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Properties

Label 547.2.a.b
Level $547$
Weight $2$
Character orbit 547.a
Self dual yes
Analytic conductor $4.368$
Analytic rank $1$
Dimension $18$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [547,2,Mod(1,547)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(547, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("547.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 547 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 547.a (trivial)

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: yes
Analytic conductor: \(4.36781699056\)
Analytic rank: \(1\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 4 x^{17} - 18 x^{16} + 84 x^{15} + 116 x^{14} - 708 x^{13} - 282 x^{12} + 3104 x^{11} + \cdots + 328 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{9} - 1) q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{3} - 2) q^{5} + ( - \beta_{16} - \beta_{14} + \cdots + \beta_1) q^{6}+ \cdots + (\beta_{17} - \beta_{15} + \beta_{14} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{9} - 1) q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{3} - 2) q^{5} + ( - \beta_{16} - \beta_{14} + \cdots + \beta_1) q^{6}+ \cdots + (3 \beta_{17} - \beta_{15} - \beta_{14} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 4 q^{2} - 10 q^{3} + 16 q^{4} - 27 q^{5} - 3 q^{6} - 11 q^{7} - 12 q^{8} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 4 q^{2} - 10 q^{3} + 16 q^{4} - 27 q^{5} - 3 q^{6} - 11 q^{7} - 12 q^{8} + 14 q^{9} - 5 q^{10} + 2 q^{11} - 32 q^{12} - 25 q^{13} - 7 q^{14} + 9 q^{15} + 8 q^{16} - 30 q^{17} - 10 q^{18} + 4 q^{19} - 41 q^{20} - 16 q^{21} - 24 q^{22} - 26 q^{23} - 12 q^{24} + 31 q^{25} - 18 q^{26} - 37 q^{27} - 16 q^{28} - 18 q^{29} + 8 q^{30} - 5 q^{31} - 28 q^{32} - 10 q^{33} + 5 q^{34} - 9 q^{35} + 31 q^{36} - 18 q^{37} - 45 q^{38} + 7 q^{39} + 7 q^{40} - 17 q^{41} + 4 q^{42} + 8 q^{43} + 12 q^{44} - 44 q^{45} + 30 q^{46} - 52 q^{47} - 7 q^{48} + 29 q^{49} + 13 q^{50} + 19 q^{51} - 14 q^{52} - 60 q^{53} + 11 q^{54} + 11 q^{55} + 7 q^{56} + 4 q^{57} + 14 q^{58} - 8 q^{59} + 86 q^{60} - 26 q^{61} + 4 q^{62} - q^{63} + 44 q^{64} - 6 q^{65} + 18 q^{66} + 12 q^{67} - 61 q^{68} - 38 q^{69} + 35 q^{70} - q^{71} + 28 q^{72} - 2 q^{73} + 16 q^{74} - 17 q^{75} + 66 q^{76} - 73 q^{77} + 115 q^{78} + 18 q^{79} - 32 q^{80} + 18 q^{81} + 44 q^{82} - 43 q^{83} + 41 q^{84} + 51 q^{85} + 4 q^{86} + 3 q^{87} - 17 q^{88} - 28 q^{89} + 58 q^{90} - q^{91} - 68 q^{92} - 60 q^{93} + 78 q^{94} - 18 q^{95} + 29 q^{96} - 34 q^{97} + 34 q^{98} + 15 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 4 x^{17} - 18 x^{16} + 84 x^{15} + 116 x^{14} - 708 x^{13} - 282 x^{12} + 3104 x^{11} + \cdots + 328 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 73030826 \nu^{17} - 256120405 \nu^{16} - 1356797574 \nu^{15} + 5113465236 \nu^{14} + \cdots - 59655036146 ) / 2338032863 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 197952229 \nu^{17} - 3918595304 \nu^{16} + 22600044356 \nu^{15} + 63480744292 \nu^{14} + \cdots + 1867879847532 ) / 4676065726 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 377442591 \nu^{17} - 3691417402 \nu^{16} + 2195758068 \nu^{15} + 67864507870 \nu^{14} + \cdots + 875081238676 ) / 4676065726 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 280914143 \nu^{17} - 701368576 \nu^{16} - 6767277373 \nu^{15} + 16268509010 \nu^{14} + \cdots - 245184253071 ) / 2338032863 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 918896525 \nu^{17} - 5543735306 \nu^{16} - 8092976468 \nu^{15} + 105501268198 \nu^{14} + \cdots + 742560783748 ) / 4676065726 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1002969289 \nu^{17} - 5357577374 \nu^{16} - 11353710084 \nu^{15} + 102147035436 \nu^{14} + \cdots + 677849390104 ) / 4676065726 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 685736770 \nu^{17} + 2668474422 \nu^{16} + 11910611780 \nu^{15} - 53542188699 \nu^{14} + \cdots - 42018281445 ) / 2338032863 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1386154783 \nu^{17} + 6430527688 \nu^{16} + 19874972614 \nu^{15} - 125741499426 \nu^{14} + \cdots - 374413808574 ) / 4676065726 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1665088329 \nu^{17} - 6609375872 \nu^{16} - 28840683926 \nu^{15} + 133345072732 \nu^{14} + \cdots - 107954465010 ) / 4676065726 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1832524509 \nu^{17} - 6639698786 \nu^{16} - 33972458906 \nu^{15} + 135523924258 \nu^{14} + \cdots - 180269480248 ) / 4676065726 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1216895826 \nu^{17} - 5844979198 \nu^{16} - 17002403395 \nu^{15} + 114758539026 \nu^{14} + \cdots + 306757426577 ) / 2338032863 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 2694174873 \nu^{17} + 8898905406 \nu^{16} + 54157234818 \nu^{15} - 186122321980 \nu^{14} + \cdots + 952441749148 ) / 4676065726 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 1410676829 \nu^{17} - 5054336188 \nu^{16} - 26608639913 \nu^{15} + 103627914575 \nu^{14} + \cdots - 361693721127 ) / 2338032863 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 1422023073 \nu^{17} - 4606595223 \nu^{16} - 28914727965 \nu^{15} + 96834645667 \nu^{14} + \cdots - 485286282032 ) / 2338032863 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 3038182645 \nu^{17} - 11663900722 \nu^{16} - 54084465652 \nu^{15} + 236680213784 \nu^{14} + \cdots - 317577900352 ) / 4676065726 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{14} - \beta_{11} + \beta_{10} + \beta_{7} - \beta_{6} + \beta_{2} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{15} + 2\beta_{11} - \beta_{10} + \beta_{9} - \beta_{5} - \beta_{3} + 8\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{17} - \beta_{16} - \beta_{15} - 9 \beta_{14} - \beta_{13} + \beta_{12} - 5 \beta_{11} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2 \beta_{17} + \beta_{16} - 11 \beta_{15} - 2 \beta_{13} - 3 \beta_{12} + 24 \beta_{11} - 13 \beta_{10} + \cdots + 87 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 13 \beta_{17} - 14 \beta_{16} - 11 \beta_{15} - 69 \beta_{14} - 16 \beta_{13} + 18 \beta_{12} + \cdots + 16 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 19 \beta_{17} + 10 \beta_{16} - 88 \beta_{15} - 5 \beta_{14} - 35 \beta_{13} - 27 \beta_{12} + \cdots + 532 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 129 \beta_{17} - 142 \beta_{16} - 90 \beta_{15} - 504 \beta_{14} - 177 \beta_{13} + 215 \beta_{12} + \cdots + 177 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 110 \beta_{17} + 52 \beta_{16} - 631 \beta_{15} - 100 \beta_{14} - 405 \beta_{13} - 123 \beta_{12} + \cdots + 3349 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1143 \beta_{17} - 1271 \beta_{16} - 668 \beta_{15} - 3612 \beta_{14} - 1672 \beta_{13} + 2131 \beta_{12} + \cdots + 1668 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 363 \beta_{17} + 9 \beta_{16} - 4324 \beta_{15} - 1303 \beta_{14} - 3942 \beta_{13} + 112 \beta_{12} + \cdots + 21527 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 9516 \beta_{17} - 10684 \beta_{16} - 4773 \beta_{15} - 25720 \beta_{14} - 14522 \beta_{13} + \cdots + 14428 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 1271 \beta_{17} - 3636 \beta_{16} - 29054 \beta_{15} - 14050 \beta_{14} - 35039 \beta_{13} + \cdots + 140724 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 76316 \beta_{17} - 86672 \beta_{16} - 33603 \beta_{15} - 183102 \beta_{14} - 120050 \beta_{13} + \cdots + 118635 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 39322 \beta_{17} - 57635 \beta_{16} - 193666 \beta_{15} - 136266 \beta_{14} - 295219 \beta_{13} + \cdots + 933162 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 597910 \beta_{17} - 688126 \beta_{16} - 235431 \beta_{15} - 1307213 \beta_{14} - 962160 \beta_{13} + \cdots + 944762 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.72204
2.59964
2.35947
1.98431
1.74487
1.35726
1.04467
0.826129
0.763493
0.523506
−0.735255
−0.924759
−0.957552
−1.15793
−1.52216
−1.87675
−2.24960
−2.50138
−2.72204 −1.76734 5.40952 0.469688 4.81078 1.03831 −9.28087 0.123492 −1.27851
1.2 −2.59964 2.09484 4.75812 −3.02323 −5.44583 −0.561390 −7.17011 1.38836 7.85930
1.3 −2.35947 −3.09007 3.56710 −4.18174 7.29093 −2.82979 −3.69754 6.54852 9.86669
1.4 −1.98431 −0.150114 1.93749 2.87852 0.297872 −2.68467 0.124041 −2.97747 −5.71188
1.5 −1.74487 −1.27304 1.04455 −3.61409 2.22129 4.28084 1.66712 −1.37936 6.30610
1.6 −1.35726 0.387927 −0.157846 −1.46929 −0.526517 0.194696 2.92876 −2.84951 1.99421
1.7 −1.04467 −2.71791 −0.908666 0.714085 2.83932 −2.03236 3.03859 4.38705 −0.745983
1.8 −0.826129 2.26041 −1.31751 −0.786316 −1.86739 −5.06179 2.74069 2.10946 0.649598
1.9 −0.763493 1.83524 −1.41708 −1.51218 −1.40119 1.20167 2.60892 0.368111 1.15454
1.10 −0.523506 −3.08736 −1.72594 −1.35183 1.61625 3.60927 1.95055 6.53179 0.707691
1.11 0.735255 0.544167 −1.45940 0.962787 0.400102 −3.25298 −2.54354 −2.70388 0.707894
1.12 0.924759 2.22306 −1.14482 −3.95421 2.05580 −3.08028 −2.90820 1.94201 −3.65669
1.13 0.957552 0.564538 −1.08309 −4.10274 0.540575 4.97202 −2.95222 −2.68130 −3.92859
1.14 1.15793 −0.220625 −0.659200 −0.421419 −0.255468 −0.645304 −3.07917 −2.95132 −0.487974
1.15 1.52216 −2.58636 0.316965 1.24712 −3.93684 −0.899316 −2.56184 3.68924 1.89831
1.16 1.87675 −1.13919 1.52220 −1.30620 −2.13798 −1.71403 −0.896704 −1.70224 −2.45142
1.17 2.24960 −0.790850 3.06069 −3.96974 −1.77909 −4.97706 2.38611 −2.37456 −8.93031
1.18 2.50138 −3.08733 4.25691 −3.57921 −7.72259 1.44216 5.64540 6.53160 −8.95298
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(547\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 547.2.a.b ✓ 18
3.b odd 2 1 4923.2.a.l 18
4.b odd 2 1 8752.2.a.s 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
547.2.a.b ✓ 18 1.a even 1 1 trivial
4923.2.a.l 18 3.b odd 2 1
8752.2.a.s 18 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{18} + 4 T_{2}^{17} - 18 T_{2}^{16} - 84 T_{2}^{15} + 116 T_{2}^{14} + 708 T_{2}^{13} - 282 T_{2}^{12} + \cdots + 328 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(547))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + 4 T^{17} + \cdots + 328 \) Copy content Toggle raw display
$3$ \( T^{18} + 10 T^{17} + \cdots - 32 \) Copy content Toggle raw display
$5$ \( T^{18} + 27 T^{17} + \cdots - 15872 \) Copy content Toggle raw display
$7$ \( T^{18} + 11 T^{17} + \cdots - 58576 \) Copy content Toggle raw display
$11$ \( T^{18} - 2 T^{17} + \cdots - 82432 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots - 160401143 \) Copy content Toggle raw display
$17$ \( T^{18} + 30 T^{17} + \cdots - 5744344 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots - 584801324 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 8189636608 \) Copy content Toggle raw display
$29$ \( T^{18} + 18 T^{17} + \cdots - 13479926 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 1756132711264 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 5114786922536 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 341990380184 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 23810212304 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 35899903339 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots - 971885299094 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots - 1458822574688 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 75258329816 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 12533716163723 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots - 368406703332424 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots - 12033371727622 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 1079280536 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots - 94\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 4077961316512 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots - 28869767458 \) Copy content Toggle raw display
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