LMFDB - Newform orbit 845.2.m.h

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [845,2,Mod(316,845)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(845, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("845.316");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 845 = 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	845.m (of order $6$, degree $2$, not 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:	$6.74735897080$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{6})$
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} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36 $ x^12 - 16x^8 - 24x^7 + 96x^5 + 304x^4 + 384x^3 + 288x^2 + 144*x + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 65)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

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

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36 $ x^12 - 16*x^8 - 24*x^7 + 96*x^5 + 304*x^4 + 384*x^3 + 288*x^2 + 144*x + 36 :

$\beta_{1}$	$=$	$ ( 473 \nu^{11} - 516 \nu^{10} + 1118 \nu^{9} - 3014 \nu^{8} - 4564 \nu^{7} - 5676 \nu^{6} + \cdots - 246168 ) / 103944 $ (473v^11 - 516v^10 + 1118v^9 - 3014v^8 - 4564v^7 - 5676v^6 - 2470v^5 + 66640v^4 + 113100v^3 + 96048v^2 + 54216*v - 246168) / 103944
$\beta_{2}$	$=$	$ ( 962 \nu^{11} - 2392 \nu^{10} + 2887 \nu^{9} - 2220 \nu^{8} - 13990 \nu^{7} + 14442 \nu^{6} + \cdots + 15984 ) / 103944 $ (962v^11 - 2392v^10 + 2887v^9 - 2220v^8 - 13990v^7 + 14442v^6 + 11380v^5 + 57708v^4 + 98118v^3 - 132080v^2 + 49284*v + 15984) / 103944
$\beta_{3}$	$=$	$ ( 748 \nu^{11} - 816 \nu^{10} + 1768 \nu^{9} - 4162 \nu^{8} - 6311 \nu^{7} - 8976 \nu^{6} + \cdots - 98004 ) / 51972 $ (748v^11 - 816v^10 + 1768v^9 - 4162v^8 - 6311v^7 - 8976v^6 - 7532v^5 + 93902v^4 + 164352v^3 + 141012v^2 + 80298*v - 98004) / 51972
$\beta_{4}$	$=$	$ ( 962 \nu^{11} - 2392 \nu^{10} + 2887 \nu^{9} - 2220 \nu^{8} - 13990 \nu^{7} + 14442 \nu^{6} + \cdots + 15984 ) / 34648 $ (962v^11 - 2392v^10 + 2887v^9 - 2220v^8 - 13990v^7 + 14442v^6 + 11380v^5 + 57708v^4 + 98118v^3 - 97432v^2 + 49284*v + 15984) / 34648
$\beta_{5}$	$=$	$ ( - 52 \nu^{11} + 126 \nu^{10} - 161 \nu^{9} + 120 \nu^{8} + 776 \nu^{7} - 718 \nu^{6} - 536 \nu^{5} + \cdots - 864 ) / 1464 $ (-52v^11 + 126v^10 - 161v^9 + 120v^8 + 776v^7 - 718v^6 - 536v^5 - 3060v^4 - 5274v^3 + 2576v^2 - 2664*v - 864) / 1464
$\beta_{6}$	$=$	$ ( 5790 \nu^{11} - 1824 \nu^{10} + 473 \nu^{9} - 516 \nu^{8} - 91522 \nu^{7} - 112790 \nu^{6} + \cdots + 404496 ) / 103944 $ (5790v^11 - 1824v^10 + 473v^9 - 516v^8 - 91522v^7 - 112790v^6 + 39212v^5 + 550164v^4 + 1582586v^3 + 1735504v^2 + 1080204*v + 404496) / 103944
$\beta_{7}$	$=$	$ ( - 118 \nu^{11} + 90 \nu^{10} - 53 \nu^{9} + 32 \nu^{8} + 1866 \nu^{7} + 1416 \nu^{6} - 1364 \nu^{5} + \cdots - 5172 ) / 1704 $ (-118v^11 + 90v^10 - 53v^9 + 32v^8 + 1866v^7 + 1416v^6 - 1364v^5 - 10408v^4 - 27758v^3 - 22776v^2 - 12468*v - 5172) / 1704
$\beta_{8}$	$=$	$ ( - 14137 \nu^{11} + 3520 \nu^{10} - 77 \nu^{9} + 84 \nu^{8} + 226010 \nu^{7} + 282250 \nu^{6} + \cdots - 1015848 ) / 103944 $ (-14137v^11 + 3520v^10 - 77v^9 + 84v^8 + 226010v^7 + 282250v^6 - 85550v^5 - 1356228v^4 - 3952074v^3 - 4346412v^2 - 2709180*v - 1015848) / 103944
$\beta_{9}$	$=$	$ ( - 7882 \nu^{11} + 5055 \nu^{10} - 4456 \nu^{9} + 2886 \nu^{8} + 127101 \nu^{7} + 103246 \nu^{6} + \cdots - 498240 ) / 51972 $ (-7882v^11 + 5055v^10 - 4456v^9 + 2886v^8 + 127101v^7 + 103246v^6 - 43336v^5 - 703200v^4 - 1980128v^3 - 1894964v^2 - 1337298*v - 498240) / 51972
$\beta_{10}$	$=$	$ ( 18668 \nu^{11} - 14130 \nu^{10} + 7753 \nu^{9} - 5308 \nu^{8} - 292962 \nu^{7} - 224016 \nu^{6} + \cdots + 541068 ) / 103944 $ (18668v^11 - 14130v^10 + 7753v^9 - 5308v^8 - 292962v^7 - 224016v^6 + 204208v^5 + 1716416v^4 + 4321654v^3 + 3548568v^2 + 1943844*v + 541068) / 103944
$\beta_{11}$	$=$	$ ( - 18707 \nu^{11} + 14250 \nu^{10} - 8297 \nu^{9} + 5114 \nu^{8} + 295450 \nu^{7} + 224484 \nu^{6} + \cdots - 548532 ) / 103944 $ (-18707v^11 + 14250v^10 - 8297v^9 + 5114v^8 + 295450v^7 + 224484v^6 - 214310v^5 - 1678984v^4 - 4388958v^3 - 3601656v^2 - 1971828*v - 548532) / 103944

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

$\nu$	$=$	$ ( 2 \beta_{11} + \beta_{10} + \beta_{9} - 2 \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} + 4 \beta_{4} + \cdots + 1 ) / 6 $ (2b11 + b10 + b9 - 2b8 - b7 - b6 + b5 + 4b4 - 2b3 - 2b2 + 4b1 + 1) / 6
$\nu^{2}$	$=$	$ \beta_{4} - 3\beta_{2} $ b4 - 3*b2
$\nu^{3}$	$=$	$ ( - 8 \beta_{11} - 4 \beta_{10} - 2 \beta_{9} + 4 \beta_{8} + 10 \beta_{7} + 5 \beta_{6} + 4 \beta_{5} + \cdots + 5 ) / 3 $ (-8b11 - 4b10 - 2b9 + 4b8 + 10b7 + 5b6 + 4b5 + 4b4 + 2b3 - 5b2 - 4*b1 + 5) / 3
$\nu^{4}$	$=$	$ -6\beta_{11} - \beta_{10} + 13\beta_{7} + 13 $ -6b11 - b10 + 13b7 + 13
$\nu^{5}$	$=$	$ ( - 19 \beta_{11} - 8 \beta_{10} + 16 \beta_{9} - 38 \beta_{8} + 29 \beta_{7} - 58 \beta_{6} - 8 \beta_{5} + \cdots + 58 ) / 3 $ (-19b11 - 8b10 + 16b9 - 38b8 + 29b7 - 58b6 - 8b5 + 19b4 - 8b3 - 29b2 + 19*b1 + 58) / 3
$\nu^{6}$	$=$	$ 8\beta_{9} - 32\beta_{8} - 62\beta_{6} + 32\beta_{4} - 62\beta_{2} $ 8b9 - 32b8 - 62b6 + 32b4 - 62*b2
$\nu^{7}$	$=$	$ ( - 94 \beta_{11} - 35 \beta_{10} + 35 \beta_{9} - 94 \beta_{8} + 155 \beta_{7} - 155 \beta_{6} + \cdots - 155 ) / 3 $ (-94b11 - 35b10 + 35b9 - 94b8 + 155b7 - 155b6 + 35b5 + 188b4 + 70b3 - 310b2 - 188*b1 - 155) / 3
$\nu^{8}$	$=$	$ -166\beta_{11} - 48\beta_{10} + 306\beta_{7} + 48\beta_{3} - 166\beta_1 $ -166b11 - 48b10 + 306b7 + 48b3 - 166*b1
$\nu^{9}$	$=$	$ ( - 944 \beta_{11} - 328 \beta_{10} + 164 \beta_{9} - 472 \beta_{8} + 1612 \beta_{7} - 806 \beta_{6} + \cdots + 806 ) / 3 $ (-944b11 - 328b10 + 164b9 - 472b8 + 1612b7 - 806b6 - 328b5 - 472b4 + 164b3 + 806b2 - 472*b1 + 806) / 3
$\nu^{10}$	$=$	$ 262\beta_{9} - 852\beta_{8} - 1534\beta_{6} - 262\beta_{5} $ 262b9 - 852b8 - 1534b6 - 262b5
$\nu^{11}$	$=$	$ ( 2386 \beta_{11} + 800 \beta_{10} + 1600 \beta_{9} - 4772 \beta_{8} - 4142 \beta_{7} - 8284 \beta_{6} + \cdots - 8284 ) / 3 $ (2386b11 + 800b10 + 1600b9 - 4772b8 - 4142b7 - 8284b6 - 800b5 + 2386b4 + 800b3 - 4142b2 - 2386*b1 - 8284) / 3

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/845\mathbb{Z}\right)^\times$.

$n$	$171$	$677$
$\chi(n)$	$-\beta_{7}$	$1$

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} $

316.1

−0.180407	−	0.673288i
2.17840	−	0.583700i
−0.403293	−	1.50511i
−1.50511	+	0.403293i
0.583700	+	2.17840i
−0.673288	+	0.180407i
−0.180407	+	0.673288i
2.17840	+	0.583700i
−0.403293	+	1.50511i
−1.50511	−	0.403293i
0.583700	−	2.17840i
−0.673288	−	0.180407i

−2.17731

−

1.25707i

−0.757068

1.31128i

2.16044

3.74200i

−

1.00000i

3.29674

−

1.90337i

−2.87597

1.66044i

−

5.83502i

0.353695

0.612617i

−1.25707

2.17731i

316.2

−1.80664

−

1.04307i

1.54307

−

2.67267i

1.17597

2.03684i

1.00000i

−5.57553

3.21903i

1.17081

−

0.675970i

−

0.734191i

−3.26210

−

5.65012i

1.04307

−

1.80664i

316.3

−0.495361

−

0.285997i

0.214003

−

0.370665i

−0.836412

−

1.44871i

−

1.00000i

−0.212018

0.122408i

2.31473

−

1.33641i

2.10083i

1.40841

2.43943i

−0.285997

0.495361i

316.4

0.495361

0.285997i

0.214003

−

0.370665i

−0.836412

−

1.44871i

1.00000i

0.212018

−

0.122408i

−2.31473

1.33641i

−

2.10083i

1.40841

2.43943i

−0.285997

0.495361i

316.5

1.80664

1.04307i

1.54307

−

2.67267i

1.17597

2.03684i

−

1.00000i

5.57553

−

3.21903i

−1.17081

0.675970i

0.734191i

−3.26210

−

5.65012i

1.04307

−

1.80664i

316.6

2.17731

1.25707i

−0.757068

1.31128i

2.16044

3.74200i

1.00000i

−3.29674

1.90337i

2.87597

−

1.66044i

5.83502i

0.353695

0.612617i

−1.25707

2.17731i

361.1

−2.17731

1.25707i

−0.757068

−

1.31128i

2.16044

−

3.74200i

1.00000i

3.29674

1.90337i

−2.87597

−

1.66044i

5.83502i

0.353695

−

0.612617i

−1.25707

−

2.17731i

361.2

−1.80664

1.04307i

1.54307

2.67267i

1.17597

−

2.03684i

−

1.00000i

−5.57553

−

3.21903i

1.17081

0.675970i

0.734191i

−3.26210

5.65012i

1.04307

1.80664i

361.3

−0.495361

0.285997i

0.214003

0.370665i

−0.836412

1.44871i

1.00000i

−0.212018

−

0.122408i

2.31473

1.33641i

−

2.10083i

1.40841

−

2.43943i

−0.285997

−

0.495361i

361.4

0.495361

−

0.285997i

0.214003

0.370665i

−0.836412

1.44871i

−

1.00000i

0.212018

0.122408i

−2.31473

−

1.33641i

2.10083i

1.40841

−

2.43943i

−0.285997

−

0.495361i

361.5

1.80664

−

1.04307i

1.54307

2.67267i

1.17597

−

2.03684i

1.00000i

5.57553

3.21903i

−1.17081

−

0.675970i

−

0.734191i

−3.26210

5.65012i

1.04307

1.80664i

361.6

2.17731

−

1.25707i

−0.757068

−

1.31128i

2.16044

−

3.74200i

−

1.00000i

−3.29674

−

1.90337i

2.87597

1.66044i

−

5.83502i

0.353695

−

0.612617i

−1.25707

−

2.17731i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	845.2.m.h		12
13.b	even	2	1	inner	845.2.m.h		12
13.c	even	3	1		65.2.c.a	✓	6
13.c	even	3	1	inner	845.2.m.h		12
13.d	odd	4	1		845.2.e.i		6
13.d	odd	4	1		845.2.e.k		6
13.e	even	6	1		65.2.c.a	✓	6
13.e	even	6	1	inner	845.2.m.h		12
13.f	odd	12	1		845.2.a.i		3
13.f	odd	12	1		845.2.a.k		3
13.f	odd	12	1		845.2.e.i		6
13.f	odd	12	1		845.2.e.k		6
39.h	odd	6	1		585.2.b.g		6
39.i	odd	6	1		585.2.b.g		6
39.k	even	12	1		7605.2.a.bs		3
39.k	even	12	1		7605.2.a.cc		3
52.i	odd	6	1		1040.2.k.d		6
52.j	odd	6	1		1040.2.k.d		6
65.l	even	6	1		325.2.c.g		6
65.n	even	6	1		325.2.c.g		6
65.q	odd	12	1		325.2.d.e		6
65.q	odd	12	1		325.2.d.f		6
65.r	odd	12	1		325.2.d.e		6
65.r	odd	12	1		325.2.d.f		6
65.s	odd	12	1		4225.2.a.bc		3
65.s	odd	12	1		4225.2.a.be		3

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
65.2.c.a	✓	6	13.c	even	3	1
65.2.c.a	✓	6	13.e	even	6	1
325.2.c.g		6	65.l	even	6	1
325.2.c.g		6	65.n	even	6	1
325.2.d.e		6	65.q	odd	12	1
325.2.d.e		6	65.r	odd	12	1
325.2.d.f		6	65.q	odd	12	1
325.2.d.f		6	65.r	odd	12	1
585.2.b.g		6	39.h	odd	6	1
585.2.b.g		6	39.i	odd	6	1
845.2.a.i		3	13.f	odd	12	1
845.2.a.k		3	13.f	odd	12	1
845.2.e.i		6	13.d	odd	4	1
845.2.e.i		6	13.f	odd	12	1
845.2.e.k		6	13.d	odd	4	1
845.2.e.k		6	13.f	odd	12	1
845.2.m.h		12	1.a	even	1	1	trivial
845.2.m.h		12	13.b	even	2	1	inner
845.2.m.h		12	13.c	even	3	1	inner
845.2.m.h		12	13.e	even	6	1	inner
1040.2.k.d		6	52.i	odd	6	1
1040.2.k.d		6	52.j	odd	6	1
4225.2.a.bc		3	65.s	odd	12	1
4225.2.a.be		3	65.s	odd	12	1
7605.2.a.bs		3	39.k	even	12	1
7605.2.a.cc		3	39.k	even	12	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{12} - 11T_{2}^{10} + 90T_{2}^{8} - 323T_{2}^{6} + 862T_{2}^{4} - 279T_{2}^{2} + 81 $ T2^12 - 11*T2^10 + 90*T2^8 - 323*T2^6 + 862*T2^4 - 279*T2^2 + 81 acting on $S_{2}^{\mathrm{new}}(845, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} - 11 T^{10} + \cdots + 81 $ T^12 - 11T^10 + 90T^8 - 323T^6 + 862T^4 - 279*T^2 + 81
$3$	$ (T^{6} - 2 T^{5} + 8 T^{4} + \cdots + 4)^{2} $ (T^6 - 2T^5 + 8T^4 + 4T^3 + 20T^2 - 8*T + 4)^2
$5$	$ (T^{2} + 1)^{6} $ (T^2 + 1)^6
$7$	$ T^{12} - 20 T^{10} + \cdots + 20736 $ T^12 - 20T^10 + 288T^8 - 1952T^6 + 9664T^4 - 16128*T^2 + 20736
$11$	$ T^{12} - 48 T^{10} + \cdots + 8503056 $ T^12 - 48T^10 + 1620T^8 - 27000T^6 + 327888T^4 - 1994544*T^2 + 8503056
$13$	$ T^{12} $ T^12
$17$	$ (T^{6} + 4 T^{5} + \cdots + 9216)^{2} $ (T^6 + 4T^5 + 48T^4 + 64T^3 + 1408T^2 + 3072*T + 9216)^2
$19$	$ T^{12} - 44 T^{10} + \cdots + 1296 $ T^12 - 44T^10 + 1740T^8 - 8552T^6 + 36832T^4 - 7056*T^2 + 1296
$23$	$ (T^{6} + 8 T^{5} + 48 T^{4} + \cdots + 36)^{2} $ (T^6 + 8T^5 + 48T^4 + 116T^3 + 208T^2 + 96*T + 36)^2
$29$	$ (T^{6} - 2 T^{5} + \cdots + 144)^{2} $ (T^6 - 2T^5 + 12T^4 - 8T^3 + 88T^2 - 96*T + 144)^2
$31$	$ (T^{6} + 56 T^{4} + \cdots + 36)^{2} $ (T^6 + 56T^4 + 604T^2 + 36)^2
$37$	$ T^{12} + \cdots + 2702336256 $ T^12 - 152T^10 + 16848T^8 - 846944T^6 + 31235968T^4 - 325211904*T^2 + 2702336256
$41$	$ T^{12} - 92 T^{10} + \cdots + 26873856 $ T^12 - 92T^10 + 6240T^8 - 194240T^6 + 4469248T^4 - 11529216*T^2 + 26873856
$43$	$ (T^{6} + 12 T^{5} + 132 T^{4} + \cdots + 4)^{2} $ (T^6 + 12T^5 + 132T^4 + 140T^3 + 120T^2 + 24*T + 4)^2
$47$	$ (T^{6} + 20 T^{4} + \cdots + 144)^{2} $ (T^6 + 20T^4 + 112T^2 + 144)^2
$53$	$ (T^{3} - 6 T^{2} - 60 T - 72)^{4} $ (T^3 - 6T^2 - 60T - 72)^4
$59$	$ T^{12} - 84 T^{10} + \cdots + 104976 $ T^12 - 84T^10 + 6588T^8 - 38664T^6 + 191808T^4 - 151632*T^2 + 104976
$61$	$ (T^{6} - 6 T^{5} + \cdots + 5776)^{2} $ (T^6 - 6T^5 + 48T^4 - 80T^3 + 600T^2 - 912*T + 5776)^2
$67$	$ T^{12} + \cdots + 136048896 $ T^12 - 156T^10 + 21168T^8 - 470880T^6 + 8216640T^4 - 36951552*T^2 + 136048896
$71$	$ T^{12} - 44 T^{10} + \cdots + 1296 $ T^12 - 44T^10 + 1740T^8 - 8552T^6 + 36832T^4 - 7056*T^2 + 1296
$73$	$ (T^{6} + 296 T^{4} + \cdots + 266256)^{2} $ (T^6 + 296T^4 + 23344T^2 + 266256)^2
$79$	$ (T^{3} - 12 T^{2} + \cdots + 32)^{4} $ (T^3 - 12T^2 - 48T + 32)^4
$83$	$ (T^{6} + 156 T^{4} + \cdots + 1296)^{2} $ (T^6 + 156T^4 + 5760T^2 + 1296)^2
$89$	$ T^{12} + \cdots + 6879707136 $ T^12 - 192T^10 + 27648T^8 - 1603584T^6 + 69009408T^4 - 764411904*T^2 + 6879707136
$97$	$ T^{12} - 140 T^{10} + \cdots + 331776 $ T^12 - 140T^10 + 18144T^8 - 202688T^6 + 2039296T^4 - 838656*T^2 + 331776
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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:	\( N \)	\(=\)	\( 845 = 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	845.m (of order \(6\), degree \(2\), not minimal)

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

Introduction

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

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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	845.2.m.h

Level	$845$
Weight	$2$
Character orbit	845.m
Analytic conductor	$6.747$
Analytic rank	$0$
Dimension	$12$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(6.74735897080\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36 \) x^12 - 16x^8 - 24x^7 + 96x^5 + 304x^4 + 384x^3 + 288x^2 + 144*x + 36
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{2}\cdot 3^{2} \)
Twist minimal:	no (minimal twist has level 65)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( ( 473 \nu^{11} - 516 \nu^{10} + 1118 \nu^{9} - 3014 \nu^{8} - 4564 \nu^{7} - 5676 \nu^{6} + \cdots - 246168 ) / 103944 \) (473v^11 - 516v^10 + 1118v^9 - 3014v^8 - 4564v^7 - 5676v^6 - 2470v^5 + 66640v^4 + 113100v^3 + 96048v^2 + 54216*v - 246168) / 103944
\(\beta_{2}\)	\(=\)	\( ( 962 \nu^{11} - 2392 \nu^{10} + 2887 \nu^{9} - 2220 \nu^{8} - 13990 \nu^{7} + 14442 \nu^{6} + \cdots + 15984 ) / 103944 \) (962v^11 - 2392v^10 + 2887v^9 - 2220v^8 - 13990v^7 + 14442v^6 + 11380v^5 + 57708v^4 + 98118v^3 - 132080v^2 + 49284*v + 15984) / 103944
\(\beta_{3}\)	\(=\)	\( ( 748 \nu^{11} - 816 \nu^{10} + 1768 \nu^{9} - 4162 \nu^{8} - 6311 \nu^{7} - 8976 \nu^{6} + \cdots - 98004 ) / 51972 \) (748v^11 - 816v^10 + 1768v^9 - 4162v^8 - 6311v^7 - 8976v^6 - 7532v^5 + 93902v^4 + 164352v^3 + 141012v^2 + 80298*v - 98004) / 51972
\(\beta_{4}\)	\(=\)	\( ( 962 \nu^{11} - 2392 \nu^{10} + 2887 \nu^{9} - 2220 \nu^{8} - 13990 \nu^{7} + 14442 \nu^{6} + \cdots + 15984 ) / 34648 \) (962v^11 - 2392v^10 + 2887v^9 - 2220v^8 - 13990v^7 + 14442v^6 + 11380v^5 + 57708v^4 + 98118v^3 - 97432v^2 + 49284*v + 15984) / 34648
\(\beta_{5}\)	\(=\)	\( ( - 52 \nu^{11} + 126 \nu^{10} - 161 \nu^{9} + 120 \nu^{8} + 776 \nu^{7} - 718 \nu^{6} - 536 \nu^{5} + \cdots - 864 ) / 1464 \) (-52v^11 + 126v^10 - 161v^9 + 120v^8 + 776v^7 - 718v^6 - 536v^5 - 3060v^4 - 5274v^3 + 2576v^2 - 2664*v - 864) / 1464
\(\beta_{6}\)	\(=\)	\( ( 5790 \nu^{11} - 1824 \nu^{10} + 473 \nu^{9} - 516 \nu^{8} - 91522 \nu^{7} - 112790 \nu^{6} + \cdots + 404496 ) / 103944 \) (5790v^11 - 1824v^10 + 473v^9 - 516v^8 - 91522v^7 - 112790v^6 + 39212v^5 + 550164v^4 + 1582586v^3 + 1735504v^2 + 1080204*v + 404496) / 103944
\(\beta_{7}\)	\(=\)	\( ( - 118 \nu^{11} + 90 \nu^{10} - 53 \nu^{9} + 32 \nu^{8} + 1866 \nu^{7} + 1416 \nu^{6} - 1364 \nu^{5} + \cdots - 5172 ) / 1704 \) (-118v^11 + 90v^10 - 53v^9 + 32v^8 + 1866v^7 + 1416v^6 - 1364v^5 - 10408v^4 - 27758v^3 - 22776v^2 - 12468*v - 5172) / 1704
\(\beta_{8}\)	\(=\)	\( ( - 14137 \nu^{11} + 3520 \nu^{10} - 77 \nu^{9} + 84 \nu^{8} + 226010 \nu^{7} + 282250 \nu^{6} + \cdots - 1015848 ) / 103944 \) (-14137v^11 + 3520v^10 - 77v^9 + 84v^8 + 226010v^7 + 282250v^6 - 85550v^5 - 1356228v^4 - 3952074v^3 - 4346412v^2 - 2709180*v - 1015848) / 103944
\(\beta_{9}\)	\(=\)	\( ( - 7882 \nu^{11} + 5055 \nu^{10} - 4456 \nu^{9} + 2886 \nu^{8} + 127101 \nu^{7} + 103246 \nu^{6} + \cdots - 498240 ) / 51972 \) (-7882v^11 + 5055v^10 - 4456v^9 + 2886v^8 + 127101v^7 + 103246v^6 - 43336v^5 - 703200v^4 - 1980128v^3 - 1894964v^2 - 1337298*v - 498240) / 51972
\(\beta_{10}\)	\(=\)	\( ( 18668 \nu^{11} - 14130 \nu^{10} + 7753 \nu^{9} - 5308 \nu^{8} - 292962 \nu^{7} - 224016 \nu^{6} + \cdots + 541068 ) / 103944 \) (18668v^11 - 14130v^10 + 7753v^9 - 5308v^8 - 292962v^7 - 224016v^6 + 204208v^5 + 1716416v^4 + 4321654v^3 + 3548568v^2 + 1943844*v + 541068) / 103944
\(\beta_{11}\)	\(=\)	\( ( - 18707 \nu^{11} + 14250 \nu^{10} - 8297 \nu^{9} + 5114 \nu^{8} + 295450 \nu^{7} + 224484 \nu^{6} + \cdots - 548532 ) / 103944 \) (-18707v^11 + 14250v^10 - 8297v^9 + 5114v^8 + 295450v^7 + 224484v^6 - 214310v^5 - 1678984v^4 - 4388958v^3 - 3601656v^2 - 1971828*v - 548532) / 103944

\(\nu\)	\(=\)	\( ( 2 \beta_{11} + \beta_{10} + \beta_{9} - 2 \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} + 4 \beta_{4} + \cdots + 1 ) / 6 \) (2b11 + b10 + b9 - 2b8 - b7 - b6 + b5 + 4b4 - 2b3 - 2b2 + 4b1 + 1) / 6
\(\nu^{2}\)	\(=\)	\( \beta_{4} - 3\beta_{2} \) b4 - 3*b2
\(\nu^{3}\)	\(=\)	\( ( - 8 \beta_{11} - 4 \beta_{10} - 2 \beta_{9} + 4 \beta_{8} + 10 \beta_{7} + 5 \beta_{6} + 4 \beta_{5} + \cdots + 5 ) / 3 \) (-8b11 - 4b10 - 2b9 + 4b8 + 10b7 + 5b6 + 4b5 + 4b4 + 2b3 - 5b2 - 4*b1 + 5) / 3
\(\nu^{4}\)	\(=\)	\( -6\beta_{11} - \beta_{10} + 13\beta_{7} + 13 \) -6b11 - b10 + 13b7 + 13
\(\nu^{5}\)	\(=\)	\( ( - 19 \beta_{11} - 8 \beta_{10} + 16 \beta_{9} - 38 \beta_{8} + 29 \beta_{7} - 58 \beta_{6} - 8 \beta_{5} + \cdots + 58 ) / 3 \) (-19b11 - 8b10 + 16b9 - 38b8 + 29b7 - 58b6 - 8b5 + 19b4 - 8b3 - 29b2 + 19*b1 + 58) / 3
\(\nu^{6}\)	\(=\)	\( 8\beta_{9} - 32\beta_{8} - 62\beta_{6} + 32\beta_{4} - 62\beta_{2} \) 8b9 - 32b8 - 62b6 + 32b4 - 62*b2
\(\nu^{7}\)	\(=\)	\( ( - 94 \beta_{11} - 35 \beta_{10} + 35 \beta_{9} - 94 \beta_{8} + 155 \beta_{7} - 155 \beta_{6} + \cdots - 155 ) / 3 \) (-94b11 - 35b10 + 35b9 - 94b8 + 155b7 - 155b6 + 35b5 + 188b4 + 70b3 - 310b2 - 188*b1 - 155) / 3
\(\nu^{8}\)	\(=\)	\( -166\beta_{11} - 48\beta_{10} + 306\beta_{7} + 48\beta_{3} - 166\beta_1 \) -166b11 - 48b10 + 306b7 + 48b3 - 166*b1
\(\nu^{9}\)	\(=\)	\( ( - 944 \beta_{11} - 328 \beta_{10} + 164 \beta_{9} - 472 \beta_{8} + 1612 \beta_{7} - 806 \beta_{6} + \cdots + 806 ) / 3 \) (-944b11 - 328b10 + 164b9 - 472b8 + 1612b7 - 806b6 - 328b5 - 472b4 + 164b3 + 806b2 - 472*b1 + 806) / 3
\(\nu^{10}\)	\(=\)	\( 262\beta_{9} - 852\beta_{8} - 1534\beta_{6} - 262\beta_{5} \) 262b9 - 852b8 - 1534b6 - 262b5
\(\nu^{11}\)	\(=\)	\( ( 2386 \beta_{11} + 800 \beta_{10} + 1600 \beta_{9} - 4772 \beta_{8} - 4142 \beta_{7} - 8284 \beta_{6} + \cdots - 8284 ) / 3 \) (2386b11 + 800b10 + 1600b9 - 4772b8 - 4142b7 - 8284b6 - 800b5 + 2386b4 + 800b3 - 4142b2 - 2386*b1 - 8284) / 3

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 316.6
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} - 11 T^{10} + \cdots + 81 \) T^12 - 11T^10 + 90T^8 - 323T^6 + 862T^4 - 279*T^2 + 81
$3$	\( (T^{6} - 2 T^{5} + 8 T^{4} + \cdots + 4)^{2} \) (T^6 - 2T^5 + 8T^4 + 4T^3 + 20T^2 - 8*T + 4)^2
$5$	\( (T^{2} + 1)^{6} \) (T^2 + 1)^6
$7$	\( T^{12} - 20 T^{10} + \cdots + 20736 \) T^12 - 20T^10 + 288T^8 - 1952T^6 + 9664T^4 - 16128*T^2 + 20736
$11$	\( T^{12} - 48 T^{10} + \cdots + 8503056 \) T^12 - 48T^10 + 1620T^8 - 27000T^6 + 327888T^4 - 1994544*T^2 + 8503056
$13$	\( T^{12} \) T^12
$17$	\( (T^{6} + 4 T^{5} + \cdots + 9216)^{2} \) (T^6 + 4T^5 + 48T^4 + 64T^3 + 1408T^2 + 3072*T + 9216)^2
$19$	\( T^{12} - 44 T^{10} + \cdots + 1296 \) T^12 - 44T^10 + 1740T^8 - 8552T^6 + 36832T^4 - 7056*T^2 + 1296
$23$	\( (T^{6} + 8 T^{5} + 48 T^{4} + \cdots + 36)^{2} \) (T^6 + 8T^5 + 48T^4 + 116T^3 + 208T^2 + 96*T + 36)^2
$29$	\( (T^{6} - 2 T^{5} + \cdots + 144)^{2} \) (T^6 - 2T^5 + 12T^4 - 8T^3 + 88T^2 - 96*T + 144)^2
$31$	\( (T^{6} + 56 T^{4} + \cdots + 36)^{2} \) (T^6 + 56T^4 + 604T^2 + 36)^2
$37$	\( T^{12} + \cdots + 2702336256 \) T^12 - 152T^10 + 16848T^8 - 846944T^6 + 31235968T^4 - 325211904*T^2 + 2702336256
$41$	\( T^{12} - 92 T^{10} + \cdots + 26873856 \) T^12 - 92T^10 + 6240T^8 - 194240T^6 + 4469248T^4 - 11529216*T^2 + 26873856
$43$	\( (T^{6} + 12 T^{5} + 132 T^{4} + \cdots + 4)^{2} \) (T^6 + 12T^5 + 132T^4 + 140T^3 + 120T^2 + 24*T + 4)^2
$47$	\( (T^{6} + 20 T^{4} + \cdots + 144)^{2} \) (T^6 + 20T^4 + 112T^2 + 144)^2
$53$	\( (T^{3} - 6 T^{2} - 60 T - 72)^{4} \) (T^3 - 6T^2 - 60T - 72)^4
$59$	\( T^{12} - 84 T^{10} + \cdots + 104976 \) T^12 - 84T^10 + 6588T^8 - 38664T^6 + 191808T^4 - 151632*T^2 + 104976
$61$	\( (T^{6} - 6 T^{5} + \cdots + 5776)^{2} \) (T^6 - 6T^5 + 48T^4 - 80T^3 + 600T^2 - 912*T + 5776)^2
$67$	\( T^{12} + \cdots + 136048896 \) T^12 - 156T^10 + 21168T^8 - 470880T^6 + 8216640T^4 - 36951552*T^2 + 136048896
$71$	\( T^{12} - 44 T^{10} + \cdots + 1296 \) T^12 - 44T^10 + 1740T^8 - 8552T^6 + 36832T^4 - 7056*T^2 + 1296
$73$	\( (T^{6} + 296 T^{4} + \cdots + 266256)^{2} \) (T^6 + 296T^4 + 23344T^2 + 266256)^2
$79$	\( (T^{3} - 12 T^{2} + \cdots + 32)^{4} \) (T^3 - 12T^2 - 48T + 32)^4
$83$	\( (T^{6} + 156 T^{4} + \cdots + 1296)^{2} \) (T^6 + 156T^4 + 5760T^2 + 1296)^2
$89$	\( T^{12} + \cdots + 6879707136 \) T^12 - 192T^10 + 27648T^8 - 1603584T^6 + 69009408T^4 - 764411904*T^2 + 6879707136
$97$	\( T^{12} - 140 T^{10} + \cdots + 331776 \) T^12 - 140T^10 + 18144T^8 - 202688T^6 + 2039296T^4 - 838656*T^2 + 331776
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\(n\)	\(171\)	\(677\)
\(\chi(n)\)	\(-\beta_{7}\)	\(1\)