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Properties

Label 2015.4.a.h
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.

\(\operatorname{Tr}(f)(q) = \) \( 52 q + 5 q^{2} + 7 q^{3} + 241 q^{4} + 260 q^{5} + 99 q^{6} + 62 q^{7} + 81 q^{8} + 637 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 52 q + 5 q^{2} + 7 q^{3} + 241 q^{4} + 260 q^{5} + 99 q^{6} + 62 q^{7} + 81 q^{8} + 637 q^{9} + 25 q^{10} + 97 q^{11} - 130 q^{12} + 676 q^{13} + 206 q^{14} + 35 q^{15} + 1133 q^{16} + 374 q^{17} + 324 q^{18} + 477 q^{19} + 1205 q^{20} + 648 q^{21} + 557 q^{22} + 380 q^{23} + 1520 q^{24} + 1300 q^{25} + 65 q^{26} + 340 q^{27} + 515 q^{28} + 761 q^{29} + 495 q^{30} - 1612 q^{31} + 1408 q^{32} + 778 q^{33} + 558 q^{34} + 310 q^{35} + 4150 q^{36} + 1413 q^{37} + 240 q^{38} + 91 q^{39} + 405 q^{40} + 1524 q^{41} - 53 q^{42} + 777 q^{43} + 792 q^{44} + 3185 q^{45} + 2129 q^{46} + 78 q^{47} - 2817 q^{48} + 4626 q^{49} + 125 q^{50} + 708 q^{51} + 3133 q^{52} + 793 q^{53} + 2278 q^{54} + 485 q^{55} + 2766 q^{56} + 880 q^{57} + 3025 q^{58} + 2455 q^{59} - 650 q^{60} + 1513 q^{61} - 155 q^{62} + 1686 q^{63} + 6165 q^{64} + 3380 q^{65} - 997 q^{66} + 3023 q^{67} + 1938 q^{68} + 3192 q^{69} + 1030 q^{70} + 3214 q^{71} + 3916 q^{72} + 1770 q^{73} - 765 q^{74} + 175 q^{75} + 2972 q^{76} + 4484 q^{77} + 1287 q^{78} - 74 q^{79} + 5665 q^{80} + 11464 q^{81} + 3128 q^{82} + 525 q^{83} + 7765 q^{84} + 1870 q^{85} - 2888 q^{86} + 1464 q^{87} + 4093 q^{88} + 6798 q^{89} + 1620 q^{90} + 806 q^{91} + 4913 q^{92} - 217 q^{93} + 5386 q^{94} + 2385 q^{95} + 6294 q^{96} + 6908 q^{97} - 3765 q^{98} + 1977 q^{99}+O(q^{100}) \) 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 −5.48967 −9.84573 22.1365 5.00000 54.0498 −28.3650 −77.6047 69.9385 −27.4484
1.2 −5.42889 −0.577247 21.4728 5.00000 3.13381 19.5868 −73.1422 −26.6668 −27.1444
1.3 −5.36205 −4.98664 20.7516 5.00000 26.7386 −23.2305 −68.3744 −2.13339 −26.8102
1.4 −4.96515 2.19193 16.6527 5.00000 −10.8833 0.471577 −42.9618 −22.1954 −24.8257
1.5 −4.75220 9.28672 14.5834 5.00000 −44.1323 21.4487 −31.2856 59.2432 −23.7610
1.6 −4.70006 −8.25390 14.0906 5.00000 38.7938 25.9615 −28.6262 41.1268 −23.5003
1.7 −4.63581 5.28096 13.4908 5.00000 −24.4816 −10.6661 −25.4542 0.888581 −23.1791
1.8 −4.07082 −2.32727 8.57161 5.00000 9.47393 14.7830 −2.32692 −21.5838 −20.3541
1.9 −4.04163 1.49784 8.33474 5.00000 −6.05371 −15.4149 −1.35287 −24.7565 −20.2081
1.10 −3.93158 −3.56210 7.45733 5.00000 14.0047 −27.0432 2.13355 −14.3115 −19.6579
1.11 −3.78684 −9.31940 6.34016 5.00000 35.2911 31.6057 6.28554 59.8512 −18.9342
1.12 −3.47523 1.11172 4.07725 5.00000 −3.86348 29.1391 13.6325 −25.7641 −17.3762
1.13 −3.44005 7.53148 3.83397 5.00000 −25.9087 4.73606 14.3314 29.7232 −17.2003
1.14 −3.23483 7.09846 2.46413 5.00000 −22.9623 −22.3166 17.9076 23.3881 −16.1742
1.15 −3.21558 −8.51274 2.33995 5.00000 27.3734 −20.0616 18.2003 45.4667 −16.0779
1.16 −2.72959 −0.219351 −0.549347 5.00000 0.598738 −22.7154 23.3362 −26.9519 −13.6479
1.17 −2.51754 8.91355 −1.66200 5.00000 −22.4402 32.5604 24.3245 52.4513 −12.5877
1.18 −2.22963 −4.21401 −3.02873 5.00000 9.39569 14.5312 24.5900 −9.24216 −11.1482
1.19 −1.71255 −5.07265 −5.06717 5.00000 8.68718 12.4207 22.3782 −1.26820 −8.56276
1.20 −1.52474 5.11468 −5.67515 5.00000 −7.79858 −4.65534 20.8511 −0.840061 −7.62372
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.h ✓ 52
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2015.4.a.h ✓ 52 1.a even 1 1 trivial