LMFDB - Newform orbit 1323.2.c.e

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1323,2,Mod(1322,1323)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1323, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1323.1322");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1323 = 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1323.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:	$10.5642081874$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 12x^{14} + 106x^{12} - 384x^{10} + 1005x^{8} - 1200x^{6} + 1030x^{4} - 252x^{2} + 49 $ x^16 - 12x^14 + 106x^12 - 384x^10 + 1005x^8 - 1200x^6 + 1030x^4 - 252*x^2 + 49
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8} $
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 $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_{7} q^{2} + ( - \beta_{4} - \beta_1 - 1) q^{4} + ( - \beta_{10} - \beta_{3}) q^{5} + ( - \beta_{15} - \beta_{14} + \cdots + 2 \beta_{7}) q^{8}+O(q^{10}) $ q - b7 * q^2 + (-b4 - b1 - 1) * q^4 + (-b10 - b3) * q^5 + (-b15 - b14 + b13 + 2b7) q^8	$ q - \beta_{7} q^{2} + ( - \beta_{4} - \beta_1 - 1) q^{4} + ( - \beta_{10} - \beta_{3}) q^{5} + ( - \beta_{15} - \beta_{14} + \cdots + 2 \beta_{7}) q^{8}+ \cdots + ( - 2 \beta_{9} + 2 \beta_{8}) q^{97}+O(q^{100}) $ q - b7 * q^2 + (-b4 - b1 - 1) * q^4 + (-b10 - b3) * q^5 + (-b15 - b14 + b13 + 2b7) q^8 + (-b9 - b8 - 2b2) q^10 + b15 * q^11 + (b12 - b9 - b2) * q^13 + (2b6 + 3b4 + 2b1 + 3) q^16 + (-2b11 - b5) q^17 + (-b12 + b9 + b8 - 2b2) q^19 + (-b11 - 2b10 - 3b5 - b3) * q^20 + (-b1 + 1) * q^22 + (b15 + b14 + 2b13 - b7) q^23 + (2b6 - b4 - b1 + 2) q^25 + (-b11 - b10 - 2b5 + 2b3) * q^26 + (b15 - 2b14 - 3b7) * q^29 + (-b12 - b9 - 3b8 - b2) q^31 + (2b14 - 3b13 - 6b7) q^32 + (-b12 - b9 + 4b8 + 2b2) * q^34 + (b6 - 4b4 - 1) q^37 + (2b11 + b10 - 2b3) * q^38 + (-4b12 + 3b8 + 4b2) q^40 + (-3b11 + b5 + 5b3) * q^41 + (-2b6 + 3b4 - b1 + 2) * q^43 + (b15 - b14 + b7) * q^44 + (b6 + 2b4 - 3b1 - 5) * q^46 + (2b10 + 3b3) * q^47 + (-b15 + b14 - b13 + b7) * q^50 + (-3b12 + 4b9 + 4b8 + 6b2) * q^52 + (2b14 + 3b13 + 3b7) q^53 + (3b12 + 5b9 + b8) * q^55 + (2b6 - b4 - 2b1 - 6) * q^58 + (-b11 + b10 - b5 + 4b3) q^59 + (b12 - 5b9 - b2) q^61 + (-2b11 + b10 - 3b5) * q^62 + (-b6 - 8b4 - 4b1 - 11) * q^64 + (b15 - 2b13 - 3b7) * q^65 + (2b6 + 3b4 + 6) * q^67 + (b11 + b10 + 5b5) q^68 + (2b15 + 2b14 + 3b13) q^71 + (-4b12 - b9) q^73 + (b14 + 3b13 + 5b7) * q^74 + (b12 - b9 + b8 - 3b2) q^76 + (b6 + b4 + 2b1 - 4) q^79 + (5b11 + 5b5 - 6b3) q^80 + (-4b12 + 6b9 + b8 - 2b2) q^82 + (2b11 + 2b10 + 2b5) q^83 + (-2b6 + 3b4 - 3b1 - 2) q^85 + (-b15 - 3b14 - b13 - 3b7) * q^86 + (b6 + 2b4 - b1 + 7) q^88 + (2b11 + b10 + b5 - 3b3) * q^89 + (-b15 + b13 + 7b7) q^92 + (-b12 + 4b9 + 2b8 + 5b2) q^94 + (-b15 - 5b14 - 2b13 - 3b7) q^95 + (-2b9 + 2b8) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{4}+O(q^{10}) $ 16 * q - 16 * q^4	$ 16 q - 16 q^{4} + 48 q^{16} + 16 q^{22} + 32 q^{25} - 16 q^{37} + 32 q^{43} - 80 q^{46} - 96 q^{58} - 176 q^{64} + 96 q^{67} - 64 q^{79} - 32 q^{85} + 112 q^{88}+O(q^{100}) $ 16 * q - 16 * q^4 + 48 * q^16 + 16 * q^22 + 32 * q^25 - 16 * q^37 + 32 * q^43 - 80 * q^46 - 96 * q^58 - 176 * q^64 + 96 * q^67 - 64 * q^79 - 32 * q^85 + 112 * q^88

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 12x^{14} + 106x^{12} - 384x^{10} + 1005x^{8} - 1200x^{6} + 1030x^{4} - 252x^{2} + 49 $ x^16 - 12*x^14 + 106*x^12 - 384*x^10 + 1005*x^8 - 1200*x^6 + 1030*x^4 - 252*x^2 + 49 :

$\beta_{1}$	$=$	$ ( 878446 \nu^{14} - 9796819 \nu^{12} + 83290551 \nu^{10} - 251050620 \nu^{8} + 538652244 \nu^{6} + \cdots - 272833414 ) / 198602691 $ (878446v^14 - 9796819v^12 + 83290551v^10 - 251050620v^8 + 538652244v^6 - 279900636v^4 + 68773075*v^2 - 272833414) / 198602691
$\beta_{2}$	$=$	$ ( 12378 \nu^{14} - 136713 \nu^{12} + 1187651 \nu^{10} - 3705756 \nu^{8} + 9677932 \nu^{6} + \cdots - 1678362 ) / 2797221 $ (12378v^14 - 136713v^12 + 1187651v^10 - 3705756v^8 + 9677932v^6 - 9033702v^4 + 11793401*v^2 - 1678362) / 2797221
$\beta_{3}$	$=$	$ ( 514545 \nu^{15} - 6964488 \nu^{13} + 63470648 \nu^{11} - 275969081 \nu^{9} + \cdots - 498124634 \nu ) / 198602691 $ (514545v^15 - 6964488v^13 + 63470648v^11 - 275969081v^9 + 775230370v^7 - 1299818308v^5 + 1148028542v^3 - 498124634v) / 198602691
$\beta_{4}$	$=$	$ ( - 5624 \nu^{14} + 62439 \nu^{12} - 533244 \nu^{10} + 1607280 \nu^{8} - 3592238 \nu^{6} + \cdots - 1259979 ) / 932407 $ (-5624v^14 + 62439v^12 - 533244v^10 + 1607280v^8 - 3592238v^6 + 1791984v^4 - 440300*v^2 - 1259979) / 932407
$\beta_{5}$	$=$	$ ( 878642 \nu^{15} - 9751721 \nu^{13} + 83806886 \nu^{11} - 257079648 \nu^{9} + \cdots - 195998558 \nu ) / 198602691 $ (878642v^15 - 9751721v^13 + 83806886v^11 - 257079648v^9 + 612892708v^7 - 460646739v^5 + 453052273v^3 - 195998558v) / 198602691
$\beta_{6}$	$=$	$ ( 1711368 \nu^{14} - 18874018 \nu^{12} + 162264708 \nu^{10} - 489090960 \nu^{8} + 1147767525 \nu^{6} + \cdots + 618727571 ) / 198602691 $ (1711368v^14 - 18874018v^12 + 162264708v^10 - 489090960v^8 + 1147767525v^6 - 545295888v^4 + 133982100*v^2 + 618727571) / 198602691
$\beta_{7}$	$=$	$ ( - 1114268 \nu^{15} + 13051750 \nu^{13} - 114609720 \nu^{11} + 397588491 \nu^{9} + \cdots + 57182020 \nu ) / 198602691 $ (-1114268v^15 + 13051750v^13 - 114609720v^11 + 397588491v^9 - 1028539320v^7 + 1110627150v^5 - 1045904084v^3 + 57182020v) / 198602691
$\beta_{8}$	$=$	$ ( - 2228536 \nu^{14} + 26103500 \nu^{12} - 229219440 \nu^{10} + 795176982 \nu^{8} + \cdots + 312966731 ) / 198602691 $ (-2228536v^14 + 26103500v^12 - 229219440v^10 + 795176982v^8 - 2057078640v^6 + 2221254300v^4 - 2091808168*v^2 + 312966731) / 198602691
$\beta_{9}$	$=$	$ ( - 38598 \nu^{14} + 462014 \nu^{12} - 4073324 \nu^{10} + 14662382 \nu^{8} - 38003215 \nu^{6} + \cdots + 5317207 ) / 2797221 $ (-38598v^14 + 462014v^12 - 4073324v^10 + 14662382v^8 - 38003215v^6 + 44294824v^4 - 34225904*v^2 + 5317207) / 2797221
$\beta_{10}$	$=$	$ ( 1696759 \nu^{15} - 22910949 \nu^{13} + 209908028 \nu^{11} - 912484061 \nu^{9} + \cdots - 1618480304 \nu ) / 198602691 $ (1696759v^15 - 22910949v^13 + 209908028v^11 - 912484061v^9 + 2597720626v^7 - 4156802779v^5 + 3723863718v^3 - 1618480304v) / 198602691
$\beta_{11}$	$=$	$ ( - 1992910 \nu^{15} + 22803471 \nu^{13} - 198416606 \nu^{11} + 654668139 \nu^{9} + \cdots + 650385960 \nu ) / 198602691 $ (-1992910v^15 + 22803471v^13 - 198416606v^11 + 654668139v^9 - 1641432028v^7 + 1571273889v^5 - 1498956357v^3 + 650385960v) / 198602691
$\beta_{12}$	$=$	$ ( -28\nu^{14} + 343\nu^{12} - 3036\nu^{10} + 11298\nu^{8} - 29154\nu^{6} + 34818\nu^{4} - 25084\nu^{2} + 3955 ) / 1491 $ (-28v^14 + 343v^12 - 3036v^10 + 11298v^8 - 29154v^6 + 34818v^4 - 25084*v^2 + 3955) / 1491
$\beta_{13}$	$=$	$ ( 3578038 \nu^{15} - 42545475 \nu^{13} + 373599324 \nu^{11} - 1321216260 \nu^{9} + \cdots - 186399234 \nu ) / 198602691 $ (3578038v^15 - 42545475v^13 + 373599324v^11 - 1321216260v^9 + 3352783644v^7 - 3620369655v^5 + 2763402976v^3 - 186399234v) / 198602691
$\beta_{14}$	$=$	$ ( 8544409 \nu^{15} - 101186975 \nu^{13} + 888540684 \nu^{11} - 3127305423 \nu^{9} + \cdots - 443317994 \nu ) / 198602691 $ (8544409v^15 - 101186975v^13 + 888540684v^11 - 3127305423v^9 + 7974009804v^7 - 8610416355v^5 + 6475452802v^3 - 443317994v) / 198602691
$\beta_{15}$	$=$	$ ( - 11651979 \nu^{15} + 136952000 \nu^{13} - 1202599680 \nu^{11} + 4191620109 \nu^{9} + \cdots + 600010880 \nu ) / 198602691 $ (-11651979v^15 + 136952000v^13 - 1202599680v^11 + 4191620109v^9 - 10792462080v^7 + 11653809600v^5 - 9788871639v^3 + 600010880v) / 198602691

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

$\nu$	$=$	$ ( \beta_{11} - \beta_{7} + \beta_{5} ) / 2 $ (b11 - b7 + b5) / 2
$\nu^{2}$	$=$	$ ( \beta_{12} - 3\beta_{8} + \beta_{4} - 3\beta_{2} + \beta _1 + 3 ) / 2 $ (b12 - 3b8 + b4 - 3b2 + b1 + 3) / 2
$\nu^{3}$	$=$	$ \beta_{15} + \beta_{14} - \beta_{13} - 6\beta_{7} $ b15 + b14 - b13 - 6*b7
$\nu^{4}$	$=$	$ ( 9\beta_{12} - 6\beta_{9} - 17\beta_{8} - 2\beta_{6} - 9\beta_{4} - 24\beta_{2} - 8\beta _1 - 17 ) / 2 $ (9b12 - 6b9 - 17b8 - 2b6 - 9b4 - 24b2 - 8*b1 - 17) / 2
$\nu^{5}$	$=$	$ ( 8\beta_{15} + 10\beta_{14} - 11\beta_{13} - 26\beta_{11} - 9\beta_{10} - 42\beta_{7} - 50\beta_{5} + 15\beta_{3} ) / 2 $ (8b15 + 10b14 - 11b13 - 26b11 - 9b10 - 42b7 - 50b5 + 15b3) / 2
$\nu^{6}$	$=$	$ -21\beta_{6} - 74\beta_{4} - 60\beta _1 - 117 $ -21b6 - 74b4 - 60*b1 - 117
$\nu^{7}$	$=$	$ ( - 60 \beta_{15} - 81 \beta_{14} + 95 \beta_{13} - 191 \beta_{11} - 74 \beta_{10} + 311 \beta_{7} + \cdots + 137 \beta_{3} ) / 2 $ (-60b15 - 81b14 + 95b13 - 191b11 - 74b10 + 311b7 - 371b5 + 137b3) / 2
$\nu^{8}$	$=$	$ ( -582\beta_{12} + 528\beta_{9} + 859\beta_{8} - 176\beta_{6} - 582\beta_{4} + 1356\beta_{2} - 452\beta _1 - 859 ) / 2 $ (-582b12 + 528b9 + 859b8 - 176b6 - 582b4 + 1356b2 - 452*b1 - 859) / 2
$\nu^{9}$	$=$	$ -452\beta_{15} - 628\beta_{14} + 758\beta_{13} + 2345\beta_{7} $ -452b15 - 628b14 + 758b13 + 2345b7
$\nu^{10}$	$=$	$ ( - 4489 \beta_{12} + 4158 \beta_{9} + 6453 \beta_{8} + 1386 \beta_{6} + 4489 \beta_{4} + 10275 \beta_{2} + \cdots + 6453 ) / 2 $ (-4489b12 + 4158b9 + 6453b8 + 1386b6 + 4489b4 + 10275b2 + 3425*b1 + 6453) / 2
$\nu^{11}$	$=$	$ ( - 3425 \beta_{15} - 4811 \beta_{14} + 5875 \beta_{13} + 10942 \beta_{11} + 4489 \beta_{10} + \cdots - 8647 \beta_{3} ) / 2 $ (-3425b15 - 4811b14 + 5875b13 + 10942b11 + 4489b10 + 17792b7 + 21217b5 - 8647b3) / 2
$\nu^{12}$	$=$	$ 10686\beta_{6} + 34353\beta_{4} + 26028\beta _1 + 48887 $ 10686b6 + 34353b4 + 26028*b1 + 48887
$\nu^{13}$	$=$	$ ( 26028 \beta_{15} + 36714 \beta_{14} - 45039 \beta_{13} + 83240 \beta_{11} + 34353 \beta_{10} + \cdots - 66411 \beta_{3} ) / 2 $ (26028b15 + 36714b14 - 45039b13 + 83240b11 + 34353b10 - 135296b7 + 161324b5 - 66411b3) / 2
$\nu^{14}$	$=$	$ ( 262088 \beta_{12} - 245259 \beta_{9} - 371535 \beta_{8} + 81753 \beta_{6} + 262088 \beta_{4} + \cdots + 371535 ) / 2 $ (262088b12 - 245259b9 - 371535b8 + 81753b6 + 262088b4 - 594114b2 + 198038*b1 + 371535) / 2
$\nu^{15}$	$=$	$ 198038\beta_{15} + 279791\beta_{14} - 343841\beta_{13} - 1029699\beta_{7} $ 198038b15 + 279791b14 - 343841b13 - 1029699b7

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1323\mathbb{Z}\right)^\times$.

$n$	$785$	$1081$
$\chi(n)$	$-1$	$-1$

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} $

1322.1

−2.38976	−	1.37973i
2.38976	−	1.37973i
−1.47769	−	0.853147i
1.47769	−	0.853147i
0.954123	−	0.550863i
−0.954123	−	0.550863i
−0.441700	−	0.255016i
0.441700	−	0.255016i
−0.441700	+	0.255016i
0.441700	+	0.255016i
0.954123	+	0.550863i
−0.954123	+	0.550863i
−1.47769	+	0.853147i
1.47769	+	0.853147i
−2.38976	+	1.37973i
2.38976	+	1.37973i

−

2.75946i

−5.61463

−2.24426

9.97442i

6.19294i

1322.2

−

2.75946i

−5.61463

2.24426

9.97442i

−

6.19294i

1322.3

−

1.70629i

−0.911441

−0.829297

−

1.85740i

1.41503i

1322.4

−

1.70629i

−0.911441

0.829297

−

1.85740i

−

1.41503i

1322.5

−

1.10173i

0.786199

−2.47687

−

3.06963i

2.72883i

1322.6

−

1.10173i

0.786199

2.47687

−

3.06963i

−

2.72883i

1322.7

−

0.510032i

1.73987

−4.01755

−

1.90745i

2.04908i

1322.8

−

0.510032i

1.73987

4.01755

−

1.90745i

−

2.04908i

1322.9

0.510032i

1.73987

−4.01755

1.90745i

−

2.04908i

1322.10

0.510032i

1.73987

4.01755

1.90745i

2.04908i

1322.11

1.10173i

0.786199

−2.47687

3.06963i

−

2.72883i

1322.12

1.10173i

0.786199

2.47687

3.06963i

2.72883i

1322.13

1.70629i

−0.911441

−0.829297

1.85740i

−

1.41503i

1322.14

1.70629i

−0.911441

0.829297

1.85740i

1.41503i

1322.15

2.75946i

−5.61463

−2.24426

−

9.97442i

−

6.19294i

1322.16

2.75946i

−5.61463

2.24426

−

9.97442i

6.19294i

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	1323.2.c.e	✓	16
3.b	odd	2	1	inner	1323.2.c.e	✓	16
7.b	odd	2	1	inner	1323.2.c.e	✓	16
21.c	even	2	1	inner	1323.2.c.e	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1323.2.c.e	✓	16	1.a	even	1	1	trivial
1323.2.c.e	✓	16	3.b	odd	2	1	inner
1323.2.c.e	✓	16	7.b	odd	2	1	inner
1323.2.c.e	✓	16	21.c	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{8} + 12T_{2}^{6} + 38T_{2}^{4} + 36T_{2}^{2} + 7 $ T2^8 + 12*T2^6 + 38*T2^4 + 36*T2^2 + 7 acting on $S_{2}^{\mathrm{new}}(1323, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{8} + 12 T^{6} + 38 T^{4} + \cdots + 7)^{2} $ (T^8 + 12T^6 + 38T^4 + 36*T^2 + 7)^2
$3$	$ T^{16} $ T^16
$5$	$ (T^{8} - 28 T^{6} + \cdots + 343)^{2} $ (T^8 - 28T^6 + 230T^4 - 644*T^2 + 343)^2
$7$	$ T^{16} $ T^16
$11$	$ (T^{8} + 36 T^{6} + 326 T^{4} + \cdots + 7)^{2} $ (T^8 + 36T^6 + 326T^4 + 204*T^2 + 7)^2
$13$	$ (T^{8} + 40 T^{6} + \cdots + 16)^{2} $ (T^8 + 40T^6 + 392T^4 + 608*T^2 + 16)^2
$17$	$ (T^{8} - 80 T^{6} + \cdots + 67228)^{2} $ (T^8 - 80T^6 + 2180T^4 - 22960*T^2 + 67228)^2
$19$	$ (T^{8} + 76 T^{6} + \cdots + 9409)^{2} $ (T^8 + 76T^6 + 1442T^4 + 8084*T^2 + 9409)^2
$23$	$ (T^{8} + 124 T^{6} + \cdots + 55447)^{2} $ (T^8 + 124T^6 + 4122T^4 + 40420*T^2 + 55447)^2
$29$	$ (T^{8} + 168 T^{6} + \cdots + 887152)^{2} $ (T^8 + 168T^6 + 8936T^4 + 159840*T^2 + 887152)^2
$31$	$ (T^{8} + 148 T^{6} + \cdots + 1681)^{2} $ (T^8 + 148T^6 + 6710T^4 + 94388*T^2 + 1681)^2
$37$	$ (T^{4} + 4 T^{3} + \cdots + 487)^{4} $ (T^4 + 4T^3 - 70T^2 - 244*T + 487)^4
$41$	$ (T^{8} - 340 T^{6} + \cdots + 22104103)^{2} $ (T^8 - 340T^6 + 39290T^4 - 1755260*T^2 + 22104103)^2
$43$	$ (T^{4} - 8 T^{3} - 72 T^{2} + \cdots - 98)^{4} $ (T^4 - 8T^3 - 72T^2 + 280*T - 98)^4
$47$	$ (T^{8} - 160 T^{6} + \cdots + 263452)^{2} $ (T^8 - 160T^6 + 8036T^4 - 135152*T^2 + 263452)^2
$53$	$ (T^{8} + 200 T^{6} + \cdots + 4171888)^{2} $ (T^8 + 200T^6 + 14232T^4 + 419168*T^2 + 4171888)^2
$59$	$ (T^{8} - 280 T^{6} + \cdots + 3294172)^{2} $ (T^8 - 280T^6 + 20924T^4 - 519104*T^2 + 3294172)^2
$61$	$ (T^{8} + 232 T^{6} + \cdots + 2972176)^{2} $ (T^8 + 232T^6 + 16904T^4 + 437600*T^2 + 2972176)^2
$67$	$ (T^{4} - 24 T^{3} + \cdots - 3708)^{4} $ (T^4 - 24T^3 + 132T^2 + 432*T - 3708)^4
$71$	$ (T^{8} + 276 T^{6} + \cdots + 15463)^{2} $ (T^8 + 276T^6 + 19382T^4 + 171756*T^2 + 15463)^2
$73$	$ (T^{8} + 392 T^{6} + \cdots + 78039556)^{2} $ (T^8 + 392T^6 + 56084T^4 + 3466000*T^2 + 78039556)^2
$79$	$ (T^{4} + 16 T^{3} + \cdots - 1538)^{4} $ (T^4 + 16T^3 + 32T^2 - 424*T - 1538)^4
$83$	$ (T^{8} - 160 T^{6} + \cdots + 351232)^{2} $ (T^8 - 160T^6 + 7616T^4 - 112640*T^2 + 351232)^2
$89$	$ (T^{8} - 220 T^{6} + \cdots + 794983)^{2} $ (T^8 - 220T^6 + 14378T^4 - 267620*T^2 + 794983)^2
$97$	$ (T^{8} + 80 T^{6} + \cdots + 256)^{2} $ (T^8 + 80T^6 + 1568T^4 + 4864*T^2 + 256)^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:	\( N \)	\(=\)	\( 1323 = 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1323.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{7} q^{2} + ( - \beta_{4} - \beta_1 - 1) q^{4} + ( - \beta_{10} - \beta_{3}) q^{5} + ( - \beta_{15} - \beta_{14} + \cdots + 2 \beta_{7}) q^{8}+O(q^{10}) \) q - b7 * q^2 + (-b4 - b1 - 1) * q^4 + (-b10 - b3) * q^5 + (-b15 - b14 + b13 + 2b7) q^8	\( q - \beta_{7} q^{2} + ( - \beta_{4} - \beta_1 - 1) q^{4} + ( - \beta_{10} - \beta_{3}) q^{5} + ( - \beta_{15} - \beta_{14} + \cdots + 2 \beta_{7}) q^{8}+ \cdots + ( - 2 \beta_{9} + 2 \beta_{8}) q^{97}+O(q^{100}) \) q - b7 * q^2 + (-b4 - b1 - 1) * q^4 + (-b10 - b3) * q^5 + (-b15 - b14 + b13 + 2b7) q^8 + (-b9 - b8 - 2b2) q^10 + b15 * q^11 + (b12 - b9 - b2) * q^13 + (2b6 + 3b4 + 2b1 + 3) q^16 + (-2b11 - b5) q^17 + (-b12 + b9 + b8 - 2b2) q^19 + (-b11 - 2b10 - 3b5 - b3) * q^20 + (-b1 + 1) * q^22 + (b15 + b14 + 2b13 - b7) q^23 + (2b6 - b4 - b1 + 2) q^25 + (-b11 - b10 - 2b5 + 2b3) * q^26 + (b15 - 2b14 - 3b7) * q^29 + (-b12 - b9 - 3b8 - b2) q^31 + (2b14 - 3b13 - 6b7) q^32 + (-b12 - b9 + 4b8 + 2b2) * q^34 + (b6 - 4b4 - 1) q^37 + (2b11 + b10 - 2b3) * q^38 + (-4b12 + 3b8 + 4b2) q^40 + (-3b11 + b5 + 5b3) * q^41 + (-2b6 + 3b4 - b1 + 2) * q^43 + (b15 - b14 + b7) * q^44 + (b6 + 2b4 - 3b1 - 5) * q^46 + (2b10 + 3b3) * q^47 + (-b15 + b14 - b13 + b7) * q^50 + (-3b12 + 4b9 + 4b8 + 6b2) * q^52 + (2b14 + 3b13 + 3b7) q^53 + (3b12 + 5b9 + b8) * q^55 + (2b6 - b4 - 2b1 - 6) * q^58 + (-b11 + b10 - b5 + 4b3) q^59 + (b12 - 5b9 - b2) q^61 + (-2b11 + b10 - 3b5) * q^62 + (-b6 - 8b4 - 4b1 - 11) * q^64 + (b15 - 2b13 - 3b7) * q^65 + (2b6 + 3b4 + 6) * q^67 + (b11 + b10 + 5b5) q^68 + (2b15 + 2b14 + 3b13) q^71 + (-4b12 - b9) q^73 + (b14 + 3b13 + 5b7) * q^74 + (b12 - b9 + b8 - 3b2) q^76 + (b6 + b4 + 2b1 - 4) q^79 + (5b11 + 5b5 - 6b3) q^80 + (-4b12 + 6b9 + b8 - 2b2) q^82 + (2b11 + 2b10 + 2b5) q^83 + (-2b6 + 3b4 - 3b1 - 2) q^85 + (-b15 - 3b14 - b13 - 3b7) * q^86 + (b6 + 2b4 - b1 + 7) q^88 + (2b11 + b10 + b5 - 3b3) * q^89 + (-b15 + b13 + 7b7) q^92 + (-b12 + 4b9 + 2b8 + 5b2) q^94 + (-b15 - 5b14 - 2b13 - 3b7) q^95 + (-2b9 + 2b8) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{4}+O(q^{10}) \) 16 * q - 16 * q^4	\( 16 q - 16 q^{4} + 48 q^{16} + 16 q^{22} + 32 q^{25} - 16 q^{37} + 32 q^{43} - 80 q^{46} - 96 q^{58} - 176 q^{64} + 96 q^{67} - 64 q^{79} - 32 q^{85} + 112 q^{88}+O(q^{100}) \) 16 * q - 16 * q^4 + 48 * q^16 + 16 * q^22 + 32 * q^25 - 16 * q^37 + 32 * q^43 - 80 * q^46 - 96 * q^58 - 176 * q^64 + 96 * q^67 - 64 * q^79 - 32 * q^85 + 112 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

Properties

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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	1323.2.c.e

Level	$1323$
Weight	$2$
Character orbit	1323.c
Analytic conductor	$10.564$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(10.5642081874\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 12x^{14} + 106x^{12} - 384x^{10} + 1005x^{8} - 1200x^{6} + 1030x^{4} - 252x^{2} + 49 \) x^16 - 12x^14 + 106x^12 - 384x^10 + 1005x^8 - 1200x^6 + 1030x^4 - 252*x^2 + 49
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{8} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( 878446 \nu^{14} - 9796819 \nu^{12} + 83290551 \nu^{10} - 251050620 \nu^{8} + 538652244 \nu^{6} + \cdots - 272833414 ) / 198602691 \) (878446v^14 - 9796819v^12 + 83290551v^10 - 251050620v^8 + 538652244v^6 - 279900636v^4 + 68773075*v^2 - 272833414) / 198602691
\(\beta_{2}\)	\(=\)	\( ( 12378 \nu^{14} - 136713 \nu^{12} + 1187651 \nu^{10} - 3705756 \nu^{8} + 9677932 \nu^{6} + \cdots - 1678362 ) / 2797221 \) (12378v^14 - 136713v^12 + 1187651v^10 - 3705756v^8 + 9677932v^6 - 9033702v^4 + 11793401*v^2 - 1678362) / 2797221
\(\beta_{3}\)	\(=\)	\( ( 514545 \nu^{15} - 6964488 \nu^{13} + 63470648 \nu^{11} - 275969081 \nu^{9} + \cdots - 498124634 \nu ) / 198602691 \) (514545v^15 - 6964488v^13 + 63470648v^11 - 275969081v^9 + 775230370v^7 - 1299818308v^5 + 1148028542v^3 - 498124634v) / 198602691
\(\beta_{4}\)	\(=\)	\( ( - 5624 \nu^{14} + 62439 \nu^{12} - 533244 \nu^{10} + 1607280 \nu^{8} - 3592238 \nu^{6} + \cdots - 1259979 ) / 932407 \) (-5624v^14 + 62439v^12 - 533244v^10 + 1607280v^8 - 3592238v^6 + 1791984v^4 - 440300*v^2 - 1259979) / 932407
\(\beta_{5}\)	\(=\)	\( ( 878642 \nu^{15} - 9751721 \nu^{13} + 83806886 \nu^{11} - 257079648 \nu^{9} + \cdots - 195998558 \nu ) / 198602691 \) (878642v^15 - 9751721v^13 + 83806886v^11 - 257079648v^9 + 612892708v^7 - 460646739v^5 + 453052273v^3 - 195998558v) / 198602691
\(\beta_{6}\)	\(=\)	\( ( 1711368 \nu^{14} - 18874018 \nu^{12} + 162264708 \nu^{10} - 489090960 \nu^{8} + 1147767525 \nu^{6} + \cdots + 618727571 ) / 198602691 \) (1711368v^14 - 18874018v^12 + 162264708v^10 - 489090960v^8 + 1147767525v^6 - 545295888v^4 + 133982100*v^2 + 618727571) / 198602691
\(\beta_{7}\)	\(=\)	\( ( - 1114268 \nu^{15} + 13051750 \nu^{13} - 114609720 \nu^{11} + 397588491 \nu^{9} + \cdots + 57182020 \nu ) / 198602691 \) (-1114268v^15 + 13051750v^13 - 114609720v^11 + 397588491v^9 - 1028539320v^7 + 1110627150v^5 - 1045904084v^3 + 57182020v) / 198602691
\(\beta_{8}\)	\(=\)	\( ( - 2228536 \nu^{14} + 26103500 \nu^{12} - 229219440 \nu^{10} + 795176982 \nu^{8} + \cdots + 312966731 ) / 198602691 \) (-2228536v^14 + 26103500v^12 - 229219440v^10 + 795176982v^8 - 2057078640v^6 + 2221254300v^4 - 2091808168*v^2 + 312966731) / 198602691
\(\beta_{9}\)	\(=\)	\( ( - 38598 \nu^{14} + 462014 \nu^{12} - 4073324 \nu^{10} + 14662382 \nu^{8} - 38003215 \nu^{6} + \cdots + 5317207 ) / 2797221 \) (-38598v^14 + 462014v^12 - 4073324v^10 + 14662382v^8 - 38003215v^6 + 44294824v^4 - 34225904*v^2 + 5317207) / 2797221
\(\beta_{10}\)	\(=\)	\( ( 1696759 \nu^{15} - 22910949 \nu^{13} + 209908028 \nu^{11} - 912484061 \nu^{9} + \cdots - 1618480304 \nu ) / 198602691 \) (1696759v^15 - 22910949v^13 + 209908028v^11 - 912484061v^9 + 2597720626v^7 - 4156802779v^5 + 3723863718v^3 - 1618480304v) / 198602691
\(\beta_{11}\)	\(=\)	\( ( - 1992910 \nu^{15} + 22803471 \nu^{13} - 198416606 \nu^{11} + 654668139 \nu^{9} + \cdots + 650385960 \nu ) / 198602691 \) (-1992910v^15 + 22803471v^13 - 198416606v^11 + 654668139v^9 - 1641432028v^7 + 1571273889v^5 - 1498956357v^3 + 650385960v) / 198602691
\(\beta_{12}\)	\(=\)	\( ( -28\nu^{14} + 343\nu^{12} - 3036\nu^{10} + 11298\nu^{8} - 29154\nu^{6} + 34818\nu^{4} - 25084\nu^{2} + 3955 ) / 1491 \) (-28v^14 + 343v^12 - 3036v^10 + 11298v^8 - 29154v^6 + 34818v^4 - 25084*v^2 + 3955) / 1491
\(\beta_{13}\)	\(=\)	\( ( 3578038 \nu^{15} - 42545475 \nu^{13} + 373599324 \nu^{11} - 1321216260 \nu^{9} + \cdots - 186399234 \nu ) / 198602691 \) (3578038v^15 - 42545475v^13 + 373599324v^11 - 1321216260v^9 + 3352783644v^7 - 3620369655v^5 + 2763402976v^3 - 186399234v) / 198602691
\(\beta_{14}\)	\(=\)	\( ( 8544409 \nu^{15} - 101186975 \nu^{13} + 888540684 \nu^{11} - 3127305423 \nu^{9} + \cdots - 443317994 \nu ) / 198602691 \) (8544409v^15 - 101186975v^13 + 888540684v^11 - 3127305423v^9 + 7974009804v^7 - 8610416355v^5 + 6475452802v^3 - 443317994v) / 198602691
\(\beta_{15}\)	\(=\)	\( ( - 11651979 \nu^{15} + 136952000 \nu^{13} - 1202599680 \nu^{11} + 4191620109 \nu^{9} + \cdots + 600010880 \nu ) / 198602691 \) (-11651979v^15 + 136952000v^13 - 1202599680v^11 + 4191620109v^9 - 10792462080v^7 + 11653809600v^5 - 9788871639v^3 + 600010880v) / 198602691

\(\nu\)	\(=\)	\( ( \beta_{11} - \beta_{7} + \beta_{5} ) / 2 \) (b11 - b7 + b5) / 2
\(\nu^{2}\)	\(=\)	\( ( \beta_{12} - 3\beta_{8} + \beta_{4} - 3\beta_{2} + \beta _1 + 3 ) / 2 \) (b12 - 3b8 + b4 - 3b2 + b1 + 3) / 2
\(\nu^{3}\)	\(=\)	\( \beta_{15} + \beta_{14} - \beta_{13} - 6\beta_{7} \) b15 + b14 - b13 - 6*b7
\(\nu^{4}\)	\(=\)	\( ( 9\beta_{12} - 6\beta_{9} - 17\beta_{8} - 2\beta_{6} - 9\beta_{4} - 24\beta_{2} - 8\beta _1 - 17 ) / 2 \) (9b12 - 6b9 - 17b8 - 2b6 - 9b4 - 24b2 - 8*b1 - 17) / 2
\(\nu^{5}\)	\(=\)	\( ( 8\beta_{15} + 10\beta_{14} - 11\beta_{13} - 26\beta_{11} - 9\beta_{10} - 42\beta_{7} - 50\beta_{5} + 15\beta_{3} ) / 2 \) (8b15 + 10b14 - 11b13 - 26b11 - 9b10 - 42b7 - 50b5 + 15b3) / 2
\(\nu^{6}\)	\(=\)	\( -21\beta_{6} - 74\beta_{4} - 60\beta _1 - 117 \) -21b6 - 74b4 - 60*b1 - 117
\(\nu^{7}\)	\(=\)	\( ( - 60 \beta_{15} - 81 \beta_{14} + 95 \beta_{13} - 191 \beta_{11} - 74 \beta_{10} + 311 \beta_{7} + \cdots + 137 \beta_{3} ) / 2 \) (-60b15 - 81b14 + 95b13 - 191b11 - 74b10 + 311b7 - 371b5 + 137b3) / 2
\(\nu^{8}\)	\(=\)	\( ( -582\beta_{12} + 528\beta_{9} + 859\beta_{8} - 176\beta_{6} - 582\beta_{4} + 1356\beta_{2} - 452\beta _1 - 859 ) / 2 \) (-582b12 + 528b9 + 859b8 - 176b6 - 582b4 + 1356b2 - 452*b1 - 859) / 2
\(\nu^{9}\)	\(=\)	\( -452\beta_{15} - 628\beta_{14} + 758\beta_{13} + 2345\beta_{7} \) -452b15 - 628b14 + 758b13 + 2345b7
\(\nu^{10}\)	\(=\)	\( ( - 4489 \beta_{12} + 4158 \beta_{9} + 6453 \beta_{8} + 1386 \beta_{6} + 4489 \beta_{4} + 10275 \beta_{2} + \cdots + 6453 ) / 2 \) (-4489b12 + 4158b9 + 6453b8 + 1386b6 + 4489b4 + 10275b2 + 3425*b1 + 6453) / 2
\(\nu^{11}\)	\(=\)	\( ( - 3425 \beta_{15} - 4811 \beta_{14} + 5875 \beta_{13} + 10942 \beta_{11} + 4489 \beta_{10} + \cdots - 8647 \beta_{3} ) / 2 \) (-3425b15 - 4811b14 + 5875b13 + 10942b11 + 4489b10 + 17792b7 + 21217b5 - 8647b3) / 2
\(\nu^{12}\)	\(=\)	\( 10686\beta_{6} + 34353\beta_{4} + 26028\beta _1 + 48887 \) 10686b6 + 34353b4 + 26028*b1 + 48887
\(\nu^{13}\)	\(=\)	\( ( 26028 \beta_{15} + 36714 \beta_{14} - 45039 \beta_{13} + 83240 \beta_{11} + 34353 \beta_{10} + \cdots - 66411 \beta_{3} ) / 2 \) (26028b15 + 36714b14 - 45039b13 + 83240b11 + 34353b10 - 135296b7 + 161324b5 - 66411b3) / 2
\(\nu^{14}\)	\(=\)	\( ( 262088 \beta_{12} - 245259 \beta_{9} - 371535 \beta_{8} + 81753 \beta_{6} + 262088 \beta_{4} + \cdots + 371535 ) / 2 \) (262088b12 - 245259b9 - 371535b8 + 81753b6 + 262088b4 - 594114b2 + 198038*b1 + 371535) / 2
\(\nu^{15}\)	\(=\)	\( 198038\beta_{15} + 279791\beta_{14} - 343841\beta_{13} - 1029699\beta_{7} \) 198038b15 + 279791b14 - 343841b13 - 1029699b7

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1322.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{8} + 12 T^{6} + 38 T^{4} + \cdots + 7)^{2} \) (T^8 + 12T^6 + 38T^4 + 36*T^2 + 7)^2
$3$	\( T^{16} \) T^16
$5$	\( (T^{8} - 28 T^{6} + \cdots + 343)^{2} \) (T^8 - 28T^6 + 230T^4 - 644*T^2 + 343)^2
$7$	\( T^{16} \) T^16
$11$	\( (T^{8} + 36 T^{6} + 326 T^{4} + \cdots + 7)^{2} \) (T^8 + 36T^6 + 326T^4 + 204*T^2 + 7)^2
$13$	\( (T^{8} + 40 T^{6} + \cdots + 16)^{2} \) (T^8 + 40T^6 + 392T^4 + 608*T^2 + 16)^2
$17$	\( (T^{8} - 80 T^{6} + \cdots + 67228)^{2} \) (T^8 - 80T^6 + 2180T^4 - 22960*T^2 + 67228)^2
$19$	\( (T^{8} + 76 T^{6} + \cdots + 9409)^{2} \) (T^8 + 76T^6 + 1442T^4 + 8084*T^2 + 9409)^2
$23$	\( (T^{8} + 124 T^{6} + \cdots + 55447)^{2} \) (T^8 + 124T^6 + 4122T^4 + 40420*T^2 + 55447)^2
$29$	\( (T^{8} + 168 T^{6} + \cdots + 887152)^{2} \) (T^8 + 168T^6 + 8936T^4 + 159840*T^2 + 887152)^2
$31$	\( (T^{8} + 148 T^{6} + \cdots + 1681)^{2} \) (T^8 + 148T^6 + 6710T^4 + 94388*T^2 + 1681)^2
$37$	\( (T^{4} + 4 T^{3} + \cdots + 487)^{4} \) (T^4 + 4T^3 - 70T^2 - 244*T + 487)^4
$41$	\( (T^{8} - 340 T^{6} + \cdots + 22104103)^{2} \) (T^8 - 340T^6 + 39290T^4 - 1755260*T^2 + 22104103)^2
$43$	\( (T^{4} - 8 T^{3} - 72 T^{2} + \cdots - 98)^{4} \) (T^4 - 8T^3 - 72T^2 + 280*T - 98)^4
$47$	\( (T^{8} - 160 T^{6} + \cdots + 263452)^{2} \) (T^8 - 160T^6 + 8036T^4 - 135152*T^2 + 263452)^2
$53$	\( (T^{8} + 200 T^{6} + \cdots + 4171888)^{2} \) (T^8 + 200T^6 + 14232T^4 + 419168*T^2 + 4171888)^2
$59$	\( (T^{8} - 280 T^{6} + \cdots + 3294172)^{2} \) (T^8 - 280T^6 + 20924T^4 - 519104*T^2 + 3294172)^2
$61$	\( (T^{8} + 232 T^{6} + \cdots + 2972176)^{2} \) (T^8 + 232T^6 + 16904T^4 + 437600*T^2 + 2972176)^2
$67$	\( (T^{4} - 24 T^{3} + \cdots - 3708)^{4} \) (T^4 - 24T^3 + 132T^2 + 432*T - 3708)^4
$71$	\( (T^{8} + 276 T^{6} + \cdots + 15463)^{2} \) (T^8 + 276T^6 + 19382T^4 + 171756*T^2 + 15463)^2
$73$	\( (T^{8} + 392 T^{6} + \cdots + 78039556)^{2} \) (T^8 + 392T^6 + 56084T^4 + 3466000*T^2 + 78039556)^2
$79$	\( (T^{4} + 16 T^{3} + \cdots - 1538)^{4} \) (T^4 + 16T^3 + 32T^2 - 424*T - 1538)^4
$83$	\( (T^{8} - 160 T^{6} + \cdots + 351232)^{2} \) (T^8 - 160T^6 + 7616T^4 - 112640*T^2 + 351232)^2
$89$	\( (T^{8} - 220 T^{6} + \cdots + 794983)^{2} \) (T^8 - 220T^6 + 14378T^4 - 267620*T^2 + 794983)^2
$97$	\( (T^{8} + 80 T^{6} + \cdots + 256)^{2} \) (T^8 + 80T^6 + 1568T^4 + 4864*T^2 + 256)^2
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\(n\)	\(785\)	\(1081\)
\(\chi(n)\)	\(-1\)	\(-1\)