# Properties

 Label 153.2.d.a Level $153$ Weight $2$ Character orbit 153.d Analytic conductor $1.222$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$153 = 3^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 153.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.22171115093$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 51) 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.

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{4} + 4 i q^{7} + 3 q^{8} +O(q^{10})$$ $$q - q^{2} - q^{4} + 4 i q^{7} + 3 q^{8} + 4 i q^{11} + 2 q^{13} -4 i q^{14} - q^{16} + ( -1 + 4 i ) q^{17} -4 q^{19} -4 i q^{22} -4 i q^{23} + 5 q^{25} -2 q^{26} -4 i q^{28} + 4 i q^{31} -5 q^{32} + ( 1 - 4 i ) q^{34} -8 i q^{37} + 4 q^{38} -8 i q^{41} -4 q^{43} -4 i q^{44} + 4 i q^{46} + 8 q^{47} -9 q^{49} -5 q^{50} -2 q^{52} -6 q^{53} + 12 i q^{56} + 12 q^{59} + 8 i q^{61} -4 i q^{62} + 7 q^{64} + 12 q^{67} + ( 1 - 4 i ) q^{68} + 12 i q^{71} + 8 i q^{74} + 4 q^{76} -16 q^{77} + 4 i q^{79} + 8 i q^{82} -12 q^{83} + 4 q^{86} + 12 i q^{88} + 10 q^{89} + 8 i q^{91} + 4 i q^{92} -8 q^{94} -16 i q^{97} + 9 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 2q^{4} + 6q^{8} + O(q^{10})$$ $$2q - 2q^{2} - 2q^{4} + 6q^{8} + 4q^{13} - 2q^{16} - 2q^{17} - 8q^{19} + 10q^{25} - 4q^{26} - 10q^{32} + 2q^{34} + 8q^{38} - 8q^{43} + 16q^{47} - 18q^{49} - 10q^{50} - 4q^{52} - 12q^{53} + 24q^{59} + 14q^{64} + 24q^{67} + 2q^{68} + 8q^{76} - 32q^{77} - 24q^{83} + 8q^{86} + 20q^{89} - 16q^{94} + 18q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/153\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$137$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
118.1
 − 1.00000i 1.00000i
−1.00000 0 −1.00000 0 0 4.00000i 3.00000 0 0
118.2 −1.00000 0 −1.00000 0 0 4.00000i 3.00000 0 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 153.2.d.a 2
3.b odd 2 1 51.2.d.b 2
4.b odd 2 1 2448.2.c.j 2
12.b even 2 1 816.2.c.c 2
15.d odd 2 1 1275.2.g.a 2
15.e even 4 1 1275.2.d.b 2
15.e even 4 1 1275.2.d.d 2
17.b even 2 1 inner 153.2.d.a 2
17.c even 4 1 2601.2.a.i 1
17.c even 4 1 2601.2.a.j 1
24.f even 2 1 3264.2.c.d 2
24.h odd 2 1 3264.2.c.e 2
51.c odd 2 1 51.2.d.b 2
51.f odd 4 1 867.2.a.a 1
51.f odd 4 1 867.2.a.b 1
51.g odd 8 4 867.2.e.d 4
51.i even 16 8 867.2.h.d 8
68.d odd 2 1 2448.2.c.j 2
204.h even 2 1 816.2.c.c 2
255.h odd 2 1 1275.2.g.a 2
255.o even 4 1 1275.2.d.b 2
255.o even 4 1 1275.2.d.d 2
408.b odd 2 1 3264.2.c.e 2
408.h even 2 1 3264.2.c.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.2.d.b 2 3.b odd 2 1
51.2.d.b 2 51.c odd 2 1
153.2.d.a 2 1.a even 1 1 trivial
153.2.d.a 2 17.b even 2 1 inner
816.2.c.c 2 12.b even 2 1
816.2.c.c 2 204.h even 2 1
867.2.a.a 1 51.f odd 4 1
867.2.a.b 1 51.f odd 4 1
867.2.e.d 4 51.g odd 8 4
867.2.h.d 8 51.i even 16 8
1275.2.d.b 2 15.e even 4 1
1275.2.d.b 2 255.o even 4 1
1275.2.d.d 2 15.e even 4 1
1275.2.d.d 2 255.o even 4 1
1275.2.g.a 2 15.d odd 2 1
1275.2.g.a 2 255.h odd 2 1
2448.2.c.j 2 4.b odd 2 1
2448.2.c.j 2 68.d odd 2 1
2601.2.a.i 1 17.c even 4 1
2601.2.a.j 1 17.c even 4 1
3264.2.c.d 2 24.f even 2 1
3264.2.c.d 2 408.h even 2 1
3264.2.c.e 2 24.h odd 2 1
3264.2.c.e 2 408.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(153, [\chi])$$.

## Hecke characteristic polynomials

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