LMFDB - Newform orbit 63.6.c.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [63,6,Mod(62,63)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(63, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("63.62");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 63 = 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	63.c (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:	$10.1041806482$
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} - 4x^{7} + 456x^{6} - 612x^{5} + 66744x^{4} + 32004x^{3} + 3729464x^{2} + 2503804x + 66026577 $ x^8 - 4x^7 + 456x^6 - 612x^5 + 66744x^4 + 32004x^3 + 3729464x^2 + 2503804*x + 66026577
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{6} $
Twist minimal:	yes
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_{4} - \beta_{2}) q^{2} - 2 \beta_1 q^{4} + \beta_{6} q^{5} + (\beta_{5} + 2 \beta_1 - 14) q^{7} + (4 \beta_{4} + 4 \beta_{2}) q^{8}+O(q^{10}) $ q + (b4 - b2) * q^2 - 2b1 q^4 + b6 * q^5 + (b5 + 2b1 - 14) q^7 + (4b4 + 4b2) * q^8	\( q + (\beta_{4} - \beta_{2}) q^{2} - 2 \beta_1 q^{4} + \beta_{6} q^{5} + (\beta_{5} + 2 \beta_1 - 14) q^{7} + (4 \beta_{4} + 4 \beta_{2}) q^{8} + (2 \beta_{5} + \beta_{3} - \beta_1) q^{10} + (4 \beta_{4} + 53 \beta_{2}) q^{11} + (4 \beta_{5} + \beta_{3} - 2 \beta_1) q^{13} + ( - \beta_{7} - 8 \beta_{6} + 21 \beta_{4} - 31 \beta_{2}) q^{14} + ( - 64 \beta_1 - 16) q^{16} + ( - 2 \beta_{7} + 13 \beta_{6}) q^{17} + (6 \beta_{5} - 6 \beta_{3} - 3 \beta_1) q^{19} + ( - 4 \beta_{7} - 4 \beta_{6}) q^{20} + (49 \beta_1 + 670) q^{22} + (158 \beta_{4} + 845 \beta_{2}) q^{23} + (20 \beta_1 + 1171) q^{25} + ( - 6 \beta_{7} - 52 \beta_{6}) q^{26} + ( - 14 \beta_{3} + 28 \beta_1 - 1260) q^{28} + (143 \beta_{4} - 850 \beta_{2}) q^{29} + (4 \beta_{5} + 24 \beta_{3} - 2 \beta_1) q^{31} + ( - 784 \beta_{4} + 1296 \beta_{2}) q^{32} + ( - 6 \beta_{5} + \beta_{3} + 3 \beta_1) q^{34} + (5 \beta_{7} - 9 \beta_{6} + 140 \beta_{4} - 2148 \beta_{2}) q^{35} + ( - 526 \beta_1 - 4288) q^{37} + (6 \beta_{7} + 72 \beta_{6}) q^{38} + ( - 8 \beta_{5} + 4 \beta_{3} + 4 \beta_1) q^{40} + (26 \beta_{7} + 111 \beta_{6}) q^{41} + (613 \beta_1 + 4052) q^{43} + (1484 \beta_{4} + 144 \beta_{2}) q^{44} + (687 \beta_1 + 8986) q^{46} + (38 \beta_{7} + 82 \beta_{6}) q^{47} + ( - 28 \beta_{5} + 35 \beta_{3} - 126 \beta_1 - 13265) q^{49} + (1451 \beta_{4} - 1531 \beta_{2}) q^{50} + ( - 72 \beta_{5} - 56 \beta_{3} + 36 \beta_1) q^{52} + ( - 6091 \beta_{4} + 326 \beta_{2}) q^{53} + ( - 106 \beta_{5} + 4 \beta_{3} + 53 \beta_1) q^{55} + ( - 4 \beta_{7} + 24 \beta_{6} - 196 \beta_{4} - 236 \beta_{2}) q^{56} + ( - 993 \beta_1 - 14474) q^{58} + (28 \beta_{7} - 328 \beta_{6}) q^{59} + ( - 172 \beta_{5} - 2 \beta_{3} + 86 \beta_1) q^{61} + ( - 52 \beta_{7} - 512 \beta_{6}) q^{62} + (32 \beta_1 + 31744) q^{64} + (4856 \beta_{4} - 8952 \beta_{2}) q^{65} + (2548 \beta_1 + 19076) q^{67} + ( - 60 \beta_{7} + 444 \beta_{6}) q^{68} + (62 \beta_{5} + 21 \beta_{3} - 2319 \beta_1 - 32592) q^{70} + (7634 \beta_{4} + 11765 \beta_{2}) q^{71} + (36 \beta_{5} - 151 \beta_{3} - 18 \beta_1) q^{73} + ( - 11652 \beta_{4} + 13756 \beta_{2}) q^{74} + (432 \beta_{5} - 84 \beta_{3} - 216 \beta_1) q^{76} + ( - 4 \beta_{7} + 367 \beta_{6} - 1911 \beta_{4} - 922 \beta_{2}) q^{77} + (1708 \beta_1 - 11824) q^{79} + ( - 128 \beta_{7} - 144 \beta_{6}) q^{80} + (638 \beta_{5} + 267 \beta_{3} - 319 \beta_1) q^{82} + (22 \beta_{7} + 798 \beta_{6}) q^{83} + ( - 3996 \beta_1 + 59400) q^{85} + (12634 \beta_{4} - 15086 \beta_{2}) q^{86} + (228 \beta_1 - 3256) q^{88} + ( - 54 \beta_{7} - 1155 \beta_{6}) q^{89} + (34 \beta_{5} + 56 \beta_{3} - 2445 \beta_1 - 62664) q^{91} + (23660 \beta_{4} + 5688 \beta_{2}) q^{92} + (772 \beta_{5} + 310 \beta_{3} - 386 \beta_1) q^{94} + ( - 24936 \beta_{4} - 10728 \beta_{2}) q^{95} + ( - 484 \beta_{5} + 163 \beta_{3} + 242 \beta_1) q^{97} + ( - 42 \beta_{7} - 476 \beta_{6} - 15225 \beta_{4} + \cdots + 15785 \beta_{2}) q^{98}+O(q^{100}) \) q + (b4 - b2) * q^2 - 2b1 q^4 + b6 * q^5 + (b5 + 2b1 - 14) q^7 + (4b4 + 4b2) * q^8 + (2b5 + b3 - b1) q^10 + (4b4 + 53b2) * q^11 + (4b5 + b3 - 2b1) * q^13 + (-b7 - 8b6 + 21b4 - 31b2) q^14 + (-64b1 - 16) q^16 + (-2b7 + 13b6) * q^17 + (6b5 - 6b3 - 3b1) q^19 + (-4b7 - 4b6) * q^20 + (49b1 + 670) q^22 + (158b4 + 845b2) * q^23 + (20b1 + 1171) q^25 + (-6b7 - 52b6) * q^26 + (-14b3 + 28b1 - 1260) * q^28 + (143b4 - 850b2) * q^29 + (4b5 + 24b3 - 2b1) q^31 + (-784b4 + 1296b2) * q^32 + (-6b5 + b3 + 3b1) * q^34 + (5b7 - 9b6 + 140b4 - 2148b2) * q^35 + (-526b1 - 4288) q^37 + (6b7 + 72b6) * q^38 + (-8b5 + 4b3 + 4b1) q^40 + (26b7 + 111b6) * q^41 + (613b1 + 4052) q^43 + (1484b4 + 144b2) * q^44 + (687b1 + 8986) q^46 + (38b7 + 82b6) * q^47 + (-28b5 + 35b3 - 126b1 - 13265) q^49 + (1451b4 - 1531b2) * q^50 + (-72b5 - 56b3 + 36b1) q^52 + (-6091b4 + 326b2) * q^53 + (-106b5 + 4b3 + 53b1) q^55 + (-4b7 + 24b6 - 196b4 - 236b2) * q^56 + (-993b1 - 14474) q^58 + (28b7 - 328b6) * q^59 + (-172b5 - 2b3 + 86b1) q^61 + (-52b7 - 512b6) * q^62 + (32b1 + 31744) q^64 + (4856b4 - 8952b2) * q^65 + (2548b1 + 19076) q^67 + (-60b7 + 444b6) * q^68 + (62b5 + 21b3 - 2319b1 - 32592) q^70 + (7634b4 + 11765b2) * q^71 + (36b5 - 151b3 - 18b1) q^73 + (-11652b4 + 13756b2) * q^74 + (432b5 - 84b3 - 216b1) q^76 + (-4b7 + 367b6 - 1911b4 - 922b2) * q^77 + (1708b1 - 11824) q^79 + (-128b7 - 144b6) * q^80 + (638b5 + 267b3 - 319b1) q^82 + (22b7 + 798b6) * q^83 + (-3996b1 + 59400) q^85 + (12634b4 - 15086b2) * q^86 + (228b1 - 3256) q^88 + (-54b7 - 1155b6) * q^89 + (34b5 + 56b3 - 2445b1 - 62664) q^91 + (23660b4 + 5688b2) * q^92 + (772b5 + 310b3 - 386b1) q^94 + (-24936b4 - 10728b2) * q^95 + (-484b5 + 163b3 + 242b1) q^97 + (-42b7 - 476b6 - 15225b4 + 15785b2) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 112 q^{7}+O(q^{10}) $ 8 * q - 112 * q^7	$ 8 q - 112 q^{7} - 128 q^{16} + 5360 q^{22} + 9368 q^{25} - 10080 q^{28} - 34304 q^{37} + 32416 q^{43} + 71888 q^{46} - 106120 q^{49} - 115792 q^{58} + 253952 q^{64} + 152608 q^{67} - 260736 q^{70} - 94592 q^{79} + 475200 q^{85} - 26048 q^{88} - 501312 q^{91}+O(q^{100}) $ 8 * q - 112 * q^7 - 128 * q^16 + 5360 * q^22 + 9368 * q^25 - 10080 * q^28 - 34304 * q^37 + 32416 * q^43 + 71888 * q^46 - 106120 * q^49 - 115792 * q^58 + 253952 * q^64 + 152608 * q^67 - 260736 * q^70 - 94592 * q^79 + 475200 * q^85 - 26048 * q^88 - 501312 * q^91

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} - 4x^{7} + 456x^{6} - 612x^{5} + 66744x^{4} + 32004x^{3} + 3729464x^{2} + 2503804x + 66026577 $ x^8 - 4*x^7 + 456*x^6 - 612*x^5 + 66744*x^4 + 32004*x^3 + 3729464*x^2 + 2503804*x + 66026577 :

$\beta_{1}$	$=$	$ ( - 7276 \nu^{7} - 6016 \nu^{6} - 2355636 \nu^{5} - 9413506 \nu^{4} - 185550156 \nu^{3} - 611944144 \nu^{2} - 3939559436 \nu + 62846012472 ) / 4466767135 $ (-7276v^7 - 6016v^6 - 2355636v^5 - 9413506v^4 - 185550156v^3 - 611944144v^2 - 3939559436*v + 62846012472) / 4466767135
$\beta_{2}$	$=$	$ ( 106 \nu^{7} - 597 \nu^{6} + 49014 \nu^{5} - 159684 \nu^{4} + 6374124 \nu^{3} - 8686185 \nu^{2} + 195744008 \nu - 314036202 ) / 51360255 $ (106v^7 - 597v^6 + 49014v^5 - 159684v^4 + 6374124v^3 - 8686185v^2 + 195744008*v - 314036202) / 51360255
$\beta_{3}$	$=$	$ ( 136314924 \nu^{7} + 13885773036 \nu^{6} + 99207113184 \nu^{5} + 4638877777212 \nu^{4} + 40666053647532 \nu^{3} + \cdots + 87\!\cdots\!64 ) / 46771518670585 $ (136314924v^7 + 13885773036v^6 + 99207113184v^5 + 4638877777212v^4 + 40666053647532v^3 + 388661146589760v^2 + 3722129576222472*v + 8741415421523664) / 46771518670585
$\beta_{4}$	$=$	$ ( - 4026892 \nu^{7} + 41326635 \nu^{6} - 1675985952 \nu^{5} + 10635929286 \nu^{4} - 189591871494 \nu^{3} + 683245379841 \nu^{2} + \cdots + 17714021093796 ) / 1287289504695 $ (-4026892v^7 + 41326635v^6 - 1675985952v^5 + 10635929286v^4 - 189591871494v^3 + 683245379841v^2 - 5417930459474*v + 17714021093796) / 1287289504695
$\beta_{5}$	$=$	$ ( 603178436 \nu^{7} - 2349790824 \nu^{6} + 303972103626 \nu^{5} - 442753391709 \nu^{4} + 52921024233816 \nu^{3} + \cdots + 26\!\cdots\!78 ) / 140314556011755 $ (603178436v^7 - 2349790824v^6 + 303972103626v^5 - 442753391709v^4 + 52921024233816v^3 + 16548578396334v^2 + 4540318229573026*v + 2628436460918178) / 140314556011755
$\beta_{6}$	$=$	$ ( - 174176 \nu^{7} - 1612878 \nu^{6} - 68934144 \nu^{5} - 1127164206 \nu^{4} - 7806037224 \nu^{3} - 184380040050 \nu^{2} + \cdots - 5955704489118 ) / 13400301405 $ (-174176v^7 - 1612878v^6 - 68934144v^5 - 1127164206v^4 - 7806037224v^3 - 184380040050v^2 - 378525797788*v - 5955704489118) / 13400301405
$\beta_{7}$	$=$	$ ( - 3252328 \nu^{7} - 23469492 \nu^{6} - 1033111860 \nu^{5} - 12230717712 \nu^{4} - 98965955544 \nu^{3} - 1566104155968 \nu^{2} + \cdots - 45975274320660 ) / 13400301405 $ (-3252328v^7 - 23469492v^6 - 1033111860v^5 - 12230717712v^4 - 98965955544v^3 - 1566104155968v^2 - 3648413057660*v - 45975274320660) / 13400301405

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

$\nu$	$=$	$ ( 6\beta_{5} + 12\beta_{4} - \beta_{3} - 9\beta _1 + 36 ) / 72 $ (6b5 + 12b4 - b3 - 9*b1 + 36) / 72
$\nu^{2}$	$=$	$ ( -\beta_{7} + 2\beta_{6} + 6\beta_{5} + 6\beta_{4} - 4\beta_{3} + 18\beta_{2} + 153\beta _1 - 4032 ) / 36 $ (-b7 + 2b6 + 6b5 + 6b4 - 4b3 + 18b2 + 153b1 - 4032) / 36
$\nu^{3}$	$=$	$ ( -3\beta_{7} + 33\beta_{6} - 402\beta_{5} - 2010\beta_{4} + 142\beta_{3} - 1404\beta_{2} + 1458\beta _1 - 16074 ) / 36 $ (-3b7 + 33b6 - 402b5 - 2010b4 + 142b3 - 1404b2 + 1458*b1 - 16074) / 36
$\nu^{4}$	$=$	$ ( 133 \beta_{7} - 716 \beta_{6} - 1302 \beta_{5} - 5352 \beta_{4} + 634 \beta_{3} - 7524 \beta_{2} - 15660 \beta _1 + 291960 ) / 18 $ (133b7 - 716b6 - 1302b5 - 5352b4 + 634b3 - 7524b2 - 15660*b1 + 291960) / 18
$\nu^{5}$	$=$	$ ( 4320 \beta_{7} - 33480 \beta_{6} + 121326 \beta_{5} + 959808 \beta_{4} - 50539 \beta_{3} + 969300 \beta_{2} - 761877 \beta _1 + 10552716 ) / 72 $ (4320b7 - 33480b6 + 121326b5 + 959808b4 - 50539b3 + 969300b2 - 761877*b1 + 10552716) / 72
$\nu^{6}$	$=$	$ ( - 59335 \beta_{7} + 387032 \beta_{6} + 727620 \beta_{5} + 5222850 \beta_{4} - 328975 \beta_{3} + 6235722 \beta_{2} + 4954770 \beta _1 - 87159168 ) / 36 $ (-59335b7 + 387032b6 + 727620b5 + 5222850b4 - 328975b3 + 6235722b2 + 4954770*b1 - 87159168) / 36
$\nu^{7}$	$=$	$ ( - 833784 \beta_{7} + 5942559 \beta_{6} - 8749824 \beta_{5} - 98335740 \beta_{4} + 3798511 \beta_{3} - 108304182 \beta_{2} + 90041283 \beta _1 - 1341413874 ) / 36 $ (-833784b7 + 5942559b6 - 8749824b5 - 98335740b4 + 3798511b3 - 108304182b2 + 90041283*b1 - 1341413874) / 36

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/63\mathbb{Z}\right)^\times$.

$n$	$10$	$29$
$\chi(n)$	$-1$	$-1$

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} $

62.1

−0.822876	+	5.87982i
−0.822876	−	7.29403i
1.82288	−	14.2575i
1.82288	+	12.8433i
1.82288	+	14.2575i
1.82288	−	12.8433i
−0.822876	−	5.87982i
−0.822876	+	7.29403i

−

7.98430i

−31.7490

−67.9227

25.6863

127.072i

−

2.00393i

542.315i

62.2

−

7.98430i

−31.7490

67.9227

25.6863

−

127.072i

−

2.00393i

−

542.315i

62.3

−

0.500983i

31.7490

−63.0754

−53.6863

−

118.003i

−

31.9372i

31.5997i

62.4

−

0.500983i

31.7490

63.0754

−53.6863

118.003i

−

31.9372i

−

31.5997i

62.5

0.500983i

31.7490

−63.0754

−53.6863

118.003i

31.9372i

−

31.5997i

62.6

0.500983i

31.7490

63.0754

−53.6863

−

118.003i

31.9372i

31.5997i

62.7

7.98430i

−31.7490

−67.9227

25.6863

−

127.072i

2.00393i

−

542.315i

62.8

7.98430i

−31.7490

67.9227

25.6863

127.072i

2.00393i

542.315i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
7.b	odd	2	1	inner
21.c	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	63.6.c.b	✓	8
3.b	odd	2	1	inner	63.6.c.b	✓	8
4.b	odd	2	1		1008.6.k.b		8
7.b	odd	2	1	inner	63.6.c.b	✓	8
12.b	even	2	1		1008.6.k.b		8
21.c	even	2	1	inner	63.6.c.b	✓	8
28.d	even	2	1		1008.6.k.b		8
84.h	odd	2	1		1008.6.k.b		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
63.6.c.b	✓	8	1.a	even	1	1	trivial
63.6.c.b	✓	8	3.b	odd	2	1	inner
63.6.c.b	✓	8	7.b	odd	2	1	inner
63.6.c.b	✓	8	21.c	even	2	1	inner
1008.6.k.b		8	4.b	odd	2	1
1008.6.k.b		8	12.b	even	2	1
1008.6.k.b		8	28.d	even	2	1
1008.6.k.b		8	84.h	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{4} + 64T_{2}^{2} + 16 $ T2^4 + 64*T2^2 + 16 acting on $S_{6}^{\mathrm{new}}(63, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{4} + 64 T^{2} + 16)^{2} $ (T^4 + 64*T^2 + 16)^2
$3$	$ T^{8} $ T^8
$5$	$ (T^{4} - 8592 T^{2} + 18354816)^{2} $ (T^4 - 8592*T^2 + 18354816)^2
$7$	$ (T^{4} + 56 T^{3} + 28098 T^{2} + \cdots + 282475249)^{2} $ (T^4 + 56T^3 + 28098T^2 + 941192*T + 282475249)^2
$11$	$ (T^{4} + 79228 T^{2} + \cdots + 1523965444)^{2} $ (T^4 + 79228*T^2 + 1523965444)^2
$13$	$ (T^{4} + 676128 T^{2} + \cdots + 26504354304)^{2} $ (T^4 + 676128*T^2 + 26504354304)^2
$17$	$ (T^{4} - 3795984 T^{2} + \cdots + 13380660864)^{2} $ (T^4 - 3795984*T^2 + 13380660864)^2
$19$	$ (T^{4} + 6287328 T^{2} + \cdots + 5001393682944)^{2} $ (T^4 + 6287328*T^2 + 5001393682944)^2
$23$	$ (T^{4} + 20891404 T^{2} + \cdots + 91145170812004)^{2} $ (T^4 + 20891404*T^2 + 91145170812004)^2
$29$	$ (T^{4} + 20966164 T^{2} + \cdots + 95002410498724)^{2} $ (T^4 + 20966164*T^2 + 95002410498724)^2
$31$	$ (T^{4} + 90530688 T^{2} + \cdots + 19\!\cdots\!04)^{2} $ (T^4 + 90530688*T^2 + 1951802383048704)^2
$37$	$ (T^{2} + 8576 T - 51335408)^{4} $ (T^2 + 8576*T - 51335408)^4
$41$	$ (T^{4} - 450270096 T^{2} + \cdots + 22\!\cdots\!04)^{2} $ (T^4 - 450270096*T^2 + 22953464070309504)^2
$43$	$ (T^{2} - 8104 T - 78275084)^{4} $ (T^2 - 8104*T - 78275084)^4
$47$	$ (T^{4} - 815119296 T^{2} + \cdots + 14\!\cdots\!36)^{2} $ (T^4 - 815119296*T^2 + 145506138984769536)^2
$53$	$ (T^{4} + 1338585844 T^{2} + \cdots + 44\!\cdots\!36)^{2} $ (T^4 + 1338585844*T^2 + 443978603020033636)^2
$59$	$ (T^{4} - 1412814336 T^{2} + \cdots + 10\!\cdots\!76)^{2} $ (T^4 - 1412814336*T^2 + 109800240831922176)^2
$61$	$ (T^{4} + 893736192 T^{2} + \cdots + 19\!\cdots\!44)^{2} $ (T^4 + 893736192*T^2 + 196515007572639744)^2
$67$	$ (T^{2} - 38152 T - 1272166832)^{4} $ (T^2 - 38152*T - 1272166832)^4
$71$	$ (T^{4} + 5973632716 T^{2} + \cdots + 78\!\cdots\!64)^{2} $ (T^4 + 5973632716*T^2 + 789983112998043364)^2
$73$	$ (T^{4} + 3510489888 T^{2} + \cdots + 30\!\cdots\!84)^{2} $ (T^4 + 3510489888*T^2 + 3023776020698601984)^2
$79$	$ (T^{2} + 23648 T - 595343552)^{4} $ (T^2 + 23648*T - 595343552)^4
$83$	$ (T^{4} - 5607969216 T^{2} + \cdots + 59\!\cdots\!96)^{2} $ (T^4 - 5607969216*T^2 + 5998768611579242496)^2
$89$	$ (T^{4} - 12592946448 T^{2} + \cdots + 19\!\cdots\!84)^{2} $ (T^4 - 12592946448*T^2 + 19415684921199801984)^2
$97$	$ (T^{4} + 10358370336 T^{2} + \cdots + 21\!\cdots\!56)^{2} $ (T^4 + 10358370336*T^2 + 2142510971815328256)^2
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 63 = 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	63.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + (\beta_{4} - \beta_{2}) q^{2} - 2 \beta_1 q^{4} + \beta_{6} q^{5} + (\beta_{5} + 2 \beta_1 - 14) q^{7} + (4 \beta_{4} + 4 \beta_{2}) q^{8}+O(q^{10}) \) q + (b4 - b2) * q^2 - 2b1 q^4 + b6 * q^5 + (b5 + 2b1 - 14) q^7 + (4b4 + 4b2) * q^8	\( q + (\beta_{4} - \beta_{2}) q^{2} - 2 \beta_1 q^{4} + \beta_{6} q^{5} + (\beta_{5} + 2 \beta_1 - 14) q^{7} + (4 \beta_{4} + 4 \beta_{2}) q^{8} + (2 \beta_{5} + \beta_{3} - \beta_1) q^{10} + (4 \beta_{4} + 53 \beta_{2}) q^{11} + (4 \beta_{5} + \beta_{3} - 2 \beta_1) q^{13} + ( - \beta_{7} - 8 \beta_{6} + 21 \beta_{4} - 31 \beta_{2}) q^{14} + ( - 64 \beta_1 - 16) q^{16} + ( - 2 \beta_{7} + 13 \beta_{6}) q^{17} + (6 \beta_{5} - 6 \beta_{3} - 3 \beta_1) q^{19} + ( - 4 \beta_{7} - 4 \beta_{6}) q^{20} + (49 \beta_1 + 670) q^{22} + (158 \beta_{4} + 845 \beta_{2}) q^{23} + (20 \beta_1 + 1171) q^{25} + ( - 6 \beta_{7} - 52 \beta_{6}) q^{26} + ( - 14 \beta_{3} + 28 \beta_1 - 1260) q^{28} + (143 \beta_{4} - 850 \beta_{2}) q^{29} + (4 \beta_{5} + 24 \beta_{3} - 2 \beta_1) q^{31} + ( - 784 \beta_{4} + 1296 \beta_{2}) q^{32} + ( - 6 \beta_{5} + \beta_{3} + 3 \beta_1) q^{34} + (5 \beta_{7} - 9 \beta_{6} + 140 \beta_{4} - 2148 \beta_{2}) q^{35} + ( - 526 \beta_1 - 4288) q^{37} + (6 \beta_{7} + 72 \beta_{6}) q^{38} + ( - 8 \beta_{5} + 4 \beta_{3} + 4 \beta_1) q^{40} + (26 \beta_{7} + 111 \beta_{6}) q^{41} + (613 \beta_1 + 4052) q^{43} + (1484 \beta_{4} + 144 \beta_{2}) q^{44} + (687 \beta_1 + 8986) q^{46} + (38 \beta_{7} + 82 \beta_{6}) q^{47} + ( - 28 \beta_{5} + 35 \beta_{3} - 126 \beta_1 - 13265) q^{49} + (1451 \beta_{4} - 1531 \beta_{2}) q^{50} + ( - 72 \beta_{5} - 56 \beta_{3} + 36 \beta_1) q^{52} + ( - 6091 \beta_{4} + 326 \beta_{2}) q^{53} + ( - 106 \beta_{5} + 4 \beta_{3} + 53 \beta_1) q^{55} + ( - 4 \beta_{7} + 24 \beta_{6} - 196 \beta_{4} - 236 \beta_{2}) q^{56} + ( - 993 \beta_1 - 14474) q^{58} + (28 \beta_{7} - 328 \beta_{6}) q^{59} + ( - 172 \beta_{5} - 2 \beta_{3} + 86 \beta_1) q^{61} + ( - 52 \beta_{7} - 512 \beta_{6}) q^{62} + (32 \beta_1 + 31744) q^{64} + (4856 \beta_{4} - 8952 \beta_{2}) q^{65} + (2548 \beta_1 + 19076) q^{67} + ( - 60 \beta_{7} + 444 \beta_{6}) q^{68} + (62 \beta_{5} + 21 \beta_{3} - 2319 \beta_1 - 32592) q^{70} + (7634 \beta_{4} + 11765 \beta_{2}) q^{71} + (36 \beta_{5} - 151 \beta_{3} - 18 \beta_1) q^{73} + ( - 11652 \beta_{4} + 13756 \beta_{2}) q^{74} + (432 \beta_{5} - 84 \beta_{3} - 216 \beta_1) q^{76} + ( - 4 \beta_{7} + 367 \beta_{6} - 1911 \beta_{4} - 922 \beta_{2}) q^{77} + (1708 \beta_1 - 11824) q^{79} + ( - 128 \beta_{7} - 144 \beta_{6}) q^{80} + (638 \beta_{5} + 267 \beta_{3} - 319 \beta_1) q^{82} + (22 \beta_{7} + 798 \beta_{6}) q^{83} + ( - 3996 \beta_1 + 59400) q^{85} + (12634 \beta_{4} - 15086 \beta_{2}) q^{86} + (228 \beta_1 - 3256) q^{88} + ( - 54 \beta_{7} - 1155 \beta_{6}) q^{89} + (34 \beta_{5} + 56 \beta_{3} - 2445 \beta_1 - 62664) q^{91} + (23660 \beta_{4} + 5688 \beta_{2}) q^{92} + (772 \beta_{5} + 310 \beta_{3} - 386 \beta_1) q^{94} + ( - 24936 \beta_{4} - 10728 \beta_{2}) q^{95} + ( - 484 \beta_{5} + 163 \beta_{3} + 242 \beta_1) q^{97} + ( - 42 \beta_{7} - 476 \beta_{6} - 15225 \beta_{4} + \cdots + 15785 \beta_{2}) q^{98}+O(q^{100}) \) q + (b4 - b2) * q^2 - 2b1 q^4 + b6 * q^5 + (b5 + 2b1 - 14) q^7 + (4b4 + 4b2) * q^8 + (2b5 + b3 - b1) q^10 + (4b4 + 53b2) * q^11 + (4b5 + b3 - 2b1) * q^13 + (-b7 - 8b6 + 21b4 - 31b2) q^14 + (-64b1 - 16) q^16 + (-2b7 + 13b6) * q^17 + (6b5 - 6b3 - 3b1) q^19 + (-4b7 - 4b6) * q^20 + (49b1 + 670) q^22 + (158b4 + 845b2) * q^23 + (20b1 + 1171) q^25 + (-6b7 - 52b6) * q^26 + (-14b3 + 28b1 - 1260) * q^28 + (143b4 - 850b2) * q^29 + (4b5 + 24b3 - 2b1) q^31 + (-784b4 + 1296b2) * q^32 + (-6b5 + b3 + 3b1) * q^34 + (5b7 - 9b6 + 140b4 - 2148b2) * q^35 + (-526b1 - 4288) q^37 + (6b7 + 72b6) * q^38 + (-8b5 + 4b3 + 4b1) q^40 + (26b7 + 111b6) * q^41 + (613b1 + 4052) q^43 + (1484b4 + 144b2) * q^44 + (687b1 + 8986) q^46 + (38b7 + 82b6) * q^47 + (-28b5 + 35b3 - 126b1 - 13265) q^49 + (1451b4 - 1531b2) * q^50 + (-72b5 - 56b3 + 36b1) q^52 + (-6091b4 + 326b2) * q^53 + (-106b5 + 4b3 + 53b1) q^55 + (-4b7 + 24b6 - 196b4 - 236b2) * q^56 + (-993b1 - 14474) q^58 + (28b7 - 328b6) * q^59 + (-172b5 - 2b3 + 86b1) q^61 + (-52b7 - 512b6) * q^62 + (32b1 + 31744) q^64 + (4856b4 - 8952b2) * q^65 + (2548b1 + 19076) q^67 + (-60b7 + 444b6) * q^68 + (62b5 + 21b3 - 2319b1 - 32592) q^70 + (7634b4 + 11765b2) * q^71 + (36b5 - 151b3 - 18b1) q^73 + (-11652b4 + 13756b2) * q^74 + (432b5 - 84b3 - 216b1) q^76 + (-4b7 + 367b6 - 1911b4 - 922b2) * q^77 + (1708b1 - 11824) q^79 + (-128b7 - 144b6) * q^80 + (638b5 + 267b3 - 319b1) q^82 + (22b7 + 798b6) * q^83 + (-3996b1 + 59400) q^85 + (12634b4 - 15086b2) * q^86 + (228b1 - 3256) q^88 + (-54b7 - 1155b6) * q^89 + (34b5 + 56b3 - 2445b1 - 62664) q^91 + (23660b4 + 5688b2) * q^92 + (772b5 + 310b3 - 386b1) q^94 + (-24936b4 - 10728b2) * q^95 + (-484b5 + 163b3 + 242b1) q^97 + (-42b7 - 476b6 - 15225b4 + 15785b2) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 112 q^{7}+O(q^{10}) \) 8 * q - 112 * q^7	\( 8 q - 112 q^{7} - 128 q^{16} + 5360 q^{22} + 9368 q^{25} - 10080 q^{28} - 34304 q^{37} + 32416 q^{43} + 71888 q^{46} - 106120 q^{49} - 115792 q^{58} + 253952 q^{64} + 152608 q^{67} - 260736 q^{70} - 94592 q^{79} + 475200 q^{85} - 26048 q^{88} - 501312 q^{91}+O(q^{100}) \) 8 * q - 112 * q^7 - 128 * q^16 + 5360 * q^22 + 9368 * q^25 - 10080 * q^28 - 34304 * q^37 + 32416 * q^43 + 71888 * q^46 - 106120 * q^49 - 115792 * q^58 + 253952 * q^64 + 152608 * q^67 - 260736 * q^70 - 94592 * q^79 + 475200 * q^85 - 26048 * q^88 - 501312 * q^91

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	63.6.c.b

Level	$63$
Weight	$6$
Character orbit	63.c
Analytic conductor	$10.104$
Analytic rank	$0$
Dimension	$8$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(10.1041806482\)
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:	\( x^{8} - 4x^{7} + 456x^{6} - 612x^{5} + 66744x^{4} + 32004x^{3} + 3729464x^{2} + 2503804x + 66026577 \) x^8 - 4x^7 + 456x^6 - 612x^5 + 66744x^4 + 32004x^3 + 3729464x^2 + 2503804*x + 66026577
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{10}\cdot 3^{6} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( - 7276 \nu^{7} - 6016 \nu^{6} - 2355636 \nu^{5} - 9413506 \nu^{4} - 185550156 \nu^{3} - 611944144 \nu^{2} - 3939559436 \nu + 62846012472 ) / 4466767135 \) (-7276v^7 - 6016v^6 - 2355636v^5 - 9413506v^4 - 185550156v^3 - 611944144v^2 - 3939559436*v + 62846012472) / 4466767135
\(\beta_{2}\)	\(=\)	\( ( 106 \nu^{7} - 597 \nu^{6} + 49014 \nu^{5} - 159684 \nu^{4} + 6374124 \nu^{3} - 8686185 \nu^{2} + 195744008 \nu - 314036202 ) / 51360255 \) (106v^7 - 597v^6 + 49014v^5 - 159684v^4 + 6374124v^3 - 8686185v^2 + 195744008*v - 314036202) / 51360255
\(\beta_{3}\)	\(=\)	\( ( 136314924 \nu^{7} + 13885773036 \nu^{6} + 99207113184 \nu^{5} + 4638877777212 \nu^{4} + 40666053647532 \nu^{3} + \cdots + 87\!\cdots\!64 ) / 46771518670585 \) (136314924v^7 + 13885773036v^6 + 99207113184v^5 + 4638877777212v^4 + 40666053647532v^3 + 388661146589760v^2 + 3722129576222472*v + 8741415421523664) / 46771518670585
\(\beta_{4}\)	\(=\)	\( ( - 4026892 \nu^{7} + 41326635 \nu^{6} - 1675985952 \nu^{5} + 10635929286 \nu^{4} - 189591871494 \nu^{3} + 683245379841 \nu^{2} + \cdots + 17714021093796 ) / 1287289504695 \) (-4026892v^7 + 41326635v^6 - 1675985952v^5 + 10635929286v^4 - 189591871494v^3 + 683245379841v^2 - 5417930459474*v + 17714021093796) / 1287289504695
\(\beta_{5}\)	\(=\)	\( ( 603178436 \nu^{7} - 2349790824 \nu^{6} + 303972103626 \nu^{5} - 442753391709 \nu^{4} + 52921024233816 \nu^{3} + \cdots + 26\!\cdots\!78 ) / 140314556011755 \) (603178436v^7 - 2349790824v^6 + 303972103626v^5 - 442753391709v^4 + 52921024233816v^3 + 16548578396334v^2 + 4540318229573026*v + 2628436460918178) / 140314556011755
\(\beta_{6}\)	\(=\)	\( ( - 174176 \nu^{7} - 1612878 \nu^{6} - 68934144 \nu^{5} - 1127164206 \nu^{4} - 7806037224 \nu^{3} - 184380040050 \nu^{2} + \cdots - 5955704489118 ) / 13400301405 \) (-174176v^7 - 1612878v^6 - 68934144v^5 - 1127164206v^4 - 7806037224v^3 - 184380040050v^2 - 378525797788*v - 5955704489118) / 13400301405
\(\beta_{7}\)	\(=\)	\( ( - 3252328 \nu^{7} - 23469492 \nu^{6} - 1033111860 \nu^{5} - 12230717712 \nu^{4} - 98965955544 \nu^{3} - 1566104155968 \nu^{2} + \cdots - 45975274320660 ) / 13400301405 \) (-3252328v^7 - 23469492v^6 - 1033111860v^5 - 12230717712v^4 - 98965955544v^3 - 1566104155968v^2 - 3648413057660*v - 45975274320660) / 13400301405

\(\nu\)	\(=\)	\( ( 6\beta_{5} + 12\beta_{4} - \beta_{3} - 9\beta _1 + 36 ) / 72 \) (6b5 + 12b4 - b3 - 9*b1 + 36) / 72
\(\nu^{2}\)	\(=\)	\( ( -\beta_{7} + 2\beta_{6} + 6\beta_{5} + 6\beta_{4} - 4\beta_{3} + 18\beta_{2} + 153\beta _1 - 4032 ) / 36 \) (-b7 + 2b6 + 6b5 + 6b4 - 4b3 + 18b2 + 153b1 - 4032) / 36
\(\nu^{3}\)	\(=\)	\( ( -3\beta_{7} + 33\beta_{6} - 402\beta_{5} - 2010\beta_{4} + 142\beta_{3} - 1404\beta_{2} + 1458\beta _1 - 16074 ) / 36 \) (-3b7 + 33b6 - 402b5 - 2010b4 + 142b3 - 1404b2 + 1458*b1 - 16074) / 36
\(\nu^{4}\)	\(=\)	\( ( 133 \beta_{7} - 716 \beta_{6} - 1302 \beta_{5} - 5352 \beta_{4} + 634 \beta_{3} - 7524 \beta_{2} - 15660 \beta _1 + 291960 ) / 18 \) (133b7 - 716b6 - 1302b5 - 5352b4 + 634b3 - 7524b2 - 15660*b1 + 291960) / 18
\(\nu^{5}\)	\(=\)	\( ( 4320 \beta_{7} - 33480 \beta_{6} + 121326 \beta_{5} + 959808 \beta_{4} - 50539 \beta_{3} + 969300 \beta_{2} - 761877 \beta _1 + 10552716 ) / 72 \) (4320b7 - 33480b6 + 121326b5 + 959808b4 - 50539b3 + 969300b2 - 761877*b1 + 10552716) / 72
\(\nu^{6}\)	\(=\)	\( ( - 59335 \beta_{7} + 387032 \beta_{6} + 727620 \beta_{5} + 5222850 \beta_{4} - 328975 \beta_{3} + 6235722 \beta_{2} + 4954770 \beta _1 - 87159168 ) / 36 \) (-59335b7 + 387032b6 + 727620b5 + 5222850b4 - 328975b3 + 6235722b2 + 4954770*b1 - 87159168) / 36
\(\nu^{7}\)	\(=\)	\( ( - 833784 \beta_{7} + 5942559 \beta_{6} - 8749824 \beta_{5} - 98335740 \beta_{4} + 3798511 \beta_{3} - 108304182 \beta_{2} + 90041283 \beta _1 - 1341413874 ) / 36 \) (-833784b7 + 5942559b6 - 8749824b5 - 98335740b4 + 3798511b3 - 108304182b2 + 90041283*b1 - 1341413874) / 36

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 62.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{4} + 64 T^{2} + 16)^{2} \) (T^4 + 64*T^2 + 16)^2
$3$	\( T^{8} \) T^8
$5$	\( (T^{4} - 8592 T^{2} + 18354816)^{2} \) (T^4 - 8592*T^2 + 18354816)^2
$7$	\( (T^{4} + 56 T^{3} + 28098 T^{2} + \cdots + 282475249)^{2} \) (T^4 + 56T^3 + 28098T^2 + 941192*T + 282475249)^2
$11$	\( (T^{4} + 79228 T^{2} + \cdots + 1523965444)^{2} \) (T^4 + 79228*T^2 + 1523965444)^2
$13$	\( (T^{4} + 676128 T^{2} + \cdots + 26504354304)^{2} \) (T^4 + 676128*T^2 + 26504354304)^2
$17$	\( (T^{4} - 3795984 T^{2} + \cdots + 13380660864)^{2} \) (T^4 - 3795984*T^2 + 13380660864)^2
$19$	\( (T^{4} + 6287328 T^{2} + \cdots + 5001393682944)^{2} \) (T^4 + 6287328*T^2 + 5001393682944)^2
$23$	\( (T^{4} + 20891404 T^{2} + \cdots + 91145170812004)^{2} \) (T^4 + 20891404*T^2 + 91145170812004)^2
$29$	\( (T^{4} + 20966164 T^{2} + \cdots + 95002410498724)^{2} \) (T^4 + 20966164*T^2 + 95002410498724)^2
$31$	\( (T^{4} + 90530688 T^{2} + \cdots + 19\!\cdots\!04)^{2} \) (T^4 + 90530688*T^2 + 1951802383048704)^2
$37$	\( (T^{2} + 8576 T - 51335408)^{4} \) (T^2 + 8576*T - 51335408)^4
$41$	\( (T^{4} - 450270096 T^{2} + \cdots + 22\!\cdots\!04)^{2} \) (T^4 - 450270096*T^2 + 22953464070309504)^2
$43$	\( (T^{2} - 8104 T - 78275084)^{4} \) (T^2 - 8104*T - 78275084)^4
$47$	\( (T^{4} - 815119296 T^{2} + \cdots + 14\!\cdots\!36)^{2} \) (T^4 - 815119296*T^2 + 145506138984769536)^2
$53$	\( (T^{4} + 1338585844 T^{2} + \cdots + 44\!\cdots\!36)^{2} \) (T^4 + 1338585844*T^2 + 443978603020033636)^2
$59$	\( (T^{4} - 1412814336 T^{2} + \cdots + 10\!\cdots\!76)^{2} \) (T^4 - 1412814336*T^2 + 109800240831922176)^2
$61$	\( (T^{4} + 893736192 T^{2} + \cdots + 19\!\cdots\!44)^{2} \) (T^4 + 893736192*T^2 + 196515007572639744)^2
$67$	\( (T^{2} - 38152 T - 1272166832)^{4} \) (T^2 - 38152*T - 1272166832)^4
$71$	\( (T^{4} + 5973632716 T^{2} + \cdots + 78\!\cdots\!64)^{2} \) (T^4 + 5973632716*T^2 + 789983112998043364)^2
$73$	\( (T^{4} + 3510489888 T^{2} + \cdots + 30\!\cdots\!84)^{2} \) (T^4 + 3510489888*T^2 + 3023776020698601984)^2
$79$	\( (T^{2} + 23648 T - 595343552)^{4} \) (T^2 + 23648*T - 595343552)^4
$83$	\( (T^{4} - 5607969216 T^{2} + \cdots + 59\!\cdots\!96)^{2} \) (T^4 - 5607969216*T^2 + 5998768611579242496)^2
$89$	\( (T^{4} - 12592946448 T^{2} + \cdots + 19\!\cdots\!84)^{2} \) (T^4 - 12592946448*T^2 + 19415684921199801984)^2
$97$	\( (T^{4} + 10358370336 T^{2} + \cdots + 21\!\cdots\!56)^{2} \) (T^4 + 10358370336*T^2 + 2142510971815328256)^2
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\(n\)	\(10\)	\(29\)
\(\chi(n)\)	\(-1\)	\(-1\)