LMFDB - Newform orbit 225.2.k.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,2,Mod(49,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 3]))

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

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

chi := DirichletCharacter("225.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	225.k (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:	$1.79663404548$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
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} - 406x^{10} + 1167x^{8} - 1842x^{6} + 2023x^{4} - 441x^{2} + 81 $ x^16 - 12x^14 + 102x^12 - 406x^10 + 1167x^8 - 1842x^6 + 2023x^4 - 441*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-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_1 q^{2} + (\beta_{14} - \beta_{10}) q^{3} + (\beta_{11} + \beta_{9} + \beta_{6} + \cdots + 1) q^{4} + (\beta_{13} + \beta_{11} - \beta_{6} + \cdots + 2) q^{6} + (\beta_{15} + \beta_{14} + \cdots + \beta_1) q^{7}+ \cdots + ( - 3 \beta_{13} + 6 \beta_{11} + \cdots + 6) q^{99}+O(q^{100}) $ q + b1 * q^2 + (b14 - b10) * q^3 + (b11 + b9 + b6 + b5 - b3 + 1) * q^4 + (b13 + b11 - b6 - b5 + 2b2 + 2) q^6 + (b15 + b14 + b7 - b4 + b1) * q^7 + (-b15 - b14 - 3b12 + 2b10 - 3b7 + 2b4) * q^8 + (-b9 - 2b6 - b5 - b3) q^9 + (-2b13 + b11 + b5 - b3 - b2) q^11 + (-b15 - b14 + 2b12 + b8 + b7 - b4 + b1) q^12 + (-2b14 + 2b12 - b8 - 2b4) q^13 + (-b13 - b11 - b5 + b2) * q^14 + (-2b11 + b6 + 2b5 - b3 - 2b2 - 1) q^16 + (2b15 - b14 + 3b12 - b10 + b8 + 3b7 - b4 - b1) q^17 + (b14 - 2b12 - b10 + b8 + 2b7 - 2b1) q^18 + (b13 + b11 - 2b9 + b2 - 1) q^19 + (b11 + b9 + b6 - 2b3 + 2b2 - 1) * q^21 + (b15 - b14 + b10 - b8 - b4) * q^22 + (b15 + b14 + 3b12 + b10 + b4) q^23 + (2b13 + 3b9 - b5 + 6) * q^24 + (b13 + b11 + b9 - 2b5 - b2 - 2) q^26 + (-3b15 - b12 + 2b10 - b8 - 2b7 + 3b4 - b1) * q^27 + (2b15 + b12 - 2b10 + 2b8 + b7 - 2b4 - 2b1) q^28 + (-b13 - b11 + 2b5 - b3 - 2b2) * q^29 + (2b13 - 2b11 - b9 + b6 - 2b5 + b3 - 2b2 - 1) * q^31 + (3b14 - 3b12 - 2b8 + 3b4) * q^32 + (-2b15 + b14 + 5b12 - 3b8 + 4b7 + b4 + 3b1) q^33 + (-b13 + b11 + 4b3) q^34 + (-b9 - 5b6 - b5 + 2b3 + 3b2) q^36 + (b15 + b14 - b12 - 2b10 + 3b8 - b7 - 2b4 - 3b1) * q^37 + (b15 + b14 - 2b10 + 6b7 + b4 - b1) * q^38 + (b11 - 2b9 + 4b6 + b5 + 4b3 - b2 - 7) q^39 + (b13 + b11 - 2b9 + 3b6 + b5 + 2b3 - b2 - 2) q^41 + (b14 - 3b12 - 2b10 - 3b8) q^42 + (2b10 - b7 - 2b4 + 4b1) q^43 + (-b13 - b11 + b9 + 2b5 + b2 - 5) q^44 + (-4b13 - 4b11 + 3b9 + 2b5 - 2b2) q^46 + (-b15 - b14 - b10 - 6b7 + 2b4 - 2b1) q^47 + (-2b15 - 2b14 - 5b12 + 3b10 + 2b8 - 10b7 + b4 + 2b1) q^48 + (-b13 - 2b11 + 2b9 - 3b6 - 2b5 - 2b3 + b2 + 2) q^49 + (-2b13 - 2b11 - 3b9 + 5b6 + 2b5 + 3b3 - b2 - 4) * q^51 + (-2b15 - 2b14 + b10 - 5b7 + b4) q^52 + (3b15 - 3b12 - 3b10 - b8 - 3b7 - 3b4 + b1) q^53 + (-3b11 - b9 - 2b6 + 2b5 - b3 - 3b2 - 3) * q^54 + (-6b6 + 3b3 + 6) * q^56 + (-2b15 - 5b12 + 2b10 + 2b8 - b7 + 4b4 - b1) q^57 + (2b15 + 3b14 - 6b12 + 2b10 - b8 + 3b4) q^58 + (3b13 - 2b9 + 6b6 + 2b3 - 3b2 - 2) q^59 + (2b13 + b11 - 4b6 - 3b5 + b3 + 3b2 + 4) * q^61 + (-3b15 - 3b12 + 3b10 - 3b7 + 3b4) q^62 + (-3b15 - 4b14 - 3b12 + 2b10) * q^63 + (-b13 - b11 - 3b9 - b5 - 2b2 - 7) * q^64 + (b11 + b9 + 7b6 + 2b5 + b3 - b2 - 4) * q^66 + (-b15 - 2b12 - b10 - 4b8) * q^67 + (-2b15 - 2b14 + 9b12 - 2b10 + 2b8 - 2b4) * q^68 + (-5b13 + 3b9 + 3b6 + b5 - 3b3 + 3) * q^69 + (3b13 + 3b11 - b9 + 3b2 - 1) q^71 + (3b15 + b14 + 6b12 - 5b10 + 3b8 - 6b4 - 3b1) * q^72 + (-3b15 + 2b14 - 2b12 + b10 + b8 - 2b7 + b4 - b1) * q^73 + (5b13 + 2b11 - 12b6 - 7b5 + 2b3 + 7b2 + 12) * q^74 + (-b13 - b11 - 3b9 - 2b6 - b5 + 3b3 + b2 - 3) q^76 + (2b15 - 4b14 - 3b12 + 2b10 + 3b8 - 4b4) * q^77 + (b15 - 4b14 + 11b12 + b10 + 4b7 - 2b4 - 3b1) q^78 + (3b13 - b11 + b6 - 2b5 + b3 + 2b2 - 1) q^79 + (-2b13 + 4b9 - 4b6 + 2b5 - 2b3 - 3b2 + 6) * q^81 + (-2b14 + 3b12 + 2b10 - b8 + 3b7 + 2b4 + b1) q^82 + (9b7 + 3b1) * q^83 + (-b13 - 4b11 - 4b9 + 2b6 - b5 + 2b3 - 2b2 - 8) q^84 + (5b9 + 6b6 - 5b3 + 5) q^86 + (-2b15 - b12 + 2b10 - 5b7 + b4 + 3b1) * q^87 + (4b15 + 4b14 - 5b10 + 3b7 + b4 - 3b1) q^88 + (3b13 + 3b11 - 3b5 + 3) q^89 + (2b13 + 2b11 - 4b9 - 3b5 - b2 - 6) * q^91 + (b15 + b14 + b10 + 3b7 - 2b4) * q^92 + (b15 - b14 + b12 - b10 + 2b8 - 7b7 + b4 - 4b1) q^93 + (b13 + 2b11 + 4b9 + 2b5 - 4b3 - b2 + 4) * q^94 + (2b11 + 3b9 + b6 + 2b5 - 6b3 - 2b2 + 7) q^96 + (2b15 + 2b14 - b10 + 7b7 - b4 + 3b1) * q^97 + (3b14 - 3b12 - 3b10 + b8 - 3b7 - 3b4 - b1) q^98 + (-3b13 + 6b11 + b9 - 4b6 + 4b5 - 2b3 + 3b2 + 6) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{4} + 16 q^{6} - 10 q^{9} + 2 q^{11} + 6 q^{14} - 8 q^{16} - 8 q^{19} - 30 q^{21} + 66 q^{24} - 40 q^{26} + 2 q^{29} + 8 q^{31} + 18 q^{34} - 28 q^{36} - 50 q^{39} + 10 q^{41} - 88 q^{44} - 6 q^{49}+ \cdots + 34 q^{99}+O(q^{100}) $ 16 * q + 8 * q^4 + 16 * q^6 - 10 * q^9 + 2 * q^11 + 6 * q^14 - 8 * q^16 - 8 * q^19 - 30 * q^21 + 66 * q^24 - 40 * q^26 + 2 * q^29 + 8 * q^31 + 18 * q^34 - 28 * q^36 - 50 * q^39 + 10 * q^41 - 88 * q^44 - 6 * q^49 + 22 * q^51 - 52 * q^54 + 60 * q^56 + 34 * q^59 + 26 * q^61 - 76 * q^64 - 16 * q^66 + 54 * q^69 - 32 * q^71 + 80 * q^74 - 22 * q^76 - 14 * q^79 + 34 * q^81 - 54 * q^84 + 68 * q^86 + 36 * q^89 - 68 * q^91 + 6 * q^94 + 68 * q^96 + 34 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 12x^{14} + 102x^{12} - 406x^{10} + 1167x^{8} - 1842x^{6} + 2023x^{4} - 441x^{2} + 81 $ x^16 - 12*x^14 + 102*x^12 - 406*x^10 + 1167*x^8 - 1842*x^6 + 2023*x^4 - 441*x^2 + 81 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( 3457 \nu^{14} - 28693 \nu^{12} + 210280 \nu^{10} - 306752 \nu^{8} + 709118 \nu^{6} + \cdots - 7606359 ) / 3976425 $ (3457v^14 - 28693v^12 + 210280v^10 - 306752v^8 + 709118v^6 - 1299110v^4 + 7466556*v^2 - 7606359) / 3976425
$\beta_{3}$	$=$	$ ( -26\nu^{14} + 74\nu^{12} + 10\nu^{10} - 10439\nu^{8} + 38576\nu^{6} - 94895\nu^{4} + 82467\nu^{2} - 18063 ) / 29025 $ (-26v^14 + 74v^12 + 10v^10 - 10439v^8 + 38576v^6 - 94895v^4 + 82467*v^2 - 18063) / 29025
$\beta_{4}$	$=$	$ ( 5917 \nu^{15} - 90503 \nu^{13} + 784955 \nu^{11} - 3838362 \nu^{9} + 10468603 \nu^{7} + \cdots - 16863264 \nu ) / 3976425 $ (5917v^15 - 90503v^13 + 784955v^11 - 3838362v^9 + 10468603v^7 - 20515435v^5 + 19722086v^3 - 16863264v) / 3976425
$\beta_{5}$	$=$	$ ( - 7559 \nu^{14} + 69641 \nu^{12} - 572135 \nu^{10} + 1494574 \nu^{8} - 4803841 \nu^{6} + \cdots - 5812992 ) / 3976425 $ (-7559v^14 + 69641v^12 - 572135v^10 + 1494574v^8 - 4803841v^6 + 3626995v^4 - 7975497*v^2 - 5812992) / 3976425
$\beta_{6}$	$=$	$ ( 11836 \nu^{14} - 138574 \nu^{12} + 1170340 \nu^{10} - 4500371 \nu^{8} + 12811274 \nu^{6} + \cdots - 814212 ) / 3976425 $ (11836v^14 - 138574v^12 + 1170340v^10 - 4500371v^8 + 12811274v^6 - 19088030v^4 + 21981813*v^2 - 814212) / 3976425
$\beta_{7}$	$=$	$ ( 3766 \nu^{15} - 41414 \nu^{13} + 332215 \nu^{11} - 1090526 \nu^{9} + 2480389 \nu^{7} + \cdots + 2721243 \nu ) / 1325475 $ (3766v^15 - 41414v^13 + 332215v^11 - 1090526v^9 + 2480389v^7 - 2137205v^5 + 467253v^3 + 2721243v) / 1325475
$\beta_{8}$	$=$	$ ( - 11836 \nu^{15} + 138574 \nu^{13} - 1170340 \nu^{11} + 4500371 \nu^{9} - 12811274 \nu^{7} + \cdots + 4790637 \nu ) / 3976425 $ (-11836v^15 + 138574v^13 - 1170340v^11 + 4500371v^9 - 12811274v^7 + 19088030v^5 - 21981813v^3 + 4790637v) / 3976425
$\beta_{9}$	$=$	$ ( - 14896 \nu^{14} + 165889 \nu^{12} - 1314040 \nu^{10} + 4313456 \nu^{8} - 9114389 \nu^{6} + \cdots - 5535918 ) / 3976425 $ (-14896v^14 + 165889v^12 - 1314040v^10 + 4313456v^8 - 9114389v^6 + 8453480v^4 - 1848168*v^2 - 5535918) / 3976425
$\beta_{10}$	$=$	$ ( 13477 \nu^{15} - 168383 \nu^{13} + 1451855 \nu^{11} - 6027522 \nu^{9} + 17277208 \nu^{7} + \cdots + 11503071 \nu ) / 3976425 $ (13477v^15 - 168383v^13 + 1451855v^11 - 6027522v^9 + 17277208v^7 - 24805735v^5 + 20660066v^3 + 11503071v) / 3976425
$\beta_{11}$	$=$	$ ( - 3323 \nu^{14} + 38066 \nu^{12} - 324695 \nu^{10} + 1252588 \nu^{8} - 3846136 \nu^{6} + \cdots + 1468098 ) / 795285 $ (-3323v^14 + 38066v^12 - 324695v^10 + 1252588v^8 - 3846136v^6 + 6436600v^4 - 8169474*v^2 + 1468098) / 795285
$\beta_{12}$	$=$	$ ( 19253 \nu^{15} - 245932 \nu^{13} + 2129695 \nu^{11} - 9130758 \nu^{9} + 26781707 \nu^{7} + \cdots - 10338741 \nu ) / 3976425 $ (19253v^15 - 245932v^13 + 2129695v^11 - 9130758v^9 + 26781707v^7 - 44578415v^5 + 47402299v^3 - 10338741v) / 3976425
$\beta_{13}$	$=$	$ ( 572 \nu^{14} - 6758 \nu^{12} + 56330 \nu^{10} - 215542 \nu^{8} + 579583 \nu^{6} - 869185 \nu^{4} + \cdots - 212679 ) / 92475 $ (572v^14 - 6758v^12 + 56330v^10 - 215542v^8 + 579583v^6 - 869185v^4 + 809301*v^2 - 212679) / 92475
$\beta_{14}$	$=$	$ ( 43568 \nu^{15} - 518857 \nu^{13} + 4432420 \nu^{11} - 17623398 \nu^{9} + 51780332 \nu^{7} + \cdots - 6887691 \nu ) / 3976425 $ (43568v^15 - 518857v^13 + 4432420v^11 - 17623398v^9 + 51780332v^7 - 81131165v^5 + 89205019v^3 - 6887691v) / 3976425
$\beta_{15}$	$=$	$ ( - 49089 \nu^{15} + 557311 \nu^{13} - 4656460 \nu^{11} + 17097154 \nu^{9} - 47712236 \nu^{7} + \cdots - 564507 \nu ) / 3976425 $ (-49089v^15 + 557311v^13 - 4656460v^11 + 17097154v^9 - 47712236v^7 + 67106795v^5 - 70902062v^3 - 564507v) / 3976425

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{11} + \beta_{9} + 3\beta_{6} + \beta_{5} - \beta_{3} + 1 $ b11 + b9 + 3*b6 + b5 - b3 + 1
$\nu^{3}$	$=$	$ -\beta_{15} - \beta_{14} - 3\beta_{12} + 2\beta_{10} - 4\beta_{8} - 3\beta_{7} + 2\beta_{4} + 4\beta_1 $ -b15 - b14 - 3b12 + 2b10 - 4b8 - 3b7 + 2b4 + 4b1
$\nu^{4}$	$=$	$ -6\beta_{13} - 2\beta_{11} + 15\beta_{6} + 8\beta_{5} - 7\beta_{3} - 8\beta_{2} - 15 $ -6b13 - 2b11 + 15b6 + 8b5 - 7b3 - 8b2 - 15
$\nu^{5}$	$=$	$ 8\beta_{15} + 11\beta_{14} - 27\beta_{12} + 8\beta_{10} - 22\beta_{8} + 11\beta_{4} $ 8b15 + 11b14 - 27b12 + 8b10 - 22b8 + 11b4
$\nu^{6}$	$=$	$ -57\beta_{13} - 57\beta_{11} - 49\beta_{9} + 19\beta_{5} - 38\beta_{2} - 139 $ -57b13 - 57b11 - 49b9 + 19b5 - 38*b2 - 139
$\nu^{7}$	$=$	$ 144\beta_{15} + 144\beta_{14} - 87\beta_{10} + 204\beta_{7} - 57\beta_{4} - 139\beta_1 $ 144b15 + 144b14 - 87b10 + 204b7 - 57b4 - 139b1
$\nu^{8}$	$=$	$ -144\beta_{13} - 253\beta_{11} - 343\beta_{9} - 588\beta_{6} - 253\beta_{5} + 343\beta_{3} + 144\beta_{2} - 343 $ -144b13 - 253b11 - 343b9 - 588b6 - 253b5 + 343b3 + 144*b2 - 343
$\nu^{9}$	$=$	$ 631 \beta_{15} + 397 \beta_{14} + 1461 \beta_{12} - 1028 \beta_{10} + 931 \beta_{8} + 1461 \beta_{7} + \cdots - 931 \beta_1 $ 631b15 + 397b14 + 1461b12 - 1028b10 + 931b8 + 1461b7 - 1028b4 - 931b1
$\nu^{10}$	$=$	$ 1725\beta_{13} + 1028\beta_{11} - 3984\beta_{6} - 2753\beta_{5} + 2392\beta_{3} + 2753\beta_{2} + 3984 $ 1725b13 + 1028b11 - 3984b6 - 2753b5 + 2392b3 + 2753b2 + 3984
$\nu^{11}$	$=$	$ -2753\beta_{15} - 4448\beta_{14} + 10260\beta_{12} - 2753\beta_{10} + 6376\beta_{8} - 4448\beta_{4} $ -2753b15 - 4448b14 + 10260b12 - 2753b10 + 6376b8 - 4448b4
$\nu^{12}$	$=$	$ 19083\beta_{13} + 19083\beta_{11} + 16636\beta_{9} - 7201\beta_{5} + 11882\beta_{2} + 44023 $ 19083b13 + 19083b11 + 16636b9 - 7201b5 + 11882*b2 + 44023
$\nu^{13}$	$=$	$ -50121\beta_{15} - 50121\beta_{14} + 31038\beta_{10} - 71511\beta_{7} + 19083\beta_{4} + 44023\beta_1 $ -50121b15 - 50121b14 + 31038b10 - 71511b7 + 19083b4 + 44023b1
$\nu^{14}$	$=$	$ 50121 \beta_{13} + 82189 \beta_{11} + 115534 \beta_{9} + 189318 \beta_{6} + 82189 \beta_{5} + \cdots + 115534 $ 50121b13 + 82189b11 + 115534b9 + 189318b6 + 82189b5 - 115534b3 - 50121*b2 + 115534
$\nu^{15}$	$=$	$ - 215776 \beta_{15} - 132310 \beta_{14} - 496965 \beta_{12} + 348086 \beta_{10} - 304852 \beta_{8} + \cdots + 304852 \beta_1 $ -215776b15 - 132310b14 - 496965b12 + 348086b10 - 304852b8 - 496965b7 + 348086b4 + 304852b1

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$-\beta_{6}$	$-1$

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} $

49.1

−2.28087	+	1.31686i
−1.41485	+	0.816862i
−1.27588	+	0.736627i
−0.409850	+	0.236627i
0.409850	−	0.236627i
1.27588	−	0.736627i
1.41485	−	0.816862i
2.28087	−	1.31686i
−2.28087	−	1.31686i
−1.41485	−	0.816862i
−1.27588	−	0.736627i
−0.409850	−	0.236627i
0.409850	+	0.236627i
1.27588	+	0.736627i
1.41485	+	0.816862i
2.28087	+	1.31686i

−2.28087

1.31686i

−0.238330

−

1.71558i

2.46825

−

4.27513i

2.80278

3.59916i

1.55662

−

0.898714i

7.73393i

−2.88640

0.817746i

49.2

−1.41485

0.816862i

−1.36657

−

1.06419i

0.334526

−

0.579416i

2.80278

0.389365i

−0.437645

0.252674i

−

2.17440i

0.735010

2.90857i

49.3

−1.27588

0.736627i

−0.350156

1.69629i

0.0852394

−

0.147639i

−0.802776

−

2.42219i

−3.34791

1.93291i

−

2.69535i

−2.75478

−

1.18793i

49.4

−0.409850

0.236627i

1.64411

0.544899i

−0.888015

1.53809i

−0.802776

0.165713i

−2.21967

1.28153i

−

1.78702i

2.40617

1.79175i

49.5

0.409850

−

0.236627i

−1.64411

−

0.544899i

−0.888015

1.53809i

−0.802776

0.165713i

2.21967

−

1.28153i

1.78702i

2.40617

1.79175i

49.6

1.27588

−

0.736627i

0.350156

−

1.69629i

0.0852394

−

0.147639i

−0.802776

−

2.42219i

3.34791

−

1.93291i

2.69535i

−2.75478

−

1.18793i

49.7

1.41485

−

0.816862i

1.36657

1.06419i

0.334526

−

0.579416i

2.80278

0.389365i

0.437645

−

0.252674i

2.17440i

0.735010

2.90857i

49.8

2.28087

−

1.31686i

0.238330

1.71558i

2.46825

−

4.27513i

2.80278

3.59916i

−1.55662

0.898714i

−

7.73393i

−2.88640

0.817746i

124.1

−2.28087

−

1.31686i

−0.238330

1.71558i

2.46825

4.27513i

2.80278

−

3.59916i

1.55662

0.898714i

−

7.73393i

−2.88640

−

0.817746i

124.2

−1.41485

−

0.816862i

−1.36657

1.06419i

0.334526

0.579416i

2.80278

−

0.389365i

−0.437645

−

0.252674i

2.17440i

0.735010

−

2.90857i

124.3

−1.27588

−

0.736627i

−0.350156

−

1.69629i

0.0852394

0.147639i

−0.802776

2.42219i

−3.34791

−

1.93291i

2.69535i

−2.75478

1.18793i

124.4

−0.409850

−

0.236627i

1.64411

−

0.544899i

−0.888015

−

1.53809i

−0.802776

−

0.165713i

−2.21967

−

1.28153i

1.78702i

2.40617

−

1.79175i

124.5

0.409850

0.236627i

−1.64411

0.544899i

−0.888015

−

1.53809i

−0.802776

−

0.165713i

2.21967

1.28153i

−

1.78702i

2.40617

−

1.79175i

124.6

1.27588

0.736627i

0.350156

1.69629i

0.0852394

0.147639i

−0.802776

2.42219i

3.34791

1.93291i

−

2.69535i

−2.75478

1.18793i

124.7

1.41485

0.816862i

1.36657

−

1.06419i

0.334526

0.579416i

2.80278

−

0.389365i

0.437645

0.252674i

−

2.17440i

0.735010

−

2.90857i

124.8

2.28087

1.31686i

0.238330

−

1.71558i

2.46825

4.27513i

2.80278

−

3.59916i

−1.55662

−

0.898714i

7.73393i

−2.88640

−

0.817746i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
9.c	even	3	1	inner
45.j	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	225.2.k.c		16
3.b	odd	2	1		675.2.k.c		16
5.b	even	2	1	inner	225.2.k.c		16
5.c	odd	4	1		225.2.e.c	✓	8
5.c	odd	4	1		225.2.e.e	yes	8
9.c	even	3	1	inner	225.2.k.c		16
9.c	even	3	1		2025.2.b.n		8
9.d	odd	6	1		675.2.k.c		16
9.d	odd	6	1		2025.2.b.o		8
15.d	odd	2	1		675.2.k.c		16
15.e	even	4	1		675.2.e.c		8
15.e	even	4	1		675.2.e.e		8
45.h	odd	6	1		675.2.k.c		16
45.h	odd	6	1		2025.2.b.o		8
45.j	even	6	1	inner	225.2.k.c		16
45.j	even	6	1		2025.2.b.n		8
45.k	odd	12	1		225.2.e.c	✓	8
45.k	odd	12	1		225.2.e.e	yes	8
45.k	odd	12	1		2025.2.a.q		4
45.k	odd	12	1		2025.2.a.y		4
45.l	even	12	1		675.2.e.c		8
45.l	even	12	1		675.2.e.e		8
45.l	even	12	1		2025.2.a.p		4
45.l	even	12	1		2025.2.a.z		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
225.2.e.c	✓	8	5.c	odd	4	1
225.2.e.c	✓	8	45.k	odd	12	1
225.2.e.e	yes	8	5.c	odd	4	1
225.2.e.e	yes	8	45.k	odd	12	1
225.2.k.c		16	1.a	even	1	1	trivial
225.2.k.c		16	5.b	even	2	1	inner
225.2.k.c		16	9.c	even	3	1	inner
225.2.k.c		16	45.j	even	6	1	inner
675.2.e.c		8	15.e	even	4	1
675.2.e.c		8	45.l	even	12	1
675.2.e.e		8	15.e	even	4	1
675.2.e.e		8	45.l	even	12	1
675.2.k.c		16	3.b	odd	2	1
675.2.k.c		16	9.d	odd	6	1
675.2.k.c		16	15.d	odd	2	1
675.2.k.c		16	45.h	odd	6	1
2025.2.a.p		4	45.l	even	12	1
2025.2.a.q		4	45.k	odd	12	1
2025.2.a.y		4	45.k	odd	12	1
2025.2.a.z		4	45.l	even	12	1
2025.2.b.n		8	9.c	even	3	1
2025.2.b.n		8	45.j	even	6	1
2025.2.b.o		8	9.d	odd	6	1
2025.2.b.o		8	45.h	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{16} - 12T_{2}^{14} + 102T_{2}^{12} - 406T_{2}^{10} + 1167T_{2}^{8} - 1842T_{2}^{6} + 2023T_{2}^{4} - 441T_{2}^{2} + 81 $ T2^16 - 12*T2^14 + 102*T2^12 - 406*T2^10 + 1167*T2^8 - 1842*T2^6 + 2023*T2^4 - 441*T2^2 + 81 acting on $S_{2}^{\mathrm{new}}(225, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} - 12 T^{14} + \cdots + 81 $ T^16 - 12T^14 + 102T^12 - 406T^10 + 1167T^8 - 1842T^6 + 2023T^4 - 441*T^2 + 81
$3$	$ T^{16} + 5 T^{14} + \cdots + 6561 $ T^16 + 5T^14 + 4T^12 + 15T^10 + 135T^8 + 135T^6 + 324T^4 + 3645*T^2 + 6561
$5$	$ T^{16} $ T^16
$7$	$ T^{16} - 25 T^{14} + \cdots + 6561 $ T^16 - 25T^14 + 451T^12 - 3630T^10 + 21195T^8 - 58590T^6 + 115506T^4 - 29160*T^2 + 6561
$11$	$ (T^{8} - T^{7} + 26 T^{6} + \cdots + 81)^{2} $ (T^8 - T^7 + 26T^6 + 107T^5 + 593T^4 + 1007T^3 + 1456T^2 + 369T + 81)^2
$13$	$ T^{16} + \cdots + 131079601 $ T^16 - 64T^14 + 2962T^12 - 59686T^10 + 862027T^8 - 5843158T^6 + 28554859T^4 - 73788805*T^2 + 131079601
$17$	$ (T^{8} + 81 T^{6} + \cdots + 91809)^{2} $ (T^8 + 81T^6 + 2214T^4 + 24220*T^2 + 91809)^2
$19$	$ (T^{4} + 2 T^{3} - 27 T^{2} + \cdots - 25)^{4} $ (T^4 + 2T^3 - 27T^2 - 80*T - 25)^4
$23$	$ T^{16} + \cdots + 3486784401 $ T^16 - 111T^14 + 9288T^12 - 281097T^10 + 6056127T^8 - 71156961T^6 + 592799472T^4 - 1640558367*T^2 + 3486784401
$29$	$ (T^{8} - T^{7} + \cdots + 16641)^{2} $ (T^8 - T^7 + 41T^6 - 244T^5 + 1871T^4 - 5938T^3 + 15004T^2 - 18318T + 16641)^2
$31$	$ (T^{8} - 4 T^{7} + \cdots + 59049)^{2} $ (T^8 - 4T^7 + 58T^6 + 114T^5 + 1629T^4 + 810T^3 + 10935T^2 + 6561*T + 59049)^2
$37$	$ (T^{8} + 199 T^{6} + \cdots + 418609)^{2} $ (T^8 + 199T^6 + 9513T^4 + 124903*T^2 + 418609)^2
$41$	$ (T^{8} - 5 T^{7} + \cdots + 42849)^{2} $ (T^8 - 5T^7 + 50T^6 - 197T^5 + 1637T^4 - 6095T^3 + 20746T^2 - 33327*T + 42849)^2
$43$	$ T^{16} + \cdots + 205144679041 $ T^16 - 196T^14 + 25738T^12 - 1927312T^10 + 105636307T^8 - 3356926096T^6 + 71980515082T^4 - 126271170052*T^2 + 205144679041
$47$	$ T^{16} + \cdots + 21071715921 $ T^16 - 186T^14 + 26709T^12 - 1294114T^10 + 45982884T^8 - 627705066T^6 + 6325951549T^4 - 12546845874*T^2 + 21071715921
$53$	$ (T^{8} + 228 T^{6} + \cdots + 221841)^{2} $ (T^8 + 228T^6 + 13614T^4 + 113053*T^2 + 221841)^2
$59$	$ (T^{8} - 17 T^{7} + \cdots + 5349969)^{2} $ (T^8 - 17T^7 + 287T^6 - 1862T^5 + 17855T^4 - 76814T^3 + 840022T^2 - 2114082*T + 5349969)^2
$61$	$ (T^{8} - 13 T^{7} + 172 T^{6} + \cdots + 1)^{2} $ (T^8 - 13T^7 + 172T^6 - 143T^5 + 1193T^4 - 299T^3 + 8278T^2 - 91*T + 1)^2
$67$	$ T^{16} + \cdots + 3486784401 $ T^16 - 217T^14 + 35107T^12 - 2455086T^10 + 127775907T^8 - 843115662T^6 + 4549304898T^4 - 4281288696*T^2 + 3486784401
$71$	$ (T^{4} + 8 T^{3} + \cdots + 381)^{4} $ (T^4 + 8T^3 - 40T^2 - 263*T + 381)^4
$73$	$ (T^{8} + 196 T^{6} + \cdots + 12769)^{2} $ (T^8 + 196T^6 + 8478T^4 + 79777*T^2 + 12769)^2
$79$	$ (T^{8} + 7 T^{7} + \cdots + 42849)^{2} $ (T^8 + 7T^7 + 82T^6 - 93T^5 + 1365T^4 - 621T^3 + 11592T^2 - 14283*T + 42849)^2
$83$	$ T^{16} + \cdots + 282429536481 $ T^16 - 324T^14 + 77274T^12 - 8029206T^10 + 613578159T^8 - 12762024174T^6 + 209121502059T^4 - 251435897361*T^2 + 282429536481
$89$	$ (T^{4} - 9 T^{3} + \cdots + 2025)^{4} $ (T^4 - 9T^3 - 99T^2 + 405*T + 2025)^4
$97$	$ T^{16} + \cdots + 824843587681 $ T^16 - 199T^14 + 28828T^12 - 1783213T^10 + 79268227T^8 - 1580980129T^6 + 22726478692T^4 - 163756440163*T^2 + 824843587681
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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:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	225.k (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_1 q^{2} + (\beta_{14} - \beta_{10}) q^{3} + (\beta_{11} + \beta_{9} + \beta_{6} + \cdots + 1) q^{4} + (\beta_{13} + \beta_{11} - \beta_{6} + \cdots + 2) q^{6} + (\beta_{15} + \beta_{14} + \cdots + \beta_1) q^{7}+ \cdots + ( - 3 \beta_{13} + 6 \beta_{11} + \cdots + 6) q^{99}+O(q^{100}) \) q + b1 * q^2 + (b14 - b10) * q^3 + (b11 + b9 + b6 + b5 - b3 + 1) * q^4 + (b13 + b11 - b6 - b5 + 2b2 + 2) q^6 + (b15 + b14 + b7 - b4 + b1) * q^7 + (-b15 - b14 - 3b12 + 2b10 - 3b7 + 2b4) * q^8 + (-b9 - 2b6 - b5 - b3) q^9 + (-2b13 + b11 + b5 - b3 - b2) q^11 + (-b15 - b14 + 2b12 + b8 + b7 - b4 + b1) q^12 + (-2b14 + 2b12 - b8 - 2b4) q^13 + (-b13 - b11 - b5 + b2) * q^14 + (-2b11 + b6 + 2b5 - b3 - 2b2 - 1) q^16 + (2b15 - b14 + 3b12 - b10 + b8 + 3b7 - b4 - b1) q^17 + (b14 - 2b12 - b10 + b8 + 2b7 - 2b1) q^18 + (b13 + b11 - 2b9 + b2 - 1) q^19 + (b11 + b9 + b6 - 2b3 + 2b2 - 1) * q^21 + (b15 - b14 + b10 - b8 - b4) * q^22 + (b15 + b14 + 3b12 + b10 + b4) q^23 + (2b13 + 3b9 - b5 + 6) * q^24 + (b13 + b11 + b9 - 2b5 - b2 - 2) q^26 + (-3b15 - b12 + 2b10 - b8 - 2b7 + 3b4 - b1) * q^27 + (2b15 + b12 - 2b10 + 2b8 + b7 - 2b4 - 2b1) q^28 + (-b13 - b11 + 2b5 - b3 - 2b2) * q^29 + (2b13 - 2b11 - b9 + b6 - 2b5 + b3 - 2b2 - 1) * q^31 + (3b14 - 3b12 - 2b8 + 3b4) * q^32 + (-2b15 + b14 + 5b12 - 3b8 + 4b7 + b4 + 3b1) q^33 + (-b13 + b11 + 4b3) q^34 + (-b9 - 5b6 - b5 + 2b3 + 3b2) q^36 + (b15 + b14 - b12 - 2b10 + 3b8 - b7 - 2b4 - 3b1) * q^37 + (b15 + b14 - 2b10 + 6b7 + b4 - b1) * q^38 + (b11 - 2b9 + 4b6 + b5 + 4b3 - b2 - 7) q^39 + (b13 + b11 - 2b9 + 3b6 + b5 + 2b3 - b2 - 2) q^41 + (b14 - 3b12 - 2b10 - 3b8) q^42 + (2b10 - b7 - 2b4 + 4b1) q^43 + (-b13 - b11 + b9 + 2b5 + b2 - 5) q^44 + (-4b13 - 4b11 + 3b9 + 2b5 - 2b2) q^46 + (-b15 - b14 - b10 - 6b7 + 2b4 - 2b1) q^47 + (-2b15 - 2b14 - 5b12 + 3b10 + 2b8 - 10b7 + b4 + 2b1) q^48 + (-b13 - 2b11 + 2b9 - 3b6 - 2b5 - 2b3 + b2 + 2) q^49 + (-2b13 - 2b11 - 3b9 + 5b6 + 2b5 + 3b3 - b2 - 4) * q^51 + (-2b15 - 2b14 + b10 - 5b7 + b4) q^52 + (3b15 - 3b12 - 3b10 - b8 - 3b7 - 3b4 + b1) q^53 + (-3b11 - b9 - 2b6 + 2b5 - b3 - 3b2 - 3) * q^54 + (-6b6 + 3b3 + 6) * q^56 + (-2b15 - 5b12 + 2b10 + 2b8 - b7 + 4b4 - b1) q^57 + (2b15 + 3b14 - 6b12 + 2b10 - b8 + 3b4) q^58 + (3b13 - 2b9 + 6b6 + 2b3 - 3b2 - 2) q^59 + (2b13 + b11 - 4b6 - 3b5 + b3 + 3b2 + 4) * q^61 + (-3b15 - 3b12 + 3b10 - 3b7 + 3b4) q^62 + (-3b15 - 4b14 - 3b12 + 2b10) * q^63 + (-b13 - b11 - 3b9 - b5 - 2b2 - 7) * q^64 + (b11 + b9 + 7b6 + 2b5 + b3 - b2 - 4) * q^66 + (-b15 - 2b12 - b10 - 4b8) * q^67 + (-2b15 - 2b14 + 9b12 - 2b10 + 2b8 - 2b4) * q^68 + (-5b13 + 3b9 + 3b6 + b5 - 3b3 + 3) * q^69 + (3b13 + 3b11 - b9 + 3b2 - 1) q^71 + (3b15 + b14 + 6b12 - 5b10 + 3b8 - 6b4 - 3b1) * q^72 + (-3b15 + 2b14 - 2b12 + b10 + b8 - 2b7 + b4 - b1) * q^73 + (5b13 + 2b11 - 12b6 - 7b5 + 2b3 + 7b2 + 12) * q^74 + (-b13 - b11 - 3b9 - 2b6 - b5 + 3b3 + b2 - 3) q^76 + (2b15 - 4b14 - 3b12 + 2b10 + 3b8 - 4b4) * q^77 + (b15 - 4b14 + 11b12 + b10 + 4b7 - 2b4 - 3b1) q^78 + (3b13 - b11 + b6 - 2b5 + b3 + 2b2 - 1) q^79 + (-2b13 + 4b9 - 4b6 + 2b5 - 2b3 - 3b2 + 6) * q^81 + (-2b14 + 3b12 + 2b10 - b8 + 3b7 + 2b4 + b1) q^82 + (9b7 + 3b1) * q^83 + (-b13 - 4b11 - 4b9 + 2b6 - b5 + 2b3 - 2b2 - 8) q^84 + (5b9 + 6b6 - 5b3 + 5) q^86 + (-2b15 - b12 + 2b10 - 5b7 + b4 + 3b1) * q^87 + (4b15 + 4b14 - 5b10 + 3b7 + b4 - 3b1) q^88 + (3b13 + 3b11 - 3b5 + 3) q^89 + (2b13 + 2b11 - 4b9 - 3b5 - b2 - 6) * q^91 + (b15 + b14 + b10 + 3b7 - 2b4) * q^92 + (b15 - b14 + b12 - b10 + 2b8 - 7b7 + b4 - 4b1) q^93 + (b13 + 2b11 + 4b9 + 2b5 - 4b3 - b2 + 4) * q^94 + (2b11 + 3b9 + b6 + 2b5 - 6b3 - 2b2 + 7) q^96 + (2b15 + 2b14 - b10 + 7b7 - b4 + 3b1) * q^97 + (3b14 - 3b12 - 3b10 + b8 - 3b7 - 3b4 - b1) q^98 + (-3b13 + 6b11 + b9 - 4b6 + 4b5 - 2b3 + 3b2 + 6) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{4} + 16 q^{6} - 10 q^{9} + 2 q^{11} + 6 q^{14} - 8 q^{16} - 8 q^{19} - 30 q^{21} + 66 q^{24} - 40 q^{26} + 2 q^{29} + 8 q^{31} + 18 q^{34} - 28 q^{36} - 50 q^{39} + 10 q^{41} - 88 q^{44} - 6 q^{49}+ \cdots + 34 q^{99}+O(q^{100}) \) 16 * q + 8 * q^4 + 16 * q^6 - 10 * q^9 + 2 * q^11 + 6 * q^14 - 8 * q^16 - 8 * q^19 - 30 * q^21 + 66 * q^24 - 40 * q^26 + 2 * q^29 + 8 * q^31 + 18 * q^34 - 28 * q^36 - 50 * q^39 + 10 * q^41 - 88 * q^44 - 6 * q^49 + 22 * q^51 - 52 * q^54 + 60 * q^56 + 34 * q^59 + 26 * q^61 - 76 * q^64 - 16 * q^66 + 54 * q^69 - 32 * q^71 + 80 * q^74 - 22 * q^76 - 14 * q^79 + 34 * q^81 - 54 * q^84 + 68 * q^86 + 36 * q^89 - 68 * q^91 + 6 * q^94 + 68 * q^96 + 34 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	225.2.k.c

Level	$225$
Weight	$2$
Character orbit	225.k
Analytic conductor	$1.797$
Analytic rank	$0$
Dimension	$16$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.79663404548\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 12x^{14} + 102x^{12} - 406x^{10} + 1167x^{8} - 1842x^{6} + 2023x^{4} - 441x^{2} + 81 \) x^16 - 12x^14 + 102x^12 - 406x^10 + 1167x^8 - 1842x^6 + 2023x^4 - 441*x^2 + 81
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( 3457 \nu^{14} - 28693 \nu^{12} + 210280 \nu^{10} - 306752 \nu^{8} + 709118 \nu^{6} + \cdots - 7606359 ) / 3976425 \) (3457v^14 - 28693v^12 + 210280v^10 - 306752v^8 + 709118v^6 - 1299110v^4 + 7466556*v^2 - 7606359) / 3976425
\(\beta_{3}\)	\(=\)	\( ( -26\nu^{14} + 74\nu^{12} + 10\nu^{10} - 10439\nu^{8} + 38576\nu^{6} - 94895\nu^{4} + 82467\nu^{2} - 18063 ) / 29025 \) (-26v^14 + 74v^12 + 10v^10 - 10439v^8 + 38576v^6 - 94895v^4 + 82467*v^2 - 18063) / 29025
\(\beta_{4}\)	\(=\)	\( ( 5917 \nu^{15} - 90503 \nu^{13} + 784955 \nu^{11} - 3838362 \nu^{9} + 10468603 \nu^{7} + \cdots - 16863264 \nu ) / 3976425 \) (5917v^15 - 90503v^13 + 784955v^11 - 3838362v^9 + 10468603v^7 - 20515435v^5 + 19722086v^3 - 16863264v) / 3976425
\(\beta_{5}\)	\(=\)	\( ( - 7559 \nu^{14} + 69641 \nu^{12} - 572135 \nu^{10} + 1494574 \nu^{8} - 4803841 \nu^{6} + \cdots - 5812992 ) / 3976425 \) (-7559v^14 + 69641v^12 - 572135v^10 + 1494574v^8 - 4803841v^6 + 3626995v^4 - 7975497*v^2 - 5812992) / 3976425
\(\beta_{6}\)	\(=\)	\( ( 11836 \nu^{14} - 138574 \nu^{12} + 1170340 \nu^{10} - 4500371 \nu^{8} + 12811274 \nu^{6} + \cdots - 814212 ) / 3976425 \) (11836v^14 - 138574v^12 + 1170340v^10 - 4500371v^8 + 12811274v^6 - 19088030v^4 + 21981813*v^2 - 814212) / 3976425
\(\beta_{7}\)	\(=\)	\( ( 3766 \nu^{15} - 41414 \nu^{13} + 332215 \nu^{11} - 1090526 \nu^{9} + 2480389 \nu^{7} + \cdots + 2721243 \nu ) / 1325475 \) (3766v^15 - 41414v^13 + 332215v^11 - 1090526v^9 + 2480389v^7 - 2137205v^5 + 467253v^3 + 2721243v) / 1325475
\(\beta_{8}\)	\(=\)	\( ( - 11836 \nu^{15} + 138574 \nu^{13} - 1170340 \nu^{11} + 4500371 \nu^{9} - 12811274 \nu^{7} + \cdots + 4790637 \nu ) / 3976425 \) (-11836v^15 + 138574v^13 - 1170340v^11 + 4500371v^9 - 12811274v^7 + 19088030v^5 - 21981813v^3 + 4790637v) / 3976425
\(\beta_{9}\)	\(=\)	\( ( - 14896 \nu^{14} + 165889 \nu^{12} - 1314040 \nu^{10} + 4313456 \nu^{8} - 9114389 \nu^{6} + \cdots - 5535918 ) / 3976425 \) (-14896v^14 + 165889v^12 - 1314040v^10 + 4313456v^8 - 9114389v^6 + 8453480v^4 - 1848168*v^2 - 5535918) / 3976425
\(\beta_{10}\)	\(=\)	\( ( 13477 \nu^{15} - 168383 \nu^{13} + 1451855 \nu^{11} - 6027522 \nu^{9} + 17277208 \nu^{7} + \cdots + 11503071 \nu ) / 3976425 \) (13477v^15 - 168383v^13 + 1451855v^11 - 6027522v^9 + 17277208v^7 - 24805735v^5 + 20660066v^3 + 11503071v) / 3976425
\(\beta_{11}\)	\(=\)	\( ( - 3323 \nu^{14} + 38066 \nu^{12} - 324695 \nu^{10} + 1252588 \nu^{8} - 3846136 \nu^{6} + \cdots + 1468098 ) / 795285 \) (-3323v^14 + 38066v^12 - 324695v^10 + 1252588v^8 - 3846136v^6 + 6436600v^4 - 8169474*v^2 + 1468098) / 795285
\(\beta_{12}\)	\(=\)	\( ( 19253 \nu^{15} - 245932 \nu^{13} + 2129695 \nu^{11} - 9130758 \nu^{9} + 26781707 \nu^{7} + \cdots - 10338741 \nu ) / 3976425 \) (19253v^15 - 245932v^13 + 2129695v^11 - 9130758v^9 + 26781707v^7 - 44578415v^5 + 47402299v^3 - 10338741v) / 3976425
\(\beta_{13}\)	\(=\)	\( ( 572 \nu^{14} - 6758 \nu^{12} + 56330 \nu^{10} - 215542 \nu^{8} + 579583 \nu^{6} - 869185 \nu^{4} + \cdots - 212679 ) / 92475 \) (572v^14 - 6758v^12 + 56330v^10 - 215542v^8 + 579583v^6 - 869185v^4 + 809301*v^2 - 212679) / 92475
\(\beta_{14}\)	\(=\)	\( ( 43568 \nu^{15} - 518857 \nu^{13} + 4432420 \nu^{11} - 17623398 \nu^{9} + 51780332 \nu^{7} + \cdots - 6887691 \nu ) / 3976425 \) (43568v^15 - 518857v^13 + 4432420v^11 - 17623398v^9 + 51780332v^7 - 81131165v^5 + 89205019v^3 - 6887691v) / 3976425
\(\beta_{15}\)	\(=\)	\( ( - 49089 \nu^{15} + 557311 \nu^{13} - 4656460 \nu^{11} + 17097154 \nu^{9} - 47712236 \nu^{7} + \cdots - 564507 \nu ) / 3976425 \) (-49089v^15 + 557311v^13 - 4656460v^11 + 17097154v^9 - 47712236v^7 + 67106795v^5 - 70902062v^3 - 564507v) / 3976425

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{11} + \beta_{9} + 3\beta_{6} + \beta_{5} - \beta_{3} + 1 \) b11 + b9 + 3*b6 + b5 - b3 + 1
\(\nu^{3}\)	\(=\)	\( -\beta_{15} - \beta_{14} - 3\beta_{12} + 2\beta_{10} - 4\beta_{8} - 3\beta_{7} + 2\beta_{4} + 4\beta_1 \) -b15 - b14 - 3b12 + 2b10 - 4b8 - 3b7 + 2b4 + 4b1
\(\nu^{4}\)	\(=\)	\( -6\beta_{13} - 2\beta_{11} + 15\beta_{6} + 8\beta_{5} - 7\beta_{3} - 8\beta_{2} - 15 \) -6b13 - 2b11 + 15b6 + 8b5 - 7b3 - 8b2 - 15
\(\nu^{5}\)	\(=\)	\( 8\beta_{15} + 11\beta_{14} - 27\beta_{12} + 8\beta_{10} - 22\beta_{8} + 11\beta_{4} \) 8b15 + 11b14 - 27b12 + 8b10 - 22b8 + 11b4
\(\nu^{6}\)	\(=\)	\( -57\beta_{13} - 57\beta_{11} - 49\beta_{9} + 19\beta_{5} - 38\beta_{2} - 139 \) -57b13 - 57b11 - 49b9 + 19b5 - 38*b2 - 139
\(\nu^{7}\)	\(=\)	\( 144\beta_{15} + 144\beta_{14} - 87\beta_{10} + 204\beta_{7} - 57\beta_{4} - 139\beta_1 \) 144b15 + 144b14 - 87b10 + 204b7 - 57b4 - 139b1
\(\nu^{8}\)	\(=\)	\( -144\beta_{13} - 253\beta_{11} - 343\beta_{9} - 588\beta_{6} - 253\beta_{5} + 343\beta_{3} + 144\beta_{2} - 343 \) -144b13 - 253b11 - 343b9 - 588b6 - 253b5 + 343b3 + 144*b2 - 343
\(\nu^{9}\)	\(=\)	\( 631 \beta_{15} + 397 \beta_{14} + 1461 \beta_{12} - 1028 \beta_{10} + 931 \beta_{8} + 1461 \beta_{7} + \cdots - 931 \beta_1 \) 631b15 + 397b14 + 1461b12 - 1028b10 + 931b8 + 1461b7 - 1028b4 - 931b1
\(\nu^{10}\)	\(=\)	\( 1725\beta_{13} + 1028\beta_{11} - 3984\beta_{6} - 2753\beta_{5} + 2392\beta_{3} + 2753\beta_{2} + 3984 \) 1725b13 + 1028b11 - 3984b6 - 2753b5 + 2392b3 + 2753b2 + 3984
\(\nu^{11}\)	\(=\)	\( -2753\beta_{15} - 4448\beta_{14} + 10260\beta_{12} - 2753\beta_{10} + 6376\beta_{8} - 4448\beta_{4} \) -2753b15 - 4448b14 + 10260b12 - 2753b10 + 6376b8 - 4448b4
\(\nu^{12}\)	\(=\)	\( 19083\beta_{13} + 19083\beta_{11} + 16636\beta_{9} - 7201\beta_{5} + 11882\beta_{2} + 44023 \) 19083b13 + 19083b11 + 16636b9 - 7201b5 + 11882*b2 + 44023
\(\nu^{13}\)	\(=\)	\( -50121\beta_{15} - 50121\beta_{14} + 31038\beta_{10} - 71511\beta_{7} + 19083\beta_{4} + 44023\beta_1 \) -50121b15 - 50121b14 + 31038b10 - 71511b7 + 19083b4 + 44023b1
\(\nu^{14}\)	\(=\)	\( 50121 \beta_{13} + 82189 \beta_{11} + 115534 \beta_{9} + 189318 \beta_{6} + 82189 \beta_{5} + \cdots + 115534 \) 50121b13 + 82189b11 + 115534b9 + 189318b6 + 82189b5 - 115534b3 - 50121*b2 + 115534
\(\nu^{15}\)	\(=\)	\( - 215776 \beta_{15} - 132310 \beta_{14} - 496965 \beta_{12} + 348086 \beta_{10} - 304852 \beta_{8} + \cdots + 304852 \beta_1 \) -215776b15 - 132310b14 - 496965b12 + 348086b10 - 304852b8 - 496965b7 + 348086b4 + 304852b1

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 49.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} - 12 T^{14} + \cdots + 81 \) T^16 - 12T^14 + 102T^12 - 406T^10 + 1167T^8 - 1842T^6 + 2023T^4 - 441*T^2 + 81
$3$	\( T^{16} + 5 T^{14} + \cdots + 6561 \) T^16 + 5T^14 + 4T^12 + 15T^10 + 135T^8 + 135T^6 + 324T^4 + 3645*T^2 + 6561
$5$	\( T^{16} \) T^16
$7$	\( T^{16} - 25 T^{14} + \cdots + 6561 \) T^16 - 25T^14 + 451T^12 - 3630T^10 + 21195T^8 - 58590T^6 + 115506T^4 - 29160*T^2 + 6561
$11$	\( (T^{8} - T^{7} + 26 T^{6} + \cdots + 81)^{2} \) (T^8 - T^7 + 26T^6 + 107T^5 + 593T^4 + 1007T^3 + 1456T^2 + 369T + 81)^2
$13$	\( T^{16} + \cdots + 131079601 \) T^16 - 64T^14 + 2962T^12 - 59686T^10 + 862027T^8 - 5843158T^6 + 28554859T^4 - 73788805*T^2 + 131079601
$17$	\( (T^{8} + 81 T^{6} + \cdots + 91809)^{2} \) (T^8 + 81T^6 + 2214T^4 + 24220*T^2 + 91809)^2
$19$	\( (T^{4} + 2 T^{3} - 27 T^{2} + \cdots - 25)^{4} \) (T^4 + 2T^3 - 27T^2 - 80*T - 25)^4
$23$	\( T^{16} + \cdots + 3486784401 \) T^16 - 111T^14 + 9288T^12 - 281097T^10 + 6056127T^8 - 71156961T^6 + 592799472T^4 - 1640558367*T^2 + 3486784401
$29$	\( (T^{8} - T^{7} + \cdots + 16641)^{2} \) (T^8 - T^7 + 41T^6 - 244T^5 + 1871T^4 - 5938T^3 + 15004T^2 - 18318T + 16641)^2
$31$	\( (T^{8} - 4 T^{7} + \cdots + 59049)^{2} \) (T^8 - 4T^7 + 58T^6 + 114T^5 + 1629T^4 + 810T^3 + 10935T^2 + 6561*T + 59049)^2
$37$	\( (T^{8} + 199 T^{6} + \cdots + 418609)^{2} \) (T^8 + 199T^6 + 9513T^4 + 124903*T^2 + 418609)^2
$41$	\( (T^{8} - 5 T^{7} + \cdots + 42849)^{2} \) (T^8 - 5T^7 + 50T^6 - 197T^5 + 1637T^4 - 6095T^3 + 20746T^2 - 33327*T + 42849)^2
$43$	\( T^{16} + \cdots + 205144679041 \) T^16 - 196T^14 + 25738T^12 - 1927312T^10 + 105636307T^8 - 3356926096T^6 + 71980515082T^4 - 126271170052*T^2 + 205144679041
$47$	\( T^{16} + \cdots + 21071715921 \) T^16 - 186T^14 + 26709T^12 - 1294114T^10 + 45982884T^8 - 627705066T^6 + 6325951549T^4 - 12546845874*T^2 + 21071715921
$53$	\( (T^{8} + 228 T^{6} + \cdots + 221841)^{2} \) (T^8 + 228T^6 + 13614T^4 + 113053*T^2 + 221841)^2
$59$	\( (T^{8} - 17 T^{7} + \cdots + 5349969)^{2} \) (T^8 - 17T^7 + 287T^6 - 1862T^5 + 17855T^4 - 76814T^3 + 840022T^2 - 2114082*T + 5349969)^2
$61$	\( (T^{8} - 13 T^{7} + 172 T^{6} + \cdots + 1)^{2} \) (T^8 - 13T^7 + 172T^6 - 143T^5 + 1193T^4 - 299T^3 + 8278T^2 - 91*T + 1)^2
$67$	\( T^{16} + \cdots + 3486784401 \) T^16 - 217T^14 + 35107T^12 - 2455086T^10 + 127775907T^8 - 843115662T^6 + 4549304898T^4 - 4281288696*T^2 + 3486784401
$71$	\( (T^{4} + 8 T^{3} + \cdots + 381)^{4} \) (T^4 + 8T^3 - 40T^2 - 263*T + 381)^4
$73$	\( (T^{8} + 196 T^{6} + \cdots + 12769)^{2} \) (T^8 + 196T^6 + 8478T^4 + 79777*T^2 + 12769)^2
$79$	\( (T^{8} + 7 T^{7} + \cdots + 42849)^{2} \) (T^8 + 7T^7 + 82T^6 - 93T^5 + 1365T^4 - 621T^3 + 11592T^2 - 14283*T + 42849)^2
$83$	\( T^{16} + \cdots + 282429536481 \) T^16 - 324T^14 + 77274T^12 - 8029206T^10 + 613578159T^8 - 12762024174T^6 + 209121502059T^4 - 251435897361*T^2 + 282429536481
$89$	\( (T^{4} - 9 T^{3} + \cdots + 2025)^{4} \) (T^4 - 9T^3 - 99T^2 + 405*T + 2025)^4
$97$	\( T^{16} + \cdots + 824843587681 \) T^16 - 199T^14 + 28828T^12 - 1783213T^10 + 79268227T^8 - 1580980129T^6 + 22726478692T^4 - 163756440163*T^2 + 824843587681
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\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(-\beta_{6}\)	\(-1\)