# Properties

 Label 1650.2.c.m Level 1650 Weight 2 Character orbit 1650.c Analytic conductor 13.175 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$1650 = 2 \cdot 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 1650.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.1753163335$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 66) 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.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1650\mathbb{Z}\right)^\times$$.

 $$n$$ $$551$$ $$727$$ $$1201$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 1.00000i − 1.00000i
1.00000i 1.00000i −1.00000 0 1.00000 2.00000i 1.00000i −1.00000 0
199.2 1.00000i 1.00000i −1.00000 0 1.00000 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1650.2.c.m 2
3.b odd 2 1 4950.2.c.d 2
5.b even 2 1 inner 1650.2.c.m 2
5.c odd 4 1 66.2.a.c 1
5.c odd 4 1 1650.2.a.c 1
15.d odd 2 1 4950.2.c.d 2
15.e even 4 1 198.2.a.c 1
15.e even 4 1 4950.2.a.bo 1
20.e even 4 1 528.2.a.a 1
35.f even 4 1 3234.2.a.s 1
40.i odd 4 1 2112.2.a.n 1
40.k even 4 1 2112.2.a.bd 1
45.k odd 12 2 1782.2.e.l 2
45.l even 12 2 1782.2.e.n 2
55.e even 4 1 726.2.a.d 1
55.k odd 20 4 726.2.e.e 4
55.l even 20 4 726.2.e.m 4
60.l odd 4 1 1584.2.a.s 1
105.k odd 4 1 9702.2.a.a 1
120.q odd 4 1 6336.2.a.d 1
120.w even 4 1 6336.2.a.c 1
165.l odd 4 1 2178.2.a.m 1
220.i odd 4 1 5808.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.2.a.c 1 5.c odd 4 1
198.2.a.c 1 15.e even 4 1
528.2.a.a 1 20.e even 4 1
726.2.a.d 1 55.e even 4 1
726.2.e.e 4 55.k odd 20 4
726.2.e.m 4 55.l even 20 4
1584.2.a.s 1 60.l odd 4 1
1650.2.a.c 1 5.c odd 4 1
1650.2.c.m 2 1.a even 1 1 trivial
1650.2.c.m 2 5.b even 2 1 inner
1782.2.e.l 2 45.k odd 12 2
1782.2.e.n 2 45.l even 12 2
2112.2.a.n 1 40.i odd 4 1
2112.2.a.bd 1 40.k even 4 1
2178.2.a.m 1 165.l odd 4 1
3234.2.a.s 1 35.f even 4 1
4950.2.a.bo 1 15.e even 4 1
4950.2.c.d 2 3.b odd 2 1
4950.2.c.d 2 15.d odd 2 1
5808.2.a.b 1 220.i odd 4 1
6336.2.a.c 1 120.w even 4 1
6336.2.a.d 1 120.q odd 4 1
9702.2.a.a 1 105.k odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1650, [\chi])$$:

 $$T_{7}^{2} + 4$$ $$T_{13}^{2} + 16$$ $$T_{17}^{2} + 4$$ $$T_{19}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$1 + T^{2}$$
$5$ 1
$7$ $$1 - 10 T^{2} + 49 T^{4}$$
$11$ $$( 1 - T )^{2}$$
$13$ $$( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} )$$
$17$ $$( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} )$$
$19$ $$( 1 + 19 T^{2} )^{2}$$
$23$ $$1 - 10 T^{2} + 529 T^{4}$$
$29$ $$( 1 + 10 T + 29 T^{2} )^{2}$$
$31$ $$( 1 + 8 T + 31 T^{2} )^{2}$$
$37$ $$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$
$41$ $$( 1 - 2 T + 41 T^{2} )^{2}$$
$43$ $$1 - 70 T^{2} + 1849 T^{4}$$
$47$ $$1 - 90 T^{2} + 2209 T^{4}$$
$53$ $$( 1 - 14 T + 53 T^{2} )( 1 + 14 T + 53 T^{2} )$$
$59$ $$( 1 + 59 T^{2} )^{2}$$
$61$ $$( 1 + 8 T + 61 T^{2} )^{2}$$
$67$ $$1 + 10 T^{2} + 4489 T^{4}$$
$71$ $$( 1 - 2 T + 71 T^{2} )^{2}$$
$73$ $$( 1 - 16 T + 73 T^{2} )( 1 + 16 T + 73 T^{2} )$$
$79$ $$( 1 + 10 T + 79 T^{2} )^{2}$$
$83$ $$1 - 150 T^{2} + 6889 T^{4}$$
$89$ $$( 1 + 10 T + 89 T^{2} )^{2}$$
$97$ $$1 - 190 T^{2} + 9409 T^{4}$$