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Properties

Label 765.2.t.e
Level $765$
Weight $2$
Character orbit 765.t
Analytic conductor $6.109$
Analytic rank $0$
Dimension $12$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [765,2,Mod(64,765)]
 
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, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("765.64");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 765 = 3^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 765.t (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: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 24x^{10} + 188x^{8} + 572x^{6} + 776x^{4} + 464x^{2} + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 85)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( - \beta_{6} + \beta_1 + 1) q^{4} + \beta_{11} q^{5} + \beta_{8} q^{7} + (\beta_{3} + \beta_{2}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + ( - \beta_{6} + \beta_1 + 1) q^{4} + \beta_{11} q^{5} + \beta_{8} q^{7} + (\beta_{3} + \beta_{2}) q^{8} + ( - \beta_{11} - \beta_{9} + \cdots + \beta_1) q^{10}+ \cdots + (3 \beta_{11} + \beta_{8} + \cdots - 3 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{4} - 4 q^{10} - 16 q^{11} + 16 q^{14} + 4 q^{16} + 32 q^{20} - 4 q^{29} + 4 q^{31} - 20 q^{35} + 24 q^{40} - 16 q^{41} - 28 q^{44} - 4 q^{46} + 40 q^{50} - 12 q^{55} - 4 q^{56} - 48 q^{61} - 76 q^{64} - 8 q^{65} + 40 q^{71} - 28 q^{74} - 4 q^{79} - 12 q^{80} - 52 q^{85} + 16 q^{86} + 16 q^{89} + 36 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 24x^{10} + 188x^{8} + 572x^{6} + 776x^{4} + 464x^{2} + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 6 \nu^{11} - 7 \nu^{10} - 142 \nu^{9} - 160 \nu^{8} - 1078 \nu^{7} - 1136 \nu^{6} - 3016 \nu^{5} + \cdots - 920 ) / 40 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 8\nu^{10} + 185\nu^{8} + 1344\nu^{6} + 3440\nu^{4} + 3438\nu^{2} + 1080 ) / 20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -9\nu^{10} - 210\nu^{8} - 1552\nu^{6} - 4110\nu^{4} - 4204\nu^{2} - 1320 ) / 20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -11\nu^{11} - 256\nu^{9} - 1883\nu^{7} - 4948\nu^{5} - 5096\nu^{3} - 1666\nu ) / 20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -11\nu^{11} - 256\nu^{9} - 1883\nu^{7} - 4948\nu^{5} - 5096\nu^{3} - 1646\nu ) / 20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 6 \nu^{11} + 25 \nu^{10} - 142 \nu^{9} + 580 \nu^{8} - 1078 \nu^{7} + 4240 \nu^{6} - 3016 \nu^{5} + \cdots + 3560 ) / 40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 11 \nu^{11} + 26 \nu^{10} - 254 \nu^{9} + 610 \nu^{8} - 1838 \nu^{7} + 4558 \nu^{6} - 4642 \nu^{5} + \cdots + 4420 ) / 40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11 \nu^{11} + 26 \nu^{10} + 254 \nu^{9} + 610 \nu^{8} + 1838 \nu^{7} + 4558 \nu^{6} + 4642 \nu^{5} + \cdots + 4420 ) / 40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 15 \nu^{11} - 36 \nu^{10} + 352 \nu^{9} - 840 \nu^{8} + 2630 \nu^{7} - 6208 \nu^{6} + 7126 \nu^{5} + \cdots - 5560 ) / 40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 15 \nu^{11} + 36 \nu^{10} + 352 \nu^{9} + 840 \nu^{8} + 2630 \nu^{7} + 6208 \nu^{6} + 7126 \nu^{5} + \cdots + 5560 ) / 40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 34 \nu^{11} - 25 \nu^{10} + 796 \nu^{9} - 580 \nu^{8} + 5922 \nu^{7} - 4240 \nu^{6} + 15928 \nu^{5} + \cdots - 3560 ) / 40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{5} - \beta_{4} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - 2\beta_{2} - \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{11} + 3\beta_{6} - 8\beta_{5} + 10\beta_{4} + \beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{10} - \beta_{9} - 12\beta_{6} + 4\beta_{3} + 24\beta_{2} + 12\beta _1 + 34 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 22 \beta_{11} - 5 \beta_{10} - 5 \beta_{9} + \beta_{8} - \beta_{7} - 40 \beta_{6} + 82 \beta_{5} + \cdots - 18 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -24\beta_{10} + 24\beta_{9} + 6\beta_{8} + 6\beta_{7} + 140\beta_{6} - 68\beta_{3} - 268\beta_{2} - 140\beta _1 - 350 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 220 \beta_{11} + 98 \beta_{10} + 98 \beta_{9} - 30 \beta_{8} + 30 \beta_{7} + 476 \beta_{6} + \cdots + 256 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 384 \beta_{10} - 384 \beta_{9} - 128 \beta_{8} - 128 \beta_{7} - 1630 \beta_{6} + 928 \beta_{3} + \cdots + 3776 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 2224 \beta_{11} - 1440 \beta_{10} - 1440 \beta_{9} + 512 \beta_{8} - 512 \beta_{7} + \cdots - 3326 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 5278 \beta_{10} + 5278 \beta_{9} + 1952 \beta_{8} + 1952 \beta_{7} + 18904 \beta_{6} - 11756 \beta_{3} + \cdots - 41556 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 23068 \beta_{11} + 18986 \beta_{10} + 18986 \beta_{9} - 7230 \beta_{8} + 7230 \beta_{7} + \cdots + 41216 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(307\) \(496\) \(596\)
\(\chi(n)\) \(-1\) \(\beta_{4}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1
1.38621i
0.803330i
− 0.767611i
− 1.23239i
− 2.80333i
− 3.38621i
− 1.38621i
− 0.803330i
0.767611i
1.23239i
2.80333i
3.38621i
−2.38621 0 3.69399 0.518989 + 2.17501i 0 −0.155559 − 0.155559i −4.04223 0 −1.23842 − 5.19002i
64.2 −1.80333 0 1.25200 1.70211 − 1.45011i 0 −2.20051 − 2.20051i 1.34889 0 −3.06947 + 2.61503i
64.3 −0.232389 0 −1.94600 −0.898299 − 2.04770i 0 1.46067 + 1.46067i 0.917007 0 0.208755 + 0.475863i
64.4 0.232389 0 −1.94600 −2.04770 − 0.898299i 0 −1.46067 − 1.46067i −0.917007 0 −0.475863 − 0.208755i
64.5 1.80333 0 1.25200 −1.45011 + 1.70211i 0 2.20051 + 2.20051i −1.34889 0 −2.61503 + 3.06947i
64.6 2.38621 0 3.69399 2.17501 + 0.518989i 0 0.155559 + 0.155559i 4.04223 0 5.19002 + 1.23842i
514.1 −2.38621 0 3.69399 0.518989 − 2.17501i 0 −0.155559 + 0.155559i −4.04223 0 −1.23842 + 5.19002i
514.2 −1.80333 0 1.25200 1.70211 + 1.45011i 0 −2.20051 + 2.20051i 1.34889 0 −3.06947 − 2.61503i
514.3 −0.232389 0 −1.94600 −0.898299 + 2.04770i 0 1.46067 − 1.46067i 0.917007 0 0.208755 − 0.475863i
514.4 0.232389 0 −1.94600 −2.04770 + 0.898299i 0 −1.46067 + 1.46067i −0.917007 0 −0.475863 + 0.208755i
514.5 1.80333 0 1.25200 −1.45011 − 1.70211i 0 2.20051 − 2.20051i −1.34889 0 −2.61503 − 3.06947i
514.6 2.38621 0 3.69399 2.17501 − 0.518989i 0 0.155559 − 0.155559i 4.04223 0 5.19002 − 1.23842i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
17.c even 4 1 inner
85.j 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.t.e 12
3.b odd 2 1 85.2.j.c ✓ 12
5.b even 2 1 inner 765.2.t.e 12
15.d odd 2 1 85.2.j.c ✓ 12
15.e even 4 2 425.2.e.d 12
17.c even 4 1 inner 765.2.t.e 12
51.f odd 4 1 85.2.j.c ✓ 12
51.g odd 8 2 1445.2.b.f 12
85.j even 4 1 inner 765.2.t.e 12
255.i odd 4 1 85.2.j.c ✓ 12
255.k even 4 1 425.2.e.d 12
255.r even 4 1 425.2.e.d 12
255.v even 8 2 7225.2.a.bp 12
255.y odd 8 2 1445.2.b.f 12
255.ba even 8 2 7225.2.a.bp 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
85.2.j.c ✓ 12 3.b odd 2 1
85.2.j.c ✓ 12 15.d odd 2 1
85.2.j.c ✓ 12 51.f odd 4 1
85.2.j.c ✓ 12 255.i odd 4 1
425.2.e.d 12 15.e even 4 2
425.2.e.d 12 255.k even 4 1
425.2.e.d 12 255.r even 4 1
765.2.t.e 12 1.a even 1 1 trivial
765.2.t.e 12 5.b even 2 1 inner
765.2.t.e 12 17.c even 4 1 inner
765.2.t.e 12 85.j even 4 1 inner
1445.2.b.f 12 51.g odd 8 2
1445.2.b.f 12 255.y odd 8 2
7225.2.a.bp 12 255.v even 8 2
7225.2.a.bp 12 255.ba even 8 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 9T_{2}^{4} + 19T_{2}^{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(765, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - 9 T^{4} + 19 T^{2} - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - 8 T^{9} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( T^{12} + 112 T^{8} + \cdots + 4 \) Copy content Toggle raw display
$11$ \( (T^{6} + 8 T^{5} + 32 T^{4} + \cdots + 2)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 22 T^{4} + \cdots + 100)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} - 70 T^{10} + \cdots + 24137569 \) Copy content Toggle raw display
$19$ \( (T^{2} + 16)^{6} \) Copy content Toggle raw display
$23$ \( T^{12} + 4444 T^{8} + \cdots + 2500 \) Copy content Toggle raw display
$29$ \( (T^{6} + 2 T^{5} + \cdots + 2312)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 2 T^{5} + \cdots + 6498)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + 1084 T^{8} + \cdots + 40000 \) Copy content Toggle raw display
$41$ \( (T^{6} + 8 T^{5} + \cdots + 2592)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} - 66 T^{4} + \cdots - 1444)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 82 T^{4} + \cdots + 13924)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} - 156 T^{4} + \cdots - 110224)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 36)^{6} \) Copy content Toggle raw display
$61$ \( (T^{2} + 8 T + 32)^{6} \) Copy content Toggle raw display
$67$ \( (T^{6} + 214 T^{4} + \cdots + 93636)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 20 T^{5} + \cdots + 13122)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + 21952 T^{8} + \cdots + 262144 \) Copy content Toggle raw display
$79$ \( (T^{6} + 2 T^{5} + \cdots + 114242)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} - 58 T^{4} + \cdots - 2916)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} - 4 T^{2} + \cdots + 1590)^{4} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 13286025000000 \) Copy content Toggle raw display
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