# Properties

 Label 45.6.a.e Level $45$ Weight $6$ Character orbit 45.a Self dual yes Analytic conductor $7.217$ 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] = [45,6,Mod(1,45)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("45.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$45 = 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 45.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: $$7.21727189158$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{409})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 102$$ x^2 - x - 102 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 $$\beta = \frac{1}{2}(1 + \sqrt{409})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + (\beta + 70) q^{4} - 25 q^{5} + (16 \beta - 64) q^{7} + (39 \beta + 102) q^{8}+O(q^{10})$$ q + b * q^2 + (b + 70) * q^4 - 25 * q^5 + (16*b - 64) * q^7 + (39*b + 102) * q^8 $$q + \beta q^{2} + (\beta + 70) q^{4} - 25 q^{5} + (16 \beta - 64) q^{7} + (39 \beta + 102) q^{8} - 25 \beta q^{10} + ( - 32 \beta - 108) q^{11} + ( - 16 \beta + 446) q^{13} + ( - 48 \beta + 1632) q^{14} + (109 \beta + 1738) q^{16} + ( - 80 \beta - 978) q^{17} + ( - 208 \beta + 836) q^{19} + ( - 25 \beta - 1750) q^{20} + ( - 140 \beta - 3264) q^{22} + ( - 48 \beta + 1632) q^{23} + 625 q^{25} + (430 \beta - 1632) q^{26} + (1072 \beta - 2848) q^{28} + ( - 64 \beta - 942) q^{29} + ( - 176 \beta + 1424) q^{31} + (599 \beta + 7854) q^{32} + ( - 1058 \beta - 8160) q^{34} + ( - 400 \beta + 1600) q^{35} + (816 \beta + 3926) q^{37} + (628 \beta - 21216) q^{38} + ( - 975 \beta - 2550) q^{40} + ( - 544 \beta + 4086) q^{41} + ( - 64 \beta - 8188) q^{43} + ( - 2380 \beta - 10824) q^{44} + (1584 \beta - 4896) q^{46} + ( - 1232 \beta + 10296) q^{47} + ( - 1792 \beta + 13401) q^{49} + 625 \beta q^{50} + ( - 690 \beta + 29588) q^{52} + (2272 \beta + 6042) q^{53} + (800 \beta + 2700) q^{55} + ( - 240 \beta + 57120) q^{56} + ( - 1006 \beta - 6528) q^{58} + (3232 \beta - 1164) q^{59} + (1568 \beta + 9326) q^{61} + (1248 \beta - 17952) q^{62} + (4965 \beta + 5482) q^{64} + (400 \beta - 11150) q^{65} + ( - 1280 \beta - 5812) q^{67} + ( - 6658 \beta - 76620) q^{68} + (1200 \beta - 40800) q^{70} + (3200 \beta + 18888) q^{71} + (608 \beta + 29258) q^{73} + (4742 \beta + 83232) q^{74} + ( - 13932 \beta + 37304) q^{76} + ( - 192 \beta - 45312) q^{77} + (3760 \beta + 51920) q^{79} + ( - 2725 \beta - 43450) q^{80} + (3542 \beta - 55488) q^{82} + ( - 4032 \beta + 63060) q^{83} + (2000 \beta + 24450) q^{85} + ( - 8252 \beta - 6528) q^{86} + ( - 8724 \beta - 138312) q^{88} + ( - 7392 \beta - 48186) q^{89} + (7904 \beta - 54656) q^{91} + ( - 1776 \beta + 109344) q^{92} + (9064 \beta - 125664) q^{94} + (5200 \beta - 20900) q^{95} + ( - 12480 \beta - 6142) q^{97} + (11609 \beta - 182784) q^{98} +O(q^{100})$$ q + b * q^2 + (b + 70) * q^4 - 25 * q^5 + (16*b - 64) * q^7 + (39*b + 102) * q^8 - 25*b * q^10 + (-32*b - 108) * q^11 + (-16*b + 446) * q^13 + (-48*b + 1632) * q^14 + (109*b + 1738) * q^16 + (-80*b - 978) * q^17 + (-208*b + 836) * q^19 + (-25*b - 1750) * q^20 + (-140*b - 3264) * q^22 + (-48*b + 1632) * q^23 + 625 * q^25 + (430*b - 1632) * q^26 + (1072*b - 2848) * q^28 + (-64*b - 942) * q^29 + (-176*b + 1424) * q^31 + (599*b + 7854) * q^32 + (-1058*b - 8160) * q^34 + (-400*b + 1600) * q^35 + (816*b + 3926) * q^37 + (628*b - 21216) * q^38 + (-975*b - 2550) * q^40 + (-544*b + 4086) * q^41 + (-64*b - 8188) * q^43 + (-2380*b - 10824) * q^44 + (1584*b - 4896) * q^46 + (-1232*b + 10296) * q^47 + (-1792*b + 13401) * q^49 + 625*b * q^50 + (-690*b + 29588) * q^52 + (2272*b + 6042) * q^53 + (800*b + 2700) * q^55 + (-240*b + 57120) * q^56 + (-1006*b - 6528) * q^58 + (3232*b - 1164) * q^59 + (1568*b + 9326) * q^61 + (1248*b - 17952) * q^62 + (4965*b + 5482) * q^64 + (400*b - 11150) * q^65 + (-1280*b - 5812) * q^67 + (-6658*b - 76620) * q^68 + (1200*b - 40800) * q^70 + (3200*b + 18888) * q^71 + (608*b + 29258) * q^73 + (4742*b + 83232) * q^74 + (-13932*b + 37304) * q^76 + (-192*b - 45312) * q^77 + (3760*b + 51920) * q^79 + (-2725*b - 43450) * q^80 + (3542*b - 55488) * q^82 + (-4032*b + 63060) * q^83 + (2000*b + 24450) * q^85 + (-8252*b - 6528) * q^86 + (-8724*b - 138312) * q^88 + (-7392*b - 48186) * q^89 + (7904*b - 54656) * q^91 + (-1776*b + 109344) * q^92 + (9064*b - 125664) * q^94 + (5200*b - 20900) * q^95 + (-12480*b - 6142) * q^97 + (11609*b - 182784) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + 141 q^{4} - 50 q^{5} - 112 q^{7} + 243 q^{8}+O(q^{10})$$ 2 * q + q^2 + 141 * q^4 - 50 * q^5 - 112 * q^7 + 243 * q^8 $$2 q + q^{2} + 141 q^{4} - 50 q^{5} - 112 q^{7} + 243 q^{8} - 25 q^{10} - 248 q^{11} + 876 q^{13} + 3216 q^{14} + 3585 q^{16} - 2036 q^{17} + 1464 q^{19} - 3525 q^{20} - 6668 q^{22} + 3216 q^{23} + 1250 q^{25} - 2834 q^{26} - 4624 q^{28} - 1948 q^{29} + 2672 q^{31} + 16307 q^{32} - 17378 q^{34} + 2800 q^{35} + 8668 q^{37} - 41804 q^{38} - 6075 q^{40} + 7628 q^{41} - 16440 q^{43} - 24028 q^{44} - 8208 q^{46} + 19360 q^{47} + 25010 q^{49} + 625 q^{50} + 58486 q^{52} + 14356 q^{53} + 6200 q^{55} + 114000 q^{56} - 14062 q^{58} + 904 q^{59} + 20220 q^{61} - 34656 q^{62} + 15929 q^{64} - 21900 q^{65} - 12904 q^{67} - 159898 q^{68} - 80400 q^{70} + 40976 q^{71} + 59124 q^{73} + 171206 q^{74} + 60676 q^{76} - 90816 q^{77} + 107600 q^{79} - 89625 q^{80} - 107434 q^{82} + 122088 q^{83} + 50900 q^{85} - 21308 q^{86} - 285348 q^{88} - 103764 q^{89} - 101408 q^{91} + 216912 q^{92} - 242264 q^{94} - 36600 q^{95} - 24764 q^{97} - 353959 q^{98}+O(q^{100})$$ 2 * q + q^2 + 141 * q^4 - 50 * q^5 - 112 * q^7 + 243 * q^8 - 25 * q^10 - 248 * q^11 + 876 * q^13 + 3216 * q^14 + 3585 * q^16 - 2036 * q^17 + 1464 * q^19 - 3525 * q^20 - 6668 * q^22 + 3216 * q^23 + 1250 * q^25 - 2834 * q^26 - 4624 * q^28 - 1948 * q^29 + 2672 * q^31 + 16307 * q^32 - 17378 * q^34 + 2800 * q^35 + 8668 * q^37 - 41804 * q^38 - 6075 * q^40 + 7628 * q^41 - 16440 * q^43 - 24028 * q^44 - 8208 * q^46 + 19360 * q^47 + 25010 * q^49 + 625 * q^50 + 58486 * q^52 + 14356 * q^53 + 6200 * q^55 + 114000 * q^56 - 14062 * q^58 + 904 * q^59 + 20220 * q^61 - 34656 * q^62 + 15929 * q^64 - 21900 * q^65 - 12904 * q^67 - 159898 * q^68 - 80400 * q^70 + 40976 * q^71 + 59124 * q^73 + 171206 * q^74 + 60676 * q^76 - 90816 * q^77 + 107600 * q^79 - 89625 * q^80 - 107434 * q^82 + 122088 * q^83 + 50900 * q^85 - 21308 * q^86 - 285348 * q^88 - 103764 * q^89 - 101408 * q^91 + 216912 * q^92 - 242264 * q^94 - 36600 * q^95 - 24764 * q^97 - 353959 * 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
 −9.61187 10.6119
−9.61187 0 60.3881 −25.0000 0 −217.790 −272.863 0 240.297
1.2 10.6119 0 80.6119 −25.0000 0 105.790 515.863 0 −265.297
 $$n$$: 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 45.6.a.e 2
3.b odd 2 1 15.6.a.c 2
4.b odd 2 1 720.6.a.bd 2
5.b even 2 1 225.6.a.m 2
5.c odd 4 2 225.6.b.g 4
12.b even 2 1 240.6.a.q 2
15.d odd 2 1 75.6.a.h 2
15.e even 4 2 75.6.b.e 4
21.c even 2 1 735.6.a.g 2
24.f even 2 1 960.6.a.bf 2
24.h odd 2 1 960.6.a.bj 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.6.a.c 2 3.b odd 2 1
45.6.a.e 2 1.a even 1 1 trivial
75.6.a.h 2 15.d odd 2 1
75.6.b.e 4 15.e even 4 2
225.6.a.m 2 5.b even 2 1
225.6.b.g 4 5.c odd 4 2
240.6.a.q 2 12.b even 2 1
720.6.a.bd 2 4.b odd 2 1
735.6.a.g 2 21.c even 2 1
960.6.a.bf 2 24.f even 2 1
960.6.a.bj 2 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - T_{2} - 102$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(45))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T - 102$$
$3$ $$T^{2}$$
$5$ $$(T + 25)^{2}$$
$7$ $$T^{2} + 112T - 23040$$
$11$ $$T^{2} + 248T - 89328$$
$13$ $$T^{2} - 876T + 165668$$
$17$ $$T^{2} + 2036 T + 381924$$
$19$ $$T^{2} - 1464 T - 3887920$$
$23$ $$T^{2} - 3216 T + 2350080$$
$29$ $$T^{2} + 1948 T + 529860$$
$31$ $$T^{2} - 2672 T - 1382400$$
$37$ $$T^{2} - 8668 T - 49300220$$
$41$ $$T^{2} - 7628 T - 15712860$$
$43$ $$T^{2} + 16440 T + 67149584$$
$47$ $$T^{2} - 19360 T - 61495104$$
$53$ $$T^{2} - 14356 T - 476289180$$
$59$ $$T^{2} - 904 T - 1067881200$$
$61$ $$T^{2} - 20220 T - 149182204$$
$67$ $$T^{2} + 12904 T - 125898096$$
$71$ $$T^{2} - 40976 T - 627281856$$
$73$ $$T^{2} - 59124 T + 836113700$$
$79$ $$T^{2} - 107600 T + 1448870400$$
$83$ $$T^{2} - 122088 T + 2064089232$$
$89$ $$T^{2} + 103764 T - 2895368220$$
$97$ $$T^{2} + 24764 T - 15772164476$$