# Properties

 Label 1183.2.c.f Level $1183$ Weight $2$ Character orbit 1183.c Analytic conductor $9.446$ 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] = [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: $$6$$ Coefficient field: 6.0.399424.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8$$ x^6 - 2*x^5 + 3*x^4 - 6*x^3 + 6*x^2 - 8*x + 8 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{5} q^{2} + ( - \beta_{3} - \beta_1 - 1) q^{3} + ( - \beta_1 - 1) q^{4} + (\beta_{5} - \beta_{2}) q^{5} + (2 \beta_{5} - 2 \beta_{2}) q^{6} - \beta_{2} q^{7} + ( - \beta_{4} + \beta_{2}) q^{8} + (2 \beta_{3} + 3) q^{9}+O(q^{10})$$ q - b5 * q^2 + (-b3 - b1 - 1) * q^3 + (-b1 - 1) * q^4 + (b5 - b2) * q^5 + (2*b5 - 2*b2) * q^6 - b2 * q^7 + (-b4 + b2) * q^8 + (2*b3 + 3) * q^9 $$q - \beta_{5} q^{2} + ( - \beta_{3} - \beta_1 - 1) q^{3} + ( - \beta_1 - 1) q^{4} + (\beta_{5} - \beta_{2}) q^{5} + (2 \beta_{5} - 2 \beta_{2}) q^{6} - \beta_{2} q^{7} + ( - \beta_{4} + \beta_{2}) q^{8} + (2 \beta_{3} + 3) q^{9} + (\beta_{3} + \beta_1 + 3) q^{10} + ( - \beta_{5} - \beta_{4} + \beta_{2}) q^{11} + 4 q^{12} + \beta_{3} q^{14} + ( - 3 \beta_{5} - \beta_{4} + 3 \beta_{2}) q^{15} + ( - 2 \beta_{3} - \beta_1 - 1) q^{16} + (\beta_{3} - \beta_1 - 1) q^{17} + ( - 3 \beta_{5} - 2 \beta_{4} + 6 \beta_{2}) q^{18} + (\beta_{5} + \beta_{2}) q^{19} - 2 \beta_{5} q^{20} + ( - \beta_{5} - \beta_{4} + \beta_{2}) q^{21} + ( - 2 \beta_{3} - 2) q^{22} + ( - 2 \beta_{3} + \beta_1 - 4) q^{23} - 4 \beta_{2} q^{24} + ( - 2 \beta_{3} - \beta_1 + 1) q^{25} + ( - 4 \beta_{3} - 4) q^{27} + ( - \beta_{4} + \beta_{2}) q^{28} + (\beta_1 + 8) q^{29} + ( - 4 \beta_{3} - 2 \beta_1 - 8) q^{30} + (\beta_{5} + 2 \beta_{4} + \beta_{2}) q^{31} + (2 \beta_{5} - \beta_{4} - 3 \beta_{2}) q^{32} + (2 \beta_{5} - 6 \beta_{2}) q^{33} + (2 \beta_{5} - 2 \beta_{4} + 4 \beta_{2}) q^{34} + ( - \beta_{3} - 1) q^{35} + ( - 4 \beta_{3} - \beta_1 - 1) q^{36} + (3 \beta_{5} - \beta_{4} - \beta_{2}) q^{37} + ( - \beta_{3} + \beta_1 + 3) q^{38} + 2 \beta_{3} q^{40} + ( - 2 \beta_{5} - 2 \beta_{4}) q^{41} + ( - 2 \beta_{3} - 2) q^{42} + ( - 2 \beta_{3} + 3 \beta_1 - 4) q^{43} - 4 \beta_{2} q^{44} + (5 \beta_{5} + 2 \beta_{4} - 9 \beta_{2}) q^{45} + (3 \beta_{5} + 3 \beta_{4} - 7 \beta_{2}) q^{46} + (\beta_{5} + 4 \beta_{4} - 3 \beta_{2}) q^{47} + (4 \beta_{3} + 8) q^{48} - q^{49} + (\beta_{4} - 5 \beta_{2}) q^{50} + ( - 2 \beta_{3} + 2) q^{51} + ( - 2 \beta_{3} - 3 \beta_1 + 2) q^{53} + (4 \beta_{5} + 4 \beta_{4} - 12 \beta_{2}) q^{54} + (3 \beta_{3} + \beta_1 + 3) q^{55} + (\beta_1 + 1) q^{56} + ( - \beta_{5} + \beta_{4} + \beta_{2}) q^{57} + ( - 9 \beta_{5} + \beta_{4} - \beta_{2}) q^{58} + (2 \beta_{5} - 4 \beta_{4} - 2 \beta_{2}) q^{59} + (4 \beta_{5} - 4 \beta_{2}) q^{60} - 2 q^{61} + (\beta_{3} - \beta_1 + 1) q^{62} + (2 \beta_{5} - 3 \beta_{2}) q^{63} + ( - 2 \beta_{3} + \beta_1 + 5) q^{64} + (6 \beta_{3} + 2 \beta_1 + 6) q^{66} + (6 \beta_{5} + 4 \beta_{4} + 2 \beta_{2}) q^{67} + ( - 4 \beta_{3} + 2 \beta_1 + 6) q^{68} + (9 \beta_{3} + 5 \beta_1 + 5) q^{69} + (\beta_{5} + \beta_{4} - 3 \beta_{2}) q^{70} + ( - 3 \beta_{5} - \beta_{4} + 3 \beta_{2}) q^{71} + ( - 4 \beta_{5} - \beta_{4} + \beta_{2}) q^{72} + ( - \beta_{5} + 4 \beta_{4} - 3 \beta_{2}) q^{73} + (4 \beta_1 + 10) q^{74} + (2 \beta_{3} - 2 \beta_1 + 6) q^{75} + ( - 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{2}) q^{76} + (\beta_{3} + \beta_1 + 1) q^{77} + ( - 4 \beta_{3} - \beta_1 - 6) q^{79} + ( - 4 \beta_{5} - 2 \beta_{4} + 6 \beta_{2}) q^{80} + (6 \beta_{3} + 4 \beta_1 + 3) q^{81} + ( - 2 \beta_{3} - 4) q^{82} + (9 \beta_{5} + 4 \beta_{4} + \beta_{2}) q^{83} - 4 \beta_{2} q^{84} + ( - \beta_{5} + \beta_{4} - 3 \beta_{2}) q^{85} + (\beta_{5} + 5 \beta_{4} - 9 \beta_{2}) q^{86} + ( - 7 \beta_{3} - 7 \beta_1 - 11) q^{87} - 4 q^{88} + (5 \beta_{5} + 2 \beta_{4} - \beta_{2}) q^{89} + (11 \beta_{3} + 3 \beta_1 + 13) q^{90} + (6 \beta_{3} + 2 \beta_1 - 2) q^{92} + (\beta_{5} + 3 \beta_{4} + 7 \beta_{2}) q^{93} + (7 \beta_{3} - 3 \beta_1 - 1) q^{94} + ( - \beta_1 - 2) q^{95} + ( - 8 \beta_{5} - 4 \beta_{4} + 4 \beta_{2}) q^{96} + (\beta_{5} + 3 \beta_{2}) q^{97} + \beta_{5} q^{98} + ( - 7 \beta_{5} - 3 \beta_{4} + 7 \beta_{2}) q^{99}+O(q^{100})$$ q - b5 * q^2 + (-b3 - b1 - 1) * q^3 + (-b1 - 1) * q^4 + (b5 - b2) * q^5 + (2*b5 - 2*b2) * q^6 - b2 * q^7 + (-b4 + b2) * q^8 + (2*b3 + 3) * q^9 + (b3 + b1 + 3) * q^10 + (-b5 - b4 + b2) * q^11 + 4 * q^12 + b3 * q^14 + (-3*b5 - b4 + 3*b2) * q^15 + (-2*b3 - b1 - 1) * q^16 + (b3 - b1 - 1) * q^17 + (-3*b5 - 2*b4 + 6*b2) * q^18 + (b5 + b2) * q^19 - 2*b5 * q^20 + (-b5 - b4 + b2) * q^21 + (-2*b3 - 2) * q^22 + (-2*b3 + b1 - 4) * q^23 - 4*b2 * q^24 + (-2*b3 - b1 + 1) * q^25 + (-4*b3 - 4) * q^27 + (-b4 + b2) * q^28 + (b1 + 8) * q^29 + (-4*b3 - 2*b1 - 8) * q^30 + (b5 + 2*b4 + b2) * q^31 + (2*b5 - b4 - 3*b2) * q^32 + (2*b5 - 6*b2) * q^33 + (2*b5 - 2*b4 + 4*b2) * q^34 + (-b3 - 1) * q^35 + (-4*b3 - b1 - 1) * q^36 + (3*b5 - b4 - b2) * q^37 + (-b3 + b1 + 3) * q^38 + 2*b3 * q^40 + (-2*b5 - 2*b4) * q^41 + (-2*b3 - 2) * q^42 + (-2*b3 + 3*b1 - 4) * q^43 - 4*b2 * q^44 + (5*b5 + 2*b4 - 9*b2) * q^45 + (3*b5 + 3*b4 - 7*b2) * q^46 + (b5 + 4*b4 - 3*b2) * q^47 + (4*b3 + 8) * q^48 - q^49 + (b4 - 5*b2) * q^50 + (-2*b3 + 2) * q^51 + (-2*b3 - 3*b1 + 2) * q^53 + (4*b5 + 4*b4 - 12*b2) * q^54 + (3*b3 + b1 + 3) * q^55 + (b1 + 1) * q^56 + (-b5 + b4 + b2) * q^57 + (-9*b5 + b4 - b2) * q^58 + (2*b5 - 4*b4 - 2*b2) * q^59 + (4*b5 - 4*b2) * q^60 - 2 * q^61 + (b3 - b1 + 1) * q^62 + (2*b5 - 3*b2) * q^63 + (-2*b3 + b1 + 5) * q^64 + (6*b3 + 2*b1 + 6) * q^66 + (6*b5 + 4*b4 + 2*b2) * q^67 + (-4*b3 + 2*b1 + 6) * q^68 + (9*b3 + 5*b1 + 5) * q^69 + (b5 + b4 - 3*b2) * q^70 + (-3*b5 - b4 + 3*b2) * q^71 + (-4*b5 - b4 + b2) * q^72 + (-b5 + 4*b4 - 3*b2) * q^73 + (4*b1 + 10) * q^74 + (2*b3 - 2*b1 + 6) * q^75 + (-2*b5 + 2*b4 - 2*b2) * q^76 + (b3 + b1 + 1) * q^77 + (-4*b3 - b1 - 6) * q^79 + (-4*b5 - 2*b4 + 6*b2) * q^80 + (6*b3 + 4*b1 + 3) * q^81 + (-2*b3 - 4) * q^82 + (9*b5 + 4*b4 + b2) * q^83 - 4*b2 * q^84 + (-b5 + b4 - 3*b2) * q^85 + (b5 + 5*b4 - 9*b2) * q^86 + (-7*b3 - 7*b1 - 11) * q^87 - 4 * q^88 + (5*b5 + 2*b4 - b2) * q^89 + (11*b3 + 3*b1 + 13) * q^90 + (6*b3 + 2*b1 - 2) * q^92 + (b5 + 3*b4 + 7*b2) * q^93 + (7*b3 - 3*b1 - 1) * q^94 + (-b1 - 2) * q^95 + (-8*b5 - 4*b4 + 4*b2) * q^96 + (b5 + 3*b2) * q^97 + b5 * q^98 + (-7*b5 - 3*b4 + 7*b2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 4 q^{3} - 6 q^{4} + 14 q^{9}+O(q^{10})$$ 6 * q - 4 * q^3 - 6 * q^4 + 14 * q^9 $$6 q - 4 q^{3} - 6 q^{4} + 14 q^{9} + 16 q^{10} + 24 q^{12} - 2 q^{14} - 2 q^{16} - 8 q^{17} - 8 q^{22} - 20 q^{23} + 10 q^{25} - 16 q^{27} + 48 q^{29} - 40 q^{30} - 4 q^{35} + 2 q^{36} + 20 q^{38} - 4 q^{40} - 8 q^{42} - 20 q^{43} + 40 q^{48} - 6 q^{49} + 16 q^{51} + 16 q^{53} + 12 q^{55} + 6 q^{56} - 12 q^{61} + 4 q^{62} + 34 q^{64} + 24 q^{66} + 44 q^{68} + 12 q^{69} + 60 q^{74} + 32 q^{75} + 4 q^{77} - 28 q^{79} + 6 q^{81} - 20 q^{82} - 52 q^{87} - 24 q^{88} + 56 q^{90} - 24 q^{92} - 20 q^{94} - 12 q^{95}+O(q^{100})$$ 6 * q - 4 * q^3 - 6 * q^4 + 14 * q^9 + 16 * q^10 + 24 * q^12 - 2 * q^14 - 2 * q^16 - 8 * q^17 - 8 * q^22 - 20 * q^23 + 10 * q^25 - 16 * q^27 + 48 * q^29 - 40 * q^30 - 4 * q^35 + 2 * q^36 + 20 * q^38 - 4 * q^40 - 8 * q^42 - 20 * q^43 + 40 * q^48 - 6 * q^49 + 16 * q^51 + 16 * q^53 + 12 * q^55 + 6 * q^56 - 12 * q^61 + 4 * q^62 + 34 * q^64 + 24 * q^66 + 44 * q^68 + 12 * q^69 + 60 * q^74 + 32 * q^75 + 4 * q^77 - 28 * q^79 + 6 * q^81 - 20 * q^82 - 52 * q^87 - 24 * q^88 + 56 * q^90 - 24 * q^92 - 20 * q^94 - 12 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{4} + 2\nu^{3} - \nu^{2} + 2\nu - 2 ) / 2$$ (-v^4 + 2*v^3 - v^2 + 2*v - 2) / 2 $$\beta_{2}$$ $$=$$ $$( -\nu^{5} - 3\nu^{3} + 4\nu^{2} - 2\nu + 8 ) / 4$$ (-v^5 - 3*v^3 + 4*v^2 - 2*v + 8) / 4 $$\beta_{3}$$ $$=$$ $$( -\nu^{5} + 2\nu^{4} - 3\nu^{3} + 6\nu^{2} - 2\nu + 4 ) / 4$$ (-v^5 + 2*v^4 - 3*v^3 + 6*v^2 - 2*v + 4) / 4 $$\beta_{4}$$ $$=$$ $$( \nu^{5} - 2\nu^{4} + 3\nu^{3} - 6\nu^{2} + 10\nu - 8 ) / 4$$ (v^5 - 2*v^4 + 3*v^3 - 6*v^2 + 10*v - 8) / 4 $$\beta_{5}$$ $$=$$ $$( 3\nu^{5} - 2\nu^{4} + 5\nu^{3} - 6\nu^{2} + 2\nu - 12 ) / 4$$ (3*v^5 - 2*v^4 + 5*v^3 - 6*v^2 + 2*v - 12) / 4
 $$\nu$$ $$=$$ $$( \beta_{4} + \beta_{3} + 1 ) / 2$$ (b4 + b3 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{5} + 2\beta_{3} + \beta_{2} + \beta_1 ) / 2$$ (b5 + 2*b3 + b2 + b1) / 2 $$\nu^{3}$$ $$=$$ $$( -\beta_{4} + \beta_{3} - 2\beta_{2} + 2\beta _1 + 3 ) / 2$$ (-b4 + b3 - 2*b2 + 2*b1 + 3) / 2 $$\nu^{4}$$ $$=$$ $$( -\beta_{5} + 2\beta_{3} - 5\beta_{2} - \beta _1 + 4 ) / 2$$ (-b5 + 2*b3 - 5*b2 - b1 + 4) / 2 $$\nu^{5}$$ $$=$$ $$( 4\beta_{5} + \beta_{4} + 3\beta_{3} + 2\beta_{2} - 2\beta _1 + 5 ) / 2$$ (4*b5 + b4 + 3*b3 + 2*b2 - 2*b1 + 5) / 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
 −0.671462 − 1.24464i 1.40680 + 0.144584i 0.264658 + 1.38923i 0.264658 − 1.38923i 1.40680 − 0.144584i −0.671462 + 1.24464i
2.34292i −1.14637 −3.48929 1.34292i 2.68585i 1.00000i 3.48929i −1.68585 3.14637
337.2 1.81361i −3.10278 −1.28917 2.81361i 5.62721i 1.00000i 1.28917i 6.62721 5.10278
337.3 0.470683i 2.24914 1.77846 0.529317i 1.05863i 1.00000i 1.77846i 2.05863 −0.249141
337.4 0.470683i 2.24914 1.77846 0.529317i 1.05863i 1.00000i 1.77846i 2.05863 −0.249141
337.5 1.81361i −3.10278 −1.28917 2.81361i 5.62721i 1.00000i 1.28917i 6.62721 5.10278
337.6 2.34292i −1.14637 −3.48929 1.34292i 2.68585i 1.00000i 3.48929i −1.68585 3.14637
 $$n$$: 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 1183.2.c.f 6
13.b even 2 1 inner 1183.2.c.f 6
13.d odd 4 1 91.2.a.d 3
13.d odd 4 1 1183.2.a.i 3
39.f even 4 1 819.2.a.i 3
52.f even 4 1 1456.2.a.t 3
65.g odd 4 1 2275.2.a.m 3
91.i even 4 1 637.2.a.j 3
91.i even 4 1 8281.2.a.bg 3
91.z odd 12 2 637.2.e.j 6
91.bb even 12 2 637.2.e.i 6
104.j odd 4 1 5824.2.a.by 3
104.m even 4 1 5824.2.a.bs 3
273.o odd 4 1 5733.2.a.x 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.a.d 3 13.d odd 4 1
637.2.a.j 3 91.i even 4 1
637.2.e.i 6 91.bb even 12 2
637.2.e.j 6 91.z odd 12 2
819.2.a.i 3 39.f even 4 1
1183.2.a.i 3 13.d odd 4 1
1183.2.c.f 6 1.a even 1 1 trivial
1183.2.c.f 6 13.b even 2 1 inner
1456.2.a.t 3 52.f even 4 1
2275.2.a.m 3 65.g odd 4 1
5733.2.a.x 3 273.o odd 4 1
5824.2.a.bs 3 104.m even 4 1
5824.2.a.by 3 104.j odd 4 1
8281.2.a.bg 3 91.i even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + 9T_{2}^{4} + 20T_{2}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(1183, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 9 T^{4} + 20 T^{2} + 4$$
$3$ $$(T^{3} + 2 T^{2} - 6 T - 8)^{2}$$
$5$ $$T^{6} + 10 T^{4} + 17 T^{2} + 4$$
$7$ $$(T^{2} + 1)^{3}$$
$11$ $$T^{6} + 16 T^{4} + 68 T^{2} + 64$$
$13$ $$T^{6}$$
$17$ $$(T^{3} + 4 T^{2} - 10 T + 4)^{2}$$
$19$ $$T^{6} + 14 T^{4} + 33 T^{2} + 16$$
$23$ $$(T^{3} + 10 T^{2} + T - 136)^{2}$$
$29$ $$(T^{3} - 24 T^{2} + 185 T - 454)^{2}$$
$31$ $$T^{6} + 54 T^{4} + 233 T^{2} + \cdots + 256$$
$37$ $$T^{6} + 116 T^{4} + 3364 T^{2} + \cdots + 15376$$
$41$ $$T^{6} + 60 T^{4} + 752 T^{2} + \cdots + 64$$
$43$ $$(T^{3} + 10 T^{2} - 71 T - 628)^{2}$$
$47$ $$T^{6} + 222 T^{4} + 14945 T^{2} + \cdots + 295936$$
$53$ $$(T^{3} - 8 T^{2} - 35 T - 22)^{2}$$
$59$ $$T^{6} + 328 T^{4} + 29840 T^{2} + \cdots + 473344$$
$61$ $$(T + 2)^{6}$$
$67$ $$T^{6} + 392 T^{4} + 38800 T^{2} + \cdots + 952576$$
$71$ $$T^{6} + 80 T^{4} + 292 T^{2} + \cdots + 256$$
$73$ $$T^{6} + 298 T^{4} + 15281 T^{2} + \cdots + 75076$$
$79$ $$(T^{3} + 14 T^{2} + 5 T - 16)^{2}$$
$83$ $$T^{6} + 686 T^{4} + \cdots + 10679824$$
$89$ $$T^{6} + 194 T^{4} + 10713 T^{2} + \cdots + 178084$$
$97$ $$T^{6} + 42 T^{4} + 401 T^{2} + \cdots + 484$$