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Properties

Label 2015.4.a.g
Level $2015$
Weight $4$
Character orbit 2015.a
Self dual yes
Analytic conductor $118.889$
Analytic rank $0$
Dimension $52$
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] = [2015,4,Mod(1,2015)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2015, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2015.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2015 = 5 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2015.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: \(118.888848662\)
Analytic rank: \(0\)
Dimension: \(52\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 52 q + q^{2} + 13 q^{3} + 241 q^{4} + 260 q^{5} + 29 q^{6} - 6 q^{7} + 33 q^{8} + 637 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 52 q + q^{2} + 13 q^{3} + 241 q^{4} + 260 q^{5} + 29 q^{6} - 6 q^{7} + 33 q^{8} + 637 q^{9} + 5 q^{10} + 225 q^{11} + 260 q^{12} - 676 q^{13} + 206 q^{14} + 65 q^{15} + 1133 q^{16} + 48 q^{17} + 144 q^{18} - 51 q^{19} + 1205 q^{20} + 496 q^{21} + 171 q^{22} + 38 q^{23} + 518 q^{24} + 1300 q^{25} - 13 q^{26} + 376 q^{27} + 101 q^{28} + 1109 q^{29} + 145 q^{30} + 1612 q^{31} + 300 q^{32} + 322 q^{33} + 850 q^{34} - 30 q^{35} + 3078 q^{36} + 555 q^{37} + 236 q^{38} - 169 q^{39} + 165 q^{40} + 1048 q^{41} + 1633 q^{42} + 1615 q^{43} + 4092 q^{44} + 3185 q^{45} + 679 q^{46} + 66 q^{47} + 4187 q^{48} + 5058 q^{49} + 25 q^{50} + 108 q^{51} - 3133 q^{52} + 1871 q^{53} + 1052 q^{54} + 1125 q^{55} + 2766 q^{56} + 3044 q^{57} + 2681 q^{58} + 883 q^{59} + 1300 q^{60} + 4209 q^{61} + 31 q^{62} - 786 q^{63} + 5529 q^{64} - 3380 q^{65} + 3879 q^{66} + 203 q^{67} + 2904 q^{68} + 4072 q^{69} + 1030 q^{70} + 2118 q^{71} + 1032 q^{72} + 1500 q^{73} + 5933 q^{74} + 325 q^{75} + 1026 q^{76} - 924 q^{77} - 377 q^{78} + 4422 q^{79} + 5665 q^{80} + 9688 q^{81} + 414 q^{82} + 2723 q^{83} + 6941 q^{84} + 240 q^{85} + 5884 q^{86} + 264 q^{87} + 6995 q^{88} + 6318 q^{89} + 720 q^{90} + 78 q^{91} - 29 q^{92} + 403 q^{93} + 4818 q^{94} - 255 q^{95} + 12058 q^{96} + 5316 q^{97} + 4633 q^{98} + 5425 q^{99}+O(q^{100}) \) 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 −5.51365 −4.09405 22.4003 5.00000 22.5731 1.36887 −79.3983 −10.2388 −27.5682
1.2 −5.25783 1.11962 19.6447 5.00000 −5.88679 −8.36677 −61.2259 −25.7464 −26.2891
1.3 −5.24891 7.80149 19.5510 5.00000 −40.9493 21.7487 −60.6303 33.8632 −26.2445
1.4 −5.22415 −3.79694 19.2918 5.00000 19.8358 −29.2235 −58.9898 −12.5832 −26.1208
1.5 −4.92290 9.95497 16.2350 5.00000 −49.0074 −22.8992 −40.5400 72.1015 −24.6145
1.6 −4.86800 −10.2643 15.6974 5.00000 49.9665 −2.22019 −37.4712 78.3552 −24.3400
1.7 −4.77134 3.82224 14.7657 5.00000 −18.2372 −13.6823 −32.2815 −12.3905 −23.8567
1.8 −4.50504 7.24796 12.2954 5.00000 −32.6524 28.1105 −19.3510 25.5330 −22.5252
1.9 −4.02110 −2.28061 8.16921 5.00000 9.17056 9.19591 −0.680393 −21.7988 −20.1055
1.10 −3.99913 2.07865 7.99306 5.00000 −8.31281 10.0423 0.0277684 −22.6792 −19.9957
1.11 −3.86222 −7.81527 6.91678 5.00000 30.1843 −18.3433 4.18364 34.0785 −19.3111
1.12 −3.72936 2.09223 5.90810 5.00000 −7.80269 −24.8983 7.80144 −22.6226 −18.6468
1.13 −3.30845 −6.11978 2.94581 5.00000 20.2469 26.7005 16.7215 10.4516 −16.5422
1.14 −3.08411 −7.42027 1.51173 5.00000 22.8849 31.5045 20.0105 28.0605 −15.4205
1.15 −3.01441 5.87533 1.08664 5.00000 −17.7106 −32.1337 20.8397 7.51948 −15.0720
1.16 −2.79569 −6.84932 −0.184097 5.00000 19.1486 10.4445 22.8802 19.9132 −13.9785
1.17 −2.56347 5.79548 −1.42863 5.00000 −14.8565 11.0247 24.1700 6.58758 −12.8173
1.18 −2.02907 8.91392 −3.88288 5.00000 −18.0870 28.9989 24.1112 52.4580 −10.1453
1.19 −1.94767 −0.204973 −4.20660 5.00000 0.399220 7.04965 23.7744 −26.9580 −9.73833
1.20 −1.68859 8.60461 −5.14865 5.00000 −14.5297 −1.62336 22.2027 47.0394 −8.44297
See all 52 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.52
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(13\) \(1\)
\(31\) \(-1\)

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 2015.4.a.g ✓ 52
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2015.4.a.g ✓ 52 1.a even 1 1 trivial