# Properties

 Label 210.3.h.a Level $210$ Weight $3$ Character orbit 210.h Analytic conductor $5.722$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$210 = 2 \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 210.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.72208555157$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 8 x^{15} + 96 x^{14} - 532 x^{13} + 3236 x^{12} - 12864 x^{11} + 49526 x^{10} - 141436 x^{9} + 362298 x^{8} - 722060 x^{7} + 1208164 x^{6} - 1570812 x^{5} + \cdots + 33750$$ x^16 - 8*x^15 + 96*x^14 - 532*x^13 + 3236*x^12 - 12864*x^11 + 49526*x^10 - 141436*x^9 + 362298*x^8 - 722060*x^7 + 1208164*x^6 - 1570812*x^5 + 1591101*x^4 - 1183860*x^3 + 619650*x^2 - 202500*x + 33750 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{12}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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.

 $$f(q)$$ $$=$$ $$q - \beta_{5} q^{2} - \beta_1 q^{3} - 2 q^{4} + \beta_{8} q^{5} - \beta_{4} q^{6} + ( - \beta_{6} + \beta_{5} - \beta_{2}) q^{7} + 2 \beta_{5} q^{8} + 3 q^{9}+O(q^{10})$$ q - b5 * q^2 - b1 * q^3 - 2 * q^4 + b8 * q^5 - b4 * q^6 + (-b6 + b5 - b2) * q^7 + 2*b5 * q^8 + 3 * q^9 $$q - \beta_{5} q^{2} - \beta_1 q^{3} - 2 q^{4} + \beta_{8} q^{5} - \beta_{4} q^{6} + ( - \beta_{6} + \beta_{5} - \beta_{2}) q^{7} + 2 \beta_{5} q^{8} + 3 q^{9} + (\beta_{11} + \beta_{6} - \beta_{5} - \beta_1) q^{10} + (\beta_{13} - \beta_{12} - \beta_{3} + 5) q^{11} + 2 \beta_1 q^{12} + (\beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} - \beta_{2} + 3 \beta_1) q^{13} + (\beta_{15} - \beta_{10} - \beta_{9} + \beta_{8} - \beta_{3} - \beta_1 + 1) q^{14} + (\beta_{13} + \beta_{5} - \beta_{3} - 2) q^{15} + 4 q^{16} + ( - \beta_{9} - \beta_{8} - 2 \beta_{2} + \beta_1) q^{17} - 3 \beta_{5} q^{18} + (2 \beta_{15} + 2 \beta_{11} - 2 \beta_{10} - 3 \beta_{9} + 3 \beta_{8} - \beta_{7} + \beta_{6} + \cdots - 4 \beta_1) q^{19}+ \cdots + (3 \beta_{13} - 3 \beta_{12} - 3 \beta_{3} + 15) q^{99}+O(q^{100})$$ q - b5 * q^2 - b1 * q^3 - 2 * q^4 + b8 * q^5 - b4 * q^6 + (-b6 + b5 - b2) * q^7 + 2*b5 * q^8 + 3 * q^9 + (b11 + b6 - b5 - b1) * q^10 + (b13 - b12 - b3 + 5) * q^11 + 2*b1 * q^12 + (b9 + b8 + b7 + b6 - b2 + 3*b1) * q^13 + (b15 - b10 - b9 + b8 - b3 - b1 + 1) * q^14 + (b13 + b5 - b3 - 2) * q^15 + 4 * q^16 + (-b9 - b8 - 2*b2 + b1) * q^17 - 3*b5 * q^18 + (2*b15 + 2*b11 - 2*b10 - 3*b9 + 3*b8 - b7 + b6 - 2*b5 - b4 - 4*b1) * q^19 - 2*b8 * q^20 + (-b14 + b13 - b12 - b11 - b9 + b8 + b5 + b4 - b1 + 1) * q^21 + (-b15 - b10 - 6*b5) * q^22 + (2*b15 + 3*b13 + 3*b12 + 2*b10 - 2*b7 + 2*b6 - 6*b5 - 2*b1) * q^23 + 2*b4 * q^24 + (b15 + b14 - 2*b13 + b7 - b6 - 3*b5 - 2*b3 + b1 + 2) * q^25 + (b15 + 2*b11 - b10 + 2*b9 - 2*b8 - b7 + b6 - 2*b5 + 4*b4 + b1) * q^26 - 3*b1 * q^27 + (2*b6 - 2*b5 + 2*b2) * q^28 + (2*b15 + 4*b14 - 2*b10 + 2*b7 - 2*b6 + 4*b3 + 2*b1 + 2) * q^29 + (-b15 - b14 - b7 + b6 + b5 - b3 - b1 + 2) * q^30 + (-2*b15 + 4*b11 + 2*b10 + 3*b9 - 3*b8 - 2*b7 + 2*b6 - 4*b5 - 9*b4 + b1) * q^31 - 4*b5 * q^32 + (2*b9 + 2*b8 - b7 - b6 - b2 - 6*b1) * q^33 + (2*b15 - 2*b11 - 2*b10 + b7 - b6 + 2*b5 + 4*b4 + b1) * q^34 + (2*b15 + 2*b14 + 5*b12 - 3*b11 + 2*b7 + b5 + 5*b4 + 2*b3 + 5*b1 + 1) * q^35 - 6 * q^36 + (-3*b15 - 4*b13 - 4*b12 - 3*b10 + 7*b7 - 7*b6 + 12*b5 + 7*b1) * q^37 + (-2*b9 - 2*b8 + 3*b7 + 3*b6 + 4*b2 + b1) * q^38 + (-2*b15 - 4*b14 + 2*b13 - 2*b12 + 2*b10 - 2*b7 + 2*b6 - 5*b3 - 2*b1 - 9) * q^39 + (-2*b11 - 2*b6 + 2*b5 + 2*b1) * q^40 + (-b15 + 2*b11 + b10 + b9 - b8 - b7 + b6 - 2*b5 + 2*b4) * q^41 + (-b15 - b13 - b12 - b10 + b9 + b8 + 3*b7 - b6 + b2 + b1) * q^42 + (-2*b15 - 2*b13 - 2*b12 - 2*b10 - 4*b7 + 4*b6 + 8*b5 - 4*b1) * q^43 + (-2*b13 + 2*b12 + 2*b3 - 10) * q^44 + 3*b8 * q^45 + (-3*b15 - 6*b14 + 4*b13 - 4*b12 + 3*b10 - 3*b7 + 3*b6 - 6*b3 - 3*b1 - 12) * q^46 + (-4*b9 - 4*b8 - 5*b7 - 5*b6 - 2*b2 + 13*b1) * q^47 - 4*b1 * q^48 + (-b15 - 2*b14 - 2*b13 + 2*b12 + b10 - 2*b9 + 2*b8 - b7 + b6 - 12*b4 + 4*b3 - 3*b1 + 17) * q^49 + (2*b15 + 2*b14 + 2*b13 - 3*b7 + 3*b6 - 5*b5 - 3*b1 - 8) * q^50 + (-2*b15 - 4*b14 + b13 - b12 + 2*b10 - 2*b7 + 2*b6 - 2*b1 - 2) * q^51 + (-2*b9 - 2*b8 - 2*b7 - 2*b6 + 2*b2 - 6*b1) * q^52 + (4*b15 - b13 - b12 + 4*b10 + 9*b5) * q^53 - 3*b4 * q^54 + (-3*b11 + 8*b8 + 5*b7 + 2*b6 + 3*b5 - 5*b4 + 5*b2 - 17*b1) * q^55 + (-2*b15 + 2*b10 + 2*b9 - 2*b8 + 2*b3 + 2*b1 - 2) * q^56 + (-3*b15 - b13 - b12 - 3*b10 + 3*b7 - 3*b6 + b5 + 3*b1) * q^57 + (4*b13 + 4*b12 - 2*b5) * q^58 + (-2*b15 - 8*b11 + 2*b10 + 6*b9 - 6*b8 + 4*b7 - 4*b6 + 8*b5 + 18*b4 + 10*b1) * q^59 + (-2*b13 - 2*b5 + 2*b3 + 4) * q^60 + (-6*b15 + 6*b10) * q^61 + (-4*b9 - 4*b8 - 3*b7 - 3*b6 - 4*b2 + 13*b1) * q^62 + (-3*b6 + 3*b5 - 3*b2) * q^63 - 8 * q^64 + (6*b15 + 6*b14 - 3*b13 + 5*b12 - 4*b7 + 4*b6 + 21*b5 + 4*b3 - 4*b1 + 24) * q^65 + (b15 + 4*b11 - b10 - 2*b9 + 2*b8 - 2*b7 + 2*b6 - 4*b5 - 8*b4 - 4*b1) * q^66 + (-2*b15 - 6*b13 - 6*b12 - 2*b10 + 9*b7 - 9*b6 - 10*b5 + 9*b1) * q^67 + (2*b9 + 2*b8 + 4*b2 - 2*b1) * q^68 + (-4*b15 - 4*b11 + 4*b10 - b9 + b8 + 2*b7 - 2*b6 + 4*b5 - 8*b4 + b1) * q^69 + (-5*b14 + 4*b13 + 5*b10 + 5*b9 + b8 - 5*b7 + 5*b6 - b5 + 5*b4 - 4*b3 - 10*b1 - 3) * q^70 + (4*b15 + 8*b14 - b13 + b12 - 4*b10 + 4*b7 - 4*b6 + 11*b3 + 4*b1 - 27) * q^71 + 6*b5 * q^72 + (7*b9 + 7*b8 - 7*b7 - 7*b6 - 7*b2 + 3*b1) * q^73 + (4*b15 + 8*b14 - 6*b13 + 6*b12 - 4*b10 + 4*b7 - 4*b6 + 4*b1 + 16) * q^74 + (-3*b11 + 5*b9 - b8 + 5*b7 + 2*b6 + 3*b5 + 5*b2 + 3*b1) * q^75 + (-4*b15 - 4*b11 + 4*b10 + 6*b9 - 6*b8 + 2*b7 - 2*b6 + 4*b5 + 2*b4 + 8*b1) * q^76 + (2*b15 + b13 + b12 + 2*b10 - 5*b9 - 5*b8 + 4*b7 - 8*b6 + 15*b5 - 9*b1) * q^77 + (-2*b15 - 4*b13 - 4*b12 - 2*b10 + b7 - b6 + 8*b5 + b1) * q^78 + (2*b15 + 4*b14 - 2*b10 + 2*b7 - 2*b6 + 8*b3 + 2*b1 - 40) * q^79 + 4*b8 * q^80 + 9 * q^81 + (-2*b9 - 2*b8 - b7 - b6 - 2*b2 - 7*b1) * q^82 + (-4*b9 - 4*b8 + 3*b7 + 3*b6 + 10*b2 + 5*b1) * q^83 + (2*b14 - 2*b13 + 2*b12 + 2*b11 + 2*b9 - 2*b8 - 2*b5 - 2*b4 + 2*b1 - 2) * q^84 + (5*b15 + 5*b14 + 10*b12 - 5*b7 + 5*b6 + 15*b5 + 10*b3 - 5*b1 - 25) * q^85 + (2*b15 + 4*b14 - 4*b13 + 4*b12 - 2*b10 + 2*b7 - 2*b6 + 16*b3 + 2*b1 + 28) * q^86 + (-6*b7 - 6*b6 - 4*b1) * q^87 + (2*b15 + 2*b10 + 12*b5) * q^88 + (-6*b15 + 12*b11 + 6*b10 + 7*b9 - 7*b8 - 6*b7 + 6*b6 - 12*b5 + b1) * q^89 + (3*b11 + 3*b6 - 3*b5 - 3*b1) * q^90 + (-6*b15 - 8*b14 - 4*b13 + 4*b12 - 2*b11 + 6*b10 + 9*b9 - 9*b8 - 3*b7 + 3*b6 + 2*b5 + 15*b4 - 6*b3 + 6*b1 + 22) * q^91 + (-4*b15 - 6*b13 - 6*b12 - 4*b10 + 4*b7 - 4*b6 + 12*b5 + 4*b1) * q^92 + (b13 + b12 - 6*b7 + 6*b6 - 25*b5 - 6*b1) * q^93 + (2*b15 - 8*b11 - 2*b10 - 10*b9 + 10*b8 + 4*b7 - 4*b6 + 8*b5 + 14*b4 - 6*b1) * q^94 + (2*b15 - 8*b14 + 8*b13 + 10*b10 - 13*b7 + 13*b6 - 34*b5 - 11*b3 - 13*b1 - 35) * q^95 - 4*b4 * q^96 + (-3*b9 - 3*b8 - 5*b7 - 5*b6 - 5*b2 + 33*b1) * q^97 + (2*b15 - 2*b13 - 2*b12 + 2*b10 + 6*b7 - 2*b6 - 11*b5 + 26*b1) * q^98 + (3*b13 - 3*b12 - 3*b3 + 15) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 32 q^{4} + 48 q^{9}+O(q^{10})$$ 16 * q - 32 * q^4 + 48 * q^9 $$16 q - 32 q^{4} + 48 q^{9} + 96 q^{11} + 16 q^{14} - 24 q^{15} + 64 q^{16} + 24 q^{21} + 24 q^{25} + 64 q^{29} + 24 q^{30} - 8 q^{35} - 96 q^{36} - 144 q^{39} - 192 q^{44} - 176 q^{46} + 224 q^{49} - 96 q^{50} - 48 q^{51} - 32 q^{56} + 48 q^{60} - 128 q^{64} + 368 q^{65} - 56 q^{70} - 384 q^{71} + 224 q^{74} - 608 q^{79} + 144 q^{81} - 48 q^{84} - 440 q^{85} + 416 q^{86} + 224 q^{91} - 560 q^{95} + 288 q^{99}+O(q^{100})$$ 16 * q - 32 * q^4 + 48 * q^9 + 96 * q^11 + 16 * q^14 - 24 * q^15 + 64 * q^16 + 24 * q^21 + 24 * q^25 + 64 * q^29 + 24 * q^30 - 8 * q^35 - 96 * q^36 - 144 * q^39 - 192 * q^44 - 176 * q^46 + 224 * q^49 - 96 * q^50 - 48 * q^51 - 32 * q^56 + 48 * q^60 - 128 * q^64 + 368 * q^65 - 56 * q^70 - 384 * q^71 + 224 * q^74 - 608 * q^79 + 144 * q^81 - 48 * q^84 - 440 * q^85 + 416 * q^86 + 224 * q^91 - 560 * q^95 + 288 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 8 x^{15} + 96 x^{14} - 532 x^{13} + 3236 x^{12} - 12864 x^{11} + 49526 x^{10} - 141436 x^{9} + 362298 x^{8} - 722060 x^{7} + 1208164 x^{6} - 1570812 x^{5} + \cdots + 33750$$ :

 $$\beta_{1}$$ $$=$$ $$( - 1304 \nu^{14} + 9128 \nu^{13} - 115446 \nu^{12} + 574012 \nu^{11} - 3591842 \nu^{10} + 12914984 \nu^{9} - 49995515 \nu^{8} + 128433344 \nu^{7} + \cdots - 99070875 ) / 560625$$ (-1304*v^14 + 9128*v^13 - 115446*v^12 + 574012*v^11 - 3591842*v^10 + 12914984*v^9 - 49995515*v^8 + 128433344*v^7 - 320844898*v^6 + 561338382*v^5 - 866468654*v^4 + 924341564*v^3 - 755470905*v^2 + 368877150*v - 99070875) / 560625 $$\beta_{2}$$ $$=$$ $$( - 19306 \nu^{14} + 135142 \nu^{13} - 1709079 \nu^{12} + 8497628 \nu^{11} - 53165468 \nu^{10} + 191153301 \nu^{9} - 739735340 \nu^{8} + \cdots - 1406219625 ) / 3027375$$ (-19306*v^14 + 135142*v^13 - 1709079*v^12 + 8497628*v^11 - 53165468*v^10 + 191153301*v^9 - 739735340*v^8 + 1899973946*v^7 - 4742613417*v^6 + 8293098988*v^5 - 12773014036*v^4 + 13602334971*v^3 - 11039578230*v^2 + 5354640900*v - 1406219625) / 3027375 $$\beta_{3}$$ $$=$$ $$( - 181 \nu^{14} + 1267 \nu^{13} - 16034 \nu^{12} + 79733 \nu^{11} - 499338 \nu^{10} + 1796001 \nu^{9} - 6961130 \nu^{8} + 17893811 \nu^{7} - 44790412 \nu^{6} + \cdots - 14271525 ) / 26325$$ (-181*v^14 + 1267*v^13 - 16034*v^12 + 79733*v^11 - 499338*v^10 + 1796001*v^9 - 6961130*v^8 + 17893811*v^7 - 44790412*v^6 + 78463393*v^5 - 121536126*v^4 + 129995121*v^3 - 106932015*v^2 + 52505910*v - 14271525) / 26325 $$\beta_{4}$$ $$=$$ $$( - 1984 \nu^{15} + 14880 \nu^{14} - 171958 \nu^{13} + 892047 \nu^{12} - 4997626 \nu^{11} + 18170922 \nu^{10} - 58997141 \nu^{9} + 142860075 \nu^{8} + \cdots - 575687250 ) / 85899825$$ (-1984*v^15 + 14880*v^14 - 171958*v^13 + 892047*v^12 - 4997626*v^11 + 18170922*v^10 - 58997141*v^9 + 142860075*v^8 - 233442698*v^7 + 259430001*v^6 + 304752040*v^5 - 1154379162*v^4 + 2711291745*v^3 - 2854944621*v^2 + 2020897980*v - 575687250) / 85899825 $$\beta_{5}$$ $$=$$ $$( 5163308 \nu^{15} - 38724810 \nu^{14} + 475369020 \nu^{13} - 2502572345 \nu^{12} + 15373400596 \nu^{11} - 58317525310 \nu^{10} + \cdots - 200352316500 ) / 1213801875$$ (5163308*v^15 - 38724810*v^14 + 475369020*v^13 - 2502572345*v^12 + 15373400596*v^11 - 58317525310*v^10 + 223952035489*v^9 - 608926339653*v^8 + 1529887949640*v^7 - 2869994011087*v^6 + 4569918940190*v^5 - 5417408634332*v^4 + 4875031263099*v^3 - 3000039760455*v^2 + 1143288079650*v - 200352316500) / 1213801875 $$\beta_{6}$$ $$=$$ $$( - 399728914 \nu^{15} + 3232898035 \nu^{14} - 38445891245 \nu^{13} + 214537975455 \nu^{12} - 1293560948733 \nu^{11} + \cdots + 33279515835750 ) / 83752329375$$ (-399728914*v^15 + 3232898035*v^14 - 38445891245*v^13 + 214537975455*v^12 - 1293560948733*v^11 + 5161735252520*v^10 - 19664320389392*v^9 + 56145887333099*v^8 - 141581224758025*v^7 + 279967659388431*v^6 - 454977456719385*v^5 + 575402543557786*v^4 - 544225821024222*v^3 + 368154613738290*v^2 - 155333941686450*v + 33279515835750) / 83752329375 $$\beta_{7}$$ $$=$$ $$( 399728914 \nu^{15} - 2568229811 \nu^{14} + 33793213677 \nu^{13} - 155694952029 \nu^{12} + 1000987616561 \nu^{11} + \cdots + 16014851920125 ) / 83752329375$$ (399728914*v^15 - 2568229811*v^14 + 33793213677*v^13 - 155694952029*v^12 + 1000987616561*v^11 - 3331119827068*v^10 + 13082276658338*v^9 - 30671013312184*v^8 + 76145779518861*v^7 - 116580706899693*v^6 + 169214780706043*v^5 - 134875076277512*v^4 + 74756402522688*v^3 + 14002064235165*v^2 - 30555343699200*v + 16014851920125) / 83752329375 $$\beta_{8}$$ $$=$$ $$( - 447494447 \nu^{15} + 3483298596 \nu^{14} - 42073363272 \nu^{13} + 228052000574 \nu^{12} - 1387056367957 \nu^{11} + \cdots + 27118230861375 ) / 83752329375$$ (-447494447*v^15 + 3483298596*v^14 - 42073363272*v^13 + 228052000574*v^12 - 1387056367957*v^11 + 5398872205078*v^10 - 20631957152977*v^9 + 57534505613712*v^8 - 144656309564376*v^7 + 279021438726355*v^6 - 448287901742858*v^5 + 550515081942494*v^4 - 507632737721937*v^3 + 330359822998215*v^2 - 133431969127950*v + 27118230861375) / 83752329375 $$\beta_{9}$$ $$=$$ $$( 447494447 \nu^{15} - 3034312245 \nu^{14} + 38930458815 \nu^{13} - 188284929725 \nu^{12} + 1189311700804 \nu^{11} + \cdots + 8909332197750 ) / 83752329375$$ (447494447*v^15 - 3034312245*v^14 + 38930458815*v^13 - 188284929725*v^12 + 1189311700804*v^11 - 4160752761880*v^10 + 16179113496331*v^9 - 40281239595702*v^8 + 100313526189915*v^7 - 168073717530868*v^6 + 253982897346575*v^5 - 249628471095518*v^4 + 185854130658771*v^3 - 65010730303320*v^2 + 2797068934350*v + 8909332197750) / 83752329375 $$\beta_{10}$$ $$=$$ $$( - 483208394 \nu^{15} + 3624062955 \nu^{14} - 44453885935 \nu^{13} + 233985303760 \nu^{12} - 1435876781253 \nu^{11} + \cdots + 16389579233250 ) / 83752329375$$ (-483208394*v^15 + 3624062955*v^14 - 44453885935*v^13 + 233985303760*v^12 - 1435876781253*v^11 + 5444406856130*v^10 - 20874093726577*v^9 + 56702007359379*v^8 - 142064690854595*v^7 + 265942990325966*v^6 - 421119834507795*v^5 + 496610692750426*v^4 - 440957572810407*v^3 + 267075981404040*v^2 - 98295840754200*v + 16389579233250) / 83752329375 $$\beta_{11}$$ $$=$$ $$( - 567713312 \nu^{15} + 4257849840 \nu^{14} - 52266459230 \nu^{13} + 275154595755 \nu^{12} - 1690195150544 \nu^{11} + \cdots + 21472387601625 ) / 83752329375$$ (-567713312*v^15 + 4257849840*v^14 - 52266459230*v^13 + 275154595755*v^12 - 1690195150544*v^11 + 6411443031015*v^10 - 24618050055196*v^9 + 66931017082167*v^8 - 168101562328660*v^7 + 315263632646343*v^6 - 501506314001410*v^5 + 593939025308073*v^4 - 532857064795086*v^3 + 326697174195345*v^2 - 123640459408350*v + 21472387601625) / 83752329375 $$\beta_{12}$$ $$=$$ $$( 34162567 \nu^{15} - 260736947 \nu^{14} + 3176839329 \nu^{13} - 16958406448 \nu^{12} + 103706772500 \nu^{11} - 398328867651 \nu^{10} + \cdots - 1667961659250 ) / 3641405625$$ (34162567*v^15 - 260736947*v^14 + 3176839329*v^13 - 16958406448*v^12 + 103706772500*v^11 - 398328867651*v^10 + 1526616776483*v^9 - 4202958874492*v^8 + 10569496137537*v^7 - 20109267310697*v^6 + 32195169162241*v^5 - 38877338622975*v^4 + 35479953139413*v^3 - 22492379099160*v^2 + 8834413324050*v - 1667961659250) / 3641405625 $$\beta_{13}$$ $$=$$ $$( 34162567 \nu^{15} - 251701558 \nu^{14} + 3113591606 \nu^{13} - 16157410637 \nu^{12} + 99723018033 \nu^{11} - 373359357129 \nu^{10} + \cdots - 947112736500 ) / 3641405625$$ (34162567*v^15 - 251701558*v^14 + 3113591606*v^13 - 16157410637*v^12 + 99723018033*v^11 - 373359357129*v^10 + 1436779569089*v^9 - 3854434623527*v^8 + 9673185200158*v^7 - 17863512405754*v^6 + 28258784429004*v^5 - 32775290830761*v^4 + 28949936815989*v^3 - 17118376991730*v^2 + 6194900930400*v - 947112736500) / 3641405625 $$\beta_{14}$$ $$=$$ $$( - 1650986060 \nu^{15} + 13415356753 \nu^{14} - 159200027621 \nu^{13} + 891510559422 \nu^{12} - 5368094554979 \nu^{11} + \cdots + 143830007146125 ) / 83752329375$$ (-1650986060*v^15 + 13415356753*v^14 - 159200027621*v^13 + 891510559422*v^12 - 5368094554979*v^11 + 21484837848419*v^10 - 81782053320718*v^9 + 234176501346290*v^8 - 590312762276383*v^7 + 1170962757989076*v^6 - 1903359133567699*v^5 + 2416998273055543*v^4 - 2288527543048278*v^3 + 1559133346137285*v^2 - 660049681351050*v + 143830007146125) / 83752329375 $$\beta_{15}$$ $$=$$ $$( 2019305898 \nu^{15} - 15144794235 \nu^{14} + 185881965095 \nu^{13} - 978536727220 \nu^{12} + 6009893091901 \nu^{11} + \cdots - 77444936604000 ) / 83752329375$$ (2019305898*v^15 - 15144794235*v^14 + 185881965095*v^13 - 978536727220*v^12 + 6009893091901*v^11 - 22795839307010*v^10 + 87512193219009*v^9 - 237899077894293*v^8 + 597373089091315*v^7 - 1120169176032722*v^6 + 1781788381868015*v^5 - 2110164042242842*v^4 + 1894947099433719*v^3 - 1163391720079530*v^2 + 442484852310900*v - 77444936604000) / 83752329375
 $$\nu$$ $$=$$ $$( \beta_{15} - \beta_{10} - 2\beta_{9} + 2\beta_{8} - 2\beta_{5} - 2\beta _1 + 2 ) / 4$$ (b15 - b10 - 2*b9 + 2*b8 - 2*b5 - 2*b1 + 2) / 4 $$\nu^{2}$$ $$=$$ $$( - \beta_{15} - 4 \beta_{14} + 2 \beta_{13} - 2 \beta_{12} + \beta_{10} - 2 \beta_{9} + 2 \beta_{8} + 4 \beta_{6} - 2 \beta_{5} - 6 \beta_{3} + 2 \beta_{2} - 4 \beta _1 - 32 ) / 4$$ (-b15 - 4*b14 + 2*b13 - 2*b12 + b10 - 2*b9 + 2*b8 + 4*b6 - 2*b5 - 6*b3 + 2*b2 - 4*b1 - 32) / 4 $$\nu^{3}$$ $$=$$ $$( - 19 \beta_{15} - 6 \beta_{14} - 3 \beta_{13} - 9 \beta_{12} - 2 \beta_{11} + 13 \beta_{10} + 32 \beta_{9} - 32 \beta_{8} + \beta_{7} + 5 \beta_{6} + 45 \beta_{5} + 19 \beta_{4} - 9 \beta_{3} + 3 \beta_{2} + 30 \beta _1 - 49 ) / 4$$ (-19*b15 - 6*b14 - 3*b13 - 9*b12 - 2*b11 + 13*b10 + 32*b9 - 32*b8 + b7 + 5*b6 + 45*b5 + 19*b4 - 9*b3 + 3*b2 + 30*b1 - 49) / 4 $$\nu^{4}$$ $$=$$ $$( \beta_{15} + 68 \beta_{14} - 36 \beta_{13} + 12 \beta_{12} - 4 \beta_{11} - 13 \beta_{10} + 70 \beta_{9} - 62 \beta_{8} - 24 \beta_{7} - 96 \beta_{6} + 92 \beta_{5} + 38 \beta_{4} + 112 \beta_{3} - 48 \beta_{2} + 42 \beta _1 + 498 ) / 4$$ (b15 + 68*b14 - 36*b13 + 12*b12 - 4*b11 - 13*b10 + 70*b9 - 62*b8 - 24*b7 - 96*b6 + 92*b5 + 38*b4 + 112*b3 - 48*b2 + 42*b1 + 498) / 4 $$\nu^{5}$$ $$=$$ $$( 344 \beta_{15} + 180 \beta_{14} + 85 \beta_{13} + 215 \beta_{12} - 4 \beta_{11} - 244 \beta_{10} - 566 \beta_{9} + 586 \beta_{8} - 13 \beta_{7} - 297 \beta_{6} - 989 \beta_{5} - 507 \beta_{4} + 295 \beta_{3} - 125 \beta_{2} + \cdots + 1327 ) / 4$$ (344*b15 + 180*b14 + 85*b13 + 215*b12 - 4*b11 - 244*b10 - 566*b9 + 586*b8 - 13*b7 - 297*b6 - 989*b5 - 507*b4 + 295*b3 - 125*b2 - 584*b1 + 1327) / 4 $$\nu^{6}$$ $$=$$ $$( 120 \beta_{15} - 605 \beta_{14} + 365 \beta_{13} + 115 \beta_{12} - \beta_{11} + 45 \beta_{10} - 931 \beta_{9} + 963 \beta_{8} + 489 \beta_{7} + 943 \beta_{6} - 1599 \beta_{5} - 808 \beta_{4} - 1040 \beta_{3} + 494 \beta_{2} + \cdots - 4337 ) / 2$$ (120*b15 - 605*b14 + 365*b13 + 115*b12 - b11 + 45*b10 - 931*b9 + 963*b8 + 489*b7 + 943*b6 - 1599*b5 - 808*b4 - 1040*b3 + 494*b2 - 49*b1 - 4337) / 2 $$\nu^{7}$$ $$=$$ $$( - 6370 \beta_{15} - 4872 \beta_{14} - 2107 \beta_{13} - 4319 \beta_{12} + 846 \beta_{11} + 5292 \beta_{10} + 10854 \beta_{9} - 10700 \beta_{8} + 879 \beta_{7} + 10237 \beta_{6} + 19715 \beta_{5} + \cdots - 35061 ) / 4$$ (-6370*b15 - 4872*b14 - 2107*b13 - 4319*b12 + 846*b11 + 5292*b10 + 10854*b9 - 10700*b8 + 879*b7 + 10237*b6 + 19715*b5 + 11633*b4 - 8323*b3 + 3899*b2 + 14498*b1 - 35061) / 4 $$\nu^{8}$$ $$=$$ $$( - 8803 \beta_{15} + 21908 \beta_{14} - 16804 \beta_{13} - 13436 \beta_{12} + 3384 \beta_{11} + 2923 \beta_{10} + 49054 \beta_{9} - 55150 \beta_{8} - 29010 \beta_{7} - 31574 \beta_{6} + 94000 \beta_{5} + \cdots + 154534 ) / 4$$ (-8803*b15 + 21908*b14 - 16804*b13 - 13436*b12 + 3384*b11 + 2923*b10 + 49054*b9 - 55150*b8 - 29010*b7 - 31574*b6 + 94000*b5 + 54162*b4 + 38712*b3 - 18376*b2 - 6676*b1 + 154534) / 4 $$\nu^{9}$$ $$=$$ $$( 122447 \beta_{15} + 128586 \beta_{14} + 47961 \beta_{13} + 76947 \beta_{12} - 27482 \beta_{11} - 118643 \beta_{10} - 212674 \beta_{9} + 184402 \beta_{8} - 45245 \beta_{7} - 295393 \beta_{6} + \cdots + 911441 ) / 4$$ (122447*b15 + 128586*b14 + 47961*b13 + 76947*b12 - 27482*b11 - 118643*b10 - 212674*b9 + 184402*b8 - 45245*b7 - 295393*b6 - 361575*b5 - 245591*b4 + 225399*b3 - 106617*b2 - 394764*b1 + 911441) / 4 $$\nu^{10}$$ $$=$$ $$( 254281 \beta_{15} - 381496 \beta_{14} + 415248 \beta_{13} + 442932 \beta_{12} - 162784 \beta_{11} - 188791 \beta_{10} - 1304300 \beta_{9} + 1489792 \beta_{8} + 762798 \beta_{7} + \cdots - 2653032 ) / 4$$ (254281*b15 - 381496*b14 + 415248*b13 + 442932*b12 - 162784*b11 - 188791*b10 - 1304300*b9 + 1489792*b8 + 762798*b7 + 389310*b6 - 2535724*b5 - 1645672*b4 - 689624*b3 + 312672*b2 + 112548*b1 - 2653032) / 4 $$\nu^{11}$$ $$=$$ $$( - 2442238 \beta_{15} - 3332538 \beta_{14} - 1012319 \beta_{13} - 1151557 \beta_{12} + 638966 \beta_{11} + 2625080 \beta_{10} + 4021312 \beta_{9} - 2740472 \beta_{8} + \cdots - 23347091 ) / 4$$ (-2442238*b15 - 3332538*b14 - 1012319*b13 - 1151557*b12 + 638966*b11 + 2625080*b10 + 4021312*b9 - 2740472*b8 + 1822547*b7 + 7762281*b6 + 5946325*b5 + 4632093*b4 - 5953937*b3 + 2741299*b2 + 10726000*b1 - 23347091) / 4 $$\nu^{12}$$ $$=$$ $$( - 3483639 \beta_{15} + 2928985 \beta_{14} - 5314483 \beta_{13} - 6157373 \beta_{12} + 2840117 \beta_{11} + 3619731 \beta_{10} + 17283722 \beta_{9} - 19271586 \beta_{8} + \cdots + 20089762 ) / 2$$ (-3483639*b15 + 2928985*b14 - 5314483*b13 - 6157373*b12 + 2840117*b11 + 3619731*b10 + 17283722*b9 - 19271586*b8 - 9291408*b7 - 83516*b6 + 32596899*b5 + 23412401*b4 + 5408710*b3 - 2290588*b2 + 1662038*b1 + 20089762) / 2 $$\nu^{13}$$ $$=$$ $$( 49916468 \beta_{15} + 84671340 \beta_{14} + 19129253 \beta_{13} + 10768693 \beta_{12} - 11306262 \beta_{11} - 56045370 \beta_{10} - 68231238 \beta_{9} + 25058836 \beta_{8} + \cdots + 587892255 ) / 4$$ (49916468*b15 + 84671340*b14 + 19129253*b13 + 10768693*b12 - 11306262*b11 - 56045370*b10 - 68231238*b9 + 25058836*b8 - 63490515*b7 - 191572397*b6 - 77428447*b5 - 70485751*b4 + 153439325*b3 - 68120299*b2 - 280857766*b1 + 587892255) / 4 $$\nu^{14}$$ $$=$$ $$( 189382655 \beta_{15} - 60250246 \beta_{14} + 274871156 \beta_{13} + 313288780 \beta_{12} - 170967462 \beta_{11} - 233773547 \beta_{10} - 901930838 \beta_{9} + \cdots - 403879466 ) / 4$$ (189382655*b15 - 60250246*b14 + 274871156*b13 + 313288780*b12 - 170967462*b11 - 233773547*b10 - 901930838*b9 + 961013426*b8 + 423040290*b7 - 215339516*b6 - 1622428532*b5 - 1262524236*b4 - 115088724*b3 + 42760214*b2 - 306867574*b1 - 403879466) / 4 $$\nu^{15}$$ $$=$$ $$( - 1023202897 \beta_{15} - 2101665624 \beta_{14} - 288089139 \beta_{13} + 136058607 \beta_{12} + 111268180 \beta_{11} + 1130711311 \beta_{10} + 868910138 \beta_{9} + \cdots - 14495944639 ) / 4$$ (-1023202897*b15 - 2101665624*b14 - 288089139*b13 + 136058607*b12 + 111268180*b11 + 1130711311*b10 + 868910138*b9 + 391402090*b8 + 2010114319*b7 + 4490400413*b6 + 292095141*b5 + 502005055*b4 - 3847971231*b3 + 1651106253*b2 + 7018115652*b1 - 14495944639) / 4

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/210\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$71$$ $$127$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
139.1
 0.5 − 1.83656i 0.5 + 3.55177i 0.5 − 0.442923i 0.5 − 4.10071i 0.5 + 2.68650i 0.5 − 0.971291i 0.5 − 4.96598i 0.5 + 0.422343i 0.5 + 1.83656i 0.5 − 3.55177i 0.5 + 0.442923i 0.5 + 4.10071i 0.5 − 2.68650i 0.5 + 0.971291i 0.5 + 4.96598i 0.5 − 0.422343i
1.41421i −1.73205 −2.00000 −4.91728 0.905717i 2.44949i 1.91369 + 6.73333i 2.82843i 3.00000 −1.28088 + 6.95409i
139.2 1.41421i −1.73205 −2.00000 1.38028 + 4.80571i 2.44949i −5.24961 4.63050i 2.82843i 3.00000 6.79630 1.95201i
139.3 1.41421i −1.73205 −2.00000 2.40341 4.38447i 2.44949i −6.94781 0.853218i 2.82843i 3.00000 −6.20058 3.39894i
139.4 1.41421i −1.73205 −2.00000 4.59769 1.96501i 2.44949i 6.81963 + 1.57881i 2.82843i 3.00000 −2.77894 6.50212i
139.5 1.41421i 1.73205 −2.00000 −4.59769 + 1.96501i 2.44949i −6.81963 + 1.57881i 2.82843i 3.00000 2.77894 + 6.50212i
139.6 1.41421i 1.73205 −2.00000 −2.40341 + 4.38447i 2.44949i 6.94781 0.853218i 2.82843i 3.00000 6.20058 + 3.39894i
139.7 1.41421i 1.73205 −2.00000 −1.38028 4.80571i 2.44949i 5.24961 4.63050i 2.82843i 3.00000 −6.79630 + 1.95201i
139.8 1.41421i 1.73205 −2.00000 4.91728 + 0.905717i 2.44949i −1.91369 + 6.73333i 2.82843i 3.00000 1.28088 6.95409i
139.9 1.41421i −1.73205 −2.00000 −4.91728 + 0.905717i 2.44949i 1.91369 6.73333i 2.82843i 3.00000 −1.28088 6.95409i
139.10 1.41421i −1.73205 −2.00000 1.38028 4.80571i 2.44949i −5.24961 + 4.63050i 2.82843i 3.00000 6.79630 + 1.95201i
139.11 1.41421i −1.73205 −2.00000 2.40341 + 4.38447i 2.44949i −6.94781 + 0.853218i 2.82843i 3.00000 −6.20058 + 3.39894i
139.12 1.41421i −1.73205 −2.00000 4.59769 + 1.96501i 2.44949i 6.81963 1.57881i 2.82843i 3.00000 −2.77894 + 6.50212i
139.13 1.41421i 1.73205 −2.00000 −4.59769 1.96501i 2.44949i −6.81963 1.57881i 2.82843i 3.00000 2.77894 6.50212i
139.14 1.41421i 1.73205 −2.00000 −2.40341 4.38447i 2.44949i 6.94781 + 0.853218i 2.82843i 3.00000 6.20058 3.39894i
139.15 1.41421i 1.73205 −2.00000 −1.38028 + 4.80571i 2.44949i 5.24961 + 4.63050i 2.82843i 3.00000 −6.79630 1.95201i
139.16 1.41421i 1.73205 −2.00000 4.91728 0.905717i 2.44949i −1.91369 6.73333i 2.82843i 3.00000 1.28088 + 6.95409i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 139.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.b odd 2 1 inner
35.c odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 210.3.h.a 16
3.b odd 2 1 630.3.h.e 16
4.b odd 2 1 1680.3.bd.a 16
5.b even 2 1 inner 210.3.h.a 16
5.c odd 4 2 1050.3.f.e 16
7.b odd 2 1 inner 210.3.h.a 16
15.d odd 2 1 630.3.h.e 16
20.d odd 2 1 1680.3.bd.a 16
21.c even 2 1 630.3.h.e 16
28.d even 2 1 1680.3.bd.a 16
35.c odd 2 1 inner 210.3.h.a 16
35.f even 4 2 1050.3.f.e 16
105.g even 2 1 630.3.h.e 16
140.c even 2 1 1680.3.bd.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.h.a 16 1.a even 1 1 trivial
210.3.h.a 16 5.b even 2 1 inner
210.3.h.a 16 7.b odd 2 1 inner
210.3.h.a 16 35.c odd 2 1 inner
630.3.h.e 16 3.b odd 2 1
630.3.h.e 16 15.d odd 2 1
630.3.h.e 16 21.c even 2 1
630.3.h.e 16 105.g even 2 1
1050.3.f.e 16 5.c odd 4 2
1050.3.f.e 16 35.f even 4 2
1680.3.bd.a 16 4.b odd 2 1
1680.3.bd.a 16 20.d odd 2 1
1680.3.bd.a 16 28.d even 2 1
1680.3.bd.a 16 140.c even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(210, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 2)^{8}$$
$3$ $$(T^{2} - 3)^{8}$$
$5$ $$T^{16} - 12 T^{14} + \cdots + 152587890625$$
$7$ $$T^{16} - 112 T^{14} + \cdots + 33232930569601$$
$11$ $$(T^{4} - 24 T^{3} + 28 T^{2} + 1776 T - 4844)^{4}$$
$13$ $$(T^{8} - 1044 T^{6} + 313732 T^{4} + \cdots + 586802176)^{2}$$
$17$ $$(T^{8} - 996 T^{6} + 270436 T^{4} + \cdots + 338560000)^{2}$$
$19$ $$(T^{8} + 1796 T^{6} + \cdots + 2149991424)^{2}$$
$23$ $$(T^{8} + 2932 T^{6} + \cdots + 11186869824)^{2}$$
$29$ $$(T^{4} - 16 T^{3} - 2088 T^{2} + \cdots + 19600)^{4}$$
$31$ $$(T^{8} + 3740 T^{6} + 1609684 T^{4} + \cdots + 747256896)^{2}$$
$37$ $$(T^{8} + 8536 T^{6} + \cdots + 3617786594304)^{2}$$
$41$ $$(T^{8} + 932 T^{6} + 280612 T^{4} + \cdots + 1141899264)^{2}$$
$43$ $$(T^{8} + 12880 T^{6} + \cdots + 9151544623104)^{2}$$
$47$ $$(T^{8} - 10904 T^{6} + \cdots + 14545741676544)^{2}$$
$53$ $$(T^{8} + 15604 T^{6} + \cdots + 38389325511744)^{2}$$
$59$ $$(T^{8} + 16112 T^{6} + \cdots + 203235065856)^{2}$$
$61$ $$(T^{8} + 13824 T^{6} + \cdots + 72666906624)^{2}$$
$67$ $$(T^{8} + 17224 T^{6} + \cdots + 404129746944)^{2}$$
$71$ $$(T^{4} + 96 T^{3} - 6116 T^{2} + \cdots - 18715436)^{4}$$
$73$ $$(T^{8} - 24020 T^{6} + \cdots + 105841627140096)^{2}$$
$79$ $$(T^{4} + 152 T^{3} + 3952 T^{2} + \cdots - 12612096)^{4}$$
$83$ $$(T^{8} - 15480 T^{6} + \cdots + 13129665286144)^{2}$$
$89$ $$(T^{8} + 24860 T^{6} + \cdots + 135150669186624)^{2}$$
$97$ $$(T^{8} - 20468 T^{6} + \cdots + 5898136817664)^{2}$$