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Properties

Label 363.8.a.o
Level $363$
Weight $8$
Character orbit 363.a
Self dual yes
Analytic conductor $113.396$
Analytic rank $1$
Dimension $14$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [363,8,Mod(1,363)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(363, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("363.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 363.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(113.395764251\)
Analytic rank: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 2 x^{13} - 1289 x^{12} + 2366 x^{11} + 623758 x^{10} - 908404 x^{9} - 141535137 x^{8} + \cdots - 20874968128476 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{4}\cdot 11^{8} \)
Twist minimal: no (minimal twist has level 33)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 2) q^{2} + 27 q^{3} + ( - \beta_{3} + 3 \beta_{2} + \cdots + 63) q^{4}+ \cdots + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 2) q^{2} + 27 q^{3} + ( - \beta_{3} + 3 \beta_{2} + \cdots + 63) q^{4}+ \cdots + (69 \beta_{13} + 529 \beta_{12} + \cdots + 5197524) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 23 q^{2} + 378 q^{3} + 845 q^{4} + 69 q^{5} - 621 q^{6} - 3278 q^{7} - 4602 q^{8} + 10206 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 23 q^{2} + 378 q^{3} + 845 q^{4} + 69 q^{5} - 621 q^{6} - 3278 q^{7} - 4602 q^{8} + 10206 q^{9} - 5320 q^{10} + 22815 q^{12} - 32188 q^{13} - 7794 q^{14} + 1863 q^{15} + 86137 q^{16} - 42917 q^{17} - 16767 q^{18} - 50419 q^{19} + 54475 q^{20} - 88506 q^{21} - 83492 q^{23} - 124254 q^{24} + 336991 q^{25} + 234897 q^{26} + 275562 q^{27} - 697184 q^{28} - 227028 q^{29} - 143640 q^{30} - 36481 q^{31} - 151749 q^{32} - 614562 q^{34} + 258783 q^{35} + 616005 q^{36} + 634524 q^{37} - 662891 q^{38} - 869076 q^{39} - 1585604 q^{40} - 1286544 q^{41} - 210438 q^{42} - 2632644 q^{43} + 50301 q^{45} - 1290627 q^{46} - 1692245 q^{47} + 2325699 q^{48} + 842030 q^{49} - 7870914 q^{50} - 1158759 q^{51} - 6419781 q^{52} + 1211317 q^{53} - 452709 q^{54} - 9100953 q^{56} - 1361313 q^{57} - 6265711 q^{58} + 2551943 q^{59} + 1470825 q^{60} - 10221163 q^{61} - 5723657 q^{62} - 2389662 q^{63} + 6349332 q^{64} - 7684087 q^{65} - 3570631 q^{67} - 13112925 q^{68} - 2254284 q^{69} + 3559265 q^{70} + 4528437 q^{71} - 3354858 q^{72} - 13052276 q^{73} - 4896205 q^{74} + 9098757 q^{75} - 9845214 q^{76} + 6342219 q^{78} - 14911090 q^{79} - 9395382 q^{80} + 7440174 q^{81} + 17478001 q^{82} - 7068900 q^{83} - 18823968 q^{84} - 24423120 q^{85} + 12792479 q^{86} - 6129756 q^{87} + 4800832 q^{89} - 3878280 q^{90} - 15295476 q^{91} - 3195753 q^{92} - 984987 q^{93} + 43320002 q^{94} - 7614177 q^{95} - 4097223 q^{96} - 11578773 q^{97} + 70266089 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 2 x^{13} - 1289 x^{12} + 2366 x^{11} + 623758 x^{10} - 908404 x^{9} - 141535137 x^{8} + \cdots - 20874968128476 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 35\!\cdots\!37 \nu^{13} + \cdots + 14\!\cdots\!08 ) / 37\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 35\!\cdots\!37 \nu^{13} + \cdots - 14\!\cdots\!88 ) / 34\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 79\!\cdots\!07 \nu^{13} + \cdots - 30\!\cdots\!00 ) / 37\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 25\!\cdots\!39 \nu^{13} + \cdots + 10\!\cdots\!16 ) / 46\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 21\!\cdots\!38 \nu^{13} + \cdots - 85\!\cdots\!16 ) / 37\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 80\!\cdots\!84 \nu^{13} + \cdots - 32\!\cdots\!00 ) / 46\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 12\!\cdots\!77 \nu^{13} + \cdots - 51\!\cdots\!00 ) / 46\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 47\!\cdots\!68 \nu^{13} + \cdots - 18\!\cdots\!76 ) / 79\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 38\!\cdots\!91 \nu^{13} + \cdots - 15\!\cdots\!40 ) / 46\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 59\!\cdots\!30 \nu^{13} + \cdots - 23\!\cdots\!68 ) / 46\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 60\!\cdots\!03 \nu^{13} + \cdots - 24\!\cdots\!48 ) / 46\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 21\!\cdots\!99 \nu^{13} + \cdots + 86\!\cdots\!08 ) / 11\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 25\!\cdots\!75 \nu^{13} + \cdots + 10\!\cdots\!36 ) / 46\!\cdots\!60 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{2} + 11\beta _1 + 6 ) / 11 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{5} - 11\beta_{3} + 34\beta_{2} + 5\beta _1 + 2048 ) / 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 8 \beta_{11} - 11 \beta_{9} - 3 \beta_{6} - 44 \beta_{5} + 3 \beta_{4} + 28 \beta_{3} + 434 \beta_{2} + \cdots + 1966 ) / 11 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 4 \beta_{13} - 20 \beta_{12} - 54 \beta_{11} - 48 \beta_{10} + 41 \beta_{9} + 10 \beta_{8} + \cdots + 660867 ) / 11 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 129 \beta_{13} - 599 \beta_{12} + 4140 \beta_{11} + 283 \beta_{10} - 6242 \beta_{9} + 69 \beta_{8} + \cdots + 271216 ) / 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 4601 \beta_{13} - 9497 \beta_{12} - 48717 \beta_{11} - 29956 \beta_{10} + 31642 \beta_{9} + \cdots + 245716204 ) / 11 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 106001 \beta_{13} - 474499 \beta_{12} + 1914049 \beta_{11} + 193660 \beta_{10} - 2996355 \beta_{9} + \cdots - 112916885 ) / 11 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 2999094 \beta_{13} - 3473758 \beta_{12} - 30632377 \beta_{11} - 14806483 \beta_{10} + \cdots + 96875703895 ) / 11 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 62891659 \beta_{13} - 278093805 \beta_{12} + 875028144 \beta_{11} + 102385091 \beta_{10} + \cdots - 150501939236 ) / 11 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 1644384541 \beta_{13} - 1087762821 \beta_{12} - 16835403571 \beta_{11} - 6856109246 \beta_{10} + \cdots + 39551442279388 ) / 11 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 33097132321 \beta_{13} - 145446696835 \beta_{12} + 402116205513 \beta_{11} + 50403671912 \beta_{10} + \cdots - 109846672762315 ) / 11 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 837207003886 \beta_{13} - 263927554574 \beta_{12} - 8669125502635 \beta_{11} - 3105503087775 \beta_{10} + \cdots + 16\!\cdots\!00 ) / 11 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 16503022222038 \beta_{13} - 71893984142466 \beta_{12} + 186149158629240 \beta_{11} + \cdots - 67\!\cdots\!24 ) / 11 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
20.5853
18.3201
17.7685
10.5217
7.06502
9.29307
0.521602
−5.25044
−4.39172
−5.24169
−11.5294
−15.3746
−18.6486
−21.6389
−20.9673 27.0000 311.626 470.315 −566.116 128.739 −3850.12 729.000 −9861.22
1.2 −20.9382 27.0000 310.406 −37.4879 −565.330 −99.4551 −3819.26 729.000 784.928
1.3 −18.1505 27.0000 201.440 −525.838 −490.063 −598.199 −1332.98 729.000 9544.22
1.4 −13.1397 27.0000 44.6520 512.527 −354.772 −1046.89 1095.17 729.000 −6734.46
1.5 −9.68306 27.0000 −34.2384 −195.133 −261.443 946.401 1570.96 729.000 1889.48
1.6 −9.67503 27.0000 −34.3938 124.761 −261.226 −573.194 1571.16 729.000 −1207.07
1.7 −3.13964 27.0000 −118.143 −324.899 −84.7702 −1656.56 772.799 729.000 1020.07
1.8 2.63240 27.0000 −121.070 254.701 71.0749 1403.03 −655.654 729.000 670.476
1.9 4.00975 27.0000 −111.922 −437.630 108.263 377.744 −962.027 729.000 −1754.79
1.10 4.85972 27.0000 −104.383 273.765 131.213 −474.800 −1129.32 729.000 1330.42
1.11 8.91136 27.0000 −48.5876 −136.704 240.607 −36.0476 −1573.64 729.000 −1218.22
1.12 14.9926 27.0000 96.7790 3.84180 404.801 972.861 −468.085 729.000 57.5987
1.13 16.0306 27.0000 128.979 322.621 432.825 −992.956 15.6903 729.000 5171.79
1.14 21.2569 27.0000 323.856 −235.840 573.936 −1628.68 4163.29 729.000 −5013.22
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(11\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.8.a.o 14
11.b odd 2 1 363.8.a.r 14
11.c even 5 2 33.8.e.b ✓ 28
33.h odd 10 2 99.8.f.b 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.8.e.b ✓ 28 11.c even 5 2
99.8.f.b 28 33.h odd 10 2
363.8.a.o 14 1.a even 1 1 trivial
363.8.a.r 14 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} + 23 T_{2}^{13} - 1054 T_{2}^{12} - 24801 T_{2}^{11} + 395219 T_{2}^{10} + \cdots - 71920119992320 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(363))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + \cdots - 71920119992320 \) Copy content Toggle raw display
$3$ \( (T - 27)^{14} \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots - 45\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots - 10\!\cdots\!91 \) Copy content Toggle raw display
$11$ \( T^{14} \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 19\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 35\!\cdots\!44 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots - 16\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots - 24\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots - 11\!\cdots\!20 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots - 14\!\cdots\!51 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 51\!\cdots\!04 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 19\!\cdots\!24 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 13\!\cdots\!80 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 96\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 21\!\cdots\!41 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots - 31\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 23\!\cdots\!80 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots - 39\!\cdots\!20 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots - 49\!\cdots\!80 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 81\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots - 53\!\cdots\!75 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots - 17\!\cdots\!99 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots - 83\!\cdots\!80 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots - 10\!\cdots\!99 \) Copy content Toggle raw display
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