Properties

Label 1397.1.l.a
Level $1397$
Weight $1$
Character orbit 1397.l
Analytic conductor $0.697$
Analytic rank $0$
Dimension $20$
Projective image $D_{25}$
CM discriminant -127
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1397 = 11 \cdot 127 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1397.l (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.697193822648\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{50})\)
Defining polynomial: \(x^{20} - x^{15} + x^{10} - x^{5} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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.

\(f(q)\) \(=\) \( q + ( \zeta_{50}^{16} - \zeta_{50}^{19} ) q^{2} + ( -\zeta_{50}^{7} + \zeta_{50}^{10} - \zeta_{50}^{13} ) q^{4} + ( -\zeta_{50} + \zeta_{50}^{4} - \zeta_{50}^{7} - \zeta_{50}^{23} ) q^{8} -\zeta_{50}^{5} q^{9} +O(q^{10})\) \( q + ( \zeta_{50}^{16} - \zeta_{50}^{19} ) q^{2} + ( -\zeta_{50}^{7} + \zeta_{50}^{10} - \zeta_{50}^{13} ) q^{4} + ( -\zeta_{50} + \zeta_{50}^{4} - \zeta_{50}^{7} - \zeta_{50}^{23} ) q^{8} -\zeta_{50}^{5} q^{9} -\zeta_{50}^{13} q^{11} + ( -\zeta_{50}^{17} + \zeta_{50}^{18} ) q^{13} + ( -\zeta_{50} + \zeta_{50}^{14} - \zeta_{50}^{17} + \zeta_{50}^{20} - \zeta_{50}^{23} ) q^{16} + ( \zeta_{50}^{16} + \zeta_{50}^{24} ) q^{17} + ( -\zeta_{50}^{21} + \zeta_{50}^{24} ) q^{18} + ( -\zeta_{50}^{9} - \zeta_{50}^{21} ) q^{19} + ( \zeta_{50}^{4} - \zeta_{50}^{7} ) q^{22} -\zeta_{50}^{15} q^{25} + ( \zeta_{50}^{8} - \zeta_{50}^{9} - \zeta_{50}^{11} + \zeta_{50}^{12} ) q^{26} + ( \zeta_{50}^{16} - \zeta_{50}^{19} ) q^{31} + ( -\zeta_{50}^{5} + \zeta_{50}^{8} - \zeta_{50}^{11} + \zeta_{50}^{14} - \zeta_{50}^{17} + \zeta_{50}^{20} ) q^{32} + ( -\zeta_{50}^{7} + \zeta_{50}^{10} - \zeta_{50}^{15} + \zeta_{50}^{18} ) q^{34} + ( \zeta_{50}^{12} - \zeta_{50}^{15} + \zeta_{50}^{18} ) q^{36} + ( \zeta_{50}^{6} + \zeta_{50}^{14} ) q^{37} + ( 1 - \zeta_{50}^{3} + \zeta_{50}^{12} - \zeta_{50}^{15} ) q^{38} + ( -\zeta_{50}^{13} - \zeta_{50}^{17} ) q^{41} + ( -\zeta_{50} + \zeta_{50}^{20} - \zeta_{50}^{23} ) q^{44} + ( \zeta_{50}^{2} - \zeta_{50}^{3} ) q^{47} + \zeta_{50}^{20} q^{49} + ( \zeta_{50}^{6} - \zeta_{50}^{9} ) q^{50} + ( 1 + \zeta_{50}^{2} - \zeta_{50}^{3} - \zeta_{50}^{5} + \zeta_{50}^{6} + \zeta_{50}^{24} ) q^{52} + ( -\zeta_{50}^{19} - \zeta_{50}^{21} ) q^{61} + ( -\zeta_{50}^{7} + 2 \zeta_{50}^{10} - \zeta_{50}^{13} ) q^{62} + ( \zeta_{50}^{2} - \zeta_{50}^{5} + \zeta_{50}^{8} - \zeta_{50}^{11} + \zeta_{50}^{14} - \zeta_{50}^{21} + \zeta_{50}^{24} ) q^{64} + ( -\zeta_{50} + \zeta_{50}^{4} + \zeta_{50}^{6} - \zeta_{50}^{9} + \zeta_{50}^{12} - \zeta_{50}^{23} ) q^{68} + ( \zeta_{50}^{4} - \zeta_{50}^{11} ) q^{71} + ( -\zeta_{50}^{3} + \zeta_{50}^{6} - \zeta_{50}^{9} + \zeta_{50}^{12} ) q^{72} + ( -\zeta_{50}^{3} - \zeta_{50}^{17} ) q^{73} + ( 1 - \zeta_{50}^{5} + \zeta_{50}^{8} + \zeta_{50}^{22} ) q^{74} + ( -\zeta_{50}^{3} + \zeta_{50}^{6} - \zeta_{50}^{9} + \zeta_{50}^{16} - \zeta_{50}^{19} + \zeta_{50}^{22} ) q^{76} + ( \zeta_{50}^{2} + \zeta_{50}^{8} ) q^{79} + \zeta_{50}^{10} q^{81} + ( \zeta_{50}^{4} - \zeta_{50}^{7} + \zeta_{50}^{8} - \zeta_{50}^{11} ) q^{82} + ( -\zeta_{50}^{11} + \zeta_{50}^{14} - \zeta_{50}^{17} + \zeta_{50}^{20} ) q^{88} + ( \zeta_{50}^{18} - \zeta_{50}^{19} - \zeta_{50}^{21} + \zeta_{50}^{22} ) q^{94} + ( -\zeta_{50}^{11} + \zeta_{50}^{14} ) q^{98} + \zeta_{50}^{18} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q - 5q^{4} - 5q^{9} + O(q^{10}) \) \( 20q - 5q^{4} - 5q^{9} - 5q^{16} - 5q^{25} - 10q^{32} - 10q^{34} - 5q^{36} + 15q^{38} - 5q^{44} - 5q^{49} + 15q^{52} - 10q^{62} - 5q^{64} + 15q^{74} - 5q^{81} - 5q^{88} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1397\mathbb{Z}\right)^\times\).

\(n\) \(255\) \(892\)
\(\chi(n)\) \(\zeta_{50}^{10}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
126.1
−0.968583 0.248690i
−0.0627905 0.998027i
−0.535827 + 0.844328i
0.929776 0.368125i
0.637424 + 0.770513i
0.187381 0.982287i
−0.876307 0.481754i
0.992115 0.125333i
−0.728969 + 0.684547i
0.425779 + 0.904827i
−0.968583 + 0.248690i
−0.0627905 + 0.998027i
−0.535827 0.844328i
0.929776 + 0.368125i
0.637424 0.770513i
0.187381 + 0.982287i
−0.876307 + 0.481754i
0.992115 + 0.125333i
−0.728969 0.684547i
0.425779 0.904827i
−0.574633 1.76854i 0 −1.98851 + 1.44474i 0 0 0 2.19334 + 1.59355i 0.309017 + 0.951057i 0
126.2 −0.393950 1.21245i 0 −0.505828 + 0.367505i 0 0 0 −0.386520 0.280823i 0.309017 + 0.951057i 0
126.3 0.0388067 + 0.119435i 0 0.796258 0.578516i 0 0 0 0.201592 + 0.146465i 0.309017 + 0.951057i 0
126.4 0.331159 + 1.01920i 0 −0.120092 + 0.0872517i 0 0 0 0.738289 + 0.536399i 0.309017 + 0.951057i 0
126.5 0.598617 + 1.84235i 0 −2.22691 + 1.61795i 0 0 0 −2.74670 1.99559i 0.309017 + 0.951057i 0
1015.1 −1.41789 + 1.03016i 0 0.640176 1.97026i 0 0 0 0.580394 + 1.78627i −0.809017 + 0.587785i 0
1015.2 −1.17950 + 0.856954i 0 0.347824 1.07049i 0 0 0 0.0565777 + 0.174128i −0.809017 + 0.587785i 0
1015.3 0.303189 0.220280i 0 −0.265616 + 0.817483i 0 0 0 0.215351 + 0.662783i −0.809017 + 0.587785i 0
1015.4 0.688925 0.500534i 0 −0.0849327 + 0.261396i 0 0 0 0.335471 + 1.03247i −0.809017 + 0.587785i 0
1015.5 1.60528 1.16630i 0 0.907634 2.79341i 0 0 0 −1.18779 3.65565i −0.809017 + 0.587785i 0
1142.1 −0.574633 + 1.76854i 0 −1.98851 1.44474i 0 0 0 2.19334 1.59355i 0.309017 0.951057i 0
1142.2 −0.393950 + 1.21245i 0 −0.505828 0.367505i 0 0 0 −0.386520 + 0.280823i 0.309017 0.951057i 0
1142.3 0.0388067 0.119435i 0 0.796258 + 0.578516i 0 0 0 0.201592 0.146465i 0.309017 0.951057i 0
1142.4 0.331159 1.01920i 0 −0.120092 0.0872517i 0 0 0 0.738289 0.536399i 0.309017 0.951057i 0
1142.5 0.598617 1.84235i 0 −2.22691 1.61795i 0 0 0 −2.74670 + 1.99559i 0.309017 0.951057i 0
1269.1 −1.41789 1.03016i 0 0.640176 + 1.97026i 0 0 0 0.580394 1.78627i −0.809017 0.587785i 0
1269.2 −1.17950 0.856954i 0 0.347824 + 1.07049i 0 0 0 0.0565777 0.174128i −0.809017 0.587785i 0
1269.3 0.303189 + 0.220280i 0 −0.265616 0.817483i 0 0 0 0.215351 0.662783i −0.809017 0.587785i 0
1269.4 0.688925 + 0.500534i 0 −0.0849327 0.261396i 0 0 0 0.335471 1.03247i −0.809017 0.587785i 0
1269.5 1.60528 + 1.16630i 0 0.907634 + 2.79341i 0 0 0 −1.18779 + 3.65565i −0.809017 0.587785i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1269.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
127.b odd 2 1 CM by \(\Q(\sqrt{-127}) \)
11.c even 5 1 inner
1397.l odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1397.1.l.a 20
11.c even 5 1 inner 1397.1.l.a 20
127.b odd 2 1 CM 1397.1.l.a 20
1397.l odd 10 1 inner 1397.1.l.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1397.1.l.a 20 1.a even 1 1 trivial
1397.1.l.a 20 11.c even 5 1 inner
1397.1.l.a 20 127.b odd 2 1 CM
1397.1.l.a 20 1397.l odd 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1397, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 10 T + 100 T^{2} - 390 T^{3} + 800 T^{4} - 752 T^{5} + 740 T^{6} - 475 T^{7} + 510 T^{8} - 175 T^{9} + 374 T^{10} + 30 T^{11} + 275 T^{12} + 20 T^{13} + 75 T^{14} + 2 T^{15} + 20 T^{16} + 5 T^{18} + T^{20} \)
$3$ \( T^{20} \)
$5$ \( T^{20} \)
$7$ \( T^{20} \)
$11$ \( 1 + T^{5} + T^{10} + T^{15} + T^{20} \)
$13$ \( 1 + 15 T + 100 T^{2} + 260 T^{3} + 675 T^{4} + 498 T^{5} + 715 T^{6} + 150 T^{7} + 485 T^{8} + 200 T^{9} + 374 T^{10} + 5 T^{11} + 275 T^{12} - 5 T^{13} + 75 T^{14} + 2 T^{15} + 20 T^{16} + 5 T^{18} + T^{20} \)
$17$ \( 1 - 10 T + 100 T^{2} - 390 T^{3} + 800 T^{4} - 752 T^{5} + 740 T^{6} - 475 T^{7} + 510 T^{8} - 175 T^{9} + 374 T^{10} + 30 T^{11} + 275 T^{12} + 20 T^{13} + 75 T^{14} + 2 T^{15} + 20 T^{16} + 5 T^{18} + T^{20} \)
$19$ \( 1 + 15 T + 100 T^{2} + 260 T^{3} + 675 T^{4} + 498 T^{5} + 715 T^{6} + 150 T^{7} + 485 T^{8} + 200 T^{9} + 374 T^{10} + 5 T^{11} + 275 T^{12} - 5 T^{13} + 75 T^{14} + 2 T^{15} + 20 T^{16} + 5 T^{18} + T^{20} \)
$23$ \( T^{20} \)
$29$ \( T^{20} \)
$31$ \( 1 - 10 T + 100 T^{2} - 390 T^{3} + 800 T^{4} - 752 T^{5} + 740 T^{6} - 475 T^{7} + 510 T^{8} - 175 T^{9} + 374 T^{10} + 30 T^{11} + 275 T^{12} + 20 T^{13} + 75 T^{14} + 2 T^{15} + 20 T^{16} + 5 T^{18} + T^{20} \)
$37$ \( 1 + 15 T + 100 T^{2} + 260 T^{3} + 675 T^{4} + 498 T^{5} + 715 T^{6} + 150 T^{7} + 485 T^{8} + 200 T^{9} + 374 T^{10} + 5 T^{11} + 275 T^{12} - 5 T^{13} + 75 T^{14} + 2 T^{15} + 20 T^{16} + 5 T^{18} + T^{20} \)
$41$ \( 1 - 10 T + 100 T^{2} - 390 T^{3} + 800 T^{4} - 752 T^{5} + 740 T^{6} - 475 T^{7} + 510 T^{8} - 175 T^{9} + 374 T^{10} + 30 T^{11} + 275 T^{12} + 20 T^{13} + 75 T^{14} + 2 T^{15} + 20 T^{16} + 5 T^{18} + T^{20} \)
$43$ \( T^{20} \)
$47$ \( 1 - 10 T + 100 T^{2} - 390 T^{3} + 800 T^{4} - 752 T^{5} + 740 T^{6} - 475 T^{7} + 510 T^{8} - 175 T^{9} + 374 T^{10} + 30 T^{11} + 275 T^{12} + 20 T^{13} + 75 T^{14} + 2 T^{15} + 20 T^{16} + 5 T^{18} + T^{20} \)
$53$ \( T^{20} \)
$59$ \( T^{20} \)
$61$ \( 1 - 10 T + 100 T^{2} - 390 T^{3} + 800 T^{4} - 752 T^{5} + 740 T^{6} - 475 T^{7} + 510 T^{8} - 175 T^{9} + 374 T^{10} + 30 T^{11} + 275 T^{12} + 20 T^{13} + 75 T^{14} + 2 T^{15} + 20 T^{16} + 5 T^{18} + T^{20} \)
$67$ \( T^{20} \)
$71$ \( 1 - 10 T + 100 T^{2} - 390 T^{3} + 800 T^{4} - 752 T^{5} + 740 T^{6} - 475 T^{7} + 510 T^{8} - 175 T^{9} + 374 T^{10} + 30 T^{11} + 275 T^{12} + 20 T^{13} + 75 T^{14} + 2 T^{15} + 20 T^{16} + 5 T^{18} + T^{20} \)
$73$ \( 1 - 10 T + 100 T^{2} - 390 T^{3} + 800 T^{4} - 752 T^{5} + 740 T^{6} - 475 T^{7} + 510 T^{8} - 175 T^{9} + 374 T^{10} + 30 T^{11} + 275 T^{12} + 20 T^{13} + 75 T^{14} + 2 T^{15} + 20 T^{16} + 5 T^{18} + T^{20} \)
$79$ \( 1 + 15 T + 100 T^{2} + 260 T^{3} + 675 T^{4} + 498 T^{5} + 715 T^{6} + 150 T^{7} + 485 T^{8} + 200 T^{9} + 374 T^{10} + 5 T^{11} + 275 T^{12} - 5 T^{13} + 75 T^{14} + 2 T^{15} + 20 T^{16} + 5 T^{18} + T^{20} \)
$83$ \( T^{20} \)
$89$ \( T^{20} \)
$97$ \( T^{20} \)
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