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Properties

Label 123.4.d.a
Level $123$
Weight $4$
Character orbit 123.d
Analytic conductor $7.257$
Analytic rank $0$
Dimension $20$
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] = [123,4,Mod(40,123)] 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("123.40"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(123, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 123 = 3 \cdot 41 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 123.d (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: \(7.25723493071\)
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: \( x^{20} + 116 x^{18} + 5618 x^{16} + 148400 x^{14} + 2342961 x^{12} + 22779360 x^{10} + 135500500 x^{8} + \cdots + 331822656 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{6} \)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + \beta_{5} q^{3} + ( - \beta_{3} - \beta_{2} + 3) q^{4} + ( - \beta_{6} + \beta_{2} - 1) q^{5} - \beta_1 q^{6} + ( - \beta_{18} - \beta_{5} - \beta_1) q^{7} + (\beta_{4} - \beta_{3} - 3 \beta_{2} + 6) q^{8}+ \cdots + ( - 9 \beta_{16} + 9 \beta_{13} - 9 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 4 q^{2} + 72 q^{4} - 16 q^{5} + 132 q^{8} - 180 q^{9} - 140 q^{10} + 152 q^{16} - 36 q^{18} + 140 q^{20} + 84 q^{21} + 280 q^{23} + 736 q^{25} - 408 q^{31} + 1600 q^{32} - 12 q^{33} - 648 q^{36}+ \cdots + 2264 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 116 x^{18} + 5618 x^{16} + 148400 x^{14} + 2342961 x^{12} + 22779360 x^{10} + 135500500 x^{8} + \cdots + 331822656 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 3\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 137162467 \nu^{18} - 14993438833 \nu^{16} - 672492149581 \nu^{14} + \cdots - 21\!\cdots\!20 ) / 872836226095104 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 137162467 \nu^{18} + 14993438833 \nu^{16} + 672492149581 \nu^{14} + 16102864139083 \nu^{12} + \cdots + 31\!\cdots\!64 ) / 872836226095104 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1551517067 \nu^{18} - 171795308969 \nu^{16} - 7852812728741 \nu^{14} + \cdots - 34\!\cdots\!88 ) / 872836226095104 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 16689297135 \nu^{19} - 1901256363509 \nu^{17} - 89967131279681 \nu^{15} + \cdots - 46\!\cdots\!32 \nu ) / 73\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 67921583399 \nu^{18} - 7031403851885 \nu^{16} - 293565084990953 \nu^{14} + \cdots - 14\!\cdots\!12 ) / 872836226095104 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 60689478721 \nu^{18} + 6296517420295 \nu^{16} + 263703692898727 \nu^{14} + \cdots + 14\!\cdots\!84 ) / 218209056523776 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 17137241831 \nu^{18} - 1794735778097 \nu^{16} - 76173310098161 \nu^{14} + \cdots - 67\!\cdots\!68 ) / 54552264130944 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 63047337803 \nu^{18} + 6535337170913 \nu^{16} + 273357426365717 \nu^{14} + \cdots + 14\!\cdots\!80 ) / 145472704349184 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 6767280197 \nu^{19} - 778203635175 \nu^{17} - 37249905919659 \nu^{15} + \cdots - 21\!\cdots\!32 \nu ) / 33\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1688198020 \nu^{18} - 175460834443 \nu^{16} - 7367483792378 \nu^{14} + \cdots - 46\!\cdots\!36 ) / 3030681340608 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 187378807915 \nu^{18} - 19497494881633 \nu^{16} - 820050665446069 \nu^{14} + \cdots - 56\!\cdots\!96 ) / 145472704349184 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 521682538403 \nu^{19} - 53647364867681 \nu^{17} + \cdots - 47\!\cdots\!12 \nu ) / 24\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 6559170518771 \nu^{19} - 695195731675809 \nu^{17} + \cdots - 41\!\cdots\!56 \nu ) / 22\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 8250750769381 \nu^{19} - 828994208303607 \nu^{17} + \cdots + 20\!\cdots\!48 \nu ) / 22\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 25305155807 \nu^{19} - 2631619155669 \nu^{17} - 110596433255409 \nu^{15} + \cdots - 71\!\cdots\!96 \nu ) / 489639834150912 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 4038356165629 \nu^{19} - 420851720522703 \nu^{17} + \cdots - 13\!\cdots\!76 \nu ) / 73\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 4071734759899 \nu^{19} - 424654233249721 \nu^{17} + \cdots - 14\!\cdots\!16 \nu ) / 73\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 18409240255245 \nu^{19} + \cdots + 37\!\cdots\!00 \nu ) / 22\!\cdots\!12 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} - 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{18} + 3\beta_{17} + 6\beta_{5} - 19\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{9} + \beta_{7} - 2\beta_{6} + \beta_{4} - 27\beta_{3} - 36\beta_{2} + 204 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 9 \beta_{19} + 102 \beta_{18} - 99 \beta_{17} - 6 \beta_{15} - 15 \beta_{14} - 12 \beta_{13} + \cdots + 433 \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 6 \beta_{11} + 45 \beta_{9} - 4 \beta_{8} - 47 \beta_{7} + 56 \beta_{6} - 49 \beta_{4} + 711 \beta_{3} + \cdots - 4594 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 351 \beta_{19} - 3114 \beta_{18} + 2817 \beta_{17} - 66 \beta_{16} + 300 \beta_{15} + 861 \beta_{14} + \cdots - 10891 \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 18 \beta_{12} - 324 \beta_{11} - 1555 \beta_{9} + 252 \beta_{8} + 1695 \beta_{7} - 1008 \beta_{6} + \cdots + 114564 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 10755 \beta_{19} + 92406 \beta_{18} - 78141 \beta_{17} + 4746 \beta_{16} - 11604 \beta_{15} + \cdots + 289783 \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1258 \beta_{12} + 12596 \beta_{11} + 49479 \beta_{9} - 10856 \beta_{8} - 55675 \beta_{7} + \cdots - 3037932 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 309303 \beta_{19} - 2721210 \beta_{18} + 2175897 \beta_{17} - 221058 \beta_{16} + 407184 \beta_{15} + \cdots - 7989079 \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 58250 \beta_{12} - 434580 \beta_{11} - 1524831 \beta_{9} + 401372 \beta_{8} + 1752339 \beta_{7} + \cdots + 83768648 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 8763759 \beta_{19} + 80146122 \beta_{18} - 61321413 \beta_{17} + 8585322 \beta_{16} + \cdots + 225686335 \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 2267306 \beta_{12} + 14179372 \beta_{11} + 46358903 \beta_{9} - 13730768 \beta_{8} - 53984307 \beta_{7} + \cdots - 2370958460 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 248804271 \beta_{19} - 2367490794 \beta_{18} + 1751247849 \beta_{17} - 303500010 \beta_{16} + \cdots - 6488638207 \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 80539810 \beta_{12} - 449053836 \beta_{11} - 1401161559 \beta_{9} + 449428692 \beta_{8} + \cdots + 68335602072 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 7119252927 \beta_{19} + 70195866618 \beta_{18} - 50620927437 \beta_{17} + 10164281610 \beta_{16} + \cdots + 188992430287 \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 2711530394 \beta_{12} + 13978545196 \beta_{11} + 42246873255 \beta_{9} - 14321124824 \beta_{8} + \cdots - 1995227044836 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 205607405439 \beta_{19} - 2088884923962 \beta_{18} + 1478167114929 \beta_{17} + \cdots - 5558202554911 \beta_1 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/123\mathbb{Z}\right)^\times\).

\(n\) \(83\) \(88\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
40.1
− 4.75880i
4.75880i
− 3.47692i
3.47692i
− 3.47574i
3.47574i
− 1.38830i
1.38830i
− 1.26394i
1.26394i
1.02108i
− 1.02108i
2.40019i
− 2.40019i
3.04563i
− 3.04563i
4.39773i
− 4.39773i
5.49907i
− 5.49907i
−4.75880 − 3.00000i 14.6461 17.5127 14.2764i 19.4729i −31.6277 −9.00000 −83.3392
40.2 −4.75880 3.00000i 14.6461 17.5127 − 14.2764i − 19.4729i −31.6277 −9.00000 −83.3392
40.3 −3.47692 − 3.00000i 4.08898 −0.336022 10.4308i − 16.1851i 13.5983 −9.00000 1.16832
40.4 −3.47692 3.00000i 4.08898 −0.336022 − 10.4308i 16.1851i 13.5983 −9.00000 1.16832
40.5 −3.47574 − 3.00000i 4.08076 −19.6824 10.4272i 28.0811i 13.6223 −9.00000 68.4110
40.6 −3.47574 3.00000i 4.08076 −19.6824 − 10.4272i − 28.0811i 13.6223 −9.00000 68.4110
40.7 −1.38830 − 3.00000i −6.07262 −6.03337 4.16491i − 11.7247i 19.5370 −9.00000 8.37614
40.8 −1.38830 3.00000i −6.07262 −6.03337 − 4.16491i 11.7247i 19.5370 −9.00000 8.37614
40.9 −1.26394 − 3.00000i −6.40245 15.2627 3.79183i 0.605375i 18.2039 −9.00000 −19.2911
40.10 −1.26394 3.00000i −6.40245 15.2627 − 3.79183i − 0.605375i 18.2039 −9.00000 −19.2911
40.11 1.02108 − 3.00000i −6.95739 0.0766992 − 3.06325i 28.6326i −15.2728 −9.00000 0.0783164
40.12 1.02108 3.00000i −6.95739 0.0766992 3.06325i − 28.6326i −15.2728 −9.00000 0.0783164
40.13 2.40019 − 3.00000i −2.23908 −0.548817 − 7.20057i − 21.4561i −24.5758 −9.00000 −1.31727
40.14 2.40019 3.00000i −2.23908 −0.548817 7.20057i 21.4561i −24.5758 −9.00000 −1.31727
40.15 3.04563 − 3.00000i 1.27586 −20.1252 − 9.13689i − 0.579469i −20.4792 −9.00000 −61.2940
40.16 3.04563 3.00000i 1.27586 −20.1252 9.13689i 0.579469i −20.4792 −9.00000 −61.2940
40.17 4.39773 − 3.00000i 11.3400 13.7041 − 13.1932i 9.84660i 14.6884 −9.00000 60.2668
40.18 4.39773 3.00000i 11.3400 13.7041 13.1932i − 9.84660i 14.6884 −9.00000 60.2668
40.19 5.49907 − 3.00000i 22.2398 −7.83021 − 16.4972i − 22.6933i 78.3056 −9.00000 −43.0589
40.20 5.49907 3.00000i 22.2398 −7.83021 16.4972i 22.6933i 78.3056 −9.00000 −43.0589
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 40.20
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 123.4.d.a ✓ 20
3.b odd 2 1 369.4.d.c 20
41.b even 2 1 inner 123.4.d.a ✓ 20
123.b odd 2 1 369.4.d.c 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
123.4.d.a ✓ 20 1.a even 1 1 trivial
123.4.d.a ✓ 20 41.b even 2 1 inner
369.4.d.c 20 3.b odd 2 1
369.4.d.c 20 123.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(123, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{10} - 2 T^{9} + \cdots - 18216)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 9)^{10} \) Copy content Toggle raw display
$5$ \( (T^{10} + 8 T^{9} + \cdots + 969552)^{2} \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 24\!\cdots\!96 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 54\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 80\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 21\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 31\!\cdots\!16 \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots - 91\!\cdots\!56)^{2} \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 88\!\cdots\!96 \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 72\!\cdots\!12)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 46\!\cdots\!32)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 24\!\cdots\!01 \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 97\!\cdots\!72)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 15\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 57\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots + 18\!\cdots\!96)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 25\!\cdots\!12)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 42\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 18\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 21\!\cdots\!56)^{2} \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 68\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots - 42\!\cdots\!68)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 42\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 78\!\cdots\!36 \) Copy content Toggle raw display
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