LMFDB - Newform orbit 200.6.f.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [200,6,Mod(149,200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(200, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("200.149");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$200 = 2^{3} \cdot 5^{2}$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	200.f (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:	$32.0767639626$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.12220785438976.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} + 5x^{6} + 116x^{4} + 320x^{2} + 4096$ x^8 + 5x^6 + 116x^4 + 320*x^2 + 4096
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$2^{15}$
Twist minimal:	no (minimal twist has level 8)
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.

$f(q)$	$=$	$q + \beta_{2} q^{2} + ( - \beta_{5} - \beta_{2}) q^{3} + ( - \beta_{6} - \beta_{4} - 5) q^{4} + ( - 2 \beta_{6} + 6 \beta_{4} + \cdots - 29) q^{6} + ( - 2 \beta_{7} + 2 \beta_{5} + \cdots - 6 \beta_1) q^{7}+ \cdots + ( - 3456 \beta_{6} + \cdots - 3456 \beta_{3}) q^{99}+O(q^{100})$ q + b2 * q^2 + (-b5 - b2) * q^3 + (-b6 - b4 - 5) * q^4 + (-2b6 + 6b4 + 3b3 - 29) q^6 + (-2b7 + 2b5 - 18b2 - 6b1) * q^7 + (3b7 - 7b5 - 5b2 - 33b1) * q^8 + (8b6 - 8b4 - 24b3 + 41) q^9 + (16b6 - 5b4 + 16b3) q^11 + (-5b7 + 17b5 - 21b2 - 185b1) * q^12 + (-3b7 + 29b5 + 47b2 - 3b1) * q^13 + (16b6 + 16b4 - 4b3 + 596) q^14 + (-4b6 + 28b4 - 24b3 - 828) q^16 + (-4b7 + 4b5 - 36b2 - 13b1) * q^17 + (16b7 + 48b5 + 25b2 + 592b1) * q^18 + (48b6 - 7b4 + 48b3) q^19 + (24b6 - 160b4 + 24b3) q^21 + (-43b7 - 41b5 - 21b2 + 721b1) * q^22 + (-26b7 + 26b5 - 234b2 + 370b1) * q^23 + (52b6 - 44b4 - 216b3 + 1948) q^24 + (28b6 - 180b4 - 78b3 + 1402) q^26 + (24b7 - 46b5 - 190b2 + 24b1) * q^27 + (-44b7 + 124b5 + 596b2 + 548b1) * q^28 + (-82b6 - 256b4 - 82b3) q^29 + (-48b6 + 48b4 + 144b3 - 3232) q^31 + (4b7 + 204b5 - 796b2 - 428b1) * q^32 + (-36b7 + 36b5 - 324b2 - 186b1) * q^33 + (32b6 + 32b4 - 9b3 + 1193) q^34 + (183b6 - 329b4 + 384b3 - 2541) q^36 + (57b7 + 217b5 - 125b2 + 57b1) * q^37 + (-137b7 - 83b5 - 55b2 + 2059b1) * q^38 + (-268b6 + 268b4 + 804b3 - 8776) q^39 + (-272b6 + 272b4 + 816b3 - 1142) q^41 + (88b7 - 824b5 - 184b2 + 3064b1) * q^42 + (-48b7 - 469b5 - 181b2 - 48b1) * q^43 + (-274b6 + 398b4 + 1016b3 + 7278) q^44 + (208b6 + 208b4 + 396b3 + 7300) q^46 + (-108b7 + 108b5 - 972b2 + 7164b1) * q^47 + (156b7 + 532b5 + 1852b2 + 4044b1) * q^48 + (-192b6 + 192b4 + 576b3 - 2457) q^49 + (48b6 - 322b4 + 48b3) q^51 + (202b7 - 610b5 + 1194b2 + 4018b1) * q^52 + (131b7 + 1059b5 + 273b2 + 131b1) * q^53 + (148b6 + 324b4 + 66b3 - 5822) q^54 + (-400b6 - 1040b4 + 352b3 + 10192) q^56 + (-76b7 + 76b5 - 684b2 + 482b1) * q^57 + (502b7 - 1198b5 - 174b2 - 34b1) * q^58 + (-256b6 - 1427b4 - 256b3) q^59 + (1314b6 - 2016b4 + 1314b3) q^61 + (-96b7 - 288b5 - 3136b2 - 3552b1) * q^62 + (-178b7 + 178b5 - 1602b2 - 18646b1) * q^63 + (1424b6 - 240b4 - 1056b3 + 10480) q^64 + (288b6 + 288b4 - 150b3 + 10806) q^66 + (792b7 - 567b5 - 5319b2 + 792b1) * q^67 + (-87b7 + 251b5 + 1193b2 + 1101b1) * q^68 + (312b6 - 1184b4 + 312b3) q^69 + (228b6 - 228b4 - 684b3 + 51672) q^71 + (-421b7 - 2431b5 - 3053b2 + 10775b1) * q^72 + (604b7 - 604b5 + 5436b2 + 3185b1) * q^73 + (1004b6 - 1188b4 - 822b3 - 4366) q^74 + (-742b6 + 1018b4 + 2856b3 + 24986) q^76 + (836b7 + 1764b5 - 3252b2 + 836b1) * q^77 + (-536b7 - 1608b5 - 8240b2 - 19832b1) * q^78 + (344b6 - 344b4 - 1032b3 + 61968) q^79 + (-1288b6 + 1288b4 + 3864b3 + 7421) q^81 + (-544b7 - 1632b5 - 598b2 - 20128b1) * q^82 + (704b7 - 1865b5 - 6089b2 + 704b1) * q^83 + (-1936b6 + 3952b4 + 5184b3 - 49168) q^84 + (-1418b6 + 2718b4 + 1551b3 - 4625) q^86 + (-942b7 + 942b5 - 8478b2 - 35658b1) * q^87 + (-866b7 - 1606b5 + 7950b2 - 22858b1) * q^88 + (2504b6 - 2504b4 - 7512b3 + 21158) q^89 + (-96b6 + 5168b4 - 96b3) q^91 + (-1020b7 + 268b5 + 7300b2 + 4884b1) * q^92 + (-144b7 + 4720b5 + 5584b2 - 144b1) * q^93 + (864b6 + 864b4 + 7272b3 + 24696) q^94 + (368b6 - 5136b4 + 1824b3 - 28912) q^96 + (-548b7 + 548b5 - 4932b2 + 14091b1) * q^97 + (-384b7 - 1152b5 - 2073b2 - 14208b1) * q^98 + (-3456b6 - 1821b4 - 3456b3) q^99
$\operatorname{Tr}(f)(q)$	$=$	$8 q - 40 q^{4} - 232 q^{6} + 328 q^{9} + 4768 q^{14} - 6624 q^{16} + 15584 q^{24} + 11216 q^{26} - 25856 q^{31} + 9544 q^{34} - 20328 q^{36} - 70208 q^{39} - 9136 q^{41} + 58224 q^{44} + 58400 q^{46} - 19656 q^{49}+ \cdots - 231296 q^{96}+O(q^{100})$ 8 * q - 40 * q^4 - 232 * q^6 + 328 * q^9 + 4768 * q^14 - 6624 * q^16 + 15584 * q^24 + 11216 * q^26 - 25856 * q^31 + 9544 * q^34 - 20328 * q^36 - 70208 * q^39 - 9136 * q^41 + 58224 * q^44 + 58400 * q^46 - 19656 * q^49 - 46576 * q^54 + 81536 * q^56 + 83840 * q^64 + 86448 * q^66 + 413376 * q^71 - 34928 * q^74 + 199888 * q^76 + 495744 * q^79 + 59368 * q^81 - 393344 * q^84 - 37000 * q^86 + 169264 * q^89 + 197568 * q^94 - 231296 * q^96

Basis of coefficient ring in terms of a root $\nu$ of $x^{8} + 5x^{6} + 116x^{4} + 320x^{2} + 4096$ x^8 + 5*x^6 + 116*x^4 + 320*x^2 + 4096 :

$\beta_{1}$	$=$	$( 5\nu^{7} - 7\nu^{5} + 164\nu^{3} - 832\nu ) / 2304$ (5v^7 - 7v^5 + 164v^3 - 832v) / 2304
$\beta_{2}$	$=$	$( \nu^{7} + 5\nu^{5} + 116\nu^{3} + 320\nu ) / 256$ (v^7 + 5v^5 + 116v^3 + 320*v) / 256
$\beta_{3}$	$=$	$( -5\nu^{6} + 7\nu^{4} - 164\nu^{2} + 976 ) / 144$ (-5v^6 + 7v^4 - 164*v^2 + 976) / 144
$\beta_{4}$	$=$	$( \nu^{6} - 11\nu^{4} + 100\nu^{2} - 608 ) / 48$ (v^6 - 11v^4 + 100v^2 - 608) / 48
$\beta_{5}$	$=$	$( -\nu^{7} - 21\nu^{5} - 68\nu^{3} - 1280\nu ) / 128$ (-v^7 - 21v^5 - 68v^3 - 1280*v) / 128
$\beta_{6}$	$=$	$( \nu^{6} + 13\nu^{4} + 124\nu^{2} + 664 ) / 24$ (v^6 + 13v^4 + 124v^2 + 664) / 24
$\beta_{7}$	$=$	$( -\nu^{7} - 13\nu^{5} - 76\nu^{3} + 368\nu ) / 72$ (-v^7 - 13v^5 - 76v^3 + 368*v) / 72

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

$\nu$	$=$	$( \beta_{7} - \beta_{5} + \beta_{2} + \beta_1 ) / 16$ (b7 - b5 + b2 + b1) / 16
$\nu^{2}$	$=$	$( \beta_{6} + 3\beta_{4} + 3\beta_{3} - 10 ) / 8$ (b6 + 3b4 + 3b3 - 10) / 8
$\nu^{3}$	$=$	$( -\beta_{7} + 9\beta_{5} + 55\beta_{2} - 73\beta_1 ) / 16$ (-b7 + 9b5 + 55b2 - 73*b1) / 16
$\nu^{4}$	$=$	$( 7\beta_{6} - 19\beta_{4} - 3\beta_{3} - 414 ) / 8$ (7b6 - 19b4 - 3*b3 - 414) / 8
$\nu^{5}$	$=$	$( -63\beta_{7} - 41\beta_{5} - 151\beta_{2} - 279\beta_1 ) / 16$ (-63b7 - 41b5 - 151b2 - 279b1) / 16
$\nu^{6}$	$=$	$( -23\beta_{6} - 125\beta_{4} - 333\beta_{3} + 1310 ) / 8$ (-23b6 - 125b4 - 333*b3 + 1310) / 8
$\nu^{7}$	$=$	$( 111\beta_{7} - 519\beta_{5} - 1849\beta_{2} + 9543\beta_1 ) / 16$ (111b7 - 519b5 - 1849b2 + 9543b1) / 16

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/200\mathbb{Z}\right)^\times$ .

$n$	$101$	$151$	$177$
$\chi(n)$	$-1$	$1$	$-1$

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}

149.1

2.10784	−	1.88600i
2.10784	+	1.88600i
1.51888	−	2.38600i
1.51888	+	2.38600i
−1.51888	−	2.38600i
−1.51888	+	2.38600i
−2.10784	−	1.88600i
−2.10784	+	1.88600i

−4.21569

−

3.77200i

−3.25452

3.54400

31.8031i

13.7200

12.2760i

112.704i

105.021

−

147.440i

−232.408

149.2

−4.21569

3.77200i

−3.25452

3.54400

−

31.8031i

13.7200

−

12.2760i

−

112.704i

105.021

147.440i

−232.408

149.3

−3.03776

−

4.77200i

23.6095

−13.5440

28.9924i

−71.7200

−

112.665i

160.704i

179.495

−

23.4400i

314.408

149.4

−3.03776

4.77200i

23.6095

−13.5440

−

28.9924i

−71.7200

112.665i

−

160.704i

179.495

23.4400i

314.408

149.5

3.03776

−

4.77200i

−23.6095

−13.5440

−

28.9924i

−71.7200

112.665i

160.704i

−179.495

−

23.4400i

314.408

149.6

3.03776

4.77200i

−23.6095

−13.5440

28.9924i

−71.7200

−

112.665i

−

160.704i

−179.495

23.4400i

314.408

149.7

4.21569

−

3.77200i

3.25452

3.54400

−

31.8031i

13.7200

−

12.2760i

112.704i

−105.021

−

147.440i

−232.408

149.8

4.21569

3.77200i

3.25452

3.54400

31.8031i

13.7200

12.2760i

−

112.704i

−105.021

147.440i

−232.408

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
8.b	even	2	1	inner
40.f	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	200.6.f.a		8
4.b	odd	2	1		800.6.f.a		8
5.b	even	2	1	inner	200.6.f.a		8
5.c	odd	4	1		8.6.b.a	✓	4
5.c	odd	4	1		200.6.d.a		4
8.b	even	2	1	inner	200.6.f.a		8
8.d	odd	2	1		800.6.f.a		8
15.e	even	4	1		72.6.d.b		4
20.d	odd	2	1		800.6.f.a		8
20.e	even	4	1		32.6.b.a		4
20.e	even	4	1		800.6.d.a		4
40.e	odd	2	1		800.6.f.a		8
40.f	even	2	1	inner	200.6.f.a		8
40.i	odd	4	1		8.6.b.a	✓	4
40.i	odd	4	1		200.6.d.a		4
40.k	even	4	1		32.6.b.a		4
40.k	even	4	1		800.6.d.a		4
60.l	odd	4	1		288.6.d.b		4
80.i	odd	4	1		256.6.a.k		4
80.j	even	4	1		256.6.a.n		4
80.s	even	4	1		256.6.a.n		4
80.t	odd	4	1		256.6.a.k		4
120.q	odd	4	1		288.6.d.b		4
120.w	even	4	1		72.6.d.b		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
8.6.b.a	✓	4	5.c	odd	4	1
8.6.b.a	✓	4	40.i	odd	4	1
32.6.b.a		4	20.e	even	4	1
32.6.b.a		4	40.k	even	4	1
72.6.d.b		4	15.e	even	4	1
72.6.d.b		4	120.w	even	4	1
200.6.d.a		4	5.c	odd	4	1
200.6.d.a		4	40.i	odd	4	1
200.6.f.a		8	1.a	even	1	1	trivial
200.6.f.a		8	5.b	even	2	1	inner
200.6.f.a		8	8.b	even	2	1	inner
200.6.f.a		8	40.f	even	2	1	inner
256.6.a.k		4	80.i	odd	4	1
256.6.a.k		4	80.t	odd	4	1
256.6.a.n		4	80.j	even	4	1
256.6.a.n		4	80.s	even	4	1
288.6.d.b		4	60.l	odd	4	1
288.6.d.b		4	120.q	odd	4	1
800.6.d.a		4	20.e	even	4	1
800.6.d.a		4	40.k	even	4	1
800.6.f.a		8	4.b	odd	2	1
800.6.f.a		8	8.d	odd	2	1
800.6.f.a		8	20.d	odd	2	1
800.6.f.a		8	40.e	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{4} - 568T_{3}^{2} + 5904$ T3^4 - 568*T3^2 + 5904 acting on $S_{6}^{\mathrm{new}}(200, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{8} + 20 T^{6} + \cdots + 1048576$ T^8 + 20T^6 + 1856T^4 + 20480*T^2 + 1048576
$3$	$(T^{4} - 568 T^{2} + 5904)^{2}$ (T^4 - 568*T^2 + 5904)^2
$5$	$T^{8}$ T^8
$7$	$(T^{4} + 38528 T^{2} + 328044544)^{2}$ (T^4 + 38528*T^2 + 328044544)^2
$11$	$(T^{4} + 347768 T^{2} + 5520765456)^{2}$ (T^4 + 347768*T^2 + 5520765456)^2
$13$	$(T^{4} - 590944 T^{2} + 7999305984)^{2}$ (T^4 - 590944*T^2 + 7999305984)^2
$17$	$(T^{4} + 154504 T^{2} + 5220351504)^{2}$ (T^4 + 154504*T^2 + 5220351504)^2
$19$	$(T^{4} + 3109816 T^{2} + 120994976016)^{2}$ (T^4 + 3109816*T^2 + 120994976016)^2
$23$	$(T^{4} + \cdots + 7936705990656)^{2}$ (T^4 + 6998656*T^2 + 7936705990656)^2
$29$	$(T^{4} + \cdots + 535633608132864)^{2}$ (T^4 + 50789216*T^2 + 535633608132864)^2
$31$	$(T^{2} + 6464 T + 7754752)^{4}$ (T^2 + 6464*T + 7754752)^4
$37$	$(T^{4} + \cdots + 306881230162176)^{2}$ (T^4 - 36113248*T^2 + 306881230162176)^2
$41$	$(T^{2} + 2284 T - 85109148)^{4}$ (T^2 + 2284*T - 85109148)^4
$43$	$(T^{4} + \cdots + 23\!\cdots\!56)^{2}$ (T^4 - 121686904*T^2 + 2359818970365456)^2
$47$	$(T^{4} + \cdots + 17\!\cdots\!64)^{2}$ (T^4 + 483273216*T^2 + 17595542840868864)^2
$53$	$(T^{4} + \cdots + 76\!\cdots\!44)^{2}$ (T^4 - 633629792*T^2 + 76254379567167744)^2
$59$	$(T^{4} + \cdots + 22\!\cdots\!36)^{2}$ (T^4 + 1322273016*T^2 + 223644487439595536)^2
$61$	$(T^{4} + \cdots + 41\!\cdots\!16)^{2}$ (T^4 + 4119483744*T^2 + 4156588216540866816)^2
$67$	$(T^{4} + \cdots + 32\!\cdots\!84)^{2}$ (T^4 - 4033664568*T^2 + 3228993094632474384)^2
$71$	$(T^{2} - 103344 T + 2609278272)^{4}$ (T^2 - 103344*T + 2609278272)^4
$73$	$(T^{4} + \cdots + 25\!\cdots\!56)^{2}$ (T^4 + 3608600776*T^2 + 2574549625669379856)^2
$79$	$(T^{2} - 123936 T + 3701816576)^{4}$ (T^2 - 123936*T + 3701816576)^4
$83$	$(T^{4} + \cdots + 72\!\cdots\!56)^{2}$ (T^4 - 5708307384*T^2 + 7255334391065309456)^2
$89$	$(T^{2} - 42316 T - 6875717724)^{4}$ (T^2 - 42316*T - 6875717724)^4
$97$	$(T^{4} + \cdots + 61\!\cdots\!04)^{2}$ (T^4 + 4045463048*T^2 + 613573070945138704)^2
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 149.8
Significant digits:
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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	200.6.f.a

Level	$200$
Weight	$6$
Character orbit	200.f
Analytic conductor	$32.077$
Analytic rank	$0$
Dimension	$8$
Inner twists	$4$