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Properties

Label 1716.2.z.f
Level $1716$
Weight $2$
Character orbit 1716.z
Analytic conductor $13.702$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1716,2,Mod(157,1716)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1716, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 8, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1716.157");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1716 = 2^{2} \cdot 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1716.z (of order \(5\), degree \(4\), 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: \(13.7023289869\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 6 x^{19} + 41 x^{18} - 146 x^{17} + 650 x^{16} - 1400 x^{15} + 5756 x^{14} - 2122 x^{13} + \cdots + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{8} q^{3} + (\beta_{16} + \beta_{8} - \beta_{4} + 1) q^{5} + (\beta_{17} - \beta_{9} + \cdots + \beta_{2}) q^{7}+ \cdots - \beta_{10} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{8} q^{3} + (\beta_{16} + \beta_{8} - \beta_{4} + 1) q^{5} + (\beta_{17} - \beta_{9} + \cdots + \beta_{2}) q^{7}+ \cdots + (\beta_{14} + \beta_{11} + \beta_{9} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 5 q^{3} + 4 q^{5} - q^{7} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 5 q^{3} + 4 q^{5} - q^{7} - 5 q^{9} - 24 q^{11} - 5 q^{13} - 4 q^{15} - 6 q^{17} - 16 q^{19} + 6 q^{21} - 34 q^{23} + 13 q^{25} + 5 q^{27} + 4 q^{29} - 12 q^{31} - 11 q^{33} - 20 q^{37} + 5 q^{39} + 24 q^{41} - 32 q^{43} - 16 q^{45} - 6 q^{47} + 6 q^{49} - 9 q^{51} + 3 q^{53} - 20 q^{55} - 14 q^{57} - 61 q^{59} + 18 q^{61} - q^{63} - 16 q^{65} - 32 q^{67} - 6 q^{69} + 16 q^{71} + 17 q^{73} + 37 q^{75} + 22 q^{77} - 41 q^{79} - 5 q^{81} + 58 q^{83} + 42 q^{85} - 4 q^{87} - 6 q^{89} - q^{91} + 12 q^{93} + 55 q^{95} - 62 q^{97} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 6 x^{19} + 41 x^{18} - 146 x^{17} + 650 x^{16} - 1400 x^{15} + 5756 x^{14} - 2122 x^{13} + \cdots + 4096 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 16\!\cdots\!79 \nu^{19} + \cdots + 31\!\cdots\!60 ) / 34\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 19\!\cdots\!77 \nu^{19} + \cdots + 16\!\cdots\!56 ) / 68\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 28\!\cdots\!61 \nu^{19} + \cdots - 84\!\cdots\!56 ) / 34\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 19\!\cdots\!21 \nu^{19} + \cdots - 57\!\cdots\!48 ) / 13\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 37\!\cdots\!21 \nu^{19} + \cdots + 15\!\cdots\!08 ) / 13\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 22\!\cdots\!11 \nu^{19} + \cdots - 11\!\cdots\!52 ) / 68\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 45\!\cdots\!87 \nu^{19} + \cdots + 94\!\cdots\!92 ) / 13\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 45\!\cdots\!65 \nu^{19} + \cdots - 23\!\cdots\!56 ) / 13\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 41\!\cdots\!97 \nu^{19} + \cdots + 76\!\cdots\!20 ) / 68\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 47\!\cdots\!52 \nu^{19} + \cdots + 14\!\cdots\!64 ) / 68\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 11\!\cdots\!31 \nu^{19} + \cdots + 13\!\cdots\!04 ) / 13\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 83\!\cdots\!11 \nu^{19} + \cdots + 20\!\cdots\!88 ) / 68\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 85\!\cdots\!73 \nu^{19} + \cdots + 67\!\cdots\!48 ) / 68\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 18\!\cdots\!73 \nu^{19} + \cdots + 99\!\cdots\!24 ) / 13\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 23\!\cdots\!71 \nu^{19} + \cdots + 63\!\cdots\!56 ) / 13\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 23\!\cdots\!71 \nu^{19} + \cdots - 58\!\cdots\!00 ) / 13\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 11\!\cdots\!64 \nu^{19} + \cdots + 23\!\cdots\!32 ) / 68\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 24\!\cdots\!03 \nu^{19} + \cdots + 40\!\cdots\!88 ) / 13\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{19} - \beta_{18} + \beta_{16} - \beta_{15} + \beta_{14} - \beta_{13} + 2 \beta_{11} - 4 \beta_{10} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4 \beta_{19} - 2 \beta_{17} - 4 \beta_{16} + 4 \beta_{15} - 2 \beta_{14} - 2 \beta_{13} - 2 \beta_{12} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 18 \beta_{19} + \beta_{18} - 22 \beta_{17} - 38 \beta_{16} + 30 \beta_{15} - 19 \beta_{14} - 18 \beta_{12} + \cdots + 42 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4 \beta_{19} - 36 \beta_{18} - 82 \beta_{17} - 50 \beta_{16} + 6 \beta_{15} - 6 \beta_{14} + 46 \beta_{13} + \cdots + 218 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 146 \beta_{19} - 96 \beta_{18} - 7 \beta_{17} + 148 \beta_{16} - 155 \beta_{15} - 3 \beta_{14} + \cdots - 257 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 240 \beta_{19} - 38 \beta_{18} + 628 \beta_{17} + 38 \beta_{16} - 116 \beta_{15} - 590 \beta_{14} + \cdots - 7479 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2604 \beta_{19} + 2560 \beta_{18} + 2484 \beta_{17} - 2007 \beta_{16} - 44 \beta_{15} - 1872 \beta_{14} + \cdots - 31882 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 26940 \beta_{19} + 27138 \beta_{18} + 18110 \beta_{17} + 4724 \beta_{16} - 3500 \beta_{15} + \cdots - 25507 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 100093 \beta_{19} + 83493 \beta_{18} + 105290 \beta_{17} + 121890 \beta_{16} - 56057 \beta_{15} + \cdots + 288115 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 31704 \beta_{19} - 124184 \beta_{18} + 315612 \beta_{17} + 718592 \beta_{16} - 402980 \beta_{15} + \cdots + 1378628 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1590767 \beta_{19} - 1403914 \beta_{18} - 22196 \beta_{17} + 1403914 \beta_{16} - 633137 \beta_{15} + \cdots + 3759583 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 10500188 \beta_{19} - 4140504 \beta_{18} - 6948332 \beta_{17} - 6966482 \beta_{16} + 6359684 \beta_{15} + \cdots + 16076722 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 37275037 \beta_{19} - 13355504 \beta_{18} - 47939673 \beta_{17} - 56120074 \beta_{16} + 38786174 \beta_{15} + \cdots + 89771522 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 61945948 \beta_{19} - 63851950 \beta_{18} - 172569116 \beta_{17} - 170663114 \beta_{16} + \cdots + 204964017 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 101095619 \beta_{19} - 185408564 \beta_{18} - 263453331 \beta_{17} - 286873027 \beta_{16} + \cdots - 1367421120 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 1539708418 \beta_{19} + 411936758 \beta_{18} + 828416166 \beta_{17} - 411936758 \beta_{16} + \cdots - 14427740004 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 11533892862 \beta_{19} + 8682042135 \beta_{18} + 9505915651 \beta_{17} + 1273216601 \beta_{16} + \cdots - 60062192579 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 61882952940 \beta_{19} + 52187619270 \beta_{18} + 58261936430 \beta_{17} + 36754751610 \beta_{16} + \cdots - 97337281780 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1716\mathbb{Z}\right)^\times\).

\(n\) \(859\) \(925\) \(937\) \(1145\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{9}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
157.1
0.269117 + 0.828255i
0.636359 + 1.95851i
1.23039 + 3.78674i
−1.06605 − 3.28095i
−0.687849 − 2.11698i
1.13529 + 0.824835i
3.54295 + 2.57410i
−1.24982 − 0.908046i
−0.694526 − 0.504603i
−0.115857 − 0.0841749i
1.13529 − 0.824835i
3.54295 − 2.57410i
−1.24982 + 0.908046i
−0.694526 + 0.504603i
−0.115857 + 0.0841749i
0.269117 − 0.828255i
0.636359 − 1.95851i
1.23039 − 3.78674i
−1.06605 + 3.28095i
−0.687849 + 2.11698i
0 −0.309017 − 0.951057i 0 −3.24609 − 2.35843i 0 −0.788917 + 2.42804i 0 −0.809017 + 0.587785i 0
157.2 0 −0.309017 − 0.951057i 0 −0.439009 − 0.318959i 0 0.972664 − 2.99355i 0 −0.809017 + 0.587785i 0
157.3 0 −0.309017 − 0.951057i 0 0.554126 + 0.402596i 0 −0.256299 + 0.788808i 0 −0.809017 + 0.587785i 0
157.4 0 −0.309017 − 0.951057i 0 2.84595 + 2.06771i 0 0.644587 − 1.98383i 0 −0.809017 + 0.587785i 0
157.5 0 −0.309017 − 0.951057i 0 3.52109 + 2.55822i 0 −1.38105 + 4.25044i 0 −0.809017 + 0.587785i 0
313.1 0 0.809017 + 0.587785i 0 −0.786258 + 2.41985i 0 −3.55835 + 2.58529i 0 0.309017 + 0.951057i 0
313.2 0 0.809017 + 0.587785i 0 −0.671939 + 2.06802i 0 0.699083 − 0.507914i 0 0.309017 + 0.951057i 0
313.3 0 0.809017 + 0.587785i 0 −0.465225 + 1.43182i 0 1.83500 − 1.33320i 0 0.309017 + 0.951057i 0
313.4 0 0.809017 + 0.587785i 0 −0.343051 + 1.05580i 0 4.04943 − 2.94208i 0 0.309017 + 0.951057i 0
313.5 0 0.809017 + 0.587785i 0 1.03041 − 3.17126i 0 −2.71614 + 1.97339i 0 0.309017 + 0.951057i 0
625.1 0 0.809017 − 0.587785i 0 −0.786258 − 2.41985i 0 −3.55835 − 2.58529i 0 0.309017 − 0.951057i 0
625.2 0 0.809017 − 0.587785i 0 −0.671939 − 2.06802i 0 0.699083 + 0.507914i 0 0.309017 − 0.951057i 0
625.3 0 0.809017 − 0.587785i 0 −0.465225 − 1.43182i 0 1.83500 + 1.33320i 0 0.309017 − 0.951057i 0
625.4 0 0.809017 − 0.587785i 0 −0.343051 − 1.05580i 0 4.04943 + 2.94208i 0 0.309017 − 0.951057i 0
625.5 0 0.809017 − 0.587785i 0 1.03041 + 3.17126i 0 −2.71614 − 1.97339i 0 0.309017 − 0.951057i 0
1093.1 0 −0.309017 + 0.951057i 0 −3.24609 + 2.35843i 0 −0.788917 − 2.42804i 0 −0.809017 − 0.587785i 0
1093.2 0 −0.309017 + 0.951057i 0 −0.439009 + 0.318959i 0 0.972664 + 2.99355i 0 −0.809017 − 0.587785i 0
1093.3 0 −0.309017 + 0.951057i 0 0.554126 − 0.402596i 0 −0.256299 − 0.788808i 0 −0.809017 − 0.587785i 0
1093.4 0 −0.309017 + 0.951057i 0 2.84595 − 2.06771i 0 0.644587 + 1.98383i 0 −0.809017 − 0.587785i 0
1093.5 0 −0.309017 + 0.951057i 0 3.52109 − 2.55822i 0 −1.38105 − 4.25044i 0 −0.809017 − 0.587785i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 157.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1716.2.z.f ✓ 20
11.c even 5 1 inner 1716.2.z.f ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1716.2.z.f ✓ 20 1.a even 1 1 trivial
1716.2.z.f ✓ 20 11.c even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{20} - 4 T_{5}^{19} + 14 T_{5}^{18} - 19 T_{5}^{17} + 296 T_{5}^{16} - 728 T_{5}^{15} + \cdots + 495616 \) acting on \(S_{2}^{\mathrm{new}}(1716, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{5} \) Copy content Toggle raw display
$5$ \( T^{20} - 4 T^{19} + \cdots + 495616 \) Copy content Toggle raw display
$7$ \( T^{20} + T^{19} + \cdots + 81018001 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 25937424601 \) Copy content Toggle raw display
$13$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{5} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 176331841 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 466516801 \) Copy content Toggle raw display
$23$ \( (T^{10} + 17 T^{9} + \cdots - 1648880)^{2} \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 843263521 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 744932979025 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 1047101958400 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 556294867497216 \) Copy content Toggle raw display
$43$ \( (T^{10} + 16 T^{9} + \cdots + 88997936)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 4210166393161 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 5791971025 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 12\!\cdots\!41 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 175101239401 \) Copy content Toggle raw display
$67$ \( (T^{10} + 16 T^{9} + \cdots - 306256)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 23\!\cdots\!61 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 88985276416 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 29\!\cdots\!25 \) Copy content Toggle raw display
$89$ \( (T^{10} + 3 T^{9} + \cdots - 8669744)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 60\!\cdots\!96 \) Copy content Toggle raw display
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