# Properties

 Label 75.3.d.b Level $75$ Weight $3$ Character orbit 75.d Analytic conductor $2.044$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 75.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.04360198270$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{4}$$ 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 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.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{2} + ( \beta_{1} - \beta_{3} ) q^{3} + q^{4} + ( 5 - \beta_{2} ) q^{6} -3 \beta_{1} q^{7} + 3 \beta_{3} q^{8} + ( 1 - 2 \beta_{2} ) q^{9} +O(q^{10})$$ $$q -\beta_{3} q^{2} + ( \beta_{1} - \beta_{3} ) q^{3} + q^{4} + ( 5 - \beta_{2} ) q^{6} -3 \beta_{1} q^{7} + 3 \beta_{3} q^{8} + ( 1 - 2 \beta_{2} ) q^{9} -\beta_{2} q^{11} + ( \beta_{1} - \beta_{3} ) q^{12} -8 \beta_{1} q^{13} + 3 \beta_{2} q^{14} -19 q^{16} + 2 \beta_{3} q^{17} + ( 10 \beta_{1} - \beta_{3} ) q^{18} + 2 q^{19} + ( 12 + 3 \beta_{2} ) q^{21} + 5 \beta_{1} q^{22} -6 \beta_{3} q^{23} + ( -15 + 3 \beta_{2} ) q^{24} + 8 \beta_{2} q^{26} + ( 11 \beta_{1} + 7 \beta_{3} ) q^{27} -3 \beta_{1} q^{28} -7 \beta_{2} q^{29} -18 q^{31} + 7 \beta_{3} q^{32} + ( 5 \beta_{1} + 4 \beta_{3} ) q^{33} -10 q^{34} + ( 1 - 2 \beta_{2} ) q^{36} -8 \beta_{1} q^{37} -2 \beta_{3} q^{38} + ( 32 + 8 \beta_{2} ) q^{39} -14 \beta_{2} q^{41} + ( -15 \beta_{1} - 12 \beta_{3} ) q^{42} -8 \beta_{1} q^{43} -\beta_{2} q^{44} + 30 q^{46} -22 \beta_{3} q^{47} + ( -19 \beta_{1} + 19 \beta_{3} ) q^{48} + 13 q^{49} + ( -10 + 2 \beta_{2} ) q^{51} -8 \beta_{1} q^{52} -2 \beta_{3} q^{53} + ( -35 - 11 \beta_{2} ) q^{54} -9 \beta_{2} q^{56} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{57} + 35 \beta_{1} q^{58} + \beta_{2} q^{59} + 82 q^{61} + 18 \beta_{3} q^{62} + ( -3 \beta_{1} - 24 \beta_{3} ) q^{63} + 41 q^{64} + ( -20 - 5 \beta_{2} ) q^{66} + 12 \beta_{1} q^{67} + 2 \beta_{3} q^{68} + ( 30 - 6 \beta_{2} ) q^{69} + 28 \beta_{2} q^{71} + ( -30 \beta_{1} + 3 \beta_{3} ) q^{72} + 37 \beta_{1} q^{73} + 8 \beta_{2} q^{74} + 2 q^{76} -12 \beta_{3} q^{77} + ( -40 \beta_{1} - 32 \beta_{3} ) q^{78} -138 q^{79} + ( -79 - 4 \beta_{2} ) q^{81} + 70 \beta_{1} q^{82} + 42 \beta_{3} q^{83} + ( 12 + 3 \beta_{2} ) q^{84} + 8 \beta_{2} q^{86} + ( 35 \beta_{1} + 28 \beta_{3} ) q^{87} -15 \beta_{1} q^{88} + 24 \beta_{2} q^{89} -96 q^{91} -6 \beta_{3} q^{92} + ( -18 \beta_{1} + 18 \beta_{3} ) q^{93} + 110 q^{94} + ( -35 + 7 \beta_{2} ) q^{96} -83 \beta_{1} q^{97} -13 \beta_{3} q^{98} + ( -40 - \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{4} + 20q^{6} + 4q^{9} + O(q^{10})$$ $$4q + 4q^{4} + 20q^{6} + 4q^{9} - 76q^{16} + 8q^{19} + 48q^{21} - 60q^{24} - 72q^{31} - 40q^{34} + 4q^{36} + 128q^{39} + 120q^{46} + 52q^{49} - 40q^{51} - 140q^{54} + 328q^{61} + 164q^{64} - 80q^{66} + 120q^{69} + 8q^{76} - 552q^{79} - 316q^{81} + 48q^{84} - 384q^{91} + 440q^{94} - 140q^{96} - 160q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu^{3} + 4 \nu$$ $$\beta_{2}$$ $$=$$ $$2 \nu^{3} + 8 \nu$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{2} + 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 3$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{2} + 2 \beta_{1}$$$$)/2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
74.1
 − 0.618034i 0.618034i 1.61803i − 1.61803i
−2.23607 −2.23607 2.00000i 1.00000 0 5.00000 + 4.47214i 6.00000i 6.70820 1.00000 + 8.94427i 0
74.2 −2.23607 −2.23607 + 2.00000i 1.00000 0 5.00000 4.47214i 6.00000i 6.70820 1.00000 8.94427i 0
74.3 2.23607 2.23607 2.00000i 1.00000 0 5.00000 4.47214i 6.00000i −6.70820 1.00000 8.94427i 0
74.4 2.23607 2.23607 + 2.00000i 1.00000 0 5.00000 + 4.47214i 6.00000i −6.70820 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
5.b even 2 1 inner
15.d 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.d.b 4
3.b odd 2 1 inner 75.3.d.b 4
4.b odd 2 1 1200.3.c.f 4
5.b even 2 1 inner 75.3.d.b 4
5.c odd 4 1 15.3.c.a 2
5.c odd 4 1 75.3.c.e 2
12.b even 2 1 1200.3.c.f 4
15.d odd 2 1 inner 75.3.d.b 4
15.e even 4 1 15.3.c.a 2
15.e even 4 1 75.3.c.e 2
20.d odd 2 1 1200.3.c.f 4
20.e even 4 1 240.3.l.b 2
20.e even 4 1 1200.3.l.g 2
40.i odd 4 1 960.3.l.c 2
40.k even 4 1 960.3.l.b 2
45.k odd 12 2 405.3.i.b 4
45.l even 12 2 405.3.i.b 4
60.h even 2 1 1200.3.c.f 4
60.l odd 4 1 240.3.l.b 2
60.l odd 4 1 1200.3.l.g 2
120.q odd 4 1 960.3.l.b 2
120.w even 4 1 960.3.l.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.3.c.a 2 5.c odd 4 1
15.3.c.a 2 15.e even 4 1
75.3.c.e 2 5.c odd 4 1
75.3.c.e 2 15.e even 4 1
75.3.d.b 4 1.a even 1 1 trivial
75.3.d.b 4 3.b odd 2 1 inner
75.3.d.b 4 5.b even 2 1 inner
75.3.d.b 4 15.d odd 2 1 inner
240.3.l.b 2 20.e even 4 1
240.3.l.b 2 60.l odd 4 1
405.3.i.b 4 45.k odd 12 2
405.3.i.b 4 45.l even 12 2
960.3.l.b 2 40.k even 4 1
960.3.l.b 2 120.q odd 4 1
960.3.l.c 2 40.i odd 4 1
960.3.l.c 2 120.w even 4 1
1200.3.c.f 4 4.b odd 2 1
1200.3.c.f 4 12.b even 2 1
1200.3.c.f 4 20.d odd 2 1
1200.3.c.f 4 60.h even 2 1
1200.3.l.g 2 20.e even 4 1
1200.3.l.g 2 60.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 5$$ acting on $$S_{3}^{\mathrm{new}}(75, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -5 + T^{2} )^{2}$$
$3$ $$81 - 2 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 36 + T^{2} )^{2}$$
$11$ $$( 20 + T^{2} )^{2}$$
$13$ $$( 256 + T^{2} )^{2}$$
$17$ $$( -20 + T^{2} )^{2}$$
$19$ $$( -2 + T )^{4}$$
$23$ $$( -180 + T^{2} )^{2}$$
$29$ $$( 980 + T^{2} )^{2}$$
$31$ $$( 18 + T )^{4}$$
$37$ $$( 256 + T^{2} )^{2}$$
$41$ $$( 3920 + T^{2} )^{2}$$
$43$ $$( 256 + T^{2} )^{2}$$
$47$ $$( -2420 + T^{2} )^{2}$$
$53$ $$( -20 + T^{2} )^{2}$$
$59$ $$( 20 + T^{2} )^{2}$$
$61$ $$( -82 + T )^{4}$$
$67$ $$( 576 + T^{2} )^{2}$$
$71$ $$( 15680 + T^{2} )^{2}$$
$73$ $$( 5476 + T^{2} )^{2}$$
$79$ $$( 138 + T )^{4}$$
$83$ $$( -8820 + T^{2} )^{2}$$
$89$ $$( 11520 + T^{2} )^{2}$$
$97$ $$( 27556 + T^{2} )^{2}$$