# Properties

 Label 75.3.c.f Level $75$ Weight $3$ Character orbit 75.c Analytic conductor $2.044$ Analytic rank $0$ Dimension $2$ 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(26,75)]

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, 0]))

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

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

chi := DirichletCharacter("75.26");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 75.c (of order $$2$$, degree $$1$$, 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: $$2$$ Coefficient field: $$\Q(\sqrt{-11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 3$$ x^2 - x + 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{-11})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - 2 \beta + 1) q^{2} + ( - \beta + 3) q^{3} - 7 q^{4} + ( - 5 \beta - 3) q^{6} + (6 \beta - 3) q^{8} + ( - 5 \beta + 6) q^{9}+O(q^{10})$$ q + (-2*b + 1) * q^2 + (-b + 3) * q^3 - 7 * q^4 + (-5*b - 3) * q^6 + (6*b - 3) * q^8 + (-5*b + 6) * q^9 $$q + ( - 2 \beta + 1) q^{2} + ( - \beta + 3) q^{3} - 7 q^{4} + ( - 5 \beta - 3) q^{6} + (6 \beta - 3) q^{8} + ( - 5 \beta + 6) q^{9} + (10 \beta - 5) q^{11} + (7 \beta - 21) q^{12} + 10 q^{13} + 5 q^{16} + ( - 2 \beta + 1) q^{17} + ( - 7 \beta - 24) q^{18} + 7 q^{19} + 55 q^{22} + ( - 12 \beta + 6) q^{23} + (15 \beta + 9) q^{24} + ( - 20 \beta + 10) q^{26} + ( - 16 \beta + 3) q^{27} + (20 \beta - 10) q^{29} + 42 q^{31} + (14 \beta - 7) q^{32} + (25 \beta + 15) q^{33} - 11 q^{34} + (35 \beta - 42) q^{36} - 40 q^{37} + ( - 14 \beta + 7) q^{38} + ( - 10 \beta + 30) q^{39} + ( - 10 \beta + 5) q^{41} - 50 q^{43} + ( - 70 \beta + 35) q^{44} - 66 q^{46} + (28 \beta - 14) q^{47} + ( - 5 \beta + 15) q^{48} - 49 q^{49} + ( - 5 \beta - 3) q^{51} - 70 q^{52} + (28 \beta - 14) q^{53} + (10 \beta - 93) q^{54} + ( - 7 \beta + 21) q^{57} + 110 q^{58} + (40 \beta - 20) q^{59} - 8 q^{61} + ( - 84 \beta + 42) q^{62} + 97 q^{64} + ( - 55 \beta + 165) q^{66} + 45 q^{67} + (14 \beta - 7) q^{68} + ( - 30 \beta - 18) q^{69} + (20 \beta - 10) q^{71} + (21 \beta + 72) q^{72} - 35 q^{73} + (80 \beta - 40) q^{74} - 49 q^{76} + ( - 50 \beta - 30) q^{78} + 12 q^{79} + ( - 35 \beta - 39) q^{81} - 55 q^{82} + ( - 42 \beta + 21) q^{83} + (100 \beta - 50) q^{86} + (50 \beta + 30) q^{87} - 165 q^{88} + ( - 90 \beta + 45) q^{89} + (84 \beta - 42) q^{92} + ( - 42 \beta + 126) q^{93} + 154 q^{94} + (35 \beta + 21) q^{96} - 70 q^{97} + (98 \beta - 49) q^{98} + (35 \beta + 120) q^{99}+O(q^{100})$$ q + (-2*b + 1) * q^2 + (-b + 3) * q^3 - 7 * q^4 + (-5*b - 3) * q^6 + (6*b - 3) * q^8 + (-5*b + 6) * q^9 + (10*b - 5) * q^11 + (7*b - 21) * q^12 + 10 * q^13 + 5 * q^16 + (-2*b + 1) * q^17 + (-7*b - 24) * q^18 + 7 * q^19 + 55 * q^22 + (-12*b + 6) * q^23 + (15*b + 9) * q^24 + (-20*b + 10) * q^26 + (-16*b + 3) * q^27 + (20*b - 10) * q^29 + 42 * q^31 + (14*b - 7) * q^32 + (25*b + 15) * q^33 - 11 * q^34 + (35*b - 42) * q^36 - 40 * q^37 + (-14*b + 7) * q^38 + (-10*b + 30) * q^39 + (-10*b + 5) * q^41 - 50 * q^43 + (-70*b + 35) * q^44 - 66 * q^46 + (28*b - 14) * q^47 + (-5*b + 15) * q^48 - 49 * q^49 + (-5*b - 3) * q^51 - 70 * q^52 + (28*b - 14) * q^53 + (10*b - 93) * q^54 + (-7*b + 21) * q^57 + 110 * q^58 + (40*b - 20) * q^59 - 8 * q^61 + (-84*b + 42) * q^62 + 97 * q^64 + (-55*b + 165) * q^66 + 45 * q^67 + (14*b - 7) * q^68 + (-30*b - 18) * q^69 + (20*b - 10) * q^71 + (21*b + 72) * q^72 - 35 * q^73 + (80*b - 40) * q^74 - 49 * q^76 + (-50*b - 30) * q^78 + 12 * q^79 + (-35*b - 39) * q^81 - 55 * q^82 + (-42*b + 21) * q^83 + (100*b - 50) * q^86 + (50*b + 30) * q^87 - 165 * q^88 + (-90*b + 45) * q^89 + (84*b - 42) * q^92 + (-42*b + 126) * q^93 + 154 * q^94 + (35*b + 21) * q^96 - 70 * q^97 + (98*b - 49) * q^98 + (35*b + 120) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 5 q^{3} - 14 q^{4} - 11 q^{6} + 7 q^{9}+O(q^{10})$$ 2 * q + 5 * q^3 - 14 * q^4 - 11 * q^6 + 7 * q^9 $$2 q + 5 q^{3} - 14 q^{4} - 11 q^{6} + 7 q^{9} - 35 q^{12} + 20 q^{13} + 10 q^{16} - 55 q^{18} + 14 q^{19} + 110 q^{22} + 33 q^{24} - 10 q^{27} + 84 q^{31} + 55 q^{33} - 22 q^{34} - 49 q^{36} - 80 q^{37} + 50 q^{39} - 100 q^{43} - 132 q^{46} + 25 q^{48} - 98 q^{49} - 11 q^{51} - 140 q^{52} - 176 q^{54} + 35 q^{57} + 220 q^{58} - 16 q^{61} + 194 q^{64} + 275 q^{66} + 90 q^{67} - 66 q^{69} + 165 q^{72} - 70 q^{73} - 98 q^{76} - 110 q^{78} + 24 q^{79} - 113 q^{81} - 110 q^{82} + 110 q^{87} - 330 q^{88} + 210 q^{93} + 308 q^{94} + 77 q^{96} - 140 q^{97} + 275 q^{99}+O(q^{100})$$ 2 * q + 5 * q^3 - 14 * q^4 - 11 * q^6 + 7 * q^9 - 35 * q^12 + 20 * q^13 + 10 * q^16 - 55 * q^18 + 14 * q^19 + 110 * q^22 + 33 * q^24 - 10 * q^27 + 84 * q^31 + 55 * q^33 - 22 * q^34 - 49 * q^36 - 80 * q^37 + 50 * q^39 - 100 * q^43 - 132 * q^46 + 25 * q^48 - 98 * q^49 - 11 * q^51 - 140 * q^52 - 176 * q^54 + 35 * q^57 + 220 * q^58 - 16 * q^61 + 194 * q^64 + 275 * q^66 + 90 * q^67 - 66 * q^69 + 165 * q^72 - 70 * q^73 - 98 * q^76 - 110 * q^78 + 24 * q^79 - 113 * q^81 - 110 * q^82 + 110 * q^87 - 330 * q^88 + 210 * q^93 + 308 * q^94 + 77 * q^96 - 140 * q^97 + 275 * q^99

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
26.1
 0.5 + 1.65831i 0.5 − 1.65831i
3.31662i 2.50000 1.65831i −7.00000 0 −5.50000 8.29156i 0 9.94987i 3.50000 8.29156i 0
26.2 3.31662i 2.50000 + 1.65831i −7.00000 0 −5.50000 + 8.29156i 0 9.94987i 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

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(75, [\chi])$$:

 $$T_{2}^{2} + 11$$ T2^2 + 11 $$T_{7}$$ T7 $$T_{13} - 10$$ T13 - 10

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 11$$
$3$ $$T^{2} - 5T + 9$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 275$$
$13$ $$(T - 10)^{2}$$
$17$ $$T^{2} + 11$$
$19$ $$(T - 7)^{2}$$
$23$ $$T^{2} + 396$$
$29$ $$T^{2} + 1100$$
$31$ $$(T - 42)^{2}$$
$37$ $$(T + 40)^{2}$$
$41$ $$T^{2} + 275$$
$43$ $$(T + 50)^{2}$$
$47$ $$T^{2} + 2156$$
$53$ $$T^{2} + 2156$$
$59$ $$T^{2} + 4400$$
$61$ $$(T + 8)^{2}$$
$67$ $$(T - 45)^{2}$$
$71$ $$T^{2} + 1100$$
$73$ $$(T + 35)^{2}$$
$79$ $$(T - 12)^{2}$$
$83$ $$T^{2} + 4851$$
$89$ $$T^{2} + 22275$$
$97$ $$(T + 70)^{2}$$