Properties

Label 546.8.a.e
Level $546$
Weight $8$
Character orbit 546.a
Self dual yes
Analytic conductor $170.562$
Analytic rank $1$
Dimension $3$
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] = [546,8,Mod(1,546)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(546, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("546.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 546.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: \(170.562223914\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 33506x + 97248 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 8 q^{2} + 27 q^{3} + 64 q^{4} + (\beta_{2} - 126) q^{5} + 216 q^{6} - 343 q^{7} + 512 q^{8} + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 8 q^{2} + 27 q^{3} + 64 q^{4} + (\beta_{2} - 126) q^{5} + 216 q^{6} - 343 q^{7} + 512 q^{8} + 729 q^{9} + (8 \beta_{2} - 1008) q^{10} + (4 \beta_{2} - 13 \beta_1 - 2023) q^{11} + 1728 q^{12} + 2197 q^{13} - 2744 q^{14} + (27 \beta_{2} - 3402) q^{15} + 4096 q^{16} + ( - 44 \beta_{2} - 9 \beta_1 - 4077) q^{17} + 5832 q^{18} + ( - 109 \beta_{2} + 148 \beta_1 - 1612) q^{19} + (64 \beta_{2} - 8064) q^{20} - 9261 q^{21} + (32 \beta_{2} - 104 \beta_1 - 16184) q^{22} + ( - 85 \beta_{2} + 34 \beta_1 + 37202) q^{23} + 13824 q^{24} + ( - 443 \beta_{2} - 10 \beta_1 + 13973) q^{25} + 17576 q^{26} + 19683 q^{27} - 21952 q^{28} + (77 \beta_{2} - 168 \beta_1 + 35138) q^{29} + (216 \beta_{2} - 27216) q^{30} + ( - 215 \beta_{2} + 755 \beta_1 - 87687) q^{31} + 32768 q^{32} + (108 \beta_{2} - 351 \beta_1 - 54621) q^{33} + ( - 352 \beta_{2} - 72 \beta_1 - 32616) q^{34} + ( - 343 \beta_{2} + 43218) q^{35} + 46656 q^{36} + ( - 864 \beta_{2} + 129 \beta_1 - 195323) q^{37} + ( - 872 \beta_{2} + 1184 \beta_1 - 12896) q^{38} + 59319 q^{39} + (512 \beta_{2} - 64512) q^{40} + (536 \beta_{2} + 386 \beta_1 - 216536) q^{41} - 74088 q^{42} + (527 \beta_{2} - 14 \beta_1 - 25394) q^{43} + (256 \beta_{2} - 832 \beta_1 - 129472) q^{44} + (729 \beta_{2} - 91854) q^{45} + ( - 680 \beta_{2} + 272 \beta_1 + 297616) q^{46} + (1063 \beta_{2} - 1755 \beta_1 - 423073) q^{47} + 110592 q^{48} + 117649 q^{49} + ( - 3544 \beta_{2} - 80 \beta_1 + 111784) q^{50} + ( - 1188 \beta_{2} - 243 \beta_1 - 110079) q^{51} + 140608 q^{52} + (2983 \beta_{2} - 5635 \beta_1 + 108837) q^{53} + 157464 q^{54} + ( - 1172 \beta_{2} - 885 \beta_1 + 158437) q^{55} - 175616 q^{56} + ( - 2943 \beta_{2} + 3996 \beta_1 - 43524) q^{57} + (616 \beta_{2} - 1344 \beta_1 + 281104) q^{58} + (3524 \beta_{2} - 4680 \beta_1 - 1144772) q^{59} + (1728 \beta_{2} - 217728) q^{60} + ( - 2470 \beta_{2} + 4897 \beta_1 - 483499) q^{61} + ( - 1720 \beta_{2} + 6040 \beta_1 - 701496) q^{62} - 250047 q^{63} + 262144 q^{64} + (2197 \beta_{2} - 276822) q^{65} + (864 \beta_{2} - 2808 \beta_1 - 436968) q^{66} + (4866 \beta_{2} - 9488 \beta_1 - 434124) q^{67} + ( - 2816 \beta_{2} - 576 \beta_1 - 260928) q^{68} + ( - 2295 \beta_{2} + 918 \beta_1 + 1004454) q^{69} + ( - 2744 \beta_{2} + 345744) q^{70} + ( - 13388 \beta_{2} + 1100 \beta_1 - 1728692) q^{71} + 373248 q^{72} + (6511 \beta_{2} + 9738 \beta_1 - 1220980) q^{73} + ( - 6912 \beta_{2} + 1032 \beta_1 - 1562584) q^{74} + ( - 11961 \beta_{2} - 270 \beta_1 + 377271) q^{75} + ( - 6976 \beta_{2} + 9472 \beta_1 - 103168) q^{76} + ( - 1372 \beta_{2} + 4459 \beta_1 + 693889) q^{77} + 474552 q^{78} + (11521 \beta_{2} - 3531 \beta_1 - 983417) q^{79} + (4096 \beta_{2} - 516096) q^{80} + 531441 q^{81} + (4288 \beta_{2} + 3088 \beta_1 - 1732288) q^{82} + ( - 5169 \beta_{2} - 11333 \beta_1 - 4550075) q^{83} - 592704 q^{84} + (11338 \beta_{2} - 145 \beta_1 - 3117923) q^{85} + (4216 \beta_{2} - 112 \beta_1 - 203152) q^{86} + (2079 \beta_{2} - 4536 \beta_1 + 948726) q^{87} + (2048 \beta_{2} - 6656 \beta_1 - 1035776) q^{88} + (11553 \beta_{2} + 13335 \beta_1 + 2126511) q^{89} + (5832 \beta_{2} - 734832) q^{90} - 753571 q^{91} + ( - 5440 \beta_{2} + 2176 \beta_1 + 2380928) q^{92} + ( - 5805 \beta_{2} + 20385 \beta_1 - 2367549) q^{93} + (8504 \beta_{2} - 14040 \beta_1 - 3384584) q^{94} + (8817 \beta_{2} + 10710 \beta_1 - 3535882) q^{95} + 884736 q^{96} + ( - 38671 \beta_{2} + 19319 \beta_1 - 6010745) q^{97} + 941192 q^{98} + (2916 \beta_{2} - 9477 \beta_1 - 1474767) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 24 q^{2} + 81 q^{3} + 192 q^{4} - 378 q^{5} + 648 q^{6} - 1029 q^{7} + 1536 q^{8} + 2187 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 24 q^{2} + 81 q^{3} + 192 q^{4} - 378 q^{5} + 648 q^{6} - 1029 q^{7} + 1536 q^{8} + 2187 q^{9} - 3024 q^{10} - 6069 q^{11} + 5184 q^{12} + 6591 q^{13} - 8232 q^{14} - 10206 q^{15} + 12288 q^{16} - 12231 q^{17} + 17496 q^{18} - 4836 q^{19} - 24192 q^{20} - 27783 q^{21} - 48552 q^{22} + 111606 q^{23} + 41472 q^{24} + 41919 q^{25} + 52728 q^{26} + 59049 q^{27} - 65856 q^{28} + 105414 q^{29} - 81648 q^{30} - 263061 q^{31} + 98304 q^{32} - 163863 q^{33} - 97848 q^{34} + 129654 q^{35} + 139968 q^{36} - 585969 q^{37} - 38688 q^{38} + 177957 q^{39} - 193536 q^{40} - 649608 q^{41} - 222264 q^{42} - 76182 q^{43} - 388416 q^{44} - 275562 q^{45} + 892848 q^{46} - 1269219 q^{47} + 331776 q^{48} + 352947 q^{49} + 335352 q^{50} - 330237 q^{51} + 421824 q^{52} + 326511 q^{53} + 472392 q^{54} + 475311 q^{55} - 526848 q^{56} - 130572 q^{57} + 843312 q^{58} - 3434316 q^{59} - 653184 q^{60} - 1450497 q^{61} - 2104488 q^{62} - 750141 q^{63} + 786432 q^{64} - 830466 q^{65} - 1310904 q^{66} - 1302372 q^{67} - 782784 q^{68} + 3013362 q^{69} + 1037232 q^{70} - 5186076 q^{71} + 1119744 q^{72} - 3662940 q^{73} - 4687752 q^{74} + 1131813 q^{75} - 309504 q^{76} + 2081667 q^{77} + 1423656 q^{78} - 2950251 q^{79} - 1548288 q^{80} + 1594323 q^{81} - 5196864 q^{82} - 13650225 q^{83} - 1778112 q^{84} - 9353769 q^{85} - 609456 q^{86} + 2846178 q^{87} - 3107328 q^{88} + 6379533 q^{89} - 2204496 q^{90} - 2260713 q^{91} + 7142784 q^{92} - 7102647 q^{93} - 10153752 q^{94} - 10607646 q^{95} + 2654208 q^{96} - 18032235 q^{97} + 2823576 q^{98} - 4424301 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 33506x + 97248 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{2} + 163\nu - 22392 ) / 114 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{2} + 179\nu + 22278 ) / 114 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -163\beta_{2} + 179\beta _1 + 67013 ) / 3 \) Copy content Toggle raw display

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
−183.985
182.082
2.90288
8.00000 27.0000 64.0000 −516.403 216.000 −343.000 512.000 729.000 −4131.22
1.2 8.00000 27.0000 64.0000 64.4978 216.000 −343.000 512.000 729.000 515.983
1.3 8.00000 27.0000 64.0000 73.9052 216.000 −343.000 512.000 729.000 591.241
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(7\) \(1\)
\(13\) \(-1\)

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 546.8.a.e 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.8.a.e 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{3} + 378T_{5}^{2} - 66705T_{5} + 2461550 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(546))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 8)^{3} \) Copy content Toggle raw display
$3$ \( (T - 27)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} + 378 T^{2} - 66705 T + 2461550 \) Copy content Toggle raw display
$7$ \( (T + 343)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} + 6069 T^{2} + \cdots - 11019191384 \) Copy content Toggle raw display
$13$ \( (T - 2197)^{3} \) Copy content Toggle raw display
$17$ \( T^{3} + 12231 T^{2} + \cdots - 2540015942296 \) Copy content Toggle raw display
$19$ \( T^{3} + 4836 T^{2} + \cdots + 26372444700364 \) Copy content Toggle raw display
$23$ \( T^{3} - 111606 T^{2} + \cdots - 29333382239720 \) Copy content Toggle raw display
$29$ \( T^{3} - 105414 T^{2} + \cdots + 26262769167062 \) Copy content Toggle raw display
$31$ \( T^{3} + 263061 T^{2} + \cdots - 48\!\cdots\!72 \) Copy content Toggle raw display
$37$ \( T^{3} + 585969 T^{2} + \cdots - 14\!\cdots\!12 \) Copy content Toggle raw display
$41$ \( T^{3} + 649608 T^{2} + \cdots - 20\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( T^{3} + 76182 T^{2} + \cdots + 13\!\cdots\!52 \) Copy content Toggle raw display
$47$ \( T^{3} + 1269219 T^{2} + \cdots - 59\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{3} - 326511 T^{2} + \cdots - 33\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{3} + 3434316 T^{2} + \cdots - 17\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( T^{3} + 1450497 T^{2} + \cdots - 47\!\cdots\!60 \) Copy content Toggle raw display
$67$ \( T^{3} + 1302372 T^{2} + \cdots - 53\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{3} + 5186076 T^{2} + \cdots - 59\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{3} + 3662940 T^{2} + \cdots - 33\!\cdots\!34 \) Copy content Toggle raw display
$79$ \( T^{3} + 2950251 T^{2} + \cdots - 40\!\cdots\!32 \) Copy content Toggle raw display
$83$ \( T^{3} + 13650225 T^{2} + \cdots + 22\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{3} - 6379533 T^{2} + \cdots + 77\!\cdots\!80 \) Copy content Toggle raw display
$97$ \( T^{3} + 18032235 T^{2} + \cdots - 67\!\cdots\!20 \) Copy content Toggle raw display
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