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Properties

Label 309.5.d.a
Level $309$
Weight $5$
Character orbit 309.d
Analytic conductor $31.941$
Analytic rank $0$
Dimension $70$
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] = [309,5,Mod(205,309)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(309, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("309.205");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 309 = 3 \cdot 103 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 309.d (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: \(31.9413185929\)
Analytic rank: \(0\)
Dimension: \(70\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 70 q + 576 q^{4} - 56 q^{7} - 180 q^{8} - 1890 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 70 q + 576 q^{4} - 56 q^{7} - 180 q^{8} - 1890 q^{9} + 708 q^{13} - 204 q^{14} + 108 q^{15} + 4584 q^{16} - 312 q^{17} + 492 q^{19} + 1008 q^{23} - 9262 q^{25} + 1410 q^{26} - 4046 q^{28} + 204 q^{29} + 2664 q^{30} + 330 q^{32} + 3384 q^{33} + 5136 q^{34} - 15552 q^{36} + 5130 q^{38} - 5724 q^{41} - 9006 q^{46} + 28518 q^{49} - 4332 q^{50} + 14116 q^{52} + 5672 q^{55} - 22986 q^{56} - 16772 q^{58} + 10008 q^{59} + 4608 q^{60} - 7572 q^{61} + 1512 q^{63} + 27464 q^{64} + 13896 q^{66} - 7014 q^{68} + 4860 q^{72} + 9204 q^{76} - 2920 q^{79} + 51030 q^{81} + 19642 q^{82} + 11244 q^{83} + 40192 q^{91} + 57660 q^{92} + 8424 q^{93} + 40776 q^{97} + 55530 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
205.1 −7.59283 − 5.19615i 41.6511 − 26.7003i 39.4535i −67.6225 −194.765 −27.0000 202.731i
205.2 −7.59283 5.19615i 41.6511 26.7003i − 39.4535i −67.6225 −194.765 −27.0000 − 202.731i
205.3 −7.58315 − 5.19615i 41.5041 25.6025i 39.4032i 28.9690 −193.402 −27.0000 − 194.147i
205.4 −7.58315 5.19615i 41.5041 − 25.6025i − 39.4032i 28.9690 −193.402 −27.0000 194.147i
205.5 −7.26619 − 5.19615i 36.7975 − 39.3102i 37.7562i 68.7223 −151.118 −27.0000 285.635i
205.6 −7.26619 5.19615i 36.7975 39.3102i − 37.7562i 68.7223 −151.118 −27.0000 − 285.635i
205.7 −6.96722 5.19615i 32.5421 − 16.2306i − 36.2027i −56.0739 −115.253 −27.0000 113.082i
205.8 −6.96722 − 5.19615i 32.5421 16.2306i 36.2027i −56.0739 −115.253 −27.0000 − 113.082i
205.9 −5.92548 5.19615i 19.1113 − 11.4883i − 30.7897i 51.1980 −18.4360 −27.0000 68.0740i
205.10 −5.92548 − 5.19615i 19.1113 11.4883i 30.7897i 51.1980 −18.4360 −27.0000 − 68.0740i
205.11 −5.84937 − 5.19615i 18.2151 37.7904i 30.3942i 2.46368 −12.9570 −27.0000 − 221.050i
205.12 −5.84937 5.19615i 18.2151 − 37.7904i − 30.3942i 2.46368 −12.9570 −27.0000 221.050i
205.13 −5.61051 − 5.19615i 15.4779 − 29.7955i 29.1531i −22.7015 2.92946 −27.0000 167.168i
205.14 −5.61051 5.19615i 15.4779 29.7955i − 29.1531i −22.7015 2.92946 −27.0000 − 167.168i
205.15 −5.55400 − 5.19615i 14.8469 − 23.9796i 28.8594i 55.9663 6.40432 −27.0000 133.183i
205.16 −5.55400 5.19615i 14.8469 23.9796i − 28.8594i 55.9663 6.40432 −27.0000 − 133.183i
205.17 −4.60923 − 5.19615i 5.24504 − 15.8200i 23.9503i −61.7901 49.5721 −27.0000 72.9179i
205.18 −4.60923 5.19615i 5.24504 15.8200i − 23.9503i −61.7901 49.5721 −27.0000 − 72.9179i
205.19 −4.53164 − 5.19615i 4.53577 17.0544i 23.5471i −70.0890 51.9518 −27.0000 − 77.2843i
205.20 −4.53164 5.19615i 4.53577 − 17.0544i − 23.5471i −70.0890 51.9518 −27.0000 77.2843i
See all 70 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 205.70
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 309.5.d.a ✓ 70
103.b odd 2 1 inner 309.5.d.a ✓ 70
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
309.5.d.a ✓ 70 1.a even 1 1 trivial
309.5.d.a ✓ 70 103.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(309, [\chi])\).