# Properties

 Label 75.4.a.d 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: $$\Q(\sqrt{19})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 19$$ x^2 - 19 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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 $$\beta = \sqrt{19}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta - 1) q^{2} + 3 q^{3} + ( - 2 \beta + 12) q^{4} + (3 \beta - 3) q^{6} + (4 \beta + 13) q^{7} + (6 \beta - 42) q^{8} + 9 q^{9}+O(q^{10})$$ q + (b - 1) * q^2 + 3 * q^3 + (-2*b + 12) * q^4 + (3*b - 3) * q^6 + (4*b + 13) * q^7 + (6*b - 42) * q^8 + 9 * q^9 $$q + (\beta - 1) q^{2} + 3 q^{3} + ( - 2 \beta + 12) q^{4} + (3 \beta - 3) q^{6} + (4 \beta + 13) q^{7} + (6 \beta - 42) q^{8} + 9 q^{9} + (4 \beta + 14) q^{11} + ( - 6 \beta + 36) q^{12} + ( - 16 \beta + 9) q^{13} + (9 \beta + 63) q^{14} + ( - 32 \beta + 60) q^{16} + ( - 20 \beta - 34) q^{17} + (9 \beta - 9) q^{18} + ( - 4 \beta + 3) q^{19} + (12 \beta + 39) q^{21} + (10 \beta + 62) q^{22} + ( - 12 \beta + 66) q^{23} + (18 \beta - 126) q^{24} + (25 \beta - 313) q^{26} + 27 q^{27} + (22 \beta + 4) q^{28} + ( - 28 \beta + 46) q^{29} + (28 \beta + 61) q^{31} + (44 \beta - 332) q^{32} + (12 \beta + 42) q^{33} + ( - 14 \beta - 346) q^{34} + ( - 18 \beta + 108) q^{36} + (24 \beta - 142) q^{37} + (7 \beta - 79) q^{38} + ( - 48 \beta + 27) q^{39} + ( - 52 \beta + 196) q^{41} + (27 \beta + 189) q^{42} + ( - 4 \beta + 345) q^{43} + (20 \beta + 16) q^{44} + (78 \beta - 294) q^{46} + ( - 32 \beta - 310) q^{47} + ( - 96 \beta + 180) q^{48} + (104 \beta + 130) q^{49} + ( - 60 \beta - 102) q^{51} + ( - 210 \beta + 716) q^{52} + (28 \beta - 424) q^{53} + (27 \beta - 27) q^{54} + ( - 90 \beta - 90) q^{56} + ( - 12 \beta + 9) q^{57} + (74 \beta - 578) q^{58} + (64 \beta + 62) q^{59} + (56 \beta + 375) q^{61} + (33 \beta + 471) q^{62} + (36 \beta + 117) q^{63} + ( - 120 \beta + 688) q^{64} + (30 \beta + 186) q^{66} + (100 \beta - 179) q^{67} + ( - 172 \beta + 352) q^{68} + ( - 36 \beta + 198) q^{69} + (20 \beta + 412) q^{71} + (54 \beta - 378) q^{72} + (8 \beta - 54) q^{73} + ( - 166 \beta + 598) q^{74} + ( - 54 \beta + 188) q^{76} + (108 \beta + 486) q^{77} + (75 \beta - 939) q^{78} + (160 \beta - 440) q^{79} + 81 q^{81} + (248 \beta - 1184) q^{82} + (192 \beta + 78) q^{83} + (66 \beta + 12) q^{84} + (349 \beta - 421) q^{86} + ( - 84 \beta + 138) q^{87} + ( - 84 \beta - 132) q^{88} + ( - 144 \beta - 432) q^{89} + ( - 172 \beta - 1099) q^{91} + ( - 276 \beta + 1248) q^{92} + (84 \beta + 183) q^{93} + ( - 278 \beta - 298) q^{94} + (132 \beta - 996) q^{96} + 521 q^{97} + (26 \beta + 1846) q^{98} + (36 \beta + 126) q^{99}+O(q^{100})$$ q + (b - 1) * q^2 + 3 * q^3 + (-2*b + 12) * q^4 + (3*b - 3) * q^6 + (4*b + 13) * q^7 + (6*b - 42) * q^8 + 9 * q^9 + (4*b + 14) * q^11 + (-6*b + 36) * q^12 + (-16*b + 9) * q^13 + (9*b + 63) * q^14 + (-32*b + 60) * q^16 + (-20*b - 34) * q^17 + (9*b - 9) * q^18 + (-4*b + 3) * q^19 + (12*b + 39) * q^21 + (10*b + 62) * q^22 + (-12*b + 66) * q^23 + (18*b - 126) * q^24 + (25*b - 313) * q^26 + 27 * q^27 + (22*b + 4) * q^28 + (-28*b + 46) * q^29 + (28*b + 61) * q^31 + (44*b - 332) * q^32 + (12*b + 42) * q^33 + (-14*b - 346) * q^34 + (-18*b + 108) * q^36 + (24*b - 142) * q^37 + (7*b - 79) * q^38 + (-48*b + 27) * q^39 + (-52*b + 196) * q^41 + (27*b + 189) * q^42 + (-4*b + 345) * q^43 + (20*b + 16) * q^44 + (78*b - 294) * q^46 + (-32*b - 310) * q^47 + (-96*b + 180) * q^48 + (104*b + 130) * q^49 + (-60*b - 102) * q^51 + (-210*b + 716) * q^52 + (28*b - 424) * q^53 + (27*b - 27) * q^54 + (-90*b - 90) * q^56 + (-12*b + 9) * q^57 + (74*b - 578) * q^58 + (64*b + 62) * q^59 + (56*b + 375) * q^61 + (33*b + 471) * q^62 + (36*b + 117) * q^63 + (-120*b + 688) * q^64 + (30*b + 186) * q^66 + (100*b - 179) * q^67 + (-172*b + 352) * q^68 + (-36*b + 198) * q^69 + (20*b + 412) * q^71 + (54*b - 378) * q^72 + (8*b - 54) * q^73 + (-166*b + 598) * q^74 + (-54*b + 188) * q^76 + (108*b + 486) * q^77 + (75*b - 939) * q^78 + (160*b - 440) * q^79 + 81 * q^81 + (248*b - 1184) * q^82 + (192*b + 78) * q^83 + (66*b + 12) * q^84 + (349*b - 421) * q^86 + (-84*b + 138) * q^87 + (-84*b - 132) * q^88 + (-144*b - 432) * q^89 + (-172*b - 1099) * q^91 + (-276*b + 1248) * q^92 + (84*b + 183) * q^93 + (-278*b - 298) * q^94 + (132*b - 996) * q^96 + 521 * q^97 + (26*b + 1846) * q^98 + (36*b + 126) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 6 q^{3} + 24 q^{4} - 6 q^{6} + 26 q^{7} - 84 q^{8} + 18 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + 6 * q^3 + 24 * q^4 - 6 * q^6 + 26 * q^7 - 84 * q^8 + 18 * q^9 $$2 q - 2 q^{2} + 6 q^{3} + 24 q^{4} - 6 q^{6} + 26 q^{7} - 84 q^{8} + 18 q^{9} + 28 q^{11} + 72 q^{12} + 18 q^{13} + 126 q^{14} + 120 q^{16} - 68 q^{17} - 18 q^{18} + 6 q^{19} + 78 q^{21} + 124 q^{22} + 132 q^{23} - 252 q^{24} - 626 q^{26} + 54 q^{27} + 8 q^{28} + 92 q^{29} + 122 q^{31} - 664 q^{32} + 84 q^{33} - 692 q^{34} + 216 q^{36} - 284 q^{37} - 158 q^{38} + 54 q^{39} + 392 q^{41} + 378 q^{42} + 690 q^{43} + 32 q^{44} - 588 q^{46} - 620 q^{47} + 360 q^{48} + 260 q^{49} - 204 q^{51} + 1432 q^{52} - 848 q^{53} - 54 q^{54} - 180 q^{56} + 18 q^{57} - 1156 q^{58} + 124 q^{59} + 750 q^{61} + 942 q^{62} + 234 q^{63} + 1376 q^{64} + 372 q^{66} - 358 q^{67} + 704 q^{68} + 396 q^{69} + 824 q^{71} - 756 q^{72} - 108 q^{73} + 1196 q^{74} + 376 q^{76} + 972 q^{77} - 1878 q^{78} - 880 q^{79} + 162 q^{81} - 2368 q^{82} + 156 q^{83} + 24 q^{84} - 842 q^{86} + 276 q^{87} - 264 q^{88} - 864 q^{89} - 2198 q^{91} + 2496 q^{92} + 366 q^{93} - 596 q^{94} - 1992 q^{96} + 1042 q^{97} + 3692 q^{98} + 252 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 + 6 * q^3 + 24 * q^4 - 6 * q^6 + 26 * q^7 - 84 * q^8 + 18 * q^9 + 28 * q^11 + 72 * q^12 + 18 * q^13 + 126 * q^14 + 120 * q^16 - 68 * q^17 - 18 * q^18 + 6 * q^19 + 78 * q^21 + 124 * q^22 + 132 * q^23 - 252 * q^24 - 626 * q^26 + 54 * q^27 + 8 * q^28 + 92 * q^29 + 122 * q^31 - 664 * q^32 + 84 * q^33 - 692 * q^34 + 216 * q^36 - 284 * q^37 - 158 * q^38 + 54 * q^39 + 392 * q^41 + 378 * q^42 + 690 * q^43 + 32 * q^44 - 588 * q^46 - 620 * q^47 + 360 * q^48 + 260 * q^49 - 204 * q^51 + 1432 * q^52 - 848 * q^53 - 54 * q^54 - 180 * q^56 + 18 * q^57 - 1156 * q^58 + 124 * q^59 + 750 * q^61 + 942 * q^62 + 234 * q^63 + 1376 * q^64 + 372 * q^66 - 358 * q^67 + 704 * q^68 + 396 * q^69 + 824 * q^71 - 756 * q^72 - 108 * q^73 + 1196 * q^74 + 376 * q^76 + 972 * q^77 - 1878 * q^78 - 880 * q^79 + 162 * q^81 - 2368 * q^82 + 156 * q^83 + 24 * q^84 - 842 * q^86 + 276 * q^87 - 264 * q^88 - 864 * q^89 - 2198 * q^91 + 2496 * q^92 + 366 * q^93 - 596 * q^94 - 1992 * q^96 + 1042 * q^97 + 3692 * q^98 + 252 * 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
 −4.35890 4.35890
−5.35890 3.00000 20.7178 0 −16.0767 −4.43560 −68.1534 9.00000 0
1.2 3.35890 3.00000 3.28220 0 10.0767 30.4356 −15.8466 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.d 2
3.b odd 2 1 225.4.a.n 2
4.b odd 2 1 1200.4.a.bl 2
5.b even 2 1 75.4.a.e yes 2
5.c odd 4 2 75.4.b.c 4
15.d odd 2 1 225.4.a.j 2
15.e even 4 2 225.4.b.h 4
20.d odd 2 1 1200.4.a.bu 2
20.e even 4 2 1200.4.f.v 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.4.a.d 2 1.a even 1 1 trivial
75.4.a.e yes 2 5.b even 2 1
75.4.b.c 4 5.c odd 4 2
225.4.a.j 2 15.d odd 2 1
225.4.a.n 2 3.b odd 2 1
225.4.b.h 4 15.e even 4 2
1200.4.a.bl 2 4.b odd 2 1
1200.4.a.bu 2 20.d odd 2 1
1200.4.f.v 4 20.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 2T_{2} - 18$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(75))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T - 18$$
$3$ $$(T - 3)^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 26T - 135$$
$11$ $$T^{2} - 28T - 108$$
$13$ $$T^{2} - 18T - 4783$$
$17$ $$T^{2} + 68T - 6444$$
$19$ $$T^{2} - 6T - 295$$
$23$ $$T^{2} - 132T + 1620$$
$29$ $$T^{2} - 92T - 12780$$
$31$ $$T^{2} - 122T - 11175$$
$37$ $$T^{2} + 284T + 9220$$
$41$ $$T^{2} - 392T - 12960$$
$43$ $$T^{2} - 690T + 118721$$
$47$ $$T^{2} + 620T + 76644$$
$53$ $$T^{2} + 848T + 164880$$
$59$ $$T^{2} - 124T - 73980$$
$61$ $$T^{2} - 750T + 81041$$
$67$ $$T^{2} + 358T - 157959$$
$71$ $$T^{2} - 824T + 162144$$
$73$ $$T^{2} + 108T + 1700$$
$79$ $$T^{2} + 880T - 292800$$
$83$ $$T^{2} - 156T - 694332$$
$89$ $$T^{2} + 864T - 207360$$
$97$ $$(T - 521)^{2}$$