# Properties

 Label 1183.2.c.e 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: $$\Q(\zeta_{12})$$ 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} + (\beta_{3} + 1) q^{3} - q^{4} - \beta_{2} q^{5} + ( - \beta_{2} - 3 \beta_1) q^{6} + \beta_1 q^{7} - \beta_{2} q^{8} + (2 \beta_{3} + 1) q^{9}+O(q^{10})$$ q - b2 * q^2 + (b3 + 1) * q^3 - q^4 - b2 * q^5 + (-b2 - 3*b1) * q^6 + b1 * q^7 - b2 * q^8 + (2*b3 + 1) * q^9 $$q - \beta_{2} q^{2} + (\beta_{3} + 1) q^{3} - q^{4} - \beta_{2} q^{5} + ( - \beta_{2} - 3 \beta_1) q^{6} + \beta_1 q^{7} - \beta_{2} q^{8} + (2 \beta_{3} + 1) q^{9} - 3 q^{10} + ( - \beta_{2} + 3 \beta_1) q^{11} + ( - \beta_{3} - 1) q^{12} + \beta_{3} q^{14} + ( - \beta_{2} - 3 \beta_1) q^{15} - 5 q^{16} + (\beta_{3} + 6) q^{17} + ( - \beta_{2} - 6 \beta_1) q^{18} - 2 \beta_1 q^{19} + \beta_{2} q^{20} + (\beta_{2} + \beta_1) q^{21} + (3 \beta_{3} - 3) q^{22} + ( - \beta_{3} - 3) q^{23} + ( - \beta_{2} - 3 \beta_1) q^{24} + 2 q^{25} + 4 q^{27} - \beta_1 q^{28} - 3 q^{29} + ( - 3 \beta_{3} - 3) q^{30} + ( - 3 \beta_{2} + \beta_1) q^{31} + 3 \beta_{2} q^{32} + 2 \beta_{2} q^{33} + ( - 6 \beta_{2} - 3 \beta_1) q^{34} + \beta_{3} q^{35} + ( - 2 \beta_{3} - 1) q^{36} - 7 \beta_1 q^{37} - 2 \beta_{3} q^{38} - 3 q^{40} + 3 \beta_{2} q^{41} + (\beta_{3} + 3) q^{42} + (3 \beta_{3} - 5) q^{43} + (\beta_{2} - 3 \beta_1) q^{44} + ( - \beta_{2} - 6 \beta_1) q^{45} + (3 \beta_{2} + 3 \beta_1) q^{46} + (4 \beta_{2} + 6 \beta_1) q^{47} + ( - 5 \beta_{3} - 5) q^{48} - q^{49} - 2 \beta_{2} q^{50} + (7 \beta_{3} + 9) q^{51} + ( - 4 \beta_{3} - 3) q^{53} - 4 \beta_{2} q^{54} + (3 \beta_{3} - 3) q^{55} + \beta_{3} q^{56} + ( - 2 \beta_{2} - 2 \beta_1) q^{57} + 3 \beta_{2} q^{58} + ( - \beta_{2} + 9 \beta_1) q^{59} + (\beta_{2} + 3 \beta_1) q^{60} + (3 \beta_{3} - 10) q^{61} + (\beta_{3} - 9) q^{62} + (2 \beta_{2} + \beta_1) q^{63} - q^{64} + 6 q^{66} + (3 \beta_{2} + \beta_1) q^{67} + ( - \beta_{3} - 6) q^{68} + ( - 4 \beta_{3} - 6) q^{69} - 3 \beta_1 q^{70} - 6 \beta_1 q^{71} + ( - \beta_{2} - 6 \beta_1) q^{72} + ( - 3 \beta_{2} + 2 \beta_1) q^{73} - 7 \beta_{3} q^{74} + (2 \beta_{3} + 2) q^{75} + 2 \beta_1 q^{76} + (\beta_{3} - 3) q^{77} + (3 \beta_{3} + 11) q^{79} + 5 \beta_{2} q^{80} + ( - 2 \beta_{3} + 1) q^{81} + 9 q^{82} + (3 \beta_{2} - 3 \beta_1) q^{83} + ( - \beta_{2} - \beta_1) q^{84} + ( - 6 \beta_{2} - 3 \beta_1) q^{85} + (5 \beta_{2} - 9 \beta_1) q^{86} + ( - 3 \beta_{3} - 3) q^{87} + (3 \beta_{3} - 3) q^{88} + (4 \beta_{2} + 6 \beta_1) q^{89} + ( - 6 \beta_{3} - 3) q^{90} + (\beta_{3} + 3) q^{92} + ( - 2 \beta_{2} - 8 \beta_1) q^{93} + (6 \beta_{3} + 12) q^{94} - 2 \beta_{3} q^{95} + (3 \beta_{2} + 9 \beta_1) q^{96} + ( - 6 \beta_{2} + 4 \beta_1) q^{97} + \beta_{2} q^{98} + (5 \beta_{2} - 3 \beta_1) q^{99}+O(q^{100})$$ q - b2 * q^2 + (b3 + 1) * q^3 - q^4 - b2 * q^5 + (-b2 - 3*b1) * q^6 + b1 * q^7 - b2 * q^8 + (2*b3 + 1) * q^9 - 3 * q^10 + (-b2 + 3*b1) * q^11 + (-b3 - 1) * q^12 + b3 * q^14 + (-b2 - 3*b1) * q^15 - 5 * q^16 + (b3 + 6) * q^17 + (-b2 - 6*b1) * q^18 - 2*b1 * q^19 + b2 * q^20 + (b2 + b1) * q^21 + (3*b3 - 3) * q^22 + (-b3 - 3) * q^23 + (-b2 - 3*b1) * q^24 + 2 * q^25 + 4 * q^27 - b1 * q^28 - 3 * q^29 + (-3*b3 - 3) * q^30 + (-3*b2 + b1) * q^31 + 3*b2 * q^32 + 2*b2 * q^33 + (-6*b2 - 3*b1) * q^34 + b3 * q^35 + (-2*b3 - 1) * q^36 - 7*b1 * q^37 - 2*b3 * q^38 - 3 * q^40 + 3*b2 * q^41 + (b3 + 3) * q^42 + (3*b3 - 5) * q^43 + (b2 - 3*b1) * q^44 + (-b2 - 6*b1) * q^45 + (3*b2 + 3*b1) * q^46 + (4*b2 + 6*b1) * q^47 + (-5*b3 - 5) * q^48 - q^49 - 2*b2 * q^50 + (7*b3 + 9) * q^51 + (-4*b3 - 3) * q^53 - 4*b2 * q^54 + (3*b3 - 3) * q^55 + b3 * q^56 + (-2*b2 - 2*b1) * q^57 + 3*b2 * q^58 + (-b2 + 9*b1) * q^59 + (b2 + 3*b1) * q^60 + (3*b3 - 10) * q^61 + (b3 - 9) * q^62 + (2*b2 + b1) * q^63 - q^64 + 6 * q^66 + (3*b2 + b1) * q^67 + (-b3 - 6) * q^68 + (-4*b3 - 6) * q^69 - 3*b1 * q^70 - 6*b1 * q^71 + (-b2 - 6*b1) * q^72 + (-3*b2 + 2*b1) * q^73 - 7*b3 * q^74 + (2*b3 + 2) * q^75 + 2*b1 * q^76 + (b3 - 3) * q^77 + (3*b3 + 11) * q^79 + 5*b2 * q^80 + (-2*b3 + 1) * q^81 + 9 * q^82 + (3*b2 - 3*b1) * q^83 + (-b2 - b1) * q^84 + (-6*b2 - 3*b1) * q^85 + (5*b2 - 9*b1) * q^86 + (-3*b3 - 3) * q^87 + (3*b3 - 3) * q^88 + (4*b2 + 6*b1) * q^89 + (-6*b3 - 3) * q^90 + (b3 + 3) * q^92 + (-2*b2 - 8*b1) * q^93 + (6*b3 + 12) * q^94 - 2*b3 * q^95 + (3*b2 + 9*b1) * q^96 + (-6*b2 + 4*b1) * q^97 + b2 * q^98 + (5*b2 - 3*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{3} - 4 q^{4} + 4 q^{9}+O(q^{10})$$ 4 * q + 4 * q^3 - 4 * q^4 + 4 * q^9 $$4 q + 4 q^{3} - 4 q^{4} + 4 q^{9} - 12 q^{10} - 4 q^{12} - 20 q^{16} + 24 q^{17} - 12 q^{22} - 12 q^{23} + 8 q^{25} + 16 q^{27} - 12 q^{29} - 12 q^{30} - 4 q^{36} - 12 q^{40} + 12 q^{42} - 20 q^{43} - 20 q^{48} - 4 q^{49} + 36 q^{51} - 12 q^{53} - 12 q^{55} - 40 q^{61} - 36 q^{62} - 4 q^{64} + 24 q^{66} - 24 q^{68} - 24 q^{69} + 8 q^{75} - 12 q^{77} + 44 q^{79} + 4 q^{81} + 36 q^{82} - 12 q^{87} - 12 q^{88} - 12 q^{90} + 12 q^{92} + 48 q^{94}+O(q^{100})$$ 4 * q + 4 * q^3 - 4 * q^4 + 4 * q^9 - 12 * q^10 - 4 * q^12 - 20 * q^16 + 24 * q^17 - 12 * q^22 - 12 * q^23 + 8 * q^25 + 16 * q^27 - 12 * q^29 - 12 * q^30 - 4 * q^36 - 12 * q^40 + 12 * q^42 - 20 * q^43 - 20 * q^48 - 4 * q^49 + 36 * q^51 - 12 * q^53 - 12 * q^55 - 40 * q^61 - 36 * q^62 - 4 * q^64 + 24 * q^66 - 24 * q^68 - 24 * q^69 + 8 * q^75 - 12 * q^77 + 44 * q^79 + 4 * q^81 + 36 * q^82 - 12 * q^87 - 12 * q^88 - 12 * q^90 + 12 * q^92 + 48 * q^94

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{12}^{3}$$ v^3 $$\beta_{2}$$ $$=$$ $$2\zeta_{12}^{2} - 1$$ 2*v^2 - 1 $$\beta_{3}$$ $$=$$ $$-\zeta_{12}^{3} + 2\zeta_{12}$$ -v^3 + 2*v
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + \beta_1 ) / 2$$ (b3 + b1) / 2 $$\zeta_{12}^{2}$$ $$=$$ $$( \beta_{2} + 1 ) / 2$$ (b2 + 1) / 2 $$\zeta_{12}^{3}$$ $$=$$ $$\beta_1$$ b1

## 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
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
1.73205i −0.732051 −1.00000 1.73205i 1.26795i 1.00000i 1.73205i −2.46410 −3.00000
337.2 1.73205i 2.73205 −1.00000 1.73205i 4.73205i 1.00000i 1.73205i 4.46410 −3.00000
337.3 1.73205i −0.732051 −1.00000 1.73205i 1.26795i 1.00000i 1.73205i −2.46410 −3.00000
337.4 1.73205i 2.73205 −1.00000 1.73205i 4.73205i 1.00000i 1.73205i 4.46410 −3.00000
 $$n$$: 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 1183.2.c.e 4
13.b even 2 1 inner 1183.2.c.e 4
13.d odd 4 1 1183.2.a.e 2
13.d odd 4 1 1183.2.a.f 2
13.f odd 12 2 91.2.f.b 4
39.k even 12 2 819.2.o.b 4
52.l even 12 2 1456.2.s.o 4
91.i even 4 1 8281.2.a.r 2
91.i even 4 1 8281.2.a.t 2
91.w even 12 2 637.2.g.d 4
91.x odd 12 2 637.2.h.d 4
91.ba even 12 2 637.2.h.e 4
91.bc even 12 2 637.2.f.d 4
91.bd odd 12 2 637.2.g.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.f.b 4 13.f odd 12 2
637.2.f.d 4 91.bc even 12 2
637.2.g.d 4 91.w even 12 2
637.2.g.e 4 91.bd odd 12 2
637.2.h.d 4 91.x odd 12 2
637.2.h.e 4 91.ba even 12 2
819.2.o.b 4 39.k even 12 2
1183.2.a.e 2 13.d odd 4 1
1183.2.a.f 2 13.d odd 4 1
1183.2.c.e 4 1.a even 1 1 trivial
1183.2.c.e 4 13.b even 2 1 inner
1456.2.s.o 4 52.l even 12 2
8281.2.a.r 2 91.i even 4 1
8281.2.a.t 2 91.i even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 3$$ acting on $$S_{2}^{\mathrm{new}}(1183, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 3)^{2}$$
$3$ $$(T^{2} - 2 T - 2)^{2}$$
$5$ $$(T^{2} + 3)^{2}$$
$7$ $$(T^{2} + 1)^{2}$$
$11$ $$T^{4} + 24T^{2} + 36$$
$13$ $$T^{4}$$
$17$ $$(T^{2} - 12 T + 33)^{2}$$
$19$ $$(T^{2} + 4)^{2}$$
$23$ $$(T^{2} + 6 T + 6)^{2}$$
$29$ $$(T + 3)^{4}$$
$31$ $$T^{4} + 56T^{2} + 676$$
$37$ $$(T^{2} + 49)^{2}$$
$41$ $$(T^{2} + 27)^{2}$$
$43$ $$(T^{2} + 10 T - 2)^{2}$$
$47$ $$T^{4} + 168T^{2} + 144$$
$53$ $$(T^{2} + 6 T - 39)^{2}$$
$59$ $$T^{4} + 168T^{2} + 6084$$
$61$ $$(T^{2} + 20 T + 73)^{2}$$
$67$ $$T^{4} + 56T^{2} + 676$$
$71$ $$(T^{2} + 36)^{2}$$
$73$ $$T^{4} + 62T^{2} + 529$$
$79$ $$(T^{2} - 22 T + 94)^{2}$$
$83$ $$T^{4} + 72T^{2} + 324$$
$89$ $$T^{4} + 168T^{2} + 144$$
$97$ $$T^{4} + 248T^{2} + 8464$$