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Properties

Label 273.4.k.d
Level $273$
Weight $4$
Character orbit 273.k
Analytic conductor $16.108$
Analytic rank $0$
Dimension $20$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [273,4,Mod(22,273)] 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("273.22"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(273, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 273.k (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(16.1075214316\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 5 x^{19} + 81 x^{18} - 194 x^{17} + 3136 x^{16} - 5035 x^{15} + 81800 x^{14} - 49122 x^{13} + \cdots + 43877376 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

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

\(f(q)\) \(=\) \( q + (\beta_{5} - \beta_{4} + \beta_1 - 1) q^{2} - 3 \beta_{5} q^{3} + ( - \beta_{12} + 6 \beta_{5} + \cdots - 6) q^{4} + (\beta_{6} + 2) q^{5} + ( - 3 \beta_{5} - 3 \beta_1 + 3) q^{6} + ( - 7 \beta_{5} + 7) q^{7}+ \cdots + (9 \beta_{19} - 18 \beta_{3} - 36) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 5 q^{2} - 30 q^{3} - 57 q^{4} + 30 q^{5} + 15 q^{6} + 70 q^{7} - 144 q^{8} - 90 q^{9} + 27 q^{10} + 40 q^{11} + 342 q^{12} + 166 q^{13} + 70 q^{14} - 45 q^{15} - 165 q^{16} - 46 q^{17} - 90 q^{18}+ \cdots - 720 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 5 x^{19} + 81 x^{18} - 194 x^{17} + 3136 x^{16} - 5035 x^{15} + 81800 x^{14} - 49122 x^{13} + \cdots + 43877376 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 60\!\cdots\!26 \nu^{19} + \cdots + 72\!\cdots\!52 ) / 33\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 16\!\cdots\!91 \nu^{19} + \cdots - 17\!\cdots\!88 ) / 13\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 30\!\cdots\!07 \nu^{19} + \cdots - 16\!\cdots\!16 ) / 13\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 14\!\cdots\!94 \nu^{19} + \cdots + 92\!\cdots\!68 ) / 28\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 69\!\cdots\!21 \nu^{19} + \cdots - 15\!\cdots\!48 ) / 64\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 14\!\cdots\!66 \nu^{19} + \cdots - 29\!\cdots\!20 ) / 12\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 21\!\cdots\!19 \nu^{19} + \cdots + 87\!\cdots\!64 ) / 64\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 10\!\cdots\!91 \nu^{19} + \cdots - 28\!\cdots\!44 ) / 26\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 56\!\cdots\!75 \nu^{19} + \cdots + 55\!\cdots\!32 ) / 12\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 19\!\cdots\!62 \nu^{19} + \cdots - 19\!\cdots\!84 ) / 33\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 18\!\cdots\!73 \nu^{19} + \cdots + 11\!\cdots\!40 ) / 28\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 22\!\cdots\!94 \nu^{19} + \cdots + 25\!\cdots\!48 ) / 27\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 58\!\cdots\!82 \nu^{19} + \cdots + 12\!\cdots\!64 ) / 66\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 58\!\cdots\!78 \nu^{19} + \cdots + 10\!\cdots\!16 ) / 66\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 29\!\cdots\!67 \nu^{19} + \cdots - 53\!\cdots\!88 ) / 16\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 34\!\cdots\!35 \nu^{19} + \cdots - 35\!\cdots\!44 ) / 15\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 66\!\cdots\!25 \nu^{19} + \cdots - 48\!\cdots\!76 ) / 26\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 40\!\cdots\!99 \nu^{19} + \cdots - 69\!\cdots\!72 ) / 12\!\cdots\!08 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{12} - 13\beta_{5} - \beta_{4} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{8} + \beta_{6} - 21\beta_{4} - 2\beta_{3} - 35 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{18} - \beta_{15} - \beta_{13} - 29 \beta_{12} + 2 \beta_{11} - \beta_{10} - \beta_{7} + \cdots - 272 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4 \beta_{19} - 2 \beta_{18} + \beta_{17} - 5 \beta_{16} + 4 \beta_{14} - 5 \beta_{13} - 92 \beta_{12} + \cdots + 516 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 20 \beta_{19} + 33 \beta_{17} - 61 \beta_{16} + 49 \beta_{15} + 33 \beta_{10} + 49 \beta_{9} + \cdots + 8288 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 107 \beta_{18} + 229 \beta_{15} - 236 \beta_{14} + 340 \beta_{13} + 3455 \beta_{12} - 1137 \beta_{11} + \cdots + 21245 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1260 \beta_{19} + 972 \beta_{18} - 989 \beta_{17} + 2769 \beta_{16} - 1260 \beta_{14} + 2769 \beta_{13} + \cdots - 56434 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 10212 \beta_{19} - 2764 \beta_{17} + 16016 \beta_{16} - 9534 \beta_{15} - 4194 \beta_{10} + \cdots - 1130810 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 29496 \beta_{18} - 66508 \beta_{15} + 56440 \beta_{14} - 111676 \beta_{13} - 809768 \beta_{12} + \cdots - 5203672 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 393380 \beta_{19} - 149049 \beta_{18} + 125423 \beta_{17} - 655147 \beta_{16} + 393380 \beta_{14} + \cdots + 12559265 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 2221388 \beta_{19} + 1096319 \beta_{17} - 4240811 \beta_{16} + 2270904 \beta_{15} + 934462 \beta_{10} + \cdots + 223342879 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 5118507 \beta_{18} + 12442935 \beta_{15} - 14359872 \beta_{14} + 25010365 \beta_{13} + 143214676 \beta_{12} + \cdots + 811397975 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 82231028 \beta_{19} + 30554915 \beta_{18} - 39284280 \beta_{17} + 155600932 \beta_{16} + \cdots - 2165026162 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 510158616 \beta_{19} - 210314433 \beta_{17} + 919020729 \beta_{16} - 427982040 \beta_{15} + \cdots - 40362637374 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 1018913980 \beta_{18} - 2568344116 \beta_{15} + 2948067668 \beta_{14} - 5590344470 \beta_{13} + \cdots - 164893955129 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 17866807308 \beta_{19} - 5900924792 \beta_{18} + 8030708560 \beta_{17} - 33012516116 \beta_{16} + \cdots + 436075868021 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 103850074652 \beta_{19} + 51455624653 \beta_{17} - 198154314169 \beta_{16} + 86327009901 \beta_{15} + \cdots + 7961298497213 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 200570795526 \beta_{18} + 494816852300 \beta_{15} - 620912942416 \beta_{14} + 1169199482889 \beta_{13} + \cdots + 31169946009609 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/273\mathbb{Z}\right)^\times\).

\(n\) \(92\) \(106\) \(157\)
\(\chi(n)\) \(1\) \(-1 + \beta_{5}\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1
2.93385 + 5.08158i
2.48265 + 4.30008i
2.24715 + 3.89219i
1.46299 + 2.53397i
0.268322 + 0.464748i
−0.143388 − 0.248356i
−1.22699 − 2.12520i
−1.25980 − 2.18205i
−2.06332 − 3.57378i
−2.20146 − 3.81305i
2.93385 − 5.08158i
2.48265 − 4.30008i
2.24715 − 3.89219i
1.46299 − 2.53397i
0.268322 − 0.464748i
−0.143388 + 0.248356i
−1.22699 + 2.12520i
−1.25980 + 2.18205i
−2.06332 + 3.57378i
−2.20146 + 3.81305i
−2.43385 + 4.21555i −1.50000 + 2.59808i −7.84726 − 13.5919i −16.7247 −7.30155 − 12.6467i 3.50000 + 6.06218i 37.4546 −4.50000 − 7.79423i 40.7054 − 70.5038i
22.2 −1.98265 + 3.43405i −1.50000 + 2.59808i −3.86181 − 6.68886i 17.1191 −5.94795 − 10.3022i 3.50000 + 6.06218i −1.09589 −4.50000 − 7.79423i −33.9413 + 58.7881i
22.3 −1.74715 + 3.02616i −1.50000 + 2.59808i −2.10510 − 3.64613i 3.30759 −5.24146 − 9.07848i 3.50000 + 6.06218i −13.2428 −4.50000 − 7.79423i −5.77887 + 10.0093i
22.4 −0.962990 + 1.66795i −1.50000 + 2.59808i 2.14530 + 3.71577i 5.67918 −2.88897 − 5.00384i 3.50000 + 6.06218i −23.6715 −4.50000 − 7.79423i −5.46900 + 9.47258i
22.5 0.231678 − 0.401278i −1.50000 + 2.59808i 3.89265 + 6.74227i −1.72939 0.695034 + 1.20383i 3.50000 + 6.06218i 7.31421 −4.50000 − 7.79423i −0.400662 + 0.693967i
22.6 0.643388 − 1.11438i −1.50000 + 2.59808i 3.17210 + 5.49424i −0.373462 1.93016 + 3.34314i 3.50000 + 6.06218i 18.4578 −4.50000 − 7.79423i −0.240281 + 0.416179i
22.7 1.72699 − 2.99123i −1.50000 + 2.59808i −1.96497 − 3.40342i 14.2940 5.18096 + 8.97369i 3.50000 + 6.06218i 14.0579 −4.50000 − 7.79423i 24.6855 − 42.7565i
22.8 1.75980 − 3.04807i −1.50000 + 2.59808i −2.19383 − 3.79982i −10.3397 5.27941 + 9.14421i 3.50000 + 6.06218i 12.7141 −4.50000 − 7.79423i −18.1959 + 31.5162i
22.9 2.56332 − 4.43981i −1.50000 + 2.59808i −9.14126 − 15.8331i −14.1720 7.68997 + 13.3194i 3.50000 + 6.06218i −52.7149 −4.50000 − 7.79423i −36.3276 + 62.9212i
22.10 2.70146 − 4.67907i −1.50000 + 2.59808i −10.5958 − 18.3525i 17.9394 8.10439 + 14.0372i 3.50000 + 6.06218i −71.2736 −4.50000 − 7.79423i 48.4627 − 83.9398i
211.1 −2.43385 − 4.21555i −1.50000 − 2.59808i −7.84726 + 13.5919i −16.7247 −7.30155 + 12.6467i 3.50000 − 6.06218i 37.4546 −4.50000 + 7.79423i 40.7054 + 70.5038i
211.2 −1.98265 − 3.43405i −1.50000 − 2.59808i −3.86181 + 6.68886i 17.1191 −5.94795 + 10.3022i 3.50000 − 6.06218i −1.09589 −4.50000 + 7.79423i −33.9413 − 58.7881i
211.3 −1.74715 − 3.02616i −1.50000 − 2.59808i −2.10510 + 3.64613i 3.30759 −5.24146 + 9.07848i 3.50000 − 6.06218i −13.2428 −4.50000 + 7.79423i −5.77887 − 10.0093i
211.4 −0.962990 − 1.66795i −1.50000 − 2.59808i 2.14530 − 3.71577i 5.67918 −2.88897 + 5.00384i 3.50000 − 6.06218i −23.6715 −4.50000 + 7.79423i −5.46900 − 9.47258i
211.5 0.231678 + 0.401278i −1.50000 − 2.59808i 3.89265 − 6.74227i −1.72939 0.695034 − 1.20383i 3.50000 − 6.06218i 7.31421 −4.50000 + 7.79423i −0.400662 − 0.693967i
211.6 0.643388 + 1.11438i −1.50000 − 2.59808i 3.17210 − 5.49424i −0.373462 1.93016 − 3.34314i 3.50000 − 6.06218i 18.4578 −4.50000 + 7.79423i −0.240281 − 0.416179i
211.7 1.72699 + 2.99123i −1.50000 − 2.59808i −1.96497 + 3.40342i 14.2940 5.18096 − 8.97369i 3.50000 − 6.06218i 14.0579 −4.50000 + 7.79423i 24.6855 + 42.7565i
211.8 1.75980 + 3.04807i −1.50000 − 2.59808i −2.19383 + 3.79982i −10.3397 5.27941 − 9.14421i 3.50000 − 6.06218i 12.7141 −4.50000 + 7.79423i −18.1959 − 31.5162i
211.9 2.56332 + 4.43981i −1.50000 − 2.59808i −9.14126 + 15.8331i −14.1720 7.68997 − 13.3194i 3.50000 − 6.06218i −52.7149 −4.50000 + 7.79423i −36.3276 − 62.9212i
211.10 2.70146 + 4.67907i −1.50000 − 2.59808i −10.5958 + 18.3525i 17.9394 8.10439 − 14.0372i 3.50000 − 6.06218i −71.2736 −4.50000 + 7.79423i 48.4627 + 83.9398i
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 22.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.4.k.d ✓ 20
13.c even 3 1 inner 273.4.k.d ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.4.k.d ✓ 20 1.a even 1 1 trivial
273.4.k.d ✓ 20 13.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} - 5 T_{2}^{19} + 81 T_{2}^{18} - 262 T_{2}^{17} + 3425 T_{2}^{16} - 9476 T_{2}^{15} + \cdots + 680166400 \) acting on \(S_{4}^{\mathrm{new}}(273, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 680166400 \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 9)^{10} \) Copy content Toggle raw display
$5$ \( (T^{10} - 15 T^{9} + \cdots - 130520544)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 7 T + 49)^{10} \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 94\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 26\!\cdots\!49 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 78\!\cdots\!29 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 16\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 49\!\cdots\!16 \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots - 79\!\cdots\!80)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 55\!\cdots\!04 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 48\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots - 56\!\cdots\!36)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots + 48\!\cdots\!33)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 69\!\cdots\!25 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 11\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots - 39\!\cdots\!52)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots - 16\!\cdots\!76)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots + 10\!\cdots\!64)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 90\!\cdots\!64 \) Copy content Toggle raw display
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