# Properties

 Label 935.1.y.a Level $935$ Weight $1$ Character orbit 935.y Analytic conductor $0.467$ Analytic rank $0$ Dimension $16$ Projective image $D_{16}$ CM discriminant -55 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$935 = 5 \cdot 11 \cdot 17$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 935.y (of order $$8$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.466625786812$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{8})$$ Coefficient field: $$\Q(\zeta_{32})$$ Defining polynomial: $$x^{16} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{16}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{16} - \cdots)$$

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{32}^{11} - \zeta_{32}^{13} ) q^{2} + ( -\zeta_{32}^{6} + \zeta_{32}^{8} - \zeta_{32}^{10} ) q^{4} + \zeta_{32}^{14} q^{5} + ( -\zeta_{32} - \zeta_{32}^{3} ) q^{7} + ( \zeta_{32} - \zeta_{32}^{3} + \zeta_{32}^{5} - \zeta_{32}^{7} ) q^{8} + \zeta_{32}^{12} q^{9} +O(q^{10})$$ $$q + ( \zeta_{32}^{11} - \zeta_{32}^{13} ) q^{2} + ( -\zeta_{32}^{6} + \zeta_{32}^{8} - \zeta_{32}^{10} ) q^{4} + \zeta_{32}^{14} q^{5} + ( -\zeta_{32} - \zeta_{32}^{3} ) q^{7} + ( \zeta_{32} - \zeta_{32}^{3} + \zeta_{32}^{5} - \zeta_{32}^{7} ) q^{8} + \zeta_{32}^{12} q^{9} + ( -\zeta_{32}^{9} + \zeta_{32}^{11} ) q^{10} + \zeta_{32}^{10} q^{11} + ( -\zeta_{32}^{5} - \zeta_{32}^{11} ) q^{13} + ( -1 - \zeta_{32}^{12} ) q^{14} + ( -1 + \zeta_{32}^{2} - \zeta_{32}^{4} + \zeta_{32}^{12} - \zeta_{32}^{14} ) q^{16} + \zeta_{32}^{13} q^{17} + ( -\zeta_{32}^{7} + \zeta_{32}^{9} ) q^{18} + ( \zeta_{32}^{4} - \zeta_{32}^{6} + \zeta_{32}^{8} ) q^{20} + ( -\zeta_{32}^{5} + \zeta_{32}^{7} ) q^{22} -\zeta_{32}^{12} q^{25} + ( 1 - \zeta_{32}^{2} + \zeta_{32}^{6} - \zeta_{32}^{8} ) q^{26} + ( \zeta_{32}^{7} + \zeta_{32}^{13} ) q^{28} + ( -\zeta_{32}^{2} - \zeta_{32}^{10} ) q^{31} + ( -\zeta_{32} - \zeta_{32}^{7} + \zeta_{32}^{9} - \zeta_{32}^{11} + \zeta_{32}^{13} - \zeta_{32}^{15} ) q^{32} + ( -\zeta_{32}^{8} + \zeta_{32}^{10} ) q^{34} + ( \zeta_{32} - \zeta_{32}^{15} ) q^{35} + ( \zeta_{32}^{2} - \zeta_{32}^{4} + \zeta_{32}^{6} ) q^{36} + ( \zeta_{32} - \zeta_{32}^{3} + \zeta_{32}^{5} + \zeta_{32}^{15} ) q^{40} + ( -\zeta_{32}^{3} + \zeta_{32}^{5} ) q^{43} + ( 1 - \zeta_{32}^{2} + \zeta_{32}^{4} ) q^{44} -\zeta_{32}^{10} q^{45} + ( \zeta_{32}^{2} + \zeta_{32}^{4} + \zeta_{32}^{6} ) q^{49} + ( \zeta_{32}^{7} - \zeta_{32}^{9} ) q^{50} + ( -\zeta_{32} + \zeta_{32}^{3} - \zeta_{32}^{5} + \zeta_{32}^{11} - \zeta_{32}^{13} + \zeta_{32}^{15} ) q^{52} -\zeta_{32}^{8} q^{55} + ( -\zeta_{32}^{2} + \zeta_{32}^{10} ) q^{56} + ( \zeta_{32}^{5} - \zeta_{32}^{7} - \zeta_{32}^{13} + \zeta_{32}^{15} ) q^{62} + ( -\zeta_{32}^{13} - \zeta_{32}^{15} ) q^{63} + ( \zeta_{32}^{2} - \zeta_{32}^{4} + \zeta_{32}^{6} - \zeta_{32}^{8} + \zeta_{32}^{10} - \zeta_{32}^{12} + \zeta_{32}^{14} ) q^{64} + ( \zeta_{32}^{3} + \zeta_{32}^{9} ) q^{65} + ( \zeta_{32}^{3} - \zeta_{32}^{5} + \zeta_{32}^{7} ) q^{68} + ( \zeta_{32}^{10} - \zeta_{32}^{14} ) q^{70} + ( -1 - \zeta_{32}^{12} ) q^{71} + ( -\zeta_{32} + \zeta_{32}^{3} + \zeta_{32}^{13} - \zeta_{32}^{15} ) q^{72} + ( \zeta_{32}^{3} - \zeta_{32}^{9} ) q^{73} + ( -\zeta_{32}^{11} - \zeta_{32}^{13} ) q^{77} + ( -1 + \zeta_{32}^{2} - \zeta_{32}^{10} + \zeta_{32}^{12} - \zeta_{32}^{14} ) q^{80} -\zeta_{32}^{8} q^{81} + ( -\zeta_{32}^{9} + \zeta_{32}^{15} ) q^{83} -\zeta_{32}^{11} q^{85} + ( -2 + \zeta_{32}^{2} - \zeta_{32}^{14} ) q^{86} + ( \zeta_{32} + \zeta_{32}^{11} - \zeta_{32}^{13} + \zeta_{32}^{15} ) q^{88} + ( -\zeta_{32}^{2} - \zeta_{32}^{14} ) q^{89} + ( \zeta_{32}^{5} - \zeta_{32}^{7} ) q^{90} + ( \zeta_{32}^{6} + \zeta_{32}^{8} + \zeta_{32}^{12} + \zeta_{32}^{14} ) q^{91} + ( \zeta_{32}^{3} + \zeta_{32}^{13} ) q^{98} -\zeta_{32}^{6} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + O(q^{10})$$ $$16q - 16q^{14} - 16q^{16} + 16q^{26} + 16q^{44} - 16q^{71} - 16q^{80} - 32q^{86} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/935\mathbb{Z}\right)^\times$$.

 $$n$$ $$496$$ $$562$$ $$596$$ $$\chi(n)$$ $$\zeta_{32}^{12}$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
219.1
 0.195090 − 0.980785i −0.980785 − 0.195090i 0.980785 + 0.195090i −0.195090 + 0.980785i 0.555570 + 0.831470i −0.831470 + 0.555570i 0.831470 − 0.555570i −0.555570 − 0.831470i 0.555570 − 0.831470i −0.831470 − 0.555570i 0.831470 + 0.555570i −0.555570 + 0.831470i 0.195090 + 0.980785i −0.980785 + 0.195090i 0.980785 − 0.195090i −0.195090 − 0.980785i
−1.38704 1.38704i 0 2.84776i 0.923880 0.382683i 0 0.360480 + 0.149316i 2.56292 2.56292i −0.707107 + 0.707107i −1.81225 0.750661i
219.2 −0.275899 0.275899i 0 0.847759i −0.923880 + 0.382683i 0 1.81225 + 0.750661i −0.509796 + 0.509796i −0.707107 + 0.707107i 0.360480 + 0.149316i
219.3 0.275899 + 0.275899i 0 0.847759i −0.923880 + 0.382683i 0 −1.81225 0.750661i 0.509796 0.509796i −0.707107 + 0.707107i −0.360480 0.149316i
219.4 1.38704 + 1.38704i 0 2.84776i 0.923880 0.382683i 0 −0.360480 0.149316i −2.56292 + 2.56292i −0.707107 + 0.707107i 1.81225 + 0.750661i
274.1 −1.17588 1.17588i 0 1.76537i 0.382683 + 0.923880i 0 0.425215 1.02656i 0.899976 0.899976i 0.707107 0.707107i 0.636379 1.53636i
274.2 −0.785695 0.785695i 0 0.234633i −0.382683 0.923880i 0 0.636379 1.53636i −0.601345 + 0.601345i 0.707107 0.707107i −0.425215 + 1.02656i
274.3 0.785695 + 0.785695i 0 0.234633i −0.382683 0.923880i 0 −0.636379 + 1.53636i 0.601345 0.601345i 0.707107 0.707107i 0.425215 1.02656i
274.4 1.17588 + 1.17588i 0 1.76537i 0.382683 + 0.923880i 0 −0.425215 + 1.02656i −0.899976 + 0.899976i 0.707107 0.707107i −0.636379 + 1.53636i
604.1 −1.17588 + 1.17588i 0 1.76537i 0.382683 0.923880i 0 0.425215 + 1.02656i 0.899976 + 0.899976i 0.707107 + 0.707107i 0.636379 + 1.53636i
604.2 −0.785695 + 0.785695i 0 0.234633i −0.382683 + 0.923880i 0 0.636379 + 1.53636i −0.601345 0.601345i 0.707107 + 0.707107i −0.425215 1.02656i
604.3 0.785695 0.785695i 0 0.234633i −0.382683 + 0.923880i 0 −0.636379 1.53636i 0.601345 + 0.601345i 0.707107 + 0.707107i 0.425215 + 1.02656i
604.4 1.17588 1.17588i 0 1.76537i 0.382683 0.923880i 0 −0.425215 1.02656i −0.899976 0.899976i 0.707107 + 0.707107i −0.636379 1.53636i
824.1 −1.38704 + 1.38704i 0 2.84776i 0.923880 + 0.382683i 0 0.360480 0.149316i 2.56292 + 2.56292i −0.707107 0.707107i −1.81225 + 0.750661i
824.2 −0.275899 + 0.275899i 0 0.847759i −0.923880 0.382683i 0 1.81225 0.750661i −0.509796 0.509796i −0.707107 0.707107i 0.360480 0.149316i
824.3 0.275899 0.275899i 0 0.847759i −0.923880 0.382683i 0 −1.81225 + 0.750661i 0.509796 + 0.509796i −0.707107 0.707107i −0.360480 + 0.149316i
824.4 1.38704 1.38704i 0 2.84776i 0.923880 + 0.382683i 0 −0.360480 + 0.149316i −2.56292 2.56292i −0.707107 0.707107i 1.81225 0.750661i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 824.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
55.d odd 2 1 CM by $$\Q(\sqrt{-55})$$
5.b even 2 1 inner
11.b odd 2 1 inner
17.d even 8 1 inner
85.m even 8 1 inner
187.i odd 8 1 inner
935.y odd 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 935.1.y.a 16
5.b even 2 1 inner 935.1.y.a 16
11.b odd 2 1 inner 935.1.y.a 16
17.d even 8 1 inner 935.1.y.a 16
55.d odd 2 1 CM 935.1.y.a 16
85.m even 8 1 inner 935.1.y.a 16
187.i odd 8 1 inner 935.1.y.a 16
935.y odd 8 1 inner 935.1.y.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
935.1.y.a 16 1.a even 1 1 trivial
935.1.y.a 16 5.b even 2 1 inner
935.1.y.a 16 11.b odd 2 1 inner
935.1.y.a 16 17.d even 8 1 inner
935.1.y.a 16 55.d odd 2 1 CM
935.1.y.a 16 85.m even 8 1 inner
935.1.y.a 16 187.i odd 8 1 inner
935.1.y.a 16 935.y odd 8 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(935, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 176 T^{4} + 148 T^{8} + 24 T^{12} + T^{16}$$
$3$ $$T^{16}$$
$5$ $$( 1 + T^{8} )^{2}$$
$7$ $$4 - 32 T^{2} + 128 T^{4} + 192 T^{6} + 140 T^{8} + 16 T^{10} + T^{16}$$
$11$ $$( 1 + T^{8} )^{2}$$
$13$ $$( 2 + 16 T^{2} + 20 T^{4} + 8 T^{6} + T^{8} )^{2}$$
$17$ $$1 + T^{16}$$
$19$ $$T^{16}$$
$23$ $$T^{16}$$
$29$ $$T^{16}$$
$31$ $$( 16 + T^{8} )^{2}$$
$37$ $$T^{16}$$
$41$ $$T^{16}$$
$43$ $$4 + 176 T^{4} + 148 T^{8} + 24 T^{12} + T^{16}$$
$47$ $$T^{16}$$
$53$ $$T^{16}$$
$59$ $$T^{16}$$
$61$ $$T^{16}$$
$67$ $$T^{16}$$
$71$ $$( 2 + 4 T + 6 T^{2} + 4 T^{3} + T^{4} )^{4}$$
$73$ $$4 + 32 T^{2} + 128 T^{4} - 192 T^{6} + 140 T^{8} - 16 T^{10} + T^{16}$$
$79$ $$T^{16}$$
$83$ $$4 + 176 T^{4} + 148 T^{8} + 24 T^{12} + T^{16}$$
$89$ $$( 2 + 4 T^{2} + T^{4} )^{4}$$
$97$ $$T^{16}$$