# 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

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,3,Mod(74,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("75.74");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 75.d (of order $$2$$, degree $$1$$, not 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: $$4$$ Coefficient field: $$\Q(i, \sqrt{11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 5x^{2} + 9$$ 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

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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} - \beta_1) q^{2} + \beta_1 q^{3} + 7 q^{4} + ( - \beta_{3} - 6) q^{6} + (3 \beta_{2} - 3 \beta_1) q^{8} + (\beta_{3} - 3) q^{9}+O(q^{10})$$ q + (b2 - b1) * q^2 + b1 * q^3 + 7 * q^4 + (-b3 - 6) * q^6 + (3*b2 - 3*b1) * q^8 + (b3 - 3) * q^9 $$q + (\beta_{2} - \beta_1) q^{2} + \beta_1 q^{3} + 7 q^{4} + ( - \beta_{3} - 6) q^{6} + (3 \beta_{2} - 3 \beta_1) q^{8} + (\beta_{3} - 3) q^{9} + (2 \beta_{3} + 1) q^{11} + 7 \beta_1 q^{12} + (2 \beta_{2} + 2 \beta_1) q^{13} + 5 q^{16} + (\beta_{2} - \beta_1) q^{17} + ( - 9 \beta_{2} - 2 \beta_1) q^{18} - 7 q^{19} + ( - 11 \beta_{2} - 11 \beta_1) q^{22} + ( - 6 \beta_{2} + 6 \beta_1) q^{23} + ( - 3 \beta_{3} - 18) q^{24} + ( - 4 \beta_{3} - 2) q^{26} + (9 \beta_{2} - 7 \beta_1) q^{27} + ( - 4 \beta_{3} - 2) q^{29} + 42 q^{31} + ( - 7 \beta_{2} + 7 \beta_1) q^{32} + (18 \beta_{2} - 7 \beta_1) q^{33} + 11 q^{34} + (7 \beta_{3} - 21) q^{36} + (8 \beta_{2} + 8 \beta_1) q^{37} + ( - 7 \beta_{2} + 7 \beta_1) q^{38} + (2 \beta_{3} - 24) q^{39} + ( - 2 \beta_{3} - 1) q^{41} + ( - 10 \beta_{2} - 10 \beta_1) q^{43} + (14 \beta_{3} + 7) q^{44} - 66 q^{46} + ( - 14 \beta_{2} + 14 \beta_1) q^{47} + 5 \beta_1 q^{48} + 49 q^{49} + ( - \beta_{3} - 6) q^{51} + (14 \beta_{2} + 14 \beta_1) q^{52} + (14 \beta_{2} - 14 \beta_1) q^{53} + ( - 2 \beta_{3} + 87) q^{54} - 7 \beta_1 q^{57} + (22 \beta_{2} + 22 \beta_1) q^{58} + ( - 8 \beta_{3} - 4) q^{59} - 8 q^{61} + (42 \beta_{2} - 42 \beta_1) q^{62} - 97 q^{64} + ( - 11 \beta_{3} + 132) q^{66} + ( - 9 \beta_{2} - 9 \beta_1) q^{67} + (7 \beta_{2} - 7 \beta_1) q^{68} + (6 \beta_{3} + 36) q^{69} + (4 \beta_{3} + 2) q^{71} + ( - 27 \beta_{2} - 6 \beta_1) q^{72} + ( - 7 \beta_{2} - 7 \beta_1) q^{73} + ( - 16 \beta_{3} - 8) q^{74} - 49 q^{76} + ( - 36 \beta_{2} + 14 \beta_1) q^{78} - 12 q^{79} + ( - 7 \beta_{3} - 60) q^{81} + (11 \beta_{2} + 11 \beta_1) q^{82} + ( - 21 \beta_{2} + 21 \beta_1) q^{83} + (20 \beta_{3} + 10) q^{86} + ( - 36 \beta_{2} + 14 \beta_1) q^{87} + ( - 33 \beta_{2} - 33 \beta_1) q^{88} + (18 \beta_{3} + 9) q^{89} + ( - 42 \beta_{2} + 42 \beta_1) q^{92} + 42 \beta_1 q^{93} - 154 q^{94} + (7 \beta_{3} + 42) q^{96} + (14 \beta_{2} + 14 \beta_1) q^{97} + (49 \beta_{2} - 49 \beta_1) q^{98} + ( - 7 \beta_{3} - 141) q^{99}+O(q^{100})$$ q + (b2 - b1) * q^2 + b1 * q^3 + 7 * q^4 + (-b3 - 6) * q^6 + (3*b2 - 3*b1) * q^8 + (b3 - 3) * q^9 + (2*b3 + 1) * q^11 + 7*b1 * q^12 + (2*b2 + 2*b1) * q^13 + 5 * q^16 + (b2 - b1) * q^17 + (-9*b2 - 2*b1) * q^18 - 7 * q^19 + (-11*b2 - 11*b1) * q^22 + (-6*b2 + 6*b1) * q^23 + (-3*b3 - 18) * q^24 + (-4*b3 - 2) * q^26 + (9*b2 - 7*b1) * q^27 + (-4*b3 - 2) * q^29 + 42 * q^31 + (-7*b2 + 7*b1) * q^32 + (18*b2 - 7*b1) * q^33 + 11 * q^34 + (7*b3 - 21) * q^36 + (8*b2 + 8*b1) * q^37 + (-7*b2 + 7*b1) * q^38 + (2*b3 - 24) * q^39 + (-2*b3 - 1) * q^41 + (-10*b2 - 10*b1) * q^43 + (14*b3 + 7) * q^44 - 66 * q^46 + (-14*b2 + 14*b1) * q^47 + 5*b1 * q^48 + 49 * q^49 + (-b3 - 6) * q^51 + (14*b2 + 14*b1) * q^52 + (14*b2 - 14*b1) * q^53 + (-2*b3 + 87) * q^54 - 7*b1 * q^57 + (22*b2 + 22*b1) * q^58 + (-8*b3 - 4) * q^59 - 8 * q^61 + (42*b2 - 42*b1) * q^62 - 97 * q^64 + (-11*b3 + 132) * q^66 + (-9*b2 - 9*b1) * q^67 + (7*b2 - 7*b1) * q^68 + (6*b3 + 36) * q^69 + (4*b3 + 2) * q^71 + (-27*b2 - 6*b1) * q^72 + (-7*b2 - 7*b1) * q^73 + (-16*b3 - 8) * q^74 - 49 * q^76 + (-36*b2 + 14*b1) * q^78 - 12 * q^79 + (-7*b3 - 60) * q^81 + (11*b2 + 11*b1) * q^82 + (-21*b2 + 21*b1) * q^83 + (20*b3 + 10) * q^86 + (-36*b2 + 14*b1) * q^87 + (-33*b2 - 33*b1) * q^88 + (18*b3 + 9) * q^89 + (-42*b2 + 42*b1) * q^92 + 42*b1 * q^93 - 154 * q^94 + (7*b3 + 42) * q^96 + (14*b2 + 14*b1) * q^97 + (49*b2 - 49*b1) * q^98 + (-7*b3 - 141) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 28 q^{4} - 22 q^{6} - 14 q^{9}+O(q^{10})$$ 4 * q + 28 * q^4 - 22 * q^6 - 14 * q^9 $$4 q + 28 q^{4} - 22 q^{6} - 14 q^{9} + 20 q^{16} - 28 q^{19} - 66 q^{24} + 168 q^{31} + 44 q^{34} - 98 q^{36} - 100 q^{39} - 264 q^{46} + 196 q^{49} - 22 q^{51} + 352 q^{54} - 32 q^{61} - 388 q^{64} + 550 q^{66} + 132 q^{69} - 196 q^{76} - 48 q^{79} - 226 q^{81} - 616 q^{94} + 154 q^{96} - 550 q^{99}+O(q^{100})$$ 4 * q + 28 * q^4 - 22 * q^6 - 14 * q^9 + 20 * q^16 - 28 * q^19 - 66 * q^24 + 168 * q^31 + 44 * q^34 - 98 * q^36 - 100 * q^39 - 264 * q^46 + 196 * q^49 - 22 * q^51 + 352 * q^54 - 32 * q^61 - 388 * q^64 + 550 * q^66 + 132 * q^69 - 196 * q^76 - 48 * q^79 - 226 * q^81 - 616 * q^94 + 154 * q^96 - 550 * q^99

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

 $$\beta_{1}$$ $$=$$ $$( -2\nu^{3} + \nu ) / 3$$ (-2*v^3 + v) / 3 $$\beta_{2}$$ $$=$$ $$-\nu^{3} + 3\nu$$ -v^3 + 3*v $$\beta_{3}$$ $$=$$ $$5\nu^{2} - 13$$ 5*v^2 - 13
 $$\nu$$ $$=$$ $$( 2\beta_{2} - 3\beta_1 ) / 5$$ (2*b2 - 3*b1) / 5 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 13 ) / 5$$ (b3 + 13) / 5 $$\nu^{3}$$ $$=$$ $$( \beta_{2} - 9\beta_1 ) / 5$$ (b2 - 9*b1) / 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
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

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