# Properties

 Label 1183.2.c.d Level $1183$ Weight $2$ Character orbit 1183.c Analytic conductor $9.446$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1183,2,Mod(337,1183)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1183, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1183.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1183 = 7 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1183.c (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: $$9.44630255912$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 91) 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} - \beta_{3} q^{3} + (\beta_{2} + 3 \beta_1) q^{5} - 2 \beta_1 q^{6} - \beta_1 q^{7} + 2 \beta_{2} q^{8} - q^{9}+O(q^{10})$$ q + b2 * q^2 - b3 * q^3 + (b2 + 3*b1) * q^5 - 2*b1 * q^6 - b1 * q^7 + 2*b2 * q^8 - q^9 $$q + \beta_{2} q^{2} - \beta_{3} q^{3} + (\beta_{2} + 3 \beta_1) q^{5} - 2 \beta_1 q^{6} - \beta_1 q^{7} + 2 \beta_{2} q^{8} - q^{9} + ( - 3 \beta_{3} - 2) q^{10} + 3 \beta_{2} q^{11} + \beta_{3} q^{14} + ( - 3 \beta_{2} - 2 \beta_1) q^{15} - 4 q^{16} + \beta_{3} q^{17} - \beta_{2} q^{18} + (3 \beta_{2} - 3 \beta_1) q^{19} + \beta_{2} q^{21} - 6 q^{22} + ( - 2 \beta_{3} + 3) q^{23} - 4 \beta_1 q^{24} + ( - 6 \beta_{3} - 6) q^{25} + 4 \beta_{3} q^{27} + (2 \beta_{3} + 3) q^{29} + (2 \beta_{3} + 6) q^{30} + ( - 3 \beta_{2} - \beta_1) q^{31} - 6 \beta_1 q^{33} + 2 \beta_1 q^{34} + (\beta_{3} + 3) q^{35} + (3 \beta_{2} + 2 \beta_1) q^{37} + (3 \beta_{3} - 6) q^{38} + ( - 6 \beta_{3} - 4) q^{40} + ( - 2 \beta_{2} + 6 \beta_1) q^{41} - 2 q^{42} + 5 q^{43} + ( - \beta_{2} - 3 \beta_1) q^{45} + (3 \beta_{2} - 4 \beta_1) q^{46} + ( - \beta_{2} - 3 \beta_1) q^{47} + 4 \beta_{3} q^{48} - q^{49} + ( - 6 \beta_{2} - 12 \beta_1) q^{50} - 2 q^{51} + ( - 2 \beta_{3} - 3) q^{53} + 8 \beta_1 q^{54} + ( - 9 \beta_{3} - 6) q^{55} + 2 \beta_{3} q^{56} + (3 \beta_{2} - 6 \beta_1) q^{57} + (3 \beta_{2} + 4 \beta_1) q^{58} + ( - 4 \beta_{2} - 6 \beta_1) q^{59} + 6 q^{61} + (\beta_{3} + 6) q^{62} + \beta_1 q^{63} - 8 q^{64} + 6 \beta_{3} q^{66} + (6 \beta_{2} - 6 \beta_1) q^{67} + ( - 3 \beta_{3} + 4) q^{69} + (3 \beta_{2} + 2 \beta_1) q^{70} + (5 \beta_{2} - 6 \beta_1) q^{71} - 2 \beta_{2} q^{72} + ( - 3 \beta_{2} + 5 \beta_1) q^{73} + ( - 2 \beta_{3} - 6) q^{74} + (6 \beta_{3} + 12) q^{75} + 3 \beta_{3} q^{77} + ( - 6 \beta_{3} + 7) q^{79} + ( - 4 \beta_{2} - 12 \beta_1) q^{80} - 5 q^{81} + ( - 6 \beta_{3} + 4) q^{82} + ( - 3 \beta_{2} + 9 \beta_1) q^{83} + (3 \beta_{2} + 2 \beta_1) q^{85} + 5 \beta_{2} q^{86} + ( - 3 \beta_{3} - 4) q^{87} - 12 q^{88} + ( - \beta_{2} - 3 \beta_1) q^{89} + (3 \beta_{3} + 2) q^{90} + (\beta_{2} + 6 \beta_1) q^{93} + (3 \beta_{3} + 2) q^{94} + ( - 6 \beta_{3} + 3) q^{95} + ( - 9 \beta_{2} - \beta_1) q^{97} - \beta_{2} q^{98} - 3 \beta_{2} q^{99}+O(q^{100})$$ q + b2 * q^2 - b3 * q^3 + (b2 + 3*b1) * q^5 - 2*b1 * q^6 - b1 * q^7 + 2*b2 * q^8 - q^9 + (-3*b3 - 2) * q^10 + 3*b2 * q^11 + b3 * q^14 + (-3*b2 - 2*b1) * q^15 - 4 * q^16 + b3 * q^17 - b2 * q^18 + (3*b2 - 3*b1) * q^19 + b2 * q^21 - 6 * q^22 + (-2*b3 + 3) * q^23 - 4*b1 * q^24 + (-6*b3 - 6) * q^25 + 4*b3 * q^27 + (2*b3 + 3) * q^29 + (2*b3 + 6) * q^30 + (-3*b2 - b1) * q^31 - 6*b1 * q^33 + 2*b1 * q^34 + (b3 + 3) * q^35 + (3*b2 + 2*b1) * q^37 + (3*b3 - 6) * q^38 + (-6*b3 - 4) * q^40 + (-2*b2 + 6*b1) * q^41 - 2 * q^42 + 5 * q^43 + (-b2 - 3*b1) * q^45 + (3*b2 - 4*b1) * q^46 + (-b2 - 3*b1) * q^47 + 4*b3 * q^48 - q^49 + (-6*b2 - 12*b1) * q^50 - 2 * q^51 + (-2*b3 - 3) * q^53 + 8*b1 * q^54 + (-9*b3 - 6) * q^55 + 2*b3 * q^56 + (3*b2 - 6*b1) * q^57 + (3*b2 + 4*b1) * q^58 + (-4*b2 - 6*b1) * q^59 + 6 * q^61 + (b3 + 6) * q^62 + b1 * q^63 - 8 * q^64 + 6*b3 * q^66 + (6*b2 - 6*b1) * q^67 + (-3*b3 + 4) * q^69 + (3*b2 + 2*b1) * q^70 + (5*b2 - 6*b1) * q^71 - 2*b2 * q^72 + (-3*b2 + 5*b1) * q^73 + (-2*b3 - 6) * q^74 + (6*b3 + 12) * q^75 + 3*b3 * q^77 + (-6*b3 + 7) * q^79 + (-4*b2 - 12*b1) * q^80 - 5 * q^81 + (-6*b3 + 4) * q^82 + (-3*b2 + 9*b1) * q^83 + (3*b2 + 2*b1) * q^85 + 5*b2 * q^86 + (-3*b3 - 4) * q^87 - 12 * q^88 + (-b2 - 3*b1) * q^89 + (3*b3 + 2) * q^90 + (b2 + 6*b1) * q^93 + (3*b3 + 2) * q^94 + (-6*b3 + 3) * q^95 + (-9*b2 - b1) * q^97 - b2 * q^98 - 3*b2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^9 $$4 q - 4 q^{9} - 8 q^{10} - 16 q^{16} - 24 q^{22} + 12 q^{23} - 24 q^{25} + 12 q^{29} + 24 q^{30} + 12 q^{35} - 24 q^{38} - 16 q^{40} - 8 q^{42} + 20 q^{43} - 4 q^{49} - 8 q^{51} - 12 q^{53} - 24 q^{55} + 24 q^{61} + 24 q^{62} - 32 q^{64} + 16 q^{69} - 24 q^{74} + 48 q^{75} + 28 q^{79} - 20 q^{81} + 16 q^{82} - 16 q^{87} - 48 q^{88} + 8 q^{90} + 8 q^{94} + 12 q^{95}+O(q^{100})$$ 4 * q - 4 * q^9 - 8 * q^10 - 16 * q^16 - 24 * q^22 + 12 * q^23 - 24 * q^25 + 12 * q^29 + 24 * q^30 + 12 * q^35 - 24 * q^38 - 16 * q^40 - 8 * q^42 + 20 * q^43 - 4 * q^49 - 8 * q^51 - 12 * q^53 - 24 * q^55 + 24 * q^61 + 24 * q^62 - 32 * q^64 + 16 * q^69 - 24 * q^74 + 48 * q^75 + 28 * q^79 - 20 * q^81 + 16 * q^82 - 16 * q^87 - 48 * q^88 + 8 * q^90 + 8 * q^94 + 12 * q^95

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{8}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{8}^{3} + \zeta_{8}$$ v^3 + v $$\beta_{3}$$ $$=$$ $$-\zeta_{8}^{3} + \zeta_{8}$$ -v^3 + v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 2$$ (b3 + b2) / 2 $$\zeta_{8}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{8}^{3}$$ $$=$$ $$( -\beta_{3} + \beta_{2} ) / 2$$ (-b3 + b2) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1183\mathbb{Z}\right)^\times$$.

 $$n$$ $$339$$ $$1016$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
 0.707107 − 0.707107i −0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i
1.41421i −1.41421 0 4.41421i 2.00000i 1.00000i 2.82843i −1.00000 −6.24264
337.2 1.41421i 1.41421 0 1.58579i 2.00000i 1.00000i 2.82843i −1.00000 2.24264
337.3 1.41421i −1.41421 0 4.41421i 2.00000i 1.00000i 2.82843i −1.00000 −6.24264
337.4 1.41421i 1.41421 0 1.58579i 2.00000i 1.00000i 2.82843i −1.00000 2.24264
 $$n$$: 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 1183.2.c.d 4
13.b even 2 1 inner 1183.2.c.d 4
13.d odd 4 1 91.2.a.c 2
13.d odd 4 1 1183.2.a.d 2
39.f even 4 1 819.2.a.h 2
52.f even 4 1 1456.2.a.q 2
65.g odd 4 1 2275.2.a.j 2
91.i even 4 1 637.2.a.g 2
91.i even 4 1 8281.2.a.v 2
91.z odd 12 2 637.2.e.f 4
91.bb even 12 2 637.2.e.g 4
104.j odd 4 1 5824.2.a.bl 2
104.m even 4 1 5824.2.a.bk 2
273.o odd 4 1 5733.2.a.s 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.a.c 2 13.d odd 4 1
637.2.a.g 2 91.i even 4 1
637.2.e.f 4 91.z odd 12 2
637.2.e.g 4 91.bb even 12 2
819.2.a.h 2 39.f even 4 1
1183.2.a.d 2 13.d odd 4 1
1183.2.c.d 4 1.a even 1 1 trivial
1183.2.c.d 4 13.b even 2 1 inner
1456.2.a.q 2 52.f even 4 1
2275.2.a.j 2 65.g odd 4 1
5733.2.a.s 2 273.o odd 4 1
5824.2.a.bk 2 104.m even 4 1
5824.2.a.bl 2 104.j odd 4 1
8281.2.a.v 2 91.i even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 2$$ acting on $$S_{2}^{\mathrm{new}}(1183, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 2)^{2}$$
$3$ $$(T^{2} - 2)^{2}$$
$5$ $$T^{4} + 22T^{2} + 49$$
$7$ $$(T^{2} + 1)^{2}$$
$11$ $$(T^{2} + 18)^{2}$$
$13$ $$T^{4}$$
$17$ $$(T^{2} - 2)^{2}$$
$19$ $$T^{4} + 54T^{2} + 81$$
$23$ $$(T^{2} - 6 T + 1)^{2}$$
$29$ $$(T^{2} - 6 T + 1)^{2}$$
$31$ $$T^{4} + 38T^{2} + 289$$
$37$ $$T^{4} + 44T^{2} + 196$$
$41$ $$T^{4} + 88T^{2} + 784$$
$43$ $$(T - 5)^{4}$$
$47$ $$T^{4} + 22T^{2} + 49$$
$53$ $$(T^{2} + 6 T + 1)^{2}$$
$59$ $$T^{4} + 136T^{2} + 16$$
$61$ $$(T - 6)^{4}$$
$67$ $$T^{4} + 216T^{2} + 1296$$
$71$ $$T^{4} + 172T^{2} + 196$$
$73$ $$T^{4} + 86T^{2} + 49$$
$79$ $$(T^{2} - 14 T - 23)^{2}$$
$83$ $$T^{4} + 198T^{2} + 3969$$
$89$ $$T^{4} + 22T^{2} + 49$$
$97$ $$T^{4} + 326 T^{2} + 25921$$