Properties

 Label 75.3.d.c 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: $$\Q(i, \sqrt{11})$$ Defining polynomial: $$x^{4} - 5 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$5^{2}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} + \beta_{2} ) q^{2} + \beta_{1} q^{3} + 7 q^{4} + ( -6 - \beta_{3} ) q^{6} + ( -3 \beta_{1} + 3 \beta_{2} ) q^{8} + ( -3 + \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{1} + \beta_{2} ) q^{2} + \beta_{1} q^{3} + 7 q^{4} + ( -6 - \beta_{3} ) q^{6} + ( -3 \beta_{1} + 3 \beta_{2} ) q^{8} + ( -3 + \beta_{3} ) q^{9} + ( 1 + 2 \beta_{3} ) q^{11} + 7 \beta_{1} q^{12} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{13} + 5 q^{16} + ( -\beta_{1} + \beta_{2} ) q^{17} + ( -2 \beta_{1} - 9 \beta_{2} ) q^{18} -7 q^{19} + ( -11 \beta_{1} - 11 \beta_{2} ) q^{22} + ( 6 \beta_{1} - 6 \beta_{2} ) q^{23} + ( -18 - 3 \beta_{3} ) q^{24} + ( -2 - 4 \beta_{3} ) q^{26} + ( -7 \beta_{1} + 9 \beta_{2} ) q^{27} + ( -2 - 4 \beta_{3} ) q^{29} + 42 q^{31} + ( 7 \beta_{1} - 7 \beta_{2} ) q^{32} + ( -7 \beta_{1} + 18 \beta_{2} ) q^{33} + 11 q^{34} + ( -21 + 7 \beta_{3} ) q^{36} + ( 8 \beta_{1} + 8 \beta_{2} ) q^{37} + ( 7 \beta_{1} - 7 \beta_{2} ) q^{38} + ( -24 + 2 \beta_{3} ) q^{39} + ( -1 - 2 \beta_{3} ) q^{41} + ( -10 \beta_{1} - 10 \beta_{2} ) q^{43} + ( 7 + 14 \beta_{3} ) q^{44} -66 q^{46} + ( 14 \beta_{1} - 14 \beta_{2} ) q^{47} + 5 \beta_{1} q^{48} + 49 q^{49} + ( -6 - \beta_{3} ) q^{51} + ( 14 \beta_{1} + 14 \beta_{2} ) q^{52} + ( -14 \beta_{1} + 14 \beta_{2} ) q^{53} + ( 87 - 2 \beta_{3} ) q^{54} -7 \beta_{1} q^{57} + ( 22 \beta_{1} + 22 \beta_{2} ) q^{58} + ( -4 - 8 \beta_{3} ) q^{59} -8 q^{61} + ( -42 \beta_{1} + 42 \beta_{2} ) q^{62} -97 q^{64} + ( 132 - 11 \beta_{3} ) q^{66} + ( -9 \beta_{1} - 9 \beta_{2} ) q^{67} + ( -7 \beta_{1} + 7 \beta_{2} ) q^{68} + ( 36 + 6 \beta_{3} ) q^{69} + ( 2 + 4 \beta_{3} ) q^{71} + ( -6 \beta_{1} - 27 \beta_{2} ) q^{72} + ( -7 \beta_{1} - 7 \beta_{2} ) q^{73} + ( -8 - 16 \beta_{3} ) q^{74} -49 q^{76} + ( 14 \beta_{1} - 36 \beta_{2} ) q^{78} -12 q^{79} + ( -60 - 7 \beta_{3} ) q^{81} + ( 11 \beta_{1} + 11 \beta_{2} ) q^{82} + ( 21 \beta_{1} - 21 \beta_{2} ) q^{83} + ( 10 + 20 \beta_{3} ) q^{86} + ( 14 \beta_{1} - 36 \beta_{2} ) q^{87} + ( -33 \beta_{1} - 33 \beta_{2} ) q^{88} + ( 9 + 18 \beta_{3} ) q^{89} + ( 42 \beta_{1} - 42 \beta_{2} ) q^{92} + 42 \beta_{1} q^{93} -154 q^{94} + ( 42 + 7 \beta_{3} ) q^{96} + ( 14 \beta_{1} + 14 \beta_{2} ) q^{97} + ( -49 \beta_{1} + 49 \beta_{2} ) q^{98} + ( -141 - 7 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 28q^{4} - 22q^{6} - 14q^{9} + O(q^{10})$$ $$4q + 28q^{4} - 22q^{6} - 14q^{9} + 20q^{16} - 28q^{19} - 66q^{24} + 168q^{31} + 44q^{34} - 98q^{36} - 100q^{39} - 264q^{46} + 196q^{49} - 22q^{51} + 352q^{54} - 32q^{61} - 388q^{64} + 550q^{66} + 132q^{69} - 196q^{76} - 48q^{79} - 226q^{81} - 616q^{94} + 154q^{96} - 550q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 5 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-2 \nu^{3} + \nu$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$-\nu^{3} + 3 \nu$$ $$\beta_{3}$$ $$=$$ $$5 \nu^{2} - 13$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$2 \beta_{2} - 3 \beta_{1}$$$$)/5$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 13$$$$)/5$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{2} - 9 \beta_{1}$$$$)/5$$

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
 −1.65831 + 0.500000i −1.65831 − 0.500000i 1.65831 + 0.500000i 1.65831 − 0.500000i
−3.31662 1.65831 2.50000i 7.00000 0 −5.50000 + 8.29156i 0 −9.94987 −3.50000 8.29156i 0
74.2 −3.31662 1.65831 + 2.50000i 7.00000 0 −5.50000 8.29156i 0 −9.94987 −3.50000 + 8.29156i 0
74.3 3.31662 −1.65831 2.50000i 7.00000 0 −5.50000 8.29156i 0 9.94987 −3.50000 + 8.29156i 0
74.4 3.31662 −1.65831 + 2.50000i 7.00000 0 −5.50000 + 8.29156i 0 9.94987 −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
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.c 4
3.b odd 2 1 inner 75.3.d.c 4
4.b odd 2 1 1200.3.c.d 4
5.b even 2 1 inner 75.3.d.c 4
5.c odd 4 1 75.3.c.c 2
5.c odd 4 1 75.3.c.f yes 2
12.b even 2 1 1200.3.c.d 4
15.d odd 2 1 inner 75.3.d.c 4
15.e even 4 1 75.3.c.c 2
15.e even 4 1 75.3.c.f yes 2
20.d odd 2 1 1200.3.c.d 4
20.e even 4 1 1200.3.l.f 2
20.e even 4 1 1200.3.l.s 2
60.h even 2 1 1200.3.c.d 4
60.l odd 4 1 1200.3.l.f 2
60.l odd 4 1 1200.3.l.s 2

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 11$$ acting on $$S_{3}^{\mathrm{new}}(75, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -11 + T^{2} )^{2}$$
$3$ $$81 + 7 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 275 + T^{2} )^{2}$$
$13$ $$( 100 + T^{2} )^{2}$$
$17$ $$( -11 + T^{2} )^{2}$$
$19$ $$( 7 + T )^{4}$$
$23$ $$( -396 + T^{2} )^{2}$$
$29$ $$( 1100 + T^{2} )^{2}$$
$31$ $$( -42 + T )^{4}$$
$37$ $$( 1600 + T^{2} )^{2}$$
$41$ $$( 275 + T^{2} )^{2}$$
$43$ $$( 2500 + T^{2} )^{2}$$
$47$ $$( -2156 + T^{2} )^{2}$$
$53$ $$( -2156 + T^{2} )^{2}$$
$59$ $$( 4400 + T^{2} )^{2}$$
$61$ $$( 8 + T )^{4}$$
$67$ $$( 2025 + T^{2} )^{2}$$
$71$ $$( 1100 + T^{2} )^{2}$$
$73$ $$( 1225 + T^{2} )^{2}$$
$79$ $$( 12 + T )^{4}$$
$83$ $$( -4851 + T^{2} )^{2}$$
$89$ $$( 22275 + T^{2} )^{2}$$
$97$ $$( 4900 + T^{2} )^{2}$$