LMFDB - Newform orbit 160.4.d.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [160,4,Mod(81,160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(160, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("160.81");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 160 = 2^{5} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	160.d (of order $2$, degree $1$, 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:	$9.44030560092$
Analytic rank:	$0$
Dimension:	$12$
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} - 4x^{11} + 7x^{10} - 12x^{9} + 21x^{8} - 68x^{6} + 336x^{4} - 768x^{3} + 1792x^{2} - 4096x + 4096 $ x^12 - 4x^11 + 7x^10 - 12x^9 + 21x^8 - 68x^6 + 336x^4 - 768x^3 + 1792x^2 - 4096*x + 4096
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{30}\cdot 5^{4} $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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_1 q^{3} + \beta_{2} q^{5} + (\beta_{4} - 2) q^{7} + ( - \beta_{8} - 9) q^{9} + (\beta_{3} + 2 \beta_{2} + \beta_1) q^{11} + ( - \beta_{6} + \beta_1) q^{13} + ( - \beta_{9} + 5) q^{15} + ( - \beta_{9} + \beta_{7} + \beta_{4}) q^{17}+ \cdots + (20 \beta_{11} - 18 \beta_{10} + \cdots - 5 \beta_1) q^{99}+O(q^{100}) $ q + b1 * q^3 + b2 * q^5 + (b4 - 2) * q^7 + (-b8 - 9) * q^9 + (b3 + 2b2 + b1) q^11 + (-b6 + b1) * q^13 + (-b9 + 5) * q^15 + (-b9 + b7 + b4) * q^17 + (-2b10 - b3 - 6b2 + b1) * q^19 + (-2b11 - b10 - b3 - 2b2 - 5b1) q^21 + (-2b9 - b8 - b5 - b4 - 51) q^23 - 25 * q^25 + (-3b11 + b10 - 2b6 + 4b2 - 13b1) * q^27 + (b10 - 2b6 + 3b3 - 5b1) q^29 + (-2b8 + 2b7 + 2b4 + 22) q^31 + (-7b9 - b7 - 2b5 - b4 - 20) * q^33 + (b11 + b10 - 2b6 + b3 - 2b2 - b1) * q^35 + (2b11 - 4b10 + 4b6 + 2b3 + 2b2 + 20b1) * q^37 + (2b9 + 2b7 + 2b5 - 50) q^39 + (8b9 + b8 + 8b4 + 6) * q^41 + (b11 + b10 - 2b6 - 2b3 + 24b2 + 22b1) * q^43 + (2b11 - 3b10 + b6 - 3b3 - 9b2 + 8b1) q^45 + (2b9 - 4b8 - 4b7 + b4 + 80) q^47 + (-6b9 - 3b8 - 2b7 + 4b5 + 6b4 + 113) q^49 + (-2b11 + 4b6 - 20b2) q^51 + (-2b11 + 10b10 - 3b6 + 24b2 + 17b1) q^53 + (-b9 - 5b8 - 2b7 + b5 - b4 - 36) * q^55 + (13b9 + 4b8 + 3b7 - 13b4 - 62) * q^57 + (10b11 - 8b10 + 12b6 - b3 + 18b2 + 7b1) q^59 + (-2b11 + 6b10 + 2b6 + 4b3 + 18b2 - 8b1) * q^61 + (-2b9 + 19b8 - 5b5 - 5b4 + 105) * q^63 + (-b9 + b7 + 2b5 - 7b4 - 2) * q^65 + (-b11 + 11b10 - 6b6 + 6b3 - 40b2 - 24b1) q^67 + (4b11 - b10 - 6b6 - 7b3 - 54b2 - 91b1) q^69 + (10b9 + 18b8 - 4b7 - 2b5 - 4b4 + 132) q^71 + (11b9 + 6b8 - 3b7 - 6b5 - 3b4 + 34) q^73 - 25b1 q^75 + (2b11 - 6b10 + 11b6 - 20b3 - 18b2 + 11b1) * q^77 + (8b9 - 8b8 + 2b7 + 10b5 - 8b4 - 168) q^79 + (-16b9 + 3b8 - 4b5 - 32b4 + 197) * q^81 + (-7b11 + 5b10 + 14b6 - 4b3 - 52b2 + 26b1) * q^83 + (6b11 - 4b10 + 3b6 + 6b3 - 11b1) q^85 + (-12b9 + 7b8 - b5 + 4b4 + 165) q^87 + (14b9 - 12b8 - 6b7 + 12b5 - 14b4 - 34) q^89 + (-6b11 + 4b10 - 4b6 - 10b3 + 120b2 + 10b1) * q^91 + (-14b11 + 8b10 + 2b6 + 6b3 + 40b2 - 46b1) * q^93 + (3b9 + 5b8 + 5b5 + 5b4 + 130) * q^95 + (-7b9 + 12b8 - b7 - 2b5 + 31b4 - 122) * q^97 + (20b11 - 18b10 - 8b6 - b3 - 190b2 - 5b1) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 28 q^{7} - 108 q^{9} + 60 q^{15} - 604 q^{23} - 300 q^{25} + 264 q^{31} - 232 q^{33} - 600 q^{39} + 40 q^{41} + 940 q^{47} + 1308 q^{49} - 440 q^{55} - 680 q^{57} + 1300 q^{63} + 1592 q^{71} + 432 q^{73}+ \cdots - 1584 q^{97}+O(q^{100}) $ 12 * q - 28 * q^7 - 108 * q^9 + 60 * q^15 - 604 * q^23 - 300 * q^25 + 264 * q^31 - 232 * q^33 - 600 * q^39 + 40 * q^41 + 940 * q^47 + 1308 * q^49 - 440 * q^55 - 680 * q^57 + 1300 * q^63 + 1592 * q^71 + 432 * q^73 - 2016 * q^79 + 2508 * q^81 + 1968 * q^87 - 424 * q^89 + 1520 * q^95 - 1584 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 4x^{11} + 7x^{10} - 12x^{9} + 21x^{8} - 68x^{6} + 336x^{4} - 768x^{3} + 1792x^{2} - 4096x + 4096 $ x^12 - 4*x^11 + 7*x^10 - 12*x^9 + 21*x^8 - 68*x^6 + 336*x^4 - 768*x^3 + 1792*x^2 - 4096*x + 4096 :

$\beta_{1}$	$=$	$ ( - 93 \nu^{11} + 204 \nu^{10} - 379 \nu^{9} + 388 \nu^{8} - 817 \nu^{7} - 1672 \nu^{6} + \cdots + 191488 ) / 15360 $ (-93v^11 + 204v^10 - 379v^9 + 388v^8 - 817v^7 - 1672v^6 + 4708v^5 + 4640v^4 - 20048v^3 + 34432v^2 - 96256*v + 191488) / 15360
$\beta_{2}$	$=$	$ ( 23 \nu^{11} - 60 \nu^{10} + 81 \nu^{9} - 116 \nu^{8} + 51 \nu^{7} + 480 \nu^{6} - 684 \nu^{5} + \cdots - 30720 ) / 3072 $ (23v^11 - 60v^10 + 81v^9 - 116v^8 + 51v^7 + 480v^6 - 684v^5 - 1024v^4 + 5616v^3 - 11520v^2 + 22528*v - 30720) / 3072
$\beta_{3}$	$=$	$ ( 177 \nu^{11} + 228 \nu^{10} + 263 \nu^{9} - 2420 \nu^{8} + 3797 \nu^{7} + 2696 \nu^{6} + \cdots - 502784 ) / 15360 $ (177v^11 + 228v^10 + 263v^9 - 2420v^8 + 3797v^7 + 2696v^6 + 4492v^5 - 544v^4 - 53360v^3 - 66176v^2 + 321536*v - 502784) / 15360
$\beta_{4}$	$=$	$ ( 163 \nu^{11} - 188 \nu^{10} + 5 \nu^{9} - 564 \nu^{8} + 1231 \nu^{7} - 1456 \nu^{6} - 956 \nu^{5} + \cdots - 168960 ) / 15360 $ (163v^11 - 188v^10 + 5v^9 - 564v^8 + 1231v^7 - 1456v^6 - 956v^5 - 16704v^4 + 8240v^3 + 28672v^2 - 9728*v - 168960) / 15360
$\beta_{5}$	$=$	$ ( - 5 \nu^{11} - 28 \nu^{10} + 109 \nu^{9} - 212 \nu^{8} + 263 \nu^{7} + 208 \nu^{6} - 284 \nu^{5} + \cdots - 36352 ) / 512 $ (-5v^11 - 28v^10 + 109v^9 - 212v^8 + 263v^7 + 208v^6 - 284v^5 + 1088v^4 - 848v^3 - 7680v^2 + 23040*v - 36352) / 512
$\beta_{6}$	$=$	$ ( - 165 \nu^{11} + 636 \nu^{10} - 307 \nu^{9} + 3028 \nu^{8} - 5209 \nu^{7} - 2488 \nu^{6} + \cdots + 934912 ) / 15360 $ (-165v^11 + 636v^10 - 307v^9 + 3028v^8 - 5209v^7 - 2488v^6 + 18436v^5 + 224v^4 - 48848v^3 - 21632v^2 - 489472*v + 934912) / 15360
$\beta_{7}$	$=$	$ ( 89 \nu^{11} - 244 \nu^{10} - 545 \nu^{9} + 1188 \nu^{8} + 1373 \nu^{7} + 112 \nu^{6} - 3508 \nu^{5} + \cdots - 49920 ) / 3840 $ (89v^11 - 244v^10 - 545v^9 + 1188v^8 + 1373v^7 + 112v^6 - 3508v^5 - 13632v^4 + 33040v^3 + 22016v^2 - 131584*v - 49920) / 3840
$\beta_{8}$	$=$	$ ( - 211 \nu^{11} + 476 \nu^{10} - 725 \nu^{9} + 1428 \nu^{8} - 1087 \nu^{7} - 2288 \nu^{6} + \cdots + 399360 ) / 7680 $ (-211v^11 + 476v^10 - 725v^9 + 1428v^8 - 1087v^7 - 2288v^6 + 11132v^5 + 11328v^4 - 54320v^3 + 121856v^2 - 193024*v + 399360) / 7680
$\beta_{9}$	$=$	$ ( 97 \nu^{11} - 212 \nu^{10} + 215 \nu^{9} - 636 \nu^{8} + 709 \nu^{7} + 2096 \nu^{6} - 4244 \nu^{5} + \cdots - 184320 ) / 3072 $ (97v^11 - 212v^10 + 215v^9 - 636v^8 + 709v^7 + 2096v^6 - 4244v^5 - 10176v^4 + 16400v^3 - 38912v^2 + 122368*v - 184320) / 3072
$\beta_{10}$	$=$	$ ( - 271 \nu^{11} + 852 \nu^{10} - 1401 \nu^{9} + 1948 \nu^{8} - 2475 \nu^{7} - 2376 \nu^{6} + \cdots + 605184 ) / 7680 $ (-271v^11 + 852v^10 - 1401v^9 + 1948v^8 - 2475v^7 - 2376v^6 + 8460v^5 + 13856v^4 - 87408v^3 + 191616v^2 - 450560*v + 605184) / 7680
$\beta_{11}$	$=$	$ ( - 1113 \nu^{11} + 1884 \nu^{10} - 2719 \nu^{9} + 7348 \nu^{8} - 11677 \nu^{7} - 17512 \nu^{6} + \cdots + 2219008 ) / 15360 $ (-1113v^11 + 1884v^10 - 2719v^9 + 7348v^8 - 11677v^7 - 17512v^6 + 19348v^5 + 98720v^4 - 236048v^3 + 380032v^2 - 1202176*v + 2219008) / 15360

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

$\nu$	$=$	$ ( \beta_{11} - 4 \beta_{10} + 3 \beta_{9} + 5 \beta_{8} - \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + \cdots + 52 ) / 160 $ (b11 - 4b10 + 3b9 + 5b8 - b7 - 2b6 - 2b5 - 3b4 + b3 - 8b2 - 6b1 + 52) / 160
$\nu^{2}$	$=$	$ ( - 2 \beta_{11} + 3 \beta_{10} - 3 \beta_{9} + 10 \beta_{8} - 3 \beta_{7} - 6 \beta_{6} - \beta_{5} + \cdots + 31 ) / 160 $ (-2b11 + 3b10 - 3b9 + 10b8 - 3b7 - 6b6 - b5 + 11b4 - 7b3 + 12b2 - 33b1 + 31) / 160
$\nu^{3}$	$=$	$ ( - 8 \beta_{11} + 2 \beta_{10} - 11 \beta_{9} - 5 \beta_{8} - \beta_{7} - 4 \beta_{6} - 2 \beta_{5} + \cdots + 102 ) / 80 $ (-8b11 + 2b10 - 11b9 - 5b8 - b7 - 4b6 - 2b5 - 13b4 - 8b3 - 8b2 + 8b1 + 102) / 80
$\nu^{4}$	$=$	$ ( 2 \beta_{11} + \beta_{10} - 11 \beta_{9} - 2 \beta_{8} + \beta_{7} - 6 \beta_{6} - \beta_{5} + \cdots + 35 ) / 32 $ (2b11 + b10 - 11b9 - 2b8 + b7 - 6b6 - b5 - 5b4 + 7b3 + 44b2 - 3b1 + 35) / 32
$\nu^{5}$	$=$	$ ( - 29 \beta_{11} - 24 \beta_{10} - 45 \beta_{9} + 85 \beta_{8} + 3 \beta_{7} + 18 \beta_{6} + \cdots - 1536 ) / 160 $ (-29b11 - 24b10 - 45b9 + 85b8 + 3b7 + 18b6 + 6b5 + 29b4 + 11b3 + 312b2 + 294*b1 - 1536) / 160
$\nu^{6}$	$=$	$ ( - 32 \beta_{11} + 83 \beta_{10} + 2 \beta_{9} + 40 \beta_{8} + 8 \beta_{7} + 4 \beta_{6} + 16 \beta_{5} + \cdots - 66 ) / 80 $ (-32b11 + 83b10 + 2b9 + 40b8 + 8b7 + 4b6 + 16b5 - 126b4 - 7b3 + 352b2 - 73*b1 - 66) / 80
$\nu^{7}$	$=$	$ ( - 73 \beta_{11} - 8 \beta_{10} - 143 \beta_{9} - 5 \beta_{8} + 141 \beta_{7} + 26 \beta_{6} + \cdots + 11928 ) / 160 $ (-73b11 - 8b10 - 143b9 - 5b8 + 141b7 + 26b6 + 102b5 - 177b4 + 147b3 - 904b2 - 622*b1 + 11928) / 160
$\nu^{8}$	$=$	$ ( 46 \beta_{11} - 91 \beta_{10} + 39 \beta_{9} + 138 \beta_{8} + 23 \beta_{7} + 46 \beta_{6} + \cdots - 1671 ) / 32 $ (46b11 - 91b10 + 39b9 + 138b8 + 23b7 + 46b6 - 31b5 - 79b4 + 43b3 - 236b2 - 735*b1 - 1671) / 32
$\nu^{9}$	$=$	$ ( - 108 \beta_{11} + 22 \beta_{10} - 217 \beta_{9} + 285 \beta_{8} - 195 \beta_{7} + 356 \beta_{6} + \cdots + 2310 ) / 80 $ (-108b11 + 22b10 - 217b9 + 285b8 - 195b7 + 356b6 + 110b5 + 545b4 - 388b3 - 1288b2 - 4192*b1 + 2310) / 80
$\nu^{10}$	$=$	$ ( - 1422 \beta_{11} + 903 \beta_{10} - 1489 \beta_{9} - 1930 \beta_{8} - 685 \beta_{7} + 334 \beta_{6} + \cdots + 22885 ) / 160 $ (-1422b11 + 903b10 - 1489b9 - 1930b8 - 685b7 + 334b6 - 815b5 - 3295b4 + 133b3 - 11628b2 - 9053*b1 + 22885) / 160
$\nu^{11}$	$=$	$ ( 1217 \beta_{11} + 412 \beta_{10} - 3995 \beta_{9} - 585 \beta_{8} - 203 \beta_{7} - 594 \beta_{6} + \cdots + 68196 ) / 160 $ (1217b11 + 412b10 - 3995b9 - 585b8 - 203b7 - 594b6 - 2186b5 + 5771b4 + 5437b3 + 18296b2 - 14682*b1 + 68196) / 160

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/160\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$101$
$\chi(n)$	$1$	$1$	$-1$

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} $

81.1

1.98839	+	0.215211i
1.23537	+	1.57285i
−1.86176	−	0.730647i
1.71681	−	1.02595i
−0.428316	+	1.95360i
−0.650488	−	1.89126i
−0.650488	+	1.89126i
−0.428316	−	1.95360i
1.71681	+	1.02595i
−1.86176	+	0.730647i
1.23537	−	1.57285i
1.98839	−	0.215211i

−

9.57890i

5.00000i

−21.5703

−64.7554

81.2

−

7.99849i

−

5.00000i

−9.93501

−36.9759

81.3

−

6.25785i

5.00000i

34.6280

−12.1606

81.4

−

4.24443i

−

5.00000i

14.6308

8.98481

81.5

−

1.51777i

5.00000i

−5.13620

24.6964

81.6

−

0.888401i

5.00000i

−26.6173

26.2107

81.7

0.888401i

−

5.00000i

−26.6173

26.2107

81.8

1.51777i

−

5.00000i

−5.13620

24.6964

81.9

4.24443i

5.00000i

14.6308

8.98481

81.10

6.25785i

−

5.00000i

34.6280

−12.1606

81.11

7.99849i

5.00000i

−9.93501

−36.9759

81.12

9.57890i

−

5.00000i

−21.5703

−64.7554

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	160.4.d.a		12
3.b	odd	2	1		1440.4.k.c		12
4.b	odd	2	1		40.4.d.a	✓	12
5.b	even	2	1		800.4.d.d		12
5.c	odd	4	1		800.4.f.b		12
5.c	odd	4	1		800.4.f.c		12
8.b	even	2	1	inner	160.4.d.a		12
8.d	odd	2	1		40.4.d.a	✓	12
12.b	even	2	1		360.4.k.c		12
16.e	even	4	1		1280.4.a.ba		6
16.e	even	4	1		1280.4.a.bd		6
16.f	odd	4	1		1280.4.a.bb		6
16.f	odd	4	1		1280.4.a.bc		6
20.d	odd	2	1		200.4.d.b		12
20.e	even	4	1		200.4.f.b		12
20.e	even	4	1		200.4.f.c		12
24.f	even	2	1		360.4.k.c		12
24.h	odd	2	1		1440.4.k.c		12
40.e	odd	2	1		200.4.d.b		12
40.f	even	2	1		800.4.d.d		12
40.i	odd	4	1		800.4.f.b		12
40.i	odd	4	1		800.4.f.c		12
40.k	even	4	1		200.4.f.b		12
40.k	even	4	1		200.4.f.c		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
40.4.d.a	✓	12	4.b	odd	2	1
40.4.d.a	✓	12	8.d	odd	2	1
160.4.d.a		12	1.a	even	1	1	trivial
160.4.d.a		12	8.b	even	2	1	inner
200.4.d.b		12	20.d	odd	2	1
200.4.d.b		12	40.e	odd	2	1
200.4.f.b		12	20.e	even	4	1
200.4.f.b		12	40.k	even	4	1
200.4.f.c		12	20.e	even	4	1
200.4.f.c		12	40.k	even	4	1
360.4.k.c		12	12.b	even	2	1
360.4.k.c		12	24.f	even	2	1
800.4.d.d		12	5.b	even	2	1
800.4.d.d		12	40.f	even	2	1
800.4.f.b		12	5.c	odd	4	1
800.4.f.b		12	40.i	odd	4	1
800.4.f.c		12	5.c	odd	4	1
800.4.f.c		12	40.i	odd	4	1
1280.4.a.ba		6	16.e	even	4	1
1280.4.a.bb		6	16.f	odd	4	1
1280.4.a.bc		6	16.f	odd	4	1
1280.4.a.bd		6	16.e	even	4	1
1440.4.k.c		12	3.b	odd	2	1
1440.4.k.c		12	24.h	odd	2	1

Hecke kernels

This newform subspace is the entire newspace $S_{4}^{\mathrm{new}}(160, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} $ T^12
$3$	$ T^{12} + 216 T^{10} + \cdots + 7529536 $ T^12 + 216T^10 + 16140T^8 + 493760T^6 + 5547312T^4 + 13618560*T^2 + 7529536
$5$	$ (T^{2} + 25)^{6} $ (T^2 + 25)^6
$7$	$ (T^{6} + 14 T^{5} + \cdots + 14843128)^{2} $ (T^6 + 14T^5 - 1258T^4 - 23408T^3 + 166612T^2 + 4186552*T + 14843128)^2
$11$	$ T^{12} + \cdots + 26\!\cdots\!00 $ T^12 + 10072T^10 + 35501552T^8 + 52214287616T^6 + 31474418953984T^4 + 5990669494016000*T^2 + 264112121858560000
$13$	$ T^{12} + \cdots + 24\!\cdots\!00 $ T^12 + 11144T^10 + 32345968T^8 + 34395950848T^6 + 13454415339264T^4 + 1388736007833600*T^2 + 24897810332160000
$17$	$ (T^{6} - 14500 T^{4} + \cdots - 7473839808)^{2} $ (T^6 - 14500T^4 - 102400T^3 + 43622832T^2 + 94617600T - 7473839808)^2
$19$	$ T^{12} + \cdots + 17\!\cdots\!24 $ T^12 + 46312T^10 + 741344240T^8 + 4896268004096T^6 + 12430900456140544T^4 + 9115013767034955776*T^2 + 1772608512783726874624
$23$	$ (T^{6} + 302 T^{5} + \cdots + 881168216)^{2} $ (T^6 + 302T^5 + 12942T^4 - 1768464T^3 - 67297772T^2 + 572113656*T + 881168216)^2
$29$	$ T^{12} + \cdots + 11\!\cdots\!00 $ T^12 + 92448T^10 + 1615955712T^8 + 9116845195264T^6 + 13691728754049024T^4 + 7210490578757222400*T^2 + 1127350354638274560000
$31$	$ (T^{6} + \cdots - 1437816300032)^{2} $ (T^6 - 132T^5 - 67624T^4 + 3324032T^3 + 598242624T^2 - 14378815744*T - 1437816300032)^2
$37$	$ T^{12} + \cdots + 21\!\cdots\!04 $ T^12 + 310360T^10 + 36606865904T^8 + 2053126014070016T^6 + 55340213982459145984T^4 + 627584725682702925780992*T^2 + 2111537125247224677443375104
$41$	$ (T^{6} + \cdots - 71667547865600)^{2} $ (T^6 - 20T^5 - 236520T^4 + 12143488T^3 + 12020264768T^2 - 248429610240*T - 71667547865600)^2
$43$	$ T^{12} + \cdots + 25\!\cdots\!04 $ T^12 + 338808T^10 + 39929204300T^8 + 2034641371605440T^6 + 45914998106297190192T^4 + 370442702927050601924480*T^2 + 258883621191661288765504
$47$	$ (T^{6} + \cdots - 72048375466472)^{2} $ (T^6 - 470T^5 - 170594T^4 + 47634640T^3 + 12457446740T^2 - 87202852120*T - 72048375466472)^2
$53$	$ T^{12} + \cdots + 86\!\cdots\!76 $ T^12 + 939912T^10 + 310944242544T^8 + 45493129845755648T^6 + 2987689602671252217600T^4 + 85513918002291204562028544*T^2 + 863790141396234424108547608576
$59$	$ T^{12} + \cdots + 10\!\cdots\!24 $ T^12 + 1889384T^10 + 1352048492016T^8 + 457619069256882944T^6 + 74713872967072499920640T^4 + 5209976767746147315458992128*T^2 + 105566813302722494834806277607424
$61$	$ T^{12} + \cdots + 61\!\cdots\!00 $ T^12 + 627448T^10 + 149193877616T^8 + 16827225536180480T^6 + 903747807700361707264T^4 + 19280170271395134962432000*T^2 + 61689356907795919619791360000
$67$	$ T^{12} + \cdots + 87\!\cdots\!36 $ T^12 + 1320888T^10 + 574605283404T^8 + 96965422362500032T^6 + 5040655716162487748400T^4 + 25238935930218138617498496*T^2 + 8788552197592891481600694336
$71$	$ (T^{6} + \cdots - 36\!\cdots\!48)^{2} $ (T^6 - 796T^5 - 994216T^4 + 646173568T^3 + 299720944960T^2 - 104342070638336*T - 36004201763291648)^2
$73$	$ (T^{6} + \cdots - 378730163491776)^{2} $ (T^6 - 216T^5 - 955764T^4 + 84325184T^3 + 66833909040T^2 - 4911943646592*T - 378730163491776)^2
$79$	$ (T^{6} + \cdots - 44\!\cdots\!00)^{2} $ (T^6 + 1008T^5 - 1164496T^4 - 914985472T^3 + 463896097536T^2 + 167913125703680*T - 44192643752652800)^2
$83$	$ T^{12} + \cdots + 78\!\cdots\!24 $ T^12 + 4256648T^10 + 6904943050860T^8 + 5310168953691257920T^6 + 1926040234196767308278832T^4 + 264990803482057714540774899840*T^2 + 784704573068492718516575676009024
$89$	$ (T^{6} + \cdots - 62\!\cdots\!00)^{2} $ (T^6 + 212T^5 - 2683156T^4 - 1209707808T^3 + 951729455216T^2 + 249244879280960*T - 62769827445803200)^2
$97$	$ (T^{6} + \cdots - 33\!\cdots\!28)^{2} $ (T^6 + 792T^5 - 1545556T^4 - 849246016T^3 + 358920178224T^2 + 113285581236608*T - 33531890854420928)^2
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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:	\( N \)	\(=\)	\( 160 = 2^{5} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	160.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_1 q^{3} + \beta_{2} q^{5} + (\beta_{4} - 2) q^{7} + ( - \beta_{8} - 9) q^{9} + (\beta_{3} + 2 \beta_{2} + \beta_1) q^{11} + ( - \beta_{6} + \beta_1) q^{13} + ( - \beta_{9} + 5) q^{15} + ( - \beta_{9} + \beta_{7} + \beta_{4}) q^{17}+ \cdots + (20 \beta_{11} - 18 \beta_{10} + \cdots - 5 \beta_1) q^{99}+O(q^{100}) \) q + b1 * q^3 + b2 * q^5 + (b4 - 2) * q^7 + (-b8 - 9) * q^9 + (b3 + 2b2 + b1) q^11 + (-b6 + b1) * q^13 + (-b9 + 5) * q^15 + (-b9 + b7 + b4) * q^17 + (-2b10 - b3 - 6b2 + b1) * q^19 + (-2b11 - b10 - b3 - 2b2 - 5b1) q^21 + (-2b9 - b8 - b5 - b4 - 51) q^23 - 25 * q^25 + (-3b11 + b10 - 2b6 + 4b2 - 13b1) * q^27 + (b10 - 2b6 + 3b3 - 5b1) q^29 + (-2b8 + 2b7 + 2b4 + 22) q^31 + (-7b9 - b7 - 2b5 - b4 - 20) * q^33 + (b11 + b10 - 2b6 + b3 - 2b2 - b1) * q^35 + (2b11 - 4b10 + 4b6 + 2b3 + 2b2 + 20b1) * q^37 + (2b9 + 2b7 + 2b5 - 50) q^39 + (8b9 + b8 + 8b4 + 6) * q^41 + (b11 + b10 - 2b6 - 2b3 + 24b2 + 22b1) * q^43 + (2b11 - 3b10 + b6 - 3b3 - 9b2 + 8b1) q^45 + (2b9 - 4b8 - 4b7 + b4 + 80) q^47 + (-6b9 - 3b8 - 2b7 + 4b5 + 6b4 + 113) q^49 + (-2b11 + 4b6 - 20b2) q^51 + (-2b11 + 10b10 - 3b6 + 24b2 + 17b1) q^53 + (-b9 - 5b8 - 2b7 + b5 - b4 - 36) * q^55 + (13b9 + 4b8 + 3b7 - 13b4 - 62) * q^57 + (10b11 - 8b10 + 12b6 - b3 + 18b2 + 7b1) q^59 + (-2b11 + 6b10 + 2b6 + 4b3 + 18b2 - 8b1) * q^61 + (-2b9 + 19b8 - 5b5 - 5b4 + 105) * q^63 + (-b9 + b7 + 2b5 - 7b4 - 2) * q^65 + (-b11 + 11b10 - 6b6 + 6b3 - 40b2 - 24b1) q^67 + (4b11 - b10 - 6b6 - 7b3 - 54b2 - 91b1) q^69 + (10b9 + 18b8 - 4b7 - 2b5 - 4b4 + 132) q^71 + (11b9 + 6b8 - 3b7 - 6b5 - 3b4 + 34) q^73 - 25b1 q^75 + (2b11 - 6b10 + 11b6 - 20b3 - 18b2 + 11b1) * q^77 + (8b9 - 8b8 + 2b7 + 10b5 - 8b4 - 168) q^79 + (-16b9 + 3b8 - 4b5 - 32b4 + 197) * q^81 + (-7b11 + 5b10 + 14b6 - 4b3 - 52b2 + 26b1) * q^83 + (6b11 - 4b10 + 3b6 + 6b3 - 11b1) q^85 + (-12b9 + 7b8 - b5 + 4b4 + 165) q^87 + (14b9 - 12b8 - 6b7 + 12b5 - 14b4 - 34) q^89 + (-6b11 + 4b10 - 4b6 - 10b3 + 120b2 + 10b1) * q^91 + (-14b11 + 8b10 + 2b6 + 6b3 + 40b2 - 46b1) * q^93 + (3b9 + 5b8 + 5b5 + 5b4 + 130) * q^95 + (-7b9 + 12b8 - b7 - 2b5 + 31b4 - 122) * q^97 + (20b11 - 18b10 - 8b6 - b3 - 190b2 - 5b1) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 28 q^{7} - 108 q^{9} + 60 q^{15} - 604 q^{23} - 300 q^{25} + 264 q^{31} - 232 q^{33} - 600 q^{39} + 40 q^{41} + 940 q^{47} + 1308 q^{49} - 440 q^{55} - 680 q^{57} + 1300 q^{63} + 1592 q^{71} + 432 q^{73}+ \cdots - 1584 q^{97}+O(q^{100}) \) 12 * q - 28 * q^7 - 108 * q^9 + 60 * q^15 - 604 * q^23 - 300 * q^25 + 264 * q^31 - 232 * q^33 - 600 * q^39 + 40 * q^41 + 940 * q^47 + 1308 * q^49 - 440 * q^55 - 680 * q^57 + 1300 * q^63 + 1592 * q^71 + 432 * q^73 - 2016 * q^79 + 2508 * q^81 + 1968 * q^87 - 424 * q^89 + 1520 * q^95 - 1584 * q^97

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	160.4.d.a

Level	$160$
Weight	$4$
Character orbit	160.d
Analytic conductor	$9.440$
Analytic rank	$0$
Dimension	$12$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(9.44030560092\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - 4x^{11} + 7x^{10} - 12x^{9} + 21x^{8} - 68x^{6} + 336x^{4} - 768x^{3} + 1792x^{2} - 4096x + 4096 \) x^12 - 4x^11 + 7x^10 - 12x^9 + 21x^8 - 68x^6 + 336x^4 - 768x^3 + 1792x^2 - 4096*x + 4096
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{30}\cdot 5^{4} \)
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( - 93 \nu^{11} + 204 \nu^{10} - 379 \nu^{9} + 388 \nu^{8} - 817 \nu^{7} - 1672 \nu^{6} + \cdots + 191488 ) / 15360 \) (-93v^11 + 204v^10 - 379v^9 + 388v^8 - 817v^7 - 1672v^6 + 4708v^5 + 4640v^4 - 20048v^3 + 34432v^2 - 96256*v + 191488) / 15360
\(\beta_{2}\)	\(=\)	\( ( 23 \nu^{11} - 60 \nu^{10} + 81 \nu^{9} - 116 \nu^{8} + 51 \nu^{7} + 480 \nu^{6} - 684 \nu^{5} + \cdots - 30720 ) / 3072 \) (23v^11 - 60v^10 + 81v^9 - 116v^8 + 51v^7 + 480v^6 - 684v^5 - 1024v^4 + 5616v^3 - 11520v^2 + 22528*v - 30720) / 3072
\(\beta_{3}\)	\(=\)	\( ( 177 \nu^{11} + 228 \nu^{10} + 263 \nu^{9} - 2420 \nu^{8} + 3797 \nu^{7} + 2696 \nu^{6} + \cdots - 502784 ) / 15360 \) (177v^11 + 228v^10 + 263v^9 - 2420v^8 + 3797v^7 + 2696v^6 + 4492v^5 - 544v^4 - 53360v^3 - 66176v^2 + 321536*v - 502784) / 15360
\(\beta_{4}\)	\(=\)	\( ( 163 \nu^{11} - 188 \nu^{10} + 5 \nu^{9} - 564 \nu^{8} + 1231 \nu^{7} - 1456 \nu^{6} - 956 \nu^{5} + \cdots - 168960 ) / 15360 \) (163v^11 - 188v^10 + 5v^9 - 564v^8 + 1231v^7 - 1456v^6 - 956v^5 - 16704v^4 + 8240v^3 + 28672v^2 - 9728*v - 168960) / 15360
\(\beta_{5}\)	\(=\)	\( ( - 5 \nu^{11} - 28 \nu^{10} + 109 \nu^{9} - 212 \nu^{8} + 263 \nu^{7} + 208 \nu^{6} - 284 \nu^{5} + \cdots - 36352 ) / 512 \) (-5v^11 - 28v^10 + 109v^9 - 212v^8 + 263v^7 + 208v^6 - 284v^5 + 1088v^4 - 848v^3 - 7680v^2 + 23040*v - 36352) / 512
\(\beta_{6}\)	\(=\)	\( ( - 165 \nu^{11} + 636 \nu^{10} - 307 \nu^{9} + 3028 \nu^{8} - 5209 \nu^{7} - 2488 \nu^{6} + \cdots + 934912 ) / 15360 \) (-165v^11 + 636v^10 - 307v^9 + 3028v^8 - 5209v^7 - 2488v^6 + 18436v^5 + 224v^4 - 48848v^3 - 21632v^2 - 489472*v + 934912) / 15360
\(\beta_{7}\)	\(=\)	\( ( 89 \nu^{11} - 244 \nu^{10} - 545 \nu^{9} + 1188 \nu^{8} + 1373 \nu^{7} + 112 \nu^{6} - 3508 \nu^{5} + \cdots - 49920 ) / 3840 \) (89v^11 - 244v^10 - 545v^9 + 1188v^8 + 1373v^7 + 112v^6 - 3508v^5 - 13632v^4 + 33040v^3 + 22016v^2 - 131584*v - 49920) / 3840
\(\beta_{8}\)	\(=\)	\( ( - 211 \nu^{11} + 476 \nu^{10} - 725 \nu^{9} + 1428 \nu^{8} - 1087 \nu^{7} - 2288 \nu^{6} + \cdots + 399360 ) / 7680 \) (-211v^11 + 476v^10 - 725v^9 + 1428v^8 - 1087v^7 - 2288v^6 + 11132v^5 + 11328v^4 - 54320v^3 + 121856v^2 - 193024*v + 399360) / 7680
\(\beta_{9}\)	\(=\)	\( ( 97 \nu^{11} - 212 \nu^{10} + 215 \nu^{9} - 636 \nu^{8} + 709 \nu^{7} + 2096 \nu^{6} - 4244 \nu^{5} + \cdots - 184320 ) / 3072 \) (97v^11 - 212v^10 + 215v^9 - 636v^8 + 709v^7 + 2096v^6 - 4244v^5 - 10176v^4 + 16400v^3 - 38912v^2 + 122368*v - 184320) / 3072
\(\beta_{10}\)	\(=\)	\( ( - 271 \nu^{11} + 852 \nu^{10} - 1401 \nu^{9} + 1948 \nu^{8} - 2475 \nu^{7} - 2376 \nu^{6} + \cdots + 605184 ) / 7680 \) (-271v^11 + 852v^10 - 1401v^9 + 1948v^8 - 2475v^7 - 2376v^6 + 8460v^5 + 13856v^4 - 87408v^3 + 191616v^2 - 450560*v + 605184) / 7680
\(\beta_{11}\)	\(=\)	\( ( - 1113 \nu^{11} + 1884 \nu^{10} - 2719 \nu^{9} + 7348 \nu^{8} - 11677 \nu^{7} - 17512 \nu^{6} + \cdots + 2219008 ) / 15360 \) (-1113v^11 + 1884v^10 - 2719v^9 + 7348v^8 - 11677v^7 - 17512v^6 + 19348v^5 + 98720v^4 - 236048v^3 + 380032v^2 - 1202176*v + 2219008) / 15360

\(\nu\)	\(=\)	\( ( \beta_{11} - 4 \beta_{10} + 3 \beta_{9} + 5 \beta_{8} - \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + \cdots + 52 ) / 160 \) (b11 - 4b10 + 3b9 + 5b8 - b7 - 2b6 - 2b5 - 3b4 + b3 - 8b2 - 6b1 + 52) / 160
\(\nu^{2}\)	\(=\)	\( ( - 2 \beta_{11} + 3 \beta_{10} - 3 \beta_{9} + 10 \beta_{8} - 3 \beta_{7} - 6 \beta_{6} - \beta_{5} + \cdots + 31 ) / 160 \) (-2b11 + 3b10 - 3b9 + 10b8 - 3b7 - 6b6 - b5 + 11b4 - 7b3 + 12b2 - 33b1 + 31) / 160
\(\nu^{3}\)	\(=\)	\( ( - 8 \beta_{11} + 2 \beta_{10} - 11 \beta_{9} - 5 \beta_{8} - \beta_{7} - 4 \beta_{6} - 2 \beta_{5} + \cdots + 102 ) / 80 \) (-8b11 + 2b10 - 11b9 - 5b8 - b7 - 4b6 - 2b5 - 13b4 - 8b3 - 8b2 + 8b1 + 102) / 80
\(\nu^{4}\)	\(=\)	\( ( 2 \beta_{11} + \beta_{10} - 11 \beta_{9} - 2 \beta_{8} + \beta_{7} - 6 \beta_{6} - \beta_{5} + \cdots + 35 ) / 32 \) (2b11 + b10 - 11b9 - 2b8 + b7 - 6b6 - b5 - 5b4 + 7b3 + 44b2 - 3b1 + 35) / 32
\(\nu^{5}\)	\(=\)	\( ( - 29 \beta_{11} - 24 \beta_{10} - 45 \beta_{9} + 85 \beta_{8} + 3 \beta_{7} + 18 \beta_{6} + \cdots - 1536 ) / 160 \) (-29b11 - 24b10 - 45b9 + 85b8 + 3b7 + 18b6 + 6b5 + 29b4 + 11b3 + 312b2 + 294*b1 - 1536) / 160
\(\nu^{6}\)	\(=\)	\( ( - 32 \beta_{11} + 83 \beta_{10} + 2 \beta_{9} + 40 \beta_{8} + 8 \beta_{7} + 4 \beta_{6} + 16 \beta_{5} + \cdots - 66 ) / 80 \) (-32b11 + 83b10 + 2b9 + 40b8 + 8b7 + 4b6 + 16b5 - 126b4 - 7b3 + 352b2 - 73*b1 - 66) / 80
\(\nu^{7}\)	\(=\)	\( ( - 73 \beta_{11} - 8 \beta_{10} - 143 \beta_{9} - 5 \beta_{8} + 141 \beta_{7} + 26 \beta_{6} + \cdots + 11928 ) / 160 \) (-73b11 - 8b10 - 143b9 - 5b8 + 141b7 + 26b6 + 102b5 - 177b4 + 147b3 - 904b2 - 622*b1 + 11928) / 160
\(\nu^{8}\)	\(=\)	\( ( 46 \beta_{11} - 91 \beta_{10} + 39 \beta_{9} + 138 \beta_{8} + 23 \beta_{7} + 46 \beta_{6} + \cdots - 1671 ) / 32 \) (46b11 - 91b10 + 39b9 + 138b8 + 23b7 + 46b6 - 31b5 - 79b4 + 43b3 - 236b2 - 735*b1 - 1671) / 32
\(\nu^{9}\)	\(=\)	\( ( - 108 \beta_{11} + 22 \beta_{10} - 217 \beta_{9} + 285 \beta_{8} - 195 \beta_{7} + 356 \beta_{6} + \cdots + 2310 ) / 80 \) (-108b11 + 22b10 - 217b9 + 285b8 - 195b7 + 356b6 + 110b5 + 545b4 - 388b3 - 1288b2 - 4192*b1 + 2310) / 80
\(\nu^{10}\)	\(=\)	\( ( - 1422 \beta_{11} + 903 \beta_{10} - 1489 \beta_{9} - 1930 \beta_{8} - 685 \beta_{7} + 334 \beta_{6} + \cdots + 22885 ) / 160 \) (-1422b11 + 903b10 - 1489b9 - 1930b8 - 685b7 + 334b6 - 815b5 - 3295b4 + 133b3 - 11628b2 - 9053*b1 + 22885) / 160
\(\nu^{11}\)	\(=\)	\( ( 1217 \beta_{11} + 412 \beta_{10} - 3995 \beta_{9} - 585 \beta_{8} - 203 \beta_{7} - 594 \beta_{6} + \cdots + 68196 ) / 160 \) (1217b11 + 412b10 - 3995b9 - 585b8 - 203b7 - 594b6 - 2186b5 + 5771b4 + 5437b3 + 18296b2 - 14682*b1 + 68196) / 160

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 81.12
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} \) T^12
$3$	\( T^{12} + 216 T^{10} + \cdots + 7529536 \) T^12 + 216T^10 + 16140T^8 + 493760T^6 + 5547312T^4 + 13618560*T^2 + 7529536
$5$	\( (T^{2} + 25)^{6} \) (T^2 + 25)^6
$7$	\( (T^{6} + 14 T^{5} + \cdots + 14843128)^{2} \) (T^6 + 14T^5 - 1258T^4 - 23408T^3 + 166612T^2 + 4186552*T + 14843128)^2
$11$	\( T^{12} + \cdots + 26\!\cdots\!00 \) T^12 + 10072T^10 + 35501552T^8 + 52214287616T^6 + 31474418953984T^4 + 5990669494016000*T^2 + 264112121858560000
$13$	\( T^{12} + \cdots + 24\!\cdots\!00 \) T^12 + 11144T^10 + 32345968T^8 + 34395950848T^6 + 13454415339264T^4 + 1388736007833600*T^2 + 24897810332160000
$17$	\( (T^{6} - 14500 T^{4} + \cdots - 7473839808)^{2} \) (T^6 - 14500T^4 - 102400T^3 + 43622832T^2 + 94617600T - 7473839808)^2
$19$	\( T^{12} + \cdots + 17\!\cdots\!24 \) T^12 + 46312T^10 + 741344240T^8 + 4896268004096T^6 + 12430900456140544T^4 + 9115013767034955776*T^2 + 1772608512783726874624
$23$	\( (T^{6} + 302 T^{5} + \cdots + 881168216)^{2} \) (T^6 + 302T^5 + 12942T^4 - 1768464T^3 - 67297772T^2 + 572113656*T + 881168216)^2
$29$	\( T^{12} + \cdots + 11\!\cdots\!00 \) T^12 + 92448T^10 + 1615955712T^8 + 9116845195264T^6 + 13691728754049024T^4 + 7210490578757222400*T^2 + 1127350354638274560000
$31$	\( (T^{6} + \cdots - 1437816300032)^{2} \) (T^6 - 132T^5 - 67624T^4 + 3324032T^3 + 598242624T^2 - 14378815744*T - 1437816300032)^2
$37$	\( T^{12} + \cdots + 21\!\cdots\!04 \) T^12 + 310360T^10 + 36606865904T^8 + 2053126014070016T^6 + 55340213982459145984T^4 + 627584725682702925780992*T^2 + 2111537125247224677443375104
$41$	\( (T^{6} + \cdots - 71667547865600)^{2} \) (T^6 - 20T^5 - 236520T^4 + 12143488T^3 + 12020264768T^2 - 248429610240*T - 71667547865600)^2
$43$	\( T^{12} + \cdots + 25\!\cdots\!04 \) T^12 + 338808T^10 + 39929204300T^8 + 2034641371605440T^6 + 45914998106297190192T^4 + 370442702927050601924480*T^2 + 258883621191661288765504
$47$	\( (T^{6} + \cdots - 72048375466472)^{2} \) (T^6 - 470T^5 - 170594T^4 + 47634640T^3 + 12457446740T^2 - 87202852120*T - 72048375466472)^2
$53$	\( T^{12} + \cdots + 86\!\cdots\!76 \) T^12 + 939912T^10 + 310944242544T^8 + 45493129845755648T^6 + 2987689602671252217600T^4 + 85513918002291204562028544*T^2 + 863790141396234424108547608576
$59$	\( T^{12} + \cdots + 10\!\cdots\!24 \) T^12 + 1889384T^10 + 1352048492016T^8 + 457619069256882944T^6 + 74713872967072499920640T^4 + 5209976767746147315458992128*T^2 + 105566813302722494834806277607424
$61$	\( T^{12} + \cdots + 61\!\cdots\!00 \) T^12 + 627448T^10 + 149193877616T^8 + 16827225536180480T^6 + 903747807700361707264T^4 + 19280170271395134962432000*T^2 + 61689356907795919619791360000
$67$	\( T^{12} + \cdots + 87\!\cdots\!36 \) T^12 + 1320888T^10 + 574605283404T^8 + 96965422362500032T^6 + 5040655716162487748400T^4 + 25238935930218138617498496*T^2 + 8788552197592891481600694336
$71$	\( (T^{6} + \cdots - 36\!\cdots\!48)^{2} \) (T^6 - 796T^5 - 994216T^4 + 646173568T^3 + 299720944960T^2 - 104342070638336*T - 36004201763291648)^2
$73$	\( (T^{6} + \cdots - 378730163491776)^{2} \) (T^6 - 216T^5 - 955764T^4 + 84325184T^3 + 66833909040T^2 - 4911943646592*T - 378730163491776)^2
$79$	\( (T^{6} + \cdots - 44\!\cdots\!00)^{2} \) (T^6 + 1008T^5 - 1164496T^4 - 914985472T^3 + 463896097536T^2 + 167913125703680*T - 44192643752652800)^2
$83$	\( T^{12} + \cdots + 78\!\cdots\!24 \) T^12 + 4256648T^10 + 6904943050860T^8 + 5310168953691257920T^6 + 1926040234196767308278832T^4 + 264990803482057714540774899840*T^2 + 784704573068492718516575676009024
$89$	\( (T^{6} + \cdots - 62\!\cdots\!00)^{2} \) (T^6 + 212T^5 - 2683156T^4 - 1209707808T^3 + 951729455216T^2 + 249244879280960*T - 62769827445803200)^2
$97$	\( (T^{6} + \cdots - 33\!\cdots\!28)^{2} \) (T^6 + 792T^5 - 1545556T^4 - 849246016T^3 + 358920178224T^2 + 113285581236608*T - 33531890854420928)^2
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\(n\)	\(31\)	\(97\)	\(101\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1\)