Properties

Label 799.1.h.b
Level $799$
Weight $1$
Character orbit 799.h
Analytic conductor $0.399$
Analytic rank $0$
Dimension $16$
Projective image $D_{40}$
CM discriminant -47
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 799 = 17 \cdot 47 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 799.h (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.398752945094\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\Q(\zeta_{40})\)
Defining polynomial: \(x^{16} - x^{12} + x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{40}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{40} + \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( \zeta_{40}^{11} - \zeta_{40}^{19} ) q^{2} + ( -\zeta_{40}^{7} + \zeta_{40}^{8} ) q^{3} + ( -\zeta_{40}^{2} + \zeta_{40}^{10} - \zeta_{40}^{18} ) q^{4} + ( -\zeta_{40}^{6} + \zeta_{40}^{7} - \zeta_{40}^{18} + \zeta_{40}^{19} ) q^{6} + ( -\zeta_{40}^{11} - \zeta_{40}^{14} ) q^{7} + ( -\zeta_{40} + \zeta_{40}^{9} - \zeta_{40}^{13} - \zeta_{40}^{17} ) q^{8} + ( \zeta_{40}^{14} - \zeta_{40}^{15} + \zeta_{40}^{16} ) q^{9} +O(q^{10})\) \( q + ( \zeta_{40}^{11} - \zeta_{40}^{19} ) q^{2} + ( -\zeta_{40}^{7} + \zeta_{40}^{8} ) q^{3} + ( -\zeta_{40}^{2} + \zeta_{40}^{10} - \zeta_{40}^{18} ) q^{4} + ( -\zeta_{40}^{6} + \zeta_{40}^{7} - \zeta_{40}^{18} + \zeta_{40}^{19} ) q^{6} + ( -\zeta_{40}^{11} - \zeta_{40}^{14} ) q^{7} + ( -\zeta_{40} + \zeta_{40}^{9} - \zeta_{40}^{13} - \zeta_{40}^{17} ) q^{8} + ( \zeta_{40}^{14} - \zeta_{40}^{15} + \zeta_{40}^{16} ) q^{9} + ( -\zeta_{40}^{5} + \zeta_{40}^{6} + \zeta_{40}^{9} - \zeta_{40}^{10} - \zeta_{40}^{17} + \zeta_{40}^{18} ) q^{12} + ( \zeta_{40}^{2} + \zeta_{40}^{5} - \zeta_{40}^{10} - \zeta_{40}^{13} ) q^{14} + ( -1 + \zeta_{40}^{4} + \zeta_{40}^{8} - \zeta_{40}^{12} - \zeta_{40}^{16} ) q^{16} + \zeta_{40}^{12} q^{17} + ( -\zeta_{40}^{5} + \zeta_{40}^{6} - \zeta_{40}^{7} + \zeta_{40}^{13} - \zeta_{40}^{14} + \zeta_{40}^{15} ) q^{18} + ( -\zeta_{40} + \zeta_{40}^{2} + \zeta_{40}^{18} - \zeta_{40}^{19} ) q^{21} + ( -1 + \zeta_{40} - \zeta_{40}^{4} + \zeta_{40}^{5} + \zeta_{40}^{8} - \zeta_{40}^{9} - \zeta_{40}^{16} + \zeta_{40}^{17} ) q^{24} + \zeta_{40}^{15} q^{25} + ( \zeta_{40} - \zeta_{40}^{2} + \zeta_{40}^{3} - \zeta_{40}^{4} ) q^{27} + ( \zeta_{40} + \zeta_{40}^{4} - \zeta_{40}^{9} - \zeta_{40}^{12} + \zeta_{40}^{13} + \zeta_{40}^{16} ) q^{28} + ( \zeta_{40}^{3} + \zeta_{40}^{7} - \zeta_{40}^{11} + \zeta_{40}^{19} ) q^{32} + ( -\zeta_{40}^{3} + \zeta_{40}^{11} ) q^{34} + ( -\zeta_{40}^{4} + \zeta_{40}^{5} - \zeta_{40}^{6} + \zeta_{40}^{12} - \zeta_{40}^{13} + \zeta_{40}^{14} - \zeta_{40}^{16} + \zeta_{40}^{17} - \zeta_{40}^{18} ) q^{36} + ( \zeta_{40}^{17} + \zeta_{40}^{18} ) q^{37} + ( -1 + \zeta_{40} - \zeta_{40}^{9} + \zeta_{40}^{10} - \zeta_{40}^{12} + \zeta_{40}^{13} + \zeta_{40}^{17} - \zeta_{40}^{18} ) q^{42} + \zeta_{40}^{10} q^{47} + ( 1 - \zeta_{40}^{3} + \zeta_{40}^{4} + \zeta_{40}^{7} - \zeta_{40}^{8} - \zeta_{40}^{11} + \zeta_{40}^{12} - \zeta_{40}^{15} + \zeta_{40}^{16} + \zeta_{40}^{19} ) q^{48} + ( -\zeta_{40}^{2} - \zeta_{40}^{5} - \zeta_{40}^{8} ) q^{49} + ( -\zeta_{40}^{6} + \zeta_{40}^{14} ) q^{50} + ( -1 - \zeta_{40}^{19} ) q^{51} + ( \zeta_{40}^{4} + \zeta_{40}^{6} ) q^{53} + ( 1 - \zeta_{40} + \zeta_{40}^{2} - \zeta_{40}^{3} + \zeta_{40}^{12} - \zeta_{40}^{13} + \zeta_{40}^{14} - \zeta_{40}^{15} ) q^{54} + ( 1 + \zeta_{40}^{3} - \zeta_{40}^{4} - \zeta_{40}^{7} - \zeta_{40}^{8} - \zeta_{40}^{11} + \zeta_{40}^{12} + \zeta_{40}^{15} ) q^{56} + ( \zeta_{40}^{11} + \zeta_{40}^{19} ) q^{59} + ( -\zeta_{40}^{9} - \zeta_{40}^{16} ) q^{61} + ( \zeta_{40}^{5} - \zeta_{40}^{6} + \zeta_{40}^{7} + \zeta_{40}^{8} - \zeta_{40}^{9} + \zeta_{40}^{10} ) q^{63} + ( \zeta_{40}^{2} - \zeta_{40}^{10} + \zeta_{40}^{18} ) q^{64} + ( -\zeta_{40}^{2} + \zeta_{40}^{10} - \zeta_{40}^{14} ) q^{68} + ( \zeta_{40}^{3} - \zeta_{40}^{12} ) q^{71} + ( -\zeta_{40}^{3} + \zeta_{40}^{4} - \zeta_{40}^{5} + \zeta_{40}^{7} - \zeta_{40}^{8} + \zeta_{40}^{9} + \zeta_{40}^{11} - \zeta_{40}^{12} + \zeta_{40}^{13} - \zeta_{40}^{15} + \zeta_{40}^{16} - \zeta_{40}^{17} ) q^{72} + ( -\zeta_{40}^{8} - \zeta_{40}^{9} + \zeta_{40}^{16} + \zeta_{40}^{17} ) q^{74} + ( \zeta_{40}^{2} - \zeta_{40}^{3} ) q^{75} + ( \zeta_{40}^{8} - \zeta_{40}^{17} ) q^{79} + ( -\zeta_{40}^{8} + \zeta_{40}^{9} - \zeta_{40}^{10} + \zeta_{40}^{11} - \zeta_{40}^{12} ) q^{81} + ( -1 - \zeta_{40}^{10} ) q^{83} + ( 2 - \zeta_{40} + \zeta_{40}^{3} - \zeta_{40}^{4} - \zeta_{40}^{8} + \zeta_{40}^{9} - \zeta_{40}^{11} + \zeta_{40}^{12} + \zeta_{40}^{16} - \zeta_{40}^{17} + \zeta_{40}^{19} ) q^{84} + ( -\zeta_{40}^{4} - \zeta_{40}^{16} ) q^{89} + ( -\zeta_{40} + \zeta_{40}^{9} ) q^{94} + ( \zeta_{40}^{6} - \zeta_{40}^{7} - \zeta_{40}^{10} + \zeta_{40}^{11} - \zeta_{40}^{14} + \zeta_{40}^{15} + \zeta_{40}^{18} - \zeta_{40}^{19} ) q^{96} + ( -\zeta_{40} - \zeta_{40}^{14} ) q^{97} + ( -\zeta_{40} - \zeta_{40}^{4} - \zeta_{40}^{7} - \zeta_{40}^{13} - \zeta_{40}^{16} - \zeta_{40}^{19} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 4q^{3} - 4q^{9} + O(q^{10}) \) \( 16q - 4q^{3} - 4q^{9} - 16q^{16} + 4q^{17} - 20q^{24} - 4q^{27} - 4q^{28} + 4q^{36} - 20q^{42} + 24q^{48} + 4q^{49} - 16q^{51} + 4q^{53} + 20q^{54} + 20q^{56} + 4q^{61} - 4q^{63} - 4q^{71} - 4q^{79} - 16q^{83} + 32q^{84} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/799\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(377\)
\(\chi(n)\) \(-1\) \(-\zeta_{40}^{15}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
93.1
−0.453990 0.891007i
−0.987688 + 0.156434i
−0.156434 + 0.987688i
0.891007 + 0.453990i
−0.891007 0.453990i
0.156434 0.987688i
0.987688 0.156434i
0.453990 + 0.891007i
−0.891007 + 0.453990i
0.156434 + 0.987688i
0.987688 + 0.156434i
0.453990 0.891007i
−0.453990 + 0.891007i
−0.987688 0.156434i
−0.156434 0.987688i
0.891007 0.453990i
−1.34500 + 1.34500i −0.652583 + 1.57547i 2.61803i 0 −1.24129 2.99673i 1.84206 0.763007i 2.17625 + 2.17625i −1.34915 1.34915i 0
93.2 −0.831254 + 0.831254i 0.763007 1.84206i 0.381966i 0 0.896969 + 2.16547i 0.431351 0.178671i −0.513743 0.513743i −2.10391 2.10391i 0
93.3 0.831254 0.831254i −0.581990 + 1.40505i 0.381966i 0 0.684170 + 1.65173i −1.57547 + 0.652583i 0.513743 + 0.513743i −0.928339 0.928339i 0
93.4 1.34500 1.34500i 0.178671 0.431351i 2.61803i 0 −0.339853 0.820478i −1.40505 + 0.581990i −2.17625 2.17625i 0.552967 + 0.552967i 0
281.1 −1.34500 + 1.34500i −1.79671 0.744220i 2.61803i 0 3.41754 1.41559i −0.497066 1.20002i 2.17625 + 2.17625i 1.96718 + 1.96718i 0
281.2 −0.831254 + 0.831254i 1.20002 + 0.497066i 0.381966i 0 −1.41071 + 0.584336i 0.399903 + 0.965451i −0.513743 0.513743i 0.485875 + 0.485875i 0
281.3 0.831254 0.831254i −0.144974 0.0600500i 0.381966i 0 −0.170427 + 0.0705930i 0.744220 + 1.79671i 0.513743 + 0.513743i −0.689695 0.689695i 0
281.4 1.34500 1.34500i −0.965451 0.399903i 2.61803i 0 −1.83640 + 0.760661i 0.0600500 + 0.144974i −2.17625 2.17625i 0.0650673 + 0.0650673i 0
563.1 −1.34500 1.34500i −1.79671 + 0.744220i 2.61803i 0 3.41754 + 1.41559i −0.497066 + 1.20002i 2.17625 2.17625i 1.96718 1.96718i 0
563.2 −0.831254 0.831254i 1.20002 0.497066i 0.381966i 0 −1.41071 0.584336i 0.399903 0.965451i −0.513743 + 0.513743i 0.485875 0.485875i 0
563.3 0.831254 + 0.831254i −0.144974 + 0.0600500i 0.381966i 0 −0.170427 0.0705930i 0.744220 1.79671i 0.513743 0.513743i −0.689695 + 0.689695i 0
563.4 1.34500 + 1.34500i −0.965451 + 0.399903i 2.61803i 0 −1.83640 0.760661i 0.0600500 0.144974i −2.17625 + 2.17625i 0.0650673 0.0650673i 0
610.1 −1.34500 1.34500i −0.652583 1.57547i 2.61803i 0 −1.24129 + 2.99673i 1.84206 + 0.763007i 2.17625 2.17625i −1.34915 + 1.34915i 0
610.2 −0.831254 0.831254i 0.763007 + 1.84206i 0.381966i 0 0.896969 2.16547i 0.431351 + 0.178671i −0.513743 + 0.513743i −2.10391 + 2.10391i 0
610.3 0.831254 + 0.831254i −0.581990 1.40505i 0.381966i 0 0.684170 1.65173i −1.57547 0.652583i 0.513743 0.513743i −0.928339 + 0.928339i 0
610.4 1.34500 + 1.34500i 0.178671 + 0.431351i 2.61803i 0 −0.339853 + 0.820478i −1.40505 0.581990i −2.17625 + 2.17625i 0.552967 0.552967i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 610.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
47.b odd 2 1 CM by \(\Q(\sqrt{-47}) \)
17.d even 8 1 inner
799.h odd 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 799.1.h.b 16
17.d even 8 1 inner 799.1.h.b 16
47.b odd 2 1 CM 799.1.h.b 16
799.h odd 8 1 inner 799.1.h.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
799.1.h.b 16 1.a even 1 1 trivial
799.1.h.b 16 17.d even 8 1 inner
799.1.h.b 16 47.b odd 2 1 CM
799.1.h.b 16 799.h odd 8 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 15 T_{2}^{4} + 25 \) acting on \(S_{1}^{\mathrm{new}}(799, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 25 + 15 T^{4} + T^{8} )^{2} \)
$3$ \( 1 + 12 T + 46 T^{2} + 48 T^{3} + 194 T^{4} + 328 T^{5} + 212 T^{6} + 4 T^{7} - 69 T^{8} - 40 T^{9} + 32 T^{10} + 44 T^{11} + 34 T^{12} + 20 T^{13} + 10 T^{14} + 4 T^{15} + T^{16} \)
$5$ \( T^{16} \)
$7$ \( 1 - 8 T + 58 T^{2} - 136 T^{3} + 82 T^{4} + 52 T^{5} + 28 T^{6} + 72 T^{7} + 75 T^{8} + 52 T^{9} + 16 T^{10} - 16 T^{11} + 2 T^{12} + 4 T^{13} - 2 T^{14} + T^{16} \)
$11$ \( T^{16} \)
$13$ \( T^{16} \)
$17$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \)
$19$ \( T^{16} \)
$23$ \( T^{16} \)
$29$ \( T^{16} \)
$31$ \( T^{16} \)
$37$ \( 1 + 8 T + 58 T^{2} + 136 T^{3} + 82 T^{4} - 52 T^{5} + 28 T^{6} - 72 T^{7} + 75 T^{8} - 52 T^{9} + 16 T^{10} + 16 T^{11} + 2 T^{12} - 4 T^{13} - 2 T^{14} + T^{16} \)
$41$ \( T^{16} \)
$43$ \( T^{16} \)
$47$ \( ( 1 + T^{2} )^{8} \)
$53$ \( ( 1 - 6 T + 18 T^{2} - 20 T^{3} + 11 T^{4} + 2 T^{6} - 2 T^{7} + T^{8} )^{2} \)
$59$ \( ( 1 + 7 T^{4} + T^{8} )^{2} \)
$61$ \( 1 - 12 T + 46 T^{2} - 48 T^{3} + 194 T^{4} - 328 T^{5} + 212 T^{6} - 4 T^{7} - 69 T^{8} + 40 T^{9} + 32 T^{10} - 44 T^{11} + 34 T^{12} - 20 T^{13} + 10 T^{14} - 4 T^{15} + T^{16} \)
$67$ \( T^{16} \)
$71$ \( 1 + 12 T + 46 T^{2} + 48 T^{3} + 194 T^{4} + 328 T^{5} + 212 T^{6} + 4 T^{7} - 69 T^{8} - 40 T^{9} + 32 T^{10} + 44 T^{11} + 34 T^{12} + 20 T^{13} + 10 T^{14} + 4 T^{15} + T^{16} \)
$73$ \( T^{16} \)
$79$ \( 1 + 12 T + 46 T^{2} + 48 T^{3} + 194 T^{4} + 328 T^{5} + 212 T^{6} + 4 T^{7} - 69 T^{8} - 40 T^{9} + 32 T^{10} + 44 T^{11} + 34 T^{12} + 20 T^{13} + 10 T^{14} + 4 T^{15} + T^{16} \)
$83$ \( ( 2 + 2 T + T^{2} )^{8} \)
$89$ \( ( 5 + 5 T^{2} + T^{4} )^{4} \)
$97$ \( 1 + 8 T + 58 T^{2} + 136 T^{3} + 82 T^{4} - 52 T^{5} + 28 T^{6} - 72 T^{7} + 75 T^{8} - 52 T^{9} + 16 T^{10} + 16 T^{11} + 2 T^{12} - 4 T^{13} - 2 T^{14} + T^{16} \)
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