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Properties

Label 845.2.bi.a
Level $845$
Weight $2$
Character orbit 845.bi
Analytic conductor $6.747$
Analytic rank $0$
Dimension $4272$
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(7,845)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(845, base_ring=CyclotomicField(156))
 
chi = DirichletCharacter(H, H._module([39, 107]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("845.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.bi (of order \(156\), degree \(48\), 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: \(4272\)
Relative dimension: \(89\) over \(\Q(\zeta_{156})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{156}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 4272 q - 46 q^{2} - 50 q^{3} - 174 q^{4} - 52 q^{5} - 96 q^{6} - 50 q^{7} - 52 q^{8} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 4272 q - 46 q^{2} - 50 q^{3} - 174 q^{4} - 52 q^{5} - 96 q^{6} - 50 q^{7} - 52 q^{8} - 12 q^{9} - 50 q^{10} - 88 q^{11} - 44 q^{12} - 48 q^{13} - 136 q^{15} + 66 q^{16} - 56 q^{17} - 52 q^{18} + 20 q^{19} - 52 q^{20} - 108 q^{21} - 34 q^{22} - 64 q^{23} - 32 q^{24} - 62 q^{25} - 80 q^{26} - 32 q^{27} - 70 q^{28} - 338 q^{30} - 104 q^{31} - 100 q^{32} - 70 q^{33} - 2 q^{34} - 80 q^{35} - 140 q^{36} - 48 q^{37} - 96 q^{38} - 4 q^{39} - 20 q^{40} - 114 q^{41} + 416 q^{42} - 50 q^{43} + 36 q^{44} + 78 q^{45} - 108 q^{46} - 142 q^{47} - 12 q^{48} - 14 q^{49} - 88 q^{50} - 104 q^{51} - 22 q^{52} + 28 q^{53} + 48 q^{54} - 198 q^{55} - 104 q^{56} + 26 q^{57} - 52 q^{58} - 400 q^{59} - 288 q^{60} - 96 q^{61} + 4 q^{62} - 146 q^{63} + 316 q^{64} - 38 q^{65} - 104 q^{66} - 278 q^{67} - 74 q^{68} + 8 q^{69} + 168 q^{70} - 88 q^{71} - 56 q^{72} - 52 q^{73} - 148 q^{74} - 212 q^{75} - 40 q^{76} - 56 q^{77} - 120 q^{78} - 50 q^{80} - 242 q^{81} - 100 q^{82} + 420 q^{83} + 40 q^{84} - 26 q^{85} - 164 q^{86} - 210 q^{87} - 742 q^{88} + 6 q^{89} - 34 q^{90} - 112 q^{91} + 92 q^{92} - 84 q^{93} - 56 q^{94} - 30 q^{95} - 160 q^{96} + 38 q^{97} - 22 q^{98} - 60 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −2.71599 + 0.554473i −1.18483 + 0.853560i 5.22920 − 2.22795i 2.20987 − 0.341309i 2.74471 − 2.97522i 1.24282 − 2.61919i −8.40447 + 5.80118i −0.274744 + 0.822959i −5.81273 + 2.15230i
7.2 −2.67822 + 0.546762i −0.937691 + 0.675519i 5.03393 − 2.14476i 0.439961 + 2.19236i 2.14199 − 2.32188i −1.31015 + 2.76109i −7.81010 + 5.39092i −0.527066 + 1.57875i −2.37701 − 5.63105i
7.3 −2.66980 + 0.545044i 2.08854 − 1.50460i 4.99081 − 2.12639i −0.623794 − 2.14730i −4.75591 + 5.15532i −1.61729 + 3.40837i −7.68046 + 5.30144i 1.14817 − 3.43919i 2.83578 + 5.39286i
7.4 −2.66595 + 0.544257i 1.53498 − 1.10581i 4.97109 − 2.11799i −1.90247 + 1.17499i −3.49034 + 3.78346i 1.62666 − 3.42811i −7.62137 + 5.26065i 0.183349 − 0.549197i 4.43239 − 4.16789i
7.5 −2.46638 + 0.503515i −1.80828 + 1.30270i 3.98953 − 1.69978i −2.12995 − 0.680680i 3.80398 − 4.12344i −1.44786 + 3.05129i −4.84052 + 3.34117i 0.622856 − 1.86568i 5.59599 + 0.606354i
7.6 −2.46147 + 0.502512i −2.55634 + 1.84161i 3.96635 − 1.68990i 1.01671 − 1.99156i 5.36693 − 5.81765i −0.400531 + 0.844101i −4.77878 + 3.29855i 2.19337 − 6.56994i −1.50181 + 5.41307i
7.7 −2.45629 + 0.501456i 1.46041 − 1.05209i 3.94196 − 1.67951i 1.28993 + 1.82649i −3.05962 + 3.31657i −0.871304 + 1.83623i −4.71404 + 3.25387i 0.0759039 − 0.227360i −4.08436 − 3.83955i
7.8 −2.43405 + 0.496915i 1.94881 − 1.40393i 3.83771 − 1.63510i 1.98929 − 1.02113i −4.04586 + 4.38563i 1.27771 − 2.69271i −4.43967 + 3.06448i 0.876816 − 2.62639i −4.33462 + 3.47400i
7.9 −2.42158 + 0.494370i 0.0159137 − 0.0114643i 3.77971 − 1.61038i −0.458813 − 2.18849i −0.0328687 + 0.0356291i 0.186952 − 0.393993i −4.28870 + 2.96028i −0.949882 + 2.84524i 2.19298 + 5.07279i
7.10 −2.32551 + 0.474757i −1.37969 + 0.993936i 3.34265 − 1.42417i −0.958124 + 2.02040i 2.73660 − 2.96643i 1.07805 − 2.27194i −3.19058 + 2.20230i −0.0343754 + 0.102967i 1.26893 − 5.15333i
7.11 −2.22587 + 0.454415i −0.0782619 + 0.0563804i 2.90805 − 1.23900i −2.16915 − 0.542943i 0.148581 − 0.161059i 0.584345 − 1.23148i −2.17064 + 1.49829i −0.947058 + 2.83678i 5.07497 + 0.222827i
7.12 −2.11582 + 0.431949i 0.520044 − 0.374643i 2.45018 − 1.04392i −1.35231 + 1.78080i −0.938494 + 1.01731i −0.731436 + 1.54147i −1.17881 + 0.813673i −0.819916 + 2.45595i 2.09204 − 4.35199i
7.13 −2.10490 + 0.429719i 2.49647 − 1.79847i 2.40600 − 1.02510i −1.53135 + 1.62941i −4.48198 + 4.85838i −1.48456 + 3.12864i −1.08782 + 0.750869i 2.04784 − 6.13403i 2.52314 − 4.08780i
7.14 −2.02163 + 0.412719i −1.36596 + 0.984050i 2.07671 − 0.884801i 2.03586 + 0.924800i 2.35534 − 2.55315i 1.30115 − 2.74212i −0.436984 + 0.301628i −0.0524999 + 0.157257i −4.49745 − 1.02937i
7.15 −1.97119 + 0.402422i −0.495771 + 0.357157i 1.88370 − 0.802571i 2.03879 − 0.918322i 0.833533 − 0.903535i −0.821982 + 1.73229i −0.0787295 + 0.0543431i −0.831776 + 2.49147i −3.64931 + 2.63065i
7.16 −1.95272 + 0.398651i −2.77960 + 2.00244i 1.81423 − 0.772973i 0.714897 + 2.11871i 4.62951 − 5.01830i 0.000400641 0 0.000844333i 0.0458637 − 0.0316574i 2.76640 − 8.28638i −2.24062 − 3.85225i
7.17 −1.83914 + 0.375464i 1.66375 − 1.19858i 1.40152 − 0.597131i 1.99858 + 1.00283i −2.60986 + 2.82904i −0.226061 + 0.476413i 0.736217 − 0.508174i 0.381478 − 1.14267i −4.05221 − 1.09395i
7.18 −1.78521 + 0.364454i −1.54641 + 1.11405i 1.21420 − 0.517323i 1.99322 + 1.01345i 2.35466 − 2.55240i −1.75884 + 3.70669i 1.01994 − 0.704017i 0.200288 − 0.599935i −3.92768 − 1.08278i
7.19 −1.73786 + 0.354787i 0.330372 − 0.238002i 1.05434 − 0.449211i 0.699302 − 2.12391i −0.489701 + 0.530826i 1.77353 − 3.73764i 1.24655 − 0.860431i −0.897504 + 2.68835i −0.461756 + 3.93916i
7.20 −1.70332 + 0.347735i 2.19937 − 1.58444i 0.940423 − 0.400677i 1.09578 − 1.94917i −3.19527 + 3.46361i −0.162880 + 0.343263i 1.39893 − 0.965609i 1.37678 − 4.12395i −1.18867 + 3.70111i
See next 80 embeddings (of 4272 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.89
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
845.bi even 156 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 845.2.bi.a ✓ 4272
5.c odd 4 1 845.2.bn.a yes 4272
169.l odd 156 1 845.2.bn.a yes 4272
845.bi even 156 1 inner 845.2.bi.a ✓ 4272
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
845.2.bi.a ✓ 4272 1.a even 1 1 trivial
845.2.bi.a ✓ 4272 845.bi even 156 1 inner
845.2.bn.a yes 4272 5.c odd 4 1
845.2.bn.a yes 4272 169.l odd 156 1

Hecke kernels

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