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Properties

Label 2015.4.a.f
Level $2015$
Weight $4$
Character orbit 2015.a
Self dual yes
Analytic conductor $118.889$
Analytic rank $0$
Dimension $51$
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: \(51\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 51 q + 5 q^{2} + 13 q^{3} + 233 q^{4} - 255 q^{5} + 59 q^{6} + 62 q^{7} + 81 q^{8} + 508 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 51 q + 5 q^{2} + 13 q^{3} + 233 q^{4} - 255 q^{5} + 59 q^{6} + 62 q^{7} + 81 q^{8} + 508 q^{9} - 25 q^{10} + 19 q^{11} + 260 q^{12} + 663 q^{13} + 26 q^{14} - 65 q^{15} + 1069 q^{16} + 40 q^{17} + 324 q^{18} + 191 q^{19} - 1165 q^{20} + 176 q^{21} + 171 q^{22} + 70 q^{23} + 346 q^{24} + 1275 q^{25} + 65 q^{26} + 508 q^{27} + 837 q^{28} - 531 q^{29} - 295 q^{30} + 1581 q^{31} + 1408 q^{32} + 730 q^{33} + 886 q^{34} - 310 q^{35} + 1886 q^{36} + 1601 q^{37} - 86 q^{38} + 169 q^{39} - 405 q^{40} + 86 q^{41} - 53 q^{42} + 1927 q^{43} - 1088 q^{44} - 2540 q^{45} + 503 q^{46} + 146 q^{47} - 275 q^{48} + 3333 q^{49} + 125 q^{50} - 72 q^{51} + 3029 q^{52} + 447 q^{53} + 3994 q^{54} - 95 q^{55} + 174 q^{56} + 1024 q^{57} + 3889 q^{58} + 61 q^{59} - 1300 q^{60} - 279 q^{61} + 155 q^{62} + 1686 q^{63} + 5605 q^{64} - 3315 q^{65} - 3993 q^{66} + 2535 q^{67} - 1080 q^{68} + 1440 q^{69} - 130 q^{70} - 590 q^{71} + 4880 q^{72} + 4524 q^{73} + 1121 q^{74} + 325 q^{75} - 912 q^{76} - 1308 q^{77} + 767 q^{78} + 2030 q^{79} - 5345 q^{80} + 7435 q^{81} + 3344 q^{82} + 3511 q^{83} + 4131 q^{84} - 200 q^{85} - 3108 q^{86} - 1528 q^{87} - 1017 q^{88} - 890 q^{89} - 1620 q^{90} + 806 q^{91} - 3105 q^{92} + 403 q^{93} - 802 q^{94} - 955 q^{95} - 78 q^{96} + 6996 q^{97} + 7243 q^{98} + 247 q^{99}+O(q^{100}) \) 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 −5.53789 −0.382342 22.6682 −5.00000 2.11737 17.1098 −81.2310 −26.8538 27.6894
1.2 −5.18438 6.96703 18.8778 −5.00000 −36.1198 3.40636 −56.3949 21.5395 25.9219
1.3 −5.15609 −5.81377 18.5853 −5.00000 29.9764 19.6006 −54.5789 6.79996 25.7805
1.4 −5.02038 4.59889 17.2042 −5.00000 −23.0882 11.8368 −46.2084 −5.85022 25.1019
1.5 −4.95647 −2.11640 16.5666 −5.00000 10.4899 −24.7129 −42.4603 −22.5208 24.7824
1.6 −4.53718 −5.37384 12.5860 −5.00000 24.3821 −14.6417 −20.8074 1.87820 22.6859
1.7 −4.51275 9.10415 12.3649 −5.00000 −41.0847 −15.6381 −19.6976 55.8855 22.5637
1.8 −4.44384 −10.2737 11.7477 −5.00000 45.6547 9.79491 −16.6543 78.5487 22.2192
1.9 −4.12859 −5.13772 9.04524 −5.00000 21.2115 −16.7340 −4.31536 −0.603836 20.6429
1.10 −4.01585 0.696499 8.12705 −5.00000 −2.79704 −11.0336 −0.510222 −26.5149 20.0793
1.11 −3.67662 7.22598 5.51754 −5.00000 −26.5672 −22.2158 9.12707 25.2148 18.3831
1.12 −3.59742 3.08040 4.94143 −5.00000 −11.0815 36.8237 11.0030 −17.5111 17.9871
1.13 −3.41712 8.22093 3.67668 −5.00000 −28.0919 34.1297 14.7733 40.5836 17.0856
1.14 −3.24066 −4.97602 2.50190 −5.00000 16.1256 28.9139 17.8175 −2.23918 16.2033
1.15 −2.33423 3.05075 −2.55136 −5.00000 −7.12115 14.6287 24.6293 −17.6929 11.6712
1.16 −2.28425 −7.95039 −2.78218 −5.00000 18.1607 −5.14495 24.6292 36.2087 11.4213
1.17 −2.23724 3.34496 −2.99476 −5.00000 −7.48349 −5.69208 24.5979 −15.8112 11.1862
1.18 −2.17542 −1.21986 −3.26755 −5.00000 2.65372 −10.5635 24.5116 −25.5119 10.8771
1.19 −2.09448 5.25321 −3.61314 −5.00000 −11.0028 −8.01817 24.3235 0.596264 10.4724
1.20 −1.79957 −7.64933 −4.76154 −5.00000 13.7655 −31.9841 22.9653 31.5123 8.99786
See all 51 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.51
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.f ✓ 51
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2015.4.a.f ✓ 51 1.a even 1 1 trivial