# Properties

 Label 14.7.d.a Level $14$ Weight $7$ Character orbit 14.d Analytic conductor $3.221$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$14 = 2 \cdot 7$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 14.d (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.22075717068$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 2x^{7} + 285x^{6} + 282x^{5} + 62091x^{4} + 29260x^{3} + 4838750x^{2} + 2401000x + 294122500$$ x^8 - 2*x^7 + 285*x^6 + 282*x^5 + 62091*x^4 + 29260*x^3 + 4838750*x^2 + 2401000*x + 294122500 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{6}\cdot 3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} + \beta_{2}) q^{2} + ( - \beta_{6} + 2 \beta_{3} - \beta_{2}) q^{3} - 32 \beta_1 q^{4} + (\beta_{7} - 3 \beta_{6} - 3 \beta_{4} + 6 \beta_{3} + 5 \beta_{2} - 28 \beta_1 - 28) q^{5} + (4 \beta_{5} + 4 \beta_{4} - \beta_{3} - 2 \beta_{2} + 56 \beta_1 - 28) q^{6} + (5 \beta_{7} - 3 \beta_{6} + 3 \beta_{5} + 5 \beta_{4} - 13 \beta_{3} - 6 \beta_{2} + \cdots + 74) q^{7}+ \cdots + (7 \beta_{7} + 7 \beta_{6} - 14 \beta_{5} - 7 \beta_{4} - 48 \beta_{3} + 55 \beta_{2} + \cdots + 189) q^{9}+O(q^{10})$$ q + (-b3 + b2) * q^2 + (-b6 + 2*b3 - b2) * q^3 - 32*b1 * q^4 + (b7 - 3*b6 - 3*b4 + 6*b3 + 5*b2 - 28*b1 - 28) * q^5 + (4*b5 + 4*b4 - b3 - 2*b2 + 56*b1 - 28) * q^6 + (5*b7 - 3*b6 + 3*b5 + 5*b4 - 13*b3 - 6*b2 + 15*b1 + 74) * q^7 + 32*b3 * q^8 + (7*b7 + 7*b6 - 14*b5 - 7*b4 - 48*b3 + 55*b2 - 189*b1 + 189) * q^9 $$q + ( - \beta_{3} + \beta_{2}) q^{2} + ( - \beta_{6} + 2 \beta_{3} - \beta_{2}) q^{3} - 32 \beta_1 q^{4} + (\beta_{7} - 3 \beta_{6} - 3 \beta_{4} + 6 \beta_{3} + 5 \beta_{2} - 28 \beta_1 - 28) q^{5} + (4 \beta_{5} + 4 \beta_{4} - \beta_{3} - 2 \beta_{2} + 56 \beta_1 - 28) q^{6} + (5 \beta_{7} - 3 \beta_{6} + 3 \beta_{5} + 5 \beta_{4} - 13 \beta_{3} - 6 \beta_{2} + \cdots + 74) q^{7}+ \cdots + ( - 9702 \beta_{7} - 13902 \beta_{6} + 4851 \beta_{5} + \cdots - 578241) q^{99}+O(q^{100})$$ q + (-b3 + b2) * q^2 + (-b6 + 2*b3 - b2) * q^3 - 32*b1 * q^4 + (b7 - 3*b6 - 3*b4 + 6*b3 + 5*b2 - 28*b1 - 28) * q^5 + (4*b5 + 4*b4 - b3 - 2*b2 + 56*b1 - 28) * q^6 + (5*b7 - 3*b6 + 3*b5 + 5*b4 - 13*b3 - 6*b2 + 15*b1 + 74) * q^7 + 32*b3 * q^8 + (7*b7 + 7*b6 - 14*b5 - 7*b4 - 48*b3 + 55*b2 - 189*b1 + 189) * q^9 + (-16*b7 - 8*b6 + 16*b5 + 46*b3 - 23*b2 + 168*b1 - 336) * q^10 + (-21*b7 - 14*b6 - 21*b5 - 28*b4 - 75*b2 - 339*b1) * q^11 + (32*b6 + 32*b4 - 32*b3 - 32*b2) * q^12 + (-25*b5 - 41*b4 - 8*b3 + 41*b2 + 1330*b1 - 665) * q^13 + (-12*b7 + 52*b6 + 32*b5 + 44*b4 - 113*b3 + 69*b2 - 484*b1 + 500) * q^14 + (70*b7 + 112*b6 - 35*b5 + 56*b4 - 57*b3 - 35*b2 + 3393) * q^15 + (1024*b1 - 1024) * q^16 + (-73*b7 - 93*b6 + 73*b5 + 336*b3 - 168*b2 + 1442*b1 - 2884) * q^17 + (-56*b6 - 112*b4 + 210*b2 - 1704*b1) * q^18 + (53*b7 - 2*b6 - 2*b4 - 385*b3 - 438*b2 - 2667*b1 - 2667) * q^19 + (-32*b5 + 96*b4 + 192*b3 - 352*b2 + 1792*b1 - 896) * q^20 + (-24*b7 - 204*b6 + 15*b5 - 101*b4 + 796*b3 + 859*b2 - 5091*b1 + 3765) * q^21 + (56*b7 - 280*b6 - 28*b5 - 140*b4 + 423*b3 - 28*b2 + 3156) * q^22 + (-56*b7 - 91*b6 + 112*b5 + 91*b4 - 1545*b3 + 1489*b2 + 1032*b1 - 1032) * q^23 + (128*b7 + 128*b6 - 128*b5 + 64*b3 - 32*b2 - 896*b1 + 1792) * q^24 + (112*b7 + 252*b6 + 112*b5 + 504*b4 - 344*b2 + 1166*b1) * q^25 + (-64*b7 - 264*b6 - 264*b4 - 656*b3 - 592*b2 - 392*b1 - 392) * q^26 + (213*b5 - 180*b4 + 903*b3 - 2019*b2 + 8946*b1 - 4473) * q^27 + (96*b7 + 256*b6 - 256*b5 + 96*b4 + 64*b3 + 512*b2 - 2848*b1 + 480) * q^28 + (-686*b7 + 154*b6 + 343*b5 + 77*b4 + 912*b3 + 343*b2 - 3789) * q^29 + (84*b7 + 364*b6 - 168*b5 - 364*b4 - 3435*b3 + 3519*b2 - 2412*b1 + 2412) * q^30 + (573*b7 + 582*b6 - 573*b5 + 6138*b3 - 3069*b2 + 259*b1 - 518) * q^31 - 1024*b2 * q^32 + (-468*b7 + 312*b6 + 312*b4 - 4572*b3 - 4104*b2 + 273*b1 + 273) * q^33 + (80*b5 + 664*b4 + 1495*b3 - 3070*b2 + 11760*b1 - 5880) * q^34 + (49*b7 + 154*b6 - 343*b5 - 763*b4 - 1491*b3 + 7161*b2 + 3115*b1 + 10696) * q^35 + (-448*b7 - 448*b6 + 224*b5 - 224*b4 + 1536*b3 + 224*b2 - 6048) * q^36 + (28*b7 - 700*b6 - 56*b5 + 700*b4 - 8754*b3 + 8782*b2 + 1531*b1 - 1531) * q^37 + (-220*b7 + 204*b6 + 220*b5 + 5118*b3 - 2559*b2 - 12964*b1 + 25928) * q^38 + (105*b7 - 385*b6 + 105*b5 - 770*b4 - 4332*b2 + 25191*b1) * q^39 + (512*b7 + 256*b6 + 256*b4 - 736*b3 - 1248*b2 + 5376*b1 + 5376) * q^40 + (-533*b5 - 873*b4 + 264*b3 + 5*b2 - 86030*b1 + 43015) * q^41 + (-368*b7 - 440*b6 + 720*b5 + 472*b4 + 1087*b3 + 3642*b2 + 24712*b1 - 51848) * q^42 + (2408*b7 - 420*b6 - 1204*b5 - 210*b4 + 5910*b3 - 1204*b2 - 37172) * q^43 + (-672*b7 - 448*b6 + 1344*b5 + 448*b4 - 3744*b3 + 3072*b2 + 10848*b1 - 10848) * q^44 + (-1251*b7 - 2463*b6 + 1251*b5 + 14976*b3 - 7488*b2 + 14343*b1 - 28686) * q^45 + (140*b7 + 588*b6 + 140*b5 + 1176*b4 - 1375*b2 - 48516*b1) * q^46 + (1327*b7 - 660*b6 - 660*b4 - 9303*b3 - 10630*b2 + 26159*b1 + 26159) * q^47 + (-1024*b4 - 1024*b3 + 2048*b2) * q^48 + (-338*b7 - 486*b6 + 1277*b5 + 2231*b4 + 5552*b3 + 10193*b2 + 49092*b1 - 20492) * q^49 + (1120*b7 + 2912*b6 - 560*b5 + 1456*b4 - 1082*b3 - 560*b2 + 4848) * q^50 + (805*b7 + 2758*b6 - 1610*b5 - 2758*b4 - 20391*b3 + 21196*b2 - 63423*b1 + 63423) * q^51 + (-800*b7 - 1312*b6 + 800*b5 + 512*b3 - 256*b2 - 21280*b1 + 42560) * q^52 + (-1736*b7 - 1260*b6 - 1736*b5 - 2520*b4 + 4726*b2 + 69621*b1) * q^53 + (-1572*b7 + 132*b6 + 132*b4 - 5079*b3 - 3507*b2 + 32172*b1 + 32172) * q^54 + (-594*b5 + 3609*b4 + 15177*b3 - 29760*b2 - 54852*b1 + 27426) * q^55 + (1024*b7 - 256*b6 - 640*b5 - 1664*b4 + 2848*b3 + 384*b2 - 512*b1 - 15488) * q^56 + (-2856*b7 + 1736*b6 + 1428*b5 + 868*b4 - 672*b3 + 1428*b2 - 10161) * q^57 + (1680*b7 - 1064*b6 - 3360*b5 + 1064*b4 + 6078*b3 - 4398*b2 + 42456*b1 - 42456) * q^58 + (-926*b7 + 4749*b6 + 926*b5 - 12366*b3 + 6183*b2 + 69622*b1 - 139244) * q^59 + (-1120*b7 - 1792*b6 - 1120*b5 - 3584*b4 + 4064*b2 - 108576*b1) * q^60 + (-904*b7 + 736*b6 + 736*b4 + 3986*b3 + 4890*b2 - 82943*b1 - 82943) * q^61 + (-36*b5 - 4620*b4 - 305*b3 + 646*b2 + 187320*b1 - 93660) * q^62 + (1601*b7 + 5345*b6 - 382*b5 + 2623*b4 - 1392*b3 - 11815*b2 - 24639*b1 + 164343) * q^63 + 32768 * q^64 + (-1897*b7 - 637*b6 + 3794*b5 + 637*b4 + 7140*b3 - 9037*b2 - 2079*b1 + 2079) * q^65 + (3120*b7 - 624*b6 - 3120*b5 + 1950*b3 - 975*b2 - 139440*b1 + 278880) * q^66 + (3010*b7 + 3703*b6 + 3010*b5 + 7406*b4 - 1115*b2 + 162202*b1) * q^67 + (2336*b7 + 2976*b6 + 2976*b4 - 5376*b3 - 7712*b2 + 46144*b1 + 46144) * q^68 + (5697*b5 + 3489*b4 - 16290*b3 + 26883*b2 - 21924*b1 + 10962) * q^69 + (-1876*b7 - 4228*b6 - 420*b5 - 5040*b4 - 13930*b3 + 12705*b2 - 48132*b1 - 172536) * q^70 + (-2324*b7 - 15568*b6 + 1162*b5 - 7784*b4 + 19422*b3 + 1162*b2 + 23766) * q^71 + (-1792*b6 + 1792*b4 + 6720*b3 - 6720*b2 + 54528*b1 - 54528) * q^72 + (5434*b7 - 2770*b6 - 5434*b5 - 1168*b3 + 584*b2 + 134757*b1 - 269514) * q^73 + (2912*b7 + 2688*b6 + 2912*b5 + 5376*b4 - 5087*b2 - 289536*b1) * q^74 + (-2076*b7 - 5594*b6 - 5594*b4 + 32042*b3 + 34118*b2 - 170184*b1 - 170184) * q^75 + (-1696*b5 + 64*b4 - 12320*b3 + 26336*b2 + 170688*b1 - 85344) * q^76 + (-1785*b7 - 4501*b6 - 5628*b5 - 4781*b4 - 13188*b3 - 35133*b2 - 369810*b1 + 366933) * q^77 + (-3920*b7 - 2240*b6 + 1960*b5 - 1120*b4 - 26976*b3 + 1960*b2 + 140304) * q^78 + (-2380*b7 - 3983*b6 + 4760*b5 + 3983*b4 + 68235*b3 - 70615*b2 - 17524*b1 + 17524) * q^79 + (-1024*b7 + 3072*b6 + 1024*b5 - 12288*b3 + 6144*b2 - 28672*b1 + 57344) * q^80 + (2457*b7 - 3507*b6 + 2457*b5 - 7014*b4 - 3870*b2 + 294480*b1) * q^81 + (-1360*b7 - 5624*b6 - 5624*b4 + 43208*b3 + 44568*b2 + 5544*b1 + 5544) * q^82 + (-10142*b5 - 3054*b4 - 31428*b3 + 72998*b2 - 76916*b1 + 38458) * q^83 + (480*b7 + 3296*b6 + 288*b5 + 6528*b4 + 27200*b3 - 53440*b2 + 42432*b1 - 162912) * q^84 + (7924*b7 + 23772*b6 - 3962*b5 + 11886*b4 - 92058*b3 - 3962*b2 + 273873) * q^85 + (-5656*b7 + 3976*b6 + 11312*b5 - 3976*b4 + 29318*b3 - 34974*b2 + 143256*b1 - 143256) * q^86 + (897*b7 + 11889*b6 - 897*b5 - 109032*b3 + 54516*b2 + 171423*b1 - 342846) * q^87 + (-896*b7 + 4480*b6 - 896*b5 + 8960*b4 - 11744*b2 - 100992*b1) * q^88 + (3072*b7 + 8184*b6 + 8184*b4 + 30180*b3 + 27108*b2 + 61593*b1 + 61593) * q^89 + (4848*b5 + 14856*b4 + 14382*b3 - 33612*b2 + 489552*b1 - 244776) * q^90 + (-6810*b7 - 6722*b6 + 5959*b5 - 13901*b4 - 25330*b3 - 31889*b2 + 44432*b1 + 257003) * q^91 + (3584*b7 + 5824*b6 - 1792*b5 + 2912*b4 + 49440*b3 - 1792*b2 + 33024) * q^92 + (11676*b7 + 7924*b6 - 23352*b5 - 7924*b4 + 130938*b3 - 119262*b2 + 5841*b1 - 5841) * q^93 + (-7948*b7 + 2668*b6 + 7948*b5 - 58946*b3 + 29473*b2 - 316260*b1 + 632520) * q^94 + (-9478*b7 + 3745*b6 - 9478*b5 + 7490*b4 + 37925*b2 + 181410*b1) * q^95 + (-4096*b7 - 4096*b6 - 4096*b4 - 1024*b3 + 3072*b2 - 28672*b1 - 28672) * q^96 + (-1165*b5 - 30605*b4 + 316*b3 + 533*b2 - 668374*b1 + 334187) * q^97 + (5168*b7 + 12680*b6 + 592*b5 + 15976*b4 - 28057*b3 - 26279*b2 + 184232*b1 - 533656) * q^98 + (-9702*b7 - 13902*b6 + 4851*b5 - 6951*b4 + 12492*b3 + 4851*b2 - 578241) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 128 q^{4} - 336 q^{5} + 652 q^{7} + 756 q^{9}+O(q^{10})$$ 8 * q - 128 * q^4 - 336 * q^5 + 652 * q^7 + 756 * q^9 $$8 q - 128 q^{4} - 336 q^{5} + 652 q^{7} + 756 q^{9} - 2016 q^{10} - 1356 q^{11} + 2064 q^{14} + 27144 q^{15} - 4096 q^{16} - 17304 q^{17} - 6816 q^{18} - 32004 q^{19} + 9756 q^{21} + 25248 q^{22} - 4128 q^{23} + 10752 q^{24} + 4664 q^{25} - 4704 q^{26} - 7552 q^{28} - 30312 q^{29} + 9648 q^{30} - 3108 q^{31} + 3276 q^{33} + 98028 q^{35} - 48384 q^{36} - 6124 q^{37} + 155568 q^{38} + 100764 q^{39} + 64512 q^{40} - 315936 q^{42} - 297376 q^{43} - 43392 q^{44} - 172116 q^{45} - 194064 q^{46} + 313908 q^{47} + 32432 q^{49} + 38784 q^{50} + 253692 q^{51} + 255360 q^{52} + 278484 q^{53} + 386064 q^{54} - 125952 q^{56} - 81288 q^{57} - 169824 q^{58} - 835464 q^{59} - 434304 q^{60} - 995316 q^{61} + 1216188 q^{63} + 262144 q^{64} + 8316 q^{65} + 1673280 q^{66} + 648808 q^{67} + 553728 q^{68} - 1572816 q^{70} + 190128 q^{71} - 218112 q^{72} - 1617084 q^{73} - 1158144 q^{74} - 2042208 q^{75} + 1456224 q^{77} + 1122432 q^{78} + 70096 q^{79} + 344064 q^{80} + 1177920 q^{81} + 66528 q^{82} - 1133568 q^{84} + 2190984 q^{85} - 573024 q^{86} - 2057076 q^{87} - 403968 q^{88} + 739116 q^{89} + 2233752 q^{91} + 264192 q^{92} - 23364 q^{93} + 3795120 q^{94} + 725640 q^{95} - 344064 q^{96} - 3532320 q^{98} - 4625928 q^{99}+O(q^{100})$$ 8 * q - 128 * q^4 - 336 * q^5 + 652 * q^7 + 756 * q^9 - 2016 * q^10 - 1356 * q^11 + 2064 * q^14 + 27144 * q^15 - 4096 * q^16 - 17304 * q^17 - 6816 * q^18 - 32004 * q^19 + 9756 * q^21 + 25248 * q^22 - 4128 * q^23 + 10752 * q^24 + 4664 * q^25 - 4704 * q^26 - 7552 * q^28 - 30312 * q^29 + 9648 * q^30 - 3108 * q^31 + 3276 * q^33 + 98028 * q^35 - 48384 * q^36 - 6124 * q^37 + 155568 * q^38 + 100764 * q^39 + 64512 * q^40 - 315936 * q^42 - 297376 * q^43 - 43392 * q^44 - 172116 * q^45 - 194064 * q^46 + 313908 * q^47 + 32432 * q^49 + 38784 * q^50 + 253692 * q^51 + 255360 * q^52 + 278484 * q^53 + 386064 * q^54 - 125952 * q^56 - 81288 * q^57 - 169824 * q^58 - 835464 * q^59 - 434304 * q^60 - 995316 * q^61 + 1216188 * q^63 + 262144 * q^64 + 8316 * q^65 + 1673280 * q^66 + 648808 * q^67 + 553728 * q^68 - 1572816 * q^70 + 190128 * q^71 - 218112 * q^72 - 1617084 * q^73 - 1158144 * q^74 - 2042208 * q^75 + 1456224 * q^77 + 1122432 * q^78 + 70096 * q^79 + 344064 * q^80 + 1177920 * q^81 + 66528 * q^82 - 1133568 * q^84 + 2190984 * q^85 - 573024 * q^86 - 2057076 * q^87 - 403968 * q^88 + 739116 * q^89 + 2233752 * q^91 + 264192 * q^92 - 23364 * q^93 + 3795120 * q^94 + 725640 * q^95 - 344064 * q^96 - 3532320 * q^98 - 4625928 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2x^{7} + 285x^{6} + 282x^{5} + 62091x^{4} + 29260x^{3} + 4838750x^{2} + 2401000x + 294122500$$ :

 $$\beta_{1}$$ $$=$$ $$( - 10737582 \nu^{7} + 884171859 \nu^{6} - 3763944460 \nu^{5} + 189328804101 \nu^{4} - 195625686972 \nu^{3} + \cdots + 41\!\cdots\!50 ) / 33\!\cdots\!50$$ (-10737582*v^7 + 884171859*v^6 - 3763944460*v^5 + 189328804101*v^4 - 195625686972*v^3 + 53809403263525*v^2 - 14874332445000*v + 4192507708849250) / 3300562030403750 $$\beta_{2}$$ $$=$$ $$( 512915233 \nu^{7} + 35396730954 \nu^{6} - 150684878760 \nu^{5} + 14474316021931 \nu^{4} - 7831633340232 \nu^{3} + \cdots + 16\!\cdots\!00 ) / 13\!\cdots\!00$$ (512915233*v^7 + 35396730954*v^6 - 150684878760*v^5 + 14474316021931*v^4 - 7831633340232*v^3 + 2154193159086150*v^2 - 1819725640383000*v + 167841880378945500) / 13202248121615000 $$\beta_{3}$$ $$=$$ $$( 14893 \nu^{7} - 236741 \nu^{6} + 3320715 \nu^{5} + 8132426 \nu^{4} + 331237078 \nu^{3} + 640160500 \nu^{2} + 15842998500 \nu + 846958238000 ) / 106489495000$$ (14893*v^7 - 236741*v^6 + 3320715*v^5 + 8132426*v^4 + 331237078*v^3 + 640160500*v^2 + 15842998500*v + 846958238000) / 106489495000 $$\beta_{4}$$ $$=$$ $$( 10421298761 \nu^{7} - 383100933507 \nu^{6} + 14696779653705 \nu^{5} - 107198920144948 \nu^{4} + \cdots - 61\!\cdots\!00 ) / 52\!\cdots\!00$$ (10421298761*v^7 - 383100933507*v^6 + 14696779653705*v^5 - 107198920144948*v^4 + 2158265060065106*v^3 - 9549742899008200*v^2 + 266704552642937500*v - 612573340664299000) / 52808992486460000 $$\beta_{5}$$ $$=$$ $$( - 4954059529 \nu^{7} + 53601861423 \nu^{6} - 2878465568645 \nu^{5} + 14314185841272 \nu^{4} - 544906891545834 \nu^{3} + \cdots + 11\!\cdots\!00 ) / 75\!\cdots\!00$$ (-4954059529*v^7 + 53601861423*v^6 - 2878465568645*v^5 + 14314185841272*v^4 - 544906891545834*v^3 + 1463392228774000*v^2 - 71979345426775500*v + 114695716432661000) / 7544141783780000 $$\beta_{6}$$ $$=$$ $$( - 98330981563 \nu^{7} + 465055977756 \nu^{6} - 28111572790890 \nu^{5} + 2750456475509 \nu^{4} + \cdots - 20\!\cdots\!00 ) / 52\!\cdots\!00$$ (-98330981563*v^7 + 465055977756*v^6 - 28111572790890*v^5 + 2750456475509*v^4 - 4249900685065948*v^3 + 772188416527850*v^2 - 232412592113240000*v - 202807001375555500) / 52808992486460000 $$\beta_{7}$$ $$=$$ $$( 112677227274 \nu^{7} + 39166570437 \nu^{6} + 18385233372395 \nu^{5} + 149484497566193 \nu^{4} + \cdots + 11\!\cdots\!00 ) / 52\!\cdots\!00$$ (112677227274*v^7 + 39166570437*v^6 + 18385233372395*v^5 + 149484497566193*v^4 + 3722665506101454*v^3 + 14974789454892050*v^2 - 118348469342092500*v + 1162007798458181500) / 52808992486460000
 $$\nu$$ $$=$$ $$( -2\beta_{7} - 2\beta_{6} - 2\beta_{5} - 4\beta_{4} + \beta_{2} + 6\beta_1 ) / 12$$ (-2*b7 - 2*b6 - 2*b5 - 4*b4 + b2 + 6*b1) / 12 $$\nu^{2}$$ $$=$$ $$( \beta_{7} + 3\beta_{6} - 2\beta_{5} - 3\beta_{4} + 54\beta_{3} - 53\beta_{2} + 849\beta _1 - 849 ) / 6$$ (b7 + 3*b6 - 2*b5 - 3*b4 + 54*b3 - 53*b2 + 849*b1 - 849) / 6 $$\nu^{3}$$ $$=$$ $$( 286\beta_{7} + 434\beta_{6} - 143\beta_{5} + 217\beta_{4} + 477\beta_{3} - 143\beta_{2} - 1911 ) / 6$$ (286*b7 + 434*b6 - 143*b5 + 217*b4 + 477*b3 - 143*b2 - 1911) / 6 $$\nu^{4}$$ $$=$$ $$( 427\beta_{7} + 1137\beta_{6} + 427\beta_{5} + 2274\beta_{4} + 15064\beta_{2} - 139071\beta_1 ) / 6$$ (427*b7 + 1137*b6 + 427*b5 + 2274*b4 + 15064*b2 - 139071*b1) / 6 $$\nu^{5}$$ $$=$$ $$( - 23747 \beta_{7} - 45681 \beta_{6} + 47494 \beta_{5} + 45681 \beta_{4} - 132588 \beta_{3} + 108841 \beta_{2} - 644823 \beta _1 + 644823 ) / 6$$ (-23747*b7 - 45681*b6 + 47494*b5 + 45681*b4 - 132588*b3 + 108841*b2 - 644823*b1 + 644823) / 6 $$\nu^{6}$$ $$=$$ $$( - 260622 \beta_{7} - 658058 \beta_{6} + 130311 \beta_{5} - 329029 \beta_{4} - 3745254 \beta_{3} + 130311 \beta_{2} + 25540707 ) / 6$$ (-260622*b7 - 658058*b6 + 130311*b5 - 329029*b4 - 3745254*b3 + 130311*b2 + 25540707) / 6 $$\nu^{7}$$ $$=$$ $$( - 4421299 \beta_{7} - 9613689 \beta_{6} - 4421299 \beta_{5} - 19227378 \beta_{4} - 25496068 \beta_{2} + 180033087 \beta_1 ) / 6$$ (-4421299*b7 - 9613689*b6 - 4421299*b5 - 19227378*b4 - 25496068*b2 + 180033087*b1) / 6

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/14\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$1 - \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 4.65421 − 8.06134i −4.86132 + 8.42006i −6.30576 + 10.9219i 7.51287 − 13.0127i 4.65421 + 8.06134i −4.86132 − 8.42006i −6.30576 − 10.9219i 7.51287 + 13.0127i
−2.82843 4.89898i −12.7609 7.36750i −16.0000 + 27.7128i −106.741 + 61.6269i 83.3537i −309.691 + 147.446i 181.019 −255.940 443.301i 603.818 + 348.614i
3.2 −2.82843 4.89898i 27.6101 + 15.9407i −16.0000 + 27.7128i 111.836 64.5687i 180.349i 298.743 168.528i 181.019 143.713 + 248.917i −632.642 365.256i
3.3 2.82843 + 4.89898i −36.7384 21.2109i −16.0000 + 27.7128i −162.347 + 93.7310i 239.974i 141.244 312.569i −181.019 535.305 + 927.175i −918.372 530.222i
3.4 2.82843 + 4.89898i 21.8891 + 12.6377i −16.0000 + 27.7128i −10.7486 + 6.20573i 142.979i 195.705 + 281.689i −181.019 −45.0775 78.0766i −60.8035 35.1049i
5.1 −2.82843 + 4.89898i −12.7609 + 7.36750i −16.0000 27.7128i −106.741 61.6269i 83.3537i −309.691 147.446i 181.019 −255.940 + 443.301i 603.818 348.614i
5.2 −2.82843 + 4.89898i 27.6101 15.9407i −16.0000 27.7128i 111.836 + 64.5687i 180.349i 298.743 + 168.528i 181.019 143.713 248.917i −632.642 + 365.256i
5.3 2.82843 4.89898i −36.7384 + 21.2109i −16.0000 27.7128i −162.347 93.7310i 239.974i 141.244 + 312.569i −181.019 535.305 927.175i −918.372 + 530.222i
5.4 2.82843 4.89898i 21.8891 12.6377i −16.0000 27.7128i −10.7486 6.20573i 142.979i 195.705 281.689i −181.019 −45.0775 + 78.0766i −60.8035 + 35.1049i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 5.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 14.7.d.a 8
3.b odd 2 1 126.7.n.c 8
4.b odd 2 1 112.7.s.c 8
7.b odd 2 1 98.7.d.c 8
7.c even 3 1 98.7.b.c 8
7.c even 3 1 98.7.d.c 8
7.d odd 6 1 inner 14.7.d.a 8
7.d odd 6 1 98.7.b.c 8
21.g even 6 1 126.7.n.c 8
28.f even 6 1 112.7.s.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.7.d.a 8 1.a even 1 1 trivial
14.7.d.a 8 7.d odd 6 1 inner
98.7.b.c 8 7.c even 3 1
98.7.b.c 8 7.d odd 6 1
98.7.d.c 8 7.b odd 2 1
98.7.d.c 8 7.c even 3 1
112.7.s.c 8 4.b odd 2 1
112.7.s.c 8 28.f even 6 1
126.7.n.c 8 3.b odd 2 1
126.7.n.c 8 21.g even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{7}^{\mathrm{new}}(14, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} + 32 T^{2} + 1024)^{2}$$
$3$ $$T^{8} - 1836 T^{6} + \cdots + 253716712209$$
$5$ $$T^{8} + 336 T^{7} + \cdots + 13\!\cdots\!25$$
$7$ $$T^{8} - 652 T^{7} + \cdots + 19\!\cdots\!01$$
$11$ $$T^{8} + 1356 T^{7} + \cdots + 57\!\cdots\!09$$
$13$ $$T^{8} + 11814792 T^{6} + \cdots + 76\!\cdots\!64$$
$17$ $$T^{8} + 17304 T^{7} + \cdots + 12\!\cdots\!41$$
$19$ $$T^{8} + 32004 T^{7} + \cdots + 47\!\cdots\!25$$
$23$ $$T^{8} + 4128 T^{7} + \cdots + 11\!\cdots\!09$$
$29$ $$(T^{4} + 15156 T^{3} + \cdots - 14\!\cdots\!56)^{2}$$
$31$ $$T^{8} + 3108 T^{7} + \cdots + 12\!\cdots\!21$$
$37$ $$T^{8} + 6124 T^{7} + \cdots + 23\!\cdots\!61$$
$41$ $$T^{8} + 25142408520 T^{6} + \cdots + 50\!\cdots\!84$$
$43$ $$(T^{4} + 148688 T^{3} + \cdots - 56\!\cdots\!48)^{2}$$
$47$ $$T^{8} - 313908 T^{7} + \cdots + 12\!\cdots\!41$$
$53$ $$T^{8} - 278484 T^{7} + \cdots + 50\!\cdots\!09$$
$59$ $$T^{8} + 835464 T^{7} + \cdots + 27\!\cdots\!49$$
$61$ $$T^{8} + 995316 T^{7} + \cdots + 35\!\cdots\!41$$
$67$ $$T^{8} - 648808 T^{7} + \cdots + 49\!\cdots\!69$$
$71$ $$(T^{4} - 95064 T^{3} + \cdots + 56\!\cdots\!56)^{2}$$
$73$ $$T^{8} + 1617084 T^{7} + \cdots + 79\!\cdots\!89$$
$79$ $$T^{8} - 70096 T^{7} + \cdots + 83\!\cdots\!25$$
$83$ $$T^{8} + 1103755415712 T^{6} + \cdots + 34\!\cdots\!84$$
$89$ $$T^{8} - 739116 T^{7} + \cdots + 11\!\cdots\!01$$
$97$ $$T^{8} + 4399256051592 T^{6} + \cdots + 11\!\cdots\!36$$