# Properties

 Label 75.4.b.c Level $75$ Weight $4$ Character orbit 75.b Analytic conductor $4.425$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,4,Mod(49,75)]

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, 1]))

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

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

chi := DirichletCharacter("75.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 75.b (of order $$2$$, degree $$1$$, not 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: $$4.42514325043$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{19})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 9x^{2} + 25$$ x^4 - 9*x^2 + 25 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} - \beta_1) q^{2} - 3 \beta_1 q^{3} + (2 \beta_{2} - 12) q^{4} + (3 \beta_{2} - 3) q^{6} + (4 \beta_{3} + 13 \beta_1) q^{7} + ( - 6 \beta_{3} + 42 \beta_1) q^{8} - 9 q^{9}+O(q^{10})$$ q + (b3 - b1) * q^2 - 3*b1 * q^3 + (2*b2 - 12) * q^4 + (3*b2 - 3) * q^6 + (4*b3 + 13*b1) * q^7 + (-6*b3 + 42*b1) * q^8 - 9 * q^9 $$q + (\beta_{3} - \beta_1) q^{2} - 3 \beta_1 q^{3} + (2 \beta_{2} - 12) q^{4} + (3 \beta_{2} - 3) q^{6} + (4 \beta_{3} + 13 \beta_1) q^{7} + ( - 6 \beta_{3} + 42 \beta_1) q^{8} - 9 q^{9} + (4 \beta_{2} + 14) q^{11} + ( - 6 \beta_{3} + 36 \beta_1) q^{12} + (16 \beta_{3} - 9 \beta_1) q^{13} + ( - 9 \beta_{2} - 63) q^{14} + ( - 32 \beta_{2} + 60) q^{16} + ( - 20 \beta_{3} - 34 \beta_1) q^{17} + ( - 9 \beta_{3} + 9 \beta_1) q^{18} + (4 \beta_{2} - 3) q^{19} + (12 \beta_{2} + 39) q^{21} + (10 \beta_{3} + 62 \beta_1) q^{22} + (12 \beta_{3} - 66 \beta_1) q^{23} + ( - 18 \beta_{2} + 126) q^{24} + (25 \beta_{2} - 313) q^{26} + 27 \beta_1 q^{27} + ( - 22 \beta_{3} - 4 \beta_1) q^{28} + (28 \beta_{2} - 46) q^{29} + (28 \beta_{2} + 61) q^{31} + (44 \beta_{3} - 332 \beta_1) q^{32} + ( - 12 \beta_{3} - 42 \beta_1) q^{33} + (14 \beta_{2} + 346) q^{34} + ( - 18 \beta_{2} + 108) q^{36} + (24 \beta_{3} - 142 \beta_1) q^{37} + ( - 7 \beta_{3} + 79 \beta_1) q^{38} + (48 \beta_{2} - 27) q^{39} + ( - 52 \beta_{2} + 196) q^{41} + (27 \beta_{3} + 189 \beta_1) q^{42} + (4 \beta_{3} - 345 \beta_1) q^{43} + ( - 20 \beta_{2} - 16) q^{44} + (78 \beta_{2} - 294) q^{46} + ( - 32 \beta_{3} - 310 \beta_1) q^{47} + (96 \beta_{3} - 180 \beta_1) q^{48} + ( - 104 \beta_{2} - 130) q^{49} + ( - 60 \beta_{2} - 102) q^{51} + ( - 210 \beta_{3} + 716 \beta_1) q^{52} + ( - 28 \beta_{3} + 424 \beta_1) q^{53} + ( - 27 \beta_{2} + 27) q^{54} + ( - 90 \beta_{2} - 90) q^{56} + ( - 12 \beta_{3} + 9 \beta_1) q^{57} + ( - 74 \beta_{3} + 578 \beta_1) q^{58} + ( - 64 \beta_{2} - 62) q^{59} + (56 \beta_{2} + 375) q^{61} + (33 \beta_{3} + 471 \beta_1) q^{62} + ( - 36 \beta_{3} - 117 \beta_1) q^{63} + (120 \beta_{2} - 688) q^{64} + (30 \beta_{2} + 186) q^{66} + (100 \beta_{3} - 179 \beta_1) q^{67} + (172 \beta_{3} - 352 \beta_1) q^{68} + (36 \beta_{2} - 198) q^{69} + (20 \beta_{2} + 412) q^{71} + (54 \beta_{3} - 378 \beta_1) q^{72} + ( - 8 \beta_{3} + 54 \beta_1) q^{73} + (166 \beta_{2} - 598) q^{74} + ( - 54 \beta_{2} + 188) q^{76} + (108 \beta_{3} + 486 \beta_1) q^{77} + ( - 75 \beta_{3} + 939 \beta_1) q^{78} + ( - 160 \beta_{2} + 440) q^{79} + 81 q^{81} + (248 \beta_{3} - 1184 \beta_1) q^{82} + ( - 192 \beta_{3} - 78 \beta_1) q^{83} + ( - 66 \beta_{2} - 12) q^{84} + (349 \beta_{2} - 421) q^{86} + ( - 84 \beta_{3} + 138 \beta_1) q^{87} + (84 \beta_{3} + 132 \beta_1) q^{88} + (144 \beta_{2} + 432) q^{89} + ( - 172 \beta_{2} - 1099) q^{91} + ( - 276 \beta_{3} + 1248 \beta_1) q^{92} + ( - 84 \beta_{3} - 183 \beta_1) q^{93} + (278 \beta_{2} + 298) q^{94} + (132 \beta_{2} - 996) q^{96} + 521 \beta_1 q^{97} + ( - 26 \beta_{3} - 1846 \beta_1) q^{98} + ( - 36 \beta_{2} - 126) q^{99}+O(q^{100})$$ q + (b3 - b1) * q^2 - 3*b1 * q^3 + (2*b2 - 12) * q^4 + (3*b2 - 3) * q^6 + (4*b3 + 13*b1) * q^7 + (-6*b3 + 42*b1) * q^8 - 9 * q^9 + (4*b2 + 14) * q^11 + (-6*b3 + 36*b1) * q^12 + (16*b3 - 9*b1) * q^13 + (-9*b2 - 63) * q^14 + (-32*b2 + 60) * q^16 + (-20*b3 - 34*b1) * q^17 + (-9*b3 + 9*b1) * q^18 + (4*b2 - 3) * q^19 + (12*b2 + 39) * q^21 + (10*b3 + 62*b1) * q^22 + (12*b3 - 66*b1) * q^23 + (-18*b2 + 126) * q^24 + (25*b2 - 313) * q^26 + 27*b1 * q^27 + (-22*b3 - 4*b1) * q^28 + (28*b2 - 46) * q^29 + (28*b2 + 61) * q^31 + (44*b3 - 332*b1) * q^32 + (-12*b3 - 42*b1) * q^33 + (14*b2 + 346) * q^34 + (-18*b2 + 108) * q^36 + (24*b3 - 142*b1) * q^37 + (-7*b3 + 79*b1) * q^38 + (48*b2 - 27) * q^39 + (-52*b2 + 196) * q^41 + (27*b3 + 189*b1) * q^42 + (4*b3 - 345*b1) * q^43 + (-20*b2 - 16) * q^44 + (78*b2 - 294) * q^46 + (-32*b3 - 310*b1) * q^47 + (96*b3 - 180*b1) * q^48 + (-104*b2 - 130) * q^49 + (-60*b2 - 102) * q^51 + (-210*b3 + 716*b1) * q^52 + (-28*b3 + 424*b1) * q^53 + (-27*b2 + 27) * q^54 + (-90*b2 - 90) * q^56 + (-12*b3 + 9*b1) * q^57 + (-74*b3 + 578*b1) * q^58 + (-64*b2 - 62) * q^59 + (56*b2 + 375) * q^61 + (33*b3 + 471*b1) * q^62 + (-36*b3 - 117*b1) * q^63 + (120*b2 - 688) * q^64 + (30*b2 + 186) * q^66 + (100*b3 - 179*b1) * q^67 + (172*b3 - 352*b1) * q^68 + (36*b2 - 198) * q^69 + (20*b2 + 412) * q^71 + (54*b3 - 378*b1) * q^72 + (-8*b3 + 54*b1) * q^73 + (166*b2 - 598) * q^74 + (-54*b2 + 188) * q^76 + (108*b3 + 486*b1) * q^77 + (-75*b3 + 939*b1) * q^78 + (-160*b2 + 440) * q^79 + 81 * q^81 + (248*b3 - 1184*b1) * q^82 + (-192*b3 - 78*b1) * q^83 + (-66*b2 - 12) * q^84 + (349*b2 - 421) * q^86 + (-84*b3 + 138*b1) * q^87 + (84*b3 + 132*b1) * q^88 + (144*b2 + 432) * q^89 + (-172*b2 - 1099) * q^91 + (-276*b3 + 1248*b1) * q^92 + (-84*b3 - 183*b1) * q^93 + (278*b2 + 298) * q^94 + (132*b2 - 996) * q^96 + 521*b1 * q^97 + (-26*b3 - 1846*b1) * q^98 + (-36*b2 - 126) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 48 q^{4} - 12 q^{6} - 36 q^{9}+O(q^{10})$$ 4 * q - 48 * q^4 - 12 * q^6 - 36 * q^9 $$4 q - 48 q^{4} - 12 q^{6} - 36 q^{9} + 56 q^{11} - 252 q^{14} + 240 q^{16} - 12 q^{19} + 156 q^{21} + 504 q^{24} - 1252 q^{26} - 184 q^{29} + 244 q^{31} + 1384 q^{34} + 432 q^{36} - 108 q^{39} + 784 q^{41} - 64 q^{44} - 1176 q^{46} - 520 q^{49} - 408 q^{51} + 108 q^{54} - 360 q^{56} - 248 q^{59} + 1500 q^{61} - 2752 q^{64} + 744 q^{66} - 792 q^{69} + 1648 q^{71} - 2392 q^{74} + 752 q^{76} + 1760 q^{79} + 324 q^{81} - 48 q^{84} - 1684 q^{86} + 1728 q^{89} - 4396 q^{91} + 1192 q^{94} - 3984 q^{96} - 504 q^{99}+O(q^{100})$$ 4 * q - 48 * q^4 - 12 * q^6 - 36 * q^9 + 56 * q^11 - 252 * q^14 + 240 * q^16 - 12 * q^19 + 156 * q^21 + 504 * q^24 - 1252 * q^26 - 184 * q^29 + 244 * q^31 + 1384 * q^34 + 432 * q^36 - 108 * q^39 + 784 * q^41 - 64 * q^44 - 1176 * q^46 - 520 * q^49 - 408 * q^51 + 108 * q^54 - 360 * q^56 - 248 * q^59 + 1500 * q^61 - 2752 * q^64 + 744 * q^66 - 792 * q^69 + 1648 * q^71 - 2392 * q^74 + 752 * q^76 + 1760 * q^79 + 324 * q^81 - 48 * q^84 - 1684 * q^86 + 1728 * q^89 - 4396 * q^91 + 1192 * q^94 - 3984 * q^96 - 504 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 9x^{2} + 25$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} - 4\nu ) / 5$$ (v^3 - 4*v) / 5 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + 14\nu ) / 5$$ (-v^3 + 14*v) / 5 $$\beta_{3}$$ $$=$$ $$2\nu^{2} - 9$$ 2*v^2 - 9
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 2$$ (b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 9 ) / 2$$ (b3 + 9) / 2 $$\nu^{3}$$ $$=$$ $$2\beta_{2} + 7\beta_1$$ 2*b2 + 7*b1

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 −2.17945 + 0.500000i 2.17945 − 0.500000i 2.17945 + 0.500000i −2.17945 − 0.500000i
5.35890i 3.00000i −20.7178 0 −16.0767 4.43560i 68.1534i −9.00000 0
49.2 3.35890i 3.00000i −3.28220 0 10.0767 30.4356i 15.8466i −9.00000 0
49.3 3.35890i 3.00000i −3.28220 0 10.0767 30.4356i 15.8466i −9.00000 0
49.4 5.35890i 3.00000i −20.7178 0 −16.0767 4.43560i 68.1534i −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
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 75.4.b.c 4
3.b odd 2 1 225.4.b.h 4
4.b odd 2 1 1200.4.f.v 4
5.b even 2 1 inner 75.4.b.c 4
5.c odd 4 1 75.4.a.d 2
5.c odd 4 1 75.4.a.e yes 2
15.d odd 2 1 225.4.b.h 4
15.e even 4 1 225.4.a.j 2
15.e even 4 1 225.4.a.n 2
20.d odd 2 1 1200.4.f.v 4
20.e even 4 1 1200.4.a.bl 2
20.e even 4 1 1200.4.a.bu 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 40T_{2}^{2} + 324$$ acting on $$S_{4}^{\mathrm{new}}(75, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 40T^{2} + 324$$
$3$ $$(T^{2} + 9)^{2}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 946 T^{2} + 18225$$
$11$ $$(T^{2} - 28 T - 108)^{2}$$
$13$ $$T^{4} + 9890 T^{2} + 22877089$$
$17$ $$T^{4} + 17512 T^{2} + 41525136$$
$19$ $$(T^{2} + 6 T - 295)^{2}$$
$23$ $$T^{4} + 14184 T^{2} + 2624400$$
$29$ $$(T^{2} + 92 T - 12780)^{2}$$
$31$ $$(T^{2} - 122 T - 11175)^{2}$$
$37$ $$T^{4} + 62216 T^{2} + 85008400$$
$41$ $$(T^{2} - 392 T - 12960)^{2}$$
$43$ $$T^{4} + \cdots + 14094675841$$
$47$ $$T^{4} + \cdots + 5874302736$$
$53$ $$T^{4} + \cdots + 27185414400$$
$59$ $$(T^{2} + 124 T - 73980)^{2}$$
$61$ $$(T^{2} - 750 T + 81041)^{2}$$
$67$ $$T^{4} + \cdots + 24951045681$$
$71$ $$(T^{2} - 824 T + 162144)^{2}$$
$73$ $$T^{4} + 8264 T^{2} + 2890000$$
$79$ $$(T^{2} - 880 T - 292800)^{2}$$
$83$ $$T^{4} + \cdots + 482096926224$$
$89$ $$(T^{2} - 864 T - 207360)^{2}$$
$97$ $$(T^{2} + 271441)^{2}$$
show more
show less