# Properties

 Label 51.2.d.b Level $51$ Weight $2$ Character orbit 51.d Analytic conductor $0.407$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$51 = 3 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 51.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.407237050309$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + q^{2} + i q^{3} - q^{4} + i q^{6} -4 i q^{7} -3 q^{8} - q^{9} +O(q^{10})$$ $$q + q^{2} + i q^{3} - q^{4} + i q^{6} -4 i q^{7} -3 q^{8} - q^{9} + 4 i q^{11} -i q^{12} + 2 q^{13} -4 i q^{14} - q^{16} + ( 1 + 4 i ) q^{17} - q^{18} -4 q^{19} + 4 q^{21} + 4 i q^{22} -4 i q^{23} -3 i q^{24} + 5 q^{25} + 2 q^{26} -i q^{27} + 4 i q^{28} -4 i q^{31} + 5 q^{32} -4 q^{33} + ( 1 + 4 i ) q^{34} + q^{36} + 8 i q^{37} -4 q^{38} + 2 i q^{39} -8 i q^{41} + 4 q^{42} -4 q^{43} -4 i q^{44} -4 i q^{46} -8 q^{47} -i q^{48} -9 q^{49} + 5 q^{50} + ( -4 + i ) q^{51} -2 q^{52} + 6 q^{53} -i q^{54} + 12 i q^{56} -4 i q^{57} -12 q^{59} -8 i q^{61} -4 i q^{62} + 4 i q^{63} + 7 q^{64} -4 q^{66} + 12 q^{67} + ( -1 - 4 i ) q^{68} + 4 q^{69} + 12 i q^{71} + 3 q^{72} + 8 i q^{74} + 5 i q^{75} + 4 q^{76} + 16 q^{77} + 2 i q^{78} -4 i q^{79} + q^{81} -8 i q^{82} + 12 q^{83} -4 q^{84} -4 q^{86} -12 i q^{88} -10 q^{89} -8 i q^{91} + 4 i q^{92} + 4 q^{93} -8 q^{94} + 5 i q^{96} + 16 i q^{97} -9 q^{98} -4 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} - 2q^{4} - 6q^{8} - 2q^{9} + O(q^{10})$$ $$2q + 2q^{2} - 2q^{4} - 6q^{8} - 2q^{9} + 4q^{13} - 2q^{16} + 2q^{17} - 2q^{18} - 8q^{19} + 8q^{21} + 10q^{25} + 4q^{26} + 10q^{32} - 8q^{33} + 2q^{34} + 2q^{36} - 8q^{38} + 8q^{42} - 8q^{43} - 16q^{47} - 18q^{49} + 10q^{50} - 8q^{51} - 4q^{52} + 12q^{53} - 24q^{59} + 14q^{64} - 8q^{66} + 24q^{67} - 2q^{68} + 8q^{69} + 6q^{72} + 8q^{76} + 32q^{77} + 2q^{81} + 24q^{83} - 8q^{84} - 8q^{86} - 20q^{89} + 8q^{93} - 16q^{94} - 18q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/51\mathbb{Z}\right)^\times$$.

 $$n$$ $$35$$ $$37$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
16.1
 − 1.00000i 1.00000i
1.00000 1.00000i −1.00000 0 1.00000i 4.00000i −3.00000 −1.00000 0
16.2 1.00000 1.00000i −1.00000 0 1.00000i 4.00000i −3.00000 −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
17.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 51.2.d.b 2
3.b odd 2 1 153.2.d.a 2
4.b odd 2 1 816.2.c.c 2
5.b even 2 1 1275.2.g.a 2
5.c odd 4 1 1275.2.d.b 2
5.c odd 4 1 1275.2.d.d 2
8.b even 2 1 3264.2.c.e 2
8.d odd 2 1 3264.2.c.d 2
12.b even 2 1 2448.2.c.j 2
17.b even 2 1 inner 51.2.d.b 2
17.c even 4 1 867.2.a.a 1
17.c even 4 1 867.2.a.b 1
17.d even 8 4 867.2.e.d 4
17.e odd 16 8 867.2.h.d 8
51.c odd 2 1 153.2.d.a 2
51.f odd 4 1 2601.2.a.i 1
51.f odd 4 1 2601.2.a.j 1
68.d odd 2 1 816.2.c.c 2
85.c even 2 1 1275.2.g.a 2
85.g odd 4 1 1275.2.d.b 2
85.g odd 4 1 1275.2.d.d 2
136.e odd 2 1 3264.2.c.d 2
136.h even 2 1 3264.2.c.e 2
204.h even 2 1 2448.2.c.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.2.d.b 2 1.a even 1 1 trivial
51.2.d.b 2 17.b even 2 1 inner
153.2.d.a 2 3.b odd 2 1
153.2.d.a 2 51.c odd 2 1
816.2.c.c 2 4.b odd 2 1
816.2.c.c 2 68.d odd 2 1
867.2.a.a 1 17.c even 4 1
867.2.a.b 1 17.c even 4 1
867.2.e.d 4 17.d even 8 4
867.2.h.d 8 17.e odd 16 8
1275.2.d.b 2 5.c odd 4 1
1275.2.d.b 2 85.g odd 4 1
1275.2.d.d 2 5.c odd 4 1
1275.2.d.d 2 85.g odd 4 1
1275.2.g.a 2 5.b even 2 1
1275.2.g.a 2 85.c even 2 1
2448.2.c.j 2 12.b even 2 1
2448.2.c.j 2 204.h even 2 1
2601.2.a.i 1 51.f odd 4 1
2601.2.a.j 1 51.f odd 4 1
3264.2.c.d 2 8.d odd 2 1
3264.2.c.d 2 136.e odd 2 1
3264.2.c.e 2 8.b even 2 1
3264.2.c.e 2 136.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} - 1$$ acting on $$S_{2}^{\mathrm{new}}(51, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$16 + T^{2}$$
$11$ $$16 + T^{2}$$
$13$ $$( -2 + T )^{2}$$
$17$ $$17 - 2 T + T^{2}$$
$19$ $$( 4 + T )^{2}$$
$23$ $$16 + T^{2}$$
$29$ $$T^{2}$$
$31$ $$16 + T^{2}$$
$37$ $$64 + T^{2}$$
$41$ $$64 + T^{2}$$
$43$ $$( 4 + T )^{2}$$
$47$ $$( 8 + T )^{2}$$
$53$ $$( -6 + T )^{2}$$
$59$ $$( 12 + T )^{2}$$
$61$ $$64 + T^{2}$$
$67$ $$( -12 + T )^{2}$$
$71$ $$144 + T^{2}$$
$73$ $$T^{2}$$
$79$ $$16 + T^{2}$$
$83$ $$( -12 + T )^{2}$$
$89$ $$( 10 + T )^{2}$$
$97$ $$256 + T^{2}$$