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Properties

Label 345.2.s.b
Level $345$
Weight $2$
Character orbit 345.s
Analytic conductor $2.755$
Analytic rank $0$
Dimension $160$
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] = [345,2,Mod(11,345)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(345, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 0, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("345.11");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 345 = 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 345.s (of order \(22\), degree \(10\), 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: \(2.75483886973\)
Analytic rank: \(0\)
Dimension: \(160\)
Relative dimension: \(16\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 160 q + 14 q^{4} + 16 q^{5} - q^{6} - 22 q^{8} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 160 q + 14 q^{4} + 16 q^{5} - q^{6} - 22 q^{8} + 24 q^{9} - 8 q^{11} - 3 q^{12} - 12 q^{14} + 2 q^{16} + 36 q^{17} + 11 q^{18} - 14 q^{20} - 49 q^{21} + 4 q^{23} - 66 q^{24} - 16 q^{25} - 27 q^{27} + q^{30} + 8 q^{31} - 110 q^{32} + 26 q^{33} - 11 q^{34} - 29 q^{36} + 76 q^{38} - 70 q^{39} - 77 q^{40} + 22 q^{41} + 30 q^{42} - 44 q^{43} + 28 q^{44} - 2 q^{45} - 82 q^{46} + 23 q^{48} - 34 q^{49} + 6 q^{51} + 2 q^{52} - 10 q^{53} + 62 q^{54} - 14 q^{55} - 134 q^{56} + 67 q^{57} - 2 q^{58} + 3 q^{60} + 22 q^{61} + 275 q^{62} - 45 q^{63} - 8 q^{64} - 6 q^{66} + 22 q^{67} - 16 q^{68} + 12 q^{69} + 12 q^{70} + 104 q^{72} + 10 q^{73} - 24 q^{74} + 44 q^{76} - 264 q^{77} + 116 q^{78} + 22 q^{79} - 13 q^{80} + 28 q^{81} + 70 q^{82} - 2 q^{83} - 258 q^{84} + 8 q^{85} + 186 q^{86} - 171 q^{87} + 44 q^{89} - 11 q^{90} - 40 q^{92} - 142 q^{93} - 30 q^{94} + 13 q^{96} - 88 q^{98} - 80 q^{99}+O(q^{100}) \) 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 −1.36242 − 2.11997i 1.31339 + 1.12916i −1.80726 + 3.95735i 0.959493 − 0.281733i 0.604397 − 4.32275i −0.618805 + 0.536197i 5.86299 − 0.842970i 0.449990 + 2.96606i −1.90450 − 1.65026i
11.2 −1.20534 − 1.87554i −1.70809 − 0.287108i −1.23399 + 2.70207i 0.959493 − 0.281733i 1.52034 + 3.54966i −2.85846 + 2.47687i 2.14168 − 0.307927i 2.83514 + 0.980811i −1.68492 − 1.45999i
11.3 −1.15818 − 1.80216i −0.573123 − 1.63448i −1.07557 + 2.35516i 0.959493 − 0.281733i −2.28181 + 2.92587i 1.78777 − 1.54912i 1.24921 − 0.179610i −2.34306 + 1.87352i −1.61899 − 1.40286i
11.4 −0.790269 − 1.22968i 1.71529 − 0.240363i −0.0567635 + 0.124295i 0.959493 − 0.281733i −1.65111 − 1.91931i −0.148193 + 0.128410i −2.69599 + 0.387625i 2.88445 − 0.824585i −1.10470 − 0.957227i
11.5 −0.628776 − 0.978395i −0.496086 + 1.65949i 0.268933 − 0.588880i 0.959493 − 0.281733i 1.93556 − 0.558078i −3.68436 + 3.19251i −3.04762 + 0.438182i −2.50780 − 1.64650i −0.878952 − 0.761617i
11.6 −0.561180 − 0.873212i −1.72280 + 0.178810i 0.383253 − 0.839206i 0.959493 − 0.281733i 1.12294 + 1.40402i 2.04970 − 1.77608i −3.00273 + 0.431727i 2.93605 − 0.616105i −0.784460 − 0.679739i
11.7 −0.390950 − 0.608331i 0.738125 − 1.56690i 0.613606 − 1.34361i 0.959493 − 0.281733i −1.24176 + 0.163555i 2.49752 − 2.16411i −2.48878 + 0.357832i −1.91034 − 2.31313i −0.546501 − 0.473546i
11.8 −0.0518555 − 0.0806888i 1.28253 + 1.16410i 0.827008 − 1.81090i 0.959493 − 0.281733i 0.0274237 − 0.163850i −1.36836 + 1.18569i −0.378881 + 0.0544749i 0.289745 + 2.98598i −0.0724877 − 0.0628109i
11.9 0.0422559 + 0.0657515i −0.823843 + 1.52358i 0.828292 − 1.81371i 0.959493 − 0.281733i −0.134990 + 0.0102112i 2.31213 − 2.00347i 0.308981 − 0.0444248i −1.64257 − 2.51037i 0.0590686 + 0.0511832i
11.10 0.337922 + 0.525816i −0.636905 − 1.61070i 0.668538 − 1.46389i 0.959493 − 0.281733i 0.631708 − 0.879185i −0.135295 + 0.117234i 2.23301 − 0.321058i −2.18870 + 2.05173i 0.472373 + 0.409314i
11.11 0.649940 + 1.01133i 1.33735 − 1.10068i 0.230470 − 0.504660i 0.959493 − 0.281733i 1.98234 + 0.637128i −1.59910 + 1.38562i 3.04003 − 0.437090i 0.577026 − 2.94398i 0.908537 + 0.787252i
11.12 0.885498 + 1.37786i 0.945956 + 1.45092i −0.283565 + 0.620921i 0.959493 − 0.281733i −1.16152 + 2.58818i 0.203828 − 0.176618i 2.13575 − 0.307075i −1.21033 + 2.74501i 1.23782 + 1.07257i
11.13 1.07740 + 1.67647i −1.65997 − 0.494465i −0.818925 + 1.79320i 0.959493 − 0.281733i −0.959501 − 3.31563i 3.35433 − 2.90655i 0.0565286 − 0.00812759i 2.51101 + 1.64159i 1.50607 + 1.30502i
11.14 1.34961 + 2.10004i −0.404092 + 1.68425i −1.75788 + 3.84922i 0.959493 − 0.281733i −4.08237 + 1.42448i −1.00507 + 0.870896i −5.51415 + 0.792815i −2.67342 − 1.36119i 1.88659 + 1.63474i
11.15 1.37512 + 2.13973i −0.985278 − 1.42451i −1.85666 + 4.06552i 0.959493 − 0.281733i 1.69319 − 4.06710i −2.74460 + 2.37821i −6.21702 + 0.893873i −1.05845 + 2.80708i 1.92225 + 1.66564i
11.16 1.50149 + 2.33637i 1.67754 − 0.431107i −2.37331 + 5.19681i 0.959493 − 0.281733i 3.52604 + 3.27205i 1.95695 − 1.69571i −10.2072 + 1.46758i 2.62829 − 1.44640i 2.09890 + 1.81871i
56.1 −1.97803 + 1.71397i −1.72966 + 0.0910124i 0.690266 − 4.80091i −0.415415 + 0.909632i 3.26532 − 3.14461i −1.35762 + 4.62364i 4.03320 + 6.27578i 2.98343 − 0.314841i −0.737380 − 2.51128i
56.2 −1.80712 + 1.56588i 1.68066 − 0.418780i 0.529079 − 3.67982i −0.415415 + 0.909632i −2.38140 + 3.38850i 0.714940 − 2.43486i 2.22053 + 3.45521i 2.64925 − 1.40765i −0.673669 − 2.29430i
56.3 −1.69938 + 1.47252i 0.795291 + 1.53867i 0.434940 − 3.02508i −0.415415 + 0.909632i −3.61722 − 1.44370i −0.620144 + 2.11201i 1.28399 + 1.99792i −1.73503 + 2.44738i −0.633503 − 2.15751i
56.4 −1.30797 + 1.13336i −0.174911 − 1.72320i 0.141646 − 0.985173i −0.415415 + 0.909632i 2.18179 + 2.05565i 0.542849 − 1.84878i −0.940078 − 1.46279i −2.93881 + 0.602811i −0.487593 − 1.66059i
See next 80 embeddings (of 160 total)
\(n\): 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
69.g even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 345.2.s.b yes 160
3.b odd 2 1 345.2.s.a ✓ 160
23.d odd 22 1 345.2.s.a ✓ 160
69.g even 22 1 inner 345.2.s.b yes 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
345.2.s.a ✓ 160 3.b odd 2 1
345.2.s.a ✓ 160 23.d odd 22 1
345.2.s.b yes 160 1.a even 1 1 trivial
345.2.s.b yes 160 69.g even 22 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{160} - 23 T_{2}^{158} + 22 T_{2}^{157} + 317 T_{2}^{156} - 484 T_{2}^{155} - 3282 T_{2}^{154} + \cdots + 8068889929 \) acting on \(S_{2}^{\mathrm{new}}(345, [\chi])\). Copy content Toggle raw display