Properties

Label 1375.1.bz.a
Level $1375$
Weight $1$
Character orbit 1375.bz
Analytic conductor $0.686$
Analytic rank $0$
Dimension $20$
Projective image $D_{50}$
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.bz (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: \(D_{50}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{50} - \cdots)\)

$q$-expansion

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

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

Character values

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

\(n\) \(376\) \(1002\)
\(\chi(n)\) \(-1\) \(-\zeta_{50}^{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
54.1
0.425779 + 0.904827i
−0.0627905 + 0.998027i
−0.0627905 0.998027i
−0.728969 0.684547i
−0.876307 0.481754i
−0.535827 0.844328i
0.637424 + 0.770513i
0.992115 + 0.125333i
0.992115 0.125333i
0.929776 + 0.368125i
−0.968583 0.248690i
−0.876307 + 0.481754i
−0.728969 + 0.684547i
−0.968583 + 0.248690i
0.929776 0.368125i
0.425779 0.904827i
0.187381 0.982287i
0.637424 0.770513i
−0.535827 + 0.844328i
0.187381 + 0.982287i
0 −0.250172 + 1.98031i 0.187381 + 0.982287i 0.809017 + 0.587785i 0 0 0 −2.89047 0.742148i 0
109.1 0 1.39436 1.15352i −0.968583 0.248690i −0.309017 0.951057i 0 0 0 0.426264 2.23455i 0
164.1 0 1.39436 + 1.15352i −0.968583 + 0.248690i −0.309017 + 0.951057i 0 0 0 0.426264 + 2.23455i 0
219.1 0 −1.52794 + 0.718995i 0.992115 0.125333i 0.809017 0.587785i 0 0 0 1.18023 1.42665i 0
329.1 0 0.503997 0.536702i 0.425779 0.904827i 0.809017 + 0.587785i 0 0 0 0.0287541 + 0.457034i 0
384.1 0 −1.36639 + 0.0859661i 0.637424 + 0.770513i −0.309017 0.951057i 0 0 0 0.867524 0.109594i 0
439.1 0 0.0623382 + 0.242791i 0.929776 + 0.368125i −0.309017 + 0.951057i 0 0 0 0.821245 0.451483i 0
494.1 0 1.51373 + 0.288760i −0.876307 0.481754i 0.809017 0.587785i 0 0 0 1.27822 + 0.506084i 0
604.1 0 1.51373 0.288760i −0.876307 + 0.481754i 0.809017 + 0.587785i 0 0 0 1.27822 0.506084i 0
659.1 0 −0.813516 0.516273i −0.0627905 0.998027i −0.309017 0.951057i 0 0 0 −0.0305086 0.0648341i 0
714.1 0 0.723208 1.82662i −0.535827 0.844328i −0.309017 + 0.951057i 0 0 0 −2.08452 1.95750i 0
769.1 0 0.503997 + 0.536702i 0.425779 + 0.904827i 0.809017 0.587785i 0 0 0 0.0287541 0.457034i 0
879.1 0 −1.52794 0.718995i 0.992115 + 0.125333i 0.809017 + 0.587785i 0 0 0 1.18023 + 1.42665i 0
934.1 0 0.723208 + 1.82662i −0.535827 + 0.844328i −0.309017 0.951057i 0 0 0 −2.08452 + 1.95750i 0
989.1 0 −0.813516 + 0.516273i −0.0627905 + 0.998027i −0.309017 + 0.951057i 0 0 0 −0.0305086 + 0.0648341i 0
1044.1 0 −0.250172 1.98031i 0.187381 0.982287i 0.809017 0.587785i 0 0 0 −2.89047 + 0.742148i 0
1154.1 0 −0.239615 0.435857i −0.728969 0.684547i 0.809017 + 0.587785i 0 0 0 0.403270 0.635452i 0
1209.1 0 0.0623382 0.242791i 0.929776 0.368125i −0.309017 0.951057i 0 0 0 0.821245 + 0.451483i 0
1264.1 0 −1.36639 0.0859661i 0.637424 0.770513i −0.309017 + 0.951057i 0 0 0 0.867524 + 0.109594i 0
1319.1 0 −0.239615 + 0.435857i −0.728969 + 0.684547i 0.809017 0.587785i 0 0 0 0.403270 + 0.635452i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1319.1
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.h even 50 1 inner
1375.bz 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.bz.a 20
11.b odd 2 1 CM 1375.1.bz.a 20
125.h even 50 1 inner 1375.1.bz.a 20
1375.bz odd 50 1 inner 1375.1.bz.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1375.1.bz.a 20 1.a even 1 1 trivial
1375.1.bz.a 20 11.b odd 2 1 CM
1375.1.bz.a 20 125.h even 50 1 inner
1375.1.bz.a 20 1375.bz odd 50 1 inner

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \)
$3$ \( 5 + 75 T^{2} + 125 T^{3} + 250 T^{4} - 5 T^{5} + 675 T^{7} + 625 T^{8} - 375 T^{9} - 375 T^{10} + 150 T^{11} + 25 T^{12} - 50 T^{13} + 25 T^{14} - 5 T^{16} + 5 T^{17} + T^{20} \)
$5$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5} \)
$7$ \( T^{20} \)
$11$ \( 1 - T^{5} + T^{10} - T^{15} + T^{20} \)
$13$ \( T^{20} \)
$17$ \( T^{20} \)
$19$ \( T^{20} \)
$23$ \( 5 + 25 T + 25 T^{2} - 125 T^{3} - 5 T^{5} + 650 T^{6} + 750 T^{7} + 750 T^{8} - 250 T^{9} - 375 T^{10} - 525 T^{11} + 25 T^{12} + 75 T^{13} + 150 T^{14} - 5 T^{16} - 20 T^{17} + T^{20} \)
$29$ \( T^{20} \)
$31$ \( 1 + 15 T + 85 T^{2} + 175 T^{3} + 600 T^{4} + 747 T^{5} + 360 T^{6} + 640 T^{7} + 800 T^{8} + 100 T^{9} + 379 T^{10} + 505 T^{11} - 5 T^{12} + 75 T^{13} + 150 T^{14} - 2 T^{15} + 5 T^{16} + 20 T^{17} + T^{20} \)
$37$ \( 3125 - 1875 T^{5} + 750 T^{10} - 50 T^{15} + T^{20} \)
$41$ \( T^{20} \)
$43$ \( T^{20} \)
$47$ \( 3125 - 1875 T^{5} + 750 T^{10} - 50 T^{15} + T^{20} \)
$53$ \( 5 - 25 T + 25 T^{2} + 125 T^{3} + 5 T^{5} + 650 T^{6} - 750 T^{7} + 750 T^{8} + 250 T^{9} - 375 T^{10} + 525 T^{11} + 25 T^{12} - 75 T^{13} + 150 T^{14} - 5 T^{16} + 20 T^{17} + T^{20} \)
$59$ \( 1 - 10 T - 5 T^{2} + 130 T^{3} + 660 T^{4} + 998 T^{5} + 1315 T^{6} + 1270 T^{7} + 805 T^{8} + 85 T^{9} - 246 T^{10} - 45 T^{11} + 165 T^{12} + 235 T^{13} + 195 T^{14} + 122 T^{15} + 70 T^{16} + 35 T^{17} + 15 T^{18} + 5 T^{19} + T^{20} \)
$61$ \( T^{20} \)
$67$ \( 5 + 25 T - 50 T^{3} + 75 T^{4} - 240 T^{5} + 1025 T^{6} - 1675 T^{7} + 1525 T^{8} - 675 T^{9} + 135 T^{10} + 100 T^{12} - 200 T^{13} + 175 T^{14} - 120 T^{15} + 70 T^{16} - 35 T^{17} + 15 T^{18} - 5 T^{19} + T^{20} \)
$71$ \( 1 - 15 T + 85 T^{2} - 175 T^{3} + 600 T^{4} - 747 T^{5} + 360 T^{6} - 640 T^{7} + 800 T^{8} - 100 T^{9} + 379 T^{10} - 505 T^{11} - 5 T^{12} - 75 T^{13} + 150 T^{14} + 2 T^{15} + 5 T^{16} - 20 T^{17} + T^{20} \)
$73$ \( T^{20} \)
$79$ \( T^{20} \)
$83$ \( T^{20} \)
$89$ \( 1 - 10 T + 110 T^{2} - 575 T^{3} + 1850 T^{4} - 3628 T^{5} + 4160 T^{6} - 2685 T^{7} + 1050 T^{8} - 525 T^{9} + 379 T^{10} - 145 T^{11} + 45 T^{12} - 50 T^{13} + 25 T^{14} - 2 T^{15} + 5 T^{16} - 5 T^{17} + T^{20} \)
$97$ \( 5 - 50 T + 300 T^{2} - 1125 T^{3} + 2750 T^{4} - 4380 T^{5} + 4400 T^{6} - 2525 T^{7} + 500 T^{8} + 375 T^{9} - 375 T^{10} + 125 T^{11} + 25 T^{12} - 50 T^{13} + 25 T^{14} - 5 T^{16} + 5 T^{17} + T^{20} \)
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