# Properties

 Label 304.5.e.f Level $304$ Weight $5$ Character orbit 304.e Analytic conductor $31.424$ Analytic rank $0$ Dimension $20$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$304 = 2^{4} \cdot 19$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 304.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$31.4244687775$$ Analytic rank: $$0$$ Dimension: $$20$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} + \cdots)$$ Defining polynomial: $$x^{20} + 996 x^{18} + 408854 x^{16} + 89661524 x^{14} + 11414409521 x^{12} + 861580608848 x^{10} + 37911701888064 x^{8} + 915101881952256 x^{6} + 10365057927217152 x^{4} + 34336742150504448 x^{2} + 34273490123096064$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{50}$$ Twist minimal: no (minimal twist has level 152) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} -\beta_{3} q^{5} + ( -2 + \beta_{5} ) q^{7} + ( -19 + \beta_{2} - \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} -\beta_{3} q^{5} + ( -2 + \beta_{5} ) q^{7} + ( -19 + \beta_{2} - \beta_{3} ) q^{9} + ( 1 - \beta_{3} - \beta_{6} ) q^{11} + \beta_{13} q^{13} + ( -4 \beta_{1} + \beta_{10} ) q^{15} + ( 11 - \beta_{3} - \beta_{7} ) q^{17} + ( 30 + \beta_{3} - \beta_{16} ) q^{19} + ( 2 \beta_{1} - \beta_{12} ) q^{21} + ( -29 - 3 \beta_{3} - \beta_{9} ) q^{23} + ( 71 + 3 \beta_{3} + \beta_{4} + \beta_{6} - \beta_{8} ) q^{25} + ( -13 \beta_{1} + \beta_{12} - \beta_{15} + \beta_{16} + \beta_{17} ) q^{27} + ( \beta_{1} - \beta_{10} - \beta_{12} - \beta_{13} - \beta_{14} - \beta_{19} ) q^{29} + ( -20 \beta_{1} - \beta_{10} - \beta_{11} - 2 \beta_{13} - 2 \beta_{14} - \beta_{15} + \beta_{16} + 2 \beta_{17} - \beta_{19} ) q^{31} + ( 2 \beta_{1} + 3 \beta_{10} - \beta_{12} + \beta_{13} + \beta_{14} - \beta_{17} + \beta_{18} + \beta_{19} ) q^{33} + ( 7 - \beta_{2} + \beta_{3} + 2 \beta_{5} + \beta_{7} + \beta_{8} - \beta_{15} - \beta_{16} ) q^{35} + ( -16 \beta_{1} - \beta_{10} + \beta_{11} + 2 \beta_{12} - \beta_{14} - \beta_{15} + \beta_{16} + \beta_{18} - \beta_{19} ) q^{37} + ( 27 - 3 \beta_{2} + \beta_{3} + 2 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{15} - \beta_{16} ) q^{39} + ( 26 \beta_{1} - 2 \beta_{10} + \beta_{11} - 2 \beta_{12} + 4 \beta_{13} + 2 \beta_{14} - \beta_{15} + \beta_{16} - \beta_{17} - 2 \beta_{18} ) q^{41} + ( 61 + \beta_{2} + 17 \beta_{3} + 4 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + \beta_{8} + 2 \beta_{9} - 2 \beta_{15} - 2 \beta_{16} ) q^{43} + ( 369 - 5 \beta_{2} + 22 \beta_{3} + 3 \beta_{4} - 6 \beta_{5} + 7 \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{9} - 2 \beta_{15} - 2 \beta_{16} ) q^{45} + ( 191 - 4 \beta_{2} + \beta_{3} - 7 \beta_{5} - 6 \beta_{6} + 4 \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{15} - \beta_{16} ) q^{47} + ( -136 + 10 \beta_{2} - 13 \beta_{3} - 4 \beta_{4} - 10 \beta_{5} - 3 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{15} - 2 \beta_{16} ) q^{49} + ( 15 \beta_{1} + 4 \beta_{10} + 3 \beta_{11} + 4 \beta_{12} - 6 \beta_{13} + 6 \beta_{14} - 4 \beta_{15} + 4 \beta_{16} - 2 \beta_{17} + 3 \beta_{19} ) q^{51} + ( 17 \beta_{1} - 10 \beta_{10} - 3 \beta_{11} - 2 \beta_{12} - 5 \beta_{15} + 5 \beta_{16} + 4 \beta_{17} + \beta_{18} + 2 \beta_{19} ) q^{53} + ( 513 - 13 \beta_{2} + 27 \beta_{3} - 3 \beta_{5} - 7 \beta_{6} - 3 \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{15} - 2 \beta_{16} ) q^{55} + ( -43 + 45 \beta_{1} + 6 \beta_{2} - 28 \beta_{3} + 3 \beta_{4} + 10 \beta_{5} - 5 \beta_{6} - 3 \beta_{7} - \beta_{8} - 2 \beta_{9} - 4 \beta_{10} + 3 \beta_{12} - 4 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} + 2 \beta_{16} - 3 \beta_{17} - \beta_{18} + 2 \beta_{19} ) q^{57} + ( 43 \beta_{1} + 6 \beta_{10} - \beta_{11} + 5 \beta_{12} - 2 \beta_{13} - \beta_{15} + \beta_{16} - 5 \beta_{17} - \beta_{19} ) q^{59} + ( 25 + 9 \beta_{2} + 38 \beta_{3} - \beta_{4} - 20 \beta_{5} + 11 \beta_{6} + 3 \beta_{7} + \beta_{8} - 4 \beta_{15} - 4 \beta_{16} ) q^{61} + ( -317 + 19 \beta_{2} + 11 \beta_{3} - 4 \beta_{4} + 21 \beta_{5} + 15 \beta_{6} - 3 \beta_{7} + \beta_{8} + \beta_{9} - 3 \beta_{15} - 3 \beta_{16} ) q^{63} + ( -4 \beta_{1} + 7 \beta_{10} - 3 \beta_{11} + 5 \beta_{12} + 15 \beta_{13} - 5 \beta_{14} - \beta_{15} + \beta_{16} - 2 \beta_{17} - \beta_{18} - \beta_{19} ) q^{65} + ( 85 \beta_{1} - 10 \beta_{10} - \beta_{11} - 6 \beta_{12} - 14 \beta_{13} - 5 \beta_{15} + 5 \beta_{16} + 3 \beta_{19} ) q^{67} + ( -75 \beta_{1} + 17 \beta_{10} + 6 \beta_{11} + 4 \beta_{12} - 20 \beta_{13} + 7 \beta_{14} - 6 \beta_{15} + 6 \beta_{16} - 4 \beta_{17} - 2 \beta_{18} + \beta_{19} ) q^{69} + ( -70 \beta_{1} + 14 \beta_{10} + 3 \beta_{11} - 6 \beta_{12} + 10 \beta_{13} - \beta_{15} + \beta_{16} - 4 \beta_{17} - 4 \beta_{18} - \beta_{19} ) q^{71} + ( 62 - 13 \beta_{2} - 42 \beta_{3} - 11 \beta_{4} + 18 \beta_{5} - 3 \beta_{6} + 6 \beta_{7} + \beta_{8} - 2 \beta_{9} - 2 \beta_{15} - 2 \beta_{16} ) q^{73} + ( 93 \beta_{1} - 26 \beta_{10} - 6 \beta_{11} - 3 \beta_{12} + 20 \beta_{13} + 2 \beta_{14} - 6 \beta_{15} + 6 \beta_{16} + \beta_{17} - 4 \beta_{18} + 2 \beta_{19} ) q^{75} + ( 475 - 31 \beta_{2} + 22 \beta_{3} - 9 \beta_{4} + 8 \beta_{5} + 7 \beta_{6} - 5 \beta_{7} + \beta_{8} - 4 \beta_{9} - 4 \beta_{15} - 4 \beta_{16} ) q^{77} + ( 36 \beta_{1} - 7 \beta_{10} - 6 \beta_{11} - 2 \beta_{12} + 20 \beta_{13} - 14 \beta_{14} - 2 \beta_{15} + 2 \beta_{16} + 6 \beta_{17} + 8 \beta_{18} - 2 \beta_{19} ) q^{79} + ( -239 + 8 \beta_{2} + 34 \beta_{3} - 6 \beta_{4} + 46 \beta_{5} - 22 \beta_{6} + 10 \beta_{7} + 2 \beta_{9} - 2 \beta_{15} - 2 \beta_{16} ) q^{81} + ( 834 - 27 \beta_{2} - 2 \beta_{3} - 12 \beta_{4} - 30 \beta_{5} + 33 \beta_{6} - \beta_{7} - 3 \beta_{8} + 2 \beta_{9} - 3 \beta_{15} - 3 \beta_{16} ) q^{83} + ( 528 - 10 \beta_{2} - \beta_{3} + 14 \beta_{4} - 38 \beta_{5} - 26 \beta_{6} + 12 \beta_{7} - 2 \beta_{9} - 6 \beta_{15} - 6 \beta_{16} ) q^{85} + ( -131 + 40 \beta_{2} - 87 \beta_{3} + 16 \beta_{4} - 64 \beta_{5} - 4 \beta_{6} - 12 \beta_{7} + \beta_{9} - 3 \beta_{15} - 3 \beta_{16} ) q^{87} + ( 84 \beta_{1} - 8 \beta_{10} - 13 \beta_{11} - 2 \beta_{12} - 8 \beta_{13} - 20 \beta_{14} + \beta_{15} - \beta_{16} + 9 \beta_{17} + 2 \beta_{18} - 6 \beta_{19} ) q^{89} + ( -25 \beta_{1} + 13 \beta_{11} + 7 \beta_{12} + 2 \beta_{13} - 11 \beta_{17} - 4 \beta_{18} - 7 \beta_{19} ) q^{91} + ( 1844 - 60 \beta_{2} + 138 \beta_{3} + 6 \beta_{4} - 2 \beta_{5} - 38 \beta_{6} - 6 \beta_{7} + 2 \beta_{9} - 6 \beta_{15} - 6 \beta_{16} ) q^{93} + ( -749 - 166 \beta_{1} + 37 \beta_{2} - 137 \beta_{3} - 12 \beta_{4} + 43 \beta_{5} - 21 \beta_{6} - 9 \beta_{7} - \beta_{8} - \beta_{9} - 10 \beta_{10} + 3 \beta_{11} - 2 \beta_{12} - 18 \beta_{13} - \beta_{15} + 3 \beta_{16} + 8 \beta_{17} + 4 \beta_{18} - 5 \beta_{19} ) q^{95} + ( -130 \beta_{1} + \beta_{10} + 7 \beta_{11} + 3 \beta_{12} - 23 \beta_{13} - 7 \beta_{14} - 7 \beta_{15} + 7 \beta_{16} + 16 \beta_{17} + 3 \beta_{18} + \beta_{19} ) q^{97} + ( -11 + 44 \beta_{2} + 159 \beta_{3} + 4 \beta_{4} - 90 \beta_{5} + 35 \beta_{6} + 8 \beta_{7} - 2 \beta_{8} + 4 \beta_{9} - 4 \beta_{15} - 4 \beta_{16} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q - 32q^{7} - 372q^{9} + O(q^{10})$$ $$20q - 32q^{7} - 372q^{9} + 24q^{11} + 216q^{17} + 596q^{19} - 576q^{23} + 1412q^{25} + 144q^{35} + 520q^{39} + 1256q^{43} + 7232q^{45} + 3768q^{47} - 2740q^{49} + 10128q^{55} - 728q^{57} + 352q^{61} - 6104q^{63} + 1352q^{73} + 9288q^{77} - 4220q^{81} + 16104q^{83} + 10232q^{85} - 2936q^{87} + 36432q^{93} - 14232q^{95} - 760q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} + 996 x^{18} + 408854 x^{16} + 89661524 x^{14} + 11414409521 x^{12} + 861580608848 x^{10} + 37911701888064 x^{8} + 915101881952256 x^{6} + 10365057927217152 x^{4} + 34336742150504448 x^{2} + 34273490123096064$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$7015591482736478697048953 \nu^{18} + 6877057136442363798278307204 \nu^{16} + 2762453645622673838324578791526 \nu^{14} + 587790123766048552377507145752628 \nu^{12} + 71688787769461832364673746163452457 \nu^{10} + 5087982217402337932788988323231547792 \nu^{8} + 204916299226419925363498434451520373312 \nu^{6} + 4353828936944978660336592305349348956160 \nu^{4} + 40244551137216999503608585033015821139968 \nu^{2} + 76456656887018284541945328020963796713472$$$$)/$$$$57\!\cdots\!32$$ $$\beta_{3}$$ $$=$$ $$($$$$7015591482736478697048953 \nu^{18} + 6877057136442363798278307204 \nu^{16} + 2762453645622673838324578791526 \nu^{14} + 587790123766048552377507145752628 \nu^{12} + 71688787769461832364673746163452457 \nu^{10} + 5087982217402337932788988323231547792 \nu^{8} + 204916299226419925363498434451520373312 \nu^{6} + 4353828936944978660336592305349348956160 \nu^{4} + 40186971497569650307959196026290575835136 \nu^{2} + 70698692922283364977006427348439266230272$$$$)/$$$$57\!\cdots\!32$$ $$\beta_{4}$$ $$=$$ $$($$$$10455416804468796827256871 \nu^{18} + 10093865115949232203052077308 \nu^{16} + 3969323674535653396041654595418 \nu^{14} + 819248809664970853004336184445004 \nu^{12} + 95555342365459310547859333569292151 \nu^{10} + 6348511732789677193101321543582364784 \nu^{8} + 232436557284241687911141751343697094080 \nu^{6} + 4357172388027681444166990554565277116416 \nu^{4} + 34960748522456071674010884825018728644608 \nu^{2} + 58090622717697780565593410239714409054208$$$$)/$$$$57\!\cdots\!32$$ $$\beta_{5}$$ $$=$$ $$($$$$-643312172309955074890481 \nu^{18} - 637580256755232951101678052 \nu^{16} - 259993917040271693274733929142 \nu^{14} - 56499853652885085742913376980116 \nu^{12} - 7101429163025354149803831900898849 \nu^{10} - 526273740917645490897535374871063696 \nu^{8} - 22528723381674710988047329081062250560 \nu^{6} - 519193082079399066235804477752172296192 \nu^{4} - 5277393403135015289567544539634859573248 \nu^{2} - 9398646894548195730109601850383344336896$$$$)/$$$$30\!\cdots\!28$$ $$\beta_{6}$$ $$=$$ $$($$$$-326858064707069045267699 \nu^{18} - 321672121357164915533942613 \nu^{16} - 129922657869745666360708045222 \nu^{14} - 27862778610788824562264188853698 \nu^{12} - 3438018037140310115727303132963703 \nu^{10} - 248350846952221396150448127622415497 \nu^{8} - 10273561523110636706368379785287355344 \nu^{6} - 226929819092999084567903673382426546752 \nu^{4} - 2203607703171522522839869736233199622144 \nu^{2} - 3955144028429171873970587124361719152640$$$$)/$$$$89\!\cdots\!88$$ $$\beta_{7}$$ $$=$$ $$($$$$18765241665041430289977025 \nu^{18} + 18863078260110627805736419716 \nu^{16} + 7835356762863573048101522959382 \nu^{14} + 1744536228692261486845893770227988 \nu^{12} + 226347285305475167107772619976906865 \nu^{10} + 17469228706108904660959373065879595504 \nu^{8} + 785408101220240112321856690491480183360 \nu^{6} + 19100515377661570656509089954774492237824 \nu^{4} + 204926522714218558841067465735083517149184 \nu^{2} + 380377480317245530016520067415011954262016$$$$)/$$$$28\!\cdots\!16$$ $$\beta_{8}$$ $$=$$ $$($$$$67669402085462715252962657 \nu^{18} + 63431418977529811606584931620 \nu^{16} + 23883932186162958841250597643862 \nu^{14} + 4606325962317218964453922339110484 \nu^{12} + 479054233294477260474837587630685073 \nu^{10} + 25580063507462217511580478649168985872 \nu^{8} + 554392432559250613299773920959651654720 \nu^{6} - 1471744953938229746830417462633450924032 \nu^{4} - 156342028813703659660921085022631958347776 \nu^{2} - 322250322855149837384824380948872964341760$$$$)/$$$$57\!\cdots\!32$$ $$\beta_{9}$$ $$=$$ $$($$$$-137240621828284286426961595 \nu^{18} - 137545212550475095237072867020 \nu^{16} - 56905001939647026820676839758866 \nu^{14} - 12600755952704034603443858077324508 \nu^{12} - 1622614672608434508954758940494290379 \nu^{10} - 123928134836511151628463012123413744816 \nu^{8} - 5491696571065831659876244594123487144640 \nu^{6} - 130950749090552781663472710565650872567808 \nu^{4} - 1369635074731126216680733994200038630162432 \nu^{2} - 2486395785099896932935915635491112963014656$$$$)/$$$$57\!\cdots\!32$$ $$\beta_{10}$$ $$=$$ $$($$$$-7015591482736478697048953 \nu^{19} - 6877057136442363798278307204 \nu^{17} - 2762453645622673838324578791526 \nu^{15} - 587790123766048552377507145752628 \nu^{13} - 71688787769461832364673746163452457 \nu^{11} - 5087982217402337932788988323231547792 \nu^{9} - 204916299226419925363498434451520373312 \nu^{7} - 4353828936944978660336592305349348956160 \nu^{5} - 40186971497569650307959196026290575835136 \nu^{3} - 70468374363693968194408871321538285010944 \nu$$$$)/$$$$57\!\cdots\!32$$ $$\beta_{11}$$ $$=$$ $$($$$$4757507011033628632965739 \nu^{19} + 4500420772583206945097889144 \nu^{17} + 1717351867500227807304918591074 \nu^{15} + 338160805602856785259840544633060 \nu^{13} + 36423013119501783511210139625351659 \nu^{11} + 2081934755712703856445202528146229724 \nu^{9} + 54203015331270985665855140593184110464 \nu^{7} + 262711646049447987545955679869328945920 \nu^{5} - 8603216675179839177307242219963959205888 \nu^{3} - 33103571513741873814383925595772966535168 \nu$$$$)/$$$$32\!\cdots\!68$$ $$\beta_{12}$$ $$=$$ $$($$$$643312172309955074890481 \nu^{19} + 637580256755232951101678052 \nu^{17} + 259993917040271693274733929142 \nu^{15} + 56499853652885085742913376980116 \nu^{13} + 7101429163025354149803831900898849 \nu^{11} + 526273740917645490897535374871063696 \nu^{9} + 22528723381674710988047329081062250560 \nu^{7} + 519193082079399066235804477752172296192 \nu^{5} + 5277393403135015289567544539634859573248 \nu^{3} + 9410768923947637666035789009693922295808 \nu$$$$)/$$$$30\!\cdots\!28$$ $$\beta_{13}$$ $$=$$ $$($$$$130314032331384017607966419 \nu^{19} + 129726445798216282516267191276 \nu^{17} + 53204034049504430945268855975202 \nu^{15} + 11648138211081514032945870078408508 \nu^{13} + 1478068970425135060871726718929145187 \nu^{11} + 110827155338421622564330993768844607472 \nu^{9} + 4806122491228111583350687714922428161216 \nu^{7} + 112014512994327709551867824101809028872192 \nu^{5} + 1144268233184962788658118007184657299603456 \nu^{3} + 1993204154558833521222249060232778861248512 \nu$$$$)/$$$$51\!\cdots\!88$$ $$\beta_{14}$$ $$=$$ $$($$$$128472238193535293 \nu^{19} + 127364482636332577140 \nu^{17} + 51949520568053402047678 \nu^{15} + 11290193140987889665694404 \nu^{13} + 1418607308373456959300506669 \nu^{11} + 105004302444107628034030635568 \nu^{9} + 4481857070326235379379724216640 \nu^{7} + 102698886644271549417058259963904 \nu^{5} + 1035521777419187138093400013799424 \nu^{3} + 1856441532162924812077690391101440 \nu$$$$)/$$$$48\!\cdots\!96$$ $$\beta_{15}$$ $$=$$ $$($$$$-$$$$20\!\cdots\!75$$$$\nu^{19} +$$$$34\!\cdots\!62$$$$\nu^{18} -$$$$19\!\cdots\!80$$$$\nu^{17} +$$$$33\!\cdots\!84$$$$\nu^{16} -$$$$82\!\cdots\!10$$$$\nu^{15} +$$$$13\!\cdots\!40$$$$\nu^{14} -$$$$18\!\cdots\!80$$$$\nu^{13} +$$$$28\!\cdots\!44$$$$\nu^{12} -$$$$22\!\cdots\!55$$$$\nu^{11} +$$$$35\!\cdots\!42$$$$\nu^{10} -$$$$17\!\cdots\!36$$$$\nu^{9} +$$$$25\!\cdots\!20$$$$\nu^{8} -$$$$75\!\cdots\!16$$$$\nu^{7} +$$$$10\!\cdots\!84$$$$\nu^{6} -$$$$17\!\cdots\!64$$$$\nu^{5} +$$$$21\!\cdots\!96$$$$\nu^{4} -$$$$18\!\cdots\!48$$$$\nu^{3} +$$$$19\!\cdots\!40$$$$\nu^{2} -$$$$33\!\cdots\!80$$$$\nu +$$$$33\!\cdots\!00$$$$)/$$$$34\!\cdots\!92$$ $$\beta_{16}$$ $$=$$ $$($$$$20\!\cdots\!75$$$$\nu^{19} +$$$$34\!\cdots\!62$$$$\nu^{18} +$$$$19\!\cdots\!80$$$$\nu^{17} +$$$$33\!\cdots\!84$$$$\nu^{16} +$$$$82\!\cdots\!10$$$$\nu^{15} +$$$$13\!\cdots\!40$$$$\nu^{14} +$$$$18\!\cdots\!80$$$$\nu^{13} +$$$$28\!\cdots\!44$$$$\nu^{12} +$$$$22\!\cdots\!55$$$$\nu^{11} +$$$$35\!\cdots\!42$$$$\nu^{10} +$$$$17\!\cdots\!36$$$$\nu^{9} +$$$$25\!\cdots\!20$$$$\nu^{8} +$$$$75\!\cdots\!16$$$$\nu^{7} +$$$$10\!\cdots\!84$$$$\nu^{6} +$$$$17\!\cdots\!64$$$$\nu^{5} +$$$$21\!\cdots\!96$$$$\nu^{4} +$$$$18\!\cdots\!48$$$$\nu^{3} +$$$$19\!\cdots\!40$$$$\nu^{2} +$$$$33\!\cdots\!80$$$$\nu +$$$$33\!\cdots\!00$$$$)/$$$$34\!\cdots\!92$$ $$\beta_{17}$$ $$=$$ $$($$$$-3701240537101493545783253 \nu^{19} - 3687971553150258167061489921 \nu^{17} - 1514390436249429079633618897186 \nu^{15} - 332098957116204928345990160763778 \nu^{13} - 42235227375263661857820924817733257 \nu^{11} - 3176315381116064873338789709726693197 \nu^{9} - 138283683002747593502755136004565159824 \nu^{7} - 3239537350237690091105302813972978443072 \nu^{5} - 33369741090976583217740716963051308398592 \nu^{3} - 60047837806541658463437230735853237633024 \nu$$$$)/$$$$26\!\cdots\!64$$ $$\beta_{18}$$ $$=$$ $$($$$$11302110714735679635473341 \nu^{19} + 11181902569463184778985391060 \nu^{17} + 4548496388545692553789759981886 \nu^{15} + 984878840499038787219994418783492 \nu^{13} + 123124481428295173942584040944498221 \nu^{11} + 9051020523449177721835247968625761424 \nu^{9} + 382839349800472436520069213796808318784 \nu^{7} + 8676395530632321361793554972273649068032 \nu^{5} + 86481873506557731474767753474810644463616 \nu^{3} + 156288098797745245063055883287489138393088 \nu$$$$)/$$$$58\!\cdots\!92$$ $$\beta_{19}$$ $$=$$ $$($$$$-1558346050875267305097550919 \nu^{19} - 1549799964337629468243093550716 \nu^{17} - 634783652976433794767906566229530 \nu^{15} - 138731991940756946637712262099617996 \nu^{13} - 17562698831876423820043913344275387799 \nu^{11} - 1312802631246926504637542608690968487792 \nu^{9} - 56717491026505220960927230734516393393600 \nu^{7} - 1317109410863299598962976095493158704064512 \nu^{5} - 13449774943388775506925322098877529727369216 \nu^{3} - 24244918478947441021245712897205773283622912 \nu$$$$)/$$$$51\!\cdots\!88$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{3} + \beta_{2} - 100$$ $$\nu^{3}$$ $$=$$ $$\beta_{17} + \beta_{16} - \beta_{15} + \beta_{12} - 175 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-2 \beta_{16} - 2 \beta_{15} + 2 \beta_{9} + 10 \beta_{7} - 22 \beta_{6} + 46 \beta_{5} - 6 \beta_{4} + 277 \beta_{3} - 235 \beta_{2} + 17500$$ $$\nu^{5}$$ $$=$$ $$-14 \beta_{19} + 28 \beta_{18} - 241 \beta_{17} - 291 \beta_{16} + 291 \beta_{15} - 182 \beta_{14} + 112 \beta_{13} - 333 \beta_{12} - 42 \beta_{11} - 56 \beta_{10} + 35457 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$234 \beta_{16} + 234 \beta_{15} - 898 \beta_{9} + 146 \beta_{8} - 3504 \beta_{7} + 7428 \beta_{6} - 15102 \beta_{5} + 2540 \beta_{4} - 65087 \beta_{3} + 53061 \beta_{2} - 3550438$$ $$\nu^{7}$$ $$=$$ $$7980 \beta_{19} - 11004 \beta_{18} + 51293 \beta_{17} + 76025 \beta_{16} - 76025 \beta_{15} + 80286 \beta_{14} - 50292 \beta_{13} + 89693 \beta_{12} + 16776 \beta_{11} + 19352 \beta_{10} - 7612981 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$19822 \beta_{16} + 19822 \beta_{15} + 282914 \beta_{9} - 51140 \beta_{8} + 1007638 \beta_{7} - 1946034 \beta_{6} + 3990734 \beta_{5} - 848226 \beta_{4} + 14782537 \beta_{3} - 12056847 \beta_{2} + 763911296$$ $$\nu^{9}$$ $$=$$ $$-3084394 \beta_{19} + 3297452 \beta_{18} - 10737017 \beta_{17} - 19356063 \beta_{16} + 19356063 \beta_{15} - 26691478 \beta_{14} + 15782024 \beta_{13} - 22579949 \beta_{12} - 5027238 \beta_{11} - 4973608 \beta_{10} + 1682376793 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$-20655942 \beta_{16} - 20655942 \beta_{15} - 78374274 \beta_{9} + 12438198 \beta_{8} - 269994396 \beta_{7} + 473039808 \beta_{6} - 987799614 \beta_{5} + 258458408 \beta_{4} - 3332520819 \beta_{3} + 2767681961 \beta_{2} - 169264445898$$ $$\nu^{11}$$ $$=$$ $$1002420040 \beta_{19} - 904682252 \beta_{18} + 2263300229 \beta_{17} + 4867692981 \beta_{16} - 4867692981 \beta_{15} + 7971477598 \beta_{14} - 4357837964 \beta_{13} + 5524787901 \beta_{12} + 1354858020 \beta_{11} + 1121455560 \beta_{10} - 378339049933 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$8124839966 \beta_{16} + 8124839966 \beta_{15} + 20445235362 \beta_{9} - 2525244936 \beta_{8} + 69769361570 \beta_{7} - 112419964174 \beta_{6} + 238504786638 \beta_{5} - 74592580126 \beta_{4} + 749991958877 \beta_{3} - 641018771763 \beta_{2} + 38178353707492$$ $$\nu^{13}$$ $$=$$ $$-296492631078 \beta_{19} + 239102205084 \beta_{18} - 483370031553 \beta_{17} - 1214351170587 \beta_{16} + 1214351170587 \beta_{15} - 2247946454262 \beta_{14} + 1135087529280 \beta_{13} - 1334534104781 \beta_{12} - 347130966498 \beta_{11} - 229720092280 \beta_{10} + 86120707333457 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$-2588688905206 \beta_{16} - 2588688905206 \beta_{15} - 5162871005442 \beta_{9} + 434472103386 \beta_{8} - 17650004264584 \beta_{7} + 26634416601980 \beta_{6} - 57068752184190 \beta_{5} + 20732752086820 \beta_{4} - 168670159110951 \beta_{3} + 149558138250445 \beta_{2} - 8717464596913006$$ $$\nu^{15}$$ $$=$$ $$82864254888292 \beta_{19} - 62001344303324 \beta_{18} + 104708928870253 \beta_{17} + 301217752722353 \beta_{16} - 301217752722353 \beta_{15} + 611303201709022 \beta_{14} - 286792511901284 \beta_{13} + 320615605771293 \beta_{12} + 86534071446720 \beta_{11} + 42330726885944 \beta_{10} - 19784608593454117 \beta_{1}$$ $$\nu^{16}$$ $$=$$ $$753426470023374 \beta_{16} + 753426470023374 \beta_{15} + 1278672947086114 \beta_{9} - 56212123079244 \beta_{8} + 4405966879677102 \beta_{7} - 6337351958218474 \beta_{6} + 13629792170532686 \beta_{5} - 5606209189458714 \beta_{4} + 37908510198095089 \beta_{3} - 35105909214237847 \beta_{2} + 2008866173108398856$$ $$\nu^{17}$$ $$=$$ $$-22329133763830626 \beta_{19} + 15895335735737676 \beta_{18} - 22996076804617417 \beta_{17} - 74403090182156887 \beta_{16} + 74403090182156887 \beta_{15} - 162129919648675734 \beta_{14} + 71216899835392440 \beta_{13} - 76894598649961133 \beta_{12} - 21227211060023454 \beta_{11} - 6582272105105160 \beta_{10} + 4578835265815550089 \beta_{1}$$ $$\nu^{18}$$ $$=$$ $$-208852207134875302 \beta_{16} - 208852207134875302 \beta_{15} - 312791878087319938 \beta_{9} + 1322948179765630 \beta_{8} - 1090327720286818420 \beta_{7} + 1517867296186625848 \beta_{6} - 3259895053042985278 \beta_{5} + 1485058056506920160 \beta_{4} - 8513985980351598555 \beta_{3} + 8282349403950101937 \beta_{2} - 466312840155402405202$$ $$\nu^{19}$$ $$=$$ $$5870109736134725248 \beta_{19} - 4043567626778405228 \beta_{18} + 5115280504404559381 \beta_{17} + 18320644702455574253 \beta_{16} - 18320644702455574253 \beta_{15} + 42231845263939501726 \beta_{14} - 17500651753194701564 \beta_{13} + 18444614177517895037 \beta_{12} + 5155672118462968668 \beta_{11} + 647878097703634152 \beta_{10} - 1066249619082234149309 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/304\mathbb{Z}\right)^\times$$.

 $$n$$ $$97$$ $$191$$ $$229$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
113.1
 − 15.6226i − 14.8343i − 13.8615i − 11.4302i − 9.35107i − 6.69459i − 6.40102i − 5.59979i − 1.51452i − 1.48358i 1.48358i 1.51452i 5.59979i 6.40102i 6.69459i 9.35107i 11.4302i 13.8615i 14.8343i 15.6226i
0 15.6226i 0 6.30407 0 28.3216 0 −163.066 0
113.2 0 14.8343i 0 −45.4605 0 2.86483 0 −139.057 0
113.3 0 13.8615i 0 16.6305 0 34.1005 0 −111.141 0
113.4 0 11.4302i 0 −9.10929 0 −75.9168 0 −49.6505 0
113.5 0 9.35107i 0 21.0112 0 9.13929 0 −6.44248 0
113.6 0 6.69459i 0 2.80752 0 −75.3813 0 36.1825 0
113.7 0 6.40102i 0 47.0652 0 −32.2009 0 40.0269 0
113.8 0 5.59979i 0 −33.0135 0 43.4389 0 49.6423 0
113.9 0 1.51452i 0 15.8064 0 74.7269 0 78.7062 0
113.10 0 1.48358i 0 −22.0416 0 −25.0930 0 78.7990 0
113.11 0 1.48358i 0 −22.0416 0 −25.0930 0 78.7990 0
113.12 0 1.51452i 0 15.8064 0 74.7269 0 78.7062 0
113.13 0 5.59979i 0 −33.0135 0 43.4389 0 49.6423 0
113.14 0 6.40102i 0 47.0652 0 −32.2009 0 40.0269 0
113.15 0 6.69459i 0 2.80752 0 −75.3813 0 36.1825 0
113.16 0 9.35107i 0 21.0112 0 9.13929 0 −6.44248 0
113.17 0 11.4302i 0 −9.10929 0 −75.9168 0 −49.6505 0
113.18 0 13.8615i 0 16.6305 0 34.1005 0 −111.141 0
113.19 0 14.8343i 0 −45.4605 0 2.86483 0 −139.057 0
113.20 0 15.6226i 0 6.30407 0 28.3216 0 −163.066 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 113.20 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 304.5.e.f 20
4.b odd 2 1 152.5.e.a 20
12.b even 2 1 1368.5.o.a 20
19.b odd 2 1 inner 304.5.e.f 20
76.d even 2 1 152.5.e.a 20
228.b odd 2 1 1368.5.o.a 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.5.e.a 20 4.b odd 2 1
152.5.e.a 20 76.d even 2 1
304.5.e.f 20 1.a even 1 1 trivial
304.5.e.f 20 19.b odd 2 1 inner
1368.5.o.a 20 12.b even 2 1
1368.5.o.a 20 228.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{5}^{\mathrm{new}}(304, [\chi])$$:

 $$37\!\cdots\!64$$$$T_{3}^{8} +$$$$91\!\cdots\!56$$$$T_{3}^{6} +$$$$10\!\cdots\!52$$$$T_{3}^{4} +$$$$34\!\cdots\!48$$$$T_{3}^{2} +$$$$34\!\cdots\!64$$">$$T_{3}^{20} + \cdots$$ $$13\!\cdots\!76$$">$$T_{5}^{10} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{20}$$
$3$ $$34273490123096064 + 34336742150504448 T^{2} + 10365057927217152 T^{4} + 915101881952256 T^{6} + 37911701888064 T^{8} + 861580608848 T^{10} + 11414409521 T^{12} + 89661524 T^{14} + 408854 T^{16} + 996 T^{18} + T^{20}$$
$5$ $$( 1386394827776 - 692665586688 T + 49729423104 T^{2} + 10460647104 T^{3} - 1014011772 T^{4} - 21809988 T^{5} + 3312217 T^{6} + 6348 T^{7} - 3478 T^{8} + T^{10} )^{2}$$
$7$ $$( 379548178285972 - 175416055532068 T + 14056176863697 T^{2} + 448564540100 T^{3} - 43036780564 T^{4} - 222923532 T^{5} + 38575206 T^{6} - 62884 T^{7} - 11192 T^{8} + 16 T^{9} + T^{10} )^{2}$$
$11$ $$( -2031616987934346432 - 292289686518966720 T + 135106809603568 T^{2} + 680836260992736 T^{3} - 9403338070324 T^{4} - 91792327164 T^{5} + 1603952041 T^{6} + 2858856 T^{7} - 76582 T^{8} - 12 T^{9} + T^{10} )^{2}$$
$13$ $$13\!\cdots\!76$$$$+$$$$27\!\cdots\!84$$$$T^{2} +$$$$20\!\cdots\!32$$$$T^{4} +$$$$67\!\cdots\!60$$$$T^{6} +$$$$11\!\cdots\!92$$$$T^{8} +$$$$10\!\cdots\!20$$$$T^{10} + 54026753546998783025 T^{12} + 1659471421806676 T^{14} + 29199462294 T^{16} + 269268 T^{18} + T^{20}$$
$17$ $$( -$$$$60\!\cdots\!04$$$$-$$$$12\!\cdots\!08$$$$T + 50916404245781354177 T^{2} + 118096386737724840 T^{3} - 3316369162731068 T^{4} - 5297897700048 T^{5} + 65069392294 T^{6} + 47417592 T^{7} - 455536 T^{8} - 108 T^{9} + T^{10} )^{2}$$
$19$ $$14\!\cdots\!01$$$$-$$$$64\!\cdots\!76$$$$T -$$$$40\!\cdots\!14$$$$T^{2} +$$$$39\!\cdots\!64$$$$T^{3} +$$$$27\!\cdots\!73$$$$T^{4} -$$$$73\!\cdots\!68$$$$T^{5} +$$$$16\!\cdots\!68$$$$T^{6} +$$$$40\!\cdots\!56$$$$T^{7} -$$$$16\!\cdots\!90$$$$T^{8} -$$$$17\!\cdots\!84$$$$T^{9} +$$$$15\!\cdots\!24$$$$T^{10} -$$$$13\!\cdots\!04$$$$T^{11} -$$$$95\!\cdots\!90$$$$T^{12} + 1835385763696324496 T^{13} + 5854554058559528 T^{14} - 19509735805168 T^{15} + 5624982813 T^{16} + 61177004 T^{17} - 4874 T^{18} - 596 T^{19} + T^{20}$$
$23$ $$( -$$$$14\!\cdots\!76$$$$+$$$$17\!\cdots\!80$$$$T +$$$$19\!\cdots\!12$$$$T^{2} - 17728035758374341792 T^{3} - 267777552636991476 T^{4} + 128956835039820 T^{5} + 1156546270585 T^{6} - 347529444 T^{7} - 1860894 T^{8} + 288 T^{9} + T^{10} )^{2}$$
$29$ $$13\!\cdots\!36$$$$+$$$$14\!\cdots\!44$$$$T^{2} +$$$$52\!\cdots\!76$$$$T^{4} +$$$$97\!\cdots\!48$$$$T^{6} +$$$$96\!\cdots\!40$$$$T^{8} +$$$$52\!\cdots\!48$$$$T^{10} +$$$$15\!\cdots\!05$$$$T^{12} + 23703926857224791700 T^{14} + 18545962830518 T^{16} + 6992228 T^{18} + T^{20}$$
$31$ $$46\!\cdots\!04$$$$+$$$$31\!\cdots\!92$$$$T^{2} +$$$$71\!\cdots\!36$$$$T^{4} +$$$$73\!\cdots\!40$$$$T^{6} +$$$$36\!\cdots\!44$$$$T^{8} +$$$$94\!\cdots\!16$$$$T^{10} +$$$$13\!\cdots\!12$$$$T^{12} +$$$$10\!\cdots\!84$$$$T^{14} + 45561932062368 T^{16} + 10576576 T^{18} + T^{20}$$
$37$ $$92\!\cdots\!16$$$$+$$$$15\!\cdots\!92$$$$T^{2} +$$$$72\!\cdots\!44$$$$T^{4} +$$$$15\!\cdots\!92$$$$T^{6} +$$$$17\!\cdots\!00$$$$T^{8} +$$$$12\!\cdots\!48$$$$T^{10} +$$$$54\!\cdots\!64$$$$T^{12} +$$$$15\!\cdots\!52$$$$T^{14} + 259991932206496 T^{16} + 24738912 T^{18} + T^{20}$$
$41$ $$92\!\cdots\!76$$$$+$$$$85\!\cdots\!84$$$$T^{2} +$$$$21\!\cdots\!24$$$$T^{4} +$$$$27\!\cdots\!64$$$$T^{6} +$$$$20\!\cdots\!44$$$$T^{8} +$$$$92\!\cdots\!28$$$$T^{10} +$$$$27\!\cdots\!84$$$$T^{12} +$$$$52\!\cdots\!72$$$$T^{14} + 602781986607520 T^{16} + 38072064 T^{18} + T^{20}$$
$43$ $$( -$$$$33\!\cdots\!64$$$$+$$$$21\!\cdots\!36$$$$T +$$$$12\!\cdots\!28$$$$T^{2} -$$$$42\!\cdots\!32$$$$T^{3} -$$$$14\!\cdots\!76$$$$T^{4} + 19231539442393580 T^{5} + 76457212744857 T^{6} + 1318105800 T^{7} - 15002766 T^{8} - 628 T^{9} + T^{10} )^{2}$$
$47$ $$($$$$62\!\cdots\!92$$$$+$$$$39\!\cdots\!20$$$$T +$$$$37\!\cdots\!16$$$$T^{2} -$$$$38\!\cdots\!48$$$$T^{3} -$$$$54\!\cdots\!92$$$$T^{4} + 15882726837278436 T^{5} + 178366758881241 T^{6} + 18149754456 T^{7} - 22076694 T^{8} - 1884 T^{9} + T^{10} )^{2}$$
$53$ $$74\!\cdots\!04$$$$+$$$$14\!\cdots\!16$$$$T^{2} +$$$$11\!\cdots\!12$$$$T^{4} +$$$$51\!\cdots\!12$$$$T^{6} +$$$$13\!\cdots\!88$$$$T^{8} +$$$$22\!\cdots\!12$$$$T^{10} +$$$$22\!\cdots\!57$$$$T^{12} +$$$$14\!\cdots\!60$$$$T^{14} + 5374400844546678 T^{16} + 112629156 T^{18} + T^{20}$$
$59$ $$12\!\cdots\!84$$$$+$$$$22\!\cdots\!28$$$$T^{2} +$$$$14\!\cdots\!60$$$$T^{4} +$$$$47\!\cdots\!76$$$$T^{6} +$$$$89\!\cdots\!92$$$$T^{8} +$$$$10\!\cdots\!84$$$$T^{10} +$$$$75\!\cdots\!65$$$$T^{12} +$$$$34\!\cdots\!96$$$$T^{14} + 9773135547054070 T^{16} + 151851092 T^{18} + T^{20}$$
$61$ $$($$$$33\!\cdots\!16$$$$+$$$$78\!\cdots\!44$$$$T +$$$$79\!\cdots\!72$$$$T^{2} -$$$$31\!\cdots\!12$$$$T^{3} -$$$$41\!\cdots\!16$$$$T^{4} + 429508413434760236 T^{5} + 701629109312505 T^{6} - 18373594036 T^{7} - 47007734 T^{8} - 176 T^{9} + T^{10} )^{2}$$
$67$ $$14\!\cdots\!96$$$$+$$$$13\!\cdots\!28$$$$T^{2} +$$$$51\!\cdots\!76$$$$T^{4} +$$$$10\!\cdots\!72$$$$T^{6} +$$$$12\!\cdots\!32$$$$T^{8} +$$$$98\!\cdots\!52$$$$T^{10} +$$$$47\!\cdots\!49$$$$T^{12} +$$$$14\!\cdots\!56$$$$T^{14} + 25885226684709942 T^{16} + 250778052 T^{18} + T^{20}$$
$71$ $$33\!\cdots\!56$$$$+$$$$31\!\cdots\!96$$$$T^{2} +$$$$23\!\cdots\!16$$$$T^{4} +$$$$51\!\cdots\!16$$$$T^{6} +$$$$54\!\cdots\!56$$$$T^{8} +$$$$33\!\cdots\!20$$$$T^{10} +$$$$12\!\cdots\!16$$$$T^{12} +$$$$28\!\cdots\!32$$$$T^{14} + 39912243786558048 T^{16} + 308190992 T^{18} + T^{20}$$
$73$ $$( -$$$$80\!\cdots\!68$$$$-$$$$10\!\cdots\!92$$$$T +$$$$13\!\cdots\!89$$$$T^{2} +$$$$16\!\cdots\!08$$$$T^{3} -$$$$86\!\cdots\!92$$$$T^{4} - 1572939432953769880 T^{5} + 1421930361133878 T^{6} + 31026206416 T^{7} - 74320640 T^{8} - 676 T^{9} + T^{10} )^{2}$$
$79$ $$51\!\cdots\!44$$$$+$$$$57\!\cdots\!68$$$$T^{2} +$$$$20\!\cdots\!68$$$$T^{4} +$$$$27\!\cdots\!76$$$$T^{6} +$$$$17\!\cdots\!20$$$$T^{8} +$$$$64\!\cdots\!00$$$$T^{10} +$$$$13\!\cdots\!60$$$$T^{12} +$$$$17\!\cdots\!08$$$$T^{14} + 133855162872163744 T^{16} + 562672512 T^{18} + T^{20}$$
$83$ $$( -$$$$22\!\cdots\!00$$$$-$$$$25\!\cdots\!60$$$$T +$$$$19\!\cdots\!24$$$$T^{2} +$$$$10\!\cdots\!04$$$$T^{3} -$$$$31\!\cdots\!92$$$$T^{4} - 65861727412334954304 T^{5} + 12943499108385008 T^{6} + 1313612886816 T^{7} - 195647124 T^{8} - 8052 T^{9} + T^{10} )^{2}$$
$89$ $$27\!\cdots\!24$$$$+$$$$10\!\cdots\!48$$$$T^{2} +$$$$12\!\cdots\!08$$$$T^{4} +$$$$51\!\cdots\!72$$$$T^{6} +$$$$61\!\cdots\!24$$$$T^{8} +$$$$31\!\cdots\!56$$$$T^{10} +$$$$85\!\cdots\!52$$$$T^{12} +$$$$13\!\cdots\!56$$$$T^{14} + 114495085948674624 T^{16} + 529225648 T^{18} + T^{20}$$
$97$ $$45\!\cdots\!24$$$$+$$$$91\!\cdots\!84$$$$T^{2} +$$$$28\!\cdots\!92$$$$T^{4} +$$$$25\!\cdots\!36$$$$T^{6} +$$$$92\!\cdots\!72$$$$T^{8} +$$$$17\!\cdots\!64$$$$T^{10} +$$$$19\!\cdots\!84$$$$T^{12} +$$$$13\!\cdots\!96$$$$T^{14} + 528164108708799072 T^{16} + 1129821904 T^{18} + T^{20}$$