# Properties

 Label 136.4.k.a Level $136$ Weight $4$ Character orbit 136.k Analytic conductor $8.024$ Analytic rank $0$ Dimension $14$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$136 = 2^{3} \cdot 17$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 136.k (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.02425976078$$ Analytic rank: $$0$$ Dimension: $$14$$ Relative dimension: $$7$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{14} + \cdots)$$ Defining polynomial: $$x^{14} + 142x^{12} + 7785x^{10} + 208489x^{8} + 2822686x^{6} + 17977041x^{4} + 45020416x^{2} + 31719424$$ x^14 + 142*x^12 + 7785*x^10 + 208489*x^8 + 2822686*x^6 + 17977041*x^4 + 45020416*x^2 + 31719424 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{17}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{13}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + (\beta_{7} - \beta_{5} + 1) q^{5} + (\beta_{9} - \beta_{5} - 1) q^{7} + (\beta_{12} - 13 \beta_{5} + \beta_{2} - \beta_1) q^{9}+O(q^{10})$$ q + b1 * q^3 + (b7 - b5 + 1) * q^5 + (b9 - b5 - 1) * q^7 + (b12 - 13*b5 + b2 - b1) * q^9 $$q + \beta_1 q^{3} + (\beta_{7} - \beta_{5} + 1) q^{5} + (\beta_{9} - \beta_{5} - 1) q^{7} + (\beta_{12} - 13 \beta_{5} + \beta_{2} - \beta_1) q^{9} + (\beta_{11} + \beta_{6} + \beta_{5} - 2 \beta_{2} + 1) q^{11} + ( - \beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} - \beta_{4} + \beta_{2} + \beta_1 + 5) q^{13} + (\beta_{13} + \beta_{12} + \beta_{11} + \beta_{10} - 3 \beta_{9} + 3 \beta_{8} + \beta_{7} - \beta_{6} - 5 \beta_{5} + \cdots + 2 \beta_1) q^{15}+ \cdots + ( - 3 \beta_{13} - \beta_{12} - 22 \beta_{10} - 24 \beta_{8} + 82 \beta_{7} + \cdots + 187) q^{99}+O(q^{100})$$ q + b1 * q^3 + (b7 - b5 + 1) * q^5 + (b9 - b5 - 1) * q^7 + (b12 - 13*b5 + b2 - b1) * q^9 + (b11 + b6 + b5 - 2*b2 + 1) * q^11 + (-b9 - b8 - b7 - b6 - b4 + b2 + b1 + 5) * q^13 + (b13 + b12 + b11 + b10 - 3*b9 + 3*b8 + b7 - b6 - 5*b5 - 2*b2 + 2*b1) * q^15 + (-b12 + b11 - b10 - b8 + 2*b6 + 15*b5 - b4 + b3 - b2 - 2*b1 - 3) * q^17 + (b13 + 3*b12 - b9 + b8 - 2*b7 + 2*b6 - 31*b5 + b2 - b1) * q^19 + (2*b11 - 2*b10 - b9 - b8 + 6*b7 + 6*b6 + b3 - 6*b2 - 6*b1 - 2) * q^21 + (b13 - b12 - b11 + 3*b9 - b6 + 15*b5 - b4 - b3 - 3*b2 + 15) * q^23 + (b13 + 3*b12 - 5*b9 + 5*b8 + 4*b7 - 4*b6 - 24*b5 - 2*b2 + 2*b1) * q^25 + (-2*b12 + 3*b11 + 3*b6 + 39*b5 - 2*b3 - 11*b2 + 39) * q^27 + (-2*b13 + b12 - 2*b10 - 6*b8 - 7*b7 - 17*b5 - 2*b4 - b3 + 2*b1 + 17) * q^29 + (b13 + 3*b12 + 3*b10 + 3*b8 + 5*b7 - 39*b5 + b4 - 3*b3 - 11*b1 + 39) * q^31 + (8*b9 + 8*b8 - 2*b7 - 2*b6 + b4 + 4*b3 + 6*b2 + 6*b1 - 67) * q^33 + (-b11 + b10 - b9 - b8 + 7*b7 + 7*b6 + 2*b3 + 19*b2 + 19*b1 - 40) * q^35 + (-b13 - 3*b12 - 4*b8 + 7*b7 - b4 + 3*b3 - 6*b1) * q^37 + (2*b12 + 3*b10 - 8*b8 - 19*b7 - 33*b5 - 2*b3 + 17*b1 + 33) * q^39 + (-b13 + 3*b12 + 2*b11 - 6*b9 - 12*b6 - 62*b5 + b4 + 3*b3 - 10*b2 - 62) * q^41 + (-b13 - 5*b12 + 8*b9 - 8*b8 + 2*b7 - 2*b6 + 65*b5 + 12*b2 - 12*b1) * q^43 + (b12 - 6*b11 + 6*b9 - 19*b6 - 49*b5 + b3 + 14*b2 - 49) * q^45 + (-4*b9 - 4*b8 + 6*b7 + 6*b6 + 3*b4 - 3*b3 + 11*b2 + 11*b1 + 47) * q^47 + (-b13 - 8*b12 - 2*b11 - 2*b10 - b9 + b8 + 8*b7 - 8*b6 + 52*b5 - 7*b2 + 7*b1) * q^49 + (b13 - b12 - 3*b11 + 5*b9 + 13*b8 - 12*b7 + 5*b6 + 76*b5 + 3*b4 + b3 + 34*b2 - 6*b1 - 26) * q^51 + (-3*b13 - b12 - 4*b11 - 4*b10 - 7*b7 + 7*b6 + 7*b5 + 13*b2 - 13*b1) * q^53 + (-4*b11 + 4*b10 + b9 + b8 + 6*b7 + 6*b6 + 3*b4 - 11*b3 - 21*b2 - 21*b1 + 109) * q^55 + (-3*b13 - 8*b12 + 10*b9 - 4*b6 + 45*b5 + 3*b4 - 8*b3 - 82*b2 + 45) * q^57 + (-5*b13 - 5*b12 - 5*b11 - 5*b10 + 6*b9 - 6*b8 - 9*b7 + 9*b6 + 127*b5 - 9*b2 + 9*b1) * q^59 + (-b13 + b12 - 6*b11 - 12*b9 + 17*b6 + 64*b5 + b4 + b3 + 20*b2 + 64) * q^61 + (6*b13 - 4*b12 + 9*b10 + 31*b8 - 9*b7 + 166*b5 + 6*b4 + 4*b3 + 17*b1 - 166) * q^63 + (-3*b13 - 6*b12 - 2*b10 - 8*b8 - 16*b7 + 77*b5 - 3*b4 + 6*b3 - 50*b1 - 77) * q^65 + (-3*b11 + 3*b10 - 7*b9 - 7*b8 - 21*b7 - 21*b6 - 3*b4 + b3 + b2 + b1 - 35) * q^67 + (2*b11 - 2*b10 - 15*b9 - 15*b8 + 12*b7 + 12*b6 + b4 + 4*b3 + 29*b2 + 29*b1 - 145) * q^69 + (2*b13 - 6*b12 - 4*b10 - 11*b8 + 16*b7 - 21*b5 + 2*b4 + 6*b3 - 46*b1 + 21) * q^71 + (b13 - b12 - 6*b10 - 22*b8 - 42*b7 - 46*b5 + b4 + b3 + 18*b1 + 46) * q^73 + (3*b13 - 3*b12 - 4*b11 - 18*b9 - 64*b6 - 97*b5 - 3*b4 - 3*b3 - 41*b2 - 97) * q^75 + (b13 + 4*b12 - 2*b11 - 2*b10 - 7*b9 + 7*b8 + 20*b7 - 20*b6 + 29*b5 + 97*b2 - 97*b1) * q^77 + (-2*b13 - 8*b12 + 5*b11 + 15*b9 + 5*b6 + 36*b5 + 2*b4 - 8*b3 + 45*b2 + 36) * q^79 + (-6*b11 + 6*b10 + 24*b9 + 24*b8 - 12*b7 - 12*b6 + 3*b4 - 6*b3 + 78*b2 + 78*b1 - 46) * q^81 + (-9*b13 + 11*b12 - 3*b11 - 3*b10 - 10*b9 + 10*b8 + 21*b7 - 21*b6 - 321*b5 - 15*b2 + 15*b1) * q^83 + (-3*b13 + 2*b11 + 10*b10 + 12*b9 - 9*b7 + 38*b6 + 80*b5 + 2*b4 - 9*b3 + 15*b2 - 61*b1 + 293) * q^85 + (-3*b13 - 3*b12 + 8*b11 + 8*b10 + 13*b9 - 13*b8 - 62*b7 + 62*b6 - 59*b5 - 77*b2 + 77*b1) * q^87 + (-6*b11 + 6*b10 - 3*b9 - 3*b8 + 40*b7 + 40*b6 - 7*b4 + 18*b3 + b2 + b1 - 289) * q^89 + (-2*b13 + 20*b12 + 7*b11 + 2*b9 + 23*b6 - 237*b5 + 2*b4 + 20*b3 - 87*b2 - 237) * q^91 + (3*b13 - 8*b12 + 10*b11 + 10*b10 - b9 + b8 + 28*b7 - 28*b6 + 431*b5 - 97*b2 + 97*b1) * q^93 + (3*b13 + 3*b12 - 10*b11 + 22*b9 - 58*b6 + 205*b5 - 3*b4 + 3*b3 - 8*b2 + 205) * q^95 + (5*b13 + 10*b12 - 4*b10 - 16*b8 - 2*b7 + 52*b5 + 5*b4 - 10*b3 + 96*b1 - 52) * q^97 + (-3*b13 - b12 - 22*b10 - 24*b8 + 82*b7 - 187*b5 - 3*b4 + b3 - 207*b1 + 187) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$14 q - 4 q^{3} + 8 q^{5} - 10 q^{7}+O(q^{10})$$ 14 * q - 4 * q^3 + 8 * q^5 - 10 * q^7 $$14 q - 4 q^{3} + 8 q^{5} - 10 q^{7} + 16 q^{11} + 72 q^{13} - 40 q^{17} - 60 q^{21} + 246 q^{23} + 572 q^{27} + 260 q^{29} + 566 q^{31} - 904 q^{33} - 804 q^{35} - 28 q^{37} + 476 q^{39} - 786 q^{41} - 604 q^{45} + 448 q^{47} - 380 q^{51} + 1612 q^{55} + 1004 q^{57} + 660 q^{61} - 2250 q^{63} - 796 q^{65} - 284 q^{67} - 2532 q^{69} + 326 q^{71} + 730 q^{73} - 864 q^{75} + 342 q^{79} - 950 q^{81} + 4148 q^{85} - 4516 q^{89} - 3112 q^{91} + 3356 q^{95} - 1194 q^{97} + 2876 q^{99}+O(q^{100})$$ 14 * q - 4 * q^3 + 8 * q^5 - 10 * q^7 + 16 * q^11 + 72 * q^13 - 40 * q^17 - 60 * q^21 + 246 * q^23 + 572 * q^27 + 260 * q^29 + 566 * q^31 - 904 * q^33 - 804 * q^35 - 28 * q^37 + 476 * q^39 - 786 * q^41 - 604 * q^45 + 448 * q^47 - 380 * q^51 + 1612 * q^55 + 1004 * q^57 + 660 * q^61 - 2250 * q^63 - 796 * q^65 - 284 * q^67 - 2532 * q^69 + 326 * q^71 + 730 * q^73 - 864 * q^75 + 342 * q^79 - 950 * q^81 + 4148 * q^85 - 4516 * q^89 - 3112 * q^91 + 3356 * q^95 - 1194 * q^97 + 2876 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{14} + 142x^{12} + 7785x^{10} + 208489x^{8} + 2822686x^{6} + 17977041x^{4} + 45020416x^{2} + 31719424$$ :

 $$\beta_{1}$$ $$=$$ $$( - 95 \nu^{12} - 10342 \nu^{10} - 385555 \nu^{8} - 5793399 \nu^{6} - 30510566 \nu^{4} - 44912651 \nu^{2} - 49770288 \nu - 72523264 ) / 49770288$$ (-95*v^12 - 10342*v^10 - 385555*v^8 - 5793399*v^6 - 30510566*v^4 - 44912651*v^2 - 49770288*v - 72523264) / 49770288 $$\beta_{2}$$ $$=$$ $$( - 95 \nu^{12} - 10342 \nu^{10} - 385555 \nu^{8} - 5793399 \nu^{6} - 30510566 \nu^{4} - 44912651 \nu^{2} + 49770288 \nu - 72523264 ) / 49770288$$ (-95*v^12 - 10342*v^10 - 385555*v^8 - 5793399*v^6 - 30510566*v^4 - 44912651*v^2 + 49770288*v - 72523264) / 49770288 $$\beta_{3}$$ $$=$$ $$( 95 \nu^{12} + 10342 \nu^{10} + 385555 \nu^{8} + 5793399 \nu^{6} + 30510566 \nu^{4} + 94682939 \nu^{2} + 1067929024 ) / 24885144$$ (95*v^12 + 10342*v^10 + 385555*v^8 + 5793399*v^6 + 30510566*v^4 + 94682939*v^2 + 1067929024) / 24885144 $$\beta_{4}$$ $$=$$ $$( 1007 \nu^{12} + 137938 \nu^{10} + 7222375 \nu^{8} + 181591239 \nu^{6} + 2215243778 \nu^{4} + 11287190879 \nu^{2} + 14732140048 ) / 49770288$$ (1007*v^12 + 137938*v^10 + 7222375*v^8 + 181591239*v^6 + 2215243778*v^4 + 11287190879*v^2 + 14732140048) / 49770288 $$\beta_{5}$$ $$=$$ $$( - 12877 \nu^{13} - 1293494 \nu^{11} - 42001301 \nu^{9} - 513267093 \nu^{7} - 3719304454 \nu^{5} - 59654849245 \nu^{3} + \cdots - 326779846400 \nu ) / 280306262016$$ (-12877*v^13 - 1293494*v^11 - 42001301*v^9 - 513267093*v^7 - 3719304454*v^5 - 59654849245*v^3 - 326779846400*v) / 280306262016 $$\beta_{6}$$ $$=$$ $$( 37343 \nu^{13} - 735680 \nu^{12} + 4283314 \nu^{11} - 90690688 \nu^{10} + 189294199 \nu^{9} - 4086250432 \nu^{8} + 4496161911 \nu^{7} + \cdots - 2300930351104 ) / 140153131008$$ (37343*v^13 - 735680*v^12 + 4283314*v^11 - 90690688*v^10 + 189294199*v^9 - 4086250432*v^8 + 4496161911*v^7 - 83167854528*v^6 + 64252033058*v^5 - 755514286208*v^4 + 448012527887*v^3 - 2593135354304*v^2 + 635731943680*v - 2300930351104) / 140153131008 $$\beta_{7}$$ $$=$$ $$( - 37343 \nu^{13} - 735680 \nu^{12} - 4283314 \nu^{11} - 90690688 \nu^{10} - 189294199 \nu^{9} - 4086250432 \nu^{8} - 4496161911 \nu^{7} + \cdots - 2300930351104 ) / 140153131008$$ (-37343*v^13 - 735680*v^12 - 4283314*v^11 - 90690688*v^10 - 189294199*v^9 - 4086250432*v^8 - 4496161911*v^7 - 83167854528*v^6 - 64252033058*v^5 - 755514286208*v^4 - 448012527887*v^3 - 2593135354304*v^2 - 635731943680*v - 2300930351104) / 140153131008 $$\beta_{8}$$ $$=$$ $$( - 92817 \nu^{13} + 461824 \nu^{12} - 13906542 \nu^{11} + 48728064 \nu^{10} - 800065401 \nu^{9} + 1710021632 \nu^{8} - 21974038905 \nu^{7} + \cdots - 1927881990144 ) / 93435420672$$ (-92817*v^13 + 461824*v^12 - 13906542*v^11 + 48728064*v^10 - 800065401*v^9 + 1710021632*v^8 - 21974038905*v^7 + 21967784960*v^6 - 287930684670*v^5 + 35986655232*v^4 - 1542482092641*v^3 - 835378734080*v^2 - 2314546490112*v - 1927881990144) / 93435420672 $$\beta_{9}$$ $$=$$ $$( 92817 \nu^{13} + 461824 \nu^{12} + 13906542 \nu^{11} + 48728064 \nu^{10} + 800065401 \nu^{9} + 1710021632 \nu^{8} + 21974038905 \nu^{7} + \cdots - 1927881990144 ) / 93435420672$$ (92817*v^13 + 461824*v^12 + 13906542*v^11 + 48728064*v^10 + 800065401*v^9 + 1710021632*v^8 + 21974038905*v^7 + 21967784960*v^6 + 287930684670*v^5 + 35986655232*v^4 + 1542482092641*v^3 - 835378734080*v^2 + 2314546490112*v - 1927881990144) / 93435420672 $$\beta_{10}$$ $$=$$ $$( 42065 \nu^{13} - 219264 \nu^{12} + 4674286 \nu^{11} - 29660928 \nu^{10} + 176604985 \nu^{9} - 1512193152 \nu^{8} + 2499780921 \nu^{7} + \cdots - 1040149929984 ) / 25482387456$$ (42065*v^13 - 219264*v^12 + 4674286*v^11 - 29660928*v^10 + 176604985*v^9 - 1512193152*v^8 + 2499780921*v^7 - 35924307072*v^6 + 4525404542*v^5 - 387336232704*v^4 - 107924826079*v^3 - 1472992448640*v^2 + 46064263936*v - 1040149929984) / 25482387456 $$\beta_{11}$$ $$=$$ $$( 42065 \nu^{13} + 219264 \nu^{12} + 4674286 \nu^{11} + 29660928 \nu^{10} + 176604985 \nu^{9} + 1512193152 \nu^{8} + 2499780921 \nu^{7} + \cdots + 1040149929984 ) / 25482387456$$ (42065*v^13 + 219264*v^12 + 4674286*v^11 + 29660928*v^10 + 176604985*v^9 + 1512193152*v^8 + 2499780921*v^7 + 35924307072*v^6 + 4525404542*v^5 + 387336232704*v^4 - 107924826079*v^3 + 1472992448640*v^2 + 46064263936*v + 1040149929984) / 25482387456 $$\beta_{12}$$ $$=$$ $$( 23125 \nu^{13} + 2698022 \nu^{11} + 110951645 \nu^{9} + 1863590109 \nu^{7} + 8120784886 \nu^{5} - 78345744539 \nu^{3} + \cdots - 533954347264 \nu ) / 11679427584$$ (23125*v^13 + 2698022*v^11 + 110951645*v^9 + 1863590109*v^7 + 8120784886*v^5 - 78345744539*v^3 - 533954347264*v) / 11679427584 $$\beta_{13}$$ $$=$$ $$( 1072661 \nu^{13} + 152053158 \nu^{11} + 8248400029 \nu^{9} + 214633695901 \nu^{7} + 2716254447222 \nu^{5} + 14628546330917 \nu^{3} + \cdots + 21378896606976 \nu ) / 93435420672$$ (1072661*v^13 + 152053158*v^11 + 8248400029*v^9 + 214633695901*v^7 + 2716254447222*v^5 + 14628546330917*v^3 + 21378896606976*v) / 93435420672
 $$\nu$$ $$=$$ $$( \beta_{2} - \beta_1 ) / 2$$ (b2 - b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + \beta_{2} + \beta _1 - 40 ) / 2$$ (b3 + b2 + b1 - 40) / 2 $$\nu^{3}$$ $$=$$ $$( -4\beta_{12} + 3\beta_{11} + 3\beta_{10} - 3\beta_{7} + 3\beta_{6} + 78\beta_{5} - 65\beta_{2} + 65\beta_1 ) / 4$$ (-4*b12 + 3*b11 + 3*b10 - 3*b7 + 3*b6 + 78*b5 - 65*b2 + 65*b1) / 4 $$\nu^{4}$$ $$=$$ $$( 6 \beta_{11} - 6 \beta_{10} - 24 \beta_{9} - 24 \beta_{8} + 12 \beta_{7} + 12 \beta_{6} - 3 \beta_{4} - 75 \beta_{3} - 159 \beta_{2} - 159 \beta _1 + 2557 ) / 4$$ (6*b11 - 6*b10 - 24*b9 - 24*b8 + 12*b7 + 12*b6 - 3*b4 - 75*b3 - 159*b2 - 159*b1 + 2557) / 4 $$\nu^{5}$$ $$=$$ $$( - 15 \beta_{13} + 213 \beta_{12} - 168 \beta_{11} - 168 \beta_{10} + 84 \beta_{9} - 84 \beta_{8} + 270 \beta_{7} - 270 \beta_{6} - 6141 \beta_{5} + 2453 \beta_{2} - 2453 \beta_1 ) / 4$$ (-15*b13 + 213*b12 - 168*b11 - 168*b10 + 84*b9 - 84*b8 + 270*b7 - 270*b6 - 6141*b5 + 2453*b2 - 2453*b1) / 4 $$\nu^{6}$$ $$=$$ $$( - 597 \beta_{11} + 597 \beta_{10} + 1704 \beta_{9} + 1704 \beta_{8} - 1095 \beta_{7} - 1095 \beta_{6} + 285 \beta_{4} + 2801 \beta_{3} + 9044 \beta_{2} + 9044 \beta _1 - 95105 ) / 4$$ (-597*b11 + 597*b10 + 1704*b9 + 1704*b8 - 1095*b7 - 1095*b6 + 285*b4 + 2801*b3 + 9044*b2 + 9044*b1 - 95105) / 4 $$\nu^{7}$$ $$=$$ $$( 690 \beta_{13} - 5276 \beta_{12} + 4125 \beta_{11} + 4125 \beta_{10} - 3702 \beta_{9} + 3702 \beta_{8} - 7917 \beta_{7} + 7917 \beta_{6} + 173220 \beta_{5} - 50236 \beta_{2} + 50236 \beta_1 ) / 2$$ (690*b13 - 5276*b12 + 4125*b11 + 4125*b10 - 3702*b9 + 3702*b8 - 7917*b7 + 7917*b6 + 173220*b5 - 50236*b2 + 50236*b1) / 2 $$\nu^{8}$$ $$=$$ $$( 19455 \beta_{11} - 19455 \beta_{10} - 46698 \beta_{9} - 46698 \beta_{8} + 36309 \beta_{7} + 36309 \beta_{6} - 8607 \beta_{4} - 55578 \beta_{3} - 234267 \beta_{2} - 234267 \beta _1 + 1926935 ) / 2$$ (19455*b11 - 19455*b10 - 46698*b9 - 46698*b8 + 36309*b7 + 36309*b6 - 8607*b4 - 55578*b3 - 234267*b2 - 234267*b1 + 1926935) / 2 $$\nu^{9}$$ $$=$$ $$( - 89832 \beta_{13} + 513192 \beta_{12} - 395841 \beta_{11} - 395841 \beta_{10} + 471372 \beta_{9} - 471372 \beta_{8} + 814257 \beta_{7} - 814257 \beta_{6} - 17837898 \beta_{5} + \cdots - 4342823 \beta_1 ) / 4$$ (-89832*b13 + 513192*b12 - 395841*b11 - 395841*b10 + 471372*b9 - 471372*b8 + 814257*b7 - 814257*b6 - 17837898*b5 + 4342823*b2 - 4342823*b1) / 4 $$\nu^{10}$$ $$=$$ $$( - 2175864 \beta_{11} + 2175864 \beta_{10} + 4713684 \beta_{9} + 4713684 \beta_{8} - 4195866 \beta_{7} - 4195866 \beta_{6} + 904089 \beta_{4} + 4668077 \beta_{3} + \cdots - 165378299 ) / 4$$ (-2175864*b11 + 2175864*b10 + 4713684*b9 + 4713684*b8 - 4195866*b7 - 4195866*b6 + 904089*b4 + 4668077*b3 + 23395775*b2 + 23395775*b1 - 165378299) / 4 $$\nu^{11}$$ $$=$$ $$( 5099955 \beta_{13} - 24777805 \beta_{12} + 18901530 \beta_{11} + 18901530 \beta_{10} - 26372424 \beta_{9} + 26372424 \beta_{8} - 39876168 \beta_{7} + 39876168 \beta_{6} + \cdots + 194596091 \beta_1 ) / 4$$ (5099955*b13 - 24777805*b12 + 18901530*b11 + 18901530*b10 - 26372424*b9 + 26372424*b8 - 39876168*b7 + 39876168*b6 + 886965669*b5 - 194596091*b2 + 194596091*b1) / 4 $$\nu^{12}$$ $$=$$ $$( 113436171 \beta_{11} - 113436171 \beta_{10} - 230308668 \beta_{9} - 230308668 \beta_{8} + 224979561 \beta_{7} + 224979561 \beta_{6} - 44976123 \beta_{4} + \cdots + 7376127727 ) / 4$$ (113436171*b11 - 113436171*b10 - 230308668*b9 - 230308668*b8 + 224979561*b7 + 224979561*b6 - 44976123*b4 - 204729939*b3 - 1147864896*b2 - 1147864896*b1 + 7376127727) / 4 $$\nu^{13}$$ $$=$$ $$( - 134977842 \beta_{13} + 596319498 \beta_{12} - 450871848 \beta_{11} - 450871848 \beta_{10} + 691237980 \beta_{9} - 691237980 \beta_{8} + 958358088 \beta_{7} + \cdots - 4464323335 \beta_1 ) / 2$$ (-134977842*b13 + 596319498*b12 - 450871848*b11 - 450871848*b10 + 691237980*b9 - 691237980*b8 + 958358088*b7 - 958358088*b6 - 21698382942*b5 + 4464323335*b2 - 4464323335*b1) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/136\mathbb{Z}\right)^\times$$.

 $$n$$ $$69$$ $$103$$ $$105$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\beta_{5}$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
81.1
 6.91794i 5.35029i 1.84168i 1.06483i − 3.07347i − 4.53742i − 5.56386i − 6.91794i − 5.35029i − 1.84168i − 1.06483i 3.07347i 4.53742i 5.56386i
0 −6.91794 6.91794i 0 8.19992 + 8.19992i 0 −18.1123 + 18.1123i 0 68.7157i 0
81.2 0 −5.35029 5.35029i 0 −11.0962 11.0962i 0 19.2933 19.2933i 0 30.2512i 0
81.3 0 −1.84168 1.84168i 0 4.90158 + 4.90158i 0 6.14401 6.14401i 0 20.2164i 0
81.4 0 −1.06483 1.06483i 0 −0.550686 0.550686i 0 −11.3140 + 11.3140i 0 24.7323i 0
81.5 0 3.07347 + 3.07347i 0 −8.67626 8.67626i 0 9.02315 9.02315i 0 8.10760i 0
81.6 0 4.53742 + 4.53742i 0 14.5849 + 14.5849i 0 9.46047 9.46047i 0 14.1763i 0
81.7 0 5.56386 + 5.56386i 0 −3.36317 3.36317i 0 −19.4947 + 19.4947i 0 34.9130i 0
89.1 0 −6.91794 + 6.91794i 0 8.19992 8.19992i 0 −18.1123 18.1123i 0 68.7157i 0
89.2 0 −5.35029 + 5.35029i 0 −11.0962 + 11.0962i 0 19.2933 + 19.2933i 0 30.2512i 0
89.3 0 −1.84168 + 1.84168i 0 4.90158 4.90158i 0 6.14401 + 6.14401i 0 20.2164i 0
89.4 0 −1.06483 + 1.06483i 0 −0.550686 + 0.550686i 0 −11.3140 11.3140i 0 24.7323i 0
89.5 0 3.07347 3.07347i 0 −8.67626 + 8.67626i 0 9.02315 + 9.02315i 0 8.10760i 0
89.6 0 4.53742 4.53742i 0 14.5849 14.5849i 0 9.46047 + 9.46047i 0 14.1763i 0
89.7 0 5.56386 5.56386i 0 −3.36317 + 3.36317i 0 −19.4947 19.4947i 0 34.9130i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 89.7 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.c even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 136.4.k.a 14
4.b odd 2 1 272.4.o.g 14
17.c even 4 1 inner 136.4.k.a 14
17.d even 8 2 2312.4.a.m 14
68.f odd 4 1 272.4.o.g 14

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.4.k.a 14 1.a even 1 1 trivial
136.4.k.a 14 17.c even 4 1 inner
272.4.o.g 14 4.b odd 2 1
272.4.o.g 14 68.f odd 4 1
2312.4.a.m 14 17.d even 8 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{14} + 4 T_{3}^{13} + 8 T_{3}^{12} - 252 T_{3}^{11} + 8148 T_{3}^{10} + 14304 T_{3}^{9} + 23784 T_{3}^{8} - 796464 T_{3}^{7} + 16150080 T_{3}^{6} + 1753536 T_{3}^{5} - 12967392 T_{3}^{4} + \cdots + 4060086272$$ acting on $$S_{4}^{\mathrm{new}}(136, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{14}$$
$3$ $$T^{14} + 4 T^{13} + \cdots + 4060086272$$
$5$ $$T^{14} - 8 T^{13} + \cdots + 1398380605952$$
$7$ $$T^{14} + 10 T^{13} + \cdots + 20\!\cdots\!28$$
$11$ $$T^{14} - 16 T^{13} + \cdots + 17\!\cdots\!68$$
$13$ $$(T^{7} - 36 T^{6} - 8876 T^{5} + \cdots - 2332419584)^{2}$$
$17$ $$T^{14} + 40 T^{13} + \cdots + 69\!\cdots\!17$$
$19$ $$T^{14} + 67636 T^{12} + \cdots + 90\!\cdots\!04$$
$23$ $$T^{14} - 246 T^{13} + \cdots + 25\!\cdots\!00$$
$29$ $$T^{14} - 260 T^{13} + \cdots + 54\!\cdots\!68$$
$31$ $$T^{14} - 566 T^{13} + \cdots + 44\!\cdots\!72$$
$37$ $$T^{14} + 28 T^{13} + \cdots + 65\!\cdots\!08$$
$41$ $$T^{14} + 786 T^{13} + \cdots + 45\!\cdots\!48$$
$43$ $$T^{14} + 427000 T^{12} + \cdots + 15\!\cdots\!44$$
$47$ $$(T^{7} - 224 T^{6} + \cdots + 65\!\cdots\!32)^{2}$$
$53$ $$T^{14} + 524980 T^{12} + \cdots + 32\!\cdots\!36$$
$59$ $$T^{14} + 873928 T^{12} + \cdots + 20\!\cdots\!04$$
$61$ $$T^{14} - 660 T^{13} + \cdots + 39\!\cdots\!88$$
$67$ $$(T^{7} + 142 T^{6} + \cdots - 13\!\cdots\!24)^{2}$$
$71$ $$T^{14} - 326 T^{13} + \cdots + 43\!\cdots\!72$$
$73$ $$T^{14} - 730 T^{13} + \cdots + 31\!\cdots\!72$$
$79$ $$T^{14} - 342 T^{13} + \cdots + 10\!\cdots\!68$$
$83$ $$T^{14} + 4438952 T^{12} + \cdots + 46\!\cdots\!36$$
$89$ $$(T^{7} + 2258 T^{6} + \cdots + 98\!\cdots\!04)^{2}$$
$97$ $$T^{14} + 1194 T^{13} + \cdots + 53\!\cdots\!92$$