# Properties

 Label 1275.2.g.a Level $1275$ Weight $2$ Character orbit 1275.g Analytic conductor $10.181$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1275 = 3 \cdot 5^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1275.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.1809262577$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 51) 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + i q^{3} - q^{4} -i q^{6} -4 i q^{7} + 3 q^{8} - q^{9} +O(q^{10})$$ $$q - q^{2} + i q^{3} - q^{4} -i q^{6} -4 i q^{7} + 3 q^{8} - q^{9} -4 i q^{11} -i q^{12} -2 q^{13} + 4 i q^{14} - q^{16} + ( -1 + 4 i ) q^{17} + q^{18} -4 q^{19} + 4 q^{21} + 4 i q^{22} -4 i q^{23} + 3 i q^{24} + 2 q^{26} -i q^{27} + 4 i q^{28} + 4 i q^{31} -5 q^{32} + 4 q^{33} + ( 1 - 4 i ) q^{34} + q^{36} + 8 i q^{37} + 4 q^{38} -2 i q^{39} + 8 i q^{41} -4 q^{42} + 4 q^{43} + 4 i q^{44} + 4 i q^{46} + 8 q^{47} -i q^{48} -9 q^{49} + ( -4 - i ) q^{51} + 2 q^{52} -6 q^{53} + i q^{54} -12 i q^{56} -4 i q^{57} -12 q^{59} + 8 i q^{61} -4 i q^{62} + 4 i q^{63} + 7 q^{64} -4 q^{66} -12 q^{67} + ( 1 - 4 i ) q^{68} + 4 q^{69} -12 i q^{71} -3 q^{72} -8 i q^{74} + 4 q^{76} -16 q^{77} + 2 i q^{78} + 4 i q^{79} + q^{81} -8 i q^{82} -12 q^{83} -4 q^{84} -4 q^{86} -12 i q^{88} -10 q^{89} + 8 i q^{91} + 4 i q^{92} -4 q^{93} -8 q^{94} -5 i q^{96} + 16 i q^{97} + 9 q^{98} + 4 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 2q^{4} + 6q^{8} - 2q^{9} + O(q^{10})$$ $$2q - 2q^{2} - 2q^{4} + 6q^{8} - 2q^{9} - 4q^{13} - 2q^{16} - 2q^{17} + 2q^{18} - 8q^{19} + 8q^{21} + 4q^{26} - 10q^{32} + 8q^{33} + 2q^{34} + 2q^{36} + 8q^{38} - 8q^{42} + 8q^{43} + 16q^{47} - 18q^{49} - 8q^{51} + 4q^{52} - 12q^{53} - 24q^{59} + 14q^{64} - 8q^{66} - 24q^{67} + 2q^{68} + 8q^{69} - 6q^{72} + 8q^{76} - 32q^{77} + 2q^{81} - 24q^{83} - 8q^{84} - 8q^{86} - 20q^{89} - 8q^{93} - 16q^{94} + 18q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1275\mathbb{Z}\right)^\times$$.

 $$n$$ $$52$$ $$751$$ $$851$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
526.1
 − 1.00000i 1.00000i
−1.00000 1.00000i −1.00000 0 1.00000i 4.00000i 3.00000 −1.00000 0
526.2 −1.00000 1.00000i −1.00000 0 1.00000i 4.00000i 3.00000 −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
17.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1275.2.g.a 2
5.b even 2 1 51.2.d.b 2
5.c odd 4 1 1275.2.d.b 2
5.c odd 4 1 1275.2.d.d 2
15.d odd 2 1 153.2.d.a 2
17.b even 2 1 inner 1275.2.g.a 2
20.d odd 2 1 816.2.c.c 2
40.e odd 2 1 3264.2.c.d 2
40.f even 2 1 3264.2.c.e 2
60.h even 2 1 2448.2.c.j 2
85.c even 2 1 51.2.d.b 2
85.g odd 4 1 1275.2.d.b 2
85.g odd 4 1 1275.2.d.d 2
85.j even 4 1 867.2.a.a 1
85.j even 4 1 867.2.a.b 1
85.m even 8 4 867.2.e.d 4
85.p odd 16 8 867.2.h.d 8
255.h odd 2 1 153.2.d.a 2
255.i odd 4 1 2601.2.a.i 1
255.i odd 4 1 2601.2.a.j 1
340.d odd 2 1 816.2.c.c 2
680.h even 2 1 3264.2.c.e 2
680.k odd 2 1 3264.2.c.d 2
1020.b even 2 1 2448.2.c.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.2.d.b 2 5.b even 2 1
51.2.d.b 2 85.c even 2 1
153.2.d.a 2 15.d odd 2 1
153.2.d.a 2 255.h odd 2 1
816.2.c.c 2 20.d odd 2 1
816.2.c.c 2 340.d odd 2 1
867.2.a.a 1 85.j even 4 1
867.2.a.b 1 85.j even 4 1
867.2.e.d 4 85.m even 8 4
867.2.h.d 8 85.p odd 16 8
1275.2.d.b 2 5.c odd 4 1
1275.2.d.b 2 85.g odd 4 1
1275.2.d.d 2 5.c odd 4 1
1275.2.d.d 2 85.g odd 4 1
1275.2.g.a 2 1.a even 1 1 trivial
1275.2.g.a 2 17.b even 2 1 inner
2448.2.c.j 2 60.h even 2 1
2448.2.c.j 2 1020.b even 2 1
2601.2.a.i 1 255.i odd 4 1
2601.2.a.j 1 255.i odd 4 1
3264.2.c.d 2 40.e odd 2 1
3264.2.c.d 2 680.k odd 2 1
3264.2.c.e 2 40.f even 2 1
3264.2.c.e 2 680.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(1275, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$16 + T^{2}$$
$11$ $$16 + T^{2}$$
$13$ $$( 2 + T )^{2}$$
$17$ $$17 + 2 T + T^{2}$$
$19$ $$( 4 + T )^{2}$$
$23$ $$16 + T^{2}$$
$29$ $$T^{2}$$
$31$ $$16 + T^{2}$$
$37$ $$64 + T^{2}$$
$41$ $$64 + T^{2}$$
$43$ $$( -4 + T )^{2}$$
$47$ $$( -8 + T )^{2}$$
$53$ $$( 6 + T )^{2}$$
$59$ $$( 12 + T )^{2}$$
$61$ $$64 + T^{2}$$
$67$ $$( 12 + T )^{2}$$
$71$ $$144 + T^{2}$$
$73$ $$T^{2}$$
$79$ $$16 + T^{2}$$
$83$ $$( 12 + T )^{2}$$
$89$ $$( 10 + T )^{2}$$
$97$ $$256 + T^{2}$$