LMFDB - Newform orbit 3168.2.h.i

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3168,2,Mod(2287,3168)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3168, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 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("3168.2287");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3168 = 2^{5} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3168.h (of order $2$, degree $1$, not 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:	$25.2966073603$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	16.0.3342602057661458415616.4
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 20x^{14} + 164x^{12} - 666x^{10} + 1300x^{8} - 924x^{6} + 273x^{4} + 404x^{2} + 64 $ x^16 - 20x^14 + 164x^12 - 666x^10 + 1300x^8 - 924x^6 + 273x^4 + 404*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{26} $
Twist minimal:	no (minimal twist has level 792)
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^{5} - \beta_{14} q^{7} + ( - \beta_{13} - \beta_{7}) q^{11} + (\beta_{14} - \beta_{11}) q^{13} - \beta_{7} q^{17} - \beta_{8} q^{19} + \beta_{5} q^{23} + q^{25} + \beta_{2} q^{29} + \beta_{12} q^{31}+ \cdots + ( - 3 \beta_{3} - 1) q^{97}+O(q^{100}) $ q + b4 * q^5 - b14 * q^7 + (-b13 - b7) * q^11 + (b14 - b11) * q^13 - b7 * q^17 - b8 * q^19 + b5 * q^23 + q^25 + b2 * q^29 + b12 * q^31 - 2b9 q^35 - b10 * q^37 + (2b9 - b7) q^41 + b6 * q^43 - 3b4 q^47 + (b3 - 2) * q^49 + (-2b5 - b4) q^53 + (-b14 - b11 - b10) * q^55 - b15 * q^59 + (-b14 - 3b11) q^61 + (2b9 + 2b7) * q^65 + (-2b3 + 2) q^67 + (2b5 + 3b4) * q^71 + (-b8 + b6) * q^73 + (-b5 - 2b4 + b2 - b1) q^77 + (-3b14 - 2b11) * q^79 + (-b9 + b7) * q^83 - 2b11 q^85 + (b15 + 2b13 + b9 + b7) q^89 - 4 * q^91 + (-2b2 + 2b1) * q^95 + (-3b3 - 1) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{25} - 32 q^{49} + 32 q^{67} - 64 q^{91} - 16 q^{97}+O(q^{100}) $ 16 * q + 16 * q^25 - 32 * q^49 + 32 * q^67 - 64 * q^91 - 16 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 20x^{14} + 164x^{12} - 666x^{10} + 1300x^{8} - 924x^{6} + 273x^{4} + 404x^{2} + 64 $ x^16 - 20*x^14 + 164*x^12 - 666*x^10 + 1300*x^8 - 924*x^6 + 273*x^4 + 404*x^2 + 64 :

$\beta_{1}$	$=$	$ ( 1017 \nu^{14} - 18889 \nu^{12} + 141809 \nu^{10} - 504639 \nu^{8} + 779299 \nu^{6} - 250803 \nu^{4} + \cdots + 1865592 ) / 347684 $ (1017v^14 - 18889v^12 + 141809v^10 - 504639v^8 + 779299v^6 - 250803v^4 - 305344*v^2 + 1865592) / 347684
$\beta_{2}$	$=$	$ ( - 2447 \nu^{14} + 49511 \nu^{12} - 412949 \nu^{10} + 1685873 \nu^{8} - 3080369 \nu^{6} + \cdots - 360024 ) / 347684 $ (-2447v^14 + 49511v^12 - 412949v^10 + 1685873v^8 - 3080369v^6 + 944715v^4 + 1161424*v^2 - 360024) / 347684
$\beta_{3}$	$=$	$ ( 800 \nu^{14} - 15844 \nu^{12} + 126372 \nu^{10} - 485558 \nu^{8} + 824496 \nu^{6} - 258554 \nu^{4} + \cdots + 310223 ) / 86921 $ (800v^14 - 15844v^12 + 126372v^10 - 485558v^8 + 824496v^6 - 258554v^4 - 316168*v^2 + 310223) / 86921
$\beta_{4}$	$=$	$ ( 6891 \nu^{15} - 135786 \nu^{13} + 1092346 \nu^{11} - 4305788 \nu^{9} + 7949022 \nu^{7} + \cdots + 3564012 \nu ) / 695368 $ (6891v^15 - 135786v^13 + 1092346v^11 - 4305788v^9 + 7949022v^7 - 4808686v^5 + 1379637v^3 + 3564012v) / 695368
$\beta_{5}$	$=$	$ ( 9337 \nu^{15} - 192870 \nu^{13} + 1637078 \nu^{11} - 6938020 \nu^{9} + 14314690 \nu^{7} + \cdots + 5304612 \nu ) / 695368 $ (9337v^15 - 192870v^13 + 1637078v^11 - 6938020v^9 + 14314690v^7 - 10690050v^5 + 197287v^3 + 5304612v) / 695368
$\beta_{6}$	$=$	$ ( 3522 \nu^{15} - 68884 \nu^{13} + 545053 \nu^{11} - 2077259 \nu^{9} + 3556830 \nu^{7} + \cdots + 1812748 \nu ) / 173842 $ (3522v^15 - 68884v^13 + 545053v^11 - 2077259v^9 + 3556830v^7 - 1765419v^5 + 1060981v^3 + 1812748v) / 173842
$\beta_{7}$	$=$	$ ( - 13987 \nu^{14} + 283685 \nu^{12} - 2375450 \nu^{10} + 10006389 \nu^{8} - 21090917 \nu^{6} + \cdots - 3647880 ) / 173842 $ (-13987v^14 + 283685v^12 - 2375450v^10 + 10006389v^8 - 21090917v^6 + 18839156v^4 - 8439840*v^2 - 3647880) / 173842
$\beta_{8}$	$=$	$ ( 11595 \nu^{15} - 231812 \nu^{13} + 1903314 \nu^{11} - 7753568 \nu^{9} + 15174808 \nu^{7} + \cdots + 6116828 \nu ) / 347684 $ (11595v^15 - 231812v^13 + 1903314v^11 - 7753568v^9 + 15174808v^7 - 10393614v^5 + 1236901v^3 + 6116828v) / 347684
$\beta_{9}$	$=$	$ ( 25585 \nu^{14} - 518343 \nu^{12} + 4328246 \nu^{10} - 18126731 \nu^{8} + 37755575 \nu^{6} + \cdots + 6522868 ) / 173842 $ (25585v^14 - 518343v^12 + 4328246v^10 - 18126731v^8 + 37755575v^6 - 32957448v^4 + 15266672*v^2 + 6522868) / 173842
$\beta_{10}$	$=$	$ ( - 38555 \nu^{14} + 783139 \nu^{12} - 6561887 \nu^{10} + 27628481 \nu^{8} - 58067193 \nu^{6} + \cdots - 10043576 ) / 173842 $ (-38555v^14 + 783139v^12 - 6561887v^10 + 27628481v^8 - 58067193v^6 + 51545801v^4 - 23316488*v^2 - 10043576) / 173842
$\beta_{11}$	$=$	$ ( - 56101 \nu^{15} + 1136722 \nu^{13} - 9499560 \nu^{11} + 39874286 \nu^{9} - 83528306 \nu^{7} + \cdots - 14653004 \nu ) / 347684 $ (-56101v^15 + 1136722v^13 - 9499560v^11 + 39874286v^9 - 83528306v^7 + 74078776v^5 - 34501685v^3 - 14653004v) / 347684
$\beta_{12}$	$=$	$ ( 141343 \nu^{14} - 2858839 \nu^{12} + 23840107 \nu^{10} - 99747945 \nu^{8} + 207692533 \nu^{6} + \cdots + 35860056 ) / 347684 $ (141343v^14 - 2858839v^12 + 23840107v^10 - 99747945v^8 + 207692533v^6 - 181431825v^4 + 83859256*v^2 + 35860056) / 347684
$\beta_{13}$	$=$	$ ( 155031 \nu^{15} - 23196 \nu^{14} - 3141792 \nu^{13} + 469316 \nu^{12} + 26258800 \nu^{11} + \cdots - 5749976 ) / 695368 $ (155031v^15 - 23196v^14 - 3141792v^13 + 469316v^12 + 26258800v^11 - 3905592v^10 - 110207086v^9 + 16240684v^8 + 230644536v^7 - 33329316v^6 - 203744128v^5 + 28236584v^4 + 94637175v^3 - 13653664v^2 + 40253716*v - 5749976) / 695368
$\beta_{14}$	$=$	$ ( - 204751 \nu^{15} + 4145926 \nu^{13} - 34610826 \nu^{11} + 144971388 \nu^{9} - 302181482 \nu^{7} + \cdots - 51983964 \nu ) / 695368 $ (-204751v^15 + 4145926v^13 - 34610826v^11 + 144971388v^9 - 302181482v^7 + 264139390v^5 - 121650617v^3 - 51983964v) / 695368
$\beta_{15}$	$=$	$ ( 101947 \nu^{15} - 2063569 \nu^{13} + 17216979 \nu^{11} - 72051334 \nu^{9} + 149956769 \nu^{7} + \cdots + 25677692 \nu ) / 86921 $ (101947v^15 - 2063569v^13 + 17216979v^11 - 72051334v^9 + 149956769v^7 - 130649461v^5 + 59995857v^3 + 25677692v) / 86921

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

$\nu$	$=$	$ ( \beta_{14} + 2\beta_{13} + \beta_{11} + \beta_{9} + \beta_{7} + \beta_{4} ) / 4 $ (b14 + 2*b13 + b11 + b9 + b7 + b4) / 4
$\nu^{2}$	$=$	$ ( \beta_{10} + \beta_{9} - \beta_{7} - 2\beta _1 + 10 ) / 4 $ (b10 + b9 - b7 - 2*b1 + 10) / 4
$\nu^{3}$	$=$	$ ( \beta_{15} + 8 \beta_{14} + 14 \beta_{13} + 12 \beta_{11} + 7 \beta_{9} - 3 \beta_{8} + \cdots + 16 \beta_{4} ) / 8 $ (b15 + 8b14 + 14b13 + 12b11 + 7b9 - 3b8 + 7b7 - 3b6 + 16b4) / 8
$\nu^{4}$	$=$	$ ( -2\beta_{12} + 7\beta_{10} + 9\beta_{9} - 13\beta_{7} + 2\beta_{3} - 8\beta _1 + 36 ) / 4 $ (-2b12 + 7b10 + 9b9 - 13b7 + 2b3 - 8b1 + 36) / 4
$\nu^{5}$	$=$	$ ( 5 \beta_{15} + 32 \beta_{14} + 42 \beta_{13} + 36 \beta_{11} + 21 \beta_{9} - 25 \beta_{8} + \cdots + 122 \beta_{4} ) / 8 $ (5b15 + 32b14 + 42b13 + 36b11 + 21b9 - 25b8 + 21b7 - 25b6 + 10b5 + 122b4) / 8
$\nu^{6}$	$=$	$ ( -20\beta_{12} + 42\beta_{10} + 71\beta_{9} - 87\beta_{7} + 9\beta_{3} + 4\beta_{2} - 20\beta _1 + 79 ) / 4 $ (-20b12 + 42b10 + 71b9 - 87b7 + 9b3 + 4b2 - 20*b1 + 79) / 4
$\nu^{7}$	$=$	$ ( 11 \beta_{15} + 56 \beta_{14} + 22 \beta_{13} + 8 \beta_{11} + 11 \beta_{9} - 175 \beta_{8} + \cdots + 754 \beta_{4} ) / 8 $ (11b15 + 56b14 + 22b13 + 8b11 + 11b9 - 175b8 + 11b7 - 147b6 + 98b5 + 754b4) / 8
$\nu^{8}$	$=$	$ ( -142\beta_{12} + 213\beta_{10} + 465\beta_{9} - 453\beta_{7} + 6\beta_{3} + 16\beta_{2} + 48\beta _1 - 264 ) / 4 $ (-142b12 + 213b10 + 465b9 - 453b7 + 6b3 + 16b2 + 48*b1 - 264) / 4
$\nu^{9}$	$=$	$ ( - 81 \beta_{15} - 608 \beta_{14} - 998 \beta_{13} - 860 \beta_{11} - 499 \beta_{9} + \cdots + 3908 \beta_{4} ) / 8 $ (-81b15 - 608b14 - 998b13 - 860b11 - 499b9 - 1017b8 - 499b7 - 705b6 + 684b5 + 3908b4) / 8
$\nu^{10}$	$=$	$ ( - 810 \beta_{12} + 883 \beta_{10} + 2511 \beta_{9} - 1923 \beta_{7} - 340 \beta_{3} - 96 \beta_{2} + \cdots - 4950 ) / 4 $ (-810b12 + 883b10 + 2511b9 - 1923b7 - 340b3 - 96b2 + 1128*b1 - 4950) / 4
$\nu^{11}$	$=$	$ ( - 1357 \beta_{15} - 8816 \beta_{14} - 10474 \beta_{13} - 8260 \beta_{11} - 5237 \beta_{9} + \cdots + 16454 \beta_{4} ) / 8 $ (-1357b15 - 8816b14 - 10474b13 - 8260b11 - 5237b9 - 4807b8 - 5237b7 - 2695b6 + 3718b5 + 16454b4) / 8
$\nu^{12}$	$=$	$ ( - 3584 \beta_{12} + 2632 \beta_{10} + 10561 \beta_{9} - 5981 \beta_{7} - 4199 \beta_{3} - 2024 \beta_{2} + \cdots - 41469 ) / 4 $ (-3584b12 + 2632b10 + 10561b9 - 5981b7 - 4199b3 - 2024b2 + 10120*b1 - 41469) / 4
$\nu^{13}$	$=$	$ ( - 12063 \beta_{15} - 74328 \beta_{14} - 74454 \beta_{13} - 54976 \beta_{11} - 37227 \beta_{9} + \cdots + 46594 \beta_{4} ) / 8 $ (-12063b15 - 74328b14 - 74454b13 - 54976b11 - 37227b9 - 16081b8 - 37227b7 - 6357b6 + 14482b5 + 46594b4) / 8
$\nu^{14}$	$=$	$ ( - 9250 \beta_{12} + 1295 \beta_{10} + 24871 \beta_{9} - 4567 \beta_{7} - 34006 \beta_{3} + \cdots - 266984 ) / 4 $ (-9250b12 + 1295b10 + 24871b9 - 4567b7 - 34006b3 - 19332b2 + 68612*b1 - 266984) / 4
$\nu^{15}$	$=$	$ ( - 82603 \beta_{15} - 491936 \beta_{14} - 431322 \beta_{13} - 298844 \beta_{11} - 215661 \beta_{9} + \cdots - 36304 \beta_{4} ) / 8 $ (-82603b15 - 491936b14 - 431322b13 - 298844b11 - 215661b9 - 5319b8 - 215661b7 + 14153b6 + 17320b5 - 36304b4) / 8

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3168\mathbb{Z}\right)^\times$.

$n$	$353$	$991$	$1189$	$1729$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

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} $

2287.1

2.34517	−	0.500000i
0.0131233	−	0.500000i
0.946412	−	0.500000i
−2.14576	−	0.500000i
2.14576	−	0.500000i
−0.946412	−	0.500000i
−0.0131233	−	0.500000i
−2.34517	−	0.500000i
2.34517	+	0.500000i
0.0131233	+	0.500000i
0.946412	+	0.500000i
−2.14576	+	0.500000i
2.14576	+	0.500000i
−0.946412	+	0.500000i
−0.0131233	+	0.500000i
−2.34517	+	0.500000i

−

2.00000i

−3.02045

2287.2

−

2.00000i

−3.02045

2287.3

−

2.00000i

−0.936426

2287.4

−

2.00000i

−0.936426

2287.5

−

2.00000i

0.936426

2287.6

−

2.00000i

0.936426

2287.7

−

2.00000i

3.02045

2287.8

−

2.00000i

3.02045

2287.9

2.00000i

−3.02045

2287.10

2.00000i

−3.02045

2287.11

2.00000i

−0.936426

2287.12

2.00000i

−0.936426

2287.13

2.00000i

0.936426

2287.14

2.00000i

0.936426

2287.15

2.00000i

3.02045

2287.16

2.00000i

3.02045

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
8.d	odd	2	1	inner
11.b	odd	2	1	inner
24.f	even	2	1	inner
33.d	even	2	1	inner
88.g	even	2	1	inner
264.p	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3168.2.h.i		16
3.b	odd	2	1	inner	3168.2.h.i		16
4.b	odd	2	1		792.2.h.i	✓	16
8.b	even	2	1		792.2.h.i	✓	16
8.d	odd	2	1	inner	3168.2.h.i		16
11.b	odd	2	1	inner	3168.2.h.i		16
12.b	even	2	1		792.2.h.i	✓	16
24.f	even	2	1	inner	3168.2.h.i		16
24.h	odd	2	1		792.2.h.i	✓	16
33.d	even	2	1	inner	3168.2.h.i		16
44.c	even	2	1		792.2.h.i	✓	16
88.b	odd	2	1		792.2.h.i	✓	16
88.g	even	2	1	inner	3168.2.h.i		16
132.d	odd	2	1		792.2.h.i	✓	16
264.m	even	2	1		792.2.h.i	✓	16
264.p	odd	2	1	inner	3168.2.h.i		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
792.2.h.i	✓	16	4.b	odd	2	1
792.2.h.i	✓	16	8.b	even	2	1
792.2.h.i	✓	16	12.b	even	2	1
792.2.h.i	✓	16	24.h	odd	2	1
792.2.h.i	✓	16	44.c	even	2	1
792.2.h.i	✓	16	88.b	odd	2	1
792.2.h.i	✓	16	132.d	odd	2	1
792.2.h.i	✓	16	264.m	even	2	1
3168.2.h.i		16	1.a	even	1	1	trivial
3168.2.h.i		16	3.b	odd	2	1	inner
3168.2.h.i		16	8.d	odd	2	1	inner
3168.2.h.i		16	11.b	odd	2	1	inner
3168.2.h.i		16	24.f	even	2	1	inner
3168.2.h.i		16	33.d	even	2	1	inner
3168.2.h.i		16	88.g	even	2	1	inner
3168.2.h.i		16	264.p	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 $S_{2}^{\mathrm{new}}(3168, [\chi])$:

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

$ T_{7}^{4} - 10T_{7}^{2} + 8 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ T^{16} $ T^16
$5$	$ (T^{2} + 4)^{8} $ (T^2 + 4)^8
$7$	$ (T^{4} - 10 T^{2} + 8)^{4} $ (T^4 - 10*T^2 + 8)^4
$11$	$ (T^{8} - 16 T^{6} + \cdots + 14641)^{2} $ (T^8 - 16T^6 + 238T^4 - 1936*T^2 + 14641)^2
$13$	$ (T^{4} - 20 T^{2} + 32)^{4} $ (T^4 - 20*T^2 + 32)^4
$17$	$ (T^{4} + 14 T^{2} + 32)^{4} $ (T^4 + 14*T^2 + 32)^4
$19$	$ (T^{4} + 58 T^{2} + 416)^{4} $ (T^4 + 58*T^2 + 416)^4
$23$	$ (T^{4} + 36 T^{2} + 256)^{4} $ (T^4 + 36*T^2 + 256)^4
$29$	$ (T^{4} - 46 T^{2} + 104)^{4} $ (T^4 - 46*T^2 + 104)^4
$31$	$ (T^{4} + 72 T^{2} + 208)^{4} $ (T^4 + 72*T^2 + 208)^4
$37$	$ (T^{4} + 60 T^{2} + 832)^{4} $ (T^4 + 60*T^2 + 832)^4
$41$	$ (T^{4} + 62 T^{2} + 128)^{4} $ (T^4 + 62*T^2 + 128)^4
$43$	$ (T^{4} + 122 T^{2} + 1664)^{4} $ (T^4 + 122*T^2 + 1664)^4
$47$	$ (T^{2} + 36)^{8} $ (T^2 + 36)^8
$53$	$ (T^{2} + 68)^{8} $ (T^2 + 68)^8
$59$	$ (T^{4} - 236 T^{2} + 13312)^{4} $ (T^4 - 236*T^2 + 13312)^4
$61$	$ (T^{4} - 148 T^{2} + 5408)^{4} $ (T^4 - 148*T^2 + 5408)^4
$67$	$ (T^{2} - 4 T - 64)^{8} $ (T^2 - 4*T - 64)^8
$71$	$ (T^{4} + 168 T^{2} + 2704)^{4} $ (T^4 + 168*T^2 + 2704)^4
$73$	$ (T^{4} + 184 T^{2} + 1664)^{4} $ (T^4 + 184*T^2 + 1664)^4
$79$	$ (T^{4} - 170 T^{2} + 2312)^{4} $ (T^4 - 170*T^2 + 2312)^4
$83$	$ (T^{4} + 28 T^{2} + 128)^{4} $ (T^4 + 28*T^2 + 128)^4
$89$	$ (T^{4} - 288 T^{2} + 3328)^{4} $ (T^4 - 288*T^2 + 3328)^4
$97$	$ (T^{2} + 2 T - 152)^{8} $ (T^2 + 2*T - 152)^8
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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:	\( N \)	\(=\)	\( 3168 = 2^{5} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3168.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_{4} q^{5} - \beta_{14} q^{7} + ( - \beta_{13} - \beta_{7}) q^{11} + (\beta_{14} - \beta_{11}) q^{13} - \beta_{7} q^{17} - \beta_{8} q^{19} + \beta_{5} q^{23} + q^{25} + \beta_{2} q^{29} + \beta_{12} q^{31}+ \cdots + ( - 3 \beta_{3} - 1) q^{97}+O(q^{100}) \) q + b4 * q^5 - b14 * q^7 + (-b13 - b7) * q^11 + (b14 - b11) * q^13 - b7 * q^17 - b8 * q^19 + b5 * q^23 + q^25 + b2 * q^29 + b12 * q^31 - 2b9 q^35 - b10 * q^37 + (2b9 - b7) q^41 + b6 * q^43 - 3b4 q^47 + (b3 - 2) * q^49 + (-2b5 - b4) q^53 + (-b14 - b11 - b10) * q^55 - b15 * q^59 + (-b14 - 3b11) q^61 + (2b9 + 2b7) * q^65 + (-2b3 + 2) q^67 + (2b5 + 3b4) * q^71 + (-b8 + b6) * q^73 + (-b5 - 2b4 + b2 - b1) q^77 + (-3b14 - 2b11) * q^79 + (-b9 + b7) * q^83 - 2b11 q^85 + (b15 + 2b13 + b9 + b7) q^89 - 4 * q^91 + (-2b2 + 2b1) * q^95 + (-3b3 - 1) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{25} - 32 q^{49} + 32 q^{67} - 64 q^{91} - 16 q^{97}+O(q^{100}) \) 16 * q + 16 * q^25 - 32 * q^49 + 32 * q^67 - 64 * q^91 - 16 * q^97

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Newspace parameters

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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	3168.2.h.i

Level	$3168$
Weight	$2$
Character orbit	3168.h
Analytic conductor	$25.297$
Analytic rank	$0$
Dimension	$16$
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(25.2966073603\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	16.0.3342602057661458415616.4
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 20x^{14} + 164x^{12} - 666x^{10} + 1300x^{8} - 924x^{6} + 273x^{4} + 404x^{2} + 64 \) x^16 - 20x^14 + 164x^12 - 666x^10 + 1300x^8 - 924x^6 + 273x^4 + 404*x^2 + 64
Coefficient ring:	\(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index:	\( 2^{26} \)
Twist minimal:	no (minimal twist has level 792)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( 1017 \nu^{14} - 18889 \nu^{12} + 141809 \nu^{10} - 504639 \nu^{8} + 779299 \nu^{6} - 250803 \nu^{4} + \cdots + 1865592 ) / 347684 \) (1017v^14 - 18889v^12 + 141809v^10 - 504639v^8 + 779299v^6 - 250803v^4 - 305344*v^2 + 1865592) / 347684
\(\beta_{2}\)	\(=\)	\( ( - 2447 \nu^{14} + 49511 \nu^{12} - 412949 \nu^{10} + 1685873 \nu^{8} - 3080369 \nu^{6} + \cdots - 360024 ) / 347684 \) (-2447v^14 + 49511v^12 - 412949v^10 + 1685873v^8 - 3080369v^6 + 944715v^4 + 1161424*v^2 - 360024) / 347684
\(\beta_{3}\)	\(=\)	\( ( 800 \nu^{14} - 15844 \nu^{12} + 126372 \nu^{10} - 485558 \nu^{8} + 824496 \nu^{6} - 258554 \nu^{4} + \cdots + 310223 ) / 86921 \) (800v^14 - 15844v^12 + 126372v^10 - 485558v^8 + 824496v^6 - 258554v^4 - 316168*v^2 + 310223) / 86921
\(\beta_{4}\)	\(=\)	\( ( 6891 \nu^{15} - 135786 \nu^{13} + 1092346 \nu^{11} - 4305788 \nu^{9} + 7949022 \nu^{7} + \cdots + 3564012 \nu ) / 695368 \) (6891v^15 - 135786v^13 + 1092346v^11 - 4305788v^9 + 7949022v^7 - 4808686v^5 + 1379637v^3 + 3564012v) / 695368
\(\beta_{5}\)	\(=\)	\( ( 9337 \nu^{15} - 192870 \nu^{13} + 1637078 \nu^{11} - 6938020 \nu^{9} + 14314690 \nu^{7} + \cdots + 5304612 \nu ) / 695368 \) (9337v^15 - 192870v^13 + 1637078v^11 - 6938020v^9 + 14314690v^7 - 10690050v^5 + 197287v^3 + 5304612v) / 695368
\(\beta_{6}\)	\(=\)	\( ( 3522 \nu^{15} - 68884 \nu^{13} + 545053 \nu^{11} - 2077259 \nu^{9} + 3556830 \nu^{7} + \cdots + 1812748 \nu ) / 173842 \) (3522v^15 - 68884v^13 + 545053v^11 - 2077259v^9 + 3556830v^7 - 1765419v^5 + 1060981v^3 + 1812748v) / 173842
\(\beta_{7}\)	\(=\)	\( ( - 13987 \nu^{14} + 283685 \nu^{12} - 2375450 \nu^{10} + 10006389 \nu^{8} - 21090917 \nu^{6} + \cdots - 3647880 ) / 173842 \) (-13987v^14 + 283685v^12 - 2375450v^10 + 10006389v^8 - 21090917v^6 + 18839156v^4 - 8439840*v^2 - 3647880) / 173842
\(\beta_{8}\)	\(=\)	\( ( 11595 \nu^{15} - 231812 \nu^{13} + 1903314 \nu^{11} - 7753568 \nu^{9} + 15174808 \nu^{7} + \cdots + 6116828 \nu ) / 347684 \) (11595v^15 - 231812v^13 + 1903314v^11 - 7753568v^9 + 15174808v^7 - 10393614v^5 + 1236901v^3 + 6116828v) / 347684
\(\beta_{9}\)	\(=\)	\( ( 25585 \nu^{14} - 518343 \nu^{12} + 4328246 \nu^{10} - 18126731 \nu^{8} + 37755575 \nu^{6} + \cdots + 6522868 ) / 173842 \) (25585v^14 - 518343v^12 + 4328246v^10 - 18126731v^8 + 37755575v^6 - 32957448v^4 + 15266672*v^2 + 6522868) / 173842
\(\beta_{10}\)	\(=\)	\( ( - 38555 \nu^{14} + 783139 \nu^{12} - 6561887 \nu^{10} + 27628481 \nu^{8} - 58067193 \nu^{6} + \cdots - 10043576 ) / 173842 \) (-38555v^14 + 783139v^12 - 6561887v^10 + 27628481v^8 - 58067193v^6 + 51545801v^4 - 23316488*v^2 - 10043576) / 173842
\(\beta_{11}\)	\(=\)	\( ( - 56101 \nu^{15} + 1136722 \nu^{13} - 9499560 \nu^{11} + 39874286 \nu^{9} - 83528306 \nu^{7} + \cdots - 14653004 \nu ) / 347684 \) (-56101v^15 + 1136722v^13 - 9499560v^11 + 39874286v^9 - 83528306v^7 + 74078776v^5 - 34501685v^3 - 14653004v) / 347684
\(\beta_{12}\)	\(=\)	\( ( 141343 \nu^{14} - 2858839 \nu^{12} + 23840107 \nu^{10} - 99747945 \nu^{8} + 207692533 \nu^{6} + \cdots + 35860056 ) / 347684 \) (141343v^14 - 2858839v^12 + 23840107v^10 - 99747945v^8 + 207692533v^6 - 181431825v^4 + 83859256*v^2 + 35860056) / 347684
\(\beta_{13}\)	\(=\)	\( ( 155031 \nu^{15} - 23196 \nu^{14} - 3141792 \nu^{13} + 469316 \nu^{12} + 26258800 \nu^{11} + \cdots - 5749976 ) / 695368 \) (155031v^15 - 23196v^14 - 3141792v^13 + 469316v^12 + 26258800v^11 - 3905592v^10 - 110207086v^9 + 16240684v^8 + 230644536v^7 - 33329316v^6 - 203744128v^5 + 28236584v^4 + 94637175v^3 - 13653664v^2 + 40253716*v - 5749976) / 695368
\(\beta_{14}\)	\(=\)	\( ( - 204751 \nu^{15} + 4145926 \nu^{13} - 34610826 \nu^{11} + 144971388 \nu^{9} - 302181482 \nu^{7} + \cdots - 51983964 \nu ) / 695368 \) (-204751v^15 + 4145926v^13 - 34610826v^11 + 144971388v^9 - 302181482v^7 + 264139390v^5 - 121650617v^3 - 51983964v) / 695368
\(\beta_{15}\)	\(=\)	\( ( 101947 \nu^{15} - 2063569 \nu^{13} + 17216979 \nu^{11} - 72051334 \nu^{9} + 149956769 \nu^{7} + \cdots + 25677692 \nu ) / 86921 \) (101947v^15 - 2063569v^13 + 17216979v^11 - 72051334v^9 + 149956769v^7 - 130649461v^5 + 59995857v^3 + 25677692v) / 86921

\(\nu\)	\(=\)	\( ( \beta_{14} + 2\beta_{13} + \beta_{11} + \beta_{9} + \beta_{7} + \beta_{4} ) / 4 \) (b14 + 2*b13 + b11 + b9 + b7 + b4) / 4
\(\nu^{2}\)	\(=\)	\( ( \beta_{10} + \beta_{9} - \beta_{7} - 2\beta _1 + 10 ) / 4 \) (b10 + b9 - b7 - 2*b1 + 10) / 4
\(\nu^{3}\)	\(=\)	\( ( \beta_{15} + 8 \beta_{14} + 14 \beta_{13} + 12 \beta_{11} + 7 \beta_{9} - 3 \beta_{8} + \cdots + 16 \beta_{4} ) / 8 \) (b15 + 8b14 + 14b13 + 12b11 + 7b9 - 3b8 + 7b7 - 3b6 + 16b4) / 8
\(\nu^{4}\)	\(=\)	\( ( -2\beta_{12} + 7\beta_{10} + 9\beta_{9} - 13\beta_{7} + 2\beta_{3} - 8\beta _1 + 36 ) / 4 \) (-2b12 + 7b10 + 9b9 - 13b7 + 2b3 - 8b1 + 36) / 4
\(\nu^{5}\)	\(=\)	\( ( 5 \beta_{15} + 32 \beta_{14} + 42 \beta_{13} + 36 \beta_{11} + 21 \beta_{9} - 25 \beta_{8} + \cdots + 122 \beta_{4} ) / 8 \) (5b15 + 32b14 + 42b13 + 36b11 + 21b9 - 25b8 + 21b7 - 25b6 + 10b5 + 122b4) / 8
\(\nu^{6}\)	\(=\)	\( ( -20\beta_{12} + 42\beta_{10} + 71\beta_{9} - 87\beta_{7} + 9\beta_{3} + 4\beta_{2} - 20\beta _1 + 79 ) / 4 \) (-20b12 + 42b10 + 71b9 - 87b7 + 9b3 + 4b2 - 20*b1 + 79) / 4
\(\nu^{7}\)	\(=\)	\( ( 11 \beta_{15} + 56 \beta_{14} + 22 \beta_{13} + 8 \beta_{11} + 11 \beta_{9} - 175 \beta_{8} + \cdots + 754 \beta_{4} ) / 8 \) (11b15 + 56b14 + 22b13 + 8b11 + 11b9 - 175b8 + 11b7 - 147b6 + 98b5 + 754b4) / 8
\(\nu^{8}\)	\(=\)	\( ( -142\beta_{12} + 213\beta_{10} + 465\beta_{9} - 453\beta_{7} + 6\beta_{3} + 16\beta_{2} + 48\beta _1 - 264 ) / 4 \) (-142b12 + 213b10 + 465b9 - 453b7 + 6b3 + 16b2 + 48*b1 - 264) / 4
\(\nu^{9}\)	\(=\)	\( ( - 81 \beta_{15} - 608 \beta_{14} - 998 \beta_{13} - 860 \beta_{11} - 499 \beta_{9} + \cdots + 3908 \beta_{4} ) / 8 \) (-81b15 - 608b14 - 998b13 - 860b11 - 499b9 - 1017b8 - 499b7 - 705b6 + 684b5 + 3908b4) / 8
\(\nu^{10}\)	\(=\)	\( ( - 810 \beta_{12} + 883 \beta_{10} + 2511 \beta_{9} - 1923 \beta_{7} - 340 \beta_{3} - 96 \beta_{2} + \cdots - 4950 ) / 4 \) (-810b12 + 883b10 + 2511b9 - 1923b7 - 340b3 - 96b2 + 1128*b1 - 4950) / 4
\(\nu^{11}\)	\(=\)	\( ( - 1357 \beta_{15} - 8816 \beta_{14} - 10474 \beta_{13} - 8260 \beta_{11} - 5237 \beta_{9} + \cdots + 16454 \beta_{4} ) / 8 \) (-1357b15 - 8816b14 - 10474b13 - 8260b11 - 5237b9 - 4807b8 - 5237b7 - 2695b6 + 3718b5 + 16454b4) / 8
\(\nu^{12}\)	\(=\)	\( ( - 3584 \beta_{12} + 2632 \beta_{10} + 10561 \beta_{9} - 5981 \beta_{7} - 4199 \beta_{3} - 2024 \beta_{2} + \cdots - 41469 ) / 4 \) (-3584b12 + 2632b10 + 10561b9 - 5981b7 - 4199b3 - 2024b2 + 10120*b1 - 41469) / 4
\(\nu^{13}\)	\(=\)	\( ( - 12063 \beta_{15} - 74328 \beta_{14} - 74454 \beta_{13} - 54976 \beta_{11} - 37227 \beta_{9} + \cdots + 46594 \beta_{4} ) / 8 \) (-12063b15 - 74328b14 - 74454b13 - 54976b11 - 37227b9 - 16081b8 - 37227b7 - 6357b6 + 14482b5 + 46594b4) / 8
\(\nu^{14}\)	\(=\)	\( ( - 9250 \beta_{12} + 1295 \beta_{10} + 24871 \beta_{9} - 4567 \beta_{7} - 34006 \beta_{3} + \cdots - 266984 ) / 4 \) (-9250b12 + 1295b10 + 24871b9 - 4567b7 - 34006b3 - 19332b2 + 68612*b1 - 266984) / 4
\(\nu^{15}\)	\(=\)	\( ( - 82603 \beta_{15} - 491936 \beta_{14} - 431322 \beta_{13} - 298844 \beta_{11} - 215661 \beta_{9} + \cdots - 36304 \beta_{4} ) / 8 \) (-82603b15 - 491936b14 - 431322b13 - 298844b11 - 215661b9 - 5319b8 - 215661b7 + 14153b6 + 17320b5 - 36304b4) / 8

\(n\)	\(353\)	\(991\)	\(1189\)	\(1729\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 2287.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( T^{16} \) T^16
$5$	\( (T^{2} + 4)^{8} \) (T^2 + 4)^8
$7$	\( (T^{4} - 10 T^{2} + 8)^{4} \) (T^4 - 10*T^2 + 8)^4
$11$	\( (T^{8} - 16 T^{6} + \cdots + 14641)^{2} \) (T^8 - 16T^6 + 238T^4 - 1936*T^2 + 14641)^2
$13$	\( (T^{4} - 20 T^{2} + 32)^{4} \) (T^4 - 20*T^2 + 32)^4
$17$	\( (T^{4} + 14 T^{2} + 32)^{4} \) (T^4 + 14*T^2 + 32)^4
$19$	\( (T^{4} + 58 T^{2} + 416)^{4} \) (T^4 + 58*T^2 + 416)^4
$23$	\( (T^{4} + 36 T^{2} + 256)^{4} \) (T^4 + 36*T^2 + 256)^4
$29$	\( (T^{4} - 46 T^{2} + 104)^{4} \) (T^4 - 46*T^2 + 104)^4
$31$	\( (T^{4} + 72 T^{2} + 208)^{4} \) (T^4 + 72*T^2 + 208)^4
$37$	\( (T^{4} + 60 T^{2} + 832)^{4} \) (T^4 + 60*T^2 + 832)^4
$41$	\( (T^{4} + 62 T^{2} + 128)^{4} \) (T^4 + 62*T^2 + 128)^4
$43$	\( (T^{4} + 122 T^{2} + 1664)^{4} \) (T^4 + 122*T^2 + 1664)^4
$47$	\( (T^{2} + 36)^{8} \) (T^2 + 36)^8
$53$	\( (T^{2} + 68)^{8} \) (T^2 + 68)^8
$59$	\( (T^{4} - 236 T^{2} + 13312)^{4} \) (T^4 - 236*T^2 + 13312)^4
$61$	\( (T^{4} - 148 T^{2} + 5408)^{4} \) (T^4 - 148*T^2 + 5408)^4
$67$	\( (T^{2} - 4 T - 64)^{8} \) (T^2 - 4*T - 64)^8
$71$	\( (T^{4} + 168 T^{2} + 2704)^{4} \) (T^4 + 168*T^2 + 2704)^4
$73$	\( (T^{4} + 184 T^{2} + 1664)^{4} \) (T^4 + 184*T^2 + 1664)^4
$79$	\( (T^{4} - 170 T^{2} + 2312)^{4} \) (T^4 - 170*T^2 + 2312)^4
$83$	\( (T^{4} + 28 T^{2} + 128)^{4} \) (T^4 + 28*T^2 + 128)^4
$89$	\( (T^{4} - 288 T^{2} + 3328)^{4} \) (T^4 - 288*T^2 + 3328)^4
$97$	\( (T^{2} + 2 T - 152)^{8} \) (T^2 + 2*T - 152)^8
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