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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 $4$

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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.

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

Display 100 coefficients 10 coefficients

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:

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 \(S_{5}^{\mathrm{new}}(45, [\chi])\).