LMFDB - Newform orbit 2112.2.m.m

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2112,2,Mod(1121,2112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2112, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 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("2112.1121");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2112 = 2^{6} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2112.m (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:	$16.8644049069$
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} - 40x^{12} + 604x^{8} - 1440x^{4} + 1296 $ x^16 - 40x^12 + 604x^8 - 1440*x^4 + 1296
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{28} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-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_{5} q^{3} + \beta_{10} q^{5} + \beta_{8} q^{7} - 3 q^{9}+O(q^{10}) $ q - b5 * q^3 + b10 * q^5 + b8 * q^7 - 3 * q^9	$ q - \beta_{5} q^{3} + \beta_{10} q^{5} + \beta_{8} q^{7} - 3 q^{9} - \beta_{2} q^{11} + \beta_{11} q^{13} + \beta_{12} q^{15} + \beta_{3} q^{17} - \beta_{9} q^{21} + (\beta_{13} - \beta_{12}) q^{23} + ( - \beta_1 + 5) q^{25} + 3 \beta_{5} q^{27} - \beta_{4} q^{33} + ( - \beta_{7} - 2 \beta_{2}) q^{35} - \beta_{14} q^{39} - 2 \beta_{4} q^{41} - 3 \beta_{10} q^{45} + ( - \beta_{13} - \beta_{12}) q^{47} + (3 \beta_1 - 7) q^{49} - \beta_{7} q^{51} - \beta_{15} q^{53} + (\beta_{14} + 2 \beta_{8}) q^{55} + ( - 2 \beta_{11} + \beta_{9}) q^{61} - 3 \beta_{8} q^{63} + ( - 2 \beta_{4} + \beta_{3}) q^{65} + \beta_{6} q^{67} + (\beta_{15} + 2 \beta_{10}) q^{69} + ( - 2 \beta_{13} + \beta_{12}) q^{71} + ( - 3 \beta_{6} - 5 \beta_{5}) q^{75} + (\beta_{15} - 2 \beta_{10}) q^{77} + (\beta_{14} - 2 \beta_{8}) q^{79} + 9 q^{81} - \beta_{7} q^{83} + (\beta_{11} + 3 \beta_{9}) q^{85} + (5 \beta_{6} - 2 \beta_{5}) q^{91} - 5 \beta_1 q^{97} + 3 \beta_{2} q^{99}+O(q^{100}) $ q - b5 * q^3 + b10 * q^5 + b8 * q^7 - 3 * q^9 - b2 * q^11 + b11 * q^13 + b12 * q^15 + b3 * q^17 - b9 * q^21 + (b13 - b12) * q^23 + (-b1 + 5) * q^25 + 3b5 q^27 - b4 * q^33 + (-b7 - 2b2) q^35 - b14 * q^39 - 2b4 q^41 - 3b10 q^45 + (-b13 - b12) * q^47 + (3b1 - 7) q^49 - b7 * q^51 - b15 * q^53 + (b14 + 2b8) q^55 + (-2b11 + b9) q^61 - 3b8 q^63 + (-2b4 + b3) q^65 + b6 * q^67 + (b15 + 2b10) q^69 + (-2b13 + b12) q^71 + (-3b6 - 5b5) * q^75 + (b15 - 2b10) q^77 + (b14 - 2b8) q^79 + 9 * q^81 - b7 * q^83 + (b11 + 3b9) q^85 + (5b6 - 2b5) * q^91 - 5b1 q^97 + 3b2 q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 48 q^{9}+O(q^{10}) $ 16 * q - 48 * q^9	$ 16 q - 48 q^{9} + 80 q^{25} - 112 q^{49} + 144 q^{81}+O(q^{100}) $ 16 * q - 48 * q^9 + 80 * q^25 - 112 * q^49 + 144 * q^81

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 40x^{12} + 604x^{8} - 1440x^{4} + 1296 $ x^16 - 40*x^12 + 604*x^8 - 1440*x^4 + 1296 :

$\beta_{1}$	$=$	$ ( 5\nu^{12} - 128\nu^{8} + 320\nu^{4} + 14544 ) / 4284 $ (5v^12 - 128v^8 + 320*v^4 + 14544) / 4284
$\beta_{2}$	$=$	$ ( 2\nu^{12} - 75\nu^{8} + 1080\nu^{4} - 1370 ) / 238 $ (2v^12 - 75v^8 + 1080*v^4 - 1370) / 238
$\beta_{3}$	$=$	$ ( 2\nu^{14} - 89\nu^{10} + 1514\nu^{6} - 5724\nu^{2} ) / 756 $ (2v^14 - 89v^10 + 1514v^6 - 5724v^2) / 756
$\beta_{4}$	$=$	$ ( -10\nu^{14} + 409\nu^{10} - 6454\nu^{6} + 24156\nu^{2} ) / 3672 $ (-10v^14 + 409v^10 - 6454v^6 + 24156v^2) / 3672
$\beta_{5}$	$=$	$ ( 5\nu^{14} - 191\nu^{10} + 2714\nu^{6} - 2844\nu^{2} ) / 1512 $ (5v^14 - 191v^10 + 2714v^6 - 2844v^2) / 1512
$\beta_{6}$	$=$	$ ( -7\nu^{14} + 271\nu^{10} - 3814\nu^{6} + 3996\nu^{2} ) / 1836 $ (-7v^14 + 271v^10 - 3814v^6 + 3996v^2) / 1836
$\beta_{7}$	$=$	$ ( -\nu^{12} + 40\nu^{8} - 568\nu^{4} + 720 ) / 36 $ (-v^12 + 40v^8 - 568v^4 + 720) / 36
$\beta_{8}$	$=$	$ ( - 139 \nu^{15} - 372 \nu^{13} + 5272 \nu^{11} + 14664 \nu^{9} - 68872 \nu^{7} - 203736 \nu^{5} + \cdots + 367632 \nu ) / 154224 $ (-139v^15 - 372v^13 + 5272v^11 + 14664v^9 - 68872v^7 - 203736v^5 - 49608v^3 + 367632v) / 154224
$\beta_{9}$	$=$	$ ( 29\nu^{15} - 62\nu^{13} - 1028\nu^{11} + 2444\nu^{9} + 13280\nu^{7} - 38240\nu^{5} + 9528\nu^{3} + 69840\nu ) / 17136 $ (29v^15 - 62v^13 - 1028v^11 + 2444v^9 + 13280v^7 - 38240v^5 + 9528v^3 + 69840v) / 17136
$\beta_{10}$	$=$	$ ( 449 \nu^{15} - 372 \nu^{13} - 17492 \nu^{11} + 14664 \nu^{9} + 251504 \nu^{7} - 203736 \nu^{5} + \cdots + 59184 \nu ) / 154224 $ (449v^15 - 372v^13 - 17492v^11 + 14664v^9 + 251504v^7 - 203736v^5 - 385272v^3 + 59184v) / 154224
$\beta_{11}$	$=$	$ ( 77 \nu^{15} - 186 \nu^{13} - 2828 \nu^{11} + 7332 \nu^{9} + 36752 \nu^{7} - 107376 \nu^{5} + \cdots + 194832 \nu ) / 22032 $ (77v^15 - 186v^13 - 2828v^11 + 7332v^9 + 36752v^7 - 107376v^5 + 26424v^3 + 194832v) / 22032
$\beta_{12}$	$=$	$ ( 253 \nu^{15} + 246 \nu^{13} - 9904 \nu^{11} - 8868 \nu^{9} + 144712 \nu^{7} + 118560 \nu^{5} + \cdots - 34992 \nu ) / 51408 $ (253v^15 + 246v^13 - 9904v^11 - 8868v^9 + 144712v^7 + 118560v^5 - 221976v^3 - 34992v) / 51408
$\beta_{13}$	$=$	$ ( - 449 \nu^{15} - 372 \nu^{13} + 17492 \nu^{11} + 14664 \nu^{9} - 251504 \nu^{7} - 203736 \nu^{5} + \cdots + 59184 \nu ) / 77112 $ (-449v^15 - 372v^13 + 17492v^11 + 14664v^9 - 251504v^7 - 203736v^5 + 385272v^3 + 59184v) / 77112
$\beta_{14}$	$=$	$ ( 313 \nu^{15} + 744 \nu^{13} - 11440 \nu^{11} - 29328 \nu^{9} + 148552 \nu^{7} + 433176 \nu^{5} + \cdots - 786672 \nu ) / 51408 $ (313v^15 + 744v^13 - 11440v^11 - 29328v^9 + 148552v^7 + 433176v^5 + 106776v^3 - 786672v) / 51408
$\beta_{15}$	$=$	$ ( 281 \nu^{15} - 264 \nu^{13} - 10988 \nu^{11} + 9696 \nu^{9} + 159968 \nu^{7} - 130728 \nu^{5} + \cdots + 38448 \nu ) / 22032 $ (281v^15 - 264v^13 - 10988v^11 + 9696v^9 + 159968v^7 - 130728v^5 - 245304v^3 + 38448v) / 22032

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

$\nu$	$=$	$ ( -\beta_{13} + \beta_{11} - 2\beta_{10} - \beta_{9} + 2\beta_{8} ) / 8 $ (-b13 + b11 - 2b10 - b9 + 2b8) / 8
$\nu^{2}$	$=$	$ ( \beta_{6} + 2\beta_{5} + 2\beta_{4} + \beta_{3} ) / 4 $ (b6 + 2b5 + 2b4 + b3) / 4
$\nu^{3}$	$=$	$ ( \beta_{15} + 2\beta_{13} + 2\beta_{12} + \beta_{11} - 5\beta_{10} - \beta_{9} - 2\beta_{8} ) / 4 $ (b15 + 2b13 + 2b12 + b11 - 5b10 - b9 - 2b8) / 4
$\nu^{4}$	$=$	$ ( \beta_{7} + 4\beta_{2} - 5\beta _1 + 20 ) / 2 $ (b7 + 4b2 - 5b1 + 20) / 2
$\nu^{5}$	$=$	$ ( 4\beta_{14} - \beta_{13} + 7\beta_{11} - 2\beta_{10} - 15\beta_{9} + 26\beta_{8} ) / 4 $ (4b14 - b13 + 7b11 - 2b10 - 15b9 + 26*b8) / 4
$\nu^{6}$	$=$	$ 20\beta_{6} + 24\beta_{5} + 6\beta_{4} + 5\beta_{3} $ 20b6 + 24b5 + 6b4 + 5b3
$\nu^{7}$	$=$	$ ( 25 \beta_{15} + 5 \beta_{14} + 43 \beta_{13} + 50 \beta_{12} - 7 \beta_{11} - 111 \beta_{10} + \cdots + 29 \beta_{8} ) / 4 $ (25b15 + 5b14 + 43b13 + 50b12 - 7b11 - 111b10 + 17b9 + 29b8) / 4
$\nu^{8}$	$=$	$ 20\beta_{7} + 70\beta_{2} - 28\beta _1 + 98 $ 20b7 + 70b2 - 28*b1 + 98
$\nu^{9}$	$=$	$ ( - 31 \beta_{15} + 39 \beta_{14} + 53 \beta_{13} + 62 \beta_{12} + 67 \beta_{11} + 137 \beta_{10} + \cdots + 251 \beta_{8} ) / 2 $ (-31b15 + 39b14 + 53b13 + 62b12 + 67b11 + 137b10 - 145b9 + 251b8) / 2
$\nu^{10}$	$=$	$ 589\beta_{6} + 682\beta_{5} + 22\beta_{4} + 19\beta_{3} $ 589b6 + 682b5 + 22b4 + 19b3
$\nu^{11}$	$=$	$ ( 195 \beta_{15} + 157 \beta_{14} + 337 \beta_{13} + 390 \beta_{12} - 271 \beta_{11} + \cdots + 1013 \beta_{8} ) / 2 $ (195b15 + 157b14 + 337b13 + 390b12 - 271b11 - 869b10 + 585b9 + 1013b8) / 2
$\nu^{12}$	$=$	$ 480\beta_{7} + 1664\beta_{2} + 300\beta _1 - 1040 $ 480b7 + 1664b2 + 300*b1 - 1040
$\nu^{13}$	$=$	$ - 611 \beta_{15} + 221 \beta_{14} + 1058 \beta_{13} + 1222 \beta_{12} + 382 \beta_{11} + \cdots + 1427 \beta_{8} $ -611b15 + 221b14 + 1058b13 + 1222b12 + 382b11 + 2727b10 - 824b9 + 1427b8
$\nu^{14}$	$=$	$ 11786\beta_{6} + 13612\beta_{5} - 2132\beta_{4} - 1846\beta_{3} $ 11786b6 + 13612b5 - 2132b4 - 1846b3
$\nu^{15}$	$=$	$ 512 \beta_{15} + 2358 \beta_{14} + 886 \beta_{13} + 1024 \beta_{12} - 4084 \beta_{11} + \cdots + 15242 \beta_{8} $ 512b15 + 2358b14 + 886b13 + 1024b12 - 4084b11 - 2284b10 + 8800b9 + 15242b8

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2112\mathbb{Z}\right)^\times$.

$n$	$133$	$1409$	$1729$	$2047$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

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} $

1121.1

−0.317378	−	2.15205i
2.15205	+	0.317378i
1.11398	+	0.164287i
0.164287	+	1.11398i
−0.164287	−	1.11398i
−1.11398	−	0.164287i
−2.15205	−	0.317378i
0.317378	+	2.15205i
2.15205	−	0.317378i
−0.317378	+	2.15205i
0.164287	−	1.11398i
1.11398	−	0.164287i
−1.11398	+	0.164287i
−0.164287	+	1.11398i
0.317378	−	2.15205i
−2.15205	+	0.317378i

−

1.73205i

−3.66935

−

4.93886i

−3.00000

1121.2

−

1.73205i

−3.66935

4.93886i

−3.00000

1121.3

−

1.73205i

−2.55654

−

1.89939i

−3.00000

1121.4

−

1.73205i

−2.55654

1.89939i

−3.00000

1121.5

−

1.73205i

2.55654

−

1.89939i

−3.00000

1121.6

−

1.73205i

2.55654

1.89939i

−3.00000

1121.7

−

1.73205i

3.66935

−

4.93886i

−3.00000

1121.8

−

1.73205i

3.66935

4.93886i

−3.00000

1121.9

1.73205i

−3.66935

−

4.93886i

−3.00000

1121.10

1.73205i

−3.66935

4.93886i

−3.00000

1121.11

1.73205i

−2.55654

−

1.89939i

−3.00000

1121.12

1.73205i

−2.55654

1.89939i

−3.00000

1121.13

1.73205i

2.55654

−

1.89939i

−3.00000

1121.14

1.73205i

2.55654

1.89939i

−3.00000

1121.15

1.73205i

3.66935

−

4.93886i

−3.00000

1121.16

1.73205i

3.66935

4.93886i

−3.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
264.p	odd	2	1	CM by $\Q(\sqrt{-66}) $
3.b	odd	2	1	inner
4.b	odd	2	1	inner
8.b	even	2	1	inner
8.d	odd	2	1	inner
11.b	odd	2	1	inner
12.b	even	2	1	inner
24.f	even	2	1	inner
24.h	odd	2	1	inner
33.d	even	2	1	inner
44.c	even	2	1	inner
88.b	odd	2	1	inner
88.g	even	2	1	inner
132.d	odd	2	1	inner
264.m	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2112.2.m.m	✓	16
3.b	odd	2	1	inner	2112.2.m.m	✓	16
4.b	odd	2	1	inner	2112.2.m.m	✓	16
8.b	even	2	1	inner	2112.2.m.m	✓	16
8.d	odd	2	1	inner	2112.2.m.m	✓	16
11.b	odd	2	1	inner	2112.2.m.m	✓	16
12.b	even	2	1	inner	2112.2.m.m	✓	16
24.f	even	2	1	inner	2112.2.m.m	✓	16
24.h	odd	2	1	inner	2112.2.m.m	✓	16
33.d	even	2	1	inner	2112.2.m.m	✓	16
44.c	even	2	1	inner	2112.2.m.m	✓	16
88.b	odd	2	1	inner	2112.2.m.m	✓	16
88.g	even	2	1	inner	2112.2.m.m	✓	16
132.d	odd	2	1	inner	2112.2.m.m	✓	16
264.m	even	2	1	inner	2112.2.m.m	✓	16
264.p	odd	2	1	CM	2112.2.m.m	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2112.2.m.m	✓	16	1.a	even	1	1	trivial
2112.2.m.m	✓	16	3.b	odd	2	1	inner
2112.2.m.m	✓	16	4.b	odd	2	1	inner
2112.2.m.m	✓	16	8.b	even	2	1	inner
2112.2.m.m	✓	16	8.d	odd	2	1	inner
2112.2.m.m	✓	16	11.b	odd	2	1	inner
2112.2.m.m	✓	16	12.b	even	2	1	inner
2112.2.m.m	✓	16	24.f	even	2	1	inner
2112.2.m.m	✓	16	24.h	odd	2	1	inner
2112.2.m.m	✓	16	33.d	even	2	1	inner
2112.2.m.m	✓	16	44.c	even	2	1	inner
2112.2.m.m	✓	16	88.b	odd	2	1	inner
2112.2.m.m	✓	16	88.g	even	2	1	inner
2112.2.m.m	✓	16	132.d	odd	2	1	inner
2112.2.m.m	✓	16	264.m	even	2	1	inner
2112.2.m.m	✓	16	264.p	odd	2	1	CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(2112, [\chi])$:

$ T_{5}^{4} - 20T_{5}^{2} + 88 $

$ T_{17}^{2} - 44 $

$ T_{167} $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{2} + 3)^{8} $ (T^2 + 3)^8
$5$	$ (T^{4} - 20 T^{2} + 88)^{4} $ (T^4 - 20*T^2 + 88)^4
$7$	$ (T^{4} + 28 T^{2} + 88)^{4} $ (T^4 + 28*T^2 + 88)^4
$11$	$ (T^{2} + 11)^{8} $ (T^2 + 11)^8
$13$	$ (T^{4} - 52 T^{2} + 88)^{4} $ (T^4 - 52*T^2 + 88)^4
$17$	$ (T^{2} - 44)^{8} $ (T^2 - 44)^8
$19$	$ T^{16} $ T^16
$23$	$ (T^{4} + 92 T^{2} + 88)^{4} $ (T^4 + 92*T^2 + 88)^4
$29$	$ T^{16} $ T^16
$31$	$ T^{16} $ T^16
$37$	$ T^{16} $ T^16
$41$	$ (T^{2} - 132)^{8} $ (T^2 - 132)^8
$43$	$ T^{16} $ T^16
$47$	$ (T^{4} + 188 T^{2} + 88)^{4} $ (T^4 + 188*T^2 + 88)^4
$53$	$ (T^{4} - 212 T^{2} + 10648)^{4} $ (T^4 - 212*T^2 + 10648)^4
$59$	$ T^{16} $ T^16
$61$	$ (T^{4} - 244 T^{2} + 14872)^{4} $ (T^4 - 244*T^2 + 14872)^4
$67$	$ (T^{2} + 4)^{8} $ (T^2 + 4)^8
$71$	$ (T^{4} + 284 T^{2} + 14872)^{4} $ (T^4 + 284*T^2 + 14872)^4
$73$	$ T^{16} $ T^16
$79$	$ (T^{4} + 316 T^{2} + 14872)^{4} $ (T^4 + 316*T^2 + 14872)^4
$83$	$ (T^{2} + 132)^{8} $ (T^2 + 132)^8
$89$	$ T^{16} $ T^16
$97$	$ (T^{2} - 300)^{8} $ (T^2 - 300)^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:	\( N \)	\(=\)	\( 2112 = 2^{6} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2112.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{5} q^{3} + \beta_{10} q^{5} + \beta_{8} q^{7} - 3 q^{9}+O(q^{10}) \) q - b5 * q^3 + b10 * q^5 + b8 * q^7 - 3 * q^9	\( q - \beta_{5} q^{3} + \beta_{10} q^{5} + \beta_{8} q^{7} - 3 q^{9} - \beta_{2} q^{11} + \beta_{11} q^{13} + \beta_{12} q^{15} + \beta_{3} q^{17} - \beta_{9} q^{21} + (\beta_{13} - \beta_{12}) q^{23} + ( - \beta_1 + 5) q^{25} + 3 \beta_{5} q^{27} - \beta_{4} q^{33} + ( - \beta_{7} - 2 \beta_{2}) q^{35} - \beta_{14} q^{39} - 2 \beta_{4} q^{41} - 3 \beta_{10} q^{45} + ( - \beta_{13} - \beta_{12}) q^{47} + (3 \beta_1 - 7) q^{49} - \beta_{7} q^{51} - \beta_{15} q^{53} + (\beta_{14} + 2 \beta_{8}) q^{55} + ( - 2 \beta_{11} + \beta_{9}) q^{61} - 3 \beta_{8} q^{63} + ( - 2 \beta_{4} + \beta_{3}) q^{65} + \beta_{6} q^{67} + (\beta_{15} + 2 \beta_{10}) q^{69} + ( - 2 \beta_{13} + \beta_{12}) q^{71} + ( - 3 \beta_{6} - 5 \beta_{5}) q^{75} + (\beta_{15} - 2 \beta_{10}) q^{77} + (\beta_{14} - 2 \beta_{8}) q^{79} + 9 q^{81} - \beta_{7} q^{83} + (\beta_{11} + 3 \beta_{9}) q^{85} + (5 \beta_{6} - 2 \beta_{5}) q^{91} - 5 \beta_1 q^{97} + 3 \beta_{2} q^{99}+O(q^{100}) \) q - b5 * q^3 + b10 * q^5 + b8 * q^7 - 3 * q^9 - b2 * q^11 + b11 * q^13 + b12 * q^15 + b3 * q^17 - b9 * q^21 + (b13 - b12) * q^23 + (-b1 + 5) * q^25 + 3b5 q^27 - b4 * q^33 + (-b7 - 2b2) q^35 - b14 * q^39 - 2b4 q^41 - 3b10 q^45 + (-b13 - b12) * q^47 + (3b1 - 7) q^49 - b7 * q^51 - b15 * q^53 + (b14 + 2b8) q^55 + (-2b11 + b9) q^61 - 3b8 q^63 + (-2b4 + b3) q^65 + b6 * q^67 + (b15 + 2b10) q^69 + (-2b13 + b12) q^71 + (-3b6 - 5b5) * q^75 + (b15 - 2b10) q^77 + (b14 - 2b8) q^79 + 9 * q^81 - b7 * q^83 + (b11 + 3b9) q^85 + (5b6 - 2b5) * q^91 - 5b1 q^97 + 3b2 q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 48 q^{9}+O(q^{10}) \) 16 * q - 48 * q^9	\( 16 q - 48 q^{9} + 80 q^{25} - 112 q^{49} + 144 q^{81}+O(q^{100}) \) 16 * q - 48 * q^9 + 80 * q^25 - 112 * q^49 + 144 * q^81

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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	2112.2.m.m

Level	$2112$
Weight	$2$
Character orbit	2112.m
Analytic conductor	$16.864$
Analytic rank	$0$
Dimension	$16$
CM discriminant	-264
Inner twists	$16$

Self dual:	no
Analytic conductor:	\(16.8644049069\)
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:	\( x^{16} - 40x^{12} + 604x^{8} - 1440x^{4} + 1296 \) x^16 - 40x^12 + 604x^8 - 1440*x^4 + 1296
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{28} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

\(\beta_{1}\)	\(=\)	\( ( 5\nu^{12} - 128\nu^{8} + 320\nu^{4} + 14544 ) / 4284 \) (5v^12 - 128v^8 + 320*v^4 + 14544) / 4284
\(\beta_{2}\)	\(=\)	\( ( 2\nu^{12} - 75\nu^{8} + 1080\nu^{4} - 1370 ) / 238 \) (2v^12 - 75v^8 + 1080*v^4 - 1370) / 238
\(\beta_{3}\)	\(=\)	\( ( 2\nu^{14} - 89\nu^{10} + 1514\nu^{6} - 5724\nu^{2} ) / 756 \) (2v^14 - 89v^10 + 1514v^6 - 5724v^2) / 756
\(\beta_{4}\)	\(=\)	\( ( -10\nu^{14} + 409\nu^{10} - 6454\nu^{6} + 24156\nu^{2} ) / 3672 \) (-10v^14 + 409v^10 - 6454v^6 + 24156v^2) / 3672
\(\beta_{5}\)	\(=\)	\( ( 5\nu^{14} - 191\nu^{10} + 2714\nu^{6} - 2844\nu^{2} ) / 1512 \) (5v^14 - 191v^10 + 2714v^6 - 2844v^2) / 1512
\(\beta_{6}\)	\(=\)	\( ( -7\nu^{14} + 271\nu^{10} - 3814\nu^{6} + 3996\nu^{2} ) / 1836 \) (-7v^14 + 271v^10 - 3814v^6 + 3996v^2) / 1836
\(\beta_{7}\)	\(=\)	\( ( -\nu^{12} + 40\nu^{8} - 568\nu^{4} + 720 ) / 36 \) (-v^12 + 40v^8 - 568v^4 + 720) / 36
\(\beta_{8}\)	\(=\)	\( ( - 139 \nu^{15} - 372 \nu^{13} + 5272 \nu^{11} + 14664 \nu^{9} - 68872 \nu^{7} - 203736 \nu^{5} + \cdots + 367632 \nu ) / 154224 \) (-139v^15 - 372v^13 + 5272v^11 + 14664v^9 - 68872v^7 - 203736v^5 - 49608v^3 + 367632v) / 154224
\(\beta_{9}\)	\(=\)	\( ( 29\nu^{15} - 62\nu^{13} - 1028\nu^{11} + 2444\nu^{9} + 13280\nu^{7} - 38240\nu^{5} + 9528\nu^{3} + 69840\nu ) / 17136 \) (29v^15 - 62v^13 - 1028v^11 + 2444v^9 + 13280v^7 - 38240v^5 + 9528v^3 + 69840v) / 17136
\(\beta_{10}\)	\(=\)	\( ( 449 \nu^{15} - 372 \nu^{13} - 17492 \nu^{11} + 14664 \nu^{9} + 251504 \nu^{7} - 203736 \nu^{5} + \cdots + 59184 \nu ) / 154224 \) (449v^15 - 372v^13 - 17492v^11 + 14664v^9 + 251504v^7 - 203736v^5 - 385272v^3 + 59184v) / 154224
\(\beta_{11}\)	\(=\)	\( ( 77 \nu^{15} - 186 \nu^{13} - 2828 \nu^{11} + 7332 \nu^{9} + 36752 \nu^{7} - 107376 \nu^{5} + \cdots + 194832 \nu ) / 22032 \) (77v^15 - 186v^13 - 2828v^11 + 7332v^9 + 36752v^7 - 107376v^5 + 26424v^3 + 194832v) / 22032
\(\beta_{12}\)	\(=\)	\( ( 253 \nu^{15} + 246 \nu^{13} - 9904 \nu^{11} - 8868 \nu^{9} + 144712 \nu^{7} + 118560 \nu^{5} + \cdots - 34992 \nu ) / 51408 \) (253v^15 + 246v^13 - 9904v^11 - 8868v^9 + 144712v^7 + 118560v^5 - 221976v^3 - 34992v) / 51408
\(\beta_{13}\)	\(=\)	\( ( - 449 \nu^{15} - 372 \nu^{13} + 17492 \nu^{11} + 14664 \nu^{9} - 251504 \nu^{7} - 203736 \nu^{5} + \cdots + 59184 \nu ) / 77112 \) (-449v^15 - 372v^13 + 17492v^11 + 14664v^9 - 251504v^7 - 203736v^5 + 385272v^3 + 59184v) / 77112
\(\beta_{14}\)	\(=\)	\( ( 313 \nu^{15} + 744 \nu^{13} - 11440 \nu^{11} - 29328 \nu^{9} + 148552 \nu^{7} + 433176 \nu^{5} + \cdots - 786672 \nu ) / 51408 \) (313v^15 + 744v^13 - 11440v^11 - 29328v^9 + 148552v^7 + 433176v^5 + 106776v^3 - 786672v) / 51408
\(\beta_{15}\)	\(=\)	\( ( 281 \nu^{15} - 264 \nu^{13} - 10988 \nu^{11} + 9696 \nu^{9} + 159968 \nu^{7} - 130728 \nu^{5} + \cdots + 38448 \nu ) / 22032 \) (281v^15 - 264v^13 - 10988v^11 + 9696v^9 + 159968v^7 - 130728v^5 - 245304v^3 + 38448v) / 22032

\(\nu\)	\(=\)	\( ( -\beta_{13} + \beta_{11} - 2\beta_{10} - \beta_{9} + 2\beta_{8} ) / 8 \) (-b13 + b11 - 2b10 - b9 + 2b8) / 8
\(\nu^{2}\)	\(=\)	\( ( \beta_{6} + 2\beta_{5} + 2\beta_{4} + \beta_{3} ) / 4 \) (b6 + 2b5 + 2b4 + b3) / 4
\(\nu^{3}\)	\(=\)	\( ( \beta_{15} + 2\beta_{13} + 2\beta_{12} + \beta_{11} - 5\beta_{10} - \beta_{9} - 2\beta_{8} ) / 4 \) (b15 + 2b13 + 2b12 + b11 - 5b10 - b9 - 2b8) / 4
\(\nu^{4}\)	\(=\)	\( ( \beta_{7} + 4\beta_{2} - 5\beta _1 + 20 ) / 2 \) (b7 + 4b2 - 5b1 + 20) / 2
\(\nu^{5}\)	\(=\)	\( ( 4\beta_{14} - \beta_{13} + 7\beta_{11} - 2\beta_{10} - 15\beta_{9} + 26\beta_{8} ) / 4 \) (4b14 - b13 + 7b11 - 2b10 - 15b9 + 26*b8) / 4
\(\nu^{6}\)	\(=\)	\( 20\beta_{6} + 24\beta_{5} + 6\beta_{4} + 5\beta_{3} \) 20b6 + 24b5 + 6b4 + 5b3
\(\nu^{7}\)	\(=\)	\( ( 25 \beta_{15} + 5 \beta_{14} + 43 \beta_{13} + 50 \beta_{12} - 7 \beta_{11} - 111 \beta_{10} + \cdots + 29 \beta_{8} ) / 4 \) (25b15 + 5b14 + 43b13 + 50b12 - 7b11 - 111b10 + 17b9 + 29b8) / 4
\(\nu^{8}\)	\(=\)	\( 20\beta_{7} + 70\beta_{2} - 28\beta _1 + 98 \) 20b7 + 70b2 - 28*b1 + 98
\(\nu^{9}\)	\(=\)	\( ( - 31 \beta_{15} + 39 \beta_{14} + 53 \beta_{13} + 62 \beta_{12} + 67 \beta_{11} + 137 \beta_{10} + \cdots + 251 \beta_{8} ) / 2 \) (-31b15 + 39b14 + 53b13 + 62b12 + 67b11 + 137b10 - 145b9 + 251b8) / 2
\(\nu^{10}\)	\(=\)	\( 589\beta_{6} + 682\beta_{5} + 22\beta_{4} + 19\beta_{3} \) 589b6 + 682b5 + 22b4 + 19b3
\(\nu^{11}\)	\(=\)	\( ( 195 \beta_{15} + 157 \beta_{14} + 337 \beta_{13} + 390 \beta_{12} - 271 \beta_{11} + \cdots + 1013 \beta_{8} ) / 2 \) (195b15 + 157b14 + 337b13 + 390b12 - 271b11 - 869b10 + 585b9 + 1013b8) / 2
\(\nu^{12}\)	\(=\)	\( 480\beta_{7} + 1664\beta_{2} + 300\beta _1 - 1040 \) 480b7 + 1664b2 + 300*b1 - 1040
\(\nu^{13}\)	\(=\)	\( - 611 \beta_{15} + 221 \beta_{14} + 1058 \beta_{13} + 1222 \beta_{12} + 382 \beta_{11} + \cdots + 1427 \beta_{8} \) -611b15 + 221b14 + 1058b13 + 1222b12 + 382b11 + 2727b10 - 824b9 + 1427b8
\(\nu^{14}\)	\(=\)	\( 11786\beta_{6} + 13612\beta_{5} - 2132\beta_{4} - 1846\beta_{3} \) 11786b6 + 13612b5 - 2132b4 - 1846b3
\(\nu^{15}\)	\(=\)	\( 512 \beta_{15} + 2358 \beta_{14} + 886 \beta_{13} + 1024 \beta_{12} - 4084 \beta_{11} + \cdots + 15242 \beta_{8} \) 512b15 + 2358b14 + 886b13 + 1024b12 - 4084b11 - 2284b10 + 8800b9 + 15242b8

\(n\)	\(133\)	\(1409\)	\(1729\)	\(2047\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1121.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{2} + 3)^{8} \) (T^2 + 3)^8
$5$	\( (T^{4} - 20 T^{2} + 88)^{4} \) (T^4 - 20*T^2 + 88)^4
$7$	\( (T^{4} + 28 T^{2} + 88)^{4} \) (T^4 + 28*T^2 + 88)^4
$11$	\( (T^{2} + 11)^{8} \) (T^2 + 11)^8
$13$	\( (T^{4} - 52 T^{2} + 88)^{4} \) (T^4 - 52*T^2 + 88)^4
$17$	\( (T^{2} - 44)^{8} \) (T^2 - 44)^8
$19$	\( T^{16} \) T^16
$23$	\( (T^{4} + 92 T^{2} + 88)^{4} \) (T^4 + 92*T^2 + 88)^4
$29$	\( T^{16} \) T^16
$31$	\( T^{16} \) T^16
$37$	\( T^{16} \) T^16
$41$	\( (T^{2} - 132)^{8} \) (T^2 - 132)^8
$43$	\( T^{16} \) T^16
$47$	\( (T^{4} + 188 T^{2} + 88)^{4} \) (T^4 + 188*T^2 + 88)^4
$53$	\( (T^{4} - 212 T^{2} + 10648)^{4} \) (T^4 - 212*T^2 + 10648)^4
$59$	\( T^{16} \) T^16
$61$	\( (T^{4} - 244 T^{2} + 14872)^{4} \) (T^4 - 244*T^2 + 14872)^4
$67$	\( (T^{2} + 4)^{8} \) (T^2 + 4)^8
$71$	\( (T^{4} + 284 T^{2} + 14872)^{4} \) (T^4 + 284*T^2 + 14872)^4
$73$	\( T^{16} \) T^16
$79$	\( (T^{4} + 316 T^{2} + 14872)^{4} \) (T^4 + 316*T^2 + 14872)^4
$83$	\( (T^{2} + 132)^{8} \) (T^2 + 132)^8
$89$	\( T^{16} \) T^16
$97$	\( (T^{2} - 300)^{8} \) (T^2 - 300)^8
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