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Properties

Label 45.5.i.a
Level $45$
Weight $5$
Character orbit 45.i
Analytic conductor $4.652$
Analytic rank $0$
Dimension $32$
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] = [45,5,Mod(11,45)]
 
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([1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("45.11");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 45.i (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: \(32\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 8 q^{3} + 128 q^{4} + 98 q^{6} - 26 q^{7} - 220 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 8 q^{3} + 128 q^{4} + 98 q^{6} - 26 q^{7} - 220 q^{9} - 738 q^{11} - 214 q^{12} + 10 q^{13} + 1998 q^{14} - 50 q^{15} - 1024 q^{16} - 2504 q^{18} + 508 q^{19} - 1620 q^{21} + 672 q^{22} + 1998 q^{23} + 4164 q^{24} + 2000 q^{25} + 2158 q^{27} - 1664 q^{28} - 270 q^{29} + 2000 q^{30} - 1472 q^{31} - 6048 q^{32} - 1354 q^{33} - 594 q^{34} + 4498 q^{36} + 2068 q^{37} - 7164 q^{38} - 4616 q^{39} + 1650 q^{40} - 13428 q^{41} - 10230 q^{42} + 568 q^{43} - 100 q^{45} + 5724 q^{46} + 12528 q^{47} + 25844 q^{48} - 1674 q^{49} + 1454 q^{51} + 1534 q^{52} + 8456 q^{54} + 21294 q^{56} - 12512 q^{57} - 8190 q^{58} - 4374 q^{59} - 5050 q^{60} - 4478 q^{61} + 5604 q^{63} - 37216 q^{64} - 8550 q^{65} - 19472 q^{66} + 12838 q^{67} + 65736 q^{68} + 8706 q^{69} + 2400 q^{70} - 36372 q^{72} + 29236 q^{73} - 20628 q^{74} + 1000 q^{75} + 6190 q^{76} - 31806 q^{77} - 42430 q^{78} - 3512 q^{79} - 3088 q^{81} + 34692 q^{82} - 43074 q^{83} - 19350 q^{84} - 4350 q^{85} - 10746 q^{86} + 69622 q^{87} - 16128 q^{88} - 26750 q^{90} - 48904 q^{91} + 27036 q^{92} + 27030 q^{93} + 5118 q^{94} + 6300 q^{95} + 53566 q^{96} + 23266 q^{97} + 53512 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −6.31571 − 3.64638i −7.77630 + 4.53091i 18.5921 + 32.2025i 9.68246 − 5.59017i 65.6342 − 0.260606i −25.8824 + 44.8297i − 154.491i 39.9417 − 70.4675i −81.5354
11.2 −5.76237 − 3.32690i 8.89517 − 1.36966i 14.1366 + 24.4853i −9.68246 + 5.59017i −55.8139 − 21.7009i −43.3865 + 75.1475i − 81.6633i 77.2481 − 24.3666i 74.3918
11.3 −5.11159 − 2.95118i 1.28176 + 8.90826i 9.41890 + 16.3140i −9.68246 + 5.59017i 19.7380 − 49.3181i 36.1606 − 62.6320i − 16.7497i −77.7142 + 22.8364i 65.9903
11.4 −3.37336 − 1.94761i 8.40237 − 3.22494i −0.413632 − 0.716431i 9.68246 − 5.59017i −34.6251 − 5.48565i 15.5174 − 26.8770i 65.5459i 60.1995 − 54.1943i −43.5499
11.5 −3.23824 − 1.86960i 3.54924 + 8.27060i −1.00919 − 1.74797i 9.68246 − 5.59017i 3.96942 − 33.4179i −19.7075 + 34.1344i 67.3743i −55.8058 + 58.7087i −41.8055
11.6 −3.23643 − 1.86855i −8.96800 + 0.758233i −1.01703 − 1.76154i −9.68246 + 5.59017i 30.4411 + 14.3032i 8.02923 − 13.9070i 67.3951i 79.8502 − 13.5997i 41.7821
11.7 −2.15054 − 1.24161i −6.81039 − 5.88376i −4.91680 − 8.51614i 9.68246 − 5.59017i 7.34064 + 21.1091i −15.7848 + 27.3401i 64.1506i 11.7628 + 80.1414i −27.7633
11.8 −1.06605 − 0.615486i 1.63702 − 8.84987i −7.24235 − 12.5441i −9.68246 + 5.59017i −7.19213 + 8.42687i −21.1314 + 36.6007i 37.5258i −75.6403 − 28.9748i 13.7627
11.9 0.817210 + 0.471817i −6.73136 + 5.97401i −7.55478 − 13.0853i 9.68246 − 5.59017i −8.31957 + 1.70605i 31.6912 − 54.8908i − 29.3560i 9.62249 − 80.4264i 10.5501
11.10 0.904609 + 0.522276i −2.25893 + 8.71190i −7.45446 − 12.9115i −9.68246 + 5.59017i −6.59347 + 6.70108i −36.4983 + 63.2168i − 32.2860i −70.7944 − 39.3592i −11.6784
11.11 2.59623 + 1.49894i 4.94975 − 7.51665i −3.50638 − 6.07323i 9.68246 − 5.59017i 24.1177 − 12.0956i −1.63719 + 2.83570i − 68.9893i −31.9999 − 74.4110i 33.5172
11.12 3.10572 + 1.79309i −5.44105 − 7.16903i −1.56968 − 2.71877i −9.68246 + 5.59017i −4.04367 − 32.0212i 45.9054 − 79.5105i − 68.6371i −21.7900 + 78.0141i −40.0946
11.13 4.78725 + 2.76392i 4.38356 + 7.86031i 7.27848 + 12.6067i 9.68246 − 5.59017i −0.740053 + 49.7450i −0.820328 + 1.42085i − 7.97692i −42.5688 + 68.9122i 61.8031
11.14 5.22878 + 3.01884i 8.75515 − 2.08503i 10.2268 + 17.7133i −9.68246 + 5.59017i 52.0731 + 15.5282i −8.13139 + 14.0840i 26.8889i 72.3053 − 36.5095i −67.5033
11.15 5.93733 + 3.42792i −5.90111 + 6.79536i 15.5013 + 26.8490i −9.68246 + 5.59017i −58.3308 + 20.1178i 12.5523 − 21.7412i 102.855i −11.3538 − 80.2003i −76.6506
11.16 6.87716 + 3.97053i −1.96687 − 8.78245i 23.5302 + 40.7555i 9.68246 − 5.59017i 21.3445 − 68.2078i 10.1236 − 17.5346i 246.652i −73.2629 + 34.5478i 88.7837
41.1 −6.31571 + 3.64638i −7.77630 − 4.53091i 18.5921 − 32.2025i 9.68246 + 5.59017i 65.6342 + 0.260606i −25.8824 − 44.8297i 154.491i 39.9417 + 70.4675i −81.5354
41.2 −5.76237 + 3.32690i 8.89517 + 1.36966i 14.1366 − 24.4853i −9.68246 − 5.59017i −55.8139 + 21.7009i −43.3865 − 75.1475i 81.6633i 77.2481 + 24.3666i 74.3918
41.3 −5.11159 + 2.95118i 1.28176 − 8.90826i 9.41890 − 16.3140i −9.68246 − 5.59017i 19.7380 + 49.3181i 36.1606 + 62.6320i 16.7497i −77.7142 − 22.8364i 65.9903
41.4 −3.37336 + 1.94761i 8.40237 + 3.22494i −0.413632 + 0.716431i 9.68246 + 5.59017i −34.6251 + 5.48565i 15.5174 + 26.8770i − 65.5459i 60.1995 + 54.1943i −43.5499
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d 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.i.a ✓ 32
3.b odd 2 1 135.5.i.a 32
9.c even 3 1 135.5.i.a 32
9.c even 3 1 405.5.c.b 32
9.d odd 6 1 inner 45.5.i.a ✓ 32
9.d odd 6 1 405.5.c.b 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.5.i.a ✓ 32 1.a even 1 1 trivial
45.5.i.a ✓ 32 9.d odd 6 1 inner
135.5.i.a 32 3.b odd 2 1
135.5.i.a 32 9.c even 3 1
405.5.c.b 32 9.c even 3 1
405.5.c.b 32 9.d odd 6 1

Hecke kernels

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