↑

Properties

Label 363.3.g.a
Level $363$
Weight $3$
Character orbit 363.g
Analytic conductor $9.891$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [363,3,Mod(40,363)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(363, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 7]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("363.40");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 363.g (of order \(10\), degree \(4\), not minimal)

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: no
Analytic conductor: \(9.89103359628\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 3 x^{14} - 4 x^{13} + 77 x^{12} + 88 x^{11} - 577 x^{10} + 578 x^{9} + 1520 x^{8} + \cdots + 83521 \) 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

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

\(f(q)\) \(=\) \( q + ( - \beta_{8} - \beta_{7} - \beta_{4} + \cdots - 1) q^{2}+ \cdots - 3 \beta_{5} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{8} - \beta_{7} - \beta_{4} + \cdots - 1) q^{2}+ \cdots + (47 \beta_{15} + 20 \beta_{14} + \cdots + 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 10 q^{2} - 10 q^{4} + 6 q^{5} + 20 q^{7} + 50 q^{8} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 10 q^{2} - 10 q^{4} + 6 q^{5} + 20 q^{7} + 50 q^{8} - 12 q^{9} - 24 q^{12} - 10 q^{13} + 28 q^{14} + 6 q^{15} + 6 q^{16} + 50 q^{17} - 70 q^{19} + 12 q^{20} + 132 q^{23} - 42 q^{25} - 44 q^{26} - 90 q^{28} + 80 q^{29} - 120 q^{30} - 30 q^{31} - 368 q^{34} + 170 q^{35} - 30 q^{36} + 134 q^{37} - 10 q^{38} - 120 q^{39} - 370 q^{40} + 150 q^{41} + 186 q^{42} - 12 q^{45} - 80 q^{46} + 110 q^{47} + 24 q^{48} - 140 q^{49} + 350 q^{50} - 90 q^{51} - 40 q^{52} - 278 q^{53} + 524 q^{56} - 240 q^{57} - 220 q^{58} + 156 q^{60} - 260 q^{61} + 770 q^{62} - 60 q^{63} + 172 q^{64} + 36 q^{67} + 290 q^{68} - 120 q^{69} - 290 q^{70} - 86 q^{71} - 120 q^{72} - 140 q^{73} + 700 q^{74} + 252 q^{75} - 312 q^{78} - 380 q^{79} + 674 q^{80} - 36 q^{81} + 124 q^{82} + 620 q^{83} - 540 q^{84} - 450 q^{85} - 774 q^{86} + 76 q^{89} - 120 q^{90} + 6 q^{91} + 90 q^{92} + 24 q^{93} - 330 q^{94} + 550 q^{95} - 360 q^{96} + 246 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 3 x^{14} - 4 x^{13} + 77 x^{12} + 88 x^{11} - 577 x^{10} + 578 x^{9} + 1520 x^{8} + \cdots + 83521 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 138203247537734 \nu^{15} - 291050501651580 \nu^{14} - 300665341738367 \nu^{13} + \cdots - 10\!\cdots\!14 ) / 10\!\cdots\!99 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1196620054805 \nu^{15} + 7612440593330 \nu^{14} - 13694873409440 \nu^{13} + \cdots + 87\!\cdots\!85 ) / 78\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 79\!\cdots\!69 \nu^{15} + \cdots + 55\!\cdots\!82 ) / 41\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 12\!\cdots\!15 \nu^{15} + \cdots - 57\!\cdots\!30 ) / 41\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 13\!\cdots\!90 \nu^{15} + \cdots - 14\!\cdots\!47 ) / 41\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 17\!\cdots\!19 \nu^{15} + \cdots - 10\!\cdots\!48 ) / 41\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 20\!\cdots\!52 \nu^{15} + \cdots - 12\!\cdots\!26 ) / 41\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 24\!\cdots\!14 \nu^{15} + \cdots + 29\!\cdots\!54 ) / 41\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 29\!\cdots\!12 \nu^{15} + \cdots + 19\!\cdots\!73 ) / 41\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 31\!\cdots\!70 \nu^{15} + \cdots - 62\!\cdots\!81 ) / 41\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 20\!\cdots\!33 \nu^{15} + \cdots - 91\!\cdots\!68 ) / 24\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 35\!\cdots\!76 \nu^{15} + \cdots - 19\!\cdots\!69 ) / 41\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 47\!\cdots\!48 \nu^{15} + \cdots + 91\!\cdots\!78 ) / 41\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 57\!\cdots\!67 \nu^{15} + \cdots + 27\!\cdots\!46 ) / 41\!\cdots\!94 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{15} + \beta_{10} + 3\beta_{8} - 3\beta_{7} - 2\beta_{5} + \beta_{4} - \beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 2 \beta_{15} - \beta_{12} - \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} - 3 \beta_{7} + 6 \beta_{6} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4 \beta_{15} - 9 \beta_{14} - 2 \beta_{13} - 2 \beta_{9} - 27 \beta_{8} - 27 \beta_{7} + 13 \beta_{6} + \cdots - 42 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 4 \beta_{15} - 36 \beta_{14} + 13 \beta_{13} - 13 \beta_{12} - 9 \beta_{11} + 23 \beta_{10} + \cdots - 73 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 7 \beta_{15} + 7 \beta_{14} + 44 \beta_{13} - 76 \beta_{12} - 44 \beta_{11} - 142 \beta_{10} + \cdots + 209 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 449 \beta_{15} + 59 \beta_{14} - 59 \beta_{13} + 157 \beta_{12} + 98 \beta_{11} - 504 \beta_{10} + \cdots - 59 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 11 \beta_{15} + 378 \beta_{14} + 602 \beta_{12} + 390 \beta_{11} + 591 \beta_{10} + 602 \beta_{9} + \cdots + 1028 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 2504 \beta_{15} + 4164 \beta_{14} - 201 \beta_{13} - 367 \beta_{9} + 3660 \beta_{8} + 3660 \beta_{7} + \cdots + 11389 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2504 \beta_{15} + 4696 \beta_{14} - 6668 \beta_{13} + 6668 \beta_{12} + 4164 \beta_{11} + 1972 \beta_{10} + \cdots - 11612 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 22394 \beta_{15} - 22394 \beta_{14} - 3312 \beta_{13} + 5504 \beta_{12} + 3312 \beta_{11} + \cdots - 76812 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 102120 \beta_{15} - 39824 \beta_{14} + 39824 \beta_{13} - 104234 \beta_{12} - 64410 \beta_{11} + \cdots + 39824 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 300533 \beta_{15} - 247371 \beta_{14} - 99867 \beta_{12} - 62296 \beta_{11} - 400400 \beta_{10} + \cdots - 695314 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 594635 \beta_{15} - 966540 \beta_{14} + 338104 \beta_{13} + 547904 \beta_{9} + 786321 \beta_{8} + \cdots - 1445411 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 594635 \beta_{15} + 1832695 \beta_{14} + 1561175 \beta_{13} - 1561175 \beta_{12} - 966540 \beta_{11} + \cdots + 11063785 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(122\) \(244\)
\(\chi(n)\) \(1\) \(-\beta_{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
40.1
1.64608 − 1.06057i
0.988132 − 0.846795i
−1.29715 − 0.104262i
−1.95510 + 0.109518i
2.24350 + 2.23726i
1.60675 + 1.36085i
−0.797732 − 1.94863i
−1.43448 − 2.82504i
2.24350 − 2.23726i
1.60675 − 1.36085i
−0.797732 + 1.94863i
−1.43448 + 2.82504i
1.64608 + 1.06057i
0.988132 + 0.846795i
−1.29715 + 0.104262i
−1.95510 − 0.109518i
−2.14608 + 2.95383i 0.535233 − 1.64728i −2.88336 − 8.87407i −4.42535 + 3.21521i 3.71712 + 5.11618i 6.81008 − 2.21273i 18.5106 + 6.01447i −2.42705 − 1.76336i − 19.9718i
40.2 −1.48813 + 2.04824i −0.535233 + 1.64728i −0.744674 − 2.29187i 6.83369 − 4.96497i −2.57752 − 3.54765i −2.32099 + 0.754134i −3.82892 − 1.24409i −2.42705 − 1.76336i 21.3855i
40.3 0.797149 − 1.09718i −0.535233 + 1.64728i 0.667707 + 2.05499i −1.85613 + 1.34856i 1.38070 + 1.90037i 9.29312 − 3.01952i 7.94622 + 2.58188i −2.42705 − 1.76336i 3.11151i
40.4 1.45510 − 2.00277i 0.535233 − 1.64728i −0.657709 − 2.02422i 2.06583 − 1.50091i −2.52030 − 3.46890i 0.162060 − 0.0526565i 4.40651 + 1.43176i −2.42705 − 1.76336i − 6.32135i
94.1 −2.74350 + 0.891416i 1.40126 + 1.01807i 3.49608 − 2.54005i −0.136315 + 0.419536i −4.75187 − 1.54398i −6.21995 − 8.56103i −0.544939 + 0.750044i 0.927051 + 2.85317i − 1.27251i
94.2 −2.10675 + 0.684524i −1.40126 − 1.01807i 0.733748 − 0.533099i 2.68691 − 8.26946i 3.64900 + 1.18563i 3.92164 + 5.39767i 4.02726 − 5.54305i 0.927051 + 2.85317i 19.2609i
94.3 0.297732 − 0.0967389i −1.40126 − 1.01807i −3.15678 + 2.29354i −2.29149 + 7.05247i −0.515686 − 0.167557i −5.89378 − 8.11209i −1.45403 + 2.00130i 0.927051 + 2.85317i 2.32142i
94.4 0.934478 − 0.303630i 1.40126 + 1.01807i −2.45501 + 1.78367i 0.122858 − 0.378117i 1.61856 + 0.525903i 4.24781 + 5.84661i −4.06274 + 5.59188i 0.927051 + 2.85317i − 0.390645i
112.1 −2.74350 − 0.891416i 1.40126 − 1.01807i 3.49608 + 2.54005i −0.136315 − 0.419536i −4.75187 + 1.54398i −6.21995 + 8.56103i −0.544939 − 0.750044i 0.927051 − 2.85317i 1.27251i
112.2 −2.10675 − 0.684524i −1.40126 + 1.01807i 0.733748 + 0.533099i 2.68691 + 8.26946i 3.64900 − 1.18563i 3.92164 − 5.39767i 4.02726 + 5.54305i 0.927051 − 2.85317i − 19.2609i
112.3 0.297732 + 0.0967389i −1.40126 + 1.01807i −3.15678 − 2.29354i −2.29149 − 7.05247i −0.515686 + 0.167557i −5.89378 + 8.11209i −1.45403 − 2.00130i 0.927051 − 2.85317i − 2.32142i
112.4 0.934478 + 0.303630i 1.40126 − 1.01807i −2.45501 − 1.78367i 0.122858 + 0.378117i 1.61856 − 0.525903i 4.24781 − 5.84661i −4.06274 − 5.59188i 0.927051 − 2.85317i 0.390645i
118.1 −2.14608 − 2.95383i 0.535233 + 1.64728i −2.88336 + 8.87407i −4.42535 − 3.21521i 3.71712 − 5.11618i 6.81008 + 2.21273i 18.5106 − 6.01447i −2.42705 + 1.76336i 19.9718i
118.2 −1.48813 − 2.04824i −0.535233 − 1.64728i −0.744674 + 2.29187i 6.83369 + 4.96497i −2.57752 + 3.54765i −2.32099 − 0.754134i −3.82892 + 1.24409i −2.42705 + 1.76336i − 21.3855i
118.3 0.797149 + 1.09718i −0.535233 − 1.64728i 0.667707 − 2.05499i −1.85613 − 1.34856i 1.38070 − 1.90037i 9.29312 + 3.01952i 7.94622 − 2.58188i −2.42705 + 1.76336i − 3.11151i
118.4 1.45510 + 2.00277i 0.535233 + 1.64728i −0.657709 + 2.02422i 2.06583 + 1.50091i −2.52030 + 3.46890i 0.162060 + 0.0526565i 4.40651 − 1.43176i −2.42705 + 1.76336i 6.32135i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 40.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d 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.g.a 16
11.b odd 2 1 363.3.g.g 16
11.c even 5 1 33.3.g.a ✓ 16
11.c even 5 1 363.3.c.e 16
11.c even 5 1 363.3.g.f 16
11.c even 5 1 363.3.g.g 16
11.d odd 10 1 33.3.g.a ✓ 16
11.d odd 10 1 363.3.c.e 16
11.d odd 10 1 inner 363.3.g.a 16
11.d odd 10 1 363.3.g.f 16
33.f even 10 1 99.3.k.c 16
33.f even 10 1 1089.3.c.m 16
33.h odd 10 1 99.3.k.c 16
33.h odd 10 1 1089.3.c.m 16
44.g even 10 1 528.3.bf.b 16
44.h odd 10 1 528.3.bf.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.3.g.a ✓ 16 11.c even 5 1
33.3.g.a ✓ 16 11.d odd 10 1
99.3.k.c 16 33.f even 10 1
99.3.k.c 16 33.h odd 10 1
363.3.c.e 16 11.c even 5 1
363.3.c.e 16 11.d odd 10 1
363.3.g.a 16 1.a even 1 1 trivial
363.3.g.a 16 11.d odd 10 1 inner
363.3.g.f 16 11.c even 5 1
363.3.g.f 16 11.d odd 10 1
363.3.g.g 16 11.b odd 2 1
363.3.g.g 16 11.c even 5 1
528.3.bf.b 16 44.g even 10 1
528.3.bf.b 16 44.h odd 10 1
1089.3.c.m 16 33.f even 10 1
1089.3.c.m 16 33.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 10 T_{2}^{15} + 47 T_{2}^{14} + 120 T_{2}^{13} + 195 T_{2}^{12} + 500 T_{2}^{11} + \cdots + 3721 \) acting on \(S_{3}^{\mathrm{new}}(363, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 10 T^{15} + \cdots + 3721 \) Copy content Toggle raw display
$3$ \( (T^{8} + 3 T^{6} + 9 T^{4} + \cdots + 81)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} - 6 T^{15} + \cdots + 9369721 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 22159001881 \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 47\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 18\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 99\!\cdots\!96 \) Copy content Toggle raw display
$23$ \( (T^{8} - 66 T^{7} + \cdots + 255717136)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 59\!\cdots\!41 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 30\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 30\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 41\!\cdots\!41 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 34\!\cdots\!41 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( (T^{8} - 18 T^{7} + \cdots + 47006885776)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 40\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 63\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 14\!\cdots\!81 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 16\!\cdots\!41 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots - 15121642690304)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 29\!\cdots\!01 \) Copy content Toggle raw display
show more
show less