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Properties

Label 2015.4.a.d
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 5 q^{2} - 17 q^{3} + 149 q^{4} - 200 q^{5} - 35 q^{6} - 20 q^{7} - 39 q^{8} + 247 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 5 q^{2} - 17 q^{3} + 149 q^{4} - 200 q^{5} - 35 q^{6} - 20 q^{7} - 39 q^{8} + 247 q^{9} + 25 q^{10} + 127 q^{11} - 76 q^{12} - 520 q^{13} + 138 q^{14} + 85 q^{15} + 413 q^{16} - 264 q^{17} - 126 q^{18} - q^{19} - 745 q^{20} + 176 q^{21} - 191 q^{22} - 106 q^{23} + 31 q^{24} + 1000 q^{25} + 65 q^{26} - 344 q^{27} + 255 q^{28} + 107 q^{29} + 175 q^{30} + 1240 q^{31} - 372 q^{32} - 386 q^{33} - 6 q^{34} + 100 q^{35} + 790 q^{36} - 741 q^{37} - 318 q^{38} + 221 q^{39} + 195 q^{40} + 1232 q^{41} - 1180 q^{42} - 615 q^{43} - 152 q^{44} - 1235 q^{45} - 329 q^{46} - 784 q^{47} - 1089 q^{48} - 516 q^{49} - 125 q^{50} - 200 q^{51} - 1937 q^{52} - 1503 q^{53} + 1658 q^{54} - 635 q^{55} + 1518 q^{56} - 1704 q^{57} - 1035 q^{58} - 107 q^{59} + 380 q^{60} - 857 q^{61} - 155 q^{62} - 2636 q^{63} - 215 q^{64} + 2600 q^{65} - 1785 q^{66} - 2689 q^{67} - 2639 q^{68} + 2544 q^{69} - 690 q^{70} + 1554 q^{71} - 420 q^{72} - 1968 q^{73} - 27 q^{74} - 425 q^{75} - 110 q^{76} - 1040 q^{77} + 455 q^{78} - 3182 q^{79} - 2065 q^{80} - 1576 q^{81} - 386 q^{82} + 317 q^{83} - 617 q^{84} + 1320 q^{85} + 347 q^{86} - 216 q^{87} - 4081 q^{88} + 3610 q^{89} + 630 q^{90} + 260 q^{91} - 4965 q^{92} - 527 q^{93} - 2942 q^{94} + 5 q^{95} + 1002 q^{96} - 3318 q^{97} + 1659 q^{98} + 5943 q^{99}+O(q^{100}) \) 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 −5.34747 −7.64905 20.5954 −5.00000 40.9031 −21.7958 −67.3535 31.5080 26.7373
1.2 −5.25776 3.13707 19.6441 −5.00000 −16.4940 −24.7609 −61.2219 −17.1588 26.2888
1.3 −5.15631 4.91805 18.5876 −5.00000 −25.3590 12.2185 −54.5927 −2.81278 25.7816
1.4 −5.02622 −1.71462 17.2629 −5.00000 8.61805 11.1771 −46.5576 −24.0601 25.1311
1.5 −4.54911 −3.80302 12.6944 −5.00000 17.3004 13.9869 −21.3554 −12.5370 22.7456
1.6 −4.33607 −8.60031 10.8015 −5.00000 37.2915 8.52424 −12.1475 46.9653 21.6803
1.7 −4.22774 8.50426 9.87377 −5.00000 −35.9538 15.5602 −7.92183 45.3225 21.1387
1.8 −4.00428 7.37620 8.03426 −5.00000 −29.5364 13.1790 −0.137173 27.4083 20.0214
1.9 −3.76849 3.72438 6.20150 −5.00000 −14.0353 −30.0034 6.77763 −13.1290 18.8424
1.10 −3.41025 −3.08447 3.62984 −5.00000 10.5188 −22.1095 14.9034 −17.4860 17.0513
1.11 −3.01342 −1.70930 1.08068 −5.00000 5.15084 2.97604 20.8508 −24.0783 15.0671
1.12 −2.63218 −9.87725 −1.07165 −5.00000 25.9987 1.27875 23.8782 70.5601 13.1609
1.13 −2.43322 0.799530 −2.07942 −5.00000 −1.94544 19.2630 24.5255 −26.3608 12.1661
1.14 −2.37712 4.38088 −2.34931 −5.00000 −10.4139 13.5532 24.6015 −7.80791 11.8856
1.15 −2.31939 8.67980 −2.62043 −5.00000 −20.1318 −21.6973 24.6329 48.3389 11.5969
1.16 −1.58814 −4.85275 −5.47781 −5.00000 7.70685 19.2971 21.4047 −3.45085 7.94071
1.17 −1.47350 −9.15185 −5.82879 −5.00000 13.4853 −16.2914 20.3768 56.7563 7.36752
1.18 −0.829109 −3.61767 −7.31258 −5.00000 2.99944 −4.39822 12.6958 −13.9125 4.14555
1.19 −0.566764 −0.796068 −7.67878 −5.00000 0.451182 −20.1270 8.88617 −26.3663 2.83382
1.20 −0.497250 4.81569 −7.75274 −5.00000 −2.39460 27.1189 7.83306 −3.80913 2.48625
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.d ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2015.4.a.d ✓ 40 1.a even 1 1 trivial