Properties

Label 1057.1.bq.a
Level $1057$
Weight $1$
Character orbit 1057.bq
Analytic conductor $0.528$
Analytic rank $0$
Dimension $20$
Projective image $D_{25}$
CM discriminant -7
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1057 = 7 \cdot 151 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1057.bq (of order \(50\), degree \(20\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.527511718353\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\Q(\zeta_{50})\)
Defining polynomial: \(x^{20} - x^{15} + x^{10} - x^{5} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{25}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{25} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( \zeta_{50}^{12} - \zeta_{50}^{23} ) q^{2} + ( \zeta_{50}^{10} - \zeta_{50}^{21} + \zeta_{50}^{24} ) q^{4} -\zeta_{50}^{23} q^{7} + ( \zeta_{50}^{8} - \zeta_{50}^{11} - \zeta_{50}^{19} + \zeta_{50}^{22} ) q^{8} -\zeta_{50}^{3} q^{9} +O(q^{10})\) \( q + ( \zeta_{50}^{12} - \zeta_{50}^{23} ) q^{2} + ( \zeta_{50}^{10} - \zeta_{50}^{21} + \zeta_{50}^{24} ) q^{4} -\zeta_{50}^{23} q^{7} + ( \zeta_{50}^{8} - \zeta_{50}^{11} - \zeta_{50}^{19} + \zeta_{50}^{22} ) q^{8} -\zeta_{50}^{3} q^{9} + ( \zeta_{50}^{6} + \zeta_{50}^{16} ) q^{11} + ( \zeta_{50}^{10} - \zeta_{50}^{21} ) q^{14} + ( \zeta_{50}^{6} - \zeta_{50}^{9} - \zeta_{50}^{17} + \zeta_{50}^{20} - \zeta_{50}^{23} ) q^{16} + ( -\zeta_{50} - \zeta_{50}^{15} ) q^{18} + ( -\zeta_{50}^{3} + \zeta_{50}^{4} + \zeta_{50}^{14} + \zeta_{50}^{18} ) q^{22} + ( -\zeta_{50}^{5} - \zeta_{50}^{15} ) q^{23} + \zeta_{50}^{6} q^{25} + ( \zeta_{50}^{8} - \zeta_{50}^{19} + \zeta_{50}^{22} ) q^{28} + ( \zeta_{50}^{12} + \zeta_{50}^{20} ) q^{29} + ( \zeta_{50}^{4} - \zeta_{50}^{7} + \zeta_{50}^{10} - \zeta_{50}^{15} + \zeta_{50}^{18} - \zeta_{50}^{21} ) q^{32} + ( \zeta_{50}^{2} - \zeta_{50}^{13} + \zeta_{50}^{24} ) q^{36} + ( -\zeta_{50}^{9} + \zeta_{50}^{24} ) q^{37} + ( -\zeta_{50}^{13} + \zeta_{50}^{16} ) q^{43} + ( -\zeta_{50} + \zeta_{50}^{2} - \zeta_{50}^{5} + \zeta_{50}^{12} - \zeta_{50}^{15} + \zeta_{50}^{16} ) q^{44} + ( \zeta_{50}^{2} - \zeta_{50}^{3} - \zeta_{50}^{13} - \zeta_{50}^{17} ) q^{46} -\zeta_{50}^{21} q^{49} + ( \zeta_{50}^{4} + \zeta_{50}^{18} ) q^{50} + ( -\zeta_{50}^{7} + \zeta_{50}^{20} ) q^{53} + ( \zeta_{50}^{6} - \zeta_{50}^{9} - \zeta_{50}^{17} + \zeta_{50}^{20} ) q^{56} + ( -\zeta_{50}^{7} + \zeta_{50}^{10} + \zeta_{50}^{18} + \zeta_{50}^{24} ) q^{58} -\zeta_{50} q^{63} + ( \zeta_{50}^{2} - \zeta_{50}^{5} + \zeta_{50}^{8} - \zeta_{50}^{13} + \zeta_{50}^{16} - \zeta_{50}^{19} + \zeta_{50}^{22} ) q^{64} + ( -\zeta_{50}^{5} + \zeta_{50}^{14} ) q^{67} + ( -\zeta_{50} + \zeta_{50}^{20} ) q^{71} + ( 1 - \zeta_{50}^{11} + \zeta_{50}^{14} + \zeta_{50}^{22} ) q^{72} + ( -\zeta_{50}^{7} - \zeta_{50}^{11} - \zeta_{50}^{21} + \zeta_{50}^{22} ) q^{74} + ( \zeta_{50}^{4} + \zeta_{50}^{14} ) q^{77} + ( 1 - \zeta_{50}^{11} ) q^{79} + \zeta_{50}^{6} q^{81} + ( 1 - \zeta_{50}^{3} - \zeta_{50}^{11} + \zeta_{50}^{14} ) q^{86} + ( 1 + \zeta_{50}^{2} - \zeta_{50}^{3} + \zeta_{50}^{10} - \zeta_{50}^{13} + \zeta_{50}^{14} - \zeta_{50}^{17} + \zeta_{50}^{24} ) q^{88} + ( 1 - \zeta_{50} + \zeta_{50}^{4} - \zeta_{50}^{11} + \zeta_{50}^{14} - \zeta_{50}^{15} ) q^{92} + ( \zeta_{50}^{8} - \zeta_{50}^{19} ) q^{98} + ( -\zeta_{50}^{9} - \zeta_{50}^{19} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q - 5q^{4} + O(q^{10}) \) \( 20q - 5q^{4} - 5q^{14} - 5q^{16} - 5q^{18} - 10q^{23} - 5q^{29} - 10q^{32} - 10q^{44} - 5q^{53} - 5q^{56} - 5q^{58} - 5q^{64} - 5q^{67} - 5q^{71} + 20q^{72} + 20q^{79} + 20q^{86} + 15q^{88} + 15q^{92} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1057\mathbb{Z}\right)^\times\).

\(n\) \(605\) \(610\)
\(\chi(n)\) \(-1\) \(-\zeta_{50}^{13}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
20.1
0.637424 0.770513i
0.992115 0.125333i
−0.535827 0.844328i
−0.728969 0.684547i
−0.968583 + 0.248690i
−0.968583 0.248690i
0.637424 + 0.770513i
0.425779 + 0.904827i
−0.0627905 0.998027i
0.992115 + 0.125333i
−0.876307 + 0.481754i
0.929776 + 0.368125i
−0.728969 + 0.684547i
−0.876307 0.481754i
0.425779 0.904827i
−0.0627905 + 0.998027i
−0.535827 + 0.844328i
0.929776 0.368125i
0.187381 + 0.982287i
0.187381 0.982287i
−0.613161 + 1.88711i 0 −2.37622 1.72642i 0 0 −0.187381 + 0.982287i 3.10969 2.25932i 0.876307 + 0.481754i 0
125.1 1.03137 0.749337i 0 0.193209 0.594636i 0 0 0.968583 + 0.248690i 0.147638 + 0.454382i −0.929776 + 0.368125i 0
160.1 0.450527 1.38658i 0 −0.910614 0.661600i 0 0 −0.425779 0.904827i −0.148122 + 0.107617i −0.992115 + 0.125333i 0
195.1 −0.866986 0.629902i 0 0.0458709 + 0.141176i 0 0 0.0627905 0.998027i −0.282001 + 0.867911i −0.637424 + 0.770513i 0
223.1 −0.115808 + 0.356420i 0 0.695393 + 0.505233i 0 0 0.876307 + 0.481754i −0.563797 + 0.409622i 0.728969 0.684547i 0
237.1 −0.115808 0.356420i 0 0.695393 0.505233i 0 0 0.876307 0.481754i −0.563797 0.409622i 0.728969 + 0.684547i 0
370.1 −0.613161 1.88711i 0 −2.37622 + 1.72642i 0 0 −0.187381 0.982287i 3.10969 + 2.25932i 0.876307 0.481754i 0
412.1 −0.101597 + 0.0738147i 0 −0.304144 + 0.936058i 0 0 −0.637424 0.770513i −0.0770013 0.236986i 0.968583 + 0.248690i 0
426.1 −0.263146 0.809880i 0 0.222357 0.161552i 0 0 −0.992115 0.125333i −0.878275 0.638104i −0.187381 0.982287i 0
482.1 1.03137 + 0.749337i 0 0.193209 + 0.594636i 0 0 0.968583 0.248690i 0.147638 0.454382i −0.929776 0.368125i 0
503.1 1.50441 + 1.09302i 0 0.759544 + 2.33764i 0 0 0.535827 + 0.844328i −0.837780 + 2.57842i 0.0627905 0.998027i 0
531.1 0.541587 1.66683i 0 −1.67600 1.21769i 0 0 0.728969 0.684547i −1.51949 + 1.10397i −0.425779 0.904827i 0
580.1 −0.866986 + 0.629902i 0 0.0458709 0.141176i 0 0 0.0627905 + 0.998027i −0.282001 0.867911i −0.637424 0.770513i 0
601.1 1.50441 1.09302i 0 0.759544 2.33764i 0 0 0.535827 0.844328i −0.837780 2.57842i 0.0627905 + 0.998027i 0
685.1 −0.101597 0.0738147i 0 −0.304144 0.936058i 0 0 −0.637424 + 0.770513i −0.0770013 + 0.236986i 0.968583 0.248690i 0
727.1 −0.263146 + 0.809880i 0 0.222357 + 0.161552i 0 0 −0.992115 + 0.125333i −0.878275 + 0.638104i −0.187381 + 0.982287i 0
839.1 0.450527 + 1.38658i 0 −0.910614 + 0.661600i 0 0 −0.425779 + 0.904827i −0.148122 0.107617i −0.992115 0.125333i 0
846.1 0.541587 + 1.66683i 0 −1.67600 + 1.21769i 0 0 0.728969 + 0.684547i −1.51949 1.10397i −0.425779 + 0.904827i 0
853.1 −1.56720 1.13864i 0 0.850604 + 2.61789i 0 0 −0.929776 0.368125i 1.04914 3.22894i 0.535827 + 0.844328i 0
1000.1 −1.56720 + 1.13864i 0 0.850604 2.61789i 0 0 −0.929776 + 0.368125i 1.04914 + 3.22894i 0.535827 0.844328i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1000.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
151.h even 25 1 inner
1057.bq odd 50 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1057.1.bq.a 20
7.b odd 2 1 CM 1057.1.bq.a 20
151.h even 25 1 inner 1057.1.bq.a 20
1057.bq odd 50 1 inner 1057.1.bq.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1057.1.bq.a 20 1.a even 1 1 trivial
1057.1.bq.a 20 7.b odd 2 1 CM
1057.1.bq.a 20 151.h even 25 1 inner
1057.1.bq.a 20 1057.bq odd 50 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1057, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 15 T + 100 T^{2} + 260 T^{3} + 675 T^{4} + 498 T^{5} + 715 T^{6} + 150 T^{7} + 485 T^{8} + 200 T^{9} + 374 T^{10} + 5 T^{11} + 275 T^{12} - 5 T^{13} + 75 T^{14} + 2 T^{15} + 20 T^{16} + 5 T^{18} + T^{20} \)
$3$ \( T^{20} \)
$5$ \( T^{20} \)
$7$ \( 1 + T^{5} + T^{10} + T^{15} + T^{20} \)
$11$ \( 1 - 7 T^{5} + 124 T^{10} - 18 T^{15} + T^{20} \)
$13$ \( T^{20} \)
$17$ \( T^{20} \)
$19$ \( T^{20} \)
$23$ \( ( 1 + 3 T + 4 T^{2} + 2 T^{3} + T^{4} )^{5} \)
$29$ \( 1 - 10 T - 5 T^{2} + 130 T^{3} + 660 T^{4} + 998 T^{5} + 1315 T^{6} + 1270 T^{7} + 805 T^{8} + 85 T^{9} - 246 T^{10} - 45 T^{11} + 165 T^{12} + 235 T^{13} + 195 T^{14} + 122 T^{15} + 70 T^{16} + 35 T^{17} + 15 T^{18} + 5 T^{19} + T^{20} \)
$31$ \( T^{20} \)
$37$ \( 1 - 7 T^{5} + 124 T^{10} - 18 T^{15} + T^{20} \)
$41$ \( T^{20} \)
$43$ \( 1 + 10 T + 85 T^{2} + 200 T^{3} - 25 T^{4} - 122 T^{5} + 385 T^{6} - 615 T^{7} + 675 T^{8} - 225 T^{9} - 246 T^{10} + 145 T^{11} + 20 T^{12} - 75 T^{13} + 25 T^{14} + 2 T^{15} + 5 T^{16} + 5 T^{17} + T^{20} \)
$47$ \( T^{20} \)
$53$ \( 1 - 10 T - 5 T^{2} + 130 T^{3} + 660 T^{4} + 998 T^{5} + 1315 T^{6} + 1270 T^{7} + 805 T^{8} + 85 T^{9} - 246 T^{10} - 45 T^{11} + 165 T^{12} + 235 T^{13} + 195 T^{14} + 122 T^{15} + 70 T^{16} + 35 T^{17} + 15 T^{18} + 5 T^{19} + T^{20} \)
$59$ \( T^{20} \)
$61$ \( T^{20} \)
$67$ \( 1 - 10 T + 70 T^{2} - 45 T^{3} - 340 T^{4} - 252 T^{5} + 665 T^{6} + 1520 T^{7} + 1655 T^{8} + 1235 T^{9} + 629 T^{10} + 205 T^{11} + 165 T^{12} + 185 T^{13} + 170 T^{14} + 122 T^{15} + 70 T^{16} + 35 T^{17} + 15 T^{18} + 5 T^{19} + T^{20} \)
$71$ \( 1 - 10 T + 70 T^{2} - 45 T^{3} - 340 T^{4} - 252 T^{5} + 665 T^{6} + 1520 T^{7} + 1655 T^{8} + 1235 T^{9} + 629 T^{10} + 205 T^{11} + 165 T^{12} + 185 T^{13} + 170 T^{14} + 122 T^{15} + 70 T^{16} + 35 T^{17} + 15 T^{18} + 5 T^{19} + T^{20} \)
$73$ \( T^{20} \)
$79$ \( 1 - 10 T + 120 T^{2} - 795 T^{3} + 3685 T^{4} - 12752 T^{5} + 33965 T^{6} - 71205 T^{7} + 119580 T^{8} - 162965 T^{9} + 181754 T^{10} - 166595 T^{11} + 125515 T^{12} - 77415 T^{13} + 38745 T^{14} - 15503 T^{15} + 4845 T^{16} - 1140 T^{17} + 190 T^{18} - 20 T^{19} + T^{20} \)
$83$ \( T^{20} \)
$89$ \( T^{20} \)
$97$ \( T^{20} \)
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