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Properties

Label 51.3.f.a
Level $51$
Weight $3$
Character orbit 51.f
Analytic conductor $1.390$
Analytic rank $0$
Dimension $20$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [51,3,Mod(38,51)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(51, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 3])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("51.38"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 51 = 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 51.f (of order \(4\), degree \(2\), 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: \(1.38964934824\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
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: \( x^{20} + 62 x^{18} + 1545 x^{16} + 20120 x^{14} + 149608 x^{12} + 655792 x^{10} + 1690896 x^{8} + \cdots + 36864 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{11} q^{3} + ( - \beta_{2} + 1) q^{4} - \beta_{9} q^{5} + (\beta_{16} - \beta_{12} + \cdots + \beta_{4}) q^{6} + \beta_{14} q^{7} + \beta_{6} q^{8} + ( - \beta_{17} + \beta_{14} + \beta_{5}) q^{9}+ \cdots + ( - 3 \beta_{19} - 3 \beta_{18} + \cdots - 41) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 6 q^{3} + 24 q^{4} - 2 q^{6} - 4 q^{7} - 16 q^{10} - 42 q^{12} - 12 q^{13} - 64 q^{16} - 4 q^{18} + 88 q^{21} - 40 q^{22} - 82 q^{24} + 54 q^{27} - 160 q^{28} + 48 q^{31} + 264 q^{33} + 152 q^{34}+ \cdots - 864 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 62 x^{18} + 1545 x^{16} + 20120 x^{14} + 149608 x^{12} + 655792 x^{10} + 1690896 x^{8} + \cdots + 36864 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 7919745 \nu^{18} + 492814953 \nu^{16} + 12258696597 \nu^{14} + 157229046205 \nu^{12} + \cdots - 122390688608 ) / 222142804448 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 7919745 \nu^{18} + 492814953 \nu^{16} + 12258696597 \nu^{14} + 157229046205 \nu^{12} + \cdots + 544037724736 ) / 111071402224 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 259771135 \nu^{19} + 15345514850 \nu^{17} + 354036168087 \nu^{15} + \cdots - 101444380980992 \nu ) / 21325709227008 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 259771135 \nu^{19} - 15345514850 \nu^{17} - 354036168087 \nu^{15} + \cdots + 122770090208000 \nu ) / 21325709227008 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 69697255 \nu^{19} + 3517955978 \nu^{17} + 59827449759 \nu^{15} + 276263253968 \nu^{13} + \cdots - 93175577147648 \nu ) / 5331427306752 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 41389488 \nu^{18} + 2486765337 \nu^{16} + 59177259790 \nu^{14} + 719152463223 \nu^{12} + \cdots - 1348192531616 ) / 222142804448 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2593455673 \nu^{19} - 7345321076 \nu^{18} - 155511977494 \nu^{17} - 437372846768 \nu^{16} + \cdots - 98029092962304 ) / 7108569742336 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2593455673 \nu^{19} - 7345321076 \nu^{18} + 155511977494 \nu^{17} - 437372846768 \nu^{16} + \cdots - 98029092962304 ) / 7108569742336 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 8816446379 \nu^{19} + 25800119448 \nu^{18} + 518080984306 \nu^{17} + 1543647316224 \nu^{16} + \cdots + 418359861341184 ) / 21325709227008 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 8816446379 \nu^{19} + 25800119448 \nu^{18} - 518080984306 \nu^{17} + \cdots + 418359861341184 ) / 21325709227008 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 2437279993 \nu^{19} - 1199581490 \nu^{18} - 145535543582 \nu^{17} - 71484048784 \nu^{16} + \cdots - 30829072092672 ) / 3554284871168 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 2437279993 \nu^{19} + 1199581490 \nu^{18} - 145535543582 \nu^{17} + 71484048784 \nu^{16} + \cdots + 30829072092672 ) / 3554284871168 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 3679521206 \nu^{19} - 14062788663 \nu^{18} - 220023677584 \nu^{17} + \cdots - 205356277688064 ) / 5331427306752 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 17039951203 \nu^{19} - 22675769088 \nu^{18} - 1017770035514 \nu^{17} + \cdots - 478465624624128 ) / 21325709227008 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 17039951203 \nu^{19} + 22675769088 \nu^{18} - 1017770035514 \nu^{17} + \cdots + 478465624624128 ) / 21325709227008 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 22295734363 \nu^{19} + 34215191424 \nu^{18} + 1327464055154 \nu^{17} + \cdots + 527337831865344 ) / 21325709227008 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 7465697701 \nu^{19} - 15524016456 \nu^{18} - 445682449662 \nu^{17} + \cdots - 407281304748032 ) / 7108569742336 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 22498451843 \nu^{19} + 29481505056 \nu^{18} - 1346630642818 \nu^{17} + \cdots + 668513821006848 ) / 21325709227008 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 8027987995 \nu^{19} + 8060025303 \nu^{18} + 479104480322 \nu^{17} + 481261121016 \nu^{16} + \cdots + 148213015652352 ) / 5331427306752 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{4} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 2\beta _1 - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{16} - \beta_{13} + \beta_{8} + 3\beta_{5} - 12\beta_{4} - 16\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{18} + \beta_{15} - \beta_{14} + \beta_{13} - \beta_{12} + \beta_{11} - 2 \beta_{8} - \beta_{7} + \cdots + 70 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 2 \beta_{19} - 6 \beta_{18} + 32 \beta_{16} + 6 \beta_{15} + 4 \beta_{14} + 26 \beta_{13} + \cdots - 5 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 39 \beta_{18} - 12 \beta_{17} - 6 \beta_{16} - 41 \beta_{15} + 41 \beta_{14} - 33 \beta_{13} + \cdots - 1010 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 82 \beta_{19} + 202 \beta_{18} - 782 \beta_{16} - 216 \beta_{15} - 134 \beta_{14} - 580 \beta_{13} + \cdots + 161 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1039 \beta_{18} + 432 \beta_{17} + 216 \beta_{16} + 1133 \beta_{15} - 1133 \beta_{14} + 823 \beta_{13} + \cdots + 16038 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 2266 \beta_{19} - 5012 \beta_{18} + 17184 \beta_{16} + 5522 \beta_{15} + 3256 \beta_{14} + \cdots - 3879 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 23847 \beta_{18} - 11044 \beta_{17} - 5522 \beta_{16} - 26505 \beta_{15} + 26505 \beta_{14} + \cdots - 269474 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 53010 \beta_{19} + 110666 \beta_{18} - 357430 \beta_{16} - 123668 \beta_{15} - 70658 \beta_{14} + \cdots + 84161 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 508251 \beta_{18} + 247336 \beta_{17} + 123668 \beta_{16} + 570337 \beta_{15} - 570337 \beta_{14} + \cdots + 4707254 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 1140674 \beta_{19} - 2306656 \beta_{18} + 7205372 \beta_{16} + 2595542 \beta_{15} + \cdots - 1736319 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 10396763 \beta_{18} - 5191084 \beta_{17} - 2595542 \beta_{16} - 11722445 \beta_{15} + 11722445 \beta_{14} + \cdots - 84565186 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 23444890 \beta_{19} + 46559830 \beta_{18} - 142548930 \beta_{16} - 52567664 \beta_{15} + \cdots + 34837385 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 207588935 \beta_{18} + 105135328 \beta_{17} + 52567664 \beta_{16} + 234598829 \beta_{15} + \cdots + 1550104598 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 469197658 \beta_{19} - 922032164 \beta_{18} + 2787561552 \beta_{16} + 1042657738 \beta_{15} + \cdots - 687433335 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 4083726991 \beta_{18} - 2085315476 \beta_{17} - 1042657738 \beta_{16} - 4619930689 \beta_{15} + \cdots - 28820126914 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 9239861378 \beta_{19} + 18044580930 \beta_{18} - 54112628254 \beta_{16} - 20419229772 \beta_{15} + \cdots + 13424650241 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/51\mathbb{Z}\right)^\times\).

\(n\) \(35\) \(37\)
\(\chi(n)\) \(-1\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
38.1
− 4.37647i
− 3.55509i
− 3.52557i
− 2.28414i
− 1.20840i
− 0.791600i
0.284142i
1.52557i
1.55509i
2.37647i
4.37647i
3.55509i
3.52557i
2.28414i
1.20840i
0.791600i
− 0.284142i
− 1.52557i
− 1.55509i
− 2.37647i
−3.37647 −1.14794 + 2.77168i 7.40053 −1.81992 − 1.81992i 3.87599 − 9.35849i −8.06294 − 8.06294i −11.4818 −6.36445 − 6.36347i 6.14491 + 6.14491i
38.2 −2.55509 −2.60936 − 1.48028i 2.52849 5.68059 + 5.68059i 6.66716 + 3.78226i 1.53652 + 1.53652i 3.75983 4.61753 + 7.72518i −14.5144 − 14.5144i
38.3 −2.52557 2.88989 + 0.805308i 2.37853 −0.0213954 − 0.0213954i −7.29864 − 2.03387i 6.89994 + 6.89994i 4.09515 7.70296 + 4.65451i 0.0540358 + 0.0540358i
38.4 −1.28414 0.662833 − 2.92586i −2.35098 −2.50753 − 2.50753i −0.851171 + 3.75722i −3.77510 − 3.77510i 8.15556 −8.12131 − 3.87871i 3.22003 + 3.22003i
38.5 −0.208400 −2.91484 + 0.709732i −3.95657 −5.25648 − 5.25648i 0.607451 − 0.147908i 2.40158 + 2.40158i 1.65815 7.99256 − 4.13751i 1.09545 + 1.09545i
38.6 0.208400 −0.709732 + 2.91484i −3.95657 5.25648 + 5.25648i −0.147908 + 0.607451i 2.40158 + 2.40158i −1.65815 −7.99256 − 4.13751i 1.09545 + 1.09545i
38.7 1.28414 2.92586 − 0.662833i −2.35098 2.50753 + 2.50753i 3.75722 − 0.851171i −3.77510 − 3.77510i −8.15556 8.12131 − 3.87871i 3.22003 + 3.22003i
38.8 2.52557 −0.805308 − 2.88989i 2.37853 0.0213954 + 0.0213954i −2.03387 − 7.29864i 6.89994 + 6.89994i −4.09515 −7.70296 + 4.65451i 0.0540358 + 0.0540358i
38.9 2.55509 1.48028 + 2.60936i 2.52849 −5.68059 − 5.68059i 3.78226 + 6.66716i 1.53652 + 1.53652i −3.75983 −4.61753 + 7.72518i −14.5144 − 14.5144i
38.10 3.37647 −2.77168 + 1.14794i 7.40053 1.81992 + 1.81992i −9.35849 + 3.87599i −8.06294 − 8.06294i 11.4818 6.36445 − 6.36347i 6.14491 + 6.14491i
47.1 −3.37647 −1.14794 − 2.77168i 7.40053 −1.81992 + 1.81992i 3.87599 + 9.35849i −8.06294 + 8.06294i −11.4818 −6.36445 + 6.36347i 6.14491 − 6.14491i
47.2 −2.55509 −2.60936 + 1.48028i 2.52849 5.68059 − 5.68059i 6.66716 − 3.78226i 1.53652 − 1.53652i 3.75983 4.61753 − 7.72518i −14.5144 + 14.5144i
47.3 −2.52557 2.88989 − 0.805308i 2.37853 −0.0213954 + 0.0213954i −7.29864 + 2.03387i 6.89994 − 6.89994i 4.09515 7.70296 − 4.65451i 0.0540358 − 0.0540358i
47.4 −1.28414 0.662833 + 2.92586i −2.35098 −2.50753 + 2.50753i −0.851171 − 3.75722i −3.77510 + 3.77510i 8.15556 −8.12131 + 3.87871i 3.22003 − 3.22003i
47.5 −0.208400 −2.91484 − 0.709732i −3.95657 −5.25648 + 5.25648i 0.607451 + 0.147908i 2.40158 − 2.40158i 1.65815 7.99256 + 4.13751i 1.09545 − 1.09545i
47.6 0.208400 −0.709732 − 2.91484i −3.95657 5.25648 − 5.25648i −0.147908 − 0.607451i 2.40158 − 2.40158i −1.65815 −7.99256 + 4.13751i 1.09545 − 1.09545i
47.7 1.28414 2.92586 + 0.662833i −2.35098 2.50753 − 2.50753i 3.75722 + 0.851171i −3.77510 + 3.77510i −8.15556 8.12131 + 3.87871i 3.22003 − 3.22003i
47.8 2.52557 −0.805308 + 2.88989i 2.37853 0.0213954 − 0.0213954i −2.03387 + 7.29864i 6.89994 − 6.89994i −4.09515 −7.70296 − 4.65451i 0.0540358 − 0.0540358i
47.9 2.55509 1.48028 − 2.60936i 2.52849 −5.68059 + 5.68059i 3.78226 − 6.66716i 1.53652 − 1.53652i −3.75983 −4.61753 − 7.72518i −14.5144 + 14.5144i
47.10 3.37647 −2.77168 − 1.14794i 7.40053 1.81992 − 1.81992i −9.35849 − 3.87599i −8.06294 + 8.06294i 11.4818 6.36445 + 6.36347i 6.14491 − 6.14491i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 38.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
17.c even 4 1 inner
51.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 51.3.f.a ✓ 20
3.b odd 2 1 inner 51.3.f.a ✓ 20
17.c even 4 1 inner 51.3.f.a ✓ 20
51.f odd 4 1 inner 51.3.f.a ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.3.f.a ✓ 20 1.a even 1 1 trivial
51.3.f.a ✓ 20 3.b odd 2 1 inner
51.3.f.a ✓ 20 17.c even 4 1 inner
51.3.f.a ✓ 20 51.f odd 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(51, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{10} - 26 T^{8} + \cdots - 34)^{2} \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 3486784401 \) Copy content Toggle raw display
$5$ \( T^{20} + 7421 T^{16} + \cdots + 73984 \) Copy content Toggle raw display
$7$ \( (T^{10} + 2 T^{9} + \cdots + 19220000)^{2} \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{5} + 3 T^{4} + \cdots + 4880)^{4} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 40\!\cdots\!01 \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 7428023193600)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 76\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 99\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 29159379617792)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 532966176800)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 1268056166400)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots + 15\!\cdots\!24)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots - 10647782293504)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots - 47\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 94518380438048)^{2} \) Copy content Toggle raw display
$67$ \( (T^{5} + 102 T^{4} + \cdots - 507289600)^{4} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 515965452800)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 12\!\cdots\!92)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots - 33\!\cdots\!96)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 39\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 55\!\cdots\!00)^{2} \) Copy content Toggle raw display
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