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Properties

Label 309.3.f.b
Level $309$
Weight $3$
Character orbit 309.f
Analytic conductor $8.420$
Analytic rank $0$
Dimension $36$
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] = [309,3,Mod(160,309)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(309, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("309.160");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 309 = 3 \cdot 103 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 309.f (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: \(8.41964016873\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 40 q^{4} + 9 q^{5} - 14 q^{7} - 24 q^{8} - 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 40 q^{4} + 9 q^{5} - 14 q^{7} - 24 q^{8} - 108 q^{9} - 21 q^{11} + 120 q^{12} - 20 q^{13} + 92 q^{14} + 9 q^{15} - 116 q^{16} + 6 q^{17} + 11 q^{19} + 15 q^{20} + 42 q^{21} - 4 q^{23} + 185 q^{25} + 120 q^{26} - 33 q^{28} - 85 q^{29} - 24 q^{30} + 83 q^{32} - 21 q^{33} + 48 q^{34} - 51 q^{35} + 120 q^{36} - 102 q^{38} + 213 q^{40} - 44 q^{41} - 120 q^{43} + 294 q^{44} - 27 q^{45} - 33 q^{46} - 66 q^{47} - 348 q^{48} - 228 q^{49} - 4 q^{50} - 18 q^{51} + 52 q^{52} + 3 q^{53} - 50 q^{55} - 221 q^{56} + 33 q^{57} + 89 q^{58} + 316 q^{59} - 15 q^{60} + 10 q^{61} - 114 q^{62} + 42 q^{63} + 792 q^{64} - 135 q^{65} - 186 q^{66} - 126 q^{67} - 128 q^{68} - 591 q^{70} - 18 q^{71} + 72 q^{72} + 129 q^{74} + 555 q^{75} + 358 q^{76} - 279 q^{77} + 360 q^{78} - 190 q^{79} + 324 q^{81} - 208 q^{82} + 170 q^{83} - 99 q^{84} + 210 q^{85} - 288 q^{86} - 255 q^{87} - 6 q^{88} + 358 q^{91} - 22 q^{92} - 258 q^{93} - 249 q^{96} + 163 q^{97} + 552 q^{98} + 63 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
160.1 −1.88399 − 3.26317i − 1.73205i −5.09887 + 8.83151i 5.84515 + 3.37470i −5.65198 + 3.26317i 1.60661 − 2.78274i 23.3530 −3.00000 − 25.4317i
160.2 −1.79491 − 3.10887i − 1.73205i −4.44339 + 7.69618i −7.71249 − 4.45281i −5.38473 + 3.10887i −2.07878 + 3.60056i 17.5427 −3.00000 31.9695i
160.3 −1.61543 − 2.79801i − 1.73205i −3.21925 + 5.57591i 1.06351 + 0.614015i −4.84630 + 2.79801i −5.46922 + 9.47297i 7.87848 −3.00000 − 3.96760i
160.4 −1.15287 − 1.99683i − 1.73205i −0.658216 + 1.14006i 7.65972 + 4.42234i −3.45861 + 1.99683i 0.258444 − 0.447639i −6.18761 −3.00000 − 20.3935i
160.5 −1.12496 − 1.94849i − 1.73205i −0.531078 + 0.919853i 1.20657 + 0.696614i −3.37489 + 1.94849i −4.13833 + 7.16779i −6.60993 −3.00000 − 3.13466i
160.6 −1.07471 − 1.86145i − 1.73205i −0.309992 + 0.536921i −1.34421 − 0.776082i −3.22412 + 1.86145i 2.42391 − 4.19834i −7.26506 −3.00000 3.33625i
160.7 −0.640219 − 1.10889i − 1.73205i 1.18024 − 2.04424i −7.70807 − 4.45026i −1.92066 + 1.10889i 3.34006 − 5.78515i −8.14420 −3.00000 11.3965i
160.8 −0.505065 − 0.874799i − 1.73205i 1.48982 − 2.58044i 2.06504 + 1.19225i −1.51520 + 0.874799i −2.23527 + 3.87160i −7.05034 −3.00000 − 2.40866i
160.9 −0.0390288 − 0.0675999i − 1.73205i 1.99695 − 3.45882i 3.54483 + 2.04661i −0.117086 + 0.0675999i 6.13127 − 10.6197i −0.623985 −3.00000 − 0.319506i
160.10 0.121998 + 0.211307i − 1.73205i 1.97023 − 3.41254i −5.68171 − 3.28034i 0.365995 − 0.211307i −6.45871 + 11.1868i 1.93745 −3.00000 − 1.60078i
160.11 0.296921 + 0.514282i − 1.73205i 1.82368 − 3.15870i 0.829708 + 0.479032i 0.890763 − 0.514282i −0.783231 + 1.35660i 4.54132 −3.00000 0.568939i
160.12 0.684583 + 1.18573i − 1.73205i 1.06269 − 1.84064i −2.56435 − 1.48053i 2.05375 − 1.18573i 2.37670 − 4.11656i 8.38667 −3.00000 − 4.05417i
160.13 0.787500 + 1.36399i − 1.73205i 0.759687 − 1.31582i 7.80019 + 4.50344i 2.36250 − 1.36399i −6.00599 + 10.4027i 8.69302 −3.00000 14.1858i
160.14 1.21604 + 2.10625i − 1.73205i −0.957521 + 1.65848i 6.93236 + 4.00240i 3.64813 − 2.10625i 4.46878 − 7.74016i 5.07080 −3.00000 19.4684i
160.15 1.32893 + 2.30177i − 1.73205i −1.53210 + 2.65368i −0.682667 − 0.394138i 3.98678 − 2.30177i −0.655427 + 1.13523i 2.48722 −3.00000 − 2.09512i
160.16 1.61330 + 2.79432i − 1.73205i −3.20548 + 5.55206i −4.49163 − 2.59324i 4.83990 − 2.79432i −4.35368 + 7.54080i −7.77924 −3.00000 − 16.7347i
160.17 1.81265 + 3.13961i − 1.73205i −4.57143 + 7.91794i −7.11623 − 4.10856i 5.43796 − 3.13961i 6.16030 − 10.6700i −18.6444 −3.00000 − 29.7896i
160.18 1.96926 + 3.41086i − 1.73205i −5.75596 + 9.96962i 4.85428 + 2.80262i 5.90778 − 3.41086i −1.58743 + 2.74951i −29.5859 −3.00000 22.0763i
253.1 −1.88399 + 3.26317i 1.73205i −5.09887 − 8.83151i 5.84515 − 3.37470i −5.65198 − 3.26317i 1.60661 + 2.78274i 23.3530 −3.00000 25.4317i
253.2 −1.79491 + 3.10887i 1.73205i −4.44339 − 7.69618i −7.71249 + 4.45281i −5.38473 − 3.10887i −2.07878 − 3.60056i 17.5427 −3.00000 − 31.9695i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 160.18
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
103.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 309.3.f.b ✓ 36
103.d odd 6 1 inner 309.3.f.b ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
309.3.f.b ✓ 36 1.a even 1 1 trivial
309.3.f.b ✓ 36 103.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} + 56 T_{2}^{34} + 8 T_{2}^{33} + 1861 T_{2}^{32} + 393 T_{2}^{31} + 40964 T_{2}^{30} + \cdots + 20958084 \) acting on \(S_{3}^{\mathrm{new}}(309, [\chi])\). Copy content Toggle raw display