Properties

Label 900.3.p.d
Level $900$
Weight $3$
Character orbit 900.p
Analytic conductor $24.523$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 900.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 2 x^{15} + 10 x^{14} - 12 x^{13} + 6 x^{12} - 27 x^{11} - 585 x^{10} + 3618 x^{9} - 12150 x^{8} + 32562 x^{7} - 47385 x^{6} - 19683 x^{5} + 39366 x^{4} - 708588 x^{3} + 5314410 x^{2} - 9565938 x + 43046721\)
Coefficient ring: \(\Z[a_1, \ldots, a_{41}]\)
Coefficient ring index: \( 3^{10} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{1} - \beta_{4} ) q^{3} -\beta_{10} q^{7} + ( 1 - \beta_{3} + \beta_{15} ) q^{9} +O(q^{10})\) \( q + ( \beta_{1} - \beta_{4} ) q^{3} -\beta_{10} q^{7} + ( 1 - \beta_{3} + \beta_{15} ) q^{9} + \beta_{11} q^{11} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{13} + ( 1 + \beta_{1} - \beta_{3} + \beta_{5} - \beta_{7} + \beta_{8} + \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} ) q^{17} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} - \beta_{8} + \beta_{9} + \beta_{11} + \beta_{13} - \beta_{15} ) q^{19} + ( 2 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} - \beta_{11} - 2 \beta_{12} - \beta_{13} ) q^{21} + ( -3 + 2 \beta_{2} + 3 \beta_{3} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} - \beta_{13} + \beta_{15} ) q^{23} + ( -1 + 4 \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{11} - \beta_{14} - \beta_{15} ) q^{27} + ( \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{10} - \beta_{11} - \beta_{15} ) q^{29} + ( \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{12} - 2 \beta_{13} - \beta_{14} ) q^{31} + ( -4 + \beta_{1} + 4 \beta_{3} - \beta_{5} - \beta_{6} - 2 \beta_{9} - \beta_{11} - \beta_{12} - 2 \beta_{13} ) q^{33} + ( 1 + \beta_{2} + 3 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} + \beta_{10} + 2 \beta_{11} - \beta_{12} + 2 \beta_{14} + \beta_{15} ) q^{37} + ( 5 + \beta_{2} - 8 \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} - 4 \beta_{8} - 3 \beta_{10} + \beta_{11} + 2 \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{39} + ( 3 + 2 \beta_{2} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - 4 \beta_{10} - \beta_{12} - \beta_{13} + \beta_{15} ) q^{41} + ( -5 + 6 \beta_{1} + \beta_{2} + 7 \beta_{3} - 2 \beta_{4} - \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{9} - \beta_{12} - 2 \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{43} + ( 5 - \beta_{2} + 5 \beta_{3} - 3 \beta_{4} - \beta_{7} + 5 \beta_{8} - \beta_{9} + 2 \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{47} + ( -3 - 7 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} + 8 \beta_{4} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{11} + \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{49} + ( 5 + 3 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} + 4 \beta_{8} + 2 \beta_{11} + \beta_{12} - \beta_{13} + 2 \beta_{14} ) q^{51} + ( -1 + 7 \beta_{1} - 5 \beta_{2} + 9 \beta_{3} - 5 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} - \beta_{7} + 5 \beta_{8} - \beta_{9} - 2 \beta_{11} - 3 \beta_{12} - 3 \beta_{13} - 4 \beta_{15} ) q^{53} + ( 7 + \beta_{1} - \beta_{2} - 6 \beta_{3} - \beta_{5} - 3 \beta_{6} - \beta_{7} - 5 \beta_{8} - \beta_{9} - 6 \beta_{10} + 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{57} + ( 1 - 10 \beta_{1} - 2 \beta_{2} + \beta_{4} + 2 \beta_{5} - \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{12} + 2 \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{59} + ( -7 + 5 \beta_{1} + \beta_{2} + 8 \beta_{3} - 3 \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{8} - 2 \beta_{9} + \beta_{10} - 2 \beta_{12} - \beta_{13} - 2 \beta_{14} - 2 \beta_{15} ) q^{61} + ( 4 + 2 \beta_{1} + 7 \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} - 4 \beta_{7} + \beta_{8} - \beta_{9} - 3 \beta_{10} + 3 \beta_{11} - \beta_{12} - 2 \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{63} + ( 3 + 7 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 8 \beta_{4} - \beta_{7} + 3 \beta_{8} - \beta_{9} + 2 \beta_{10} - 4 \beta_{11} - 3 \beta_{12} - \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{67} + ( -7 - 7 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} + 4 \beta_{4} - 3 \beta_{5} + 6 \beta_{6} + 4 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + 3 \beta_{10} - \beta_{11} + 3 \beta_{13} + 4 \beta_{14} - 3 \beta_{15} ) q^{69} + ( -8 + 6 \beta_{1} - 5 \beta_{2} + 22 \beta_{3} - 5 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} - 3 \beta_{8} - \beta_{9} + 4 \beta_{10} - \beta_{12} - 2 \beta_{13} - 4 \beta_{15} ) q^{71} + ( -7 + 4 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} - 11 \beta_{4} - \beta_{5} - 3 \beta_{6} - 2 \beta_{7} + 3 \beta_{8} - 3 \beta_{9} + \beta_{11} - 4 \beta_{12} - 4 \beta_{13} - 4 \beta_{14} + 6 \beta_{15} ) q^{73} + ( -29 + 10 \beta_{1} + 2 \beta_{2} + 14 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{9} - 5 \beta_{10} - 2 \beta_{13} - 2 \beta_{15} ) q^{77} + ( -1 + 14 \beta_{1} - 8 \beta_{3} - 2 \beta_{4} + \beta_{5} + 2 \beta_{7} - 5 \beta_{8} + \beta_{9} - 5 \beta_{10} + 2 \beta_{11} + 5 \beta_{12} + 3 \beta_{13} - 6 \beta_{14} + 4 \beta_{15} ) q^{79} + ( -5 + 2 \beta_{1} - 5 \beta_{3} - 3 \beta_{4} - \beta_{5} - 2 \beta_{7} - \beta_{8} - 3 \beta_{9} + 6 \beta_{10} - 2 \beta_{11} - \beta_{12} - 2 \beta_{13} + \beta_{15} ) q^{81} + ( 10 + 9 \beta_{1} - \beta_{2} + 13 \beta_{3} + 4 \beta_{4} - 5 \beta_{6} - 2 \beta_{7} - 5 \beta_{8} - 3 \beta_{9} - 4 \beta_{10} + \beta_{11} - 3 \beta_{12} - 4 \beta_{13} + 4 \beta_{15} ) q^{83} + ( -29 + 5 \beta_{1} + 3 \beta_{2} + 11 \beta_{3} - 3 \beta_{4} + \beta_{5} - 5 \beta_{6} - 3 \beta_{7} + 12 \beta_{8} - \beta_{9} + 6 \beta_{10} - 5 \beta_{11} - 5 \beta_{12} - \beta_{13} - 3 \beta_{14} + 3 \beta_{15} ) q^{87} + ( 13 + 9 \beta_{1} - 23 \beta_{3} - 16 \beta_{4} - 3 \beta_{5} - 4 \beta_{6} + 4 \beta_{7} - 5 \beta_{8} + 2 \beta_{9} + 3 \beta_{10} + 2 \beta_{12} + 3 \beta_{13} + 2 \beta_{14} + 3 \beta_{15} ) q^{89} + ( -9 + 3 \beta_{1} + 5 \beta_{2} + 3 \beta_{3} - 9 \beta_{4} - 2 \beta_{5} - \beta_{6} - 3 \beta_{7} - \beta_{9} - \beta_{10} - 2 \beta_{11} + \beta_{12} - 3 \beta_{13} - 2 \beta_{14} + 6 \beta_{15} ) q^{91} + ( 11 + 3 \beta_{1} + \beta_{2} - 5 \beta_{3} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 8 \beta_{8} - 3 \beta_{9} + 6 \beta_{10} + \beta_{11} - \beta_{12} - 5 \beta_{13} + 4 \beta_{14} - \beta_{15} ) q^{93} + ( -3 - 15 \beta_{1} + 11 \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} - 6 \beta_{8} - 2 \beta_{9} - 3 \beta_{10} + 4 \beta_{11} + 6 \beta_{12} - 2 \beta_{13} + 5 \beta_{14} - 4 \beta_{15} ) q^{97} + ( 13 - 4 \beta_{1} + \beta_{2} + 6 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 11 \beta_{6} - \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + 6 \beta_{10} + 3 \beta_{11} + 2 \beta_{12} + 4 \beta_{13} + 8 \beta_{14} - \beta_{15} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 2q^{3} + q^{7} + 14q^{9} + O(q^{10}) \) \( 16q - 2q^{3} + q^{7} + 14q^{9} + 10q^{13} + 2q^{19} + q^{21} - 27q^{23} + 16q^{27} + 9q^{29} + 8q^{31} - 36q^{33} + 22q^{37} + 19q^{39} + 54q^{41} - 44q^{43} + 108q^{47} - 45q^{49} + 90q^{51} + 68q^{57} + 9q^{59} - 55q^{61} + 107q^{63} + 28q^{67} - 147q^{69} - 86q^{73} - 342q^{77} + 11q^{79} - 130q^{81} + 306q^{83} - 375q^{87} - 134q^{91} + 83q^{93} - 41q^{97} + 252q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 2 x^{15} + 10 x^{14} - 12 x^{13} + 6 x^{12} - 27 x^{11} - 585 x^{10} + 3618 x^{9} - 12150 x^{8} + 32562 x^{7} - 47385 x^{6} - 19683 x^{5} + 39366 x^{4} - 708588 x^{3} + 5314410 x^{2} - 9565938 x + 43046721\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 1 \)
\(\beta_{3}\)\(=\)\((\)\(-4099 \nu^{15} + 12833 \nu^{14} - 9679 \nu^{13} - 10653 \nu^{12} + 361803 \nu^{11} - 1741122 \nu^{10} + 5995773 \nu^{9} - 18242955 \nu^{8} + 47971845 \nu^{7} - 18730683 \nu^{6} - 93873330 \nu^{5} + 981387819 \nu^{4} - 4192459317 \nu^{3} + 7909790697 \nu^{2} - 22770652527 \nu + 54664552701\)\()/ 36541883160 \)
\(\beta_{4}\)\(=\)\((\)\(-515 \nu^{15} - 3479 \nu^{14} + 6649 \nu^{13} - 42933 \nu^{12} + 205755 \nu^{11} - 399762 \nu^{10} + 379197 \nu^{9} + 203445 \nu^{8} - 12748995 \nu^{7} + 32011605 \nu^{6} - 100078578 \nu^{5} + 447899787 \nu^{4} - 556143165 \nu^{3} + 109653993 \nu^{2} + 2343123369 \nu - 19605389931\)\()/ 4060209240 \)
\(\beta_{5}\)\(=\)\((\)\(-2405 \nu^{15} - 16133 \nu^{14} + 254923 \nu^{13} - 203007 \nu^{12} + 344697 \nu^{11} - 1882494 \nu^{10} + 4435191 \nu^{9} - 72418725 \nu^{8} + 174037815 \nu^{7} + 186610635 \nu^{6} - 329197446 \nu^{5} + 746307189 \nu^{4} - 6883715907 \nu^{3} + 10269743031 \nu^{2} - 181098618129 \nu + 504904556547\)\()/ 18270941580 \)
\(\beta_{6}\)\(=\)\((\)\(1817 \nu^{15} + 25193 \nu^{14} - 60625 \nu^{13} + 28569 \nu^{12} + 179409 \nu^{11} - 2954178 \nu^{10} + 11589093 \nu^{9} - 34750755 \nu^{8} + 100605645 \nu^{7} - 235959561 \nu^{6} - 31692546 \nu^{5} + 823057767 \nu^{4} - 7560614277 \nu^{3} + 28262918115 \nu^{2} - 43981525719 \nu + 140184038421\)\()/ 9135470790 \)
\(\beta_{7}\)\(=\)\((\)\(-4759 \nu^{15} - 14899 \nu^{14} + 34049 \nu^{13} - 262149 \nu^{12} + 1530723 \nu^{11} - 1992870 \nu^{10} + 14112189 \nu^{9} - 55496475 \nu^{8} + 77787945 \nu^{7} + 127040157 \nu^{6} - 640649574 \nu^{5} + 4115630007 \nu^{4} - 11258262657 \nu^{3} + 23096248713 \nu^{2} - 107371808199 \nu + 100045362573\)\()/ 18270941580 \)
\(\beta_{8}\)\(=\)\((\)\(655 \nu^{15} - 1301 \nu^{14} + 10591 \nu^{13} - 9867 \nu^{12} - 49035 \nu^{11} + 109782 \nu^{10} - 306657 \nu^{9} + 1430595 \nu^{8} - 5501925 \nu^{7} + 9259515 \nu^{6} + 10003338 \nu^{5} - 5863347 \nu^{4} + 344485305 \nu^{3} - 63595773 \nu^{2} + 841979691 \nu + 4565609631\)\()/ 2030104620 \)
\(\beta_{9}\)\(=\)\((\)\(6551 \nu^{15} - 50857 \nu^{14} + 172367 \nu^{13} + 206985 \nu^{12} - 535767 \nu^{11} - 1056834 \nu^{10} + 12814623 \nu^{9} - 44711325 \nu^{8} + 53738235 \nu^{7} + 102553047 \nu^{6} - 329063310 \nu^{5} + 449435061 \nu^{4} - 3568311387 \nu^{3} + 37692807219 \nu^{2} - 144191103561 \nu + 281616431751\)\()/ 18270941580 \)
\(\beta_{10}\)\(=\)\((\)\(10129 \nu^{15} + 32833 \nu^{14} - 98447 \nu^{13} - 6897 \nu^{12} - 298569 \nu^{11} - 2217510 \nu^{10} - 3017727 \nu^{9} + 22298625 \nu^{8} + 74682405 \nu^{7} - 177858747 \nu^{6} - 39629898 \nu^{5} - 320550777 \nu^{4} - 4866011577 \nu^{3} + 1835774361 \nu^{2} + 61719963417 \nu + 70544009781\)\()/ 18270941580 \)
\(\beta_{11}\)\(=\)\((\)\(27817 \nu^{15} - 106331 \nu^{14} + 379645 \nu^{13} - 604497 \nu^{12} - 150969 \nu^{11} + 1825038 \nu^{10} - 10942983 \nu^{9} + 86884785 \nu^{8} - 259185015 \nu^{7} + 408639249 \nu^{6} - 1381765554 \nu^{5} - 47823129 \nu^{4} - 5093232129 \nu^{3} - 8131933035 \nu^{2} + 53563406949 \nu + 71366680449\)\()/ 36541883160 \)
\(\beta_{12}\)\(=\)\((\)\(10603 \nu^{15} - 25049 \nu^{14} + 120439 \nu^{13} - 320619 \nu^{12} - 840315 \nu^{11} + 705834 \nu^{10} - 6072021 \nu^{9} + 23644035 \nu^{8} - 53084565 \nu^{7} + 247543371 \nu^{6} - 99649926 \nu^{5} - 1594080243 \nu^{4} + 4989857013 \nu^{3} - 3183508737 \nu^{2} + 15092392959 \nu + 94768153443\)\()/ 12180627720 \)
\(\beta_{13}\)\(=\)\((\)\(31943 \nu^{15} - 51853 \nu^{14} - 346237 \nu^{13} + 1172577 \nu^{12} - 1145463 \nu^{11} + 5186322 \nu^{10} - 20517417 \nu^{9} + 182796615 \nu^{8} - 398722905 \nu^{7} - 295527609 \nu^{6} + 3473622306 \nu^{5} - 5074480791 \nu^{4} + 16152401241 \nu^{3} - 24362495469 \nu^{2} + 282647427291 \nu - 896524492329\)\()/ 36541883160 \)
\(\beta_{14}\)\(=\)\((\)\(-11429 \nu^{15} - 14033 \nu^{14} + 1207 \nu^{13} + 50037 \nu^{12} - 164451 \nu^{11} + 3564810 \nu^{10} - 8984133 \nu^{9} + 12611835 \nu^{8} - 25324245 \nu^{7} + 59595507 \nu^{6} + 372987018 \nu^{5} - 619902963 \nu^{4} + 8382576357 \nu^{3} - 29633681601 \nu^{2} + 10449724383 \nu - 107787395061\)\()/ 12180627720 \)
\(\beta_{15}\)\(=\)\((\)\(18241 \nu^{15} - 46679 \nu^{14} + 216169 \nu^{13} - 945129 \nu^{12} + 2042403 \nu^{11} - 1221210 \nu^{10} - 6302331 \nu^{9} + 76406625 \nu^{8} - 195528735 \nu^{7} + 550802997 \nu^{6} - 2016870354 \nu^{5} + 2902107447 \nu^{4} - 947519937 \nu^{3} - 18905304987 \nu^{2} + 99007989741 \nu - 90699441147\)\()/ 18270941580 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 1\)
\(\nu^{3}\)\(=\)\(\beta_{15} + \beta_{14} - 2 \beta_{11} + \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} + \beta_{4} - 4 \beta_{3} + 1\)
\(\nu^{4}\)\(=\)\(-\beta_{15} + 2 \beta_{14} - \beta_{13} - 2 \beta_{12} - 2 \beta_{11} + 6 \beta_{10} + 2 \beta_{9} + 12 \beta_{8} + 3 \beta_{7} - \beta_{6} - 2 \beta_{5} + 5 \beta_{4} - 4 \beta_{3} - \beta_{2} - 5 \beta_{1} + 12\)
\(\nu^{5}\)\(=\)\(-3 \beta_{15} + 4 \beta_{14} + 9 \beta_{13} + 6 \beta_{12} + \beta_{11} - 3 \beta_{10} - 8 \beta_{9} + 10 \beta_{8} + 5 \beta_{7} + 4 \beta_{6} + 6 \beta_{5} + \beta_{4} + 39 \beta_{3} + \beta_{2} + 9 \beta_{1} - 4\)
\(\nu^{6}\)\(=\)\(-5 \beta_{15} - 42 \beta_{14} + 4 \beta_{13} + 20 \beta_{12} - 26 \beta_{11} + 6 \beta_{10} + 3 \beta_{9} - 19 \beta_{8} + 34 \beta_{7} - 63 \beta_{6} - 7 \beta_{5} - 39 \beta_{4} + 163 \beta_{3} + 9 \beta_{2} + 8 \beta_{1} + 103\)
\(\nu^{7}\)\(=\)\(56 \beta_{15} + 119 \beta_{14} + 23 \beta_{13} + 19 \beta_{12} - 38 \beta_{11} + 105 \beta_{10} + 68 \beta_{9} - 45 \beta_{8} + 15 \beta_{7} + 23 \beta_{6} + 19 \beta_{5} - 97 \beta_{4} + 272 \beta_{3} + 50 \beta_{2} + 262 \beta_{1} - 1416\)
\(\nu^{8}\)\(=\)\(159 \beta_{15} + 343 \beta_{14} + 93 \beta_{13} + 30 \beta_{12} + 154 \beta_{11} + 159 \beta_{10} + 133 \beta_{9} - 14 \beta_{8} - 106 \beta_{7} + 55 \beta_{6} - 105 \beta_{5} - 335 \beta_{4} + 2565 \beta_{3} + 289 \beta_{2} - 1227 \beta_{1} - 166\)
\(\nu^{9}\)\(=\)\(598 \beta_{15} + 417 \beta_{14} + 247 \beta_{13} + 425 \beta_{12} + 37 \beta_{11} - 966 \beta_{10} + 390 \beta_{9} - 172 \beta_{8} - 2 \beta_{7} + 468 \beta_{6} - 655 \beta_{5} - 1965 \beta_{4} + 7111 \beta_{3} - 1359 \beta_{2} + 1637 \beta_{1} - 392\)
\(\nu^{10}\)\(=\)\(398 \beta_{15} - 826 \beta_{14} - 283 \beta_{13} - 1079 \beta_{12} + 1168 \beta_{11} - 1542 \beta_{10} - 1201 \beta_{9} + 1098 \beta_{8} + 3624 \beta_{7} - 1354 \beta_{6} - 998 \beta_{5} - 8251 \beta_{4} - 817 \beta_{3} + 1427 \beta_{2} + 4780 \beta_{1} + 4299\)
\(\nu^{11}\)\(=\)\(8079 \beta_{15} - 674 \beta_{14} + 5475 \beta_{13} - 4353 \beta_{12} - 4391 \beta_{11} - 8886 \beta_{10} - 3341 \beta_{9} - 10247 \beta_{8} - 3049 \beta_{7} + 3709 \beta_{6} + 8634 \beta_{5} - 7427 \beta_{4} - 15516 \beta_{3} + 6805 \beta_{2} + 14316 \beta_{1} + 21488\)
\(\nu^{12}\)\(=\)\(2812 \beta_{15} - 21111 \beta_{14} + 679 \beta_{13} - 26791 \beta_{12} - 13094 \beta_{11} + 9591 \beta_{10} + 20361 \beta_{9} + 5030 \beta_{8} - 9875 \beta_{7} - 29808 \beta_{6} + 3233 \beta_{5} - 7266 \beta_{4} + 11593 \beta_{3} - 153 \beta_{2} + 18134 \beta_{1} + 85504\)
\(\nu^{13}\)\(=\)\(24662 \beta_{15} + 16967 \beta_{14} + 6404 \beta_{13} - 46376 \beta_{12} + 30967 \beta_{11} - 33429 \beta_{10} - 5359 \beta_{9} + 104382 \beta_{8} - 68448 \beta_{7} + 19463 \beta_{6} + 44344 \beta_{5} - 35764 \beta_{4} + 277490 \beta_{3} + 25808 \beta_{2} + 166222 \beta_{1} - 1055946\)
\(\nu^{14}\)\(=\)\(31551 \beta_{15} - 325664 \beta_{14} - 63420 \beta_{13} - 134610 \beta_{12} - 120851 \beta_{11} - 118209 \beta_{10} - 170651 \beta_{9} + 387868 \beta_{8} - 17197 \beta_{7} - 217871 \beta_{6} - 2796 \beta_{5} - 363521 \beta_{4} + 699858 \beta_{3} + 67753 \beta_{2} - 697197 \beta_{1} - 138784\)
\(\nu^{15}\)\(=\)\(229072 \beta_{15} - 680946 \beta_{14} + 140782 \beta_{13} - 165328 \beta_{12} - 714482 \beta_{11} + 140595 \beta_{10} + 40629 \beta_{9} + 576044 \beta_{8} + 440548 \beta_{7} - 105066 \beta_{6} + 211457 \beta_{5} - 374502 \beta_{4} - 4767002 \beta_{3} - 916596 \beta_{2} + 537533 \beta_{1} - 1621616\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(\beta_{3}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
−1.82249 + 2.38297i
0.127146 + 2.99730i
1.13333 + 2.77769i
−2.99949 + 0.0553819i
2.94519 + 0.570848i
−1.69925 2.47235i
2.47109 1.70110i
0.844479 2.87869i
−1.82249 2.38297i
0.127146 2.99730i
1.13333 2.77769i
−2.99949 0.0553819i
2.94519 0.570848i
−1.69925 + 2.47235i
2.47109 + 1.70110i
0.844479 + 2.87869i
0 −2.97496 0.386835i 0 0 0 2.32731 4.03103i 0 8.70072 + 2.30163i 0
101.2 0 −2.53217 + 1.60876i 0 0 0 −4.69530 + 8.13249i 0 3.82376 8.14732i 0
101.3 0 −1.83888 + 2.37034i 0 0 0 4.13058 7.15437i 0 −2.23701 8.71756i 0
101.4 0 −1.54771 2.56994i 0 0 0 −0.725042 + 1.25581i 0 −4.20921 + 7.95503i 0
101.5 0 0.978225 + 2.83603i 0 0 0 −2.71262 + 4.69840i 0 −7.08615 + 5.54856i 0
101.6 0 1.29149 2.70777i 0 0 0 1.73252 3.00081i 0 −5.66408 6.99415i 0
101.7 0 2.70874 + 1.28947i 0 0 0 5.40503 9.36178i 0 5.67451 + 6.98569i 0
101.8 0 2.91526 0.708005i 0 0 0 −4.96248 + 8.59526i 0 7.99746 4.12804i 0
401.1 0 −2.97496 + 0.386835i 0 0 0 2.32731 + 4.03103i 0 8.70072 2.30163i 0
401.2 0 −2.53217 1.60876i 0 0 0 −4.69530 8.13249i 0 3.82376 + 8.14732i 0
401.3 0 −1.83888 2.37034i 0 0 0 4.13058 + 7.15437i 0 −2.23701 + 8.71756i 0
401.4 0 −1.54771 + 2.56994i 0 0 0 −0.725042 1.25581i 0 −4.20921 7.95503i 0
401.5 0 0.978225 2.83603i 0 0 0 −2.71262 4.69840i 0 −7.08615 5.54856i 0
401.6 0 1.29149 + 2.70777i 0 0 0 1.73252 + 3.00081i 0 −5.66408 + 6.99415i 0
401.7 0 2.70874 1.28947i 0 0 0 5.40503 + 9.36178i 0 5.67451 6.98569i 0
401.8 0 2.91526 + 0.708005i 0 0 0 −4.96248 8.59526i 0 7.99746 + 4.12804i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 401.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.3.p.d 16
3.b odd 2 1 2700.3.p.e 16
5.b even 2 1 900.3.p.e yes 16
5.c odd 4 2 900.3.u.d 32
9.c even 3 1 2700.3.p.e 16
9.d odd 6 1 inner 900.3.p.d 16
15.d odd 2 1 2700.3.p.d 16
15.e even 4 2 2700.3.u.d 32
45.h odd 6 1 900.3.p.e yes 16
45.j even 6 1 2700.3.p.d 16
45.k odd 12 2 2700.3.u.d 32
45.l even 12 2 900.3.u.d 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
900.3.p.d 16 1.a even 1 1 trivial
900.3.p.d 16 9.d odd 6 1 inner
900.3.p.e yes 16 5.b even 2 1
900.3.p.e yes 16 45.h odd 6 1
900.3.u.d 32 5.c odd 4 2
900.3.u.d 32 45.l even 12 2
2700.3.p.d 16 15.d odd 2 1
2700.3.p.d 16 45.j even 6 1
2700.3.p.e 16 3.b odd 2 1
2700.3.p.e 16 9.c even 3 1
2700.3.u.d 32 15.e even 4 2
2700.3.u.d 32 45.k odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(11\!\cdots\!76\)\( \)">\(T_{7}^{16} - \cdots\) acting on \(S_{3}^{\mathrm{new}}(900, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( 43046721 + 9565938 T - 2657205 T^{2} - 1062882 T^{3} + 216513 T^{4} - 6561 T^{5} - 69255 T^{6} + 10449 T^{7} + 13608 T^{8} + 1161 T^{9} - 855 T^{10} - 9 T^{11} + 33 T^{12} - 18 T^{13} - 5 T^{14} + 2 T^{15} + T^{16} \)
$5$ \( T^{16} \)
$7$ \( 1115296517776 + 406181614664 T + 455222811324 T^{2} - 75179664908 T^{3} + 76104497504 T^{4} - 5819046060 T^{5} + 3960781930 T^{6} - 209228176 T^{7} + 147133197 T^{8} - 3758746 T^{9} + 2596750 T^{10} - 36030 T^{11} + 33299 T^{12} - 218 T^{13} + 219 T^{14} - T^{15} + T^{16} \)
$11$ \( 318893306317056 + 248178559423104 T - 4510587772800 T^{2} - 53615322357408 T^{3} + 12933127758168 T^{4} + 476311200228 T^{5} - 274475233773 T^{6} - 2405918700 T^{7} + 4006444842 T^{8} - 26430624 T^{9} - 31731156 T^{10} + 205335 T^{11} + 182736 T^{12} - 510 T^{14} + T^{16} \)
$13$ \( 206194665870400 + 134293446830000 T + 125674151178900 T^{2} - 10684156872500 T^{3} + 10322576120180 T^{4} - 411902570100 T^{5} + 433304095960 T^{6} - 27006053500 T^{7} + 9769191561 T^{8} - 150937105 T^{9} + 81848419 T^{10} - 2070750 T^{11} + 425357 T^{12} - 3350 T^{13} + 786 T^{14} - 10 T^{15} + T^{16} \)
$17$ \( 477684734055939216 + 54394394047926912 T^{2} + 2082124583926632 T^{4} + 35523015814395 T^{6} + 291643132581 T^{8} + 1203277950 T^{10} + 2537163 T^{12} + 2586 T^{14} + T^{16} \)
$19$ \( ( 261665566 + 21046769 T - 37435013 T^{2} - 249250 T^{3} + 521206 T^{4} + 3350 T^{5} - 1478 T^{6} - T^{7} + T^{8} )^{2} \)
$23$ \( 7418978556121798416 - 3359842647846181416 T + 122252368904599212 T^{2} + 174328179790739760 T^{3} + 11482159246547796 T^{4} - 1794709470750300 T^{5} - 96879096749070 T^{6} + 11857193822358 T^{7} + 662290348293 T^{8} - 43281243459 T^{9} - 1385893530 T^{10} + 80723385 T^{11} + 2871531 T^{12} - 56295 T^{13} - 1842 T^{14} + 27 T^{15} + T^{16} \)
$29$ \( 387425055353648400 + 1932664865564327400 T + 3408412584313032300 T^{2} + 971365060523106000 T^{3} + 86437879141486320 T^{4} - 3828243288335760 T^{5} - 574476459568380 T^{6} + 15931396514790 T^{7} + 3256236158481 T^{8} + 27915595542 T^{9} - 5134786236 T^{10} - 51536898 T^{11} + 6258033 T^{12} + 25974 T^{13} - 2859 T^{14} - 9 T^{15} + T^{16} \)
$31$ \( \)\(39\!\cdots\!16\)\( - 49621504665149746472 T + 13060086801728375316 T^{2} - 1749405959576516296 T^{3} + 330311652010907168 T^{4} - 35803585069286856 T^{5} + 3472657299214858 T^{6} - 205752461033474 T^{7} + 10662938626023 T^{8} - 317471725031 T^{9} + 11441084983 T^{10} - 211781184 T^{11} + 9702503 T^{12} - 80854 T^{13} + 3486 T^{14} - 8 T^{15} + T^{16} \)
$37$ \( ( 38426783044 - 61687919954 T - 3055526594 T^{2} + 197674444 T^{3} + 11148637 T^{4} - 32303 T^{5} - 6197 T^{6} - 11 T^{7} + T^{8} )^{2} \)
$41$ \( \)\(56\!\cdots\!00\)\( - \)\(30\!\cdots\!00\)\( T - \)\(64\!\cdots\!25\)\( T^{2} + 38143394611895088450 T^{3} + 6985018130497634295 T^{4} + 9611742628755090 T^{5} - 19560264289276455 T^{6} - 152659882502235 T^{7} + 42227929956276 T^{8} + 825432763707 T^{9} - 25781044176 T^{10} - 606518523 T^{11} + 13626063 T^{12} + 261954 T^{13} - 3879 T^{14} - 54 T^{15} + T^{16} \)
$43$ \( \)\(10\!\cdots\!41\)\( + \)\(31\!\cdots\!94\)\( T + \)\(76\!\cdots\!08\)\( T^{2} + \)\(78\!\cdots\!48\)\( T^{3} + 68720124849093042212 T^{4} + 924332773393954266 T^{5} + 119080640836286020 T^{6} + 958627860748892 T^{7} + 145040704176249 T^{8} + 195062945342 T^{9} + 90069854020 T^{10} + 379952016 T^{11} + 35031572 T^{12} + 116548 T^{13} + 8148 T^{14} + 44 T^{15} + T^{16} \)
$47$ \( 1967471862670688256 + 2854352473464766464 T + 1587213465631948800 T^{2} + 300129187534405632 T^{3} - 29166540224643072 T^{4} - 11261360418558336 T^{5} + 1743932525024016 T^{6} - 30462942379428 T^{7} - 5166516605463 T^{8} + 135614382354 T^{9} + 15319302579 T^{10} - 684793926 T^{11} + 398250 T^{12} + 335988 T^{13} + 777 T^{14} - 108 T^{15} + T^{16} \)
$53$ \( \)\(32\!\cdots\!76\)\( + \)\(86\!\cdots\!24\)\( T^{2} + \)\(50\!\cdots\!24\)\( T^{4} + 9435375883430093556 T^{6} + 7022528013405429 T^{8} + 2370865183119 T^{10} + 396048474 T^{12} + 32016 T^{14} + T^{16} \)
$59$ \( \)\(68\!\cdots\!01\)\( - \)\(23\!\cdots\!02\)\( T + \)\(20\!\cdots\!24\)\( T^{2} + \)\(19\!\cdots\!88\)\( T^{3} + \)\(36\!\cdots\!16\)\( T^{4} - \)\(71\!\cdots\!86\)\( T^{5} - 20796796545938784117 T^{6} + 201354836066365635 T^{7} + 7944372702979425 T^{8} - 21535087257666 T^{9} - 1605889013139 T^{10} + 1467084069 T^{11} + 237279780 T^{12} + 167886 T^{13} - 18627 T^{14} - 9 T^{15} + T^{16} \)
$61$ \( \)\(22\!\cdots\!16\)\( - \)\(20\!\cdots\!28\)\( T + \)\(28\!\cdots\!48\)\( T^{2} - \)\(25\!\cdots\!88\)\( T^{3} + 79908378718009989632 T^{4} - 231063452519165712 T^{5} + 149638401245403388 T^{6} + 813665334188062 T^{7} + 148469902088607 T^{8} + 1131238827559 T^{9} + 105322505464 T^{10} + 992838105 T^{11} + 39819839 T^{12} + 258593 T^{13} + 8820 T^{14} + 55 T^{15} + T^{16} \)
$67$ \( \)\(10\!\cdots\!76\)\( - \)\(26\!\cdots\!62\)\( T + \)\(46\!\cdots\!99\)\( T^{2} - \)\(46\!\cdots\!45\)\( T^{3} + \)\(33\!\cdots\!34\)\( T^{4} - 17320216304255911911 T^{5} + 708144286903314895 T^{6} - 22029107616353908 T^{7} + 594788149314612 T^{8} - 13058810099587 T^{9} + 278368405750 T^{10} - 4703525109 T^{11} + 77921834 T^{12} - 759200 T^{13} + 8874 T^{14} - 28 T^{15} + T^{16} \)
$71$ \( \)\(41\!\cdots\!56\)\( + \)\(81\!\cdots\!84\)\( T^{2} + \)\(42\!\cdots\!68\)\( T^{4} + 58348337181884984880 T^{6} + 29728587656303181 T^{8} + 7007465271735 T^{10} + 819681282 T^{12} + 46188 T^{14} + T^{16} \)
$73$ \( ( 1737226523997346 - 24225460061393 T - 1228494619592 T^{2} + 10163976976 T^{3} + 308714368 T^{4} - 1231886 T^{5} - 30767 T^{6} + 43 T^{7} + T^{8} )^{2} \)
$79$ \( \)\(20\!\cdots\!00\)\( - \)\(16\!\cdots\!00\)\( T + \)\(27\!\cdots\!00\)\( T^{2} + \)\(67\!\cdots\!00\)\( T^{3} + \)\(86\!\cdots\!00\)\( T^{4} + \)\(71\!\cdots\!00\)\( T^{5} + 91398436026446206960 T^{6} + 1240210604784495430 T^{7} + 60431026981700151 T^{8} + 261599052224329 T^{9} + 7902689781340 T^{10} + 12904336611 T^{11} + 760718339 T^{12} + 519521 T^{13} + 32130 T^{14} - 11 T^{15} + T^{16} \)
$83$ \( \)\(17\!\cdots\!81\)\( + \)\(37\!\cdots\!27\)\( T + \)\(19\!\cdots\!57\)\( T^{2} - \)\(18\!\cdots\!82\)\( T^{3} - \)\(78\!\cdots\!40\)\( T^{4} + \)\(77\!\cdots\!39\)\( T^{5} + \)\(45\!\cdots\!39\)\( T^{6} - 3901548176410442436 T^{7} - 87712995912043113 T^{8} + 721552869940077 T^{9} + 12613716527751 T^{10} - 129448880406 T^{11} - 359146062 T^{12} + 6897852 T^{13} + 8670 T^{14} - 306 T^{15} + T^{16} \)
$89$ \( \)\(24\!\cdots\!21\)\( + \)\(14\!\cdots\!69\)\( T^{2} + \)\(13\!\cdots\!23\)\( T^{4} + \)\(10\!\cdots\!75\)\( T^{6} + 283155444795983256 T^{8} + 36886290735375 T^{10} + 2439497007 T^{12} + 79173 T^{14} + T^{16} \)
$97$ \( \)\(29\!\cdots\!21\)\( - \)\(11\!\cdots\!29\)\( T + \)\(12\!\cdots\!88\)\( T^{2} + \)\(21\!\cdots\!37\)\( T^{3} + \)\(22\!\cdots\!84\)\( T^{4} + \)\(38\!\cdots\!16\)\( T^{5} + \)\(75\!\cdots\!84\)\( T^{6} + 2279449030004579150 T^{7} + 165303805429238697 T^{8} + 225844856885522 T^{9} + 16673880500836 T^{10} + 17840489712 T^{11} + 1216174604 T^{12} + 602971 T^{13} + 42132 T^{14} + 41 T^{15} + T^{16} \)
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