# Properties

 Label 1148.2.d.a Level $1148$ Weight $2$ Character orbit 1148.d Analytic conductor $9.167$ Analytic rank $0$ Dimension $20$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1148 = 2^{2} \cdot 7 \cdot 41$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1148.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.16682615204$$ Analytic rank: $$0$$ Dimension: $$20$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} + \cdots)$$ Defining polynomial: $$x^{20} + 39 x^{18} + 531 x^{16} + 3488 x^{14} + 12661 x^{12} + 27027 x^{10} + 34540 x^{8} + 25909 x^{6} + 10677 x^{4} + 2092 x^{2} + 144$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{10}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + \beta_{12} q^{3} + \beta_{2} q^{5} -\beta_{9} q^{7} + ( -1 - \beta_{7} - \beta_{11} ) q^{9} +O(q^{10})$$ $$q + \beta_{12} q^{3} + \beta_{2} q^{5} -\beta_{9} q^{7} + ( -1 - \beta_{7} - \beta_{11} ) q^{9} + \beta_{19} q^{11} + \beta_{15} q^{13} + ( -\beta_{9} - \beta_{10} - \beta_{14} - \beta_{17} + \beta_{18} + \beta_{19} ) q^{15} + ( -\beta_{12} - \beta_{16} + \beta_{18} ) q^{17} + ( -\beta_{9} + \beta_{12} + \beta_{13} - \beta_{14} + \beta_{16} + \beta_{19} ) q^{19} -\beta_{4} q^{21} + ( -\beta_{1} + \beta_{8} - \beta_{11} ) q^{23} + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{5} - \beta_{7} ) q^{25} + ( -2 \beta_{9} - \beta_{12} - \beta_{15} - \beta_{17} + \beta_{18} ) q^{27} + ( -\beta_{9} + \beta_{13} - 2 \beta_{14} - \beta_{17} + \beta_{19} ) q^{29} + ( 1 - \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{31} + ( -\beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{11} ) q^{33} + \beta_{14} q^{35} + ( -1 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{6} - \beta_{7} - \beta_{8} ) q^{37} + ( -1 + \beta_{1} + \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{8} - \beta_{11} ) q^{39} + ( -2 + \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} - \beta_{15} - \beta_{16} - \beta_{17} ) q^{41} + ( \beta_{1} - \beta_{3} - \beta_{4} - \beta_{11} ) q^{43} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} + \beta_{11} ) q^{45} + ( \beta_{9} + 2 \beta_{10} - \beta_{13} - \beta_{14} + \beta_{15} + \beta_{17} - \beta_{19} ) q^{47} - q^{49} + ( 3 - 2 \beta_{2} + \beta_{7} + 2 \beta_{8} + \beta_{11} ) q^{51} + ( \beta_{9} + 2 \beta_{10} - 2 \beta_{13} + 2 \beta_{14} + \beta_{15} - \beta_{16} - \beta_{18} - \beta_{19} ) q^{53} + ( -2 \beta_{9} + \beta_{17} + \beta_{18} ) q^{55} + ( -1 - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} - \beta_{11} ) q^{57} + ( 1 - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{8} - \beta_{11} ) q^{59} + ( -\beta_{1} - \beta_{5} - \beta_{8} - \beta_{11} ) q^{61} + ( \beta_{9} - \beta_{13} + \beta_{15} ) q^{63} + ( -2 \beta_{9} + \beta_{12} + \beta_{13} - \beta_{15} + \beta_{16} - \beta_{18} ) q^{65} + ( -2 \beta_{9} - \beta_{12} - 2 \beta_{13} - \beta_{17} - \beta_{19} ) q^{67} + ( -\beta_{12} + \beta_{16} - \beta_{17} ) q^{69} + ( \beta_{9} + \beta_{10} - \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} - \beta_{16} - 3 \beta_{17} - \beta_{19} ) q^{71} + ( 2 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} + \beta_{7} + \beta_{8} + \beta_{11} ) q^{73} + ( -3 \beta_{9} - 2 \beta_{10} - \beta_{12} + \beta_{13} - 5 \beta_{14} + 2 \beta_{16} - \beta_{17} + 2 \beta_{18} + 3 \beta_{19} ) q^{75} -\beta_{3} q^{77} + ( 2 \beta_{9} - \beta_{10} - \beta_{13} + \beta_{14} + 2 \beta_{15} + \beta_{18} ) q^{79} + ( 2 - 2 \beta_{2} - \beta_{3} - 3 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} + \beta_{11} ) q^{81} + ( -4 + \beta_{1} - \beta_{3} + \beta_{6} + \beta_{7} - 2 \beta_{11} ) q^{83} + ( \beta_{10} + \beta_{12} - \beta_{14} - \beta_{15} - \beta_{16} + 2 \beta_{17} ) q^{85} + ( 3 - 3 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{8} ) q^{87} + ( \beta_{9} + 3 \beta_{13} - \beta_{14} - \beta_{15} + 2 \beta_{17} + \beta_{18} + \beta_{19} ) q^{89} + \beta_{11} q^{91} + ( 4 \beta_{9} + \beta_{10} - \beta_{12} - 2 \beta_{13} + 3 \beta_{14} + \beta_{15} - \beta_{16} - \beta_{19} ) q^{93} + ( -3 \beta_{9} - 2 \beta_{10} - 2 \beta_{14} + \beta_{15} + \beta_{16} - 2 \beta_{17} + \beta_{18} + 2 \beta_{19} ) q^{95} + ( 2 \beta_{9} - 2 \beta_{10} + \beta_{12} - 2 \beta_{14} + 2 \beta_{15} + 2 \beta_{16} + 2 \beta_{19} ) q^{97} + ( -6 \beta_{9} - \beta_{12} + \beta_{13} - 5 \beta_{15} - \beta_{16} - 2 \beta_{17} + \beta_{18} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q + 4q^{5} - 20q^{9} + O(q^{10})$$ $$20q + 4q^{5} - 20q^{9} + 4q^{21} + 8q^{31} + 20q^{37} + 4q^{39} - 16q^{41} + 20q^{43} - 4q^{45} - 20q^{49} + 52q^{51} - 36q^{57} + 20q^{59} - 4q^{61} - 12q^{73} + 8q^{77} + 20q^{81} - 48q^{83} + 44q^{87} - 4q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} + 39 x^{18} + 531 x^{16} + 3488 x^{14} + 12661 x^{12} + 27027 x^{10} + 34540 x^{8} + 25909 x^{6} + 10677 x^{4} + 2092 x^{2} + 144$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-31787625 \nu^{18} - 1234487653 \nu^{16} - 16663895421 \nu^{14} - 107644253652 \nu^{12} - 377905031721 \nu^{10} - 752256795415 \nu^{8} - 827469231082 \nu^{6} - 449405330741 \nu^{4} - 91003651153 \nu^{2} - 2443071016$$$$)/ 1308220948$$ $$\beta_{2}$$ $$=$$ $$($$$$78534511 \nu^{18} + 3023650107 \nu^{16} + 40148207403 \nu^{14} + 252303406944 \nu^{12} + 850222236003 \nu^{10} + 1608764602793 \nu^{8} + 1701990453470 \nu^{6} + 959850263683 \nu^{4} + 259552032227 \nu^{2} + 25131533044$$$$)/ 1308220948$$ $$\beta_{3}$$ $$=$$ $$($$$$65716921 \nu^{18} + 2470157524 \nu^{16} + 31428482111 \nu^{14} + 185642737294 \nu^{12} + 580415674738 \nu^{10} + 1021258449659 \nu^{8} + 1039060026781 \nu^{6} + 615624085834 \nu^{4} + 198329385512 \nu^{2} + 21827912202$$$$)/ 654110474$$ $$\beta_{4}$$ $$=$$ $$($$$$-132705459 \nu^{18} - 5017740939 \nu^{16} - 64447009489 \nu^{14} - 384270456442 \nu^{12} - 1199136434943 \nu^{10} - 2027852552593 \nu^{8} - 1797941640618 \nu^{6} - 735262452025 \nu^{4} - 95505255211 \nu^{2} - 1710402836$$$$)/ 1308220948$$ $$\beta_{5}$$ $$=$$ $$($$$$-157161803 \nu^{18} - 5956564083 \nu^{16} - 76935253097 \nu^{14} - 464683022950 \nu^{12} - 1491603036323 \nu^{10} - 2672786542173 \nu^{8} - 2654195785182 \nu^{6} - 1365361763397 \nu^{4} - 317901763039 \nu^{2} - 32192745056$$$$)/ 1308220948$$ $$\beta_{6}$$ $$=$$ $$($$$$-81400012 \nu^{18} - 3072480759 \nu^{16} - 39346942920 \nu^{14} - 233733680863 \nu^{12} - 726779307974 \nu^{10} - 1225255445524 \nu^{8} - 1078804919153 \nu^{6} - 426880885365 \nu^{4} - 44535356886 \nu^{2} + 2586015382$$$$)/ 654110474$$ $$\beta_{7}$$ $$=$$ $$($$$$254818261 \nu^{18} + 9591778621 \nu^{16} + 122242189893 \nu^{14} + 721325370166 \nu^{12} + 2227511191483 \nu^{10} + 3743842788823 \nu^{8} + 3332423755998 \nu^{6} + 1393871028825 \nu^{4} + 188404688943 \nu^{2} - 469951060$$$$)/ 1308220948$$ $$\beta_{8}$$ $$=$$ $$($$$$-147573535 \nu^{18} - 5598394419 \nu^{16} - 72436339599 \nu^{14} - 438766175071 \nu^{12} - 1414547618202 \nu^{10} - 2551719292225 \nu^{8} - 2564146241390 \nu^{6} - 1351987979829 \nu^{4} - 324713448428 \nu^{2} - 25607577556$$$$)/ 654110474$$ $$\beta_{9}$$ $$=$$ $$($$$$5623603 \nu^{19} + 216227373 \nu^{17} + 2866746153 \nu^{15} + 18022264568 \nu^{13} + 61108447099 \nu^{11} + 117555643485 \nu^{9} + 128042385988 \nu^{7} + 74782499719 \nu^{5} + 20531370135 \nu^{3} + 1935913456 \nu$$$$)/73358184$$ $$\beta_{10}$$ $$=$$ $$($$$$608503607 \nu^{19} + 22567526841 \nu^{17} + 279469541481 \nu^{15} + 1569995546788 \nu^{13} + 4472860961675 \nu^{11} + 6537075118413 \nu^{9} + 4304068140716 \nu^{7} + 380040790175 \nu^{5} - 708066333741 \nu^{3} - 155209856212 \nu$$$$)/ 7849325688$$ $$\beta_{11}$$ $$=$$ $$($$$$-313679163 \nu^{18} - 11920654724 \nu^{16} - 154696614366 \nu^{14} - 940624815713 \nu^{12} - 3043090741214 \nu^{10} - 5496457422545 \nu^{8} - 5502820095955 \nu^{6} - 2858193854909 \nu^{4} - 656411574460 \nu^{2} - 47870220358$$$$)/ 654110474$$ $$\beta_{12}$$ $$=$$ $$($$$$-232671212 \nu^{19} - 8702946798 \nu^{17} - 109604242404 \nu^{15} - 634457380336 \nu^{13} - 1904709690599 \nu^{11} - 3075295705011 \nu^{9} - 2581683687998 \nu^{7} - 986465427488 \nu^{5} - 122383016340 \nu^{3} - 9014188499 \nu$$$$)/ 1962331422$$ $$\beta_{13}$$ $$=$$ $$($$$$588668419 \nu^{19} + 22208188653 \nu^{17} + 284385522525 \nu^{15} + 1694539204220 \nu^{13} + 5340504255169 \nu^{11} + 9386941785531 \nu^{9} + 9276688883788 \nu^{7} + 5039065631803 \nu^{5} + 1432979660859 \nu^{3} + 182489439166 \nu$$$$)/ 3924662844$$ $$\beta_{14}$$ $$=$$ $$($$$$302557282 \nu^{19} + 11425963938 \nu^{17} + 146552844963 \nu^{15} + 874594463891 \nu^{13} + 2752866200317 \nu^{11} + 4778882242827 \nu^{9} + 4501150860358 \nu^{7} + 2095442065027 \nu^{5} + 372197567676 \nu^{3} + 9058558051 \nu$$$$)/ 1962331422$$ $$\beta_{15}$$ $$=$$ $$($$$$33910453 \nu^{19} + 1287273639 \nu^{17} + 16680088383 \nu^{15} + 101368174544 \nu^{13} + 329355900889 \nu^{11} + 605696469483 \nu^{9} + 638274527368 \nu^{7} + 375346743901 \nu^{5} + 111629493429 \nu^{3} + 11249472208 \nu$$$$)/ 191446968$$ $$\beta_{16}$$ $$=$$ $$($$$$-751498207 \nu^{19} - 28683213891 \nu^{17} - 375155900355 \nu^{15} - 2308300695836 \nu^{13} - 7588073741251 \nu^{11} - 13995056124285 \nu^{9} - 14425085367430 \nu^{7} - 7845373364095 \nu^{5} - 1955054872167 \nu^{3} - 152958371824 \nu$$$$)/ 3924662844$$ $$\beta_{17}$$ $$=$$ $$($$$$-1850657971 \nu^{19} - 70481463165 \nu^{17} - 918223917393 \nu^{15} - 5616400889096 \nu^{13} - 18316670673547 \nu^{11} - 33435543106221 \nu^{9} - 33999444335596 \nu^{7} - 18221748989695 \nu^{5} - 4607030945895 \nu^{3} - 465025308544 \nu$$$$)/ 7849325688$$ $$\beta_{18}$$ $$=$$ $$($$$$-2134858519 \nu^{19} - 81836573685 \nu^{17} - 1079672604993 \nu^{15} - 6749307948548 \nu^{13} - 22810245655159 \nu^{11} - 44073411770829 \nu^{9} - 48956823461800 \nu^{7} - 29879878728139 \nu^{5} - 8831088758991 \nu^{3} - 905372335252 \nu$$$$)/ 7849325688$$ $$\beta_{19}$$ $$=$$ $$($$$$1147322905 \nu^{19} + 43601198779 \nu^{17} + 565803981419 \nu^{15} + 3439887938000 \nu^{13} + 11123703697949 \nu^{11} + 20064460456303 \nu^{9} + 20003724846728 \nu^{7} + 10244168446377 \nu^{5} + 2226446831473 \nu^{3} + 125108902800 \nu$$$$)/ 2616441896$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{14} + \beta_{10} + \beta_{9}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_{1} - 9$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$5 \beta_{19} - \beta_{18} + \beta_{17} + 2 \beta_{16} - 2 \beta_{15} + 2 \beta_{14} + \beta_{13} - 5 \beta_{12} - 15 \beta_{10} - 18 \beta_{9}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$4 \beta_{11} - 14 \beta_{7} - 8 \beta_{6} + 12 \beta_{5} - 28 \beta_{4} + 20 \beta_{3} - 15 \beta_{2} - 47 \beta_{1} + 111$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-95 \beta_{19} + 29 \beta_{18} - 11 \beta_{17} - 22 \beta_{16} + 32 \beta_{15} + 15 \beta_{14} + 9 \beta_{13} + 119 \beta_{12} + 254 \beta_{10} + 363 \beta_{9}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-102 \beta_{11} - 20 \beta_{8} + 193 \beta_{7} + 63 \beta_{6} - 189 \beta_{5} + 559 \beta_{4} - 389 \beta_{3} + 224 \beta_{2} + 969 \beta_{1} - 1860$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$1772 \beta_{19} - 650 \beta_{18} + 78 \beta_{17} + 276 \beta_{16} - 558 \beta_{15} - 573 \beta_{14} - 418 \beta_{13} - 2394 \beta_{12} - 4629 \beta_{10} - 7275 \beta_{9}$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$2146 \beta_{11} + 616 \beta_{8} - 3045 \beta_{7} - 573 \beta_{6} + 3341 \beta_{5} - 10743 \beta_{4} + 7543 \beta_{3} - 3701 \beta_{2} - 19172 \beta_{1} + 34091$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$-33621 \beta_{19} + 13255 \beta_{18} - 213 \beta_{17} - 4234 \beta_{16} + 10268 \beta_{15} + 12974 \beta_{14} + 9957 \beta_{13} + 46559 \beta_{12} + 87061 \beta_{10} + 142784 \beta_{9}$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$-42642 \beta_{11} - 13978 \beta_{8} + 53164 \beta_{7} + 6392 \beta_{6} - 61972 \beta_{5} + 205870 \beta_{4} - 145630 \beta_{3} + 66091 \beta_{2} + 372887 \beta_{1} - 643495$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$642815 \beta_{19} - 260737 \beta_{18} - 6873 \beta_{17} + 73534 \beta_{16} - 193624 \beta_{15} - 264057 \beta_{14} - 206351 \beta_{13} - 897873 \beta_{12} - 1658276 \beta_{10} - 2769101 \beta_{9}$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$($$$$830018 \beta_{11} + 286758 \beta_{8} - 979163 \beta_{7} - 88163 \beta_{6} + 1173239 \beta_{5} - 3948889 \beta_{4} + 2804025 \beta_{3} - 1229500 \beta_{2} - 7200091 \beta_{1} + 12276920$$$$)/2$$ $$\nu^{13}$$ $$=$$ $$($$$$-12325824 \beta_{19} + 5056778 \beta_{18} + 221408 \beta_{17} - 1354084 \beta_{16} + 3690506 \beta_{15} + 5187285 \beta_{14} + 4082134 \beta_{13} + 17271626 \beta_{12} + 31745793 \beta_{10} + 53408601 \beta_{9}$$$$)/2$$ $$\nu^{14}$$ $$=$$ $$($$$$-16028518 \beta_{11} - 5657626 \beta_{8} + 18478287 \beta_{7} + 1428069 \beta_{6} - 22402389 \beta_{5} + 75804203 \beta_{4} - 53918615 \beta_{3} + 23301041 \beta_{2} + 138605658 \beta_{1} - 235189781$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$($$$$236604049 \beta_{19} - 97516461 \beta_{18} - 4967495 \beta_{17} + 25563982 \beta_{16} - 70663378 \beta_{15} - 100542516 \beta_{14} - 79343629 \beta_{13} - 331954503 \beta_{12} - 608974529 \beta_{10} - 1027679528 \beta_{9}$$$$)/2$$ $$\nu^{16}$$ $$=$$ $$($$$$308556528 \beta_{11} + 109875106 \beta_{8} - 352359700 \beta_{7} - 25372692 \beta_{6} + 429279674 \beta_{5} - 1455728826 \beta_{4} + 1036192852 \beta_{3} - 445104571 \beta_{2} - 2664882997 \beta_{1} + 4512893735$$$$)/2$$ $$\nu^{17}$$ $$=$$ $$($$$$-4543748283 \beta_{19} + 1876213613 \beta_{18} + 101080219 \beta_{17} - 487607498 \beta_{16} + 1355597420 \beta_{15} + 1938402667 \beta_{14} + 1531436353 \beta_{13} + 6378011399 \beta_{12} + 11691490754 \beta_{10} + 19754909633 \beta_{9}$$$$)/2$$ $$\nu^{18}$$ $$=$$ $$($$$$-5932354398 \beta_{11} - 2120124070 \beta_{8} + 6748252039 \beta_{7} + 471320859 \beta_{6} - 8237938373 \beta_{5} + 27960355095 \beta_{4} - 19908290521 \beta_{3} + 8530611348 \beta_{2} + 51209516351 \beta_{1} - 86651605338$$$$)/2$$ $$\nu^{19}$$ $$=$$ $$($$$$87273104296 \beta_{19} - 36064473216 \beta_{18} - 1986303818 \beta_{17} + 9339696968 \beta_{16} - 26026104824 \beta_{15} - 37290917315 \beta_{14} - 29475338054 \beta_{13} - 122528756046 \beta_{12} - 224536092865 \beta_{10} - 379590218363 \beta_{9}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1148\mathbb{Z}\right)^\times$$.

 $$n$$ $$493$$ $$575$$ $$785$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1065.1
 0.544821i 1.52399i − 1.02366i − 1.40239i 1.20144i − 4.38287i 1.97172i 0.379831i − 0.911088i − 2.80197i 2.80197i 0.911088i − 0.379831i − 1.97172i 4.38287i − 1.20144i 1.40239i 1.02366i − 1.52399i − 0.544821i
0 3.32626i 0 −0.885775 0 1.00000i 0 −8.06402 0
1065.2 0 2.64112i 0 0.508610 0 1.00000i 0 −3.97553 0
1065.3 0 2.57093i 0 −2.38500 0 1.00000i 0 −3.60969 0
1065.4 0 2.39349i 0 3.99395 0 1.00000i 0 −2.72882 0
1065.5 0 2.23543i 0 0.468853 0 1.00000i 0 −1.99714 0
1065.6 0 1.40464i 0 −1.35828 0 1.00000i 0 1.02697 0
1065.7 0 1.24486i 0 3.14516 0 1.00000i 0 1.45033 0
1065.8 0 1.03684i 0 2.44552 0 1.00000i 0 1.92496 0
1065.9 0 0.162086i 0 −1.05958 0 1.00000i 0 2.97373 0
1065.10 0 0.0281596i 0 −2.87346 0 1.00000i 0 2.99921 0
1065.11 0 0.0281596i 0 −2.87346 0 1.00000i 0 2.99921 0
1065.12 0 0.162086i 0 −1.05958 0 1.00000i 0 2.97373 0
1065.13 0 1.03684i 0 2.44552 0 1.00000i 0 1.92496 0
1065.14 0 1.24486i 0 3.14516 0 1.00000i 0 1.45033 0
1065.15 0 1.40464i 0 −1.35828 0 1.00000i 0 1.02697 0
1065.16 0 2.23543i 0 0.468853 0 1.00000i 0 −1.99714 0
1065.17 0 2.39349i 0 3.99395 0 1.00000i 0 −2.72882 0
1065.18 0 2.57093i 0 −2.38500 0 1.00000i 0 −3.60969 0
1065.19 0 2.64112i 0 0.508610 0 1.00000i 0 −3.97553 0
1065.20 0 3.32626i 0 −0.885775 0 1.00000i 0 −8.06402 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1065.20 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1148.2.d.a 20
41.b even 2 1 inner 1148.2.d.a 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1148.2.d.a 20 1.a even 1 1 trivial
1148.2.d.a 20 41.b even 2 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(1148, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{20}$$
$3$ $$1 + 1302 T^{2} + 51697 T^{4} + 140592 T^{6} + 156436 T^{8} + 90770 T^{10} + 29994 T^{12} + 5836 T^{14} + 660 T^{16} + 40 T^{18} + T^{20}$$
$5$ $$( -64 + 96 T + 304 T^{2} - 216 T^{3} - 548 T^{4} - 50 T^{5} + 185 T^{6} + 26 T^{7} - 23 T^{8} - 2 T^{9} + T^{10} )^{2}$$
$7$ $$( 1 + T^{2} )^{10}$$
$11$ $$2985984 + 24079360 T^{2} + 66966784 T^{4} + 81498304 T^{6} + 48836368 T^{8} + 14760780 T^{10} + 2182065 T^{12} + 163964 T^{14} + 6470 T^{16} + 128 T^{18} + T^{20}$$
$13$ $$4609609 + 179786820 T^{2} + 294259388 T^{4} + 195143726 T^{6} + 67927982 T^{8} + 13643274 T^{10} + 1639461 T^{12} + 118078 T^{14} + 4943 T^{16} + 110 T^{18} + T^{20}$$
$17$ $$16643322081 + 32586712300 T^{2} + 24791222432 T^{4} + 9442091662 T^{6} + 1954147722 T^{8} + 229664254 T^{10} + 15853305 T^{12} + 649198 T^{14} + 15395 T^{16} + 194 T^{18} + T^{20}$$
$19$ $$273529 + 53304706 T^{2} + 219610553 T^{4} + 314718232 T^{6} + 184671800 T^{8} + 43327978 T^{10} + 4953590 T^{12} + 300480 T^{14} + 9800 T^{16} + 160 T^{18} + T^{20}$$
$23$ $$( -27743 - 170426 T - 277183 T^{2} - 165080 T^{3} - 18896 T^{4} + 14610 T^{5} + 3878 T^{6} - 232 T^{7} - 116 T^{8} + T^{10} )^{2}$$
$29$ $$1220370927616 + 1850123874304 T^{2} + 863961903104 T^{4} + 193229865152 T^{6} + 23980520480 T^{8} + 1758826420 T^{10} + 78316601 T^{12} + 2116902 T^{14} + 33615 T^{16} + 286 T^{18} + T^{20}$$
$31$ $$( 24000 - 107200 T + 98240 T^{2} + 40912 T^{3} - 29380 T^{4} - 4932 T^{5} + 2701 T^{6} + 244 T^{7} - 91 T^{8} - 4 T^{9} + T^{10} )^{2}$$
$37$ $$( 1664144 + 6275024 T + 6687340 T^{2} + 1586068 T^{3} - 443347 T^{4} - 110272 T^{5} + 14570 T^{6} + 1966 T^{7} - 211 T^{8} - 10 T^{9} + T^{10} )^{2}$$
$41$ $$13422659310152401 + 5238110950303376 T + 654763868787922 T^{2} - 96598119844976 T^{3} - 40979149287107 T^{4} - 5331238945216 T^{5} + 21860087096 T^{6} + 116576838976 T^{7} + 17507040098 T^{8} - 123990560 T^{9} - 291350676 T^{10} - 3024160 T^{11} + 10414658 T^{12} + 1691456 T^{13} + 7736 T^{14} - 46016 T^{15} - 8627 T^{16} - 496 T^{17} + 82 T^{18} + 16 T^{19} + T^{20}$$
$43$ $$( 261201 + 1223116 T - 1795644 T^{2} + 523326 T^{3} + 115662 T^{4} - 60710 T^{5} + 1633 T^{6} + 1622 T^{7} - 133 T^{8} - 10 T^{9} + T^{10} )^{2}$$
$47$ $$190650259603456 + 167985802772480 T^{2} + 55955419226368 T^{4} + 8895782440112 T^{6} + 720755873033 T^{8} + 32085370368 T^{10} + 829612070 T^{12} + 12713962 T^{14} + 112881 T^{16} + 530 T^{18} + T^{20}$$
$53$ $$1684097961984 + 9087159933952 T^{2} + 8746094594304 T^{4} + 3269180478144 T^{6} + 527203771216 T^{8} + 34168749068 T^{10} + 1066471529 T^{12} + 17313550 T^{14} + 146767 T^{16} + 614 T^{18} + T^{20}$$
$59$ $$( -4392448 - 9527424 T + 3792000 T^{2} + 1698984 T^{3} - 375152 T^{4} - 90970 T^{5} + 13389 T^{6} + 1698 T^{7} - 195 T^{8} - 10 T^{9} + T^{10} )^{2}$$
$61$ $$( 1741312 + 1067520 T - 1396192 T^{2} - 958184 T^{3} - 39824 T^{4} + 73430 T^{5} + 11973 T^{6} - 816 T^{7} - 214 T^{8} + 2 T^{9} + T^{10} )^{2}$$
$67$ $$11529990701056 + 39788076072960 T^{2} + 24104335024128 T^{4} + 5173631600704 T^{6} + 489013162656 T^{8} + 23888903220 T^{10} + 660162065 T^{12} + 10681790 T^{14} + 99703 T^{16} + 494 T^{18} + T^{20}$$
$71$ $$620545852771139584 + 922949312533266432 T^{2} + 102131927166604288 T^{4} + 4845231633687104 T^{6} + 126821309427424 T^{8} + 2014999098836 T^{10} + 20193019681 T^{12} + 128079420 T^{14} + 497798 T^{16} + 1080 T^{18} + T^{20}$$
$73$ $$( -5926912 - 13600768 T - 11233600 T^{2} - 3627400 T^{3} - 64772 T^{4} + 179698 T^{5} + 19565 T^{6} - 2224 T^{7} - 298 T^{8} + 6 T^{9} + T^{10} )^{2}$$
$79$ $$89369947930624 + 159463646429184 T^{2} + 84561408032768 T^{4} + 14613555314688 T^{6} + 1188588032000 T^{8} + 51916943360 T^{10} + 1281389056 T^{12} + 18154368 T^{14} + 144784 T^{16} + 600 T^{18} + T^{20}$$
$83$ $$( 897075264 + 132826688 T - 94761872 T^{2} - 16718016 T^{3} + 2796328 T^{4} + 628348 T^{5} - 13595 T^{6} - 7492 T^{7} - 190 T^{8} + 24 T^{9} + T^{10} )^{2}$$
$89$ $$14390342339821009 + 13424449636343378 T^{2} + 3626940778439013 T^{4} + 410920569901528 T^{6} + 21794205060488 T^{8} + 615569776590 T^{10} + 9797758758 T^{12} + 88016492 T^{14} + 428020 T^{16} + 1044 T^{18} + T^{20}$$
$97$ $$69228477606528601 + 34989824475346190 T^{2} + 5919551758337089 T^{4} + 478329604347576 T^{6} + 21081199613660 T^{8} + 538509540394 T^{10} + 8181382506 T^{12} + 73657372 T^{14} + 376796 T^{16} + 984 T^{18} + T^{20}$$