LMFDB - Newform orbit 624.4.q.m

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [624,4,Mod(289,624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(624, base_ring=CyclotomicField(6))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("624.289");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	624.q (of order $3$, 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:	$36.8171918436$
Analytic rank:	$0$
Dimension:	$10$
Relative dimension:	$5$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} + 120 x^{8} - 979 x^{7} + 14252 x^{6} - 68003 x^{5} + 352315 x^{4} - 602502 x^{3} + \cdots + 12873744 $ x^10 - x^9 + 120x^8 - 979x^7 + 14252x^6 - 68003x^5 + 352315x^4 - 602502x^3 + 2179576x^2 - 1643304x + 12873744
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 312)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{9}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + ( - 3 \beta_{2} + 3) q^{3} + ( - \beta_{8} - \beta_1 - 2) q^{5} + (\beta_{7} + \beta_{2}) q^{7} - 9 \beta_{2} q^{9} + (\beta_{5} - \beta_{3} + 9 \beta_{2} - 9) q^{11} + (\beta_{9} - \beta_{8} - \beta_{7} + \cdots - 1) q^{13}+ \cdots + ( - 9 \beta_{9} + 9 \beta_{6} + \cdots + 81) q^{99}+O(q^{100}) $ q + (-3b2 + 3) q^3 + (-b8 - b1 - 2) * q^5 + (b7 + b2) * q^7 - 9b2 q^9 + (b5 - b3 + 9b2 - 9) q^11 + (b9 - b8 - b7 - b4 - b3 - 4b2 - b1 - 1) q^13 + (6b2 - 3b1 - 6) * q^15 + (b9 + 4b8 + 4b7 - 2b2) q^17 + (b9 + 3b8 - 2b6 + 2b3 - 32b2) * q^19 + (3b7 + 3b4 + 3) * q^21 + (-2b5 - b4 + 2b3 - 4b2 + 3b1 + 4) * q^23 + (-5b9 + 8b8 + 3b7 + 3b6 - 5b5 + 3b4 + 8b1 + 32) q^25 - 27 * q^27 + (-2b5 - 4b4 - 3b3 + 25b2 - 6b1 - 25) q^29 + (b9 - 12b8 - b7 + 3b6 + b5 - b4 - 12b1 + 42) q^31 + (-3b9 + 3b6 - 3b3 + 27b2) * q^33 + (2b9 - 6b8 - 3b7 + b6 - b3 + 47b2) * q^35 + (-6b5 + 7b4 - 2b3 - 52b2 + 2b1 + 52) q^37 + (3b9 + 3b6 + 3b5 - 3b4 - 3b3 + 3b2 - 3b1 - 15) q^39 + (2b5 - 3b4 - 11b3 + 31b2 + 3b1 - 31) q^41 + (2b9 - 7b8 - 15b7 - 3b6 + 3b3 - 54b2) * q^43 + (9b8 + 18b2) * q^45 + (b9 + 17b8 + 6b6 + b5 + 17b1 + 30) q^47 + (-7b5 + 9b4 - 6b3 - 58b2 + 5b1 + 58) q^49 + (3b9 + 12b8 + 12b7 + 3b5 + 12b4 + 12b1 - 6) * q^51 + (-15b9 + 4b8 - 5b7 + 7b6 - 15b5 - 5b4 + 4b1 + 113) q^53 + (b5 + 10b4 + 5b3 - 15b2 + 42b1 + 15) * q^55 + (3b9 + 9b8 - 6b6 + 3b5 + 9b1 - 96) q^57 + (-5b9 + 20b8 - 7b7 + 8b6 - 8b3 + 8b2) * q^59 + (b9 - 37b8 + 14b7 + 16b6 - 16b3 - 21b2) q^61 + (9b4 - 9b2 + 9) * q^63 + (-3b9 + 45b8 + 19b7 + 3b6 - 10b5 + 3b4 + 3b3 - 51b2 + 16b1 + 134) q^65 + (4b5 + 11b4 + b3 + 222b2 + 5b1 - 222) * q^67 + (6b9 - 9b8 + 3b7 - 6b6 + 6b3 - 12b2) * q^69 + (14b9 - b8 + 23b7 - 2b6 + 2b3 - 172b2) q^71 + (-16b9 + 19b8 + 30b7 + 21b6 - 16b5 + 30b4 + 19b1 + 46) q^73 + (-15b5 + 9b4 + 9b3 - 96b2 + 24b1 + 96) q^75 + (6b9 - 5b8 - 2b7 - b6 + 6b5 - 2b4 - 5b1 + 21) * q^77 + (-5b9 - 54b8 - 13b7 + 13b6 - 5b5 - 13b4 - 54b1 + 22) q^79 + (81b2 - 81) q^81 + (-2b9 - 46b8 + 23b7 + b6 - 2b5 + 23b4 - 46b1 + 217) * q^83 + (32b9 - 13b8 - 11b7 - 7b6 + 7b3 - 443b2) * q^85 + (6b9 + 18b8 + 12b7 + 9b6 - 9b3 + 75b2) * q^87 + (21b5 + 53b4 - 16b3 + 26b2 + 34b1 - 26) q^89 + (-5b9 - b8 - 14b7 - 6b6 - 6b5 - 47b4 + 5b3 - 270b2 + 5b1 + 94) * q^91 + (3b5 - 3b4 + 9b3 - 126b2 - 36b1 + 126) q^93 + (13b9 + 51b8 - 2b7 - 370b2) * q^95 + (19b9 + 39b8 - 5b7 + 22b6 - 22b3 + 519b2) * q^97 + (-9b9 + 9b6 - 9b5 + 81) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 15 q^{3} - 22 q^{5} + 4 q^{7} - 45 q^{9} - 46 q^{11} - 31 q^{13} - 33 q^{15} - 11 q^{17} - 158 q^{19} + 24 q^{21} + 26 q^{23} + 340 q^{25} - 270 q^{27} - 125 q^{29} + 396 q^{31} + 138 q^{33} + 230 q^{35}+ \cdots + 828 q^{99}+O(q^{100}) $ 10 * q + 15 * q^3 - 22 * q^5 + 4 * q^7 - 45 * q^9 - 46 * q^11 - 31 * q^13 - 33 * q^15 - 11 * q^17 - 158 * q^19 + 24 * q^21 + 26 * q^23 + 340 * q^25 - 270 * q^27 - 125 * q^29 + 396 * q^31 + 138 * q^33 + 230 * q^35 + 261 * q^37 - 141 * q^39 - 151 * q^41 - 264 * q^43 + 99 * q^45 + 332 * q^47 + 293 * q^49 - 66 * q^51 + 1178 * q^53 + 106 * q^55 - 948 * q^57 + 72 * q^59 - 157 * q^61 + 36 * q^63 + 1137 * q^65 - 1120 * q^67 - 78 * q^69 - 898 * q^71 + 470 * q^73 + 510 * q^75 + 192 * q^77 + 148 * q^79 - 405 * q^81 + 2036 * q^83 - 2249 * q^85 + 375 * q^87 - 170 * q^89 - 334 * q^91 + 594 * q^93 - 1810 * q^95 + 2620 * q^97 + 828 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $ x^{10} - x^{9} + 120 x^{8} - 979 x^{7} + 14252 x^{6} - 68003 x^{5} + 352315 x^{4} - 602502 x^{3} + \cdots + 12873744 $ x^10 - x^9 + 120*x^8 - 979*x^7 + 14252*x^6 - 68003*x^5 + 352315*x^4 - 602502*x^3 + 2179576*x^2 - 1643304*x + 12873744 :

$\beta_{1}$	$=$	$ ( - 22705040150959 \nu^{9} + \cdots - 22\!\cdots\!00 ) / 52\!\cdots\!10 $ (-22705040150959v^9 - 8869585653476875v^8 - 36227560450782926v^7 - 975658460292756135v^6 + 4070581174274678818v^5 - 77939722727738700965v^4 + 57889277434541639377v^3 - 346979687375955796360v^2 - 3997870794130462705476*v - 2295159629608930944600) / 529633358081702752410
$\beta_{2}$	$=$	$ ( - 40025142237939 \nu^{9} - 229951117700615 \nu^{8} + \cdots - 42\!\cdots\!84 ) / 56\!\cdots\!04 $ (-40025142237939v^9 - 229951117700615v^8 - 4277560778258186v^7 + 6441434294656781v^6 - 284569200354047262v^5 - 1370243297138987323v^4 + 6952490734722105077v^3 - 84866115938377551592v^2 + 42242054389784226904*v - 42984454325870852184) / 564942248620482935904
$\beta_{3}$	$=$	$ ( - 963657775984429 \nu^{9} + \cdots - 45\!\cdots\!20 ) / 84\!\cdots\!60 $ (-963657775984429v^9 - 145362637221139225v^8 - 643804378886399606v^7 - 15513913850264246445v^6 + 60860760783084152158v^5 - 1267589213100904025285v^4 + 1030515799973497806187v^3 - 6824666737090956015640v^2 - 29435766973011663729816*v - 45841454617938201934920) / 8474133729307244038560
$\beta_{4}$	$=$	$ ( 277783668470921 \nu^{9} + \cdots + 64\!\cdots\!36 ) / 42\!\cdots\!28 $ (277783668470921v^9 + 11777718211073405v^8 + 76649267496543046v^7 + 1096273016646869313v^6 - 2955655859533788062v^5 + 88411848058020536209v^4 - 111102555443935722983v^3 + 937160377390304661944v^2 + 90828678625312702392*v + 6491409211931651093736) / 423706686465362201928
$\beta_{5}$	$=$	$ ( - 13\!\cdots\!19 \nu^{9} + \cdots - 23\!\cdots\!80 ) / 42\!\cdots\!80 $ (-13456608753972919v^9 - 230740255320803275v^8 - 2022008496500234066v^7 - 15027782914947416775v^6 - 21373502961772140662v^5 - 1864861736321704130015v^4 + 3284552393793583373857v^3 - 33778899629413458510280v^2 - 13691474641143730310376*v - 238628028254677566200280) / 4237066864653622019280
$\beta_{6}$	$=$	$ ( 428509431163265 \nu^{9} - 701042128200067 \nu^{8} + \cdots - 80\!\cdots\!48 ) / 92\!\cdots\!80 $ (428509431163265v^9 - 701042128200067v^8 + 43684601335207750v^7 - 474041278574202543v^6 + 5459475219966950770v^5 - 28089310741738821791v^4 + 85902258922480635985v^3 - 172746738389094099424v^2 + 145100349127095440520*v - 806615013952046724048) / 92110149231600478680
$\beta_{7}$	$=$	$ ( - 21\!\cdots\!85 \nu^{9} + \cdots + 13\!\cdots\!00 ) / 42\!\cdots\!28 $ (-2160657980322085v^9 - 9111756698245417v^8 - 242775469015864046v^7 + 1086889017670940715v^6 - 17805936391761408218v^5 + 18407725755053378515v^4 - 99971499469251139733v^3 - 280229591797830532408v^2 - 642624256537207467672*v + 134757620759394336600) / 423706686465362201928
$\beta_{8}$	$=$	$ ( 13\!\cdots\!41 \nu^{9} + \cdots - 14\!\cdots\!04 ) / 21\!\cdots\!40 $ (13996180976437241v^9 + 11472529310888549v^8 + 1640803771857192454v^7 - 11288352468351524439v^6 + 170666701033324449538v^5 - 650104486983261855263v^4 + 3378915008284192194697v^3 - 4527454640568076512832v^2 + 20960160166435720393464*v - 14982074891156746469304) / 2118533432326811009640
$\beta_{9}$	$=$	$ ( 54\!\cdots\!49 \nu^{9} + \cdots - 12\!\cdots\!56 ) / 42\!\cdots\!80 $ (54795576648662849v^9 + 163649163798954461v^8 + 6202709581898459566v^7 - 30416800124638828671v^6 + 543855871581332333002v^5 - 823339788169032143207v^4 + 4369595356394762876713v^3 + 17246701678456595422472v^2 + 27577859512382122572216*v - 12892723108336589891256) / 4237066864653622019280

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

$\nu$	$=$	$ ( \beta_{3} - \beta_{2} - \beta _1 + 1 ) / 4 $ (b3 - b2 - b1 + 1) / 4
$\nu^{2}$	$=$	$ -\beta_{9} + 2\beta_{6} - 2\beta_{3} - 48\beta_{2} $ -b9 + 2b6 - 2b3 - 48*b2
$\nu^{3}$	$=$	$ ( 22\beta_{9} + 41\beta_{8} - 22\beta_{7} - 125\beta_{6} + 22\beta_{5} - 22\beta_{4} + 41\beta _1 + 1023 ) / 4 $ (22b9 + 41b8 - 22b7 - 125b6 + 22b5 - 22b4 + 41*b1 + 1023) / 4
$\nu^{4}$	$=$	$ -119\beta_{5} + 15\beta_{4} + 338\beta_{3} + 4964\beta_{2} - 55\beta _1 - 4964 $ -119b5 + 15b4 + 338b3 + 4964b2 - 55*b1 - 4964
$\nu^{5}$	$=$	$ ( -4338\beta_{9} - 4201\beta_{8} + 2558\beta_{7} + 17457\beta_{6} - 17457\beta_{3} - 193395\beta_{2} ) / 4 $ (-4338b9 - 4201b8 + 2558b7 + 17457b6 - 17457b3 - 193395b2) / 4
$\nu^{6}$	$=$	$ 15638 \beta_{9} + 10225 \beta_{8} - 4165 \beta_{7} - 51201 \beta_{6} + 15638 \beta_{5} - 4165 \beta_{4} + \cdots + 661687 $ 15638b9 + 10225b8 - 4165b7 - 51201b6 + 15638b5 - 4165b4 + 10225*b1 + 661687
$\nu^{7}$	$=$	$ ( -690566\beta_{5} + 310606\beta_{4} + 2528873\beta_{3} + 30110771\beta_{2} - 561021\beta _1 - 30110771 ) / 4 $ (-690566b5 + 310606b4 + 2528873b3 + 30110771b2 - 561021*b1 - 30110771) / 4
$\nu^{8}$	$=$	$ -2209435\beta_{9} - 1591140\beta_{8} + 742465\beta_{7} + 7589745\beta_{6} - 7589745\beta_{3} - 94565381\beta_{2} $ -2209435b9 - 1591140b8 + 742465b7 + 7589745b6 - 7589745b3 - 94565381b2
$\nu^{9}$	$=$	$ ( 103985970 \beta_{9} + 80137621 \beta_{8} - 41751750 \beta_{7} - 370168021 \beta_{6} + 103985970 \beta_{5} + \cdots + 4498166639 ) / 4 $ (103985970b9 + 80137621b8 - 41751750b7 - 370168021b6 + 103985970b5 - 41751750b4 + 80137621*b1 + 4498166639) / 4

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/624\mathbb{Z}\right)^\times$.

$n$	$79$	$145$	$209$	$469$
$\chi(n)$	$1$	$-\beta_{2}$	$1$	$1$

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} $

289.1

−1.19909	−	2.07689i
2.57048	+	4.45220i
−6.06267	−	10.5009i
3.45416	+	5.98279i
1.73712	+	3.00877i
−1.19909	+	2.07689i
2.57048	−	4.45220i
−6.06267	+	10.5009i
3.45416	−	5.98279i
1.73712	−	3.00877i

1.50000

2.59808i

−21.8246

2.84961

−

4.93567i

−4.50000

7.79423i

289.2

1.50000

2.59808i

−6.40898

−14.7066

25.4726i

−4.50000

7.79423i

289.3

1.50000

2.59808i

−3.53523

0.928779

−

1.60869i

−4.50000

7.79423i

289.4

1.50000

2.59808i

5.41254

11.2944

−

19.5625i

−4.50000

7.79423i

289.5

1.50000

2.59808i

15.3563

1.63381

−

2.82985i

−4.50000

7.79423i

529.1

1.50000

−

2.59808i

−21.8246

2.84961

4.93567i

−4.50000

−

7.79423i

529.2

1.50000

−

2.59808i

−6.40898

−14.7066

−

25.4726i

−4.50000

−

7.79423i

529.3

1.50000

−

2.59808i

−3.53523

0.928779

1.60869i

−4.50000

−

7.79423i

529.4

1.50000

−

2.59808i

5.41254

11.2944

19.5625i

−4.50000

−

7.79423i

529.5

1.50000

−

2.59808i

15.3563

1.63381

2.82985i

−4.50000

−

7.79423i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	624.4.q.m		10
4.b	odd	2	1		312.4.q.d	✓	10
13.c	even	3	1	inner	624.4.q.m		10
52.j	odd	6	1		312.4.q.d	✓	10

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
312.4.q.d	✓	10	4.b	odd	2	1
312.4.q.d	✓	10	52.j	odd	6	1
624.4.q.m		10	1.a	even	1	1	trivial
624.4.q.m		10	13.c	even	3	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(624, [\chi])$:

$ T_{5}^{5} + 11T_{5}^{4} - 337T_{5}^{3} - 1843T_{5}^{2} + 9652T_{5} + 41100 $

$ T_{7}^{10} - 4 T_{7}^{9} + 719 T_{7}^{8} - 11984 T_{7}^{7} + 547477 T_{7}^{6} - 5367218 T_{7}^{5} + \cdots + 528264256 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{10} $ T^10
$3$	$ (T^{2} - 3 T + 9)^{5} $ (T^2 - 3*T + 9)^5
$5$	$ (T^{5} + 11 T^{4} + \cdots + 41100)^{2} $ (T^5 + 11T^4 - 337T^3 - 1843T^2 + 9652T + 41100)^2
$7$	$ T^{10} + \cdots + 528264256 $ T^10 - 4T^9 + 719T^8 - 11984T^7 + 547477T^6 - 5367218T^5 + 38178112T^4 - 142839544T^3 + 390517344T^2 - 544169184*T + 528264256
$11$	$ T^{10} + \cdots + 32799353693184 $ T^10 + 46T^9 + 4116T^8 + 77184T^7 + 7482928T^6 + 137347104T^5 + 7708969152T^4 - 11630963968T^3 + 651176631040T^2 + 2338386405888*T + 32799353693184
$13$	$ T^{10} + \cdots + 51\!\cdots\!57 $ T^10 + 31T^9 + 5355T^8 + 258674T^7 + 15125669T^6 + 880234641T^5 + 33231094793T^4 + 1248569991266T^3 + 56787094142415T^2 + 722240638796911*T + 51185893014090757
$17$	$ T^{10} + \cdots + 37\!\cdots\!76 $ T^10 + 11T^9 + 15962T^8 - 196093T^7 + 191821994T^6 - 2082263569T^5 + 941386610773T^4 - 18725156490156T^3 + 3487105209524832T^2 - 36067902504929856*T + 373773981198571776
$19$	$ T^{10} + \cdots + 78\!\cdots\!00 $ T^10 + 158T^9 + 24656T^8 + 1520008T^7 + 145709152T^6 + 8777602336T^5 + 594390214720T^4 + 21784699564544T^3 + 657604630262784T^2 + 8205132763176960*T + 78005183123046400
$23$	$ T^{10} + \cdots + 14\!\cdots\!36 $ T^10 - 26T^9 + 16980T^8 + 58496T^7 + 221939504T^6 - 1643583840T^5 + 795230469824T^4 - 30035825545984T^3 + 2583070856974080T^2 - 58046270772164096*T + 1424683480884315136
$29$	$ T^{10} + \cdots + 73\!\cdots\!00 $ T^10 + 125T^9 + 59674T^8 + 3481085T^7 + 2107419982T^6 + 126376297505T^5 + 34188221506681T^4 + 613090353668660T^3 + 277420165538717236T^2 + 10688100744491781840*T + 733665039823353459600
$31$	$ (T^{5} - 198 T^{4} + \cdots - 106542921664)^{2} $ (T^5 - 198T^4 - 52711T^3 + 10406284T^2 + 541147152T - 106542921664)^2
$37$	$ T^{10} + \cdots + 92\!\cdots\!76 $ T^10 - 261T^9 + 190318T^8 - 42612961T^7 + 24950195338T^6 - 5010150910357T^5 + 1272425556502357T^4 - 85268526046091056T^3 + 11409380620191041836T^2 + 89661156992953888800*T + 92373285472979428700176
$41$	$ T^{10} + \cdots + 95\!\cdots\!96 $ T^10 + 151T^9 + 197910T^8 + 98861043T^7 + 47639138854T^6 + 12931580087967T^5 + 2655773403406689T^4 + 362667595842820424T^3 + 37129305495533536048T^2 + 2321885142633242289792*T + 95444245599783262730496
$43$	$ T^{10} + \cdots + 12\!\cdots\!16 $ T^10 + 264T^9 + 204911T^8 + 32077492T^7 + 25760636701T^6 + 3920324529122T^5 + 1316897640651952T^4 - 19012774974564008T^3 + 6010936611223003040T^2 + 167030325994761639648*T + 12934440817116588145216
$47$	$ (T^{5} - 166 T^{4} + \cdots - 485225223264)^{2} $ (T^5 - 166T^4 - 187984T^3 + 21309520T^2 + 7858733040T - 485225223264)^2
$53$	$ (T^{5} + \cdots - 2698227619776)^{2} $ (T^5 - 589T^4 - 371877T^3 + 155969729T^2 + 41809791192T - 2698227619776)^2
$59$	$ T^{10} + \cdots + 17\!\cdots\!84 $ T^10 - 72T^9 + 364868T^8 + 67680032T^7 + 97415521168T^6 + 16071298582016T^5 + 11690231891444224T^4 + 2364845353597636608T^3 + 1014532195061152235520T^2 + 127042850713453235601408*T + 17403787108171374437597184
$61$	$ T^{10} + \cdots + 20\!\cdots\!25 $ T^10 + 157T^9 + 1086531T^8 + 253472722T^7 + 933672954389T^6 + 197201164340927T^5 + 277965460314970381T^4 + 48650293592592662082T^3 + 61014631805937384491371T^2 + 10291116674153576659527885*T + 2057008578829461524195559225
$67$	$ T^{10} + \cdots + 11\!\cdots\!96 $ T^10 + 1120T^9 + 871143T^8 + 385281484T^7 + 131383537181T^6 + 27775624868690T^5 + 5182790183550448T^4 + 624020887097474328T^3 + 106797559564779578272T^2 + 9808360806893139282912*T + 1156965707525116781906496
$71$	$ T^{10} + \cdots + 44\!\cdots\!00 $ T^10 + 898T^9 + 1254708T^8 + 844560448T^7 + 914097477232T^6 + 552512984521056T^5 + 322066820693693376T^4 + 93616951054109195520T^3 + 22144820606530662433536T^2 + 322250671865884626777600*T + 4427086282282241516160000
$73$	$ (T^{5} + \cdots - 107667757737387)^{2} $ (T^5 - 235T^4 - 1465762T^3 + 227012374T^2 + 403548940065T - 107667757737387)^2
$79$	$ (T^{5} + \cdots - 48889825273536)^{2} $ (T^5 - 74T^4 - 1314351T^3 - 19749316T^2 + 299506426544T - 48889825273536)^2
$83$	$ (T^{5} + \cdots - 48722741338752)^{2} $ (T^5 - 1018T^4 - 1045412T^3 + 745044776T^2 + 366167229792T - 48722741338752)^2
$89$	$ T^{10} + \cdots + 30\!\cdots\!00 $ T^10 + 170T^9 + 3180976T^8 - 506395048T^7 + 7530213063616T^6 - 1325509189010464T^5 + 7684072668842301376T^4 - 3523292780793446874880T^3 + 5789708119710213515008000T^2 - 1332177379885529541192192000*T + 306095262694052883801384960000
$97$	$ T^{10} + \cdots + 13\!\cdots\!44 $ T^10 - 2620T^9 + 7225639T^8 - 7478595792T^7 + 12267711589453T^6 - 8432839557479422T^5 + 14348055594809221768T^4 - 5462713004178531430296T^3 + 6034028824063599479823712T^2 + 1259916331214034087103386656*T + 1310907333543397169203215252544
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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:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	624.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - 3 \beta_{2} + 3) q^{3} + ( - \beta_{8} - \beta_1 - 2) q^{5} + (\beta_{7} + \beta_{2}) q^{7} - 9 \beta_{2} q^{9} + (\beta_{5} - \beta_{3} + 9 \beta_{2} - 9) q^{11} + (\beta_{9} - \beta_{8} - \beta_{7} + \cdots - 1) q^{13}+ \cdots + ( - 9 \beta_{9} + 9 \beta_{6} + \cdots + 81) q^{99}+O(q^{100}) \) q + (-3b2 + 3) q^3 + (-b8 - b1 - 2) * q^5 + (b7 + b2) * q^7 - 9b2 q^9 + (b5 - b3 + 9b2 - 9) q^11 + (b9 - b8 - b7 - b4 - b3 - 4b2 - b1 - 1) q^13 + (6b2 - 3b1 - 6) * q^15 + (b9 + 4b8 + 4b7 - 2b2) q^17 + (b9 + 3b8 - 2b6 + 2b3 - 32b2) * q^19 + (3b7 + 3b4 + 3) * q^21 + (-2b5 - b4 + 2b3 - 4b2 + 3b1 + 4) * q^23 + (-5b9 + 8b8 + 3b7 + 3b6 - 5b5 + 3b4 + 8b1 + 32) q^25 - 27 * q^27 + (-2b5 - 4b4 - 3b3 + 25b2 - 6b1 - 25) q^29 + (b9 - 12b8 - b7 + 3b6 + b5 - b4 - 12b1 + 42) q^31 + (-3b9 + 3b6 - 3b3 + 27b2) * q^33 + (2b9 - 6b8 - 3b7 + b6 - b3 + 47b2) * q^35 + (-6b5 + 7b4 - 2b3 - 52b2 + 2b1 + 52) q^37 + (3b9 + 3b6 + 3b5 - 3b4 - 3b3 + 3b2 - 3b1 - 15) q^39 + (2b5 - 3b4 - 11b3 + 31b2 + 3b1 - 31) q^41 + (2b9 - 7b8 - 15b7 - 3b6 + 3b3 - 54b2) * q^43 + (9b8 + 18b2) * q^45 + (b9 + 17b8 + 6b6 + b5 + 17b1 + 30) q^47 + (-7b5 + 9b4 - 6b3 - 58b2 + 5b1 + 58) q^49 + (3b9 + 12b8 + 12b7 + 3b5 + 12b4 + 12b1 - 6) * q^51 + (-15b9 + 4b8 - 5b7 + 7b6 - 15b5 - 5b4 + 4b1 + 113) q^53 + (b5 + 10b4 + 5b3 - 15b2 + 42b1 + 15) * q^55 + (3b9 + 9b8 - 6b6 + 3b5 + 9b1 - 96) q^57 + (-5b9 + 20b8 - 7b7 + 8b6 - 8b3 + 8b2) * q^59 + (b9 - 37b8 + 14b7 + 16b6 - 16b3 - 21b2) q^61 + (9b4 - 9b2 + 9) * q^63 + (-3b9 + 45b8 + 19b7 + 3b6 - 10b5 + 3b4 + 3b3 - 51b2 + 16b1 + 134) q^65 + (4b5 + 11b4 + b3 + 222b2 + 5b1 - 222) * q^67 + (6b9 - 9b8 + 3b7 - 6b6 + 6b3 - 12b2) * q^69 + (14b9 - b8 + 23b7 - 2b6 + 2b3 - 172b2) q^71 + (-16b9 + 19b8 + 30b7 + 21b6 - 16b5 + 30b4 + 19b1 + 46) q^73 + (-15b5 + 9b4 + 9b3 - 96b2 + 24b1 + 96) q^75 + (6b9 - 5b8 - 2b7 - b6 + 6b5 - 2b4 - 5b1 + 21) * q^77 + (-5b9 - 54b8 - 13b7 + 13b6 - 5b5 - 13b4 - 54b1 + 22) q^79 + (81b2 - 81) q^81 + (-2b9 - 46b8 + 23b7 + b6 - 2b5 + 23b4 - 46b1 + 217) * q^83 + (32b9 - 13b8 - 11b7 - 7b6 + 7b3 - 443b2) * q^85 + (6b9 + 18b8 + 12b7 + 9b6 - 9b3 + 75b2) * q^87 + (21b5 + 53b4 - 16b3 + 26b2 + 34b1 - 26) q^89 + (-5b9 - b8 - 14b7 - 6b6 - 6b5 - 47b4 + 5b3 - 270b2 + 5b1 + 94) * q^91 + (3b5 - 3b4 + 9b3 - 126b2 - 36b1 + 126) q^93 + (13b9 + 51b8 - 2b7 - 370b2) * q^95 + (19b9 + 39b8 - 5b7 + 22b6 - 22b3 + 519b2) * q^97 + (-9b9 + 9b6 - 9b5 + 81) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 15 q^{3} - 22 q^{5} + 4 q^{7} - 45 q^{9} - 46 q^{11} - 31 q^{13} - 33 q^{15} - 11 q^{17} - 158 q^{19} + 24 q^{21} + 26 q^{23} + 340 q^{25} - 270 q^{27} - 125 q^{29} + 396 q^{31} + 138 q^{33} + 230 q^{35}+ \cdots + 828 q^{99}+O(q^{100}) \) 10 * q + 15 * q^3 - 22 * q^5 + 4 * q^7 - 45 * q^9 - 46 * q^11 - 31 * q^13 - 33 * q^15 - 11 * q^17 - 158 * q^19 + 24 * q^21 + 26 * q^23 + 340 * q^25 - 270 * q^27 - 125 * q^29 + 396 * q^31 + 138 * q^33 + 230 * q^35 + 261 * q^37 - 141 * q^39 - 151 * q^41 - 264 * q^43 + 99 * q^45 + 332 * q^47 + 293 * q^49 - 66 * q^51 + 1178 * q^53 + 106 * q^55 - 948 * q^57 + 72 * q^59 - 157 * q^61 + 36 * q^63 + 1137 * q^65 - 1120 * q^67 - 78 * q^69 - 898 * q^71 + 470 * q^73 + 510 * q^75 + 192 * q^77 + 148 * q^79 - 405 * q^81 + 2036 * q^83 - 2249 * q^85 + 375 * q^87 - 170 * q^89 - 334 * q^91 + 594 * q^93 - 1810 * q^95 + 2620 * q^97 + 828 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

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	624.4.q.m

Level	$624$
Weight	$4$
Character orbit	624.q
Analytic conductor	$36.817$
Analytic rank	$0$
Dimension	$10$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(36.8171918436\)
Analytic rank:	\(0\)
Dimension:	\(10\)
Relative dimension:	\(5\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{10} - x^{9} + 120 x^{8} - 979 x^{7} + 14252 x^{6} - 68003 x^{5} + 352315 x^{4} - 602502 x^{3} + \cdots + 12873744 \) x^10 - x^9 + 120x^8 - 979x^7 + 14252x^6 - 68003x^5 + 352315x^4 - 602502x^3 + 2179576x^2 - 1643304x + 12873744
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{10} \)
Twist minimal:	no (minimal twist has level 312)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

\(\beta_{1}\)	\(=\)	\( ( - 22705040150959 \nu^{9} + \cdots - 22\!\cdots\!00 ) / 52\!\cdots\!10 \) (-22705040150959v^9 - 8869585653476875v^8 - 36227560450782926v^7 - 975658460292756135v^6 + 4070581174274678818v^5 - 77939722727738700965v^4 + 57889277434541639377v^3 - 346979687375955796360v^2 - 3997870794130462705476*v - 2295159629608930944600) / 529633358081702752410
\(\beta_{2}\)	\(=\)	\( ( - 40025142237939 \nu^{9} - 229951117700615 \nu^{8} + \cdots - 42\!\cdots\!84 ) / 56\!\cdots\!04 \) (-40025142237939v^9 - 229951117700615v^8 - 4277560778258186v^7 + 6441434294656781v^6 - 284569200354047262v^5 - 1370243297138987323v^4 + 6952490734722105077v^3 - 84866115938377551592v^2 + 42242054389784226904*v - 42984454325870852184) / 564942248620482935904
\(\beta_{3}\)	\(=\)	\( ( - 963657775984429 \nu^{9} + \cdots - 45\!\cdots\!20 ) / 84\!\cdots\!60 \) (-963657775984429v^9 - 145362637221139225v^8 - 643804378886399606v^7 - 15513913850264246445v^6 + 60860760783084152158v^5 - 1267589213100904025285v^4 + 1030515799973497806187v^3 - 6824666737090956015640v^2 - 29435766973011663729816*v - 45841454617938201934920) / 8474133729307244038560
\(\beta_{4}\)	\(=\)	\( ( 277783668470921 \nu^{9} + \cdots + 64\!\cdots\!36 ) / 42\!\cdots\!28 \) (277783668470921v^9 + 11777718211073405v^8 + 76649267496543046v^7 + 1096273016646869313v^6 - 2955655859533788062v^5 + 88411848058020536209v^4 - 111102555443935722983v^3 + 937160377390304661944v^2 + 90828678625312702392*v + 6491409211931651093736) / 423706686465362201928
\(\beta_{5}\)	\(=\)	\( ( - 13\!\cdots\!19 \nu^{9} + \cdots - 23\!\cdots\!80 ) / 42\!\cdots\!80 \) (-13456608753972919v^9 - 230740255320803275v^8 - 2022008496500234066v^7 - 15027782914947416775v^6 - 21373502961772140662v^5 - 1864861736321704130015v^4 + 3284552393793583373857v^3 - 33778899629413458510280v^2 - 13691474641143730310376*v - 238628028254677566200280) / 4237066864653622019280
\(\beta_{6}\)	\(=\)	\( ( 428509431163265 \nu^{9} - 701042128200067 \nu^{8} + \cdots - 80\!\cdots\!48 ) / 92\!\cdots\!80 \) (428509431163265v^9 - 701042128200067v^8 + 43684601335207750v^7 - 474041278574202543v^6 + 5459475219966950770v^5 - 28089310741738821791v^4 + 85902258922480635985v^3 - 172746738389094099424v^2 + 145100349127095440520*v - 806615013952046724048) / 92110149231600478680
\(\beta_{7}\)	\(=\)	\( ( - 21\!\cdots\!85 \nu^{9} + \cdots + 13\!\cdots\!00 ) / 42\!\cdots\!28 \) (-2160657980322085v^9 - 9111756698245417v^8 - 242775469015864046v^7 + 1086889017670940715v^6 - 17805936391761408218v^5 + 18407725755053378515v^4 - 99971499469251139733v^3 - 280229591797830532408v^2 - 642624256537207467672*v + 134757620759394336600) / 423706686465362201928
\(\beta_{8}\)	\(=\)	\( ( 13\!\cdots\!41 \nu^{9} + \cdots - 14\!\cdots\!04 ) / 21\!\cdots\!40 \) (13996180976437241v^9 + 11472529310888549v^8 + 1640803771857192454v^7 - 11288352468351524439v^6 + 170666701033324449538v^5 - 650104486983261855263v^4 + 3378915008284192194697v^3 - 4527454640568076512832v^2 + 20960160166435720393464*v - 14982074891156746469304) / 2118533432326811009640
\(\beta_{9}\)	\(=\)	\( ( 54\!\cdots\!49 \nu^{9} + \cdots - 12\!\cdots\!56 ) / 42\!\cdots\!80 \) (54795576648662849v^9 + 163649163798954461v^8 + 6202709581898459566v^7 - 30416800124638828671v^6 + 543855871581332333002v^5 - 823339788169032143207v^4 + 4369595356394762876713v^3 + 17246701678456595422472v^2 + 27577859512382122572216*v - 12892723108336589891256) / 4237066864653622019280

\(\nu\)	\(=\)	\( ( \beta_{3} - \beta_{2} - \beta _1 + 1 ) / 4 \) (b3 - b2 - b1 + 1) / 4
\(\nu^{2}\)	\(=\)	\( -\beta_{9} + 2\beta_{6} - 2\beta_{3} - 48\beta_{2} \) -b9 + 2b6 - 2b3 - 48*b2
\(\nu^{3}\)	\(=\)	\( ( 22\beta_{9} + 41\beta_{8} - 22\beta_{7} - 125\beta_{6} + 22\beta_{5} - 22\beta_{4} + 41\beta _1 + 1023 ) / 4 \) (22b9 + 41b8 - 22b7 - 125b6 + 22b5 - 22b4 + 41*b1 + 1023) / 4
\(\nu^{4}\)	\(=\)	\( -119\beta_{5} + 15\beta_{4} + 338\beta_{3} + 4964\beta_{2} - 55\beta _1 - 4964 \) -119b5 + 15b4 + 338b3 + 4964b2 - 55*b1 - 4964
\(\nu^{5}\)	\(=\)	\( ( -4338\beta_{9} - 4201\beta_{8} + 2558\beta_{7} + 17457\beta_{6} - 17457\beta_{3} - 193395\beta_{2} ) / 4 \) (-4338b9 - 4201b8 + 2558b7 + 17457b6 - 17457b3 - 193395b2) / 4
\(\nu^{6}\)	\(=\)	\( 15638 \beta_{9} + 10225 \beta_{8} - 4165 \beta_{7} - 51201 \beta_{6} + 15638 \beta_{5} - 4165 \beta_{4} + \cdots + 661687 \) 15638b9 + 10225b8 - 4165b7 - 51201b6 + 15638b5 - 4165b4 + 10225*b1 + 661687
\(\nu^{7}\)	\(=\)	\( ( -690566\beta_{5} + 310606\beta_{4} + 2528873\beta_{3} + 30110771\beta_{2} - 561021\beta _1 - 30110771 ) / 4 \) (-690566b5 + 310606b4 + 2528873b3 + 30110771b2 - 561021*b1 - 30110771) / 4
\(\nu^{8}\)	\(=\)	\( -2209435\beta_{9} - 1591140\beta_{8} + 742465\beta_{7} + 7589745\beta_{6} - 7589745\beta_{3} - 94565381\beta_{2} \) -2209435b9 - 1591140b8 + 742465b7 + 7589745b6 - 7589745b3 - 94565381b2
\(\nu^{9}\)	\(=\)	\( ( 103985970 \beta_{9} + 80137621 \beta_{8} - 41751750 \beta_{7} - 370168021 \beta_{6} + 103985970 \beta_{5} + \cdots + 4498166639 ) / 4 \) (103985970b9 + 80137621b8 - 41751750b7 - 370168021b6 + 103985970b5 - 41751750b4 + 80137621*b1 + 4498166639) / 4

\(n\)	\(79\)	\(145\)	\(209\)	\(469\)
\(\chi(n)\)	\(1\)	\(-\beta_{2}\)	\(1\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 289.5
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{10} \) T^10
$3$	\( (T^{2} - 3 T + 9)^{5} \) (T^2 - 3*T + 9)^5
$5$	\( (T^{5} + 11 T^{4} + \cdots + 41100)^{2} \) (T^5 + 11T^4 - 337T^3 - 1843T^2 + 9652T + 41100)^2
$7$	\( T^{10} + \cdots + 528264256 \) T^10 - 4T^9 + 719T^8 - 11984T^7 + 547477T^6 - 5367218T^5 + 38178112T^4 - 142839544T^3 + 390517344T^2 - 544169184*T + 528264256
$11$	\( T^{10} + \cdots + 32799353693184 \) T^10 + 46T^9 + 4116T^8 + 77184T^7 + 7482928T^6 + 137347104T^5 + 7708969152T^4 - 11630963968T^3 + 651176631040T^2 + 2338386405888*T + 32799353693184
$13$	\( T^{10} + \cdots + 51\!\cdots\!57 \) T^10 + 31T^9 + 5355T^8 + 258674T^7 + 15125669T^6 + 880234641T^5 + 33231094793T^4 + 1248569991266T^3 + 56787094142415T^2 + 722240638796911*T + 51185893014090757
$17$	\( T^{10} + \cdots + 37\!\cdots\!76 \) T^10 + 11T^9 + 15962T^8 - 196093T^7 + 191821994T^6 - 2082263569T^5 + 941386610773T^4 - 18725156490156T^3 + 3487105209524832T^2 - 36067902504929856*T + 373773981198571776
$19$	\( T^{10} + \cdots + 78\!\cdots\!00 \) T^10 + 158T^9 + 24656T^8 + 1520008T^7 + 145709152T^6 + 8777602336T^5 + 594390214720T^4 + 21784699564544T^3 + 657604630262784T^2 + 8205132763176960*T + 78005183123046400
$23$	\( T^{10} + \cdots + 14\!\cdots\!36 \) T^10 - 26T^9 + 16980T^8 + 58496T^7 + 221939504T^6 - 1643583840T^5 + 795230469824T^4 - 30035825545984T^3 + 2583070856974080T^2 - 58046270772164096*T + 1424683480884315136
$29$	\( T^{10} + \cdots + 73\!\cdots\!00 \) T^10 + 125T^9 + 59674T^8 + 3481085T^7 + 2107419982T^6 + 126376297505T^5 + 34188221506681T^4 + 613090353668660T^3 + 277420165538717236T^2 + 10688100744491781840*T + 733665039823353459600
$31$	\( (T^{5} - 198 T^{4} + \cdots - 106542921664)^{2} \) (T^5 - 198T^4 - 52711T^3 + 10406284T^2 + 541147152T - 106542921664)^2
$37$	\( T^{10} + \cdots + 92\!\cdots\!76 \) T^10 - 261T^9 + 190318T^8 - 42612961T^7 + 24950195338T^6 - 5010150910357T^5 + 1272425556502357T^4 - 85268526046091056T^3 + 11409380620191041836T^2 + 89661156992953888800*T + 92373285472979428700176
$41$	\( T^{10} + \cdots + 95\!\cdots\!96 \) T^10 + 151T^9 + 197910T^8 + 98861043T^7 + 47639138854T^6 + 12931580087967T^5 + 2655773403406689T^4 + 362667595842820424T^3 + 37129305495533536048T^2 + 2321885142633242289792*T + 95444245599783262730496
$43$	\( T^{10} + \cdots + 12\!\cdots\!16 \) T^10 + 264T^9 + 204911T^8 + 32077492T^7 + 25760636701T^6 + 3920324529122T^5 + 1316897640651952T^4 - 19012774974564008T^3 + 6010936611223003040T^2 + 167030325994761639648*T + 12934440817116588145216
$47$	\( (T^{5} - 166 T^{4} + \cdots - 485225223264)^{2} \) (T^5 - 166T^4 - 187984T^3 + 21309520T^2 + 7858733040T - 485225223264)^2
$53$	\( (T^{5} + \cdots - 2698227619776)^{2} \) (T^5 - 589T^4 - 371877T^3 + 155969729T^2 + 41809791192T - 2698227619776)^2
$59$	\( T^{10} + \cdots + 17\!\cdots\!84 \) T^10 - 72T^9 + 364868T^8 + 67680032T^7 + 97415521168T^6 + 16071298582016T^5 + 11690231891444224T^4 + 2364845353597636608T^3 + 1014532195061152235520T^2 + 127042850713453235601408*T + 17403787108171374437597184
$61$	\( T^{10} + \cdots + 20\!\cdots\!25 \) T^10 + 157T^9 + 1086531T^8 + 253472722T^7 + 933672954389T^6 + 197201164340927T^5 + 277965460314970381T^4 + 48650293592592662082T^3 + 61014631805937384491371T^2 + 10291116674153576659527885*T + 2057008578829461524195559225
$67$	\( T^{10} + \cdots + 11\!\cdots\!96 \) T^10 + 1120T^9 + 871143T^8 + 385281484T^7 + 131383537181T^6 + 27775624868690T^5 + 5182790183550448T^4 + 624020887097474328T^3 + 106797559564779578272T^2 + 9808360806893139282912*T + 1156965707525116781906496
$71$	\( T^{10} + \cdots + 44\!\cdots\!00 \) T^10 + 898T^9 + 1254708T^8 + 844560448T^7 + 914097477232T^6 + 552512984521056T^5 + 322066820693693376T^4 + 93616951054109195520T^3 + 22144820606530662433536T^2 + 322250671865884626777600*T + 4427086282282241516160000
$73$	\( (T^{5} + \cdots - 107667757737387)^{2} \) (T^5 - 235T^4 - 1465762T^3 + 227012374T^2 + 403548940065T - 107667757737387)^2
$79$	\( (T^{5} + \cdots - 48889825273536)^{2} \) (T^5 - 74T^4 - 1314351T^3 - 19749316T^2 + 299506426544T - 48889825273536)^2
$83$	\( (T^{5} + \cdots - 48722741338752)^{2} \) (T^5 - 1018T^4 - 1045412T^3 + 745044776T^2 + 366167229792T - 48722741338752)^2
$89$	\( T^{10} + \cdots + 30\!\cdots\!00 \) T^10 + 170T^9 + 3180976T^8 - 506395048T^7 + 7530213063616T^6 - 1325509189010464T^5 + 7684072668842301376T^4 - 3523292780793446874880T^3 + 5789708119710213515008000T^2 - 1332177379885529541192192000*T + 306095262694052883801384960000
$97$	\( T^{10} + \cdots + 13\!\cdots\!44 \) T^10 - 2620T^9 + 7225639T^8 - 7478595792T^7 + 12267711589453T^6 - 8432839557479422T^5 + 14348055594809221768T^4 - 5462713004178531430296T^3 + 6034028824063599479823712T^2 + 1259916331214034087103386656*T + 1310907333543397169203215252544
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