# Properties

 Label 1950.2.f.d Level $1950$ Weight $2$ Character orbit 1950.f 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.f (of order $$2$$, degree $$1$$, not 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: $$\Z[a_1, a_2, a_3]$$ 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.

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

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
649.1
 − 1.00000i 1.00000i
−1.00000 1.00000i 1.00000 0 1.00000i 2.00000 −1.00000 −1.00000 0
649.2 −1.00000 1.00000i 1.00000 0 1.00000i 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

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.d 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.f.d 2
5.b even 2 1 1950.2.f.g 2
5.c odd 4 1 78.2.b.a 2
5.c odd 4 1 1950.2.b.c 2
13.b even 2 1 1950.2.f.g 2
15.e even 4 1 234.2.b.a 2
20.e even 4 1 624.2.c.a 2
35.f even 4 1 3822.2.c.d 2
40.i odd 4 1 2496.2.c.f 2
40.k even 4 1 2496.2.c.m 2
60.l odd 4 1 1872.2.c.b 2
65.d even 2 1 inner 1950.2.f.d 2
65.f even 4 1 1014.2.a.g 1
65.h odd 4 1 78.2.b.a 2
65.h odd 4 1 1950.2.b.c 2
65.k even 4 1 1014.2.a.b 1
65.o even 12 2 1014.2.e.e 2
65.q odd 12 2 1014.2.i.c 4
65.r odd 12 2 1014.2.i.c 4
65.t even 12 2 1014.2.e.b 2
195.j odd 4 1 3042.2.a.n 1
195.s even 4 1 234.2.b.a 2
195.u odd 4 1 3042.2.a.c 1
260.l odd 4 1 8112.2.a.j 1
260.p even 4 1 624.2.c.a 2
260.s odd 4 1 8112.2.a.g 1
455.s even 4 1 3822.2.c.d 2
520.bc even 4 1 2496.2.c.m 2
520.bg odd 4 1 2496.2.c.f 2
780.w odd 4 1 1872.2.c.b 2

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

## Hecke kernels

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

 $$T_{7} - 2$$ T7 - 2 $$T_{11}$$ T11 $$T_{19}^{2} + 36$$ T19^2 + 36 $$T_{37} + 8$$ T37 + 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$(T - 2)^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 4T + 13$$
$17$ $$T^{2} + 4$$
$19$ $$T^{2} + 36$$
$23$ $$T^{2} + 16$$
$29$ $$(T - 10)^{2}$$
$31$ $$T^{2} + 100$$
$37$ $$(T + 8)^{2}$$
$41$ $$T^{2} + 100$$
$43$ $$T^{2} + 16$$
$47$ $$(T - 12)^{2}$$
$53$ $$T^{2} + 36$$
$59$ $$T^{2} + 16$$
$61$ $$(T - 2)^{2}$$
$67$ $$(T - 2)^{2}$$
$71$ $$T^{2}$$
$73$ $$(T + 4)^{2}$$
$79$ $$T^{2}$$
$83$ $$(T + 4)^{2}$$
$89$ $$T^{2} + 36$$
$97$ $$(T - 12)^{2}$$