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Properties

Label 546.4.l.h
Level $546$
Weight $4$
Character orbit 546.l
Analytic conductor $32.215$
Analytic rank $0$
Dimension $10$
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(211,546)]
 
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, 0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("546.211");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 546.l (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: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 218 x^{8} - 1187 x^{7} + 37612 x^{6} - 176472 x^{5} + 2657151 x^{4} - 12165606 x^{3} + \cdots + 1979894016 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{3} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \beta_{3} q^{2} + 3 \beta_{3} q^{3} + (4 \beta_{3} - 4) q^{4} + (\beta_{2} - 2) q^{5} + ( - 6 \beta_{3} + 6) q^{6} + ( - 7 \beta_{3} + 7) q^{7} + 8 q^{8} + (9 \beta_{3} - 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_{3} q^{2} + 3 \beta_{3} q^{3} + (4 \beta_{3} - 4) q^{4} + (\beta_{2} - 2) q^{5} + ( - 6 \beta_{3} + 6) q^{6} + ( - 7 \beta_{3} + 7) q^{7} + 8 q^{8} + (9 \beta_{3} - 9) q^{9} + (4 \beta_{3} - 2 \beta_1) q^{10} + (\beta_{9} - 4 \beta_{3} - \beta_1) q^{11} - 12 q^{12} + (\beta_{9} - \beta_{6} - \beta_{5} + \cdots - \beta_1) q^{13}+ \cdots + (9 \beta_{7} + 9 \beta_{2} + 36) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 10 q^{2} + 15 q^{3} - 20 q^{4} - 18 q^{5} + 30 q^{6} + 35 q^{7} + 80 q^{8} - 45 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 10 q^{2} + 15 q^{3} - 20 q^{4} - 18 q^{5} + 30 q^{6} + 35 q^{7} + 80 q^{8} - 45 q^{9} + 18 q^{10} - 23 q^{11} - 120 q^{12} - 16 q^{13} - 140 q^{14} - 27 q^{15} - 80 q^{16} - 14 q^{17} + 180 q^{18} + 16 q^{19} + 36 q^{20} + 210 q^{21} - 46 q^{22} - 47 q^{23} + 120 q^{24} - 348 q^{25} - 50 q^{26} - 270 q^{27} + 140 q^{28} - 149 q^{29} - 54 q^{30} + 350 q^{31} - 160 q^{32} + 69 q^{33} + 56 q^{34} - 63 q^{35} - 180 q^{36} + 187 q^{37} - 64 q^{38} + 75 q^{39} - 144 q^{40} + 358 q^{41} - 210 q^{42} + 575 q^{43} + 184 q^{44} + 81 q^{45} - 94 q^{46} + 332 q^{47} + 240 q^{48} - 245 q^{49} + 348 q^{50} - 84 q^{51} + 164 q^{52} - 94 q^{53} + 270 q^{54} - 530 q^{55} + 280 q^{56} + 96 q^{57} - 298 q^{58} - 329 q^{59} + 216 q^{60} - 197 q^{61} - 350 q^{62} + 315 q^{63} + 640 q^{64} + 169 q^{65} - 276 q^{66} + 231 q^{67} - 56 q^{68} + 141 q^{69} + 252 q^{70} + 38 q^{71} - 360 q^{72} - 754 q^{73} + 374 q^{74} - 522 q^{75} + 64 q^{76} - 322 q^{77} - 246 q^{78} - 304 q^{79} + 144 q^{80} - 405 q^{81} + 716 q^{82} + 360 q^{83} - 420 q^{84} - 748 q^{85} - 2300 q^{86} + 447 q^{87} - 184 q^{88} + 1052 q^{89} - 324 q^{90} - 287 q^{91} + 376 q^{92} + 525 q^{93} - 332 q^{94} + 2272 q^{95} - 960 q^{96} - 1415 q^{97} - 490 q^{98} + 414 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + 218 x^{8} - 1187 x^{7} + 37612 x^{6} - 176472 x^{5} + 2657151 x^{4} - 12165606 x^{3} + \cdots + 1979894016 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2087315101 \nu^{9} + 65004637490 \nu^{8} - 709244146510 \nu^{7} + 10225419141385 \nu^{6} + \cdots + 97\!\cdots\!88 ) / 20\!\cdots\!59 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 8143072008139 \nu^{9} - 11582967294587 \nu^{8} + \cdots + 73\!\cdots\!36 ) / 34\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 608388106031 \nu^{9} - 15015738291515 \nu^{8} + 163831764932485 \nu^{7} + \cdots - 16\!\cdots\!92 ) / 18\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 93227252809367 \nu^{9} - 364874105326490 \nu^{8} + \cdots - 13\!\cdots\!72 ) / 64\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2788594780523 \nu^{9} + 32552612045213 \nu^{8} - 355170806855587 \nu^{7} + \cdots + 20\!\cdots\!17 ) / 18\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 953059197149 \nu^{9} - 8558405450270 \nu^{8} + 93377937381730 \nu^{7} + \cdots - 56\!\cdots\!43 ) / 62\!\cdots\!77 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 16\!\cdots\!51 \nu^{9} + \cdots + 41\!\cdots\!44 ) / 10\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 663219885876457 \nu^{9} + \cdots + 14\!\cdots\!00 ) / 34\!\cdots\!32 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + 88\beta_{3} + 4\beta_{2} - 4\beta _1 - 88 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 11\beta_{7} + 10\beta_{6} - 2\beta_{4} - 125\beta_{2} + 315 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 215\beta_{9} - 237\beta_{8} - 150\beta_{5} - 10534\beta_{3} + 953\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 2874 \beta_{9} + 2635 \beta_{8} - 2874 \beta_{7} - 2635 \beta_{6} + 118 \beta_{5} - 118 \beta_{4} + \cdots - 75101 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 41324\beta_{7} + 45635\beta_{6} + 20849\beta_{4} - 190538\beta_{2} + 1536155 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 576799\beta_{9} - 546248\beta_{8} - 85919\beta_{5} - 15033597\beta_{3} + 3107926\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 7730116 \beta_{9} + 8348934 \beta_{8} - 7730116 \beta_{7} - 8348934 \beta_{6} + 3083292 \beta_{5} + \cdots - 249791114 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 107623629\beta_{7} + 104855435\beta_{6} + 22321607\beta_{4} - 538223016\beta_{2} + 2857706302 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(157\) \(365\) \(379\)
\(\chi(n)\) \(1\) \(1\) \(-1 + \beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
211.1
−6.73237 − 11.6608i
−4.16673 − 7.21699i
2.60409 + 4.51041i
3.84697 + 6.66315i
4.94804 + 8.57026i
−6.73237 + 11.6608i
−4.16673 + 7.21699i
2.60409 − 4.51041i
3.84697 − 6.66315i
4.94804 − 8.57026i
−1.00000 − 1.73205i 1.50000 + 2.59808i −2.00000 + 3.46410i −15.4647 3.00000 − 5.19615i 3.50000 − 6.06218i 8.00000 −4.50000 + 7.79423i 15.4647 + 26.7857i
211.2 −1.00000 − 1.73205i 1.50000 + 2.59808i −2.00000 + 3.46410i −10.3335 3.00000 − 5.19615i 3.50000 − 6.06218i 8.00000 −4.50000 + 7.79423i 10.3335 + 17.8981i
211.3 −1.00000 − 1.73205i 1.50000 + 2.59808i −2.00000 + 3.46410i 3.20817 3.00000 − 5.19615i 3.50000 − 6.06218i 8.00000 −4.50000 + 7.79423i −3.20817 − 5.55672i
211.4 −1.00000 − 1.73205i 1.50000 + 2.59808i −2.00000 + 3.46410i 5.69395 3.00000 − 5.19615i 3.50000 − 6.06218i 8.00000 −4.50000 + 7.79423i −5.69395 − 9.86220i
211.5 −1.00000 − 1.73205i 1.50000 + 2.59808i −2.00000 + 3.46410i 7.89608 3.00000 − 5.19615i 3.50000 − 6.06218i 8.00000 −4.50000 + 7.79423i −7.89608 − 13.6764i
295.1 −1.00000 + 1.73205i 1.50000 − 2.59808i −2.00000 − 3.46410i −15.4647 3.00000 + 5.19615i 3.50000 + 6.06218i 8.00000 −4.50000 − 7.79423i 15.4647 − 26.7857i
295.2 −1.00000 + 1.73205i 1.50000 − 2.59808i −2.00000 − 3.46410i −10.3335 3.00000 + 5.19615i 3.50000 + 6.06218i 8.00000 −4.50000 − 7.79423i 10.3335 − 17.8981i
295.3 −1.00000 + 1.73205i 1.50000 − 2.59808i −2.00000 − 3.46410i 3.20817 3.00000 + 5.19615i 3.50000 + 6.06218i 8.00000 −4.50000 − 7.79423i −3.20817 + 5.55672i
295.4 −1.00000 + 1.73205i 1.50000 − 2.59808i −2.00000 − 3.46410i 5.69395 3.00000 + 5.19615i 3.50000 + 6.06218i 8.00000 −4.50000 − 7.79423i −5.69395 + 9.86220i
295.5 −1.00000 + 1.73205i 1.50000 − 2.59808i −2.00000 − 3.46410i 7.89608 3.00000 + 5.19615i 3.50000 + 6.06218i 8.00000 −4.50000 − 7.79423i −7.89608 + 13.6764i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 211.5
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 546.4.l.h ✓ 10
13.c even 3 1 inner 546.4.l.h ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.4.l.h ✓ 10 1.a even 1 1 trivial
546.4.l.h ✓ 10 13.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{5} + 9T_{5}^{4} - 185T_{5}^{3} - 544T_{5}^{2} + 10431T_{5} - 23050 \) acting on \(S_{4}^{\mathrm{new}}(546, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 4)^{5} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3 T + 9)^{5} \) Copy content Toggle raw display
$5$ \( (T^{5} + 9 T^{4} + \cdots - 23050)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 7 T + 49)^{5} \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 7022440000 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 51\!\cdots\!57 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 96939994247721 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 25\!\cdots\!44 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 53\!\cdots\!01 \) Copy content Toggle raw display
$31$ \( (T^{5} - 175 T^{4} + \cdots - 16181858784)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 12\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 73\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 25\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( (T^{5} + \cdots + 1841915955960)^{2} \) Copy content Toggle raw display
$53$ \( (T^{5} + \cdots - 25812278069949)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 10\!\cdots\!81 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 80\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 28\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots + 47676538558200)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots + 83341633624940)^{2} \) Copy content Toggle raw display
$83$ \( (T^{5} + \cdots - 21364496386848)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 80\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
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