# Properties

 Label 75.3.c.c Level $75$ Weight $3$ Character orbit 75.c Analytic conductor $2.044$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 75.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.04360198270$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-11})$$ Defining polynomial: $$x^{2} - x + 3$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{-11})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - 2 \beta ) q^{2} + ( -2 - \beta ) q^{3} -7 q^{4} + ( -8 + 5 \beta ) q^{6} + ( -3 + 6 \beta ) q^{8} + ( 1 + 5 \beta ) q^{9} +O(q^{10})$$ $$q + ( 1 - 2 \beta ) q^{2} + ( -2 - \beta ) q^{3} -7 q^{4} + ( -8 + 5 \beta ) q^{6} + ( -3 + 6 \beta ) q^{8} + ( 1 + 5 \beta ) q^{9} + ( 5 - 10 \beta ) q^{11} + ( 14 + 7 \beta ) q^{12} -10 q^{13} + 5 q^{16} + ( 1 - 2 \beta ) q^{17} + ( 31 - 7 \beta ) q^{18} + 7 q^{19} -55 q^{22} + ( 6 - 12 \beta ) q^{23} + ( 24 - 15 \beta ) q^{24} + ( -10 + 20 \beta ) q^{26} + ( 13 - 16 \beta ) q^{27} + ( 10 - 20 \beta ) q^{29} + 42 q^{31} + ( -7 + 14 \beta ) q^{32} + ( -40 + 25 \beta ) q^{33} -11 q^{34} + ( -7 - 35 \beta ) q^{36} + 40 q^{37} + ( 7 - 14 \beta ) q^{38} + ( 20 + 10 \beta ) q^{39} + ( -5 + 10 \beta ) q^{41} + 50 q^{43} + ( -35 + 70 \beta ) q^{44} -66 q^{46} + ( -14 + 28 \beta ) q^{47} + ( -10 - 5 \beta ) q^{48} -49 q^{49} + ( -8 + 5 \beta ) q^{51} + 70 q^{52} + ( -14 + 28 \beta ) q^{53} + ( -83 - 10 \beta ) q^{54} + ( -14 - 7 \beta ) q^{57} -110 q^{58} + ( 20 - 40 \beta ) q^{59} -8 q^{61} + ( 42 - 84 \beta ) q^{62} + 97 q^{64} + ( 110 + 55 \beta ) q^{66} -45 q^{67} + ( -7 + 14 \beta ) q^{68} + ( -48 + 30 \beta ) q^{69} + ( 10 - 20 \beta ) q^{71} + ( -93 + 21 \beta ) q^{72} + 35 q^{73} + ( 40 - 80 \beta ) q^{74} -49 q^{76} + ( 80 - 50 \beta ) q^{78} + 12 q^{79} + ( -74 + 35 \beta ) q^{81} + 55 q^{82} + ( 21 - 42 \beta ) q^{83} + ( 50 - 100 \beta ) q^{86} + ( -80 + 50 \beta ) q^{87} + 165 q^{88} + ( -45 + 90 \beta ) q^{89} + ( -42 + 84 \beta ) q^{92} + ( -84 - 42 \beta ) q^{93} + 154 q^{94} + ( 56 - 35 \beta ) q^{96} + 70 q^{97} + ( -49 + 98 \beta ) q^{98} + ( 155 - 35 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 5q^{3} - 14q^{4} - 11q^{6} + 7q^{9} + O(q^{10})$$ $$2q - 5q^{3} - 14q^{4} - 11q^{6} + 7q^{9} + 35q^{12} - 20q^{13} + 10q^{16} + 55q^{18} + 14q^{19} - 110q^{22} + 33q^{24} + 10q^{27} + 84q^{31} - 55q^{33} - 22q^{34} - 49q^{36} + 80q^{37} + 50q^{39} + 100q^{43} - 132q^{46} - 25q^{48} - 98q^{49} - 11q^{51} + 140q^{52} - 176q^{54} - 35q^{57} - 220q^{58} - 16q^{61} + 194q^{64} + 275q^{66} - 90q^{67} - 66q^{69} - 165q^{72} + 70q^{73} - 98q^{76} + 110q^{78} + 24q^{79} - 113q^{81} + 110q^{82} - 110q^{87} + 330q^{88} - 210q^{93} + 308q^{94} + 77q^{96} + 140q^{97} + 275q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$$.

 $$n$$ $$26$$ $$52$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
26.1
 0.5 + 1.65831i 0.5 − 1.65831i
3.31662i −2.50000 1.65831i −7.00000 0 −5.50000 + 8.29156i 0 9.94987i 3.50000 + 8.29156i 0
26.2 3.31662i −2.50000 + 1.65831i −7.00000 0 −5.50000 8.29156i 0 9.94987i 3.50000 8.29156i 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
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.3.c.c 2
3.b odd 2 1 inner 75.3.c.c 2
4.b odd 2 1 1200.3.l.s 2
5.b even 2 1 75.3.c.f yes 2
5.c odd 4 2 75.3.d.c 4
12.b even 2 1 1200.3.l.s 2
15.d odd 2 1 75.3.c.f yes 2
15.e even 4 2 75.3.d.c 4
20.d odd 2 1 1200.3.l.f 2
20.e even 4 2 1200.3.c.d 4
60.h even 2 1 1200.3.l.f 2
60.l odd 4 2 1200.3.c.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.3.c.c 2 1.a even 1 1 trivial
75.3.c.c 2 3.b odd 2 1 inner
75.3.c.f yes 2 5.b even 2 1
75.3.c.f yes 2 15.d odd 2 1
75.3.d.c 4 5.c odd 4 2
75.3.d.c 4 15.e even 4 2
1200.3.c.d 4 20.e even 4 2
1200.3.c.d 4 60.l odd 4 2
1200.3.l.f 2 20.d odd 2 1
1200.3.l.f 2 60.h even 2 1
1200.3.l.s 2 4.b odd 2 1
1200.3.l.s 2 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(75, [\chi])$$:

 $$T_{2}^{2} + 11$$ $$T_{7}$$ $$T_{13} + 10$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$11 + T^{2}$$
$3$ $$9 + 5 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$275 + T^{2}$$
$13$ $$( 10 + T )^{2}$$
$17$ $$11 + T^{2}$$
$19$ $$( -7 + T )^{2}$$
$23$ $$396 + T^{2}$$
$29$ $$1100 + T^{2}$$
$31$ $$( -42 + T )^{2}$$
$37$ $$( -40 + T )^{2}$$
$41$ $$275 + T^{2}$$
$43$ $$( -50 + T )^{2}$$
$47$ $$2156 + T^{2}$$
$53$ $$2156 + T^{2}$$
$59$ $$4400 + T^{2}$$
$61$ $$( 8 + T )^{2}$$
$67$ $$( 45 + T )^{2}$$
$71$ $$1100 + T^{2}$$
$73$ $$( -35 + T )^{2}$$
$79$ $$( -12 + T )^{2}$$
$83$ $$4851 + T^{2}$$
$89$ $$22275 + T^{2}$$
$97$ $$( -70 + T )^{2}$$