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Properties

Label 2015.4.a.c
Level $2015$
Weight $4$
Character orbit 2015.a
Self dual yes
Analytic conductor $118.889$
Analytic rank $1$
Dimension $40$
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: \(1\)
Dimension: \(40\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 9 q^{2} + q^{3} + 149 q^{4} - 200 q^{5} + 11 q^{6} - 64 q^{7} - 87 q^{8} + 247 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 9 q^{2} + q^{3} + 149 q^{4} - 200 q^{5} + 11 q^{6} - 64 q^{7} - 87 q^{8} + 247 q^{9} + 45 q^{10} - q^{11} - 226 q^{12} + 520 q^{13} + 138 q^{14} - 5 q^{15} + 413 q^{16} + 6 q^{17} - 306 q^{18} - 265 q^{19} - 745 q^{20} - 344 q^{21} - 313 q^{22} + 24 q^{23} - 321 q^{24} + 1000 q^{25} - 117 q^{26} + 28 q^{27} - 675 q^{28} - 9 q^{29} - 55 q^{30} - 1240 q^{31} - 160 q^{32} - 986 q^{33} - 298 q^{34} + 320 q^{35} + 238 q^{36} - 1403 q^{37} - 314 q^{38} + 13 q^{39} + 435 q^{40} - 84 q^{41} + 744 q^{42} - 421 q^{43} + 812 q^{44} - 1235 q^{45} - 527 q^{46} - 460 q^{47} - 1573 q^{48} + 748 q^{49} - 225 q^{50} - 160 q^{51} + 1937 q^{52} - 649 q^{53} + 1184 q^{54} + 5 q^{55} + 1518 q^{56} - 468 q^{57} - 1111 q^{58} + 49 q^{59} + 1130 q^{60} - 161 q^{61} + 279 q^{62} - 244 q^{63} + 421 q^{64} - 2600 q^{65} + 2027 q^{66} - 1981 q^{67} + 2041 q^{68} + 1664 q^{69} - 690 q^{70} - 1510 q^{71} - 876 q^{72} - 3562 q^{73} - 1005 q^{74} + 25 q^{75} - 2776 q^{76} - 1096 q^{77} + 143 q^{78} - 1094 q^{79} - 2065 q^{80} + 712 q^{81} + 228 q^{82} - 817 q^{83} - 1837 q^{84} - 30 q^{85} + 1125 q^{86} - 1288 q^{87} - 4011 q^{88} - 2326 q^{89} + 1530 q^{90} - 832 q^{91} + 4425 q^{92} - 31 q^{93} - 2590 q^{94} + 1325 q^{95} - 3706 q^{96} - 7166 q^{97} - 7363 q^{98} - 2161 q^{99}+O(q^{100}) \) 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 −5.37200 7.54523 20.8584 −5.00000 −40.5329 −10.2525 −69.0752 29.9304 26.8600
1.2 −5.37128 0.548410 20.8507 −5.00000 −2.94567 −31.6748 −69.0248 −26.6992 26.8564
1.3 −5.36511 −9.10973 20.7844 −5.00000 48.8746 4.74988 −68.5894 55.9871 26.8255
1.4 −4.73109 −0.361979 14.3832 −5.00000 1.71256 19.9138 −30.1994 −26.8690 23.6554
1.5 −4.49232 3.54801 12.1809 −5.00000 −15.9388 9.29021 −18.7821 −14.4116 22.4616
1.6 −4.45556 −3.10163 11.8520 −5.00000 13.8195 −28.9167 −17.1628 −17.3799 22.2778
1.7 −4.40388 −3.73959 11.3941 −5.00000 16.4687 28.9489 −14.9473 −13.0154 22.0194
1.8 −3.93110 −9.15441 7.45353 −5.00000 35.9869 −24.8983 2.14824 56.8033 19.6555
1.9 −3.75281 7.14209 6.08361 −5.00000 −26.8029 2.21929 7.19186 24.0094 18.7641
1.10 −3.52283 1.79907 4.41035 −5.00000 −6.33783 −1.45908 12.6457 −23.7633 17.6142
1.11 −3.21179 −3.50672 2.31559 −5.00000 11.2628 10.7876 18.2571 −14.7029 16.0589
1.12 −3.06130 −6.09476 1.37157 −5.00000 18.6579 −0.141560 20.2916 10.1461 15.3065
1.13 −2.91217 4.66418 0.480737 −5.00000 −13.5829 −33.5642 21.8974 −5.24544 14.5609
1.14 −2.71216 9.79344 −0.644172 −5.00000 −26.5614 −17.0607 23.4444 68.9116 13.5608
1.15 −2.01521 7.28595 −3.93893 −5.00000 −14.6827 23.8997 24.0594 26.0851 10.0761
1.16 −1.92910 −8.94718 −4.27859 −5.00000 17.2600 24.8032 23.6866 53.0520 9.64548
1.17 −1.66495 −2.23181 −5.22795 −5.00000 3.71584 −0.535064 22.0239 −22.0190 8.32474
1.18 −1.28053 −0.0479436 −6.36024 −5.00000 0.0613932 −24.4791 18.3887 −26.9977 6.40266
1.19 −0.592810 2.83155 −7.64858 −5.00000 −1.67857 21.1735 9.27663 −18.9823 2.96405
1.20 −0.137956 −8.67576 −7.98097 −5.00000 1.19688 21.2726 2.20468 48.2687 0.689782
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.40
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.c ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2015.4.a.c ✓ 40 1.a even 1 1 trivial