# Properties

 Label 2475.4.a.n Level $2475$ Weight $4$ Character orbit 2475.a Self dual yes Analytic conductor $146.030$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,4,Mod(1,2475)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2475.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2475 = 3^{2} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2475.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: $$146.029727264$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 165) 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 $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} + (\beta - 4) q^{4} + ( - 4 \beta + 4) q^{7} + (11 \beta - 4) q^{8} +O(q^{10})$$ q - b * q^2 + (b - 4) * q^4 + (-4*b + 4) * q^7 + (11*b - 4) * q^8 $$q - \beta q^{2} + (\beta - 4) q^{4} + ( - 4 \beta + 4) q^{7} + (11 \beta - 4) q^{8} + 11 q^{11} + (2 \beta + 44) q^{13} + 16 q^{14} + ( - 15 \beta - 12) q^{16} + (44 \beta - 30) q^{17} + ( - 22 \beta - 74) q^{19} - 11 \beta q^{22} + ( - 60 \beta - 32) q^{23} + ( - 46 \beta - 8) q^{26} + (16 \beta - 32) q^{28} + ( - 34 \beta + 96) q^{29} + ( - 12 \beta + 36) q^{31} + ( - 61 \beta + 92) q^{32} + ( - 14 \beta - 176) q^{34} + (112 \beta + 130) q^{37} + (96 \beta + 88) q^{38} + (154 \beta - 96) q^{41} + (124 \beta + 196) q^{43} + (11 \beta - 44) q^{44} + (92 \beta + 240) q^{46} + (216 \beta + 4) q^{47} + ( - 16 \beta - 263) q^{49} + (38 \beta - 168) q^{52} + ( - 196 \beta + 334) q^{53} + (16 \beta - 192) q^{56} + ( - 62 \beta + 136) q^{58} + ( - 240 \beta - 4) q^{59} + (364 \beta - 146) q^{61} + ( - 24 \beta + 48) q^{62} + (89 \beta + 340) q^{64} + ( - 16 \beta + 380) q^{67} + ( - 162 \beta + 296) q^{68} + ( - 44 \beta - 1008) q^{71} + ( - 58 \beta + 272) q^{73} + ( - 242 \beta - 448) q^{74} + ( - 8 \beta + 208) q^{76} + ( - 44 \beta + 44) q^{77} + ( - 306 \beta + 474) q^{79} + ( - 58 \beta - 616) q^{82} + ( - 426 \beta + 70) q^{83} + ( - 320 \beta - 496) q^{86} + (121 \beta - 44) q^{88} + (128 \beta - 186) q^{89} + ( - 176 \beta + 144) q^{91} + (148 \beta - 112) q^{92} + ( - 220 \beta - 864) q^{94} + ( - 428 \beta + 298) q^{97} + (279 \beta + 64) q^{98} +O(q^{100})$$ q - b * q^2 + (b - 4) * q^4 + (-4*b + 4) * q^7 + (11*b - 4) * q^8 + 11 * q^11 + (2*b + 44) * q^13 + 16 * q^14 + (-15*b - 12) * q^16 + (44*b - 30) * q^17 + (-22*b - 74) * q^19 - 11*b * q^22 + (-60*b - 32) * q^23 + (-46*b - 8) * q^26 + (16*b - 32) * q^28 + (-34*b + 96) * q^29 + (-12*b + 36) * q^31 + (-61*b + 92) * q^32 + (-14*b - 176) * q^34 + (112*b + 130) * q^37 + (96*b + 88) * q^38 + (154*b - 96) * q^41 + (124*b + 196) * q^43 + (11*b - 44) * q^44 + (92*b + 240) * q^46 + (216*b + 4) * q^47 + (-16*b - 263) * q^49 + (38*b - 168) * q^52 + (-196*b + 334) * q^53 + (16*b - 192) * q^56 + (-62*b + 136) * q^58 + (-240*b - 4) * q^59 + (364*b - 146) * q^61 + (-24*b + 48) * q^62 + (89*b + 340) * q^64 + (-16*b + 380) * q^67 + (-162*b + 296) * q^68 + (-44*b - 1008) * q^71 + (-58*b + 272) * q^73 + (-242*b - 448) * q^74 + (-8*b + 208) * q^76 + (-44*b + 44) * q^77 + (-306*b + 474) * q^79 + (-58*b - 616) * q^82 + (-426*b + 70) * q^83 + (-320*b - 496) * q^86 + (121*b - 44) * q^88 + (128*b - 186) * q^89 + (-176*b + 144) * q^91 + (148*b - 112) * q^92 + (-220*b - 864) * q^94 + (-428*b + 298) * q^97 + (279*b + 64) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - 7 q^{4} + 4 q^{7} + 3 q^{8}+O(q^{10})$$ 2 * q - q^2 - 7 * q^4 + 4 * q^7 + 3 * q^8 $$2 q - q^{2} - 7 q^{4} + 4 q^{7} + 3 q^{8} + 22 q^{11} + 90 q^{13} + 32 q^{14} - 39 q^{16} - 16 q^{17} - 170 q^{19} - 11 q^{22} - 124 q^{23} - 62 q^{26} - 48 q^{28} + 158 q^{29} + 60 q^{31} + 123 q^{32} - 366 q^{34} + 372 q^{37} + 272 q^{38} - 38 q^{41} + 516 q^{43} - 77 q^{44} + 572 q^{46} + 224 q^{47} - 542 q^{49} - 298 q^{52} + 472 q^{53} - 368 q^{56} + 210 q^{58} - 248 q^{59} + 72 q^{61} + 72 q^{62} + 769 q^{64} + 744 q^{67} + 430 q^{68} - 2060 q^{71} + 486 q^{73} - 1138 q^{74} + 408 q^{76} + 44 q^{77} + 642 q^{79} - 1290 q^{82} - 286 q^{83} - 1312 q^{86} + 33 q^{88} - 244 q^{89} + 112 q^{91} - 76 q^{92} - 1948 q^{94} + 168 q^{97} + 407 q^{98}+O(q^{100})$$ 2 * q - q^2 - 7 * q^4 + 4 * q^7 + 3 * q^8 + 22 * q^11 + 90 * q^13 + 32 * q^14 - 39 * q^16 - 16 * q^17 - 170 * q^19 - 11 * q^22 - 124 * q^23 - 62 * q^26 - 48 * q^28 + 158 * q^29 + 60 * q^31 + 123 * q^32 - 366 * q^34 + 372 * q^37 + 272 * q^38 - 38 * q^41 + 516 * q^43 - 77 * q^44 + 572 * q^46 + 224 * q^47 - 542 * q^49 - 298 * q^52 + 472 * q^53 - 368 * q^56 + 210 * q^58 - 248 * q^59 + 72 * q^61 + 72 * q^62 + 769 * q^64 + 744 * q^67 + 430 * q^68 - 2060 * q^71 + 486 * q^73 - 1138 * q^74 + 408 * q^76 + 44 * q^77 + 642 * q^79 - 1290 * q^82 - 286 * q^83 - 1312 * q^86 + 33 * q^88 - 244 * q^89 + 112 * q^91 - 76 * q^92 - 1948 * q^94 + 168 * q^97 + 407 * q^98

## 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
 2.56155 −1.56155
−2.56155 0 −1.43845 0 0 −6.24621 24.1771 0 0
1.2 1.56155 0 −5.56155 0 0 10.2462 −21.1771 0 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
$$3$$ $$-1$$
$$5$$ $$+1$$
$$11$$ $$-1$$

## 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 2475.4.a.n 2
3.b odd 2 1 825.4.a.m 2
5.b even 2 1 495.4.a.d 2
15.d odd 2 1 165.4.a.c 2
15.e even 4 2 825.4.c.j 4
165.d even 2 1 1815.4.a.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.a.c 2 15.d odd 2 1
495.4.a.d 2 5.b even 2 1
825.4.a.m 2 3.b odd 2 1
825.4.c.j 4 15.e even 4 2
1815.4.a.n 2 165.d even 2 1
2475.4.a.n 2 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(2475))$$:

 $$T_{2}^{2} + T_{2} - 4$$ T2^2 + T2 - 4 $$T_{7}^{2} - 4T_{7} - 64$$ T7^2 - 4*T7 - 64 $$T_{29}^{2} - 158T_{29} + 1328$$ T29^2 - 158*T29 + 1328

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T - 4$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 4T - 64$$
$11$ $$(T - 11)^{2}$$
$13$ $$T^{2} - 90T + 2008$$
$17$ $$T^{2} + 16T - 8164$$
$19$ $$T^{2} + 170T + 5168$$
$23$ $$T^{2} + 124T - 11456$$
$29$ $$T^{2} - 158T + 1328$$
$31$ $$T^{2} - 60T + 288$$
$37$ $$T^{2} - 372T - 18716$$
$41$ $$T^{2} + 38T - 100432$$
$43$ $$T^{2} - 516T + 1216$$
$47$ $$T^{2} - 224T - 185744$$
$53$ $$T^{2} - 472T - 107572$$
$59$ $$T^{2} + 248T - 229424$$
$61$ $$T^{2} - 72T - 561812$$
$67$ $$T^{2} - 744T + 137296$$
$71$ $$T^{2} + 2060 T + 1052672$$
$73$ $$T^{2} - 486T + 44752$$
$79$ $$T^{2} - 642T - 294912$$
$83$ $$T^{2} + 286T - 750824$$
$89$ $$T^{2} + 244T - 54748$$
$97$ $$T^{2} - 168T - 771476$$
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