# Properties

 Label 75.3.c.d Level $75$ Weight $3$ Character orbit 75.c Analytic conductor $2.044$ Analytic rank $0$ Dimension $2$ CM discriminant -15 Inner twists $4$

# 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: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{2} + 3 i q^{3} + 3 q^{4} - 3 q^{6} + 7 i q^{8} - 9 q^{9}+O(q^{10})$$ q + i * q^2 + 3*i * q^3 + 3 * q^4 - 3 * q^6 + 7*i * q^8 - 9 * q^9 $$q + i q^{2} + 3 i q^{3} + 3 q^{4} - 3 q^{6} + 7 i q^{8} - 9 q^{9} + 9 i q^{12} + 5 q^{16} - 14 i q^{17} - 9 i q^{18} + 22 q^{19} - 34 i q^{23} - 21 q^{24} - 27 i q^{27} + 2 q^{31} + 33 i q^{32} + 14 q^{34} - 27 q^{36} + 22 i q^{38} + 34 q^{46} - 14 i q^{47} + 15 i q^{48} - 49 q^{49} + 42 q^{51} + 86 i q^{53} + 27 q^{54} + 66 i q^{57} - 118 q^{61} + 2 i q^{62} - 13 q^{64} - 42 i q^{68} + 102 q^{69} - 63 i q^{72} + 66 q^{76} - 98 q^{79} + 81 q^{81} - 154 i q^{83} - 102 i q^{92} + 6 i q^{93} + 14 q^{94} - 99 q^{96} - 49 i q^{98} +O(q^{100})$$ q + i * q^2 + 3*i * q^3 + 3 * q^4 - 3 * q^6 + 7*i * q^8 - 9 * q^9 + 9*i * q^12 + 5 * q^16 - 14*i * q^17 - 9*i * q^18 + 22 * q^19 - 34*i * q^23 - 21 * q^24 - 27*i * q^27 + 2 * q^31 + 33*i * q^32 + 14 * q^34 - 27 * q^36 + 22*i * q^38 + 34 * q^46 - 14*i * q^47 + 15*i * q^48 - 49 * q^49 + 42 * q^51 + 86*i * q^53 + 27 * q^54 + 66*i * q^57 - 118 * q^61 + 2*i * q^62 - 13 * q^64 - 42*i * q^68 + 102 * q^69 - 63*i * q^72 + 66 * q^76 - 98 * q^79 + 81 * q^81 - 154*i * q^83 - 102*i * q^92 + 6*i * q^93 + 14 * q^94 - 99 * q^96 - 49*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{4} - 6 q^{6} - 18 q^{9}+O(q^{10})$$ 2 * q + 6 * q^4 - 6 * q^6 - 18 * q^9 $$2 q + 6 q^{4} - 6 q^{6} - 18 q^{9} + 10 q^{16} + 44 q^{19} - 42 q^{24} + 4 q^{31} + 28 q^{34} - 54 q^{36} + 68 q^{46} - 98 q^{49} + 84 q^{51} + 54 q^{54} - 236 q^{61} - 26 q^{64} + 204 q^{69} + 132 q^{76} - 196 q^{79} + 162 q^{81} + 28 q^{94} - 198 q^{96}+O(q^{100})$$ 2 * q + 6 * q^4 - 6 * q^6 - 18 * q^9 + 10 * q^16 + 44 * q^19 - 42 * q^24 + 4 * q^31 + 28 * q^34 - 54 * q^36 + 68 * q^46 - 98 * q^49 + 84 * q^51 + 54 * q^54 - 236 * q^61 - 26 * q^64 + 204 * q^69 + 132 * q^76 - 196 * q^79 + 162 * q^81 + 28 * q^94 - 198 * q^96

## 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
 − 1.00000i 1.00000i
1.00000i 3.00000i 3.00000 0 −3.00000 0 7.00000i −9.00000 0
26.2 1.00000i 3.00000i 3.00000 0 −3.00000 0 7.00000i −9.00000 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
15.d odd 2 1 CM by $$\Q(\sqrt{-15})$$
3.b odd 2 1 inner
5.b even 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.d 2
3.b odd 2 1 inner 75.3.c.d 2
4.b odd 2 1 1200.3.l.l 2
5.b even 2 1 inner 75.3.c.d 2
5.c odd 4 1 15.3.d.a 1
5.c odd 4 1 15.3.d.b yes 1
12.b even 2 1 1200.3.l.l 2
15.d odd 2 1 CM 75.3.c.d 2
15.e even 4 1 15.3.d.a 1
15.e even 4 1 15.3.d.b yes 1
20.d odd 2 1 1200.3.l.l 2
20.e even 4 1 240.3.c.a 1
20.e even 4 1 240.3.c.b 1
40.i odd 4 1 960.3.c.b 1
40.i odd 4 1 960.3.c.c 1
40.k even 4 1 960.3.c.a 1
40.k even 4 1 960.3.c.d 1
45.k odd 12 2 405.3.h.a 2
45.k odd 12 2 405.3.h.b 2
45.l even 12 2 405.3.h.a 2
45.l even 12 2 405.3.h.b 2
60.h even 2 1 1200.3.l.l 2
60.l odd 4 1 240.3.c.a 1
60.l odd 4 1 240.3.c.b 1
120.q odd 4 1 960.3.c.a 1
120.q odd 4 1 960.3.c.d 1
120.w even 4 1 960.3.c.b 1
120.w even 4 1 960.3.c.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.3.d.a 1 5.c odd 4 1
15.3.d.a 1 15.e even 4 1
15.3.d.b yes 1 5.c odd 4 1
15.3.d.b yes 1 15.e even 4 1
75.3.c.d 2 1.a even 1 1 trivial
75.3.c.d 2 3.b odd 2 1 inner
75.3.c.d 2 5.b even 2 1 inner
75.3.c.d 2 15.d odd 2 1 CM
240.3.c.a 1 20.e even 4 1
240.3.c.a 1 60.l odd 4 1
240.3.c.b 1 20.e even 4 1
240.3.c.b 1 60.l odd 4 1
405.3.h.a 2 45.k odd 12 2
405.3.h.a 2 45.l even 12 2
405.3.h.b 2 45.k odd 12 2
405.3.h.b 2 45.l even 12 2
960.3.c.a 1 40.k even 4 1
960.3.c.a 1 120.q odd 4 1
960.3.c.b 1 40.i odd 4 1
960.3.c.b 1 120.w even 4 1
960.3.c.c 1 40.i odd 4 1
960.3.c.c 1 120.w even 4 1
960.3.c.d 1 40.k even 4 1
960.3.c.d 1 120.q odd 4 1
1200.3.l.l 2 4.b odd 2 1
1200.3.l.l 2 12.b even 2 1
1200.3.l.l 2 20.d odd 2 1
1200.3.l.l 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 $$S_{3}^{\mathrm{new}}(75, [\chi])$$:

 $$T_{2}^{2} + 1$$ T2^2 + 1 $$T_{7}$$ T7 $$T_{13}$$ T13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2} + 9$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 196$$
$19$ $$(T - 22)^{2}$$
$23$ $$T^{2} + 1156$$
$29$ $$T^{2}$$
$31$ $$(T - 2)^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2} + 196$$
$53$ $$T^{2} + 7396$$
$59$ $$T^{2}$$
$61$ $$(T + 118)^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2}$$
$79$ $$(T + 98)^{2}$$
$83$ $$T^{2} + 23716$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$