# Properties

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

## Newform invariants

 Self dual: no Analytic conductor: $$10.1809262577$$ 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 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.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
424.1
 − 1.00000i 1.00000i
1.00000i 1.00000 1.00000 0 1.00000i 4.00000 3.00000i 1.00000 0
424.2 1.00000i 1.00000 1.00000 0 1.00000i 4.00000 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
85.c 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.d.d 2
5.b even 2 1 1275.2.d.b 2
5.c odd 4 1 51.2.d.b 2
5.c odd 4 1 1275.2.g.a 2
15.e even 4 1 153.2.d.a 2
17.b even 2 1 1275.2.d.b 2
20.e even 4 1 816.2.c.c 2
40.i odd 4 1 3264.2.c.e 2
40.k even 4 1 3264.2.c.d 2
60.l odd 4 1 2448.2.c.j 2
85.c even 2 1 inner 1275.2.d.d 2
85.f odd 4 1 867.2.a.a 1
85.g odd 4 1 51.2.d.b 2
85.g odd 4 1 1275.2.g.a 2
85.i odd 4 1 867.2.a.b 1
85.k odd 8 2 867.2.e.d 4
85.n odd 8 2 867.2.e.d 4
85.o even 16 4 867.2.h.d 8
85.r even 16 4 867.2.h.d 8
255.k even 4 1 2601.2.a.i 1
255.o even 4 1 153.2.d.a 2
255.r even 4 1 2601.2.a.j 1
340.r even 4 1 816.2.c.c 2
680.u even 4 1 3264.2.c.d 2
680.bi odd 4 1 3264.2.c.e 2
1020.x odd 4 1 2448.2.c.j 2

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

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$T^{2}$$
$7$ $$( -4 + T )^{2}$$
$11$ $$16 + T^{2}$$
$13$ $$4 + T^{2}$$
$17$ $$17 + 8 T + T^{2}$$
$19$ $$( -4 + T )^{2}$$
$23$ $$( 4 + T )^{2}$$
$29$ $$T^{2}$$
$31$ $$16 + T^{2}$$
$37$ $$( 8 + T )^{2}$$
$41$ $$64 + T^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$64 + T^{2}$$
$53$ $$36 + T^{2}$$
$59$ $$( -12 + T )^{2}$$
$61$ $$64 + T^{2}$$
$67$ $$144 + T^{2}$$
$71$ $$144 + T^{2}$$
$73$ $$T^{2}$$
$79$ $$16 + T^{2}$$
$83$ $$144 + T^{2}$$
$89$ $$( -10 + T )^{2}$$
$97$ $$( 16 + T )^{2}$$