LMFDB - Newform orbit 4788.2.i.h

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4788,2,Mod(3457,4788)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4788, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4788.3457");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4788 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4788.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:	$38.2323724878$
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} + 48x^{14} + 860x^{12} + 7192x^{10} + 28448x^{8} + 45264x^{6} + 12088x^{4} + 784x^{2} + 4 $ x^16 + 48x^14 + 860x^12 + 7192x^10 + 28448x^8 + 45264x^6 + 12088x^4 + 784*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{17} $
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_{12} q^{5} + ( - \beta_{10} - \beta_{3} + 1) q^{7} + \beta_{5} q^{11} + ( - \beta_{6} + \beta_{2}) q^{13} + (\beta_{12} + \beta_{4}) q^{17} + ( - \beta_{11} - \beta_{10} - \beta_{6}) q^{19} - \beta_{7} q^{23}+ \cdots + ( - \beta_{8} + \beta_{7} - \beta_{5}) q^{95}+O(q^{100}) $ q - b12 * q^5 + (-b10 - b3 + 1) * q^7 + b5 * q^11 + (-b6 + b2) * q^13 + (b12 + b4) * q^17 + (-b11 - b10 - b6) * q^19 - b7 * q^23 + (-b3 + 1) * q^25 + b15 * q^29 + (b6 + b2) * q^31 + (-b12 + b7 + b4) * q^35 - b13 * q^37 + b1 * q^41 + (3b3 + 4) q^43 + (3b12 - 3b4) * q^47 + (-2b11 - 4b3 - 1) * q^49 + b8 * q^53 + (-3b11 + b10) q^55 + b9 * q^59 + (-b11 + b10) * q^61 + (-b15 - b8) * q^65 - 2b14 q^67 + b8 * q^71 + (-5b11 + 3b10) * q^73 + (-b7 + 2b5 + 2b4) * q^77 - b13 * q^79 + 3b12 q^83 + (5b3 + 6) q^85 - b1 * q^89 + (-2b14 - b13 + b6 + b2) q^91 + (-b8 + b7 - b5) * q^95
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{7} + 16 q^{25} + 64 q^{43} - 16 q^{49} + 96 q^{85}+O(q^{100}) $ 16 * q + 16 * q^7 + 16 * q^25 + 64 * q^43 - 16 * q^49 + 96 * q^85

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} + 48x^{14} + 860x^{12} + 7192x^{10} + 28448x^{8} + 45264x^{6} + 12088x^{4} + 784x^{2} + 4 $ x^16 + 48*x^14 + 860*x^12 + 7192*x^10 + 28448*x^8 + 45264*x^6 + 12088*x^4 + 784*x^2 + 4 :

$\beta_{1}$	$=$	$ ( - 5621 \nu^{14} - 264143 \nu^{12} - 4587660 \nu^{10} - 36675050 \nu^{8} - 136487358 \nu^{6} + \cdots + 49768688 ) / 5153652 $ (-5621v^14 - 264143v^12 - 4587660v^10 - 36675050v^8 - 136487358v^6 - 203991090v^4 - 53001380*v^2 + 49768688) / 5153652
$\beta_{2}$	$=$	$ ( - 4715 \nu^{14} - 225881 \nu^{12} - 4043958 \nu^{10} - 33963278 \nu^{8} - 137351826 \nu^{6} + \cdots - 12503284 ) / 2122092 $ (-4715v^14 - 225881v^12 - 4043958v^10 - 33963278v^8 - 137351826v^6 - 239653518v^4 - 119976296*v^2 - 12503284) / 2122092
$\beta_{3}$	$=$	$ ( - 1307 \nu^{14} - 62723 \nu^{12} - 1123548 \nu^{10} - 9394838 \nu^{8} - 37167174 \nu^{6} + \cdots - 513400 ) / 303156 $ (-1307v^14 - 62723v^12 - 1123548v^10 - 9394838v^8 - 37167174v^6 - 59150814v^4 - 15528428*v^2 - 513400) / 303156
$\beta_{4}$	$=$	$ ( 36907 \nu^{15} + 1789930 \nu^{13} + 32617926 \nu^{11} + 281022442 \nu^{9} + 1178309178 \nu^{7} + \cdots + 135825596 \nu ) / 4244184 $ (36907v^15 + 1789930v^13 + 32617926v^11 + 281022442v^9 + 1178309178v^7 + 2164203840v^5 + 1180351936v^3 + 135825596v) / 4244184
$\beta_{5}$	$=$	$ ( - 584417 \nu^{14} - 28003475 \nu^{12} - 500274870 \nu^{10} - 4161686408 \nu^{8} + \cdots - 179455828 ) / 36075564 $ (-584417v^14 - 28003475v^12 - 500274870v^10 - 4161686408v^8 - 16282347570v^6 - 25131698142v^4 - 5141894840*v^2 - 179455828) / 36075564
$\beta_{6}$	$=$	$ ( - 20563 \nu^{14} - 984604 \nu^{12} - 17568924 \nu^{10} - 145853581 \nu^{8} - 568402110 \nu^{6} + \cdots - 3309428 ) / 1061046 $ (-20563v^14 - 984604v^12 - 17568924v^10 - 145853581v^8 - 568402110v^6 - 868366848v^4 - 158627692*v^2 - 3309428) / 1061046
$\beta_{7}$	$=$	$ ( - 752323 \nu^{14} - 35994757 \nu^{12} - 641491374 \nu^{10} - 5314271974 \nu^{8} + \cdots + 37865728 ) / 36075564 $ (-752323v^14 - 35994757v^12 - 641491374v^10 - 5314271974v^8 - 20618465814v^6 - 31065217950v^4 - 4646804488*v^2 + 37865728) / 36075564
$\beta_{8}$	$=$	$ ( - 8864 \nu^{15} - 426770 \nu^{13} - 7684881 \nu^{11} - 64850468 \nu^{9} - 261355302 \nu^{7} + \cdots - 31128568 \nu ) / 606312 $ (-8864v^15 - 426770v^13 - 7684881v^11 - 64850468v^9 - 261355302v^7 - 438171000v^5 - 169932614v^3 - 31128568v) / 606312
$\beta_{9}$	$=$	$ ( - 109649 \nu^{14} - 5270348 \nu^{12} - 94607160 \nu^{10} - 793219535 \nu^{8} - 3147603318 \nu^{6} + \cdots - 42589426 ) / 2576826 $ (-109649v^14 - 5270348v^12 - 94607160v^10 - 793219535v^8 - 3147603318v^6 - 5022944502v^4 - 1319200352*v^2 - 42589426) / 2576826
$\beta_{10}$	$=$	$ ( 403567 \nu^{15} + 19362424 \nu^{13} + 346645500 \nu^{11} + 2894893780 \nu^{9} + \cdots + 173827424 \nu ) / 10307304 $ (403567v^15 + 19362424v^13 + 346645500v^11 + 2894893780v^9 + 11417518998v^7 + 18016980960v^5 + 4470885100v^3 + 173827424v) / 10307304
$\beta_{11}$	$=$	$ ( 480349 \nu^{15} + 23054302 \nu^{13} + 412993974 \nu^{11} + 3453054772 \nu^{9} + \cdots + 392545352 \nu ) / 10307304 $ (480349v^15 + 23054302v^13 + 412993974v^11 + 3453054772v^9 + 13654773270v^7 + 21719618148v^5 + 5805909328v^3 + 392545352v) / 10307304
$\beta_{12}$	$=$	$ ( - 253499 \nu^{15} - 12154460 \nu^{13} - 217360203 \nu^{11} - 1811514836 \nu^{9} + \cdots - 104900536 \nu ) / 4244184 $ (-253499v^15 - 12154460v^13 - 217360203v^11 - 1811514836v^9 - 7114075632v^7 - 11091454764v^5 - 2477732150v^3 - 104900536v) / 4244184
$\beta_{13}$	$=$	$ ( 1240375 \nu^{15} + 59629117 \nu^{13} + 1071080967 \nu^{11} + 8998400482 \nu^{9} + \cdots + 1478547368 \nu ) / 12025188 $ (1240375v^15 + 59629117v^13 + 1071080967v^11 + 8998400482v^9 + 35928092988v^7 + 58623187386v^5 + 18678254470v^3 + 1478547368v) / 12025188
$\beta_{14}$	$=$	$ ( - 12603541 \nu^{15} - 604487902 \nu^{13} - 10816013568 \nu^{11} - 90234830716 \nu^{9} + \cdots - 6558814088 \nu ) / 72151128 $ (-12603541v^15 - 604487902v^13 - 10816013568v^11 - 90234830716v^9 - 355153559202v^7 - 557302576476v^5 - 132275790124v^3 - 6558814088v) / 72151128
$\beta_{15}$	$=$	$ ( 127043 \nu^{15} + 6096770 \nu^{13} + 109194729 \nu^{11} + 912574814 \nu^{9} + 3604738788 \nu^{7} + \cdots + 80946700 \nu ) / 606312 $ (127043v^15 + 6096770v^13 + 109194729v^11 + 912574814v^9 + 3604738788v^7 + 5713316412v^5 + 1476512402v^3 + 80946700v) / 606312

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

$\nu$	$=$	$ ( \beta_{13} - \beta_{11} - \beta_{10} - 2\beta_{4} ) / 4 $ (b13 - b11 - b10 - 2*b4) / 4
$\nu^{2}$	$=$	$ ( -\beta_{6} + \beta_{5} + \beta_{3} - \beta_{2} + \beta _1 - 12 ) / 2 $ (-b6 + b5 + b3 - b2 + b1 - 12) / 2
$\nu^{3}$	$=$	$ ( -6\beta_{15} - 11\beta_{13} - 10\beta_{12} + 19\beta_{11} + 17\beta_{10} + 28\beta_{4} ) / 4 $ (-6b15 - 11b13 - 10b12 + 19b11 + 17b10 + 28b4) / 4
$\nu^{4}$	$=$	$ \beta_{9} + 2\beta_{7} + 10\beta_{6} - 15\beta_{5} - 11\beta_{3} + 9\beta_{2} - 7\beta _1 + 73 $ b9 + 2b7 + 10b6 - 15b5 - 11b3 + 9b2 - 7b1 + 73
$\nu^{5}$	$=$	$ ( 65 \beta_{15} - 14 \beta_{14} + 72 \beta_{13} + 126 \beta_{12} - 177 \beta_{11} - 153 \beta_{10} + \cdots - 207 \beta_{4} ) / 2 $ (65b15 - 14b14 + 72b13 + 126b12 - 177b11 - 153b10 - 5b8 - 207b4) / 2
$\nu^{6}$	$=$	$ -28\beta_{9} - 53\beta_{7} - 170\beta_{6} + 290\beta_{5} + 268\beta_{3} - 169\beta_{2} + 104\beta _1 - 1041 $ -28b9 - 53b7 - 170b6 + 290b5 + 268b3 - 169b2 + 104*b1 - 1041
$\nu^{7}$	$=$	$ ( - 1183 \beta_{15} + 453 \beta_{14} - 1050 \beta_{13} - 2630 \beta_{12} + 3272 \beta_{11} + \cdots + 3245 \beta_{4} ) / 2 $ (-1183b15 + 453b14 - 1050b13 - 2630b12 + 3272b11 + 2551b10 + 161b8 + 3245b4) / 2
$\nu^{8}$	$=$	$ 610\beta_{9} + 1174\beta_{7} + 2742\beta_{6} - 5170\beta_{5} - 5804\beta_{3} + 3140\beta_{2} - 1632\beta _1 + 16116 $ 610b9 + 1174b7 + 2742b6 - 5170b5 - 5804b3 + 3140b2 - 1632*b1 + 16116
$\nu^{9}$	$=$	$ ( 20706 \beta_{15} - 10560 \beta_{14} + 16307 \beta_{13} + 51168 \beta_{12} - 59701 \beta_{11} + \cdots - 52964 \beta_{4} ) / 2 $ (20706b15 - 10560b14 + 16307b13 + 51168b12 - 59701b11 - 41485b10 - 3966b8 - 52964b4) / 2
$\nu^{10}$	$=$	$ - 12120 \beta_{9} - 24168 \beta_{7} - 43653 \beta_{6} + 90421 \beta_{5} + 115957 \beta_{3} + \cdots - 260926 $ -12120b9 - 24168b7 - 43653b6 + 90421b5 + 115957b3 - 57603b2 + 26571*b1 - 260926
$\nu^{11}$	$=$	$ ( - 359480 \beta_{15} + 217378 \beta_{14} - 263135 \beta_{13} - 959878 \beta_{12} + \cdots + 886482 \beta_{4} ) / 2 $ (-359480b15 + 217378b14 - 263135b13 - 959878b12 + 1079039b11 + 673635b10 + 86526b8 + 886482b4) / 2
$\nu^{12}$	$=$	$ 230080 \beta_{9} + 475114 \beta_{7} + 698076 \beta_{6} - 1576116 \beta_{5} - 2211682 \beta_{3} + \cdots + 4337374 $ 230080b9 + 475114b7 + 698076b6 - 1576116b5 - 2211682b3 + 1046042b2 - 443058*b1 + 4337374
$\nu^{13}$	$=$	$ 3121313 \beta_{15} - 2106124 \beta_{14} + 2176409 \beta_{13} + 8813206 \beta_{12} + \cdots - 7536449 \beta_{4} $ 3121313b15 - 2106124b14 + 2176409b13 + 8813206b12 - 9685764b11 - 5515474b10 - 879177b8 - 7536449b4
$\nu^{14}$	$=$	$ - 4256486 \beta_{9} - 9047168 \beta_{7} - 11296978 \beta_{6} + 27497108 \beta_{5} + 41047822 \beta_{3} + \cdots - 73321926 $ -4256486b9 - 9047168b7 - 11296978b6 + 27497108b5 + 41047822b3 - 18847964b2 + 7505290*b1 - 73321926
$\nu^{15}$	$=$	$ - 54331355 \beta_{15} + 39468547 \beta_{14} - 36620811 \beta_{13} - 159698772 \beta_{12} + \cdots + 129464559 \beta_{4} $ -54331355b15 + 39468547b14 - 36620811b13 - 159698772b12 + 173022417b11 + 91385856b10 + 17079147b8 + 129464559b4

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4788\mathbb{Z}\right)^\times$.

$n$	$533$	$1009$	$2395$	$4105$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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} $

3457.1

	−	3.58985i
	−	0.466174i
		0.299193i
	−	2.82448i
		2.34875i
	−	1.92240i
	−	0.0746361i
		4.19650i
		0.0746361i
	−	4.19650i
	−	2.34875i
		1.92240i
	−	0.299193i
		2.82448i
		3.58985i
		0.466174i

−

2.32685i

−0.414214

−

2.61313i

3457.2

−

2.32685i

−0.414214

−

2.61313i

3457.3

−

2.32685i

−0.414214

2.61313i

3457.4

−

2.32685i

−0.414214

2.61313i

3457.5

−

1.60804i

2.41421

−

1.08239i

3457.6

−

1.60804i

2.41421

−

1.08239i

3457.7

−

1.60804i

2.41421

1.08239i

3457.8

−

1.60804i

2.41421

1.08239i

3457.9

1.60804i

2.41421

−

1.08239i

3457.10

1.60804i

2.41421

−

1.08239i

3457.11

1.60804i

2.41421

1.08239i

3457.12

1.60804i

2.41421

1.08239i

3457.13

2.32685i

−0.414214

−

2.61313i

3457.14

2.32685i

−0.414214

−

2.61313i

3457.15

2.32685i

−0.414214

2.61313i

3457.16

2.32685i

−0.414214

2.61313i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
7.b	odd	2	1	inner
19.b	odd	2	1	inner
21.c	even	2	1	inner
57.d	even	2	1	inner
133.c	even	2	1	inner
399.h	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	4788.2.i.h	✓	16
3.b	odd	2	1	inner	4788.2.i.h	✓	16
7.b	odd	2	1	inner	4788.2.i.h	✓	16
19.b	odd	2	1	inner	4788.2.i.h	✓	16
21.c	even	2	1	inner	4788.2.i.h	✓	16
57.d	even	2	1	inner	4788.2.i.h	✓	16
133.c	even	2	1	inner	4788.2.i.h	✓	16
399.h	odd	2	1	inner	4788.2.i.h	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
4788.2.i.h	✓	16	1.a	even	1	1	trivial
4788.2.i.h	✓	16	3.b	odd	2	1	inner
4788.2.i.h	✓	16	7.b	odd	2	1	inner
4788.2.i.h	✓	16	19.b	odd	2	1	inner
4788.2.i.h	✓	16	21.c	even	2	1	inner
4788.2.i.h	✓	16	57.d	even	2	1	inner
4788.2.i.h	✓	16	133.c	even	2	1	inner
4788.2.i.h	✓	16	399.h	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(4788, [\chi])$:

$ T_{5}^{4} + 8T_{5}^{2} + 14 $

$ T_{13}^{4} - 44T_{13}^{2} + 356 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ T^{16} $ T^16
$5$	$ (T^{4} + 8 T^{2} + 14)^{4} $ (T^4 + 8*T^2 + 14)^4
$7$	$ (T^{4} - 4 T^{3} + 10 T^{2} + \cdots + 49)^{4} $ (T^4 - 4T^3 + 10T^2 - 28*T + 49)^4
$11$	$ (T^{4} - 24 T^{2} + 112)^{4} $ (T^4 - 24*T^2 + 112)^4
$13$	$ (T^{4} - 44 T^{2} + 356)^{4} $ (T^4 - 44*T^2 + 356)^4
$17$	$ (T^{4} + 32 T^{2} + 14)^{4} $ (T^4 + 32*T^2 + 14)^4
$19$	$ (T^{8} - 12 T^{6} + \cdots + 130321)^{2} $ (T^8 - 12T^6 + 246T^4 - 4332*T^2 + 130321)^2
$23$	$ (T^{4} - 40 T^{2} + 112)^{4} $ (T^4 - 40*T^2 + 112)^4
$29$	$ (T^{4} + 96 T^{2} + 1246)^{4} $ (T^4 + 96*T^2 + 1246)^4
$31$	$ (T^{4} - 68 T^{2} + 356)^{4} $ (T^4 - 68*T^2 + 356)^4
$37$	$ (T^{4} + 112 T^{2} + 2848)^{4} $ (T^4 + 112*T^2 + 2848)^4
$41$	$ (T^{4} - 200 T^{2} + 9968)^{4} $ (T^4 - 200*T^2 + 9968)^4
$43$	$ (T^{2} - 8 T - 2)^{8} $ (T^2 - 8*T - 2)^8
$47$	$ (T^{4} + 144 T^{2} + 1134)^{4} $ (T^4 + 144*T^2 + 1134)^4
$53$	$ (T^{4} + 104 T^{2} + 1246)^{4} $ (T^4 + 104*T^2 + 1246)^4
$59$	$ (T^{4} - 400 T^{2} + 39872)^{4} $ (T^4 - 400*T^2 + 39872)^4
$61$	$ (T^{4} + 16 T^{2} + 32)^{4} $ (T^4 + 16*T^2 + 32)^4
$67$	$ (T^{4} + 224 T^{2} + 11392)^{4} $ (T^4 + 224*T^2 + 11392)^4
$71$	$ (T^{4} + 104 T^{2} + 1246)^{4} $ (T^4 + 104*T^2 + 1246)^4
$73$	$ (T^{4} + 272 T^{2} + 16928)^{4} $ (T^4 + 272*T^2 + 16928)^4
$79$	$ (T^{4} + 112 T^{2} + 2848)^{4} $ (T^4 + 112*T^2 + 2848)^4
$83$	$ (T^{4} + 72 T^{2} + 1134)^{4} $ (T^4 + 72*T^2 + 1134)^4
$89$	$ (T^{4} - 200 T^{2} + 9968)^{4} $ (T^4 - 200*T^2 + 9968)^4
$97$	$ T^{16} $ T^16
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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:	\( N \)	\(=\)	\( 4788 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4788.i (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{12} q^{5} + ( - \beta_{10} - \beta_{3} + 1) q^{7} + \beta_{5} q^{11} + ( - \beta_{6} + \beta_{2}) q^{13} + (\beta_{12} + \beta_{4}) q^{17} + ( - \beta_{11} - \beta_{10} - \beta_{6}) q^{19} - \beta_{7} q^{23}+ \cdots + ( - \beta_{8} + \beta_{7} - \beta_{5}) q^{95}+O(q^{100}) \) q - b12 * q^5 + (-b10 - b3 + 1) * q^7 + b5 * q^11 + (-b6 + b2) * q^13 + (b12 + b4) * q^17 + (-b11 - b10 - b6) * q^19 - b7 * q^23 + (-b3 + 1) * q^25 + b15 * q^29 + (b6 + b2) * q^31 + (-b12 + b7 + b4) * q^35 - b13 * q^37 + b1 * q^41 + (3b3 + 4) q^43 + (3b12 - 3b4) * q^47 + (-2b11 - 4b3 - 1) * q^49 + b8 * q^53 + (-3b11 + b10) q^55 + b9 * q^59 + (-b11 + b10) * q^61 + (-b15 - b8) * q^65 - 2b14 q^67 + b8 * q^71 + (-5b11 + 3b10) * q^73 + (-b7 + 2b5 + 2b4) * q^77 - b13 * q^79 + 3b12 q^83 + (5b3 + 6) q^85 - b1 * q^89 + (-2b14 - b13 + b6 + b2) q^91 + (-b8 + b7 - b5) * q^95
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{7} + 16 q^{25} + 64 q^{43} - 16 q^{49} + 96 q^{85}+O(q^{100}) \) 16 * q + 16 * q^7 + 16 * q^25 + 64 * q^43 - 16 * q^49 + 96 * q^85

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Newspace parameters

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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	4788.2.i.h

Level	$4788$
Weight	$2$
Character orbit	4788.i
Analytic conductor	$38.232$
Analytic rank	$0$
Dimension	$16$
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(38.2323724878\)
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:	\( x^{16} + 48x^{14} + 860x^{12} + 7192x^{10} + 28448x^{8} + 45264x^{6} + 12088x^{4} + 784x^{2} + 4 \) x^16 + 48x^14 + 860x^12 + 7192x^10 + 28448x^8 + 45264x^6 + 12088x^4 + 784*x^2 + 4
Coefficient ring:	\(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index:	\( 2^{17} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( - 5621 \nu^{14} - 264143 \nu^{12} - 4587660 \nu^{10} - 36675050 \nu^{8} - 136487358 \nu^{6} + \cdots + 49768688 ) / 5153652 \) (-5621v^14 - 264143v^12 - 4587660v^10 - 36675050v^8 - 136487358v^6 - 203991090v^4 - 53001380*v^2 + 49768688) / 5153652
\(\beta_{2}\)	\(=\)	\( ( - 4715 \nu^{14} - 225881 \nu^{12} - 4043958 \nu^{10} - 33963278 \nu^{8} - 137351826 \nu^{6} + \cdots - 12503284 ) / 2122092 \) (-4715v^14 - 225881v^12 - 4043958v^10 - 33963278v^8 - 137351826v^6 - 239653518v^4 - 119976296*v^2 - 12503284) / 2122092
\(\beta_{3}\)	\(=\)	\( ( - 1307 \nu^{14} - 62723 \nu^{12} - 1123548 \nu^{10} - 9394838 \nu^{8} - 37167174 \nu^{6} + \cdots - 513400 ) / 303156 \) (-1307v^14 - 62723v^12 - 1123548v^10 - 9394838v^8 - 37167174v^6 - 59150814v^4 - 15528428*v^2 - 513400) / 303156
\(\beta_{4}\)	\(=\)	\( ( 36907 \nu^{15} + 1789930 \nu^{13} + 32617926 \nu^{11} + 281022442 \nu^{9} + 1178309178 \nu^{7} + \cdots + 135825596 \nu ) / 4244184 \) (36907v^15 + 1789930v^13 + 32617926v^11 + 281022442v^9 + 1178309178v^7 + 2164203840v^5 + 1180351936v^3 + 135825596v) / 4244184
\(\beta_{5}\)	\(=\)	\( ( - 584417 \nu^{14} - 28003475 \nu^{12} - 500274870 \nu^{10} - 4161686408 \nu^{8} + \cdots - 179455828 ) / 36075564 \) (-584417v^14 - 28003475v^12 - 500274870v^10 - 4161686408v^8 - 16282347570v^6 - 25131698142v^4 - 5141894840*v^2 - 179455828) / 36075564
\(\beta_{6}\)	\(=\)	\( ( - 20563 \nu^{14} - 984604 \nu^{12} - 17568924 \nu^{10} - 145853581 \nu^{8} - 568402110 \nu^{6} + \cdots - 3309428 ) / 1061046 \) (-20563v^14 - 984604v^12 - 17568924v^10 - 145853581v^8 - 568402110v^6 - 868366848v^4 - 158627692*v^2 - 3309428) / 1061046
\(\beta_{7}\)	\(=\)	\( ( - 752323 \nu^{14} - 35994757 \nu^{12} - 641491374 \nu^{10} - 5314271974 \nu^{8} + \cdots + 37865728 ) / 36075564 \) (-752323v^14 - 35994757v^12 - 641491374v^10 - 5314271974v^8 - 20618465814v^6 - 31065217950v^4 - 4646804488*v^2 + 37865728) / 36075564
\(\beta_{8}\)	\(=\)	\( ( - 8864 \nu^{15} - 426770 \nu^{13} - 7684881 \nu^{11} - 64850468 \nu^{9} - 261355302 \nu^{7} + \cdots - 31128568 \nu ) / 606312 \) (-8864v^15 - 426770v^13 - 7684881v^11 - 64850468v^9 - 261355302v^7 - 438171000v^5 - 169932614v^3 - 31128568v) / 606312
\(\beta_{9}\)	\(=\)	\( ( - 109649 \nu^{14} - 5270348 \nu^{12} - 94607160 \nu^{10} - 793219535 \nu^{8} - 3147603318 \nu^{6} + \cdots - 42589426 ) / 2576826 \) (-109649v^14 - 5270348v^12 - 94607160v^10 - 793219535v^8 - 3147603318v^6 - 5022944502v^4 - 1319200352*v^2 - 42589426) / 2576826
\(\beta_{10}\)	\(=\)	\( ( 403567 \nu^{15} + 19362424 \nu^{13} + 346645500 \nu^{11} + 2894893780 \nu^{9} + \cdots + 173827424 \nu ) / 10307304 \) (403567v^15 + 19362424v^13 + 346645500v^11 + 2894893780v^9 + 11417518998v^7 + 18016980960v^5 + 4470885100v^3 + 173827424v) / 10307304
\(\beta_{11}\)	\(=\)	\( ( 480349 \nu^{15} + 23054302 \nu^{13} + 412993974 \nu^{11} + 3453054772 \nu^{9} + \cdots + 392545352 \nu ) / 10307304 \) (480349v^15 + 23054302v^13 + 412993974v^11 + 3453054772v^9 + 13654773270v^7 + 21719618148v^5 + 5805909328v^3 + 392545352v) / 10307304
\(\beta_{12}\)	\(=\)	\( ( - 253499 \nu^{15} - 12154460 \nu^{13} - 217360203 \nu^{11} - 1811514836 \nu^{9} + \cdots - 104900536 \nu ) / 4244184 \) (-253499v^15 - 12154460v^13 - 217360203v^11 - 1811514836v^9 - 7114075632v^7 - 11091454764v^5 - 2477732150v^3 - 104900536v) / 4244184
\(\beta_{13}\)	\(=\)	\( ( 1240375 \nu^{15} + 59629117 \nu^{13} + 1071080967 \nu^{11} + 8998400482 \nu^{9} + \cdots + 1478547368 \nu ) / 12025188 \) (1240375v^15 + 59629117v^13 + 1071080967v^11 + 8998400482v^9 + 35928092988v^7 + 58623187386v^5 + 18678254470v^3 + 1478547368v) / 12025188
\(\beta_{14}\)	\(=\)	\( ( - 12603541 \nu^{15} - 604487902 \nu^{13} - 10816013568 \nu^{11} - 90234830716 \nu^{9} + \cdots - 6558814088 \nu ) / 72151128 \) (-12603541v^15 - 604487902v^13 - 10816013568v^11 - 90234830716v^9 - 355153559202v^7 - 557302576476v^5 - 132275790124v^3 - 6558814088v) / 72151128
\(\beta_{15}\)	\(=\)	\( ( 127043 \nu^{15} + 6096770 \nu^{13} + 109194729 \nu^{11} + 912574814 \nu^{9} + 3604738788 \nu^{7} + \cdots + 80946700 \nu ) / 606312 \) (127043v^15 + 6096770v^13 + 109194729v^11 + 912574814v^9 + 3604738788v^7 + 5713316412v^5 + 1476512402v^3 + 80946700v) / 606312

\(\nu\)	\(=\)	\( ( \beta_{13} - \beta_{11} - \beta_{10} - 2\beta_{4} ) / 4 \) (b13 - b11 - b10 - 2*b4) / 4
\(\nu^{2}\)	\(=\)	\( ( -\beta_{6} + \beta_{5} + \beta_{3} - \beta_{2} + \beta _1 - 12 ) / 2 \) (-b6 + b5 + b3 - b2 + b1 - 12) / 2
\(\nu^{3}\)	\(=\)	\( ( -6\beta_{15} - 11\beta_{13} - 10\beta_{12} + 19\beta_{11} + 17\beta_{10} + 28\beta_{4} ) / 4 \) (-6b15 - 11b13 - 10b12 + 19b11 + 17b10 + 28b4) / 4
\(\nu^{4}\)	\(=\)	\( \beta_{9} + 2\beta_{7} + 10\beta_{6} - 15\beta_{5} - 11\beta_{3} + 9\beta_{2} - 7\beta _1 + 73 \) b9 + 2b7 + 10b6 - 15b5 - 11b3 + 9b2 - 7b1 + 73
\(\nu^{5}\)	\(=\)	\( ( 65 \beta_{15} - 14 \beta_{14} + 72 \beta_{13} + 126 \beta_{12} - 177 \beta_{11} - 153 \beta_{10} + \cdots - 207 \beta_{4} ) / 2 \) (65b15 - 14b14 + 72b13 + 126b12 - 177b11 - 153b10 - 5b8 - 207b4) / 2
\(\nu^{6}\)	\(=\)	\( -28\beta_{9} - 53\beta_{7} - 170\beta_{6} + 290\beta_{5} + 268\beta_{3} - 169\beta_{2} + 104\beta _1 - 1041 \) -28b9 - 53b7 - 170b6 + 290b5 + 268b3 - 169b2 + 104*b1 - 1041
\(\nu^{7}\)	\(=\)	\( ( - 1183 \beta_{15} + 453 \beta_{14} - 1050 \beta_{13} - 2630 \beta_{12} + 3272 \beta_{11} + \cdots + 3245 \beta_{4} ) / 2 \) (-1183b15 + 453b14 - 1050b13 - 2630b12 + 3272b11 + 2551b10 + 161b8 + 3245b4) / 2
\(\nu^{8}\)	\(=\)	\( 610\beta_{9} + 1174\beta_{7} + 2742\beta_{6} - 5170\beta_{5} - 5804\beta_{3} + 3140\beta_{2} - 1632\beta _1 + 16116 \) 610b9 + 1174b7 + 2742b6 - 5170b5 - 5804b3 + 3140b2 - 1632*b1 + 16116
\(\nu^{9}\)	\(=\)	\( ( 20706 \beta_{15} - 10560 \beta_{14} + 16307 \beta_{13} + 51168 \beta_{12} - 59701 \beta_{11} + \cdots - 52964 \beta_{4} ) / 2 \) (20706b15 - 10560b14 + 16307b13 + 51168b12 - 59701b11 - 41485b10 - 3966b8 - 52964b4) / 2
\(\nu^{10}\)	\(=\)	\( - 12120 \beta_{9} - 24168 \beta_{7} - 43653 \beta_{6} + 90421 \beta_{5} + 115957 \beta_{3} + \cdots - 260926 \) -12120b9 - 24168b7 - 43653b6 + 90421b5 + 115957b3 - 57603b2 + 26571*b1 - 260926
\(\nu^{11}\)	\(=\)	\( ( - 359480 \beta_{15} + 217378 \beta_{14} - 263135 \beta_{13} - 959878 \beta_{12} + \cdots + 886482 \beta_{4} ) / 2 \) (-359480b15 + 217378b14 - 263135b13 - 959878b12 + 1079039b11 + 673635b10 + 86526b8 + 886482b4) / 2
\(\nu^{12}\)	\(=\)	\( 230080 \beta_{9} + 475114 \beta_{7} + 698076 \beta_{6} - 1576116 \beta_{5} - 2211682 \beta_{3} + \cdots + 4337374 \) 230080b9 + 475114b7 + 698076b6 - 1576116b5 - 2211682b3 + 1046042b2 - 443058*b1 + 4337374
\(\nu^{13}\)	\(=\)	\( 3121313 \beta_{15} - 2106124 \beta_{14} + 2176409 \beta_{13} + 8813206 \beta_{12} + \cdots - 7536449 \beta_{4} \) 3121313b15 - 2106124b14 + 2176409b13 + 8813206b12 - 9685764b11 - 5515474b10 - 879177b8 - 7536449b4
\(\nu^{14}\)	\(=\)	\( - 4256486 \beta_{9} - 9047168 \beta_{7} - 11296978 \beta_{6} + 27497108 \beta_{5} + 41047822 \beta_{3} + \cdots - 73321926 \) -4256486b9 - 9047168b7 - 11296978b6 + 27497108b5 + 41047822b3 - 18847964b2 + 7505290*b1 - 73321926
\(\nu^{15}\)	\(=\)	\( - 54331355 \beta_{15} + 39468547 \beta_{14} - 36620811 \beta_{13} - 159698772 \beta_{12} + \cdots + 129464559 \beta_{4} \) -54331355b15 + 39468547b14 - 36620811b13 - 159698772b12 + 173022417b11 + 91385856b10 + 17079147b8 + 129464559b4

\(n\)	\(533\)	\(1009\)	\(2395\)	\(4105\)
\(\chi(n)\)	\(1\)	\(-1\)	\(1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 3457.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( T^{16} \) T^16
$5$	\( (T^{4} + 8 T^{2} + 14)^{4} \) (T^4 + 8*T^2 + 14)^4
$7$	\( (T^{4} - 4 T^{3} + 10 T^{2} + \cdots + 49)^{4} \) (T^4 - 4T^3 + 10T^2 - 28*T + 49)^4
$11$	\( (T^{4} - 24 T^{2} + 112)^{4} \) (T^4 - 24*T^2 + 112)^4
$13$	\( (T^{4} - 44 T^{2} + 356)^{4} \) (T^4 - 44*T^2 + 356)^4
$17$	\( (T^{4} + 32 T^{2} + 14)^{4} \) (T^4 + 32*T^2 + 14)^4
$19$	\( (T^{8} - 12 T^{6} + \cdots + 130321)^{2} \) (T^8 - 12T^6 + 246T^4 - 4332*T^2 + 130321)^2
$23$	\( (T^{4} - 40 T^{2} + 112)^{4} \) (T^4 - 40*T^2 + 112)^4
$29$	\( (T^{4} + 96 T^{2} + 1246)^{4} \) (T^4 + 96*T^2 + 1246)^4
$31$	\( (T^{4} - 68 T^{2} + 356)^{4} \) (T^4 - 68*T^2 + 356)^4
$37$	\( (T^{4} + 112 T^{2} + 2848)^{4} \) (T^4 + 112*T^2 + 2848)^4
$41$	\( (T^{4} - 200 T^{2} + 9968)^{4} \) (T^4 - 200*T^2 + 9968)^4
$43$	\( (T^{2} - 8 T - 2)^{8} \) (T^2 - 8*T - 2)^8
$47$	\( (T^{4} + 144 T^{2} + 1134)^{4} \) (T^4 + 144*T^2 + 1134)^4
$53$	\( (T^{4} + 104 T^{2} + 1246)^{4} \) (T^4 + 104*T^2 + 1246)^4
$59$	\( (T^{4} - 400 T^{2} + 39872)^{4} \) (T^4 - 400*T^2 + 39872)^4
$61$	\( (T^{4} + 16 T^{2} + 32)^{4} \) (T^4 + 16*T^2 + 32)^4
$67$	\( (T^{4} + 224 T^{2} + 11392)^{4} \) (T^4 + 224*T^2 + 11392)^4
$71$	\( (T^{4} + 104 T^{2} + 1246)^{4} \) (T^4 + 104*T^2 + 1246)^4
$73$	\( (T^{4} + 272 T^{2} + 16928)^{4} \) (T^4 + 272*T^2 + 16928)^4
$79$	\( (T^{4} + 112 T^{2} + 2848)^{4} \) (T^4 + 112*T^2 + 2848)^4
$83$	\( (T^{4} + 72 T^{2} + 1134)^{4} \) (T^4 + 72*T^2 + 1134)^4
$89$	\( (T^{4} - 200 T^{2} + 9968)^{4} \) (T^4 - 200*T^2 + 9968)^4
$97$	\( T^{16} \) T^16
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