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Properties

Label 2015.4.a.a
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.

\(\operatorname{Tr}(f)(q) = \) \( 37 q - 9 q^{2} - 17 q^{3} + 117 q^{4} + 185 q^{5} - 21 q^{6} - 64 q^{7} - 87 q^{8} + 196 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 37 q - 9 q^{2} - 17 q^{3} + 117 q^{4} + 185 q^{5} - 21 q^{6} - 64 q^{7} - 87 q^{8} + 196 q^{9} - 45 q^{10} - 191 q^{11} - 76 q^{12} + 481 q^{13} - 242 q^{14} - 85 q^{15} + 125 q^{16} - 272 q^{17} - 306 q^{18} - 359 q^{19} + 585 q^{20} - 680 q^{21} - 191 q^{22} - 218 q^{23} + 171 q^{24} + 925 q^{25} - 117 q^{26} - 476 q^{27} - 997 q^{28} - 921 q^{29} - 105 q^{30} + 1147 q^{31} - 160 q^{32} - 938 q^{33} - 1250 q^{34} - 320 q^{35} + 182 q^{36} - 1651 q^{37} - 596 q^{38} - 221 q^{39} - 435 q^{40} - 1750 q^{41} + 72 q^{42} - 487 q^{43} - 364 q^{44} + 980 q^{45} - 1625 q^{46} - 784 q^{47} + 13 q^{48} - 11 q^{49} - 225 q^{50} - 1652 q^{51} + 1521 q^{52} - 719 q^{53} - 960 q^{54} - 955 q^{55} - 2610 q^{56} - 612 q^{57} + 137 q^{58} - 2017 q^{59} - 380 q^{60} - 1493 q^{61} - 279 q^{62} - 244 q^{63} - 1963 q^{64} + 2405 q^{65} - 1641 q^{66} - 2645 q^{67} + 163 q^{68} - 3656 q^{69} - 1210 q^{70} - 1546 q^{71} - 5448 q^{72} - 4272 q^{73} + 2937 q^{74} - 425 q^{75} - 5612 q^{76} - 656 q^{77} - 273 q^{78} - 3974 q^{79} + 625 q^{80} - 767 q^{81} - 1300 q^{82} + 937 q^{83} - 3067 q^{84} - 1360 q^{85} - 2617 q^{86} - 384 q^{87} - 2069 q^{88} - 2298 q^{89} - 1530 q^{90} - 832 q^{91} - 769 q^{92} - 527 q^{93} - 162 q^{94} - 1795 q^{95} + 3378 q^{96} - 7254 q^{97} + 453 q^{98} - 6339 q^{99}+O(q^{100}) \) 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 −5.38750 −6.86341 21.0251 5.00000 36.9766 18.6598 −70.1729 20.1064 −26.9375
1.2 −4.96976 2.72744 16.6985 5.00000 −13.5547 −12.7075 −43.2293 −19.5611 −24.8488
1.3 −4.86537 −4.89480 15.6719 5.00000 23.8150 −1.63143 −37.3265 −3.04093 −24.3269
1.4 −4.84266 8.40673 15.4514 5.00000 −40.7109 −34.6611 −36.0844 43.6730 −24.2133
1.5 −4.63717 5.15733 13.5033 5.00000 −23.9154 23.1029 −25.5198 −0.401905 −23.1858
1.6 −4.40111 −6.56412 11.3698 5.00000 28.8894 −4.71905 −14.8309 16.0876 −22.0056
1.7 −4.18959 −1.46124 9.55262 5.00000 6.12197 −14.7687 −6.50485 −24.8648 −20.9479
1.8 −3.96708 8.06512 7.73773 5.00000 −31.9950 13.2183 1.04044 38.0462 −19.8354
1.9 −3.54448 3.28311 4.56332 5.00000 −11.6369 11.4824 12.1812 −16.2212 −17.7224
1.10 −3.36493 −9.24784 3.32273 5.00000 31.1183 −17.5101 15.7387 58.5226 −16.8246
1.11 −3.19452 −4.58066 2.20495 5.00000 14.6330 19.2955 18.5124 −6.01755 −15.9726
1.12 −2.28340 1.50746 −2.78609 5.00000 −3.44214 −31.4455 24.6289 −24.7276 −11.4170
1.13 −1.94811 −0.00735910 −4.20487 5.00000 0.0143363 11.7688 23.7764 −26.9999 −9.74055
1.14 −1.85971 −2.47626 −4.54148 5.00000 4.60512 30.9633 23.3235 −20.8681 −9.29854
1.15 −1.77485 4.74748 −4.84990 5.00000 −8.42608 6.57312 22.8067 −4.46142 −8.87427
1.16 −1.61847 −8.90868 −5.38056 5.00000 14.4184 4.82365 21.6560 52.3645 −8.09234
1.17 −0.994297 8.56579 −7.01137 5.00000 −8.51695 −7.22465 14.9258 46.3728 −4.97149
1.18 −0.612379 −1.60947 −7.62499 5.00000 0.985605 −21.6794 9.56842 −24.4096 −3.06190
1.19 −0.587872 6.65878 −7.65441 5.00000 −3.91451 −14.9932 9.20279 17.3394 −2.93936
1.20 −0.174346 −8.11460 −7.96960 5.00000 1.41475 8.48848 2.78423 38.8468 −0.871729
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.a ✓ 37
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2015.4.a.a ✓ 37 1.a even 1 1 trivial