# Properties

 Label 338.2.a.g Level $338$ Weight $2$ Character orbit 338.a Self dual yes Analytic conductor $2.699$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [338,2,Mod(1,338)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(338, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("338.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 338.a (trivial)

## 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: yes Analytic conductor: $$2.69894358832$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 2x + 1$$ x^3 - x^2 - 2*x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + ( - \beta_{2} - \beta_1 + 1) q^{3} + q^{4} - 2 \beta_1 q^{5} + (\beta_{2} + \beta_1 - 1) q^{6} + ( - 2 \beta_{2} + 2 \beta_1) q^{7} - q^{8} + ( - \beta_1 + 3) q^{9}+O(q^{10})$$ q - q^2 + (-b2 - b1 + 1) * q^3 + q^4 - 2*b1 * q^5 + (b2 + b1 - 1) * q^6 + (-2*b2 + 2*b1) * q^7 - q^8 + (-b1 + 3) * q^9 $$q - q^{2} + ( - \beta_{2} - \beta_1 + 1) q^{3} + q^{4} - 2 \beta_1 q^{5} + (\beta_{2} + \beta_1 - 1) q^{6} + ( - 2 \beta_{2} + 2 \beta_1) q^{7} - q^{8} + ( - \beta_1 + 3) q^{9} + 2 \beta_1 q^{10} + (\beta_{2} - 2 \beta_1) q^{11} + ( - \beta_{2} - \beta_1 + 1) q^{12} + (2 \beta_{2} - 2 \beta_1) q^{14} + (4 \beta_{2} - 2 \beta_1 + 6) q^{15} + q^{16} + ( - \beta_{2} + 4 \beta_1) q^{17} + (\beta_1 - 3) q^{18} + ( - 2 \beta_{2} + 3 \beta_1 - 2) q^{19} - 2 \beta_1 q^{20} + ( - 6 \beta_{2} + 4 \beta_1 - 2) q^{21} + ( - \beta_{2} + 2 \beta_1) q^{22} + (4 \beta_{2} - 2 \beta_1 + 2) q^{23} + (\beta_{2} + \beta_1 - 1) q^{24} + (4 \beta_{2} + 3) q^{25} + (2 \beta_{2} - \beta_1 + 3) q^{27} + ( - 2 \beta_{2} + 2 \beta_1) q^{28} + ( - 4 \beta_{2} + 4 \beta_1 - 6) q^{29} + ( - 4 \beta_{2} + 2 \beta_1 - 6) q^{30} + (2 \beta_{2} + 6) q^{31} - q^{32} + (5 \beta_{2} - 3 \beta_1 + 4) q^{33} + (\beta_{2} - 4 \beta_1) q^{34} - 4 q^{35} + ( - \beta_1 + 3) q^{36} + (2 \beta_{2} - 2 \beta_1 + 6) q^{37} + (2 \beta_{2} - 3 \beta_1 + 2) q^{38} + 2 \beta_1 q^{40} + ( - 4 \beta_{2} + \beta_1 - 4) q^{41} + (6 \beta_{2} - 4 \beta_1 + 2) q^{42} + (2 \beta_{2} - 5 \beta_1 + 6) q^{43} + (\beta_{2} - 2 \beta_1) q^{44} + (2 \beta_{2} - 6 \beta_1 + 4) q^{45} + ( - 4 \beta_{2} + 2 \beta_1 - 2) q^{46} + (4 \beta_{2} + 2) q^{47} + ( - \beta_{2} - \beta_1 + 1) q^{48} + ( - 8 \beta_{2} + 4 \beta_1 - 3) q^{49} + ( - 4 \beta_{2} - 3) q^{50} + ( - 9 \beta_{2} + 5 \beta_1 - 10) q^{51} + ( - 4 \beta_{2} + 2 \beta_1 - 2) q^{53} + ( - 2 \beta_{2} + \beta_1 - 3) q^{54} + (2 \beta_{2} + 6) q^{55} + (2 \beta_{2} - 2 \beta_1) q^{56} + ( - 6 \beta_{2} + 7 \beta_1 - 7) q^{57} + (4 \beta_{2} - 4 \beta_1 + 6) q^{58} + (3 \beta_{2} + 3 \beta_1 + 1) q^{59} + (4 \beta_{2} - 2 \beta_1 + 6) q^{60} + (8 \beta_{2} - 8 \beta_1 + 4) q^{61} + ( - 2 \beta_{2} - 6) q^{62} + ( - 6 \beta_{2} + 6 \beta_1 - 2) q^{63} + q^{64} + ( - 5 \beta_{2} + 3 \beta_1 - 4) q^{66} + (3 \beta_{2} + 3 \beta_1 - 7) q^{67} + ( - \beta_{2} + 4 \beta_1) q^{68} + (6 \beta_{2} - 8 \beta_1) q^{69} + 4 q^{70} + ( - 4 \beta_{2} + 2 \beta_1 + 2) q^{71} + (\beta_1 - 3) q^{72} + (7 \beta_{2} - 3 \beta_1 + 3) q^{73} + ( - 2 \beta_{2} + 2 \beta_1 - 6) q^{74} + (\beta_{2} - 7 \beta_1 - 5) q^{75} + ( - 2 \beta_{2} + 3 \beta_1 - 2) q^{76} + (4 \beta_{2} - 2 \beta_1 - 4) q^{77} + ( - 6 \beta_{2} - 8) q^{79} - 2 \beta_1 q^{80} + (\beta_{2} - 3 \beta_1 - 7) q^{81} + (4 \beta_{2} - \beta_1 + 4) q^{82} + (2 \beta_{2} - 3 \beta_1 + 2) q^{83} + ( - 6 \beta_{2} + 4 \beta_1 - 2) q^{84} + ( - 6 \beta_{2} - 14) q^{85} + ( - 2 \beta_{2} + 5 \beta_1 - 6) q^{86} + ( - 6 \beta_{2} + 14 \beta_1 - 10) q^{87} + ( - \beta_{2} + 2 \beta_1) q^{88} + (3 \beta_{2} + 5 \beta_1 - 9) q^{89} + ( - 2 \beta_{2} + 6 \beta_1 - 4) q^{90} + (4 \beta_{2} - 2 \beta_1 + 2) q^{92} + ( - 4 \beta_{2} - 8 \beta_1 + 2) q^{93} + ( - 4 \beta_{2} - 2) q^{94} + ( - 2 \beta_{2} + 4 \beta_1 - 8) q^{95} + (\beta_{2} + \beta_1 - 1) q^{96} + (3 \beta_{2} - 8 \beta_1 - 4) q^{97} + (8 \beta_{2} - 4 \beta_1 + 3) q^{98} + (4 \beta_{2} - 6 \beta_1 + 3) q^{99}+O(q^{100})$$ q - q^2 + (-b2 - b1 + 1) * q^3 + q^4 - 2*b1 * q^5 + (b2 + b1 - 1) * q^6 + (-2*b2 + 2*b1) * q^7 - q^8 + (-b1 + 3) * q^9 + 2*b1 * q^10 + (b2 - 2*b1) * q^11 + (-b2 - b1 + 1) * q^12 + (2*b2 - 2*b1) * q^14 + (4*b2 - 2*b1 + 6) * q^15 + q^16 + (-b2 + 4*b1) * q^17 + (b1 - 3) * q^18 + (-2*b2 + 3*b1 - 2) * q^19 - 2*b1 * q^20 + (-6*b2 + 4*b1 - 2) * q^21 + (-b2 + 2*b1) * q^22 + (4*b2 - 2*b1 + 2) * q^23 + (b2 + b1 - 1) * q^24 + (4*b2 + 3) * q^25 + (2*b2 - b1 + 3) * q^27 + (-2*b2 + 2*b1) * q^28 + (-4*b2 + 4*b1 - 6) * q^29 + (-4*b2 + 2*b1 - 6) * q^30 + (2*b2 + 6) * q^31 - q^32 + (5*b2 - 3*b1 + 4) * q^33 + (b2 - 4*b1) * q^34 - 4 * q^35 + (-b1 + 3) * q^36 + (2*b2 - 2*b1 + 6) * q^37 + (2*b2 - 3*b1 + 2) * q^38 + 2*b1 * q^40 + (-4*b2 + b1 - 4) * q^41 + (6*b2 - 4*b1 + 2) * q^42 + (2*b2 - 5*b1 + 6) * q^43 + (b2 - 2*b1) * q^44 + (2*b2 - 6*b1 + 4) * q^45 + (-4*b2 + 2*b1 - 2) * q^46 + (4*b2 + 2) * q^47 + (-b2 - b1 + 1) * q^48 + (-8*b2 + 4*b1 - 3) * q^49 + (-4*b2 - 3) * q^50 + (-9*b2 + 5*b1 - 10) * q^51 + (-4*b2 + 2*b1 - 2) * q^53 + (-2*b2 + b1 - 3) * q^54 + (2*b2 + 6) * q^55 + (2*b2 - 2*b1) * q^56 + (-6*b2 + 7*b1 - 7) * q^57 + (4*b2 - 4*b1 + 6) * q^58 + (3*b2 + 3*b1 + 1) * q^59 + (4*b2 - 2*b1 + 6) * q^60 + (8*b2 - 8*b1 + 4) * q^61 + (-2*b2 - 6) * q^62 + (-6*b2 + 6*b1 - 2) * q^63 + q^64 + (-5*b2 + 3*b1 - 4) * q^66 + (3*b2 + 3*b1 - 7) * q^67 + (-b2 + 4*b1) * q^68 + (6*b2 - 8*b1) * q^69 + 4 * q^70 + (-4*b2 + 2*b1 + 2) * q^71 + (b1 - 3) * q^72 + (7*b2 - 3*b1 + 3) * q^73 + (-2*b2 + 2*b1 - 6) * q^74 + (b2 - 7*b1 - 5) * q^75 + (-2*b2 + 3*b1 - 2) * q^76 + (4*b2 - 2*b1 - 4) * q^77 + (-6*b2 - 8) * q^79 - 2*b1 * q^80 + (b2 - 3*b1 - 7) * q^81 + (4*b2 - b1 + 4) * q^82 + (2*b2 - 3*b1 + 2) * q^83 + (-6*b2 + 4*b1 - 2) * q^84 + (-6*b2 - 14) * q^85 + (-2*b2 + 5*b1 - 6) * q^86 + (-6*b2 + 14*b1 - 10) * q^87 + (-b2 + 2*b1) * q^88 + (3*b2 + 5*b1 - 9) * q^89 + (-2*b2 + 6*b1 - 4) * q^90 + (4*b2 - 2*b1 + 2) * q^92 + (-4*b2 - 8*b1 + 2) * q^93 + (-4*b2 - 2) * q^94 + (-2*b2 + 4*b1 - 8) * q^95 + (b2 + b1 - 1) * q^96 + (3*b2 - 8*b1 - 4) * q^97 + (8*b2 - 4*b1 + 3) * q^98 + (4*b2 - 6*b1 + 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} - 2 q^{5} - 3 q^{6} + 4 q^{7} - 3 q^{8} + 8 q^{9}+O(q^{10})$$ 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 - 2 * q^5 - 3 * q^6 + 4 * q^7 - 3 * q^8 + 8 * q^9 $$3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} - 2 q^{5} - 3 q^{6} + 4 q^{7} - 3 q^{8} + 8 q^{9} + 2 q^{10} - 3 q^{11} + 3 q^{12} - 4 q^{14} + 12 q^{15} + 3 q^{16} + 5 q^{17} - 8 q^{18} - q^{19} - 2 q^{20} + 4 q^{21} + 3 q^{22} - 3 q^{24} + 5 q^{25} + 6 q^{27} + 4 q^{28} - 10 q^{29} - 12 q^{30} + 16 q^{31} - 3 q^{32} + 4 q^{33} - 5 q^{34} - 12 q^{35} + 8 q^{36} + 14 q^{37} + q^{38} + 2 q^{40} - 7 q^{41} - 4 q^{42} + 11 q^{43} - 3 q^{44} + 4 q^{45} + 2 q^{47} + 3 q^{48} + 3 q^{49} - 5 q^{50} - 16 q^{51} - 6 q^{54} + 16 q^{55} - 4 q^{56} - 8 q^{57} + 10 q^{58} + 3 q^{59} + 12 q^{60} - 4 q^{61} - 16 q^{62} + 6 q^{63} + 3 q^{64} - 4 q^{66} - 21 q^{67} + 5 q^{68} - 14 q^{69} + 12 q^{70} + 12 q^{71} - 8 q^{72} - q^{73} - 14 q^{74} - 23 q^{75} - q^{76} - 18 q^{77} - 18 q^{79} - 2 q^{80} - 25 q^{81} + 7 q^{82} + q^{83} + 4 q^{84} - 36 q^{85} - 11 q^{86} - 10 q^{87} + 3 q^{88} - 25 q^{89} - 4 q^{90} + 2 q^{93} - 2 q^{94} - 18 q^{95} - 3 q^{96} - 23 q^{97} - 3 q^{98} - q^{99}+O(q^{100})$$ 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 - 2 * q^5 - 3 * q^6 + 4 * q^7 - 3 * q^8 + 8 * q^9 + 2 * q^10 - 3 * q^11 + 3 * q^12 - 4 * q^14 + 12 * q^15 + 3 * q^16 + 5 * q^17 - 8 * q^18 - q^19 - 2 * q^20 + 4 * q^21 + 3 * q^22 - 3 * q^24 + 5 * q^25 + 6 * q^27 + 4 * q^28 - 10 * q^29 - 12 * q^30 + 16 * q^31 - 3 * q^32 + 4 * q^33 - 5 * q^34 - 12 * q^35 + 8 * q^36 + 14 * q^37 + q^38 + 2 * q^40 - 7 * q^41 - 4 * q^42 + 11 * q^43 - 3 * q^44 + 4 * q^45 + 2 * q^47 + 3 * q^48 + 3 * q^49 - 5 * q^50 - 16 * q^51 - 6 * q^54 + 16 * q^55 - 4 * q^56 - 8 * q^57 + 10 * q^58 + 3 * q^59 + 12 * q^60 - 4 * q^61 - 16 * q^62 + 6 * q^63 + 3 * q^64 - 4 * q^66 - 21 * q^67 + 5 * q^68 - 14 * q^69 + 12 * q^70 + 12 * q^71 - 8 * q^72 - q^73 - 14 * q^74 - 23 * q^75 - q^76 - 18 * q^77 - 18 * q^79 - 2 * q^80 - 25 * q^81 + 7 * q^82 + q^83 + 4 * q^84 - 36 * q^85 - 11 * q^86 - 10 * q^87 + 3 * q^88 - 25 * q^89 - 4 * q^90 + 2 * q^93 - 2 * q^94 - 18 * q^95 - 3 * q^96 - 23 * q^97 - 3 * q^98 - q^99

Basis of coefficient ring in terms of $$\nu = \zeta_{14} + \zeta_{14}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 1.80194 0.445042 −1.24698
−1.00000 −2.04892 1.00000 −3.60388 2.04892 1.10992 −1.00000 1.19806 3.60388
1.2 −1.00000 2.35690 1.00000 −0.890084 −2.35690 4.49396 −1.00000 2.55496 0.890084
1.3 −1.00000 2.69202 1.00000 2.49396 −2.69202 −1.60388 −1.00000 4.24698 −2.49396
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$13$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 338.2.a.g 3
3.b odd 2 1 3042.2.a.bi 3
4.b odd 2 1 2704.2.a.v 3
5.b even 2 1 8450.2.a.bx 3
13.b even 2 1 338.2.a.h yes 3
13.c even 3 2 338.2.c.i 6
13.d odd 4 2 338.2.b.d 6
13.e even 6 2 338.2.c.h 6
13.f odd 12 4 338.2.e.e 12
39.d odd 2 1 3042.2.a.z 3
39.f even 4 2 3042.2.b.n 6
52.b odd 2 1 2704.2.a.w 3
52.f even 4 2 2704.2.f.m 6
65.d even 2 1 8450.2.a.bn 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
338.2.a.g 3 1.a even 1 1 trivial
338.2.a.h yes 3 13.b even 2 1
338.2.b.d 6 13.d odd 4 2
338.2.c.h 6 13.e even 6 2
338.2.c.i 6 13.c even 3 2
338.2.e.e 12 13.f odd 12 4
2704.2.a.v 3 4.b odd 2 1
2704.2.a.w 3 52.b odd 2 1
2704.2.f.m 6 52.f even 4 2
3042.2.a.z 3 39.d odd 2 1
3042.2.a.bi 3 3.b odd 2 1
3042.2.b.n 6 39.f even 4 2
8450.2.a.bn 3 65.d even 2 1
8450.2.a.bx 3 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(338))$$:

 $$T_{3}^{3} - 3T_{3}^{2} - 4T_{3} + 13$$ T3^3 - 3*T3^2 - 4*T3 + 13 $$T_{5}^{3} + 2T_{5}^{2} - 8T_{5} - 8$$ T5^3 + 2*T5^2 - 8*T5 - 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{3}$$
$3$ $$T^{3} - 3 T^{2} + \cdots + 13$$
$5$ $$T^{3} + 2 T^{2} + \cdots - 8$$
$7$ $$T^{3} - 4 T^{2} + \cdots + 8$$
$11$ $$T^{3} + 3 T^{2} + \cdots - 13$$
$13$ $$T^{3}$$
$17$ $$T^{3} - 5 T^{2} + \cdots + 97$$
$19$ $$T^{3} + T^{2} + \cdots + 13$$
$23$ $$T^{3} - 28T + 56$$
$29$ $$T^{3} + 10 T^{2} + \cdots - 104$$
$31$ $$T^{3} - 16 T^{2} + \cdots - 104$$
$37$ $$T^{3} - 14 T^{2} + \cdots - 56$$
$41$ $$T^{3} + 7 T^{2} + \cdots - 91$$
$43$ $$T^{3} - 11 T^{2} + \cdots + 1$$
$47$ $$T^{3} - 2 T^{2} + \cdots + 8$$
$53$ $$T^{3} - 28T - 56$$
$59$ $$T^{3} - 3 T^{2} + \cdots - 127$$
$61$ $$T^{3} + 4 T^{2} + \cdots - 64$$
$67$ $$T^{3} + 21 T^{2} + \cdots - 287$$
$71$ $$T^{3} - 12 T^{2} + \cdots - 8$$
$73$ $$T^{3} + T^{2} + \cdots + 251$$
$79$ $$T^{3} + 18 T^{2} + \cdots - 232$$
$83$ $$T^{3} - T^{2} + \cdots - 13$$
$89$ $$T^{3} + 25 T^{2} + \cdots - 757$$
$97$ $$T^{3} + 23 T^{2} + \cdots - 883$$