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Properties

Label 363.8.a.q
Level $363$
Weight $8$
Character orbit 363.a
Self dual yes
Analytic conductor $113.396$
Analytic rank $0$
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: \(0\)
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: \( x^{14} - 2 x^{13} - 1447 x^{12} + 2312 x^{11} + 800984 x^{10} - 1116196 x^{9} - 213596799 x^{8} + \cdots - 84027524573916 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{4}\cdot 5\cdot 11^{8} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} - 27 q^{3} + ( - \beta_{3} + \beta_{2} - \beta_1 + 82) q^{4} + ( - \beta_{7} - 4 \beta_{2} - 7) q^{5} + (27 \beta_1 - 27) q^{6} + (\beta_{10} - \beta_{8} - \beta_{7} + \cdots - 42) q^{7}+ \cdots + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} - 27 q^{3} + ( - \beta_{3} + \beta_{2} - \beta_1 + 82) q^{4} + ( - \beta_{7} - 4 \beta_{2} - 7) q^{5} + (27 \beta_1 - 27) q^{6} + (\beta_{10} - \beta_{8} - \beta_{7} + \cdots - 42) q^{7}+ \cdots + ( - 211 \beta_{13} - 398 \beta_{12} + \cdots - 469122) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 9 q^{2} - 378 q^{3} + 1129 q^{4} - 71 q^{5} - 243 q^{6} - 534 q^{7} + 3384 q^{8} + 10206 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 9 q^{2} - 378 q^{3} + 1129 q^{4} - 71 q^{5} - 243 q^{6} - 534 q^{7} + 3384 q^{8} + 10206 q^{9} - 676 q^{10} - 30483 q^{12} + 12320 q^{13} - 39924 q^{14} + 1917 q^{15} + 121041 q^{16} + 33141 q^{17} + 6561 q^{18} - 45019 q^{19} + 84003 q^{20} + 14418 q^{21} + 69254 q^{23} - 91368 q^{24} + 79811 q^{25} - 371495 q^{26} - 275562 q^{27} - 7064 q^{28} + 411600 q^{29} + 18252 q^{30} + 412799 q^{31} + 1363161 q^{32} - 771854 q^{34} + 850847 q^{35} + 823041 q^{36} - 831248 q^{37} + 948531 q^{38} - 332640 q^{39} - 50612 q^{40} + 616814 q^{41} + 1077948 q^{42} - 99892 q^{43} - 51759 q^{45} - 250599 q^{46} - 1139331 q^{47} - 3268107 q^{48} + 2854014 q^{49} - 3832250 q^{50} - 894807 q^{51} - 3588929 q^{52} + 2582463 q^{53} - 177147 q^{54} - 3153909 q^{56} + 1215513 q^{57} - 5034575 q^{58} - 1606585 q^{59} - 2268081 q^{60} - 3071551 q^{61} + 1169699 q^{62} - 389286 q^{63} + 19996212 q^{64} + 2447897 q^{65} - 1617843 q^{67} + 9075543 q^{68} - 1869858 q^{69} + 781473 q^{70} - 571759 q^{71} + 2466936 q^{72} + 7998216 q^{73} - 2349001 q^{74} - 2154897 q^{75} - 3267874 q^{76} + 10030365 q^{78} - 422038 q^{79} + 5233418 q^{80} + 7440174 q^{81} - 26251615 q^{82} + 7152906 q^{83} + 190728 q^{84} + 11341160 q^{85} + 12629343 q^{86} - 11113200 q^{87} - 20120730 q^{89} - 492804 q^{90} + 14374304 q^{91} + 21153333 q^{92} - 11145573 q^{93} - 37707710 q^{94} + 18573421 q^{95} - 36805347 q^{96} + 21339043 q^{97} - 8477109 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 2 x^{13} - 1447 x^{12} + 2312 x^{11} + 800984 x^{10} - 1116196 x^{9} - 213596799 x^{8} + \cdots - 84027524573916 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 34\!\cdots\!79 \nu^{13} + \cdots - 19\!\cdots\!72 ) / 73\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 34\!\cdots\!79 \nu^{13} + \cdots - 23\!\cdots\!72 ) / 66\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 26\!\cdots\!15 \nu^{13} + \cdots + 13\!\cdots\!96 ) / 14\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 28\!\cdots\!09 \nu^{13} + \cdots - 64\!\cdots\!64 ) / 14\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 64\!\cdots\!82 \nu^{13} + \cdots + 21\!\cdots\!24 ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 11\!\cdots\!41 \nu^{13} + \cdots - 18\!\cdots\!72 ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 32\!\cdots\!46 \nu^{13} + \cdots + 14\!\cdots\!08 ) / 53\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 49\!\cdots\!88 \nu^{13} + \cdots - 30\!\cdots\!44 ) / 73\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 12\!\cdots\!54 \nu^{13} + \cdots - 59\!\cdots\!28 ) / 24\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 30\!\cdots\!83 \nu^{13} + \cdots - 16\!\cdots\!04 ) / 53\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 93\!\cdots\!14 \nu^{13} + \cdots - 14\!\cdots\!52 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 71\!\cdots\!21 \nu^{13} + \cdots - 73\!\cdots\!48 ) / 53\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 47\!\cdots\!01 \nu^{13} + \cdots + 30\!\cdots\!88 ) / 26\!\cdots\!00 \) 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
\(\nu^{2}\)\(=\) \( ( 2\beta_{8} - 11\beta_{3} + 12\beta_{2} - \beta _1 + 2288 ) / 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 11\beta_{8} + 6\beta_{7} + 3\beta_{5} - 11\beta_{4} - 32\beta_{3} + 404\beta_{2} - 3739\beta _1 + 2931 ) / 11 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 5 \beta_{13} - 26 \beta_{12} + 46 \beta_{11} + 2 \beta_{10} - 14 \beta_{9} + 1078 \beta_{8} + \cdots + 784166 ) / 11 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 73 \beta_{13} - 135 \beta_{12} + 48 \beta_{11} - 419 \beta_{10} + 24 \beta_{9} + 7987 \beta_{8} + \cdots + 2058286 ) / 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2035 \beta_{13} - 18820 \beta_{12} + 27116 \beta_{11} - 1542 \beta_{10} - 4862 \beta_{9} + \cdots + 312600917 ) / 11 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 28240 \beta_{13} - 123204 \beta_{12} + 62402 \beta_{11} - 351258 \beta_{10} + 32168 \beta_{9} + \cdots + 1405512893 ) / 11 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1396023 \beta_{13} - 10234339 \beta_{12} + 13907326 \beta_{11} - 3218137 \beta_{10} + \cdots + 133120355136 ) / 11 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 113053 \beta_{13} - 81465339 \beta_{12} + 52810280 \beta_{11} - 221037131 \beta_{10} + \cdots + 861373891690 ) / 11 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 1061849351 \beta_{13} - 5112809178 \beta_{12} + 6957687324 \beta_{11} - 2877246120 \beta_{10} + \cdots + 58645511996897 ) / 11 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 8716850648 \beta_{13} - 47878706744 \beta_{12} + 37789112922 \beta_{11} - 128084205766 \beta_{10} + \cdots + 490081991244962 ) / 11 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 729393568486 \beta_{13} - 2489133759981 \beta_{12} + 3452370816488 \beta_{11} - 1999710217615 \beta_{10} + \cdots + 26\!\cdots\!22 ) / 11 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 8736079741422 \beta_{13} - 26618521960756 \beta_{12} + 24551028047040 \beta_{11} + \cdots + 26\!\cdots\!86 ) / 11 \) 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
−18.9926
−18.4731
−20.6916
−12.6254
−9.86607
−5.14934
−2.31980
1.02978
10.2620
8.80924
13.3291
13.3895
22.3437
20.9545
−19.6106 −27.0000 256.576 −313.530 529.487 −1115.06 −2521.46 729.000 6148.53
1.2 −19.0911 −27.0000 236.470 432.914 515.460 1654.16 −2070.81 729.000 −8264.81
1.3 −19.0735 −27.0000 235.800 −0.254042 514.986 784.701 −2056.13 729.000 4.84548
1.4 −11.0074 −27.0000 −6.83704 406.651 297.200 −1248.44 1484.21 729.000 −4476.18
1.5 −8.24804 −27.0000 −59.9699 −332.737 222.697 −118.093 1550.38 729.000 2744.43
1.6 −5.76738 −27.0000 −94.7374 −461.996 155.719 625.834 1284.61 729.000 2664.50
1.7 −2.93783 −27.0000 −119.369 198.475 79.3214 245.012 726.728 729.000 −583.086
1.8 2.64781 −27.0000 −120.989 −149.877 −71.4909 748.741 −659.276 729.000 −396.845
1.9 9.64401 −27.0000 −34.9930 107.891 −260.388 −1541.58 −1571.91 729.000 1040.50
1.10 10.4273 −27.0000 −19.2720 230.478 −281.536 −141.344 −1535.65 729.000 2403.26
1.11 12.7110 −27.0000 33.5707 −57.1188 −343.198 1189.25 −1200.29 729.000 −726.039
1.12 15.0076 −27.0000 97.2268 −196.088 −405.204 −1206.21 −461.831 729.000 −2942.80
1.13 21.7256 −27.0000 344.002 −305.496 −586.592 −1168.09 4692.79 729.000 −6637.09
1.14 22.5726 −27.0000 381.522 369.687 −609.460 757.126 5722.64 729.000 8344.78
\(n\): 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.q 14
11.b odd 2 1 363.8.a.p 14
11.c even 5 2 33.8.e.a ✓ 28
33.h odd 10 2 99.8.f.c 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.8.e.a ✓ 28 11.c even 5 2
99.8.f.c 28 33.h odd 10 2
363.8.a.p 14 11.b odd 2 1
363.8.a.q 14 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} - 9 T_{2}^{13} - 1420 T_{2}^{12} + 11127 T_{2}^{11} + 771385 T_{2}^{10} + \cdots - 273623765626880 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(363))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + \cdots - 273623765626880 \) Copy content Toggle raw display
$3$ \( (T + 27)^{14} \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 20\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots - 67\!\cdots\!39 \) Copy content Toggle raw display
$11$ \( T^{14} \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots - 12\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 23\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots - 96\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots - 26\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots - 38\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 20\!\cdots\!69 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots - 93\!\cdots\!04 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots - 85\!\cdots\!36 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots - 28\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots - 62\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots - 28\!\cdots\!49 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 52\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots - 10\!\cdots\!80 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 22\!\cdots\!20 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots - 98\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots - 60\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots - 43\!\cdots\!75 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 23\!\cdots\!59 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots - 16\!\cdots\!39 \) Copy content Toggle raw display
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