LMFDB - Newform orbit 110.4.b.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [110,4,Mod(89,110)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(110, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

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

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

chi := DirichletCharacter("110.89");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$110 = 2 \cdot 5 \cdot 11$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	110.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:	$6.49021010063$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} + 151x^{6} + 7935x^{4} + 171721x^{2} + 1308736$ x^8 + 151x^6 + 7935x^4 + 171721*x^2 + 1308736
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$2^{3}$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/110\mathbb{Z}\right)^\times$ .

$n$	$67$	$101$
$\chi(n)$	$-1$	$1$

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}

89.1

	−	7.97576i
	−	4.48251i
		4.82127i
		6.63700i
	−	6.63700i
	−	4.82127i
		4.48251i
		7.97576i

−

2.00000i

−

5.97576i

−4.00000

1.78092

−

11.0376i

−11.9515

−

8.09941i

8.00000i

−8.70973

−22.0752

−

3.56184i

89.2

−

2.00000i

−

2.48251i

−4.00000

3.44238

10.6372i

−4.96501

31.7569i

8.00000i

20.8372

21.2744

−

6.88475i

89.3

−

2.00000i

6.82127i

−4.00000

−8.37442

−

7.40737i

13.6425

−

13.6360i

8.00000i

−19.5297

−14.8147

16.7488i

89.4

−

2.00000i

8.63700i

−4.00000

11.1511

0.807757i

17.2740

0.978517i

8.00000i

−47.5977

1.61551

−

22.3022i

89.5

2.00000i

−

8.63700i

−4.00000

11.1511

−

0.807757i

17.2740

−

0.978517i

−

8.00000i

−47.5977

1.61551

22.3022i

89.6

2.00000i

−

6.82127i

−4.00000

−8.37442

7.40737i

13.6425

13.6360i

−

8.00000i

−19.5297

−14.8147

−

16.7488i

89.7

2.00000i

2.48251i

−4.00000

3.44238

−

10.6372i

−4.96501

−

31.7569i

−

8.00000i

20.8372

21.2744

6.88475i

89.8

2.00000i

5.97576i

−4.00000

1.78092

11.0376i

−11.9515

8.09941i

−

8.00000i

−8.70973

−22.0752

3.56184i

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	110.4.b.c	✓	8
3.b	odd	2	1		990.4.c.i		8
4.b	odd	2	1		880.4.b.h		8
5.b	even	2	1	inner	110.4.b.c	✓	8
5.c	odd	4	1		550.4.a.ba		4
5.c	odd	4	1		550.4.a.bb		4
15.d	odd	2	1		990.4.c.i		8
20.d	odd	2	1		880.4.b.h		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
110.4.b.c	✓	8	1.a	even	1	1	trivial
110.4.b.c	✓	8	5.b	even	2	1	inner
550.4.a.ba		4	5.c	odd	4	1
550.4.a.bb		4	5.c	odd	4	1
880.4.b.h		8	4.b	odd	2	1
880.4.b.h		8	20.d	odd	2	1
990.4.c.i		8	3.b	odd	2	1
990.4.c.i		8	15.d	odd	2	1

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{8} + 163T_{3}^{6} + 8763T_{3}^{4} + 171997T_{3}^{2} + 763876$ T3^8 + 163*T3^6 + 8763*T3^4 + 171997*T3^2 + 763876 acting on $S_{4}^{\mathrm{new}}(110, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$(T^{2} + 4)^{4}$ (T^2 + 4)^4
$3$	$T^{8} + 163 T^{6} + \cdots + 763876$ T^8 + 163T^6 + 8763T^4 + 171997*T^2 + 763876
$5$	$T^{8} - 16 T^{7} + \cdots + 244140625$ T^8 - 16T^7 + 209T^6 - 2234T^5 + 11840T^4 - 279250T^3 + 3265625T^2 - 31250000*T + 244140625
$7$	$T^{8} + 1261 T^{6} + \cdots + 11778624$ T^8 + 1261T^6 + 267084T^4 + 12556080*T^2 + 11778624
$11$	$(T + 11)^{8}$ (T + 11)^8
$13$	$T^{8} + \cdots + 5498837401600$ T^8 + 13408T^6 + 47839488T^4 + 36256301056*T^2 + 5498837401600
$17$	$T^{8} + \cdots + 136200103430400$ T^8 + 30105T^6 + 236217240T^4 + 441671312016*T^2 + 136200103430400
$19$	$(T^{4} + 151 T^{3} + \cdots - 390272)^{2}$ (T^4 + 151T^3 + 2220T^2 - 119552*T - 390272)^2
$23$	$T^{8} + \cdots + 73\!\cdots\!84$ T^8 + 64986T^6 + 994991985T^4 + 4847227794244*T^2 + 7391206792848384
$29$	$(T^{4} + 29 T^{3} + \cdots + 15788512)^{2}$ (T^4 + 29T^3 - 17130T^2 + 58688*T + 15788512)^2
$31$	$(T^{4} - 511 T^{3} + \cdots - 256907000)^{2}$ (T^4 - 511T^3 + 67767T^2 + 392735*T - 256907000)^2
$37$	$T^{8} + \cdots + 49077590691600$ T^8 + 194703T^6 + 7228831887T^4 + 1448823197809*T^2 + 49077590691600
$41$	$(T^{4} - 226 T^{3} + \cdots + 3452698624)^{2}$ (T^4 - 226T^3 - 134544T^2 + 13735232*T + 3452698624)^2
$43$	$T^{8} + \cdots + 16\!\cdots\!00$ T^8 + 269824T^6 + 26292862944T^4 + 1095898833709056*T^2 + 16487163248545440000
$47$	$T^{8} + \cdots + 75\!\cdots\!00$ T^8 + 567580T^6 + 108225043440T^4 + 7330539898913344*T^2 + 75467271486050406400
$53$	$T^{8} + \cdots + 15\!\cdots\!76$ T^8 + 711289T^6 + 152041942800T^4 + 9112812573302784*T^2 + 159780252562837733376
$59$	$(T^{4} - 134 T^{3} + \cdots + 1066184592)^{2}$ (T^4 - 134T^3 - 68559T^2 + 4972920*T + 1066184592)^2
$61$	$(T^{4} + 245 T^{3} + \cdots + 22378892000)^{2}$ (T^4 + 245T^3 - 492522T^2 - 138649760*T + 22378892000)^2
$67$	$T^{8} + \cdots + 80\!\cdots\!44$ T^8 + 895162T^6 + 269663552865T^4 + 30335062433568948*T^2 + 804541791894773511744
$71$	$(T^{4} - 1017 T^{3} + \cdots - 50573201976)^{2}$ (T^4 - 1017T^3 - 280557T^2 + 335339349*T - 50573201976)^2
$73$	$T^{8} + \cdots + 16\!\cdots\!00$ T^8 + 1592980T^6 + 730644379200T^4 + 102562384514789376*T^2 + 165298096162735718400
$79$	$(T^{4} - 136 T^{3} + \cdots + 145164269184)^{2}$ (T^4 - 136T^3 - 834540T^2 + 78707856*T + 145164269184)^2
$83$	$T^{8} + \cdots + 40\!\cdots\!00$ T^8 + 1637340T^6 + 803514985200T^4 + 132931053420019264*T^2 + 4098391266026535321600
$89$	$(T^{4} + \cdots - 1272726658650)^{2}$ (T^4 - 2891T^3 + 1443513T^2 + 1811309163*T - 1272726658650)^2
$97$	$T^{8} + \cdots + 77\!\cdots\!96$ T^8 + 3204198T^6 + 3184951292649T^4 + 989336764126928448*T^2 + 77112581132978751799296
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Classical	Maass
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

$f(q)$	$=$	$q + \beta_{2} q^{2} + ( - \beta_{2} + \beta_1) q^{3} - 4 q^{4} + (\beta_{5} + \beta_{2} + 2) q^{5} + ( - \beta_{3} + 3) q^{6} + (\beta_{6} + \beta_{5} - \beta_{2} - \beta_1) q^{7} - 4 \beta_{2} q^{8} + (\beta_{4} + \beta_{3} - 13) q^{9}+ \cdots + ( - 11 \beta_{4} - 11 \beta_{3} + 143) q^{99}+O(q^{100})$ q + b2 * q^2 + (-b2 + b1) * q^3 - 4 * q^4 + (b5 + b2 + 2) * q^5 + (-b3 + 3) * q^6 + (b6 + b5 - b2 - b1) * q^7 - 4b2 q^8 + (b4 + b3 - 13) * q^9 + (b7 - b6 + b5 + b4 + 2b2 - 3) q^10 - 11 * q^11 + (4b2 - 4b1) * q^12 + (-4b7 - 2b3 - 8b2 - 2) q^13 + (-2b6 + 2b5 + 2b4 + 6) q^14 + (-4b7 - b5 + 2b4 - 5b3 - 3b2 + 3b1 - 2) q^15 + 16 * q^16 + (-6b7 - 3b6 - 3b5 - 3b3 + 20b2 - b1 - 3) q^17 + (-2b7 - 2b6 - 2b5 - b3 - 15b2 + 6b1 - 1) q^18 + (-3b6 + 3b5 + b4 + 3b3 - 36) q^19 + (-4b5 - 4b2 - 8) * q^20 + (b6 - b5 + 3b4 - 3b3 + 28) * q^21 - 11b2 q^22 + (6b7 + 5b6 + 5b5 + 3b3 - 25b2 - 12b1 + 3) * q^23 + (4b3 - 12) q^24 + (4b7 + b6 + 4b5 - b4 - 5b3 - 22b2 + 5b1 - 22) q^25 + (-8b6 + 8b5 + 32) * q^26 + (6b7 + 3b3 + 44b2 + 3b1 + 3) * q^27 + (-4b6 - 4b5 + 4b2 + 4b1) * q^28 + (-3b6 + 3b5 - b4 + 7b3 - 4) q^29 + (-5b7 - 11b6 + 3b5 - b4 - 5b3 + b2 - 8b1 + 6) q^30 + (7b4 - 7b3 + 126) * q^31 + 16b2 q^32 + (11b2 - 11b1) * q^33 + (-6b6 + 6b5 - 6b4 + 4b3 - 82) * q^34 + (4b7 + 5b6 + b5 - 7b4 - 27b2 + 2b1 - 133) q^35 + (-4b4 - 4b3 + 52) * q^36 + (-2b7 + 6b6 + 6b5 - b3 + 20b2 + 29b1 - 1) q^37 + (4b7 - 2b6 - 2b5 + 2b3 - 40b2 + 14b1 + 2) * q^38 + (28b6 - 28b5 + 4b4 + 10b3 - 34) * q^39 + (-4b7 + 4b6 - 4b5 - 4b4 - 8b2 + 12) q^40 + (-12b6 + 12b5 - 10b4 + 4b3 + 56) * q^41 + (-8b7 - 6b6 - 6b5 - 4b3 + 28b2 - 6b1 - 4) * q^42 + (6b7 + 14b6 + 14b5 + 3b3 + 55b2 - 20b1 + 3) * q^43 + 44 * q^44 + (6b7 + 18b6 - 10b5 + 5b4 + 10b3 - 51b2 - 8b1 - 70) q^45 + (2b6 - 2b5 + 10b4 + 7b3 + 117) * q^46 + (26b7 + 8b6 + 8b5 + 13b3 + 79b2 + 18b1 + 13) * q^47 + (-16b2 + 16b1) * q^48 + (11b6 - 11b5 - 25b4 - b3 + 21) q^49 + (5b7 + 5b6 - b5 + 5b4 - 5b3 - 16b2 - 30b1 + 88) * q^50 + (39b6 - 39b5 - 7b4 - 5b3 + 100) * q^51 + (16b7 + 8b3 + 32b2 + 8) q^52 + (-22b7 - 27b6 - 27b5 - 11b3 - 8b2 - 7b1 - 11) * q^53 + (12b6 - 12b5 - 3b3 - 179) q^54 + (-11b5 - 11b2 - 22) * q^55 + (8b6 - 8b5 - 8b4 - 24) q^56 + (-14b7 + b6 + b5 - 7b3 + 166b2 - 47b1 - 7) * q^57 + (8b7 + 2b6 + 2b5 + 4b3 - 10b2 + 26b1 + 4) * q^58 + (-13b6 + 13b5 + 10b4 - 8b3 + 32) * q^59 + (16b7 + 4b5 - 8b4 + 20b3 + 12b2 - 12b1 + 8) * q^60 + (29b6 - 29b5 - 25b4 + 31b3 - 52) * q^61 + (-14b7 - 14b6 - 14b5 - 7b3 + 126b2 - 14b1 - 7) * q^62 + (14b7 + 16b6 + 16b5 + 7b3 - 113b2 - 16b1 + 7) * q^63 - 64 * q^64 + (-10b7 - 10b6 + 2b5 - 10b4 + 10b3 - 218b2 + 60b1 - 176) q^65 + (11b3 - 33) q^66 + (-6b7 + b6 + b5 - 3b3 + 221b2 + 20b1 - 3) * q^67 + (24b7 + 12b6 + 12b5 + 12b3 - 80b2 + 4b1 + 12) * q^68 + (-37b6 + 37b5 + 2b4 + 2b3 + 354) * q^69 + (10b7 + 16b6 + 12b5 + 6b4 - 126b2 - 22b1 + 114) * q^70 + (-42b6 + 42b5 + b4 - b3 + 254) * q^71 + (8b7 + 8b6 + 8b5 + 4b3 + 60b2 - 24b1 + 4) * q^72 + (36b7 - 10b6 - 10b5 + 18b3 + 102b2 - 6b1 + 18) * q^73 + (-16b6 + 16b5 + 12b4 - 35b3 - 103) * q^74 + (-30b7 - 39b6 + 12b5 + 11b4 - 5b3 - 271b2 + 8b1 - 271) q^75 + (12b6 - 12b5 - 4b4 - 12b3 + 144) * q^76 + (-11b6 - 11b5 + 11b2 + 11b1) * q^77 + (-64b7 - 8b6 - 8b5 - 32b3 - 48b2 + 48b1 - 32) * q^78 + (10b6 - 10b5 - 52b3 + 8) q^79 + (16b5 + 16b2 + 32) * q^80 + (-42b6 + 42b5 + 24b4 - 20b3 - 315) * q^81 + (44b7 + 20b6 + 20b5 + 22b3 + 62b2 - 4b1 + 22) * q^82 + (-46b7 - 8b6 - 8b5 - 23b3 - 49b2 - 38b1 - 23) * q^83 + (-4b6 + 4b5 - 12b4 + 12b3 - 112) * q^84 + (14b7 - 55b6 + 29b5 + 23b4 + 35b3 - 184b2 + 72b1 + 68) q^85 + (-16b6 + 16b5 + 28b4 + 6b3 - 186) * q^86 + (-14b7 + 13b6 + 13b5 - 7b3 + 252b2 - 11b1 - 7) * q^87 + 44b2 q^88 + (-40b6 + 40b5 + 47b4 - 65b3 + 702) * q^89 + (-38b7 - 6b6 - 14b5 + 8b4 - 15b3 - 85b2 + 38b1 + 197) q^90 + (-8b6 + 8b5 - 24b4 + 20b3 - 436) * q^91 + (-24b7 - 20b6 - 20b5 - 12b3 + 100b2 + 48b1 - 12) * q^92 + (14b7 - 28b6 - 28b5 + 7b3 - 266b2 + 91b1 + 7) * q^93 + (36b6 - 36b5 + 16b4 - 26b3 - 326) * q^94 + (28b7 + 43b6 - 33b5 + 4b4 + 5b3 - 148b2 + 23b1 + 169) q^95 + (-16b3 + 48) q^96 + (-40b7 - 37b6 - 37b5 - 20b3 + 108b2 - 98b1 - 20) * q^97 + (28b7 + 50b6 + 50b5 + 14b3 + 47b2 - 54b1 + 14) * q^98 + (-11b4 - 11b3 + 143) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$8 q - 32 q^{4} + 16 q^{5} + 28 q^{6} - 110 q^{9} - 28 q^{10} - 88 q^{11} + 44 q^{14} + 8 q^{15} + 128 q^{16} - 302 q^{19} - 64 q^{20} + 230 q^{21} - 112 q^{24} - 162 q^{25} + 256 q^{26} - 58 q^{29} + 80 q^{30}+ \cdots + 1210 q^{99}+O(q^{100})$ 8 * q - 32 * q^4 + 16 * q^5 + 28 * q^6 - 110 * q^9 - 28 * q^10 - 88 * q^11 + 44 * q^14 + 8 * q^15 + 128 * q^16 - 302 * q^19 - 64 * q^20 + 230 * q^21 - 112 * q^24 - 162 * q^25 + 256 * q^26 - 58 * q^29 + 80 * q^30 + 1022 * q^31 - 660 * q^34 - 1058 * q^35 + 440 * q^36 - 320 * q^39 + 112 * q^40 + 452 * q^41 + 352 * q^44 - 622 * q^45 + 888 * q^46 + 222 * q^49 + 704 * q^50 + 834 * q^51 - 1420 * q^54 - 176 * q^55 - 176 * q^56 + 268 * q^59 - 32 * q^60 - 490 * q^61 - 512 * q^64 - 1408 * q^65 - 308 * q^66 + 2820 * q^69 + 880 * q^70 + 2034 * q^71 - 708 * q^74 - 2110 * q^75 + 1208 * q^76 + 272 * q^79 + 256 * q^80 - 2488 * q^81 - 920 * q^84 + 330 * q^85 - 1568 * q^86 + 5782 * q^89 + 1696 * q^90 - 3520 * q^91 - 2536 * q^94 + 1268 * q^95 + 448 * q^96 + 1210 * q^99

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( 8\nu^{7} + 1065\nu^{5} + 43460\nu^{3} + 522203\nu ) / 31460$ (8v^7 + 1065v^5 + 43460v^3 + 522203v) / 31460
$\beta_{3}$	$=$	$( \nu^{6} + 140\nu^{4} + 5955\nu^{2} + 72996 ) / 220$ (v^6 + 140v^4 + 5955v^2 + 72996) / 220
$\beta_{4}$	$=$	$( \nu^{6} + 140\nu^{4} + 6175\nu^{2} + 81356 ) / 220$ (v^6 + 140v^4 + 6175v^2 + 81356) / 220
$\beta_{5}$	$=$	$( - 80 \nu^{7} + 1001 \nu^{6} - 10650 \nu^{5} + 124410 \nu^{4} - 450330 \nu^{3} + 4686825 \nu^{2} + \cdots + 52336856 ) / 188760$ (-80v^7 + 1001v^6 - 10650v^5 + 124410v^4 - 450330v^3 + 4686825v^2 - 5929880*v + 52336856) / 188760
$\beta_{6}$	$=$	$( - 80 \nu^{7} - 1001 \nu^{6} - 10650 \nu^{5} - 124410 \nu^{4} - 450330 \nu^{3} - 4686825 \nu^{2} + \cdots - 52336856 ) / 188760$ (-80v^7 - 1001v^6 - 10650v^5 - 124410v^4 - 450330v^3 - 4686825v^2 - 5929880*v - 52336856) / 188760
$\beta_{7}$	$=$	$( - 323 \nu^{7} - 429 \nu^{6} - 40050 \nu^{5} - 60060 \nu^{4} - 1499085 \nu^{3} - 2554695 \nu^{2} + \cdots - 31409664 ) / 188760$ (-323v^7 - 429v^6 - 40050v^5 - 60060v^4 - 1499085v^3 - 2554695v^2 - 16450478*v - 31409664) / 188760

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$ -expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	110.4.b.c

Level	$110$
Weight	$4$
Character orbit	110.b
Analytic conductor	$6.490$
Analytic rank	$0$
Dimension	$8$
Inner twists	$2$

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{4} - \beta_{3} - 38$ b4 - b3 - 38
$\nu^{3}$	$=$	$-6\beta_{6} - 6\beta_{5} - 20\beta_{2} - 45\beta_1$ -6b6 - 6b5 - 20b2 - 45b1
$\nu^{4}$	$=$	$6\beta_{6} - 6\beta_{5} - 81\beta_{4} + 95\beta_{3} + 1760$ 6b6 - 6b5 - 81b4 + 95b3 + 1760
$\nu^{5}$	$=$	$64\beta_{7} + 520\beta_{6} + 520\beta_{5} + 32\beta_{3} + 2164\beta_{2} + 2329\beta _1 + 32$ 64b7 + 520b6 + 520b5 + 32b3 + 2164b2 + 2329b1 + 32
$\nu^{6}$	$=$	$-840\beta_{6} + 840\beta_{5} + 5385\beta_{4} - 7125\beta_{3} - 93106$ -840b6 + 840b5 + 5385b4 - 7125b3 - 93106
$\nu^{7}$	$=$	$-8520\beta_{7} - 36630\beta_{6} - 36630\beta_{5} - 4260\beta_{3} - 175500\beta_{2} - 130861\beta _1 - 4260$ -8520b7 - 36630b6 - 36630b5 - 4260b3 - 175500b2 - 130861b1 - 4260

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 89.8
Significant digits:
Format: