LMFDB - Newform orbit 525.3.l.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [525,3,Mod(43,525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(525, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("525.43");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 525 = 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	525.l (of order $4$, degree $2$, 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:	$14.3052138789$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
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} + 79x^{12} + 705x^{8} + 1264x^{4} + 256 $ x^16 + 79x^12 + 705x^8 + 1264*x^4 + 256
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-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_{9} - \beta_{5}) q^{2} + \beta_{13} q^{3} + (\beta_{12} + 2 \beta_{3}) q^{4} + (\beta_{8} - \beta_{7} - 2) q^{6} + ( - \beta_{5} + \beta_{2}) q^{7} + (2 \beta_{15} + \beta_{14} + \cdots - \beta_{11}) q^{8}+ \cdots - 3 \beta_{6} q^{9}+O(q^{10}) $ q + (b9 - b5) * q^2 + b13 * q^3 + (b12 + 2b3) q^4 + (b8 - b7 - 2) * q^6 + (-b5 + b2) * q^7 + (2b15 + b14 + 2b13 - b11) * q^8 - 3b6 q^9	$ q + (\beta_{9} - \beta_{5}) q^{2} + \beta_{13} q^{3} + (\beta_{12} + 2 \beta_{3}) q^{4} + (\beta_{8} - \beta_{7} - 2) q^{6} + ( - \beta_{5} + \beta_{2}) q^{7} + (2 \beta_{15} + \beta_{14} + \cdots - \beta_{11}) q^{8}+ \cdots + (3 \beta_{12} + 9 \beta_{6} + \cdots + 6 \beta_{3}) q^{99}+O(q^{100}) $ q + (b9 - b5) * q^2 + b13 * q^3 + (b12 + 2b3) q^4 + (b8 - b7 - 2) * q^6 + (-b5 + b2) * q^7 + (2b15 + b14 + 2b13 - b11) * q^8 - 3b6 q^9 + (b8 - 2b7 + 3b1 - 3) * q^11 + (b10 - 2b9 + 4b5) * q^12 + (5b15 + 3b14 + 2b13 - 2b11) * q^13 + (b6 - b4 + 2b3) q^14 + (-2b7 + 3b1 - 1) * q^16 + (-3b10 - 3b9 + 2b5 + b2) q^17 + (-3b14 - 3b13) * q^18 + (6b6 - 2b4 - 6b3) q^19 + (b8 - b7 + b1) * q^21 + (5b10 - 3b9 + 8b5 - 10b2) * q^22 + (-2b15 + b14 + 6b13 + b11) * q^23 + (b12 - 2b6 + b4 - 5b3) * q^24 + (7b8 - 22b7 + 7b1 + 2) q^26 + 3b9 q^27 + (3b15 + 3b14 + 3b13 - b11) q^28 + (3b12 - 2b6 + 5b4 + 11b3) * q^29 + (-b7 + 6b1 + 7) q^31 + (-3b10 + 9b9 + 3b5 - 7b2) * q^32 + (-b15 - 4b14 - 2b13 + 9b11) q^33 + (-17b6 - 4b4 + 5b3) q^34 + (-3b8 + 6b7) * q^36 + (-3b10 + 10b9 - 4b5 - 7b2) * q^37 + (-4b15 - 12b14 - 10b13) q^38 + (2b12 + 5b6 + 2b4 - 13b3) * q^39 + (11b8 - 16b7 + 11b1 - 6) q^41 + (2b10 - 2b9 + 2b5 - 3b2) * q^42 + (-2b15 - 24b14 + b13 - 2b11) q^43 + (-4b12 + 10b6 + 3b4 - 28b3) * q^44 + (4b8 - 3b1 - 19) * q^46 + (-14b10 + 6b9 + 2b5 + 6b2) * q^47 + (-2b15 - 2b14 + 9b11) q^48 + 7b6 q^49 + (-5b8 - 4b7 + b1 + 12) * q^51 + (9b10 - 34b9 + 45b5 - 28b2) * q^52 + (-7b15 - 9b14 - 24b13 + 12b11) * q^53 + (3b12 + 6b6 + 3b3) q^54 + (6b8 - 8b7 - 1) * q^56 + (-6b10 + 2b9 - 6b5 - 6b2) * q^57 + (6b15 + 28b14 + 36b13 + 7b11) * q^58 + (-13b12 - 38b6 + 3b4 + 4b3) * q^59 + (18b8 + 5b7 - 4b1 - 29) q^61 + (7b10 + 17b9 - 4b5 - 19b2) * q^62 + (-3b14 + 3b11) * q^63 + (12b12 + 17b6 - 8b4 + 16b3) * q^64 + (-3b8 + 18b7 - 10b1 + 1) q^66 + (-3b10 + 24b9 + 10b5 + 3b2) * q^67 + (-3b15 + 6b14 - 3b13 - 13b11) * q^68 + (-3b12 - 20b6 - b4 + 3b3) q^69 + (-5b8 + 26b7 + 13b1 + 11) q^71 + (-6b10 + 6b9 - 3b5 + 3b2) * q^72 + (10b15 - 28b14 - 14b13 + 12b11) * q^73 + (13b12 + 29b6 + 4b4 + 16b3) * q^74 + (-14b8 + 10b7 - 12b1 + 60) q^76 + (-3b10 - 18b9 + 6b5 - b2) q^77 + (-15b15 - 36b14 - 13b13 + 21b11) * q^78 + (-13b12 + 17b6 - 15b4 + 32b3) * q^79 - 9 * q^81 + (27b10 - 38b9 + 43b5 - 38b2) * q^82 + (b15 + 54b14 + 13b13 + 3b11) q^83 + (-b6 + b4 - 9b3) q^84 + (-b8 + 27b7 + 44) q^86 + (8b10 - 14b9 + 17b5 + 15b2) * q^87 + (-11b15 - 38b14 - 36b13 - 7b11) * q^88 + (-22b12 + 48b6 + 8b4 - 16b3) * q^89 + (11b8 - 18b7 - 3b1) q^91 + (5b10 - 9b9 + 11b5 + 9b2) * q^92 + (-b15 - b14 + 12b13 + 18b11) * q^93 + (20b12 - 26b6 - 20b4 + 52b3) * q^94 + (-6b8 - 3b7 - 7b1 - 17) q^96 + (-4b10 + 14b9 - 32b5 + 8b2) * q^97 + (7b14 + 7b13) * q^98 + (3b12 + 9b6 - 9b4 + 6b3) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 36 q^{6}+O(q^{10}) $ 16 * q - 36 * q^6	$ 16 q - 36 q^{6} - 48 q^{11} - 20 q^{16} - 88 q^{26} + 128 q^{31} + 36 q^{36} - 136 q^{41} - 300 q^{46} + 144 q^{51} - 56 q^{56} - 368 q^{61} + 108 q^{66} + 416 q^{71} + 936 q^{76} - 144 q^{81} + 916 q^{86} - 112 q^{91} - 348 q^{96}+O(q^{100}) $ 16 * q - 36 * q^6 - 48 * q^11 - 20 * q^16 - 88 * q^26 + 128 * q^31 + 36 * q^36 - 136 * q^41 - 300 * q^46 + 144 * q^51 - 56 * q^56 - 368 * q^61 + 108 * q^66 + 416 * q^71 + 936 * q^76 - 144 * q^81 + 916 * q^86 - 112 * q^91 - 348 * q^96

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} + 79x^{12} + 705x^{8} + 1264x^{4} + 256 $ x^16 + 79*x^12 + 705*x^8 + 1264*x^4 + 256 :

$\beta_{1}$	$=$	$ ( -5\nu^{12} - 267\nu^{8} + 4731\nu^{4} + 5648 ) / 7696 $ (-5v^12 - 267v^8 + 4731*v^4 + 5648) / 7696
$\beta_{2}$	$=$	$ ( -19\nu^{13} - 1429\nu^{9} - 8011\nu^{5} - 4408\nu ) / 7696 $ (-19v^13 - 1429v^9 - 8011v^5 - 4408v) / 7696
$\beta_{3}$	$=$	$ ( -35\nu^{14} - 2461\nu^{10} - 1811\nu^{6} + 83936\nu^{2} ) / 30784 $ (-35v^14 - 2461v^10 - 1811v^6 + 83936v^2) / 30784
$\beta_{4}$	$=$	$ ( 9\nu^{14} + 747\nu^{10} + 8741\nu^{6} + 11412\nu^{2} ) / 7696 $ (9v^14 + 747v^10 + 8741v^6 + 11412v^2) / 7696
$\beta_{5}$	$=$	$ ( -41\nu^{13} - 3255\nu^{9} - 30233\nu^{5} - 70784\nu ) / 15392 $ (-41v^13 - 3255v^9 - 30233v^5 - 70784v) / 15392
$\beta_{6}$	$=$	$ ( 41\nu^{14} + 3255\nu^{10} + 30233\nu^{6} + 70784\nu^{2} ) / 30784 $ (41v^14 + 3255v^10 + 30233v^6 + 70784v^2) / 30784
$\beta_{7}$	$=$	$ ( -19\nu^{12} - 1429\nu^{8} - 8011\nu^{4} + 3288 ) / 3848 $ (-19v^12 - 1429v^8 - 8011*v^4 + 3288) / 3848
$\beta_{8}$	$=$	$ ( 41\nu^{12} + 3255\nu^{8} + 30233\nu^{4} + 47696 ) / 7696 $ (41v^12 + 3255v^8 + 30233*v^4 + 47696) / 7696
$\beta_{9}$	$=$	$ ( -79\nu^{13} - 6113\nu^{9} - 46255\nu^{5} - 48816\nu ) / 15392 $ (-79v^13 - 6113v^9 - 46255v^5 - 48816v) / 15392
$\beta_{10}$	$=$	$ ( 115\nu^{13} + 9101\nu^{9} + 81219\nu^{5} + 94464\nu ) / 15392 $ (115v^13 + 9101v^9 + 81219v^5 + 94464v) / 15392
$\beta_{11}$	$=$	$ ( 117\nu^{15} + 8971\nu^{11} + 62277\nu^{7} + 57632\nu^{3} ) / 61568 $ (117v^15 + 8971v^11 + 62277v^7 + 57632v^3) / 61568
$\beta_{12}$	$=$	$ ( -179\nu^{14} - 14117\nu^{10} - 124203\nu^{6} - 197816\nu^{2} ) / 15392 $ (-179v^14 - 14117v^10 - 124203v^6 - 197816v^2) / 15392
$\beta_{13}$	$=$	$ ( -217\nu^{15} - 16975\nu^{11} - 140225\nu^{7} - 191240\nu^{3} ) / 30784 $ (-217v^15 - 16975v^11 - 140225v^7 - 191240v^3) / 30784
$\beta_{14}$	$=$	$ ( \nu^{15} + 79\nu^{11} + 705\nu^{7} + 1264\nu^{3} ) / 128 $ (v^15 + 79v^11 + 705v^7 + 1264*v^3) / 128
$\beta_{15}$	$=$	$ ( -787\nu^{15} - 61917\nu^{11} - 533587\nu^{7} - 752976\nu^{3} ) / 61568 $ (-787v^15 - 61917v^11 - 533587v^7 - 752976v^3) / 61568

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

$\nu$	$=$	$ ( \beta_{9} - \beta_{5} - \beta_{2} ) / 2 $ (b9 - b5 - b2) / 2
$\nu^{2}$	$=$	$ ( \beta_{12} + 7\beta_{6} + \beta_{4} - \beta_{3} ) / 2 $ (b12 + 7*b6 + b4 - b3) / 2
$\nu^{3}$	$=$	$ -\beta_{15} + \beta_{14} + 4\beta_{13} + 4\beta_{11} $ -b15 + b14 + 4b13 + 4b11
$\nu^{4}$	$=$	$ ( 7\beta_{8} + 9\beta_{7} - 11\beta _1 - 43 ) / 2 $ (7b8 + 9b7 - 11*b1 - 43) / 2
$\nu^{5}$	$=$	$ ( -22\beta_{10} - 67\beta_{9} + 9\beta_{5} + 63\beta_{2} ) / 2 $ (-22b10 - 67b9 + 9b5 + 63b2) / 2
$\nu^{6}$	$=$	$ -27\beta_{12} - 162\beta_{6} - 49\beta_{4} + 36\beta_{3} $ -27b12 - 162b6 - 49b4 + 36b3
$\nu^{7}$	$=$	$ ( 196\beta_{15} - 63\beta_{14} - 563\beta_{13} - 511\beta_{11} ) / 2 $ (196b15 - 63b14 - 563b13 - 511b11) / 2
$\nu^{8}$	$=$	$ ( -439\beta_{8} - 583\beta_{7} + 831\beta _1 + 2609 ) / 2 $ (-439b8 - 583b7 + 831*b1 + 2609) / 2
$\nu^{9}$	$=$	$ 831\beta_{10} + 2352\beta_{9} - 251\beta_{5} - 2104\beta_{2} $ 831b10 + 2352b9 - 251b5 - 2104b2
$\nu^{10}$	$=$	$ ( 3625\beta_{12} + 21461\beta_{6} + 6949\beta_{4} - 4791\beta_{3} ) / 2 $ (3625b12 + 21461b6 + 6949b4 - 4791b3) / 2
$\nu^{11}$	$=$	$ ( -13898\beta_{15} + 4127\beta_{14} + 39173\beta_{13} + 34857\beta_{11} ) / 2 $ (-13898b15 + 4127b14 + 39173b13 + 34857b11) / 2
$\nu^{12}$	$=$	$ 15033\beta_{8} + 19824\beta_{7} - 28931\beta _1 - 88874 $ 15033b8 + 19824b7 - 28931*b1 - 88874
$\nu^{13}$	$=$	$ ( -115724\beta_{10} - 325773\beta_{9} + 34193\beta_{5} + 289345\beta_{2} ) / 2 $ (-115724b10 - 325773b9 + 34193b5 + 289345b2) / 2
$\nu^{14}$	$=$	$ ( -249697\beta_{12} - 1475463\beta_{6} - 481145\beta_{4} + 328993\beta_{3} ) / 2 $ (-249697b12 - 1475463b6 - 481145b4 + 328993b3) / 2
$\nu^{15}$	$=$	$ 481145\beta_{15} - 141945\beta_{14} - 1353932\beta_{13} - 1201780\beta_{11} $ 481145b15 - 141945b14 - 1353932b13 - 1201780b11

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/525\mathbb{Z}\right)^\times$.

$n$	$127$	$176$	$451$
$\chi(n)$	$\beta_{6}$	$1$	$1$

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} $

43.1

−0.851440	+	0.851440i
−2.03837	+	2.03837i
−0.490587	+	0.490587i
1.17448	−	1.17448i
−1.17448	+	1.17448i
0.490587	−	0.490587i
2.03837	−	2.03837i
0.851440	−	0.851440i
−0.851440	−	0.851440i
−2.03837	−	2.03837i
−0.490587	−	0.490587i
1.17448	+	1.17448i
−1.17448	−	1.17448i
0.490587	+	0.490587i
2.03837	+	2.03837i
0.851440	+	0.851440i

−2.39923

2.39923i

1.22474

1.22474i

−

7.51257i

−5.87688

−1.87083

1.87083i

8.42746

8.42746i

3.00000i

43.2

−1.71533

1.71533i

1.22474

1.22474i

−

1.88473i

−4.20169

1.87083

−

1.87083i

−3.62839

−

3.62839i

3.00000i

43.3

−0.813629

0.813629i

−1.22474

−

1.22474i

2.67602i

1.99298

−1.87083

1.87083i

−5.43180

−

5.43180i

3.00000i

43.4

−0.373305

0.373305i

1.22474

1.22474i

3.72129i

−0.914408

−1.87083

1.87083i

−2.88240

−

2.88240i

3.00000i

43.5

0.373305

−

0.373305i

−1.22474

−

1.22474i

3.72129i

−0.914408

1.87083

−

1.87083i

2.88240

2.88240i

3.00000i

43.6

0.813629

−

0.813629i

1.22474

1.22474i

2.67602i

1.99298

1.87083

−

1.87083i

5.43180

5.43180i

3.00000i

43.7

1.71533

−

1.71533i

−1.22474

−

1.22474i

−

1.88473i

−4.20169

−1.87083

1.87083i

3.62839

3.62839i

3.00000i

43.8

2.39923

−

2.39923i

−1.22474

−

1.22474i

−

7.51257i

−5.87688

1.87083

−

1.87083i

−8.42746

−

8.42746i

3.00000i

232.1

−2.39923

−

2.39923i

1.22474

−

1.22474i

7.51257i

−5.87688

−1.87083

−

1.87083i

8.42746

−

8.42746i

−

3.00000i

232.2

−1.71533

−

1.71533i

1.22474

−

1.22474i

1.88473i

−4.20169

1.87083

1.87083i

−3.62839

3.62839i

−

3.00000i

232.3

−0.813629

−

0.813629i

−1.22474

1.22474i

−

2.67602i

1.99298

−1.87083

−

1.87083i

−5.43180

5.43180i

−

3.00000i

232.4

−0.373305

−

0.373305i

1.22474

−

1.22474i

−

3.72129i

−0.914408

−1.87083

−

1.87083i

−2.88240

2.88240i

−

3.00000i

232.5

0.373305

0.373305i

−1.22474

1.22474i

−

3.72129i

−0.914408

1.87083

1.87083i

2.88240

−

2.88240i

−

3.00000i

232.6

0.813629

0.813629i

1.22474

−

1.22474i

−

2.67602i

1.99298

1.87083

1.87083i

5.43180

−

5.43180i

−

3.00000i

232.7

1.71533

1.71533i

−1.22474

1.22474i

1.88473i

−4.20169

−1.87083

−

1.87083i

3.62839

−

3.62839i

−

3.00000i

232.8

2.39923

2.39923i

−1.22474

1.22474i

7.51257i

−5.87688

1.87083

1.87083i

−8.42746

8.42746i

−

3.00000i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
5.c	odd	4	2	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	525.3.l.c	✓	16
5.b	even	2	1	inner	525.3.l.c	✓	16
5.c	odd	4	2	inner	525.3.l.c	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
525.3.l.c	✓	16	1.a	even	1	1	trivial
525.3.l.c	✓	16	5.b	even	2	1	inner
525.3.l.c	✓	16	5.c	odd	4	2	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{16} + 169T_{2}^{12} + 4896T_{2}^{8} + 8425T_{2}^{4} + 625 $ T2^16 + 169*T2^12 + 4896*T2^8 + 8425*T2^4 + 625 acting on $S_{3}^{\mathrm{new}}(525, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} + 169 T^{12} + \cdots + 625 $ T^16 + 169T^12 + 4896T^8 + 8425*T^4 + 625
$3$	$ (T^{4} + 9)^{4} $ (T^4 + 9)^4
$5$	$ T^{16} $ T^16
$7$	$ (T^{4} + 49)^{4} $ (T^4 + 49)^4
$11$	$ (T^{4} + 12 T^{3} + \cdots + 4596)^{4} $ (T^4 + 12T^3 - 261T^2 - 2832*T + 4596)^4
$13$	$ T^{16} + \cdots + 62\!\cdots\!00 $ T^16 + 579394T^12 + 85320129201T^8 + 4343741835410800*T^4 + 62798657533248160000
$17$	$ T^{16} + \cdots + 151439211172096 $ T^16 + 182242T^12 + 7693470609T^8 + 78321816253552*T^4 + 151439211172096
$19$	$ (T^{8} + 1348 T^{6} + \cdots + 69488896)^{2} $ (T^8 + 1348T^6 + 282624T^4 + 9882688*T^2 + 69488896)^2
$23$	$ T^{16} + \cdots + 215074265121 $ T^16 + 231204T^12 + 10187662950T^8 + 7664510023524*T^4 + 215074265121
$29$	$ (T^{8} + 5524 T^{6} + \cdots + 1176284209)^{2} $ (T^8 + 5524T^6 + 9181014T^4 + 4798150900*T^2 + 1176284209)^2
$31$	$ (T^{4} - 32 T^{3} + \cdots + 9388)^{4} $ (T^4 - 32T^3 - 981T^2 + 9964*T + 9388)^4
$37$	$ T^{16} + \cdots + 21\!\cdots\!76 $ T^16 + 2690482T^12 + 2201321977329T^8 + 581854161361891072*T^4 + 21286679572602987544576
$41$	$ (T^{4} + 34 T^{3} + \cdots + 5609200)^{4} $ (T^4 + 34T^3 - 4911T^2 - 65300*T + 5609200)^4
$43$	$ T^{16} + \cdots + 17\!\cdots\!25 $ T^16 + 26040484T^12 + 168602737968486T^8 + 308071436790713731300*T^4 + 17872329258609286829550625
$47$	$ T^{16} + \cdots + 18\!\cdots\!36 $ T^16 + 36094864T^12 + 414314727075840T^8 + 1747908528516437807104*T^4 + 1857298741965309011479822336
$53$	$ T^{16} + \cdots + 17\!\cdots\!56 $ T^16 + 67994194T^12 + 392338841773905T^8 + 596325395518841496064*T^4 + 17146238973928781324025856
$59$	$ (T^{8} + \cdots + 23406824563600)^{2} $ (T^8 + 12286T^6 + 43553961T^4 + 56510110780*T^2 + 23406824563600)^2
$61$	$ (T^{4} + 92 T^{3} + \cdots - 4268864)^{4} $ (T^4 + 92T^3 - 4785T^2 - 417112*T - 4268864)^4
$67$	$ T^{16} + \cdots + 29\!\cdots\!56 $ T^16 + 51958594T^12 + 365116379907585T^8 + 511088207489596977664*T^4 + 29630323495728613891833856
$71$	$ (T^{4} - 104 T^{3} + \cdots - 32872460)^{4} $ (T^4 - 104T^3 - 12681T^2 + 1635400*T - 32872460)^4
$73$	$ T^{16} + \cdots + 35\!\cdots\!56 $ T^16 + 128311024T^12 + 4665165002707200T^8 + 34578847248097412644864*T^4 + 35872743114954887328018989056
$79$	$ (T^{8} + \cdots + 28081563033616)^{2} $ (T^8 + 50362T^6 + 837572049T^4 + 4604255789032*T^2 + 28081563033616)^2
$83$	$ T^{16} + \cdots + 11\!\cdots\!96 $ T^16 + 480057874T^12 + 60554898348919665T^8 + 1595308266646093524557344*T^4 + 11312160453934851907552645386496
$89$	$ (T^{8} + \cdots + 3050304166144)^{2} $ (T^8 + 25300T^6 + 73385856T^4 + 34631661376*T^2 + 3050304166144)^2
$97$	$ T^{16} + \cdots + 55\!\cdots\!56 $ T^16 + 95674432T^12 + 663673321833984T^8 + 2645605169276403712*T^4 + 552695442234692141056
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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:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	525.l (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + (\beta_{9} - \beta_{5}) q^{2} + \beta_{13} q^{3} + (\beta_{12} + 2 \beta_{3}) q^{4} + (\beta_{8} - \beta_{7} - 2) q^{6} + ( - \beta_{5} + \beta_{2}) q^{7} + (2 \beta_{15} + \beta_{14} + \cdots - \beta_{11}) q^{8}+ \cdots - 3 \beta_{6} q^{9}+O(q^{10}) \) q + (b9 - b5) * q^2 + b13 * q^3 + (b12 + 2b3) q^4 + (b8 - b7 - 2) * q^6 + (-b5 + b2) * q^7 + (2b15 + b14 + 2b13 - b11) * q^8 - 3b6 q^9	\( q + (\beta_{9} - \beta_{5}) q^{2} + \beta_{13} q^{3} + (\beta_{12} + 2 \beta_{3}) q^{4} + (\beta_{8} - \beta_{7} - 2) q^{6} + ( - \beta_{5} + \beta_{2}) q^{7} + (2 \beta_{15} + \beta_{14} + \cdots - \beta_{11}) q^{8}+ \cdots + (3 \beta_{12} + 9 \beta_{6} + \cdots + 6 \beta_{3}) q^{99}+O(q^{100}) \) q + (b9 - b5) * q^2 + b13 * q^3 + (b12 + 2b3) q^4 + (b8 - b7 - 2) * q^6 + (-b5 + b2) * q^7 + (2b15 + b14 + 2b13 - b11) * q^8 - 3b6 q^9 + (b8 - 2b7 + 3b1 - 3) * q^11 + (b10 - 2b9 + 4b5) * q^12 + (5b15 + 3b14 + 2b13 - 2b11) * q^13 + (b6 - b4 + 2b3) q^14 + (-2b7 + 3b1 - 1) * q^16 + (-3b10 - 3b9 + 2b5 + b2) q^17 + (-3b14 - 3b13) * q^18 + (6b6 - 2b4 - 6b3) q^19 + (b8 - b7 + b1) * q^21 + (5b10 - 3b9 + 8b5 - 10b2) * q^22 + (-2b15 + b14 + 6b13 + b11) * q^23 + (b12 - 2b6 + b4 - 5b3) * q^24 + (7b8 - 22b7 + 7b1 + 2) q^26 + 3b9 q^27 + (3b15 + 3b14 + 3b13 - b11) q^28 + (3b12 - 2b6 + 5b4 + 11b3) * q^29 + (-b7 + 6b1 + 7) q^31 + (-3b10 + 9b9 + 3b5 - 7b2) * q^32 + (-b15 - 4b14 - 2b13 + 9b11) q^33 + (-17b6 - 4b4 + 5b3) q^34 + (-3b8 + 6b7) * q^36 + (-3b10 + 10b9 - 4b5 - 7b2) * q^37 + (-4b15 - 12b14 - 10b13) q^38 + (2b12 + 5b6 + 2b4 - 13b3) * q^39 + (11b8 - 16b7 + 11b1 - 6) q^41 + (2b10 - 2b9 + 2b5 - 3b2) * q^42 + (-2b15 - 24b14 + b13 - 2b11) q^43 + (-4b12 + 10b6 + 3b4 - 28b3) * q^44 + (4b8 - 3b1 - 19) * q^46 + (-14b10 + 6b9 + 2b5 + 6b2) * q^47 + (-2b15 - 2b14 + 9b11) q^48 + 7b6 q^49 + (-5b8 - 4b7 + b1 + 12) * q^51 + (9b10 - 34b9 + 45b5 - 28b2) * q^52 + (-7b15 - 9b14 - 24b13 + 12b11) * q^53 + (3b12 + 6b6 + 3b3) q^54 + (6b8 - 8b7 - 1) * q^56 + (-6b10 + 2b9 - 6b5 - 6b2) * q^57 + (6b15 + 28b14 + 36b13 + 7b11) * q^58 + (-13b12 - 38b6 + 3b4 + 4b3) * q^59 + (18b8 + 5b7 - 4b1 - 29) q^61 + (7b10 + 17b9 - 4b5 - 19b2) * q^62 + (-3b14 + 3b11) * q^63 + (12b12 + 17b6 - 8b4 + 16b3) * q^64 + (-3b8 + 18b7 - 10b1 + 1) q^66 + (-3b10 + 24b9 + 10b5 + 3b2) * q^67 + (-3b15 + 6b14 - 3b13 - 13b11) * q^68 + (-3b12 - 20b6 - b4 + 3b3) q^69 + (-5b8 + 26b7 + 13b1 + 11) q^71 + (-6b10 + 6b9 - 3b5 + 3b2) * q^72 + (10b15 - 28b14 - 14b13 + 12b11) * q^73 + (13b12 + 29b6 + 4b4 + 16b3) * q^74 + (-14b8 + 10b7 - 12b1 + 60) q^76 + (-3b10 - 18b9 + 6b5 - b2) q^77 + (-15b15 - 36b14 - 13b13 + 21b11) * q^78 + (-13b12 + 17b6 - 15b4 + 32b3) * q^79 - 9 * q^81 + (27b10 - 38b9 + 43b5 - 38b2) * q^82 + (b15 + 54b14 + 13b13 + 3b11) q^83 + (-b6 + b4 - 9b3) q^84 + (-b8 + 27b7 + 44) q^86 + (8b10 - 14b9 + 17b5 + 15b2) * q^87 + (-11b15 - 38b14 - 36b13 - 7b11) * q^88 + (-22b12 + 48b6 + 8b4 - 16b3) * q^89 + (11b8 - 18b7 - 3b1) q^91 + (5b10 - 9b9 + 11b5 + 9b2) * q^92 + (-b15 - b14 + 12b13 + 18b11) * q^93 + (20b12 - 26b6 - 20b4 + 52b3) * q^94 + (-6b8 - 3b7 - 7b1 - 17) q^96 + (-4b10 + 14b9 - 32b5 + 8b2) * q^97 + (7b14 + 7b13) * q^98 + (3b12 + 9b6 - 9b4 + 6b3) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 36 q^{6}+O(q^{10}) \) 16 * q - 36 * q^6	\( 16 q - 36 q^{6} - 48 q^{11} - 20 q^{16} - 88 q^{26} + 128 q^{31} + 36 q^{36} - 136 q^{41} - 300 q^{46} + 144 q^{51} - 56 q^{56} - 368 q^{61} + 108 q^{66} + 416 q^{71} + 936 q^{76} - 144 q^{81} + 916 q^{86} - 112 q^{91} - 348 q^{96}+O(q^{100}) \) 16 * q - 36 * q^6 - 48 * q^11 - 20 * q^16 - 88 * q^26 + 128 * q^31 + 36 * q^36 - 136 * q^41 - 300 * q^46 + 144 * q^51 - 56 * q^56 - 368 * q^61 + 108 * q^66 + 416 * q^71 + 936 * q^76 - 144 * q^81 + 916 * q^86 - 112 * q^91 - 348 * q^96

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	525.3.l.c

Level	$525$
Weight	$3$
Character orbit	525.l
Analytic conductor	$14.305$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(14.3052138789\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} + 79x^{12} + 705x^{8} + 1264x^{4} + 256 \) x^16 + 79x^12 + 705x^8 + 1264*x^4 + 256
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2^{4} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

\(\beta_{1}\)	\(=\)	\( ( -5\nu^{12} - 267\nu^{8} + 4731\nu^{4} + 5648 ) / 7696 \) (-5v^12 - 267v^8 + 4731*v^4 + 5648) / 7696
\(\beta_{2}\)	\(=\)	\( ( -19\nu^{13} - 1429\nu^{9} - 8011\nu^{5} - 4408\nu ) / 7696 \) (-19v^13 - 1429v^9 - 8011v^5 - 4408v) / 7696
\(\beta_{3}\)	\(=\)	\( ( -35\nu^{14} - 2461\nu^{10} - 1811\nu^{6} + 83936\nu^{2} ) / 30784 \) (-35v^14 - 2461v^10 - 1811v^6 + 83936v^2) / 30784
\(\beta_{4}\)	\(=\)	\( ( 9\nu^{14} + 747\nu^{10} + 8741\nu^{6} + 11412\nu^{2} ) / 7696 \) (9v^14 + 747v^10 + 8741v^6 + 11412v^2) / 7696
\(\beta_{5}\)	\(=\)	\( ( -41\nu^{13} - 3255\nu^{9} - 30233\nu^{5} - 70784\nu ) / 15392 \) (-41v^13 - 3255v^9 - 30233v^5 - 70784v) / 15392
\(\beta_{6}\)	\(=\)	\( ( 41\nu^{14} + 3255\nu^{10} + 30233\nu^{6} + 70784\nu^{2} ) / 30784 \) (41v^14 + 3255v^10 + 30233v^6 + 70784v^2) / 30784
\(\beta_{7}\)	\(=\)	\( ( -19\nu^{12} - 1429\nu^{8} - 8011\nu^{4} + 3288 ) / 3848 \) (-19v^12 - 1429v^8 - 8011*v^4 + 3288) / 3848
\(\beta_{8}\)	\(=\)	\( ( 41\nu^{12} + 3255\nu^{8} + 30233\nu^{4} + 47696 ) / 7696 \) (41v^12 + 3255v^8 + 30233*v^4 + 47696) / 7696
\(\beta_{9}\)	\(=\)	\( ( -79\nu^{13} - 6113\nu^{9} - 46255\nu^{5} - 48816\nu ) / 15392 \) (-79v^13 - 6113v^9 - 46255v^5 - 48816v) / 15392
\(\beta_{10}\)	\(=\)	\( ( 115\nu^{13} + 9101\nu^{9} + 81219\nu^{5} + 94464\nu ) / 15392 \) (115v^13 + 9101v^9 + 81219v^5 + 94464v) / 15392
\(\beta_{11}\)	\(=\)	\( ( 117\nu^{15} + 8971\nu^{11} + 62277\nu^{7} + 57632\nu^{3} ) / 61568 \) (117v^15 + 8971v^11 + 62277v^7 + 57632v^3) / 61568
\(\beta_{12}\)	\(=\)	\( ( -179\nu^{14} - 14117\nu^{10} - 124203\nu^{6} - 197816\nu^{2} ) / 15392 \) (-179v^14 - 14117v^10 - 124203v^6 - 197816v^2) / 15392
\(\beta_{13}\)	\(=\)	\( ( -217\nu^{15} - 16975\nu^{11} - 140225\nu^{7} - 191240\nu^{3} ) / 30784 \) (-217v^15 - 16975v^11 - 140225v^7 - 191240v^3) / 30784
\(\beta_{14}\)	\(=\)	\( ( \nu^{15} + 79\nu^{11} + 705\nu^{7} + 1264\nu^{3} ) / 128 \) (v^15 + 79v^11 + 705v^7 + 1264*v^3) / 128
\(\beta_{15}\)	\(=\)	\( ( -787\nu^{15} - 61917\nu^{11} - 533587\nu^{7} - 752976\nu^{3} ) / 61568 \) (-787v^15 - 61917v^11 - 533587v^7 - 752976v^3) / 61568

\(\nu\)	\(=\)	\( ( \beta_{9} - \beta_{5} - \beta_{2} ) / 2 \) (b9 - b5 - b2) / 2
\(\nu^{2}\)	\(=\)	\( ( \beta_{12} + 7\beta_{6} + \beta_{4} - \beta_{3} ) / 2 \) (b12 + 7*b6 + b4 - b3) / 2
\(\nu^{3}\)	\(=\)	\( -\beta_{15} + \beta_{14} + 4\beta_{13} + 4\beta_{11} \) -b15 + b14 + 4b13 + 4b11
\(\nu^{4}\)	\(=\)	\( ( 7\beta_{8} + 9\beta_{7} - 11\beta _1 - 43 ) / 2 \) (7b8 + 9b7 - 11*b1 - 43) / 2
\(\nu^{5}\)	\(=\)	\( ( -22\beta_{10} - 67\beta_{9} + 9\beta_{5} + 63\beta_{2} ) / 2 \) (-22b10 - 67b9 + 9b5 + 63b2) / 2
\(\nu^{6}\)	\(=\)	\( -27\beta_{12} - 162\beta_{6} - 49\beta_{4} + 36\beta_{3} \) -27b12 - 162b6 - 49b4 + 36b3
\(\nu^{7}\)	\(=\)	\( ( 196\beta_{15} - 63\beta_{14} - 563\beta_{13} - 511\beta_{11} ) / 2 \) (196b15 - 63b14 - 563b13 - 511b11) / 2
\(\nu^{8}\)	\(=\)	\( ( -439\beta_{8} - 583\beta_{7} + 831\beta _1 + 2609 ) / 2 \) (-439b8 - 583b7 + 831*b1 + 2609) / 2
\(\nu^{9}\)	\(=\)	\( 831\beta_{10} + 2352\beta_{9} - 251\beta_{5} - 2104\beta_{2} \) 831b10 + 2352b9 - 251b5 - 2104b2
\(\nu^{10}\)	\(=\)	\( ( 3625\beta_{12} + 21461\beta_{6} + 6949\beta_{4} - 4791\beta_{3} ) / 2 \) (3625b12 + 21461b6 + 6949b4 - 4791b3) / 2
\(\nu^{11}\)	\(=\)	\( ( -13898\beta_{15} + 4127\beta_{14} + 39173\beta_{13} + 34857\beta_{11} ) / 2 \) (-13898b15 + 4127b14 + 39173b13 + 34857b11) / 2
\(\nu^{12}\)	\(=\)	\( 15033\beta_{8} + 19824\beta_{7} - 28931\beta _1 - 88874 \) 15033b8 + 19824b7 - 28931*b1 - 88874
\(\nu^{13}\)	\(=\)	\( ( -115724\beta_{10} - 325773\beta_{9} + 34193\beta_{5} + 289345\beta_{2} ) / 2 \) (-115724b10 - 325773b9 + 34193b5 + 289345b2) / 2
\(\nu^{14}\)	\(=\)	\( ( -249697\beta_{12} - 1475463\beta_{6} - 481145\beta_{4} + 328993\beta_{3} ) / 2 \) (-249697b12 - 1475463b6 - 481145b4 + 328993b3) / 2
\(\nu^{15}\)	\(=\)	\( 481145\beta_{15} - 141945\beta_{14} - 1353932\beta_{13} - 1201780\beta_{11} \) 481145b15 - 141945b14 - 1353932b13 - 1201780b11

\(n\)	\(127\)	\(176\)	\(451\)
\(\chi(n)\)	\(\beta_{6}\)	\(1\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 43.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} + 169 T^{12} + \cdots + 625 \) T^16 + 169T^12 + 4896T^8 + 8425*T^4 + 625
$3$	\( (T^{4} + 9)^{4} \) (T^4 + 9)^4
$5$	\( T^{16} \) T^16
$7$	\( (T^{4} + 49)^{4} \) (T^4 + 49)^4
$11$	\( (T^{4} + 12 T^{3} + \cdots + 4596)^{4} \) (T^4 + 12T^3 - 261T^2 - 2832*T + 4596)^4
$13$	\( T^{16} + \cdots + 62\!\cdots\!00 \) T^16 + 579394T^12 + 85320129201T^8 + 4343741835410800*T^4 + 62798657533248160000
$17$	\( T^{16} + \cdots + 151439211172096 \) T^16 + 182242T^12 + 7693470609T^8 + 78321816253552*T^4 + 151439211172096
$19$	\( (T^{8} + 1348 T^{6} + \cdots + 69488896)^{2} \) (T^8 + 1348T^6 + 282624T^4 + 9882688*T^2 + 69488896)^2
$23$	\( T^{16} + \cdots + 215074265121 \) T^16 + 231204T^12 + 10187662950T^8 + 7664510023524*T^4 + 215074265121
$29$	\( (T^{8} + 5524 T^{6} + \cdots + 1176284209)^{2} \) (T^8 + 5524T^6 + 9181014T^4 + 4798150900*T^2 + 1176284209)^2
$31$	\( (T^{4} - 32 T^{3} + \cdots + 9388)^{4} \) (T^4 - 32T^3 - 981T^2 + 9964*T + 9388)^4
$37$	\( T^{16} + \cdots + 21\!\cdots\!76 \) T^16 + 2690482T^12 + 2201321977329T^8 + 581854161361891072*T^4 + 21286679572602987544576
$41$	\( (T^{4} + 34 T^{3} + \cdots + 5609200)^{4} \) (T^4 + 34T^3 - 4911T^2 - 65300*T + 5609200)^4
$43$	\( T^{16} + \cdots + 17\!\cdots\!25 \) T^16 + 26040484T^12 + 168602737968486T^8 + 308071436790713731300*T^4 + 17872329258609286829550625
$47$	\( T^{16} + \cdots + 18\!\cdots\!36 \) T^16 + 36094864T^12 + 414314727075840T^8 + 1747908528516437807104*T^4 + 1857298741965309011479822336
$53$	\( T^{16} + \cdots + 17\!\cdots\!56 \) T^16 + 67994194T^12 + 392338841773905T^8 + 596325395518841496064*T^4 + 17146238973928781324025856
$59$	\( (T^{8} + \cdots + 23406824563600)^{2} \) (T^8 + 12286T^6 + 43553961T^4 + 56510110780*T^2 + 23406824563600)^2
$61$	\( (T^{4} + 92 T^{3} + \cdots - 4268864)^{4} \) (T^4 + 92T^3 - 4785T^2 - 417112*T - 4268864)^4
$67$	\( T^{16} + \cdots + 29\!\cdots\!56 \) T^16 + 51958594T^12 + 365116379907585T^8 + 511088207489596977664*T^4 + 29630323495728613891833856
$71$	\( (T^{4} - 104 T^{3} + \cdots - 32872460)^{4} \) (T^4 - 104T^3 - 12681T^2 + 1635400*T - 32872460)^4
$73$	\( T^{16} + \cdots + 35\!\cdots\!56 \) T^16 + 128311024T^12 + 4665165002707200T^8 + 34578847248097412644864*T^4 + 35872743114954887328018989056
$79$	\( (T^{8} + \cdots + 28081563033616)^{2} \) (T^8 + 50362T^6 + 837572049T^4 + 4604255789032*T^2 + 28081563033616)^2
$83$	\( T^{16} + \cdots + 11\!\cdots\!96 \) T^16 + 480057874T^12 + 60554898348919665T^8 + 1595308266646093524557344*T^4 + 11312160453934851907552645386496
$89$	\( (T^{8} + \cdots + 3050304166144)^{2} \) (T^8 + 25300T^6 + 73385856T^4 + 34631661376*T^2 + 3050304166144)^2
$97$	\( T^{16} + \cdots + 55\!\cdots\!56 \) T^16 + 95674432T^12 + 663673321833984T^8 + 2645605169276403712*T^4 + 552695442234692141056
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