# Properties

 Label 630.3.h.e Level $630$ Weight $3$ Character orbit 630.h Analytic conductor $17.166$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$630 = 2 \cdot 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 630.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$17.1662566547$$ 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: $$2^{14}$$ Twist minimal: no (minimal twist has level 210) 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.

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

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

 $$\beta_{1}$$ $$=$$ $$( - 61322 \nu^{14} + 429254 \nu^{13} - 5428353 \nu^{12} + 26989816 \nu^{11} - 168847606 \nu^{10} + 607061937 \nu^{9} - 2348797795 \nu^{8} + \cdots - 4356184500 ) / 15136875$$ (-61322*v^14 + 429254*v^13 - 5428353*v^12 + 26989816*v^11 - 168847606*v^10 + 607061937*v^9 - 2348797795*v^8 + 6032169442*v^7 - 15050254839*v^6 + 26309358626*v^5 - 40470416522*v^4 + 43054452627*v^3 - 34800176715*v^2 + 16813521450*v - 4356184500) / 15136875 $$\beta_{2}$$ $$=$$ $$( 2608 \nu^{14} - 18256 \nu^{13} + 230892 \nu^{12} - 1148024 \nu^{11} + 7183684 \nu^{10} - 25829968 \nu^{9} + 99991030 \nu^{8} - 256866688 \nu^{7} + \cdots + 198141750 ) / 560625$$ (2608*v^14 - 18256*v^13 + 230892*v^12 - 1148024*v^11 + 7183684*v^10 - 25829968*v^9 + 99991030*v^8 - 256866688*v^7 + 641689796*v^6 - 1122676764*v^5 + 1732937308*v^4 - 1848683128*v^3 + 1510941810*v^2 - 737754300*v + 198141750) / 560625 $$\beta_{3}$$ $$=$$ $$( - 181 \nu^{14} + 1267 \nu^{13} - 16034 \nu^{12} + 79733 \nu^{11} - 499338 \nu^{10} + 1796001 \nu^{9} - 6961130 \nu^{8} + 17893811 \nu^{7} - 44790412 \nu^{6} + \cdots - 14271525 ) / 26325$$ (-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 $$\beta_{6}$$ $$=$$ $$( 399728914 \nu^{15} - 3427703899 \nu^{14} + 39809532293 \nu^{13} - 231784568841 \nu^{12} + 1379313175425 \nu^{11} + \cdots - 48079812922875 ) / 83752329375$$ (399728914*v^15 - 3427703899*v^14 + 39809532293*v^13 - 231784568841*v^12 + 1379313175425*v^11 - 5698324120742*v^10 + 21593702764136*v^9 - 63614767314464*v^8 + 160768010451529*v^7 - 327898999545549*v^6 + 538836358944747*v^5 - 704845162247500*v^4 + 682314131611746*v^3 - 481015167707145*v^2 + 210440868002100*v - 48079812922875) / 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 $$\beta_{9}$$ $$=$$ $$( - 447494447 \nu^{15} + 3229118109 \nu^{14} - 40294099863 \nu^{13} + 205531523111 \nu^{12} - 1275063927496 \nu^{11} + \cdots + 5890964889375 ) / 83752329375$$ (-447494447*v^15 + 3229118109*v^14 - 40294099863*v^13 + 205531523111*v^12 - 1275063927496*v^11 + 4697341630102*v^10 - 18108495871075*v^9 + 47750119577067*v^8 - 119500311883419*v^7 + 216005057687986*v^6 - 337841799571937*v^5 + 379071089785232*v^4 - 323942441246295*v^3 + 177871284272175*v^2 - 57903995250000*v + 5890964889375) / 83752329375 $$\beta_{10}$$ $$=$$ $$( - 567713312 \nu^{15} + 4257849840 \nu^{14} - 52266459230 \nu^{13} + 275154595755 \nu^{12} - 1690195150544 \nu^{11} + \cdots + 21472387601625 ) / 83752329375$$ (-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 $$\beta_{11}$$ $$=$$ $$( - 785739041 \nu^{15} + 6191755645 \nu^{14} - 74430945615 \nu^{13} + 407289941690 \nu^{12} - 2471007994192 \nu^{11} + \cdots + 53163415249875 ) / 83752329375$$ (-785739041*v^15 + 6191755645*v^14 - 74430945615*v^13 + 407289941690*v^12 - 2471007994192*v^11 + 9698152824195*v^10 - 37041568233853*v^9 + 104136934094681*v^8 - 262285196856855*v^7 + 510444488303149*v^6 - 824347792956905*v^5 + 1023621407018139*v^4 - 954127232794023*v^3 + 630185273249535*v^2 - 258298432768800*v + 53163415249875) / 83752329375 $$\beta_{12}$$ $$=$$ $$( - 785739041 \nu^{15} + 5983941698 \nu^{14} - 72976247986 \nu^{13} + 388867038037 \nu^{12} - 2379381641451 \nu^{11} + \cdots + 36667642356000 ) / 83752329375$$ (-785739041*v^15 + 5983941698*v^14 - 72976247986*v^13 + 388867038037*v^12 - 2379381641451*v^11 + 9123854082189*v^10 - 34975312463791*v^9 + 96120876322486*v^8 - 241670045297138*v^7 + 458792125489460*v^6 - 733810944092454*v^5 + 883274307797217*v^4 - 803936857355271*v^3 + 506583224778645*v^2 - 197589647714850*v + 36667642356000) / 83752329375 $$\beta_{13}$$ $$=$$ $$( 1154058879 \nu^{15} - 8813843083 \nu^{14} + 107361432781 \nu^{13} - 573407815907 \nu^{12} + 3506125623632 \nu^{11} + \cdots - 56954262763125 ) / 83752329375$$ (1154058879*v^15 - 8813843083*v^14 + 107361432781*v^13 - 573407815907*v^12 + 3506125623632*v^11 - 13473032026674*v^10 + 51634595604495*v^9 - 142225168653269*v^8 + 357702043010303*v^7 - 680969930375392*v^6 + 1090651975806169*v^5 - 1318392265732044*v^4 + 1204683444700185*v^3 - 765704845680075*v^2 + 302197871360250*v - 56954262763125) / 83752329375 $$\beta_{14}$$ $$=$$ $$( 1154058879 \nu^{15} - 8497040102 \nu^{14} + 105143811914 \nu^{13} - 545327879638 \nu^{12} + 3366475077289 \nu^{11} + \cdots - 32182844707125 ) / 83752329375$$ (1154058879*v^15 - 8497040102*v^14 + 105143811914*v^13 - 545327879638*v^12 + 3366475077289*v^11 - 12597785646336*v^10 + 48485640413619*v^9 - 130006171709434*v^8 + 326274236023012*v^7 - 602150482364495*v^6 + 952406364501896*v^5 - 1103451322671288*v^4 + 974123016762939*v^3 - 574645179613905*v^2 + 207779843745900*v - 32182844707125) / 83752329375 $$\beta_{15}$$ $$=$$ $$( 1607525398 \nu^{15} - 12056440485 \nu^{14} + 147968387895 \nu^{13} - 778938507295 \nu^{12} + 4783649577701 \nu^{11} + \cdots - 60825320086500 ) / 83752329375$$ (1607525398*v^15 - 12056440485*v^14 + 147968387895*v^13 - 778938507295*v^12 + 4783649577701*v^11 - 18144032327960*v^10 + 69645833921534*v^9 - 189316460062893*v^8 + 475281158498115*v^7 - 891085669944347*v^6 + 1416778515061015*v^5 - 1677188563265542*v^4 + 1504329493275894*v^3 - 922236594213180*v^2 + 349444728687150*v - 60825320086500) / 83752329375
 $$\nu$$ $$=$$ $$( \beta_{15} + \beta_{9} - \beta_{8} - 2\beta_{5} + 2 ) / 4$$ (b15 + b9 - b8 - 2*b5 + 2) / 4 $$\nu^{2}$$ $$=$$ $$( \beta_{15} - 2 \beta_{14} + 2 \beta_{13} + \beta_{9} - \beta_{8} - 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{3} - 4 \beta_{2} + 2 \beta _1 - 32 ) / 4$$ (b15 - 2*b14 + 2*b13 + b9 - b8 - 2*b7 - 2*b6 - 2*b5 - 2*b3 - 4*b2 + 2*b1 - 32) / 4 $$\nu^{3}$$ $$=$$ $$( - 13 \beta_{15} - 6 \beta_{14} + 3 \beta_{12} + 3 \beta_{11} - 2 \beta_{10} - 19 \beta_{9} + 19 \beta_{8} - \beta_{7} - 5 \beta_{6} + 45 \beta_{5} - 19 \beta_{4} - 3 \beta_{3} - 9 \beta_{2} + 3 \beta _1 - 52 ) / 4$$ (-13*b15 - 6*b14 + 3*b12 + 3*b11 - 2*b10 - 19*b9 + 19*b8 - b7 - 5*b6 + 45*b5 - 19*b4 - 3*b3 - 9*b2 + 3*b1 - 52) / 4 $$\nu^{4}$$ $$=$$ $$( - 27 \beta_{15} + 28 \beta_{14} - 40 \beta_{13} - 4 \beta_{12} + 16 \beta_{11} - 4 \beta_{10} - 43 \beta_{9} + 35 \beta_{8} + 64 \beta_{7} + 56 \beta_{6} + 92 \beta_{5} - 38 \beta_{4} + 44 \beta_{3} + 114 \beta_{2} - 48 \beta _1 + 502 ) / 4$$ (-27*b15 + 28*b14 - 40*b13 - 4*b12 + 16*b11 - 4*b10 - 43*b9 + 35*b8 + 64*b7 + 56*b6 + 92*b5 - 38*b4 + 44*b3 + 114*b2 - 48*b1 + 502) / 4 $$\nu^{5}$$ $$=$$ $$( 204 \beta_{15} + 140 \beta_{14} - 40 \beta_{13} - 125 \beta_{12} - 75 \beta_{11} - 4 \beta_{10} + 362 \beta_{9} - 382 \beta_{8} + 53 \beta_{7} + 257 \beta_{6} - 989 \beta_{5} + 507 \beta_{4} + 115 \beta_{3} + 410 \beta_{2} - 125 \beta _1 + 1452 ) / 4$$ (204*b15 + 140*b14 - 40*b13 - 125*b12 - 75*b11 - 4*b10 + 362*b9 - 382*b8 + 53*b7 + 257*b6 - 989*b5 + 507*b4 + 115*b3 + 410*b2 - 125*b1 + 1452) / 4 $$\nu^{6}$$ $$=$$ $$( 680 \beta_{15} - 440 \beta_{14} + 770 \beta_{13} + 40 \beta_{12} - 670 \beta_{11} - 2 \beta_{10} + 1182 \beta_{9} - 1246 \beta_{8} - 1748 \beta_{7} - 1116 \beta_{6} - 3198 \beta_{5} + 1616 \beta_{4} - 870 \beta_{3} + \cdots - 8714 ) / 4$$ (680*b15 - 440*b14 + 770*b13 + 40*b12 - 670*b11 - 2*b10 + 1182*b9 - 1246*b8 - 1748*b7 - 1116*b6 - 3198*b5 + 1616*b4 - 870*b3 - 2609*b2 + 988*b1 - 8714) / 4 $$\nu^{7}$$ $$=$$ $$( - 3395 \beta_{15} - 2975 \beta_{14} + 1897 \beta_{13} + 4004 \beta_{12} + 1344 \beta_{11} + 846 \beta_{10} - 7459 \beta_{9} + 7305 \beta_{8} - 2776 \beta_{7} - 8340 \beta_{6} + 19715 \beta_{5} - 11633 \beta_{4} + \cdots - 39065 ) / 4$$ (-3395*b15 - 2975*b14 + 1897*b13 + 4004*b12 + 1344*b11 + 846*b10 - 7459*b9 + 7305*b8 - 2776*b7 - 8340*b6 + 19715*b5 - 11633*b4 - 3451*b3 - 14000*b2 + 3899*b1 - 39065) / 4 $$\nu^{8}$$ $$=$$ $$( - 16817 \beta_{15} + 8014 \beta_{14} - 13894 \beta_{13} + 2910 \beta_{12} + 21450 \beta_{11} + 3384 \beta_{10} - 32237 \beta_{9} + 38333 \beta_{8} + 42904 \beta_{7} + 17680 \beta_{6} + 94000 \beta_{5} + \cdots + 151624 ) / 4$$ (-16817*b15 + 8014*b14 - 13894*b13 + 2910*b12 + 21450*b11 + 3384*b10 - 32237*b9 + 38333*b8 + 42904*b7 + 17680*b6 + 94000*b5 - 54162*b4 + 16804*b3 + 51614*b2 - 18376*b1 + 151624) / 4 $$\nu^{9}$$ $$=$$ $$( 56252 \beta_{15} + 66195 \beta_{14} - 62391 \beta_{13} - 110352 \beta_{12} - 10752 \beta_{11} - 27482 \beta_{10} + 156422 \beta_{9} - 128150 \beta_{8} + 107636 \beta_{7} + 233002 \beta_{6} + \cdots + 1021793 ) / 4$$ (56252*b15 + 66195*b14 - 62391*b13 - 110352*b12 - 10752*b11 - 27482*b10 + 156422*b9 - 128150*b8 + 107636*b7 + 233002*b6 - 361575*b5 + 245591*b4 + 96813*b3 + 416895*b2 - 106617*b1 + 1021793) / 4 $$\nu^{10}$$ $$=$$ $$( 412284 \beta_{15} - 158003 \beta_{14} + 223493 \beta_{13} - 191755 \beta_{12} - 600935 \beta_{11} - 162784 \beta_{10} + 892016 \beta_{9} - 1077508 \beta_{8} - 986291 \beta_{7} + \cdots - 2461277 ) / 4$$ (412284*b15 - 158003*b14 + 223493*b13 - 191755*b12 - 600935*b11 - 162784*b10 + 892016*b9 - 1077508*b8 - 986291*b7 - 165817*b6 - 2535724*b5 + 1645672*b4 - 308128*b3 - 853818*b2 + 312672*b1 - 2461277) / 4 $$\nu^{11}$$ $$=$$ $$( - 867390 \beta_{15} - 1574848 \beta_{14} + 1757690 \beta_{13} + 2770009 \beta_{12} - 423291 \beta_{11} + 638966 \beta_{10} - 3153922 \beta_{9} + 1873082 \beta_{8} - 3580237 \beta_{7} + \cdots - 26117100 ) / 4$$ (-867390*b15 - 1574848*b14 + 1757690*b13 + 2770009*b12 - 423291*b11 + 638966*b10 - 3153922*b9 + 1873082*b8 - 3580237*b7 - 6004591*b6 + 5946325*b5 - 4632093*b4 - 2621399*b3 - 11443762*b2 + 2741299*b1 - 26117100) / 4 $$\nu^{12}$$ $$=$$ $$( - 10032355 \beta_{15} + 3065077 \beta_{14} - 2792893 \beta_{13} + 7836073 \beta_{12} + 15379823 \beta_{11} + 5680234 \beta_{10} - 24535089 \beta_{9} + 28510817 \beta_{8} + \cdots + 32343451 ) / 4$$ (-10032355*b15 + 3065077*b14 - 2792893*b13 + 7836073*b12 + 15379823*b11 + 5680234*b10 - 24535089*b9 + 28510817*b8 + 21375709*b7 - 2625861*b6 + 65193798*b5 - 46824802*b4 + 4959450*b3 + 9316825*b2 - 4581176*b1 + 32343451) / 4 $$\nu^{13}$$ $$=$$ $$( 10645249 \beta_{15} + 39271219 \beta_{14} - 45400121 \beta_{13} - 64529374 \beta_{12} + 28502526 \beta_{11} - 11306262 \beta_{10} + 57585989 \beta_{9} - 14413587 \beta_{8} + \cdots + 652421629 ) / 4$$ (10645249*b15 + 39271219*b14 - 45400121*b13 - 64529374*b12 + 28502526*b11 - 11306262*b10 + 57585989*b9 - 14413587*b8 + 108890636*b7 + 146172276*b6 - 77428447*b5 + 70485751*b4 + 68767985*b3 + 296508706*b2 - 68120299*b1 + 652421629) / 4 $$\nu^{14}$$ $$=$$ $$( 241703224 \beta_{15} - 52320569 \beta_{14} + 7929677 \beta_{13} - 266941479 \beta_{12} - 365609349 \beta_{11} - 170967462 \beta_{10} + 660227614 \beta_{9} - 719310202 \beta_{8} + \cdots - 136937987 ) / 4$$ (241703224*b15 - 52320569*b14 + 7929677*b13 - 266941479*b12 - 365609349*b11 - 170967462*b10 + 660227614*b9 - 719310202*b8 - 430969967*b7 + 223269193*b6 - 1622428532*b5 + 1262524236*b4 - 54838478*b3 + 74908310*b2 + 42760214*b1 - 136937987) / 4 $$\nu^{15}$$ $$=$$ $$( - 26124292 \beta_{15} - 997078605 \beta_{14} + 1104587019 \beta_{13} + 1392676158 \beta_{12} - 1133137212 \beta_{11} + 111268180 \beta_{10} - 842785846 \beta_{9} + \cdots - 15888620797 ) / 4$$ (-26124292*b15 - 997078605*b14 + 1104587019*b13 + 1392676158*b12 - 1133137212*b11 + 111268180*b10 - 842785846*b9 - 417526382*b8 - 3114701338*b7 - 3385813394*b6 + 292095141*b5 - 502005055*b4 - 1746305607*b3 - 7325958375*b2 + 1651106253*b1 - 15888620797) / 4

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/630\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$281$$ $$451$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
559.1
 0.5 + 0.422343i 0.5 − 4.10071i 0.5 − 0.442923i 0.5 + 3.55177i 0.5 − 4.96598i 0.5 − 0.971291i 0.5 + 2.68650i 0.5 − 1.83656i 0.5 − 0.422343i 0.5 + 4.10071i 0.5 + 0.442923i 0.5 − 3.55177i 0.5 + 4.96598i 0.5 + 0.971291i 0.5 − 2.68650i 0.5 + 1.83656i
1.41421i 0 −2.00000 −4.91728 + 0.905717i 0 −1.91369 6.73333i 2.82843i 0 1.28088 + 6.95409i
559.2 1.41421i 0 −2.00000 −4.59769 1.96501i 0 6.81963 1.57881i 2.82843i 0 −2.77894 + 6.50212i
559.3 1.41421i 0 −2.00000 −2.40341 4.38447i 0 −6.94781 + 0.853218i 2.82843i 0 −6.20058 + 3.39894i
559.4 1.41421i 0 −2.00000 −1.38028 + 4.80571i 0 −5.24961 + 4.63050i 2.82843i 0 6.79630 + 1.95201i
559.5 1.41421i 0 −2.00000 1.38028 4.80571i 0 5.24961 + 4.63050i 2.82843i 0 −6.79630 1.95201i
559.6 1.41421i 0 −2.00000 2.40341 + 4.38447i 0 6.94781 + 0.853218i 2.82843i 0 6.20058 3.39894i
559.7 1.41421i 0 −2.00000 4.59769 + 1.96501i 0 −6.81963 1.57881i 2.82843i 0 2.77894 6.50212i
559.8 1.41421i 0 −2.00000 4.91728 0.905717i 0 1.91369 6.73333i 2.82843i 0 −1.28088 6.95409i
559.9 1.41421i 0 −2.00000 −4.91728 0.905717i 0 −1.91369 + 6.73333i 2.82843i 0 1.28088 6.95409i
559.10 1.41421i 0 −2.00000 −4.59769 + 1.96501i 0 6.81963 + 1.57881i 2.82843i 0 −2.77894 6.50212i
559.11 1.41421i 0 −2.00000 −2.40341 + 4.38447i 0 −6.94781 0.853218i 2.82843i 0 −6.20058 3.39894i
559.12 1.41421i 0 −2.00000 −1.38028 4.80571i 0 −5.24961 4.63050i 2.82843i 0 6.79630 1.95201i
559.13 1.41421i 0 −2.00000 1.38028 + 4.80571i 0 5.24961 4.63050i 2.82843i 0 −6.79630 + 1.95201i
559.14 1.41421i 0 −2.00000 2.40341 4.38447i 0 6.94781 0.853218i 2.82843i 0 6.20058 + 3.39894i
559.15 1.41421i 0 −2.00000 4.59769 1.96501i 0 −6.81963 + 1.57881i 2.82843i 0 2.77894 + 6.50212i
559.16 1.41421i 0 −2.00000 4.91728 + 0.905717i 0 1.91369 + 6.73333i 2.82843i 0 −1.28088 + 6.95409i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 559.16 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 630.3.h.e 16
3.b odd 2 1 210.3.h.a 16
5.b even 2 1 inner 630.3.h.e 16
7.b odd 2 1 inner 630.3.h.e 16
12.b even 2 1 1680.3.bd.a 16
15.d odd 2 1 210.3.h.a 16
15.e even 4 2 1050.3.f.e 16
21.c even 2 1 210.3.h.a 16
35.c odd 2 1 inner 630.3.h.e 16
60.h even 2 1 1680.3.bd.a 16
84.h odd 2 1 1680.3.bd.a 16
105.g even 2 1 210.3.h.a 16
105.k odd 4 2 1050.3.f.e 16
420.o odd 2 1 1680.3.bd.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.h.a 16 3.b odd 2 1
210.3.h.a 16 15.d odd 2 1
210.3.h.a 16 21.c even 2 1
210.3.h.a 16 105.g even 2 1
630.3.h.e 16 1.a even 1 1 trivial
630.3.h.e 16 5.b even 2 1 inner
630.3.h.e 16 7.b odd 2 1 inner
630.3.h.e 16 35.c odd 2 1 inner
1050.3.f.e 16 15.e even 4 2
1050.3.f.e 16 105.k odd 4 2
1680.3.bd.a 16 12.b even 2 1
1680.3.bd.a 16 60.h even 2 1
1680.3.bd.a 16 84.h odd 2 1
1680.3.bd.a 16 420.o odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(630, [\chi])$$:

 $$T_{11}^{4} + 24T_{11}^{3} + 28T_{11}^{2} - 1776T_{11} - 4844$$ T11^4 + 24*T11^3 + 28*T11^2 - 1776*T11 - 4844 $$T_{13}^{8} - 1044T_{13}^{6} + 313732T_{13}^{4} - 31875456T_{13}^{2} + 586802176$$ T13^8 - 1044*T13^6 + 313732*T13^4 - 31875456*T13^2 + 586802176

## Hecke characteristic polynomials

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