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Properties

Label 666.4.s.d
Level $666$
Weight $4$
Character orbit 666.s
Analytic conductor $39.295$
Analytic rank $0$
Dimension $20$
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] = [666,4,Mod(307,666)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(666, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("666.307");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 666.s (of order \(6\), degree \(2\), 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: \(39.2952720638\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 346 x^{18} + 50697 x^{16} + 4104768 x^{14} + 200532432 x^{12} + 6039270720 x^{10} + \cdots + 1118416232704 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{18}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 74)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{11} q^{2} + 4 \beta_{6} q^{4} + ( - \beta_{10} - \beta_{6} + 1) q^{5} + ( - \beta_{18} + \beta_{14} + \beta_{8} + \cdots - 1) q^{7}+ \cdots + ( - 4 \beta_{12} - 4 \beta_{11}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{11} q^{2} + 4 \beta_{6} q^{4} + ( - \beta_{10} - \beta_{6} + 1) q^{5} + ( - \beta_{18} + \beta_{14} + \beta_{8} + \cdots - 1) q^{7}+ \cdots + (36 \beta_{19} - 16 \beta_{18} + \cdots + 17) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 40 q^{4} + 18 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 40 q^{4} + 18 q^{5} - 2 q^{7} - 16 q^{10} + 16 q^{11} - 150 q^{13} - 160 q^{16} - 90 q^{17} + 162 q^{19} + 72 q^{20} + 532 q^{25} - 528 q^{26} + 8 q^{28} - 488 q^{34} + 342 q^{35} - 112 q^{37} - 144 q^{38} - 32 q^{40} + 498 q^{41} + 32 q^{44} - 424 q^{47} + 84 q^{49} - 1008 q^{50} - 600 q^{52} + 142 q^{53} - 540 q^{55} + 224 q^{58} - 1590 q^{59} - 1542 q^{61} - 8 q^{62} - 1280 q^{64} + 694 q^{65} + 62 q^{67} - 368 q^{70} + 178 q^{71} - 528 q^{73} + 560 q^{74} + 648 q^{76} - 3468 q^{77} - 3474 q^{79} - 938 q^{83} - 1100 q^{85} + 2120 q^{86} - 510 q^{89} + 666 q^{91} - 1344 q^{92} + 264 q^{94} - 4126 q^{95} + 816 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 346 x^{18} + 50697 x^{16} + 4104768 x^{14} + 200532432 x^{12} + 6039270720 x^{10} + \cdots + 1118416232704 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 17184 \nu^{18} + 5549863 \nu^{16} + 741687905 \nu^{14} + 53039703547 \nu^{12} + \cdots + 24\!\cdots\!00 ) / 856978891847424 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3111595001479 \nu^{18} + 968351882231062 \nu^{16} + \cdots + 93\!\cdots\!52 ) / 13\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 25858167 \nu^{18} + 8050739056 \nu^{16} + 1031897132831 \nu^{14} + 70375153812298 \nu^{12} + \cdots + 83\!\cdots\!96 ) / 46\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 8070670183116 \nu^{18} + \cdots - 26\!\cdots\!08 ) / 66\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1137896675 \nu^{19} + 384625762766 \nu^{17} + 54753313374787 \nu^{15} + \cdots + 45\!\cdots\!24 ) / 90\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 847723382320573 \nu^{19} + \cdots + 45\!\cdots\!00 ) / 10\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 847723382320573 \nu^{19} + \cdots + 45\!\cdots\!00 ) / 10\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 69\!\cdots\!11 \nu^{19} + \cdots - 24\!\cdots\!24 ) / 21\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 69\!\cdots\!11 \nu^{19} + \cdots + 24\!\cdots\!24 ) / 21\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 76\!\cdots\!23 \nu^{19} + \cdots + 23\!\cdots\!68 ) / 47\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 76\!\cdots\!23 \nu^{19} + \cdots - 23\!\cdots\!68 ) / 47\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 12\!\cdots\!39 \nu^{19} + \cdots + 12\!\cdots\!88 ) / 35\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 591265350194 \nu^{19} + 2017790345511 \nu^{18} + 184042545216882 \nu^{17} + \cdots + 65\!\cdots\!68 ) / 72\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 46\!\cdots\!31 \nu^{19} + \cdots + 10\!\cdots\!16 ) / 28\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 46\!\cdots\!31 \nu^{19} + \cdots - 10\!\cdots\!16 ) / 28\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 18\!\cdots\!81 \nu^{19} + \cdots + 21\!\cdots\!12 ) / 10\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 20\!\cdots\!15 \nu^{19} + \cdots + 48\!\cdots\!56 ) / 10\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 20\!\cdots\!15 \nu^{19} + \cdots - 48\!\cdots\!56 ) / 10\!\cdots\!80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + \beta_{7} + \beta_{5} - 2\beta_{4} - \beta_{3} - 35 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 10 \beta_{19} - 10 \beta_{18} - 12 \beta_{17} - \beta_{16} - \beta_{15} + 32 \beta_{14} + 10 \beta_{13} + \cdots - 31 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 18 \beta_{19} - 18 \beta_{18} + 63 \beta_{12} - 63 \beta_{11} + 6 \beta_{10} - 6 \beta_{9} - 53 \beta_{8} + \cdots + 1842 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1100 \beta_{19} + 1100 \beta_{18} + 1128 \beta_{17} + 17 \beta_{16} + 17 \beta_{15} - 3748 \beta_{14} + \cdots + 5057 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 1974 \beta_{19} + 1974 \beta_{18} + 60 \beta_{16} - 60 \beta_{15} - 8631 \beta_{12} + 8631 \beta_{11} + \cdots - 111490 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 97956 \beta_{19} - 97956 \beta_{18} - 90744 \beta_{17} + 2103 \beta_{16} + 2103 \beta_{15} + \cdots - 521361 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 179598 \beta_{19} - 179598 \beta_{18} - 9222 \beta_{16} + 9222 \beta_{15} + 895419 \beta_{12} + \cdots + 7419740 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 8250220 \beta_{19} + 8250220 \beta_{18} + 7048872 \beta_{17} - 351923 \beta_{16} - 351923 \beta_{15} + \cdots + 46622605 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 15483054 \beta_{19} + 15483054 \beta_{18} + 1043268 \beta_{16} - 1043268 \beta_{15} + \cdots - 528154590 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 681819932 \beta_{19} - 681819932 \beta_{18} - 544782600 \beta_{17} + 38507695 \beta_{16} + \cdots - 3941341073 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1305733614 \beta_{19} - 1305733614 \beta_{18} - 104415330 \beta_{16} + 104415330 \beta_{15} + \cdots + 39393611536 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 55973553228 \beta_{19} + 55973553228 \beta_{18} + 42315645288 \beta_{17} - 3702226683 \beta_{16} + \cdots + 325429268229 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 108956680782 \beta_{19} + 108956680782 \beta_{18} + 9784553520 \beta_{16} - 9784553520 \beta_{15} + \cdots - 3033275040866 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 4586257361116 \beta_{19} - 4586257361116 \beta_{18} - 3314535601416 \beta_{17} + \cdots - 26619356988121 ) / 2 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 9041754045102 \beta_{19} - 9041754045102 \beta_{18} - 880632614766 \beta_{16} + 880632614766 \beta_{15} + \cdots + 238630955366916 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 375766067065196 \beta_{19} + 375766067065196 \beta_{18} + 261970896361128 \beta_{17} + \cdots + 21\!\cdots\!25 ) / 2 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 748080482854734 \beta_{19} + 748080482854734 \beta_{18} + 77193638799228 \beta_{16} + \cdots - 19\!\cdots\!42 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 30\!\cdots\!44 \beta_{19} + \cdots - 17\!\cdots\!81 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/666\mathbb{Z}\right)^\times\).

\(n\) \(371\) \(631\)
\(\chi(n)\) \(1\) \(\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
− 7.08776i
5.08624i
1.08183i
− 5.89146i
7.81115i
− 6.77277i
9.07921i
− 0.522574i
3.11082i
− 5.89468i
7.08776i
− 5.08624i
− 1.08183i
5.89146i
− 7.81115i
6.77277i
− 9.07921i
0.522574i
− 3.11082i
5.89468i
−1.73205 + 1.00000i 0 2.00000 − 3.46410i −13.1273 − 7.57908i 0 −5.28959 + 9.16184i 8.00000i 0 30.3163
307.2 −1.73205 + 1.00000i 0 2.00000 − 3.46410i −9.68300 − 5.59048i 0 −10.2076 + 17.6800i 8.00000i 0 22.3619
307.3 −1.73205 + 1.00000i 0 2.00000 − 3.46410i −2.09330 − 1.20857i 0 17.5687 − 30.4298i 8.00000i 0 4.83427
307.4 −1.73205 + 1.00000i 0 2.00000 − 3.46410i 12.2374 + 7.06527i 0 0.0476865 − 0.0825955i 8.00000i 0 −28.2611
307.5 −1.73205 + 1.00000i 0 2.00000 − 3.46410i 18.8983 + 10.9109i 0 −2.61919 + 4.53657i 8.00000i 0 −43.6437
307.6 1.73205 − 1.00000i 0 2.00000 − 3.46410i −14.5934 − 8.42549i 0 −7.56088 + 13.0958i − 8.00000i 0 −33.7019
307.7 1.73205 − 1.00000i 0 2.00000 − 3.46410i −4.92746 − 2.84487i 0 4.51268 − 7.81619i − 8.00000i 0 −11.3795
307.8 1.73205 − 1.00000i 0 2.00000 − 3.46410i −2.77235 − 1.60061i 0 8.06150 − 13.9629i − 8.00000i 0 −6.40246
307.9 1.73205 − 1.00000i 0 2.00000 − 3.46410i 12.2401 + 7.06683i 0 −13.5058 + 23.3927i − 8.00000i 0 28.2673
307.10 1.73205 − 1.00000i 0 2.00000 − 3.46410i 12.8210 + 7.40221i 0 7.99249 − 13.8434i − 8.00000i 0 29.6089
397.1 −1.73205 − 1.00000i 0 2.00000 + 3.46410i −13.1273 + 7.57908i 0 −5.28959 − 9.16184i − 8.00000i 0 30.3163
397.2 −1.73205 − 1.00000i 0 2.00000 + 3.46410i −9.68300 + 5.59048i 0 −10.2076 − 17.6800i − 8.00000i 0 22.3619
397.3 −1.73205 − 1.00000i 0 2.00000 + 3.46410i −2.09330 + 1.20857i 0 17.5687 + 30.4298i − 8.00000i 0 4.83427
397.4 −1.73205 − 1.00000i 0 2.00000 + 3.46410i 12.2374 − 7.06527i 0 0.0476865 + 0.0825955i − 8.00000i 0 −28.2611
397.5 −1.73205 − 1.00000i 0 2.00000 + 3.46410i 18.8983 − 10.9109i 0 −2.61919 − 4.53657i − 8.00000i 0 −43.6437
397.6 1.73205 + 1.00000i 0 2.00000 + 3.46410i −14.5934 + 8.42549i 0 −7.56088 − 13.0958i 8.00000i 0 −33.7019
397.7 1.73205 + 1.00000i 0 2.00000 + 3.46410i −4.92746 + 2.84487i 0 4.51268 + 7.81619i 8.00000i 0 −11.3795
397.8 1.73205 + 1.00000i 0 2.00000 + 3.46410i −2.77235 + 1.60061i 0 8.06150 + 13.9629i 8.00000i 0 −6.40246
397.9 1.73205 + 1.00000i 0 2.00000 + 3.46410i 12.2401 − 7.06683i 0 −13.5058 − 23.3927i 8.00000i 0 28.2673
397.10 1.73205 + 1.00000i 0 2.00000 + 3.46410i 12.8210 − 7.40221i 0 7.99249 + 13.8434i 8.00000i 0 29.6089
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 666.4.s.d 20
3.b odd 2 1 74.4.e.a ✓ 20
37.e even 6 1 inner 666.4.s.d 20
111.h odd 6 1 74.4.e.a ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.4.e.a ✓ 20 3.b odd 2 1
74.4.e.a ✓ 20 111.h odd 6 1
666.4.s.d 20 1.a even 1 1 trivial
666.4.s.d 20 37.e even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{20} - 18 T_{5}^{19} - 729 T_{5}^{18} + 15066 T_{5}^{17} + 383175 T_{5}^{16} + \cdots + 65\!\cdots\!25 \) acting on \(S_{4}^{\mathrm{new}}(666, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 4 T^{2} + 16)^{5} \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 65\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 12\!\cdots\!84 \) Copy content Toggle raw display
$11$ \( (T^{10} + \cdots - 1734696000000)^{2} \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 15\!\cdots\!81 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 37\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 59\!\cdots\!84 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 80\!\cdots\!36 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 11\!\cdots\!49 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 40\!\cdots\!09 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 55\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots + 49\!\cdots\!76)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 14\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 19\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 65\!\cdots\!69 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 17\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 60\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 35\!\cdots\!04 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 55\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 12\!\cdots\!25 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 12\!\cdots\!04 \) Copy content Toggle raw display
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