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Properties

Label 87.2.g.a
Level $87$
Weight $2$
Character orbit 87.g
Analytic conductor $0.695$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [87,2,Mod(7,87)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(87, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("87.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 87 = 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 87.g (of order \(7\), degree \(6\), minimal)

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: no
Analytic conductor: \(0.694698497585\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{7})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 6 x^{17} + 18 x^{16} - 37 x^{15} + 71 x^{14} - 83 x^{13} + 225 x^{12} - 237 x^{11} + 485 x^{10} + \cdots + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{9} - \beta_{3}) q^{2} - \beta_{10} q^{3} + ( - \beta_{16} + 2 \beta_{12} + \cdots + 2 \beta_1) q^{4}+ \cdots + ( - \beta_{12} - \beta_{11} + \beta_{10} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{9} - \beta_{3}) q^{2} - \beta_{10} q^{3} + ( - \beta_{16} + 2 \beta_{12} + \cdots + 2 \beta_1) q^{4}+ \cdots + ( - 2 \beta_{12} + 2 \beta_{10} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 4 q^{2} - 3 q^{3} - 6 q^{4} - q^{5} - 4 q^{6} - 4 q^{7} - 15 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 4 q^{2} - 3 q^{3} - 6 q^{4} - q^{5} - 4 q^{6} - 4 q^{7} - 15 q^{8} - 3 q^{9} - 14 q^{10} + 26 q^{11} + 22 q^{12} + 9 q^{13} - 10 q^{14} - q^{15} - 14 q^{16} + 4 q^{17} - 4 q^{18} - 10 q^{19} - q^{20} - 4 q^{21} - 8 q^{22} - 8 q^{23} - 8 q^{24} + 16 q^{25} + 5 q^{26} - 3 q^{27} + 80 q^{28} + 8 q^{29} - 12 q^{31} + 9 q^{32} - 16 q^{33} - 22 q^{34} + 9 q^{35} - 6 q^{36} - 16 q^{37} - 32 q^{38} + 2 q^{39} + 33 q^{40} + 24 q^{41} - 3 q^{42} - 31 q^{43} - 52 q^{44} + 6 q^{45} - 44 q^{46} + 5 q^{47} - 47 q^{49} - 7 q^{50} + 11 q^{51} + 80 q^{52} + 5 q^{53} + 3 q^{54} - 17 q^{55} + 45 q^{56} + 18 q^{57} + 54 q^{58} - 32 q^{59} + 27 q^{60} - 28 q^{61} + 69 q^{62} + 3 q^{63} - 75 q^{64} + 22 q^{65} + 34 q^{66} + 6 q^{67} + 38 q^{68} + 20 q^{69} - 12 q^{70} + 46 q^{71} - 8 q^{72} - q^{73} - 35 q^{74} + 2 q^{75} - 45 q^{76} - 36 q^{77} - 51 q^{78} - 15 q^{79} - 86 q^{80} - 3 q^{81} + 47 q^{82} - 16 q^{83} - 25 q^{84} + 19 q^{85} + 116 q^{86} + 43 q^{87} + 54 q^{88} - 72 q^{89} - 14 q^{90} - 47 q^{91} - 121 q^{92} - 12 q^{93} - 22 q^{94} - 72 q^{95} - 12 q^{96} + 43 q^{97} + 31 q^{98} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 6 x^{17} + 18 x^{16} - 37 x^{15} + 71 x^{14} - 83 x^{13} + 225 x^{12} - 237 x^{11} + 485 x^{10} + \cdots + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 27\!\cdots\!79 \nu^{17} + \cdots - 31\!\cdots\!20 ) / 14\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 35\!\cdots\!37 \nu^{17} + \cdots + 67\!\cdots\!32 ) / 14\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 62\!\cdots\!91 \nu^{17} + \cdots - 33\!\cdots\!84 ) / 14\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 80\!\cdots\!03 \nu^{17} + \cdots + 26\!\cdots\!20 ) / 14\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 85\!\cdots\!61 \nu^{17} + \cdots + 36\!\cdots\!20 ) / 14\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 21\!\cdots\!01 \nu^{17} + \cdots - 39\!\cdots\!56 ) / 28\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 15\!\cdots\!25 \nu^{17} + \cdots + 20\!\cdots\!56 ) / 14\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 31\!\cdots\!54 \nu^{17} + \cdots - 69\!\cdots\!68 ) / 14\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 81\!\cdots\!35 \nu^{17} + \cdots - 44\!\cdots\!32 ) / 28\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 52\!\cdots\!81 \nu^{17} + \cdots - 17\!\cdots\!96 ) / 14\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 56\!\cdots\!55 \nu^{17} + \cdots - 36\!\cdots\!08 ) / 14\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 86\!\cdots\!99 \nu^{17} + \cdots - 47\!\cdots\!08 ) / 14\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 12\!\cdots\!21 \nu^{17} + \cdots + 82\!\cdots\!88 ) / 14\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 14\!\cdots\!27 \nu^{17} + \cdots + 59\!\cdots\!32 ) / 14\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 17\!\cdots\!45 \nu^{17} + \cdots - 11\!\cdots\!32 ) / 14\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 39\!\cdots\!75 \nu^{17} + \cdots + 74\!\cdots\!88 ) / 28\!\cdots\!12 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{17} + \beta_{16} - \beta_{12} + 2\beta_{11} - \beta_{7} + \beta_{6} - \beta_{5} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2 \beta_{17} + \beta_{16} + \beta_{15} - 2 \beta_{14} + 2 \beta_{13} - \beta_{12} + 2 \beta_{10} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3 \beta_{17} + \beta_{16} + 8 \beta_{15} - 8 \beta_{14} + 11 \beta_{13} - \beta_{12} + 8 \beta_{10} + \cdots + 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4 \beta_{17} + 21 \beta_{15} - 13 \beta_{14} + 34 \beta_{13} + 2 \beta_{12} + 2 \beta_{11} + 9 \beta_{10} + \cdots + 19 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 18 \beta_{16} + 41 \beta_{15} - 18 \beta_{14} + 87 \beta_{13} + 24 \beta_{12} + 4 \beta_{11} + \cdots + 41 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 63 \beta_{17} - 143 \beta_{16} + 63 \beta_{15} + 143 \beta_{13} + 173 \beta_{12} - 30 \beta_{10} + \cdots + 106 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 457 \beta_{17} - 648 \beta_{16} + 226 \beta_{14} + 788 \beta_{12} - 54 \beta_{11} - 266 \beta_{10} + \cdots + 403 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 2007 \beta_{17} - 2007 \beta_{16} - 754 \beta_{15} + 1504 \beta_{14} - 1504 \beta_{13} + 2211 \beta_{12} + \cdots + 1111 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 6360 \beta_{17} - 4816 \beta_{16} - 4816 \beta_{15} + 6360 \beta_{14} - 8884 \beta_{13} + 4816 \beta_{12} + \cdots + 1012 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 15550 \beta_{17} - 8236 \beta_{16} - 20051 \beta_{15} + 20051 \beta_{14} - 35601 \beta_{13} + \cdots - 6409 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 26842 \beta_{17} - 63696 \beta_{15} + 49755 \beta_{14} - 113451 \beta_{13} - 10364 \beta_{12} + \cdots - 41106 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 86625 \beta_{16} - 159543 \beta_{15} + 86625 \beta_{14} - 288853 \beta_{13} - 125125 \beta_{12} + \cdots - 159543 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 279122 \beta_{17} + 509987 \beta_{16} - 279122 \beta_{15} - 509987 \beta_{13} - 633169 \beta_{12} + \cdots - 521858 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 1630935 \beta_{17} + 2049943 \beta_{16} - 895790 \beta_{14} - 2371999 \beta_{12} + 182724 \beta_{11} + \cdots - 1448211 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 6534172 \beta_{17} + 6534172 \beta_{16} + 2871755 \beta_{15} - 5209837 \beta_{14} + 5209837 \beta_{13} + \cdots - 2972357 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 20832904 \beta_{17} + 16642785 \beta_{16} + 16642785 \beta_{15} - 20832904 \beta_{14} + 30023648 \beta_{13} + \cdots - 1971815 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/87\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(59\)
\(\chi(n)\) \(\beta_{12}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
−1.38228 + 0.665671i
1.03105 − 0.496527i
2.87569 − 1.38486i
1.10857 + 1.39010i
0.102196 + 0.128149i
−1.05678 − 1.32516i
−1.38228 − 0.665671i
1.03105 + 0.496527i
2.87569 + 1.38486i
1.10857 − 1.39010i
0.102196 − 0.128149i
−1.05678 + 1.32516i
0.491931 + 2.15529i
0.183119 + 0.802295i
−0.353498 − 1.54877i
0.491931 − 2.15529i
0.183119 − 0.802295i
−0.353498 + 1.54877i
−0.563916 − 2.47068i −0.900969 − 0.433884i −3.98431 + 1.91874i −0.242440 − 1.06220i −0.563916 + 2.47068i −1.55919 − 0.750867i 3.82730 + 4.79928i 0.623490 + 0.781831i −2.48764 + 1.19798i
7.2 0.0321271 + 0.140758i −0.900969 − 0.433884i 1.78316 − 0.858723i −0.345850 − 1.51527i 0.0321271 − 0.140758i 1.37625 + 0.662766i 0.358196 + 0.449164i 0.623490 + 0.781831i 0.202175 − 0.0973624i
7.3 0.487716 + 2.13682i −0.900969 − 0.433884i −2.52621 + 1.21656i 0.533332 + 2.33668i 0.487716 − 2.13682i −1.21803 − 0.586571i −1.09855 − 1.37753i 0.623490 + 0.781831i −4.73296 + 2.27927i
16.1 −2.50290 − 1.20533i 0.623490 − 0.781831i 3.56470 + 4.46999i 2.28635 + 1.10105i −2.50290 + 1.20533i 0.527912 − 0.661981i −2.29793 − 10.0679i −0.222521 − 0.974928i −4.39538 − 5.51163i
16.2 −1.04865 − 0.505001i 0.623490 − 0.781831i −0.402348 − 0.504528i −2.80239 − 1.34956i −1.04865 + 0.505001i 1.08876 − 1.36527i 0.685121 + 3.00171i −0.222521 − 0.974928i 2.25719 + 2.83042i
16.3 0.626118 + 0.301523i 0.623490 − 0.781831i −0.945872 − 1.18609i 1.81798 + 0.875492i 0.626118 − 0.301523i −1.49319 + 1.87240i −0.543873 − 2.38286i −0.222521 − 0.974928i 0.874288 + 1.09632i
25.1 −0.563916 + 2.47068i −0.900969 + 0.433884i −3.98431 − 1.91874i −0.242440 + 1.06220i −0.563916 − 2.47068i −1.55919 + 0.750867i 3.82730 − 4.79928i 0.623490 − 0.781831i −2.48764 − 1.19798i
25.2 0.0321271 − 0.140758i −0.900969 + 0.433884i 1.78316 + 0.858723i −0.345850 + 1.51527i 0.0321271 + 0.140758i 1.37625 − 0.662766i 0.358196 − 0.449164i 0.623490 − 0.781831i 0.202175 + 0.0973624i
25.3 0.487716 − 2.13682i −0.900969 + 0.433884i −2.52621 − 1.21656i 0.533332 − 2.33668i 0.487716 + 2.13682i −1.21803 + 0.586571i −1.09855 + 1.37753i 0.623490 − 0.781831i −4.73296 − 2.27927i
49.1 −2.50290 + 1.20533i 0.623490 + 0.781831i 3.56470 − 4.46999i 2.28635 − 1.10105i −2.50290 − 1.20533i 0.527912 + 0.661981i −2.29793 + 10.0679i −0.222521 + 0.974928i −4.39538 + 5.51163i
49.2 −1.04865 + 0.505001i 0.623490 + 0.781831i −0.402348 + 0.504528i −2.80239 + 1.34956i −1.04865 − 0.505001i 1.08876 + 1.36527i 0.685121 − 3.00171i −0.222521 + 0.974928i 2.25719 − 2.83042i
49.3 0.626118 − 0.301523i 0.623490 + 0.781831i −0.945872 + 1.18609i 1.81798 − 0.875492i 0.626118 + 0.301523i −1.49319 − 1.87240i −0.543873 + 2.38286i −0.222521 + 0.974928i 0.874288 − 1.09632i
52.1 −0.754870 + 0.946578i −0.222521 + 0.974928i 0.118862 + 0.520769i 1.12131 − 1.40607i −0.754870 − 0.946578i −0.951706 + 4.16970i −2.76431 − 1.33122i −0.900969 − 0.433884i 0.484517 + 2.12281i
52.2 0.110403 − 0.138441i −0.222521 + 0.974928i 0.438065 + 1.91929i −2.15193 + 2.69844i 0.110403 + 0.138441i 0.844514 − 3.70006i 0.633146 + 0.304907i −0.900969 − 0.433884i 0.135995 + 0.595831i
52.3 1.61397 − 2.02385i −0.222521 + 0.974928i −1.04604 − 4.58301i −0.716354 + 0.898279i 1.61397 + 2.02385i −0.615328 + 2.69593i −6.29911 − 3.03349i −0.900969 − 0.433884i 0.661812 + 2.89959i
82.1 −0.754870 − 0.946578i −0.222521 − 0.974928i 0.118862 − 0.520769i 1.12131 + 1.40607i −0.754870 + 0.946578i −0.951706 − 4.16970i −2.76431 + 1.33122i −0.900969 + 0.433884i 0.484517 − 2.12281i
82.2 0.110403 + 0.138441i −0.222521 − 0.974928i 0.438065 − 1.91929i −2.15193 − 2.69844i 0.110403 − 0.138441i 0.844514 + 3.70006i 0.633146 − 0.304907i −0.900969 + 0.433884i 0.135995 − 0.595831i
82.3 1.61397 + 2.02385i −0.222521 − 0.974928i −1.04604 + 4.58301i −0.716354 − 0.898279i 1.61397 − 2.02385i −0.615328 − 2.69593i −6.29911 + 3.03349i −0.900969 + 0.433884i 0.661812 − 2.89959i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.d even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 87.2.g.a ✓ 18
3.b odd 2 1 261.2.k.c 18
29.d even 7 1 inner 87.2.g.a ✓ 18
29.d even 7 1 2523.2.a.r 9
29.e even 14 1 2523.2.a.o 9
87.h odd 14 1 7569.2.a.bm 9
87.j odd 14 1 261.2.k.c 18
87.j odd 14 1 7569.2.a.bj 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
87.2.g.a ✓ 18 1.a even 1 1 trivial
87.2.g.a ✓ 18 29.d even 7 1 inner
261.2.k.c 18 3.b odd 2 1
261.2.k.c 18 87.j odd 14 1
2523.2.a.o 9 29.e even 14 1
2523.2.a.r 9 29.d even 7 1
7569.2.a.bj 9 87.j odd 14 1
7569.2.a.bm 9 87.h odd 14 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{18} + 4 T_{2}^{17} + 14 T_{2}^{16} + 45 T_{2}^{15} + 126 T_{2}^{14} + 309 T_{2}^{13} + 828 T_{2}^{12} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(87, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + 4 T^{17} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{6} + T^{5} + T^{4} + \cdots + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{18} + T^{17} + \cdots + 212521 \) Copy content Toggle raw display
$7$ \( T^{18} + 4 T^{17} + \cdots + 322624 \) Copy content Toggle raw display
$11$ \( T^{18} - 26 T^{17} + \cdots + 1236544 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 2180609809 \) Copy content Toggle raw display
$17$ \( (T^{9} - 2 T^{8} + \cdots - 86143)^{2} \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 589467701824 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 880427584 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 14507145975869 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 177848896 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 18850466209 \) Copy content Toggle raw display
$41$ \( (T^{9} - 12 T^{8} + \cdots - 3560291)^{2} \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 271326784 \) Copy content Toggle raw display
$47$ \( T^{18} - 5 T^{17} + \cdots + 817216 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 5317548372361 \) Copy content Toggle raw display
$59$ \( (T^{9} + 16 T^{8} + \cdots + 897224)^{2} \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 522207124321 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 2088855616 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 80\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 61148882089 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 8744494144 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 645862966336 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 21\!\cdots\!41 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 40807578862561 \) Copy content Toggle raw display
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