LMFDB - Newform orbit 1840.2.i.c

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} + 8x^{14} + 44x^{12} + 162x^{10} + 404x^{8} - 84x^{6} - 79x^{4} - 4x^{2} + 16 $ x^16 + 8x^14 + 44x^12 + 162x^10 + 404x^8 - 84x^6 - 79x^4 - 4*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
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_{3} q^{3} + \beta_{5} q^{5} + (\beta_{14} - \beta_{10}) q^{7} + ( - \beta_{4} - \beta_1 + 1) q^{9}+O(q^{10}) $ q + b3 * q^3 + b5 * q^5 + (b14 - b10) * q^7 + (-b4 - b1 + 1) * q^9	$ q + \beta_{3} q^{3} + \beta_{5} q^{5} + (\beta_{14} - \beta_{10}) q^{7} + ( - \beta_{4} - \beta_1 + 1) q^{9} + ( - \beta_{13} + \beta_{11}) q^{11} + (2 \beta_{8} - \beta_{4} - \beta_1) q^{13} - \beta_{10} q^{15} + ( - \beta_{15} + \beta_{12} + \beta_{6}) q^{17} + (\beta_{14} - 2 \beta_{11} + \beta_{10}) q^{19} + (\beta_{6} - 3 \beta_{5}) q^{21} + ( - \beta_{14} + \beta_{13} + \cdots + \beta_{7}) q^{23}+ \cdots + (\beta_{14} + 2 \beta_{11} + 3 \beta_{10}) q^{99}+O(q^{100}) $ q + b3 * q^3 + b5 * q^5 + (b14 - b10) * q^7 + (-b4 - b1 + 1) * q^9 + (-b13 + b11) * q^11 + (2b8 - b4 - b1) q^13 - b10 * q^15 + (-b15 + b12 + b6) * q^17 + (b14 - 2b11 + b10) q^19 + (b6 - 3b5) q^21 + (-b14 + b13 - b11 - b10 + b9 + b7) * q^23 - q^25 + (-b9 - 2b7) q^27 + (b8 - b1 - 5) * q^29 + (-b9 + b7 + b3 + b2) * q^31 + (-b15 - b12 + b5) * q^33 + (b9 - b3) * q^35 + (-2b12 + 2b6 + 2b5) q^37 + (-b9 + 2b7 - 2b3) * q^39 + (-b4 + 2b1 + 1) q^41 + (-2b13 - 4b11 + 2b10) q^43 + (-b12 + b6 + b5) * q^45 + (-b7 - 4b3 - 5b2) * q^47 + (b8 - 3b4 + b1 + 3) q^49 + (-b14 + 2b13 + 2b11 - b10) * q^51 + (-2b15 + 2b6) * q^53 + (-b7 + b3 + b2) * q^55 + (2b12 + b6 - b5) q^57 - 2b2 q^59 + (3b15 - b12 + 5b6) * q^61 + (3b14 + b13 + b11 + b10) q^63 + (2b15 - b12 + b6) q^65 + (-2b14 + 4b13 + 2b11 - 2b10) * q^67 + (b15 - b12 - b8 + b6 - 2b5 - 2b4 - b1 + 1) * q^69 + (-2b9 + 4b7 - 3b3 - 4b2) * q^71 + (3b8 - 3b1 + 5) * q^73 - b3 * q^75 + (2b8 + 2b4 - 3b1 - 3) q^77 + (2b14 - 2b13 + 2b11) q^79 + (2b8 - b1 + 2) q^81 + (-2b14 + 2b13 - 4b11) q^83 + (b8 - b4 + b1) * q^85 + (b7 - 6b3 - b2) q^87 + (2b12 + 2b5) * q^89 + (4b14 - 3b13 + 5b11) q^91 + (-b8 - 2b1 - 2) q^93 + (b9 - b3 - 2b2) q^95 + (b15 + b12 - b6 - 2b5) q^97 + (b14 + 2b11 + 3b10) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 8 q^{13} - 16 q^{25} - 84 q^{29} + 24 q^{41} + 36 q^{49} - 12 q^{69} + 68 q^{73} - 48 q^{77} + 32 q^{81} + 4 q^{85} - 52 q^{93}+O(q^{100}) $ 16 * q - 8 * q^13 - 16 * q^25 - 84 * q^29 + 24 * q^41 + 36 * q^49 - 12 * q^69 + 68 * q^73 - 48 * q^77 + 32 * q^81 + 4 * q^85 - 52 * q^93

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} + 8x^{14} + 44x^{12} + 162x^{10} + 404x^{8} - 84x^{6} - 79x^{4} - 4x^{2} + 16 $ x^16 + 8*x^14 + 44*x^12 + 162*x^10 + 404*x^8 - 84*x^6 - 79*x^4 - 4*x^2 + 16 :

$\beta_{1}$	$=$	$ ( -35\nu^{14} - 828\nu^{12} - 1561\nu^{10} - 5803\nu^{8} + 1072\nu^{6} + 1099\nu^{4} + 84\nu^{2} - 1863616 ) / 1193292 $ (-35v^14 - 828v^12 - 1561v^10 - 5803v^8 + 1072v^6 + 1099v^4 + 84*v^2 - 1863616) / 1193292
$\beta_{2}$	$=$	$ ( 293 \nu^{14} + 3459 \nu^{12} + 21907 \nu^{10} + 97195 \nu^{8} + 303239 \nu^{6} + 441599 \nu^{4} + \cdots - 48044 ) / 44196 $ (293v^14 + 3459v^12 + 21907v^10 + 97195v^8 + 303239v^6 + 441599v^4 - 58158*v^2 - 48044) / 44196
$\beta_{3}$	$=$	$ ( 11062 \nu^{14} + 74178 \nu^{12} + 374036 \nu^{10} + 1177769 \nu^{8} + 2201194 \nu^{6} + \cdots + 511628 ) / 1193292 $ (11062v^14 + 74178v^12 + 374036v^10 + 1177769v^8 + 2201194v^6 - 6433136v^4 + 749091*v^2 + 511628) / 1193292
$\beta_{4}$	$=$	$ ( 4767 \nu^{14} + 38377 \nu^{12} + 211135 \nu^{10} + 774900 \nu^{8} + 1930469 \nu^{6} - 401601 \nu^{4} + \cdots + 204280 ) / 397764 $ (4767v^14 + 38377v^12 + 211135v^10 + 774900v^8 + 1930469v^6 - 401601v^4 - 775295*v^2 + 204280) / 397764
$\beta_{5}$	$=$	$ ( - 85 \nu^{15} - 942 \nu^{13} - 5966 \nu^{11} - 26360 \nu^{9} - 82582 \nu^{7} - 120262 \nu^{5} + \cdots + 25120 \nu ) / 18792 $ (-85v^15 - 942v^13 - 5966v^11 - 26360v^9 - 82582v^7 - 120262v^5 - 27027v^3 + 25120v) / 18792
$\beta_{6}$	$=$	$ ( - 8459 \nu^{15} - 93414 \nu^{13} - 591622 \nu^{11} - 2609053 \nu^{9} - 8189294 \nu^{7} + \cdots + 2491040 \nu ) / 1193292 $ (-8459v^15 - 93414v^13 - 591622v^11 - 2609053v^9 - 8189294v^7 - 11925854v^5 - 2680074v^3 + 2491040v) / 1193292
$\beta_{7}$	$=$	$ ( - 23473 \nu^{14} - 177744 \nu^{12} - 956294 \nu^{10} - 3387989 \nu^{8} - 8017342 \nu^{6} + \cdots - 261674 ) / 596646 $ (-23473v^14 - 177744v^12 - 956294v^10 - 3387989v^8 - 8017342v^6 + 5490332v^4 - 549150*v^2 - 261674) / 596646
$\beta_{8}$	$=$	$ ( - 50557 \nu^{14} - 408714 \nu^{12} - 2246003 \nu^{10} - 8289539 \nu^{8} - 20627170 \nu^{6} + \cdots - 989684 ) / 1193292 $ (-50557v^14 - 408714v^12 - 2246003v^10 - 8289539v^8 - 20627170v^6 + 4292285v^4 + 8284338*v^2 - 989684) / 1193292
$\beta_{9}$	$=$	$ ( 59564 \nu^{14} + 502149 \nu^{12} + 2837758 \nu^{10} + 10883380 \nu^{8} + 28818305 \nu^{6} + \cdots - 1501916 ) / 1193292 $ (59564v^14 + 502149v^12 + 2837758v^10 + 10883380v^8 + 28818305v^6 + 7636250v^4 - 1353924*v^2 - 1501916) / 1193292
$\beta_{10}$	$=$	$ ( - 30172 \nu^{15} - 227943 \nu^{13} - 1226342 \nu^{11} - 4346207 \nu^{9} - 10267249 \nu^{7} + \cdots + 428758 \nu ) / 1193292 $ (-30172v^15 - 227943v^13 - 1226342v^11 - 4346207v^9 - 10267249v^7 + 7033190v^5 - 703227v^3 + 428758v) / 1193292
$\beta_{11}$	$=$	$ ( 65371 \nu^{15} + 523038 \nu^{13} + 2877980 \nu^{11} + 10593224 \nu^{9} + 26421490 \nu^{7} + \cdots - 2648236 \nu ) / 2386584 $ (65371v^15 + 523038v^13 + 2877980v^11 + 10593224v^9 + 26421490v^7 - 5493308v^5 - 5166507v^3 - 2648236v) / 2386584
$\beta_{12}$	$=$	$ ( 65371 \nu^{15} + 523038 \nu^{13} + 2877980 \nu^{11} + 10593224 \nu^{9} + 26421490 \nu^{7} + \cdots + 2124932 \nu ) / 2386584 $ (65371v^15 + 523038v^13 + 2877980v^11 + 10593224v^9 + 26421490v^7 - 5493308v^5 - 5166507v^3 + 2124932v) / 2386584
$\beta_{13}$	$=$	$ ( 88865 \nu^{15} + 643824 \nu^{13} + 3399880 \nu^{11} + 11651146 \nu^{9} + 26182376 \nu^{7} + \cdots + 450832 \nu ) / 2386584 $ (88865v^15 + 643824v^13 + 3399880v^11 + 11651146v^9 + 26182376v^7 - 30534400v^5 + 8769561v^3 + 450832v) / 2386584
$\beta_{14}$	$=$	$ ( 107167 \nu^{15} + 882966 \nu^{13} + 4914446 \nu^{11} + 18437858 \nu^{9} + 47147158 \nu^{7} + \cdots - 5926288 \nu ) / 2386584 $ (107167v^15 + 882966v^13 + 4914446v^11 + 18437858v^9 + 47147158v^7 - 3938v^5 - 14640789v^3 - 5926288v) / 2386584
$\beta_{15}$	$=$	$ ( - 243817 \nu^{15} - 1984836 \nu^{13} - 11009036 \nu^{11} - 41094428 \nu^{9} - 104586772 \nu^{7} + \cdots - 4378520 \nu ) / 2386584 $ (-243817v^15 - 1984836v^13 - 11009036v^11 - 41094428v^9 - 104586772v^7 + 4294748v^5 + 14968749v^3 - 4378520v) / 2386584

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

$\nu$	$=$	$ ( \beta_{12} - \beta_{11} ) / 2 $ (b12 - b11) / 2
$\nu^{2}$	$=$	$ ( -\beta_{9} - \beta_{7} - \beta_{4} + \beta_{3} + 2\beta_{2} - \beta _1 - 1 ) / 2 $ (-b9 - b7 - b4 + b3 + 2*b2 - b1 - 1) / 2
$\nu^{3}$	$=$	$ ( -\beta_{14} + \beta_{13} + 3\beta_{11} + 3\beta_{10} + 3\beta_{6} - 5\beta_{5} ) / 2 $ (-b14 + b13 + 3b11 + 3b10 + 3b6 - 5b5) / 2
$\nu^{4}$	$=$	$ ( 2\beta_{9} + 2\beta_{8} + 4\beta_{7} + 7\beta_{4} + 7\beta_{3} - \beta_{2} - 4\beta _1 - 8 ) / 2 $ (2b9 + 2b8 + 4b7 + 7b4 + 7b3 - b2 - 4b1 - 8) / 2
$\nu^{5}$	$=$	$ ( -2\beta_{15} - 9\beta_{14} - 11\beta_{13} - 7\beta_{12} + 12\beta_{11} - 19\beta_{10} - 11\beta_{6} + 19\beta_{5} ) / 2 $ (-2b15 - 9b14 - 11b13 - 7b12 + 12b11 - 19b10 - 11b6 + 19b5) / 2
$\nu^{6}$	$=$	$ ( 11\beta_{9} - 11\beta_{7} - 78\beta_{3} - 39\beta_{2} + 7\beta _1 + 11 ) / 2 $ (11b9 - 11b7 - 78b3 - 39b2 + 7*b1 + 11) / 2
$\nu^{7}$	$=$	$ ( - 2 \beta_{15} + 48 \beta_{14} + 50 \beta_{13} - 7 \beta_{12} - 73 \beta_{11} + 80 \beta_{10} + \cdots + 80 \beta_{5} ) / 2 $ (-2b15 + 48b14 + 50b13 - 7b12 - 73b11 + 80b10 - 50b6 + 80b5) / 2
$\nu^{8}$	$=$	$ ( 9\beta_{9} - 48\beta_{8} + 57\beta_{7} - 171\beta_{4} + 171\beta_{3} + 34\beta_{2} + 57\beta _1 + 137 ) / 2 $ (9b9 - 48b8 + 57b7 - 171b4 + 171b3 + 34b2 + 57*b1 + 137) / 2
$\nu^{9}$	$=$	$ ( 57\beta_{14} - 57\beta_{13} - 203\beta_{11} - 203\beta_{10} + 381\beta_{6} - 595\beta_{5} ) / 2 $ (57b14 - 57b13 - 203b11 - 203b10 + 381b6 - 595b5) / 2
$\nu^{10}$	$=$	$ ( -422\beta_{9} + 162\beta_{8} - 260\beta_{7} + 577\beta_{4} + 577\beta_{3} + 821\beta_{2} + 260\beta _1 + 244 ) / 2 $ (-422b9 + 162b8 - 260b7 + 577b4 + 577b3 + 821b2 + 260*b1 + 244) / 2
$\nu^{11}$	$=$	$ ( 422 \beta_{15} - 739 \beta_{14} - 317 \beta_{13} + 1503 \beta_{12} + 1576 \beta_{11} + \cdots + 73 \beta_{5} ) / 2 $ (422b15 - 739b14 - 317b13 + 1503b12 + 1576b11 - 73b10 - 317b6 + 73b5) / 2
$\nu^{12}$	$=$	$ ( 317\beta_{9} - 317\beta_{7} - 2258\beta_{3} - 1129\beta_{2} - 4167\beta _1 - 6507 ) / 2 $ (317b9 - 317b7 - 2258b3 - 1129b2 - 4167*b1 - 6507) / 2
$\nu^{13}$	$=$	$ ( - 2242 \beta_{15} - 796 \beta_{14} + 1446 \beta_{13} - 7985 \beta_{12} + 3485 \beta_{11} + \cdots + 4500 \beta_{5} ) / 2 $ (-2242b15 - 796b14 + 1446b13 - 7985b12 + 3485b11 + 4500b10 - 1446b6 + 4500b5) / 2
$\nu^{14}$	$=$	$ ( 10227 \beta_{9} + 796 \beta_{8} + 9431 \beta_{7} + 2835 \beta_{4} - 2835 \beta_{3} - 16766 \beta_{2} + \cdots + 13931 ) / 2 $ (10227b9 + 796b8 + 9431b7 + 2835b4 - 2835b3 - 16766b2 + 9431*b1 + 13931) / 2
$\nu^{15}$	$=$	$ ( 9431\beta_{14} - 9431\beta_{13} - 33589\beta_{11} - 33589\beta_{10} - 16693\beta_{6} + 26067\beta_{5} ) / 2 $ (9431b14 - 9431b13 - 33589b11 - 33589b10 - 16693b6 + 26067b5) / 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

−0.605201	−	2.04824i
0.605201	+	2.04824i
0.215960	−	0.625946i
−0.215960	+	0.625946i
−1.47123	+	1.54824i
1.47123	−	1.54824i
0.650065	+	0.125946i
−0.650065	−	0.125946i
−0.650065	+	0.125946i
0.650065	−	0.125946i
1.47123	+	1.54824i
−1.47123	−	1.54824i
−0.215960	−	0.625946i
0.215960	+	0.625946i
0.605201	−	2.04824i
−0.605201	+	2.04824i

−

2.94245i

−

1.00000i

−1.89011

−5.65803

1471.2

−

2.94245i

1.00000i

1.89011

−5.65803

1471.3

−

1.30013i

−

1.00000i

−1.10639

1.30966

1471.4

−

1.30013i

1.00000i

1.10639

1.30966

1471.5

−

1.21040i

−

1.00000i

−4.59480

1.53493

1471.6

−

1.21040i

1.00000i

4.59480

1.53493

1471.7

−

0.431920i

−

1.00000i

−3.33035

2.81344

1471.8

−

0.431920i

1.00000i

3.33035

2.81344

1471.9

0.431920i

−

1.00000i

3.33035

2.81344

1471.10

0.431920i

1.00000i

−3.33035

2.81344

1471.11

1.21040i

−

1.00000i

4.59480

1.53493

1471.12

1.21040i

1.00000i

−4.59480

1.53493

1471.13

1.30013i

−

1.00000i

1.10639

1.30966

1471.14

1.30013i

1.00000i

−1.10639

1.30966

1471.15

2.94245i

−

1.00000i

1.89011

−5.65803

1471.16

2.94245i

1.00000i

−1.89011

−5.65803

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.c	✓	16
4.b	odd	2	1	inner	1840.2.i.c	✓	16
23.b	odd	2	1	inner	1840.2.i.c	✓	16
92.b	even	2	1	inner	1840.2.i.c	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1840.2.i.c	✓	16	1.a	even	1	1	trivial
1840.2.i.c	✓	16	4.b	odd	2	1	inner
1840.2.i.c	✓	16	23.b	odd	2	1	inner
1840.2.i.c	✓	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} + 12T_{3}^{6} + 32T_{3}^{4} + 27T_{3}^{2} + 4 $ T3^8 + 12*T3^6 + 32*T3^4 + 27*T3^2 + 4 acting on $S_{2}^{\mathrm{new}}(1840, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{8} + 12 T^{6} + 32 T^{4} + \cdots + 4)^{2} $ (T^8 + 12T^6 + 32T^4 + 27*T^2 + 4)^2
$5$	$ (T^{2} + 1)^{8} $ (T^2 + 1)^8
$7$	$ (T^{8} - 37 T^{6} + \cdots + 1024)^{2} $ (T^8 - 37T^6 + 393T^4 - 1264*T^2 + 1024)^2
$11$	$ (T^{8} - 29 T^{6} + \cdots + 1024)^{2} $ (T^8 - 29T^6 + 261T^4 - 908*T^2 + 1024)^2
$13$	$ (T^{4} + 2 T^{3} - 36 T^{2} + \cdots - 2)^{4} $ (T^4 + 2T^3 - 36T^2 - 37*T - 2)^4
$17$	$ (T^{8} + 67 T^{6} + \cdots + 1024)^{2} $ (T^8 + 67T^6 + 1173T^4 + 3364*T^2 + 1024)^2
$19$	$ (T^{8} - 73 T^{6} + \cdots + 16)^{2} $ (T^8 - 73T^6 + 537T^4 - 904*T^2 + 16)^2
$23$	$ (T^{8} + 16 T^{6} + \cdots + 279841)^{2} $ (T^8 + 16T^6 + 510T^4 + 8464*T^2 + 279841)^2
$29$	$ (T^{4} + 21 T^{3} + \cdots + 376)^{4} $ (T^4 + 21T^3 + 152T^2 + 432*T + 376)^4
$31$	$ (T^{8} + 56 T^{6} + \cdots + 13924)^{2} $ (T^8 + 56T^6 + 1020T^4 + 7067*T^2 + 13924)^2
$37$	$ (T^{8} + 176 T^{6} + \cdots + 262144)^{2} $ (T^8 + 176T^6 + 6720T^4 + 76352*T^2 + 262144)^2
$41$	$ (T^{4} - 6 T^{3} + \cdots + 118)^{4} $ (T^4 - 6T^3 - 34T^2 + 27*T + 118)^4
$43$	$ (T^{8} - 220 T^{6} + \cdots + 5683456)^{2} $ (T^8 - 220T^6 + 16512T^4 - 513472*T^2 + 5683456)^2
$47$	$ (T^{8} + 283 T^{6} + \cdots + 11999296)^{2} $ (T^8 + 283T^6 + 25296T^4 + 923008*T^2 + 11999296)^2
$53$	$ (T^{8} + 252 T^{6} + \cdots + 82944)^{2} $ (T^8 + 252T^6 + 17136T^4 + 191808*T^2 + 82944)^2
$59$	$ (T^{2} + 12)^{8} $ (T^2 + 12)^8
$61$	$ (T^{8} + 395 T^{6} + \cdots + 19254544)^{2} $ (T^8 + 395T^6 + 48705T^4 + 1949144*T^2 + 19254544)^2
$67$	$ (T^{8} - 348 T^{6} + \cdots + 5308416)^{2} $ (T^8 - 348T^6 + 29376T^4 - 801792*T^2 + 5308416)^2
$71$	$ (T^{8} + 384 T^{6} + \cdots + 69455556)^{2} $ (T^8 + 384T^6 + 53244T^4 + 3182139*T^2 + 69455556)^2
$73$	$ (T^{4} - 17 T^{3} + \cdots + 376)^{4} $ (T^4 - 17T^3 - 12T^2 + 544*T + 376)^4
$79$	$ (T^{8} - 112 T^{6} + \cdots + 16384)^{2} $ (T^8 - 112T^6 + 2880T^4 - 15040*T^2 + 16384)^2
$83$	$ (T^{8} - 200 T^{6} + \cdots + 4096)^{2} $ (T^8 - 200T^6 + 10128T^4 - 130304*T^2 + 4096)^2
$89$	$ (T^{8} + 144 T^{6} + \cdots + 82944)^{2} $ (T^8 + 144T^6 + 4608T^4 + 46656*T^2 + 82944)^2
$97$	$ (T^{8} + 111 T^{6} + \cdots + 82944)^{2} $ (T^8 + 111T^6 + 3537T^4 + 34128*T^2 + 82944)^2
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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)

\(f(q)\)	\(=\)	\( q + \beta_{3} q^{3} + \beta_{5} q^{5} + (\beta_{14} - \beta_{10}) q^{7} + ( - \beta_{4} - \beta_1 + 1) q^{9}+O(q^{10}) \) q + b3 * q^3 + b5 * q^5 + (b14 - b10) * q^7 + (-b4 - b1 + 1) * q^9	\( q + \beta_{3} q^{3} + \beta_{5} q^{5} + (\beta_{14} - \beta_{10}) q^{7} + ( - \beta_{4} - \beta_1 + 1) q^{9} + ( - \beta_{13} + \beta_{11}) q^{11} + (2 \beta_{8} - \beta_{4} - \beta_1) q^{13} - \beta_{10} q^{15} + ( - \beta_{15} + \beta_{12} + \beta_{6}) q^{17} + (\beta_{14} - 2 \beta_{11} + \beta_{10}) q^{19} + (\beta_{6} - 3 \beta_{5}) q^{21} + ( - \beta_{14} + \beta_{13} + \cdots + \beta_{7}) q^{23}+ \cdots + (\beta_{14} + 2 \beta_{11} + 3 \beta_{10}) q^{99}+O(q^{100}) \) q + b3 * q^3 + b5 * q^5 + (b14 - b10) * q^7 + (-b4 - b1 + 1) * q^9 + (-b13 + b11) * q^11 + (2b8 - b4 - b1) q^13 - b10 * q^15 + (-b15 + b12 + b6) * q^17 + (b14 - 2b11 + b10) q^19 + (b6 - 3b5) q^21 + (-b14 + b13 - b11 - b10 + b9 + b7) * q^23 - q^25 + (-b9 - 2b7) q^27 + (b8 - b1 - 5) * q^29 + (-b9 + b7 + b3 + b2) * q^31 + (-b15 - b12 + b5) * q^33 + (b9 - b3) * q^35 + (-2b12 + 2b6 + 2b5) q^37 + (-b9 + 2b7 - 2b3) * q^39 + (-b4 + 2b1 + 1) q^41 + (-2b13 - 4b11 + 2b10) q^43 + (-b12 + b6 + b5) * q^45 + (-b7 - 4b3 - 5b2) * q^47 + (b8 - 3b4 + b1 + 3) q^49 + (-b14 + 2b13 + 2b11 - b10) * q^51 + (-2b15 + 2b6) * q^53 + (-b7 + b3 + b2) * q^55 + (2b12 + b6 - b5) q^57 - 2b2 q^59 + (3b15 - b12 + 5b6) * q^61 + (3b14 + b13 + b11 + b10) q^63 + (2b15 - b12 + b6) q^65 + (-2b14 + 4b13 + 2b11 - 2b10) * q^67 + (b15 - b12 - b8 + b6 - 2b5 - 2b4 - b1 + 1) * q^69 + (-2b9 + 4b7 - 3b3 - 4b2) * q^71 + (3b8 - 3b1 + 5) * q^73 - b3 * q^75 + (2b8 + 2b4 - 3b1 - 3) q^77 + (2b14 - 2b13 + 2b11) q^79 + (2b8 - b1 + 2) q^81 + (-2b14 + 2b13 - 4b11) q^83 + (b8 - b4 + b1) * q^85 + (b7 - 6b3 - b2) q^87 + (2b12 + 2b5) * q^89 + (4b14 - 3b13 + 5b11) q^91 + (-b8 - 2b1 - 2) q^93 + (b9 - b3 - 2b2) q^95 + (b15 + b12 - b6 - 2b5) q^97 + (b14 + 2b11 + 3b10) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 8 q^{13} - 16 q^{25} - 84 q^{29} + 24 q^{41} + 36 q^{49} - 12 q^{69} + 68 q^{73} - 48 q^{77} + 32 q^{81} + 4 q^{85} - 52 q^{93}+O(q^{100}) \) 16 * q - 8 * q^13 - 16 * q^25 - 84 * q^29 + 24 * q^41 + 36 * q^49 - 12 * q^69 + 68 * q^73 - 48 * q^77 + 32 * q^81 + 4 * q^85 - 52 * q^93

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	1840.2.i.c

Level	$1840$
Weight	$2$
Character orbit	1840.i
Analytic conductor	$14.692$
Analytic rank	$0$
Dimension	$16$
CM	no
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:	\( x^{16} + 8x^{14} + 44x^{12} + 162x^{10} + 404x^{8} - 84x^{6} - 79x^{4} - 4x^{2} + 16 \) x^16 + 8x^14 + 44x^12 + 162x^10 + 404x^8 - 84x^6 - 79x^4 - 4*x^2 + 16
Coefficient ring:	\(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index:	\( 2^{8} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( -35\nu^{14} - 828\nu^{12} - 1561\nu^{10} - 5803\nu^{8} + 1072\nu^{6} + 1099\nu^{4} + 84\nu^{2} - 1863616 ) / 1193292 \) (-35v^14 - 828v^12 - 1561v^10 - 5803v^8 + 1072v^6 + 1099v^4 + 84*v^2 - 1863616) / 1193292
\(\beta_{2}\)	\(=\)	\( ( 293 \nu^{14} + 3459 \nu^{12} + 21907 \nu^{10} + 97195 \nu^{8} + 303239 \nu^{6} + 441599 \nu^{4} + \cdots - 48044 ) / 44196 \) (293v^14 + 3459v^12 + 21907v^10 + 97195v^8 + 303239v^6 + 441599v^4 - 58158*v^2 - 48044) / 44196
\(\beta_{3}\)	\(=\)	\( ( 11062 \nu^{14} + 74178 \nu^{12} + 374036 \nu^{10} + 1177769 \nu^{8} + 2201194 \nu^{6} + \cdots + 511628 ) / 1193292 \) (11062v^14 + 74178v^12 + 374036v^10 + 1177769v^8 + 2201194v^6 - 6433136v^4 + 749091*v^2 + 511628) / 1193292
\(\beta_{4}\)	\(=\)	\( ( 4767 \nu^{14} + 38377 \nu^{12} + 211135 \nu^{10} + 774900 \nu^{8} + 1930469 \nu^{6} - 401601 \nu^{4} + \cdots + 204280 ) / 397764 \) (4767v^14 + 38377v^12 + 211135v^10 + 774900v^8 + 1930469v^6 - 401601v^4 - 775295*v^2 + 204280) / 397764
\(\beta_{5}\)	\(=\)	\( ( - 85 \nu^{15} - 942 \nu^{13} - 5966 \nu^{11} - 26360 \nu^{9} - 82582 \nu^{7} - 120262 \nu^{5} + \cdots + 25120 \nu ) / 18792 \) (-85v^15 - 942v^13 - 5966v^11 - 26360v^9 - 82582v^7 - 120262v^5 - 27027v^3 + 25120v) / 18792
\(\beta_{6}\)	\(=\)	\( ( - 8459 \nu^{15} - 93414 \nu^{13} - 591622 \nu^{11} - 2609053 \nu^{9} - 8189294 \nu^{7} + \cdots + 2491040 \nu ) / 1193292 \) (-8459v^15 - 93414v^13 - 591622v^11 - 2609053v^9 - 8189294v^7 - 11925854v^5 - 2680074v^3 + 2491040v) / 1193292
\(\beta_{7}\)	\(=\)	\( ( - 23473 \nu^{14} - 177744 \nu^{12} - 956294 \nu^{10} - 3387989 \nu^{8} - 8017342 \nu^{6} + \cdots - 261674 ) / 596646 \) (-23473v^14 - 177744v^12 - 956294v^10 - 3387989v^8 - 8017342v^6 + 5490332v^4 - 549150*v^2 - 261674) / 596646
\(\beta_{8}\)	\(=\)	\( ( - 50557 \nu^{14} - 408714 \nu^{12} - 2246003 \nu^{10} - 8289539 \nu^{8} - 20627170 \nu^{6} + \cdots - 989684 ) / 1193292 \) (-50557v^14 - 408714v^12 - 2246003v^10 - 8289539v^8 - 20627170v^6 + 4292285v^4 + 8284338*v^2 - 989684) / 1193292
\(\beta_{9}\)	\(=\)	\( ( 59564 \nu^{14} + 502149 \nu^{12} + 2837758 \nu^{10} + 10883380 \nu^{8} + 28818305 \nu^{6} + \cdots - 1501916 ) / 1193292 \) (59564v^14 + 502149v^12 + 2837758v^10 + 10883380v^8 + 28818305v^6 + 7636250v^4 - 1353924*v^2 - 1501916) / 1193292
\(\beta_{10}\)	\(=\)	\( ( - 30172 \nu^{15} - 227943 \nu^{13} - 1226342 \nu^{11} - 4346207 \nu^{9} - 10267249 \nu^{7} + \cdots + 428758 \nu ) / 1193292 \) (-30172v^15 - 227943v^13 - 1226342v^11 - 4346207v^9 - 10267249v^7 + 7033190v^5 - 703227v^3 + 428758v) / 1193292
\(\beta_{11}\)	\(=\)	\( ( 65371 \nu^{15} + 523038 \nu^{13} + 2877980 \nu^{11} + 10593224 \nu^{9} + 26421490 \nu^{7} + \cdots - 2648236 \nu ) / 2386584 \) (65371v^15 + 523038v^13 + 2877980v^11 + 10593224v^9 + 26421490v^7 - 5493308v^5 - 5166507v^3 - 2648236v) / 2386584
\(\beta_{12}\)	\(=\)	\( ( 65371 \nu^{15} + 523038 \nu^{13} + 2877980 \nu^{11} + 10593224 \nu^{9} + 26421490 \nu^{7} + \cdots + 2124932 \nu ) / 2386584 \) (65371v^15 + 523038v^13 + 2877980v^11 + 10593224v^9 + 26421490v^7 - 5493308v^5 - 5166507v^3 + 2124932v) / 2386584
\(\beta_{13}\)	\(=\)	\( ( 88865 \nu^{15} + 643824 \nu^{13} + 3399880 \nu^{11} + 11651146 \nu^{9} + 26182376 \nu^{7} + \cdots + 450832 \nu ) / 2386584 \) (88865v^15 + 643824v^13 + 3399880v^11 + 11651146v^9 + 26182376v^7 - 30534400v^5 + 8769561v^3 + 450832v) / 2386584
\(\beta_{14}\)	\(=\)	\( ( 107167 \nu^{15} + 882966 \nu^{13} + 4914446 \nu^{11} + 18437858 \nu^{9} + 47147158 \nu^{7} + \cdots - 5926288 \nu ) / 2386584 \) (107167v^15 + 882966v^13 + 4914446v^11 + 18437858v^9 + 47147158v^7 - 3938v^5 - 14640789v^3 - 5926288v) / 2386584
\(\beta_{15}\)	\(=\)	\( ( - 243817 \nu^{15} - 1984836 \nu^{13} - 11009036 \nu^{11} - 41094428 \nu^{9} - 104586772 \nu^{7} + \cdots - 4378520 \nu ) / 2386584 \) (-243817v^15 - 1984836v^13 - 11009036v^11 - 41094428v^9 - 104586772v^7 + 4294748v^5 + 14968749v^3 - 4378520v) / 2386584

\(\nu\)	\(=\)	\( ( \beta_{12} - \beta_{11} ) / 2 \) (b12 - b11) / 2
\(\nu^{2}\)	\(=\)	\( ( -\beta_{9} - \beta_{7} - \beta_{4} + \beta_{3} + 2\beta_{2} - \beta _1 - 1 ) / 2 \) (-b9 - b7 - b4 + b3 + 2*b2 - b1 - 1) / 2
\(\nu^{3}\)	\(=\)	\( ( -\beta_{14} + \beta_{13} + 3\beta_{11} + 3\beta_{10} + 3\beta_{6} - 5\beta_{5} ) / 2 \) (-b14 + b13 + 3b11 + 3b10 + 3b6 - 5b5) / 2
\(\nu^{4}\)	\(=\)	\( ( 2\beta_{9} + 2\beta_{8} + 4\beta_{7} + 7\beta_{4} + 7\beta_{3} - \beta_{2} - 4\beta _1 - 8 ) / 2 \) (2b9 + 2b8 + 4b7 + 7b4 + 7b3 - b2 - 4b1 - 8) / 2
\(\nu^{5}\)	\(=\)	\( ( -2\beta_{15} - 9\beta_{14} - 11\beta_{13} - 7\beta_{12} + 12\beta_{11} - 19\beta_{10} - 11\beta_{6} + 19\beta_{5} ) / 2 \) (-2b15 - 9b14 - 11b13 - 7b12 + 12b11 - 19b10 - 11b6 + 19b5) / 2
\(\nu^{6}\)	\(=\)	\( ( 11\beta_{9} - 11\beta_{7} - 78\beta_{3} - 39\beta_{2} + 7\beta _1 + 11 ) / 2 \) (11b9 - 11b7 - 78b3 - 39b2 + 7*b1 + 11) / 2
\(\nu^{7}\)	\(=\)	\( ( - 2 \beta_{15} + 48 \beta_{14} + 50 \beta_{13} - 7 \beta_{12} - 73 \beta_{11} + 80 \beta_{10} + \cdots + 80 \beta_{5} ) / 2 \) (-2b15 + 48b14 + 50b13 - 7b12 - 73b11 + 80b10 - 50b6 + 80b5) / 2
\(\nu^{8}\)	\(=\)	\( ( 9\beta_{9} - 48\beta_{8} + 57\beta_{7} - 171\beta_{4} + 171\beta_{3} + 34\beta_{2} + 57\beta _1 + 137 ) / 2 \) (9b9 - 48b8 + 57b7 - 171b4 + 171b3 + 34b2 + 57*b1 + 137) / 2
\(\nu^{9}\)	\(=\)	\( ( 57\beta_{14} - 57\beta_{13} - 203\beta_{11} - 203\beta_{10} + 381\beta_{6} - 595\beta_{5} ) / 2 \) (57b14 - 57b13 - 203b11 - 203b10 + 381b6 - 595b5) / 2
\(\nu^{10}\)	\(=\)	\( ( -422\beta_{9} + 162\beta_{8} - 260\beta_{7} + 577\beta_{4} + 577\beta_{3} + 821\beta_{2} + 260\beta _1 + 244 ) / 2 \) (-422b9 + 162b8 - 260b7 + 577b4 + 577b3 + 821b2 + 260*b1 + 244) / 2
\(\nu^{11}\)	\(=\)	\( ( 422 \beta_{15} - 739 \beta_{14} - 317 \beta_{13} + 1503 \beta_{12} + 1576 \beta_{11} + \cdots + 73 \beta_{5} ) / 2 \) (422b15 - 739b14 - 317b13 + 1503b12 + 1576b11 - 73b10 - 317b6 + 73b5) / 2
\(\nu^{12}\)	\(=\)	\( ( 317\beta_{9} - 317\beta_{7} - 2258\beta_{3} - 1129\beta_{2} - 4167\beta _1 - 6507 ) / 2 \) (317b9 - 317b7 - 2258b3 - 1129b2 - 4167*b1 - 6507) / 2
\(\nu^{13}\)	\(=\)	\( ( - 2242 \beta_{15} - 796 \beta_{14} + 1446 \beta_{13} - 7985 \beta_{12} + 3485 \beta_{11} + \cdots + 4500 \beta_{5} ) / 2 \) (-2242b15 - 796b14 + 1446b13 - 7985b12 + 3485b11 + 4500b10 - 1446b6 + 4500b5) / 2
\(\nu^{14}\)	\(=\)	\( ( 10227 \beta_{9} + 796 \beta_{8} + 9431 \beta_{7} + 2835 \beta_{4} - 2835 \beta_{3} - 16766 \beta_{2} + \cdots + 13931 ) / 2 \) (10227b9 + 796b8 + 9431b7 + 2835b4 - 2835b3 - 16766b2 + 9431*b1 + 13931) / 2
\(\nu^{15}\)	\(=\)	\( ( 9431\beta_{14} - 9431\beta_{13} - 33589\beta_{11} - 33589\beta_{10} - 16693\beta_{6} + 26067\beta_{5} ) / 2 \) (9431b14 - 9431b13 - 33589b11 - 33589b10 - 16693b6 + 26067b5) / 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
$3$	\( (T^{8} + 12 T^{6} + 32 T^{4} + \cdots + 4)^{2} \) (T^8 + 12T^6 + 32T^4 + 27*T^2 + 4)^2
$5$	\( (T^{2} + 1)^{8} \) (T^2 + 1)^8
$7$	\( (T^{8} - 37 T^{6} + \cdots + 1024)^{2} \) (T^8 - 37T^6 + 393T^4 - 1264*T^2 + 1024)^2
$11$	\( (T^{8} - 29 T^{6} + \cdots + 1024)^{2} \) (T^8 - 29T^6 + 261T^4 - 908*T^2 + 1024)^2
$13$	\( (T^{4} + 2 T^{3} - 36 T^{2} + \cdots - 2)^{4} \) (T^4 + 2T^3 - 36T^2 - 37*T - 2)^4
$17$	\( (T^{8} + 67 T^{6} + \cdots + 1024)^{2} \) (T^8 + 67T^6 + 1173T^4 + 3364*T^2 + 1024)^2
$19$	\( (T^{8} - 73 T^{6} + \cdots + 16)^{2} \) (T^8 - 73T^6 + 537T^4 - 904*T^2 + 16)^2
$23$	\( (T^{8} + 16 T^{6} + \cdots + 279841)^{2} \) (T^8 + 16T^6 + 510T^4 + 8464*T^2 + 279841)^2
$29$	\( (T^{4} + 21 T^{3} + \cdots + 376)^{4} \) (T^4 + 21T^3 + 152T^2 + 432*T + 376)^4
$31$	\( (T^{8} + 56 T^{6} + \cdots + 13924)^{2} \) (T^8 + 56T^6 + 1020T^4 + 7067*T^2 + 13924)^2
$37$	\( (T^{8} + 176 T^{6} + \cdots + 262144)^{2} \) (T^8 + 176T^6 + 6720T^4 + 76352*T^2 + 262144)^2
$41$	\( (T^{4} - 6 T^{3} + \cdots + 118)^{4} \) (T^4 - 6T^3 - 34T^2 + 27*T + 118)^4
$43$	\( (T^{8} - 220 T^{6} + \cdots + 5683456)^{2} \) (T^8 - 220T^6 + 16512T^4 - 513472*T^2 + 5683456)^2
$47$	\( (T^{8} + 283 T^{6} + \cdots + 11999296)^{2} \) (T^8 + 283T^6 + 25296T^4 + 923008*T^2 + 11999296)^2
$53$	\( (T^{8} + 252 T^{6} + \cdots + 82944)^{2} \) (T^8 + 252T^6 + 17136T^4 + 191808*T^2 + 82944)^2
$59$	\( (T^{2} + 12)^{8} \) (T^2 + 12)^8
$61$	\( (T^{8} + 395 T^{6} + \cdots + 19254544)^{2} \) (T^8 + 395T^6 + 48705T^4 + 1949144*T^2 + 19254544)^2
$67$	\( (T^{8} - 348 T^{6} + \cdots + 5308416)^{2} \) (T^8 - 348T^6 + 29376T^4 - 801792*T^2 + 5308416)^2
$71$	\( (T^{8} + 384 T^{6} + \cdots + 69455556)^{2} \) (T^8 + 384T^6 + 53244T^4 + 3182139*T^2 + 69455556)^2
$73$	\( (T^{4} - 17 T^{3} + \cdots + 376)^{4} \) (T^4 - 17T^3 - 12T^2 + 544*T + 376)^4
$79$	\( (T^{8} - 112 T^{6} + \cdots + 16384)^{2} \) (T^8 - 112T^6 + 2880T^4 - 15040*T^2 + 16384)^2
$83$	\( (T^{8} - 200 T^{6} + \cdots + 4096)^{2} \) (T^8 - 200T^6 + 10128T^4 - 130304*T^2 + 4096)^2
$89$	\( (T^{8} + 144 T^{6} + \cdots + 82944)^{2} \) (T^8 + 144T^6 + 4608T^4 + 46656*T^2 + 82944)^2
$97$	\( (T^{8} + 111 T^{6} + \cdots + 82944)^{2} \) (T^8 + 111T^6 + 3537T^4 + 34128*T^2 + 82944)^2
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