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 $4$

Related objects

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: Complex embeddings Normalized embeddings Satake parameters Satake angles

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

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