Properties

 Label 5225.2.a.ba Level $5225$ Weight $2$ Character orbit 5225.a Self dual yes Analytic conductor $41.722$ Analytic rank $1$ Dimension $20$ Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5225,2,Mod(1,5225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5225, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5225 = 5^{2} \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5225.a (trivial)

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: yes Analytic conductor: $$41.7218350561$$ Analytic rank: $$1$$ Dimension: $$20$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{20} - 26 x^{18} + 281 x^{16} - 1640 x^{14} + 5623 x^{12} - 11551 x^{10} + 13894 x^{8} - 9095 x^{6} + \cdots + 4$$ x^20 - 26*x^18 + 281*x^16 - 1640*x^14 + 5623*x^12 - 11551*x^10 + 13894*x^8 - 9095*x^6 + 2753*x^4 - 276*x^2 + 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: no (minimal twist has level 1045) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

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

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} - \beta_{6} q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{14} - \beta_{2} - 1) q^{6} - \beta_{3} q^{7} + (\beta_{7} + \beta_{6}) q^{8} + ( - \beta_{11} + 1) q^{9}+O(q^{10})$$ q + b1 * q^2 - b6 * q^3 + (b2 + 1) * q^4 + (b14 - b2 - 1) * q^6 - b3 * q^7 + (b7 + b6) * q^8 + (-b11 + 1) * q^9 $$q + \beta_1 q^{2} - \beta_{6} q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{14} - \beta_{2} - 1) q^{6} - \beta_{3} q^{7} + (\beta_{7} + \beta_{6}) q^{8} + ( - \beta_{11} + 1) q^{9} + q^{11} + (\beta_{4} - \beta_1) q^{12} + ( - \beta_{18} + \beta_{6} + \beta_{3}) q^{13} + ( - \beta_{17} + \beta_{15} + \beta_{9} + \cdots - 1) q^{14}+ \cdots + ( - \beta_{11} + 1) q^{99}+O(q^{100})$$ q + b1 * q^2 - b6 * q^3 + (b2 + 1) * q^4 + (b14 - b2 - 1) * q^6 - b3 * q^7 + (b7 + b6) * q^8 + (-b11 + 1) * q^9 + q^11 + (b4 - b1) * q^12 + (-b18 + b6 + b3) * q^13 + (-b17 + b15 + b9 - b2 - 1) * q^14 + (-b19 + b17 - b14 + b5 + b2 - 1) * q^16 + (-b16 + b13 - b10 - b4 - b1) * q^17 + (-b13 + b10 - b7 + b1) * q^18 - q^19 + (b17 - b15 - b14 + b11 - b9 + b2 - 2) * q^21 + b1 * q^22 + (b18 + b16 - b8 - b7 + b3 - 2*b1) * q^23 + (b19 - b12 + b11 - b2 - 2) * q^24 + (-b15 - b14 + b12 - b9 - b5 + b2 + 1) * q^26 + (b18 + b16 - b13 + b8 - b7 - b4 + b3) * q^27 + (-b16 + b10 + b8 - b7 - b6 - b3) * q^28 + (-b14 + b11 - b9 - 3) * q^29 + (b19 - b17 + 2*b14 + b9 - 2*b5 - 3*b2 - 2) * q^31 + (b18 + b16 + b13 - b10 - b6 - b4 - 2*b1) * q^32 - b6 * q^33 + (-b14 + b12 - b5 - 1) * q^34 + (b19 - b17 - b15 - b14 + b11) * q^36 + (b18 + b16 + b13 - b8 + 2*b7 + b6 + b4 - 3*b1) * q^37 - b1 * q^38 + (-2*b17 + 2*b15 + b14 + b11 + b9 - b5 - b2 - 2) * q^39 + (b19 + b17 - b15 - b12 + b11 - b9 + 2*b5 - 3) * q^41 + (b16 + b13 - 2*b10 - b8 + b7 + 2*b6 - b4 + 3*b3 - 4*b1) * q^42 + (-b16 - b13 + b10 - b7 + b6 + b4 + 2*b1) * q^43 + (b2 + 1) * q^44 + (b19 + b17 - 2*b15 - b14 - b12 - 2*b9 - b2 - 2) * q^46 + (-b16 - 2*b13 + b10 + 2*b8 - 2*b7 + b4 + 2*b1) * q^47 + (b18 - b7 - b4 - b3) * q^48 + (-b15 - 3*b14 + b12 - b9 + 2*b5 + 4*b2 + 1) * q^49 + (-b14 + b11 + b9 - b5 - b2 - 2) * q^51 + (-b10 - b8 - b7 - 2*b4 + 2*b3 - b1) * q^52 + (-b18 + b16 + 2*b10 - b8 + 2*b7 + 2*b6 + 2*b4 - b3 + b1) * q^53 + (-b19 + 2*b17 - 2*b14 - b11 - b9 + b5 + 2*b2) * q^54 + (-b17 + b15 + 2*b14 - b11 + b9 - b5 - 3*b2 - 2) * q^56 + b6 * q^57 + (b16 + b13 - b10 - b8 - b7 + b6 - b4 + 2*b3 - 6*b1) * q^58 + (-b19 + 2*b17 - b11 + b9 - b5 - b2 - 2) * q^59 + (-2*b19 - b17 + b15 - b14 + b9 + b5 + b2 - 1) * q^61 + (-b18 - 2*b16 + b8 - 2*b7 - 3*b6 + b4 - b3 - b1) * q^62 + (-b18 - 2*b16 + b13 + b10 + b7 + 3*b6 + b4 - b3 + b1) * q^63 + (-b17 + 2*b15 + 2*b14 - 2*b5 - 3*b2 - 2) * q^64 + (b14 - b2 - 1) * q^66 + (-b18 - b16 - 2*b13 + 2*b8 - b6 - 2*b4 - b3 + 3*b1) * q^67 + (-b18 + b16 - b13 + b8 - b7 + b6 - b4 + b3) * q^68 + (-2*b19 - b17 + b15 + b12 - 2*b11 + 2*b9 + b5 + b2 + 1) * q^69 + (b17 - b14 + b11 + b9 + b5 - 3) * q^71 + (-2*b18 - b16 - b8 + b6 + b4 - b1) * q^72 + (2*b18 + b16 + 2*b13 - 2*b10 + b8 - b7 - 2*b4 + 2*b3 - 2*b1) * q^73 + (-2*b19 + 2*b17 + b15 + b14 - 2*b12 + b11 + 2*b5 + b2 - 5) * q^74 + (-b2 - 1) * q^76 - b3 * q^77 + (-b16 + 2*b13 + b8 - b6 - 3*b3 - b1) * q^78 + (b19 + b17 - b14 - b12 + 2*b11 + b5 - 2*b2 - 7) * q^79 + (-2*b19 + b17 + 2*b15 + 2*b12 - 2*b11 + 2*b5 - 2) * q^81 + (b18 + b16 - 2*b13 + b10 - b8 - b6 + 3*b4 - 2*b1) * q^82 + (b18 - b8 + b7 + 2*b6 + b4 - 2*b1) * q^83 + (-b19 + 3*b17 - 2*b15 - 2*b14 + b12 - b11 - 3*b9 + 3*b2 + 1) * q^84 + (3*b19 - b17 - 2*b15 - b12 + 5) * q^86 + (-2*b18 - b16 + b13 + b10 + 2*b7 + 4*b6 - 2*b3) * q^87 + (b7 + b6) * q^88 + (-2*b19 - 2*b17 + b15 - b14 + 2*b12 - b11 + b9 - b5 + 3*b2) * q^89 + (2*b15 + 3*b14 - b12 + b9 + b5 - 4*b2 - 5) * q^91 + (-2*b18 - 2*b13 + b10 - b8 - 2*b7 + b6 + b4 + 3*b3 - 3*b1) * q^92 + (-b18 - b16 + b13 - b10 - b8 + b7 + 3*b6 + b3 + b1) * q^93 + (3*b19 - b17 + 2*b14 - b12 + b9 - 4*b2 - 1) * q^94 + (-2*b19 - b17 + b15 - 2*b14 + 2*b12 - 3*b11 + b9 + 2*b2 + 5) * q^96 + (-b18 - b16 + b7 + 2*b6 + 3*b4 + 2*b3 - b1) * q^97 + (-2*b18 - 3*b13 + 2*b10 - b8 + 4*b6 - b4 + b3 + 3*b1) * q^98 + (-b11 + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20 q + 12 q^{4} - 8 q^{6} + 10 q^{9}+O(q^{10})$$ 20 * q + 12 * q^4 - 8 * q^6 + 10 * q^9 $$20 q + 12 q^{4} - 8 q^{6} + 10 q^{9} + 20 q^{11} - 24 q^{14} - 4 q^{16} - 20 q^{19} - 30 q^{21} - 38 q^{24} + 8 q^{26} - 50 q^{29} - 50 q^{31} - 28 q^{34} - 12 q^{36} - 48 q^{39} - 34 q^{41} + 12 q^{44} - 36 q^{46} + 6 q^{49} - 40 q^{51} + 6 q^{54} - 40 q^{56} - 30 q^{59} - 14 q^{61} - 36 q^{64} - 8 q^{66} + 12 q^{69} - 40 q^{71} - 50 q^{74} - 12 q^{76} - 106 q^{79} + 30 q^{84} + 56 q^{86} - 36 q^{89} - 56 q^{91} - 28 q^{94} + 66 q^{96} + 10 q^{99}+O(q^{100})$$ 20 * q + 12 * q^4 - 8 * q^6 + 10 * q^9 + 20 * q^11 - 24 * q^14 - 4 * q^16 - 20 * q^19 - 30 * q^21 - 38 * q^24 + 8 * q^26 - 50 * q^29 - 50 * q^31 - 28 * q^34 - 12 * q^36 - 48 * q^39 - 34 * q^41 + 12 * q^44 - 36 * q^46 + 6 * q^49 - 40 * q^51 + 6 * q^54 - 40 * q^56 - 30 * q^59 - 14 * q^61 - 36 * q^64 - 8 * q^66 + 12 * q^69 - 40 * q^71 - 50 * q^74 - 12 * q^76 - 106 * q^79 + 30 * q^84 + 56 * q^86 - 36 * q^89 - 56 * q^91 - 28 * q^94 + 66 * q^96 + 10 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} - 26 x^{18} + 281 x^{16} - 1640 x^{14} + 5623 x^{12} - 11551 x^{10} + 13894 x^{8} - 9095 x^{6} + \cdots + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3 $$\beta_{3}$$ $$=$$ $$( 61 \nu^{19} - 1299 \nu^{17} + 10262 \nu^{15} - 33228 \nu^{13} + 5459 \nu^{11} + 240718 \nu^{9} + \cdots - 1604 \nu ) / 1076$$ (61*v^19 - 1299*v^17 + 10262*v^15 - 33228*v^13 + 5459*v^11 + 240718*v^9 - 592650*v^7 + 522039*v^5 - 131958*v^3 - 1604*v) / 1076 $$\beta_{4}$$ $$=$$ $$( - 57 \nu^{19} + 1205 \nu^{17} - 9104 \nu^{15} + 23376 \nu^{13} + 49969 \nu^{11} - 428112 \nu^{9} + \cdots - 27024 \nu ) / 1076$$ (-57*v^19 + 1205*v^17 - 9104*v^15 + 23376*v^13 + 49969*v^11 - 428112*v^9 + 946122*v^7 - 866947*v^5 + 288912*v^3 - 27024*v) / 1076 $$\beta_{5}$$ $$=$$ $$( 127 \nu^{18} - 3119 \nu^{16} + 31252 \nu^{14} - 164582 \nu^{12} + 489083 \nu^{10} - 819526 \nu^{8} + \cdots - 1064 ) / 538$$ (127*v^18 - 3119*v^16 + 31252*v^14 - 164582*v^12 + 489083*v^10 - 819526*v^8 + 735502*v^6 - 319949*v^4 + 59110*v^2 - 1064) / 538 $$\beta_{6}$$ $$=$$ $$( - 424 \nu^{19} + 10233 \nu^{17} - 99883 \nu^{15} + 504698 \nu^{13} - 1398132 \nu^{11} + \cdots - 13202 \nu ) / 1076$$ (-424*v^19 + 10233*v^17 - 99883*v^15 + 504698*v^13 - 1398132*v^11 + 2051391*v^9 - 1360162*v^7 + 189834*v^5 + 79343*v^3 - 13202*v) / 1076 $$\beta_{7}$$ $$=$$ $$( 424 \nu^{19} - 10233 \nu^{17} + 99883 \nu^{15} - 504698 \nu^{13} + 1398132 \nu^{11} - 2051391 \nu^{9} + \cdots + 8898 \nu ) / 1076$$ (424*v^19 - 10233*v^17 + 99883*v^15 - 504698*v^13 + 1398132*v^11 - 2051391*v^9 + 1360162*v^7 - 189834*v^5 - 78267*v^3 + 8898*v) / 1076 $$\beta_{8}$$ $$=$$ $$( 117 \nu^{19} - 3153 \nu^{17} + 35620 \nu^{15} - 219576 \nu^{13} + 805123 \nu^{11} - 1793688 \nu^{9} + \cdots - 62380 \nu ) / 1076$$ (117*v^19 - 3153*v^17 + 35620*v^15 - 219576*v^13 + 805123*v^11 - 1793688*v^9 + 2372890*v^7 - 1728577*v^5 + 585360*v^3 - 62380*v) / 1076 $$\beta_{9}$$ $$=$$ $$( - 365 \nu^{18} + 8981 \nu^{16} - 90200 \nu^{14} + 476396 \nu^{12} - 1420387 \nu^{10} + 2386344 \nu^{8} + \cdots + 8116 ) / 1076$$ (-365*v^18 + 8981*v^16 - 90200*v^14 + 476396*v^12 - 1420387*v^10 + 2386344*v^8 - 2136542*v^6 + 912841*v^4 - 168752*v^2 + 8116) / 1076 $$\beta_{10}$$ $$=$$ $$( 563 \nu^{19} - 13634 \nu^{17} + 133533 \nu^{15} - 676778 \nu^{13} + 1878649 \nu^{11} + \cdots - 2730 \nu ) / 1076$$ (563*v^19 - 13634*v^17 + 133533*v^15 - 676778*v^13 + 1878649*v^11 - 2756833*v^9 + 1827220*v^7 - 280745*v^5 - 55629*v^3 - 2730*v) / 1076 $$\beta_{11}$$ $$=$$ $$( 119 \nu^{18} - 2931 \nu^{16} + 29474 \nu^{14} - 155907 \nu^{12} + 465652 \nu^{10} - 783140 \nu^{8} + \cdots - 29 ) / 269$$ (119*v^18 - 2931*v^16 + 29474*v^14 - 155907*v^12 + 465652*v^10 - 783140*v^8 + 697023*v^6 - 282189*v^4 + 36260*v^2 - 29) / 269 $$\beta_{12}$$ $$=$$ $$( 284 \nu^{18} - 6943 \nu^{16} + 69037 \nu^{14} - 358938 \nu^{12} + 1043638 \nu^{10} - 1682291 \nu^{8} + \cdots - 1100 ) / 538$$ (284*v^18 - 6943*v^16 + 69037*v^14 - 358938*v^12 + 1043638*v^10 - 1682291*v^8 + 1400840*v^6 - 516802*v^4 + 68197*v^2 - 1100) / 538 $$\beta_{13}$$ $$=$$ $$( 615 \nu^{19} - 15125 \nu^{17} + 151546 \nu^{15} - 795708 \nu^{13} + 2343125 \nu^{11} + \cdots - 11744 \nu ) / 1076$$ (615*v^19 - 15125*v^17 + 151546*v^15 - 795708*v^13 + 2343125*v^11 - 3838002*v^9 + 3255150*v^7 - 1219667*v^5 + 167678*v^3 - 11744*v) / 1076 $$\beta_{14}$$ $$=$$ $$( 791 \nu^{18} - 19261 \nu^{16} + 190662 \nu^{14} - 986020 \nu^{12} + 2846233 \nu^{10} - 4530894 \nu^{8} + \cdots - 3848 ) / 1076$$ (791*v^18 - 19261*v^16 + 190662*v^14 - 986020*v^12 + 2846233*v^10 - 4530894*v^8 + 3666446*v^6 - 1246615*v^4 + 131302*v^2 - 3848) / 1076 $$\beta_{15}$$ $$=$$ $$( - 288 \nu^{18} + 7037 \nu^{16} - 69926 \nu^{14} + 363141 \nu^{12} - 1052798 \nu^{10} + 1682461 \nu^{8} + \cdots - 669 ) / 269$$ (-288*v^18 + 7037*v^16 - 69926*v^14 + 363141*v^12 - 1052798*v^10 + 1682461*v^8 - 1362379*v^6 + 453906*v^4 - 37927*v^2 - 669) / 269 $$\beta_{16}$$ $$=$$ $$( 784 \nu^{19} - 19769 \nu^{17} + 205179 \nu^{15} - 1134214 \nu^{13} + 3612724 \nu^{11} + \cdots - 56238 \nu ) / 1076$$ (784*v^19 - 19769*v^17 + 205179*v^15 - 1134214*v^13 + 3612724*v^11 - 6712859*v^9 + 7054894*v^7 - 3875330*v^5 + 939761*v^3 - 56238*v) / 1076 $$\beta_{17}$$ $$=$$ $$( - 615 \nu^{18} + 15125 \nu^{16} - 151546 \nu^{14} + 795708 \nu^{12} - 2343125 \nu^{10} + 3838002 \nu^{8} + \cdots + 3674 ) / 538$$ (-615*v^18 + 15125*v^16 - 151546*v^14 + 795708*v^12 - 2343125*v^10 + 3838002*v^8 - 3254612*v^6 + 1214287*v^4 - 153152*v^2 + 3674) / 538 $$\beta_{18}$$ $$=$$ $$( - 1317 \nu^{19} + 32698 \nu^{17} - 332179 \nu^{15} + 1781218 \nu^{13} - 5425363 \nu^{11} + \cdots + 40090 \nu ) / 1076$$ (-1317*v^19 + 32698*v^17 - 332179*v^15 + 1781218*v^13 - 5425363*v^11 + 9417307*v^9 - 8896864*v^7 + 4138215*v^5 - 803421*v^3 + 40090*v) / 1076 $$\beta_{19}$$ $$=$$ $$( - 1767 \nu^{18} + 43273 \nu^{16} - 431250 \nu^{14} + 2248272 \nu^{12} - 6554317 \nu^{10} + \cdots + 460 ) / 1076$$ (-1767*v^18 + 43273*v^16 - 431250*v^14 + 2248272*v^12 - 6554317*v^10 + 10567846*v^8 - 8704666*v^6 + 3034215*v^4 - 311854*v^2 + 460) / 1076
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{7} + \beta_{6} + 4\beta_1$$ b7 + b6 + 4*b1 $$\nu^{4}$$ $$=$$ $$-\beta_{19} + \beta_{17} - \beta_{14} + \beta_{5} + 7\beta_{2} + 13$$ -b19 + b17 - b14 + b5 + 7*b2 + 13 $$\nu^{5}$$ $$=$$ $$\beta_{18} + \beta_{16} + \beta_{13} - \beta_{10} + 8\beta_{7} + 7\beta_{6} - \beta_{4} + 18\beta_1$$ b18 + b16 + b13 - b10 + 8*b7 + 7*b6 - b4 + 18*b1 $$\nu^{6}$$ $$=$$ $$-10\beta_{19} + 9\beta_{17} + 2\beta_{15} - 8\beta_{14} + 8\beta_{5} + 43\beta_{2} + 64$$ -10*b19 + 9*b17 + 2*b15 - 8*b14 + 8*b5 + 43*b2 + 64 $$\nu^{7}$$ $$=$$ $$11 \beta_{18} + 10 \beta_{16} + 13 \beta_{13} - 12 \beta_{10} + \beta_{8} + 54 \beta_{7} + \cdots + 87 \beta_1$$ 11*b18 + 10*b16 + 13*b13 - 12*b10 + b8 + 54*b7 + 43*b6 - 11*b4 + 87*b1 $$\nu^{8}$$ $$=$$ $$- 77 \beta_{19} + 64 \beta_{17} + 26 \beta_{15} - 51 \beta_{14} + \beta_{11} + 2 \beta_{9} + 52 \beta_{5} + \cdots + 337$$ -77*b19 + 64*b17 + 26*b15 - 51*b14 + b11 + 2*b9 + 52*b5 + 257*b2 + 337 $$\nu^{9}$$ $$=$$ $$90 \beta_{18} + 75 \beta_{16} + 116 \beta_{13} - 103 \beta_{10} + 15 \beta_{8} + 350 \beta_{7} + \cdots + 445 \beta_1$$ 90*b18 + 75*b16 + 116*b13 - 103*b10 + 15*b8 + 350*b7 + 256*b6 - 89*b4 - 5*b3 + 445*b1 $$\nu^{10}$$ $$=$$ $$- 542 \beta_{19} + 422 \beta_{17} + 239 \beta_{15} - 303 \beta_{14} - \beta_{12} + 14 \beta_{11} + \cdots + 1851$$ -542*b19 + 422*b17 + 239*b15 - 303*b14 - b12 + 14*b11 + 33*b9 + 324*b5 + 1533*b2 + 1851 $$\nu^{11}$$ $$=$$ $$662 \beta_{18} + 510 \beta_{16} + 893 \beta_{13} - 772 \beta_{10} + 151 \beta_{8} + 2241 \beta_{7} + \cdots + 2386 \beta_1$$ 662*b18 + 510*b16 + 893*b13 - 772*b10 + 151*b8 + 2241*b7 + 1513*b6 - 639*b4 - 87*b3 + 2386*b1 $$\nu^{12}$$ $$=$$ $$- 3662 \beta_{19} + 2695 \beta_{17} + 1913 \beta_{15} - 1747 \beta_{14} - 23 \beta_{12} + 133 \beta_{11} + \cdots + 10476$$ -3662*b19 + 2695*b17 + 1913*b15 - 1747*b14 - 23*b12 + 133*b11 + 359*b9 + 2010*b5 + 9190*b2 + 10476 $$\nu^{13}$$ $$=$$ $$4631 \beta_{18} + 3326 \beta_{16} + 6393 \beta_{13} - 5403 \beta_{10} + 1282 \beta_{8} + \cdots + 13300 \beta_1$$ 4631*b18 + 3326*b16 + 6393*b13 - 5403*b10 + 1282*b8 + 14292*b7 + 8950*b6 - 4322*b4 - 979*b3 + 13300*b1 $$\nu^{14}$$ $$=$$ $$- 24212 \beta_{19} + 16954 \beta_{17} + 14252 \beta_{15} - 9919 \beta_{14} - 309 \beta_{12} + \cdots + 60653$$ -24212*b19 + 16954*b17 + 14252*b15 - 9919*b14 - 309*b12 + 1081*b11 + 3251*b9 + 12530*b5 + 55455*b2 + 60653 $$\nu^{15}$$ $$=$$ $$31515 \beta_{18} + 21270 \beta_{16} + 43969 \beta_{13} - 36402 \beta_{10} + 9936 \beta_{8} + \cdots + 76488 \beta_1$$ 31515*b18 + 21270*b16 + 43969*b13 - 36402*b10 + 9936*b8 + 91034*b7 + 53153*b6 - 28286*b4 - 9072*b3 + 76488*b1 $$\nu^{16}$$ $$=$$ $$- 158093 \beta_{19} + 105910 \beta_{17} + 101750 \beta_{15} - 55820 \beta_{14} - 3229 \beta_{12} + \cdots + 357443$$ -158093*b19 + 105910*b17 + 101750*b15 - 55820*b14 - 3229*b12 + 8116*b11 + 26575*b9 + 78580*b5 + 336819*b2 + 357443 $$\nu^{17}$$ $$=$$ $$210889 \beta_{18} + 134747 \beta_{16} + 295289 \beta_{13} - 239877 \beta_{10} + 72913 \beta_{8} + \cdots + 450757 \beta_1$$ 210889*b18 + 134747*b16 + 295289*b13 - 239877*b10 + 72913*b8 + 579693*b7 + 317288*b6 - 181671*b4 - 75387*b3 + 450757*b1 $$\nu^{18}$$ $$=$$ $$- 1024436 \beta_{19} + 659783 \beta_{17} + 706661 \beta_{15} - 312428 \beta_{14} - 29218 \beta_{12} + \cdots + 2136195$$ -1024436*b19 + 659783*b17 + 706661*b15 - 312428*b14 - 29218*b12 + 58206*b11 + 203712*b9 + 495293*b5 + 2058073*b2 + 2136195 $$\nu^{19}$$ $$=$$ $$1395662 \beta_{18} + 849942 \beta_{16} + 1953793 \beta_{13} - 1559922 \beta_{10} + 516502 \beta_{8} + \cdots + 2706467 \beta_1$$ 1395662*b18 + 849942*b16 + 1953793*b13 - 1559922*b10 + 516502*b8 + 3691782*b7 + 1904426*b6 - 1154361*b4 - 584942*b3 + 2706467*b1

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.52711 −2.33380 −2.04146 −2.00304 −1.51095 −1.23709 −1.22468 −0.712399 −0.386431 −0.131596 0.131596 0.386431 0.712399 1.22468 1.23709 1.51095 2.00304 2.04146 2.33380 2.52711
−2.52711 0.473551 4.38628 0 −1.19672 1.17543 −6.03039 −2.77575 0
1.2 −2.33380 1.97214 3.44661 0 −4.60257 4.79282 −3.37609 0.889329 0
1.3 −2.04146 −1.84076 2.16755 0 3.75784 2.14151 −0.342047 0.388411 0
1.4 −2.00304 2.62978 2.01218 0 −5.26757 −4.24477 −0.0243959 3.91575 0
1.5 −1.51095 −0.900785 0.282982 0 1.36104 0.603482 2.59434 −2.18859 0
1.6 −1.23709 2.07277 −0.469598 0 −2.56421 −0.314194 3.05513 1.29637 0
1.7 −1.22468 −1.51450 −0.500160 0 1.85478 0.966830 3.06189 −0.706295 0
1.8 −0.712399 −3.32227 −1.49249 0 2.36678 2.70450 2.48804 8.03747 0
1.9 −0.386431 −0.261531 −1.85067 0 0.101064 −4.13791 1.48802 −2.93160 0
1.10 −0.131596 −1.44045 −1.98268 0 0.189557 −0.456875 0.524103 −0.925103 0
1.11 0.131596 1.44045 −1.98268 0 0.189557 0.456875 −0.524103 −0.925103 0
1.12 0.386431 0.261531 −1.85067 0 0.101064 4.13791 −1.48802 −2.93160 0
1.13 0.712399 3.32227 −1.49249 0 2.36678 −2.70450 −2.48804 8.03747 0
1.14 1.22468 1.51450 −0.500160 0 1.85478 −0.966830 −3.06189 −0.706295 0
1.15 1.23709 −2.07277 −0.469598 0 −2.56421 0.314194 −3.05513 1.29637 0
1.16 1.51095 0.900785 0.282982 0 1.36104 −0.603482 −2.59434 −2.18859 0
1.17 2.00304 −2.62978 2.01218 0 −5.26757 4.24477 0.0243959 3.91575 0
1.18 2.04146 1.84076 2.16755 0 3.75784 −2.14151 0.342047 0.388411 0
1.19 2.33380 −1.97214 3.44661 0 −4.60257 −4.79282 3.37609 0.889329 0
1.20 2.52711 −0.473551 4.38628 0 −1.19672 −1.17543 6.03039 −2.77575 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.20 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$-1$$
$$11$$ $$-1$$
$$19$$ $$+1$$

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5225.2.a.ba 20
5.b even 2 1 inner 5225.2.a.ba 20
5.c odd 4 2 1045.2.b.c 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.b.c 20 5.c odd 4 2
5225.2.a.ba 20 1.a even 1 1 trivial
5225.2.a.ba 20 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5225))$$:

 $$T_{2}^{20} - 26 T_{2}^{18} + 281 T_{2}^{16} - 1640 T_{2}^{14} + 5623 T_{2}^{12} - 11551 T_{2}^{10} + \cdots + 4$$ T2^20 - 26*T2^18 + 281*T2^16 - 1640*T2^14 + 5623*T2^12 - 11551*T2^10 + 13894*T2^8 - 9095*T2^6 + 2753*T2^4 - 276*T2^2 + 4 $$T_{7}^{20} - 73 T_{7}^{18} + 2053 T_{7}^{16} - 28024 T_{7}^{14} + 194005 T_{7}^{12} - 670233 T_{7}^{10} + \cdots + 2304$$ T7^20 - 73*T7^18 + 2053*T7^16 - 28024*T7^14 + 194005*T7^12 - 670233*T7^10 + 1099880*T7^8 - 856532*T7^6 + 304176*T7^4 - 46016*T7^2 + 2304

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{20} - 26 T^{18} + \cdots + 4$$
$3$ $$T^{20} - 35 T^{18} + \cdots + 256$$
$5$ $$T^{20}$$
$7$ $$T^{20} - 73 T^{18} + \cdots + 2304$$
$11$ $$(T - 1)^{20}$$
$13$ $$T^{20} - 137 T^{18} + \cdots + 33732864$$
$17$ $$T^{20} - 157 T^{18} + \cdots + 7054336$$
$19$ $$(T + 1)^{20}$$
$23$ $$T^{20} + \cdots + 172764736$$
$29$ $$(T^{10} + 25 T^{9} + \cdots + 4194192)^{2}$$
$31$ $$(T^{10} + 25 T^{9} + \cdots - 7456)^{2}$$
$37$ $$T^{20} + \cdots + 61309721664$$
$41$ $$(T^{10} + 17 T^{9} + \cdots + 166848)^{2}$$
$43$ $$T^{20} + \cdots + 7783474176$$
$47$ $$T^{20} + \cdots + 47\!\cdots\!44$$
$53$ $$T^{20} + \cdots + 58\!\cdots\!36$$
$59$ $$(T^{10} + 15 T^{9} + \cdots + 86423136)^{2}$$
$61$ $$(T^{10} + 7 T^{9} + \cdots - 26701648)^{2}$$
$67$ $$T^{20} + \cdots + 345472702914816$$
$71$ $$(T^{10} + 20 T^{9} + \cdots - 400032)^{2}$$
$73$ $$T^{20} + \cdots + 27\!\cdots\!84$$
$79$ $$(T^{10} + 53 T^{9} + \cdots - 204893824)^{2}$$
$83$ $$T^{20} + \cdots + 5917659525376$$
$89$ $$(T^{10} + 18 T^{9} + \cdots + 14189256)^{2}$$
$97$ $$T^{20} + \cdots + 19285658703936$$