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Properties

Label 308.3.e.a
Level $308$
Weight $3$
Character orbit 308.e
Analytic conductor $8.392$
Analytic rank $0$
Dimension $60$
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] = [308,3,Mod(155,308)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(308, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("308.155");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 308 = 2^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 308.e (of order \(2\), degree \(1\), 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: \(8.39239214230\)
Analytic rank: \(0\)
Dimension: \(60\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 8 q^{4} + 8 q^{5} - 12 q^{6} + 36 q^{8} - 180 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q - 8 q^{4} + 8 q^{5} - 12 q^{6} + 36 q^{8} - 180 q^{9} + 40 q^{10} - 60 q^{12} - 24 q^{13} + 56 q^{16} - 40 q^{17} - 56 q^{18} + 84 q^{20} - 212 q^{24} + 420 q^{25} + 108 q^{26} + 40 q^{29} - 176 q^{30} + 60 q^{32} - 56 q^{34} + 128 q^{36} + 40 q^{37} + 252 q^{38} - 160 q^{40} - 200 q^{41} - 140 q^{42} - 72 q^{45} - 120 q^{46} + 100 q^{48} - 420 q^{49} + 24 q^{50} - 36 q^{52} - 200 q^{53} + 440 q^{54} - 32 q^{57} + 192 q^{58} + 376 q^{60} + 328 q^{61} - 156 q^{62} + 616 q^{64} + 272 q^{65} - 588 q^{68} - 144 q^{69} - 84 q^{70} - 612 q^{72} + 312 q^{73} - 640 q^{74} - 456 q^{76} - 120 q^{78} - 180 q^{80} + 380 q^{81} - 240 q^{82} + 196 q^{84} + 432 q^{85} - 424 q^{86} + 132 q^{88} - 536 q^{89} + 672 q^{90} + 152 q^{92} - 144 q^{93} + 652 q^{94} + 372 q^{96} - 312 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
155.1 −1.99565 − 0.131830i − 5.15426i 3.96524 + 0.526172i 4.66231 −0.679483 + 10.2861i 2.64575i −7.84387 − 1.57279i −17.5664 −9.30435 − 0.614631i
155.2 −1.99565 + 0.131830i 5.15426i 3.96524 − 0.526172i 4.66231 −0.679483 − 10.2861i − 2.64575i −7.84387 + 1.57279i −17.5664 −9.30435 + 0.614631i
155.3 −1.97358 − 0.323995i − 0.0102461i 3.79005 + 1.27886i −4.71811 −0.00331970 + 0.0202216i − 2.64575i −7.06564 − 3.75190i 8.99990 9.31157 + 1.52865i
155.4 −1.97358 + 0.323995i 0.0102461i 3.79005 − 1.27886i −4.71811 −0.00331970 − 0.0202216i 2.64575i −7.06564 + 3.75190i 8.99990 9.31157 − 1.52865i
155.5 −1.94458 − 0.467575i − 2.00506i 3.56275 + 1.81847i −0.696725 −0.937514 + 3.89898i 2.64575i −6.07776 − 5.20200i 4.97975 1.35483 + 0.325771i
155.6 −1.94458 + 0.467575i 2.00506i 3.56275 − 1.81847i −0.696725 −0.937514 − 3.89898i − 2.64575i −6.07776 + 5.20200i 4.97975 1.35483 − 0.325771i
155.7 −1.88804 − 0.659769i − 0.524746i 3.12941 + 2.49134i 7.62274 −0.346211 + 0.990744i − 2.64575i −4.26475 − 6.76845i 8.72464 −14.3921 − 5.02925i
155.8 −1.88804 + 0.659769i 0.524746i 3.12941 − 2.49134i 7.62274 −0.346211 − 0.990744i 2.64575i −4.26475 + 6.76845i 8.72464 −14.3921 + 5.02925i
155.9 −1.70363 − 1.04768i 4.38006i 1.80472 + 3.56973i −8.05590 4.58892 − 7.46201i − 2.64575i 0.665372 − 7.97228i −10.1849 13.7243 + 8.44003i
155.10 −1.70363 + 1.04768i − 4.38006i 1.80472 − 3.56973i −8.05590 4.58892 + 7.46201i 2.64575i 0.665372 + 7.97228i −10.1849 13.7243 − 8.44003i
155.11 −1.49882 − 1.32421i − 4.58991i 0.492936 + 3.96951i 0.165282 −6.07800 + 6.87946i − 2.64575i 4.51764 − 6.60234i −12.0672 −0.247728 − 0.218868i
155.12 −1.49882 + 1.32421i 4.58991i 0.492936 − 3.96951i 0.165282 −6.07800 − 6.87946i 2.64575i 4.51764 + 6.60234i −12.0672 −0.247728 + 0.218868i
155.13 −1.43238 − 1.39582i 1.34646i 0.103397 + 3.99866i −1.87029 1.87941 − 1.92864i 2.64575i 5.43329 − 5.87191i 7.18704 2.67895 + 2.61058i
155.14 −1.43238 + 1.39582i − 1.34646i 0.103397 − 3.99866i −1.87029 1.87941 + 1.92864i − 2.64575i 5.43329 + 5.87191i 7.18704 2.67895 − 2.61058i
155.15 −1.24203 − 1.56760i − 4.06133i −0.914744 + 3.89400i 8.25854 −6.36655 + 5.04428i 2.64575i 7.24037 − 3.40250i −7.49442 −10.2573 − 12.9461i
155.16 −1.24203 + 1.56760i 4.06133i −0.914744 − 3.89400i 8.25854 −6.36655 − 5.04428i − 2.64575i 7.24037 + 3.40250i −7.49442 −10.2573 + 12.9461i
155.17 −1.18753 − 1.60928i − 0.0872955i −1.17954 + 3.82213i −7.56267 −0.140483 + 0.103666i 2.64575i 7.55160 − 2.64070i 8.99238 8.98090 + 12.1704i
155.18 −1.18753 + 1.60928i 0.0872955i −1.17954 − 3.82213i −7.56267 −0.140483 − 0.103666i − 2.64575i 7.55160 + 2.64070i 8.99238 8.98090 − 12.1704i
155.19 −1.06123 − 1.69523i 3.28921i −1.74760 + 3.59804i 0.315415 5.57597 − 3.49060i − 2.64575i 7.95410 − 0.855766i −1.81893 −0.334726 − 0.534700i
155.20 −1.06123 + 1.69523i − 3.28921i −1.74760 − 3.59804i 0.315415 5.57597 + 3.49060i 2.64575i 7.95410 + 0.855766i −1.81893 −0.334726 + 0.534700i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 155.60
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 308.3.e.a ✓ 60
4.b odd 2 1 inner 308.3.e.a ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
308.3.e.a ✓ 60 1.a even 1 1 trivial
308.3.e.a ✓ 60 4.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(308, [\chi])\).