# Properties

 Label 75.3.c.e 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: $$\Q(\sqrt{-5})$$ Defining polynomial: $$x^{2} + 5$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + ( 2 - \beta ) q^{3} - q^{4} + ( 5 + 2 \beta ) q^{6} + 6 q^{7} + 3 \beta q^{8} + ( -1 - 4 \beta ) q^{9} +O(q^{10})$$ $$q + \beta q^{2} + ( 2 - \beta ) q^{3} - q^{4} + ( 5 + 2 \beta ) q^{6} + 6 q^{7} + 3 \beta q^{8} + ( -1 - 4 \beta ) q^{9} + 2 \beta q^{11} + ( -2 + \beta ) q^{12} -16 q^{13} + 6 \beta q^{14} -19 q^{16} -2 \beta q^{17} + ( 20 - \beta ) q^{18} -2 q^{19} + ( 12 - 6 \beta ) q^{21} -10 q^{22} -6 \beta q^{23} + ( 15 + 6 \beta ) q^{24} -16 \beta q^{26} + ( -22 - 7 \beta ) q^{27} -6 q^{28} -14 \beta q^{29} -18 q^{31} -7 \beta q^{32} + ( 10 + 4 \beta ) q^{33} + 10 q^{34} + ( 1 + 4 \beta ) q^{36} + 16 q^{37} -2 \beta q^{38} + ( -32 + 16 \beta ) q^{39} + 28 \beta q^{41} + ( 30 + 12 \beta ) q^{42} -16 q^{43} -2 \beta q^{44} + 30 q^{46} + 22 \beta q^{47} + ( -38 + 19 \beta ) q^{48} -13 q^{49} + ( -10 - 4 \beta ) q^{51} + 16 q^{52} -2 \beta q^{53} + ( 35 - 22 \beta ) q^{54} + 18 \beta q^{56} + ( -4 + 2 \beta ) q^{57} + 70 q^{58} + 2 \beta q^{59} + 82 q^{61} -18 \beta q^{62} + ( -6 - 24 \beta ) q^{63} -41 q^{64} + ( -20 + 10 \beta ) q^{66} -24 q^{67} + 2 \beta q^{68} + ( -30 - 12 \beta ) q^{69} -56 \beta q^{71} + ( 60 - 3 \beta ) q^{72} + 74 q^{73} + 16 \beta q^{74} + 2 q^{76} + 12 \beta q^{77} + ( -80 - 32 \beta ) q^{78} + 138 q^{79} + ( -79 + 8 \beta ) q^{81} -140 q^{82} + 42 \beta q^{83} + ( -12 + 6 \beta ) q^{84} -16 \beta q^{86} + ( -70 - 28 \beta ) q^{87} -30 q^{88} + 48 \beta q^{89} -96 q^{91} + 6 \beta q^{92} + ( -36 + 18 \beta ) q^{93} -110 q^{94} + ( -35 - 14 \beta ) q^{96} + 166 q^{97} -13 \beta q^{98} + ( 40 - 2 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{3} - 2q^{4} + 10q^{6} + 12q^{7} - 2q^{9} + O(q^{10})$$ $$2q + 4q^{3} - 2q^{4} + 10q^{6} + 12q^{7} - 2q^{9} - 4q^{12} - 32q^{13} - 38q^{16} + 40q^{18} - 4q^{19} + 24q^{21} - 20q^{22} + 30q^{24} - 44q^{27} - 12q^{28} - 36q^{31} + 20q^{33} + 20q^{34} + 2q^{36} + 32q^{37} - 64q^{39} + 60q^{42} - 32q^{43} + 60q^{46} - 76q^{48} - 26q^{49} - 20q^{51} + 32q^{52} + 70q^{54} - 8q^{57} + 140q^{58} + 164q^{61} - 12q^{63} - 82q^{64} - 40q^{66} - 48q^{67} - 60q^{69} + 120q^{72} + 148q^{73} + 4q^{76} - 160q^{78} + 276q^{79} - 158q^{81} - 280q^{82} - 24q^{84} - 140q^{87} - 60q^{88} - 192q^{91} - 72q^{93} - 220q^{94} - 70q^{96} + 332q^{97} + 80q^{99} + O(q^{100})$$

## 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
 − 2.23607i 2.23607i
2.23607i 2.00000 + 2.23607i −1.00000 0 5.00000 4.47214i 6.00000 6.70820i −1.00000 + 8.94427i 0
26.2 2.23607i 2.00000 2.23607i −1.00000 0 5.00000 + 4.47214i 6.00000 6.70820i −1.00000 8.94427i 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.e 2
3.b odd 2 1 inner 75.3.c.e 2
4.b odd 2 1 1200.3.l.g 2
5.b even 2 1 15.3.c.a 2
5.c odd 4 2 75.3.d.b 4
12.b even 2 1 1200.3.l.g 2
15.d odd 2 1 15.3.c.a 2
15.e even 4 2 75.3.d.b 4
20.d odd 2 1 240.3.l.b 2
20.e even 4 2 1200.3.c.f 4
40.e odd 2 1 960.3.l.b 2
40.f even 2 1 960.3.l.c 2
45.h odd 6 2 405.3.i.b 4
45.j even 6 2 405.3.i.b 4
60.h even 2 1 240.3.l.b 2
60.l odd 4 2 1200.3.c.f 4
120.i odd 2 1 960.3.l.c 2
120.m even 2 1 960.3.l.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.3.c.a 2 5.b even 2 1
15.3.c.a 2 15.d odd 2 1
75.3.c.e 2 1.a even 1 1 trivial
75.3.c.e 2 3.b odd 2 1 inner
75.3.d.b 4 5.c odd 4 2
75.3.d.b 4 15.e even 4 2
240.3.l.b 2 20.d odd 2 1
240.3.l.b 2 60.h even 2 1
405.3.i.b 4 45.h odd 6 2
405.3.i.b 4 45.j even 6 2
960.3.l.b 2 40.e odd 2 1
960.3.l.b 2 120.m even 2 1
960.3.l.c 2 40.f even 2 1
960.3.l.c 2 120.i odd 2 1
1200.3.c.f 4 20.e even 4 2
1200.3.c.f 4 60.l odd 4 2
1200.3.l.g 2 4.b odd 2 1
1200.3.l.g 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 $$S_{3}^{\mathrm{new}}(75, [\chi])$$:

 $$T_{2}^{2} + 5$$ $$T_{7} - 6$$ $$T_{13} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$5 + T^{2}$$
$3$ $$9 - 4 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$( -6 + T )^{2}$$
$11$ $$20 + T^{2}$$
$13$ $$( 16 + T )^{2}$$
$17$ $$20 + T^{2}$$
$19$ $$( 2 + T )^{2}$$
$23$ $$180 + T^{2}$$
$29$ $$980 + T^{2}$$
$31$ $$( 18 + T )^{2}$$
$37$ $$( -16 + T )^{2}$$
$41$ $$3920 + T^{2}$$
$43$ $$( 16 + T )^{2}$$
$47$ $$2420 + T^{2}$$
$53$ $$20 + T^{2}$$
$59$ $$20 + T^{2}$$
$61$ $$( -82 + T )^{2}$$
$67$ $$( 24 + T )^{2}$$
$71$ $$15680 + T^{2}$$
$73$ $$( -74 + T )^{2}$$
$79$ $$( -138 + T )^{2}$$
$83$ $$8820 + T^{2}$$
$89$ $$11520 + T^{2}$$
$97$ $$( -166 + T )^{2}$$