# Properties

 Label 1950.2.z.p Level $1950$ Weight $2$ Character orbit 1950.z Analytic conductor $15.571$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1950.z (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$15.5708283941$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.3317760000.2 Defining polynomial: $$x^{8} - 25 x^{4} + 625$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + \beta_{1} q^{3} + \beta_{2} q^{4} + \beta_{2} q^{6} + ( 2 \beta_{1} - 2 \beta_{3} + \beta_{5} ) q^{7} + \beta_{3} q^{8} + \beta_{2} q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + \beta_{1} q^{3} + \beta_{2} q^{4} + \beta_{2} q^{6} + ( 2 \beta_{1} - 2 \beta_{3} + \beta_{5} ) q^{7} + \beta_{3} q^{8} + \beta_{2} q^{9} + ( -1 + \beta_{2} ) q^{11} + \beta_{3} q^{12} + ( -\beta_{1} - \beta_{3} + \beta_{5} ) q^{13} + ( 2 + \beta_{7} ) q^{14} + ( -1 + \beta_{2} ) q^{16} + ( 2 \beta_{1} - 2 \beta_{3} + \beta_{5} ) q^{17} + \beta_{3} q^{18} + ( 2 \beta_{2} - \beta_{4} ) q^{19} + ( 2 + \beta_{7} ) q^{21} + ( -\beta_{1} + \beta_{3} ) q^{22} + ( -\beta_{1} - \beta_{6} ) q^{23} + ( -1 + \beta_{2} ) q^{24} + ( 1 - 2 \beta_{2} + \beta_{7} ) q^{26} + \beta_{3} q^{27} + ( 2 \beta_{1} + \beta_{6} ) q^{28} + ( -2 + 2 \beta_{2} - 2 \beta_{4} + 2 \beta_{7} ) q^{29} + ( -4 - \beta_{7} ) q^{31} + ( -\beta_{1} + \beta_{3} ) q^{32} + ( -\beta_{1} + \beta_{3} ) q^{33} + ( 2 + \beta_{7} ) q^{34} + ( -1 + \beta_{2} ) q^{36} + ( -3 \beta_{1} - \beta_{6} ) q^{37} + ( 2 \beta_{3} + \beta_{5} - \beta_{6} ) q^{38} + ( 1 - 2 \beta_{2} + \beta_{7} ) q^{39} + ( -2 + 2 \beta_{2} + \beta_{4} - \beta_{7} ) q^{41} + ( 2 \beta_{1} + \beta_{6} ) q^{42} + \beta_{5} q^{43} - q^{44} + ( -\beta_{2} - \beta_{4} ) q^{46} + 6 \beta_{3} q^{47} + ( -\beta_{1} + \beta_{3} ) q^{48} + ( 7 - 7 \beta_{2} - 4 \beta_{4} + 4 \beta_{7} ) q^{49} + ( 2 + \beta_{7} ) q^{51} + ( \beta_{1} - 2 \beta_{3} + \beta_{6} ) q^{52} + ( 2 \beta_{3} - 3 \beta_{5} + 3 \beta_{6} ) q^{53} + ( -1 + \beta_{2} ) q^{54} + ( 2 \beta_{2} + \beta_{4} ) q^{56} + ( 2 \beta_{3} + \beta_{5} - \beta_{6} ) q^{57} + ( -2 \beta_{1} + 2 \beta_{3} + 2 \beta_{5} ) q^{58} + 10 \beta_{2} q^{59} + ( -3 \beta_{2} - \beta_{4} ) q^{61} + ( -4 \beta_{1} - \beta_{6} ) q^{62} + ( 2 \beta_{1} + \beta_{6} ) q^{63} - q^{64} - q^{66} + ( 2 \beta_{1} + \beta_{6} ) q^{68} + ( -\beta_{2} - \beta_{4} ) q^{69} + ( -\beta_{2} + 3 \beta_{4} ) q^{71} + ( -\beta_{1} + \beta_{3} ) q^{72} + ( 3 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} ) q^{73} + ( -3 \beta_{2} - \beta_{4} ) q^{74} + ( -2 + 2 \beta_{2} - \beta_{4} + \beta_{7} ) q^{76} + ( 2 \beta_{3} - \beta_{5} + \beta_{6} ) q^{77} + ( \beta_{1} - 2 \beta_{3} + \beta_{6} ) q^{78} + ( 4 - 3 \beta_{7} ) q^{79} + ( -1 + \beta_{2} ) q^{81} + ( -2 \beta_{1} + 2 \beta_{3} - \beta_{5} ) q^{82} -9 \beta_{3} q^{83} + ( 2 \beta_{2} + \beta_{4} ) q^{84} + \beta_{7} q^{86} + ( -2 \beta_{1} + 2 \beta_{3} + 2 \beta_{5} ) q^{87} -\beta_{1} q^{88} + ( -2 + 2 \beta_{2} - 3 \beta_{4} + 3 \beta_{7} ) q^{89} + ( 8 - 12 \beta_{2} - 3 \beta_{4} + \beta_{7} ) q^{91} + ( -\beta_{3} + \beta_{5} - \beta_{6} ) q^{92} + ( -4 \beta_{1} - \beta_{6} ) q^{93} + ( -6 + 6 \beta_{2} ) q^{94} - q^{96} + ( -\beta_{1} + \beta_{3} - 2 \beta_{5} ) q^{97} + ( 7 \beta_{1} - 7 \beta_{3} + 4 \beta_{5} ) q^{98} - q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{4} + 4q^{6} + 4q^{9} + O(q^{10})$$ $$8q + 4q^{4} + 4q^{6} + 4q^{9} - 4q^{11} + 16q^{14} - 4q^{16} + 8q^{19} + 16q^{21} - 4q^{24} - 8q^{29} - 32q^{31} + 16q^{34} - 4q^{36} - 8q^{41} - 8q^{44} - 4q^{46} + 28q^{49} + 16q^{51} - 4q^{54} + 8q^{56} + 40q^{59} - 12q^{61} - 8q^{64} - 8q^{66} - 4q^{69} - 4q^{71} - 12q^{74} - 8q^{76} + 32q^{79} - 4q^{81} + 8q^{84} - 8q^{89} + 16q^{91} - 24q^{94} - 8q^{96} - 8q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 25 x^{4} + 625$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2}$$$$/5$$ $$\beta_{2}$$ $$=$$ $$\nu^{4}$$$$/25$$ $$\beta_{3}$$ $$=$$ $$\nu^{6}$$$$/125$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{7} + 125 \nu$$$$)/125$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} + 125 \nu$$$$)/125$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{7} + 5 \nu^{5} + 25 \nu^{3}$$$$)/125$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{5} + 5 \nu^{3} + 25 \nu$$$$)/25$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} + \beta_{4}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$5 \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$($$$$5 \beta_{7} + 5 \beta_{6} - 5 \beta_{5}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$25 \beta_{2}$$ $$\nu^{5}$$ $$=$$ $$($$$$-25 \beta_{7} + 25 \beta_{6} + 25 \beta_{4}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$125 \beta_{3}$$ $$\nu^{7}$$ $$=$$ $$($$$$-125 \beta_{5} + 125 \beta_{4}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times$$.

 $$n$$ $$301$$ $$1301$$ $$1327$$ $$\chi(n)$$ $$-\beta_{2}$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1699.1
 −0.578737 + 2.15988i 0.578737 − 2.15988i −2.15988 − 0.578737i 2.15988 + 0.578737i −0.578737 − 2.15988i 0.578737 + 2.15988i −2.15988 + 0.578737i 2.15988 − 0.578737i
−0.866025 0.500000i −0.866025 0.500000i 0.500000 + 0.866025i 0 0.500000 + 0.866025i −4.47066 + 2.58114i 1.00000i 0.500000 + 0.866025i 0
1699.2 −0.866025 0.500000i −0.866025 0.500000i 0.500000 + 0.866025i 0 0.500000 + 0.866025i 1.00656 0.581139i 1.00000i 0.500000 + 0.866025i 0
1699.3 0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i 0 0.500000 + 0.866025i −1.00656 + 0.581139i 1.00000i 0.500000 + 0.866025i 0
1699.4 0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i 0 0.500000 + 0.866025i 4.47066 2.58114i 1.00000i 0.500000 + 0.866025i 0
1849.1 −0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 0.866025i 0 0.500000 0.866025i −4.47066 2.58114i 1.00000i 0.500000 0.866025i 0
1849.2 −0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 0.866025i 0 0.500000 0.866025i 1.00656 + 0.581139i 1.00000i 0.500000 0.866025i 0
1849.3 0.866025 0.500000i 0.866025 0.500000i 0.500000 0.866025i 0 0.500000 0.866025i −1.00656 0.581139i 1.00000i 0.500000 0.866025i 0
1849.4 0.866025 0.500000i 0.866025 0.500000i 0.500000 0.866025i 0 0.500000 0.866025i 4.47066 + 2.58114i 1.00000i 0.500000 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1849.4 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
13.c even 3 1 inner
65.n even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1950.2.z.p 8
5.b even 2 1 inner 1950.2.z.p 8
5.c odd 4 1 1950.2.i.bc 4
5.c odd 4 1 1950.2.i.bd yes 4
13.c even 3 1 inner 1950.2.z.p 8
65.n even 6 1 inner 1950.2.z.p 8
65.q odd 12 1 1950.2.i.bc 4
65.q odd 12 1 1950.2.i.bd yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1950.2.i.bc 4 5.c odd 4 1
1950.2.i.bc 4 65.q odd 12 1
1950.2.i.bd yes 4 5.c odd 4 1
1950.2.i.bd yes 4 65.q odd 12 1
1950.2.z.p 8 1.a even 1 1 trivial
1950.2.z.p 8 5.b even 2 1 inner
1950.2.z.p 8 13.c even 3 1 inner
1950.2.z.p 8 65.n even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1950, [\chi])$$:

 $$T_{7}^{8} - 28 T_{7}^{6} + 748 T_{7}^{4} - 1008 T_{7}^{2} + 1296$$ $$T_{11}^{2} + T_{11} + 1$$ $$T_{17}^{8} - 28 T_{17}^{6} + 748 T_{17}^{4} - 1008 T_{17}^{2} + 1296$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} )^{2}$$
$3$ $$( 1 - T^{2} + T^{4} )^{2}$$
$5$ $$T^{8}$$
$7$ $$1296 - 1008 T^{2} + 748 T^{4} - 28 T^{6} + T^{8}$$
$11$ $$( 1 + T + T^{2} )^{4}$$
$13$ $$28561 - 2366 T^{2} + 27 T^{4} - 14 T^{6} + T^{8}$$
$17$ $$1296 - 1008 T^{2} + 748 T^{4} - 28 T^{6} + T^{8}$$
$19$ $$( 36 + 24 T + 22 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$23$ $$6561 - 1782 T^{2} + 403 T^{4} - 22 T^{6} + T^{8}$$
$29$ $$( 1296 - 144 T + 52 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$31$ $$( 6 + 8 T + T^{2} )^{4}$$
$37$ $$1 - 38 T^{2} + 1443 T^{4} - 38 T^{6} + T^{8}$$
$41$ $$( 36 - 24 T + 22 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$43$ $$( 100 - 10 T^{2} + T^{4} )^{2}$$
$47$ $$( 36 + T^{2} )^{4}$$
$53$ $$( 7396 + 188 T^{2} + T^{4} )^{2}$$
$59$ $$( 100 - 10 T + T^{2} )^{4}$$
$61$ $$( 1 - 6 T + 37 T^{2} + 6 T^{3} + T^{4} )^{2}$$
$67$ $$T^{8}$$
$71$ $$( 7921 - 178 T + 93 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$73$ $$( 961 + 98 T^{2} + T^{4} )^{2}$$
$79$ $$( -74 - 8 T + T^{2} )^{4}$$
$83$ $$( 81 + T^{2} )^{4}$$
$89$ $$( 7396 - 344 T + 102 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$97$ $$2313441 - 124722 T^{2} + 5203 T^{4} - 82 T^{6} + T^{8}$$