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Properties

Label 87.2.g.b
Level $87$
Weight $2$
Character orbit 87.g
Analytic conductor $0.695$
Analytic rank $0$
Dimension $18$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [87,2,Mod(7,87)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://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);
 
Copy content sage: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")
 
Level: \( N \) \(=\) \( 87 = 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 87.g (of order \(7\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [18,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content 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: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 5 x^{17} + 15 x^{16} - 32 x^{15} + 66 x^{14} - 115 x^{13} + 272 x^{12} - 387 x^{11} + 762 x^{10} + \cdots + 49 \) 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

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + \beta_{14} q^{3} + ( - \beta_{16} - \beta_{10}) q^{4} + (\beta_{15} - \beta_{14} + \beta_{6}) q^{5} + \beta_{2} q^{6} + (\beta_{17} - \beta_{15} + \cdots + \beta_{7}) q^{7} + ( - \beta_{17} + \beta_{15} - \beta_{14} + \cdots + 1) q^{8}+ \cdots + (\beta_{16} - \beta_{15} + \cdots - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 2 q^{2} + 3 q^{3} - 6 q^{4} - 7 q^{5} + 2 q^{6} - 4 q^{7} - 3 q^{8} - 3 q^{9} + 6 q^{10} - 6 q^{11} - 22 q^{12} - 11 q^{13} - 2 q^{14} + 7 q^{15} + 18 q^{16} - 32 q^{17} - 2 q^{18} + 2 q^{19} + 51 q^{20}+ \cdots - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 5 x^{17} + 15 x^{16} - 32 x^{15} + 66 x^{14} - 115 x^{13} + 272 x^{12} - 387 x^{11} + 762 x^{10} + \cdots + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 54\!\cdots\!15 \nu^{17} + \cdots + 68\!\cdots\!88 ) / 40\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 81\!\cdots\!66 \nu^{17} + \cdots + 32\!\cdots\!77 ) / 40\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 91\!\cdots\!83 \nu^{17} + \cdots - 21\!\cdots\!89 ) / 40\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 12\!\cdots\!20 \nu^{17} + \cdots - 25\!\cdots\!15 ) / 40\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 14\!\cdots\!12 \nu^{17} + \cdots - 42\!\cdots\!36 ) / 40\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 25\!\cdots\!78 \nu^{17} + \cdots - 32\!\cdots\!30 ) / 40\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 42\!\cdots\!36 \nu^{17} + \cdots + 12\!\cdots\!08 ) / 40\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 43\!\cdots\!61 \nu^{17} + \cdots - 38\!\cdots\!27 ) / 40\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 52\!\cdots\!35 \nu^{17} + \cdots - 82\!\cdots\!85 ) / 40\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 66\!\cdots\!43 \nu^{17} + \cdots - 64\!\cdots\!38 ) / 40\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 67\!\cdots\!73 \nu^{17} + \cdots - 93\!\cdots\!10 ) / 40\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 69\!\cdots\!59 \nu^{17} + \cdots + 14\!\cdots\!56 ) / 40\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 80\!\cdots\!81 \nu^{17} + \cdots + 27\!\cdots\!53 ) / 40\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 16\!\cdots\!34 \nu^{17} + \cdots + 14\!\cdots\!30 ) / 40\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 17\!\cdots\!42 \nu^{17} + \cdots + 25\!\cdots\!99 ) / 40\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 20\!\cdots\!28 \nu^{17} + \cdots - 28\!\cdots\!10 ) / 40\!\cdots\!51 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + 3\beta_{6} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{17} - \beta_{13} + \beta_{12} + \beta_{8} - 4\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 6 \beta_{17} + \beta_{15} - \beta_{14} - 6 \beta_{13} + \beta_{11} - \beta_{10} - 13 \beta_{9} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 8 \beta_{17} - \beta_{16} + 8 \beta_{15} - 12 \beta_{14} - 9 \beta_{13} - 8 \beta_{12} + 9 \beta_{11} + \cdots + 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 9 \beta_{17} - 12 \beta_{16} + 36 \beta_{15} - 66 \beta_{14} - 12 \beta_{13} - 10 \beta_{12} + \cdots - 12 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 69 \beta_{16} + 69 \beta_{15} - 48 \beta_{14} - 14 \beta_{13} - 15 \beta_{12} + 57 \beta_{11} + \cdots - 51 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 70 \beta_{17} - 228 \beta_{16} + 109 \beta_{15} - 53 \beta_{14} - 53 \beta_{12} + 70 \beta_{11} + \cdots - 82 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 403 \beta_{17} - 403 \beta_{16} + 138 \beta_{15} + 138 \beta_{13} - 267 \beta_{12} - 267 \beta_{10} + \cdots - 157 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1511 \beta_{17} - 526 \beta_{16} + 507 \beta_{14} + 887 \beta_{13} - 1179 \beta_{12} - 526 \beta_{11} + \cdots - 507 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 3635 \beta_{17} - 1182 \beta_{15} + 3368 \beta_{14} + 3635 \beta_{13} - 2864 \beta_{11} + 1957 \beta_{10} + \cdots - 1957 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 6839 \beta_{17} + 3897 \beta_{16} - 6839 \beta_{15} + 15110 \beta_{14} + 10324 \beta_{13} + 6180 \beta_{12} + \cdots - 6180 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 9432 \beta_{17} + 20453 \beta_{16} - 26020 \beta_{15} + 36895 \beta_{14} + 20453 \beta_{13} + \cdots + 15769 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 71907 \beta_{16} - 71907 \beta_{15} + 67760 \beta_{14} + 28631 \beta_{13} + 33966 \beta_{12} + \cdots + 34562 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 72329 \beta_{17} + 186043 \beta_{16} - 146526 \beta_{15} + 91955 \beta_{14} + 91955 \beta_{12} + \cdots + 97230 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 377353 \beta_{17} + 377353 \beta_{16} - 209123 \beta_{15} - 209123 \beta_{13} + 243349 \beta_{12} + \cdots + 260120 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 1330846 \beta_{17} + 541637 \beta_{16} - 701975 \beta_{14} - 1051632 \beta_{13} + 462397 \beta_{12} + \cdots + 701975 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(31\) \(59\)
\(\chi(n)\) \(-\beta_{10}\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
−1.61697 + 0.778692i
0.824998 − 0.397298i
2.41546 − 1.16323i
1.29273 + 1.62103i
0.719749 + 0.902536i
−1.23500 − 1.54863i
−1.61697 − 0.778692i
0.824998 + 0.397298i
2.41546 + 1.16323i
1.29273 − 1.62103i
0.719749 − 0.902536i
−1.23500 + 1.54863i
0.488787 + 2.14152i
0.0185039 + 0.0810709i
−0.408260 − 1.78870i
0.488787 − 2.14152i
0.0185039 − 0.0810709i
−0.408260 + 1.78870i
−0.399359 − 1.74971i 0.900969 + 0.433884i −1.10004 + 0.529753i −0.299361 − 1.31158i 0.399359 − 1.74971i 1.00367 + 0.483344i −0.871733 − 1.09312i 0.623490 + 0.781831i −2.17533 + 1.04759i
7.2 0.203758 + 0.892721i 0.900969 + 0.433884i 1.04650 − 0.503970i −0.0634200 − 0.277861i −0.203758 + 0.892721i −4.54015 − 2.18642i 1.80497 + 2.26336i 0.623490 + 0.781831i 0.235130 − 0.113233i
7.3 0.596570 + 2.61374i 0.900969 + 0.433884i −4.67383 + 2.25080i −0.692177 − 3.03263i −0.596570 + 2.61374i 2.13550 + 1.02840i −5.32817 − 6.68131i 0.623490 + 0.781831i 7.51358 − 3.61835i
16.1 −1.86804 − 0.899602i −0.623490 + 0.781831i 1.43332 + 1.79733i −1.70773 − 0.822397i 1.86804 − 0.899602i −2.93833 + 3.68455i −0.137887 − 0.604122i −0.222521 − 0.974928i 2.45027 + 3.07255i
16.2 −1.04007 − 0.500870i −0.623490 + 0.781831i −0.416111 − 0.521786i 3.10796 + 1.49671i 1.04007 − 0.500870i 2.81664 − 3.53195i 0.685187 + 3.00200i −0.222521 − 0.974928i −2.48283 − 3.11337i
16.3 1.78462 + 0.859427i −0.623490 + 0.781831i 1.19927 + 1.50384i −1.09830 − 0.528911i −1.78462 + 0.859427i 0.245180 − 0.307445i −0.0337262 − 0.147764i −0.222521 − 0.974928i −1.50548 − 1.88781i
25.1 −0.399359 + 1.74971i 0.900969 − 0.433884i −1.10004 − 0.529753i −0.299361 + 1.31158i 0.399359 + 1.74971i 1.00367 − 0.483344i −0.871733 + 1.09312i 0.623490 − 0.781831i −2.17533 − 1.04759i
25.2 0.203758 − 0.892721i 0.900969 − 0.433884i 1.04650 + 0.503970i −0.0634200 + 0.277861i −0.203758 − 0.892721i −4.54015 + 2.18642i 1.80497 − 2.26336i 0.623490 − 0.781831i 0.235130 + 0.113233i
25.3 0.596570 − 2.61374i 0.900969 − 0.433884i −4.67383 − 2.25080i −0.692177 + 3.03263i −0.596570 − 2.61374i 2.13550 − 1.02840i −5.32817 + 6.68131i 0.623490 − 0.781831i 7.51358 + 3.61835i
49.1 −1.86804 + 0.899602i −0.623490 − 0.781831i 1.43332 − 1.79733i −1.70773 + 0.822397i 1.86804 + 0.899602i −2.93833 − 3.68455i −0.137887 + 0.604122i −0.222521 + 0.974928i 2.45027 − 3.07255i
49.2 −1.04007 + 0.500870i −0.623490 − 0.781831i −0.416111 + 0.521786i 3.10796 − 1.49671i 1.04007 + 0.500870i 2.81664 + 3.53195i 0.685187 − 3.00200i −0.222521 + 0.974928i −2.48283 + 3.11337i
49.3 1.78462 − 0.859427i −0.623490 − 0.781831i 1.19927 − 1.50384i −1.09830 + 0.528911i −1.78462 − 0.859427i 0.245180 + 0.307445i −0.0337262 + 0.147764i −0.222521 + 0.974928i −1.50548 + 1.88781i
52.1 −1.36955 + 1.71736i 0.222521 − 0.974928i −0.628623 − 2.75418i −2.54240 + 3.18806i 1.36955 + 1.71736i −0.811607 + 3.55588i 1.63274 + 0.786286i −0.900969 − 0.433884i −1.99312 − 8.73243i
52.2 −0.0518468 + 0.0650138i 0.222521 − 0.974928i 0.443503 + 1.94311i 1.20357 − 1.50923i 0.0518468 + 0.0650138i −0.0765305 + 0.335302i −0.299165 − 0.144070i −0.900969 − 0.433884i 0.0357195 + 0.156498i
52.3 1.14392 − 1.43443i 0.222521 − 0.974928i −0.303995 − 1.33189i −1.40816 + 1.76577i −1.14392 − 1.43443i 0.165617 − 0.725615i 1.04778 + 0.504584i −0.900969 − 0.433884i 0.922058 + 4.03980i
82.1 −1.36955 − 1.71736i 0.222521 + 0.974928i −0.628623 + 2.75418i −2.54240 − 3.18806i 1.36955 − 1.71736i −0.811607 − 3.55588i 1.63274 − 0.786286i −0.900969 + 0.433884i −1.99312 + 8.73243i
82.2 −0.0518468 − 0.0650138i 0.222521 + 0.974928i 0.443503 − 1.94311i 1.20357 + 1.50923i 0.0518468 − 0.0650138i −0.0765305 − 0.335302i −0.299165 + 0.144070i −0.900969 + 0.433884i 0.0357195 − 0.156498i
82.3 1.14392 + 1.43443i 0.222521 + 0.974928i −0.303995 + 1.33189i −1.40816 − 1.76577i −1.14392 + 1.43443i 0.165617 + 0.725615i 1.04778 − 0.504584i −0.900969 + 0.433884i 0.922058 − 4.03980i
\(n\): 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.b ✓ 18
3.b odd 2 1 261.2.k.b 18
29.d even 7 1 inner 87.2.g.b ✓ 18
29.d even 7 1 2523.2.a.p 9
29.e even 14 1 2523.2.a.q 9
87.h odd 14 1 7569.2.a.bk 9
87.j odd 14 1 261.2.k.b 18
87.j odd 14 1 7569.2.a.bl 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
87.2.g.b ✓ 18 1.a even 1 1 trivial
87.2.g.b ✓ 18 29.d even 7 1 inner
261.2.k.b 18 3.b odd 2 1
261.2.k.b 18 87.j odd 14 1
2523.2.a.p 9 29.d even 7 1
2523.2.a.q 9 29.e even 14 1
7569.2.a.bk 9 87.h odd 14 1
7569.2.a.bl 9 87.j odd 14 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{18} + 2 T_{2}^{17} + 8 T_{2}^{16} + 17 T_{2}^{15} + 38 T_{2}^{14} + 25 T_{2}^{13} + 76 T_{2}^{12} + \cdots + 49 \) acting on \(S_{2}^{\mathrm{new}}(87, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + 2 T^{17} + \cdots + 49 \) Copy content Toggle raw display
$3$ \( (T^{6} - T^{5} + T^{4} + \cdots + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{18} + 7 T^{17} + \cdots + 28561 \) Copy content Toggle raw display
$7$ \( T^{18} + 4 T^{17} + \cdots + 10816 \) Copy content Toggle raw display
$11$ \( T^{18} + 6 T^{17} + \cdots + 56190016 \) Copy content Toggle raw display
$13$ \( T^{18} + 11 T^{17} + \cdots + 707281 \) Copy content Toggle raw display
$17$ \( (T^{9} + 16 T^{8} + \cdots - 271727)^{2} \) Copy content Toggle raw display
$19$ \( T^{18} - 2 T^{17} + \cdots + 64 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 180848704 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 14507145975869 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 216775910464 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 12378120049 \) Copy content Toggle raw display
$41$ \( (T^{9} + 34 T^{8} + \cdots + 583681)^{2} \) Copy content Toggle raw display
$43$ \( T^{18} + 3 T^{17} + \cdots + 59845696 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 49056352192576 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 1650985998649 \) Copy content Toggle raw display
$59$ \( (T^{9} - 10 T^{8} + \cdots - 56161784)^{2} \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 413051721481 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 26\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 437259497536 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 1649870887729 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 979940416 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 22138078932544 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 56\!\cdots\!09 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 328495941544849 \) Copy content Toggle raw display
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