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Properties

Label 309.3.h.b
Level $309$
Weight $3$
Character orbit 309.h
Analytic conductor $8.420$
Analytic rank $0$
Dimension $132$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [309,3,Mod(56,309)]
 
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([3, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("309.56");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 309 = 3 \cdot 103 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 309.h (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: \(132\)
Relative dimension: \(66\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 132 q + 4 q^{3} + 130 q^{4} + 7 q^{6} + 16 q^{7} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 132 q + 4 q^{3} + 130 q^{4} + 7 q^{6} + 16 q^{7} - 24 q^{9} - 12 q^{12} - 36 q^{13} - 35 q^{15} - 170 q^{16} + 21 q^{18} - 12 q^{19} - 23 q^{21} - 44 q^{22} + 282 q^{24} + 350 q^{25} + 178 q^{27} + 126 q^{28} - 40 q^{30} + 164 q^{31} + 19 q^{33} - 108 q^{34} - 79 q^{36} - 276 q^{37} - 26 q^{39} + 36 q^{40} + 216 q^{42} + 26 q^{43} - 29 q^{45} + 46 q^{46} - 72 q^{48} - 126 q^{49} + 12 q^{51} - 210 q^{52} - 184 q^{54} + 64 q^{55} - 72 q^{57} + 174 q^{58} + 72 q^{60} - 924 q^{61} + 163 q^{63} - 660 q^{64} - 344 q^{66} - 290 q^{67} + 610 q^{69} - 146 q^{70} + 290 q^{72} + 392 q^{73} - 74 q^{75} + 96 q^{76} + 10 q^{78} - 272 q^{79} - 80 q^{81} + 138 q^{82} - 95 q^{84} - 204 q^{85} + 19 q^{87} - 94 q^{88} - 350 q^{90} + 182 q^{91} - 644 q^{93} - 460 q^{94} + 421 q^{96} + 106 q^{97} + 144 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
56.1 −3.42718 − 1.97868i 1.01724 + 2.82227i 5.83036 + 10.0985i −4.15706 + 2.40008i 2.09812 − 11.6852i −1.76855 − 3.06322i − 30.3163i −6.93044 + 5.74186i 18.9960
56.2 −3.29185 − 1.90055i −0.694708 − 2.91846i 5.22418 + 9.04855i 1.04114 − 0.601101i −3.25979 + 10.9274i −1.40224 − 2.42874i − 24.5109i −8.03476 + 4.05495i −4.56969
56.3 −3.10566 − 1.79305i −1.58262 + 2.54859i 4.43007 + 7.67311i 7.71502 − 4.45427i 9.48483 − 5.07733i −0.388626 − 0.673119i − 17.4290i −3.99063 − 8.06690i −31.9469
56.4 −3.03097 − 1.74993i 2.46539 + 1.70935i 4.12451 + 7.14386i 3.52313 − 2.03408i −4.48128 − 9.49523i 4.15103 + 7.18980i − 14.8710i 3.15628 + 8.42840i −14.2380
56.5 −3.02487 − 1.74641i −2.91991 − 0.688587i 4.09988 + 7.10121i 0.137819 − 0.0795701i 7.62978 + 7.18223i 5.34164 + 9.25199i − 14.6690i 8.05170 + 4.02122i −0.555847
56.6 −3.02281 − 1.74522i −2.59626 + 1.50315i 4.09159 + 7.08683i −5.15242 + 2.97475i 10.4713 − 0.0126834i −0.264097 − 0.457430i − 14.6011i 4.48111 − 7.80511i 20.7664
56.7 −2.98874 − 1.72555i 2.66657 − 1.37455i 3.95503 + 6.85032i −4.83210 + 2.78982i −10.3415 − 0.493117i −4.13243 − 7.15757i − 13.4940i 5.22120 − 7.33069i 19.2559
56.8 −2.90959 − 1.67986i 2.39065 − 1.81240i 3.64383 + 6.31129i 6.06396 − 3.50103i −10.0004 + 1.25741i −0.542682 − 0.939953i − 11.0456i 2.43040 − 8.66563i −23.5249
56.9 −2.63615 − 1.52198i 2.91284 + 0.717873i 2.63288 + 4.56027i −0.207893 + 0.120027i −6.58611 − 6.32573i 0.166391 + 0.288198i − 3.85291i 7.96932 + 4.18211i 0.730719
56.10 −2.59878 − 1.50041i −2.58801 − 1.51730i 2.50243 + 4.33434i 3.96669 − 2.29017i 4.44910 + 7.82619i −5.21197 − 9.02740i − 3.01540i 4.39560 + 7.85358i −13.7447
56.11 −2.56837 − 1.48285i 0.399984 − 2.97322i 2.39768 + 4.15290i −5.34104 + 3.08365i −5.43613 + 7.04320i 3.56733 + 6.17880i − 2.35878i −8.68003 − 2.37848i 18.2903
56.12 −2.33934 − 1.35062i −2.51577 − 1.63430i 1.64833 + 2.85499i −6.90311 + 3.98551i 3.67792 + 7.22100i −3.99827 − 6.92521i 1.89990i 3.65816 + 8.22301i 21.5316
56.13 −2.31647 − 1.33741i 1.15432 + 2.76903i 1.57735 + 2.73205i 2.62472 − 1.51538i 1.02940 − 7.95818i −5.55733 − 9.62558i 2.26104i −6.33510 + 6.39269i −8.10678
56.14 −2.29626 − 1.32575i −0.598933 + 2.93961i 1.51521 + 2.62443i −0.718976 + 0.415101i 5.27248 − 5.95607i 5.99029 + 10.3755i 2.57082i −8.28256 − 3.52125i 2.20128
56.15 −2.19752 − 1.26874i −1.83672 + 2.37202i 1.21941 + 2.11208i −2.52384 + 1.45714i 7.04571 − 2.88225i −3.76679 − 6.52428i 3.96148i −2.25295 − 8.71345i 7.39493
56.16 −2.07535 − 1.19820i 1.30407 + 2.70174i 0.871373 + 1.50926i −8.36030 + 4.82682i 0.530834 − 7.16958i 3.36572 + 5.82960i 5.40929i −5.59881 + 7.04652i 23.1340
56.17 −1.93245 − 1.11570i −1.01796 − 2.82201i 0.489585 + 0.847987i 5.27721 − 3.04680i −1.18136 + 6.58915i 1.87290 + 3.24397i 6.74069i −6.92750 + 5.74541i −13.5973
56.18 −1.93161 − 1.11522i 1.64007 − 2.51200i 0.487414 + 0.844226i 4.69649 − 2.71152i −5.96941 + 3.02318i 3.46148 + 5.99546i 6.74744i −3.62033 − 8.23973i −12.0957
56.19 −1.85777 − 1.07258i −2.97012 + 0.422392i 0.300870 + 0.521122i 5.70915 − 3.29618i 5.97084 + 2.40099i 0.319293 + 0.553032i 7.28983i 8.64317 − 2.50911i −14.1417
56.20 −1.49022 − 0.860382i 2.91672 + 0.701938i −0.519487 − 0.899778i −2.42530 + 1.40025i −3.74264 − 3.55554i −1.47557 − 2.55576i 8.67088i 8.01457 + 4.09472i 4.81899
See next 80 embeddings (of 132 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 56.66
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
103.c even 3 1 inner
309.h 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.h.b ✓ 132
3.b odd 2 1 inner 309.3.h.b ✓ 132
103.c even 3 1 inner 309.3.h.b ✓ 132
309.h odd 6 1 inner 309.3.h.b ✓ 132
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
309.3.h.b ✓ 132 1.a even 1 1 trivial
309.3.h.b ✓ 132 3.b odd 2 1 inner
309.3.h.b ✓ 132 103.c even 3 1 inner
309.3.h.b ✓ 132 309.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{132} - 197 T_{2}^{130} + 20365 T_{2}^{128} - 1448924 T_{2}^{126} + 79060653 T_{2}^{124} + \cdots + 31\!\cdots\!89 \) acting on \(S_{3}^{\mathrm{new}}(309, [\chi])\). Copy content Toggle raw display