Properties

Label 765.1.bz.a
Level $765$
Weight $1$
Character orbit 765.bz
Analytic conductor $0.382$
Analytic rank $0$
Dimension $16$
Projective image $D_{16}$
CM discriminant -15
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 765 = 3^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 765.bz (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.381784734664\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{16})\)
Coefficient field: \(\Q(\zeta_{32})\)
Defining polynomial: \(x^{16} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{16}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{16} - \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + ( \zeta_{32} - \zeta_{32}^{3} ) q^{2} + ( \zeta_{32}^{2} - \zeta_{32}^{4} + \zeta_{32}^{6} ) q^{4} + \zeta_{32}^{11} q^{5} + ( \zeta_{32}^{3} - \zeta_{32}^{5} + \zeta_{32}^{7} - \zeta_{32}^{9} ) q^{8} +O(q^{10})\) \( q + ( \zeta_{32} - \zeta_{32}^{3} ) q^{2} + ( \zeta_{32}^{2} - \zeta_{32}^{4} + \zeta_{32}^{6} ) q^{4} + \zeta_{32}^{11} q^{5} + ( \zeta_{32}^{3} - \zeta_{32}^{5} + \zeta_{32}^{7} - \zeta_{32}^{9} ) q^{8} + ( \zeta_{32}^{12} - \zeta_{32}^{14} ) q^{10} + ( \zeta_{32}^{4} - \zeta_{32}^{6} + \zeta_{32}^{8} - \zeta_{32}^{10} + \zeta_{32}^{12} ) q^{16} + \zeta_{32}^{15} q^{17} + ( \zeta_{32}^{8} - \zeta_{32}^{12} ) q^{19} + ( -\zeta_{32} + \zeta_{32}^{13} - \zeta_{32}^{15} ) q^{20} + ( \zeta_{32}^{5} - \zeta_{32}^{13} ) q^{23} -\zeta_{32}^{6} q^{25} + ( -\zeta_{32}^{4} + \zeta_{32}^{10} ) q^{31} + ( \zeta_{32}^{5} - \zeta_{32}^{7} + \zeta_{32}^{9} - \zeta_{32}^{11} + \zeta_{32}^{13} - \zeta_{32}^{15} ) q^{32} + ( -1 + \zeta_{32}^{2} ) q^{34} + ( \zeta_{32}^{9} - \zeta_{32}^{11} - \zeta_{32}^{13} + \zeta_{32}^{15} ) q^{38} + ( 1 - \zeta_{32}^{2} + \zeta_{32}^{4} + \zeta_{32}^{14} ) q^{40} + ( -1 + \zeta_{32}^{6} - \zeta_{32}^{8} - \zeta_{32}^{14} ) q^{46} + ( -\zeta_{32} - \zeta_{32}^{7} ) q^{47} -\zeta_{32}^{10} q^{49} + ( -\zeta_{32}^{7} + \zeta_{32}^{9} ) q^{50} + ( -\zeta_{32}^{9} - \zeta_{32}^{11} ) q^{53} + ( -\zeta_{32}^{2} - \zeta_{32}^{8} ) q^{61} + ( -\zeta_{32}^{5} + \zeta_{32}^{7} + \zeta_{32}^{11} - \zeta_{32}^{13} ) q^{62} + ( 1 - \zeta_{32}^{2} + \zeta_{32}^{6} - \zeta_{32}^{8} + \zeta_{32}^{10} - \zeta_{32}^{12} + \zeta_{32}^{14} ) q^{64} + ( -\zeta_{32} + \zeta_{32}^{3} - \zeta_{32}^{5} ) q^{68} + ( -1 + \zeta_{32}^{2} + \zeta_{32}^{10} - \zeta_{32}^{12} ) q^{76} + ( -\zeta_{32}^{4} + \zeta_{32}^{14} ) q^{79} + ( \zeta_{32} - \zeta_{32}^{3} + \zeta_{32}^{5} - \zeta_{32}^{7} + \zeta_{32}^{15} ) q^{80} + ( \zeta_{32} + \zeta_{32}^{3} ) q^{83} -\zeta_{32}^{10} q^{85} + ( -\zeta_{32} + \zeta_{32}^{3} + \zeta_{32}^{7} - \zeta_{32}^{9} + \zeta_{32}^{11} - \zeta_{32}^{15} ) q^{92} + ( -\zeta_{32}^{2} + \zeta_{32}^{4} - \zeta_{32}^{8} + \zeta_{32}^{10} ) q^{94} + ( -\zeta_{32}^{3} + \zeta_{32}^{7} ) q^{95} + ( -\zeta_{32}^{11} + \zeta_{32}^{13} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q - 16q^{34} + 16q^{40} - 16q^{46} + 16q^{64} - 16q^{76} + O(q^{100}) \)

Character values

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

\(n\) \(307\) \(496\) \(596\)
\(\chi(n)\) \(-1\) \(\zeta_{32}^{14}\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
−0.831470 + 0.555570i
0.831470 0.555570i
−0.555570 + 0.831470i
0.555570 0.831470i
−0.195090 + 0.980785i
0.195090 0.980785i
−0.980785 0.195090i
0.980785 + 0.195090i
−0.831470 0.555570i
0.831470 + 0.555570i
−0.555570 0.831470i
0.555570 + 0.831470i
−0.195090 0.980785i
0.195090 + 0.980785i
−0.980785 + 0.195090i
0.980785 0.195090i
−1.02656 0.425215i 0 0.165911 + 0.165911i −0.980785 + 0.195090i 0 0 0.325446 + 0.785695i 0 1.08979 + 0.216773i
109.2 1.02656 + 0.425215i 0 0.165911 + 0.165911i 0.980785 0.195090i 0 0 −0.325446 0.785695i 0 1.08979 + 0.216773i
199.1 −1.53636 + 0.636379i 0 1.24830 1.24830i 0.195090 0.980785i 0 0 −0.487064 + 1.17588i 0 0.324423 + 1.63099i
199.2 1.53636 0.636379i 0 1.24830 1.24830i −0.195090 + 0.980785i 0 0 0.487064 1.17588i 0 0.324423 + 1.63099i
244.1 −0.750661 + 1.81225i 0 −2.01367 2.01367i 0.831470 + 0.555570i 0 0 3.34861 1.38704i 0 −1.63099 + 1.08979i
244.2 0.750661 1.81225i 0 −2.01367 2.01367i −0.831470 0.555570i 0 0 −3.34861 + 1.38704i 0 −1.63099 + 1.08979i
334.1 −0.149316 + 0.360480i 0 0.599456 + 0.599456i 0.555570 0.831470i 0 0 −0.666080 + 0.275899i 0 0.216773 + 0.324423i
334.2 0.149316 0.360480i 0 0.599456 + 0.599456i −0.555570 + 0.831470i 0 0 0.666080 0.275899i 0 0.216773 + 0.324423i
379.1 −1.02656 + 0.425215i 0 0.165911 0.165911i −0.980785 0.195090i 0 0 0.325446 0.785695i 0 1.08979 0.216773i
379.2 1.02656 0.425215i 0 0.165911 0.165911i 0.980785 + 0.195090i 0 0 −0.325446 + 0.785695i 0 1.08979 0.216773i
469.1 −1.53636 0.636379i 0 1.24830 + 1.24830i 0.195090 + 0.980785i 0 0 −0.487064 1.17588i 0 0.324423 1.63099i
469.2 1.53636 + 0.636379i 0 1.24830 + 1.24830i −0.195090 0.980785i 0 0 0.487064 + 1.17588i 0 0.324423 1.63099i
649.1 −0.750661 1.81225i 0 −2.01367 + 2.01367i 0.831470 0.555570i 0 0 3.34861 + 1.38704i 0 −1.63099 1.08979i
649.2 0.750661 + 1.81225i 0 −2.01367 + 2.01367i −0.831470 + 0.555570i 0 0 −3.34861 1.38704i 0 −1.63099 1.08979i
694.1 −0.149316 0.360480i 0 0.599456 0.599456i 0.555570 + 0.831470i 0 0 −0.666080 0.275899i 0 0.216773 0.324423i
694.2 0.149316 + 0.360480i 0 0.599456 0.599456i −0.555570 0.831470i 0 0 0.666080 + 0.275899i 0 0.216773 0.324423i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 694.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
3.b odd 2 1 inner
5.b even 2 1 inner
17.e odd 16 1 inner
51.i even 16 1 inner
85.p odd 16 1 inner
255.be even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 765.1.bz.a 16
3.b odd 2 1 inner 765.1.bz.a 16
5.b even 2 1 inner 765.1.bz.a 16
5.c odd 4 2 3825.1.ck.a 16
15.d odd 2 1 CM 765.1.bz.a 16
15.e even 4 2 3825.1.ck.a 16
17.e odd 16 1 inner 765.1.bz.a 16
51.i even 16 1 inner 765.1.bz.a 16
85.o even 16 1 3825.1.ck.a 16
85.p odd 16 1 inner 765.1.bz.a 16
85.r even 16 1 3825.1.ck.a 16
255.bc odd 16 1 3825.1.ck.a 16
255.be even 16 1 inner 765.1.bz.a 16
255.bj odd 16 1 3825.1.ck.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
765.1.bz.a 16 1.a even 1 1 trivial
765.1.bz.a 16 3.b odd 2 1 inner
765.1.bz.a 16 5.b even 2 1 inner
765.1.bz.a 16 15.d odd 2 1 CM
765.1.bz.a 16 17.e odd 16 1 inner
765.1.bz.a 16 51.i even 16 1 inner
765.1.bz.a 16 85.p odd 16 1 inner
765.1.bz.a 16 255.be even 16 1 inner
3825.1.ck.a 16 5.c odd 4 2
3825.1.ck.a 16 15.e even 4 2
3825.1.ck.a 16 85.o even 16 1
3825.1.ck.a 16 85.r even 16 1
3825.1.ck.a 16 255.bc odd 16 1
3825.1.ck.a 16 255.bj odd 16 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(765, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + 32 T^{2} + 128 T^{4} - 192 T^{6} + 140 T^{8} - 16 T^{10} + T^{16} \)
$3$ \( T^{16} \)
$5$ \( 1 + T^{16} \)
$7$ \( T^{16} \)
$11$ \( T^{16} \)
$13$ \( T^{16} \)
$17$ \( 1 + T^{16} \)
$19$ \( ( 2 - 4 T + 2 T^{2} + T^{4} )^{4} \)
$23$ \( 256 + T^{16} \)
$29$ \( T^{16} \)
$31$ \( ( 2 + 8 T + 20 T^{2} + 16 T^{3} + 2 T^{4} + T^{8} )^{2} \)
$37$ \( T^{16} \)
$41$ \( T^{16} \)
$43$ \( T^{16} \)
$47$ \( 4 + 176 T^{4} + 148 T^{8} + 24 T^{12} + T^{16} \)
$53$ \( 4 + 32 T^{2} + 128 T^{4} - 192 T^{6} + 140 T^{8} - 16 T^{10} + T^{16} \)
$59$ \( T^{16} \)
$61$ \( ( 2 + 8 T + 4 T^{2} - 8 T^{3} + 6 T^{4} + 4 T^{6} + T^{8} )^{2} \)
$67$ \( T^{16} \)
$71$ \( T^{16} \)
$73$ \( T^{16} \)
$79$ \( ( 2 - 8 T + 12 T^{2} + 2 T^{4} + 8 T^{5} + T^{8} )^{2} \)
$83$ \( 4 - 32 T^{2} + 128 T^{4} + 192 T^{6} + 140 T^{8} + 16 T^{10} + T^{16} \)
$89$ \( T^{16} \)
$97$ \( T^{16} \)
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