LMFDB - Newform orbit 184.2.h.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [184,2,Mod(91,184)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(184, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("184.91");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 184 = 2^{3} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	184.h (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:	$1.46924739719$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.64974433091584.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4x^{11} + 8x^{10} - 10x^{9} + 24x^{7} - 46x^{6} + 48x^{5} - 80x^{3} + 128x^{2} - 128x + 64 $ x^12 - 4x^11 + 8x^10 - 10x^9 + 24x^7 - 46x^6 + 48x^5 - 80x^3 + 128x^2 - 128*x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{6}\cdot 23 $
Twist minimal:	yes
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 $1,\beta_1,\ldots,\beta_{11}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_{2} q^{2} + ( - \beta_{7} - \beta_{2} + 1) q^{3} + (\beta_{7} - \beta_{4} + \beta_{2} - 1) q^{4} - \beta_{10} q^{5} + (\beta_{7} - \beta_{4} + 1) q^{6} - \beta_{6} q^{7} + ( - \beta_{9} + \beta_{4} + \beta_{2} + \cdots - 1) q^{8}+ \cdots + (2 \beta_{11} + \beta_{10} + \cdots - \beta_{5}) q^{99}+O(q^{100}) $ q - b2 * q^2 + (-b7 - b2 + 1) * q^3 + (b7 - b4 + b2 - 1) * q^4 - b10 * q^5 + (b7 - b4 + 1) * q^6 - b6 * q^7 + (-b9 + b4 + b2 + b1 - 1) * q^8 + (-b7 - b2 - b1) * q^9 + (-b8 + b5) * q^10 + (b8 - b5 - b3) * q^11 + (-b9 + b7 + b1 - 2) * q^12 + (b9 + b4 - b2 - b1) * q^13 + (b11 + b10 + b6 - 2b5 - b3) q^14 + (-b10 + b5) * q^15 + 2b9 q^16 + (-b8 - b3) * q^17 + (-b9 - b4 + b2 + b1 + 1) * q^18 + (-2b11 - b10 - b6 + b5) q^19 + (-b11 + b10) * q^20 + (b10 - 2b5) q^21 + (b11 - b10 + b6 + b3) * q^22 + (b10 + b9 - b7 + b5 + b4 - b1) * q^23 + (b9 + 2b7 - b4 + 3b2 + b1 - 3) * q^24 + (-b7 - b2 + b1 + 1) * q^25 + (-2b7 - b2 - 2b1) * q^26 + (2b7 + 2b2 - b1 - 1) * q^27 + (b11 + b10 + b8 + b6 + b5 + b3) * q^28 + (-b9 + b7 - 3b4 + 2b2 + 2b1) q^29 + (-b11 - b10 - b8 + b5) * q^30 + (-3b9 - 2b7 + 3b4 - b2) q^31 + (-2b7 + 2b4 - 2b2 - 4b1 + 2) * q^32 + (2b11 + b10 + b8 + b6 - b5 + b3) q^33 + (2b10 + b6 + b3) q^34 + (3b7 + 3b2 + 5b1 - 4) q^35 + (2b7 + 2b1) * q^36 + (-2b10 - b6 + b5) q^37 + (-b8 - b6 - b5 + b3) * q^38 + (b9 - b7 + b4 - b1) * q^39 + (b8 - b6 - b5 + b3) * q^40 + (-3b7 - 3b2 - 3b1 + 3) q^41 + (2b11 + 2b10 + b8 - b5) * q^42 + (2b8 + 2b3) * q^43 + (-b11 - b10 - b8 - b6 + 3b5 - b3) q^44 + (b10 + b6 + b5) * q^45 + (-b11 - b10 + b8 - 3b7 - b5 + b4 - 2b2 - 2b1 + 3) q^46 + (3b9 + 4b7 - 3b4 - b2) q^47 + (-2b7 + 4b4 - 2b1) q^48 + (-b7 - b2 - b1 + 7) * q^49 + (b9 + 2b7 - b4 - b1 + 1) q^50 + (-b8 + b5 + b3) * q^51 + (-2b9 - b7 - b4 + b2 + 2b1 + 3) * q^52 + (-b10 + b6 - b5) * q^53 + (-b9 - 3b7 + 2b4 - b2 + b1 - 2) * q^54 + (2b9 - 3b7 + 3b2 - b1) q^55 + (-2b11 - 4b10 + b8 - b6 + b5 - b3) * q^56 + (-2b11 - b10 - 2b8 - b6 + 3b5 + 2b3) * q^57 + (-b9 + 4b7 + b4 + 2b2 + 3b1 - 1) q^58 + (2b7 + 2b2 - 8) * q^59 + (-b11 + b10 - b8 - b6 + b5 + b3) * q^60 + 2b10 q^61 + (3b9 + b7 - 4b4 + b2 + 3b1) q^62 + (3b10 + b6 - b5) q^63 + (-2b9 - 4b7 - 2b4 - 2b2 + 2b1 + 2) q^64 + (2b11 + b10 - b8 + b6 - b5 - b3) q^65 + (-2b10 + b8 + b5 - 2b3) * q^66 + (2b11 + b10 + b8 + b6 - 2b5 - b3) * q^67 + (-b11 - b10 + 2b8 - 2b6) * q^68 + (b9 - 3b7 + b6 - b5 + 3b4 - 2b1) q^69 + (5b9 + 2b7 + 3b4 + b2 - 5b1 - 3) * q^70 + (-3b9 - b7 + b4 + b1) q^71 + (2b9 + 2b7 - 2b1 - 4) q^72 + (-4b1 - 3) q^73 + (-2b8 + b6 - b3) q^74 + (-b1 + 3) * q^75 + (2b11 + 4b10 - 2b5 - 2b3) * q^76 + (-8b9 + b7 + 2b4 - 3b2 + 3b1) * q^77 + (-3b7 + b4 - 2b2 - 2b1 + 3) q^78 + (b10 + b6 - 3b5) q^79 + (2b11 - 2b5 - 2b3) q^80 + (3b7 + 3b2 + 5b1 - 5) q^81 + (-3b9 - 3b4 + 3b1 + 3) q^82 + (-2b11 - b10 - b8 - b6 + 2b5 + b3) * q^83 + (b11 - b10 + 2b8 + 2b6 - 2b5 - 2b3) * q^84 + (4b9 + 5b7 - 2b4 - 3b2 - b1) * q^85 + (-4b10 - 2b6 - 2b3) q^86 + (-3b9 + 2b7 - b4 - b2 + 2b1) q^87 + (-2b11 - b8 + b6 - b5 + b3) q^88 + (2b11 + b10 - 2b8 + b6 - b5 - 2b3) q^89 + (-2b11 - 2b10 + b8 - b6 + b5 + b3) * q^90 + (-3b8 + b5 - b3) q^91 + (b11 - b10 - b9 - b8 - b7 - b6 + b5 - 2b4 + b3 - 2b2 + b1 + 4) * q^92 + (3b9 + 2b7 - 5b4 + 3b2 + b1) * q^93 + (-3b9 + b7 + 2b4 + b2 - 3b1 - 6) q^94 + (-4b9 + 4b7 - 8b4 + 4b2 + 6b1) q^95 + (2b9 - 6b7 - 4b2 - 2b1 + 4) * q^96 + (-2b11 - b10 + 3b8 - b6 - b5 - b3) * q^97 + (-b9 - b4 - 6b2 + b1 + 1) q^98 + (2b11 + b10 + b6 - b5) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{2} + 4 q^{3} - 8 q^{4} + 12 q^{6} - 4 q^{8} - 8 q^{9} - 20 q^{12} + 12 q^{18} - 20 q^{24} + 4 q^{25} - 12 q^{26} + 4 q^{27} + 16 q^{32} - 24 q^{35} + 8 q^{36} + 12 q^{41} + 20 q^{46} + 8 q^{48}+ \cdots - 16 q^{98}+O(q^{100}) $ 12 * q - 4 * q^2 + 4 * q^3 - 8 * q^4 + 12 * q^6 - 4 * q^8 - 8 * q^9 - 20 * q^12 + 12 * q^18 - 20 * q^24 + 4 * q^25 - 12 * q^26 + 4 * q^27 + 16 * q^32 - 24 * q^35 + 8 * q^36 + 12 * q^41 + 20 * q^46 + 8 * q^48 + 76 * q^49 + 16 * q^50 + 32 * q^52 - 32 * q^54 + 16 * q^58 - 80 * q^59 - 8 * q^62 - 8 * q^64 - 12 * q^70 - 40 * q^72 - 36 * q^73 + 36 * q^75 + 20 * q^78 - 36 * q^81 + 24 * q^82 + 28 * q^92 - 56 * q^94 + 8 * q^96 - 16 * q^98

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 4x^{11} + 8x^{10} - 10x^{9} + 24x^{7} - 46x^{6} + 48x^{5} - 80x^{3} + 128x^{2} - 128x + 64 $ x^12 - 4*x^11 + 8*x^10 - 10*x^9 + 24*x^7 - 46*x^6 + 48*x^5 - 80*x^3 + 128*x^2 - 128*x + 64 :

$\beta_{1}$	$=$	$ ( 2\nu^{11} - \nu^{10} - 4\nu^{9} + 4\nu^{8} - 14\nu^{7} + 28\nu^{5} - 18\nu^{4} + 32\nu^{3} + 16\nu^{2} - 64\nu ) / 16 $ (2v^11 - v^10 - 4v^9 + 4v^8 - 14v^7 + 28v^5 - 18v^4 + 32v^3 + 16v^2 - 64*v) / 16
$\beta_{2}$	$=$	$ ( - 2 \nu^{11} + 3 \nu^{10} - 4 \nu^{9} + 4 \nu^{8} + 10 \nu^{7} - 16 \nu^{6} + 20 \nu^{5} - 26 \nu^{4} + \cdots + 96 ) / 16 $ (-2v^11 + 3v^10 - 4v^9 + 4v^8 + 10v^7 - 16v^6 + 20v^5 - 26v^4 - 32v^3 + 48v^2 - 64*v + 96) / 16
$\beta_{3}$	$=$	$ ( 2 \nu^{11} - 3 \nu^{10} + 8 \nu^{9} - 16 \nu^{8} - 2 \nu^{7} + 24 \nu^{6} - 60 \nu^{5} + 90 \nu^{4} + \cdots - 224 ) / 16 $ (2v^11 - 3v^10 + 8v^9 - 16v^8 - 2v^7 + 24v^6 - 60v^5 + 90v^4 + 8v^3 - 88v^2 + 176*v - 224) / 16
$\beta_{4}$	$=$	$ ( - 2 \nu^{11} + 5 \nu^{10} - 12 \nu^{9} + 12 \nu^{8} + 6 \nu^{7} - 32 \nu^{6} + 68 \nu^{5} - 54 \nu^{4} + \cdots + 144 ) / 16 $ (-2v^11 + 5v^10 - 12v^9 + 12v^8 + 6v^7 - 32v^6 + 68v^5 - 54v^4 - 32v^3 + 112v^2 - 192*v + 144) / 16
$\beta_{5}$	$=$	$ ( - 3 \nu^{10} + 8 \nu^{9} - 6 \nu^{8} - 2 \nu^{7} + 24 \nu^{6} - 44 \nu^{5} + 18 \nu^{4} + 24 \nu^{3} + \cdots - 32 ) / 8 $ (-3v^10 + 8v^9 - 6v^8 - 2v^7 + 24v^6 - 44v^5 + 18v^4 + 24v^3 - 76v^2 + 112v - 32) / 8
$\beta_{6}$	$=$	$ ( - 2 \nu^{11} + 7 \nu^{10} - 12 \nu^{9} + 12 \nu^{8} + 10 \nu^{7} - 48 \nu^{6} + 68 \nu^{5} + \cdots + 128 ) / 16 $ (-2v^11 + 7v^10 - 12v^9 + 12v^8 + 10v^7 - 48v^6 + 68v^5 - 50v^4 - 48v^3 + 176v^2 - 176*v + 128) / 16
$\beta_{7}$	$=$	$ ( 5 \nu^{11} - 14 \nu^{10} + 24 \nu^{9} - 22 \nu^{8} - 28 \nu^{7} + 88 \nu^{6} - 126 \nu^{5} + 92 \nu^{4} + \cdots - 256 ) / 16 $ (5v^11 - 14v^10 + 24v^9 - 22v^8 - 28v^7 + 88v^6 - 126v^5 + 92v^4 + 112v^3 - 264v^2 + 320*v - 256) / 16
$\beta_{8}$	$=$	$ ( - 4 \nu^{11} + 7 \nu^{10} - 14 \nu^{9} + 16 \nu^{8} + 18 \nu^{7} - 44 \nu^{6} + 80 \nu^{5} - 74 \nu^{4} + \cdots + 208 ) / 8 $ (-4v^11 + 7v^10 - 14v^9 + 16v^8 + 18v^7 - 44v^6 + 80v^5 - 74v^4 - 68v^3 + 144v^2 - 208*v + 208) / 8
$\beta_{9}$	$=$	$ ( - 8 \nu^{11} + 23 \nu^{10} - 36 \nu^{9} + 24 \nu^{8} + 50 \nu^{7} - 144 \nu^{6} + 184 \nu^{5} + \cdots + 256 ) / 16 $ (-8v^11 + 23v^10 - 36v^9 + 24v^8 + 50v^7 - 144v^6 + 184v^5 - 82v^4 - 192v^3 + 416v^2 - 448*v + 256) / 16
$\beta_{10}$	$=$	$ ( 6 \nu^{11} - 21 \nu^{10} + 40 \nu^{9} - 40 \nu^{8} - 30 \nu^{7} + 136 \nu^{6} - 228 \nu^{5} + \cdots - 416 ) / 16 $ (6v^11 - 21v^10 + 40v^9 - 40v^8 - 30v^7 + 136v^6 - 228v^5 + 166v^4 + 152v^3 - 440v^2 + 592*v - 416) / 16
$\beta_{11}$	$=$	$ ( - 18 \nu^{11} + 49 \nu^{10} - 76 \nu^{9} + 64 \nu^{8} + 102 \nu^{7} - 304 \nu^{6} + 396 \nu^{5} + \cdots + 704 ) / 16 $ (-18v^11 + 49v^10 - 76v^9 + 64v^8 + 102v^7 - 304v^6 + 396v^5 - 238v^4 - 416v^3 + 888v^2 - 976*v + 704) / 16

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

$\nu$	$=$	$ ( 2\beta_{11} + \beta_{10} + 9\beta_{8} + \beta_{6} + 9\beta_{5} - 23\beta_{4} + 6\beta_{3} + 23\beta _1 + 23 ) / 46 $ (2b11 + b10 + 9b8 + b6 + 9b5 - 23b4 + 6b3 + 23b1 + 23) / 46
$\nu^{2}$	$=$	$ ( -4\beta_{11} - 2\beta_{10} + 5\beta_{8} + 21\beta_{6} + 5\beta_{5} + 11\beta_{3} ) / 46 $ (-4b11 - 2b10 + 5b8 + 21b6 + 5b5 + 11b3) / 46
$\nu^{3}$	$=$	$ ( - 19 \beta_{11} + 2 \beta_{10} + 23 \beta_{9} - 5 \beta_{8} - 46 \beta_{7} + 2 \beta_{6} + 18 \beta_{5} + \cdots + 23 ) / 46 $ (-19b11 + 2b10 + 23b9 - 5b8 - 46b7 + 2b6 + 18b5 + 23b4 - 11b3 - 69b2 - 23*b1 + 23) / 46
$\nu^{4}$	$=$	$ \beta_{4} - 2\beta_{2} - \beta _1 + 3 $ b4 - 2*b2 - b1 + 3
$\nu^{5}$	$=$	$ ( - 8 \beta_{11} - 50 \beta_{10} + 46 \beta_{9} + 33 \beta_{8} + 92 \beta_{7} - 27 \beta_{6} + 33 \beta_{5} + \cdots + 92 ) / 46 $ (-8b11 - 50b10 + 46b9 + 33b8 + 92b7 - 27b6 + 33b5 - 92b4 - b3 + 46*b1 + 92) / 46
$\nu^{6}$	$=$	$ ( -15\beta_{11} - 42\beta_{10} + 13\beta_{8} + 4\beta_{6} + 36\beta_{5} + 47\beta_{3} ) / 23 $ (-15b11 - 42b10 + 13b8 + 4b6 + 36b5 + 47b3) / 23
$\nu^{7}$	$=$	$ ( - 54 \beta_{11} - 27 \beta_{10} + 46 \beta_{9} + 10 \beta_{8} - 92 \beta_{7} - 4 \beta_{6} + 10 \beta_{5} + \cdots + 23 ) / 23 $ (-54b11 - 27b10 + 46b9 + 10b8 - 92b7 - 4b6 + 10b5 - 23b4 - b3 - 92b2 - 69b1 + 23) / 23
$\nu^{8}$	$=$	$ -2\beta_{9} - 4\beta_{7} + 4\beta_{4} - 12\beta_{2} - 6\beta _1 + 4 $ -2b9 - 4b7 + 4b4 - 12b2 - 6*b1 + 4
$\nu^{9}$	$=$	$ ( - 7 \beta_{11} - 130 \beta_{10} + 69 \beta_{9} + 3 \beta_{8} + 230 \beta_{7} - 38 \beta_{6} + \cdots + 69 ) / 23 $ (-7b11 - 130b10 + 69b9 + 3b8 + 230b7 - 38b6 + 26b5 - 115b4 - 67b3 - 23b2 - 69*b1 + 69) / 23
$\nu^{10}$	$=$	$ ( -32\beta_{11} - 246\beta_{10} - 52\beta_{8} - 108\beta_{6} + 224\beta_{5} + 134\beta_{3} ) / 23 $ (-32b11 - 246b10 - 52b8 - 108b6 + 224b5 + 134b3) / 23
$\nu^{11}$	$=$	$ ( - 152 \beta_{11} - 214 \beta_{10} + 46 \beta_{9} - 17 \beta_{8} - 276 \beta_{7} - 53 \beta_{6} + \cdots + 276 ) / 23 $ (-152b11 - 214b10 + 46b9 - 17b8 - 276b7 - 53b6 - 17b5 - 276b4 + 73b3 - 138b1 + 276) / 23

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/184\mathbb{Z}\right)^\times$.

$n$	$47$	$93$	$97$
$\chi(n)$	$-1$	$-1$	$-1$

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} $

91.1

0.303172	+	1.38134i
−1.38134	+	0.303172i
0.303172	−	1.38134i
−1.38134	−	0.303172i
1.35381	−	0.408906i
0.408906	+	1.35381i
1.35381	+	0.408906i
0.408906	−	1.35381i
−0.0955343	−	1.41098i
1.41098	−	0.0955343i
−0.0955343	+	1.41098i
1.41098	+	0.0955343i

−1.24060

−

0.678910i

−1.48119

1.07816

1.68451i

−1.35782

1.83757

1.00560i

3.63235

−0.193937

−

2.82177i

−0.806063

1.68451

0.921837i

91.2

−1.24060

−

0.678910i

−1.48119

1.07816

1.68451i

1.35782

1.83757

1.00560i

−3.63235

−0.193937

−

2.82177i

−0.806063

−1.68451

−

0.921837i

91.3

−1.24060

0.678910i

−1.48119

1.07816

−

1.68451i

−1.35782

1.83757

−

1.00560i

3.63235

−0.193937

2.82177i

−0.806063

1.68451

−

0.921837i

91.4

−1.24060

0.678910i

−1.48119

1.07816

−

1.68451i

1.35782

1.83757

−

1.00560i

−3.63235

−0.193937

2.82177i

−0.806063

−1.68451

0.921837i

91.5

−0.344446

−

1.37163i

0.311108

−1.76271

0.944902i

−2.74325

−0.107160

−

0.426723i

−3.33118

1.90321

2.09232i

−2.90321

0.944902

3.76271i

91.6

−0.344446

−

1.37163i

0.311108

−1.76271

0.944902i

2.74325

−0.107160

−

0.426723i

3.33118

1.90321

2.09232i

−2.90321

−0.944902

−

3.76271i

91.7

−0.344446

1.37163i

0.311108

−1.76271

−

0.944902i

−2.74325

−0.107160

0.426723i

−3.33118

1.90321

−

2.09232i

−2.90321

0.944902

−

3.76271i

91.8

−0.344446

1.37163i

0.311108

−1.76271

−

0.944902i

2.74325

−0.107160

0.426723i

3.33118

1.90321

−

2.09232i

−2.90321

−0.944902

3.76271i

91.9

0.585043

−

1.28753i

2.17009

−1.31545

−

1.50652i

−2.57505

1.26959

−

2.79404i

3.96349

−2.70928

0.812297i

1.70928

−1.50652

3.31545i

91.10

0.585043

−

1.28753i

2.17009

−1.31545

−

1.50652i

2.57505

1.26959

−

2.79404i

−3.96349

−2.70928

0.812297i

1.70928

1.50652

−

3.31545i

91.11

0.585043

1.28753i

2.17009

−1.31545

1.50652i

−2.57505

1.26959

2.79404i

3.96349

−2.70928

−

0.812297i

1.70928

−1.50652

−

3.31545i

91.12

0.585043

1.28753i

2.17009

−1.31545

1.50652i

2.57505

1.26959

2.79404i

−3.96349

−2.70928

−

0.812297i

1.70928

1.50652

3.31545i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	inner
23.b	odd	2	1	inner
184.h	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	184.2.h.c	✓	12
4.b	odd	2	1		736.2.h.c		12
8.b	even	2	1		736.2.h.c		12
8.d	odd	2	1	inner	184.2.h.c	✓	12
23.b	odd	2	1	inner	184.2.h.c	✓	12
92.b	even	2	1		736.2.h.c		12
184.e	odd	2	1		736.2.h.c		12
184.h	even	2	1	inner	184.2.h.c	✓	12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
184.2.h.c	✓	12	1.a	even	1	1	trivial
184.2.h.c	✓	12	8.d	odd	2	1	inner
184.2.h.c	✓	12	23.b	odd	2	1	inner
184.2.h.c	✓	12	184.h	even	2	1	inner
736.2.h.c		12	4.b	odd	2	1
736.2.h.c		12	8.b	even	2	1
736.2.h.c		12	92.b	even	2	1
736.2.h.c		12	184.e	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{3} - T_{3}^{2} - 3T_{3} + 1 $ T3^3 - T3^2 - 3*T3 + 1 acting on $S_{2}^{\mathrm{new}}(184, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{6} + 2 T^{5} + 4 T^{4} + \cdots + 8)^{2} $ (T^6 + 2T^5 + 4T^4 + 6T^3 + 8T^2 + 8*T + 8)^2
$3$	$ (T^{3} - T^{2} - 3 T + 1)^{4} $ (T^3 - T^2 - 3*T + 1)^4
$5$	$ (T^{6} - 16 T^{4} + \cdots - 92)^{2} $ (T^6 - 16T^4 + 76T^2 - 92)^2
$7$	$ (T^{6} - 40 T^{4} + \cdots - 2300)^{2} $ (T^6 - 40T^4 + 528T^2 - 2300)^2
$11$	$ (T^{6} + 56 T^{4} + \cdots + 2116)^{2} $ (T^6 + 56T^4 + 832T^2 + 2116)^2
$13$	$ (T^{6} + 21 T^{4} + \cdots + 23)^{2} $ (T^6 + 21T^4 + 83T^2 + 23)^2
$17$	$ (T^{6} + 52 T^{4} + \cdots + 2116)^{2} $ (T^6 + 52T^4 + 692T^2 + 2116)^2
$19$	$ (T^{6} + 108 T^{4} + \cdots + 33856)^{2} $ (T^6 + 108T^4 + 3440T^2 + 33856)^2
$23$	$ T^{12} + \cdots + 148035889 $ T^12 - 38T^10 - 449T^8 + 37996T^6 - 237521T^4 - 10633958*T^2 + 148035889
$29$	$ (T^{6} + 77 T^{4} + \cdots + 14375)^{2} $ (T^6 + 77T^4 + 1883T^2 + 14375)^2
$31$	$ (T^{6} + 109 T^{4} + \cdots + 23)^{2} $ (T^6 + 109T^4 + 1207T^2 + 23)^2
$37$	$ (T^{6} - 84 T^{4} + \cdots - 15548)^{2} $ (T^6 - 84T^4 + 2052T^2 - 15548)^2
$41$	$ (T^{3} - 3 T^{2} - 45 T + 27)^{4} $ (T^3 - 3T^2 - 45T + 27)^4
$43$	$ (T^{6} + 208 T^{4} + \cdots + 135424)^{2} $ (T^6 + 208T^4 + 11072T^2 + 135424)^2
$47$	$ (T^{6} + 157 T^{4} + \cdots + 103247)^{2} $ (T^6 + 157T^4 + 7423T^2 + 103247)^2
$53$	$ (T^{6} - 84 T^{4} + \cdots - 1472)^{2} $ (T^6 - 84T^4 + 1008T^2 - 1472)^2
$59$	$ (T^{3} + 20 T^{2} + \cdots + 208)^{4} $ (T^3 + 20T^2 + 120T + 208)^4
$61$	$ (T^{6} - 64 T^{4} + \cdots - 5888)^{2} $ (T^6 - 64T^4 + 1216T^2 - 5888)^2
$67$	$ (T^{6} + 164 T^{4} + \cdots + 2116)^{2} $ (T^6 + 164T^4 + 1944T^2 + 2116)^2
$71$	$ (T^{6} + 81 T^{4} + \cdots + 12167)^{2} $ (T^6 + 81T^4 + 1931T^2 + 12167)^2
$73$	$ (T^{3} + 9 T^{2} - 37 T - 37)^{4} $ (T^3 + 9T^2 - 37T - 37)^4
$79$	$ (T^{6} - 212 T^{4} + \cdots - 1472)^{2} $ (T^6 - 212T^4 + 9008T^2 - 1472)^2
$83$	$ (T^{6} + 164 T^{4} + \cdots + 2116)^{2} $ (T^6 + 164T^4 + 1944T^2 + 2116)^2
$89$	$ (T^{6} + 300 T^{4} + \cdots + 33856)^{2} $ (T^6 + 300T^4 + 6320T^2 + 33856)^2
$97$	$ (T^{6} + 368 T^{4} + \cdots + 357604)^{2} $ (T^6 + 368T^4 + 34876T^2 + 357604)^2
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Overview	Random
Universe	Knowledge

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:	\( N \)	\(=\)	\( 184 = 2^{3} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	184.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{2} q^{2} + ( - \beta_{7} - \beta_{2} + 1) q^{3} + (\beta_{7} - \beta_{4} + \beta_{2} - 1) q^{4} - \beta_{10} q^{5} + (\beta_{7} - \beta_{4} + 1) q^{6} - \beta_{6} q^{7} + ( - \beta_{9} + \beta_{4} + \beta_{2} + \cdots - 1) q^{8}+ \cdots + (2 \beta_{11} + \beta_{10} + \cdots - \beta_{5}) q^{99}+O(q^{100}) \) q - b2 * q^2 + (-b7 - b2 + 1) * q^3 + (b7 - b4 + b2 - 1) * q^4 - b10 * q^5 + (b7 - b4 + 1) * q^6 - b6 * q^7 + (-b9 + b4 + b2 + b1 - 1) * q^8 + (-b7 - b2 - b1) * q^9 + (-b8 + b5) * q^10 + (b8 - b5 - b3) * q^11 + (-b9 + b7 + b1 - 2) * q^12 + (b9 + b4 - b2 - b1) * q^13 + (b11 + b10 + b6 - 2b5 - b3) q^14 + (-b10 + b5) * q^15 + 2b9 q^16 + (-b8 - b3) * q^17 + (-b9 - b4 + b2 + b1 + 1) * q^18 + (-2b11 - b10 - b6 + b5) q^19 + (-b11 + b10) * q^20 + (b10 - 2b5) q^21 + (b11 - b10 + b6 + b3) * q^22 + (b10 + b9 - b7 + b5 + b4 - b1) * q^23 + (b9 + 2b7 - b4 + 3b2 + b1 - 3) * q^24 + (-b7 - b2 + b1 + 1) * q^25 + (-2b7 - b2 - 2b1) * q^26 + (2b7 + 2b2 - b1 - 1) * q^27 + (b11 + b10 + b8 + b6 + b5 + b3) * q^28 + (-b9 + b7 - 3b4 + 2b2 + 2b1) q^29 + (-b11 - b10 - b8 + b5) * q^30 + (-3b9 - 2b7 + 3b4 - b2) q^31 + (-2b7 + 2b4 - 2b2 - 4b1 + 2) * q^32 + (2b11 + b10 + b8 + b6 - b5 + b3) q^33 + (2b10 + b6 + b3) q^34 + (3b7 + 3b2 + 5b1 - 4) q^35 + (2b7 + 2b1) * q^36 + (-2b10 - b6 + b5) q^37 + (-b8 - b6 - b5 + b3) * q^38 + (b9 - b7 + b4 - b1) * q^39 + (b8 - b6 - b5 + b3) * q^40 + (-3b7 - 3b2 - 3b1 + 3) q^41 + (2b11 + 2b10 + b8 - b5) * q^42 + (2b8 + 2b3) * q^43 + (-b11 - b10 - b8 - b6 + 3b5 - b3) q^44 + (b10 + b6 + b5) * q^45 + (-b11 - b10 + b8 - 3b7 - b5 + b4 - 2b2 - 2b1 + 3) q^46 + (3b9 + 4b7 - 3b4 - b2) q^47 + (-2b7 + 4b4 - 2b1) q^48 + (-b7 - b2 - b1 + 7) * q^49 + (b9 + 2b7 - b4 - b1 + 1) q^50 + (-b8 + b5 + b3) * q^51 + (-2b9 - b7 - b4 + b2 + 2b1 + 3) * q^52 + (-b10 + b6 - b5) * q^53 + (-b9 - 3b7 + 2b4 - b2 + b1 - 2) * q^54 + (2b9 - 3b7 + 3b2 - b1) q^55 + (-2b11 - 4b10 + b8 - b6 + b5 - b3) * q^56 + (-2b11 - b10 - 2b8 - b6 + 3b5 + 2b3) * q^57 + (-b9 + 4b7 + b4 + 2b2 + 3b1 - 1) q^58 + (2b7 + 2b2 - 8) * q^59 + (-b11 + b10 - b8 - b6 + b5 + b3) * q^60 + 2b10 q^61 + (3b9 + b7 - 4b4 + b2 + 3b1) q^62 + (3b10 + b6 - b5) q^63 + (-2b9 - 4b7 - 2b4 - 2b2 + 2b1 + 2) q^64 + (2b11 + b10 - b8 + b6 - b5 - b3) q^65 + (-2b10 + b8 + b5 - 2b3) * q^66 + (2b11 + b10 + b8 + b6 - 2b5 - b3) * q^67 + (-b11 - b10 + 2b8 - 2b6) * q^68 + (b9 - 3b7 + b6 - b5 + 3b4 - 2b1) q^69 + (5b9 + 2b7 + 3b4 + b2 - 5b1 - 3) * q^70 + (-3b9 - b7 + b4 + b1) q^71 + (2b9 + 2b7 - 2b1 - 4) q^72 + (-4b1 - 3) q^73 + (-2b8 + b6 - b3) q^74 + (-b1 + 3) * q^75 + (2b11 + 4b10 - 2b5 - 2b3) * q^76 + (-8b9 + b7 + 2b4 - 3b2 + 3b1) * q^77 + (-3b7 + b4 - 2b2 - 2b1 + 3) q^78 + (b10 + b6 - 3b5) q^79 + (2b11 - 2b5 - 2b3) q^80 + (3b7 + 3b2 + 5b1 - 5) q^81 + (-3b9 - 3b4 + 3b1 + 3) q^82 + (-2b11 - b10 - b8 - b6 + 2b5 + b3) * q^83 + (b11 - b10 + 2b8 + 2b6 - 2b5 - 2b3) * q^84 + (4b9 + 5b7 - 2b4 - 3b2 - b1) * q^85 + (-4b10 - 2b6 - 2b3) q^86 + (-3b9 + 2b7 - b4 - b2 + 2b1) q^87 + (-2b11 - b8 + b6 - b5 + b3) q^88 + (2b11 + b10 - 2b8 + b6 - b5 - 2b3) q^89 + (-2b11 - 2b10 + b8 - b6 + b5 + b3) * q^90 + (-3b8 + b5 - b3) q^91 + (b11 - b10 - b9 - b8 - b7 - b6 + b5 - 2b4 + b3 - 2b2 + b1 + 4) * q^92 + (3b9 + 2b7 - 5b4 + 3b2 + b1) * q^93 + (-3b9 + b7 + 2b4 + b2 - 3b1 - 6) q^94 + (-4b9 + 4b7 - 8b4 + 4b2 + 6b1) q^95 + (2b9 - 6b7 - 4b2 - 2b1 + 4) * q^96 + (-2b11 - b10 + 3b8 - b6 - b5 - b3) * q^97 + (-b9 - b4 - 6b2 + b1 + 1) q^98 + (2b11 + b10 + b6 - b5) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{2} + 4 q^{3} - 8 q^{4} + 12 q^{6} - 4 q^{8} - 8 q^{9} - 20 q^{12} + 12 q^{18} - 20 q^{24} + 4 q^{25} - 12 q^{26} + 4 q^{27} + 16 q^{32} - 24 q^{35} + 8 q^{36} + 12 q^{41} + 20 q^{46} + 8 q^{48}+ \cdots - 16 q^{98}+O(q^{100}) \) 12 * q - 4 * q^2 + 4 * q^3 - 8 * q^4 + 12 * q^6 - 4 * q^8 - 8 * q^9 - 20 * q^12 + 12 * q^18 - 20 * q^24 + 4 * q^25 - 12 * q^26 + 4 * q^27 + 16 * q^32 - 24 * q^35 + 8 * q^36 + 12 * q^41 + 20 * q^46 + 8 * q^48 + 76 * q^49 + 16 * q^50 + 32 * q^52 - 32 * q^54 + 16 * q^58 - 80 * q^59 - 8 * q^62 - 8 * q^64 - 12 * q^70 - 40 * q^72 - 36 * q^73 + 36 * q^75 + 20 * q^78 - 36 * q^81 + 24 * q^82 + 28 * q^92 - 56 * q^94 + 8 * q^96 - 16 * q^98

Introduction

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Newspace parameters

Newform invariants

$q$-expansion

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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	184.2.h.c

Level	$184$
Weight	$2$
Character orbit	184.h
Analytic conductor	$1.469$
Analytic rank	$0$
Dimension	$12$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.46924739719\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Coefficient field:	12.0.64974433091584.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - 4x^{11} + 8x^{10} - 10x^{9} + 24x^{7} - 46x^{6} + 48x^{5} - 80x^{3} + 128x^{2} - 128x + 64 \) x^12 - 4x^11 + 8x^10 - 10x^9 + 24x^7 - 46x^6 + 48x^5 - 80x^3 + 128x^2 - 128*x + 64
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{6}\cdot 23 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( 2\nu^{11} - \nu^{10} - 4\nu^{9} + 4\nu^{8} - 14\nu^{7} + 28\nu^{5} - 18\nu^{4} + 32\nu^{3} + 16\nu^{2} - 64\nu ) / 16 \) (2v^11 - v^10 - 4v^9 + 4v^8 - 14v^7 + 28v^5 - 18v^4 + 32v^3 + 16v^2 - 64*v) / 16
\(\beta_{2}\)	\(=\)	\( ( - 2 \nu^{11} + 3 \nu^{10} - 4 \nu^{9} + 4 \nu^{8} + 10 \nu^{7} - 16 \nu^{6} + 20 \nu^{5} - 26 \nu^{4} + \cdots + 96 ) / 16 \) (-2v^11 + 3v^10 - 4v^9 + 4v^8 + 10v^7 - 16v^6 + 20v^5 - 26v^4 - 32v^3 + 48v^2 - 64*v + 96) / 16
\(\beta_{3}\)	\(=\)	\( ( 2 \nu^{11} - 3 \nu^{10} + 8 \nu^{9} - 16 \nu^{8} - 2 \nu^{7} + 24 \nu^{6} - 60 \nu^{5} + 90 \nu^{4} + \cdots - 224 ) / 16 \) (2v^11 - 3v^10 + 8v^9 - 16v^8 - 2v^7 + 24v^6 - 60v^5 + 90v^4 + 8v^3 - 88v^2 + 176*v - 224) / 16
\(\beta_{4}\)	\(=\)	\( ( - 2 \nu^{11} + 5 \nu^{10} - 12 \nu^{9} + 12 \nu^{8} + 6 \nu^{7} - 32 \nu^{6} + 68 \nu^{5} - 54 \nu^{4} + \cdots + 144 ) / 16 \) (-2v^11 + 5v^10 - 12v^9 + 12v^8 + 6v^7 - 32v^6 + 68v^5 - 54v^4 - 32v^3 + 112v^2 - 192*v + 144) / 16
\(\beta_{5}\)	\(=\)	\( ( - 3 \nu^{10} + 8 \nu^{9} - 6 \nu^{8} - 2 \nu^{7} + 24 \nu^{6} - 44 \nu^{5} + 18 \nu^{4} + 24 \nu^{3} + \cdots - 32 ) / 8 \) (-3v^10 + 8v^9 - 6v^8 - 2v^7 + 24v^6 - 44v^5 + 18v^4 + 24v^3 - 76v^2 + 112v - 32) / 8
\(\beta_{6}\)	\(=\)	\( ( - 2 \nu^{11} + 7 \nu^{10} - 12 \nu^{9} + 12 \nu^{8} + 10 \nu^{7} - 48 \nu^{6} + 68 \nu^{5} + \cdots + 128 ) / 16 \) (-2v^11 + 7v^10 - 12v^9 + 12v^8 + 10v^7 - 48v^6 + 68v^5 - 50v^4 - 48v^3 + 176v^2 - 176*v + 128) / 16
\(\beta_{7}\)	\(=\)	\( ( 5 \nu^{11} - 14 \nu^{10} + 24 \nu^{9} - 22 \nu^{8} - 28 \nu^{7} + 88 \nu^{6} - 126 \nu^{5} + 92 \nu^{4} + \cdots - 256 ) / 16 \) (5v^11 - 14v^10 + 24v^9 - 22v^8 - 28v^7 + 88v^6 - 126v^5 + 92v^4 + 112v^3 - 264v^2 + 320*v - 256) / 16
\(\beta_{8}\)	\(=\)	\( ( - 4 \nu^{11} + 7 \nu^{10} - 14 \nu^{9} + 16 \nu^{8} + 18 \nu^{7} - 44 \nu^{6} + 80 \nu^{5} - 74 \nu^{4} + \cdots + 208 ) / 8 \) (-4v^11 + 7v^10 - 14v^9 + 16v^8 + 18v^7 - 44v^6 + 80v^5 - 74v^4 - 68v^3 + 144v^2 - 208*v + 208) / 8
\(\beta_{9}\)	\(=\)	\( ( - 8 \nu^{11} + 23 \nu^{10} - 36 \nu^{9} + 24 \nu^{8} + 50 \nu^{7} - 144 \nu^{6} + 184 \nu^{5} + \cdots + 256 ) / 16 \) (-8v^11 + 23v^10 - 36v^9 + 24v^8 + 50v^7 - 144v^6 + 184v^5 - 82v^4 - 192v^3 + 416v^2 - 448*v + 256) / 16
\(\beta_{10}\)	\(=\)	\( ( 6 \nu^{11} - 21 \nu^{10} + 40 \nu^{9} - 40 \nu^{8} - 30 \nu^{7} + 136 \nu^{6} - 228 \nu^{5} + \cdots - 416 ) / 16 \) (6v^11 - 21v^10 + 40v^9 - 40v^8 - 30v^7 + 136v^6 - 228v^5 + 166v^4 + 152v^3 - 440v^2 + 592*v - 416) / 16
\(\beta_{11}\)	\(=\)	\( ( - 18 \nu^{11} + 49 \nu^{10} - 76 \nu^{9} + 64 \nu^{8} + 102 \nu^{7} - 304 \nu^{6} + 396 \nu^{5} + \cdots + 704 ) / 16 \) (-18v^11 + 49v^10 - 76v^9 + 64v^8 + 102v^7 - 304v^6 + 396v^5 - 238v^4 - 416v^3 + 888v^2 - 976*v + 704) / 16

\(\nu\)	\(=\)	\( ( 2\beta_{11} + \beta_{10} + 9\beta_{8} + \beta_{6} + 9\beta_{5} - 23\beta_{4} + 6\beta_{3} + 23\beta _1 + 23 ) / 46 \) (2b11 + b10 + 9b8 + b6 + 9b5 - 23b4 + 6b3 + 23b1 + 23) / 46
\(\nu^{2}\)	\(=\)	\( ( -4\beta_{11} - 2\beta_{10} + 5\beta_{8} + 21\beta_{6} + 5\beta_{5} + 11\beta_{3} ) / 46 \) (-4b11 - 2b10 + 5b8 + 21b6 + 5b5 + 11b3) / 46
\(\nu^{3}\)	\(=\)	\( ( - 19 \beta_{11} + 2 \beta_{10} + 23 \beta_{9} - 5 \beta_{8} - 46 \beta_{7} + 2 \beta_{6} + 18 \beta_{5} + \cdots + 23 ) / 46 \) (-19b11 + 2b10 + 23b9 - 5b8 - 46b7 + 2b6 + 18b5 + 23b4 - 11b3 - 69b2 - 23*b1 + 23) / 46
\(\nu^{4}\)	\(=\)	\( \beta_{4} - 2\beta_{2} - \beta _1 + 3 \) b4 - 2*b2 - b1 + 3
\(\nu^{5}\)	\(=\)	\( ( - 8 \beta_{11} - 50 \beta_{10} + 46 \beta_{9} + 33 \beta_{8} + 92 \beta_{7} - 27 \beta_{6} + 33 \beta_{5} + \cdots + 92 ) / 46 \) (-8b11 - 50b10 + 46b9 + 33b8 + 92b7 - 27b6 + 33b5 - 92b4 - b3 + 46*b1 + 92) / 46
\(\nu^{6}\)	\(=\)	\( ( -15\beta_{11} - 42\beta_{10} + 13\beta_{8} + 4\beta_{6} + 36\beta_{5} + 47\beta_{3} ) / 23 \) (-15b11 - 42b10 + 13b8 + 4b6 + 36b5 + 47b3) / 23
\(\nu^{7}\)	\(=\)	\( ( - 54 \beta_{11} - 27 \beta_{10} + 46 \beta_{9} + 10 \beta_{8} - 92 \beta_{7} - 4 \beta_{6} + 10 \beta_{5} + \cdots + 23 ) / 23 \) (-54b11 - 27b10 + 46b9 + 10b8 - 92b7 - 4b6 + 10b5 - 23b4 - b3 - 92b2 - 69b1 + 23) / 23
\(\nu^{8}\)	\(=\)	\( -2\beta_{9} - 4\beta_{7} + 4\beta_{4} - 12\beta_{2} - 6\beta _1 + 4 \) -2b9 - 4b7 + 4b4 - 12b2 - 6*b1 + 4
\(\nu^{9}\)	\(=\)	\( ( - 7 \beta_{11} - 130 \beta_{10} + 69 \beta_{9} + 3 \beta_{8} + 230 \beta_{7} - 38 \beta_{6} + \cdots + 69 ) / 23 \) (-7b11 - 130b10 + 69b9 + 3b8 + 230b7 - 38b6 + 26b5 - 115b4 - 67b3 - 23b2 - 69*b1 + 69) / 23
\(\nu^{10}\)	\(=\)	\( ( -32\beta_{11} - 246\beta_{10} - 52\beta_{8} - 108\beta_{6} + 224\beta_{5} + 134\beta_{3} ) / 23 \) (-32b11 - 246b10 - 52b8 - 108b6 + 224b5 + 134b3) / 23
\(\nu^{11}\)	\(=\)	\( ( - 152 \beta_{11} - 214 \beta_{10} + 46 \beta_{9} - 17 \beta_{8} - 276 \beta_{7} - 53 \beta_{6} + \cdots + 276 ) / 23 \) (-152b11 - 214b10 + 46b9 - 17b8 - 276b7 - 53b6 - 17b5 - 276b4 + 73b3 - 138b1 + 276) / 23

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 91.12
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{6} + 2 T^{5} + 4 T^{4} + \cdots + 8)^{2} \) (T^6 + 2T^5 + 4T^4 + 6T^3 + 8T^2 + 8*T + 8)^2
$3$	\( (T^{3} - T^{2} - 3 T + 1)^{4} \) (T^3 - T^2 - 3*T + 1)^4
$5$	\( (T^{6} - 16 T^{4} + \cdots - 92)^{2} \) (T^6 - 16T^4 + 76T^2 - 92)^2
$7$	\( (T^{6} - 40 T^{4} + \cdots - 2300)^{2} \) (T^6 - 40T^4 + 528T^2 - 2300)^2
$11$	\( (T^{6} + 56 T^{4} + \cdots + 2116)^{2} \) (T^6 + 56T^4 + 832T^2 + 2116)^2
$13$	\( (T^{6} + 21 T^{4} + \cdots + 23)^{2} \) (T^6 + 21T^4 + 83T^2 + 23)^2
$17$	\( (T^{6} + 52 T^{4} + \cdots + 2116)^{2} \) (T^6 + 52T^4 + 692T^2 + 2116)^2
$19$	\( (T^{6} + 108 T^{4} + \cdots + 33856)^{2} \) (T^6 + 108T^4 + 3440T^2 + 33856)^2
$23$	\( T^{12} + \cdots + 148035889 \) T^12 - 38T^10 - 449T^8 + 37996T^6 - 237521T^4 - 10633958*T^2 + 148035889
$29$	\( (T^{6} + 77 T^{4} + \cdots + 14375)^{2} \) (T^6 + 77T^4 + 1883T^2 + 14375)^2
$31$	\( (T^{6} + 109 T^{4} + \cdots + 23)^{2} \) (T^6 + 109T^4 + 1207T^2 + 23)^2
$37$	\( (T^{6} - 84 T^{4} + \cdots - 15548)^{2} \) (T^6 - 84T^4 + 2052T^2 - 15548)^2
$41$	\( (T^{3} - 3 T^{2} - 45 T + 27)^{4} \) (T^3 - 3T^2 - 45T + 27)^4
$43$	\( (T^{6} + 208 T^{4} + \cdots + 135424)^{2} \) (T^6 + 208T^4 + 11072T^2 + 135424)^2
$47$	\( (T^{6} + 157 T^{4} + \cdots + 103247)^{2} \) (T^6 + 157T^4 + 7423T^2 + 103247)^2
$53$	\( (T^{6} - 84 T^{4} + \cdots - 1472)^{2} \) (T^6 - 84T^4 + 1008T^2 - 1472)^2
$59$	\( (T^{3} + 20 T^{2} + \cdots + 208)^{4} \) (T^3 + 20T^2 + 120T + 208)^4
$61$	\( (T^{6} - 64 T^{4} + \cdots - 5888)^{2} \) (T^6 - 64T^4 + 1216T^2 - 5888)^2
$67$	\( (T^{6} + 164 T^{4} + \cdots + 2116)^{2} \) (T^6 + 164T^4 + 1944T^2 + 2116)^2
$71$	\( (T^{6} + 81 T^{4} + \cdots + 12167)^{2} \) (T^6 + 81T^4 + 1931T^2 + 12167)^2
$73$	\( (T^{3} + 9 T^{2} - 37 T - 37)^{4} \) (T^3 + 9T^2 - 37T - 37)^4
$79$	\( (T^{6} - 212 T^{4} + \cdots - 1472)^{2} \) (T^6 - 212T^4 + 9008T^2 - 1472)^2
$83$	\( (T^{6} + 164 T^{4} + \cdots + 2116)^{2} \) (T^6 + 164T^4 + 1944T^2 + 2116)^2
$89$	\( (T^{6} + 300 T^{4} + \cdots + 33856)^{2} \) (T^6 + 300T^4 + 6320T^2 + 33856)^2
$97$	\( (T^{6} + 368 T^{4} + \cdots + 357604)^{2} \) (T^6 + 368T^4 + 34876T^2 + 357604)^2
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\(n\)	\(47\)	\(93\)	\(97\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)