# 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

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: $$\Q(i, \sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 3x^{2} + 1$$ 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

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.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{2} + ( - \beta_{3} + \beta_1) q^{3} + q^{4} + ( - \beta_{2} + 5) q^{6} - 3 \beta_1 q^{7} + 3 \beta_{3} q^{8} + ( - 2 \beta_{2} + 1) q^{9}+O(q^{10})$$ q - b3 * q^2 + (-b3 + b1) * q^3 + q^4 + (-b2 + 5) * q^6 - 3*b1 * q^7 + 3*b3 * q^8 + (-2*b2 + 1) * q^9 $$q - \beta_{3} q^{2} + ( - \beta_{3} + \beta_1) q^{3} + q^{4} + ( - \beta_{2} + 5) q^{6} - 3 \beta_1 q^{7} + 3 \beta_{3} q^{8} + ( - 2 \beta_{2} + 1) q^{9} - \beta_{2} q^{11} + ( - \beta_{3} + \beta_1) q^{12} - 8 \beta_1 q^{13} + 3 \beta_{2} q^{14} - 19 q^{16} + 2 \beta_{3} q^{17} + ( - \beta_{3} + 10 \beta_1) q^{18} + 2 q^{19} + (3 \beta_{2} + 12) q^{21} + 5 \beta_1 q^{22} - 6 \beta_{3} q^{23} + (3 \beta_{2} - 15) q^{24} + 8 \beta_{2} q^{26} + (7 \beta_{3} + 11 \beta_1) q^{27} - 3 \beta_1 q^{28} - 7 \beta_{2} q^{29} - 18 q^{31} + 7 \beta_{3} q^{32} + (4 \beta_{3} + 5 \beta_1) q^{33} - 10 q^{34} + ( - 2 \beta_{2} + 1) q^{36} - 8 \beta_1 q^{37} - 2 \beta_{3} q^{38} + (8 \beta_{2} + 32) q^{39} - 14 \beta_{2} q^{41} + ( - 12 \beta_{3} - 15 \beta_1) q^{42} - 8 \beta_1 q^{43} - \beta_{2} q^{44} + 30 q^{46} - 22 \beta_{3} q^{47} + (19 \beta_{3} - 19 \beta_1) q^{48} + 13 q^{49} + (2 \beta_{2} - 10) q^{51} - 8 \beta_1 q^{52} - 2 \beta_{3} q^{53} + ( - 11 \beta_{2} - 35) q^{54} - 9 \beta_{2} q^{56} + ( - 2 \beta_{3} + 2 \beta_1) q^{57} + 35 \beta_1 q^{58} + \beta_{2} q^{59} + 82 q^{61} + 18 \beta_{3} q^{62} + ( - 24 \beta_{3} - 3 \beta_1) q^{63} + 41 q^{64} + ( - 5 \beta_{2} - 20) q^{66} + 12 \beta_1 q^{67} + 2 \beta_{3} q^{68} + ( - 6 \beta_{2} + 30) q^{69} + 28 \beta_{2} q^{71} + (3 \beta_{3} - 30 \beta_1) q^{72} + 37 \beta_1 q^{73} + 8 \beta_{2} q^{74} + 2 q^{76} - 12 \beta_{3} q^{77} + ( - 32 \beta_{3} - 40 \beta_1) q^{78} - 138 q^{79} + ( - 4 \beta_{2} - 79) q^{81} + 70 \beta_1 q^{82} + 42 \beta_{3} q^{83} + (3 \beta_{2} + 12) q^{84} + 8 \beta_{2} q^{86} + (28 \beta_{3} + 35 \beta_1) q^{87} - 15 \beta_1 q^{88} + 24 \beta_{2} q^{89} - 96 q^{91} - 6 \beta_{3} q^{92} + (18 \beta_{3} - 18 \beta_1) q^{93} + 110 q^{94} + (7 \beta_{2} - 35) q^{96} - 83 \beta_1 q^{97} - 13 \beta_{3} q^{98} + ( - \beta_{2} - 40) q^{99}+O(q^{100})$$ q - b3 * q^2 + (-b3 + b1) * q^3 + q^4 + (-b2 + 5) * q^6 - 3*b1 * q^7 + 3*b3 * q^8 + (-2*b2 + 1) * q^9 - b2 * q^11 + (-b3 + b1) * q^12 - 8*b1 * q^13 + 3*b2 * q^14 - 19 * q^16 + 2*b3 * q^17 + (-b3 + 10*b1) * q^18 + 2 * q^19 + (3*b2 + 12) * q^21 + 5*b1 * q^22 - 6*b3 * q^23 + (3*b2 - 15) * q^24 + 8*b2 * q^26 + (7*b3 + 11*b1) * q^27 - 3*b1 * q^28 - 7*b2 * q^29 - 18 * q^31 + 7*b3 * q^32 + (4*b3 + 5*b1) * q^33 - 10 * q^34 + (-2*b2 + 1) * q^36 - 8*b1 * q^37 - 2*b3 * q^38 + (8*b2 + 32) * q^39 - 14*b2 * q^41 + (-12*b3 - 15*b1) * q^42 - 8*b1 * q^43 - b2 * q^44 + 30 * q^46 - 22*b3 * q^47 + (19*b3 - 19*b1) * q^48 + 13 * q^49 + (2*b2 - 10) * q^51 - 8*b1 * q^52 - 2*b3 * q^53 + (-11*b2 - 35) * q^54 - 9*b2 * q^56 + (-2*b3 + 2*b1) * q^57 + 35*b1 * q^58 + b2 * q^59 + 82 * q^61 + 18*b3 * q^62 + (-24*b3 - 3*b1) * q^63 + 41 * q^64 + (-5*b2 - 20) * q^66 + 12*b1 * q^67 + 2*b3 * q^68 + (-6*b2 + 30) * q^69 + 28*b2 * q^71 + (3*b3 - 30*b1) * q^72 + 37*b1 * q^73 + 8*b2 * q^74 + 2 * q^76 - 12*b3 * q^77 + (-32*b3 - 40*b1) * q^78 - 138 * q^79 + (-4*b2 - 79) * q^81 + 70*b1 * q^82 + 42*b3 * q^83 + (3*b2 + 12) * q^84 + 8*b2 * q^86 + (28*b3 + 35*b1) * q^87 - 15*b1 * q^88 + 24*b2 * q^89 - 96 * q^91 - 6*b3 * q^92 + (18*b3 - 18*b1) * q^93 + 110 * q^94 + (7*b2 - 35) * q^96 - 83*b1 * q^97 - 13*b3 * q^98 + (-b2 - 40) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{4} + 20 q^{6} + 4 q^{9}+O(q^{10})$$ 4 * q + 4 * q^4 + 20 * q^6 + 4 * q^9 $$4 q + 4 q^{4} + 20 q^{6} + 4 q^{9} - 76 q^{16} + 8 q^{19} + 48 q^{21} - 60 q^{24} - 72 q^{31} - 40 q^{34} + 4 q^{36} + 128 q^{39} + 120 q^{46} + 52 q^{49} - 40 q^{51} - 140 q^{54} + 328 q^{61} + 164 q^{64} - 80 q^{66} + 120 q^{69} + 8 q^{76} - 552 q^{79} - 316 q^{81} + 48 q^{84} - 384 q^{91} + 440 q^{94} - 140 q^{96} - 160 q^{99}+O(q^{100})$$ 4 * q + 4 * q^4 + 20 * q^6 + 4 * q^9 - 76 * q^16 + 8 * q^19 + 48 * q^21 - 60 * q^24 - 72 * q^31 - 40 * q^34 + 4 * q^36 + 128 * q^39 + 120 * q^46 + 52 * q^49 - 40 * q^51 - 140 * q^54 + 328 * q^61 + 164 * q^64 - 80 * q^66 + 120 * q^69 + 8 * q^76 - 552 * q^79 - 316 * q^81 + 48 * q^84 - 384 * q^91 + 440 * q^94 - 140 * q^96 - 160 * q^99

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

 $$\beta_{1}$$ $$=$$ $$2\nu^{3} + 4\nu$$ 2*v^3 + 4*v $$\beta_{2}$$ $$=$$ $$2\nu^{3} + 8\nu$$ 2*v^3 + 8*v $$\beta_{3}$$ $$=$$ $$2\nu^{2} + 3$$ 2*v^2 + 3
 $$\nu$$ $$=$$ $$( \beta_{2} - \beta_1 ) / 4$$ (b2 - b1) / 4 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - 3 ) / 2$$ (b3 - 3) / 2 $$\nu^{3}$$ $$=$$ $$( -\beta_{2} + 2\beta_1 ) / 2$$ (-b2 + 2*b1) / 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.

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
 − 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

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