# Properties

 Label 45.5.h.a Level $45$ Weight $5$ Character orbit 45.h Analytic conductor $4.652$ Analytic rank $0$ Dimension $44$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [45,5,Mod(14,45)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([5, 3]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("45.14");

S:= CuspForms(chi, 5);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$45 = 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 45.h (of order $$6$$, degree $$2$$, 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: $$4.65164833877$$ Analytic rank: $$0$$ Dimension: $$44$$ Relative dimension: $$22$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$44 q - 162 q^{4} + 6 q^{5} - 36 q^{6} - 126 q^{9}+O(q^{10})$$ 44 * q - 162 * q^4 + 6 * q^5 - 36 * q^6 - 126 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$44 q - 162 q^{4} + 6 q^{5} - 36 q^{6} - 126 q^{9} + 28 q^{10} + 228 q^{11} + 282 q^{14} + 450 q^{15} - 1058 q^{16} - 8 q^{19} - 2196 q^{20} + 1056 q^{21} + 2970 q^{24} - 148 q^{25} + 2370 q^{29} + 1068 q^{30} - 1112 q^{31} - 436 q^{34} - 648 q^{36} - 6420 q^{39} - 850 q^{40} + 1830 q^{41} - 1944 q^{45} - 5668 q^{46} + 5396 q^{49} + 13524 q^{50} - 1500 q^{51} + 17766 q^{54} - 3072 q^{55} - 8250 q^{56} - 27348 q^{59} - 28722 q^{60} + 2782 q^{61} + 8348 q^{64} - 24906 q^{65} + 27240 q^{66} + 19662 q^{69} + 4782 q^{70} + 64728 q^{74} + 28446 q^{75} - 1776 q^{76} + 7900 q^{79} - 47754 q^{81} - 113166 q^{84} + 6766 q^{85} + 9192 q^{86} + 10674 q^{90} + 1560 q^{91} - 18826 q^{94} + 58746 q^{95} + 1902 q^{96} + 44568 q^{99}+O(q^{100})$$ 44 * q - 162 * q^4 + 6 * q^5 - 36 * q^6 - 126 * q^9 + 28 * q^10 + 228 * q^11 + 282 * q^14 + 450 * q^15 - 1058 * q^16 - 8 * q^19 - 2196 * q^20 + 1056 * q^21 + 2970 * q^24 - 148 * q^25 + 2370 * q^29 + 1068 * q^30 - 1112 * q^31 - 436 * q^34 - 648 * q^36 - 6420 * q^39 - 850 * q^40 + 1830 * q^41 - 1944 * q^45 - 5668 * q^46 + 5396 * q^49 + 13524 * q^50 - 1500 * q^51 + 17766 * q^54 - 3072 * q^55 - 8250 * q^56 - 27348 * q^59 - 28722 * q^60 + 2782 * q^61 + 8348 * q^64 - 24906 * q^65 + 27240 * q^66 + 19662 * q^69 + 4782 * q^70 + 64728 * q^74 + 28446 * q^75 - 1776 * q^76 + 7900 * q^79 - 47754 * q^81 - 113166 * q^84 + 6766 * q^85 + 9192 * q^86 + 10674 * q^90 + 1560 * q^91 - 18826 * q^94 + 58746 * q^95 + 1902 * q^96 + 44568 * q^99

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
14.1 −3.75365 6.50151i 3.49115 8.29529i −20.1797 + 34.9523i 2.78274 + 24.8446i −67.0364 + 8.43986i −60.4872 + 34.9223i 182.874 −56.6237 57.9202i 151.082 111.350i
14.2 −3.64116 6.30667i 4.56543 + 7.75608i −18.5161 + 32.0708i 21.9161 12.0286i 32.2916 57.0338i 38.4567 22.2030i 153.163 −39.3137 + 70.8197i −155.660 94.4195i
14.3 −3.12350 5.41005i −7.73715 4.59745i −11.5125 + 19.9402i −11.8313 22.0232i −0.705517 + 56.2185i −10.1314 + 5.84936i 43.8846 38.7269 + 71.1423i −82.1915 + 132.797i
14.4 −2.87361 4.97724i −8.30470 + 3.46872i −8.51525 + 14.7488i 7.19048 + 23.9436i 41.1291 + 31.3667i 41.4151 23.9110i 5.92246 56.9359 57.6134i 98.5104 104.593i
14.5 −2.61690 4.53261i 8.42379 3.16855i −5.69638 + 9.86641i −19.7616 15.3127i −36.4061 29.8900i 50.0684 28.9070i −24.1135 60.9206 53.3825i −17.6925 + 129.644i
14.6 −2.30921 3.99967i 4.25182 + 7.93234i −2.66491 + 4.61577i −21.0948 + 13.4168i 21.9084 35.3233i −45.6166 + 26.3368i −49.2794 −44.8440 + 67.4538i 102.375 + 53.3902i
14.7 −1.64811 2.85461i −5.28472 + 7.28504i 2.56747 4.44699i 15.0736 19.9446i 29.5057 + 3.07927i −64.7475 + 37.3820i −69.6654 −25.1435 76.9987i −81.7769 10.1586i
14.8 −1.59606 2.76446i −0.340063 8.99357i 2.90517 5.03190i 24.0861 6.69764i −24.3196 + 15.2944i 21.0893 12.1759i −69.6213 −80.7687 + 6.11676i −56.9583 55.8953i
14.9 −1.02189 1.76997i 8.99991 + 0.0407214i 5.91148 10.2390i 15.4498 + 19.6546i −9.12485 15.9711i −2.09732 + 1.21089i −56.8640 80.9967 + 0.732978i 19.0000 47.4305i
14.10 −0.547257 0.947877i −6.76074 5.94074i 7.40102 12.8189i −17.4860 + 17.8673i −1.93123 + 9.65947i −16.9207 + 9.76914i −33.7133 10.4152 + 80.3276i 26.5053 + 6.79660i
14.11 −0.316502 0.548197i −4.90909 + 7.54326i 7.79965 13.5094i −24.0216 6.92551i 5.68893 + 0.303692i 73.2178 42.2723i −20.0025 −32.8017 74.0611i 3.80633 + 15.3605i
14.12 0.316502 + 0.548197i 4.90909 7.54326i 7.79965 13.5094i −18.0085 17.3406i 5.68893 + 0.303692i −73.2178 + 42.2723i 20.0025 −32.8017 74.0611i 3.80633 15.3605i
14.13 0.547257 + 0.947877i 6.76074 + 5.94074i 7.40102 12.8189i 6.73050 24.0770i −1.93123 + 9.65947i 16.9207 9.76914i 33.7133 10.4152 + 80.3276i 26.5053 6.79660i
14.14 1.02189 + 1.76997i −8.99991 0.0407214i 5.91148 10.2390i 24.7463 + 3.55264i −9.12485 15.9711i 2.09732 1.21089i 56.8640 80.9967 + 0.732978i 19.0000 + 47.4305i
14.15 1.59606 + 2.76446i 0.340063 + 8.99357i 2.90517 5.03190i 6.24274 + 24.2080i −24.3196 + 15.2944i −21.0893 + 12.1759i 69.6213 −80.7687 + 6.11676i −56.9583 + 55.8953i
14.16 1.64811 + 2.85461i 5.28472 7.28504i 2.56747 4.44699i −9.73566 + 23.0264i 29.5057 + 3.07927i 64.7475 37.3820i 69.6654 −25.1435 76.9987i −81.7769 + 10.1586i
14.17 2.30921 + 3.99967i −4.25182 7.93234i −2.66491 + 4.61577i 1.07184 24.9770i 21.9084 35.3233i 45.6166 26.3368i 49.2794 −44.8440 + 67.4538i 102.375 53.3902i
14.18 2.61690 + 4.53261i −8.42379 + 3.16855i −5.69638 + 9.86641i −23.1420 9.45767i −36.4061 29.8900i −50.0684 + 28.9070i 24.1135 60.9206 53.3825i −17.6925 129.644i
14.19 2.87361 + 4.97724i 8.30470 3.46872i −8.51525 + 14.7488i 24.3310 5.74467i 41.1291 + 31.3667i −41.4151 + 23.9110i −5.92246 56.9359 57.6134i 98.5104 + 104.593i
14.20 3.12350 + 5.41005i 7.73715 + 4.59745i −11.5125 + 19.9402i −24.9883 + 0.765363i −0.705517 + 56.2185i 10.1314 5.84936i −43.8846 38.7269 + 71.1423i −82.1915 132.797i
See all 44 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 14.22 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
9.d odd 6 1 inner
45.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 45.5.h.a 44
3.b odd 2 1 135.5.h.a 44
5.b even 2 1 inner 45.5.h.a 44
9.c even 3 1 135.5.h.a 44
9.c even 3 1 405.5.d.a 44
9.d odd 6 1 inner 45.5.h.a 44
9.d odd 6 1 405.5.d.a 44
15.d odd 2 1 135.5.h.a 44
45.h odd 6 1 inner 45.5.h.a 44
45.h odd 6 1 405.5.d.a 44
45.j even 6 1 135.5.h.a 44
45.j even 6 1 405.5.d.a 44

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.5.h.a 44 1.a even 1 1 trivial
45.5.h.a 44 5.b even 2 1 inner
45.5.h.a 44 9.d odd 6 1 inner
45.5.h.a 44 45.h odd 6 1 inner
135.5.h.a 44 3.b odd 2 1
135.5.h.a 44 9.c even 3 1
135.5.h.a 44 15.d odd 2 1
135.5.h.a 44 45.j even 6 1
405.5.d.a 44 9.c even 3 1
405.5.d.a 44 9.d odd 6 1
405.5.d.a 44 45.h odd 6 1
405.5.d.a 44 45.j even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{5}^{\mathrm{new}}(45, [\chi])$$.