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Properties

Label 765.2.k.c
Level $765$
Weight $2$
Character orbit 765.k
Analytic conductor $6.109$
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] = [765,2,Mod(361,765)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(765, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("765.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 765 = 3^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 765.k (of order \(4\), degree \(2\), 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: \(6.10855575463\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 26x^{14} + 277x^{12} + 1566x^{10} + 5064x^{8} + 9342x^{6} + 9109x^{4} + 3770x^{2} + 289 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 255)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{10} + \beta_{9} - \beta_{8} + \cdots - 1) q^{4}+ \cdots + ( - \beta_{7} + \beta_{5} + \cdots - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{10} + \beta_{9} - \beta_{8} + \cdots - 1) q^{4}+ \cdots + (5 \beta_{14} + 4 \beta_{13} + \cdots - 8) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 20 q^{4} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 20 q^{4} + 4 q^{7} - 4 q^{11} - 16 q^{13} + 16 q^{14} - 4 q^{16} + 12 q^{17} - 8 q^{20} - 20 q^{22} - 20 q^{23} - 8 q^{28} - 20 q^{29} - 12 q^{31} + 12 q^{34} - 4 q^{37} + 68 q^{38} + 12 q^{40} + 4 q^{41} + 44 q^{44} + 76 q^{46} - 40 q^{47} - 4 q^{50} + 72 q^{52} + 28 q^{56} + 20 q^{58} - 44 q^{61} + 36 q^{62} - 16 q^{64} + 4 q^{65} - 24 q^{67} - 48 q^{68} + 16 q^{71} + 12 q^{73} - 32 q^{74} - 24 q^{79} - 28 q^{82} + 8 q^{85} - 16 q^{86} + 116 q^{88} + 40 q^{89} + 84 q^{91} - 24 q^{92} - 16 q^{95} + 40 q^{97} - 56 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 26x^{14} + 277x^{12} + 1566x^{10} + 5064x^{8} + 9342x^{6} + 9109x^{4} + 3770x^{2} + 289 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{12} - 58\nu^{10} - 412\nu^{8} - 1328\nu^{6} - 1928\nu^{4} - 1030\nu^{2} - 55 ) / 24 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 9\nu^{15} + 217\nu^{13} + 2102\nu^{11} + 10524\nu^{9} + 28984\nu^{7} + 42666\nu^{5} + 29315\nu^{3} + 5863\nu ) / 408 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 10 \nu^{15} + 17 \nu^{14} - 226 \nu^{13} + 374 \nu^{12} - 2022 \nu^{11} + 3196 \nu^{10} + \cdots + 1530 ) / 408 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 20\nu^{15} + 469\nu^{13} + 4350\nu^{11} + 20372\nu^{9} + 50688\nu^{7} + 63760\nu^{5} + 33566\nu^{3} + 4017\nu ) / 408 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 53 \nu^{15} + 17 \nu^{14} + 1191 \nu^{13} + 459 \nu^{12} + 10482 \nu^{11} + 4862 \nu^{10} + \cdots + 1173 ) / 816 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 21 \nu^{15} - 17 \nu^{14} + 478 \nu^{13} - 374 \nu^{12} + 4270 \nu^{11} - 3196 \nu^{10} + 19082 \nu^{9} + \cdots - 1530 ) / 408 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 50 \nu^{15} + 51 \nu^{14} + 1130 \nu^{13} + 1139 \nu^{12} + 10008 \nu^{11} + 9962 \nu^{10} + \cdots + 3621 ) / 816 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 3\nu^{14} + 70\nu^{12} + 644\nu^{10} + 2976\nu^{8} + 7240\nu^{6} + 8742\nu^{4} + 4199\nu^{2} + 340 ) / 24 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 50 \nu^{15} - 51 \nu^{14} + 1130 \nu^{13} - 1139 \nu^{12} + 10008 \nu^{11} - 9962 \nu^{10} + \cdots - 3621 ) / 816 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 10 \nu^{15} + 51 \nu^{14} - 243 \nu^{13} + 1207 \nu^{12} - 2328 \nu^{11} + 11322 \nu^{10} + \cdots + 7259 ) / 408 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 3 \nu^{15} - 10 \nu^{14} - 67 \nu^{13} - 226 \nu^{12} - 586 \nu^{11} - 2004 \nu^{10} - 2564 \nu^{9} + \cdots - 850 ) / 48 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 31 \nu^{15} - 204 \nu^{14} - 653 \nu^{13} - 4590 \nu^{12} - 5306 \nu^{11} - 40460 \nu^{10} + \cdots - 17510 ) / 816 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 3 \nu^{15} - 10 \nu^{14} + 67 \nu^{13} - 226 \nu^{12} + 586 \nu^{11} - 2004 \nu^{10} + 2564 \nu^{9} + \cdots - 850 ) / 48 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 28 \nu^{15} - 119 \nu^{14} - 609 \nu^{13} - 2686 \nu^{12} - 5138 \nu^{11} - 23766 \nu^{10} + \cdots - 11322 ) / 408 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + \beta_{9} - \beta_{8} + \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + \beta_{5} - \beta_{4} - \beta_{3} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{13} + \beta_{12} + \beta_{11} - 6\beta_{10} - 8\beta_{9} + 6\beta_{8} + 2\beta_{5} + 2\beta_{4} - 7\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{15} + 3 \beta_{13} - 2 \beta_{12} + \beta_{10} + 7 \beta_{7} + \beta_{6} - 12 \beta_{5} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2 \beta_{15} - \beta_{14} + 13 \beta_{13} - 12 \beta_{12} - 10 \beta_{11} + 40 \beta_{10} + 56 \beta_{9} + \cdots - 89 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 11 \beta_{15} - 3 \beta_{14} - 37 \beta_{13} + 29 \beta_{12} - 9 \beta_{10} + 2 \beta_{8} - 44 \beta_{7} + \cdots + 11 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 26 \beta_{15} + 17 \beta_{14} - 121 \beta_{13} + 112 \beta_{12} + 80 \beta_{11} - 285 \beta_{10} + \cdots + 577 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 95 \beta_{15} + 49 \beta_{14} + 339 \beta_{13} - 293 \beta_{12} + 55 \beta_{10} - 40 \beta_{8} + \cdots - 95 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 244 \beta_{15} - 189 \beta_{14} + 1006 \beta_{13} - 951 \beta_{12} - 593 \beta_{11} + 2082 \beta_{10} + \cdots - 3925 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 762 \beta_{15} - 535 \beta_{14} - 2794 \beta_{13} + 2567 \beta_{12} - 270 \beta_{10} + 492 \beta_{8} + \cdots + 762 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 2032 \beta_{15} + 1762 \beta_{14} - 7944 \beta_{13} + 7674 \beta_{12} + 4262 \beta_{11} - 15306 \beta_{10} + \cdots + 27411 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 5912 \beta_{15} + 4934 \beta_{14} + 21912 \beta_{13} - 20934 \beta_{12} + 1008 \beta_{10} + \cdots - 5912 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 16000 \beta_{15} - 14992 \beta_{14} + 60978 \beta_{13} - 59970 \beta_{12} - 30290 \beta_{11} + \cdots - 194245 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 44978 \beta_{15} - 41648 \beta_{14} - 167238 \beta_{13} + 163908 \beta_{12} - 1314 \beta_{10} + \cdots + 44978 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/765\mathbb{Z}\right)^\times\).

\(n\) \(307\) \(496\) \(596\)
\(\chi(n)\) \(1\) \(-\beta_{5}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
− 2.69140i
− 2.31326i
− 1.67959i
− 0.952323i
0.312109i
1.48305i
1.88794i
1.95348i
− 1.95348i
− 1.88794i
− 1.48305i
− 0.312109i
0.952323i
1.67959i
2.31326i
2.69140i
− 2.69140i 0 −5.24366 0.707107 − 0.707107i 0 −1.47826 − 1.47826i 8.72999i 0 −1.90311 − 1.90311i
361.2 − 2.31326i 0 −3.35118 −0.707107 + 0.707107i 0 2.10381 + 2.10381i 3.12563i 0 1.63572 + 1.63572i
361.3 − 1.67959i 0 −0.821013 0.707107 − 0.707107i 0 3.02569 + 3.02569i − 1.98021i 0 −1.18765 − 1.18765i
361.4 − 0.952323i 0 1.09308 −0.707107 + 0.707107i 0 1.01478 + 1.01478i − 2.94561i 0 0.673394 + 0.673394i
361.5 0.312109i 0 1.90259 −0.707107 + 0.707107i 0 −3.20742 − 3.20742i 1.21803i 0 −0.220694 − 0.220694i
361.6 1.48305i 0 −0.199441 0.707107 − 0.707107i 0 2.85528 + 2.85528i 2.67032i 0 1.04868 + 1.04868i
361.7 1.88794i 0 −1.56432 0.707107 − 0.707107i 0 −3.40271 − 3.40271i 0.822542i 0 1.33498 + 1.33498i
361.8 1.95348i 0 −1.81606 −0.707107 + 0.707107i 0 1.08884 + 1.08884i 0.359313i 0 −1.38132 − 1.38132i
676.1 − 1.95348i 0 −1.81606 −0.707107 − 0.707107i 0 1.08884 − 1.08884i − 0.359313i 0 −1.38132 + 1.38132i
676.2 − 1.88794i 0 −1.56432 0.707107 + 0.707107i 0 −3.40271 + 3.40271i − 0.822542i 0 1.33498 − 1.33498i
676.3 − 1.48305i 0 −0.199441 0.707107 + 0.707107i 0 2.85528 − 2.85528i − 2.67032i 0 1.04868 − 1.04868i
676.4 − 0.312109i 0 1.90259 −0.707107 − 0.707107i 0 −3.20742 + 3.20742i − 1.21803i 0 −0.220694 + 0.220694i
676.5 0.952323i 0 1.09308 −0.707107 − 0.707107i 0 1.01478 − 1.01478i 2.94561i 0 0.673394 − 0.673394i
676.6 1.67959i 0 −0.821013 0.707107 + 0.707107i 0 3.02569 − 3.02569i 1.98021i 0 −1.18765 + 1.18765i
676.7 2.31326i 0 −3.35118 −0.707107 − 0.707107i 0 2.10381 − 2.10381i − 3.12563i 0 1.63572 − 1.63572i
676.8 2.69140i 0 −5.24366 0.707107 + 0.707107i 0 −1.47826 + 1.47826i − 8.72999i 0 −1.90311 + 1.90311i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 361.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 765.2.k.c 16
3.b odd 2 1 255.2.j.b ✓ 16
17.c even 4 1 inner 765.2.k.c 16
51.f odd 4 1 255.2.j.b ✓ 16
51.g odd 8 1 4335.2.a.bf 8
51.g odd 8 1 4335.2.a.bg 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
255.2.j.b ✓ 16 3.b odd 2 1
255.2.j.b ✓ 16 51.f odd 4 1
765.2.k.c 16 1.a even 1 1 trivial
765.2.k.c 16 17.c even 4 1 inner
4335.2.a.bf 8 51.g odd 8 1
4335.2.a.bg 8 51.g odd 8 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 26T_{2}^{14} + 277T_{2}^{12} + 1566T_{2}^{10} + 5064T_{2}^{8} + 9342T_{2}^{6} + 9109T_{2}^{4} + 3770T_{2}^{2} + 289 \) acting on \(S_{2}^{\mathrm{new}}(765, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 26 T^{14} + \cdots + 289 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{4} + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{16} - 4 T^{15} + \cdots + 26873856 \) Copy content Toggle raw display
$11$ \( T^{16} + 4 T^{15} + \cdots + 13075456 \) Copy content Toggle raw display
$13$ \( (T^{8} + 8 T^{7} + \cdots + 3008)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 6975757441 \) Copy content Toggle raw display
$19$ \( T^{16} + 148 T^{14} + \cdots + 64770304 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 1212153856 \) Copy content Toggle raw display
$29$ \( T^{16} + 20 T^{15} + \cdots + 58736896 \) Copy content Toggle raw display
$31$ \( T^{16} + 12 T^{15} + \cdots + 51609856 \) Copy content Toggle raw display
$37$ \( T^{16} + 4 T^{15} + \cdots + 1679616 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 2567651584 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 339738624 \) Copy content Toggle raw display
$47$ \( (T^{8} + 20 T^{7} + \cdots - 10172)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + 324 T^{14} + \cdots + 56130064 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 54426148798464 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 468929126656 \) Copy content Toggle raw display
$67$ \( (T^{8} + 12 T^{7} + \cdots - 43136)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 2120308752384 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 1473951396096 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 24526065664 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 285104738304 \) Copy content Toggle raw display
$89$ \( (T^{8} - 20 T^{7} + \cdots - 2544064)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} - 40 T^{15} + \cdots + 1048576 \) Copy content Toggle raw display
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