LMFDB - Newform orbit 372.2.j.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [372,2,Mod(97,372)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("372.97"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(372, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 0, 6])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 372 = 2^{2} \cdot 3 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	372.j (of order $5$, degree $4$, minimal)

Newform invariants

comment:select newform

sage:traces = [12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$2.97043495519$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$3$ over $\Q(\zeta_{5})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} + 13 x^{10} - 29 x^{9} + 148 x^{8} - 21 x^{7} + 897 x^{6} + 81 x^{5} + 6453 x^{4} + \cdots + 1296 $ x^12 - 3x^11 + 13x^10 - 29x^9 + 148x^8 - 21x^7 + 897x^6 + 81x^5 + 6453x^4 - 7236x^3 + 11448x^2 - 5832*x + 1296
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 5 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{11}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + (\beta_{8} + \beta_{7} + \beta_{6} - 1) q^{3} - \beta_{3} q^{5} + ( - \beta_{7} - \beta_{6} + \beta_{2}) q^{7} - \beta_{6} q^{9} + (\beta_{9} + \beta_{4}) q^{11} + (\beta_{11} + \beta_{9} + \cdots - \beta_{7}) q^{13}+ \cdots + ( - \beta_{11} + \beta_{10} + \beta_{4}) q^{99}+O(q^{100}) $ q + (b8 + b7 + b6 - 1) * q^3 - b3 * q^5 + (-b7 - b6 + b2) * q^7 - b6 * q^9 + (b9 + b4) * q^11 + (b11 + b9 - b8 - b7) * q^13 + (b5 + b3 + b2 + b1) * q^15 - b10 * q^17 + (b8 + b7 + b5 - 1) * q^19 + (-b8 + b5 + 1) * q^21 + (b9 - b5 - b3 - b1) * q^23 + (-b4 - b3 + 1) * q^25 - b8 * q^27 + (-b5 - 2b3 - 2b2) * q^29 + (b10 + b8 + b7 + b6 + 2b5 + b4 + b3 + b2 + b1 - 1) q^31 + (b11 - b4) * q^33 + (-b11 + 6b8 - b5 + b2 - b1) q^35 + (b11 - b10 + b8 + b6 - b4 - b2 - b1 - 1) * q^37 + (b10 - b7 + 1) * q^39 + (-b11 - b10 - b9 - b5 + b3 + b2) * q^41 + (-b11 - b8 + 6b7 + b5 + b4 + 1) q^43 - b1 * q^45 + (b10 - 6b8 - 6b7 - 6b6 + b4 - b3 - b1 + 6) q^47 + (-b11 - b9 + b6 + b5 + 3b3 + b2 + 3b1 - 1) * q^49 + b11 * q^51 + (-b10 - b5 - b3 + b1) * q^53 + (b11 - b9 - 2b5 - b4 - b2 - 2b1) * q^55 + (-b8 - b6 + b3) * q^57 + (6b6 - 2b5 - b3 - 2b2 - b1 - 6) q^59 + (-b8 - b6 - b4 + b3 + b2 + b1) * q^61 + (b8 + b6 + b3 - 1) * q^63 + (b10 - 4b5 + b4 - 2b3 - 4b2 - 2b1) * q^65 + (-b11 + b10 - b8 - b6 + 2b4 + b3 + 2b2 + 2b1 - 6) q^67 + (-b9 + b5 - b4 + b1) * q^69 + (b5 + b3 + b1) * q^71 + (-b11 - b9 - b7 - b6 + 2b5 - b4 - 2b2 + 2b1) q^73 + (-b11 - b9 + b8 + b7 + b6 + b5 + b3 + b2 + b1 - 1) * q^75 + (-b5 + b3 - b2 + b1) * q^77 + (b10 - b9 - b7 + 6b6 + b5 + b3 + 1) q^79 - b7 * q^81 + (-b11 - 6b8 - 6b7 + 3b5 + b4 + 6) q^83 + (-b9 + b5 + b3 - 2b1) q^85 + (b3 + 2b2 + 2b1) * q^87 + (-b9 - 3b5 - b4 - 2b2 - 3b1) q^89 + (b7 - 4b5 - 2b3 - 2b2) q^91 + (b9 - b6 + b3 - b1) * q^93 + (-b10 - b9 + 6b7 + 2b5 - b4 + b3 + b2) * q^95 + (b11 + 7b8 + b7 + b6 - 3b5 - 3b2 - 3b1) * q^97 + (-b11 + b10 + b4) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{3} + 2 q^{5} - 3 q^{7} - 3 q^{9} - 2 q^{11} - 5 q^{13} + 2 q^{15} - q^{17} - 8 q^{19} + 7 q^{21} + 3 q^{23} + 18 q^{25} - 3 q^{27} - 6 q^{31} + 3 q^{33} + 21 q^{35} - 10 q^{37} + 10 q^{39}+ \cdots - 2 q^{99}+O(q^{100}) $ 12 * q - 3 * q^3 + 2 * q^5 - 3 * q^7 - 3 * q^9 - 2 * q^11 - 5 * q^13 + 2 * q^15 - q^17 - 8 * q^19 + 7 * q^21 + 3 * q^23 + 18 * q^25 - 3 * q^27 - 6 * q^31 + 3 * q^33 + 21 * q^35 - 10 * q^37 + 10 * q^39 + q^41 + 22 * q^43 - 3 * q^45 + 14 * q^47 - 6 * q^49 - q^51 + 6 * q^53 - 4 * q^55 - 8 * q^57 - 57 * q^59 + 2 * q^61 - 8 * q^63 - 9 * q^65 - 74 * q^67 + 3 * q^69 - q^71 - 7 * q^73 - 2 * q^75 + 22 * q^79 - 3 * q^81 + 27 * q^83 - 12 * q^85 + 10 * q^87 - 7 * q^89 + 9 * q^91 - 6 * q^93 + 16 * q^95 + 14 * q^97 - 2 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 3 x^{11} + 13 x^{10} - 29 x^{9} + 148 x^{8} - 21 x^{7} + 897 x^{6} + 81 x^{5} + 6453 x^{4} + \cdots + 1296 $ x^12 - 3*x^11 + 13*x^10 - 29*x^9 + 148*x^8 - 21*x^7 + 897*x^6 + 81*x^5 + 6453*x^4 - 7236*x^3 + 11448*x^2 - 5832*x + 1296 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( 460834567 \nu^{11} - 947475468 \nu^{10} + 2822343043 \nu^{9} + 8527874974 \nu^{8} + \cdots + 27975633786576 ) / 44525669308560 $ (460834567v^11 - 947475468v^10 + 2822343043v^9 + 8527874974v^8 - 35816800538v^7 + 276104133831v^6 - 429932639592v^5 + 1870407303219v^4 - 2361342813948v^3 - 285754079664v^2 - 43869807088920*v + 27975633786576) / 44525669308560
$\beta_{3}$	$=$	$ ( 270690548 \nu^{11} - 2159218725 \nu^{10} + 6717463707 \nu^{9} - 23081777314 \nu^{8} + \cdots - 4579861436064 ) / 7420944884760 $ (270690548v^11 - 2159218725v^10 + 6717463707v^9 - 23081777314v^8 + 68133166128v^7 - 188842792826v^6 + 231735728077v^5 - 1262138017359v^4 + 773312870988v^3 - 11150756200116v^2 + 6940214665200*v - 4579861436064) / 7420944884760
$\beta_{4}$	$=$	$ ( - 62399137 \nu^{11} + 488600566 \nu^{10} - 1884753323 \nu^{9} + 7348817586 \nu^{8} + \cdots + 8946376547472 ) / 1349262706320 $ (-62399137v^11 + 488600566v^10 - 1884753323v^9 + 7348817586v^8 - 21047898682v^7 + 59889636459v^6 - 97681638538v^5 + 391149587241v^4 - 232991577972v^3 + 3516666012504v^2 - 2191045726440*v + 8946376547472) / 1349262706320
$\beta_{5}$	$=$	$ ( - 6827099144 \nu^{11} + 19828580379 \nu^{10} - 84006360167 \nu^{9} + 178695981064 \nu^{8} + \cdots + 12212237403648 ) / 22262834654280 $ (-6827099144v^11 + 19828580379v^10 - 84006360167v^9 + 178695981064v^8 - 930476902928v^7 - 49642891134v^6 - 5761523963247v^5 - 1098360591381v^4 - 42069950812278v^3 + 50135146986816v^2 - 57277722180816*v + 12212237403648) / 22262834654280
$\beta_{6}$	$=$	$ ( 9423022688 \nu^{11} - 21441968920 \nu^{10} + 102670714565 \nu^{9} - 189261297785 \nu^{8} + \cdots + 2322653864400 ) / 22262834654280 $ (9423022688v^11 - 21441968920v^10 + 102670714565v^9 - 189261297785v^8 + 1215911376760v^7 + 732593426480v^6 + 8502094242270v^5 + 6524788800975v^4 + 61905125997045v^3 - 26115041358090v^2 + 57739616745408*v + 2322653864400) / 22262834654280
$\beta_{7}$	$=$	$ ( - 19229120353 \nu^{11} + 46118140626 \nu^{10} - 224224191649 \nu^{9} + \cdots + 39885935747904 ) / 44525669308560 $ (-19229120353v^11 + 46118140626v^10 - 224224191649v^9 + 432758895188v^8 - 2618480639026v^7 - 1084160082213v^6 - 18204759362034v^5 - 12120124946217v^4 - 131984655621606v^3 + 57280547661732v^2 - 187054967107872*v + 39885935747904) / 44525669308560
$\beta_{8}$	$=$	$ ( - 10601531102 \nu^{11} + 30992521662 \nu^{10} - 131342248151 \nu^{9} + \cdots + 41007485391264 ) / 22262834654280 $ (-10601531102v^11 + 30992521662v^10 - 131342248151v^9 + 287292010837v^8 - 1499781271154v^7 + 18232654758v^6 - 8943045020016v^5 - 1553931203493v^4 - 64625266149129v^3 + 74392740441108v^2 - 87914059455348*v + 41007485391264) / 22262834654280
$\beta_{9}$	$=$	$ ( - 21976359115 \nu^{11} + 62129478294 \nu^{10} - 310502944321 \nu^{9} + \cdots - 5723380100304 ) / 44525669308560 $ (-21976359115v^11 + 62129478294v^10 - 310502944321v^9 + 608724569642v^8 - 3385270210294v^7 + 144074678193v^6 - 23207362830186v^5 - 15368661038853v^4 - 180649837685304v^3 + 82096870173048v^2 - 445186832743656*v - 5723380100304) / 44525669308560
$\beta_{10}$	$=$	$ ( - 17498898295 \nu^{11} + 39685451257 \nu^{10} - 190109677708 \nu^{9} + 350451630546 \nu^{8} + \cdots - 4294492778592 ) / 7420944884760 $ (-17498898295v^11 + 39685451257v^10 - 190109677708v^9 + 350451630546v^8 - 2248664462202v^7 - 1454113159481v^6 - 15720124532793v^5 - 12076124366694v^4 - 114618212599302v^3 + 48388733444484v^2 - 119898984215448*v - 4294492778592) / 7420944884760
$\beta_{11}$	$=$	$ ( - 58902125149 \nu^{11} + 172724256948 \nu^{10} - 732603207715 \nu^{9} + \cdots + 252535874400288 ) / 22262834654280 $ (-58902125149v^11 + 172724256948v^10 - 732603207715v^9 + 1615879495280v^8 - 8417000009950v^7 + 339935730075v^6 - 50392218875580v^5 - 4785789690135v^4 - 363614830121790v^3 + 416120494923720v^2 - 528143883482664*v + 252535874400288) / 22262834654280

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{10} + \beta_{9} - 6\beta_{7} - \beta_{5} + \beta_{4} $ b10 + b9 - 6*b7 - b5 + b4
$\nu^{3}$	$=$	$ \beta_{11} + \beta_{10} + \beta_{9} - 6 \beta_{8} - 6 \beta_{7} - 6 \beta_{6} - 10 \beta_{5} + \beta_{4} + \cdots + 6 $ b11 + b10 + b9 - 6b8 - 6b7 - 6b6 - 10b5 + b4 - 9b3 - 10b2 - 9*b1 + 6
$\nu^{4}$	$=$	$ 11\beta_{11} + \beta_{9} - 60\beta_{8} - 6\beta_{7} - 6\beta_{6} - 4\beta_{5} + \beta_{4} - 20\beta_{2} - 4\beta_1 $ 11b11 + b9 - 60b8 - 6b7 - 6b6 - 4b5 + b4 - 20b2 - 4*b1
$\nu^{5}$	$=$	$ 14\beta_{11} - 14\beta_{10} - 24\beta_{8} - 24\beta_{6} + 7\beta_{4} + 91\beta_{3} - 23\beta_{2} - 23\beta _1 - 96 $ 14b11 - 14b10 - 24b8 - 24b6 + 7b4 + 91b3 - 23b2 - 23b1 - 96
$\nu^{6}$	$=$	$ -128\beta_{10} - 30\beta_{9} + 138\beta_{7} - 546\beta_{6} + 96\beta_{5} + 96\beta_{3} - 215\beta _1 - 138 $ -128b10 - 30b9 + 138b7 - 546b6 + 96b5 + 96b3 - 215*b1 - 138
$\nu^{7}$	$=$	$ - 194 \beta_{11} - 341 \beta_{10} - 341 \beta_{9} + 576 \beta_{8} + 1866 \beta_{7} + 1409 \beta_{5} + \cdots - 576 $ -194b11 - 341b10 - 341b9 + 576b8 + 1866b7 + 1409b5 - 147b4 + 422b3 + 422*b2 - 576
$\nu^{8}$	$=$	$ - 1603 \beta_{11} - 569 \beta_{10} - 1603 \beta_{9} + 8454 \beta_{8} + 8454 \beta_{7} + 5922 \beta_{6} + \cdots - 5922 $ -1603b11 - 569b10 - 1603b9 + 8454b8 + 8454b7 + 5922b6 + 4492b5 - 569b4 + 2765b3 + 4492b2 + 2765*b1 - 5922
$\nu^{9}$	$=$	$ - 5061 \beta_{11} - 2761 \beta_{9} + 26952 \beta_{8} + 10362 \beta_{7} + 10362 \beta_{6} + \cdots + 7000 \beta_1 $ -5061b11 - 2761b9 + 26952b8 + 10362b7 + 10362b6 + 7000b5 - 2761b4 + 18324b2 + 7000*b1
$\nu^{10}$	$=$	$ - 9300 \beta_{11} + 9300 \beta_{10} + 42000 \beta_{8} + 42000 \beta_{6} - 11785 \beta_{4} + \cdots + 67944 $ -9300b11 + 9300b10 + 42000b8 + 42000b6 - 11785b4 - 35275b3 + 27945b2 + 27945b1 + 67944
$\nu^{11}$	$=$	$ 72520 \beta_{10} + 39730 \beta_{9} - 167670 \beta_{7} + 211650 \beta_{6} - 109630 \beta_{5} + \cdots + 167670 $ 72520b10 + 39730b9 - 167670b7 + 211650b6 - 109630b5 - 109630b3 + 136089*b1 + 167670

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/372\mathbb{Z}\right)^\times$.

$n$	$125$	$187$	$313$
$\chi(n)$	$1$	$1$	$-\beta_{6}$

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} $

97.1

−2.04360	−	1.48476i
0.316459	+	0.229921i
3.03616	+	2.20590i
0.814352	+	2.50632i
0.358338	+	1.10285i
−0.981706	−	3.02138i
0.814352	−	2.50632i
0.358338	−	1.10285i
−0.981706	+	3.02138i
−2.04360	+	1.48476i
0.316459	−	0.229921i
3.03616	−	2.20590i

0.309017

−

0.951057i

−2.52603

−2.54360

1.84803i

−0.809017

−

0.587785i

97.2

0.309017

−

0.951057i

0.391165

−0.183541

0.133351i

−0.809017

−

0.587785i

97.3

0.309017

−

0.951057i

3.75289

2.53616

−

1.84262i

−0.809017

−

0.587785i

109.1

−0.809017

−

0.587785i

−2.63530

0.314352

−

0.967475i

0.309017

0.951057i

109.2

−0.809017

−

0.587785i

−1.15961

−0.141662

0.435992i

0.309017

0.951057i

109.3

−0.809017

−

0.587785i

3.17687

−1.48171

4.56022i

0.309017

0.951057i

157.1

−0.809017

0.587785i

−2.63530

0.314352

0.967475i

0.309017

−

0.951057i

157.2

−0.809017

0.587785i

−1.15961

−0.141662

−

0.435992i

0.309017

−

0.951057i

157.3

−0.809017

0.587785i

3.17687

−1.48171

−

4.56022i

0.309017

−

0.951057i

349.1

0.309017

0.951057i

−2.52603

−2.54360

−

1.84803i

−0.809017

0.587785i

349.2

0.309017

0.951057i

0.391165

−0.183541

−

0.133351i

−0.809017

0.587785i

349.3

0.309017

0.951057i

3.75289

2.53616

1.84262i

−0.809017

0.587785i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
31.d	even	5	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	372.2.j.c	✓	12
3.b	odd	2	1		1116.2.m.f		12
31.d	even	5	1	inner	372.2.j.c	✓	12
93.l	odd	10	1		1116.2.m.f		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
372.2.j.c	✓	12	1.a	even	1	1	trivial
372.2.j.c	✓	12	31.d	even	5	1	inner
1116.2.m.f		12	3.b	odd	2	1
1116.2.m.f		12	93.l	odd	10	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{6} - T_{5}^{5} - 19T_{5}^{4} + 3T_{5}^{3} + 99T_{5}^{2} + 54T_{5} - 36 $ T5^6 - T5^5 - 19*T5^4 + 3*T5^3 + 99*T5^2 + 54*T5 - 36 acting on $S_{2}^{\mathrm{new}}(372, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} $ T^12
$3$	$ (T^{4} + T^{3} + T^{2} + \cdots + 1)^{3} $ (T^4 + T^3 + T^2 + T + 1)^3
$5$	$ (T^{6} - T^{5} - 19 T^{4} + \cdots - 36)^{2} $ (T^6 - T^5 - 19T^4 + 3T^3 + 99T^2 + 54T - 36)^2
$7$	$ T^{12} + 3 T^{11} + \cdots + 25 $ T^12 + 3T^11 + 18T^10 - 14T^9 - 25T^8 + 276T^7 + 2193T^6 + 303T^5 + 2351T^4 + 1285T^3 + 765T^2 + 200*T + 25
$11$	$ T^{12} + 2 T^{11} + \cdots + 810000 $ T^12 + 2T^11 + 49T^10 + 93T^9 + 976T^8 + 2295T^7 + 3495T^6 + 21825T^5 + 193275T^4 + 600750T^3 + 1120500T^2 + 1215000*T + 810000
$13$	$ T^{12} + 5 T^{11} + \cdots + 203401 $ T^12 + 5T^11 + 67T^10 + 174T^9 + 1414T^8 + 2201T^7 + 14549T^6 - 7647T^5 + 111462T^4 - 159138T^3 + 2357639T^2 - 1115323*T + 203401
$17$	$ T^{12} + T^{11} + \cdots + 810000 $ T^12 + T^11 + T^10 + 31T^9 + 766T^8 + 3825T^7 + 21705T^6 + 73800T^5 + 258525T^4 + 560250T^3 + 837000T^2 + 810000
$19$	$ T^{12} + 8 T^{11} + \cdots + 21025 $ T^12 + 8T^11 + 53T^10 + 161T^9 + 260T^8 + 111T^7 + 1458T^6 + 5803T^5 + 17136T^4 + 24785T^3 + 24585T^2 + 10875*T + 21025
$23$	$ T^{12} - 3 T^{11} + \cdots + 104976 $ T^12 - 3T^11 + 76T^10 - 275T^9 + 2575T^8 - 5223T^7 + 10944T^6 + 81027T^5 + 698625T^4 - 826200T^3 + 2146176T^2 - 752328*T + 104976
$29$	$ T^{12} + 55 T^{10} + \cdots + 20250000 $ T^12 + 55T^10 + 250T^9 + 2125T^8 + 3750T^7 + 99375T^6 + 675000T^5 + 5653125T^4 + 16031250T^3 + 20925000*T^2 + 20250000
$31$	$ T^{12} + \cdots + 887503681 $ T^12 + 6T^11 + 102T^10 + 667T^9 + 4758T^8 + 34662T^7 + 159043T^6 + 1074522T^5 + 4572438T^4 + 19870597T^3 + 94199142T^2 + 171774906*T + 887503681
$37$	$ (T^{6} + 5 T^{5} + \cdots + 3376)^{2} $ (T^6 + 5T^5 - 69T^4 - 72T^3 + 1704T^2 - 4376*T + 3376)^2
$41$	$ T^{12} + \cdots + 4902800400 $ T^12 - T^11 + 97T^10 + 197T^9 + 5410T^8 - 37497T^7 + 339267T^6 - 2200284T^5 + 20575161T^4 - 82809540T^3 + 387707040T^2 - 1543941000T + 4902800400
$43$	$ T^{12} - 22 T^{11} + \cdots + 92756161 $ T^12 - 22T^11 + 303T^10 - 3531T^9 + 44173T^8 - 283259T^7 + 1038567T^6 - 2402826T^5 + 34836023T^4 + 77037456T^3 + 204213298T^2 + 203859377*T + 92756161
$47$	$ T^{12} + \cdots + 1881824400 $ T^12 - 14T^11 + 141T^10 - 1279T^9 + 15706T^8 - 48705T^7 + 225075T^6 - 1241235T^5 + 15034365T^4 + 132864300T^3 + 716585400T^2 + 1042421400*T + 1881824400
$53$	$ T^{12} - 6 T^{11} + \cdots + 2509056 $ T^12 - 6T^11 - 11T^10 + 110T^9 + 4855T^8 - 51546T^7 + 324561T^6 - 1217394T^5 + 3892545T^4 - 7268940T^3 + 27193104T^2 + 4504896*T + 2509056
$59$	$ T^{12} + \cdots + 29549610000 $ T^12 + 57T^11 + 1708T^10 + 33649T^9 + 472795T^8 + 4848549T^7 + 36598668T^6 + 203440167T^5 + 846771651T^4 + 2700869130T^3 + 6922521900T^2 + 13722777000*T + 29549610000
$61$	$ (T^{6} - T^{5} - 84 T^{4} + \cdots + 319)^{2} $ (T^6 - T^5 - 84T^4 + 3T^3 + 1284T^2 + 1519T + 319)^2
$67$	$ (T^{6} + 37 T^{5} + \cdots - 46336)^{2} $ (T^6 + 37T^5 + 401T^4 + 216T^3 - 17024T^2 - 67520*T - 46336)^2
$71$	$ T^{12} + T^{11} + \cdots + 1296 $ T^12 + T^11 + 37T^10 + 7T^9 + 508T^8 + 267T^7 + 723T^6 - 108T^5 + 7893T^4 - 12258T^3 + 16632T^2 + 7776T + 1296
$73$	$ T^{12} + \cdots + 223629681025 $ T^12 + 7T^11 + 388T^10 + 2664T^9 + 66785T^8 + 215334T^7 + 3200793T^6 - 28126213T^5 + 207930961T^4 - 1651163535T^3 + 24843620865T^2 - 114364926800*T + 223629681025
$79$	$ T^{12} + \cdots + 11727807025 $ T^12 - 22T^11 + 414T^10 - 6768T^9 + 97876T^8 - 956870T^7 + 9159825T^6 - 72198360T^5 + 440113415T^4 - 1937966250T^3 + 6068981875T^2 - 11852887750*T + 11727807025
$83$	$ T^{12} + \cdots + 314586374400 $ T^12 - 27T^11 + 613T^10 - 6829T^9 + 57910T^8 - 241599T^7 + 3598563T^6 - 54866952T^5 + 824617161T^4 - 4624453080T^3 + 18406016640T^2 - 61079832000*T + 314586374400
$89$	$ T^{12} + \cdots + 29847399696 $ T^12 + 7T^11 + 126T^10 + 1745T^9 + 24505T^8 + 66147T^7 + 2380494T^6 + 30321297T^5 + 371240145T^4 + 2421339480T^3 + 9805591296T^2 + 17821988712*T + 29847399696
$97$	$ T^{12} + \cdots + 424002834025 $ T^12 - 14T^11 + 27T^10 + 463T^9 + 58025T^8 - 429413T^7 + 5903227T^6 - 80859336T^5 + 1122150741T^4 - 1153204000T^3 + 51086241310T^2 - 72554945875*T + 424002834025
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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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 372 = 2^{2} \cdot 3 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	372.j (of order \(5\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\( q + (\beta_{8} + \beta_{7} + \beta_{6} - 1) q^{3} - \beta_{3} q^{5} + ( - \beta_{7} - \beta_{6} + \beta_{2}) q^{7} - \beta_{6} q^{9} + (\beta_{9} + \beta_{4}) q^{11} + (\beta_{11} + \beta_{9} + \cdots - \beta_{7}) q^{13}+ \cdots + ( - \beta_{11} + \beta_{10} + \beta_{4}) q^{99}+O(q^{100}) \) q + (b8 + b7 + b6 - 1) * q^3 - b3 * q^5 + (-b7 - b6 + b2) * q^7 - b6 * q^9 + (b9 + b4) * q^11 + (b11 + b9 - b8 - b7) * q^13 + (b5 + b3 + b2 + b1) * q^15 - b10 * q^17 + (b8 + b7 + b5 - 1) * q^19 + (-b8 + b5 + 1) * q^21 + (b9 - b5 - b3 - b1) * q^23 + (-b4 - b3 + 1) * q^25 - b8 * q^27 + (-b5 - 2b3 - 2b2) * q^29 + (b10 + b8 + b7 + b6 + 2b5 + b4 + b3 + b2 + b1 - 1) q^31 + (b11 - b4) * q^33 + (-b11 + 6b8 - b5 + b2 - b1) q^35 + (b11 - b10 + b8 + b6 - b4 - b2 - b1 - 1) * q^37 + (b10 - b7 + 1) * q^39 + (-b11 - b10 - b9 - b5 + b3 + b2) * q^41 + (-b11 - b8 + 6b7 + b5 + b4 + 1) q^43 - b1 * q^45 + (b10 - 6b8 - 6b7 - 6b6 + b4 - b3 - b1 + 6) q^47 + (-b11 - b9 + b6 + b5 + 3b3 + b2 + 3b1 - 1) * q^49 + b11 * q^51 + (-b10 - b5 - b3 + b1) * q^53 + (b11 - b9 - 2b5 - b4 - b2 - 2b1) * q^55 + (-b8 - b6 + b3) * q^57 + (6b6 - 2b5 - b3 - 2b2 - b1 - 6) q^59 + (-b8 - b6 - b4 + b3 + b2 + b1) * q^61 + (b8 + b6 + b3 - 1) * q^63 + (b10 - 4b5 + b4 - 2b3 - 4b2 - 2b1) * q^65 + (-b11 + b10 - b8 - b6 + 2b4 + b3 + 2b2 + 2b1 - 6) q^67 + (-b9 + b5 - b4 + b1) * q^69 + (b5 + b3 + b1) * q^71 + (-b11 - b9 - b7 - b6 + 2b5 - b4 - 2b2 + 2b1) q^73 + (-b11 - b9 + b8 + b7 + b6 + b5 + b3 + b2 + b1 - 1) * q^75 + (-b5 + b3 - b2 + b1) * q^77 + (b10 - b9 - b7 + 6b6 + b5 + b3 + 1) q^79 - b7 * q^81 + (-b11 - 6b8 - 6b7 + 3b5 + b4 + 6) q^83 + (-b9 + b5 + b3 - 2b1) q^85 + (b3 + 2b2 + 2b1) * q^87 + (-b9 - 3b5 - b4 - 2b2 - 3b1) q^89 + (b7 - 4b5 - 2b3 - 2b2) q^91 + (b9 - b6 + b3 - b1) * q^93 + (-b10 - b9 + 6b7 + 2b5 - b4 + b3 + b2) * q^95 + (b11 + 7b8 + b7 + b6 - 3b5 - 3b2 - 3b1) * q^97 + (-b11 + b10 + b4) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{3} + 2 q^{5} - 3 q^{7} - 3 q^{9} - 2 q^{11} - 5 q^{13} + 2 q^{15} - q^{17} - 8 q^{19} + 7 q^{21} + 3 q^{23} + 18 q^{25} - 3 q^{27} - 6 q^{31} + 3 q^{33} + 21 q^{35} - 10 q^{37} + 10 q^{39}+ \cdots - 2 q^{99}+O(q^{100}) \) 12 * q - 3 * q^3 + 2 * q^5 - 3 * q^7 - 3 * q^9 - 2 * q^11 - 5 * q^13 + 2 * q^15 - q^17 - 8 * q^19 + 7 * q^21 + 3 * q^23 + 18 * q^25 - 3 * q^27 - 6 * q^31 + 3 * q^33 + 21 * q^35 - 10 * q^37 + 10 * q^39 + q^41 + 22 * q^43 - 3 * q^45 + 14 * q^47 - 6 * q^49 - q^51 + 6 * q^53 - 4 * q^55 - 8 * q^57 - 57 * q^59 + 2 * q^61 - 8 * q^63 - 9 * q^65 - 74 * q^67 + 3 * q^69 - q^71 - 7 * q^73 - 2 * q^75 + 22 * q^79 - 3 * q^81 + 27 * q^83 - 12 * q^85 + 10 * q^87 - 7 * q^89 + 9 * q^91 - 6 * q^93 + 16 * q^95 + 14 * q^97 - 2 * q^99

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	372.2.j.c

Level	$372$
Weight	$2$
Character orbit	372.j
Analytic conductor	$2.970$
Analytic rank	$0$
Dimension	$12$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(2.97043495519\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(3\) over \(\Q(\zeta_{5})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - 3 x^{11} + 13 x^{10} - 29 x^{9} + 148 x^{8} - 21 x^{7} + 897 x^{6} + 81 x^{5} + 6453 x^{4} + \cdots + 1296 \) x^12 - 3x^11 + 13x^10 - 29x^9 + 148x^8 - 21x^7 + 897x^6 + 81x^5 + 6453x^4 - 7236x^3 + 11448x^2 - 5832*x + 1296
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 5 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( 460834567 \nu^{11} - 947475468 \nu^{10} + 2822343043 \nu^{9} + 8527874974 \nu^{8} + \cdots + 27975633786576 ) / 44525669308560 \) (460834567v^11 - 947475468v^10 + 2822343043v^9 + 8527874974v^8 - 35816800538v^7 + 276104133831v^6 - 429932639592v^5 + 1870407303219v^4 - 2361342813948v^3 - 285754079664v^2 - 43869807088920*v + 27975633786576) / 44525669308560
\(\beta_{3}\)	\(=\)	\( ( 270690548 \nu^{11} - 2159218725 \nu^{10} + 6717463707 \nu^{9} - 23081777314 \nu^{8} + \cdots - 4579861436064 ) / 7420944884760 \) (270690548v^11 - 2159218725v^10 + 6717463707v^9 - 23081777314v^8 + 68133166128v^7 - 188842792826v^6 + 231735728077v^5 - 1262138017359v^4 + 773312870988v^3 - 11150756200116v^2 + 6940214665200*v - 4579861436064) / 7420944884760
\(\beta_{4}\)	\(=\)	\( ( - 62399137 \nu^{11} + 488600566 \nu^{10} - 1884753323 \nu^{9} + 7348817586 \nu^{8} + \cdots + 8946376547472 ) / 1349262706320 \) (-62399137v^11 + 488600566v^10 - 1884753323v^9 + 7348817586v^8 - 21047898682v^7 + 59889636459v^6 - 97681638538v^5 + 391149587241v^4 - 232991577972v^3 + 3516666012504v^2 - 2191045726440*v + 8946376547472) / 1349262706320
\(\beta_{5}\)	\(=\)	\( ( - 6827099144 \nu^{11} + 19828580379 \nu^{10} - 84006360167 \nu^{9} + 178695981064 \nu^{8} + \cdots + 12212237403648 ) / 22262834654280 \) (-6827099144v^11 + 19828580379v^10 - 84006360167v^9 + 178695981064v^8 - 930476902928v^7 - 49642891134v^6 - 5761523963247v^5 - 1098360591381v^4 - 42069950812278v^3 + 50135146986816v^2 - 57277722180816*v + 12212237403648) / 22262834654280
\(\beta_{6}\)	\(=\)	\( ( 9423022688 \nu^{11} - 21441968920 \nu^{10} + 102670714565 \nu^{9} - 189261297785 \nu^{8} + \cdots + 2322653864400 ) / 22262834654280 \) (9423022688v^11 - 21441968920v^10 + 102670714565v^9 - 189261297785v^8 + 1215911376760v^7 + 732593426480v^6 + 8502094242270v^5 + 6524788800975v^4 + 61905125997045v^3 - 26115041358090v^2 + 57739616745408*v + 2322653864400) / 22262834654280
\(\beta_{7}\)	\(=\)	\( ( - 19229120353 \nu^{11} + 46118140626 \nu^{10} - 224224191649 \nu^{9} + \cdots + 39885935747904 ) / 44525669308560 \) (-19229120353v^11 + 46118140626v^10 - 224224191649v^9 + 432758895188v^8 - 2618480639026v^7 - 1084160082213v^6 - 18204759362034v^5 - 12120124946217v^4 - 131984655621606v^3 + 57280547661732v^2 - 187054967107872*v + 39885935747904) / 44525669308560
\(\beta_{8}\)	\(=\)	\( ( - 10601531102 \nu^{11} + 30992521662 \nu^{10} - 131342248151 \nu^{9} + \cdots + 41007485391264 ) / 22262834654280 \) (-10601531102v^11 + 30992521662v^10 - 131342248151v^9 + 287292010837v^8 - 1499781271154v^7 + 18232654758v^6 - 8943045020016v^5 - 1553931203493v^4 - 64625266149129v^3 + 74392740441108v^2 - 87914059455348*v + 41007485391264) / 22262834654280
\(\beta_{9}\)	\(=\)	\( ( - 21976359115 \nu^{11} + 62129478294 \nu^{10} - 310502944321 \nu^{9} + \cdots - 5723380100304 ) / 44525669308560 \) (-21976359115v^11 + 62129478294v^10 - 310502944321v^9 + 608724569642v^8 - 3385270210294v^7 + 144074678193v^6 - 23207362830186v^5 - 15368661038853v^4 - 180649837685304v^3 + 82096870173048v^2 - 445186832743656*v - 5723380100304) / 44525669308560
\(\beta_{10}\)	\(=\)	\( ( - 17498898295 \nu^{11} + 39685451257 \nu^{10} - 190109677708 \nu^{9} + 350451630546 \nu^{8} + \cdots - 4294492778592 ) / 7420944884760 \) (-17498898295v^11 + 39685451257v^10 - 190109677708v^9 + 350451630546v^8 - 2248664462202v^7 - 1454113159481v^6 - 15720124532793v^5 - 12076124366694v^4 - 114618212599302v^3 + 48388733444484v^2 - 119898984215448*v - 4294492778592) / 7420944884760
\(\beta_{11}\)	\(=\)	\( ( - 58902125149 \nu^{11} + 172724256948 \nu^{10} - 732603207715 \nu^{9} + \cdots + 252535874400288 ) / 22262834654280 \) (-58902125149v^11 + 172724256948v^10 - 732603207715v^9 + 1615879495280v^8 - 8417000009950v^7 + 339935730075v^6 - 50392218875580v^5 - 4785789690135v^4 - 363614830121790v^3 + 416120494923720v^2 - 528143883482664*v + 252535874400288) / 22262834654280

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{10} + \beta_{9} - 6\beta_{7} - \beta_{5} + \beta_{4} \) b10 + b9 - 6*b7 - b5 + b4
\(\nu^{3}\)	\(=\)	\( \beta_{11} + \beta_{10} + \beta_{9} - 6 \beta_{8} - 6 \beta_{7} - 6 \beta_{6} - 10 \beta_{5} + \beta_{4} + \cdots + 6 \) b11 + b10 + b9 - 6b8 - 6b7 - 6b6 - 10b5 + b4 - 9b3 - 10b2 - 9*b1 + 6
\(\nu^{4}\)	\(=\)	\( 11\beta_{11} + \beta_{9} - 60\beta_{8} - 6\beta_{7} - 6\beta_{6} - 4\beta_{5} + \beta_{4} - 20\beta_{2} - 4\beta_1 \) 11b11 + b9 - 60b8 - 6b7 - 6b6 - 4b5 + b4 - 20b2 - 4*b1
\(\nu^{5}\)	\(=\)	\( 14\beta_{11} - 14\beta_{10} - 24\beta_{8} - 24\beta_{6} + 7\beta_{4} + 91\beta_{3} - 23\beta_{2} - 23\beta _1 - 96 \) 14b11 - 14b10 - 24b8 - 24b6 + 7b4 + 91b3 - 23b2 - 23b1 - 96
\(\nu^{6}\)	\(=\)	\( -128\beta_{10} - 30\beta_{9} + 138\beta_{7} - 546\beta_{6} + 96\beta_{5} + 96\beta_{3} - 215\beta _1 - 138 \) -128b10 - 30b9 + 138b7 - 546b6 + 96b5 + 96b3 - 215*b1 - 138
\(\nu^{7}\)	\(=\)	\( - 194 \beta_{11} - 341 \beta_{10} - 341 \beta_{9} + 576 \beta_{8} + 1866 \beta_{7} + 1409 \beta_{5} + \cdots - 576 \) -194b11 - 341b10 - 341b9 + 576b8 + 1866b7 + 1409b5 - 147b4 + 422b3 + 422*b2 - 576
\(\nu^{8}\)	\(=\)	\( - 1603 \beta_{11} - 569 \beta_{10} - 1603 \beta_{9} + 8454 \beta_{8} + 8454 \beta_{7} + 5922 \beta_{6} + \cdots - 5922 \) -1603b11 - 569b10 - 1603b9 + 8454b8 + 8454b7 + 5922b6 + 4492b5 - 569b4 + 2765b3 + 4492b2 + 2765*b1 - 5922
\(\nu^{9}\)	\(=\)	\( - 5061 \beta_{11} - 2761 \beta_{9} + 26952 \beta_{8} + 10362 \beta_{7} + 10362 \beta_{6} + \cdots + 7000 \beta_1 \) -5061b11 - 2761b9 + 26952b8 + 10362b7 + 10362b6 + 7000b5 - 2761b4 + 18324b2 + 7000*b1
\(\nu^{10}\)	\(=\)	\( - 9300 \beta_{11} + 9300 \beta_{10} + 42000 \beta_{8} + 42000 \beta_{6} - 11785 \beta_{4} + \cdots + 67944 \) -9300b11 + 9300b10 + 42000b8 + 42000b6 - 11785b4 - 35275b3 + 27945b2 + 27945b1 + 67944
\(\nu^{11}\)	\(=\)	\( 72520 \beta_{10} + 39730 \beta_{9} - 167670 \beta_{7} + 211650 \beta_{6} - 109630 \beta_{5} + \cdots + 167670 \) 72520b10 + 39730b9 - 167670b7 + 211650b6 - 109630b5 - 109630b3 + 136089*b1 + 167670

\(n\)	\(125\)	\(187\)	\(313\)
\(\chi(n)\)	\(1\)	\(1\)	\(-\beta_{6}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 97.3
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} \) T^12
$3$	\( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{3} \) (T^4 + T^3 + T^2 + T + 1)^3
$5$	\( (T^{6} - T^{5} - 19 T^{4} + \cdots - 36)^{2} \) (T^6 - T^5 - 19T^4 + 3T^3 + 99T^2 + 54T - 36)^2
$7$	\( T^{12} + 3 T^{11} + \cdots + 25 \) T^12 + 3T^11 + 18T^10 - 14T^9 - 25T^8 + 276T^7 + 2193T^6 + 303T^5 + 2351T^4 + 1285T^3 + 765T^2 + 200*T + 25
$11$	\( T^{12} + 2 T^{11} + \cdots + 810000 \) T^12 + 2T^11 + 49T^10 + 93T^9 + 976T^8 + 2295T^7 + 3495T^6 + 21825T^5 + 193275T^4 + 600750T^3 + 1120500T^2 + 1215000*T + 810000
$13$	\( T^{12} + 5 T^{11} + \cdots + 203401 \) T^12 + 5T^11 + 67T^10 + 174T^9 + 1414T^8 + 2201T^7 + 14549T^6 - 7647T^5 + 111462T^4 - 159138T^3 + 2357639T^2 - 1115323*T + 203401
$17$	\( T^{12} + T^{11} + \cdots + 810000 \) T^12 + T^11 + T^10 + 31T^9 + 766T^8 + 3825T^7 + 21705T^6 + 73800T^5 + 258525T^4 + 560250T^3 + 837000T^2 + 810000
$19$	\( T^{12} + 8 T^{11} + \cdots + 21025 \) T^12 + 8T^11 + 53T^10 + 161T^9 + 260T^8 + 111T^7 + 1458T^6 + 5803T^5 + 17136T^4 + 24785T^3 + 24585T^2 + 10875*T + 21025
$23$	\( T^{12} - 3 T^{11} + \cdots + 104976 \) T^12 - 3T^11 + 76T^10 - 275T^9 + 2575T^8 - 5223T^7 + 10944T^6 + 81027T^5 + 698625T^4 - 826200T^3 + 2146176T^2 - 752328*T + 104976
$29$	\( T^{12} + 55 T^{10} + \cdots + 20250000 \) T^12 + 55T^10 + 250T^9 + 2125T^8 + 3750T^7 + 99375T^6 + 675000T^5 + 5653125T^4 + 16031250T^3 + 20925000*T^2 + 20250000
$31$	\( T^{12} + \cdots + 887503681 \) T^12 + 6T^11 + 102T^10 + 667T^9 + 4758T^8 + 34662T^7 + 159043T^6 + 1074522T^5 + 4572438T^4 + 19870597T^3 + 94199142T^2 + 171774906*T + 887503681
$37$	\( (T^{6} + 5 T^{5} + \cdots + 3376)^{2} \) (T^6 + 5T^5 - 69T^4 - 72T^3 + 1704T^2 - 4376*T + 3376)^2
$41$	\( T^{12} + \cdots + 4902800400 \) T^12 - T^11 + 97T^10 + 197T^9 + 5410T^8 - 37497T^7 + 339267T^6 - 2200284T^5 + 20575161T^4 - 82809540T^3 + 387707040T^2 - 1543941000T + 4902800400
$43$	\( T^{12} - 22 T^{11} + \cdots + 92756161 \) T^12 - 22T^11 + 303T^10 - 3531T^9 + 44173T^8 - 283259T^7 + 1038567T^6 - 2402826T^5 + 34836023T^4 + 77037456T^3 + 204213298T^2 + 203859377*T + 92756161
$47$	\( T^{12} + \cdots + 1881824400 \) T^12 - 14T^11 + 141T^10 - 1279T^9 + 15706T^8 - 48705T^7 + 225075T^6 - 1241235T^5 + 15034365T^4 + 132864300T^3 + 716585400T^2 + 1042421400*T + 1881824400
$53$	\( T^{12} - 6 T^{11} + \cdots + 2509056 \) T^12 - 6T^11 - 11T^10 + 110T^9 + 4855T^8 - 51546T^7 + 324561T^6 - 1217394T^5 + 3892545T^4 - 7268940T^3 + 27193104T^2 + 4504896*T + 2509056
$59$	\( T^{12} + \cdots + 29549610000 \) T^12 + 57T^11 + 1708T^10 + 33649T^9 + 472795T^8 + 4848549T^7 + 36598668T^6 + 203440167T^5 + 846771651T^4 + 2700869130T^3 + 6922521900T^2 + 13722777000*T + 29549610000
$61$	\( (T^{6} - T^{5} - 84 T^{4} + \cdots + 319)^{2} \) (T^6 - T^5 - 84T^4 + 3T^3 + 1284T^2 + 1519T + 319)^2
$67$	\( (T^{6} + 37 T^{5} + \cdots - 46336)^{2} \) (T^6 + 37T^5 + 401T^4 + 216T^3 - 17024T^2 - 67520*T - 46336)^2
$71$	\( T^{12} + T^{11} + \cdots + 1296 \) T^12 + T^11 + 37T^10 + 7T^9 + 508T^8 + 267T^7 + 723T^6 - 108T^5 + 7893T^4 - 12258T^3 + 16632T^2 + 7776T + 1296
$73$	\( T^{12} + \cdots + 223629681025 \) T^12 + 7T^11 + 388T^10 + 2664T^9 + 66785T^8 + 215334T^7 + 3200793T^6 - 28126213T^5 + 207930961T^4 - 1651163535T^3 + 24843620865T^2 - 114364926800*T + 223629681025
$79$	\( T^{12} + \cdots + 11727807025 \) T^12 - 22T^11 + 414T^10 - 6768T^9 + 97876T^8 - 956870T^7 + 9159825T^6 - 72198360T^5 + 440113415T^4 - 1937966250T^3 + 6068981875T^2 - 11852887750*T + 11727807025
$83$	\( T^{12} + \cdots + 314586374400 \) T^12 - 27T^11 + 613T^10 - 6829T^9 + 57910T^8 - 241599T^7 + 3598563T^6 - 54866952T^5 + 824617161T^4 - 4624453080T^3 + 18406016640T^2 - 61079832000*T + 314586374400
$89$	\( T^{12} + \cdots + 29847399696 \) T^12 + 7T^11 + 126T^10 + 1745T^9 + 24505T^8 + 66147T^7 + 2380494T^6 + 30321297T^5 + 371240145T^4 + 2421339480T^3 + 9805591296T^2 + 17821988712*T + 29847399696
$97$	\( T^{12} + \cdots + 424002834025 \) T^12 - 14T^11 + 27T^10 + 463T^9 + 58025T^8 - 429413T^7 + 5903227T^6 - 80859336T^5 + 1122150741T^4 - 1153204000T^3 + 51086241310T^2 - 72554945875*T + 424002834025
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