# Properties

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

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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: $$\Q(i, \sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 3x^{2} + 1$$ x^4 + 3*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} + \beta_1) q^{2} + (\beta_{2} - 1) q^{3} + 3 \beta_{2} q^{4} + ( - \beta_{3} + \beta_1) q^{5} + (2 \beta_{3} - 3 \beta_1) q^{6} + \beta_{3} q^{7} + (\beta_{3} - 4 \beta_1) q^{8} + ( - 3 \beta_{2} - 1) q^{9}+O(q^{10})$$ q + (-b3 + b1) * q^2 + (b2 - 1) * q^3 + 3*b2 * q^4 + (-b3 + b1) * q^5 + (2*b3 - 3*b1) * q^6 + b3 * q^7 + (b3 - 4*b1) * q^8 + (-3*b2 - 1) * q^9 $$q + ( - \beta_{3} + \beta_1) q^{2} + (\beta_{2} - 1) q^{3} + 3 \beta_{2} q^{4} + ( - \beta_{3} + \beta_1) q^{5} + (2 \beta_{3} - 3 \beta_1) q^{6} + \beta_{3} q^{7} + (\beta_{3} - 4 \beta_1) q^{8} + ( - 3 \beta_{2} - 1) q^{9} + (3 \beta_{2} - 2) q^{10} + ( - 3 \beta_{3} - 3 \beta_1) q^{11} + ( - 6 \beta_{2} + 3) q^{12} + ( - \beta_{2} + 1) q^{14} + (2 \beta_{3} - 3 \beta_1) q^{15} + ( - 3 \beta_{2} + 5) q^{16} + ( - 4 \beta_{2} - 5) q^{17} + ( - 2 \beta_{3} + 5 \beta_1) q^{18} + ( - 3 \beta_{3} - 3 \beta_1) q^{19} + (3 \beta_{3} - 6 \beta_1) q^{20} + ( - \beta_{3} + \beta_1) q^{21} - 3 \beta_{2} q^{22} + ( - 4 \beta_{2} - 2) q^{23} + ( - 5 \beta_{3} + 9 \beta_1) q^{24} + (3 \beta_{2} + 3) q^{25} + (2 \beta_{2} + 1) q^{27} + 3 \beta_1 q^{28} + ( - 5 \beta_{2} - 1) q^{29} + ( - 8 \beta_{2} + 5) q^{30} + ( - 5 \beta_{3} - 6 \beta_1) q^{31} + ( - 6 \beta_{3} + 3 \beta_1) q^{32} + 3 \beta_1 q^{33} + (\beta_{3} + 3 \beta_1) q^{34} + ( - \beta_{2} + 1) q^{35} + (6 \beta_{2} - 9) q^{36} + 4 \beta_{3} q^{37} - 3 \beta_{2} q^{38} + ( - 9 \beta_{2} + 5) q^{40} + ( - 4 \beta_{3} - 2 \beta_1) q^{41} + (3 \beta_{2} - 2) q^{42} + (9 \beta_{2} + 2) q^{43} - 9 \beta_{3} q^{44} + ( - 2 \beta_{3} + 5 \beta_1) q^{45} + ( - 2 \beta_{3} + 6 \beta_1) q^{46} + (\beta_{3} + 2 \beta_1) q^{47} + (11 \beta_{2} - 8) q^{48} - q^{49} - 3 \beta_1 q^{50} + (3 \beta_{2} + 1) q^{51} + (2 \beta_{2} + 7) q^{53} + (\beta_{3} - 3 \beta_1) q^{54} - 3 \beta_{2} q^{55} + (4 \beta_{2} - 1) q^{56} + 3 \beta_1 q^{57} + ( - 4 \beta_{3} + 9 \beta_1) q^{58} + (\beta_{3} + 2 \beta_1) q^{59} + ( - 9 \beta_{3} + 15 \beta_1) q^{60} - 6 q^{61} + ( - 7 \beta_{2} + 1) q^{62} + ( - \beta_{3} - 3 \beta_1) q^{63} + (6 \beta_{2} + 1) q^{64} + (6 \beta_{2} - 3) q^{66} + (3 \beta_{3} - 6 \beta_1) q^{67} + ( - 3 \beta_{2} - 12) q^{68} + (6 \beta_{2} - 2) q^{69} + ( - 2 \beta_{3} + 3 \beta_1) q^{70} + ( - 2 \beta_{3} - 10 \beta_1) q^{71} + (11 \beta_{3} - 11 \beta_1) q^{72} - 2 \beta_{3} q^{73} + ( - 4 \beta_{2} + 4) q^{74} - 3 \beta_{2} q^{75} - 9 \beta_{3} q^{76} + (3 \beta_{2} + 3) q^{77} + 4 q^{79} + ( - 8 \beta_{3} + 11 \beta_1) q^{80} + (6 \beta_{2} + 4) q^{81} - 2 q^{82} + (3 \beta_{3} + 6 \beta_1) q^{83} + (3 \beta_{3} - 6 \beta_1) q^{84} + (\beta_{3} + 3 \beta_1) q^{85} + (7 \beta_{3} - 16 \beta_1) q^{86} + (9 \beta_{2} - 4) q^{87} + (3 \beta_{2} - 9) q^{88} + (13 \beta_{3} + 5 \beta_1) q^{89} + (12 \beta_{2} - 7) q^{90} + (6 \beta_{2} - 12) q^{92} + ( - \beta_{3} + 7 \beta_1) q^{93} + (3 \beta_{2} - 1) q^{94} - 3 \beta_{2} q^{95} + (9 \beta_{3} - 12 \beta_1) q^{96} + ( - 14 \beta_{3} + 3 \beta_1) q^{97} + (\beta_{3} - \beta_1) q^{98} + (12 \beta_{3} + 3 \beta_1) q^{99}+O(q^{100})$$ q + (-b3 + b1) * q^2 + (b2 - 1) * q^3 + 3*b2 * q^4 + (-b3 + b1) * q^5 + (2*b3 - 3*b1) * q^6 + b3 * q^7 + (b3 - 4*b1) * q^8 + (-3*b2 - 1) * q^9 + (3*b2 - 2) * q^10 + (-3*b3 - 3*b1) * q^11 + (-6*b2 + 3) * q^12 + (-b2 + 1) * q^14 + (2*b3 - 3*b1) * q^15 + (-3*b2 + 5) * q^16 + (-4*b2 - 5) * q^17 + (-2*b3 + 5*b1) * q^18 + (-3*b3 - 3*b1) * q^19 + (3*b3 - 6*b1) * q^20 + (-b3 + b1) * q^21 - 3*b2 * q^22 + (-4*b2 - 2) * q^23 + (-5*b3 + 9*b1) * q^24 + (3*b2 + 3) * q^25 + (2*b2 + 1) * q^27 + 3*b1 * q^28 + (-5*b2 - 1) * q^29 + (-8*b2 + 5) * q^30 + (-5*b3 - 6*b1) * q^31 + (-6*b3 + 3*b1) * q^32 + 3*b1 * q^33 + (b3 + 3*b1) * q^34 + (-b2 + 1) * q^35 + (6*b2 - 9) * q^36 + 4*b3 * q^37 - 3*b2 * q^38 + (-9*b2 + 5) * q^40 + (-4*b3 - 2*b1) * q^41 + (3*b2 - 2) * q^42 + (9*b2 + 2) * q^43 - 9*b3 * q^44 + (-2*b3 + 5*b1) * q^45 + (-2*b3 + 6*b1) * q^46 + (b3 + 2*b1) * q^47 + (11*b2 - 8) * q^48 - q^49 - 3*b1 * q^50 + (3*b2 + 1) * q^51 + (2*b2 + 7) * q^53 + (b3 - 3*b1) * q^54 - 3*b2 * q^55 + (4*b2 - 1) * q^56 + 3*b1 * q^57 + (-4*b3 + 9*b1) * q^58 + (b3 + 2*b1) * q^59 + (-9*b3 + 15*b1) * q^60 - 6 * q^61 + (-7*b2 + 1) * q^62 + (-b3 - 3*b1) * q^63 + (6*b2 + 1) * q^64 + (6*b2 - 3) * q^66 + (3*b3 - 6*b1) * q^67 + (-3*b2 - 12) * q^68 + (6*b2 - 2) * q^69 + (-2*b3 + 3*b1) * q^70 + (-2*b3 - 10*b1) * q^71 + (11*b3 - 11*b1) * q^72 - 2*b3 * q^73 + (-4*b2 + 4) * q^74 - 3*b2 * q^75 - 9*b3 * q^76 + (3*b2 + 3) * q^77 + 4 * q^79 + (-8*b3 + 11*b1) * q^80 + (6*b2 + 4) * q^81 - 2 * q^82 + (3*b3 + 6*b1) * q^83 + (3*b3 - 6*b1) * q^84 + (b3 + 3*b1) * q^85 + (7*b3 - 16*b1) * q^86 + (9*b2 - 4) * q^87 + (3*b2 - 9) * q^88 + (13*b3 + 5*b1) * q^89 + (12*b2 - 7) * q^90 + (6*b2 - 12) * q^92 + (-b3 + 7*b1) * q^93 + (3*b2 - 1) * q^94 - 3*b2 * q^95 + (9*b3 - 12*b1) * q^96 + (-14*b3 + 3*b1) * q^97 + (b3 - b1) * q^98 + (12*b3 + 3*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 6 q^{3} - 6 q^{4} + 2 q^{9}+O(q^{10})$$ 4 * q - 6 * q^3 - 6 * q^4 + 2 * q^9 $$4 q - 6 q^{3} - 6 q^{4} + 2 q^{9} - 14 q^{10} + 24 q^{12} + 6 q^{14} + 26 q^{16} - 12 q^{17} + 6 q^{22} + 6 q^{25} + 6 q^{29} + 36 q^{30} + 6 q^{35} - 48 q^{36} + 6 q^{38} + 38 q^{40} - 14 q^{42} - 10 q^{43} - 54 q^{48} - 4 q^{49} - 2 q^{51} + 24 q^{53} + 6 q^{55} - 12 q^{56} - 24 q^{61} + 18 q^{62} - 8 q^{64} - 24 q^{66} - 42 q^{68} - 20 q^{69} + 24 q^{74} + 6 q^{75} + 6 q^{77} + 16 q^{79} + 4 q^{81} - 8 q^{82} - 34 q^{87} - 42 q^{88} - 52 q^{90} - 60 q^{92} - 10 q^{94} + 6 q^{95}+O(q^{100})$$ 4 * q - 6 * q^3 - 6 * q^4 + 2 * q^9 - 14 * q^10 + 24 * q^12 + 6 * q^14 + 26 * q^16 - 12 * q^17 + 6 * q^22 + 6 * q^25 + 6 * q^29 + 36 * q^30 + 6 * q^35 - 48 * q^36 + 6 * q^38 + 38 * q^40 - 14 * q^42 - 10 * q^43 - 54 * q^48 - 4 * q^49 - 2 * q^51 + 24 * q^53 + 6 * q^55 - 12 * q^56 - 24 * q^61 + 18 * q^62 - 8 * q^64 - 24 * q^66 - 42 * q^68 - 20 * q^69 + 24 * q^74 + 6 * q^75 + 6 * q^77 + 16 * q^79 + 4 * q^81 - 8 * q^82 - 34 * q^87 - 42 * q^88 - 52 * q^90 - 60 * q^92 - 10 * q^94 + 6 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + 1$$ v^2 + 1 $$\beta_{3}$$ $$=$$ $$\nu^{3} + 2\nu$$ v^3 + 2*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - 1$$ b2 - 1 $$\nu^{3}$$ $$=$$ $$\beta_{3} - 2\beta_1$$ b3 - 2*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
 − 1.61803i 0.618034i − 0.618034i 1.61803i
2.61803i −2.61803 −4.85410 2.61803i 6.85410i 1.00000i 7.47214i 3.85410 −6.85410
337.2 0.381966i −0.381966 1.85410 0.381966i 0.145898i 1.00000i 1.47214i −2.85410 −0.145898
337.3 0.381966i −0.381966 1.85410 0.381966i 0.145898i 1.00000i 1.47214i −2.85410 −0.145898
337.4 2.61803i −2.61803 −4.85410 2.61803i 6.85410i 1.00000i 7.47214i 3.85410 −6.85410
 $$n$$: 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.c 4
13.b even 2 1 inner 1183.2.c.c 4
13.d odd 4 1 1183.2.a.c 2
13.d odd 4 1 1183.2.a.g 2
13.f odd 12 2 91.2.f.a 4
39.k even 12 2 819.2.o.c 4
52.l even 12 2 1456.2.s.h 4
91.i even 4 1 8281.2.a.n 2
91.i even 4 1 8281.2.a.bb 2
91.w even 12 2 637.2.g.c 4
91.x odd 12 2 637.2.h.g 4
91.ba even 12 2 637.2.h.f 4
91.bc even 12 2 637.2.f.c 4
91.bd odd 12 2 637.2.g.b 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 7T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(1183, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 7T^{2} + 1$$
$3$ $$(T^{2} + 3 T + 1)^{2}$$
$5$ $$T^{4} + 7T^{2} + 1$$
$7$ $$(T^{2} + 1)^{2}$$
$11$ $$T^{4} + 27T^{2} + 81$$
$13$ $$T^{4}$$
$17$ $$(T^{2} + 6 T - 11)^{2}$$
$19$ $$T^{4} + 27T^{2} + 81$$
$23$ $$(T^{2} - 20)^{2}$$
$29$ $$(T^{2} - 3 T - 29)^{2}$$
$31$ $$T^{4} + 98T^{2} + 1681$$
$37$ $$(T^{2} + 16)^{2}$$
$41$ $$T^{4} + 28T^{2} + 16$$
$43$ $$(T^{2} + 5 T - 95)^{2}$$
$47$ $$(T^{2} + 5)^{2}$$
$53$ $$(T^{2} - 12 T + 31)^{2}$$
$59$ $$(T^{2} + 5)^{2}$$
$61$ $$(T + 6)^{4}$$
$67$ $$T^{4} + 162T^{2} + 81$$
$71$ $$T^{4} + 268 T^{2} + 13456$$
$73$ $$(T^{2} + 4)^{2}$$
$79$ $$(T - 4)^{4}$$
$83$ $$(T^{2} + 45)^{2}$$
$89$ $$T^{4} + 283T^{2} + 6241$$
$97$ $$T^{4} + 503 T^{2} + 52441$$
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