LMFDB - Newform orbit 1100.6.b.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1100,6,Mod(749,1100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1100, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1100.749");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1100 = 2^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1100.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:	$176.422201794$
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} + 1202x^{6} + 449473x^{4} + 53281872x^{2} + 1947986496 $ x^8 + 1202x^6 + 449473x^4 + 53281872*x^2 + 1947986496
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{7}\cdot 5^{4} $
Twist minimal:	no (minimal twist has level 220)
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 $1,\beta_1,\ldots,\beta_{7}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_1 q^{3} + ( - \beta_{4} - \beta_{2} + 3 \beta_1) q^{7} + (\beta_{7} - \beta_{5} - 58) q^{9}+O(q^{10}) $ q + b1 * q^3 + (-b4 - b2 + 3b1) q^7 + (b7 - b5 - 58) * q^9	$ q + \beta_1 q^{3} + ( - \beta_{4} - \beta_{2} + 3 \beta_1) q^{7} + (\beta_{7} - \beta_{5} - 58) q^{9} + 121 q^{11} + ( - 3 \beta_{6} - 3 \beta_{4} + \cdots - 12 \beta_1) q^{13}+ \cdots + (121 \beta_{7} - 121 \beta_{5} - 7018) q^{99}+O(q^{100}) $ q + b1 * q^3 + (-b4 - b2 + 3b1) q^7 + (b7 - b5 - 58) * q^9 + 121 * q^11 + (-3b6 - 3b4 - 8b2 - 12b1) * q^13 + (-4b6 + 4b4 + 22b2 + 9b1) * q^17 + (-7b7 - b5 - 4b3 - 213) * q^19 + (-4b7 - 5b5 + 5b3 - 761) q^21 + (3b6 + 7b4 + 102b2 - 84b1) * q^23 + (-9b6 - 15b4 - 42b2 + 35b1) * q^27 + (-6b7 - 5b5 - 23b3 + 3829) q^29 + (-4b7 - 43b5 - 4b3 - 2375) q^31 + 121b1 q^33 + (10b6 + 53b4 - 497b2 + 195b1) * q^37 + (-48b7 + 66b5 + 2b3 + 3666) q^39 + (-34b7 + 56b5 - 40b3 - 2482) q^41 + (24b6 - 42b4 + 139b2 + 330b1) * q^43 + (-35b6 + 95b4 - 406b2 - 228b1) * q^47 + (-55b7 - 45b5 - 24b3 + 3810) q^49 + (17b7 + 79b5 - 62b3 - 3773) q^51 + (-38b6 - 69b4 + 485b2 + 249b1) * q^53 + (15b6 + 105b4 - 474b2 + 301b1) * q^57 + (-62b7 + 168b5 + 51b3 + 5628) q^59 + (38b7 - 99b5 - 42b3 - 21085) q^61 + (-56b6 - 12b4 + 1949b2 - 16b1) * q^63 + (133b6 - 249b4 - 908b2 + 558b1) * q^67 + (-20b7 + 38b5 - 112b3 + 24662) q^69 + (60b7 + 219b5 - 186b3 - 5241) q^71 + (107b6 + 5b4 + 644b2 - 684b1) * q^73 + (-121b4 - 121b2 + 363b1) q^77 + (364b7 - 330b5 + 24b3 + 2746) q^79 + (128b7 - 128b5 + 48b3 - 23615) q^81 + (94b6 - 412b4 + 3047b2 + 1296b1) * q^83 + (58b6 - 225b4 - 6124b2 + 3957b1) * q^87 + (175b7 - 191b5 - 422b3 + 37543) q^89 + (-294b7 + 152b5 - 397b3 - 22032) q^91 + (-324b6 + 411b4 + 1584b2 - 4885b1) * q^93 + (-284b6 + 8b4 + 403b2 + 4050b1) * q^97 + (121b7 - 121b5 - 7018) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 460 q^{9}+O(q^{10}) $ 8 * q - 460 * q^9	$ 8 q - 460 q^{9} + 968 q^{11} - 1732 q^{19} - 6104 q^{21} + 30608 q^{29} - 19016 q^{31} + 29136 q^{39} - 19992 q^{41} + 30260 q^{49} - 30116 q^{51} + 44776 q^{59} - 168528 q^{61} + 197216 q^{69} - 41688 q^{71} + 23424 q^{79} - 188408 q^{81} + 301044 q^{89} - 177432 q^{91} - 55660 q^{99}+O(q^{100}) $ 8 * q - 460 * q^9 + 968 * q^11 - 1732 * q^19 - 6104 * q^21 + 30608 * q^29 - 19016 * q^31 + 29136 * q^39 - 19992 * q^41 + 30260 * q^49 - 30116 * q^51 + 44776 * q^59 - 168528 * q^61 + 197216 * q^69 - 41688 * q^71 + 23424 * q^79 - 188408 * q^81 + 301044 * q^89 - 177432 * q^91 - 55660 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} + 1202x^{6} + 449473x^{4} + 53281872x^{2} + 1947986496 $ x^8 + 1202*x^6 + 449473*x^4 + 53281872*x^2 + 1947986496 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( \nu^{7} + 1202\nu^{5} + 405337\nu^{3} + 26756136\nu ) / 2118528 $ (v^7 + 1202v^5 + 405337v^3 + 26756136*v) / 2118528
$\beta_{3}$	$=$	$ ( \nu^{4} + 601\nu^{2} + 44136 ) / 48 $ (v^4 + 601*v^2 + 44136) / 48
$\beta_{4}$	$=$	$ ( -121\nu^{7} - 128891\nu^{5} - 39451714\nu^{3} - 2667829104\nu ) / 19596384 $ (-121v^7 - 128891v^5 - 39451714v^3 - 2667829104v) / 19596384
$\beta_{5}$	$=$	$ ( \nu^{6} + 1001\nu^{4} + 279352\nu^{2} + 16269408 ) / 10656 $ (v^6 + 1001v^4 + 279352v^2 + 16269408) / 10656
$\beta_{6}$	$=$	$ ( 317\nu^{7} + 325864\nu^{5} + 92156867\nu^{3} + 4618824120\nu ) / 39192768 $ (317v^7 + 325864v^5 + 92156867v^3 + 4618824120v) / 39192768
$\beta_{7}$	$=$	$ ( \nu^{6} + 1001\nu^{4} + 290008\nu^{2} + 19476864 ) / 10656 $ (v^6 + 1001v^4 + 290008v^2 + 19476864) / 10656

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{7} - \beta_{5} - 301 $ b7 - b5 - 301
$\nu^{3}$	$=$	$ -9\beta_{6} - 15\beta_{4} - 42\beta_{2} - 451\beta_1 $ -9b6 - 15b4 - 42b2 - 451b1
$\nu^{4}$	$=$	$ -601\beta_{7} + 601\beta_{5} + 48\beta_{3} + 136765 $ -601b7 + 601b5 + 48*b3 + 136765
$\nu^{5}$	$=$	$ 5217\beta_{6} + 9879\beta_{4} + 39834\beta_{2} + 227011\beta_1 $ 5217b6 + 9879b4 + 39834b2 + 227011b1
$\nu^{6}$	$=$	$ 322249\beta_{7} - 311593\beta_{5} - 48048\beta_{3} - 69086221 $ 322249b7 - 311593b5 - 48048*b3 - 69086221
$\nu^{7}$	$=$	$ -2622801\beta_{6} - 5794503\beta_{4} - 28737786\beta_{2} - 116816371\beta_1 $ -2622801b6 - 5794503b4 - 28737786b2 - 116816371b1

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1100\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$551$
$\chi(n)$	$1$	$-1$	$1$

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} $

749.1

	−	23.2360i
	−	22.1144i
	−	9.84560i
	−	8.72399i
		8.72399i
		9.84560i
		22.1144i
		23.2360i

−

23.2360i

63.7039i

−296.911

749.2

−

22.1144i

−

130.428i

−246.045

749.3

−

9.84560i

−

11.6329i

146.064

749.4

−

8.72399i

−

175.764i

166.892

749.5

8.72399i

175.764i

166.892

749.6

9.84560i

11.6329i

146.064

749.7

22.1144i

130.428i

−246.045

749.8

23.2360i

−

63.7039i

−296.911

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	1100.6.b.f		8
5.b	even	2	1	inner	1100.6.b.f		8
5.c	odd	4	1		220.6.a.d	✓	4
5.c	odd	4	1		1100.6.a.e		4
20.e	even	4	1		880.6.a.m		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
220.6.a.d	✓	4	5.c	odd	4	1
880.6.a.m		4	20.e	even	4	1
1100.6.a.e		4	5.c	odd	4	1
1100.6.b.f		8	1.a	even	1	1	trivial
1100.6.b.f		8	5.b	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{8} + 1202T_{3}^{6} + 449473T_{3}^{4} + 53281872T_{3}^{2} + 1947986496 $ T3^8 + 1202*T3^6 + 449473*T3^4 + 53281872*T3^2 + 1947986496 acting on $S_{6}^{\mathrm{new}}(1100, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} $ T^8
$3$	$ T^{8} + \cdots + 1947986496 $ T^8 + 1202T^6 + 449473T^4 + 53281872*T^2 + 1947986496
$5$	$ T^{8} $ T^8
$7$	$ T^{8} + \cdots + 288611034965136 $ T^8 + 52098T^6 + 726971289T^4 + 2230143545512*T^2 + 288611034965136
$11$	$ (T - 121)^{8} $ (T - 121)^8
$13$	$ T^{8} + \cdots + 12\!\cdots\!96 $ T^8 + 2554648T^6 + 2314635232848T^4 + 897411407862454528*T^2 + 126867955352776033616896
$17$	$ T^{8} + \cdots + 18\!\cdots\!36 $ T^8 + 4763698T^6 + 7183628125089T^4 + 3612427237170262512*T^2 + 180324057883216817756736
$19$	$ (T^{4} + 866 T^{3} + \cdots - 313652697344)^{2} $ (T^4 + 866T^3 - 2942079T^2 + 1756161056*T - 313652697344)^2
$23$	$ T^{8} + \cdots + 15\!\cdots\!56 $ T^8 + 16106776T^6 + 75473755061712T^4 + 84370694948753059584*T^2 + 1542702963017761573856256
$29$	$ (T^{4} - 15304 T^{3} + \cdots + 379396794636)^{2} $ (T^4 - 15304T^3 + 59704767T^2 - 9535309476*T + 379396794636)^2
$31$	$ (T^{4} + \cdots - 659542417773824)^{2} $ (T^4 + 9508T^3 - 47235957T^2 - 490458471392*T - 659542417773824)^2
$37$	$ T^{8} + \cdots + 91\!\cdots\!96 $ T^8 + 304136866T^6 + 27527740169938617T^4 + 895354168982584874323624*T^2 + 9196975884822590323014421246096
$41$	$ (T^{4} + \cdots - 89\!\cdots\!04)^{2} $ (T^4 + 9996T^3 - 239152368T^2 - 3295301399856*T - 8919552853517904)^2
$43$	$ T^{8} + \cdots + 86\!\cdots\!76 $ T^8 + 329438728T^6 + 32222357579299344T^4 + 1103089552977951843100672*T^2 + 8694681938536146846700590923776
$47$	$ T^{8} + \cdots + 17\!\cdots\!36 $ T^8 + 797103928T^6 + 210569664388553808T^4 + 19091943729146671375949568*T^2 + 170863697699096730202467379086336
$53$	$ T^{8} + \cdots + 10\!\cdots\!00 $ T^8 + 688551250T^6 + 78283653883475625T^4 + 1779621418034021120625000*T^2 + 10128838452765445767058256250000
$59$	$ (T^{4} + \cdots + 65\!\cdots\!36)^{2} $ (T^4 - 22388T^3 - 1200263052T^2 + 28765744848672*T + 6584040820057536)^2
$61$	$ (T^{4} + \cdots - 89\!\cdots\!64)^{2} $ (T^4 + 84264T^3 + 2132242287T^2 + 12025832898916*T - 89719715422740564)^2
$67$	$ T^{8} + \cdots + 19\!\cdots\!56 $ T^8 + 7377106416T^6 + 15713120740890194496T^4 + 11071323334078227710830944256*T^2 + 1974552348652969139497742345607315456
$71$	$ (T^{4} + \cdots + 29\!\cdots\!24)^{2} $ (T^4 + 20844T^3 - 5784550605T^2 - 177645621979008*T + 299902343459716224)^2
$73$	$ T^{8} + \cdots + 73\!\cdots\!76 $ T^8 + 3348926296T^6 + 1169909541995107152T^4 + 102451247389850466458523904*T^2 + 735029033072224976305871461221376
$79$	$ (T^{4} + \cdots + 30\!\cdots\!24)^{2} $ (T^4 - 11712T^3 - 11252746500T^2 + 58430635896416*T + 30080987808105157824)^2
$83$	$ T^{8} + \cdots + 38\!\cdots\!96 $ T^8 + 14968480816T^6 + 62179327109933709792T^4 + 70976776350586290882157741824*T^2 + 3889625763050006818483229172348324096
$89$	$ (T^{4} + \cdots - 45\!\cdots\!44)^{2} $ (T^4 - 150522T^3 - 6154950747T^2 + 1561790691431748*T - 45266502083678714844)^2
$97$	$ T^{8} + \cdots + 57\!\cdots\!36 $ T^8 + 39103796248T^6 + 455487864829627494288T^4 + 1654985403846579201305773458688*T^2 + 575838766634769477349287853660656910336
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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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1100.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_1 q^{3} + ( - \beta_{4} - \beta_{2} + 3 \beta_1) q^{7} + (\beta_{7} - \beta_{5} - 58) q^{9}+O(q^{10}) \) q + b1 * q^3 + (-b4 - b2 + 3b1) q^7 + (b7 - b5 - 58) * q^9	\( q + \beta_1 q^{3} + ( - \beta_{4} - \beta_{2} + 3 \beta_1) q^{7} + (\beta_{7} - \beta_{5} - 58) q^{9} + 121 q^{11} + ( - 3 \beta_{6} - 3 \beta_{4} + \cdots - 12 \beta_1) q^{13}+ \cdots + (121 \beta_{7} - 121 \beta_{5} - 7018) q^{99}+O(q^{100}) \) q + b1 * q^3 + (-b4 - b2 + 3b1) q^7 + (b7 - b5 - 58) * q^9 + 121 * q^11 + (-3b6 - 3b4 - 8b2 - 12b1) * q^13 + (-4b6 + 4b4 + 22b2 + 9b1) * q^17 + (-7b7 - b5 - 4b3 - 213) * q^19 + (-4b7 - 5b5 + 5b3 - 761) q^21 + (3b6 + 7b4 + 102b2 - 84b1) * q^23 + (-9b6 - 15b4 - 42b2 + 35b1) * q^27 + (-6b7 - 5b5 - 23b3 + 3829) q^29 + (-4b7 - 43b5 - 4b3 - 2375) q^31 + 121b1 q^33 + (10b6 + 53b4 - 497b2 + 195b1) * q^37 + (-48b7 + 66b5 + 2b3 + 3666) q^39 + (-34b7 + 56b5 - 40b3 - 2482) q^41 + (24b6 - 42b4 + 139b2 + 330b1) * q^43 + (-35b6 + 95b4 - 406b2 - 228b1) * q^47 + (-55b7 - 45b5 - 24b3 + 3810) q^49 + (17b7 + 79b5 - 62b3 - 3773) q^51 + (-38b6 - 69b4 + 485b2 + 249b1) * q^53 + (15b6 + 105b4 - 474b2 + 301b1) * q^57 + (-62b7 + 168b5 + 51b3 + 5628) q^59 + (38b7 - 99b5 - 42b3 - 21085) q^61 + (-56b6 - 12b4 + 1949b2 - 16b1) * q^63 + (133b6 - 249b4 - 908b2 + 558b1) * q^67 + (-20b7 + 38b5 - 112b3 + 24662) q^69 + (60b7 + 219b5 - 186b3 - 5241) q^71 + (107b6 + 5b4 + 644b2 - 684b1) * q^73 + (-121b4 - 121b2 + 363b1) q^77 + (364b7 - 330b5 + 24b3 + 2746) q^79 + (128b7 - 128b5 + 48b3 - 23615) q^81 + (94b6 - 412b4 + 3047b2 + 1296b1) * q^83 + (58b6 - 225b4 - 6124b2 + 3957b1) * q^87 + (175b7 - 191b5 - 422b3 + 37543) q^89 + (-294b7 + 152b5 - 397b3 - 22032) q^91 + (-324b6 + 411b4 + 1584b2 - 4885b1) * q^93 + (-284b6 + 8b4 + 403b2 + 4050b1) * q^97 + (121b7 - 121b5 - 7018) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 460 q^{9}+O(q^{10}) \) 8 * q - 460 * q^9	\( 8 q - 460 q^{9} + 968 q^{11} - 1732 q^{19} - 6104 q^{21} + 30608 q^{29} - 19016 q^{31} + 29136 q^{39} - 19992 q^{41} + 30260 q^{49} - 30116 q^{51} + 44776 q^{59} - 168528 q^{61} + 197216 q^{69} - 41688 q^{71} + 23424 q^{79} - 188408 q^{81} + 301044 q^{89} - 177432 q^{91} - 55660 q^{99}+O(q^{100}) \) 8 * q - 460 * q^9 + 968 * q^11 - 1732 * q^19 - 6104 * q^21 + 30608 * q^29 - 19016 * q^31 + 29136 * q^39 - 19992 * q^41 + 30260 * q^49 - 30116 * q^51 + 44776 * q^59 - 168528 * q^61 + 197216 * q^69 - 41688 * q^71 + 23424 * q^79 - 188408 * q^81 + 301044 * q^89 - 177432 * q^91 - 55660 * q^99

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