LMFDB - Newform orbit 1840.2.i.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,2,Mod(1471,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1840.1471");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1840.i (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:	$14.6924739719$
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} + 400x^{12} - 2398x^{10} + 7128x^{8} - 9200x^{6} + 4705x^{4} + 2696x^{2} + 784 $ x^16 - 32x^14 + 400x^12 - 2398x^10 + 7128x^8 - 9200x^6 + 4705x^4 + 2696*x^2 + 784
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
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_1 q^{3} + \beta_{8} q^{5} + ( - \beta_{10} + \beta_{3}) q^{7} + ( - \beta_{6} - 2) q^{9} + (\beta_{12} - 2 \beta_{10}) q^{11} + (\beta_{6} + 1) q^{13} - \beta_{3} q^{15} + \beta_{13} q^{17} + (\beta_{15} + \beta_{12} - \beta_{10}) q^{19}+ \cdots + ( - 3 \beta_{15} - \beta_{12} + \cdots + 4 \beta_{3}) q^{99}+O(q^{100}) $ q + b1 * q^3 + b8 * q^5 + (-b10 + b3) * q^7 + (-b6 - 2) * q^9 + (b12 - 2b10) q^11 + (b6 + 1) * q^13 - b3 * q^15 + b13 * q^17 + (b15 + b12 - b10) * q^19 + (b13 + 4b8 - b4) q^21 + (b12 - b10 + b9 + b3 - b1) * q^23 - q^25 + (b11 + b9 - b1) * q^27 + (-b6 - b2 + 5) * q^29 + (3b9 + b5) q^31 + (b14 - b13 - 3b8 - b4) q^33 + (b5 + b1) * q^35 + (b14 + b13 + b8) * q^37 + (-b11 - b9 + 3b1) q^39 + (-b7 + b6 - b2) * q^41 + (2b10 - 2b3) * q^43 + (-b13 - 2b8) q^45 + (b9 + b1) * q^47 + (b6 - 2b2 - 1) q^49 + (b15 + b12 - b10 - 2b3) q^51 + (-b14 + b13 + b8) * q^53 + (b9 + 2b5) q^55 + (2b14 - 3b13 - b4) * q^57 + (-b11 - b9 - 5b5 - b1) q^59 + (-b14 - b8 - 2b4) q^61 + (b12 + b10 - 3b3) q^63 + (b13 + b8) * q^65 + (b15 - b12 - 4b10 - b3) q^67 + (b14 + 3b8 - b7 + 2b6 - b2 + 6) * q^69 + (-b11 - b9 - b5 - 2b1) q^71 + (-b7 - 2b6 + b2 - 2) q^73 - b1 * q^75 + (-b6 - b2 + 2) * q^77 + (4b12 + 2b10 - 2b3) q^79 + (-2b7 + b6 + b2 - 1) q^81 + (-b15 + b12 + 4b10 + b3) q^83 - b6 * q^85 + (2b11 + b9 + 4b5 + 3b1) q^87 + (-2b13 - 4b8) * q^89 + (-b12 + 2b3) q^91 + (-3b7 + 3b6 - 2b2 + 4) q^93 + (b11 + b9) * q^95 + (b14 + 2b13 - b8 - 2b4) * q^97 + (-3b15 - b12 + 3b10 + 4b3) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 32 q^{9} + 16 q^{13} - 16 q^{25} + 72 q^{29} - 4 q^{41} - 32 q^{49} + 92 q^{69} - 20 q^{73} + 24 q^{77} + 60 q^{93}+O(q^{100}) $ 16 * q - 32 * q^9 + 16 * q^13 - 16 * q^25 + 72 * q^29 - 4 * q^41 - 32 * q^49 + 92 * q^69 - 20 * q^73 + 24 * q^77 + 60 * q^93

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 32x^{14} + 400x^{12} - 2398x^{10} + 7128x^{8} - 9200x^{6} + 4705x^{4} + 2696x^{2} + 784 $ x^16 - 32*x^14 + 400*x^12 - 2398*x^10 + 7128*x^8 - 9200*x^6 + 4705*x^4 + 2696*x^2 + 784 :

$\beta_{1}$	$=$	$ ( - 7141 \nu^{14} + 639364 \nu^{12} - 14917580 \nu^{10} + 151546566 \nu^{8} - 729577504 \nu^{6} + \cdots - 188557796 ) / 547177568 $ (-7141v^14 + 639364v^12 - 14917580v^10 + 151546566v^8 - 729577504v^6 + 1633121252v^4 - 989294405*v^2 - 188557796) / 547177568
$\beta_{2}$	$=$	$ ( - 5547 \nu^{14} + 173356 \nu^{12} - 2080516 \nu^{10} + 11582258 \nu^{8} - 29821744 \nu^{6} + \cdots + 5858020 ) / 18868192 $ (-5547v^14 + 173356v^12 - 2080516v^10 + 11582258v^8 - 29821744v^6 + 24310636v^4 + 12535373*v^2 + 5858020) / 18868192
$\beta_{3}$	$=$	$ ( - 165589 \nu^{15} + 5706972 \nu^{13} - 79149040 \nu^{11} + 555881878 \nu^{9} + \cdots - 262952740 \nu ) / 1094355136 $ (-165589v^15 + 5706972v^13 - 79149040v^11 + 555881878v^9 - 2108936952v^7 + 4187604864v^5 - 3971651317v^3 - 262952740v) / 1094355136
$\beta_{4}$	$=$	$ ( - 7343 \nu^{15} + 214244 \nu^{13} - 2331136 \nu^{11} + 10946626 \nu^{9} - 19800248 \nu^{7} + \cdots - 11846476 \nu ) / 37736384 $ (-7343v^15 + 214244v^13 - 2331136v^11 + 10946626v^9 - 19800248v^7 - 235120v^5 - 16298863v^3 - 11846476v) / 37736384
$\beta_{5}$	$=$	$ ( 971 \nu^{14} - 32540 \nu^{12} + 432756 \nu^{10} - 2840762 \nu^{8} + 9652416 \nu^{6} + \cdots + 1863596 ) / 2088464 $ (971v^14 - 32540v^12 + 432756v^10 - 2840762v^8 + 9652416v^6 - 15727740v^4 + 10040651*v^2 + 1863596) / 2088464
$\beta_{6}$	$=$	$ ( - 9687 \nu^{14} + 298300 \nu^{12} - 3520980 \nu^{10} + 19188922 \nu^{8} - 48266192 \nu^{6} + \cdots - 80965164 ) / 18868192 $ (-9687v^14 + 298300v^12 - 3520980v^10 + 19188922v^8 - 48266192v^6 + 39042108v^4 + 20161121*v^2 - 80965164) / 18868192
$\beta_{7}$	$=$	$ ( - 20175 \nu^{14} + 647232 \nu^{12} - 8072756 \nu^{10} + 47481542 \nu^{8} - 129690636 \nu^{6} + \cdots - 86948092 ) / 18868192 $ (-20175v^14 + 647232v^12 - 8072756v^10 + 47481542v^8 - 129690636v^6 + 107683996v^4 + 55334005*v^2 - 86948092) / 18868192
$\beta_{8}$	$=$	$ ( - 9589 \nu^{15} + 302708 \nu^{13} - 3710656 \nu^{11} + 21553958 \nu^{9} - 60743728 \nu^{7} + \cdots - 37094388 \nu ) / 18868192 $ (-9589v^15 + 302708v^13 - 3710656v^11 + 21553958v^9 - 60743728v^7 + 69774352v^5 - 30384773v^3 - 37094388v) / 18868192
$\beta_{9}$	$=$	$ ( 760831 \nu^{14} - 24021912 \nu^{12} + 294082508 \nu^{10} - 1701925438 \nu^{8} + 4773358420 \nu^{6} + \cdots + 692456380 ) / 547177568 $ (760831v^14 - 24021912v^12 + 294082508v^10 - 1701925438v^8 + 4773358420v^6 - 5691404292v^4 + 3858584675*v^2 + 692456380) / 547177568
$\beta_{10}$	$=$	$ ( - 345 \nu^{15} + 11264 \nu^{13} - 145224 \nu^{11} + 918310 \nu^{9} - 3009724 \nu^{7} + \cdots - 189996 \nu ) / 425488 $ (-345v^15 + 11264v^13 - 145224v^11 + 918310v^9 - 3009724v^7 + 4829752v^5 - 3773569v^3 - 189996v) / 425488
$\beta_{11}$	$=$	$ ( - 169285 \nu^{14} + 5459985 \nu^{12} - 69284106 \nu^{10} + 428229415 \nu^{8} - 1354547795 \nu^{6} + \cdots - 239476132 ) / 68397196 $ (-169285v^14 + 5459985v^12 - 69284106v^10 + 428229415v^8 - 1354547795v^6 + 2000680458v^4 - 1302558772*v^2 - 239476132) / 68397196
$\beta_{12}$	$=$	$ ( 1996395 \nu^{15} - 64300516 \nu^{13} + 811056936 \nu^{11} - 4930874706 \nu^{9} + \cdots + 361303484 \nu ) / 1094355136 $ (1996395v^15 - 64300516v^13 + 811056936v^11 - 4930874706v^9 + 14996450792v^7 - 20448941960v^5 + 12816032259v^3 + 361303484v) / 1094355136
$\beta_{13}$	$=$	$ ( 80267 \nu^{15} - 2562948 \nu^{13} + 31894496 \nu^{11} - 189379626 \nu^{9} + 550748136 \nu^{7} + \cdots + 336396572 \nu ) / 37736384 $ (80267v^15 - 2562948v^13 + 31894496v^11 - 189379626v^9 + 550748136v^7 - 668515696v^5 + 265063979v^3 + 336396572v) / 37736384
$\beta_{14}$	$=$	$ ( 81585 \nu^{15} - 2642028 \nu^{13} + 33591704 \nu^{11} - 206610582 \nu^{9} + 635359128 \nu^{7} + \cdots + 390867284 \nu ) / 37736384 $ (81585v^15 - 2642028v^13 + 33591704v^11 - 206610582v^9 + 635359128v^7 - 838029688v^5 + 290130617v^3 + 390867284v) / 37736384
$\beta_{15}$	$=$	$ ( - 2530093 \nu^{15} + 81970636 \nu^{13} - 1045043272 \nu^{11} + 6493543742 \nu^{9} + \cdots - 1062631524 \nu ) / 547177568 $ (-2530093v^15 + 81970636v^13 - 1045043272v^11 + 6493543742v^9 - 20714457512v^7 + 31723436072v^5 - 23466355989v^3 - 1062631524v) / 547177568

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

$\nu$	$=$	$ ( \beta_{10} - \beta_{8} - 2\beta_{3} ) / 2 $ (b10 - b8 - 2*b3) / 2
$\nu^{2}$	$=$	$ ( 2\beta_{6} + \beta_{5} - 2\beta_{2} + 2\beta _1 + 9 ) / 2 $ (2b6 + b5 - 2b2 + 2*b1 + 9) / 2
$\nu^{3}$	$=$	$ ( -\beta_{15} - 3\beta_{13} + 2\beta_{12} + 13\beta_{10} - 14\beta_{8} + 3\beta_{4} - 14\beta_{3} ) / 2 $ (-b15 - 3b13 + 2b12 + 13b10 - 14b8 + 3b4 - 14b3) / 2
$\nu^{4}$	$=$	$ ( 2\beta_{11} - 4\beta_{9} + 14\beta_{6} + 25\beta_{5} - 22\beta_{2} + 30\beta _1 + 67 ) / 2 $ (2b11 - 4b9 + 14b6 + 25b5 - 22b2 + 30b1 + 67) / 2
$\nu^{5}$	$=$	$ ( -14\beta_{15} - 40\beta_{13} + 12\beta_{12} + 127\beta_{10} - 191\beta_{8} + 60\beta_{4} - 106\beta_{3} ) / 2 $ (-14b15 - 40b13 + 12b12 + 127b10 - 191b8 + 60b4 - 106*b3) / 2
$\nu^{6}$	$=$	$ ( 47\beta_{11} - 46\beta_{9} + 8\beta_{7} + 91\beta_{6} + 402\beta_{5} - 183\beta_{2} + 394\beta _1 + 484 ) / 2 $ (47b11 - 46b9 + 8b7 + 91b6 + 402b5 - 183b2 + 394*b1 + 484) / 2
$\nu^{7}$	$=$	$ ( - 122 \beta_{15} - 28 \beta_{14} - 462 \beta_{13} + 56 \beta_{12} + 953 \beta_{10} + \cdots - 706 \beta_{3} ) / 2 $ (-122b15 - 28b14 - 462b13 + 56b12 + 953b10 - 2379b8 + 854b4 - 706b3) / 2
$\nu^{8}$	$=$	$ ( 712\beta_{11} - 448\beta_{9} + 80\beta_{7} + 470\beta_{6} + 5259\beta_{5} - 1118\beta_{2} + 4734\beta _1 + 2731 ) / 2 $ (712b11 - 448b9 + 80b7 + 470b6 + 5259b5 - 1118b2 + 4734*b1 + 2731) / 2
$\nu^{9}$	$=$	$ ( - 607 \beta_{15} - 528 \beta_{14} - 5061 \beta_{13} + 126 \beta_{12} + 4259 \beta_{10} + \cdots - 2826 \beta_{3} ) / 2 $ (-607b15 - 528b14 - 5061b13 + 126b12 + 4259b10 - 27394b8 + 10413b4 - 2826b3) / 2
$\nu^{10}$	$=$	$ ( 8714 \beta_{11} - 4332 \beta_{9} + 280 \beta_{7} + 118 \beta_{6} + 60643 \beta_{5} - 1502 \beta_{2} + \cdots + 2249 ) / 2 $ (8714b11 - 4332b9 + 280b7 + 118b6 + 60643b5 - 1502b2 + 52546*b1 + 2249) / 2
$\nu^{11}$	$=$	$ ( 2634 \beta_{15} - 6688 \beta_{14} - 52580 \beta_{13} - 2468 \beta_{12} - 24711 \beta_{10} + \cdots + 21314 \beta_{3} ) / 2 $ (2634b15 - 6688b14 - 52580b13 - 2468b12 - 24711b10 - 291829b8 + 113784b4 + 21314b3) / 2
$\nu^{12}$	$=$	$ ( 92865 \beta_{11} - 41314 \beta_{9} - 4744 \beta_{7} - 48755 \beta_{6} + 630646 \beta_{5} + 98679 \beta_{2} + \cdots - 261836 ) / 2 $ (92865b11 - 41314b9 - 4744b7 - 48755b6 + 630646b5 + 98679b2 + 537406*b1 - 261836) / 2
$\nu^{13}$	$=$	$ ( 125478 \beta_{15} - 69836 \beta_{14} - 507910 \beta_{13} - 62296 \beta_{12} - 998513 \beta_{10} + \cdots + 756434 \beta_{3} ) / 2 $ (125478b15 - 69836b14 - 507910b13 - 62296b12 - 998513b10 - 2850433b8 + 1123486b4 + 756434b3) / 2
$\nu^{14}$	$=$	$ ( 876628 \beta_{11} - 372008 \beta_{9} - 129248 \beta_{7} - 1009946 \beta_{6} + 5895185 \beta_{5} + \cdots - 5610087 ) / 2 $ (876628b11 - 372008b9 - 129248b7 - 1009946b6 + 5895185b5 + 2188986b2 + 4989146*b1 - 5610087) / 2
$\nu^{15}$	$=$	$ ( 2251575 \beta_{15} - 626000 \beta_{14} - 4417867 \beta_{13} - 1002078 \beta_{12} + \cdots + 13032306 \beta_{3} ) / 2 $ (2251575b15 - 626000b14 - 4417867b13 - 1002078b12 - 17543547b10 - 24904974b8 + 9858843b4 + 13032306b3) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times$.

$n$	$737$	$1151$	$1201$	$1381$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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} $

1471.1

−2.25010	+	0.500000i
2.25010	−	0.500000i
−3.23546	+	0.500000i
3.23546	−	0.500000i
−1.19389	+	0.500000i
1.19389	−	0.500000i
0.208533	+	0.500000i
−0.208533	−	0.500000i
−0.208533	+	0.500000i
0.208533	−	0.500000i
1.19389	+	0.500000i
−1.19389	−	0.500000i
3.23546	+	0.500000i
−3.23546	−	0.500000i
2.25010	+	0.500000i
−2.25010	−	0.500000i

−

3.11612i

−

1.00000i

1.38407

−6.71022

1471.2

−

3.11612i

1.00000i

−1.38407

−6.71022

1471.3

−

2.36943i

−

1.00000i

4.10148

−2.61421

1471.4

−

2.36943i

1.00000i

−4.10148

−2.61421

1471.5

−

2.05992i

−

1.00000i

0.327869

−1.24327

1471.6

−

2.05992i

1.00000i

−0.327869

−1.24327

1471.7

−

0.657492i

−

1.00000i

−1.07456

2.56770

1471.8

−

0.657492i

1.00000i

1.07456

2.56770

1471.9

0.657492i

−

1.00000i

1.07456

2.56770

1471.10

0.657492i

1.00000i

−1.07456

2.56770

1471.11

2.05992i

−

1.00000i

−0.327869

−1.24327

1471.12

2.05992i

1.00000i

0.327869

−1.24327

1471.13

2.36943i

−

1.00000i

−4.10148

−2.61421

1471.14

2.36943i

1.00000i

4.10148

−2.61421

1471.15

3.11612i

−

1.00000i

−1.38407

−6.71022

1471.16

3.11612i

1.00000i

1.38407

−6.71022

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
23.b	odd	2	1	inner
92.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1840.2.i.a	✓	16
4.b	odd	2	1	inner	1840.2.i.a	✓	16
23.b	odd	2	1	inner	1840.2.i.a	✓	16
92.b	even	2	1	inner	1840.2.i.a	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1840.2.i.a	✓	16	1.a	even	1	1	trivial
1840.2.i.a	✓	16	4.b	odd	2	1	inner
1840.2.i.a	✓	16	23.b	odd	2	1	inner
1840.2.i.a	✓	16	92.b	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{8} + 20T_{3}^{6} + 128T_{3}^{4} + 283T_{3}^{2} + 100 $ T3^8 + 20*T3^6 + 128*T3^4 + 283*T3^2 + 100 acting on $S_{2}^{\mathrm{new}}(1840, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{8} + 20 T^{6} + \cdots + 100)^{2} $ (T^8 + 20T^6 + 128T^4 + 283*T^2 + 100)^2
$5$	$ (T^{2} + 1)^{8} $ (T^2 + 1)^8
$7$	$ (T^{8} - 20 T^{6} + 56 T^{4} + \cdots + 4)^{2} $ (T^8 - 20T^6 + 56T^4 - 43*T^2 + 4)^2
$11$	$ (T^{8} - 65 T^{6} + \cdots + 784)^{2} $ (T^8 - 65T^6 + 1121T^4 - 2728*T^2 + 784)^2
$13$	$ (T^{4} - 4 T^{3} - 16 T^{2} + \cdots - 8)^{4} $ (T^4 - 4T^3 - 16T^2 + 37*T - 8)^4
$17$	$ (T^{8} + 44 T^{6} + \cdots + 100)^{2} $ (T^8 + 44T^6 + 504T^4 + 449*T^2 + 100)^2
$19$	$ (T^{8} - 133 T^{6} + \cdots + 462400)^{2} $ (T^8 - 133T^6 + 5549T^4 - 88508*T^2 + 462400)^2
$23$	$ T^{16} + \cdots + 78310985281 $ T^16 + 36T^14 + 900T^12 + 26268T^10 + 598070T^8 + 13895772T^6 + 251856900T^4 + 5329292004*T^2 + 78310985281
$29$	$ (T^{4} - 18 T^{3} + \cdots - 1172)^{4} $ (T^4 - 18T^3 + 59T^2 + 330*T - 1172)^4
$31$	$ (T^{8} + 255 T^{6} + \cdots + 6487209)^{2} $ (T^8 + 255T^6 + 21816T^4 + 701487*T^2 + 6487209)^2
$37$	$ (T^{8} + 137 T^{6} + \cdots + 153664)^{2} $ (T^8 + 137T^6 + 5328T^4 + 62720*T^2 + 153664)^2
$41$	$ (T^{4} + T^{3} - 58 T^{2} + \cdots - 89)^{4} $ (T^4 + T^3 - 58T^2 + 173T - 89)^4
$43$	$ (T^{8} - 80 T^{6} + \cdots + 1024)^{2} $ (T^8 - 80T^6 + 896T^4 - 2752*T^2 + 1024)^2
$47$	$ (T^{8} + 43 T^{6} + \cdots + 64)^{2} $ (T^8 + 43T^6 + 224T^4 + 320*T^2 + 64)^2
$53$	$ (T^{8} + 129 T^{6} + \cdots + 4096)^{2} $ (T^8 + 129T^6 + 3512T^4 + 11280*T^2 + 4096)^2
$59$	$ (T^{8} + 415 T^{6} + \cdots + 462400)^{2} $ (T^8 + 415T^6 + 45884T^4 + 582992*T^2 + 462400)^2
$61$	$ (T^{8} + 247 T^{6} + \cdots + 4129024)^{2} $ (T^8 + 247T^6 + 17757T^4 + 479620*T^2 + 4129024)^2
$67$	$ (T^{8} - 383 T^{6} + \cdots + 20070400)^{2} $ (T^8 - 383T^6 + 47972T^4 - 2169088*T^2 + 20070400)^2
$71$	$ (T^{8} + 191 T^{6} + \cdots + 1466521)^{2} $ (T^8 + 191T^6 + 11360T^4 + 228535*T^2 + 1466521)^2
$73$	$ (T^{4} + 5 T^{3} + \cdots - 1640)^{4} $ (T^4 + 5T^3 - 136T^2 - 968*T - 1640)^4
$79$	$ (T^{8} - 640 T^{6} + \cdots + 333865984)^{2} $ (T^8 - 640T^6 + 139328T^4 - 11789504*T^2 + 333865984)^2
$83$	$ (T^{8} - 383 T^{6} + \cdots + 20070400)^{2} $ (T^8 - 383T^6 + 47972T^4 - 2169088*T^2 + 20070400)^2
$89$	$ (T^{8} + 240 T^{6} + \cdots + 802816)^{2} $ (T^8 + 240T^6 + 11840T^4 + 194112*T^2 + 802816)^2
$97$	$ (T^{8} + 431 T^{6} + \cdots + 1993744)^{2} $ (T^8 + 431T^6 + 48753T^4 + 860696*T^2 + 1993744)^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:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1840.i (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_1 q^{3} + \beta_{8} q^{5} + ( - \beta_{10} + \beta_{3}) q^{7} + ( - \beta_{6} - 2) q^{9} + (\beta_{12} - 2 \beta_{10}) q^{11} + (\beta_{6} + 1) q^{13} - \beta_{3} q^{15} + \beta_{13} q^{17} + (\beta_{15} + \beta_{12} - \beta_{10}) q^{19}+ \cdots + ( - 3 \beta_{15} - \beta_{12} + \cdots + 4 \beta_{3}) q^{99}+O(q^{100}) \) q + b1 * q^3 + b8 * q^5 + (-b10 + b3) * q^7 + (-b6 - 2) * q^9 + (b12 - 2b10) q^11 + (b6 + 1) * q^13 - b3 * q^15 + b13 * q^17 + (b15 + b12 - b10) * q^19 + (b13 + 4b8 - b4) q^21 + (b12 - b10 + b9 + b3 - b1) * q^23 - q^25 + (b11 + b9 - b1) * q^27 + (-b6 - b2 + 5) * q^29 + (3b9 + b5) q^31 + (b14 - b13 - 3b8 - b4) q^33 + (b5 + b1) * q^35 + (b14 + b13 + b8) * q^37 + (-b11 - b9 + 3b1) q^39 + (-b7 + b6 - b2) * q^41 + (2b10 - 2b3) * q^43 + (-b13 - 2b8) q^45 + (b9 + b1) * q^47 + (b6 - 2b2 - 1) q^49 + (b15 + b12 - b10 - 2b3) q^51 + (-b14 + b13 + b8) * q^53 + (b9 + 2b5) q^55 + (2b14 - 3b13 - b4) * q^57 + (-b11 - b9 - 5b5 - b1) q^59 + (-b14 - b8 - 2b4) q^61 + (b12 + b10 - 3b3) q^63 + (b13 + b8) * q^65 + (b15 - b12 - 4b10 - b3) q^67 + (b14 + 3b8 - b7 + 2b6 - b2 + 6) * q^69 + (-b11 - b9 - b5 - 2b1) q^71 + (-b7 - 2b6 + b2 - 2) q^73 - b1 * q^75 + (-b6 - b2 + 2) * q^77 + (4b12 + 2b10 - 2b3) q^79 + (-2b7 + b6 + b2 - 1) q^81 + (-b15 + b12 + 4b10 + b3) q^83 - b6 * q^85 + (2b11 + b9 + 4b5 + 3b1) q^87 + (-2b13 - 4b8) * q^89 + (-b12 + 2b3) q^91 + (-3b7 + 3b6 - 2b2 + 4) q^93 + (b11 + b9) * q^95 + (b14 + 2b13 - b8 - 2b4) * q^97 + (-3b15 - b12 + 3b10 + 4b3) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 32 q^{9} + 16 q^{13} - 16 q^{25} + 72 q^{29} - 4 q^{41} - 32 q^{49} + 92 q^{69} - 20 q^{73} + 24 q^{77} + 60 q^{93}+O(q^{100}) \) 16 * q - 32 * q^9 + 16 * q^13 - 16 * q^25 + 72 * q^29 - 4 * q^41 - 32 * q^49 + 92 * q^69 - 20 * q^73 + 24 * q^77 + 60 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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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	1840.2.i.a

Level	$1840$
Weight	$2$
Character orbit	1840.i
Analytic conductor	$14.692$
Analytic rank	$0$
Dimension	$16$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(14.6924739719\)
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:	\( x^{16} - 32x^{14} + 400x^{12} - 2398x^{10} + 7128x^{8} - 9200x^{6} + 4705x^{4} + 2696x^{2} + 784 \) x^16 - 32x^14 + 400x^12 - 2398x^10 + 7128x^8 - 9200x^6 + 4705x^4 + 2696*x^2 + 784
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{8} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( - 7141 \nu^{14} + 639364 \nu^{12} - 14917580 \nu^{10} + 151546566 \nu^{8} - 729577504 \nu^{6} + \cdots - 188557796 ) / 547177568 \) (-7141v^14 + 639364v^12 - 14917580v^10 + 151546566v^8 - 729577504v^6 + 1633121252v^4 - 989294405*v^2 - 188557796) / 547177568
\(\beta_{2}\)	\(=\)	\( ( - 5547 \nu^{14} + 173356 \nu^{12} - 2080516 \nu^{10} + 11582258 \nu^{8} - 29821744 \nu^{6} + \cdots + 5858020 ) / 18868192 \) (-5547v^14 + 173356v^12 - 2080516v^10 + 11582258v^8 - 29821744v^6 + 24310636v^4 + 12535373*v^2 + 5858020) / 18868192
\(\beta_{3}\)	\(=\)	\( ( - 165589 \nu^{15} + 5706972 \nu^{13} - 79149040 \nu^{11} + 555881878 \nu^{9} + \cdots - 262952740 \nu ) / 1094355136 \) (-165589v^15 + 5706972v^13 - 79149040v^11 + 555881878v^9 - 2108936952v^7 + 4187604864v^5 - 3971651317v^3 - 262952740v) / 1094355136
\(\beta_{4}\)	\(=\)	\( ( - 7343 \nu^{15} + 214244 \nu^{13} - 2331136 \nu^{11} + 10946626 \nu^{9} - 19800248 \nu^{7} + \cdots - 11846476 \nu ) / 37736384 \) (-7343v^15 + 214244v^13 - 2331136v^11 + 10946626v^9 - 19800248v^7 - 235120v^5 - 16298863v^3 - 11846476v) / 37736384
\(\beta_{5}\)	\(=\)	\( ( 971 \nu^{14} - 32540 \nu^{12} + 432756 \nu^{10} - 2840762 \nu^{8} + 9652416 \nu^{6} + \cdots + 1863596 ) / 2088464 \) (971v^14 - 32540v^12 + 432756v^10 - 2840762v^8 + 9652416v^6 - 15727740v^4 + 10040651*v^2 + 1863596) / 2088464
\(\beta_{6}\)	\(=\)	\( ( - 9687 \nu^{14} + 298300 \nu^{12} - 3520980 \nu^{10} + 19188922 \nu^{8} - 48266192 \nu^{6} + \cdots - 80965164 ) / 18868192 \) (-9687v^14 + 298300v^12 - 3520980v^10 + 19188922v^8 - 48266192v^6 + 39042108v^4 + 20161121*v^2 - 80965164) / 18868192
\(\beta_{7}\)	\(=\)	\( ( - 20175 \nu^{14} + 647232 \nu^{12} - 8072756 \nu^{10} + 47481542 \nu^{8} - 129690636 \nu^{6} + \cdots - 86948092 ) / 18868192 \) (-20175v^14 + 647232v^12 - 8072756v^10 + 47481542v^8 - 129690636v^6 + 107683996v^4 + 55334005*v^2 - 86948092) / 18868192
\(\beta_{8}\)	\(=\)	\( ( - 9589 \nu^{15} + 302708 \nu^{13} - 3710656 \nu^{11} + 21553958 \nu^{9} - 60743728 \nu^{7} + \cdots - 37094388 \nu ) / 18868192 \) (-9589v^15 + 302708v^13 - 3710656v^11 + 21553958v^9 - 60743728v^7 + 69774352v^5 - 30384773v^3 - 37094388v) / 18868192
\(\beta_{9}\)	\(=\)	\( ( 760831 \nu^{14} - 24021912 \nu^{12} + 294082508 \nu^{10} - 1701925438 \nu^{8} + 4773358420 \nu^{6} + \cdots + 692456380 ) / 547177568 \) (760831v^14 - 24021912v^12 + 294082508v^10 - 1701925438v^8 + 4773358420v^6 - 5691404292v^4 + 3858584675*v^2 + 692456380) / 547177568
\(\beta_{10}\)	\(=\)	\( ( - 345 \nu^{15} + 11264 \nu^{13} - 145224 \nu^{11} + 918310 \nu^{9} - 3009724 \nu^{7} + \cdots - 189996 \nu ) / 425488 \) (-345v^15 + 11264v^13 - 145224v^11 + 918310v^9 - 3009724v^7 + 4829752v^5 - 3773569v^3 - 189996v) / 425488
\(\beta_{11}\)	\(=\)	\( ( - 169285 \nu^{14} + 5459985 \nu^{12} - 69284106 \nu^{10} + 428229415 \nu^{8} - 1354547795 \nu^{6} + \cdots - 239476132 ) / 68397196 \) (-169285v^14 + 5459985v^12 - 69284106v^10 + 428229415v^8 - 1354547795v^6 + 2000680458v^4 - 1302558772*v^2 - 239476132) / 68397196
\(\beta_{12}\)	\(=\)	\( ( 1996395 \nu^{15} - 64300516 \nu^{13} + 811056936 \nu^{11} - 4930874706 \nu^{9} + \cdots + 361303484 \nu ) / 1094355136 \) (1996395v^15 - 64300516v^13 + 811056936v^11 - 4930874706v^9 + 14996450792v^7 - 20448941960v^5 + 12816032259v^3 + 361303484v) / 1094355136
\(\beta_{13}\)	\(=\)	\( ( 80267 \nu^{15} - 2562948 \nu^{13} + 31894496 \nu^{11} - 189379626 \nu^{9} + 550748136 \nu^{7} + \cdots + 336396572 \nu ) / 37736384 \) (80267v^15 - 2562948v^13 + 31894496v^11 - 189379626v^9 + 550748136v^7 - 668515696v^5 + 265063979v^3 + 336396572v) / 37736384
\(\beta_{14}\)	\(=\)	\( ( 81585 \nu^{15} - 2642028 \nu^{13} + 33591704 \nu^{11} - 206610582 \nu^{9} + 635359128 \nu^{7} + \cdots + 390867284 \nu ) / 37736384 \) (81585v^15 - 2642028v^13 + 33591704v^11 - 206610582v^9 + 635359128v^7 - 838029688v^5 + 290130617v^3 + 390867284v) / 37736384
\(\beta_{15}\)	\(=\)	\( ( - 2530093 \nu^{15} + 81970636 \nu^{13} - 1045043272 \nu^{11} + 6493543742 \nu^{9} + \cdots - 1062631524 \nu ) / 547177568 \) (-2530093v^15 + 81970636v^13 - 1045043272v^11 + 6493543742v^9 - 20714457512v^7 + 31723436072v^5 - 23466355989v^3 - 1062631524v) / 547177568

\(\nu\)	\(=\)	\( ( \beta_{10} - \beta_{8} - 2\beta_{3} ) / 2 \) (b10 - b8 - 2*b3) / 2
\(\nu^{2}\)	\(=\)	\( ( 2\beta_{6} + \beta_{5} - 2\beta_{2} + 2\beta _1 + 9 ) / 2 \) (2b6 + b5 - 2b2 + 2*b1 + 9) / 2
\(\nu^{3}\)	\(=\)	\( ( -\beta_{15} - 3\beta_{13} + 2\beta_{12} + 13\beta_{10} - 14\beta_{8} + 3\beta_{4} - 14\beta_{3} ) / 2 \) (-b15 - 3b13 + 2b12 + 13b10 - 14b8 + 3b4 - 14b3) / 2
\(\nu^{4}\)	\(=\)	\( ( 2\beta_{11} - 4\beta_{9} + 14\beta_{6} + 25\beta_{5} - 22\beta_{2} + 30\beta _1 + 67 ) / 2 \) (2b11 - 4b9 + 14b6 + 25b5 - 22b2 + 30b1 + 67) / 2
\(\nu^{5}\)	\(=\)	\( ( -14\beta_{15} - 40\beta_{13} + 12\beta_{12} + 127\beta_{10} - 191\beta_{8} + 60\beta_{4} - 106\beta_{3} ) / 2 \) (-14b15 - 40b13 + 12b12 + 127b10 - 191b8 + 60b4 - 106*b3) / 2
\(\nu^{6}\)	\(=\)	\( ( 47\beta_{11} - 46\beta_{9} + 8\beta_{7} + 91\beta_{6} + 402\beta_{5} - 183\beta_{2} + 394\beta _1 + 484 ) / 2 \) (47b11 - 46b9 + 8b7 + 91b6 + 402b5 - 183b2 + 394*b1 + 484) / 2
\(\nu^{7}\)	\(=\)	\( ( - 122 \beta_{15} - 28 \beta_{14} - 462 \beta_{13} + 56 \beta_{12} + 953 \beta_{10} + \cdots - 706 \beta_{3} ) / 2 \) (-122b15 - 28b14 - 462b13 + 56b12 + 953b10 - 2379b8 + 854b4 - 706b3) / 2
\(\nu^{8}\)	\(=\)	\( ( 712\beta_{11} - 448\beta_{9} + 80\beta_{7} + 470\beta_{6} + 5259\beta_{5} - 1118\beta_{2} + 4734\beta _1 + 2731 ) / 2 \) (712b11 - 448b9 + 80b7 + 470b6 + 5259b5 - 1118b2 + 4734*b1 + 2731) / 2
\(\nu^{9}\)	\(=\)	\( ( - 607 \beta_{15} - 528 \beta_{14} - 5061 \beta_{13} + 126 \beta_{12} + 4259 \beta_{10} + \cdots - 2826 \beta_{3} ) / 2 \) (-607b15 - 528b14 - 5061b13 + 126b12 + 4259b10 - 27394b8 + 10413b4 - 2826b3) / 2
\(\nu^{10}\)	\(=\)	\( ( 8714 \beta_{11} - 4332 \beta_{9} + 280 \beta_{7} + 118 \beta_{6} + 60643 \beta_{5} - 1502 \beta_{2} + \cdots + 2249 ) / 2 \) (8714b11 - 4332b9 + 280b7 + 118b6 + 60643b5 - 1502b2 + 52546*b1 + 2249) / 2
\(\nu^{11}\)	\(=\)	\( ( 2634 \beta_{15} - 6688 \beta_{14} - 52580 \beta_{13} - 2468 \beta_{12} - 24711 \beta_{10} + \cdots + 21314 \beta_{3} ) / 2 \) (2634b15 - 6688b14 - 52580b13 - 2468b12 - 24711b10 - 291829b8 + 113784b4 + 21314b3) / 2
\(\nu^{12}\)	\(=\)	\( ( 92865 \beta_{11} - 41314 \beta_{9} - 4744 \beta_{7} - 48755 \beta_{6} + 630646 \beta_{5} + 98679 \beta_{2} + \cdots - 261836 ) / 2 \) (92865b11 - 41314b9 - 4744b7 - 48755b6 + 630646b5 + 98679b2 + 537406*b1 - 261836) / 2
\(\nu^{13}\)	\(=\)	\( ( 125478 \beta_{15} - 69836 \beta_{14} - 507910 \beta_{13} - 62296 \beta_{12} - 998513 \beta_{10} + \cdots + 756434 \beta_{3} ) / 2 \) (125478b15 - 69836b14 - 507910b13 - 62296b12 - 998513b10 - 2850433b8 + 1123486b4 + 756434b3) / 2
\(\nu^{14}\)	\(=\)	\( ( 876628 \beta_{11} - 372008 \beta_{9} - 129248 \beta_{7} - 1009946 \beta_{6} + 5895185 \beta_{5} + \cdots - 5610087 ) / 2 \) (876628b11 - 372008b9 - 129248b7 - 1009946b6 + 5895185b5 + 2188986b2 + 4989146*b1 - 5610087) / 2
\(\nu^{15}\)	\(=\)	\( ( 2251575 \beta_{15} - 626000 \beta_{14} - 4417867 \beta_{13} - 1002078 \beta_{12} + \cdots + 13032306 \beta_{3} ) / 2 \) (2251575b15 - 626000b14 - 4417867b13 - 1002078b12 - 17543547b10 - 24904974b8 + 9858843b4 + 13032306b3) / 2

\(n\)	\(737\)	\(1151\)	\(1201\)	\(1381\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-1\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1471.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{8} + 20 T^{6} + \cdots + 100)^{2} \) (T^8 + 20T^6 + 128T^4 + 283*T^2 + 100)^2
$5$	\( (T^{2} + 1)^{8} \) (T^2 + 1)^8
$7$	\( (T^{8} - 20 T^{6} + 56 T^{4} + \cdots + 4)^{2} \) (T^8 - 20T^6 + 56T^4 - 43*T^2 + 4)^2
$11$	\( (T^{8} - 65 T^{6} + \cdots + 784)^{2} \) (T^8 - 65T^6 + 1121T^4 - 2728*T^2 + 784)^2
$13$	\( (T^{4} - 4 T^{3} - 16 T^{2} + \cdots - 8)^{4} \) (T^4 - 4T^3 - 16T^2 + 37*T - 8)^4
$17$	\( (T^{8} + 44 T^{6} + \cdots + 100)^{2} \) (T^8 + 44T^6 + 504T^4 + 449*T^2 + 100)^2
$19$	\( (T^{8} - 133 T^{6} + \cdots + 462400)^{2} \) (T^8 - 133T^6 + 5549T^4 - 88508*T^2 + 462400)^2
$23$	\( T^{16} + \cdots + 78310985281 \) T^16 + 36T^14 + 900T^12 + 26268T^10 + 598070T^8 + 13895772T^6 + 251856900T^4 + 5329292004*T^2 + 78310985281
$29$	\( (T^{4} - 18 T^{3} + \cdots - 1172)^{4} \) (T^4 - 18T^3 + 59T^2 + 330*T - 1172)^4
$31$	\( (T^{8} + 255 T^{6} + \cdots + 6487209)^{2} \) (T^8 + 255T^6 + 21816T^4 + 701487*T^2 + 6487209)^2
$37$	\( (T^{8} + 137 T^{6} + \cdots + 153664)^{2} \) (T^8 + 137T^6 + 5328T^4 + 62720*T^2 + 153664)^2
$41$	\( (T^{4} + T^{3} - 58 T^{2} + \cdots - 89)^{4} \) (T^4 + T^3 - 58T^2 + 173T - 89)^4
$43$	\( (T^{8} - 80 T^{6} + \cdots + 1024)^{2} \) (T^8 - 80T^6 + 896T^4 - 2752*T^2 + 1024)^2
$47$	\( (T^{8} + 43 T^{6} + \cdots + 64)^{2} \) (T^8 + 43T^6 + 224T^4 + 320*T^2 + 64)^2
$53$	\( (T^{8} + 129 T^{6} + \cdots + 4096)^{2} \) (T^8 + 129T^6 + 3512T^4 + 11280*T^2 + 4096)^2
$59$	\( (T^{8} + 415 T^{6} + \cdots + 462400)^{2} \) (T^8 + 415T^6 + 45884T^4 + 582992*T^2 + 462400)^2
$61$	\( (T^{8} + 247 T^{6} + \cdots + 4129024)^{2} \) (T^8 + 247T^6 + 17757T^4 + 479620*T^2 + 4129024)^2
$67$	\( (T^{8} - 383 T^{6} + \cdots + 20070400)^{2} \) (T^8 - 383T^6 + 47972T^4 - 2169088*T^2 + 20070400)^2
$71$	\( (T^{8} + 191 T^{6} + \cdots + 1466521)^{2} \) (T^8 + 191T^6 + 11360T^4 + 228535*T^2 + 1466521)^2
$73$	\( (T^{4} + 5 T^{3} + \cdots - 1640)^{4} \) (T^4 + 5T^3 - 136T^2 - 968*T - 1640)^4
$79$	\( (T^{8} - 640 T^{6} + \cdots + 333865984)^{2} \) (T^8 - 640T^6 + 139328T^4 - 11789504*T^2 + 333865984)^2
$83$	\( (T^{8} - 383 T^{6} + \cdots + 20070400)^{2} \) (T^8 - 383T^6 + 47972T^4 - 2169088*T^2 + 20070400)^2
$89$	\( (T^{8} + 240 T^{6} + \cdots + 802816)^{2} \) (T^8 + 240T^6 + 11840T^4 + 194112*T^2 + 802816)^2
$97$	\( (T^{8} + 431 T^{6} + \cdots + 1993744)^{2} \) (T^8 + 431T^6 + 48753T^4 + 860696*T^2 + 1993744)^2
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