Properties

Label 42.4.e.b
Level $42$
Weight $4$
Character orbit 42.e
Analytic conductor $2.478$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 42.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.47808022024\)
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: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 2 - 2 \zeta_{6} ) q^{2} + 3 \zeta_{6} q^{3} -4 \zeta_{6} q^{4} + ( 15 - 15 \zeta_{6} ) q^{5} + 6 q^{6} + ( 14 + 7 \zeta_{6} ) q^{7} -8 q^{8} + ( -9 + 9 \zeta_{6} ) q^{9} +O(q^{10})\) \( q + ( 2 - 2 \zeta_{6} ) q^{2} + 3 \zeta_{6} q^{3} -4 \zeta_{6} q^{4} + ( 15 - 15 \zeta_{6} ) q^{5} + 6 q^{6} + ( 14 + 7 \zeta_{6} ) q^{7} -8 q^{8} + ( -9 + 9 \zeta_{6} ) q^{9} -30 \zeta_{6} q^{10} + 9 \zeta_{6} q^{11} + ( 12 - 12 \zeta_{6} ) q^{12} -88 q^{13} + ( 42 - 28 \zeta_{6} ) q^{14} + 45 q^{15} + ( -16 + 16 \zeta_{6} ) q^{16} + 84 \zeta_{6} q^{17} + 18 \zeta_{6} q^{18} + ( -104 + 104 \zeta_{6} ) q^{19} -60 q^{20} + ( -21 + 63 \zeta_{6} ) q^{21} + 18 q^{22} + ( 84 - 84 \zeta_{6} ) q^{23} -24 \zeta_{6} q^{24} -100 \zeta_{6} q^{25} + ( -176 + 176 \zeta_{6} ) q^{26} -27 q^{27} + ( 28 - 84 \zeta_{6} ) q^{28} + 51 q^{29} + ( 90 - 90 \zeta_{6} ) q^{30} -185 \zeta_{6} q^{31} + 32 \zeta_{6} q^{32} + ( -27 + 27 \zeta_{6} ) q^{33} + 168 q^{34} + ( 315 - 210 \zeta_{6} ) q^{35} + 36 q^{36} + ( -44 + 44 \zeta_{6} ) q^{37} + 208 \zeta_{6} q^{38} -264 \zeta_{6} q^{39} + ( -120 + 120 \zeta_{6} ) q^{40} -168 q^{41} + ( 84 + 42 \zeta_{6} ) q^{42} + 326 q^{43} + ( 36 - 36 \zeta_{6} ) q^{44} + 135 \zeta_{6} q^{45} -168 \zeta_{6} q^{46} + ( 138 - 138 \zeta_{6} ) q^{47} -48 q^{48} + ( 147 + 245 \zeta_{6} ) q^{49} -200 q^{50} + ( -252 + 252 \zeta_{6} ) q^{51} + 352 \zeta_{6} q^{52} -639 \zeta_{6} q^{53} + ( -54 + 54 \zeta_{6} ) q^{54} + 135 q^{55} + ( -112 - 56 \zeta_{6} ) q^{56} -312 q^{57} + ( 102 - 102 \zeta_{6} ) q^{58} -159 \zeta_{6} q^{59} -180 \zeta_{6} q^{60} + ( -722 + 722 \zeta_{6} ) q^{61} -370 q^{62} + ( -189 + 126 \zeta_{6} ) q^{63} + 64 q^{64} + ( -1320 + 1320 \zeta_{6} ) q^{65} + 54 \zeta_{6} q^{66} + 166 \zeta_{6} q^{67} + ( 336 - 336 \zeta_{6} ) q^{68} + 252 q^{69} + ( 210 - 630 \zeta_{6} ) q^{70} + 1086 q^{71} + ( 72 - 72 \zeta_{6} ) q^{72} -218 \zeta_{6} q^{73} + 88 \zeta_{6} q^{74} + ( 300 - 300 \zeta_{6} ) q^{75} + 416 q^{76} + ( -63 + 189 \zeta_{6} ) q^{77} -528 q^{78} + ( 583 - 583 \zeta_{6} ) q^{79} + 240 \zeta_{6} q^{80} -81 \zeta_{6} q^{81} + ( -336 + 336 \zeta_{6} ) q^{82} -597 q^{83} + ( 252 - 168 \zeta_{6} ) q^{84} + 1260 q^{85} + ( 652 - 652 \zeta_{6} ) q^{86} + 153 \zeta_{6} q^{87} -72 \zeta_{6} q^{88} + ( 1038 - 1038 \zeta_{6} ) q^{89} + 270 q^{90} + ( -1232 - 616 \zeta_{6} ) q^{91} -336 q^{92} + ( 555 - 555 \zeta_{6} ) q^{93} -276 \zeta_{6} q^{94} + 1560 \zeta_{6} q^{95} + ( -96 + 96 \zeta_{6} ) q^{96} -169 q^{97} + ( 784 - 294 \zeta_{6} ) q^{98} -81 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 3q^{3} - 4q^{4} + 15q^{5} + 12q^{6} + 35q^{7} - 16q^{8} - 9q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + 3q^{3} - 4q^{4} + 15q^{5} + 12q^{6} + 35q^{7} - 16q^{8} - 9q^{9} - 30q^{10} + 9q^{11} + 12q^{12} - 176q^{13} + 56q^{14} + 90q^{15} - 16q^{16} + 84q^{17} + 18q^{18} - 104q^{19} - 120q^{20} + 21q^{21} + 36q^{22} + 84q^{23} - 24q^{24} - 100q^{25} - 176q^{26} - 54q^{27} - 28q^{28} + 102q^{29} + 90q^{30} - 185q^{31} + 32q^{32} - 27q^{33} + 336q^{34} + 420q^{35} + 72q^{36} - 44q^{37} + 208q^{38} - 264q^{39} - 120q^{40} - 336q^{41} + 210q^{42} + 652q^{43} + 36q^{44} + 135q^{45} - 168q^{46} + 138q^{47} - 96q^{48} + 539q^{49} - 400q^{50} - 252q^{51} + 352q^{52} - 639q^{53} - 54q^{54} + 270q^{55} - 280q^{56} - 624q^{57} + 102q^{58} - 159q^{59} - 180q^{60} - 722q^{61} - 740q^{62} - 252q^{63} + 128q^{64} - 1320q^{65} + 54q^{66} + 166q^{67} + 336q^{68} + 504q^{69} - 210q^{70} + 2172q^{71} + 72q^{72} - 218q^{73} + 88q^{74} + 300q^{75} + 832q^{76} + 63q^{77} - 1056q^{78} + 583q^{79} + 240q^{80} - 81q^{81} - 336q^{82} - 1194q^{83} + 336q^{84} + 2520q^{85} + 652q^{86} + 153q^{87} - 72q^{88} + 1038q^{89} + 540q^{90} - 3080q^{91} - 672q^{92} + 555q^{93} - 276q^{94} + 1560q^{95} - 96q^{96} - 338q^{97} + 1274q^{98} - 162q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/42\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(31\)
\(\chi(n)\) \(1\) \(-1 + \zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
0.500000 0.866025i
0.500000 + 0.866025i
1.00000 + 1.73205i 1.50000 2.59808i −2.00000 + 3.46410i 7.50000 + 12.9904i 6.00000 17.5000 6.06218i −8.00000 −4.50000 7.79423i −15.0000 + 25.9808i
37.1 1.00000 1.73205i 1.50000 + 2.59808i −2.00000 3.46410i 7.50000 12.9904i 6.00000 17.5000 + 6.06218i −8.00000 −4.50000 + 7.79423i −15.0000 25.9808i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 42.4.e.b 2
3.b odd 2 1 126.4.g.a 2
4.b odd 2 1 336.4.q.d 2
7.b odd 2 1 294.4.e.e 2
7.c even 3 1 inner 42.4.e.b 2
7.c even 3 1 294.4.a.a 1
7.d odd 6 1 294.4.a.g 1
7.d odd 6 1 294.4.e.e 2
21.c even 2 1 882.4.g.l 2
21.g even 6 1 882.4.a.h 1
21.g even 6 1 882.4.g.l 2
21.h odd 6 1 126.4.g.a 2
21.h odd 6 1 882.4.a.r 1
28.f even 6 1 2352.4.a.q 1
28.g odd 6 1 336.4.q.d 2
28.g odd 6 1 2352.4.a.u 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.e.b 2 1.a even 1 1 trivial
42.4.e.b 2 7.c even 3 1 inner
126.4.g.a 2 3.b odd 2 1
126.4.g.a 2 21.h odd 6 1
294.4.a.a 1 7.c even 3 1
294.4.a.g 1 7.d odd 6 1
294.4.e.e 2 7.b odd 2 1
294.4.e.e 2 7.d odd 6 1
336.4.q.d 2 4.b odd 2 1
336.4.q.d 2 28.g odd 6 1
882.4.a.h 1 21.g even 6 1
882.4.a.r 1 21.h odd 6 1
882.4.g.l 2 21.c even 2 1
882.4.g.l 2 21.g even 6 1
2352.4.a.q 1 28.f even 6 1
2352.4.a.u 1 28.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 15 T_{5} + 225 \) acting on \(S_{4}^{\mathrm{new}}(42, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 - 2 T + T^{2} \)
$3$ \( 9 - 3 T + T^{2} \)
$5$ \( 225 - 15 T + T^{2} \)
$7$ \( 343 - 35 T + T^{2} \)
$11$ \( 81 - 9 T + T^{2} \)
$13$ \( ( 88 + T )^{2} \)
$17$ \( 7056 - 84 T + T^{2} \)
$19$ \( 10816 + 104 T + T^{2} \)
$23$ \( 7056 - 84 T + T^{2} \)
$29$ \( ( -51 + T )^{2} \)
$31$ \( 34225 + 185 T + T^{2} \)
$37$ \( 1936 + 44 T + T^{2} \)
$41$ \( ( 168 + T )^{2} \)
$43$ \( ( -326 + T )^{2} \)
$47$ \( 19044 - 138 T + T^{2} \)
$53$ \( 408321 + 639 T + T^{2} \)
$59$ \( 25281 + 159 T + T^{2} \)
$61$ \( 521284 + 722 T + T^{2} \)
$67$ \( 27556 - 166 T + T^{2} \)
$71$ \( ( -1086 + T )^{2} \)
$73$ \( 47524 + 218 T + T^{2} \)
$79$ \( 339889 - 583 T + T^{2} \)
$83$ \( ( 597 + T )^{2} \)
$89$ \( 1077444 - 1038 T + T^{2} \)
$97$ \( ( 169 + T )^{2} \)
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