LMFDB - Newform orbit 650.2.l.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [650,2,Mod(131,650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(650, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([4, 0]))

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

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

chi := DirichletCharacter("650.131");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$650 = 2 \cdot 5^{2} \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	650.l (of order $5$ , degree $4$ , 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:	$5.19027613138$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$5$ over $\Q(\zeta_{5})$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{20} - 9 x^{19} + 43 x^{18} - 132 x^{17} + 291 x^{16} - 480 x^{15} + 624 x^{14} - 653 x^{13} + \cdots + 31$ x^20 - 9x^19 + 43x^18 - 132x^17 + 291x^16 - 480x^15 + 624x^14 - 653x^13 + 552x^12 - 246x^11 - 32x^10 + 229x^9 - 103x^8 + 657x^7 - 46x^6 - 270x^5 + 1406x^4 + 588x^3 + 548x^2 + 126*x + 31
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$5^{2}$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

$q$ -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 $1,\beta_1,\ldots,\beta_{19}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + \beta_{10} q^{2} - \beta_{2} q^{3} - \beta_{13} q^{4} + (\beta_{18} - \beta_{17} + \beta_{11} + \cdots - 1) q^{5} - \beta_{6} q^{6} + (\beta_{12} - \beta_{9} - \beta_{8} + \cdots + 1) q^{7} + ( - \beta_{13} - \beta_{11} - \beta_{10} + 1) q^{8}+ \cdots + ( - 2 \beta_{18} + \beta_{17} + \cdots + 3) q^{99}+O(q^{100})$ q + b10 * q^2 - b2 * q^3 - b13 * q^4 + (b18 - b17 + b11 + b1 - 1) * q^5 - b6 * q^6 + (b12 - b9 - b8 + b7 - b4 - b2 + 1) * q^7 + (-b13 - b11 - b10 + 1) * q^8 + (-b18 + b17 + b14 - b13 - b11 - b10 - b9 - b8 + b7 - b5 - b4 - b2 - b1 + 1) * q^9 + (-b17 - b15 + b11 - b10 - b7 + b6 + b4 + b3 + b2 + b1) * q^10 + (-b19 + b18 - b17 + b16 - b15 - b14 + b11 + b10 + b8 - b6 - b3 + 2b1 - 2) q^11 + (-b6 - b4) * q^12 + b11 * q^13 + (-b18 - b16 + b12 - b8 + b6 - b5 + b3 + 1) * q^14 + (b19 - b16 + b14 + 2b13 + b11 + b9 - b7 + b6 - b1) q^15 - b11 * q^16 + (b17 - b13 - b11 - b10 - b5 - 2b4 - 2b2 + 1) * q^17 + (-b18 + b17 - b16 + b15 + b14 + b12 - 2b11 - b10 - b9 - b8 + b7 - b5 - b4 - b2 - b1 + 2) q^18 + (-b16 - b7 - b5 + b3) * q^19 + (-b15 + b13 + b11 + b10 + b8 - b7 + b4 + b2 + b1 - 2) * q^20 + (b19 - b16 - b15 + 2b14 + 4b13 - b12 + b11 + 2b9 + b7 + b6 - b2 - b1 - 1) q^21 + (-b19 + b16 - b15 - b13 - b12 + b11 + b1 - 1) * q^22 + (b19 - b18 + b17 - 2b16 + b13 + 2b12 - 2b10 - 2b8 - b7 + b6 - 2b5 - b4 + b3 - b2 - b1 + 1) q^23 + (b7 - b4 - b2) * q^24 + (2b19 + b18 + b17 + b16 - b15 - b14 + b13 + 2b11 + 2b10 - b9 + 2b8 - 3b6 + b5 - b3 + 2b2 - b1 - 4) * q^25 + q^26 + (-b19 - b17 - b15 + b14 + b13 + b12 + b11 - b10 - b7 - b5 - b4 + 2b1 - 2) q^27 + (-b16 + b14 - b12 + b9) * q^28 + (-b16 - b15 + b14 - b13 - 3b12 - b11 + b9 - b8 + b6 + b3 - 3b2 + 2) * q^29 + (b19 + b18 + 2b13 + 2b11 + 2b10 + b9 + b6 + b5 + b4 - b2 - 1) q^30 + (2b18 - b17 + 2b16 - b13 - b11 + 2b7 - b6 + b5 - b4 - 2b3 - b2) * q^31 - q^32 + (2b19 - b17 - b16 + 2b13 + 2b11 + b10 + b9 - b7 + b5 + b3 - 1) q^33 + (-b19 + b15 + b14 - b12 - 2b11 + 2b7 - 2b2 + 1) q^34 + (2b19 + 2b18 - b15 - 2b13 - 2b12 + b11 + b9 + b8 - 2b7 - b6 + b2) q^35 + (-b16 + b15 + b14 + b13 + b12 - b11 - b8 + b7 - b5 - b4 - b2 - b1 + 1) * q^36 + (2b19 + b17 - 2b15 + b14 + b13 - 3b12 + b10 + 2b9 + 2b7 + b6 + 2b5 + b4 - 2b2 - b1 - 2) q^37 + (b18 + b14 - b12 + b8 - b6 - 1) * q^38 - b7 * q^39 + (-b19 + b18 + b16 + b11 + b10 + b8 - b6 - b3 + b1 - 2) * q^40 + (2b19 + 2b18 + 2b17 - 2b15 - b14 + 4b13 + 5b11 + 4b10 + b9 + 2b8 + b7 - 4b6 + b5 - 2b4 - 2b3 - b2 - 2b1 - 4) * q^41 + (b19 + b18 + b17 + 4b13 + 4b11 + 4b10 + b9 + 2b5 + b4 + b2 - 4) * q^42 + (3b17 + 3b15 + b14 - b12 - 3b11 + b9 + b6 - b5 - b2 - 2b1 + 2) * q^43 + (-b19 - b18 + b17 - b13 - b11 - b10 - b9 + 1) * q^44 + (-b16 + b15 - 2b14 - 2b13 - 3b11 - 2b7 + 2b6 + 2b4 + b2 + 4) * q^45 + (-2b16 + b15 + 2b14 + 3b13 - 2b12 + 2b9 - b8 + b6 + b3 - 2b2 + 1) * q^46 + (-b19 + 3b16 - b15 + b14 + b13 + 3b11 + b9 + b1 - 3) * q^47 + b7 * q^48 + (-b18 - b16 - b14 - b12 - b11 - b10 + b9 - b8 - b7 + b6 + b5 + b4 - b1 + 2) * q^49 + (b18 + 2b16 + b15 - b14 - b13 + b12 - b9 + b8 + b7 - b6 - b5 - 2b4 - b3 + b1) * q^50 + (-b17 - b15 + b14 - 3b11 - 4b10 - b8 + 2b7 - b5 - 2b4 - 2b2 + 3) q^51 + b10 * q^52 + (b19 - b16 - b14 - 7b13 - b12 - b11 - b9 - b8 - 2b7 - b6 + b3 + 3b2 - b1 + 2) q^53 + (-b19 - b15 + b14 + 2b13 + 2b11 + b9 + b7 + b6 - 2b2 + b1 - 2) q^54 + (b19 + b16 - b15 - 2b13 - b12 + 3b11 + b10 + b8 + 2b7 + 2b5 + b2) * q^55 + (b18 + b5) * q^56 + (b12 - b9 - b8 + b1 - 1) * q^57 + (b17 - b16 - b13 - b11 + b10 - 2b9 - b7 - 2b6 + b5 + b4 + b3 + b2 - 1) * q^58 + (-2b19 - 2b18 + 2b17 + 2b15 + 2b12 - 2b11 - 2b9 - 2b8 + 4b7 + b6 - 2b5 - b4 - 2b3 - 4b2 - 2b1 + 4) q^59 + (-b17 - b14 + 2b11 + b10 + b9 - 2b7 + b5 + 2b4 + b3 + 2b2 + b1) * q^60 + (b19 + 2b18 + b17 + 3b16 - 3b15 - 3b14 + 3b13 - b12 + 3b11 + 6b10 + 3b8 - b7 - 2b6 + b5 + 3b4 - 2b3 + b2 + 2b1 - 9) * q^61 + (b19 - 2b18 - b15 - b14 - b13 + b12 - b10 - 2b8 - b7 + 3b6 + b4 + 2b3 + b2 + 1) * q^62 + (-2b19 - b18 + 2b15 + 2b13 - 2b12 + 3b11 + 2b10 + 2b9 - b8 + 4b6 + 2b5 + 3b4 + 3) * q^63 - b10 * q^64 + (b19 - b16 + b13 + b10 - 1) * q^65 + (b19 + b18 - 2b17 - b15 - b14 + b13 + 3b11 + b10 + b9 + b8 - b6 + b5 + 2b1 - 2) q^66 + (-b18 - b17 + b16 - b13 - b11 + 2b9 + b7 - b6 - 2b4 - b3 - 2b2) q^67 + (b12 - b9 - 2b6 + 2b2 - b1 - 1) * q^68 + (-b19 + b17 + b16 - b13 - b11 + b7 + 3b6 + b5 + b4 - b3 + b2) q^69 + (b18 - b17 + b16 - 2b13 - b12 - 2b11 - b9 + b8 - b7 + b6 + b5 + b4 + b3 + 2b1 + 1) q^70 + (2b19 - 3b16 - b15 - b14 - b13 + b12 - b9 + 3b2 - 2b1) * q^71 + (b19 - b16 + b14 + b13 + b11 + b9 - b8 + b6 + b3 - b1) * q^72 + (-2b18 + b16 + b15 + b14 + 2b13 - b11 + b10 + b8 + 3b7 + 2b6 + b4 + 2b3 + 3b2 - b1 - 2) * q^73 + (b17 + b15 - 2b14 + b12 + b10 - b9 - b7 - 2b6 + 2b5 + b4 + 3b2 + b1 - 2) * q^74 + (-b19 - 3b18 - b17 - 3b16 + 2b15 + 2b14 - b13 + 2b12 - 3b11 - 5b10 - 2b9 - 2b8 - 2b7 - 4b5 + 2b3 + b2 - b1 + 2) * q^75 + (b18 + b16 + b12 - b9 + b8 - b6 + b2 - 1) * q^76 + (b19 + 2b18 + b17 + 2b16 - 2b15 - b14 + b13 + b12 + 2b11 + 3b10 + 2b8 + b7 - 2b6 - b5 - 2b4 - 2b3 + b1 - 5) q^77 - b2 * q^78 + (-3b19 + b16 + 2b15 - 2b14 + 4b13 + 4b12 + 2b11 - 2b9 + b8 - 2b7 - 3b6 - b3 + 3b2 + 3b1 - 3) q^79 + (-b19 + b16 - b13 - b10 + 1) * q^80 + (-3b19 + 2b18 - b17 + 3b16 - 2b10 + b9 + 3b7 - b6 - b5 - 2b4 - 3b3 - 2b2 + 2) * q^81 + (2b18 + 2b17 + 2b16 + 2b15 - b14 - b12 + 2b11 + 4b10 + b9 + 2b7 - b6 + b5 - 2b4 - b2 + 1) * q^82 + (3b19 - b18 + 2b17 - b16 + 3b13 + 3b11 + 4b10 - b9 - b7 - b6 + 2b5 + b4 + b3 + b2 - 4) * q^83 + (-b19 - b17 + b15 - 2b14 + b12 + 3b11 + b9 - 2b7 + b6 + b5 + b4 + b3 + 2b2 + b1 + 1) * q^84 + (-2b19 - 3b17 + b15 - b13 - b12 - b11 - b10 + b9 + 2b8 - 3b7 - b6 + 2b5 + 3b4 - b3 + 5b2 + 2b1 + 1) * q^85 + (-b19 - b17 + 3b15 + b14 - b12 - 3b11 - b10 + b5 + b4 - 2b1 + 3) q^86 + (-2b18 - b17 - 2b13 + 2b12 - 5b11 - 2b10 - 2b9 - 2b8 + 3b7 + b6 - 2b5 - b4 + b3 - 3b2 + b1 + 2) * q^87 + (-b19 + b17 + b15 + b12 - 2b11 - b9 + b7 - b6 - b5 - b4 - b3 - b2 - b1 + 1) q^88 + (2b19 + 2b17 + 2b16 + b15 - b14 - 2b13 - 2b12 - b11 + 3b10 + 2b8 + 2b5 - b4 + 2b2 - 3b1 + 1) * q^89 + (b18 - b17 - 2b13 - 2b12 - 2b11 + b10 - 2b7 + b6 + 2b4) q^90 + (b14 - b12 + b7 - b6 - b4 - b3 - b2) * q^91 + (b18 - b17 - b16 + 3b13 + 3b11 + 2b10 - b7 - b6 + 2b5 + b4 + b3 + b2 - 2) * q^92 + (-b18 - b16 + b14 - b12 - 2b11 - 2b10 + b9 + b8 + 2b7 + b6 - b5 - 2b4 - 3b2 - 2b1 + 3) * q^93 + (-b19 - 3b18 + b17 + b13 + b11 - b10 + b9 + b5 + 1) q^94 + (-2b17 + 2b16 - b14 - b11 - b10 + 2b8 - b7 + 2b6 + 2b5 + 3b4 + 3b2 + 2b1 - 2) * q^95 + b2 * q^96 + (-2b19 + 5b16 - 2b15 - 3b14 - 5b13 + b12 + 2b11 - 3b9 + b8 - b7 - 2b6 - b3 + 2b2 + 2b1 - 3) * q^97 + (b19 + b17 - b16 - b14 + b13 - b12 + b10 - b8 - b7 + b5 - b2 - b1) * q^98 + (-2b18 + b17 - 2b16 + b15 + b14 + 2b12 - 7b11 - 6b10 - 2b9 - b8 - 2b7 + b6 - b5 + 2b4 + b2 - b1 + 3) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$20 q + 5 q^{2} - 4 q^{3} - 5 q^{4} - 10 q^{5} + 4 q^{6} + 8 q^{7} + 5 q^{8} - 7 q^{9} + 10 q^{10} + q^{12} + 5 q^{13} + 2 q^{14} + 5 q^{15} - 5 q^{16} + 2 q^{18} - 4 q^{19} + 5 q^{20} + 3 q^{21} - 5 q^{22}+ \cdots - 24 q^{99}+O(q^{100})$ 20 * q + 5 * q^2 - 4 * q^3 - 5 * q^4 - 10 * q^5 + 4 * q^6 + 8 * q^7 + 5 * q^8 - 7 * q^9 + 10 * q^10 + q^12 + 5 * q^13 + 2 * q^14 + 5 * q^15 - 5 * q^16 + 2 * q^18 - 4 * q^19 + 5 * q^20 + 3 * q^21 - 5 * q^22 - 17 * q^23 - 6 * q^24 + 20 * q^26 - 19 * q^27 - 2 * q^28 + 9 * q^29 + 10 * q^30 - 2 * q^31 - 20 * q^32 - 3 * q^33 - 5 * q^34 + 30 * q^35 - 7 * q^36 - 19 * q^37 - q^38 - q^39 - 5 * q^40 + 33 * q^41 - 3 * q^42 + 2 * q^43 + 5 * q^44 + 35 * q^45 + 2 * q^46 - 13 * q^47 + q^48 - 4 * q^49 + 15 * q^50 + 12 * q^51 + 5 * q^52 - 7 * q^53 - 16 * q^54 + 35 * q^55 + 2 * q^56 - 18 * q^57 - 9 * q^58 + 11 * q^59 + 20 * q^60 - 18 * q^61 - 13 * q^62 + 44 * q^63 - 5 * q^64 - 10 * q^65 + 8 * q^66 - 27 * q^67 - 10 * q^68 - 8 * q^69 + 20 * q^70 - 7 * q^71 - 8 * q^72 - 30 * q^73 - 16 * q^74 - 60 * q^75 + 6 * q^76 + q^77 - 4 * q^78 - q^79 + 10 * q^80 + 28 * q^81 + 52 * q^82 - 6 * q^83 + 18 * q^84 + 25 * q^85 - 7 * q^86 - 37 * q^87 + 29 * q^89 - 20 * q^90 + 2 * q^91 - 2 * q^92 + 8 * q^93 + 13 * q^94 - 20 * q^95 + 4 * q^96 - 15 * q^97 - 11 * q^98 - 24 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{20} - 9 x^{19} + 43 x^{18} - 132 x^{17} + 291 x^{16} - 480 x^{15} + 624 x^{14} - 653 x^{13} + \cdots + 31$ x^20 - 9*x^19 + 43*x^18 - 132*x^17 + 291*x^16 - 480*x^15 + 624*x^14 - 653*x^13 + 552*x^12 - 246*x^11 - 32*x^10 + 229*x^9 - 103*x^8 + 657*x^7 - 46*x^6 - 270*x^5 + 1406*x^4 + 588*x^3 + 548*x^2 + 126*x + 31 :

$\beta_{1}$	$=$	$( - 41602842506613 \nu^{19} + 310044321766184 \nu^{18} + \cdots - 23\!\cdots\!85 ) / 12\!\cdots\!75$ (-41602842506613v^19 + 310044321766184v^18 - 1235156190048984v^17 + 2856768941816893v^16 - 3849284928136890v^15 + 506821330122457v^14 + 10104897397756629v^13 - 28506245331512719v^12 + 48232839836568728v^11 - 65167088856405200v^10 + 58180900544574833v^9 - 43529683881420744v^8 + 2082044331028989v^7 - 13904379836415348v^6 - 70805732523916000v^5 + 1927183082142398v^4 - 96341924874289869v^3 - 119364847290490686v^2 - 35611840253610823*v - 237827016998753985) / 120068399564401375
$\beta_{2}$	$=$	$( 19\!\cdots\!17 \nu^{19} + \cdots + 62\!\cdots\!62 ) / 36\!\cdots\!75$ (19369594621552617v^19 + 44452029990991953v^18 - 1095541953863065843v^17 + 6396849052529030898v^16 - 20693666048580473423v^15 + 45202709134806087598v^14 - 69538882866964789828v^13 + 79190750298361939788v^12 - 65518979311351672313v^11 + 35304911843029648328v^10 + 27574403080226784613v^9 - 68533990014035328318v^8 + 81424097631665958153v^7 - 13001514540292920133v^6 + 134528128526772698008v^5 + 20156337940221778272v^4 - 91705248881358297932v^3 + 327276970569641352052v^2 + 243024031693372580348*v + 62418463225191002662) / 36620861867142419375
$\beta_{3}$	$=$	$( - 11\!\cdots\!06 \nu^{19} + \cdots + 12\!\cdots\!78 ) / 36\!\cdots\!75$ (-110654844234684506v^19 + 1522640167977398800v^18 - 9639867200717775239v^17 + 38600836689567958643v^16 - 108400062350305149957v^15 + 227488117102831677176v^14 - 369140976666469182820v^13 + 479237093191604198159v^12 - 506035016966446756908v^11 + 422915181375022821447v^10 - 216175818155550067174v^9 + 2927774378256825595v^8 + 135230078785585071904v^7 - 156161474035785764413v^6 + 354920323477653421757v^5 - 92933476211852680021v^4 - 260010421592426725650v^3 + 751925275166353703426v^2 - 47773525599257511622*v + 121214551398202610778) / 36620861867142419375
$\beta_{4}$	$=$	$( - 17\!\cdots\!19 \nu^{19} + \cdots - 60\!\cdots\!03 ) / 36\!\cdots\!75$ (-178473969757869219v^19 + 1360896523732960120v^18 - 5389245104657902346v^17 + 12354482065930527747v^16 - 16637702735946612738v^15 + 6172436805927563859v^14 + 21885071232092456190v^13 - 56844730714543401279v^12 + 78984284856763994443v^11 - 96214895247432044472v^10 + 52298272817931021374v^9 + 5884588410380663285v^8 - 85182636414102309219v^7 - 49879354516949752877v^6 - 173824299125121883687v^5 + 118285452831027888136v^4 - 189670346703558520770v^3 - 460122808365780196531v^2 - 195238861356499143263*v - 60108438458050350003) / 36620861867142419375
$\beta_{5}$	$=$	$( 18\!\cdots\!64 \nu^{19} + \cdots - 23\!\cdots\!10 ) / 36\!\cdots\!75$ (182488095527517464v^19 - 1712065588817433488v^18 + 8531528948442122307v^17 - 27554502710458018481v^16 + 64406310526665148905v^15 - 113912294838184097294v^14 + 159796638190741048723v^13 - 181464953351619608622v^12 + 167694562398812795896v^11 - 101126727163020386355v^10 + 21131137623024303911v^9 + 46998706379631136458v^8 - 39546673821272040807v^7 + 135961926649884162686v^6 - 43334387303607168040v^5 - 41220023439728030156v^4 + 331156572801592583007v^3 - 40799788478623202903v^2 + 148835929608518351274*v - 236650300861027610) / 36620861867142419375
$\beta_{6}$	$=$	$( - 32\!\cdots\!76 \nu^{19} + \cdots + 20\!\cdots\!67 ) / 36\!\cdots\!75$ (-320089331962986276v^19 + 2905658299476807999v^18 - 13857214452964627632v^17 + 42090837881248787785v^16 - 90365222489298940443v^15 + 141714153586666368616v^14 - 168490954935020909644v^13 + 152137908670287426117v^12 - 98511497715680803985v^11 - 5428748541186617872v^10 + 84428416938231831906v^9 - 107089476636787202604v^8 + 22837055484317808317v^7 - 172338441135422736405v^6 + 13424213287704867023v^5 + 154653692775589334004v^4 - 464653854859317643776v^3 - 239819509052437266352v^2 + 72061396268581859655*v + 2065481488189801067) / 36620861867142419375
$\beta_{7}$	$=$	$( 32\!\cdots\!58 \nu^{19} + \cdots + 98\!\cdots\!48 ) / 36\!\cdots\!75$ (326610730058317758v^19 - 2582727669496209003v^18 + 10750186860443860988v^17 - 27102256622118424313v^16 + 45131705179072560313v^15 - 45658795015881280603v^14 + 19731629165733907888v^13 + 24659063536627944042v^12 - 65679034092147911857v^11 + 124687053698608121962v^10 - 102541068172170875563v^9 + 57784097660654234118v^8 + 47598600060077486177v^7 + 188463873729595427473v^6 + 193197975433532098402v^5 - 131169315635335730442v^4 + 307214034142919920497v^3 + 763319383807481242893v^2 + 319567832429179703147*v + 98608893891905036748) / 36620861867142419375
$\beta_{8}$	$=$	$( 56\!\cdots\!52 \nu^{19} + \cdots + 28\!\cdots\!12 ) / 60\!\cdots\!75$ (5657456831187352v^19 - 45497109524755852v^18 + 193149594822040222v^17 - 501195778034566362v^16 + 867789383859676797v^15 - 932928502245280792v^14 + 445607878321145722v^13 + 538431821112002818v^12 - 1596846000464450408v^11 + 2920747629900746653v^10 - 2745429921999437447v^9 + 1899853122026275712v^8 + 440289573405687963v^7 + 2793590198906525827v^6 + 3408617607063095188v^5 - 2887204639823450888v^4 + 7072305065342703543v^3 + 10425500486157247897v^2 + 3159861441239764993*v + 2820229327264739812) / 600341997822006875
$\beta_{9}$	$=$	$( 35\!\cdots\!83 \nu^{19} + \cdots + 13\!\cdots\!94 ) / 36\!\cdots\!75$ (359308380695361283v^19 - 3413444929657614187v^18 + 17206253355961692631v^17 - 56337396619110067290v^16 + 133730413844158696619v^15 - 240951918406980521628v^14 + 345249239772564113927v^13 - 402644018751612850176v^12 + 386438775944444519130v^11 - 258695496029820845029v^10 + 88367421392291841077v^9 + 72864600718080517302v^8 - 92397973967518371986v^7 + 297144040173937318725v^6 - 153003869714777947684v^5 + 5279185727540863643v^4 + 544125570023460976533v^3 - 40403733611886003694v^2 + 246202335451609295535*v + 1363158179401108394) / 36620861867142419375
$\beta_{10}$	$=$	$( 39\!\cdots\!01 \nu^{19} + \cdots + 35\!\cdots\!68 ) / 36\!\cdots\!75$ (394265844560795601v^19 - 3651005445323914954v^18 + 17859781295186238767v^17 - 56237920356507603160v^16 + 127002936243802491713v^15 - 214424145374710285661v^14 + 283266127789396963274v^13 - 298585896013450169507v^12 + 250917631086340745950v^11 - 113931734088830955888v^10 - 24820416182935705456v^9 + 117942127520351184609v^8 - 64112292181504836127v^7 + 250709538966083699430v^6 - 57546777541229114868v^5 - 135760104262150879859v^4 + 620601677250560384846v^3 + 96569443243411027042v^2 + 114853166793971460230*v + 35121163634527260268) / 36620861867142419375
$\beta_{11}$	$=$	$( - 40\!\cdots\!11 \nu^{19} + \cdots + 23\!\cdots\!92 ) / 36\!\cdots\!75$ (-404302253391919711v^19 + 3748280542381481544v^18 - 18365681998295178752v^17 + 57989619596313132575v^16 - 131570645069998140723v^15 + 224003665347848900826v^14 - 300374113086505416534v^13 + 325201899640430942622v^12 - 286867404301799460685v^11 + 154109130185915708378v^10 - 12073511758902690209v^9 - 92124487427624339799v^8 + 52541418483658698287v^7 - 260764185792749362920v^6 + 75947052797516303178v^5 + 119712389467126158719v^4 - 622581337092519894861v^3 - 102909914074306729957v^2 - 116230423209088708170*v + 23691080522490628692) / 36620861867142419375
$\beta_{12}$	$=$	$( 62\!\cdots\!48 \nu^{19} + \cdots + 75\!\cdots\!37 ) / 36\!\cdots\!75$ (624084643935581048v^19 - 5533024374646203184v^18 + 26119526492610363030v^17 - 79061223974335429906v^16 + 171702989622039324562v^15 - 278137448124714279578v^14 + 354582184716582712899v^13 - 362065807134245872995v^12 + 294994047614019815101v^11 - 105854369896517238157v^10 - 53967114029257174468v^9 + 168765809448251044699v^8 - 76269505276659509645v^7 + 432408294224288960741v^6 + 2871393716713729033v^5 - 156797451434590313577v^4 + 870625572797648926011v^3 + 450362408030109622410v^2 + 395551285991001912614*v + 75708043905675008737) / 36620861867142419375
$\beta_{13}$	$=$	$( - 65\!\cdots\!33 \nu^{19} + \cdots - 20\!\cdots\!74 ) / 36\!\cdots\!75$ (-651279716856324633v^19 + 6030279581784843782v^18 - 29495507632775769681v^17 + 92869269998738798475v^16 - 209717230812558034219v^15 + 354070442426960293003v^14 - 467764311000304180977v^13 + 493098767249630024591v^12 - 414390750713984603080v^11 + 187984575515145024109v^10 + 42022842172041329998v^9 - 198947119530163417172v^8 + 117929334712009803661v^7 - 438425092515559633760v^6 + 128914605131516695209v^5 + 192749002346174919507v^4 - 1006108302161957090583v^3 - 166011503966304906721v^2 - 187878066322965725235*v - 20891993958289238774) / 36620861867142419375
$\beta_{14}$	$=$	$( 66\!\cdots\!95 \nu^{19} + \cdots + 12\!\cdots\!63 ) / 36\!\cdots\!75$ (666335683188011195v^19 - 5709059415228998162v^18 + 26012554087327864424v^17 - 75135752473999851476v^16 + 153743787922300425313v^15 - 229373946099199287530v^14 + 262904740362021362312v^13 - 232159493058567962644v^12 + 151832724835180897956v^11 + 19526585310190430852v^10 - 106134251972608694425v^9 + 135812893841112595712v^8 + 22918761843362873541v^7 + 371743731608642227476v^6 + 168506359897610692922v^5 - 256533957328826185440v^4 + 869608429569850574113v^3 + 771967701100381494579v^2 + 394630492575112495994*v + 121284434633157510163) / 36620861867142419375
$\beta_{15}$	$=$	$( 84\!\cdots\!50 \nu^{19} + \cdots + 28\!\cdots\!73 ) / 36\!\cdots\!75$ (842452289132895550v^19 - 7778258260714327737v^18 + 37893524381495494659v^17 - 118754090462088876266v^16 + 266689677767364901603v^15 - 447367361152022368460v^14 + 586331397428760408107v^13 - 611440244499983654864v^12 + 504086470334973374431v^11 - 210143279854570066083v^10 - 80827299702658895080v^9 + 270218531069788821287v^8 - 166323149362338656494v^7 + 569098592774219931016v^6 - 160029413141837246468v^5 - 267356097678094919960v^4 + 1252987870985600086193v^3 + 211564666247616393474v^2 + 243779811402746260144*v + 28159482863768573073) / 36620861867142419375
$\beta_{16}$	$=$	$( 12\!\cdots\!88 \nu^{19} + \cdots + 19\!\cdots\!80 ) / 36\!\cdots\!75$ (1238035798888681588v^19 - 11494516973248926431v^18 + 56304643536761729859v^17 - 177655632253587224672v^16 + 402250826865195360090v^15 - 682113292308209947153v^14 + 906790610003631111436v^13 - 965547106757744849714v^12 + 824232633507088102717v^11 - 397572349058333767385v^10 - 46749622273090401063v^9 + 353535382726521313046v^8 - 226995382055888573684v^7 + 819590520427033869832v^6 - 261547144973812256910v^5 - 395155099139387134647v^4 + 1863815165091171318724v^3 + 195339014815946732364v^2 + 271335294024385348248*v + 19364264559738389380) / 36620861867142419375
$\beta_{17}$	$=$	$( - 12\!\cdots\!05 \nu^{19} + \cdots + 32\!\cdots\!78 ) / 36\!\cdots\!75$ (-1253247999670636105v^19 + 11610367643098181108v^18 - 56777300709440545151v^17 + 178824608214333554279v^16 - 404312764409637630087v^15 + 685081367759293185285v^14 - 911933166553041524513v^13 + 975743698075680432481v^12 - 842542181321837694469v^11 + 421667068544592037787v^10 + 17683149804826577025v^9 - 326917220006371091308v^8 + 206069054451745096291v^7 - 837156893012004663304v^6 + 267528308147222815322v^5 + 369565678549375214970v^4 - 1859899256278851645312v^3 - 298701075729276823471v^2 - 354762987843461811681*v + 3279427597950083978) / 36620861867142419375
$\beta_{18}$	$=$	$( - 13\!\cdots\!75 \nu^{19} + \cdots + 47\!\cdots\!12 ) / 36\!\cdots\!75$ (-1333135568151749575v^19 + 12324879006973956322v^18 - 60089506242849248874v^17 + 188473169923884844321v^16 - 423747563665888626808v^15 + 712591508063927728985v^14 - 939037474477499485152v^13 + 991104984163592612859v^12 - 838245760583407679816v^11 + 389161725391720625073v^10 + 64487152196498640605v^9 - 362961216707007498422v^8 + 208681783718268382059v^7 - 862271959026664835896v^6 + 241415455079993939423v^5 + 451289502302428274560v^4 - 1963794182708570511598v^3 - 357158842500125990444v^2 - 320459591063983828809*v + 4782182444167496812) / 36620861867142419375
$\beta_{19}$	$=$	$( 20\!\cdots\!76 \nu^{19} + \cdots - 60\!\cdots\!65 ) / 36\!\cdots\!75$ (2008912186430334476v^19 - 18602756754533967982v^18 + 90927774021226170248v^17 - 286195290825081057384v^16 + 646474350223676482865v^15 - 1093852761678256372191v^14 + 1452855434476551117012v^13 - 1548954438063815427633v^12 + 1329223933288034880004v^11 - 650745504538007664945v^10 - 51417807904168061376v^9 + 537563148548248853462v^8 - 328491448384310352298v^7 + 1325419408910747163554v^6 - 398311244310308303680v^5 - 625988755514018114209v^4 + 3006722938015229979433v^3 + 473018201262830635458v^2 + 571844643079483253076*v - 6022615783840749765) / 36620861867142419375

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

$\nu$	$=$	$( \beta_{19} + 3 \beta_{18} - \beta_{17} - \beta_{16} + \beta_{15} + 2 \beta_{13} + 2 \beta_{11} + \cdots - 1 ) / 5$ (b19 + 3b18 - b17 - b16 + b15 + 2b13 + 2b11 + 4b10 + b8 + b7 - 2b6 - b4 - 2b3 - b2 + 2*b1 - 1) / 5
$\nu^{2}$	$=$	$( \beta_{19} + 4 \beta_{18} - 6 \beta_{17} - 3 \beta_{16} + \beta_{15} + 2 \beta_{14} + 6 \beta_{13} + \cdots - 5 ) / 5$ (b19 + 4b18 - 6b17 - 3b16 + b15 + 2b14 + 6b13 - 6b12 + 3b11 + 7b10 + 3b9 + 3b8 + 3b7 - b6 + 2b5 + 2b4 - b3 - 3b2 + 2*b1 - 5) / 5
$\nu^{3}$	$=$	$( 4 \beta_{18} - 15 \beta_{17} - 3 \beta_{16} - 2 \beta_{14} + 11 \beta_{13} - 4 \beta_{12} + 9 \beta_{11} + \cdots - 11 ) / 5$ (4b18 - 15b17 - 3b16 - 2b14 + 11b13 - 4b12 + 9b11 + 7b10 + 2b9 + 3b8 - 2b7 - b6 + 8b5 + 2b4 + 4b3 + 2*b2 - 11) / 5
$\nu^{4}$	$=$	$( 7 \beta_{19} + 3 \beta_{18} - 17 \beta_{17} + 4 \beta_{16} - 8 \beta_{15} - 16 \beta_{14} + 17 \beta_{13} + \cdots - 30 ) / 5$ (7b19 + 3b18 - 17b17 + 4b16 - 8b15 - 16b14 + 17b13 + 18b12 + 36b11 + 9b10 - 4b9 - 9b8 - 19b7 - 7b6 + 9b5 - b4 + 18b3 + 24b2 - 11b1 - 30) / 5
$\nu^{5}$	$=$	$( 54 \beta_{19} + 26 \beta_{17} + 15 \beta_{16} - 36 \beta_{15} - 24 \beta_{14} + 15 \beta_{13} + \cdots - 51 ) / 5$ (54b19 + 26b17 + 15b16 - 36b15 - 24b14 + 15b13 + 47b12 + 91b11 - 10b10 - 6b9 - 40b8 - 35b7 - 5b6 + b5 - 10b4 + 40b3 + 35b2 - 47*b1 - 51) / 5
$\nu^{6}$	$=$	$( 158 \beta_{19} - 48 \beta_{18} + 157 \beta_{17} - 29 \beta_{16} - 62 \beta_{15} + \beta_{14} - 17 \beta_{13} + \cdots + 10 ) / 5$ (158b19 - 48b18 + 157b17 - 29b16 - 62b15 + b14 - 17b13 + 62b12 + 99b11 - 134b10 + 14b9 - 61b8 - 56b7 + 32b6 - 39b5 - 14b4 + 57b3 + b2 - 109*b1 + 10) / 5
$\nu^{7}$	$=$	$( 223 \beta_{19} - 224 \beta_{18} + 362 \beta_{17} - 232 \beta_{16} + 63 \beta_{15} + 119 \beta_{14} + \cdots + 329 ) / 5$ (223b19 - 224b18 + 362b17 - 232b16 + 63b15 + 119b14 - 16b13 - 7b12 - 217b11 - 362b10 + 106b9 - 13b8 - 88b7 + 176b6 - 161b5 + 8b4 + 51b3 - 147b2 - 129*b1 + 329) / 5
$\nu^{8}$	$=$	$( - 186 \beta_{19} - 449 \beta_{18} + 351 \beta_{17} - 557 \beta_{16} + 734 \beta_{15} + 443 \beta_{14} + \cdots + 1190 ) / 5$ (-186b19 - 449b18 + 351b17 - 557b16 + 734b15 + 443b14 + 189b13 - 329b12 - 1393b11 - 472b10 + 302b9 + 127b8 - 8b7 + 416b6 - 322b5 + 63b4 - 64b3 - 517b2 + 103*b1 + 1190) / 5
$\nu^{9}$	$=$	$( - 1828 \beta_{19} + 52 \beta_{18} - 667 \beta_{17} - 434 \beta_{16} + 2272 \beta_{15} + 1007 \beta_{14} + \cdots + 2144 ) / 5$ (-1828b19 + 52b18 - 667b17 - 434b16 + 2272b15 + 1007b14 + 888b13 - 991b12 - 3610b11 + 381b10 + 313b9 + 309b8 + 689b7 + 47b6 - 83b5 + 121b4 - 753b3 - 1089b2 + 754*b1 + 2144) / 5
$\nu^{10}$	$=$	$- 948 \beta_{19} + 583 \beta_{18} - 747 \beta_{17} + 297 \beta_{16} + 671 \beta_{15} + 258 \beta_{14} + \cdots - 12$ -948b19 + 583b18 - 747b17 + 297b16 + 671b15 + 258b14 + 404b13 - 256b12 - 899b11 + 745b10 - 177b9 + 55b8 + 563b7 - 512b6 + 269b5 + 5b4 - 570b3 - 284b2 + 208*b1 - 12
$\nu^{11}$	$=$	$( - 5169 \beta_{19} + 7923 \beta_{18} - 7946 \beta_{17} + 6234 \beta_{16} - 1904 \beta_{15} + \cdots - 12246 ) / 5$ (-5169b19 + 7923b18 - 7946b17 + 6234b16 - 1904b15 - 715b14 + 862b13 + 1730b12 + 4382b11 + 8929b10 - 5045b9 - 1249b8 + 5071b7 - 7252b6 + 3695b5 - 406b4 - 5047b3 + 744b2 - 2388*b1 - 12246) / 5
$\nu^{12}$	$=$	$( 8941 \beta_{19} + 5134 \beta_{18} - 3671 \beta_{17} + 10732 \beta_{16} - 24589 \beta_{15} - 9478 \beta_{14} + \cdots - 35965 ) / 5$ (8941b19 + 5134b18 - 3671b17 + 10732b16 - 24589b15 - 9478b14 - 12209b13 + 14549b12 + 35993b11 + 4542b10 - 13167b9 - 8057b8 - 2162b7 - 4321b6 + 1942b5 - 538b4 + 2679b3 + 12467b2 - 15373*b1 - 35965) / 5
$\nu^{13}$	$=$	$( 56200 \beta_{19} - 31776 \beta_{18} + 37730 \beta_{17} + 2157 \beta_{16} - 67675 \beta_{15} + \cdots - 38451 ) / 5$ (56200b19 - 31776b18 + 37730b17 + 2157b16 - 67675b15 - 28532b14 - 53254b13 + 41271b12 + 83509b11 - 42068b10 - 19093b9 - 23672b8 - 38662b7 + 32154b6 - 17792b5 + 1517b4 + 40984b3 + 42157b2 - 37130*b1 - 38451) / 5
$\nu^{14}$	$=$	$( 128682 \beta_{19} - 126572 \beta_{18} + 155413 \beta_{17} - 43481 \beta_{16} - 80008 \beta_{15} + \cdots + 87835 ) / 5$ (128682b19 - 126572b18 + 155413b17 - 43481b16 - 80008b15 - 40831b14 - 119723b13 + 50753b12 + 50856b11 - 169261b10 + 6571b9 - 29714b8 - 104124b7 + 125108b6 - 70516b5 + 6639b4 + 109053b3 + 66764b2 - 30406*b1 + 87835) / 5
$\nu^{15}$	$=$	$( 86219 \beta_{19} - 216840 \beta_{18} + 301811 \beta_{17} - 140695 \beta_{16} + 120284 \beta_{15} + \cdots + 501014 ) / 5$ (86219b19 - 216840b18 + 301811b17 - 140695b16 + 120284b15 + 32561b14 - 111930b13 - 91898b12 - 336389b11 - 312310b10 + 131129b9 + 64060b8 - 83900b7 + 208505b6 - 125044b5 + 9515b4 + 84990b3 - 43445b2 + 127388*b1 + 501014) / 5
$\nu^{16}$	$=$	$( - 506172 \beta_{19} + 11457 \beta_{18} + 65602 \beta_{17} - 204689 \beta_{16} + 838468 \beta_{15} + \cdots + 1116355 ) / 5$ (-506172b19 + 11457b18 + 65602b17 - 204689b16 + 838468b15 + 339541b14 + 280398b13 - 638243b12 - 1418306b11 - 28934b10 + 403459b9 + 436789b8 + 364124b7 - 39443b6 + 5966b5 - 7034b4 - 390823b3 - 550869b2 + 639856*b1 + 1116355) / 5
$\nu^{17}$	$=$	$( - 2274617 \beta_{19} + 1250846 \beta_{18} - 1652148 \beta_{17} + 172408 \beta_{16} + 2025878 \beta_{15} + \cdots + 755289 ) / 5$ (-2274617b19 + 1250846b18 - 1652148b17 + 172408b16 + 2025878b15 + 873024b14 + 1500949b13 - 1592062b12 - 2676782b11 + 1791328b10 + 571176b9 + 1110372b8 + 1663027b7 - 1302379b6 + 770374b5 - 72202b4 - 1745804b3 - 1493892b2 + 1491571*b1 + 755289) / 5
$\nu^{18}$	$=$	$( - 4661261 \beta_{19} + 4121881 \beta_{18} - 6034594 \beta_{17} + 1944033 \beta_{16} + 1685959 \beta_{15} + \cdots - 4084120 ) / 5$ (-4661261b19 + 4121881b18 - 6034594b17 + 1944033b16 + 1685959b15 + 723223b14 + 3231329b13 - 1485004b12 - 272793b11 + 6288253b10 - 501578b9 + 1110502b8 + 3284527b7 - 4108289b6 + 2610828b5 - 205152b4 - 3359369b3 - 1462247b2 + 1443608*b1 - 4084120) / 5
$\nu^{19}$	$=$	$( - 1916403 \beta_{19} + 6797252 \beta_{18} - 10457397 \beta_{17} + 5838451 \beta_{16} - 6671978 \beta_{15} + \cdots - 18425246 ) / 5$ (-1916403b19 + 6797252b18 - 10457397b17 + 5838451b16 - 6671978b15 - 3020278b14 + 1917253b13 + 4445224b12 + 15713770b11 + 10745551b10 - 5002837b9 - 2723536b8 + 1202344b7 - 6409918b6 + 4430742b5 - 324694b4 - 918713b3 + 4201996b2 - 3592951*b1 - 18425246) / 5

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/650\mathbb{Z}\right)^\times$ .

$n$	$27$	$301$
$\chi(n)$	$-\beta_{13}$	$1$

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}

131.1

−0.885825	−	0.248389i
0.704508	+	1.87902i
−0.277264	−	1.40948i
1.66610	−	1.78167i
−0.634573	+	0.972731i
1.91485	+	0.738251i
−0.149850	−	0.244564i
1.02648	−	1.17527i
1.21043	+	1.15658i
−0.0748646	+	0.476059i
1.91485	−	0.738251i
−0.149850	+	0.244564i
1.02648	+	1.17527i
1.21043	−	1.15658i
−0.0748646	−	0.476059i
−0.885825	+	0.248389i
0.704508	−	1.87902i
−0.277264	+	1.40948i
1.66610	+	1.78167i
−0.634573	−	0.972731i

−0.309017

−

0.951057i

−1.69698

1.23293i

−0.809017

0.587785i

−1.53931

1.62189i

1.69698

1.23293i

−3.44863

0.809017

0.587785i

0.432586

−

1.33136i

2.01818

0.962774i

131.2

−0.309017

−

0.951057i

−1.47354

1.07059i

−0.809017

0.587785i

1.58287

1.57940i

1.47354

1.07059i

2.37656

0.809017

0.587785i

0.0981028

−

0.301929i

1.01296

−

1.99347i

131.3

−0.309017

−

0.951057i

−0.286240

0.207966i

−0.809017

0.587785i

−2.22306

0.240824i

0.286240

0.207966i

2.80842

0.809017

0.587785i

−0.888367

2.73411i

0.916001

2.03984i

131.4

−0.309017

−

0.951057i

−0.126245

0.0917222i

−0.809017

0.587785i

−1.23420

−

1.86460i

0.126245

0.0917222i

−2.41858

0.809017

0.587785i

−0.919526

2.83001i

−1.39195

1.74999i

131.5

−0.309017

−

0.951057i

1.46497

−

1.06437i

−0.809017

0.587785i

−0.204340

2.22671i

−1.46497

−

1.06437i

2.68223

0.809017

0.587785i

0.0862215

−

0.265362i

2.18087

−

0.493753i

261.1

0.809017

−

0.587785i

−0.777911

2.39416i

0.309017

−

0.951057i

2.08891

0.797773i

0.777911

2.39416i

1.47068

−0.309017

−

0.951057i

−2.69982

−

1.96153i

2.15889

−

0.582420i

261.2

0.809017

−

0.587785i

−0.311946

0.960072i

0.309017

−

0.951057i

−2.22385

0.233444i

0.311946

0.960072i

−0.400732

−0.309017

−

0.951057i

1.60262

1.16437i

−1.66192

1.49601i

261.3

0.809017

−

0.587785i

−0.0885960

0.272670i

0.309017

−

0.951057i

−0.642903

−

2.14165i

0.0885960

0.272670i

3.26367

−0.309017

−

0.951057i

2.36055

1.71504i

−1.77895

−

1.35474i

261.4

0.809017

−

0.587785i

0.370600

−

1.14059i

0.309017

−

0.951057i

1.06050

1.96859i

−0.370600

−

1.14059i

1.46282

−0.309017

−

0.951057i

1.26345

0.917949i

2.01507

0.969274i

261.5

0.809017

−

0.587785i

0.925886

−

2.84959i

0.309017

−

0.951057i

−1.66463

1.49299i

−0.925886

−

2.84959i

−3.79644

−0.309017

−

0.951057i

−4.83582

−

3.51343i

−0.469156

2.18630i

391.1

0.809017

0.587785i

−0.777911

−

2.39416i

0.309017

0.951057i

2.08891

−

0.797773i

0.777911

−

2.39416i

1.47068

−0.309017

0.951057i

−2.69982

1.96153i

2.15889

0.582420i

391.2

0.809017

0.587785i

−0.311946

−

0.960072i

0.309017

0.951057i

−2.22385

−

0.233444i

0.311946

−

0.960072i

−0.400732

−0.309017

0.951057i

1.60262

−

1.16437i

−1.66192

−

1.49601i

391.3

0.809017

0.587785i

−0.0885960

−

0.272670i

0.309017

0.951057i

−0.642903

2.14165i

0.0885960

−

0.272670i

3.26367

−0.309017

0.951057i

2.36055

−

1.71504i

−1.77895

1.35474i

391.4

0.809017

0.587785i

0.370600

1.14059i

0.309017

0.951057i

1.06050

−

1.96859i

−0.370600

1.14059i

1.46282

−0.309017

0.951057i

1.26345

−

0.917949i

2.01507

−

0.969274i

391.5

0.809017

0.587785i

0.925886

2.84959i

0.309017

0.951057i

−1.66463

−

1.49299i

−0.925886

2.84959i

−3.79644

−0.309017

0.951057i

−4.83582

3.51343i

−0.469156

−

2.18630i

521.1

−0.309017

0.951057i

−1.69698

−

1.23293i

−0.809017

−

0.587785i

−1.53931

−

1.62189i

1.69698

−

1.23293i

−3.44863

0.809017

−

0.587785i

0.432586

1.33136i

2.01818

−

0.962774i

521.2

−0.309017

0.951057i

−1.47354

−

1.07059i

−0.809017

−

0.587785i

1.58287

−

1.57940i

1.47354

−

1.07059i

2.37656

0.809017

−

0.587785i

0.0981028

0.301929i

1.01296

1.99347i

521.3

−0.309017

0.951057i

−0.286240

−

0.207966i

−0.809017

−

0.587785i

−2.22306

−

0.240824i

0.286240

−

0.207966i

2.80842

0.809017

−

0.587785i

−0.888367

−

2.73411i

0.916001

−

2.03984i

521.4

−0.309017

0.951057i

−0.126245

−

0.0917222i

−0.809017

−

0.587785i

−1.23420

1.86460i

0.126245

−

0.0917222i

−2.41858

0.809017

−

0.587785i

−0.919526

−

2.83001i

−1.39195

−

1.74999i

521.5

−0.309017

0.951057i

1.46497

1.06437i

−0.809017

−

0.587785i

−0.204340

−

2.22671i

−1.46497

1.06437i

2.68223

0.809017

−

0.587785i

0.0862215

0.265362i

2.18087

0.493753i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
25.d	even	5	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	650.2.l.b	✓	20
25.d	even	5	1	inner	650.2.l.b	✓	20

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
650.2.l.b	✓	20	1.a	even	1	1	trivial
650.2.l.b	✓	20	25.d	even	5	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{20} + 4 T_{3}^{19} + 19 T_{3}^{18} + 59 T_{3}^{17} + 174 T_{3}^{16} + 367 T_{3}^{15} + 676 T_{3}^{14} + \cdots + 1$ T3^20 + 4*T3^19 + 19*T3^18 + 59*T3^17 + 174*T3^16 + 367*T3^15 + 676*T3^14 + 921*T3^13 + 1936*T3^12 + 4031*T3^11 + 7499*T3^10 + 10048*T3^9 + 13823*T3^8 + 13513*T3^7 + 12058*T3^6 + 7188*T3^5 + 2943*T3^4 + 818*T3^3 + 158*T3^2 + 18*T3 + 1 acting on $S_{2}^{\mathrm{new}}(650, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$(T^{4} - T^{3} + T^{2} + \cdots + 1)^{5}$ (T^4 - T^3 + T^2 - T + 1)^5
$3$	$T^{20} + 4 T^{19} + \cdots + 1$ T^20 + 4T^19 + 19T^18 + 59T^17 + 174T^16 + 367T^15 + 676T^14 + 921T^13 + 1936T^12 + 4031T^11 + 7499T^10 + 10048T^9 + 13823T^8 + 13513T^7 + 12058T^6 + 7188T^5 + 2943T^4 + 818T^3 + 158T^2 + 18*T + 1
$5$	$T^{20} + 10 T^{19} + \cdots + 9765625$ T^20 + 10T^19 + 50T^18 + 165T^17 + 410T^16 + 900T^15 + 2025T^14 + 4725T^13 + 10150T^12 + 19375T^11 + 39125T^10 + 96875T^9 + 253750T^8 + 590625T^7 + 1265625T^6 + 2812500T^5 + 6406250T^4 + 12890625T^3 + 19531250T^2 + 19531250*T + 9765625
$7$	$(T^{10} - 4 T^{9} + \cdots + 1595)^{2}$ (T^10 - 4T^9 - 26T^8 + 130T^7 + 135T^6 - 1310T^5 + 890T^4 + 3840T^3 - 6065T^2 + 1025*T + 1595)^2
$11$	$T^{20} + 10 T^{18} + \cdots + 121$ T^20 + 10T^18 - 35T^17 + 525T^16 + 2182T^15 + 14975T^14 + 10140T^13 + 101885T^12 - 27775T^11 + 367134T^10 - 200650T^9 + 406570T^8 - 75995T^7 + 117050T^6 - 399002T^5 + 658900T^4 - 301280T^3 + 60355T^2 - 2200T + 121
$13$	$(T^{4} - T^{3} + T^{2} + \cdots + 1)^{5}$ (T^4 - T^3 + T^2 - T + 1)^5
$17$	$T^{20} + \cdots + 100180081$ T^20 - 5T^18 + 30T^17 + 925T^16 + 3932T^15 + 14325T^14 + 35105T^13 + 298580T^12 + 1879910T^11 + 9473324T^10 + 37735050T^9 + 140158665T^8 + 391473720T^7 + 814901385T^6 + 1226992713T^5 + 1451147250T^4 + 1204454965T^3 + 804756225T^2 + 307376390T + 100180081
$19$	$T^{20} + 4 T^{19} + \cdots + 297025$ T^20 + 4T^19 + 45T^18 + 165T^17 + 890T^16 + 2247T^15 + 3813T^14 - 6400T^13 + 29175T^12 + 86670T^11 + 158088T^10 - 56393T^9 + 451570T^8 + 184995T^7 + 2061865T^6 + 1435639T^5 + 2753426T^4 + 2729510T^3 + 2378515T^2 - 209825*T + 297025
$23$	$T^{20} + 17 T^{19} + \cdots + 4116841$ T^20 + 17T^19 + 212T^18 + 2037T^17 + 17132T^16 + 107264T^15 + 516751T^14 + 1562346T^13 + 4118346T^12 + 19707841T^11 + 144196056T^10 + 415375576T^9 + 1002306141T^8 + 2668670361T^7 + 12573534691T^6 - 17326251216T^5 + 40590785517T^4 - 27910902543T^3 + 8443453327T^2 - 126441193*T + 4116841
$29$	$T^{20} + \cdots + 32042790025$ T^20 - 9T^19 + 135T^18 - 835T^17 + 8920T^16 - 64787T^15 + 658543T^14 - 5046470T^13 + 45095850T^12 - 293730455T^11 + 1685195303T^10 - 6080998627T^9 + 13238243505T^8 - 3491628530T^7 + 15028413075T^6 - 28872511384T^5 + 104975650171T^4 - 149219044190T^3 + 149861892085T^2 - 86401238375*T + 32042790025
$31$	$T^{20} + 2 T^{19} + \cdots + 3222025$ T^20 + 2T^19 + 14T^18 - 287T^17 + 2886T^16 + 9530T^15 + 257635T^14 - 1286110T^13 + 3742640T^12 + 6642785T^11 + 46612790T^10 + 131441915T^9 + 395291160T^8 + 765301425T^7 + 1227978050T^6 + 1173629825T^5 + 928463150T^4 + 443006175T^3 + 166024375T^2 + 33342125*T + 3222025
$37$	$T^{20} + \cdots + 20992822321$ T^20 + 19T^19 + 373T^18 + 4966T^17 + 59667T^16 + 576648T^15 + 4796629T^14 + 33084203T^13 + 197113061T^12 + 986393737T^11 + 4261607994T^10 + 14966356328T^9 + 41767438841T^8 + 80680798147T^7 + 98024086344T^6 - 7102330623T^5 - 3453751333T^4 + 167768965929T^3 + 261999272678T^2 + 116129837501*T + 20992822321
$41$	$T^{20} + \cdots + 34279795461025$ T^20 - 33T^19 + 749T^18 - 13202T^17 + 197006T^16 - 2427575T^15 + 25196675T^14 - 218132990T^13 + 1604757480T^12 - 9969223425T^11 + 52394982985T^10 - 220290578585T^9 + 718872624860T^8 - 1631031197575T^7 + 3883826942375T^6 - 19502756994500T^5 + 113629774710550T^4 - 256502021041775T^3 + 275059780175100T^2 - 89929577103875*T + 34279795461025
$43$	$(T^{10} - T^{9} + \cdots + 3448001)^{2}$ (T^10 - T^9 - 234T^8 + 25T^7 + 15101T^6 + 9735T^5 - 311267T^4 - 611788T^3 + 1363605T^2 + 4705067T + 3448001)^2
$47$	$T^{20} + \cdots + 2715556321$ T^20 + 13T^19 + 201T^18 + 1267T^17 + 9259T^16 + 57574T^15 + 464969T^14 + 1501568T^13 + 16765821T^12 + 121238142T^11 + 672867616T^10 + 1318212939T^9 + 3675512203T^8 - 6989321689T^7 + 19225442172T^6 + 15925728414T^5 + 78162133983T^4 + 45231292486T^3 + 16519663582T^2 + 4090661389*T + 2715556321
$53$	$T^{20} + \cdots + 10\!\cdots\!41$ T^20 + 7T^19 + 252T^18 + 3392T^17 + 46767T^16 + 333544T^15 + 5739821T^14 + 25295726T^13 + 220876011T^12 + 1141146256T^11 + 8726786111T^10 - 23186167349T^9 + 234415526451T^8 + 721024402756T^7 + 5516803763736T^6 - 15238218920291T^5 + 140985144763087T^4 - 420374128537823T^3 + 1933341750924637T^2 - 3948590864281778*T + 10511791675566841
$59$	$T^{20} + \cdots + 2003937516025$ T^20 - 11T^19 + 150T^18 - 1725T^17 + 32965T^16 - 80693T^15 + 3147763T^14 - 22357795T^13 + 282683465T^12 - 2208161540T^11 + 17351689533T^10 - 109638176433T^9 + 571220124535T^8 - 2326851193150T^7 + 7254770475850T^6 - 16365835218866T^5 + 26212638861356T^4 - 28892031293715T^3 + 22478617436460T^2 - 9748889421650*T + 2003937516025
$61$	$T^{20} + \cdots + 99\!\cdots\!61$ T^20 + 18T^19 + 396T^18 + 3407T^17 + 48959T^16 + 731084T^15 + 14644484T^14 + 204136538T^13 + 2436342621T^12 + 21893902322T^11 + 180917308996T^10 + 1161596422284T^9 + 5710638591853T^8 + 16697136048696T^7 + 64133124554752T^6 + 162646028058899T^5 + 200870035967968T^4 - 604933026181014T^3 + 4094456846957207T^2 + 2550234412258539*T + 9955116676159561
$67$	$T^{20} + \cdots + 57274699680025$ T^20 + 27T^19 + 694T^18 + 11163T^17 + 163931T^16 + 1899485T^15 + 19863270T^14 + 175445875T^13 + 1373439280T^12 + 9178276600T^11 + 53280259040T^10 + 255166611215T^9 + 995735209885T^8 + 2634105749150T^7 + 3109922089375T^6 - 6228475873400T^5 - 8068767601575T^4 + 24551971437000T^3 + 46938369518500T^2 - 93830017991250*T + 57274699680025
$71$	$T^{20} + \cdots + 45\!\cdots\!25$ T^20 + 7T^19 + 284T^18 + 2423T^17 + 43696T^16 + 329020T^15 + 4084575T^14 + 27647275T^13 + 342643870T^12 + 2175087775T^11 + 24515580650T^10 + 172636153275T^9 + 1472860551525T^8 + 6236140421125T^7 + 34075038197375T^6 + 111013102861750T^5 + 1698497482600500T^4 + 9240379783751875T^3 + 39187352340098750T^2 + 23588188309759375*T + 453632524330125625
$73$	$T^{20} + \cdots + 83\!\cdots\!21$ T^20 + 30T^19 + 500T^18 + 4145T^17 + 40720T^16 + 751126T^15 + 20192860T^14 + 297115425T^13 + 3120279115T^12 + 20345811540T^11 + 111221640866T^10 + 480640119315T^9 + 4732046176025T^8 + 23020266331585T^7 + 130318180164635T^6 + 609988081850511T^5 + 5133723023425920T^4 + 3195103736330550T^3 + 33326948751673280T^2 + 11427825240999355*T + 83321558347158121
$79$	$T^{20} + \cdots + 44\!\cdots\!25$ T^20 + T^19 + 60T^18 - 850T^17 + 19780T^16 + 165893T^15 + 3343208T^14 - 3978770T^13 + 116561490T^12 + 216110375T^11 + 29954988353T^10 + 123250826808T^9 + 1348514843845T^8 - 2257866381120T^7 + 64290940715520T^6 + 312607286157636T^5 + 5822186266009251T^4 + 16336808817689525T^3 + 82300024628980665T^2 + 12550471662772925T + 4481250777201025
$83$	$T^{20} + \cdots + 10538049025$ T^20 + 6T^19 + 56T^18 + 701T^17 + 22881T^16 - 198530T^15 + 1178365T^14 + 10064285T^13 + 169833325T^12 - 1570409550T^11 + 12607790765T^10 - 74148160635T^9 + 842742852635T^8 - 1845452750400T^7 + 6412135882600T^6 - 5122549750800T^5 + 10057416276125T^4 + 2391958861950T^3 + 3529352466150T^2 + 309228683050*T + 10538049025
$89$	$T^{20} + \cdots + 46\!\cdots\!25$ T^20 - 29T^19 + 485T^18 - 6135T^17 + 114825T^16 - 874492T^15 + 5130983T^14 - 29648570T^13 + 787246545T^12 + 3865087310T^11 + 39327708438T^10 - 50914455507T^9 + 918368053825T^8 - 76281440885T^7 + 6684000551360T^6 - 30983069845944T^5 + 330763145711881T^4 - 921964621038420T^3 + 2818999487287460T^2 - 3303609687932775*T + 4612514461095025
$97$	$T^{20} + \cdots + 16832713467361$ T^20 + 15T^19 + 435T^18 + 7205T^17 + 116075T^16 + 1104152T^15 + 12579255T^14 + 77310925T^13 + 782711015T^12 + 4056152510T^11 + 26936753289T^10 + 254639979500T^9 + 2075071528855T^8 + 10463653579580T^7 + 37196509918010T^6 + 90807720228638T^5 + 151626992304010T^4 + 167042545629785T^3 + 124756381463990T^2 + 60049296373165*T + 16832713467361
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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 $\F_{q}$

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 131.5
Significant digits:
Format:

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	650.2.l.b

Level	$650$
Weight	$2$
Character orbit	650.l
Analytic conductor	$5.190$
Analytic rank	$0$
Dimension	$20$
Inner twists	$2$