# Properties

 Label 4010.2.a.m Level 4010 Weight 2 Character orbit 4010.a Self dual yes Analytic conductor 32.020 Analytic rank 0 Dimension 20 CM no Inner twists 1

# Learn more about

## Newspace parameters

 Level: $$N$$ = $$4010 = 2 \cdot 5 \cdot 401$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 4010.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$32.0200112105$$ Analytic rank: $$0$$ Dimension: $$20$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q - q^{2} + \beta_{1} q^{3} + q^{4} + q^{5} -\beta_{1} q^{6} + ( 1 + \beta_{16} ) q^{7} - q^{8} + ( 1 + \beta_{2} ) q^{9} +O(q^{10})$$ $$q - q^{2} + \beta_{1} q^{3} + q^{4} + q^{5} -\beta_{1} q^{6} + ( 1 + \beta_{16} ) q^{7} - q^{8} + ( 1 + \beta_{2} ) q^{9} - q^{10} + ( 1 + \beta_{15} ) q^{11} + \beta_{1} q^{12} + ( -\beta_{4} + \beta_{16} ) q^{13} + ( -1 - \beta_{16} ) q^{14} + \beta_{1} q^{15} + q^{16} + ( 1 + \beta_{1} - \beta_{2} - \beta_{7} - \beta_{8} + \beta_{15} - \beta_{18} + \beta_{19} ) q^{17} + ( -1 - \beta_{2} ) q^{18} + ( 1 - \beta_{3} + \beta_{6} ) q^{19} + q^{20} + ( 1 - \beta_{3} + \beta_{6} + \beta_{8} - \beta_{9} - \beta_{12} ) q^{21} + ( -1 - \beta_{15} ) q^{22} + ( 1 - \beta_{2} - \beta_{4} - \beta_{7} + \beta_{15} - \beta_{17} + \beta_{19} ) q^{23} -\beta_{1} q^{24} + q^{25} + ( \beta_{4} - \beta_{16} ) q^{26} + ( 2 \beta_{1} - \beta_{6} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{13} - \beta_{14} - 2 \beta_{18} + \beta_{19} ) q^{27} + ( 1 + \beta_{16} ) q^{28} + ( 2 - \beta_{2} - \beta_{5} - \beta_{9} + \beta_{15} ) q^{29} -\beta_{1} q^{30} + ( 1 - \beta_{2} - \beta_{5} - \beta_{6} - \beta_{8} + \beta_{10} + \beta_{11} ) q^{31} - q^{32} + ( -3 + \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{5} + \beta_{7} + \beta_{9} - \beta_{11} - \beta_{15} + \beta_{18} - \beta_{19} ) q^{33} + ( -1 - \beta_{1} + \beta_{2} + \beta_{7} + \beta_{8} - \beta_{15} + \beta_{18} - \beta_{19} ) q^{34} + ( 1 + \beta_{16} ) q^{35} + ( 1 + \beta_{2} ) q^{36} + ( 2 + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} + \beta_{16} + \beta_{18} ) q^{37} + ( -1 + \beta_{3} - \beta_{6} ) q^{38} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{8} - \beta_{12} + \beta_{17} + \beta_{18} - \beta_{19} ) q^{39} - q^{40} + ( \beta_{2} + \beta_{3} - \beta_{6} + \beta_{7} - \beta_{11} + \beta_{12} - \beta_{15} - \beta_{19} ) q^{41} + ( -1 + \beta_{3} - \beta_{6} - \beta_{8} + \beta_{9} + \beta_{12} ) q^{42} + ( -1 + 3 \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} - 2 \beta_{15} + \beta_{16} + \beta_{17} + 2 \beta_{18} - 2 \beta_{19} ) q^{43} + ( 1 + \beta_{15} ) q^{44} + ( 1 + \beta_{2} ) q^{45} + ( -1 + \beta_{2} + \beta_{4} + \beta_{7} - \beta_{15} + \beta_{17} - \beta_{19} ) q^{46} + ( 1 + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{13} - \beta_{14} - \beta_{18} + \beta_{19} ) q^{47} + \beta_{1} q^{48} + ( 1 + \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - 2 \beta_{6} + \beta_{7} + \beta_{10} - \beta_{13} - \beta_{15} + \beta_{16} - \beta_{18} ) q^{49} - q^{50} + ( 2 \beta_{2} + \beta_{3} + \beta_{6} + \beta_{8} + 2 \beta_{14} - \beta_{15} - \beta_{16} + \beta_{18} - \beta_{19} ) q^{51} + ( -\beta_{4} + \beta_{16} ) q^{52} + ( 3 - \beta_{2} - 2 \beta_{3} - \beta_{5} + \beta_{6} - \beta_{10} + \beta_{11} - \beta_{12} - \beta_{14} + 2 \beta_{15} - \beta_{16} - \beta_{18} + 2 \beta_{19} ) q^{53} + ( -2 \beta_{1} + \beta_{6} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{13} + \beta_{14} + 2 \beta_{18} - \beta_{19} ) q^{54} + ( 1 + \beta_{15} ) q^{55} + ( -1 - \beta_{16} ) q^{56} + ( 2 \beta_{1} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{13} + \beta_{14} - \beta_{15} + \beta_{17} - \beta_{19} ) q^{57} + ( -2 + \beta_{2} + \beta_{5} + \beta_{9} - \beta_{15} ) q^{58} + ( 4 - \beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{6} - 2 \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} + \beta_{13} + \beta_{14} + 2 \beta_{15} - \beta_{16} - \beta_{17} - \beta_{18} + \beta_{19} ) q^{59} + \beta_{1} q^{60} + ( 1 - \beta_{3} - \beta_{6} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{12} - \beta_{14} + \beta_{16} + \beta_{18} ) q^{61} + ( -1 + \beta_{2} + \beta_{5} + \beta_{6} + \beta_{8} - \beta_{10} - \beta_{11} ) q^{62} + ( -2 + \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{10} - \beta_{11} - 2 \beta_{13} - \beta_{14} - 3 \beta_{15} + \beta_{16} + \beta_{18} - 2 \beta_{19} ) q^{63} + q^{64} + ( -\beta_{4} + \beta_{16} ) q^{65} + ( 3 - \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{5} - \beta_{7} - \beta_{9} + \beta_{11} + \beta_{15} - \beta_{18} + \beta_{19} ) q^{66} + ( 3 - \beta_{1} + \beta_{2} - \beta_{5} + \beta_{7} - \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} - \beta_{14} - \beta_{15} ) q^{67} + ( 1 + \beta_{1} - \beta_{2} - \beta_{7} - \beta_{8} + \beta_{15} - \beta_{18} + \beta_{19} ) q^{68} + ( -2 + 2 \beta_{2} + \beta_{3} + \beta_{5} - \beta_{7} + 2 \beta_{9} - \beta_{11} - \beta_{14} - 2 \beta_{15} - \beta_{17} + 2 \beta_{18} - \beta_{19} ) q^{69} + ( -1 - \beta_{16} ) q^{70} + ( 3 - \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{10} + \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} + \beta_{17} + \beta_{18} + \beta_{19} ) q^{71} + ( -1 - \beta_{2} ) q^{72} + ( 2 - 3 \beta_{2} - 2 \beta_{3} + \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{11} + \beta_{13} + 2 \beta_{15} - \beta_{17} - \beta_{18} + \beta_{19} ) q^{73} + ( -2 - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} - \beta_{16} - \beta_{18} ) q^{74} + \beta_{1} q^{75} + ( 1 - \beta_{3} + \beta_{6} ) q^{76} + ( -4 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{6} - \beta_{7} + 2 \beta_{9} + 2 \beta_{10} + \beta_{11} - \beta_{13} - \beta_{15} + \beta_{16} - \beta_{17} - 2 \beta_{19} ) q^{77} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{8} + \beta_{12} - \beta_{17} - \beta_{18} + \beta_{19} ) q^{78} + ( 4 - \beta_{2} + \beta_{6} - \beta_{10} + \beta_{11} + \beta_{15} - 2 \beta_{18} + \beta_{19} ) q^{79} + q^{80} + ( -1 + \beta_{1} + 4 \beta_{2} - \beta_{3} + 2 \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} - 2 \beta_{14} - 3 \beta_{15} - 2 \beta_{16} - \beta_{18} + \beta_{19} ) q^{81} + ( -\beta_{2} - \beta_{3} + \beta_{6} - \beta_{7} + \beta_{11} - \beta_{12} + \beta_{15} + \beta_{19} ) q^{82} + ( -1 + \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} - 2 \beta_{9} + \beta_{10} + \beta_{12} + 2 \beta_{14} - \beta_{15} + \beta_{18} - \beta_{19} ) q^{83} + ( 1 - \beta_{3} + \beta_{6} + \beta_{8} - \beta_{9} - \beta_{12} ) q^{84} + ( 1 + \beta_{1} - \beta_{2} - \beta_{7} - \beta_{8} + \beta_{15} - \beta_{18} + \beta_{19} ) q^{85} + ( 1 - 3 \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} + 2 \beta_{15} - \beta_{16} - \beta_{17} - 2 \beta_{18} + 2 \beta_{19} ) q^{86} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - 3 \beta_{9} + \beta_{10} - \beta_{12} + \beta_{13} + 2 \beta_{14} - \beta_{16} - \beta_{17} + 2 \beta_{18} - \beta_{19} ) q^{87} + ( -1 - \beta_{15} ) q^{88} + ( \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{8} - 2 \beta_{10} + \beta_{12} + \beta_{14} + \beta_{18} - \beta_{19} ) q^{89} + ( -1 - \beta_{2} ) q^{90} + ( 3 + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} - 2 \beta_{11} - \beta_{12} - 3 \beta_{13} - \beta_{14} - 2 \beta_{15} - \beta_{17} - 2 \beta_{18} + \beta_{19} ) q^{91} + ( 1 - \beta_{2} - \beta_{4} - \beta_{7} + \beta_{15} - \beta_{17} + \beta_{19} ) q^{92} + ( 5 + 2 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{8} - 2 \beta_{9} + 2 \beta_{11} - \beta_{12} + \beta_{13} + 3 \beta_{15} - 2 \beta_{16} + \beta_{17} - 2 \beta_{18} + 2 \beta_{19} ) q^{93} + ( -1 - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{13} + \beta_{14} + \beta_{18} - \beta_{19} ) q^{94} + ( 1 - \beta_{3} + \beta_{6} ) q^{95} -\beta_{1} q^{96} + ( 1 + \beta_{1} - 3 \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} + 2 \beta_{10} + \beta_{11} + \beta_{13} + \beta_{14} + \beta_{15} + \beta_{18} + \beta_{19} ) q^{97} + ( -1 - \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{6} - \beta_{7} - \beta_{10} + \beta_{13} + \beta_{15} - \beta_{16} + \beta_{18} ) q^{98} + ( 5 - \beta_{1} + 2 \beta_{2} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{12} + \beta_{13} - \beta_{14} + 2 \beta_{15} + \beta_{17} - 2 \beta_{18} + 2 \beta_{19} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q - 20q^{2} + 4q^{3} + 20q^{4} + 20q^{5} - 4q^{6} + 11q^{7} - 20q^{8} + 26q^{9} + O(q^{10})$$ $$20q - 20q^{2} + 4q^{3} + 20q^{4} + 20q^{5} - 4q^{6} + 11q^{7} - 20q^{8} + 26q^{9} - 20q^{10} + 10q^{11} + 4q^{12} - 9q^{13} - 11q^{14} + 4q^{15} + 20q^{16} - 11q^{17} - 26q^{18} + 17q^{19} + 20q^{20} - 2q^{21} - 10q^{22} - 3q^{23} - 4q^{24} + 20q^{25} + 9q^{26} - 2q^{27} + 11q^{28} + 6q^{29} - 4q^{30} + 28q^{31} - 20q^{32} + 2q^{33} + 11q^{34} + 11q^{35} + 26q^{36} + 33q^{37} - 17q^{38} + 36q^{39} - 20q^{40} + 32q^{41} + 2q^{42} + 30q^{43} + 10q^{44} + 26q^{45} + 3q^{46} + 13q^{47} + 4q^{48} + 43q^{49} - 20q^{50} + 43q^{51} - 9q^{52} + 2q^{54} + 10q^{55} - 11q^{56} + 19q^{57} - 6q^{58} + 52q^{59} + 4q^{60} + 25q^{61} - 28q^{62} + 16q^{63} + 20q^{64} - 9q^{65} - 2q^{66} + 40q^{67} - 11q^{68} + 39q^{69} - 11q^{70} + 25q^{71} - 26q^{72} - 5q^{73} - 33q^{74} + 4q^{75} + 17q^{76} - 9q^{77} - 36q^{78} + 40q^{79} + 20q^{80} + 48q^{81} - 32q^{82} + 10q^{83} - 2q^{84} - 11q^{85} - 30q^{86} + 10q^{87} - 10q^{88} + 17q^{89} - 26q^{90} + 88q^{91} - 3q^{92} - 4q^{93} - 13q^{94} + 17q^{95} - 4q^{96} - 2q^{97} - 43q^{98} + 50q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} - 4 x^{19} - 35 x^{18} + 154 x^{17} + 460 x^{16} - 2392 x^{15} - 2591 x^{14} + 19157 x^{13} + 2776 x^{12} - 83577 x^{11} + 34362 x^{10} + 190617 x^{9} - 150697 x^{8} - 189098 x^{7} + 211360 x^{6} + 37365 x^{5} - 68876 x^{4} - 8984 x^{3} + 7920 x^{2} + 1972 x + 104$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ $$\beta_{3}$$ $$=$$ $$($$$$116014587172107068127 \nu^{19} + 640111674902866325002 \nu^{18} - 7785496031020130083777 \nu^{17} - 22774025091904881148580 \nu^{16} + 196621751166453406494308 \nu^{15} + 307904447861665559694004 \nu^{14} - 2523901525854569513964825 \nu^{13} - 1859977858221430827617547 \nu^{12} + 18123369656041119825105522 \nu^{11} + 3393190587424715294223925 \nu^{10} - 73746841119611087873283396 \nu^{9} + 14421889160306297367063167 \nu^{8} + 160504605873151474966194043 \nu^{7} - 75240595290174938068450304 \nu^{6} - 154547417955549848542242656 \nu^{5} + 100582635779098028690027499 \nu^{4} + 30317438886744979785281938 \nu^{3} - 15302329716260491274790272 \nu^{2} - 4243524215589906115557284 \nu - 192689250572281045604052$$$$)/$$$$12\!\cdots\!76$$ $$\beta_{4}$$ $$=$$ $$($$$$381890322228470127811 \nu^{19} - 3713157145230569889126 \nu^{18} - 6105654787245623330365 \nu^{17} + 139473444254698661552340 \nu^{16} - 103781999398782250499212 \nu^{15} - 2080239065107749043928048 \nu^{14} + 3353365447882748541181147 \nu^{13} + 15512929663601058950255437 \nu^{12} - 33787734343004053799434818 \nu^{11} - 58676991135624483297555223 \nu^{10} + 165865851676723905907971088 \nu^{9} + 91242894083313732643391463 \nu^{8} - 409808669434183097900157517 \nu^{7} + 27830621819326022712984952 \nu^{6} + 441432831515493023022840972 \nu^{5} - 175380389696309148203994441 \nu^{4} - 111351583060334801827646554 \nu^{3} + 30692107501390467607144180 \nu^{2} + 13944746847373487468891484 \nu + 849481945617866039571448$$$$)/$$$$12\!\cdots\!76$$ $$\beta_{5}$$ $$=$$ $$($$$$158358215677478871834 \nu^{19} - 165510479941526953732 \nu^{18} - 7052680498361920376066 \nu^{17} + 6947979307521970282602 \nu^{16} + 131042000829305332961906 \nu^{15} - 122430844130049582041055 \nu^{14} - 1316500543950236316611309 \nu^{13} + 1176238060510791095985110 \nu^{12} + 7734560140578400203662932 \nu^{11} - 6679433655930631630668038 \nu^{10} - 26717309844529810196229795 \nu^{9} + 22585670582814874853777716 \nu^{8} + 51173281785468624732206197 \nu^{7} - 42812984088725050790439808 \nu^{6} - 45795645395669983298002057 \nu^{5} + 37348188371556988720349338 \nu^{4} + 11232664459172077565024340 \nu^{3} - 5912694544531963846562872 \nu^{2} - 1845600069658615177331918 \nu - 97930463161670898984784$$$$)/$$$$31\!\cdots\!19$$ $$\beta_{6}$$ $$=$$ $$($$$$-999626326932643253811 \nu^{19} + 2885395428797447040394 \nu^{18} + 38333738780464460561397 \nu^{17} - 111572241785130709379400 \nu^{16} - 589292053025298126485872 \nu^{15} + 1747121076757043457794012 \nu^{14} + 4617869516112734577765049 \nu^{13} - 14199801235848615666407733 \nu^{12} - 19257689277831527744719262 \nu^{11} + 63680802429345329998519419 \nu^{10} + 39549692162843203014824044 \nu^{9} - 153750683910395021484611183 \nu^{8} - 27583030355109342007533899 \nu^{7} + 176608360040356555042609360 \nu^{6} - 7317961243969399947208616 \nu^{5} - 68619860824489316187794735 \nu^{4} - 9142653565077423376943002 \nu^{3} + 10241607847011599675503092 \nu^{2} + 3538350924370565587810236 \nu + 308889410401021481347968$$$$)/$$$$12\!\cdots\!76$$ $$\beta_{7}$$ $$=$$ $$($$$$296221760166791911206 \nu^{19} - 1244183378690323911445 \nu^{18} - 10071449640775877619681 \nu^{17} + 47466929437924614832456 \nu^{16} + 125023760355703914577064 \nu^{15} - 727150517520315475409018 \nu^{14} - 596980868306248709499010 \nu^{13} + 5694219644280705889090321 \nu^{12} - 491328919934575895320239 \nu^{11} - 23857680923758730001730199 \nu^{10} + 15509572235364018390928239 \nu^{9} + 49857169321253674143592880 \nu^{8} - 54931703343995084463191829 \nu^{7} - 36975255093475330305977433 \nu^{6} + 67677154972691604550812769 \nu^{5} - 10652934185109545792409197 \nu^{4} - 14032845569575371093974027 \nu^{3} + 2035241542476548031276896 \nu^{2} + 1244849369564183836427184 \nu + 93583911115562606944987$$$$)/$$$$31\!\cdots\!19$$ $$\beta_{8}$$ $$=$$ $$($$$$-466793766632834090905 \nu^{19} + 2145576029452580751067 \nu^{18} + 15259569747340112833082 \nu^{17} - 81685678338935034438956 \nu^{16} - 173438948868376514272501 \nu^{15} + 1246937604319414623253837 \nu^{14} + 573279854859614309256400 \nu^{13} - 9703424217813463228470465 \nu^{12} + 3735812431564036078098954 \nu^{11} + 40154851697151314424725489 \nu^{10} - 37519361493098511832066873 \nu^{9} - 81409465321846185964993384 \nu^{8} + 117185728932173939708309293 \nu^{7} + 52681976363325771101082567 \nu^{6} - 139293281261527691706815030 \nu^{5} + 30085096393798773969719460 \nu^{4} + 31635282983204416494482189 \nu^{3} - 5656087873568691359199317 \nu^{2} - 3225994613560182407576658 \nu - 218947124132096447481041$$$$)/$$$$31\!\cdots\!19$$ $$\beta_{9}$$ $$=$$ $$($$$$1884753628400079777355 \nu^{19} - 8888942055680230084478 \nu^{18} - 60790240391370671408381 \nu^{17} + 337841240953512241566252 \nu^{16} + 668813633126194984703628 \nu^{15} - 5143190390781541573654824 \nu^{14} - 1830244294205914165632057 \nu^{13} + 39837967846741924857698677 \nu^{12} - 18909251345769732840700930 \nu^{11} - 163388966210012147429801151 \nu^{10} + 167764158738860977503275800 \nu^{9} + 324059764027864699964818695 \nu^{8} - 508326460010982070729117693 \nu^{7} - 187674378859922962869132764 \nu^{6} + 591889518781073264019445620 \nu^{5} - 154815115746377582396262601 \nu^{4} - 125681247228007480688403482 \nu^{3} + 27708583065443687293291992 \nu^{2} + 12710119782824726532399896 \nu + 768531102416619265207312$$$$)/$$$$12\!\cdots\!76$$ $$\beta_{10}$$ $$=$$ $$($$$$-476462524614547076662 \nu^{19} + 2261150940496645137064 \nu^{18} + 15358368994970655094252 \nu^{17} - 86004232101683687400687 \nu^{16} - 168702378818310521319570 \nu^{15} + 1310730952554843700891871 \nu^{14} + 456367020084764491730201 \nu^{13} - 10170243128634954925798368 \nu^{12} + 4839230598661196559956655 \nu^{11} + 41844531593334313944906169 \nu^{10} - 42739873769995057284338591 \nu^{9} - 83625563617360858132525289 \nu^{8} + 129606374627203981236496536 \nu^{7} + 50366895624856063009545583 \nu^{6} - 151689817123212191455930115 \nu^{5} + 36907829246212195311101221 \nu^{4} + 33496443289456748286697156 \nu^{3} - 6701212743343895922814451 \nu^{2} - 3401240547623802135591091 \nu - 224710702256973187588725$$$$)/$$$$31\!\cdots\!19$$ $$\beta_{11}$$ $$=$$ $$($$$$1207434850304812365853 \nu^{19} - 2247181342857484764768 \nu^{18} - 50414302737871564246701 \nu^{17} + 89232430953300415098274 \nu^{16} + 869790392702661565032060 \nu^{15} - 1455508542425966469741592 \nu^{14} - 8027508521515750685593553 \nu^{13} + 12603138058775366850027407 \nu^{12} + 42848505233762839661010308 \nu^{11} - 62509875650115713349464691 \nu^{10} - 133118622709114169808015760 \nu^{9} + 178427262859425888212445389 \nu^{8} + 227872220297451973399592059 \nu^{7} - 276670047108631459193912154 \nu^{6} - 184388049715590943196801386 \nu^{5} + 195466303198729993721916239 \nu^{4} + 48538361927323472489944564 \nu^{3} - 30774867882955314066751758 \nu^{2} - 8868469585737039803341956 \nu - 436036784036438682904452$$$$)/$$$$63\!\cdots\!38$$ $$\beta_{12}$$ $$=$$ $$($$$$-2640649899689473379501 \nu^{19} + 10649506491248448238214 \nu^{18} + 91288581180531005559555 \nu^{17} - 406992850767994113620008 \nu^{16} - 1172351233751431179109504 \nu^{15} + 6252075359393404533340596 \nu^{14} + 6216476889098677147808871 \nu^{13} - 49189944811880306907760159 \nu^{12} - 2749870479204769967432866 \nu^{11} + 207921164446610883661717853 \nu^{10} - 107367204487314058684431104 \nu^{9} - 443335161158479647005455289 \nu^{8} + 420010142839881237390997231 \nu^{7} + 354916157103821177416797104 \nu^{6} - 536002055774682948870807056 \nu^{5} + 53641870346786670161614791 \nu^{4} + 114546479174603106008396202 \nu^{3} - 13211705517797761836745184 \nu^{2} - 9609884254476182545264624 \nu - 604059387067058925497476$$$$)/$$$$12\!\cdots\!76$$ $$\beta_{13}$$ $$=$$ $$($$$$906457072838216964082 \nu^{19} - 3628977296486894225234 \nu^{18} - 31390865586963716408176 \nu^{17} + 138618825668018039168042 \nu^{16} + 404589169952395976706645 \nu^{15} - 2127899778284846061265956 \nu^{14} - 2169161541444692884268581 \nu^{13} + 16723705361063596849228437 \nu^{12} + 1249533456063875953741397 \nu^{11} - 70558132248270910576003880 \nu^{10} + 35339158994767751030520578 \nu^{9} + 149861361592849943117898630 \nu^{8} - 139857317203626593295263391 \nu^{7} - 118400990854284243160166845 \nu^{6} + 177362450703633048197133085 \nu^{5} - 20381918000793002939944745 \nu^{4} - 34766170116891192798108945 \nu^{3} + 4483077563939623557009137 \nu^{2} + 2631491299065404623481847 \nu + 140031249837851789305768$$$$)/$$$$31\!\cdots\!19$$ $$\beta_{14}$$ $$=$$ $$($$$$3631258242295806765453 \nu^{19} - 17564545293762038295530 \nu^{18} - 115636723222934464488163 \nu^{17} + 667188840556892290729636 \nu^{16} + 1231316305717842238789600 \nu^{15} - 10147240693829979483090548 \nu^{14} - 2634367907510142021186183 \nu^{13} + 78462784018346905024421023 \nu^{12} - 43613024108359836061648198 \nu^{11} - 320693015892015725084839045 \nu^{10} + 354889209213996383882036604 \nu^{9} + 630419774302201966751461757 \nu^{8} - 1053311329079702402249544959 \nu^{7} - 346966847568919206376515488 \nu^{6} + 1218727562507744425200435480 \nu^{5} - 330977420585354991276338559 \nu^{4} - 264382287138187932221509550 \nu^{3} + 59277512190399136368455356 \nu^{2} + 27270917625149273833516268 \nu + 1592074908429308896508416$$$$)/$$$$12\!\cdots\!76$$ $$\beta_{15}$$ $$=$$ $$($$$$-3927785207346788767135 \nu^{19} + 14925956319362018390438 \nu^{18} + 138695996226101670224337 \nu^{17} - 571419830443223297042128 \nu^{16} - 1855859584849898459966444 \nu^{15} + 8803688782975893353907588 \nu^{14} + 10989958517098690230175977 \nu^{13} - 69620842627828531311759201 \nu^{12} - 18108209528281622156357818 \nu^{11} + 297161588029706448826147683 \nu^{10} - 98015177947681939839931512 \nu^{9} - 647802148768359369079590651 \nu^{8} + 481313367239573210807895753 \nu^{7} + 560935202377639861594338668 \nu^{6} - 647285058589129120834044824 \nu^{5} + 9139608244597743615157309 \nu^{4} + 130813845334220221247135166 \nu^{3} - 6832345975076365549363456 \nu^{2} - 8895736270030353418450328 \nu - 611989356765519066222632$$$$)/$$$$12\!\cdots\!76$$ $$\beta_{16}$$ $$=$$ $$($$$$1264472658655511227494 \nu^{19} - 5614620561958718180829 \nu^{18} - 41989787193908721570809 \nu^{17} + 213893582685869905990658 \nu^{16} + 495060092922914141164018 \nu^{15} - 3268651599263988205081725 \nu^{14} - 1946728139186136232579558 \nu^{13} + 25485667188491878171747223 \nu^{12} - 6940209720507964758129297 \nu^{11} - 105875503238112161972589831 \nu^{10} + 87543514595202460841969015 \nu^{9} + 216735518039565914836472611 \nu^{8} - 284185995548277354367416141 \nu^{7} - 146866351638555918297252989 \nu^{6} + 341092711768412943785063238 \nu^{5} - 69210761944566981329902851 \nu^{4} - 74014035244757325495950890 \nu^{3} + 13193929518238796506121678 \nu^{2} + 7120300586889498935989802 \nu + 434790140985479929088453$$$$)/$$$$31\!\cdots\!19$$ $$\beta_{17}$$ $$=$$ $$($$$$5418404036695537037889 \nu^{19} - 21028882424189033473846 \nu^{18} - 189937094677161838202411 \nu^{17} + 804798273735627353154448 \nu^{16} + 2506202389660641475350108 \nu^{15} - 12391881937746586803605320 \nu^{14} - 14315694482819121711656027 \nu^{13} + 97889157905570962471010043 \nu^{12} + 18120464240992583538283722 \nu^{11} - 416911015802940054516558769 \nu^{10} + 165875733340926319740473736 \nu^{9} + 904241211612978215609104833 \nu^{8} - 737414111761402929820796123 \nu^{7} - 768988415983968580414341056 \nu^{6} + 975470236198143672379935044 \nu^{5} - 36563629108135351662405247 \nu^{4} - 209919950187223911336578014 \nu^{3} + 14691710720940172080593128 \nu^{2} + 16294502047601695803672512 \nu + 1112816409198791104770760$$$$)/$$$$12\!\cdots\!76$$ $$\beta_{18}$$ $$=$$ $$($$$$2758753361192571885413 \nu^{19} - 11431752151240422468314 \nu^{18} - 94397143893885790930495 \nu^{17} + 436634595061272592855124 \nu^{16} + 1187274336422850598138538 \nu^{15} - 6700748010287355896296002 \nu^{14} - 5911219384637382066178013 \nu^{13} + 52626949058217068420362635 \nu^{12} - 1813842703208030192357358 \nu^{11} - 221689602467094470043371235 \nu^{10} + 132771934441679099734661132 \nu^{9} + 468939029513702380312779877 \nu^{8} - 486641734722183014335399537 \nu^{7} - 364231038870957598427223512 \nu^{6} + 609995251439267990222801910 \nu^{5} - 74734627074977725735521141 \nu^{4} - 132983362883294444683458648 \nu^{3} + 16388938799762510418203468 \nu^{2} + 11773889608624996970981568 \nu + 788415911390967734805626$$$$)/$$$$63\!\cdots\!38$$ $$\beta_{19}$$ $$=$$ $$($$$$11634694858096714461999 \nu^{19} - 48406217720396724392326 \nu^{18} - 397193027005254290163733 \nu^{17} + 1848029398972936980518076 \nu^{16} + 4972099334320587284612548 \nu^{15} - 28340846153083168985306712 \nu^{14} - 24389190301742277875629049 \nu^{13} + 222332963227085352961603569 \nu^{12} - 11918526908097317367907938 \nu^{11} - 934616652952177256655492303 \nu^{10} + 578082175331391735988689832 \nu^{9} + 1967670775033323362634671215 \nu^{8} - 2091395692889556420271389561 \nu^{7} - 1501016739616879718371822192 \nu^{6} + 2605018497515346382262540644 \nu^{5} - 357276643406505762646451513 \nu^{4} - 558302259460451814779557586 \nu^{3} + 75869399959125235523461804 \nu^{2} + 49110444353489173637753464 \nu + 3071590555772710668815584$$$$)/$$$$12\!\cdots\!76$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{19} - 2 \beta_{18} - \beta_{14} - \beta_{13} - \beta_{10} + \beta_{9} - \beta_{8} - \beta_{6} + 8 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{19} - \beta_{18} - 2 \beta_{16} - 3 \beta_{15} - 2 \beta_{14} - \beta_{13} - \beta_{12} - \beta_{11} - \beta_{10} + 2 \beta_{9} + \beta_{8} + \beta_{7} + 2 \beta_{6} - \beta_{3} + 13 \beta_{2} + \beta_{1} + 26$$ $$\nu^{5}$$ $$=$$ $$16 \beta_{19} - 28 \beta_{18} - \beta_{17} - 2 \beta_{16} + 2 \beta_{15} - 14 \beta_{14} - 14 \beta_{13} + \beta_{11} - 12 \beta_{10} + 15 \beta_{9} - 14 \beta_{8} - 14 \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - 2 \beta_{2} + 72 \beta_{1} + 3$$ $$\nu^{6}$$ $$=$$ $$15 \beta_{19} - 16 \beta_{18} - 32 \beta_{16} - 45 \beta_{15} - 32 \beta_{14} - 15 \beta_{13} - 13 \beta_{12} - 15 \beta_{11} - 13 \beta_{10} + 37 \beta_{9} + 10 \beta_{8} + 16 \beta_{7} + 24 \beta_{6} + \beta_{5} + 4 \beta_{4} - 14 \beta_{3} + 142 \beta_{2} + 22 \beta_{1} + 190$$ $$\nu^{7}$$ $$=$$ $$197 \beta_{19} - 324 \beta_{18} - 13 \beta_{17} - 40 \beta_{16} + 34 \beta_{15} - 162 \beta_{14} - 158 \beta_{13} + \beta_{12} + 14 \beta_{11} - 132 \beta_{10} + 180 \beta_{9} - 153 \beta_{8} + 7 \beta_{7} - 150 \beta_{6} - 11 \beta_{5} + 16 \beta_{4} - 17 \beta_{3} - 14 \beta_{2} + 681 \beta_{1} + 56$$ $$\nu^{8}$$ $$=$$ $$179 \beta_{19} - 212 \beta_{18} + \beta_{17} - 394 \beta_{16} - 537 \beta_{15} - 400 \beta_{14} - 188 \beta_{13} - 133 \beta_{12} - 181 \beta_{11} - 152 \beta_{10} + 521 \beta_{9} + 69 \beta_{8} + 202 \beta_{7} + 230 \beta_{6} + 33 \beta_{5} + 75 \beta_{4} - 142 \beta_{3} + 1504 \beta_{2} + 345 \beta_{1} + 1474$$ $$\nu^{9}$$ $$=$$ $$2242 \beta_{19} - 3548 \beta_{18} - 130 \beta_{17} - 589 \beta_{16} + 458 \beta_{15} - 1775 \beta_{14} - 1666 \beta_{13} + 27 \beta_{12} + 154 \beta_{11} - 1434 \beta_{10} + 2046 \beta_{9} - 1569 \beta_{8} + 136 \beta_{7} - 1467 \beta_{6} - 56 \beta_{5} + 192 \beta_{4} - 217 \beta_{3} + 59 \beta_{2} + 6643 \beta_{1} + 792$$ $$\nu^{10}$$ $$=$$ $$2022 \beta_{19} - 2637 \beta_{18} + \beta_{17} - 4475 \beta_{16} - 5936 \beta_{15} - 4580 \beta_{14} - 2215 \beta_{13} - 1242 \beta_{12} - 2029 \beta_{11} - 1758 \beta_{10} + 6612 \beta_{9} + 296 \beta_{8} + 2331 \beta_{7} + 2070 \beta_{6} + 662 \beta_{5} + 1023 \beta_{4} - 1228 \beta_{3} + 15831 \beta_{2} + 4760 \beta_{1} + 11839$$ $$\nu^{11}$$ $$=$$ $$24694 \beta_{19} - 38052 \beta_{18} - 1220 \beta_{17} - 7703 \beta_{16} + 5632 \beta_{15} - 19042 \beta_{14} - 17090 \beta_{13} + 557 \beta_{12} + 1576 \beta_{11} - 15534 \beta_{10} + 22922 \beta_{9} - 15828 \beta_{8} + 1847 \beta_{7} - 13777 \beta_{6} + 406 \beta_{5} + 2133 \beta_{4} - 2405 \beta_{3} + 3310 \beta_{2} + 66233 \beta_{1} + 10042$$ $$\nu^{12}$$ $$=$$ $$22669 \beta_{19} - 31860 \beta_{18} - 353 \beta_{17} - 49411 \beta_{16} - 63248 \beta_{15} - 50359 \beta_{14} - 25164 \beta_{13} - 10933 \beta_{12} - 21906 \beta_{11} - 20358 \beta_{10} + 79596 \beta_{9} - 1148 \beta_{8} + 25731 \beta_{7} + 18378 \beta_{6} + 10727 \beta_{5} + 12516 \beta_{4} - 9242 \beta_{3} + 166584 \beta_{2} + 61363 \beta_{1} + 97185$$ $$\nu^{13}$$ $$=$$ $$267779 \beta_{19} - 404748 \beta_{18} - 11574 \beta_{17} - 94960 \beta_{16} + 65865 \beta_{15} - 202402 \beta_{14} - 173219 \beta_{13} + 9779 \beta_{12} + 15735 \beta_{11} - 168085 \beta_{10} + 255934 \beta_{9} - 159750 \beta_{8} + 21890 \beta_{7} - 126656 \beta_{6} + 16685 \beta_{5} + 23371 \beta_{4} - 24056 \beta_{3} + 64731 \beta_{2} + 671333 \beta_{1} + 119688$$ $$\nu^{14}$$ $$=$$ $$255922 \beta_{19} - 378970 \beta_{18} - 9832 \beta_{17} - 540829 \beta_{16} - 659313 \beta_{15} - 542483 \beta_{14} - 278787 \beta_{13} - 90757 \beta_{12} - 230996 \beta_{11} - 236055 \beta_{10} + 929323 \beta_{9} - 52810 \beta_{8} + 277141 \beta_{7} + 164453 \beta_{6} + 154678 \beta_{5} + 146344 \beta_{4} - 56115 \beta_{3} + 1755226 \beta_{2} + 758547 \beta_{1} + 809827$$ $$\nu^{15}$$ $$=$$ $$2881358 \beta_{19} - 4293131 \beta_{18} - 115183 \beta_{17} - 1132664 \beta_{16} + 746471 \beta_{15} - 2142698 \beta_{14} - 1746963 \beta_{13} + 153024 \beta_{12} + 156000 \beta_{11} - 1817868 \beta_{10} + 2856511 \beta_{9} - 1623913 \beta_{8} + 243658 \beta_{7} - 1148814 \beta_{6} + 315763 \beta_{5} + 258149 \beta_{4} - 218500 \beta_{3} + 996727 \beta_{2} + 6892946 \beta_{1} + 1369666$$ $$\nu^{16}$$ $$=$$ $$2916094 \beta_{19} - 4463886 \beta_{18} - 185101 \beta_{17} - 5912648 \beta_{16} - 6771585 \beta_{15} - 5783070 \beta_{14} - 3034233 \beta_{13} - 694271 \beta_{12} - 2396598 \beta_{11} - 2735823 \beta_{10} + 10646296 \beta_{9} - 945847 \beta_{8} + 2942256 \beta_{7} + 1500291 \beta_{6} + 2076332 \beta_{5} + 1675575 \beta_{4} - 163863 \beta_{3} + 18529245 \beta_{2} + 9118033 \beta_{1} + 6824720$$ $$\nu^{17}$$ $$=$$ $$30888593 \beta_{19} - 45526694 \beta_{18} - 1217038 \beta_{17} - 13245715 \beta_{16} + 8285536 \beta_{15} - 22650576 \beta_{14} - 17593238 \beta_{13} + 2205910 \beta_{12} + 1546697 \beta_{11} - 19660193 \beta_{10} + 31889147 \beta_{9} - 16664782 \beta_{8} + 2631900 \beta_{7} - 10308512 \beta_{6} + 4841333 \beta_{5} + 2890837 \beta_{4} - 1760551 \beta_{3} + 13762848 \beta_{2} + 71512985 \beta_{1} + 15231619$$ $$\nu^{18}$$ $$=$$ $$33465540 \beta_{19} - 52204864 \beta_{18} - 2926840 \beta_{17} - 64734513 \beta_{16} - 68796280 \beta_{15} - 61350429 \beta_{14} - 32615584 \beta_{13} - 4563528 \beta_{12} - 24577691 \beta_{11} - 31644668 \beta_{10} + 120478540 \beta_{9} - 13742616 \beta_{8} + 30965684 \beta_{7} + 14047429 \beta_{6} + 26590683 \beta_{5} + 18977414 \beta_{4} + 2574068 \beta_{3} + 196036179 \beta_{2} + 107483928 \beta_{1} + 58044469$$ $$\nu^{19}$$ $$=$$ $$330626168 \beta_{19} - 483298046 \beta_{18} - 13558852 \beta_{17} - 152965455 \beta_{16} + 90657185 \beta_{15} - 239435481 \beta_{14} - 177270201 \beta_{13} + 29974767 \beta_{12} + 15376435 \beta_{11} - 212705635 \beta_{10} + 356064094 \beta_{9} - 172693043 \beta_{8} + 28054584 \beta_{7} - 91494216 \beta_{6} + 67104395 \beta_{5} + 32782410 \beta_{4} - 11477953 \beta_{3} + 178706955 \beta_{2} + 748301832 \beta_{1} + 165879535$$

## 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
 −3.16561 −2.99276 −2.78165 −2.20278 −1.97232 −1.72088 −0.330073 −0.323343 −0.241588 −0.0780369 0.622776 1.02602 1.03584 1.30183 1.96074 2.32194 2.35112 2.72853 3.13315 3.32710
−1.00000 −3.16561 1.00000 1.00000 3.16561 0.575814 −1.00000 7.02106 −1.00000
1.2 −1.00000 −2.99276 1.00000 1.00000 2.99276 0.839081 −1.00000 5.95662 −1.00000
1.3 −1.00000 −2.78165 1.00000 1.00000 2.78165 −3.35211 −1.00000 4.73759 −1.00000
1.4 −1.00000 −2.20278 1.00000 1.00000 2.20278 4.44205 −1.00000 1.85224 −1.00000
1.5 −1.00000 −1.97232 1.00000 1.00000 1.97232 4.46416 −1.00000 0.890065 −1.00000
1.6 −1.00000 −1.72088 1.00000 1.00000 1.72088 −0.251451 −1.00000 −0.0385865 −1.00000
1.7 −1.00000 −0.330073 1.00000 1.00000 0.330073 −4.66703 −1.00000 −2.89105 −1.00000
1.8 −1.00000 −0.323343 1.00000 1.00000 0.323343 4.27073 −1.00000 −2.89545 −1.00000
1.9 −1.00000 −0.241588 1.00000 1.00000 0.241588 0.516163 −1.00000 −2.94164 −1.00000
1.10 −1.00000 −0.0780369 1.00000 1.00000 0.0780369 −1.81697 −1.00000 −2.99391 −1.00000
1.11 −1.00000 0.622776 1.00000 1.00000 −0.622776 3.30687 −1.00000 −2.61215 −1.00000
1.12 −1.00000 1.02602 1.00000 1.00000 −1.02602 −3.81201 −1.00000 −1.94728 −1.00000
1.13 −1.00000 1.03584 1.00000 1.00000 −1.03584 −1.66553 −1.00000 −1.92705 −1.00000
1.14 −1.00000 1.30183 1.00000 1.00000 −1.30183 5.09963 −1.00000 −1.30525 −1.00000
1.15 −1.00000 1.96074 1.00000 1.00000 −1.96074 −2.08760 −1.00000 0.844520 −1.00000
1.16 −1.00000 2.32194 1.00000 1.00000 −2.32194 1.00226 −1.00000 2.39142 −1.00000
1.17 −1.00000 2.35112 1.00000 1.00000 −2.35112 2.99774 −1.00000 2.52776 −1.00000
1.18 −1.00000 2.72853 1.00000 1.00000 −2.72853 0.894798 −1.00000 4.44487 −1.00000
1.19 −1.00000 3.13315 1.00000 1.00000 −3.13315 3.12083 −1.00000 6.81662 −1.00000
1.20 −1.00000 3.32710 1.00000 1.00000 −3.32710 −2.87743 −1.00000 8.06959 −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.20 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4010.2.a.m 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4010.2.a.m 20 1.a even 1 1 trivial

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$5$$ $$-1$$
$$401$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4010))$$:

 $$T_{3}^{20} - \cdots$$ $$T_{7}^{20} - \cdots$$ $$T_{11}^{20} - \cdots$$