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Properties

Label 309.2.m.a
Level $309$
Weight $2$
Character orbit 309.m
Analytic conductor $2.467$
Analytic rank $0$
Dimension $256$
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] = [309,2,Mod(4,309)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(309, base_ring=CyclotomicField(102))
 
chi = DirichletCharacter(H, H._module([0, 88]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("309.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 309 = 3 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 309.m (of order \(51\), degree \(32\), 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: \(2.46737742246\)
Analytic rank: \(0\)
Dimension: \(256\)
Relative dimension: \(8\) over \(\Q(\zeta_{51})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{51}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 256 q - q^{2} + 16 q^{3} + 3 q^{4} - q^{5} + q^{6} - 4 q^{7} + 57 q^{8} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 256 q - q^{2} + 16 q^{3} + 3 q^{4} - q^{5} + q^{6} - 4 q^{7} + 57 q^{8} - 16 q^{9} - 58 q^{10} - 32 q^{11} - 37 q^{12} + 10 q^{13} + 57 q^{14} + q^{15} - 7 q^{16} - q^{18} - 15 q^{19} + 46 q^{20} - 30 q^{21} + 54 q^{22} - 21 q^{23} - 6 q^{24} - 27 q^{25} + 14 q^{26} + 16 q^{27} - 49 q^{28} + 2 q^{29} - 10 q^{30} - 92 q^{31} + 8 q^{32} + 32 q^{33} - 35 q^{34} + 58 q^{35} + 3 q^{36} + 4 q^{37} - 51 q^{38} - 10 q^{39} + 41 q^{40} - 216 q^{41} + 28 q^{42} + 3 q^{43} - 133 q^{44} - q^{45} - 36 q^{46} + 2 q^{47} + 7 q^{48} + 26 q^{49} - 60 q^{50} - 34 q^{51} + 62 q^{52} - 7 q^{53} + q^{54} + 16 q^{55} + 19 q^{56} - 36 q^{57} - 32 q^{58} + 73 q^{59} + 5 q^{60} - 58 q^{61} - 20 q^{62} - 4 q^{63} - 269 q^{64} + 41 q^{65} - 20 q^{66} - 10 q^{67} - 8 q^{68} + 4 q^{69} + 89 q^{70} - 178 q^{71} + 6 q^{72} - 84 q^{73} - 239 q^{74} - 7 q^{75} + 58 q^{76} + 33 q^{77} - 65 q^{78} - 42 q^{79} - 24 q^{80} - 16 q^{81} - 198 q^{82} + 31 q^{83} - 104 q^{84} + 130 q^{85} - 86 q^{86} + 32 q^{87} + 66 q^{88} + 73 q^{89} + 10 q^{90} - 27 q^{91} + 146 q^{92} + 92 q^{93} - 446 q^{94} + 48 q^{95} + 196 q^{96} + 100 q^{97} + 363 q^{98} + 36 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −2.35704 + 0.291865i −0.445738 + 0.895163i 3.53088 − 0.888052i −2.62788 + 1.20902i 0.789358 − 2.24004i 0.482979 + 3.11140i −3.63392 + 1.40779i −0.602635 − 0.798017i 5.84115 − 3.61669i
4.2 −2.07388 + 0.256802i −0.445738 + 0.895163i 2.29545 − 0.577330i 2.84109 − 1.30711i 0.694529 − 1.97093i 0.199393 + 1.28451i −0.715026 + 0.277003i −0.602635 − 0.798017i −5.55642 + 3.44039i
4.3 −0.828238 + 0.102558i −0.445738 + 0.895163i −1.26413 + 0.317942i 1.49463 − 0.687639i 0.277371 − 0.787123i 0.252699 + 1.62792i 2.57081 − 0.995938i −0.602635 − 0.798017i −1.16739 + 0.722815i
4.4 −0.532681 + 0.0659602i −0.445738 + 0.895163i −1.66020 + 0.417556i −3.11723 + 1.43415i 0.178391 − 0.506238i 0.0298225 + 0.192120i 1.85782 − 0.719723i −0.602635 − 0.798017i 1.56589 − 0.969560i
4.5 −0.00449171 0.000556194i −0.445738 + 0.895163i −1.93957 + 0.487822i 1.89417 − 0.871455i 0.00150424 − 0.00426873i −0.709062 − 4.56786i 0.0168814 − 0.00653991i −0.602635 − 0.798017i −0.00802335 + 0.00496785i
4.6 1.13366 − 0.140377i −0.445738 + 0.895163i −0.674123 + 0.169549i −2.39887 + 1.10365i −0.379654 + 1.07738i −0.168774 − 1.08726i −2.87078 + 1.11214i −0.602635 − 0.798017i −2.56456 + 1.58791i
4.7 1.16645 − 0.144437i −0.445738 + 0.895163i −0.599858 + 0.150870i 2.33592 − 1.07469i −0.390635 + 1.10854i 0.718672 + 4.62976i −2.86988 + 1.11180i −0.602635 − 0.798017i 2.56950 − 1.59097i
4.8 2.50382 − 0.310040i −0.445738 + 0.895163i 4.23338 − 1.06474i 0.00758076 − 0.00348770i −0.838511 + 2.37952i 0.0996018 + 0.641645i 5.56434 − 2.15564i −0.602635 − 0.798017i 0.0178995 − 0.0110829i
7.1 −1.75151 + 2.04437i −0.0922684 + 0.995734i −0.804884 − 5.18515i 2.00807 + 0.248653i −1.87404 − 1.93267i −3.52552 − 1.89316i 7.43245 + 4.60198i −0.982973 − 0.183750i −4.02549 + 3.66972i
7.2 −1.31101 + 1.53022i −0.0922684 + 0.995734i −0.316035 − 2.03593i 1.63118 + 0.201984i −1.40273 − 1.44661i 2.18043 + 1.17087i 0.103334 + 0.0639817i −0.982973 − 0.183750i −2.44758 + 2.23126i
7.3 −0.883903 + 1.03170i −0.0922684 + 0.995734i 0.0236712 + 0.152492i −1.68521 − 0.208674i −0.945739 − 0.975325i −1.07039 − 0.574789i −2.48839 − 1.54074i −0.982973 − 0.183750i 1.70485 − 1.55417i
7.4 0.000702385 0 0.000819827i −0.0922684 + 0.995734i 0.306783 + 1.97633i 3.00003 + 0.371484i 0.000751522 0 0.000775033i 1.33098 + 0.714720i 0.00367146 + 0.00227327i −0.982973 − 0.183750i 0.00241173 − 0.00219858i
7.5 0.168103 − 0.196211i −0.0922684 + 0.995734i 0.296543 + 1.91036i −2.13958 − 0.264938i 0.179863 + 0.185490i −4.24131 − 2.27754i 0.864032 + 0.534986i −0.982973 − 0.183750i −0.411654 + 0.375272i
7.6 0.763579 − 0.891254i −0.0922684 + 0.995734i 0.0955038 + 0.615245i −2.38821 − 0.295725i 0.816997 + 0.842556i 2.90637 + 1.56069i 2.61693 + 1.62033i −0.982973 − 0.183750i −2.08715 + 1.90269i
7.7 1.11372 − 1.29994i −0.0922684 + 0.995734i −0.142684 − 0.919187i 3.18216 + 0.394037i 1.19163 + 1.22891i −2.54541 − 1.36686i 1.55698 + 0.964040i −0.982973 − 0.183750i 4.05625 − 3.69776i
7.8 1.24971 − 1.45866i −0.0922684 + 0.995734i −0.259149 − 1.66947i −0.374543 − 0.0463785i 1.33713 + 1.37896i 2.57538 + 1.38295i 0.507142 + 0.314009i −0.982973 − 0.183750i −0.535719 + 0.488372i
16.1 −2.33087 + 0.586237i 0.602635 + 0.798017i 3.32725 − 1.78670i −2.51462 + 2.93507i −1.87249 − 1.50679i −1.45865 + 0.464031i −3.15560 + 2.87671i −0.273663 + 0.961826i 4.14059 − 8.31543i
16.2 −1.97759 + 0.497384i 0.602635 + 0.798017i 1.90145 − 1.02106i 1.40956 − 1.64524i −1.58869 − 1.27841i −4.22482 + 1.34401i −0.238488 + 0.217411i −0.273663 + 0.961826i −1.96921 + 3.95471i
16.3 −1.41833 + 0.356724i 0.602635 + 0.798017i 0.122379 − 0.0657159i 0.971494 − 1.13393i −1.13941 − 0.916876i 2.65609 − 0.844964i 2.01147 − 1.83370i −0.273663 + 0.961826i −0.973396 + 1.95484i
16.4 −0.682812 + 0.171734i 0.602635 + 0.798017i −1.32529 + 0.711663i −1.71238 + 1.99870i −0.548533 − 0.441403i −0.0851646 + 0.0270928i 1.82334 − 1.66220i −0.273663 + 0.961826i 0.825991 − 1.65881i
See next 80 embeddings (of 256 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
103.g even 51 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 309.2.m.a ✓ 256
3.b odd 2 1 927.2.ba.b 256
103.g even 51 1 inner 309.2.m.a ✓ 256
309.n odd 102 1 927.2.ba.b 256
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
309.2.m.a ✓ 256 1.a even 1 1 trivial
309.2.m.a ✓ 256 103.g even 51 1 inner
927.2.ba.b 256 3.b odd 2 1
927.2.ba.b 256 309.n odd 102 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{256} + T_{2}^{255} - 9 T_{2}^{254} - 31 T_{2}^{253} + 227 T_{2}^{251} + 1291 T_{2}^{250} + \cdots + 22005481 \) acting on \(S_{2}^{\mathrm{new}}(309, [\chi])\). Copy content Toggle raw display