# Properties

 Label 1950.2.y.a 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: $$x^{4} - x^{2} + 1$$ 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.

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

## Character values

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

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

## 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 −2.36603 + 4.09808i 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 −0.633975 + 1.09808i 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 −2.36603 4.09808i 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 −0.633975 1.09808i 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.a 4
5.b even 2 1 1950.2.y.h 4
5.c odd 4 1 78.2.i.b 4
5.c odd 4 1 1950.2.bc.c 4
13.e even 6 1 1950.2.y.h 4
15.e even 4 1 234.2.l.a 4
20.e even 4 1 624.2.bv.d 4
60.l odd 4 1 1872.2.by.k 4
65.f even 4 1 1014.2.e.j 4
65.h odd 4 1 1014.2.i.f 4
65.k even 4 1 1014.2.e.h 4
65.l even 6 1 inner 1950.2.y.a 4
65.o even 12 1 1014.2.a.j 2
65.o even 12 1 1014.2.e.h 4
65.q odd 12 1 1014.2.b.d 4
65.q odd 12 1 1014.2.i.f 4
65.r odd 12 1 78.2.i.b 4
65.r odd 12 1 1014.2.b.d 4
65.r odd 12 1 1950.2.bc.c 4
65.t even 12 1 1014.2.a.h 2
65.t even 12 1 1014.2.e.j 4
195.bc odd 12 1 3042.2.a.v 2
195.bf even 12 1 234.2.l.a 4
195.bf even 12 1 3042.2.b.l 4
195.bl even 12 1 3042.2.b.l 4
195.bn odd 12 1 3042.2.a.s 2
260.be odd 12 1 8112.2.a.bx 2
260.bg even 12 1 624.2.bv.d 4
260.bl odd 12 1 8112.2.a.bq 2
780.cw odd 12 1 1872.2.by.k 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} + 6T_{7}^{3} + 30T_{7}^{2} + 36T_{7} + 36$$ acting on $$S_{2}^{\mathrm{new}}(1950, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + T + 1)^{2}$$
$3$ $$T^{4} - T^{2} + 1$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 6 T^{3} + 30 T^{2} + 36 T + 36$$
$11$ $$T^{4} - 6 T^{3} + 6 T^{2} + 36 T + 36$$
$13$ $$T^{4} + 6 T^{3} + 23 T^{2} + 78 T + 169$$
$17$ $$(T^{2} - 9 T + 27)^{2}$$
$19$ $$T^{4} - 6 T^{3} + 6 T^{2} + 36 T + 36$$
$23$ $$T^{4} + 18 T^{3} + 126 T^{2} + \cdots + 324$$
$29$ $$(T^{2} + 3 T + 9)^{2}$$
$31$ $$T^{4} + 96T^{2} + 576$$
$37$ $$(T^{2} + 3 T + 9)^{2}$$
$41$ $$T^{4} - 12 T^{3} + 51 T^{2} - 36 T + 9$$
$43$ $$T^{4} - 18 T^{3} + 134 T^{2} + \cdots + 676$$
$47$ $$(T^{2} + 6 T + 6)^{2}$$
$53$ $$(T^{2} + 9)^{2}$$
$59$ $$(T^{2} - 24 T + 192)^{2}$$
$61$ $$T^{4} + 20 T^{3} + 327 T^{2} + \cdots + 5329$$
$67$ $$T^{4} - 18 T^{3} + 246 T^{2} + \cdots + 6084$$
$71$ $$T^{4} - 18 T^{3} + 126 T^{2} + \cdots + 324$$
$73$ $$(T^{2} - 147)^{2}$$
$79$ $$(T^{2} - 4 T - 104)^{2}$$
$83$ $$(T^{2} + 6 T - 66)^{2}$$
$89$ $$T^{4} - 12 T^{3} + 24 T^{2} + \cdots + 576$$
$97$ $$(T^{2} - 6 T + 36)^{2}$$