# Properties

 Label 75.3.f.c Level $75$ Weight $3$ Character orbit 75.f Analytic conductor $2.044$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 75.f (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.04360198270$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{1} + \beta_{2} ) q^{2} + \beta_{3} q^{3} + ( 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{4} + ( -3 - \beta_{1} + \beta_{3} ) q^{6} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{7} + ( -3 + 3 \beta_{2} + \beta_{3} ) q^{8} -3 \beta_{2} q^{9} +O(q^{10})$$ $$q + ( 1 + \beta_{1} + \beta_{2} ) q^{2} + \beta_{3} q^{3} + ( 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{4} + ( -3 - \beta_{1} + \beta_{3} ) q^{6} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{7} + ( -3 + 3 \beta_{2} + \beta_{3} ) q^{8} -3 \beta_{2} q^{9} + ( 4 - 3 \beta_{1} + 3 \beta_{3} ) q^{11} + ( -6 - \beta_{1} - 6 \beta_{2} ) q^{12} + ( 8 - 8 \beta_{2} - 2 \beta_{3} ) q^{13} + ( \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{14} + ( -5 + 4 \beta_{1} - 4 \beta_{3} ) q^{16} + ( 10 - 6 \beta_{1} + 10 \beta_{2} ) q^{17} + ( 3 - 3 \beta_{2} - 3 \beta_{3} ) q^{18} + ( -6 \beta_{1} - 6 \beta_{2} - 6 \beta_{3} ) q^{19} + ( -6 + \beta_{1} - \beta_{3} ) q^{21} + ( -5 - 2 \beta_{1} - 5 \beta_{2} ) q^{22} + ( -14 + 14 \beta_{2} + 2 \beta_{3} ) q^{23} + ( -3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{24} + ( 22 + 10 \beta_{1} - 10 \beta_{3} ) q^{26} + 3 \beta_{1} q^{27} + ( -11 + 11 \beta_{2} - 2 \beta_{3} ) q^{28} + ( -7 \beta_{1} + 18 \beta_{2} - 7 \beta_{3} ) q^{29} + ( -4 - 6 \beta_{1} + 6 \beta_{3} ) q^{31} + ( 19 + 7 \beta_{1} + 19 \beta_{2} ) q^{32} + ( 9 - 9 \beta_{2} + 4 \beta_{3} ) q^{33} + ( 4 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{34} + ( 3 + 6 \beta_{1} - 6 \beta_{3} ) q^{36} + ( -16 - 18 \beta_{1} - 16 \beta_{2} ) q^{37} + ( 24 - 24 \beta_{2} - 18 \beta_{3} ) q^{38} + ( 8 \beta_{1} + 6 \beta_{2} + 8 \beta_{3} ) q^{39} + ( -14 - 6 \beta_{1} + 6 \beta_{3} ) q^{41} + ( -3 - 4 \beta_{1} - 3 \beta_{2} ) q^{42} + ( 2 - 2 \beta_{2} + 20 \beta_{3} ) q^{43} + ( 5 \beta_{1} - 32 \beta_{2} + 5 \beta_{3} ) q^{44} + ( -34 - 16 \beta_{1} + 16 \beta_{3} ) q^{46} + ( -32 + 10 \beta_{1} - 32 \beta_{2} ) q^{47} + ( -12 + 12 \beta_{2} - 5 \beta_{3} ) q^{48} + ( -4 \beta_{1} - 35 \beta_{2} - 4 \beta_{3} ) q^{49} + ( 18 - 10 \beta_{1} + 10 \beta_{3} ) q^{51} + ( 20 + 34 \beta_{1} + 20 \beta_{2} ) q^{52} + ( -14 + 14 \beta_{2} + 12 \beta_{3} ) q^{53} + ( 3 \beta_{1} + 9 \beta_{2} + 3 \beta_{3} ) q^{54} + ( -5 \beta_{1} + 5 \beta_{3} ) q^{56} + ( 18 + 6 \beta_{1} + 18 \beta_{2} ) q^{57} + ( 3 - 3 \beta_{2} + 4 \beta_{3} ) q^{58} + ( 31 \beta_{1} + 36 \beta_{2} + 31 \beta_{3} ) q^{59} + ( 50 + 18 \beta_{1} - 18 \beta_{3} ) q^{61} + ( -22 - 16 \beta_{1} - 22 \beta_{2} ) q^{62} + ( -3 + 3 \beta_{2} - 6 \beta_{3} ) q^{63} + ( 10 \beta_{1} + 79 \beta_{2} + 10 \beta_{3} ) q^{64} + ( 6 + 5 \beta_{1} - 5 \beta_{3} ) q^{66} + ( 50 + 4 \beta_{1} + 50 \beta_{2} ) q^{67} + ( 26 - 26 \beta_{2} + 34 \beta_{3} ) q^{68} + ( -14 \beta_{1} - 6 \beta_{2} - 14 \beta_{3} ) q^{69} -68 q^{71} + ( 9 + 3 \beta_{1} + 9 \beta_{2} ) q^{72} + ( -19 + 19 \beta_{2} - 48 \beta_{3} ) q^{73} + ( -34 \beta_{1} - 86 \beta_{2} - 34 \beta_{3} ) q^{74} + ( 78 + 18 \beta_{1} - 18 \beta_{3} ) q^{76} + ( -22 + 14 \beta_{1} - 22 \beta_{2} ) q^{77} + ( -30 + 30 \beta_{2} + 22 \beta_{3} ) q^{78} + ( 10 \beta_{1} + 10 \beta_{3} ) q^{79} -9 q^{81} + ( -32 - 26 \beta_{1} - 32 \beta_{2} ) q^{82} + ( 4 - 4 \beta_{2} - 14 \beta_{3} ) q^{83} + ( -11 \beta_{1} + 6 \beta_{2} - 11 \beta_{3} ) q^{84} + ( -56 - 18 \beta_{1} + 18 \beta_{3} ) q^{86} + ( 21 - 18 \beta_{1} + 21 \beta_{2} ) q^{87} + ( -3 + 3 \beta_{2} - 14 \beta_{3} ) q^{88} + ( -36 \beta_{1} - 6 \beta_{2} - 36 \beta_{3} ) q^{89} + ( -4 + 14 \beta_{1} - 14 \beta_{3} ) q^{91} + ( -26 - 58 \beta_{1} - 26 \beta_{2} ) q^{92} + ( 18 - 18 \beta_{2} - 4 \beta_{3} ) q^{93} + ( -22 \beta_{1} - 34 \beta_{2} - 22 \beta_{3} ) q^{94} + ( -21 - 19 \beta_{1} + 19 \beta_{3} ) q^{96} + ( 5 - 16 \beta_{1} + 5 \beta_{2} ) q^{97} + ( 47 - 47 \beta_{2} - 43 \beta_{3} ) q^{98} + ( 9 \beta_{1} - 12 \beta_{2} + 9 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{2} - 12q^{6} - 4q^{7} - 12q^{8} + O(q^{10})$$ $$4q + 4q^{2} - 12q^{6} - 4q^{7} - 12q^{8} + 16q^{11} - 24q^{12} + 32q^{13} - 20q^{16} + 40q^{17} + 12q^{18} - 24q^{21} - 20q^{22} - 56q^{23} + 88q^{26} - 44q^{28} - 16q^{31} + 76q^{32} + 36q^{33} + 12q^{36} - 64q^{37} + 96q^{38} - 56q^{41} - 12q^{42} + 8q^{43} - 136q^{46} - 128q^{47} - 48q^{48} + 72q^{51} + 80q^{52} - 56q^{53} + 72q^{57} + 12q^{58} + 200q^{61} - 88q^{62} - 12q^{63} + 24q^{66} + 200q^{67} + 104q^{68} - 272q^{71} + 36q^{72} - 76q^{73} + 312q^{76} - 88q^{77} - 120q^{78} - 36q^{81} - 128q^{82} + 16q^{83} - 224q^{86} + 84q^{87} - 12q^{88} - 16q^{91} - 104q^{92} + 72q^{93} - 84q^{96} + 20q^{97} + 188q^{98} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$3 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{3}$$

## Character values

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

 $$n$$ $$26$$ $$52$$ $$\chi(n)$$ $$1$$ $$\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 −1.22474 − 1.22474i 1.22474 + 1.22474i −1.22474 + 1.22474i 1.22474 − 1.22474i
−0.224745 0.224745i 1.22474 1.22474i 3.89898i 0 −0.550510 −3.44949 3.44949i −1.77526 + 1.77526i 3.00000i 0
7.2 2.22474 + 2.22474i −1.22474 + 1.22474i 5.89898i 0 −5.44949 1.44949 + 1.44949i −4.22474 + 4.22474i 3.00000i 0
43.1 −0.224745 + 0.224745i 1.22474 + 1.22474i 3.89898i 0 −0.550510 −3.44949 + 3.44949i −1.77526 1.77526i 3.00000i 0
43.2 2.22474 2.22474i −1.22474 1.22474i 5.89898i 0 −5.44949 1.44949 1.44949i −4.22474 4.22474i 3.00000i 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
5.c odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.3.f.c 4
3.b odd 2 1 225.3.g.a 4
4.b odd 2 1 1200.3.bg.k 4
5.b even 2 1 15.3.f.a 4
5.c odd 4 1 15.3.f.a 4
5.c odd 4 1 inner 75.3.f.c 4
15.d odd 2 1 45.3.g.b 4
15.e even 4 1 45.3.g.b 4
15.e even 4 1 225.3.g.a 4
20.d odd 2 1 240.3.bg.a 4
20.e even 4 1 240.3.bg.a 4
20.e even 4 1 1200.3.bg.k 4
40.e odd 2 1 960.3.bg.h 4
40.f even 2 1 960.3.bg.i 4
40.i odd 4 1 960.3.bg.i 4
40.k even 4 1 960.3.bg.h 4
45.h odd 6 2 405.3.l.f 8
45.j even 6 2 405.3.l.h 8
45.k odd 12 2 405.3.l.h 8
45.l even 12 2 405.3.l.f 8
60.h even 2 1 720.3.bh.k 4
60.l odd 4 1 720.3.bh.k 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.3.f.a 4 5.b even 2 1
15.3.f.a 4 5.c odd 4 1
45.3.g.b 4 15.d odd 2 1
45.3.g.b 4 15.e even 4 1
75.3.f.c 4 1.a even 1 1 trivial
75.3.f.c 4 5.c odd 4 1 inner
225.3.g.a 4 3.b odd 2 1
225.3.g.a 4 15.e even 4 1
240.3.bg.a 4 20.d odd 2 1
240.3.bg.a 4 20.e even 4 1
405.3.l.f 8 45.h odd 6 2
405.3.l.f 8 45.l even 12 2
405.3.l.h 8 45.j even 6 2
405.3.l.h 8 45.k odd 12 2
720.3.bh.k 4 60.h even 2 1
720.3.bh.k 4 60.l odd 4 1
960.3.bg.h 4 40.e odd 2 1
960.3.bg.h 4 40.k even 4 1
960.3.bg.i 4 40.f even 2 1
960.3.bg.i 4 40.i odd 4 1
1200.3.bg.k 4 4.b odd 2 1
1200.3.bg.k 4 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 4 T_{2}^{3} + 8 T_{2}^{2} + 4 T_{2} + 1$$ acting on $$S_{3}^{\mathrm{new}}(75, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 4 T + 8 T^{2} - 4 T^{3} + T^{4}$$
$3$ $$9 + T^{4}$$
$5$ $$T^{4}$$
$7$ $$100 - 40 T + 8 T^{2} + 4 T^{3} + T^{4}$$
$11$ $$( -38 - 8 T + T^{2} )^{2}$$
$13$ $$13456 - 3712 T + 512 T^{2} - 32 T^{3} + T^{4}$$
$17$ $$8464 - 3680 T + 800 T^{2} - 40 T^{3} + T^{4}$$
$19$ $$32400 + 504 T^{2} + T^{4}$$
$23$ $$144400 + 21280 T + 1568 T^{2} + 56 T^{3} + T^{4}$$
$29$ $$900 + 1236 T^{2} + T^{4}$$
$31$ $$( -200 + 8 T + T^{2} )^{2}$$
$37$ $$211600 - 29440 T + 2048 T^{2} + 64 T^{3} + T^{4}$$
$41$ $$( -20 + 28 T + T^{2} )^{2}$$
$43$ $$1420864 + 9536 T + 32 T^{2} - 8 T^{3} + T^{4}$$
$47$ $$3055504 + 223744 T + 8192 T^{2} + 128 T^{3} + T^{4}$$
$53$ $$1600 - 2240 T + 1568 T^{2} + 56 T^{3} + T^{4}$$
$59$ $$19980900 + 14124 T^{2} + T^{4}$$
$61$ $$( 556 - 100 T + T^{2} )^{2}$$
$67$ $$24522304 - 990400 T + 20000 T^{2} - 200 T^{3} + T^{4}$$
$71$ $$( 68 + T )^{4}$$
$73$ $$38316100 - 470440 T + 2888 T^{2} + 76 T^{3} + T^{4}$$
$79$ $$( 600 + T^{2} )^{2}$$
$83$ $$309136 + 8896 T + 128 T^{2} - 16 T^{3} + T^{4}$$
$89$ $$59907600 + 15624 T^{2} + T^{4}$$
$97$ $$515524 + 14360 T + 200 T^{2} - 20 T^{3} + T^{4}$$
show more
show less