# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1950.y (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: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 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 primitive root of unity $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{24}^{4} - 1) q^{2} - \zeta_{24}^{2} q^{3} - \zeta_{24}^{4} q^{4} + ( - \zeta_{24}^{6} + \zeta_{24}^{2}) q^{6} + (\zeta_{24}^{7} - \zeta_{24}^{5} - \zeta_{24}^{4} - \zeta_{24}^{3} + \zeta_{24}) q^{7} + q^{8} + \zeta_{24}^{4} q^{9} +O(q^{10})$$ q + (z^4 - 1) * q^2 - z^2 * q^3 - z^4 * q^4 + (-z^6 + z^2) * q^6 + (z^7 - z^5 - z^4 - z^3 + z) * q^7 + q^8 + z^4 * q^9 $$q + (\zeta_{24}^{4} - 1) q^{2} - \zeta_{24}^{2} q^{3} - \zeta_{24}^{4} q^{4} + ( - \zeta_{24}^{6} + \zeta_{24}^{2}) q^{6} + (\zeta_{24}^{7} - \zeta_{24}^{5} - \zeta_{24}^{4} - \zeta_{24}^{3} + \zeta_{24}) q^{7} + q^{8} + \zeta_{24}^{4} q^{9} + (\zeta_{24}^{7} - \zeta_{24}^{5} - \zeta_{24}^{4} - \zeta_{24}^{3} - 2 \zeta_{24}^{2} - 1) q^{11} + \zeta_{24}^{6} q^{12} + (4 \zeta_{24}^{5} - 3 \zeta_{24}) q^{13} + ( - \zeta_{24}^{7} + \zeta_{24}^{5} + 1) q^{14} + (\zeta_{24}^{4} - 1) q^{16} + (3 \zeta_{24}^{7} - \zeta_{24}^{6} + 3 \zeta_{24}^{5} - 3 \zeta_{24}^{3} + \zeta_{24}^{2} - 3 \zeta_{24}) q^{17} - q^{18} + ( - 2 \zeta_{24}^{7} - \zeta_{24}^{5} + \zeta_{24}^{4} + \zeta_{24}^{3} + 2 \zeta_{24} - 2) q^{19} + (\zeta_{24}^{7} + \zeta_{24}^{6} - \zeta_{24}^{3} + \zeta_{24}) q^{21} + ( - \zeta_{24}^{7} - 2 \zeta_{24}^{6} - \zeta_{24}^{4} + 2 \zeta_{24}^{2} + \zeta_{24} + 2) q^{22} + ( - 3 \zeta_{24}^{7} + 3 \zeta_{24}^{5} - \zeta_{24}^{4} + 4 \zeta_{24}^{3} + \zeta_{24}^{2} + \zeta_{24} - 1) q^{23} - \zeta_{24}^{2} q^{24} + ( - 3 \zeta_{24}^{5} - \zeta_{24}) q^{26} - \zeta_{24}^{6} q^{27} + (\zeta_{24}^{4} + \zeta_{24}^{3} - \zeta_{24} - 1) q^{28} + ( - 5 \zeta_{24}^{7} - 2 \zeta_{24}^{6} - 5 \zeta_{24}^{5} + \zeta_{24}^{4} + \zeta_{24}^{3} + \zeta_{24}^{2} + 4 \zeta_{24} - 1) q^{29} + (5 \zeta_{24}^{7} + \zeta_{24}^{6} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{4} - 3 \zeta_{24}^{3} + 3 \zeta_{24} - 2) q^{31} - \zeta_{24}^{4} q^{32} + (\zeta_{24}^{7} + \zeta_{24}^{6} + 2 \zeta_{24}^{4} + \zeta_{24}^{2} + \zeta_{24}) q^{33} + ( - 3 \zeta_{24}^{7} + \zeta_{24}^{6} - 3 \zeta_{24}^{5}) q^{34} + ( - \zeta_{24}^{4} + 1) q^{36} + (2 \zeta_{24}^{7} + 4 \zeta_{24}^{6} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{4} - 5 \zeta_{24}^{3} - 2 \zeta_{24}^{2} + \cdots - 4) q^{37} + \cdots + ( - 2 \zeta_{24}^{6} - \zeta_{24}^{5} - 2 \zeta_{24}^{4} - \zeta_{24}^{3} + \zeta_{24} + 1) q^{99} +O(q^{100})$$ q + (z^4 - 1) * q^2 - z^2 * q^3 - z^4 * q^4 + (-z^6 + z^2) * q^6 + (z^7 - z^5 - z^4 - z^3 + z) * q^7 + q^8 + z^4 * q^9 + (z^7 - z^5 - z^4 - z^3 - 2*z^2 - 1) * q^11 + z^6 * q^12 + (4*z^5 - 3*z) * q^13 + (-z^7 + z^5 + 1) * q^14 + (z^4 - 1) * q^16 + (3*z^7 - z^6 + 3*z^5 - 3*z^3 + z^2 - 3*z) * q^17 - q^18 + (-2*z^7 - z^5 + z^4 + z^3 + 2*z - 2) * q^19 + (z^7 + z^6 - z^3 + z) * q^21 + (-z^7 - 2*z^6 - z^4 + 2*z^2 + z + 2) * q^22 + (-3*z^7 + 3*z^5 - z^4 + 4*z^3 + z^2 + z - 1) * q^23 - z^2 * q^24 + (-3*z^5 - z) * q^26 - z^6 * q^27 + (z^4 + z^3 - z - 1) * q^28 + (-5*z^7 - 2*z^6 - 5*z^5 + z^4 + z^3 + z^2 + 4*z - 1) * q^29 + (5*z^7 + z^6 + 2*z^5 + 4*z^4 - 3*z^3 + 3*z - 2) * q^31 - z^4 * q^32 + (z^7 + z^6 + 2*z^4 + z^2 + z) * q^33 + (-3*z^7 + z^6 - 3*z^5) * q^34 + (-z^4 + 1) * q^36 + (2*z^7 + 4*z^6 + 2*z^5 + 4*z^4 - 5*z^3 - 2*z^2 + 3*z - 4) * q^37 + (z^7 + 2*z^5 - 2*z^4 + z^3 - z + 1) * q^38 + (-4*z^7 + 3*z^3) * q^39 + (4*z^7 - 4*z^5 - 3*z^3 + 5*z^2 + z) * q^41 + (-z^7 + z^5 - z^2 - z) * q^42 + (3*z^7 - z^6 + 5*z^5 - 2*z^4 - 5*z^3 + z^2 - 3*z + 4) * q^43 + (2*z^6 + z^5 + 2*z^4 + z^3 - z - 1) * q^44 + (4*z^7 + z^6 + z^5 - z^4 - z^3 - z^2 - 4*z + 2) * q^46 + (-4*z^7 + 2*z^6 + z^5 + 3*z^3 - 4*z^2 + 3*z + 4) * q^47 + (-z^6 + z^2) * q^48 + (-2*z^6 - 4*z^4 + 2*z^3 + z^2 - 2*z + 4) * q^49 + (-3*z^7 + 3*z^3 + 3*z - 1) * q^51 + (-z^5 + 4*z) * q^52 + (-z^7 - z^6 - 2*z^5 + 4*z^4 - z^3 + z - 2) * q^53 + z^2 * q^54 + (z^7 - z^5 - z^4 - z^3 + z) * q^56 + (z^7 - z^6 + z^5 - 2*z^3 + 2*z^2 - 2*z) * q^57 + (z^7 + z^6 + 4*z^5 - z^4 + 4*z^3 + z^2 + z) * q^58 + (-6*z^7 + 3*z^6 - 4*z^5 - z^4 + 4*z^3 - 3*z^2 + 6*z + 2) * q^59 + (3*z^7 - 4*z^6 - 3*z^5 - 4*z^4 - 3*z^3 - 4*z^2 + 3*z) * q^61 + (-3*z^7 + 3*z^5 - 2*z^4 - 2*z^3 - z^2 - 5*z - 2) * q^62 + (-z^4 - z^3 + z + 1) * q^63 + q^64 + (z^6 + z^5 - z^3 - 2*z^2 - z - 2) * q^66 + (-4*z^6 - 6*z^4 - 4*z^3 + 2*z^2 + 4*z + 6) * q^67 + (3*z^3 - z^2 + 3*z) * q^68 + (-3*z^7 + z^6 - z^5 - z^4 - z^3 + z^2 - 3*z) * q^69 + (-4*z^7 - z^6 - z^5 - 5*z^4 + z^3 + z^2 + 4*z + 10) * q^71 + z^4 * q^72 + (-4*z^7 + 2*z^6 + 4*z^5 - 4*z^2 - 3) * q^73 + (-5*z^7 - 2*z^6 + 3*z^5 - 4*z^4 + 3*z^3 - 2*z^2 - 5*z) * q^74 + (z^7 - z^5 + z^4 - 2*z^3 - z + 1) * q^76 + (z^7 + z^6 + 2*z^5 + 4*z^4 + z^3 - z - 2) * q^77 + (3*z^7 + z^3) * q^78 + (-3*z^7 - 3*z^5 + 6*z^3 + 6*z + 1) * q^79 + (z^4 - 1) * q^81 + (-3*z^7 + 5*z^6 + z^5 - z^3 - 5*z^2 + 3*z) * q^82 + (-4*z^7 + 2*z^6 + 3*z^5 + z^3 - 4*z^2 + z - 1) * q^83 + (-z^6 - z^5 + z^3 + z^2) * q^84 + (-5*z^7 + z^6 - 3*z^5 + 4*z^4 + 2*z^3 - 2*z - 2) * q^86 + (5*z^7 - z^6 + 4*z^5 + z^4 - 4*z^3 + z^2 - 5*z - 2) * q^87 + (z^7 - z^5 - z^4 - z^3 - 2*z^2 - 1) * q^88 + (-6*z^7 + 6*z^5 - 2*z^4 + 5*z^3 + z^2 - z - 2) * q^89 + (3*z^6 - z^5 - 4*z^4 + z^2 + 4*z + 3) * q^91 + (-z^7 - z^6 - 4*z^5 + 2*z^4 - 3*z^3 + 3*z - 1) * q^92 + (-2*z^7 - 4*z^6 - 2*z^5 - z^4 - 3*z^3 + 2*z^2 + 5*z + 1) * q^93 + (3*z^7 - 4*z^6 + 3*z^5 + 4*z^4 + z^3 + 2*z^2 - 4*z - 4) * q^94 + z^6 * q^96 + (8*z^7 - z^6 - 2*z^5 - 2*z^3 - z^2 + 8*z) * q^97 + (2*z^7 + z^6 - 2*z^5 + 4*z^4 - 2*z^3 + z^2 + 2*z) * q^98 + (-2*z^6 - z^5 - 2*z^4 - z^3 + z + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 4 q^{2} - 4 q^{4} - 4 q^{7} + 8 q^{8} + 4 q^{9}+O(q^{10})$$ 8 * q - 4 * q^2 - 4 * q^4 - 4 * q^7 + 8 * q^8 + 4 * q^9 $$8 q - 4 q^{2} - 4 q^{4} - 4 q^{7} + 8 q^{8} + 4 q^{9} - 12 q^{11} + 8 q^{14} - 4 q^{16} - 8 q^{18} - 12 q^{19} + 12 q^{22} - 12 q^{23} - 4 q^{28} - 4 q^{29} - 4 q^{32} + 8 q^{33} + 4 q^{36} - 16 q^{37} + 24 q^{43} + 12 q^{46} + 32 q^{47} + 16 q^{49} - 8 q^{51} - 4 q^{56} - 4 q^{58} + 12 q^{59} - 16 q^{61} - 24 q^{62} + 4 q^{63} + 8 q^{64} - 16 q^{66} + 24 q^{67} - 4 q^{69} + 60 q^{71} + 4 q^{72} - 24 q^{73} - 16 q^{74} + 12 q^{76} + 8 q^{79} - 4 q^{81} - 8 q^{83} - 12 q^{87} - 12 q^{88} - 24 q^{89} + 8 q^{91} + 4 q^{93} - 16 q^{94} + 16 q^{98}+O(q^{100})$$ 8 * q - 4 * q^2 - 4 * q^4 - 4 * q^7 + 8 * q^8 + 4 * q^9 - 12 * q^11 + 8 * q^14 - 4 * q^16 - 8 * q^18 - 12 * q^19 + 12 * q^22 - 12 * q^23 - 4 * q^28 - 4 * q^29 - 4 * q^32 + 8 * q^33 + 4 * q^36 - 16 * q^37 + 24 * q^43 + 12 * q^46 + 32 * q^47 + 16 * q^49 - 8 * q^51 - 4 * q^56 - 4 * q^58 + 12 * q^59 - 16 * q^61 - 24 * q^62 + 4 * q^63 + 8 * q^64 - 16 * q^66 + 24 * q^67 - 4 * q^69 + 60 * q^71 + 4 * q^72 - 24 * q^73 - 16 * q^74 + 12 * q^76 + 8 * q^79 - 4 * q^81 - 8 * q^83 - 12 * q^87 - 12 * q^88 - 24 * q^89 + 8 * q^91 + 4 * q^93 - 16 * q^94 + 16 * q^98

## Character values

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

 $$n$$ $$301$$ $$1301$$ $$1327$$ $$\chi(n)$$ $$\zeta_{24}^{4}$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 0.965926 − 0.258819i −0.965926 + 0.258819i 0.258819 + 0.965926i −0.258819 − 0.965926i 0.965926 + 0.258819i −0.965926 − 0.258819i 0.258819 − 0.965926i −0.258819 + 0.965926i
−0.500000 0.866025i −0.866025 + 0.500000i −0.500000 + 0.866025i 0 0.866025 + 0.500000i −0.758819 + 1.31431i 1.00000 0.500000 0.866025i 0
49.2 −0.500000 0.866025i −0.866025 + 0.500000i −0.500000 + 0.866025i 0 0.866025 + 0.500000i −0.241181 + 0.417738i 1.00000 0.500000 0.866025i 0
49.3 −0.500000 0.866025i 0.866025 0.500000i −0.500000 + 0.866025i 0 −0.866025 0.500000i −1.46593 + 2.53906i 1.00000 0.500000 0.866025i 0
49.4 −0.500000 0.866025i 0.866025 0.500000i −0.500000 + 0.866025i 0 −0.866025 0.500000i 0.465926 0.807007i 1.00000 0.500000 0.866025i 0
199.1 −0.500000 + 0.866025i −0.866025 0.500000i −0.500000 0.866025i 0 0.866025 0.500000i −0.758819 1.31431i 1.00000 0.500000 + 0.866025i 0
199.2 −0.500000 + 0.866025i −0.866025 0.500000i −0.500000 0.866025i 0 0.866025 0.500000i −0.241181 0.417738i 1.00000 0.500000 + 0.866025i 0
199.3 −0.500000 + 0.866025i 0.866025 + 0.500000i −0.500000 0.866025i 0 −0.866025 + 0.500000i −1.46593 2.53906i 1.00000 0.500000 + 0.866025i 0
199.4 −0.500000 + 0.866025i 0.866025 + 0.500000i −0.500000 0.866025i 0 −0.866025 + 0.500000i 0.465926 + 0.807007i 1.00000 0.500000 + 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 199.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.l 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.y.i 8
5.b even 2 1 1950.2.y.l 8
5.c odd 4 1 1950.2.bc.e 8
5.c odd 4 1 1950.2.bc.f yes 8
13.e even 6 1 1950.2.y.l 8
65.l even 6 1 inner 1950.2.y.i 8
65.r odd 12 1 1950.2.bc.e 8
65.r odd 12 1 1950.2.bc.f yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1950.2.y.i 8 1.a even 1 1 trivial
1950.2.y.i 8 65.l even 6 1 inner
1950.2.y.l 8 5.b even 2 1
1950.2.y.l 8 13.e even 6 1
1950.2.bc.e 8 5.c odd 4 1
1950.2.bc.e 8 65.r odd 12 1
1950.2.bc.f yes 8 5.c odd 4 1
1950.2.bc.f yes 8 65.r odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{8} + 4T_{7}^{7} + 14T_{7}^{6} + 16T_{7}^{5} + 22T_{7}^{4} + 8T_{7}^{3} + 20T_{7}^{2} + 8T_{7} + 4$$ acting on $$S_{2}^{\mathrm{new}}(1950, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + T + 1)^{4}$$
$3$ $$(T^{4} - T^{2} + 1)^{2}$$
$5$ $$T^{8}$$
$7$ $$T^{8} + 4 T^{7} + 14 T^{6} + 16 T^{5} + \cdots + 4$$
$11$ $$T^{8} + 12 T^{7} + 54 T^{6} + \cdots + 529$$
$13$ $$T^{8} + 191 T^{4} + 28561$$
$17$ $$T^{8} - 38 T^{6} + 1398 T^{4} + \cdots + 2116$$
$19$ $$T^{8} + 12 T^{7} + 54 T^{6} + \cdots + 324$$
$23$ $$T^{8} + 12 T^{7} + 12 T^{6} + \cdots + 58081$$
$29$ $$T^{8} + 4 T^{7} + 100 T^{6} + \cdots + 341056$$
$31$ $$T^{8} + 204 T^{6} + 12128 T^{4} + \cdots + 145924$$
$37$ $$T^{8} + 16 T^{7} + 260 T^{6} + \cdots + 16056049$$
$41$ $$T^{8} - 102 T^{6} + 10550 T^{4} + \cdots + 21316$$
$43$ $$T^{8} - 24 T^{7} + 186 T^{6} + \cdots + 386884$$
$47$ $$(T^{4} - 16 T^{3} + 20 T^{2} + 544 T - 1724)^{2}$$
$53$ $$T^{8} + 76 T^{6} + 1824 T^{4} + \cdots + 9604$$
$59$ $$T^{8} - 12 T^{7} - 64 T^{6} + \cdots + 5740816$$
$61$ $$T^{8} + 16 T^{7} + 292 T^{6} + \cdots + 8567329$$
$67$ $$T^{8} - 24 T^{7} + 448 T^{6} + \cdots + 20647936$$
$71$ $$T^{8} - 60 T^{7} + 1596 T^{6} + \cdots + 6115729$$
$73$ $$(T^{4} + 12 T^{3} - 34 T^{2} - 804 T - 2231)^{2}$$
$79$ $$(T^{4} - 4 T^{3} - 102 T^{2} + 212 T + 622)^{2}$$
$83$ $$(T^{4} + 4 T^{3} - 70 T^{2} - 340 T - 263)^{2}$$
$89$ $$T^{8} + 24 T^{7} + 138 T^{6} + \cdots + 5080516$$
$97$ $$T^{8} + 214 T^{6} + \cdots + 61606801$$