Properties

Label 979.1.v.a
Level $979$
Weight $1$
Character orbit 979.v
Analytic conductor $0.489$
Analytic rank $0$
Dimension $20$
Projective image $D_{44}$
CM discriminant -11
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 979 = 11 \cdot 89 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 979.v (of order \(44\), degree \(20\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.488584647368\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\Q(\zeta_{44})\)
Defining polynomial: \(x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{44}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{44} - \cdots)\)

$q$-expansion

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/979\mathbb{Z}\right)^\times\).

\(n\) \(90\) \(804\)
\(\chi(n)\) \(-1\) \(-\zeta_{44}^{9}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
−0.755750 + 0.654861i
0.540641 + 0.841254i
−0.755750 0.654861i
−0.281733 + 0.959493i
0.909632 + 0.415415i
0.909632 0.415415i
−0.989821 0.142315i
0.281733 + 0.959493i
0.989821 0.142315i
0.540641 0.841254i
−0.540641 + 0.841254i
−0.989821 + 0.142315i
−0.281733 0.959493i
0.989821 + 0.142315i
−0.909632 + 0.415415i
−0.909632 0.415415i
0.281733 0.959493i
0.755750 + 0.654861i
−0.540641 0.841254i
0.755750 0.654861i
0 0.142315 0.0101786i −0.654861 0.755750i −0.368991 1.25667i 0 0 0 −0.969672 + 0.139418i 0
21.1 0 −0.415415 + 0.0903680i 0.841254 0.540641i −1.27155 1.10181i 0 0 0 −0.745229 + 0.340335i 0
98.1 0 0.142315 + 0.0101786i −0.654861 + 0.755750i −0.368991 + 1.25667i 0 0 0 −0.969672 0.139418i 0
109.1 0 −0.841254 0.459359i −0.959493 0.281733i 1.74557 + 0.797176i 0 0 0 −0.0439442 0.0683785i 0
131.1 0 0.654861 + 0.244250i 0.415415 0.909632i 0.822373 0.118239i 0 0 0 −0.386565 0.334961i 0
142.1 0 0.654861 0.244250i 0.415415 + 0.909632i 0.822373 + 0.118239i 0 0 0 −0.386565 + 0.334961i 0
285.1 0 0.959493 + 0.718267i −0.142315 + 0.989821i −0.153882 0.239446i 0 0 0 0.122986 + 0.418852i 0
307.1 0 −0.841254 1.54064i −0.959493 + 0.281733i −1.74557 + 0.797176i 0 0 0 −1.12523 + 1.75089i 0
351.1 0 0.959493 + 1.28173i −0.142315 0.989821i 0.153882 0.239446i 0 0 0 −0.440479 + 1.50013i 0
373.1 0 −0.415415 0.0903680i 0.841254 + 0.540641i −1.27155 + 1.10181i 0 0 0 −0.745229 0.340335i 0
428.1 0 −0.415415 + 1.90963i 0.841254 + 0.540641i 1.27155 1.10181i 0 0 0 −2.56449 1.17116i 0
450.1 0 0.959493 0.718267i −0.142315 0.989821i −0.153882 + 0.239446i 0 0 0 0.122986 0.418852i 0
494.1 0 −0.841254 + 0.459359i −0.959493 + 0.281733i 1.74557 0.797176i 0 0 0 −0.0439442 + 0.0683785i 0
516.1 0 0.959493 1.28173i −0.142315 + 0.989821i 0.153882 + 0.239446i 0 0 0 −0.440479 1.50013i 0
659.1 0 0.654861 + 1.75575i 0.415415 + 0.909632i −0.822373 0.118239i 0 0 0 −1.89806 + 1.64468i 0
670.1 0 0.654861 1.75575i 0.415415 0.909632i −0.822373 + 0.118239i 0 0 0 −1.89806 1.64468i 0
692.1 0 −0.841254 + 1.54064i −0.959493 0.281733i −1.74557 0.797176i 0 0 0 −1.12523 1.75089i 0
703.1 0 0.142315 1.98982i −0.654861 + 0.755750i 0.368991 1.25667i 0 0 0 −2.94931 0.424047i 0
780.1 0 −0.415415 1.90963i 0.841254 0.540641i 1.27155 + 1.10181i 0 0 0 −2.56449 + 1.17116i 0
791.1 0 0.142315 + 1.98982i −0.654861 0.755750i 0.368991 + 1.25667i 0 0 0 −2.94931 + 0.424047i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 791.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
89.g even 44 1 inner
979.v odd 44 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 979.1.v.a 20
11.b odd 2 1 CM 979.1.v.a 20
89.g even 44 1 inner 979.1.v.a 20
979.v odd 44 1 inner 979.1.v.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
979.1.v.a 20 1.a even 1 1 trivial
979.1.v.a 20 11.b odd 2 1 CM
979.1.v.a 20 89.g even 44 1 inner
979.1.v.a 20 979.v odd 44 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(979, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \)
$3$ \( 1 - 12 T + 17 T^{2} + 154 T^{3} - 135 T^{4} - 404 T^{5} + 554 T^{6} - 22 T^{7} - 90 T^{8} + 46 T^{9} + 230 T^{10} - 252 T^{11} + 368 T^{12} - 242 T^{13} + 223 T^{14} - 102 T^{15} + 73 T^{16} - 22 T^{17} + 13 T^{18} - 2 T^{19} + T^{20} \)
$5$ \( 1 + 8 T^{2} + 130 T^{4} - 335 T^{6} + 125 T^{8} + 120 T^{10} + 36 T^{12} - 9 T^{14} + 16 T^{16} - 4 T^{18} + T^{20} \)
$7$ \( T^{20} \)
$11$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
$13$ \( T^{20} \)
$17$ \( T^{20} \)
$19$ \( T^{20} \)
$23$ \( 1 + 12 T + 105 T^{2} + 484 T^{3} + 1218 T^{4} + 1702 T^{5} + 1324 T^{6} + 484 T^{7} - 178 T^{8} - 420 T^{9} - 331 T^{10} - 122 T^{11} + 93 T^{12} - 8 T^{14} - 8 T^{15} - 4 T^{16} + 2 T^{18} + 2 T^{19} + T^{20} \)
$29$ \( T^{20} \)
$31$ \( 1 - 12 T + 50 T^{2} - 44 T^{3} + 250 T^{4} - 74 T^{5} - 51 T^{6} - 638 T^{7} + 713 T^{8} - 394 T^{9} + 824 T^{10} - 626 T^{11} + 214 T^{12} - 286 T^{13} + 179 T^{14} - 36 T^{15} + 40 T^{16} - 22 T^{17} + 2 T^{18} - 2 T^{19} + T^{20} \)
$37$ \( 1 + 10 T + 50 T^{2} - 25 T^{4} - 52 T^{5} + 730 T^{6} - 748 T^{7} + 383 T^{8} + 2 T^{9} + 472 T^{10} - 472 T^{11} + 236 T^{12} + 58 T^{14} - 58 T^{15} + 29 T^{16} + 2 T^{18} - 2 T^{19} + T^{20} \)
$41$ \( T^{20} \)
$43$ \( T^{20} \)
$47$ \( ( 11 + 22 T - 11 T^{3} + 33 T^{4} + 11 T^{7} + T^{10} )^{2} \)
$53$ \( 1 - 14 T^{2} + 119 T^{4} + 303 T^{6} + 257 T^{8} - 34 T^{10} + 14 T^{12} - 42 T^{14} + 16 T^{16} - 4 T^{18} + T^{20} \)
$59$ \( 1 - 10 T + 17 T^{2} + 44 T^{3} + 338 T^{4} + 316 T^{5} + 400 T^{6} + 110 T^{7} - 90 T^{8} + 460 T^{9} + 505 T^{10} + 274 T^{11} + 93 T^{12} - 44 T^{13} - 52 T^{14} - 30 T^{15} - 4 T^{16} + 2 T^{18} + 2 T^{19} + T^{20} \)
$61$ \( T^{20} \)
$67$ \( 121 - 121 T^{2} + 847 T^{4} + 1573 T^{6} + 1452 T^{8} + 462 T^{10} + 22 T^{12} + T^{20} \)
$71$ \( 1 - 14 T^{2} + 119 T^{4} + 303 T^{6} + 257 T^{8} - 34 T^{10} + 14 T^{12} - 42 T^{14} + 16 T^{16} - 4 T^{18} + T^{20} \)
$73$ \( T^{20} \)
$79$ \( T^{20} \)
$83$ \( T^{20} \)
$89$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \)
$97$ \( 121 - 605 T^{2} + 1089 T^{4} + 484 T^{8} + 462 T^{10} + 330 T^{12} + 165 T^{14} + 55 T^{16} + 11 T^{18} + T^{20} \)
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