LMFDB - Newform orbit 1300.2.i.i

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1300,2,Mod(601,1300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1300, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("1300.601");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1300 = 2^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1300.i (of order $3$, 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:	$10.3805522628$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 6x^{10} + 28x^{8} + 46x^{6} + 58x^{4} + 8x^{2} + 1 $ x^12 + 6x^10 + 28x^8 + 46x^6 + 58x^4 + 8*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 260)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-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 $1,\beta_1,\ldots,\beta_{11}$ 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_{9} q^{7} + ( - \beta_{6} + \beta_{5} - \beta_{4}) q^{9}+O(q^{10}) $ q - b3 * q^3 + b9 * q^7 + (-b6 + b5 - b4) * q^9	$ q - \beta_{3} q^{3} + \beta_{9} q^{7} + ( - \beta_{6} + \beta_{5} - \beta_{4}) q^{9} + (\beta_{5} + 1) q^{11} + ( - \beta_{11} - \beta_{7} + \cdots + \beta_{2}) q^{13}+ \cdots + ( - \beta_{4} - 1) q^{99}+O(q^{100}) $ q - b3 * q^3 + b9 * q^7 + (-b6 + b5 - b4) * q^9 + (b5 + 1) * q^11 + (-b11 - b7 + b3 + b2) * q^13 + (-2b10 + b9 - 2b3) * q^17 + (b8 - b6 + 2b5 - b4) q^19 + q^21 + (-b7 + b3 - b2) * q^23 + (-2b11 - 2b10 + b9 - 2b7 + b2) q^27 + (b8 - b6 + 2b5 - b1 + 2) q^29 + (b1 - 2) * q^31 + (-b10 - b3) * q^33 + (-2b7 - b2) q^37 + (-b8 + 2b6 - 3b5 + b4 + b1) * q^39 + (b8 - 2b6 + b5 - b1 + 1) q^41 + (-b11 - 2b10 + 2b9 - 2b3) q^43 + (-2b11 + 2b10 + 2b9 - 2b7 + 2b2) q^47 + (-2b8 - b6 + b5 + 2b1 + 1) * q^49 + (-2b4 - 7) q^51 + (b11 - 2b10 + b9 + b7 + b2) q^53 + (-3b11 - 5b10 + b9 - 3b7 + b2) q^57 + (b8 + 2b6 - b5 + 2b4) * q^59 + (b8 + b6 - 4b5 + b4) q^61 + (-b3 - 3b2) q^63 + (-b7 - 5b3 - b2) q^67 + (-b8 + 2b6 - 5b5 + 2b4) q^69 + (-3b8 + b6 + 2b5 + b4) * q^71 + (3b10 + 3b9 + 3b2) q^73 + (b9 + b2) * q^77 + (-b4 - 2b1 - 1) q^79 + (-2b8 + b6 - 4b5 + 2b1 - 4) q^81 + (b11 + b10 + 2b9 + b7 + 2b2) * q^83 + (-3b11 - 5b10 + b9 - 5b3) q^87 + (b8 + 2b6 - b5 - b1 - 1) q^89 + (b8 + b5 - b4 - 2b1 + 5) q^91 + (-b7 + b3) * q^93 + (2b11 - 2b10 + 3b9 - 2b3) * q^97 + (-b4 - 1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{9}+O(q^{10}) $ 12 * q - 4 * q^9	$ 12 q - 4 q^{9} + 6 q^{11} - 10 q^{19} + 12 q^{21} + 10 q^{29} - 24 q^{31} + 18 q^{39} + 2 q^{41} + 4 q^{49} - 76 q^{51} + 2 q^{59} + 22 q^{61} + 26 q^{69} - 14 q^{71} - 8 q^{79} - 22 q^{81} - 2 q^{89} + 58 q^{91} - 8 q^{99}+O(q^{100}) $ 12 * q - 4 * q^9 + 6 * q^11 - 10 * q^19 + 12 * q^21 + 10 * q^29 - 24 * q^31 + 18 * q^39 + 2 * q^41 + 4 * q^49 - 76 * q^51 + 2 * q^59 + 22 * q^61 + 26 * q^69 - 14 * q^71 - 8 * q^79 - 22 * q^81 - 2 * q^89 + 58 * q^91 - 8 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} + 6x^{10} + 28x^{8} + 46x^{6} + 58x^{4} + 8x^{2} + 1 $ x^12 + 6*x^10 + 28*x^8 + 46*x^6 + 58*x^4 + 8*x^2 + 1 :

$\beta_{1}$	$=$	$ ( -9\nu^{10} - 42\nu^{8} - 196\nu^{6} - 87\nu^{4} - 12\nu^{2} + 732 ) / 197 $ (-9v^10 - 42v^8 - 196v^6 - 87v^4 - 12*v^2 + 732) / 197
$\beta_{2}$	$=$	$ ( 9\nu^{11} + 42\nu^{9} + 196\nu^{7} + 87\nu^{5} + 12\nu^{3} - 732\nu ) / 394 $ (9v^11 + 42v^9 + 196v^7 + 87v^5 + 12v^3 - 732v) / 394
$\beta_{3}$	$=$	$ ( -9\nu^{11} - 42\nu^{9} - 196\nu^{7} - 87\nu^{5} - 12\nu^{3} + 1520\nu ) / 394 $ (-9v^11 - 42v^9 - 196v^7 - 87v^5 - 12v^3 + 1520v) / 394
$\beta_{4}$	$=$	$ ( 24\nu^{10} + 112\nu^{8} + 457\nu^{6} + 232\nu^{4} + 32\nu^{2} - 376 ) / 197 $ (24v^10 + 112v^8 + 457v^6 + 232v^4 + 32*v^2 - 376) / 197
$\beta_{5}$	$=$	$ ( 56\nu^{10} + 327\nu^{8} + 1526\nu^{6} + 2380\nu^{4} + 3161\nu^{2} + 42 ) / 394 $ (56v^10 + 327v^8 + 1526v^6 + 2380v^4 + 3161*v^2 + 42) / 394
$\beta_{6}$	$=$	$ ( 96\nu^{10} + 645\nu^{8} + 3010\nu^{6} + 5656\nu^{4} + 6235\nu^{2} + 860 ) / 394 $ (96v^10 + 645v^8 + 3010v^6 + 5656v^4 + 6235*v^2 + 860) / 394
$\beta_{7}$	$=$	$ ( -75\nu^{11} - 350\nu^{9} - 1502\nu^{7} - 725\nu^{5} - 100\nu^{3} + 4130\nu ) / 394 $ (-75v^11 - 350v^9 - 1502v^7 - 725v^5 - 100v^3 + 4130v) / 394
$\beta_{8}$	$=$	$ ( -112\nu^{10} - 654\nu^{8} - 3052\nu^{6} - 4760\nu^{4} - 5928\nu^{2} - 84 ) / 197 $ (-112v^10 - 654v^8 - 3052v^6 - 4760v^4 - 5928*v^2 - 84) / 197
$\beta_{9}$	$=$	$ ( 103\nu^{11} + 612\nu^{9} + 2856\nu^{7} + 4673\nu^{5} + 5916\nu^{3} + 816\nu ) / 394 $ (103v^11 + 612v^9 + 2856v^7 + 4673v^5 + 5916v^3 + 816v) / 394
$\beta_{10}$	$=$	$ ( 112\nu^{11} + 654\nu^{9} + 3052\nu^{7} + 4760\nu^{5} + 6125\nu^{3} + 84\nu ) / 197 $ (112v^11 + 654v^9 + 3052v^7 + 4760v^5 + 6125v^3 + 84v) / 197
$\beta_{11}$	$=$	$ ( -573\nu^{11} - 3462\nu^{9} - 16156\nu^{7} - 26815\nu^{5} - 33466\nu^{3} - 4616\nu ) / 394 $ (-573v^11 - 3462v^9 - 16156v^7 - 26815v^5 - 33466v^3 - 4616v) / 394

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

$\nu$	$=$	$ ( \beta_{3} + \beta_{2} ) / 2 $ (b3 + b2) / 2
$\nu^{2}$	$=$	$ ( \beta_{8} + 4\beta_{5} ) / 2 $ (b8 + 4*b5) / 2
$\nu^{3}$	$=$	$ \beta_{10} - 2\beta_{9} - 2\beta_{2} $ b10 - 2b9 - 2b2
$\nu^{4}$	$=$	$ ( -4\beta_{8} - \beta_{6} - 13\beta_{5} + 4\beta _1 - 13 ) / 2 $ (-4b8 - b6 - 13b5 + 4*b1 - 13) / 2
$\nu^{5}$	$=$	$ ( \beta_{11} - 5\beta_{10} + 16\beta_{9} - 5\beta_{3} ) / 2 $ (b11 - 5b10 + 16b9 - 5*b3) / 2
$\nu^{6}$	$=$	$ -3\beta_{4} - 8\beta _1 + 24 $ -3b4 - 8b1 + 24
$\nu^{7}$	$=$	$ ( 6\beta_{7} + 15\beta_{3} + 65\beta_{2} ) / 2 $ (6b7 + 15b3 + 65*b2) / 2
$\nu^{8}$	$=$	$ ( 65\beta_{8} + 28\beta_{6} + 188\beta_{5} + 28\beta_{4} ) / 2 $ (65b8 + 28b6 + 188b5 + 28b4) / 2
$\nu^{9}$	$=$	$ -14\beta_{11} + 26\beta_{10} - 133\beta_{9} - 14\beta_{7} - 133\beta_{2} $ -14b11 + 26b10 - 133b9 - 14b7 - 133*b2
$\nu^{10}$	$=$	$ ( -266\beta_{8} - 121\beta_{6} - 757\beta_{5} + 266\beta _1 - 757 ) / 2 $ (-266b8 - 121b6 - 757b5 + 266b1 - 757) / 2
$\nu^{11}$	$=$	$ ( 121\beta_{11} - 197\beta_{10} + 1092\beta_{9} - 197\beta_{3} ) / 2 $ (121b11 - 197b10 + 1092b9 - 197b3) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1300\mathbb{Z}\right)^\times$.

$n$	$301$	$651$	$677$
$\chi(n)$	$\beta_{5}$	$1$	$1$

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} $

601.1

0.660662	+	1.14430i
0.186544	+	0.323103i
−1.01426	−	1.75675i
1.01426	+	1.75675i
−0.186544	−	0.323103i
−0.660662	−	1.14430i
0.660662	−	1.14430i
0.186544	−	0.323103i
−1.01426	+	1.75675i
1.01426	−	1.75675i
−0.186544	+	0.323103i
−0.660662	+	1.14430i

−1.48920

−

2.57937i

−0.167875

0.290769i

−2.93543

5.08432i

601.2

−0.720209

−

1.24744i

−0.347122

0.601232i

0.462598

−

0.801244i

601.3

−0.116546

−

0.201864i

−2.14507

3.71537i

1.47283

−

2.55102i

601.4

0.116546

0.201864i

2.14507

−

3.71537i

1.47283

−

2.55102i

601.5

0.720209

1.24744i

0.347122

−

0.601232i

0.462598

−

0.801244i

601.6

1.48920

2.57937i

0.167875

−

0.290769i

−2.93543

5.08432i

1101.1

−1.48920

2.57937i

−0.167875

−

0.290769i

−2.93543

−

5.08432i

1101.2

−0.720209

1.24744i

−0.347122

−

0.601232i

0.462598

0.801244i

1101.3

−0.116546

0.201864i

−2.14507

−

3.71537i

1.47283

2.55102i

1101.4

0.116546

−

0.201864i

2.14507

3.71537i

1.47283

2.55102i

1101.5

0.720209

−

1.24744i

0.347122

0.601232i

0.462598

0.801244i

1101.6

1.48920

−

2.57937i

0.167875

0.290769i

−2.93543

−

5.08432i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
13.c	even	3	1	inner
65.n	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1300.2.i.i		12
5.b	even	2	1	inner	1300.2.i.i		12
5.c	odd	4	2		260.2.ba.a	✓	12
13.c	even	3	1	inner	1300.2.i.i		12
15.e	even	4	2		2340.2.de.a		12
20.e	even	4	2		1040.2.dh.c		12
65.n	even	6	1	inner	1300.2.i.i		12
65.o	even	12	2		3380.2.d.c		12
65.q	odd	12	2		260.2.ba.a	✓	12
65.q	odd	12	2		3380.2.c.a		6
65.r	odd	12	2		3380.2.c.b		6
65.t	even	12	2		3380.2.d.c		12
195.bl	even	12	2		2340.2.de.a		12
260.bj	even	12	2		1040.2.dh.c		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
260.2.ba.a	✓	12	5.c	odd	4	2
260.2.ba.a	✓	12	65.q	odd	12	2
1040.2.dh.c		12	20.e	even	4	2
1040.2.dh.c		12	260.bj	even	12	2
1300.2.i.i		12	1.a	even	1	1	trivial
1300.2.i.i		12	5.b	even	2	1	inner
1300.2.i.i		12	13.c	even	3	1	inner
1300.2.i.i		12	65.n	even	6	1	inner
2340.2.de.a		12	15.e	even	4	2
2340.2.de.a		12	195.bl	even	12	2
3380.2.c.a		6	65.q	odd	12	2
3380.2.c.b		6	65.r	odd	12	2
3380.2.d.c		12	65.o	even	12	2
3380.2.d.c		12	65.t	even	12	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1300, [\chi])$:

$ T_{3}^{12} + 11T_{3}^{10} + 102T_{3}^{8} + 207T_{3}^{6} + 350T_{3}^{4} + 19T_{3}^{2} + 1 $

$ T_{19}^{6} + 5T_{19}^{5} + 42T_{19}^{4} + 21T_{19}^{3} + 554T_{19}^{2} + 901T_{19} + 2809 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} $ T^12
$3$	$ T^{12} + 11 T^{10} + \cdots + 1 $ T^12 + 11T^10 + 102T^8 + 207T^6 + 350T^4 + 19*T^2 + 1
$5$	$ T^{12} $ T^12
$7$	$ T^{12} + 19 T^{10} + \cdots + 1 $ T^12 + 19T^10 + 350T^8 + 207T^6 + 102T^4 + 11*T^2 + 1
$11$	$ (T^{2} - T + 1)^{6} $ (T^2 - T + 1)^6
$13$	$ T^{12} - 6 T^{10} + \cdots + 4826809 $ T^12 - 6T^10 - 57T^8 + 844T^6 - 9633T^4 - 171366*T^2 + 4826809
$17$	$ T^{12} + 51 T^{10} + \cdots + 4879681 $ T^12 + 51T^10 + 1918T^8 + 30415T^6 + 353830T^4 + 1508747*T^2 + 4879681
$19$	$ (T^{6} + 5 T^{5} + \cdots + 2809)^{2} $ (T^6 + 5T^5 + 42T^4 + 21T^3 + 554T^2 + 901*T + 2809)^2
$23$	$ T^{12} + 59 T^{10} + \cdots + 5764801 $ T^12 + 59T^10 + 2550T^8 + 50127T^6 + 725102T^4 + 2235331*T^2 + 5764801
$29$	$ (T^{6} - 5 T^{5} + \cdots + 2809)^{2} $ (T^6 - 5T^5 + 42T^4 - 21T^3 + 554T^2 - 901*T + 2809)^2
$31$	$ (T^{3} + 6 T^{2} - 4 T - 16)^{4} $ (T^3 + 6T^2 - 4T - 16)^4
$37$	$ T^{12} + \cdots + 607573201 $ T^12 + 107T^10 + 8110T^8 + 307975T^6 + 8511478T^4 + 82303011*T^2 + 607573201
$41$	$ (T^{6} - T^{5} + \cdots + 32041)^{2} $ (T^6 - T^5 + 78T^4 + 435T^3 + 5750T^2 + 13783T + 32041)^2
$43$	$ T^{12} + \cdots + 163047361 $ T^12 + 99T^10 + 7086T^8 + 243247T^6 + 6107094T^4 + 34667835*T^2 + 163047361
$47$	$ (T^{6} - 204 T^{4} + \cdots - 87616)^{2} $ (T^6 - 204T^4 + 9008T^2 - 87616)^2
$53$	$ (T^{6} - 80 T^{4} + \cdots - 16384)^{2} $ (T^6 - 80T^4 + 2032T^2 - 16384)^2
$59$	$ (T^{6} - T^{5} + \cdots + 11881)^{2} $ (T^6 - T^5 + 126T^4 - 93T^3 + 15734T^2 - 13625T + 11881)^2
$61$	$ (T^{6} - 11 T^{5} + \cdots + 51529)^{2} $ (T^6 - 11T^5 + 130T^4 - 355T^3 + 2578T^2 - 2043*T + 51529)^2
$67$	$ T^{12} + \cdots + 123657019201 $ T^12 + 299T^10 + 65478T^8 + 6449679T^6 + 467166878T^4 + 8412499027*T^2 + 123657019201
$71$	$ (T^{6} + 7 T^{5} + \cdots + 18769)^{2} $ (T^6 + 7T^5 + 162T^4 - 1065T^3 + 11810T^2 - 15481*T + 18769)^2
$73$	$ (T^{6} - 324 T^{4} + \cdots - 746496)^{2} $ (T^6 - 324T^4 + 29808T^2 - 746496)^2
$79$	$ (T^{3} + 2 T^{2} + \cdots - 512)^{4} $ (T^3 + 2T^2 - 108T - 512)^4
$83$	$ (T^{6} - 128 T^{4} + \cdots - 3136)^{2} $ (T^6 - 128T^4 + 2784T^2 - 3136)^2
$89$	$ (T^{6} + T^{5} + \cdots + 11881)^{2} $ (T^6 + T^5 + 126T^4 + 93T^3 + 15734T^2 + 13625T + 11881)^2
$97$	$ T^{12} + \cdots + 4499860561 $ T^12 + 299T^10 + 71758T^8 + 5141095T^6 + 291218230T^4 + 1183510083*T^2 + 4499860561
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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:	\( N \)	\(=\)	\( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1300.i (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{3} q^{3} + \beta_{9} q^{7} + ( - \beta_{6} + \beta_{5} - \beta_{4}) q^{9}+O(q^{10}) \) q - b3 * q^3 + b9 * q^7 + (-b6 + b5 - b4) * q^9	\( q - \beta_{3} q^{3} + \beta_{9} q^{7} + ( - \beta_{6} + \beta_{5} - \beta_{4}) q^{9} + (\beta_{5} + 1) q^{11} + ( - \beta_{11} - \beta_{7} + \cdots + \beta_{2}) q^{13}+ \cdots + ( - \beta_{4} - 1) q^{99}+O(q^{100}) \) q - b3 * q^3 + b9 * q^7 + (-b6 + b5 - b4) * q^9 + (b5 + 1) * q^11 + (-b11 - b7 + b3 + b2) * q^13 + (-2b10 + b9 - 2b3) * q^17 + (b8 - b6 + 2b5 - b4) q^19 + q^21 + (-b7 + b3 - b2) * q^23 + (-2b11 - 2b10 + b9 - 2b7 + b2) q^27 + (b8 - b6 + 2b5 - b1 + 2) q^29 + (b1 - 2) * q^31 + (-b10 - b3) * q^33 + (-2b7 - b2) q^37 + (-b8 + 2b6 - 3b5 + b4 + b1) * q^39 + (b8 - 2b6 + b5 - b1 + 1) q^41 + (-b11 - 2b10 + 2b9 - 2b3) q^43 + (-2b11 + 2b10 + 2b9 - 2b7 + 2b2) q^47 + (-2b8 - b6 + b5 + 2b1 + 1) * q^49 + (-2b4 - 7) q^51 + (b11 - 2b10 + b9 + b7 + b2) q^53 + (-3b11 - 5b10 + b9 - 3b7 + b2) q^57 + (b8 + 2b6 - b5 + 2b4) * q^59 + (b8 + b6 - 4b5 + b4) q^61 + (-b3 - 3b2) q^63 + (-b7 - 5b3 - b2) q^67 + (-b8 + 2b6 - 5b5 + 2b4) q^69 + (-3b8 + b6 + 2b5 + b4) * q^71 + (3b10 + 3b9 + 3b2) q^73 + (b9 + b2) * q^77 + (-b4 - 2b1 - 1) q^79 + (-2b8 + b6 - 4b5 + 2b1 - 4) q^81 + (b11 + b10 + 2b9 + b7 + 2b2) * q^83 + (-3b11 - 5b10 + b9 - 5b3) q^87 + (b8 + 2b6 - b5 - b1 - 1) q^89 + (b8 + b5 - b4 - 2b1 + 5) q^91 + (-b7 + b3) * q^93 + (2b11 - 2b10 + 3b9 - 2b3) * q^97 + (-b4 - 1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{9}+O(q^{10}) \) 12 * q - 4 * q^9	\( 12 q - 4 q^{9} + 6 q^{11} - 10 q^{19} + 12 q^{21} + 10 q^{29} - 24 q^{31} + 18 q^{39} + 2 q^{41} + 4 q^{49} - 76 q^{51} + 2 q^{59} + 22 q^{61} + 26 q^{69} - 14 q^{71} - 8 q^{79} - 22 q^{81} - 2 q^{89} + 58 q^{91} - 8 q^{99}+O(q^{100}) \) 12 * q - 4 * q^9 + 6 * q^11 - 10 * q^19 + 12 * q^21 + 10 * q^29 - 24 * q^31 + 18 * q^39 + 2 * q^41 + 4 * q^49 - 76 * q^51 + 2 * q^59 + 22 * q^61 + 26 * q^69 - 14 * q^71 - 8 * q^79 - 22 * q^81 - 2 * q^89 + 58 * q^91 - 8 * q^99

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	1300.2.i.i

Level	$1300$
Weight	$2$
Character orbit	1300.i
Analytic conductor	$10.381$
Analytic rank	$0$
Dimension	$12$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(10.3805522628\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} + 6x^{10} + 28x^{8} + 46x^{6} + 58x^{4} + 8x^{2} + 1 \) x^12 + 6x^10 + 28x^8 + 46x^6 + 58x^4 + 8*x^2 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{8} \)
Twist minimal:	no (minimal twist has level 260)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

\(\beta_{1}\)	\(=\)	\( ( -9\nu^{10} - 42\nu^{8} - 196\nu^{6} - 87\nu^{4} - 12\nu^{2} + 732 ) / 197 \) (-9v^10 - 42v^8 - 196v^6 - 87v^4 - 12*v^2 + 732) / 197
\(\beta_{2}\)	\(=\)	\( ( 9\nu^{11} + 42\nu^{9} + 196\nu^{7} + 87\nu^{5} + 12\nu^{3} - 732\nu ) / 394 \) (9v^11 + 42v^9 + 196v^7 + 87v^5 + 12v^3 - 732v) / 394
\(\beta_{3}\)	\(=\)	\( ( -9\nu^{11} - 42\nu^{9} - 196\nu^{7} - 87\nu^{5} - 12\nu^{3} + 1520\nu ) / 394 \) (-9v^11 - 42v^9 - 196v^7 - 87v^5 - 12v^3 + 1520v) / 394
\(\beta_{4}\)	\(=\)	\( ( 24\nu^{10} + 112\nu^{8} + 457\nu^{6} + 232\nu^{4} + 32\nu^{2} - 376 ) / 197 \) (24v^10 + 112v^8 + 457v^6 + 232v^4 + 32*v^2 - 376) / 197
\(\beta_{5}\)	\(=\)	\( ( 56\nu^{10} + 327\nu^{8} + 1526\nu^{6} + 2380\nu^{4} + 3161\nu^{2} + 42 ) / 394 \) (56v^10 + 327v^8 + 1526v^6 + 2380v^4 + 3161*v^2 + 42) / 394
\(\beta_{6}\)	\(=\)	\( ( 96\nu^{10} + 645\nu^{8} + 3010\nu^{6} + 5656\nu^{4} + 6235\nu^{2} + 860 ) / 394 \) (96v^10 + 645v^8 + 3010v^6 + 5656v^4 + 6235*v^2 + 860) / 394
\(\beta_{7}\)	\(=\)	\( ( -75\nu^{11} - 350\nu^{9} - 1502\nu^{7} - 725\nu^{5} - 100\nu^{3} + 4130\nu ) / 394 \) (-75v^11 - 350v^9 - 1502v^7 - 725v^5 - 100v^3 + 4130v) / 394
\(\beta_{8}\)	\(=\)	\( ( -112\nu^{10} - 654\nu^{8} - 3052\nu^{6} - 4760\nu^{4} - 5928\nu^{2} - 84 ) / 197 \) (-112v^10 - 654v^8 - 3052v^6 - 4760v^4 - 5928*v^2 - 84) / 197
\(\beta_{9}\)	\(=\)	\( ( 103\nu^{11} + 612\nu^{9} + 2856\nu^{7} + 4673\nu^{5} + 5916\nu^{3} + 816\nu ) / 394 \) (103v^11 + 612v^9 + 2856v^7 + 4673v^5 + 5916v^3 + 816v) / 394
\(\beta_{10}\)	\(=\)	\( ( 112\nu^{11} + 654\nu^{9} + 3052\nu^{7} + 4760\nu^{5} + 6125\nu^{3} + 84\nu ) / 197 \) (112v^11 + 654v^9 + 3052v^7 + 4760v^5 + 6125v^3 + 84v) / 197
\(\beta_{11}\)	\(=\)	\( ( -573\nu^{11} - 3462\nu^{9} - 16156\nu^{7} - 26815\nu^{5} - 33466\nu^{3} - 4616\nu ) / 394 \) (-573v^11 - 3462v^9 - 16156v^7 - 26815v^5 - 33466v^3 - 4616v) / 394

\(\nu\)	\(=\)	\( ( \beta_{3} + \beta_{2} ) / 2 \) (b3 + b2) / 2
\(\nu^{2}\)	\(=\)	\( ( \beta_{8} + 4\beta_{5} ) / 2 \) (b8 + 4*b5) / 2
\(\nu^{3}\)	\(=\)	\( \beta_{10} - 2\beta_{9} - 2\beta_{2} \) b10 - 2b9 - 2b2
\(\nu^{4}\)	\(=\)	\( ( -4\beta_{8} - \beta_{6} - 13\beta_{5} + 4\beta _1 - 13 ) / 2 \) (-4b8 - b6 - 13b5 + 4*b1 - 13) / 2
\(\nu^{5}\)	\(=\)	\( ( \beta_{11} - 5\beta_{10} + 16\beta_{9} - 5\beta_{3} ) / 2 \) (b11 - 5b10 + 16b9 - 5*b3) / 2
\(\nu^{6}\)	\(=\)	\( -3\beta_{4} - 8\beta _1 + 24 \) -3b4 - 8b1 + 24
\(\nu^{7}\)	\(=\)	\( ( 6\beta_{7} + 15\beta_{3} + 65\beta_{2} ) / 2 \) (6b7 + 15b3 + 65*b2) / 2
\(\nu^{8}\)	\(=\)	\( ( 65\beta_{8} + 28\beta_{6} + 188\beta_{5} + 28\beta_{4} ) / 2 \) (65b8 + 28b6 + 188b5 + 28b4) / 2
\(\nu^{9}\)	\(=\)	\( -14\beta_{11} + 26\beta_{10} - 133\beta_{9} - 14\beta_{7} - 133\beta_{2} \) -14b11 + 26b10 - 133b9 - 14b7 - 133*b2
\(\nu^{10}\)	\(=\)	\( ( -266\beta_{8} - 121\beta_{6} - 757\beta_{5} + 266\beta _1 - 757 ) / 2 \) (-266b8 - 121b6 - 757b5 + 266b1 - 757) / 2
\(\nu^{11}\)	\(=\)	\( ( 121\beta_{11} - 197\beta_{10} + 1092\beta_{9} - 197\beta_{3} ) / 2 \) (121b11 - 197b10 + 1092b9 - 197b3) / 2

\(n\)	\(301\)	\(651\)	\(677\)
\(\chi(n)\)	\(\beta_{5}\)	\(1\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 601.6
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} \) T^12
$3$	\( T^{12} + 11 T^{10} + \cdots + 1 \) T^12 + 11T^10 + 102T^8 + 207T^6 + 350T^4 + 19*T^2 + 1
$5$	\( T^{12} \) T^12
$7$	\( T^{12} + 19 T^{10} + \cdots + 1 \) T^12 + 19T^10 + 350T^8 + 207T^6 + 102T^4 + 11*T^2 + 1
$11$	\( (T^{2} - T + 1)^{6} \) (T^2 - T + 1)^6
$13$	\( T^{12} - 6 T^{10} + \cdots + 4826809 \) T^12 - 6T^10 - 57T^8 + 844T^6 - 9633T^4 - 171366*T^2 + 4826809
$17$	\( T^{12} + 51 T^{10} + \cdots + 4879681 \) T^12 + 51T^10 + 1918T^8 + 30415T^6 + 353830T^4 + 1508747*T^2 + 4879681
$19$	\( (T^{6} + 5 T^{5} + \cdots + 2809)^{2} \) (T^6 + 5T^5 + 42T^4 + 21T^3 + 554T^2 + 901*T + 2809)^2
$23$	\( T^{12} + 59 T^{10} + \cdots + 5764801 \) T^12 + 59T^10 + 2550T^8 + 50127T^6 + 725102T^4 + 2235331*T^2 + 5764801
$29$	\( (T^{6} - 5 T^{5} + \cdots + 2809)^{2} \) (T^6 - 5T^5 + 42T^4 - 21T^3 + 554T^2 - 901*T + 2809)^2
$31$	\( (T^{3} + 6 T^{2} - 4 T - 16)^{4} \) (T^3 + 6T^2 - 4T - 16)^4
$37$	\( T^{12} + \cdots + 607573201 \) T^12 + 107T^10 + 8110T^8 + 307975T^6 + 8511478T^4 + 82303011*T^2 + 607573201
$41$	\( (T^{6} - T^{5} + \cdots + 32041)^{2} \) (T^6 - T^5 + 78T^4 + 435T^3 + 5750T^2 + 13783T + 32041)^2
$43$	\( T^{12} + \cdots + 163047361 \) T^12 + 99T^10 + 7086T^8 + 243247T^6 + 6107094T^4 + 34667835*T^2 + 163047361
$47$	\( (T^{6} - 204 T^{4} + \cdots - 87616)^{2} \) (T^6 - 204T^4 + 9008T^2 - 87616)^2
$53$	\( (T^{6} - 80 T^{4} + \cdots - 16384)^{2} \) (T^6 - 80T^4 + 2032T^2 - 16384)^2
$59$	\( (T^{6} - T^{5} + \cdots + 11881)^{2} \) (T^6 - T^5 + 126T^4 - 93T^3 + 15734T^2 - 13625T + 11881)^2
$61$	\( (T^{6} - 11 T^{5} + \cdots + 51529)^{2} \) (T^6 - 11T^5 + 130T^4 - 355T^3 + 2578T^2 - 2043*T + 51529)^2
$67$	\( T^{12} + \cdots + 123657019201 \) T^12 + 299T^10 + 65478T^8 + 6449679T^6 + 467166878T^4 + 8412499027*T^2 + 123657019201
$71$	\( (T^{6} + 7 T^{5} + \cdots + 18769)^{2} \) (T^6 + 7T^5 + 162T^4 - 1065T^3 + 11810T^2 - 15481*T + 18769)^2
$73$	\( (T^{6} - 324 T^{4} + \cdots - 746496)^{2} \) (T^6 - 324T^4 + 29808T^2 - 746496)^2
$79$	\( (T^{3} + 2 T^{2} + \cdots - 512)^{4} \) (T^3 + 2T^2 - 108T - 512)^4
$83$	\( (T^{6} - 128 T^{4} + \cdots - 3136)^{2} \) (T^6 - 128T^4 + 2784T^2 - 3136)^2
$89$	\( (T^{6} + T^{5} + \cdots + 11881)^{2} \) (T^6 + T^5 + 126T^4 + 93T^3 + 15734T^2 + 13625T + 11881)^2
$97$	\( T^{12} + \cdots + 4499860561 \) T^12 + 299T^10 + 71758T^8 + 5141095T^6 + 291218230T^4 + 1183510083*T^2 + 4499860561
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