# Properties

 Label 338.4.e.g Level $338$ Weight $4$ Character orbit 338.e Analytic conductor $19.943$ Analytic rank $0$ Dimension $8$ 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: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.45979465625856.49 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 109x^{6} + 8965x^{4} - 317844x^{2} + 8503056$$ x^8 - 109*x^6 + 8965*x^4 - 317844*x^2 + 8503056 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{2}$$ 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 basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{7} q^{2} + ( - \beta_{6} - \beta_{4} - 2 \beta_{2} + 1) q^{3} + 4 \beta_{2} q^{4} + ( - 5 \beta_{7} + \beta_{5} + \cdots - \beta_1) q^{5}+ \cdots + ( - 3 \beta_{6} - 31 \beta_{2}) q^{9}+O(q^{10})$$ q + b7 * q^2 + (-b6 - b4 - 2*b2 + 1) * q^3 + 4*b2 * q^4 + (-5*b7 + b5 - 5*b3 - b1) * q^5 + (-2*b3 - 2*b1) * q^6 + (-2*b3 - b1) * q^7 + (4*b7 + 4*b3) * q^8 + (-3*b6 - 31*b2) * q^9 $$q + \beta_{7} q^{2} + ( - \beta_{6} - \beta_{4} - 2 \beta_{2} + 1) q^{3} + 4 \beta_{2} q^{4} + ( - 5 \beta_{7} + \beta_{5} + \cdots - \beta_1) q^{5}+ \cdots + ( - 393 \beta_{7} + 150 \beta_{5} + \cdots - 150 \beta_1) q^{99}+O(q^{100})$$ q + b7 * q^2 + (-b6 - b4 - 2*b2 + 1) * q^3 + 4*b2 * q^4 + (-5*b7 + b5 - 5*b3 - b1) * q^5 + (-2*b3 - 2*b1) * q^6 + (-2*b3 - b1) * q^7 + (4*b7 + 4*b3) * q^8 + (-3*b6 - 31*b2) * q^9 + (-2*b6 - 2*b4 - 20*b2 + 18) * q^10 + (-3*b7 - 6*b5) * q^11 + (-4*b4 + 4) * q^12 + (-2*b4 + 6) * q^14 + (-37*b7 + 11*b5) * q^15 + (16*b2 - 16) * q^16 + (5*b6 - 4*b2) * q^17 + (-31*b7 + 6*b5 - 31*b3 - 6*b1) * q^18 + (27*b3 - 12*b1) * q^19 + (-20*b3 - 4*b1) * q^20 + (-31*b7 + 5*b5 - 31*b3 - 5*b1) * q^21 + (12*b6 - 12*b2) * q^22 + (10*b6 + 10*b4 - 44*b2 + 54) * q^23 + (8*b7 - 8*b5) * q^24 + (19*b4 - 10) * q^25 + (7*b4 - 163) * q^27 + (8*b7 - 4*b5) * q^28 + (14*b6 + 14*b4 + 194*b2 - 180) * q^29 + (-22*b6 - 148*b2) * q^30 + (-103*b7 + 2*b5 - 103*b3 - 2*b1) * q^31 + 16*b3 * q^32 - 156*b3 * q^33 + (-4*b7 - 10*b5 - 4*b3 + 10*b1) * q^34 + (-13*b6 - 94*b2) * q^35 + (-12*b6 - 12*b4 - 124*b2 + 112) * q^36 + (-b7 + 17*b5) * q^37 + (-24*b4 - 132) * q^38 + (-8*b4 + 72) * q^40 + (-68*b7 - 32*b5) * q^41 + (-10*b6 - 10*b4 - 124*b2 + 114) * q^42 + (25*b6 + 122*b2) * q^43 + (-12*b7 - 24*b5 - 12*b3 + 24*b1) * q^44 + (236*b3 + 58*b1) * q^45 + (-44*b3 + 20*b1) * q^46 + (-114*b7 - 9*b5 - 114*b3 + 9*b1) * q^47 + (16*b6 + 32*b2) * q^48 + (-7*b6 - 7*b4 + 273*b2 - 280) * q^49 + (-29*b7 + 38*b5) * q^50 + (-b4 + 261) * q^51 + (12*b4 + 18) * q^53 + (-170*b7 + 14*b5) * q^54 + (-48*b6 - 48*b4 - 264*b2 + 216) * q^55 + (8*b6 + 32*b2) * q^56 + (-270*b7 - 42*b5 - 270*b3 + 42*b1) * q^57 + (194*b3 + 28*b1) * q^58 + (-161*b3 + 20*b1) * q^59 + (-148*b7 + 44*b5 - 148*b3 - 44*b1) * q^60 + (-46*b6 + 118*b2) * q^61 + (-4*b6 - 4*b4 - 412*b2 + 408) * q^62 + (-143*b7 + 40*b5) * q^63 - 64 * q^64 + 624 * q^66 + (-11*b7 - 50*b5) * q^67 + (20*b6 + 20*b4 - 16*b2 + 36) * q^68 + (-34*b6 + 452*b2) * q^69 + (-94*b7 + 26*b5 - 94*b3 - 26*b1) * q^70 + (-30*b3 - 45*b1) * q^71 + (-124*b3 - 24*b1) * q^72 + (250*b7 - 50*b5 + 250*b3 + 50*b1) * q^73 + (-34*b6 - 4*b2) * q^74 + (48*b6 + 48*b4 + 1084*b2 - 1036) * q^75 + (-108*b7 - 48*b5) * q^76 + (-12*b4 + 288) * q^77 + (-74*b4 - 398) * q^79 + (80*b7 - 16*b5) * q^80 + (96*b6 + 96*b4 - 119*b2 + 215) * q^81 + (64*b6 - 272*b2) * q^82 + (-335*b7 + 88*b5 - 335*b3 - 88*b1) * q^83 + (-124*b3 - 20*b1) * q^84 + (-115*b3 - 41*b1) * q^85 + (122*b7 - 50*b5 + 122*b3 + 50*b1) * q^86 + (208*b6 + 1144*b2) * q^87 + (48*b6 + 48*b4 - 48*b2 + 96) * q^88 + (210*b7 - 138*b5) * q^89 + (116*b4 - 828) * q^90 + (40*b4 + 216) * q^92 + (-260*b7 + 208*b5) * q^93 + (18*b6 + 18*b4 - 456*b2 + 474) * q^94 + (-54*b6 - 108*b2) * q^95 + (32*b7 - 32*b5 + 32*b3 + 32*b1) * q^96 + (108*b3 - 132*b1) * q^97 + (273*b3 - 14*b1) * q^98 + (-393*b7 + 150*b5 - 393*b3 - 150*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 6 q^{3} + 16 q^{4} - 118 q^{9}+O(q^{10})$$ 8 * q + 6 * q^3 + 16 * q^4 - 118 * q^9 $$8 q + 6 q^{3} + 16 q^{4} - 118 q^{9} + 76 q^{10} + 48 q^{12} + 56 q^{14} - 64 q^{16} - 26 q^{17} - 72 q^{22} + 196 q^{23} - 156 q^{25} - 1332 q^{27} - 748 q^{29} - 548 q^{30} - 350 q^{35} + 472 q^{36} - 960 q^{38} + 608 q^{40} + 476 q^{42} + 438 q^{43} + 96 q^{48} - 1106 q^{49} + 2092 q^{51} + 96 q^{53} + 960 q^{55} + 112 q^{56} + 564 q^{61} + 1640 q^{62} - 512 q^{64} + 4992 q^{66} + 104 q^{68} + 1876 q^{69} + 52 q^{74} - 4240 q^{75} + 2352 q^{77} - 2888 q^{79} + 668 q^{81} - 1216 q^{82} + 4160 q^{87} + 288 q^{88} - 7088 q^{90} + 1568 q^{92} + 1860 q^{94} - 324 q^{95}+O(q^{100})$$ 8 * q + 6 * q^3 + 16 * q^4 - 118 * q^9 + 76 * q^10 + 48 * q^12 + 56 * q^14 - 64 * q^16 - 26 * q^17 - 72 * q^22 + 196 * q^23 - 156 * q^25 - 1332 * q^27 - 748 * q^29 - 548 * q^30 - 350 * q^35 + 472 * q^36 - 960 * q^38 + 608 * q^40 + 476 * q^42 + 438 * q^43 + 96 * q^48 - 1106 * q^49 + 2092 * q^51 + 96 * q^53 + 960 * q^55 + 112 * q^56 + 564 * q^61 + 1640 * q^62 - 512 * q^64 + 4992 * q^66 + 104 * q^68 + 1876 * q^69 + 52 * q^74 - 4240 * q^75 + 2352 * q^77 - 2888 * q^79 + 668 * q^81 - 1216 * q^82 + 4160 * q^87 + 288 * q^88 - 7088 * q^90 + 1568 * q^92 + 1860 * q^94 - 324 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 109x^{6} + 8965x^{4} - 317844x^{2} + 8503056$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -109\nu^{6} + 8965\nu^{4} - 977185\nu^{2} + 34644996 ) / 26141940$$ (-109*v^6 + 8965*v^4 - 977185*v^2 + 34644996) / 26141940 $$\beta_{3}$$ $$=$$ $$( \nu^{7} + 175231\nu ) / 242055$$ (v^7 + 175231*v) / 242055 $$\beta_{4}$$ $$=$$ $$( -\nu^{6} - 175231 ) / 8965$$ (-v^6 - 175231) / 8965 $$\beta_{5}$$ $$=$$ $$( -109\nu^{7} + 8965\nu^{5} - 977185\nu^{3} + 34644996\nu ) / 26141940$$ (-109*v^7 + 8965*v^5 - 977185*v^3 + 34644996*v) / 26141940 $$\beta_{6}$$ $$=$$ $$( -\nu^{6} + 163\nu^{4} - 8965\nu^{2} + 317844 ) / 8802$$ (-v^6 + 163*v^4 - 8965*v^2 + 317844) / 8802 $$\beta_{7}$$ $$=$$ $$( 3079\nu^{7} - 493075\nu^{5} + 27603235\nu^{3} - 978641676\nu ) / 705832380$$ (3079*v^7 - 493075*v^5 + 27603235*v^3 - 978641676*v) / 705832380
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{6} + \beta_{4} - 54\beta_{2} + 55$$ b6 + b4 - 54*b2 + 55 $$\nu^{3}$$ $$=$$ $$-27\beta_{7} - 55\beta_{5} - 27\beta_{3} + 55\beta_1$$ -27*b7 - 55*b5 - 27*b3 + 55*b1 $$\nu^{4}$$ $$=$$ $$109\beta_{6} - 2970\beta_{2}$$ 109*b6 - 2970*b2 $$\nu^{5}$$ $$=$$ $$-2943\beta_{7} - 3079\beta_{5}$$ -2943*b7 - 3079*b5 $$\nu^{6}$$ $$=$$ $$-8965\beta_{4} - 175231$$ -8965*b4 - 175231 $$\nu^{7}$$ $$=$$ $$242055\beta_{3} - 175231\beta_1$$ 242055*b3 - 175231*b1

## Character values

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

 $$n$$ $$171$$ $$\chi(n)$$ $$\beta_{2}$$

## 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
 −6.81169 − 3.93273i 5.94566 + 3.43273i 6.81169 + 3.93273i −5.94566 − 3.43273i −6.81169 + 3.93273i 5.94566 − 3.43273i 6.81169 − 3.93273i −5.94566 + 3.43273i
−1.73205 + 1.00000i −2.93273 5.07964i 2.00000 3.46410i 2.13454i 10.1593 + 5.86546i 3.34759 + 1.93273i 8.00000i −3.70181 + 6.41172i 2.13454 + 3.69713i
23.2 −1.73205 + 1.00000i 4.43273 + 7.67771i 2.00000 3.46410i 16.8655i −15.3554 8.86546i −9.40976 5.43273i 8.00000i −25.7982 + 44.6838i 16.8655 + 29.2118i
23.3 1.73205 1.00000i −2.93273 5.07964i 2.00000 3.46410i 2.13454i −10.1593 5.86546i −3.34759 1.93273i 8.00000i −3.70181 + 6.41172i 2.13454 + 3.69713i
23.4 1.73205 1.00000i 4.43273 + 7.67771i 2.00000 3.46410i 16.8655i 15.3554 + 8.86546i 9.40976 + 5.43273i 8.00000i −25.7982 + 44.6838i 16.8655 + 29.2118i
147.1 −1.73205 1.00000i −2.93273 + 5.07964i 2.00000 + 3.46410i 2.13454i 10.1593 5.86546i 3.34759 1.93273i 8.00000i −3.70181 6.41172i 2.13454 3.69713i
147.2 −1.73205 1.00000i 4.43273 7.67771i 2.00000 + 3.46410i 16.8655i −15.3554 + 8.86546i −9.40976 + 5.43273i 8.00000i −25.7982 44.6838i 16.8655 29.2118i
147.3 1.73205 + 1.00000i −2.93273 + 5.07964i 2.00000 + 3.46410i 2.13454i −10.1593 + 5.86546i −3.34759 + 1.93273i 8.00000i −3.70181 6.41172i 2.13454 3.69713i
147.4 1.73205 + 1.00000i 4.43273 7.67771i 2.00000 + 3.46410i 16.8655i 15.3554 8.86546i 9.40976 5.43273i 8.00000i −25.7982 44.6838i 16.8655 29.2118i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 23.4 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.g 8
13.b even 2 1 inner 338.4.e.g 8
13.c even 3 1 26.4.b.a 4
13.c even 3 1 inner 338.4.e.g 8
13.d odd 4 1 338.4.c.h 4
13.d odd 4 1 338.4.c.i 4
13.e even 6 1 26.4.b.a 4
13.e even 6 1 inner 338.4.e.g 8
13.f odd 12 1 338.4.a.f 2
13.f odd 12 1 338.4.a.i 2
13.f odd 12 1 338.4.c.h 4
13.f odd 12 1 338.4.c.i 4
39.h odd 6 1 234.4.b.b 4
39.i odd 6 1 234.4.b.b 4
52.i odd 6 1 208.4.f.d 4
52.j odd 6 1 208.4.f.d 4
65.l even 6 1 650.4.d.d 4
65.n even 6 1 650.4.d.d 4
65.q odd 12 1 650.4.c.e 4
65.q odd 12 1 650.4.c.f 4
65.r odd 12 1 650.4.c.e 4
65.r odd 12 1 650.4.c.f 4
104.n odd 6 1 832.4.f.h 4
104.p odd 6 1 832.4.f.h 4
104.r even 6 1 832.4.f.j 4
104.s even 6 1 832.4.f.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.4.b.a 4 13.c even 3 1
26.4.b.a 4 13.e even 6 1
208.4.f.d 4 52.i odd 6 1
208.4.f.d 4 52.j odd 6 1
234.4.b.b 4 39.h odd 6 1
234.4.b.b 4 39.i odd 6 1
338.4.a.f 2 13.f odd 12 1
338.4.a.i 2 13.f odd 12 1
338.4.c.h 4 13.d odd 4 1
338.4.c.h 4 13.f odd 12 1
338.4.c.i 4 13.d odd 4 1
338.4.c.i 4 13.f odd 12 1
338.4.e.g 8 1.a even 1 1 trivial
338.4.e.g 8 13.b even 2 1 inner
338.4.e.g 8 13.c even 3 1 inner
338.4.e.g 8 13.e even 6 1 inner
650.4.c.e 4 65.q odd 12 1
650.4.c.e 4 65.r odd 12 1
650.4.c.f 4 65.q odd 12 1
650.4.c.f 4 65.r odd 12 1
650.4.d.d 4 65.l even 6 1
650.4.d.d 4 65.n even 6 1
832.4.f.h 4 104.n odd 6 1
832.4.f.h 4 104.p odd 6 1
832.4.f.j 4 104.r even 6 1
832.4.f.j 4 104.s even 6 1

## Hecke kernels

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

 $$T_{3}^{4} - 3T_{3}^{3} + 61T_{3}^{2} + 156T_{3} + 2704$$ T3^4 - 3*T3^3 + 61*T3^2 + 156*T3 + 2704 $$T_{5}^{4} + 289T_{5}^{2} + 1296$$ T5^4 + 289*T5^2 + 1296

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - 4 T^{2} + 16)^{2}$$
$3$ $$(T^{4} - 3 T^{3} + \cdots + 2704)^{2}$$
$5$ $$(T^{4} + 289 T^{2} + 1296)^{2}$$
$7$ $$T^{8} - 133 T^{6} + \cdots + 3111696$$
$11$ $$T^{8} + \cdots + 12280707219456$$
$13$ $$T^{8}$$
$17$ $$(T^{4} + 13 T^{3} + \cdots + 1726596)^{2}$$
$19$ $$T^{8} + \cdots + 314741094011136$$
$23$ $$(T^{4} - 98 T^{3} + \cdots + 9144576)^{2}$$
$29$ $$(T^{4} + 374 T^{3} + \cdots + 592240896)^{2}$$
$31$ $$(T^{4} + 84484 T^{2} + 1747908864)^{2}$$
$37$ $$T^{8} + \cdots + 59\!\cdots\!16$$
$41$ $$T^{8} + \cdots + 11\!\cdots\!16$$
$43$ $$(T^{4} - 219 T^{3} + \cdots + 480311056)^{2}$$
$47$ $$(T^{4} + 116901 T^{2} + 2466314244)^{2}$$
$53$ $$(T^{2} - 24 T - 7668)^{4}$$
$59$ $$T^{8} + \cdots + 61\!\cdots\!76$$
$61$ $$(T^{4} - 282 T^{3} + \cdots + 9008287744)^{2}$$
$67$ $$T^{8} + \cdots + 31\!\cdots\!36$$
$71$ $$T^{8} + \cdots + 13\!\cdots\!00$$
$73$ $$(T^{4} + 722500 T^{2} + 8100000000)^{2}$$
$79$ $$(T^{2} + 722 T - 166752)^{4}$$
$83$ $$(T^{4} + 1623976 T^{2} + 797271696)^{2}$$
$89$ $$T^{8} + \cdots + 68\!\cdots\!16$$
$97$ $$T^{8} + \cdots + 56\!\cdots\!56$$