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Properties

Label 87.3.d.c
Level $87$
Weight $3$
Character orbit 87.d
Analytic conductor $2.371$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [87,3,Mod(86,87)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(87, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("87.86");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 87 = 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 87.d (of order \(2\), degree \(1\), 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: \(2.37057829993\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 22x^{10} + 332x^{8} + 3614x^{6} + 26892x^{4} + 144342x^{2} + 531441 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{8} q^{2} + \beta_{6} q^{3} + \beta_{5} q^{4} - \beta_{9} q^{5} + ( - \beta_{7} - \beta_{2} - 1) q^{6} + (\beta_{5} + \beta_{2}) q^{7} + (\beta_{10} + \beta_{6} - \beta_1) q^{8} + ( - \beta_{5} - \beta_{3} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{8} q^{2} + \beta_{6} q^{3} + \beta_{5} q^{4} - \beta_{9} q^{5} + ( - \beta_{7} - \beta_{2} - 1) q^{6} + (\beta_{5} + \beta_{2}) q^{7} + (\beta_{10} + \beta_{6} - \beta_1) q^{8} + ( - \beta_{5} - \beta_{3} - 3) q^{9} + ( - \beta_{11} + \beta_{10} + \cdots + \beta_1) q^{10}+ \cdots + (3 \beta_{11} - 4 \beta_{10} + \cdots + 14 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{4} - 12 q^{6} + 4 q^{7} - 44 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 4 q^{4} - 12 q^{6} + 4 q^{7} - 44 q^{9} + 24 q^{13} - 52 q^{16} + 120 q^{22} + 80 q^{24} - 144 q^{25} + 84 q^{28} + 12 q^{30} + 76 q^{33} + 4 q^{34} - 128 q^{36} - 108 q^{42} + 140 q^{45} - 296 q^{49} + 148 q^{51} - 208 q^{52} + 20 q^{54} + 152 q^{57} + 272 q^{58} + 48 q^{63} - 348 q^{64} + 136 q^{67} - 444 q^{78} - 360 q^{81} + 636 q^{82} + 20 q^{87} + 96 q^{88} + 632 q^{91} + 128 q^{93} - 108 q^{94} - 132 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 22x^{10} + 332x^{8} + 3614x^{6} + 26892x^{4} + 144342x^{2} + 531441 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{10} - 125\nu^{8} - 2446\nu^{6} - 27559\nu^{4} - 189054\nu^{2} - 918540 ) / 59049 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{10} + 125\nu^{8} + 2446\nu^{6} + 27559\nu^{4} + 248103\nu^{2} + 1154736 ) / 59049 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 29\nu^{11} - 3655\nu^{9} - 52013\nu^{7} - 421613\nu^{5} - 4375215\nu^{3} - 9887427\nu ) / 4251528 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7\nu^{10} + 73\nu^{8} + 542\nu^{6} + 4967\nu^{4} - 6075\nu^{2} - 32805 ) / 59049 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{11} + 22\nu^{9} + 332\nu^{7} + 3614\nu^{5} + 26892\nu^{3} + 144342\nu ) / 59049 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 25\nu^{10} + 469\nu^{8} + 7247\nu^{6} + 61271\nu^{4} + 361341\nu^{2} + 1371249 ) / 104976 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 193\nu^{11} + 2221\nu^{9} + 26087\nu^{7} + 110495\nu^{5} + 227205\nu^{3} - 1410615\nu ) / 8503056 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 251\nu^{10} + 6575\nu^{8} + 73693\nu^{6} + 620293\nu^{4} + 3187431\nu^{2} + 5268483 ) / 944784 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -331\nu^{11} - 5743\nu^{9} - 43229\nu^{7} - 258821\nu^{5} - 65367\nu^{3} + 9008253\nu ) / 8503056 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 233\nu^{11} + 13469\nu^{9} + 214975\nu^{7} + 2070103\nu^{5} + 14260941\nu^{3} + 54279153\nu ) / 4251528 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} + \beta_{10} + \beta_{8} + 5\beta_{6} - 2\beta_{4} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{9} - 5\beta_{7} + 10\beta_{5} - 4\beta_{3} - 12\beta_{2} - 32 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 6\beta_{11} - 15\beta_{10} - 51\beta_{8} + 6\beta_{6} + 21\beta_{4} - 35\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -12\beta_{9} + 84\beta_{7} - 123\beta_{5} - 50\beta_{3} + 16\beta_{2} + 128 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 56\beta_{11} + 286\beta_{10} + 700\beta_{8} - 460\beta_{6} - 5\beta_{4} + 211\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 515\beta_{9} - 593\beta_{7} + 34\beta_{5} + 272\beta_{3} + 261\beta_{2} + 3634 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -339\beta_{11} - 3468\beta_{10} - 4998\beta_{8} + 264\beta_{6} - 1092\beta_{4} + 3988\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -3732\beta_{9} + 3228\beta_{7} + 10509\beta_{5} + 4741\beta_{3} + 5422\beta_{2} - 23887 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -5926\beta_{11} + 8662\beta_{10} + 34978\beta_{8} + 49817\beta_{6} + 3574\beta_{4} - 41180\beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(31\) \(59\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
86.1
0.521442 − 2.95434i
0.521442 + 2.95434i
−2.08152 − 2.16039i
−2.08152 + 2.16039i
−1.84265 − 2.36741i
−1.84265 + 2.36741i
1.84265 − 2.36741i
1.84265 + 2.36741i
2.08152 − 2.16039i
2.08152 + 2.16039i
−0.521442 − 2.95434i
−0.521442 + 2.95434i
−2.96916 −0.521442 − 2.95434i 4.81591 − 4.07603i 1.54824 + 8.77189i −0.280578 −2.42256 −8.45620 + 3.08103i 12.1024i
86.2 −2.96916 −0.521442 + 2.95434i 4.81591 4.07603i 1.54824 − 8.77189i −0.280578 −2.42256 −8.45620 − 3.08103i − 12.1024i
86.3 −2.03882 2.08152 − 2.16039i 0.156804 7.93416i −4.24385 + 4.40465i 6.64451 7.83560 −0.334549 − 8.99378i − 16.1763i
86.4 −2.03882 2.08152 + 2.16039i 0.156804 − 7.93416i −4.24385 − 4.40465i 6.64451 7.83560 −0.334549 + 8.99378i 16.1763i
86.5 −0.165191 1.84265 − 2.36741i −3.97271 − 5.60670i −0.304390 + 0.391075i −5.36393 1.31702 −2.20925 − 8.72463i 0.926178i
86.6 −0.165191 1.84265 + 2.36741i −3.97271 5.60670i −0.304390 − 0.391075i −5.36393 1.31702 −2.20925 + 8.72463i − 0.926178i
86.7 0.165191 −1.84265 − 2.36741i −3.97271 5.60670i −0.304390 − 0.391075i −5.36393 −1.31702 −2.20925 + 8.72463i 0.926178i
86.8 0.165191 −1.84265 + 2.36741i −3.97271 − 5.60670i −0.304390 + 0.391075i −5.36393 −1.31702 −2.20925 − 8.72463i − 0.926178i
86.9 2.03882 −2.08152 − 2.16039i 0.156804 − 7.93416i −4.24385 − 4.40465i 6.64451 −7.83560 −0.334549 + 8.99378i − 16.1763i
86.10 2.03882 −2.08152 + 2.16039i 0.156804 7.93416i −4.24385 + 4.40465i 6.64451 −7.83560 −0.334549 − 8.99378i 16.1763i
86.11 2.96916 0.521442 − 2.95434i 4.81591 4.07603i 1.54824 − 8.77189i −0.280578 2.42256 −8.45620 − 3.08103i 12.1024i
86.12 2.96916 0.521442 + 2.95434i 4.81591 − 4.07603i 1.54824 + 8.77189i −0.280578 2.42256 −8.45620 + 3.08103i − 12.1024i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 86.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
29.b even 2 1 inner
87.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 87.3.d.c ✓ 12
3.b odd 2 1 inner 87.3.d.c ✓ 12
29.b even 2 1 inner 87.3.d.c ✓ 12
87.d odd 2 1 inner 87.3.d.c ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
87.3.d.c ✓ 12 1.a even 1 1 trivial
87.3.d.c ✓ 12 3.b odd 2 1 inner
87.3.d.c ✓ 12 29.b even 2 1 inner
87.3.d.c ✓ 12 87.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 13T_{2}^{4} + 37T_{2}^{2} - 1 \) acting on \(S_{3}^{\mathrm{new}}(87, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - 13 T^{4} + 37 T^{2} - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} + 22 T^{10} + \cdots + 531441 \) Copy content Toggle raw display
$5$ \( (T^{6} + 111 T^{4} + \cdots + 32877)^{2} \) Copy content Toggle raw display
$7$ \( (T^{3} - T^{2} - 36 T - 10)^{4} \) Copy content Toggle raw display
$11$ \( (T^{6} - 180 T^{4} + \cdots - 16900)^{2} \) Copy content Toggle raw display
$13$ \( (T^{3} - 6 T^{2} + \cdots - 620)^{4} \) Copy content Toggle raw display
$17$ \( (T^{6} - 265 T^{4} + \cdots - 26244)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 941 T^{4} + \cdots + 29589300)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + 924 T^{4} + \cdots + 13150800)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 35\!\cdots\!41 \) Copy content Toggle raw display
$31$ \( (T^{6} + 4356 T^{4} + \cdots + 555621300)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 6105 T^{4} + \cdots + 3981536208)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 5769 T^{4} + \cdots - 1694968900)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 8601 T^{4} + \cdots + 16526643237)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 7933 T^{4} + \cdots - 5492736769)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 4010 T^{4} + \cdots + 266303700)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 23141 T^{4} + \cdots + 419953570452)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 8952 T^{4} + \cdots + 8219250000)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} - 34 T^{2} + \cdots + 169000)^{4} \) Copy content Toggle raw display
$71$ \( (T^{6} + 12468 T^{4} + \cdots + 19391643648)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 16256 T^{4} + \cdots + 51206059008)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 15516 T^{4} + \cdots + 6657592500)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots + 2992188373200)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 37032 T^{4} + \cdots - 416154010000)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 19080 T^{4} + \cdots + 505516752)^{2} \) Copy content Toggle raw display
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