Properties

Label 371.1.n.a
Level $371$
Weight $1$
Character orbit 371.n
Analytic conductor $0.185$
Analytic rank $0$
Dimension $12$
Projective image $D_{13}$
CM discriminant -7
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 371 = 7 \cdot 53 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 371.n (of order \(26\), degree \(12\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.185153119687\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\Q(\zeta_{26})\)
Defining polynomial: \(x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{13}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{13} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( -\zeta_{26} + \zeta_{26}^{6} ) q^{2} + ( \zeta_{26}^{2} - \zeta_{26}^{7} + \zeta_{26}^{12} ) q^{4} + \zeta_{26}^{10} q^{7} + ( 1 - \zeta_{26}^{3} - \zeta_{26}^{5} + \zeta_{26}^{8} ) q^{8} + \zeta_{26}^{2} q^{9} +O(q^{10})\) \( q + ( -\zeta_{26} + \zeta_{26}^{6} ) q^{2} + ( \zeta_{26}^{2} - \zeta_{26}^{7} + \zeta_{26}^{12} ) q^{4} + \zeta_{26}^{10} q^{7} + ( 1 - \zeta_{26}^{3} - \zeta_{26}^{5} + \zeta_{26}^{8} ) q^{8} + \zeta_{26}^{2} q^{9} + ( \zeta_{26}^{6} + \zeta_{26}^{10} ) q^{11} + ( -\zeta_{26}^{3} - \zeta_{26}^{11} ) q^{14} + ( -\zeta_{26} + \zeta_{26}^{4} + \zeta_{26}^{6} - \zeta_{26}^{9} - \zeta_{26}^{11} ) q^{16} + ( -\zeta_{26}^{3} + \zeta_{26}^{8} ) q^{18} + ( -\zeta_{26}^{3} - \zeta_{26}^{7} - \zeta_{26}^{11} + \zeta_{26}^{12} ) q^{22} + ( -\zeta_{26}^{5} + \zeta_{26}^{8} ) q^{23} + \zeta_{26}^{4} q^{25} + ( \zeta_{26}^{4} - \zeta_{26}^{9} + \zeta_{26}^{12} ) q^{28} + ( -\zeta_{26} - \zeta_{26}^{9} ) q^{29} + ( \zeta_{26}^{2} + \zeta_{26}^{4} - \zeta_{26}^{5} - \zeta_{26}^{7} + \zeta_{26}^{10} + \zeta_{26}^{12} ) q^{32} + ( -\zeta_{26} + \zeta_{26}^{4} - \zeta_{26}^{9} ) q^{36} + ( -\zeta_{26}^{7} + \zeta_{26}^{8} ) q^{37} + ( 1 - \zeta_{26}^{11} ) q^{43} + ( 1 + \zeta_{26}^{4} - \zeta_{26}^{5} + \zeta_{26}^{8} - \zeta_{26}^{9} + \zeta_{26}^{12} ) q^{44} + ( -\zeta_{26} + \zeta_{26}^{6} - \zeta_{26}^{9} - \zeta_{26}^{11} ) q^{46} -\zeta_{26}^{7} q^{49} + ( -\zeta_{26}^{5} + \zeta_{26}^{10} ) q^{50} -\zeta_{26}^{11} q^{53} + ( 1 + \zeta_{26}^{2} - \zeta_{26}^{5} + \zeta_{26}^{10} ) q^{56} + ( 2 \zeta_{26}^{2} - \zeta_{26}^{7} + \zeta_{26}^{10} ) q^{58} + \zeta_{26}^{12} q^{63} + ( 1 - \zeta_{26}^{3} - \zeta_{26}^{5} + \zeta_{26}^{6} + \zeta_{26}^{8} + \zeta_{26}^{10} - \zeta_{26}^{11} ) q^{64} + ( -\zeta_{26}^{5} - \zeta_{26}^{9} ) q^{67} + ( \zeta_{26}^{2} - \zeta_{26}^{9} ) q^{71} + ( \zeta_{26}^{2} - \zeta_{26}^{5} - \zeta_{26}^{7} + \zeta_{26}^{10} ) q^{72} + ( 1 - \zeta_{26} + \zeta_{26}^{8} - \zeta_{26}^{9} ) q^{74} + ( -\zeta_{26}^{3} - \zeta_{26}^{7} ) q^{77} + ( 1 + \zeta_{26}^{6} ) q^{79} + \zeta_{26}^{4} q^{81} + ( -\zeta_{26} + \zeta_{26}^{4} + \zeta_{26}^{6} + \zeta_{26}^{12} ) q^{86} + ( 1 - \zeta_{26} + \zeta_{26}^{2} - \zeta_{26}^{5} + \zeta_{26}^{6} - \zeta_{26}^{9} + \zeta_{26}^{10} - \zeta_{26}^{11} ) q^{88} + ( \zeta_{26}^{2} + \zeta_{26}^{4} - 2 \zeta_{26}^{7} + \zeta_{26}^{10} + \zeta_{26}^{12} ) q^{92} + ( 1 + \zeta_{26}^{8} ) q^{98} + ( \zeta_{26}^{8} + \zeta_{26}^{12} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 2q^{2} - 3q^{4} - q^{7} + 9q^{8} - q^{9} + O(q^{10}) \) \( 12q - 2q^{2} - 3q^{4} - q^{7} + 9q^{8} - q^{9} - 2q^{11} - 2q^{14} - 5q^{16} - 2q^{18} - 4q^{22} - 2q^{23} - q^{25} - 3q^{28} - 2q^{29} - 6q^{32} - 3q^{36} - 2q^{37} + 11q^{43} + 7q^{44} - 4q^{46} - q^{49} - 2q^{50} - q^{53} + 9q^{56} - 4q^{58} - q^{63} + 6q^{64} - 2q^{67} - 2q^{71} - 4q^{72} + 9q^{74} - 2q^{77} + 11q^{79} - q^{81} - 4q^{86} + 5q^{88} - 6q^{92} + 11q^{98} - 2q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/371\mathbb{Z}\right)^\times\).

\(n\) \(213\) \(267\)
\(\chi(n)\) \(-1\) \(-\zeta_{26}^{7}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
−0.885456 + 0.464723i
−0.568065 0.822984i
0.354605 0.935016i
0.354605 + 0.935016i
0.970942 0.239316i
0.748511 0.663123i
0.970942 + 0.239316i
0.748511 + 0.663123i
−0.120537 0.992709i
−0.885456 0.464723i
−0.568065 + 0.822984i
−0.120537 + 0.992709i
−0.0854858 0.704039i 0 0.482579 0.118945i 0 0 0.120537 + 0.992709i −0.376485 0.992709i 0.568065 0.822984i 0
69.1 1.45352 + 0.358261i 0 1.09892 + 0.576756i 0 0 −0.970942 0.239316i 0.270132 + 0.239316i −0.354605 + 0.935016i 0
97.1 0.213460 + 0.112032i 0 −0.535051 0.775155i 0 0 0.885456 + 0.464723i −0.0564276 0.464723i −0.748511 0.663123i 0
153.1 0.213460 0.112032i 0 −0.535051 + 0.775155i 0 0 0.885456 0.464723i −0.0564276 + 0.464723i −0.748511 + 0.663123i 0
174.1 −0.850405 0.753393i 0 0.0350509 + 0.288670i 0 0 −0.748511 0.663123i −0.457721 + 0.663123i 0.885456 0.464723i 0
195.1 −1.10312 + 1.59814i 0 −0.982579 2.59085i 0 0 0.568065 0.822984i 3.33898 + 0.822984i 0.120537 0.992709i 0
258.1 −0.850405 + 0.753393i 0 0.0350509 0.288670i 0 0 −0.748511 + 0.663123i −0.457721 0.663123i 0.885456 + 0.464723i 0
293.1 −1.10312 1.59814i 0 −0.982579 + 2.59085i 0 0 0.568065 + 0.822984i 3.33898 0.822984i 0.120537 + 0.992709i 0
307.1 −0.627974 + 1.65583i 0 −1.59892 1.41652i 0 0 −0.354605 + 0.935016i 1.78152 0.935016i −0.970942 + 0.239316i 0
314.1 −0.0854858 + 0.704039i 0 0.482579 + 0.118945i 0 0 0.120537 0.992709i −0.376485 + 0.992709i 0.568065 + 0.822984i 0
328.1 1.45352 0.358261i 0 1.09892 0.576756i 0 0 −0.970942 + 0.239316i 0.270132 0.239316i −0.354605 0.935016i 0
342.1 −0.627974 1.65583i 0 −1.59892 + 1.41652i 0 0 −0.354605 0.935016i 1.78152 + 0.935016i −0.970942 0.239316i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 342.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
53.d even 13 1 inner
371.n odd 26 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 371.1.n.a 12
3.b odd 2 1 3339.1.ck.a 12
7.b odd 2 1 CM 371.1.n.a 12
7.c even 3 2 2597.1.bf.a 24
7.d odd 6 2 2597.1.bf.a 24
21.c even 2 1 3339.1.ck.a 12
53.d even 13 1 inner 371.1.n.a 12
159.j odd 26 1 3339.1.ck.a 12
371.n odd 26 1 inner 371.1.n.a 12
371.q even 39 2 2597.1.bf.a 24
371.v odd 78 2 2597.1.bf.a 24
1113.bd even 26 1 3339.1.ck.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
371.1.n.a 12 1.a even 1 1 trivial
371.1.n.a 12 7.b odd 2 1 CM
371.1.n.a 12 53.d even 13 1 inner
371.1.n.a 12 371.n odd 26 1 inner
2597.1.bf.a 24 7.c even 3 2
2597.1.bf.a 24 7.d odd 6 2
2597.1.bf.a 24 371.q even 39 2
2597.1.bf.a 24 371.v odd 78 2
3339.1.ck.a 12 3.b odd 2 1
3339.1.ck.a 12 21.c even 2 1
3339.1.ck.a 12 159.j odd 26 1
3339.1.ck.a 12 1113.bd even 26 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(371, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 6 T + 10 T^{2} + 5 T^{3} + 35 T^{4} + 24 T^{5} + 12 T^{6} - 20 T^{7} - 10 T^{8} - 5 T^{9} + 4 T^{10} + 2 T^{11} + T^{12} \)
$3$ \( T^{12} \)
$5$ \( T^{12} \)
$7$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
$11$ \( 1 - 6 T + 10 T^{2} + 5 T^{3} + 35 T^{4} + 24 T^{5} + 12 T^{6} - 20 T^{7} - 10 T^{8} - 5 T^{9} + 4 T^{10} + 2 T^{11} + T^{12} \)
$13$ \( T^{12} \)
$17$ \( T^{12} \)
$19$ \( T^{12} \)
$23$ \( ( -1 + 3 T + 6 T^{2} - 4 T^{3} - 5 T^{4} + T^{5} + T^{6} )^{2} \)
$29$ \( 1 + 7 T + 10 T^{2} - 8 T^{3} + 22 T^{4} + 11 T^{5} + 38 T^{6} + 19 T^{7} + 16 T^{8} + 8 T^{9} + 4 T^{10} + 2 T^{11} + T^{12} \)
$31$ \( T^{12} \)
$37$ \( 1 + 7 T + 10 T^{2} - 8 T^{3} + 22 T^{4} + 11 T^{5} + 38 T^{6} + 19 T^{7} + 16 T^{8} + 8 T^{9} + 4 T^{10} + 2 T^{11} + T^{12} \)
$41$ \( T^{12} \)
$43$ \( 1 - 6 T + 36 T^{2} - 125 T^{3} + 295 T^{4} - 496 T^{5} + 610 T^{6} - 553 T^{7} + 367 T^{8} - 174 T^{9} + 56 T^{10} - 11 T^{11} + T^{12} \)
$47$ \( T^{12} \)
$53$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
$59$ \( T^{12} \)
$61$ \( T^{12} \)
$67$ \( 1 + 7 T + 36 T^{2} + 96 T^{3} + 139 T^{4} + 115 T^{5} + 64 T^{6} + 32 T^{7} + 16 T^{8} + 8 T^{9} + 4 T^{10} + 2 T^{11} + T^{12} \)
$71$ \( 1 - 6 T + 10 T^{2} + 5 T^{3} + 35 T^{4} + 24 T^{5} + 12 T^{6} - 20 T^{7} - 10 T^{8} - 5 T^{9} + 4 T^{10} + 2 T^{11} + T^{12} \)
$73$ \( T^{12} \)
$79$ \( 1 - 6 T + 36 T^{2} - 125 T^{3} + 295 T^{4} - 496 T^{5} + 610 T^{6} - 553 T^{7} + 367 T^{8} - 174 T^{9} + 56 T^{10} - 11 T^{11} + T^{12} \)
$83$ \( T^{12} \)
$89$ \( T^{12} \)
$97$ \( T^{12} \)
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