 q^{76} - 1444 q^{77} - 920 q^{79} + 176 q^{80} - 81 q^{81} + 624 q^{82} - 1056 q^{83} + 228 q^{84} - 561 q^{85} + 1492 q^{86} + 570 q^{87} - 304 q^{88} + 870 q^{89} + 792 q^{90} - 416 q^{92} + 876 q^{93} - 82 q^{94} - 990 q^{95} + 192 q^{96} - 346 q^{97} - 36 q^{98} - 1368 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 - 3 * q^3 - 4 * q^4 - 22 * q^5 - 6 * q^6 + 19 * q^7 + 16 * q^8 + 18 * q^9 + 22 * q^10 - 38 * q^11 + 24 * q^12 - 76 * q^14 + 33 * q^15 - 16 * q^16 + 51 * q^17 - 72 * q^18 + 90 * q^19 + 44 * q^20 - 114 * q^21 - 76 * q^22 + 52 * q^23 - 24 * q^24 - 8 * q^25 - 270 * q^27 + 76 * q^28 + 190 * q^29 + 66 * q^30 - 584 * q^31 - 32 * q^32 - 114 * q^33 - 204 * q^34 - 209 * q^35 + 72 * q^36 - 441 * q^37 - 360 * q^38 - 176 * q^40 + 312 * q^41 + 114 * q^42 - 373 * q^43 + 304 * q^44 - 198 * q^45 + 104 * q^46 + 82 * q^47 - 48 * q^48 - 18 * q^49 + 8 * q^50 - 306 * q^51 + 936 * q^53 + 270 * q^54 + 418 * q^55 + 152 * q^56 - 540 * q^57 + 380 * q^58 + 530 * q^59 - 264 * q^60 - 592 * q^61 + 584 * q^62 - 342 * q^63 + 128 * q^64 + 456 * q^66 - 206 * q^67 + 204 * q^68 + 156 * q^69 + 836 * q^70 - 863 * q^71 + 144 * q^72 + 644 * q^73 - 882 * q^74 + 12 * q^75 + 360 * q^76 - 1444 * q^77 - 920 * q^79 + 176 * q^80 - 81 * q^81 + 624 * q^82 - 1056 * q^83 + 228 * q^84 - 561 * q^85 + 1492 * q^86 + 570 * q^87 - 304 * q^88 + 870 * q^89 + 792 * q^90 - 416 * q^92 + 876 * q^93 - 82 * q^94 - 990 * q^95 + 192 * q^96 - 346 * q^97 - 36 * q^98 - 1368 * 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 −1.50000 + 2.59808i −2.00000 3.46410i −11.0000 −3.00000 5.19615i 9.50000 + 16.4545i 8.00000 9.00000 + 15.5885i 11.0000 19.0526i
315.1 −1.00000 1.73205i −1.50000 2.59808i −2.00000 + 3.46410i −11.0000 −3.00000 + 5.19615i 9.50000 16.4545i 8.00000 9.00000 15.5885i 11.0000 + 19.0526i
 $$n$$: 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.b 2
13.b even 2 1 338.4.c.f 2
13.c even 3 1 338.4.a.e 1
13.c even 3 1 inner 338.4.c.b 2
13.d odd 4 2 338.4.e.b 4
13.e even 6 1 26.4.a.a 1
13.e even 6 1 338.4.c.f 2
13.f odd 12 2 338.4.b.c 2
13.f odd 12 2 338.4.e.b 4
39.h odd 6 1 234.4.a.g 1
52.i odd 6 1 208.4.a.c 1
65.l even 6 1 650.4.a.f 1
65.r odd 12 2 650.4.b.b 2
91.t odd 6 1 1274.4.a.b 1
104.p odd 6 1 832.4.a.m 1
104.s even 6 1 832.4.a.e 1
156.r even 6 1 1872.4.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.4.a.a 1 13.e even 6 1
208.4.a.c 1 52.i odd 6 1
234.4.a.g 1 39.h odd 6 1
338.4.a.e 1 13.c even 3 1
338.4.b.c 2 13.f odd 12 2
338.4.c.b 2 1.a even 1 1 trivial
338.4.c.b 2 13.c even 3 1 inner
338.4.c.f 2 13.b even 2 1
338.4.c.f 2 13.e even 6 1
338.4.e.b 4 13.d odd 4 2
338.4.e.b 4 13.f odd 12 2
650.4.a.f 1 65.l even 6 1
650.4.b.b 2 65.r odd 12 2
832.4.a.e 1 104.s even 6 1
832.4.a.m 1 104.p odd 6 1
1274.4.a.b 1 91.t odd 6 1
1872.4.a.c 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 $$S_{4}^{\mathrm{new}}(338, [\chi])$$:

 $$T_{3}^{2} + 3T_{3} + 9$$ T3^2 + 3*T3 + 9 $$T_{5} + 11$$ T5 + 11

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T + 4$$
$3$ $$T^{2} + 3T + 9$$
$5$ $$(T + 11)^{2}$$
$7$ $$T^{2} - 19T + 361$$
$11$ $$T^{2} + 38T + 1444$$
$13$ $$T^{2}$$
$17$ $$T^{2} - 51T + 2601$$
$19$ $$T^{2} - 90T + 8100$$
$23$ $$T^{2} - 52T + 2704$$
$29$ $$T^{2} - 190T + 36100$$
$31$ $$(T + 292)^{2}$$
$37$ $$T^{2} + 441T + 194481$$
$41$ $$T^{2} - 312T + 97344$$
$43$ $$T^{2} + 373T + 139129$$
$47$ $$(T - 41)^{2}$$
$53$ $$(T - 468)^{2}$$
$59$ $$T^{2} - 530T + 280900$$
$61$ $$T^{2} + 592T + 350464$$
$67$ $$T^{2} + 206T + 42436$$
$71$ $$T^{2} + 863T + 744769$$
$73$ $$(T - 322)^{2}$$
$79$ $$(T + 460)^{2}$$
$83$ $$(T + 528)^{2}$$
$89$ $$T^{2} - 870T + 756900$$
$97$ $$T^{2} + 346T + 119716$$