Properties

Label 8015.2.a.o
Level $8015$
Weight $2$
Character orbit 8015.a
Self dual yes
Analytic conductor $64.000$
Analytic rank $0$
Dimension $73$
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] = [8015,2,Mod(1,8015)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8015, 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("8015.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8015 = 5 \cdot 7 \cdot 229 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8015.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: \(64.0000972201\)
Analytic rank: \(0\)
Dimension: \(73\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 73 q + 7 q^{2} + 14 q^{3} + 95 q^{4} + 73 q^{5} - q^{6} + 73 q^{7} + 18 q^{8} + 111 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 73 q + 7 q^{2} + 14 q^{3} + 95 q^{4} + 73 q^{5} - q^{6} + 73 q^{7} + 18 q^{8} + 111 q^{9} + 7 q^{10} + 27 q^{11} + 21 q^{12} + 21 q^{13} + 7 q^{14} + 14 q^{15} + 135 q^{16} + 23 q^{17} + 41 q^{18} + 26 q^{19} + 95 q^{20} + 14 q^{21} + 48 q^{22} + 16 q^{23} - q^{24} + 73 q^{25} + 7 q^{26} + 44 q^{27} + 95 q^{28} + 66 q^{29} - q^{30} + 23 q^{31} + 3 q^{32} + 77 q^{33} + 29 q^{34} + 73 q^{35} + 142 q^{36} + 66 q^{37} - 12 q^{38} + 53 q^{39} + 18 q^{40} + 50 q^{41} - q^{42} + 43 q^{43} + 37 q^{44} + 111 q^{45} + 65 q^{46} + 28 q^{47} - 20 q^{48} + 73 q^{49} + 7 q^{50} + 71 q^{51} + 29 q^{52} - 7 q^{53} - 16 q^{54} + 27 q^{55} + 18 q^{56} + 33 q^{57} + 48 q^{58} + 16 q^{59} + 21 q^{60} + 42 q^{61} - 3 q^{62} + 111 q^{63} + 216 q^{64} + 21 q^{65} - 53 q^{66} + 48 q^{67} + 13 q^{68} + 73 q^{69} + 7 q^{70} + 68 q^{71} + 18 q^{72} + 65 q^{73} + 4 q^{74} + 14 q^{75} + 37 q^{76} + 27 q^{77} + 60 q^{78} + 116 q^{79} + 135 q^{80} + 177 q^{81} + 20 q^{82} + 40 q^{83} + 21 q^{84} + 23 q^{85} + 35 q^{86} - 14 q^{87} + 47 q^{88} + 59 q^{89} + 41 q^{90} + 21 q^{91} + 3 q^{92} + 37 q^{93} - 11 q^{94} + 26 q^{95} - 23 q^{96} + 70 q^{97} + 7 q^{98} + 49 q^{99}+O(q^{100}) \) 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 −2.77376 0.883907 5.69373 1.00000 −2.45174 1.00000 −10.2455 −2.21871 −2.77376
1.2 −2.76326 3.03127 5.63562 1.00000 −8.37621 1.00000 −10.0462 6.18862 −2.76326
1.3 −2.72942 −2.22505 5.44974 1.00000 6.07311 1.00000 −9.41580 1.95086 −2.72942
1.4 −2.69749 −3.25136 5.27643 1.00000 8.77051 1.00000 −8.83812 7.57137 −2.69749
1.5 −2.68177 −0.319561 5.19187 1.00000 0.856987 1.00000 −8.55985 −2.89788 −2.68177
1.6 −2.45360 1.56371 4.02015 1.00000 −3.83672 1.00000 −4.95665 −0.554809 −2.45360
1.7 −2.45327 3.30020 4.01854 1.00000 −8.09629 1.00000 −4.95203 7.89132 −2.45327
1.8 −2.45229 −0.606235 4.01370 1.00000 1.48666 1.00000 −4.93818 −2.63248 −2.45229
1.9 −2.41178 −1.36759 3.81666 1.00000 3.29833 1.00000 −4.38139 −1.12969 −2.41178
1.10 −2.29631 0.975796 3.27305 1.00000 −2.24073 1.00000 −2.92333 −2.04782 −2.29631
1.11 −2.19612 2.45439 2.82293 1.00000 −5.39013 1.00000 −1.80725 3.02403 −2.19612
1.12 −2.19191 −0.935723 2.80448 1.00000 2.05102 1.00000 −1.76336 −2.12442 −2.19191
1.13 −2.08852 −0.140980 2.36194 1.00000 0.294440 1.00000 −0.755913 −2.98012 −2.08852
1.14 −2.01224 2.54989 2.04911 1.00000 −5.13098 1.00000 −0.0988175 3.50192 −2.01224
1.15 −1.86693 −0.178602 1.48541 1.00000 0.333437 1.00000 0.960700 −2.96810 −1.86693
1.16 −1.84781 −2.80168 1.41441 1.00000 5.17697 1.00000 1.08207 4.84939 −1.84781
1.17 −1.63497 2.59836 0.673111 1.00000 −4.24822 1.00000 2.16942 3.75146 −1.63497
1.18 −1.53607 −1.95086 0.359523 1.00000 2.99667 1.00000 2.51989 0.805862 −1.53607
1.19 −1.50054 2.96680 0.251626 1.00000 −4.45181 1.00000 2.62351 5.80192 −1.50054
1.20 −1.40112 2.16021 −0.0368596 1.00000 −3.02672 1.00000 2.85389 1.66651 −1.40112
See all 73 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.73
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(7\) \(-1\)
\(229\) \(-1\)

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 8015.2.a.o 73
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8015.2.a.o 73 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8015))\):

\( T_{2}^{73} - 7 T_{2}^{72} - 96 T_{2}^{71} + 771 T_{2}^{70} + 4193 T_{2}^{69} - 40402 T_{2}^{68} - 107553 T_{2}^{67} + 1340585 T_{2}^{66} + 1686113 T_{2}^{65} - 31625519 T_{2}^{64} - 12629340 T_{2}^{63} + \cdots + 31509048 \) Copy content Toggle raw display
\( T_{3}^{73} - 14 T_{3}^{72} - 67 T_{3}^{71} + 1810 T_{3}^{70} - 982 T_{3}^{69} - 108249 T_{3}^{68} + 294790 T_{3}^{67} + 3940592 T_{3}^{66} - 17099479 T_{3}^{65} - 96004886 T_{3}^{64} + 588675495 T_{3}^{63} + \cdots - 451619323904 \) Copy content Toggle raw display