# Properties

 Label 1950.2.bc.c Level $1950$ Weight $2$ Character orbit 1950.bc 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.bc (of order $$6$$, degree $$2$$, 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: $$\Z[a_1, a_2]$$ 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.

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

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
751.1
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
−0.866025 + 0.500000i −0.500000 0.866025i 0.500000 0.866025i 0 0.866025 + 0.500000i 4.09808 + 2.36603i 1.00000i −0.500000 + 0.866025i 0
751.2 0.866025 0.500000i −0.500000 0.866025i 0.500000 0.866025i 0 −0.866025 0.500000i −1.09808 0.633975i 1.00000i −0.500000 + 0.866025i 0
901.1 −0.866025 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 0 0.866025 0.500000i 4.09808 2.36603i 1.00000i −0.500000 0.866025i 0
901.2 0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 0 −0.866025 + 0.500000i −1.09808 + 0.633975i 1.00000i −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
13.e 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.bc.c 4
5.b even 2 1 78.2.i.b 4
5.c odd 4 1 1950.2.y.a 4
5.c odd 4 1 1950.2.y.h 4
13.e even 6 1 inner 1950.2.bc.c 4
15.d odd 2 1 234.2.l.a 4
20.d odd 2 1 624.2.bv.d 4
60.h even 2 1 1872.2.by.k 4
65.d even 2 1 1014.2.i.f 4
65.g odd 4 1 1014.2.e.h 4
65.g odd 4 1 1014.2.e.j 4
65.l even 6 1 78.2.i.b 4
65.l even 6 1 1014.2.b.d 4
65.n even 6 1 1014.2.b.d 4
65.n even 6 1 1014.2.i.f 4
65.r odd 12 1 1950.2.y.a 4
65.r odd 12 1 1950.2.y.h 4
65.s odd 12 1 1014.2.a.h 2
65.s odd 12 1 1014.2.a.j 2
65.s odd 12 1 1014.2.e.h 4
65.s odd 12 1 1014.2.e.j 4
195.x odd 6 1 3042.2.b.l 4
195.y odd 6 1 234.2.l.a 4
195.y odd 6 1 3042.2.b.l 4
195.bh even 12 1 3042.2.a.s 2
195.bh even 12 1 3042.2.a.v 2
260.w odd 6 1 624.2.bv.d 4
260.bc even 12 1 8112.2.a.bq 2
260.bc even 12 1 8112.2.a.bx 2
780.cb even 6 1 1872.2.by.k 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.i.b 4 5.b even 2 1
78.2.i.b 4 65.l even 6 1
234.2.l.a 4 15.d odd 2 1
234.2.l.a 4 195.y odd 6 1
624.2.bv.d 4 20.d odd 2 1
624.2.bv.d 4 260.w odd 6 1
1014.2.a.h 2 65.s odd 12 1
1014.2.a.j 2 65.s odd 12 1
1014.2.b.d 4 65.l even 6 1
1014.2.b.d 4 65.n even 6 1
1014.2.e.h 4 65.g odd 4 1
1014.2.e.h 4 65.s odd 12 1
1014.2.e.j 4 65.g odd 4 1
1014.2.e.j 4 65.s odd 12 1
1014.2.i.f 4 65.d even 2 1
1014.2.i.f 4 65.n even 6 1
1872.2.by.k 4 60.h even 2 1
1872.2.by.k 4 780.cb even 6 1
1950.2.y.a 4 5.c odd 4 1
1950.2.y.a 4 65.r odd 12 1
1950.2.y.h 4 5.c odd 4 1
1950.2.y.h 4 65.r odd 12 1
1950.2.bc.c 4 1.a even 1 1 trivial
1950.2.bc.c 4 13.e even 6 1 inner
3042.2.a.s 2 195.bh even 12 1
3042.2.a.v 2 195.bh even 12 1
3042.2.b.l 4 195.x odd 6 1
3042.2.b.l 4 195.y odd 6 1
8112.2.a.bq 2 260.bc even 12 1
8112.2.a.bx 2 260.bc even 12 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{2} + 1$$
$3$ $$(T^{2} + T + 1)^{2}$$
$5$ $$T^{4}$$
$7$ $$T^{4} - 6 T^{3} + 6 T^{2} + 36 T + 36$$
$11$ $$T^{4} - 6 T^{3} + 6 T^{2} + 36 T + 36$$
$13$ $$T^{4} - 4 T^{3} + 3 T^{2} - 52 T + 169$$
$17$ $$T^{4} + 27T^{2} + 729$$
$19$ $$T^{4} + 6 T^{3} + 6 T^{2} - 36 T + 36$$
$23$ $$T^{4} - 6 T^{3} + 54 T^{2} + 108 T + 324$$
$29$ $$(T^{2} - 3 T + 9)^{2}$$
$31$ $$T^{4} + 96T^{2} + 576$$
$37$ $$T^{4} - 9T^{2} + 81$$
$41$ $$T^{4} - 12 T^{3} + 51 T^{2} - 36 T + 9$$
$43$ $$T^{4} + 2 T^{3} + 30 T^{2} - 52 T + 676$$
$47$ $$T^{4} + 24T^{2} + 36$$
$53$ $$(T + 3)^{4}$$
$59$ $$(T^{2} + 24 T + 192)^{2}$$
$61$ $$T^{4} + 20 T^{3} + 327 T^{2} + \cdots + 5329$$
$67$ $$T^{4} - 6 T^{3} - 66 T^{2} + \cdots + 6084$$
$71$ $$T^{4} - 18 T^{3} + 126 T^{2} + \cdots + 324$$
$73$ $$(T^{2} + 147)^{2}$$
$79$ $$(T^{2} + 4 T - 104)^{2}$$
$83$ $$T^{4} + 168T^{2} + 4356$$
$89$ $$T^{4} + 12 T^{3} + 24 T^{2} + \cdots + 576$$
$97$ $$T^{4} - 36T^{2} + 1296$$