# Properties

 Label 1950.2.z.m 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: 8.0.1731891456.1 Defining polynomial: $$x^{8} - 9 x^{6} + 65 x^{4} - 144 x^{2} + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{3} + \beta_{7} ) q^{2} + ( -\beta_{3} - \beta_{7} ) q^{3} -\beta_{2} q^{4} + \beta_{2} q^{6} + ( \beta_{1} + \beta_{3} ) q^{7} + \beta_{7} q^{8} -\beta_{2} q^{9} +O(q^{10})$$ $$q + ( \beta_{3} + \beta_{7} ) q^{2} + ( -\beta_{3} - \beta_{7} ) q^{3} -\beta_{2} q^{4} + \beta_{2} q^{6} + ( \beta_{1} + \beta_{3} ) q^{7} + \beta_{7} q^{8} -\beta_{2} q^{9} + ( -2 - \beta_{2} - \beta_{4} + \beta_{5} ) q^{11} -\beta_{7} q^{12} + ( -\beta_{1} - \beta_{6} - \beta_{7} ) q^{13} + ( 1 - \beta_{4} ) q^{14} + ( -1 - \beta_{2} ) q^{16} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{17} + \beta_{7} q^{18} + ( 4 \beta_{2} - \beta_{5} ) q^{19} + ( -1 + \beta_{4} ) q^{21} + ( \beta_{1} - 2 \beta_{3} ) q^{22} + ( 3 \beta_{1} - 3 \beta_{6} ) q^{23} + ( 1 + \beta_{2} ) q^{24} + ( 1 + 2 \beta_{4} - \beta_{5} ) q^{26} -\beta_{7} q^{27} + ( \beta_{1} + \beta_{3} - \beta_{6} + \beta_{7} ) q^{28} + ( -4 - 2 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} ) q^{29} + 4 q^{31} -\beta_{3} q^{32} + ( -\beta_{1} + 2 \beta_{3} ) q^{33} + ( -2 - 2 \beta_{4} ) q^{34} + ( -1 - \beta_{2} ) q^{36} + ( \beta_{1} - 2 \beta_{3} - \beta_{6} - 2 \beta_{7} ) q^{37} + ( -\beta_{6} - 5 \beta_{7} ) q^{38} + ( -1 - 2 \beta_{4} + \beta_{5} ) q^{39} + ( -2 - 2 \beta_{4} + 2 \beta_{5} ) q^{41} + ( -\beta_{1} - \beta_{3} + \beta_{6} - \beta_{7} ) q^{42} + ( -\beta_{1} + 3 \beta_{3} ) q^{43} + ( -2 - \beta_{4} ) q^{44} + ( -3 \beta_{2} - 3 \beta_{5} ) q^{46} -4 \beta_{7} q^{47} + \beta_{3} q^{48} + ( -2 + \beta_{2} - 3 \beta_{4} + 3 \beta_{5} ) q^{49} + ( 2 + 2 \beta_{4} ) q^{51} + ( -2 \beta_{1} + \beta_{3} + \beta_{6} ) q^{52} + ( -4 \beta_{6} - 6 \beta_{7} ) q^{53} + ( 1 + \beta_{2} ) q^{54} + ( -2 \beta_{2} - \beta_{5} ) q^{56} + ( \beta_{6} + 5 \beta_{7} ) q^{57} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{58} + ( 4 \beta_{2} + 4 \beta_{5} ) q^{59} + ( 3 \beta_{2} - 3 \beta_{5} ) q^{61} + ( 4 \beta_{3} + 4 \beta_{7} ) q^{62} + ( \beta_{1} + \beta_{3} - \beta_{6} + \beta_{7} ) q^{63} - q^{64} + ( 2 + \beta_{4} ) q^{66} + ( 5 \beta_{1} - \beta_{3} - 5 \beta_{6} - \beta_{7} ) q^{67} + ( 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{6} - 2 \beta_{7} ) q^{68} + ( 3 \beta_{2} + 3 \beta_{5} ) q^{69} + ( 11 \beta_{2} + \beta_{5} ) q^{71} -\beta_{3} q^{72} + 9 \beta_{7} q^{73} + ( \beta_{2} - \beta_{5} ) q^{74} + ( 5 + 4 \beta_{2} + \beta_{4} - \beta_{5} ) q^{76} -2 \beta_{7} q^{77} + ( 2 \beta_{1} - \beta_{3} - \beta_{6} ) q^{78} + ( 3 - 5 \beta_{4} ) q^{79} + ( -1 - \beta_{2} ) q^{81} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{82} + ( -\beta_{6} + 4 \beta_{7} ) q^{83} + ( 2 \beta_{2} + \beta_{5} ) q^{84} + ( 3 + \beta_{4} ) q^{86} + ( -2 \beta_{1} + 4 \beta_{3} ) q^{87} + ( \beta_{1} - 2 \beta_{3} - \beta_{6} - 2 \beta_{7} ) q^{88} + ( 4 \beta_{2} - 4 \beta_{4} + 4 \beta_{5} ) q^{89} + ( -4 - 10 \beta_{2} + 2 \beta_{4} - 3 \beta_{5} ) q^{91} -3 \beta_{6} q^{92} + ( -4 \beta_{3} - 4 \beta_{7} ) q^{93} + ( 4 + 4 \beta_{2} ) q^{94} + q^{96} + ( 5 \beta_{1} - 10 \beta_{3} ) q^{97} + ( 3 \beta_{1} - 2 \beta_{3} ) q^{98} + ( -2 - \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 4 q^{4} - 4 q^{6} + 4 q^{9} + O(q^{10})$$ $$8 q + 4 q^{4} - 4 q^{6} + 4 q^{9} - 6 q^{11} + 12 q^{14} - 4 q^{16} - 18 q^{19} - 12 q^{21} + 4 q^{24} - 2 q^{26} - 12 q^{29} + 32 q^{31} - 8 q^{34} - 4 q^{36} + 2 q^{39} - 4 q^{41} - 12 q^{44} + 6 q^{46} - 2 q^{49} + 8 q^{51} + 4 q^{54} + 6 q^{56} - 8 q^{59} - 18 q^{61} - 8 q^{64} + 12 q^{66} - 6 q^{69} - 42 q^{71} - 6 q^{74} + 18 q^{76} + 44 q^{79} - 4 q^{81} - 6 q^{84} + 20 q^{86} + 8 q^{89} - 6 q^{91} + 16 q^{94} + 8 q^{96} - 12 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 9 x^{6} + 65 x^{4} - 144 x^{2} + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$9 \nu^{6} - 65 \nu^{4} + 585 \nu^{2} - 1296$$$$)/1040$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{7} - 181 \nu$$$$)/260$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{6} + 116$$$$)/65$$ $$\beta_{5}$$ $$=$$ $$($$$$-29 \nu^{6} + 325 \nu^{4} - 1885 \nu^{2} + 4176$$$$)/1040$$ $$\beta_{6}$$ $$=$$ $$($$$$9 \nu^{7} - 65 \nu^{5} + 585 \nu^{3} - 256 \nu$$$$)/1040$$ $$\beta_{7}$$ $$=$$ $$($$$$9 \nu^{7} - 65 \nu^{5} + 377 \nu^{3} - 256 \nu$$$$)/832$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} - \beta_{4} + 5 \beta_{2} + 4$$ $$\nu^{3}$$ $$=$$ $$-4 \beta_{7} + 5 \beta_{6}$$ $$\nu^{4}$$ $$=$$ $$9 \beta_{5} + 29 \beta_{2}$$ $$\nu^{5}$$ $$=$$ $$-36 \beta_{7} + 29 \beta_{6} - 36 \beta_{3} - 29 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$65 \beta_{4} - 116$$ $$\nu^{7}$$ $$=$$ $$-260 \beta_{3} - 181 \beta_{1}$$

## 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
 −2.21837 + 1.28078i 1.35234 − 0.780776i −1.35234 + 0.780776i 2.21837 − 1.28078i −2.21837 − 1.28078i 1.35234 + 0.780776i −1.35234 − 0.780776i 2.21837 + 1.28078i
−0.866025 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i 0 −0.500000 0.866025i −3.08440 + 1.78078i 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 0.486319 0.280776i 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 −0.486319 + 0.280776i 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 3.08440 1.78078i 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 −3.08440 1.78078i 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 0.486319 + 0.280776i 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 −0.486319 0.280776i 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 3.08440 + 1.78078i 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.m 8
5.b even 2 1 inner 1950.2.z.m 8
5.c odd 4 1 1950.2.i.z 4
5.c odd 4 1 1950.2.i.bg yes 4
13.c even 3 1 inner 1950.2.z.m 8
65.n even 6 1 inner 1950.2.z.m 8
65.q odd 12 1 1950.2.i.z 4
65.q odd 12 1 1950.2.i.bg yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1950.2.i.z 4 5.c odd 4 1
1950.2.i.z 4 65.q odd 12 1
1950.2.i.bg yes 4 5.c odd 4 1
1950.2.i.bg yes 4 65.q odd 12 1
1950.2.z.m 8 1.a even 1 1 trivial
1950.2.z.m 8 5.b even 2 1 inner
1950.2.z.m 8 13.c even 3 1 inner
1950.2.z.m 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 $$S_{2}^{\mathrm{new}}(1950, [\chi])$$:

 $$T_{7}^{8} - 13 T_{7}^{6} + 165 T_{7}^{4} - 52 T_{7}^{2} + 16$$ $$T_{11}^{4} + 3 T_{11}^{3} + 11 T_{11}^{2} - 6 T_{11} + 4$$ $$T_{17}^{8} - 36 T_{17}^{6} + 1040 T_{17}^{4} - 9216 T_{17}^{2} + 65536$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} )^{2}$$
$3$ $$( 1 - T^{2} + T^{4} )^{2}$$
$5$ $$T^{8}$$
$7$ $$16 - 52 T^{2} + 165 T^{4} - 13 T^{6} + T^{8}$$
$11$ $$( 4 - 6 T + 11 T^{2} + 3 T^{3} + T^{4} )^{2}$$
$13$ $$28561 + 4225 T^{2} + 456 T^{4} + 25 T^{6} + T^{8}$$
$17$ $$65536 - 9216 T^{2} + 1040 T^{4} - 36 T^{6} + T^{8}$$
$19$ $$( 256 + 144 T + 65 T^{2} + 9 T^{3} + T^{4} )^{2}$$
$23$ $$1679616 - 104976 T^{2} + 5265 T^{4} - 81 T^{6} + T^{8}$$
$29$ $$( 64 - 48 T + 44 T^{2} + 6 T^{3} + T^{4} )^{2}$$
$31$ $$( -4 + T )^{8}$$
$37$ $$16 - 52 T^{2} + 165 T^{4} - 13 T^{6} + T^{8}$$
$41$ $$( 256 - 32 T + 20 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$43$ $$16 - 84 T^{2} + 437 T^{4} - 21 T^{6} + T^{8}$$
$47$ $$( 16 + T^{2} )^{4}$$
$53$ $$( 2704 + 168 T^{2} + T^{4} )^{2}$$
$59$ $$( 4096 - 256 T + 80 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$61$ $$( 324 - 162 T + 99 T^{2} + 9 T^{3} + T^{4} )^{2}$$
$67$ $$116985856 - 2347072 T^{2} + 36273 T^{4} - 217 T^{6} + T^{8}$$
$71$ $$( 11236 + 2226 T + 335 T^{2} + 21 T^{3} + T^{4} )^{2}$$
$73$ $$( 81 + T^{2} )^{4}$$
$79$ $$( -76 - 11 T + T^{2} )^{4}$$
$83$ $$( 256 + 49 T^{2} + T^{4} )^{2}$$
$89$ $$( 4096 + 256 T + 80 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$97$ $$6250000 - 812500 T^{2} + 103125 T^{4} - 325 T^{6} + T^{8}$$