Properties

Label 1375.1.ch.a
Level $1375$
Weight $1$
Character orbit 1375.ch
Analytic conductor $0.686$
Analytic rank $0$
Dimension $20$
Projective image $D_{25}$
CM discriminant -11
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1375 = 5^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1375.ch (of order \(50\), degree \(20\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.686214392370\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\Q(\zeta_{50})\)
Defining polynomial: \(x^{20} - x^{15} + x^{10} - x^{5} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
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.

\(f(q)\) \(=\) \( q + ( \zeta_{50}^{4} + \zeta_{50}^{18} ) q^{3} -\zeta_{50}^{21} q^{4} -\zeta_{50}^{5} q^{5} + ( \zeta_{50}^{8} - \zeta_{50}^{11} + \zeta_{50}^{22} ) q^{9} +O(q^{10})\) \( q + ( \zeta_{50}^{4} + \zeta_{50}^{18} ) q^{3} -\zeta_{50}^{21} q^{4} -\zeta_{50}^{5} q^{5} + ( \zeta_{50}^{8} - \zeta_{50}^{11} + \zeta_{50}^{22} ) q^{9} -\zeta_{50}^{23} q^{11} + ( 1 + \zeta_{50}^{14} ) q^{12} + ( -\zeta_{50}^{9} - \zeta_{50}^{23} ) q^{15} -\zeta_{50}^{17} q^{16} -\zeta_{50} q^{20} + ( \zeta_{50}^{2} + \zeta_{50}^{24} ) q^{23} + \zeta_{50}^{10} q^{25} + ( -\zeta_{50} + \zeta_{50}^{4} + \zeta_{50}^{12} - \zeta_{50}^{15} ) q^{27} + ( \zeta_{50}^{16} - \zeta_{50}^{17} ) q^{31} + ( \zeta_{50}^{2} + \zeta_{50}^{16} ) q^{33} + ( \zeta_{50}^{4} - \zeta_{50}^{7} + \zeta_{50}^{18} ) q^{36} + ( \zeta_{50}^{2} - \zeta_{50}^{7} ) q^{37} -\zeta_{50}^{19} q^{44} + ( \zeta_{50}^{2} - \zeta_{50}^{13} + \zeta_{50}^{16} ) q^{45} + ( -\zeta_{50} - \zeta_{50}^{11} ) q^{47} + ( \zeta_{50}^{10} - \zeta_{50}^{21} ) q^{48} + \zeta_{50}^{10} q^{49} + ( \zeta_{50}^{14} + \zeta_{50}^{18} ) q^{53} -\zeta_{50}^{3} q^{55} + ( -\zeta_{50}^{15} + \zeta_{50}^{24} ) q^{59} + ( -\zeta_{50}^{5} - \zeta_{50}^{19} ) q^{60} -\zeta_{50}^{13} q^{64} + ( \zeta_{50}^{8} - \zeta_{50}^{15} ) q^{67} + ( -\zeta_{50}^{3} + \zeta_{50}^{6} - \zeta_{50}^{17} + \zeta_{50}^{20} ) q^{69} + ( -\zeta_{50}^{13} + \zeta_{50}^{24} ) q^{71} + ( -\zeta_{50}^{3} + \zeta_{50}^{14} ) q^{75} + \zeta_{50}^{22} q^{80} + ( -\zeta_{50}^{5} + \zeta_{50}^{8} + \zeta_{50}^{16} - \zeta_{50}^{19} + \zeta_{50}^{22} ) q^{81} + ( \zeta_{50}^{12} + \zeta_{50}^{24} ) q^{89} + ( \zeta_{50}^{20} - \zeta_{50}^{23} ) q^{92} + ( -\zeta_{50}^{9} + \zeta_{50}^{10} + \zeta_{50}^{20} - \zeta_{50}^{21} ) q^{93} + ( -\zeta_{50}^{9} + \zeta_{50}^{18} ) q^{97} + ( \zeta_{50}^{6} - \zeta_{50}^{9} + \zeta_{50}^{20} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q - 5q^{5} + O(q^{10}) \) \( 20q - 5q^{5} + 20q^{12} - 5q^{25} - 5q^{27} - 5q^{48} - 5q^{49} - 5q^{59} - 5q^{60} - 5q^{67} - 5q^{69} - 5q^{81} - 5q^{92} - 10q^{93} - 5q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1375\mathbb{Z}\right)^\times\).

\(n\) \(376\) \(1002\)
\(\chi(n)\) \(-1\) \(-\zeta_{50}^{21}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
21.1
0.992115 + 0.125333i
0.992115 0.125333i
0.637424 0.770513i
−0.535827 + 0.844328i
−0.876307 + 0.481754i
−0.728969 + 0.684547i
−0.0627905 + 0.998027i
−0.0627905 0.998027i
0.425779 0.904827i
0.187381 0.982287i
−0.535827 0.844328i
0.637424 + 0.770513i
0.187381 + 0.982287i
0.425779 + 0.904827i
0.929776 + 0.368125i
−0.968583 0.248690i
−0.728969 0.684547i
−0.876307 0.481754i
−0.968583 + 0.248690i
0.929776 0.368125i
0 0.238883 + 1.25227i 0.876307 0.481754i −0.809017 0.587785i 0 0 0 −0.581331 + 0.230165i 0
131.1 0 0.238883 1.25227i 0.876307 + 0.481754i −0.809017 + 0.587785i 0 0 0 −0.581331 0.230165i 0
186.1 0 −1.92189 + 0.493458i −0.929776 0.368125i 0.309017 0.951057i 0 0 0 2.57386 1.41499i 0
241.1 0 0.0915446 + 1.45506i −0.637424 0.770513i 0.309017 + 0.951057i 0 0 0 −1.11671 + 0.141073i 0
296.1 0 −1.35556 1.27295i −0.425779 + 0.904827i −0.809017 0.587785i 0 0 0 0.154335 + 2.45309i 0
406.1 0 −0.456288 0.969661i −0.992115 + 0.125333i −0.809017 + 0.587785i 0 0 0 −0.0946201 + 0.114376i 0
461.1 0 0.542804 0.656137i 0.968583 0.248690i 0.309017 0.951057i 0 0 0 0.0515014 + 0.269980i 0
516.1 0 0.542804 + 0.656137i 0.968583 + 0.248690i 0.309017 + 0.951057i 0 0 0 0.0515014 0.269980i 0
571.1 0 −0.124591 0.0157395i −0.187381 0.982287i −0.809017 0.587785i 0 0 0 −0.953308 0.244768i 0
681.1 0 1.69755 + 0.933237i 0.728969 0.684547i −0.809017 + 0.587785i 0 0 0 1.47492 + 2.32411i 0
736.1 0 0.0915446 1.45506i −0.637424 + 0.770513i 0.309017 0.951057i 0 0 0 −1.11671 0.141073i 0
791.1 0 −1.92189 0.493458i −0.929776 + 0.368125i 0.309017 + 0.951057i 0 0 0 2.57386 + 1.41499i 0
846.1 0 1.69755 0.933237i 0.728969 + 0.684547i −0.809017 0.587785i 0 0 0 1.47492 2.32411i 0
956.1 0 −0.124591 + 0.0157395i −0.187381 + 0.982287i −0.809017 + 0.587785i 0 0 0 −0.953308 + 0.244768i 0
1011.1 0 0.939097 + 1.47978i 0.0627905 0.998027i 0.309017 0.951057i 0 0 0 −0.882067 + 1.87449i 0
1066.1 0 0.348445 0.137959i 0.535827 0.844328i 0.309017 + 0.951057i 0 0 0 −0.626587 + 0.588404i 0
1121.1 0 −0.456288 + 0.969661i −0.992115 0.125333i −0.809017 0.587785i 0 0 0 −0.0946201 0.114376i 0
1231.1 0 −1.35556 + 1.27295i −0.425779 0.904827i −0.809017 + 0.587785i 0 0 0 0.154335 2.45309i 0
1286.1 0 0.348445 + 0.137959i 0.535827 + 0.844328i 0.309017 0.951057i 0 0 0 −0.626587 0.588404i 0
1341.1 0 0.939097 1.47978i 0.0627905 + 0.998027i 0.309017 + 0.951057i 0 0 0 −0.882067 1.87449i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1341.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
125.g even 25 1 inner
1375.ch odd 50 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1375.1.ch.a 20
11.b odd 2 1 CM 1375.1.ch.a 20
125.g even 25 1 inner 1375.1.ch.a 20
1375.ch odd 50 1 inner 1375.1.ch.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1375.1.ch.a 20 1.a even 1 1 trivial
1375.1.ch.a 20 11.b odd 2 1 CM
1375.1.ch.a 20 125.g even 25 1 inner
1375.1.ch.a 20 1375.ch odd 50 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1375, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \)
$3$ \( 1 + 10 T - 15 T^{2} - 175 T^{3} + 600 T^{4} - 747 T^{5} + 1010 T^{6} - 715 T^{7} + 925 T^{8} - 225 T^{9} + 379 T^{10} + 120 T^{11} + 45 T^{12} + 50 T^{13} + 25 T^{14} + 2 T^{15} + 5 T^{16} + 5 T^{17} + T^{20} \)
$5$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
$7$ \( T^{20} \)
$11$ \( 1 + T^{5} + T^{10} + T^{15} + T^{20} \)
$13$ \( T^{20} \)
$17$ \( T^{20} \)
$19$ \( T^{20} \)
$23$ \( 1 - 15 T + 85 T^{2} - 175 T^{3} + 600 T^{4} - 747 T^{5} + 360 T^{6} - 640 T^{7} + 800 T^{8} - 100 T^{9} + 379 T^{10} - 505 T^{11} - 5 T^{12} - 75 T^{13} + 150 T^{14} + 2 T^{15} + 5 T^{16} - 20 T^{17} + T^{20} \)
$29$ \( T^{20} \)
$31$ \( 1 - 15 T + 85 T^{2} - 175 T^{3} + 600 T^{4} - 747 T^{5} + 360 T^{6} - 640 T^{7} + 800 T^{8} - 100 T^{9} + 379 T^{10} - 505 T^{11} - 5 T^{12} - 75 T^{13} + 150 T^{14} + 2 T^{15} + 5 T^{16} - 20 T^{17} + T^{20} \)
$37$ \( 1 - 7 T^{5} + 124 T^{10} - 18 T^{15} + T^{20} \)
$41$ \( T^{20} \)
$43$ \( T^{20} \)
$47$ \( 1 - 7 T^{5} + 124 T^{10} - 18 T^{15} + T^{20} \)
$53$ \( 1 - 15 T + 85 T^{2} - 175 T^{3} + 600 T^{4} - 747 T^{5} + 360 T^{6} - 640 T^{7} + 800 T^{8} - 100 T^{9} + 379 T^{10} - 505 T^{11} - 5 T^{12} - 75 T^{13} + 150 T^{14} + 2 T^{15} + 5 T^{16} - 20 T^{17} + T^{20} \)
$59$ \( 1 - 10 T - 5 T^{2} + 130 T^{3} + 660 T^{4} + 998 T^{5} + 1315 T^{6} + 1270 T^{7} + 805 T^{8} + 85 T^{9} - 246 T^{10} - 45 T^{11} + 165 T^{12} + 235 T^{13} + 195 T^{14} + 122 T^{15} + 70 T^{16} + 35 T^{17} + 15 T^{18} + 5 T^{19} + T^{20} \)
$61$ \( T^{20} \)
$67$ \( 1 + 15 T + 120 T^{2} + 380 T^{3} + 435 T^{4} - 252 T^{5} - 735 T^{6} - 905 T^{7} - 645 T^{8} + 85 T^{9} + 629 T^{10} + 730 T^{11} + 540 T^{12} + 310 T^{13} + 195 T^{14} + 122 T^{15} + 70 T^{16} + 35 T^{17} + 15 T^{18} + 5 T^{19} + T^{20} \)
$71$ \( 1 - 15 T + 85 T^{2} - 175 T^{3} + 600 T^{4} - 747 T^{5} + 360 T^{6} - 640 T^{7} + 800 T^{8} - 100 T^{9} + 379 T^{10} - 505 T^{11} - 5 T^{12} - 75 T^{13} + 150 T^{14} + 2 T^{15} + 5 T^{16} - 20 T^{17} + T^{20} \)
$73$ \( T^{20} \)
$79$ \( T^{20} \)
$83$ \( T^{20} \)
$89$ \( 1 + 10 T + 110 T^{2} + 575 T^{3} + 1850 T^{4} + 3628 T^{5} + 4160 T^{6} + 2685 T^{7} + 1050 T^{8} + 525 T^{9} + 379 T^{10} + 145 T^{11} + 45 T^{12} + 50 T^{13} + 25 T^{14} + 2 T^{15} + 5 T^{16} + 5 T^{17} + T^{20} \)
$97$ \( 1 + 10 T + 110 T^{2} + 575 T^{3} + 1850 T^{4} + 3628 T^{5} + 4160 T^{6} + 2685 T^{7} + 1050 T^{8} + 525 T^{9} + 379 T^{10} + 145 T^{11} + 45 T^{12} + 50 T^{13} + 25 T^{14} + 2 T^{15} + 5 T^{16} + 5 T^{17} + T^{20} \)
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