# Properties

 Label 1950.2.a.h Level $1950$ Weight $2$ Character orbit 1950.a Self dual yes Analytic conductor $15.571$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1950.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$15.5708283941$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 390) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
−1.00000 1.00000 1.00000 0 −1.00000 −2.00000 −1.00000 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$1$$
$$13$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1950.2.a.h 1
3.b odd 2 1 5850.2.a.bi 1
5.b even 2 1 390.2.a.e 1
5.c odd 4 2 1950.2.e.f 2
15.d odd 2 1 1170.2.a.e 1
15.e even 4 2 5850.2.e.i 2
20.d odd 2 1 3120.2.a.o 1
60.h even 2 1 9360.2.a.bh 1
65.d even 2 1 5070.2.a.e 1
65.g odd 4 2 5070.2.b.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.a.e 1 5.b even 2 1
1170.2.a.e 1 15.d odd 2 1
1950.2.a.h 1 1.a even 1 1 trivial
1950.2.e.f 2 5.c odd 4 2
3120.2.a.o 1 20.d odd 2 1
5070.2.a.e 1 65.d even 2 1
5070.2.b.e 2 65.g odd 4 2
5850.2.a.bi 1 3.b odd 2 1
5850.2.e.i 2 15.e even 4 2
9360.2.a.bh 1 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1950))$$:

 $$T_{7} + 2$$ T7 + 2 $$T_{11} - 4$$ T11 - 4 $$T_{17} + 8$$ T17 + 8 $$T_{23} + 6$$ T23 + 6 $$T_{31}$$ T31

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T - 1$$
$5$ $$T$$
$7$ $$T + 2$$
$11$ $$T - 4$$
$13$ $$T - 1$$
$17$ $$T + 8$$
$19$ $$T + 6$$
$23$ $$T + 6$$
$29$ $$T + 4$$
$31$ $$T$$
$37$ $$T - 2$$
$41$ $$T + 2$$
$43$ $$T - 4$$
$47$ $$T$$
$53$ $$T - 10$$
$59$ $$T - 4$$
$61$ $$T + 10$$
$67$ $$T + 12$$
$71$ $$T + 8$$
$73$ $$T - 8$$
$79$ $$T - 8$$
$83$ $$T + 12$$
$89$ $$T + 14$$
$97$ $$T - 16$$