# Properties

 Label 14.14.c.b Level $14$ Weight $14$ Character orbit 14.c Analytic conductor $15.012$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$14 = 2 \cdot 7$$ Weight: $$k$$ $$=$$ $$14$$ Character orbit: $$[\chi]$$ $$=$$ 14.c (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$15.0123300533$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ Defining polynomial: $$x^{8} + 209077 x^{6} - 47718852 x^{5} + 40973427094 x^{4} - 4988457209802 x^{3} + 1142094021456771 x^{2} + 65369216338084710 x + 7506311351102577225$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{14}\cdot 3\cdot 7^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 64 + 64 \beta_{1} ) q^{2} + ( -46 \beta_{1} - \beta_{2} ) q^{3} + 4096 \beta_{1} q^{4} + ( 444 + 443 \beta_{1} - 11 \beta_{2} - 11 \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{5} + ( 2944 + 64 \beta_{3} ) q^{6} + ( -79898 - 92720 \beta_{1} - 72 \beta_{2} - 10 \beta_{3} - 7 \beta_{5} + 6 \beta_{6} + 10 \beta_{7} ) q^{7} -262144 q^{8} + ( 149845 + 149818 \beta_{1} - 350 \beta_{2} - 311 \beta_{3} - 12 \beta_{5} + 66 \beta_{6} - 27 \beta_{7} ) q^{9} +O(q^{10})$$ $$q + ( 64 + 64 \beta_{1} ) q^{2} + ( -46 \beta_{1} - \beta_{2} ) q^{3} + 4096 \beta_{1} q^{4} + ( 444 + 443 \beta_{1} - 11 \beta_{2} - 11 \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{5} + ( 2944 + 64 \beta_{3} ) q^{6} + ( -79898 - 92720 \beta_{1} - 72 \beta_{2} - 10 \beta_{3} - 7 \beta_{5} + 6 \beta_{6} + 10 \beta_{7} ) q^{7} -262144 q^{8} + ( 149845 + 149818 \beta_{1} - 350 \beta_{2} - 311 \beta_{3} - 12 \beta_{5} + 66 \beta_{6} - 27 \beta_{7} ) q^{9} + ( 64 + 28352 \beta_{1} - 704 \beta_{2} - 64 \beta_{4} - 64 \beta_{7} ) q^{10} + ( -51 + 2183894 \beta_{1} + 4314 \beta_{2} - 68 \beta_{3} - 85 \beta_{4} - 68 \beta_{6} + 51 \beta_{7} ) q^{11} + ( 188416 + 188416 \beta_{1} + 4096 \beta_{2} + 4096 \beta_{3} ) q^{12} + ( 177215 - 455 \beta_{2} - 4602 \beta_{3} - 572 \beta_{4} - 572 \beta_{5} - 117 \beta_{6} + 455 \beta_{7} ) q^{13} + ( 821632 - 5112832 \beta_{1} - 640 \beta_{2} + 3968 \beta_{3} + 448 \beta_{4} + 1024 \beta_{6} - 384 \beta_{7} ) q^{14} + ( -17118177 - 81 \beta_{2} - 17577 \beta_{3} - 474 \beta_{4} - 474 \beta_{5} - 393 \beta_{6} + 81 \beta_{7} ) q^{15} + ( -16777216 - 16777216 \beta_{1} ) q^{16} + ( 5590 + 1966943 \beta_{1} - 39731 \beta_{2} + 3507 \beta_{3} + 1424 \beta_{4} + 3507 \beta_{6} - 5590 \beta_{7} ) q^{17} + ( 4224 + 9588352 \beta_{1} - 19904 \beta_{2} + 2496 \beta_{3} + 768 \beta_{4} + 2496 \beta_{6} - 4224 \beta_{7} ) q^{18} + ( -53933779 - 53932442 \beta_{1} - 14834 \beta_{2} - 10144 \beta_{3} - 6027 \beta_{5} + 3353 \beta_{6} + 1337 \beta_{7} ) q^{19} + ( -1814528 + 45056 \beta_{3} - 4096 \beta_{4} - 4096 \beta_{5} - 4096 \beta_{6} ) q^{20} + ( -89985826 - 76246917 \beta_{1} + 167571 \beta_{2} - 114775 \beta_{3} - 5103 \beta_{4} - 924 \beta_{5} + 5592 \beta_{6} - 480 \beta_{7} ) q^{21} + ( -139773568 - 4352 \beta_{2} - 280448 \beta_{3} - 5440 \beta_{4} - 5440 \beta_{5} - 1088 \beta_{6} + 4352 \beta_{7} ) q^{22} + ( -15324598 - 15326912 \beta_{1} + 289534 \beta_{2} + 307851 \beta_{3} - 16003 \beta_{5} + 20631 \beta_{6} - 2314 \beta_{7} ) q^{23} + ( 12058624 \beta_{1} + 262144 \beta_{2} ) q^{24} + ( 56056 + 331887768 \beta_{1} - 196826 \beta_{2} + 6090 \beta_{3} - 43876 \beta_{4} + 6090 \beta_{6} - 56056 \beta_{7} ) q^{25} + ( 11363392 + 11370880 \beta_{1} - 294528 \beta_{2} - 265408 \beta_{3} - 36608 \beta_{5} + 21632 \beta_{6} + 7488 \beta_{7} ) q^{26} + ( -408307114 - 24612 \beta_{2} + 576422 \beta_{3} - 29697 \beta_{4} - 29697 \beta_{5} - 5085 \beta_{6} + 24612 \beta_{7} ) q^{27} + ( 379846656 + 52559872 \beta_{1} + 253952 \beta_{2} + 294912 \beta_{3} + 28672 \beta_{4} + 28672 \beta_{5} + 40960 \beta_{6} - 65536 \beta_{7} ) q^{28} + ( -791103373 - 11411 \beta_{2} - 822080 \beta_{3} + 76850 \beta_{4} + 76850 \beta_{5} + 88261 \beta_{6} + 11411 \beta_{7} ) q^{29} + ( -1095583296 - 1095558144 \beta_{1} - 1124928 \beta_{2} - 1119744 \beta_{3} - 30336 \beta_{5} - 19968 \beta_{6} + 25152 \beta_{7} ) q^{30} + ( -179334 - 1529739620 \beta_{1} - 2500440 \beta_{2} - 88319 \beta_{3} + 2696 \beta_{4} - 88319 \beta_{6} + 179334 \beta_{7} ) q^{31} -1073741824 \beta_{1} q^{32} + ( 6310891299 + 6311028483 \beta_{1} + 4426134 \beta_{2} + 4213824 \beta_{3} + 75126 \beta_{5} - 349494 \beta_{6} + 137184 \beta_{7} ) q^{33} + ( -125659904 + 224448 \beta_{2} + 2767232 \beta_{3} + 91136 \beta_{4} + 91136 \beta_{5} - 133312 \beta_{6} - 224448 \beta_{7} ) q^{34} + ( 5042864253 - 296311862 \beta_{1} + 1601852 \beta_{2} - 2638195 \beta_{3} + 166551 \beta_{4} + 282044 \beta_{5} - 72352 \beta_{6} + 239449 \beta_{7} ) q^{35} + ( -613494784 + 159744 \beta_{2} + 1433600 \beta_{3} + 49152 \beta_{4} + 49152 \beta_{5} - 110592 \beta_{6} - 159744 \beta_{7} ) q^{36} + ( -991140065 - 991113389 \beta_{1} - 8919351 \beta_{2} - 9373662 \beta_{3} + 427635 \beta_{5} - 480987 \beta_{6} + 26676 \beta_{7} ) q^{37} + ( 214592 - 3451676288 \beta_{1} - 649216 \beta_{2} + 300160 \beta_{3} + 385728 \beta_{4} + 300160 \beta_{6} - 214592 \beta_{7} ) q^{38} + ( -710463 + 5889676228 \beta_{1} + 6805657 \beta_{2} - 457704 \beta_{3} - 204945 \beta_{4} - 457704 \beta_{6} + 710463 \beta_{7} ) q^{39} + ( -116391936 - 116129792 \beta_{1} + 2883584 \beta_{2} + 2883584 \beta_{3} - 262144 \beta_{5} - 262144 \beta_{6} + 262144 \beta_{7} ) q^{40} + ( 10806812441 - 536809 \beta_{2} + 18966454 \beta_{3} + 13440 \beta_{4} + 13440 \beta_{5} + 550249 \beta_{6} + 536809 \beta_{7} ) q^{41} + ( -878963008 - 5759123584 \beta_{1} - 7345600 \beta_{2} - 18070144 \beta_{3} - 267456 \beta_{4} - 326592 \beta_{5} + 327168 \beta_{6} - 357888 \beta_{7} ) q^{42} + ( -13620832758 - 212498 \beta_{2} + 26534270 \beta_{3} + 331278 \beta_{4} + 331278 \beta_{5} + 543776 \beta_{6} + 212498 \beta_{7} ) q^{43} + ( -8945299456 - 8945229824 \beta_{1} - 17948672 \beta_{2} - 17670144 \beta_{3} - 348160 \beta_{5} + 208896 \beta_{6} + 69632 \beta_{7} ) q^{44} + ( -24480 + 26241373590 \beta_{1} - 326973 \beta_{2} - 864549 \beta_{3} - 1704618 \beta_{4} - 864549 \beta_{6} + 24480 \beta_{7} ) q^{45} + ( 1320384 - 980922368 \beta_{1} + 19702464 \beta_{2} + 1172288 \beta_{3} + 1024192 \beta_{4} + 1172288 \beta_{6} - 1320384 \beta_{7} ) q^{46} + ( -781449895 - 783690780 \beta_{1} + 282815 \beta_{2} + 978916 \beta_{3} + 1544784 \beta_{5} + 2936986 \beta_{6} - 2240885 \beta_{7} ) q^{47} + ( -771751936 - 16777216 \beta_{3} ) q^{48} + ( 27980814470 + 18208405040 \beta_{1} + 57148217 \beta_{2} + 15191358 \beta_{3} - 1284290 \beta_{4} - 2543296 \beta_{5} - 1473451 \beta_{6} + 870891 \beta_{7} ) q^{49} + ( -21240427392 + 389760 \beta_{2} + 12986624 \beta_{3} - 2808064 \beta_{4} - 2808064 \beta_{5} - 3197824 \beta_{6} - 389760 \beta_{7} ) q^{50} + ( -60368799447 - 60368897838 \beta_{1} - 101552388 \beta_{2} - 98760336 \beta_{3} - 2693661 \beta_{5} + 2890443 \beta_{6} - 98391 \beta_{7} ) q^{51} + ( 1384448 + 727736320 \beta_{1} - 16986112 \beta_{2} + 1863680 \beta_{3} + 2342912 \beta_{4} + 1863680 \beta_{6} - 1384448 \beta_{7} ) q^{52} + ( -768145 - 37338164571 \beta_{1} + 137444132 \beta_{2} + 164941 \beta_{3} + 1098027 \beta_{4} + 164941 \beta_{6} + 768145 \beta_{7} ) q^{53} + ( -26130405568 - 26130080128 \beta_{1} + 36891008 \beta_{2} + 38466176 \beta_{3} - 1900608 \beta_{5} + 1249728 \beta_{6} + 325440 \beta_{7} ) q^{54} + ( 21922508839 + 2260251 \beta_{2} + 116943428 \beta_{3} + 71813 \beta_{4} + 71813 \beta_{5} - 2188438 \beta_{6} - 2260251 \beta_{7} ) q^{55} + ( 20944781312 + 24305991680 \beta_{1} + 18874368 \beta_{2} + 2621440 \beta_{3} + 1835008 \beta_{5} - 1572864 \beta_{6} - 2621440 \beta_{7} ) q^{56} + ( -16086173855 - 3491982 \beta_{2} - 118798736 \beta_{3} - 1676484 \beta_{4} - 1676484 \beta_{5} + 1815498 \beta_{6} + 3491982 \beta_{7} ) q^{57} + ( -50624236864 - 50629885568 \beta_{1} - 52613120 \beta_{2} - 51882816 \beta_{3} + 4918400 \beta_{5} + 6379008 \beta_{6} - 5648704 \beta_{7} ) q^{58} + ( 5651426 + 216452053366 \beta_{1} - 248302717 \beta_{2} + 1898946 \beta_{3} - 1853534 \beta_{4} + 1898946 \beta_{6} - 5651426 \beta_{7} ) q^{59} + ( -1277952 - 70115721216 \beta_{1} - 71663616 \beta_{2} + 331776 \beta_{3} + 1941504 \beta_{4} + 331776 \beta_{6} + 1277952 \beta_{7} ) q^{60} + ( -119501358733 - 119502118273 \beta_{1} - 50626425 \beta_{2} - 50698994 \beta_{3} + 832109 \beta_{5} + 686971 \beta_{6} - 759540 \beta_{7} ) q^{61} + ( 97897683264 - 5652416 \beta_{2} + 154375744 \beta_{3} + 172544 \beta_{4} + 172544 \beta_{5} + 5824960 \beta_{6} + 5652416 \beta_{7} ) q^{62} + ( 260047483353 + 281944012508 \beta_{1} + 151787429 \beta_{2} - 29983743 \beta_{3} + 5854758 \beta_{4} - 75411 \beta_{5} + 3614847 \beta_{6} + 4820619 \beta_{7} ) q^{63} + 68719476736 q^{64} + ( -274860395097 - 274849545582 \beta_{1} + 199074120 \beta_{2} + 185929911 \beta_{3} + 2294694 \beta_{5} - 23993724 \beta_{6} + 10849515 \beta_{7} ) q^{65} + ( -22367616 + 403905822912 \beta_{1} + 269684736 \beta_{2} - 13587840 \beta_{3} - 4808064 \beta_{4} - 13587840 \beta_{6} + 22367616 \beta_{7} ) q^{66} + ( 693304 - 473922451134 \beta_{1} - 98379179 \beta_{2} - 3291746 \beta_{3} - 7276796 \beta_{4} - 3291746 \beta_{6} - 693304 \beta_{7} ) q^{67} + ( -8065130496 - 8056598528 \beta_{1} + 177102848 \beta_{2} + 162738176 \beta_{3} + 5832704 \beta_{5} - 22896640 \beta_{6} + 8531968 \beta_{7} ) q^{68} + ( 437773445049 + 2535624 \beta_{2} - 273628641 \beta_{3} - 4955169 \beta_{4} - 4955169 \beta_{5} - 7490793 \beta_{6} - 2535624 \beta_{7} ) q^{69} + ( 341717965568 + 322758636928 \beta_{1} - 168844480 \beta_{2} - 271363008 \beta_{3} - 7391552 \beta_{4} + 10659264 \beta_{5} + 10694208 \beta_{6} + 4630528 \beta_{7} ) q^{70} + ( 80087447034 + 31584322 \beta_{2} + 420976584 \beta_{3} + 45749452 \beta_{4} + 45749452 \beta_{5} + 14165130 \beta_{6} - 31584322 \beta_{7} ) q^{71} + ( -39280967680 - 39273889792 \beta_{1} + 91750400 \beta_{2} + 81526784 \beta_{3} + 3145728 \beta_{5} - 17301504 \beta_{6} + 7077888 \beta_{7} ) q^{72} + ( 8306036 + 741704169099 \beta_{1} + 115366922 \beta_{2} + 13431138 \beta_{3} + 18556240 \beta_{4} + 13431138 \beta_{6} - 8306036 \beta_{7} ) q^{73} + ( -30783168 - 63431256896 \beta_{1} - 599914368 \beta_{2} - 29075904 \beta_{3} - 27368640 \beta_{4} - 29075904 \beta_{6} + 30783168 \beta_{7} ) q^{74} + ( -333099128354 - 333070769996 \beta_{1} - 566952482 \beta_{2} - 573247946 \beta_{3} - 22062894 \beta_{5} - 34653822 \beta_{6} + 28358358 \beta_{7} ) q^{75} + ( 220926492672 + 19210240 \beta_{2} + 60760064 \beta_{3} + 24686592 \beta_{4} + 24686592 \beta_{5} + 5476352 \beta_{6} - 19210240 \beta_{7} ) q^{76} + ( 988143463762 + 239417266207 \beta_{1} - 729844724 \beta_{2} + 718223800 \beta_{3} + 20579461 \beta_{4} + 20075790 \beta_{5} - 6925961 \beta_{6} - 22285228 \beta_{7} ) q^{77} + ( -376968571648 - 29293056 \beta_{2} - 464855104 \beta_{3} - 13116480 \beta_{4} - 13116480 \beta_{5} + 16176576 \beta_{6} + 29293056 \beta_{7} ) q^{78} + ( -1626309774286 - 1626307079524 \beta_{1} + 334886160 \beta_{2} + 368374531 \beta_{3} - 36183133 \beta_{5} + 30793609 \beta_{6} + 2694762 \beta_{7} ) q^{79} + ( -16777216 - 7432306688 \beta_{1} + 184549376 \beta_{2} + 16777216 \beta_{4} + 16777216 \beta_{7} ) q^{80} + ( 122559216 - 612232784743 \beta_{1} + 90160151 \beta_{2} + 70042011 \beta_{3} + 17524806 \beta_{4} + 70042011 \beta_{6} - 122559216 \beta_{7} ) q^{81} + ( 691705567936 + 691670352000 \beta_{1} + 1213853056 \beta_{2} + 1248208832 \beta_{3} + 860160 \beta_{5} + 69571712 \beta_{6} - 35215936 \beta_{7} ) q^{82} + ( -423648444804 - 16441880 \beta_{2} - 2314217022 \beta_{3} - 28388594 \beta_{4} - 28388594 \beta_{5} - 11946714 \beta_{6} + 16441880 \beta_{7} ) q^{83} + ( 312328310784 - 56276537344 \beta_{1} - 1156489216 \beta_{2} - 686370816 \beta_{3} + 3784704 \beta_{4} - 17117184 \beta_{5} - 1966080 \beta_{6} - 20938752 \beta_{7} ) q^{84} + ( -2085891475202 - 81103773 \beta_{2} - 508815769 \beta_{3} - 134889019 \beta_{4} - 134889019 \beta_{5} - 53785246 \beta_{6} + 81103773 \beta_{7} ) q^{85} + ( -871684894976 - 871719696640 \beta_{1} + 1698193280 \beta_{2} + 1711793152 \beta_{3} + 21201792 \beta_{5} + 48401536 \beta_{6} - 34801664 \beta_{7} ) q^{86} + ( 30202941 + 1319608349844 \beta_{1} + 2523611667 \beta_{2} - 7302834 \beta_{3} - 44808609 \beta_{4} - 7302834 \beta_{6} - 30202941 \beta_{7} ) q^{87} + ( 13369344 - 572494708736 \beta_{1} - 1130889216 \beta_{2} + 17825792 \beta_{3} + 22282240 \beta_{4} + 17825792 \beta_{6} - 13369344 \beta_{7} ) q^{88} + ( 2395850454569 + 2395798833069 \beta_{1} - 1819243402 \beta_{2} - 1754789096 \beta_{3} - 12832806 \beta_{5} + 116075806 \beta_{6} - 51621500 \beta_{7} ) q^{89} + ( -1679503240896 - 55331136 \beta_{2} - 34404864 \beta_{3} - 109095552 \beta_{4} - 109095552 \beta_{5} - 53764416 \beta_{6} + 55331136 \beta_{7} ) q^{90} + ( 3580508064406 + 3141683766992 \beta_{1} + 975723496 \beta_{2} + 2578368725 \beta_{3} - 100696687 \beta_{4} - 84227409 \beta_{5} + 13439485 \beta_{6} + 44337634 \beta_{7} ) q^{91} + ( 62854057984 + 75026432 \beta_{2} - 1185931264 \beta_{3} + 65548288 \beta_{4} + 65548288 \beta_{5} - 9478144 \beta_{6} - 75026432 \beta_{7} ) q^{92} + ( -3548429596009 - 3548548250131 \beta_{1} + 578815643 \beta_{2} + 658459064 \beta_{3} + 39010701 \beta_{5} + 198297543 \beta_{6} - 118654122 \beta_{7} ) q^{93} + ( 187967104 - 50156209920 \beta_{1} + 62650624 \beta_{2} + 44550464 \beta_{3} - 98866176 \beta_{4} + 44550464 \beta_{6} - 187967104 \beta_{7} ) q^{94} + ( -321524675 - 3595183601140 \beta_{1} + 2025630673 \beta_{2} - 133760211 \beta_{3} + 54004253 \beta_{4} - 133760211 \beta_{6} + 321524675 \beta_{7} ) q^{95} + ( -49392123904 - 49392123904 \beta_{1} - 1073741824 \beta_{2} - 1073741824 \beta_{3} ) q^{96} + ( -5570480839187 + 31967299 \beta_{2} - 937819484 \beta_{3} + 159969002 \beta_{4} + 159969002 \beta_{5} + 128001703 \beta_{6} - 31967299 \beta_{7} ) q^{97} + ( 625395639680 + 1790827863104 \beta_{1} + 972246912 \beta_{2} - 2685238976 \beta_{3} + 80576384 \beta_{4} - 82194560 \beta_{5} - 38563840 \beta_{6} + 94300864 \beta_{7} ) q^{98} + ( 3091481063682 + 127291050 \beta_{2} + 6304084647 \beta_{3} + 53046351 \beta_{4} + 53046351 \beta_{5} - 74244699 \beta_{6} - 127291050 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 256q^{2} + 182q^{3} - 16384q^{4} + 1792q^{5} + 23296q^{6} - 268352q^{7} - 2097152q^{8} + 599840q^{9} + O(q^{10})$$ $$8q + 256q^{2} + 182q^{3} - 16384q^{4} + 1792q^{5} + 23296q^{6} - 268352q^{7} - 2097152q^{8} + 599840q^{9} - 114688q^{10} - 8726914q^{11} + 745472q^{12} + 1438416q^{13} + 27004544q^{14} - 136873212q^{15} - 67108864q^{16} - 7943068q^{17} - 38389760q^{18} - 215706806q^{19} - 14680064q^{20} - 414124312q^{21} - 1117044992q^{22} - 61927978q^{23} - 47710208q^{24} - 1327844792q^{25} + 46029312q^{26} - 3268643812q^{27} + 2827460608q^{28} - 6325846064q^{29} - 4379942784q^{30} + 6113775570q^{31} + 4294967296q^{32} + 25235960652q^{33} - 1016712704q^{34} + 41542225382q^{35} - 4913889280q^{36} - 3945652880q^{37} + 13805235584q^{38} - 23545599116q^{39} - 469762048q^{40} + 86378579952q^{41} + 16061065216q^{42} - 109074124256q^{43} - 35745439744q^{44} - 104964468168q^{45} + 3963390592q^{46} - 3141202722q^{47} - 6106906624q^{48} + 151080461672q^{49} - 169964133376q^{50} - 241278267462q^{51} - 2945875968q^{52} + 149625680376q^{53} - 104596601984q^{54} + 174912009748q^{55} + 70346866688q^{56} - 128207489960q^{57} - 202427074048q^{58} - 866297313938q^{59} + 280316338176q^{60} - 477908594184q^{61} + 782563272960q^{62} + 953051390564q^{63} + 549755813888q^{64} - 1099748343120q^{65} - 1615101481728q^{66} + 1895501016278q^{67} - 32534806528q^{68} + 3503301895632q^{69} + 1443396653440q^{70} + 638832672128q^{71} - 157244456960q^{72} - 2966596192756q^{73} + 252521784320q^{74} - 1331079867376q^{75} + 1767070154752q^{76} + 6943031627392q^{77} - 3013836686848q^{78} - 6505959677634q^{79} + 30064771072q^{80} + 2449216493684q^{81} + 2764114558464q^{82} - 3379817135968q^{83} + 2724161355776q^{84} - 16684556982464q^{85} - 3490371976192q^{86} - 5273311164492q^{87} + 2287708143616q^{88} + 9586601667468q^{89} - 13435451925504q^{90} + 16069253407200q^{91} + 507313995776q^{92} - 14195747226896q^{93} + 201036974208q^{94} + 14384410136978q^{95} - 195421011968q^{96} - 44560735311568q^{97} - 2146516623104q^{98} + 24706419985464q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 209077 x^{6} - 47718852 x^{5} + 40973427094 x^{4} - 4988457209802 x^{3} + 1142094021456771 x^{2} + 65369216338084710 x + 7506311351102577225$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$36343564573613107681343 \nu^{7} - 3911745461420541052708455 \nu^{6} + 7122353933684804467943766146 \nu^{5} - 2500869301396565104975855035246 \nu^{4} + 1582452394760617117871338276154072 \nu^{3} - 330212894100361428024557552701671966 \nu^{2} + 40284736421255101261261116490164523803 \nu - 2240733124886924787542123968883350179225$$$$)/$$$$44\!\cdots\!80$$ $$\beta_{2}$$ $$=$$ $$($$$$-19245665182455297508653899719 \nu^{7} + 3626754114613992386988321873015 \nu^{6} - 6603452778174386406803937952075618 \nu^{5} + 1242412722272850209753755293685910478 \nu^{4} - 1467162367920194256697258163799496773976 \nu^{3} + 306155138208347124589081494569525382543678 \nu^{2} - 119920255574041349256887334773044499481407859 \nu + 2077483865088801616780561787398094760449644425$$$$)/$$$$65\!\cdots\!40$$ $$\beta_{3}$$ $$=$$ $$($$$$-158991540582276108146 \nu^{7} - 23003238574461279327135 \nu^{6} - 31158034105189152444784412 \nu^{5} + 3793446897148813713877484196 \nu^{4} - 4871566787094386785283230406789 \nu^{3} - 49709687876945726097309999179580 \nu^{2} - 5708136264639463982283216865573050 \nu - 17141819188993673802166662001668558819$$$$)/$$$$12\!\cdots\!59$$ $$\beta_{4}$$ $$=$$ $$($$$$8357847539494883782444119938677 \nu^{7} - 2711102675721777438597462081953445 \nu^{6} + 2937718815225000051364748287717864294 \nu^{5} - 558396945369105603640746344718536159274 \nu^{4} + 734951905785638046580119699784378625501848 \nu^{3} - 140566228738276471279871633654911284908895354 \nu^{2} + 35241573308346689975671016155995864658055561337 \nu + 1689218913382682802490892114189434471535294584005$$$$)/$$$$17\!\cdots\!80$$ $$\beta_{5}$$ $$=$$ $$($$$$-2097626037983599656116149039181 \nu^{7} - 1059283614420369301514289862491195 \nu^{6} - 803058927386536749689935908095479262 \nu^{5} - 156052317685030352686392832488180810918 \nu^{4} - 74687307372785502399468249861516149236604 \nu^{3} - 14964076009625927965745708625191621697650238 \nu^{2} - 2937208244110434333794801303587910539057978161 \nu - 852663352606556587181068376121206441976199517805$$$$)/$$$$44\!\cdots\!20$$ $$\beta_{6}$$ $$=$$ $$($$$$-6464778813206104496442177092891 \nu^{7} + 1573890822543971847263263268648235 \nu^{6} - 1177549124262279025567790566482846362 \nu^{5} + 548707706412438305726265026786898495222 \nu^{4} - 326450603258116348786176364081627046144744 \nu^{3} + 64782854440792042507622958272063554835776902 \nu^{2} - 8063403640505881184070111586175454490966758231 \nu - 1240593512581668349677606742851186702956568567435$$$$)/$$$$58\!\cdots\!60$$ $$\beta_{7}$$ $$=$$ $$($$$$53788658455044938453977448125 \nu^{7} - 2725367993960370052607819346585 \nu^{6} + 9720699835571375649835452974167054 \nu^{5} - 4050600765548118313433609688040201050 \nu^{4} + 1963931153151053437571443822532051271076 \nu^{3} - 440286613224709046625831074301596092166610 \nu^{2} + 50204275011954926977722253127963660258748225 \nu - 9280248143498228005836989600864226782346158835$$$$)/$$$$42\!\cdots\!24$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{7} - \beta_{6} - 3 \beta_{4} - \beta_{3} - 50 \beta_{2} - 24 \beta_{1} + 1$$$$)/336$$ $$\nu^{2}$$ $$=$$ $$($$$$890 \beta_{7} - 1345 \beta_{6} - 435 \beta_{5} - 262 \beta_{3} + 193 \beta_{2} - 35125512 \beta_{1} - 35126402$$$$)/336$$ $$\nu^{3}$$ $$=$$ $$($$$$-7549 \beta_{7} + 30461 \beta_{6} + 38010 \beta_{5} + 38010 \beta_{4} - 329923 \beta_{3} + 7549 \beta_{2} + 375640003$$$$)/21$$ $$\nu^{4}$$ $$=$$ $$($$$$-305067991 \beta_{7} + 71270609 \beta_{6} - 162526773 \beta_{4} + 71270609 \beta_{3} - 1138193126 \beta_{2} + 6422803061640 \beta_{1} + 305067991$$$$)/336$$ $$\nu^{5}$$ $$=$$ $$($$$$117654471422 \beta_{7} - 105997118719 \beta_{6} - 129311824125 \beta_{5} + 985682722142 \beta_{3} + 974025369439 \beta_{2} - 2094718013991480 \beta_{1} - 2094835668462902$$$$)/336$$ $$\nu^{6}$$ $$=$$ $$($$$$96184064357 \beta_{7} + 518498001245 \beta_{6} + 422313936888 \beta_{5} + 422313936888 \beta_{4} - 3377595081934 \beta_{3} - 96184064357 \beta_{2} + 12410821217365270$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$-28435930876253209 \beta_{7} - 405888753844057 \beta_{6} - 29247708383941323 \beta_{4} - 405888753844057 \beta_{3} - 218446718466434690 \beta_{2} + 574735439159406401160 \beta_{1} + 28435930876253209$$$$)/336$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
9.1
 −35.6648 − 61.7733i −248.013 − 429.572i 169.399 + 293.408i 114.279 + 197.937i −35.6648 + 61.7733i −248.013 + 429.572i 169.399 − 293.408i 114.279 − 197.937i
32.0000 55.4256i −746.854 1293.59i −2048.00 3547.24i −7178.20 + 12433.0i −95597.3 −302505. 73348.4i −262144. −318420. + 551520.i 459405. + 795712.i
9.2 32.0000 55.4256i −244.702 423.837i −2048.00 3547.24i 34096.3 59056.5i −31321.9 100530. + 294589.i −262144. 677403. 1.17330e6i −2.18216e6 3.77962e6i
9.3 32.0000 55.4256i 194.343 + 336.611i −2048.00 3547.24i −12951.1 + 22432.0i 24875.9 286520. 121636.i −262144. 721623. 1.24989e6i 828871. + 1.43565e6i
9.4 32.0000 55.4256i 888.214 + 1538.43i −2048.00 3547.24i −13071.0 + 22639.6i 113691. −218721. + 221472.i −262144. −780686. + 1.35219e6i 836543. + 1.44893e6i
11.1 32.0000 + 55.4256i −746.854 + 1293.59i −2048.00 + 3547.24i −7178.20 12433.0i −95597.3 −302505. + 73348.4i −262144. −318420. 551520.i 459405. 795712.i
11.2 32.0000 + 55.4256i −244.702 + 423.837i −2048.00 + 3547.24i 34096.3 + 59056.5i −31321.9 100530. 294589.i −262144. 677403. + 1.17330e6i −2.18216e6 + 3.77962e6i
11.3 32.0000 + 55.4256i 194.343 336.611i −2048.00 + 3547.24i −12951.1 22432.0i 24875.9 286520. + 121636.i −262144. 721623. + 1.24989e6i 828871. 1.43565e6i
11.4 32.0000 + 55.4256i 888.214 1538.43i −2048.00 + 3547.24i −13071.0 22639.6i 113691. −218721. 221472.i −262144. −780686. 1.35219e6i 836543. 1.44893e6i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 11.4 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 14.14.c.b 8
3.b odd 2 1 126.14.g.b 8
7.b odd 2 1 98.14.c.o 8
7.c even 3 1 inner 14.14.c.b 8
7.c even 3 1 98.14.a.h 4
7.d odd 6 1 98.14.a.j 4
7.d odd 6 1 98.14.c.o 8
21.h odd 6 1 126.14.g.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.14.c.b 8 1.a even 1 1 trivial
14.14.c.b 8 7.c even 3 1 inner
98.14.a.h 4 7.c even 3 1
98.14.a.j 4 7.d odd 6 1
98.14.c.o 8 7.b odd 2 1
98.14.c.o 8 7.d odd 6 1
126.14.g.b 8 3.b odd 2 1
126.14.g.b 8 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$77\!\cdots\!57$$$$T_{3}^{4} +$$$$79\!\cdots\!48$$$$T_{3}^{3} +$$$$14\!\cdots\!24$$$$T_{3}^{2} -$$$$10\!\cdots\!70$$$$T_{3} +$$$$25\!\cdots\!25$$">$$T_{3}^{8} - \cdots$$ acting on $$S_{14}^{\mathrm{new}}(14, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 4096 - 64 T + T^{2} )^{4}$$
$3$ $$25\!\cdots\!25$$$$-$$$$10\!\cdots\!70$$$$T + 1495307469814735824 T^{2} + 796865962371948 T^{3} + 7705719718857 T^{4} + 949684092 T^{5} + 2905288 T^{6} - 182 T^{7} + T^{8}$$
$5$ $$43\!\cdots\!25$$$$+$$$$57\!\cdots\!00$$$$T +$$$$55\!\cdots\!50$$$$T^{2} +$$$$26\!\cdots\!80$$$$T^{3} + 10139379896094153831 T^{4} + 180372900060528 T^{5} + 3106934278 T^{6} - 1792 T^{7} + T^{8}$$
$7$ $$88\!\cdots\!01$$$$+$$$$24\!\cdots\!36$$$$T -$$$$37\!\cdots\!16$$$$T^{2} -$$$$65\!\cdots\!28$$$$T^{3} +$$$$40\!\cdots\!06$$$$T^{4} - 6763360828435904 T^{5} - 39533832884 T^{6} + 268352 T^{7} + T^{8}$$
$11$ $$48\!\cdots\!25$$$$-$$$$19\!\cdots\!30$$$$T +$$$$88\!\cdots\!04$$$$T^{2} +$$$$28\!\cdots\!04$$$$T^{3} +$$$$20\!\cdots\!61$$$$T^{4} -$$$$13\!\cdots\!96$$$$T^{5} + 112179301147984 T^{6} + 8726914 T^{7} + T^{8}$$
$13$ $$($$$$29\!\cdots\!00$$$$-$$$$24\!\cdots\!20$$$$T - 549797025415320 T^{2} - 719208 T^{3} + T^{4} )^{2}$$
$17$ $$61\!\cdots\!41$$$$-$$$$76\!\cdots\!84$$$$T +$$$$30\!\cdots\!38$$$$T^{2} +$$$$25\!\cdots\!84$$$$T^{3} +$$$$66\!\cdots\!23$$$$T^{4} +$$$$17\!\cdots\!68$$$$T^{5} + 27198159130142842 T^{6} + 7943068 T^{7} + T^{8}$$
$19$ $$12\!\cdots\!25$$$$+$$$$35\!\cdots\!10$$$$T +$$$$85\!\cdots\!84$$$$T^{2} +$$$$53\!\cdots\!00$$$$T^{3} +$$$$39\!\cdots\!37$$$$T^{4} +$$$$11\!\cdots\!24$$$$T^{5} + 84562681108154656 T^{6} + 215706806 T^{7} + T^{8}$$
$23$ $$12\!\cdots\!09$$$$-$$$$23\!\cdots\!50$$$$T +$$$$53\!\cdots\!76$$$$T^{2} +$$$$17\!\cdots\!32$$$$T^{3} +$$$$67\!\cdots\!61$$$$T^{4} +$$$$37\!\cdots\!24$$$$T^{5} + 825289508152995676 T^{6} + 61927978 T^{7} + T^{8}$$
$29$ $$( -$$$$94\!\cdots\!96$$$$-$$$$87\!\cdots\!92$$$$T - 18126451463113919640 T^{2} + 3162923032 T^{3} + T^{4} )^{2}$$
$31$ $$15\!\cdots\!25$$$$-$$$$92\!\cdots\!10$$$$T +$$$$45\!\cdots\!24$$$$T^{2} -$$$$10\!\cdots\!44$$$$T^{3} +$$$$25\!\cdots\!09$$$$T^{4} -$$$$31\!\cdots\!96$$$$T^{5} + 63485338461969276948 T^{6} - 6113775570 T^{7} + T^{8}$$
$37$ $$82\!\cdots\!69$$$$+$$$$46\!\cdots\!04$$$$T +$$$$19\!\cdots\!06$$$$T^{2} -$$$$12\!\cdots\!92$$$$T^{3} +$$$$32\!\cdots\!59$$$$T^{4} -$$$$56\!\cdots\!64$$$$T^{5} +$$$$61\!\cdots\!66$$$$T^{6} + 3945652880 T^{7} + T^{8}$$
$41$ $$( -$$$$96\!\cdots\!16$$$$+$$$$79\!\cdots\!04$$$$T -$$$$11\!\cdots\!00$$$$T^{2} - 43189289976 T^{3} + T^{4} )^{2}$$
$43$ $$($$$$22\!\cdots\!00$$$$-$$$$67\!\cdots\!80$$$$T -$$$$15\!\cdots\!20$$$$T^{2} + 54537062128 T^{3} + T^{4} )^{2}$$
$47$ $$26\!\cdots\!21$$$$+$$$$26\!\cdots\!98$$$$T +$$$$20\!\cdots\!00$$$$T^{2} +$$$$61\!\cdots\!28$$$$T^{3} +$$$$14\!\cdots\!49$$$$T^{4} +$$$$99\!\cdots\!12$$$$T^{5} +$$$$11\!\cdots\!00$$$$T^{6} + 3141202722 T^{7} + T^{8}$$
$53$ $$91\!\cdots\!21$$$$+$$$$18\!\cdots\!32$$$$T +$$$$51\!\cdots\!22$$$$T^{2} -$$$$20\!\cdots\!48$$$$T^{3} +$$$$30\!\cdots\!03$$$$T^{4} -$$$$47\!\cdots\!24$$$$T^{5} +$$$$71\!\cdots\!78$$$$T^{6} - 149625680376 T^{7} + T^{8}$$
$59$ $$11\!\cdots\!25$$$$+$$$$83\!\cdots\!50$$$$T +$$$$68\!\cdots\!60$$$$T^{2} +$$$$13\!\cdots\!60$$$$T^{3} +$$$$83\!\cdots\!29$$$$T^{4} +$$$$21\!\cdots\!76$$$$T^{5} +$$$$68\!\cdots\!32$$$$T^{6} + 866297313938 T^{7} + T^{8}$$
$61$ $$90\!\cdots\!41$$$$+$$$$44\!\cdots\!04$$$$T +$$$$14\!\cdots\!70$$$$T^{2} +$$$$26\!\cdots\!36$$$$T^{3} +$$$$34\!\cdots\!59$$$$T^{4} +$$$$27\!\cdots\!76$$$$T^{5} +$$$$15\!\cdots\!70$$$$T^{6} + 477908594184 T^{7} + T^{8}$$
$67$ $$52\!\cdots\!21$$$$-$$$$71\!\cdots\!78$$$$T +$$$$69\!\cdots\!92$$$$T^{2} -$$$$29\!\cdots\!00$$$$T^{3} +$$$$90\!\cdots\!09$$$$T^{4} -$$$$17\!\cdots\!80$$$$T^{5} +$$$$23\!\cdots\!92$$$$T^{6} - 1895501016278 T^{7} + T^{8}$$
$71$ $$($$$$19\!\cdots\!56$$$$+$$$$12\!\cdots\!64$$$$T -$$$$34\!\cdots\!52$$$$T^{2} - 319416336064 T^{3} + T^{4} )^{2}$$
$73$ $$76\!\cdots\!25$$$$+$$$$82\!\cdots\!80$$$$T +$$$$64\!\cdots\!06$$$$T^{2} +$$$$20\!\cdots\!36$$$$T^{3} +$$$$46\!\cdots\!07$$$$T^{4} +$$$$62\!\cdots\!36$$$$T^{5} +$$$$60\!\cdots\!82$$$$T^{6} + 2966596192756 T^{7} + T^{8}$$
$79$ $$10\!\cdots\!25$$$$-$$$$22\!\cdots\!50$$$$T +$$$$89\!\cdots\!00$$$$T^{2} +$$$$13\!\cdots\!00$$$$T^{3} +$$$$13\!\cdots\!85$$$$T^{4} +$$$$72\!\cdots\!20$$$$T^{5} +$$$$29\!\cdots\!96$$$$T^{6} + 6505959677634 T^{7} + T^{8}$$
$83$ $$( -$$$$29\!\cdots\!64$$$$-$$$$45\!\cdots\!44$$$$T -$$$$15\!\cdots\!12$$$$T^{2} + 1689908567984 T^{3} + T^{4} )^{2}$$
$89$ $$25\!\cdots\!61$$$$-$$$$10\!\cdots\!08$$$$T +$$$$68\!\cdots\!02$$$$T^{2} -$$$$20\!\cdots\!00$$$$T^{3} +$$$$10\!\cdots\!39$$$$T^{4} -$$$$28\!\cdots\!80$$$$T^{5} +$$$$76\!\cdots\!62$$$$T^{6} - 9586601667468 T^{7} + T^{8}$$
$97$ $$( -$$$$20\!\cdots\!80$$$$-$$$$12\!\cdots\!68$$$$T +$$$$12\!\cdots\!20$$$$T^{2} + 22280367655784 T^{3} + T^{4} )^{2}$$