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Properties

Label 1028.6.a.a
Level $1028$
Weight $6$
Character orbit 1028.a
Self dual yes
Analytic conductor $164.875$
Analytic rank $1$
Dimension $49$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1028,6,Mod(1,1028)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1028, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1028.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1028 = 2^{2} \cdot 257 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1028.a (trivial)

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: yes
Analytic conductor: \(164.874566768\)
Analytic rank: \(1\)
Dimension: \(49\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 49 q - 27 q^{3} - 89 q^{5} + 88 q^{7} + 3430 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 49 q - 27 q^{3} - 89 q^{5} + 88 q^{7} + 3430 q^{9} - 484 q^{11} - 2922 q^{13} - 244 q^{15} - 3638 q^{17} - 3567 q^{19} - 3811 q^{21} + 3712 q^{23} + 13568 q^{25} - 8814 q^{27} - 5911 q^{29} - 3539 q^{31} - 17215 q^{33} - 10524 q^{35} - 31531 q^{37} - 14986 q^{39} - 35365 q^{41} - 44830 q^{43} - 34403 q^{45} + 12410 q^{47} + 68891 q^{49} - 48608 q^{51} - 79861 q^{53} - 34265 q^{55} - 109824 q^{57} - 51564 q^{59} - 66228 q^{61} + 24981 q^{63} - 31903 q^{65} - 80135 q^{67} + 50903 q^{69} + 162703 q^{71} - 58717 q^{73} + 207827 q^{75} + 89720 q^{77} + 125647 q^{79} + 420081 q^{81} - 28722 q^{83} - 125617 q^{85} + 65848 q^{87} + 93837 q^{89} - 66437 q^{91} - 278711 q^{93} - 83170 q^{95} - 347060 q^{97} + 7934 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 0 −30.6737 0 42.6130 0 15.7852 0 697.875 0
1.2 0 −29.3150 0 39.9945 0 −134.016 0 616.369 0
1.3 0 −28.2719 0 −73.9896 0 49.6674 0 556.301 0
1.4 0 −27.0766 0 −61.5587 0 221.022 0 490.140 0
1.5 0 −26.1755 0 12.4225 0 151.898 0 442.157 0
1.6 0 −24.5283 0 5.59326 0 131.433 0 358.638 0
1.7 0 −22.2940 0 −24.1219 0 −222.167 0 254.020 0
1.8 0 −20.5636 0 62.7910 0 15.3467 0 179.863 0
1.9 0 −19.7457 0 −10.8196 0 −70.7802 0 146.891 0
1.10 0 −19.7180 0 9.18763 0 −4.07607 0 145.799 0
1.11 0 −17.1264 0 −52.0430 0 −180.678 0 50.3147 0
1.12 0 −16.7147 0 60.4053 0 191.832 0 36.3820 0
1.13 0 −15.1206 0 38.6175 0 −225.318 0 −14.3669 0
1.14 0 −13.6293 0 −91.2960 0 −63.7695 0 −57.2414 0
1.15 0 −13.2828 0 109.418 0 −15.8096 0 −66.5679 0
1.16 0 −12.7314 0 −83.6517 0 121.595 0 −80.9122 0
1.17 0 −9.78147 0 −19.4476 0 160.887 0 −147.323 0
1.18 0 −9.49742 0 −80.8595 0 −210.734 0 −152.799 0
1.19 0 −9.24112 0 59.2564 0 −65.2931 0 −157.602 0
1.20 0 −5.91999 0 −46.7531 0 29.6970 0 −207.954 0
See all 49 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.49
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(257\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1028.6.a.a ✓ 49
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1028.6.a.a ✓ 49 1.a even 1 1 trivial