# Properties

 Label 7.18.a.b Level 7 Weight 18 Character orbit 7.a Self dual Yes Analytic conductor 12.826 Analytic rank 0 Dimension 5 CM No Inner twists 1

# Related objects

## 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: $$0$$ Dimension: $$5$$ Coefficient field: $$\mathbb{Q}[x]/(x^{5} - \cdots)$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{5}\cdot 3^{3}\cdot 7^{2}$$ Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( 119 + \beta_{1} ) q^{2}$$ $$+ ( -349 - 13 \beta_{1} - \beta_{3} ) q^{3}$$ $$+ ( 106915 + 49 \beta_{1} + \beta_{2} - 4 \beta_{3} + \beta_{4} ) q^{4}$$ $$+ ( 322754 - 492 \beta_{1} + 2 \beta_{2} - 32 \beta_{3} + 5 \beta_{4} ) q^{5}$$ $$+ ( -2878611 + 4501 \beta_{1} - 24 \beta_{2} - 245 \beta_{3} - 47 \beta_{4} ) q^{6}$$ $$+ 5764801 q^{7}$$ $$+ ( 8318416 + 162302 \beta_{1} + 39 \beta_{2} + 1671 \beta_{3} + 552 \beta_{4} ) q^{8}$$ $$+ ( 110523540 + 23523 \beta_{1} + 378 \beta_{2} + 3003 \beta_{3} - 1455 \beta_{4} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ ( 119 + \beta_{1} ) q^{2}$$ $$+ ( -349 - 13 \beta_{1} - \beta_{3} ) q^{3}$$ $$+ ( 106915 + 49 \beta_{1} + \beta_{2} - 4 \beta_{3} + \beta_{4} ) q^{4}$$ $$+ ( 322754 - 492 \beta_{1} + 2 \beta_{2} - 32 \beta_{3} + 5 \beta_{4} ) q^{5}$$ $$+ ( -2878611 + 4501 \beta_{1} - 24 \beta_{2} - 245 \beta_{3} - 47 \beta_{4} ) q^{6}$$ $$+ 5764801 q^{7}$$ $$+ ( 8318416 + 162302 \beta_{1} + 39 \beta_{2} + 1671 \beta_{3} + 552 \beta_{4} ) q^{8}$$ $$+ ( 110523540 + 23523 \beta_{1} + 378 \beta_{2} + 3003 \beta_{3} - 1455 \beta_{4} ) q^{9}$$ $$+ ( -69388065 + 927595 \beta_{1} - 560 \beta_{2} + 2015 \beta_{3} + 565 \beta_{4} ) q^{10}$$ $$+ ( 153269839 - 209457 \beta_{1} - 110 \beta_{2} + 167 \beta_{3} + 7789 \beta_{4} ) q^{11}$$ $$+ ( 728983004 - 5440024 \beta_{1} - 666 \beta_{2} - 59398 \beta_{3} - 25812 \beta_{4} ) q^{12}$$ $$+ ( 122253920 - 1585486 \beta_{1} - 2446 \beta_{2} - 36458 \beta_{3} + 32285 \beta_{4} ) q^{13}$$ $$+ ( 686011319 + 5764801 \beta_{1} ) q^{14}$$ $$+ ( 8238431559 - 17573957 \beta_{1} + 9822 \beta_{2} + 69163 \beta_{3} - 98165 \beta_{4} ) q^{15}$$ $$+ ( 23202285874 + 7168812 \beta_{1} + 87873 \beta_{2} + 1078923 \beta_{3} + 261222 \beta_{4} ) q^{16}$$ $$+ ( 1764599746 + 16397248 \beta_{1} - 152524 \beta_{2} - 405416 \beta_{3} - 175678 \beta_{4} ) q^{17}$$ $$+ ( 18094213089 + 104171379 \beta_{1} - 62568 \beta_{2} - 149862 \beta_{3} - 162570 \beta_{4} ) q^{18}$$ $$+ ( 3671317451 + 40799035 \beta_{1} - 247484 \beta_{2} - 3414817 \beta_{3} - 165782 \beta_{4} ) q^{19}$$ $$+ ( 156998019122 - 139260006 \beta_{1} + 749576 \beta_{2} + 661954 \beta_{3} + 308030 \beta_{4} ) q^{20}$$ $$+ ( -2011915549 - 74942413 \beta_{1} - 5764801 \beta_{3} ) q^{21}$$ $$+ ( -28301399910 + 401053538 \beta_{1} + 357368 \beta_{2} + 9611530 \beta_{3} + 2008742 \beta_{4} ) q^{22}$$ $$+ ( -43430390707 - 809661639 \beta_{1} + 284378 \beta_{2} + 7338025 \beta_{3} + 1152449 \beta_{4} ) q^{23}$$ $$+ ( -750308226348 - 167312248 \beta_{1} - 4780830 \beta_{2} + 5354438 \beta_{3} - 8991940 \beta_{4} ) q^{24}$$ $$+ ( -72189613694 - 1036569713 \beta_{1} + 2342458 \beta_{2} + 7496087 \beta_{3} - 1403855 \beta_{4} ) q^{25}$$ $$+ ( -335993741927 + 1063084277 \beta_{1} + 430944 \beta_{2} + 27877845 \beta_{3} + 5649207 \beta_{4} ) q^{26}$$ $$+ ( -804588407370 + 776508822 \beta_{1} + 2134728 \beta_{2} - 89714466 \beta_{3} + 14419764 \beta_{4} ) q^{27}$$ $$+ ( 616343698915 + 282475249 \beta_{1} + 5764801 \beta_{2} - 23059204 \beta_{3} + 5764801 \beta_{4} ) q^{28}$$ $$+ ( 417516287538 - 238179628 \beta_{1} + 9584324 \beta_{2} - 108225380 \beta_{3} - 18724246 \beta_{4} ) q^{29}$$ $$+ ( -2963534441880 + 7448549640 \beta_{1} - 24254280 \beta_{2} - 3017640 \beta_{3} - 40154400 \beta_{4} ) q^{30}$$ $$+ ( -2417943384676 - 5491609572 \beta_{1} - 21062316 \beta_{2} + 107887332 \beta_{3} + 23835282 \beta_{4} ) q^{31}$$ $$+ ( 3198824238018 + 18028310616 \beta_{1} + 29393967 \beta_{2} + 540026889 \beta_{3} + 77749158 \beta_{4} ) q^{32}$$ $$+ ( 1235754019572 - 13424030764 \beta_{1} - 7096584 \beta_{2} + 389245748 \beta_{3} - 53134420 \beta_{4} ) q^{33}$$ $$+ ( 3918474271656 - 25086134404 \beta_{1} + 5084768 \beta_{2} - 678516002 \beta_{3} - 101541238 \beta_{4} ) q^{34}$$ $$+ ( 1860612581954 - 2836282092 \beta_{1} + 11529602 \beta_{2} - 184473632 \beta_{3} + 28824005 \beta_{4} ) q^{35}$$ $$+ ( 10993503642231 - 5787601131 \beta_{1} + 43650225 \beta_{2} - 1160651280 \beta_{3} + 220904301 \beta_{4} ) q^{36}$$ $$+ ( 3368357391548 + 8754001278 \beta_{1} - 39481320 \beta_{2} + 622465734 \beta_{3} + 24668412 \beta_{4} ) q^{37}$$ $$+ ( 9836556428359 - 26208197713 \beta_{1} + 704136 \beta_{2} - 1839748215 \beta_{3} - 209835141 \beta_{4} ) q^{38}$$ $$+ ( 14862914701093 - 42945662351 \beta_{1} - 13282254 \beta_{2} + 2287257745 \beta_{3} - 249316515 \beta_{4} ) q^{39}$$ $$+ ( -3508600429260 + 158940563780 \beta_{1} - 64883920 \beta_{2} + 2273031700 \beta_{3} + 160563020 \beta_{4} ) q^{40}$$ $$+ ( -6532831756800 - 8165306114 \beta_{1} + 5766808 \beta_{2} - 132422842 \beta_{3} + 375634108 \beta_{4} ) q^{41}$$ $$+ ( -16594619571411 + 25947369301 \beta_{1} - 138355224 \beta_{2} - 1412376245 \beta_{3} - 270945647 \beta_{4} ) q^{42}$$ $$+ ( 516409433069 + 59424810821 \beta_{1} - 104814106 \beta_{2} - 600172595 \beta_{3} - 529798273 \beta_{4} ) q^{43}$$ $$+ ( 65659790005396 + 48316151460 \beta_{1} + 652247728 \beta_{2} + 4214406452 \beta_{3} + 412530988 \beta_{4} ) q^{44}$$ $$+ ( -17890037072028 - 57264728106 \beta_{1} + 326566206 \beta_{2} - 8607326286 \beta_{3} + 1062054555 \beta_{4} ) q^{45}$$ $$+ ( -186905127768480 + 61649063776 \beta_{1} - 656773400 \beta_{2} + 7282492208 \beta_{3} - 127578728 \beta_{4} ) q^{46}$$ $$+ ( 14502781027898 - 84727738142 \beta_{1} + 302097480 \beta_{2} - 2036762790 \beta_{3} - 656214348 \beta_{4} ) q^{47}$$ $$+ ( -222547695603772 - 1013983129360 \beta_{1} - 486867618 \beta_{2} - 9390647086 \beta_{3} - 873992052 \beta_{4} ) q^{48}$$ $$+ 33232930569601 q^{49}$$ $$+ ( -241463030531965 + 258764863945 \beta_{1} - 1144203720 \beta_{2} + 9231804870 \beta_{3} - 367556550 \beta_{4} ) q^{50}$$ $$+ ( 18418419535632 + 869572564376 \beta_{1} - 368871780 \beta_{2} - 12517450048 \beta_{3} + 907502534 \beta_{4} ) q^{51}$$ $$+ ( 180113853940970 - 70301682326 \beta_{1} + 2080709716 \beta_{2} + 16107887414 \beta_{3} - 437277290 \beta_{4} ) q^{52}$$ $$+ ( 220545222127012 - 283879045370 \beta_{1} + 238867676 \beta_{2} + 8416776166 \beta_{3} + 289156070 \beta_{4} ) q^{53}$$ $$+ ( 84975280207182 + 259628160798 \beta_{1} + 746753040 \beta_{2} - 9138701358 \beta_{3} + 2640683574 \beta_{4} ) q^{54}$$ $$+ ( 352692519360498 - 280086081454 \beta_{1} + 808428764 \beta_{2} + 19520738146 \beta_{3} + 89554790 \beta_{4} ) q^{55}$$ $$+ ( 47954012875216 + 935638731902 \beta_{1} + 224827239 \beta_{2} + 9632982471 \beta_{3} + 3182170152 \beta_{4} ) q^{56}$$ $$+ ( 559231057555057 + 1594299803173 \beta_{1} - 939684042 \beta_{2} - 12227768051 \beta_{3} - 5871998745 \beta_{4} ) q^{57}$$ $$+ ( 2372620589856 + 1623036892524 \beta_{1} - 3141008832 \beta_{2} - 34380039318 \beta_{3} - 5974636242 \beta_{4} ) q^{58}$$ $$+ ( 33225756706741 - 874090288795 \beta_{1} - 5115150744 \beta_{2} + 37022044137 \beta_{3} + 1319925828 \beta_{4} ) q^{59}$$ $$+ ( 235789457542392 - 5902100987816 \beta_{1} + 4158512256 \beta_{2} - 132090033416 \beta_{3} + 119013160 \beta_{4} ) q^{60}$$ $$+ ( 749300201510918 + 1944360805584 \beta_{1} + 31413450 \beta_{2} + 7426079484 \beta_{3} + 5646911097 \beta_{4} ) q^{61}$$ $$+ ( -1521401052266066 - 4688500839922 \beta_{1} - 1788340608 \beta_{2} + 41161145418 \beta_{3} - 2306854386 \beta_{4} ) q^{62}$$ $$+ ( 637146213915540 + 135605413923 \beta_{1} + 2179094778 \beta_{2} + 17311697403 \beta_{3} - 8387785455 \beta_{4} ) q^{63}$$ $$+ ( 1335667155061222 + 5536229676528 \beta_{1} + 16931885169 \beta_{2} + 92120121375 \beta_{3} + 34907729970 \beta_{4} ) q^{64}$$ $$+ ( 1506626011034485 - 1572402587355 \beta_{1} + 1860077110 \beta_{2} + 98182907405 \beta_{3} - 6263390105 \beta_{4} ) q^{65}$$ $$+ ( -2887431768598200 - 2068090695960 \beta_{1} - 12698335584 \beta_{2} + 94816648200 \beta_{3} - 18029327112 \beta_{4} ) q^{66}$$ $$+ ( -757858474494685 + 3144234110763 \beta_{1} - 3580762290 \beta_{2} - 192073594629 \beta_{3} + 6733967619 \beta_{4} ) q^{67}$$ $$+ ( -5335627098771530 + 3669770866906 \beta_{1} - 20062375018 \beta_{2} - 154802091668 \beta_{3} - 52694960638 \beta_{4} ) q^{68}$$ $$+ ( 933273711985494 - 7568116608338 \beta_{1} + 17689378044 \beta_{2} + 209443844590 \beta_{3} + 41735757814 \beta_{4} ) q^{69}$$ $$+ ( -400008386500065 + 5347400583595 \beta_{1} - 3228288560 \beta_{2} + 11616074015 \beta_{3} + 3257112565 \beta_{4} ) q^{70}$$ $$+ ( -776849931671214 - 5199907668174 \beta_{1} - 5359474020 \beta_{2} - 271449957822 \beta_{3} - 26705809722 \beta_{4} ) q^{71}$$ $$+ ( -2269567221802632 + 15594273572934 \beta_{1} + 3892548807 \beta_{2} + 34741548975 \beta_{3} + 55177552128 \beta_{4} ) q^{72}$$ $$+ ( 1174574130486158 + 6188961506164 \beta_{1} - 3621879392 \beta_{2} - 127638944764 \beta_{3} - 31656815024 \beta_{4} ) q^{73}$$ $$+ ( 2320360224709504 - 4517641891780 \beta_{1} + 18877540176 \beta_{2} + 102820627890 \beta_{3} + 23228545302 \beta_{4} ) q^{74}$$ $$+ ( 1318026233379521 - 12098149114783 \beta_{1} + 30691034928 \beta_{2} + 63975323717 \beta_{3} + 59323329720 \beta_{4} ) q^{75}$$ $$+ ( -5052338894150452 + 6953059428576 \beta_{1} - 29142092550 \beta_{2} - 233322950418 \beta_{3} - 127426166868 \beta_{4} ) q^{76}$$ $$+ ( 883570121137039 - 1207477923057 \beta_{1} - 634128110 \beta_{2} + 962721767 \beta_{3} + 44902034989 \beta_{4} ) q^{77}$$ $$+ ( -8019343916142924 - 1013563127884 \beta_{1} - 35231890584 \beta_{2} + 539789616332 \beta_{3} - 41880160756 \beta_{4} ) q^{78}$$ $$+ ( 8684471390609876 - 16059889034700 \beta_{1} - 4229721312 \beta_{2} - 238754303292 \beta_{3} - 57705245808 \beta_{4} ) q^{79}$$ $$+ ( 14428413311570216 - 9416232925368 \beta_{1} + 99721613408 \beta_{2} + 12931789672 \beta_{3} + 219542874200 \beta_{4} ) q^{80}$$ $$+ ( 3338120577086118 - 5830675510815 \beta_{1} - 79801000578 \beta_{2} + 1327493913129 \beta_{3} - 167584553829 \beta_{4} ) q^{81}$$ $$+ ( -2579548298658084 + 7408174992192 \beta_{1} + 17204559696 \beta_{2} + 435187031922 \beta_{3} + 97908957270 \beta_{4} ) q^{82}$$ $$+ ( 7011972583073859 - 2347593003469 \beta_{1} - 12867694228 \beta_{2} - 1163616412313 \beta_{3} + 127579455758 \beta_{4} ) q^{83}$$ $$+ ( 4202441950442204 - 31360655795224 \beta_{1} - 3839357466 \beta_{2} - 342417649798 \beta_{3} - 148801043412 \beta_{4} ) q^{84}$$ $$+ ( -14487186412261638 + 30921222576374 \beta_{1} - 98252108224 \beta_{2} - 480779936306 \beta_{3} - 53653916320 \beta_{4} ) q^{85}$$ $$+ ( 13394054589677158 - 33138240775282 \beta_{1} + 18660372648 \beta_{2} - 1223614437378 \beta_{3} - 150777695406 \beta_{4} ) q^{86}$$ $$+ ( 20720983896121392 - 3467038879512 \beta_{1} + 58529041956 \beta_{2} - 1671773395392 \beta_{3} - 125560989414 \beta_{4} ) q^{87}$$ $$+ ( 21978323706088872 + 99005283546952 \beta_{1} + 52750501408 \beta_{2} + 1516273358504 \beta_{3} + 275824299736 \beta_{4} ) q^{88}$$ $$+ ( -11675583295651192 + 44827695797702 \beta_{1} + 130845881260 \beta_{2} + 1363894843142 \beta_{3} + 29864446126 \beta_{4} ) q^{89}$$ $$+ ( -14309834050121445 + 100447078770135 \beta_{1} - 87886905120 \beta_{2} - 486052098885 \beta_{3} + 71318116665 \beta_{4} ) q^{90}$$ $$+ ( 704769520269920 - 9140011278286 \beta_{1} - 14100703246 \beta_{2} - 210173114858 \beta_{3} + 186116600285 \beta_{4} ) q^{91}$$ $$+ ( -3213768361458088 - 211207563961800 \beta_{1} + 120254205632 \beta_{2} - 444962130344 \beta_{3} - 108540938872 \beta_{4} ) q^{92}$$ $$+ ( -3232575199481402 + 26814717176926 \beta_{1} + 50803411140 \beta_{2} + 4920634145998 \beta_{3} + 432633551274 \beta_{4} ) q^{93}$$ $$+ ( -17152115963731254 + 50376397140970 \beta_{1} - 165859266128 \beta_{2} - 442283461378 \beta_{3} - 237889501574 \beta_{4} ) q^{94}$$ $$+ ( -1194353511189347 + 50015288904681 \beta_{1} - 162721600166 \beta_{2} + 38257311641 \beta_{3} - 360637083455 \beta_{4} ) q^{95}$$ $$+ ( -154397584973681460 - 189813659015712 \beta_{1} - 535073755806 \beta_{2} - 1366318863426 \beta_{3} - 577660939932 \beta_{4} ) q^{96}$$ $$+ ( 9508537489517528 + 102890785456382 \beta_{1} + 361667667536 \beta_{2} + 1497406355158 \beta_{3} - 391538962360 \beta_{4} ) q^{97}$$ $$+ ( 3954718737782519 + 33232930569601 \beta_{1} ) q^{98}$$ $$+ ( -66004492991018733 - 12397248317229 \beta_{1} + 379384066314 \beta_{2} - 3354485585973 \beta_{3} + 916271207193 \beta_{4} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5q$$ $$\mathstrut +\mathstrut 597q^{2}$$ $$\mathstrut -\mathstrut 1770q^{3}$$ $$\mathstrut +\mathstrut 534677q^{4}$$ $$\mathstrut +\mathstrut 1612824q^{5}$$ $$\mathstrut -\mathstrut 14383854q^{6}$$ $$\mathstrut +\mathstrut 28824005q^{7}$$ $$\mathstrut +\mathstrut 41916039q^{8}$$ $$\mathstrut +\mathstrut 552658077q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$5q$$ $$\mathstrut +\mathstrut 597q^{2}$$ $$\mathstrut -\mathstrut 1770q^{3}$$ $$\mathstrut +\mathstrut 534677q^{4}$$ $$\mathstrut +\mathstrut 1612824q^{5}$$ $$\mathstrut -\mathstrut 14383854q^{6}$$ $$\mathstrut +\mathstrut 28824005q^{7}$$ $$\mathstrut +\mathstrut 41916039q^{8}$$ $$\mathstrut +\mathstrut 552658077q^{9}$$ $$\mathstrut -\mathstrut 345084900q^{10}$$ $$\mathstrut +\mathstrut 765945912q^{11}$$ $$\mathstrut +\mathstrut 3634044078q^{12}$$ $$\mathstrut +\mathstrut 608204548q^{13}$$ $$\mathstrut +\mathstrut 3441586197q^{14}$$ $$\mathstrut +\mathstrut 41156724744q^{15}$$ $$\mathstrut +\mathstrut 116025034769q^{16}$$ $$\mathstrut +\mathstrut 8856152334q^{17}$$ $$\mathstrut +\mathstrut 90679358061q^{18}$$ $$\mathstrut +\mathstrut 18441763546q^{19}$$ $$\mathstrut +\mathstrut 784710030552q^{20}$$ $$\mathstrut -\mathstrut 10203697770q^{21}$$ $$\mathstrut -\mathstrut 140711201256q^{22}$$ $$\mathstrut -\mathstrut 218776878696q^{23}$$ $$\mathstrut -\mathstrut 3751889532894q^{24}$$ $$\mathstrut -\mathstrut 363036196609q^{25}$$ $$\mathstrut -\mathstrut 1677859982400q^{26}$$ $$\mathstrut -\mathstrut 4021274734668q^{27}$$ $$\mathstrut +\mathstrut 3082306504277q^{28}$$ $$\mathstrut +\mathstrut 2087156686674q^{29}$$ $$\mathstrut -\mathstrut 14802803892720q^{30}$$ $$\mathstrut -\mathstrut 12100718234660q^{31}$$ $$\mathstrut +\mathstrut 16029734494815q^{32}$$ $$\mathstrut +\mathstrut 6151440714912q^{33}$$ $$\mathstrut +\mathstrut 19542664353462q^{34}$$ $$\mathstrut +\mathstrut 9297609408024q^{35}$$ $$\mathstrut +\mathstrut 54957458168325q^{36}$$ $$\mathstrut +\mathstrut 16858800794026q^{37}$$ $$\mathstrut +\mathstrut 49131784416030q^{38}$$ $$\mathstrut +\mathstrut 74225922854496q^{39}$$ $$\mathstrut -\mathstrut 17226943156560q^{40}$$ $$\mathstrut -\mathstrut 32679617238786q^{41}$$ $$\mathstrut -\mathstrut 82920055923054q^{42}$$ $$\mathstrut +\mathstrut 2700646991248q^{43}$$ $$\mathstrut +\mathstrut 328390888489968q^{44}$$ $$\mathstrut -\mathstrut 89554636513368q^{45}$$ $$\mathstrut -\mathstrut 934408564817712q^{46}$$ $$\mathstrut +\mathstrut 72344569802340q^{47}$$ $$\mathstrut -\mathstrut 1114757827879362q^{48}$$ $$\mathstrut +\mathstrut 166164652848005q^{49}$$ $$\mathstrut -\mathstrut 1206805301442465q^{50}$$ $$\mathstrut +\mathstrut 93846313005588q^{51}$$ $$\mathstrut +\mathstrut 900407522478772q^{52}$$ $$\mathstrut +\mathstrut 1102150036344942q^{53}$$ $$\mathstrut +\mathstrut 425408583919932q^{54}$$ $$\mathstrut +\mathstrut 1762881466153488q^{55}$$ $$\mathstrut +\mathstrut 241637623543239q^{56}$$ $$\mathstrut +\mathstrut 2799346250520276q^{57}$$ $$\mathstrut +\mathstrut 15137889518826q^{58}$$ $$\mathstrut +\mathstrut 164356451065122q^{59}$$ $$\mathstrut +\mathstrut 1167267096771552q^{60}$$ $$\mathstrut +\mathstrut 3750393534081568q^{61}$$ $$\mathstrut -\mathstrut 7616424461183148q^{62}$$ $$\mathstrut +\mathstrut 3185963834947677q^{63}$$ $$\mathstrut +\mathstrut 6689352066227393q^{64}$$ $$\mathstrut +\mathstrut 7529870820155880q^{65}$$ $$\mathstrut -\mathstrut 14441400503014176q^{66}$$ $$\mathstrut -\mathstrut 3782791201197452q^{67}$$ $$\mathstrut -\mathstrut 26670706415203410q^{68}$$ $$\mathstrut +\mathstrut 4651070975625744q^{69}$$ $$\mathstrut -\mathstrut 1989345776604900q^{70}$$ $$\mathstrut -\mathstrut 3894420716406000q^{71}$$ $$\mathstrut -\mathstrut 11316579733409625q^{72}$$ $$\mathstrut +\mathstrut 5885320144516618q^{73}$$ $$\mathstrut +\mathstrut 11592671721146322q^{74}$$ $$\mathstrut +\mathstrut 6565928157933906q^{75}$$ $$\mathstrut -\mathstrut 25247751597093326q^{76}$$ $$\mathstrut +\mathstrut 4415525759443512q^{77}$$ $$\mathstrut -\mathstrut 40099299793127064q^{78}$$ $$\mathstrut +\mathstrut 43390368978234280q^{79}$$ $$\mathstrut +\mathstrut 72123460802732256q^{80}$$ $$\mathstrut +\mathstrut 16677438473389329q^{81}$$ $$\mathstrut -\mathstrut 12883198921542810q^{82}$$ $$\mathstrut +\mathstrut 35056612240074642q^{83}$$ $$\mathstrut +\mathstrut 20949540934898478q^{84}$$ $$\mathstrut -\mathstrut 72373519639835328q^{85}$$ $$\mathstrut +\mathstrut 66904881205136496q^{86}$$ $$\mathstrut +\mathstrut 103599288996180588q^{87}$$ $$\mathstrut +\mathstrut 110088558971776416q^{88}$$ $$\mathstrut -\mathstrut 58289826944373966q^{89}$$ $$\mathstrut -\mathstrut 71347471630924500q^{90}$$ $$\mathstrut +\mathstrut 3506178186514948q^{91}$$ $$\mathstrut -\mathstrut 16491269563372704q^{92}$$ $$\mathstrut -\mathstrut 16113403536918888q^{93}$$ $$\mathstrut -\mathstrut 85659528801383844q^{94}$$ $$\mathstrut -\mathstrut 5872171066415592q^{95}$$ $$\mathstrut -\mathstrut 772366271041943550q^{96}$$ $$\mathstrut +\mathstrut 47745465198885454q^{97}$$ $$\mathstrut +\mathstrut 19840059550051797q^{98}$$ $$\mathstrut -\mathstrut 330042831191860392q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5}\mathstrut -\mathstrut$$ $$2$$ $$x^{4}\mathstrut -\mathstrut$$ $$559376$$ $$x^{3}\mathstrut +\mathstrut$$ $$70948970$$ $$x^{2}\mathstrut +\mathstrut$$ $$30882981215$$ $$x\mathstrut +\mathstrut$$ $$584478460232$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$431 \nu^{4} - 30823 \nu^{3} - 199426527 \nu^{2} + 52955726287 \nu + 61684487176$$$$)/46343808$$ $$\beta_{3}$$ $$=$$ $$($$$$57 \nu^{4} + 7871 \nu^{3} - 28704009 \nu^{2} + 1193532057 \nu + 1032327997240$$$$)/46343808$$ $$\beta_{4}$$ $$=$$ $$($$$$-29 \nu^{4} + 8901 \nu^{3} + 18707757 \nu^{2} - 5631802621 \nu - 900760238232$$$$)/6620544$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$4$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$189$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$223826$$ $$\nu^{3}$$ $$=$$ $$195$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$3099$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$318$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$449436$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$42077489$$ $$\nu^{4}$$ $$=$$ $$476652$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$1629201$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$547491$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$178177202$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$100413463809$$

## 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
 −774.082 −170.283 −19.9879 375.387 590.967
−655.082 15677.8 298061. 1.36270e6 −1.02702e7 5.76480e6 −1.09391e8 1.16652e8 −8.92682e8
1.2 −51.2830 1738.66 −128442. 190715. −89163.6 5.76480e6 1.33087e7 −1.26117e8 −9.78045e6
1.3 99.0121 −21601.2 −121269. −991914. −2.13878e6 5.76480e6 −2.49848e7 3.37472e8 −9.82115e7
1.4 494.387 16699.6 113346. 421257. 8.25605e6 5.76480e6 −8.76337e6 1.49736e8 2.08264e8
1.5 709.967 −14284.8 372981. 630065. −1.01417e7 5.76480e6 1.71747e8 7.49154e7 4.47325e8
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$T_{2}^{5}$$ $$\mathstrut -\mathstrut 597 T_{2}^{4}$$ $$\mathstrut -\mathstrut 416814 T_{2}^{3}$$ $$\mathstrut +\mathstrut 253624680 T_{2}^{2}$$ $$\mathstrut -\mathstrut 8750693376 T_{2}$$ $$\mathstrut -\mathstrut$$$$11\!\cdots\!40$$ acting on $$S_{18}^{\mathrm{new}}(\Gamma_0(7))$$.