LMFDB - Newform orbit 197.2.b.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [197,2,Mod(196,197)] 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("197.196"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(197, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 197 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	197.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$1.57305291982$
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} + 24x^{14} + 228x^{12} + 1095x^{10} + 2834x^{8} + 3942x^{6} + 2795x^{4} + 925x^{2} + 112 $ x^16 + 24x^14 + 228x^12 + 1095x^10 + 2834x^8 + 3942x^6 + 2795x^4 + 925*x^2 + 112
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
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_1 q^{2} + \beta_{9} q^{3} + (\beta_{2} - 1) q^{4} - \beta_{10} q^{5} + ( - \beta_{12} + \beta_{8} + \beta_{2} - 1) q^{6} + \beta_{5} q^{7} + (\beta_{3} - \beta_1) q^{8} + ( - \beta_{12} - \beta_{5} - 1) q^{9}+ \cdots + ( - \beta_{15} - \beta_{14} + \cdots + \beta_1) q^{99}+O(q^{100}) $ q + b1 * q^2 + b9 * q^3 + (b2 - 1) * q^4 - b10 * q^5 + (-b12 + b8 + b2 - 1) * q^6 + b5 * q^7 + (b3 - b1) * q^8 + (-b12 - b5 - 1) * q^9 + (b12 - b8 + b5 - b4 + 1) * q^10 + b14 * q^11 + (-b15 - b10 + b3 - b1) * q^12 + (b10 + b6) * q^13 + (b13 + b10 + b1) * q^14 + (b12 + b11 - b8 - b7 + b5 - b2 + 1) * q^15 + (b12 + b11 - b2 + 1) * q^16 + (-b14 - b9 - b6 - b3 + b1) * q^17 + (-b15 + b14 - b13 - 2b10 - b9 + b3 - b1) q^18 + (-b7 - 1) * q^19 + (b15 + b13 + b10 + b9) * q^20 + (b15 - b14 + b10 - b6 - b3) * q^21 + (-b12 - b11 + 2b8 + b4 + b2 - 1) q^22 + (b12 - b4 - b2 + 1) * q^23 + (3b12 + 2b11 - 2b8 - b7 + b5 - 2b4 - 2b2 + 4) q^24 + (-b12 - 2b11 + b8 + b7 + b4 - 1) q^25 + (-b12 + b11 + b7 - 2b5 + b2 - 1) q^26 + (-b15 + b14 - b13 - b10 + b6 + b3 - 2b1) q^27 + (-b12 + b7 - 3b5 + b4 - 2) q^28 + (b12 + b11 + b7 - b5 - b4 + 2) * q^29 + (b15 + b13 + b10 + b9 - b3 + 2b1) q^30 + (-b13 - b9 - b3) * q^31 + (b15 - b14 - b6 - b3 + b1) * q^32 + (-b12 - b11 + b7 - b5 + 2b4 + b2 - 2) q^33 + (b12 - b11 - 2b8 - b7 + b5 - 1) q^34 + (-b14 + b10 - b9 - b1) * q^35 + (4b12 + b11 - 3b8 - 2b7 + 4b5 - 2b4 - 3b2 + 3) * q^36 + (-2b12 - 2b11 + b7 - 2b5 + b4 - 1) q^37 + (b6 + b3 - 2b1) q^38 + (-2b12 - b11 + 3b8 + 2b7 + b4 + 3b2 - 1) * q^39 + (-3b12 - b11 + 2b8 + 2b7 - 3b5 + b2) * q^40 + (-b11 - b7 + 2b4 - 2b2 - 2) * q^41 + (-4b12 - 2b11 + 3b8 + 2b4 + b2 - 2) * q^42 + (b8 + 2b4) q^43 + (-b15 + b14 - b10 - b9 + b6 + b3 - b1) * q^44 + (b15 - b14 + b10 - 2b3 + 2b1) * q^45 + (b15 - b14 + 2b10 - 2b3 + 2b1) q^46 + (2b12 + 2b11 - 3b8 - b7 + b5 - 2b4 + 1) * q^47 + (b15 - b14 + b13 + 2b10 + b9 - b6 - 2b3 + 3b1) q^48 + (-b12 - b11 + 2b8 + b5 - 1) q^49 + (-b15 + b9 + b6 + b3) * q^50 + (-b11 - b7 - b4 - b2 + 2) * q^51 + (-b15 + b14 - 2b13 - 2b10 - 2b9 - 2b1) * q^52 + (b12 - 2b11 + b7 + b2) q^53 + (4b12 + 2b11 - 2b8 - b7 + 4b5 - 2b4 - 2b2 + 6) * q^54 + (-2b8 + b5 - b2 + 2) q^55 + (-b15 + b14 - b13 - 3b10 - b6 - b1) q^56 + (-b15 + 2b14 - b13 - b9 + 2b6 + b3 - 2b1) q^57 + (b15 - b14 - b13 - b9 - 2b6 - 3b3 + 2b1) q^58 + (2b11 - b7 - 2b5 - b2 - 1) * q^59 + (-4b12 + 2b8 - 3b5 + 2b4 + 3b2 - 6) q^60 + (b12 - b8 + b7 - b5 - 2b2) q^61 + (-b11 - b7 + 4b5 + 2b2 - 1) * q^62 + (-b12 - b5 - b4 + 2b2 - 3) q^63 + (-b12 + 2b8 + b5 + b4 - 2) q^64 + (2b12 + 2b11 - b7 - b2 + 3) * q^65 + (-b15 + b14 - b13 - 3b10 + 2b9 + 2b3 - 4b1) * q^66 + (-b15 - 2b14 + b10 - b3 + b1) q^67 + (b15 - b14 + b13 + 3b10 + 2b9 + b3 - 3b1) q^68 + (b15 + 2b14 + b13 + b9 + b6 - b3 + 2b1) * q^69 + (b12 + b11 - 2b8 - b5 - 3b2 + 4) * q^70 + (-b15 + b14 + b13 + b10 + b9 + b3 - b1) * q^71 + (2b15 + b14 + 2b13 + 5b10 + 2b9 + b6 - b3 + 3b1) q^72 + (2b15 - 2b14 + b10 - b3 - 2b1) q^73 + (-2b15 + 2b14 - 2b13 - 3b10 + b9 + b6 + 3b3 - 2b1) * q^74 + (b15 - b14 - 2b10 + b9 - b6 + b3) q^75 + (b12 + 2b11 - b8 - b7 - b5 - b4 - 3b2 + 4) * q^76 + (b15 - b13 + 2b10 - b9) q^77 + (-2b15 - b14 - 2b10 - 3b9 - b6 + b3 - b1) q^78 + (b15 - b14 + 2b13 - b9 - 2b6 - b1) * q^79 + (-b15 + b14 - b13 - 3b10 - 2b9 - b6 + b3 + b1) * q^80 + (2b12 + b11 - 3b8 + b7 - b4 - 3b2 + 2) q^81 + (-b10 + 3b9 + 2b6) * q^82 + (b12 + b11 + b7 + b5 - 2b4 - b2 + 1) q^83 + (-2b15 - b14 - 2b10 - 3b9 + 2b3 + b1) * q^84 + (b12 - b7 + b5 - b4 + 2b2) q^85 + (-b14 - 2b10 + b9 - b3 + b1) q^86 + (2b13 - b10 + 2b9 - b6 - b3 + 3b1) q^87 + (3b12 - b2 + 4) q^88 + (b13 + 2b10 - 2b9 - b6 + b3 + b1) * q^89 + (-5b12 - 2b11 + 2b8 + b7 - b5 + b4 + 6b2 - 8) * q^90 + (b13 - 2b9 - 2b3 + 3b1) q^91 + (-4b12 - 2b11 + 3b8 + b7 - 2b5 + 4b2 - 7) q^92 + (b12 - b11 + 2b7 - b5 + 2b2 + 1) * q^93 + (2b15 + b14 + b13 + 3b10 + b9 - b6 - 2b1) q^94 + (b15 + b14 + b13 + b3 + b1) * q^95 + (-b12 + b11 - b7 - 3b5 - b4 + 2b2 - 3) * q^96 + (-b12 + 2b11 - b7 + 1) q^97 + (-b15 - b14 + b13 + b10 - 2b9 + b6 + 2b1) * q^98 + (-b15 - b14 - b10 - 2b6 + 2b3 + b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{4} - 10 q^{6} + 2 q^{7} - 18 q^{9} + 4 q^{10} + 14 q^{15} + 16 q^{16} - 14 q^{19} + 4 q^{22} + 8 q^{23} + 40 q^{24} - 4 q^{25} - 22 q^{26} - 32 q^{28} + 20 q^{29} - 20 q^{33} - 24 q^{34} + 26 q^{36}+ \cdots + 18 q^{97}+O(q^{100}) $ 16 * q - 16 * q^4 - 10 * q^6 + 2 * q^7 - 18 * q^9 + 4 * q^10 + 14 * q^15 + 16 * q^16 - 14 * q^19 + 4 * q^22 + 8 * q^23 + 40 * q^24 - 4 * q^25 - 22 * q^26 - 32 * q^28 + 20 * q^29 - 20 * q^33 - 24 * q^34 + 26 * q^36 - 14 * q^37 + 6 * q^39 + 2 * q^40 - 14 * q^41 + 2 * q^42 + 22 * q^43 - 14 * q^47 - 2 * q^49 + 26 * q^51 - 2 * q^53 + 78 * q^54 + 22 * q^55 - 18 * q^59 - 74 * q^60 - 10 * q^61 - 6 * q^62 - 58 * q^63 - 10 * q^64 + 50 * q^65 + 50 * q^70 + 50 * q^76 + 4 * q^81 - 4 * q^85 + 64 * q^88 - 112 * q^90 - 100 * q^92 + 10 * q^93 - 60 * q^96 + 18 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} + 24x^{14} + 228x^{12} + 1095x^{10} + 2834x^{8} + 3942x^{6} + 2795x^{4} + 925x^{2} + 112 $ x^16 + 24*x^14 + 228*x^12 + 1095*x^10 + 2834*x^8 + 3942*x^6 + 2795*x^4 + 925*x^2 + 112 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ \nu^{2} + 3 $ v^2 + 3
$\beta_{3}$	$=$	$ \nu^{3} + 5\nu $ v^3 + 5*v
$\beta_{4}$	$=$	$ ( 3\nu^{14} + 53\nu^{12} + 287\nu^{10} + 210\nu^{8} - 2212\nu^{6} - 4310\nu^{4} - 413\nu^{2} + 760 ) / 92 $ (3v^14 + 53v^12 + 287v^10 + 210v^8 - 2212v^6 - 4310v^4 - 413*v^2 + 760) / 92
$\beta_{5}$	$=$	$ ( -4\nu^{14} - 86\nu^{12} - 697\nu^{10} - 2626\nu^{8} - 4587\nu^{6} - 3277\nu^{4} - 630\nu^{2} + 37 ) / 23 $ (-4v^14 - 86v^12 - 697v^10 - 2626v^8 - 4587v^6 - 3277v^4 - 630*v^2 + 37) / 23
$\beta_{6}$	$=$	$ ( \nu^{15} + 10\nu^{13} - 73\nu^{11} - 1333\nu^{9} - 6196\nu^{7} - 11434\nu^{5} - 7973\nu^{3} - 1740\nu ) / 23 $ (v^15 + 10v^13 - 73v^11 - 1333v^9 - 6196v^7 - 11434v^5 - 7973v^3 - 1740*v) / 23
$\beta_{7}$	$=$	$ ( -\nu^{14} - 10\nu^{12} + 73\nu^{10} + 1333\nu^{8} + 6196\nu^{6} + 11434\nu^{4} + 7950\nu^{2} + 1648 ) / 23 $ (-v^14 - 10v^12 + 73v^10 + 1333v^8 + 6196v^6 + 11434v^4 + 7950v^2 + 1648) / 23
$\beta_{8}$	$=$	$ ( 13\nu^{14} + 291\nu^{12} + 2501\nu^{10} + 10340\nu^{8} + 21250\nu^{6} + 20776\nu^{4} + 8545\nu^{2} + 1162 ) / 46 $ (13v^14 + 291v^12 + 2501v^10 + 10340v^8 + 21250v^6 + 20776v^4 + 8545*v^2 + 1162) / 46
$\beta_{9}$	$=$	$ ( 19\nu^{15} + 443\nu^{13} + 4041\nu^{11} + 18304\nu^{9} + 43460\nu^{7} + 53050\nu^{5} + 29891\nu^{3} + 5718\nu ) / 92 $ (19v^15 + 443v^13 + 4041v^11 + 18304v^9 + 43460v^7 + 53050v^5 + 29891v^3 + 5718v) / 92
$\beta_{10}$	$=$	$ ( 8\nu^{15} + 195\nu^{13} + 1877\nu^{11} + 9047\nu^{9} + 22836\nu^{7} + 28680\nu^{5} + 15152\nu^{3} + 2479\nu ) / 46 $ (8v^15 + 195v^13 + 1877v^11 + 9047v^9 + 22836v^7 + 28680v^5 + 15152v^3 + 2479v) / 46
$\beta_{11}$	$=$	$ ( -39\nu^{14} - 873\nu^{12} - 7503\nu^{10} - 31066\nu^{8} - 64348\nu^{6} - 64674\nu^{4} - 28395\nu^{2} - 4084 ) / 92 $ (-39v^14 - 873v^12 - 7503v^10 - 31066v^8 - 64348v^6 - 64674v^4 - 28395*v^2 - 4084) / 92
$\beta_{12}$	$=$	$ ( 39\nu^{14} + 873\nu^{12} + 7503\nu^{10} + 31066\nu^{8} + 64348\nu^{6} + 64766\nu^{4} + 29039\nu^{2} + 4636 ) / 92 $ (39v^14 + 873v^12 + 7503v^10 + 31066v^8 + 64348v^6 + 64766v^4 + 29039*v^2 + 4636) / 92
$\beta_{13}$	$=$	$ ( -16\nu^{15} - 367\nu^{13} - 3271\nu^{11} - 14299\nu^{9} - 32010\nu^{7} - 35234\nu^{5} - 16412\nu^{3} - 2451\nu ) / 46 $ (-16v^15 - 367v^13 - 3271v^11 - 14299v^9 - 32010v^7 - 35234v^5 - 16412v^3 - 2451v) / 46
$\beta_{14}$	$=$	$ ( - 45 \nu^{15} - 1025 \nu^{13} - 9043 \nu^{11} - 38984 \nu^{9} - 85960 \nu^{7} - 94602 \nu^{5} + \cdots - 8410 \nu ) / 92 $ (-45v^15 - 1025v^13 - 9043v^11 - 38984v^9 - 85960v^7 - 94602v^5 - 47073v^3 - 8410v) / 92
$\beta_{15}$	$=$	$ ( - 41 \nu^{15} - 985 \nu^{13} - 9335 \nu^{11} - 44316 \nu^{9} - 110744 \nu^{7} - 140246 \nu^{5} + \cdots - 13898 \nu ) / 92 $ (-41v^15 - 985v^13 - 9335v^11 - 44316v^9 - 110744v^7 - 140246v^5 - 78137v^3 - 13898v) / 92

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{2} - 3 $ b2 - 3
$\nu^{3}$	$=$	$ \beta_{3} - 5\beta_1 $ b3 - 5*b1
$\nu^{4}$	$=$	$ \beta_{12} + \beta_{11} - 7\beta_{2} + 15 $ b12 + b11 - 7*b2 + 15
$\nu^{5}$	$=$	$ \beta_{15} - \beta_{14} - \beta_{6} - 9\beta_{3} + 29\beta_1 $ b15 - b14 - b6 - 9b3 + 29b1
$\nu^{6}$	$=$	$ -11\beta_{12} - 10\beta_{11} + 2\beta_{8} + \beta_{5} + \beta_{4} + 46\beta_{2} - 88 $ -11b12 - 10b11 + 2b8 + b5 + b4 + 46b2 - 88
$\nu^{7}$	$=$	$ -11\beta_{15} + 9\beta_{14} + \beta_{13} - \beta_{10} - 2\beta_{9} + 10\beta_{6} + 65\beta_{3} - 176\beta_1 $ -11b15 + 9b14 + b13 - b10 - 2b9 + 10b6 + 65b3 - 176b1
$\nu^{8}$	$=$	$ 92\beta_{12} + 77\beta_{11} - 29\beta_{8} - 13\beta_{5} - 13\beta_{4} - 301\beta_{2} + 546 $ 92b12 + 77b11 - 29b8 - 13b5 - 13b4 - 301b2 + 546
$\nu^{9}$	$=$	$ 92\beta_{15} - 63\beta_{14} - 13\beta_{13} + 15\beta_{10} + 31\beta_{9} - 77\beta_{6} - 441\beta_{3} + 1091\beta_1 $ 92b15 - 63b14 - 13b13 + 15b10 + 31b9 - 77b6 - 441b3 + 1091b1
$\nu^{10}$	$=$	$ -700\beta_{12} - 547\beta_{11} + 286\beta_{8} + 2\beta_{7} + 114\beta_{5} + 121\beta_{4} + 1969\beta_{2} - 3466 $ -700b12 - 547b11 + 286b8 + 2b7 + 114b5 + 121b4 + 1969*b2 - 3466
$\nu^{11}$	$=$	$ - 700 \beta_{15} + 414 \beta_{14} + 114 \beta_{13} - 160 \beta_{10} - 318 \beta_{9} + 545 \beta_{6} + \cdots - 6849 \beta_1 $ -700b15 + 414b14 + 114b13 - 160b10 - 318b9 + 545b6 + 2928b3 - 6849b1
$\nu^{12}$	$=$	$ 5092 \beta_{12} + 3759 \beta_{11} - 2409 \beta_{8} - 41 \beta_{7} - 841 \beta_{5} - 991 \beta_{4} + \cdots + 22239 $ 5092b12 + 3759b11 - 2409b8 - 41b7 - 841b5 - 991b4 - 12878*b2 + 22239
$\nu^{13}$	$=$	$ 5092 \beta_{15} - 2683 \beta_{14} - 841 \beta_{13} + 1483 \beta_{10} + 2751 \beta_{9} + \cdots + 43371 \beta_1 $ 5092b15 - 2683b14 - 841b13 + 1483b10 + 2751b9 - 3718b6 - 19279b3 + 43371b1
$\nu^{14}$	$=$	$ - 36106 \beta_{12} - 25406 \beta_{11} + 18703 \beta_{8} + 533 \beta_{7} + 5599 \beta_{5} + \cdots - 143530 $ -36106b12 - 25406b11 + 18703b8 + 533b7 + 5599b5 + 7610b4 + 84212*b2 - 143530
$\nu^{15}$	$=$	$ - 36106 \beta_{15} + 17403 \beta_{14} + 5599 \beta_{13} - 12711 \beta_{10} - 21793 \beta_{9} + \cdots - 276419 \beta_1 $ -36106b15 + 17403b14 + 5599b13 - 12711b10 - 21793b9 + 24873b6 + 126488b3 - 276419b1

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/197\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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} $

196.1

	−	2.57304i
	−	2.52499i
	−	2.21923i
	−	1.52415i
	−	1.49265i
	−	0.887983i
	−	0.661670i
	−	0.549122i
		0.549122i
		0.661670i
		0.887983i
		1.49265i
		1.52415i
		2.21923i
		2.52499i
		2.57304i

−

2.57304i

−

3.19092i

−4.62054

2.59506i

−8.21037

2.31895

6.74277i

−7.18197

6.67720

196.2

−

2.52499i

−

0.319666i

−4.37558

−

2.38853i

−0.807154

−2.39004

5.99832i

2.89781

−6.03103

196.3

−

2.21923i

2.10111i

−2.92500

−

0.563008i

4.66286

4.64533

2.05279i

−1.41466

−1.24945

196.4

−

1.52415i

−

1.55676i

−0.323023

−

0.503395i

−2.37273

−2.46528

−

2.55596i

0.576498

−0.767247

196.5

−

1.49265i

0.148869i

−0.227997

2.55955i

0.222210

1.45621

−

2.64498i

2.97784

3.82051

196.6

−

0.887983i

2.19265i

1.21149

2.66980i

1.94703

−3.36955

−

2.85175i

−1.80769

2.37074

196.7

−

0.661670i

1.70174i

1.56219

−

3.88654i

1.12599

−0.705857

−

2.35700i

0.104097

−2.57161

196.8

−

0.549122i

−

2.85516i

1.69847

−

0.453660i

−1.56783

1.51024

−

2.03091i

−5.15192

−0.249115

196.9