Properties

Label 1412.1.r.a
Level $1412$
Weight $1$
Character orbit 1412.r
Analytic conductor $0.705$
Analytic rank $0$
Dimension $20$
Projective image $D_{44}$
CM discriminant -4
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1412 = 2^{2} \cdot 353 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1412.r (of order \(44\), degree \(20\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.704679797838\)
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: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + \zeta_{44}^{14} q^{2} -\zeta_{44}^{6} q^{4} + ( -\zeta_{44} + \zeta_{44}^{4} ) q^{5} -\zeta_{44}^{20} q^{8} -\zeta_{44}^{5} q^{9} +O(q^{10})\) \( q + \zeta_{44}^{14} q^{2} -\zeta_{44}^{6} q^{4} + ( -\zeta_{44} + \zeta_{44}^{4} ) q^{5} -\zeta_{44}^{20} q^{8} -\zeta_{44}^{5} q^{9} + ( -\zeta_{44}^{15} + \zeta_{44}^{18} ) q^{10} + ( \zeta_{44}^{15} + \zeta_{44}^{20} ) q^{13} + \zeta_{44}^{12} q^{16} + ( -\zeta_{44}^{4} + \zeta_{44}^{14} ) q^{17} -\zeta_{44}^{19} q^{18} + ( \zeta_{44}^{7} - \zeta_{44}^{10} ) q^{20} + ( \zeta_{44}^{2} - \zeta_{44}^{5} + \zeta_{44}^{8} ) q^{25} + ( -\zeta_{44}^{7} - \zeta_{44}^{12} ) q^{26} + ( \zeta_{44}^{15} + \zeta_{44}^{17} ) q^{29} -\zeta_{44}^{4} q^{32} + ( -\zeta_{44}^{6} - \zeta_{44}^{18} ) q^{34} + \zeta_{44}^{11} q^{36} + ( -1 + \zeta_{44}^{7} ) q^{37} + ( \zeta_{44}^{2} + \zeta_{44}^{21} ) q^{40} + ( \zeta_{44}^{5} + \zeta_{44}^{9} ) q^{41} + ( \zeta_{44}^{6} - \zeta_{44}^{9} ) q^{45} -\zeta_{44}^{11} q^{49} + ( -1 + \zeta_{44}^{16} - \zeta_{44}^{19} ) q^{50} + ( \zeta_{44}^{4} - \zeta_{44}^{21} ) q^{52} + ( -\zeta_{44}^{2} - \zeta_{44}^{3} ) q^{53} + ( -\zeta_{44}^{7} - \zeta_{44}^{9} ) q^{58} + ( -\zeta_{44}^{11} - \zeta_{44}^{13} ) q^{61} -\zeta_{44}^{18} q^{64} + ( -\zeta_{44}^{2} - \zeta_{44}^{16} + \zeta_{44}^{19} - \zeta_{44}^{21} ) q^{65} + ( \zeta_{44}^{10} - \zeta_{44}^{20} ) q^{68} -\zeta_{44}^{3} q^{72} + ( -\zeta_{44}^{14} + \zeta_{44}^{21} ) q^{74} + ( -\zeta_{44}^{13} + \zeta_{44}^{16} ) q^{80} + \zeta_{44}^{10} q^{81} + ( -\zeta_{44} + \zeta_{44}^{19} ) q^{82} + ( \zeta_{44}^{5} - \zeta_{44}^{8} - \zeta_{44}^{15} + \zeta_{44}^{18} ) q^{85} + ( -\zeta_{44}^{17} + \zeta_{44}^{20} ) q^{89} + ( \zeta_{44} + \zeta_{44}^{20} ) q^{90} + ( -\zeta_{44}^{10} + \zeta_{44}^{16} ) q^{97} + \zeta_{44}^{3} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 2q^{2} - 2q^{4} - 2q^{5} + 2q^{8} + O(q^{10}) \) \( 20q + 2q^{2} - 2q^{4} - 2q^{5} + 2q^{8} + 2q^{10} - 2q^{13} - 2q^{16} + 4q^{17} - 2q^{20} + 2q^{26} + 2q^{32} - 4q^{34} - 20q^{37} + 2q^{40} + 2q^{45} - 22q^{50} - 2q^{52} - 2q^{53} - 2q^{64} + 4q^{68} - 2q^{74} - 2q^{80} + 2q^{81} + 4q^{85} - 2q^{89} - 2q^{90} - 4q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1412\mathbb{Z}\right)^\times\).

\(n\) \(707\) \(709\)
\(\chi(n)\) \(-1\) \(-\zeta_{44}^{5}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
35.1
0.909632 0.415415i
−0.989821 + 0.142315i
−0.540641 + 0.841254i
−0.281733 + 0.959493i
−0.281733 0.959493i
0.989821 + 0.142315i
−0.989821 0.142315i
0.281733 + 0.959493i
0.281733 0.959493i
0.540641 0.841254i
0.989821 0.142315i
−0.909632 + 0.415415i
−0.909632 0.415415i
0.755750 + 0.654861i
0.540641 + 0.841254i
−0.755750 + 0.654861i
0.755750 0.654861i
−0.540641 0.841254i
−0.755750 0.654861i
0.909632 + 0.415415i
0.959493 + 0.281733i 0 0.841254 + 0.540641i −1.05195 0.574406i 0 0 0.654861 + 0.755750i 0.540641 + 0.841254i −0.847507 0.847507i
135.1 −0.415415 0.909632i 0 −0.654861 + 0.755750i 1.83107 0.682956i 0 0 0.959493 + 0.281733i 0.755750 0.654861i −1.38189 1.38189i
171.1 0.142315 0.989821i 0 −0.959493 0.281733i −0.114220 0.0855040i 0 0 −0.415415 + 0.909632i 0.281733 + 0.959493i −0.100889 + 0.100889i
191.1 0.654861 + 0.755750i 0 −0.142315 + 0.989821i 0.697148 0.0498610i 0 0 −0.841254 + 0.540641i 0.989821 0.142315i 0.494217 + 0.494217i
207.1 0.654861 0.755750i 0 −0.142315 0.989821i 0.697148 + 0.0498610i 0 0 −0.841254 0.540641i 0.989821 + 0.142315i 0.494217 0.494217i
319.1 −0.415415 + 0.909632i 0 −0.654861 0.755750i −0.148568 + 0.398326i 0 0 0.959493 0.281733i −0.755750 0.654861i −0.300613 0.300613i
387.1 −0.415415 + 0.909632i 0 −0.654861 0.755750i 1.83107 + 0.682956i 0 0 0.959493 0.281733i 0.755750 + 0.654861i −1.38189 + 1.38189i
499.1 0.654861 0.755750i 0 −0.142315 0.989821i 0.133682 1.86912i 0 0 −0.841254 0.540641i −0.989821 0.142315i −1.32505 1.32505i
515.1 0.654861 + 0.755750i 0 −0.142315 + 0.989821i 0.133682 + 1.86912i 0 0 −0.841254 + 0.540641i −0.989821 + 0.142315i −1.32505 + 1.32505i
535.1 0.142315 0.989821i 0 −0.959493 0.281733i −1.19550 + 1.59700i 0 0 −0.415415 + 0.909632i −0.281733 0.959493i 1.41061 + 1.41061i
571.1 −0.415415 0.909632i 0 −0.654861 + 0.755750i −0.148568 0.398326i 0 0 0.959493 + 0.281733i −0.755750 + 0.654861i −0.300613 + 0.300613i
671.1 0.959493 + 0.281733i 0 0.841254 + 0.540641i 0.767317 1.40524i 0 0 0.654861 + 0.755750i −0.540641 0.841254i 1.13214 1.13214i
827.1 0.959493 0.281733i 0 0.841254 0.540641i 0.767317 + 1.40524i 0 0 0.654861 0.755750i −0.540641 + 0.841254i 1.13214 + 1.13214i
971.1 −0.841254 0.540641i 0 0.415415 + 0.909632i −1.71524 0.373128i 0 0 0.142315 0.989821i 0.909632 + 0.415415i 1.24123 + 1.24123i
995.1 0.142315 + 0.989821i 0 −0.959493 + 0.281733i −1.19550 1.59700i 0 0 −0.415415 0.909632i −0.281733 + 0.959493i 1.41061 1.41061i
1055.1 −0.841254 + 0.540641i 0 0.415415 0.909632i −0.203743 0.936593i 0 0 0.142315 + 0.989821i −0.909632 + 0.415415i 0.677760 + 0.677760i
1063.1 −0.841254 + 0.540641i 0 0.415415 0.909632i −1.71524 + 0.373128i 0 0 0.142315 + 0.989821i 0.909632 0.415415i 1.24123 1.24123i
1123.1 0.142315 + 0.989821i 0 −0.959493 + 0.281733i −0.114220 + 0.0855040i 0 0 −0.415415 0.909632i 0.281733 0.959493i −0.100889 0.100889i
1147.1 −0.841254 0.540641i 0 0.415415 + 0.909632i −0.203743 + 0.936593i 0 0 0.142315 0.989821i −0.909632 0.415415i 0.677760 0.677760i
1291.1 0.959493 0.281733i 0 0.841254 0.540641i −1.05195 + 0.574406i 0 0 0.654861 0.755750i 0.540641 0.841254i −0.847507 + 0.847507i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1291.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
353.i even 44 1 inner
1412.r odd 44 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1412.1.r.a 20
4.b odd 2 1 CM 1412.1.r.a 20
353.i even 44 1 inner 1412.1.r.a 20
1412.r odd 44 1 inner 1412.1.r.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1412.1.r.a 20 1.a even 1 1 trivial
1412.1.r.a 20 4.b odd 2 1 CM
1412.1.r.a 20 353.i even 44 1 inner
1412.1.r.a 20 1412.r odd 44 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1412, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \)
$3$ \( T^{20} \)
$5$ \( 1 + 12 T + 61 T^{2} + 66 T^{3} + 63 T^{4} - 454 T^{5} - 403 T^{6} - 176 T^{7} + 328 T^{8} + 658 T^{9} + 494 T^{10} + 164 T^{11} + 148 T^{12} + 66 T^{13} - 8 T^{14} - 8 T^{15} - 4 T^{16} + 2 T^{18} + 2 T^{19} + T^{20} \)
$7$ \( T^{20} \)
$11$ \( T^{20} \)
$13$ \( 1 + 12 T + 94 T^{2} + 330 T^{3} + 514 T^{4} + 382 T^{5} + 125 T^{6} - 66 T^{7} - 79 T^{8} - 530 T^{9} - 188 T^{10} + 76 T^{11} + 170 T^{12} + 132 T^{13} + 47 T^{14} - 8 T^{15} - 4 T^{16} + 2 T^{18} + 2 T^{19} + T^{20} \)
$17$ \( ( 1 - 6 T + 14 T^{2} - 7 T^{3} + 9 T^{4} + 12 T^{5} - 6 T^{6} + 3 T^{7} + 4 T^{8} - 2 T^{9} + T^{10} )^{2} \)
$19$ \( T^{20} \)
$23$ \( T^{20} \)
$29$ \( 121 + 605 T^{2} + 484 T^{4} - 968 T^{6} + 484 T^{8} + 99 T^{10} + 165 T^{12} + 22 T^{16} + T^{20} \)
$31$ \( T^{20} \)
$37$ \( 1 + 10 T + 105 T^{2} + 660 T^{3} + 2945 T^{4} + 9892 T^{5} + 25942 T^{6} + 54384 T^{7} + 92530 T^{8} + 128988 T^{9} + 148070 T^{10} + 140152 T^{11} + 109136 T^{12} + 69498 T^{13} + 35819 T^{14} + 14704 T^{15} + 4693 T^{16} + 1122 T^{17} + 189 T^{18} + 20 T^{19} + T^{20} \)
$41$ \( 1 + 19 T^{2} + 119 T^{4} - 203 T^{6} + 444 T^{8} - 474 T^{10} + 234 T^{12} - 64 T^{14} + 16 T^{16} - 4 T^{18} + T^{20} \)
$43$ \( T^{20} \)
$47$ \( T^{20} \)
$53$ \( 1 + 12 T + 105 T^{2} + 484 T^{3} + 1218 T^{4} + 1702 T^{5} + 1324 T^{6} + 484 T^{7} - 178 T^{8} - 420 T^{9} - 331 T^{10} - 122 T^{11} + 93 T^{12} - 8 T^{14} - 8 T^{15} - 4 T^{16} + 2 T^{18} + 2 T^{19} + T^{20} \)
$59$ \( T^{20} \)
$61$ \( 121 - 605 T^{2} + 1089 T^{4} + 484 T^{8} + 462 T^{10} + 330 T^{12} + 165 T^{14} + 55 T^{16} + 11 T^{18} + T^{20} \)
$67$ \( T^{20} \)
$71$ \( T^{20} \)
$73$ \( T^{20} \)
$79$ \( T^{20} \)
$83$ \( T^{20} \)
$89$ \( 1 - 10 T + 17 T^{2} + 44 T^{3} + 338 T^{4} + 316 T^{5} + 400 T^{6} + 110 T^{7} - 90 T^{8} + 460 T^{9} + 505 T^{10} + 274 T^{11} + 93 T^{12} - 44 T^{13} - 52 T^{14} - 30 T^{15} - 4 T^{16} + 2 T^{18} + 2 T^{19} + T^{20} \)
$97$ \( ( 1 + 6 T + 25 T^{2} + 51 T^{3} + 53 T^{4} + 32 T^{5} + 16 T^{6} + 8 T^{7} + 4 T^{8} + 2 T^{9} + T^{10} )^{2} \)
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