Properties

Label 3.19.b.b
Level $3$
Weight $19$
Character orbit 3.b
Analytic conductor $6.162$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 19 \)
Character orbit: \([\chi]\) \(=\) 3.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.16158413129\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.601940665.1
Defining polynomial: \( x^{4} - x^{3} + 123x^{2} - 1744x + 16016 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{11}\cdot 3^{11} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 11 \beta_1 + 3969) q^{3} + (7 \beta_{3} + 7 \beta_{2} - 263384) q^{4} + ( - 9 \beta_{3} + 45 \beta_{2} + 706 \beta_1) q^{5} + ( - 297 \beta_{3} - 9 \beta_{2} + 1467 \beta_1 + 5674536) q^{6} + ( - 287 \beta_{3} - 287 \beta_{2} - 23936038) q^{7} + ( - 2016 \beta_{3} + 10080 \beta_{2} + 229552 \beta_1) q^{8} + (567 \beta_{3} + 13041 \beta_{2} - 294354 \beta_1 - 221335335) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + 11 \beta_1 + 3969) q^{3} + (7 \beta_{3} + 7 \beta_{2} - 263384) q^{4} + ( - 9 \beta_{3} + 45 \beta_{2} + 706 \beta_1) q^{5} + ( - 297 \beta_{3} - 9 \beta_{2} + 1467 \beta_1 + 5674536) q^{6} + ( - 287 \beta_{3} - 287 \beta_{2} - 23936038) q^{7} + ( - 2016 \beta_{3} + 10080 \beta_{2} + 229552 \beta_1) q^{8} + (567 \beta_{3} + 13041 \beta_{2} - 294354 \beta_1 - 221335335) q^{9} + ( - 14230 \beta_{3} - 14230 \beta_{2} + 365284080) q^{10} + (14427 \beta_{3} - 72135 \beta_{2} + 3151562 \beta_1) q^{11} + (11907 \beta_{3} - 156221 \beta_{2} - 10912336 \beta_1 + 1780792776) q^{12} + (271492 \beta_{3} + 271492 \beta_{2} - 1356555382) q^{13} + (82656 \beta_{3} - 413280 \beta_{2} + 14575246 \beta_1) q^{14} + (48573 \beta_{3} + 353115 \beta_{2} - 8111862 \beta_1 - 17071955280) q^{15} + ( - 1852368 \beta_{3} - 1852368 \beta_{2} + \cdots + 50306002048) q^{16}+ \cdots + (8675986498263 \beta_{3} + \cdots + 75\!\cdots\!40) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 15876 q^{3} - 1053536 q^{4} + 22698144 q^{6} - 95744152 q^{7} - 885341340 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 15876 q^{3} - 1053536 q^{4} + 22698144 q^{6} - 95744152 q^{7} - 885341340 q^{9} + 1461136320 q^{10} + 7123171104 q^{12} - 5426221528 q^{13} - 68287821120 q^{15} + 201224008192 q^{16} - 624067623360 q^{18} + 191416649480 q^{19} - 843499414296 q^{21} + 6661732766400 q^{22} - 16917300997632 q^{24} + 11407599454180 q^{25} - 4632207691356 q^{27} + 5750860980032 q^{28} - 17181499602240 q^{30} + 35728415085608 q^{31} + 12242871023040 q^{33} - 97283346838272 q^{34} + 412657454022048 q^{36} - 475299833502232 q^{37} + 416909545005096 q^{39} - 967003294602240 q^{40} + 149151729948480 q^{42} + 15\!\cdots\!92 q^{43}+ \cdots + 30\!\cdots\!60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 123x^{2} - 1744x + 16016 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -36\nu^{3} + 216\nu^{2} - 11592\nu + 59904 ) / 169 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 882\nu^{3} + 22086\nu^{2} + 201870\nu + 229788 ) / 169 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4770\nu^{3} + 1242\nu^{2} - 139662\nu + 5911308 ) / 169 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - 108\beta _1 + 1944 ) / 7776 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 17\beta_{2} + 284\beta _1 - 158760 ) / 2592 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -38\beta_{3} - 2\beta_{2} + 423\beta _1 + 1181952 ) / 972 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
−6.07949 12.9551i
6.57949 5.90892i
6.57949 + 5.90892i
−6.07949 + 12.9551i
932.767i −4234.02 + 19222.2i −607911. 1.14247e6i 1.79299e7 + 3.94936e6i −9.81043e6 3.22520e8i −3.51567e8 1.62775e8i 1.06566e9
2.2 425.442i 12172.0 15468.1i 81143.0 787628.i −6.58078e6 5.17849e6i −3.80616e7 1.46049e8i −9.11041e7 3.76556e8i −3.35090e8
2.3 425.442i 12172.0 + 15468.1i 81143.0 787628.i −6.58078e6 + 5.17849e6i −3.80616e7 1.46049e8i −9.11041e7 + 3.76556e8i −3.35090e8
2.4 932.767i −4234.02 19222.2i −607911. 1.14247e6i 1.79299e7 3.94936e6i −9.81043e6 3.22520e8i −3.51567e8 + 1.62775e8i 1.06566e9
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3.19.b.b 4
3.b odd 2 1 inner 3.19.b.b 4
4.b odd 2 1 48.19.e.b 4
12.b even 2 1 48.19.e.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.19.b.b 4 1.a even 1 1 trivial
3.19.b.b 4 3.b odd 2 1 inner
48.19.e.b 4 4.b odd 2 1
48.19.e.b 4 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 1051056T_{2}^{2} + 157480796160 \) acting on \(S_{19}^{\mathrm{new}}(3, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 1051056 T^{2} + \cdots + 157480796160 \) Copy content Toggle raw display
$3$ \( T^{4} - 15876 T^{3} + \cdots + 15\!\cdots\!21 \) Copy content Toggle raw display
$5$ \( T^{4} + 1925594804160 T^{2} + \cdots + 80\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{2} + 47872076 T + 373401093446500)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 44\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{2} + 2713110764 T - 17\!\cdots\!40)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 38\!\cdots\!40 \) Copy content Toggle raw display
$19$ \( (T^{2} - 95708324740 T - 79\!\cdots\!64)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 12\!\cdots\!60 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{2} - 17864207542804 T - 34\!\cdots\!96)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 237649916751116 T + 14\!\cdots\!40)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{2} - 799803662287396 T - 34\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 66\!\cdots\!40 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 14\!\cdots\!40 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{2} + \cdots - 11\!\cdots\!96)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + \cdots + 84\!\cdots\!80)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 58\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{2} + \cdots - 64\!\cdots\!60)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + \cdots + 11\!\cdots\!36)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 24\!\cdots\!40 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 66\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{2} + \cdots + 74\!\cdots\!20)^{2} \) Copy content Toggle raw display
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