Properties

Label 1900.2.i.g
Level $1900$
Weight $2$
Character orbit 1900.i
Analytic conductor $15.172$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1900.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(15.1715763840\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{3})\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 380)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(257269301703 \nu^{18} + 4826223210105 \nu^{16} + 61286511972345 \nu^{14} + 438571056316700 \nu^{12} + 2323089270554580 \nu^{10} + 7266718287702885 \nu^{8} + 16741977055075822 \nu^{6} + 10869496174421115 \nu^{4} + 2170497176992320 \nu^{2} - 63024290250218436\)\()/ 16690142463557537 \)
\(\beta_{3}\)\(=\)\((\)\(71197437104086 \nu^{18} + 1324128474975807 \nu^{16} + 16814642029945023 \nu^{14} + 120040501810627383 \nu^{12} + 637365600209189772 \nu^{10} + 1993705675325673459 \nu^{8} + 4509311556838210134 \nu^{6} + 2982168202054308141 \nu^{4} + 595500247666250688 \nu^{2} - 5542150149868056414\)\()/ 1385281824475275571 \)
\(\beta_{4}\)\(=\)\((\)\(-58379368812683 \nu^{18} - 1151122140944668 \nu^{16} - 14928136974663543 \nu^{14} - 112486124646259822 \nu^{12} - 618424582627382742 \nu^{10} - 2140435717198619993 \nu^{8} - 5336438184716201168 \nu^{6} - 6768804320649661609 \nu^{4} - 6579414048167166051 \nu^{2} - 238859269757341696\)\()/ 1068169117667682368 \)
\(\beta_{5}\)\(=\)\((\)\(4665182447608185 \nu^{19} + 95269022977644980 \nu^{17} + 1244505919725087325 \nu^{15} + 9664608257577091770 \nu^{13} + 53667142267812748210 \nu^{11} + 193803756970732068819 \nu^{9} + 468118864763077635440 \nu^{7} + 594892885857002960195 \nu^{5} + 67319780802499040193 \nu^{3} + 21000148240688394240 \nu\)\()/ \)\(35\!\cdots\!76\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-6366481800502185 \nu^{18} - 137623959156754484 \nu^{16} - 1868654652948801741 \nu^{14} - 15323421807441242074 \nu^{12} - 89902370458361971154 \nu^{10} - 349276396009867994211 \nu^{8} - 944354362189538749232 \nu^{6} - 1468016942861277361043 \nu^{4} - 1259578493369127028273 \nu^{2} - 250780428775971323584\)\()/ 88658036766417636544 \)
\(\beta_{7}\)\(=\)\((\)\(108802723986440 \nu^{18} + 2093074375676352 \nu^{16} + 26579215713710528 \nu^{14} + 192433368253136232 \nu^{12} + 1007495594987342592 \nu^{10} + 3151487442894034624 \nu^{8} + 6782358791543175575 \nu^{6} + 4713968444633542976 \nu^{4} + 941318257748307968 \nu^{2} - 4955504751551263591\)\()/ 1385281824475275571 \)
\(\beta_{8}\)\(=\)\((\)\(-123113231093090 \nu^{18} - 2391532877467530 \nu^{16} - 30369235310188170 \nu^{14} - 220945270298193633 \nu^{12} - 1151157774093599880 \nu^{10} - 3600868617089492610 \nu^{8} - 7533337923107099056 \nu^{6} - 5386149030199971390 \nu^{4} - 1075543988176763520 \nu^{2} + 2918597117977935675\)\()/ 1385281824475275571 \)
\(\beta_{9}\)\(=\)\((\)\(-135352029273473 \nu^{19} - 2546138925304077 \nu^{17} - 32332523162663053 \nu^{15} - 232191507992026067 \nu^{13} - 1225576970068606692 \nu^{11} - 3833654906967465449 \nu^{9} - 8380018149669050523 \nu^{7} - 5734348807198092151 \nu^{5} - 1145074960070594368 \nu^{3} - 339468898031171989 \nu\)\()/ 5541127297901102284 \)
\(\beta_{10}\)\(=\)\((\)\(149437719142871 \nu^{19} + 2750374974599151 \nu^{17} + 34926044957117039 \nu^{15} + 247970499250483465 \nu^{13} + 1323885430768152396 \nu^{11} + 4141167794335228387 \nu^{9} + 9657228077683790013 \nu^{7} + 6194324001019140413 \nu^{5} + 1236926030594408384 \nu^{3} - 22508069497503397645 \nu\)\()/ 5541127297901102284 \)
\(\beta_{11}\)\(=\)\((\)\(54263059985435 \nu^{18} + 1073902569582988 \nu^{16} + 13947552783106023 \nu^{14} + 105468987745192622 \nu^{12} + 581255154298509462 \nu^{10} + 2024168224595373833 \nu^{8} + 5068566551834988016 \nu^{6} + 6594892381858923769 \nu^{4} + 6277643813918368339 \nu^{2} + 1247247913760836672\)\()/ 267042279416920592 \)
\(\beta_{12}\)\(=\)\((\)\(9174325379691717 \nu^{18} + 183358087220694436 \nu^{16} + 2387658497653929081 \nu^{14} + 18206740636302532114 \nu^{12} + 100804631860325583978 \nu^{10} + 356441852003984207511 \nu^{8} + 901408307418290504720 \nu^{6} + 1230925670964922837447 \nu^{4} + 1127871072781709528461 \nu^{2} + 224153346284272802752\)\()/ 44329018383208818272 \)
\(\beta_{13}\)\(=\)\((\)\(21472706330772813 \nu^{18} + 429531870127860356 \nu^{16} + 5597440381244918081 \nu^{14} + 42637019206442019714 \nu^{12} + 235405967494108868890 \nu^{10} + 823296813298088269295 \nu^{8} + 2040723648133655490800 \nu^{6} + 2590582498362261966815 \nu^{4} + 2114820325419353216181 \nu^{2} + 91430995209659510784\)\()/ 88658036766417636544 \)
\(\beta_{14}\)\(=\)\((\)\(58379368812683 \nu^{19} + 1151122140944668 \nu^{17} + 14928136974663543 \nu^{15} + 112486124646259822 \nu^{13} + 618424582627382742 \nu^{11} + 2140435717198619993 \nu^{9} + 5336438184716201168 \nu^{7} + 6768804320649661609 \nu^{5} + 6579414048167166051 \nu^{3} + 1307028387425024064 \nu\)\()/ 1068169117667682368 \)
\(\beta_{15}\)\(=\)\((\)\(-352957477246353 \nu^{19} - 6732287676656781 \nu^{17} - 85490954590084109 \nu^{15} - 617058244498298531 \nu^{13} - 3240568160043291876 \nu^{11} - 10136629792755534697 \nu^{9} - 21944735732755401673 \nu^{7} - 15162285696465178103 \nu^{5} - 3027711475567210304 \nu^{3} + 15112667902972457477 \nu\)\()/ 2770563648950551142 \)
\(\beta_{16}\)\(=\)\((\)\(-395664181329051 \nu^{19} - 7533440729534211 \nu^{17} - 95664515577493379 \nu^{15} - 689861039846870731 \nu^{13} - 3626200978955352156 \nu^{11} - 11342905028514213607 \nu^{9} - 24723903923897988125 \nu^{7} - 16966622061419083193 \nu^{5} - 3388014006947935424 \nu^{3} + 31115827382409820137 \nu\)\()/ 2770563648950551142 \)
\(\beta_{17}\)\(=\)\((\)\(-86955964488571567 \nu^{19} - 1710572564381717100 \nu^{17} - 22161310421172271211 \nu^{15} - 166719685296363775190 \nu^{13} - 915935506268344505246 \nu^{11} - 3168119711944513996773 \nu^{9} - 7897531353133828196624 \nu^{7} - 10015947541588282623477 \nu^{5} - 9108096864853737363623 \nu^{3} - 353436183667271585792 \nu\)\()/ \)\(35\!\cdots\!76\)\( \)
\(\beta_{18}\)\(=\)\((\)\(-44403267011197975 \nu^{19} - 905638902720961484 \nu^{17} - 11924107064037836787 \nu^{15} - 92792312405128286118 \nu^{13} - 522399676445550876270 \nu^{11} - 1902277734940496224477 \nu^{9} - 4916800551459821697744 \nu^{7} - 7052914632463443149421 \nu^{5} - 6287552361196403142031 \nu^{3} - 1250374162009682137408 \nu\)\()/ \)\(17\!\cdots\!88\)\( \)
\(\beta_{19}\)\(=\)\((\)\(-99815889287337571 \nu^{19} - 2000882759717695292 \nu^{17} - 26141423587608532335 \nu^{15} - 200169113650494448030 \nu^{13} - 1113670060332052229382 \nu^{11} - 3956950094181766287265 \nu^{9} - 10054026219204032528144 \nu^{7} - 13679166208473456186225 \nu^{5} - 12640273317629698848059 \nu^{3} - 2512480201371985676864 \nu\)\()/ \)\(17\!\cdots\!88\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{11} - 4 \beta_{4} - \beta_{2}\)
\(\nu^{3}\)\(=\)\(\beta_{19} - \beta_{18} + \beta_{16} - \beta_{15} + 6 \beta_{14} - 6 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-2 \beta_{13} - \beta_{12} + 9 \beta_{11} + 2 \beta_{7} - \beta_{6} + 25 \beta_{4} - 25\)
\(\nu^{5}\)\(=\)\(-10 \beta_{19} + 11 \beta_{18} + 3 \beta_{17} - 40 \beta_{14} + 3 \beta_{10} + 3 \beta_{9} + 3 \beta_{5}\)
\(\nu^{6}\)\(=\)\(-16 \beta_{8} - 27 \beta_{7} - 8 \beta_{3} + 72 \beta_{2} + 177\)
\(\nu^{7}\)\(=\)\(-88 \beta_{16} + 99 \beta_{15} - 40 \beta_{10} - 46 \beta_{9} + 283 \beta_{1}\)
\(\nu^{8}\)\(=\)\(279 \beta_{13} + 47 \beta_{12} - 569 \beta_{11} + 174 \beta_{8} + 174 \beta_{6} - 1329 \beta_{4} + 47 \beta_{3} - 569 \beta_{2}\)
\(\nu^{9}\)\(=\)\(743 \beta_{19} - 848 \beta_{18} - 395 \beta_{17} + 743 \beta_{16} - 848 \beta_{15} + 2083 \beta_{14} - 511 \beta_{5} - 2083 \beta_{1}\)
\(\nu^{10}\)\(=\)\(-2613 \beta_{13} - 221 \beta_{12} + 4522 \beta_{11} + 2613 \beta_{7} - 1649 \beta_{6} + 10318 \beta_{4} - 10318\)
\(\nu^{11}\)\(=\)\(-6171 \beta_{19} + 7135 \beta_{18} + 3519 \beta_{17} - 15785 \beta_{14} + 3519 \beta_{10} + 5005 \beta_{9} + 5005 \beta_{5}\)
\(\nu^{12}\)\(=\)\(-14695 \beta_{8} - 23316 \beta_{7} - 644 \beta_{3} + 36226 \beta_{2} + 81771\)
\(\nu^{13}\)\(=\)\(-50921 \beta_{16} + 59542 \beta_{15} - 30034 \beta_{10} - 45988 \beta_{9} + 122286 \beta_{1}\)
\(\nu^{14}\)\(=\)\(202439 \beta_{13} - 2400 \beta_{12} - 292291 \beta_{11} + 126943 \beta_{8} + 126943 \beta_{6} - 656616 \beta_{4} - 2400 \beta_{3} - 292291 \beta_{2}\)
\(\nu^{15}\)\(=\)\(419234 \beta_{19} - 494730 \beta_{18} - 251486 \beta_{17} + 419234 \beta_{16} - 494730 \beta_{15} + 963263 \beta_{14} - 407278 \beta_{5} - 963263 \beta_{1}\)
\(\nu^{16}\)\(=\)\(-1728520 \beta_{13} + 68340 \beta_{12} + 2371957 \beta_{11} + 1728520 \beta_{7} - 1077998 \beta_{6} + 5318130 \beta_{4} - 5318130\)
\(\nu^{17}\)\(=\)\(-3449955 \beta_{19} + 4100477 \beta_{18} + 2087656 \beta_{17} - 7683002 \beta_{14} + 2087656 \beta_{10} + 3525380 \beta_{9} + 3525380 \beta_{5}\)
\(\nu^{18}\)\(=\)\(-9062991 \beta_{8} - 14601192 \beta_{7} + 862627 \beta_{3} + 19333911 \beta_{2} + 43320969\)
\(\nu^{19}\)\(=\)\(-28396902 \beta_{16} + 33935103 \beta_{15} - 17263355 \beta_{10} - 30065011 \beta_{9} + 61849398 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\) \(951\)
\(\chi(n)\) \(1\) \(-1 + \beta_{4}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
201.1
1.43632 + 2.48777i
1.21562 + 2.10552i
1.00667 + 1.74361i
0.628167 + 1.08802i
0.226426 + 0.392182i
−0.226426 0.392182i
−0.628167 1.08802i
−1.00667 1.74361i
−1.21562 2.10552i
−1.43632 2.48777i
1.43632 2.48777i
1.21562 2.10552i
1.00667 1.74361i
0.628167 1.08802i
0.226426 0.392182i
−0.226426 + 0.392182i
−0.628167 + 1.08802i
−1.00667 + 1.74361i
−1.21562 + 2.10552i
−1.43632 + 2.48777i
0 −1.43632 + 2.48777i 0 0 0 3.54568 0 −2.62601 4.54838i 0
201.2 0 −1.21562 + 2.10552i 0 0 0 0.663818 0 −1.45548 2.52097i 0
201.3 0 −1.00667 + 1.74361i 0 0 0 −1.34403 0 −0.526784 0.912416i 0
201.4 0 −0.628167 + 1.08802i 0 0 0 −4.97100 0 0.710812 + 1.23116i 0
201.5 0 −0.226426 + 0.392182i 0 0 0 2.54366 0 1.39746 + 2.42048i 0
201.6 0 0.226426 0.392182i 0 0 0 −2.54366 0 1.39746 + 2.42048i 0
201.7 0 0.628167 1.08802i 0 0 0 4.97100 0 0.710812 + 1.23116i 0
201.8 0 1.00667 1.74361i 0 0 0 1.34403 0 −0.526784 0.912416i 0
201.9 0 1.21562 2.10552i 0 0 0 −0.663818 0 −1.45548 2.52097i 0
201.10 0 1.43632 2.48777i 0 0 0 −3.54568 0 −2.62601 4.54838i 0
501.1 0 −1.43632 2.48777i 0 0 0 3.54568 0 −2.62601 + 4.54838i 0
501.2 0 −1.21562 2.10552i 0 0 0 0.663818 0 −1.45548 + 2.52097i 0
501.3 0 −1.00667 1.74361i 0 0 0 −1.34403 0 −0.526784 + 0.912416i 0
501.4 0 −0.628167 1.08802i 0 0 0 −4.97100 0 0.710812 1.23116i 0
501.5 0 −0.226426 0.392182i 0 0 0 2.54366 0 1.39746 2.42048i 0
501.6 0 0.226426 + 0.392182i 0 0 0 −2.54366 0 1.39746 2.42048i 0
501.7 0 0.628167 + 1.08802i 0 0 0 4.97100 0 0.710812 1.23116i 0
501.8 0 1.00667 + 1.74361i 0 0 0 1.34403 0 −0.526784 + 0.912416i 0
501.9 0 1.21562 + 2.10552i 0 0 0 −0.663818 0 −1.45548 + 2.52097i 0
501.10 0 1.43632 + 2.48777i 0 0 0 −3.54568 0 −2.62601 + 4.54838i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 501.10
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 1900.2.i.g 20
5.b even 2 1 inner 1900.2.i.g 20
5.c odd 4 2 380.2.r.a 20
15.e even 4 2 3420.2.bj.c 20
19.c even 3 1 inner 1900.2.i.g 20
95.i even 6 1 inner 1900.2.i.g 20
95.m odd 12 2 380.2.r.a 20
285.v even 12 2 3420.2.bj.c 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.r.a 20 5.c odd 4 2
380.2.r.a 20 95.m odd 12 2
1900.2.i.g 20 1.a even 1 1 trivial
1900.2.i.g 20 5.b even 2 1 inner
1900.2.i.g 20 19.c even 3 1 inner
1900.2.i.g 20 95.i even 6 1 inner
3420.2.bj.c 20 15.e even 4 2
3420.2.bj.c 20 285.v even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{20} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(1900, [\chi])\).

Hecke characteristic polynomials

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