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Properties

Label 845.2.be.a
Level $845$
Weight $2$
Character orbit 845.be
Analytic conductor $6.747$
Analytic rank $0$
Dimension $2136$
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(18,845)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(845, base_ring=CyclotomicField(52))
 
chi = DirichletCharacter(H, H._module([39, 31]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("845.18");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.be (of order \(52\), 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: \(2136\)
Relative dimension: \(89\) over \(\Q(\zeta_{52})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{52}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 2136 q - 26 q^{2} - 22 q^{3} + 174 q^{4} - 20 q^{5} - 48 q^{6} - 22 q^{7} - 26 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 2136 q - 26 q^{2} - 22 q^{3} + 174 q^{4} - 20 q^{5} - 48 q^{6} - 22 q^{7} - 26 q^{8} - 34 q^{10} - 56 q^{11} - 34 q^{12} - 36 q^{13} + 82 q^{15} - 210 q^{16} - 40 q^{17} - 44 q^{18} + 4 q^{19} - 20 q^{20} - 36 q^{21} - 68 q^{22} + 16 q^{23} - 4 q^{24} - 16 q^{25} - 64 q^{26} - 34 q^{27} - 14 q^{28} - 166 q^{30} - 64 q^{31} - 26 q^{32} - 26 q^{33} + 2 q^{34} - 22 q^{35} - 52 q^{36} + 18 q^{37} + 18 q^{38} + 4 q^{39} - 46 q^{40} - 54 q^{41} + 226 q^{42} - 34 q^{43} + 12 q^{44} - 180 q^{45} - 60 q^{46} + 76 q^{47} + 6 q^{48} - 22 q^{49} - 62 q^{50} - 52 q^{51} + 16 q^{52} - 94 q^{53} - 12 q^{54} + 114 q^{55} - 52 q^{56} - 104 q^{57} - 50 q^{58} - 200 q^{59} + 198 q^{60} - 36 q^{61} + 2 q^{62} + 104 q^{63} + 134 q^{64} - 28 q^{65} - 28 q^{66} + 182 q^{67} - 28 q^{68} + 16 q^{69} - 162 q^{70} - 44 q^{71} - 4 q^{72} - 26 q^{73} + 130 q^{74} + 128 q^{75} - 68 q^{76} - 22 q^{77} - 30 q^{78} - 4 q^{80} + 110 q^{81} - 68 q^{82} + 174 q^{83} + 20 q^{84} - 64 q^{85} - 4 q^{86} + 150 q^{87} - 182 q^{88} - 18 q^{89} + 16 q^{90} - 80 q^{91} - 158 q^{92} - 6 q^{93} + 104 q^{94} + 6 q^{95} - 92 q^{96} - 182 q^{97} - 26 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
18.1 −1.28245 − 2.44350i −2.76072 − 1.66891i −3.18988 + 4.62135i 0.849275 + 2.06851i −0.537510 + 8.88611i 0.200968 − 1.65512i 9.90416 + 1.20258i 3.44213 + 6.55844i 3.96525 − 4.72796i
18.2 −1.26220 − 2.40493i 1.33307 + 0.805868i −3.05438 + 4.42504i −0.525466 − 2.17345i 0.255451 − 4.22310i −0.136038 + 1.12038i 9.10469 + 1.10551i −0.266520 − 0.507812i −4.56374 + 4.00704i
18.3 −1.23819 − 2.35918i −1.36243 − 0.823618i −2.89647 + 4.19625i −2.19950 + 0.402731i −0.256110 + 4.23401i −0.328544 + 2.70581i 8.19621 + 0.995200i −0.216299 − 0.412123i 3.67352 + 4.69035i
18.4 −1.23731 − 2.35750i −0.975228 − 0.589546i −2.89073 + 4.18795i −1.18159 − 1.89838i −0.183193 + 3.02855i 0.348054 − 2.86648i 8.16369 + 0.991252i −0.790664 − 1.50648i −3.01343 + 5.13448i
18.5 −1.20296 − 2.29206i 0.715961 + 0.432813i −2.67027 + 3.86856i 0.753266 + 2.10537i 0.130758 − 2.16168i −0.152026 + 1.25205i 6.93982 + 0.842646i −1.06890 − 2.03661i 3.91948 − 4.25922i
18.6 −1.16452 − 2.21881i −0.125290 − 0.0757405i −2.43088 + 3.52174i 2.22830 + 0.186193i −0.0221508 + 0.366196i 0.489185 − 4.02880i 5.66974 + 0.688431i −1.38421 − 2.63739i −2.18178 − 5.16101i
18.7 −1.15439 − 2.19951i 2.79370 + 1.68885i −2.36911 + 3.43225i 1.39239 + 1.74964i 0.489620 − 8.09439i −0.163273 + 1.34468i 5.35229 + 0.649885i 3.55838 + 6.77993i 2.24099 − 5.08236i
18.8 −1.12895 − 2.15103i 1.64057 + 0.991758i −2.21627 + 3.21082i 1.98088 − 1.03736i 0.281185 − 4.64855i 0.394522 − 3.24918i 4.58547 + 0.556776i 0.313711 + 0.597727i −4.46770 − 3.08980i
18.9 −1.12574 − 2.14492i 1.37622 + 0.831957i −2.19728 + 3.18330i −2.07293 + 0.838424i 0.235209 − 3.88847i −0.273667 + 2.25385i 4.49205 + 0.545433i −0.192328 − 0.366450i 4.13195 + 3.50243i
18.10 −1.10775 − 2.11065i −2.39065 − 1.44520i −2.09159 + 3.03019i 0.982462 − 2.00867i −0.402054 + 6.64675i −0.250067 + 2.05949i 3.98001 + 0.483261i 2.23245 + 4.25357i −5.32793 + 0.151482i
18.11 −1.02574 − 1.95439i 1.44745 + 0.875015i −1.63136 + 2.36344i 0.787909 − 2.09265i 0.225407 − 3.72643i −0.435838 + 3.58945i 1.91020 + 0.231940i −0.0647048 − 0.123285i −4.89805 + 0.606643i
18.12 −1.02136 − 1.94603i −0.654469 − 0.395640i −1.60774 + 2.32921i −1.46874 + 1.68606i −0.101482 + 1.67770i 0.180835 − 1.48931i 1.81128 + 0.219929i −1.12237 − 2.13850i 4.78123 + 1.13615i
18.13 −1.00771 − 1.92002i −0.391217 − 0.236499i −1.53489 + 2.22367i 2.06947 + 0.846935i −0.0598517 + 0.989467i −0.241406 + 1.98816i 1.51103 + 0.183472i −1.29705 − 2.47132i −0.459282 − 4.82689i
18.14 −0.960874 − 1.83079i −2.27595 − 1.37586i −1.29239 + 1.87235i −1.86340 − 1.23601i −0.332012 + 5.48881i −0.338989 + 2.79182i 0.564611 + 0.0685562i 1.89278 + 3.60639i −0.472385 + 4.59916i
18.15 −0.951312 − 1.81257i −1.81201 − 1.09540i −1.24430 + 1.80268i 0.0142307 + 2.23602i −0.261704 + 4.32648i 0.365771 − 3.01239i 0.386950 + 0.0469843i 0.689324 + 1.31340i 4.03942 − 2.15295i
18.16 −0.922593 − 1.75785i 2.82664 + 1.70877i −1.10274 + 1.59760i 0.299000 − 2.21599i 0.395919 − 6.54532i 0.366054 − 3.01472i −0.115824 − 0.0140636i 3.67586 + 7.00377i −4.17124 + 1.51886i
18.17 −0.916194 − 1.74566i −0.625757 − 0.378283i −1.07179 + 1.55276i 1.11893 − 1.93597i −0.0870399 + 1.43894i −0.0864046 + 0.711606i −0.221655 − 0.0269138i −1.14570 − 2.18294i −4.40471 − 0.179557i
18.18 −0.887331 − 1.69067i −2.10342 − 1.27156i −0.934873 + 1.35440i 2.14653 + 0.626417i −0.283359 + 4.68448i −0.145268 + 1.19639i −0.671528 − 0.0815383i 1.41334 + 2.69289i −0.845622 − 4.18491i
18.19 −0.843059 − 1.60631i 2.37685 + 1.43685i −0.733368 + 1.06247i −2.06455 − 0.858852i 0.304218 − 5.02932i −0.240993 + 1.98476i −1.27684 − 0.155036i 2.19068 + 4.17400i 0.360953 + 4.04038i
18.20 −0.831729 − 1.58473i 0.256464 + 0.155038i −0.683458 + 0.990160i −1.72205 − 1.42637i 0.0323842 − 0.535375i 0.367115 − 3.02346i −1.41578 − 0.171906i −1.35243 − 2.57684i −0.828124 + 3.91534i
See next 80 embeddings (of 2136 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 18.89
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
845.be even 52 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 845.2.be.a yes 2136
5.c odd 4 1 845.2.z.a ✓ 2136
169.j odd 52 1 845.2.z.a ✓ 2136
845.be even 52 1 inner 845.2.be.a yes 2136
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
845.2.z.a ✓ 2136 5.c odd 4 1
845.2.z.a ✓ 2136 169.j odd 52 1
845.2.be.a yes 2136 1.a even 1 1 trivial
845.2.be.a yes 2136 845.be even 52 1 inner

Hecke kernels

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