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Properties

Label 945.2.dh.a
Level $945$
Weight $2$
Character orbit 945.dh
Analytic conductor $7.546$
Analytic rank $0$
Dimension $1680$
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] = [945,2,Mod(23,945)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(945, base_ring=CyclotomicField(36))
 
chi = DirichletCharacter(H, H._module([22, 27, 12]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.23");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.dh (of order \(36\), degree \(12\), 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: \(7.54586299101\)
Analytic rank: \(0\)
Dimension: \(1680\)
Relative dimension: \(140\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1680 q - 6 q^{2} - 6 q^{3} - 6 q^{5} - 36 q^{6} - 12 q^{7} - 36 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1680 q - 6 q^{2} - 6 q^{3} - 6 q^{5} - 36 q^{6} - 12 q^{7} - 36 q^{8} - 12 q^{10} - 24 q^{11} - 6 q^{12} - 24 q^{13} - 24 q^{15} - 12 q^{16} - 6 q^{18} - 24 q^{20} - 24 q^{21} - 24 q^{22} - 30 q^{23} - 6 q^{25} - 24 q^{27} - 24 q^{28} + 30 q^{30} - 12 q^{31} - 18 q^{32} - 6 q^{33} - 72 q^{35} - 84 q^{36} + 6 q^{37} + 42 q^{38} + 90 q^{40} - 72 q^{41} + 36 q^{42} - 24 q^{43} - 6 q^{45} + 12 q^{46} - 6 q^{47} - 96 q^{48} - 72 q^{50} - 12 q^{51} - 30 q^{52} + 90 q^{53} - 48 q^{55} - 180 q^{56} - 24 q^{57} - 6 q^{58} - 102 q^{60} - 84 q^{61} - 36 q^{62} - 54 q^{63} - 78 q^{65} - 12 q^{66} - 6 q^{67} - 246 q^{68} - 12 q^{70} - 72 q^{71} - 126 q^{72} + 6 q^{73} + 66 q^{75} - 12 q^{77} - 84 q^{78} - 24 q^{81} - 12 q^{82} - 24 q^{83} - 24 q^{85} + 204 q^{86} - 102 q^{87} - 78 q^{88} + 30 q^{90} - 12 q^{91} + 108 q^{92} - 78 q^{93} + 114 q^{95} + 204 q^{96} - 24 q^{97} - 18 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 −2.80394 − 0.245313i −1.01491 + 1.40355i 5.83228 + 1.02839i 1.39044 − 1.75119i 3.19006 − 3.68649i 1.13563 − 2.38963i −10.6636 − 2.85730i −0.939900 − 2.84896i −4.32830 + 4.56914i
23.2 −2.72361 − 0.238285i −0.480717 + 1.66400i 5.39166 + 0.950694i −2.22793 − 0.190602i 1.70579 − 4.41755i 0.207444 + 2.63761i −9.17652 − 2.45884i −2.53782 − 1.59983i 6.02259 + 1.05001i
23.3 −2.71580 − 0.237602i −0.385641 − 1.68857i 5.34950 + 0.943261i −2.23430 + 0.0889899i 0.646117 + 4.67746i −2.63903 + 0.188514i −9.03749 − 2.42159i −2.70256 + 1.30237i 6.08905 + 0.289194i
23.4 −2.66969 − 0.233567i −1.65681 − 0.504950i 5.10305 + 0.899806i 2.02833 + 0.941217i 4.30523 + 1.73503i 0.990785 + 2.45323i −8.23624 − 2.20690i 2.49005 + 1.67321i −5.19516 − 2.98650i
23.5 −2.62161 − 0.229361i 1.26014 − 1.18830i 4.85062 + 0.855295i −1.23958 − 1.86104i −3.57614 + 2.82623i 2.45496 − 0.986484i −7.43636 − 1.99257i 0.175891 − 2.99484i 2.82284 + 5.16322i
23.6 −2.59821 − 0.227314i 1.73149 + 0.0442062i 4.72939 + 0.833919i 1.87704 + 1.21521i −4.48871 − 0.508447i −1.12556 − 2.39439i −7.05984 − 1.89168i 2.99609 + 0.153085i −4.60069 − 3.58404i
23.7 −2.56556 − 0.224457i 1.33773 − 1.10022i 4.56209 + 0.804419i −0.0795769 + 2.23465i −3.67897 + 2.52242i −1.20986 + 2.35292i −6.54853 − 1.75467i 0.579022 − 2.94359i 0.705743 − 5.71527i
23.8 −2.50713 − 0.219346i 1.49881 + 0.868089i 4.26798 + 0.752561i −1.92170 − 1.14327i −3.56729 − 2.50517i −2.05754 − 1.66329i −5.67342 − 1.52019i 1.49284 + 2.60220i 4.56719 + 3.28784i
23.9 −2.49837 − 0.218579i −1.51700 − 0.835890i 4.22445 + 0.744884i −1.86141 + 1.23901i 3.60731 + 2.41994i 2.21158 − 1.45221i −5.54649 − 1.48618i 1.60258 + 2.53609i 4.92131 − 2.68865i
23.10 −2.49097 − 0.217932i −1.63108 + 0.582742i 4.18782 + 0.738426i −0.254064 + 2.22159i 4.18996 − 1.09613i −2.46044 − 0.972748i −5.44024 − 1.45771i 2.32082 − 1.90099i 1.11702 − 5.47854i
23.11 −2.48260 − 0.217199i 1.12387 + 1.31792i 4.14650 + 0.731140i 2.20390 + 0.377907i −2.50386 − 3.51597i 2.56627 + 0.643622i −5.32096 − 1.42575i −0.473839 + 2.96234i −5.38932 − 1.41688i
23.12 −2.46123 − 0.215330i 1.70887 + 0.282404i 4.04166 + 0.712654i −1.67451 + 1.48190i −4.14512 − 1.06303i 2.41463 + 1.08145i −5.02110 − 1.34540i 2.84050 + 0.965184i 4.44044 − 3.28672i
23.13 −2.42040 − 0.211758i −0.627239 − 1.61449i 3.84388 + 0.677780i 0.167498 − 2.22979i 1.17629 + 4.04053i 2.12238 + 1.57972i −4.46650 − 1.19679i −2.21314 + 2.02534i −0.877587 + 5.36151i
23.14 −2.40755 − 0.210633i 0.556775 − 1.64012i 3.78231 + 0.666923i 2.15475 − 0.597538i −1.68593 + 3.83140i −0.611342 + 2.57415i −4.29683 − 1.15133i −2.38000 − 1.82636i −5.31353 + 0.984739i
23.15 −2.39696 − 0.209707i 0.248707 + 1.71410i 3.73185 + 0.658026i −0.789023 + 2.09223i −0.236682 − 4.16080i 0.464853 − 2.60459i −4.15885 − 1.11436i −2.87629 + 0.852618i 2.33002 − 4.84955i
23.16 −2.39285 − 0.209347i −1.72745 + 0.126182i 3.71227 + 0.654574i −0.228622 − 2.22435i 4.15993 + 0.0597017i −2.36767 + 1.18074i −4.10558 − 1.10009i 2.96816 − 0.435946i 0.0813954 + 5.37039i
23.17 −2.30600 − 0.201749i 1.25173 − 1.19715i 3.30733 + 0.583172i 1.58744 − 1.57481i −3.12802 + 2.50810i −1.91488 − 1.82571i −3.03719 − 0.813812i 0.133654 − 2.99702i −3.97836 + 3.31126i
23.18 −2.28494 − 0.199907i 0.493029 + 1.66040i 3.21139 + 0.566255i 1.67331 − 1.48325i −0.794619 − 3.89248i −2.63322 + 0.257216i −2.79362 − 0.748547i −2.51384 + 1.63725i −4.11993 + 3.05464i
23.19 −2.17073 − 0.189914i 0.693633 + 1.58710i 2.70638 + 0.477207i −0.984060 − 2.00789i −1.20428 − 3.57688i 1.35780 + 2.27077i −1.57463 − 0.421922i −2.03775 + 2.20172i 1.75480 + 4.54547i
23.20 −2.12186 − 0.185639i −0.656366 + 1.60287i 2.49822 + 0.440503i 0.720006 + 2.11698i 1.69027 − 3.27922i 2.62985 − 0.289648i −1.10432 − 0.295902i −2.13837 − 2.10414i −1.13476 − 4.62559i
See next 80 embeddings (of 1680 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.140
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
189.bf odd 18 1 inner
945.dh even 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 945.2.dh.a ✓ 1680
5.c odd 4 1 inner 945.2.dh.a ✓ 1680
7.c even 3 1 945.2.do.a yes 1680
27.f odd 18 1 945.2.do.a yes 1680
35.l odd 12 1 945.2.do.a yes 1680
135.q even 36 1 945.2.do.a yes 1680
189.bf odd 18 1 inner 945.2.dh.a ✓ 1680
945.dh even 36 1 inner 945.2.dh.a ✓ 1680
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
945.2.dh.a ✓ 1680 1.a even 1 1 trivial
945.2.dh.a ✓ 1680 5.c odd 4 1 inner
945.2.dh.a ✓ 1680 189.bf odd 18 1 inner
945.2.dh.a ✓ 1680 945.dh even 36 1 inner
945.2.do.a yes 1680 7.c even 3 1
945.2.do.a yes 1680 27.f odd 18 1
945.2.do.a yes 1680 35.l odd 12 1
945.2.do.a yes 1680 135.q even 36 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(945, [\chi])\).