LMFDB - Newform orbit 675.2.k.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [675,2,Mod(199,675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(675, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 3]))

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

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

chi := DirichletCharacter("675.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	675.k (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:	$5.38990213644$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
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} - 12x^{14} + 102x^{12} - 406x^{10} + 1167x^{8} - 1842x^{6} + 2023x^{4} - 441x^{2} + 81 $ x^16 - 12x^14 + 102x^12 - 406x^10 + 1167x^8 - 1842x^6 + 2023x^4 - 441*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 225)
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 $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_1 q^{2} + (\beta_{11} + \beta_{7} + \beta_{3} + 1) q^{4} - \beta_{2} q^{7} + ( - \beta_{15} + \beta_{14} + \cdots + \beta_1) q^{8}+O(q^{10}) $ q + b1 * q^2 + (b11 + b7 + b3 + 1) * q^4 - b2 * q^7 + (-b15 + b14 - b13 - b9 - b8 + b4 - b2 + b1) * q^8	$ q + \beta_1 q^{2} + (\beta_{11} + \beta_{7} + \beta_{3} + 1) q^{4} - \beta_{2} q^{7} + ( - \beta_{15} + \beta_{14} + \cdots + \beta_1) q^{8}+ \cdots + (3 \beta_{15} - 6 \beta_{13} + \cdots - 4 \beta_1) q^{98}+O(q^{100}) $ q + b1 * q^2 + (b11 + b7 + b3 + 1) * q^4 - b2 * q^7 + (-b15 + b14 - b13 - b9 - b8 + b4 - b2 + b1) * q^8 + (b10 - 2b7 - b6 + 2b5 - 1) * q^11 + (2b14 + b9) q^13 + (b12 + b11 + b10 + b7 + 2b6 + b5 + b3 + 1) q^14 + (2b10 + b6 + 3b3 + 1) * q^16 + (-b15 - 2b14 + 2b13 + 2b8 - 2b4 - b2) * q^17 + (-2b11 + b5 - 2) q^19 + (b15 + b14 + 2b9) q^22 + (-b15 + b14 + 5b13 - b9) q^23 + (-2b12 - 3b11 - b5 + 1) * q^26 + (2b14 + b13 - 2b9 + b8 + 2b4 + 2b1) * q^28 + (-b10 - b7 - 2b6 + b5 - 2b3 - 2) * q^29 + (2b12 + b11 + 2b10 - 2b7 - b6 + 2b5 + b3 + 1) * q^31 + (3b14 - 2b9) * q^32 + (-b10 + b7 - b5 - 5b3) q^34 + (-b15 + b14 + 3b13 - 4b9 + 3b8 + b4 - b2 + 4b1) * q^37 + (5b8 - 2b4 - b2) * q^38 + (-b12 + b11 - b10 - b7 + b6 - b5 + b3 + 1) * q^41 + (b8 - 2b4 - 2b2 - 2b1) q^43 + (2b12 + b11 + b5 + 6) q^44 + (-2b12 + b11 - 4b5 + 2) * q^46 + (-5b8 - b4 - 2b2) * q^47 + (-b12 + b11 - b10 - 2b7 - b5 + b3 + 1) q^49 + (3b8 - b4 + b2 - b1) q^52 + (-3b14 - 6b13 - b9 - 6b8 - 3b4 + b1) * q^53 + (-6b6 + 3b3 - 6) * q^56 + (2b15 - 3b14 + b13 + 3b9) q^58 + (-3b12 - b11 - 3b10 + 3b6 - 3b5 - b3 - 1) * q^59 + (-b10 - 2b7 + b6 + 2b5 - 2b3 + 1) q^61 + (3b14 + 3b4) * q^62 + (b12 - 2b11 - b5 - 5) q^64 + (-b15 + 3b13 + 3b9) * q^67 + (2b15 - 2b14 + 5b13 + 4b9) * q^68 + (b11 - 3b5 + 4) q^71 + (-2b15 - 3b14 + b13 - 3b9 + b8 - 3b4 - 2b2 + 3b1) * q^73 + (2b10 + 5b7 - 5b6 - 5b5 + 4b3 - 5) q^74 + (-b12 - 4b11 - b10 - b7 - b5 - 4b3 - 4) * q^76 + (-2b15 - 4b14 - 5b13 + b9) q^77 + (b10 - 3b7 - 3b6 + 3b5 - 3) q^79 + (2b15 - 5b13 + 3b9 - 5b8 + 2b2 - 3b1) * q^82 + (9b8 + 3b1) * q^83 + (-5b11 + 6b6 - 5b3 - 5) q^86 + (b8 + 5b4 + b2 + 2b1) * q^88 + (-3b12 - 3b11 - 3b5 - 3) q^89 + (3b12 - b11 + 2b5 - 5) * q^91 + (2b8 + b4 + 2b2 - 2b1) q^92 + (b12 + 5b11 + b10 + 2b7 + 3b6 + b5 + 5b3 + 5) * q^94 + (-5b8 + b4 - b2 - 2b1) * q^97 + (3b15 - 6b13 + 4b9 - 6b8 + 3b2 - 4b1) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{4}+O(q^{10}) $ 16 * q + 8 * q^4	$ 16 q + 8 q^{4} - 2 q^{11} - 6 q^{14} - 8 q^{16} - 8 q^{19} + 40 q^{26} - 2 q^{29} + 8 q^{31} + 18 q^{34} - 10 q^{41} + 88 q^{44} - 6 q^{49} - 60 q^{56} - 34 q^{59} + 26 q^{61} - 76 q^{64} + 32 q^{71} - 80 q^{74} - 22 q^{76} - 14 q^{79} - 68 q^{86} - 36 q^{89} - 68 q^{91} + 6 q^{94}+O(q^{100}) $ 16 * q + 8 * q^4 - 2 * q^11 - 6 * q^14 - 8 * q^16 - 8 * q^19 + 40 * q^26 - 2 * q^29 + 8 * q^31 + 18 * q^34 - 10 * q^41 + 88 * q^44 - 6 * q^49 - 60 * q^56 - 34 * q^59 + 26 * q^61 - 76 * q^64 + 32 * q^71 - 80 * q^74 - 22 * q^76 - 14 * q^79 - 68 * q^86 - 36 * q^89 - 68 * q^91 + 6 * q^94

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 12x^{14} + 102x^{12} - 406x^{10} + 1167x^{8} - 1842x^{6} + 2023x^{4} - 441x^{2} + 81 $ x^16 - 12*x^14 + 102*x^12 - 406*x^10 + 1167*x^8 - 1842*x^6 + 2023*x^4 - 441*x^2 + 81 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( - 28 \nu^{15} + 943 \nu^{13} - 2470 \nu^{11} + 8108 \nu^{9} + 208132 \nu^{7} + 15890 \nu^{5} + \cdots + 4310244 \nu ) / 795285 $ (-28v^15 + 943v^13 - 2470v^11 + 8108v^9 + 208132v^7 + 15890v^5 - 3474v^3 + 4310244v) / 795285
$\beta_{3}$	$=$	$ ( 26\nu^{14} - 74\nu^{12} - 10\nu^{10} + 10439\nu^{8} - 38576\nu^{6} + 94895\nu^{4} - 82467\nu^{2} + 18063 ) / 29025 $ (26v^14 - 74v^12 - 10v^10 + 10439v^8 - 38576v^6 + 94895v^4 - 82467*v^2 + 18063) / 29025
$\beta_{4}$	$=$	$ ( 1540 \nu^{15} - 16519 \nu^{13} + 135850 \nu^{11} - 445940 \nu^{9} + 1153589 \nu^{7} + \cdots + 2158308 \nu ) / 795285 $ (1540v^15 - 16519v^13 + 135850v^11 - 445940v^9 + 1153589v^7 - 873950v^5 + 191070v^3 + 2158308v) / 795285
$\beta_{5}$	$=$	$ ( 266 \nu^{14} - 2999 \nu^{12} + 23465 \nu^{10} - 77026 \nu^{8} + 148849 \nu^{6} - 150955 \nu^{4} + \cdots - 126387 ) / 92475 $ (266v^14 - 2999v^12 + 23465v^10 - 77026v^8 + 148849v^6 - 150955v^4 + 33003*v^2 - 126387) / 92475
$\beta_{6}$	$=$	$ ( - 11836 \nu^{14} + 138574 \nu^{12} - 1170340 \nu^{10} + 4500371 \nu^{8} - 12811274 \nu^{6} + \cdots + 814212 ) / 3976425 $ (-11836v^14 + 138574v^12 - 1170340v^10 + 4500371v^8 - 12811274v^6 + 19088030v^4 - 21981813*v^2 + 814212) / 3976425
$\beta_{7}$	$=$	$ ( - 12338 \nu^{14} + 121397 \nu^{12} - 1025270 \nu^{10} + 3257143 \nu^{8} - 11223247 \nu^{6} + \cdots + 713286 ) / 3976425 $ (-12338v^14 + 121397v^12 - 1025270v^10 + 3257143v^8 - 11223247v^6 + 16721965v^4 - 26841054*v^2 + 713286) / 3976425
$\beta_{8}$	$=$	$ ( 3766 \nu^{15} - 41414 \nu^{13} + 332215 \nu^{11} - 1090526 \nu^{9} + 2480389 \nu^{7} + \cdots + 2721243 \nu ) / 1325475 $ (3766v^15 - 41414v^13 + 332215v^11 - 1090526v^9 + 2480389v^7 - 2137205v^5 + 467253v^3 + 2721243v) / 1325475
$\beta_{9}$	$=$	$ ( - 11836 \nu^{15} + 138574 \nu^{13} - 1170340 \nu^{11} + 4500371 \nu^{9} - 12811274 \nu^{7} + \cdots + 4790637 \nu ) / 3976425 $ (-11836v^15 + 138574v^13 - 1170340v^11 + 4500371v^9 - 12811274v^7 + 19088030v^5 - 21981813v^3 + 4790637v) / 3976425
$\beta_{10}$	$=$	$ ( 13873 \nu^{14} - 220432 \nu^{12} + 2012770 \nu^{10} - 10392128 \nu^{8} + 31813907 \nu^{6} + \cdots - 12812391 ) / 3976425 $ (13873v^14 - 220432v^12 + 2012770v^10 - 10392128v^8 + 31813907v^6 - 59345540v^4 + 58685109*v^2 - 12812391) / 3976425
$\beta_{11}$	$=$	$ ( - 14896 \nu^{14} + 165889 \nu^{12} - 1314040 \nu^{10} + 4313456 \nu^{8} - 9114389 \nu^{6} + \cdots - 5535918 ) / 3976425 $ (-14896v^14 + 165889v^12 - 1314040v^10 + 4313456v^8 - 9114389v^6 + 8453480v^4 - 1848168*v^2 - 5535918) / 3976425
$\beta_{12}$	$=$	$ ( 18998 \nu^{14} - 206837 \nu^{12} + 1675895 \nu^{10} - 5501278 \nu^{8} + 13209112 \nu^{6} + \cdots + 14978844 ) / 3976425 $ (18998v^14 - 206837v^12 + 1675895v^10 - 5501278v^8 + 13209112v^6 - 10781365v^4 + 2357109*v^2 + 14978844) / 3976425
$\beta_{13}$	$=$	$ ( 19253 \nu^{15} - 245932 \nu^{13} + 2129695 \nu^{11} - 9130758 \nu^{9} + 26781707 \nu^{7} + \cdots - 10338741 \nu ) / 3976425 $ (19253v^15 - 245932v^13 + 2129695v^11 - 9130758v^9 + 26781707v^7 - 44578415v^5 + 47402299v^3 - 10338741v) / 3976425
$\beta_{14}$	$=$	$ ( 30232 \nu^{15} - 363428 \nu^{13} + 3087680 \nu^{11} - 12331002 \nu^{9} + 35467228 \nu^{7} + \cdots - 13412214 \nu ) / 3976425 $ (30232v^15 - 363428v^13 + 3087680v^11 - 12331002v^9 + 35467228v^7 - 57068185v^5 + 61524806v^3 - 13412214v) / 3976425
$\beta_{15}$	$=$	$ ( 66701 \nu^{15} - 773434 \nu^{13} + 6504640 \nu^{11} - 24700761 \nu^{9} + 70028009 \nu^{7} + \cdots - 26067942 \nu ) / 3976425 $ (66701v^15 - 773434v^13 + 6504640v^11 - 24700761v^9 + 70028009v^7 - 105967505v^5 + 119626108v^3 - 26067942v) / 3976425

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{11} + \beta_{7} - 2\beta_{6} + \beta_{3} + 1 $ b11 + b7 - 2*b6 + b3 + 1
$\nu^{3}$	$=$	$ -\beta_{15} + \beta_{14} - \beta_{13} - 5\beta_{9} - \beta_{8} + \beta_{4} - \beta_{2} + 5\beta_1 $ -b15 + b14 - b13 - 5b9 - b8 + b4 - b2 + 5b1
$\nu^{4}$	$=$	$ 2\beta_{10} + 6\beta_{7} - 7\beta_{6} - 6\beta_{5} + 9\beta_{3} - 7 $ 2b10 + 6b7 - 7b6 - 6b5 + 9*b3 - 7
$\nu^{5}$	$=$	$ -8\beta_{15} + 11\beta_{14} - 8\beta_{13} - 30\beta_{9} $ -8b15 + 11b14 - 8b13 - 30b9
$\nu^{6}$	$=$	$ -19\beta_{12} - 68\beta_{11} - 57\beta_{5} - 101 $ -19b12 - 68b11 - 57*b5 - 101
$\nu^{7}$	$=$	$ 60\beta_{8} - 87\beta_{4} + 57\beta_{2} - 196\beta_1 $ 60b8 - 87b4 + 57b2 - 196b1
$\nu^{8}$	$=$	$ -144\beta_{12} - 487\beta_{11} - 144\beta_{10} - 253\beta_{7} + 191\beta_{6} - 144\beta_{5} - 487\beta_{3} - 487 $ -144b12 - 487b11 - 144b10 - 253b7 + 191b6 - 144b5 - 487*b3 - 487
$\nu^{9}$	$=$	$ 397 \beta_{15} - 631 \beta_{14} + 433 \beta_{13} + 1328 \beta_{9} + 433 \beta_{8} - 631 \beta_{4} + \cdots - 1328 \beta_1 $ 397b15 - 631b14 + 433b13 + 1328b9 + 433b8 - 631b4 + 397b2 - 1328b1
$\nu^{10}$	$=$	$ -1028\beta_{10} - 1725\beta_{7} + 1231\beta_{6} + 1725\beta_{5} - 3420\beta_{3} + 1231 $ -1028b10 - 1725b7 + 1231b6 + 1725b5 - 3420*b3 + 1231
$\nu^{11}$	$=$	$ 2753\beta_{15} - 4448\beta_{14} + 3059\beta_{13} + 9129\beta_{9} $ 2753b15 - 4448b14 + 3059b13 + 9129b9
$\nu^{12}$	$=$	$ 7201\beta_{12} + 23837\beta_{11} + 19083\beta_{5} + 32141 $ 7201b12 + 23837b11 + 19083*b5 + 32141
$\nu^{13}$	$=$	$ -21390\beta_{8} + 31038\beta_{4} - 19083\beta_{2} + 63106\beta_1 $ -21390b8 + 31038b4 - 19083b2 + 63106b1
$\nu^{14}$	$=$	$ 50121 \beta_{12} + 165655 \beta_{11} + 50121 \beta_{10} + 82189 \beta_{7} - 57008 \beta_{6} + \cdots + 165655 $ 50121b12 + 165655b11 + 50121b10 + 82189b7 - 57008b6 + 50121b5 + 165655*b3 + 165655
$\nu^{15}$	$=$	$ - 132310 \beta_{15} + 215776 \beta_{14} - 148879 \beta_{13} - 437162 \beta_{9} - 148879 \beta_{8} + \cdots + 437162 \beta_1 $ -132310b15 + 215776b14 - 148879b13 - 437162b9 - 148879b8 + 215776b4 - 132310b2 + 437162b1

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/675\mathbb{Z}\right)^\times$.

$n$	$326$	$352$
$\chi(n)$	$\beta_{6}$	$-1$

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} $

199.1

−2.28087	+	1.31686i
−1.41485	+	0.816862i
−1.27588	+	0.736627i
−0.409850	+	0.236627i
0.409850	−	0.236627i
1.27588	−	0.736627i
1.41485	−	0.816862i
2.28087	−	1.31686i
−2.28087	−	1.31686i
−1.41485	−	0.816862i
−1.27588	−	0.736627i
−0.409850	−	0.236627i
0.409850	+	0.236627i
1.27588	+	0.736627i
1.41485	+	0.816862i
2.28087	+	1.31686i

−2.28087

1.31686i

2.46825

−

4.27513i

−1.55662

0.898714i

7.73393i

199.2

−1.41485

0.816862i

0.334526

−

0.579416i

0.437645

−

0.252674i

−

2.17440i

199.3

−1.27588

0.736627i

0.0852394

−

0.147639i

3.34791

−

1.93291i

−

2.69535i

199.4

−0.409850

0.236627i

−0.888015

1.53809i

2.21967

−

1.28153i

−

1.78702i

199.5

0.409850

−

0.236627i

−0.888015

1.53809i

−2.21967

1.28153i

1.78702i

199.6

1.27588

−

0.736627i

0.0852394

−

0.147639i

−3.34791

1.93291i

2.69535i

199.7

1.41485

−

0.816862i

0.334526

−

0.579416i

−0.437645

0.252674i

2.17440i

199.8

2.28087

−

1.31686i

2.46825

−

4.27513i

1.55662

−

0.898714i

−

7.73393i

424.1

−2.28087

−

1.31686i

2.46825

4.27513i

−1.55662

−

0.898714i

−

7.73393i

424.2

−1.41485

−

0.816862i

0.334526

0.579416i

0.437645

0.252674i

2.17440i

424.3

−1.27588

−

0.736627i

0.0852394

0.147639i

3.34791

1.93291i

2.69535i

424.4

−0.409850

−

0.236627i

−0.888015

−

1.53809i

2.21967

1.28153i

1.78702i

424.5

0.409850

0.236627i

−0.888015

−

1.53809i

−2.21967

−

1.28153i

−

1.78702i

424.6

1.27588

0.736627i

0.0852394

0.147639i

−3.34791

−

1.93291i

−

2.69535i

424.7

1.41485

0.816862i

0.334526

0.579416i

−0.437645

−

0.252674i

−

2.17440i

424.8

2.28087

1.31686i

2.46825

4.27513i

1.55662

0.898714i

7.73393i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
9.c	even	3	1	inner
45.j	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	675.2.k.c		16
3.b	odd	2	1		225.2.k.c		16
5.b	even	2	1	inner	675.2.k.c		16
5.c	odd	4	1		675.2.e.c		8
5.c	odd	4	1		675.2.e.e		8
9.c	even	3	1	inner	675.2.k.c		16
9.c	even	3	1		2025.2.b.o		8
9.d	odd	6	1		225.2.k.c		16
9.d	odd	6	1		2025.2.b.n		8
15.d	odd	2	1		225.2.k.c		16
15.e	even	4	1		225.2.e.c	✓	8
15.e	even	4	1		225.2.e.e	yes	8
45.h	odd	6	1		225.2.k.c		16
45.h	odd	6	1		2025.2.b.n		8
45.j	even	6	1	inner	675.2.k.c		16
45.j	even	6	1		2025.2.b.o		8
45.k	odd	12	1		675.2.e.c		8
45.k	odd	12	1		675.2.e.e		8
45.k	odd	12	1		2025.2.a.p		4
45.k	odd	12	1		2025.2.a.z		4
45.l	even	12	1		225.2.e.c	✓	8
45.l	even	12	1		225.2.e.e	yes	8
45.l	even	12	1		2025.2.a.q		4
45.l	even	12	1		2025.2.a.y		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
225.2.e.c	✓	8	15.e	even	4	1
225.2.e.c	✓	8	45.l	even	12	1
225.2.e.e	yes	8	15.e	even	4	1
225.2.e.e	yes	8	45.l	even	12	1
225.2.k.c		16	3.b	odd	2	1
225.2.k.c		16	9.d	odd	6	1
225.2.k.c		16	15.d	odd	2	1
225.2.k.c		16	45.h	odd	6	1
675.2.e.c		8	5.c	odd	4	1
675.2.e.c		8	45.k	odd	12	1
675.2.e.e		8	5.c	odd	4	1
675.2.e.e		8	45.k	odd	12	1
675.2.k.c		16	1.a	even	1	1	trivial
675.2.k.c		16	5.b	even	2	1	inner
675.2.k.c		16	9.c	even	3	1	inner
675.2.k.c		16	45.j	even	6	1	inner
2025.2.a.p		4	45.k	odd	12	1
2025.2.a.q		4	45.l	even	12	1
2025.2.a.y		4	45.l	even	12	1
2025.2.a.z		4	45.k	odd	12	1
2025.2.b.n		8	9.d	odd	6	1
2025.2.b.n		8	45.h	odd	6	1
2025.2.b.o		8	9.c	even	3	1
2025.2.b.o		8	45.j	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{16} - 12T_{2}^{14} + 102T_{2}^{12} - 406T_{2}^{10} + 1167T_{2}^{8} - 1842T_{2}^{6} + 2023T_{2}^{4} - 441T_{2}^{2} + 81 $ T2^16 - 12*T2^14 + 102*T2^12 - 406*T2^10 + 1167*T2^8 - 1842*T2^6 + 2023*T2^4 - 441*T2^2 + 81 acting on $S_{2}^{\mathrm{new}}(675, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} - 12 T^{14} + \cdots + 81 $ T^16 - 12T^14 + 102T^12 - 406T^10 + 1167T^8 - 1842T^6 + 2023T^4 - 441*T^2 + 81
$3$	$ T^{16} $ T^16
$5$	$ T^{16} $ T^16
$7$	$ T^{16} - 25 T^{14} + \cdots + 6561 $ T^16 - 25T^14 + 451T^12 - 3630T^10 + 21195T^8 - 58590T^6 + 115506T^4 - 29160*T^2 + 6561
$11$	$ (T^{8} + T^{7} + 26 T^{6} + \cdots + 81)^{2} $ (T^8 + T^7 + 26T^6 - 107T^5 + 593T^4 - 1007T^3 + 1456T^2 - 369T + 81)^2
$13$	$ T^{16} + \cdots + 131079601 $ T^16 - 64T^14 + 2962T^12 - 59686T^10 + 862027T^8 - 5843158T^6 + 28554859T^4 - 73788805*T^2 + 131079601
$17$	$ (T^{8} + 81 T^{6} + \cdots + 91809)^{2} $ (T^8 + 81T^6 + 2214T^4 + 24220*T^2 + 91809)^2
$19$	$ (T^{4} + 2 T^{3} - 27 T^{2} + \cdots - 25)^{4} $ (T^4 + 2T^3 - 27T^2 - 80*T - 25)^4
$23$	$ T^{16} + \cdots + 3486784401 $ T^16 - 111T^14 + 9288T^12 - 281097T^10 + 6056127T^8 - 71156961T^6 + 592799472T^4 - 1640558367*T^2 + 3486784401
$29$	$ (T^{8} + T^{7} + \cdots + 16641)^{2} $ (T^8 + T^7 + 41T^6 + 244T^5 + 1871T^4 + 5938T^3 + 15004T^2 + 18318T + 16641)^2
$31$	$ (T^{8} - 4 T^{7} + \cdots + 59049)^{2} $ (T^8 - 4T^7 + 58T^6 + 114T^5 + 1629T^4 + 810T^3 + 10935T^2 + 6561*T + 59049)^2
$37$	$ (T^{8} + 199 T^{6} + \cdots + 418609)^{2} $ (T^8 + 199T^6 + 9513T^4 + 124903*T^2 + 418609)^2
$41$	$ (T^{8} + 5 T^{7} + \cdots + 42849)^{2} $ (T^8 + 5T^7 + 50T^6 + 197T^5 + 1637T^4 + 6095T^3 + 20746T^2 + 33327*T + 42849)^2
$43$	$ T^{16} + \cdots + 205144679041 $ T^16 - 196T^14 + 25738T^12 - 1927312T^10 + 105636307T^8 - 3356926096T^6 + 71980515082T^4 - 126271170052*T^2 + 205144679041
$47$	$ T^{16} + \cdots + 21071715921 $ T^16 - 186T^14 + 26709T^12 - 1294114T^10 + 45982884T^8 - 627705066T^6 + 6325951549T^4 - 12546845874*T^2 + 21071715921
$53$	$ (T^{8} + 228 T^{6} + \cdots + 221841)^{2} $ (T^8 + 228T^6 + 13614T^4 + 113053*T^2 + 221841)^2
$59$	$ (T^{8} + 17 T^{7} + \cdots + 5349969)^{2} $ (T^8 + 17T^7 + 287T^6 + 1862T^5 + 17855T^4 + 76814T^3 + 840022T^2 + 2114082*T + 5349969)^2
$61$	$ (T^{8} - 13 T^{7} + 172 T^{6} + \cdots + 1)^{2} $ (T^8 - 13T^7 + 172T^6 - 143T^5 + 1193T^4 - 299T^3 + 8278T^2 - 91*T + 1)^2
$67$	$ T^{16} + \cdots + 3486784401 $ T^16 - 217T^14 + 35107T^12 - 2455086T^10 + 127775907T^8 - 843115662T^6 + 4549304898T^4 - 4281288696*T^2 + 3486784401
$71$	$ (T^{4} - 8 T^{3} + \cdots + 381)^{4} $ (T^4 - 8T^3 - 40T^2 + 263*T + 381)^4
$73$	$ (T^{8} + 196 T^{6} + \cdots + 12769)^{2} $ (T^8 + 196T^6 + 8478T^4 + 79777*T^2 + 12769)^2
$79$	$ (T^{8} + 7 T^{7} + \cdots + 42849)^{2} $ (T^8 + 7T^7 + 82T^6 - 93T^5 + 1365T^4 - 621T^3 + 11592T^2 - 14283*T + 42849)^2
$83$	$ T^{16} + \cdots + 282429536481 $ T^16 - 324T^14 + 77274T^12 - 8029206T^10 + 613578159T^8 - 12762024174T^6 + 209121502059T^4 - 251435897361*T^2 + 282429536481
$89$	$ (T^{4} + 9 T^{3} + \cdots + 2025)^{4} $ (T^4 + 9T^3 - 99T^2 - 405*T + 2025)^4
$97$	$ T^{16} + \cdots + 824843587681 $ T^16 - 199T^14 + 28828T^12 - 1783213T^10 + 79268227T^8 - 1580980129T^6 + 22726478692T^4 - 163756440163*T^2 + 824843587681
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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:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	675.k (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_1 q^{2} + (\beta_{11} + \beta_{7} + \beta_{3} + 1) q^{4} - \beta_{2} q^{7} + ( - \beta_{15} + \beta_{14} + \cdots + \beta_1) q^{8}+O(q^{10}) \) q + b1 * q^2 + (b11 + b7 + b3 + 1) * q^4 - b2 * q^7 + (-b15 + b14 - b13 - b9 - b8 + b4 - b2 + b1) * q^8	\( q + \beta_1 q^{2} + (\beta_{11} + \beta_{7} + \beta_{3} + 1) q^{4} - \beta_{2} q^{7} + ( - \beta_{15} + \beta_{14} + \cdots + \beta_1) q^{8}+ \cdots + (3 \beta_{15} - 6 \beta_{13} + \cdots - 4 \beta_1) q^{98}+O(q^{100}) \) q + b1 * q^2 + (b11 + b7 + b3 + 1) * q^4 - b2 * q^7 + (-b15 + b14 - b13 - b9 - b8 + b4 - b2 + b1) * q^8 + (b10 - 2b7 - b6 + 2b5 - 1) * q^11 + (2b14 + b9) q^13 + (b12 + b11 + b10 + b7 + 2b6 + b5 + b3 + 1) q^14 + (2b10 + b6 + 3b3 + 1) * q^16 + (-b15 - 2b14 + 2b13 + 2b8 - 2b4 - b2) * q^17 + (-2b11 + b5 - 2) q^19 + (b15 + b14 + 2b9) q^22 + (-b15 + b14 + 5b13 - b9) q^23 + (-2b12 - 3b11 - b5 + 1) * q^26 + (2b14 + b13 - 2b9 + b8 + 2b4 + 2b1) * q^28 + (-b10 - b7 - 2b6 + b5 - 2b3 - 2) * q^29 + (2b12 + b11 + 2b10 - 2b7 - b6 + 2b5 + b3 + 1) * q^31 + (3b14 - 2b9) * q^32 + (-b10 + b7 - b5 - 5b3) q^34 + (-b15 + b14 + 3b13 - 4b9 + 3b8 + b4 - b2 + 4b1) * q^37 + (5b8 - 2b4 - b2) * q^38 + (-b12 + b11 - b10 - b7 + b6 - b5 + b3 + 1) * q^41 + (b8 - 2b4 - 2b2 - 2b1) q^43 + (2b12 + b11 + b5 + 6) q^44 + (-2b12 + b11 - 4b5 + 2) * q^46 + (-5b8 - b4 - 2b2) * q^47 + (-b12 + b11 - b10 - 2b7 - b5 + b3 + 1) q^49 + (3b8 - b4 + b2 - b1) q^52 + (-3b14 - 6b13 - b9 - 6b8 - 3b4 + b1) * q^53 + (-6b6 + 3b3 - 6) * q^56 + (2b15 - 3b14 + b13 + 3b9) q^58 + (-3b12 - b11 - 3b10 + 3b6 - 3b5 - b3 - 1) * q^59 + (-b10 - 2b7 + b6 + 2b5 - 2b3 + 1) q^61 + (3b14 + 3b4) * q^62 + (b12 - 2b11 - b5 - 5) q^64 + (-b15 + 3b13 + 3b9) * q^67 + (2b15 - 2b14 + 5b13 + 4b9) * q^68 + (b11 - 3b5 + 4) q^71 + (-2b15 - 3b14 + b13 - 3b9 + b8 - 3b4 - 2b2 + 3b1) * q^73 + (2b10 + 5b7 - 5b6 - 5b5 + 4b3 - 5) q^74 + (-b12 - 4b11 - b10 - b7 - b5 - 4b3 - 4) * q^76 + (-2b15 - 4b14 - 5b13 + b9) q^77 + (b10 - 3b7 - 3b6 + 3b5 - 3) q^79 + (2b15 - 5b13 + 3b9 - 5b8 + 2b2 - 3b1) * q^82 + (9b8 + 3b1) * q^83 + (-5b11 + 6b6 - 5b3 - 5) q^86 + (b8 + 5b4 + b2 + 2b1) * q^88 + (-3b12 - 3b11 - 3b5 - 3) q^89 + (3b12 - b11 + 2b5 - 5) * q^91 + (2b8 + b4 + 2b2 - 2b1) q^92 + (b12 + 5b11 + b10 + 2b7 + 3b6 + b5 + 5b3 + 5) * q^94 + (-5b8 + b4 - b2 - 2b1) * q^97 + (3b15 - 6b13 + 4b9 - 6b8 + 3b2 - 4b1) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{4}+O(q^{10}) \) 16 * q + 8 * q^4	\( 16 q + 8 q^{4} - 2 q^{11} - 6 q^{14} - 8 q^{16} - 8 q^{19} + 40 q^{26} - 2 q^{29} + 8 q^{31} + 18 q^{34} - 10 q^{41} + 88 q^{44} - 6 q^{49} - 60 q^{56} - 34 q^{59} + 26 q^{61} - 76 q^{64} + 32 q^{71} - 80 q^{74} - 22 q^{76} - 14 q^{79} - 68 q^{86} - 36 q^{89} - 68 q^{91} + 6 q^{94}+O(q^{100}) \) 16 * q + 8 * q^4 - 2 * q^11 - 6 * q^14 - 8 * q^16 - 8 * q^19 + 40 * q^26 - 2 * q^29 + 8 * q^31 + 18 * q^34 - 10 * q^41 + 88 * q^44 - 6 * q^49 - 60 * q^56 - 34 * q^59 + 26 * q^61 - 76 * q^64 + 32 * q^71 - 80 * q^74 - 22 * q^76 - 14 * q^79 - 68 * q^86 - 36 * q^89 - 68 * q^91 + 6 * q^94

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	675.2.k.c

Level	$675$
Weight	$2$
Character orbit	675.k
Analytic conductor	$5.390$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(5.38990213644\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 12x^{14} + 102x^{12} - 406x^{10} + 1167x^{8} - 1842x^{6} + 2023x^{4} - 441x^{2} + 81 \) x^16 - 12x^14 + 102x^12 - 406x^10 + 1167x^8 - 1842x^6 + 2023x^4 - 441*x^2 + 81
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 3^{2} \)
Twist minimal:	no (minimal twist has level 225)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( - 28 \nu^{15} + 943 \nu^{13} - 2470 \nu^{11} + 8108 \nu^{9} + 208132 \nu^{7} + 15890 \nu^{5} + \cdots + 4310244 \nu ) / 795285 \) (-28v^15 + 943v^13 - 2470v^11 + 8108v^9 + 208132v^7 + 15890v^5 - 3474v^3 + 4310244v) / 795285
\(\beta_{3}\)	\(=\)	\( ( 26\nu^{14} - 74\nu^{12} - 10\nu^{10} + 10439\nu^{8} - 38576\nu^{6} + 94895\nu^{4} - 82467\nu^{2} + 18063 ) / 29025 \) (26v^14 - 74v^12 - 10v^10 + 10439v^8 - 38576v^6 + 94895v^4 - 82467*v^2 + 18063) / 29025
\(\beta_{4}\)	\(=\)	\( ( 1540 \nu^{15} - 16519 \nu^{13} + 135850 \nu^{11} - 445940 \nu^{9} + 1153589 \nu^{7} + \cdots + 2158308 \nu ) / 795285 \) (1540v^15 - 16519v^13 + 135850v^11 - 445940v^9 + 1153589v^7 - 873950v^5 + 191070v^3 + 2158308v) / 795285
\(\beta_{5}\)	\(=\)	\( ( 266 \nu^{14} - 2999 \nu^{12} + 23465 \nu^{10} - 77026 \nu^{8} + 148849 \nu^{6} - 150955 \nu^{4} + \cdots - 126387 ) / 92475 \) (266v^14 - 2999v^12 + 23465v^10 - 77026v^8 + 148849v^6 - 150955v^4 + 33003*v^2 - 126387) / 92475
\(\beta_{6}\)	\(=\)	\( ( - 11836 \nu^{14} + 138574 \nu^{12} - 1170340 \nu^{10} + 4500371 \nu^{8} - 12811274 \nu^{6} + \cdots + 814212 ) / 3976425 \) (-11836v^14 + 138574v^12 - 1170340v^10 + 4500371v^8 - 12811274v^6 + 19088030v^4 - 21981813*v^2 + 814212) / 3976425
\(\beta_{7}\)	\(=\)	\( ( - 12338 \nu^{14} + 121397 \nu^{12} - 1025270 \nu^{10} + 3257143 \nu^{8} - 11223247 \nu^{6} + \cdots + 713286 ) / 3976425 \) (-12338v^14 + 121397v^12 - 1025270v^10 + 3257143v^8 - 11223247v^6 + 16721965v^4 - 26841054*v^2 + 713286) / 3976425
\(\beta_{8}\)	\(=\)	\( ( 3766 \nu^{15} - 41414 \nu^{13} + 332215 \nu^{11} - 1090526 \nu^{9} + 2480389 \nu^{7} + \cdots + 2721243 \nu ) / 1325475 \) (3766v^15 - 41414v^13 + 332215v^11 - 1090526v^9 + 2480389v^7 - 2137205v^5 + 467253v^3 + 2721243v) / 1325475
\(\beta_{9}\)	\(=\)	\( ( - 11836 \nu^{15} + 138574 \nu^{13} - 1170340 \nu^{11} + 4500371 \nu^{9} - 12811274 \nu^{7} + \cdots + 4790637 \nu ) / 3976425 \) (-11836v^15 + 138574v^13 - 1170340v^11 + 4500371v^9 - 12811274v^7 + 19088030v^5 - 21981813v^3 + 4790637v) / 3976425
\(\beta_{10}\)	\(=\)	\( ( 13873 \nu^{14} - 220432 \nu^{12} + 2012770 \nu^{10} - 10392128 \nu^{8} + 31813907 \nu^{6} + \cdots - 12812391 ) / 3976425 \) (13873v^14 - 220432v^12 + 2012770v^10 - 10392128v^8 + 31813907v^6 - 59345540v^4 + 58685109*v^2 - 12812391) / 3976425
\(\beta_{11}\)	\(=\)	\( ( - 14896 \nu^{14} + 165889 \nu^{12} - 1314040 \nu^{10} + 4313456 \nu^{8} - 9114389 \nu^{6} + \cdots - 5535918 ) / 3976425 \) (-14896v^14 + 165889v^12 - 1314040v^10 + 4313456v^8 - 9114389v^6 + 8453480v^4 - 1848168*v^2 - 5535918) / 3976425
\(\beta_{12}\)	\(=\)	\( ( 18998 \nu^{14} - 206837 \nu^{12} + 1675895 \nu^{10} - 5501278 \nu^{8} + 13209112 \nu^{6} + \cdots + 14978844 ) / 3976425 \) (18998v^14 - 206837v^12 + 1675895v^10 - 5501278v^8 + 13209112v^6 - 10781365v^4 + 2357109*v^2 + 14978844) / 3976425
\(\beta_{13}\)	\(=\)	\( ( 19253 \nu^{15} - 245932 \nu^{13} + 2129695 \nu^{11} - 9130758 \nu^{9} + 26781707 \nu^{7} + \cdots - 10338741 \nu ) / 3976425 \) (19253v^15 - 245932v^13 + 2129695v^11 - 9130758v^9 + 26781707v^7 - 44578415v^5 + 47402299v^3 - 10338741v) / 3976425
\(\beta_{14}\)	\(=\)	\( ( 30232 \nu^{15} - 363428 \nu^{13} + 3087680 \nu^{11} - 12331002 \nu^{9} + 35467228 \nu^{7} + \cdots - 13412214 \nu ) / 3976425 \) (30232v^15 - 363428v^13 + 3087680v^11 - 12331002v^9 + 35467228v^7 - 57068185v^5 + 61524806v^3 - 13412214v) / 3976425
\(\beta_{15}\)	\(=\)	\( ( 66701 \nu^{15} - 773434 \nu^{13} + 6504640 \nu^{11} - 24700761 \nu^{9} + 70028009 \nu^{7} + \cdots - 26067942 \nu ) / 3976425 \) (66701v^15 - 773434v^13 + 6504640v^11 - 24700761v^9 + 70028009v^7 - 105967505v^5 + 119626108v^3 - 26067942v) / 3976425

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{11} + \beta_{7} - 2\beta_{6} + \beta_{3} + 1 \) b11 + b7 - 2*b6 + b3 + 1
\(\nu^{3}\)	\(=\)	\( -\beta_{15} + \beta_{14} - \beta_{13} - 5\beta_{9} - \beta_{8} + \beta_{4} - \beta_{2} + 5\beta_1 \) -b15 + b14 - b13 - 5b9 - b8 + b4 - b2 + 5b1
\(\nu^{4}\)	\(=\)	\( 2\beta_{10} + 6\beta_{7} - 7\beta_{6} - 6\beta_{5} + 9\beta_{3} - 7 \) 2b10 + 6b7 - 7b6 - 6b5 + 9*b3 - 7
\(\nu^{5}\)	\(=\)	\( -8\beta_{15} + 11\beta_{14} - 8\beta_{13} - 30\beta_{9} \) -8b15 + 11b14 - 8b13 - 30b9
\(\nu^{6}\)	\(=\)	\( -19\beta_{12} - 68\beta_{11} - 57\beta_{5} - 101 \) -19b12 - 68b11 - 57*b5 - 101
\(\nu^{7}\)	\(=\)	\( 60\beta_{8} - 87\beta_{4} + 57\beta_{2} - 196\beta_1 \) 60b8 - 87b4 + 57b2 - 196b1
\(\nu^{8}\)	\(=\)	\( -144\beta_{12} - 487\beta_{11} - 144\beta_{10} - 253\beta_{7} + 191\beta_{6} - 144\beta_{5} - 487\beta_{3} - 487 \) -144b12 - 487b11 - 144b10 - 253b7 + 191b6 - 144b5 - 487*b3 - 487
\(\nu^{9}\)	\(=\)	\( 397 \beta_{15} - 631 \beta_{14} + 433 \beta_{13} + 1328 \beta_{9} + 433 \beta_{8} - 631 \beta_{4} + \cdots - 1328 \beta_1 \) 397b15 - 631b14 + 433b13 + 1328b9 + 433b8 - 631b4 + 397b2 - 1328b1
\(\nu^{10}\)	\(=\)	\( -1028\beta_{10} - 1725\beta_{7} + 1231\beta_{6} + 1725\beta_{5} - 3420\beta_{3} + 1231 \) -1028b10 - 1725b7 + 1231b6 + 1725b5 - 3420*b3 + 1231
\(\nu^{11}\)	\(=\)	\( 2753\beta_{15} - 4448\beta_{14} + 3059\beta_{13} + 9129\beta_{9} \) 2753b15 - 4448b14 + 3059b13 + 9129b9
\(\nu^{12}\)	\(=\)	\( 7201\beta_{12} + 23837\beta_{11} + 19083\beta_{5} + 32141 \) 7201b12 + 23837b11 + 19083*b5 + 32141
\(\nu^{13}\)	\(=\)	\( -21390\beta_{8} + 31038\beta_{4} - 19083\beta_{2} + 63106\beta_1 \) -21390b8 + 31038b4 - 19083b2 + 63106b1
\(\nu^{14}\)	\(=\)	\( 50121 \beta_{12} + 165655 \beta_{11} + 50121 \beta_{10} + 82189 \beta_{7} - 57008 \beta_{6} + \cdots + 165655 \) 50121b12 + 165655b11 + 50121b10 + 82189b7 - 57008b6 + 50121b5 + 165655*b3 + 165655
\(\nu^{15}\)	\(=\)	\( - 132310 \beta_{15} + 215776 \beta_{14} - 148879 \beta_{13} - 437162 \beta_{9} - 148879 \beta_{8} + \cdots + 437162 \beta_1 \) -132310b15 + 215776b14 - 148879b13 - 437162b9 - 148879b8 + 215776b4 - 132310b2 + 437162b1

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 199.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} - 12 T^{14} + \cdots + 81 \) T^16 - 12T^14 + 102T^12 - 406T^10 + 1167T^8 - 1842T^6 + 2023T^4 - 441*T^2 + 81
$3$	\( T^{16} \) T^16
$5$	\( T^{16} \) T^16
$7$	\( T^{16} - 25 T^{14} + \cdots + 6561 \) T^16 - 25T^14 + 451T^12 - 3630T^10 + 21195T^8 - 58590T^6 + 115506T^4 - 29160*T^2 + 6561
$11$	\( (T^{8} + T^{7} + 26 T^{6} + \cdots + 81)^{2} \) (T^8 + T^7 + 26T^6 - 107T^5 + 593T^4 - 1007T^3 + 1456T^2 - 369T + 81)^2
$13$	\( T^{16} + \cdots + 131079601 \) T^16 - 64T^14 + 2962T^12 - 59686T^10 + 862027T^8 - 5843158T^6 + 28554859T^4 - 73788805*T^2 + 131079601
$17$	\( (T^{8} + 81 T^{6} + \cdots + 91809)^{2} \) (T^8 + 81T^6 + 2214T^4 + 24220*T^2 + 91809)^2
$19$	\( (T^{4} + 2 T^{3} - 27 T^{2} + \cdots - 25)^{4} \) (T^4 + 2T^3 - 27T^2 - 80*T - 25)^4
$23$	\( T^{16} + \cdots + 3486784401 \) T^16 - 111T^14 + 9288T^12 - 281097T^10 + 6056127T^8 - 71156961T^6 + 592799472T^4 - 1640558367*T^2 + 3486784401
$29$	\( (T^{8} + T^{7} + \cdots + 16641)^{2} \) (T^8 + T^7 + 41T^6 + 244T^5 + 1871T^4 + 5938T^3 + 15004T^2 + 18318T + 16641)^2
$31$	\( (T^{8} - 4 T^{7} + \cdots + 59049)^{2} \) (T^8 - 4T^7 + 58T^6 + 114T^5 + 1629T^4 + 810T^3 + 10935T^2 + 6561*T + 59049)^2
$37$	\( (T^{8} + 199 T^{6} + \cdots + 418609)^{2} \) (T^8 + 199T^6 + 9513T^4 + 124903*T^2 + 418609)^2
$41$	\( (T^{8} + 5 T^{7} + \cdots + 42849)^{2} \) (T^8 + 5T^7 + 50T^6 + 197T^5 + 1637T^4 + 6095T^3 + 20746T^2 + 33327*T + 42849)^2
$43$	\( T^{16} + \cdots + 205144679041 \) T^16 - 196T^14 + 25738T^12 - 1927312T^10 + 105636307T^8 - 3356926096T^6 + 71980515082T^4 - 126271170052*T^2 + 205144679041
$47$	\( T^{16} + \cdots + 21071715921 \) T^16 - 186T^14 + 26709T^12 - 1294114T^10 + 45982884T^8 - 627705066T^6 + 6325951549T^4 - 12546845874*T^2 + 21071715921
$53$	\( (T^{8} + 228 T^{6} + \cdots + 221841)^{2} \) (T^8 + 228T^6 + 13614T^4 + 113053*T^2 + 221841)^2
$59$	\( (T^{8} + 17 T^{7} + \cdots + 5349969)^{2} \) (T^8 + 17T^7 + 287T^6 + 1862T^5 + 17855T^4 + 76814T^3 + 840022T^2 + 2114082*T + 5349969)^2
$61$	\( (T^{8} - 13 T^{7} + 172 T^{6} + \cdots + 1)^{2} \) (T^8 - 13T^7 + 172T^6 - 143T^5 + 1193T^4 - 299T^3 + 8278T^2 - 91*T + 1)^2
$67$	\( T^{16} + \cdots + 3486784401 \) T^16 - 217T^14 + 35107T^12 - 2455086T^10 + 127775907T^8 - 843115662T^6 + 4549304898T^4 - 4281288696*T^2 + 3486784401
$71$	\( (T^{4} - 8 T^{3} + \cdots + 381)^{4} \) (T^4 - 8T^3 - 40T^2 + 263*T + 381)^4
$73$	\( (T^{8} + 196 T^{6} + \cdots + 12769)^{2} \) (T^8 + 196T^6 + 8478T^4 + 79777*T^2 + 12769)^2
$79$	\( (T^{8} + 7 T^{7} + \cdots + 42849)^{2} \) (T^8 + 7T^7 + 82T^6 - 93T^5 + 1365T^4 - 621T^3 + 11592T^2 - 14283*T + 42849)^2
$83$	\( T^{16} + \cdots + 282429536481 \) T^16 - 324T^14 + 77274T^12 - 8029206T^10 + 613578159T^8 - 12762024174T^6 + 209121502059T^4 - 251435897361*T^2 + 282429536481
$89$	\( (T^{4} + 9 T^{3} + \cdots + 2025)^{4} \) (T^4 + 9T^3 - 99T^2 - 405*T + 2025)^4
$97$	\( T^{16} + \cdots + 824843587681 \) T^16 - 199T^14 + 28828T^12 - 1783213T^10 + 79268227T^8 - 1580980129T^6 + 22726478692T^4 - 163756440163*T^2 + 824843587681
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\(n\)	\(326\)	\(352\)
\(\chi(n)\)	\(\beta_{6}\)	\(-1\)