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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [20,11,Mod(11,20)] 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("20.11"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(20, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 20.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.7071450535\)
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: \( x^{20} - x^{19} + 84 x^{18} - 86 x^{17} - 51565 x^{16} + 188134 x^{15} - 12946328 x^{14} + \cdots + 11\!\cdots\!25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{94}\cdot 3^{4}\cdot 5^{29} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + (\beta_{2} - \beta_1) q^{3} + (\beta_{3} - \beta_{2} - \beta_1 - 32) q^{4} + (\beta_{4} + 2 \beta_1) q^{5} + ( - \beta_{8} + \beta_{4} + \beta_{3} + \cdots - 739) q^{6} + ( - \beta_{6} + 4 \beta_{3} + \cdots - 10) q^{7}+ \cdots + ( - 32799 \beta_{19} + 26828 \beta_{18} + \cdots + 9603344) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 22 q^{2} - 644 q^{4} - 14784 q^{6} + 3448 q^{8} - 414868 q^{9} - 31250 q^{10} + 1329640 q^{12} - 278864 q^{13} - 2240504 q^{14} + 4261360 q^{16} - 1921656 q^{17} - 3556082 q^{18} - 1187500 q^{20}+ \cdots + 38416891998 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - x^{19} + 84 x^{18} - 86 x^{17} - 51565 x^{16} + 188134 x^{15} - 12946328 x^{14} + \cdots + 11\!\cdots\!25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 29\!\cdots\!84 \nu^{19} + \cdots - 38\!\cdots\!75 ) / 10\!\cdots\!15 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 33\!\cdots\!39 \nu^{19} + \cdots + 53\!\cdots\!40 ) / 24\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 69\!\cdots\!65 \nu^{19} + \cdots + 13\!\cdots\!00 ) / 34\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 26\!\cdots\!76 \nu^{19} + \cdots - 33\!\cdots\!75 ) / 14\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11\!\cdots\!57 \nu^{19} + \cdots - 63\!\cdots\!40 ) / 24\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 64\!\cdots\!57 \nu^{19} + \cdots + 28\!\cdots\!50 ) / 82\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 13\!\cdots\!45 \nu^{19} + \cdots + 69\!\cdots\!60 ) / 12\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 70\!\cdots\!57 \nu^{19} + \cdots + 18\!\cdots\!00 ) / 61\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 10\!\cdots\!87 \nu^{19} + \cdots - 22\!\cdots\!20 ) / 61\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 21\!\cdots\!24 \nu^{19} + \cdots - 14\!\cdots\!45 ) / 12\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 82\!\cdots\!63 \nu^{19} + \cdots - 13\!\cdots\!30 ) / 35\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 33\!\cdots\!71 \nu^{19} + \cdots + 24\!\cdots\!80 ) / 11\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 21\!\cdots\!04 \nu^{19} + \cdots - 31\!\cdots\!05 ) / 61\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 91\!\cdots\!49 \nu^{19} + \cdots - 78\!\cdots\!40 ) / 24\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 49\!\cdots\!22 \nu^{19} + \cdots - 24\!\cdots\!15 ) / 12\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 69\!\cdots\!11 \nu^{19} + \cdots + 77\!\cdots\!15 ) / 12\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 13\!\cdots\!55 \nu^{19} + \cdots + 21\!\cdots\!60 ) / 24\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 18\!\cdots\!65 \nu^{19} + \cdots - 14\!\cdots\!80 ) / 24\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 17\!\cdots\!45 \nu^{19} + \cdots - 24\!\cdots\!50 ) / 22\!\cdots\!20 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} - 623\beta_1 ) / 1250 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{9} + 619\beta_{3} - 675\beta_{2} + 633\beta _1 - 20632 ) / 2500 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 15 \beta_{19} - 15 \beta_{16} + 610 \beta_{15} - 30 \beta_{13} + 15 \beta_{10} - 21 \beta_{9} + \cdots + 3761 ) / 5000 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 1495 \beta_{19} + 1175 \beta_{18} + 2625 \beta_{17} + 3280 \beta_{16} + 5455 \beta_{15} + \cdots + 110217379 ) / 10000 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 2965 \beta_{19} + 6575 \beta_{18} + 2625 \beta_{17} + 5460 \beta_{16} - 4165 \beta_{15} + \cdots - 69594271 ) / 2000 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 129360 \beta_{19} - 826175 \beta_{18} - 463125 \beta_{17} - 317710 \beta_{16} + \cdots + 24855714571 ) / 10000 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 5860815 \beta_{19} + 8491100 \beta_{18} + 114000 \beta_{17} + 13611860 \beta_{16} + \cdots - 251574266952 ) / 10000 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 83867960 \beta_{19} + 14307300 \beta_{18} - 6304500 \beta_{17} + 43201190 \beta_{16} + \cdots - 9705580637361 ) / 5000 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 3293850425 \beta_{19} - 432657750 \beta_{18} - 2727291750 \beta_{17} + 6280359300 \beta_{16} + \cdots + 238415301771326 ) / 10000 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 1040701415 \beta_{19} + 5069226800 \beta_{18} + 249825500 \beta_{17} + 9149902760 \beta_{16} + \cdots - 735576109152838 ) / 2000 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 22585590495 \beta_{19} + 568484609750 \beta_{18} + 602744039750 \beta_{17} + 126199061920 \beta_{16} + \cdots + 12\!\cdots\!44 ) / 10000 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 4490857413325 \beta_{19} - 5369267873375 \beta_{18} - 1997166635125 \beta_{17} + \cdots - 51\!\cdots\!99 ) / 5000 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 104775524271145 \beta_{19} - 153232175321875 \beta_{18} - 19407191189125 \beta_{17} + \cdots - 15\!\cdots\!33 ) / 10000 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 13\!\cdots\!95 \beta_{19} + \cdots - 23\!\cdots\!36 ) / 10000 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 273154966312280 \beta_{19} + \cdots + 90\!\cdots\!97 ) / 2000 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 82\!\cdots\!65 \beta_{19} + \cdots + 31\!\cdots\!06 ) / 10000 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 12\!\cdots\!90 \beta_{19} + \cdots - 23\!\cdots\!87 ) / 10000 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 35\!\cdots\!30 \beta_{19} + \cdots + 30\!\cdots\!13 ) / 10000 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 33\!\cdots\!25 \beta_{19} + \cdots - 33\!\cdots\!59 ) / 10000 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/20\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(17\)
\(\chi(n)\) \(-1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
−15.1364 − 2.79258i
−15.1364 + 2.79258i
−16.5762 − 5.67923i
−16.5762 + 5.67923i
−7.66078 − 13.6916i
−7.66078 + 13.6916i
−8.12576 − 14.6168i
−8.12576 + 14.6168i
−2.00318 − 15.9954i
−2.00318 + 15.9954i
4.14605 − 15.6062i
4.14605 + 15.6062i
5.61235 − 15.2006i
5.61235 + 15.2006i
10.5132 − 10.4323i
10.5132 + 10.4323i
13.6469 − 4.79406i
13.6469 + 4.79406i
16.0838 − 4.09984i
16.0838 + 4.09984i
−31.5088 − 5.58516i − 275.626i 961.612 + 351.964i 1397.54 −1539.42 + 8684.66i 19327.7i −28333.5 − 16460.7i −16920.9 −44034.9 − 7805.50i
11.2 −31.5088 + 5.58516i 275.626i 961.612 − 351.964i 1397.54 −1539.42 − 8684.66i − 19327.7i −28333.5 + 16460.7i −16920.9 −44034.9 + 7805.50i
11.3 −29.9163 − 11.3585i − 137.165i 765.971 + 679.606i −1397.54 −1557.98 + 4103.46i − 24910.8i −15195.7 − 29031.6i 40234.9 41809.3 + 15873.9i
11.4 −29.9163 + 11.3585i 137.165i 765.971 − 679.606i −1397.54 −1557.98 − 4103.46i 24910.8i −15195.7 + 29031.6i 40234.9 41809.3 − 15873.9i
11.5 −16.5576 − 27.3833i − 17.1314i −475.690 + 906.805i 1397.54 −469.114 + 283.655i − 2883.30i 32707.6 − 1988.60i 58755.5 −23140.0 − 38269.3i
11.6 −16.5576 + 27.3833i 17.1314i −475.690 − 906.805i 1397.54 −469.114 − 283.655i 2883.30i 32707.6 + 1988.60i 58755.5 −23140.0 + 38269.3i
11.7 −13.0154 − 29.2335i − 448.707i −685.197 + 760.974i −1397.54 −13117.3 + 5840.12i 19334.7i 31164.1 + 10126.3i −142289. 18189.6 + 40855.1i
11.8 −13.0154 + 29.2335i 448.707i −685.197 − 760.974i −1397.54 −13117.3 − 5840.12i − 19334.7i 31164.1 − 10126.3i −142289. 18189.6 − 40855.1i
11.9 −0.770283 − 31.9907i 80.5620i −1022.81 + 49.2838i −1397.54 2577.24 − 62.0555i − 345.112i 2364.48 + 32682.6i 52558.8 1076.50 + 44708.4i
11.10 −0.770283 + 31.9907i − 80.5620i −1022.81 − 49.2838i −1397.54 2577.24 + 62.0555i 345.112i 2364.48 − 32682.6i 52558.8 1076.50 − 44708.4i
11.11 7.05603 − 31.2124i 442.423i −924.425 − 440.471i 1397.54 13809.1 + 3121.75i − 2455.96i −20270.9 + 25745.5i −136689. 9861.10 − 43620.6i
11.12 7.05603 + 31.2124i − 442.423i −924.425 + 440.471i 1397.54 13809.1 − 3121.75i 2455.96i −20270.9 − 25745.5i −136689. 9861.10 + 43620.6i
11.13 9.98863 − 30.4011i − 330.781i −824.454 − 607.331i 1397.54 −10056.1 − 3304.05i − 29449.0i −26698.7 + 18997.9i −50366.8 13959.5 − 42486.8i
11.14 9.98863 + 30.4011i 330.781i −824.454 + 607.331i 1397.54 −10056.1 + 3304.05i 29449.0i −26698.7 − 18997.9i −50366.8 13959.5 + 42486.8i
11.15 24.2624 − 20.8647i 208.777i 153.329 − 1012.46i −1397.54 4356.06 + 5065.43i − 17557.3i −17404.4 − 27763.8i 15461.2 −33907.8 + 29159.3i
11.16 24.2624 + 20.8647i − 208.777i 153.329 + 1012.46i −1397.54 4356.06 − 5065.43i 17557.3i −17404.4 + 27763.8i 15461.2 −33907.8 − 29159.3i
11.17 30.5298 − 9.58811i − 321.971i 840.136 − 585.446i −1397.54 −3087.10 − 9829.71i − 9880.78i 20035.9 − 25928.9i −44616.5 −42666.7 + 13399.8i
11.18 30.5298 + 9.58811i 321.971i 840.136 + 585.446i −1397.54 −3087.10 + 9829.71i 9880.78i 20035.9 + 25928.9i −44616.5 −42666.7 − 13399.8i
11.19 30.9316 − 8.19968i 206.425i 889.530 − 507.259i 1397.54 1692.62 + 6385.07i 1527.38i 23355.3 − 22984.2i 16437.6 43228.3 − 11459.4i
11.20 30.9316 + 8.19968i − 206.425i 889.530 + 507.259i 1397.54 1692.62 − 6385.07i − 1527.38i 23355.3 + 22984.2i 16437.6 43228.3 + 11459.4i
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 11.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 20.11.b.a ✓ 20
3.b odd 2 1 180.11.c.a 20
4.b odd 2 1 inner 20.11.b.a ✓ 20
5.b even 2 1 100.11.b.e 20
5.c odd 4 2 100.11.d.c 40
8.b even 2 1 320.11.b.d 20
8.d odd 2 1 320.11.b.d 20
12.b even 2 1 180.11.c.a 20
20.d odd 2 1 100.11.b.e 20
20.e even 4 2 100.11.d.c 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.11.b.a ✓ 20 1.a even 1 1 trivial
20.11.b.a ✓ 20 4.b odd 2 1 inner
100.11.b.e 20 5.b even 2 1
100.11.b.e 20 20.d odd 2 1
100.11.d.c 40 5.c odd 4 2
100.11.d.c 40 20.e even 4 2
180.11.c.a 20 3.b odd 2 1
180.11.c.a 20 12.b even 2 1
320.11.b.d 20 8.b even 2 1
320.11.b.d 20 8.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(20, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( (T^{2} - 1953125)^{10} \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots + 82\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 79\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots - 19\!\cdots\!56)^{2} \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 44\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots - 54\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots - 10\!\cdots\!16)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 64\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots - 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 11\!\cdots\!44)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 78\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots - 36\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 10\!\cdots\!24)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 47\!\cdots\!00)^{2} \) Copy content Toggle raw display
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