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Properties

Label 786.2.e.g
Level $786$
Weight $2$
Character orbit 786.e
Analytic conductor $6.276$
Analytic rank $0$
Dimension $16$
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] = [786,2,Mod(61,786)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(786, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("786.61");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 786 = 2 \cdot 3 \cdot 131 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 786.e (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: \(6.27624159887\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{15} + 4 x^{14} - 5 x^{13} + 89 x^{12} + 21 x^{11} + 817 x^{10} - 296 x^{9} + 3720 x^{8} + \cdots + 961 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{7} q^{2} + (\beta_{7} + \beta_{4} + \beta_{3} + 1) q^{3} + ( - \beta_{7} - \beta_{4} - \beta_{3} - 1) q^{4} + ( - \beta_{9} - \beta_{7}) q^{5} - \beta_{3} q^{6} + \beta_{11} q^{7} + \beta_{3} q^{8} + \beta_{4} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{2} + (\beta_{7} + \beta_{4} + \beta_{3} + 1) q^{3} + ( - \beta_{7} - \beta_{4} - \beta_{3} - 1) q^{4} + ( - \beta_{9} - \beta_{7}) q^{5} - \beta_{3} q^{6} + \beta_{11} q^{7} + \beta_{3} q^{8} + \beta_{4} q^{9} + ( - \beta_{10} + \beta_{7} + \beta_1) q^{10} + ( - \beta_{14} - \beta_{13} + \cdots + \beta_{2}) q^{11}+ \cdots + (\beta_{13} + \beta_{7} + \beta_{4} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{2} + 4 q^{3} - 4 q^{4} + 8 q^{5} + 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{2} + 4 q^{3} - 4 q^{4} + 8 q^{5} + 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9} - 2 q^{10} + 8 q^{11} + 4 q^{12} - 14 q^{13} + 6 q^{14} + 2 q^{15} - 4 q^{16} - 10 q^{17} + 16 q^{18} - 16 q^{19} - 2 q^{20} - 6 q^{21} - 12 q^{22} + 20 q^{23} - 16 q^{24} - 12 q^{25} + 16 q^{26} + 4 q^{27} + 6 q^{28} - 12 q^{29} - 8 q^{30} + 28 q^{31} + 16 q^{32} + 12 q^{33} - 4 q^{35} - 4 q^{36} - 28 q^{37} + 14 q^{38} + 4 q^{39} + 8 q^{40} + 6 q^{41} + 4 q^{42} + 32 q^{43} - 12 q^{44} - 12 q^{45} - 10 q^{46} - 4 q^{47} + 4 q^{48} + 18 q^{49} - 12 q^{50} - 20 q^{51} - 4 q^{52} - 20 q^{53} + 4 q^{54} - 14 q^{55} - 4 q^{56} + 6 q^{57} + 18 q^{58} - 2 q^{59} + 12 q^{60} + 48 q^{61} + 8 q^{62} - 4 q^{63} - 4 q^{64} + 12 q^{65} - 8 q^{66} - 6 q^{67} + 20 q^{68} + 20 q^{69} - 4 q^{70} - 4 q^{72} - 28 q^{73} + 12 q^{74} + 2 q^{75} - 6 q^{76} - 10 q^{77} - 16 q^{78} - 32 q^{79} - 12 q^{80} - 4 q^{81} + 6 q^{82} + 60 q^{83} + 4 q^{84} - 26 q^{85} - 28 q^{86} - 18 q^{87} + 8 q^{88} + 40 q^{89} + 8 q^{90} + 16 q^{91} - 20 q^{92} - 28 q^{93} - 4 q^{94} - 48 q^{95} + 4 q^{96} + 10 q^{97} + 18 q^{98} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{15} + 4 x^{14} - 5 x^{13} + 89 x^{12} + 21 x^{11} + 817 x^{10} - 296 x^{9} + 3720 x^{8} + \cdots + 961 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 36\!\cdots\!95 \nu^{15} + \cdots + 31\!\cdots\!57 ) / 39\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 32\!\cdots\!11 \nu^{15} + \cdots - 25\!\cdots\!68 ) / 19\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 79\!\cdots\!48 \nu^{15} + \cdots - 65\!\cdots\!64 ) / 39\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 90\!\cdots\!13 \nu^{15} + \cdots - 19\!\cdots\!43 ) / 39\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 10\!\cdots\!13 \nu^{15} + \cdots + 16\!\cdots\!62 ) / 39\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 14\!\cdots\!98 \nu^{15} + \cdots + 80\!\cdots\!46 ) / 39\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 15\!\cdots\!22 \nu^{15} + \cdots - 33\!\cdots\!55 ) / 39\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 16\!\cdots\!54 \nu^{15} + \cdots - 99\!\cdots\!34 ) / 39\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 16\!\cdots\!03 \nu^{15} + \cdots + 50\!\cdots\!82 ) / 39\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 18\!\cdots\!10 \nu^{15} + \cdots - 43\!\cdots\!07 ) / 39\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 25\!\cdots\!89 \nu^{15} + \cdots + 83\!\cdots\!80 ) / 39\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 30\!\cdots\!27 \nu^{15} + \cdots + 86\!\cdots\!10 ) / 39\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 15\!\cdots\!82 \nu^{15} + \cdots + 51\!\cdots\!42 ) / 19\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 19\!\cdots\!87 \nu^{15} + \cdots + 54\!\cdots\!13 ) / 19\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 19\!\cdots\!51 \nu^{15} + \cdots - 37\!\cdots\!54 ) / 19\!\cdots\!91 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{15} - \beta_{13} - \beta_{10} + \beta_{5} - \beta_{4} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{15} - \beta_{12} + \beta_{11} + \beta_{8} + \beta_{7} + 2 \beta_{6} - 3 \beta_{5} - \beta_{4} + \cdots + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3 \beta_{15} + 7 \beta_{14} + 7 \beta_{13} - \beta_{11} + 6 \beta_{10} - 5 \beta_{9} + 5 \beta_{8} + \cdots + 11 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 11 \beta_{14} + 15 \beta_{13} + 11 \beta_{12} + 20 \beta_{11} + 6 \beta_{10} - 15 \beta_{9} + 6 \beta_{8} + \cdots + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 20 \beta_{15} + 20 \beta_{14} + 55 \beta_{13} - 5 \beta_{12} - 9 \beta_{11} + 32 \beta_{10} - 57 \beta_{9} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 134 \beta_{15} + 134 \beta_{13} + 193 \beta_{12} + 89 \beta_{10} - 107 \beta_{9} + 193 \beta_{7} + \cdots - 273 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 756 \beta_{15} - 459 \beta_{14} - 71 \beta_{12} + 71 \beta_{11} - 493 \beta_{8} - 1044 \beta_{7} + \cdots - 1044 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1002 \beta_{15} - 1671 \beta_{14} - 1671 \beta_{13} - 1906 \beta_{11} - 1015 \beta_{10} + 1490 \beta_{9} + \cdots - 3915 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 8380 \beta_{14} - 13541 \beta_{13} - 21 \beta_{12} - 277 \beta_{11} - 8211 \beta_{10} + 13454 \beta_{9} + \cdots - 8211 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 6341 \beta_{15} - 6341 \beta_{14} - 16671 \beta_{13} - 9650 \beta_{12} - 5908 \beta_{11} + \cdots + 6395 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 94183 \beta_{15} - 94183 \beta_{13} - 11247 \beta_{12} - 57800 \beta_{10} + 92397 \beta_{9} + \cdots + 94563 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 252075 \beta_{15} + 155587 \beta_{14} - 123116 \beta_{12} + 123116 \beta_{11} + 148296 \beta_{8} + \cdots + 581353 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 660521 \beta_{15} + 1068540 \beta_{14} + 1068540 \beta_{13} + 204329 \beta_{11} + 649401 \beta_{10} + \cdots + 1773306 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 3041349 \beta_{14} + 4920736 \beta_{13} + 1305026 \beta_{12} + 2114553 \beta_{11} + 2888417 \beta_{10} + \cdots + 2888417 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 3773367 \beta_{15} + 3773367 \beta_{14} + 9877743 \beta_{13} + 1517952 \beta_{12} + 934580 \beta_{11} + \cdots - 673909 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/786\mathbb{Z}\right)^\times\).

\(n\) \(133\) \(263\)
\(\chi(n)\) \(-1 - \beta_{3} - \beta_{4} - \beta_{7}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1
0.355083 + 1.09283i
−0.591903 − 1.82169i
0.562761 + 1.73200i
−0.634958 − 1.95420i
0.355083 − 1.09283i
−0.591903 + 1.82169i
0.562761 − 1.73200i
−0.634958 + 1.95420i
2.76608 − 2.00968i
−0.321657 + 0.233698i
−2.39278 + 1.73846i
0.757375 − 0.550265i
2.76608 + 2.00968i
−0.321657 − 0.233698i
−2.39278 − 1.73846i
0.757375 + 0.550265i
0.309017 − 0.951057i 0.809017 + 0.587785i −0.809017 − 0.587785i −1.04512 + 3.21653i 0.809017 − 0.587785i −0.0437379 + 0.134611i −0.809017 + 0.587785i 0.309017 + 0.951057i 2.73615 + 1.98793i
61.2 0.309017 − 0.951057i 0.809017 + 0.587785i −0.809017 − 0.587785i −0.389690 + 1.19934i 0.809017 − 0.587785i 0.314251 − 0.967165i −0.809017 + 0.587785i 0.309017 + 0.951057i 1.02022 + 0.741234i
61.3 0.309017 − 0.951057i 0.809017 + 0.587785i −0.809017 − 0.587785i −0.0445202 + 0.137019i 0.809017 − 0.587785i −1.37784 + 4.24057i −0.809017 + 0.587785i 0.309017 + 0.951057i 0.116555 + 0.0846825i
61.4 0.309017 − 0.951057i 0.809017 + 0.587785i −0.809017 − 0.587785i 1.24326 − 3.82635i 0.809017 − 0.587785i 0.107332 − 0.330334i −0.809017 + 0.587785i 0.309017 + 0.951057i −3.25489 − 2.36482i
451.1 0.309017 + 0.951057i 0.809017 − 0.587785i −0.809017 + 0.587785i −1.04512 − 3.21653i 0.809017 + 0.587785i −0.0437379 − 0.134611i −0.809017 − 0.587785i 0.309017 − 0.951057i 2.73615 − 1.98793i
451.2 0.309017 + 0.951057i 0.809017 − 0.587785i −0.809017 + 0.587785i −0.389690 − 1.19934i 0.809017 + 0.587785i 0.314251 + 0.967165i −0.809017 − 0.587785i 0.309017 − 0.951057i 1.02022 − 0.741234i
451.3 0.309017 + 0.951057i 0.809017 − 0.587785i −0.809017 + 0.587785i −0.0445202 − 0.137019i 0.809017 + 0.587785i −1.37784 − 4.24057i −0.809017 − 0.587785i 0.309017 − 0.951057i 0.116555 − 0.0846825i
451.4 0.309017 + 0.951057i 0.809017 − 0.587785i −0.809017 + 0.587785i 1.24326 + 3.82635i 0.809017 + 0.587785i 0.107332 + 0.330334i −0.809017 − 0.587785i 0.309017 − 0.951057i −3.25489 + 2.36482i
577.1 −0.809017 − 0.587785i −0.309017 − 0.951057i 0.309017 + 0.951057i −0.871577 − 0.633238i −0.309017 + 0.951057i −1.82645 − 1.32700i 0.309017 − 0.951057i −0.809017 + 0.587785i 0.332913 + 1.02460i
577.2 −0.809017 − 0.587785i −0.309017 − 0.951057i 0.309017 + 0.951057i −0.417204 − 0.303116i −0.309017 + 0.951057i 2.03118 + 1.47574i 0.309017 − 0.951057i −0.809017 + 0.587785i 0.159358 + 0.490452i
577.3 −0.809017 − 0.587785i −0.309017 − 0.951057i 0.309017 + 0.951057i 2.68330 + 1.94953i −0.309017 + 0.951057i −3.54363 − 2.57460i 0.309017 − 0.951057i −0.809017 + 0.587785i −1.02493 − 3.15441i
577.4 −0.809017 − 0.587785i −0.309017 − 0.951057i 0.309017 + 0.951057i 2.84155 + 2.06451i −0.309017 + 0.951057i 2.33890 + 1.69931i 0.309017 − 0.951057i −0.809017 + 0.587785i −1.08538 − 3.34044i
613.1 −0.809017 + 0.587785i −0.309017 + 0.951057i 0.309017 − 0.951057i −0.871577 + 0.633238i −0.309017 − 0.951057i −1.82645 + 1.32700i 0.309017 + 0.951057i −0.809017 − 0.587785i 0.332913 − 1.02460i
613.2 −0.809017 + 0.587785i −0.309017 + 0.951057i 0.309017 − 0.951057i −0.417204 + 0.303116i −0.309017 − 0.951057i 2.03118 − 1.47574i 0.309017 + 0.951057i −0.809017 − 0.587785i 0.159358 − 0.490452i
613.3 −0.809017 + 0.587785i −0.309017 + 0.951057i 0.309017 − 0.951057i 2.68330 − 1.94953i −0.309017 − 0.951057i −3.54363 + 2.57460i 0.309017 + 0.951057i −0.809017 − 0.587785i −1.02493 + 3.15441i
613.4 −0.809017 + 0.587785i −0.309017 + 0.951057i 0.309017 − 0.951057i 2.84155 − 2.06451i −0.309017 − 0.951057i 2.33890 − 1.69931i 0.309017 + 0.951057i −0.809017 − 0.587785i −1.08538 + 3.34044i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 61.4
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 786.2.e.g ✓ 16
131.c even 5 1 inner 786.2.e.g ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
786.2.e.g ✓ 16 1.a even 1 1 trivial
786.2.e.g ✓ 16 131.c even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} - 8 T_{5}^{15} + 48 T_{5}^{14} - 184 T_{5}^{13} + 626 T_{5}^{12} - 1524 T_{5}^{11} + \cdots + 256 \) acting on \(S_{2}^{\mathrm{new}}(786, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{16} - 8 T^{15} + \cdots + 256 \) Copy content Toggle raw display
$7$ \( T^{16} + 4 T^{15} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( T^{16} - 8 T^{15} + \cdots + 121 \) Copy content Toggle raw display
$13$ \( T^{16} + 14 T^{15} + \cdots + 92416 \) Copy content Toggle raw display
$17$ \( T^{16} + 10 T^{15} + \cdots + 6241 \) Copy content Toggle raw display
$19$ \( (T^{8} + 8 T^{7} - 22 T^{6} + \cdots + 41)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 156150016 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 10952878336 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 351037696 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 1809991936 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 7283769025 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 1495120008001 \) Copy content Toggle raw display
$47$ \( (T^{8} + 2 T^{7} + \cdots - 15056)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 43104403456 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 1229413046521 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 545021921536 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 722117550625 \) Copy content Toggle raw display
$71$ \( (T^{8} - 320 T^{6} + \cdots - 675536)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 38755434496 \) Copy content Toggle raw display
$79$ \( (T^{8} + 16 T^{7} + \cdots - 3492656)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 9615301731025 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 70218768752896 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 39\!\cdots\!56 \) Copy content Toggle raw display
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