# 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 $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,3,Mod(7,75)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("75.7");

S:= CuspForms(chi, 3);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$2.04360198270$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 9$$ 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

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + \beta_1 + 1) q^{2} + \beta_{3} q^{3} + (2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{4} + (\beta_{3} - \beta_1 - 3) q^{6} + ( - \beta_{2} + 2 \beta_1 - 1) q^{7} + (\beta_{3} + 3 \beta_{2} - 3) q^{8} - 3 \beta_{2} q^{9}+O(q^{10})$$ q + (b2 + b1 + 1) * q^2 + b3 * q^3 + (2*b3 + b2 + 2*b1) * q^4 + (b3 - b1 - 3) * q^6 + (-b2 + 2*b1 - 1) * q^7 + (b3 + 3*b2 - 3) * q^8 - 3*b2 * q^9 $$q + (\beta_{2} + \beta_1 + 1) q^{2} + \beta_{3} q^{3} + (2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{4} + (\beta_{3} - \beta_1 - 3) q^{6} + ( - \beta_{2} + 2 \beta_1 - 1) q^{7} + (\beta_{3} + 3 \beta_{2} - 3) q^{8} - 3 \beta_{2} q^{9} + (3 \beta_{3} - 3 \beta_1 + 4) q^{11} + ( - 6 \beta_{2} - \beta_1 - 6) q^{12} + ( - 2 \beta_{3} - 8 \beta_{2} + 8) q^{13} + (\beta_{3} + 4 \beta_{2} + \beta_1) q^{14} + ( - 4 \beta_{3} + 4 \beta_1 - 5) q^{16} + (10 \beta_{2} - 6 \beta_1 + 10) q^{17} + ( - 3 \beta_{3} - 3 \beta_{2} + 3) q^{18} + ( - 6 \beta_{3} - 6 \beta_{2} - 6 \beta_1) q^{19} + ( - \beta_{3} + \beta_1 - 6) q^{21} + ( - 5 \beta_{2} - 2 \beta_1 - 5) q^{22} + (2 \beta_{3} + 14 \beta_{2} - 14) q^{23} + ( - 3 \beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{24} + ( - 10 \beta_{3} + 10 \beta_1 + 22) q^{26} + 3 \beta_1 q^{27} + ( - 2 \beta_{3} + 11 \beta_{2} - 11) q^{28} + ( - 7 \beta_{3} + 18 \beta_{2} - 7 \beta_1) q^{29} + (6 \beta_{3} - 6 \beta_1 - 4) q^{31} + (19 \beta_{2} + 7 \beta_1 + 19) q^{32} + (4 \beta_{3} - 9 \beta_{2} + 9) q^{33} + (4 \beta_{3} + 2 \beta_{2} + 4 \beta_1) q^{34} + ( - 6 \beta_{3} + 6 \beta_1 + 3) q^{36} + ( - 16 \beta_{2} - 18 \beta_1 - 16) q^{37} + ( - 18 \beta_{3} - 24 \beta_{2} + 24) q^{38} + (8 \beta_{3} + 6 \beta_{2} + 8 \beta_1) q^{39} + (6 \beta_{3} - 6 \beta_1 - 14) q^{41} + ( - 3 \beta_{2} - 4 \beta_1 - 3) q^{42} + (20 \beta_{3} - 2 \beta_{2} + 2) q^{43} + (5 \beta_{3} - 32 \beta_{2} + 5 \beta_1) q^{44} + (16 \beta_{3} - 16 \beta_1 - 34) q^{46} + ( - 32 \beta_{2} + 10 \beta_1 - 32) q^{47} + ( - 5 \beta_{3} + 12 \beta_{2} - 12) q^{48} + ( - 4 \beta_{3} - 35 \beta_{2} - 4 \beta_1) q^{49} + (10 \beta_{3} - 10 \beta_1 + 18) q^{51} + (20 \beta_{2} + 34 \beta_1 + 20) q^{52} + (12 \beta_{3} + 14 \beta_{2} - 14) q^{53} + (3 \beta_{3} + 9 \beta_{2} + 3 \beta_1) q^{54} + (5 \beta_{3} - 5 \beta_1) q^{56} + (18 \beta_{2} + 6 \beta_1 + 18) q^{57} + (4 \beta_{3} - 3 \beta_{2} + 3) q^{58} + (31 \beta_{3} + 36 \beta_{2} + 31 \beta_1) q^{59} + ( - 18 \beta_{3} + 18 \beta_1 + 50) q^{61} + ( - 22 \beta_{2} - 16 \beta_1 - 22) q^{62} + ( - 6 \beta_{3} + 3 \beta_{2} - 3) q^{63} + (10 \beta_{3} + 79 \beta_{2} + 10 \beta_1) q^{64} + ( - 5 \beta_{3} + 5 \beta_1 + 6) q^{66} + (50 \beta_{2} + 4 \beta_1 + 50) q^{67} + (34 \beta_{3} - 26 \beta_{2} + 26) q^{68} + ( - 14 \beta_{3} - 6 \beta_{2} - 14 \beta_1) q^{69} - 68 q^{71} + (9 \beta_{2} + 3 \beta_1 + 9) q^{72} + ( - 48 \beta_{3} + 19 \beta_{2} - 19) q^{73} + ( - 34 \beta_{3} - 86 \beta_{2} - 34 \beta_1) q^{74} + ( - 18 \beta_{3} + 18 \beta_1 + 78) q^{76} + ( - 22 \beta_{2} + 14 \beta_1 - 22) q^{77} + (22 \beta_{3} + 30 \beta_{2} - 30) q^{78} + (10 \beta_{3} + 10 \beta_1) q^{79} - 9 q^{81} + ( - 32 \beta_{2} - 26 \beta_1 - 32) q^{82} + ( - 14 \beta_{3} - 4 \beta_{2} + 4) q^{83} + ( - 11 \beta_{3} + 6 \beta_{2} - 11 \beta_1) q^{84} + (18 \beta_{3} - 18 \beta_1 - 56) q^{86} + (21 \beta_{2} - 18 \beta_1 + 21) q^{87} + ( - 14 \beta_{3} + 3 \beta_{2} - 3) q^{88} + ( - 36 \beta_{3} - 6 \beta_{2} - 36 \beta_1) q^{89} + ( - 14 \beta_{3} + 14 \beta_1 - 4) q^{91} + ( - 26 \beta_{2} - 58 \beta_1 - 26) q^{92} + ( - 4 \beta_{3} - 18 \beta_{2} + 18) q^{93} + ( - 22 \beta_{3} - 34 \beta_{2} - 22 \beta_1) q^{94} + (19 \beta_{3} - 19 \beta_1 - 21) q^{96} + (5 \beta_{2} - 16 \beta_1 + 5) q^{97} + ( - 43 \beta_{3} - 47 \beta_{2} + 47) q^{98} + (9 \beta_{3} - 12 \beta_{2} + 9 \beta_1) q^{99}+O(q^{100})$$ q + (b2 + b1 + 1) * q^2 + b3 * q^3 + (2*b3 + b2 + 2*b1) * q^4 + (b3 - b1 - 3) * q^6 + (-b2 + 2*b1 - 1) * q^7 + (b3 + 3*b2 - 3) * q^8 - 3*b2 * q^9 + (3*b3 - 3*b1 + 4) * q^11 + (-6*b2 - b1 - 6) * q^12 + (-2*b3 - 8*b2 + 8) * q^13 + (b3 + 4*b2 + b1) * q^14 + (-4*b3 + 4*b1 - 5) * q^16 + (10*b2 - 6*b1 + 10) * q^17 + (-3*b3 - 3*b2 + 3) * q^18 + (-6*b3 - 6*b2 - 6*b1) * q^19 + (-b3 + b1 - 6) * q^21 + (-5*b2 - 2*b1 - 5) * q^22 + (2*b3 + 14*b2 - 14) * q^23 + (-3*b3 - 3*b2 - 3*b1) * q^24 + (-10*b3 + 10*b1 + 22) * q^26 + 3*b1 * q^27 + (-2*b3 + 11*b2 - 11) * q^28 + (-7*b3 + 18*b2 - 7*b1) * q^29 + (6*b3 - 6*b1 - 4) * q^31 + (19*b2 + 7*b1 + 19) * q^32 + (4*b3 - 9*b2 + 9) * q^33 + (4*b3 + 2*b2 + 4*b1) * q^34 + (-6*b3 + 6*b1 + 3) * q^36 + (-16*b2 - 18*b1 - 16) * q^37 + (-18*b3 - 24*b2 + 24) * q^38 + (8*b3 + 6*b2 + 8*b1) * q^39 + (6*b3 - 6*b1 - 14) * q^41 + (-3*b2 - 4*b1 - 3) * q^42 + (20*b3 - 2*b2 + 2) * q^43 + (5*b3 - 32*b2 + 5*b1) * q^44 + (16*b3 - 16*b1 - 34) * q^46 + (-32*b2 + 10*b1 - 32) * q^47 + (-5*b3 + 12*b2 - 12) * q^48 + (-4*b3 - 35*b2 - 4*b1) * q^49 + (10*b3 - 10*b1 + 18) * q^51 + (20*b2 + 34*b1 + 20) * q^52 + (12*b3 + 14*b2 - 14) * q^53 + (3*b3 + 9*b2 + 3*b1) * q^54 + (5*b3 - 5*b1) * q^56 + (18*b2 + 6*b1 + 18) * q^57 + (4*b3 - 3*b2 + 3) * q^58 + (31*b3 + 36*b2 + 31*b1) * q^59 + (-18*b3 + 18*b1 + 50) * q^61 + (-22*b2 - 16*b1 - 22) * q^62 + (-6*b3 + 3*b2 - 3) * q^63 + (10*b3 + 79*b2 + 10*b1) * q^64 + (-5*b3 + 5*b1 + 6) * q^66 + (50*b2 + 4*b1 + 50) * q^67 + (34*b3 - 26*b2 + 26) * q^68 + (-14*b3 - 6*b2 - 14*b1) * q^69 - 68 * q^71 + (9*b2 + 3*b1 + 9) * q^72 + (-48*b3 + 19*b2 - 19) * q^73 + (-34*b3 - 86*b2 - 34*b1) * q^74 + (-18*b3 + 18*b1 + 78) * q^76 + (-22*b2 + 14*b1 - 22) * q^77 + (22*b3 + 30*b2 - 30) * q^78 + (10*b3 + 10*b1) * q^79 - 9 * q^81 + (-32*b2 - 26*b1 - 32) * q^82 + (-14*b3 - 4*b2 + 4) * q^83 + (-11*b3 + 6*b2 - 11*b1) * q^84 + (18*b3 - 18*b1 - 56) * q^86 + (21*b2 - 18*b1 + 21) * q^87 + (-14*b3 + 3*b2 - 3) * q^88 + (-36*b3 - 6*b2 - 36*b1) * q^89 + (-14*b3 + 14*b1 - 4) * q^91 + (-26*b2 - 58*b1 - 26) * q^92 + (-4*b3 - 18*b2 + 18) * q^93 + (-22*b3 - 34*b2 - 22*b1) * q^94 + (19*b3 - 19*b1 - 21) * q^96 + (5*b2 - 16*b1 + 5) * q^97 + (-43*b3 - 47*b2 + 47) * q^98 + (9*b3 - 12*b2 + 9*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{2} - 12 q^{6} - 4 q^{7} - 12 q^{8}+O(q^{10})$$ 4 * q + 4 * q^2 - 12 * q^6 - 4 * q^7 - 12 * q^8 $$4 q + 4 q^{2} - 12 q^{6} - 4 q^{7} - 12 q^{8} + 16 q^{11} - 24 q^{12} + 32 q^{13} - 20 q^{16} + 40 q^{17} + 12 q^{18} - 24 q^{21} - 20 q^{22} - 56 q^{23} + 88 q^{26} - 44 q^{28} - 16 q^{31} + 76 q^{32} + 36 q^{33} + 12 q^{36} - 64 q^{37} + 96 q^{38} - 56 q^{41} - 12 q^{42} + 8 q^{43} - 136 q^{46} - 128 q^{47} - 48 q^{48} + 72 q^{51} + 80 q^{52} - 56 q^{53} + 72 q^{57} + 12 q^{58} + 200 q^{61} - 88 q^{62} - 12 q^{63} + 24 q^{66} + 200 q^{67} + 104 q^{68} - 272 q^{71} + 36 q^{72} - 76 q^{73} + 312 q^{76} - 88 q^{77} - 120 q^{78} - 36 q^{81} - 128 q^{82} + 16 q^{83} - 224 q^{86} + 84 q^{87} - 12 q^{88} - 16 q^{91} - 104 q^{92} + 72 q^{93} - 84 q^{96} + 20 q^{97} + 188 q^{98}+O(q^{100})$$ 4 * q + 4 * q^2 - 12 * q^6 - 4 * q^7 - 12 * q^8 + 16 * q^11 - 24 * q^12 + 32 * q^13 - 20 * q^16 + 40 * q^17 + 12 * q^18 - 24 * q^21 - 20 * q^22 - 56 * q^23 + 88 * q^26 - 44 * q^28 - 16 * q^31 + 76 * q^32 + 36 * q^33 + 12 * q^36 - 64 * q^37 + 96 * q^38 - 56 * q^41 - 12 * q^42 + 8 * q^43 - 136 * q^46 - 128 * q^47 - 48 * q^48 + 72 * q^51 + 80 * q^52 - 56 * q^53 + 72 * q^57 + 12 * q^58 + 200 * q^61 - 88 * q^62 - 12 * q^63 + 24 * q^66 + 200 * q^67 + 104 * q^68 - 272 * q^71 + 36 * q^72 - 76 * q^73 + 312 * q^76 - 88 * q^77 - 120 * q^78 - 36 * q^81 - 128 * q^82 + 16 * q^83 - 224 * q^86 + 84 * q^87 - 12 * q^88 - 16 * q^91 - 104 * q^92 + 72 * q^93 - 84 * q^96 + 20 * q^97 + 188 * q^98

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 3$$ (v^2) / 3 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 3$$ (v^3) / 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$3\beta_{2}$$ 3*b2 $$\nu^{3}$$ $$=$$ $$3\beta_{3}$$ 3*b3

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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 $$T_{2}^{4} - 4T_{2}^{3} + 8T_{2}^{2} + 4T_{2} + 1$$ acting on $$S_{3}^{\mathrm{new}}(75, [\chi])$$.

## Hecke characteristic polynomials

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