# Properties

 Label 1950.2.y.b Level $1950$ Weight $2$ Character orbit 1950.y Analytic conductor $15.571$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 78) 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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{12}^{2} ) q^{2} + \zeta_{12} q^{3} -\zeta_{12}^{2} q^{4} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{6} + ( \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{7} + q^{8} + \zeta_{12}^{2} q^{9} +O(q^{10})$$ $$q + ( -1 + \zeta_{12}^{2} ) q^{2} + \zeta_{12} q^{3} -\zeta_{12}^{2} q^{4} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{6} + ( \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{7} + q^{8} + \zeta_{12}^{2} q^{9} + ( -1 - 3 \zeta_{12} - \zeta_{12}^{2} ) q^{11} -\zeta_{12}^{3} q^{12} + ( -1 - 3 \zeta_{12}^{2} ) q^{13} + ( 1 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{14} + ( -1 + \zeta_{12}^{2} ) q^{16} + ( -2 + 4 \zeta_{12} + \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{17} - q^{18} + ( -2 + 3 \zeta_{12} + \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{19} + ( -1 + 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{21} + ( 2 + 3 \zeta_{12} - \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{22} + ( 3 + \zeta_{12} + 3 \zeta_{12}^{2} ) q^{23} + \zeta_{12} q^{24} + ( 4 - \zeta_{12}^{2} ) q^{26} + \zeta_{12}^{3} q^{27} + ( -1 + \zeta_{12} + \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{28} + ( -1 + 2 \zeta_{12} + \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{29} + ( -2 + 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{31} -\zeta_{12}^{2} q^{32} + ( -\zeta_{12} - 3 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{33} + ( 1 - 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{34} + ( 1 - \zeta_{12}^{2} ) q^{36} + ( 7 + 2 \zeta_{12} - 7 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{37} + ( 1 - 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{38} + ( -\zeta_{12} - 3 \zeta_{12}^{3} ) q^{39} + ( 6 + \zeta_{12} + 6 \zeta_{12}^{2} ) q^{41} + ( -1 + \zeta_{12} - \zeta_{12}^{2} ) q^{42} + ( 10 - \zeta_{12} - 5 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{43} + ( -1 + 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{44} + ( -6 - \zeta_{12} + 3 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{46} + ( 3 + 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{47} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{48} + ( 3 + 2 \zeta_{12} - 3 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{49} + ( 4 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{51} + ( -3 + 4 \zeta_{12}^{2} ) q^{52} + ( 2 - 4 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{53} -\zeta_{12} q^{54} + ( \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{56} + ( 3 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{57} + ( 2 \zeta_{12} - \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{58} + ( -8 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{59} + ( -3 \zeta_{12} + 4 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{61} + ( -2 - 2 \zeta_{12} - 2 \zeta_{12}^{2} ) q^{62} + ( 1 - \zeta_{12} - \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{63} + q^{64} + ( 3 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{66} + ( -1 + 7 \zeta_{12} + \zeta_{12}^{2} - 14 \zeta_{12}^{3} ) q^{67} + ( 1 - 4 \zeta_{12} + \zeta_{12}^{2} ) q^{68} + ( 3 \zeta_{12} + \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{69} + ( 2 - 3 \zeta_{12} - \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{71} + \zeta_{12}^{2} q^{72} + ( -8 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{73} + ( 2 \zeta_{12} + 7 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{74} + ( 1 - 3 \zeta_{12} + \zeta_{12}^{2} ) q^{76} + ( 2 - 4 \zeta_{12}^{2} ) q^{77} + ( 4 \zeta_{12} - \zeta_{12}^{3} ) q^{78} + ( 6 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{79} + ( -1 + \zeta_{12}^{2} ) q^{81} + ( -12 - \zeta_{12} + 6 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{82} + ( -5 - 6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{83} + ( 2 - \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{84} + ( -5 + 10 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{86} + ( 4 - \zeta_{12} - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{87} + ( -1 - 3 \zeta_{12} - \zeta_{12}^{2} ) q^{88} + ( 2 - 6 \zeta_{12} + 2 \zeta_{12}^{2} ) q^{89} + ( -3 + 2 \zeta_{12} + 4 \zeta_{12}^{2} - 7 \zeta_{12}^{3} ) q^{91} + ( 3 - 6 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{92} + ( -2 - 2 \zeta_{12} + 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{93} + ( -3 - 3 \zeta_{12} + 3 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{94} -\zeta_{12}^{3} q^{96} + 6 \zeta_{12}^{2} q^{97} + ( 2 \zeta_{12} + 3 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{98} + ( 1 - 2 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{2} - 2q^{4} - 2q^{7} + 4q^{8} + 2q^{9} + O(q^{10})$$ $$4q - 2q^{2} - 2q^{4} - 2q^{7} + 4q^{8} + 2q^{9} - 6q^{11} - 10q^{13} + 4q^{14} - 2q^{16} - 6q^{17} - 4q^{18} - 6q^{19} + 6q^{22} + 18q^{23} + 14q^{26} - 2q^{28} - 2q^{29} - 2q^{32} - 6q^{33} + 2q^{36} + 14q^{37} + 36q^{41} - 6q^{42} + 30q^{43} - 18q^{46} + 12q^{47} + 6q^{49} + 16q^{51} - 4q^{52} - 2q^{56} + 12q^{57} - 2q^{58} + 8q^{61} - 12q^{62} + 2q^{63} + 4q^{64} + 12q^{66} - 2q^{67} + 6q^{68} + 2q^{69} + 6q^{71} + 2q^{72} - 32q^{73} + 14q^{74} + 6q^{76} + 24q^{79} - 2q^{81} - 36q^{82} - 20q^{83} + 6q^{84} + 12q^{87} - 6q^{88} + 12q^{89} - 4q^{91} - 4q^{93} - 6q^{94} + 12q^{97} + 6q^{98} + O(q^{100})$$

## Character values

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

 $$n$$ $$301$$ $$1301$$ $$1327$$ $$\chi(n)$$ $$\zeta_{12}^{2}$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
−0.500000 0.866025i −0.866025 + 0.500000i −0.500000 + 0.866025i 0 0.866025 + 0.500000i −1.36603 + 2.36603i 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 0.366025 0.633975i 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 −1.36603 2.36603i 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 0.366025 + 0.633975i 1.00000 0.500000 + 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 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.b 4
5.b even 2 1 1950.2.y.g 4
5.c odd 4 1 78.2.i.a 4
5.c odd 4 1 1950.2.bc.d 4
13.e even 6 1 1950.2.y.g 4
15.e even 4 1 234.2.l.c 4
20.e even 4 1 624.2.bv.e 4
60.l odd 4 1 1872.2.by.h 4
65.f even 4 1 1014.2.e.i 4
65.h odd 4 1 1014.2.i.a 4
65.k even 4 1 1014.2.e.g 4
65.l even 6 1 inner 1950.2.y.b 4
65.o even 12 1 1014.2.a.k 2
65.o even 12 1 1014.2.e.g 4
65.q odd 12 1 1014.2.b.e 4
65.q odd 12 1 1014.2.i.a 4
65.r odd 12 1 78.2.i.a 4
65.r odd 12 1 1014.2.b.e 4
65.r odd 12 1 1950.2.bc.d 4
65.t even 12 1 1014.2.a.i 2
65.t even 12 1 1014.2.e.i 4
195.bc odd 12 1 3042.2.a.y 2
195.bf even 12 1 234.2.l.c 4
195.bf even 12 1 3042.2.b.i 4
195.bl even 12 1 3042.2.b.i 4
195.bn odd 12 1 3042.2.a.p 2
260.be odd 12 1 8112.2.a.bp 2
260.bg even 12 1 624.2.bv.e 4
260.bl odd 12 1 8112.2.a.bj 2
780.cw odd 12 1 1872.2.by.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.i.a 4 5.c odd 4 1
78.2.i.a 4 65.r odd 12 1
234.2.l.c 4 15.e even 4 1
234.2.l.c 4 195.bf even 12 1
624.2.bv.e 4 20.e even 4 1
624.2.bv.e 4 260.bg even 12 1
1014.2.a.i 2 65.t even 12 1
1014.2.a.k 2 65.o even 12 1
1014.2.b.e 4 65.q odd 12 1
1014.2.b.e 4 65.r odd 12 1
1014.2.e.g 4 65.k even 4 1
1014.2.e.g 4 65.o even 12 1
1014.2.e.i 4 65.f even 4 1
1014.2.e.i 4 65.t even 12 1
1014.2.i.a 4 65.h odd 4 1
1014.2.i.a 4 65.q odd 12 1
1872.2.by.h 4 60.l odd 4 1
1872.2.by.h 4 780.cw odd 12 1
1950.2.y.b 4 1.a even 1 1 trivial
1950.2.y.b 4 65.l even 6 1 inner
1950.2.y.g 4 5.b even 2 1
1950.2.y.g 4 13.e even 6 1
1950.2.bc.d 4 5.c odd 4 1
1950.2.bc.d 4 65.r odd 12 1
3042.2.a.p 2 195.bn odd 12 1
3042.2.a.y 2 195.bc odd 12 1
3042.2.b.i 4 195.bf even 12 1
3042.2.b.i 4 195.bl even 12 1
8112.2.a.bj 2 260.bl odd 12 1
8112.2.a.bp 2 260.be odd 12 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} )^{2}$$
$3$ $$1 - T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$4 - 4 T + 6 T^{2} + 2 T^{3} + T^{4}$$
$11$ $$36 - 36 T + 6 T^{2} + 6 T^{3} + T^{4}$$
$13$ $$( 13 + 5 T + T^{2} )^{2}$$
$17$ $$169 - 78 T - T^{2} + 6 T^{3} + T^{4}$$
$19$ $$36 - 36 T + 6 T^{2} + 6 T^{3} + T^{4}$$
$23$ $$676 - 468 T + 134 T^{2} - 18 T^{3} + T^{4}$$
$29$ $$121 - 22 T + 15 T^{2} + 2 T^{3} + T^{4}$$
$31$ $$64 + 32 T^{2} + T^{4}$$
$37$ $$1369 - 518 T + 159 T^{2} - 14 T^{3} + T^{4}$$
$41$ $$11449 - 3852 T + 539 T^{2} - 36 T^{3} + T^{4}$$
$43$ $$5476 - 2220 T + 374 T^{2} - 30 T^{3} + T^{4}$$
$47$ $$( -18 - 6 T + T^{2} )^{2}$$
$53$ $$9 + 42 T^{2} + T^{4}$$
$59$ $$4096 - 64 T^{2} + T^{4}$$
$61$ $$121 + 88 T + 75 T^{2} - 8 T^{3} + T^{4}$$
$67$ $$21316 - 292 T + 150 T^{2} + 2 T^{3} + T^{4}$$
$71$ $$36 + 36 T + 6 T^{2} - 6 T^{3} + T^{4}$$
$73$ $$( 61 + 16 T + T^{2} )^{2}$$
$79$ $$( 24 - 12 T + T^{2} )^{2}$$
$83$ $$( -2 + 10 T + T^{2} )^{2}$$
$89$ $$576 + 288 T + 24 T^{2} - 12 T^{3} + T^{4}$$
$97$ $$( 36 - 6 T + T^{2} )^{2}$$