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Properties

Label 310.2.n.a
Level $310$
Weight $2$
Character orbit 310.n
Analytic conductor $2.475$
Analytic rank $0$
Dimension $64$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [310,2,Mod(39,310)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(310, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("310.39");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 310 = 2 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 310.n (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: \(2.47536246266\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(16\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 16 q^{4} + 4 q^{5} - 16 q^{6} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q + 16 q^{4} + 4 q^{5} - 16 q^{6} + 8 q^{9} - 4 q^{10} + 4 q^{11} - 14 q^{15} - 16 q^{16} + 16 q^{19} - 4 q^{20} + 32 q^{21} - 4 q^{24} - 8 q^{25} - 48 q^{26} - 40 q^{29} - 8 q^{30} - 60 q^{31} - 20 q^{34} - 8 q^{35} + 72 q^{36} - 8 q^{39} - 6 q^{40} + 28 q^{41} - 4 q^{44} - 22 q^{45} + 36 q^{46} + 32 q^{49} + 20 q^{50} - 4 q^{51} - 40 q^{54} + 32 q^{59} + 14 q^{60} - 64 q^{61} + 16 q^{64} + 48 q^{65} - 4 q^{66} - 20 q^{70} - 20 q^{71} - 4 q^{74} + 62 q^{75} + 24 q^{76} - 116 q^{79} - 6 q^{80} - 116 q^{81} + 8 q^{84} + 8 q^{85} - 24 q^{86} + 60 q^{89} - 82 q^{90} + 28 q^{91} + 16 q^{94} - 90 q^{95} + 4 q^{96} + 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
39.1 −0.951057 − 0.309017i −2.19957 + 0.714683i 0.809017 + 0.587785i −1.55188 + 1.60986i 2.31276 −1.07079 + 1.47381i −0.587785 − 0.809017i 1.90028 − 1.38063i 1.97340 − 1.05151i
39.2 −0.951057 − 0.309017i −1.64442 + 0.534303i 0.809017 + 0.587785i 1.53763 − 1.62349i 1.72904 −0.410273 + 0.564692i −0.587785 − 0.809017i −0.00842805 + 0.00612334i −1.96405 + 1.06887i
39.3 −0.951057 − 0.309017i −1.50125 + 0.487785i 0.809017 + 0.587785i 1.27806 + 1.83482i 1.57851 1.54238 − 2.12290i −0.587785 − 0.809017i −0.411240 + 0.298783i −0.648521 − 2.13996i
39.4 −0.951057 − 0.309017i −0.143676 + 0.0466831i 0.809017 + 0.587785i −0.736590 − 2.11126i 0.151070 −2.30332 + 3.17025i −0.587785 − 0.809017i −2.40859 + 1.74994i 0.0481224 + 2.23555i
39.5 −0.951057 − 0.309017i 0.170738 − 0.0554763i 0.809017 + 0.587785i −1.94448 − 1.10408i −0.179525 2.60575 − 3.58651i −0.587785 − 0.809017i −2.40098 + 1.74441i 1.50813 + 1.65092i
39.6 −0.951057 − 0.309017i 1.77693 − 0.577361i 0.809017 + 0.587785i 2.10827 − 0.745128i −1.86838 1.47699 − 2.03290i −0.587785 − 0.809017i 0.397101 − 0.288511i −2.23534 + 0.0571688i
39.7 −0.951057 − 0.309017i 2.65851 − 0.863802i 0.809017 + 0.587785i 0.130564 + 2.23225i −2.79532 −1.24493 + 1.71351i −0.587785 − 0.809017i 3.89447 − 2.82950i 0.565630 − 2.16335i
39.8 −0.951057 − 0.309017i 2.78484 − 0.904849i 0.809017 + 0.587785i −1.43960 − 1.71101i −2.92815 −0.595802 + 0.820051i −0.587785 − 0.809017i 4.50952 − 3.27636i 0.840407 + 2.07213i
39.9 0.951057 + 0.309017i −2.78484 + 0.904849i 0.809017 + 0.587785i −1.43960 + 1.71101i −2.92815 0.595802 − 0.820051i 0.587785 + 0.809017i 4.50952 − 3.27636i −1.89787 + 1.18241i
39.10 0.951057 + 0.309017i −2.65851 + 0.863802i 0.809017 + 0.587785i 0.130564 − 2.23225i −2.79532 1.24493 − 1.71351i 0.587785 + 0.809017i 3.89447 − 2.82950i 0.813978 − 2.08265i
39.11 0.951057 + 0.309017i −1.77693 + 0.577361i 0.809017 + 0.587785i 2.10827 + 0.745128i −1.86838 −1.47699 + 2.03290i 0.587785 + 0.809017i 0.397101 − 0.288511i 1.77482 + 1.36015i
39.12 0.951057 + 0.309017i −0.170738 + 0.0554763i 0.809017 + 0.587785i −1.94448 + 1.10408i −0.179525 −2.60575 + 3.58651i 0.587785 + 0.809017i −2.40098 + 1.74441i −2.19049 + 0.449163i
39.13 0.951057 + 0.309017i 0.143676 − 0.0466831i 0.809017 + 0.587785i −0.736590 + 2.11126i 0.151070 2.30332 − 3.17025i 0.587785 + 0.809017i −2.40859 + 1.74994i −1.35296 + 1.78031i
39.14 0.951057 + 0.309017i 1.50125 − 0.487785i 0.809017 + 0.587785i 1.27806 − 1.83482i 1.57851 −1.54238 + 2.12290i 0.587785 + 0.809017i −0.411240 + 0.298783i 1.78250 − 1.35007i
39.15 0.951057 + 0.309017i 1.64442 − 0.534303i 0.809017 + 0.587785i 1.53763 + 1.62349i 1.72904 0.410273 − 0.564692i 0.587785 + 0.809017i −0.00842805 + 0.00612334i 0.960685 + 2.01918i
39.16 0.951057 + 0.309017i 2.19957 − 0.714683i 0.809017 + 0.587785i −1.55188 − 1.60986i 2.31276 1.07079 − 1.47381i 0.587785 + 0.809017i 1.90028 − 1.38063i −0.978454 − 2.01063i
109.1 −0.587785 − 0.809017i −1.59835 + 2.19994i −0.309017 + 0.951057i 2.01257 + 0.974446i 2.71928 0.956338 + 0.310733i 0.951057 − 0.309017i −1.35796 − 4.17938i −0.394618 − 2.20097i
109.2 −0.587785 − 0.809017i −1.29655 + 1.78455i −0.309017 + 0.951057i −1.59654 + 1.56558i 2.20583 −3.31306 − 1.07648i 0.951057 − 0.309017i −0.576523 − 1.77435i 2.20501 + 0.371406i
109.3 −0.587785 − 0.809017i −0.641784 + 0.883340i −0.309017 + 0.951057i −1.04522 − 1.97674i 1.09187 −1.70139 − 0.552816i 0.951057 − 0.309017i 0.558648 + 1.71934i −0.984857 + 2.00750i
109.4 −0.587785 − 0.809017i −0.00820391 + 0.0112917i −0.309017 + 0.951057i 1.91563 − 1.15341i 0.0139573 1.51995 + 0.493860i 0.951057 − 0.309017i 0.926991 + 2.85298i −2.05911 − 0.871822i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 39.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
31.d even 5 1 inner
155.n even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 310.2.n.a ✓ 64
5.b even 2 1 inner 310.2.n.a ✓ 64
31.d even 5 1 inner 310.2.n.a ✓ 64
155.n even 10 1 inner 310.2.n.a ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
310.2.n.a ✓ 64 1.a even 1 1 trivial
310.2.n.a ✓ 64 5.b even 2 1 inner
310.2.n.a ✓ 64 31.d even 5 1 inner
310.2.n.a ✓ 64 155.n even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(310, [\chi])\).