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Properties

Label 1028.6.a.b
Level $1028$
Weight $6$
Character orbit 1028.a
Self dual yes
Analytic conductor $164.875$
Analytic rank $0$
Dimension $57$
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: \(0\)
Dimension: \(57\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 57 q + 27 q^{3} + 61 q^{5} + 88 q^{7} + 5374 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 57 q + 27 q^{3} + 61 q^{5} + 88 q^{7} + 5374 q^{9} + 484 q^{11} + 3162 q^{13} - 244 q^{15} + 3298 q^{17} + 2931 q^{19} + 5009 q^{21} + 3712 q^{23} + 48568 q^{25} + 8682 q^{27} + 10909 q^{29} - 3539 q^{31} + 13277 q^{33} + 13976 q^{35} + 42395 q^{37} - 14986 q^{39} - 8469 q^{41} + 69808 q^{43} + 26347 q^{45} + 12410 q^{47} + 203347 q^{49} + 24220 q^{51} + 49353 q^{53} - 34265 q^{55} + 98112 q^{57} - 9792 q^{59} + 157032 q^{61} + 24981 q^{63} + 16143 q^{65} + 20201 q^{67} - 113743 q^{69} - 229673 q^{71} - 19561 q^{73} - 482061 q^{75} + 21204 q^{77} - 10925 q^{79} + 323569 q^{81} - 62846 q^{83} + 218157 q^{85} - 169628 q^{87} + 69901 q^{89} + 56581 q^{91} + 177641 q^{93} - 192910 q^{95} + 277990 q^{97} + 424882 q^{99}+O(q^{100}) \) 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 −29.5134 0 −107.184 0 −142.282 0 628.041 0
1.2 0 −29.1254 0 83.8021 0 130.827 0 605.290 0
1.3 0 −26.8472 0 −27.7983 0 −148.747 0 477.771 0
1.4 0 −26.5164 0 −49.2199 0 −86.7278 0 460.120 0
1.5 0 −25.3624 0 63.2925 0 −82.1666 0 400.249 0
1.6 0 −24.6173 0 −7.87155 0 165.885 0 363.009 0
1.7 0 −23.7025 0 79.6527 0 −163.638 0 318.808 0
1.8 0 −22.4232 0 91.9325 0 −225.282 0 259.802 0
1.9 0 −21.1395 0 −58.6028 0 77.5956 0 203.880 0
1.10 0 −21.1022 0 70.8764 0 79.9641 0 202.303 0
1.11 0 −20.8847 0 110.347 0 192.813 0 193.172 0
1.12 0 −20.0123 0 −46.6500 0 25.6441 0 157.493 0
1.13 0 −19.0570 0 −9.45543 0 −38.2403 0 120.170 0
1.14 0 −18.4132 0 −106.970 0 −50.3746 0 96.0457 0
1.15 0 −17.2042 0 −92.6944 0 209.257 0 52.9843 0
1.16 0 −16.4054 0 −78.9904 0 44.7523 0 26.1387 0
1.17 0 −15.1335 0 45.8699 0 223.098 0 −13.9785 0
1.18 0 −13.0981 0 23.8997 0 161.931 0 −71.4392 0
1.19 0 −11.7981 0 −24.5614 0 −200.710 0 −103.804 0
1.20 0 −9.91744 0 8.20543 0 107.816 0 −144.644 0
See all 57 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.57
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.b ✓ 57
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1028.6.a.b ✓ 57 1.a even 1 1 trivial