LMFDB - Newform orbit 2300.2.i.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2300,2,Mod(1057,2300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2300, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1, 2]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2300.1057");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2300 = 2^{2} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2300.i (of order $4$, degree $2$, 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:	$18.3655924649$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
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} + 74x^{12} + 1357x^{8} + 3177x^{4} + 1296 $ x^16 + 74x^12 + 1357x^8 + 3177*x^4 + 1296
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-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_1 q^{3} + (\beta_{9} - \beta_{5} - \beta_1) q^{7} + (\beta_{11} - \beta_{7} + \beta_{6}) q^{9}+O(q^{10}) $ q + b1 * q^3 + (b9 - b5 - b1) * q^7 + (b11 - b7 + b6) * q^9	$ q + \beta_1 q^{3} + (\beta_{9} - \beta_{5} - \beta_1) q^{7} + (\beta_{11} - \beta_{7} + \beta_{6}) q^{9} + (\beta_{13} + 2 \beta_{6}) q^{11} + ( - \beta_{9} + 2 \beta_{5}) q^{13} + ( - 2 \beta_{5} + 2 \beta_1) q^{17} + ( - \beta_{4} + 4) q^{19} + ( - 2 \beta_{11} - 4 \beta_{6}) q^{21} + ( - \beta_{15} + \beta_{12} + \cdots + \beta_1) q^{23}+ \cdots + (\beta_{4} + 2 \beta_{2} - 2) q^{99}+O(q^{100}) $ q + b1 * q^3 + (b9 - b5 - b1) * q^7 + (b11 - b7 + b6) * q^9 + (b13 + 2b6) q^11 + (-b9 + 2b5) q^13 + (-2b5 + 2b1) * q^17 + (-b4 + 4) * q^19 + (-2b11 - 4b6) * q^21 + (-b15 + b12 - b10 - b9 - b8 - b5 + b1) * q^23 + (b15 - b14 + b12) * q^27 + (2b11 - b7 + 2b6) * q^29 + (b4 + b2) * q^31 + (2b14 - 2b10) * q^33 + (2b9 + 4b8) * q^37 + (b11 + 2b7 + b6) q^39 + b3 * q^41 + (-b15 + 2b14 + 3b12 + b10) * q^43 + (3b12 - b10) q^47 + (b13 + 2b11 + 2b7 + 3b6) q^49 + (2b11 - 4b7 + 6b6) q^51 + (-2b14 - 2b10) * q^53 + (-2b8 + 4b1) * q^57 + (b13 + 2b11 - 4b7 + 3b6) q^59 + (4b11 + 2b6) * q^61 + (-b15 + 4b14 - b12 + 3b10) * q^63 + (2b9 + 2b1) * q^67 + (-b13 + 3b11 - 2b7 + 2b6 + 2b2 - 4) * q^69 + (2b4 + b3 + b2) q^71 + (3b9 + 2b8 - b1) * q^73 + (-b15 - 2b14 + 5b12 + b10) * q^77 + (-3b4 + 2b3 - 2) * q^79 + (b4 + b3 - 2b2 + 5) q^81 + (3b15 - 2b14 + 3b12 - 3b10) * q^83 + (3b15 - 3b14 - 5b10) q^87 + (-2b3 + 4b2 + 6) * q^89 + (-3b13 - 4b7 - 10b6) q^91 + (-b9 + b8 + 2b5 - b1) q^93 + (-2b8 - 4b5 + 2b1) q^97 + (b4 + 2b2 - 2) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 68 q^{19} - 4 q^{41} - 56 q^{69} - 8 q^{71} - 28 q^{79} + 64 q^{81} + 120 q^{89} - 28 q^{99}+O(q^{100}) $ 16 * q + 68 * q^19 - 4 * q^41 - 56 * q^69 - 8 * q^71 - 28 * q^79 + 64 * q^81 + 120 * q^89 - 28 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} + 74x^{12} + 1357x^{8} + 3177x^{4} + 1296 $ x^16 + 74*x^12 + 1357*x^8 + 3177*x^4 + 1296 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( -347\nu^{12} - 23572\nu^{8} - 353096\nu^{4} - 57267 ) / 179019 $ (-347v^12 - 23572v^8 - 353096*v^4 - 57267) / 179019
$\beta_{3}$	$=$	$ ( -389\nu^{12} - 26941\nu^{8} - 443813\nu^{4} - 690507 ) / 179019 $ (-389v^12 - 26941v^8 - 443813*v^4 - 690507) / 179019
$\beta_{4}$	$=$	$ ( -683\nu^{12} - 50524\nu^{8} - 899813\nu^{4} - 1184769 ) / 179019 $ (-683v^12 - 50524v^8 - 899813*v^4 - 1184769) / 179019
$\beta_{5}$	$=$	$ ( -305\nu^{13} - 20203\nu^{9} - 262379\nu^{5} + 1113030\nu ) / 537057 $ (-305v^13 - 20203v^9 - 262379v^5 + 1113030v) / 537057
$\beta_{6}$	$=$	$ ( 767\nu^{14} + 57262\nu^{10} + 1081247\nu^{6} + 3525363\nu^{2} ) / 2148228 $ (767v^14 + 57262v^10 + 1081247v^6 + 3525363v^2) / 2148228
$\beta_{7}$	$=$	$ ( -1987\nu^{14} - 138074\nu^{10} - 2130763\nu^{6} + 926757\nu^{2} ) / 2148228 $ (-1987v^14 - 138074v^10 - 2130763v^6 + 926757v^2) / 2148228
$\beta_{8}$	$=$	$ ( -683\nu^{13} - 50524\nu^{9} - 899813\nu^{5} - 1184769\nu ) / 358038 $ (-683v^13 - 50524v^9 - 899813v^5 - 1184769v) / 358038
$\beta_{9}$	$=$	$ ( 2911\nu^{13} + 212192\nu^{9} + 3768499\nu^{5} + 7275915\nu ) / 1074114 $ (2911v^13 + 212192v^9 + 3768499v^5 + 7275915v) / 1074114
$\beta_{10}$	$=$	$ ( -767\nu^{15} - 57262\nu^{11} - 1081247\nu^{7} - 3525363\nu^{3} ) / 2148228 $ (-767v^15 - 57262v^11 - 1081247v^7 - 3525363v^3) / 2148228
$\beta_{11}$	$=$	$ ( -1685\nu^{14} - 122374\nu^{10} - 2151917\nu^{6} - 3675489\nu^{2} ) / 716076 $ (-1685v^14 - 122374v^10 - 2151917v^6 - 3675489v^2) / 716076
$\beta_{12}$	$=$	$ ( -1691\nu^{15} - 131380\nu^{11} - 2718983\nu^{7} - 11728035\nu^{3} ) / 3222342 $ (-1691v^15 - 131380v^11 - 2718983v^7 - 11728035v^3) / 3222342
$\beta_{13}$	$=$	$ ( -1017\nu^{14} - 74474\nu^{10} - 1317181\nu^{6} - 2015009\nu^{2} ) / 238692 $ (-1017v^14 - 74474v^10 - 1317181v^6 - 2015009v^2) / 238692
$\beta_{14}$	$=$	$ ( -1017\nu^{15} - 74474\nu^{11} - 1317181\nu^{7} - 2015009\nu^{3} ) / 477384 $ (-1017v^15 - 74474v^11 - 1317181v^7 - 2015009v^3) / 477384
$\beta_{15}$	$=$	$ ( -48307\nu^{15} - 3546710\nu^{11} - 63612847\nu^{7} - 121516803\nu^{3} ) / 12889368 $ (-48307v^15 - 3546710v^11 - 63612847v^7 - 121516803v^3) / 12889368

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{11} - \beta_{7} + 4\beta_{6} $ b11 - b7 + 4*b6
$\nu^{3}$	$=$	$ \beta_{15} - \beta_{14} + \beta_{12} - 6\beta_{10} $ b15 - b14 + b12 - 6*b10
$\nu^{4}$	$=$	$ \beta_{4} - 8\beta_{3} + 7\beta_{2} - 22 $ b4 - 8b3 + 7b2 - 22
$\nu^{5}$	$=$	$ 9\beta_{9} + 11\beta_{8} + 6\beta_{5} - 37\beta_1 $ 9b9 + 11b8 + 6b5 - 37b1
$\nu^{6}$	$=$	$ 11\beta_{13} - 57\beta_{11} + 43\beta_{7} - 133\beta_{6} $ 11b13 - 57b11 + 43b7 - 133b6
$\nu^{7}$	$=$	$ -71\beta_{15} + 93\beta_{14} - 29\beta_{12} + 233\beta_{10} $ -71b15 + 93b14 - 29b12 + 233b10
$\nu^{8}$	$=$	$ -93\beta_{4} + 397\beta_{3} - 262\beta_{2} + 832 $ -93b4 + 397b3 - 262*b2 + 832
$\nu^{9}$	$=$	$ -532\beta_{9} - 718\beta_{8} - 127\beta_{5} + 1491\beta_1 $ -532b9 - 718b8 - 127b5 + 1491b1
$\nu^{10}$	$=$	$ -718\beta_{13} + 2741\beta_{11} - 1618\beta_{7} + 5305\beta_{6} $ -718b13 + 2741b11 - 1618b7 + 5305b6
$\nu^{11}$	$=$	$ 3864\beta_{15} - 5300\beta_{14} + 495\beta_{12} - 9664\beta_{10} $ 3864b15 - 5300b14 + 495b12 - 9664b10
$\nu^{12}$	$=$	$ 5300\beta_{4} - 18828\beta_{3} + 10159\beta_{2} - 34297 $ 5300b4 - 18828b3 + 10159*b2 - 34297
$\nu^{13}$	$=$	$ 27497\beta_{9} + 38097\beta_{8} + 1490\beta_{5} - 63284\beta_1 $ 27497b9 + 38097b8 + 1490b5 - 63284b1
$\nu^{14}$	$=$	$ 38097\beta_{13} - 128878\beta_{11} + 64774\beta_{7} - 224149\beta_{6} $ 38097b13 - 128878b11 + 64774b7 - 224149b6
$\nu^{15}$	$=$	$ -192982\beta_{15} + 269176\beta_{14} - 670\beta_{12} + 417801\beta_{10} $ -192982b15 + 269176b14 - 670b12 + 417801b10

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2300\mathbb{Z}\right)^\times$.

$n$	$277$	$1151$	$1201$
$\chi(n)$	$-\beta_{6}$	$1$	$-1$

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} $

1057.1

−1.84123	+	1.84123i
−1.58699	+	1.58699i
−0.854787	+	0.854787i
−0.600551	+	0.600551i
0.600551	−	0.600551i
0.854787	−	0.854787i
1.58699	−	1.58699i
1.84123	−	1.84123i
−1.84123	−	1.84123i
−1.58699	−	1.58699i
−0.854787	−	0.854787i
−0.600551	−	0.600551i
0.600551	+	0.600551i
0.854787	+	0.854787i
1.58699	+	1.58699i
1.84123	+	1.84123i

−1.84123

1.84123i

2.81318

−

2.81318i

−

3.78026i

1057.2

−1.58699

1.58699i

−0.216666

0.216666i

−

2.03710i

1057.3

−0.854787

0.854787i

3.24270

−

3.24270i

1.53868i

1057.4

−0.600551

0.600551i

−1.01189

1.01189i

2.27868i

1057.5

0.600551

−

0.600551i

1.01189

−

1.01189i

2.27868i

1057.6

0.854787

−

0.854787i

−3.24270

3.24270i

1.53868i

1057.7

1.58699

−

1.58699i

0.216666

−

0.216666i

−

2.03710i

1057.8

1.84123

−

1.84123i

−2.81318

2.81318i

−

3.78026i

1793.1

−1.84123

−

1.84123i

2.81318

2.81318i

3.78026i

1793.2

−1.58699

−

1.58699i

−0.216666

−

0.216666i

2.03710i

1793.3

−0.854787

−

0.854787i

3.24270

3.24270i

−

1.53868i

1793.4

−0.600551

−

0.600551i

−1.01189

−

1.01189i

−

2.27868i

1793.5

0.600551

0.600551i

1.01189

1.01189i

−

2.27868i

1793.6

0.854787

0.854787i

−3.24270

−

3.24270i

−

1.53868i

1793.7

1.58699

1.58699i

0.216666

0.216666i

2.03710i

1793.8

1.84123

1.84123i

−2.81318

−

2.81318i

3.78026i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
115.e	even	4	2	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2300.2.i.f	yes	16
5.b	even	2	1	inner	2300.2.i.f	yes	16
5.c	odd	4	2		2300.2.i.e	✓	16
23.b	odd	2	1		2300.2.i.e	✓	16
115.c	odd	2	1		2300.2.i.e	✓	16
115.e	even	4	2	inner	2300.2.i.f	yes	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2300.2.i.e	✓	16	5.c	odd	4	2
2300.2.i.e	✓	16	23.b	odd	2	1
2300.2.i.e	✓	16	115.c	odd	2	1
2300.2.i.f	yes	16	1.a	even	1	1	trivial
2300.2.i.f	yes	16	5.b	even	2	1	inner
2300.2.i.f	yes	16	115.e	even	4	2	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(2300, [\chi])$:

$ T_{3}^{16} + 74T_{3}^{12} + 1357T_{3}^{8} + 3177T_{3}^{4} + 1296 $

$ T_{19}^{4} - 17T_{19}^{3} + 94T_{19}^{2} - 192T_{19} + 120 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ T^{16} + 74 T^{12} + \cdots + 1296 $ T^16 + 74T^12 + 1357T^8 + 3177*T^4 + 1296
$5$	$ T^{16} $ T^16
$7$	$ T^{16} + 697 T^{12} + \cdots + 4096 $ T^16 + 697T^12 + 113712T^8 + 465664*T^4 + 4096
$11$	$ (T^{8} + 41 T^{6} + \cdots + 576)^{2} $ (T^8 + 41T^6 + 472T^4 + 1488*T^2 + 576)^2
$13$	$ T^{16} + 2507 T^{12} + \cdots + 50625 $ T^16 + 2507T^12 + 428671T^8 + 5287374*T^4 + 50625
$17$	$ T^{16} + \cdots + 6879707136 $ T^16 + 2720T^12 + 1711360T^8 + 316182528*T^4 + 6879707136
$19$	$ (T^{4} - 17 T^{3} + \cdots + 120)^{4} $ (T^4 - 17T^3 + 94T^2 - 192*T + 120)^4
$23$	$ T^{16} + \cdots + 78310985281 $ T^16 + 16T^14 + 260T^12 - 16080T^10 - 346042T^8 - 8506320T^6 + 72758660T^4 + 2368574224*T^2 + 78310985281
$29$	$ (T^{8} + 79 T^{6} + \cdots + 25)^{2} $ (T^8 + 79T^6 + 999T^4 + 514*T^2 + 25)^2
$31$	$ (T^{4} - 27 T^{2} + \cdots - 36)^{4} $ (T^4 - 27T^2 - 63T - 36)^4
$37$	$ T^{16} + \cdots + 907401035776 $ T^16 + 17968T^12 + 95936256T^8 + 150918160384*T^4 + 907401035776
$41$	$ (T^{4} + T^{3} - 11 T^{2} + \cdots + 15)^{4} $ (T^4 + T^3 - 11T^2 - 12T + 15)^4
$43$	$ T^{16} + \cdots + 37341681094656 $ T^16 + 12377T^12 + 50449312T^8 + 75558069504*T^4 + 37341681094656
$47$	$ T^{16} + \cdots + 4430766096 $ T^16 + 9626T^12 + 12890341T^8 + 4596369561*T^4 + 4430766096
$53$	$ T^{16} + 6368 T^{12} + \cdots + 5308416 $ T^16 + 6368T^12 + 4032256T^8 + 9252864*T^4 + 5308416
$59$	$ (T^{8} + 335 T^{6} + \cdots + 20304036)^{2} $ (T^8 + 335T^6 + 35407T^4 + 1453029*T^2 + 20304036)^2
$61$	$ (T^{8} + 368 T^{6} + \cdots + 21678336)^{2} $ (T^8 + 368T^6 + 39904T^4 + 1641216*T^2 + 21678336)^2
$67$	$ T^{16} + \cdots + 319794774016 $ T^16 + 14128T^12 + 36628224T^8 + 7422140416*T^4 + 319794774016
$71$	$ (T^{4} + 2 T^{3} + \cdots + 240)^{4} $ (T^4 + 2T^3 - 101T^2 - 171*T + 240)^4
$73$	$ T^{16} + \cdots + 1816891022241 $ T^16 + 11579T^12 + 39058111T^8 + 31097085678*T^4 + 1816891022241
$79$	$ (T^{4} + 7 T^{3} + \cdots + 4296)^{4} $ (T^4 + 7T^3 - 128T^2 - 516*T + 4296)^4
$83$	$ T^{16} + \cdots + 131135844519936 $ T^16 + 62649T^12 + 1281259296T^8 + 8520250975488*T^4 + 131135844519936
$89$	$ (T^{4} - 30 T^{3} + \cdots - 864)^{4} $ (T^4 - 30T^3 + 192T^2 + 504*T - 864)^4
$97$	$ T^{16} + \cdots + 4111875506176 $ T^16 + 57568T^12 + 146194176T^8 + 84244148224*T^4 + 4111875506176
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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:	\( N \)	\(=\)	\( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2300.i (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_1 q^{3} + (\beta_{9} - \beta_{5} - \beta_1) q^{7} + (\beta_{11} - \beta_{7} + \beta_{6}) q^{9}+O(q^{10}) \) q + b1 * q^3 + (b9 - b5 - b1) * q^7 + (b11 - b7 + b6) * q^9	\( q + \beta_1 q^{3} + (\beta_{9} - \beta_{5} - \beta_1) q^{7} + (\beta_{11} - \beta_{7} + \beta_{6}) q^{9} + (\beta_{13} + 2 \beta_{6}) q^{11} + ( - \beta_{9} + 2 \beta_{5}) q^{13} + ( - 2 \beta_{5} + 2 \beta_1) q^{17} + ( - \beta_{4} + 4) q^{19} + ( - 2 \beta_{11} - 4 \beta_{6}) q^{21} + ( - \beta_{15} + \beta_{12} + \cdots + \beta_1) q^{23}+ \cdots + (\beta_{4} + 2 \beta_{2} - 2) q^{99}+O(q^{100}) \) q + b1 * q^3 + (b9 - b5 - b1) * q^7 + (b11 - b7 + b6) * q^9 + (b13 + 2b6) q^11 + (-b9 + 2b5) q^13 + (-2b5 + 2b1) * q^17 + (-b4 + 4) * q^19 + (-2b11 - 4b6) * q^21 + (-b15 + b12 - b10 - b9 - b8 - b5 + b1) * q^23 + (b15 - b14 + b12) * q^27 + (2b11 - b7 + 2b6) * q^29 + (b4 + b2) * q^31 + (2b14 - 2b10) * q^33 + (2b9 + 4b8) * q^37 + (b11 + 2b7 + b6) q^39 + b3 * q^41 + (-b15 + 2b14 + 3b12 + b10) * q^43 + (3b12 - b10) q^47 + (b13 + 2b11 + 2b7 + 3b6) q^49 + (2b11 - 4b7 + 6b6) q^51 + (-2b14 - 2b10) * q^53 + (-2b8 + 4b1) * q^57 + (b13 + 2b11 - 4b7 + 3b6) q^59 + (4b11 + 2b6) * q^61 + (-b15 + 4b14 - b12 + 3b10) * q^63 + (2b9 + 2b1) * q^67 + (-b13 + 3b11 - 2b7 + 2b6 + 2b2 - 4) * q^69 + (2b4 + b3 + b2) q^71 + (3b9 + 2b8 - b1) * q^73 + (-b15 - 2b14 + 5b12 + b10) * q^77 + (-3b4 + 2b3 - 2) * q^79 + (b4 + b3 - 2b2 + 5) q^81 + (3b15 - 2b14 + 3b12 - 3b10) * q^83 + (3b15 - 3b14 - 5b10) q^87 + (-2b3 + 4b2 + 6) * q^89 + (-3b13 - 4b7 - 10b6) q^91 + (-b9 + b8 + 2b5 - b1) q^93 + (-2b8 - 4b5 + 2b1) q^97 + (b4 + 2b2 - 2) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 68 q^{19} - 4 q^{41} - 56 q^{69} - 8 q^{71} - 28 q^{79} + 64 q^{81} + 120 q^{89} - 28 q^{99}+O(q^{100}) \) 16 * q + 68 * q^19 - 4 * q^41 - 56 * q^69 - 8 * q^71 - 28 * q^79 + 64 * q^81 + 120 * q^89 - 28 * q^99

Introduction

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2300.2.i.f

Level	$2300$
Weight	$2$
Character orbit	2300.i
Analytic conductor	$18.366$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(18.3655924649\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} + 74x^{12} + 1357x^{8} + 3177x^{4} + 1296 \) x^16 + 74x^12 + 1357x^8 + 3177*x^4 + 1296
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( -347\nu^{12} - 23572\nu^{8} - 353096\nu^{4} - 57267 ) / 179019 \) (-347v^12 - 23572v^8 - 353096*v^4 - 57267) / 179019
\(\beta_{3}\)	\(=\)	\( ( -389\nu^{12} - 26941\nu^{8} - 443813\nu^{4} - 690507 ) / 179019 \) (-389v^12 - 26941v^8 - 443813*v^4 - 690507) / 179019
\(\beta_{4}\)	\(=\)	\( ( -683\nu^{12} - 50524\nu^{8} - 899813\nu^{4} - 1184769 ) / 179019 \) (-683v^12 - 50524v^8 - 899813*v^4 - 1184769) / 179019
\(\beta_{5}\)	\(=\)	\( ( -305\nu^{13} - 20203\nu^{9} - 262379\nu^{5} + 1113030\nu ) / 537057 \) (-305v^13 - 20203v^9 - 262379v^5 + 1113030v) / 537057
\(\beta_{6}\)	\(=\)	\( ( 767\nu^{14} + 57262\nu^{10} + 1081247\nu^{6} + 3525363\nu^{2} ) / 2148228 \) (767v^14 + 57262v^10 + 1081247v^6 + 3525363v^2) / 2148228
\(\beta_{7}\)	\(=\)	\( ( -1987\nu^{14} - 138074\nu^{10} - 2130763\nu^{6} + 926757\nu^{2} ) / 2148228 \) (-1987v^14 - 138074v^10 - 2130763v^6 + 926757v^2) / 2148228
\(\beta_{8}\)	\(=\)	\( ( -683\nu^{13} - 50524\nu^{9} - 899813\nu^{5} - 1184769\nu ) / 358038 \) (-683v^13 - 50524v^9 - 899813v^5 - 1184769v) / 358038
\(\beta_{9}\)	\(=\)	\( ( 2911\nu^{13} + 212192\nu^{9} + 3768499\nu^{5} + 7275915\nu ) / 1074114 \) (2911v^13 + 212192v^9 + 3768499v^5 + 7275915v) / 1074114
\(\beta_{10}\)	\(=\)	\( ( -767\nu^{15} - 57262\nu^{11} - 1081247\nu^{7} - 3525363\nu^{3} ) / 2148228 \) (-767v^15 - 57262v^11 - 1081247v^7 - 3525363v^3) / 2148228
\(\beta_{11}\)	\(=\)	\( ( -1685\nu^{14} - 122374\nu^{10} - 2151917\nu^{6} - 3675489\nu^{2} ) / 716076 \) (-1685v^14 - 122374v^10 - 2151917v^6 - 3675489v^2) / 716076
\(\beta_{12}\)	\(=\)	\( ( -1691\nu^{15} - 131380\nu^{11} - 2718983\nu^{7} - 11728035\nu^{3} ) / 3222342 \) (-1691v^15 - 131380v^11 - 2718983v^7 - 11728035v^3) / 3222342
\(\beta_{13}\)	\(=\)	\( ( -1017\nu^{14} - 74474\nu^{10} - 1317181\nu^{6} - 2015009\nu^{2} ) / 238692 \) (-1017v^14 - 74474v^10 - 1317181v^6 - 2015009v^2) / 238692
\(\beta_{14}\)	\(=\)	\( ( -1017\nu^{15} - 74474\nu^{11} - 1317181\nu^{7} - 2015009\nu^{3} ) / 477384 \) (-1017v^15 - 74474v^11 - 1317181v^7 - 2015009v^3) / 477384
\(\beta_{15}\)	\(=\)	\( ( -48307\nu^{15} - 3546710\nu^{11} - 63612847\nu^{7} - 121516803\nu^{3} ) / 12889368 \) (-48307v^15 - 3546710v^11 - 63612847v^7 - 121516803v^3) / 12889368

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{11} - \beta_{7} + 4\beta_{6} \) b11 - b7 + 4*b6
\(\nu^{3}\)	\(=\)	\( \beta_{15} - \beta_{14} + \beta_{12} - 6\beta_{10} \) b15 - b14 + b12 - 6*b10
\(\nu^{4}\)	\(=\)	\( \beta_{4} - 8\beta_{3} + 7\beta_{2} - 22 \) b4 - 8b3 + 7b2 - 22
\(\nu^{5}\)	\(=\)	\( 9\beta_{9} + 11\beta_{8} + 6\beta_{5} - 37\beta_1 \) 9b9 + 11b8 + 6b5 - 37b1
\(\nu^{6}\)	\(=\)	\( 11\beta_{13} - 57\beta_{11} + 43\beta_{7} - 133\beta_{6} \) 11b13 - 57b11 + 43b7 - 133b6
\(\nu^{7}\)	\(=\)	\( -71\beta_{15} + 93\beta_{14} - 29\beta_{12} + 233\beta_{10} \) -71b15 + 93b14 - 29b12 + 233b10
\(\nu^{8}\)	\(=\)	\( -93\beta_{4} + 397\beta_{3} - 262\beta_{2} + 832 \) -93b4 + 397b3 - 262*b2 + 832
\(\nu^{9}\)	\(=\)	\( -532\beta_{9} - 718\beta_{8} - 127\beta_{5} + 1491\beta_1 \) -532b9 - 718b8 - 127b5 + 1491b1
\(\nu^{10}\)	\(=\)	\( -718\beta_{13} + 2741\beta_{11} - 1618\beta_{7} + 5305\beta_{6} \) -718b13 + 2741b11 - 1618b7 + 5305b6
\(\nu^{11}\)	\(=\)	\( 3864\beta_{15} - 5300\beta_{14} + 495\beta_{12} - 9664\beta_{10} \) 3864b15 - 5300b14 + 495b12 - 9664b10
\(\nu^{12}\)	\(=\)	\( 5300\beta_{4} - 18828\beta_{3} + 10159\beta_{2} - 34297 \) 5300b4 - 18828b3 + 10159*b2 - 34297
\(\nu^{13}\)	\(=\)	\( 27497\beta_{9} + 38097\beta_{8} + 1490\beta_{5} - 63284\beta_1 \) 27497b9 + 38097b8 + 1490b5 - 63284b1
\(\nu^{14}\)	\(=\)	\( 38097\beta_{13} - 128878\beta_{11} + 64774\beta_{7} - 224149\beta_{6} \) 38097b13 - 128878b11 + 64774b7 - 224149b6
\(\nu^{15}\)	\(=\)	\( -192982\beta_{15} + 269176\beta_{14} - 670\beta_{12} + 417801\beta_{10} \) -192982b15 + 269176b14 - 670b12 + 417801b10

\(n\)	\(277\)	\(1151\)	\(1201\)
\(\chi(n)\)	\(-\beta_{6}\)	\(1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1057.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( T^{16} + 74 T^{12} + \cdots + 1296 \) T^16 + 74T^12 + 1357T^8 + 3177*T^4 + 1296
$5$	\( T^{16} \) T^16
$7$	\( T^{16} + 697 T^{12} + \cdots + 4096 \) T^16 + 697T^12 + 113712T^8 + 465664*T^4 + 4096
$11$	\( (T^{8} + 41 T^{6} + \cdots + 576)^{2} \) (T^8 + 41T^6 + 472T^4 + 1488*T^2 + 576)^2
$13$	\( T^{16} + 2507 T^{12} + \cdots + 50625 \) T^16 + 2507T^12 + 428671T^8 + 5287374*T^4 + 50625
$17$	\( T^{16} + \cdots + 6879707136 \) T^16 + 2720T^12 + 1711360T^8 + 316182528*T^4 + 6879707136
$19$	\( (T^{4} - 17 T^{3} + \cdots + 120)^{4} \) (T^4 - 17T^3 + 94T^2 - 192*T + 120)^4
$23$	\( T^{16} + \cdots + 78310985281 \) T^16 + 16T^14 + 260T^12 - 16080T^10 - 346042T^8 - 8506320T^6 + 72758660T^4 + 2368574224*T^2 + 78310985281
$29$	\( (T^{8} + 79 T^{6} + \cdots + 25)^{2} \) (T^8 + 79T^6 + 999T^4 + 514*T^2 + 25)^2
$31$	\( (T^{4} - 27 T^{2} + \cdots - 36)^{4} \) (T^4 - 27T^2 - 63T - 36)^4
$37$	\( T^{16} + \cdots + 907401035776 \) T^16 + 17968T^12 + 95936256T^8 + 150918160384*T^4 + 907401035776
$41$	\( (T^{4} + T^{3} - 11 T^{2} + \cdots + 15)^{4} \) (T^4 + T^3 - 11T^2 - 12T + 15)^4
$43$	\( T^{16} + \cdots + 37341681094656 \) T^16 + 12377T^12 + 50449312T^8 + 75558069504*T^4 + 37341681094656
$47$	\( T^{16} + \cdots + 4430766096 \) T^16 + 9626T^12 + 12890341T^8 + 4596369561*T^4 + 4430766096
$53$	\( T^{16} + 6368 T^{12} + \cdots + 5308416 \) T^16 + 6368T^12 + 4032256T^8 + 9252864*T^4 + 5308416
$59$	\( (T^{8} + 335 T^{6} + \cdots + 20304036)^{2} \) (T^8 + 335T^6 + 35407T^4 + 1453029*T^2 + 20304036)^2
$61$	\( (T^{8} + 368 T^{6} + \cdots + 21678336)^{2} \) (T^8 + 368T^6 + 39904T^4 + 1641216*T^2 + 21678336)^2
$67$	\( T^{16} + \cdots + 319794774016 \) T^16 + 14128T^12 + 36628224T^8 + 7422140416*T^4 + 319794774016
$71$	\( (T^{4} + 2 T^{3} + \cdots + 240)^{4} \) (T^4 + 2T^3 - 101T^2 - 171*T + 240)^4
$73$	\( T^{16} + \cdots + 1816891022241 \) T^16 + 11579T^12 + 39058111T^8 + 31097085678*T^4 + 1816891022241
$79$	\( (T^{4} + 7 T^{3} + \cdots + 4296)^{4} \) (T^4 + 7T^3 - 128T^2 - 516*T + 4296)^4
$83$	\( T^{16} + \cdots + 131135844519936 \) T^16 + 62649T^12 + 1281259296T^8 + 8520250975488*T^4 + 131135844519936
$89$	\( (T^{4} - 30 T^{3} + \cdots - 864)^{4} \) (T^4 - 30T^3 + 192T^2 + 504*T - 864)^4
$97$	\( T^{16} + \cdots + 4111875506176 \) T^16 + 57568T^12 + 146194176T^8 + 84244148224*T^4 + 4111875506176
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