Properties

Label 380.2.r.a
Level $380$
Weight $2$
Character orbit 380.r
Analytic conductor $3.034$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.r (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.03431527681\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 20 x^{18} + 261 x^{16} - 1994 x^{14} + 11074 x^{12} - 39211 x^{10} + 99376 x^{8} - 134299 x^{6} + 124617 x^{4} - 24768 x^{2} + 4096\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( \beta_{1} - \beta_{11} ) q^{3} + ( -\beta_{7} + \beta_{13} ) q^{5} + ( \beta_{15} - \beta_{18} ) q^{7} + ( 1 - \beta_{3} + \beta_{10} ) q^{9} +O(q^{10})\) \( q + ( \beta_{1} - \beta_{11} ) q^{3} + ( -\beta_{7} + \beta_{13} ) q^{5} + ( \beta_{15} - \beta_{18} ) q^{7} + ( 1 - \beta_{3} + \beta_{10} ) q^{9} -\beta_{6} q^{11} + ( \beta_{9} + \beta_{16} ) q^{13} + ( -1 + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{8} + \beta_{9} + \beta_{12} ) q^{15} + ( \beta_{7} - \beta_{9} - \beta_{13} - \beta_{14} + \beta_{17} - \beta_{18} ) q^{17} + ( 1 - \beta_{7} - \beta_{8} - \beta_{10} + \beta_{12} ) q^{19} + ( 2 \beta_{2} - \beta_{3} + \beta_{7} + 2 \beta_{10} - \beta_{12} - \beta_{13} + \beta_{14} ) q^{21} + ( \beta_{1} + \beta_{9} - 2 \beta_{16} ) q^{23} + ( 1 + \beta_{1} - \beta_{3} + \beta_{4} - \beta_{7} + \beta_{9} - \beta_{16} ) q^{25} + ( \beta_{15} + \beta_{16} - \beta_{18} - \beta_{19} ) q^{27} + ( -2 + 2 \beta_{3} + \beta_{5} - \beta_{10} + \beta_{12} ) q^{29} + ( 1 + 2 \beta_{2} - \beta_{5} + \beta_{6} - \beta_{8} - \beta_{13} + \beta_{14} ) q^{31} + ( -\beta_{1} - 2 \beta_{9} + \beta_{11} + 2 \beta_{17} + \beta_{19} ) q^{33} + ( -2 \beta_{2} + \beta_{4} - \beta_{6} - \beta_{7} - 2 \beta_{10} + \beta_{12} + \beta_{13} - \beta_{14} + \beta_{19} ) q^{35} + ( -\beta_{11} + \beta_{16} - \beta_{17} - \beta_{19} ) q^{37} + ( -1 + \beta_{2} + \beta_{5} + \beta_{8} + \beta_{13} - \beta_{14} ) q^{39} + ( 3 \beta_{3} - \beta_{4} + \beta_{6} + 2 \beta_{7} - 2 \beta_{13} + 2 \beta_{14} ) q^{41} + ( -\beta_{7} + 3 \beta_{9} + \beta_{13} + \beta_{14} - 3 \beta_{17} ) q^{43} + ( -2 + \beta_{5} - \beta_{8} + \beta_{13} + \beta_{14} - \beta_{15} - \beta_{16} - \beta_{17} + \beta_{18} + \beta_{19} ) q^{45} + ( -\beta_{1} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{16} ) q^{47} + ( -3 - 2 \beta_{2} + \beta_{5} - 3 \beta_{6} + 2 \beta_{8} + 2 \beta_{13} - 2 \beta_{14} ) q^{49} + ( 4 - 4 \beta_{3} + \beta_{4} - 3 \beta_{5} - \beta_{7} - \beta_{8} + 2 \beta_{10} - 3 \beta_{12} ) q^{51} + ( \beta_{1} - \beta_{7} + \beta_{8} - 2 \beta_{9} - \beta_{16} ) q^{53} + ( \beta_{1} - \beta_{3} - \beta_{4} + \beta_{6} - \beta_{9} - \beta_{11} + \beta_{12} - 2 \beta_{14} + \beta_{17} + \beta_{18} ) q^{55} + ( -\beta_{1} + 2 \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{11} - \beta_{13} - \beta_{14} - 2 \beta_{15} - \beta_{16} + 2 \beta_{17} + \beta_{18} + \beta_{19} ) q^{57} + ( -\beta_{2} - \beta_{10} - \beta_{12} ) q^{59} + ( 1 - \beta_{3} - 2 \beta_{5} - 2 \beta_{12} ) q^{61} + ( -3 \beta_{1} - \beta_{7} + \beta_{8} + 2 \beta_{16} ) q^{63} + ( -1 - 2 \beta_{2} + \beta_{11} + \beta_{13} - \beta_{15} + 2 \beta_{16} + \beta_{18} - 2 \beta_{19} ) q^{65} + ( \beta_{1} - \beta_{7} + \beta_{8} + \beta_{9} + 3 \beta_{15} - \beta_{16} ) q^{67} + ( 3 - 3 \beta_{2} - 2 \beta_{5} + 3 \beta_{6} + \beta_{8} + \beta_{13} - \beta_{14} ) q^{69} + ( 2 \beta_{2} + 3 \beta_{7} + 2 \beta_{10} - \beta_{12} - 3 \beta_{13} + 3 \beta_{14} ) q^{71} + ( -\beta_{1} - \beta_{9} + \beta_{11} + \beta_{17} - 2 \beta_{19} ) q^{73} + ( 2 - 2 \beta_{2} + \beta_{6} + 2 \beta_{8} - 2 \beta_{11} + \beta_{13} - 2 \beta_{14} + \beta_{16} - \beta_{17} - \beta_{19} ) q^{75} + ( \beta_{8} - \beta_{13} - \beta_{14} - 2 \beta_{15} - \beta_{16} - 2 \beta_{17} + 2 \beta_{18} + \beta_{19} ) q^{77} + ( \beta_{2} - \beta_{3} - 3 \beta_{4} + 3 \beta_{6} + \beta_{7} + \beta_{10} + \beta_{12} - \beta_{13} + \beta_{14} ) q^{79} + ( 2 \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} - 2 \beta_{12} - \beta_{13} + \beta_{14} ) q^{81} + ( -2 \beta_{8} + 2 \beta_{13} + 2 \beta_{14} - \beta_{15} + 2 \beta_{16} - \beta_{17} + \beta_{18} - 2 \beta_{19} ) q^{83} + ( -4 - 2 \beta_{1} + 4 \beta_{3} + \beta_{5} - \beta_{7} - 2 \beta_{8} - 3 \beta_{10} + \beta_{12} + \beta_{16} ) q^{85} + ( -2 \beta_{8} + 2 \beta_{11} + 2 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} - \beta_{16} - 2 \beta_{17} + 2 \beta_{18} + \beta_{19} ) q^{87} + ( -3 + 3 \beta_{3} + 3 \beta_{4} - \beta_{5} - 3 \beta_{7} - 3 \beta_{8} - \beta_{10} - \beta_{12} ) q^{89} + ( -1 + \beta_{3} - 3 \beta_{4} + 3 \beta_{7} + 3 \beta_{8} ) q^{91} + ( -4 \beta_{1} + 3 \beta_{7} - 2 \beta_{9} + 4 \beta_{11} - 3 \beta_{13} - 3 \beta_{14} + 2 \beta_{17} + \beta_{18} + \beta_{19} ) q^{93} + ( -2 - 2 \beta_{1} + \beta_{6} - \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{14} + 2 \beta_{15} + 2 \beta_{17} - \beta_{18} ) q^{95} + ( -2 \beta_{7} + \beta_{9} + 2 \beta_{13} + 2 \beta_{14} - \beta_{17} - \beta_{18} + \beta_{19} ) q^{97} + ( -2 + 2 \beta_{3} + \beta_{5} - 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{10} + \beta_{12} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + q^{5} + 10q^{9} + O(q^{10}) \) \( 20q + q^{5} + 10q^{9} - 5q^{15} + 14q^{19} - 8q^{21} + 9q^{25} - 16q^{29} + 8q^{31} - 2q^{35} - 8q^{39} + 26q^{41} - 32q^{45} - 44q^{49} + 26q^{51} - 12q^{55} + 4q^{59} + 2q^{61} - 18q^{65} + 48q^{69} - 2q^{71} + 46q^{75} - 16q^{79} + 26q^{81} - 39q^{85} - 40q^{89} - 4q^{91} - 43q^{95} - 20q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 20 x^{18} + 261 x^{16} - 1994 x^{14} + 11074 x^{12} - 39211 x^{10} + 99376 x^{8} - 134299 x^{6} + 124617 x^{4} - 24768 x^{2} + 4096\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(257269301703 \nu^{18} - 4826223210105 \nu^{16} + 61286511972345 \nu^{14} - 438571056316700 \nu^{12} + 2323089270554580 \nu^{10} - 7266718287702885 \nu^{8} + 16741977055075822 \nu^{6} - 10869496174421115 \nu^{4} + 2170497176992320 \nu^{2} + 63024290250218436\)\()/ 16690142463557537 \)
\(\beta_{3}\)\(=\)\((\)\(58379368812683 \nu^{18} - 1151122140944668 \nu^{16} + 14928136974663543 \nu^{14} - 112486124646259822 \nu^{12} + 618424582627382742 \nu^{10} - 2140435717198619993 \nu^{8} + 5336438184716201168 \nu^{6} - 6768804320649661609 \nu^{4} + 6579414048167166051 \nu^{2} - 238859269757341696\)\()/ 1068169117667682368 \)
\(\beta_{4}\)\(=\)\((\)\(6366481800502185 \nu^{18} - 137623959156754484 \nu^{16} + 1868654652948801741 \nu^{14} - 15323421807441242074 \nu^{12} + 89902370458361971154 \nu^{10} - 349276396009867994211 \nu^{8} + 944354362189538749232 \nu^{6} - 1468016942861277361043 \nu^{4} + 1259578493369127028273 \nu^{2} - 250780428775971323584\)\()/ 88658036766417636544 \)
\(\beta_{5}\)\(=\)\((\)\(-108802723986440 \nu^{18} + 2093074375676352 \nu^{16} - 26579215713710528 \nu^{14} + 192433368253136232 \nu^{12} - 1007495594987342592 \nu^{10} + 3151487442894034624 \nu^{8} - 6782358791543175575 \nu^{6} + 4713968444633542976 \nu^{4} - 941318257748307968 \nu^{2} - 4955504751551263591\)\()/ 1385281824475275571 \)
\(\beta_{6}\)\(=\)\((\)\(-123113231093090 \nu^{18} + 2391532877467530 \nu^{16} - 30369235310188170 \nu^{14} + 220945270298193633 \nu^{12} - 1151157774093599880 \nu^{10} + 3600868617089492610 \nu^{8} - 7533337923107099056 \nu^{6} + 5386149030199971390 \nu^{4} - 1075543988176763520 \nu^{2} - 2918597117977935675\)\()/ 1385281824475275571 \)
\(\beta_{7}\)\(=\)\((\)\(112685518955184 \nu^{19} - 9174325379691717 \nu^{18} - 1633888394360592 \nu^{17} + 183358087220694436 \nu^{16} + 20748174355631888 \nu^{15} - 2387658497653929081 \nu^{14} - 126231930067659184 \nu^{13} + 18206740636302532114 \nu^{12} + 786467685596365632 \nu^{11} - 100804631860325583978 \nu^{10} - 2460103098942103504 \nu^{9} + 356441852003984207511 \nu^{8} + 10217679424117915920 \nu^{7} - 901408307418290504720 \nu^{6} - 3679801550568386096 \nu^{5} + 1230925670964922837447 \nu^{4} + 734808564190512128 \nu^{3} - 1127871072781709528461 \nu^{2} + 182780307164276557072 \nu + 224153346284272802752\)\()/ 88658036766417636544 \)
\(\beta_{8}\)\(=\)\((\)\(-112685518955184 \nu^{19} - 9174325379691717 \nu^{18} + 1633888394360592 \nu^{17} + 183358087220694436 \nu^{16} - 20748174355631888 \nu^{15} - 2387658497653929081 \nu^{14} + 126231930067659184 \nu^{13} + 18206740636302532114 \nu^{12} - 786467685596365632 \nu^{11} - 100804631860325583978 \nu^{10} + 2460103098942103504 \nu^{9} + 356441852003984207511 \nu^{8} - 10217679424117915920 \nu^{7} - 901408307418290504720 \nu^{6} + 3679801550568386096 \nu^{5} + 1230925670964922837447 \nu^{4} - 734808564190512128 \nu^{3} - 1127871072781709528461 \nu^{2} - 182780307164276557072 \nu + 224153346284272802752\)\()/ 88658036766417636544 \)
\(\beta_{9}\)\(=\)\((\)\(149437719142871 \nu^{19} - 2750374974599151 \nu^{17} + 34926044957117039 \nu^{15} - 247970499250483465 \nu^{13} + 1323885430768152396 \nu^{11} - 4141167794335228387 \nu^{9} + 9657228077683790013 \nu^{7} - 6194324001019140413 \nu^{5} + 1236926030594408384 \nu^{3} + 22508069497503397645 \nu\)\()/ 5541127297901102284 \)
\(\beta_{10}\)\(=\)\((\)\(54263059985435 \nu^{18} - 1073902569582988 \nu^{16} + 13947552783106023 \nu^{14} - 105468987745192622 \nu^{12} + 581255154298509462 \nu^{10} - 2024168224595373833 \nu^{8} + 5068566551834988016 \nu^{6} - 6594892381858923769 \nu^{4} + 6277643813918368339 \nu^{2} - 1247247913760836672\)\()/ 267042279416920592 \)
\(\beta_{11}\)\(=\)\((\)\(58379368812683 \nu^{19} - 1151122140944668 \nu^{17} + 14928136974663543 \nu^{15} - 112486124646259822 \nu^{13} + 618424582627382742 \nu^{11} - 2140435717198619993 \nu^{9} + 5336438184716201168 \nu^{7} - 6768804320649661609 \nu^{5} + 6579414048167166051 \nu^{3} - 238859269757341696 \nu\)\()/ 1068169117667682368 \)
\(\beta_{12}\)\(=\)\((\)\(21472706330772813 \nu^{18} - 429531870127860356 \nu^{16} + 5597440381244918081 \nu^{14} - 42637019206442019714 \nu^{12} + 235405967494108868890 \nu^{10} - 823296813298088269295 \nu^{8} + 2040723648133655490800 \nu^{6} - 2590582498362261966815 \nu^{4} + 2114820325419353216181 \nu^{2} - 91430995209659510784\)\()/ 88658036766417636544 \)
\(\beta_{13}\)\(=\)\((\)\(41145391020481691 \nu^{19} + 9113271949323008 \nu^{18} - 807651770702036060 \nu^{17} - 169488444796903296 \nu^{16} + 10458402250723591943 \nu^{15} + 2152274179832962944 \nu^{14} - 78527538519393341710 \nu^{13} - 15365184231760305024 \nu^{12} + 431134182000265878518 \nu^{11} + 81582796826776290816 \nu^{10} - 1487157977486890963977 \nu^{9} - 255194326441686202752 \nu^{8} + 3714706244185375280592 \nu^{7} + 577191879275290897152 \nu^{6} - 4710527327865639831641 \nu^{5} - 381717529862951442048 \nu^{4} + 4520388542025619161715 \nu^{3} + 76224031701280088064 \nu^{2} - 166218017713291595776 \nu + 709395219183111220992\)\()/ \)\(35\!\cdots\!76\)\( \)
\(\beta_{14}\)\(=\)\((\)\(40694648944660955 \nu^{19} - 45810573468089876 \nu^{18} - 801116217124593692 \nu^{17} + 902920793679681040 \nu^{16} + 10375409553301064391 \nu^{15} - 11702908170448679268 \nu^{14} - 78022610799122704974 \nu^{13} + 88192146776970433480 \nu^{12} + 427988311257880415990 \nu^{11} - 484801324268078626728 \nu^{10} - 1477317565091122549961 \nu^{9} + 1680961734457623032796 \nu^{8} + 3673835526488903616912 \nu^{7} - 4182825108948452916032 \nu^{6} - 4695808121663366287257 \nu^{5} + 5305420213722642791836 \nu^{4} + 4517449307768857113203 \nu^{3} - 4587708322828118201908 \nu^{2} - 897339246370397824064 \nu + 187218165953979990016\)\()/ \)\(35\!\cdots\!76\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-352957477246353 \nu^{19} + 6732287676656781 \nu^{17} - 85490954590084109 \nu^{15} + 617058244498298531 \nu^{13} - 3240568160043291876 \nu^{11} + 10136629792755534697 \nu^{9} - 21944735732755401673 \nu^{7} + 15162285696465178103 \nu^{5} - 3027711475567210304 \nu^{3} - 15112667902972457477 \nu\)\()/ 2770563648950551142 \)
\(\beta_{16}\)\(=\)\((\)\(395664181329051 \nu^{19} - 7533440729534211 \nu^{17} + 95664515577493379 \nu^{15} - 689861039846870731 \nu^{13} + 3626200978955352156 \nu^{11} - 11342905028514213607 \nu^{9} + 24723903923897988125 \nu^{7} - 16966622061419083193 \nu^{5} + 3388014006947935424 \nu^{3} + 31115827382409820137 \nu\)\()/ 2770563648950551142 \)
\(\beta_{17}\)\(=\)\((\)\(86955964488571567 \nu^{19} - 1710572564381717100 \nu^{17} + 22161310421172271211 \nu^{15} - 166719685296363775190 \nu^{13} + 915935506268344505246 \nu^{11} - 3168119711944513996773 \nu^{9} + 7897531353133828196624 \nu^{7} - 10015947541588282623477 \nu^{5} + 9108096864853737363623 \nu^{3} - 353436183667271585792 \nu\)\()/ \)\(35\!\cdots\!76\)\( \)
\(\beta_{18}\)\(=\)\((\)\(44403267011197975 \nu^{19} - 905638902720961484 \nu^{17} + 11924107064037836787 \nu^{15} - 92792312405128286118 \nu^{13} + 522399676445550876270 \nu^{11} - 1902277734940496224477 \nu^{9} + 4916800551459821697744 \nu^{7} - 7052914632463443149421 \nu^{5} + 6287552361196403142031 \nu^{3} - 1250374162009682137408 \nu\)\()/ \)\(17\!\cdots\!88\)\( \)
\(\beta_{19}\)\(=\)\((\)\(-99815889287337571 \nu^{19} + 2000882759717695292 \nu^{17} - 26141423587608532335 \nu^{15} + 200169113650494448030 \nu^{13} - 1113670060332052229382 \nu^{11} + 3956950094181766287265 \nu^{9} - 10054026219204032528144 \nu^{7} + 13679166208473456186225 \nu^{5} - 12640273317629698848059 \nu^{3} + 2512480201371985676864 \nu\)\()/ \)\(17\!\cdots\!88\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{10} + 4 \beta_{3} - \beta_{2}\)
\(\nu^{3}\)\(=\)\(\beta_{19} + \beta_{18} - \beta_{16} - \beta_{15} + 6 \beta_{11}\)
\(\nu^{4}\)\(=\)\(2 \beta_{12} - 9 \beta_{10} - \beta_{8} - \beta_{7} + 2 \beta_{5} - \beta_{4} + 25 \beta_{3} - 25\)
\(\nu^{5}\)\(=\)\(10 \beta_{19} + 11 \beta_{18} + 3 \beta_{14} + 3 \beta_{13} + 40 \beta_{11} - 3 \beta_{7} - 40 \beta_{1}\)
\(\nu^{6}\)\(=\)\(8 \beta_{14} - 8 \beta_{13} - 8 \beta_{8} - 16 \beta_{6} + 27 \beta_{5} + 72 \beta_{2} - 177\)
\(\nu^{7}\)\(=\)\(88 \beta_{16} + 99 \beta_{15} + 6 \beta_{9} + 46 \beta_{8} - 46 \beta_{7} - 283 \beta_{1}\)
\(\nu^{8}\)\(=\)\(47 \beta_{14} - 47 \beta_{13} - 279 \beta_{12} + 569 \beta_{10} + 47 \beta_{7} - 174 \beta_{6} + 174 \beta_{4} - 1329 \beta_{3} + 569 \beta_{2}\)
\(\nu^{9}\)\(=\)\(-743 \beta_{19} - 848 \beta_{18} + 116 \beta_{17} + 743 \beta_{16} + 848 \beta_{15} - 511 \beta_{14} - 511 \beta_{13} - 2083 \beta_{11} + 511 \beta_{8}\)
\(\nu^{10}\)\(=\)\(-2613 \beta_{12} + 4522 \beta_{10} + 221 \beta_{8} + 221 \beta_{7} - 2613 \beta_{5} + 1649 \beta_{4} - 10318 \beta_{3} + 10318\)
\(\nu^{11}\)\(=\)\(-6171 \beta_{19} - 7135 \beta_{18} + 1486 \beta_{17} - 5005 \beta_{14} - 5005 \beta_{13} - 15785 \beta_{11} - 1486 \beta_{9} + 5005 \beta_{7} + 15785 \beta_{1}\)
\(\nu^{12}\)\(=\)\(-644 \beta_{14} + 644 \beta_{13} + 644 \beta_{8} + 14695 \beta_{6} - 23316 \beta_{5} - 36226 \beta_{2} + 81771\)
\(\nu^{13}\)\(=\)\(-50921 \beta_{16} - 59542 \beta_{15} - 15954 \beta_{9} - 45988 \beta_{8} + 45988 \beta_{7} + 122286 \beta_{1}\)
\(\nu^{14}\)\(=\)\(2400 \beta_{14} - 2400 \beta_{13} + 202439 \beta_{12} - 292291 \beta_{10} + 2400 \beta_{7} + 126943 \beta_{6} - 126943 \beta_{4} + 656616 \beta_{3} - 292291 \beta_{2}\)
\(\nu^{15}\)\(=\)\(419234 \beta_{19} + 494730 \beta_{18} - 155792 \beta_{17} - 419234 \beta_{16} - 494730 \beta_{15} + 407278 \beta_{14} + 407278 \beta_{13} + 963263 \beta_{11} - 407278 \beta_{8}\)
\(\nu^{16}\)\(=\)\(1728520 \beta_{12} - 2371957 \beta_{10} + 68340 \beta_{8} + 68340 \beta_{7} + 1728520 \beta_{5} - 1077998 \beta_{4} + 5318130 \beta_{3} - 5318130\)
\(\nu^{17}\)\(=\)\(3449955 \beta_{19} + 4100477 \beta_{18} - 1437724 \beta_{17} + 3525380 \beta_{14} + 3525380 \beta_{13} + 7683002 \beta_{11} + 1437724 \beta_{9} - 3525380 \beta_{7} - 7683002 \beta_{1}\)
\(\nu^{18}\)\(=\)\(-862627 \beta_{14} + 862627 \beta_{13} + 862627 \beta_{8} - 9062991 \beta_{6} + 14601192 \beta_{5} + 19333911 \beta_{2} - 43320969\)
\(\nu^{19}\)\(=\)\(28396902 \beta_{16} + 33935103 \beta_{15} + 12801656 \beta_{9} + 30065011 \beta_{8} - 30065011 \beta_{7} - 61849398 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(-1 + \beta_{3}\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−2.48777 + 1.43632i
−2.10552 + 1.21562i
−1.74361 + 1.00667i
−1.08802 + 0.628167i
−0.392182 + 0.226426i
0.392182 0.226426i
1.08802 0.628167i
1.74361 1.00667i
2.10552 1.21562i
2.48777 1.43632i
−2.48777 1.43632i
−2.10552 1.21562i
−1.74361 1.00667i
−1.08802 0.628167i
−0.392182 0.226426i
0.392182 + 0.226426i
1.08802 + 0.628167i
1.74361 + 1.00667i
2.10552 + 1.21562i
2.48777 + 1.43632i
0 −2.48777 1.43632i 0 1.38776 + 1.75332i 0 3.54568i 0 2.62601 + 4.54838i 0
49.2 0 −2.10552 1.21562i 0 −0.896156 2.04863i 0 0.663818i 0 1.45548 + 2.52097i 0
49.3 0 −1.74361 1.00667i 0 −1.95659 + 1.08248i 0 1.34403i 0 0.526784 + 0.912416i 0
49.4 0 −1.08802 0.628167i 0 1.87507 + 1.21824i 0 4.97100i 0 −0.710812 1.23116i 0
49.5 0 −0.392182 0.226426i 0 1.82467 1.29251i 0 2.54366i 0 −1.39746 2.42048i 0
49.6 0 0.392182 + 0.226426i 0 0.207009 2.22647i 0 2.54366i 0 −1.39746 2.42048i 0
49.7 0 1.08802 + 0.628167i 0 −1.99256 1.01474i 0 4.97100i 0 −0.710812 1.23116i 0
49.8 0 1.74361 + 1.00667i 0 0.0408382 + 2.23570i 0 1.34403i 0 0.526784 + 0.912416i 0
49.9 0 2.10552 + 1.21562i 0 2.22225 0.248224i 0 0.663818i 0 1.45548 + 2.52097i 0
49.10 0 2.48777 + 1.43632i 0 −2.21230 0.325180i 0 3.54568i 0 2.62601 + 4.54838i 0
349.1 0 −2.48777 + 1.43632i 0 1.38776 1.75332i 0 3.54568i 0 2.62601 4.54838i 0
349.2 0 −2.10552 + 1.21562i 0 −0.896156 + 2.04863i 0 0.663818i 0 1.45548 2.52097i 0
349.3 0 −1.74361 + 1.00667i 0 −1.95659 1.08248i 0 1.34403i 0 0.526784 0.912416i 0
349.4 0 −1.08802 + 0.628167i 0 1.87507 1.21824i 0 4.97100i 0 −0.710812 + 1.23116i 0
349.5 0 −0.392182 + 0.226426i 0 1.82467 + 1.29251i 0 2.54366i 0 −1.39746 + 2.42048i 0
349.6 0 0.392182 0.226426i 0 0.207009 + 2.22647i 0 2.54366i 0 −1.39746 + 2.42048i 0
349.7 0 1.08802 0.628167i 0 −1.99256 + 1.01474i 0 4.97100i 0 −0.710812 + 1.23116i 0
349.8 0 1.74361 1.00667i 0 0.0408382 2.23570i 0 1.34403i 0 0.526784 0.912416i 0
349.9 0 2.10552 1.21562i 0 2.22225 + 0.248224i 0 0.663818i 0 1.45548 2.52097i 0
349.10 0 2.48777 1.43632i 0 −2.21230 + 0.325180i 0 3.54568i 0 2.62601 4.54838i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 349.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
19.c even 3 1 inner
95.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 380.2.r.a 20
3.b odd 2 1 3420.2.bj.c 20
5.b even 2 1 inner 380.2.r.a 20
5.c odd 4 2 1900.2.i.g 20
15.d odd 2 1 3420.2.bj.c 20
19.c even 3 1 inner 380.2.r.a 20
57.h odd 6 1 3420.2.bj.c 20
95.i even 6 1 inner 380.2.r.a 20
95.m odd 12 2 1900.2.i.g 20
285.n odd 6 1 3420.2.bj.c 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.r.a 20 1.a even 1 1 trivial
380.2.r.a 20 5.b even 2 1 inner
380.2.r.a 20 19.c even 3 1 inner
380.2.r.a 20 95.i even 6 1 inner
1900.2.i.g 20 5.c odd 4 2
1900.2.i.g 20 95.m odd 12 2
3420.2.bj.c 20 3.b odd 2 1
3420.2.bj.c 20 15.d odd 2 1
3420.2.bj.c 20 57.h odd 6 1
3420.2.bj.c 20 285.n odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(380, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \)
$3$ \( 4096 - 24768 T^{2} + 124617 T^{4} - 134299 T^{6} + 99376 T^{8} - 39211 T^{10} + 11074 T^{12} - 1994 T^{14} + 261 T^{16} - 20 T^{18} + T^{20} \)
$5$ \( 9765625 - 1953125 T - 1562500 T^{2} + 859375 T^{3} - 78125 T^{4} - 31250 T^{5} + 100000 T^{6} - 43250 T^{7} - 10075 T^{8} + 5385 T^{9} - 1324 T^{10} + 1077 T^{11} - 403 T^{12} - 346 T^{13} + 160 T^{14} - 10 T^{15} - 5 T^{16} + 11 T^{17} - 4 T^{18} - T^{19} + T^{20} \)
$7$ \( ( 1600 + 4956 T^{2} + 3285 T^{4} + 651 T^{6} + 46 T^{8} + T^{10} )^{2} \)
$11$ \( ( 148 + 107 T - 29 T^{2} - 27 T^{3} + T^{5} )^{4} \)
$13$ \( 13051691536 - 8019357580 T^{2} + 3073157905 T^{4} - 747179442 T^{6} + 134047248 T^{8} - 17014504 T^{10} + 1608521 T^{12} - 101388 T^{14} + 4368 T^{16} - 78 T^{18} + T^{20} \)
$17$ \( 9721171216 - 8855399740 T^{2} + 5691063605 T^{4} - 1664210705 T^{6} + 343225592 T^{8} - 43575681 T^{10} + 3975100 T^{12} - 200240 T^{14} + 7069 T^{16} - 98 T^{18} + T^{20} \)
$19$ \( ( 2476099 - 912247 T + 329232 T^{2} - 5054 T^{3} - 8626 T^{4} + 5658 T^{5} - 454 T^{6} - 14 T^{7} + 48 T^{8} - 7 T^{9} + T^{10} )^{2} \)
$23$ \( 39691260010000 - 19222228809900 T^{2} + 6588380920801 T^{4} - 1123194038130 T^{6} + 138088662735 T^{8} - 5384649772 T^{10} + 144063556 T^{12} - 2393045 T^{14} + 29086 T^{16} - 211 T^{18} + T^{20} \)
$29$ \( ( 28561 + 6591 T + 14196 T^{2} + 1807 T^{3} + 4819 T^{4} + 595 T^{5} + 757 T^{6} + 38 T^{7} + 78 T^{8} + 8 T^{9} + T^{10} )^{2} \)
$31$ \( ( 388 + 909 T + 251 T^{2} - 93 T^{3} - 2 T^{4} + T^{5} )^{4} \)
$37$ \( ( 518400 + 223884 T^{2} + 36133 T^{4} + 2655 T^{6} + 86 T^{8} + T^{10} )^{2} \)
$41$ \( ( 1368900 + 2341170 T + 4851081 T^{2} - 1488504 T^{3} + 474949 T^{4} - 65504 T^{5} + 11702 T^{6} - 1227 T^{7} + 186 T^{8} - 13 T^{9} + T^{10} )^{2} \)
$43$ \( 122963703210000 - 54989755299900 T^{2} + 18727060639581 T^{4} - 2180566687815 T^{6} + 177900973222 T^{8} - 7914771733 T^{10} + 251687408 T^{12} - 4244448 T^{14} + 50823 T^{16} - 266 T^{18} + T^{20} \)
$47$ \( 2998219536 - 3159366444 T^{2} + 2310165441 T^{4} - 851140494 T^{6} + 224594797 T^{8} - 28541648 T^{10} + 2567980 T^{12} - 127453 T^{14} + 4528 T^{16} - 81 T^{18} + T^{20} \)
$53$ \( 591687332741376 - 250730707883184 T^{2} + 75923593126281 T^{4} - 11228617995345 T^{6} + 1202856445708 T^{8} - 35026544273 T^{10} + 704471884 T^{12} - 8107180 T^{14} + 67789 T^{16} - 318 T^{18} + T^{20} \)
$59$ \( ( 1 - 25 T + 724 T^{2} + 2359 T^{3} + 11249 T^{4} - 5643 T^{5} + 3537 T^{6} - 82 T^{7} + 62 T^{8} - 2 T^{9} + T^{10} )^{2} \)
$61$ \( ( 3073009 - 2666313 T + 2807787 T^{2} + 29238 T^{3} + 251165 T^{4} - 30859 T^{5} + 11757 T^{6} - 450 T^{7} + 115 T^{8} - T^{9} + T^{10} )^{2} \)
$67$ \( 805854925357056 - 1428297897590784 T^{2} + 2432500867768320 T^{4} - 171872785142784 T^{6} + 8943639356416 T^{8} - 178788433664 T^{10} + 2502852784 T^{12} - 20777032 T^{14} + 125425 T^{16} - 435 T^{18} + T^{20} \)
$71$ \( ( 640140601 + 154943324 T + 49647856 T^{2} + 5308606 T^{3} + 1203311 T^{4} + 91293 T^{5} + 20925 T^{6} + 797 T^{7} + 164 T^{8} + T^{9} + T^{10} )^{2} \)
$73$ \( 25628906250000 - 10168593187500 T^{2} + 2771171274321 T^{4} - 377531500050 T^{6} + 36582797191 T^{8} - 2142404556 T^{10} + 90647500 T^{12} - 2274613 T^{14} + 39310 T^{16} - 227 T^{18} + T^{20} \)
$79$ \( ( 233967616 + 34752512 T + 20641536 T^{2} + 2258944 T^{3} + 1240304 T^{4} + 129732 T^{5} + 28025 T^{6} + 832 T^{7} + 213 T^{8} + 8 T^{9} + T^{10} )^{2} \)
$83$ \( ( 54464400 + 12000024 T^{2} + 881113 T^{4} + 26698 T^{6} + 323 T^{8} + T^{10} )^{2} \)
$89$ \( ( 5314993216 - 32952608 T + 195732832 T^{2} + 12731096 T^{3} + 5699336 T^{4} + 302862 T^{5} + 60333 T^{6} + 3784 T^{7} + 479 T^{8} + 20 T^{9} + T^{10} )^{2} \)
$97$ \( 385136700010000 - 578707509531900 T^{2} + 834276242546061 T^{4} - 51542252873283 T^{6} + 2110546422524 T^{8} - 48785561539 T^{10} + 815187090 T^{12} - 8826414 T^{14} + 69709 T^{16} - 328 T^{18} + T^{20} \)
show more
show less