Properties

Label 21.5.f.b
Level $21$
Weight $5$
Character orbit 21.f
Analytic conductor $2.171$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 21.f (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.17076922476\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 5 \zeta_{6} q^{2} + ( 3 + 3 \zeta_{6} ) q^{3} + ( -9 + 9 \zeta_{6} ) q^{4} + ( -2 + \zeta_{6} ) q^{5} + ( -15 + 30 \zeta_{6} ) q^{6} + ( -35 - 21 \zeta_{6} ) q^{7} + 35 q^{8} + 27 \zeta_{6} q^{9} +O(q^{10})\) \( q + 5 \zeta_{6} q^{2} + ( 3 + 3 \zeta_{6} ) q^{3} + ( -9 + 9 \zeta_{6} ) q^{4} + ( -2 + \zeta_{6} ) q^{5} + ( -15 + 30 \zeta_{6} ) q^{6} + ( -35 - 21 \zeta_{6} ) q^{7} + 35 q^{8} + 27 \zeta_{6} q^{9} + ( -5 - 5 \zeta_{6} ) q^{10} + ( 149 - 149 \zeta_{6} ) q^{11} + ( -54 + 27 \zeta_{6} ) q^{12} + ( 24 - 48 \zeta_{6} ) q^{13} + ( 105 - 280 \zeta_{6} ) q^{14} -9 q^{15} + 319 \zeta_{6} q^{16} + ( -154 - 154 \zeta_{6} ) q^{17} + ( -135 + 135 \zeta_{6} ) q^{18} + ( -412 + 206 \zeta_{6} ) q^{19} + ( 9 - 18 \zeta_{6} ) q^{20} + ( -42 - 231 \zeta_{6} ) q^{21} + 745 q^{22} + 560 \zeta_{6} q^{23} + ( 105 + 105 \zeta_{6} ) q^{24} + ( -622 + 622 \zeta_{6} ) q^{25} + ( 240 - 120 \zeta_{6} ) q^{26} + ( -81 + 162 \zeta_{6} ) q^{27} + ( 504 - 315 \zeta_{6} ) q^{28} + 235 q^{29} -45 \zeta_{6} q^{30} + ( -743 - 743 \zeta_{6} ) q^{31} + ( -1035 + 1035 \zeta_{6} ) q^{32} + ( 894 - 447 \zeta_{6} ) q^{33} + ( 770 - 1540 \zeta_{6} ) q^{34} + ( 91 - 14 \zeta_{6} ) q^{35} -243 q^{36} -1970 \zeta_{6} q^{37} + ( -1030 - 1030 \zeta_{6} ) q^{38} + ( 216 - 216 \zeta_{6} ) q^{39} + ( -70 + 35 \zeta_{6} ) q^{40} + ( -1626 + 3252 \zeta_{6} ) q^{41} + ( 1155 - 1365 \zeta_{6} ) q^{42} + 2798 q^{43} + 1341 \zeta_{6} q^{44} + ( -27 - 27 \zeta_{6} ) q^{45} + ( -2800 + 2800 \zeta_{6} ) q^{46} + ( 4768 - 2384 \zeta_{6} ) q^{47} + ( -957 + 1914 \zeta_{6} ) q^{48} + ( 784 + 1911 \zeta_{6} ) q^{49} -3110 q^{50} -1386 \zeta_{6} q^{51} + ( 216 + 216 \zeta_{6} ) q^{52} + ( -901 + 901 \zeta_{6} ) q^{53} + ( -810 + 405 \zeta_{6} ) q^{54} + ( -149 + 298 \zeta_{6} ) q^{55} + ( -1225 - 735 \zeta_{6} ) q^{56} -1854 q^{57} + 1175 \zeta_{6} q^{58} + ( 797 + 797 \zeta_{6} ) q^{59} + ( 81 - 81 \zeta_{6} ) q^{60} + ( -4120 + 2060 \zeta_{6} ) q^{61} + ( 3715 - 7430 \zeta_{6} ) q^{62} + ( 567 - 1512 \zeta_{6} ) q^{63} -71 q^{64} + 72 \zeta_{6} q^{65} + ( 2235 + 2235 \zeta_{6} ) q^{66} + ( 4156 - 4156 \zeta_{6} ) q^{67} + ( 2772 - 1386 \zeta_{6} ) q^{68} + ( -1680 + 3360 \zeta_{6} ) q^{69} + ( 70 + 385 \zeta_{6} ) q^{70} + 484 q^{71} + 945 \zeta_{6} q^{72} + ( -548 - 548 \zeta_{6} ) q^{73} + ( 9850 - 9850 \zeta_{6} ) q^{74} + ( -3732 + 1866 \zeta_{6} ) q^{75} + ( 1854 - 3708 \zeta_{6} ) q^{76} + ( -8344 + 5215 \zeta_{6} ) q^{77} + 1080 q^{78} -4325 \zeta_{6} q^{79} + ( -319 - 319 \zeta_{6} ) q^{80} + ( -729 + 729 \zeta_{6} ) q^{81} + ( -16260 + 8130 \zeta_{6} ) q^{82} + ( 753 - 1506 \zeta_{6} ) q^{83} + ( 2457 - 378 \zeta_{6} ) q^{84} + 462 q^{85} + 13990 \zeta_{6} q^{86} + ( 705 + 705 \zeta_{6} ) q^{87} + ( 5215 - 5215 \zeta_{6} ) q^{88} + ( 2284 - 1142 \zeta_{6} ) q^{89} + ( 135 - 270 \zeta_{6} ) q^{90} + ( -1848 + 2184 \zeta_{6} ) q^{91} -5040 q^{92} -6687 \zeta_{6} q^{93} + ( 11920 + 11920 \zeta_{6} ) q^{94} + ( 618 - 618 \zeta_{6} ) q^{95} + ( -6210 + 3105 \zeta_{6} ) q^{96} + ( -3147 + 6294 \zeta_{6} ) q^{97} + ( -9555 + 13475 \zeta_{6} ) q^{98} + 4023 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 5q^{2} + 9q^{3} - 9q^{4} - 3q^{5} - 91q^{7} + 70q^{8} + 27q^{9} + O(q^{10}) \) \( 2q + 5q^{2} + 9q^{3} - 9q^{4} - 3q^{5} - 91q^{7} + 70q^{8} + 27q^{9} - 15q^{10} + 149q^{11} - 81q^{12} - 70q^{14} - 18q^{15} + 319q^{16} - 462q^{17} - 135q^{18} - 618q^{19} - 315q^{21} + 1490q^{22} + 560q^{23} + 315q^{24} - 622q^{25} + 360q^{26} + 693q^{28} + 470q^{29} - 45q^{30} - 2229q^{31} - 1035q^{32} + 1341q^{33} + 168q^{35} - 486q^{36} - 1970q^{37} - 3090q^{38} + 216q^{39} - 105q^{40} + 945q^{42} + 5596q^{43} + 1341q^{44} - 81q^{45} - 2800q^{46} + 7152q^{47} + 3479q^{49} - 6220q^{50} - 1386q^{51} + 648q^{52} - 901q^{53} - 1215q^{54} - 3185q^{56} - 3708q^{57} + 1175q^{58} + 2391q^{59} + 81q^{60} - 6180q^{61} - 378q^{63} - 142q^{64} + 72q^{65} + 6705q^{66} + 4156q^{67} + 4158q^{68} + 525q^{70} + 968q^{71} + 945q^{72} - 1644q^{73} + 9850q^{74} - 5598q^{75} - 11473q^{77} + 2160q^{78} - 4325q^{79} - 957q^{80} - 729q^{81} - 24390q^{82} + 4536q^{84} + 924q^{85} + 13990q^{86} + 2115q^{87} + 5215q^{88} + 3426q^{89} - 1512q^{91} - 10080q^{92} - 6687q^{93} + 35760q^{94} + 618q^{95} - 9315q^{96} - 5635q^{98} + 8046q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(8\) \(10\)
\(\chi(n)\) \(1\) \(\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
0.500000 + 0.866025i
0.500000 0.866025i
2.50000 + 4.33013i 4.50000 + 2.59808i −4.50000 + 7.79423i −1.50000 + 0.866025i 25.9808i −45.5000 18.1865i 35.0000 13.5000 + 23.3827i −7.50000 4.33013i
19.1 2.50000 4.33013i 4.50000 2.59808i −4.50000 7.79423i −1.50000 0.866025i 25.9808i −45.5000 + 18.1865i 35.0000 13.5000 23.3827i −7.50000 + 4.33013i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.5.f.b 2
3.b odd 2 1 63.5.m.a 2
4.b odd 2 1 336.5.bh.a 2
7.b odd 2 1 147.5.f.b 2
7.c even 3 1 147.5.d.a 2
7.c even 3 1 147.5.f.b 2
7.d odd 6 1 inner 21.5.f.b 2
7.d odd 6 1 147.5.d.a 2
21.g even 6 1 63.5.m.a 2
21.g even 6 1 441.5.d.c 2
21.h odd 6 1 441.5.d.c 2
28.f even 6 1 336.5.bh.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.5.f.b 2 1.a even 1 1 trivial
21.5.f.b 2 7.d odd 6 1 inner
63.5.m.a 2 3.b odd 2 1
63.5.m.a 2 21.g even 6 1
147.5.d.a 2 7.c even 3 1
147.5.d.a 2 7.d odd 6 1
147.5.f.b 2 7.b odd 2 1
147.5.f.b 2 7.c even 3 1
336.5.bh.a 2 4.b odd 2 1
336.5.bh.a 2 28.f even 6 1
441.5.d.c 2 21.g even 6 1
441.5.d.c 2 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 5 T_{2} + 25 \) acting on \(S_{5}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 25 - 5 T + T^{2} \)
$3$ \( 27 - 9 T + T^{2} \)
$5$ \( 3 + 3 T + T^{2} \)
$7$ \( 2401 + 91 T + T^{2} \)
$11$ \( 22201 - 149 T + T^{2} \)
$13$ \( 1728 + T^{2} \)
$17$ \( 71148 + 462 T + T^{2} \)
$19$ \( 127308 + 618 T + T^{2} \)
$23$ \( 313600 - 560 T + T^{2} \)
$29$ \( ( -235 + T )^{2} \)
$31$ \( 1656147 + 2229 T + T^{2} \)
$37$ \( 3880900 + 1970 T + T^{2} \)
$41$ \( 7931628 + T^{2} \)
$43$ \( ( -2798 + T )^{2} \)
$47$ \( 17050368 - 7152 T + T^{2} \)
$53$ \( 811801 + 901 T + T^{2} \)
$59$ \( 1905627 - 2391 T + T^{2} \)
$61$ \( 12730800 + 6180 T + T^{2} \)
$67$ \( 17272336 - 4156 T + T^{2} \)
$71$ \( ( -484 + T )^{2} \)
$73$ \( 900912 + 1644 T + T^{2} \)
$79$ \( 18705625 + 4325 T + T^{2} \)
$83$ \( 1701027 + T^{2} \)
$89$ \( 3912492 - 3426 T + T^{2} \)
$97$ \( 29710827 + T^{2} \)
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