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Properties

Label 888.2.c.c
Level $888$
Weight $2$
Character orbit 888.c
Analytic conductor $7.091$
Analytic rank $0$
Dimension $20$
CM discriminant -296
Inner twists $8$

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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] = [888,2,Mod(443,888)] 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("888.443"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(888, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 888.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,0,0,-40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.09071569949\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 110x^{15} + 12005x^{10} - 3058x^{5} + 243 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{37}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - \beta_{12} q^{3} - 2 q^{4} + \beta_{16} q^{5} - \beta_{7} q^{6} - 2 \beta_{2} q^{8} - \beta_{3} q^{9} + \beta_{10} q^{10} + ( - \beta_{18} + \beta_{3}) q^{11} + 2 \beta_{12} q^{12}+ \cdots + ( - 2 \beta_{9} + \beta_{8} + \beta_{5} + 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 40 q^{4} + 80 q^{16} - 100 q^{25} + 140 q^{49} - 160 q^{64} - 20 q^{75} + 40 q^{78} - 80 q^{90} + 100 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 110x^{15} + 12005x^{10} - 3058x^{5} + 243 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -36\nu^{15} + 22\nu^{10} - 4\nu^{5} - 23527669 ) / 3867925 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 67\nu^{15} - 7359\nu^{10} + 803663\nu^{5} - 102432 ) / 35925 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 9444\nu^{16} - 1037218\nu^{11} + 113202321\nu^{6} - 7047284\nu ) / 2320755 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 68807\nu^{16} - 7563014\nu^{11} + 825594748\nu^{6} - 155830447\nu ) / 11603775 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -11\nu^{15} + 1209\nu^{10} - 131857\nu^{5} + 16806 ) / 969 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -10154\nu^{17} + 1114558\nu^{12} - 121625881\nu^{7} + 2075159\nu^{2} ) / 1832175 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 167779\nu^{18} - 18367733\nu^{13} + 2004463331\nu^{8} + 558356816\nu^{3} ) / 104433975 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 9444\nu^{17} - 1037218\nu^{12} + 113202321\nu^{7} - 9368039\nu^{2} ) / 773585 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 5008\nu^{17} - 550451\nu^{12} + 60073277\nu^{7} - 9831463\nu^{2} ) / 366435 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 117677 \nu^{19} - 137835 \nu^{16} + 13007974 \nu^{14} + 15169140 \nu^{11} + \cdots + 484160625 \nu ) / 62660385 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 118649 \nu^{18} + 84996 \nu^{17} - 13008568 \nu^{13} - 9334962 \nu^{12} + 1419713566 \nu^{8} + \cdots - 84312351 \nu^{2} ) / 20886795 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -151816\nu^{18} + 16681877\nu^{13} - 1820606189\nu^{8} + 249968296\nu^{3} ) / 20886795 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 318137\nu^{19} - 35048719\nu^{14} + 3825069358\nu^{9} - 1615718042\nu^{4} ) / 62660385 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 1595059 \nu^{18} - 424494 \nu^{17} - 175246268 \nu^{13} + 46674513 \nu^{12} + \cdots + 1104704199 \nu^{2} ) / 104433975 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 1595059 \nu^{18} + 1427766 \nu^{17} - 175246268 \nu^{13} - 156878832 \nu^{12} + \cdots - 2327692461 \nu^{2} ) / 104433975 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 4374178 \nu^{19} - 1274940 \nu^{16} + 480576731 \nu^{14} + 140024430 \nu^{11} + \cdots + 1577987190 \nu ) / 313301925 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 4374178 \nu^{19} - 2549880 \nu^{16} - 480576731 \nu^{14} + 280048860 \nu^{11} + \cdots + 3155974380 \nu ) / 313301925 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 993679 \nu^{19} - 137835 \nu^{16} + 109124033 \nu^{14} + 15169140 \nu^{11} + \cdots + 484160625 \nu ) / 62660385 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 123484 \nu^{19} + 15315 \nu^{16} + 13570223 \nu^{14} - 1685460 \nu^{11} - 1480999226 \nu^{9} + \cdots - 53795625 \nu ) / 6962265 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{17} + \beta_{16} + 3\beta_{3} ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{15} + 3\beta_{14} + 3\beta_{9} + \beta_{8} ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{15} - 4\beta_{14} - 15\beta_{12} - 9\beta_{11} + 3\beta_{8} + 21\beta_{7} - 2\beta_{6} ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 8\beta_{19} + 25\beta_{18} + 15\beta_{17} - 30\beta_{16} - 24\beta_{13} - 17\beta_{10} ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 11\beta_{5} + 67\beta_{2} + 9\beta _1 + 55 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -37\beta_{19} + 37\beta_{18} + 45\beta_{17} + 45\beta_{16} + 37\beta_{10} + 105\beta_{4} + 42\beta_{3} ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -109\beta_{15} + 109\beta_{14} - 39\beta_{9} + 429\beta_{8} + 500\beta_{6} ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -519\beta_{15} - 1038\beta_{14} - 3192\beta_{12} - 105\beta_{11} + 35\beta_{8} + 732\beta_{7} - 519\beta_{6} ) / 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -2163\beta_{19} + 783\beta_{18} - 325\beta_{17} + 650\beta_{16} - 3519\beta_{13} - 2946\beta_{10} ) / 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 1208\beta_{5} + 7348\beta_{2} - 979\beta _1 - 5955 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 10089 \beta_{19} + 10089 \beta_{18} - 4967 \beta_{17} - 4967 \beta_{16} + 10089 \beta_{10} + \cdots - 23523 \beta_{3} ) / 6 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 11370\beta_{15} - 11370\beta_{14} - 21459\beta_{9} + 19067\beta_{8} + 25470\beta_{6} ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 33167 \beta_{15} - 66334 \beta_{14} - 168918 \beta_{12} + 98487 \beta_{11} - 32829 \beta_{8} + \cdots - 33167 \beta_{6} ) / 6 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 332542 \beta_{19} - 214409 \beta_{18} - 215085 \beta_{17} + 430170 \beta_{16} + \cdots - 118133 \beta_{10} ) / 6 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 737\beta_{5} + 4483\beta_{2} - 215484\beta _1 - 1310738 ) / 2 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 221042 \beta_{19} + 221042 \beta_{18} - 1623573 \beta_{17} - 1623573 \beta_{16} + \cdots - 3586659 \beta_{3} ) / 6 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 3801068\beta_{15} - 3801068\beta_{14} - 4243152\beta_{9} - 952617\beta_{8} - 398695\beta_{6} ) / 6 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 1288095 \beta_{15} + 2576190 \beta_{14} + 9846231 \beta_{12} + 6033165 \beta_{11} + \cdots + 1288095 \beta_{6} ) / 3 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 10588584 \beta_{19} - 32908455 \beta_{18} - 19711855 \beta_{17} + 39423710 \beta_{16} + \cdots + 22319871 \beta_{10} ) / 6 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/888\mathbb{Z}\right)^\times\).

\(n\) \(223\) \(409\) \(445\) \(593\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
443.1
−2.33616 − 1.04103i
−0.582201 − 0.345921i
0.674337 − 0.0623530i
1.27809 + 2.21537i
2.50189 − 0.530951i
0.267683 + 0.622065i
−0.508900 + 0.446810i
0.268163 − 2.54352i
−1.71199 + 1.90012i
0.149080 − 0.660601i
−0.582201 + 0.345921i
−2.33616 + 1.04103i
1.27809 − 2.21537i
0.674337 + 0.0623530i
0.267683 − 0.622065i
2.50189 + 0.530951i
0.268163 + 2.54352i
−0.508900 − 0.446810i
0.149080 + 0.660601i
−1.71199 − 1.90012i
− 1.41421i −1.72023 − 0.202039i −2.00000 − 2.77390i −0.285727 + 2.43277i 0 2.82843i 2.91836 + 0.695107i −3.92288
443.2 − 1.41421i −1.72023 + 0.202039i −2.00000 − 2.77390i 0.285727 + 2.43277i 0 2.82843i 2.91836 − 0.695107i −3.92288
443.3 − 1.41421i −0.723730 − 1.57360i −2.00000 4.30603i −2.22541 + 1.02351i 0 2.82843i −1.95243 + 2.27772i 6.08965
443.4 − 1.41421i −0.723730 + 1.57360i −2.00000 4.30603i 2.22541 + 1.02351i 0 2.82843i −1.95243 − 2.27772i 6.08965
443.5 − 1.41421i −0.339428 − 1.69847i −2.00000 0.182228i −2.40199 + 0.480024i 0 2.82843i −2.76958 + 1.15302i 0.257709
443.6 − 1.41421i −0.339428 + 1.69847i −2.00000 0.182228i 2.40199 + 0.480024i 0 2.82843i −2.76958 − 1.15302i 0.257709
443.7 − 1.41421i 1.27294 − 1.17458i −2.00000 − 4.19341i −1.66110 − 1.80020i 0 2.82843i 0.240737 − 2.99033i −5.93038
443.8 − 1.41421i 1.27294 + 1.17458i −2.00000 − 4.19341i 1.66110 − 1.80020i 0 2.82843i 0.240737 + 2.99033i −5.93038
443.9 − 1.41421i 1.51045 − 0.847671i −2.00000 2.47905i −1.19879 − 2.13610i 0 2.82843i 1.56291 − 2.56073i 3.50590
443.10 − 1.41421i 1.51045 + 0.847671i −2.00000 2.47905i 1.19879 − 2.13610i 0 2.82843i 1.56291 + 2.56073i 3.50590
443.11 1.41421i −1.72023 − 0.202039i −2.00000 2.77390i 0.285727 − 2.43277i 0 − 2.82843i 2.91836 + 0.695107i −3.92288
443.12 1.41421i −1.72023 + 0.202039i −2.00000 2.77390i −0.285727 − 2.43277i 0 − 2.82843i 2.91836 − 0.695107i −3.92288
443.13 1.41421i −0.723730 − 1.57360i −2.00000 − 4.30603i 2.22541 − 1.02351i 0 − 2.82843i −1.95243 + 2.27772i 6.08965
443.14 1.41421i −0.723730 + 1.57360i −2.00000 − 4.30603i −2.22541 − 1.02351i 0 − 2.82843i −1.95243 − 2.27772i 6.08965
443.15 1.41421i −0.339428 − 1.69847i −2.00000 − 0.182228i 2.40199 − 0.480024i 0 − 2.82843i −2.76958 + 1.15302i 0.257709
443.16 1.41421i −0.339428 + 1.69847i −2.00000 − 0.182228i −2.40199 − 0.480024i 0 − 2.82843i −2.76958 − 1.15302i 0.257709
443.17 1.41421i 1.27294 − 1.17458i −2.00000 4.19341i 1.66110 + 1.80020i 0 − 2.82843i 0.240737 − 2.99033i −5.93038
443.18 1.41421i 1.27294 + 1.17458i −2.00000 4.19341i −1.66110 + 1.80020i 0 − 2.82843i 0.240737 + 2.99033i −5.93038
443.19 1.41421i 1.51045 − 0.847671i −2.00000 − 2.47905i 1.19879 + 2.13610i 0 − 2.82843i 1.56291 − 2.56073i 3.50590
443.20 1.41421i 1.51045 + 0.847671i −2.00000 − 2.47905i −1.19879 + 2.13610i 0 − 2.82843i 1.56291 + 2.56073i 3.50590
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 443.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
296.h odd 2 1 CM by \(\Q(\sqrt{-74}) \)
3.b odd 2 1 inner
8.d odd 2 1 inner
24.f even 2 1 inner
37.b even 2 1 inner
111.d odd 2 1 inner
888.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 888.2.c.c ✓ 20
3.b odd 2 1 inner 888.2.c.c ✓ 20
8.d odd 2 1 inner 888.2.c.c ✓ 20
24.f even 2 1 inner 888.2.c.c ✓ 20
37.b even 2 1 inner 888.2.c.c ✓ 20
111.d odd 2 1 inner 888.2.c.c ✓ 20
296.h odd 2 1 CM 888.2.c.c ✓ 20
888.c even 2 1 inner 888.2.c.c ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
888.2.c.c ✓ 20 1.a even 1 1 trivial
888.2.c.c ✓ 20 3.b odd 2 1 inner
888.2.c.c ✓ 20 8.d odd 2 1 inner
888.2.c.c ✓ 20 24.f even 2 1 inner
888.2.c.c ✓ 20 37.b even 2 1 inner
888.2.c.c ✓ 20 111.d odd 2 1 inner
888.2.c.c ✓ 20 296.h odd 2 1 CM
888.2.c.c ✓ 20 888.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} + 50T_{5}^{8} + 875T_{5}^{6} + 6250T_{5}^{4} + 15625T_{5}^{2} + 512 \) acting on \(S_{2}^{\mathrm{new}}(888, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2)^{10} \) Copy content Toggle raw display
$3$ \( (T^{10} + 26 T^{5} + 243)^{2} \) Copy content Toggle raw display
$5$ \( (T^{10} + 50 T^{8} + \cdots + 512)^{2} \) Copy content Toggle raw display
$7$ \( T^{20} \) Copy content Toggle raw display
$11$ \( (T^{10} + 110 T^{8} + \cdots + 7400)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} - 130 T^{8} + \cdots - 1069300)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} \) Copy content Toggle raw display
$19$ \( T^{20} \) Copy content Toggle raw display
$23$ \( (T^{10} + 230 T^{8} + \cdots + 9800)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + 290 T^{8} + \cdots + 79581728)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} - 310 T^{8} + \cdots - 46911412)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 37)^{10} \) Copy content Toggle raw display
$41$ \( (T^{10} + 410 T^{8} + \cdots + 198060704)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} \) Copy content Toggle raw display
$47$ \( T^{20} \) Copy content Toggle raw display
$53$ \( T^{20} \) Copy content Toggle raw display
$59$ \( T^{20} \) Copy content Toggle raw display
$61$ \( (T^{10} - 610 T^{8} + \cdots - 1267964692)^{2} \) Copy content Toggle raw display
$67$ \( (T^{5} - 335 T^{3} + \cdots + 72262)^{4} \) Copy content Toggle raw display
$71$ \( T^{20} \) Copy content Toggle raw display
$73$ \( (T^{5} - 365 T^{3} + \cdots + 12346)^{4} \) Copy content Toggle raw display
$79$ \( (T^{10} - 790 T^{8} + \cdots - 3103151668)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 296)^{10} \) Copy content Toggle raw display
$89$ \( T^{20} \) Copy content Toggle raw display
$97$ \( T^{20} \) Copy content Toggle raw display
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