# Properties

 Label 7.12.a.b Level $7$ Weight $12$ Character orbit 7.a Self dual yes Analytic conductor $5.378$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7$$ Weight: $$k$$ $$=$$ $$12$$ Character orbit: $$[\chi]$$ $$=$$ 7.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$5.37840226392$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: $$\mathbb{Q}[x]/(x^{3} - \cdots)$$ Defining polynomial: $$x^{3} - x^{2} - 818x - 4704$$ x^3 - x^2 - 818*x - 4704 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + 26) q^{2} + (10 \beta_{2} - 11 \beta_1 - 47) q^{3} + (9 \beta_{2} + 21 \beta_1 + 1841) q^{4} + (2 \beta_{2} + 177 \beta_1 + 1735) q^{5} + ( - 118 \beta_{2} - 538 \beta_1 + 21206) q^{6} - 16807 q^{7} + ( - 549 \beta_{2} + 1617 \beta_1 + 42057) q^{8} + ( - 356 \beta_{2} - 5138 \beta_1 + 217675) q^{9}+O(q^{10})$$ q + (b2 + 26) * q^2 + (10*b2 - 11*b1 - 47) * q^3 + (9*b2 + 21*b1 + 1841) * q^4 + (2*b2 + 177*b1 + 1735) * q^5 + (-118*b2 - 538*b1 + 21206) * q^6 - 16807 * q^7 + (-549*b2 + 1617*b1 + 42057) * q^8 + (-356*b2 - 5138*b1 + 217675) * q^9 $$q + (\beta_{2} + 26) q^{2} + (10 \beta_{2} - 11 \beta_1 - 47) q^{3} + (9 \beta_{2} + 21 \beta_1 + 1841) q^{4} + (2 \beta_{2} + 177 \beta_1 + 1735) q^{5} + ( - 118 \beta_{2} - 538 \beta_1 + 21206) q^{6} - 16807 q^{7} + ( - 549 \beta_{2} + 1617 \beta_1 + 42057) q^{8} + ( - 356 \beta_{2} - 5138 \beta_1 + 217675) q^{9} + (108 \beta_{2} + 12078 \beta_1 + 207650) q^{10} + (7436 \beta_{2} - 5082 \beta_1 - 345566) q^{11} + (7574 \beta_{2} - 16534 \beta_1 - 206038) q^{12} + ( - 9450 \beta_{2} + 2331 \beta_1 - 629447) q^{13} + ( - 16807 \beta_{2} - 436982) q^{14} + ( - 46064 \beta_{2} + 30856 \beta_1 - 2745320) q^{15} + (18405 \beta_{2} + 55419 \beta_1 - 3014629) q^{16} + ( - 61548 \beta_{2} + 41898 \beta_1 + 5259228) q^{17} + (269969 \beta_{2} - 356860 \beta_1 - 15994) q^{18} + (32958 \beta_{2} - 48081 \beta_1 + 4108411) q^{19} + (93016 \beta_{2} + 461076 \beta_1 + 12845420) q^{20} + ( - 168070 \beta_{2} + 184877 \beta_1 + 789929) q^{21} + ( - 426240 \beta_{2} - 189420 \beta_1 + 10424828) q^{22} + ( - 508416 \beta_{2} - 553728 \beta_1 + 2071536) q^{23} + (55674 \beta_{2} + 136566 \beta_1 - 39034602) q^{24} + (752292 \beta_{2} + 894642 \beta_1 + 23523905) q^{25} + ( - 489776 \beta_{2} - 39942 \beta_1 - 44672530) q^{26} + (2358700 \beta_{2} - 1820426 \beta_1 + 68737630) q^{27} + ( - 151263 \beta_{2} - 352947 \beta_1 - 30941687) q^{28} + ( - 487060 \beta_{2} + 1978326 \beta_1 - 41787380) q^{29} + ( - 2239936 \beta_{2} + 1130864 \beta_1 - 192166960) q^{30} + (2285244 \beta_{2} - 1586946 \beta_1 + 54294270) q^{31} + ( - 2701933 \beta_{2} + 843381 \beta_1 - 56498267) q^{32} + ( - 4467440 \beta_{2} + 478984 \beta_1 + 260786056) q^{33} + (5928462 \beta_{2} + 1556556 \beta_1 - 24059760) q^{34} + ( - 33614 \beta_{2} - 2974839 \beta_1 - 29160145) q^{35} + ( - 664639 \beta_{2} - 8074507 \beta_1 + 106445633) q^{36} + (290844 \beta_{2} + 4496814 \beta_1 + 190636332) q^{37} + (3980854 \beta_{2} - 2577390 \beta_1 + 170305298) q^{38} + ( - 3547544 \beta_{2} + 9969652 \beta_1 - 218031884) q^{39} + (6893280 \beta_{2} + 8570760 \beta_1 + 614243160) q^{40} + ( - 8852564 \beta_{2} - 10082058 \beta_1 + 291582704) q^{41} + (1983226 \beta_{2} + 9042166 \beta_1 - 356409242) q^{42} + (6422724 \beta_{2} - 5533374 \beta_1 + 153695726) q^{43} + (4146760 \beta_{2} - 11423664 \beta_1 - 557812864) q^{44} + ( - 21318326 \beta_{2} + 19245589 \beta_1 - 1683632765) q^{45} + (15698160 \beta_{2} - 48330240 \beta_1 - 2068068768) q^{46} + (10946196 \beta_{2} - 16537302 \beta_1 + 906078186) q^{47} + ( - 56721706 \beta_{2} + 44317274 \beta_1 - 293602054) q^{48} + 282475249 q^{49} + (2683163 \beta_{2} + 76633788 \beta_1 + 3817809970) q^{50} + (61025532 \beta_{2} - 30400482 \beta_1 - 2268747354) q^{51} + ( - 16633260 \beta_{2} - 17775240 \beta_1 - 1481257456) q^{52} + (5811360 \beta_{2} + 35890512 \beta_1 + 45042798) q^{53} + (45023564 \beta_{2} - 74256268 \beta_1 + 7760065748) q^{54} + ( - 21771576 \beta_{2} - 22013916 \beta_1 - 1372508420) q^{55} + (9227043 \beta_{2} - 27176919 \beta_1 - 706851999) q^{56} + (45646660 \beta_{2} - 67201022 \beta_1 + 1281870250) q^{57} + ( - 51312294 \beta_{2} + 124297908 \beta_1 - 906512128) q^{58} + ( - 38340978 \beta_{2} + 18634647 \beta_1 + 1959822699) q^{59} + ( - 69926752 \beta_{2} - 33332992 \beta_1 - 5573417920) q^{60} + ( - 86546430 \beta_{2} + 161957553 \beta_1 + 727406287) q^{61} + (29727636 \beta_{2} - 59922204 \beta_1 + 7354453620) q^{62} + (5983292 \beta_{2} + 86354366 \beta_1 - 3658463725) q^{63} + ( - 55849275 \beta_{2} - 112888797 \beta_1 - 3232443437) q^{64} + (8072036 \beta_{2} - 174519534 \beta_1 - 1719178370) q^{65} + (332421680 \beta_{2} - 61245328 \beta_1 - 7150983376) q^{66} + ( - 93370464 \beta_{2} + 167269200 \beta_1 - 8984462204) q^{67} + ( - 12802314 \beta_{2} + 144536406 \beta_1 + 9024578094) q^{68} + (408916320 \beta_{2} - 52478544 \beta_1 - 3026416656) q^{69} + ( - 1815156 \beta_{2} - 202994946 \beta_1 - 3489973550) q^{70} + ( - 42991032 \beta_{2} - 137467932 \beta_1 + 5461196988) q^{71} + ( - 362481453 \beta_{2} + 167825385 \beta_1 - 6456858111) q^{72} + ( - 270287784 \beta_{2} - 21412500 \beta_1 + 1915034550) q^{73} + (145220658 \beta_{2} + 311891076 \beta_1 + 9857216352) q^{74} + ( - 365830594 \beta_{2} - 193367569 \beta_1 + 2076074915) q^{75} + (58329306 \beta_{2} + 6805302 \beta_1 + 6531137942) q^{76} + ( - 124976852 \beta_{2} + 85413174 \beta_1 + 5807927762) q^{77} + ( - 247450504 \beta_{2} + 603437912 \beta_1 - 8273854792) q^{78} + (157585464 \beta_{2} + 83144796 \beta_1 - 18218095492) q^{79} + (229423792 \beta_{2} - 216713088 \beta_1 + 19370420960) q^{80} + (503282836 \beta_{2} - 959146454 \beta_1 + 38967760843) q^{81} + (532814814 \beta_{2} - 871483788 \beta_1 - 29754512984) q^{82} + (172922022 \beta_{2} + 178290147 \beta_1 + 41304138687) q^{83} + ( - 127296218 \beta_{2} + 277886938 \beta_1 + 3462880666) q^{84} + (184300308 \beta_{2} + 606290538 \beta_1 + 15457826310) q^{85} + (94309784 \beta_{2} - 241392228 \beta_1 + 19751865220) q^{86} + ( - 934092524 \beta_{2} + 1146223210 \beta_1 - 39234097598) q^{87} + (347444712 \beta_{2} - 301795032 \beta_1 - 32605313976) q^{88} + (167810960 \beta_{2} + 773412936 \beta_1 - 30235729070) q^{89} + ( - 1494431524 \beta_{2} + 861015206 \beta_1 - 95295623830) q^{90} + (158826150 \beta_{2} - 39177117 \beta_1 + 10579115729) q^{91} + ( - 858729360 \beta_{2} - 1822760016 \beta_1 - 50201377296) q^{92} + (240899352 \beta_{2} - 1625398068 \beta_1 + 72986656380) q^{93} + (868828572 \beta_{2} - 894666420 \beta_1 + 44142260220) q^{94} + ( - 193320160 \beta_{2} + 818040720 \beta_1 - 6262897520) q^{95} + (157791130 \beta_{2} + 1542731638 \beta_1 - 70849794218) q^{96} + ( - 599814684 \beta_{2} - 661659054 \beta_1 - 24166196124) q^{97} + (282475249 \beta_{2} + 7344356474) q^{98} + (2809727452 \beta_{2} - 706080242 \beta_1 - 58566600902) q^{99}+O(q^{100})$$ q + (b2 + 26) * q^2 + (10*b2 - 11*b1 - 47) * q^3 + (9*b2 + 21*b1 + 1841) * q^4 + (2*b2 + 177*b1 + 1735) * q^5 + (-118*b2 - 538*b1 + 21206) * q^6 - 16807 * q^7 + (-549*b2 + 1617*b1 + 42057) * q^8 + (-356*b2 - 5138*b1 + 217675) * q^9 + (108*b2 + 12078*b1 + 207650) * q^10 + (7436*b2 - 5082*b1 - 345566) * q^11 + (7574*b2 - 16534*b1 - 206038) * q^12 + (-9450*b2 + 2331*b1 - 629447) * q^13 + (-16807*b2 - 436982) * q^14 + (-46064*b2 + 30856*b1 - 2745320) * q^15 + (18405*b2 + 55419*b1 - 3014629) * q^16 + (-61548*b2 + 41898*b1 + 5259228) * q^17 + (269969*b2 - 356860*b1 - 15994) * q^18 + (32958*b2 - 48081*b1 + 4108411) * q^19 + (93016*b2 + 461076*b1 + 12845420) * q^20 + (-168070*b2 + 184877*b1 + 789929) * q^21 + (-426240*b2 - 189420*b1 + 10424828) * q^22 + (-508416*b2 - 553728*b1 + 2071536) * q^23 + (55674*b2 + 136566*b1 - 39034602) * q^24 + (752292*b2 + 894642*b1 + 23523905) * q^25 + (-489776*b2 - 39942*b1 - 44672530) * q^26 + (2358700*b2 - 1820426*b1 + 68737630) * q^27 + (-151263*b2 - 352947*b1 - 30941687) * q^28 + (-487060*b2 + 1978326*b1 - 41787380) * q^29 + (-2239936*b2 + 1130864*b1 - 192166960) * q^30 + (2285244*b2 - 1586946*b1 + 54294270) * q^31 + (-2701933*b2 + 843381*b1 - 56498267) * q^32 + (-4467440*b2 + 478984*b1 + 260786056) * q^33 + (5928462*b2 + 1556556*b1 - 24059760) * q^34 + (-33614*b2 - 2974839*b1 - 29160145) * q^35 + (-664639*b2 - 8074507*b1 + 106445633) * q^36 + (290844*b2 + 4496814*b1 + 190636332) * q^37 + (3980854*b2 - 2577390*b1 + 170305298) * q^38 + (-3547544*b2 + 9969652*b1 - 218031884) * q^39 + (6893280*b2 + 8570760*b1 + 614243160) * q^40 + (-8852564*b2 - 10082058*b1 + 291582704) * q^41 + (1983226*b2 + 9042166*b1 - 356409242) * q^42 + (6422724*b2 - 5533374*b1 + 153695726) * q^43 + (4146760*b2 - 11423664*b1 - 557812864) * q^44 + (-21318326*b2 + 19245589*b1 - 1683632765) * q^45 + (15698160*b2 - 48330240*b1 - 2068068768) * q^46 + (10946196*b2 - 16537302*b1 + 906078186) * q^47 + (-56721706*b2 + 44317274*b1 - 293602054) * q^48 + 282475249 * q^49 + (2683163*b2 + 76633788*b1 + 3817809970) * q^50 + (61025532*b2 - 30400482*b1 - 2268747354) * q^51 + (-16633260*b2 - 17775240*b1 - 1481257456) * q^52 + (5811360*b2 + 35890512*b1 + 45042798) * q^53 + (45023564*b2 - 74256268*b1 + 7760065748) * q^54 + (-21771576*b2 - 22013916*b1 - 1372508420) * q^55 + (9227043*b2 - 27176919*b1 - 706851999) * q^56 + (45646660*b2 - 67201022*b1 + 1281870250) * q^57 + (-51312294*b2 + 124297908*b1 - 906512128) * q^58 + (-38340978*b2 + 18634647*b1 + 1959822699) * q^59 + (-69926752*b2 - 33332992*b1 - 5573417920) * q^60 + (-86546430*b2 + 161957553*b1 + 727406287) * q^61 + (29727636*b2 - 59922204*b1 + 7354453620) * q^62 + (5983292*b2 + 86354366*b1 - 3658463725) * q^63 + (-55849275*b2 - 112888797*b1 - 3232443437) * q^64 + (8072036*b2 - 174519534*b1 - 1719178370) * q^65 + (332421680*b2 - 61245328*b1 - 7150983376) * q^66 + (-93370464*b2 + 167269200*b1 - 8984462204) * q^67 + (-12802314*b2 + 144536406*b1 + 9024578094) * q^68 + (408916320*b2 - 52478544*b1 - 3026416656) * q^69 + (-1815156*b2 - 202994946*b1 - 3489973550) * q^70 + (-42991032*b2 - 137467932*b1 + 5461196988) * q^71 + (-362481453*b2 + 167825385*b1 - 6456858111) * q^72 + (-270287784*b2 - 21412500*b1 + 1915034550) * q^73 + (145220658*b2 + 311891076*b1 + 9857216352) * q^74 + (-365830594*b2 - 193367569*b1 + 2076074915) * q^75 + (58329306*b2 + 6805302*b1 + 6531137942) * q^76 + (-124976852*b2 + 85413174*b1 + 5807927762) * q^77 + (-247450504*b2 + 603437912*b1 - 8273854792) * q^78 + (157585464*b2 + 83144796*b1 - 18218095492) * q^79 + (229423792*b2 - 216713088*b1 + 19370420960) * q^80 + (503282836*b2 - 959146454*b1 + 38967760843) * q^81 + (532814814*b2 - 871483788*b1 - 29754512984) * q^82 + (172922022*b2 + 178290147*b1 + 41304138687) * q^83 + (-127296218*b2 + 277886938*b1 + 3462880666) * q^84 + (184300308*b2 + 606290538*b1 + 15457826310) * q^85 + (94309784*b2 - 241392228*b1 + 19751865220) * q^86 + (-934092524*b2 + 1146223210*b1 - 39234097598) * q^87 + (347444712*b2 - 301795032*b1 - 32605313976) * q^88 + (167810960*b2 + 773412936*b1 - 30235729070) * q^89 + (-1494431524*b2 + 861015206*b1 - 95295623830) * q^90 + (158826150*b2 - 39177117*b1 + 10579115729) * q^91 + (-858729360*b2 - 1822760016*b1 - 50201377296) * q^92 + (240899352*b2 - 1625398068*b1 + 72986656380) * q^93 + (868828572*b2 - 894666420*b1 + 44142260220) * q^94 + (-193320160*b2 + 818040720*b1 - 6262897520) * q^95 + (157791130*b2 + 1542731638*b1 - 70849794218) * q^96 + (-599814684*b2 - 661659054*b1 - 24166196124) * q^97 + (282475249*b2 + 7344356474) * q^98 + (2809727452*b2 - 706080242*b1 - 58566600902) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 77 q^{2} - 140 q^{3} + 5493 q^{4} + 5026 q^{5} + 64274 q^{6} - 50421 q^{7} + 125103 q^{8} + 658519 q^{9}+O(q^{10})$$ 3 * q + 77 * q^2 - 140 * q^3 + 5493 * q^4 + 5026 * q^5 + 64274 * q^6 - 50421 * q^7 + 125103 * q^8 + 658519 * q^9 $$3 q + 77 q^{2} - 140 q^{3} + 5493 q^{4} + 5026 q^{5} + 64274 q^{6} - 50421 q^{7} + 125103 q^{8} + 658519 q^{9} + 610764 q^{10} - 1039052 q^{11} - 609154 q^{12} - 1881222 q^{13} - 1294139 q^{14} - 8220752 q^{15} - 9117711 q^{16} + 15797334 q^{17} + 38909 q^{18} + 12340356 q^{19} + 37982168 q^{20} + 2352980 q^{21} + 31890144 q^{22} + 7276752 q^{23} - 117296046 q^{24} + 68924781 q^{25} - 133487872 q^{26} + 205674616 q^{27} - 92320851 q^{28} - 126853406 q^{29} - 575391808 q^{30} + 162184512 q^{31} - 167636249 q^{32} + 786346624 q^{33} - 79664298 q^{34} - 84471982 q^{35} + 328076045 q^{36} + 567121338 q^{37} + 509512430 q^{38} - 660517760 q^{39} + 1827265440 q^{40} + 893682734 q^{41} - 1080253118 q^{42} + 460197828 q^{43} - 1666161688 q^{44} - 5048825558 q^{45} - 6171574224 q^{46} + 2723825664 q^{47} - 868401730 q^{48} + 847425747 q^{49} + 11374112959 q^{50} - 6836867112 q^{51} - 4409363868 q^{52} + 93426522 q^{53} + 23309429948 q^{54} - 4073739768 q^{55} - 2102606121 q^{56} + 3867165112 q^{57} - 2792521998 q^{58} + 5899174428 q^{59} - 16616994016 q^{60} + 2106807738 q^{61} + 22093555428 q^{62} - 11067728833 q^{63} - 9528592239 q^{64} - 4991087612 q^{65} - 21724126480 q^{66} - 27027285348 q^{67} + 26942000190 q^{68} - 9435687744 q^{69} - 10265110548 q^{70} + 16564049928 q^{71} - 19175918265 q^{72} + 6036803934 q^{73} + 29114537322 q^{74} + 6787422908 q^{75} + 19528279218 q^{76} + 17463346964 q^{77} - 25177551784 q^{78} - 54895016736 q^{79} + 58098552176 q^{80} + 117359146147 q^{81} - 88924869978 q^{82} + 123561203892 q^{83} + 10238051278 q^{84} + 45582888084 q^{85} + 59402678104 q^{86} - 117914423480 q^{87} - 97861591608 q^{88} - 91648411106 q^{89} - 285253455172 q^{90} + 31617698154 q^{91} - 147922642512 q^{92} + 220344467856 q^{93} + 132452618508 q^{94} - 19413413120 q^{95} - 214249905422 q^{96} - 71237114634 q^{97} + 21750594173 q^{98} - 177803449916 q^{99}+O(q^{100})$$ 3 * q + 77 * q^2 - 140 * q^3 + 5493 * q^4 + 5026 * q^5 + 64274 * q^6 - 50421 * q^7 + 125103 * q^8 + 658519 * q^9 + 610764 * q^10 - 1039052 * q^11 - 609154 * q^12 - 1881222 * q^13 - 1294139 * q^14 - 8220752 * q^15 - 9117711 * q^16 + 15797334 * q^17 + 38909 * q^18 + 12340356 * q^19 + 37982168 * q^20 + 2352980 * q^21 + 31890144 * q^22 + 7276752 * q^23 - 117296046 * q^24 + 68924781 * q^25 - 133487872 * q^26 + 205674616 * q^27 - 92320851 * q^28 - 126853406 * q^29 - 575391808 * q^30 + 162184512 * q^31 - 167636249 * q^32 + 786346624 * q^33 - 79664298 * q^34 - 84471982 * q^35 + 328076045 * q^36 + 567121338 * q^37 + 509512430 * q^38 - 660517760 * q^39 + 1827265440 * q^40 + 893682734 * q^41 - 1080253118 * q^42 + 460197828 * q^43 - 1666161688 * q^44 - 5048825558 * q^45 - 6171574224 * q^46 + 2723825664 * q^47 - 868401730 * q^48 + 847425747 * q^49 + 11374112959 * q^50 - 6836867112 * q^51 - 4409363868 * q^52 + 93426522 * q^53 + 23309429948 * q^54 - 4073739768 * q^55 - 2102606121 * q^56 + 3867165112 * q^57 - 2792521998 * q^58 + 5899174428 * q^59 - 16616994016 * q^60 + 2106807738 * q^61 + 22093555428 * q^62 - 11067728833 * q^63 - 9528592239 * q^64 - 4991087612 * q^65 - 21724126480 * q^66 - 27027285348 * q^67 + 26942000190 * q^68 - 9435687744 * q^69 - 10265110548 * q^70 + 16564049928 * q^71 - 19175918265 * q^72 + 6036803934 * q^73 + 29114537322 * q^74 + 6787422908 * q^75 + 19528279218 * q^76 + 17463346964 * q^77 - 25177551784 * q^78 - 54895016736 * q^79 + 58098552176 * q^80 + 117359146147 * q^81 - 88924869978 * q^82 + 123561203892 * q^83 + 10238051278 * q^84 + 45582888084 * q^85 + 59402678104 * q^86 - 117914423480 * q^87 - 97861591608 * q^88 - 91648411106 * q^89 - 285253455172 * q^90 + 31617698154 * q^91 - 147922642512 * q^92 + 220344467856 * q^93 + 132452618508 * q^94 - 19413413120 * q^95 - 214249905422 * q^96 - 71237114634 * q^97 + 21750594173 * q^98 - 177803449916 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 818x - 4704$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu - 1$$ 2*v - 1 $$\beta_{2}$$ $$=$$ $$( \nu^{2} - 5\nu - 546 ) / 6$$ (v^2 - 5*v - 546) / 6
 $$\nu$$ $$=$$ $$( \beta _1 + 1 ) / 2$$ (b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( 12\beta_{2} + 5\beta _1 + 1097 ) / 2$$ (12*b2 + 5*b1 + 1097) / 2

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −6.06890 −24.5296 31.5985
−53.8040 −700.524 846.870 −749.998 37691.0 −16807.0 64625.6 313587. 40352.9
1.2 55.7251 800.902 1057.28 −7066.04 44630.3 −16807.0 −55207.8 464297. −393755.
1.3 75.0789 −240.378 3588.85 12842.0 −18047.3 −16807.0 115685. −119365. 964166.
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$7$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7.12.a.b 3
3.b odd 2 1 63.12.a.d 3
4.b odd 2 1 112.12.a.h 3
5.b even 2 1 175.12.a.b 3
5.c odd 4 2 175.12.b.b 6
7.b odd 2 1 49.12.a.d 3
7.c even 3 2 49.12.c.g 6
7.d odd 6 2 49.12.c.f 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.12.a.b 3 1.a even 1 1 trivial
49.12.a.d 3 7.b odd 2 1
49.12.c.f 6 7.d odd 6 2
49.12.c.g 6 7.c even 3 2
63.12.a.d 3 3.b odd 2 1
112.12.a.h 3 4.b odd 2 1
175.12.a.b 3 5.b even 2 1
175.12.b.b 6 5.c odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} - 77T_{2}^{2} - 2854T_{2} + 225104$$ acting on $$S_{12}^{\mathrm{new}}(\Gamma_0(7))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - 77 T^{2} - 2854 T + 225104$$
$3$ $$T^{3} + 140 T^{2} + \cdots - 134864352$$
$5$ $$T^{3} - 5026 T^{2} + \cdots - 68056486400$$
$7$ $$(T + 16807)^{3}$$
$11$ $$T^{3} + 1039052 T^{2} + \cdots - 33\!\cdots\!52$$
$13$ $$T^{3} + 1881222 T^{2} + \cdots - 91\!\cdots\!44$$
$17$ $$T^{3} - 15797334 T^{2} + \cdots - 62\!\cdots\!52$$
$19$ $$T^{3} - 12340356 T^{2} + \cdots - 43\!\cdots\!00$$
$23$ $$T^{3} - 7276752 T^{2} + \cdots + 42\!\cdots\!72$$
$29$ $$T^{3} + 126853406 T^{2} + \cdots - 25\!\cdots\!60$$
$31$ $$T^{3} - 162184512 T^{2} + \cdots + 14\!\cdots\!00$$
$37$ $$T^{3} - 567121338 T^{2} + \cdots + 13\!\cdots\!84$$
$41$ $$T^{3} - 893682734 T^{2} + \cdots + 46\!\cdots\!72$$
$43$ $$T^{3} - 460197828 T^{2} + \cdots + 22\!\cdots\!64$$
$47$ $$T^{3} - 2723825664 T^{2} + \cdots - 21\!\cdots\!48$$
$53$ $$T^{3} - 93426522 T^{2} + \cdots - 36\!\cdots\!88$$
$59$ $$T^{3} - 5899174428 T^{2} + \cdots + 66\!\cdots\!60$$
$61$ $$T^{3} - 2106807738 T^{2} + \cdots + 35\!\cdots\!68$$
$67$ $$T^{3} + 27027285348 T^{2} + \cdots + 23\!\cdots\!64$$
$71$ $$T^{3} - 16564049928 T^{2} + \cdots + 61\!\cdots\!92$$
$73$ $$T^{3} - 6036803934 T^{2} + \cdots - 15\!\cdots\!96$$
$79$ $$T^{3} + 54895016736 T^{2} + \cdots + 29\!\cdots\!20$$
$83$ $$T^{3} - 123561203892 T^{2} + \cdots - 57\!\cdots\!72$$
$89$ $$T^{3} + 91648411106 T^{2} + \cdots - 89\!\cdots\!00$$
$97$ $$T^{3} + 71237114634 T^{2} + \cdots - 27\!\cdots\!84$$