Properties

Label 712.1.s.a
Level $712$
Weight $1$
Character orbit 712.s
Analytic conductor $0.355$
Analytic rank $0$
Dimension $10$
Projective image $D_{11}$
CM discriminant -8
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 712 = 2^{3} \cdot 89 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 712.s (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.355334288995\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\Q(\zeta_{22})\)
Defining polynomial: \(x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{11}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{11} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{22}^{5} q^{2} + ( \zeta_{22}^{4} - \zeta_{22}^{9} ) q^{3} + \zeta_{22}^{10} q^{4} + ( -\zeta_{22}^{3} - \zeta_{22}^{9} ) q^{6} + \zeta_{22}^{4} q^{8} + ( \zeta_{22}^{2} - \zeta_{22}^{7} + \zeta_{22}^{8} ) q^{9} +O(q^{10})\) \( q -\zeta_{22}^{5} q^{2} + ( \zeta_{22}^{4} - \zeta_{22}^{9} ) q^{3} + \zeta_{22}^{10} q^{4} + ( -\zeta_{22}^{3} - \zeta_{22}^{9} ) q^{6} + \zeta_{22}^{4} q^{8} + ( \zeta_{22}^{2} - \zeta_{22}^{7} + \zeta_{22}^{8} ) q^{9} + ( 1 - \zeta_{22}^{3} ) q^{11} + ( -\zeta_{22}^{3} + \zeta_{22}^{8} ) q^{12} -\zeta_{22}^{9} q^{16} + ( 1 - \zeta_{22} ) q^{17} + ( -\zeta_{22} + \zeta_{22}^{2} - \zeta_{22}^{7} ) q^{18} + ( \zeta_{22}^{6} - \zeta_{22}^{9} ) q^{19} + ( -\zeta_{22}^{5} + \zeta_{22}^{8} ) q^{22} + ( \zeta_{22}^{2} + \zeta_{22}^{8} ) q^{24} + \zeta_{22}^{8} q^{25} + ( 1 - \zeta_{22} - \zeta_{22}^{5} + \zeta_{22}^{6} ) q^{27} -\zeta_{22}^{3} q^{32} + ( -\zeta_{22} + \zeta_{22}^{4} - \zeta_{22}^{7} - \zeta_{22}^{9} ) q^{33} + ( -\zeta_{22}^{5} + \zeta_{22}^{6} ) q^{34} + ( -\zeta_{22} + \zeta_{22}^{6} - \zeta_{22}^{7} ) q^{36} + ( 1 - \zeta_{22}^{3} ) q^{38} + 2 \zeta_{22}^{10} q^{41} + ( -\zeta_{22} + \zeta_{22}^{2} ) q^{43} + ( \zeta_{22}^{2} + \zeta_{22}^{10} ) q^{44} + ( \zeta_{22}^{2} - \zeta_{22}^{7} ) q^{48} + \zeta_{22}^{8} q^{49} + \zeta_{22}^{2} q^{50} + ( \zeta_{22}^{4} - \zeta_{22}^{5} - \zeta_{22}^{9} + \zeta_{22}^{10} ) q^{51} + ( 1 - \zeta_{22}^{5} + \zeta_{22}^{6} + \zeta_{22}^{10} ) q^{54} + ( \zeta_{22}^{2} + \zeta_{22}^{4} - \zeta_{22}^{7} + \zeta_{22}^{10} ) q^{57} + ( \zeta_{22}^{4} - \zeta_{22}^{5} ) q^{59} + \zeta_{22}^{8} q^{64} + ( -\zeta_{22} - \zeta_{22}^{3} + \zeta_{22}^{6} - \zeta_{22}^{9} ) q^{66} + ( -\zeta_{22}^{3} + \zeta_{22}^{10} ) q^{67} + ( 1 + \zeta_{22}^{10} ) q^{68} + ( 1 - \zeta_{22} + \zeta_{22}^{6} ) q^{72} + ( -\zeta_{22} + \zeta_{22}^{6} ) q^{73} + ( -\zeta_{22} + \zeta_{22}^{6} ) q^{75} + ( -\zeta_{22}^{5} + \zeta_{22}^{8} ) q^{76} + ( -\zeta_{22}^{3} + \zeta_{22}^{4} - \zeta_{22}^{5} - \zeta_{22}^{9} + \zeta_{22}^{10} ) q^{81} + 2 \zeta_{22}^{4} q^{82} + ( -\zeta_{22}^{5} - \zeta_{22}^{7} ) q^{83} + ( \zeta_{22}^{6} - \zeta_{22}^{7} ) q^{86} + ( \zeta_{22}^{4} - \zeta_{22}^{7} ) q^{88} -\zeta_{22}^{3} q^{89} + ( -\zeta_{22} - \zeta_{22}^{7} ) q^{96} + ( \zeta_{22}^{2} + \zeta_{22}^{6} ) q^{97} + \zeta_{22}^{2} q^{98} + ( 1 + \zeta_{22}^{2} - \zeta_{22}^{5} - \zeta_{22}^{7} + \zeta_{22}^{8} + \zeta_{22}^{10} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - q^{2} - 2q^{3} - q^{4} - 2q^{6} - q^{8} - 3q^{9} + O(q^{10}) \) \( 10q - q^{2} - 2q^{3} - q^{4} - 2q^{6} - q^{8} - 3q^{9} + 9q^{11} - 2q^{12} - q^{16} + 9q^{17} - 3q^{18} - 2q^{19} - 2q^{22} - 2q^{24} - q^{25} + 7q^{27} - q^{32} - 4q^{33} - 2q^{34} - 3q^{36} + 9q^{38} - 2q^{41} - 2q^{43} - 2q^{44} - 2q^{48} - q^{49} - q^{50} - 4q^{51} + 7q^{54} - 4q^{57} - 2q^{59} - q^{64} - 4q^{66} - 2q^{67} + 9q^{68} + 8q^{72} - 2q^{73} - 2q^{75} - 2q^{76} - 5q^{81} - 2q^{82} - 2q^{83} - 2q^{86} - 2q^{88} - q^{89} - 2q^{96} - 2q^{97} - q^{98} + 5q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/712\mathbb{Z}\right)^\times\).

\(n\) \(357\) \(535\) \(537\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{22}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
−0.415415 + 0.909632i
−0.841254 0.540641i
0.142315 0.989821i
0.654861 0.755750i
0.959493 0.281733i
0.959493 + 0.281733i
0.654861 + 0.755750i
0.142315 + 0.989821i
−0.841254 + 0.540641i
−0.415415 0.909632i
0.841254 + 0.540641i −0.797176 + 1.74557i 0.415415 + 0.909632i 0 −1.61435 + 1.03748i 0 −0.142315 + 0.989821i −1.75667 2.02730i 0
91.1 −0.959493 + 0.281733i −0.239446 0.153882i 0.841254 0.540641i 0 0.273100 + 0.0801894i 0 −0.654861 + 0.755750i −0.381761 0.835939i 0
275.1 −0.654861 + 0.755750i −0.118239 + 0.822373i −0.142315 0.989821i 0 −0.544078 0.627899i 0 0.841254 + 0.540641i 0.297176 + 0.0872586i 0
283.1 0.415415 0.909632i −1.10181 + 1.27155i −0.654861 0.755750i 0 0.698939 + 1.53046i 0 −0.959493 + 0.281733i −0.260554 1.81219i 0
299.1 −0.142315 + 0.989821i 1.25667 0.368991i −0.959493 0.281733i 0 0.186393 + 1.29639i 0 0.415415 0.909632i 0.601808 0.386758i 0
331.1 −0.142315 0.989821i 1.25667 + 0.368991i −0.959493 + 0.281733i 0 0.186393 1.29639i 0 0.415415 + 0.909632i 0.601808 + 0.386758i 0
395.1 0.415415 + 0.909632i −1.10181 1.27155i −0.654861 + 0.755750i 0 0.698939 1.53046i 0 −0.959493 0.281733i −0.260554 + 1.81219i 0
523.1 −0.654861 0.755750i −0.118239 0.822373i −0.142315 + 0.989821i 0 −0.544078 + 0.627899i 0 0.841254 0.540641i 0.297176 0.0872586i 0
579.1 −0.959493 0.281733i −0.239446 + 0.153882i 0.841254 + 0.540641i 0 0.273100 0.0801894i 0 −0.654861 0.755750i −0.381761 + 0.835939i 0
627.1 0.841254 0.540641i −0.797176 1.74557i 0.415415 0.909632i 0 −1.61435 1.03748i 0 −0.142315 0.989821i −1.75667 + 2.02730i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 627.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
89.e even 11 1 inner
712.s odd 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 712.1.s.a 10
4.b odd 2 1 2848.1.bu.a 10
8.b even 2 1 2848.1.bu.a 10
8.d odd 2 1 CM 712.1.s.a 10
89.e even 11 1 inner 712.1.s.a 10
356.l odd 22 1 2848.1.bu.a 10
712.s odd 22 1 inner 712.1.s.a 10
712.x even 22 1 2848.1.bu.a 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
712.1.s.a 10 1.a even 1 1 trivial
712.1.s.a 10 8.d odd 2 1 CM
712.1.s.a 10 89.e even 11 1 inner
712.1.s.a 10 712.s odd 22 1 inner
2848.1.bu.a 10 4.b odd 2 1
2848.1.bu.a 10 8.b even 2 1
2848.1.bu.a 10 356.l odd 22 1
2848.1.bu.a 10 712.x even 22 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(712, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
$3$ \( 1 + 6 T + 14 T^{2} + 7 T^{3} + 9 T^{4} - 12 T^{5} - 6 T^{6} - 3 T^{7} + 4 T^{8} + 2 T^{9} + T^{10} \)
$5$ \( T^{10} \)
$7$ \( T^{10} \)
$11$ \( 1 - 5 T + 25 T^{2} - 70 T^{3} + 130 T^{4} - 166 T^{5} + 148 T^{6} - 91 T^{7} + 37 T^{8} - 9 T^{9} + T^{10} \)
$13$ \( T^{10} \)
$17$ \( 1 - 5 T + 25 T^{2} - 70 T^{3} + 130 T^{4} - 166 T^{5} + 148 T^{6} - 91 T^{7} + 37 T^{8} - 9 T^{9} + T^{10} \)
$19$ \( 1 - 5 T + 14 T^{2} - 4 T^{3} - 2 T^{4} - T^{5} + 5 T^{6} + 8 T^{7} + 4 T^{8} + 2 T^{9} + T^{10} \)
$23$ \( T^{10} \)
$29$ \( T^{10} \)
$31$ \( T^{10} \)
$37$ \( T^{10} \)
$41$ \( 1024 + 512 T + 256 T^{2} + 128 T^{3} + 64 T^{4} + 32 T^{5} + 16 T^{6} + 8 T^{7} + 4 T^{8} + 2 T^{9} + T^{10} \)
$43$ \( 1 + 6 T + 25 T^{2} + 51 T^{3} + 53 T^{4} + 32 T^{5} + 16 T^{6} + 8 T^{7} + 4 T^{8} + 2 T^{9} + T^{10} \)
$47$ \( T^{10} \)
$53$ \( T^{10} \)
$59$ \( 1 - 5 T + 3 T^{2} + 7 T^{3} + 20 T^{4} + 10 T^{5} + 16 T^{6} + 8 T^{7} + 4 T^{8} + 2 T^{9} + T^{10} \)
$61$ \( T^{10} \)
$67$ \( 1 - 5 T + 14 T^{2} - 4 T^{3} - 2 T^{4} - T^{5} + 5 T^{6} + 8 T^{7} + 4 T^{8} + 2 T^{9} + T^{10} \)
$71$ \( T^{10} \)
$73$ \( 1 + 6 T + 25 T^{2} + 51 T^{3} + 53 T^{4} + 32 T^{5} + 16 T^{6} + 8 T^{7} + 4 T^{8} + 2 T^{9} + T^{10} \)
$79$ \( T^{10} \)
$83$ \( 1 - 5 T + 14 T^{2} - 4 T^{3} - 2 T^{4} - T^{5} + 5 T^{6} + 8 T^{7} + 4 T^{8} + 2 T^{9} + T^{10} \)
$89$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
$97$ \( 1 - 5 T + 3 T^{2} + 7 T^{3} + 20 T^{4} + 10 T^{5} + 16 T^{6} + 8 T^{7} + 4 T^{8} + 2 T^{9} + T^{10} \)
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