Properties

Label 126.4.g.e
Level $126$
Weight $4$
Character orbit 126.g
Analytic conductor $7.434$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 126.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.43424066072\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{193})\)
Defining polynomial: \(x^{4} - x^{3} + 49 x^{2} + 48 x + 2304\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 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.

\(f(q)\) \(=\) \( q -2 \beta_{2} q^{2} + ( -4 + 4 \beta_{2} ) q^{4} + ( \beta_{1} + 3 \beta_{2} ) q^{5} + ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{7} + 8 q^{8} +O(q^{10})\) \( q -2 \beta_{2} q^{2} + ( -4 + 4 \beta_{2} ) q^{4} + ( \beta_{1} + 3 \beta_{2} ) q^{5} + ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{7} + 8 q^{8} + ( 8 - 2 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} ) q^{10} + ( -16 + 7 \beta_{1} + 9 \beta_{2} + 7 \beta_{3} ) q^{11} + ( -27 + 5 \beta_{3} ) q^{13} + ( -6 + 6 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{14} -16 \beta_{2} q^{16} + ( 44 + 10 \beta_{1} - 54 \beta_{2} + 10 \beta_{3} ) q^{17} + ( 3 \beta_{1} + 58 \beta_{2} ) q^{19} + ( -16 + 4 \beta_{3} ) q^{20} + ( 32 - 14 \beta_{3} ) q^{22} + ( 14 \beta_{1} + 54 \beta_{2} ) q^{23} + ( 61 + 7 \beta_{1} - 68 \beta_{2} + 7 \beta_{3} ) q^{25} + ( 10 \beta_{1} + 44 \beta_{2} ) q^{26} + ( 4 - 4 \beta_{1} + 12 \beta_{2} - 12 \beta_{3} ) q^{28} + ( -40 + 7 \beta_{3} ) q^{29} + ( -63 + 28 \beta_{1} + 35 \beta_{2} + 28 \beta_{3} ) q^{31} + ( -32 + 32 \beta_{2} ) q^{32} + ( -88 - 20 \beta_{3} ) q^{34} + ( 108 - 10 \beta_{1} - 138 \beta_{2} - 9 \beta_{3} ) q^{35} + ( -21 \beta_{1} - 134 \beta_{2} ) q^{37} + ( 122 - 6 \beta_{1} - 116 \beta_{2} - 6 \beta_{3} ) q^{38} + ( 8 \beta_{1} + 24 \beta_{2} ) q^{40} -168 q^{41} + ( 143 + 21 \beta_{3} ) q^{43} + ( -28 \beta_{1} - 36 \beta_{2} ) q^{44} + ( 136 - 28 \beta_{1} - 108 \beta_{2} - 28 \beta_{3} ) q^{46} + ( -24 \beta_{1} + 348 \beta_{2} ) q^{47} + ( -149 + 2 \beta_{1} + 379 \beta_{2} + 13 \beta_{3} ) q^{49} + ( -122 - 14 \beta_{3} ) q^{50} + ( 108 - 20 \beta_{1} - 88 \beta_{2} - 20 \beta_{3} ) q^{52} + ( -156 - 63 \beta_{1} + 219 \beta_{2} - 63 \beta_{3} ) q^{53} + ( -400 + 37 \beta_{3} ) q^{55} + ( 16 - 16 \beta_{1} - 8 \beta_{2} + 8 \beta_{3} ) q^{56} + ( 14 \beta_{1} + 66 \beta_{2} ) q^{58} + ( 412 - 61 \beta_{1} - 351 \beta_{2} - 61 \beta_{3} ) q^{59} + ( -34 \beta_{1} + 220 \beta_{2} ) q^{61} + ( 126 - 56 \beta_{3} ) q^{62} + 64 q^{64} + ( -42 \beta_{1} - 306 \beta_{2} ) q^{65} + ( 503 + 35 \beta_{1} - 538 \beta_{2} + 35 \beta_{3} ) q^{67} + ( -40 \beta_{1} + 216 \beta_{2} ) q^{68} + ( -296 + 2 \beta_{1} + 78 \beta_{2} + 20 \beta_{3} ) q^{70} + ( -812 - 28 \beta_{3} ) q^{71} + ( 143 - 97 \beta_{1} - 46 \beta_{2} - 97 \beta_{3} ) q^{73} + ( -310 + 42 \beta_{1} + 268 \beta_{2} + 42 \beta_{3} ) q^{74} + ( -244 + 12 \beta_{3} ) q^{76} + ( 1024 + 5 \beta_{1} - 309 \beta_{2} - 34 \beta_{3} ) q^{77} + ( 98 \beta_{1} - 311 \beta_{2} ) q^{79} + ( 64 - 16 \beta_{1} - 48 \beta_{2} - 16 \beta_{3} ) q^{80} + 336 \beta_{2} q^{82} + ( -188 + 89 \beta_{3} ) q^{83} + ( -304 - 14 \beta_{3} ) q^{85} + ( 42 \beta_{1} - 328 \beta_{2} ) q^{86} + ( -128 + 56 \beta_{1} + 72 \beta_{2} + 56 \beta_{3} ) q^{88} + ( 54 \beta_{1} + 1170 \beta_{2} ) q^{89} + ( 186 + 59 \beta_{1} + 502 \beta_{2} - 12 \beta_{3} ) q^{91} + ( -272 + 56 \beta_{3} ) q^{92} + ( 648 + 48 \beta_{1} - 696 \beta_{2} + 48 \beta_{3} ) q^{94} + ( -388 + 70 \beta_{1} + 318 \beta_{2} + 70 \beta_{3} ) q^{95} + ( 46 - 155 \beta_{3} ) q^{97} + ( 762 + 22 \beta_{1} - 486 \beta_{2} - 4 \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} - 8q^{4} + 7q^{5} + 6q^{7} + 32q^{8} + O(q^{10}) \) \( 4q - 4q^{2} - 8q^{4} + 7q^{5} + 6q^{7} + 32q^{8} + 14q^{10} - 25q^{11} - 98q^{13} - 18q^{14} - 32q^{16} + 98q^{17} + 119q^{19} - 56q^{20} + 100q^{22} + 122q^{23} + 129q^{25} + 98q^{26} + 12q^{28} - 146q^{29} - 98q^{31} - 64q^{32} - 392q^{34} + 128q^{35} - 289q^{37} + 238q^{38} + 56q^{40} - 672q^{41} + 614q^{43} - 100q^{44} + 244q^{46} + 672q^{47} + 190q^{49} - 516q^{50} + 196q^{52} - 375q^{53} - 1526q^{55} + 48q^{56} + 146q^{58} + 763q^{59} + 406q^{61} + 392q^{62} + 256q^{64} - 654q^{65} + 1041q^{67} + 392q^{68} - 986q^{70} - 3304q^{71} + 189q^{73} - 578q^{74} - 952q^{76} + 3415q^{77} - 524q^{79} + 112q^{80} + 672q^{82} - 574q^{83} - 1244q^{85} - 614q^{86} - 200q^{88} + 2394q^{89} + 1783q^{91} - 976q^{92} + 1344q^{94} - 706q^{95} - 126q^{97} + 2090q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 49 x^{2} + 48 x + 2304\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 49 \nu^{2} - 49 \nu + 2304 \)\()/2352\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 97 \)\()/49\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 48 \beta_{2} + \beta_{1} - 49\)
\(\nu^{3}\)\(=\)\(49 \beta_{3} - 97\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/126\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\)
\(\chi(n)\) \(1\) \(-\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
−3.22311 + 5.58259i
3.72311 6.44862i
−3.22311 5.58259i
3.72311 + 6.44862i
−1.00000 + 1.73205i 0 −2.00000 3.46410i −1.72311 + 2.98452i 0 15.3924 10.2992i 8.00000 0 −3.44622 5.96903i
37.2 −1.00000 + 1.73205i 0 −2.00000 3.46410i 5.22311 9.04669i 0 −12.3924 + 13.7633i 8.00000 0 10.4462 + 18.0934i
109.1 −1.00000 1.73205i 0 −2.00000 + 3.46410i −1.72311 2.98452i 0 15.3924 + 10.2992i 8.00000 0 −3.44622 + 5.96903i
109.2 −1.00000 1.73205i 0 −2.00000 + 3.46410i 5.22311 + 9.04669i 0 −12.3924 13.7633i 8.00000 0 10.4462 18.0934i
\(n\): 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 126.4.g.e 4
3.b odd 2 1 126.4.g.f yes 4
7.b odd 2 1 882.4.g.z 4
7.c even 3 1 inner 126.4.g.e 4
7.c even 3 1 882.4.a.bd 2
7.d odd 6 1 882.4.a.bh 2
7.d odd 6 1 882.4.g.z 4
21.c even 2 1 882.4.g.bj 4
21.g even 6 1 882.4.a.u 2
21.g even 6 1 882.4.g.bj 4
21.h odd 6 1 126.4.g.f yes 4
21.h odd 6 1 882.4.a.ba 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.4.g.e 4 1.a even 1 1 trivial
126.4.g.e 4 7.c even 3 1 inner
126.4.g.f yes 4 3.b odd 2 1
126.4.g.f yes 4 21.h odd 6 1
882.4.a.u 2 21.g even 6 1
882.4.a.ba 2 21.h odd 6 1
882.4.a.bd 2 7.c even 3 1
882.4.a.bh 2 7.d odd 6 1
882.4.g.z 4 7.b odd 2 1
882.4.g.z 4 7.d odd 6 1
882.4.g.bj 4 21.c even 2 1
882.4.g.bj 4 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 7 T_{5}^{3} + 85 T_{5}^{2} + 252 T_{5} + 1296 \) acting on \(S_{4}^{\mathrm{new}}(126, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 + 2 T + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( 1296 + 252 T + 85 T^{2} - 7 T^{3} + T^{4} \)
$7$ \( 117649 - 2058 T - 77 T^{2} - 6 T^{3} + T^{4} \)
$11$ \( 4875264 - 55200 T + 2833 T^{2} + 25 T^{3} + T^{4} \)
$13$ \( ( -606 + 49 T + T^{2} )^{2} \)
$17$ \( 5875776 + 237552 T + 12028 T^{2} - 98 T^{3} + T^{4} \)
$19$ \( 9647236 - 369614 T + 11055 T^{2} - 119 T^{3} + T^{4} \)
$23$ \( 32901696 + 699792 T + 20620 T^{2} - 122 T^{3} + T^{4} \)
$29$ \( ( -1032 + 73 T + T^{2} )^{2} \)
$31$ \( 1255072329 - 3471846 T + 45031 T^{2} + 98 T^{3} + T^{4} \)
$37$ \( 158404 - 115022 T + 83919 T^{2} + 289 T^{3} + T^{4} \)
$41$ \( ( 168 + T )^{4} \)
$43$ \( ( 2284 - 307 T + T^{2} )^{2} \)
$47$ \( 7242690816 - 57189888 T + 366480 T^{2} - 672 T^{3} + T^{4} \)
$53$ \( 24444697104 - 58630500 T + 296973 T^{2} + 375 T^{3} + T^{4} \)
$59$ \( 1155728016 + 25938948 T + 616165 T^{2} - 763 T^{3} + T^{4} \)
$61$ \( 212226624 + 5914608 T + 179404 T^{2} - 406 T^{3} + T^{4} \)
$67$ \( 44865170596 - 220498374 T + 871867 T^{2} - 1041 T^{3} + T^{4} \)
$71$ \( ( 644448 + 1652 T + T^{2} )^{2} \)
$73$ \( 198073062916 + 84115206 T + 480775 T^{2} - 189 T^{3} + T^{4} \)
$79$ \( 155826773001 - 206848476 T + 669325 T^{2} + 524 T^{3} + T^{4} \)
$83$ \( ( -361596 + 287 T + T^{2} )^{2} \)
$89$ \( 1669553420544 - 3093316128 T + 4439124 T^{2} - 2394 T^{3} + T^{4} \)
$97$ \( ( -1158214 + 63 T + T^{2} )^{2} \)
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