# Properties

 Label 75.4.a.f Level $75$ Weight $4$ Character orbit 75.a Self dual yes Analytic conductor $4.425$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,4,Mod(1,75)]

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([0, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("75.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 75.a (trivial)

## 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: yes Analytic conductor: $$4.42514325043$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{41})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 10$$ x^2 - x - 10 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 $$\beta = \frac{1}{2}(1 + \sqrt{41})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{2} + 3 q^{3} + (3 \beta + 3) q^{4} + (3 \beta + 3) q^{6} + ( - 6 \beta + 6) q^{7} + (\beta + 25) q^{8} + 9 q^{9}+O(q^{10})$$ q + (b + 1) * q^2 + 3 * q^3 + (3*b + 3) * q^4 + (3*b + 3) * q^6 + (-6*b + 6) * q^7 + (b + 25) * q^8 + 9 * q^9 $$q + (\beta + 1) q^{2} + 3 q^{3} + (3 \beta + 3) q^{4} + (3 \beta + 3) q^{6} + ( - 6 \beta + 6) q^{7} + (\beta + 25) q^{8} + 9 q^{9} + ( - 6 \beta - 18) q^{11} + (9 \beta + 9) q^{12} + ( - 6 \beta + 42) q^{13} + ( - 6 \beta - 54) q^{14} + (3 \beta + 11) q^{16} + (10 \beta + 46) q^{17} + (9 \beta + 9) q^{18} + ( - 24 \beta + 40) q^{19} + ( - 18 \beta + 18) q^{21} + ( - 30 \beta - 78) q^{22} + (8 \beta - 28) q^{23} + (3 \beta + 75) q^{24} + (30 \beta - 18) q^{26} + 27 q^{27} + ( - 18 \beta - 162) q^{28} + (42 \beta - 180) q^{29} + ( - 12 \beta + 32) q^{31} + (9 \beta - 159) q^{32} + ( - 18 \beta - 54) q^{33} + (66 \beta + 146) q^{34} + (27 \beta + 27) q^{36} + (54 \beta + 126) q^{37} + ( - 8 \beta - 200) q^{38} + ( - 18 \beta + 126) q^{39} + ( - 12 \beta - 198) q^{41} + ( - 18 \beta - 162) q^{42} + (96 \beta + 12) q^{43} + ( - 90 \beta - 234) q^{44} + ( - 12 \beta + 52) q^{46} + ( - 92 \beta + 136) q^{47} + (9 \beta + 33) q^{48} + ( - 36 \beta + 53) q^{49} + (30 \beta + 138) q^{51} + (90 \beta - 54) q^{52} + ( - 82 \beta + 242) q^{53} + (27 \beta + 27) q^{54} + ( - 150 \beta + 90) q^{56} + ( - 72 \beta + 120) q^{57} + ( - 96 \beta + 240) q^{58} + ( - 6 \beta - 90) q^{59} + (96 \beta + 122) q^{61} + (8 \beta - 88) q^{62} + ( - 54 \beta + 54) q^{63} + ( - 165 \beta - 157) q^{64} + ( - 90 \beta - 234) q^{66} + (60 \beta + 336) q^{67} + (198 \beta + 438) q^{68} + (24 \beta - 84) q^{69} + (180 \beta - 108) q^{71} + (9 \beta + 225) q^{72} + (108 \beta + 612) q^{73} + (234 \beta + 666) q^{74} + ( - 24 \beta - 600) q^{76} + (108 \beta + 252) q^{77} + (90 \beta - 54) q^{78} + (300 \beta + 40) q^{79} + 81 q^{81} + ( - 222 \beta - 318) q^{82} + ( - 208 \beta - 388) q^{83} + ( - 54 \beta - 486) q^{84} + (204 \beta + 972) q^{86} + (126 \beta - 540) q^{87} + ( - 174 \beta - 510) q^{88} + ( - 144 \beta + 630) q^{89} + ( - 252 \beta + 612) q^{91} + ( - 36 \beta + 156) q^{92} + ( - 36 \beta + 96) q^{93} + ( - 48 \beta - 784) q^{94} + (27 \beta - 477) q^{96} + ( - 240 \beta - 264) q^{97} + ( - 19 \beta - 307) q^{98} + ( - 54 \beta - 162) q^{99}+O(q^{100})$$ q + (b + 1) * q^2 + 3 * q^3 + (3*b + 3) * q^4 + (3*b + 3) * q^6 + (-6*b + 6) * q^7 + (b + 25) * q^8 + 9 * q^9 + (-6*b - 18) * q^11 + (9*b + 9) * q^12 + (-6*b + 42) * q^13 + (-6*b - 54) * q^14 + (3*b + 11) * q^16 + (10*b + 46) * q^17 + (9*b + 9) * q^18 + (-24*b + 40) * q^19 + (-18*b + 18) * q^21 + (-30*b - 78) * q^22 + (8*b - 28) * q^23 + (3*b + 75) * q^24 + (30*b - 18) * q^26 + 27 * q^27 + (-18*b - 162) * q^28 + (42*b - 180) * q^29 + (-12*b + 32) * q^31 + (9*b - 159) * q^32 + (-18*b - 54) * q^33 + (66*b + 146) * q^34 + (27*b + 27) * q^36 + (54*b + 126) * q^37 + (-8*b - 200) * q^38 + (-18*b + 126) * q^39 + (-12*b - 198) * q^41 + (-18*b - 162) * q^42 + (96*b + 12) * q^43 + (-90*b - 234) * q^44 + (-12*b + 52) * q^46 + (-92*b + 136) * q^47 + (9*b + 33) * q^48 + (-36*b + 53) * q^49 + (30*b + 138) * q^51 + (90*b - 54) * q^52 + (-82*b + 242) * q^53 + (27*b + 27) * q^54 + (-150*b + 90) * q^56 + (-72*b + 120) * q^57 + (-96*b + 240) * q^58 + (-6*b - 90) * q^59 + (96*b + 122) * q^61 + (8*b - 88) * q^62 + (-54*b + 54) * q^63 + (-165*b - 157) * q^64 + (-90*b - 234) * q^66 + (60*b + 336) * q^67 + (198*b + 438) * q^68 + (24*b - 84) * q^69 + (180*b - 108) * q^71 + (9*b + 225) * q^72 + (108*b + 612) * q^73 + (234*b + 666) * q^74 + (-24*b - 600) * q^76 + (108*b + 252) * q^77 + (90*b - 54) * q^78 + (300*b + 40) * q^79 + 81 * q^81 + (-222*b - 318) * q^82 + (-208*b - 388) * q^83 + (-54*b - 486) * q^84 + (204*b + 972) * q^86 + (126*b - 540) * q^87 + (-174*b - 510) * q^88 + (-144*b + 630) * q^89 + (-252*b + 612) * q^91 + (-36*b + 156) * q^92 + (-36*b + 96) * q^93 + (-48*b - 784) * q^94 + (27*b - 477) * q^96 + (-240*b - 264) * q^97 + (-19*b - 307) * q^98 + (-54*b - 162) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{2} + 6 q^{3} + 9 q^{4} + 9 q^{6} + 6 q^{7} + 51 q^{8} + 18 q^{9}+O(q^{10})$$ 2 * q + 3 * q^2 + 6 * q^3 + 9 * q^4 + 9 * q^6 + 6 * q^7 + 51 * q^8 + 18 * q^9 $$2 q + 3 q^{2} + 6 q^{3} + 9 q^{4} + 9 q^{6} + 6 q^{7} + 51 q^{8} + 18 q^{9} - 42 q^{11} + 27 q^{12} + 78 q^{13} - 114 q^{14} + 25 q^{16} + 102 q^{17} + 27 q^{18} + 56 q^{19} + 18 q^{21} - 186 q^{22} - 48 q^{23} + 153 q^{24} - 6 q^{26} + 54 q^{27} - 342 q^{28} - 318 q^{29} + 52 q^{31} - 309 q^{32} - 126 q^{33} + 358 q^{34} + 81 q^{36} + 306 q^{37} - 408 q^{38} + 234 q^{39} - 408 q^{41} - 342 q^{42} + 120 q^{43} - 558 q^{44} + 92 q^{46} + 180 q^{47} + 75 q^{48} + 70 q^{49} + 306 q^{51} - 18 q^{52} + 402 q^{53} + 81 q^{54} + 30 q^{56} + 168 q^{57} + 384 q^{58} - 186 q^{59} + 340 q^{61} - 168 q^{62} + 54 q^{63} - 479 q^{64} - 558 q^{66} + 732 q^{67} + 1074 q^{68} - 144 q^{69} - 36 q^{71} + 459 q^{72} + 1332 q^{73} + 1566 q^{74} - 1224 q^{76} + 612 q^{77} - 18 q^{78} + 380 q^{79} + 162 q^{81} - 858 q^{82} - 984 q^{83} - 1026 q^{84} + 2148 q^{86} - 954 q^{87} - 1194 q^{88} + 1116 q^{89} + 972 q^{91} + 276 q^{92} + 156 q^{93} - 1616 q^{94} - 927 q^{96} - 768 q^{97} - 633 q^{98} - 378 q^{99}+O(q^{100})$$ 2 * q + 3 * q^2 + 6 * q^3 + 9 * q^4 + 9 * q^6 + 6 * q^7 + 51 * q^8 + 18 * q^9 - 42 * q^11 + 27 * q^12 + 78 * q^13 - 114 * q^14 + 25 * q^16 + 102 * q^17 + 27 * q^18 + 56 * q^19 + 18 * q^21 - 186 * q^22 - 48 * q^23 + 153 * q^24 - 6 * q^26 + 54 * q^27 - 342 * q^28 - 318 * q^29 + 52 * q^31 - 309 * q^32 - 126 * q^33 + 358 * q^34 + 81 * q^36 + 306 * q^37 - 408 * q^38 + 234 * q^39 - 408 * q^41 - 342 * q^42 + 120 * q^43 - 558 * q^44 + 92 * q^46 + 180 * q^47 + 75 * q^48 + 70 * q^49 + 306 * q^51 - 18 * q^52 + 402 * q^53 + 81 * q^54 + 30 * q^56 + 168 * q^57 + 384 * q^58 - 186 * q^59 + 340 * q^61 - 168 * q^62 + 54 * q^63 - 479 * q^64 - 558 * q^66 + 732 * q^67 + 1074 * q^68 - 144 * q^69 - 36 * q^71 + 459 * q^72 + 1332 * q^73 + 1566 * q^74 - 1224 * q^76 + 612 * q^77 - 18 * q^78 + 380 * q^79 + 162 * q^81 - 858 * q^82 - 984 * q^83 - 1026 * q^84 + 2148 * q^86 - 954 * q^87 - 1194 * q^88 + 1116 * q^89 + 972 * q^91 + 276 * q^92 + 156 * q^93 - 1616 * q^94 - 927 * q^96 - 768 * q^97 - 633 * q^98 - 378 * q^99

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.70156 3.70156
−1.70156 3.00000 −5.10469 0 −5.10469 22.2094 22.2984 9.00000 0
1.2 4.70156 3.00000 14.1047 0 14.1047 −16.2094 28.7016 9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.4.a.f 2
3.b odd 2 1 225.4.a.i 2
4.b odd 2 1 1200.4.a.bn 2
5.b even 2 1 75.4.a.c 2
5.c odd 4 2 15.4.b.a 4
15.d odd 2 1 225.4.a.o 2
15.e even 4 2 45.4.b.b 4
20.d odd 2 1 1200.4.a.bt 2
20.e even 4 2 240.4.f.f 4
40.i odd 4 2 960.4.f.q 4
40.k even 4 2 960.4.f.p 4
60.l odd 4 2 720.4.f.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.b.a 4 5.c odd 4 2
45.4.b.b 4 15.e even 4 2
75.4.a.c 2 5.b even 2 1
75.4.a.f 2 1.a even 1 1 trivial
225.4.a.i 2 3.b odd 2 1
225.4.a.o 2 15.d odd 2 1
240.4.f.f 4 20.e even 4 2
720.4.f.j 4 60.l odd 4 2
960.4.f.p 4 40.k even 4 2
960.4.f.q 4 40.i odd 4 2
1200.4.a.bn 2 4.b odd 2 1
1200.4.a.bt 2 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 3T_{2} - 8$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(75))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 3T - 8$$
$3$ $$(T - 3)^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 6T - 360$$
$11$ $$T^{2} + 42T + 72$$
$13$ $$T^{2} - 78T + 1152$$
$17$ $$T^{2} - 102T + 1576$$
$19$ $$T^{2} - 56T - 5120$$
$23$ $$T^{2} + 48T - 80$$
$29$ $$T^{2} + 318T + 7200$$
$31$ $$T^{2} - 52T - 800$$
$37$ $$T^{2} - 306T - 6480$$
$41$ $$T^{2} + 408T + 40140$$
$43$ $$T^{2} - 120T - 90864$$
$47$ $$T^{2} - 180T - 78656$$
$53$ $$T^{2} - 402T - 28520$$
$59$ $$T^{2} + 186T + 8280$$
$61$ $$T^{2} - 340T - 65564$$
$67$ $$T^{2} - 732T + 97056$$
$71$ $$T^{2} + 36T - 331776$$
$73$ $$T^{2} - 1332 T + 324000$$
$79$ $$T^{2} - 380T - 886400$$
$83$ $$T^{2} + 984T - 201392$$
$89$ $$T^{2} - 1116T + 98820$$
$97$ $$T^{2} + 768T - 442944$$