LMFDB - Newform orbit 735.2.p.e

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [735,2,Mod(374,735)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(735, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("735.374");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 735 = 3 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	735.p (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$5.86900454856$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	16.0.721389578983833600000000.5
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 8x^{14} + 44x^{12} + 128x^{10} + 223x^{8} - 464x^{6} - 724x^{4} + 784x^{2} + 2401 $ x^16 + 8x^14 + 44x^12 + 128x^10 + 223x^8 - 464x^6 - 724x^4 + 784*x^2 + 2401
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 7^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

$q$-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_{7} - \beta_{2}) q^{2} + ( - \beta_{3} + 1) q^{3} + ( - \beta_{9} - \beta_{8} - 2 \beta_{3} - 2) q^{4} + \beta_{11} q^{5} + (2 \beta_{7} - \beta_{2}) q^{6} + (\beta_{14} + \beta_{6} + 2 \beta_{2}) q^{8} - 3 \beta_{3} q^{9}+O(q^{10}) $ q + (b7 - b2) * q^2 + (-b3 + 1) * q^3 + (-b9 - b8 - 2b3 - 2) q^4 + b11 * q^5 + (2b7 - b2) q^6 + (b14 + b6 + 2b2) q^8 - 3b3 q^9	$ q + (\beta_{7} - \beta_{2}) q^{2} + ( - \beta_{3} + 1) q^{3} + ( - \beta_{9} - \beta_{8} - 2 \beta_{3} - 2) q^{4} + \beta_{11} q^{5} + (2 \beta_{7} - \beta_{2}) q^{6} + (\beta_{14} + \beta_{6} + 2 \beta_{2}) q^{8} - 3 \beta_{3} q^{9} + ( - \beta_{14} - \beta_{5}) q^{10} + ( - \beta_{9} - 2 \beta_{8} - 2 \beta_{3} - 4) q^{12} + (\beta_{11} - \beta_1) q^{15} + (\beta_{11} + 2 \beta_{9} + \cdots + 2 \beta_1) q^{16}+ \cdots + ( - 3 \beta_{14} - 4 \beta_{7} + \cdots - 4 \beta_{2}) q^{96}+O(q^{100}) $ q + (b7 - b2) * q^2 + (-b3 + 1) * q^3 + (-b9 - b8 - 2b3 - 2) q^4 + b11 * q^5 + (2b7 - b2) q^6 + (b14 + b6 + 2b2) q^8 - 3b3 q^9 + (-b14 - b5) * q^10 + (-b9 - 2b8 - 2b3 - 4) * q^12 + (b11 - b1) * q^15 + (b11 + 2b9 + 4b3 + 2b1) q^16 + (b12 - b8) * q^17 + 3b7 q^18 + (-2b6 + b5 + b4) q^19 + (-b13 + b12 - 2b11 - 2b9 - b8 - 2b1) q^20 + (b5 - b4) * q^23 + (2b14 - 2b7 + b6 + 4b2) q^24 + (5b3 + 5) q^25 + (-6b3 - 3) q^27 + (-b14 - b5 + b4) * q^30 + (b15 + b10 - b7) * q^31 + (-b14 - 4b7 - b5 - 2b4) * q^32 + (-b15 + b14 - b7 - b6 + 2b4 + b2) q^34 + (-3b8 - 6) q^36 + (-b12 - b9 + 2b8 + b1) q^38 + (2b14 + b10 + b7 + 2b5 + 2b4 + b2) q^40 - 3b1 q^45 + (2b13 - b12 + 6b11 + b9 + 3b1) q^46 + (b13 + 2b9 + 3b8) * q^47 + (3b11 + 4b9 + 2b8 + 8b3 + 3b1 + 4) q^48 - 5b2 q^50 + (-b13 + 2b12 - b8) q^51 + (b14 + b6 + b5 + 2b4) q^53 + (3b7 + 3b2) * q^54 + (-b14 - 3b6 + 2b5 + b4) * q^57 + (-2b13 + b12 - 4b11 - 3b9 - 2b8 - 2b1) q^60 + (-b10 - 2b7 - 2b2) * q^61 + (2b13 - 2b12 + 2b9 + b8 + 2b3 + 1) * q^62 + (b13 + b12 + 2b11 + b8 - 2b1 + 8) * q^64 + (-2b13 - 5b11 + 2b8 + 3b3 + 6) * q^68 + (-b14 + b6 - 3b4) q^69 + (3b14 - 6b7 + 6b2) q^72 + (5b3 + 10) q^75 + (b15 + 5b7 - b4 - 3b2) * q^76 + (b13 - 2b12 + 4b9 + b8) * q^79 + (-2b12 + 2b9 + 5b3 + 4b1 - 5) * q^80 + (-9b3 - 9) q^81 + (b13 - b12 - 4b9 - 2b8) * q^83 + (-b13 - b12 + 4b8) q^85 + 3b4 q^90 + (-b15 - 6b14 - 2b10 + b7 - 6b5 - 3b4 + b2) * q^92 + (2b15 + b10 - 2b7) * q^93 + (-b15 - 4b14 - b10 + 4b7 - b6 - 2b5 - 6b2) * q^94 + (b15 - b10 + b7) * q^95 + (-3b14 - 4b7 - 3b5 - 3b4 - 4b2) q^96
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 24 q^{3} - 16 q^{4} + 24 q^{9}+O(q^{10}) $ 16 * q + 24 * q^3 - 16 * q^4 + 24 * q^9	$ 16 q + 24 q^{3} - 16 q^{4} + 24 q^{9} - 48 q^{12} - 32 q^{16} + 40 q^{25} - 96 q^{36} + 128 q^{64} + 72 q^{68} + 120 q^{75} - 120 q^{80} - 72 q^{81}+O(q^{100}) $ 16 * q + 24 * q^3 - 16 * q^4 + 24 * q^9 - 48 * q^12 - 32 * q^16 + 40 * q^25 - 96 * q^36 + 128 * q^64 + 72 * q^68 + 120 * q^75 - 120 * q^80 - 72 * q^81

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} + 8x^{14} + 44x^{12} + 128x^{10} + 223x^{8} - 464x^{6} - 724x^{4} + 784x^{2} + 2401 $ x^16 + 8*x^14 + 44*x^12 + 128*x^10 + 223*x^8 - 464*x^6 - 724*x^4 + 784*x^2 + 2401 :

$\beta_{1}$	$=$	$ ( - 1072 \nu^{14} + 6557 \nu^{12} + 25896 \nu^{10} + 258070 \nu^{8} - 224432 \nu^{6} + \cdots + 3985660 ) / 12647439 $ (-1072v^14 + 6557v^12 + 25896v^10 + 258070v^8 - 224432v^6 - 1523133v^4 - 17690204*v^2 + 3985660) / 12647439
$\beta_{2}$	$=$	$ ( 76 \nu^{15} - 419 \nu^{13} - 3897 \nu^{11} - 29632 \nu^{9} - 84316 \nu^{7} - 154695 \nu^{5} + \cdots + 801395 \nu ) / 743967 $ (76v^15 - 419v^13 - 3897v^11 - 29632v^9 - 84316v^7 - 154695v^5 + 714863v^3 + 801395v) / 743967
$\beta_{3}$	$=$	$ ( 1576 \nu^{14} + 7268 \nu^{12} + 28704 \nu^{10} - 12139 \nu^{8} - 248768 \nu^{6} - 1688292 \nu^{4} + \cdots + 202027 ) / 4215813 $ (1576v^14 + 7268v^12 + 28704v^10 - 12139v^8 - 248768v^6 - 1688292v^4 + 614768*v^2 + 202027) / 4215813
$\beta_{4}$	$=$	$ ( 4262 \nu^{15} + 13915 \nu^{13} - 51774 \nu^{11} - 895085 \nu^{9} - 4570117 \nu^{7} + \cdots + 28614824 \nu ) / 12647439 $ (4262v^15 + 13915v^13 - 51774v^11 - 895085v^9 - 4570117v^7 - 14385558v^5 - 11627102v^3 + 28614824v) / 12647439
$\beta_{5}$	$=$	$ ( 1576 \nu^{15} + 7268 \nu^{13} + 28704 \nu^{11} - 12139 \nu^{9} - 248768 \nu^{7} + \cdots + 12849466 \nu ) / 4215813 $ (1576v^15 + 7268v^13 + 28704v^11 - 12139v^9 - 248768v^7 - 1688292v^5 + 614768v^3 + 12849466v) / 4215813
$\beta_{6}$	$=$	$ ( - 5194 \nu^{15} - 29693 \nu^{13} - 223998 \nu^{11} - 822251 \nu^{9} - 3077509 \nu^{7} + \cdots + 40050101 \nu ) / 12647439 $ (-5194v^15 - 29693v^13 - 223998v^11 - 822251v^9 - 3077509v^7 - 4255806v^5 - 15315710v^3 + 40050101v) / 12647439
$\beta_{7}$	$=$	$ ( - 6608 \nu^{15} - 68366 \nu^{13} - 407226 \nu^{11} - 1526905 \nu^{9} - 3697816 \nu^{7} + \cdots - 2218720 \nu ) / 12647439 $ (-6608v^15 - 68366v^13 - 407226v^11 - 1526905v^9 - 3697816v^7 - 2072601v^5 + 7833569v^3 - 2218720v) / 12647439
$\beta_{8}$	$=$	$ ( - 99 \nu^{14} - 878 \nu^{12} - 4941 \nu^{10} - 14598 \nu^{8} - 24658 \nu^{6} + 52506 \nu^{4} + \cdots - 26950 ) / 106281 $ (-99v^14 - 878v^12 - 4941v^10 - 14598v^8 - 24658v^6 + 52506v^4 + 187002*v^2 - 26950) / 106281
$\beta_{9}$	$=$	$ ( - 1902 \nu^{14} - 12910 \nu^{12} - 74946 \nu^{10} - 228534 \nu^{8} - 526307 \nu^{6} + \cdots - 864164 ) / 1806777 $ (-1902v^14 - 12910v^12 - 74946v^10 - 228534v^8 - 526307v^6 + 64908v^4 - 2013897*v^2 - 864164) / 1806777
$\beta_{10}$	$=$	$ ( 11908 \nu^{15} + 10036 \nu^{13} - 44223 \nu^{11} - 1527802 \nu^{9} - 4033300 \nu^{7} + \cdots - 4991875 \nu ) / 12647439 $ (11908v^15 + 10036v^13 - 44223v^11 - 1527802v^9 - 4033300v^7 - 16815984v^5 + 44335097v^3 - 4991875v) / 12647439
$\beta_{11}$	$=$	$ ( - 18208 \nu^{14} - 148024 \nu^{12} - 798060 \nu^{10} - 2425142 \nu^{8} - 4325648 \nu^{6} + \cdots - 14731850 ) / 12647439 $ (-18208v^14 - 148024v^12 - 798060v^10 - 2425142v^8 - 4325648v^6 + 7862397v^4 + 13958560*v^2 - 14731850) / 12647439
$\beta_{12}$	$=$	$ ( - 3006 \nu^{14} - 15880 \nu^{12} - 92121 \nu^{10} - 160884 \nu^{8} - 234062 \nu^{6} + \cdots + 5487706 ) / 1806777 $ (-3006v^14 - 15880v^12 - 92121v^10 - 160884v^8 - 234062v^6 + 2551788v^4 - 1008309*v^2 + 5487706) / 1806777
$\beta_{13}$	$=$	$ ( 4026 \nu^{14} + 28154 \nu^{12} + 132972 \nu^{10} + 275532 \nu^{8} + 80773 \nu^{6} + \cdots + 13393072 ) / 1806777 $ (4026v^14 + 28154v^12 + 132972v^10 + 275532v^8 + 80773v^6 - 2887164v^4 + 1205730*v^2 + 13393072) / 1806777
$\beta_{14}$	$=$	$ ( - 29401 \nu^{15} - 198821 \nu^{13} - 998676 \nu^{11} - 2095238 \nu^{9} - 1382458 \nu^{7} + \cdots - 42597415 \nu ) / 12647439 $ (-29401v^15 - 198821v^13 - 998676v^11 - 2095238v^9 - 1382458v^7 + 23877903v^5 + 14381254v^3 - 42597415v) / 12647439
$\beta_{15}$	$=$	$ ( 34448 \nu^{15} + 269378 \nu^{13} + 1399308 \nu^{11} + 3980116 \nu^{9} + 5538928 \nu^{7} + \cdots + 81635470 \nu ) / 12647439 $ (34448v^15 + 269378v^13 + 1399308v^11 + 3980116v^9 + 5538928v^7 - 17282721v^5 - 16395422v^3 + 81635470v) / 12647439

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

$\nu$	$=$	$ ( \beta_{6} + 2\beta_{5} - \beta_{4} ) / 7 $ (b6 + 2*b5 - b4) / 7
$\nu^{2}$	$=$	$ ( 3\beta_{13} - 2\beta_{12} + \beta_{9} + 4\beta_{8} - 14\beta_{3} - 14 ) / 7 $ (3b13 - 2b12 + b9 + 4b8 - 14b3 - 14) / 7
$\nu^{3}$	$=$	$ ( 3\beta_{15} + 2\beta_{10} + 5\beta_{7} + 6\beta_{6} - 9\beta_{5} - 6\beta_{4} + 4\beta_{2} ) / 7 $ (3b15 + 2b10 + 5b7 + 6b6 - 9b5 - 6b4 + 4*b2) / 7
$\nu^{4}$	$=$	$ ( -4\beta_{13} + 12\beta_{12} + 21\beta_{11} - 20\beta_{9} - 24\beta_{8} + 42\beta_{3} ) / 7 $ (-4b13 + 12b12 + 21b11 - 20b9 - 24b8 + 42b3) / 7
$\nu^{5}$	$=$	$ ( -13\beta_{15} - 18\beta_{10} - 38\beta_{7} - 30\beta_{6} + 10\beta_{5} + 30\beta_{4} + 62\beta_{2} ) / 7 $ (-13b15 - 18b10 - 38b7 - 30b6 + 10b5 + 30b4 + 62*b2) / 7
$\nu^{6}$	$=$	$ ( -25\beta_{13} - 2\beta_{12} - 126\beta_{11} + 92\beta_{9} + 46\beta_{8} - 126\beta _1 + 140 ) / 7 $ (-25b13 - 2b12 - 126b11 + 92b9 + 46b8 - 126b1 + 140) / 7
$\nu^{7}$	$=$	$ ( -7\beta_{15} + 42\beta_{14} + 77\beta_{10} - 126\beta_{7} + 64\beta_{6} + 65\beta_{5} - \beta_{4} - 189\beta_{2} ) / 7 $ (-7b15 + 42b14 + 77b10 - 126b7 + 64b6 + 65b5 - b4 - 189*b2) / 7
$\nu^{8}$	$=$	$ ( 24\beta_{13} - 72\beta_{12} - 216\beta_{9} + 312\beta_{8} - 161\beta_{3} + 588\beta _1 - 161 ) / 7 $ (24b13 - 72b12 - 216b9 + 312b8 - 161b3 + 588b1 - 161) / 7
$\nu^{9}$	$=$	$ ( 360 \beta_{15} + 168 \beta_{14} - 12 \beta_{10} + 1104 \beta_{7} - 218 \beta_{6} + 75 \beta_{5} + \cdots - 780 \beta_{2} ) / 7 $ (360b15 + 168b14 - 12b10 + 1104b7 - 218b6 + 75b5 - 118b4 - 780b2) / 7
$\nu^{10}$	$=$	$ ( 409\beta_{13} - 303\beta_{12} + 2520\beta_{11} - 727\beta_{9} - 2166\beta_{8} - 1526\beta_{3} ) / 7 $ (409b13 - 303b12 + 2520b11 - 727b9 - 2166b8 - 1526b3) / 7
$\nu^{11}$	$=$	$ ( - 1595 \beta_{15} - 2772 \beta_{14} - 1341 \beta_{10} - 325 \beta_{7} + 957 \beta_{6} + \cdots + 5872 \beta_{2} ) / 7 $ (-1595b15 - 2772b14 - 1341b10 - 325b7 + 957b6 - 1551b5 - 1881b4 + 5872b2) / 7
$\nu^{12}$	$=$	$ ( 584\beta_{13} + 2476\beta_{12} - 9387\beta_{11} + 7568\beta_{9} + 3784\beta_{8} - 9387\beta _1 - 15246 ) / 7 $ (584b13 + 2476b12 - 9387b11 + 7568b9 + 3784b8 - 9387b1 - 15246) / 7
$\nu^{13}$	$=$	$ ( 1925 \beta_{15} + 8736 \beta_{14} + 5915 \beta_{10} - 19530 \beta_{7} + 4082 \beta_{6} + \cdots - 2205 \beta_{2} ) / 7 $ (1925b15 + 8736b14 + 5915b10 - 19530b7 + 4082b6 - 4940b5 + 9022b4 - 2205b2) / 7
$\nu^{14}$	$=$	$ ( -19359\beta_{13} + 6865\beta_{12} - 30617\beta_{9} + 4393\beta_{8} + 95732\beta_{3} + 27930\beta _1 + 95732 ) / 7 $ (-19359b13 + 6865b12 - 30617b9 + 4393b8 + 95732b3 + 27930b1 + 95732) / 7
$\nu^{15}$	$=$	$ ( 6744 \beta_{15} + 18123 \beta_{14} - 10855 \beta_{10} + 41942 \beta_{7} - 70463 \beta_{6} + \cdots - 67763 \beta_{2} ) / 7 $ (6744b15 + 18123b14 - 10855b10 + 41942b7 - 70463b6 + 78510b5 + 34217b4 - 67763b2) / 7

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/735\mathbb{Z}\right)^\times$.

$n$	$346$	$442$	$491$
$\chi(n)$	$-\beta_{3}$	$-1$	$-1$

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} $

374.1

1.22796	+	0.279124i
−1.36434	−	1.59774i
−0.372250	+	1.20300i
0.701515	+	1.98043i
−0.701515	−	1.98043i
0.372250	−	1.20300i
1.36434	+	1.59774i
−1.22796	−	0.279124i
1.22796	−	0.279124i
−1.36434	+	1.59774i
−0.372250	−	1.20300i
0.701515	−	1.98043i
−0.701515	+	1.98043i
0.372250	+	1.20300i
1.36434	−	1.59774i
−1.22796	+	0.279124i

−1.36434

−

2.36311i

1.50000

0.866025i

−2.72286

4.71613i

1.93649

−

1.11803i

−

4.72622i

9.40228

1.50000

2.59808i

−5.28407

−

3.05076i

374.2

−1.22796

−

2.12688i

1.50000

0.866025i

−2.01575

3.49139i

−1.93649

1.11803i

−

4.25377i

4.98920

1.50000

2.59808i

4.75585

2.74579i

374.3

−0.701515

−

1.21506i

1.50000

0.866025i

0.0157530

−

0.0272850i

−1.93649

1.11803i

−

2.43012i

−2.85026

1.50000

2.59808i

2.71696

1.56864i

374.4

−0.372250

−

0.644756i

1.50000

0.866025i

0.722860

−

1.25203i

1.93649

−

1.11803i

−

1.28951i

−2.56534

1.50000

2.59808i

−1.44172

−

0.832376i

374.5

0.372250

0.644756i

1.50000

0.866025i

0.722860

−

1.25203i

1.93649

−

1.11803i

1.28951i

2.56534

1.50000

2.59808i

1.44172

0.832376i

374.6

0.701515

1.21506i

1.50000

0.866025i

0.0157530

−

0.0272850i

−1.93649

1.11803i

2.43012i

2.85026

1.50000

2.59808i

−2.71696

−

1.56864i

374.7

1.22796

2.12688i

1.50000

0.866025i

−2.01575

3.49139i

−1.93649

1.11803i

4.25377i

−4.98920

1.50000

2.59808i

−4.75585

−

2.74579i

374.8

1.36434

2.36311i

1.50000

0.866025i

−2.72286

4.71613i

1.93649

−

1.11803i

4.72622i

−9.40228

1.50000

2.59808i

5.28407

3.05076i

509.1

−1.36434

2.36311i

1.50000

−

0.866025i

−2.72286

−

4.71613i

1.93649

1.11803i

4.72622i

9.40228

1.50000

−

2.59808i

−5.28407

3.05076i

509.2

−1.22796

2.12688i

1.50000

−

0.866025i

−2.01575

−

3.49139i

−1.93649

−

1.11803i

4.25377i

4.98920

1.50000

−

2.59808i

4.75585

−

2.74579i

509.3

−0.701515

1.21506i

1.50000

−

0.866025i

0.0157530

0.0272850i

−1.93649

−

1.11803i

2.43012i

−2.85026

1.50000

−

2.59808i

2.71696

−

1.56864i

509.4

−0.372250

0.644756i

1.50000

−

0.866025i

0.722860

1.25203i

1.93649

1.11803i

1.28951i

−2.56534

1.50000

−

2.59808i

−1.44172

0.832376i

509.5

0.372250

−

0.644756i

1.50000

−

0.866025i

0.722860

1.25203i

1.93649

1.11803i

−

1.28951i

2.56534

1.50000

−

2.59808i

1.44172

−

0.832376i

509.6

0.701515

−

1.21506i

1.50000

−

0.866025i

0.0157530

0.0272850i

−1.93649

−

1.11803i

−

2.43012i

2.85026

1.50000

−

2.59808i

−2.71696

1.56864i

509.7

1.22796

−

2.12688i

1.50000

−

0.866025i

−2.01575

−

3.49139i

−1.93649

−

1.11803i

−

4.25377i

−4.98920

1.50000

−

2.59808i

−4.75585

2.74579i

509.8

1.36434

−

2.36311i

1.50000

−

0.866025i

−2.72286

−

4.71613i

1.93649

1.11803i

−

4.72622i

−9.40228

1.50000

−

2.59808i

5.28407

−

3.05076i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
15.d	odd	2	1	CM by $\Q(\sqrt{-15}) $
7.d	odd	6	1	inner
21.c	even	2	1	inner
21.h	odd	6	1	inner
35.c	odd	2	1	inner
35.j	even	6	1	inner
105.p	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	735.2.p.e		16
3.b	odd	2	1		735.2.p.d		16
5.b	even	2	1		735.2.p.d		16
7.b	odd	2	1		735.2.p.d		16
7.c	even	3	1		735.2.g.a	✓	16
7.c	even	3	1		735.2.p.d		16
7.d	odd	6	1		735.2.g.a	✓	16
7.d	odd	6	1	inner	735.2.p.e		16
15.d	odd	2	1	CM	735.2.p.e		16
21.c	even	2	1	inner	735.2.p.e		16
21.g	even	6	1		735.2.g.a	✓	16
21.g	even	6	1		735.2.p.d		16
21.h	odd	6	1		735.2.g.a	✓	16
21.h	odd	6	1	inner	735.2.p.e		16
35.c	odd	2	1	inner	735.2.p.e		16
35.i	odd	6	1		735.2.g.a	✓	16
35.i	odd	6	1		735.2.p.d		16
35.j	even	6	1		735.2.g.a	✓	16
35.j	even	6	1	inner	735.2.p.e		16
105.g	even	2	1		735.2.p.d		16
105.o	odd	6	1		735.2.g.a	✓	16
105.o	odd	6	1		735.2.p.d		16
105.p	even	6	1		735.2.g.a	✓	16
105.p	even	6	1	inner	735.2.p.e		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
735.2.g.a	✓	16	7.c	even	3	1
735.2.g.a	✓	16	7.d	odd	6	1
735.2.g.a	✓	16	21.g	even	6	1
735.2.g.a	✓	16	21.h	odd	6	1
735.2.g.a	✓	16	35.i	odd	6	1
735.2.g.a	✓	16	35.j	even	6	1
735.2.g.a	✓	16	105.o	odd	6	1
735.2.g.a	✓	16	105.p	even	6	1
735.2.p.d		16	3.b	odd	2	1
735.2.p.d		16	5.b	even	2	1
735.2.p.d		16	7.b	odd	2	1
735.2.p.d		16	7.c	even	3	1
735.2.p.d		16	21.g	even	6	1
735.2.p.d		16	35.i	odd	6	1
735.2.p.d		16	105.g	even	2	1
735.2.p.d		16	105.o	odd	6	1
735.2.p.e		16	1.a	even	1	1	trivial
735.2.p.e		16	7.d	odd	6	1	inner
735.2.p.e		16	15.d	odd	2	1	CM
735.2.p.e		16	21.c	even	2	1	inner
735.2.p.e		16	21.h	odd	6	1	inner
735.2.p.e		16	35.c	odd	2	1	inner
735.2.p.e		16	35.j	even	6	1	inner
735.2.p.e		16	105.p	even	6	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(735, [\chi])$:

$ T_{2}^{16} + 16T_{2}^{14} + 176T_{2}^{12} + 1024T_{2}^{10} + 4303T_{2}^{8} + 8672T_{2}^{6} + 12464T_{2}^{4} + 6272T_{2}^{2} + 2401 $

$ T_{13} $

$ T_{257}^{2} + 12T_{257} + 48 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} + 16 T^{14} + \cdots + 2401 $ T^16 + 16T^14 + 176T^12 + 1024T^10 + 4303T^8 + 8672T^6 + 12464T^4 + 6272*T^2 + 2401
$3$	$ (T^{2} - 3 T + 3)^{8} $ (T^2 - 3*T + 3)^8
$5$	$ (T^{4} - 5 T^{2} + 25)^{4} $ (T^4 - 5*T^2 + 25)^4
$7$	$ T^{16} $ T^16
$11$	$ T^{16} $ T^16
$13$	$ T^{16} $ T^16
$17$	$ (T^{8} - 68 T^{6} + \cdots + 38416)^{2} $ (T^8 - 68T^6 + 4428T^4 - 13328*T^2 + 38416)^2
$19$	$ T^{16} + \cdots + 3208542736 $ T^16 - 152T^14 + 15884T^12 - 877952T^10 + 35390668T^8 - 775131904T^6 + 11634775856T^4 - 6216339136*T^2 + 3208542736
$23$	$ T^{16} + 184 T^{14} + \cdots + 92236816 $ T^16 + 184T^14 + 23276T^12 + 1557376T^10 + 76107148T^8 + 2056095488T^6 + 37795577264T^4 + 1869629888*T^2 + 92236816
$29$	$ T^{16} $ T^16
$31$	$ T^{16} + \cdots + 13533010268176 $ T^16 - 248T^14 + 42284T^12 - 3813248T^10 + 247518988T^8 - 7336681216T^6 + 156495867056T^4 - 1753485866944*T^2 + 13533010268176
$37$	$ T^{16} $ T^16
$41$	$ T^{16} $ T^16
$43$	$ T^{16} $ T^16
$47$	$ (T^{8} - 188 T^{6} + \cdots + 38416)^{2} $ (T^8 - 188T^6 + 35148T^4 - 36848*T^2 + 38416)^2
$53$	$ T^{16} + \cdots + 9993716528656 $ T^16 + 424T^14 + 123596T^12 + 19056256T^10 + 2143049548T^8 + 131141788928T^6 + 5496475513904T^4 + 7530279649088*T^2 + 9993716528656
$59$	$ T^{16} $ T^16
$61$	$ T^{16} + \cdots + 17\!\cdots\!76 $ T^16 - 488T^14 + 163724T^12 - 29053568T^10 + 3724052428T^8 - 229262884096T^6 + 10062361004336T^4 - 152590515805504*T^2 + 1765371482472976
$67$	$ T^{16} $ T^16
$71$	$ T^{16} $ T^16
$73$	$ T^{16} $ T^16
$79$	$ (T^{8} + 316 T^{6} + \cdots + 92236816)^{2} $ (T^8 + 316T^6 + 90252T^4 + 3034864*T^2 + 92236816)^2
$83$	$ (T^{4} + 332 T^{2} + 23716)^{4} $ (T^4 + 332*T^2 + 23716)^4
$89$	$ T^{16} $ T^16
$97$	$ T^{16} $ T^16
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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:	\( N \)	\(=\)	\( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	735.p (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + (\beta_{7} - \beta_{2}) q^{2} + ( - \beta_{3} + 1) q^{3} + ( - \beta_{9} - \beta_{8} - 2 \beta_{3} - 2) q^{4} + \beta_{11} q^{5} + (2 \beta_{7} - \beta_{2}) q^{6} + (\beta_{14} + \beta_{6} + 2 \beta_{2}) q^{8} - 3 \beta_{3} q^{9}+O(q^{10}) \) q + (b7 - b2) * q^2 + (-b3 + 1) * q^3 + (-b9 - b8 - 2b3 - 2) q^4 + b11 * q^5 + (2b7 - b2) q^6 + (b14 + b6 + 2b2) q^8 - 3b3 q^9	\( q + (\beta_{7} - \beta_{2}) q^{2} + ( - \beta_{3} + 1) q^{3} + ( - \beta_{9} - \beta_{8} - 2 \beta_{3} - 2) q^{4} + \beta_{11} q^{5} + (2 \beta_{7} - \beta_{2}) q^{6} + (\beta_{14} + \beta_{6} + 2 \beta_{2}) q^{8} - 3 \beta_{3} q^{9} + ( - \beta_{14} - \beta_{5}) q^{10} + ( - \beta_{9} - 2 \beta_{8} - 2 \beta_{3} - 4) q^{12} + (\beta_{11} - \beta_1) q^{15} + (\beta_{11} + 2 \beta_{9} + \cdots + 2 \beta_1) q^{16}+ \cdots + ( - 3 \beta_{14} - 4 \beta_{7} + \cdots - 4 \beta_{2}) q^{96}+O(q^{100}) \) q + (b7 - b2) * q^2 + (-b3 + 1) * q^3 + (-b9 - b8 - 2b3 - 2) q^4 + b11 * q^5 + (2b7 - b2) q^6 + (b14 + b6 + 2b2) q^8 - 3b3 q^9 + (-b14 - b5) * q^10 + (-b9 - 2b8 - 2b3 - 4) * q^12 + (b11 - b1) * q^15 + (b11 + 2b9 + 4b3 + 2b1) q^16 + (b12 - b8) * q^17 + 3b7 q^18 + (-2b6 + b5 + b4) q^19 + (-b13 + b12 - 2b11 - 2b9 - b8 - 2b1) q^20 + (b5 - b4) * q^23 + (2b14 - 2b7 + b6 + 4b2) q^24 + (5b3 + 5) q^25 + (-6b3 - 3) q^27 + (-b14 - b5 + b4) * q^30 + (b15 + b10 - b7) * q^31 + (-b14 - 4b7 - b5 - 2b4) * q^32 + (-b15 + b14 - b7 - b6 + 2b4 + b2) q^34 + (-3b8 - 6) q^36 + (-b12 - b9 + 2b8 + b1) q^38 + (2b14 + b10 + b7 + 2b5 + 2b4 + b2) q^40 - 3b1 q^45 + (2b13 - b12 + 6b11 + b9 + 3b1) q^46 + (b13 + 2b9 + 3b8) * q^47 + (3b11 + 4b9 + 2b8 + 8b3 + 3b1 + 4) q^48 - 5b2 q^50 + (-b13 + 2b12 - b8) q^51 + (b14 + b6 + b5 + 2b4) q^53 + (3b7 + 3b2) * q^54 + (-b14 - 3b6 + 2b5 + b4) * q^57 + (-2b13 + b12 - 4b11 - 3b9 - 2b8 - 2b1) q^60 + (-b10 - 2b7 - 2b2) * q^61 + (2b13 - 2b12 + 2b9 + b8 + 2b3 + 1) * q^62 + (b13 + b12 + 2b11 + b8 - 2b1 + 8) * q^64 + (-2b13 - 5b11 + 2b8 + 3b3 + 6) * q^68 + (-b14 + b6 - 3b4) q^69 + (3b14 - 6b7 + 6b2) q^72 + (5b3 + 10) q^75 + (b15 + 5b7 - b4 - 3b2) * q^76 + (b13 - 2b12 + 4b9 + b8) * q^79 + (-2b12 + 2b9 + 5b3 + 4b1 - 5) * q^80 + (-9b3 - 9) q^81 + (b13 - b12 - 4b9 - 2b8) * q^83 + (-b13 - b12 + 4b8) q^85 + 3b4 q^90 + (-b15 - 6b14 - 2b10 + b7 - 6b5 - 3b4 + b2) * q^92 + (2b15 + b10 - 2b7) * q^93 + (-b15 - 4b14 - b10 + 4b7 - b6 - 2b5 - 6b2) * q^94 + (b15 - b10 + b7) * q^95 + (-3b14 - 4b7 - 3b5 - 3b4 - 4b2) q^96
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 24 q^{3} - 16 q^{4} + 24 q^{9}+O(q^{10}) \) 16 * q + 24 * q^3 - 16 * q^4 + 24 * q^9	\( 16 q + 24 q^{3} - 16 q^{4} + 24 q^{9} - 48 q^{12} - 32 q^{16} + 40 q^{25} - 96 q^{36} + 128 q^{64} + 72 q^{68} + 120 q^{75} - 120 q^{80} - 72 q^{81}+O(q^{100}) \) 16 * q + 24 * q^3 - 16 * q^4 + 24 * q^9 - 48 * q^12 - 32 * q^16 + 40 * q^25 - 96 * q^36 + 128 * q^64 + 72 * q^68 + 120 * q^75 - 120 * q^80 - 72 * q^81

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	735.2.p.e

Level	$735$
Weight	$2$
Character orbit	735.p
Analytic conductor	$5.869$
Analytic rank	$0$
Dimension	$16$
CM discriminant	-15
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(5.86900454856\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{6})\)
Coefficient field:	16.0.721389578983833600000000.5
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} + 8x^{14} + 44x^{12} + 128x^{10} + 223x^{8} - 464x^{6} - 724x^{4} + 784x^{2} + 2401 \) x^16 + 8x^14 + 44x^12 + 128x^10 + 223x^8 - 464x^6 - 724x^4 + 784*x^2 + 2401
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 7^{6} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

\(\beta_{1}\)	\(=\)	\( ( - 1072 \nu^{14} + 6557 \nu^{12} + 25896 \nu^{10} + 258070 \nu^{8} - 224432 \nu^{6} + \cdots + 3985660 ) / 12647439 \) (-1072v^14 + 6557v^12 + 25896v^10 + 258070v^8 - 224432v^6 - 1523133v^4 - 17690204*v^2 + 3985660) / 12647439
\(\beta_{2}\)	\(=\)	\( ( 76 \nu^{15} - 419 \nu^{13} - 3897 \nu^{11} - 29632 \nu^{9} - 84316 \nu^{7} - 154695 \nu^{5} + \cdots + 801395 \nu ) / 743967 \) (76v^15 - 419v^13 - 3897v^11 - 29632v^9 - 84316v^7 - 154695v^5 + 714863v^3 + 801395v) / 743967
\(\beta_{3}\)	\(=\)	\( ( 1576 \nu^{14} + 7268 \nu^{12} + 28704 \nu^{10} - 12139 \nu^{8} - 248768 \nu^{6} - 1688292 \nu^{4} + \cdots + 202027 ) / 4215813 \) (1576v^14 + 7268v^12 + 28704v^10 - 12139v^8 - 248768v^6 - 1688292v^4 + 614768*v^2 + 202027) / 4215813
\(\beta_{4}\)	\(=\)	\( ( 4262 \nu^{15} + 13915 \nu^{13} - 51774 \nu^{11} - 895085 \nu^{9} - 4570117 \nu^{7} + \cdots + 28614824 \nu ) / 12647439 \) (4262v^15 + 13915v^13 - 51774v^11 - 895085v^9 - 4570117v^7 - 14385558v^5 - 11627102v^3 + 28614824v) / 12647439
\(\beta_{5}\)	\(=\)	\( ( 1576 \nu^{15} + 7268 \nu^{13} + 28704 \nu^{11} - 12139 \nu^{9} - 248768 \nu^{7} + \cdots + 12849466 \nu ) / 4215813 \) (1576v^15 + 7268v^13 + 28704v^11 - 12139v^9 - 248768v^7 - 1688292v^5 + 614768v^3 + 12849466v) / 4215813
\(\beta_{6}\)	\(=\)	\( ( - 5194 \nu^{15} - 29693 \nu^{13} - 223998 \nu^{11} - 822251 \nu^{9} - 3077509 \nu^{7} + \cdots + 40050101 \nu ) / 12647439 \) (-5194v^15 - 29693v^13 - 223998v^11 - 822251v^9 - 3077509v^7 - 4255806v^5 - 15315710v^3 + 40050101v) / 12647439
\(\beta_{7}\)	\(=\)	\( ( - 6608 \nu^{15} - 68366 \nu^{13} - 407226 \nu^{11} - 1526905 \nu^{9} - 3697816 \nu^{7} + \cdots - 2218720 \nu ) / 12647439 \) (-6608v^15 - 68366v^13 - 407226v^11 - 1526905v^9 - 3697816v^7 - 2072601v^5 + 7833569v^3 - 2218720v) / 12647439
\(\beta_{8}\)	\(=\)	\( ( - 99 \nu^{14} - 878 \nu^{12} - 4941 \nu^{10} - 14598 \nu^{8} - 24658 \nu^{6} + 52506 \nu^{4} + \cdots - 26950 ) / 106281 \) (-99v^14 - 878v^12 - 4941v^10 - 14598v^8 - 24658v^6 + 52506v^4 + 187002*v^2 - 26950) / 106281
\(\beta_{9}\)	\(=\)	\( ( - 1902 \nu^{14} - 12910 \nu^{12} - 74946 \nu^{10} - 228534 \nu^{8} - 526307 \nu^{6} + \cdots - 864164 ) / 1806777 \) (-1902v^14 - 12910v^12 - 74946v^10 - 228534v^8 - 526307v^6 + 64908v^4 - 2013897*v^2 - 864164) / 1806777
\(\beta_{10}\)	\(=\)	\( ( 11908 \nu^{15} + 10036 \nu^{13} - 44223 \nu^{11} - 1527802 \nu^{9} - 4033300 \nu^{7} + \cdots - 4991875 \nu ) / 12647439 \) (11908v^15 + 10036v^13 - 44223v^11 - 1527802v^9 - 4033300v^7 - 16815984v^5 + 44335097v^3 - 4991875v) / 12647439
\(\beta_{11}\)	\(=\)	\( ( - 18208 \nu^{14} - 148024 \nu^{12} - 798060 \nu^{10} - 2425142 \nu^{8} - 4325648 \nu^{6} + \cdots - 14731850 ) / 12647439 \) (-18208v^14 - 148024v^12 - 798060v^10 - 2425142v^8 - 4325648v^6 + 7862397v^4 + 13958560*v^2 - 14731850) / 12647439
\(\beta_{12}\)	\(=\)	\( ( - 3006 \nu^{14} - 15880 \nu^{12} - 92121 \nu^{10} - 160884 \nu^{8} - 234062 \nu^{6} + \cdots + 5487706 ) / 1806777 \) (-3006v^14 - 15880v^12 - 92121v^10 - 160884v^8 - 234062v^6 + 2551788v^4 - 1008309*v^2 + 5487706) / 1806777
\(\beta_{13}\)	\(=\)	\( ( 4026 \nu^{14} + 28154 \nu^{12} + 132972 \nu^{10} + 275532 \nu^{8} + 80773 \nu^{6} + \cdots + 13393072 ) / 1806777 \) (4026v^14 + 28154v^12 + 132972v^10 + 275532v^8 + 80773v^6 - 2887164v^4 + 1205730*v^2 + 13393072) / 1806777
\(\beta_{14}\)	\(=\)	\( ( - 29401 \nu^{15} - 198821 \nu^{13} - 998676 \nu^{11} - 2095238 \nu^{9} - 1382458 \nu^{7} + \cdots - 42597415 \nu ) / 12647439 \) (-29401v^15 - 198821v^13 - 998676v^11 - 2095238v^9 - 1382458v^7 + 23877903v^5 + 14381254v^3 - 42597415v) / 12647439
\(\beta_{15}\)	\(=\)	\( ( 34448 \nu^{15} + 269378 \nu^{13} + 1399308 \nu^{11} + 3980116 \nu^{9} + 5538928 \nu^{7} + \cdots + 81635470 \nu ) / 12647439 \) (34448v^15 + 269378v^13 + 1399308v^11 + 3980116v^9 + 5538928v^7 - 17282721v^5 - 16395422v^3 + 81635470v) / 12647439

\(\nu\)	\(=\)	\( ( \beta_{6} + 2\beta_{5} - \beta_{4} ) / 7 \) (b6 + 2*b5 - b4) / 7
\(\nu^{2}\)	\(=\)	\( ( 3\beta_{13} - 2\beta_{12} + \beta_{9} + 4\beta_{8} - 14\beta_{3} - 14 ) / 7 \) (3b13 - 2b12 + b9 + 4b8 - 14b3 - 14) / 7
\(\nu^{3}\)	\(=\)	\( ( 3\beta_{15} + 2\beta_{10} + 5\beta_{7} + 6\beta_{6} - 9\beta_{5} - 6\beta_{4} + 4\beta_{2} ) / 7 \) (3b15 + 2b10 + 5b7 + 6b6 - 9b5 - 6b4 + 4*b2) / 7
\(\nu^{4}\)	\(=\)	\( ( -4\beta_{13} + 12\beta_{12} + 21\beta_{11} - 20\beta_{9} - 24\beta_{8} + 42\beta_{3} ) / 7 \) (-4b13 + 12b12 + 21b11 - 20b9 - 24b8 + 42b3) / 7
\(\nu^{5}\)	\(=\)	\( ( -13\beta_{15} - 18\beta_{10} - 38\beta_{7} - 30\beta_{6} + 10\beta_{5} + 30\beta_{4} + 62\beta_{2} ) / 7 \) (-13b15 - 18b10 - 38b7 - 30b6 + 10b5 + 30b4 + 62*b2) / 7
\(\nu^{6}\)	\(=\)	\( ( -25\beta_{13} - 2\beta_{12} - 126\beta_{11} + 92\beta_{9} + 46\beta_{8} - 126\beta _1 + 140 ) / 7 \) (-25b13 - 2b12 - 126b11 + 92b9 + 46b8 - 126b1 + 140) / 7
\(\nu^{7}\)	\(=\)	\( ( -7\beta_{15} + 42\beta_{14} + 77\beta_{10} - 126\beta_{7} + 64\beta_{6} + 65\beta_{5} - \beta_{4} - 189\beta_{2} ) / 7 \) (-7b15 + 42b14 + 77b10 - 126b7 + 64b6 + 65b5 - b4 - 189*b2) / 7
\(\nu^{8}\)	\(=\)	\( ( 24\beta_{13} - 72\beta_{12} - 216\beta_{9} + 312\beta_{8} - 161\beta_{3} + 588\beta _1 - 161 ) / 7 \) (24b13 - 72b12 - 216b9 + 312b8 - 161b3 + 588b1 - 161) / 7
\(\nu^{9}\)	\(=\)	\( ( 360 \beta_{15} + 168 \beta_{14} - 12 \beta_{10} + 1104 \beta_{7} - 218 \beta_{6} + 75 \beta_{5} + \cdots - 780 \beta_{2} ) / 7 \) (360b15 + 168b14 - 12b10 + 1104b7 - 218b6 + 75b5 - 118b4 - 780b2) / 7
\(\nu^{10}\)	\(=\)	\( ( 409\beta_{13} - 303\beta_{12} + 2520\beta_{11} - 727\beta_{9} - 2166\beta_{8} - 1526\beta_{3} ) / 7 \) (409b13 - 303b12 + 2520b11 - 727b9 - 2166b8 - 1526b3) / 7
\(\nu^{11}\)	\(=\)	\( ( - 1595 \beta_{15} - 2772 \beta_{14} - 1341 \beta_{10} - 325 \beta_{7} + 957 \beta_{6} + \cdots + 5872 \beta_{2} ) / 7 \) (-1595b15 - 2772b14 - 1341b10 - 325b7 + 957b6 - 1551b5 - 1881b4 + 5872b2) / 7
\(\nu^{12}\)	\(=\)	\( ( 584\beta_{13} + 2476\beta_{12} - 9387\beta_{11} + 7568\beta_{9} + 3784\beta_{8} - 9387\beta _1 - 15246 ) / 7 \) (584b13 + 2476b12 - 9387b11 + 7568b9 + 3784b8 - 9387b1 - 15246) / 7
\(\nu^{13}\)	\(=\)	\( ( 1925 \beta_{15} + 8736 \beta_{14} + 5915 \beta_{10} - 19530 \beta_{7} + 4082 \beta_{6} + \cdots - 2205 \beta_{2} ) / 7 \) (1925b15 + 8736b14 + 5915b10 - 19530b7 + 4082b6 - 4940b5 + 9022b4 - 2205b2) / 7
\(\nu^{14}\)	\(=\)	\( ( -19359\beta_{13} + 6865\beta_{12} - 30617\beta_{9} + 4393\beta_{8} + 95732\beta_{3} + 27930\beta _1 + 95732 ) / 7 \) (-19359b13 + 6865b12 - 30617b9 + 4393b8 + 95732b3 + 27930b1 + 95732) / 7
\(\nu^{15}\)	\(=\)	\( ( 6744 \beta_{15} + 18123 \beta_{14} - 10855 \beta_{10} + 41942 \beta_{7} - 70463 \beta_{6} + \cdots - 67763 \beta_{2} ) / 7 \) (6744b15 + 18123b14 - 10855b10 + 41942b7 - 70463b6 + 78510b5 + 34217b4 - 67763b2) / 7

\(n\)	\(346\)	\(442\)	\(491\)
\(\chi(n)\)	\(-\beta_{3}\)	\(-1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 374.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} + 16 T^{14} + \cdots + 2401 \) T^16 + 16T^14 + 176T^12 + 1024T^10 + 4303T^8 + 8672T^6 + 12464T^4 + 6272*T^2 + 2401
$3$	\( (T^{2} - 3 T + 3)^{8} \) (T^2 - 3*T + 3)^8
$5$	\( (T^{4} - 5 T^{2} + 25)^{4} \) (T^4 - 5*T^2 + 25)^4
$7$	\( T^{16} \) T^16
$11$	\( T^{16} \) T^16
$13$	\( T^{16} \) T^16
$17$	\( (T^{8} - 68 T^{6} + \cdots + 38416)^{2} \) (T^8 - 68T^6 + 4428T^4 - 13328*T^2 + 38416)^2
$19$	\( T^{16} + \cdots + 3208542736 \) T^16 - 152T^14 + 15884T^12 - 877952T^10 + 35390668T^8 - 775131904T^6 + 11634775856T^4 - 6216339136*T^2 + 3208542736
$23$	\( T^{16} + 184 T^{14} + \cdots + 92236816 \) T^16 + 184T^14 + 23276T^12 + 1557376T^10 + 76107148T^8 + 2056095488T^6 + 37795577264T^4 + 1869629888*T^2 + 92236816
$29$	\( T^{16} \) T^16
$31$	\( T^{16} + \cdots + 13533010268176 \) T^16 - 248T^14 + 42284T^12 - 3813248T^10 + 247518988T^8 - 7336681216T^6 + 156495867056T^4 - 1753485866944*T^2 + 13533010268176
$37$	\( T^{16} \) T^16
$41$	\( T^{16} \) T^16
$43$	\( T^{16} \) T^16
$47$	\( (T^{8} - 188 T^{6} + \cdots + 38416)^{2} \) (T^8 - 188T^6 + 35148T^4 - 36848*T^2 + 38416)^2
$53$	\( T^{16} + \cdots + 9993716528656 \) T^16 + 424T^14 + 123596T^12 + 19056256T^10 + 2143049548T^8 + 131141788928T^6 + 5496475513904T^4 + 7530279649088*T^2 + 9993716528656
$59$	\( T^{16} \) T^16
$61$	\( T^{16} + \cdots + 17\!\cdots\!76 \) T^16 - 488T^14 + 163724T^12 - 29053568T^10 + 3724052428T^8 - 229262884096T^6 + 10062361004336T^4 - 152590515805504*T^2 + 1765371482472976
$67$	\( T^{16} \) T^16
$71$	\( T^{16} \) T^16
$73$	\( T^{16} \) T^16
$79$	\( (T^{8} + 316 T^{6} + \cdots + 92236816)^{2} \) (T^8 + 316T^6 + 90252T^4 + 3034864*T^2 + 92236816)^2
$83$	\( (T^{4} + 332 T^{2} + 23716)^{4} \) (T^4 + 332*T^2 + 23716)^4
$89$	\( T^{16} \) T^16
$97$	\( T^{16} \) T^16
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