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Properties

Label 546.4.i.e
Level $546$
Weight $4$
Character orbit 546.i
Analytic conductor $32.215$
Analytic rank $0$
Dimension $12$
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] = [546,4,Mod(79,546)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(546, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("546.79");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 546.i (of order \(3\), 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: \(32.2150428631\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 373 x^{10} - 3270 x^{9} + 108613 x^{8} - 867771 x^{7} + 15135417 x^{6} - 142511676 x^{5} + \cdots + 291450979044 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \beta_1 - 2) q^{2} - 3 \beta_1 q^{3} - 4 \beta_1 q^{4} + ( - \beta_{3} + \beta_1 - 1) q^{5} + 6 q^{6} + (\beta_{11} + \beta_{4} - 3 \beta_1) q^{7} + 8 q^{8} + (9 \beta_1 - 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (2 \beta_1 - 2) q^{2} - 3 \beta_1 q^{3} - 4 \beta_1 q^{4} + ( - \beta_{3} + \beta_1 - 1) q^{5} + 6 q^{6} + (\beta_{11} + \beta_{4} - 3 \beta_1) q^{7} + 8 q^{8} + (9 \beta_1 - 9) q^{9} + ( - 2 \beta_{4} + 2 \beta_{3} - 2 \beta_1) q^{10} + (\beta_{9} - \beta_{7} + \cdots - 3 \beta_1) q^{11}+ \cdots + ( - 9 \beta_{11} + 9 \beta_{10} + \cdots + 36) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{2} - 18 q^{3} - 24 q^{4} - 7 q^{5} + 72 q^{6} - 11 q^{7} + 96 q^{8} - 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{2} - 18 q^{3} - 24 q^{4} - 7 q^{5} + 72 q^{6} - 11 q^{7} + 96 q^{8} - 54 q^{9} - 14 q^{10} - 20 q^{11} - 72 q^{12} - 156 q^{13} + 62 q^{14} + 42 q^{15} - 96 q^{16} - 190 q^{17} - 108 q^{18} - 126 q^{19} + 56 q^{20} - 60 q^{21} + 80 q^{22} + 92 q^{23} - 144 q^{24} - 115 q^{25} + 156 q^{26} + 324 q^{27} - 80 q^{28} + 382 q^{29} - 42 q^{30} - 272 q^{31} - 192 q^{32} - 60 q^{33} + 760 q^{34} - 389 q^{35} + 432 q^{36} - 365 q^{37} - 252 q^{38} + 234 q^{39} - 56 q^{40} + 832 q^{41} - 66 q^{42} + 36 q^{43} - 80 q^{44} - 63 q^{45} + 184 q^{46} - 283 q^{47} + 576 q^{48} - 39 q^{49} + 460 q^{50} - 570 q^{51} + 312 q^{52} + 38 q^{53} - 324 q^{54} + 1552 q^{55} - 88 q^{56} + 756 q^{57} - 382 q^{58} - 474 q^{59} - 84 q^{60} - 852 q^{61} + 1088 q^{62} + 279 q^{63} + 768 q^{64} + 91 q^{65} - 120 q^{66} + 798 q^{67} - 760 q^{68} - 552 q^{69} - 1126 q^{70} + 3556 q^{71} - 432 q^{72} - 1214 q^{73} - 730 q^{74} - 345 q^{75} + 1008 q^{76} + 687 q^{77} - 936 q^{78} + 100 q^{79} - 112 q^{80} - 486 q^{81} - 832 q^{82} + 1500 q^{83} + 372 q^{84} + 1752 q^{85} - 36 q^{86} - 573 q^{87} - 160 q^{88} - 1547 q^{89} + 252 q^{90} + 143 q^{91} - 736 q^{92} - 816 q^{93} - 566 q^{94} - 255 q^{95} - 576 q^{96} - 1714 q^{97} + 300 q^{98} + 360 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 373 x^{10} - 3270 x^{9} + 108613 x^{8} - 867771 x^{7} + 15135417 x^{6} - 142511676 x^{5} + \cdots + 291450979044 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 80\!\cdots\!37 \nu^{11} + \cdots - 33\!\cdots\!52 ) / 84\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 30\!\cdots\!87 \nu^{11} + \cdots - 83\!\cdots\!80 ) / 88\!\cdots\!85 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 60\!\cdots\!01 \nu^{11} + \cdots - 69\!\cdots\!92 ) / 16\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 11\!\cdots\!48 \nu^{11} + \cdots - 67\!\cdots\!11 ) / 32\!\cdots\!35 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 13\!\cdots\!25 \nu^{11} + \cdots + 40\!\cdots\!88 ) / 35\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 47\!\cdots\!09 \nu^{11} + \cdots + 56\!\cdots\!28 ) / 11\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 13\!\cdots\!86 \nu^{11} + \cdots + 77\!\cdots\!27 ) / 32\!\cdots\!35 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 25\!\cdots\!07 \nu^{11} + \cdots - 30\!\cdots\!44 ) / 58\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 48\!\cdots\!91 \nu^{11} + \cdots - 14\!\cdots\!76 ) / 58\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 17\!\cdots\!07 \nu^{11} + \cdots - 17\!\cdots\!38 ) / 88\!\cdots\!85 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 17\!\cdots\!38 \nu^{11} + \cdots - 83\!\cdots\!22 ) / 88\!\cdots\!85 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -9\beta_{11} + 6\beta_{10} + 2\beta_{8} - 3\beta_{6} - 6\beta_{5} + \beta_{3} - 3\beta_{2} + 248\beta _1 - 242 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 110 \beta_{11} - 122 \beta_{10} + 26 \beta_{9} - 26 \beta_{8} - 12 \beta_{7} - 12 \beta_{5} + 75 \beta_{4} + \cdots + 711 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 524 \beta_{11} + 524 \beta_{10} - 673 \beta_{9} - 251 \beta_{7} + 251 \beta_{6} + 1732 \beta_{5} + \cdots - 26472 \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 35650 \beta_{11} + 29025 \beta_{10} + 12308 \beta_{8} + 3066 \beta_{6} - 29025 \beta_{5} + \cdots - 265095 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 409928 \beta_{11} - 579226 \beta_{10} + 263023 \beta_{9} - 263023 \beta_{8} + 28229 \beta_{7} + \cdots + 6287165 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2574799 \beta_{11} + 2574799 \beta_{10} - 4639907 \beta_{9} + 753915 \beta_{7} - 753915 \beta_{6} + \cdots - 97671319 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 187590205 \beta_{11} + 133505345 \beta_{10} + 90448213 \beta_{8} - 142190 \beta_{6} + \cdots - 1774875452 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2547118350 \beta_{11} - 3440525779 \beta_{10} + 1597523108 \beta_{9} - 1597523108 \beta_{8} + \cdots + 27762633675 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 17225570189 \beta_{11} + 17225570189 \beta_{10} - 29759264272 \beta_{9} + 1385785225 \beta_{7} + \cdots - 590299334586 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1086451922686 \beta_{11} + 790596020571 \beta_{10} + 527685083963 \beta_{8} + 56894186790 \beta_{6} + \cdots - 8847107033901 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/546\mathbb{Z}\right)^\times\).

\(n\) \(157\) \(365\) \(379\)
\(\chi(n)\) \(-1 + \beta_{1}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
2.48464 + 4.30353i
−6.27823 − 10.8742i
2.39953 + 4.15611i
6.28653 + 10.8886i
−8.91400 − 15.4395i
4.02152 + 6.96549i
2.48464 − 4.30353i
−6.27823 + 10.8742i
2.39953 − 4.15611i
6.28653 − 10.8886i
−8.91400 + 15.4395i
4.02152 − 6.96549i
−1.00000 + 1.73205i −1.50000 − 2.59808i −2.00000 − 3.46410i −8.91731 + 15.4452i 6.00000 2.44803 − 18.3578i 8.00000 −4.50000 + 7.79423i −17.8346 − 30.8905i
79.2 −1.00000 + 1.73205i −1.50000 − 2.59808i −2.00000 − 3.46410i −5.84896 + 10.1307i 6.00000 16.9054 + 7.56350i 8.00000 −4.50000 + 7.79423i −11.6979 − 20.2614i
79.3 −1.00000 + 1.73205i −1.50000 − 2.59808i −2.00000 − 3.46410i −2.67198 + 4.62800i 6.00000 −3.62709 + 18.1616i 8.00000 −4.50000 + 7.79423i −5.34396 − 9.25600i
79.4 −1.00000 + 1.73205i −1.50000 − 2.59808i −2.00000 − 3.46410i 1.03417 − 1.79124i 6.00000 −15.1072 − 10.7131i 8.00000 −4.50000 + 7.79423i 2.06835 + 3.58249i
79.5 −1.00000 + 1.73205i −1.50000 − 2.59808i −2.00000 − 3.46410i 4.10101 − 7.10316i 6.00000 12.2270 − 13.9105i 8.00000 −4.50000 + 7.79423i 8.20202 + 14.2063i
79.6 −1.00000 + 1.73205i −1.50000 − 2.59808i −2.00000 − 3.46410i 8.80306 − 15.2474i 6.00000 −18.3461 + 2.53379i 8.00000 −4.50000 + 7.79423i 17.6061 + 30.4947i
235.1 −1.00000 − 1.73205i −1.50000 + 2.59808i −2.00000 + 3.46410i −8.91731 − 15.4452i 6.00000 2.44803 + 18.3578i 8.00000 −4.50000 − 7.79423i −17.8346 + 30.8905i
235.2 −1.00000 − 1.73205i −1.50000 + 2.59808i −2.00000 + 3.46410i −5.84896 − 10.1307i 6.00000 16.9054 − 7.56350i 8.00000 −4.50000 − 7.79423i −11.6979 + 20.2614i
235.3 −1.00000 − 1.73205i −1.50000 + 2.59808i −2.00000 + 3.46410i −2.67198 − 4.62800i 6.00000 −3.62709 − 18.1616i 8.00000 −4.50000 − 7.79423i −5.34396 + 9.25600i
235.4 −1.00000 − 1.73205i −1.50000 + 2.59808i −2.00000 + 3.46410i 1.03417 + 1.79124i 6.00000 −15.1072 + 10.7131i 8.00000 −4.50000 − 7.79423i 2.06835 − 3.58249i
235.5 −1.00000 − 1.73205i −1.50000 + 2.59808i −2.00000 + 3.46410i 4.10101 + 7.10316i 6.00000 12.2270 + 13.9105i 8.00000 −4.50000 − 7.79423i 8.20202 − 14.2063i
235.6 −1.00000 − 1.73205i −1.50000 + 2.59808i −2.00000 + 3.46410i 8.80306 + 15.2474i 6.00000 −18.3461 − 2.53379i 8.00000 −4.50000 − 7.79423i 17.6061 − 30.4947i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.4.i.e ✓ 12
7.c even 3 1 inner 546.4.i.e ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.4.i.e ✓ 12 1.a even 1 1 trivial
546.4.i.e ✓ 12 7.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 7 T_{5}^{11} + 457 T_{5}^{10} + 2146 T_{5}^{9} + 152989 T_{5}^{8} + 697720 T_{5}^{7} + \cdots + 110889000000 \) acting on \(S_{4}^{\mathrm{new}}(546, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 4)^{6} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 9)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 110889000000 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 16\!\cdots\!49 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 37\!\cdots\!61 \) Copy content Toggle raw display
$13$ \( (T + 13)^{12} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 41\!\cdots\!89 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 45066235233321 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 24\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 2481780157353)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 31\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 107775128854008)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 29145825344040)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 27\!\cdots\!49 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 73\!\cdots\!61 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 13\!\cdots\!09 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 10\!\cdots\!89 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 91\!\cdots\!93)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 54\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 11\!\cdots\!84)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 61\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 97\!\cdots\!04)^{2} \) Copy content Toggle raw display
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