# Properties

 Label 338.4.e.a Level $338$ Weight $4$ Character orbit 338.e Analytic conductor $19.943$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (4 \beta_{2} - 4) q^{3} + 4 \beta_{2} q^{4} + 9 \beta_{3} q^{5} + (4 \beta_{3} - 4 \beta_1) q^{6} + ( - 10 \beta_{3} + 10 \beta_1) q^{7} + 4 \beta_{3} q^{8} + 11 \beta_{2} q^{9}+O(q^{10})$$ q + b1 * q^2 + (4*b2 - 4) * q^3 + 4*b2 * q^4 + 9*b3 * q^5 + (4*b3 - 4*b1) * q^6 + (-10*b3 + 10*b1) * q^7 + 4*b3 * q^8 + 11*b2 * q^9 $$q + \beta_1 q^{2} + (4 \beta_{2} - 4) q^{3} + 4 \beta_{2} q^{4} + 9 \beta_{3} q^{5} + (4 \beta_{3} - 4 \beta_1) q^{6} + ( - 10 \beta_{3} + 10 \beta_1) q^{7} + 4 \beta_{3} q^{8} + 11 \beta_{2} q^{9} + (36 \beta_{2} - 36) q^{10} + 24 \beta_1 q^{11} - 16 q^{12} + 40 q^{14} - 36 \beta_1 q^{15} + (16 \beta_{2} - 16) q^{16} + 66 \beta_{2} q^{17} + 11 \beta_{3} q^{18} + ( - 8 \beta_{3} + 8 \beta_1) q^{19} + (36 \beta_{3} - 36 \beta_1) q^{20} + 40 \beta_{3} q^{21} + 96 \beta_{2} q^{22} + ( - 168 \beta_{2} + 168) q^{23} - 16 \beta_1 q^{24} - 199 q^{25} - 152 q^{27} + 40 \beta_1 q^{28} + (6 \beta_{2} - 6) q^{29} - 144 \beta_{2} q^{30} - 10 \beta_{3} q^{31} + (16 \beta_{3} - 16 \beta_1) q^{32} + (96 \beta_{3} - 96 \beta_1) q^{33} + 66 \beta_{3} q^{34} + 360 \beta_{2} q^{35} + (44 \beta_{2} - 44) q^{36} - 127 \beta_1 q^{37} + 32 q^{38} - 144 q^{40} - 195 \beta_1 q^{41} + (160 \beta_{2} - 160) q^{42} - 124 \beta_{2} q^{43} + 96 \beta_{3} q^{44} + (99 \beta_{3} - 99 \beta_1) q^{45} + ( - 168 \beta_{3} + 168 \beta_1) q^{46} - 234 \beta_{3} q^{47} - 64 \beta_{2} q^{48} + ( - 57 \beta_{2} + 57) q^{49} - 199 \beta_1 q^{50} - 264 q^{51} + 558 q^{53} - 152 \beta_1 q^{54} + (864 \beta_{2} - 864) q^{55} + 160 \beta_{2} q^{56} + 32 \beta_{3} q^{57} + (6 \beta_{3} - 6 \beta_1) q^{58} + (48 \beta_{3} - 48 \beta_1) q^{59} - 144 \beta_{3} q^{60} + 826 \beta_{2} q^{61} + ( - 40 \beta_{2} + 40) q^{62} + 110 \beta_1 q^{63} - 64 q^{64} - 384 q^{66} - 80 \beta_1 q^{67} + (264 \beta_{2} - 264) q^{68} + 672 \beta_{2} q^{69} + 360 \beta_{3} q^{70} + ( - 210 \beta_{3} + 210 \beta_1) q^{71} + (44 \beta_{3} - 44 \beta_1) q^{72} + 181 \beta_{3} q^{73} - 508 \beta_{2} q^{74} + ( - 796 \beta_{2} + 796) q^{75} + 32 \beta_1 q^{76} + 960 q^{77} + 776 q^{79} - 144 \beta_1 q^{80} + ( - 311 \beta_{2} + 311) q^{81} - 780 \beta_{2} q^{82} + (160 \beta_{3} - 160 \beta_1) q^{84} + (594 \beta_{3} - 594 \beta_1) q^{85} - 124 \beta_{3} q^{86} - 24 \beta_{2} q^{87} + (384 \beta_{2} - 384) q^{88} - 813 \beta_1 q^{89} - 396 q^{90} + 672 q^{92} + 40 \beta_1 q^{93} + ( - 936 \beta_{2} + 936) q^{94} + 288 \beta_{2} q^{95} - 64 \beta_{3} q^{96} + ( - 647 \beta_{3} + 647 \beta_1) q^{97} + ( - 57 \beta_{3} + 57 \beta_1) q^{98} + 264 \beta_{3} q^{99}+O(q^{100})$$ q + b1 * q^2 + (4*b2 - 4) * q^3 + 4*b2 * q^4 + 9*b3 * q^5 + (4*b3 - 4*b1) * q^6 + (-10*b3 + 10*b1) * q^7 + 4*b3 * q^8 + 11*b2 * q^9 + (36*b2 - 36) * q^10 + 24*b1 * q^11 - 16 * q^12 + 40 * q^14 - 36*b1 * q^15 + (16*b2 - 16) * q^16 + 66*b2 * q^17 + 11*b3 * q^18 + (-8*b3 + 8*b1) * q^19 + (36*b3 - 36*b1) * q^20 + 40*b3 * q^21 + 96*b2 * q^22 + (-168*b2 + 168) * q^23 - 16*b1 * q^24 - 199 * q^25 - 152 * q^27 + 40*b1 * q^28 + (6*b2 - 6) * q^29 - 144*b2 * q^30 - 10*b3 * q^31 + (16*b3 - 16*b1) * q^32 + (96*b3 - 96*b1) * q^33 + 66*b3 * q^34 + 360*b2 * q^35 + (44*b2 - 44) * q^36 - 127*b1 * q^37 + 32 * q^38 - 144 * q^40 - 195*b1 * q^41 + (160*b2 - 160) * q^42 - 124*b2 * q^43 + 96*b3 * q^44 + (99*b3 - 99*b1) * q^45 + (-168*b3 + 168*b1) * q^46 - 234*b3 * q^47 - 64*b2 * q^48 + (-57*b2 + 57) * q^49 - 199*b1 * q^50 - 264 * q^51 + 558 * q^53 - 152*b1 * q^54 + (864*b2 - 864) * q^55 + 160*b2 * q^56 + 32*b3 * q^57 + (6*b3 - 6*b1) * q^58 + (48*b3 - 48*b1) * q^59 - 144*b3 * q^60 + 826*b2 * q^61 + (-40*b2 + 40) * q^62 + 110*b1 * q^63 - 64 * q^64 - 384 * q^66 - 80*b1 * q^67 + (264*b2 - 264) * q^68 + 672*b2 * q^69 + 360*b3 * q^70 + (-210*b3 + 210*b1) * q^71 + (44*b3 - 44*b1) * q^72 + 181*b3 * q^73 - 508*b2 * q^74 + (-796*b2 + 796) * q^75 + 32*b1 * q^76 + 960 * q^77 + 776 * q^79 - 144*b1 * q^80 + (-311*b2 + 311) * q^81 - 780*b2 * q^82 + (160*b3 - 160*b1) * q^84 + (594*b3 - 594*b1) * q^85 - 124*b3 * q^86 - 24*b2 * q^87 + (384*b2 - 384) * q^88 - 813*b1 * q^89 - 396 * q^90 + 672 * q^92 + 40*b1 * q^93 + (-936*b2 + 936) * q^94 + 288*b2 * q^95 - 64*b3 * q^96 + (-647*b3 + 647*b1) * q^97 + (-57*b3 + 57*b1) * q^98 + 264*b3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{3} + 8 q^{4} + 22 q^{9}+O(q^{10})$$ 4 * q - 8 * q^3 + 8 * q^4 + 22 * q^9 $$4 q - 8 q^{3} + 8 q^{4} + 22 q^{9} - 72 q^{10} - 64 q^{12} + 160 q^{14} - 32 q^{16} + 132 q^{17} + 192 q^{22} + 336 q^{23} - 796 q^{25} - 608 q^{27} - 12 q^{29} - 288 q^{30} + 720 q^{35} - 88 q^{36} + 128 q^{38} - 576 q^{40} - 320 q^{42} - 248 q^{43} - 128 q^{48} + 114 q^{49} - 1056 q^{51} + 2232 q^{53} - 1728 q^{55} + 320 q^{56} + 1652 q^{61} + 80 q^{62} - 256 q^{64} - 1536 q^{66} - 528 q^{68} + 1344 q^{69} - 1016 q^{74} + 1592 q^{75} + 3840 q^{77} + 3104 q^{79} + 622 q^{81} - 1560 q^{82} - 48 q^{87} - 768 q^{88} - 1584 q^{90} + 2688 q^{92} + 1872 q^{94} + 576 q^{95}+O(q^{100})$$ 4 * q - 8 * q^3 + 8 * q^4 + 22 * q^9 - 72 * q^10 - 64 * q^12 + 160 * q^14 - 32 * q^16 + 132 * q^17 + 192 * q^22 + 336 * q^23 - 796 * q^25 - 608 * q^27 - 12 * q^29 - 288 * q^30 + 720 * q^35 - 88 * q^36 + 128 * q^38 - 576 * q^40 - 320 * q^42 - 248 * q^43 - 128 * q^48 + 114 * q^49 - 1056 * q^51 + 2232 * q^53 - 1728 * q^55 + 320 * q^56 + 1652 * q^61 + 80 * q^62 - 256 * q^64 - 1536 * q^66 - 528 * q^68 + 1344 * q^69 - 1016 * q^74 + 1592 * q^75 + 3840 * q^77 + 3104 * q^79 + 622 * q^81 - 1560 * q^82 - 48 * q^87 - 768 * q^88 - 1584 * q^90 + 2688 * q^92 + 1872 * q^94 + 576 * q^95

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$2\zeta_{12}$$ 2*v $$\beta_{2}$$ $$=$$ $$\zeta_{12}^{2}$$ v^2 $$\beta_{3}$$ $$=$$ $$2\zeta_{12}^{3}$$ 2*v^3
 $$\zeta_{12}$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\zeta_{12}^{2}$$ $$=$$ $$\beta_{2}$$ b2 $$\zeta_{12}^{3}$$ $$=$$ $$( \beta_{3} ) / 2$$ (b3) / 2

## 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
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
−1.73205 + 1.00000i −2.00000 3.46410i 2.00000 3.46410i 18.0000i 6.92820 + 4.00000i −17.3205 10.0000i 8.00000i 5.50000 9.52628i −18.0000 31.1769i
23.2 1.73205 1.00000i −2.00000 3.46410i 2.00000 3.46410i 18.0000i −6.92820 4.00000i 17.3205 + 10.0000i 8.00000i 5.50000 9.52628i −18.0000 31.1769i
147.1 −1.73205 1.00000i −2.00000 + 3.46410i 2.00000 + 3.46410i 18.0000i 6.92820 4.00000i −17.3205 + 10.0000i 8.00000i 5.50000 + 9.52628i −18.0000 + 31.1769i
147.2 1.73205 + 1.00000i −2.00000 + 3.46410i 2.00000 + 3.46410i 18.0000i −6.92820 + 4.00000i 17.3205 10.0000i 8.00000i 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.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.a 4
13.b even 2 1 inner 338.4.e.a 4
13.c even 3 1 338.4.b.d 2
13.c even 3 1 inner 338.4.e.a 4
13.d odd 4 1 338.4.c.a 2
13.d odd 4 1 338.4.c.e 2
13.e even 6 1 338.4.b.d 2
13.e even 6 1 inner 338.4.e.a 4
13.f odd 12 1 26.4.a.c 1
13.f odd 12 1 338.4.a.c 1
13.f odd 12 1 338.4.c.a 2
13.f odd 12 1 338.4.c.e 2
39.k even 12 1 234.4.a.e 1
52.l even 12 1 208.4.a.b 1
65.o even 12 1 650.4.b.f 2
65.s odd 12 1 650.4.a.b 1
65.t even 12 1 650.4.b.f 2
91.bc even 12 1 1274.4.a.d 1
104.u even 12 1 832.4.a.o 1
104.x odd 12 1 832.4.a.d 1
156.v odd 12 1 1872.4.a.q 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.4.a.c 1 13.f odd 12 1
208.4.a.b 1 52.l even 12 1
234.4.a.e 1 39.k even 12 1
338.4.a.c 1 13.f odd 12 1
338.4.b.d 2 13.c even 3 1
338.4.b.d 2 13.e even 6 1
338.4.c.a 2 13.d odd 4 1
338.4.c.a 2 13.f odd 12 1
338.4.c.e 2 13.d odd 4 1
338.4.c.e 2 13.f odd 12 1
338.4.e.a 4 1.a even 1 1 trivial
338.4.e.a 4 13.b even 2 1 inner
338.4.e.a 4 13.c even 3 1 inner
338.4.e.a 4 13.e even 6 1 inner
650.4.a.b 1 65.s odd 12 1
650.4.b.f 2 65.o even 12 1
650.4.b.f 2 65.t even 12 1
832.4.a.d 1 104.x odd 12 1
832.4.a.o 1 104.u even 12 1
1274.4.a.d 1 91.bc even 12 1
1872.4.a.q 1 156.v odd 12 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 $$T_{5}^{2} + 324$$ T5^2 + 324

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 4T^{2} + 16$$
$3$ $$(T^{2} + 4 T + 16)^{2}$$
$5$ $$(T^{2} + 324)^{2}$$
$7$ $$T^{4} - 400 T^{2} + 160000$$
$11$ $$T^{4} - 2304 T^{2} + 5308416$$
$13$ $$T^{4}$$
$17$ $$(T^{2} - 66 T + 4356)^{2}$$
$19$ $$T^{4} - 256 T^{2} + 65536$$
$23$ $$(T^{2} - 168 T + 28224)^{2}$$
$29$ $$(T^{2} + 6 T + 36)^{2}$$
$31$ $$(T^{2} + 400)^{2}$$
$37$ $$T^{4} + \cdots + 4162314256$$
$41$ $$T^{4} + \cdots + 23134410000$$
$43$ $$(T^{2} + 124 T + 15376)^{2}$$
$47$ $$(T^{2} + 219024)^{2}$$
$53$ $$(T - 558)^{4}$$
$59$ $$T^{4} - 9216 T^{2} + 84934656$$
$61$ $$(T^{2} - 826 T + 682276)^{2}$$
$67$ $$T^{4} - 25600 T^{2} + 655360000$$
$71$ $$T^{4} + \cdots + 31116960000$$
$73$ $$(T^{2} + 131044)^{2}$$
$79$ $$(T - 776)^{4}$$
$83$ $$T^{4}$$
$89$ $$T^{4} + \cdots + 6990080303376$$
$97$ $$T^{4} + \cdots + 2803735918096$$