# Properties

 Label 605.2.m.b Level $605$ Weight $2$ Character orbit 605.m Analytic conductor $4.831$ Analytic rank $0$ Dimension $16$ CM no Inner twists $16$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [605,2,Mod(112,605)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(605, base_ring=CyclotomicField(20))

chi = DirichletCharacter(H, H._module([5, 2]))

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

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

chi := DirichletCharacter("605.112");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$605 = 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 605.m (of order $$20$$, degree $$8$$, 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: $$4.83094932229$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$2$$ over $$\Q(\zeta_{20})$$ Coefficient field: $$\Q(\zeta_{40})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} - x^{12} + x^{8} - x^{4} + 1$$ x^16 - x^12 + x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$5^{4}$$ Twist minimal: no (minimal twist has level 55) Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{15} + \beta_{12} + \cdots + \beta_{9}) q^{2}+ \cdots - \beta_{7} q^{9}+O(q^{10})$$ q + (b15 + b12 + b11 + b9) * q^2 + (b6 + b1) * q^3 + (-3*b7 + 3*b5 - 3*b3 + 3*b1) * q^4 + (-2*b6 + 2*b4 - b3 - 2*b2 + 2) * q^5 + (-b14 + b12) * q^6 + b13 * q^8 - b7 * q^9 $$q + (\beta_{15} + \beta_{12} + \cdots + \beta_{9}) q^{2}+ \cdots + 7 \beta_{10} q^{98}+O(q^{100})$$ q + (b15 + b12 + b11 + b9) * q^2 + (b6 + b1) * q^3 + (-3*b7 + 3*b5 - 3*b3 + 3*b1) * q^4 + (-2*b6 + 2*b4 - b3 - 2*b2 + 2) * q^5 + (-b14 + b12) * q^6 + b13 * q^8 - b7 * q^9 + (b10 + 2*b9) * q^10 + (3*b5 + 3) * q^12 + (-2*b14 - 2*b13 + 2*b10 - 2*b8) * q^13 + (-3*b7 + 3*b5 + b4 - 3*b3 + 3*b1) * q^15 + (b6 - b4 + b2 - 1) * q^16 + 2*b12 * q^17 - b8 * q^18 + (-2*b13 - 2*b11) * q^19 + (-6*b7 - 3*b2) * q^20 + (b5 - 1) * q^23 + (b15 + b14 + b13 + b12 + b11 - b10 + b9 + b8) * q^24 + (-3*b6 - 4*b1) * q^25 - 10*b4 * q^26 + (-4*b6 + 4*b4 + 4*b3 - 4*b2 + 4) * q^27 + (2*b15 + 2*b8) * q^29 + (3*b13 - b11) * q^30 - 2*b2 * q^31 - 3*b9 * q^32 + 10*b5 * q^34 - 3*b6 * q^36 + (3*b7 - 3*b5 + 3*b4 + 3*b3 - 3*b1) * q^37 + (10*b6 - 10*b4 + 10*b3 + 10*b2 - 10) * q^38 + (-2*b14 - 2*b12) * q^39 + (-b15 - 2*b8) * q^40 + (-2*b13 + 2*b11) * q^41 + (-2*b5 - 1) * q^45 + (-b15 + b14 + b13 - b12 - b11 - b10 - b9 + b8) * q^46 + (-3*b6 + 3*b1) * q^47 + (b7 - b5 - b4 + b3 - b1) * q^48 - 7*b3 * q^49 + (4*b14 - 3*b12) * q^50 + (-2*b15 + 2*b8) * q^51 + 6*b11 * q^52 + (-b7 + b2) * q^53 + (-4*b10 + 4*b9) * q^54 + (-4*b15 - 4*b12 - 4*b11 - 4*b9) * q^57 + (10*b6 + 10*b1) * q^58 + (-6*b7 + 6*b5 - 6*b3 + 6*b1) * q^59 + (-9*b6 + 9*b4 + 3*b3 - 9*b2 + 9) * q^60 + (2*b14 - 2*b12) * q^61 - 2*b15 * q^62 + 13*b7 * q^64 + (4*b10 - 2*b9) * q^65 + (3*b5 + 3) * q^67 + (6*b14 + 6*b13 - 6*b10 + 6*b8) * q^68 - 2*b1 * q^69 + (-8*b6 + 8*b4 - 8*b2 + 8) * q^71 - b12 * q^72 - 2*b8 * q^73 + (-3*b13 - 3*b11) * q^74 + (-7*b7 - b2) * q^75 + (-6*b10 - 6*b9) * q^76 + (-10*b5 + 10) * q^78 + (-2*b15 - 2*b14 - 2*b13 - 2*b12 - 2*b11 + 2*b10 - 2*b9 - 2*b8) * q^79 + (2*b6 + b1) * q^80 + 5*b4 * q^81 + (10*b6 - 10*b4 - 10*b3 + 10*b2 - 10) * q^82 + 4*b14 * q^83 + (2*b13 - 4*b11) * q^85 - 4*b9 * q^87 - 6*b5 * q^89 + (-b15 - 2*b14 - 2*b13 - b12 - b11 + 2*b10 - b9 - 2*b8) * q^90 + (3*b7 - 3*b5 + 3*b4 + 3*b3 - 3*b1) * q^92 + (-2*b6 + 2*b4 - 2*b3 - 2*b2 + 2) * q^93 + (-3*b14 - 3*b12) * q^94 + (6*b15 + 2*b8) * q^95 + (-3*b13 + 3*b11) * q^96 + (7*b7 + 7*b2) * q^97 + 7*b10 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 4 q^{3} + 8 q^{5}+O(q^{10})$$ 16 * q + 4 * q^3 + 8 * q^5 $$16 q + 4 q^{3} + 8 q^{5} + 48 q^{12} - 4 q^{15} - 4 q^{16} - 12 q^{20} - 16 q^{23} - 12 q^{25} + 40 q^{26} + 16 q^{27} - 8 q^{31} - 12 q^{36} - 12 q^{37} - 40 q^{38} - 16 q^{45} - 12 q^{47} + 4 q^{48} + 4 q^{53} + 40 q^{58} + 36 q^{60} + 48 q^{67} + 32 q^{71} - 4 q^{75} + 160 q^{78} + 8 q^{80} - 20 q^{81} - 40 q^{82} - 12 q^{92} + 8 q^{93} + 28 q^{97}+O(q^{100})$$ 16 * q + 4 * q^3 + 8 * q^5 + 48 * q^12 - 4 * q^15 - 4 * q^16 - 12 * q^20 - 16 * q^23 - 12 * q^25 + 40 * q^26 + 16 * q^27 - 8 * q^31 - 12 * q^36 - 12 * q^37 - 40 * q^38 - 16 * q^45 - 12 * q^47 + 4 * q^48 + 4 * q^53 + 40 * q^58 + 36 * q^60 + 48 * q^67 + 32 * q^71 - 4 * q^75 + 160 * q^78 + 8 * q^80 - 20 * q^81 - 40 * q^82 - 12 * q^92 + 8 * q^93 + 28 * q^97

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{40}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{40}^{4}$$ v^4 $$\beta_{3}$$ $$=$$ $$\zeta_{40}^{6}$$ v^6 $$\beta_{4}$$ $$=$$ $$\zeta_{40}^{8}$$ v^8 $$\beta_{5}$$ $$=$$ $$\zeta_{40}^{10}$$ v^10 $$\beta_{6}$$ $$=$$ $$\zeta_{40}^{12}$$ v^12 $$\beta_{7}$$ $$=$$ $$\zeta_{40}^{14}$$ v^14 $$\beta_{8}$$ $$=$$ $$\zeta_{40}^{13} + 2\zeta_{40}^{5} - 2\zeta_{40}$$ v^13 + 2*v^5 - 2*v $$\beta_{9}$$ $$=$$ $$-\zeta_{40}^{15} - 2\zeta_{40}^{7} + 2\zeta_{40}^{3}$$ -v^15 - 2*v^7 + 2*v^3 $$\beta_{10}$$ $$=$$ $$-2\zeta_{40}^{9} + \zeta_{40}^{5} - 2\zeta_{40}$$ -2*v^9 + v^5 - 2*v $$\beta_{11}$$ $$=$$ $$-2\zeta_{40}^{11} + \zeta_{40}^{7} - 2\zeta_{40}^{3}$$ -2*v^11 + v^7 - 2*v^3 $$\beta_{12}$$ $$=$$ $$2\zeta_{40}^{15} - \zeta_{40}^{11} + 2\zeta_{40}^{7}$$ 2*v^15 - v^11 + 2*v^7 $$\beta_{13}$$ $$=$$ $$-\zeta_{40}^{13} - \zeta_{40}^{9} + \zeta_{40}^{5} + \zeta_{40}$$ -v^13 - v^9 + v^5 + v $$\beta_{14}$$ $$=$$ $$2\zeta_{40}^{13} - 2\zeta_{40}^{9} - \zeta_{40}$$ 2*v^13 - 2*v^9 - v $$\beta_{15}$$ $$=$$ $$-2\zeta_{40}^{15} + 2\zeta_{40}^{11} + \zeta_{40}^{3}$$ -2*v^15 + 2*v^11 + v^3
 $$\zeta_{40}$$ $$=$$ $$( \beta_{14} + 2\beta_{13} - 2\beta_{10} ) / 5$$ (b14 + 2*b13 - 2*b10) / 5 $$\zeta_{40}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{40}^{3}$$ $$=$$ $$( \beta_{15} + 2\beta_{12} + 2\beta_{9} ) / 5$$ (b15 + 2*b12 + 2*b9) / 5 $$\zeta_{40}^{4}$$ $$=$$ $$\beta_{2}$$ b2 $$\zeta_{40}^{5}$$ $$=$$ $$( 2\beta_{13} - \beta_{10} + 2\beta_{8} ) / 5$$ (2*b13 - b10 + 2*b8) / 5 $$\zeta_{40}^{6}$$ $$=$$ $$\beta_{3}$$ b3 $$\zeta_{40}^{7}$$ $$=$$ $$( 2\beta_{15} + 2\beta_{12} + \beta_{11} ) / 5$$ (2*b15 + 2*b12 + b11) / 5 $$\zeta_{40}^{8}$$ $$=$$ $$\beta_{4}$$ b4 $$\zeta_{40}^{9}$$ $$=$$ $$( -\beta_{14} - \beta_{13} - \beta_{10} + \beta_{8} ) / 5$$ (-b14 - b13 - b10 + b8) / 5 $$\zeta_{40}^{10}$$ $$=$$ $$\beta_{5}$$ b5 $$\zeta_{40}^{11}$$ $$=$$ $$( -\beta_{12} - 2\beta_{11} - 2\beta_{9} ) / 5$$ (-b12 - 2*b11 - 2*b9) / 5 $$\zeta_{40}^{12}$$ $$=$$ $$\beta_{6}$$ b6 $$\zeta_{40}^{13}$$ $$=$$ $$( 2\beta_{14} - 2\beta_{10} + \beta_{8} ) / 5$$ (2*b14 - 2*b10 + b8) / 5 $$\zeta_{40}^{14}$$ $$=$$ $$\beta_{7}$$ b7 $$\zeta_{40}^{15}$$ $$=$$ $$( -2\beta_{15} - 2\beta_{11} - \beta_{9} ) / 5$$ (-2*b15 - 2*b11 - b9) / 5

## Character values

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

 $$n$$ $$122$$ $$486$$ $$\chi(n)$$ $$\beta_{5}$$ $$-\beta_{4}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
112.1
 0.453990 − 0.891007i −0.453990 + 0.891007i −0.987688 + 0.156434i 0.987688 − 0.156434i 0.891007 + 0.453990i −0.891007 − 0.453990i −0.987688 − 0.156434i 0.987688 + 0.156434i −0.156434 + 0.987688i 0.156434 − 0.987688i 0.891007 − 0.453990i −0.891007 + 0.453990i 0.453990 + 0.891007i −0.453990 − 0.891007i −0.156434 − 0.987688i 0.156434 + 0.987688i
−1.01515 1.99235i 0.221232 1.39680i −1.76336 + 2.42705i −1.56909 1.59310i −3.00750 + 0.977198i 0 2.20854 + 0.349798i 0.951057 + 0.309017i −1.58114 + 4.74342i
112.2 1.01515 + 1.99235i 0.221232 1.39680i −1.76336 + 2.42705i −1.56909 1.59310i 3.00750 0.977198i 0 −2.20854 0.349798i 0.951057 + 0.309017i 1.58114 4.74342i
118.1 −2.20854 0.349798i 0.642040 1.26007i 2.85317 + 0.927051i 1.03025 + 1.98459i −1.85874 + 2.55834i 0 −1.99235 1.01515i 0.587785 + 0.809017i −1.58114 4.74342i
118.2 2.20854 + 0.349798i 0.642040 1.26007i 2.85317 + 0.927051i 1.03025 + 1.98459i 1.85874 2.55834i 0 1.99235 + 1.01515i 0.587785 + 0.809017i 1.58114 + 4.74342i
233.1 −1.99235 + 1.01515i 1.39680 + 0.221232i 1.76336 2.42705i 0.333023 2.21113i −3.00750 + 0.977198i 0 −0.349798 + 2.20854i −0.951057 0.309017i 1.58114 + 4.74342i
233.2 1.99235 1.01515i 1.39680 + 0.221232i 1.76336 2.42705i 0.333023 2.21113i 3.00750 0.977198i 0 0.349798 2.20854i −0.951057 0.309017i −1.58114 4.74342i
282.1 −2.20854 + 0.349798i 0.642040 + 1.26007i 2.85317 0.927051i 1.03025 1.98459i −1.85874 2.55834i 0 −1.99235 + 1.01515i 0.587785 0.809017i −1.58114 + 4.74342i
282.2 2.20854 0.349798i 0.642040 + 1.26007i 2.85317 0.927051i 1.03025 1.98459i 1.85874 + 2.55834i 0 1.99235 1.01515i 0.587785 0.809017i 1.58114 4.74342i
403.1 −0.349798 2.20854i −1.26007 + 0.642040i −2.85317 + 0.927051i 2.20582 0.366554i 1.85874 + 2.55834i 0 1.01515 + 1.99235i −0.587785 + 0.809017i −1.58114 4.74342i
403.2 0.349798 + 2.20854i −1.26007 + 0.642040i −2.85317 + 0.927051i 2.20582 0.366554i −1.85874 2.55834i 0 −1.01515 1.99235i −0.587785 + 0.809017i 1.58114 + 4.74342i
457.1 −1.99235 1.01515i 1.39680 0.221232i 1.76336 + 2.42705i 0.333023 + 2.21113i −3.00750 0.977198i 0 −0.349798 2.20854i −0.951057 + 0.309017i 1.58114 4.74342i
457.2 1.99235 + 1.01515i 1.39680 0.221232i 1.76336 + 2.42705i 0.333023 + 2.21113i 3.00750 + 0.977198i 0 0.349798 + 2.20854i −0.951057 + 0.309017i −1.58114 + 4.74342i
578.1 −1.01515 + 1.99235i 0.221232 + 1.39680i −1.76336 2.42705i −1.56909 + 1.59310i −3.00750 0.977198i 0 2.20854 0.349798i 0.951057 0.309017i −1.58114 4.74342i
578.2 1.01515 1.99235i 0.221232 + 1.39680i −1.76336 2.42705i −1.56909 + 1.59310i 3.00750 + 0.977198i 0 −2.20854 + 0.349798i 0.951057 0.309017i 1.58114 + 4.74342i
602.1 −0.349798 + 2.20854i −1.26007 0.642040i −2.85317 0.927051i 2.20582 + 0.366554i 1.85874 2.55834i 0 1.01515 1.99235i −0.587785 0.809017i −1.58114 + 4.74342i
602.2 0.349798 2.20854i −1.26007 0.642040i −2.85317 0.927051i 2.20582 + 0.366554i −1.85874 + 2.55834i 0 −1.01515 + 1.99235i −0.587785 0.809017i 1.58114 4.74342i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 112.2 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.c odd 4 1 inner
11.b odd 2 1 inner
11.c even 5 3 inner
11.d odd 10 3 inner
55.e even 4 1 inner
55.k odd 20 3 inner
55.l even 20 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.m.b 16
5.c odd 4 1 inner 605.2.m.b 16
11.b odd 2 1 inner 605.2.m.b 16
11.c even 5 1 55.2.e.a 4
11.c even 5 3 inner 605.2.m.b 16
11.d odd 10 1 55.2.e.a 4
11.d odd 10 3 inner 605.2.m.b 16
33.f even 10 1 495.2.k.b 4
33.h odd 10 1 495.2.k.b 4
44.g even 10 1 880.2.bd.e 4
44.h odd 10 1 880.2.bd.e 4
55.e even 4 1 inner 605.2.m.b 16
55.h odd 10 1 275.2.e.b 4
55.j even 10 1 275.2.e.b 4
55.k odd 20 1 55.2.e.a 4
55.k odd 20 1 275.2.e.b 4
55.k odd 20 3 inner 605.2.m.b 16
55.l even 20 1 55.2.e.a 4
55.l even 20 1 275.2.e.b 4
55.l even 20 3 inner 605.2.m.b 16
165.u odd 20 1 495.2.k.b 4
165.v even 20 1 495.2.k.b 4
220.v even 20 1 880.2.bd.e 4
220.w odd 20 1 880.2.bd.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.e.a 4 11.c even 5 1
55.2.e.a 4 11.d odd 10 1
55.2.e.a 4 55.k odd 20 1
55.2.e.a 4 55.l even 20 1
275.2.e.b 4 55.h odd 10 1
275.2.e.b 4 55.j even 10 1
275.2.e.b 4 55.k odd 20 1
275.2.e.b 4 55.l even 20 1
495.2.k.b 4 33.f even 10 1
495.2.k.b 4 33.h odd 10 1
495.2.k.b 4 165.u odd 20 1
495.2.k.b 4 165.v even 20 1
605.2.m.b 16 1.a even 1 1 trivial
605.2.m.b 16 5.c odd 4 1 inner
605.2.m.b 16 11.b odd 2 1 inner
605.2.m.b 16 11.c even 5 3 inner
605.2.m.b 16 11.d odd 10 3 inner
605.2.m.b 16 55.e even 4 1 inner
605.2.m.b 16 55.k odd 20 3 inner
605.2.m.b 16 55.l even 20 3 inner
880.2.bd.e 4 44.g even 10 1
880.2.bd.e 4 44.h odd 10 1
880.2.bd.e 4 220.v even 20 1
880.2.bd.e 4 220.w odd 20 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{16} - 25T_{2}^{12} + 625T_{2}^{8} - 15625T_{2}^{4} + 390625$$ acting on $$S_{2}^{\mathrm{new}}(605, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} - 25 T^{12} + \cdots + 390625$$
$3$ $$(T^{8} - 2 T^{7} + 2 T^{6} + \cdots + 16)^{2}$$
$5$ $$(T^{8} - 4 T^{7} + \cdots + 625)^{2}$$
$7$ $$T^{16}$$
$11$ $$T^{16}$$
$13$ $$T^{16} + \cdots + 25600000000$$
$17$ $$T^{16} + \cdots + 25600000000$$
$19$ $$(T^{8} + 40 T^{6} + \cdots + 2560000)^{2}$$
$23$ $$(T^{2} + 2 T + 2)^{8}$$
$29$ $$(T^{8} + 40 T^{6} + \cdots + 2560000)^{2}$$
$31$ $$(T^{4} + 2 T^{3} + 4 T^{2} + \cdots + 16)^{4}$$
$37$ $$(T^{8} + 6 T^{7} + \cdots + 104976)^{2}$$
$41$ $$(T^{8} - 40 T^{6} + \cdots + 2560000)^{2}$$
$43$ $$T^{16}$$
$47$ $$(T^{8} + 6 T^{7} + \cdots + 104976)^{2}$$
$53$ $$(T^{8} - 2 T^{7} + 2 T^{6} + \cdots + 16)^{2}$$
$59$ $$(T^{8} - 36 T^{6} + \cdots + 1679616)^{2}$$
$61$ $$(T^{8} - 40 T^{6} + \cdots + 2560000)^{2}$$
$67$ $$(T^{2} - 6 T + 18)^{8}$$
$71$ $$(T^{4} - 8 T^{3} + \cdots + 4096)^{4}$$
$73$ $$T^{16} + \cdots + 25600000000$$
$79$ $$(T^{8} + 40 T^{6} + \cdots + 2560000)^{2}$$
$83$ $$T^{16} + \cdots + 16\!\cdots\!00$$
$89$ $$(T^{2} + 36)^{8}$$
$97$ $$(T^{8} - 14 T^{7} + \cdots + 92236816)^{2}$$