# Properties

 Label 4368.2.h.o Level $4368$ Weight $2$ Character orbit 4368.h Analytic conductor $34.879$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4368,2,Mod(337,4368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 1]))

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

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

chi := DirichletCharacter("4368.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4368.h (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: $$34.8786556029$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.350464.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2$$ x^6 - 2*x^5 + 2*x^4 + 2*x^3 + 4*x^2 - 4*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 273) 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 a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_1) q^{5} + \beta_{4} q^{7} + q^{9}+O(q^{10})$$ q - q^3 + (b5 - b4 - b3 + b1) * q^5 + b4 * q^7 + q^9 $$q - q^{3} + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_1) q^{5} + \beta_{4} q^{7} + q^{9} + (3 \beta_{4} - \beta_1) q^{11} + (2 \beta_{4} + \beta_{3} + \beta_1) q^{13} + ( - \beta_{5} + \beta_{4} + \beta_{3} - \beta_1) q^{15} + (\beta_{5} + \beta_{3} - 2 \beta_{2}) q^{17} + ( - 2 \beta_{4} + 2 \beta_1) q^{19} - \beta_{4} q^{21} + 2 \beta_{2} q^{23} + ( - 3 \beta_{5} - 3 \beta_{3} + 2 \beta_{2} - 3) q^{25} - q^{27} + ( - 2 \beta_{5} - 2 \beta_{3} - 2) q^{29} + (2 \beta_{5} - 4 \beta_{4} - 2 \beta_{3} + 4 \beta_1) q^{31} + ( - 3 \beta_{4} + \beta_1) q^{33} + (\beta_{5} + \beta_{3} - \beta_{2} + 1) q^{35} + ( - 2 \beta_{4} - 2 \beta_1) q^{37} + ( - 2 \beta_{4} - \beta_{3} - \beta_1) q^{39} + (\beta_{5} + 5 \beta_{4} - \beta_{3} + 3 \beta_1) q^{41} + ( - 2 \beta_{5} - 2 \beta_{3} + 4) q^{43} + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_1) q^{45} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \beta_1) q^{47} - q^{49} + ( - \beta_{5} - \beta_{3} + 2 \beta_{2}) q^{51} - 6 q^{53} + (3 \beta_{5} + 3 \beta_{3} - 2 \beta_{2} + 4) q^{55} + (2 \beta_{4} - 2 \beta_1) q^{57} + ( - \beta_{5} + 3 \beta_{4} + \beta_{3} - \beta_1) q^{59} + (3 \beta_{5} + 3 \beta_{3} - 8 \beta_{2} + 4) q^{61} + \beta_{4} q^{63} + (4 \beta_{5} - 3 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} + \beta_1 + 4) q^{65} + ( - 6 \beta_{4} - 2 \beta_1) q^{67} - 2 \beta_{2} q^{69} + (4 \beta_{5} + \beta_{4} - 4 \beta_{3} + 5 \beta_1) q^{71} + (3 \beta_{5} - 6 \beta_{4} - 3 \beta_{3} + 4 \beta_1) q^{73} + (3 \beta_{5} + 3 \beta_{3} - 2 \beta_{2} + 3) q^{75} + (\beta_{2} - 3) q^{77} + (\beta_{5} + \beta_{3} + 6) q^{79} + q^{81} + (\beta_{5} - 5 \beta_{4} - \beta_{3} - 5 \beta_1) q^{83} + (2 \beta_{5} - 8 \beta_{4} - 2 \beta_{3}) q^{85} + (2 \beta_{5} + 2 \beta_{3} + 2) q^{87} + ( - \beta_{5} - 3 \beta_{4} + \beta_{3} - 5 \beta_1) q^{89} + ( - \beta_{5} - \beta_{2} - 2) q^{91} + ( - 2 \beta_{5} + 4 \beta_{4} + 2 \beta_{3} - 4 \beta_1) q^{93} + ( - 2 \beta_{5} - 2 \beta_{3} - 4) q^{95} + (\beta_{5} - \beta_{3} + 2 \beta_1) q^{97} + (3 \beta_{4} - \beta_1) q^{99}+O(q^{100})$$ q - q^3 + (b5 - b4 - b3 + b1) * q^5 + b4 * q^7 + q^9 + (3*b4 - b1) * q^11 + (2*b4 + b3 + b1) * q^13 + (-b5 + b4 + b3 - b1) * q^15 + (b5 + b3 - 2*b2) * q^17 + (-2*b4 + 2*b1) * q^19 - b4 * q^21 + 2*b2 * q^23 + (-3*b5 - 3*b3 + 2*b2 - 3) * q^25 - q^27 + (-2*b5 - 2*b3 - 2) * q^29 + (2*b5 - 4*b4 - 2*b3 + 4*b1) * q^31 + (-3*b4 + b1) * q^33 + (b5 + b3 - b2 + 1) * q^35 + (-2*b4 - 2*b1) * q^37 + (-2*b4 - b3 - b1) * q^39 + (b5 + 5*b4 - b3 + 3*b1) * q^41 + (-2*b5 - 2*b3 + 4) * q^43 + (b5 - b4 - b3 + b1) * q^45 + (-b5 + b4 + b3 + b1) * q^47 - q^49 + (-b5 - b3 + 2*b2) * q^51 - 6 * q^53 + (3*b5 + 3*b3 - 2*b2 + 4) * q^55 + (2*b4 - 2*b1) * q^57 + (-b5 + 3*b4 + b3 - b1) * q^59 + (3*b5 + 3*b3 - 8*b2 + 4) * q^61 + b4 * q^63 + (4*b5 - 3*b4 + 2*b3 - 4*b2 + b1 + 4) * q^65 + (-6*b4 - 2*b1) * q^67 - 2*b2 * q^69 + (4*b5 + b4 - 4*b3 + 5*b1) * q^71 + (3*b5 - 6*b4 - 3*b3 + 4*b1) * q^73 + (3*b5 + 3*b3 - 2*b2 + 3) * q^75 + (b2 - 3) * q^77 + (b5 + b3 + 6) * q^79 + q^81 + (b5 - 5*b4 - b3 - 5*b1) * q^83 + (2*b5 - 8*b4 - 2*b3) * q^85 + (2*b5 + 2*b3 + 2) * q^87 + (-b5 - 3*b4 + b3 - 5*b1) * q^89 + (-b5 - b2 - 2) * q^91 + (-2*b5 + 4*b4 + 2*b3 - 4*b1) * q^93 + (-2*b5 - 2*b3 - 4) * q^95 + (b5 - b3 + 2*b1) * q^97 + (3*b4 - b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{3} + 6 q^{9}+O(q^{10})$$ 6 * q - 6 * q^3 + 6 * q^9 $$6 q - 6 q^{3} + 6 q^{9} - 2 q^{13} - 4 q^{23} - 10 q^{25} - 6 q^{27} - 4 q^{29} + 4 q^{35} + 2 q^{39} + 32 q^{43} - 6 q^{49} - 36 q^{53} + 16 q^{55} + 28 q^{61} + 20 q^{65} + 4 q^{69} + 10 q^{75} - 20 q^{77} + 32 q^{79} + 6 q^{81} + 4 q^{87} - 8 q^{91} - 16 q^{95}+O(q^{100})$$ 6 * q - 6 * q^3 + 6 * q^9 - 2 * q^13 - 4 * q^23 - 10 * q^25 - 6 * q^27 - 4 * q^29 + 4 * q^35 + 2 * q^39 + 32 * q^43 - 6 * q^49 - 36 * q^53 + 16 * q^55 + 28 * q^61 + 20 * q^65 + 4 * q^69 + 10 * q^75 - 20 * q^77 + 32 * q^79 + 6 * q^81 + 4 * q^87 - 8 * q^91 - 16 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2$$ :

 $$\beta_{1}$$ $$=$$ $$( -3\nu^{5} + \nu^{4} + 11\nu^{3} - 26\nu^{2} + 6\nu - 1 ) / 23$$ (-3*v^5 + v^4 + 11*v^3 - 26*v^2 + 6*v - 1) / 23 $$\beta_{2}$$ $$=$$ $$( -4\nu^{5} + 9\nu^{4} - 16\nu^{3} - 4\nu^{2} + 8\nu - 9 ) / 23$$ (-4*v^5 + 9*v^4 - 16*v^3 - 4*v^2 + 8*v - 9) / 23 $$\beta_{3}$$ $$=$$ $$( 6\nu^{5} - 2\nu^{4} + \nu^{3} + 6\nu^{2} + 80\nu + 2 ) / 23$$ (6*v^5 - 2*v^4 + v^3 + 6*v^2 + 80*v + 2) / 23 $$\beta_{4}$$ $$=$$ $$( 7\nu^{5} - 10\nu^{4} + 5\nu^{3} + 30\nu^{2} + 32\nu - 13 ) / 23$$ (7*v^5 - 10*v^4 + 5*v^3 + 30*v^2 + 32*v - 13) / 23 $$\beta_{5}$$ $$=$$ $$( -16\nu^{5} + 36\nu^{4} - 41\nu^{3} - 16\nu^{2} - 60\nu + 56 ) / 23$$ (-16*v^5 + 36*v^4 - 41*v^3 - 16*v^2 - 60*v + 56) / 23
 $$\nu$$ $$=$$ $$( \beta_{4} + \beta_{2} + \beta _1 + 1 ) / 2$$ (b4 + b2 + b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{5} + 4\beta_{4} - \beta_{3} + 2\beta_1 ) / 2$$ (b5 + 4*b4 - b3 + 2*b1) / 2 $$\nu^{3}$$ $$=$$ $$\beta_{5} + 2\beta_{4} - 2\beta_{2} + 2\beta _1 - 2$$ b5 + 2*b4 - 2*b2 + 2*b1 - 2 $$\nu^{4}$$ $$=$$ $$2\beta_{5} + 2\beta_{3} - 5\beta_{2} - 7$$ 2*b5 + 2*b3 - 5*b2 - 7 $$\nu^{5}$$ $$=$$ $$-9\beta_{4} + 5\beta_{3} - 8\beta_{2} - 8\beta _1 - 9$$ -9*b4 + 5*b3 - 8*b2 - 8*b1 - 9

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4368\mathbb{Z}\right)^\times$$.

 $$n$$ $$1093$$ $$1249$$ $$1457$$ $$2017$$ $$3823$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$1$$

## 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
 0.403032 + 0.403032i 1.45161 − 1.45161i −0.854638 + 0.854638i −0.854638 − 0.854638i 1.45161 + 1.45161i 0.403032 − 0.403032i
0 −1.00000 0 4.15633i 0 1.00000i 0 1.00000 0
337.2 0 −1.00000 0 1.52543i 0 1.00000i 0 1.00000 0
337.3 0 −1.00000 0 0.630898i 0 1.00000i 0 1.00000 0
337.4 0 −1.00000 0 0.630898i 0 1.00000i 0 1.00000 0
337.5 0 −1.00000 0 1.52543i 0 1.00000i 0 1.00000 0
337.6 0 −1.00000 0 4.15633i 0 1.00000i 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 337.6 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 4368.2.h.o 6
4.b odd 2 1 273.2.c.b 6
12.b even 2 1 819.2.c.c 6
13.b even 2 1 inner 4368.2.h.o 6
28.d even 2 1 1911.2.c.h 6
52.b odd 2 1 273.2.c.b 6
52.f even 4 1 3549.2.a.k 3
52.f even 4 1 3549.2.a.q 3
156.h even 2 1 819.2.c.c 6
364.h even 2 1 1911.2.c.h 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.c.b 6 4.b odd 2 1
273.2.c.b 6 52.b odd 2 1
819.2.c.c 6 12.b even 2 1
819.2.c.c 6 156.h even 2 1
1911.2.c.h 6 28.d even 2 1
1911.2.c.h 6 364.h even 2 1
3549.2.a.k 3 52.f even 4 1
3549.2.a.q 3 52.f even 4 1
4368.2.h.o 6 1.a even 1 1 trivial
4368.2.h.o 6 13.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(4368, [\chi])$$:

 $$T_{5}^{6} + 20T_{5}^{4} + 48T_{5}^{2} + 16$$ T5^6 + 20*T5^4 + 48*T5^2 + 16 $$T_{11}^{6} + 44T_{11}^{4} + 384T_{11}^{2} + 400$$ T11^6 + 44*T11^4 + 384*T11^2 + 400 $$T_{17}^{3} - 16T_{17} + 16$$ T17^3 - 16*T17 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$(T + 1)^{6}$$
$5$ $$T^{6} + 20 T^{4} + 48 T^{2} + 16$$
$7$ $$(T^{2} + 1)^{3}$$
$11$ $$T^{6} + 44 T^{4} + 384 T^{2} + \cdots + 400$$
$13$ $$T^{6} + 2 T^{5} + 27 T^{4} + \cdots + 2197$$
$17$ $$(T^{3} - 16 T + 16)^{2}$$
$19$ $$T^{6} + 64 T^{4} + 512 T^{2} + \cdots + 1024$$
$23$ $$(T^{3} + 2 T^{2} - 20 T - 8)^{2}$$
$29$ $$(T^{3} + 2 T^{2} - 52 T - 40)^{2}$$
$31$ $$T^{6} + 176 T^{4} + 7936 T^{2} + \cdots + 102400$$
$37$ $$T^{6} + 48 T^{4} + 512 T^{2} + \cdots + 1024$$
$41$ $$T^{6} + 132 T^{4} + 464 T^{2} + \cdots + 400$$
$43$ $$(T^{3} - 16 T^{2} + 32 T + 128)^{2}$$
$47$ $$T^{6} + 56 T^{4} + 784 T^{2} + \cdots + 2704$$
$53$ $$(T + 6)^{6}$$
$59$ $$T^{6} + 40 T^{4} + 80 T^{2} + 16$$
$61$ $$(T^{3} - 14 T^{2} - 172 T + 2392)^{2}$$
$67$ $$T^{6} + 128 T^{4} + 3072 T^{2} + \cdots + 1024$$
$71$ $$T^{6} + 332 T^{4} + 30784 T^{2} + \cdots + 547600$$
$73$ $$T^{6} + 272 T^{4} + 6720 T^{2} + \cdots + 43264$$
$79$ $$(T^{3} - 16 T^{2} + 72 T - 80)^{2}$$
$83$ $$T^{6} + 408 T^{4} + 49616 T^{2} + \cdots + 1567504$$
$89$ $$T^{6} + 212 T^{4} + 6832 T^{2} + \cdots + 5776$$
$97$ $$T^{6} + 32 T^{4} + 256 T^{2} + \cdots + 256$$