LMFDB - Newform orbit 3060.2.be.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3060,2,Mod(361,3060)] 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("3060.361"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3060, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 0, 3])) N = Newforms(chi, 2, names="a")

Level:	$N$	$=$	$3060 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 17$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	3060.be (of order $4$ , degree $2$ , minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$24.4342230185$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{12} + 6 x^{10} - 16 x^{9} + 9 x^{8} - 72 x^{7} + 114 x^{6} - 144 x^{5} + 391 x^{4} - 484 x^{3} + \cdots + 121$ x^12 + 6x^10 - 16x^9 + 9x^8 - 72x^7 + 114x^6 - 144x^5 + 391x^4 - 484x^3 + 504x^2 - 396x + 121
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 340)
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 $1,\beta_1,\ldots,\beta_{11}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - \beta_{3} q^{5} + ( - \beta_{11} + \beta_{8} + \cdots + \beta_{2}) q^{7} + (\beta_{11} - 2 \beta_{9} - 2 \beta_{8} + \cdots + 2) q^{11} + (\beta_{10} - \beta_{8} - \beta_{4} + \cdots + 2) q^{13}+ \cdots + ( - \beta_{11} + 2 \beta_{10} + \cdots - 3) q^{97}+O(q^{100})$ q - b3 * q^5 + (-b11 + b8 + b5 + b4 + b3 + b2) * q^7 + (b11 - 2b9 - 2b8 - 2b6 - b5 + b4 - b3 - b2 + 2) q^11 + (b10 - b8 - b4 + b3 + b2 + b1 + 2) * q^13 + (b9 + b8 - b7 - b5 - 2b4 + 2b2 + 2b1 - 1) q^17 + (2b9 - 2b6 + 2b5) q^19 + (-b11 - b9 - b8 + b6 - 2b5 - b4 + b3 + b2 + 3b1 - 1) * q^23 + b6 * q^25 + (b11 + 2b9 - 2b7 - b6 + 2b5 - b4 + 3b3 + b2 + 3b1 - 1) q^29 + (-b11 + 3b9 - b7 - b6 + 3b5 + b4 + 5b3 - b2 + 2b1 - 1) * q^31 + (b10 - b9 - b8 + b7 + b4 - b3 - b2 - 2b1 + 2) q^35 + (-b11 + 2b10 - b9 - b6 - b5 + b4 + 3b3 - b2 - 2b1 - 1) q^37 + (2b11 - 4b9 + 2b8 - 2b3 - 2b2 - 2b1) * q^41 + (-b11 - b9 + b7 - 2b5 - 2b4 - 2b3) q^43 + (b10 + 2b9 - b8 - 2b7 - b4 + b3 - b2 + 3b1) q^47 + (5b11 - 3b10 - 6b9 - 3b8 - 2b7 - 7b6 - 4b5 - 5b4 - 5b3) q^49 + (b10 + 3b9 + b8 + b7 + 2b5) * q^53 + (-b10 + b9 + b8 - b7 + 2b2 + b1) q^55 + (2b9 + 2b5 - 4b4 - 4b3) * q^59 + (-b11 + 2b9 + 2b6 - 2b5 - b4 + b3 + b2 + 3b1 - 2) * q^61 + (b11 + b10 - b4 - b3 + b2 + b1) * q^65 + (3b10 - 2b9 - 3b8 + 2b7 + 7b4 - 7b3 - 5b2 - 3b1 + 6) * q^67 + (-4b11 + 2b10 + 3b9 - b7 - 2b6 + 3b5 + 4b4 - 2b3 - 4b2 - b1 - 2) * q^71 + (-b11 - 4b10 - 2b7 - 2b6 + b4 - 3b3 - b2 - b1 - 2) * q^73 + (-7b11 + 3b10 + 6b9 + 3b8 + 6b6 + 6b5 + 8b4 + 8b3) * q^77 + (-2b11 - 2b8 + b6 + b5 - 6b4 + 2b3 + 2b2 + b1 - 1) q^79 + (-5b11 + b10 + 4b9 + b8 + 2b6 + 4b5 + 4b4 + 4b3) * q^83 + (2b10 + 2b9 + 2b6 + b5 + b4 - b2 + b1) q^85 + (b10 - b9 - b8 + b7 - 4b4 + 4b3 + 3b2 + 3b1 + 6) * q^89 + (-4b11 + 2b9 + b6 + 2b4 + 4b3 + 4b2 + 4b1 - 1) * q^91 + (-2b9 + 2b4) * q^95 + (-b11 + 2b10 + b9 - 3b6 + b5 + b4 - 5b3 - b2 - 3) q^97
$\operatorname{Tr}(f)(q)$	$=$	$12 q + 12 q^{11} + 16 q^{13} + 4 q^{17} - 12 q^{23} - 4 q^{29} - 8 q^{31} + 8 q^{35} - 20 q^{37} - 8 q^{41} + 8 q^{47} + 16 q^{55} - 8 q^{61} - 4 q^{65} + 32 q^{67} - 28 q^{71} - 24 q^{79} - 4 q^{85} + 56 q^{89}+ \cdots - 44 q^{97}+O(q^{100})$ 12 * q + 12 * q^11 + 16 * q^13 + 4 * q^17 - 12 * q^23 - 4 * q^29 - 8 * q^31 + 8 * q^35 - 20 * q^37 - 8 * q^41 + 8 * q^47 + 16 * q^55 - 8 * q^61 - 4 * q^65 + 32 * q^67 - 28 * q^71 - 24 * q^79 - 4 * q^85 + 56 * q^89 - 4 * q^91 - 8 * q^95 - 44 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $x^{12} + 6 x^{10} - 16 x^{9} + 9 x^{8} - 72 x^{7} + 114 x^{6} - 144 x^{5} + 391 x^{4} - 484 x^{3} + \cdots + 121$ x^12 + 6*x^10 - 16*x^9 + 9*x^8 - 72*x^7 + 114*x^6 - 144*x^5 + 391*x^4 - 484*x^3 + 504*x^2 - 396*x + 121 :

$\beta_{1}$	$=$	$( 237029484 \nu^{11} + 452648926 \nu^{10} + 1781832384 \nu^{9} + 7020348 \nu^{8} + \cdots + 229900141235 ) / 91818516077$ (237029484v^11 + 452648926v^10 + 1781832384v^9 + 7020348v^8 - 2154195286v^7 - 11213095538v^6 - 11375060157v^5 - 11073903716v^4 - 10963179920v^3 + 46892612187v^2 - 14415761207*v + 229900141235) / 91818516077
$\beta_{2}$	$=$	$( - 6748724960 \nu^{11} - 7586462510 \nu^{10} - 49427698226 \nu^{9} + 46585030124 \nu^{8} + \cdots - 730333528452 ) / 1010003676847$ (-6748724960v^11 - 7586462510v^10 - 49427698226v^9 + 46585030124v^8 - 13653681406v^7 + 445221044856v^6 - 223649959859v^5 + 843300294187v^4 - 1376879728680v^3 + 1609214152887v^2 - 1434507604477*v - 730333528452) / 1010003676847
$\beta_{3}$	$=$	$( 1681562281 \nu^{11} + 1206883623 \nu^{10} + 10971218980 \nu^{9} - 18940257967 \nu^{8} + \cdots - 291050856503 ) / 59411980991$ (1681562281v^11 + 1206883623v^10 + 10971218980v^9 - 18940257967v^8 + 1569096424v^7 - 121943505218v^6 + 100961817419v^5 - 176541913276v^4 + 541727327456v^3 - 405063970212v^2 + 575293488258*v - 291050856503) / 59411980991
$\beta_{4}$	$=$	$( 1877462551 \nu^{11} + 702587919 \nu^{10} + 11535649657 \nu^{9} - 25605083005 \nu^{8} + \cdots - 392384744796 ) / 59411980991$ (1877462551v^11 + 702587919v^10 + 11535649657v^9 - 25605083005v^8 + 6206244710v^7 - 133897993651v^6 + 155956130923v^5 - 205296597965v^4 + 653017838202v^3 - 607285810857v^2 + 686426960951*v - 392384744796) / 59411980991
$\beta_{5}$	$=$	$( 32219944720 \nu^{11} + 37654083852 \nu^{10} + 222519008952 \nu^{9} - 272711613374 \nu^{8} + \cdots - 2812980959800 ) / 1010003676847$ (32219944720v^11 + 37654083852v^10 + 222519008952v^9 - 272711613374v^8 - 138478128814v^7 - 2325141152849v^6 + 1046088013009v^5 - 2191973179656v^4 + 9351939602630v^3 - 4037516195405v^2 + 6221724061759*v - 2812980959800) / 1010003676847
$\beta_{6}$	$=$	$( 2354972 \nu^{11} + 1834921 \nu^{10} + 15435730 \nu^{9} - 25839746 \nu^{8} + 945970 \nu^{7} + \cdots - 385736208 ) / 49800487$ (2354972v^11 + 1834921v^10 + 15435730v^9 - 25839746v^8 + 945970v^7 - 166696279v^6 + 144512890v^5 - 218703357v^4 + 739653400v^3 - 584723216v^2 + 696299706*v - 385736208) / 49800487
$\beta_{7}$	$=$	$( 73257346909 \nu^{11} + 56161613079 \nu^{10} + 490247008135 \nu^{9} - 801822345473 \nu^{8} + \cdots - 12607579566516 ) / 1010003676847$ (73257346909v^11 + 56161613079v^10 + 490247008135v^9 - 801822345473v^8 + 68761161218v^7 - 5345644673517v^6 + 4299759622066v^5 - 7543561488176v^4 + 23728481266737v^3 - 17959072699491v^2 + 25065472715263*v - 12607579566516) / 1010003676847
$\beta_{8}$	$=$	$( - 84496178552 \nu^{11} - 52566459528 \nu^{10} - 527061304371 \nu^{9} + \cdots + 17037121470248 ) / 1010003676847$ (-84496178552v^11 - 52566459528v^10 - 527061304371v^9 + 1047034493211v^8 - 30608455483v^7 + 6025893557670v^6 - 6020317331675v^5 + 7881505857241v^4 - 28201658254877v^3 + 23215477313116v^2 - 27447383123964*v + 17037121470248) / 1010003676847
$\beta_{9}$	$=$	$( 85003027341 \nu^{11} + 45952103197 \nu^{10} + 532895178107 \nu^{9} - 1072415229638 \nu^{8} + \cdots - 17011911372159 ) / 1010003676847$ (85003027341v^11 + 45952103197v^10 + 532895178107v^9 - 1072415229638v^8 + 189424835950v^7 - 5986089291485v^6 + 6451651885166v^5 - 8853691313006v^4 + 28244932267173v^3 - 25834105522797v^2 + 29532601568435*v - 17011911372159) / 1010003676847
$\beta_{10}$	$=$	$( - 86757664220 \nu^{11} - 64707732221 \nu^{10} - 564718779799 \nu^{9} + \cdots + 13315455160535 ) / 1010003676847$ (-86757664220v^11 - 64707732221v^10 - 564718779799v^9 + 981046406518v^8 - 6562149437v^7 + 6242542078232v^6 - 5279301694343v^5 + 8215251223637v^4 - 27716108570291v^3 + 21530851893947v^2 - 27108522841694*v + 13315455160535) / 1010003676847
$\beta_{11}$	$=$	$( 127836924130 \nu^{11} + 85006522887 \nu^{10} + 831492598779 \nu^{9} - 1466105836771 \nu^{8} + \cdots - 22714181280351 ) / 1010003676847$ (127836924130v^11 + 85006522887v^10 + 831492598779v^9 - 1466105836771v^8 + 234253438182v^7 - 9025207727477v^6 + 8412241513732v^5 - 13207313948854v^4 + 41084119438424v^3 - 34331893394458v^2 + 42015081262731*v - 22714181280351) / 1010003676847

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

$\nu$	$=$	$( \beta_{11} - \beta_{10} - 2\beta_{9} - \beta_{8} - 2\beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 ) / 2$ (b11 - b10 - 2b9 - b8 - 2b5 - b4 - b3 + b2 + b1) / 2
$\nu^{2}$	$=$	$-\beta_{11} + 2\beta_{10} + \beta_{9} + \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_1$ -b11 + 2*b10 + b9 + b7 + b6 + b5 + b4 + b3 + b1
$\nu^{3}$	$=$	$( \beta_{11} + \beta_{10} + 4 \beta_{9} + \beta_{8} - 6 \beta_{6} + 4 \beta_{5} - 3 \beta_{4} - \beta_{3} + \cdots + 8 ) / 2$ (b11 + b10 + 4b9 + b8 - 6b6 + 4b5 - 3b4 - b3 + b2 - 3*b1 + 8) / 2
$\nu^{4}$	$=$	$3 \beta_{11} - 7 \beta_{10} - 6 \beta_{9} + \beta_{8} - 4 \beta_{7} + 4 \beta_{6} - 6 \beta_{5} + \cdots - 1$ 3b11 - 7b10 - 6b9 + b8 - 4b7 + 4b6 - 6b5 - b4 - b3 + 6*b2 - 1
$\nu^{5}$	$=$	$( - 3 \beta_{11} + 29 \beta_{10} - 24 \beta_{9} - 27 \beta_{8} + 18 \beta_{7} + 22 \beta_{6} - 8 \beta_{5} + \cdots + 10 ) / 2$ (-3b11 + 29b10 - 24b9 - 27b8 + 18b7 + 22b6 - 8b5 + 19b4 - 9b3 - 25b2 + 11*b1 + 10) / 2
$\nu^{6}$	$=$	$- 7 \beta_{11} + 15 \beta_{10} + 44 \beta_{9} + 11 \beta_{8} + 12 \beta_{7} - 34 \beta_{6} + 31 \beta_{5} + \cdots - 1$ -7b11 + 15b10 + 44b9 + 11b8 + 12b7 - 34b6 + 31b5 - 15b4 - 13b3 + 5b2 + 6*b1 - 1
$\nu^{7}$	$=$	$( - 13 \beta_{11} - 101 \beta_{10} + 74 \beta_{9} + 137 \beta_{8} - 98 \beta_{7} + 86 \beta_{6} + \cdots + 28 ) / 2$ (-13b11 - 101b10 + 74b9 + 137b8 - 98b7 + 86b6 - 10b5 - 43b4 + 143b3 + 151b2 - 45*b1 + 28) / 2
$\nu^{8}$	$=$	$138 \beta_{11} - 16 \beta_{10} - 369 \beta_{9} - 212 \beta_{8} - 19 \beta_{7} + 56 \beta_{6} - 196 \beta_{5} + \cdots + 83$ 138b11 - 16b10 - 369b9 - 212b8 - 19b7 + 56b6 - 196b5 + 46b4 - 98b3 - 120b2 - 8*b1 + 83
$\nu^{9}$	$=$	$( - 329 \beta_{11} + 599 \beta_{10} + 572 \beta_{9} - 17 \beta_{8} + 666 \beta_{7} - 314 \beta_{6} + \cdots - 514 ) / 2$ (-329b11 + 599b10 + 572b9 - 17b8 + 666b7 - 314b6 + 550b5 + 307b4 - 705b3 - 273b2 + 371*b1 - 514) / 2
$\nu^{10}$	$=$	$- 668 \beta_{11} + 66 \beta_{10} + 1756 \beta_{9} + 1072 \beta_{8} - 120 \beta_{7} + 187 \beta_{6} + \cdots + 346$ -668b11 + 66b10 + 1756b9 + 1072b8 - 120b7 + 187b6 + 773b5 - 518b4 + 1018b3 + 677b2 - 234*b1 + 346
$\nu^{11}$	$=$	$( 4875 \beta_{11} - 4321 \beta_{10} - 8194 \beta_{9} - 2885 \beta_{8} - 4208 \beta_{7} - 36 \beta_{6} + \cdots + 1514 ) / 2$ (4875b11 - 4321b10 - 8194b9 - 2885b8 - 4208b7 - 36b6 - 5510b5 - 1919b4 + 459b3 + 787b2 - 907*b1 + 1514) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3060\mathbb{Z}\right)^\times$ .

$n$	$1261$	$1361$	$1531$	$1837$
$\chi(n)$	$\beta_{6}$	$1$	$1$	$1$

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}

361.1

−1.47591	+	1.40653i
−0.609374	−	2.33025i
0.671074	−	0.0762761i
0.150127	+	1.01316i
1.67572	−	0.131109i
−0.411629	−	1.88205i
−1.47591	−	1.40653i
−0.609374	+	2.33025i
0.671074	+	0.0762761i
0.150127	−	1.01316i
1.67572	+	0.131109i
−0.411629	+	1.88205i

−0.707107

0.707107i

−3.70031

−

3.70031i

361.2

−0.707107

0.707107i

−0.260019

−

0.260019i

361.3

−0.707107

0.707107i

2.54612

2.54612i

361.4

0.707107

−

0.707107i

−2.24258

−

2.24258i

361.5

0.707107

−

0.707107i

1.68323

1.68323i

361.6

0.707107

−

0.707107i

1.97356

1.97356i

1441.1

−0.707107

−

0.707107i

−3.70031

3.70031i

1441.2

−0.707107

−

0.707107i

−0.260019

0.260019i

1441.3

−0.707107

−

0.707107i

2.54612

−

2.54612i

1441.4

0.707107

0.707107i

−2.24258

2.24258i

1441.5

0.707107

0.707107i

1.68323

−

1.68323i

1441.6

0.707107

0.707107i

1.97356

−

1.97356i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
17.c	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3060.2.be.b		12
3.b	odd	2	1		340.2.o.a	✓	12
12.b	even	2	1		1360.2.bt.c		12
15.d	odd	2	1		1700.2.o.d		12
15.e	even	4	1		1700.2.m.c		12
15.e	even	4	1		1700.2.m.f		12
17.c	even	4	1	inner	3060.2.be.b		12
51.f	odd	4	1		340.2.o.a	✓	12
51.g	odd	8	1		5780.2.a.m		6
51.g	odd	8	1		5780.2.a.n		6
51.g	odd	8	2		5780.2.c.h		12
204.l	even	4	1		1360.2.bt.c		12
255.i	odd	4	1		1700.2.o.d		12
255.k	even	4	1		1700.2.m.c		12
255.r	even	4	1		1700.2.m.f		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
340.2.o.a	✓	12	3.b	odd	2	1
340.2.o.a	✓	12	51.f	odd	4	1
1360.2.bt.c		12	12.b	even	2	1
1360.2.bt.c		12	204.l	even	4	1
1700.2.m.c		12	15.e	even	4	1
1700.2.m.c		12	255.k	even	4	1
1700.2.m.f		12	15.e	even	4	1
1700.2.m.f		12	255.r	even	4	1
1700.2.o.d		12	15.d	odd	2	1
1700.2.o.d		12	255.i	odd	4	1
3060.2.be.b		12	1.a	even	1	1	trivial
3060.2.be.b		12	17.c	even	4	1	inner
5780.2.a.m		6	51.g	odd	8	1
5780.2.a.n		6	51.g	odd	8	1
5780.2.c.h		12	51.g	odd	8	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{7}^{12} - 44 T_{7}^{9} + 556 T_{7}^{8} - 960 T_{7}^{7} + 968 T_{7}^{6} - 4040 T_{7}^{5} + \cdots + 21316$ T7^12 - 44*T7^9 + 556*T7^8 - 960*T7^7 + 968*T7^6 - 4040*T7^5 + 45412*T7^4 - 98408*T7^3 + 100352*T7^2 + 65408*T7 + 21316 acting on $S_{2}^{\mathrm{new}}(3060, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{12}$ T^12
$3$	$T^{12}$ T^12
$5$	$(T^{4} + 1)^{3}$ (T^4 + 1)^3
$7$	$T^{12} - 44 T^{9} + \cdots + 21316$ T^12 - 44T^9 + 556T^8 - 960T^7 + 968T^6 - 4040T^5 + 45412T^4 - 98408T^3 + 100352T^2 + 65408*T + 21316
$11$	$T^{12} - 12 T^{11} + \cdots + 24964$ T^12 - 12T^11 + 72T^10 - 220T^9 + 836T^8 - 5680T^7 + 32168T^6 - 99208T^5 + 173688T^4 - 121752T^3 + 20000T^2 + 31600*T + 24964
$13$	$(T^{6} - 8 T^{5} - 2 T^{4} + \cdots + 68)^{2}$ (T^6 - 8T^5 - 2T^4 + 92T^3 - 128T^2 - 32*T + 68)^2
$17$	$T^{12} - 4 T^{11} + \cdots + 24137569$ T^12 - 4T^11 - 30T^10 + 212T^9 - 57T^8 - 2384T^7 + 9980T^6 - 40528T^5 - 16473T^4 + 1041556T^3 - 2505630T^2 - 5679428*T + 24137569
$19$	$T^{12} + 104 T^{10} + \cdots + 1183744$ T^12 + 104T^10 + 3440T^8 + 50176T^6 + 346880T^4 + 1071104*T^2 + 1183744
$23$	$T^{12} + \cdots + 220997956$ T^12 + 12T^11 + 72T^10 - 196T^9 + 584T^8 + 8528T^7 + 79496T^6 - 349792T^5 + 813516T^4 + 1450872T^3 + 17334272T^2 - 87531008*T + 220997956
$29$	$T^{12} + 4 T^{11} + \cdots + 33856$ T^12 + 4T^11 + 8T^10 - 296T^9 + 4972T^8 - 2208T^7 - 4800T^6 + 81600T^5 + 737200T^4 + 1157440T^3 + 881792T^2 - 244352*T + 33856
$31$	$T^{12} + \cdots + 1726734916$ T^12 + 8T^11 + 32T^10 - 188T^9 + 7328T^8 + 54048T^7 + 215560T^6 - 1234320T^5 + 5887184T^4 + 20737528T^3 + 90531968T^2 - 559150624*T + 1726734916
$37$	$T^{12} + \cdots + 17587003456$ T^12 + 20T^11 + 200T^10 + 840T^9 + 11100T^8 + 185344T^7 + 1839680T^6 + 7839552T^5 + 26536368T^4 + 193721792T^3 + 1908137088T^2 + 8192486016*T + 17587003456
$41$	$T^{12} + \cdots + 6066540544$ T^12 + 8T^11 + 32T^10 + 32T^9 + 28336T^8 + 221952T^7 + 869376T^6 + 1741824T^5 + 175323904T^4 + 1405282304T^3 + 5684338688T^2 + 8304730112*T + 6066540544
$43$	$T^{12} + 116 T^{10} + \cdots + 1296$ T^12 + 116T^10 + 3868T^8 + 29432T^6 + 52672T^4 + 19296*T^2 + 1296
$47$	$(T^{6} - 4 T^{5} + \cdots + 292)^{2}$ (T^6 - 4T^5 - 94T^4 + 228T^3 + 984T^2 + 960*T + 292)^2
$53$	$T^{12} + 144 T^{10} + \cdots + 135424$ T^12 + 144T^10 + 5296T^8 + 74976T^6 + 428864T^4 + 801536*T^2 + 135424
$59$	$T^{12} + \cdots + 154157056$ T^12 + 336T^10 + 41408T^8 + 2305280T^6 + 56932352T^4 + 494059520*T^2 + 154157056
$61$	$T^{12} + 8 T^{11} + \cdots + 295936$ T^12 + 8T^11 + 32T^10 - 208T^9 + 8016T^8 + 45632T^7 + 130176T^6 - 7936T^5 - 49920T^4 - 29184T^3 + 663552T^2 - 626688*T + 295936
$67$	$(T^{6} - 16 T^{5} + \cdots + 928964)^{2}$ (T^6 - 16T^5 - 186T^4 + 3972T^3 - 632T^2 - 237296*T + 928964)^2
$71$	$T^{12} + \cdots + 1942870084$ T^12 + 28T^11 + 392T^10 + 1700T^9 + 9452T^8 + 249912T^7 + 4737352T^6 + 15696552T^5 + 8381528T^4 + 139842536T^3 + 13764723200T^2 + 7313421760*T + 1942870084
$73$	$T^{12} + \cdots + 520569856$ T^12 + 48T^9 + 16320T^8 - 4096T^7 + 1152T^6 + 1856000T^5 + 45668352T^4 + 90827264T^3 + 78675968T^2 - 286203904*T + 520569856
$79$	$T^{12} + \cdots + 445969924$ T^12 + 24T^11 + 288T^10 - 812T^9 - 9672T^8 - 34072T^7 + 2297480T^6 - 21569120T^5 + 109332080T^4 - 339859400T^3 + 671537952T^2 - 773932464*T + 445969924
$83$	$T^{12} + 356 T^{10} + \cdots + 1336336$ T^12 + 356T^10 + 42276T^8 + 1957704T^6 + 29272944T^4 + 58275200*T^2 + 1336336
$89$	$(T^{6} - 28 T^{5} + \cdots + 32996)^{2}$ (T^6 - 28T^5 + 164T^4 + 804T^3 - 5064T^2 - 4976*T + 32996)^2
$97$	$T^{12} + 44 T^{11} + \cdots + 20720704$ T^12 + 44T^11 + 968T^10 + 11832T^9 + 85596T^8 + 302912T^7 + 469312T^6 - 416448T^5 + 1006256T^4 + 4632896T^3 + 8988800T^2 - 19300480*T + 20720704
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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

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 361.6
Significant digits:
Format:

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	3060.2.be.b

Level	$3060$
Weight	$2$
Character orbit	3060.be
Analytic conductor	$24.434$
Analytic rank	$0$
Dimension	$12$
Inner twists	$2$