# Properties

 Label 1275.2.b.d Level $1275$ Weight $2$ Character orbit 1275.b Analytic conductor $10.181$ Analytic rank $0$ Dimension $4$ 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.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.1809262577$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{17})$$ Defining polynomial: $$x^{4} + 9 x^{2} + 16$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + \beta_{2} q^{3} + ( -3 + \beta_{3} ) q^{4} + ( 1 - \beta_{3} ) q^{6} + ( -\beta_{1} + 4 \beta_{2} ) q^{8} - q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + \beta_{2} q^{3} + ( -3 + \beta_{3} ) q^{4} + ( 1 - \beta_{3} ) q^{6} + ( -\beta_{1} + 4 \beta_{2} ) q^{8} - q^{9} -\beta_{3} q^{11} + ( \beta_{1} - 2 \beta_{2} ) q^{12} + ( -\beta_{1} - 3 \beta_{2} ) q^{13} + ( 3 - 3 \beta_{3} ) q^{16} + \beta_{2} q^{17} -\beta_{1} q^{18} -3 \beta_{3} q^{19} -4 \beta_{2} q^{22} + ( \beta_{1} + 5 \beta_{2} ) q^{23} + ( -5 + \beta_{3} ) q^{24} + ( 2 + 2 \beta_{3} ) q^{26} -\beta_{2} q^{27} + ( 2 - 4 \beta_{3} ) q^{29} -2 \beta_{3} q^{31} + ( \beta_{1} - 4 \beta_{2} ) q^{32} + ( -\beta_{1} - \beta_{2} ) q^{33} + ( 1 - \beta_{3} ) q^{34} + ( 3 - \beta_{3} ) q^{36} + 2 \beta_{1} q^{37} -12 \beta_{2} q^{38} + ( 2 + \beta_{3} ) q^{39} + ( -2 + \beta_{3} ) q^{41} + ( 3 \beta_{1} + 3 \beta_{2} ) q^{43} + ( -4 + 2 \beta_{3} ) q^{44} -4 \beta_{3} q^{46} + ( 2 \beta_{1} - 6 \beta_{2} ) q^{47} -3 \beta_{1} q^{48} + 7 q^{49} - q^{51} + 2 \beta_{2} q^{52} + ( 4 \beta_{1} - 2 \beta_{2} ) q^{53} + ( -1 + \beta_{3} ) q^{54} + ( -3 \beta_{1} - 3 \beta_{2} ) q^{57} + ( 2 \beta_{1} - 16 \beta_{2} ) q^{58} + ( -4 + 2 \beta_{3} ) q^{59} + ( 6 - 2 \beta_{3} ) q^{61} -8 \beta_{2} q^{62} + ( -3 - \beta_{3} ) q^{64} + 4 q^{66} + 4 \beta_{2} q^{67} + ( \beta_{1} - 2 \beta_{2} ) q^{68} + ( -4 - \beta_{3} ) q^{69} + 4 \beta_{3} q^{71} + ( \beta_{1} - 4 \beta_{2} ) q^{72} + ( -4 \beta_{1} + 2 \beta_{2} ) q^{73} + ( -10 + 2 \beta_{3} ) q^{74} + ( -12 + 6 \beta_{3} ) q^{76} + ( 2 \beta_{1} + 4 \beta_{2} ) q^{78} -6 \beta_{3} q^{79} + q^{81} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{82} + ( 2 \beta_{1} + 6 \beta_{2} ) q^{83} -12 q^{86} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{87} -4 \beta_{1} q^{88} + ( -2 - 2 \beta_{3} ) q^{89} + ( 2 \beta_{1} - 6 \beta_{2} ) q^{92} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{93} + ( -16 + 8 \beta_{3} ) q^{94} + ( 5 - \beta_{3} ) q^{96} + ( -2 \beta_{1} - 8 \beta_{2} ) q^{97} + 7 \beta_{1} q^{98} + \beta_{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 10q^{4} + 2q^{6} - 4q^{9} + O(q^{10})$$ $$4q - 10q^{4} + 2q^{6} - 4q^{9} - 2q^{11} + 6q^{16} - 6q^{19} - 18q^{24} + 12q^{26} - 4q^{31} + 2q^{34} + 10q^{36} + 10q^{39} - 6q^{41} - 12q^{44} - 8q^{46} + 28q^{49} - 4q^{51} - 2q^{54} - 12q^{59} + 20q^{61} - 14q^{64} + 16q^{66} - 18q^{69} + 8q^{71} - 36q^{74} - 36q^{76} - 12q^{79} + 4q^{81} - 48q^{86} - 12q^{89} - 48q^{94} + 18q^{96} + 2q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 5 \nu$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 5$$ $$\nu^{3}$$ $$=$$ $$4 \beta_{2} - 5 \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1174.1
 − 2.56155i − 1.56155i 1.56155i 2.56155i
2.56155i 1.00000i −4.56155 0 2.56155 0 6.56155i −1.00000 0
1174.2 1.56155i 1.00000i −0.438447 0 −1.56155 0 2.43845i −1.00000 0
1174.3 1.56155i 1.00000i −0.438447 0 −1.56155 0 2.43845i −1.00000 0
1174.4 2.56155i 1.00000i −4.56155 0 2.56155 0 6.56155i −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 1275.2.b.d 4
5.b even 2 1 inner 1275.2.b.d 4
5.c odd 4 1 51.2.a.b 2
5.c odd 4 1 1275.2.a.n 2
15.e even 4 1 153.2.a.e 2
15.e even 4 1 3825.2.a.s 2
20.e even 4 1 816.2.a.m 2
35.f even 4 1 2499.2.a.o 2
40.i odd 4 1 3264.2.a.bl 2
40.k even 4 1 3264.2.a.bg 2
55.e even 4 1 6171.2.a.p 2
60.l odd 4 1 2448.2.a.v 2
65.h odd 4 1 8619.2.a.q 2
85.f odd 4 1 867.2.d.c 4
85.g odd 4 1 867.2.a.f 2
85.i odd 4 1 867.2.d.c 4
85.k odd 8 2 867.2.e.f 8
85.n odd 8 2 867.2.e.f 8
85.o even 16 4 867.2.h.j 16
85.r even 16 4 867.2.h.j 16
105.k odd 4 1 7497.2.a.v 2
120.q odd 4 1 9792.2.a.cz 2
120.w even 4 1 9792.2.a.cy 2
255.o even 4 1 2601.2.a.t 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.2.a.b 2 5.c odd 4 1
153.2.a.e 2 15.e even 4 1
816.2.a.m 2 20.e even 4 1
867.2.a.f 2 85.g odd 4 1
867.2.d.c 4 85.f odd 4 1
867.2.d.c 4 85.i odd 4 1
867.2.e.f 8 85.k odd 8 2
867.2.e.f 8 85.n odd 8 2
867.2.h.j 16 85.o even 16 4
867.2.h.j 16 85.r even 16 4
1275.2.a.n 2 5.c odd 4 1
1275.2.b.d 4 1.a even 1 1 trivial
1275.2.b.d 4 5.b even 2 1 inner
2448.2.a.v 2 60.l odd 4 1
2499.2.a.o 2 35.f even 4 1
2601.2.a.t 2 255.o even 4 1
3264.2.a.bg 2 40.k even 4 1
3264.2.a.bl 2 40.i odd 4 1
3825.2.a.s 2 15.e even 4 1
6171.2.a.p 2 55.e even 4 1
7497.2.a.v 2 105.k odd 4 1
8619.2.a.q 2 65.h odd 4 1
9792.2.a.cy 2 120.w even 4 1
9792.2.a.cz 2 120.q odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1275, [\chi])$$:

 $$T_{2}^{4} + 9 T_{2}^{2} + 16$$ $$T_{7}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 + 9 T^{2} + T^{4}$$
$3$ $$( 1 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$( -4 + T + T^{2} )^{2}$$
$13$ $$4 + 21 T^{2} + T^{4}$$
$17$ $$( 1 + T^{2} )^{2}$$
$19$ $$( -36 + 3 T + T^{2} )^{2}$$
$23$ $$256 + 49 T^{2} + T^{4}$$
$29$ $$( -68 + T^{2} )^{2}$$
$31$ $$( -16 + 2 T + T^{2} )^{2}$$
$37$ $$256 + 36 T^{2} + T^{4}$$
$41$ $$( -2 + 3 T + T^{2} )^{2}$$
$43$ $$1296 + 81 T^{2} + T^{4}$$
$47$ $$1024 + 132 T^{2} + T^{4}$$
$53$ $$2704 + 168 T^{2} + T^{4}$$
$59$ $$( -8 + 6 T + T^{2} )^{2}$$
$61$ $$( 8 - 10 T + T^{2} )^{2}$$
$67$ $$( 16 + T^{2} )^{2}$$
$71$ $$( -64 - 4 T + T^{2} )^{2}$$
$73$ $$2704 + 168 T^{2} + T^{4}$$
$79$ $$( -144 + 6 T + T^{2} )^{2}$$
$83$ $$64 + 84 T^{2} + T^{4}$$
$89$ $$( -8 + 6 T + T^{2} )^{2}$$
$97$ $$1024 + 132 T^{2} + T^{4}$$