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Properties

Label 765.2.be.a
Level $765$
Weight $2$
Character orbit 765.be
Analytic conductor $6.109$
Analytic rank $0$
Dimension $16$
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(406,765)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(765, base_ring=CyclotomicField(8))
 
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("765.406");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 765 = 3^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 765.be (of order \(8\), 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: \(6.10855575463\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 20x^{14} + 130x^{12} + 376x^{10} + 535x^{8} + 384x^{6} + 134x^{4} + 20x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 255)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{15} + \beta_{13} + \cdots - \beta_{7}) q^{2}+ \cdots + ( - \beta_{13} + \beta_{11} - \beta_{10} + \cdots - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{15} + \beta_{13} + \cdots - \beta_{7}) q^{2}+ \cdots + ( - 3 \beta_{15} + \beta_{14} + \cdots + 2 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{7} + 8 q^{16} - 8 q^{17} + 16 q^{19} + 8 q^{22} + 16 q^{23} - 40 q^{26} - 24 q^{28} + 16 q^{29} + 8 q^{40} + 16 q^{41} + 32 q^{43} - 16 q^{44} - 40 q^{46} + 8 q^{49} - 8 q^{50} - 32 q^{52} - 24 q^{53} - 16 q^{56} - 48 q^{58} + 24 q^{59} - 32 q^{61} + 24 q^{62} + 16 q^{67} + 16 q^{68} - 8 q^{70} - 16 q^{71} + 32 q^{73} + 8 q^{74} - 40 q^{76} + 8 q^{77} - 40 q^{79} + 8 q^{82} - 24 q^{83} - 8 q^{85} + 16 q^{86} + 48 q^{88} - 16 q^{91} + 88 q^{92} + 48 q^{94} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 20x^{14} + 130x^{12} + 376x^{10} + 535x^{8} + 384x^{6} + 134x^{4} + 20x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 4 \nu^{15} + 3 \nu^{14} - 85 \nu^{13} + 58 \nu^{12} - 613 \nu^{11} + 351 \nu^{10} - 2022 \nu^{9} + \cdots - 21 ) / 16 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -4\nu^{14} - 79\nu^{12} - 499\nu^{10} - 1356\nu^{8} - 1671\nu^{6} - 830\nu^{4} - 69\nu^{2} + 17 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{15} - 79\nu^{13} - 499\nu^{11} - 1356\nu^{9} - 1671\nu^{7} - 830\nu^{5} - 69\nu^{3} + 17\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4\nu^{15} + 79\nu^{13} + 499\nu^{11} + 1356\nu^{9} + 1671\nu^{7} + 830\nu^{5} + 69\nu^{3} - 25\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 12 \nu^{15} - 3 \nu^{14} + 231 \nu^{13} - 46 \nu^{12} + 1387 \nu^{11} - 123 \nu^{10} + 3476 \nu^{9} + \cdots + 87 ) / 16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 4 \nu^{15} - 7 \nu^{14} - 81 \nu^{13} - 140 \nu^{12} - 541 \nu^{11} - 907 \nu^{10} - 1652 \nu^{9} + \cdots - 47 ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{15} + 20\nu^{13} + 130\nu^{11} + 376\nu^{9} + 535\nu^{7} + 384\nu^{5} + 134\nu^{3} + 20\nu \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 17 \nu^{15} - 3 \nu^{14} + 342 \nu^{13} - 55 \nu^{12} + 2247 \nu^{11} - 294 \nu^{10} + 6595 \nu^{9} + \cdots + 12 ) / 16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 8 \nu^{15} - 12 \nu^{14} + 162 \nu^{13} - 235 \nu^{12} + 1078 \nu^{11} - 1463 \nu^{10} + 3229 \nu^{9} + \cdots - 51 ) / 8 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 27 \nu^{15} - 15 \nu^{14} + 528 \nu^{13} - 297 \nu^{12} + 3277 \nu^{11} - 1890 \nu^{10} + 8727 \nu^{9} + \cdots - 90 ) / 16 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 17 \nu^{15} + 3 \nu^{14} + 342 \nu^{13} + 55 \nu^{12} + 2247 \nu^{11} + 294 \nu^{10} + 6595 \nu^{9} + \cdots - 12 ) / 16 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 4 \nu^{15} + 25 \nu^{14} + 93 \nu^{13} + 492 \nu^{12} + 769 \nu^{11} + 3095 \nu^{10} + 2982 \nu^{9} + \cdots + 157 ) / 16 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 3 \nu^{15} - 30 \nu^{14} - 67 \nu^{13} - 583 \nu^{12} - 522 \nu^{11} - 3571 \nu^{10} - 1885 \nu^{9} + \cdots - 69 ) / 16 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 8 \nu^{15} - 12 \nu^{14} - 162 \nu^{13} - 235 \nu^{12} - 1078 \nu^{11} - 1463 \nu^{10} - 3229 \nu^{9} + \cdots - 51 ) / 8 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 27 \nu^{15} - 15 \nu^{14} - 528 \nu^{13} - 297 \nu^{12} - 3277 \nu^{11} - 1890 \nu^{10} - 8727 \nu^{9} + \cdots - 90 ) / 16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( -\beta_{4} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( - \beta_{14} + 2 \beta_{13} + \beta_{12} + 2 \beta_{11} - 4 \beta_{8} + \beta_{7} + 2 \beta_{6} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2 \beta_{15} - 4 \beta_{13} - 3 \beta_{12} + \beta_{11} - 2 \beta_{10} + \beta_{9} - \beta_{8} + \cdots + 2 \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 6 \beta_{15} + 10 \beta_{14} - 20 \beta_{13} - 13 \beta_{12} - 22 \beta_{11} - 6 \beta_{10} + \cdots + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 26 \beta_{15} - 4 \beta_{14} + 59 \beta_{13} + 42 \beta_{12} - 16 \beta_{11} + 26 \beta_{10} + \cdots - 25 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 85 \beta_{15} - 100 \beta_{14} + 205 \beta_{13} + 147 \beta_{12} + 238 \beta_{11} + 85 \beta_{10} + \cdots - 63 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 290 \beta_{15} + 64 \beta_{14} - 694 \beta_{13} - 483 \beta_{12} + 185 \beta_{11} - 290 \beta_{10} + \cdots + 272 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 984 \beta_{15} + 1032 \beta_{14} - 2162 \beta_{13} - 1613 \beta_{12} - 2578 \beta_{11} - 984 \beta_{10} + \cdots + 691 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 3146 \beta_{15} - 776 \beta_{14} + 7695 \beta_{13} + 5306 \beta_{12} - 2016 \beta_{11} + \cdots - 2917 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 10841 \beta_{15} - 10897 \beta_{14} + 23095 \beta_{13} + 17504 \beta_{12} + 27860 \beta_{11} + \cdots - 7522 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 33936 \beta_{15} + 8700 \beta_{14} - 83726 \beta_{13} - 57522 \beta_{12} + 21738 \beta_{11} + \cdots + 31318 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 117662 \beta_{15} + 116404 \beta_{14} - 247974 \beta_{13} - 189092 \beta_{12} - 300540 \beta_{11} + \cdots + 81391 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 365636 \beta_{15} - 95044 \beta_{14} + 905086 \beta_{13} + 620950 \beta_{12} - 234066 \beta_{11} + \cdots - 336814 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 1270722 \beta_{15} - 1249709 \beta_{14} + 2667780 \beta_{13} + 2039021 \beta_{12} + 3239178 \beta_{11} + \cdots - 878257 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 3938502 \beta_{15} + 1028952 \beta_{14} - 9761490 \beta_{13} - 6693517 \beta_{12} + 2520431 \beta_{11} + \cdots + 3625544 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(307\) \(496\) \(596\)
\(\chi(n)\) \(1\) \(\beta_{14}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
406.1
0.733914i
1.55182i
− 1.13761i
− 0.319701i
0.426500i
1.98771i
3.28192i
− 0.867710i
− 0.733914i
− 1.55182i
1.13761i
0.319701i
− 0.426500i
− 1.98771i
− 3.28192i
0.867710i
−1.67058 − 1.67058i 0 3.58167i 0.923880 − 0.382683i 0 −0.516070 − 0.213763i 2.64231 − 2.64231i 0 −2.18272 − 0.904111i
406.2 −1.16277 − 1.16277i 0 0.704063i −0.923880 + 0.382683i 0 1.20098 + 0.497461i −1.50687 + 1.50687i 0 1.51923 + 0.629286i
406.3 −0.0855339 − 0.0855339i 0 − 1.98537i −0.923880 + 0.382683i 0 −4.21465 − 1.74576i −0.340884 + 0.340884i 0 0.111755 + 0.0462906i
406.4 1.50467 + 1.50467i 0 2.52806i 0.923880 − 0.382683i 0 0.115526 + 0.0478525i −0.794553 + 0.794553i 0 1.96594 + 0.814321i
451.1 −0.950824 + 0.950824i 0 0.191866i −0.382683 + 0.923880i 0 −0.850283 − 2.05276i −2.08408 − 2.08408i 0 −0.514582 − 1.24231i
451.2 0.351368 − 0.351368i 0 1.75308i −0.382683 + 0.923880i 0 1.09859 + 2.65222i 1.31871 + 1.31871i 0 0.190159 + 0.459085i
451.3 0.491652 − 0.491652i 0 1.51656i 0.382683 − 0.923880i 0 −0.0728103 − 0.175780i 1.72892 + 1.72892i 0 −0.266080 − 0.642374i
451.4 1.52202 − 1.52202i 0 − 2.63308i 0.382683 − 0.923880i 0 −0.761279 − 1.83789i −0.963555 − 0.963555i 0 −0.823710 − 1.98861i
586.1 −1.67058 + 1.67058i 0 − 3.58167i 0.923880 + 0.382683i 0 −0.516070 + 0.213763i 2.64231 + 2.64231i 0 −2.18272 + 0.904111i
586.2 −1.16277 + 1.16277i 0 − 0.704063i −0.923880 − 0.382683i 0 1.20098 − 0.497461i −1.50687 − 1.50687i 0 1.51923 − 0.629286i
586.3 −0.0855339 + 0.0855339i 0 1.98537i −0.923880 − 0.382683i 0 −4.21465 + 1.74576i −0.340884 − 0.340884i 0 0.111755 − 0.0462906i
586.4 1.50467 − 1.50467i 0 − 2.52806i 0.923880 + 0.382683i 0 0.115526 − 0.0478525i −0.794553 − 0.794553i 0 1.96594 − 0.814321i
631.1 −0.950824 − 0.950824i 0 − 0.191866i −0.382683 − 0.923880i 0 −0.850283 + 2.05276i −2.08408 + 2.08408i 0 −0.514582 + 1.24231i
631.2 0.351368 + 0.351368i 0 − 1.75308i −0.382683 − 0.923880i 0 1.09859 − 2.65222i 1.31871 − 1.31871i 0 0.190159 − 0.459085i
631.3 0.491652 + 0.491652i 0 − 1.51656i 0.382683 + 0.923880i 0 −0.0728103 + 0.175780i 1.72892 − 1.72892i 0 −0.266080 + 0.642374i
631.4 1.52202 + 1.52202i 0 2.63308i 0.382683 + 0.923880i 0 −0.761279 + 1.83789i −0.963555 + 0.963555i 0 −0.823710 + 1.98861i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 406.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.d even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 765.2.be.a 16
3.b odd 2 1 255.2.w.a ✓ 16
17.d even 8 1 inner 765.2.be.a 16
51.g odd 8 1 255.2.w.a ✓ 16
51.i even 16 1 4335.2.a.bd 8
51.i even 16 1 4335.2.a.be 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
255.2.w.a ✓ 16 3.b odd 2 1
255.2.w.a ✓ 16 51.g odd 8 1
765.2.be.a 16 1.a even 1 1 trivial
765.2.be.a 16 17.d even 8 1 inner
4335.2.a.bd 8 51.i even 16 1
4335.2.a.be 8 51.i even 16 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 42T_{2}^{12} + 8T_{2}^{9} + 403T_{2}^{8} + 80T_{2}^{7} - 232T_{2}^{5} + 426T_{2}^{4} - 176T_{2}^{3} + 32T_{2}^{2} + 8T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(765, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 42 T^{12} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + 8 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{16} - 20 T^{14} + \cdots + 6561 \) Copy content Toggle raw display
$13$ \( T^{16} + 72 T^{14} + \cdots + 16 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 6975757441 \) Copy content Toggle raw display
$19$ \( T^{16} - 16 T^{15} + \cdots + 2719201 \) Copy content Toggle raw display
$23$ \( T^{16} - 16 T^{15} + \cdots + 2972176 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 921304609 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 15398824464 \) Copy content Toggle raw display
$37$ \( T^{16} - 68 T^{14} + \cdots + 36012001 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 782153089 \) Copy content Toggle raw display
$43$ \( T^{16} - 32 T^{15} + \cdots + 9048064 \) Copy content Toggle raw display
$47$ \( T^{16} + 264 T^{14} + \cdots + 18809569 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 669963531169 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 765744004624 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 247219772944 \) Copy content Toggle raw display
$67$ \( (T^{8} - 8 T^{7} + \cdots - 29327812)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 8671658573824 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 776792641 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 27\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( T^{16} + 24 T^{15} + \cdots + 73984 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 104372932624 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 107082097410304 \) Copy content Toggle raw display
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