Properties

Label 21.4.a.c
Level 21
Weight 4
Character orbit 21.a
Self dual Yes
Analytic conductor 1.239
Analytic rank 0
Dimension 2
CM No
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 21 = 3 \cdot 7 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 21.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(1.23904011012\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{57})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -1 - \beta ) q^{2} \) \( + 3 q^{3} \) \( + ( 7 + 3 \beta ) q^{4} \) \( + ( 2 + 2 \beta ) q^{5} \) \( + ( -3 - 3 \beta ) q^{6} \) \( + 7 q^{7} \) \( + ( -41 - 5 \beta ) q^{8} \) \( + 9 q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -1 - \beta ) q^{2} \) \( + 3 q^{3} \) \( + ( 7 + 3 \beta ) q^{4} \) \( + ( 2 + 2 \beta ) q^{5} \) \( + ( -3 - 3 \beta ) q^{6} \) \( + 7 q^{7} \) \( + ( -41 - 5 \beta ) q^{8} \) \( + 9 q^{9} \) \( + ( -30 - 6 \beta ) q^{10} \) \( + ( -8 + 10 \beta ) q^{11} \) \( + ( 21 + 9 \beta ) q^{12} \) \( + ( 14 - 12 \beta ) q^{13} \) \( + ( -7 - 7 \beta ) q^{14} \) \( + ( 6 + 6 \beta ) q^{15} \) \( + ( 55 + 27 \beta ) q^{16} \) \( + ( -2 - 2 \beta ) q^{17} \) \( + ( -9 - 9 \beta ) q^{18} \) \( + ( 44 - 24 \beta ) q^{19} \) \( + ( 98 + 26 \beta ) q^{20} \) \( + 21 q^{21} \) \( + ( -132 - 12 \beta ) q^{22} \) \( + ( 20 - 34 \beta ) q^{23} \) \( + ( -123 - 15 \beta ) q^{24} \) \( + ( -65 + 12 \beta ) q^{25} \) \( + ( 154 + 10 \beta ) q^{26} \) \( + 27 q^{27} \) \( + ( 49 + 21 \beta ) q^{28} \) \( + ( -138 + 24 \beta ) q^{29} \) \( + ( -90 - 18 \beta ) q^{30} \) \( + ( -16 + 72 \beta ) q^{31} \) \( + ( -105 - 69 \beta ) q^{32} \) \( + ( -24 + 30 \beta ) q^{33} \) \( + ( 30 + 6 \beta ) q^{34} \) \( + ( 14 + 14 \beta ) q^{35} \) \( + ( 63 + 27 \beta ) q^{36} \) \( + ( -106 - 36 \beta ) q^{37} \) \( + ( 292 + 4 \beta ) q^{38} \) \( + ( 42 - 36 \beta ) q^{39} \) \( + ( -222 - 102 \beta ) q^{40} \) \( + ( -210 - 30 \beta ) q^{41} \) \( + ( -21 - 21 \beta ) q^{42} \) \( + ( 212 - 48 \beta ) q^{43} \) \( + ( 364 + 76 \beta ) q^{44} \) \( + ( 18 + 18 \beta ) q^{45} \) \( + ( 456 + 48 \beta ) q^{46} \) \( + ( -40 + 68 \beta ) q^{47} \) \( + ( 165 + 81 \beta ) q^{48} \) \( + 49 q^{49} \) \( + ( -103 + 41 \beta ) q^{50} \) \( + ( -6 - 6 \beta ) q^{51} \) \( + ( -406 - 78 \beta ) q^{52} \) \( + ( -554 + 4 \beta ) q^{53} \) \( + ( -27 - 27 \beta ) q^{54} \) \( + ( 264 + 24 \beta ) q^{55} \) \( + ( -287 - 35 \beta ) q^{56} \) \( + ( 132 - 72 \beta ) q^{57} \) \( + ( -198 + 90 \beta ) q^{58} \) \( + ( 460 - 116 \beta ) q^{59} \) \( + ( 294 + 78 \beta ) q^{60} \) \( + ( -250 + 72 \beta ) q^{61} \) \( + ( -992 - 128 \beta ) q^{62} \) \( + 63 q^{63} \) \( + ( 631 + 27 \beta ) q^{64} \) \( + ( -308 - 20 \beta ) q^{65} \) \( + ( -396 - 36 \beta ) q^{66} \) \( + ( 20 + 108 \beta ) q^{67} \) \( + ( -98 - 26 \beta ) q^{68} \) \( + ( 60 - 102 \beta ) q^{69} \) \( + ( -210 - 42 \beta ) q^{70} \) \( + ( 492 - 30 \beta ) q^{71} \) \( + ( -369 - 45 \beta ) q^{72} \) \( + ( 530 + 12 \beta ) q^{73} \) \( + ( 610 + 178 \beta ) q^{74} \) \( + ( -195 + 36 \beta ) q^{75} \) \( + ( -700 - 108 \beta ) q^{76} \) \( + ( -56 + 70 \beta ) q^{77} \) \( + ( 462 + 30 \beta ) q^{78} \) \( + ( -232 - 108 \beta ) q^{79} \) \( + ( 866 + 218 \beta ) q^{80} \) \( + 81 q^{81} \) \( + ( 630 + 270 \beta ) q^{82} \) \( + ( 924 + 96 \beta ) q^{83} \) \( + ( 147 + 63 \beta ) q^{84} \) \( + ( -60 - 12 \beta ) q^{85} \) \( + ( 460 - 116 \beta ) q^{86} \) \( + ( -414 + 72 \beta ) q^{87} \) \( + ( -372 - 420 \beta ) q^{88} \) \( + ( 254 - 142 \beta ) q^{89} \) \( + ( -270 - 54 \beta ) q^{90} \) \( + ( 98 - 84 \beta ) q^{91} \) \( + ( -1288 - 280 \beta ) q^{92} \) \( + ( -48 + 216 \beta ) q^{93} \) \( + ( -912 - 96 \beta ) q^{94} \) \( + ( -584 - 8 \beta ) q^{95} \) \( + ( -315 - 207 \beta ) q^{96} \) \( + ( 266 + 276 \beta ) q^{97} \) \( + ( -49 - 49 \beta ) q^{98} \) \( + ( -72 + 90 \beta ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 3q^{2} \) \(\mathstrut +\mathstrut 6q^{3} \) \(\mathstrut +\mathstrut 17q^{4} \) \(\mathstrut +\mathstrut 6q^{5} \) \(\mathstrut -\mathstrut 9q^{6} \) \(\mathstrut +\mathstrut 14q^{7} \) \(\mathstrut -\mathstrut 87q^{8} \) \(\mathstrut +\mathstrut 18q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 3q^{2} \) \(\mathstrut +\mathstrut 6q^{3} \) \(\mathstrut +\mathstrut 17q^{4} \) \(\mathstrut +\mathstrut 6q^{5} \) \(\mathstrut -\mathstrut 9q^{6} \) \(\mathstrut +\mathstrut 14q^{7} \) \(\mathstrut -\mathstrut 87q^{8} \) \(\mathstrut +\mathstrut 18q^{9} \) \(\mathstrut -\mathstrut 66q^{10} \) \(\mathstrut -\mathstrut 6q^{11} \) \(\mathstrut +\mathstrut 51q^{12} \) \(\mathstrut +\mathstrut 16q^{13} \) \(\mathstrut -\mathstrut 21q^{14} \) \(\mathstrut +\mathstrut 18q^{15} \) \(\mathstrut +\mathstrut 137q^{16} \) \(\mathstrut -\mathstrut 6q^{17} \) \(\mathstrut -\mathstrut 27q^{18} \) \(\mathstrut +\mathstrut 64q^{19} \) \(\mathstrut +\mathstrut 222q^{20} \) \(\mathstrut +\mathstrut 42q^{21} \) \(\mathstrut -\mathstrut 276q^{22} \) \(\mathstrut +\mathstrut 6q^{23} \) \(\mathstrut -\mathstrut 261q^{24} \) \(\mathstrut -\mathstrut 118q^{25} \) \(\mathstrut +\mathstrut 318q^{26} \) \(\mathstrut +\mathstrut 54q^{27} \) \(\mathstrut +\mathstrut 119q^{28} \) \(\mathstrut -\mathstrut 252q^{29} \) \(\mathstrut -\mathstrut 198q^{30} \) \(\mathstrut +\mathstrut 40q^{31} \) \(\mathstrut -\mathstrut 279q^{32} \) \(\mathstrut -\mathstrut 18q^{33} \) \(\mathstrut +\mathstrut 66q^{34} \) \(\mathstrut +\mathstrut 42q^{35} \) \(\mathstrut +\mathstrut 153q^{36} \) \(\mathstrut -\mathstrut 248q^{37} \) \(\mathstrut +\mathstrut 588q^{38} \) \(\mathstrut +\mathstrut 48q^{39} \) \(\mathstrut -\mathstrut 546q^{40} \) \(\mathstrut -\mathstrut 450q^{41} \) \(\mathstrut -\mathstrut 63q^{42} \) \(\mathstrut +\mathstrut 376q^{43} \) \(\mathstrut +\mathstrut 804q^{44} \) \(\mathstrut +\mathstrut 54q^{45} \) \(\mathstrut +\mathstrut 960q^{46} \) \(\mathstrut -\mathstrut 12q^{47} \) \(\mathstrut +\mathstrut 411q^{48} \) \(\mathstrut +\mathstrut 98q^{49} \) \(\mathstrut -\mathstrut 165q^{50} \) \(\mathstrut -\mathstrut 18q^{51} \) \(\mathstrut -\mathstrut 890q^{52} \) \(\mathstrut -\mathstrut 1104q^{53} \) \(\mathstrut -\mathstrut 81q^{54} \) \(\mathstrut +\mathstrut 552q^{55} \) \(\mathstrut -\mathstrut 609q^{56} \) \(\mathstrut +\mathstrut 192q^{57} \) \(\mathstrut -\mathstrut 306q^{58} \) \(\mathstrut +\mathstrut 804q^{59} \) \(\mathstrut +\mathstrut 666q^{60} \) \(\mathstrut -\mathstrut 428q^{61} \) \(\mathstrut -\mathstrut 2112q^{62} \) \(\mathstrut +\mathstrut 126q^{63} \) \(\mathstrut +\mathstrut 1289q^{64} \) \(\mathstrut -\mathstrut 636q^{65} \) \(\mathstrut -\mathstrut 828q^{66} \) \(\mathstrut +\mathstrut 148q^{67} \) \(\mathstrut -\mathstrut 222q^{68} \) \(\mathstrut +\mathstrut 18q^{69} \) \(\mathstrut -\mathstrut 462q^{70} \) \(\mathstrut +\mathstrut 954q^{71} \) \(\mathstrut -\mathstrut 783q^{72} \) \(\mathstrut +\mathstrut 1072q^{73} \) \(\mathstrut +\mathstrut 1398q^{74} \) \(\mathstrut -\mathstrut 354q^{75} \) \(\mathstrut -\mathstrut 1508q^{76} \) \(\mathstrut -\mathstrut 42q^{77} \) \(\mathstrut +\mathstrut 954q^{78} \) \(\mathstrut -\mathstrut 572q^{79} \) \(\mathstrut +\mathstrut 1950q^{80} \) \(\mathstrut +\mathstrut 162q^{81} \) \(\mathstrut +\mathstrut 1530q^{82} \) \(\mathstrut +\mathstrut 1944q^{83} \) \(\mathstrut +\mathstrut 357q^{84} \) \(\mathstrut -\mathstrut 132q^{85} \) \(\mathstrut +\mathstrut 804q^{86} \) \(\mathstrut -\mathstrut 756q^{87} \) \(\mathstrut -\mathstrut 1164q^{88} \) \(\mathstrut +\mathstrut 366q^{89} \) \(\mathstrut -\mathstrut 594q^{90} \) \(\mathstrut +\mathstrut 112q^{91} \) \(\mathstrut -\mathstrut 2856q^{92} \) \(\mathstrut +\mathstrut 120q^{93} \) \(\mathstrut -\mathstrut 1920q^{94} \) \(\mathstrut -\mathstrut 1176q^{95} \) \(\mathstrut -\mathstrut 837q^{96} \) \(\mathstrut +\mathstrut 808q^{97} \) \(\mathstrut -\mathstrut 147q^{98} \) \(\mathstrut -\mathstrut 54q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
4.27492
−3.27492
−5.27492 3.00000 19.8248 10.5498 −15.8248 7.00000 −62.3746 9.00000 −55.6495
1.2 2.27492 3.00000 −2.82475 −4.54983 6.82475 7.00000 −24.6254 9.00000 −10.3505
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{2} \) \(\mathstrut +\mathstrut 3 T_{2} \) \(\mathstrut -\mathstrut 12 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(21))\).