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Properties

Label 546.4.i.i
Level $546$
Weight $4$
Character orbit 546.i
Analytic conductor $32.215$
Analytic rank $0$
Dimension $14$
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: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 842 x^{12} + 218685 x^{10} + 21911078 x^{8} + 961392250 x^{6} + 18473532408 x^{4} + \cdots + 274163125248 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2}\cdot 7 \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \beta_{4} + 2) q^{2} + 3 \beta_{4} q^{3} + 4 \beta_{4} q^{4} + ( - \beta_{9} + 2 \beta_{4} + 2) q^{5} - 6 q^{6} + (\beta_{5} - 3 \beta_{4} - 3) q^{7} - 8 q^{8} + ( - 9 \beta_{4} - 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (2 \beta_{4} + 2) q^{2} + 3 \beta_{4} q^{3} + 4 \beta_{4} q^{4} + ( - \beta_{9} + 2 \beta_{4} + 2) q^{5} - 6 q^{6} + (\beta_{5} - 3 \beta_{4} - 3) q^{7} - 8 q^{8} + ( - 9 \beta_{4} - 9) q^{9} + ( - 2 \beta_{9} + 4 \beta_{4} - 2 \beta_1) q^{10} + (\beta_{13} + \beta_{10} + \cdots + \beta_{3}) q^{11}+ \cdots + (9 \beta_{7} + 9 \beta_{6} + 9 \beta_{2} - 18) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 14 q^{2} - 21 q^{3} - 28 q^{4} + 12 q^{5} - 84 q^{6} - 14 q^{7} - 112 q^{8} - 63 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 14 q^{2} - 21 q^{3} - 28 q^{4} + 12 q^{5} - 84 q^{6} - 14 q^{7} - 112 q^{8} - 63 q^{9} - 24 q^{10} + 15 q^{11} - 84 q^{12} - 182 q^{13} + 58 q^{14} - 72 q^{15} - 112 q^{16} - 2 q^{17} + 126 q^{18} - 20 q^{19} - 96 q^{20} + 129 q^{21} + 60 q^{22} + 36 q^{23} + 168 q^{24} - 331 q^{25} - 182 q^{26} + 378 q^{27} + 172 q^{28} - 436 q^{29} - 72 q^{30} + 153 q^{31} + 224 q^{32} + 45 q^{33} - 8 q^{34} + 1045 q^{35} + 504 q^{36} + 21 q^{37} + 40 q^{38} + 273 q^{39} - 96 q^{40} - 1268 q^{41} + 84 q^{42} + 476 q^{43} + 60 q^{44} + 108 q^{45} - 72 q^{46} - 269 q^{47} + 672 q^{48} + 1556 q^{49} - 1324 q^{50} - 6 q^{51} + 364 q^{52} - 209 q^{53} + 378 q^{54} + 442 q^{55} + 112 q^{56} + 120 q^{57} - 436 q^{58} + 641 q^{59} + 144 q^{60} - 366 q^{61} + 612 q^{62} - 261 q^{63} + 896 q^{64} - 156 q^{65} - 90 q^{66} - 232 q^{67} - 8 q^{68} - 216 q^{69} + 1468 q^{70} - 1540 q^{71} + 504 q^{72} + 1662 q^{73} - 42 q^{74} - 993 q^{75} + 160 q^{76} - 34 q^{77} + 1092 q^{78} - 817 q^{79} + 192 q^{80} - 567 q^{81} - 1268 q^{82} - 650 q^{83} - 348 q^{84} + 2848 q^{85} + 476 q^{86} + 654 q^{87} - 120 q^{88} - 1901 q^{89} + 432 q^{90} + 182 q^{91} - 288 q^{92} + 459 q^{93} + 538 q^{94} + 2751 q^{95} + 672 q^{96} - 4876 q^{97} - 706 q^{98} - 270 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 842 x^{12} + 218685 x^{10} + 21911078 x^{8} + 961392250 x^{6} + 18473532408 x^{4} + \cdots + 274163125248 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 16\!\cdots\!56 \nu^{12} + \cdots + 49\!\cdots\!32 ) / 65\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 13\!\cdots\!58 \nu^{12} + \cdots - 23\!\cdots\!76 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 13\!\cdots\!87 \nu^{13} + \cdots + 99\!\cdots\!88 ) / 32\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4572048238137 \nu^{13} + \cdots - 45\!\cdots\!80 ) / 91\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 11\!\cdots\!31 \nu^{13} + \cdots + 13\!\cdots\!92 ) / 16\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11\!\cdots\!31 \nu^{13} + \cdots - 17\!\cdots\!40 ) / 16\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 11\!\cdots\!31 \nu^{13} + \cdots + 68\!\cdots\!28 ) / 16\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 11\!\cdots\!95 \nu^{13} + \cdots + 29\!\cdots\!96 ) / 54\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 84\!\cdots\!03 \nu^{13} + \cdots - 12\!\cdots\!44 ) / 32\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 97\!\cdots\!05 \nu^{13} + \cdots + 18\!\cdots\!92 ) / 32\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 11\!\cdots\!65 \nu^{13} + \cdots - 22\!\cdots\!68 ) / 32\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 40\!\cdots\!49 \nu^{13} + \cdots + 21\!\cdots\!72 ) / 82\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 18\!\cdots\!89 \nu^{13} + \cdots - 15\!\cdots\!64 ) / 32\!\cdots\!80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 15 \beta_{13} + 4 \beta_{12} - 5 \beta_{11} + 14 \beta_{10} - 24 \beta_{9} + 5 \beta_{8} + 8 \beta_{7} + \cdots - 14 ) / 42 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 41 \beta_{13} - 32 \beta_{12} - 9 \beta_{11} - 18 \beta_{9} + 9 \beta_{8} - 43 \beta_{7} - 7 \beta_{6} + \cdots - 1715 ) / 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 6357 \beta_{13} - 1964 \beta_{12} + 2497 \beta_{11} - 3850 \beta_{10} + 4056 \beta_{9} + \cdots - 16961 ) / 42 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 19750 \beta_{13} + 15948 \beta_{12} + 3802 \beta_{11} + 7604 \beta_{9} - 3802 \beta_{8} + \cdots + 567553 ) / 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 3008301 \beta_{13} + 916126 \beta_{12} - 1205837 \beta_{11} + 1399706 \beta_{10} - 930936 \beta_{9} + \cdots + 11241034 ) / 42 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1291738 \beta_{13} - 1017298 \beta_{12} - 274440 \beta_{11} - 548880 \beta_{9} + 274440 \beta_{8} + \cdots - 33380961 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 706465791 \beta_{13} - 210078391 \beta_{12} + 284913896 \beta_{11} - 292723886 \beta_{10} + \cdots - 2915804101 ) / 21 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 4152005649 \beta_{13} + 3201763946 \beta_{12} + 950241703 \beta_{11} + 1900483406 \beta_{9} + \cdots + 102580519496 ) / 14 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 658227658479 \beta_{13} + 192935738362 \beta_{12} - 266322124385 \beta_{11} + 258385128974 \beta_{10} + \cdots + 2827965618541 ) / 42 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 1912210794981 \beta_{13} - 1457012760278 \beta_{12} - 455198034703 \beta_{11} - 910396069406 \beta_{9} + \cdots - 46243526056021 ) / 14 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 305306890670388 \beta_{13} - 88824558125708 \beta_{12} + 123735726657526 \beta_{11} + \cdots - 13\!\cdots\!33 ) / 42 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 125999197640173 \beta_{13} + 95417111231736 \beta_{12} + 30582086408437 \beta_{11} + \cdots + 30\!\cdots\!88 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 14\!\cdots\!96 \beta_{13} + \cdots + 62\!\cdots\!71 ) / 42 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(157\) \(365\) \(379\)
\(\chi(n)\) \(-1 - \beta_{4}\) \(1\) \(1\)

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
14.9679i
2.83043i
8.61006i
6.15084i
− 21.4916i
− 1.90586i
− 5.69759i
− 14.9679i
− 2.83043i
− 8.61006i
− 6.15084i
21.4916i
1.90586i
5.69759i
1.00000 − 1.73205i −1.50000 − 2.59808i −2.00000 − 3.46410i −8.93425 + 15.4746i −6.00000 13.8526 − 12.2925i −8.00000 −4.50000 + 7.79423i 17.8685 + 30.9492i
79.2 1.00000 − 1.73205i −1.50000 − 2.59808i −2.00000 − 3.46410i −5.84687 + 10.1271i −6.00000 −18.2091 − 3.38047i −8.00000 −4.50000 + 7.79423i 11.6937 + 20.2541i
79.3 1.00000 − 1.73205i −1.50000 − 2.59808i −2.00000 − 3.46410i −1.11101 + 1.92432i −6.00000 −18.5202 − 0.0623525i −8.00000 −4.50000 + 7.79423i 2.22202 + 3.84865i
79.4 1.00000 − 1.73205i −1.50000 − 2.59808i −2.00000 − 3.46410i 1.31179 − 2.27209i −6.00000 15.6942 + 9.83327i −8.00000 −4.50000 + 7.79423i −2.62358 − 4.54417i
79.5 1.00000 − 1.73205i −1.50000 − 2.59808i −2.00000 − 3.46410i 1.48400 − 2.57036i −6.00000 −6.39216 + 17.3822i −8.00000 −4.50000 + 7.79423i −2.96800 − 5.14072i
79.6 1.00000 − 1.73205i −1.50000 − 2.59808i −2.00000 − 3.46410i 9.52761 − 16.5023i −6.00000 −11.1046 + 14.8219i −8.00000 −4.50000 + 7.79423i −19.0552 − 33.0046i
79.7 1.00000 − 1.73205i −1.50000 − 2.59808i −2.00000 − 3.46410i 9.56873 − 16.5735i −6.00000 17.6793 − 5.51733i −8.00000 −4.50000 + 7.79423i −19.1375 − 33.1470i
235.1 1.00000 + 1.73205i −1.50000 + 2.59808i −2.00000 + 3.46410i −8.93425 − 15.4746i −6.00000 13.8526 + 12.2925i −8.00000 −4.50000 − 7.79423i 17.8685 − 30.9492i
235.2 1.00000 + 1.73205i −1.50000 + 2.59808i −2.00000 + 3.46410i −5.84687 − 10.1271i −6.00000 −18.2091 + 3.38047i −8.00000 −4.50000 − 7.79423i 11.6937 − 20.2541i
235.3 1.00000 + 1.73205i −1.50000 + 2.59808i −2.00000 + 3.46410i −1.11101 − 1.92432i −6.00000 −18.5202 + 0.0623525i −8.00000 −4.50000 − 7.79423i 2.22202 − 3.84865i
235.4 1.00000 + 1.73205i −1.50000 + 2.59808i −2.00000 + 3.46410i 1.31179 + 2.27209i −6.00000 15.6942 − 9.83327i −8.00000 −4.50000 − 7.79423i −2.62358 + 4.54417i
235.5 1.00000 + 1.73205i −1.50000 + 2.59808i −2.00000 + 3.46410i 1.48400 + 2.57036i −6.00000 −6.39216 − 17.3822i −8.00000 −4.50000 − 7.79423i −2.96800 + 5.14072i
235.6 1.00000 + 1.73205i −1.50000 + 2.59808i −2.00000 + 3.46410i 9.52761 + 16.5023i −6.00000 −11.1046 − 14.8219i −8.00000 −4.50000 − 7.79423i −19.0552 + 33.0046i
235.7 1.00000 + 1.73205i −1.50000 + 2.59808i −2.00000 + 3.46410i 9.56873 + 16.5735i −6.00000 17.6793 + 5.51733i −8.00000 −4.50000 − 7.79423i −19.1375 + 33.1470i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.7
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.i ✓ 14
7.c even 3 1 inner 546.4.i.i ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.4.i.i ✓ 14 1.a even 1 1 trivial
546.4.i.i ✓ 14 7.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{14} - 12 T_{5}^{13} + 675 T_{5}^{12} - 3086 T_{5}^{11} + 269520 T_{5}^{10} - 1129911 T_{5}^{9} + \cdots + 1738157465664 \) acting on \(S_{4}^{\mathrm{new}}(546, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 4)^{7} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 9)^{7} \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 1738157465664 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 55\!\cdots\!07 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 77\!\cdots\!25 \) Copy content Toggle raw display
$13$ \( (T + 13)^{14} \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 50\!\cdots\!84 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T^{7} + \cdots - 26345621804295)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 27\!\cdots\!56 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{7} + \cdots - 66\!\cdots\!32)^{2} \) Copy content Toggle raw display
$43$ \( (T^{7} + \cdots + 71\!\cdots\!80)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 31\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 40\!\cdots\!21 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 21\!\cdots\!89 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 34\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 59\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( (T^{7} + \cdots + 67\!\cdots\!14)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 77\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 17\!\cdots\!84 \) Copy content Toggle raw display
$83$ \( (T^{7} + \cdots - 20\!\cdots\!80)^{2} \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{7} + \cdots + 22\!\cdots\!52)^{2} \) Copy content Toggle raw display
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