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Properties

Label 5077.2.a.b
Level $5077$
Weight $2$
Character orbit 5077.a
Self dual yes
Analytic conductor $40.540$
Analytic rank $1$
Dimension $205$
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] = [5077,2,Mod(1,5077)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5077, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5077.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5077 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5077.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: \(40.5400491062\)
Analytic rank: \(1\)
Dimension: \(205\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 205 q - 25 q^{2} - 59 q^{3} + 195 q^{4} - 44 q^{5} - 26 q^{6} - 30 q^{7} - 75 q^{8} + 186 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 205 q - 25 q^{2} - 59 q^{3} + 195 q^{4} - 44 q^{5} - 26 q^{6} - 30 q^{7} - 75 q^{8} + 186 q^{9} - 28 q^{10} - 83 q^{11} - 108 q^{12} - 36 q^{13} - 67 q^{14} - 63 q^{15} + 187 q^{16} - 72 q^{17} - 57 q^{18} - 47 q^{19} - 132 q^{20} - 35 q^{21} - 40 q^{22} - 97 q^{23} - 49 q^{24} + 175 q^{25} - 78 q^{26} - 227 q^{27} - 59 q^{28} - 46 q^{29} + 30 q^{30} - 77 q^{31} - 175 q^{32} - 74 q^{33} - 28 q^{34} - 171 q^{35} + 171 q^{36} - 52 q^{37} - 144 q^{38} - 54 q^{39} - 49 q^{40} - 107 q^{41} + 7 q^{42} - 58 q^{43} - 139 q^{44} - 89 q^{45} - 33 q^{46} - 255 q^{47} - 202 q^{48} + 171 q^{49} - 74 q^{50} - 63 q^{51} - 90 q^{52} - 82 q^{53} - 51 q^{54} - 70 q^{55} - 180 q^{56} - 70 q^{57} - 50 q^{58} - 289 q^{59} - 105 q^{60} - 20 q^{61} - 143 q^{62} - 119 q^{63} + 201 q^{64} - 92 q^{65} - 3 q^{66} - 138 q^{67} - 177 q^{68} - 67 q^{69} + 4 q^{70} - 141 q^{71} - 138 q^{72} - 71 q^{73} - 26 q^{74} - 251 q^{75} - 42 q^{76} - 149 q^{77} - 6 q^{78} - 47 q^{79} - 294 q^{80} + 193 q^{81} - 70 q^{82} - 329 q^{83} - 40 q^{84} - 45 q^{85} - 83 q^{86} - 139 q^{87} - 45 q^{88} - 163 q^{89} - 116 q^{90} - 141 q^{91} - 204 q^{92} - 91 q^{93} - 8 q^{94} - 173 q^{95} - 53 q^{96} - 147 q^{97} - 156 q^{98} - 157 q^{99}+O(q^{100}) \) 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 −2.81633 −3.09812 5.93172 −0.550838 8.72533 −3.77340 −11.0730 6.59836 1.55134
1.2 −2.81174 −2.29296 5.90586 2.02073 6.44719 2.09923 −10.9822 2.25765 −5.68176
1.3 −2.78060 0.0708990 5.73173 −3.99378 −0.197142 2.14019 −10.3764 −2.99497 11.1051
1.4 −2.77684 2.42901 5.71081 0.850994 −6.74496 0.0263654 −10.3043 2.90008 −2.36307
1.5 −2.77001 −0.453815 5.67294 −2.74837 1.25707 −3.66610 −10.1741 −2.79405 7.61300
1.6 −2.75099 0.881000 5.56794 −0.176984 −2.42362 4.01245 −9.81537 −2.22384 0.486880
1.7 −2.72893 1.44573 5.44706 1.77131 −3.94530 −0.914980 −9.40679 −0.909861 −4.83377
1.8 −2.70901 2.53540 5.33875 −3.58245 −6.86844 1.35106 −9.04473 3.42827 9.70491
1.9 −2.65566 −0.00621299 5.05254 3.61372 0.0164996 −2.37510 −8.10651 −2.99996 −9.59681
1.10 −2.63506 −2.68532 4.94352 −3.56465 7.07597 1.99537 −7.75633 4.21094 9.39304
1.11 −2.63322 −3.42055 4.93387 2.45426 9.00708 4.09681 −7.72553 8.70017 −6.46261
1.12 −2.62320 −1.94038 4.88119 −2.34759 5.09001 2.45052 −7.55796 0.765079 6.15820
1.13 −2.61107 −0.462954 4.81767 −1.81913 1.20880 2.59865 −7.35711 −2.78567 4.74986
1.14 −2.58891 3.03601 4.70247 −2.12687 −7.85998 3.03640 −6.99646 6.21738 5.50629
1.15 −2.58762 0.766219 4.69576 1.84822 −1.98268 −5.04954 −6.97561 −2.41291 −4.78248
1.16 −2.54248 2.75174 4.46418 −3.07759 −6.99622 −3.90745 −6.26513 4.57206 7.82470
1.17 −2.48215 −2.18420 4.16105 −1.67259 5.42150 2.74743 −5.36403 1.77073 4.15162
1.18 −2.47319 −1.27283 4.11668 3.46270 3.14796 5.16648 −5.23495 −1.37990 −8.56393
1.19 −2.46210 −1.32036 4.06192 0.169195 3.25085 −3.54943 −5.07665 −1.25666 −0.416575
1.20 −2.46112 −3.25745 4.05713 −3.21291 8.01698 −0.720862 −5.06284 7.61098 7.90737
See next 80 embeddings (of 205 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.205
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5077\) \(1\)

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 5077.2.a.b ✓ 205
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5077.2.a.b ✓ 205 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{205} + 25 T_{2}^{204} + 10 T_{2}^{203} - 4900 T_{2}^{202} - 31793 T_{2}^{201} + 419555 T_{2}^{200} + \cdots + 4648448 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5077))\). Copy content Toggle raw display