Properties

Label 1045.1.w.f
Level $1045$
Weight $1$
Character orbit 1045.w
Analytic conductor $0.522$
Analytic rank $0$
Dimension $16$
Projective image $D_{20}$
CM discriminant -95
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1045.w (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.521522938201\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{40})\)
Defining polynomial: \(x^{16} - x^{12} + x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{20}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{20} + \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( \zeta_{40} + \zeta_{40}^{7} ) q^{2} + ( \zeta_{40}^{11} + \zeta_{40}^{13} ) q^{3} + ( \zeta_{40}^{2} + \zeta_{40}^{8} + \zeta_{40}^{14} ) q^{4} -\zeta_{40}^{16} q^{5} + ( -1 + \zeta_{40}^{12} + \zeta_{40}^{14} + \zeta_{40}^{18} ) q^{6} + ( -\zeta_{40} + \zeta_{40}^{3} + \zeta_{40}^{9} + \zeta_{40}^{15} ) q^{8} + ( -\zeta_{40}^{2} - \zeta_{40}^{4} - \zeta_{40}^{6} ) q^{9} +O(q^{10})\) \( q + ( \zeta_{40} + \zeta_{40}^{7} ) q^{2} + ( \zeta_{40}^{11} + \zeta_{40}^{13} ) q^{3} + ( \zeta_{40}^{2} + \zeta_{40}^{8} + \zeta_{40}^{14} ) q^{4} -\zeta_{40}^{16} q^{5} + ( -1 + \zeta_{40}^{12} + \zeta_{40}^{14} + \zeta_{40}^{18} ) q^{6} + ( -\zeta_{40} + \zeta_{40}^{3} + \zeta_{40}^{9} + \zeta_{40}^{15} ) q^{8} + ( -\zeta_{40}^{2} - \zeta_{40}^{4} - \zeta_{40}^{6} ) q^{9} + ( \zeta_{40}^{3} - \zeta_{40}^{17} ) q^{10} -\zeta_{40}^{14} q^{11} + ( -\zeta_{40} - \zeta_{40}^{5} - \zeta_{40}^{7} + \zeta_{40}^{13} + \zeta_{40}^{15} + \zeta_{40}^{19} ) q^{12} + ( -\zeta_{40}^{11} + \zeta_{40}^{17} ) q^{13} + ( \zeta_{40}^{7} + \zeta_{40}^{9} ) q^{15} + ( -\zeta_{40}^{2} + \zeta_{40}^{4} - \zeta_{40}^{8} + \zeta_{40}^{10} + \zeta_{40}^{16} ) q^{16} + ( -\zeta_{40}^{3} - \zeta_{40}^{5} - \zeta_{40}^{7} - \zeta_{40}^{9} - \zeta_{40}^{11} - \zeta_{40}^{13} ) q^{18} + \zeta_{40}^{12} q^{19} + ( \zeta_{40}^{4} + \zeta_{40}^{10} - \zeta_{40}^{18} ) q^{20} + ( \zeta_{40} - \zeta_{40}^{15} ) q^{22} + ( -1 - \zeta_{40}^{2} - \zeta_{40}^{6} - \zeta_{40}^{8} - \zeta_{40}^{12} + \zeta_{40}^{16} ) q^{24} -\zeta_{40}^{12} q^{25} + ( -\zeta_{40}^{4} - \zeta_{40}^{12} ) q^{26} + ( -\zeta_{40}^{13} - \zeta_{40}^{15} - \zeta_{40}^{17} - \zeta_{40}^{19} ) q^{27} + ( \zeta_{40}^{8} + \zeta_{40}^{10} + \zeta_{40}^{14} + \zeta_{40}^{16} ) q^{30} + ( -\zeta_{40}^{3} + \zeta_{40}^{5} - \zeta_{40}^{9} + \zeta_{40}^{11} - \zeta_{40}^{15} + \zeta_{40}^{17} ) q^{32} + ( \zeta_{40}^{5} + \zeta_{40}^{7} ) q^{33} + ( 1 - \zeta_{40}^{4} - \zeta_{40}^{6} - \zeta_{40}^{8} - \zeta_{40}^{10} - \zeta_{40}^{12} - \zeta_{40}^{14} - \zeta_{40}^{16} - \zeta_{40}^{18} ) q^{36} + ( -\zeta_{40}^{17} + \zeta_{40}^{19} ) q^{37} + ( \zeta_{40}^{13} + \zeta_{40}^{19} ) q^{38} + ( \zeta_{40}^{2} + \zeta_{40}^{4} - \zeta_{40}^{8} - \zeta_{40}^{10} ) q^{39} + ( \zeta_{40}^{5} + \zeta_{40}^{11} + \zeta_{40}^{17} - \zeta_{40}^{19} ) q^{40} + ( \zeta_{40}^{2} + \zeta_{40}^{8} - \zeta_{40}^{16} ) q^{44} + ( -1 - \zeta_{40}^{2} + \zeta_{40}^{18} ) q^{45} + ( -\zeta_{40}^{3} - \zeta_{40}^{7} - \zeta_{40}^{9} - \zeta_{40}^{13} + \zeta_{40}^{17} - \zeta_{40}^{19} ) q^{48} + \zeta_{40}^{16} q^{49} + ( -\zeta_{40}^{13} - \zeta_{40}^{19} ) q^{50} + ( -\zeta_{40}^{11} - \zeta_{40}^{13} ) q^{52} + ( -\zeta_{40}^{9} + \zeta_{40}^{19} ) q^{53} + ( 2 + \zeta_{40}^{2} + \zeta_{40}^{4} + \zeta_{40}^{6} - \zeta_{40}^{14} - \zeta_{40}^{16} - \zeta_{40}^{18} ) q^{54} -\zeta_{40}^{10} q^{55} + ( -\zeta_{40}^{3} - \zeta_{40}^{5} ) q^{57} + ( -\zeta_{40} - \zeta_{40}^{3} + \zeta_{40}^{9} + \zeta_{40}^{11} + \zeta_{40}^{15} + \zeta_{40}^{17} ) q^{60} + ( -\zeta_{40}^{2} + \zeta_{40}^{10} ) q^{61} + ( \zeta_{40}^{2} - \zeta_{40}^{4} + \zeta_{40}^{6} - \zeta_{40}^{10} + \zeta_{40}^{12} - \zeta_{40}^{16} + \zeta_{40}^{18} ) q^{64} + ( -\zeta_{40}^{7} + \zeta_{40}^{13} ) q^{65} + ( \zeta_{40}^{6} + \zeta_{40}^{8} + \zeta_{40}^{12} + \zeta_{40}^{14} ) q^{66} + ( \zeta_{40}^{9} - \zeta_{40}^{11} ) q^{67} + ( \zeta_{40} + \zeta_{40}^{3} - \zeta_{40}^{9} - \zeta_{40}^{11} - \zeta_{40}^{13} - \zeta_{40}^{15} - \zeta_{40}^{17} - \zeta_{40}^{19} ) q^{72} + ( -1 + \zeta_{40}^{4} - \zeta_{40}^{6} - \zeta_{40}^{18} ) q^{74} + ( \zeta_{40}^{3} + \zeta_{40}^{5} ) q^{75} + ( -1 - \zeta_{40}^{6} + \zeta_{40}^{14} ) q^{76} + ( \zeta_{40}^{3} + \zeta_{40}^{5} - \zeta_{40}^{15} - \zeta_{40}^{17} ) q^{78} + ( 1 - \zeta_{40}^{4} + \zeta_{40}^{6} + \zeta_{40}^{12} + \zeta_{40}^{18} ) q^{80} + ( \zeta_{40}^{4} + \zeta_{40}^{6} + \zeta_{40}^{8} + \zeta_{40}^{10} + \zeta_{40}^{12} ) q^{81} + ( \zeta_{40}^{3} + \zeta_{40}^{9} + \zeta_{40}^{15} - \zeta_{40}^{17} ) q^{88} + ( -\zeta_{40} - \zeta_{40}^{3} - \zeta_{40}^{5} - \zeta_{40}^{7} - \zeta_{40}^{9} + \zeta_{40}^{19} ) q^{90} + \zeta_{40}^{8} q^{95} + ( 1 - \zeta_{40}^{4} + \zeta_{40}^{6} - \zeta_{40}^{10} - \zeta_{40}^{14} + \zeta_{40}^{18} ) q^{96} + ( -\zeta_{40} - \zeta_{40}^{7} ) q^{97} + ( -\zeta_{40}^{3} + \zeta_{40}^{17} ) q^{98} + ( -1 + \zeta_{40}^{16} + \zeta_{40}^{18} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 4q^{4} + 4q^{5} - 12q^{6} - 4q^{9} + O(q^{10}) \) \( 16q - 4q^{4} + 4q^{5} - 12q^{6} - 4q^{9} + 4q^{16} + 4q^{19} + 4q^{20} - 20q^{24} - 4q^{25} - 8q^{26} - 8q^{30} + 16q^{36} + 8q^{39} - 16q^{45} - 4q^{49} + 40q^{54} + 4q^{64} - 12q^{74} - 16q^{76} + 16q^{80} + 4q^{81} - 4q^{95} + 12q^{96} - 20q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1045\mathbb{Z}\right)^\times\).

\(n\) \(496\) \(761\) \(837\)
\(\chi(n)\) \(-1\) \(\zeta_{40}^{8}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
284.1
−0.453990 + 0.891007i
0.891007 + 0.453990i
−0.891007 0.453990i
0.453990 0.891007i
−0.453990 0.891007i
0.891007 0.453990i
−0.891007 + 0.453990i
0.453990 + 0.891007i
−0.987688 0.156434i
0.156434 0.987688i
−0.156434 + 0.987688i
0.987688 + 0.156434i
−0.987688 + 0.156434i
0.156434 + 0.987688i
−0.156434 0.987688i
0.987688 0.156434i
−0.610425 + 1.87869i −0.734572 + 0.533698i −2.34786 1.70582i −0.309017 0.951057i −0.554254 1.70582i 0 3.03979 2.20854i −0.0542543 + 0.166977i 1.97538
284.2 −0.0966818 + 0.297556i 1.44168 1.04744i 0.729825 + 0.530249i −0.309017 0.951057i 0.172288 + 0.530249i 0 −0.481456 + 0.349798i 0.672288 2.06909i 0.312869
284.3 0.0966818 0.297556i −1.44168 + 1.04744i 0.729825 + 0.530249i −0.309017 0.951057i 0.172288 + 0.530249i 0 0.481456 0.349798i 0.672288 2.06909i −0.312869
284.4 0.610425 1.87869i 0.734572 0.533698i −2.34786 1.70582i −0.309017 0.951057i −0.554254 1.70582i 0 −3.03979 + 2.20854i −0.0542543 + 0.166977i −1.97538
379.1 −0.610425 1.87869i −0.734572 0.533698i −2.34786 + 1.70582i −0.309017 + 0.951057i −0.554254 + 1.70582i 0 3.03979 + 2.20854i −0.0542543 0.166977i 1.97538
379.2 −0.0966818 0.297556i 1.44168 + 1.04744i 0.729825 0.530249i −0.309017 + 0.951057i 0.172288 0.530249i 0 −0.481456 0.349798i 0.672288 + 2.06909i 0.312869
379.3 0.0966818 + 0.297556i −1.44168 1.04744i 0.729825 0.530249i −0.309017 + 0.951057i 0.172288 0.530249i 0 0.481456 + 0.349798i 0.672288 + 2.06909i −0.312869
379.4 0.610425 + 1.87869i 0.734572 + 0.533698i −2.34786 + 1.70582i −0.309017 + 0.951057i −0.554254 + 1.70582i 0 −3.03979 2.20854i −0.0542543 0.166977i −1.97538
664.1 −1.44168 1.04744i 0.610425 1.87869i 0.672288 + 2.06909i 0.809017 0.587785i −2.84786 + 2.06909i 0 0.647354 1.99235i −2.34786 1.70582i −1.78201
664.2 −0.734572 0.533698i −0.0966818 + 0.297556i −0.0542543 0.166977i 0.809017 0.587785i 0.229825 0.166977i 0 −0.329843 + 1.01515i 0.729825 + 0.530249i −0.907981
664.3 0.734572 + 0.533698i 0.0966818 0.297556i −0.0542543 0.166977i 0.809017 0.587785i 0.229825 0.166977i 0 0.329843 1.01515i 0.729825 + 0.530249i 0.907981
664.4 1.44168 + 1.04744i −0.610425 + 1.87869i 0.672288 + 2.06909i 0.809017 0.587785i −2.84786 + 2.06909i 0 −0.647354 + 1.99235i −2.34786 1.70582i 1.78201
949.1 −1.44168 + 1.04744i 0.610425 + 1.87869i 0.672288 2.06909i 0.809017 + 0.587785i −2.84786 2.06909i 0 0.647354 + 1.99235i −2.34786 + 1.70582i −1.78201
949.2 −0.734572 + 0.533698i −0.0966818 0.297556i −0.0542543 + 0.166977i 0.809017 + 0.587785i 0.229825 + 0.166977i 0 −0.329843 1.01515i 0.729825 0.530249i −0.907981
949.3 0.734572 0.533698i 0.0966818 + 0.297556i −0.0542543 + 0.166977i 0.809017 + 0.587785i 0.229825 + 0.166977i 0 0.329843 + 1.01515i 0.729825 0.530249i 0.907981
949.4 1.44168 1.04744i −0.610425 1.87869i 0.672288 2.06909i 0.809017 + 0.587785i −2.84786 2.06909i 0 −0.647354 1.99235i −2.34786 + 1.70582i 1.78201
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 949.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
95.d odd 2 1 CM by \(\Q(\sqrt{-95}) \)
5.b even 2 1 inner
11.c even 5 1 inner
19.b odd 2 1 inner
55.j even 10 1 inner
209.m odd 10 1 inner
1045.w odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1045.1.w.f 16
5.b even 2 1 inner 1045.1.w.f 16
11.c even 5 1 inner 1045.1.w.f 16
19.b odd 2 1 inner 1045.1.w.f 16
55.j even 10 1 inner 1045.1.w.f 16
95.d odd 2 1 CM 1045.1.w.f 16
209.m odd 10 1 inner 1045.1.w.f 16
1045.w odd 10 1 inner 1045.1.w.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.1.w.f 16 1.a even 1 1 trivial
1045.1.w.f 16 5.b even 2 1 inner
1045.1.w.f 16 11.c even 5 1 inner
1045.1.w.f 16 19.b odd 2 1 inner
1045.1.w.f 16 55.j even 10 1 inner
1045.1.w.f 16 95.d odd 2 1 CM
1045.1.w.f 16 209.m odd 10 1 inner
1045.1.w.f 16 1045.w odd 10 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1045, [\chi])\):

\(T_{2}^{16} + \cdots\)
\( T_{7} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 16 T^{2} + 97 T^{4} - 32 T^{6} + 150 T^{8} + 32 T^{10} + 12 T^{12} + 4 T^{14} + T^{16} \)
$3$ \( 1 + 16 T^{2} + 97 T^{4} - 32 T^{6} + 150 T^{8} + 32 T^{10} + 12 T^{12} + 4 T^{14} + T^{16} \)
$5$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \)
$7$ \( T^{16} \)
$11$ \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
$13$ \( 1 + 16 T^{2} + 97 T^{4} - 32 T^{6} + 150 T^{8} + 32 T^{10} + 12 T^{12} + 4 T^{14} + T^{16} \)
$17$ \( T^{16} \)
$19$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \)
$23$ \( T^{16} \)
$29$ \( T^{16} \)
$31$ \( T^{16} \)
$37$ \( 1 + 16 T^{2} + 97 T^{4} - 32 T^{6} + 150 T^{8} + 32 T^{10} + 12 T^{12} + 4 T^{14} + T^{16} \)
$41$ \( T^{16} \)
$43$ \( T^{16} \)
$47$ \( T^{16} \)
$53$ \( ( 16 + 8 T^{2} + 4 T^{4} + 2 T^{6} + T^{8} )^{2} \)
$59$ \( T^{16} \)
$61$ \( ( 25 + 10 T^{4} + 5 T^{6} + T^{8} )^{2} \)
$67$ \( ( 1 - 12 T^{2} + 19 T^{4} - 8 T^{6} + T^{8} )^{2} \)
$71$ \( T^{16} \)
$73$ \( T^{16} \)
$79$ \( T^{16} \)
$83$ \( T^{16} \)
$89$ \( T^{16} \)
$97$ \( 1 + 16 T^{2} + 97 T^{4} - 32 T^{6} + 150 T^{8} + 32 T^{10} + 12 T^{12} + 4 T^{14} + T^{16} \)
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