Properties

Label 21.4.e.b
Level 21
Weight 4
Character orbit 21.e
Analytic conductor 1.239
Analytic rank 0
Dimension 6
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 21 = 3 \cdot 7 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 21.e (of order \(3\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(1.23904011012\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.9924270768.1
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{2} \) \( + ( 3 - 3 \beta_{4} ) q^{3} \) \( + ( -8 + \beta_{1} + \beta_{2} + 8 \beta_{4} + \beta_{5} ) q^{4} \) \( + ( \beta_{1} + \beta_{3} - 4 \beta_{4} - \beta_{5} ) q^{5} \) \( + 3 \beta_{2} q^{6} \) \( + ( -4 - 4 \beta_{1} - 3 \beta_{2} - \beta_{3} + 3 \beta_{4} ) q^{7} \) \( + ( 10 - 9 \beta_{2} - \beta_{3} ) q^{8} \) \( -9 \beta_{4} q^{9} \) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{2} \) \( + ( 3 - 3 \beta_{4} ) q^{3} \) \( + ( -8 + \beta_{1} + \beta_{2} + 8 \beta_{4} + \beta_{5} ) q^{4} \) \( + ( \beta_{1} + \beta_{3} - 4 \beta_{4} - \beta_{5} ) q^{5} \) \( + 3 \beta_{2} q^{6} \) \( + ( -4 - 4 \beta_{1} - 3 \beta_{2} - \beta_{3} + 3 \beta_{4} ) q^{7} \) \( + ( 10 - 9 \beta_{2} - \beta_{3} ) q^{8} \) \( -9 \beta_{4} q^{9} \) \( + ( 22 + 11 \beta_{1} + 11 \beta_{2} - 22 \beta_{4} - \beta_{5} ) q^{10} \) \( + ( -12 - \beta_{1} - \beta_{2} + 12 \beta_{4} - 3 \beta_{5} ) q^{11} \) \( + ( 3 \beta_{1} - 3 \beta_{3} + 24 \beta_{4} + 3 \beta_{5} ) q^{12} \) \( + ( 19 - 5 \beta_{2} + \beta_{3} ) q^{13} \) \( + ( -64 - 6 \beta_{1} + \beta_{2} + 3 \beta_{3} + 22 \beta_{4} + \beta_{5} ) q^{14} \) \( + ( -12 - 3 \beta_{2} + 3 \beta_{3} ) q^{15} \) \( + ( -19 \beta_{1} + \beta_{3} - 74 \beta_{4} - \beta_{5} ) q^{16} \) \( + ( -16 + 16 \beta_{4} + 4 \beta_{5} ) q^{17} \) \( + ( 9 \beta_{1} + 9 \beta_{2} ) q^{18} \) \( + ( 7 \beta_{1} - \beta_{3} + 65 \beta_{4} + \beta_{5} ) q^{19} \) \( + ( 150 + 11 \beta_{2} - 3 \beta_{3} ) q^{20} \) \( + ( -3 - 9 \beta_{1} + 3 \beta_{2} + 12 \beta_{4} - 3 \beta_{5} ) q^{21} \) \( + ( 2 + 13 \beta_{2} + \beta_{3} ) q^{22} \) \( + ( 24 \beta_{1} - 4 \beta_{3} - 80 \beta_{4} + 4 \beta_{5} ) q^{23} \) \( + ( 30 - 27 \beta_{1} - 27 \beta_{2} - 30 \beta_{4} - 3 \beta_{5} ) q^{24} \) \( + ( -53 - 29 \beta_{1} - 29 \beta_{2} + 53 \beta_{4} + \beta_{5} ) q^{25} \) \( + ( -16 \beta_{1} + 5 \beta_{3} - 86 \beta_{4} - 5 \beta_{5} ) q^{26} \) \( -27 q^{27} \) \( + ( -110 + 49 \beta_{1} - 16 \beta_{2} - \beta_{3} + 126 \beta_{4} - \beta_{5} ) q^{28} \) \( + ( 26 + 25 \beta_{2} - 5 \beta_{3} ) q^{29} \) \( + ( 33 \beta_{1} + 3 \beta_{3} - 66 \beta_{4} - 3 \beta_{5} ) q^{30} \) \( + ( 39 + 22 \beta_{1} + 22 \beta_{2} - 39 \beta_{4} - 2 \beta_{5} ) q^{31} \) \( + ( -218 + 29 \beta_{1} + 29 \beta_{2} + 218 \beta_{4} + 11 \beta_{5} ) q^{32} \) \( + ( -3 \beta_{1} + 9 \beta_{3} + 36 \beta_{4} - 9 \beta_{5} ) q^{33} \) \( + ( -24 - 48 \beta_{2} ) q^{34} \) \( + ( 100 + 24 \beta_{1} + 47 \beta_{2} - 11 \beta_{3} - 158 \beta_{4} + 4 \beta_{5} ) q^{35} \) \( + ( 72 - 9 \beta_{2} - 9 \beta_{3} ) q^{36} \) \( + ( -19 \beta_{1} + \beta_{3} - 81 \beta_{4} - \beta_{5} ) q^{37} \) \( + ( 106 - 80 \beta_{1} - 80 \beta_{2} - 106 \beta_{4} - 7 \beta_{5} ) q^{38} \) \( + ( 57 - 15 \beta_{1} - 15 \beta_{2} - 57 \beta_{4} + 3 \beta_{5} ) q^{39} \) \( + ( -75 \beta_{1} - 3 \beta_{3} + 18 \beta_{4} + 3 \beta_{5} ) q^{40} \) \( + ( 82 + 2 \beta_{2} + 14 \beta_{3} ) q^{41} \) \( + ( -126 + 3 \beta_{1} + 21 \beta_{2} - 3 \beta_{3} + 192 \beta_{4} + 12 \beta_{5} ) q^{42} \) \( + ( 143 + 69 \beta_{2} + 3 \beta_{3} ) q^{43} \) \( + ( 11 \beta_{1} + 11 \beta_{3} + 298 \beta_{4} - 11 \beta_{5} ) q^{44} \) \( + ( -36 - 9 \beta_{1} - 9 \beta_{2} + 36 \beta_{4} + 9 \beta_{5} ) q^{45} \) \( + ( 360 + 24 \beta_{1} + 24 \beta_{2} - 360 \beta_{4} - 24 \beta_{5} ) q^{46} \) \( + ( 72 \beta_{1} - 28 \beta_{3} + 46 \beta_{4} + 28 \beta_{5} ) q^{47} \) \( + ( -222 + 57 \beta_{2} + 3 \beta_{3} ) q^{48} \) \( + ( -7 - 46 \beta_{1} - 35 \beta_{2} + 25 \beta_{3} - 95 \beta_{4} - 2 \beta_{5} ) q^{49} \) \( + ( -470 - 32 \beta_{2} + 29 \beta_{3} ) q^{50} \) \( + ( -12 \beta_{3} + 48 \beta_{4} + 12 \beta_{5} ) q^{51} \) \( + ( -74 + 102 \beta_{1} + 102 \beta_{2} + 74 \beta_{4} + 24 \beta_{5} ) q^{52} \) \( + ( -154 - 69 \beta_{1} - 69 \beta_{2} + 154 \beta_{4} - 11 \beta_{5} ) q^{53} \) \( + 27 \beta_{1} q^{54} \) \( + ( -350 - 19 \beta_{2} - 25 \beta_{3} ) q^{55} \) \( + ( 454 - 81 \beta_{1} - 111 \beta_{2} + 8 \beta_{3} - 522 \beta_{4} - 33 \beta_{5} ) q^{56} \) \( + ( 195 - 21 \beta_{2} - 3 \beta_{3} ) q^{57} \) \( + ( -41 \beta_{1} - 25 \beta_{3} + 430 \beta_{4} + 25 \beta_{5} ) q^{58} \) \( + ( -358 + 69 \beta_{1} + 69 \beta_{2} + 358 \beta_{4} - 29 \beta_{5} ) q^{59} \) \( + ( 450 + 33 \beta_{1} + 33 \beta_{2} - 450 \beta_{4} - 9 \beta_{5} ) q^{60} \) \( + ( 100 \beta_{1} + 20 \beta_{3} - 10 \beta_{4} - 20 \beta_{5} ) q^{61} \) \( + ( 364 + 33 \beta_{2} - 22 \beta_{3} ) q^{62} \) \( + ( 27 + 9 \beta_{1} + 36 \beta_{2} + 9 \beta_{3} + 9 \beta_{4} - 9 \beta_{5} ) q^{63} \) \( + ( -194 - 183 \beta_{2} - 21 \beta_{3} ) q^{64} \) \( + ( -50 \beta_{1} + 18 \beta_{3} + 174 \beta_{4} - 18 \beta_{5} ) q^{65} \) \( + ( 6 + 39 \beta_{1} + 39 \beta_{2} - 6 \beta_{4} + 3 \beta_{5} ) q^{66} \) \( + ( 215 + 17 \beta_{1} + 17 \beta_{2} - 215 \beta_{4} + 47 \beta_{5} ) q^{67} \) \( + ( -24 \beta_{1} + 16 \beta_{3} - 640 \beta_{4} - 16 \beta_{5} ) q^{68} \) \( + ( -240 - 72 \beta_{2} - 12 \beta_{3} ) q^{69} \) \( + ( 360 - 7 \beta_{1} + 102 \beta_{2} - 47 \beta_{3} + 434 \beta_{4} + 23 \beta_{5} ) q^{70} \) \( + ( 66 - 120 \beta_{2} - 12 \beta_{3} ) q^{71} \) \( + ( -81 \beta_{1} + 9 \beta_{3} - 90 \beta_{4} - 9 \beta_{5} ) q^{72} \) \( + ( -363 - 101 \beta_{1} - 101 \beta_{2} + 363 \beta_{4} - 23 \beta_{5} ) q^{73} \) \( + ( -298 + 108 \beta_{1} + 108 \beta_{2} + 298 \beta_{4} + 19 \beta_{5} ) q^{74} \) \( + ( -87 \beta_{1} - 3 \beta_{3} + 159 \beta_{4} + 3 \beta_{5} ) q^{75} \) \( + ( -718 + 186 \beta_{2} + 72 \beta_{3} ) q^{76} \) \( + ( 410 - 25 \beta_{1} + 24 \beta_{2} - 5 \beta_{3} - 438 \beta_{4} + 45 \beta_{5} ) q^{77} \) \( + ( -258 + 48 \beta_{2} + 15 \beta_{3} ) q^{78} \) \( + ( 36 \beta_{1} + 48 \beta_{3} - 299 \beta_{4} - 48 \beta_{5} ) q^{79} \) \( + ( -18 + 121 \beta_{1} + 121 \beta_{2} + 18 \beta_{4} + 51 \beta_{5} ) q^{80} \) \( + ( -81 + 81 \beta_{4} ) q^{81} \) \( + ( 32 \beta_{1} - 2 \beta_{3} - 52 \beta_{4} + 2 \beta_{5} ) q^{82} \) \( + ( 156 - 51 \beta_{2} + 27 \beta_{3} ) q^{83} \) \( + ( 48 - 48 \beta_{1} - 195 \beta_{2} + 3 \beta_{3} + 330 \beta_{4} - 6 \beta_{5} ) q^{84} \) \( + ( 624 + 72 \beta_{2} ) q^{85} \) \( + ( -50 \beta_{1} - 69 \beta_{3} + 1086 \beta_{4} + 69 \beta_{5} ) q^{86} \) \( + ( 78 + 75 \beta_{1} + 75 \beta_{2} - 78 \beta_{4} - 15 \beta_{5} ) q^{87} \) \( + ( 258 - 117 \beta_{1} - 117 \beta_{2} - 258 \beta_{4} - 3 \beta_{5} ) q^{88} \) \( + ( -170 \beta_{1} + 22 \beta_{3} - 532 \beta_{4} - 22 \beta_{5} ) q^{89} \) \( + ( -198 - 99 \beta_{2} + 9 \beta_{3} ) q^{90} \) \( + ( 18 - 49 \beta_{1} - 74 \beta_{2} - 23 \beta_{3} - 287 \beta_{4} - 23 \beta_{5} ) q^{91} \) \( + ( -112 + 336 \beta_{2} - 56 \beta_{3} ) q^{92} \) \( + ( 66 \beta_{1} + 6 \beta_{3} - 117 \beta_{4} - 6 \beta_{5} ) q^{93} \) \( + ( 984 - 342 \beta_{1} - 342 \beta_{2} - 984 \beta_{4} - 72 \beta_{5} ) q^{94} \) \( + ( 246 + 2 \beta_{1} + 2 \beta_{2} - 246 \beta_{4} - 54 \beta_{5} ) q^{95} \) \( + ( 87 \beta_{1} - 33 \beta_{3} + 654 \beta_{4} + 33 \beta_{5} ) q^{96} \) \( + ( 24 + 53 \beta_{2} - 49 \beta_{3} ) q^{97} \) \( + ( -724 + 313 \beta_{1} + 157 \beta_{2} + 35 \beta_{3} + 26 \beta_{4} + 11 \beta_{5} ) q^{98} \) \( + ( 108 + 9 \beta_{2} + 27 \beta_{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(6q \) \(\mathstrut -\mathstrut q^{2} \) \(\mathstrut +\mathstrut 9q^{3} \) \(\mathstrut -\mathstrut 25q^{4} \) \(\mathstrut -\mathstrut 11q^{5} \) \(\mathstrut -\mathstrut 6q^{6} \) \(\mathstrut -\mathstrut 13q^{7} \) \(\mathstrut +\mathstrut 78q^{8} \) \(\mathstrut -\mathstrut 27q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut -\mathstrut q^{2} \) \(\mathstrut +\mathstrut 9q^{3} \) \(\mathstrut -\mathstrut 25q^{4} \) \(\mathstrut -\mathstrut 11q^{5} \) \(\mathstrut -\mathstrut 6q^{6} \) \(\mathstrut -\mathstrut 13q^{7} \) \(\mathstrut +\mathstrut 78q^{8} \) \(\mathstrut -\mathstrut 27q^{9} \) \(\mathstrut +\mathstrut 55q^{10} \) \(\mathstrut -\mathstrut 35q^{11} \) \(\mathstrut +\mathstrut 75q^{12} \) \(\mathstrut +\mathstrut 124q^{13} \) \(\mathstrut -\mathstrut 326q^{14} \) \(\mathstrut -\mathstrut 66q^{15} \) \(\mathstrut -\mathstrut 241q^{16} \) \(\mathstrut -\mathstrut 48q^{17} \) \(\mathstrut -\mathstrut 9q^{18} \) \(\mathstrut +\mathstrut 202q^{19} \) \(\mathstrut +\mathstrut 878q^{20} \) \(\mathstrut +\mathstrut 3q^{21} \) \(\mathstrut -\mathstrut 14q^{22} \) \(\mathstrut -\mathstrut 216q^{23} \) \(\mathstrut +\mathstrut 117q^{24} \) \(\mathstrut -\mathstrut 130q^{25} \) \(\mathstrut -\mathstrut 274q^{26} \) \(\mathstrut -\mathstrut 162q^{27} \) \(\mathstrut -\mathstrut 201q^{28} \) \(\mathstrut +\mathstrut 106q^{29} \) \(\mathstrut -\mathstrut 165q^{30} \) \(\mathstrut +\mathstrut 95q^{31} \) \(\mathstrut -\mathstrut 683q^{32} \) \(\mathstrut +\mathstrut 105q^{33} \) \(\mathstrut -\mathstrut 48q^{34} \) \(\mathstrut +\mathstrut 56q^{35} \) \(\mathstrut +\mathstrut 450q^{36} \) \(\mathstrut -\mathstrut 262q^{37} \) \(\mathstrut +\mathstrut 398q^{38} \) \(\mathstrut +\mathstrut 186q^{39} \) \(\mathstrut -\mathstrut 21q^{40} \) \(\mathstrut +\mathstrut 488q^{41} \) \(\mathstrut -\mathstrut 219q^{42} \) \(\mathstrut +\mathstrut 720q^{43} \) \(\mathstrut +\mathstrut 905q^{44} \) \(\mathstrut -\mathstrut 99q^{45} \) \(\mathstrut +\mathstrut 1056q^{46} \) \(\mathstrut +\mathstrut 210q^{47} \) \(\mathstrut -\mathstrut 1446q^{48} \) \(\mathstrut -\mathstrut 303q^{49} \) \(\mathstrut -\mathstrut 2756q^{50} \) \(\mathstrut +\mathstrut 144q^{51} \) \(\mathstrut -\mathstrut 324q^{52} \) \(\mathstrut -\mathstrut 393q^{53} \) \(\mathstrut +\mathstrut 27q^{54} \) \(\mathstrut -\mathstrut 2062q^{55} \) \(\mathstrut +\mathstrut 1299q^{56} \) \(\mathstrut +\mathstrut 1212q^{57} \) \(\mathstrut +\mathstrut 1249q^{58} \) \(\mathstrut -\mathstrut 1143q^{59} \) \(\mathstrut +\mathstrut 1317q^{60} \) \(\mathstrut +\mathstrut 70q^{61} \) \(\mathstrut +\mathstrut 2118q^{62} \) \(\mathstrut +\mathstrut 126q^{63} \) \(\mathstrut -\mathstrut 798q^{64} \) \(\mathstrut +\mathstrut 472q^{65} \) \(\mathstrut -\mathstrut 21q^{66} \) \(\mathstrut +\mathstrut 628q^{67} \) \(\mathstrut -\mathstrut 1944q^{68} \) \(\mathstrut -\mathstrut 1296q^{69} \) \(\mathstrut +\mathstrut 3251q^{70} \) \(\mathstrut +\mathstrut 636q^{71} \) \(\mathstrut -\mathstrut 351q^{72} \) \(\mathstrut -\mathstrut 988q^{73} \) \(\mathstrut -\mathstrut 1002q^{74} \) \(\mathstrut +\mathstrut 390q^{75} \) \(\mathstrut -\mathstrut 4680q^{76} \) \(\mathstrut +\mathstrut 1073q^{77} \) \(\mathstrut -\mathstrut 1644q^{78} \) \(\mathstrut -\mathstrut 861q^{79} \) \(\mathstrut -\mathstrut 175q^{80} \) \(\mathstrut -\mathstrut 243q^{81} \) \(\mathstrut -\mathstrut 124q^{82} \) \(\mathstrut +\mathstrut 1038q^{83} \) \(\mathstrut +\mathstrut 1620q^{84} \) \(\mathstrut +\mathstrut 3600q^{85} \) \(\mathstrut +\mathstrut 3208q^{86} \) \(\mathstrut +\mathstrut 159q^{87} \) \(\mathstrut +\mathstrut 891q^{88} \) \(\mathstrut -\mathstrut 1766q^{89} \) \(\mathstrut -\mathstrut 990q^{90} \) \(\mathstrut -\mathstrut 654q^{91} \) \(\mathstrut -\mathstrut 1344q^{92} \) \(\mathstrut -\mathstrut 285q^{93} \) \(\mathstrut +\mathstrut 3294q^{94} \) \(\mathstrut +\mathstrut 736q^{95} \) \(\mathstrut +\mathstrut 2049q^{96} \) \(\mathstrut +\mathstrut 38q^{97} \) \(\mathstrut -\mathstrut 4267q^{98} \) \(\mathstrut +\mathstrut 630q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6}\mathstrut -\mathstrut \) \(x^{5}\mathstrut +\mathstrut \) \(25\) \(x^{4}\mathstrut +\mathstrut \) \(12\) \(x^{3}\mathstrut +\mathstrut \) \(582\) \(x^{2}\mathstrut -\mathstrut \) \(144\) \(x\mathstrut +\mathstrut \) \(36\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} - 25 \nu^{4} + 625 \nu^{3} - 582 \nu^{2} + 144 \nu - 3600 \)\()/14406\)
\(\beta_{3}\)\(=\)\((\)\( -25 \nu^{5} + 625 \nu^{4} - 1219 \nu^{3} + 14550 \nu^{2} - 3600 \nu + 234060 \)\()/14406\)
\(\beta_{4}\)\(=\)\((\)\( 100 \nu^{5} - 99 \nu^{4} + 2475 \nu^{3} + 1825 \nu^{2} + 57618 \nu + 150 \)\()/14406\)
\(\beta_{5}\)\(=\)\((\)\( -1601 \nu^{5} + 1609 \nu^{4} - 40225 \nu^{3} - 14212 \nu^{2} - 936438 \nu + 231696 \)\()/14406\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5}\mathstrut +\mathstrut \) \(16\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut -\mathstrut \) \(16\)
\(\nu^{3}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(25\) \(\beta_{2}\mathstrut -\mathstrut \) \(10\)
\(\nu^{4}\)\(=\)\(-\)\(25\) \(\beta_{5}\mathstrut -\mathstrut \) \(394\) \(\beta_{4}\mathstrut +\mathstrut \) \(25\) \(\beta_{3}\mathstrut -\mathstrut \) \(43\) \(\beta_{1}\)
\(\nu^{5}\)\(=\)\(-\)\(43\) \(\beta_{5}\mathstrut -\mathstrut \) \(538\) \(\beta_{4}\mathstrut -\mathstrut \) \(637\) \(\beta_{2}\mathstrut -\mathstrut \) \(637\) \(\beta_{1}\mathstrut +\mathstrut \) \(538\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).

\(n\) \(8\) \(10\)
\(\chi(n)\) \(1\) \(-\beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
2.65415 + 4.59712i
0.124036 + 0.214837i
−2.27818 3.94593i
2.65415 4.59712i
0.124036 0.214837i
−2.27818 + 3.94593i
−2.65415 4.59712i 1.50000 2.59808i −10.0890 + 17.4746i −2.78070 4.81631i −15.9249 9.67799 15.7904i 64.6443 −4.50000 7.79423i −14.7608 + 25.5664i
4.2 −0.124036 0.214837i 1.50000 2.59808i 3.96923 6.87491i 6.21730 + 10.7687i −0.744216 −18.4385 + 1.73873i −3.95388 −4.50000 7.79423i 1.54234 2.67141i
4.3 2.27818 + 3.94593i 1.50000 2.59808i −6.38024 + 11.0509i −8.93660 15.4786i 13.6691 2.26047 + 18.3818i −21.6905 −4.50000 7.79423i 40.7184 70.5264i
16.1 −2.65415 + 4.59712i 1.50000 + 2.59808i −10.0890 17.4746i −2.78070 + 4.81631i −15.9249 9.67799 + 15.7904i 64.6443 −4.50000 + 7.79423i −14.7608 25.5664i
16.2 −0.124036 + 0.214837i 1.50000 + 2.59808i 3.96923 + 6.87491i 6.21730 10.7687i −0.744216 −18.4385 1.73873i −3.95388 −4.50000 + 7.79423i 1.54234 + 2.67141i
16.3 2.27818 3.94593i 1.50000 + 2.59808i −6.38024 11.0509i −8.93660 + 15.4786i 13.6691 2.26047 18.3818i −21.6905 −4.50000 + 7.79423i 40.7184 + 70.5264i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.3
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
7.c Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{6} \) \(\mathstrut +\mathstrut T_{2}^{5} \) \(\mathstrut +\mathstrut 25 T_{2}^{4} \) \(\mathstrut -\mathstrut 12 T_{2}^{3} \) \(\mathstrut +\mathstrut 582 T_{2}^{2} \) \(\mathstrut +\mathstrut 144 T_{2} \) \(\mathstrut +\mathstrut 36 \) acting on \(S_{4}^{\mathrm{new}}(21, [\chi])\).