Properties

 Label 605.2.b.c Level $605$ Weight $2$ Character orbit 605.b Analytic conductor $4.831$ 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] = [605,2,Mod(364,605)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(605, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("605.364");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$605 = 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 605.b (of order $$2$$, degree $$1$$, 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: $$4.83094932229$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ x^4 - x^3 - 2*x^2 - 3*x + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 55) 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 2\nu^{2} + 4\nu - 9 ) / 6$$ (v^3 + 2*v^2 + 4*v - 9) / 6 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} - 2\nu^{2} + 2\nu + 6 ) / 3$$ (-v^3 - 2*v^2 + 2*v + 6) / 3 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 2\nu + 3 ) / 2$$ (-v^3 + 2*v + 3) / 2
 $$\nu$$ $$=$$ $$( \beta_{2} + 2\beta _1 + 1 ) / 2$$ (b2 + 2*b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( 2\beta_{3} - 3\beta_{2} + 3 ) / 2$$ (2*b3 - 3*b2 + 3) / 2 $$\nu^{3}$$ $$=$$ $$-2\beta_{3} + \beta_{2} + 2\beta _1 + 4$$ -2*b3 + b2 + 2*b1 + 4

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/605\mathbb{Z}\right)^\times$$.

 $$n$$ $$122$$ $$486$$ $$\chi(n)$$ $$-1$$ $$1$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
364.1
 −1.18614 + 1.26217i 1.68614 + 0.396143i 1.68614 − 0.396143i −1.18614 − 1.26217i
2.52434i 0.792287i −4.37228 0.686141 + 2.12819i −2.00000 3.46410i 5.98844i 2.37228 5.37228 1.73205i
364.2 0.792287i 2.52434i 1.37228 −2.18614 0.469882i −2.00000 3.46410i 2.67181i −3.37228 −0.372281 + 1.73205i
364.3 0.792287i 2.52434i 1.37228 −2.18614 + 0.469882i −2.00000 3.46410i 2.67181i −3.37228 −0.372281 1.73205i
364.4 2.52434i 0.792287i −4.37228 0.686141 2.12819i −2.00000 3.46410i 5.98844i 2.37228 5.37228 + 1.73205i
 $$n$$: 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
5.b even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.b.c 4
5.b even 2 1 inner 605.2.b.c 4
5.c odd 4 2 3025.2.a.ba 4
11.b odd 2 1 55.2.b.a 4
11.c even 5 4 605.2.j.j 16
11.d odd 10 4 605.2.j.i 16
33.d even 2 1 495.2.c.a 4
44.c even 2 1 880.2.b.h 4
55.d odd 2 1 55.2.b.a 4
55.e even 4 2 275.2.a.h 4
55.h odd 10 4 605.2.j.i 16
55.j even 10 4 605.2.j.j 16
165.d even 2 1 495.2.c.a 4
165.l odd 4 2 2475.2.a.bi 4
220.g even 2 1 880.2.b.h 4
220.i odd 4 2 4400.2.a.cc 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.b.a 4 11.b odd 2 1
55.2.b.a 4 55.d odd 2 1
275.2.a.h 4 55.e even 4 2
495.2.c.a 4 33.d even 2 1
495.2.c.a 4 165.d even 2 1
605.2.b.c 4 1.a even 1 1 trivial
605.2.b.c 4 5.b even 2 1 inner
605.2.j.i 16 11.d odd 10 4
605.2.j.i 16 55.h odd 10 4
605.2.j.j 16 11.c even 5 4
605.2.j.j 16 55.j even 10 4
880.2.b.h 4 44.c even 2 1
880.2.b.h 4 220.g even 2 1
2475.2.a.bi 4 165.l odd 4 2
3025.2.a.ba 4 5.c odd 4 2
4400.2.a.cc 4 220.i odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(605, [\chi])$$:

 $$T_{2}^{4} + 7T_{2}^{2} + 4$$ T2^4 + 7*T2^2 + 4 $$T_{19} - 4$$ T19 - 4

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 7T^{2} + 4$$
$3$ $$T^{4} + 7T^{2} + 4$$
$5$ $$T^{4} + 3 T^{3} + \cdots + 25$$
$7$ $$(T^{2} + 12)^{2}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4} + 28T^{2} + 64$$
$19$ $$(T - 4)^{4}$$
$23$ $$T^{4} + 7T^{2} + 4$$
$29$ $$(T^{2} - 6 T - 24)^{2}$$
$31$ $$(T^{2} - T - 8)^{2}$$
$37$ $$T^{4} + 123T^{2} + 144$$
$41$ $$(T^{2} + 6 T - 24)^{2}$$
$43$ $$(T^{2} + 12)^{2}$$
$47$ $$(T^{2} + 44)^{2}$$
$53$ $$T^{4} + 112T^{2} + 1024$$
$59$ $$(T^{2} - 9 T + 12)^{2}$$
$61$ $$(T^{2} + 10 T - 8)^{2}$$
$67$ $$T^{4} + 87T^{2} + 36$$
$71$ $$(T^{2} + 3 T - 72)^{2}$$
$73$ $$(T^{2} + 48)^{2}$$
$79$ $$(T^{2} - 14 T + 16)^{2}$$
$83$ $$(T^{2} + 44)^{2}$$
$89$ $$(T^{2} - 3 T - 6)^{2}$$
$97$ $$T^{4} + 51T^{2} + 576$$