Properties

Label 684.3.g.b
Level $684$
Weight $3$
Character orbit 684.g
Analytic conductor $18.638$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 684.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.6376500822\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \( x^{14} - 2 x^{13} + x^{12} + 14 x^{11} - 42 x^{10} + 28 x^{9} + 132 x^{8} - 440 x^{7} + 528 x^{6} + 448 x^{5} - 2688 x^{4} + 3584 x^{3} + 1024 x^{2} - 8192 x + 16384 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 76)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{2} q^{4} - \beta_{10} q^{5} + ( - \beta_{11} + \beta_{5} + \beta_{3}) q^{7} + ( - \beta_{12} + \beta_{10} - \beta_{6} - \beta_{3} + 3) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{2} q^{4} - \beta_{10} q^{5} + ( - \beta_{11} + \beta_{5} + \beta_{3}) q^{7} + ( - \beta_{12} + \beta_{10} - \beta_{6} - \beta_{3} + 3) q^{8} + ( - \beta_{9} + \beta_{6} - 1) q^{10} + (\beta_{13} + \beta_{12} - \beta_{10} - \beta_{8} + \beta_{5} + \beta_{3} + \beta_1 - 1) q^{11} + ( - \beta_{11} - \beta_{10} - \beta_{5} - 2 \beta_{2} + 5) q^{13} + ( - \beta_{12} + 2 \beta_{11} - \beta_{10} - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{2} + \cdots - 3) q^{14}+ \cdots + ( - 4 \beta_{12} + 8 \beta_{10} + 4 \beta_{9} - 4 \beta_{8} - 8 \beta_{7} + 8 \beta_{6} + \cdots - 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 2 q^{2} + 2 q^{4} + 40 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 2 q^{2} + 2 q^{4} + 40 q^{8} - 12 q^{10} + 54 q^{13} - 30 q^{14} + 58 q^{16} - 34 q^{17} - 32 q^{20} + 36 q^{22} - 86 q^{25} + 16 q^{26} + 18 q^{28} - 54 q^{29} - 72 q^{32} - 82 q^{34} + 100 q^{37} - 148 q^{40} - 224 q^{41} + 96 q^{44} + 46 q^{46} - 220 q^{49} + 58 q^{50} - 288 q^{52} - 14 q^{53} - 12 q^{56} - 72 q^{58} + 28 q^{61} - 396 q^{62} - 118 q^{64} + 472 q^{65} - 30 q^{68} + 156 q^{70} + 70 q^{73} + 224 q^{74} - 228 q^{77} + 348 q^{80} - 400 q^{82} + 48 q^{85} + 124 q^{86} + 472 q^{88} - 126 q^{92} - 88 q^{94} + 308 q^{97} - 68 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 2 x^{13} + x^{12} + 14 x^{11} - 42 x^{10} + 28 x^{9} + 132 x^{8} - 440 x^{7} + 528 x^{6} + 448 x^{5} - 2688 x^{4} + 3584 x^{3} + 1024 x^{2} - 8192 x + 16384 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{13} + 14 \nu^{12} - 9 \nu^{11} + 94 \nu^{10} - 286 \nu^{9} + 844 \nu^{8} - 980 \nu^{7} - 1880 \nu^{6} + 8592 \nu^{5} - 8832 \nu^{4} - 3968 \nu^{3} + 59392 \nu^{2} - 119808 \nu + 114688 ) / 20480 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3 \nu^{13} - 17 \nu^{12} + 17 \nu^{11} + 63 \nu^{10} - 352 \nu^{9} + 578 \nu^{8} - 280 \nu^{7} - 2260 \nu^{6} + 6264 \nu^{5} - 5744 \nu^{4} - 5696 \nu^{3} + 32384 \nu^{2} - 47616 \nu + 24576 ) / 5120 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - \nu^{13} - 166 \nu^{12} - 129 \nu^{11} + 1194 \nu^{10} - 2006 \nu^{9} - 556 \nu^{8} + 12860 \nu^{7} - 29480 \nu^{6} + 6512 \nu^{5} + 83008 \nu^{4} - 199808 \nu^{3} + \cdots - 991232 ) / 20480 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 21 \nu^{13} + 6 \nu^{12} - 91 \nu^{11} + 246 \nu^{10} - 354 \nu^{9} - 244 \nu^{8} + 2100 \nu^{7} - 1560 \nu^{6} - 112 \nu^{5} + 3392 \nu^{4} - 10112 \nu^{3} + 4608 \nu^{2} - 13312 \nu + 172032 ) / 20480 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - \nu^{13} + 3 \nu^{11} - 16 \nu^{10} + 14 \nu^{9} + 56 \nu^{8} - 188 \nu^{7} + 176 \nu^{6} + 352 \nu^{5} - 1504 \nu^{4} + 1792 \nu^{3} + 768 \nu^{2} - 6144 \nu + 6144 ) / 1024 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - \nu^{13} + 2 \nu^{12} - \nu^{11} - 14 \nu^{10} + 42 \nu^{9} - 28 \nu^{8} - 132 \nu^{7} + 440 \nu^{6} - 528 \nu^{5} - 448 \nu^{4} + 2688 \nu^{3} - 2560 \nu^{2} - 1024 \nu + 7168 ) / 1024 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 39 \nu^{13} - 34 \nu^{12} - 151 \nu^{11} + 526 \nu^{10} - 394 \nu^{9} - 1604 \nu^{8} + 8260 \nu^{7} - 14200 \nu^{6} - 3632 \nu^{5} + 71232 \nu^{4} - 130432 \nu^{3} + \cdots - 831488 ) / 20480 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3 \nu^{13} + 9 \nu^{12} - \nu^{11} - 39 \nu^{10} + 112 \nu^{9} - 82 \nu^{8} - 328 \nu^{7} + 1012 \nu^{6} - 952 \nu^{5} - 1168 \nu^{4} + 4672 \nu^{3} - 4736 \nu^{2} - 7680 \nu + 12288 ) / 1024 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 19 \nu^{13} + 9 \nu^{12} - 139 \nu^{11} + 249 \nu^{10} + 84 \nu^{9} - 1506 \nu^{8} + 3600 \nu^{7} - 1260 \nu^{6} - 10328 \nu^{5} + 24368 \nu^{4} - 20928 \nu^{3} - 39808 \nu^{2} + \cdots - 13312 ) / 5120 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - \nu^{13} + 2 \nu^{12} + \nu^{11} - 14 \nu^{10} + 36 \nu^{9} - 28 \nu^{8} - 96 \nu^{7} + 296 \nu^{6} - 344 \nu^{5} - 224 \nu^{4} + 1600 \nu^{3} - 1984 \nu^{2} - 256 \nu + 256 ) / 256 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 19 \nu^{13} + 11 \nu^{12} + 139 \nu^{11} - 389 \nu^{10} + 396 \nu^{9} + 1146 \nu^{8} - 4560 \nu^{7} + 5820 \nu^{6} + 5848 \nu^{5} - 29168 \nu^{4} + 47808 \nu^{3} - 2432 \nu^{2} + \cdots + 95232 ) / 5120 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{12} - \beta_{10} + \beta_{6} + \beta_{3} - 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{3} - \beta_{2} - 3\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{13} + 2 \beta_{11} + 2 \beta_{9} - \beta_{8} - \beta_{7} - \beta_{5} + 3 \beta_{4} - \beta_{3} - 3 \beta_{2} + 4 \beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3 \beta_{13} - 7 \beta_{12} + \beta_{11} + 5 \beta_{10} + 4 \beta_{9} + 5 \beta_{8} - 5 \beta_{7} - 2 \beta_{5} + 3 \beta_{4} - 10 \beta_{3} + \beta_{2} + 7 \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2 \beta_{13} + 6 \beta_{12} - 2 \beta_{11} - 6 \beta_{10} + 2 \beta_{9} - 7 \beta_{8} + 5 \beta_{7} + 26 \beta_{6} + \beta_{5} - 7 \beta_{4} - \beta_{3} + 7 \beta_{2} - 8 \beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 7 \beta_{13} - 5 \beta_{12} - 13 \beta_{11} - 15 \beta_{10} + 2 \beta_{9} + 13 \beta_{8} + 11 \beta_{7} + 10 \beta_{6} - 4 \beta_{5} - 39 \beta_{4} + 32 \beta_{3} - 15 \beta_{2} + 3 \beta _1 - 64 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 26 \beta_{13} - 2 \beta_{12} + 30 \beta_{11} + 2 \beta_{10} - 14 \beta_{9} - 27 \beta_{8} + 25 \beta_{7} - 30 \beta_{6} + 21 \beta_{5} - 19 \beta_{4} + 51 \beta_{3} + 35 \beta_{2} - 88 \beta _1 + 117 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 69 \beta_{13} - 33 \beta_{12} + 55 \beta_{11} + 69 \beta_{10} - 46 \beta_{9} + 41 \beta_{8} - 81 \beta_{7} - 22 \beta_{6} + 76 \beta_{5} + 45 \beta_{4} + 32 \beta_{3} - 59 \beta_{2} + 191 \beta _1 - 136 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 74 \beta_{13} + 54 \beta_{12} + 14 \beta_{11} + 42 \beta_{10} - 86 \beta_{9} - 127 \beta_{8} - 171 \beta_{7} + 98 \beta_{6} + 105 \beta_{5} + 9 \beta_{4} - 25 \beta_{3} + 151 \beta_{2} - 32 \beta _1 + 881 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 151 \beta_{13} + 267 \beta_{12} - 189 \beta_{11} - 159 \beta_{10} - 78 \beta_{9} - 291 \beta_{8} - 53 \beta_{7} + 410 \beta_{6} - 4 \beta_{5} - 279 \beta_{4} + 480 \beta_{3} - 215 \beta_{2} + 1187 \beta _1 - 1840 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 54 \beta_{13} - 186 \beta_{12} + 110 \beta_{11} - 70 \beta_{10} - 78 \beta_{9} - 347 \beta_{8} + 57 \beta_{7} - 806 \beta_{6} - 219 \beta_{5} - 243 \beta_{4} + 283 \beta_{3} + 1123 \beta_{2} - 1704 \beta _1 - 3011 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/684\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(343\) \(533\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
343.1
1.94929 + 0.447510i
1.94929 0.447510i
1.57398 + 1.23393i
1.57398 1.23393i
0.711746 + 1.86907i
0.711746 1.86907i
0.645572 + 1.89294i
0.645572 1.89294i
−0.0607713 + 1.99908i
−0.0607713 1.99908i
−1.89728 + 0.632718i
−1.89728 0.632718i
−1.92254 + 0.551226i
−1.92254 0.551226i
−1.94929 0.447510i 0 3.59947 + 1.74465i 3.90374 0 2.64664i −6.23566 5.01164i 0 −7.60953 1.74696i
343.2 −1.94929 + 0.447510i 0 3.59947 1.74465i 3.90374 0 2.64664i −6.23566 + 5.01164i 0 −7.60953 + 1.74696i
343.3 −1.57398 1.23393i 0 0.954817 + 3.88437i −3.66290 0 1.93414i 3.29019 7.29209i 0 5.76533 + 4.51977i
343.4 −1.57398 + 1.23393i 0 0.954817 3.88437i −3.66290 0 1.93414i 3.29019 + 7.29209i 0 5.76533 4.51977i
343.5 −0.711746 1.86907i 0 −2.98683 + 2.66060i −4.97973 0 12.2628i 7.09872 + 3.68892i 0 3.54430 + 9.30745i
343.6 −0.711746 + 1.86907i 0 −2.98683 2.66060i −4.97973 0 12.2628i 7.09872 3.68892i 0 3.54430 9.30745i
343.7 −0.645572 1.89294i 0 −3.16647 + 2.44406i 2.38184 0 12.3764i 6.67066 + 4.41614i 0 −1.53765 4.50868i
343.8 −0.645572 + 1.89294i 0 −3.16647 2.44406i 2.38184 0 12.3764i 6.67066 4.41614i 0 −1.53765 + 4.50868i
343.9 0.0607713 1.99908i 0 −3.99261 0.242973i 5.82257 0 5.45132i −0.728358 + 7.96677i 0 0.353845 11.6398i
343.10 0.0607713 + 1.99908i 0 −3.99261 + 0.242973i 5.82257 0 5.45132i −0.728358 7.96677i 0 0.353845 + 11.6398i
343.11 1.89728 0.632718i 0 3.19934 2.40089i −5.79268 0 5.87536i 4.55095 6.57943i 0 −10.9903 + 3.66514i
343.12 1.89728 + 0.632718i 0 3.19934 + 2.40089i −5.79268 0 5.87536i 4.55095 + 6.57943i 0 −10.9903 3.66514i
343.13 1.92254 0.551226i 0 3.39230 2.11950i 2.32715 0 8.62924i 5.35350 5.94475i 0 4.47404 1.28279i
343.14 1.92254 + 0.551226i 0 3.39230 + 2.11950i 2.32715 0 8.62924i 5.35350 + 5.94475i 0 4.47404 + 1.28279i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 343.14
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.3.g.b 14
3.b odd 2 1 76.3.b.b 14
4.b odd 2 1 inner 684.3.g.b 14
12.b even 2 1 76.3.b.b 14
24.f even 2 1 1216.3.d.d 14
24.h odd 2 1 1216.3.d.d 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.3.b.b 14 3.b odd 2 1
76.3.b.b 14 12.b even 2 1
684.3.g.b 14 1.a even 1 1 trivial
684.3.g.b 14 4.b odd 2 1 inner
1216.3.d.d 14 24.f even 2 1
1216.3.d.d 14 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{7} - 66T_{5}^{5} + 28T_{5}^{4} + 1337T_{5}^{3} - 1348T_{5}^{2} - 8400T_{5} + 13312 \) acting on \(S_{3}^{\mathrm{new}}(684, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 2 T^{13} + T^{12} - 14 T^{11} + \cdots + 16384 \) Copy content Toggle raw display
$3$ \( T^{14} \) Copy content Toggle raw display
$5$ \( (T^{7} - 66 T^{5} + 28 T^{4} + 1337 T^{3} + \cdots + 13312)^{2} \) Copy content Toggle raw display
$7$ \( T^{14} + 453 T^{12} + \cdots + 46106276299 \) Copy content Toggle raw display
$11$ \( T^{14} + 880 T^{12} + \cdots + 94488337600 \) Copy content Toggle raw display
$13$ \( (T^{7} - 27 T^{6} - 293 T^{5} + \cdots - 1265920)^{2} \) Copy content Toggle raw display
$17$ \( (T^{7} + 17 T^{6} - 1107 T^{5} + \cdots + 69101055)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 19)^{7} \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 683970816311296 \) Copy content Toggle raw display
$29$ \( (T^{7} + 27 T^{6} - 2901 T^{5} + \cdots + 112127472)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} + 4048 T^{12} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{7} - 50 T^{6} - 1508 T^{5} + \cdots + 914894720)^{2} \) Copy content Toggle raw display
$41$ \( (T^{7} + 112 T^{6} + \cdots - 18882293760)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + 13244 T^{12} + \cdots + 58\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{14} + 8204 T^{12} + \cdots + 64\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( (T^{7} + 7 T^{6} - 6645 T^{5} + \cdots - 18228603200)^{2} \) Copy content Toggle raw display
$59$ \( T^{14} + 30513 T^{12} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{7} - 14 T^{6} + \cdots + 522558358600)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + 27889 T^{12} + \cdots + 29\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{14} + 37812 T^{12} + \cdots + 49\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{7} - 35 T^{6} - 5907 T^{5} + \cdots - 77971926925)^{2} \) Copy content Toggle raw display
$79$ \( T^{14} + 38740 T^{12} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{14} + 62788 T^{12} + \cdots + 34\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{7} - 23640 T^{5} + \cdots + 88310345728)^{2} \) Copy content Toggle raw display
$97$ \( (T^{7} - 154 T^{6} + \cdots + 4410892984320)^{2} \) Copy content Toggle raw display
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