LMFDB - Newform orbit 980.2.c.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [980,2,Mod(979,980)] 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("980.979"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(980, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 980 = 2^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	980.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	no
Analytic conductor:	$7.82533939809$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 12x^{14} + 102x^{12} - 388x^{10} + 573x^{8} + 48x^{6} - 700x^{4} + 144x^{2} + 324 $ x^16 - 12x^14 + 102x^12 - 388x^10 + 573x^8 + 48x^6 - 700x^4 + 144*x^2 + 324
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
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 $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_{4} q^{2} + ( - \beta_{12} - \beta_{10} - \beta_{8}) q^{3} + (\beta_{9} - \beta_1) q^{4} + (\beta_{15} + \beta_{2}) q^{5} + (\beta_{15} + \beta_{6} - 2 \beta_{3}) q^{6} + (\beta_{14} + \beta_{11}) q^{8}+ \cdots + (2 \beta_{13} + 6 \beta_{9} - 3 \beta_1) q^{99}+O(q^{100}) $ q - b4 * q^2 + (-b12 - b10 - b8) * q^3 + (b9 - b1) * q^4 + (b15 + b2) * q^5 + (b15 + b6 - 2b3) q^6 + (b14 + b11) * q^8 + (3b1 - 2) q^9 + (b15 - b8 - b6 - b2) * q^10 - b13 * q^11 + (-b12 - b10 - 2b2) q^12 + (b12 - b8 + b2) * q^13 + (b14 + b13 + 2b9 + b4 - b1) q^15 + (b13 - 3) * q^16 + b2 * q^17 + (3b14 - 3b11 + 2b4) q^18 + (b15 + 2b7 + 2b6 - b3) * q^19 + (-b12 - b10 - 2b8 + b7 + b6 - b3) q^20 + (2b5 - b4) q^22 + (2b14 - 3b11 - 3b5 + 2b4) * q^23 + (-2b15 - b7 + 3b6 - b3) * q^24 + (2b14 - 2b4 - 2b1 + 1) q^25 + (b15 + b7 - 2b6 - b3) q^26 + (2b12 + 5b10 + 2b8) q^27 + (b1 + 3) * q^29 + (3b14 + b11 - b9 - 2b5 + b4 + b1 - 2) * q^30 + (b15 + 2b7) q^31 + (-2b5 + 4b4) * q^32 + (2b12 - 2b8 + b2) * q^33 + (b15 - b6) * q^34 + (-3b13 - 2b9 + 2b1 - 3) q^36 + (-4b14 - b11 + b5 + 4b4) * q^37 + (3b12 - b10 + b8 - 3b2) * q^38 + (3b13 + 4b9 - 2b1) q^39 + (2b15 + b12 - b10 + b8 + b7 + b6 - 3b3 - b2) * q^40 - b3 * q^41 + (-3b14 + b11 + b5 - 3b4) * q^43 + (-b9 - 3b1) q^44 + (b15 - 3b12 + 3b8 + 3b3 - 2b2) * q^45 + (-3b13 + b9 + 5b1 - 1) * q^46 + (3b12 + b10 + 3b8) * q^47 + (b12 + 3b10 + 5b8 - b2) * q^48 + (-2b14 + 2b11 + 2b9 - b4 - 2b1 - 4) * q^50 + (b13 + 2b9 - b1) q^51 + (-2b12 - 4b10 - 3b8 + b2) q^52 + (b14 - 2b11 + 2b5 - b4) * q^53 + (b15 + 3b7 - 2b6 + 7b3) q^54 + (b15 + b12 + 3b10 + b8 + 2b7) * q^55 + (-3b14 - b11 + b5 + 3b4) * q^57 + (b14 - b11 - 3b4) q^58 + (2b15 + 4b7 - 2b6 + b3) q^59 + (-b14 + b13 - b11 + b9 + 2b4 + 3b1 - 7) * q^60 + (b15 - b3) * q^61 + (2b12 - 2b10 - b8 - b2) * q^62 + (-2b9 + 6b1) * q^64 + (b11 - b5 - 4b1 + 5) q^65 + (b15 + 2b7 - 3b6 - 2b3) q^66 + (b14 + 2b11 + 2b5 + b4) * q^67 + (-b12 - b10 - 2b8) q^68 + (-b15 + 12b3) q^69 + (-2b13 + 2b9 - b1) * q^71 + (-2b14 - 2b11 + 6b5) q^72 + (-3b12 + 3b8 + 8b2) q^73 + (-b13 - 5b9 + 3b1 + 9) * q^74 + (-3b12 - 5b10 - 3b8 + 4b6 - 2b3) q^75 + (-8b15 - 2b7) * q^76 + (6b14 + 2b11 - 6b5 + 3b4) * q^78 + (b13 - 2b9 + b1) q^79 + (-4b15 - b12 - 3b10 - b8 - 2b7 - 3b2) * q^80 + (-3b1 + 7) q^81 + (-b12 - b10) * q^82 - 6b10 q^83 + (b14 - b4 - b1 + 3) * q^85 + (b13 + 2b9 - 4b1 + 5) * q^86 + (-2b12 - b10 - 2b8) * q^87 + (-5b14 + 3b11) * q^88 - 4b3 q^89 + (-2b15 + 3b12 + 3b10 - b8 - 3b7 + 5b6 + 3b3 - b2) * q^90 + (7b14 - 5b11 + 6b5 - 2b4) * q^92 + (-b14 + 2b11 - 2b5 + b4) * q^93 + (-5b15 - 2b7 - 3b6 + 4b3) * q^94 + (3b14 + 2b13 + b11 - 4b9 + b5 + 3b4 + 2b1) q^95 + (-4b15 - 2b7 + 8b3) q^96 + (-7b12 + 7b8 + 3b2) q^97 + (2b13 + 6b9 - 3b1) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 32 q^{9} - 48 q^{16} + 16 q^{25} + 48 q^{29} - 32 q^{30} - 48 q^{36} - 16 q^{46} - 64 q^{50} - 112 q^{60} + 80 q^{65} + 144 q^{74} + 112 q^{81} + 48 q^{85} + 80 q^{86}+O(q^{100}) $ 16 * q - 32 * q^9 - 48 * q^16 + 16 * q^25 + 48 * q^29 - 32 * q^30 - 48 * q^36 - 16 * q^46 - 64 * q^50 - 112 * q^60 + 80 * q^65 + 144 * q^74 + 112 * q^81 + 48 * q^85 + 80 * q^86

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 12x^{14} + 102x^{12} - 388x^{10} + 573x^{8} + 48x^{6} - 700x^{4} + 144x^{2} + 324 $ x^16 - 12*x^14 + 102*x^12 - 388*x^10 + 573*x^8 + 48*x^6 - 700*x^4 + 144*x^2 + 324 :

$\beta_{1}$	$=$	$ ( \nu^{14} - 6\nu^{12} + 44\nu^{10} + 82\nu^{8} - 599\nu^{6} + 244\nu^{4} + 1234\nu^{2} - 288 ) / 612 $ (v^14 - 6v^12 + 44v^10 + 82v^8 - 599v^6 + 244v^4 + 1234v^2 - 288) / 612
$\beta_{2}$	$=$	$ ( 57\nu^{14} - 640\nu^{12} + 5388\nu^{10} - 18592\nu^{8} + 23169\nu^{6} + 10504\nu^{4} - 47018\nu^{2} + 7776 ) / 17136 $ (57v^14 - 640v^12 + 5388v^10 - 18592v^8 + 23169v^6 + 10504v^4 - 47018*v^2 + 7776) / 17136
$\beta_{3}$	$=$	$ ( 11\nu^{15} - 90\nu^{13} + 684\nu^{11} - 632\nu^{9} - 4581\nu^{7} + 9870\nu^{5} + 7282\nu^{3} - 16380\nu ) / 7344 $ (11v^15 - 90v^13 + 684v^11 - 632v^9 - 4581v^7 + 9870v^5 + 7282v^3 - 16380v) / 7344
$\beta_{4}$	$=$	$ ( 62 \nu^{15} - 459 \nu^{13} + 3096 \nu^{11} + 2800 \nu^{9} - 57042 \nu^{7} + 105297 \nu^{5} + \cdots - 73170 \nu ) / 25704 $ (62v^15 - 459v^13 + 3096v^11 + 2800v^9 - 57042v^7 + 105297v^5 + 13180v^3 - 73170v) / 25704
$\beta_{5}$	$=$	$ ( - 82 \nu^{15} + 1023 \nu^{13} - 8628 \nu^{11} + 33544 \nu^{9} - 43878 \nu^{7} - 40053 \nu^{5} + \cdots + 7866 \nu ) / 25704 $ (-82v^15 + 1023v^13 - 8628v^11 + 33544v^9 - 43878v^7 - 40053v^5 + 96868v^3 + 7866v) / 25704
$\beta_{6}$	$=$	$ ( - 79 \nu^{15} + 1224 \nu^{13} - 11154 \nu^{11} + 56938 \nu^{9} - 137625 \nu^{7} + 130374 \nu^{5} + \cdots - 104004 \nu ) / 25704 $ (-79v^15 + 1224v^13 - 11154v^11 + 56938v^9 - 137625v^7 + 130374v^5 + 25954v^3 - 104004v) / 25704
$\beta_{7}$	$=$	$ ( - 59 \nu^{15} + 436 \nu^{13} - 3200 \nu^{11} - 308 \nu^{9} + 34793 \nu^{7} - 55424 \nu^{5} + \cdots + 13632 \nu ) / 17136 $ (-59v^15 + 436v^13 - 3200v^11 - 308v^9 + 34793v^7 - 55424v^5 - 14458v^3 + 13632v) / 17136
$\beta_{8}$	$=$	$ ( 53\nu^{14} - 614\nu^{12} + 5116\nu^{10} - 18200\nu^{8} + 20989\nu^{6} + 10530\nu^{4} - 14738\nu^{2} - 15012 ) / 5712 $ (53v^14 - 614v^12 + 5116v^10 - 18200v^8 + 20989v^6 + 10530v^4 - 14738*v^2 - 15012) / 5712
$\beta_{9}$	$=$	$ ( 23\nu^{14} - 223\nu^{12} + 1760\nu^{10} - 4081\nu^{8} - 2795\nu^{6} + 16220\nu^{4} - 2966\nu^{2} - 10908 ) / 2142 $ (23v^14 - 223v^12 + 1760v^10 - 4081v^8 - 2795v^6 + 16220v^4 - 2966*v^2 - 10908) / 2142
$\beta_{10}$	$=$	$ ( -27\nu^{14} + 274\nu^{12} - 2212\nu^{10} + 5976\nu^{8} - 995\nu^{6} - 15654\nu^{4} + 6766\nu^{2} + 10476 ) / 2448 $ (-27v^14 + 274v^12 - 2212v^10 + 5976v^8 - 995v^6 - 15654v^4 + 6766*v^2 + 10476) / 2448
$\beta_{11}$	$=$	$ ( 62 \nu^{15} - 669 \nu^{13} + 5679 \nu^{11} - 18893 \nu^{9} + 26811 \nu^{7} - 6696 \nu^{5} + \cdots + 30780 \nu ) / 12852 $ (62v^15 - 669v^13 + 5679v^11 - 18893v^9 + 26811v^7 - 6696v^5 - 15338v^3 + 30780v) / 12852
$\beta_{12}$	$=$	$ ( 275 \nu^{14} - 2960 \nu^{12} + 24524 \nu^{10} - 78008 \nu^{8} + 74611 \nu^{6} + 55056 \nu^{4} + \cdots - 13680 ) / 17136 $ (275v^14 - 2960v^12 + 24524v^10 - 78008v^8 + 74611v^6 + 55056v^4 - 67198*v^2 - 13680) / 17136
$\beta_{13}$	$=$	$ ( -24\nu^{14} + 268\nu^{12} - 2248\nu^{10} + 7686\nu^{8} - 9440\nu^{6} - 2280\nu^{4} + 6304\nu^{2} + 1683 ) / 1071 $ (-24v^14 + 268v^12 - 2248v^10 + 7686v^8 - 9440v^6 - 2280v^4 + 6304*v^2 + 1683) / 1071
$\beta_{14}$	$=$	$ ( - 265 \nu^{15} + 2958 \nu^{13} - 24726 \nu^{11} + 84238 \nu^{9} - 99303 \nu^{7} - 25224 \nu^{5} + \cdots + 40536 \nu ) / 25704 $ (-265v^15 + 2958v^13 - 24726v^11 + 84238v^9 - 99303v^7 - 25224v^5 + 57814v^3 + 40536v) / 25704
$\beta_{15}$	$=$	$ ( 89 \nu^{15} - 960 \nu^{13} + 7908 \nu^{11} - 24920 \nu^{9} + 20649 \nu^{7} + 28032 \nu^{5} + \cdots - 13968 \nu ) / 7344 $ (89v^15 - 960v^13 + 7908v^11 - 24920v^9 + 20649v^7 + 28032v^5 - 25130v^3 - 13968v) / 7344

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

$\nu$	$=$	$ ( -\beta_{15} - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} ) / 2 $ (-b15 - b7 - b6 - b5 + b4) / 2
$\nu^{2}$	$=$	$ ( \beta_{13} - 2\beta_{10} - 2\beta_{9} + 2\beta_{8} + 2\beta _1 + 3 ) / 2 $ (b13 - 2b10 - 2b9 + 2b8 + 2b1 + 3) / 2
$\nu^{3}$	$=$	$ ( -9\beta_{15} - 6\beta_{14} + 4\beta_{11} + \beta_{7} - \beta_{6} - 3\beta_{5} + 9\beta_{4} - 2\beta_{3} ) / 2 $ (-9b15 - 6b14 + 4b11 + b7 - b6 - 3b5 + 9b4 - 2b3) / 2
$\nu^{4}$	$=$	$ ( 7\beta_{13} + 16\beta_{12} - 12\beta_{10} - 12\beta_{9} - 4\beta_{8} - 16\beta_{2} - 8\beta _1 - 15 ) / 2 $ (7b13 + 16b12 - 12b10 - 12b9 - 4b8 - 16b2 - 8*b1 - 15) / 2
$\nu^{5}$	$=$	$ ( 5\beta_{15} + 40\beta_{11} + 51\beta_{7} + 19\beta_{6} + 23\beta_{5} + 31\beta_{4} - 14\beta_{3} ) / 2 $ (5b15 + 40b11 + 51b7 + 19b6 + 23b5 + 31b4 - 14*b3) / 2
$\nu^{6}$	$=$	$ ( -25\beta_{13} + 88\beta_{12} + 22\beta_{10} + 28\beta_{9} - 134\beta_{8} - 160\beta_{2} - 156\beta _1 - 195 ) / 2 $ (-25b13 + 88b12 + 22b10 + 28b9 - 134b8 - 160b2 - 156*b1 - 195) / 2
$\nu^{7}$	$=$	$ ( 521 \beta_{15} + 384 \beta_{14} + 38 \beta_{11} + 259 \beta_{7} + 105 \beta_{6} + 287 \beta_{5} + \cdots + 122 \beta_{3} ) / 2 $ (521b15 + 384b14 + 38b11 + 259b7 + 105b6 + 287b5 - 245b4 + 122b3) / 2
$\nu^{8}$	$=$	$ ( -585\beta_{13} - 400\beta_{12} + 712\beta_{10} + 808\beta_{9} - 648\beta_{8} - 176\beta_{2} - 560\beta _1 - 219 ) / 2 $ (-585b13 - 400b12 + 712b10 + 808b9 - 648b8 - 176b2 - 560*b1 - 219) / 2
$\nu^{9}$	$=$	$ ( 2923 \beta_{15} + 2496 \beta_{14} - 1920 \beta_{11} - 1223 \beta_{7} - 503 \beta_{6} + \cdots + 1746 \beta_{3} ) / 2 $ (2923b15 + 2496b14 - 1920b11 - 1223b7 - 503b6 + 385b5 - 3139b4 + 1746b3) / 2
$\nu^{10}$	$=$	$ ( - 2371 \beta_{13} - 7548 \beta_{12} + 2930 \beta_{10} + 3308 \beta_{9} + 3310 \beta_{8} + 7692 \beta_{2} + \cdots + 9663 ) / 2 $ (-2371b13 - 7548b12 + 2930b10 + 3308b9 + 3310b8 + 7692b2 + 5200*b1 + 9663) / 2
$\nu^{11}$	$=$	$ ( - 10977 \beta_{15} - 5724 \beta_{14} - 13786 \beta_{11} - 21955 \beta_{7} - 9065 \beta_{6} + \cdots + 4598 \beta_{3} ) / 2 $ (-10977b15 - 5724b14 - 13786b11 - 21955b7 - 9065b6 - 13827b5 - 5775b4 + 4598b3) / 2
$\nu^{12}$	$=$	$ ( 17477 \beta_{13} - 24976 \beta_{12} - 22116 \beta_{10} - 24804 \beta_{9} + 56068 \beta_{8} + \cdots + 72135 ) / 2 $ (17477b13 - 24976b12 - 22116b10 - 24804b9 + 56068b8 + 57376b2 + 63512*b1 + 72135) / 2
$\nu^{13}$	$=$	$ ( - 230867 \beta_{15} - 174096 \beta_{14} + 21944 \beta_{11} - 68121 \beta_{7} - 28105 \beta_{6} + \cdots - 67354 \beta_{3} ) / 2 $ (-230867b15 - 174096b14 + 21944b11 - 68121b7 - 28105b6 - 107597b5 + 138383b4 - 67354b3) / 2
$\nu^{14}$	$=$	$ ( 239239 \beta_{13} + 263864 \beta_{12} - 301426 \beta_{10} - 338464 \beta_{9} + 162146 \beta_{8} + \cdots - 90675 ) / 2 $ (239239b13 + 263864b12 - 301426b10 - 338464b9 + 162146b8 - 71696b2 + 105456*b1 - 90675) / 2
$\nu^{15}$	$=$	$ ( - 821447 \beta_{15} - 761196 \beta_{14} + 903754 \beta_{11} + 797531 \beta_{7} + \cdots - 670646 \beta_{3} ) / 2 $ (-821447b15 - 761196b14 + 903754b11 + 797531b7 + 330681b6 + 100951b5 + 1176659b4 - 670646b3) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/980\mathbb{Z}\right)^\times$.

$n$	$101$	$197$	$491$
$\chi(n)$	$-1$	$-1$	$-1$

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} $

979.1

1.53860	−	0.297929i
0.106729	−	0.839125i
0.106729	+	0.839125i
1.53860	+	0.297929i
1.15989	+	0.169383i
−2.29695	+	1.47595i
−2.29695	−	1.47595i
1.15989	−	0.169383i
2.29695	+	1.47595i
−1.15989	+	0.169383i
−1.15989	−	0.169383i
2.29695	−	1.47595i
−0.106729	−	0.839125i
−1.53860	−	0.297929i
−1.53860	+	0.297929i
−0.106729	+	0.839125i

−1.16342

−

0.804019i

−

3.04017i

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3.53701i

0.681517

−

2.74509i

−6.24264

3.05973

0.798801i

979.2

−1.16342

−

0.804019i

3.04017i

0.707107

1.87083i

2.10100

−

0.765367i

2.44436

−

3.53701i

0.681517

−

2.74509i

−6.24264

−3.05973

−

0.798801i

979.3

−1.16342

0.804019i

−

3.04017i

0.707107

−

1.87083i

2.10100

0.765367i

2.44436

3.53701i

0.681517

2.74509i

−6.24264

−3.05973

0.798801i

979.4

−1.16342

0.804019i

3.04017i

0.707107

−

1.87083i

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−

0.765367i

−2.44436

−

3.53701i

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2.74509i

−6.24264

3.05973

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0.798801i

979.5

−0.804019

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1.16342i

−

0.870264i

−0.707107

1.87083i

−1.25928

1.84776i

−1.01249

0.699709i

2.74509

−

0.681517i

2.24264

3.16221

−

0.0205578i

979.6

−0.804019

−

1.16342i

0.870264i

−0.707107

1.87083i

1.25928

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1.84776i

1.01249

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0.699709i

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0.681517i

2.24264

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0.0205578i

979.7

−0.804019

1.16342i

−

0.870264i

−0.707107

−

1.87083i

1.25928

1.84776i

1.01249

0.699709i

2.74509

0.681517i

2.24264

−3.16221

−

0.0205578i

979.8

−0.804019

1.16342i

0.870264i

−0.707107

−

1.87083i

−1.25928

−

1.84776i

−1.01249

−

0.699709i

2.74509

0.681517i

2.24264

3.16221

0.0205578i

979.9

0.804019

−

1.16342i

−

0.870264i

−0.707107

−

1.87083i

1.25928

−

1.84776i

−1.01249

−

0.699709i

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−

0.681517i

2.24264

−1.13724

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2.95071i

979.10

0.804019

−

1.16342i

0.870264i

−0.707107

−

1.87083i

−1.25928

1.84776i

1.01249

0.699709i

−2.74509

−

0.681517i

2.24264

1.13724

2.95071i

979.11

0.804019

1.16342i

−

0.870264i

−0.707107

1.87083i

−1.25928

−

1.84776i

1.01249

−

0.699709i

−2.74509

0.681517i

2.24264

1.13724

−

2.95071i

979.12

0.804019

1.16342i

0.870264i

−0.707107

1.87083i

1.25928

1.84776i

−1.01249

0.699709i

−2.74509

0.681517i

2.24264

−1.13724

2.95071i

979.13

1.16342

−

0.804019i

−

3.04017i

0.707107

−

1.87083i

2.10100

−

0.765367i

−2.44436

−

3.53701i

−0.681517

−

2.74509i

−6.24264

1.82899

−

2.57969i

979.14

1.16342

−

0.804019i

3.04017i

0.707107

−

1.87083i

−2.10100

0.765367i

2.44436

3.53701i

−0.681517

−

2.74509i

−6.24264

−1.82899

2.57969i

979.15

1.16342

0.804019i

−

3.04017i

0.707107

1.87083i

−2.10100

−

0.765367i

2.44436

−

3.53701i

−0.681517

2.74509i

−6.24264

−1.82899

−

2.57969i

979.16

1.16342

0.804019i

3.04017i

0.707107

1.87083i

2.10100

0.765367i

−2.44436

3.53701i

−0.681517

2.74509i

−6.24264

1.82899

2.57969i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
5.b	even	2	1	inner
7.b	odd	2	1	inner
20.d	odd	2	1	inner
28.d	even	2	1	inner
35.c	odd	2	1	inner
140.c	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	980.2.c.c	✓	16
4.b	odd	2	1	inner	980.2.c.c	✓	16
5.b	even	2	1	inner	980.2.c.c	✓	16
7.b	odd	2	1	inner	980.2.c.c	✓	16
7.c	even	3	2		980.2.s.d		32
7.d	odd	6	2		980.2.s.d		32
20.d	odd	2	1	inner	980.2.c.c	✓	16
28.d	even	2	1	inner	980.2.c.c	✓	16
28.f	even	6	2		980.2.s.d		32
28.g	odd	6	2		980.2.s.d		32
35.c	odd	2	1	inner	980.2.c.c	✓	16
35.i	odd	6	2		980.2.s.d		32
35.j	even	6	2		980.2.s.d		32
140.c	even	2	1	inner	980.2.c.c	✓	16
140.p	odd	6	2		980.2.s.d		32
140.s	even	6	2		980.2.s.d		32

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
980.2.c.c	✓	16	1.a	even	1	1	trivial
980.2.c.c	✓	16	4.b	odd	2	1	inner
980.2.c.c	✓	16	5.b	even	2	1	inner
980.2.c.c	✓	16	7.b	odd	2	1	inner
980.2.c.c	✓	16	20.d	odd	2	1	inner
980.2.c.c	✓	16	28.d	even	2	1	inner
980.2.c.c	✓	16	35.c	odd	2	1	inner
980.2.c.c	✓	16	140.c	even	2	1	inner
980.2.s.d		32	7.c	even	3	2
980.2.s.d		32	7.d	odd	6	2
980.2.s.d		32	28.f	even	6	2
980.2.s.d		32	28.g	odd	6	2
980.2.s.d		32	35.i	odd	6	2
980.2.s.d		32	35.j	even	6	2
980.2.s.d		32	140.p	odd	6	2
980.2.s.d		32	140.s	even	6	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{4} + 10T_{3}^{2} + 7 $ T3^4 + 10*T3^2 + 7 acting on $S_{2}^{\mathrm{new}}(980, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{8} + 6 T^{4} + 16)^{2} $ (T^8 + 6*T^4 + 16)^2
$3$	$ (T^{4} + 10 T^{2} + 7)^{4} $ (T^4 + 10*T^2 + 7)^4
$5$	$ (T^{8} - 4 T^{6} + \cdots + 625)^{2} $ (T^8 - 4T^6 + 22T^4 - 100*T^2 + 625)^2
$7$	$ T^{16} $ T^16
$11$	$ (T^{2} + 7)^{8} $ (T^2 + 7)^8
$13$	$ (T^{4} - 26 T^{2} + 7)^{4} $ (T^4 - 26*T^2 + 7)^4
$17$	$ (T^{4} - 6 T^{2} + 7)^{4} $ (T^4 - 6*T^2 + 7)^4
$19$	$ (T^{4} - 56 T^{2} + 392)^{4} $ (T^4 - 56*T^2 + 392)^4
$23$	$ (T^{4} - 128 T^{2} + 4046)^{4} $ (T^4 - 128*T^2 + 4046)^4
$29$	$ (T^{2} - 6 T + 7)^{8} $ (T^2 - 6*T + 7)^8
$31$	$ (T^{4} - 28 T^{2} + 98)^{4} $ (T^4 - 28*T^2 + 98)^4
$37$	$ (T^{4} + 176 T^{2} + 7406)^{4} $ (T^4 + 176*T^2 + 7406)^4
$41$	$ (T^{4} + 4 T^{2} + 2)^{4} $ (T^4 + 4*T^2 + 2)^4
$43$	$ (T^{4} - 64 T^{2} + 56)^{4} $ (T^4 - 64*T^2 + 56)^4
$47$	$ (T^{4} + 90 T^{2} + 2023)^{4} $ (T^4 + 90*T^2 + 2023)^4
$53$	$ (T^{4} + 56 T^{2} + 686)^{4} $ (T^4 + 56*T^2 + 686)^4
$59$	$ (T^{4} - 140 T^{2} + 4802)^{4} $ (T^4 - 140*T^2 + 4802)^4
$61$	$ (T^{4} + 8 T^{2} + 8)^{4} $ (T^4 + 8*T^2 + 8)^4
$67$	$ (T^{4} - 88 T^{2} + 14)^{4} $ (T^4 - 88*T^2 + 14)^4
$71$	$ (T^{4} + 84 T^{2} + 196)^{4} $ (T^4 + 84*T^2 + 196)^4
$73$	$ (T^{4} - 300 T^{2} + 14812)^{4} $ (T^4 - 300*T^2 + 14812)^4
$79$	$ (T^{4} + 42 T^{2} + 49)^{4} $ (T^4 + 42*T^2 + 49)^4
$83$	$ (T^{4} + 216 T^{2} + 9072)^{4} $ (T^4 + 216*T^2 + 9072)^4
$89$	$ (T^{4} + 64 T^{2} + 512)^{4} $ (T^4 + 64*T^2 + 512)^4
$97$	$ (T^{4} - 474 T^{2} + 55447)^{4} $ (T^4 - 474*T^2 + 55447)^4
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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:	\( N \)	\(=\)	\( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	980.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{4} q^{2} + ( - \beta_{12} - \beta_{10} - \beta_{8}) q^{3} + (\beta_{9} - \beta_1) q^{4} + (\beta_{15} + \beta_{2}) q^{5} + (\beta_{15} + \beta_{6} - 2 \beta_{3}) q^{6} + (\beta_{14} + \beta_{11}) q^{8}+ \cdots + (2 \beta_{13} + 6 \beta_{9} - 3 \beta_1) q^{99}+O(q^{100}) \) q - b4 * q^2 + (-b12 - b10 - b8) * q^3 + (b9 - b1) * q^4 + (b15 + b2) * q^5 + (b15 + b6 - 2b3) q^6 + (b14 + b11) * q^8 + (3b1 - 2) q^9 + (b15 - b8 - b6 - b2) * q^10 - b13 * q^11 + (-b12 - b10 - 2b2) q^12 + (b12 - b8 + b2) * q^13 + (b14 + b13 + 2b9 + b4 - b1) q^15 + (b13 - 3) * q^16 + b2 * q^17 + (3b14 - 3b11 + 2b4) q^18 + (b15 + 2b7 + 2b6 - b3) * q^19 + (-b12 - b10 - 2b8 + b7 + b6 - b3) q^20 + (2b5 - b4) q^22 + (2b14 - 3b11 - 3b5 + 2b4) * q^23 + (-2b15 - b7 + 3b6 - b3) * q^24 + (2b14 - 2b4 - 2b1 + 1) q^25 + (b15 + b7 - 2b6 - b3) q^26 + (2b12 + 5b10 + 2b8) q^27 + (b1 + 3) * q^29 + (3b14 + b11 - b9 - 2b5 + b4 + b1 - 2) * q^30 + (b15 + 2b7) q^31 + (-2b5 + 4b4) * q^32 + (2b12 - 2b8 + b2) * q^33 + (b15 - b6) * q^34 + (-3b13 - 2b9 + 2b1 - 3) q^36 + (-4b14 - b11 + b5 + 4b4) * q^37 + (3b12 - b10 + b8 - 3b2) * q^38 + (3b13 + 4b9 - 2b1) q^39 + (2b15 + b12 - b10 + b8 + b7 + b6 - 3b3 - b2) * q^40 - b3 * q^41 + (-3b14 + b11 + b5 - 3b4) * q^43 + (-b9 - 3b1) q^44 + (b15 - 3b12 + 3b8 + 3b3 - 2b2) * q^45 + (-3b13 + b9 + 5b1 - 1) * q^46 + (3b12 + b10 + 3b8) * q^47 + (b12 + 3b10 + 5b8 - b2) * q^48 + (-2b14 + 2b11 + 2b9 - b4 - 2b1 - 4) * q^50 + (b13 + 2b9 - b1) q^51 + (-2b12 - 4b10 - 3b8 + b2) q^52 + (b14 - 2b11 + 2b5 - b4) * q^53 + (b15 + 3b7 - 2b6 + 7b3) q^54 + (b15 + b12 + 3b10 + b8 + 2b7) * q^55 + (-3b14 - b11 + b5 + 3b4) * q^57 + (b14 - b11 - 3b4) q^58 + (2b15 + 4b7 - 2b6 + b3) q^59 + (-b14 + b13 - b11 + b9 + 2b4 + 3b1 - 7) * q^60 + (b15 - b3) * q^61 + (2b12 - 2b10 - b8 - b2) * q^62 + (-2b9 + 6b1) * q^64 + (b11 - b5 - 4b1 + 5) q^65 + (b15 + 2b7 - 3b6 - 2b3) q^66 + (b14 + 2b11 + 2b5 + b4) * q^67 + (-b12 - b10 - 2b8) q^68 + (-b15 + 12b3) q^69 + (-2b13 + 2b9 - b1) * q^71 + (-2b14 - 2b11 + 6b5) q^72 + (-3b12 + 3b8 + 8b2) q^73 + (-b13 - 5b9 + 3b1 + 9) * q^74 + (-3b12 - 5b10 - 3b8 + 4b6 - 2b3) q^75 + (-8b15 - 2b7) * q^76 + (6b14 + 2b11 - 6b5 + 3b4) * q^78 + (b13 - 2b9 + b1) q^79 + (-4b15 - b12 - 3b10 - b8 - 2b7 - 3b2) * q^80 + (-3b1 + 7) q^81 + (-b12 - b10) * q^82 - 6b10 q^83 + (b14 - b4 - b1 + 3) * q^85 + (b13 + 2b9 - 4b1 + 5) * q^86 + (-2b12 - b10 - 2b8) * q^87 + (-5b14 + 3b11) * q^88 - 4b3 q^89 + (-2b15 + 3b12 + 3b10 - b8 - 3b7 + 5b6 + 3b3 - b2) * q^90 + (7b14 - 5b11 + 6b5 - 2b4) * q^92 + (-b14 + 2b11 - 2b5 + b4) * q^93 + (-5b15 - 2b7 - 3b6 + 4b3) * q^94 + (3b14 + 2b13 + b11 - 4b9 + b5 + 3b4 + 2b1) q^95 + (-4b15 - 2b7 + 8b3) q^96 + (-7b12 + 7b8 + 3b2) q^97 + (2b13 + 6b9 - 3b1) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 32 q^{9} - 48 q^{16} + 16 q^{25} + 48 q^{29} - 32 q^{30} - 48 q^{36} - 16 q^{46} - 64 q^{50} - 112 q^{60} + 80 q^{65} + 144 q^{74} + 112 q^{81} + 48 q^{85} + 80 q^{86}+O(q^{100}) \) 16 * q - 32 * q^9 - 48 * q^16 + 16 * q^25 + 48 * q^29 - 32 * q^30 - 48 * q^36 - 16 * q^46 - 64 * q^50 - 112 * q^60 + 80 * q^65 + 144 * q^74 + 112 * q^81 + 48 * q^85 + 80 * q^86

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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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	980.2.c.c

Level	$980$
Weight	$2$
Character orbit	980.c
Analytic conductor	$7.825$
Analytic rank	$0$
Dimension	$16$
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(7.82533939809\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 12x^{14} + 102x^{12} - 388x^{10} + 573x^{8} + 48x^{6} - 700x^{4} + 144x^{2} + 324 \) x^16 - 12x^14 + 102x^12 - 388x^10 + 573x^8 + 48x^6 - 700x^4 + 144*x^2 + 324
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{6} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( \nu^{14} - 6\nu^{12} + 44\nu^{10} + 82\nu^{8} - 599\nu^{6} + 244\nu^{4} + 1234\nu^{2} - 288 ) / 612 \) (v^14 - 6v^12 + 44v^10 + 82v^8 - 599v^6 + 244v^4 + 1234v^2 - 288) / 612
\(\beta_{2}\)	\(=\)	\( ( 57\nu^{14} - 640\nu^{12} + 5388\nu^{10} - 18592\nu^{8} + 23169\nu^{6} + 10504\nu^{4} - 47018\nu^{2} + 7776 ) / 17136 \) (57v^14 - 640v^12 + 5388v^10 - 18592v^8 + 23169v^6 + 10504v^4 - 47018*v^2 + 7776) / 17136
\(\beta_{3}\)	\(=\)	\( ( 11\nu^{15} - 90\nu^{13} + 684\nu^{11} - 632\nu^{9} - 4581\nu^{7} + 9870\nu^{5} + 7282\nu^{3} - 16380\nu ) / 7344 \) (11v^15 - 90v^13 + 684v^11 - 632v^9 - 4581v^7 + 9870v^5 + 7282v^3 - 16380v) / 7344
\(\beta_{4}\)	\(=\)	\( ( 62 \nu^{15} - 459 \nu^{13} + 3096 \nu^{11} + 2800 \nu^{9} - 57042 \nu^{7} + 105297 \nu^{5} + \cdots - 73170 \nu ) / 25704 \) (62v^15 - 459v^13 + 3096v^11 + 2800v^9 - 57042v^7 + 105297v^5 + 13180v^3 - 73170v) / 25704
\(\beta_{5}\)	\(=\)	\( ( - 82 \nu^{15} + 1023 \nu^{13} - 8628 \nu^{11} + 33544 \nu^{9} - 43878 \nu^{7} - 40053 \nu^{5} + \cdots + 7866 \nu ) / 25704 \) (-82v^15 + 1023v^13 - 8628v^11 + 33544v^9 - 43878v^7 - 40053v^5 + 96868v^3 + 7866v) / 25704
\(\beta_{6}\)	\(=\)	\( ( - 79 \nu^{15} + 1224 \nu^{13} - 11154 \nu^{11} + 56938 \nu^{9} - 137625 \nu^{7} + 130374 \nu^{5} + \cdots - 104004 \nu ) / 25704 \) (-79v^15 + 1224v^13 - 11154v^11 + 56938v^9 - 137625v^7 + 130374v^5 + 25954v^3 - 104004v) / 25704
\(\beta_{7}\)	\(=\)	\( ( - 59 \nu^{15} + 436 \nu^{13} - 3200 \nu^{11} - 308 \nu^{9} + 34793 \nu^{7} - 55424 \nu^{5} + \cdots + 13632 \nu ) / 17136 \) (-59v^15 + 436v^13 - 3200v^11 - 308v^9 + 34793v^7 - 55424v^5 - 14458v^3 + 13632v) / 17136
\(\beta_{8}\)	\(=\)	\( ( 53\nu^{14} - 614\nu^{12} + 5116\nu^{10} - 18200\nu^{8} + 20989\nu^{6} + 10530\nu^{4} - 14738\nu^{2} - 15012 ) / 5712 \) (53v^14 - 614v^12 + 5116v^10 - 18200v^8 + 20989v^6 + 10530v^4 - 14738*v^2 - 15012) / 5712
\(\beta_{9}\)	\(=\)	\( ( 23\nu^{14} - 223\nu^{12} + 1760\nu^{10} - 4081\nu^{8} - 2795\nu^{6} + 16220\nu^{4} - 2966\nu^{2} - 10908 ) / 2142 \) (23v^14 - 223v^12 + 1760v^10 - 4081v^8 - 2795v^6 + 16220v^4 - 2966*v^2 - 10908) / 2142
\(\beta_{10}\)	\(=\)	\( ( -27\nu^{14} + 274\nu^{12} - 2212\nu^{10} + 5976\nu^{8} - 995\nu^{6} - 15654\nu^{4} + 6766\nu^{2} + 10476 ) / 2448 \) (-27v^14 + 274v^12 - 2212v^10 + 5976v^8 - 995v^6 - 15654v^4 + 6766*v^2 + 10476) / 2448
\(\beta_{11}\)	\(=\)	\( ( 62 \nu^{15} - 669 \nu^{13} + 5679 \nu^{11} - 18893 \nu^{9} + 26811 \nu^{7} - 6696 \nu^{5} + \cdots + 30780 \nu ) / 12852 \) (62v^15 - 669v^13 + 5679v^11 - 18893v^9 + 26811v^7 - 6696v^5 - 15338v^3 + 30780v) / 12852
\(\beta_{12}\)	\(=\)	\( ( 275 \nu^{14} - 2960 \nu^{12} + 24524 \nu^{10} - 78008 \nu^{8} + 74611 \nu^{6} + 55056 \nu^{4} + \cdots - 13680 ) / 17136 \) (275v^14 - 2960v^12 + 24524v^10 - 78008v^8 + 74611v^6 + 55056v^4 - 67198*v^2 - 13680) / 17136
\(\beta_{13}\)	\(=\)	\( ( -24\nu^{14} + 268\nu^{12} - 2248\nu^{10} + 7686\nu^{8} - 9440\nu^{6} - 2280\nu^{4} + 6304\nu^{2} + 1683 ) / 1071 \) (-24v^14 + 268v^12 - 2248v^10 + 7686v^8 - 9440v^6 - 2280v^4 + 6304*v^2 + 1683) / 1071
\(\beta_{14}\)	\(=\)	\( ( - 265 \nu^{15} + 2958 \nu^{13} - 24726 \nu^{11} + 84238 \nu^{9} - 99303 \nu^{7} - 25224 \nu^{5} + \cdots + 40536 \nu ) / 25704 \) (-265v^15 + 2958v^13 - 24726v^11 + 84238v^9 - 99303v^7 - 25224v^5 + 57814v^3 + 40536v) / 25704
\(\beta_{15}\)	\(=\)	\( ( 89 \nu^{15} - 960 \nu^{13} + 7908 \nu^{11} - 24920 \nu^{9} + 20649 \nu^{7} + 28032 \nu^{5} + \cdots - 13968 \nu ) / 7344 \) (89v^15 - 960v^13 + 7908v^11 - 24920v^9 + 20649v^7 + 28032v^5 - 25130v^3 - 13968v) / 7344

\(\nu\)	\(=\)	\( ( -\beta_{15} - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} ) / 2 \) (-b15 - b7 - b6 - b5 + b4) / 2
\(\nu^{2}\)	\(=\)	\( ( \beta_{13} - 2\beta_{10} - 2\beta_{9} + 2\beta_{8} + 2\beta _1 + 3 ) / 2 \) (b13 - 2b10 - 2b9 + 2b8 + 2b1 + 3) / 2
\(\nu^{3}\)	\(=\)	\( ( -9\beta_{15} - 6\beta_{14} + 4\beta_{11} + \beta_{7} - \beta_{6} - 3\beta_{5} + 9\beta_{4} - 2\beta_{3} ) / 2 \) (-9b15 - 6b14 + 4b11 + b7 - b6 - 3b5 + 9b4 - 2b3) / 2
\(\nu^{4}\)	\(=\)	\( ( 7\beta_{13} + 16\beta_{12} - 12\beta_{10} - 12\beta_{9} - 4\beta_{8} - 16\beta_{2} - 8\beta _1 - 15 ) / 2 \) (7b13 + 16b12 - 12b10 - 12b9 - 4b8 - 16b2 - 8*b1 - 15) / 2
\(\nu^{5}\)	\(=\)	\( ( 5\beta_{15} + 40\beta_{11} + 51\beta_{7} + 19\beta_{6} + 23\beta_{5} + 31\beta_{4} - 14\beta_{3} ) / 2 \) (5b15 + 40b11 + 51b7 + 19b6 + 23b5 + 31b4 - 14*b3) / 2
\(\nu^{6}\)	\(=\)	\( ( -25\beta_{13} + 88\beta_{12} + 22\beta_{10} + 28\beta_{9} - 134\beta_{8} - 160\beta_{2} - 156\beta _1 - 195 ) / 2 \) (-25b13 + 88b12 + 22b10 + 28b9 - 134b8 - 160b2 - 156*b1 - 195) / 2
\(\nu^{7}\)	\(=\)	\( ( 521 \beta_{15} + 384 \beta_{14} + 38 \beta_{11} + 259 \beta_{7} + 105 \beta_{6} + 287 \beta_{5} + \cdots + 122 \beta_{3} ) / 2 \) (521b15 + 384b14 + 38b11 + 259b7 + 105b6 + 287b5 - 245b4 + 122b3) / 2
\(\nu^{8}\)	\(=\)	\( ( -585\beta_{13} - 400\beta_{12} + 712\beta_{10} + 808\beta_{9} - 648\beta_{8} - 176\beta_{2} - 560\beta _1 - 219 ) / 2 \) (-585b13 - 400b12 + 712b10 + 808b9 - 648b8 - 176b2 - 560*b1 - 219) / 2
\(\nu^{9}\)	\(=\)	\( ( 2923 \beta_{15} + 2496 \beta_{14} - 1920 \beta_{11} - 1223 \beta_{7} - 503 \beta_{6} + \cdots + 1746 \beta_{3} ) / 2 \) (2923b15 + 2496b14 - 1920b11 - 1223b7 - 503b6 + 385b5 - 3139b4 + 1746b3) / 2
\(\nu^{10}\)	\(=\)	\( ( - 2371 \beta_{13} - 7548 \beta_{12} + 2930 \beta_{10} + 3308 \beta_{9} + 3310 \beta_{8} + 7692 \beta_{2} + \cdots + 9663 ) / 2 \) (-2371b13 - 7548b12 + 2930b10 + 3308b9 + 3310b8 + 7692b2 + 5200*b1 + 9663) / 2
\(\nu^{11}\)	\(=\)	\( ( - 10977 \beta_{15} - 5724 \beta_{14} - 13786 \beta_{11} - 21955 \beta_{7} - 9065 \beta_{6} + \cdots + 4598 \beta_{3} ) / 2 \) (-10977b15 - 5724b14 - 13786b11 - 21955b7 - 9065b6 - 13827b5 - 5775b4 + 4598b3) / 2
\(\nu^{12}\)	\(=\)	\( ( 17477 \beta_{13} - 24976 \beta_{12} - 22116 \beta_{10} - 24804 \beta_{9} + 56068 \beta_{8} + \cdots + 72135 ) / 2 \) (17477b13 - 24976b12 - 22116b10 - 24804b9 + 56068b8 + 57376b2 + 63512*b1 + 72135) / 2
\(\nu^{13}\)	\(=\)	\( ( - 230867 \beta_{15} - 174096 \beta_{14} + 21944 \beta_{11} - 68121 \beta_{7} - 28105 \beta_{6} + \cdots - 67354 \beta_{3} ) / 2 \) (-230867b15 - 174096b14 + 21944b11 - 68121b7 - 28105b6 - 107597b5 + 138383b4 - 67354b3) / 2
\(\nu^{14}\)	\(=\)	\( ( 239239 \beta_{13} + 263864 \beta_{12} - 301426 \beta_{10} - 338464 \beta_{9} + 162146 \beta_{8} + \cdots - 90675 ) / 2 \) (239239b13 + 263864b12 - 301426b10 - 338464b9 + 162146b8 - 71696b2 + 105456*b1 - 90675) / 2
\(\nu^{15}\)	\(=\)	\( ( - 821447 \beta_{15} - 761196 \beta_{14} + 903754 \beta_{11} + 797531 \beta_{7} + \cdots - 670646 \beta_{3} ) / 2 \) (-821447b15 - 761196b14 + 903754b11 + 797531b7 + 330681b6 + 100951b5 + 1176659b4 - 670646b3) / 2

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 979.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{8} + 6 T^{4} + 16)^{2} \) (T^8 + 6*T^4 + 16)^2
$3$	\( (T^{4} + 10 T^{2} + 7)^{4} \) (T^4 + 10*T^2 + 7)^4
$5$	\( (T^{8} - 4 T^{6} + \cdots + 625)^{2} \) (T^8 - 4T^6 + 22T^4 - 100*T^2 + 625)^2
$7$	\( T^{16} \) T^16
$11$	\( (T^{2} + 7)^{8} \) (T^2 + 7)^8
$13$	\( (T^{4} - 26 T^{2} + 7)^{4} \) (T^4 - 26*T^2 + 7)^4
$17$	\( (T^{4} - 6 T^{2} + 7)^{4} \) (T^4 - 6*T^2 + 7)^4
$19$	\( (T^{4} - 56 T^{2} + 392)^{4} \) (T^4 - 56*T^2 + 392)^4
$23$	\( (T^{4} - 128 T^{2} + 4046)^{4} \) (T^4 - 128*T^2 + 4046)^4
$29$	\( (T^{2} - 6 T + 7)^{8} \) (T^2 - 6*T + 7)^8
$31$	\( (T^{4} - 28 T^{2} + 98)^{4} \) (T^4 - 28*T^2 + 98)^4
$37$	\( (T^{4} + 176 T^{2} + 7406)^{4} \) (T^4 + 176*T^2 + 7406)^4
$41$	\( (T^{4} + 4 T^{2} + 2)^{4} \) (T^4 + 4*T^2 + 2)^4
$43$	\( (T^{4} - 64 T^{2} + 56)^{4} \) (T^4 - 64*T^2 + 56)^4
$47$	\( (T^{4} + 90 T^{2} + 2023)^{4} \) (T^4 + 90*T^2 + 2023)^4
$53$	\( (T^{4} + 56 T^{2} + 686)^{4} \) (T^4 + 56*T^2 + 686)^4
$59$	\( (T^{4} - 140 T^{2} + 4802)^{4} \) (T^4 - 140*T^2 + 4802)^4
$61$	\( (T^{4} + 8 T^{2} + 8)^{4} \) (T^4 + 8*T^2 + 8)^4
$67$	\( (T^{4} - 88 T^{2} + 14)^{4} \) (T^4 - 88*T^2 + 14)^4
$71$	\( (T^{4} + 84 T^{2} + 196)^{4} \) (T^4 + 84*T^2 + 196)^4
$73$	\( (T^{4} - 300 T^{2} + 14812)^{4} \) (T^4 - 300*T^2 + 14812)^4
$79$	\( (T^{4} + 42 T^{2} + 49)^{4} \) (T^4 + 42*T^2 + 49)^4
$83$	\( (T^{4} + 216 T^{2} + 9072)^{4} \) (T^4 + 216*T^2 + 9072)^4
$89$	\( (T^{4} + 64 T^{2} + 512)^{4} \) (T^4 + 64*T^2 + 512)^4
$97$	\( (T^{4} - 474 T^{2} + 55447)^{4} \) (T^4 - 474*T^2 + 55447)^4
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\(n\)	\(101\)	\(197\)	\(491\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)