# Properties

 Label 959.1.r.a Level $959$ Weight $1$ Character orbit 959.r Analytic conductor $0.479$ Analytic rank $0$ Dimension $16$ Projective image $D_{34}$ CM discriminant -7 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$959 = 7 \cdot 137$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 959.r (of order $$34$$, degree $$16$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.478603347115$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\Q(\zeta_{34})$$ Defining polynomial: $$x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{34}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{34} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( \zeta_{34}^{10} - \zeta_{34}^{15} ) q^{2} + ( -\zeta_{34}^{3} + \zeta_{34}^{8} - \zeta_{34}^{13} ) q^{4} -\zeta_{34}^{10} q^{7} + ( -\zeta_{34} + \zeta_{34}^{6} - \zeta_{34}^{11} - \zeta_{34}^{13} ) q^{8} + \zeta_{34}^{11} q^{9} +O(q^{10})$$ $$q + ( \zeta_{34}^{10} - \zeta_{34}^{15} ) q^{2} + ( -\zeta_{34}^{3} + \zeta_{34}^{8} - \zeta_{34}^{13} ) q^{4} -\zeta_{34}^{10} q^{7} + ( -\zeta_{34} + \zeta_{34}^{6} - \zeta_{34}^{11} - \zeta_{34}^{13} ) q^{8} + \zeta_{34}^{11} q^{9} + ( -\zeta_{34}^{4} - \zeta_{34}^{12} ) q^{11} + ( \zeta_{34}^{3} - \zeta_{34}^{8} ) q^{14} + ( \zeta_{34}^{4} + \zeta_{34}^{6} - \zeta_{34}^{9} - \zeta_{34}^{11} + \zeta_{34}^{16} ) q^{16} + ( -\zeta_{34}^{4} + \zeta_{34}^{9} ) q^{18} + ( -\zeta_{34}^{2} + \zeta_{34}^{5} - \zeta_{34}^{10} - \zeta_{34}^{14} ) q^{22} + ( 1 + \zeta_{34}^{15} ) q^{23} + \zeta_{34}^{9} q^{25} + ( \zeta_{34} - \zeta_{34}^{6} + \zeta_{34}^{13} ) q^{28} + ( \zeta_{34}^{3} + \zeta_{34}^{12} ) q^{29} + ( \zeta_{34}^{2} + \zeta_{34}^{4} - \zeta_{34}^{7} - \zeta_{34}^{9} + \zeta_{34}^{14} + \zeta_{34}^{16} ) q^{32} + ( -\zeta_{34}^{2} + \zeta_{34}^{7} - \zeta_{34}^{14} ) q^{36} + ( \zeta_{34}^{8} - \zeta_{34}^{9} ) q^{37} + ( \zeta_{34}^{14} - \zeta_{34}^{16} ) q^{43} + ( -1 + \zeta_{34}^{3} + \zeta_{34}^{7} - \zeta_{34}^{8} - \zeta_{34}^{12} + \zeta_{34}^{15} ) q^{44} + ( -\zeta_{34}^{8} + \zeta_{34}^{10} + \zeta_{34}^{13} - \zeta_{34}^{15} ) q^{46} -\zeta_{34}^{3} q^{49} + ( -\zeta_{34}^{2} + \zeta_{34}^{7} ) q^{50} + ( \zeta_{34} + \zeta_{34}^{12} ) q^{53} + ( -\zeta_{34}^{4} - \zeta_{34}^{6} + \zeta_{34}^{11} - \zeta_{34}^{16} ) q^{56} + ( \zeta_{34} - \zeta_{34}^{5} + \zeta_{34}^{10} + \zeta_{34}^{13} ) q^{58} + \zeta_{34}^{4} q^{63} + ( 1 + \zeta_{34}^{2} - \zeta_{34}^{5} - \zeta_{34}^{7} - \zeta_{34}^{9} + \zeta_{34}^{12} + \zeta_{34}^{14} ) q^{64} + ( \zeta_{34}^{4} - \zeta_{34}^{16} ) q^{67} + ( -1 - \zeta_{34} ) q^{71} + ( -1 + \zeta_{34}^{5} + \zeta_{34}^{7} - \zeta_{34}^{12} ) q^{72} + ( -\zeta_{34} + \zeta_{34}^{2} + \zeta_{34}^{6} - \zeta_{34}^{7} ) q^{74} + ( -\zeta_{34}^{5} + \zeta_{34}^{14} ) q^{77} + ( -\zeta_{34} + \zeta_{34}^{5} ) q^{79} -\zeta_{34}^{5} q^{81} + ( -\zeta_{34}^{7} + \zeta_{34}^{9} + \zeta_{34}^{12} - \zeta_{34}^{14} ) q^{86} + ( -1 + \zeta_{34} + \zeta_{34}^{5} - \zeta_{34}^{6} - \zeta_{34}^{8} - \zeta_{34}^{10} + \zeta_{34}^{13} + \zeta_{34}^{15} ) q^{88} + ( \zeta_{34} - \zeta_{34}^{3} - \zeta_{34}^{6} + \zeta_{34}^{8} + \zeta_{34}^{11} - \zeta_{34}^{13} ) q^{92} + ( -\zeta_{34} - \zeta_{34}^{13} ) q^{98} + ( \zeta_{34}^{6} - \zeta_{34}^{15} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 2q^{2} - 3q^{4} + q^{7} - 4q^{8} + q^{9} + O(q^{10})$$ $$16q - 2q^{2} - 3q^{4} + q^{7} - 4q^{8} + q^{9} + 2q^{11} + 2q^{14} - 5q^{16} + 2q^{18} + 4q^{22} + 17q^{23} + q^{25} + 3q^{28} - 6q^{32} + 3q^{36} - 2q^{37} - 11q^{44} - q^{49} + 2q^{50} + 4q^{56} - q^{63} + 10q^{64} - 17q^{71} - 13q^{72} - 4q^{74} - 2q^{77} - q^{81} - 9q^{88} - 2q^{98} - 2q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/959\mathbb{Z}\right)^\times$$.

 $$n$$ $$414$$ $$549$$ $$\chi(n)$$ $$\zeta_{34}^{11}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
202.1
 0.273663 − 0.961826i −0.739009 − 0.673696i −0.932472 + 0.361242i −0.445738 + 0.895163i 0.850217 + 0.526432i −0.445738 − 0.895163i 0.273663 + 0.961826i −0.0922684 + 0.995734i −0.932472 − 0.361242i −0.0922684 − 0.995734i 0.850217 − 0.526432i 0.602635 − 0.798017i 0.602635 + 0.798017i 0.982973 + 0.183750i −0.739009 + 0.673696i 0.982973 − 0.183750i
0.0822551 + 0.165190i 0 0.582113 0.770842i 0 0 −0.932472 + 0.361242i 0.356612 + 0.0666624i −0.0922684 0.995734i 0
258.1 0.538007 0.100571i 0 −0.653136 + 0.253026i 0 0 −0.445738 0.895163i −0.791290 + 0.489946i 0.273663 0.961826i 0
377.1 −0.111208 + 1.20013i 0 −0.444966 0.0831786i 0 0 0.850217 0.526432i −0.180529 + 0.634493i 0.602635 0.798017i 0
426.1 −0.510366 + 1.79375i 0 −2.10685 1.30451i 0 0 −0.0922684 0.995734i 2.03702 1.85699i −0.932472 0.361242i 0
433.1 1.18475 1.56886i 0 −0.784029 2.75558i 0 0 −0.739009 + 0.673696i −3.41880 1.32445i 0.982973 0.183750i 0
475.1 −0.510366 1.79375i 0 −2.10685 + 1.30451i 0 0 −0.0922684 + 0.995734i 2.03702 + 1.85699i −0.932472 + 0.361242i 0
489.1 0.0822551 0.165190i 0 0.582113 + 0.770842i 0 0 −0.932472 0.361242i 0.356612 0.0666624i −0.0922684 + 0.995734i 0
510.1 −1.58561 0.614268i 0 1.39782 + 1.27428i 0 0 0.602635 + 0.798017i −0.675694 1.35698i 0.850217 0.526432i 0
552.1 −0.111208 1.20013i 0 −0.444966 + 0.0831786i 0 0 0.850217 + 0.526432i −0.180529 0.634493i 0.602635 + 0.798017i 0
566.1 −1.58561 + 0.614268i 0 1.39782 1.27428i 0 0 0.602635 0.798017i −0.675694 + 1.35698i 0.850217 + 0.526432i 0
629.1 1.18475 + 1.56886i 0 −0.784029 + 2.75558i 0 0 −0.739009 0.673696i −3.41880 + 1.32445i 0.982973 + 0.183750i 0
699.1 −1.25664 + 0.778076i 0 0.527993 1.06035i 0 0 0.982973 + 0.183750i 0.0251661 + 0.271585i −0.739009 + 0.673696i 0
734.1 −1.25664 0.778076i 0 0.527993 + 1.06035i 0 0 0.982973 0.183750i 0.0251661 0.271585i −0.739009 0.673696i 0
748.1 0.658809 + 0.600584i 0 −0.0189399 0.204394i 0 0 0.273663 0.961826i 0.647513 0.857445i −0.445738 + 0.895163i 0
762.1 0.538007 + 0.100571i 0 −0.653136 0.253026i 0 0 −0.445738 + 0.895163i −0.791290 0.489946i 0.273663 + 0.961826i 0
909.1 0.658809 0.600584i 0 −0.0189399 + 0.204394i 0 0 0.273663 + 0.961826i 0.647513 + 0.857445i −0.445738 0.895163i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 909.1 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 $$\Q(\sqrt{-7})$$
137.f even 34 1 inner
959.r odd 34 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 959.1.r.a 16
7.b odd 2 1 CM 959.1.r.a 16
137.f even 34 1 inner 959.1.r.a 16
959.r odd 34 1 inner 959.1.r.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
959.1.r.a 16 1.a even 1 1 trivial
959.1.r.a 16 7.b odd 2 1 CM
959.1.r.a 16 137.f even 34 1 inner
959.1.r.a 16 959.r odd 34 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(959, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 8 T + 47 T^{2} - 104 T^{3} + 67 T^{4} + 8 T^{5} + 4 T^{6} + 2 T^{7} + T^{8} + 9 T^{9} + 47 T^{10} + 32 T^{11} + 16 T^{12} + 8 T^{13} + 4 T^{14} + 2 T^{15} + T^{16}$$
$3$ $$T^{16}$$
$5$ $$T^{16}$$
$7$ $$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16}$$
$11$ $$1 - 9 T + 13 T^{2} + 36 T^{3} + 33 T^{4} - 25 T^{5} + 140 T^{6} - 70 T^{7} + 154 T^{8} - 77 T^{9} + 64 T^{10} - 32 T^{11} + 16 T^{12} - 8 T^{13} + 4 T^{14} - 2 T^{15} + T^{16}$$
$13$ $$T^{16}$$
$17$ $$T^{16}$$
$19$ $$T^{16}$$
$23$ $$17 - 136 T + 680 T^{2} - 2380 T^{3} + 6188 T^{4} - 12376 T^{5} + 19448 T^{6} - 24310 T^{7} + 24310 T^{8} - 19448 T^{9} + 12376 T^{10} - 6188 T^{11} + 2380 T^{12} - 680 T^{13} + 136 T^{14} - 17 T^{15} + T^{16}$$
$29$ $$17 + 34 T + 17 T^{2} - 221 T^{3} + 85 T^{5} + 68 T^{6} + 119 T^{8} + 17 T^{11} + T^{16}$$
$31$ $$T^{16}$$
$37$ $$( 1 - 4 T - 10 T^{2} + 10 T^{3} + 15 T^{4} - 6 T^{5} - 7 T^{6} + T^{7} + T^{8} )^{2}$$
$41$ $$T^{16}$$
$43$ $$17 - 17 T + 85 T^{2} + 102 T^{3} + 17 T^{4} - 255 T^{5} + 238 T^{7} - 51 T^{9} + T^{16}$$
$47$ $$T^{16}$$
$53$ $$17 - 102 T + 255 T^{2} - 238 T^{3} + 51 T^{4} + 17 T^{8} + 85 T^{9} + 17 T^{10} + T^{16}$$
$59$ $$T^{16}$$
$61$ $$T^{16}$$
$67$ $$17 - 34 T + 17 T^{2} + 221 T^{3} - 85 T^{5} + 68 T^{6} + 119 T^{8} - 17 T^{11} + T^{16}$$
$71$ $$17 + 136 T + 680 T^{2} + 2380 T^{3} + 6188 T^{4} + 12376 T^{5} + 19448 T^{6} + 24310 T^{7} + 24310 T^{8} + 19448 T^{9} + 12376 T^{10} + 6188 T^{11} + 2380 T^{12} + 680 T^{13} + 136 T^{14} + 17 T^{15} + T^{16}$$
$73$ $$T^{16}$$
$79$ $$17 + 85 T + 119 T^{2} + 17 T^{3} - 17 T^{5} + 221 T^{6} - 68 T^{7} + 34 T^{11} + T^{16}$$
$83$ $$T^{16}$$
$89$ $$T^{16}$$
$97$ $$T^{16}$$