# Properties

 Label 75.3.c.c 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.

 $$f(q)$$ $$=$$ $$q + ( - 2 \beta + 1) q^{2} + ( - \beta - 2) q^{3} - 7 q^{4} + (5 \beta - 8) q^{6} + (6 \beta - 3) q^{8} + (5 \beta + 1) q^{9}+O(q^{10})$$ q + (-2*b + 1) * q^2 + (-b - 2) * q^3 - 7 * q^4 + (5*b - 8) * q^6 + (6*b - 3) * q^8 + (5*b + 1) * q^9 $$q + ( - 2 \beta + 1) q^{2} + ( - \beta - 2) q^{3} - 7 q^{4} + (5 \beta - 8) q^{6} + (6 \beta - 3) q^{8} + (5 \beta + 1) q^{9} + ( - 10 \beta + 5) q^{11} + (7 \beta + 14) q^{12} - 10 q^{13} + 5 q^{16} + ( - 2 \beta + 1) q^{17} + ( - 7 \beta + 31) q^{18} + 7 q^{19} - 55 q^{22} + ( - 12 \beta + 6) q^{23} + ( - 15 \beta + 24) q^{24} + (20 \beta - 10) q^{26} + ( - 16 \beta + 13) q^{27} + ( - 20 \beta + 10) q^{29} + 42 q^{31} + (14 \beta - 7) q^{32} + (25 \beta - 40) q^{33} - 11 q^{34} + ( - 35 \beta - 7) q^{36} + 40 q^{37} + ( - 14 \beta + 7) q^{38} + (10 \beta + 20) q^{39} + (10 \beta - 5) q^{41} + 50 q^{43} + (70 \beta - 35) q^{44} - 66 q^{46} + (28 \beta - 14) q^{47} + ( - 5 \beta - 10) q^{48} - 49 q^{49} + (5 \beta - 8) q^{51} + 70 q^{52} + (28 \beta - 14) q^{53} + ( - 10 \beta - 83) q^{54} + ( - 7 \beta - 14) q^{57} - 110 q^{58} + ( - 40 \beta + 20) q^{59} - 8 q^{61} + ( - 84 \beta + 42) q^{62} + 97 q^{64} + (55 \beta + 110) q^{66} - 45 q^{67} + (14 \beta - 7) q^{68} + (30 \beta - 48) q^{69} + ( - 20 \beta + 10) q^{71} + (21 \beta - 93) q^{72} + 35 q^{73} + ( - 80 \beta + 40) q^{74} - 49 q^{76} + ( - 50 \beta + 80) q^{78} + 12 q^{79} + (35 \beta - 74) q^{81} + 55 q^{82} + ( - 42 \beta + 21) q^{83} + ( - 100 \beta + 50) q^{86} + (50 \beta - 80) q^{87} + 165 q^{88} + (90 \beta - 45) q^{89} + (84 \beta - 42) q^{92} + ( - 42 \beta - 84) q^{93} + 154 q^{94} + ( - 35 \beta + 56) q^{96} + 70 q^{97} + (98 \beta - 49) q^{98} + ( - 35 \beta + 155) q^{99}+O(q^{100})$$ q + (-2*b + 1) * q^2 + (-b - 2) * q^3 - 7 * q^4 + (5*b - 8) * q^6 + (6*b - 3) * q^8 + (5*b + 1) * q^9 + (-10*b + 5) * q^11 + (7*b + 14) * q^12 - 10 * q^13 + 5 * q^16 + (-2*b + 1) * q^17 + (-7*b + 31) * q^18 + 7 * q^19 - 55 * q^22 + (-12*b + 6) * q^23 + (-15*b + 24) * q^24 + (20*b - 10) * q^26 + (-16*b + 13) * q^27 + (-20*b + 10) * q^29 + 42 * q^31 + (14*b - 7) * q^32 + (25*b - 40) * q^33 - 11 * q^34 + (-35*b - 7) * q^36 + 40 * q^37 + (-14*b + 7) * q^38 + (10*b + 20) * 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 - 10) * q^48 - 49 * q^49 + (5*b - 8) * q^51 + 70 * q^52 + (28*b - 14) * q^53 + (-10*b - 83) * q^54 + (-7*b - 14) * q^57 - 110 * q^58 + (-40*b + 20) * q^59 - 8 * q^61 + (-84*b + 42) * q^62 + 97 * q^64 + (55*b + 110) * q^66 - 45 * q^67 + (14*b - 7) * q^68 + (30*b - 48) * q^69 + (-20*b + 10) * q^71 + (21*b - 93) * q^72 + 35 * q^73 + (-80*b + 40) * q^74 - 49 * q^76 + (-50*b + 80) * q^78 + 12 * q^79 + (35*b - 74) * q^81 + 55 * q^82 + (-42*b + 21) * q^83 + (-100*b + 50) * q^86 + (50*b - 80) * q^87 + 165 * q^88 + (90*b - 45) * q^89 + (84*b - 42) * q^92 + (-42*b - 84) * q^93 + 154 * q^94 + (-35*b + 56) * q^96 + 70 * q^97 + (98*b - 49) * q^98 + (-35*b + 155) * 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.c 2
3.b odd 2 1 inner 75.3.c.c 2
4.b odd 2 1 1200.3.l.s 2
5.b even 2 1 75.3.c.f yes 2
5.c odd 4 2 75.3.d.c 4
12.b even 2 1 1200.3.l.s 2
15.d odd 2 1 75.3.c.f yes 2
15.e even 4 2 75.3.d.c 4
20.d odd 2 1 1200.3.l.f 2
20.e even 4 2 1200.3.c.d 4
60.h even 2 1 1200.3.l.f 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 1.a even 1 1 trivial
75.3.c.c 2 3.b odd 2 1 inner
75.3.c.f yes 2 5.b even 2 1
75.3.c.f yes 2 15.d odd 2 1
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 20.d odd 2 1
1200.3.l.f 2 60.h even 2 1
1200.3.l.s 2 4.b odd 2 1
1200.3.l.s 2 12.b 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

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