Properties

Label 363.3.h.o
Level $363$
Weight $3$
Character orbit 363.h
Analytic conductor $9.891$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 363.h (of order \(10\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.89103359628\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \( x^{16} - 12x^{14} + 180x^{12} - 2562x^{10} + 25179x^{8} - 96398x^{6} + 239275x^{4} - 536393x^{2} + 1771561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + ( - \beta_{12} + \beta_{6} - \beta_{2}) q^{3} + ( - \beta_{15} + \beta_{14} + \beta_{13} - \beta_{12} + \beta_{10} - 2 \beta_{8} + 3 \beta_{6} - \beta_{5} + \cdots + 1) q^{4}+ \cdots + (\beta_{10} + 3 \beta_{8} + \beta_{6} - 3 \beta_{3} - 3 \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} + ( - \beta_{12} + \beta_{6} - \beta_{2}) q^{3} + ( - \beta_{15} + \beta_{14} + \beta_{13} - \beta_{12} + \beta_{10} - 2 \beta_{8} + 3 \beta_{6} - \beta_{5} + \cdots + 1) q^{4}+ \cdots + (8 \beta_{15} + 8 \beta_{14} + 31 \beta_{13} + 31 \beta_{12} + 38 \beta_{11} + \cdots + 22 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 5 q^{3} + 18 q^{4} + 32 q^{6} - 34 q^{7} + 17 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 5 q^{3} + 18 q^{4} + 32 q^{6} - 34 q^{7} + 17 q^{9} - 12 q^{10} + 106 q^{12} - 2 q^{13} - 28 q^{15} + 102 q^{16} + 42 q^{18} + 66 q^{19} - 12 q^{21} - 74 q^{24} - 176 q^{25} - 55 q^{27} - 146 q^{28} + 110 q^{30} - 126 q^{31} - 132 q^{34} + 226 q^{36} - 230 q^{37} + 136 q^{39} + 226 q^{40} - 72 q^{42} - 156 q^{43} - 72 q^{45} + 308 q^{46} + 255 q^{48} + 170 q^{49} - 169 q^{51} - 224 q^{52} + 1046 q^{54} - 259 q^{57} + 184 q^{58} - 316 q^{60} + 104 q^{61} - 108 q^{63} - 184 q^{64} + 368 q^{67} - 22 q^{69} - 52 q^{70} + 73 q^{72} + 354 q^{73} + 54 q^{75} - 900 q^{76} - 492 q^{78} + 566 q^{79} + 377 q^{81} + 200 q^{82} - 720 q^{84} - 162 q^{85} + 132 q^{87} - 774 q^{90} - 226 q^{91} - 370 q^{93} - 530 q^{94} - 67 q^{96} + 252 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 12x^{14} + 180x^{12} - 2562x^{10} + 25179x^{8} - 96398x^{6} + 239275x^{4} - 536393x^{2} + 1771561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 127407047787804 \nu^{14} - 322126035973969 \nu^{12} + \cdots + 37\!\cdots\!56 ) / 21\!\cdots\!19 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 234959048766210 \nu^{14} + \cdots - 77\!\cdots\!99 ) / 21\!\cdots\!19 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 127407047787804 \nu^{15} + 322126035973969 \nu^{13} + \cdots - 37\!\cdots\!56 \nu ) / 21\!\cdots\!19 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 234959048766210 \nu^{15} + \cdots - 77\!\cdots\!99 \nu ) / 21\!\cdots\!19 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 44\!\cdots\!44 \nu^{14} + \cdots + 20\!\cdots\!70 ) / 21\!\cdots\!19 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 29\!\cdots\!01 \nu^{15} + \cdots + 38\!\cdots\!99 ) / 47\!\cdots\!18 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 29\!\cdots\!01 \nu^{15} + \cdots - 38\!\cdots\!99 ) / 47\!\cdots\!18 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 55\!\cdots\!28 \nu^{15} + \cdots - 91\!\cdots\!65 ) / 47\!\cdots\!18 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 55\!\cdots\!28 \nu^{15} + \cdots - 91\!\cdots\!65 ) / 47\!\cdots\!18 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 30\!\cdots\!44 \nu^{15} + \cdots - 55\!\cdots\!46 \nu ) / 23\!\cdots\!09 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 72\!\cdots\!54 \nu^{15} + \cdots + 25\!\cdots\!82 ) / 47\!\cdots\!18 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 72\!\cdots\!54 \nu^{15} + \cdots - 25\!\cdots\!82 ) / 47\!\cdots\!18 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 99\!\cdots\!41 \nu^{15} + \cdots - 84\!\cdots\!42 ) / 47\!\cdots\!18 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 94\!\cdots\!95 \nu^{15} + \cdots - 74\!\cdots\!23 ) / 47\!\cdots\!18 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{13} + \beta_{12} - \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} - \beta_{3} - 8\beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} - \beta_{9} + 2\beta_{8} + 2\beta_{7} - \beta_{5} + 10\beta_{4} - \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} - \beta_{14} - 15 \beta_{13} + 15 \beta_{12} + \beta_{9} + \beta_{8} + \beta_{7} - 23 \beta_{6} + \beta_{5} + 85 \beta_{3} + 23 \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 8 \beta_{15} - 8 \beta_{14} - 41 \beta_{13} - 41 \beta_{12} - 85 \beta_{11} + 11 \beta_{10} - 3 \beta_{9} - 44 \beta_{8} - 44 \beta_{7} + 99 \beta_{5} + 3 \beta_{4} - 96 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 190 \beta_{15} + 190 \beta_{14} + 220 \beta_{13} - 220 \beta_{12} + 220 \beta_{10} + 30 \beta_{9} - 380 \beta_{8} + 633 \beta_{6} - 190 \beta_{5} + 378 \beta_{2} - 190 \beta _1 + 378 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 603 \beta_{15} + 603 \beta_{14} + 1143 \beta_{13} + 1143 \beta_{12} + 1073 \beta_{11} + 540 \beta_{10} - 1143 \beta_{9} - 540 \beta_{8} - 540 \beta_{7} - 1683 \beta_{5} - 118 \beta_{4} + 721 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2341 \beta_{10} - 2341 \beta_{9} + 588 \beta_{8} - 588 \beta_{7} + 6969 \beta_{6} - 12506 \beta_{3} - 12506 \beta_{2} - 6969 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 6381 \beta_{15} + 6381 \beta_{14} + 12816 \beta_{13} + 12816 \beta_{12} + 12827 \beta_{11} - 6381 \beta_{9} + 6446 \beta_{8} + 6446 \beta_{7} - 22393 \beta_{5} + 16012 \beta_{4} + 6446 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 28893 \beta_{15} - 28893 \beta_{14} - 38546 \beta_{13} + 38546 \beta_{12} - 28893 \beta_{10} + 38546 \beta_{8} + 19240 \beta_{7} - 77049 \beta_{6} + 28893 \beta_{5} + 77049 \beta_{3} + 28893 \beta _1 - 80060 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 48156 \beta_{15} - 48156 \beta_{14} - 134901 \beta_{13} - 134901 \beta_{12} - 211927 \beta_{11} - 77026 \beta_{10} + 125182 \beta_{9} + 260083 \beta_{5} - 205308 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 144620 \beta_{15} + 144620 \beta_{14} + 359360 \beta_{13} - 359360 \beta_{12} + 503980 \beta_{10} + 359360 \beta_{9} - 503980 \beta_{8} + 214740 \beta_{7} - 144620 \beta_{5} + \cdots + 1043432 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 219120 \beta_{10} - 219120 \beta_{9} - 1152580 \beta_{8} - 1152580 \beta_{7} + 464952 \beta_{5} - 2418485 \beta_{4} + 464952 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 2055772 \beta_{15} + 2055772 \beta_{14} + 4504525 \beta_{13} - 4504525 \beta_{12} - 2055772 \beta_{9} - 2055772 \beta_{8} - 2055772 \beta_{7} + 13903349 \beta_{6} + \cdots - 2055772 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 9398824 \beta_{15} + 9398824 \beta_{14} + 24575190 \beta_{13} + 24575190 \beta_{12} + 36032993 \beta_{11} + 3718563 \beta_{10} - 13117387 \beta_{9} + 11457803 \beta_{8} + \cdots + 32314430 \beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/363\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(244\)
\(\chi(n)\) \(-1\) \(-\beta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
245.1
−2.91048 0.945671i
−1.90610 0.619331i
1.90610 + 0.619331i
2.91048 + 0.945671i
2.10855 2.90217i
0.974642 1.34148i
−0.974642 + 1.34148i
−2.10855 + 2.90217i
2.10855 + 2.90217i
0.974642 + 1.34148i
−0.974642 1.34148i
−2.10855 2.90217i
−2.91048 + 0.945671i
−1.90610 + 0.619331i
1.90610 0.619331i
2.91048 0.945671i
−1.79877 2.47580i −2.81156 + 1.04648i −1.65793 + 5.10257i 3.90250 5.37133i 7.64823 + 5.07849i −0.946512 + 2.91306i 3.97326 1.29099i 6.80977 5.88447i −20.3180
245.2 −1.17804 1.62143i 1.88824 + 2.33121i −0.00519352 + 0.0159840i −3.22364 + 4.43696i 1.55548 5.80790i 1.72766 5.31721i −7.59238 + 2.46692i −1.86912 + 8.80377i 10.9918
245.3 1.17804 + 1.62143i −0.157363 + 2.99587i −0.00519352 + 0.0159840i 3.22364 4.43696i −5.04297 + 3.27409i 1.72766 5.31721i 7.59238 2.46692i −8.95047 0.942880i 10.9918
245.4 1.79877 + 2.47580i 2.88971 0.805978i −1.65793 + 5.10257i −3.90250 + 5.37133i 7.19336 + 5.70456i −0.946512 + 2.91306i −3.97326 + 1.29099i 7.70080 4.65808i −20.3180
251.1 −3.41170 1.10853i 1.50565 2.59480i 7.17480 + 5.21280i −1.98428 + 0.644731i −8.01326 + 7.18363i −7.17034 5.20956i −10.2655 14.1293i −4.46601 7.81376i 7.48447
251.2 −1.57700 0.512399i −2.99453 0.181006i −1.01168 0.735029i 0.664316 0.215849i 4.62964 + 1.81985i −2.11081 1.53360i 5.11736 + 7.04345i 8.93447 + 1.08406i −1.15823
251.3 1.57700 + 0.512399i −0.753215 + 2.90391i −1.01168 0.735029i −0.664316 + 0.215849i −2.67578 + 4.19352i −2.11081 1.53360i −5.11736 7.04345i −7.86534 4.37453i −1.15823
251.4 3.41170 + 1.10853i 2.93308 0.630124i 7.17480 + 5.21280i 1.98428 0.644731i 10.7053 + 1.10160i −7.17034 5.20956i 10.2655 + 14.1293i 8.20589 3.69641i 7.48447
269.1 −3.41170 + 1.10853i 1.50565 + 2.59480i 7.17480 5.21280i −1.98428 0.644731i −8.01326 7.18363i −7.17034 + 5.20956i −10.2655 + 14.1293i −4.46601 + 7.81376i 7.48447
269.2 −1.57700 + 0.512399i −2.99453 + 0.181006i −1.01168 + 0.735029i 0.664316 + 0.215849i 4.62964 1.81985i −2.11081 + 1.53360i 5.11736 7.04345i 8.93447 1.08406i −1.15823
269.3 1.57700 0.512399i −0.753215 2.90391i −1.01168 + 0.735029i −0.664316 0.215849i −2.67578 4.19352i −2.11081 + 1.53360i −5.11736 + 7.04345i −7.86534 + 4.37453i −1.15823
269.4 3.41170 1.10853i 2.93308 + 0.630124i 7.17480 5.21280i 1.98428 + 0.644731i 10.7053 1.10160i −7.17034 + 5.20956i 10.2655 14.1293i 8.20589 + 3.69641i 7.48447
323.1 −1.79877 + 2.47580i −2.81156 1.04648i −1.65793 5.10257i 3.90250 + 5.37133i 7.64823 5.07849i −0.946512 2.91306i 3.97326 + 1.29099i 6.80977 + 5.88447i −20.3180
323.2 −1.17804 + 1.62143i 1.88824 2.33121i −0.00519352 0.0159840i −3.22364 4.43696i 1.55548 + 5.80790i 1.72766 + 5.31721i −7.59238 2.46692i −1.86912 8.80377i 10.9918
323.3 1.17804 1.62143i −0.157363 2.99587i −0.00519352 0.0159840i 3.22364 + 4.43696i −5.04297 3.27409i 1.72766 + 5.31721i 7.59238 + 2.46692i −8.95047 + 0.942880i 10.9918
323.4 1.79877 2.47580i 2.88971 + 0.805978i −1.65793 5.10257i −3.90250 5.37133i 7.19336 5.70456i −0.946512 2.91306i −3.97326 1.29099i 7.70080 + 4.65808i −20.3180
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 323.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.c even 5 1 inner
33.h odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.3.h.o 16
3.b odd 2 1 inner 363.3.h.o 16
11.b odd 2 1 363.3.h.n 16
11.c even 5 2 33.3.h.b 16
11.c even 5 1 363.3.b.m 8
11.c even 5 1 inner 363.3.h.o 16
11.d odd 10 1 363.3.b.l 8
11.d odd 10 2 363.3.h.j 16
11.d odd 10 1 363.3.h.n 16
33.d even 2 1 363.3.h.n 16
33.f even 10 1 363.3.b.l 8
33.f even 10 2 363.3.h.j 16
33.f even 10 1 363.3.h.n 16
33.h odd 10 2 33.3.h.b 16
33.h odd 10 1 363.3.b.m 8
33.h odd 10 1 inner 363.3.h.o 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.3.h.b 16 11.c even 5 2
33.3.h.b 16 33.h odd 10 2
363.3.b.l 8 11.d odd 10 1
363.3.b.l 8 33.f even 10 1
363.3.b.m 8 11.c even 5 1
363.3.b.m 8 33.h odd 10 1
363.3.h.j 16 11.d odd 10 2
363.3.h.j 16 33.f even 10 2
363.3.h.n 16 11.b odd 2 1
363.3.h.n 16 11.d odd 10 1
363.3.h.n 16 33.d even 2 1
363.3.h.n 16 33.f even 10 1
363.3.h.o 16 1.a even 1 1 trivial
363.3.h.o 16 3.b odd 2 1 inner
363.3.h.o 16 11.c even 5 1 inner
363.3.h.o 16 33.h odd 10 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(363, [\chi])\):

\( T_{2}^{16} - 17 T_{2}^{14} + 175 T_{2}^{12} - 1372 T_{2}^{10} + 18829 T_{2}^{8} - 48408 T_{2}^{6} + 245930 T_{2}^{4} - 875798 T_{2}^{2} + 1771561 \) Copy content Toggle raw display
\( T_{5}^{16} + 38 T_{5}^{14} + 3020 T_{5}^{12} + 35613 T_{5}^{10} + 1364209 T_{5}^{8} - 12321408 T_{5}^{6} + 42508800 T_{5}^{4} - 28971008 T_{5}^{2} + 7929856 \) Copy content Toggle raw display
\( T_{7}^{8} + 17T_{7}^{7} + 151T_{7}^{6} + 861T_{7}^{5} + 5629T_{7}^{4} + 23148T_{7}^{3} + 72504T_{7}^{2} + 140184T_{7} + 156816 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 17 T^{14} + 175 T^{12} + \cdots + 1771561 \) Copy content Toggle raw display
$3$ \( T^{16} - 5 T^{15} + 4 T^{14} + \cdots + 43046721 \) Copy content Toggle raw display
$5$ \( T^{16} + 38 T^{14} + 3020 T^{12} + \cdots + 7929856 \) Copy content Toggle raw display
$7$ \( (T^{8} + 17 T^{7} + 151 T^{6} + \cdots + 156816)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( (T^{8} + T^{7} + 129 T^{6} + \cdots + 19749136)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} - 472 T^{14} + \cdots + 69002444446441 \) Copy content Toggle raw display
$19$ \( (T^{8} - 33 T^{7} + 1060 T^{6} + \cdots + 4224870001)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 1848 T^{6} + \cdots + 25255373616)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + 68 T^{14} + \cdots + 116101021696 \) Copy content Toggle raw display
$31$ \( (T^{8} + 63 T^{7} + 1558 T^{6} + \cdots + 9448617616)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 115 T^{7} + \cdots + 487254257296)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} - 1714 T^{14} + \cdots + 15\!\cdots\!81 \) Copy content Toggle raw display
$43$ \( (T^{4} + 39 T^{3} - 1528 T^{2} + \cdots - 684409)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} - 1406 T^{14} + \cdots + 21\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{16} - 4259 T^{14} + \cdots + 30\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{16} - 7753 T^{14} + \cdots + 26\!\cdots\!01 \) Copy content Toggle raw display
$61$ \( (T^{8} - 52 T^{7} + \cdots + 1393523586576)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 92 T^{3} - 3927 T^{2} + \cdots - 11977619)^{4} \) Copy content Toggle raw display
$71$ \( T^{16} - 2046 T^{14} + \cdots + 27\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( (T^{8} - 177 T^{7} + \cdots + 5458559358736)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} - 283 T^{7} + \cdots + 743504776736656)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} - 5211 T^{14} + \cdots + 21\!\cdots\!81 \) Copy content Toggle raw display
$89$ \( (T^{8} + 8529 T^{6} + \cdots + 3248250664131)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} - 126 T^{7} + \cdots + 9061117408561)^{2} \) Copy content Toggle raw display
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