Properties

 Label 1950.2.f.g 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.g 2
5.b even 2 1 1950.2.f.d 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.d 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.g 2
65.f even 4 1 1014.2.a.b 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.g 1
65.o even 12 2 1014.2.e.b 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.e 2
195.j odd 4 1 3042.2.a.c 1
195.s even 4 1 234.2.b.a 2
195.u odd 4 1 3042.2.a.n 1
260.l odd 4 1 8112.2.a.g 1
260.p even 4 1 624.2.c.a 2
260.s odd 4 1 8112.2.a.j 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.f even 4 1
1014.2.a.g 1 65.k even 4 1
1014.2.e.b 2 65.o even 12 2
1014.2.e.e 2 65.t 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 5.b even 2 1
1950.2.f.d 2 13.b even 2 1
1950.2.f.g 2 1.a even 1 1 trivial
1950.2.f.g 2 65.d even 2 1 inner
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.j odd 4 1
3042.2.a.n 1 195.u 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.l odd 4 1
8112.2.a.j 1 260.s 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

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