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Properties

Label 506.2.e.g
Level $506$
Weight $2$
Character orbit 506.e
Analytic conductor $4.040$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [506,2,Mod(47,506)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(506, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([8, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("506.47");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 506 = 2 \cdot 11 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 506.e (of order \(5\), degree \(4\), 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: \(4.04043034228\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 4 x^{14} - x^{13} + 31 x^{12} - 108 x^{11} + 386 x^{10} - 568 x^{9} + 968 x^{8} + \cdots + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{8} q^{2} + ( - \beta_{5} - \beta_{4} + \cdots + \beta_1) q^{3}+ \cdots + (\beta_{13} - \beta_{12} - \beta_{11} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{8} q^{2} + ( - \beta_{5} - \beta_{4} + \cdots + \beta_1) q^{3}+ \cdots + (\beta_{15} - 2 \beta_{14} - 3 \beta_{13} + \cdots - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{2} + 5 q^{3} - 4 q^{4} + 7 q^{5} + 8 q^{7} - 4 q^{8} - 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{2} + 5 q^{3} - 4 q^{4} + 7 q^{5} + 8 q^{7} - 4 q^{8} - 15 q^{9} - 18 q^{10} - 7 q^{11} - 10 q^{12} + 9 q^{13} + 8 q^{14} - 26 q^{15} - 4 q^{16} + 2 q^{17} + 10 q^{18} + 2 q^{19} + 7 q^{20} + 30 q^{21} + 8 q^{22} + 16 q^{23} + q^{25} - q^{26} + 14 q^{27} - 2 q^{28} + 6 q^{29} - 26 q^{30} - 15 q^{31} + 16 q^{32} - 35 q^{33} + 2 q^{34} + 11 q^{35} + 10 q^{36} - 19 q^{37} + 2 q^{38} + 40 q^{39} + 2 q^{40} - 4 q^{43} - 2 q^{44} - 48 q^{45} - 4 q^{46} + 23 q^{47} + 5 q^{48} + 28 q^{49} + 6 q^{50} + 32 q^{51} - q^{52} - 40 q^{53} - 16 q^{54} - 9 q^{55} - 12 q^{56} + 8 q^{57} + 6 q^{58} - 20 q^{59} + 24 q^{60} + 3 q^{61} + 25 q^{62} + 10 q^{63} - 4 q^{64} + 38 q^{65} + 10 q^{66} + 12 q^{67} + 2 q^{68} + 5 q^{69} - 24 q^{70} + 25 q^{71} - 15 q^{72} + 5 q^{73} - 19 q^{74} - 73 q^{75} - 8 q^{76} - 48 q^{77} - 60 q^{78} + 34 q^{79} + 2 q^{80} - 25 q^{81} - 5 q^{82} + 19 q^{83} - 15 q^{84} + 10 q^{85} + 21 q^{86} + 22 q^{87} + 3 q^{88} - 74 q^{89} - 3 q^{90} + 15 q^{91} - 4 q^{92} + 63 q^{93} + 23 q^{94} - 22 q^{95} + 5 q^{96} - 48 q^{97} + 8 q^{98} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 4 x^{14} - x^{13} + 31 x^{12} - 108 x^{11} + 386 x^{10} - 568 x^{9} + 968 x^{8} + \cdots + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 53\!\cdots\!50 \nu^{15} + \cdots + 69\!\cdots\!28 ) / 10\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 10\!\cdots\!43 \nu^{15} + \cdots + 70\!\cdots\!24 ) / 20\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 14\!\cdots\!63 \nu^{15} + \cdots - 12\!\cdots\!60 ) / 20\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 17\!\cdots\!72 \nu^{15} + \cdots + 23\!\cdots\!08 ) / 20\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 92\!\cdots\!25 \nu^{15} + \cdots - 48\!\cdots\!96 ) / 10\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 19\!\cdots\!63 \nu^{15} + \cdots - 88\!\cdots\!52 ) / 18\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 23\!\cdots\!04 \nu^{15} + \cdots + 87\!\cdots\!84 ) / 20\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 36\!\cdots\!58 \nu^{15} + \cdots - 17\!\cdots\!64 ) / 20\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 38\!\cdots\!89 \nu^{15} + \cdots + 25\!\cdots\!32 ) / 20\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 41\!\cdots\!57 \nu^{15} + \cdots + 56\!\cdots\!32 ) / 20\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 44\!\cdots\!64 \nu^{15} + \cdots - 26\!\cdots\!48 ) / 20\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 79\!\cdots\!63 \nu^{15} + \cdots + 69\!\cdots\!56 ) / 20\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 95\!\cdots\!90 \nu^{15} + \cdots + 52\!\cdots\!84 ) / 20\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 17\!\cdots\!04 \nu^{15} + \cdots + 10\!\cdots\!20 ) / 20\!\cdots\!88 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{14} + \beta_{12} + \beta_{9} - 3\beta_{8} - \beta_{5} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{15} - 3 \beta_{12} - \beta_{9} + 2 \beta_{8} + \beta_{6} + 6 \beta_{5} - \beta_{4} + \beta_{3} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{15} + 7 \beta_{14} + \beta_{13} + 17 \beta_{12} + 7 \beta_{11} - 9 \beta_{8} - 2 \beta_{5} + \cdots + 2 \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{13} + 2 \beta_{12} - 9 \beta_{11} - 8 \beta_{10} + \beta_{9} - 7 \beta_{8} - 7 \beta_{7} + \cdots + 23 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 6 \beta_{15} - 45 \beta_{14} + 44 \beta_{13} + 11 \beta_{12} - 11 \beta_{11} - 5 \beta_{10} - 56 \beta_{9} + \cdots - 56 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 44 \beta_{15} + 45 \beta_{14} + 25 \beta_{13} + 164 \beta_{12} - 69 \beta_{11} - 44 \beta_{10} + \cdots + 164 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 15 \beta_{15} - 361 \beta_{14} - 847 \beta_{12} - 252 \beta_{11} - 267 \beta_{9} + 832 \beta_{8} + \cdots - 359 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 94 \beta_{15} + 441 \beta_{14} - 65 \beta_{13} + 1233 \beta_{12} + 718 \beta_{11} + 277 \beta_{10} + \cdots - 277 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1787 \beta_{13} - 3241 \beta_{12} - 1002 \beta_{11} - 99 \beta_{10} + 1787 \beta_{9} - 164 \beta_{8} + \cdots + 2602 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 739 \beta_{15} - 3356 \beta_{14} + 1569 \beta_{13} - 3265 \beta_{12} + 3265 \beta_{11} + 2526 \beta_{10} + \cdots - 10681 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1569 \beta_{15} + 19218 \beta_{14} - 5607 \beta_{13} + 28054 \beta_{12} + 4038 \beta_{11} - 1569 \beta_{10} + \cdots + 28054 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 11864 \beta_{15} - 36281 \beta_{14} - 102220 \beta_{12} - 19546 \beta_{11} - 31410 \beta_{9} + \cdots - 43303 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 4871 \beta_{15} + 94880 \beta_{14} + 41886 \beta_{13} + 303472 \beta_{12} + 108819 \beta_{11} + \cdots - 13939 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 118236 \beta_{13} - 300249 \beta_{12} - 259175 \beta_{11} - 122827 \beta_{10} + 118236 \beta_{9} + \cdots + 450469 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/506\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(419\)
\(\chi(n)\) \(\beta_{12}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
−0.375603 − 1.15599i
−0.185488 − 0.570873i
0.390147 + 1.20075i
0.670944 + 2.06495i
1.91645 + 1.39239i
1.01346 + 0.736321i
−0.197607 − 0.143570i
−2.23231 − 1.62187i
1.91645 − 1.39239i
1.01346 − 0.736321i
−0.197607 + 0.143570i
−2.23231 + 1.62187i
−0.375603 + 1.15599i
−0.185488 + 0.570873i
0.390147 − 1.20075i
0.670944 − 2.06495i
−0.809017 + 0.587785i −0.916756 − 2.82148i 0.309017 − 0.951057i −0.174325 − 0.126655i 2.40010 + 1.74377i −0.406350 + 1.25062i 0.309017 + 0.951057i −4.69328 + 3.40987i 0.215478
47.2 −0.809017 + 0.587785i −0.609143 − 1.87475i 0.309017 − 0.951057i 0.323404 + 0.234966i 1.59476 + 1.15866i −0.251465 + 0.773930i 0.309017 + 0.951057i −0.716575 + 0.520622i −0.399749
47.3 −0.809017 + 0.587785i 0.322254 + 0.991797i 0.309017 − 0.951057i 1.83044 + 1.32989i −0.843673 − 0.612964i 1.00658 − 3.09794i 0.309017 + 0.951057i 1.54724 − 1.12413i −2.26254
47.4 −0.809017 + 0.587785i 0.776593 + 2.39011i 0.309017 − 0.951057i 2.56557 + 1.86400i −2.03315 − 1.47717i −0.584835 + 1.79994i 0.309017 + 0.951057i −2.68247 + 1.94893i −3.17122
93.1 0.309017 + 0.951057i −0.375417 − 0.272757i −0.809017 + 0.587785i 0.423004 − 1.30187i 0.143397 − 0.441329i −0.796565 + 0.578738i −0.809017 − 0.587785i −0.860509 − 2.64837i 1.36887
93.2 0.309017 + 0.951057i 0.182665 + 0.132714i −0.809017 + 0.587785i 0.0780897 − 0.240335i −0.0697719 + 0.214736i 3.29424 − 2.39341i −0.809017 − 0.587785i −0.911297 − 2.80469i 0.252704
93.3 0.309017 + 0.951057i 0.931145 + 0.676516i −0.809017 + 0.587785i −0.384496 + 1.18336i −0.355666 + 1.09463i −1.90739 + 1.38580i −0.809017 − 0.587785i −0.517695 − 1.59330i −1.24426
93.4 0.309017 + 0.951057i 2.18866 + 1.59015i −0.809017 + 0.587785i −1.16168 + 3.57529i −0.835993 + 2.57292i 3.64578 − 2.64882i −0.809017 − 0.587785i 1.33459 + 4.10743i −3.75928
185.1 0.309017 − 0.951057i −0.375417 + 0.272757i −0.809017 − 0.587785i 0.423004 + 1.30187i 0.143397 + 0.441329i −0.796565 − 0.578738i −0.809017 + 0.587785i −0.860509 + 2.64837i 1.36887
185.2 0.309017 − 0.951057i 0.182665 − 0.132714i −0.809017 − 0.587785i 0.0780897 + 0.240335i −0.0697719 − 0.214736i 3.29424 + 2.39341i −0.809017 + 0.587785i −0.911297 + 2.80469i 0.252704
185.3 0.309017 − 0.951057i 0.931145 − 0.676516i −0.809017 − 0.587785i −0.384496 − 1.18336i −0.355666 − 1.09463i −1.90739 − 1.38580i −0.809017 + 0.587785i −0.517695 + 1.59330i −1.24426
185.4 0.309017 − 0.951057i 2.18866 − 1.59015i −0.809017 − 0.587785i −1.16168 − 3.57529i −0.835993 − 2.57292i 3.64578 + 2.64882i −0.809017 + 0.587785i 1.33459 − 4.10743i −3.75928
323.1 −0.809017 − 0.587785i −0.916756 + 2.82148i 0.309017 + 0.951057i −0.174325 + 0.126655i 2.40010 − 1.74377i −0.406350 − 1.25062i 0.309017 − 0.951057i −4.69328 − 3.40987i 0.215478
323.2 −0.809017 − 0.587785i −0.609143 + 1.87475i 0.309017 + 0.951057i 0.323404 − 0.234966i 1.59476 − 1.15866i −0.251465 − 0.773930i 0.309017 − 0.951057i −0.716575 − 0.520622i −0.399749
323.3 −0.809017 − 0.587785i 0.322254 − 0.991797i 0.309017 + 0.951057i 1.83044 − 1.32989i −0.843673 + 0.612964i 1.00658 + 3.09794i 0.309017 − 0.951057i 1.54724 + 1.12413i −2.26254
323.4 −0.809017 − 0.587785i 0.776593 − 2.39011i 0.309017 + 0.951057i 2.56557 − 1.86400i −2.03315 + 1.47717i −0.584835 − 1.79994i 0.309017 − 0.951057i −2.68247 − 1.94893i −3.17122
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 506.2.e.g ✓ 16
11.c even 5 1 inner 506.2.e.g ✓ 16
11.c even 5 1 5566.2.a.bq 8
11.d odd 10 1 5566.2.a.bn 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
506.2.e.g ✓ 16 1.a even 1 1 trivial
506.2.e.g ✓ 16 11.c even 5 1 inner
5566.2.a.bn 8 11.d odd 10 1
5566.2.a.bq 8 11.c even 5 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(506, [\chi])\):

\( T_{3}^{16} - 5 T_{3}^{15} + 26 T_{3}^{14} - 93 T_{3}^{13} + 311 T_{3}^{12} - 771 T_{3}^{11} + 1785 T_{3}^{10} + \cdots + 25 \) Copy content Toggle raw display
\( T_{5}^{16} - 7 T_{5}^{15} + 34 T_{5}^{14} - 143 T_{5}^{13} + 533 T_{5}^{12} - 1343 T_{5}^{11} + 2469 T_{5}^{10} + \cdots + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} - 5 T^{15} + \cdots + 25 \) Copy content Toggle raw display
$5$ \( T^{16} - 7 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{16} - 8 T^{15} + \cdots + 78961 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 214358881 \) Copy content Toggle raw display
$13$ \( T^{16} - 9 T^{15} + \cdots + 16900321 \) Copy content Toggle raw display
$17$ \( T^{16} - 2 T^{15} + \cdots + 52441 \) Copy content Toggle raw display
$19$ \( T^{16} - 2 T^{15} + \cdots + 477481 \) Copy content Toggle raw display
$23$ \( (T - 1)^{16} \) Copy content Toggle raw display
$29$ \( T^{16} - 6 T^{15} + \cdots + 10995856 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 4526598400 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 1273704721 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 31086211401001 \) Copy content Toggle raw display
$43$ \( (T^{8} + 2 T^{7} + \cdots - 114224)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 389276166400 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 1252876816 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 86214103374025 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 10585117456 \) Copy content Toggle raw display
$67$ \( (T^{8} - 6 T^{7} + \cdots - 44224)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 5999437890625 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 42351993616 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 219528794521 \) Copy content Toggle raw display
$83$ \( T^{16} - 19 T^{15} + \cdots + 87254281 \) Copy content Toggle raw display
$89$ \( (T^{8} + 37 T^{7} + \cdots - 68628899)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 689532605161 \) Copy content Toggle raw display
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