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: \( x^{2} - x - 14 \) Copy content Toggle raw display
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.

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

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 \( T_{2}^{2} + 3T_{2} - 12 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(21))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 3T - 12 \) Copy content Toggle raw display
$3$ \( (T - 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 6T - 48 \) Copy content Toggle raw display
$7$ \( (T - 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 6T - 1416 \) Copy content Toggle raw display
$13$ \( T^{2} - 16T - 1988 \) Copy content Toggle raw display
$17$ \( T^{2} + 6T - 48 \) Copy content Toggle raw display
$19$ \( T^{2} - 64T - 7184 \) Copy content Toggle raw display
$23$ \( T^{2} - 6T - 16464 \) Copy content Toggle raw display
$29$ \( T^{2} + 252T + 7668 \) Copy content Toggle raw display
$31$ \( T^{2} - 40T - 73472 \) Copy content Toggle raw display
$37$ \( T^{2} + 248T - 3092 \) Copy content Toggle raw display
$41$ \( T^{2} + 450T + 37800 \) Copy content Toggle raw display
$43$ \( T^{2} - 376T + 2512 \) Copy content Toggle raw display
$47$ \( T^{2} + 12T - 65856 \) Copy content Toggle raw display
$53$ \( T^{2} + 1104 T + 304476 \) Copy content Toggle raw display
$59$ \( T^{2} - 804T - 30144 \) Copy content Toggle raw display
$61$ \( T^{2} + 428T - 28076 \) Copy content Toggle raw display
$67$ \( T^{2} - 148T - 160736 \) Copy content Toggle raw display
$71$ \( T^{2} - 954T + 214704 \) Copy content Toggle raw display
$73$ \( T^{2} - 1072 T + 285244 \) Copy content Toggle raw display
$79$ \( T^{2} + 572T - 84416 \) Copy content Toggle raw display
$83$ \( T^{2} - 1944 T + 813456 \) Copy content Toggle raw display
$89$ \( T^{2} - 366T - 253848 \) Copy content Toggle raw display
$97$ \( T^{2} - 808T - 922292 \) Copy content Toggle raw display
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