Properties

Label 4015.2.a.i
Level $4015$
Weight $2$
Character orbit 4015.a
Self dual yes
Analytic conductor $32.060$
Analytic rank $0$
Dimension $38$
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] = [4015,2,Mod(1,4015)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4015, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4015.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4015 = 5 \cdot 11 \cdot 73 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4015.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: \(32.0599364115\)
Analytic rank: \(0\)
Dimension: \(38\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 38 q + 4 q^{2} + 5 q^{3} + 50 q^{4} + 38 q^{5} + 11 q^{6} + 15 q^{8} + 63 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 38 q + 4 q^{2} + 5 q^{3} + 50 q^{4} + 38 q^{5} + 11 q^{6} + 15 q^{8} + 63 q^{9} + 4 q^{10} - 38 q^{11} + 12 q^{12} - q^{13} + 23 q^{14} + 5 q^{15} + 74 q^{16} + 26 q^{17} + 16 q^{18} - 10 q^{19} + 50 q^{20} + 21 q^{21} - 4 q^{22} + 10 q^{23} + 41 q^{24} + 38 q^{25} + 25 q^{26} + 5 q^{27} + 2 q^{28} + 28 q^{29} + 11 q^{30} + 24 q^{31} + 39 q^{32} - 5 q^{33} + 38 q^{34} + 111 q^{36} + 12 q^{37} + 19 q^{38} - 18 q^{39} + 15 q^{40} + 62 q^{41} - 17 q^{42} - 32 q^{43} - 50 q^{44} + 63 q^{45} - 9 q^{46} + 31 q^{47} + 53 q^{48} + 88 q^{49} + 4 q^{50} - 3 q^{51} - 21 q^{52} + 30 q^{53} + 49 q^{54} - 38 q^{55} + 32 q^{56} + 49 q^{57} + 12 q^{58} + 31 q^{59} + 12 q^{60} + 25 q^{61} + 12 q^{62} + 15 q^{63} + 137 q^{64} - q^{65} - 11 q^{66} + 20 q^{67} + 75 q^{68} + 92 q^{69} + 23 q^{70} + 32 q^{71} + 6 q^{72} - 38 q^{73} + 55 q^{74} + 5 q^{75} - 57 q^{76} - 17 q^{78} - 2 q^{79} + 74 q^{80} + 118 q^{81} + 14 q^{82} + 4 q^{83} + 22 q^{84} + 26 q^{85} + 5 q^{86} + 24 q^{87} - 15 q^{88} + 143 q^{89} + 16 q^{90} + 66 q^{91} + 29 q^{92} - 8 q^{93} - 7 q^{94} - 10 q^{95} + 59 q^{96} + 41 q^{97} - 10 q^{98} - 63 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −2.76981 1.77605 5.67182 1.00000 −4.91932 −0.953960 −10.1702 0.154367 −2.76981
1.2 −2.64676 −3.26170 5.00536 1.00000 8.63297 0.334745 −7.95449 7.63872 −2.64676
1.3 −2.64612 −1.80895 5.00197 1.00000 4.78670 −4.64214 −7.94360 0.272285 −2.64612
1.4 −2.40266 2.98645 3.77280 1.00000 −7.17544 4.58733 −4.25943 5.91889 −2.40266
1.5 −2.35033 0.502249 3.52405 1.00000 −1.18045 0.984719 −3.58202 −2.74775 −2.35033
1.6 −2.26509 0.846807 3.13061 1.00000 −1.91809 −3.17432 −2.56093 −2.28292 −2.26509
1.7 −1.83787 −3.02191 1.37778 1.00000 5.55388 4.06050 1.14356 6.13192 −1.83787
1.8 −1.73652 −1.80706 1.01551 1.00000 3.13799 −4.26912 1.70959 0.265453 −1.73652
1.9 −1.69384 0.407884 0.869088 1.00000 −0.690890 2.45284 1.91558 −2.83363 −1.69384
1.10 −1.61749 2.78278 0.616262 1.00000 −4.50111 −5.17171 2.23818 4.74386 −1.61749
1.11 −1.50911 −0.619433 0.277405 1.00000 0.934791 1.62740 2.59958 −2.61630 −1.50911
1.12 −1.49358 2.85677 0.230796 1.00000 −4.26684 0.0973246 2.64246 5.16116 −1.49358
1.13 −1.16964 −3.05105 −0.631953 1.00000 3.56861 −2.34807 3.07843 6.30888 −1.16964
1.14 −1.00909 −1.30445 −0.981746 1.00000 1.31630 1.28192 3.00884 −1.29840 −1.00909
1.15 −0.454453 2.93224 −1.79347 1.00000 −1.33257 4.01407 1.72395 5.59806 −0.454453
1.16 −0.445505 −0.895604 −1.80153 1.00000 0.398996 −1.53298 1.69360 −2.19789 −0.445505
1.17 −0.364998 1.30973 −1.86678 1.00000 −0.478050 −1.97253 1.41137 −1.28460 −0.364998
1.18 −0.0798923 0.0600931 −1.99362 1.00000 −0.00480098 −4.79075 0.319059 −2.99639 −0.0798923
1.19 0.0204713 2.32254 −1.99958 1.00000 0.0475455 2.78561 −0.0818767 2.39419 0.0204713
1.20 0.0631532 0.00106429 −1.99601 1.00000 6.72133e−5 0 3.26238 −0.252361 −3.00000 0.0631532
See all 38 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.38
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(11\) \(1\)
\(73\) \(1\)

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 4015.2.a.i 38
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4015.2.a.i 38 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{38} - 4 T_{2}^{37} - 55 T_{2}^{36} + 231 T_{2}^{35} + 1363 T_{2}^{34} - 6076 T_{2}^{33} - 20091 T_{2}^{32} + 96455 T_{2}^{31} + 195624 T_{2}^{30} - 1032379 T_{2}^{29} - 1318302 T_{2}^{28} + 7883197 T_{2}^{27} + \cdots + 192 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4015))\). Copy content Toggle raw display