LMFDB - Newform orbit 192.4.c.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,4,Mod(191,192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(192, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("192.191");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 192 = 2^{6} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	192.c (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:	$11.3283667211$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 16x^{10} + 103x^{8} - 260x^{6} + 259x^{4} + 356x^{2} + 169 $ x^12 - 16x^10 + 103x^8 - 260x^6 + 259x^4 + 356*x^2 + 169
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{44}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 96)
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 $1,\beta_1,\ldots,\beta_{11}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_{4} q^{3} + \beta_{8} q^{5} + \beta_1 q^{7} + (\beta_{5} - 2) q^{9}+O(q^{10}) $ q - b4 * q^3 + b8 * q^5 + b1 * q^7 + (b5 - 2) * q^9	$ q - \beta_{4} q^{3} + \beta_{8} q^{5} + \beta_1 q^{7} + (\beta_{5} - 2) q^{9} - \beta_{10} q^{11} + ( - \beta_{2} - 6) q^{13} + ( - \beta_{11} + \beta_{10} + \cdots + \beta_1) q^{15}+ \cdots + ( - 14 \beta_{11} + 14 \beta_{10} + \cdots - 22 \beta_1) q^{99}+O(q^{100}) $ q - b4 * q^3 + b8 * q^5 + b1 * q^7 + (b5 - 2) * q^9 - b10 * q^11 + (-b2 - 6) * q^13 + (-b11 + b10 + b6 + b1) * q^15 + (-b9 - b8 + b7 + b5 - 1) * q^17 + (-b6 - 3b4 + b3 - 2b1) * q^19 + (-b9 + 3b8 + b5 - b2 - 12) q^21 + (2b11 - 2b6 + 4b4 - b3) q^23 + (b8 - b7 + 4b5 - 2b2 - 12) * q^25 + (2b11 + b10 + 2b6 - b4 - 3b3 - 2b1) * q^27 + (9b8 + 2b7 + 4b5 - 2) q^29 + (2b6 + 6b4 + 3b3 - 5b1) * q^31 + (b9 + 11b8 + b7 + 2b5 - 2b2 + 6) q^33 + (-4b11 + 3b10 - 3b6 + 13b4 + 2b3) q^35 + (2b8 - 2b7 + 8b5 - b2) q^37 + (-3b11 - 3b10 - 5b6 + 4b4 - 3b3 - 6b1) * q^39 + (-3b9 + 16b8 - 3b5) q^41 + (7b6 + 21b4 + 5b3 + 10b1) * q^43 + (-3b9 - 17b8 + 2b7 - b5 + 44) q^45 + (4b11 - 2b10 + 10b6 - 34b4 - 2b3) q^47 + (-b8 + b7 - 4b5 - 2b2 - 44) * q^49 + (6b11 - 3b10 - 11b6 + 3b4 - 12b3 + 12b1) * q^51 + (2b9 - 15b8 + 2b5) q^53 + (-14b6 - 42b4 + 11b3 + 6b1) * q^55 + (3b9 - 10b8 - 2b7 + 2b5 - 73) * q^57 + (-8b11 + 4b10 + 17b6 - 43b4 + 4b3) q^59 + (-2b8 + 2b7 - 8b5 + 3b2 - 36) * q^61 + (-2b11 - 4b10 + 12b6 + 10b4 - 12b3 + 11b1) * q^63 + (b9 - 38b8 - 2b7 - 3b5 + 2) q^65 + (-19b6 - 57b4 + 14b3 - 8b1) * q^67 + (b9 + 22b8 - 4b7 - 5b5 + 4b2 - 106) * q^69 + (-2b11 + 6b10 - 16b6 + 50b4 + b3) * q^71 + (-2b8 + 2b7 - 8b5 + 10b2 + 204) * q^73 + (-3b10 + 23b6 - 6b4 - 15b3 - 18b1) q^75 + (6b9 + 22b8 - 8b7 - 10b5 + 8) * q^77 + (34b6 + 102b4 + 13b3 + 3b1) * q^79 + (13b8 - b7 - 6b5 + 12b2 + 246) q^81 + (12b11 - 17b10 - 34b6 + 90b4 - 6b3) q^83 + (-8b8 + 8b7 - 32b5 + 4b2 + 264) * q^85 + (b11 + 17b10 - 31b6 - 6b4 - 18b3 - b1) * q^87 + (8b9 - 21b8 - 3b7 + 2b5 + 3) * q^89 + (28b6 + 84b4 + 23b3 - 26b1) * q^91 + (8b9 - 27b8 - 6b7 - 8b5 - b2 + 204) * q^93 + (-2b11 + 8b10 + 18b6 - 52b4 + b3) * q^95 + (b8 - b7 + 4b5 + 12b2 - 247) * q^97 + (-14b11 + 14b10 - 31b6 - 17b4 - 9b3 - 22b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 20 q^{9}+O(q^{10}) $ 12 * q - 20 * q^9	$ 12 q - 20 q^{9} - 72 q^{13} - 136 q^{21} - 132 q^{25} + 80 q^{33} + 24 q^{37} + 544 q^{45} - 540 q^{49} - 888 q^{57} - 456 q^{61} - 1312 q^{69} + 2424 q^{73} + 2924 q^{81} + 3072 q^{85} + 2360 q^{93} - 2952 q^{97}+O(q^{100}) $ 12 * q - 20 * q^9 - 72 * q^13 - 136 * q^21 - 132 * q^25 + 80 * q^33 + 24 * q^37 + 544 * q^45 - 540 * q^49 - 888 * q^57 - 456 * q^61 - 1312 * q^69 + 2424 * q^73 + 2924 * q^81 + 3072 * q^85 + 2360 * q^93 - 2952 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 16x^{10} + 103x^{8} - 260x^{6} + 259x^{4} + 356x^{2} + 169 $ x^12 - 16*x^10 + 103*x^8 - 260*x^6 + 259*x^4 + 356*x^2 + 169 :

$\beta_{1}$	$=$	$ ( -16388\nu^{10} + 259515\nu^{8} - 1685594\nu^{6} + 4399856\nu^{4} - 5980224\nu^{2} - 2762095 ) / 217953 $ (-16388v^10 + 259515v^8 - 1685594v^6 + 4399856v^4 - 5980224*v^2 - 2762095) / 217953
$\beta_{2}$	$=$	$ ( 5904\nu^{10} - 97064\nu^{8} + 571840\nu^{6} - 896416\nu^{4} - 1055120\nu^{2} + 1560520 ) / 72651 $ (5904v^10 - 97064v^8 + 571840v^6 - 896416v^4 - 1055120*v^2 + 1560520) / 72651
$\beta_{3}$	$=$	$ ( -544\nu^{10} + 8496\nu^{8} - 53344\nu^{6} + 135616\nu^{4} - 180960\nu^{2} - 83504 ) / 3573 $ (-544v^10 + 8496v^8 - 53344v^6 + 135616v^4 - 180960*v^2 - 83504) / 3573
$\beta_{4}$	$=$	$ ( 2707 \nu^{11} + 10504 \nu^{10} - 41518 \nu^{9} - 183781 \nu^{8} + 254446 \nu^{7} + 1315990 \nu^{6} + \cdots + 2381353 ) / 629642 $ (2707v^11 + 10504v^10 - 41518v^9 - 183781v^8 + 254446v^7 + 1315990v^6 - 571012v^5 - 4058808v^4 + 500393v^3 + 5081284v^2 + 95344*v + 2381353) / 629642
$\beta_{5}$	$=$	$ ( - 2631 \nu^{11} - 7176 \nu^{10} + 57332 \nu^{9} + 97500 \nu^{8} - 464316 \nu^{7} - 531232 \nu^{6} + \cdots - 7299877 ) / 314821 $ (-2631v^11 - 7176v^10 + 57332v^9 + 97500v^8 - 464316v^7 - 531232v^6 + 1492582v^5 + 802880v^4 - 515419v^3 + 954824v^2 - 2325140*v - 7299877) / 314821
$\beta_{6}$	$=$	$ ( - 8121 \nu^{11} + 31512 \nu^{10} + 124554 \nu^{9} - 551343 \nu^{8} - 763338 \nu^{7} + \cdots + 7144059 ) / 629642 $ (-8121v^11 + 31512v^10 + 124554v^9 - 551343v^8 - 763338v^7 + 3947970v^6 + 1713036v^5 - 12176424v^4 - 1501179v^3 + 15243852v^2 - 286032*v + 7144059) / 629642
$\beta_{7}$	$=$	$ ( 18023 \nu^{11} + 43056 \nu^{10} - 102208 \nu^{9} - 585000 \nu^{8} - 549788 \nu^{7} + \cdots + 44743725 ) / 944463 $ (18023v^11 + 43056v^10 - 102208v^9 - 585000v^8 - 549788v^7 + 3187392v^6 + 5801042v^5 - 4817280v^4 + 2774507v^3 - 5728944v^2 - 1691184*v + 44743725) / 944463
$\beta_{8}$	$=$	$ ( 3815\nu^{11} - 60784\nu^{9} + 386308\nu^{7} - 931534\nu^{5} + 689195\nu^{3} + 2016192\nu ) / 72651 $ (3815v^11 - 60784v^9 + 386308v^7 - 931534v^5 + 689195v^3 + 2016192v) / 72651
$\beta_{9}$	$=$	$ ( 168575 \nu^{11} + 21528 \nu^{10} - 2577340 \nu^{9} - 292500 \nu^{8} + 15775756 \nu^{7} + \cdots + 21899631 ) / 944463 $ (168575v^11 + 21528v^10 - 2577340v^9 - 292500v^8 + 15775756v^7 + 1593696v^6 - 35075014v^5 - 2408640v^4 + 31664915v^3 - 2864472v^2 + 84598716*v + 21899631) / 944463
$\beta_{10}$	$=$	$ ( - 525547 \nu^{11} + 8893158 \nu^{9} - 62629246 \nu^{7} + 194548900 \nu^{5} - 288501633 \nu^{3} - 73378376 \nu ) / 2833389 $ (-525547v^11 + 8893158v^9 - 62629246v^7 + 194548900v^5 - 288501633v^3 - 73378376v) / 2833389
$\beta_{11}$	$=$	$ ( 1307467 \nu^{11} - 336856 \nu^{10} - 21563934 \nu^{9} + 5083299 \nu^{8} + 143863390 \nu^{7} + \cdots - 44786495 ) / 5666778 $ (1307467v^11 - 336856v^10 - 21563934v^9 + 5083299v^8 + 143863390v^7 - 30457882v^6 - 397870564v^5 + 71014216v^4 + 510897681v^3 - 97769724v^2 + 122714864*v - 44786495) / 5666778

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

$\nu$	$=$	$ ( 3\beta_{9} + 6\beta_{8} + 32\beta_{6} + 3\beta_{5} - 96\beta_{4} ) / 384 $ (3b9 + 6b8 + 32b6 + 3b5 - 96*b4) / 384
$\nu^{2}$	$=$	$ ( 4\beta_{8} - 4\beta_{7} - 8\beta_{6} + 16\beta_{5} - 24\beta_{4} + 3\beta_{3} - 24\beta _1 + 508 ) / 192 $ (4b8 - 4b7 - 8b6 + 16b5 - 24b4 + 3b3 - 24*b1 + 508) / 192
$\nu^{3}$	$=$	$ ( 24 \beta_{11} + 12 \beta_{10} + 39 \beta_{9} - 126 \beta_{8} - 12 \beta_{7} + 124 \beta_{6} + \cdots + 12 ) / 384 $ (24b11 + 12b10 + 39b9 - 126b8 - 12b7 + 124b6 + 15b5 - 396b4 - 12*b3 + 12) / 384
$\nu^{4}$	$=$	$ ( 7\beta_{8} - 7\beta_{7} - 24\beta_{6} + 28\beta_{5} - 72\beta_{4} + 42\beta_{3} + 9\beta_{2} - 120\beta _1 + 793 ) / 96 $ (7b8 - 7b7 - 24b6 + 28b5 - 72b4 + 42b3 + 9b2 - 120b1 + 793) / 96
$\nu^{5}$	$=$	$ ( 90 \beta_{11} + 72 \beta_{10} + 372 \beta_{9} - 1216 \beta_{8} - 200 \beta_{7} + 232 \beta_{6} + \cdots + 200 ) / 384 $ (90b11 + 72b10 + 372b9 - 1216b8 - 200b7 + 232b6 - 28b5 - 786b4 - 45*b3 + 200) / 384
$\nu^{6}$	$=$	$ ( - 4 \beta_{8} + 4 \beta_{7} - 80 \beta_{6} - 16 \beta_{5} - 240 \beta_{4} + 411 \beta_{3} + \cdots - 1084 ) / 96 $ (-4b8 + 4b7 - 80b6 - 16b5 - 240b4 + 411b3 + 18b2 - 912b1 - 1084) / 96
$\nu^{7}$	$=$	$ ( - 450 \beta_{11} - 252 \beta_{10} + 2622 \beta_{9} - 8728 \beta_{8} - 1652 \beta_{7} - 2764 \beta_{6} + \cdots + 1652 ) / 384 $ (-450b11 - 252b10 + 2622b9 - 8728b8 - 1652b7 - 2764b6 - 682b5 + 8742b4 + 225*b3 + 1652) / 384
$\nu^{8}$	$=$	$ ( - 34 \beta_{8} + 34 \beta_{7} - 20 \beta_{6} - 136 \beta_{5} - 60 \beta_{4} + 223 \beta_{3} + \cdots - 4118 ) / 8 $ (-34b8 + 34b7 - 20b6 - 136b5 - 60b4 + 223b3 - 33b2 - 452b1 - 4118) / 8
$\nu^{9}$	$=$	$ ( - 10680 \beta_{11} - 7176 \beta_{10} + 13431 \beta_{9} - 45018 \beta_{8} - 8904 \beta_{7} + \cdots + 8904 ) / 384 $ (-10680b11 - 7176b10 + 13431b9 - 45018b8 - 8904b7 - 46024b6 - 4377b5 + 148752b4 + 5340*b3 + 8904) / 384
$\nu^{10}$	$=$	$ ( - 9800 \beta_{8} + 9800 \beta_{7} - 1112 \beta_{6} - 39200 \beta_{5} - 3336 \beta_{4} + \cdots - 1134008 ) / 192 $ (-9800b8 + 9800b7 - 1112b6 - 39200b5 - 3336b4 + 21663b3 - 11412b2 - 42408b1 - 1134008) / 192
$\nu^{11}$	$=$	$ ( - 106956 \beta_{11} - 73404 \beta_{10} + 30693 \beta_{9} - 103482 \beta_{8} - 21252 \beta_{7} + \cdots + 21252 ) / 384 $ (-106956b11 - 73404b10 + 30693b9 - 103482b8 - 21252b7 - 436076b6 - 11811b5 + 1415184b4 + 53478*b3 + 21252) / 384

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/192\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$133$
$\chi(n)$	$-1$	$-1$	$1$

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} $

191.1

−2.59708	−	0.707107i
−2.59708	+	0.707107i
−1.64111	+	0.707107i
−1.64111	−	0.707107i
−0.248859	−	0.707107i
−0.248859	+	0.707107i
0.248859	+	0.707107i
0.248859	−	0.707107i
1.64111	−	0.707107i
1.64111	+	0.707107i
2.59708	+	0.707107i
2.59708	−	0.707107i

−5.19416

−

0.143987i

7.96714i

−

25.6706i

26.9585

1.49578i

191.2

−5.19416

0.143987i

−

7.96714i

25.6706i

26.9585

−

1.49578i

191.3

−3.28223

−

4.02827i

−

2.00080i

14.5105i

−5.45398

26.4434i

191.4

−3.28223

4.02827i

2.00080i

−

14.5105i

−5.45398

−

26.4434i

191.5

−0.497717

−

5.17226i

−

18.4532i

−

17.1600i

−26.5046

5.14865i

191.6

−0.497717

5.17226i

18.4532i

17.1600i

−26.5046

−

5.14865i

191.7

0.497717

−

5.17226i

18.4532i

−

17.1600i

−26.5046

−

5.14865i

191.8

0.497717

5.17226i

−

18.4532i

17.1600i

−26.5046

5.14865i

191.9

3.28223

−

4.02827i

2.00080i

14.5105i

−5.45398

−

26.4434i

191.10

3.28223

4.02827i

−

2.00080i

−

14.5105i

−5.45398

26.4434i

191.11

5.19416

−

0.143987i

−

7.96714i

−

25.6706i

26.9585

−

1.49578i

191.12

5.19416

0.143987i

7.96714i

25.6706i

26.9585

1.49578i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
4.b	odd	2	1	inner
12.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	192.4.c.d		12
3.b	odd	2	1	inner	192.4.c.d		12
4.b	odd	2	1	inner	192.4.c.d		12
8.b	even	2	1		96.4.c.a	✓	12
8.d	odd	2	1		96.4.c.a	✓	12
12.b	even	2	1	inner	192.4.c.d		12
16.e	even	4	1		768.4.f.d		12
16.e	even	4	1		768.4.f.i		12
16.f	odd	4	1		768.4.f.d		12
16.f	odd	4	1		768.4.f.i		12
24.f	even	2	1		96.4.c.a	✓	12
24.h	odd	2	1		96.4.c.a	✓	12
48.i	odd	4	1		768.4.f.d		12
48.i	odd	4	1		768.4.f.i		12
48.k	even	4	1		768.4.f.d		12
48.k	even	4	1		768.4.f.i		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
96.4.c.a	✓	12	8.b	even	2	1
96.4.c.a	✓	12	8.d	odd	2	1
96.4.c.a	✓	12	24.f	even	2	1
96.4.c.a	✓	12	24.h	odd	2	1
192.4.c.d		12	1.a	even	1	1	trivial
192.4.c.d		12	3.b	odd	2	1	inner
192.4.c.d		12	4.b	odd	2	1	inner
192.4.c.d		12	12.b	even	2	1	inner
768.4.f.d		12	16.e	even	4	1
768.4.f.d		12	16.f	odd	4	1
768.4.f.d		12	48.i	odd	4	1
768.4.f.d		12	48.k	even	4	1
768.4.f.i		12	16.e	even	4	1
768.4.f.i		12	16.f	odd	4	1
768.4.f.i		12	48.i	odd	4	1
768.4.f.i		12	48.k	even	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{6} + 408T_{5}^{4} + 23232T_{5}^{2} + 86528 $ T5^6 + 408*T5^4 + 23232*T5^2 + 86528 acting on $S_{4}^{\mathrm{new}}(192, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} $ T^12
$3$	$ T^{12} + \cdots + 387420489 $ T^12 + 10T^10 - 681T^8 - 16596T^6 - 496449T^4 + 5314410*T^2 + 387420489
$5$	$ (T^{6} + 408 T^{4} + \cdots + 86528)^{2} $ (T^6 + 408T^4 + 23232T^2 + 86528)^2
$7$	$ (T^{6} + 1164 T^{4} + \cdots + 40857664)^{2} $ (T^6 + 1164T^4 + 394800T^2 + 40857664)^2
$11$	$ (T^{6} - 5400 T^{4} + \cdots - 3202560512)^{2} $ (T^6 - 5400T^4 + 8526528T^2 - 3202560512)^2
$13$	$ (T^{3} + 18 T^{2} + \cdots - 133928)^{4} $ (T^3 + 18T^2 - 4500T - 133928)^4
$17$	$ (T^{6} + 24576 T^{4} + \cdots + 391378894848)^{2} $ (T^6 + 24576T^4 + 179306496T^2 + 391378894848)^2
$19$	$ (T^{6} + 8076 T^{4} + \cdots + 2518433856)^{2} $ (T^6 + 8076T^4 + 9033264T^2 + 2518433856)^2
$23$	$ (T^{6} - 26976 T^{4} + \cdots - 124728147968)^{2} $ (T^6 - 26976T^4 + 177859584T^2 - 124728147968)^2
$29$	$ (T^{6} + \cdots + 19385223967232)^{2} $ (T^6 + 118296T^4 + 2933334720T^2 + 19385223967232)^2
$31$	$ (T^{6} + 58956 T^{4} + \cdots + 708600302656)^{2} $ (T^6 + 58956T^4 + 414766128T^2 + 708600302656)^2
$37$	$ (T^{3} - 6 T^{2} + \cdots + 3249144)^{4} $ (T^3 - 6T^2 - 96756T + 3249144)^4
$41$	$ (T^{6} + \cdots + 359281246896128)^{2} $ (T^6 + 317280T^4 + 25285684224T^2 + 359281246896128)^2
$43$	$ (T^{6} + \cdots + 24182238991936)^{2} $ (T^6 + 246252T^4 + 4898641200T^2 + 24182238991936)^2
$47$	$ (T^{6} + \cdots - 165630483365888)^{2} $ (T^6 - 273792T^4 + 15725936640T^2 - 165630483365888)^2
$53$	$ (T^{6} + \cdots + 152455972372992)^{2} $ (T^6 + 187224T^4 + 9858320064T^2 + 152455972372992)^2
$59$	$ (T^{6} + \cdots - 61\!\cdots\!48)^{2} $ (T^6 - 680472T^4 + 131795888832T^2 - 6138947587240448)^2
$61$	$ (T^{3} + 114 T^{2} + \cdots - 17941928)^{4} $ (T^3 + 114T^2 - 104724T - 17941928)^4
$67$	$ (T^{6} + \cdots + 53\!\cdots\!56)^{2} $ (T^6 + 1232556T^4 + 348595981104T^2 + 5330258010235456)^2
$71$	$ (T^{6} + \cdots - 61\!\cdots\!68)^{2} $ (T^6 - 589152T^4 + 108701346816T^2 - 6162070084026368)^2
$73$	$ (T^{3} - 606 T^{2} + \cdots + 145079064)^{4} $ (T^3 - 606T^2 - 319956T + 145079064)^4
$79$	$ (T^{6} + \cdots + 76\!\cdots\!44)^{2} $ (T^6 + 2122764T^4 + 836589949488T^2 + 76931405354772544)^2
$83$	$ (T^{6} + \cdots - 22\!\cdots\!92)^{2} $ (T^6 - 3104664T^4 + 2166924624576T^2 - 227939504096072192)^2
$89$	$ (T^{6} + \cdots + 761571160915968)^{2} $ (T^6 + 1258080T^4 + 130935794688T^2 + 761571160915968)^2
$97$	$ (T^{3} + 738 T^{2} + \cdots - 38714344)^{4} $ (T^3 + 738T^2 - 581844T - 38714344)^4
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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:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	192.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q - \beta_{4} q^{3} + \beta_{8} q^{5} + \beta_1 q^{7} + (\beta_{5} - 2) q^{9}+O(q^{10}) \) q - b4 * q^3 + b8 * q^5 + b1 * q^7 + (b5 - 2) * q^9	\( q - \beta_{4} q^{3} + \beta_{8} q^{5} + \beta_1 q^{7} + (\beta_{5} - 2) q^{9} - \beta_{10} q^{11} + ( - \beta_{2} - 6) q^{13} + ( - \beta_{11} + \beta_{10} + \cdots + \beta_1) q^{15}+ \cdots + ( - 14 \beta_{11} + 14 \beta_{10} + \cdots - 22 \beta_1) q^{99}+O(q^{100}) \) q - b4 * q^3 + b8 * q^5 + b1 * q^7 + (b5 - 2) * q^9 - b10 * q^11 + (-b2 - 6) * q^13 + (-b11 + b10 + b6 + b1) * q^15 + (-b9 - b8 + b7 + b5 - 1) * q^17 + (-b6 - 3b4 + b3 - 2b1) * q^19 + (-b9 + 3b8 + b5 - b2 - 12) q^21 + (2b11 - 2b6 + 4b4 - b3) q^23 + (b8 - b7 + 4b5 - 2b2 - 12) * q^25 + (2b11 + b10 + 2b6 - b4 - 3b3 - 2b1) * q^27 + (9b8 + 2b7 + 4b5 - 2) q^29 + (2b6 + 6b4 + 3b3 - 5b1) * q^31 + (b9 + 11b8 + b7 + 2b5 - 2b2 + 6) q^33 + (-4b11 + 3b10 - 3b6 + 13b4 + 2b3) q^35 + (2b8 - 2b7 + 8b5 - b2) q^37 + (-3b11 - 3b10 - 5b6 + 4b4 - 3b3 - 6b1) * q^39 + (-3b9 + 16b8 - 3b5) q^41 + (7b6 + 21b4 + 5b3 + 10b1) * q^43 + (-3b9 - 17b8 + 2b7 - b5 + 44) q^45 + (4b11 - 2b10 + 10b6 - 34b4 - 2b3) q^47 + (-b8 + b7 - 4b5 - 2b2 - 44) * q^49 + (6b11 - 3b10 - 11b6 + 3b4 - 12b3 + 12b1) * q^51 + (2b9 - 15b8 + 2b5) q^53 + (-14b6 - 42b4 + 11b3 + 6b1) * q^55 + (3b9 - 10b8 - 2b7 + 2b5 - 73) * q^57 + (-8b11 + 4b10 + 17b6 - 43b4 + 4b3) q^59 + (-2b8 + 2b7 - 8b5 + 3b2 - 36) * q^61 + (-2b11 - 4b10 + 12b6 + 10b4 - 12b3 + 11b1) * q^63 + (b9 - 38b8 - 2b7 - 3b5 + 2) q^65 + (-19b6 - 57b4 + 14b3 - 8b1) * q^67 + (b9 + 22b8 - 4b7 - 5b5 + 4b2 - 106) * q^69 + (-2b11 + 6b10 - 16b6 + 50b4 + b3) * q^71 + (-2b8 + 2b7 - 8b5 + 10b2 + 204) * q^73 + (-3b10 + 23b6 - 6b4 - 15b3 - 18b1) q^75 + (6b9 + 22b8 - 8b7 - 10b5 + 8) * q^77 + (34b6 + 102b4 + 13b3 + 3b1) * q^79 + (13b8 - b7 - 6b5 + 12b2 + 246) q^81 + (12b11 - 17b10 - 34b6 + 90b4 - 6b3) q^83 + (-8b8 + 8b7 - 32b5 + 4b2 + 264) * q^85 + (b11 + 17b10 - 31b6 - 6b4 - 18b3 - b1) * q^87 + (8b9 - 21b8 - 3b7 + 2b5 + 3) * q^89 + (28b6 + 84b4 + 23b3 - 26b1) * q^91 + (8b9 - 27b8 - 6b7 - 8b5 - b2 + 204) * q^93 + (-2b11 + 8b10 + 18b6 - 52b4 + b3) * q^95 + (b8 - b7 + 4b5 + 12b2 - 247) * q^97 + (-14b11 + 14b10 - 31b6 - 17b4 - 9b3 - 22b1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 20 q^{9}+O(q^{10}) \) 12 * q - 20 * q^9	\( 12 q - 20 q^{9} - 72 q^{13} - 136 q^{21} - 132 q^{25} + 80 q^{33} + 24 q^{37} + 544 q^{45} - 540 q^{49} - 888 q^{57} - 456 q^{61} - 1312 q^{69} + 2424 q^{73} + 2924 q^{81} + 3072 q^{85} + 2360 q^{93} - 2952 q^{97}+O(q^{100}) \) 12 * q - 20 * q^9 - 72 * q^13 - 136 * q^21 - 132 * q^25 + 80 * q^33 + 24 * q^37 + 544 * q^45 - 540 * q^49 - 888 * q^57 - 456 * q^61 - 1312 * q^69 + 2424 * q^73 + 2924 * q^81 + 3072 * q^85 + 2360 * q^93 - 2952 * q^97

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	192.4.c.d

Level	$192$
Weight	$4$
Character orbit	192.c
Analytic conductor	$11.328$
Analytic rank	$0$
Dimension	$12$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(11.3283667211\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - 16x^{10} + 103x^{8} - 260x^{6} + 259x^{4} + 356x^{2} + 169 \) x^12 - 16x^10 + 103x^8 - 260x^6 + 259x^4 + 356*x^2 + 169
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{44}\cdot 3^{2} \)
Twist minimal:	no (minimal twist has level 96)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( -16388\nu^{10} + 259515\nu^{8} - 1685594\nu^{6} + 4399856\nu^{4} - 5980224\nu^{2} - 2762095 ) / 217953 \) (-16388v^10 + 259515v^8 - 1685594v^6 + 4399856v^4 - 5980224*v^2 - 2762095) / 217953
\(\beta_{2}\)	\(=\)	\( ( 5904\nu^{10} - 97064\nu^{8} + 571840\nu^{6} - 896416\nu^{4} - 1055120\nu^{2} + 1560520 ) / 72651 \) (5904v^10 - 97064v^8 + 571840v^6 - 896416v^4 - 1055120*v^2 + 1560520) / 72651
\(\beta_{3}\)	\(=\)	\( ( -544\nu^{10} + 8496\nu^{8} - 53344\nu^{6} + 135616\nu^{4} - 180960\nu^{2} - 83504 ) / 3573 \) (-544v^10 + 8496v^8 - 53344v^6 + 135616v^4 - 180960*v^2 - 83504) / 3573
\(\beta_{4}\)	\(=\)	\( ( 2707 \nu^{11} + 10504 \nu^{10} - 41518 \nu^{9} - 183781 \nu^{8} + 254446 \nu^{7} + 1315990 \nu^{6} + \cdots + 2381353 ) / 629642 \) (2707v^11 + 10504v^10 - 41518v^9 - 183781v^8 + 254446v^7 + 1315990v^6 - 571012v^5 - 4058808v^4 + 500393v^3 + 5081284v^2 + 95344*v + 2381353) / 629642
\(\beta_{5}\)	\(=\)	\( ( - 2631 \nu^{11} - 7176 \nu^{10} + 57332 \nu^{9} + 97500 \nu^{8} - 464316 \nu^{7} - 531232 \nu^{6} + \cdots - 7299877 ) / 314821 \) (-2631v^11 - 7176v^10 + 57332v^9 + 97500v^8 - 464316v^7 - 531232v^6 + 1492582v^5 + 802880v^4 - 515419v^3 + 954824v^2 - 2325140*v - 7299877) / 314821
\(\beta_{6}\)	\(=\)	\( ( - 8121 \nu^{11} + 31512 \nu^{10} + 124554 \nu^{9} - 551343 \nu^{8} - 763338 \nu^{7} + \cdots + 7144059 ) / 629642 \) (-8121v^11 + 31512v^10 + 124554v^9 - 551343v^8 - 763338v^7 + 3947970v^6 + 1713036v^5 - 12176424v^4 - 1501179v^3 + 15243852v^2 - 286032*v + 7144059) / 629642
\(\beta_{7}\)	\(=\)	\( ( 18023 \nu^{11} + 43056 \nu^{10} - 102208 \nu^{9} - 585000 \nu^{8} - 549788 \nu^{7} + \cdots + 44743725 ) / 944463 \) (18023v^11 + 43056v^10 - 102208v^9 - 585000v^8 - 549788v^7 + 3187392v^6 + 5801042v^5 - 4817280v^4 + 2774507v^3 - 5728944v^2 - 1691184*v + 44743725) / 944463
\(\beta_{8}\)	\(=\)	\( ( 3815\nu^{11} - 60784\nu^{9} + 386308\nu^{7} - 931534\nu^{5} + 689195\nu^{3} + 2016192\nu ) / 72651 \) (3815v^11 - 60784v^9 + 386308v^7 - 931534v^5 + 689195v^3 + 2016192v) / 72651
\(\beta_{9}\)	\(=\)	\( ( 168575 \nu^{11} + 21528 \nu^{10} - 2577340 \nu^{9} - 292500 \nu^{8} + 15775756 \nu^{7} + \cdots + 21899631 ) / 944463 \) (168575v^11 + 21528v^10 - 2577340v^9 - 292500v^8 + 15775756v^7 + 1593696v^6 - 35075014v^5 - 2408640v^4 + 31664915v^3 - 2864472v^2 + 84598716*v + 21899631) / 944463
\(\beta_{10}\)	\(=\)	\( ( - 525547 \nu^{11} + 8893158 \nu^{9} - 62629246 \nu^{7} + 194548900 \nu^{5} - 288501633 \nu^{3} - 73378376 \nu ) / 2833389 \) (-525547v^11 + 8893158v^9 - 62629246v^7 + 194548900v^5 - 288501633v^3 - 73378376v) / 2833389
\(\beta_{11}\)	\(=\)	\( ( 1307467 \nu^{11} - 336856 \nu^{10} - 21563934 \nu^{9} + 5083299 \nu^{8} + 143863390 \nu^{7} + \cdots - 44786495 ) / 5666778 \) (1307467v^11 - 336856v^10 - 21563934v^9 + 5083299v^8 + 143863390v^7 - 30457882v^6 - 397870564v^5 + 71014216v^4 + 510897681v^3 - 97769724v^2 + 122714864*v - 44786495) / 5666778

\(\nu\)	\(=\)	\( ( 3\beta_{9} + 6\beta_{8} + 32\beta_{6} + 3\beta_{5} - 96\beta_{4} ) / 384 \) (3b9 + 6b8 + 32b6 + 3b5 - 96*b4) / 384
\(\nu^{2}\)	\(=\)	\( ( 4\beta_{8} - 4\beta_{7} - 8\beta_{6} + 16\beta_{5} - 24\beta_{4} + 3\beta_{3} - 24\beta _1 + 508 ) / 192 \) (4b8 - 4b7 - 8b6 + 16b5 - 24b4 + 3b3 - 24*b1 + 508) / 192
\(\nu^{3}\)	\(=\)	\( ( 24 \beta_{11} + 12 \beta_{10} + 39 \beta_{9} - 126 \beta_{8} - 12 \beta_{7} + 124 \beta_{6} + \cdots + 12 ) / 384 \) (24b11 + 12b10 + 39b9 - 126b8 - 12b7 + 124b6 + 15b5 - 396b4 - 12*b3 + 12) / 384
\(\nu^{4}\)	\(=\)	\( ( 7\beta_{8} - 7\beta_{7} - 24\beta_{6} + 28\beta_{5} - 72\beta_{4} + 42\beta_{3} + 9\beta_{2} - 120\beta _1 + 793 ) / 96 \) (7b8 - 7b7 - 24b6 + 28b5 - 72b4 + 42b3 + 9b2 - 120b1 + 793) / 96
\(\nu^{5}\)	\(=\)	\( ( 90 \beta_{11} + 72 \beta_{10} + 372 \beta_{9} - 1216 \beta_{8} - 200 \beta_{7} + 232 \beta_{6} + \cdots + 200 ) / 384 \) (90b11 + 72b10 + 372b9 - 1216b8 - 200b7 + 232b6 - 28b5 - 786b4 - 45*b3 + 200) / 384
\(\nu^{6}\)	\(=\)	\( ( - 4 \beta_{8} + 4 \beta_{7} - 80 \beta_{6} - 16 \beta_{5} - 240 \beta_{4} + 411 \beta_{3} + \cdots - 1084 ) / 96 \) (-4b8 + 4b7 - 80b6 - 16b5 - 240b4 + 411b3 + 18b2 - 912b1 - 1084) / 96
\(\nu^{7}\)	\(=\)	\( ( - 450 \beta_{11} - 252 \beta_{10} + 2622 \beta_{9} - 8728 \beta_{8} - 1652 \beta_{7} - 2764 \beta_{6} + \cdots + 1652 ) / 384 \) (-450b11 - 252b10 + 2622b9 - 8728b8 - 1652b7 - 2764b6 - 682b5 + 8742b4 + 225*b3 + 1652) / 384
\(\nu^{8}\)	\(=\)	\( ( - 34 \beta_{8} + 34 \beta_{7} - 20 \beta_{6} - 136 \beta_{5} - 60 \beta_{4} + 223 \beta_{3} + \cdots - 4118 ) / 8 \) (-34b8 + 34b7 - 20b6 - 136b5 - 60b4 + 223b3 - 33b2 - 452b1 - 4118) / 8
\(\nu^{9}\)	\(=\)	\( ( - 10680 \beta_{11} - 7176 \beta_{10} + 13431 \beta_{9} - 45018 \beta_{8} - 8904 \beta_{7} + \cdots + 8904 ) / 384 \) (-10680b11 - 7176b10 + 13431b9 - 45018b8 - 8904b7 - 46024b6 - 4377b5 + 148752b4 + 5340*b3 + 8904) / 384
\(\nu^{10}\)	\(=\)	\( ( - 9800 \beta_{8} + 9800 \beta_{7} - 1112 \beta_{6} - 39200 \beta_{5} - 3336 \beta_{4} + \cdots - 1134008 ) / 192 \) (-9800b8 + 9800b7 - 1112b6 - 39200b5 - 3336b4 + 21663b3 - 11412b2 - 42408b1 - 1134008) / 192
\(\nu^{11}\)	\(=\)	\( ( - 106956 \beta_{11} - 73404 \beta_{10} + 30693 \beta_{9} - 103482 \beta_{8} - 21252 \beta_{7} + \cdots + 21252 ) / 384 \) (-106956b11 - 73404b10 + 30693b9 - 103482b8 - 21252b7 - 436076b6 - 11811b5 + 1415184b4 + 53478*b3 + 21252) / 384

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 191.12
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} \) T^12
$3$	\( T^{12} + \cdots + 387420489 \) T^12 + 10T^10 - 681T^8 - 16596T^6 - 496449T^4 + 5314410*T^2 + 387420489
$5$	\( (T^{6} + 408 T^{4} + \cdots + 86528)^{2} \) (T^6 + 408T^4 + 23232T^2 + 86528)^2
$7$	\( (T^{6} + 1164 T^{4} + \cdots + 40857664)^{2} \) (T^6 + 1164T^4 + 394800T^2 + 40857664)^2
$11$	\( (T^{6} - 5400 T^{4} + \cdots - 3202560512)^{2} \) (T^6 - 5400T^4 + 8526528T^2 - 3202560512)^2
$13$	\( (T^{3} + 18 T^{2} + \cdots - 133928)^{4} \) (T^3 + 18T^2 - 4500T - 133928)^4
$17$	\( (T^{6} + 24576 T^{4} + \cdots + 391378894848)^{2} \) (T^6 + 24576T^4 + 179306496T^2 + 391378894848)^2
$19$	\( (T^{6} + 8076 T^{4} + \cdots + 2518433856)^{2} \) (T^6 + 8076T^4 + 9033264T^2 + 2518433856)^2
$23$	\( (T^{6} - 26976 T^{4} + \cdots - 124728147968)^{2} \) (T^6 - 26976T^4 + 177859584T^2 - 124728147968)^2
$29$	\( (T^{6} + \cdots + 19385223967232)^{2} \) (T^6 + 118296T^4 + 2933334720T^2 + 19385223967232)^2
$31$	\( (T^{6} + 58956 T^{4} + \cdots + 708600302656)^{2} \) (T^6 + 58956T^4 + 414766128T^2 + 708600302656)^2
$37$	\( (T^{3} - 6 T^{2} + \cdots + 3249144)^{4} \) (T^3 - 6T^2 - 96756T + 3249144)^4
$41$	\( (T^{6} + \cdots + 359281246896128)^{2} \) (T^6 + 317280T^4 + 25285684224T^2 + 359281246896128)^2
$43$	\( (T^{6} + \cdots + 24182238991936)^{2} \) (T^6 + 246252T^4 + 4898641200T^2 + 24182238991936)^2
$47$	\( (T^{6} + \cdots - 165630483365888)^{2} \) (T^6 - 273792T^4 + 15725936640T^2 - 165630483365888)^2
$53$	\( (T^{6} + \cdots + 152455972372992)^{2} \) (T^6 + 187224T^4 + 9858320064T^2 + 152455972372992)^2
$59$	\( (T^{6} + \cdots - 61\!\cdots\!48)^{2} \) (T^6 - 680472T^4 + 131795888832T^2 - 6138947587240448)^2
$61$	\( (T^{3} + 114 T^{2} + \cdots - 17941928)^{4} \) (T^3 + 114T^2 - 104724T - 17941928)^4
$67$	\( (T^{6} + \cdots + 53\!\cdots\!56)^{2} \) (T^6 + 1232556T^4 + 348595981104T^2 + 5330258010235456)^2
$71$	\( (T^{6} + \cdots - 61\!\cdots\!68)^{2} \) (T^6 - 589152T^4 + 108701346816T^2 - 6162070084026368)^2
$73$	\( (T^{3} - 606 T^{2} + \cdots + 145079064)^{4} \) (T^3 - 606T^2 - 319956T + 145079064)^4
$79$	\( (T^{6} + \cdots + 76\!\cdots\!44)^{2} \) (T^6 + 2122764T^4 + 836589949488T^2 + 76931405354772544)^2
$83$	\( (T^{6} + \cdots - 22\!\cdots\!92)^{2} \) (T^6 - 3104664T^4 + 2166924624576T^2 - 227939504096072192)^2
$89$	\( (T^{6} + \cdots + 761571160915968)^{2} \) (T^6 + 1258080T^4 + 130935794688T^2 + 761571160915968)^2
$97$	\( (T^{3} + 738 T^{2} + \cdots - 38714344)^{4} \) (T^3 + 738T^2 - 581844T - 38714344)^4
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\(n\)	\(65\)	\(127\)	\(133\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(1\)