# Properties

 Label 975.2.c.e Level $975$ Weight $2$ Character orbit 975.c Analytic conductor $7.785$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$975 = 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 975.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.78541419707$$ 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 195) 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} + i q^{3} + q^{4} - q^{6} + 3 i q^{8} - q^{9} +O(q^{10})$$ $$q + i q^{2} + i q^{3} + q^{4} - q^{6} + 3 i q^{8} - q^{9} + 4 q^{11} + i q^{12} + i q^{13} - q^{16} -2 i q^{17} -i q^{18} + 4 q^{19} + 4 i q^{22} + 8 i q^{23} -3 q^{24} - q^{26} -i q^{27} + 2 q^{29} -8 q^{31} + 5 i q^{32} + 4 i q^{33} + 2 q^{34} - q^{36} -6 i q^{37} + 4 i q^{38} - q^{39} -6 q^{41} -4 i q^{43} + 4 q^{44} -8 q^{46} + 8 i q^{47} -i q^{48} + 7 q^{49} + 2 q^{51} + i q^{52} + 6 i q^{53} + q^{54} + 4 i q^{57} + 2 i q^{58} + 12 q^{59} -2 q^{61} -8 i q^{62} -7 q^{64} -4 q^{66} + 4 i q^{67} -2 i q^{68} -8 q^{69} -3 i q^{72} -6 i q^{73} + 6 q^{74} + 4 q^{76} -i q^{78} -16 q^{79} + q^{81} -6 i q^{82} -4 i q^{83} + 4 q^{86} + 2 i q^{87} + 12 i q^{88} -10 q^{89} + 8 i q^{92} -8 i q^{93} -8 q^{94} -5 q^{96} -18 i q^{97} + 7 i q^{98} -4 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{4} - 2q^{6} - 2q^{9} + O(q^{10})$$ $$2q + 2q^{4} - 2q^{6} - 2q^{9} + 8q^{11} - 2q^{16} + 8q^{19} - 6q^{24} - 2q^{26} + 4q^{29} - 16q^{31} + 4q^{34} - 2q^{36} - 2q^{39} - 12q^{41} + 8q^{44} - 16q^{46} + 14q^{49} + 4q^{51} + 2q^{54} + 24q^{59} - 4q^{61} - 14q^{64} - 8q^{66} - 16q^{69} + 12q^{74} + 8q^{76} - 32q^{79} + 2q^{81} + 8q^{86} - 20q^{89} - 16q^{94} - 10q^{96} - 8q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/975\mathbb{Z}\right)^\times$$.

 $$n$$ $$301$$ $$326$$ $$352$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
274.1
 − 1.00000i 1.00000i
1.00000i 1.00000i 1.00000 0 −1.00000 0 3.00000i −1.00000 0
274.2 1.00000i 1.00000i 1.00000 0 −1.00000 0 3.00000i −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 975.2.c.e 2
3.b odd 2 1 2925.2.c.f 2
5.b even 2 1 inner 975.2.c.e 2
5.c odd 4 1 195.2.a.a 1
5.c odd 4 1 975.2.a.i 1
15.d odd 2 1 2925.2.c.f 2
15.e even 4 1 585.2.a.g 1
15.e even 4 1 2925.2.a.d 1
20.e even 4 1 3120.2.a.k 1
35.f even 4 1 9555.2.a.b 1
60.l odd 4 1 9360.2.a.o 1
65.h odd 4 1 2535.2.a.k 1
195.s even 4 1 7605.2.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.a.a 1 5.c odd 4 1
585.2.a.g 1 15.e even 4 1
975.2.a.i 1 5.c odd 4 1
975.2.c.e 2 1.a even 1 1 trivial
975.2.c.e 2 5.b even 2 1 inner
2535.2.a.k 1 65.h odd 4 1
2925.2.a.d 1 15.e even 4 1
2925.2.c.f 2 3.b odd 2 1
2925.2.c.f 2 15.d odd 2 1
3120.2.a.k 1 20.e even 4 1
7605.2.a.h 1 195.s even 4 1
9360.2.a.o 1 60.l odd 4 1
9555.2.a.b 1 35.f even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(975, [\chi])$$:

 $$T_{2}^{2} + 1$$ $$T_{7}$$ $$T_{11} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$( -4 + T )^{2}$$
$13$ $$1 + T^{2}$$
$17$ $$4 + T^{2}$$
$19$ $$( -4 + T )^{2}$$
$23$ $$64 + T^{2}$$
$29$ $$( -2 + T )^{2}$$
$31$ $$( 8 + T )^{2}$$
$37$ $$36 + T^{2}$$
$41$ $$( 6 + T )^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$64 + T^{2}$$
$53$ $$36 + T^{2}$$
$59$ $$( -12 + T )^{2}$$
$61$ $$( 2 + T )^{2}$$
$67$ $$16 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$36 + T^{2}$$
$79$ $$( 16 + T )^{2}$$
$83$ $$16 + T^{2}$$
$89$ $$( 10 + T )^{2}$$
$97$ $$324 + T^{2}$$