LMFDB - Newform orbit 2300.2.i.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("2300.1057");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2300 = 2^{2} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2300.i (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$18.3655924649$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 18x^{14} + 146x^{12} - 798x^{10} + 3934x^{8} - 19950x^{6} + 91250x^{4} - 281250x^{2} + 390625 $ x^16 - 18x^14 + 146x^12 - 798x^10 + 3934x^8 - 19950x^6 + 91250x^4 - 281250*x^2 + 390625
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 460)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_{5} q^{3} + ( - \beta_{15} - \beta_{12} + \beta_{10}) q^{7} + (\beta_{11} - 3 \beta_{7} + \cdots + \beta_{3}) q^{9}+O(q^{10}) $ q + b5 * q^3 + (-b15 - b12 + b10) * q^7 + (b11 - 3b7 + b5 + b3) q^9	$ q + \beta_{5} q^{3} + ( - \beta_{15} - \beta_{12} + \beta_{10}) q^{7} + (\beta_{11} - 3 \beta_{7} + \cdots + \beta_{3}) q^{9}+ \cdots + (\beta_{15} - 2 \beta_{14} + \cdots + \beta_1) q^{99}+O(q^{100}) $ q + b5 * q^3 + (-b15 - b12 + b10) * q^7 + (b11 - 3b7 + b5 + b3) q^9 + (b15 - b13) * q^11 + (b7 - b5 - b2 - 1) * q^13 + (b10 - b6 + b4) * q^17 + (-b14 - b12 + b10 - b6) * q^19 + (2b14 + b12 + 3b4 - b1) * q^21 + (-b14 + b7 - b6 - b5 - b2 + b1 - 1) * q^23 + (b9 - 4b7 + 3b3 - 4) * q^27 + (-b11 - 2b7) q^29 + (-b5 + b3 - 1) * q^31 + (-b14 + b13 - b12 + b10 - b4 - b1) * q^33 + (b15 + b12 - b10 + b6 + b1) * q^37 + (-b9 + 3b7 - b5 - b3 + b2) q^39 + (-b9 - b2 - 3) * q^41 + (-2b14 - b13 - 2b12 + 2b10 - 2b4 - 2b1) q^43 + (-b11 - b9 + b8 - b3) * q^47 + (b9 - 3b7 + b5 + b3 - b2) q^49 + (-b15 + 3b14 + b13 + 3b4) * q^51 + (2b13 - b12 + b10 - b4 - b1) q^53 + (-b15 - 2b12 + 3b10 - 3b6 + 2b4 - b1) * q^57 + (-b11 - b9 + b7 + b5 + b3 + b2) * q^59 + (-b15 + b13 - 2b12 - 2b4 - 3b1) q^61 + (8b14 + 5b12 - 5b10 + 5b6 + 5b4) q^63 + (b15 - 2b10 + 3b6 - b4 + 2b1) q^67 + (-2b14 - 2b12 + 4b10 - b9 + 3b7 - b6 - b5 - b3 + b2) * q^69 + (b8 - b5 + b3) * q^71 + (-3b7 - b5 - 2b2 + 3) * q^73 + (b11 - b8 + 4b7 + 4) q^77 + (2b15 + 2b14 + 2b13 + 4b12 - 5b10 - b6 + 2b4 + 2b1) q^79 + (-b9 - b8 - 5b5 + 5b3 - b2 - 12) * q^81 + (b14 - 2b13 - 2b12 + 2b10 - b6 - 2b4 - b1) * q^83 + (b11 - b9 - b8 - 2b7 + b3 - 2) q^87 + (-2b15 + b14 - 2b13 - b12 + b10 + b6 - 2b4 - 2b1) * q^89 + (-b14 - 4b12 - 5b4 - b1) * q^91 + (-b11 - b8 + 6b7 - 3b5 - 6) * q^93 + (b15 - 2b10 - b6 - b4 - 2b1) * q^97 + (b15 - 2b14 + b13 - b12 - 2b10 - 6b6 + b4 + b1) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{3}+O(q^{10}) $ 16 * q - 4 * q^3	$ 16 q - 4 q^{3} - 12 q^{13} - 12 q^{23} - 52 q^{27} - 8 q^{31} - 48 q^{41} - 4 q^{47} + 8 q^{71} + 52 q^{73} + 64 q^{77} - 152 q^{81} - 28 q^{87} - 84 q^{93}+O(q^{100}) $ 16 * q - 4 * q^3 - 12 * q^13 - 12 * q^23 - 52 * q^27 - 8 * q^31 - 48 * q^41 - 4 * q^47 + 8 * q^71 + 52 * q^73 + 64 * q^77 - 152 * q^81 - 28 * q^87 - 84 * q^93

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 18x^{14} + 146x^{12} - 798x^{10} + 3934x^{8} - 19950x^{6} + 91250x^{4} - 281250x^{2} + 390625 $ x^16 - 18*x^14 + 146*x^12 - 798*x^10 + 3934*x^8 - 19950*x^6 + 91250*x^4 - 281250*x^2 + 390625 :

$\beta_{1}$	$=$	$ ( -\nu^{15} + 18\nu^{13} - 146\nu^{11} + 798\nu^{9} - 3934\nu^{7} + 19950\nu^{5} - 91250\nu^{3} + 203125\nu ) / 78125 $ (-v^15 + 18v^13 - 146v^11 + 798v^9 - 3934v^7 + 19950v^5 - 91250v^3 + 203125*v) / 78125
$\beta_{2}$	$=$	$ ( 9539 \nu^{14} - 106427 \nu^{12} + 622119 \nu^{10} - 2938847 \nu^{8} + 15928851 \nu^{6} + \cdots - 434828125 ) / 56781250 $ (9539v^14 - 106427v^12 + 622119v^10 - 2938847v^8 + 15928851v^6 - 75874325v^4 + 260211875*v^2 - 434828125) / 56781250
$\beta_{3}$	$=$	$ ( - 9679 \nu^{14} + 155072 \nu^{12} - 1012184 \nu^{10} + 4868567 \nu^{8} - 24161111 \nu^{6} + \cdots + 1164515625 ) / 56781250 $ (-9679v^14 + 155072v^12 - 1012184v^10 + 4868567v^8 - 24161111v^6 + 120434950v^4 - 535191250*v^2 + 1164515625) / 56781250
$\beta_{4}$	$=$	$ ( 1968 \nu^{15} - 42104 \nu^{13} + 324443 \nu^{11} - 1621369 \nu^{9} + 7917752 \nu^{7} + \cdots - 517009375 \nu ) / 56781250 $ (1968v^15 - 42104v^13 + 324443v^11 - 1621369v^9 + 7917752v^7 - 38606970v^5 + 186172875v^3 - 517009375v) / 56781250
$\beta_{5}$	$=$	$ ( 13152 \nu^{14} - 165036 \nu^{12} + 932717 \nu^{10} - 4530221 \nu^{8} + 23029618 \nu^{6} + \cdots - 766046875 ) / 56781250 $ (13152v^14 - 165036v^12 + 932717v^10 - 4530221v^8 + 23029618v^6 - 120333350v^4 + 471120625*v^2 - 766046875) / 56781250
$\beta_{6}$	$=$	$ ( 10911 \nu^{15} - 146673 \nu^{13} + 903581 \nu^{11} - 4695253 \nu^{9} + 22842699 \nu^{7} + \cdots - 851015625 \nu ) / 283906250 $ (10911v^15 - 146673v^13 + 903581v^11 - 4695253v^9 + 22842699v^7 - 104910675v^5 + 427434375v^3 - 851015625v) / 283906250
$\beta_{7}$	$=$	$ ( - 14766 \nu^{14} + 182013 \nu^{12} - 1107886 \nu^{10} + 5441493 \nu^{8} - 26865744 \nu^{6} + \cdots + 983109375 ) / 56781250 $ (-14766v^14 + 182013v^12 - 1107886v^10 + 5441493v^8 - 26865744v^6 + 137700225v^4 - 545257500*v^2 + 983109375) / 56781250
$\beta_{8}$	$=$	$ ( 67 \nu^{14} - 976 \nu^{12} + 6267 \nu^{10} - 31136 \nu^{8} + 155663 \nu^{6} - 793080 \nu^{4} + \cdots - 6756250 ) / 143750 $ (67v^14 - 976v^12 + 6267v^10 - 31136v^8 + 155663v^6 - 793080v^4 + 3243375*v^2 - 6756250) / 143750
$\beta_{9}$	$=$	$ ( - 37979 \nu^{14} + 527497 \nu^{12} - 3336559 \nu^{10} + 15924867 \nu^{8} - 79211011 \nu^{6} + \cdots + 3273890625 ) / 56781250 $ (-37979v^14 + 527497v^12 - 3336559v^10 + 15924867v^8 - 79211011v^6 + 411359675v^4 - 1676681875*v^2 + 3273890625) / 56781250
$\beta_{10}$	$=$	$ ( - 39617 \nu^{15} + 550556 \nu^{13} - 3408182 \nu^{11} + 16828941 \nu^{9} + \cdots + 3718671875 \nu ) / 283906250 $ (-39617v^15 + 550556v^13 - 3408182v^11 + 16828941v^9 - 84516503v^7 + 433599950v^5 - 1809035000v^3 + 3718671875v) / 283906250
$\beta_{11}$	$=$	$ ( - 69391 \nu^{14} + 891313 \nu^{12} - 5517661 \nu^{10} + 26866793 \nu^{8} - 135170269 \nu^{6} + \cdots + 5315421875 ) / 56781250 $ (-69391v^14 + 891313v^12 - 5517661v^10 + 26866793v^8 - 135170269v^6 + 697330925v^4 - 2806464375*v^2 + 5315421875) / 56781250
$\beta_{12}$	$=$	$ ( 62919 \nu^{15} - 763392 \nu^{13} + 4635849 \nu^{11} - 22512212 \nu^{9} + 111486021 \nu^{7} + \cdots - 3780625000 \nu ) / 283906250 $ (62919v^15 - 763392v^13 + 4635849v^11 - 22512212v^9 + 111486021v^7 - 583590450v^5 + 2298853125v^3 - 3780625000v) / 283906250
$\beta_{13}$	$=$	$ ( 64304 \nu^{15} - 864372 \nu^{13} + 5421959 \nu^{11} - 26293867 \nu^{9} + 132465636 \nu^{7} + \cdots - 5099359375 \nu ) / 283906250 $ (64304v^15 - 864372v^13 + 5421959v^11 - 26293867v^9 + 132465636v^7 - 680065650v^5 + 2682835625v^3 - 5099359375v) / 283906250
$\beta_{14}$	$=$	$ ( - 77464 \nu^{15} + 975477 \nu^{13} - 6069994 \nu^{11} + 30107397 \nu^{9} + \cdots + 5937609375 \nu ) / 283906250 $ (-77464v^15 + 975477v^13 - 6069994v^11 + 30107397v^9 - 148624876v^7 + 760999425v^5 - 3057890000v^3 + 5937609375v) / 283906250
$\beta_{15}$	$=$	$ ( - 93696 \nu^{15} + 1179753 \nu^{13} - 7247666 \nu^{11} + 35247133 \nu^{9} + \cdots + 6903703125 \nu ) / 283906250 $ (-93696v^15 + 1179753v^13 - 7247666v^11 + 35247133v^9 - 177964864v^7 + 910905475v^5 - 3668715000v^3 + 6903703125v) / 283906250

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

$\nu$	$=$	$ ( \beta_{14} + \beta_{12} + \beta_{6} - \beta_1 ) / 2 $ (b14 + b12 + b6 - b1) / 2
$\nu^{2}$	$=$	$ ( -\beta_{11} + \beta_{9} - \beta_{8} + \beta_{7} - 2\beta_{3} - \beta_{2} + 5 ) / 2 $ (-b11 + b9 - b8 + b7 - 2*b3 - b2 + 5) / 2
$\nu^{3}$	$=$	$ ( -2\beta_{15} + 5\beta_{14} - 4\beta_{13} + 5\beta_{12} - 2\beta_{10} + 3\beta_{6} + 2\beta_{4} - 3\beta_1 ) / 2 $ (-2b15 + 5b14 - 4b13 + 5b12 - 2b10 + 3b6 + 2b4 - 3b1) / 2
$\nu^{4}$	$=$	$ 3\beta_{9} - 2\beta_{8} - 5\beta_{7} - \beta_{5} - 9\beta_{3} + 2\beta_{2} + 6 $ 3b9 - 2b8 - 5b7 - b5 - 9b3 + 2*b2 + 6
$\nu^{5}$	$=$	$ ( -18\beta_{15} + 23\beta_{14} - 16\beta_{13} + 13\beta_{12} + 8\beta_{10} + 45\beta_{6} + 16\beta_{4} + 7\beta_1 ) / 2 $ (-18b15 + 23b14 - 16b13 + 13b12 + 8b10 + 45b6 + 16b4 + 7b1) / 2
$\nu^{6}$	$=$	$ ( -19\beta_{11} + 33\beta_{9} + 3\beta_{8} + 75\beta_{7} - 2\beta_{5} - 48\beta_{3} + 55\beta_{2} + 97 ) / 2 $ (-19b11 + 33b9 + 3b8 + 75b7 - 2b5 - 48b3 + 55*b2 + 97) / 2
$\nu^{7}$	$=$	$ ( - 178 \beta_{15} + 113 \beta_{14} + 14 \beta_{13} - 119 \beta_{12} + 74 \beta_{10} + 143 \beta_{6} + \cdots - 161 \beta_1 ) / 2 $ (-178b15 + 113b14 + 14b13 - 119b12 + 74b10 + 143b6 - 56b4 - 161b1) / 2
$\nu^{8}$	$=$	$ -74\beta_{11} - 29\beta_{9} - 80\beta_{8} + 368\beta_{7} + 6\beta_{5} - 38\beta_{3} + 91\beta_{2} + 25 $ -74b11 - 29b9 - 80b8 + 368b7 + 6b5 - 38b3 + 91*b2 + 25
$\nu^{9}$	$=$	$ ( - 762 \beta_{15} + 577 \beta_{14} + 126 \beta_{13} - 259 \beta_{12} + 378 \beta_{10} - 401 \beta_{6} + \cdots + 125 \beta_1 ) / 2 $ (-762b15 + 577b14 + 126b13 - 259b12 + 378b10 - 401b6 + 132b4 + 125b1) / 2
$\nu^{10}$	$=$	$ ( -81\beta_{11} - 933\beta_{9} - 801\beta_{8} + 409\beta_{7} - 1488\beta_{5} - 6\beta_{3} + 597\beta_{2} + 1269 ) / 2 $ (-81b11 - 933b9 - 801b8 + 409b7 - 1488b5 - 6b3 + 597*b2 + 1269) / 2
$\nu^{11}$	$=$	$ ( - 1224 \beta_{15} + 1397 \beta_{14} + 942 \beta_{13} + 1741 \beta_{12} + 4752 \beta_{10} + \cdots + 3319 \beta_1 ) / 2 $ (-1224b15 + 1397b14 + 942b13 + 1741b12 + 4752b10 - 1163b6 + 3702b4 + 3319b1) / 2
$\nu^{12}$	$=$	$ -1856\beta_{11} + 302\beta_{9} - 814\beta_{8} + 773\beta_{7} - 5507\beta_{5} + 1173\beta_{3} - 61\beta_{2} + 5870 $ -1856b11 + 302b9 - 814b8 + 773b7 - 5507b5 + 1173b3 - 61*b2 + 5870
$\nu^{13}$	$=$	$ ( - 9412 \beta_{15} - 4029 \beta_{14} - 874 \beta_{13} - 2827 \beta_{12} + 24278 \beta_{10} + \cdots - 17499 \beta_1 ) / 2 $ (-9412b15 - 4029b14 - 874b13 - 2827b12 + 24278b10 - 9829b6 + 4632b4 - 17499b1) / 2
$\nu^{14}$	$=$	$ ( - 22723 \beta_{11} + 15059 \beta_{9} - 24765 \beta_{8} - 14733 \beta_{7} - 34710 \beta_{5} + \cdots - 48127 ) / 2 $ (-22723b11 + 15059b9 - 24765b8 - 14733b7 - 34710b5 - 5312b3 - 5067*b2 - 48127) / 2
$\nu^{15}$	$=$	$ ( - 75136 \beta_{15} - 54855 \beta_{14} - 61992 \beta_{13} - 37383 \beta_{12} + 95840 \beta_{10} + \cdots - 12407 \beta_1 ) / 2 $ (-75136b15 - 54855b14 - 61992b13 - 37383b12 + 95840b10 - 62559b6 + 5224b4 - 12407b1) / 2

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

Character values

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

$n$	$277$	$1151$	$1201$
$\chi(n)$	$\beta_{7}$	$1$	$-1$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

1057.1

−1.06856	−	1.96422i
1.06856	+	1.96422i
1.88618	−	1.20098i
−1.88618	+	1.20098i
−2.23266	+	0.123447i
2.23266	−	0.123447i
2.19448	+	0.429243i
−2.19448	−	0.429243i
−1.06856	+	1.96422i
1.06856	−	1.96422i
1.88618	+	1.20098i
−1.88618	−	1.20098i
−2.23266	−	0.123447i
2.23266	+	0.123447i
2.19448	−	0.429243i
−2.19448	+	0.429243i

−2.03584

2.03584i

−0.641104

0.641104i

−

5.28931i

1057.2

−2.03584

2.03584i

0.641104

−

0.641104i

−

5.28931i

1057.3

−1.09396

1.09396i

−2.67082

2.67082i

0.606488i

1057.4

−1.09396

1.09396i

2.67082

−

2.67082i

0.606488i

1057.5

−0.237125

0.237125i

−1.69421

1.69421i

2.88754i

1057.6

−0.237125

0.237125i

1.69421

−

1.69421i

2.88754i

1057.7

2.36693

−

2.36693i

−2.93008

2.93008i

−

8.20472i

1057.8

2.36693

−

2.36693i

2.93008

−

2.93008i

−

8.20472i

1793.1

−2.03584

−

2.03584i

−0.641104

−

0.641104i

5.28931i

1793.2

−2.03584

−

2.03584i

0.641104

0.641104i

5.28931i

1793.3

−1.09396

−

1.09396i

−2.67082

−

2.67082i

−

0.606488i

1793.4

−1.09396

−

1.09396i

2.67082

2.67082i

−

0.606488i

1793.5

−0.237125

−

0.237125i

−1.69421

−

1.69421i

−

2.88754i

1793.6

−0.237125

−

0.237125i

1.69421

1.69421i

−

2.88754i

1793.7

2.36693

2.36693i

−2.93008

−

2.93008i

8.20472i

1793.8

2.36693

2.36693i

2.93008

2.93008i

8.20472i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2300.2.i.d		16
5.b	even	2	1		460.2.i.b	✓	16
5.c	odd	4	1		460.2.i.b	✓	16
5.c	odd	4	1	inner	2300.2.i.d		16
23.b	odd	2	1	inner	2300.2.i.d		16
115.c	odd	2	1		460.2.i.b	✓	16
115.e	even	4	1		460.2.i.b	✓	16
115.e	even	4	1	inner	2300.2.i.d		16

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

Hecke kernels

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

$ T_{3}^{8} + 2T_{3}^{7} + 2T_{3}^{6} + 6T_{3}^{5} + 110T_{3}^{4} + 270T_{3}^{3} + 338T_{3}^{2} + 130T_{3} + 25 $

$ T_{19}^{8} - 36T_{19}^{6} + 178T_{19}^{4} - 244T_{19}^{2} + 100 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{8} + 2 T^{7} + 2 T^{6} + \cdots + 25)^{2} $ (T^8 + 2T^7 + 2T^6 + 6T^5 + 110T^4 + 270T^3 + 338T^2 + 130*T + 25)^2
$5$	$ T^{16} $ T^16
$7$	$ T^{16} + 532 T^{12} + \cdots + 1336336 $ T^16 + 532T^12 + 76792T^8 + 2029264*T^4 + 1336336
$11$	$ (T^{8} + 50 T^{6} + \cdots + 2500)^{2} $ (T^8 + 50T^6 + 672T^4 + 2596*T^2 + 2500)^2
$13$	$ (T^{8} + 6 T^{7} + \cdots + 13225)^{2} $ (T^8 + 6T^7 + 18T^6 + 42T^5 + 486T^4 + 2898T^3 + 9522T^2 + 15870*T + 13225)^2
$17$	$ T^{16} + 880 T^{12} + \cdots + 16 $ T^16 + 880T^12 + 71020T^8 + 10288*T^4 + 16
$19$	$ (T^{8} - 36 T^{6} + \cdots + 100)^{2} $ (T^8 - 36T^6 + 178T^4 - 244*T^2 + 100)^2
$23$	$ T^{16} + \cdots + 78310985281 $ T^16 + 12T^15 + 72T^14 + 612T^13 + 4004T^12 + 17940T^11 + 114264T^10 + 590364T^9 + 2322310T^8 + 13578372T^7 + 60445656T^6 + 218275980T^5 + 1120483364T^4 + 3939041916T^3 + 10658584008T^2 + 40857905364*T + 78310985281
$29$	$ (T^{8} + 88 T^{6} + \cdots + 28561)^{2} $ (T^8 + 88T^6 + 2110T^4 + 16680*T^2 + 28561)^2
$31$	$ (T^{4} + 2 T^{3} - 20 T^{2} + \cdots + 11)^{4} $ (T^4 + 2T^3 - 20T^2 + 10*T + 11)^4
$37$	$ T^{16} + 1348 T^{12} + \cdots + 456976 $ T^16 + 1348T^12 + 458296T^8 + 21869584*T^4 + 456976
$41$	$ (T^{4} + 12 T^{3} + \cdots - 107)^{4} $ (T^4 + 12T^3 + 10T^2 - 188*T - 107)^4
$43$	$ T^{16} + \cdots + 127880620816 $ T^16 + 6892T^12 + 11231212T^8 + 3092373184*T^4 + 127880620816
$47$	$ (T^{8} + 2 T^{7} + \cdots + 24025)^{2} $ (T^8 + 2T^7 + 2T^6 - 474T^5 + 6086T^4 - 24162T^3 + 51842T^2 - 49910*T + 24025)^2
$53$	$ T^{16} + \cdots + 533794816 $ T^16 + 54928T^12 + 584919936T^8 + 2911069184*T^4 + 533794816
$59$	$ (T^{8} + 168 T^{6} + \cdots + 110224)^{2} $ (T^8 + 168T^6 + 9244T^4 + 167728*T^2 + 110224)^2
$61$	$ (T^{8} + 224 T^{6} + \cdots + 100)^{2} $ (T^8 + 224T^6 + 5698T^4 + 3044*T^2 + 100)^2
$67$	$ T^{16} + \cdots + 36\!\cdots\!16 $ T^16 + 38308T^12 + 498843384T^8 + 2504414676496*T^4 + 3675002150265616
$71$	$ (T^{4} - 2 T^{3} + \cdots + 169)^{4} $ (T^4 - 2T^3 - 74T^2 - 162*T + 169)^4
$73$	$ (T^{8} - 26 T^{7} + \cdots + 25)^{2} $ (T^8 - 26T^7 + 338T^6 - 1326T^5 + 2294T^4 + 3874T^3 + 3042T^2 + 390*T + 25)^2
$79$	$ (T^{8} - 424 T^{6} + \cdots + 115562500)^{2} $ (T^8 - 424T^6 + 66346T^4 - 4549996*T^2 + 115562500)^2
$83$	$ T^{16} + \cdots + 17799252215056 $ T^16 + 62144T^12 + 527018796T^8 + 1094073550768*T^4 + 17799252215056
$89$	$ (T^{8} - 260 T^{6} + \cdots + 1144900)^{2} $ (T^8 - 260T^6 + 16786T^4 - 257316*T^2 + 1144900)^2
$97$	$ T^{16} + \cdots + 583506543376 $ T^16 + 15908T^12 + 22546936T^8 + 8112583184*T^4 + 583506543376
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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:	\( N \)	\(=\)	\( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2300.i (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{5} q^{3} + ( - \beta_{15} - \beta_{12} + \beta_{10}) q^{7} + (\beta_{11} - 3 \beta_{7} + \cdots + \beta_{3}) q^{9}+O(q^{10}) \) q + b5 * q^3 + (-b15 - b12 + b10) * q^7 + (b11 - 3b7 + b5 + b3) q^9	\( q + \beta_{5} q^{3} + ( - \beta_{15} - \beta_{12} + \beta_{10}) q^{7} + (\beta_{11} - 3 \beta_{7} + \cdots + \beta_{3}) q^{9}+ \cdots + (\beta_{15} - 2 \beta_{14} + \cdots + \beta_1) q^{99}+O(q^{100}) \) q + b5 * q^3 + (-b15 - b12 + b10) * q^7 + (b11 - 3b7 + b5 + b3) q^9 + (b15 - b13) * q^11 + (b7 - b5 - b2 - 1) * q^13 + (b10 - b6 + b4) * q^17 + (-b14 - b12 + b10 - b6) * q^19 + (2b14 + b12 + 3b4 - b1) * q^21 + (-b14 + b7 - b6 - b5 - b2 + b1 - 1) * q^23 + (b9 - 4b7 + 3b3 - 4) * q^27 + (-b11 - 2b7) q^29 + (-b5 + b3 - 1) * q^31 + (-b14 + b13 - b12 + b10 - b4 - b1) * q^33 + (b15 + b12 - b10 + b6 + b1) * q^37 + (-b9 + 3b7 - b5 - b3 + b2) q^39 + (-b9 - b2 - 3) * q^41 + (-2b14 - b13 - 2b12 + 2b10 - 2b4 - 2b1) q^43 + (-b11 - b9 + b8 - b3) * q^47 + (b9 - 3b7 + b5 + b3 - b2) q^49 + (-b15 + 3b14 + b13 + 3b4) * q^51 + (2b13 - b12 + b10 - b4 - b1) q^53 + (-b15 - 2b12 + 3b10 - 3b6 + 2b4 - b1) * q^57 + (-b11 - b9 + b7 + b5 + b3 + b2) * q^59 + (-b15 + b13 - 2b12 - 2b4 - 3b1) q^61 + (8b14 + 5b12 - 5b10 + 5b6 + 5b4) q^63 + (b15 - 2b10 + 3b6 - b4 + 2b1) q^67 + (-2b14 - 2b12 + 4b10 - b9 + 3b7 - b6 - b5 - b3 + b2) * q^69 + (b8 - b5 + b3) * q^71 + (-3b7 - b5 - 2b2 + 3) * q^73 + (b11 - b8 + 4b7 + 4) q^77 + (2b15 + 2b14 + 2b13 + 4b12 - 5b10 - b6 + 2b4 + 2b1) q^79 + (-b9 - b8 - 5b5 + 5b3 - b2 - 12) * q^81 + (b14 - 2b13 - 2b12 + 2b10 - b6 - 2b4 - b1) * q^83 + (b11 - b9 - b8 - 2b7 + b3 - 2) q^87 + (-2b15 + b14 - 2b13 - b12 + b10 + b6 - 2b4 - 2b1) * q^89 + (-b14 - 4b12 - 5b4 - b1) * q^91 + (-b11 - b8 + 6b7 - 3b5 - 6) * q^93 + (b15 - 2b10 - b6 - b4 - 2b1) * q^97 + (b15 - 2b14 + b13 - b12 - 2b10 - 6b6 + b4 + b1) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{3}+O(q^{10}) \) 16 * q - 4 * q^3	\( 16 q - 4 q^{3} - 12 q^{13} - 12 q^{23} - 52 q^{27} - 8 q^{31} - 48 q^{41} - 4 q^{47} + 8 q^{71} + 52 q^{73} + 64 q^{77} - 152 q^{81} - 28 q^{87} - 84 q^{93}+O(q^{100}) \) 16 * q - 4 * q^3 - 12 * q^13 - 12 * q^23 - 52 * q^27 - 8 * q^31 - 48 * q^41 - 4 * q^47 + 8 * q^71 + 52 * q^73 + 64 * q^77 - 152 * q^81 - 28 * q^87 - 84 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

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

Label	2300.2.i.d

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

Self dual:	no
Analytic conductor:	\(18.3655924649\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 18x^{14} + 146x^{12} - 798x^{10} + 3934x^{8} - 19950x^{6} + 91250x^{4} - 281250x^{2} + 390625 \) x^16 - 18x^14 + 146x^12 - 798x^10 + 3934x^8 - 19950x^6 + 91250x^4 - 281250*x^2 + 390625
Coefficient ring:	\(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index:	\( 2^{4} \)
Twist minimal:	no (minimal twist has level 460)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

\(\beta_{1}\)	\(=\)	\( ( -\nu^{15} + 18\nu^{13} - 146\nu^{11} + 798\nu^{9} - 3934\nu^{7} + 19950\nu^{5} - 91250\nu^{3} + 203125\nu ) / 78125 \) (-v^15 + 18v^13 - 146v^11 + 798v^9 - 3934v^7 + 19950v^5 - 91250v^3 + 203125*v) / 78125
\(\beta_{2}\)	\(=\)	\( ( 9539 \nu^{14} - 106427 \nu^{12} + 622119 \nu^{10} - 2938847 \nu^{8} + 15928851 \nu^{6} + \cdots - 434828125 ) / 56781250 \) (9539v^14 - 106427v^12 + 622119v^10 - 2938847v^8 + 15928851v^6 - 75874325v^4 + 260211875*v^2 - 434828125) / 56781250
\(\beta_{3}\)	\(=\)	\( ( - 9679 \nu^{14} + 155072 \nu^{12} - 1012184 \nu^{10} + 4868567 \nu^{8} - 24161111 \nu^{6} + \cdots + 1164515625 ) / 56781250 \) (-9679v^14 + 155072v^12 - 1012184v^10 + 4868567v^8 - 24161111v^6 + 120434950v^4 - 535191250*v^2 + 1164515625) / 56781250
\(\beta_{4}\)	\(=\)	\( ( 1968 \nu^{15} - 42104 \nu^{13} + 324443 \nu^{11} - 1621369 \nu^{9} + 7917752 \nu^{7} + \cdots - 517009375 \nu ) / 56781250 \) (1968v^15 - 42104v^13 + 324443v^11 - 1621369v^9 + 7917752v^7 - 38606970v^5 + 186172875v^3 - 517009375v) / 56781250
\(\beta_{5}\)	\(=\)	\( ( 13152 \nu^{14} - 165036 \nu^{12} + 932717 \nu^{10} - 4530221 \nu^{8} + 23029618 \nu^{6} + \cdots - 766046875 ) / 56781250 \) (13152v^14 - 165036v^12 + 932717v^10 - 4530221v^8 + 23029618v^6 - 120333350v^4 + 471120625*v^2 - 766046875) / 56781250
\(\beta_{6}\)	\(=\)	\( ( 10911 \nu^{15} - 146673 \nu^{13} + 903581 \nu^{11} - 4695253 \nu^{9} + 22842699 \nu^{7} + \cdots - 851015625 \nu ) / 283906250 \) (10911v^15 - 146673v^13 + 903581v^11 - 4695253v^9 + 22842699v^7 - 104910675v^5 + 427434375v^3 - 851015625v) / 283906250
\(\beta_{7}\)	\(=\)	\( ( - 14766 \nu^{14} + 182013 \nu^{12} - 1107886 \nu^{10} + 5441493 \nu^{8} - 26865744 \nu^{6} + \cdots + 983109375 ) / 56781250 \) (-14766v^14 + 182013v^12 - 1107886v^10 + 5441493v^8 - 26865744v^6 + 137700225v^4 - 545257500*v^2 + 983109375) / 56781250
\(\beta_{8}\)	\(=\)	\( ( 67 \nu^{14} - 976 \nu^{12} + 6267 \nu^{10} - 31136 \nu^{8} + 155663 \nu^{6} - 793080 \nu^{4} + \cdots - 6756250 ) / 143750 \) (67v^14 - 976v^12 + 6267v^10 - 31136v^8 + 155663v^6 - 793080v^4 + 3243375*v^2 - 6756250) / 143750
\(\beta_{9}\)	\(=\)	\( ( - 37979 \nu^{14} + 527497 \nu^{12} - 3336559 \nu^{10} + 15924867 \nu^{8} - 79211011 \nu^{6} + \cdots + 3273890625 ) / 56781250 \) (-37979v^14 + 527497v^12 - 3336559v^10 + 15924867v^8 - 79211011v^6 + 411359675v^4 - 1676681875*v^2 + 3273890625) / 56781250
\(\beta_{10}\)	\(=\)	\( ( - 39617 \nu^{15} + 550556 \nu^{13} - 3408182 \nu^{11} + 16828941 \nu^{9} + \cdots + 3718671875 \nu ) / 283906250 \) (-39617v^15 + 550556v^13 - 3408182v^11 + 16828941v^9 - 84516503v^7 + 433599950v^5 - 1809035000v^3 + 3718671875v) / 283906250
\(\beta_{11}\)	\(=\)	\( ( - 69391 \nu^{14} + 891313 \nu^{12} - 5517661 \nu^{10} + 26866793 \nu^{8} - 135170269 \nu^{6} + \cdots + 5315421875 ) / 56781250 \) (-69391v^14 + 891313v^12 - 5517661v^10 + 26866793v^8 - 135170269v^6 + 697330925v^4 - 2806464375*v^2 + 5315421875) / 56781250
\(\beta_{12}\)	\(=\)	\( ( 62919 \nu^{15} - 763392 \nu^{13} + 4635849 \nu^{11} - 22512212 \nu^{9} + 111486021 \nu^{7} + \cdots - 3780625000 \nu ) / 283906250 \) (62919v^15 - 763392v^13 + 4635849v^11 - 22512212v^9 + 111486021v^7 - 583590450v^5 + 2298853125v^3 - 3780625000v) / 283906250
\(\beta_{13}\)	\(=\)	\( ( 64304 \nu^{15} - 864372 \nu^{13} + 5421959 \nu^{11} - 26293867 \nu^{9} + 132465636 \nu^{7} + \cdots - 5099359375 \nu ) / 283906250 \) (64304v^15 - 864372v^13 + 5421959v^11 - 26293867v^9 + 132465636v^7 - 680065650v^5 + 2682835625v^3 - 5099359375v) / 283906250
\(\beta_{14}\)	\(=\)	\( ( - 77464 \nu^{15} + 975477 \nu^{13} - 6069994 \nu^{11} + 30107397 \nu^{9} + \cdots + 5937609375 \nu ) / 283906250 \) (-77464v^15 + 975477v^13 - 6069994v^11 + 30107397v^9 - 148624876v^7 + 760999425v^5 - 3057890000v^3 + 5937609375v) / 283906250
\(\beta_{15}\)	\(=\)	\( ( - 93696 \nu^{15} + 1179753 \nu^{13} - 7247666 \nu^{11} + 35247133 \nu^{9} + \cdots + 6903703125 \nu ) / 283906250 \) (-93696v^15 + 1179753v^13 - 7247666v^11 + 35247133v^9 - 177964864v^7 + 910905475v^5 - 3668715000v^3 + 6903703125v) / 283906250

\(\nu\)	\(=\)	\( ( \beta_{14} + \beta_{12} + \beta_{6} - \beta_1 ) / 2 \) (b14 + b12 + b6 - b1) / 2
\(\nu^{2}\)	\(=\)	\( ( -\beta_{11} + \beta_{9} - \beta_{8} + \beta_{7} - 2\beta_{3} - \beta_{2} + 5 ) / 2 \) (-b11 + b9 - b8 + b7 - 2*b3 - b2 + 5) / 2
\(\nu^{3}\)	\(=\)	\( ( -2\beta_{15} + 5\beta_{14} - 4\beta_{13} + 5\beta_{12} - 2\beta_{10} + 3\beta_{6} + 2\beta_{4} - 3\beta_1 ) / 2 \) (-2b15 + 5b14 - 4b13 + 5b12 - 2b10 + 3b6 + 2b4 - 3b1) / 2
\(\nu^{4}\)	\(=\)	\( 3\beta_{9} - 2\beta_{8} - 5\beta_{7} - \beta_{5} - 9\beta_{3} + 2\beta_{2} + 6 \) 3b9 - 2b8 - 5b7 - b5 - 9b3 + 2*b2 + 6
\(\nu^{5}\)	\(=\)	\( ( -18\beta_{15} + 23\beta_{14} - 16\beta_{13} + 13\beta_{12} + 8\beta_{10} + 45\beta_{6} + 16\beta_{4} + 7\beta_1 ) / 2 \) (-18b15 + 23b14 - 16b13 + 13b12 + 8b10 + 45b6 + 16b4 + 7b1) / 2
\(\nu^{6}\)	\(=\)	\( ( -19\beta_{11} + 33\beta_{9} + 3\beta_{8} + 75\beta_{7} - 2\beta_{5} - 48\beta_{3} + 55\beta_{2} + 97 ) / 2 \) (-19b11 + 33b9 + 3b8 + 75b7 - 2b5 - 48b3 + 55*b2 + 97) / 2
\(\nu^{7}\)	\(=\)	\( ( - 178 \beta_{15} + 113 \beta_{14} + 14 \beta_{13} - 119 \beta_{12} + 74 \beta_{10} + 143 \beta_{6} + \cdots - 161 \beta_1 ) / 2 \) (-178b15 + 113b14 + 14b13 - 119b12 + 74b10 + 143b6 - 56b4 - 161b1) / 2
\(\nu^{8}\)	\(=\)	\( -74\beta_{11} - 29\beta_{9} - 80\beta_{8} + 368\beta_{7} + 6\beta_{5} - 38\beta_{3} + 91\beta_{2} + 25 \) -74b11 - 29b9 - 80b8 + 368b7 + 6b5 - 38b3 + 91*b2 + 25
\(\nu^{9}\)	\(=\)	\( ( - 762 \beta_{15} + 577 \beta_{14} + 126 \beta_{13} - 259 \beta_{12} + 378 \beta_{10} - 401 \beta_{6} + \cdots + 125 \beta_1 ) / 2 \) (-762b15 + 577b14 + 126b13 - 259b12 + 378b10 - 401b6 + 132b4 + 125b1) / 2
\(\nu^{10}\)	\(=\)	\( ( -81\beta_{11} - 933\beta_{9} - 801\beta_{8} + 409\beta_{7} - 1488\beta_{5} - 6\beta_{3} + 597\beta_{2} + 1269 ) / 2 \) (-81b11 - 933b9 - 801b8 + 409b7 - 1488b5 - 6b3 + 597*b2 + 1269) / 2
\(\nu^{11}\)	\(=\)	\( ( - 1224 \beta_{15} + 1397 \beta_{14} + 942 \beta_{13} + 1741 \beta_{12} + 4752 \beta_{10} + \cdots + 3319 \beta_1 ) / 2 \) (-1224b15 + 1397b14 + 942b13 + 1741b12 + 4752b10 - 1163b6 + 3702b4 + 3319b1) / 2
\(\nu^{12}\)	\(=\)	\( -1856\beta_{11} + 302\beta_{9} - 814\beta_{8} + 773\beta_{7} - 5507\beta_{5} + 1173\beta_{3} - 61\beta_{2} + 5870 \) -1856b11 + 302b9 - 814b8 + 773b7 - 5507b5 + 1173b3 - 61*b2 + 5870
\(\nu^{13}\)	\(=\)	\( ( - 9412 \beta_{15} - 4029 \beta_{14} - 874 \beta_{13} - 2827 \beta_{12} + 24278 \beta_{10} + \cdots - 17499 \beta_1 ) / 2 \) (-9412b15 - 4029b14 - 874b13 - 2827b12 + 24278b10 - 9829b6 + 4632b4 - 17499b1) / 2
\(\nu^{14}\)	\(=\)	\( ( - 22723 \beta_{11} + 15059 \beta_{9} - 24765 \beta_{8} - 14733 \beta_{7} - 34710 \beta_{5} + \cdots - 48127 ) / 2 \) (-22723b11 + 15059b9 - 24765b8 - 14733b7 - 34710b5 - 5312b3 - 5067*b2 - 48127) / 2
\(\nu^{15}\)	\(=\)	\( ( - 75136 \beta_{15} - 54855 \beta_{14} - 61992 \beta_{13} - 37383 \beta_{12} + 95840 \beta_{10} + \cdots - 12407 \beta_1 ) / 2 \) (-75136b15 - 54855b14 - 61992b13 - 37383b12 + 95840b10 - 62559b6 + 5224b4 - 12407b1) / 2

\(n\)	\(277\)	\(1151\)	\(1201\)
\(\chi(n)\)	\(\beta_{7}\)	\(1\)	\(-1\)

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

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{8} + 2 T^{7} + 2 T^{6} + \cdots + 25)^{2} \) (T^8 + 2T^7 + 2T^6 + 6T^5 + 110T^4 + 270T^3 + 338T^2 + 130*T + 25)^2
$5$	\( T^{16} \) T^16
$7$	\( T^{16} + 532 T^{12} + \cdots + 1336336 \) T^16 + 532T^12 + 76792T^8 + 2029264*T^4 + 1336336
$11$	\( (T^{8} + 50 T^{6} + \cdots + 2500)^{2} \) (T^8 + 50T^6 + 672T^4 + 2596*T^2 + 2500)^2
$13$	\( (T^{8} + 6 T^{7} + \cdots + 13225)^{2} \) (T^8 + 6T^7 + 18T^6 + 42T^5 + 486T^4 + 2898T^3 + 9522T^2 + 15870*T + 13225)^2
$17$	\( T^{16} + 880 T^{12} + \cdots + 16 \) T^16 + 880T^12 + 71020T^8 + 10288*T^4 + 16
$19$	\( (T^{8} - 36 T^{6} + \cdots + 100)^{2} \) (T^8 - 36T^6 + 178T^4 - 244*T^2 + 100)^2
$23$	\( T^{16} + \cdots + 78310985281 \) T^16 + 12T^15 + 72T^14 + 612T^13 + 4004T^12 + 17940T^11 + 114264T^10 + 590364T^9 + 2322310T^8 + 13578372T^7 + 60445656T^6 + 218275980T^5 + 1120483364T^4 + 3939041916T^3 + 10658584008T^2 + 40857905364*T + 78310985281
$29$	\( (T^{8} + 88 T^{6} + \cdots + 28561)^{2} \) (T^8 + 88T^6 + 2110T^4 + 16680*T^2 + 28561)^2
$31$	\( (T^{4} + 2 T^{3} - 20 T^{2} + \cdots + 11)^{4} \) (T^4 + 2T^3 - 20T^2 + 10*T + 11)^4
$37$	\( T^{16} + 1348 T^{12} + \cdots + 456976 \) T^16 + 1348T^12 + 458296T^8 + 21869584*T^4 + 456976
$41$	\( (T^{4} + 12 T^{3} + \cdots - 107)^{4} \) (T^4 + 12T^3 + 10T^2 - 188*T - 107)^4
$43$	\( T^{16} + \cdots + 127880620816 \) T^16 + 6892T^12 + 11231212T^8 + 3092373184*T^4 + 127880620816
$47$	\( (T^{8} + 2 T^{7} + \cdots + 24025)^{2} \) (T^8 + 2T^7 + 2T^6 - 474T^5 + 6086T^4 - 24162T^3 + 51842T^2 - 49910*T + 24025)^2
$53$	\( T^{16} + \cdots + 533794816 \) T^16 + 54928T^12 + 584919936T^8 + 2911069184*T^4 + 533794816
$59$	\( (T^{8} + 168 T^{6} + \cdots + 110224)^{2} \) (T^8 + 168T^6 + 9244T^4 + 167728*T^2 + 110224)^2
$61$	\( (T^{8} + 224 T^{6} + \cdots + 100)^{2} \) (T^8 + 224T^6 + 5698T^4 + 3044*T^2 + 100)^2
$67$	\( T^{16} + \cdots + 36\!\cdots\!16 \) T^16 + 38308T^12 + 498843384T^8 + 2504414676496*T^4 + 3675002150265616
$71$	\( (T^{4} - 2 T^{3} + \cdots + 169)^{4} \) (T^4 - 2T^3 - 74T^2 - 162*T + 169)^4
$73$	\( (T^{8} - 26 T^{7} + \cdots + 25)^{2} \) (T^8 - 26T^7 + 338T^6 - 1326T^5 + 2294T^4 + 3874T^3 + 3042T^2 + 390*T + 25)^2
$79$	\( (T^{8} - 424 T^{6} + \cdots + 115562500)^{2} \) (T^8 - 424T^6 + 66346T^4 - 4549996*T^2 + 115562500)^2
$83$	\( T^{16} + \cdots + 17799252215056 \) T^16 + 62144T^12 + 527018796T^8 + 1094073550768*T^4 + 17799252215056
$89$	\( (T^{8} - 260 T^{6} + \cdots + 1144900)^{2} \) (T^8 - 260T^6 + 16786T^4 - 257316*T^2 + 1144900)^2
$97$	\( T^{16} + \cdots + 583506543376 \) T^16 + 15908T^12 + 22546936T^8 + 8112583184*T^4 + 583506543376
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