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Properties

Label 308.2.v.a
Level $308$
Weight $2$
Character orbit 308.v
Analytic conductor $2.459$
Analytic rank $0$
Dimension $72$
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] = [308,2,Mod(127,308)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(308, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 0, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("308.127");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 308 = 2^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 308.v (of order \(10\), degree \(4\), 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: \(2.45939238226\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(18\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q - q^{2} - q^{4} + 18 q^{7} - 7 q^{8} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q - q^{2} - q^{4} + 18 q^{7} - 7 q^{8} + 22 q^{9} - 24 q^{10} + 4 q^{11} + 8 q^{12} + q^{14} - 17 q^{16} - 20 q^{18} - 12 q^{19} - 15 q^{20} - 3 q^{22} + 28 q^{24} - 18 q^{25} + q^{26} + 60 q^{27} - 14 q^{28} + 12 q^{30} + 44 q^{32} + 2 q^{33} - 10 q^{34} - 61 q^{36} - 12 q^{37} + 29 q^{38} + 62 q^{40} - 40 q^{43} - 16 q^{44} - 48 q^{45} - 33 q^{46} - 16 q^{48} - 18 q^{49} + 82 q^{50} - 8 q^{51} - 25 q^{52} - 24 q^{53} - 64 q^{54} - 20 q^{55} + 7 q^{56} - 10 q^{57} + 50 q^{58} - 36 q^{60} - 129 q^{62} - 12 q^{63} - q^{64} - 10 q^{66} + 6 q^{68} + 12 q^{69} - 6 q^{70} - 80 q^{71} - 83 q^{72} + 55 q^{74} + 96 q^{76} - 4 q^{77} - 22 q^{78} + 24 q^{79} - 73 q^{80} - 20 q^{81} + 28 q^{82} - 24 q^{83} + 12 q^{84} + 20 q^{85} + 13 q^{86} + 72 q^{87} + 19 q^{88} + 12 q^{89} - 60 q^{90} - 23 q^{92} + 24 q^{93} - 8 q^{94} + 8 q^{95} - 30 q^{96} + 22 q^{97} - q^{98} + 96 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1 −1.37548 − 0.328732i 0.511458 − 0.703962i 1.78387 + 0.904325i −0.909414 + 2.79889i −0.934914 + 0.800151i 0.809017 − 0.587785i −2.15639 − 1.83029i 0.693078 + 2.13307i 2.17096 − 3.55085i
127.2 −1.28551 + 0.589471i −0.388982 + 0.535388i 1.30505 − 1.51554i 0.00558488 − 0.0171885i 0.184443 − 0.917537i 0.809017 − 0.587785i −0.784284 + 2.71752i 0.791718 + 2.43666i 0.00295272 + 0.0253880i
127.3 −1.25690 − 0.648239i 0.398065 − 0.547889i 1.15957 + 1.62954i 0.623380 − 1.91857i −0.855489 + 0.430598i 0.809017 − 0.587785i −0.401131 − 2.79984i 0.785324 + 2.41698i −2.02721 + 2.00734i
127.4 −1.22147 + 0.712754i 1.28462 − 1.76812i 0.983962 − 1.74121i 1.19117 − 3.66606i −0.308879 + 3.07532i 0.809017 − 0.587785i 0.0391793 + 2.82816i −0.548967 − 1.68955i 1.15802 + 5.32698i
127.5 −0.888268 − 1.10045i −1.47977 + 2.03674i −0.421961 + 1.95498i 0.692975 − 2.13276i 3.55575 − 0.180755i 0.809017 − 0.587785i 2.52616 − 1.27220i −1.03151 − 3.17465i −2.96253 + 1.13188i
127.6 −0.732938 + 1.20946i −1.80272 + 2.48123i −0.925604 − 1.77292i 0.791257 − 2.43524i −1.67968 − 3.99891i 0.809017 − 0.587785i 2.82270 + 0.179958i −1.97966 − 6.09276i 2.36539 + 2.74188i
127.7 −0.373768 + 1.36393i 1.03113 − 1.41923i −1.72059 − 1.01959i −0.408163 + 1.25620i 1.55032 + 1.93685i 0.809017 − 0.587785i 2.03374 − 1.96568i −0.0239268 − 0.0736391i −1.56080 − 1.02623i
127.8 −0.285828 − 1.38503i 1.65473 − 2.27754i −1.83661 + 0.791758i 0.0343997 − 0.105871i −3.62742 − 1.64086i 0.809017 − 0.587785i 1.62156 + 2.31744i −1.52201 − 4.68426i −0.156467 − 0.0173836i
127.9 −0.164081 + 1.40466i −1.25298 + 1.72457i −1.94615 − 0.460957i −1.27886 + 3.93593i −2.21686 − 2.04298i 0.809017 − 0.587785i 0.966815 − 2.65806i −0.477155 − 1.46853i −5.31882 − 2.44218i
127.10 0.179271 − 1.40281i −0.385759 + 0.530952i −1.93572 − 0.502964i −0.647741 + 1.99354i 0.675667 + 0.636329i 0.809017 − 0.587785i −1.05258 + 2.62528i 0.793951 + 2.44353i 2.68043 + 1.26604i
127.11 0.546505 − 1.30435i 0.153071 − 0.210684i −1.40267 − 1.42567i 1.31046 − 4.03317i −0.191152 − 0.314798i 0.809017 − 0.587785i −2.62614 + 1.05043i 0.906094 + 2.78867i −4.54450 − 3.91344i
127.12 0.654935 − 1.25342i −1.96050 + 2.69840i −1.14212 − 1.64182i −0.457143 + 1.40694i 2.09823 + 4.22461i 0.809017 − 0.587785i −2.80590 + 0.356270i −2.51075 − 7.72730i 1.46409 + 1.49445i
127.13 0.738473 + 1.20609i 1.43106 − 1.96968i −0.909314 + 1.78133i 0.598117 − 1.84082i 3.43242 + 0.271429i 0.809017 − 0.587785i −2.81996 + 0.218751i −0.904672 − 2.78429i 2.66189 − 0.638009i
127.14 0.755572 + 1.19545i 0.134272 − 0.184810i −0.858222 + 1.80650i −0.451054 + 1.38820i 0.322384 + 0.0208792i 0.809017 − 0.587785i −2.80804 + 0.338978i 0.910925 + 2.80354i −2.00033 + 0.509671i
127.15 0.969994 − 1.02913i 1.59092 − 2.18971i −0.118222 − 1.99650i −0.529385 + 1.62928i −0.710318 − 3.76127i 0.809017 − 0.587785i −2.16934 − 1.81493i −1.33676 − 4.11414i 1.16324 + 2.12520i
127.16 1.30070 − 0.555134i −0.736653 + 1.01392i 1.38365 − 1.44413i 0.449342 − 1.38293i −0.395307 + 1.72774i 0.809017 − 0.587785i 0.998036 − 2.64649i 0.441683 + 1.35936i −0.183253 − 2.04823i
127.17 1.33634 + 0.462805i −0.972142 + 1.33804i 1.57162 + 1.23693i 0.150078 − 0.461893i −1.91836 + 1.33817i 0.809017 − 0.587785i 1.52777 + 2.38032i 0.0817642 + 0.251644i 0.414323 − 0.547791i
127.18 1.41145 − 0.0884008i 0.790194 − 1.08761i 1.98437 − 0.249546i −1.16500 + 3.58551i 1.01917 − 1.60496i 0.809017 − 0.587785i 2.77878 − 0.527641i 0.368564 + 1.13432i −1.32738 + 5.16375i
183.1 −1.39393 − 0.238677i −0.0348162 − 0.0113125i 1.88607 + 0.665397i −2.47045 + 1.79489i 0.0458312 + 0.0240786i −0.309017 − 0.951057i −2.47022 − 1.37768i −2.42597 − 1.76257i 3.87202 − 1.91230i
183.2 −1.38833 + 0.269313i −2.44243 − 0.793593i 1.85494 − 0.747794i 2.83055 − 2.05651i 3.60463 + 0.443993i −0.309017 − 0.951057i −2.37389 + 1.53775i 2.90862 + 2.11324i −3.37590 + 3.61743i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.18
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
44.g even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 308.2.v.a ✓ 72
4.b odd 2 1 308.2.v.b yes 72
11.d odd 10 1 308.2.v.b yes 72
44.g even 10 1 inner 308.2.v.a ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
308.2.v.a ✓ 72 1.a even 1 1 trivial
308.2.v.a ✓ 72 44.g even 10 1 inner
308.2.v.b yes 72 4.b odd 2 1
308.2.v.b yes 72 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{72} - 38 T_{3}^{70} - 20 T_{3}^{69} + 835 T_{3}^{68} + 760 T_{3}^{67} - 13865 T_{3}^{66} + \cdots + 93392896 \) acting on \(S_{2}^{\mathrm{new}}(308, [\chi])\). Copy content Toggle raw display