Properties

 Label 338.4.b.a Level $338$ Weight $4$ Character orbit 338.b Analytic conductor $19.943$ Analytic rank $0$ Dimension $2$ Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [338,4,Mod(337,338)]

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([1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("338.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 338.b (of order $$2$$, degree $$1$$, 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: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ 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: $\mathrm{SU}(2)[C_{2}]$

$q$-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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 i q^{2} - 3 q^{3} - 4 q^{4} - 2 i q^{5} - 6 i q^{6} - 5 i q^{7} - 8 i q^{8} - 18 q^{9} +O(q^{10})$$ q + 2*i * q^2 - 3 * q^3 - 4 * q^4 - 2*i * q^5 - 6*i * q^6 - 5*i * q^7 - 8*i * q^8 - 18 * q^9 $$q + 2 i q^{2} - 3 q^{3} - 4 q^{4} - 2 i q^{5} - 6 i q^{6} - 5 i q^{7} - 8 i q^{8} - 18 q^{9} + 4 q^{10} + 13 i q^{11} + 12 q^{12} + 10 q^{14} + 6 i q^{15} + 16 q^{16} - 27 q^{17} - 36 i q^{18} - 75 i q^{19} + 8 i q^{20} + 15 i q^{21} - 26 q^{22} + 187 q^{23} + 24 i q^{24} + 121 q^{25} + 135 q^{27} + 20 i q^{28} - 13 q^{29} - 12 q^{30} + 104 i q^{31} + 32 i q^{32} - 39 i q^{33} - 54 i q^{34} - 10 q^{35} + 72 q^{36} + 423 i q^{37} + 150 q^{38} - 16 q^{40} - 195 i q^{41} - 30 q^{42} - 199 q^{43} - 52 i q^{44} + 36 i q^{45} + 374 i q^{46} + 388 i q^{47} - 48 q^{48} + 318 q^{49} + 242 i q^{50} + 81 q^{51} + 618 q^{53} + 270 i q^{54} + 26 q^{55} - 40 q^{56} + 225 i q^{57} - 26 i q^{58} + 491 i q^{59} - 24 i q^{60} + 175 q^{61} - 208 q^{62} + 90 i q^{63} - 64 q^{64} + 78 q^{66} - 817 i q^{67} + 108 q^{68} - 561 q^{69} - 20 i q^{70} - 79 i q^{71} + 144 i q^{72} + 230 i q^{73} - 846 q^{74} - 363 q^{75} + 300 i q^{76} + 65 q^{77} + 764 q^{79} - 32 i q^{80} + 81 q^{81} + 390 q^{82} + 732 i q^{83} - 60 i q^{84} + 54 i q^{85} - 398 i q^{86} + 39 q^{87} + 104 q^{88} - 1041 i q^{89} - 72 q^{90} - 748 q^{92} - 312 i q^{93} - 776 q^{94} - 150 q^{95} - 96 i q^{96} + 97 i q^{97} + 636 i q^{98} - 234 i q^{99} +O(q^{100})$$ q + 2*i * q^2 - 3 * q^3 - 4 * q^4 - 2*i * q^5 - 6*i * q^6 - 5*i * q^7 - 8*i * q^8 - 18 * q^9 + 4 * q^10 + 13*i * q^11 + 12 * q^12 + 10 * q^14 + 6*i * q^15 + 16 * q^16 - 27 * q^17 - 36*i * q^18 - 75*i * q^19 + 8*i * q^20 + 15*i * q^21 - 26 * q^22 + 187 * q^23 + 24*i * q^24 + 121 * q^25 + 135 * q^27 + 20*i * q^28 - 13 * q^29 - 12 * q^30 + 104*i * q^31 + 32*i * q^32 - 39*i * q^33 - 54*i * q^34 - 10 * q^35 + 72 * q^36 + 423*i * q^37 + 150 * q^38 - 16 * q^40 - 195*i * q^41 - 30 * q^42 - 199 * q^43 - 52*i * q^44 + 36*i * q^45 + 374*i * q^46 + 388*i * q^47 - 48 * q^48 + 318 * q^49 + 242*i * q^50 + 81 * q^51 + 618 * q^53 + 270*i * q^54 + 26 * q^55 - 40 * q^56 + 225*i * q^57 - 26*i * q^58 + 491*i * q^59 - 24*i * q^60 + 175 * q^61 - 208 * q^62 + 90*i * q^63 - 64 * q^64 + 78 * q^66 - 817*i * q^67 + 108 * q^68 - 561 * q^69 - 20*i * q^70 - 79*i * q^71 + 144*i * q^72 + 230*i * q^73 - 846 * q^74 - 363 * q^75 + 300*i * q^76 + 65 * q^77 + 764 * q^79 - 32*i * q^80 + 81 * q^81 + 390 * q^82 + 732*i * q^83 - 60*i * q^84 + 54*i * q^85 - 398*i * q^86 + 39 * q^87 + 104 * q^88 - 1041*i * q^89 - 72 * q^90 - 748 * q^92 - 312*i * q^93 - 776 * q^94 - 150 * q^95 - 96*i * q^96 + 97*i * q^97 + 636*i * q^98 - 234*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{3} - 8 q^{4} - 36 q^{9}+O(q^{10})$$ 2 * q - 6 * q^3 - 8 * q^4 - 36 * q^9 $$2 q - 6 q^{3} - 8 q^{4} - 36 q^{9} + 8 q^{10} + 24 q^{12} + 20 q^{14} + 32 q^{16} - 54 q^{17} - 52 q^{22} + 374 q^{23} + 242 q^{25} + 270 q^{27} - 26 q^{29} - 24 q^{30} - 20 q^{35} + 144 q^{36} + 300 q^{38} - 32 q^{40} - 60 q^{42} - 398 q^{43} - 96 q^{48} + 636 q^{49} + 162 q^{51} + 1236 q^{53} + 52 q^{55} - 80 q^{56} + 350 q^{61} - 416 q^{62} - 128 q^{64} + 156 q^{66} + 216 q^{68} - 1122 q^{69} - 1692 q^{74} - 726 q^{75} + 130 q^{77} + 1528 q^{79} + 162 q^{81} + 780 q^{82} + 78 q^{87} + 208 q^{88} - 144 q^{90} - 1496 q^{92} - 1552 q^{94} - 300 q^{95}+O(q^{100})$$ 2 * q - 6 * q^3 - 8 * q^4 - 36 * q^9 + 8 * q^10 + 24 * q^12 + 20 * q^14 + 32 * q^16 - 54 * q^17 - 52 * q^22 + 374 * q^23 + 242 * q^25 + 270 * q^27 - 26 * q^29 - 24 * q^30 - 20 * q^35 + 144 * q^36 + 300 * q^38 - 32 * q^40 - 60 * q^42 - 398 * q^43 - 96 * q^48 + 636 * q^49 + 162 * q^51 + 1236 * q^53 + 52 * q^55 - 80 * q^56 + 350 * q^61 - 416 * q^62 - 128 * q^64 + 156 * q^66 + 216 * q^68 - 1122 * q^69 - 1692 * q^74 - 726 * q^75 + 130 * q^77 + 1528 * q^79 + 162 * q^81 + 780 * q^82 + 78 * q^87 + 208 * q^88 - 144 * q^90 - 1496 * q^92 - 1552 * q^94 - 300 * q^95

Character values

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

 $$n$$ $$171$$ $$\chi(n)$$ $$-1$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
337.1
 − 1.00000i 1.00000i
2.00000i −3.00000 −4.00000 2.00000i 6.00000i 5.00000i 8.00000i −18.0000 4.00000
337.2 2.00000i −3.00000 −4.00000 2.00000i 6.00000i 5.00000i 8.00000i −18.0000 4.00000
 $$n$$: 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

Twists

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

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 3$$ acting on $$S_{4}^{\mathrm{new}}(338, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 4$$
$3$ $$(T + 3)^{2}$$
$5$ $$T^{2} + 4$$
$7$ $$T^{2} + 25$$
$11$ $$T^{2} + 169$$
$13$ $$T^{2}$$
$17$ $$(T + 27)^{2}$$
$19$ $$T^{2} + 5625$$
$23$ $$(T - 187)^{2}$$
$29$ $$(T + 13)^{2}$$
$31$ $$T^{2} + 10816$$
$37$ $$T^{2} + 178929$$
$41$ $$T^{2} + 38025$$
$43$ $$(T + 199)^{2}$$
$47$ $$T^{2} + 150544$$
$53$ $$(T - 618)^{2}$$
$59$ $$T^{2} + 241081$$
$61$ $$(T - 175)^{2}$$
$67$ $$T^{2} + 667489$$
$71$ $$T^{2} + 6241$$
$73$ $$T^{2} + 52900$$
$79$ $$(T - 764)^{2}$$
$83$ $$T^{2} + 535824$$
$89$ $$T^{2} + 1083681$$
$97$ $$T^{2} + 9409$$
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