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Properties

Label 87.3.e.a
Level $87$
Weight $3$
Character orbit 87.e
Analytic conductor $2.371$
Analytic rank $0$
Dimension $8$
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] = [87,3,Mod(46,87)] 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("87.46"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(87, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 87 = 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 87.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.37057829993\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 210x^{6} + 644x^{5} + 17515x^{4} - 36108x^{3} - 683586x^{2} + 701748x + 10556010 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} - \beta_{2} - 1) q^{2} + \beta_{3} q^{3} + ( - 2 \beta_{4} - 2 \beta_{3} + \beta_{2}) q^{4} + ( - \beta_{5} - \beta_{2}) q^{5} + ( - \beta_{4} - \beta_{3} + 3 \beta_{2}) q^{6} + (\beta_{3} + \beta_1 + 1) q^{7}+ \cdots + (3 \beta_{5} + 9 \beta_{4} + 9 \beta_{2} + \cdots - 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} + 12 q^{7} + 24 q^{8} + 8 q^{10} + 20 q^{11} + 48 q^{12} + 12 q^{15} - 40 q^{16} + 48 q^{17} + 24 q^{18} - 28 q^{19} - 44 q^{20} + 12 q^{21} - 88 q^{23} - 24 q^{24} - 260 q^{25} + 96 q^{26}+ \cdots - 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 210x^{6} + 644x^{5} + 17515x^{4} - 36108x^{3} - 683586x^{2} + 701748x + 10556010 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -8\nu^{7} + 28\nu^{6} + 1694\nu^{5} - 4305\nu^{4} - 116064\nu^{3} + 178415\nu^{2} + 2753586\nu - 1406673 ) / 157251 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 11 \nu^{7} + 1178 \nu^{6} - 1659 \nu^{5} - 182257 \nu^{4} + 255190 \nu^{3} + 9957564 \nu^{2} + \cdots - 189781605 ) / 786255 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 11 \nu^{7} - 1101 \nu^{6} + 5178 \nu^{5} + 173267 \nu^{4} - 467253 \nu^{3} - 9630441 \nu^{2} + \cdots + 187415685 ) / 786255 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 9 \nu^{7} + 1108 \nu^{6} - 5894 \nu^{5} - 302537 \nu^{4} + 807435 \nu^{3} + 23533074 \nu^{2} + \cdots - 612022005 ) / 786255 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1094 \nu^{7} + 3829 \nu^{6} + 166703 \nu^{5} - 426330 \nu^{4} - 8980056 \nu^{3} + \cdots - 107510235 ) / 786255 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1094 \nu^{7} - 3829 \nu^{6} - 166703 \nu^{5} + 426330 \nu^{4} + 8980056 \nu^{3} - 13505201 \nu^{2} + \cdots + 65052465 ) / 786255 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{6} + \beta _1 + 54 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 2\beta_{6} - 57\beta_{4} - 57\beta_{3} + 4\beta_{2} + 52\beta _1 + 55 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 109\beta_{7} + 111\beta_{6} - 6\beta_{5} - 114\beta_{4} - 108\beta_{3} + 5\beta_{2} + 103\beta _1 + 2639 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 223\beta_{7} + 322\beta_{6} - 15\beta_{5} - 6333\beta_{4} - 6318\beta_{3} + 768\beta_{2} + 2413\beta _1 + 5337 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8761 \beta_{7} + 9053 \beta_{6} - 981 \beta_{5} - 19059 \beta_{4} - 17388 \beta_{3} + 1816 \beta_{2} + \cdots + 111640 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 27022 \beta_{7} + 33423 \beta_{6} - 3381 \beta_{5} - 519417 \beta_{4} - 513621 \beta_{3} + 88601 \beta_{2} + \cdots + 331265 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(31\) \(59\)
\(\chi(n)\) \(\beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
46.1
8.04816 − 1.22474i
−7.04816 − 1.22474i
7.88412 + 1.22474i
−6.88412 + 1.22474i
−7.04816 + 1.22474i
8.04816 + 1.22474i
−6.88412 − 1.22474i
7.88412 − 1.22474i
−2.22474 + 2.22474i −1.22474 + 1.22474i − 5.89898i − 8.27291i − 5.44949i 7.82342 4.22474 + 4.22474i − 3.00000i 18.4051 + 18.4051i
46.2 −2.22474 + 2.22474i −1.22474 + 1.22474i − 5.89898i 6.82342i − 5.44949i −7.27291 4.22474 + 4.22474i − 3.00000i −15.1804 − 15.1804i
46.3 0.224745 − 0.224745i 1.22474 − 1.22474i 3.89898i − 5.65938i − 0.550510i 10.1089 1.77526 + 1.77526i − 3.00000i −1.27192 − 1.27192i
46.4 0.224745 − 0.224745i 1.22474 − 1.22474i 3.89898i 9.10887i − 0.550510i −4.65938 1.77526 + 1.77526i − 3.00000i 2.04717 + 2.04717i
70.1 −2.22474 − 2.22474i −1.22474 − 1.22474i 5.89898i − 6.82342i 5.44949i −7.27291 4.22474 − 4.22474i 3.00000i −15.1804 + 15.1804i
70.2 −2.22474 − 2.22474i −1.22474 − 1.22474i 5.89898i 8.27291i 5.44949i 7.82342 4.22474 − 4.22474i 3.00000i 18.4051 − 18.4051i
70.3 0.224745 + 0.224745i 1.22474 + 1.22474i − 3.89898i − 9.10887i 0.550510i −4.65938 1.77526 − 1.77526i 3.00000i 2.04717 − 2.04717i
70.4 0.224745 + 0.224745i 1.22474 + 1.22474i − 3.89898i 5.65938i 0.550510i 10.1089 1.77526 − 1.77526i 3.00000i −1.27192 + 1.27192i
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 46.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 87.3.e.a ✓ 8
3.b odd 2 1 261.3.f.b 8
29.c odd 4 1 inner 87.3.e.a ✓ 8
87.f even 4 1 261.3.f.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
87.3.e.a ✓ 8 1.a even 1 1 trivial
87.3.e.a ✓ 8 29.c odd 4 1 inner
261.3.f.b 8 3.b odd 2 1
261.3.f.b 8 87.f even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 4T_{2}^{3} + 8T_{2}^{2} - 4T_{2} + 1 \) acting on \(S_{3}^{\mathrm{new}}(87, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 4 T^{3} + 8 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} + 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + 230 T^{6} + \cdots + 8468100 \) Copy content Toggle raw display
$7$ \( (T^{4} - 6 T^{3} + \cdots + 2680)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 20 T^{7} + \cdots + 13307904 \) Copy content Toggle raw display
$13$ \( (T^{4} + 120 T^{2} + 144)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 3857403664 \) Copy content Toggle raw display
$19$ \( T^{8} + 28 T^{7} + \cdots + 47554816 \) Copy content Toggle raw display
$23$ \( (T^{4} + 44 T^{3} + \cdots - 116000)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 500246412961 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 1775726814096 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 3857403664 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 56882250000 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 2060924390464 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 72686316816 \) Copy content Toggle raw display
$53$ \( (T^{4} + 116 T^{3} + \cdots - 703520)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 106 T^{3} + \cdots - 10046450)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 18389688422400 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 95829219777600 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 377353935360000 \) Copy content Toggle raw display
$73$ \( T^{8} - 272 T^{7} + \cdots + 7862416 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 92\!\cdots\!56 \) Copy content Toggle raw display
$83$ \( (T^{4} + 108 T^{3} + \cdots + 52600120)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 5288675282944 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 421691481976384 \) Copy content Toggle raw display
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