Properties

Label 109.4.e.a
Level $109$
Weight $4$
Character orbit 109.e
Analytic conductor $6.431$
Analytic rank $0$
Dimension $52$
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] = [109,4,Mod(46,109)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(109, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("109.46");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 109 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 109.e (of order \(6\), degree \(2\), 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: \(6.43120819063\)
Analytic rank: \(0\)
Dimension: \(52\)
Relative dimension: \(26\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 52 q - q^{3} - 198 q^{4} + 23 q^{5} + 15 q^{6} + 3 q^{7} - 169 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 52 q - q^{3} - 198 q^{4} + 23 q^{5} + 15 q^{6} + 3 q^{7} - 169 q^{9} + 120 q^{10} + 15 q^{11} + 4 q^{12} - 84 q^{13} + 225 q^{14} + 133 q^{15} + 666 q^{16} + 243 q^{18} - 434 q^{20} - 125 q^{21} + 25 q^{22} + 180 q^{24} - 655 q^{25} + 91 q^{26} + 590 q^{27} + 219 q^{28} - 664 q^{29} - 474 q^{30} + 21 q^{31} - 448 q^{34} + 531 q^{35} + 768 q^{36} - 192 q^{37} - 1416 q^{38} + 933 q^{39} - 588 q^{40} - 639 q^{42} - 232 q^{43} + 819 q^{44} - 812 q^{45} + 1040 q^{46} + 1041 q^{47} - 233 q^{48} - 467 q^{49} + 2742 q^{50} + 276 q^{51} + 1173 q^{52} - 1815 q^{53} - 3903 q^{56} - 1224 q^{57} + 420 q^{58} + 225 q^{59} - 3604 q^{60} - 1257 q^{61} - 867 q^{62} + 136 q^{63} - 1162 q^{64} + 873 q^{65} + 538 q^{66} + 843 q^{67} - 852 q^{69} + 3186 q^{70} - 744 q^{71} + 1389 q^{72} + 1912 q^{73} + 456 q^{74} + 4936 q^{75} + 5935 q^{78} - 4149 q^{79} + 2991 q^{80} + 1178 q^{81} - 3542 q^{82} - 2985 q^{83} + 3068 q^{84} - 3084 q^{85} - 1735 q^{87} + 4237 q^{88} + 3626 q^{89} - 3399 q^{91} + 2658 q^{93} - 2257 q^{94} - 2346 q^{95} - 5277 q^{96} - 928 q^{97} + 1251 q^{98} + 1050 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
46.1 5.10799i 2.61128 + 4.52286i −18.0916 3.73827 + 6.47487i 23.1027 13.3384i 15.6971 + 27.1882i 51.5478i −0.137518 + 0.238188i 33.0736 19.0951i
46.2 4.91334i −4.35312 7.53982i −16.1409 9.64430 + 16.7044i −37.0457 + 21.3883i 3.49660 + 6.05629i 39.9990i −24.3993 + 42.2608i 82.0745 47.3857i
46.3 4.82340i 2.59386 + 4.49270i −15.2652 −2.46834 4.27529i 21.6701 12.5112i −11.9912 20.7694i 35.0428i 0.0437764 0.0758230i −20.6215 + 11.9058i
46.4 4.73022i −2.45405 4.25053i −14.3749 −5.46049 9.45785i −20.1059 + 11.6082i 1.27920 + 2.21564i 30.1548i 1.45530 2.52066i −44.7377 + 25.8293i
46.5 3.81152i 0.361158 + 0.625545i −6.52767 10.5183 + 18.2183i 2.38428 1.37656i −11.5289 19.9686i 5.61183i 13.2391 22.9308i 69.4393 40.0908i
46.6 3.11861i 3.08348 + 5.34074i −1.72574 −10.5158 18.2139i 16.6557 9.61617i 6.20063 + 10.7398i 19.5670i −5.51566 + 9.55341i −56.8020 + 32.7947i
46.7 2.75335i −0.992557 1.71916i 0.419077 0.307591 + 0.532763i −4.73344 + 2.73285i −1.66075 2.87650i 23.1806i 11.5297 19.9700i 1.46688 0.846905i
46.8 2.63572i 4.87560 + 8.44479i 1.05299 3.33030 + 5.76825i 22.2581 12.8507i −3.38317 5.85982i 23.8611i −34.0429 + 58.9641i 15.2035 8.77774i
46.9 2.54381i −0.513075 0.888672i 1.52901 3.21600 + 5.57027i −2.26062 + 1.30517i 15.2344 + 26.3867i 24.2400i 12.9735 22.4708i 14.1697 8.18091i
46.10 2.29881i −4.53777 7.85965i 2.71547 −0.994942 1.72329i −18.0678 + 10.4315i −11.7430 20.3394i 24.6328i −27.6827 + 47.9478i −3.96152 + 2.28718i
46.11 0.570629i 2.87339 + 4.97685i 7.67438 1.27074 + 2.20098i 2.83993 1.63964i 8.44184 + 14.6217i 8.94425i −3.01268 + 5.21812i 1.25594 0.725119i
46.12 0.476992i 1.29609 + 2.24489i 7.77248 −4.80343 8.31978i 1.07079 0.618223i −13.4649 23.3219i 7.52335i 10.1403 17.5636i −3.96847 + 2.29120i
46.13 0.392313i −3.20948 5.55899i 7.84609 −10.7878 18.6850i −2.18086 + 1.25912i 7.93910 + 13.7509i 6.21662i −7.10158 + 12.3003i −7.33035 + 4.23218i
46.14 0.00776848i −3.57258 6.18789i 7.99994 7.08629 + 12.2738i −0.0480705 + 0.0277535i 8.53919 + 14.7903i 0.124295i −12.0266 + 20.8308i 0.0953489 0.0550497i
46.15 0.522266i 2.15477 + 3.73217i 7.72724 9.74051 + 16.8711i −1.94919 + 1.12536i −4.20111 7.27654i 8.21381i 4.21393 7.29875i −8.81119 + 5.08714i
46.16 1.65255i −0.440124 0.762317i 5.26906 −3.75307 6.50051i 1.25977 0.727329i 7.82854 + 13.5594i 21.9278i 13.1126 22.7117i 10.7424 6.20215i
46.17 1.70726i −2.85787 4.94998i 5.08526 0.376921 + 0.652847i 8.45092 4.87914i −11.1601 19.3298i 22.3400i −2.83489 + 4.91017i −1.11458 + 0.643504i
46.18 2.21870i 3.85635 + 6.67940i 3.07736 −5.87494 10.1757i −14.8196 + 8.55610i 9.25809 + 16.0355i 24.5774i −16.2429 + 28.1335i 22.5768 13.0347i
46.19 2.78760i −0.912464 1.58043i 0.229303 5.82500 + 10.0892i 4.40561 2.54358i −4.04575 7.00744i 22.9400i 11.8348 20.4985i −28.1246 + 16.2378i
46.20 3.35996i 3.92788 + 6.80329i −3.28930 3.56237 + 6.17021i −22.8588 + 13.1975i −7.86115 13.6159i 15.8277i −17.3565 + 30.0623i −20.7316 + 11.9694i
See all 52 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 46.26
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
109.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 109.4.e.a 52
109.e even 6 1 inner 109.4.e.a 52
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
109.4.e.a 52 1.a even 1 1 trivial
109.4.e.a 52 109.e even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(109, [\chi])\).