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Properties

Label 363.3.h.j
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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [363,3,Mod(245,363)]
 
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([5, 8]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("363.245");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 363.h (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: \( x^{16} - 12x^{14} + 180x^{12} - 2562x^{10} + 25179x^{8} - 96398x^{6} + 239275x^{4} - 536393x^{2} + 1771561 \) 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

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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{10} + \beta_{6} - 1) q^{3} + ( - \beta_{13} + \beta_{12} - \beta_{10} + \cdots - 1) q^{4}+ \cdots + (\beta_{14} - 2 \beta_{13} - 3 \beta_{11} + \cdots - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{10} + \beta_{6} - 1) q^{3} + ( - \beta_{13} + \beta_{12} - \beta_{10} + \cdots - 1) q^{4}+ \cdots + ( - 8 \beta_{15} - 8 \beta_{14} + \cdots - 22 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 10 q^{3} + 8 q^{4} + 33 q^{6} - 6 q^{7} - 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 10 q^{3} + 8 q^{4} + 33 q^{6} - 6 q^{7} - 28 q^{9} + 12 q^{10} + 106 q^{12} + 42 q^{13} + 82 q^{15} - 88 q^{16} + 43 q^{18} + 134 q^{19} + 12 q^{21} - 41 q^{24} + 134 q^{25} + 80 q^{27} - 264 q^{28} + 120 q^{30} + 124 q^{31} - 132 q^{34} - 219 q^{36} + 90 q^{37} + 174 q^{39} + 284 q^{40} - 102 q^{42} + 156 q^{43} - 72 q^{45} + 22 q^{46} + 30 q^{48} - 30 q^{49} - 111 q^{51} - 326 q^{52} - 1046 q^{54} - 281 q^{57} - 116 q^{58} + 54 q^{60} + 126 q^{61} + 138 q^{63} + 236 q^{64} + 368 q^{67} + 198 q^{69} - 322 q^{70} + 562 q^{72} - 24 q^{73} - 21 q^{75} + 900 q^{76} - 492 q^{78} + 314 q^{79} - 388 q^{81} - 270 q^{84} - 318 q^{85} - 132 q^{87} - 176 q^{90} + 374 q^{91} - 10 q^{93} - 990 q^{94} + 332 q^{96} + 72 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

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

\(\nu\)\(=\) \( \beta_1 \) 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
\(\nu^{4}\)\(=\) \( \beta_{15} - \beta_{14} - 15 \beta_{13} + 15 \beta_{12} + \beta_{9} + \beta_{8} + \beta_{7} + \cdots + \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 8 \beta_{15} - 8 \beta_{14} - 41 \beta_{13} - 41 \beta_{12} - 85 \beta_{11} + 11 \beta_{10} + \cdots - 96 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 190 \beta_{15} + 190 \beta_{14} + 220 \beta_{13} - 220 \beta_{12} + 220 \beta_{10} + 30 \beta_{9} + \cdots + 378 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 603 \beta_{15} + 603 \beta_{14} + 1143 \beta_{13} + 1143 \beta_{12} + 1073 \beta_{11} + 540 \beta_{10} + \cdots + 721 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2341 \beta_{10} - 2341 \beta_{9} + 588 \beta_{8} - 588 \beta_{7} + 6969 \beta_{6} - 12506 \beta_{3} + \cdots - 6969 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 6381 \beta_{15} + 6381 \beta_{14} + 12816 \beta_{13} + 12816 \beta_{12} + 12827 \beta_{11} + \cdots + 6446 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 28893 \beta_{15} - 28893 \beta_{14} - 38546 \beta_{13} + 38546 \beta_{12} - 28893 \beta_{10} + \cdots - 80060 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 48156 \beta_{15} - 48156 \beta_{14} - 134901 \beta_{13} - 134901 \beta_{12} - 211927 \beta_{11} + \cdots - 205308 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 144620 \beta_{15} + 144620 \beta_{14} + 359360 \beta_{13} - 359360 \beta_{12} + 503980 \beta_{10} + \cdots + 1043432 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 219120 \beta_{10} - 219120 \beta_{9} - 1152580 \beta_{8} - 1152580 \beta_{7} + 464952 \beta_{5} + \cdots + 464952 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 2055772 \beta_{15} + 2055772 \beta_{14} + 4504525 \beta_{13} - 4504525 \beta_{12} + \cdots - 2055772 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 9398824 \beta_{15} + 9398824 \beta_{14} + 24575190 \beta_{13} + 24575190 \beta_{12} + \cdots + 32314430 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(122\) \(244\)
\(\chi(n)\) \(-1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
245.1
−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
−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 −2.00253 − 2.23380i −2.74053 + 8.43448i 1.22635 − 1.68793i −2.26043 + 10.5218i −2.73883 + 8.42924i 16.6100 − 5.39692i −0.979734 + 8.94651i −7.48447
245.2 −0.974642 − 1.34148i −1.09751 + 2.79204i 0.386428 − 1.18930i −0.410570 + 0.565101i 4.81514 − 1.24895i −0.806259 + 2.48141i −8.28007 + 2.69036i −6.59095 − 6.12857i 1.15823
245.3 0.974642 + 1.34148i 2.52902 + 1.61371i 0.386428 − 1.18930i 0.410570 − 0.565101i 0.300138 + 4.96542i −0.806259 + 2.48141i 8.28007 − 2.69036i 3.79191 + 8.16220i 1.15823
245.4 2.10855 + 2.90217i 0.307087 − 2.98424i −2.74053 + 8.43448i −1.22635 + 1.68793i 9.30827 − 5.40120i −2.73883 + 8.42924i −16.6100 + 5.39692i −8.81140 − 1.83284i −7.48447
251.1 −2.91048 − 0.945671i −1.86408 − 2.35058i 4.34051 + 3.15356i −6.31437 + 2.05166i 3.20248 + 8.60410i −2.47800 − 1.80037i −2.45561 − 3.37986i −2.05042 + 8.76332i 20.3180
251.2 −1.90610 − 0.619331i −1.63362 + 2.51621i 0.0135968 + 0.00987866i 5.21596 − 1.69477i 4.67221 − 3.78440i 4.52308 + 3.28621i 4.69235 + 6.45847i −3.66258 − 8.22104i −10.9918
251.3 1.90610 + 0.619331i −2.89787 + 0.776113i 0.0135968 + 0.00987866i −5.21596 + 1.69477i −6.00431 − 0.315387i 4.52308 + 3.28621i −4.69235 − 6.45847i 7.79530 − 4.49815i −10.9918
251.4 2.91048 + 0.945671i 1.65950 + 2.49921i 4.34051 + 3.15356i 6.31437 − 2.05166i 2.46650 + 8.84324i −2.47800 − 1.80037i 2.45561 + 3.37986i −3.49213 + 8.29488i 20.3180
269.1 −2.91048 + 0.945671i −1.86408 + 2.35058i 4.34051 − 3.15356i −6.31437 − 2.05166i 3.20248 − 8.60410i −2.47800 + 1.80037i −2.45561 + 3.37986i −2.05042 − 8.76332i 20.3180
269.2 −1.90610 + 0.619331i −1.63362 − 2.51621i 0.0135968 − 0.00987866i 5.21596 + 1.69477i 4.67221 + 3.78440i 4.52308 − 3.28621i 4.69235 − 6.45847i −3.66258 + 8.22104i −10.9918
269.3 1.90610 − 0.619331i −2.89787 − 0.776113i 0.0135968 − 0.00987866i −5.21596 − 1.69477i −6.00431 + 0.315387i 4.52308 − 3.28621i −4.69235 + 6.45847i 7.79530 + 4.49815i −10.9918
269.4 2.91048 − 0.945671i 1.65950 − 2.49921i 4.34051 − 3.15356i 6.31437 + 2.05166i 2.46650 − 8.84324i −2.47800 + 1.80037i 2.45561 − 3.37986i −3.49213 − 8.29488i 20.3180
323.1 −2.10855 + 2.90217i −2.00253 + 2.23380i −2.74053 − 8.43448i 1.22635 + 1.68793i −2.26043 − 10.5218i −2.73883 − 8.42924i 16.6100 + 5.39692i −0.979734 − 8.94651i −7.48447
323.2 −0.974642 + 1.34148i −1.09751 − 2.79204i 0.386428 + 1.18930i −0.410570 − 0.565101i 4.81514 + 1.24895i −0.806259 − 2.48141i −8.28007 − 2.69036i −6.59095 + 6.12857i 1.15823
323.3 0.974642 − 1.34148i 2.52902 − 1.61371i 0.386428 + 1.18930i 0.410570 + 0.565101i 0.300138 − 4.96542i −0.806259 − 2.48141i 8.28007 + 2.69036i 3.79191 − 8.16220i 1.15823
323.4 2.10855 − 2.90217i 0.307087 + 2.98424i −2.74053 − 8.43448i −1.22635 − 1.68793i 9.30827 + 5.40120i −2.73883 − 8.42924i −16.6100 − 5.39692i −8.81140 + 1.83284i −7.48447
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 245.4
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.j 16
3.b odd 2 1 inner 363.3.h.j 16
11.b odd 2 1 33.3.h.b ✓ 16
11.c even 5 1 363.3.b.l 8
11.c even 5 1 inner 363.3.h.j 16
11.c even 5 2 363.3.h.n 16
11.d odd 10 1 33.3.h.b ✓ 16
11.d odd 10 1 363.3.b.m 8
11.d odd 10 2 363.3.h.o 16
33.d even 2 1 33.3.h.b ✓ 16
33.f even 10 1 33.3.h.b ✓ 16
33.f even 10 1 363.3.b.m 8
33.f even 10 2 363.3.h.o 16
33.h odd 10 1 363.3.b.l 8
33.h odd 10 1 inner 363.3.h.j 16
33.h odd 10 2 363.3.h.n 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.3.h.b ✓ 16 11.b odd 2 1
33.3.h.b ✓ 16 11.d odd 10 1
33.3.h.b ✓ 16 33.d even 2 1
33.3.h.b ✓ 16 33.f even 10 1
363.3.b.l 8 11.c even 5 1
363.3.b.l 8 33.h odd 10 1
363.3.b.m 8 11.d odd 10 1
363.3.b.m 8 33.f even 10 1
363.3.h.j 16 1.a even 1 1 trivial
363.3.h.j 16 3.b odd 2 1 inner
363.3.h.j 16 11.c even 5 1 inner
363.3.h.j 16 33.h odd 10 1 inner
363.3.h.n 16 11.c even 5 2
363.3.h.n 16 33.h odd 10 2
363.3.h.o 16 11.d odd 10 2
363.3.h.o 16 33.f even 10 2

Hecke kernels

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

\( T_{2}^{16} - 12 T_{2}^{14} + 180 T_{2}^{12} - 2562 T_{2}^{10} + 25179 T_{2}^{8} - 96398 T_{2}^{6} + \cdots + 1771561 \) Copy content Toggle raw display
\( T_{5}^{16} - 117 T_{5}^{14} + 5980 T_{5}^{12} - 142582 T_{5}^{10} + 1407569 T_{5}^{8} + 2117552 T_{5}^{6} + \cdots + 7929856 \) Copy content Toggle raw display
\( T_{7}^{8} + 3T_{7}^{7} + 61T_{7}^{6} - 181T_{7}^{5} + 259T_{7}^{4} + 5802T_{7}^{3} + 36864T_{7}^{2} + 85536T_{7} + 156816 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 12 T^{14} + \cdots + 1771561 \) Copy content Toggle raw display
$3$ \( T^{16} + 10 T^{15} + \cdots + 43046721 \) Copy content Toggle raw display
$5$ \( T^{16} - 117 T^{14} + \cdots + 7929856 \) Copy content Toggle raw display
$7$ \( (T^{8} + 3 T^{7} + \cdots + 156816)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( (T^{8} - 21 T^{7} + \cdots + 19749136)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 69002444446441 \) Copy content Toggle raw display
$19$ \( (T^{8} - 67 T^{7} + \cdots + 4224870001)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 1848 T^{6} + \cdots + 25255373616)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 116101021696 \) Copy content Toggle raw display
$31$ \( (T^{8} - 62 T^{7} + \cdots + 9448617616)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 45 T^{7} + \cdots + 487254257296)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 15\!\cdots\!81 \) Copy content Toggle raw display
$43$ \( (T^{4} - 39 T^{3} + \cdots - 684409)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 21\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 30\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 26\!\cdots\!01 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 1393523586576)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 92 T^{3} + \cdots - 11977619)^{4} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 27\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 5458559358736)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 743504776736656)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 21\!\cdots\!81 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 3248250664131)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 9061117408561)^{2} \) Copy content Toggle raw display
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