# Properties

 Label 338.4.e.d 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ 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 primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{12} q^{2} + ( - 3 \zeta_{12}^{2} + 3) q^{3} + 4 \zeta_{12}^{2} q^{4} + 2 \zeta_{12}^{3} q^{5} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}) q^{6} + ( - 5 \zeta_{12}^{3} + 5 \zeta_{12}) q^{7} + 8 \zeta_{12}^{3} q^{8} + 18 \zeta_{12}^{2} q^{9}+O(q^{10})$$ q + 2*z * q^2 + (-3*z^2 + 3) * q^3 + 4*z^2 * q^4 + 2*z^3 * q^5 + (-6*z^3 + 6*z) * q^6 + (-5*z^3 + 5*z) * q^7 + 8*z^3 * q^8 + 18*z^2 * q^9 $$q + 2 \zeta_{12} q^{2} + ( - 3 \zeta_{12}^{2} + 3) q^{3} + 4 \zeta_{12}^{2} q^{4} + 2 \zeta_{12}^{3} q^{5} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}) q^{6} + ( - 5 \zeta_{12}^{3} + 5 \zeta_{12}) q^{7} + 8 \zeta_{12}^{3} q^{8} + 18 \zeta_{12}^{2} q^{9} + (4 \zeta_{12}^{2} - 4) q^{10} + 13 \zeta_{12} q^{11} + 12 q^{12} + 10 q^{14} + 6 \zeta_{12} q^{15} + (16 \zeta_{12}^{2} - 16) q^{16} + 27 \zeta_{12}^{2} q^{17} + 36 \zeta_{12}^{3} q^{18} + ( - 75 \zeta_{12}^{3} + 75 \zeta_{12}) q^{19} + (8 \zeta_{12}^{3} - 8 \zeta_{12}) q^{20} - 15 \zeta_{12}^{3} q^{21} + 26 \zeta_{12}^{2} q^{22} + (187 \zeta_{12}^{2} - 187) q^{23} + 24 \zeta_{12} q^{24} + 121 q^{25} + 135 q^{27} + 20 \zeta_{12} q^{28} + ( - 13 \zeta_{12}^{2} + 13) q^{29} + 12 \zeta_{12}^{2} q^{30} - 104 \zeta_{12}^{3} q^{31} + (32 \zeta_{12}^{3} - 32 \zeta_{12}) q^{32} + ( - 39 \zeta_{12}^{3} + 39 \zeta_{12}) q^{33} + 54 \zeta_{12}^{3} q^{34} + 10 \zeta_{12}^{2} q^{35} + (72 \zeta_{12}^{2} - 72) q^{36} + 423 \zeta_{12} q^{37} + 150 q^{38} - 16 q^{40} - 195 \zeta_{12} q^{41} + ( - 30 \zeta_{12}^{2} + 30) q^{42} + 199 \zeta_{12}^{2} q^{43} + 52 \zeta_{12}^{3} q^{44} + (36 \zeta_{12}^{3} - 36 \zeta_{12}) q^{45} + (374 \zeta_{12}^{3} - 374 \zeta_{12}) q^{46} - 388 \zeta_{12}^{3} q^{47} + 48 \zeta_{12}^{2} q^{48} + (318 \zeta_{12}^{2} - 318) q^{49} + 242 \zeta_{12} q^{50} + 81 q^{51} + 618 q^{53} + 270 \zeta_{12} q^{54} + (26 \zeta_{12}^{2} - 26) q^{55} + 40 \zeta_{12}^{2} q^{56} - 225 \zeta_{12}^{3} q^{57} + ( - 26 \zeta_{12}^{3} + 26 \zeta_{12}) q^{58} + (491 \zeta_{12}^{3} - 491 \zeta_{12}) q^{59} + 24 \zeta_{12}^{3} q^{60} - 175 \zeta_{12}^{2} q^{61} + ( - 208 \zeta_{12}^{2} + 208) q^{62} + 90 \zeta_{12} q^{63} - 64 q^{64} + 78 q^{66} - 817 \zeta_{12} q^{67} + (108 \zeta_{12}^{2} - 108) q^{68} + 561 \zeta_{12}^{2} q^{69} + 20 \zeta_{12}^{3} q^{70} + ( - 79 \zeta_{12}^{3} + 79 \zeta_{12}) q^{71} + (144 \zeta_{12}^{3} - 144 \zeta_{12}) q^{72} - 230 \zeta_{12}^{3} q^{73} + 846 \zeta_{12}^{2} q^{74} + ( - 363 \zeta_{12}^{2} + 363) q^{75} + 300 \zeta_{12} q^{76} + 65 q^{77} + 764 q^{79} - 32 \zeta_{12} q^{80} + (81 \zeta_{12}^{2} - 81) q^{81} - 390 \zeta_{12}^{2} q^{82} - 732 \zeta_{12}^{3} q^{83} + ( - 60 \zeta_{12}^{3} + 60 \zeta_{12}) q^{84} + (54 \zeta_{12}^{3} - 54 \zeta_{12}) q^{85} + 398 \zeta_{12}^{3} q^{86} - 39 \zeta_{12}^{2} q^{87} + (104 \zeta_{12}^{2} - 104) q^{88} - 1041 \zeta_{12} q^{89} - 72 q^{90} - 748 q^{92} - 312 \zeta_{12} q^{93} + ( - 776 \zeta_{12}^{2} + 776) q^{94} + 150 \zeta_{12}^{2} q^{95} + 96 \zeta_{12}^{3} q^{96} + (97 \zeta_{12}^{3} - 97 \zeta_{12}) q^{97} + (636 \zeta_{12}^{3} - 636 \zeta_{12}) q^{98} + 234 \zeta_{12}^{3} q^{99} +O(q^{100})$$ q + 2*z * q^2 + (-3*z^2 + 3) * q^3 + 4*z^2 * q^4 + 2*z^3 * q^5 + (-6*z^3 + 6*z) * q^6 + (-5*z^3 + 5*z) * q^7 + 8*z^3 * q^8 + 18*z^2 * q^9 + (4*z^2 - 4) * q^10 + 13*z * q^11 + 12 * q^12 + 10 * q^14 + 6*z * q^15 + (16*z^2 - 16) * q^16 + 27*z^2 * q^17 + 36*z^3 * q^18 + (-75*z^3 + 75*z) * q^19 + (8*z^3 - 8*z) * q^20 - 15*z^3 * q^21 + 26*z^2 * q^22 + (187*z^2 - 187) * q^23 + 24*z * q^24 + 121 * q^25 + 135 * q^27 + 20*z * q^28 + (-13*z^2 + 13) * q^29 + 12*z^2 * q^30 - 104*z^3 * q^31 + (32*z^3 - 32*z) * q^32 + (-39*z^3 + 39*z) * q^33 + 54*z^3 * q^34 + 10*z^2 * q^35 + (72*z^2 - 72) * q^36 + 423*z * q^37 + 150 * q^38 - 16 * q^40 - 195*z * q^41 + (-30*z^2 + 30) * q^42 + 199*z^2 * q^43 + 52*z^3 * q^44 + (36*z^3 - 36*z) * q^45 + (374*z^3 - 374*z) * q^46 - 388*z^3 * q^47 + 48*z^2 * q^48 + (318*z^2 - 318) * q^49 + 242*z * q^50 + 81 * q^51 + 618 * q^53 + 270*z * q^54 + (26*z^2 - 26) * q^55 + 40*z^2 * q^56 - 225*z^3 * q^57 + (-26*z^3 + 26*z) * q^58 + (491*z^3 - 491*z) * q^59 + 24*z^3 * q^60 - 175*z^2 * q^61 + (-208*z^2 + 208) * q^62 + 90*z * q^63 - 64 * q^64 + 78 * q^66 - 817*z * q^67 + (108*z^2 - 108) * q^68 + 561*z^2 * q^69 + 20*z^3 * q^70 + (-79*z^3 + 79*z) * q^71 + (144*z^3 - 144*z) * q^72 - 230*z^3 * q^73 + 846*z^2 * q^74 + (-363*z^2 + 363) * q^75 + 300*z * q^76 + 65 * q^77 + 764 * q^79 - 32*z * q^80 + (81*z^2 - 81) * q^81 - 390*z^2 * q^82 - 732*z^3 * q^83 + (-60*z^3 + 60*z) * q^84 + (54*z^3 - 54*z) * q^85 + 398*z^3 * q^86 - 39*z^2 * q^87 + (104*z^2 - 104) * q^88 - 1041*z * q^89 - 72 * q^90 - 748 * q^92 - 312*z * q^93 + (-776*z^2 + 776) * q^94 + 150*z^2 * q^95 + 96*z^3 * q^96 + (97*z^3 - 97*z) * q^97 + (636*z^3 - 636*z) * q^98 + 234*z^3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 6 q^{3} + 8 q^{4} + 36 q^{9}+O(q^{10})$$ 4 * q + 6 * q^3 + 8 * q^4 + 36 * q^9 $$4 q + 6 q^{3} + 8 q^{4} + 36 q^{9} - 8 q^{10} + 48 q^{12} + 40 q^{14} - 32 q^{16} + 54 q^{17} + 52 q^{22} - 374 q^{23} + 484 q^{25} + 540 q^{27} + 26 q^{29} + 24 q^{30} + 20 q^{35} - 144 q^{36} + 600 q^{38} - 64 q^{40} + 60 q^{42} + 398 q^{43} + 96 q^{48} - 636 q^{49} + 324 q^{51} + 2472 q^{53} - 52 q^{55} + 80 q^{56} - 350 q^{61} + 416 q^{62} - 256 q^{64} + 312 q^{66} - 216 q^{68} + 1122 q^{69} + 1692 q^{74} + 726 q^{75} + 260 q^{77} + 3056 q^{79} - 162 q^{81} - 780 q^{82} - 78 q^{87} - 208 q^{88} - 288 q^{90} - 2992 q^{92} + 1552 q^{94} + 300 q^{95}+O(q^{100})$$ 4 * q + 6 * q^3 + 8 * q^4 + 36 * q^9 - 8 * q^10 + 48 * q^12 + 40 * q^14 - 32 * q^16 + 54 * q^17 + 52 * q^22 - 374 * q^23 + 484 * q^25 + 540 * q^27 + 26 * q^29 + 24 * q^30 + 20 * q^35 - 144 * q^36 + 600 * q^38 - 64 * q^40 + 60 * q^42 + 398 * q^43 + 96 * q^48 - 636 * q^49 + 324 * q^51 + 2472 * q^53 - 52 * q^55 + 80 * q^56 - 350 * q^61 + 416 * q^62 - 256 * q^64 + 312 * q^66 - 216 * q^68 + 1122 * q^69 + 1692 * q^74 + 726 * q^75 + 260 * q^77 + 3056 * q^79 - 162 * q^81 - 780 * q^82 - 78 * q^87 - 208 * q^88 - 288 * q^90 - 2992 * q^92 + 1552 * q^94 + 300 * q^95

## Character values

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

 $$n$$ $$171$$ $$\chi(n)$$ $$\zeta_{12}^{2}$$

## 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 1.50000 + 2.59808i 2.00000 3.46410i 2.00000i −5.19615 3.00000i −4.33013 2.50000i 8.00000i 9.00000 15.5885i −2.00000 3.46410i
23.2 1.73205 1.00000i 1.50000 + 2.59808i 2.00000 3.46410i 2.00000i 5.19615 + 3.00000i 4.33013 + 2.50000i 8.00000i 9.00000 15.5885i −2.00000 3.46410i
147.1 −1.73205 1.00000i 1.50000 2.59808i 2.00000 + 3.46410i 2.00000i −5.19615 + 3.00000i −4.33013 + 2.50000i 8.00000i 9.00000 + 15.5885i −2.00000 + 3.46410i
147.2 1.73205 + 1.00000i 1.50000 2.59808i 2.00000 + 3.46410i 2.00000i 5.19615 3.00000i 4.33013 2.50000i 8.00000i 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.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.d 4
13.b even 2 1 inner 338.4.e.d 4
13.c even 3 1 338.4.b.a 2
13.c even 3 1 inner 338.4.e.d 4
13.d odd 4 1 26.4.c.a 2
13.d odd 4 1 338.4.c.d 2
13.e even 6 1 338.4.b.a 2
13.e even 6 1 inner 338.4.e.d 4
13.f odd 12 1 26.4.c.a 2
13.f odd 12 1 338.4.a.a 1
13.f odd 12 1 338.4.a.d 1
13.f odd 12 1 338.4.c.d 2
39.f even 4 1 234.4.h.b 2
39.k even 12 1 234.4.h.b 2
52.f even 4 1 208.4.i.a 2
52.l even 12 1 208.4.i.a 2

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

## 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 $$T_{5}^{2} + 4$$ T5^2 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 4T^{2} + 16$$
$3$ $$(T^{2} - 3 T + 9)^{2}$$
$5$ $$(T^{2} + 4)^{2}$$
$7$ $$T^{4} - 25T^{2} + 625$$
$11$ $$T^{4} - 169 T^{2} + 28561$$
$13$ $$T^{4}$$
$17$ $$(T^{2} - 27 T + 729)^{2}$$
$19$ $$T^{4} - 5625 T^{2} + 31640625$$
$23$ $$(T^{2} + 187 T + 34969)^{2}$$
$29$ $$(T^{2} - 13 T + 169)^{2}$$
$31$ $$(T^{2} + 10816)^{2}$$
$37$ $$T^{4} + \cdots + 32015587041$$
$41$ $$T^{4} + \cdots + 1445900625$$
$43$ $$(T^{2} - 199 T + 39601)^{2}$$
$47$ $$(T^{2} + 150544)^{2}$$
$53$ $$(T - 618)^{4}$$
$59$ $$T^{4} + \cdots + 58120048561$$
$61$ $$(T^{2} + 175 T + 30625)^{2}$$
$67$ $$T^{4} + \cdots + 445541565121$$
$71$ $$T^{4} - 6241 T^{2} + 38950081$$
$73$ $$(T^{2} + 52900)^{2}$$
$79$ $$(T - 764)^{4}$$
$83$ $$(T^{2} + 535824)^{2}$$
$89$ $$T^{4} + \cdots + 1174364509761$$
$97$ $$T^{4} - 9409 T^{2} + 88529281$$