# Properties

 Label 950.2.f.b Level $950$ Weight $2$ Character orbit 950.f Analytic conductor $7.586$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$950 = 2 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 950.f (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.58578819202$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.3317760000.4 Defining polynomial: $$x^{8} + 17 x^{4} + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} -\beta_{5} q^{3} -\beta_{2} q^{4} + q^{6} + \beta_{3} q^{7} + \beta_{5} q^{8} -2 \beta_{2} q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} -\beta_{5} q^{3} -\beta_{2} q^{4} + q^{6} + \beta_{3} q^{7} + \beta_{5} q^{8} -2 \beta_{2} q^{9} + \beta_{1} q^{12} + \beta_{5} q^{13} -\beta_{6} q^{14} - q^{16} + \beta_{3} q^{17} + 2 \beta_{5} q^{18} + ( 2 \beta_{2} - \beta_{6} ) q^{19} -\beta_{4} q^{21} + \beta_{7} q^{23} -\beta_{2} q^{24} - q^{26} + 5 \beta_{1} q^{27} + \beta_{7} q^{28} + \beta_{6} q^{29} -\beta_{1} q^{32} -\beta_{6} q^{34} -2 q^{36} + 2 \beta_{1} q^{37} + ( -2 \beta_{5} + \beta_{7} ) q^{38} -\beta_{2} q^{39} -2 \beta_{4} q^{41} + \beta_{3} q^{42} + \beta_{4} q^{46} + 2 \beta_{3} q^{47} + \beta_{5} q^{48} + 8 \beta_{2} q^{49} -\beta_{4} q^{51} -\beta_{1} q^{52} + 9 \beta_{5} q^{53} -5 \beta_{2} q^{54} + \beta_{4} q^{56} + ( -2 \beta_{1} + \beta_{3} ) q^{57} -\beta_{7} q^{58} + 3 \beta_{6} q^{59} + 10 q^{61} + 2 \beta_{7} q^{63} + \beta_{2} q^{64} + 13 \beta_{1} q^{67} + \beta_{7} q^{68} -\beta_{6} q^{69} + 2 \beta_{4} q^{71} -2 \beta_{1} q^{72} -\beta_{7} q^{73} -2 \beta_{2} q^{74} + ( 2 + \beta_{4} ) q^{76} + \beta_{5} q^{78} + 4 \beta_{6} q^{79} - q^{81} + 2 \beta_{3} q^{82} + 2 \beta_{7} q^{83} -\beta_{6} q^{84} -\beta_{3} q^{87} + 4 \beta_{6} q^{89} + \beta_{4} q^{91} -\beta_{3} q^{92} -2 \beta_{6} q^{94} - q^{96} -8 \beta_{1} q^{97} -8 \beta_{5} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 8q^{6} + O(q^{10})$$ $$8q + 8q^{6} - 8q^{16} - 8q^{26} - 16q^{36} + 80q^{61} + 16q^{76} - 8q^{81} - 8q^{96} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 17 x^{4} + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{5} + 5 \nu$$$$)/28$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} + 33 \nu^{2}$$$$)/112$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{5} + 61 \nu$$$$)/28$$ $$\beta_{4}$$ $$=$$ $$($$$$2 \nu^{4} + 17$$$$)/7$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{7} + 13 \nu^{3}$$$$)/448$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{6} - \nu^{2}$$$$)/16$$ $$\beta_{7}$$ $$=$$ $$($$$$-11 \nu^{7} - 251 \nu^{3}$$$$)/448$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{6} + 7 \beta_{2}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-3 \beta_{7} + 11 \beta_{5}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$7 \beta_{4} - 17$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-5 \beta_{3} + 61 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-33 \beta_{6} - 7 \beta_{2}$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-13 \beta_{7} - 251 \beta_{5}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/950\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$\beta_{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
493.1
 −1.01575 + 1.72286i 1.72286 − 1.01575i −1.72286 + 1.01575i 1.01575 − 1.72286i −1.01575 − 1.72286i 1.72286 + 1.01575i −1.72286 − 1.01575i 1.01575 + 1.72286i
−0.707107 0.707107i −0.707107 + 0.707107i 1.00000i 0 1.00000 −2.73861 + 2.73861i 0.707107 0.707107i 2.00000i 0
493.2 −0.707107 0.707107i −0.707107 + 0.707107i 1.00000i 0 1.00000 2.73861 2.73861i 0.707107 0.707107i 2.00000i 0
493.3 0.707107 + 0.707107i 0.707107 0.707107i 1.00000i 0 1.00000 −2.73861 + 2.73861i −0.707107 + 0.707107i 2.00000i 0
493.4 0.707107 + 0.707107i 0.707107 0.707107i 1.00000i 0 1.00000 2.73861 2.73861i −0.707107 + 0.707107i 2.00000i 0
607.1 −0.707107 + 0.707107i −0.707107 0.707107i 1.00000i 0 1.00000 −2.73861 2.73861i 0.707107 + 0.707107i 2.00000i 0
607.2 −0.707107 + 0.707107i −0.707107 0.707107i 1.00000i 0 1.00000 2.73861 + 2.73861i 0.707107 + 0.707107i 2.00000i 0
607.3 0.707107 0.707107i 0.707107 + 0.707107i 1.00000i 0 1.00000 −2.73861 2.73861i −0.707107 0.707107i 2.00000i 0
607.4 0.707107 0.707107i 0.707107 + 0.707107i 1.00000i 0 1.00000 2.73861 + 2.73861i −0.707107 0.707107i 2.00000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 607.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner
19.b odd 2 1 inner
95.d odd 2 1 inner
95.g even 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.f.b 8
5.b even 2 1 inner 950.2.f.b 8
5.c odd 4 2 inner 950.2.f.b 8
19.b odd 2 1 inner 950.2.f.b 8
95.d odd 2 1 inner 950.2.f.b 8
95.g even 4 2 inner 950.2.f.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.f.b 8 1.a even 1 1 trivial
950.2.f.b 8 5.b even 2 1 inner
950.2.f.b 8 5.c odd 4 2 inner
950.2.f.b 8 19.b odd 2 1 inner
950.2.f.b 8 95.d odd 2 1 inner
950.2.f.b 8 95.g even 4 2 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 1$$ acting on $$S_{2}^{\mathrm{new}}(950, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{4} )^{2}$$
$3$ $$( 1 + T^{4} )^{2}$$
$5$ $$T^{8}$$
$7$ $$( 225 + T^{4} )^{2}$$
$11$ $$T^{8}$$
$13$ $$( 1 + T^{4} )^{2}$$
$17$ $$( 225 + T^{4} )^{2}$$
$19$ $$( 361 - 22 T^{2} + T^{4} )^{2}$$
$23$ $$( 225 + T^{4} )^{2}$$
$29$ $$( -15 + T^{2} )^{4}$$
$31$ $$T^{8}$$
$37$ $$( 16 + T^{4} )^{2}$$
$41$ $$( 60 + T^{2} )^{4}$$
$43$ $$T^{8}$$
$47$ $$( 3600 + T^{4} )^{2}$$
$53$ $$( 6561 + T^{4} )^{2}$$
$59$ $$( -135 + T^{2} )^{4}$$
$61$ $$( -10 + T )^{8}$$
$67$ $$( 28561 + T^{4} )^{2}$$
$71$ $$( 60 + T^{2} )^{4}$$
$73$ $$( 225 + T^{4} )^{2}$$
$79$ $$( -240 + T^{2} )^{4}$$
$83$ $$( 3600 + T^{4} )^{2}$$
$89$ $$( -240 + T^{2} )^{4}$$
$97$ $$( 4096 + T^{4} )^{2}$$