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Properties

Label 87.3.b.a
Level $87$
Weight $3$
Character orbit 87.b
Analytic conductor $2.371$
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] = [87,3,Mod(59,87)]
 
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, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("87.59");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 87 = 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 87.b (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: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 54 x^{16} + 1187 x^{14} + 13673 x^{12} + 88449 x^{10} + 318861 x^{8} + 593533 x^{6} + \cdots + 15341 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{9} q^{3} + (\beta_{2} - 2) q^{4} - \beta_{5} q^{5} - \beta_{6} q^{6} + ( - \beta_{4} - 1) q^{7} + (\beta_{3} - 2 \beta_1) q^{8} + (\beta_{15} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{9} q^{3} + (\beta_{2} - 2) q^{4} - \beta_{5} q^{5} - \beta_{6} q^{6} + ( - \beta_{4} - 1) q^{7} + (\beta_{3} - 2 \beta_1) q^{8} + (\beta_{15} - 1) q^{9} + ( - \beta_{17} + \beta_{15} + \cdots - \beta_{4}) q^{10}+ \cdots + (\beta_{17} + 3 \beta_{16} - \beta_{15} + \cdots - 16) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 2 q^{3} - 36 q^{4} + 8 q^{6} - 12 q^{7} - 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 2 q^{3} - 36 q^{4} + 8 q^{6} - 12 q^{7} - 22 q^{9} + 12 q^{10} + 18 q^{12} + 32 q^{13} + 30 q^{15} + 76 q^{16} - 50 q^{18} - 24 q^{19} + 32 q^{21} - 94 q^{22} + 38 q^{24} - 114 q^{25} - 68 q^{27} + 94 q^{28} - 88 q^{30} + 24 q^{31} - 20 q^{33} + 70 q^{34} + 168 q^{36} - 40 q^{37} + 38 q^{39} + 160 q^{40} - 118 q^{42} - 36 q^{43} + 32 q^{45} - 228 q^{46} + 94 q^{48} + 190 q^{49} + 204 q^{51} - 386 q^{52} - 32 q^{54} + 188 q^{55} - 140 q^{57} - 354 q^{60} - 8 q^{61} - 340 q^{63} + 86 q^{64} + 178 q^{66} + 136 q^{67} + 4 q^{69} + 252 q^{70} + 358 q^{72} - 68 q^{73} + 244 q^{75} + 120 q^{76} + 66 q^{78} - 96 q^{79} + 366 q^{81} - 548 q^{82} - 664 q^{84} - 320 q^{85} + 504 q^{88} + 562 q^{90} - 156 q^{91} - 40 q^{93} - 174 q^{94} - 504 q^{96} - 12 q^{97} - 198 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 54 x^{16} + 1187 x^{14} + 13673 x^{12} + 88449 x^{10} + 318861 x^{8} + 593533 x^{6} + \cdots + 15341 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 10\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 82527 \nu^{16} - 4293023 \nu^{14} - 89972898 \nu^{12} - 974869229 \nu^{10} - 5828294254 \nu^{8} + \cdots - 2990955031 ) / 67823424 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 50195 \nu^{17} + 2452217 \nu^{15} + 48044642 \nu^{13} + 493140477 \nu^{11} + \cdots + 21388148593 \nu ) / 779969376 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4442 \nu^{17} - 2173 \nu^{16} + 226026 \nu^{15} - 92333 \nu^{14} + 4607764 \nu^{13} + \cdots + 152893075 ) / 22607808 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 13326 \nu^{17} + 51813 \nu^{16} + 678078 \nu^{15} + 2953057 \nu^{14} + 13823292 \nu^{13} + \cdots + 4378110929 ) / 67823424 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 71857 \nu^{17} + 3856496 \nu^{15} + 84075604 \nu^{13} + 957697147 \nu^{11} + \cdots + 1226136346 \nu ) / 194992344 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 229225 \nu^{17} + 102166 \nu^{16} - 12428129 \nu^{15} + 5198598 \nu^{14} - 274213734 \nu^{13} + \cdots + 752877078 ) / 519979584 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 687675 \nu^{17} + 884741 \nu^{16} + 37284387 \nu^{15} + 50320849 \nu^{14} + 822641202 \nu^{13} + \cdots + 29655426809 ) / 1559938752 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 863 \nu^{17} - 46947 \nu^{15} - 1039538 \nu^{13} - 12044197 \nu^{11} - 77992018 \nu^{9} + \cdots - 62186811 \nu ) / 967104 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 2011791 \nu^{17} + 1889795 \nu^{16} + 107795443 \nu^{15} + 101471607 \nu^{14} + \cdots + 123212318751 ) / 1559938752 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 229225 \nu^{17} - 102166 \nu^{16} - 12428129 \nu^{15} - 5198598 \nu^{14} - 274213734 \nu^{13} + \cdots - 752877078 ) / 173326528 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 2646393 \nu^{17} + 1803844 \nu^{16} - 140406433 \nu^{15} + 96488404 \nu^{14} + \cdots + 84408188612 ) / 1559938752 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 1502982 \nu^{17} - 195799 \nu^{16} + 80337812 \nu^{15} - 8977751 \nu^{14} + 1741719588 \nu^{13} + \cdots - 5448751819 ) / 779969376 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 3365535 \nu^{17} - 1190848 \nu^{16} + 180944815 \nu^{15} - 65296816 \nu^{14} + \cdots - 79890926144 ) / 1559938752 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 160593 \nu^{17} + 57876 \nu^{16} + 8606957 \nu^{15} + 3033596 \nu^{14} + 187220886 \nu^{13} + \cdots + 1553629444 ) / 67823424 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 10\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{17} - \beta_{16} + \beta_{14} + \beta_{13} - \beta_{10} - 2 \beta_{9} - \beta_{7} + \beta_{6} + \cdots + 61 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{16} + \beta_{15} + \beta_{14} + \beta_{12} + 4 \beta_{11} - 3 \beta_{9} - 3 \beta_{8} + \cdots + 115 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 40 \beta_{17} + 20 \beta_{16} + \beta_{15} - 20 \beta_{14} - 22 \beta_{13} - \beta_{12} + 16 \beta_{10} + \cdots - 707 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2 \beta_{17} - 19 \beta_{16} - 25 \beta_{15} - 23 \beta_{14} + 5 \beta_{13} - 27 \beta_{12} - 106 \beta_{11} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 622 \beta_{17} - 316 \beta_{16} - 17 \beta_{15} + 316 \beta_{14} + 376 \beta_{13} + 27 \beta_{12} + \cdots + 8730 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 70 \beta_{17} + 253 \beta_{16} + 469 \beta_{15} + 393 \beta_{14} - 161 \beta_{13} + 539 \beta_{12} + \cdots + 70 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 8930 \beta_{17} + 4669 \beta_{16} + 91 \beta_{15} - 4669 \beta_{14} - 5871 \beta_{13} - 499 \beta_{12} + \cdots - 111747 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1610 \beta_{17} - 2742 \beta_{16} - 7804 \beta_{15} - 5962 \beta_{14} + 3619 \beta_{13} - 9414 \beta_{12} + \cdots - 1610 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 124018 \beta_{17} - 67352 \beta_{16} + 2548 \beta_{15} + 67352 \beta_{14} + 87562 \beta_{13} + \cdots + 1462067 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 30896 \beta_{17} + 23222 \beta_{16} + 121564 \beta_{15} + 85014 \beta_{14} - 69952 \beta_{13} + \cdots + 30896 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 1695534 \beta_{17} + 962152 \beta_{16} - 102194 \beta_{15} - 962152 \beta_{14} - 1270894 \beta_{13} + \cdots - 19404296 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 537512 \beta_{17} - 93676 \beta_{16} - 1819026 \beta_{15} - 1168700 \beta_{14} + 1243130 \beta_{13} + \cdots - 537512 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 23002360 \beta_{17} - 13683061 \beta_{16} + 2434152 \beta_{15} + 13683061 \beta_{14} + 18131171 \beta_{13} + \cdots + 260104483 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 8811872 \beta_{17} - 1926725 \beta_{16} + 26530945 \beta_{15} + 15697019 \beta_{14} - 20932208 \beta_{13} + \cdots + 8811872 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
59.1
− 3.74519i
− 3.52466i
− 3.14645i
− 2.54885i
− 2.44888i
− 1.92628i
− 0.949055i
− 0.591999i
− 0.441441i
0.441441i
0.591999i
0.949055i
1.92628i
2.44888i
2.54885i
3.14645i
3.52466i
3.74519i
− 3.74519i 0.905668 − 2.86003i −10.0264 1.22987i −10.7113 − 3.39190i 5.48015 22.5701i −7.35953 − 5.18047i 4.60608
59.2 − 3.52466i −0.379125 + 2.97595i −8.42321 − 8.55504i 10.4892 + 1.33629i −7.79951 15.5903i −8.71253 − 2.25652i −30.1536
59.3 − 3.14645i −2.99737 + 0.125689i −5.90015 7.65220i 0.395474 + 9.43106i −12.1906 5.97873i 8.96840 − 0.753472i 24.0773
59.4 − 2.54885i 2.99036 + 0.240340i −2.49663 − 1.22979i 0.612591 − 7.62197i −0.723222 − 3.83186i 8.88447 + 1.43741i −3.13454
59.5 − 2.44888i −0.668222 + 2.92463i −1.99699 5.55196i 7.16206 + 1.63639i 13.2930 − 4.90512i −8.10696 − 3.90861i 13.5961
59.6 − 1.92628i −1.38419 − 2.66158i 0.289440 − 2.71194i −5.12695 + 2.66635i −1.40218 − 8.26267i −5.16801 + 7.36829i −5.22397
59.7 − 0.949055i −2.83105 + 0.992548i 3.09929 − 4.66941i 0.941983 + 2.68682i 2.70818 − 6.73762i 7.02970 − 5.61991i −4.43152
59.8 − 0.591999i 1.87458 + 2.34221i 3.64954 5.79366i 1.38659 − 1.10975i −9.59672 − 4.52852i −1.97189 + 8.78132i 3.42984
59.9 − 0.441441i 1.48935 − 2.60419i 3.80513 7.32676i −1.14960 − 0.657462i 4.23094 − 3.44551i −4.56366 − 7.75713i 3.23433
59.10 0.441441i 1.48935 + 2.60419i 3.80513 − 7.32676i −1.14960 + 0.657462i 4.23094 3.44551i −4.56366 + 7.75713i 3.23433
59.11 0.591999i 1.87458 − 2.34221i 3.64954 − 5.79366i 1.38659 + 1.10975i −9.59672 4.52852i −1.97189 − 8.78132i 3.42984
59.12 0.949055i −2.83105 − 0.992548i 3.09929 4.66941i 0.941983 − 2.68682i 2.70818 6.73762i 7.02970 + 5.61991i −4.43152
59.13 1.92628i −1.38419 + 2.66158i 0.289440 2.71194i −5.12695 − 2.66635i −1.40218 8.26267i −5.16801 − 7.36829i −5.22397
59.14 2.44888i −0.668222 − 2.92463i −1.99699 − 5.55196i 7.16206 − 1.63639i 13.2930 4.90512i −8.10696 + 3.90861i 13.5961
59.15 2.54885i 2.99036 − 0.240340i −2.49663 1.22979i 0.612591 + 7.62197i −0.723222 3.83186i 8.88447 − 1.43741i −3.13454
59.16 3.14645i −2.99737 − 0.125689i −5.90015 − 7.65220i 0.395474 − 9.43106i −12.1906 − 5.97873i 8.96840 + 0.753472i 24.0773
59.17 3.52466i −0.379125 − 2.97595i −8.42321 8.55504i 10.4892 − 1.33629i −7.79951 − 15.5903i −8.71253 + 2.25652i −30.1536
59.18 3.74519i 0.905668 + 2.86003i −10.0264 − 1.22987i −10.7113 + 3.39190i 5.48015 − 22.5701i −7.35953 + 5.18047i 4.60608
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 59.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b 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.b.a ✓ 18
3.b odd 2 1 inner 87.3.b.a ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
87.3.b.a ✓ 18 1.a even 1 1 trivial
87.3.b.a ✓ 18 3.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(87, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + 54 T^{16} + \cdots + 15341 \) Copy content Toggle raw display
$3$ \( T^{18} + \cdots + 387420489 \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots + 87317155136 \) Copy content Toggle raw display
$7$ \( (T^{9} + 6 T^{8} + \cdots + 772360)^{2} \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots + 804411847193600 \) Copy content Toggle raw display
$13$ \( (T^{9} - 16 T^{8} + \cdots - 448069760)^{2} \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 13\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( (T^{9} + 12 T^{8} + \cdots - 1679616)^{2} \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 37\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( (T^{2} + 29)^{9} \) Copy content Toggle raw display
$31$ \( (T^{9} - 12 T^{8} + \cdots - 622914048)^{2} \) Copy content Toggle raw display
$37$ \( (T^{9} + \cdots - 23130932961280)^{2} \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{9} + \cdots + 305376704944160)^{2} \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 56\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 80\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{9} + \cdots + 6974164539392)^{2} \) Copy content Toggle raw display
$67$ \( (T^{9} + \cdots - 93\!\cdots\!80)^{2} \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{9} + \cdots + 13976230489600)^{2} \) Copy content Toggle raw display
$79$ \( (T^{9} + \cdots - 15\!\cdots\!52)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 32\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 68\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{9} + \cdots - 17\!\cdots\!40)^{2} \) Copy content Toggle raw display
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