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Properties

Label 363.3.g.e
Level $363$
Weight $3$
Character orbit 363.g
Analytic conductor $9.891$
Analytic rank $0$
Dimension $16$
Inner twists $8$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [363,3,Mod(40,363)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://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);
 
Copy content sage: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")
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 363.g (of order \(10\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,20,-4,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content 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: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} - 8 x^{14} + 44 x^{13} - 22 x^{12} + 460 x^{11} - 304 x^{10} - 5968 x^{9} + \cdots + 234256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 33)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + (\beta_{15} - \beta_{12} + \cdots + \beta_{3}) q^{2} - \beta_{8} q^{3} + (2 \beta_{13} + 5 \beta_{10} + \cdots + 5) q^{4} + ( - 3 \beta_{7} - \beta_{4}) q^{5} + ( - \beta_{12} + \beta_{11}) q^{6}+ \cdots + ( - 35 \beta_{9} + 10 \beta_{6}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 20 q^{4} - 4 q^{5} - 12 q^{9} + 96 q^{12} - 12 q^{14} + 36 q^{15} - 4 q^{16} + 92 q^{20} - 368 q^{23} - 12 q^{25} - 204 q^{26} - 80 q^{31} + 192 q^{34} + 60 q^{36} + 64 q^{37} + 120 q^{38} - 84 q^{42}+ \cdots + 376 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} - 8 x^{14} + 44 x^{13} - 22 x^{12} + 460 x^{11} - 304 x^{10} - 5968 x^{9} + \cdots + 234256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 137\nu^{15} + 88738\nu^{10} - 769016\nu^{5} - 5412462616 ) / 3125878184 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 7243 \nu^{15} + 28972 \nu^{14} - 159346 \nu^{13} + 79673 \nu^{12} + 1716957 \nu^{11} + \cdots - 848358104 ) / 1562939092 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 19473 \nu^{15} + 77892 \nu^{14} - 428406 \nu^{13} + 214203 \nu^{12} + 4679840 \nu^{11} + \cdots - 2280833544 ) / 3125878184 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 6530 \nu^{15} - 48975 \nu^{14} + 14068 \nu^{13} + 574640 \nu^{12} - 1723920 \nu^{11} + \cdots + 4589075040 ) / 1322486924 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1519 \nu^{15} + 6076 \nu^{14} - 33418 \nu^{13} + 16709 \nu^{12} + 361966 \nu^{11} + \cdots - 177917432 ) / 240452168 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -35393\nu^{15} - 33296042\nu^{10} - 9465209584\nu^{5} + 41207537360 ) / 3125878184 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 293978 \nu^{15} - 2204835 \nu^{14} + 631662 \nu^{13} + 25870064 \nu^{12} - 77610192 \nu^{11} + \cdots + 206598331104 ) / 34384660024 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 34233 \nu^{15} - 136932 \nu^{14} + 753126 \nu^{13} - 376563 \nu^{12} - 8139472 \nu^{11} + \cdots + 4009642824 ) / 3125878184 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 4429\nu^{15} + 4141592\nu^{10} + 1175369008\nu^{5} - 5117187592 ) / 284170744 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3684 \nu^{15} + 8596 \nu^{14} + 20262 \nu^{13} - 159983 \nu^{12} + 189112 \nu^{11} + \cdots + 143833184 ) / 240452168 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 475406 \nu^{15} - 3565545 \nu^{14} + 870212 \nu^{13} + 41835728 \nu^{12} - 125507184 \nu^{11} + \cdots + 334100123808 ) / 34384660024 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 324448 \nu^{15} + 2433360 \nu^{14} - 555036 \nu^{13} - 28551424 \nu^{12} + \cdots - 228011672064 ) / 17192330012 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 82776 \nu^{15} + 193144 \nu^{14} + 455268 \nu^{13} - 3601949 \nu^{12} + 4249168 \nu^{11} + \cdots + 3231795776 ) / 3125878184 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 1198805 \nu^{15} - 2258559 \nu^{14} - 9590440 \nu^{13} + 52747420 \nu^{12} + \cdots + 51059502560 ) / 34384660024 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 409215 \nu^{15} - 768446 \nu^{14} - 3273720 \nu^{13} + 18005460 \nu^{12} - 9002730 \nu^{11} + \cdots + 17429285280 ) / 8596165006 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{8} + \beta_{5} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{15} + \beta_{14} + 2\beta_{13} + \beta_{12} + \beta_{11} - 5\beta_{10} - \beta_{9} - \beta_{6} - \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{12} - 2\beta_{11} + 12\beta_{7} - 19\beta_{4} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 14 \beta_{15} + 18 \beta_{14} - 2 \beta_{13} + 2 \beta_{10} - 2 \beta_{8} + 2 \beta_{7} - 2 \beta_{5} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 20\beta_{9} + 27\beta_{6} - 137\beta _1 - 233 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -390\beta_{8} - 676\beta_{5} + 110\beta_{3} - 148\beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 538 \beta_{15} - 736 \beta_{14} - 592 \beta_{13} - 538 \beta_{12} - 736 \beta_{11} + \cdots - 736 \beta_{2} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -144\beta_{12} - 200\beta_{11} - 6368\beta_{7} + 11036\beta_{4} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 6168 \beta_{15} - 8430 \beta_{14} + 10206 \beta_{13} - 17686 \beta_{10} + 10206 \beta_{8} - 10206 \beta_{7} + \cdots - 17686 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -18636\beta_{9} - 25460\beta_{6} + 49792\beta _1 + 86260 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 260552\beta_{8} + 451292\beta_{5} - 24332\beta_{3} + 33248\beta_{2} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 293800 \beta_{15} + 401336 \beta_{14} - 104520 \beta_{13} + 293800 \beta_{12} + \cdots + 401336 \beta_{2} \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 505856\beta_{12} + 690996\beta_{11} + 2892220\beta_{7} - 5009452\beta_{4} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 2201224 \beta_{15} + 3006912 \beta_{14} - 9176312 \beta_{13} + 15893792 \beta_{10} - 9176312 \beta_{8} + \cdots + 15893792 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 12183224\beta_{9} + 16642576\beta_{6} - 10203792\beta _1 - 17673416 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(122\) \(244\)
\(\chi(n)\) \(1\) \(1 - \beta_{4} - \beta_{5} + \beta_{10}\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
1.34210 + 0.0512119i
−3.47887 + 0.305971i
2.63462 + 2.29236i
−1.11588 − 0.747434i
−1.05568 + 0.830296i
2.99431 − 1.79729i
−0.784034 + 3.40315i
0.463436 − 1.26058i
−1.05568 − 0.830296i
2.99431 + 1.79729i
−0.784034 − 3.40315i
0.463436 + 1.26058i
1.34210 − 0.0512119i
−3.47887 − 0.305971i
2.63462 − 2.29236i
−1.11588 + 0.747434i
−2.07515 + 2.85620i −0.535233 + 1.64728i −2.61555 − 8.04984i −5.01279 + 3.64201i −3.59426 − 4.94708i 2.45798 − 0.798646i 14.9889 + 4.87020i −2.42705 − 1.76336i − 21.8752i
40.2 −1.38297 + 1.90349i 0.535233 − 1.64728i −0.474619 − 1.46073i 3.39476 − 2.46644i 2.39537 + 3.29695i −6.11349 + 1.98639i −5.51390 − 1.79158i −2.42705 − 1.76336i 9.87291i
40.3 1.38297 − 1.90349i 0.535233 − 1.64728i −0.474619 − 1.46073i 3.39476 − 2.46644i −2.39537 − 3.29695i 6.11349 − 1.98639i 5.51390 + 1.79158i −2.42705 − 1.76336i − 9.87291i
40.4 2.07515 − 2.85620i −0.535233 + 1.64728i −2.61555 − 8.04984i −5.01279 + 3.64201i 3.59426 + 4.94708i −2.45798 + 0.798646i −14.9889 − 4.87020i −2.42705 − 1.76336i 21.8752i
94.1 −3.35766 + 1.09097i 1.40126 + 1.01807i 6.84760 − 4.97507i 1.91472 − 5.89289i −5.81564 − 1.88962i −1.51911 − 2.09088i −9.26367 + 12.7504i 0.927051 + 2.85317i 21.8752i
94.2 −2.23769 + 0.727070i −1.40126 − 1.01807i 1.24257 − 0.902778i −1.29668 + 3.99078i 3.87580 + 1.25932i 3.77834 + 5.20044i 3.40778 − 4.69040i 0.927051 + 2.85317i − 9.87291i
94.3 2.23769 − 0.727070i −1.40126 − 1.01807i 1.24257 − 0.902778i −1.29668 + 3.99078i −3.87580 − 1.25932i −3.77834 − 5.20044i −3.40778 + 4.69040i 0.927051 + 2.85317i 9.87291i
94.4 3.35766 − 1.09097i 1.40126 + 1.01807i 6.84760 − 4.97507i 1.91472 − 5.89289i 5.81564 + 1.88962i 1.51911 + 2.09088i 9.26367 − 12.7504i 0.927051 + 2.85317i − 21.8752i
112.1 −3.35766 − 1.09097i 1.40126 − 1.01807i 6.84760 + 4.97507i 1.91472 + 5.89289i −5.81564 + 1.88962i −1.51911 + 2.09088i −9.26367 − 12.7504i 0.927051 − 2.85317i − 21.8752i
112.2 −2.23769 − 0.727070i −1.40126 + 1.01807i 1.24257 + 0.902778i −1.29668 − 3.99078i 3.87580 − 1.25932i 3.77834 − 5.20044i 3.40778 + 4.69040i 0.927051 − 2.85317i 9.87291i
112.3 2.23769 + 0.727070i −1.40126 + 1.01807i 1.24257 + 0.902778i −1.29668 − 3.99078i −3.87580 + 1.25932i −3.77834 + 5.20044i −3.40778 − 4.69040i 0.927051 − 2.85317i − 9.87291i
112.4 3.35766 + 1.09097i 1.40126 − 1.01807i 6.84760 + 4.97507i 1.91472 + 5.89289i 5.81564 − 1.88962i 1.51911 − 2.09088i 9.26367 + 12.7504i 0.927051 − 2.85317i 21.8752i
118.1 −2.07515 − 2.85620i −0.535233 − 1.64728i −2.61555 + 8.04984i −5.01279 − 3.64201i −3.59426 + 4.94708i 2.45798 + 0.798646i 14.9889 − 4.87020i −2.42705 + 1.76336i 21.8752i
118.2 −1.38297 − 1.90349i 0.535233 + 1.64728i −0.474619 + 1.46073i 3.39476 + 2.46644i 2.39537 − 3.29695i −6.11349 − 1.98639i −5.51390 + 1.79158i −2.42705 + 1.76336i − 9.87291i
118.3 1.38297 + 1.90349i 0.535233 + 1.64728i −0.474619 + 1.46073i 3.39476 + 2.46644i −2.39537 + 3.29695i 6.11349 + 1.98639i 5.51390 − 1.79158i −2.42705 + 1.76336i 9.87291i
118.4 2.07515 + 2.85620i −0.535233 − 1.64728i −2.61555 + 8.04984i −5.01279 − 3.64201i 3.59426 − 4.94708i −2.45798 − 0.798646i −14.9889 + 4.87020i −2.42705 + 1.76336i − 21.8752i
\(n\): 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.b odd 2 1 inner
11.c even 5 3 inner
11.d odd 10 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.3.g.e 16
11.b odd 2 1 inner 363.3.g.e 16
11.c even 5 1 33.3.c.a ✓ 4
11.c even 5 3 inner 363.3.g.e 16
11.d odd 10 1 33.3.c.a ✓ 4
11.d odd 10 3 inner 363.3.g.e 16
33.f even 10 1 99.3.c.b 4
33.h odd 10 1 99.3.c.b 4
44.g even 10 1 528.3.j.c 4
44.h odd 10 1 528.3.j.c 4
55.h odd 10 1 825.3.b.a 4
55.j even 10 1 825.3.b.a 4
55.k odd 20 2 825.3.h.a 8
55.l even 20 2 825.3.h.a 8
88.k even 10 1 2112.3.j.d 4
88.l odd 10 1 2112.3.j.d 4
88.o even 10 1 2112.3.j.a 4
88.p odd 10 1 2112.3.j.a 4
132.n odd 10 1 1584.3.j.f 4
132.o even 10 1 1584.3.j.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.3.c.a ✓ 4 11.c even 5 1
33.3.c.a ✓ 4 11.d odd 10 1
99.3.c.b 4 33.f even 10 1
99.3.c.b 4 33.h odd 10 1
363.3.g.e 16 1.a even 1 1 trivial
363.3.g.e 16 11.b odd 2 1 inner
363.3.g.e 16 11.c even 5 3 inner
363.3.g.e 16 11.d odd 10 3 inner
528.3.j.c 4 44.g even 10 1
528.3.j.c 4 44.h odd 10 1
825.3.b.a 4 55.h odd 10 1
825.3.b.a 4 55.j even 10 1
825.3.h.a 8 55.k odd 20 2
825.3.h.a 8 55.l even 20 2
1584.3.j.f 4 132.n odd 10 1
1584.3.j.f 4 132.o even 10 1
2112.3.j.a 4 88.o even 10 1
2112.3.j.a 4 88.p odd 10 1
2112.3.j.d 4 88.k even 10 1
2112.3.j.d 4 88.l odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - 18 T_{2}^{14} + 255 T_{2}^{12} - 3348 T_{2}^{10} + 42669 T_{2}^{8} - 231012 T_{2}^{6} + \cdots + 22667121 \) acting on \(S_{3}^{\mathrm{new}}(363, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 18 T^{14} + \cdots + 22667121 \) Copy content Toggle raw display
$3$ \( (T^{8} + 3 T^{6} + 9 T^{4} + \cdots + 81)^{2} \) Copy content Toggle raw display
$5$ \( (T^{8} + 2 T^{7} + \cdots + 456976)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 5802782976 \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 12\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 97\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 97\!\cdots\!16 \) Copy content Toggle raw display
$23$ \( (T^{2} + 46 T + 454)^{8} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 24\!\cdots\!96 \) Copy content Toggle raw display
$31$ \( (T^{8} + 40 T^{7} + \cdots + 107049369856)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 32 T^{7} + \cdots + 3544535296)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 380291185115136 \) Copy content Toggle raw display
$43$ \( (T^{4} + 7272 T^{2} + 3843024)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} + 50 T^{7} + \cdots + 127880620816)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 14 T^{7} + \cdots + 9721171216)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 726672516714496)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 45\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( (T + 34)^{16} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 644217699525136)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 53\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 380291185115136 \) Copy content Toggle raw display
$89$ \( (T^{2} - 152 T + 4804)^{8} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 157751992324096)^{2} \) Copy content Toggle raw display
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