# Properties

 Label 675.2.b.f Level $675$ Weight $2$ Character orbit 675.b Analytic conductor $5.390$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$675 = 3^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 675.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.38990213644$$ 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: $$1$$ Twist minimal: no (minimal twist has level 27) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + 2 q^{4} + i q^{7} +O(q^{10})$$ $$q + 2 q^{4} + i q^{7} + 5 i q^{13} + 4 q^{16} + 7 q^{19} + 2 i q^{28} -4 q^{31} -11 i q^{37} + 8 i q^{43} + 6 q^{49} + 10 i q^{52} - q^{61} + 8 q^{64} -5 i q^{67} -7 i q^{73} + 14 q^{76} -17 q^{79} -5 q^{91} + 19 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{4} + O(q^{10})$$ $$2 q + 4 q^{4} + 8 q^{16} + 14 q^{19} - 8 q^{31} + 12 q^{49} - 2 q^{61} + 16 q^{64} + 28 q^{76} - 34 q^{79} - 10 q^{91} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/675\mathbb{Z}\right)^\times$$.

 $$n$$ $$326$$ $$352$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
649.1
 − 1.00000i 1.00000i
0 0 2.00000 0 0 1.00000i 0 0 0
649.2 0 0 2.00000 0 0 1.00000i 0 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
5.b even 2 1 inner
15.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 675.2.b.f 2
3.b odd 2 1 CM 675.2.b.f 2
5.b even 2 1 inner 675.2.b.f 2
5.c odd 4 1 27.2.a.a 1
5.c odd 4 1 675.2.a.e 1
15.d odd 2 1 inner 675.2.b.f 2
15.e even 4 1 27.2.a.a 1
15.e even 4 1 675.2.a.e 1
20.e even 4 1 432.2.a.e 1
35.f even 4 1 1323.2.a.i 1
40.i odd 4 1 1728.2.a.n 1
40.k even 4 1 1728.2.a.o 1
45.k odd 12 2 81.2.c.a 2
45.l even 12 2 81.2.c.a 2
55.e even 4 1 3267.2.a.f 1
60.l odd 4 1 432.2.a.e 1
65.h odd 4 1 4563.2.a.e 1
85.g odd 4 1 7803.2.a.k 1
95.g even 4 1 9747.2.a.f 1
105.k odd 4 1 1323.2.a.i 1
120.q odd 4 1 1728.2.a.o 1
120.w even 4 1 1728.2.a.n 1
135.q even 36 6 729.2.e.f 6
135.r odd 36 6 729.2.e.f 6
165.l odd 4 1 3267.2.a.f 1
180.v odd 12 2 1296.2.i.i 2
180.x even 12 2 1296.2.i.i 2
195.s even 4 1 4563.2.a.e 1
255.o even 4 1 7803.2.a.k 1
285.j odd 4 1 9747.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.2.a.a 1 5.c odd 4 1
27.2.a.a 1 15.e even 4 1
81.2.c.a 2 45.k odd 12 2
81.2.c.a 2 45.l even 12 2
432.2.a.e 1 20.e even 4 1
432.2.a.e 1 60.l odd 4 1
675.2.a.e 1 5.c odd 4 1
675.2.a.e 1 15.e even 4 1
675.2.b.f 2 1.a even 1 1 trivial
675.2.b.f 2 3.b odd 2 1 CM
675.2.b.f 2 5.b even 2 1 inner
675.2.b.f 2 15.d odd 2 1 inner
729.2.e.f 6 135.q even 36 6
729.2.e.f 6 135.r odd 36 6
1296.2.i.i 2 180.v odd 12 2
1296.2.i.i 2 180.x even 12 2
1323.2.a.i 1 35.f even 4 1
1323.2.a.i 1 105.k odd 4 1
1728.2.a.n 1 40.i odd 4 1
1728.2.a.n 1 120.w even 4 1
1728.2.a.o 1 40.k even 4 1
1728.2.a.o 1 120.q odd 4 1
3267.2.a.f 1 55.e even 4 1
3267.2.a.f 1 165.l odd 4 1
4563.2.a.e 1 65.h odd 4 1
4563.2.a.e 1 195.s even 4 1
7803.2.a.k 1 85.g odd 4 1
7803.2.a.k 1 255.o even 4 1
9747.2.a.f 1 95.g even 4 1
9747.2.a.f 1 285.j odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(675, [\chi])$$:

 $$T_{2}$$ $$T_{7}^{2} + 1$$ $$T_{11}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$T^{2}$$
$13$ $$25 + T^{2}$$
$17$ $$T^{2}$$
$19$ $$( -7 + T )^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$( 4 + T )^{2}$$
$37$ $$121 + T^{2}$$
$41$ $$T^{2}$$
$43$ $$64 + T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$( 1 + T )^{2}$$
$67$ $$25 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$49 + T^{2}$$
$79$ $$( 17 + T )^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$361 + T^{2}$$