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Properties

Label 2015.4.a.e
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.

\(\operatorname{Tr}(f)(q) = \) \( 51 q + q^{2} + 7 q^{3} + 233 q^{4} - 255 q^{5} - 59 q^{6} - 6 q^{7} + 33 q^{8} + 508 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 51 q + q^{2} + 7 q^{3} + 233 q^{4} - 255 q^{5} - 59 q^{6} - 6 q^{7} + 33 q^{8} + 508 q^{9} - 5 q^{10} + 19 q^{11} - 130 q^{12} - 663 q^{13} + 26 q^{14} - 35 q^{15} + 1069 q^{16} + 382 q^{17} + 144 q^{18} - 153 q^{19} - 1165 q^{20} - 344 q^{21} + 557 q^{22} + 492 q^{23} - 348 q^{24} + 1275 q^{25} - 13 q^{26} + 472 q^{27} + 423 q^{28} - 415 q^{29} + 295 q^{30} - 1581 q^{31} + 300 q^{32} + 274 q^{33} - 374 q^{34} + 30 q^{35} + 2958 q^{36} + 399 q^{37} + 518 q^{38} - 91 q^{39} - 165 q^{40} - 414 q^{41} + 1633 q^{42} + 649 q^{43} + 692 q^{44} - 2540 q^{45} + 665 q^{46} + 678 q^{47} + 1645 q^{48} + 4053 q^{49} + 25 q^{50} + 528 q^{51} - 3029 q^{52} + 1641 q^{53} - 288 q^{54} - 95 q^{55} + 174 q^{56} + 2276 q^{57} - 235 q^{58} - 1063 q^{59} + 650 q^{60} - 983 q^{61} - 31 q^{62} - 786 q^{63} + 4969 q^{64} + 3315 q^{65} + 4763 q^{66} + 835 q^{67} + 5922 q^{68} + 560 q^{69} - 130 q^{70} + 66 q^{71} + 4624 q^{72} - 710 q^{73} - 2745 q^{74} + 175 q^{75} + 1674 q^{76} + 4100 q^{77} + 767 q^{78} + 14 q^{79} - 5345 q^{80} + 5147 q^{81} + 4042 q^{82} - 1023 q^{83} - 4569 q^{84} - 1910 q^{85} + 3548 q^{86} + 4896 q^{87} + 7709 q^{88} + 394 q^{89} - 720 q^{90} + 78 q^{91} + 10293 q^{92} - 217 q^{93} + 1286 q^{94} + 765 q^{95} + 62 q^{96} + 4636 q^{97} - 6087 q^{98} + 5119 q^{99}+O(q^{100}) \) 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 −5.58759 −2.63090 23.2212 −5.00000 14.7004 −6.48377 −85.0497 −20.0784 27.9380
1.2 −5.48692 4.23912 22.1062 −5.00000 −23.2597 35.8149 −77.3998 −9.02988 27.4346
1.3 −5.15846 8.08769 18.6098 −5.00000 −41.7201 −31.0092 −54.7300 38.4107 25.7923
1.4 −5.08826 −6.91596 17.8903 −5.00000 35.1902 35.6755 −50.3246 20.8305 25.4413
1.5 −4.83783 2.84733 15.4046 −5.00000 −13.7749 −1.86768 −35.8220 −18.8927 24.1891
1.6 −4.63628 9.50697 13.4951 −5.00000 −44.0769 −23.3990 −25.4767 63.3824 23.1814
1.7 −4.52782 −0.956645 12.5012 −5.00000 4.33152 −16.4359 −20.3805 −26.0848 22.6391
1.8 −4.34430 −8.62666 10.8729 −5.00000 37.4768 23.9952 −12.4808 47.4192 21.7215
1.9 −4.21956 −4.60046 9.80468 −5.00000 19.4119 −5.61631 −7.61495 −5.83581 21.0978
1.10 −4.05726 −8.86180 8.46136 −5.00000 35.9546 −33.0312 −1.87187 51.5315 20.2863
1.11 −3.88341 6.79815 7.08085 −5.00000 −26.4000 −8.73959 3.56943 19.2149 19.4170
1.12 −3.69787 1.86605 5.67424 −5.00000 −6.90039 32.1623 8.60037 −23.5179 18.4893
1.13 −3.16647 −7.24584 2.02653 −5.00000 22.9437 −4.20195 18.9148 25.5022 15.8323
1.14 −3.09981 3.90579 1.60883 −5.00000 −12.1072 −18.8778 19.8114 −11.7448 15.4991
1.15 −2.93448 −0.129605 0.611156 −5.00000 0.380322 18.0122 21.6824 −26.9832 14.6724
1.16 −2.74935 −4.51897 −0.441051 −5.00000 12.4242 −19.0873 23.2074 −6.57894 13.7468
1.17 −2.57843 8.65803 −1.35171 −5.00000 −22.3241 12.6094 24.1127 47.9615 12.8921
1.18 −2.00621 9.46754 −3.97512 −5.00000 −18.9939 18.1832 24.0246 62.6343 10.0310
1.19 −1.66846 0.990882 −5.21624 −5.00000 −1.65325 −18.7481 22.0508 −26.0182 8.34231
1.20 −1.44292 −1.89928 −5.91799 −5.00000 2.74051 23.4521 20.0825 −23.3927 7.21459
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.e ✓ 51
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2015.4.a.e ✓ 51 1.a even 1 1 trivial