# Properties

 Label 45.6.b.b Level $45$ Weight $6$ Character orbit 45.b Analytic conductor $7.217$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [45,6,Mod(19,45)]

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, 1]))

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

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

chi := DirichletCharacter("45.19");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$45 = 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 45.b (of order $$2$$, degree $$1$$, 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: $$7.21727189158$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 3$$ x^2 - x + 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 5) 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 $$\beta = 2\sqrt{-11}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} - 12 q^{4} + ( - 5 \beta + 45) q^{5} - 9 \beta q^{7} - 20 \beta q^{8} +O(q^{10})$$ q - b * q^2 - 12 * q^4 + (-5*b + 45) * q^5 - 9*b * q^7 - 20*b * q^8 $$q - \beta q^{2} - 12 q^{4} + ( - 5 \beta + 45) q^{5} - 9 \beta q^{7} - 20 \beta q^{8} + ( - 45 \beta - 220) q^{10} - 252 q^{11} - 18 \beta q^{13} - 396 q^{14} - 1264 q^{16} + 104 \beta q^{17} - 220 q^{19} + (60 \beta - 540) q^{20} + 252 \beta q^{22} - 367 \beta q^{23} + ( - 450 \beta + 925) q^{25} - 792 q^{26} + 108 \beta q^{28} + 6930 q^{29} + 6752 q^{31} + 624 \beta q^{32} + 4576 q^{34} + ( - 405 \beta - 1980) q^{35} + 2106 \beta q^{37} + 220 \beta q^{38} + ( - 900 \beta - 4400) q^{40} + 198 q^{41} - 63 \beta q^{43} + 3024 q^{44} - 16148 q^{46} + 1589 \beta q^{47} + 13243 q^{49} + ( - 925 \beta - 19800) q^{50} + 216 \beta q^{52} + 878 \beta q^{53} + (1260 \beta - 11340) q^{55} - 7920 q^{56} - 6930 \beta q^{58} + 24660 q^{59} - 5698 q^{61} - 6752 \beta q^{62} - 12992 q^{64} + ( - 810 \beta - 3960) q^{65} - 6579 \beta q^{67} - 1248 \beta q^{68} + (1980 \beta - 17820) q^{70} - 53352 q^{71} + 10692 \beta q^{73} + 92664 q^{74} + 2640 q^{76} + 2268 \beta q^{77} + 51920 q^{79} + (6320 \beta - 56880) q^{80} - 198 \beta q^{82} + 9323 \beta q^{83} + (4680 \beta + 22880) q^{85} - 2772 q^{86} + 5040 \beta q^{88} + 9990 q^{89} - 7128 q^{91} + 4404 \beta q^{92} + 69916 q^{94} + (1100 \beta - 9900) q^{95} - 15264 \beta q^{97} - 13243 \beta q^{98} +O(q^{100})$$ q - b * q^2 - 12 * q^4 + (-5*b + 45) * q^5 - 9*b * q^7 - 20*b * q^8 + (-45*b - 220) * q^10 - 252 * q^11 - 18*b * q^13 - 396 * q^14 - 1264 * q^16 + 104*b * q^17 - 220 * q^19 + (60*b - 540) * q^20 + 252*b * q^22 - 367*b * q^23 + (-450*b + 925) * q^25 - 792 * q^26 + 108*b * q^28 + 6930 * q^29 + 6752 * q^31 + 624*b * q^32 + 4576 * q^34 + (-405*b - 1980) * q^35 + 2106*b * q^37 + 220*b * q^38 + (-900*b - 4400) * q^40 + 198 * q^41 - 63*b * q^43 + 3024 * q^44 - 16148 * q^46 + 1589*b * q^47 + 13243 * q^49 + (-925*b - 19800) * q^50 + 216*b * q^52 + 878*b * q^53 + (1260*b - 11340) * q^55 - 7920 * q^56 - 6930*b * q^58 + 24660 * q^59 - 5698 * q^61 - 6752*b * q^62 - 12992 * q^64 + (-810*b - 3960) * q^65 - 6579*b * q^67 - 1248*b * q^68 + (1980*b - 17820) * q^70 - 53352 * q^71 + 10692*b * q^73 + 92664 * q^74 + 2640 * q^76 + 2268*b * q^77 + 51920 * q^79 + (6320*b - 56880) * q^80 - 198*b * q^82 + 9323*b * q^83 + (4680*b + 22880) * q^85 - 2772 * q^86 + 5040*b * q^88 + 9990 * q^89 - 7128 * q^91 + 4404*b * q^92 + 69916 * q^94 + (1100*b - 9900) * q^95 - 15264*b * q^97 - 13243*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 24 q^{4} + 90 q^{5}+O(q^{10})$$ 2 * q - 24 * q^4 + 90 * q^5 $$2 q - 24 q^{4} + 90 q^{5} - 440 q^{10} - 504 q^{11} - 792 q^{14} - 2528 q^{16} - 440 q^{19} - 1080 q^{20} + 1850 q^{25} - 1584 q^{26} + 13860 q^{29} + 13504 q^{31} + 9152 q^{34} - 3960 q^{35} - 8800 q^{40} + 396 q^{41} + 6048 q^{44} - 32296 q^{46} + 26486 q^{49} - 39600 q^{50} - 22680 q^{55} - 15840 q^{56} + 49320 q^{59} - 11396 q^{61} - 25984 q^{64} - 7920 q^{65} - 35640 q^{70} - 106704 q^{71} + 185328 q^{74} + 5280 q^{76} + 103840 q^{79} - 113760 q^{80} + 45760 q^{85} - 5544 q^{86} + 19980 q^{89} - 14256 q^{91} + 139832 q^{94} - 19800 q^{95}+O(q^{100})$$ 2 * q - 24 * q^4 + 90 * q^5 - 440 * q^10 - 504 * q^11 - 792 * q^14 - 2528 * q^16 - 440 * q^19 - 1080 * q^20 + 1850 * q^25 - 1584 * q^26 + 13860 * q^29 + 13504 * q^31 + 9152 * q^34 - 3960 * q^35 - 8800 * q^40 + 396 * q^41 + 6048 * q^44 - 32296 * q^46 + 26486 * q^49 - 39600 * q^50 - 22680 * q^55 - 15840 * q^56 + 49320 * q^59 - 11396 * q^61 - 25984 * q^64 - 7920 * q^65 - 35640 * q^70 - 106704 * q^71 + 185328 * q^74 + 5280 * q^76 + 103840 * q^79 - 113760 * q^80 + 45760 * q^85 - 5544 * q^86 + 19980 * q^89 - 14256 * q^91 + 139832 * q^94 - 19800 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/45\mathbb{Z}\right)^\times$$.

 $$n$$ $$11$$ $$37$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 0.5 + 1.65831i 0.5 − 1.65831i
6.63325i 0 −12.0000 45.0000 33.1662i 0 59.6992i 132.665i 0 −220.000 298.496i
19.2 6.63325i 0 −12.0000 45.0000 + 33.1662i 0 59.6992i 132.665i 0 −220.000 + 298.496i
 $$n$$: 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 45.6.b.b 2
3.b odd 2 1 5.6.b.a 2
4.b odd 2 1 720.6.f.f 2
5.b even 2 1 inner 45.6.b.b 2
5.c odd 4 2 225.6.a.n 2
12.b even 2 1 80.6.c.a 2
15.d odd 2 1 5.6.b.a 2
15.e even 4 2 25.6.a.c 2
20.d odd 2 1 720.6.f.f 2
21.c even 2 1 245.6.b.a 2
24.f even 2 1 320.6.c.g 2
24.h odd 2 1 320.6.c.f 2
60.h even 2 1 80.6.c.a 2
60.l odd 4 2 400.6.a.t 2
105.g even 2 1 245.6.b.a 2
120.i odd 2 1 320.6.c.f 2
120.m even 2 1 320.6.c.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.6.b.a 2 3.b odd 2 1
5.6.b.a 2 15.d odd 2 1
25.6.a.c 2 15.e even 4 2
45.6.b.b 2 1.a even 1 1 trivial
45.6.b.b 2 5.b even 2 1 inner
80.6.c.a 2 12.b even 2 1
80.6.c.a 2 60.h even 2 1
225.6.a.n 2 5.c odd 4 2
245.6.b.a 2 21.c even 2 1
245.6.b.a 2 105.g even 2 1
320.6.c.f 2 24.h odd 2 1
320.6.c.f 2 120.i odd 2 1
320.6.c.g 2 24.f even 2 1
320.6.c.g 2 120.m even 2 1
400.6.a.t 2 60.l odd 4 2
720.6.f.f 2 4.b odd 2 1
720.6.f.f 2 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 44$$ acting on $$S_{6}^{\mathrm{new}}(45, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 44$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 90T + 3125$$
$7$ $$T^{2} + 3564$$
$11$ $$(T + 252)^{2}$$
$13$ $$T^{2} + 14256$$
$17$ $$T^{2} + 475904$$
$19$ $$(T + 220)^{2}$$
$23$ $$T^{2} + 5926316$$
$29$ $$(T - 6930)^{2}$$
$31$ $$(T - 6752)^{2}$$
$37$ $$T^{2} + 195150384$$
$41$ $$(T - 198)^{2}$$
$43$ $$T^{2} + 174636$$
$47$ $$T^{2} + 111096524$$
$53$ $$T^{2} + 33918896$$
$59$ $$(T - 24660)^{2}$$
$61$ $$(T + 5698)^{2}$$
$67$ $$T^{2} + 1904462604$$
$71$ $$(T + 53352)^{2}$$
$73$ $$T^{2} + 5030030016$$
$79$ $$(T - 51920)^{2}$$
$83$ $$T^{2} + 3824406476$$
$89$ $$(T - 9990)^{2}$$
$97$ $$T^{2} + 10251546624$$