LMFDB - Newform orbit 1100.6.b.d

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:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 150x^{3} + 11449x^{2} - 6848x + 2048 $ x^6 - 2x^5 + 2x^4 + 150x^3 + 11449x^2 - 6848*x + 2048
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8} $
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 $1,\beta_1,\ldots,\beta_{5}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_{4} q^{3} + (2 \beta_{5} + 7 \beta_{4} - 24 \beta_{3}) q^{7} + (5 \beta_{2} + 3 \beta_1 + 34) q^{9}+O(q^{10}) $ q + b4 * q^3 + (2b5 + 7b4 - 24b3) q^7 + (5b2 + 3b1 + 34) * q^9	$ q + \beta_{4} q^{3} + (2 \beta_{5} + 7 \beta_{4} - 24 \beta_{3}) q^{7} + (5 \beta_{2} + 3 \beta_1 + 34) q^{9} - 121 q^{11} + (12 \beta_{5} - 16 \beta_{4} - 181 \beta_{3}) q^{13} + (29 \beta_{5} - 9 \beta_{4} + 5 \beta_{3}) q^{17} + ( - 34 \beta_{2} + 47 \beta_1 + 1242) q^{19} + (75 \beta_{2} + 9 \beta_1 - 1647) q^{21} + ( - 73 \beta_{5} + \cdots + 824 \beta_{3}) q^{23}+ \cdots + ( - 605 \beta_{2} - 363 \beta_1 - 4114) q^{99}+O(q^{100}) $ q + b4 * q^3 + (2b5 + 7b4 - 24b3) q^7 + (5b2 + 3b1 + 34) * q^9 - 121 * q^11 + (12b5 - 16b4 - 181b3) q^13 + (29b5 - 9b4 + 5b3) q^17 + (-34b2 + 47b1 + 1242) * q^19 + (75b2 + 9b1 - 1647) * q^21 + (-73b5 - 40b4 + 824b3) q^23 + (-3b5 + 167b4 + 822b3) q^27 + (108b2 + 91b1 - 2830) * q^29 + (33b2 + 119b1 - 1653) * q^31 - 121b4 q^33 + (-247b5 + 237b4 - 771b3) q^37 + (197b2 - 120b1 + 2203) * q^39 + (-150b2 + 4b1 - 6184) * q^41 + (231b5 - 554b4 + 3596b3) q^43 + (211b5 - 856b4 - 4628b3) q^47 + (821b2 + 91b1 - 3066) * q^49 + (182b2 - 201b1 - 434) * q^51 + (-369b5 + 1167b4 + 5235b3) q^53 + (627b5 + 979b4 - 10038b3) q^57 + (973b2 + 172b1 + 21871) * q^59 + (968b2 - 29b1 - 9934) * q^61 + (117b5 - 1056b4 + 8874b3) q^63 + (-15b5 - 3038b4 - 10898b3) q^67 + (-1608b2 + 318b1 + 15024) * q^69 + (57b2 - 855b1 + 2379) * q^71 + (-1018b5 + 708b4 + 10417b3) q^73 + (-242b5 - 847b4 + 2904b3) q^77 + (-664b2 - 1450b1 + 37320) * q^79 + (1204b2 + 1248b1 - 25579) * q^81 + (745b5 + 1172b4 + 19612b3) q^83 + (171b5 - 5599b4 + 16026b3) q^87 + (1939b2 + 2501b1 - 31089) * q^89 + (909b2 + 172b1 - 27597) * q^91 + (873b5 - 3867b4 - 1122b3) q^93 + (-2374b5 + 4154b4 - 10887b3) q^97 + (-605b2 - 363b1 - 4114) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 208 q^{9}+O(q^{10}) $ 6 * q + 208 * q^9	$ 6 q + 208 q^{9} - 726 q^{11} + 7290 q^{19} - 9750 q^{21} - 16946 q^{29} - 10090 q^{31} + 13852 q^{39} - 37412 q^{41} - 16936 q^{49} - 1838 q^{51} + 132828 q^{59} - 57610 q^{61} + 86292 q^{69} + 16098 q^{71} + 225492 q^{79} - 153562 q^{81} - 187658 q^{89} - 164108 q^{91} - 25168 q^{99}+O(q^{100}) $ 6 * q + 208 * q^9 - 726 * q^11 + 7290 * q^19 - 9750 * q^21 - 16946 * q^29 - 10090 * q^31 + 13852 * q^39 - 37412 * q^41 - 16936 * q^49 - 1838 * q^51 + 132828 * q^59 - 57610 * q^61 + 86292 * q^69 + 16098 * q^71 + 225492 * q^79 - 153562 * q^81 - 187658 * q^89 - 164108 * q^91 - 25168 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} - 2x^{5} + 2x^{4} + 150x^{3} + 11449x^{2} - 6848x + 2048 $ x^6 - 2*x^5 + 2*x^4 + 150*x^3 + 11449*x^2 - 6848*x + 2048 :

$\beta_{1}$	$=$	$ ( -80\nu^{5} - 8381\nu^{4} - 8720\nu^{3} - 6000\nu^{2} + 5120\nu - 67928919 ) / 1561615 $ (-80v^5 - 8381v^4 - 8720v^3 - 6000v^2 + 5120*v - 67928919) / 1561615
$\beta_{2}$	$=$	$ ( -310\nu^{5} + 6564\nu^{4} - 33790\nu^{3} - 23250\nu^{2} + 19840\nu + 45155361 ) / 1561615 $ (-310v^5 + 6564v^4 - 33790v^3 - 23250v^2 + 19840*v + 45155361) / 1561615
$\beta_{3}$	$=$	$ ( 2897\nu^{5} - 4922\nu^{4} + 3450\nu^{3} + 529598\nu^{2} + 33233153\nu - 9900128 ) / 4997168 $ (2897v^5 - 4922v^4 + 3450v^3 + 529598v^2 + 33233153*v - 9900128) / 4997168
$\beta_{4}$	$=$	$ ( 198459\nu^{5} - 341494\nu^{4} + 706390\nu^{3} + 25815730\nu^{2} + 2326285571\nu - 692973856 ) / 49971680 $ (198459v^5 - 341494v^4 + 706390v^3 + 25815730v^2 + 2326285571*v - 692973856) / 49971680
$\beta_{5}$	$=$	$ ( -218709\nu^{5} + 367274\nu^{4} + 209590\nu^{3} - 30457710\nu^{2} - 2459288461\nu + 732644576 ) / 24985840 $ (-218709v^5 + 367274v^4 + 209590v^3 - 30457710v^2 - 2459288461*v + 732644576) / 24985840

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

$\nu$	$=$	$ ( 2\beta_{5} + 4\beta_{4} + 4\beta_{3} + 3\beta_{2} + 2\beta _1 + 5 ) / 16 $ (2b5 + 4b4 + 4b3 + 3b2 + 2*b1 + 5) / 16
$\nu^{2}$	$=$	$ ( 5\beta_{5} - 10\beta_{4} + 144\beta_{3} ) / 4 $ (5b5 - 10b4 + 144*b3) / 4
$\nu^{3}$	$=$	$ ( 234\beta_{5} + 388\beta_{4} + 748\beta_{3} - 311\beta_{2} - 234\beta _1 - 1185 ) / 16 $ (234b5 + 388b4 + 748b3 - 311b2 - 234*b1 - 1185) / 16
$\nu^{4}$	$=$	$ 40\beta_{2} - 155\beta _1 - 7899 $ 40b2 - 155b1 - 7899
$\nu^{5}$	$=$	$ ( -26878\beta_{5} - 39036\beta_{4} - 124476\beta_{3} - 32957\beta_{2} - 26878\beta _1 - 215995 ) / 16 $ (-26878b5 - 39036b4 - 124476b3 - 32957b2 - 26878*b1 - 215995) / 16

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

−6.98601	−	6.98601i
0.297901	−	0.297901i
7.68811	−	7.68811i
7.68811	+	7.68811i
0.297901	+	0.297901i
−6.98601	+	6.98601i

−

20.0937i

−

228.057i

−160.758

749.2

−

13.9603i

9.35268i

48.1099

749.3

−

5.13342i

−

82.4100i

216.648

749.4

5.13342i

82.4100i

216.648

749.5

13.9603i

−

9.35268i

48.1099

749.6

20.0937i

228.057i

−160.758

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.d		6
5.b	even	2	1	inner	1100.6.b.d		6
5.c	odd	4	1		220.6.a.b	✓	3
5.c	odd	4	1		1100.6.a.d		3
20.e	even	4	1		880.6.a.j		3

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
220.6.a.b	✓	3	5.c	odd	4	1
880.6.a.j		3	20.e	even	4	1
1100.6.a.d		3	5.c	odd	4	1
1100.6.b.d		6	1.a	even	1	1	trivial
1100.6.b.d		6	5.b	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{6} + 625T_{3}^{4} + 94464T_{3}^{2} + 2073600 $ T3^6 + 625*T3^4 + 94464*T3^2 + 2073600 acting on $S_{6}^{\mathrm{new}}(1100, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} + 625 T^{4} + \cdots + 2073600 $ T^6 + 625T^4 + 94464T^2 + 2073600
$5$	$ T^{6} $ T^6
$7$	$ T^{6} + \cdots + 30897202176 $ T^6 + 58889T^4 + 358365184T^2 + 30897202176
$11$	$ (T + 121)^{6} $ (T + 121)^6
$13$	$ T^{6} + \cdots + 10\!\cdots\!36 $ T^6 + 974988T^4 + 82182604848T^2 + 1019981166876736
$17$	$ T^{6} + \cdots + 24\!\cdots\!00 $ T^6 + 2868745T^4 + 2075653817304T^2 + 243389663967344400
$19$	$ (T^{3} - 3645 T^{2} + \cdots + 13139307248)^{2} $ (T^3 - 3645T^2 - 2339448T + 13139307248)^2
$23$	$ T^{6} + \cdots + 64\!\cdots\!24 $ T^6 + 29689476T^4 + 262666493610624T^2 + 641298491451656193024
$29$	$ (T^{3} + 8473 T^{2} + \cdots - 57124516620)^{2} $ (T^3 + 8473T^2 + 2034096T - 57124516620)^2
$31$	$ (T^{3} + 5045 T^{2} + \cdots - 63522375360)^{2} $ (T^3 + 5045T^2 - 15687824T - 63522375360)^2
$37$	$ T^{6} + \cdots + 24\!\cdots\!36 $ T^6 + 228030129T^4 + 12280010371517784T^2 + 24432354348686378977936
$41$	$ (T^{3} + 18706 T^{2} + \cdots + 25081114200)^{2} $ (T^3 + 18706T^2 + 88019628T + 25081114200)^2
$43$	$ T^{6} + \cdots + 23\!\cdots\!00 $ T^6 + 476812784T^4 + 67097190573510400T^2 + 2396282674799444544000000
$47$	$ T^{6} + \cdots + 14\!\cdots\!00 $ T^6 + 780775300T^4 + 21206228578917504T^2 + 146488394892948408345600
$53$	$ T^{6} + \cdots + 43\!\cdots\!84 $ T^6 + 1451892681T^4 + 47607247755834264T^2 + 43519205291582516883984
$59$	$ (T^{3} + \cdots + 27595270294560)^{2} $ (T^3 - 66414T^2 + 366946944T + 27595270294560)^2
$61$	$ (T^{3} + 28805 T^{2} + \cdots - 230105423484)^{2} $ (T^3 + 28805T^2 - 918040472T - 230105423484)^2
$67$	$ T^{6} + \cdots + 92\!\cdots\!00 $ T^6 + 7081242432T^4 + 14530031604245062656T^2 + 9229335665689712660330905600
$71$	$ (T^{3} - 8049 T^{2} + \cdots - 259369439040)^{2} $ (T^3 - 8049T^2 - 1360868112T - 259369439040)^2
$73$	$ T^{6} + \cdots + 14\!\cdots\!00 $ T^6 + 4963858892T^4 + 5084910967316013616T^2 + 1465504482261188296141761600
$79$	$ (T^{3} + \cdots + 159067707906944)^{2} $ (T^3 - 112746T^2 + 602869536T + 159067707906944)^2
$83$	$ T^{6} + \cdots + 72\!\cdots\!00 $ T^6 + 7598034532T^4 + 7302623171092861056T^2 + 7254372405367913103590400
$89$	$ (T^{3} + \cdots - 854013774895068)^{2} $ (T^3 + 93829T^2 - 9350679240T - 854013774895068)^2
$97$	$ T^{6} + \cdots + 49\!\cdots\!44 $ T^6 + 27404313816T^4 + 225362866873739408784T^2 + 490154178385503774510944729344
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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)

\(f(q)\)	\(=\)	\( q + \beta_{4} q^{3} + (2 \beta_{5} + 7 \beta_{4} - 24 \beta_{3}) q^{7} + (5 \beta_{2} + 3 \beta_1 + 34) q^{9}+O(q^{10}) \) q + b4 * q^3 + (2b5 + 7b4 - 24b3) q^7 + (5b2 + 3b1 + 34) * q^9	\( q + \beta_{4} q^{3} + (2 \beta_{5} + 7 \beta_{4} - 24 \beta_{3}) q^{7} + (5 \beta_{2} + 3 \beta_1 + 34) q^{9} - 121 q^{11} + (12 \beta_{5} - 16 \beta_{4} - 181 \beta_{3}) q^{13} + (29 \beta_{5} - 9 \beta_{4} + 5 \beta_{3}) q^{17} + ( - 34 \beta_{2} + 47 \beta_1 + 1242) q^{19} + (75 \beta_{2} + 9 \beta_1 - 1647) q^{21} + ( - 73 \beta_{5} + \cdots + 824 \beta_{3}) q^{23}+ \cdots + ( - 605 \beta_{2} - 363 \beta_1 - 4114) q^{99}+O(q^{100}) \) q + b4 * q^3 + (2b5 + 7b4 - 24b3) q^7 + (5b2 + 3b1 + 34) * q^9 - 121 * q^11 + (12b5 - 16b4 - 181b3) q^13 + (29b5 - 9b4 + 5b3) q^17 + (-34b2 + 47b1 + 1242) * q^19 + (75b2 + 9b1 - 1647) * q^21 + (-73b5 - 40b4 + 824b3) q^23 + (-3b5 + 167b4 + 822b3) q^27 + (108b2 + 91b1 - 2830) * q^29 + (33b2 + 119b1 - 1653) * q^31 - 121b4 q^33 + (-247b5 + 237b4 - 771b3) q^37 + (197b2 - 120b1 + 2203) * q^39 + (-150b2 + 4b1 - 6184) * q^41 + (231b5 - 554b4 + 3596b3) q^43 + (211b5 - 856b4 - 4628b3) q^47 + (821b2 + 91b1 - 3066) * q^49 + (182b2 - 201b1 - 434) * q^51 + (-369b5 + 1167b4 + 5235b3) q^53 + (627b5 + 979b4 - 10038b3) q^57 + (973b2 + 172b1 + 21871) * q^59 + (968b2 - 29b1 - 9934) * q^61 + (117b5 - 1056b4 + 8874b3) q^63 + (-15b5 - 3038b4 - 10898b3) q^67 + (-1608b2 + 318b1 + 15024) * q^69 + (57b2 - 855b1 + 2379) * q^71 + (-1018b5 + 708b4 + 10417b3) q^73 + (-242b5 - 847b4 + 2904b3) q^77 + (-664b2 - 1450b1 + 37320) * q^79 + (1204b2 + 1248b1 - 25579) * q^81 + (745b5 + 1172b4 + 19612b3) q^83 + (171b5 - 5599b4 + 16026b3) q^87 + (1939b2 + 2501b1 - 31089) * q^89 + (909b2 + 172b1 - 27597) * q^91 + (873b5 - 3867b4 - 1122b3) q^93 + (-2374b5 + 4154b4 - 10887b3) q^97 + (-605b2 - 363b1 - 4114) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 208 q^{9}+O(q^{10}) \) 6 * q + 208 * q^9	\( 6 q + 208 q^{9} - 726 q^{11} + 7290 q^{19} - 9750 q^{21} - 16946 q^{29} - 10090 q^{31} + 13852 q^{39} - 37412 q^{41} - 16936 q^{49} - 1838 q^{51} + 132828 q^{59} - 57610 q^{61} + 86292 q^{69} + 16098 q^{71} + 225492 q^{79} - 153562 q^{81} - 187658 q^{89} - 164108 q^{91} - 25168 q^{99}+O(q^{100}) \) 6 * q + 208 * q^9 - 726 * q^11 + 7290 * q^19 - 9750 * q^21 - 16946 * q^29 - 10090 * q^31 + 13852 * q^39 - 37412 * q^41 - 16936 * q^49 - 1838 * q^51 + 132828 * q^59 - 57610 * q^61 + 86292 * q^69 + 16098 * q^71 + 225492 * q^79 - 153562 * q^81 - 187658 * q^89 - 164108 * q^91 - 25168 * q^99

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