Properties

 Label 2550.2.d.m Level $2550$ Weight $2$ Character orbit 2550.d Analytic conductor $20.362$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2550.d (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$20.3618525154$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 102) 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.

 $$f(q)$$ $$=$$ $$q -i q^{2} + i q^{3} - q^{4} + q^{6} + i q^{8} - q^{9} +O(q^{10})$$ $$q -i q^{2} + i q^{3} - q^{4} + q^{6} + i q^{8} - q^{9} -4 q^{11} -i q^{12} -2 i q^{13} + q^{16} -i q^{17} + i q^{18} -4 q^{19} + 4 i q^{22} - q^{24} -2 q^{26} -i q^{27} + 10 q^{29} + 8 q^{31} -i q^{32} -4 i q^{33} - q^{34} + q^{36} + 2 i q^{37} + 4 i q^{38} + 2 q^{39} + 10 q^{41} + 12 i q^{43} + 4 q^{44} + i q^{48} + 7 q^{49} + q^{51} + 2 i q^{52} + 6 i q^{53} - q^{54} -4 i q^{57} -10 i q^{58} -12 q^{59} -10 q^{61} -8 i q^{62} - q^{64} -4 q^{66} + 12 i q^{67} + i q^{68} -i q^{72} + 10 i q^{73} + 2 q^{74} + 4 q^{76} -2 i q^{78} + 8 q^{79} + q^{81} -10 i q^{82} + 4 i q^{83} + 12 q^{86} + 10 i q^{87} -4 i q^{88} + 6 q^{89} + 8 i q^{93} + q^{96} + 14 i q^{97} -7 i q^{98} + 4 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + 2q^{6} - 2q^{9} + O(q^{10})$$ $$2q - 2q^{4} + 2q^{6} - 2q^{9} - 8q^{11} + 2q^{16} - 8q^{19} - 2q^{24} - 4q^{26} + 20q^{29} + 16q^{31} - 2q^{34} + 2q^{36} + 4q^{39} + 20q^{41} + 8q^{44} + 14q^{49} + 2q^{51} - 2q^{54} - 24q^{59} - 20q^{61} - 2q^{64} - 8q^{66} + 4q^{74} + 8q^{76} + 16q^{79} + 2q^{81} + 24q^{86} + 12q^{89} + 2q^{96} + 8q^{99} + O(q^{100})$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2550\mathbb{Z}\right)^\times$$.

 $$n$$ $$751$$ $$851$$ $$1327$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2449.1
 1.00000i − 1.00000i
1.00000i 1.00000i −1.00000 0 1.00000 0 1.00000i −1.00000 0
2449.2 1.00000i 1.00000i −1.00000 0 1.00000 0 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 2550.2.d.m 2
5.b even 2 1 inner 2550.2.d.m 2
5.c odd 4 1 102.2.a.c 1
5.c odd 4 1 2550.2.a.c 1
15.e even 4 1 306.2.a.b 1
15.e even 4 1 7650.2.a.ca 1
20.e even 4 1 816.2.a.b 1
35.f even 4 1 4998.2.a.be 1
40.i odd 4 1 3264.2.a.m 1
40.k even 4 1 3264.2.a.bc 1
60.l odd 4 1 2448.2.a.p 1
85.f odd 4 1 1734.2.b.b 2
85.g odd 4 1 1734.2.a.j 1
85.i odd 4 1 1734.2.b.b 2
85.k odd 8 2 1734.2.f.e 4
85.n odd 8 2 1734.2.f.e 4
120.q odd 4 1 9792.2.a.l 1
120.w even 4 1 9792.2.a.k 1
255.o even 4 1 5202.2.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
102.2.a.c 1 5.c odd 4 1
306.2.a.b 1 15.e even 4 1
816.2.a.b 1 20.e even 4 1
1734.2.a.j 1 85.g odd 4 1
1734.2.b.b 2 85.f odd 4 1
1734.2.b.b 2 85.i odd 4 1
1734.2.f.e 4 85.k odd 8 2
1734.2.f.e 4 85.n odd 8 2
2448.2.a.p 1 60.l odd 4 1
2550.2.a.c 1 5.c odd 4 1
2550.2.d.m 2 1.a even 1 1 trivial
2550.2.d.m 2 5.b even 2 1 inner
3264.2.a.m 1 40.i odd 4 1
3264.2.a.bc 1 40.k even 4 1
4998.2.a.be 1 35.f even 4 1
5202.2.a.c 1 255.o even 4 1
7650.2.a.ca 1 15.e even 4 1
9792.2.a.k 1 120.w even 4 1
9792.2.a.l 1 120.q odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2550, [\chi])$$:

 $$T_{7}$$ $$T_{11} + 4$$ $$T_{13}^{2} + 4$$ $$T_{19} + 4$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$( 4 + T )^{2}$$
$13$ $$4 + T^{2}$$
$17$ $$1 + T^{2}$$
$19$ $$( 4 + T )^{2}$$
$23$ $$T^{2}$$
$29$ $$( -10 + T )^{2}$$
$31$ $$( -8 + T )^{2}$$
$37$ $$4 + T^{2}$$
$41$ $$( -10 + T )^{2}$$
$43$ $$144 + T^{2}$$
$47$ $$T^{2}$$
$53$ $$36 + T^{2}$$
$59$ $$( 12 + T )^{2}$$
$61$ $$( 10 + T )^{2}$$
$67$ $$144 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$100 + T^{2}$$
$79$ $$( -8 + T )^{2}$$
$83$ $$16 + T^{2}$$
$89$ $$( -6 + T )^{2}$$
$97$ $$196 + T^{2}$$