Properties

Label 7.18.a.a
Level 7
Weight 18
Character orbit 7.a
Self dual Yes
Analytic conductor 12.826
Analytic rank 1
Dimension 4
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 7 \)
Weight: \( k \) = \( 18 \)
Character orbit: \([\chi]\) = 7.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(12.8255461141\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2}\cdot 7 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( 46 - \beta_{1} ) q^{2} \) \( + ( -698 - 3 \beta_{1} - \beta_{3} ) q^{3} \) \( + ( -4108 - 66 \beta_{1} - 16 \beta_{2} + 4 \beta_{3} ) q^{4} \) \( + ( 69566 + 1771 \beta_{1} + 221 \beta_{2} + 52 \beta_{3} ) q^{5} \) \( + ( 369728 + 4554 \beta_{1} - 494 \beta_{2} + 234 \beta_{3} ) q^{6} \) \( -5764801 q^{7} \) \( + ( 2212680 + 100692 \beta_{1} - 4744 \beta_{2} - 1040 \beta_{3} ) q^{8} \) \( + ( -363869 + 297750 \beta_{1} + 1069 \beta_{2} - 4259 \beta_{3} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 46 - \beta_{1} ) q^{2} \) \( + ( -698 - 3 \beta_{1} - \beta_{3} ) q^{3} \) \( + ( -4108 - 66 \beta_{1} - 16 \beta_{2} + 4 \beta_{3} ) q^{4} \) \( + ( 69566 + 1771 \beta_{1} + 221 \beta_{2} + 52 \beta_{3} ) q^{5} \) \( + ( 369728 + 4554 \beta_{1} - 494 \beta_{2} + 234 \beta_{3} ) q^{6} \) \( -5764801 q^{7} \) \( + ( 2212680 + 100692 \beta_{1} - 4744 \beta_{2} - 1040 \beta_{3} ) q^{8} \) \( + ( -363869 + 297750 \beta_{1} + 1069 \beta_{2} - 4259 \beta_{3} ) q^{9} \) \( + ( -223470916 + 3996 \beta_{1} + 127110 \beta_{2} - 12882 \beta_{3} ) q^{10} \) \( + ( 152602550 + 150010 \beta_{1} - 318461 \beta_{2} + 31331 \beta_{3} ) q^{11} \) \( + ( -457180936 - 1342212 \beta_{1} + 8280 \beta_{2} + 48064 \beta_{3} ) q^{12} \) \( + ( -2132155022 - 6849207 \beta_{1} + 462453 \beta_{2} + 36390 \beta_{3} ) q^{13} \) \( + ( -265180846 + 5764801 \beta_{1} ) q^{14} \) \( + ( -7669181630 - 15730812 \beta_{1} + 36569 \beta_{2} - 295605 \beta_{3} ) q^{15} \) \( + ( -11813586640 + 7334616 \beta_{1} + 1621936 \beta_{2} - 819520 \beta_{3} ) q^{16} \) \( + ( -11932351902 + 16746054 \beta_{1} - 8375390 \beta_{2} + 555152 \beta_{3} ) q^{17} \) \( + ( -37094047786 + 24166599 \beta_{1} + 3230084 \beta_{2} - 217708 \beta_{3} ) q^{18} \) \( + ( -35659734506 + 87270735 \beta_{1} + 9745486 \beta_{2} + 4447751 \beta_{3} ) q^{19} \) \( + ( -21929717552 + 180926576 \beta_{1} + 8823272 \beta_{2} - 667064 \beta_{3} ) q^{20} \) \( + ( 4023831098 + 17294403 \beta_{1} + 5764801 \beta_{3} ) q^{21} \) \( + ( -6589488472 - 620690880 \beta_{1} - 92539876 \beta_{2} - 15835508 \beta_{3} ) q^{22} \) \( + ( 40142497110 - 376221900 \beta_{1} + 101259227 \beta_{2} - 1768247 \beta_{3} ) q^{23} \) \( + ( 96614076816 - 345725496 \beta_{1} + 67542480 \beta_{2} - 35756928 \beta_{3} ) q^{24} \) \( + ( 480032857181 - 880135134 \beta_{1} - 202375499 \beta_{2} + 45903257 \beta_{3} ) q^{25} \) \( + ( 747361890364 + 2358591224 \beta_{1} + 64801554 \beta_{2} + 31342026 \beta_{3} ) q^{26} \) \( + ( 510541958236 + 226655958 \beta_{1} + 154539892 \beta_{2} + 21291742 \beta_{3} ) q^{27} \) \( + ( 23681802508 + 380476866 \beta_{1} + 92236816 \beta_{2} - 23059204 \beta_{3} ) q^{28} \) \( + ( 616609271414 - 1793451854 \beta_{1} - 251466414 \beta_{2} - 52681980 \beta_{3} ) q^{29} \) \( + ( 1618559678816 + 8552161056 \beta_{1} - 371026224 \beta_{2} + 129498352 \beta_{3} ) q^{30} \) \( + ( 767848596936 + 1165354878 \beta_{1} + 36824854 \beta_{2} - 347010076 \beta_{3} ) q^{31} \) \( + ( -1757218696288 + 3746015440 \beta_{1} + 928355616 \beta_{2} + 331080192 \beta_{3} ) q^{32} \) \( + ( -3715248570680 - 20689859448 \beta_{1} + 1011025532 \beta_{2} + 32560380 \beta_{3} ) q^{33} \) \( + ( -2497628429268 + 930621090 \beta_{1} - 2348848724 \beta_{2} - 407988100 \beta_{3} ) q^{34} \) \( + ( -401034146366 - 10209462571 \beta_{1} - 1274021021 \beta_{2} - 299769652 \beta_{3} ) q^{35} \) \( + ( -4730439401644 + 2936948190 \beta_{1} + 1254140576 \beta_{2} + 593882612 \beta_{3} ) q^{36} \) \( + ( -13396340222670 - 53823793086 \beta_{1} + 970827468 \beta_{2} + 99636330 \beta_{3} ) q^{37} \) \( + ( -12840137848208 + 30649317974 \beta_{1} + 6712984918 \beta_{2} - 1083101026 \beta_{3} ) q^{38} \) \( + ( -1013145140378 + 43832993064 \beta_{1} - 4952951577 \beta_{2} + 4194755033 \beta_{3} ) q^{39} \) \( + ( 5546372662080 + 37289367552 \beta_{1} - 11045688224 \beta_{2} + 1342256480 \beta_{3} ) q^{40} \) \( + ( -21181553651354 + 64936057106 \beta_{1} + 5222728372 \beta_{2} - 5513852878 \beta_{3} ) q^{41} \) \( + ( -2131408344128 - 26252903754 \beta_{1} + 2847811694 \beta_{2} - 1348963434 \beta_{3} ) q^{42} \) \( + ( 3677999420330 + 23951745822 \beta_{1} + 14847177241 \beta_{2} - 4197464791 \beta_{3} ) q^{43} \) \( + ( 59355207841024 - 64713214864 \beta_{1} - 6901008048 \beta_{2} - 514407312 \beta_{3} ) q^{44} \) \( + ( 40207328142586 - 47424881337 \beta_{1} - 35673673613 \beta_{2} + 3639668962 \beta_{3} ) q^{45} \) \( + ( 46965742536864 + 70035114480 \beta_{1} + 27822467072 \beta_{2} + 4530178336 \beta_{3} ) q^{46} \) \( + ( 27553161942252 - 188732568510 \beta_{1} + 19966217388 \beta_{2} + 7206957066 \beta_{3} ) q^{47} \) \( + ( 107239785055712 + 286313427696 \beta_{1} + 535054176 \beta_{2} + 4777199872 \beta_{3} ) q^{48} \) \( + 33232930569601 q^{49} \) \( + ( 134508395311154 - 898791510179 \beta_{1} - 62821730180 \beta_{2} - 11931745492 \beta_{3} ) q^{50} \) \( + ( -57558030674232 - 480658990842 \beta_{1} + 33598978194 \beta_{2} + 11659353696 \beta_{3} ) q^{51} \) \( + ( 17308009139744 + 145706720280 \beta_{1} + 13263495032 \beta_{2} - 19477164344 \beta_{3} ) q^{52} \) \( + ( -128935518926094 + 977472557976 \beta_{1} - 69056865178 \beta_{2} - 26283508622 \beta_{3} ) q^{53} \) \( + ( -8292875562112 - 420320658900 \beta_{1} + 65975255324 \beta_{2} - 1615353364 \beta_{3} ) q^{54} \) \( + ( -423533639787980 + 3440941442412 \beta_{1} + 48519783026 \beta_{2} + 10957896610 \beta_{3} ) q^{55} \) \( + ( -12755659876680 - 580469342292 \beta_{1} + 27348215944 \beta_{2} + 5995393040 \beta_{3} ) q^{56} \) \( + ( -588670961405126 - 1227269063334 \beta_{1} + 648193491 \beta_{2} + 35613936755 \beta_{3} ) q^{57} \) \( + ( 258428210078036 - 721279931730 \beta_{1} - 138192906332 \beta_{2} + 12331080212 \beta_{3} ) q^{58} \) \( + ( -694238182085574 + 634424349111 \beta_{1} + 88742514916 \beta_{2} + 47539009337 \beta_{3} ) q^{59} \) \( + ( 15374699252864 - 298665702720 \beta_{1} + 62906701312 \beta_{2} - 33858421632 \beta_{3} ) q^{60} \) \( + ( -833191017514130 - 1747652384013 \beta_{1} - 247703088623 \beta_{2} - 94969117000 \beta_{3} ) q^{61} \) \( + ( -101391425616336 + 654636348516 \beta_{1} - 123526715780 \beta_{2} + 73332263564 \beta_{3} ) q^{62} \) \( + ( 2097632375069 - 1716469497750 \beta_{1} - 6162572269 \beta_{2} + 24552287459 \beta_{3} ) q^{63} \) \( + ( 973464040439488 + 589887228768 \beta_{1} + 312505237952 \beta_{2} + 43069507072 \beta_{3} ) q^{64} \) \( + ( -452666387070898 - 7726187095046 \beta_{1} + 295911055643 \beta_{2} - 199066948661 \beta_{3} ) q^{65} \) \( + ( 2392330661241344 + 4280006865648 \beta_{1} + 29254910256 \beta_{2} + 101817697264 \beta_{3} ) q^{66} \) \( + ( -383759417007578 + 3732034496124 \beta_{1} + 76852088101 \beta_{2} - 174327188857 \beta_{3} ) q^{67} \) \( + ( 1388119352195736 - 650657043276 \beta_{1} + 127400099312 \beta_{2} - 46984075928 \beta_{3} ) q^{68} \) \( + ( 210132074039820 + 5546841347016 \beta_{1} - 504663532314 \beta_{2} + 40469037570 \beta_{3} ) q^{69} \) \( + ( 1288265360027716 - 23036144796 \beta_{1} - 732763855110 \beta_{2} + 74262166482 \beta_{3} ) q^{70} \) \( + ( 183086233368420 - 3423863344884 \beta_{1} - 438248307062 \beta_{2} + 413995235498 \beta_{3} ) q^{71} \) \( + ( 4237982789853384 + 667999382868 \beta_{1} + 317405322936 \beta_{2} - 82446654672 \beta_{3} ) q^{72} \) \( + ( -437441731508478 + 9594916136196 \beta_{1} + 911803706720 \beta_{2} + 25993847500 \beta_{3} ) q^{73} \) \( + ( 6082629270940068 + 12991773742662 \beta_{1} - 484719892140 \beta_{2} + 218417421252 \beta_{3} ) q^{74} \) \( + ( -5539651481592190 - 17039403154257 \beta_{1} + 94299003504 \beta_{2} + 61445857525 \beta_{3} ) q^{75} \) \( + ( 160464725874824 + 13599803547876 \beta_{1} + 1025806530952 \beta_{2} - 290586855328 \beta_{3} ) q^{76} \) \( + ( -879723332842550 - 864777798010 \beta_{1} + 1835864291261 \beta_{2} - 180616980131 \beta_{3} ) q^{77} \) \( + ( -5540632077287776 - 19955384732808 \beta_{1} + 878279194408 \beta_{2} - 1235344330584 \beta_{3} ) q^{78} \) \( + ( 460114239375032 + 8061280041324 \beta_{1} - 2540371606304 \beta_{2} + 849241518020 \beta_{3} ) q^{79} \) \( + ( -1355413631470592 - 45855252773248 \beta_{1} - 3738833009280 \beta_{2} - 646892889984 \beta_{3} ) q^{80} \) \( + ( -3316498472964197 - 41846585785998 \beta_{1} - 573843899993 \beta_{2} + 77331351559 \beta_{3} ) q^{81} \) \( + ( -9028982637044516 + 49786187502690 \beta_{1} + 365971633332 \beta_{2} + 1100122048164 \beta_{3} ) q^{82} \) \( + ( -12311671546522218 - 31702291419357 \beta_{1} + 3397711630546 \beta_{2} + 182904861695 \beta_{3} ) q^{83} \) \( + ( 2635557117033736 + 7737585079812 \beta_{1} - 47732552280 \beta_{2} - 277079395264 \beta_{3} ) q^{84} \) \( + ( -11212369924226952 + 59412832097046 \beta_{1} - 3787956376168 \beta_{2} - 402863755814 \beta_{3} ) q^{85} \) \( + ( -2985115425015064 + 29481119519320 \beta_{1} + 3588893252788 \beta_{2} + 1222056808580 \beta_{3} ) q^{86} \) \( + ( 7327365238906360 + 12945442307274 \beta_{1} + 51358775006 \beta_{2} - 383880493152 \beta_{3} ) q^{87} \) \( + ( 11816854558369344 + 15170439114336 \beta_{1} + 8504404775680 \beta_{2} + 2269216778048 \beta_{3} ) q^{88} \) \( + ( -34647233180359306 + 33233402684896 \beta_{1} + 1732334285366 \beta_{2} - 943293898310 \beta_{3} ) q^{89} \) \( + ( 8340342951389708 - 94435768357140 \beta_{1} - 11335902119986 \beta_{2} - 1545822498154 \beta_{3} ) q^{90} \) \( + ( 12291449402980622 + 39484315362807 \beta_{1} - 2665949516853 \beta_{2} - 209781108390 \beta_{3} ) q^{91} \) \( + ( -12490464306522048 + 16444717242528 \beta_{1} - 615944293184 \beta_{2} - 330688233856 \beta_{3} ) q^{92} \) \( + ( 43044950662208988 + 91177684263468 \beta_{1} + 1343326254786 \beta_{2} - 3111031327470 \beta_{3} ) q^{93} \) \( + ( 24259070652235920 - 37707215360436 \beta_{1} + 7023028101972 \beta_{2} - 325892542524 \beta_{3} ) q^{94} \) \( + ( 69152024920810166 - 75956197442780 \beta_{1} - 18211100205197 \beta_{2} + 610512836417 \beta_{3} ) q^{95} \) \( + ( -43616445940102592 - 74320749720288 \beta_{1} - 1958293424320 \beta_{2} + 2494851393024 \beta_{3} ) q^{96} \) \( + ( 18280437970873134 - 162293480683194 \beta_{1} + 14653681206752 \beta_{2} - 1599012381470 \beta_{3} ) q^{97} \) \( + ( 1528714806201646 - 33232930569601 \beta_{1} ) q^{98} \) \( + ( -14313134092646306 + 113670554924802 \beta_{1} + 27594048828775 \beta_{2} + 5506494640951 \beta_{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut +\mathstrut 186q^{2} \) \(\mathstrut -\mathstrut 2786q^{3} \) \(\mathstrut -\mathstrut 16300q^{4} \) \(\mathstrut +\mathstrut 274722q^{5} \) \(\mathstrut +\mathstrut 1469804q^{6} \) \(\mathstrut -\mathstrut 23059204q^{7} \) \(\mathstrut +\mathstrut 8649336q^{8} \) \(\mathstrut -\mathstrut 2050976q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut +\mathstrut 186q^{2} \) \(\mathstrut -\mathstrut 2786q^{3} \) \(\mathstrut -\mathstrut 16300q^{4} \) \(\mathstrut +\mathstrut 274722q^{5} \) \(\mathstrut +\mathstrut 1469804q^{6} \) \(\mathstrut -\mathstrut 23059204q^{7} \) \(\mathstrut +\mathstrut 8649336q^{8} \) \(\mathstrut -\mathstrut 2050976q^{9} \) \(\mathstrut -\mathstrut 893891656q^{10} \) \(\mathstrut +\mathstrut 610110180q^{11} \) \(\mathstrut -\mathstrut 1826039320q^{12} \) \(\mathstrut -\mathstrut 8514921674q^{13} \) \(\mathstrut -\mathstrut 1072252986q^{14} \) \(\mathstrut -\mathstrut 30645264896q^{15} \) \(\mathstrut -\mathstrut 47269015792q^{16} \) \(\mathstrut -\mathstrut 47762899716q^{17} \) \(\mathstrut -\mathstrut 148424524342q^{18} \) \(\mathstrut -\mathstrut 142813479494q^{19} \) \(\mathstrut -\mathstrut 88080723360q^{20} \) \(\mathstrut +\mathstrut 16060735586q^{21} \) \(\mathstrut -\mathstrut 25116572128q^{22} \) \(\mathstrut +\mathstrut 161322432240q^{23} \) \(\mathstrut +\mathstrut 387147758256q^{24} \) \(\mathstrut +\mathstrut 1921891698992q^{25} \) \(\mathstrut +\mathstrut 2984730379008q^{26} \) \(\mathstrut +\mathstrut 2041714521028q^{27} \) \(\mathstrut +\mathstrut 93966256300q^{28} \) \(\mathstrut +\mathstrut 2470023989364q^{29} \) \(\mathstrut +\mathstrut 6457134393152q^{30} \) \(\mathstrut +\mathstrut 3069063677988q^{31} \) \(\mathstrut -\mathstrut 7036366816032q^{32} \) \(\mathstrut -\mathstrut 14819614563824q^{33} \) \(\mathstrut -\mathstrut 9992374959252q^{34} \) \(\mathstrut -\mathstrut 1583717660322q^{35} \) \(\mathstrut -\mathstrut 18927631502956q^{36} \) \(\mathstrut -\mathstrut 53477713304508q^{37} \) \(\mathstrut -\mathstrut 51421850028780q^{38} \) \(\mathstrut -\mathstrut 4140246547640q^{39} \) \(\mathstrut +\mathstrut 22110911913216q^{40} \) \(\mathstrut -\mathstrut 84856086719628q^{41} \) \(\mathstrut -\mathstrut 8473127569004q^{42} \) \(\mathstrut +\mathstrut 14664094189676q^{43} \) \(\mathstrut +\mathstrut 237550257793824q^{44} \) \(\mathstrut +\mathstrut 160924162333018q^{45} \) \(\mathstrut +\mathstrut 187722899918496q^{46} \) \(\mathstrut +\mathstrut 110590112906028q^{47} \) \(\mathstrut +\mathstrut 428386513367456q^{48} \) \(\mathstrut +\mathstrut 132931722278404q^{49} \) \(\mathstrut +\mathstrut 539831164264974q^{50} \) \(\mathstrut -\mathstrut 229270804715244q^{51} \) \(\mathstrut +\mathstrut 68940623118416q^{52} \) \(\mathstrut -\mathstrut 517697020820328q^{53} \) \(\mathstrut -\mathstrut 32330860930648q^{54} \) \(\mathstrut -\mathstrut 1701016442036744q^{55} \) \(\mathstrut -\mathstrut 49861700822136q^{56} \) \(\mathstrut -\mathstrut 2352229307493836q^{57} \) \(\mathstrut +\mathstrut 1035155400175604q^{58} \) \(\mathstrut -\mathstrut 2778221577040518q^{59} \) \(\mathstrut +\mathstrut 62096128416896q^{60} \) \(\mathstrut -\mathstrut 3329268765288494q^{61} \) \(\mathstrut -\mathstrut 406874975162376q^{62} \) \(\mathstrut +\mathstrut 11823468495776q^{63} \) \(\mathstrut +\mathstrut 3892676387300416q^{64} \) \(\mathstrut -\mathstrut 1795213174093500q^{65} \) \(\mathstrut +\mathstrut 9560762631234080q^{66} \) \(\mathstrut -\mathstrut 1542501737022560q^{67} \) \(\mathstrut +\mathstrut 5553778722869496q^{68} \) \(\mathstrut +\mathstrut 829434613465248q^{69} \) \(\mathstrut +\mathstrut 5153107512400456q^{70} \) \(\mathstrut +\mathstrut 739192660163448q^{71} \) \(\mathstrut +\mathstrut 16950595160647800q^{72} \) \(\mathstrut -\mathstrut 1768956758306304q^{73} \) \(\mathstrut +\mathstrut 24304533536274948q^{74} \) \(\mathstrut -\mathstrut 22124527120060246q^{75} \) \(\mathstrut +\mathstrut 614659296403544q^{76} \) \(\mathstrut -\mathstrut 3517163775774180q^{77} \) \(\mathstrut -\mathstrut 22122617539685488q^{78} \) \(\mathstrut +\mathstrut 1824334397417480q^{79} \) \(\mathstrut -\mathstrut 5329944020335872q^{80} \) \(\mathstrut -\mathstrut 13182300720284792q^{81} \) \(\mathstrut -\mathstrut 36215502923183444q^{82} \) \(\mathstrut -\mathstrut 49183281603250158q^{83} \) \(\mathstrut +\mathstrut 10526753297975320q^{84} \) \(\mathstrut -\mathstrut 44968305361101900q^{85} \) \(\mathstrut -\mathstrut 11999423939098896q^{86} \) \(\mathstrut +\mathstrut 29283570071010892q^{87} \) \(\mathstrut +\mathstrut 47237077355248704q^{88} \) \(\mathstrut -\mathstrut 138655399526807016q^{89} \) \(\mathstrut +\mathstrut 33550243342273112q^{90} \) \(\mathstrut +\mathstrut 49086828981196874q^{91} \) \(\mathstrut -\mathstrut 49994746660573248q^{92} \) \(\mathstrut +\mathstrut 171997447280309016q^{93} \) \(\mathstrut +\mathstrut 97111697039664552q^{94} \) \(\mathstrut +\mathstrut 276760012078126224q^{95} \) \(\mathstrut -\mathstrut 174317142260969792q^{96} \) \(\mathstrut +\mathstrut 73446338844858924q^{97} \) \(\mathstrut +\mathstrut 6181325085945786q^{98} \) \(\mathstrut -\mathstrut 57479877480434828q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut -\mathstrut \) \(2\) \(x^{3}\mathstrut -\mathstrut \) \(2290\) \(x^{2}\mathstrut -\mathstrut \) \(4009\) \(x\mathstrut +\mathstrut \) \(1104138\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 6 \nu^{3} - 212 \nu^{2} - 6952 \nu + 207760 \)\()/11\)
\(\beta_{2}\)\(=\)\((\)\( 20 \nu^{3} - 670 \nu^{2} - 22418 \nu + 650154 \)\()/11\)
\(\beta_{3}\)\(=\)\((\)\( 92 \nu^{3} - 2950 \nu^{2} - 109274 \nu + 2842512 \)\()/11\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\)\(5\) \(\beta_{3}\mathstrut +\mathstrut \) \(41\) \(\beta_{2}\mathstrut -\mathstrut \) \(60\) \(\beta_{1}\mathstrut +\mathstrut \) \(1986\)\()/4032\)
\(\nu^{2}\)\(=\)\((\)\(103\) \(\beta_{3}\mathstrut +\mathstrut \) \(365\) \(\beta_{2}\mathstrut -\mathstrut \) \(2796\) \(\beta_{1}\mathstrut +\mathstrut \) \(4619274\)\()/4032\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(359\) \(\beta_{3}\mathstrut +\mathstrut \) \(10067\) \(\beta_{2}\mathstrut -\mathstrut \) \(26820\) \(\beta_{1}\mathstrut +\mathstrut \) \(4316790\)\()/672\)

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
−32.4553
23.8876
43.3902
−32.8226
−404.862 −14458.1 32840.9 1.27737e6 5.85352e6 −5.76480e6 3.97700e7 7.98958e7 −5.17159e8
1.2 −181.857 16535.0 −98000.2 −359290. −3.00701e6 −5.76480e6 4.16583e7 1.44267e8 6.53393e7
1.3 307.439 −5612.83 −36553.2 904274. −1.72560e6 −5.76480e6 −5.15345e7 −9.76363e7 2.78009e8
1.4 465.279 749.859 85412.5 −1.54763e6 348894. −5.76480e6 −2.12444e7 −1.28578e8 −7.20081e8
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(7\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{4} \) \(\mathstrut -\mathstrut 186 T_{2}^{3} \) \(\mathstrut -\mathstrut 236696 T_{2}^{2} \) \(\mathstrut +\mathstrut 27034368 T_{2} \) \(\mathstrut +\mathstrut 10531932160 \) acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(7))\).