Properties

Label 546.8.a.i
Level $546$
Weight $8$
Character orbit 546.a
Self dual yes
Analytic conductor $170.562$
Analytic rank $1$
Dimension $5$
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: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 86504x^{3} - 9117228x^{2} + 89606664x + 21810067776 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2}\cdot 5\cdot 13 \)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 8 q^{2} + 27 q^{3} + 64 q^{4} + ( - \beta_1 - 68) q^{5} - 216 q^{6} + 343 q^{7} - 512 q^{8} + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 8 q^{2} + 27 q^{3} + 64 q^{4} + ( - \beta_1 - 68) q^{5} - 216 q^{6} + 343 q^{7} - 512 q^{8} + 729 q^{9} + (8 \beta_1 + 544) q^{10} + ( - \beta_{2} - 6 \beta_1 - 261) q^{11} + 1728 q^{12} + 2197 q^{13} - 2744 q^{14} + ( - 27 \beta_1 - 1836) q^{15} + 4096 q^{16} + ( - 3 \beta_{4} + 2 \beta_{3} + 6 \beta_1 - 850) q^{17} - 5832 q^{18} + (9 \beta_{4} - \beta_{3} + 45 \beta_1 - 3395) q^{19} + ( - 64 \beta_1 - 4352) q^{20} + 9261 q^{21} + (8 \beta_{2} + 48 \beta_1 + 2088) q^{22} + ( - 7 \beta_{4} - 8 \beta_{3} + 7 \beta_{2} + 95 \beta_1 - 15613) q^{23} - 13824 q^{24} + ( - 15 \beta_{4} - 9 \beta_{3} + 10 \beta_{2} + 101 \beta_1 - 15910) q^{25} - 17576 q^{26} + 19683 q^{27} + 21952 q^{28} + (9 \beta_{4} + 23 \beta_{3} + 24 \beta_{2} - 83 \beta_1 - 42617) q^{29} + (216 \beta_1 + 14688) q^{30} + (20 \beta_{4} + 38 \beta_{3} - 9 \beta_{2} + 195 \beta_1 - 37205) q^{31} - 32768 q^{32} + ( - 27 \beta_{2} - 162 \beta_1 - 7047) q^{33} + (24 \beta_{4} - 16 \beta_{3} - 48 \beta_1 + 6800) q^{34} + ( - 343 \beta_1 - 23324) q^{35} + 46656 q^{36} + (7 \beta_{4} - 87 \beta_{3} + 19 \beta_{2} - 304 \beta_1 + 20214) q^{37} + ( - 72 \beta_{4} + 8 \beta_{3} - 360 \beta_1 + 27160) q^{38} + 59319 q^{39} + (512 \beta_1 + 34816) q^{40} + ( - 9 \beta_{4} - 143 \beta_{3} - 20 \beta_{2} + 30 \beta_1 - 4805) q^{41} - 74088 q^{42} + (19 \beta_{4} + 85 \beta_{3} + 12 \beta_{2} + 1893 \beta_1 - 11097) q^{43} + ( - 64 \beta_{2} - 384 \beta_1 - 16704) q^{44} + ( - 729 \beta_1 - 49572) q^{45} + (56 \beta_{4} + 64 \beta_{3} - 56 \beta_{2} - 760 \beta_1 + 124904) q^{46} + (55 \beta_{4} + 272 \beta_{3} + 38 \beta_{2} + 693 \beta_1 - 197170) q^{47} + 110592 q^{48} + 117649 q^{49} + (120 \beta_{4} + 72 \beta_{3} - 80 \beta_{2} - 808 \beta_1 + 127280) q^{50} + ( - 81 \beta_{4} + 54 \beta_{3} + 162 \beta_1 - 22950) q^{51} + 140608 q^{52} + ( - 242 \beta_{4} + 34 \beta_{3} - 103 \beta_{2} + 2903 \beta_1 - 378421) q^{53} - 157464 q^{54} + ( - 55 \beta_{4} - 99 \beta_{3} + 145 \beta_{2} + 3572 \beta_1 + 349438) q^{55} - 175616 q^{56} + (243 \beta_{4} - 27 \beta_{3} + 1215 \beta_1 - 91665) q^{57} + ( - 72 \beta_{4} - 184 \beta_{3} - 192 \beta_{2} + 664 \beta_1 + 340936) q^{58} + ( - 29 \beta_{4} - 336 \beta_{3} - 129 \beta_{2} + 5352 \beta_1 - 560499) q^{59} + ( - 1728 \beta_1 - 117504) q^{60} + ( - 217 \beta_{4} + 179 \beta_{3} + 463 \beta_{2} + 626 \beta_1 + 228260) q^{61} + ( - 160 \beta_{4} - 304 \beta_{3} + 72 \beta_{2} - 1560 \beta_1 + 297640) q^{62} + 250047 q^{63} + 262144 q^{64} + ( - 2197 \beta_1 - 149396) q^{65} + (216 \beta_{2} + 1296 \beta_1 + 56376) q^{66} + (306 \beta_{4} - 300 \beta_{3} - 122 \beta_{2} + 1658 \beta_1 + 53046) q^{67} + ( - 192 \beta_{4} + 128 \beta_{3} + 384 \beta_1 - 54400) q^{68} + ( - 189 \beta_{4} - 216 \beta_{3} + 189 \beta_{2} + 2565 \beta_1 - 421551) q^{69} + (2744 \beta_1 + 186592) q^{70} + (411 \beta_{4} - 147 \beta_{3} - 410 \beta_{2} - 3506 \beta_1 - 896737) q^{71} - 373248 q^{72} + (297 \beta_{4} - 539 \beta_{3} - 402 \beta_{2} + 8955 \beta_1 - 470217) q^{73} + ( - 56 \beta_{4} + 696 \beta_{3} - 152 \beta_{2} + 2432 \beta_1 - 161712) q^{74} + ( - 405 \beta_{4} - 243 \beta_{3} + 270 \beta_{2} + 2727 \beta_1 - 429570) q^{75} + (576 \beta_{4} - 64 \beta_{3} + 2880 \beta_1 - 217280) q^{76} + ( - 343 \beta_{2} - 2058 \beta_1 - 89523) q^{77} - 474552 q^{78} + (226 \beta_{4} + 1428 \beta_{3} - 291 \beta_{2} + 3639 \beta_1 - 816049) q^{79} + ( - 4096 \beta_1 - 278528) q^{80} + 531441 q^{81} + (72 \beta_{4} + 1144 \beta_{3} + 160 \beta_{2} - 240 \beta_1 + 38440) q^{82} + (809 \beta_{4} + 1164 \beta_{3} + 366 \beta_{2} - 9603 \beta_1 - 1746006) q^{83} + 592704 q^{84} + (165 \beta_{4} + 45 \beta_{3} + 95 \beta_{2} - 4526 \beta_1 - 343108) q^{85} + ( - 152 \beta_{4} - 680 \beta_{3} - 96 \beta_{2} - 15144 \beta_1 + 88776) q^{86} + (243 \beta_{4} + 621 \beta_{3} + 648 \beta_{2} - 2241 \beta_1 - 1150659) q^{87} + (512 \beta_{2} + 3072 \beta_1 + 133632) q^{88} + (45 \beta_{4} - 437 \beta_{3} + 257 \beta_{2} - 6533 \beta_1 - 4015464) q^{89} + (5832 \beta_1 + 396576) q^{90} + 753571 q^{91} + ( - 448 \beta_{4} - 512 \beta_{3} + 448 \beta_{2} + 6080 \beta_1 - 999232) q^{92} + (540 \beta_{4} + 1026 \beta_{3} - 243 \beta_{2} + 5265 \beta_1 - 1004535) q^{93} + ( - 440 \beta_{4} - 2176 \beta_{3} - 304 \beta_{2} + \cdots + 1577360) q^{94}+ \cdots + ( - 729 \beta_{2} - 4374 \beta_1 - 190269) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 40 q^{2} + 135 q^{3} + 320 q^{4} - 340 q^{5} - 1080 q^{6} + 1715 q^{7} - 2560 q^{8} + 3645 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 40 q^{2} + 135 q^{3} + 320 q^{4} - 340 q^{5} - 1080 q^{6} + 1715 q^{7} - 2560 q^{8} + 3645 q^{9} + 2720 q^{10} - 1303 q^{11} + 8640 q^{12} + 10985 q^{13} - 13720 q^{14} - 9180 q^{15} + 20480 q^{16} - 4247 q^{17} - 29160 q^{18} - 16984 q^{19} - 21760 q^{20} + 46305 q^{21} + 10424 q^{22} - 78072 q^{23} - 69120 q^{24} - 79555 q^{25} - 87880 q^{26} + 98415 q^{27} + 109760 q^{28} - 213142 q^{29} + 73440 q^{30} - 186027 q^{31} - 163840 q^{32} - 35181 q^{33} + 33976 q^{34} - 116620 q^{35} + 233280 q^{36} + 101025 q^{37} + 135872 q^{38} + 296595 q^{39} + 174080 q^{40} - 23976 q^{41} - 370440 q^{42} - 55528 q^{43} - 83392 q^{44} - 247860 q^{45} + 624576 q^{46} - 985981 q^{47} + 552960 q^{48} + 588245 q^{49} + 636440 q^{50} - 114669 q^{51} + 703040 q^{52} - 1891657 q^{53} - 787320 q^{54} + 1746955 q^{55} - 878080 q^{56} - 458568 q^{57} + 1705136 q^{58} - 2802208 q^{59} - 587520 q^{60} + 1140591 q^{61} + 1488216 q^{62} + 1250235 q^{63} + 1310720 q^{64} - 746980 q^{65} + 281448 q^{66} + 265168 q^{67} - 271808 q^{68} - 2107944 q^{69} + 932960 q^{70} - 4483276 q^{71} - 1866240 q^{72} - 2350578 q^{73} - 808200 q^{74} - 2147985 q^{75} - 1086976 q^{76} - 446929 q^{77} - 2372760 q^{78} - 4079889 q^{79} - 1392640 q^{80} + 2657205 q^{81} + 191808 q^{82} - 8731571 q^{83} + 2963520 q^{84} - 1715895 q^{85} + 444224 q^{86} - 5754834 q^{87} + 667136 q^{88} - 20077879 q^{89} + 1982880 q^{90} + 3767855 q^{91} - 4996608 q^{92} - 5022729 q^{93} + 7887848 q^{94} - 11580740 q^{95} - 4423680 q^{96} + 3780209 q^{97} - 4705960 q^{98} - 949887 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 2x^{4} - 86504x^{3} - 9117228x^{2} + 89606664x + 21810067776 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 69\nu^{4} - 1741\nu^{3} - 6302648\nu^{2} - 371941044\nu + 25331552820 ) / 68034708 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 283\nu^{4} + 38508\nu^{3} - 32715544\nu^{2} - 7562728596\nu + 89369189784 ) / 136069416 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2173\nu^{4} + 283072\nu^{3} + 164159976\nu^{2} - 783684612\nu - 877198268520 ) / 136069416 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3887\nu^{4} + 308060\nu^{3} + 296253744\nu^{2} + 13428544068\nu - 560044524816 ) / 136069416 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - 5\beta_{2} + 26\beta _1 + 50 ) / 130 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -325\beta_{4} + 372\beta_{3} - 365\beta_{2} - 2548\beta _1 + 2248945 ) / 65 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -48880\beta_{4} + 122076\beta_{3} - 191780\beta_{2} + 938756\beta _1 + 362232820 ) / 65 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -30919720\beta_{4} + 39754926\beta_{3} - 51655190\beta_{2} - 74888164\beta _1 + 190836284620 ) / 65 \) 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
−65.0947
−108.730
336.062
44.7350
−204.972
−8.00000 27.0000 64.0000 −428.929 −216.000 343.000 −512.000 729.000 3431.43
1.2 −8.00000 27.0000 64.0000 −114.198 −216.000 343.000 −512.000 729.000 913.581
1.3 −8.00000 27.0000 64.0000 −105.351 −216.000 343.000 −512.000 729.000 842.807
1.4 −8.00000 27.0000 64.0000 −12.1505 −216.000 343.000 −512.000 729.000 97.2036
1.5 −8.00000 27.0000 64.0000 320.628 −216.000 343.000 −512.000 729.000 −2565.02
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.i 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.8.a.i 5 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{5} + 340T_{5}^{4} - 97735T_{5}^{3} - 30126750T_{5}^{2} - 2005595100T_{5} - 20103633000 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(546))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 8)^{5} \) Copy content Toggle raw display
$3$ \( (T - 27)^{5} \) Copy content Toggle raw display
$5$ \( T^{5} + 340 T^{4} + \cdots - 20103633000 \) Copy content Toggle raw display
$7$ \( (T - 343)^{5} \) Copy content Toggle raw display
$11$ \( T^{5} + 1303 T^{4} + \cdots + 47\!\cdots\!04 \) Copy content Toggle raw display
$13$ \( (T - 2197)^{5} \) Copy content Toggle raw display
$17$ \( T^{5} + 4247 T^{4} + \cdots - 40\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( T^{5} + 16984 T^{4} + \cdots + 15\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( T^{5} + 78072 T^{4} + \cdots - 39\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( T^{5} + 213142 T^{4} + \cdots + 68\!\cdots\!28 \) Copy content Toggle raw display
$31$ \( T^{5} + 186027 T^{4} + \cdots + 31\!\cdots\!40 \) Copy content Toggle raw display
$37$ \( T^{5} - 101025 T^{4} + \cdots - 92\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{5} + 23976 T^{4} + \cdots - 92\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{5} + 55528 T^{4} + \cdots - 13\!\cdots\!48 \) Copy content Toggle raw display
$47$ \( T^{5} + 985981 T^{4} + \cdots + 81\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{5} + 1891657 T^{4} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{5} + 2802208 T^{4} + \cdots + 19\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( T^{5} - 1140591 T^{4} + \cdots - 40\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{5} - 265168 T^{4} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{5} + 4483276 T^{4} + \cdots + 10\!\cdots\!32 \) Copy content Toggle raw display
$73$ \( T^{5} + 2350578 T^{4} + \cdots - 25\!\cdots\!60 \) Copy content Toggle raw display
$79$ \( T^{5} + 4079889 T^{4} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{5} + 8731571 T^{4} + \cdots + 65\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( T^{5} + 20077879 T^{4} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{5} - 3780209 T^{4} + \cdots - 19\!\cdots\!92 \) Copy content Toggle raw display
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