# Properties

 Label 23.10.a.a Level $23$ Weight $10$ Character orbit 23.a Self dual yes Analytic conductor $11.846$ Analytic rank $1$ Dimension $7$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$23$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 23.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$11.8458242318$$ Analytic rank: $$1$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ Defining polynomial: $$x^{7} - 640x^{5} - 1455x^{4} + 114552x^{3} + 321544x^{2} - 5741296x - 13379024$$ x^7 - 640*x^5 - 1455*x^4 + 114552*x^3 + 321544*x^2 - 5741296*x - 13379024 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{7}\cdot 3^{2}$$ 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 $$1,\beta_1,\ldots,\beta_{6}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + ( - \beta_{4} - 2 \beta_1 - 13) q^{3} + (\beta_{5} + 2 \beta_{4} + 6 \beta_1 + 220) q^{4} + ( - 2 \beta_{5} + 2 \beta_{4} - \beta_{3} - 13 \beta_1 - 341) q^{5} + (\beta_{6} - 7 \beta_{5} + 12 \beta_{4} + 4 \beta_{3} - \beta_{2} - 81 \beta_1 - 1783) q^{6} + ( - 6 \beta_{6} - 8 \beta_{5} + 9 \beta_{4} + 4 \beta_{3} + 7 \beta_{2} + \cdots - 1410) q^{7}+ \cdots + (25 \beta_{6} + 40 \beta_{5} + 13 \beta_{4} - 17 \beta_{3} - 9 \beta_{2} + \cdots + 4728) q^{9}+O(q^{10})$$ q + b1 * q^2 + (-b4 - 2*b1 - 13) * q^3 + (b5 + 2*b4 + 6*b1 + 220) * q^4 + (-2*b5 + 2*b4 - b3 - 13*b1 - 341) * q^5 + (b6 - 7*b5 + 12*b4 + 4*b3 - b2 - 81*b1 - 1783) * q^6 + (-6*b6 - 8*b5 + 9*b4 + 4*b3 + 7*b2 - 146*b1 - 1410) * q^7 + (4*b6 + 30*b5 + 28*b4 - 16*b3 - 12*b2 + 96*b1 + 4988) * q^8 + (25*b6 + 40*b5 + 13*b4 - 17*b3 - 9*b2 - 95*b1 + 4728) * q^9 $$q + \beta_1 q^{2} + ( - \beta_{4} - 2 \beta_1 - 13) q^{3} + (\beta_{5} + 2 \beta_{4} + 6 \beta_1 + 220) q^{4} + ( - 2 \beta_{5} + 2 \beta_{4} - \beta_{3} - 13 \beta_1 - 341) q^{5} + (\beta_{6} - 7 \beta_{5} + 12 \beta_{4} + 4 \beta_{3} - \beta_{2} - 81 \beta_1 - 1783) q^{6} + ( - 6 \beta_{6} - 8 \beta_{5} + 9 \beta_{4} + 4 \beta_{3} + 7 \beta_{2} + \cdots - 1410) q^{7}+ \cdots + ( - 818753 \beta_{6} - 1142486 \beta_{5} + \cdots - 268003869) q^{99}+O(q^{100})$$ q + b1 * q^2 + (-b4 - 2*b1 - 13) * q^3 + (b5 + 2*b4 + 6*b1 + 220) * q^4 + (-2*b5 + 2*b4 - b3 - 13*b1 - 341) * q^5 + (b6 - 7*b5 + 12*b4 + 4*b3 - b2 - 81*b1 - 1783) * q^6 + (-6*b6 - 8*b5 + 9*b4 + 4*b3 + 7*b2 - 146*b1 - 1410) * q^7 + (4*b6 + 30*b5 + 28*b4 - 16*b3 - 12*b2 + 96*b1 + 4988) * q^8 + (25*b6 + 40*b5 + 13*b4 - 17*b3 - 9*b2 - 95*b1 + 4728) * q^9 + (-20*b6 - 23*b5 - 120*b4 + 6*b3 + 38*b2 - 880*b1 - 8718) * q^10 + (-35*b6 + 32*b5 + 67*b4 + 58*b3 - 42*b2 - 324*b1 - 11195) * q^11 + (-54*b6 - 149*b5 - 354*b4 + 36*b3 + 74*b2 - 2030*b1 - 49158) * q^12 + (102*b6 + 22*b5 + 52*b4 + 8*b3 - 103*b2 + 160*b1 - 42363) * q^13 + (110*b6 - 273*b5 - 720*b4 - 90*b3 + 132*b2 - 2342*b1 - 101748) * q^14 + (194*b5 + 481*b4 - 84*b3 - 77*b2 + 4798*b1 - 33902) * q^15 + (-32*b6 + 408*b5 + 576*b4 - 256*b3 - 432*b2 + 8424*b1 - 36288) * q^16 + (-219*b6 - 308*b5 + 582*b4 + 307*b3 + 567*b2 + 2491*b1 - 160894) * q^17 + (-491*b6 + 460*b5 + 1036*b4 - 2*b3 - 207*b2 + 10684*b1 - 71385) * q^18 + (595*b6 + 14*b5 + 1080*b4 - 46*b3 + 393*b2 + 6878*b1 - 185237) * q^19 + (480*b6 - 1154*b5 - 1444*b4 + 680*b3 + 576*b2 - 14352*b1 - 497280) * q^20 + (-1117*b6 + 1134*b5 + 3494*b4 + 215*b3 - 313*b2 + 22719*b1 - 15036) * q^21 + (1390*b6 + 1157*b5 - 448*b4 - 646*b3 - 2168*b2 + 1664*b1 - 223584) * q^22 - 279841 * q^23 + (440*b6 - 3018*b5 - 10836*b4 - 568*b3 + 2552*b2 - 62176*b1 - 660488) * q^24 + (-1385*b6 + 162*b5 + 229*b4 - 775*b3 - 1662*b2 + 9807*b1 - 190035) * q^25 + (-2319*b6 + 884*b5 - 3804*b4 + 1678*b3 - 747*b2 - 47759*b1 + 97315) * q^26 + (2485*b6 + 268*b5 - 4142*b4 - 4001*b3 + 1620*b2 - 3521*b1 + 424047) * q^27 + (-1268*b6 - 8600*b5 - 13184*b4 + 3584*b3 + 4396*b2 - 148900*b1 - 1197396) * q^28 + (629*b6 - 72*b5 + 5835*b4 + 2365*b3 + 239*b2 - 9993*b1 + 404997) * q^29 + (256*b6 + 11823*b5 + 14376*b4 - 3182*b3 - 3026*b2 + 53332*b1 + 3649874) * q^30 + (4353*b6 + 2092*b5 + 14536*b4 - 1721*b3 - 6096*b2 + 86663*b1 + 1051361) * q^31 + (-480*b6 + 11064*b5 + 24720*b4 + 4256*b3 - 5632*b2 + 32624*b1 + 3725472) * q^32 + (-8427*b6 - 10466*b5 + 14776*b4 + 7989*b3 + 7697*b2 + 40257*b1 + 98566) * q^33 + (4320*b6 - 7373*b5 - 22184*b4 - 5718*b3 + 9926*b2 - 93382*b1 + 2136122) * q^34 + (7999*b6 + 9756*b5 + 14021*b4 - 4769*b3 - 9614*b2 + 229973*b1 + 3346944) * q^35 + (1394*b6 + 12026*b5 + 42716*b4 - 4756*b3 - 12078*b2 + 217136*b1 + 5751870) * q^36 + (-10346*b6 - 22236*b5 - 47826*b4 + 289*b3 + 10318*b2 + 52245*b1 - 1918501) * q^37 + (-16540*b6 - 1776*b5 - 29040*b4 + 1448*b3 + 17644*b2 - 80190*b1 + 5358652) * q^38 + (7572*b6 - 4128*b5 - 9741*b4 - 6246*b3 + 1926*b2 + 238764*b1 + 898983) * q^39 + (-3736*b6 - 47328*b5 - 46096*b4 + 16560*b3 + 7960*b2 - 399672*b1 - 6477960) * q^40 + (18821*b6 + 136*b5 - 60259*b4 + 7997*b3 - 5629*b2 - 78681*b1 - 2248985) * q^41 + (32838*b6 + 74991*b5 + 87056*b4 - 36378*b3 - 36804*b2 + 641110*b1 + 17801876) * q^42 + (-28937*b6 - 12756*b5 - 50766*b4 + 13248*b3 - 10615*b2 + 100856*b1 - 6663521) * q^43 + (-11148*b6 + 19634*b5 + 1204*b4 - 5984*b3 - 10316*b2 - 135440*b1 + 6194060) * q^44 + (-4425*b6 - 20238*b5 + 11190*b4 + 39450*b3 + 17163*b2 - 284046*b1 - 10367847) * q^45 - 279841*b1 * q^46 + (615*b6 + 28576*b5 - 128230*b4 - 34871*b3 + 46784*b2 - 387979*b1 + 1138081) * q^47 + (3416*b6 - 124768*b5 + 75152*b4 + 38768*b3 + 53160*b2 - 1162456*b1 - 23238744) * q^48 + (3229*b6 + 104598*b5 - 167*b4 - 17859*b3 - 25636*b2 + 495963*b1 + 4160505) * q^49 + (32380*b6 + 65402*b5 + 117664*b4 - 13180*b3 - 48192*b2 - 231617*b1 + 7223264) * q^50 + (-49800*b6 + 484*b5 + 327097*b4 + 40896*b3 - 38587*b2 + 488754*b1 - 19399208) * q^51 + (25674*b6 - 34673*b5 + 29126*b4 - 20580*b3 - 25846*b2 - 58642*b1 - 14414350) * q^52 + (11331*b6 + 57146*b5 - 106474*b4 - 101854*b3 - 6991*b2 - 77638*b1 - 7720557) * q^53 + (-82001*b6 - 49181*b5 + 92308*b4 + 31492*b3 + 92169*b2 - 121199*b1 - 3707745) * q^54 + (61995*b6 + 21712*b5 - 16429*b4 - 28257*b3 + 1064*b2 + 105633*b1 - 25934000) * q^55 + (-45928*b6 - 276700*b5 - 214664*b4 + 130656*b3 + 75464*b2 - 2877720*b1 - 60019416) * q^56 + (30952*b6 + 10042*b5 - 31434*b4 - 44516*b3 - 58020*b2 + 609296*b1 - 31304710) * q^57 + (-8041*b6 - 13376*b5 - 225844*b4 - 10662*b3 + 1899*b2 + 862335*b1 - 5686979) * q^58 + (94242*b6 - 57836*b5 - 4*b4 - 4170*b3 - 62290*b2 - 1248234*b1 - 48732788) * q^59 + (34096*b6 + 261694*b5 + 306908*b4 - 93704*b3 - 139920*b2 + 4561416*b1 + 60139648) * q^60 + (-6825*b6 - 22884*b5 - 325738*b4 + 102520*b3 + 119657*b2 - 543760*b1 - 39611819) * q^61 + (-115039*b6 + 238993*b5 - 67468*b4 + 19196*b3 - 51513*b2 + 1914943*b1 + 66321729) * q^62 + (-69194*b6 - 287438*b5 + 358726*b4 + 183628*b3 + 34992*b2 - 915656*b1 - 81926934) * q^63 + (101216*b6 + 165008*b5 - 195168*b4 - 20736*b3 - 57248*b2 + 3540608*b1 + 48614688) * q^64 + (-80963*b6 + 135734*b5 + 33682*b4 + 68951*b3 + 73587*b2 - 1291193*b1 + 15262540) * q^65 + (173340*b6 - 101473*b5 - 589896*b4 - 123558*b3 + 117226*b2 + 585704*b1 + 36747126) * q^66 + (-120427*b6 + 10106*b5 - 514573*b4 - 289716*b3 + 172306*b2 + 1224662*b1 + 12534231) * q^67 + (-62160*b6 - 323380*b5 - 519080*b4 - 19976*b3 + 89872*b2 - 2547716*b1 + 8799368) * q^68 + (279841*b4 + 559682*b1 + 3637933) * q^69 + (-174460*b6 + 516652*b5 + 552544*b4 + 26000*b3 - 147092*b2 + 6066220*b1 + 169753652) * q^70 + (18882*b6 - 202688*b5 + 1438665*b4 - 9224*b3 - 332868*b2 - 2587738*b1 - 40551199) * q^71 + (229524*b6 + 575340*b5 - 53688*b4 - 183576*b3 - 190188*b2 + 5208504*b1 + 206567676) * q^72 + (130468*b6 - 181070*b5 + 155010*b4 + 242496*b3 - 255533*b2 - 3013328*b1 - 90939599) * q^73 + (164476*b6 - 520745*b5 + 206504*b4 + 163018*b3 + 302018*b2 - 8054128*b1 + 27411366) * q^74 + (-69955*b6 - 310680*b5 + 921854*b4 + 201125*b3 + 203540*b2 - 2175267*b1 - 22584123) * q^75 + (118288*b6 - 319750*b5 + 274004*b4 - 180608*b3 - 129840*b2 + 2006396*b1 + 32666472) * q^76 + (-31317*b6 + 284194*b5 - 980907*b4 - 108897*b3 + 36154*b2 - 7604083*b1 + 72767484) * q^77 + (-243729*b6 + 45411*b5 + 306732*b4 + 153948*b3 + 248193*b2 - 20943*b1 + 171460599) * q^78 + (-68728*b6 + 277466*b5 + 706738*b4 + 152002*b3 + 116700*b2 - 690906*b1 + 39480892) * q^79 + (-298832*b6 - 909432*b5 - 2024720*b4 + 147904*b3 + 284304*b2 - 12974064*b1 - 49893168) * q^80 + (46885*b6 - 136646*b5 - 1690901*b4 - 294851*b3 - 99306*b2 - 1367297*b1 - 34585647) * q^81 + (-355839*b6 - 615478*b5 - 264196*b4 + 553210*b3 + 49225*b2 - 6883495*b1 - 81441985) * q^82 + (201201*b6 + 502018*b5 + 398399*b4 - 661784*b3 + 312732*b2 - 1747014*b1 + 166374325) * q^83 + (-67620*b6 + 1965812*b5 + 1639160*b4 - 450432*b3 - 718244*b2 + 24240564*b1 + 492494156) * q^84 + (395631*b6 + 734674*b5 - 394501*b4 + 174941*b3 + 43156*b2 + 5336983*b1 - 2518638) * q^85 + (787758*b6 + 137416*b5 + 1266936*b4 - 9820*b3 - 613906*b2 - 10136932*b1 + 59840810) * q^86 + (-242785*b6 - 109576*b5 - 1010194*b4 + 82289*b3 + 4512*b2 + 4439405*b1 - 85408755) * q^87 + (-341968*b6 - 77928*b5 + 1616896*b4 + 62720*b3 + 520992*b2 + 8395960*b1 + 15252496) * q^88 + (-445786*b6 + 332478*b5 - 1292004*b4 - 170286*b3 - 543928*b2 + 2159802*b1 - 72810440) * q^89 + (197544*b6 - 1039686*b5 - 2933808*b4 + 31116*b3 + 238644*b2 - 10777056*b1 - 202690188) * q^90 + (541160*b6 + 76416*b5 + 1456783*b4 - 320464*b3 - 488885*b2 + 8772574*b1 - 241388896) * q^91 + (-279841*b5 - 559682*b4 - 1679046*b1 - 61565020) * q^92 + (183742*b6 - 1172090*b5 - 2760812*b4 + 86656*b3 + 325197*b2 - 1300232*b1 - 393988389) * q^93 + (28301*b6 - 1107251*b5 + 3618788*b4 - 57524*b3 + 648795*b2 - 391565*b1 - 315085875) * q^94 + (-210970*b6 + 888404*b5 - 377604*b4 + 1053170*b3 + 652578*b2 - 6555750*b1 + 23719816) * q^95 + (-954992*b6 - 2196760*b5 - 5649232*b4 + 794752*b3 + 1266416*b2 - 15700592*b1 - 477401744) * q^96 + (-97337*b6 - 500594*b5 + 478880*b4 - 19647*b3 - 144601*b2 + 18725941*b1 - 502801630) * q^97 + (465512*b6 + 2467342*b5 + 6582976*b4 - 672812*b3 - 1760316*b2 + 27982185*b1 + 354802284) * q^98 + (-818753*b6 - 1142486*b5 + 4184236*b4 + 375784*b3 + 341307*b2 + 16672204*b1 - 268003869) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7 q - 89 q^{3} + 1536 q^{4} - 2388 q^{5} - 12518 q^{6} - 9896 q^{7} + 34920 q^{8} + 33064 q^{9}+O(q^{10})$$ 7 * q - 89 * q^3 + 1536 * q^4 - 2388 * q^5 - 12518 * q^6 - 9896 * q^7 + 34920 * q^8 + 33064 * q^9 $$7 q - 89 q^{3} + 1536 q^{4} - 2388 q^{5} - 12518 q^{6} - 9896 q^{7} + 34920 q^{8} + 33064 q^{9} - 60820 q^{10} - 78484 q^{11} - 343492 q^{12} - 296769 q^{13} - 711120 q^{14} - 237870 q^{15} - 253440 q^{16} - 1128820 q^{17} - 499874 q^{18} - 1301252 q^{19} - 3482704 q^{20} - 108908 q^{21} - 1562088 q^{22} - 1958887 q^{23} - 4606464 q^{24} - 1320899 q^{25} + 692230 q^{26} + 2977921 q^{27} - 8371144 q^{28} + 2813849 q^{29} + 25535196 q^{30} + 7334751 q^{31} + 26028800 q^{32} + 646330 q^{33} + 14981564 q^{34} + 23410104 q^{35} + 40211900 q^{36} - 13324320 q^{37} + 37578632 q^{38} + 6304533 q^{39} - 45307920 q^{40} - 15691573 q^{41} + 124523248 q^{42} - 46474818 q^{43} + 43428040 q^{44} - 72736710 q^{45} + 8232227 q^{47} - 163054384 q^{48} + 29219031 q^{49} + 50366304 q^{50} - 136344764 q^{51} - 100922292 q^{52} - 53545400 q^{53} - 26171642 q^{54} - 181608484 q^{55} - 420111696 q^{56} - 218913370 q^{57} - 39304854 q^{58} - 341275144 q^{59} + 420822384 q^{60} - 277157656 q^{61} + 464777594 q^{62} - 574619276 q^{63} + 340566208 q^{64} + 106659278 q^{65} + 258025876 q^{66} + 89654580 q^{67} + 62700400 q^{68} + 24905849 q^{69} + 1187910040 q^{70} - 286098961 q^{71} + 1446323640 q^{72} - 637495039 q^{73} + 189880036 q^{74} - 160733159 q^{75} + 228563936 q^{76} + 511682536 q^{77} + 1199383686 q^{78} + 274469546 q^{79} - 345318560 q^{80} - 237775217 q^{81} - 570256066 q^{82} + 1164579762 q^{83} + 3447171416 q^{84} - 18639492 q^{85} + 415245796 q^{86} - 595368433 q^{87} + 103329440 q^{88} - 504153000 q^{89} - 1414126968 q^{90} - 1692320156 q^{91} - 429835776 q^{92} - 2753858687 q^{93} - 2214048622 q^{94} + 162962164 q^{95} - 3332565856 q^{96} - 3519929016 q^{97} + 2474592568 q^{98} - 1883749262 q^{99}+O(q^{100})$$ 7 * q - 89 * q^3 + 1536 * q^4 - 2388 * q^5 - 12518 * q^6 - 9896 * q^7 + 34920 * q^8 + 33064 * q^9 - 60820 * q^10 - 78484 * q^11 - 343492 * q^12 - 296769 * q^13 - 711120 * q^14 - 237870 * q^15 - 253440 * q^16 - 1128820 * q^17 - 499874 * q^18 - 1301252 * q^19 - 3482704 * q^20 - 108908 * q^21 - 1562088 * q^22 - 1958887 * q^23 - 4606464 * q^24 - 1320899 * q^25 + 692230 * q^26 + 2977921 * q^27 - 8371144 * q^28 + 2813849 * q^29 + 25535196 * q^30 + 7334751 * q^31 + 26028800 * q^32 + 646330 * q^33 + 14981564 * q^34 + 23410104 * q^35 + 40211900 * q^36 - 13324320 * q^37 + 37578632 * q^38 + 6304533 * q^39 - 45307920 * q^40 - 15691573 * q^41 + 124523248 * q^42 - 46474818 * q^43 + 43428040 * q^44 - 72736710 * q^45 + 8232227 * q^47 - 163054384 * q^48 + 29219031 * q^49 + 50366304 * q^50 - 136344764 * q^51 - 100922292 * q^52 - 53545400 * q^53 - 26171642 * q^54 - 181608484 * q^55 - 420111696 * q^56 - 218913370 * q^57 - 39304854 * q^58 - 341275144 * q^59 + 420822384 * q^60 - 277157656 * q^61 + 464777594 * q^62 - 574619276 * q^63 + 340566208 * q^64 + 106659278 * q^65 + 258025876 * q^66 + 89654580 * q^67 + 62700400 * q^68 + 24905849 * q^69 + 1187910040 * q^70 - 286098961 * q^71 + 1446323640 * q^72 - 637495039 * q^73 + 189880036 * q^74 - 160733159 * q^75 + 228563936 * q^76 + 511682536 * q^77 + 1199383686 * q^78 + 274469546 * q^79 - 345318560 * q^80 - 237775217 * q^81 - 570256066 * q^82 + 1164579762 * q^83 + 3447171416 * q^84 - 18639492 * q^85 + 415245796 * q^86 - 595368433 * q^87 + 103329440 * q^88 - 504153000 * q^89 - 1414126968 * q^90 - 1692320156 * q^91 - 429835776 * q^92 - 2753858687 * q^93 - 2214048622 * q^94 + 162962164 * q^95 - 3332565856 * q^96 - 3519929016 * q^97 + 2474592568 * q^98 - 1883749262 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{7} - 640x^{5} - 1455x^{4} + 114552x^{3} + 321544x^{2} - 5741296x - 13379024$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu$$ 2*v $$\beta_{2}$$ $$=$$ $$( 3199 \nu^{6} - 79358 \nu^{5} - 1488724 \nu^{4} + 30366663 \nu^{3} + 273707722 \nu^{2} - 2328138428 \nu - 14876298328 ) / 11419280$$ (3199*v^6 - 79358*v^5 - 1488724*v^4 + 30366663*v^3 + 273707722*v^2 - 2328138428*v - 14876298328) / 11419280 $$\beta_{3}$$ $$=$$ $$( 1391 \nu^{6} - 6842 \nu^{5} - 682136 \nu^{4} - 269473 \nu^{3} + 87308078 \nu^{2} + 269497648 \nu - 2521122692 ) / 2854820$$ (1391*v^6 - 6842*v^5 - 682136*v^4 - 269473*v^3 + 87308078*v^2 + 269497648*v - 2521122692) / 2854820 $$\beta_{4}$$ $$=$$ $$( - 7713 \nu^{6} + 105666 \nu^{5} + 3328828 \nu^{4} - 32964761 \nu^{3} - 360613014 \nu^{2} + 2265197396 \nu + 6770868456 ) / 11419280$$ (-7713*v^6 + 105666*v^5 + 3328828*v^4 - 32964761*v^3 - 360613014*v^2 + 2265197396*v + 6770868456) / 11419280 $$\beta_{5}$$ $$=$$ $$( 7713 \nu^{6} - 105666 \nu^{5} - 3328828 \nu^{4} + 32964761 \nu^{3} + 383451574 \nu^{2} - 2333713076 \nu - 10950324936 ) / 5709640$$ (7713*v^6 - 105666*v^5 - 3328828*v^4 + 32964761*v^3 + 383451574*v^2 - 2333713076*v - 10950324936) / 5709640 $$\beta_{6}$$ $$=$$ $$( - 29851 \nu^{6} + 497782 \nu^{5} + 11250276 \nu^{4} - 154091107 \nu^{3} - 1009430098 \nu^{2} + 10082064652 \nu + 17652094632 ) / 11419280$$ (-29851*v^6 + 497782*v^5 + 11250276*v^4 - 154091107*v^3 - 1009430098*v^2 + 10082064652*v + 17652094632) / 11419280
 $$\nu$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{5} + 2\beta_{4} + 6\beta _1 + 732 ) / 4$$ (b5 + 2*b4 + 6*b1 + 732) / 4 $$\nu^{3}$$ $$=$$ $$( 2\beta_{6} + 15\beta_{5} + 14\beta_{4} - 8\beta_{3} - 6\beta_{2} + 560\beta _1 + 2494 ) / 4$$ (2*b6 + 15*b5 + 14*b4 - 8*b3 - 6*b2 + 560*b1 + 2494) / 4 $$\nu^{4}$$ $$=$$ $$( -4\beta_{6} + 243\beta_{5} + 456\beta_{4} - 32\beta_{3} - 54\beta_{2} + 2205\beta _1 + 103240 ) / 2$$ (-4*b6 + 243*b5 + 456*b4 - 32*b3 - 54*b2 + 2205*b1 + 103240) / 2 $$\nu^{5}$$ $$=$$ $$( 964\beta_{6} + 9063\beta_{5} + 10258\beta_{4} - 3564\beta_{3} - 3776\beta_{2} + 192494\beta _1 + 1742612 ) / 4$$ (964*b6 + 9063*b5 + 10258*b4 - 3564*b3 - 3776*b2 + 192494*b1 + 1742612) / 4 $$\nu^{6}$$ $$=$$ $$( 1206 \beta_{6} + 223049 \beta_{5} + 374874 \beta_{4} - 42256 \beta_{3} - 72698 \beta_{2} + 2453864 \beta _1 + 71615698 ) / 4$$ (1206*b6 + 223049*b5 + 374874*b4 - 42256*b3 - 72698*b2 + 2453864*b1 + 71615698) / 4

## 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
 −15.4224 −15.0730 −10.3773 −2.27545 8.70295 13.1537 21.2915
−30.8448 107.347 439.399 −1512.00 −3311.09 8168.19 2239.35 −8159.64 46637.2
1.2 −30.1460 −193.374 396.783 568.966 5829.45 3614.33 3473.33 17710.4 −17152.1
1.3 −20.7546 245.319 −81.2472 −680.714 −5091.50 −8213.17 12312.6 40498.5 14127.9
1.4 −4.55091 −23.1179 −491.289 2146.10 105.207 615.663 4565.88 −19148.6 −9766.68
1.5 17.4059 86.8015 −209.035 −1221.93 1510.86 −4757.43 −12550.3 −12148.5 −21268.8
1.6 26.3074 −105.922 180.082 92.1465 −2786.54 2337.45 −8731.92 −8463.52 2424.14
1.7 42.5829 −206.054 1301.31 −1780.57 −8774.39 −11661.0 33611.0 22775.3 −75821.7
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.7 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$23$$ $$1$$

## 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 23.10.a.a 7
3.b odd 2 1 207.10.a.b 7
4.b odd 2 1 368.10.a.f 7

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.10.a.a 7 1.a even 1 1 trivial
207.10.a.b 7 3.b odd 2 1
368.10.a.f 7 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{7} - 2560T_{2}^{5} - 11640T_{2}^{4} + 1832832T_{2}^{3} + 10289408T_{2}^{2} - 367442944T_{2} - 1712515072$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(23))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{7} - 2560 T^{5} + \cdots - 1712515072$$
$3$ $$T^{7} + \cdots - 223029139672800$$
$5$ $$T^{7} + 2388 T^{6} + \cdots - 25\!\cdots\!80$$
$7$ $$T^{7} + 9896 T^{6} + \cdots + 19\!\cdots\!28$$
$11$ $$T^{7} + 78484 T^{6} + \cdots - 10\!\cdots\!48$$
$13$ $$T^{7} + 296769 T^{6} + \cdots - 72\!\cdots\!88$$
$17$ $$T^{7} + 1128820 T^{6} + \cdots + 48\!\cdots\!00$$
$19$ $$T^{7} + 1301252 T^{6} + \cdots - 58\!\cdots\!40$$
$23$ $$(T + 279841)^{7}$$
$29$ $$T^{7} - 2813849 T^{6} + \cdots - 15\!\cdots\!00$$
$31$ $$T^{7} - 7334751 T^{6} + \cdots - 89\!\cdots\!28$$
$37$ $$T^{7} + 13324320 T^{6} + \cdots + 12\!\cdots\!92$$
$41$ $$T^{7} + 15691573 T^{6} + \cdots + 21\!\cdots\!20$$
$43$ $$T^{7} + 46474818 T^{6} + \cdots - 77\!\cdots\!68$$
$47$ $$T^{7} - 8232227 T^{6} + \cdots - 30\!\cdots\!40$$
$53$ $$T^{7} + 53545400 T^{6} + \cdots - 18\!\cdots\!72$$
$59$ $$T^{7} + 341275144 T^{6} + \cdots - 23\!\cdots\!60$$
$61$ $$T^{7} + 277157656 T^{6} + \cdots + 16\!\cdots\!04$$
$67$ $$T^{7} - 89654580 T^{6} + \cdots - 12\!\cdots\!56$$
$71$ $$T^{7} + 286098961 T^{6} + \cdots + 17\!\cdots\!20$$
$73$ $$T^{7} + 637495039 T^{6} + \cdots + 95\!\cdots\!60$$
$79$ $$T^{7} - 274469546 T^{6} + \cdots + 26\!\cdots\!56$$
$83$ $$T^{7} - 1164579762 T^{6} + \cdots + 71\!\cdots\!08$$
$89$ $$T^{7} + 504153000 T^{6} + \cdots + 63\!\cdots\!00$$
$97$ $$T^{7} + 3519929016 T^{6} + \cdots - 62\!\cdots\!00$$