# Properties

 Label 75.3.c.e 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: $$\Q(\sqrt{-5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 5$$ x^2 + 5 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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 $$\beta = \sqrt{-5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + ( - \beta + 2) q^{3} - q^{4} + (2 \beta + 5) q^{6} + 6 q^{7} + 3 \beta q^{8} + ( - 4 \beta - 1) q^{9}+O(q^{10})$$ q + b * q^2 + (-b + 2) * q^3 - q^4 + (2*b + 5) * q^6 + 6 * q^7 + 3*b * q^8 + (-4*b - 1) * q^9 $$q + \beta q^{2} + ( - \beta + 2) q^{3} - q^{4} + (2 \beta + 5) q^{6} + 6 q^{7} + 3 \beta q^{8} + ( - 4 \beta - 1) q^{9} + 2 \beta q^{11} + (\beta - 2) q^{12} - 16 q^{13} + 6 \beta q^{14} - 19 q^{16} - 2 \beta q^{17} + ( - \beta + 20) q^{18} - 2 q^{19} + ( - 6 \beta + 12) q^{21} - 10 q^{22} - 6 \beta q^{23} + (6 \beta + 15) q^{24} - 16 \beta q^{26} + ( - 7 \beta - 22) q^{27} - 6 q^{28} - 14 \beta q^{29} - 18 q^{31} - 7 \beta q^{32} + (4 \beta + 10) q^{33} + 10 q^{34} + (4 \beta + 1) q^{36} + 16 q^{37} - 2 \beta q^{38} + (16 \beta - 32) q^{39} + 28 \beta q^{41} + (12 \beta + 30) q^{42} - 16 q^{43} - 2 \beta q^{44} + 30 q^{46} + 22 \beta q^{47} + (19 \beta - 38) q^{48} - 13 q^{49} + ( - 4 \beta - 10) q^{51} + 16 q^{52} - 2 \beta q^{53} + ( - 22 \beta + 35) q^{54} + 18 \beta q^{56} + (2 \beta - 4) q^{57} + 70 q^{58} + 2 \beta q^{59} + 82 q^{61} - 18 \beta q^{62} + ( - 24 \beta - 6) q^{63} - 41 q^{64} + (10 \beta - 20) q^{66} - 24 q^{67} + 2 \beta q^{68} + ( - 12 \beta - 30) q^{69} - 56 \beta q^{71} + ( - 3 \beta + 60) q^{72} + 74 q^{73} + 16 \beta q^{74} + 2 q^{76} + 12 \beta q^{77} + ( - 32 \beta - 80) q^{78} + 138 q^{79} + (8 \beta - 79) q^{81} - 140 q^{82} + 42 \beta q^{83} + (6 \beta - 12) q^{84} - 16 \beta q^{86} + ( - 28 \beta - 70) q^{87} - 30 q^{88} + 48 \beta q^{89} - 96 q^{91} + 6 \beta q^{92} + (18 \beta - 36) q^{93} - 110 q^{94} + ( - 14 \beta - 35) q^{96} + 166 q^{97} - 13 \beta q^{98} + ( - 2 \beta + 40) q^{99} +O(q^{100})$$ q + b * q^2 + (-b + 2) * q^3 - q^4 + (2*b + 5) * q^6 + 6 * q^7 + 3*b * q^8 + (-4*b - 1) * q^9 + 2*b * q^11 + (b - 2) * q^12 - 16 * q^13 + 6*b * q^14 - 19 * q^16 - 2*b * q^17 + (-b + 20) * q^18 - 2 * q^19 + (-6*b + 12) * q^21 - 10 * q^22 - 6*b * q^23 + (6*b + 15) * q^24 - 16*b * q^26 + (-7*b - 22) * q^27 - 6 * q^28 - 14*b * q^29 - 18 * q^31 - 7*b * q^32 + (4*b + 10) * q^33 + 10 * q^34 + (4*b + 1) * q^36 + 16 * q^37 - 2*b * q^38 + (16*b - 32) * q^39 + 28*b * q^41 + (12*b + 30) * q^42 - 16 * q^43 - 2*b * q^44 + 30 * q^46 + 22*b * q^47 + (19*b - 38) * q^48 - 13 * q^49 + (-4*b - 10) * q^51 + 16 * q^52 - 2*b * q^53 + (-22*b + 35) * q^54 + 18*b * q^56 + (2*b - 4) * q^57 + 70 * q^58 + 2*b * q^59 + 82 * q^61 - 18*b * q^62 + (-24*b - 6) * q^63 - 41 * q^64 + (10*b - 20) * q^66 - 24 * q^67 + 2*b * q^68 + (-12*b - 30) * q^69 - 56*b * q^71 + (-3*b + 60) * q^72 + 74 * q^73 + 16*b * q^74 + 2 * q^76 + 12*b * q^77 + (-32*b - 80) * q^78 + 138 * q^79 + (8*b - 79) * q^81 - 140 * q^82 + 42*b * q^83 + (6*b - 12) * q^84 - 16*b * q^86 + (-28*b - 70) * q^87 - 30 * q^88 + 48*b * q^89 - 96 * q^91 + 6*b * q^92 + (18*b - 36) * q^93 - 110 * q^94 + (-14*b - 35) * q^96 + 166 * q^97 - 13*b * q^98 + (-2*b + 40) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{3} - 2 q^{4} + 10 q^{6} + 12 q^{7} - 2 q^{9}+O(q^{10})$$ 2 * q + 4 * q^3 - 2 * q^4 + 10 * q^6 + 12 * q^7 - 2 * q^9 $$2 q + 4 q^{3} - 2 q^{4} + 10 q^{6} + 12 q^{7} - 2 q^{9} - 4 q^{12} - 32 q^{13} - 38 q^{16} + 40 q^{18} - 4 q^{19} + 24 q^{21} - 20 q^{22} + 30 q^{24} - 44 q^{27} - 12 q^{28} - 36 q^{31} + 20 q^{33} + 20 q^{34} + 2 q^{36} + 32 q^{37} - 64 q^{39} + 60 q^{42} - 32 q^{43} + 60 q^{46} - 76 q^{48} - 26 q^{49} - 20 q^{51} + 32 q^{52} + 70 q^{54} - 8 q^{57} + 140 q^{58} + 164 q^{61} - 12 q^{63} - 82 q^{64} - 40 q^{66} - 48 q^{67} - 60 q^{69} + 120 q^{72} + 148 q^{73} + 4 q^{76} - 160 q^{78} + 276 q^{79} - 158 q^{81} - 280 q^{82} - 24 q^{84} - 140 q^{87} - 60 q^{88} - 192 q^{91} - 72 q^{93} - 220 q^{94} - 70 q^{96} + 332 q^{97} + 80 q^{99}+O(q^{100})$$ 2 * q + 4 * q^3 - 2 * q^4 + 10 * q^6 + 12 * q^7 - 2 * q^9 - 4 * q^12 - 32 * q^13 - 38 * q^16 + 40 * q^18 - 4 * q^19 + 24 * q^21 - 20 * q^22 + 30 * q^24 - 44 * q^27 - 12 * q^28 - 36 * q^31 + 20 * q^33 + 20 * q^34 + 2 * q^36 + 32 * q^37 - 64 * q^39 + 60 * q^42 - 32 * q^43 + 60 * q^46 - 76 * q^48 - 26 * q^49 - 20 * q^51 + 32 * q^52 + 70 * q^54 - 8 * q^57 + 140 * q^58 + 164 * q^61 - 12 * q^63 - 82 * q^64 - 40 * q^66 - 48 * q^67 - 60 * q^69 + 120 * q^72 + 148 * q^73 + 4 * q^76 - 160 * q^78 + 276 * q^79 - 158 * q^81 - 280 * q^82 - 24 * q^84 - 140 * q^87 - 60 * q^88 - 192 * q^91 - 72 * q^93 - 220 * q^94 - 70 * q^96 + 332 * q^97 + 80 * 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
 − 2.23607i 2.23607i
2.23607i 2.00000 + 2.23607i −1.00000 0 5.00000 4.47214i 6.00000 6.70820i −1.00000 + 8.94427i 0
26.2 2.23607i 2.00000 2.23607i −1.00000 0 5.00000 + 4.47214i 6.00000 6.70820i −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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.3.c.e 2
3.b odd 2 1 inner 75.3.c.e 2
4.b odd 2 1 1200.3.l.g 2
5.b even 2 1 15.3.c.a 2
5.c odd 4 2 75.3.d.b 4
12.b even 2 1 1200.3.l.g 2
15.d odd 2 1 15.3.c.a 2
15.e even 4 2 75.3.d.b 4
20.d odd 2 1 240.3.l.b 2
20.e even 4 2 1200.3.c.f 4
40.e odd 2 1 960.3.l.b 2
40.f even 2 1 960.3.l.c 2
45.h odd 6 2 405.3.i.b 4
45.j even 6 2 405.3.i.b 4
60.h even 2 1 240.3.l.b 2
60.l odd 4 2 1200.3.c.f 4
120.i odd 2 1 960.3.l.c 2
120.m even 2 1 960.3.l.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.3.c.a 2 5.b even 2 1
15.3.c.a 2 15.d odd 2 1
75.3.c.e 2 1.a even 1 1 trivial
75.3.c.e 2 3.b odd 2 1 inner
75.3.d.b 4 5.c odd 4 2
75.3.d.b 4 15.e even 4 2
240.3.l.b 2 20.d odd 2 1
240.3.l.b 2 60.h even 2 1
405.3.i.b 4 45.h odd 6 2
405.3.i.b 4 45.j even 6 2
960.3.l.b 2 40.e odd 2 1
960.3.l.b 2 120.m even 2 1
960.3.l.c 2 40.f even 2 1
960.3.l.c 2 120.i odd 2 1
1200.3.c.f 4 20.e even 4 2
1200.3.c.f 4 60.l odd 4 2
1200.3.l.g 2 4.b odd 2 1
1200.3.l.g 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 $$S_{3}^{\mathrm{new}}(75, [\chi])$$:

 $$T_{2}^{2} + 5$$ T2^2 + 5 $$T_{7} - 6$$ T7 - 6 $$T_{13} + 16$$ T13 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 5$$
$3$ $$T^{2} - 4T + 9$$
$5$ $$T^{2}$$
$7$ $$(T - 6)^{2}$$
$11$ $$T^{2} + 20$$
$13$ $$(T + 16)^{2}$$
$17$ $$T^{2} + 20$$
$19$ $$(T + 2)^{2}$$
$23$ $$T^{2} + 180$$
$29$ $$T^{2} + 980$$
$31$ $$(T + 18)^{2}$$
$37$ $$(T - 16)^{2}$$
$41$ $$T^{2} + 3920$$
$43$ $$(T + 16)^{2}$$
$47$ $$T^{2} + 2420$$
$53$ $$T^{2} + 20$$
$59$ $$T^{2} + 20$$
$61$ $$(T - 82)^{2}$$
$67$ $$(T + 24)^{2}$$
$71$ $$T^{2} + 15680$$
$73$ $$(T - 74)^{2}$$
$79$ $$(T - 138)^{2}$$
$83$ $$T^{2} + 8820$$
$89$ $$T^{2} + 11520$$
$97$ $$(T - 166)^{2}$$