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Properties

Label 51.5.c.c
Level $51$
Weight $5$
Character orbit 51.c
Analytic conductor $5.272$
Analytic rank $0$
Dimension $20$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [51,5,Mod(50,51)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(51, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("51.50");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 51 = 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 51.c (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: \(5.27186811728\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 200 x^{18} + 22051 x^{16} + 1226808 x^{14} + 5013252 x^{12} - 3569195664 x^{10} + \cdots + 12\!\cdots\!01 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{10} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{6} q^{3} + ( - \beta_{4} - 8) q^{4} - \beta_{14} q^{5} - \beta_{10} q^{6} + \beta_{15} q^{7} + (\beta_{9} - 7 \beta_1) q^{8} + (\beta_{5} - 20) q^{9} + ( - \beta_{18} + 2 \beta_{6}) q^{10}+ \cdots + ( - 31 \beta_{19} + \cdots + 243 \beta_{6}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 168 q^{4} - 400 q^{9} - 308 q^{13} - 72 q^{15} + 600 q^{16} - 124 q^{18} - 548 q^{19} + 16 q^{21} + 3200 q^{25} - 2580 q^{30} + 4088 q^{33} - 2072 q^{34} + 4588 q^{36} - 464 q^{42} + 9028 q^{43}+ \cdots - 13736 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 200 x^{18} + 22051 x^{16} + 1226808 x^{14} + 5013252 x^{12} - 3569195664 x^{10} + \cdots + 12\!\cdots\!01 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 9676288307 \nu^{18} + 2066932568335 \nu^{16} + 303759546502772 \nu^{14} + \cdots + 16\!\cdots\!61 ) / 74\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 22296163853 \nu^{18} - 62252999998105 \nu^{16} + \cdots - 14\!\cdots\!07 ) / 74\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 10596916 \nu^{18} - 823861717 \nu^{16} - 222708163958 \nu^{14} - 20897213752134 \nu^{12} + \cdots - 50\!\cdots\!57 ) / 53\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 47899676 \nu^{18} + 5914967161 \nu^{16} + 454247668586 \nu^{14} + 3196707526746 \nu^{12} + \cdots + 33\!\cdots\!13 ) / 16\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - \nu^{18} - 200 \nu^{16} - 22051 \nu^{14} - 1226808 \nu^{12} - 5013252 \nu^{10} + \cdots - 33\!\cdots\!80 ) / 18\!\cdots\!41 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{19} + 200 \nu^{17} + 22051 \nu^{15} + 1226808 \nu^{13} + 5013252 \nu^{11} + \cdots + 37\!\cdots\!00 \nu ) / 15\!\cdots\!21 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - \nu^{19} - 200 \nu^{17} - 22051 \nu^{15} - 1226808 \nu^{13} - 5013252 \nu^{11} + \cdots + 79\!\cdots\!63 \nu ) / 15\!\cdots\!21 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 158730996001 \nu^{19} - 122002712855335 \nu^{17} + \cdots - 13\!\cdots\!69 \nu ) / 20\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 174866532347 \nu^{18} - 6054603354571 \nu^{16} - 7355327030372 \nu^{14} + \cdots + 23\!\cdots\!07 ) / 24\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 907610110121 \nu^{19} - 118035894441973 \nu^{17} + \cdots + 73\!\cdots\!01 \nu ) / 60\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 229167004 \nu^{18} - 46766958929 \nu^{16} - 2796425671090 \nu^{14} + \cdots - 21\!\cdots\!89 ) / 16\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 828988309543 \nu^{18} - 183250833369281 \nu^{16} + \cdots - 16\!\cdots\!75 ) / 37\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 633534294115 \nu^{18} - 93776242199123 \nu^{16} + \cdots - 26\!\cdots\!57 ) / 24\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 58464901223 \nu^{19} - 7551515335633 \nu^{17} - 434954518032500 \nu^{15} + \cdots - 45\!\cdots\!35 \nu ) / 15\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 18885109733 \nu^{19} - 3863813826733 \nu^{17} - 170035349894360 \nu^{15} + \cdots - 15\!\cdots\!11 \nu ) / 49\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 347883676277 \nu^{19} + 69424256467225 \nu^{17} + \cdots + 50\!\cdots\!35 \nu ) / 66\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 7780409851645 \nu^{19} + \cdots - 85\!\cdots\!47 \nu ) / 60\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 129290273341 \nu^{19} - 27416167122101 \nu^{17} + \cdots - 30\!\cdots\!55 \nu ) / 98\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 234547778099 \nu^{19} + 21510636793915 \nu^{17} + \cdots + 21\!\cdots\!69 \nu ) / 10\!\cdots\!72 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta _1 - 60 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3 \beta_{19} + 9 \beta_{18} - 15 \beta_{17} + 9 \beta_{16} - 3 \beta_{15} + 18 \beta_{14} + \cdots + 5 \beta_{6} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 45 \beta_{12} - 9 \beta_{11} + 63 \beta_{9} + 28 \beta_{5} + 523 \beta_{4} - 80 \beta_{3} - 28 \beta_{2} + \cdots - 1425 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 651 \beta_{19} - 171 \beta_{18} - 102 \beta_{17} - 486 \beta_{16} - 321 \beta_{15} + \cdots - 5306 \beta_{6} ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 4482 \beta_{13} + 1764 \beta_{12} - 540 \beta_{11} - 4950 \beta_{9} - 667 \beta_{5} - 47458 \beta_{4} + \cdots + 483618 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 42474 \beta_{19} - 10116 \beta_{18} - 48030 \beta_{17} - 193680 \beta_{16} - 113592 \beta_{15} + \cdots + 112559 \beta_{6} ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 85320 \beta_{13} + 117360 \beta_{12} + 196056 \beta_{11} + 937692 \beta_{9} - 713276 \beta_{5} + \cdots + 753171 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1025922 \beta_{19} + 6022260 \beta_{18} - 1097454 \beta_{17} + 2422242 \beta_{16} + \cdots + 41402443 \beta_{6} ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 19939770 \beta_{13} + 16676316 \beta_{12} - 32255190 \beta_{11} - 29438298 \beta_{9} - 28540027 \beta_{5} + \cdots - 3412104702 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 207475341 \beta_{19} + 96119307 \beta_{18} - 17584647 \beta_{17} - 560202507 \beta_{16} + \cdots + 2135081579 \beta_{6} ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 3049074144 \beta_{13} + 1282313043 \beta_{12} + 1412735175 \beta_{11} - 261088263 \beta_{9} + \cdots - 183613046097 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 14516668581 \beta_{19} - 24712389801 \beta_{18} + 35536646076 \beta_{17} + \cdots + 41511557974 \beta_{6} ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 104328211032 \beta_{13} - 153044906400 \beta_{12} + 195191011512 \beta_{11} + 129975854544 \beta_{9} + \cdots - 4524796184604 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 117333258888 \beta_{19} + 259120246032 \beta_{18} - 1101385341240 \beta_{17} + \cdots + 12750738869741 \beta_{6} ) / 3 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 6337818083376 \beta_{13} - 1189044059040 \beta_{12} - 30691850959800 \beta_{11} + 9229689217080 \beta_{9} + \cdots - 814310579928765 ) / 3 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 687231001272 \beta_{19} + 45116932842000 \beta_{18} + 362332230399384 \beta_{17} + \cdots + 282679244400001 \beta_{6} ) / 3 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 84541607934696 \beta_{13} - 59745797980704 \beta_{12} + \cdots - 59\!\cdots\!48 ) / 3 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 85\!\cdots\!65 \beta_{19} + \cdots + 73\!\cdots\!29 \beta_{6} ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(35\) \(37\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
50.1
1.27725 − 8.90891i
−1.27725 + 8.90891i
8.75115 − 2.10174i
−8.75115 + 2.10174i
6.04116 + 6.67116i
−6.04116 − 6.67116i
3.53470 + 8.27683i
−3.53470 − 8.27683i
5.02953 − 7.46350i
−5.02953 + 7.46350i
5.02953 + 7.46350i
−5.02953 − 7.46350i
3.53470 − 8.27683i
−3.53470 + 8.27683i
6.04116 − 6.67116i
−6.04116 + 6.67116i
8.75115 + 2.10174i
−8.75115 − 2.10174i
1.27725 + 8.90891i
−1.27725 − 8.90891i
− 7.45221i −1.27725 − 8.90891i −39.5354 18.0441 −66.3910 + 9.51836i − 41.9203i 175.391i −77.7372 + 22.7579i − 134.468i
50.2 − 7.45221i 1.27725 + 8.90891i −39.5354 −18.0441 66.3910 − 9.51836i 41.9203i 175.391i −77.7372 + 22.7579i 134.468i
50.3 − 5.60864i −8.75115 − 2.10174i −15.4569 −14.8672 −11.7879 + 49.0821i 42.9260i − 3.04611i 72.1654 + 36.7853i 83.3848i
50.4 − 5.60864i 8.75115 + 2.10174i −15.4569 14.8672 11.7879 − 49.0821i − 42.9260i − 3.04611i 72.1654 + 36.7853i − 83.3848i
50.5 − 4.92716i −6.04116 + 6.67116i −8.27693 40.8754 32.8699 + 29.7658i − 9.63955i − 38.0528i −8.00871 − 80.6031i − 201.400i
50.6 − 4.92716i 6.04116 − 6.67116i −8.27693 −40.8754 −32.8699 − 29.7658i 9.63955i − 38.0528i −8.00871 − 80.6031i 201.400i
50.7 − 2.64556i −3.53470 + 8.27683i 9.00101 −41.0631 21.8968 + 9.35126i − 79.2595i − 66.1417i −56.0118 − 58.5122i 108.635i
50.8 − 2.64556i 3.53470 − 8.27683i 9.00101 41.0631 −21.8968 − 9.35126i 79.2595i − 66.1417i −56.0118 − 58.5122i − 108.635i
50.9 − 1.93178i −5.02953 − 7.46350i 12.2682 4.62649 −14.4178 + 9.71595i − 58.0263i − 54.6080i −30.4076 + 75.0758i − 8.93735i
50.10 − 1.93178i 5.02953 + 7.46350i 12.2682 −4.62649 14.4178 − 9.71595i 58.0263i − 54.6080i −30.4076 + 75.0758i 8.93735i
50.11 1.93178i −5.02953 + 7.46350i 12.2682 4.62649 −14.4178 − 9.71595i 58.0263i 54.6080i −30.4076 − 75.0758i 8.93735i
50.12 1.93178i 5.02953 − 7.46350i 12.2682 −4.62649 14.4178 + 9.71595i − 58.0263i 54.6080i −30.4076 − 75.0758i − 8.93735i
50.13 2.64556i −3.53470 − 8.27683i 9.00101 −41.0631 21.8968 − 9.35126i 79.2595i 66.1417i −56.0118 + 58.5122i − 108.635i
50.14 2.64556i 3.53470 + 8.27683i 9.00101 41.0631 −21.8968 + 9.35126i − 79.2595i 66.1417i −56.0118 + 58.5122i 108.635i
50.15 4.92716i −6.04116 − 6.67116i −8.27693 40.8754 32.8699 − 29.7658i 9.63955i 38.0528i −8.00871 + 80.6031i 201.400i
50.16 4.92716i 6.04116 + 6.67116i −8.27693 −40.8754 −32.8699 + 29.7658i − 9.63955i 38.0528i −8.00871 + 80.6031i − 201.400i
50.17 5.60864i −8.75115 + 2.10174i −15.4569 −14.8672 −11.7879 − 49.0821i − 42.9260i 3.04611i 72.1654 − 36.7853i − 83.3848i
50.18 5.60864i 8.75115 − 2.10174i −15.4569 14.8672 11.7879 + 49.0821i 42.9260i 3.04611i 72.1654 − 36.7853i 83.3848i
50.19 7.45221i −1.27725 + 8.90891i −39.5354 18.0441 −66.3910 − 9.51836i 41.9203i − 175.391i −77.7372 − 22.7579i 134.468i
50.20 7.45221i 1.27725 − 8.90891i −39.5354 −18.0441 66.3910 + 9.51836i − 41.9203i − 175.391i −77.7372 − 22.7579i − 134.468i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 50.20
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 51.5.c.c ✓ 20
3.b odd 2 1 inner 51.5.c.c ✓ 20
17.b even 2 1 inner 51.5.c.c ✓ 20
51.c odd 2 1 inner 51.5.c.c ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.5.c.c ✓ 20 1.a even 1 1 trivial
51.5.c.c ✓ 20 3.b odd 2 1 inner
51.5.c.c ✓ 20 17.b even 2 1 inner
51.5.c.c ✓ 20 51.c odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(51, [\chi])\):

\( T_{2}^{10} + 122T_{2}^{8} + 5079T_{2}^{6} + 86726T_{2}^{4} + 555892T_{2}^{2} + 1107720 \) Copy content Toggle raw display
\( T_{5}^{10} - 3925T_{5}^{8} + 4807776T_{5}^{6} - 1882685412T_{5}^{4} + 240880604544T_{5}^{2} - 4339677233664 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{10} + 122 T^{8} + \cdots + 1107720)^{2} \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 12\!\cdots\!01 \) Copy content Toggle raw display
$5$ \( (T^{10} + \cdots - 4339677233664)^{2} \) Copy content Toggle raw display
$7$ \( (T^{10} + \cdots + 63\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} + \cdots - 5675378866696)^{2} \) Copy content Toggle raw display
$13$ \( (T^{5} + 77 T^{4} + \cdots + 29775471712)^{4} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 16\!\cdots\!01 \) Copy content Toggle raw display
$19$ \( (T^{5} + \cdots - 2581305358016)^{4} \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots - 91\!\cdots\!96)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots - 22\!\cdots\!76)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 20\!\cdots\!20)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots - 26\!\cdots\!16)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} + \cdots - 14\!\cdots\!44)^{4} \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots + 86\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots + 19\!\cdots\!80)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots + 13\!\cdots\!80)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 35\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( (T^{5} + \cdots + 11\!\cdots\!48)^{4} \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots - 51\!\cdots\!04)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 98\!\cdots\!20)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 20\!\cdots\!80)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots + 11\!\cdots\!80)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 56\!\cdots\!20)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 23\!\cdots\!80)^{2} \) Copy content Toggle raw display
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