Properties

Label 1792.1.bt.a
Level $1792$
Weight $1$
Character orbit 1792.bt
Analytic conductor $0.894$
Analytic rank $0$
Dimension $32$
Projective image $D_{64}$
CM discriminant -7
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1792 = 2^{8} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1792.bt (of order \(64\), degree \(32\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.894324502638\)
Analytic rank: \(0\)
Dimension: \(32\)
Coefficient field: \(\Q(\zeta_{64})\)
Defining polynomial: \(x^{32} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{64}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{64} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{64}^{3} q^{2} + \zeta_{64}^{6} q^{4} -\zeta_{64}^{7} q^{7} + \zeta_{64}^{9} q^{8} -\zeta_{64}^{17} q^{9} +O(q^{10})\) \( q + \zeta_{64}^{3} q^{2} + \zeta_{64}^{6} q^{4} -\zeta_{64}^{7} q^{7} + \zeta_{64}^{9} q^{8} -\zeta_{64}^{17} q^{9} + ( \zeta_{64}^{5} - \zeta_{64}^{18} ) q^{11} -\zeta_{64}^{10} q^{14} + \zeta_{64}^{12} q^{16} -\zeta_{64}^{20} q^{18} + ( \zeta_{64}^{8} - \zeta_{64}^{21} ) q^{22} + ( -\zeta_{64}^{28} - \zeta_{64}^{30} ) q^{23} + \zeta_{64}^{27} q^{25} -\zeta_{64}^{13} q^{28} + ( \zeta_{64}^{26} + \zeta_{64}^{31} ) q^{29} + \zeta_{64}^{15} q^{32} -\zeta_{64}^{23} q^{36} + ( \zeta_{64}^{13} + \zeta_{64}^{22} ) q^{37} + ( \zeta_{64}^{4} - \zeta_{64}^{11} ) q^{43} + ( \zeta_{64}^{11} - \zeta_{64}^{24} ) q^{44} + ( \zeta_{64} - \zeta_{64}^{31} ) q^{46} + \zeta_{64}^{14} q^{49} + \zeta_{64}^{30} q^{50} + ( -\zeta_{64}^{19} + \zeta_{64}^{20} ) q^{53} -\zeta_{64}^{16} q^{56} + ( -\zeta_{64}^{2} + \zeta_{64}^{29} ) q^{58} + \zeta_{64}^{24} q^{63} + \zeta_{64}^{18} q^{64} + ( -\zeta_{64}^{8} - \zeta_{64}^{25} ) q^{67} + ( -\zeta_{64} + \zeta_{64}^{29} ) q^{71} -\zeta_{64}^{26} q^{72} + ( \zeta_{64}^{16} + \zeta_{64}^{25} ) q^{74} + ( -\zeta_{64}^{12} + \zeta_{64}^{25} ) q^{77} + ( -\zeta_{64}^{9} + \zeta_{64}^{19} ) q^{79} -\zeta_{64}^{2} q^{81} + ( \zeta_{64}^{7} - \zeta_{64}^{14} ) q^{86} + ( \zeta_{64}^{14} - \zeta_{64}^{27} ) q^{88} + ( \zeta_{64}^{2} + \zeta_{64}^{4} ) q^{92} + \zeta_{64}^{17} q^{98} + ( -\zeta_{64}^{3} - \zeta_{64}^{22} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32q + O(q^{10}) \) \( 32q + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times\).

\(n\) \(1023\) \(1025\) \(1541\)
\(\chi(n)\) \(1\) \(-1\) \(\zeta_{64}^{27}\)

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
0.956940 0.290285i
−0.956940 0.290285i
0.773010 0.634393i
0.995185 0.0980171i
0.471397 0.881921i
−0.881921 + 0.471397i
0.0980171 0.995185i
0.634393 0.773010i
−0.290285 0.956940i
−0.290285 + 0.956940i
−0.634393 0.773010i
−0.0980171 0.995185i
−0.881921 0.471397i
0.471397 + 0.881921i
−0.995185 0.0980171i
−0.773010 0.634393i
−0.956940 + 0.290285i
0.956940 + 0.290285i
−0.773010 + 0.634393i
−0.995185 + 0.0980171i
0.634393 0.773010i 0 −0.195090 0.980785i 0 0 0.471397 + 0.881921i −0.881921 0.471397i −0.290285 0.956940i 0
69.1 −0.634393 0.773010i 0 −0.195090 + 0.980785i 0 0 −0.471397 + 0.881921i 0.881921 0.471397i 0.290285 0.956940i 0
125.1 −0.471397 0.881921i 0 −0.555570 + 0.831470i 0 0 −0.0980171 0.995185i 0.995185 + 0.0980171i −0.634393 0.773010i 0
181.1 0.956940 0.290285i 0 0.831470 0.555570i 0 0 −0.773010 + 0.634393i 0.634393 0.773010i 0.0980171 + 0.995185i 0
237.1 −0.995185 + 0.0980171i 0 0.980785 0.195090i 0 0 −0.290285 + 0.956940i −0.956940 + 0.290285i −0.881921 0.471397i 0
293.1 −0.0980171 + 0.995185i 0 −0.980785 0.195090i 0 0 −0.956940 + 0.290285i 0.290285 0.956940i −0.471397 0.881921i 0
349.1 −0.290285 + 0.956940i 0 −0.831470 0.555570i 0 0 0.634393 0.773010i 0.773010 0.634393i −0.995185 0.0980171i 0
405.1 −0.881921 0.471397i 0 0.555570 + 0.831470i 0 0 −0.995185 0.0980171i −0.0980171 0.995185i 0.773010 + 0.634393i 0
461.1 0.773010 + 0.634393i 0 0.195090 + 0.980785i 0 0 −0.881921 + 0.471397i −0.471397 + 0.881921i −0.956940 + 0.290285i 0
517.1 0.773010 0.634393i 0 0.195090 0.980785i 0 0 −0.881921 0.471397i −0.471397 0.881921i −0.956940 0.290285i 0
573.1 0.881921 0.471397i 0 0.555570 0.831470i 0 0 0.995185 0.0980171i 0.0980171 0.995185i −0.773010 + 0.634393i 0
629.1 0.290285 + 0.956940i 0 −0.831470 + 0.555570i 0 0 −0.634393 0.773010i −0.773010 0.634393i 0.995185 0.0980171i 0
685.1 −0.0980171 0.995185i 0 −0.980785 + 0.195090i 0 0 −0.956940 0.290285i 0.290285 + 0.956940i −0.471397 + 0.881921i 0
741.1 −0.995185 0.0980171i 0 0.980785 + 0.195090i 0 0 −0.290285 0.956940i −0.956940 0.290285i −0.881921 + 0.471397i 0
797.1 −0.956940 0.290285i 0 0.831470 + 0.555570i 0 0 0.773010 + 0.634393i −0.634393 0.773010i −0.0980171 + 0.995185i 0
853.1 0.471397 0.881921i 0 −0.555570 0.831470i 0 0 0.0980171 0.995185i −0.995185 + 0.0980171i 0.634393 0.773010i 0
909.1 −0.634393 + 0.773010i 0 −0.195090 0.980785i 0 0 −0.471397 0.881921i 0.881921 + 0.471397i 0.290285 + 0.956940i 0
965.1 0.634393 + 0.773010i 0 −0.195090 + 0.980785i 0 0 0.471397 0.881921i −0.881921 + 0.471397i −0.290285 + 0.956940i 0
1021.1 0.471397 + 0.881921i 0 −0.555570 + 0.831470i 0 0 0.0980171 + 0.995185i −0.995185 0.0980171i 0.634393 + 0.773010i 0
1077.1 −0.956940 + 0.290285i 0 0.831470 0.555570i 0 0 0.773010 0.634393i −0.634393 + 0.773010i −0.0980171 0.995185i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1749.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
256.m even 64 1 inner
1792.bt odd 64 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1792.1.bt.a 32
7.b odd 2 1 CM 1792.1.bt.a 32
256.m even 64 1 inner 1792.1.bt.a 32
1792.bt odd 64 1 inner 1792.1.bt.a 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1792.1.bt.a 32 1.a even 1 1 trivial
1792.1.bt.a 32 7.b odd 2 1 CM
1792.1.bt.a 32 256.m even 64 1 inner
1792.1.bt.a 32 1792.bt odd 64 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1792, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{32} \)
$3$ \( T^{32} \)
$5$ \( T^{32} \)
$7$ \( 1 + T^{32} \)
$11$ \( 2 - 32 T + 304 T^{2} - 832 T^{3} + 280 T^{4} + 672 T^{6} + 15104 T^{7} + 13460 T^{8} + 32 T^{9} - 192 T^{11} + 30384 T^{12} - 6272 T^{13} + 2 T^{16} + 11360 T^{17} + 208 T^{18} + 1376 T^{22} + 64 T^{27} + T^{32} \)
$13$ \( T^{32} \)
$17$ \( T^{32} \)
$19$ \( T^{32} \)
$23$ \( ( 2 + 16 T + 104 T^{2} + 352 T^{3} + 660 T^{4} + 672 T^{5} + 336 T^{6} + 64 T^{7} + 2 T^{8} + T^{16} )^{2} \)
$29$ \( 2 - 32 T + 80 T^{2} + 384 T^{3} + 1880 T^{4} - 32 T^{5} + 12736 T^{6} + 13452 T^{8} + 160 T^{9} + 336 T^{10} - 10496 T^{11} + 28864 T^{13} - 1280 T^{15} + 2 T^{16} + 2320 T^{18} + 1320 T^{20} - 96 T^{25} + T^{32} \)
$31$ \( T^{32} \)
$37$ \( 2 - 32 T + 336 T^{2} - 1280 T^{3} + 1320 T^{4} - 96 T^{5} + 384 T^{7} + 13452 T^{8} + 28864 T^{9} + 2320 T^{10} - 32 T^{13} + 12736 T^{14} - 10496 T^{15} + 2 T^{16} + 1880 T^{20} + 160 T^{21} + 80 T^{26} + T^{32} \)
$41$ \( T^{32} \)
$43$ \( 2 - 32 T + 400 T^{2} - 2464 T^{3} + 7752 T^{4} - 11648 T^{5} + 6864 T^{6} - 960 T^{7} + 4 T^{8} + 32 T^{11} + 2616 T^{12} + 20352 T^{13} + 23136 T^{14} + 2688 T^{15} + 6 T^{16} + 144 T^{22} - 448 T^{23} + 4 T^{24} + T^{32} \)
$47$ \( T^{32} \)
$53$ \( 2 + 32 T + 112 T^{2} - 160 T^{3} + 2632 T^{4} + 5936 T^{6} + 2112 T^{7} + 4 T^{8} - 32192 T^{9} + 2272 T^{11} + 3640 T^{12} + 22944 T^{14} + 6 T^{16} + 1824 T^{17} - 1632 T^{19} + 368 T^{22} + 4 T^{24} + 32 T^{27} + T^{32} \)
$59$ \( T^{32} \)
$61$ \( T^{32} \)
$67$ \( 2 + 32 T + 240 T^{2} + 224 T^{3} + 8 T^{4} - 2912 T^{5} + 9072 T^{6} - 32 T^{7} + 28 T^{8} + 21312 T^{9} + 8848 T^{10} + 56 T^{12} - 30912 T^{13} + 272 T^{14} + 70 T^{16} + 9888 T^{17} + 56 T^{20} - 480 T^{21} + 28 T^{24} + 8 T^{28} + T^{32} \)
$71$ \( 16 + 1280 T^{4} + 30144 T^{8} + 8704 T^{12} + 10368 T^{16} - 2432 T^{20} + 400 T^{24} - 32 T^{28} + T^{32} \)
$73$ \( T^{32} \)
$79$ \( 4 - 64 T^{2} + 2400 T^{4} + 10496 T^{6} + 3200 T^{8} - 28864 T^{10} + 12736 T^{12} + 1280 T^{14} + 13452 T^{16} + 32 T^{18} + 336 T^{20} - 384 T^{22} + 32 T^{26} + T^{32} \)
$83$ \( T^{32} \)
$89$ \( T^{32} \)
$97$ \( T^{32} \)
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