# Properties

 Label 975.2.a.l Level $975$ Weight $2$ Character orbit 975.a Self dual yes Analytic conductor $7.785$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$975 = 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 975.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$7.78541419707$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta ) q^{2} - q^{3} + ( 1 + 2 \beta ) q^{4} + ( -1 - \beta ) q^{6} + 2 \beta q^{7} + ( 3 + \beta ) q^{8} + q^{9} +O(q^{10})$$ $$q + ( 1 + \beta ) q^{2} - q^{3} + ( 1 + 2 \beta ) q^{4} + ( -1 - \beta ) q^{6} + 2 \beta q^{7} + ( 3 + \beta ) q^{8} + q^{9} -2 q^{11} + ( -1 - 2 \beta ) q^{12} + q^{13} + ( 4 + 2 \beta ) q^{14} + 3 q^{16} + ( -2 + 4 \beta ) q^{17} + ( 1 + \beta ) q^{18} + 2 \beta q^{19} -2 \beta q^{21} + ( -2 - 2 \beta ) q^{22} + 4 q^{23} + ( -3 - \beta ) q^{24} + ( 1 + \beta ) q^{26} - q^{27} + ( 8 + 2 \beta ) q^{28} + 2 q^{29} + ( -4 - 2 \beta ) q^{31} + ( -3 + \beta ) q^{32} + 2 q^{33} + ( 6 + 2 \beta ) q^{34} + ( 1 + 2 \beta ) q^{36} + ( 2 - 4 \beta ) q^{37} + ( 4 + 2 \beta ) q^{38} - q^{39} + ( 8 + 2 \beta ) q^{41} + ( -4 - 2 \beta ) q^{42} + ( -4 - 4 \beta ) q^{43} + ( -2 - 4 \beta ) q^{44} + ( 4 + 4 \beta ) q^{46} + ( 6 - 4 \beta ) q^{47} -3 q^{48} + q^{49} + ( 2 - 4 \beta ) q^{51} + ( 1 + 2 \beta ) q^{52} + 2 q^{53} + ( -1 - \beta ) q^{54} + ( 4 + 6 \beta ) q^{56} -2 \beta q^{57} + ( 2 + 2 \beta ) q^{58} + ( 2 - 4 \beta ) q^{59} + ( 2 - 8 \beta ) q^{61} + ( -8 - 6 \beta ) q^{62} + 2 \beta q^{63} + ( -7 - 2 \beta ) q^{64} + ( 2 + 2 \beta ) q^{66} + ( -4 + 2 \beta ) q^{67} + 14 q^{68} -4 q^{69} + 2 q^{71} + ( 3 + \beta ) q^{72} + ( -6 - 4 \beta ) q^{73} + ( -6 - 2 \beta ) q^{74} + ( 8 + 2 \beta ) q^{76} -4 \beta q^{77} + ( -1 - \beta ) q^{78} + 8 \beta q^{79} + q^{81} + ( 12 + 10 \beta ) q^{82} + ( 2 + 4 \beta ) q^{83} + ( -8 - 2 \beta ) q^{84} + ( -12 - 8 \beta ) q^{86} -2 q^{87} + ( -6 - 2 \beta ) q^{88} + ( 12 - 2 \beta ) q^{89} + 2 \beta q^{91} + ( 4 + 8 \beta ) q^{92} + ( 4 + 2 \beta ) q^{93} + ( -2 + 2 \beta ) q^{94} + ( 3 - \beta ) q^{96} + ( 2 + 4 \beta ) q^{97} + ( 1 + \beta ) q^{98} -2 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{6} + 6q^{8} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{6} + 6q^{8} + 2q^{9} - 4q^{11} - 2q^{12} + 2q^{13} + 8q^{14} + 6q^{16} - 4q^{17} + 2q^{18} - 4q^{22} + 8q^{23} - 6q^{24} + 2q^{26} - 2q^{27} + 16q^{28} + 4q^{29} - 8q^{31} - 6q^{32} + 4q^{33} + 12q^{34} + 2q^{36} + 4q^{37} + 8q^{38} - 2q^{39} + 16q^{41} - 8q^{42} - 8q^{43} - 4q^{44} + 8q^{46} + 12q^{47} - 6q^{48} + 2q^{49} + 4q^{51} + 2q^{52} + 4q^{53} - 2q^{54} + 8q^{56} + 4q^{58} + 4q^{59} + 4q^{61} - 16q^{62} - 14q^{64} + 4q^{66} - 8q^{67} + 28q^{68} - 8q^{69} + 4q^{71} + 6q^{72} - 12q^{73} - 12q^{74} + 16q^{76} - 2q^{78} + 2q^{81} + 24q^{82} + 4q^{83} - 16q^{84} - 24q^{86} - 4q^{87} - 12q^{88} + 24q^{89} + 8q^{92} + 8q^{93} - 4q^{94} + 6q^{96} + 4q^{97} + 2q^{98} - 4q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.41421 1.41421
−0.414214 −1.00000 −1.82843 0 0.414214 −2.82843 1.58579 1.00000 0
1.2 2.41421 −1.00000 3.82843 0 −2.41421 2.82843 4.41421 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$5$$ $$1$$
$$13$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.a.l 2
3.b odd 2 1 2925.2.a.v 2
5.b even 2 1 39.2.a.b 2
5.c odd 4 2 975.2.c.h 4
15.d odd 2 1 117.2.a.c 2
15.e even 4 2 2925.2.c.u 4
20.d odd 2 1 624.2.a.k 2
35.c odd 2 1 1911.2.a.h 2
40.e odd 2 1 2496.2.a.bi 2
40.f even 2 1 2496.2.a.bf 2
45.h odd 6 2 1053.2.e.e 4
45.j even 6 2 1053.2.e.m 4
55.d odd 2 1 4719.2.a.p 2
60.h even 2 1 1872.2.a.w 2
65.d even 2 1 507.2.a.h 2
65.g odd 4 2 507.2.b.e 4
65.l even 6 2 507.2.e.d 4
65.n even 6 2 507.2.e.h 4
65.s odd 12 4 507.2.j.f 8
105.g even 2 1 5733.2.a.u 2
120.i odd 2 1 7488.2.a.cl 2
120.m even 2 1 7488.2.a.co 2
195.e odd 2 1 1521.2.a.f 2
195.n even 4 2 1521.2.b.j 4
260.g odd 2 1 8112.2.a.bm 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.a.b 2 5.b even 2 1
117.2.a.c 2 15.d odd 2 1
507.2.a.h 2 65.d even 2 1
507.2.b.e 4 65.g odd 4 2
507.2.e.d 4 65.l even 6 2
507.2.e.h 4 65.n even 6 2
507.2.j.f 8 65.s odd 12 4
624.2.a.k 2 20.d odd 2 1
975.2.a.l 2 1.a even 1 1 trivial
975.2.c.h 4 5.c odd 4 2
1053.2.e.e 4 45.h odd 6 2
1053.2.e.m 4 45.j even 6 2
1521.2.a.f 2 195.e odd 2 1
1521.2.b.j 4 195.n even 4 2
1872.2.a.w 2 60.h even 2 1
1911.2.a.h 2 35.c odd 2 1
2496.2.a.bf 2 40.f even 2 1
2496.2.a.bi 2 40.e odd 2 1
2925.2.a.v 2 3.b odd 2 1
2925.2.c.u 4 15.e even 4 2
4719.2.a.p 2 55.d odd 2 1
5733.2.a.u 2 105.g even 2 1
7488.2.a.cl 2 120.i odd 2 1
7488.2.a.co 2 120.m even 2 1
8112.2.a.bm 2 260.g odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(975))$$:

 $$T_{2}^{2} - 2 T_{2} - 1$$ $$T_{7}^{2} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 - 2 T + T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$T^{2}$$
$7$ $$-8 + T^{2}$$
$11$ $$( 2 + T )^{2}$$
$13$ $$( -1 + T )^{2}$$
$17$ $$-28 + 4 T + T^{2}$$
$19$ $$-8 + T^{2}$$
$23$ $$( -4 + T )^{2}$$
$29$ $$( -2 + T )^{2}$$
$31$ $$8 + 8 T + T^{2}$$
$37$ $$-28 - 4 T + T^{2}$$
$41$ $$56 - 16 T + T^{2}$$
$43$ $$-16 + 8 T + T^{2}$$
$47$ $$4 - 12 T + T^{2}$$
$53$ $$( -2 + T )^{2}$$
$59$ $$-28 - 4 T + T^{2}$$
$61$ $$-124 - 4 T + T^{2}$$
$67$ $$8 + 8 T + T^{2}$$
$71$ $$( -2 + T )^{2}$$
$73$ $$4 + 12 T + T^{2}$$
$79$ $$-128 + T^{2}$$
$83$ $$-28 - 4 T + T^{2}$$
$89$ $$136 - 24 T + T^{2}$$
$97$ $$-28 - 4 T + T^{2}$$