# Properties

 Label 1950.2.b.c Level $1950$ Weight $2$ Character orbit 1950.b Analytic conductor $15.571$ Analytic rank $0$ Dimension $2$ 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.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$15.5708283941$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ 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: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$301$$ $$1301$$ $$1327$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1351.1
 1.00000i − 1.00000i
1.00000i −1.00000 −1.00000 0 1.00000i 2.00000i 1.00000i 1.00000 0
1351.2 1.00000i −1.00000 −1.00000 0 1.00000i 2.00000i 1.00000i 1.00000 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.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1950.2.b.c 2
5.b even 2 1 78.2.b.a 2
5.c odd 4 1 1950.2.f.d 2
5.c odd 4 1 1950.2.f.g 2
13.b even 2 1 inner 1950.2.b.c 2
15.d odd 2 1 234.2.b.a 2
20.d odd 2 1 624.2.c.a 2
35.c odd 2 1 3822.2.c.d 2
40.e odd 2 1 2496.2.c.m 2
40.f even 2 1 2496.2.c.f 2
60.h even 2 1 1872.2.c.b 2
65.d even 2 1 78.2.b.a 2
65.g odd 4 1 1014.2.a.b 1
65.g odd 4 1 1014.2.a.g 1
65.h odd 4 1 1950.2.f.d 2
65.h odd 4 1 1950.2.f.g 2
65.l even 6 2 1014.2.i.c 4
65.n even 6 2 1014.2.i.c 4
65.s odd 12 2 1014.2.e.b 2
65.s odd 12 2 1014.2.e.e 2
195.e odd 2 1 234.2.b.a 2
195.n even 4 1 3042.2.a.c 1
195.n even 4 1 3042.2.a.n 1
260.g odd 2 1 624.2.c.a 2
260.u even 4 1 8112.2.a.g 1
260.u even 4 1 8112.2.a.j 1
455.h odd 2 1 3822.2.c.d 2
520.b odd 2 1 2496.2.c.m 2
520.p even 2 1 2496.2.c.f 2
780.d even 2 1 1872.2.c.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.b.a 2 5.b even 2 1
78.2.b.a 2 65.d even 2 1
234.2.b.a 2 15.d odd 2 1
234.2.b.a 2 195.e odd 2 1
624.2.c.a 2 20.d odd 2 1
624.2.c.a 2 260.g odd 2 1
1014.2.a.b 1 65.g odd 4 1
1014.2.a.g 1 65.g odd 4 1
1014.2.e.b 2 65.s odd 12 2
1014.2.e.e 2 65.s odd 12 2
1014.2.i.c 4 65.l even 6 2
1014.2.i.c 4 65.n even 6 2
1872.2.c.b 2 60.h even 2 1
1872.2.c.b 2 780.d even 2 1
1950.2.b.c 2 1.a even 1 1 trivial
1950.2.b.c 2 13.b even 2 1 inner
1950.2.f.d 2 5.c odd 4 1
1950.2.f.d 2 65.h odd 4 1
1950.2.f.g 2 5.c odd 4 1
1950.2.f.g 2 65.h odd 4 1
2496.2.c.f 2 40.f even 2 1
2496.2.c.f 2 520.p even 2 1
2496.2.c.m 2 40.e odd 2 1
2496.2.c.m 2 520.b odd 2 1
3042.2.a.c 1 195.n even 4 1
3042.2.a.n 1 195.n even 4 1
3822.2.c.d 2 35.c odd 2 1
3822.2.c.d 2 455.h odd 2 1
8112.2.a.g 1 260.u even 4 1
8112.2.a.j 1 260.u even 4 1

## Hecke kernels

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

 $$T_{7}^{2} + 4$$ T7^2 + 4 $$T_{11}$$ T11 $$T_{17} - 2$$ T17 - 2 $$T_{19}^{2} + 36$$ T19^2 + 36

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 4$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 6T + 13$$
$17$ $$(T - 2)^{2}$$
$19$ $$T^{2} + 36$$
$23$ $$(T + 4)^{2}$$
$29$ $$(T + 10)^{2}$$
$31$ $$T^{2} + 100$$
$37$ $$T^{2} + 64$$
$41$ $$T^{2} + 100$$
$43$ $$(T + 4)^{2}$$
$47$ $$T^{2} + 144$$
$53$ $$(T - 6)^{2}$$
$59$ $$T^{2} + 16$$
$61$ $$(T - 2)^{2}$$
$67$ $$T^{2} + 4$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 16$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 16$$
$89$ $$T^{2} + 36$$
$97$ $$T^{2} + 144$$