# Properties

 Label 1850.2.d.i Level $1850$ Weight $2$ Character orbit 1850.d Analytic conductor $14.772$ Analytic rank $0$ Dimension $20$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1850 = 2 \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1850.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.7723243739$$ Analytic rank: $$0$$ Dimension: $$20$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} + \cdots)$$ Defining polynomial: $$x^{20} + 48 x^{18} + 878 x^{16} + 8102 x^{14} + 41081 x^{12} + 115688 x^{10} + 175041 x^{8} + 134990 x^{6} + 49782 x^{4} + 6932 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{12}$$ Twist minimal: no (minimal twist has level 370) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{19}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{2} -\beta_{1} q^{3} - q^{4} -\beta_{2} q^{6} + \beta_{9} q^{7} + \beta_{3} q^{8} + ( 1 - \beta_{6} - \beta_{7} ) q^{9} +O(q^{10})$$ $$q -\beta_{3} q^{2} -\beta_{1} q^{3} - q^{4} -\beta_{2} q^{6} + \beta_{9} q^{7} + \beta_{3} q^{8} + ( 1 - \beta_{6} - \beta_{7} ) q^{9} -\beta_{7} q^{11} + \beta_{1} q^{12} + \beta_{17} q^{13} + \beta_{4} q^{14} + q^{16} + ( 2 \beta_{3} - \beta_{16} ) q^{17} + ( -\beta_{3} + \beta_{16} - \beta_{18} ) q^{18} + ( -\beta_{4} - \beta_{12} - \beta_{19} ) q^{19} + ( -1 + \beta_{6} + \beta_{7} + \beta_{14} ) q^{21} -\beta_{18} q^{22} + ( -\beta_{3} - \beta_{11} - \beta_{16} - \beta_{17} - \beta_{18} ) q^{23} + \beta_{2} q^{24} + \beta_{13} q^{26} + ( -2 \beta_{1} - \beta_{5} + \beta_{8} - \beta_{9} - \beta_{15} ) q^{27} -\beta_{9} q^{28} + ( \beta_{2} - \beta_{4} - \beta_{10} + \beta_{12} ) q^{29} + ( \beta_{2} + \beta_{4} + \beta_{12} ) q^{31} -\beta_{3} q^{32} + ( -2 \beta_{1} - \beta_{9} - \beta_{15} ) q^{33} + ( 2 - \beta_{6} ) q^{34} + ( -1 + \beta_{6} + \beta_{7} ) q^{36} + ( -\beta_{3} - \beta_{8} - \beta_{15} + \beta_{16} ) q^{37} + ( -\beta_{5} + \beta_{9} + \beta_{15} ) q^{38} + ( \beta_{2} - 2 \beta_{10} - \beta_{12} ) q^{39} + ( -\beta_{6} - \beta_{13} ) q^{41} + ( \beta_{3} - \beta_{11} - \beta_{16} + \beta_{18} ) q^{42} + ( \beta_{3} - \beta_{11} - 2 \beta_{16} + \beta_{18} ) q^{43} + \beta_{7} q^{44} + ( -1 - \beta_{6} + \beta_{7} - \beta_{13} - \beta_{14} ) q^{46} + ( -\beta_{5} - 2 \beta_{8} + \beta_{9} + \beta_{15} ) q^{47} -\beta_{1} q^{48} + ( 1 - \beta_{6} + 2 \beta_{7} ) q^{49} + ( 4 \beta_{2} - \beta_{10} - \beta_{12} ) q^{51} -\beta_{17} q^{52} + ( -\beta_{5} - \beta_{15} ) q^{53} + ( -2 \beta_{2} - \beta_{4} + \beta_{10} + \beta_{12} - \beta_{19} ) q^{54} -\beta_{4} q^{56} + ( \beta_{3} - \beta_{11} - \beta_{16} - \beta_{18} ) q^{57} + ( -\beta_{1} + \beta_{5} + \beta_{8} + \beta_{9} ) q^{58} + ( -2 \beta_{2} + \beta_{4} - \beta_{10} - \beta_{19} ) q^{59} + ( -\beta_{2} - \beta_{10} - \beta_{12} - \beta_{19} ) q^{61} + ( -\beta_{1} + \beta_{5} - \beta_{9} ) q^{62} + ( 4 \beta_{1} + \beta_{5} - \beta_{8} - \beta_{9} ) q^{63} - q^{64} + ( -2 \beta_{2} - \beta_{4} - \beta_{19} ) q^{66} + ( -\beta_{1} + 2 \beta_{5} + \beta_{9} - \beta_{15} ) q^{67} + ( -2 \beta_{3} + \beta_{16} ) q^{68} + ( -3 \beta_{2} + \beta_{10} - 2 \beta_{19} ) q^{69} + ( -2 - 2 \beta_{6} - 2 \beta_{13} - 2 \beta_{14} ) q^{71} + ( \beta_{3} - \beta_{16} + \beta_{18} ) q^{72} + ( -\beta_{1} - \beta_{8} + 2 \beta_{9} ) q^{73} + ( -1 + \beta_{6} - \beta_{10} - \beta_{19} ) q^{74} + ( \beta_{4} + \beta_{12} + \beta_{19} ) q^{76} + ( 4 \beta_{1} + \beta_{5} - 2 \beta_{8} - 4 \beta_{9} - \beta_{15} ) q^{77} + ( -\beta_{1} - \beta_{5} + 2 \beta_{8} ) q^{78} + ( \beta_{2} + 2 \beta_{4} - 2 \beta_{10} ) q^{79} + ( 6 - 2 \beta_{6} - 4 \beta_{7} + 3 \beta_{13} + \beta_{14} ) q^{81} + ( \beta_{16} + \beta_{17} ) q^{82} + ( 4 \beta_{1} - \beta_{8} + \beta_{9} - \beta_{15} ) q^{83} + ( 1 - \beta_{6} - \beta_{7} - \beta_{14} ) q^{84} + ( 1 - 2 \beta_{6} - \beta_{7} - \beta_{14} ) q^{86} + ( -\beta_{3} + \beta_{11} + 2 \beta_{16} - 2 \beta_{18} ) q^{87} + \beta_{18} q^{88} + ( 2 \beta_{4} + 2 \beta_{12} ) q^{89} + ( -2 \beta_{4} - \beta_{10} - \beta_{12} - 2 \beta_{19} ) q^{91} + ( \beta_{3} + \beta_{11} + \beta_{16} + \beta_{17} + \beta_{18} ) q^{92} + ( 3 \beta_{3} - \beta_{11} - \beta_{17} ) q^{93} + ( \beta_{4} - 2 \beta_{10} + \beta_{12} + \beta_{19} ) q^{94} -\beta_{2} q^{96} + ( \beta_{3} + \beta_{11} - 2 \beta_{16} + \beta_{18} ) q^{97} + ( -\beta_{3} + \beta_{16} + 2 \beta_{18} ) q^{98} + ( 9 - 3 \beta_{6} - 2 \beta_{7} + \beta_{13} + \beta_{14} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q - 20q^{4} + 16q^{9} + O(q^{10})$$ $$20q - 20q^{4} + 16q^{9} + 20q^{16} - 24q^{21} + 4q^{26} + 36q^{34} - 16q^{36} - 8q^{41} - 20q^{46} + 16q^{49} - 20q^{64} - 40q^{71} - 16q^{74} + 116q^{81} + 24q^{84} + 20q^{86} + 164q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} + 48 x^{18} + 878 x^{16} + 8102 x^{14} + 41081 x^{12} + 115688 x^{10} + 175041 x^{8} + 134990 x^{6} + 49782 x^{4} + 6932 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$54063874 \nu^{18} + 2574618933 \nu^{16} + 46502302940 \nu^{14} + 420755397517 \nu^{12} + 2066151235918 \nu^{10} + 5497605875904 \nu^{8} + 7437802265100 \nu^{6} + 4494358644653 \nu^{4} + 938448129224 \nu^{2} + 23257566513$$$$)/ 23403378928$$ $$\beta_{2}$$ $$=$$ $$($$$$-145812415 \nu^{19} - 7053059794 \nu^{17} - 130597919303 \nu^{15} - 1227874489270 \nu^{13} - 6410875218132 \nu^{11} - 18934897902438 \nu^{9} - 31020756809919 \nu^{7} - 27121020165950 \nu^{5} - 11753192288183 \nu^{3} - 1925816411076 \nu$$$$)/ 23403378928$$ $$\beta_{3}$$ $$=$$ $$($$$$145812415 \nu^{19} + 7053059794 \nu^{17} + 130597919303 \nu^{15} + 1227874489270 \nu^{13} + 6410875218132 \nu^{11} + 18934897902438 \nu^{9} + 31020756809919 \nu^{7} + 27121020165950 \nu^{5} + 11753192288183 \nu^{3} + 1949219790004 \nu$$$$)/ 23403378928$$ $$\beta_{4}$$ $$=$$ $$($$$$613303985 \nu^{19} + 32361753535 \nu^{17} + 677080261599 \nu^{15} + 7453904939383 \nu^{13} + 47405530585298 \nu^{11} + 177779949462390 \nu^{9} + 381323657841435 \nu^{7} + 428787705696787 \nu^{5} + 214864237440851 \nu^{3} + 37340664846289 \nu$$$$)/ 93613515712$$ $$\beta_{5}$$ $$=$$ $$($$$$-390695550 \nu^{18} - 18488174953 \nu^{16} - 330462274588 \nu^{14} - 2940177837725 \nu^{12} - 14037196563830 \nu^{10} - 35515491574376 \nu^{8} - 43565309579936 \nu^{6} - 21536413636769 \nu^{4} - 3234711267008 \nu^{2} - 13582288129$$$$)/ 23403378928$$ $$\beta_{6}$$ $$=$$ $$($$$$477639065 \nu^{18} + 22592949918 \nu^{16} + 403550267529 \nu^{14} + 3586552255918 \nu^{12} + 17095732850268 \nu^{10} + 43160087966538 \nu^{8} + 52827128777497 \nu^{6} + 26146987882498 \nu^{4} + 3868578012953 \nu^{2} - 68796237176$$$$)/ 23403378928$$ $$\beta_{7}$$ $$=$$ $$($$$$-585766813 \nu^{18} - 27742187784 \nu^{16} - 496554873409 \nu^{14} - 4428063050952 \nu^{12} - 21228035322104 \nu^{10} - 54155299718346 \nu^{8} - 67702733307697 \nu^{6} - 35135705171804 \nu^{4} - 5722070892473 \nu^{2} + 139297998790$$$$)/ 23403378928$$ $$\beta_{8}$$ $$=$$ $$($$$$1757120435 \nu^{18} + 83402607063 \nu^{16} + 1498220961617 \nu^{14} + 13437365531935 \nu^{12} + 65030997765010 \nu^{10} + 168720626332266 \nu^{8} + 218102515800005 \nu^{6} + 121853365919827 \nu^{4} + 23183106793557 \nu^{2} - 67439214595$$$$)/ 46806757856$$ $$\beta_{9}$$ $$=$$ $$($$$$-4057864633 \nu^{18} - 192353961175 \nu^{16} - 3448085901127 \nu^{14} - 30824311286719 \nu^{12} - 148394248802082 \nu^{10} - 381555708941190 \nu^{8} - 485136079342291 \nu^{6} - 263169235311499 \nu^{4} - 49562210695387 \nu^{2} - 455501597545$$$$)/ 93613515712$$ $$\beta_{10}$$ $$=$$ $$($$$$2241687127 \nu^{19} + 106747249363 \nu^{17} + 1927702590245 \nu^{15} + 17435486160835 \nu^{13} + 85587766275690 \nu^{11} + 227996014942130 \nu^{9} + 311746546912209 \nu^{7} + 200221136329103 \nu^{5} + 56783144665785 \nu^{3} + 5898561406361 \nu$$$$)/ 46806757856$$ $$\beta_{11}$$ $$=$$ $$($$$$-1206828866 \nu^{19} - 56678790431 \nu^{17} - 1000449551428 \nu^{15} - 8719105962891 \nu^{13} - 40135081226378 \nu^{11} - 94311341550480 \nu^{9} - 95432421260152 \nu^{7} - 17331781380231 \nu^{5} + 16880673980776 \nu^{3} + 5276190286737 \nu$$$$)/ 23403378928$$ $$\beta_{12}$$ $$=$$ $$($$$$1208955653 \nu^{19} + 57331395806 \nu^{17} + 1028299056857 \nu^{15} + 9198988235086 \nu^{13} + 44318132548420 \nu^{11} + 113977821506138 \nu^{9} + 144442103626133 \nu^{7} + 76270079866490 \nu^{5} + 11365133627849 \nu^{3} - 1089782663136 \nu$$$$)/ 23403378928$$ $$\beta_{13}$$ $$=$$ $$($$$$-652713292 \nu^{18} - 30975328581 \nu^{16} - 556220975934 \nu^{14} - 4985087030133 \nu^{12} - 24092820792814 \nu^{10} - 62340308954316 \nu^{8} - 80121681186158 \nu^{6} - 44166472452589 \nu^{4} - 8205980137890 \nu^{2} + 39702760795$$$$)/ 11701689464$$ $$\beta_{14}$$ $$=$$ $$($$$$926692235 \nu^{18} + 43943784426 \nu^{16} + 788127547195 \nu^{14} + 7050093005546 \nu^{12} + 33968937091892 \nu^{10} + 87438358868934 \nu^{8} + 111329537278091 \nu^{6} + 60432105366038 \nu^{4} + 11258177979467 \nu^{2} + 33311845352$$$$)/ 11701689464$$ $$\beta_{15}$$ $$=$$ $$($$$$8135398299 \nu^{18} + 385482610181 \nu^{16} + 6905411302437 \nu^{14} + 61664032838653 \nu^{12} + 296309748186694 \nu^{10} + 759209179600082 \nu^{8} + 958061198786473 \nu^{6} + 510146287390145 \nu^{4} + 91317132566849 \nu^{2} - 277209785317$$$$)/ 93613515712$$ $$\beta_{16}$$ $$=$$ $$($$$$-4287569963 \nu^{19} - 204436380315 \nu^{17} - 3699591846113 \nu^{15} - 33571856076251 \nu^{13} - 165676066544522 \nu^{11} - 445371073631082 \nu^{9} - 619059523849157 \nu^{7} - 408951400960887 \nu^{5} - 118789025048309 \nu^{3} - 11559854198745 \nu$$$$)/ 46806757856$$ $$\beta_{17}$$ $$=$$ $$($$$$2291028611 \nu^{19} + 109181047801 \nu^{17} + 1974157614537 \nu^{15} + 17892174824925 \nu^{13} + 88131132456614 \nu^{11} + 236228116442074 \nu^{9} + 326966044975945 \nu^{7} + 215279291456365 \nu^{5} + 63489510306037 \nu^{3} + 6814000675891 \nu$$$$)/ 23403378928$$ $$\beta_{18}$$ $$=$$ $$($$$$-5270572201 \nu^{19} - 251903976945 \nu^{17} - 4576183967811 \nu^{15} - 41780613806905 \nu^{13} - 208273620325150 \nu^{11} - 569975672797518 \nu^{9} - 820059669238871 \nu^{7} - 580666239245893 \nu^{5} - 191185075035855 \nu^{3} - 23348494829651 \nu$$$$)/ 46806757856$$ $$\beta_{19}$$ $$=$$ $$($$$$13437406989 \nu^{19} + 638839429843 \nu^{17} + 11506290522083 \nu^{15} + 103642570792395 \nu^{13} + 505340853098474 \nu^{11} + 1330271778225534 \nu^{9} + 1777772668346447 \nu^{7} + 1090055754809815 \nu^{5} + 284419734506215 \nu^{3} + 24205664106461 \nu$$$$)/ 93613515712$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{3} + \beta_{2}$$ $$\nu^{2}$$ $$=$$ $$\beta_{7} + \beta_{6} + 2 \beta_{1} - 5$$ $$\nu^{3}$$ $$=$$ $$-\beta_{19} - 3 \beta_{18} + 3 \beta_{16} + \beta_{12} + \beta_{10} - \beta_{4} - 13 \beta_{3} - 11 \beta_{2}$$ $$\nu^{4}$$ $$=$$ $$-4 \beta_{15} + \beta_{14} + 3 \beta_{13} - 4 \beta_{9} + 4 \beta_{8} - 19 \beta_{7} - 17 \beta_{6} - 4 \beta_{5} - 36 \beta_{1} + 58$$ $$\nu^{5}$$ $$=$$ $$24 \beta_{19} + 75 \beta_{18} + 15 \beta_{17} - 65 \beta_{16} - 24 \beta_{12} + 5 \beta_{11} - 27 \beta_{10} + 22 \beta_{4} + 206 \beta_{3} + 170 \beta_{2}$$ $$\nu^{6}$$ $$=$$ $$104 \beta_{15} - 31 \beta_{14} - 90 \beta_{13} + 92 \beta_{9} - 122 \beta_{8} + 363 \beta_{7} + 305 \beta_{6} + 104 \beta_{5} + 676 \beta_{1} - 902$$ $$\nu^{7}$$ $$=$$ $$-498 \beta_{19} - 1547 \beta_{18} - 420 \beta_{17} + 1281 \beta_{16} + 499 \beta_{12} - 147 \beta_{11} + 607 \beta_{10} - 424 \beta_{4} - 3662 \beta_{3} - 3035 \beta_{2}$$ $$\nu^{8}$$ $$=$$ $$-2192 \beta_{15} + 719 \beta_{14} + 2024 \beta_{13} - 1824 \beta_{9} + 2728 \beta_{8} - 7000 \beta_{7} - 5738 \beta_{6} - 2200 \beta_{5} - 12920 \beta_{1} + 16010$$ $$\nu^{9}$$ $$=$$ $$9911 \beta_{19} + 30528 \beta_{18} + 9144 \beta_{17} - 24882 \beta_{16} - 9962 \beta_{12} + 3279 \beta_{11} - 12514 \beta_{10} + 8105 \beta_{4} + 68458 \beta_{3} + 57149 \beta_{2}$$ $$\nu^{10}$$ $$=$$ $$43718 \beta_{15} - 14996 \beta_{14} - 41531 \beta_{13} + 35354 \beta_{9} - 55684 \beta_{8} + 135528 \beta_{7} + 110060 \beta_{6} + 43988 \beta_{5} + 248850 \beta_{1} - 300055$$ $$\nu^{11}$$ $$=$$ $$-194242 \beta_{19} - 595144 \beta_{18} - 184921 \beta_{17} + 482240 \beta_{16} + 195579 \beta_{12} - 67078 \beta_{11} + 248806 \beta_{10} - 155886 \beta_{4} - 1307417 \beta_{3} - 1096608 \beta_{2}$$ $$\nu^{12}$$ $$=$$ $$-856464 \beta_{15} + 299676 \beta_{14} + 823548 \beta_{13} - 683952 \beta_{9} + 1100888 \beta_{8} - 2627280 \beta_{7} - 2125892 \beta_{6} - 862740 \beta_{5} - 4810792 \beta_{1} + 5743281$$ $$\nu^{13}$$ $$=$$ $$3783420 \beta_{19} + 11560432 \beta_{18} + 3643640 \beta_{17} - 9346116 \beta_{16} - 3812180 \beta_{12} + 1328652 \beta_{11} - 4873876 \beta_{10} + 3011556 \beta_{4} + 25194105 \beta_{3} + 21183641 \beta_{2}$$ $$\nu^{14}$$ $$=$$ $$16672504 \beta_{15} - 5883936 \beta_{14} - 16113116 \beta_{13} + 13243336 \beta_{9} - 21507272 \beta_{8} + 50946829 \beta_{7} + 41165673 \beta_{6} + 16801936 \beta_{5} + 93163134 \beta_{1} - 110816833$$ $$\nu^{15}$$ $$=$$ $$-73503269 \beta_{19} - 224302179 \beta_{18} - 71094828 \beta_{17} + 181176015 \beta_{16} + 74080725 \beta_{12} - 25985608 \beta_{11} + 94899177 \beta_{10} - 58306229 \beta_{4} - 487302033 \beta_{3} - 410202023 \beta_{2}$$ $$\nu^{16}$$ $$=$$ $$-323791056 \beta_{15} + 114685917 \beta_{14} + 313577999 \beta_{13} - 256622800 \beta_{9} + 418264848 \beta_{8} - 987978419 \beta_{7} - 797845717 \beta_{6} - 326351568 \beta_{5} - 1805540880 \beta_{1} + 2144779450$$ $$\nu^{17}$$ $$=$$ $$1426455392 \beta_{19} + 4350427347 \beta_{18} + 1381985471 \beta_{17} - 3512712533 \beta_{16} - 1437775284 \beta_{12} + 505645229 \beta_{11} - 1843266563 \beta_{10} + 1129915302 \beta_{4} + 9439739210 \beta_{3} + 7950232518 \beta_{2}$$ $$\nu^{18}$$ $$=$$ $$6282527968 \beta_{15} - 2228640711 \beta_{14} - 6089482710 \beta_{13} + 4974697420 \beta_{9} - 8119950038 \beta_{8} + 19159036519 \beta_{7} + 15468410921 \beta_{6} + 6332473288 \beta_{5} + 35003882188 \beta_{1} - 41559602026$$ $$\nu^{19}$$ $$=$$ $$-27670205198 \beta_{19} - 84367618639 \beta_{18} - 26824434004 \beta_{17} + 68111937105 \beta_{16} + 27890366919 \beta_{12} - 9818999227 \beta_{11} + 35767326379 \beta_{10} - 21905093228 \beta_{4} - 182974874698 \beta_{3} - 154136502515 \beta_{2}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1850\mathbb{Z}\right)^\times$$.

 $$n$$ $$1001$$ $$1777$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1701.1
 − 2.40359i − 0.786469i − 0.762160i 0.0120170i 0.622139i 1.37786i 1.98798i 2.76216i 2.78647i 4.40359i 2.40359i 0.786469i 0.762160i − 0.0120170i − 0.622139i − 1.37786i − 1.98798i − 2.76216i − 2.78647i − 4.40359i
1.00000i −3.40359 −1.00000 0 3.40359i 2.06225 1.00000i 8.58443 0
1701.2 1.00000i −1.78647 −1.00000 0 1.78647i 3.14934 1.00000i 0.191472 0
1701.3 1.00000i −1.76216 −1.00000 0 1.76216i −1.22131 1.00000i 0.105209 0
1701.4 1.00000i −0.987983 −1.00000 0 0.987983i −4.78937 1.00000i −2.02389 0
1701.5 1.00000i −0.377861 −1.00000 0 0.377861i 0.631751 1.00000i −2.85722 0
1701.6 1.00000i 0.377861 −1.00000 0 0.377861i −0.631751 1.00000i −2.85722 0
1701.7 1.00000i 0.987983 −1.00000 0 0.987983i 4.78937 1.00000i −2.02389 0
1701.8 1.00000i 1.76216 −1.00000 0 1.76216i 1.22131 1.00000i 0.105209 0
1701.9 1.00000i 1.78647 −1.00000 0 1.78647i −3.14934 1.00000i 0.191472 0
1701.10 1.00000i 3.40359 −1.00000 0 3.40359i −2.06225 1.00000i 8.58443 0
1701.11 1.00000i −3.40359 −1.00000 0 3.40359i 2.06225 1.00000i 8.58443 0
1701.12 1.00000i −1.78647 −1.00000 0 1.78647i 3.14934 1.00000i 0.191472 0
1701.13 1.00000i −1.76216 −1.00000 0 1.76216i −1.22131 1.00000i 0.105209 0
1701.14 1.00000i −0.987983 −1.00000 0 0.987983i −4.78937 1.00000i −2.02389 0
1701.15 1.00000i −0.377861 −1.00000 0 0.377861i 0.631751 1.00000i −2.85722 0
1701.16 1.00000i 0.377861 −1.00000 0 0.377861i −0.631751 1.00000i −2.85722 0
1701.17 1.00000i 0.987983 −1.00000 0 0.987983i 4.78937 1.00000i −2.02389 0
1701.18 1.00000i 1.76216 −1.00000 0 1.76216i 1.22131 1.00000i 0.105209 0
1701.19 1.00000i 1.78647 −1.00000 0 1.78647i −3.14934 1.00000i 0.191472 0
1701.20 1.00000i 3.40359 −1.00000 0 3.40359i −2.06225 1.00000i 8.58443 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1701.20 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
37.b even 2 1 inner
185.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1850.2.d.i 20
5.b even 2 1 inner 1850.2.d.i 20
5.c odd 4 1 370.2.c.a 10
5.c odd 4 1 370.2.c.b yes 10
15.e even 4 1 3330.2.e.c 10
15.e even 4 1 3330.2.e.d 10
37.b even 2 1 inner 1850.2.d.i 20
185.d even 2 1 inner 1850.2.d.i 20
185.h odd 4 1 370.2.c.a 10
185.h odd 4 1 370.2.c.b yes 10
555.n even 4 1 3330.2.e.c 10
555.n even 4 1 3330.2.e.d 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.c.a 10 5.c odd 4 1
370.2.c.a 10 185.h odd 4 1
370.2.c.b yes 10 5.c odd 4 1
370.2.c.b yes 10 185.h odd 4 1
1850.2.d.i 20 1.a even 1 1 trivial
1850.2.d.i 20 5.b even 2 1 inner
1850.2.d.i 20 37.b even 2 1 inner
1850.2.d.i 20 185.d even 2 1 inner
3330.2.e.c 10 15.e even 4 1
3330.2.e.c 10 555.n even 4 1
3330.2.e.d 10 15.e even 4 1
3330.2.e.d 10 555.n even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1850, [\chi])$$:

 $$T_{3}^{10} - 19 T_{3}^{8} + 103 T_{3}^{6} - 210 T_{3}^{4} + 140 T_{3}^{2} - 16$$ $$T_{7}^{10} - 39 T_{7}^{8} + 438 T_{7}^{6} - 1684 T_{7}^{4} + 2048 T_{7}^{2} - 576$$ $$T_{11}^{5} - 28 T_{11}^{3} + 51 T_{11}^{2} + 16 T_{11} - 48$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{10}$$
$3$ $$( -16 + 140 T^{2} - 210 T^{4} + 103 T^{6} - 19 T^{8} + T^{10} )^{2}$$
$5$ $$T^{20}$$
$7$ $$( -576 + 2048 T^{2} - 1684 T^{4} + 438 T^{6} - 39 T^{8} + T^{10} )^{2}$$
$11$ $$( -48 + 16 T + 51 T^{2} - 28 T^{3} + T^{5} )^{4}$$
$13$ $$( 238144 + 123200 T^{2} + 23456 T^{4} + 2041 T^{6} + 79 T^{8} + T^{10} )^{2}$$
$17$ $$( 20736 + 43648 T^{2} + 15152 T^{4} + 1736 T^{6} + 73 T^{8} + T^{10} )^{2}$$
$19$ $$( 9216 + 374336 T^{2} + 72240 T^{4} + 4668 T^{6} + 118 T^{8} + T^{10} )^{2}$$
$23$ $$( 589824 + 538624 T^{2} + 134800 T^{4} + 7561 T^{6} + 151 T^{8} + T^{10} )^{2}$$
$29$ $$( 2262016 + 1251872 T^{2} + 194729 T^{4} + 10258 T^{6} + 182 T^{8} + T^{10} )^{2}$$
$31$ $$( 60516 + 346130 T^{2} + 68053 T^{4} + 4502 T^{6} + 116 T^{8} + T^{10} )^{2}$$
$37$ $$4808584372417849 - 456622329009730 T^{2} + 27599518981613 T^{4} - 1264039131416 T^{6} + 45931223170 T^{8} - 1353604492 T^{10} + 33550930 T^{12} - 674456 T^{14} + 10757 T^{16} - 130 T^{18} + T^{20}$$
$41$ $$( -36 - 8 T + 89 T^{2} - 44 T^{3} + 2 T^{4} + T^{5} )^{4}$$
$43$ $$( 6885376 + 3045888 T^{2} + 306192 T^{4} + 12216 T^{6} + 201 T^{8} + T^{10} )^{2}$$
$47$ $$( -4596736 + 10494016 T^{2} - 992816 T^{4} + 29148 T^{6} - 310 T^{8} + T^{10} )^{2}$$
$53$ $$( -39337984 + 10468352 T^{2} - 826000 T^{4} + 22964 T^{6} - 257 T^{8} + T^{10} )^{2}$$
$59$ $$( 21827584 + 7068992 T^{2} + 590448 T^{4} + 18844 T^{6} + 238 T^{8} + T^{10} )^{2}$$
$61$ $$( 82944 + 89792 T^{2} + 28169 T^{4} + 3402 T^{6} + 150 T^{8} + T^{10} )^{2}$$
$67$ $$( -559417104 + 70672748 T^{2} - 2839394 T^{4} + 48727 T^{6} - 367 T^{8} + T^{10} )^{2}$$
$71$ $$( 4608 - 2816 T - 1896 T^{2} - 164 T^{3} + 10 T^{4} + T^{5} )^{4}$$
$73$ $$( -589824 + 785408 T^{2} - 189152 T^{4} + 10965 T^{6} - 189 T^{8} + T^{10} )^{2}$$
$79$ $$( 186486336 + 29601332 T^{2} + 1472598 T^{4} + 31019 T^{6} + 291 T^{8} + T^{10} )^{2}$$
$83$ $$( -1024 + 23736128 T^{2} - 2459520 T^{4} + 62212 T^{6} - 466 T^{8} + T^{10} )^{2}$$
$89$ $$( 1230045184 + 117901312 T^{2} + 3949632 T^{4} + 59888 T^{6} + 412 T^{8} + T^{10} )^{2}$$
$97$ $$( 1364224 + 8554304 T^{2} + 1064512 T^{4} + 36836 T^{6} + 357 T^{8} + T^{10} )^{2}$$