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Properties

Label 309.2.k.a
Level $309$
Weight $2$
Character orbit 309.k
Analytic conductor $2.467$
Analytic rank $0$
Dimension $512$
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] = [309,2,Mod(80,309)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(309, base_ring=CyclotomicField(34))
 
chi = DirichletCharacter(H, H._module([17, 25]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("309.80");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 309 = 3 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 309.k (of order \(34\), degree \(16\), 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: \(512\)
Relative dimension: \(32\) over \(\Q(\zeta_{34})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{34}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 512 q - 17 q^{3} - 2 q^{4} - 17 q^{6} - 18 q^{7} - 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 512 q - 17 q^{3} - 2 q^{4} - 17 q^{6} - 18 q^{7} - 13 q^{9} + 34 q^{10} - 17 q^{12} - 38 q^{13} - 23 q^{15} - 50 q^{16} + 5 q^{18} - 22 q^{19} - 51 q^{21} - 34 q^{22} - 17 q^{24} + 6 q^{25} - 17 q^{27} - 152 q^{28} - 21 q^{30} - 34 q^{31} - 11 q^{33} - 128 q^{34} + 149 q^{36} + 68 q^{37} + 119 q^{39} - 34 q^{40} - 17 q^{42} - 34 q^{43} - 17 q^{45} - 96 q^{46} - 187 q^{48} - 34 q^{49} - 17 q^{51} + 2 q^{52} - 17 q^{54} - 80 q^{55} - 14 q^{58} - 172 q^{60} - 38 q^{61} - 61 q^{63} + 56 q^{64} + 101 q^{66} - 136 q^{67} + 119 q^{69} - 34 q^{70} - 101 q^{72} - 136 q^{73} - 255 q^{75} - 114 q^{76} - 17 q^{78} - 54 q^{79} - 5 q^{81} - 66 q^{82} - 17 q^{84} - 17 q^{87} - 17 q^{90} + 244 q^{91} + 35 q^{93} - 34 q^{94} - 17 q^{96} + 236 q^{97} - 17 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
80.1 −2.39361 + 1.19187i 0.215968 + 1.71853i 3.10352 − 4.10973i −2.75747 − 2.51376i −2.56522 − 3.85609i −2.01098 + 0.375917i −1.54766 + 8.27927i −2.90672 + 0.742298i 9.59638 + 2.73041i
80.2 −2.19065 + 1.09081i 1.65241 + 0.519185i 2.40379 − 3.18313i 1.53354 + 1.39800i −4.18617 + 0.665115i 0.795146 − 0.148638i −0.894309 + 4.78413i 2.46089 + 1.71581i −4.88440 − 1.38973i
80.3 −2.18588 + 1.08844i −1.73180 − 0.0295124i 2.38811 − 3.16237i −0.0549963 − 0.0501358i 3.81763 − 1.82045i −0.234343 + 0.0438064i −0.880690 + 4.71128i 2.99826 + 0.102219i 0.174785 + 0.0497307i
80.4 −2.06825 + 1.02987i 1.28116 − 1.16561i 2.01178 − 2.66402i −0.817247 − 0.745019i −1.44934 + 3.73019i −2.68018 + 0.501012i −0.568174 + 3.03946i 0.282723 − 2.98665i 2.45755 + 0.699232i
80.5 −1.92197 + 0.957029i −0.921843 − 1.46636i 1.57281 − 2.08273i 3.17254 + 2.89215i 3.17510 + 1.93607i −0.294340 + 0.0550218i −0.240612 + 1.28716i −1.30041 + 2.70350i −8.86541 − 2.52243i
80.6 −1.55340 + 0.773503i −0.276968 + 1.70976i 0.609486 − 0.807090i 2.35495 + 2.14682i −0.892264 − 2.87019i −4.88838 + 0.913796i 0.315241 − 1.68639i −2.84658 − 0.947098i −5.31876 − 1.51332i
80.7 −1.52979 + 0.761744i −1.14719 + 1.29767i 0.554728 − 0.734578i −0.419808 − 0.382706i 0.766458 − 2.85903i 2.08728 − 0.390180i 0.338982 − 1.81339i −0.367921 − 2.97735i 0.933741 + 0.265672i
80.8 −1.45710 + 0.725550i 0.625602 − 1.61512i 0.391452 − 0.518366i 0.290873 + 0.265165i 0.260288 + 2.80730i 3.16390 − 0.591436i 0.403912 − 2.16074i −2.21724 − 2.02085i −0.616222 − 0.175330i
80.9 −1.39212 + 0.693195i 0.959648 + 1.44190i 0.252217 − 0.333989i −0.580105 − 0.528836i −2.33546 − 1.34208i 4.00443 − 0.748558i 0.451923 − 2.41757i −1.15815 + 2.76743i 1.17416 + 0.334078i
80.10 −1.34683 + 0.670642i −0.515798 − 1.65347i 0.158923 − 0.210448i −1.80858 − 1.64874i 1.80358 + 1.88102i −2.23084 + 0.417015i 0.480019 − 2.56787i −2.46790 + 1.70571i 3.54156 + 1.00766i
80.11 −0.983170 + 0.489560i 1.48569 + 0.890350i −0.478316 + 0.633393i −2.60061 − 2.37077i −1.89657 − 0.148030i −1.73122 + 0.323621i 0.563811 − 3.01612i 1.41455 + 2.64557i 3.71748 + 1.05771i
80.12 −0.667574 + 0.332412i −1.69211 − 0.369818i −0.870113 + 1.15222i 0.0884702 + 0.0806512i 1.25254 − 0.315597i −3.15355 + 0.589501i 0.471919 − 2.52454i 2.72647 + 1.25155i −0.0858698 − 0.0244321i
80.13 −0.651604 + 0.324460i 1.71465 − 0.244911i −0.885956 + 1.17319i 1.89995 + 1.73204i −1.03781 + 0.715921i −0.772864 + 0.144473i 0.464146 − 2.48296i 2.88004 − 0.839874i −1.79999 − 0.512142i
80.14 −0.339807 + 0.169204i 1.47167 − 0.913343i −1.11843 + 1.48104i −1.86326 − 1.69858i −0.345542 + 0.559372i 1.76047 − 0.329088i 0.268956 − 1.43879i 1.33161 − 2.68827i 0.920554 + 0.261920i
80.15 −0.161835 + 0.0805844i −0.229882 + 1.71673i −1.18557 + 1.56995i −1.29483 − 1.18039i −0.101138 − 0.296352i −0.846830 + 0.158300i 0.131793 − 0.705033i −2.89431 − 0.789289i 0.304670 + 0.0866860i
80.16 −0.0991577 + 0.0493746i 0.481425 + 1.66380i −1.19787 + 1.58624i 3.02671 + 2.75921i −0.129886 − 0.141208i 3.59005 − 0.671096i 0.0811663 − 0.434202i −2.53646 + 1.60199i −0.436356 − 0.124154i
80.17 0.0991577 − 0.0493746i −1.70396 + 0.310666i −1.19787 + 1.58624i −3.02671 − 2.75921i −0.153622 + 0.114937i 3.59005 − 0.671096i −0.0811663 + 0.434202i 2.80697 − 1.05873i −0.436356 − 0.124154i
80.18 0.161835 − 0.0805844i −1.43428 + 0.970993i −1.18557 + 1.56995i 1.29483 + 1.18039i −0.153871 + 0.272722i −0.846830 + 0.158300i −0.131793 + 0.705033i 1.11434 − 2.78536i 0.304670 + 0.0866860i
80.19 0.339807 − 0.169204i 0.161612 − 1.72449i −1.11843 + 1.48104i 1.86326 + 1.69858i −0.236874 − 0.613340i 1.76047 − 0.329088i −0.268956 + 1.43879i −2.94776 − 0.557399i 0.920554 + 0.261920i
80.20 0.651604 − 0.324460i −0.545049 − 1.64406i −0.885956 + 1.17319i −1.89995 − 1.73204i −0.888587 − 0.894428i −0.772864 + 0.144473i −0.464146 + 2.48296i −2.40584 + 1.79218i −1.79999 − 0.512142i
See next 80 embeddings (of 512 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 80.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
103.f odd 34 1 inner
309.k even 34 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 309.2.k.a ✓ 512
3.b odd 2 1 inner 309.2.k.a ✓ 512
103.f odd 34 1 inner 309.2.k.a ✓ 512
309.k even 34 1 inner 309.2.k.a ✓ 512
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
309.2.k.a ✓ 512 1.a even 1 1 trivial
309.2.k.a ✓ 512 3.b odd 2 1 inner
309.2.k.a ✓ 512 103.f odd 34 1 inner
309.2.k.a ✓ 512 309.k even 34 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(309, [\chi])\).