# Properties

 Label 14.10.c.a Level $14$ Weight $10$ Character orbit 14.c Analytic conductor $7.211$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$14 = 2 \cdot 7$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 14.c (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.21050170629$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ Defining polynomial: $$x^{6} - x^{5} + 1116 x^{4} - 3085 x^{3} + 1245325 x^{2} - 2341500 x + 4410000$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}\cdot 3\cdot 7^{2}$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -16 - 16 \beta_{2} ) q^{2} + ( 78 \beta_{2} - \beta_{3} ) q^{3} + 256 \beta_{2} q^{4} + ( -246 + 5 \beta_{1} - 246 \beta_{2} + 5 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{5} + ( 1248 - 16 \beta_{1} ) q^{6} + ( 134 - 31 \beta_{1} - 1423 \beta_{2} - \beta_{3} - 3 \beta_{4} + 8 \beta_{5} ) q^{7} + 4096 q^{8} + ( -5103 + 251 \beta_{1} - 5103 \beta_{2} + 251 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{9} +O(q^{10})$$ $$q + ( -16 - 16 \beta_{2} ) q^{2} + ( 78 \beta_{2} - \beta_{3} ) q^{3} + 256 \beta_{2} q^{4} + ( -246 + 5 \beta_{1} - 246 \beta_{2} + 5 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{5} + ( 1248 - 16 \beta_{1} ) q^{6} + ( 134 - 31 \beta_{1} - 1423 \beta_{2} - \beta_{3} - 3 \beta_{4} + 8 \beta_{5} ) q^{7} + 4096 q^{8} + ( -5103 + 251 \beta_{1} - 5103 \beta_{2} + 251 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{9} + ( 3936 \beta_{2} - 80 \beta_{3} - 16 \beta_{4} + 32 \beta_{5} ) q^{10} + ( -2475 \beta_{2} + 86 \beta_{3} - 51 \beta_{4} + 102 \beta_{5} ) q^{11} + ( -19968 + 256 \beta_{1} - 19968 \beta_{2} + 256 \beta_{3} ) q^{12} + ( 32789 + 151 \beta_{1} + 73 \beta_{4} + 73 \beta_{5} ) q^{13} + ( -24912 + 480 \beta_{1} - 2144 \beta_{2} + 496 \beta_{3} - 80 \beta_{4} - 48 \beta_{5} ) q^{14} + ( 123471 - 1294 \beta_{1} + 13 \beta_{4} + 13 \beta_{5} ) q^{15} + ( -65536 - 65536 \beta_{2} ) q^{16} + ( 101526 \beta_{2} + 2087 \beta_{3} + 145 \beta_{4} - 290 \beta_{5} ) q^{17} + ( 81648 \beta_{2} - 4016 \beta_{3} - 16 \beta_{4} + 32 \beta_{5} ) q^{18} + ( -125285 - 2136 \beta_{1} - 125285 \beta_{2} - 2136 \beta_{3} + 686 \beta_{4} - 343 \beta_{5} ) q^{19} + ( 62976 - 1280 \beta_{1} - 256 \beta_{4} - 256 \beta_{5} ) q^{20} + ( 52791 - 579 \beta_{1} + 635823 \beta_{2} - 5954 \beta_{3} - 173 \beta_{4} + 151 \beta_{5} ) q^{21} + ( -39600 + 1376 \beta_{1} - 816 \beta_{4} - 816 \beta_{5} ) q^{22} + ( -758040 + 6865 \beta_{1} - 758040 \beta_{2} + 6865 \beta_{3} + 312 \beta_{4} - 156 \beta_{5} ) q^{23} + ( 319488 \beta_{2} - 4096 \beta_{3} ) q^{24} + ( 47494 \beta_{2} + 130 \beta_{3} + 626 \beta_{4} - 1252 \beta_{5} ) q^{25} + ( -524624 - 2416 \beta_{1} - 524624 \beta_{2} - 2416 \beta_{3} - 2336 \beta_{4} + 1168 \beta_{5} ) q^{26} + ( 3567735 - 29026 \beta_{1} - 233 \beta_{4} - 233 \beta_{5} ) q^{27} + ( 364288 + 256 \beta_{1} + 398592 \beta_{2} - 7680 \beta_{3} + 2048 \beta_{4} - 1280 \beta_{5} ) q^{28} + ( -2185725 + 14197 \beta_{1} + 3387 \beta_{4} + 3387 \beta_{5} ) q^{29} + ( -1975536 + 20704 \beta_{1} - 1975536 \beta_{2} + 20704 \beta_{3} - 416 \beta_{4} + 208 \beta_{5} ) q^{30} + ( 2210729 \beta_{2} + 22330 \beta_{3} - 729 \beta_{4} + 1458 \beta_{5} ) q^{31} + 1048576 \beta_{2} q^{32} + ( 1251999 - 8020 \beta_{1} + 1251999 \beta_{2} - 8020 \beta_{3} - 2008 \beta_{4} + 1004 \beta_{5} ) q^{33} + ( 1624416 + 33392 \beta_{1} + 2320 \beta_{4} + 2320 \beta_{5} ) q^{34} + ( 2636136 + 60385 \beta_{1} - 2319945 \beta_{2} + 32614 \beta_{3} - 1775 \beta_{4} + 5452 \beta_{5} ) q^{35} + ( 1306368 - 64256 \beta_{1} - 256 \beta_{4} - 256 \beta_{5} ) q^{36} + ( -7425827 - 10488 \beta_{1} - 7425827 \beta_{2} - 10488 \beta_{3} - 8004 \beta_{4} + 4002 \beta_{5} ) q^{37} + ( 2004560 \beta_{2} + 34176 \beta_{3} - 5488 \beta_{4} + 10976 \beta_{5} ) q^{38} + ( -1052889 \beta_{2} + 6693 \beta_{3} - 1163 \beta_{4} + 2326 \beta_{5} ) q^{39} + ( -1007616 + 20480 \beta_{1} - 1007616 \beta_{2} + 20480 \beta_{3} + 8192 \beta_{4} - 4096 \beta_{5} ) q^{40} + ( 11338167 + 33597 \beta_{1} - 4477 \beta_{4} - 4477 \beta_{5} ) q^{41} + ( 9328512 - 86000 \beta_{1} - 844656 \beta_{2} + 9264 \beta_{3} + 352 \beta_{4} - 2768 \beta_{5} ) q^{42} + ( -20962888 + 64524 \beta_{1} - 14236 \beta_{4} - 14236 \beta_{5} ) q^{43} + ( 633600 - 22016 \beta_{1} + 633600 \beta_{2} - 22016 \beta_{3} + 26112 \beta_{4} - 13056 \beta_{5} ) q^{44} + ( 28849059 \beta_{2} - 246539 \beta_{3} + 18155 \beta_{4} - 36310 \beta_{5} ) q^{45} + ( 12128640 \beta_{2} - 109840 \beta_{3} - 2496 \beta_{4} + 4992 \beta_{5} ) q^{46} + ( -17543715 - 71874 \beta_{1} - 17543715 \beta_{2} - 71874 \beta_{3} + 7862 \beta_{4} - 3931 \beta_{5} ) q^{47} + ( 5111808 - 65536 \beta_{1} ) q^{48} + ( -2320038 - 181783 \beta_{1} - 23931530 \beta_{2} - 85274 \beta_{3} - 14007 \beta_{4} - 26789 \beta_{5} ) q^{49} + ( 759904 + 2080 \beta_{1} + 10016 \beta_{4} + 10016 \beta_{5} ) q^{50} + ( 32674131 - 286060 \beta_{1} + 32674131 \beta_{2} - 286060 \beta_{3} + 1046 \beta_{4} - 523 \beta_{5} ) q^{51} + ( 8393984 \beta_{2} + 38656 \beta_{3} + 18688 \beta_{4} - 37376 \beta_{5} ) q^{52} + ( 3866913 \beta_{2} + 490386 \beta_{3} - 2500 \beta_{4} + 5000 \beta_{5} ) q^{53} + ( -57083760 + 464416 \beta_{1} - 57083760 \beta_{2} + 464416 \beta_{3} + 7456 \beta_{4} - 3728 \beta_{5} ) q^{54} + ( 62775900 + 389999 \beta_{1} + 15916 \beta_{4} + 15916 \beta_{5} ) q^{55} + ( 548864 - 126976 \beta_{1} - 5828608 \beta_{2} - 4096 \beta_{3} - 12288 \beta_{4} + 32768 \beta_{5} ) q^{56} + ( -26480103 + 181474 \beta_{1} + 8310 \beta_{4} + 8310 \beta_{5} ) q^{57} + ( 34971600 - 227152 \beta_{1} + 34971600 \beta_{2} - 227152 \beta_{3} - 108384 \beta_{4} + 54192 \beta_{5} ) q^{58} + ( 4156638 \beta_{2} + 479983 \beta_{3} - 97084 \beta_{4} + 194168 \beta_{5} ) q^{59} + ( 31608576 \beta_{2} - 331264 \beta_{3} + 3328 \beta_{4} - 6656 \beta_{5} ) q^{60} + ( -53318603 - 296344 \beta_{1} - 53318603 \beta_{2} - 296344 \beta_{3} - 45308 \beta_{4} + 22654 \beta_{5} ) q^{61} + ( 35371664 + 357280 \beta_{1} - 11664 \beta_{4} - 11664 \beta_{5} ) q^{62} + ( -131462955 + 1095139 \beta_{1} - 143825463 \beta_{2} + 910603 \beta_{3} + 107026 \beta_{4} + 53857 \beta_{5} ) q^{63} + 16777216 q^{64} + ( 111244863 + 456681 \beta_{1} + 111244863 \beta_{2} + 456681 \beta_{3} + 153942 \beta_{4} - 76971 \beta_{5} ) q^{65} + ( -20031984 \beta_{2} + 128320 \beta_{3} + 16064 \beta_{4} - 32128 \beta_{5} ) q^{66} + ( 160205648 \beta_{2} + 273281 \beta_{3} + 10814 \beta_{4} - 21628 \beta_{5} ) q^{67} + ( -25990656 - 534272 \beta_{1} - 25990656 \beta_{2} - 534272 \beta_{3} - 74240 \beta_{4} + 37120 \beta_{5} ) q^{68} + ( 189196938 - 1974233 \beta_{1} - 4057 \beta_{4} - 4057 \beta_{5} ) q^{69} + ( -79297296 - 444336 \beta_{1} - 42178176 \beta_{2} - 966160 \beta_{3} - 58832 \beta_{4} - 28400 \beta_{5} ) q^{70} + ( -12174294 - 687838 \beta_{1} + 63998 \beta_{4} + 63998 \beta_{5} ) q^{71} + ( -20901888 + 1028096 \beta_{1} - 20901888 \beta_{2} + 1028096 \beta_{3} + 8192 \beta_{4} - 4096 \beta_{5} ) q^{72} + ( -83252491 \beta_{2} - 1624810 \beta_{3} + 160486 \beta_{4} - 320972 \beta_{5} ) q^{73} + ( 118813232 \beta_{2} + 167808 \beta_{3} + 64032 \beta_{4} - 128064 \beta_{5} ) q^{74} + ( 5470626 - 89554 \beta_{1} + 5470626 \beta_{2} - 89554 \beta_{3} + 22276 \beta_{4} - 11138 \beta_{5} ) q^{75} + ( 32072960 + 546816 \beta_{1} - 87808 \beta_{4} - 87808 \beta_{5} ) q^{76} + ( -254287791 - 1207305 \beta_{1} - 333233190 \beta_{2} - 1396631 \beta_{3} - 35918 \beta_{4} - 199901 \beta_{5} ) q^{77} + ( -16846224 + 107088 \beta_{1} - 18608 \beta_{4} - 18608 \beta_{5} ) q^{78} + ( 95137612 + 1081949 \beta_{1} + 95137612 \beta_{2} + 1081949 \beta_{3} - 309752 \beta_{4} + 154876 \beta_{5} ) q^{79} + ( 16121856 \beta_{2} - 327680 \beta_{3} - 65536 \beta_{4} + 131072 \beta_{5} ) q^{80} + ( 723195342 \beta_{2} - 3691439 \beta_{3} - 5149 \beta_{4} + 10298 \beta_{5} ) q^{81} + ( -181410672 - 537552 \beta_{1} - 181410672 \beta_{2} - 537552 \beta_{3} + 143264 \beta_{4} - 71632 \beta_{5} ) q^{82} + ( 382307766 + 667874 \beta_{1} + 74126 \beta_{4} + 74126 \beta_{5} ) q^{83} + ( -162770688 + 1524224 \beta_{1} - 149256192 \beta_{2} + 1376000 \beta_{3} + 38656 \beta_{4} + 5632 \beta_{5} ) q^{84} + ( -398372985 + 526418 \beta_{1} - 351908 \beta_{4} - 351908 \beta_{5} ) q^{85} + ( 335406208 - 1032384 \beta_{1} + 335406208 \beta_{2} - 1032384 \beta_{3} + 455552 \beta_{4} - 227776 \beta_{5} ) q^{86} + ( -472486995 \beta_{2} + 5261627 \beta_{3} - 46769 \beta_{4} + 93538 \beta_{5} ) q^{87} + ( -10137600 \beta_{2} + 352256 \beta_{3} - 208896 \beta_{4} + 417792 \beta_{5} ) q^{88} + ( -300776163 + 1084644 \beta_{1} - 300776163 \beta_{2} + 1084644 \beta_{3} + 486272 \beta_{4} - 243136 \beta_{5} ) q^{89} + ( 461584944 - 3944624 \beta_{1} + 290480 \beta_{4} + 290480 \beta_{5} ) q^{90} + ( -40023767 + 174867 \beta_{1} - 421551452 \beta_{2} - 1937432 \beta_{3} - 495453 \beta_{4} + 503839 \beta_{5} ) q^{91} + ( 194058240 - 1757440 \beta_{1} - 39936 \beta_{4} - 39936 \beta_{5} ) q^{92} + ( 237325281 - 1518954 \beta_{1} + 237325281 \beta_{2} - 1518954 \beta_{3} - 70904 \beta_{4} + 35452 \beta_{5} ) q^{93} + ( 280699440 \beta_{2} + 1149984 \beta_{3} - 62896 \beta_{4} + 125792 \beta_{5} ) q^{94} + ( 294681606 \beta_{2} + 3321359 \beta_{3} - 176630 \beta_{4} + 353260 \beta_{5} ) q^{95} + ( -81788928 + 1048576 \beta_{1} - 81788928 \beta_{2} + 1048576 \beta_{3} ) q^{96} + ( 102649241 + 6906215 \beta_{1} - 385911 \beta_{4} - 385911 \beta_{5} ) q^{97} + ( -345783872 + 1544144 \beta_{1} + 37120608 \beta_{2} + 2908528 \beta_{3} + 652736 \beta_{4} - 224112 \beta_{5} ) q^{98} + ( -209746629 + 1130453 \beta_{1} + 993781 \beta_{4} + 993781 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 48q^{2} - 233q^{3} - 768q^{4} - 733q^{5} + 7456q^{6} + 5012q^{7} + 24576q^{8} - 15058q^{9} + O(q^{10})$$ $$6q - 48q^{2} - 233q^{3} - 768q^{4} - 733q^{5} + 7456q^{6} + 5012q^{7} + 24576q^{8} - 15058q^{9} - 11728q^{10} + 7339q^{11} - 59648q^{12} + 197036q^{13} - 142576q^{14} + 738238q^{15} - 196608q^{16} - 306665q^{17} - 240928q^{18} - 377991q^{19} + 375296q^{20} - 1585927q^{21} - 234848q^{22} - 2267255q^{23} - 954368q^{24} - 142612q^{25} - 1576288q^{26} + 21348358q^{27} + 998144q^{28} - 13085956q^{29} - 5905904q^{30} - 6654517q^{31} - 3145728q^{32} + 3747977q^{33} + 9813280q^{34} + 22864807q^{35} + 7709696q^{36} - 22287969q^{37} - 6047856q^{38} + 3151974q^{39} - 3002368q^{40} + 68096196q^{41} + 58323776q^{42} - 125648280q^{43} + 1878784q^{44} - 86300638q^{45} - 36276080q^{46} - 52703019q^{47} + 30539776q^{48} + 57596070q^{49} + 4563584q^{50} + 97736333q^{51} - 25220608q^{52} - 12091125q^{53} - 170786864q^{54} + 377435398q^{55} + 20529152q^{56} - 158517670q^{57} + 104687648q^{58} - 12949897q^{59} - 94494464q^{60} - 160252153q^{61} + 212944544q^{62} - 356021666q^{63} + 100663296q^{64} + 334191270q^{65} + 59967632q^{66} - 480890225q^{67} - 78506240q^{68} + 1131233162q^{69} - 349171760q^{70} - 74421440q^{71} - 61677568q^{72} + 251382283q^{73} - 356607504q^{74} + 16322324q^{75} + 193531392q^{76} - 527045155q^{77} - 100863168q^{78} + 286494785q^{79} - 48037888q^{80} - 2165894587q^{81} - 544769568q^{82} + 2295182344q^{83} - 527183104q^{84} - 2389185074q^{85} + 1005186240q^{86} + 1412199358q^{87} + 30060544q^{88} - 901243845q^{89} + 2761620416q^{90} + 1026798920q^{91} + 1160834560q^{92} + 710456889q^{93} - 843248304q^{94} - 887366177q^{95} - 244318208q^{96} + 629707876q^{97} - 2185885296q^{98} - 1256218868q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 1116 x^{4} - 3085 x^{3} + 1245325 x^{2} - 2341500 x + 4410000$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-1081 \nu^{5} + 1206396 \nu^{4} + 41103264 \nu^{3} + 1346196325 \nu^{2} - 2531161500 \nu + 962136423000$$$$)/ 10405809000$$ $$\beta_{2}$$ $$=$$ $$($$$$20739 \nu^{5} - 20704 \nu^{4} + 23105664 \nu^{3} - 20388855 \nu^{2} + 25783208800 \nu - 48478416000$$$$)/ 48560442000$$ $$\beta_{3}$$ $$=$$ $$($$$$6283441 \nu^{5} - 5742096 \nu^{4} + 6408179136 \nu^{3} - 25009225165 \nu^{2} + 7150775701200 \nu - 13445117784000$$$$)/ 145681326000$$ $$\beta_{4}$$ $$=$$ $$($$$$5096423 \nu^{5} + 24024872 \nu^{4} + 5561870848 \nu^{3} + 11292574565 \nu^{2} + 6436810520600 \nu + 12258477336000$$$$)/ 24280221000$$ $$\beta_{5}$$ $$=$$ $$($$$$-1283932 \nu^{5} + 4959877 \nu^{4} - 1488519232 \nu^{3} + 9413731140 \nu^{2} - 1632210794525 \nu + 6114081939000$$$$)/ 6070055250$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{5} + 2 \beta_{4} - 15 \beta_{3} + 33 \beta_{2} - 15 \beta_{1} + 33$$$$)/84$$ $$\nu^{2}$$ $$=$$ $$($$$$34 \beta_{5} - 17 \beta_{4} - 60 \beta_{3} + 31254 \beta_{2}$$$$)/42$$ $$\nu^{3}$$ $$=$$ $$($$$$-1081 \beta_{5} - 1081 \beta_{4} + 16845 \beta_{1} + 77097$$$$)/84$$ $$\nu^{4}$$ $$=$$ $$($$$$-38929 \beta_{5} + 77858 \beta_{4} + 119145 \beta_{3} - 69550023 \beta_{2} + 119145 \beta_{1} - 69550023$$$$)/84$$ $$\nu^{5}$$ $$=$$ $$($$$$1237786 \beta_{5} - 618893 \beta_{4} + 9324660 \beta_{3} + 73839966 \beta_{2}$$$$)/42$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/14\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$-1 - \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
9.1
 16.4632 − 28.5151i 0.943118 − 1.63353i −16.9063 + 29.2826i 16.4632 + 28.5151i 0.943118 + 1.63353i −16.9063 − 29.2826i
−8.00000 + 13.8564i −135.410 234.536i −128.000 221.703i −684.785 + 1186.08i 4333.11 6327.81 + 558.972i 4096.00 −26830.1 + 46471.0i −10956.6 18977.3i
9.2 −8.00000 + 13.8564i 6.98734 + 12.1024i −128.000 221.703i 859.469 1488.64i −223.595 1802.93 + 6091.23i 4096.00 9743.85 16876.9i 13751.5 + 23818.3i
9.3 −8.00000 + 13.8564i 11.9224 + 20.6501i −128.000 221.703i −541.184 + 937.358i −381.515 −5624.74 2952.27i 4096.00 9557.22 16553.6i −8658.94 14997.7i
11.1 −8.00000 13.8564i −135.410 + 234.536i −128.000 + 221.703i −684.785 1186.08i 4333.11 6327.81 558.972i 4096.00 −26830.1 46471.0i −10956.6 + 18977.3i
11.2 −8.00000 13.8564i 6.98734 12.1024i −128.000 + 221.703i 859.469 + 1488.64i −223.595 1802.93 6091.23i 4096.00 9743.85 + 16876.9i 13751.5 23818.3i
11.3 −8.00000 13.8564i 11.9224 20.6501i −128.000 + 221.703i −541.184 937.358i −381.515 −5624.74 + 2952.27i 4096.00 9557.22 + 16553.6i −8658.94 + 14997.7i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 11.3 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 14.10.c.a 6
3.b odd 2 1 126.10.g.f 6
4.b odd 2 1 112.10.i.b 6
7.b odd 2 1 98.10.c.k 6
7.c even 3 1 inner 14.10.c.a 6
7.c even 3 1 98.10.a.j 3
7.d odd 6 1 98.10.a.i 3
7.d odd 6 1 98.10.c.k 6
21.h odd 6 1 126.10.g.f 6
28.g odd 6 1 112.10.i.b 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.10.c.a 6 1.a even 1 1 trivial
14.10.c.a 6 7.c even 3 1 inner
98.10.a.i 3 7.d odd 6 1
98.10.a.j 3 7.c even 3 1
98.10.c.k 6 7.b odd 2 1
98.10.c.k 6 7.d odd 6 1
112.10.i.b 6 4.b odd 2 1
112.10.i.b 6 28.g odd 6 1
126.10.g.f 6 3.b odd 2 1
126.10.g.f 6 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{6} + 233 T_{3}^{5} + 64198 T_{3}^{4} - 2489283 T_{3}^{3} + 77161662 T_{3}^{2} - 894217887 T_{3} + 8143799049$$ acting on $$S_{10}^{\mathrm{new}}(14, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 256 + 16 T + T^{2} )^{3}$$
$3$ $$8143799049 - 894217887 T + 77161662 T^{2} - 2489283 T^{3} + 64198 T^{4} + 233 T^{5} + T^{6}$$
$5$ $$6492888957535596225 + 6962338884679965 T + 9333499195206 T^{2} + 3093417753 T^{3} + 3269638 T^{4} + 733 T^{5} + T^{6}$$
$7$ $$65\!\cdots\!43$$$$- 8161608952727170388 T - 655260377382541 T^{2} + 108859207912 T^{3} - 16237963 T^{4} - 5012 T^{5} + T^{6}$$
$11$ $$34\!\cdots\!25$$$$-$$$$31\!\cdots\!75$$$$T + 29393401767411032406 T^{2} - 77234109908151 T^{3} + 5435780062 T^{4} - 7339 T^{5} + T^{6}$$
$13$ $$( -62445940634280 - 8685337220 T - 98518 T^{2} + T^{3} )^{2}$$
$17$ $$38\!\cdots\!89$$$$+$$$$28\!\cdots\!61$$$$T +$$$$26\!\cdots\!94$$$$T^{2} - 4548467624326611 T^{3} + 237514149958 T^{4} + 306665 T^{5} + T^{6}$$
$19$ $$13\!\cdots\!29$$$$-$$$$10\!\cdots\!25$$$$T +$$$$75\!\cdots\!82$$$$T^{2} - 185139708475646821 T^{3} + 440736113406 T^{4} + 377991 T^{5} + T^{6}$$
$23$ $$37\!\cdots\!61$$$$-$$$$18\!\cdots\!77$$$$T +$$$$23\!\cdots\!34$$$$T^{2} + 823791450941800803 T^{3} + 4831125048742 T^{4} + 2267255 T^{5} + T^{6}$$
$29$ $$( -$$$$12\!\cdots\!84$$$$- 16444066423332 T + 6542978 T^{2} + T^{3} )^{2}$$
$31$ $$11\!\cdots\!25$$$$-$$$$78\!\cdots\!95$$$$T +$$$$22\!\cdots\!06$$$$T^{2} + 70040367511099741897 T^{3} + 44052661351518 T^{4} + 6654517 T^{5} + T^{6}$$
$37$ $$45\!\cdots\!29$$$$+$$$$27\!\cdots\!85$$$$T +$$$$11\!\cdots\!62$$$$T^{2} +$$$$24\!\cdots\!41$$$$T^{3} + 368441349990006 T^{4} + 22287969 T^{5} + T^{6}$$
$41$ $$( -$$$$89\!\cdots\!08$$$$+ 318544865229276 T - 34048098 T^{2} + T^{3} )^{2}$$
$43$ $$( -$$$$54\!\cdots\!00$$$$+ 812958581920560 T + 62824140 T^{2} + T^{3} )^{2}$$
$47$ $$56\!\cdots\!09$$$$+$$$$17\!\cdots\!59$$$$T +$$$$45\!\cdots\!66$$$$T^{2} +$$$$35\!\cdots\!99$$$$T^{3} + 2019019098650814 T^{4} + 52703019 T^{5} + T^{6}$$
$53$ $$54\!\cdots\!01$$$$+$$$$15\!\cdots\!13$$$$T +$$$$47\!\cdots\!94$$$$T^{2} +$$$$38\!\cdots\!73$$$$T^{3} + 6805560938113062 T^{4} + 12091125 T^{5} + T^{6}$$
$59$ $$17\!\cdots\!25$$$$-$$$$31\!\cdots\!75$$$$T +$$$$56\!\cdots\!90$$$$T^{2} -$$$$29\!\cdots\!95$$$$T^{3} + 24394104860226214 T^{4} + 12949897 T^{5} + T^{6}$$
$61$ $$20\!\cdots\!21$$$$-$$$$21\!\cdots\!07$$$$T +$$$$30\!\cdots\!86$$$$T^{2} +$$$$86\!\cdots\!17$$$$T^{3} + 20848056568634646 T^{4} + 160252153 T^{5} + T^{6}$$
$67$ $$13\!\cdots\!89$$$$+$$$$27\!\cdots\!21$$$$T +$$$$37\!\cdots\!94$$$$T^{2} +$$$$28\!\cdots\!09$$$$T^{3} + 156596557267770038 T^{4} + 480890225 T^{5} + T^{6}$$
$71$ $$($$$$38\!\cdots\!16$$$$- 19772597378317632 T + 37210720 T^{2} + T^{3} )^{2}$$
$73$ $$70\!\cdots\!25$$$$+$$$$81\!\cdots\!85$$$$T +$$$$92\!\cdots\!06$$$$T^{2} +$$$$26\!\cdots\!97$$$$T^{3} + 160498810857408038 T^{4} - 251382283 T^{5} + T^{6}$$
$79$ $$19\!\cdots\!25$$$$+$$$$69\!\cdots\!75$$$$T +$$$$20\!\cdots\!50$$$$T^{2} +$$$$16\!\cdots\!75$$$$T^{3} + 131305608972835590 T^{4} - 286494785 T^{5} + T^{6}$$
$83$ $$( -$$$$47\!\cdots\!64$$$$+ 413638076348661360 T - 1147591172 T^{2} + T^{3} )^{2}$$
$89$ $$21\!\cdots\!69$$$$-$$$$50\!\cdots\!19$$$$T +$$$$53\!\cdots\!34$$$$T^{2} +$$$$19\!\cdots\!89$$$$T^{3} + 704138715796436838 T^{4} + 901243845 T^{5} + T^{6}$$
$97$ $$($$$$24\!\cdots\!80$$$$- 1522439324115043684 T - 314853938 T^{2} + T^{3} )^{2}$$