Properties

Label 42.5.g.a
Level $42$
Weight $5$
Character orbit 42.g
Analytic conductor $4.342$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 42.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.34153844952\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( -6 - 3 \beta_{2} ) q^{3} + 8 \beta_{2} q^{4} + ( -11 + \beta_{1} + 11 \beta_{2} + 2 \beta_{3} ) q^{5} + ( -6 \beta_{1} - 3 \beta_{3} ) q^{6} + ( -7 \beta_{1} + 35 \beta_{2} + 7 \beta_{3} ) q^{7} + 8 \beta_{3} q^{8} + ( 27 + 27 \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( -6 - 3 \beta_{2} ) q^{3} + 8 \beta_{2} q^{4} + ( -11 + \beta_{1} + 11 \beta_{2} + 2 \beta_{3} ) q^{5} + ( -6 \beta_{1} - 3 \beta_{3} ) q^{6} + ( -7 \beta_{1} + 35 \beta_{2} + 7 \beta_{3} ) q^{7} + 8 \beta_{3} q^{8} + ( 27 + 27 \beta_{2} ) q^{9} + ( -16 - 11 \beta_{1} - 8 \beta_{2} + 11 \beta_{3} ) q^{10} + ( 6 \beta_{1} + 81 \beta_{2} + 6 \beta_{3} ) q^{11} + ( 24 - 24 \beta_{2} ) q^{12} + ( 36 - 68 \beta_{1} + 72 \beta_{2} - 34 \beta_{3} ) q^{13} + ( -56 - 112 \beta_{2} + 35 \beta_{3} ) q^{14} + ( 99 - 9 \beta_{3} ) q^{15} + ( -64 - 64 \beta_{2} ) q^{16} + ( -68 + 34 \beta_{1} - 34 \beta_{2} - 34 \beta_{3} ) q^{17} + ( 27 \beta_{1} + 27 \beta_{3} ) q^{18} + ( -74 - 34 \beta_{1} + 74 \beta_{2} - 68 \beta_{3} ) q^{19} + ( -88 - 16 \beta_{1} - 176 \beta_{2} - 8 \beta_{3} ) q^{20} + ( 105 + 63 \beta_{1} - 105 \beta_{2} ) q^{21} + ( -48 + 81 \beta_{3} ) q^{22} + ( 156 + 306 \beta_{1} + 156 \beta_{2} ) q^{23} + ( 24 \beta_{1} - 24 \beta_{3} ) q^{24} + ( -66 \beta_{1} + 238 \beta_{2} - 66 \beta_{3} ) q^{25} + ( 272 + 36 \beta_{1} - 272 \beta_{2} + 72 \beta_{3} ) q^{26} + ( -81 - 162 \beta_{2} ) q^{27} + ( -280 - 56 \beta_{1} - 280 \beta_{2} - 112 \beta_{3} ) q^{28} + ( 681 + 21 \beta_{3} ) q^{29} + ( 72 + 99 \beta_{1} + 72 \beta_{2} ) q^{30} + ( -1262 - 87 \beta_{1} - 631 \beta_{2} + 87 \beta_{3} ) q^{31} + ( -64 \beta_{1} - 64 \beta_{3} ) q^{32} + ( 243 - 18 \beta_{1} - 243 \beta_{2} - 36 \beta_{3} ) q^{33} + ( 272 - 68 \beta_{1} + 544 \beta_{2} - 34 \beta_{3} ) q^{34} + ( -217 - 70 \beta_{1} - 770 \beta_{2} - 266 \beta_{3} ) q^{35} -216 q^{36} + ( 698 - 48 \beta_{1} + 698 \beta_{2} ) q^{37} + ( 544 - 74 \beta_{1} + 272 \beta_{2} + 74 \beta_{3} ) q^{38} + ( 306 \beta_{1} - 324 \beta_{2} + 306 \beta_{3} ) q^{39} + ( 64 - 88 \beta_{1} - 64 \beta_{2} - 176 \beta_{3} ) q^{40} + ( 518 - 784 \beta_{1} + 1036 \beta_{2} - 392 \beta_{3} ) q^{41} + ( 105 \beta_{1} + 504 \beta_{2} - 105 \beta_{3} ) q^{42} + ( -158 + 1140 \beta_{3} ) q^{43} + ( -648 - 48 \beta_{1} - 648 \beta_{2} ) q^{44} + ( -594 - 27 \beta_{1} - 297 \beta_{2} + 27 \beta_{3} ) q^{45} + ( 156 \beta_{1} + 2448 \beta_{2} + 156 \beta_{3} ) q^{46} + ( -1316 + 340 \beta_{1} + 1316 \beta_{2} + 680 \beta_{3} ) q^{47} + ( 192 + 384 \beta_{2} ) q^{48} + ( -49 - 490 \beta_{1} - 49 \beta_{2} - 980 \beta_{3} ) q^{49} + ( 528 + 238 \beta_{3} ) q^{50} + ( 306 - 306 \beta_{1} + 306 \beta_{2} ) q^{51} + ( -576 + 272 \beta_{1} - 288 \beta_{2} - 272 \beta_{3} ) q^{52} + ( 591 \beta_{1} + 519 \beta_{2} + 591 \beta_{3} ) q^{53} + ( -81 \beta_{1} - 162 \beta_{3} ) q^{54} + ( -939 - 294 \beta_{1} - 1878 \beta_{2} - 147 \beta_{3} ) q^{55} + ( 896 - 280 \beta_{1} + 448 \beta_{2} - 280 \beta_{3} ) q^{56} + ( 666 + 306 \beta_{3} ) q^{57} + ( -168 + 681 \beta_{1} - 168 \beta_{2} ) q^{58} + ( -322 + 656 \beta_{1} - 161 \beta_{2} - 656 \beta_{3} ) q^{59} + ( 72 \beta_{1} + 792 \beta_{2} + 72 \beta_{3} ) q^{60} + ( 848 + 164 \beta_{1} - 848 \beta_{2} + 328 \beta_{3} ) q^{61} + ( -696 - 1262 \beta_{1} - 1392 \beta_{2} - 631 \beta_{3} ) q^{62} + ( -945 - 378 \beta_{1} - 189 \beta_{3} ) q^{63} + 512 q^{64} + ( -372 + 1014 \beta_{1} - 372 \beta_{2} ) q^{65} + ( 288 + 243 \beta_{1} + 144 \beta_{2} - 243 \beta_{3} ) q^{66} + ( 486 \beta_{1} - 7300 \beta_{2} + 486 \beta_{3} ) q^{67} + ( 272 + 272 \beta_{1} - 272 \beta_{2} + 544 \beta_{3} ) q^{68} + ( -468 - 1836 \beta_{1} - 936 \beta_{2} - 918 \beta_{3} ) q^{69} + ( 2128 - 217 \beta_{1} + 1568 \beta_{2} - 770 \beta_{3} ) q^{70} + ( -2424 + 732 \beta_{3} ) q^{71} -216 \beta_{1} q^{72} + ( 7528 + 500 \beta_{1} + 3764 \beta_{2} - 500 \beta_{3} ) q^{73} + ( 698 \beta_{1} - 384 \beta_{2} + 698 \beta_{3} ) q^{74} + ( 714 + 198 \beta_{1} - 714 \beta_{2} + 396 \beta_{3} ) q^{75} + ( -592 + 544 \beta_{1} - 1184 \beta_{2} + 272 \beta_{3} ) q^{76} + ( -2499 - 777 \beta_{1} - 3171 \beta_{2} - 1134 \beta_{3} ) q^{77} + ( -2448 - 324 \beta_{3} ) q^{78} + ( 1987 + 2991 \beta_{1} + 1987 \beta_{2} ) q^{79} + ( 1408 + 64 \beta_{1} + 704 \beta_{2} - 64 \beta_{3} ) q^{80} + 729 \beta_{2} q^{81} + ( 3136 + 518 \beta_{1} - 3136 \beta_{2} + 1036 \beta_{3} ) q^{82} + ( 2741 - 1420 \beta_{1} + 5482 \beta_{2} - 710 \beta_{3} ) q^{83} + ( 840 + 1680 \beta_{2} + 504 \beta_{3} ) q^{84} + ( 306 + 1020 \beta_{3} ) q^{85} + ( -9120 - 158 \beta_{1} - 9120 \beta_{2} ) q^{86} + ( -4086 + 63 \beta_{1} - 2043 \beta_{2} - 63 \beta_{3} ) q^{87} + ( -648 \beta_{1} - 384 \beta_{2} - 648 \beta_{3} ) q^{88} + ( -5526 - 1218 \beta_{1} + 5526 \beta_{2} - 2436 \beta_{3} ) q^{89} + ( -216 - 594 \beta_{1} - 432 \beta_{2} - 297 \beta_{3} ) q^{90} + ( -2520 + 434 \beta_{1} + 4452 \beta_{2} - 1946 \beta_{3} ) q^{91} + ( -1248 + 2448 \beta_{3} ) q^{92} + ( 5679 + 783 \beta_{1} + 5679 \beta_{2} ) q^{93} + ( -5440 - 1316 \beta_{1} - 2720 \beta_{2} + 1316 \beta_{3} ) q^{94} + ( 900 \beta_{1} - 1626 \beta_{2} + 900 \beta_{3} ) q^{95} + ( 192 \beta_{1} + 384 \beta_{3} ) q^{96} + ( -1013 - 80 \beta_{1} - 2026 \beta_{2} - 40 \beta_{3} ) q^{97} + ( 7840 - 49 \beta_{1} + 3920 \beta_{2} - 49 \beta_{3} ) q^{98} + ( -2187 + 162 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 18q^{3} - 16q^{4} - 66q^{5} - 70q^{7} + 54q^{9} + O(q^{10}) \) \( 4q - 18q^{3} - 16q^{4} - 66q^{5} - 70q^{7} + 54q^{9} - 48q^{10} - 162q^{11} + 144q^{12} + 396q^{15} - 128q^{16} - 204q^{17} - 444q^{19} + 630q^{21} - 192q^{22} + 312q^{23} - 476q^{25} + 1632q^{26} - 560q^{28} + 2724q^{29} + 144q^{30} - 3786q^{31} + 1458q^{33} + 672q^{35} - 864q^{36} + 1396q^{37} + 1632q^{38} + 648q^{39} + 384q^{40} - 1008q^{42} - 632q^{43} - 1296q^{44} - 1782q^{45} - 4896q^{46} - 7896q^{47} - 98q^{49} + 2112q^{50} + 612q^{51} - 1728q^{52} - 1038q^{53} + 2688q^{56} + 2664q^{57} - 336q^{58} - 966q^{59} - 1584q^{60} + 5088q^{61} - 3780q^{63} + 2048q^{64} - 744q^{65} + 864q^{66} + 14600q^{67} + 1632q^{68} + 5376q^{70} - 9696q^{71} + 22584q^{73} + 768q^{74} + 4284q^{75} - 3654q^{77} - 9792q^{78} + 3974q^{79} + 4224q^{80} - 1458q^{81} + 18816q^{82} + 1224q^{85} - 18240q^{86} - 12258q^{87} + 768q^{88} - 33156q^{89} - 18984q^{91} - 4992q^{92} + 11358q^{93} - 16320q^{94} + 3252q^{95} + 23520q^{98} - 8748q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(\beta_{3}\)

Character values

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

\(n\) \(29\) \(31\)
\(\chi(n)\) \(1\) \(1 + \beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−0.707107 + 1.22474i
0.707107 1.22474i
−0.707107 1.22474i
0.707107 + 1.22474i
−1.41421 + 2.44949i −4.50000 + 2.59808i −4.00000 6.92820i −12.2574 7.07679i 14.6969i 12.1985 47.4573i 22.6274 13.5000 23.3827i 34.6690 20.0162i
19.2 1.41421 2.44949i −4.50000 + 2.59808i −4.00000 6.92820i −20.7426 11.9758i 14.6969i −47.1985 13.1645i −22.6274 13.5000 23.3827i −58.6690 + 33.8726i
31.1 −1.41421 2.44949i −4.50000 2.59808i −4.00000 + 6.92820i −12.2574 + 7.07679i 14.6969i 12.1985 + 47.4573i 22.6274 13.5000 + 23.3827i 34.6690 + 20.0162i
31.2 1.41421 + 2.44949i −4.50000 2.59808i −4.00000 + 6.92820i −20.7426 + 11.9758i 14.6969i −47.1985 + 13.1645i −22.6274 13.5000 + 23.3827i −58.6690 33.8726i
\(n\): 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 42.5.g.a 4
3.b odd 2 1 126.5.n.b 4
4.b odd 2 1 336.5.bh.d 4
7.b odd 2 1 294.5.g.c 4
7.c even 3 1 294.5.c.a 4
7.c even 3 1 294.5.g.c 4
7.d odd 6 1 inner 42.5.g.a 4
7.d odd 6 1 294.5.c.a 4
21.g even 6 1 126.5.n.b 4
21.g even 6 1 882.5.c.a 4
21.h odd 6 1 882.5.c.a 4
28.f even 6 1 336.5.bh.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.5.g.a 4 1.a even 1 1 trivial
42.5.g.a 4 7.d odd 6 1 inner
126.5.n.b 4 3.b odd 2 1
126.5.n.b 4 21.g even 6 1
294.5.c.a 4 7.c even 3 1
294.5.c.a 4 7.d odd 6 1
294.5.g.c 4 7.b odd 2 1
294.5.g.c 4 7.c even 3 1
336.5.bh.d 4 4.b odd 2 1
336.5.bh.d 4 28.f even 6 1
882.5.c.a 4 21.g even 6 1
882.5.c.a 4 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 66 T_{5}^{3} + 1791 T_{5}^{2} + 22374 T_{5} + 114921 \) acting on \(S_{5}^{\mathrm{new}}(42, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 64 + 8 T^{2} + T^{4} \)
$3$ \( ( 27 + 9 T + T^{2} )^{2} \)
$5$ \( 114921 + 22374 T + 1791 T^{2} + 66 T^{3} + T^{4} \)
$7$ \( 5764801 + 168070 T + 2499 T^{2} + 70 T^{3} + T^{4} \)
$11$ \( 39350529 + 1016226 T + 19971 T^{2} + 162 T^{3} + T^{4} \)
$13$ \( 569108736 + 63264 T^{2} + T^{4} \)
$17$ \( 589324176 - 4952304 T - 10404 T^{2} + 204 T^{3} + T^{4} \)
$19$ \( 128051856 - 5024304 T + 54396 T^{2} + 444 T^{3} + T^{4} \)
$23$ \( 525265461504 + 226122624 T + 822096 T^{2} - 312 T^{3} + T^{4} \)
$29$ \( ( 460233 - 1362 T + T^{2} )^{2} \)
$31$ \( 1025818531929 + 3834563022 T + 5790759 T^{2} + 3786 T^{3} + T^{4} \)
$37$ \( 219747187984 - 654405712 T + 1480044 T^{2} - 1396 T^{3} + T^{4} \)
$41$ \( 8311481425296 + 8985816 T^{2} + T^{4} \)
$43$ \( ( -10371836 + 316 T + T^{2} )^{2} \)
$47$ \( 5862054484224 + 19117542528 T + 23203440 T^{2} + 7896 T^{3} + T^{4} \)
$53$ \( 6375054362769 - 2620832706 T + 3602331 T^{2} + 1038 T^{3} + T^{4} \)
$59$ \( 105068670590601 - 9901790766 T - 9939249 T^{2} + 966 T^{3} + T^{4} \)
$61$ \( 2285563428864 - 7692079104 T + 10141056 T^{2} - 5088 T^{3} + T^{4} \)
$67$ \( 2642004409786624 - 750446307200 T + 161759568 T^{2} - 14600 T^{3} + T^{4} \)
$71$ \( ( 1589184 + 4848 T + T^{2} )^{2} \)
$73$ \( 1332475433535744 - 824385739392 T + 206515440 T^{2} - 22584 T^{3} + T^{4} \)
$79$ \( 4572529180189441 + 268723783546 T + 83413155 T^{2} - 3974 T^{3} + T^{4} \)
$83$ \( 109011202550649 + 69275286 T^{2} + T^{4} \)
$89$ \( 3136610653724304 + 1856916766512 T + 422445564 T^{2} + 33156 T^{3} + T^{4} \)
$97$ \( 9242250571449 + 6233814 T^{2} + T^{4} \)
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