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Properties

Label 2015.4.a.b
Level $2015$
Weight $4$
Character orbit 2015.a
Self dual yes
Analytic conductor $118.889$
Analytic rank $1$
Dimension $37$
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: \(37\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 37 q - 5 q^{2} + q^{3} + 117 q^{4} + 185 q^{5} - 19 q^{6} - 20 q^{7} - 39 q^{8} + 196 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 37 q - 5 q^{2} + q^{3} + 117 q^{4} + 185 q^{5} - 19 q^{6} - 20 q^{7} - 39 q^{8} + 196 q^{9} - 25 q^{10} - 15 q^{11} - 226 q^{12} - 481 q^{13} - 242 q^{14} + 5 q^{15} + 125 q^{16} + 14 q^{17} - 126 q^{18} - 279 q^{19} + 585 q^{20} - 192 q^{21} - 313 q^{22} - 8 q^{23} - 149 q^{24} + 925 q^{25} + 65 q^{26} - 104 q^{27} - 67 q^{28} - 573 q^{29} - 95 q^{30} - 1147 q^{31} - 372 q^{32} - 338 q^{33} - 606 q^{34} - 100 q^{35} - 370 q^{36} - 525 q^{37} + 8 q^{38} - 13 q^{39} - 195 q^{40} - 1418 q^{41} - 508 q^{42} - 733 q^{43} - 1088 q^{44} + 980 q^{45} + 201 q^{46} - 1140 q^{47} - 2675 q^{48} - 987 q^{49} - 125 q^{50} - 1612 q^{51} - 1521 q^{52} - 857 q^{53} - 1758 q^{54} - 75 q^{55} - 2610 q^{56} - 936 q^{57} - 231 q^{58} - 1725 q^{59} - 1130 q^{60} - 1813 q^{61} + 155 q^{62} - 2636 q^{63} - 2599 q^{64} - 2405 q^{65} - 1285 q^{66} - 2169 q^{67} - 761 q^{68} - 4536 q^{69} - 1210 q^{70} - 2778 q^{71} - 404 q^{72} - 1802 q^{73} - 4281 q^{74} + 25 q^{75} - 3158 q^{76} - 712 q^{77} + 247 q^{78} - 3054 q^{79} + 625 q^{80} - 2543 q^{81} - 2702 q^{82} - 677 q^{83} - 371 q^{84} + 70 q^{85} - 5051 q^{86} - 2760 q^{87} - 1627 q^{88} - 7014 q^{89} - 630 q^{90} + 260 q^{91} - 2075 q^{92} - 31 q^{93} - 3170 q^{94} - 1395 q^{95} - 3842 q^{96} - 2638 q^{97} - 6445 q^{98} - 4475 q^{99}+O(q^{100}) \) 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 −5.40953 3.05698 21.2630 5.00000 −16.5368 20.2796 −71.7467 −17.6549 −27.0476
1.2 −4.99033 −7.37756 16.9034 5.00000 36.8165 1.04601 −44.4310 27.4284 −24.9517
1.3 −4.90717 −3.10811 16.0803 5.00000 15.2520 21.3773 −39.6514 −17.3397 −24.5358
1.4 −4.67369 6.59335 13.8433 5.00000 −30.8153 −12.8340 −27.3100 16.4723 −23.3684
1.5 −4.52993 −2.21840 12.5203 5.00000 10.0492 −36.3127 −20.4766 −22.0787 −22.6497
1.6 −4.46523 −7.93999 11.9383 5.00000 35.4539 11.6157 −17.5854 36.0434 −22.3262
1.7 −3.76793 3.67184 6.19732 5.00000 −13.8353 30.6753 6.79238 −13.5176 −18.8397
1.8 −3.70127 −2.84917 5.69937 5.00000 10.5455 −2.15992 8.51525 −18.8823 −18.5063
1.9 −3.31859 10.2231 3.01301 5.00000 −33.9263 −3.43367 16.5497 77.5122 −16.5929
1.10 −3.19670 6.34535 2.21886 5.00000 −20.2841 −16.8844 18.4805 13.2634 −15.9835
1.11 −3.04755 3.52306 1.28754 5.00000 −10.7367 16.0178 20.4565 −14.5880 −15.2377
1.12 −2.73112 −7.03021 −0.540974 5.00000 19.2003 −18.8999 23.3264 22.4238 −13.6556
1.13 −1.99753 1.94331 −4.00986 5.00000 −3.88183 −20.4931 23.9901 −23.2235 −9.98766
1.14 −1.97206 −3.91941 −4.11096 5.00000 7.72933 12.2988 23.8836 −11.6382 −9.86032
1.15 −1.19814 −6.05369 −6.56447 5.00000 7.25315 −27.5594 17.4502 9.64716 −5.99068
1.16 −1.13009 4.53754 −6.72289 5.00000 −5.12785 16.2119 16.6383 −6.41076 −5.65047
1.17 −0.676285 8.07989 −7.54264 5.00000 −5.46431 −6.73695 10.5113 38.2846 −3.38143
1.18 −0.582622 −8.58166 −7.66055 5.00000 4.99987 24.6698 9.12419 46.6450 −2.91311
1.19 −0.442679 7.22900 −7.80404 5.00000 −3.20013 12.6090 6.99611 25.2585 −2.21339
1.20 0.0435348 −3.38102 −7.99810 5.00000 −0.147192 6.39996 −0.696474 −15.5687 0.217674
See all 37 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.37
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.b ✓ 37
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2015.4.a.b ✓ 37 1.a even 1 1 trivial