LMFDB - Newform orbit 2400.2.k.f

Newspace parameters

Level:	$ N $	$=$	$ 2400 = 2^{5} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2400.k (of order $2$, degree $1$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$19.1640964851$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.180227832610816.1
Defining polynomial:	$ x^{12} + x^{10} - 8x^{6} + 16x^{2} + 64 $ x^12 + x^10 - 8x^6 + 16x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{31}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{11}$ 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^{7} - q^{9}+O(q^{10}) $ q + b3 * q^3 - b5 * q^7 - q^9	\( q + \beta_{3} q^{3} - \beta_{5} q^{7} - q^{9} + \beta_{7} q^{11} + ( - \beta_{11} + \beta_{3}) q^{13} + ( - \beta_{9} - \beta_{8}) q^{17} + ( - \beta_{6} + \beta_{2}) q^{19} - \beta_{2} q^{21} + (2 \beta_{9} + \beta_{8} - \beta_{5}) q^{23} - \beta_{3} q^{27} + ( - 2 \beta_{7} - \beta_{6} - 2 \beta_{2}) q^{29} + ( - \beta_{10} + \beta_{4} + 3) q^{31} + ( - \beta_{9} + \beta_{5}) q^{33} + ( - \beta_{11} - 3 \beta_{3}) q^{37} + ( - \beta_{10} - 1) q^{39} + ( - 2 \beta_{10} + 2 \beta_{4}) q^{41} - 2 \beta_1 q^{43} + ( - \beta_{8} + \beta_{5}) q^{47} + (2 \beta_{4} + 1) q^{49} + ( - \beta_{7} - \beta_{6} - \beta_{2}) q^{51} + ( - 4 \beta_{3} + \beta_1) q^{53} + (\beta_{8} - \beta_{5}) q^{57} + (\beta_{7} - 2 \beta_{2}) q^{59} + ( - 2 \beta_{6} - 2 \beta_{2}) q^{61} + \beta_{5} q^{63} - 2 \beta_1 q^{67} + (2 \beta_{7} + \beta_{6} + \beta_{2}) q^{69} + ( - 2 \beta_{10} + 2 \beta_{4} - 2) q^{71} + ( - 2 \beta_{9} - 4 \beta_{8}) q^{73} + 4 \beta_{3} q^{77} + ( - \beta_{10} + \beta_{4} + 3) q^{79} + q^{81} + (2 \beta_{11} - 2 \beta_{3}) q^{83} + (2 \beta_{9} + \beta_{8}) q^{87} + ( - 2 \beta_{10} - 2 \beta_{4} + 4) q^{89} + ( - 2 \beta_{7} - 4 \beta_{6} - 4 \beta_{2}) q^{91} + (\beta_{11} + 3 \beta_{3} + \beta_1) q^{93} + (2 \beta_{9} - 2 \beta_{8}) q^{97} - \beta_{7} q^{99}+O(q^{100}) \) q + b3 * q^3 - b5 * q^7 - q^9 + b7 * q^11 + (-b11 + b3) * q^13 + (-b9 - b8) * q^17 + (-b6 + b2) * q^19 - b2 * q^21 + (2b9 + b8 - b5) q^23 - b3 * q^27 + (-2b7 - b6 - 2b2) * q^29 + (-b10 + b4 + 3) * q^31 + (-b9 + b5) * q^33 + (-b11 - 3b3) q^37 + (-b10 - 1) * q^39 + (-2b10 + 2b4) * q^41 - 2b1 q^43 + (-b8 + b5) * q^47 + (2b4 + 1) q^49 + (-b7 - b6 - b2) * q^51 + (-4b3 + b1) q^53 + (b8 - b5) * q^57 + (b7 - 2b2) q^59 + (-2b6 - 2b2) * q^61 + b5 * q^63 - 2b1 q^67 + (2b7 + b6 + b2) q^69 + (-2b10 + 2b4 - 2) * q^71 + (-2b9 - 4b8) * q^73 + 4b3 q^77 + (-b10 + b4 + 3) * q^79 + q^81 + (2b11 - 2b3) * q^83 + (2b9 + b8) q^87 + (-2b10 - 2b4 + 4) * q^89 + (-2b7 - 4b6 - 4b2) q^91 + (b11 + 3b3 + b1) q^93 + (2b9 - 2b8) * q^97 - b7 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^9	$ 12 q - 12 q^{9} + 32 q^{31} - 16 q^{39} - 8 q^{41} + 12 q^{49} - 32 q^{71} + 32 q^{79} + 12 q^{81} + 40 q^{89}+O(q^{100}) $ 12 * q - 12 * q^9 + 32 * q^31 - 16 * q^39 - 8 * q^41 + 12 * q^49 - 32 * q^71 + 32 * q^79 + 12 * q^81 + 40 * q^89

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} + x^{10} - 8x^{6} + 16x^{2} + 64 $ x^12 + x^10 - 8*x^6 + 16*x^2 + 64 :

$\beta_{1}$	$=$	$ ( \nu^{9} + \nu^{7} + 8\nu ) / 8 $ (v^9 + v^7 + 8*v) / 8
$\beta_{2}$	$=$	$ ( \nu^{8} + \nu^{6} + 4\nu^{4} - 4\nu^{2} ) / 8 $ (v^8 + v^6 + 4v^4 - 4v^2) / 8
$\beta_{3}$	$=$	$ ( \nu^{11} - \nu^{9} + 2\nu^{7} + 4\nu^{5} + 8\nu^{3} ) / 64 $ (v^11 - v^9 + 2v^7 + 4v^5 + 8*v^3) / 64
$\beta_{4}$	$=$	$ ( -\nu^{8} - \nu^{6} + 4\nu^{4} + 12\nu^{2} ) / 8 $ (-v^8 - v^6 + 4v^4 + 12v^2) / 8
$\beta_{5}$	$=$	$ ( -\nu^{11} - 3\nu^{9} + 2\nu^{7} + 4\nu^{5} + 24\nu^{3} ) / 32 $ (-v^11 - 3v^9 + 2v^7 + 4v^5 + 24v^3) / 32
$\beta_{6}$	$=$	$ ( -\nu^{8} + 3\nu^{6} + 4\nu^{2} - 16 ) / 8 $ (-v^8 + 3v^6 + 4v^2 - 16) / 8
$\beta_{7}$	$=$	$ ( \nu^{10} + \nu^{8} - 4\nu^{6} - 12\nu^{4} + 16\nu^{2} + 32 ) / 16 $ (v^10 + v^8 - 4v^6 - 12v^4 + 16*v^2 + 32) / 16
$\beta_{8}$	$=$	$ ( -\nu^{11} + \nu^{9} - 2\nu^{7} + 12\nu^{5} - 24\nu^{3} + 32\nu ) / 32 $ (-v^11 + v^9 - 2v^7 + 12v^5 - 24v^3 + 32v) / 32
$\beta_{9}$	$=$	$ ( \nu^{11} - \nu^{9} - 6\nu^{7} - 4\nu^{5} + 40\nu^{3} + 32\nu ) / 32 $ (v^11 - v^9 - 6v^7 - 4v^5 + 40v^3 + 32v) / 32
$\beta_{10}$	$=$	$ ( -\nu^{10} + \nu^{6} + 4\nu^{4} + 4\nu^{2} - 8 ) / 8 $ (-v^10 + v^6 + 4v^4 + 4v^2 - 8) / 8
$\beta_{11}$	$=$	$ ( 3\nu^{11} - 3\nu^{9} - 10\nu^{7} - 36\nu^{5} - 8\nu^{3} + 128\nu ) / 64 $ (3v^11 - 3v^9 - 10v^7 - 36v^5 - 8v^3 + 128v) / 64

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

$\nu$	$=$	$ ( \beta_{11} + \beta_{8} + \beta_{5} + \beta_{3} + \beta_1 ) / 4 $ (b11 + b8 + b5 + b3 + b1) / 4
$\nu^{2}$	$=$	$ ( \beta_{10} + 2\beta_{7} + \beta_{6} + \beta_{4} + \beta_{2} - 1 ) / 4 $ (b10 + 2*b7 + b6 + b4 + b2 - 1) / 4
$\nu^{3}$	$=$	$ ( -\beta_{11} + 2\beta_{9} - \beta_{8} + \beta_{5} - \beta_{3} + \beta_1 ) / 4 $ (-b11 + 2*b9 - b8 + b5 - b3 + b1) / 4
$\nu^{4}$	$=$	$ ( -\beta_{10} - 2\beta_{7} - \beta_{6} + 3\beta_{4} + 3\beta_{2} + 1 ) / 4 $ (-b10 - 2b7 - b6 + 3b4 + 3*b2 + 1) / 4
$\nu^{5}$	$=$	$ ( -3\beta_{11} + 2\beta_{9} + 5\beta_{8} - \beta_{5} + 13\beta_{3} - \beta_1 ) / 4 $ (-3b11 + 2b9 + 5b8 - b5 + 13b3 - b1) / 4
$\nu^{6}$	$=$	$ ( \beta_{10} + 2\beta_{7} + 9\beta_{6} - 3\beta_{4} + 5\beta_{2} + 15 ) / 4 $ (b10 + 2b7 + 9b6 - 3b4 + 5b2 + 15) / 4
$\nu^{7}$	$=$	$ ( 3\beta_{11} - 10\beta_{9} - 5\beta_{8} + 9\beta_{5} + 19\beta_{3} + 9\beta_1 ) / 4 $ (3b11 - 10b9 - 5b8 + 9b5 + 19b3 + 9b1) / 4
$\nu^{8}$	$=$	$ ( 7\beta_{10} + 14\beta_{7} - \beta_{6} - 5\beta_{4} + 19\beta_{2} - 23 ) / 4 $ (7b10 + 14b7 - b6 - 5b4 + 19b2 - 23) / 4
$\nu^{9}$	$=$	$ ( -11\beta_{11} + 10\beta_{9} - 3\beta_{8} - 17\beta_{5} - 27\beta_{3} + 15\beta_1 ) / 4 $ (-11b11 + 10b9 - 3b8 - 17b5 - 27b3 + 15b1) / 4
$\nu^{10}$	$=$	$ ( -31\beta_{10} + 2\beta_{7} + 9\beta_{6} + 13\beta_{4} + 21\beta_{2} - 17 ) / 4 $ (-31b10 + 2b7 + 9b6 + 13b4 + 21*b2 - 17) / 4
$\nu^{11}$	$=$	$ ( 3\beta_{11} + 6\beta_{9} - 5\beta_{8} - 39\beta_{5} + 147\beta_{3} - 7\beta_1 ) / 4 $ (3b11 + 6b9 - 5b8 - 39b5 + 147b3 - 7b1) / 4

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2400\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1601$	$1951$
$\chi(n)$	$1$	$-1$	$1$	$1$

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.

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

1201.1

1.37729	−	0.321037i
−0.450129	−	1.34067i
0.806504	+	1.16170i
−0.806504	+	1.16170i
0.450129	−	1.34067i
−1.37729	−	0.321037i
1.37729	+	0.321037i
−0.450129	+	1.34067i
0.806504	−	1.16170i
−0.806504	−	1.16170i
0.450129	+	1.34067i
−1.37729	+	0.321037i

−

1.00000i

−4.05705

−1.00000

1201.2

−

1.00000i

−2.64265

−1.00000

1201.3

−

1.00000i

−0.746175

−1.00000

1201.4

−

1.00000i

0.746175

−1.00000

1201.5

−

1.00000i

2.64265

−1.00000

1201.6

−

1.00000i

4.05705

−1.00000

1201.7

1.00000i

−4.05705

−1.00000

1201.8

1.00000i

−2.64265

−1.00000

1201.9

1.00000i

−0.746175

−1.00000

1201.10

1.00000i

0.746175

−1.00000

1201.11

1.00000i

2.64265

−1.00000

1201.12

1.00000i

4.05705

−1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
8.b	even	2	1	inner
40.f	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2400.2.k.f		12
3.b	odd	2	1		7200.2.k.u		12
4.b	odd	2	1		600.2.k.f		12
5.b	even	2	1	inner	2400.2.k.f		12
5.c	odd	4	1		480.2.d.a		6
5.c	odd	4	1		480.2.d.b		6
8.b	even	2	1	inner	2400.2.k.f		12
8.d	odd	2	1		600.2.k.f		12
12.b	even	2	1		1800.2.k.u		12
15.d	odd	2	1		7200.2.k.u		12
15.e	even	4	1		1440.2.d.e		6
15.e	even	4	1		1440.2.d.f		6
20.d	odd	2	1		600.2.k.f		12
20.e	even	4	1		120.2.d.a	✓	6
20.e	even	4	1		120.2.d.b	yes	6
24.f	even	2	1		1800.2.k.u		12
24.h	odd	2	1		7200.2.k.u		12
40.e	odd	2	1		600.2.k.f		12
40.f	even	2	1	inner	2400.2.k.f		12
40.i	odd	4	1		480.2.d.a		6
40.i	odd	4	1		480.2.d.b		6
40.k	even	4	1		120.2.d.a	✓	6
40.k	even	4	1		120.2.d.b	yes	6
60.h	even	2	1		1800.2.k.u		12
60.l	odd	4	1		360.2.d.e		6
60.l	odd	4	1		360.2.d.f		6
80.i	odd	4	2		3840.2.f.m		12
80.j	even	4	2		3840.2.f.l		12
80.s	even	4	2		3840.2.f.l		12
80.t	odd	4	2		3840.2.f.m		12
120.i	odd	2	1		7200.2.k.u		12
120.m	even	2	1		1800.2.k.u		12
120.q	odd	4	1		360.2.d.e		6
120.q	odd	4	1		360.2.d.f		6
120.w	even	4	1		1440.2.d.e		6
120.w	even	4	1		1440.2.d.f		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
120.2.d.a	✓	6	20.e	even	4	1
120.2.d.a	✓	6	40.k	even	4	1
120.2.d.b	yes	6	20.e	even	4	1
120.2.d.b	yes	6	40.k	even	4	1
360.2.d.e		6	60.l	odd	4	1
360.2.d.e		6	120.q	odd	4	1
360.2.d.f		6	60.l	odd	4	1
360.2.d.f		6	120.q	odd	4	1
480.2.d.a		6	5.c	odd	4	1
480.2.d.a		6	40.i	odd	4	1
480.2.d.b		6	5.c	odd	4	1
480.2.d.b		6	40.i	odd	4	1
600.2.k.f		12	4.b	odd	2	1
600.2.k.f		12	8.d	odd	2	1
600.2.k.f		12	20.d	odd	2	1
600.2.k.f		12	40.e	odd	2	1
1440.2.d.e		6	15.e	even	4	1
1440.2.d.e		6	120.w	even	4	1
1440.2.d.f		6	15.e	even	4	1
1440.2.d.f		6	120.w	even	4	1
1800.2.k.u		12	12.b	even	2	1
1800.2.k.u		12	24.f	even	2	1
1800.2.k.u		12	60.h	even	2	1
1800.2.k.u		12	120.m	even	2	1
2400.2.k.f		12	1.a	even	1	1	trivial
2400.2.k.f		12	5.b	even	2	1	inner
2400.2.k.f		12	8.b	even	2	1	inner
2400.2.k.f		12	40.f	even	2	1	inner
3840.2.f.l		12	80.j	even	4	2
3840.2.f.l		12	80.s	even	4	2
3840.2.f.m		12	80.i	odd	4	2
3840.2.f.m		12	80.t	odd	4	2
7200.2.k.u		12	3.b	odd	2	1
7200.2.k.u		12	15.d	odd	2	1
7200.2.k.u		12	24.h	odd	2	1
7200.2.k.u		12	120.i	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{6} - 24T_{7}^{4} + 128T_{7}^{2} - 64 $ T7^6 - 24*T7^4 + 128*T7^2 - 64 acting on $S_{2}^{\mathrm{new}}(2400, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} $ T^12
$3$	$ (T^{2} + 1)^{6} $ (T^2 + 1)^6
$5$	$ T^{12} $ T^12
$7$	$ (T^{6} - 24 T^{4} + 128 T^{2} - 64)^{2} $ (T^6 - 24T^4 + 128T^2 - 64)^2
$11$	$ (T^{6} + 32 T^{4} + 96 T^{2} + 64)^{2} $ (T^6 + 32T^4 + 96T^2 + 64)^2
$13$	$ (T^{6} + 48 T^{4} + 704 T^{2} + 3136)^{2} $ (T^6 + 48T^4 + 704T^2 + 3136)^2
$17$	$ (T^{6} - 36 T^{4} + 368 T^{2} - 1024)^{2} $ (T^6 - 36T^4 + 368T^2 - 1024)^2
$19$	$ (T^{6} + 60 T^{4} + 512 T^{2} + 1024)^{2} $ (T^6 + 60T^4 + 512T^2 + 1024)^2
$23$	$ (T^{6} - 92 T^{4} + 2304 T^{2} + \cdots - 16384)^{2} $ (T^6 - 92T^4 + 2304T^2 - 16384)^2
$29$	$ (T^{6} + 108 T^{4} + 3120 T^{2} + \cdots + 12544)^{2} $ (T^6 + 108T^4 + 3120T^2 + 12544)^2
$31$	$ (T^{3} - 8 T^{2} - 4 T + 64)^{4} $ (T^3 - 8T^2 - 4T + 64)^4
$37$	$ (T^{6} + 64 T^{4} + 128 T^{2} + 64)^{2} $ (T^6 + 64T^4 + 128T^2 + 64)^2
$41$	$ (T^{3} + 2 T^{2} - 100 T + 56)^{4} $ (T^3 + 2T^2 - 100T + 56)^4
$43$	$ (T^{6} + 128 T^{4} + 4096 T^{2} + \cdots + 4096)^{2} $ (T^6 + 128T^4 + 4096T^2 + 4096)^2
$47$	$ (T^{6} - 60 T^{4} + 512 T^{2} - 1024)^{2} $ (T^6 - 60T^4 + 512T^2 - 1024)^2
$53$	$ (T^{6} + 80 T^{4} + 1216 T^{2} + 64)^{2} $ (T^6 + 80T^4 + 1216T^2 + 64)^2
$59$	$ (T^{6} + 176 T^{4} + 9888 T^{2} + \cdots + 179776)^{2} $ (T^6 + 176T^4 + 9888T^2 + 179776)^2
$61$	$ (T^{6} + 176 T^{4} + 7168 T^{2} + \cdots + 65536)^{2} $ (T^6 + 176T^4 + 7168T^2 + 65536)^2
$67$	$ (T^{6} + 128 T^{4} + 4096 T^{2} + \cdots + 4096)^{2} $ (T^6 + 128T^4 + 4096T^2 + 4096)^2
$71$	$ (T^{3} + 8 T^{2} - 80 T - 128)^{4} $ (T^3 + 8T^2 - 80T - 128)^4
$73$	$ (T^{6} - 384 T^{4} + 34560 T^{2} + \cdots - 16384)^{2} $ (T^6 - 384T^4 + 34560T^2 - 16384)^2
$79$	$ (T^{3} - 8 T^{2} - 4 T + 64)^{4} $ (T^3 - 8T^2 - 4T + 64)^4
$83$	$ (T^{6} + 192 T^{4} + 11264 T^{2} + \cdots + 200704)^{2} $ (T^6 + 192T^4 + 11264T^2 + 200704)^2
$89$	$ (T^{3} - 10 T^{2} - 164 T + 1384)^{4} $ (T^3 - 10T^2 - 164T + 1384)^4
$97$	$ (T^{6} - 336 T^{4} + 28416 T^{2} + \cdots - 262144)^{2} $ (T^6 - 336T^4 + 28416T^2 - 262144)^2
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Overview	Random
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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:	\( N \)	\(=\)	\( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2400.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_{3} q^{3} - \beta_{5} q^{7} - q^{9}+O(q^{10}) \) q + b3 * q^3 - b5 * q^7 - q^9	\( q + \beta_{3} q^{3} - \beta_{5} q^{7} - q^{9} + \beta_{7} q^{11} + ( - \beta_{11} + \beta_{3}) q^{13} + ( - \beta_{9} - \beta_{8}) q^{17} + ( - \beta_{6} + \beta_{2}) q^{19} - \beta_{2} q^{21} + (2 \beta_{9} + \beta_{8} - \beta_{5}) q^{23} - \beta_{3} q^{27} + ( - 2 \beta_{7} - \beta_{6} - 2 \beta_{2}) q^{29} + ( - \beta_{10} + \beta_{4} + 3) q^{31} + ( - \beta_{9} + \beta_{5}) q^{33} + ( - \beta_{11} - 3 \beta_{3}) q^{37} + ( - \beta_{10} - 1) q^{39} + ( - 2 \beta_{10} + 2 \beta_{4}) q^{41} - 2 \beta_1 q^{43} + ( - \beta_{8} + \beta_{5}) q^{47} + (2 \beta_{4} + 1) q^{49} + ( - \beta_{7} - \beta_{6} - \beta_{2}) q^{51} + ( - 4 \beta_{3} + \beta_1) q^{53} + (\beta_{8} - \beta_{5}) q^{57} + (\beta_{7} - 2 \beta_{2}) q^{59} + ( - 2 \beta_{6} - 2 \beta_{2}) q^{61} + \beta_{5} q^{63} - 2 \beta_1 q^{67} + (2 \beta_{7} + \beta_{6} + \beta_{2}) q^{69} + ( - 2 \beta_{10} + 2 \beta_{4} - 2) q^{71} + ( - 2 \beta_{9} - 4 \beta_{8}) q^{73} + 4 \beta_{3} q^{77} + ( - \beta_{10} + \beta_{4} + 3) q^{79} + q^{81} + (2 \beta_{11} - 2 \beta_{3}) q^{83} + (2 \beta_{9} + \beta_{8}) q^{87} + ( - 2 \beta_{10} - 2 \beta_{4} + 4) q^{89} + ( - 2 \beta_{7} - 4 \beta_{6} - 4 \beta_{2}) q^{91} + (\beta_{11} + 3 \beta_{3} + \beta_1) q^{93} + (2 \beta_{9} - 2 \beta_{8}) q^{97} - \beta_{7} q^{99}+O(q^{100}) \) q + b3 * q^3 - b5 * q^7 - q^9 + b7 * q^11 + (-b11 + b3) * q^13 + (-b9 - b8) * q^17 + (-b6 + b2) * q^19 - b2 * q^21 + (2b9 + b8 - b5) q^23 - b3 * q^27 + (-2b7 - b6 - 2b2) * q^29 + (-b10 + b4 + 3) * q^31 + (-b9 + b5) * q^33 + (-b11 - 3b3) q^37 + (-b10 - 1) * q^39 + (-2b10 + 2b4) * q^41 - 2b1 q^43 + (-b8 + b5) * q^47 + (2b4 + 1) q^49 + (-b7 - b6 - b2) * q^51 + (-4b3 + b1) q^53 + (b8 - b5) * q^57 + (b7 - 2b2) q^59 + (-2b6 - 2b2) * q^61 + b5 * q^63 - 2b1 q^67 + (2b7 + b6 + b2) q^69 + (-2b10 + 2b4 - 2) * q^71 + (-2b9 - 4b8) * q^73 + 4b3 q^77 + (-b10 + b4 + 3) * q^79 + q^81 + (2b11 - 2b3) * q^83 + (2b9 + b8) q^87 + (-2b10 - 2b4 + 4) * q^89 + (-2b7 - 4b6 - 4b2) q^91 + (b11 + 3b3 + b1) q^93 + (2b9 - 2b8) * q^97 - b7 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^9	\( 12 q - 12 q^{9} + 32 q^{31} - 16 q^{39} - 8 q^{41} + 12 q^{49} - 32 q^{71} + 32 q^{79} + 12 q^{81} + 40 q^{89}+O(q^{100}) \) 12 * q - 12 * q^9 + 32 * q^31 - 16 * q^39 - 8 * q^41 + 12 * q^49 - 32 * q^71 + 32 * q^79 + 12 * q^81 + 40 * q^89

Introduction

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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	2400.2.k.f

Level	$2400$
Weight	$2$
Character orbit	2400.k
Analytic conductor	$19.164$
Analytic rank	$0$
Dimension	$12$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(19.1640964851\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Coefficient field:	12.0.180227832610816.1
Defining polynomial:	\( x^{12} + x^{10} - 8x^{6} + 16x^{2} + 64 \) x^12 + x^10 - 8x^6 + 16x^2 + 64
Coefficient ring:	\(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index:	\( 2^{14} \)
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( \nu^{9} + \nu^{7} + 8\nu ) / 8 \) (v^9 + v^7 + 8*v) / 8
\(\beta_{2}\)	\(=\)	\( ( \nu^{8} + \nu^{6} + 4\nu^{4} - 4\nu^{2} ) / 8 \) (v^8 + v^6 + 4v^4 - 4v^2) / 8
\(\beta_{3}\)	\(=\)	\( ( \nu^{11} - \nu^{9} + 2\nu^{7} + 4\nu^{5} + 8\nu^{3} ) / 64 \) (v^11 - v^9 + 2v^7 + 4v^5 + 8*v^3) / 64
\(\beta_{4}\)	\(=\)	\( ( -\nu^{8} - \nu^{6} + 4\nu^{4} + 12\nu^{2} ) / 8 \) (-v^8 - v^6 + 4v^4 + 12v^2) / 8
\(\beta_{5}\)	\(=\)	\( ( -\nu^{11} - 3\nu^{9} + 2\nu^{7} + 4\nu^{5} + 24\nu^{3} ) / 32 \) (-v^11 - 3v^9 + 2v^7 + 4v^5 + 24v^3) / 32
\(\beta_{6}\)	\(=\)	\( ( -\nu^{8} + 3\nu^{6} + 4\nu^{2} - 16 ) / 8 \) (-v^8 + 3v^6 + 4v^2 - 16) / 8
\(\beta_{7}\)	\(=\)	\( ( \nu^{10} + \nu^{8} - 4\nu^{6} - 12\nu^{4} + 16\nu^{2} + 32 ) / 16 \) (v^10 + v^8 - 4v^6 - 12v^4 + 16*v^2 + 32) / 16
\(\beta_{8}\)	\(=\)	\( ( -\nu^{11} + \nu^{9} - 2\nu^{7} + 12\nu^{5} - 24\nu^{3} + 32\nu ) / 32 \) (-v^11 + v^9 - 2v^7 + 12v^5 - 24v^3 + 32v) / 32
\(\beta_{9}\)	\(=\)	\( ( \nu^{11} - \nu^{9} - 6\nu^{7} - 4\nu^{5} + 40\nu^{3} + 32\nu ) / 32 \) (v^11 - v^9 - 6v^7 - 4v^5 + 40v^3 + 32v) / 32
\(\beta_{10}\)	\(=\)	\( ( -\nu^{10} + \nu^{6} + 4\nu^{4} + 4\nu^{2} - 8 ) / 8 \) (-v^10 + v^6 + 4v^4 + 4v^2 - 8) / 8
\(\beta_{11}\)	\(=\)	\( ( 3\nu^{11} - 3\nu^{9} - 10\nu^{7} - 36\nu^{5} - 8\nu^{3} + 128\nu ) / 64 \) (3v^11 - 3v^9 - 10v^7 - 36v^5 - 8v^3 + 128v) / 64

\(\nu\)	\(=\)	\( ( \beta_{11} + \beta_{8} + \beta_{5} + \beta_{3} + \beta_1 ) / 4 \) (b11 + b8 + b5 + b3 + b1) / 4
\(\nu^{2}\)	\(=\)	\( ( \beta_{10} + 2\beta_{7} + \beta_{6} + \beta_{4} + \beta_{2} - 1 ) / 4 \) (b10 + 2*b7 + b6 + b4 + b2 - 1) / 4
\(\nu^{3}\)	\(=\)	\( ( -\beta_{11} + 2\beta_{9} - \beta_{8} + \beta_{5} - \beta_{3} + \beta_1 ) / 4 \) (-b11 + 2*b9 - b8 + b5 - b3 + b1) / 4
\(\nu^{4}\)	\(=\)	\( ( -\beta_{10} - 2\beta_{7} - \beta_{6} + 3\beta_{4} + 3\beta_{2} + 1 ) / 4 \) (-b10 - 2b7 - b6 + 3b4 + 3*b2 + 1) / 4
\(\nu^{5}\)	\(=\)	\( ( -3\beta_{11} + 2\beta_{9} + 5\beta_{8} - \beta_{5} + 13\beta_{3} - \beta_1 ) / 4 \) (-3b11 + 2b9 + 5b8 - b5 + 13b3 - b1) / 4
\(\nu^{6}\)	\(=\)	\( ( \beta_{10} + 2\beta_{7} + 9\beta_{6} - 3\beta_{4} + 5\beta_{2} + 15 ) / 4 \) (b10 + 2b7 + 9b6 - 3b4 + 5b2 + 15) / 4
\(\nu^{7}\)	\(=\)	\( ( 3\beta_{11} - 10\beta_{9} - 5\beta_{8} + 9\beta_{5} + 19\beta_{3} + 9\beta_1 ) / 4 \) (3b11 - 10b9 - 5b8 + 9b5 + 19b3 + 9b1) / 4
\(\nu^{8}\)	\(=\)	\( ( 7\beta_{10} + 14\beta_{7} - \beta_{6} - 5\beta_{4} + 19\beta_{2} - 23 ) / 4 \) (7b10 + 14b7 - b6 - 5b4 + 19b2 - 23) / 4
\(\nu^{9}\)	\(=\)	\( ( -11\beta_{11} + 10\beta_{9} - 3\beta_{8} - 17\beta_{5} - 27\beta_{3} + 15\beta_1 ) / 4 \) (-11b11 + 10b9 - 3b8 - 17b5 - 27b3 + 15b1) / 4
\(\nu^{10}\)	\(=\)	\( ( -31\beta_{10} + 2\beta_{7} + 9\beta_{6} + 13\beta_{4} + 21\beta_{2} - 17 ) / 4 \) (-31b10 + 2b7 + 9b6 + 13b4 + 21*b2 - 17) / 4
\(\nu^{11}\)	\(=\)	\( ( 3\beta_{11} + 6\beta_{9} - 5\beta_{8} - 39\beta_{5} + 147\beta_{3} - 7\beta_1 ) / 4 \) (3b11 + 6b9 - 5b8 - 39b5 + 147b3 - 7b1) / 4

\(n\)	\(577\)	\(901\)	\(1601\)	\(1951\)
\(\chi(n)\)	\(1\)	\(-1\)	\(1\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1201.12
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} \) T^12
$3$	\( (T^{2} + 1)^{6} \) (T^2 + 1)^6
$5$	\( T^{12} \) T^12
$7$	\( (T^{6} - 24 T^{4} + 128 T^{2} - 64)^{2} \) (T^6 - 24T^4 + 128T^2 - 64)^2
$11$	\( (T^{6} + 32 T^{4} + 96 T^{2} + 64)^{2} \) (T^6 + 32T^4 + 96T^2 + 64)^2
$13$	\( (T^{6} + 48 T^{4} + 704 T^{2} + 3136)^{2} \) (T^6 + 48T^4 + 704T^2 + 3136)^2
$17$	\( (T^{6} - 36 T^{4} + 368 T^{2} - 1024)^{2} \) (T^6 - 36T^4 + 368T^2 - 1024)^2
$19$	\( (T^{6} + 60 T^{4} + 512 T^{2} + 1024)^{2} \) (T^6 + 60T^4 + 512T^2 + 1024)^2
$23$	\( (T^{6} - 92 T^{4} + 2304 T^{2} + \cdots - 16384)^{2} \) (T^6 - 92T^4 + 2304T^2 - 16384)^2
$29$	\( (T^{6} + 108 T^{4} + 3120 T^{2} + \cdots + 12544)^{2} \) (T^6 + 108T^4 + 3120T^2 + 12544)^2
$31$	\( (T^{3} - 8 T^{2} - 4 T + 64)^{4} \) (T^3 - 8T^2 - 4T + 64)^4
$37$	\( (T^{6} + 64 T^{4} + 128 T^{2} + 64)^{2} \) (T^6 + 64T^4 + 128T^2 + 64)^2
$41$	\( (T^{3} + 2 T^{2} - 100 T + 56)^{4} \) (T^3 + 2T^2 - 100T + 56)^4
$43$	\( (T^{6} + 128 T^{4} + 4096 T^{2} + \cdots + 4096)^{2} \) (T^6 + 128T^4 + 4096T^2 + 4096)^2
$47$	\( (T^{6} - 60 T^{4} + 512 T^{2} - 1024)^{2} \) (T^6 - 60T^4 + 512T^2 - 1024)^2
$53$	\( (T^{6} + 80 T^{4} + 1216 T^{2} + 64)^{2} \) (T^6 + 80T^4 + 1216T^2 + 64)^2
$59$	\( (T^{6} + 176 T^{4} + 9888 T^{2} + \cdots + 179776)^{2} \) (T^6 + 176T^4 + 9888T^2 + 179776)^2
$61$	\( (T^{6} + 176 T^{4} + 7168 T^{2} + \cdots + 65536)^{2} \) (T^6 + 176T^4 + 7168T^2 + 65536)^2
$67$	\( (T^{6} + 128 T^{4} + 4096 T^{2} + \cdots + 4096)^{2} \) (T^6 + 128T^4 + 4096T^2 + 4096)^2
$71$	\( (T^{3} + 8 T^{2} - 80 T - 128)^{4} \) (T^3 + 8T^2 - 80T - 128)^4
$73$	\( (T^{6} - 384 T^{4} + 34560 T^{2} + \cdots - 16384)^{2} \) (T^6 - 384T^4 + 34560T^2 - 16384)^2
$79$	\( (T^{3} - 8 T^{2} - 4 T + 64)^{4} \) (T^3 - 8T^2 - 4T + 64)^4
$83$	\( (T^{6} + 192 T^{4} + 11264 T^{2} + \cdots + 200704)^{2} \) (T^6 + 192T^4 + 11264T^2 + 200704)^2
$89$	\( (T^{3} - 10 T^{2} - 164 T + 1384)^{4} \) (T^3 - 10T^2 - 164T + 1384)^4
$97$	\( (T^{6} - 336 T^{4} + 28416 T^{2} + \cdots - 262144)^{2} \) (T^6 - 336T^4 + 28416T^2 - 262144)^2
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