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Properties

Label 845.2.bg.a
Level $845$
Weight $2$
Character orbit 845.bg
Analytic conductor $6.747$
Analytic rank $0$
Dimension $1440$
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] = [845,2,Mod(36,845)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(845, base_ring=CyclotomicField(78))
 
chi = DirichletCharacter(H, H._module([0, 47]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("845.36");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.bg (of order \(78\), degree \(24\), 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: \(6.74735897080\)
Analytic rank: \(0\)
Dimension: \(1440\)
Relative dimension: \(60\) over \(\Q(\zeta_{78})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{78}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1440 q - 2 q^{3} - 58 q^{4} + 18 q^{6} + 6 q^{7} + 60 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1440 q - 2 q^{3} - 58 q^{4} + 18 q^{6} + 6 q^{7} + 60 q^{9} + 2 q^{10} - 20 q^{12} + 26 q^{13} + 4 q^{14} + 6 q^{15} + 58 q^{16} + 6 q^{17} - 156 q^{18} - 12 q^{19} - 12 q^{20} + 60 q^{22} - 146 q^{23} - 144 q^{24} + 120 q^{25} - 10 q^{26} + 4 q^{27} + 18 q^{28} + 4 q^{29} - 4 q^{30} + 52 q^{31} + 20 q^{32} - 42 q^{33} + 130 q^{34} - 10 q^{35} - 56 q^{36} - 6 q^{37} - 162 q^{38} + 12 q^{40} - 12 q^{41} - 24 q^{42} + 2 q^{43} + 42 q^{46} - 156 q^{47} + 30 q^{48} + 18 q^{49} - 124 q^{52} + 48 q^{53} - 330 q^{54} - 4 q^{55} + 20 q^{56} + 78 q^{57} + 68 q^{58} - 196 q^{59} + 12 q^{61} - 118 q^{62} + 24 q^{63} + 96 q^{64} + 8 q^{65} + 44 q^{66} + 98 q^{67} + 120 q^{68} + 28 q^{69} - 156 q^{71} + 386 q^{72} - 30 q^{74} + 2 q^{75} - 392 q^{76} + 4 q^{77} - 204 q^{78} + 40 q^{79} + 40 q^{81} - 244 q^{82} + 30 q^{84} - 18 q^{85} - 30 q^{87} + 30 q^{88} - 24 q^{89} - 56 q^{90} + 76 q^{91} + 20 q^{92} - 130 q^{93} - 148 q^{94} + 16 q^{95} - 338 q^{96} + 30 q^{97} - 72 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
36.1 −0.884668 + 2.64990i 0.0711924 − 0.0871948i −4.64047 − 3.48709i 0.239316 + 0.970942i 0.168076 + 0.265791i −0.0816205 − 1.01105i 8.74741 − 6.03790i 0.597542 + 2.92696i −2.78462 − 0.224798i
36.2 −0.831245 + 2.48988i −1.32828 + 1.62685i −3.90966 − 2.93792i −0.239316 − 0.970942i −2.94654 − 4.65959i −0.252037 − 3.12203i 6.24434 − 4.31016i −0.282235 − 1.38248i 2.61646 + 0.211223i
36.3 −0.818909 + 2.45293i 2.07399 − 2.54018i −3.74738 − 2.81597i 0.239316 + 0.970942i 4.53248 + 7.16754i −0.0143236 − 0.177429i 5.71965 − 3.94799i −1.55099 − 7.59727i −2.57763 − 0.208088i
36.4 −0.779719 + 2.33554i −1.50331 + 1.84122i −3.24792 − 2.44065i 0.239316 + 0.970942i −3.12810 − 4.94669i −0.248506 − 3.07829i 4.17991 − 2.88518i −0.530080 − 2.59650i −2.45428 − 0.198130i
36.5 −0.771941 + 2.31225i 0.0529410 − 0.0648410i −3.15170 − 2.36835i −0.239316 − 0.970942i 0.109061 + 0.172466i 0.00197558 + 0.0244720i 3.89677 − 2.68974i 0.598675 + 2.93251i 2.42979 + 0.196153i
36.6 −0.769308 + 2.30436i 1.66755 − 2.04238i −3.11935 − 2.34404i −0.239316 − 0.970942i 3.42351 + 5.41385i 0.295831 + 3.66452i 3.80256 − 2.62472i −0.790507 − 3.87216i 2.42151 + 0.195484i
36.7 −0.753586 + 2.25727i 0.130140 − 0.159393i −2.92847 − 2.20060i 0.239316 + 0.970942i 0.261720 + 0.413877i 0.375180 + 4.64743i 3.25723 − 2.24831i 0.591607 + 2.89789i −2.37202 − 0.191489i
36.8 −0.700600 + 2.09855i −1.22869 + 1.50487i −2.31419 − 1.73900i −0.239316 − 0.970942i −2.29723 − 3.63278i 0.305438 + 3.78353i 1.62916 − 1.12453i −0.154882 − 0.758660i 2.20524 + 0.178025i
36.9 −0.695772 + 2.08409i 0.709219 − 0.868635i −2.26045 − 1.69862i 0.239316 + 0.970942i 1.31686 + 2.08245i −0.283028 − 3.50593i 1.49637 − 1.03287i 0.348541 + 1.70727i −2.19004 − 0.176798i
36.10 −0.620520 + 1.85868i −0.642227 + 0.786586i −1.47078 − 1.10522i −0.239316 − 0.970942i −1.06350 − 1.68179i −0.152161 − 1.88486i −0.258421 + 0.178375i 0.393816 + 1.92904i 1.95318 + 0.157677i
36.11 −0.609016 + 1.82422i 1.12141 − 1.37347i −1.35801 − 1.02048i 0.239316 + 0.970942i 1.82257 + 2.88216i 0.375473 + 4.65107i −0.476892 + 0.329174i −0.0287996 − 0.141070i −1.91696 − 0.154753i
36.12 −0.588121 + 1.76164i −1.20562 + 1.47661i −1.15860 − 0.870629i 0.239316 + 0.970942i −1.89221 − 2.99229i 0.0861505 + 1.06717i −0.841790 + 0.581046i −0.126795 − 0.621084i −1.85119 − 0.149444i
36.13 −0.564166 + 1.68988i 0.610425 − 0.747635i −0.938541 − 0.705268i −0.239316 − 0.970942i 0.919036 + 1.45334i −0.347889 − 4.30938i −1.21109 + 0.835957i 0.413738 + 2.02662i 1.77579 + 0.143357i
36.14 −0.560185 + 1.67796i −1.40547 + 1.72139i −0.902850 − 0.678448i 0.239316 + 0.970942i −2.10110 − 3.32262i −0.0832562 − 1.03131i −1.26754 + 0.874921i −0.387760 − 1.89937i −1.76326 − 0.142345i
36.15 −0.544893 + 1.63215i −2.03862 + 2.49686i −0.768135 − 0.577216i −0.239316 − 0.970942i −2.96443 − 4.68787i −0.0919441 − 1.13893i −1.47157 + 1.01575i −1.47824 − 7.24092i 1.71513 + 0.138460i
36.16 −0.488399 + 1.46293i 0.148728 − 0.182159i −0.302754 − 0.227505i 0.239316 + 0.970942i 0.193848 + 0.306546i −0.283980 − 3.51772i −2.05790 + 1.42046i 0.589015 + 2.88519i −1.53730 − 0.124104i
36.17 −0.482104 + 1.44408i 1.49379 − 1.82956i −0.254048 − 0.190905i −0.239316 − 0.970942i 1.92186 + 3.03918i 0.103140 + 1.27762i −2.10771 + 1.45484i −0.515805 − 2.52658i 1.51749 + 0.122504i
36.18 −0.461736 + 1.38307i 1.92787 − 2.36121i −0.100788 − 0.0757374i −0.239316 − 0.970942i 2.37555 + 3.75663i −0.227680 − 2.82032i −2.24871 + 1.55217i −1.25857 − 6.16486i 1.45338 + 0.117329i
36.19 −0.363850 + 1.08986i −0.622835 + 0.762834i 0.543472 + 0.408393i −0.239316 − 0.970942i −0.604766 − 0.956361i 0.369869 + 4.58165i −2.53404 + 1.74912i 0.406084 + 1.98913i 1.14527 + 0.0924556i
36.20 −0.320055 + 0.958682i 1.49409 − 1.82993i 0.782249 + 0.587822i 0.239316 + 0.970942i 1.27613 + 2.01803i 0.0105891 + 0.131169i −2.47747 + 1.71007i −0.516253 − 2.52877i −1.00742 − 0.0813273i
See next 80 embeddings (of 1440 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 36.60
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
169.k even 78 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 845.2.bg.a ✓ 1440
169.k even 78 1 inner 845.2.bg.a ✓ 1440
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
845.2.bg.a ✓ 1440 1.a even 1 1 trivial
845.2.bg.a ✓ 1440 169.k even 78 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(845, [\chi])\).