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Properties

Label 798.4.e.a
Level $798$
Weight $4$
Character orbit 798.e
Analytic conductor $47.084$
Analytic rank $0$
Dimension $40$
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] = [798,4,Mod(265,798)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(798, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("798.265");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 798.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: \(47.0835241846\)
Analytic rank: \(0\)
Dimension: \(40\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 120 q^{3} - 160 q^{4} - 10 q^{7} + 360 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 120 q^{3} - 160 q^{4} - 10 q^{7} + 360 q^{9} - 8 q^{10} + 480 q^{12} - 84 q^{13} - 8 q^{14} + 640 q^{16} + 128 q^{19} + 30 q^{21} + 204 q^{23} - 1132 q^{25} - 1080 q^{27} + 40 q^{28} + 24 q^{30} - 104 q^{31} - 296 q^{34} - 394 q^{35} - 1440 q^{36} + 384 q^{38} + 252 q^{39} + 32 q^{40} - 32 q^{41} + 24 q^{42} + 612 q^{43} - 1920 q^{48} + 50 q^{49} + 336 q^{52} + 32 q^{56} - 384 q^{57} + 336 q^{58} + 1304 q^{59} - 90 q^{63} - 2560 q^{64} - 612 q^{69} + 192 q^{70} + 1920 q^{74} + 3396 q^{75} - 512 q^{76} - 566 q^{77} + 3240 q^{81} - 120 q^{84} + 100 q^{85} - 992 q^{89} - 72 q^{90} + 880 q^{91} - 816 q^{92} + 312 q^{93} + 3440 q^{95} + 92 q^{97} + 928 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
265.1 − 2.00000i −3.00000 −4.00000 − 20.6257i 6.00000i −17.3451 − 6.49217i 8.00000i 9.00000 −41.2513
265.2 − 2.00000i −3.00000 −4.00000 − 17.7805i 6.00000i 18.3340 − 2.61970i 8.00000i 9.00000 −35.5610
265.3 − 2.00000i −3.00000 −4.00000 − 14.0192i 6.00000i −18.5089 − 0.647553i 8.00000i 9.00000 −28.0385
265.4 − 2.00000i −3.00000 −4.00000 − 13.9658i 6.00000i −2.20869 − 18.3881i 8.00000i 9.00000 −27.9317
265.5 − 2.00000i −3.00000 −4.00000 − 13.3323i 6.00000i 4.59040 + 17.9424i 8.00000i 9.00000 −26.6647
265.6 − 2.00000i −3.00000 −4.00000 − 11.4338i 6.00000i 12.7533 − 13.4296i 8.00000i 9.00000 −22.8677
265.7 − 2.00000i −3.00000 −4.00000 − 9.83057i 6.00000i −9.74124 + 15.7515i 8.00000i 9.00000 −19.6611
265.8 − 2.00000i −3.00000 −4.00000 − 8.60585i 6.00000i 8.72974 + 16.3338i 8.00000i 9.00000 −17.2117
265.9 − 2.00000i −3.00000 −4.00000 − 3.30270i 6.00000i −0.332747 − 18.5173i 8.00000i 9.00000 −6.60540
265.10 − 2.00000i −3.00000 −4.00000 3.24183i 6.00000i −15.2716 − 10.4775i 8.00000i 9.00000 6.48366
265.11 − 2.00000i −3.00000 −4.00000 3.45330i 6.00000i 14.3222 + 11.7420i 8.00000i 9.00000 6.90659
265.12 − 2.00000i −3.00000 −4.00000 3.49808i 6.00000i −15.4528 + 10.2084i 8.00000i 9.00000 6.99615
265.13 − 2.00000i −3.00000 −4.00000 7.72726i 6.00000i 10.3686 − 15.3458i 8.00000i 9.00000 15.4545
265.14 − 2.00000i −3.00000 −4.00000 8.71306i 6.00000i 16.3570 − 8.68615i 8.00000i 9.00000 17.4261
265.15 − 2.00000i −3.00000 −4.00000 8.72794i 6.00000i −3.02218 + 18.2720i 8.00000i 9.00000 17.4559
265.16 − 2.00000i −3.00000 −4.00000 9.44152i 6.00000i 18.5105 − 0.601895i 8.00000i 9.00000 18.8830
265.17 − 2.00000i −3.00000 −4.00000 14.5846i 6.00000i −9.84423 − 15.6873i 8.00000i 9.00000 29.1693
265.18 − 2.00000i −3.00000 −4.00000 14.5913i 6.00000i −13.0767 + 13.1149i 8.00000i 9.00000 29.1826
265.19 − 2.00000i −3.00000 −4.00000 18.2708i 6.00000i −16.6167 − 8.17834i 8.00000i 9.00000 36.5417
265.20 − 2.00000i −3.00000 −4.00000 18.6467i 6.00000i 12.4552 + 13.7065i 8.00000i 9.00000 37.2934
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 265.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
133.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 798.4.e.a ✓ 40
7.b odd 2 1 798.4.e.b yes 40
19.b odd 2 1 798.4.e.b yes 40
133.c even 2 1 inner 798.4.e.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.4.e.a ✓ 40 1.a even 1 1 trivial
798.4.e.a ✓ 40 133.c even 2 1 inner
798.4.e.b yes 40 7.b odd 2 1
798.4.e.b yes 40 19.b odd 2 1