# Properties

 Label 338.4.e.b 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: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{12} q^{2} + (3 \zeta_{12}^{2} - 3) q^{3} + 4 \zeta_{12}^{2} q^{4} + 11 \zeta_{12}^{3} q^{5} + (6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{6} + (19 \zeta_{12}^{3} - 19 \zeta_{12}) q^{7} + 8 \zeta_{12}^{3} q^{8} + 18 \zeta_{12}^{2} q^{9}+O(q^{10})$$ q + 2*z * q^2 + (3*z^2 - 3) * q^3 + 4*z^2 * q^4 + 11*z^3 * q^5 + (6*z^3 - 6*z) * q^6 + (19*z^3 - 19*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} + 11 \zeta_{12}^{3} q^{5} + (6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{6} + (19 \zeta_{12}^{3} - 19 \zeta_{12}) q^{7} + 8 \zeta_{12}^{3} q^{8} + 18 \zeta_{12}^{2} q^{9} + (22 \zeta_{12}^{2} - 22) q^{10} - 38 \zeta_{12} q^{11} - 12 q^{12} - 38 q^{14} - 33 \zeta_{12} q^{15} + (16 \zeta_{12}^{2} - 16) q^{16} - 51 \zeta_{12}^{2} q^{17} + 36 \zeta_{12}^{3} q^{18} + ( - 90 \zeta_{12}^{3} + 90 \zeta_{12}) q^{19} + (44 \zeta_{12}^{3} - 44 \zeta_{12}) q^{20} - 57 \zeta_{12}^{3} q^{21} - 76 \zeta_{12}^{2} q^{22} + (52 \zeta_{12}^{2} - 52) q^{23} - 24 \zeta_{12} q^{24} + 4 q^{25} - 135 q^{27} - 76 \zeta_{12} q^{28} + ( - 190 \zeta_{12}^{2} + 190) q^{29} - 66 \zeta_{12}^{2} q^{30} + 292 \zeta_{12}^{3} q^{31} + (32 \zeta_{12}^{3} - 32 \zeta_{12}) q^{32} + ( - 114 \zeta_{12}^{3} + 114 \zeta_{12}) q^{33} - 102 \zeta_{12}^{3} q^{34} - 209 \zeta_{12}^{2} q^{35} + (72 \zeta_{12}^{2} - 72) q^{36} - 441 \zeta_{12} q^{37} + 180 q^{38} - 88 q^{40} - 312 \zeta_{12} q^{41} + ( - 114 \zeta_{12}^{2} + 114) q^{42} + 373 \zeta_{12}^{2} q^{43} - 152 \zeta_{12}^{3} q^{44} + (198 \zeta_{12}^{3} - 198 \zeta_{12}) q^{45} + (104 \zeta_{12}^{3} - 104 \zeta_{12}) q^{46} + 41 \zeta_{12}^{3} q^{47} - 48 \zeta_{12}^{2} q^{48} + ( - 18 \zeta_{12}^{2} + 18) q^{49} + 8 \zeta_{12} q^{50} + 153 q^{51} + 468 q^{53} - 270 \zeta_{12} q^{54} + ( - 418 \zeta_{12}^{2} + 418) q^{55} - 152 \zeta_{12}^{2} q^{56} + 270 \zeta_{12}^{3} q^{57} + ( - 380 \zeta_{12}^{3} + 380 \zeta_{12}) q^{58} + (530 \zeta_{12}^{3} - 530 \zeta_{12}) q^{59} - 132 \zeta_{12}^{3} q^{60} - 592 \zeta_{12}^{2} q^{61} + (584 \zeta_{12}^{2} - 584) q^{62} - 342 \zeta_{12} q^{63} - 64 q^{64} + 228 q^{66} + 206 \zeta_{12} q^{67} + ( - 204 \zeta_{12}^{2} + 204) q^{68} - 156 \zeta_{12}^{2} q^{69} - 418 \zeta_{12}^{3} q^{70} + (863 \zeta_{12}^{3} - 863 \zeta_{12}) q^{71} + (144 \zeta_{12}^{3} - 144 \zeta_{12}) q^{72} + 322 \zeta_{12}^{3} q^{73} - 882 \zeta_{12}^{2} q^{74} + (12 \zeta_{12}^{2} - 12) q^{75} + 360 \zeta_{12} q^{76} + 722 q^{77} - 460 q^{79} - 176 \zeta_{12} q^{80} + (81 \zeta_{12}^{2} - 81) q^{81} - 624 \zeta_{12}^{2} q^{82} + 528 \zeta_{12}^{3} q^{83} + ( - 228 \zeta_{12}^{3} + 228 \zeta_{12}) q^{84} + ( - 561 \zeta_{12}^{3} + 561 \zeta_{12}) q^{85} + 746 \zeta_{12}^{3} q^{86} + 570 \zeta_{12}^{2} q^{87} + ( - 304 \zeta_{12}^{2} + 304) q^{88} + 870 \zeta_{12} q^{89} - 396 q^{90} - 208 q^{92} - 876 \zeta_{12} q^{93} + (82 \zeta_{12}^{2} - 82) q^{94} + 990 \zeta_{12}^{2} q^{95} - 96 \zeta_{12}^{3} q^{96} + (346 \zeta_{12}^{3} - 346 \zeta_{12}) q^{97} + ( - 36 \zeta_{12}^{3} + 36 \zeta_{12}) q^{98} - 684 \zeta_{12}^{3} q^{99} +O(q^{100})$$ q + 2*z * q^2 + (3*z^2 - 3) * q^3 + 4*z^2 * q^4 + 11*z^3 * q^5 + (6*z^3 - 6*z) * q^6 + (19*z^3 - 19*z) * q^7 + 8*z^3 * q^8 + 18*z^2 * q^9 + (22*z^2 - 22) * q^10 - 38*z * q^11 - 12 * q^12 - 38 * q^14 - 33*z * q^15 + (16*z^2 - 16) * q^16 - 51*z^2 * q^17 + 36*z^3 * q^18 + (-90*z^3 + 90*z) * q^19 + (44*z^3 - 44*z) * q^20 - 57*z^3 * q^21 - 76*z^2 * q^22 + (52*z^2 - 52) * q^23 - 24*z * q^24 + 4 * q^25 - 135 * q^27 - 76*z * q^28 + (-190*z^2 + 190) * q^29 - 66*z^2 * q^30 + 292*z^3 * q^31 + (32*z^3 - 32*z) * q^32 + (-114*z^3 + 114*z) * q^33 - 102*z^3 * q^34 - 209*z^2 * q^35 + (72*z^2 - 72) * q^36 - 441*z * q^37 + 180 * q^38 - 88 * q^40 - 312*z * q^41 + (-114*z^2 + 114) * q^42 + 373*z^2 * q^43 - 152*z^3 * q^44 + (198*z^3 - 198*z) * q^45 + (104*z^3 - 104*z) * q^46 + 41*z^3 * q^47 - 48*z^2 * q^48 + (-18*z^2 + 18) * q^49 + 8*z * q^50 + 153 * q^51 + 468 * q^53 - 270*z * q^54 + (-418*z^2 + 418) * q^55 - 152*z^2 * q^56 + 270*z^3 * q^57 + (-380*z^3 + 380*z) * q^58 + (530*z^3 - 530*z) * q^59 - 132*z^3 * q^60 - 592*z^2 * q^61 + (584*z^2 - 584) * q^62 - 342*z * q^63 - 64 * q^64 + 228 * q^66 + 206*z * q^67 + (-204*z^2 + 204) * q^68 - 156*z^2 * q^69 - 418*z^3 * q^70 + (863*z^3 - 863*z) * q^71 + (144*z^3 - 144*z) * q^72 + 322*z^3 * q^73 - 882*z^2 * q^74 + (12*z^2 - 12) * q^75 + 360*z * q^76 + 722 * q^77 - 460 * q^79 - 176*z * q^80 + (81*z^2 - 81) * q^81 - 624*z^2 * q^82 + 528*z^3 * q^83 + (-228*z^3 + 228*z) * q^84 + (-561*z^3 + 561*z) * q^85 + 746*z^3 * q^86 + 570*z^2 * q^87 + (-304*z^2 + 304) * q^88 + 870*z * q^89 - 396 * q^90 - 208 * q^92 - 876*z * q^93 + (82*z^2 - 82) * q^94 + 990*z^2 * q^95 - 96*z^3 * q^96 + (346*z^3 - 346*z) * q^97 + (-36*z^3 + 36*z) * q^98 - 684*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 $$4 q - 6 q^{3} + 8 q^{4} + 36 q^{9} - 44 q^{10} - 48 q^{12} - 152 q^{14} - 32 q^{16} - 102 q^{17} - 152 q^{22} - 104 q^{23} + 16 q^{25} - 540 q^{27} + 380 q^{29} - 132 q^{30} - 418 q^{35} - 144 q^{36} + 720 q^{38} - 352 q^{40} + 228 q^{42} + 746 q^{43} - 96 q^{48} + 36 q^{49} + 612 q^{51} + 1872 q^{53} + 836 q^{55} - 304 q^{56} - 1184 q^{61} - 1168 q^{62} - 256 q^{64} + 912 q^{66} + 408 q^{68} - 312 q^{69} - 1764 q^{74} - 24 q^{75} + 2888 q^{77} - 1840 q^{79} - 162 q^{81} - 1248 q^{82} + 1140 q^{87} + 608 q^{88} - 1584 q^{90} - 832 q^{92} - 164 q^{94} + 1980 q^{95}+O(q^{100})$$ 4 * q - 6 * q^3 + 8 * q^4 + 36 * q^9 - 44 * q^10 - 48 * q^12 - 152 * q^14 - 32 * q^16 - 102 * q^17 - 152 * q^22 - 104 * q^23 + 16 * q^25 - 540 * q^27 + 380 * q^29 - 132 * q^30 - 418 * q^35 - 144 * q^36 + 720 * q^38 - 352 * q^40 + 228 * q^42 + 746 * q^43 - 96 * q^48 + 36 * q^49 + 612 * q^51 + 1872 * q^53 + 836 * q^55 - 304 * q^56 - 1184 * q^61 - 1168 * q^62 - 256 * q^64 + 912 * q^66 + 408 * q^68 - 312 * q^69 - 1764 * q^74 - 24 * q^75 + 2888 * q^77 - 1840 * q^79 - 162 * q^81 - 1248 * q^82 + 1140 * q^87 + 608 * q^88 - 1584 * q^90 - 832 * q^92 - 164 * q^94 + 1980 * 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 11.0000i 5.19615 + 3.00000i 16.4545 + 9.50000i 8.00000i 9.00000 15.5885i −11.0000 19.0526i
23.2 1.73205 1.00000i −1.50000 2.59808i 2.00000 3.46410i 11.0000i −5.19615 3.00000i −16.4545 9.50000i 8.00000i 9.00000 15.5885i −11.0000 19.0526i
147.1 −1.73205 1.00000i −1.50000 + 2.59808i 2.00000 + 3.46410i 11.0000i 5.19615 3.00000i 16.4545 9.50000i 8.00000i 9.00000 + 15.5885i −11.0000 + 19.0526i
147.2 1.73205 + 1.00000i −1.50000 + 2.59808i 2.00000 + 3.46410i 11.0000i −5.19615 + 3.00000i −16.4545 + 9.50000i 8.00000i 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.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.b 4
13.b even 2 1 inner 338.4.e.b 4
13.c even 3 1 338.4.b.c 2
13.c even 3 1 inner 338.4.e.b 4
13.d odd 4 1 338.4.c.b 2
13.d odd 4 1 338.4.c.f 2
13.e even 6 1 338.4.b.c 2
13.e even 6 1 inner 338.4.e.b 4
13.f odd 12 1 26.4.a.a 1
13.f odd 12 1 338.4.a.e 1
13.f odd 12 1 338.4.c.b 2
13.f odd 12 1 338.4.c.f 2
39.k even 12 1 234.4.a.g 1
52.l even 12 1 208.4.a.c 1
65.o even 12 1 650.4.b.b 2
65.s odd 12 1 650.4.a.f 1
65.t even 12 1 650.4.b.b 2
91.bc even 12 1 1274.4.a.b 1
104.u even 12 1 832.4.a.m 1
104.x odd 12 1 832.4.a.e 1
156.v odd 12 1 1872.4.a.c 1

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

 $$T_{3}^{2} + 3T_{3} + 9$$ T3^2 + 3*T3 + 9 $$T_{5}^{2} + 121$$ T5^2 + 121

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 4T^{2} + 16$$
$3$ $$(T^{2} + 3 T + 9)^{2}$$
$5$ $$(T^{2} + 121)^{2}$$
$7$ $$T^{4} - 361 T^{2} + 130321$$
$11$ $$T^{4} - 1444 T^{2} + 2085136$$
$13$ $$T^{4}$$
$17$ $$(T^{2} + 51 T + 2601)^{2}$$
$19$ $$T^{4} - 8100 T^{2} + 65610000$$
$23$ $$(T^{2} + 52 T + 2704)^{2}$$
$29$ $$(T^{2} - 190 T + 36100)^{2}$$
$31$ $$(T^{2} + 85264)^{2}$$
$37$ $$T^{4} + \cdots + 37822859361$$
$41$ $$T^{4} + \cdots + 9475854336$$
$43$ $$(T^{2} - 373 T + 139129)^{2}$$
$47$ $$(T^{2} + 1681)^{2}$$
$53$ $$(T - 468)^{4}$$
$59$ $$T^{4} + \cdots + 78904810000$$
$61$ $$(T^{2} + 592 T + 350464)^{2}$$
$67$ $$T^{4} + \cdots + 1800814096$$
$71$ $$T^{4} + \cdots + 554680863361$$
$73$ $$(T^{2} + 103684)^{2}$$
$79$ $$(T + 460)^{4}$$
$83$ $$(T^{2} + 278784)^{2}$$
$89$ $$T^{4} + \cdots + 572897610000$$
$97$ $$T^{4} + \cdots + 14331920656$$