# Properties

 Label 2275.2.a.j Level $2275$ Weight $2$ Character orbit 2275.a Self dual yes Analytic conductor $18.166$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2275 = 5^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2275.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$18.1659664598$$ 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 91) 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} -\beta q^{3} -2 q^{6} - q^{7} -2 \beta q^{8} - q^{9} +O(q^{10})$$ $$q + \beta q^{2} -\beta q^{3} -2 q^{6} - q^{7} -2 \beta q^{8} - q^{9} + 3 \beta q^{11} + q^{13} -\beta q^{14} -4 q^{16} -\beta q^{17} -\beta q^{18} + ( -3 - 3 \beta ) q^{19} + \beta q^{21} + 6 q^{22} + ( 3 + 2 \beta ) q^{23} + 4 q^{24} + \beta q^{26} + 4 \beta q^{27} + ( 3 - 2 \beta ) q^{29} + ( -1 + 3 \beta ) q^{31} -6 q^{33} -2 q^{34} + ( 2 - 3 \beta ) q^{37} + ( -6 - 3 \beta ) q^{38} -\beta q^{39} + ( 6 + 2 \beta ) q^{41} + 2 q^{42} + 5 q^{43} + ( 4 + 3 \beta ) q^{46} + ( -3 + \beta ) q^{47} + 4 \beta q^{48} + q^{49} + 2 q^{51} + ( 3 - 2 \beta ) q^{53} + 8 q^{54} + 2 \beta q^{56} + ( 6 + 3 \beta ) q^{57} + ( -4 + 3 \beta ) q^{58} + ( 6 - 4 \beta ) q^{59} + 6 q^{61} + ( 6 - \beta ) q^{62} + q^{63} + 8 q^{64} -6 \beta q^{66} + ( 6 + 6 \beta ) q^{67} + ( -4 - 3 \beta ) q^{69} + ( -6 - 5 \beta ) q^{71} + 2 \beta q^{72} + ( 5 + 3 \beta ) q^{73} + ( -6 + 2 \beta ) q^{74} -3 \beta q^{77} -2 q^{78} + ( 7 + 6 \beta ) q^{79} -5 q^{81} + ( 4 + 6 \beta ) q^{82} + ( -9 - 3 \beta ) q^{83} + 5 \beta q^{86} + ( 4 - 3 \beta ) q^{87} -12 q^{88} + ( 3 - \beta ) q^{89} - q^{91} + ( -6 + \beta ) q^{93} + ( 2 - 3 \beta ) q^{94} + ( 1 - 9 \beta ) q^{97} + \beta q^{98} -3 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{6} - 2q^{7} - 2q^{9} + O(q^{10})$$ $$2q - 4q^{6} - 2q^{7} - 2q^{9} + 2q^{13} - 8q^{16} - 6q^{19} + 12q^{22} + 6q^{23} + 8q^{24} + 6q^{29} - 2q^{31} - 12q^{33} - 4q^{34} + 4q^{37} - 12q^{38} + 12q^{41} + 4q^{42} + 10q^{43} + 8q^{46} - 6q^{47} + 2q^{49} + 4q^{51} + 6q^{53} + 16q^{54} + 12q^{57} - 8q^{58} + 12q^{59} + 12q^{61} + 12q^{62} + 2q^{63} + 16q^{64} + 12q^{67} - 8q^{69} - 12q^{71} + 10q^{73} - 12q^{74} - 4q^{78} + 14q^{79} - 10q^{81} + 8q^{82} - 18q^{83} + 8q^{87} - 24q^{88} + 6q^{89} - 2q^{91} - 12q^{93} + 4q^{94} + 2q^{97} + O(q^{100})$$

## 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
−1.41421 1.41421 0 0 −2.00000 −1.00000 2.82843 −1.00000 0
1.2 1.41421 −1.41421 0 0 −2.00000 −1.00000 −2.82843 −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
$$5$$ $$1$$
$$7$$ $$1$$
$$13$$ $$-1$$

## 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 2275.2.a.j 2
5.b even 2 1 91.2.a.c 2
15.d odd 2 1 819.2.a.h 2
20.d odd 2 1 1456.2.a.q 2
35.c odd 2 1 637.2.a.g 2
35.i odd 6 2 637.2.e.g 4
35.j even 6 2 637.2.e.f 4
40.e odd 2 1 5824.2.a.bk 2
40.f even 2 1 5824.2.a.bl 2
65.d even 2 1 1183.2.a.d 2
65.g odd 4 2 1183.2.c.d 4
105.g even 2 1 5733.2.a.s 2
455.h odd 2 1 8281.2.a.v 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.a.c 2 5.b even 2 1
637.2.a.g 2 35.c odd 2 1
637.2.e.f 4 35.j even 6 2
637.2.e.g 4 35.i odd 6 2
819.2.a.h 2 15.d odd 2 1
1183.2.a.d 2 65.d even 2 1
1183.2.c.d 4 65.g odd 4 2
1456.2.a.q 2 20.d odd 2 1
2275.2.a.j 2 1.a even 1 1 trivial
5733.2.a.s 2 105.g even 2 1
5824.2.a.bk 2 40.e odd 2 1
5824.2.a.bl 2 40.f even 2 1
8281.2.a.v 2 455.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2275))$$:

 $$T_{2}^{2} - 2$$ $$T_{3}^{2} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-2 + T^{2}$$
$3$ $$-2 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$-18 + T^{2}$$
$13$ $$( -1 + T )^{2}$$
$17$ $$-2 + T^{2}$$
$19$ $$-9 + 6 T + T^{2}$$
$23$ $$1 - 6 T + T^{2}$$
$29$ $$1 - 6 T + T^{2}$$
$31$ $$-17 + 2 T + T^{2}$$
$37$ $$-14 - 4 T + T^{2}$$
$41$ $$28 - 12 T + T^{2}$$
$43$ $$( -5 + T )^{2}$$
$47$ $$7 + 6 T + T^{2}$$
$53$ $$1 - 6 T + T^{2}$$
$59$ $$4 - 12 T + T^{2}$$
$61$ $$( -6 + T )^{2}$$
$67$ $$-36 - 12 T + T^{2}$$
$71$ $$-14 + 12 T + T^{2}$$
$73$ $$7 - 10 T + T^{2}$$
$79$ $$-23 - 14 T + T^{2}$$
$83$ $$63 + 18 T + T^{2}$$
$89$ $$7 - 6 T + T^{2}$$
$97$ $$-161 - 2 T + T^{2}$$