LMFDB - Newform orbit 1520.2.g.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1520,2,Mod(1519,1520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1520, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 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("1520.1519");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1520 = 2^{4} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1520.g (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:	$12.1372611072$
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} - 32x^{14} + 380x^{12} - 1752x^{10} + 1904x^{8} + 7824x^{6} + 7352x^{4} + 2992x^{2} + 484 $ x^16 - 32x^14 + 380x^12 - 1752x^10 + 1904x^8 + 7824x^6 + 7352x^4 + 2992*x^2 + 484
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{15} $
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_{4} q^{3} + (\beta_{2} - \beta_1) q^{5} + \beta_{8} q^{7} + (\beta_{2} - 2) q^{9}+O(q^{10}) $ q + b4 * q^3 + (b2 - b1) * q^5 + b8 * q^7 + (b2 - 2) * q^9	$ q + \beta_{4} q^{3} + (\beta_{2} - \beta_1) q^{5} + \beta_{8} q^{7} + (\beta_{2} - 2) q^{9} - \beta_{6} q^{11} + \beta_{3} q^{13} + (\beta_{9} + \beta_{7} - \beta_{4}) q^{15} + (\beta_{10} - 3 \beta_1) q^{17} + ( - \beta_{13} + \beta_{6}) q^{19} + \beta_{11} q^{21} + (\beta_{14} - \beta_{8}) q^{23} + ( - 2 \beta_{10} + 1) q^{25} + \beta_{7} q^{27} + \beta_{15} q^{29} + ( - \beta_{13} + \beta_{12} - \beta_{9}) q^{31} + \beta_{5} q^{33} + (\beta_{14} - \beta_{13} + \cdots - \beta_{8}) q^{35}+ \cdots + ( - \beta_{13} - \beta_{12} + \beta_{6}) q^{99}+O(q^{100}) $ q + b4 * q^3 + (b2 - b1) * q^5 + b8 * q^7 + (b2 - 2) * q^9 - b6 * q^11 + b3 * q^13 + (b9 + b7 - b4) * q^15 + (b10 - 3b1) q^17 + (-b13 + b6) * q^19 + b11 * q^21 + (b14 - b8) * q^23 + (-2b10 + 1) q^25 + b7 * q^27 + b15 * q^29 + (-b13 + b12 - b9) * q^31 + b5 * q^33 + (b14 - b13 - b12 - b8) * q^35 + (-b5 + b3) * q^37 + (-3b13 - 3b12 + b6) * q^39 - b11 * q^41 + (b14 + b8) * q^43 + (-b10 - 2b2 + 2b1 + 3) * q^45 - 3b8 q^47 + (-2b2 + 1) q^49 + (b13 - b12 + 4b9) q^51 + (b5 + b3) * q^53 + (-b14 - b13 - b12 - b6) * q^55 + (2b10 - b5 - b3 + b1) q^57 + (b13 - b12 + b9) * q^59 + (3b2 + 3) q^61 + (b14 - 3b8) q^63 + (b11 - b5 - b3) * q^65 + (b7 - b4) * q^67 + (-b15 - b11) * q^69 + (-2b13 + 2b12 + b9) * q^71 + (2b10 + 6b1) * q^73 + (-2b13 + 2b12 - 2b9 + b4) q^75 + (-3b10 - b1) q^77 + (2b13 - 2b12 + 3b9) q^79 + (-b2 - 8) * q^81 + (-2b14 - b8) q^83 + (-3b10 + 2b2 + 3b1 - 6) q^85 + (4b14 - 2b8) * q^87 - b15 * q^89 + (-b13 + b12 - 5b9) q^91 + (5b10 - 3b1) * q^93 + (b14 + b13 + 2b12 + b9 - b8 - b7) q^95 + (b5 - b3) * q^97 + (-b13 - b12 + b6) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 32 q^{9}+O(q^{10}) $ 16 * q - 32 * q^9	$ 16 q - 32 q^{9} + 16 q^{25} + 48 q^{45} + 16 q^{49} + 48 q^{61} - 128 q^{81} - 96 q^{85}+O(q^{100}) $ 16 * q - 32 * q^9 + 16 * q^25 + 48 * q^45 + 16 * q^49 + 48 * q^61 - 128 * q^81 - 96 * q^85

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 32x^{14} + 380x^{12} - 1752x^{10} + 1904x^{8} + 7824x^{6} + 7352x^{4} + 2992x^{2} + 484 $ x^16 - 32*x^14 + 380*x^12 - 1752*x^10 + 1904*x^8 + 7824*x^6 + 7352*x^4 + 2992*x^2 + 484 :

$\beta_{1}$	$=$	$ ( 273231 \nu^{15} - 8704991 \nu^{13} + 102633488 \nu^{11} - 465108276 \nu^{9} + \cdots + 1434400396 \nu ) / 371999496 $ (273231v^15 - 8704991v^13 + 102633488v^11 - 465108276v^9 + 463305394v^7 + 2175968394v^5 + 2314047260v^3 + 1434400396v) / 371999496
$\beta_{2}$	$=$	$ ( - 9418 \nu^{14} + 299253 \nu^{12} - 3506282 \nu^{10} + 15542327 \nu^{8} - 12340648 \nu^{6} + \cdots - 18646288 ) / 768594 $ (-9418v^14 + 299253v^12 - 3506282v^10 + 15542327v^8 - 12340648v^6 - 87304466v^4 - 69461172*v^2 - 18646288) / 768594
$\beta_{3}$	$=$	$ ( - 417883 \nu^{14} + 13275704 \nu^{12} - 155512084 \nu^{10} + 689037518 \nu^{8} + \cdots - 977505188 ) / 33818136 $ (-417883v^14 + 13275704v^12 - 155512084v^10 + 689037518v^8 - 546462062v^6 - 3869146772v^4 - 3078554636*v^2 - 977505188) / 33818136
$\beta_{4}$	$=$	$ ( 607847 \nu^{14} - 19751712 \nu^{12} + 240757978 \nu^{10} - 1184238544 \nu^{8} + 1745285738 \nu^{6} + \cdots + 574684088 ) / 33818136 $ (607847v^14 - 19751712v^12 + 240757978v^10 - 1184238544v^8 + 1745285738v^6 + 3885107176v^4 + 2555420376*v^2 + 574684088) / 33818136
$\beta_{5}$	$=$	$ ( - 716283 \nu^{14} + 22762217 \nu^{12} - 266747237 \nu^{10} + 1182820446 \nu^{8} + \cdots - 1394895040 ) / 16909068 $ (-716283v^14 + 22762217v^12 - 266747237v^10 + 1182820446v^8 - 940027588v^6 - 6645946230v^4 - 5287396490*v^2 - 1394895040) / 16909068
$\beta_{6}$	$=$	$ ( - 920545 \nu^{14} + 29979829 \nu^{12} - 366750576 \nu^{10} + 1818589610 \nu^{8} + \cdots - 566216376 ) / 16909068 $ (-920545v^14 + 29979829v^12 - 366750576v^10 + 1818589610v^8 - 2755021050v^6 - 5797901006v^4 - 3190168300*v^2 - 566216376) / 16909068
$\beta_{7}$	$=$	$ ( - 353982 \nu^{14} + 11529208 \nu^{12} - 141053133 \nu^{10} + 699541134 \nu^{8} - 1060054206 \nu^{6} + \cdots - 215694204 ) / 5636356 $ (-353982v^14 + 11529208v^12 - 141053133v^10 + 699541134v^8 - 1060054206v^6 - 2228864424v^4 - 1222168758*v^2 - 215694204) / 5636356
$\beta_{8}$	$=$	$ ( - 13061002 \nu^{15} + 424638381 \nu^{13} - 5180449592 \nu^{11} + 25531213262 \nu^{9} + \cdots - 11340893344 \nu ) / 371999496 $ (-13061002v^15 + 424638381v^13 - 5180449592v^11 + 25531213262v^9 - 37894771792v^7 - 82991136530v^5 - 53288307768v^3 - 11340893344v) / 371999496
$\beta_{9}$	$=$	$ ( - 1212203 \nu^{15} + 39394852 \nu^{13} - 480280280 \nu^{11} + 2363301958 \nu^{9} + \cdots - 1093754068 \nu ) / 33818136 $ (-1212203v^15 + 39394852v^13 - 480280280v^11 + 2363301958v^9 - 3487097926v^7 - 7742464084v^5 - 5054759548v^3 - 1093754068v) / 33818136
$\beta_{10}$	$=$	$ ( - 113 \nu^{15} + 3627 \nu^{13} - 43270 \nu^{11} + 201430 \nu^{9} - 225338 \nu^{7} + \cdots - 162272 \nu ) / 2904 $ (-113v^15 + 3627v^13 - 43270v^11 + 201430v^9 - 225338v^7 - 910798v^5 - 655920v^3 - 162272v) / 2904
$\beta_{11}$	$=$	$ ( 478877 \nu^{15} - 15361992 \nu^{13} + 183103598 \nu^{11} - 850654440 \nu^{9} + \cdots + 828700840 \nu ) / 11272712 $ (478877v^15 - 15361992v^13 + 183103598v^11 - 850654440v^9 + 943754254v^7 + 3855966728v^5 + 2878061792v^3 + 828700840v) / 11272712
$\beta_{12}$	$=$	$ ( 1775295 \nu^{15} + 604356 \nu^{14} - 57983887 \nu^{13} - 19643140 \nu^{12} + 712771771 \nu^{11} + \cdots + 552888116 ) / 33818136 $ (1775295v^15 + 604356v^14 - 57983887v^13 - 19643140v^12 + 712771771v^11 + 239522302v^10 - 3575281818v^9 - 1179063414v^8 + 5663689412v^7 + 1741812188v^6 + 10563796026v^5 + 3857356908v^4 + 5328528790v^3 + 2499339172v^2 + 819989720*v + 552888116) / 33818136
$\beta_{13}$	$=$	$ ( - 1775295 \nu^{15} + 604356 \nu^{14} + 57983887 \nu^{13} - 19643140 \nu^{12} - 712771771 \nu^{11} + \cdots + 552888116 ) / 33818136 $ (-1775295v^15 + 604356v^14 + 57983887v^13 - 19643140v^12 - 712771771v^11 + 239522302v^10 + 3575281818v^9 - 1179063414v^8 - 5663689412v^7 + 1741812188v^6 - 10563796026v^5 + 3857356908v^4 - 5328528790v^3 + 2499339172v^2 - 819989720*v + 552888116) / 33818136
$\beta_{14}$	$=$	$ ( 16985579 \nu^{15} - 554706168 \nu^{13} + 6817424848 \nu^{11} - 34181649619 \nu^{9} + \cdots + 7996450550 \nu ) / 185999748 $ (16985579v^15 - 554706168v^13 + 6817424848v^11 - 34181649619v^9 + 54072323630v^7 + 101208945526v^5 + 51378699528v^3 + 7996450550v) / 185999748
$\beta_{15}$	$=$	$ ( 2258416 \nu^{15} - 72492744 \nu^{13} + 864912425 \nu^{11} - 4027203638 \nu^{9} + \cdots + 3174857356 \nu ) / 16909068 $ (2258416v^15 - 72492744v^13 + 864912425v^11 - 4027203638v^9 + 4509531406v^7 + 18205941992v^5 + 13062002142v^3 + 3174857356v) / 16909068

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

$\nu$	$=$	$ ( \beta_{9} - \beta_{8} + \beta_1 ) / 2 $ (b9 - b8 + b1) / 2
$\nu^{2}$	$=$	$ ( -\beta_{13} - \beta_{12} + 2\beta_{4} + 2\beta_{3} - 2\beta_{2} + 8 ) / 2 $ (-b13 - b12 + 2b4 + 2b3 - 2*b2 + 8) / 2
$\nu^{3}$	$=$	$ ( 2\beta_{14} + 2\beta_{13} - 2\beta_{12} - 3\beta_{11} - 3\beta_{10} + 9\beta_{9} - 10\beta_{8} + 13\beta_1 ) / 2 $ (2b14 + 2b13 - 2b12 - 3b11 - 3b10 + 9b9 - 10b8 + 13b1) / 2
$\nu^{4}$	$=$	$ -11\beta_{13} - 11\beta_{12} - 4\beta_{7} + 4\beta_{6} + 2\beta_{5} + 20\beta_{4} + 8\beta_{3} - 15\beta_{2} + 33 $ -11b13 - 11b12 - 4b7 + 4b6 + 2b5 + 20b4 + 8b3 - 15b2 + 33
$\nu^{5}$	$=$	$ ( - 10 \beta_{15} + 27 \beta_{14} + 25 \beta_{13} - 25 \beta_{12} - 50 \beta_{11} - 85 \beta_{10} + \cdots + 209 \beta_1 ) / 2 $ (-10b15 + 27b14 + 25b13 - 25b12 - 50b11 - 85b10 + 72b9 - 78b8 + 209*b1) / 2
$\nu^{6}$	$=$	$ - 173 \beta_{13} - 173 \beta_{12} - 95 \beta_{7} + 101 \beta_{6} + 21 \beta_{5} + 318 \beta_{4} + \cdots + 193 $ -173b13 - 173b12 - 95b7 + 101b6 + 21b5 + 318b4 + 50b3 - 123b2 + 193
$\nu^{7}$	$=$	$ ( - 217 \beta_{15} + 156 \beta_{14} + 141 \beta_{13} - 141 \beta_{12} - 714 \beta_{11} + \cdots + 2876 \beta_1 ) / 2 $ (-217b15 + 156b14 + 141b13 - 141b12 - 714b11 - 1470b10 + 290b9 - 312b8 + 2876*b1) / 2
$\nu^{8}$	$=$	$ - 2344 \beta_{13} - 2344 \beta_{12} - 1488 \beta_{7} + 1604 \beta_{6} + 40 \beta_{5} + 4320 \beta_{4} + \cdots - 308 $ -2344b13 - 2344b12 - 1488b7 + 1604b6 + 40b5 + 4320b4 - 56b3 - 82b2 - 308
$\nu^{9}$	$=$	$ - 1566 \beta_{15} - 709 \beta_{14} - 671 \beta_{13} + 671 \beta_{12} - 4476 \beta_{11} + \cdots + 17732 \beta_1 $ -1566b15 - 709b14 - 671b13 + 671b12 - 4476b11 - 9927b10 - 2373b9 + 2585b8 + 17732*b1
$\nu^{10}$	$=$	$ - 27531 \beta_{13} - 27531 \beta_{12} - 18430 \beta_{7} + 19958 \beta_{6} - 2946 \beta_{5} + \cdots - 37886 $ -27531b13 - 27531b12 - 18430b7 + 19958b6 - 2946b5 + 50798b4 - 9434b3 + 19696b2 - 37886
$\nu^{11}$	$=$	$ - 17721 \beta_{15} - 30436 \beta_{14} - 28199 \beta_{13} + 28199 \beta_{12} - 48213 \beta_{11} + \cdots + 189767 \beta_1 $ -17721b15 - 30436b14 - 28199b13 + 28199b12 - 48213b11 - 109879b10 - 80721b9 + 87570b8 + 189767*b1
$\nu^{12}$	$=$	$ - 268398 \beta_{13} - 268398 \beta_{12} - 183056 \beta_{7} + 198540 \beta_{6} - 76356 \beta_{5} + \cdots - 856818 $ -268398b13 - 268398b12 - 183056b7 + 198540b6 - 76356b5 + 495432b4 - 216504b3 + 482274b2 - 856818
$\nu^{13}$	$=$	$ - 152620 \beta_{15} - 601371 \beta_{14} - 555451 \beta_{13} + 555451 \beta_{12} - 407862 \beta_{11} + \cdots + 1601437 \beta_1 $ -152620b15 - 601371b14 - 555451b13 + 555451b12 - 407862b11 - 938899b10 - 1534878b9 + 1663812b8 + 1601437*b1
$\nu^{14}$	$=$	$ - 1814468 \beta_{13} - 1814468 \beta_{12} - 1249170 \beta_{7} + 1355870 \beta_{6} - 1309442 \beta_{5} + \cdots - 14218802 $ -1814468b13 - 1814468b12 - 1249170b7 + 1355870b6 - 1309442b5 + 3350008b4 - 3606552b3 + 8163506b2 - 14218802
$\nu^{15}$	$=$	$ - 647799 \beta_{15} - 9181848 \beta_{14} - 8473777 \beta_{13} + 8473777 \beta_{12} + \cdots + 6596386 \beta_1 $ -647799b15 - 9181848b14 - 8473777b13 + 8473777b12 - 1686196b11 - 3939864b10 - 23191980b9 + 25134720b8 + 6596386*b1

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times$.

$n$	$191$	$401$	$1141$	$1217$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

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} $

1519.1

−0.141739	+	0.707107i
−3.52761	+	0.707107i
3.52761	−	0.707107i
0.141739	−	0.707107i
−0.213388	+	0.707107i
−2.34315	+	0.707107i
2.34315	−	0.707107i
0.213388	−	0.707107i
2.34315	+	0.707107i
0.213388	+	0.707107i
−0.213388	−	0.707107i
−2.34315	−	0.707107i
3.52761	+	0.707107i
0.141739	+	0.707107i
−0.141739	−	0.707107i
−3.52761	−	0.707107i

−

2.59462i

−1.73205

−

1.41421i

−3.38587

−3.73205

1519.2

−

2.59462i

−1.73205

−

1.41421i

3.38587

−3.73205

1519.3

−

2.59462i

−1.73205

1.41421i

−3.38587

−3.73205

1519.4

−

2.59462i

−1.73205

1.41421i

3.38587

−3.73205

1519.5

−

1.80775i

1.73205

−

1.41421i

−2.12976

−0.267949

1519.6

−

1.80775i

1.73205

−

1.41421i

2.12976

−0.267949

1519.7

−

1.80775i

1.73205

1.41421i

−2.12976

−0.267949

1519.8

−

1.80775i

1.73205

1.41421i

2.12976

−0.267949

1519.9

1.80775i

1.73205

−

1.41421i

−2.12976

−0.267949

1519.10

1.80775i

1.73205

−

1.41421i

2.12976

−0.267949

1519.11

1.80775i

1.73205

1.41421i

−2.12976

−0.267949

1519.12

1.80775i

1.73205

1.41421i

2.12976

−0.267949

1519.13

2.59462i

−1.73205

−

1.41421i

−3.38587

−3.73205

1519.14

2.59462i

−1.73205

−

1.41421i

3.38587

−3.73205

1519.15

2.59462i

−1.73205

1.41421i

−3.38587

−3.73205

1519.16

2.59462i

−1.73205

1.41421i

3.38587

−3.73205

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
5.b	even	2	1	inner
19.b	odd	2	1	inner
20.d	odd	2	1	inner
76.d	even	2	1	inner
95.d	odd	2	1	inner
380.d	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1520.2.g.f	✓	16
4.b	odd	2	1	inner	1520.2.g.f	✓	16
5.b	even	2	1	inner	1520.2.g.f	✓	16
19.b	odd	2	1	inner	1520.2.g.f	✓	16
20.d	odd	2	1	inner	1520.2.g.f	✓	16
76.d	even	2	1	inner	1520.2.g.f	✓	16
95.d	odd	2	1	inner	1520.2.g.f	✓	16
380.d	even	2	1	inner	1520.2.g.f	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1520.2.g.f	✓	16	1.a	even	1	1	trivial
1520.2.g.f	✓	16	4.b	odd	2	1	inner
1520.2.g.f	✓	16	5.b	even	2	1	inner
1520.2.g.f	✓	16	19.b	odd	2	1	inner
1520.2.g.f	✓	16	20.d	odd	2	1	inner
1520.2.g.f	✓	16	76.d	even	2	1	inner
1520.2.g.f	✓	16	95.d	odd	2	1	inner
1520.2.g.f	✓	16	380.d	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1520, [\chi])$:

$ T_{3}^{4} + 10T_{3}^{2} + 22 $

$ T_{7}^{4} - 16T_{7}^{2} + 52 $

$ T_{31}^{4} - 60T_{31}^{2} + 792 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{4} + 10 T^{2} + 22)^{4} $ (T^4 + 10*T^2 + 22)^4
$5$	$ (T^{4} - 2 T^{2} + 25)^{4} $ (T^4 - 2*T^2 + 25)^4
$7$	$ (T^{4} - 16 T^{2} + 52)^{4} $ (T^4 - 16*T^2 + 52)^4
$11$	$ (T^{4} + 20 T^{2} + 52)^{4} $ (T^4 + 20*T^2 + 52)^4
$13$	$ (T^{4} - 46 T^{2} + 286)^{4} $ (T^4 - 46*T^2 + 286)^4
$17$	$ (T^{4} + 48 T^{2} + 144)^{4} $ (T^4 + 48*T^2 + 144)^4
$19$	$ (T^{8} + 20 T^{6} + \cdots + 130321)^{2} $ (T^8 + 20T^6 + 390T^4 + 7220*T^2 + 130321)^2
$23$	$ (T^{4} - 48 T^{2} + 468)^{4} $ (T^4 - 48*T^2 + 468)^4
$29$	$ (T^{4} + 152 T^{2} + 4576)^{4} $ (T^4 + 152*T^2 + 4576)^4
$31$	$ (T^{4} - 60 T^{2} + 792)^{4} $ (T^4 - 60*T^2 + 792)^4
$37$	$ (T^{4} - 106 T^{2} + 286)^{4} $ (T^4 - 106*T^2 + 286)^4
$41$	$ (T^{4} + 92 T^{2} + 1144)^{4} $ (T^4 + 92*T^2 + 1144)^4
$43$	$ (T^{4} - 64 T^{2} + 52)^{4} $ (T^4 - 64*T^2 + 52)^4
$47$	$ (T^{4} - 144 T^{2} + 4212)^{4} $ (T^4 - 144*T^2 + 4212)^4
$53$	$ (T^{4} - 138 T^{2} + 2574)^{4} $ (T^4 - 138*T^2 + 2574)^4
$59$	$ (T^{4} - 60 T^{2} + 792)^{4} $ (T^4 - 60*T^2 + 792)^4
$61$	$ (T^{2} - 6 T - 18)^{8} $ (T^2 - 6*T - 18)^8
$67$	$ (T^{4} + 30 T^{2} + 198)^{4} $ (T^4 + 30*T^2 + 198)^4
$71$	$ (T^{4} - 276 T^{2} + 792)^{4} $ (T^4 - 276*T^2 + 792)^4
$73$	$ (T^{4} + 192 T^{2} + 2304)^{4} $ (T^4 + 192*T^2 + 2304)^4
$79$	$ (T^{4} - 308 T^{2} + 10648)^{4} $ (T^4 - 308*T^2 + 10648)^4
$83$	$ (T^{4} - 192 T^{2} + 468)^{4} $ (T^4 - 192*T^2 + 468)^4
$89$	$ (T^{4} + 152 T^{2} + 4576)^{4} $ (T^4 + 152*T^2 + 4576)^4
$97$	$ (T^{4} - 106 T^{2} + 286)^{4} $ (T^4 - 106*T^2 + 286)^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:	\( N \)	\(=\)	\( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1520.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{4} q^{3} + (\beta_{2} - \beta_1) q^{5} + \beta_{8} q^{7} + (\beta_{2} - 2) q^{9}+O(q^{10}) \) q + b4 * q^3 + (b2 - b1) * q^5 + b8 * q^7 + (b2 - 2) * q^9	\( q + \beta_{4} q^{3} + (\beta_{2} - \beta_1) q^{5} + \beta_{8} q^{7} + (\beta_{2} - 2) q^{9} - \beta_{6} q^{11} + \beta_{3} q^{13} + (\beta_{9} + \beta_{7} - \beta_{4}) q^{15} + (\beta_{10} - 3 \beta_1) q^{17} + ( - \beta_{13} + \beta_{6}) q^{19} + \beta_{11} q^{21} + (\beta_{14} - \beta_{8}) q^{23} + ( - 2 \beta_{10} + 1) q^{25} + \beta_{7} q^{27} + \beta_{15} q^{29} + ( - \beta_{13} + \beta_{12} - \beta_{9}) q^{31} + \beta_{5} q^{33} + (\beta_{14} - \beta_{13} + \cdots - \beta_{8}) q^{35}+ \cdots + ( - \beta_{13} - \beta_{12} + \beta_{6}) q^{99}+O(q^{100}) \) q + b4 * q^3 + (b2 - b1) * q^5 + b8 * q^7 + (b2 - 2) * q^9 - b6 * q^11 + b3 * q^13 + (b9 + b7 - b4) * q^15 + (b10 - 3b1) q^17 + (-b13 + b6) * q^19 + b11 * q^21 + (b14 - b8) * q^23 + (-2b10 + 1) q^25 + b7 * q^27 + b15 * q^29 + (-b13 + b12 - b9) * q^31 + b5 * q^33 + (b14 - b13 - b12 - b8) * q^35 + (-b5 + b3) * q^37 + (-3b13 - 3b12 + b6) * q^39 - b11 * q^41 + (b14 + b8) * q^43 + (-b10 - 2b2 + 2b1 + 3) * q^45 - 3b8 q^47 + (-2b2 + 1) q^49 + (b13 - b12 + 4b9) q^51 + (b5 + b3) * q^53 + (-b14 - b13 - b12 - b6) * q^55 + (2b10 - b5 - b3 + b1) q^57 + (b13 - b12 + b9) * q^59 + (3b2 + 3) q^61 + (b14 - 3b8) q^63 + (b11 - b5 - b3) * q^65 + (b7 - b4) * q^67 + (-b15 - b11) * q^69 + (-2b13 + 2b12 + b9) * q^71 + (2b10 + 6b1) * q^73 + (-2b13 + 2b12 - 2b9 + b4) q^75 + (-3b10 - b1) q^77 + (2b13 - 2b12 + 3b9) q^79 + (-b2 - 8) * q^81 + (-2b14 - b8) q^83 + (-3b10 + 2b2 + 3b1 - 6) q^85 + (4b14 - 2b8) * q^87 - b15 * q^89 + (-b13 + b12 - 5b9) q^91 + (5b10 - 3b1) * q^93 + (b14 + b13 + 2b12 + b9 - b8 - b7) q^95 + (b5 - b3) * q^97 + (-b13 - b12 + b6) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 32 q^{9}+O(q^{10}) \) 16 * q - 32 * q^9	\( 16 q - 32 q^{9} + 16 q^{25} + 48 q^{45} + 16 q^{49} + 48 q^{61} - 128 q^{81} - 96 q^{85}+O(q^{100}) \) 16 * q - 32 * q^9 + 16 * q^25 + 48 * q^45 + 16 * q^49 + 48 * q^61 - 128 * q^81 - 96 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	1520.2.g.f

Level	$1520$
Weight	$2$
Character orbit	1520.g
Analytic conductor	$12.137$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(12.1372611072\)
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:	\( x^{16} - 32x^{14} + 380x^{12} - 1752x^{10} + 1904x^{8} + 7824x^{6} + 7352x^{4} + 2992x^{2} + 484 \) x^16 - 32x^14 + 380x^12 - 1752x^10 + 1904x^8 + 7824x^6 + 7352x^4 + 2992*x^2 + 484
Coefficient ring:	\(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index:	\( 2^{15} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( 273231 \nu^{15} - 8704991 \nu^{13} + 102633488 \nu^{11} - 465108276 \nu^{9} + \cdots + 1434400396 \nu ) / 371999496 \) (273231v^15 - 8704991v^13 + 102633488v^11 - 465108276v^9 + 463305394v^7 + 2175968394v^5 + 2314047260v^3 + 1434400396v) / 371999496
\(\beta_{2}\)	\(=\)	\( ( - 9418 \nu^{14} + 299253 \nu^{12} - 3506282 \nu^{10} + 15542327 \nu^{8} - 12340648 \nu^{6} + \cdots - 18646288 ) / 768594 \) (-9418v^14 + 299253v^12 - 3506282v^10 + 15542327v^8 - 12340648v^6 - 87304466v^4 - 69461172*v^2 - 18646288) / 768594
\(\beta_{3}\)	\(=\)	\( ( - 417883 \nu^{14} + 13275704 \nu^{12} - 155512084 \nu^{10} + 689037518 \nu^{8} + \cdots - 977505188 ) / 33818136 \) (-417883v^14 + 13275704v^12 - 155512084v^10 + 689037518v^8 - 546462062v^6 - 3869146772v^4 - 3078554636*v^2 - 977505188) / 33818136
\(\beta_{4}\)	\(=\)	\( ( 607847 \nu^{14} - 19751712 \nu^{12} + 240757978 \nu^{10} - 1184238544 \nu^{8} + 1745285738 \nu^{6} + \cdots + 574684088 ) / 33818136 \) (607847v^14 - 19751712v^12 + 240757978v^10 - 1184238544v^8 + 1745285738v^6 + 3885107176v^4 + 2555420376*v^2 + 574684088) / 33818136
\(\beta_{5}\)	\(=\)	\( ( - 716283 \nu^{14} + 22762217 \nu^{12} - 266747237 \nu^{10} + 1182820446 \nu^{8} + \cdots - 1394895040 ) / 16909068 \) (-716283v^14 + 22762217v^12 - 266747237v^10 + 1182820446v^8 - 940027588v^6 - 6645946230v^4 - 5287396490*v^2 - 1394895040) / 16909068
\(\beta_{6}\)	\(=\)	\( ( - 920545 \nu^{14} + 29979829 \nu^{12} - 366750576 \nu^{10} + 1818589610 \nu^{8} + \cdots - 566216376 ) / 16909068 \) (-920545v^14 + 29979829v^12 - 366750576v^10 + 1818589610v^8 - 2755021050v^6 - 5797901006v^4 - 3190168300*v^2 - 566216376) / 16909068
\(\beta_{7}\)	\(=\)	\( ( - 353982 \nu^{14} + 11529208 \nu^{12} - 141053133 \nu^{10} + 699541134 \nu^{8} - 1060054206 \nu^{6} + \cdots - 215694204 ) / 5636356 \) (-353982v^14 + 11529208v^12 - 141053133v^10 + 699541134v^8 - 1060054206v^6 - 2228864424v^4 - 1222168758*v^2 - 215694204) / 5636356
\(\beta_{8}\)	\(=\)	\( ( - 13061002 \nu^{15} + 424638381 \nu^{13} - 5180449592 \nu^{11} + 25531213262 \nu^{9} + \cdots - 11340893344 \nu ) / 371999496 \) (-13061002v^15 + 424638381v^13 - 5180449592v^11 + 25531213262v^9 - 37894771792v^7 - 82991136530v^5 - 53288307768v^3 - 11340893344v) / 371999496
\(\beta_{9}\)	\(=\)	\( ( - 1212203 \nu^{15} + 39394852 \nu^{13} - 480280280 \nu^{11} + 2363301958 \nu^{9} + \cdots - 1093754068 \nu ) / 33818136 \) (-1212203v^15 + 39394852v^13 - 480280280v^11 + 2363301958v^9 - 3487097926v^7 - 7742464084v^5 - 5054759548v^3 - 1093754068v) / 33818136
\(\beta_{10}\)	\(=\)	\( ( - 113 \nu^{15} + 3627 \nu^{13} - 43270 \nu^{11} + 201430 \nu^{9} - 225338 \nu^{7} + \cdots - 162272 \nu ) / 2904 \) (-113v^15 + 3627v^13 - 43270v^11 + 201430v^9 - 225338v^7 - 910798v^5 - 655920v^3 - 162272v) / 2904
\(\beta_{11}\)	\(=\)	\( ( 478877 \nu^{15} - 15361992 \nu^{13} + 183103598 \nu^{11} - 850654440 \nu^{9} + \cdots + 828700840 \nu ) / 11272712 \) (478877v^15 - 15361992v^13 + 183103598v^11 - 850654440v^9 + 943754254v^7 + 3855966728v^5 + 2878061792v^3 + 828700840v) / 11272712
\(\beta_{12}\)	\(=\)	\( ( 1775295 \nu^{15} + 604356 \nu^{14} - 57983887 \nu^{13} - 19643140 \nu^{12} + 712771771 \nu^{11} + \cdots + 552888116 ) / 33818136 \) (1775295v^15 + 604356v^14 - 57983887v^13 - 19643140v^12 + 712771771v^11 + 239522302v^10 - 3575281818v^9 - 1179063414v^8 + 5663689412v^7 + 1741812188v^6 + 10563796026v^5 + 3857356908v^4 + 5328528790v^3 + 2499339172v^2 + 819989720*v + 552888116) / 33818136
\(\beta_{13}\)	\(=\)	\( ( - 1775295 \nu^{15} + 604356 \nu^{14} + 57983887 \nu^{13} - 19643140 \nu^{12} - 712771771 \nu^{11} + \cdots + 552888116 ) / 33818136 \) (-1775295v^15 + 604356v^14 + 57983887v^13 - 19643140v^12 - 712771771v^11 + 239522302v^10 + 3575281818v^9 - 1179063414v^8 - 5663689412v^7 + 1741812188v^6 - 10563796026v^5 + 3857356908v^4 - 5328528790v^3 + 2499339172v^2 - 819989720*v + 552888116) / 33818136
\(\beta_{14}\)	\(=\)	\( ( 16985579 \nu^{15} - 554706168 \nu^{13} + 6817424848 \nu^{11} - 34181649619 \nu^{9} + \cdots + 7996450550 \nu ) / 185999748 \) (16985579v^15 - 554706168v^13 + 6817424848v^11 - 34181649619v^9 + 54072323630v^7 + 101208945526v^5 + 51378699528v^3 + 7996450550v) / 185999748
\(\beta_{15}\)	\(=\)	\( ( 2258416 \nu^{15} - 72492744 \nu^{13} + 864912425 \nu^{11} - 4027203638 \nu^{9} + \cdots + 3174857356 \nu ) / 16909068 \) (2258416v^15 - 72492744v^13 + 864912425v^11 - 4027203638v^9 + 4509531406v^7 + 18205941992v^5 + 13062002142v^3 + 3174857356v) / 16909068

\(\nu\)	\(=\)	\( ( \beta_{9} - \beta_{8} + \beta_1 ) / 2 \) (b9 - b8 + b1) / 2
\(\nu^{2}\)	\(=\)	\( ( -\beta_{13} - \beta_{12} + 2\beta_{4} + 2\beta_{3} - 2\beta_{2} + 8 ) / 2 \) (-b13 - b12 + 2b4 + 2b3 - 2*b2 + 8) / 2
\(\nu^{3}\)	\(=\)	\( ( 2\beta_{14} + 2\beta_{13} - 2\beta_{12} - 3\beta_{11} - 3\beta_{10} + 9\beta_{9} - 10\beta_{8} + 13\beta_1 ) / 2 \) (2b14 + 2b13 - 2b12 - 3b11 - 3b10 + 9b9 - 10b8 + 13b1) / 2
\(\nu^{4}\)	\(=\)	\( -11\beta_{13} - 11\beta_{12} - 4\beta_{7} + 4\beta_{6} + 2\beta_{5} + 20\beta_{4} + 8\beta_{3} - 15\beta_{2} + 33 \) -11b13 - 11b12 - 4b7 + 4b6 + 2b5 + 20b4 + 8b3 - 15b2 + 33
\(\nu^{5}\)	\(=\)	\( ( - 10 \beta_{15} + 27 \beta_{14} + 25 \beta_{13} - 25 \beta_{12} - 50 \beta_{11} - 85 \beta_{10} + \cdots + 209 \beta_1 ) / 2 \) (-10b15 + 27b14 + 25b13 - 25b12 - 50b11 - 85b10 + 72b9 - 78b8 + 209*b1) / 2
\(\nu^{6}\)	\(=\)	\( - 173 \beta_{13} - 173 \beta_{12} - 95 \beta_{7} + 101 \beta_{6} + 21 \beta_{5} + 318 \beta_{4} + \cdots + 193 \) -173b13 - 173b12 - 95b7 + 101b6 + 21b5 + 318b4 + 50b3 - 123b2 + 193
\(\nu^{7}\)	\(=\)	\( ( - 217 \beta_{15} + 156 \beta_{14} + 141 \beta_{13} - 141 \beta_{12} - 714 \beta_{11} + \cdots + 2876 \beta_1 ) / 2 \) (-217b15 + 156b14 + 141b13 - 141b12 - 714b11 - 1470b10 + 290b9 - 312b8 + 2876*b1) / 2
\(\nu^{8}\)	\(=\)	\( - 2344 \beta_{13} - 2344 \beta_{12} - 1488 \beta_{7} + 1604 \beta_{6} + 40 \beta_{5} + 4320 \beta_{4} + \cdots - 308 \) -2344b13 - 2344b12 - 1488b7 + 1604b6 + 40b5 + 4320b4 - 56b3 - 82b2 - 308
\(\nu^{9}\)	\(=\)	\( - 1566 \beta_{15} - 709 \beta_{14} - 671 \beta_{13} + 671 \beta_{12} - 4476 \beta_{11} + \cdots + 17732 \beta_1 \) -1566b15 - 709b14 - 671b13 + 671b12 - 4476b11 - 9927b10 - 2373b9 + 2585b8 + 17732*b1
\(\nu^{10}\)	\(=\)	\( - 27531 \beta_{13} - 27531 \beta_{12} - 18430 \beta_{7} + 19958 \beta_{6} - 2946 \beta_{5} + \cdots - 37886 \) -27531b13 - 27531b12 - 18430b7 + 19958b6 - 2946b5 + 50798b4 - 9434b3 + 19696b2 - 37886
\(\nu^{11}\)	\(=\)	\( - 17721 \beta_{15} - 30436 \beta_{14} - 28199 \beta_{13} + 28199 \beta_{12} - 48213 \beta_{11} + \cdots + 189767 \beta_1 \) -17721b15 - 30436b14 - 28199b13 + 28199b12 - 48213b11 - 109879b10 - 80721b9 + 87570b8 + 189767*b1
\(\nu^{12}\)	\(=\)	\( - 268398 \beta_{13} - 268398 \beta_{12} - 183056 \beta_{7} + 198540 \beta_{6} - 76356 \beta_{5} + \cdots - 856818 \) -268398b13 - 268398b12 - 183056b7 + 198540b6 - 76356b5 + 495432b4 - 216504b3 + 482274b2 - 856818
\(\nu^{13}\)	\(=\)	\( - 152620 \beta_{15} - 601371 \beta_{14} - 555451 \beta_{13} + 555451 \beta_{12} - 407862 \beta_{11} + \cdots + 1601437 \beta_1 \) -152620b15 - 601371b14 - 555451b13 + 555451b12 - 407862b11 - 938899b10 - 1534878b9 + 1663812b8 + 1601437*b1
\(\nu^{14}\)	\(=\)	\( - 1814468 \beta_{13} - 1814468 \beta_{12} - 1249170 \beta_{7} + 1355870 \beta_{6} - 1309442 \beta_{5} + \cdots - 14218802 \) -1814468b13 - 1814468b12 - 1249170b7 + 1355870b6 - 1309442b5 + 3350008b4 - 3606552b3 + 8163506b2 - 14218802
\(\nu^{15}\)	\(=\)	\( - 647799 \beta_{15} - 9181848 \beta_{14} - 8473777 \beta_{13} + 8473777 \beta_{12} + \cdots + 6596386 \beta_1 \) -647799b15 - 9181848b14 - 8473777b13 + 8473777b12 - 1686196b11 - 3939864b10 - 23191980b9 + 25134720b8 + 6596386*b1

\(n\)	\(191\)	\(401\)	\(1141\)	\(1217\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1519.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{4} + 10 T^{2} + 22)^{4} \) (T^4 + 10*T^2 + 22)^4
$5$	\( (T^{4} - 2 T^{2} + 25)^{4} \) (T^4 - 2*T^2 + 25)^4
$7$	\( (T^{4} - 16 T^{2} + 52)^{4} \) (T^4 - 16*T^2 + 52)^4
$11$	\( (T^{4} + 20 T^{2} + 52)^{4} \) (T^4 + 20*T^2 + 52)^4
$13$	\( (T^{4} - 46 T^{2} + 286)^{4} \) (T^4 - 46*T^2 + 286)^4
$17$	\( (T^{4} + 48 T^{2} + 144)^{4} \) (T^4 + 48*T^2 + 144)^4
$19$	\( (T^{8} + 20 T^{6} + \cdots + 130321)^{2} \) (T^8 + 20T^6 + 390T^4 + 7220*T^2 + 130321)^2
$23$	\( (T^{4} - 48 T^{2} + 468)^{4} \) (T^4 - 48*T^2 + 468)^4
$29$	\( (T^{4} + 152 T^{2} + 4576)^{4} \) (T^4 + 152*T^2 + 4576)^4
$31$	\( (T^{4} - 60 T^{2} + 792)^{4} \) (T^4 - 60*T^2 + 792)^4
$37$	\( (T^{4} - 106 T^{2} + 286)^{4} \) (T^4 - 106*T^2 + 286)^4
$41$	\( (T^{4} + 92 T^{2} + 1144)^{4} \) (T^4 + 92*T^2 + 1144)^4
$43$	\( (T^{4} - 64 T^{2} + 52)^{4} \) (T^4 - 64*T^2 + 52)^4
$47$	\( (T^{4} - 144 T^{2} + 4212)^{4} \) (T^4 - 144*T^2 + 4212)^4
$53$	\( (T^{4} - 138 T^{2} + 2574)^{4} \) (T^4 - 138*T^2 + 2574)^4
$59$	\( (T^{4} - 60 T^{2} + 792)^{4} \) (T^4 - 60*T^2 + 792)^4
$61$	\( (T^{2} - 6 T - 18)^{8} \) (T^2 - 6*T - 18)^8
$67$	\( (T^{4} + 30 T^{2} + 198)^{4} \) (T^4 + 30*T^2 + 198)^4
$71$	\( (T^{4} - 276 T^{2} + 792)^{4} \) (T^4 - 276*T^2 + 792)^4
$73$	\( (T^{4} + 192 T^{2} + 2304)^{4} \) (T^4 + 192*T^2 + 2304)^4
$79$	\( (T^{4} - 308 T^{2} + 10648)^{4} \) (T^4 - 308*T^2 + 10648)^4
$83$	\( (T^{4} - 192 T^{2} + 468)^{4} \) (T^4 - 192*T^2 + 468)^4
$89$	\( (T^{4} + 152 T^{2} + 4576)^{4} \) (T^4 + 152*T^2 + 4576)^4
$97$	\( (T^{4} - 106 T^{2} + 286)^{4} \) (T^4 - 106*T^2 + 286)^4
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