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

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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: \(\Q(\sqrt{57}) \)
Defining polynomial: \(x^{2} - x - 14\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{2} + 6q^{3} + 17q^{4} + 6q^{5} - 9q^{6} + 14q^{7} - 87q^{8} + 18q^{9} + O(q^{10}) \) \( 2q - 3q^{2} + 6q^{3} + 17q^{4} + 6q^{5} - 9q^{6} + 14q^{7} - 87q^{8} + 18q^{9} - 66q^{10} - 6q^{11} + 51q^{12} + 16q^{13} - 21q^{14} + 18q^{15} + 137q^{16} - 6q^{17} - 27q^{18} + 64q^{19} + 222q^{20} + 42q^{21} - 276q^{22} + 6q^{23} - 261q^{24} - 118q^{25} + 318q^{26} + 54q^{27} + 119q^{28} - 252q^{29} - 198q^{30} + 40q^{31} - 279q^{32} - 18q^{33} + 66q^{34} + 42q^{35} + 153q^{36} - 248q^{37} + 588q^{38} + 48q^{39} - 546q^{40} - 450q^{41} - 63q^{42} + 376q^{43} + 804q^{44} + 54q^{45} + 960q^{46} - 12q^{47} + 411q^{48} + 98q^{49} - 165q^{50} - 18q^{51} - 890q^{52} - 1104q^{53} - 81q^{54} + 552q^{55} - 609q^{56} + 192q^{57} - 306q^{58} + 804q^{59} + 666q^{60} - 428q^{61} - 2112q^{62} + 126q^{63} + 1289q^{64} - 636q^{65} - 828q^{66} + 148q^{67} - 222q^{68} + 18q^{69} - 462q^{70} + 954q^{71} - 783q^{72} + 1072q^{73} + 1398q^{74} - 354q^{75} - 1508q^{76} - 42q^{77} + 954q^{78} - 572q^{79} + 1950q^{80} + 162q^{81} + 1530q^{82} + 1944q^{83} + 357q^{84} - 132q^{85} + 804q^{86} - 756q^{87} - 1164q^{88} + 366q^{89} - 594q^{90} + 112q^{91} - 2856q^{92} + 120q^{93} - 1920q^{94} - 1176q^{95} - 837q^{96} + 808q^{97} - 147q^{98} - 54q^{99} + O(q^{100}) \)

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:

Atkin-Lehner signs

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

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.4.a.c 2
3.b odd 2 1 63.4.a.e 2
4.b odd 2 1 336.4.a.m 2
5.b even 2 1 525.4.a.n 2
5.c odd 4 2 525.4.d.g 4
7.b odd 2 1 147.4.a.i 2
7.c even 3 2 147.4.e.l 4
7.d odd 6 2 147.4.e.m 4
8.b even 2 1 1344.4.a.bg 2
8.d odd 2 1 1344.4.a.bo 2
12.b even 2 1 1008.4.a.ba 2
15.d odd 2 1 1575.4.a.p 2
21.c even 2 1 441.4.a.r 2
21.g even 6 2 441.4.e.p 4
21.h odd 6 2 441.4.e.q 4
28.d even 2 1 2352.4.a.bz 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.c 2 1.a even 1 1 trivial
63.4.a.e 2 3.b odd 2 1
147.4.a.i 2 7.b odd 2 1
147.4.e.l 4 7.c even 3 2
147.4.e.m 4 7.d odd 6 2
336.4.a.m 2 4.b odd 2 1
441.4.a.r 2 21.c even 2 1
441.4.e.p 4 21.g even 6 2
441.4.e.q 4 21.h odd 6 2
525.4.a.n 2 5.b even 2 1
525.4.d.g 4 5.c odd 4 2
1008.4.a.ba 2 12.b even 2 1
1344.4.a.bg 2 8.b even 2 1
1344.4.a.bo 2 8.d odd 2 1
1575.4.a.p 2 15.d odd 2 1
2352.4.a.bz 2 28.d even 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T + 4 T^{2} + 24 T^{3} + 64 T^{4} \)
$3$ \( ( 1 - 3 T )^{2} \)
$5$ \( 1 - 6 T + 202 T^{2} - 750 T^{3} + 15625 T^{4} \)
$7$ \( ( 1 - 7 T )^{2} \)
$11$ \( 1 + 6 T + 1246 T^{2} + 7986 T^{3} + 1771561 T^{4} \)
$13$ \( 1 - 16 T + 2406 T^{2} - 35152 T^{3} + 4826809 T^{4} \)
$17$ \( 1 + 6 T + 9778 T^{2} + 29478 T^{3} + 24137569 T^{4} \)
$19$ \( 1 - 64 T + 6534 T^{2} - 438976 T^{3} + 47045881 T^{4} \)
$23$ \( 1 - 6 T + 7870 T^{2} - 73002 T^{3} + 148035889 T^{4} \)
$29$ \( 1 + 252 T + 56446 T^{2} + 6146028 T^{3} + 594823321 T^{4} \)
$31$ \( 1 - 40 T - 13890 T^{2} - 1191640 T^{3} + 887503681 T^{4} \)
$37$ \( 1 + 248 T + 98214 T^{2} + 12561944 T^{3} + 2565726409 T^{4} \)
$41$ \( 1 + 450 T + 175642 T^{2} + 31014450 T^{3} + 4750104241 T^{4} \)
$43$ \( 1 - 376 T + 161526 T^{2} - 29894632 T^{3} + 6321363049 T^{4} \)
$47$ \( 1 + 12 T + 141790 T^{2} + 1245876 T^{3} + 10779215329 T^{4} \)
$53$ \( 1 + 1104 T + 602230 T^{2} + 164360208 T^{3} + 22164361129 T^{4} \)
$59$ \( 1 - 804 T + 380614 T^{2} - 165124716 T^{3} + 42180533641 T^{4} \)
$61$ \( 1 + 428 T + 425886 T^{2} + 97147868 T^{3} + 51520374361 T^{4} \)
$67$ \( 1 - 148 T + 440790 T^{2} - 44512924 T^{3} + 90458382169 T^{4} \)
$71$ \( 1 - 954 T + 930526 T^{2} - 341447094 T^{3} + 128100283921 T^{4} \)
$73$ \( 1 - 1072 T + 1063278 T^{2} - 417026224 T^{3} + 151334226289 T^{4} \)
$79$ \( 1 + 572 T + 901662 T^{2} + 282018308 T^{3} + 243087455521 T^{4} \)
$83$ \( 1 - 1944 T + 1957030 T^{2} - 1111553928 T^{3} + 326940373369 T^{4} \)
$89$ \( 1 - 366 T + 1156090 T^{2} - 258018654 T^{3} + 496981290961 T^{4} \)
$97$ \( 1 - 808 T + 903054 T^{2} - 737439784 T^{3} + 832972004929 T^{4} \)
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