# Properties

 Label 2475.2.c.o Level $2475$ Weight $2$ Character orbit 2475.c Analytic conductor $19.763$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2475 = 3^{2} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2475.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$19.7629745003$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 825) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{2} + ( -1 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{4} + ( -\zeta_{8} + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{7} + ( -\zeta_{8} - 3 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{8} +O(q^{10})$$ $$q + ( \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{2} + ( -1 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{4} + ( -\zeta_{8} + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{7} + ( -\zeta_{8} - 3 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{8} + q^{11} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{13} + q^{14} + 3 q^{16} + ( \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{17} + ( -5 - \zeta_{8} + \zeta_{8}^{3} ) q^{19} + ( \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{22} -\zeta_{8}^{2} q^{23} + ( 4 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{26} + ( -\zeta_{8} + 3 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{28} + ( 4 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{29} + ( -6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{31} + ( \zeta_{8} - 3 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{32} + ( -3 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{34} + ( -2 \zeta_{8} + 3 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{37} + ( -6 \zeta_{8} - 7 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{38} + ( 1 + 7 \zeta_{8} - 7 \zeta_{8}^{3} ) q^{41} + ( -4 \zeta_{8} - 6 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{43} + ( -1 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{44} + ( 1 + \zeta_{8} - \zeta_{8}^{3} ) q^{46} + ( -6 \zeta_{8} + \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{47} + ( 4 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{49} + ( 2 \zeta_{8} + 8 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{52} + ( -4 \zeta_{8} - 2 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{53} + ( 1 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{56} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{58} + 11 q^{59} + ( 6 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{61} + ( -6 \zeta_{8} - 12 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{62} + ( 7 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{64} + ( -4 \zeta_{8} + 6 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{67} + ( -3 \zeta_{8} - 5 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{68} + ( -5 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{71} + ( -2 \zeta_{8} - 6 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{73} + ( 1 - \zeta_{8} + \zeta_{8}^{3} ) q^{74} + ( 9 + 11 \zeta_{8} - 11 \zeta_{8}^{3} ) q^{76} + ( -\zeta_{8} + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{77} + ( -9 - 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{79} + ( 8 \zeta_{8} + 15 \zeta_{8}^{2} + 8 \zeta_{8}^{3} ) q^{82} + ( 6 \zeta_{8} - 4 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{83} + ( 14 + 10 \zeta_{8} - 10 \zeta_{8}^{3} ) q^{86} + ( -\zeta_{8} - 3 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{88} + ( -2 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{89} + ( -4 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{91} + ( 2 \zeta_{8} + \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{92} + ( 11 + 5 \zeta_{8} - 5 \zeta_{8}^{3} ) q^{94} + ( -2 \zeta_{8} - 3 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{97} + ( 6 \zeta_{8} + 8 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{4} + O(q^{10})$$ $$4q - 4q^{4} + 4q^{11} + 4q^{14} + 12q^{16} - 20q^{19} + 16q^{26} + 16q^{29} - 12q^{34} + 4q^{41} - 4q^{44} + 4q^{46} + 16q^{49} + 4q^{56} + 44q^{59} + 24q^{61} + 28q^{64} - 20q^{71} + 4q^{74} + 36q^{76} - 36q^{79} + 56q^{86} - 8q^{89} - 16q^{91} + 44q^{94} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times$$.

 $$n$$ $$551$$ $$2026$$ $$2377$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
 0.707107 − 0.707107i −0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i
2.41421i 0 −3.82843 0 0 0.414214i 4.41421i 0 0
199.2 0.414214i 0 1.82843 0 0 2.41421i 1.58579i 0 0
199.3 0.414214i 0 1.82843 0 0 2.41421i 1.58579i 0 0
199.4 2.41421i 0 −3.82843 0 0 0.414214i 4.41421i 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
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 2475.2.c.o 4
3.b odd 2 1 825.2.c.d 4
5.b even 2 1 inner 2475.2.c.o 4
5.c odd 4 1 2475.2.a.l 2
5.c odd 4 1 2475.2.a.w 2
15.d odd 2 1 825.2.c.d 4
15.e even 4 1 825.2.a.d 2
15.e even 4 1 825.2.a.f yes 2
165.l odd 4 1 9075.2.a.w 2
165.l odd 4 1 9075.2.a.ca 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.a.d 2 15.e even 4 1
825.2.a.f yes 2 15.e even 4 1
825.2.c.d 4 3.b odd 2 1
825.2.c.d 4 15.d odd 2 1
2475.2.a.l 2 5.c odd 4 1
2475.2.a.w 2 5.c odd 4 1
2475.2.c.o 4 1.a even 1 1 trivial
2475.2.c.o 4 5.b even 2 1 inner
9075.2.a.w 2 165.l odd 4 1
9075.2.a.ca 2 165.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2475, [\chi])$$:

 $$T_{2}^{4} + 6 T_{2}^{2} + 1$$ $$T_{7}^{4} + 6 T_{7}^{2} + 1$$ $$T_{29}^{2} - 8 T_{29} + 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 6 T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$1 + 6 T^{2} + T^{4}$$
$11$ $$( -1 + T )^{4}$$
$13$ $$( 8 + T^{2} )^{2}$$
$17$ $$1 + 6 T^{2} + T^{4}$$
$19$ $$( 23 + 10 T + T^{2} )^{2}$$
$23$ $$( 1 + T^{2} )^{2}$$
$29$ $$( 8 - 8 T + T^{2} )^{2}$$
$31$ $$( -72 + T^{2} )^{2}$$
$37$ $$1 + 34 T^{2} + T^{4}$$
$41$ $$( -97 - 2 T + T^{2} )^{2}$$
$43$ $$16 + 136 T^{2} + T^{4}$$
$47$ $$5041 + 146 T^{2} + T^{4}$$
$53$ $$784 + 72 T^{2} + T^{4}$$
$59$ $$( -11 + T )^{4}$$
$61$ $$( 28 - 12 T + T^{2} )^{2}$$
$67$ $$16 + 136 T^{2} + T^{4}$$
$71$ $$( 17 + 10 T + T^{2} )^{2}$$
$73$ $$784 + 88 T^{2} + T^{4}$$
$79$ $$( 63 + 18 T + T^{2} )^{2}$$
$83$ $$3136 + 176 T^{2} + T^{4}$$
$89$ $$( -28 + 4 T + T^{2} )^{2}$$
$97$ $$1 + 34 T^{2} + T^{4}$$