# Properties

 Label 338.4.c.d 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.

 $$f(q)$$ $$=$$ $$q + (2 \zeta_{6} - 2) q^{2} + ( - 3 \zeta_{6} + 3) q^{3} - 4 \zeta_{6} q^{4} - 2 q^{5} + 6 \zeta_{6} q^{6} - 5 \zeta_{6} q^{7} + 8 q^{8} + 18 \zeta_{6} q^{9} +O(q^{10})$$ q + (2*z - 2) * q^2 + (-3*z + 3) * q^3 - 4*z * q^4 - 2 * q^5 + 6*z * q^6 - 5*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} - 2 q^{5} + 6 \zeta_{6} q^{6} - 5 \zeta_{6} q^{7} + 8 q^{8} + 18 \zeta_{6} q^{9} + ( - 4 \zeta_{6} + 4) q^{10} + ( - 13 \zeta_{6} + 13) q^{11} - 12 q^{12} + 10 q^{14} + (6 \zeta_{6} - 6) q^{15} + (16 \zeta_{6} - 16) q^{16} - 27 \zeta_{6} q^{17} - 36 q^{18} + 75 \zeta_{6} q^{19} + 8 \zeta_{6} q^{20} - 15 q^{21} + 26 \zeta_{6} q^{22} + ( - 187 \zeta_{6} + 187) q^{23} + ( - 24 \zeta_{6} + 24) q^{24} - 121 q^{25} + 135 q^{27} + (20 \zeta_{6} - 20) q^{28} + ( - 13 \zeta_{6} + 13) q^{29} - 12 \zeta_{6} q^{30} + 104 q^{31} - 32 \zeta_{6} q^{32} - 39 \zeta_{6} q^{33} + 54 q^{34} + 10 \zeta_{6} q^{35} + ( - 72 \zeta_{6} + 72) q^{36} + ( - 423 \zeta_{6} + 423) q^{37} - 150 q^{38} - 16 q^{40} + ( - 195 \zeta_{6} + 195) q^{41} + ( - 30 \zeta_{6} + 30) q^{42} - 199 \zeta_{6} q^{43} - 52 q^{44} - 36 \zeta_{6} q^{45} + 374 \zeta_{6} q^{46} - 388 q^{47} + 48 \zeta_{6} q^{48} + ( - 318 \zeta_{6} + 318) q^{49} + ( - 242 \zeta_{6} + 242) q^{50} - 81 q^{51} + 618 q^{53} + (270 \zeta_{6} - 270) q^{54} + (26 \zeta_{6} - 26) q^{55} - 40 \zeta_{6} q^{56} + 225 q^{57} + 26 \zeta_{6} q^{58} + 491 \zeta_{6} q^{59} + 24 q^{60} - 175 \zeta_{6} q^{61} + (208 \zeta_{6} - 208) q^{62} + ( - 90 \zeta_{6} + 90) q^{63} + 64 q^{64} + 78 q^{66} + ( - 817 \zeta_{6} + 817) q^{67} + (108 \zeta_{6} - 108) q^{68} - 561 \zeta_{6} q^{69} - 20 q^{70} + 79 \zeta_{6} q^{71} + 144 \zeta_{6} q^{72} - 230 q^{73} + 846 \zeta_{6} q^{74} + (363 \zeta_{6} - 363) q^{75} + ( - 300 \zeta_{6} + 300) q^{76} - 65 q^{77} + 764 q^{79} + ( - 32 \zeta_{6} + 32) q^{80} + (81 \zeta_{6} - 81) q^{81} + 390 \zeta_{6} q^{82} + 732 q^{83} + 60 \zeta_{6} q^{84} + 54 \zeta_{6} q^{85} + 398 q^{86} - 39 \zeta_{6} q^{87} + ( - 104 \zeta_{6} + 104) q^{88} + (1041 \zeta_{6} - 1041) q^{89} + 72 q^{90} - 748 q^{92} + ( - 312 \zeta_{6} + 312) q^{93} + ( - 776 \zeta_{6} + 776) q^{94} - 150 \zeta_{6} q^{95} - 96 q^{96} - 97 \zeta_{6} q^{97} + 636 \zeta_{6} q^{98} + 234 q^{99} +O(q^{100})$$ q + (2*z - 2) * q^2 + (-3*z + 3) * q^3 - 4*z * q^4 - 2 * q^5 + 6*z * q^6 - 5*z * q^7 + 8 * q^8 + 18*z * q^9 + (-4*z + 4) * q^10 + (-13*z + 13) * q^11 - 12 * q^12 + 10 * q^14 + (6*z - 6) * q^15 + (16*z - 16) * q^16 - 27*z * q^17 - 36 * q^18 + 75*z * q^19 + 8*z * q^20 - 15 * q^21 + 26*z * q^22 + (-187*z + 187) * q^23 + (-24*z + 24) * q^24 - 121 * q^25 + 135 * q^27 + (20*z - 20) * q^28 + (-13*z + 13) * q^29 - 12*z * q^30 + 104 * q^31 - 32*z * q^32 - 39*z * q^33 + 54 * q^34 + 10*z * q^35 + (-72*z + 72) * q^36 + (-423*z + 423) * q^37 - 150 * q^38 - 16 * q^40 + (-195*z + 195) * q^41 + (-30*z + 30) * q^42 - 199*z * q^43 - 52 * q^44 - 36*z * q^45 + 374*z * q^46 - 388 * q^47 + 48*z * q^48 + (-318*z + 318) * q^49 + (-242*z + 242) * q^50 - 81 * q^51 + 618 * q^53 + (270*z - 270) * q^54 + (26*z - 26) * q^55 - 40*z * q^56 + 225 * q^57 + 26*z * q^58 + 491*z * q^59 + 24 * q^60 - 175*z * q^61 + (208*z - 208) * q^62 + (-90*z + 90) * q^63 + 64 * q^64 + 78 * q^66 + (-817*z + 817) * q^67 + (108*z - 108) * q^68 - 561*z * q^69 - 20 * q^70 + 79*z * q^71 + 144*z * q^72 - 230 * q^73 + 846*z * q^74 + (363*z - 363) * q^75 + (-300*z + 300) * q^76 - 65 * q^77 + 764 * q^79 + (-32*z + 32) * q^80 + (81*z - 81) * q^81 + 390*z * q^82 + 732 * q^83 + 60*z * q^84 + 54*z * q^85 + 398 * q^86 - 39*z * q^87 + (-104*z + 104) * q^88 + (1041*z - 1041) * q^89 + 72 * q^90 - 748 * q^92 + (-312*z + 312) * q^93 + (-776*z + 776) * q^94 - 150*z * q^95 - 96 * q^96 - 97*z * q^97 + 636*z * q^98 + 234 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 3 q^{3} - 4 q^{4} - 4 q^{5} + 6 q^{6} - 5 q^{7} + 16 q^{8} + 18 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + 3 * q^3 - 4 * q^4 - 4 * q^5 + 6 * q^6 - 5 * q^7 + 16 * q^8 + 18 * q^9 $$2 q - 2 q^{2} + 3 q^{3} - 4 q^{4} - 4 q^{5} + 6 q^{6} - 5 q^{7} + 16 q^{8} + 18 q^{9} + 4 q^{10} + 13 q^{11} - 24 q^{12} + 20 q^{14} - 6 q^{15} - 16 q^{16} - 27 q^{17} - 72 q^{18} + 75 q^{19} + 8 q^{20} - 30 q^{21} + 26 q^{22} + 187 q^{23} + 24 q^{24} - 242 q^{25} + 270 q^{27} - 20 q^{28} + 13 q^{29} - 12 q^{30} + 208 q^{31} - 32 q^{32} - 39 q^{33} + 108 q^{34} + 10 q^{35} + 72 q^{36} + 423 q^{37} - 300 q^{38} - 32 q^{40} + 195 q^{41} + 30 q^{42} - 199 q^{43} - 104 q^{44} - 36 q^{45} + 374 q^{46} - 776 q^{47} + 48 q^{48} + 318 q^{49} + 242 q^{50} - 162 q^{51} + 1236 q^{53} - 270 q^{54} - 26 q^{55} - 40 q^{56} + 450 q^{57} + 26 q^{58} + 491 q^{59} + 48 q^{60} - 175 q^{61} - 208 q^{62} + 90 q^{63} + 128 q^{64} + 156 q^{66} + 817 q^{67} - 108 q^{68} - 561 q^{69} - 40 q^{70} + 79 q^{71} + 144 q^{72} - 460 q^{73} + 846 q^{74} - 363 q^{75} + 300 q^{76} - 130 q^{77} + 1528 q^{79} + 32 q^{80} - 81 q^{81} + 390 q^{82} + 1464 q^{83} + 60 q^{84} + 54 q^{85} + 796 q^{86} - 39 q^{87} + 104 q^{88} - 1041 q^{89} + 144 q^{90} - 1496 q^{92} + 312 q^{93} + 776 q^{94} - 150 q^{95} - 192 q^{96} - 97 q^{97} + 636 q^{98} + 468 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 + 3 * q^3 - 4 * q^4 - 4 * q^5 + 6 * q^6 - 5 * q^7 + 16 * q^8 + 18 * q^9 + 4 * q^10 + 13 * q^11 - 24 * q^12 + 20 * q^14 - 6 * q^15 - 16 * q^16 - 27 * q^17 - 72 * q^18 + 75 * q^19 + 8 * q^20 - 30 * q^21 + 26 * q^22 + 187 * q^23 + 24 * q^24 - 242 * q^25 + 270 * q^27 - 20 * q^28 + 13 * q^29 - 12 * q^30 + 208 * q^31 - 32 * q^32 - 39 * q^33 + 108 * q^34 + 10 * q^35 + 72 * q^36 + 423 * q^37 - 300 * q^38 - 32 * q^40 + 195 * q^41 + 30 * q^42 - 199 * q^43 - 104 * q^44 - 36 * q^45 + 374 * q^46 - 776 * q^47 + 48 * q^48 + 318 * q^49 + 242 * q^50 - 162 * q^51 + 1236 * q^53 - 270 * q^54 - 26 * q^55 - 40 * q^56 + 450 * q^57 + 26 * q^58 + 491 * q^59 + 48 * q^60 - 175 * q^61 - 208 * q^62 + 90 * q^63 + 128 * q^64 + 156 * q^66 + 817 * q^67 - 108 * q^68 - 561 * q^69 - 40 * q^70 + 79 * q^71 + 144 * q^72 - 460 * q^73 + 846 * q^74 - 363 * q^75 + 300 * q^76 - 130 * q^77 + 1528 * q^79 + 32 * q^80 - 81 * q^81 + 390 * q^82 + 1464 * q^83 + 60 * q^84 + 54 * q^85 + 796 * q^86 - 39 * q^87 + 104 * q^88 - 1041 * q^89 + 144 * q^90 - 1496 * q^92 + 312 * q^93 + 776 * q^94 - 150 * q^95 - 192 * q^96 - 97 * q^97 + 636 * q^98 + 468 * 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 −2.00000 3.00000 + 5.19615i −2.50000 4.33013i 8.00000 9.00000 + 15.5885i 2.00000 3.46410i
315.1 −1.00000 1.73205i 1.50000 + 2.59808i −2.00000 + 3.46410i −2.00000 3.00000 5.19615i −2.50000 + 4.33013i 8.00000 9.00000 15.5885i 2.00000 + 3.46410i
 $$n$$: 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.d 2
13.b even 2 1 26.4.c.a 2
13.c even 3 1 338.4.a.d 1
13.c even 3 1 inner 338.4.c.d 2
13.d odd 4 2 338.4.e.d 4
13.e even 6 1 26.4.c.a 2
13.e even 6 1 338.4.a.a 1
13.f odd 12 2 338.4.b.a 2
13.f odd 12 2 338.4.e.d 4
39.d odd 2 1 234.4.h.b 2
39.h odd 6 1 234.4.h.b 2
52.b odd 2 1 208.4.i.a 2
52.i odd 6 1 208.4.i.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.4.c.a 2 13.b even 2 1
26.4.c.a 2 13.e even 6 1
208.4.i.a 2 52.b odd 2 1
208.4.i.a 2 52.i odd 6 1
234.4.h.b 2 39.d odd 2 1
234.4.h.b 2 39.h odd 6 1
338.4.a.a 1 13.e even 6 1
338.4.a.d 1 13.c even 3 1
338.4.b.a 2 13.f odd 12 2
338.4.c.d 2 1.a even 1 1 trivial
338.4.c.d 2 13.c even 3 1 inner
338.4.e.d 4 13.d odd 4 2
338.4.e.d 4 13.f odd 12 2

## 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 $$T_{5} + 2$$ T5 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T + 4$$
$3$ $$T^{2} - 3T + 9$$
$5$ $$(T + 2)^{2}$$
$7$ $$T^{2} + 5T + 25$$
$11$ $$T^{2} - 13T + 169$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 27T + 729$$
$19$ $$T^{2} - 75T + 5625$$
$23$ $$T^{2} - 187T + 34969$$
$29$ $$T^{2} - 13T + 169$$
$31$ $$(T - 104)^{2}$$
$37$ $$T^{2} - 423T + 178929$$
$41$ $$T^{2} - 195T + 38025$$
$43$ $$T^{2} + 199T + 39601$$
$47$ $$(T + 388)^{2}$$
$53$ $$(T - 618)^{2}$$
$59$ $$T^{2} - 491T + 241081$$
$61$ $$T^{2} + 175T + 30625$$
$67$ $$T^{2} - 817T + 667489$$
$71$ $$T^{2} - 79T + 6241$$
$73$ $$(T + 230)^{2}$$
$79$ $$(T - 764)^{2}$$
$83$ $$(T - 732)^{2}$$
$89$ $$T^{2} + 1041 T + 1083681$$
$97$ $$T^{2} + 97T + 9409$$